HISTORY OF HINDU MATHEMATICS A SOURCE BOOK Parts I and II BY BIBHUTIBHUSAN DATTA AND AVADHESH NARAYAN SINGH ASIA PUBLISHING HOUSK BOMBAY CALCUTTA NEW DELHI MADRAS - LONDON ' N E W YORK © 1935, 1938 AVADHESH NARAYAN SINGH Part I First Published: 1935 Part II First Published: 1938 Single Volume Edition: 1962 All Rights Reserved Part I: pp. 1-250 Part II: pp. 1-308 PRINTED IN INDIA BY SOMESHWAR DAYAL AT THE MUDRAN KALA MANDIH , LUCKNOW AND PUBLISHED BY P. S. JAYASIKGHE, ASIA PUBLISHING HOUSE, BOMBAY TRANSLITERATION Vowels Short : ^ 5 <3 **£ $ a / u r I Long : WJ k T, ^ % 1 * ?rt ^ A a i U T I e at tftf Anusvdra : - = - m Visarga : : = = b Non-aspirant : S = _ 9 Consonants Classified : 3> m 7T V -^ •s k kb £ ^ « ^ "^ §L 3T ?K 37 c ch 7 Jh ii z ^v / tb d dh n *t \ ** t th d hundred ayutas niyuta, hundred niyutas hankara, hundred hankaras vivara, hundred vivaras ksobbya, hundred kso- bbyas vlvdha, hundred vivdbas utsanga, hundred utsangas balntla, hundred bahulas ndgabala y hundred ndgabalas tip- 1 xvii. 10; the list is che same with the exception that niyuta and prayuta change places. In xxxix. 6, after nyarbuda a new term vddava intervenes. 2 Cf. Bhaskara II, L, p. z. '" Satottara ganana or SatottAra sarhjnd (names on the cen- tesimal scale). * Lalitavistara, ed. by Rajencka Lai Mitra, Calcutta, 1877, NUMERAL TERMINOLOGY II lambha, hundred titilambhas vyavasthdna-prajnapti, hundred vyavasthdna-prajfiaptis hetuhila, hundred hetuhilas karahu, hundred karahu r hetvindriya, hundred hetvindriyas samdpta- lambha, hundred samdpta-lambhas ganandgati, hundred ganandgatis niravadya, hundred niravadyas mudrd-bala, hun- dred mudrd-balas sarva-bala, hundred sarva-balas visamjni- gati, hundred visarfijnd-gatis sarvajnd, hundred sarvajnds vibhutangamdy hundred vibhutangamds tallaksana. 1 " Another interesting series of number names increasing by multiples of 10 millions is found in Kaccayana's Pali Grammar. 2 "For example: dasa (10) multiplied by dasa (10) becomes sata (ioo), sata (ioo) multiplied by ten becomes sahassa (1,000), sahassa multiplied by ten becomes dasa sahassa (io,ooo), dasa sahassa multiplied by ten becomes sata sahassa 3, (100,000), sata sahassa multiplied by ten becomes dasa sata sahassa (1,000,000), dasa sata sahassa multiplied by ten becomes koti (10,000,000). Hundred-hundred- thousand kotis give pakotiS In this manner the further terms are formed. What are their names? hundred hundred-thousands is koti, hundred-hundred- ' Thus ta//aksana—io 53 . This and the following show that the Hindus anticipated Archimedes by several centuries in the matter of evolving a series of number names which "are sufficient to exceed not only the number of a sand-heap as large as the whole earth, but one as large as the universe." Cf. 'De harenae numero* in the 1544 edition of the Opera of Archimedes; quoted by Smith and Karpinski, Hindu Arabic Numerals, Boston, 191 1, p. 16. 2 "Grammaire Palie de Kaccayana," Journ. A.siatique, Sixieme Serie, XVII, 1871, p. 411. The explanations to sutras 51 and 5 2 are quoted here. 3 Also called lakkha (Jaksd). 4 Also called koti-koti, i.e., (10,000,000)*= 10 14 . The follow- ing numbers are in the denomination kpti-koti. Compare the Jlnuyogadvara- sulfa, Sutra J42. 12 NUMERAL NOTATION thousand kotis is pakoti, hundred-hundred-thousand pa kotis is koiippakoti, hundred-hundred-thousand koti- ppakotis is nabuta, hundred-hundred-fhousand nahutas is ninnabuta, hundred-hundred-thousand ninnahutas is ak- kbobbini\ similarly we have bindu, abbuda, nir abbuda, cthaha, ababa, atata, sogandbika, uppala, kumuda, pundarika padnwa, katbdna, mabdkaibdna, asankbyeya." 1 In the Anuyogadvdra-sutra 2 (c. ioo B.C.), a Jaina canonical work written before the commencement of the Christian era; the total number of human beings in the world is given thus: "a number which when expressed in terms of the denominations, koti-koti, etc., occupies twenty-nine places (stbdnd), or it is beyond the 24th place and within the 32nd place, or it is a number obtained by multiplying sixth square (of two) by (its) fifth square, (i.e., 2 06 ), or it is a number which can be divided (by two) ninety-six times." Another big number that occurs in the Jaina works is the number representing the period of time known as Sirsaprahelikd. According to the commentator Hema Candra (b. 1089) 3 , this number is so large as to occupy 194 notational places (anka-stbdnehi). It is also stated to be (8, 400,000) 28 . Notational Places. Later on, when the idea of place-value was developed, the denominations (number names) were used to denote the places which unity would occupy in order to represent them (denominations) in writing a number on the decimal scale. For instance, according to Aryabhata I (499) the denominations are the names of 'places'. He says: "Eka (unit) dasa (ten), sata (hundred), sabasra (thousand), ayuta (ten thousand), niyuta (hundred thousand), prayuta (million), 1 Thus asankhyeya is (io) 14n =(io,ooo,ooo) 20 . 2 Sutra 142. 3 The figures within brackets after the names of authors or works denote dates after Christ. NUMERAL TERMINOLOGY 1 3 koti (ten million), arbuda (hundred million), and vrnda , (thousand million) are respectively from place to place each ten times the preceding." 1 The first use of the word 'place' for the denomination is met with in the Jaina work quoted above. In most of the mathematical works, the denomina- tions are called "names of places," and eighteen of these are generally enumerated. Sridhara (750) gives the following names: 2 eka, dasa, sat a, sahasra, ayuta, laksa, prajuta, koti, arbuda, abja, kharva, nikharva, mahd- saroja, sank*, saritd-pati, antya, madhya, pardrdha, and adds that the decuple names proceed even beyond this. Mahavira (850) gives twenty-four notational places: 3 eka, dasa, sat a, sahasra, dasa-sahasra, laksa, dasa-laksa, koti, das'a-koti, sata-koti, arbuda, nyarbuda, kharva, mahdkharva, padma, mahd-padma, ksoni, mahd-ksoni, sankha, mahd-sankha, ksiti, mahd-ksiti, ksobha, mahd- ksobha. Bhaskara II's (11 50) list agrees with that of Sridhara except for mahdsaroja and saritdpati which are replaced by their synonyms mahdpadma and jaladhi respectively. He remarks that the names of places have -been assigned for practical use by ancient writers. 4 Narayana (1356) gives a similar list in which abja, mahdsaroja and saritdpati are replaced by their synonyms saroja, mahdbja and pdrdvdra respectively. Numerals in Spoken Language. The Sanskrit names for the numbers from one to nine are: eka, '•*•> 4+4+ 1 (reading from right to left, the order being the same as that of the script). The number 10 has an entirely new sign. The question why it was not written as fj J\)( , or why the base X (4) was abandoned cannot be satisfac- torily answered. KHAROSTHI NUMERALS 23 > It is accepted by all that the Kharosthi is a foreign script brought into India from the west. The exact period at -which it -was imported is unknown. It might have been introduced at the time of the conquest of the Punjab by Darius (c. 500 B.C.) or earlier. 1 The numerals given above undoubtedly belong to this script as they proceed from right to left. The old symbols of the inscriptions of Asoka, however, seem to have undergone modification in India, especially the numbers from 4 to 19. The symbols for four and ten seem to have been coined in India, in order to introduce simplification and also to bring the Kharosthi numeral system in line with the Brahmi notation already in extensive use. The symbol sC seems to have been derived by turning the Brahmi symbol ""y"* which represents 4 in the inscrip- tions of Asoka. The inclined cross to represent 4 is found in the Nabatean numerals in use in the earlier cen- turies of the Christian Era. 2 The Nabatean numerals resemble the Kharosthi also in the use of the scale of twenty and in the method of formation of the hundreds. It is possible that the Semites might have borrowed the Kharosthi symbol for 4, although it is not unlikely, as Buhler thinks, that the symbol might have been invented independently by both nations. 1 The theory of the foreign origin of the script has to be revised in the light of the discoveries at Mohenjo-daro andHarappa, especially in view of the fact that the Mohenjo-daro alphabet ran from right to left. 2 J. Euting, Nabataische Inschriften aus A.rabien, Berlin, 1885, pp. 96-97. 24 NUMERAL NOTATION The numeral O (10) closely resembles the letter a of the Brahml alphabet. The symbol for twenty O appears to be a cursive combination of two tens. It resembles one of the early Phoenician forms found in the papyrus Blacas 1 (5th century B.C.). The mode of expressing the numbers 30, 40, etc., by the help of the symbols for 10 and 20, is the same as amongst the early Phoenicians and Aramaeans. The symbol for 100 resembles the letter ta or tra of the Brahmi script, to the right of which stands a vertical stroke. The symbols for 200, 300, etc., are formed by writing the symbols for 2, 3, etc., respectively to the right of the symbol for 100. This evidently is the use of the multiplicative principle, as is found amongst the early Phoenicians. 2 The formation of other numbers may be illustrated by the number 274 which is written with the help of the symbols for 2, 100, 20, 10 and 4 arranged as nvfii in the right to left order. The 2 on the right of 100 multiplies 100, whilst the numbers written to the left are added, thus giving 274. The ancient Kharosthl numerals are given in Table I. 1 Biihler, Palaeography, p. 77; Ojha, I.e., p. 128; see Table II(£). 2 See Table II(V). BRAHMI NUMERALS 2 J 7. BRAHMl numerals Early Occurrence and Forms. The Brahmi ins- criptions are found distributed ail over India. The Brahmi script was, thus, the national script of the ancient Hindus. It is undoubtedly an invention of the Brahmanas. The early grammatical and phonetic re- searches seem to have resulted in the perfection of this script about 1,000 B.C. or earlier. The Brahmi numerals are likewise a purely Indian invention. Attempts have been made by several writers of note to evolve a theory of a foreign origin of the numerals, but we are con- vinced that all those attempts were utter failures. 1 These theories will be dealt with at their proper places. Due to the lack of early documents, we are not in a position to say what exactly were the original forms of the Brahmi symbols. Our knowledge of these symbols goes back to the time of King Asoka (c. $oo B.C.) whose vast dominions included the whole of India and extended in the north upto Central Asia. The forms of these symbols are: 4 6 50 200 + 64 6.0 4 v, V • -a if Ki ?n 1,000 4,000 6,000 10,000 20,000 T 1* TV Tec To A number of inscriptions containing numerals and dating from the first or the second century A.D. are found in a cave in the district of Nasik in the Bombay presidency. These contain a fuller list of numerals. The forms *. are as follows: 1234 5678 9 10 20 40 70 IOO 500 ? «c< e * Z 7 / ft 1,000 2,O00 3,000 4,000 8,000 70,000 1 f f T> V V xt- -"!9 n ,A ncient Na gari Numeration; from an inscription at Nf.naghat /o»w. o//fe Bewfc*,- Branch of the Royal Asiatic Society, 1876, Vol. XII, p. 404. -^ 2 E- Senart, "The inscriptions in the caves at Nasik," El, Vol. Vlil, pp. 39 - 9 6; "The inscriptions in the cave at Karle " EI, Vol. VII, pp. 47-74. BRAHMI NUMERALS 27 Even after the invention of the zero and the place- value system, the samd numerical symbols from i to 9, continued to be employed with the zero to denote numbers. Thus the gradual development of these forms can be easily traced. This gradual change from the old system without place-value to the new system with the zero and the place-value is to be met with in India alone. Ail other nations of the world have given up their indigenous numerical symbols which, they had used without place-value and have adopted the zero and a new set of symbols, which were never in use in those countries previously. This fact alone is a strong proof of the Hindu origin of the zero and the place- value system. The numbers 1, 2 and 3 of the Brahmi notation were denoted by one, two and three horizontal 1 lines placed one below the other. These forms clearly dis- tinguish the Brahmi notation from the Kharosthi and the Semitic systems. It cannot be said why the strokes were hori- zontal in Brahmi and vertical in Kharosthi and Semitic writings, just as it cannot be said why the writing proceeded from left to right in Brahmi and from right to left in Kharosthi and Semitic writings. It appears to us that the Brahmi and the Kharosthi (Semitic) numerals have always existed side by side and it cannot be definitely said which of these is the' earlier. The difference in writing the symbols 1 to 3, seems to be due to the inherent difference between the two systems of writing. The principles upon which numeri- cal signs are formed in the two systems are quite different. Difference from other Notations. In the Brahmi 3 It has been jncorfectly stated by Smith and Karpinski that the Nanaghat forms were vertical. See Hindu Arabic Numerals, p. 28. 28 NUMERAL NOTATION there are separate signs for each of the numbers i, 4 to 9 and 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 1000, 2000, etc., while in the oldest KharosthI and in the earliest Semitic writings, the Hieroglyphic and the Phoenician, the only symbols are those for 1, 10, zo and 100. The Hieratic and the Demotic numerals, however, resemble the Brahmi in having nineteen symbols for the numbers from 1 to 100, but the principle of forma- tion of the numbers 200, 300, 400, 2,000, 3,000 and 4,000 are different, as will appear from Table 11(f). The method of formation of intermediate and higher numbers is also different in the two systems. While the Brahmi places the bigger numbers to the left, the arrangement is the reverse of this in the KharosthI and Semitic writings. Thus the number 274, is written in Brahmi with the help of the symbols for 200, 70 and 4 as (200) (70) (4), while in the KharosthI and the Semitic numerals it is written as (4) (70) (200). 1 Theories about their Origin. Quite a large number of theories have been advanced to explain the origin of the Brahmi numerals. Points of resemblance have been imagined between these numerals and those of other nations. Recourse has been taken by writers to the turning, twisting, adding on or cutting off of parts of the numerals of other nations to fit their pet theories'. It is needless to say that each of these theories had its own supporters who were quite convinced of the correctness of their explanations. We give below the outlines of some of these theories: \. Cunningham believed that writing had been known in India from the earliest known times, and 1 Compare the- same number written in KharosthI, p. 24. 2 Inscriptions of Asoka, Corpus Inscriptionum Indicarum , Vol. I, P- J*- BRAHMI NUMERALS 29 that the earliest alphabet was pictographic. He suggest- ed that the Brahmi script was derived from the early pictographic writing. The theory is evidently capable of extension to the numerical signs. Later epigraphists, however, discarded the hypothesis as it appeared too fanciful to them. Cunningham's bold hypothesis re- garding the antiquity of writing in India has been more than justified by the recent discovery of the use of a quasi-pictographic script on certain seals and in inscriptions belonging to the fourth millenium B.C. found amongst the excavations at Mohenjo-daro and Harappa. His theory has been revived by Langdon who is of opinion that the Brahmi alphabet could be derived from the pictographs of Mohenjo-daro. 1 The theory is incomplete as the writings of Mohenjo-daro have not been completely deciphered as yet. It can be called a guess only. As regards the evolution of the Brahmi numerals, it may be stated that it is at present extremely difficult to differentiate the numerical symbols from the Mohenjo-daro script. If the surmise that the figures, given on p. 19, are numerical symbols be correct, it will not be possible to develop a theory deriving the Brahmi numerals from them. 2. Bayley 2 asserted that the principles of the Brahmi system have been derived from the hieroglyphic notation of the Egyptians, and that the majority of the Indian symbols have been borrowed from Phoenician, Bactrian, and Akkadean figures or letters. As has been already remarked 3 the principles of the Brahmi and the hieroglyphic systems are entirely different and 1 Mohenjo-daro etc., Chap. xii. This view is strongly supported by Hunter, I.e., p. 490. 2 Journal of the KojaJ Astatic Soc, XV, part I, reprint, London, 1882, pp. 12 and 17. The theory was supported by Taylor, The Alphabet, London, 1883, Vol. II, pp. 265-66. 6 See pages 27-8. 50 NUMERAL, NOTATION unconnected. The reader will find the hieroglyphic and the Brahmi systems shown together in Tables 11(a), (b), (c), and convince himself of the incorrectness of Bayley's assertion. Moreover, the assumption that the Hindus borrowed from four or five different, partly very ancient and partly more modern, sources, is extremely difficult to believe. Regarding the resemblance between the Bactrian and Akkadean numbers and the Brahmi forms postu- lated by Bayley, Biihler 1 remarks that in four cases (four, six, seven and ten) the facts are absolutely against Bayley's hypothesis. Some writers have also criticized Bayley's drawings as being affected by his theory. 2 Under these circumstances his derivation has to be rejected. 3. Burnell 3 pointed out the general agreement of the principles of the Indian system with those of the Demotic notation of the Egyptians. He asserted a resemblance between the Demotic signs for 1 to 9 and the corresponding Indian symbols, and put forward the theory that the Hindus borrowed these signs and later on modified them and converted them into aksaras (letter forms). 4. Biihler 3 has put forward a modification of Burnell's theory. He states, "It seems to me probable that the Brahma numerals are derived from the Egyptian Hieratic figures, and that the Hindus effected their transformation into A.ksaras, because they were already accustomed to express numerals by words." The above theories like the one examined before are not well founded. Tables II (a), (b), (c), show the Hieratic and Demotic symbols together with those of the Brahmi. An examination of the Tables will reveal 1 Biihler, On the Origin of the Indian Brahma Alphabet, Strassburg, 1898, pp. 52, 53 foot-note.' 2 Cf. Smith and Karpinski, Hindu Arabic Numerals, pp. 30-1. 3 Biihler, I.e., p. 8i. BRAHMI NUMERALS 3 I that out of the nineteen symbols to represent the num- bers from i to ioo, only the nine of the Brahmi resem- bles the corresponding symbol of the Demotic or the Hieratic. ' There is absolutely no resemblance between any of the others. To base the derivation on a resem- blance between the Hieratic 5 and the Brahmi 7, as is sought to be done, is absurd. Likewise the changing and twisting of the Demotic and Hieratic forms to suit the theory is\ unacceptable. That there is some resemblance between these systems in the fact that each employs the same number of signs, i.e., nineteen, for the representation of numbers upto hundred, cannot be denied. There is, however, a difference in the method of formation of the hundreds and the thousands. In the Brahmi the numbers 200 and 300 or 2,000 and 3,000, are formed by adding one mdtrkd and two mdtrkds to the right of the symbol for hundred or thousand respectively, thus /v ^ = 100, jf = 200, ^/ = 3°° n = 1,000, CT = 2,000, ^f = 3,000. The numbers 400 and 4,000 are formed by connecting the symbol for 100 and 1,000 to the number V (4), thus /V p» = 400 and <^ = 4>°°°- In the Hieratic the corresponding symbols are: 4 = 2 "i — 3 __J» ^ IOO, J> = 200, 5 = 1,000, >-w 2,000. "*) — 4, 32 NUMERAL NOTATION J0 = 300, -*H = 3,000, -^ = 4°o, -^ = 4,000. It will be observed that in the Hieratic system the sign for one thousand is not used in the formation of the other thousands. The similarity in principle, even if it were complete, would not force us to conclude that one of these nations copied the other. The use of nineteen signs afforded the easiest and probably the best method of denoting numbers. It is not beyond the limits of probability that what appeared easy to the Egyptians might have also independently occurred to the Hindus. There are on the other hand some considerations which make us suggest that the Egyptians borrowed the principles of the Hieratic and the Demotic systems from outside, and probably from India — a hypothesis which is not a priori impossible as it has been shown that the numeration system of the ancient Hindus based on nineteen signs might have been perfected about 1,000 B.C. It is known that the ancient Egyptian system employed only four signs, those for 1, 10, zo and 100. Why should there be a sudden change from the old system to one containing nineteen signs cannot be adequately explained except on the hypothesis of foreign influence. Further, the cursive forms for the numbers 2, 3 and 4 are unsuited to the right to left Hieratic or Demotic script. Although these figures are connected with the earlier hieroglyphic and Phoenician figures, yet it is possible that the cursive combinations might have been formed to obtain the nineteen signs necessary for the new system, under the influence of a people with a left to right script. It may be, however, asserted that the hypothesis of an Indian origin of the Hieratic system BRAHMI NUMERALS 33 is a mere suggestion. The two points noted above, by themselves, would not be enough, unless backed by- other facts, to put forward a theory. It is expected that further discoveries will throw light on this point. Relation with Letter Forms. It was suggested by James Princep, 1 as early as 1838, that the numerals were formed after the initial letters of the number names. But knowing the pronunciation of the number names, we find this not to be the case. Other investi- gators have held that the numeral signs were formed after the letters in the order of the ancient alphabet. Although we find that letters were used to denote numbers as early as the 8th century B.C., 2 and that many systems of letter-numerals were invented in later times 3 and came into common use, yet we are forced to reject this hypothesis as resemblance between the old numerical forms and the letters in the alphabetic order cannot be shown to exist. A peculiar numerical notation, using distinct letters or syllables of the alphabet, is found to have been used in the pagination of old manuscripts as well as in some coins and a few inscriptions. ' The signs are, however, not always the same. Very frequently they are slightly differentiated, probably in order to distinguish the signs with numerical values from those with letter values. The fact that these symbols are letters is also acknow- ledged by the name aksarapalli which the Jainas occa- sionally give to this system, in order to distinguish it from the decimal notation, the ankapallt.* 1 "Examination of inscriptions from Girnar in Gujerat, and Dhauli in Cuttack," JASB, 1838. 2 The method seems to have been used by Panini. See p. 63. 3 Vide infra, pp. 64ff. 4 Buhler, I.e., p. 78. The details of the aksarapalli are given later on (pp. 72fF). 36 NUMERAL NOTATION to put forward the hypothesis that the Brahrru numerals are derived from the letters or syllables of the Bsahmi script. The Pandit, however, admitted his inability to find the key to the system, nor has it been found by any other scholar upto this time. The problem, in fact, appears to be insoluble, unless further epigraphic material is discovered to show the forms of the numeri- cal symbols anterior to Asoka. The Asokan forms as well as those of later inscriptions are in a too well developed state, and are too far away from the time of invention of those symbols, to give us the desired information regarding their origin. But of all the theories that have been advanced from time to time, that of Pandit Indraji seems to us to be the most plausible. The Hindus knew the art of writing in the fourth millennium B.C. They used numbers as lajrge as io B about 2,000 B.C., and since then their religion and their sciences have necessitated the use of large numbers. Buddha in the sixth century , B.C. is stated to have given number names as large as 10 53 and this number series was continued still further in later times. 1 All these facts reveal a condition that would have been impossible unless arithmetic had at- tained a considerable degree of progress. It is certain that the Hindus must have felt the necessity of some method of writing these numbers from the earliest known times. It would not be, therefore, against historical testimony to conclude that the Hindus invented the Brahmi number system. The conclusion is sup- ported by the use, in writing numbers, of the mdtrkd, the anundsika and the upadhmdniya signs which are found only in the Sanskrit script and in no other script, whether ancient or modern. It is further strengthened by Indian tradition, Hindu, Jaina as well as Buddhist, which 1 Cf. pp. 10-12. BRAHMI NUMERALS 37 ascribes the invention of the Brahmi script and the num- eral notation to Brahma, the Creator, and thereby claims it as a national invention of the remotest antiquity. 1 Period of Invention. The invention of the system may be assigned to the period 1,000 B.C. to 600 B.C. As the Asokan numerical figures indicate that the system was common all over India, 2 and that it has had a long history, the lower limit 1,000 B.C. is certainly not placed too early. On the other hand general con- siderations, such as the high development of the arts and the sciences, the mention of numerical signs and of 64 different scripts in ancient Buddhist literature, 3 and the use of large numbers at a very early period, all point to the date of the invention of the system as being nearer to 1,000 B.C., if not earlier. Resume. The strength of Pandit Indraji's hypo- thesis lies in the fact that out of the nineteen signs, eleven definitely resemble the letters or the signs of the Brahmi alphabet. The resemblance is too striking to be entirely accidental. Moreover, it has been found that the numerical forms closely followed the changing forms of the letters from century to century. This is especially true in the case of the tens and shows that the writers of the ancient inscriptions knew the phonetical values of these symbols. The divergence from letter forms in the case of the signs for the units may be due to the 1 Biihler {I.e., p. i, foot-note,) quotes several authorities. Of these the Ndrada Smrti and the Jaina canonical -work, the Samavd- ydnga-sutra, belong to the fourth century B.C. 2 Megasthenes speaks of mile-stones indicating distances and the halting places on the roads. Indika of Megasthenes, pp. 127-126; Biihler, I.e. 3 Related in the ~Lalitavistara, both in the Sanskrit text and the Chinese translation of 308 A.D. The Jaina Samavdydnga-sfitra {c. 300 B.C.) and Pannavand-siitra (c. 168 B.C.) each gives a list of 18 scripts; see Weber, Indische Studien, 16, 280, 399. 38 NUMERAL NOTATION fact that they were the first to be invented and were in more common use, so that they acquired special cursive forms and did not follow the changes in the forms of the corresponding letters. We may now summarize the discussion given in this section by saying that, (i) the Brahmi numerical forms were undoubtedly of Indian origin, (2) the form of the tens were derived from certain letters or signs of the alphabet, and (3) the origin of the forms of the units is doubtful. It is probable that they, too, were fashioned after the letters of the alphabet, but there appears to be no means of justifying this assertion unless the forms of these numerals anterior to Asoka are discovered. 8. THE DECIMAL PLACE-VALUE SYSTEM Important Features. The third and most im- portant of the Hindu numeral notations is the decimal place-value notation. In this system there are only ten symbols, those called anka (literally ' meaning "mark") for the numbers one to nine, and the zero symbol, ordinarily called sunya (liter- ally, "empty"). With the application of the principle of place-value these are quite sufficient for the writing of all numbers in as simple a way as possible. The scale is, of course, decimal. This system is now commonly used throughout the civilised world. Without the zero and the place-value, the Hindu numerals would have been no better than many others of the same kind, and would not have been adopted by all the civilised peoples of the world. "The importance of the creation of the zero mark," says Professor Halsted, "can never be exaggerated. This giving to airy nothing, not merely a local habitation and a name, a 'picture, a symbol, but helpful power, is the characteristic of the Hindu race whence it sprang. THE DECIMAL PLACE- VALUE SYSTEM 39 It is like coining the Nirvana into dynamos. No single mathematical creation has been more potent for the general on-go of intelligence and power." 1 Forms. A large number of scripts differing from each other are in use in different parts of India today. The forms of the numerical signs in these scripts are also different. Although all the Hindu scripts are derived from a common source — the Brahmi Script — yet the differences in the forms of the various modern Indian scripts are so great that it would have been difficult to establish any relation between them, if their previous history had not been known. The above remark applies to the numerical signs also, as will appear from a study of the numerical signs in the various vernaculars of India given in Table XV. The great divergence in the forms of the numerical symbols shows that in India, people already knew the use of the zero and the place- value principle before the different scripts came into being, and that the numeral forms were independently modified in various parts of India, just as the letters of the alphabet were modified. And as the changes in the forms in different localities were independent of each other, so there has come about a great divergence in the modern forms. That this divergence already existed in the eleventh century is testified to by Al- Biruni who says, "As in different parts of India, the letters have different shapes the numerical signs, too, which are called arika, differ." 2 Nagari Forms. The most important as well as the most widely used of the different symbols are those belonging to the Nagari script. The present forms of these symbols are: 1 G. B. Halsted, On the foundation and technique of Arithmetic, Chicago, 19 1 2, p. 20. 2 Alberuni's India, I, p. 74, i-O NUMERAL NOTATION ?, R, \> *, \, S, », ^ % °- The gradual development of these figures from the Bjrahml numerals is shown in Table XIV. Epigraphic Instances. The following is a list of inscriptions and grant plates upto the middle* of the tenth century, which contain numerals written in the decimal place-value notation. The numerals in the inscriptions and plates after this period, are always given in decimal figures. i. 595 A.p. Gurjara grant plate from Sankheda, (EI, II, p. 19). The date Sam vat 346 is given in the decimal place- value notation. *2. 646 A.D. Belhari Inscription, (J A, 1865). *3. 674 A.D. Kanheri Inscription, (J A, 1863, p. 392). 4. 8th Century Ragholi plates of Jaivardhana II, (EI, IX, p. 41). The number 30 is written in decimal figures. j. 725 A.D. Two Sanskrit Inscriptions in the British Museum, (I A, XIII, p. 250). The dates Samvat 781 (=723 A.D.) and Samvat 783 (=725 A.D.) are given in decimal figures. *6. 736 A.D. Dhiniki copper plate gfant, (IA, XII, p. 155). The date Vikrama Sam- vat 794 is given in decimal figures. 7. 753 A.D. Ciacole plates of Devendravarmana, (EI, III, p. 133). The number 20 is written in decimal figures. 8. 754 A.D. Rastrakuta grant of Dantidurga, (IA, XI, .p. 108). The date Samvat 675 is given in decimal figures. THE DECIMAL PLACE-VALUE SYSTEM 41 9. 791 A.D. Inscription of Samanta Devadatta, (I A, XIV, p. 3 5 i). The date Vikra- ma Sarhvat 847 is given in decimal figures. 10. 79 j A.D. Daulatabad plates of Sahkargana, (EI, IX, p. 197). The date Saka 71 j is given in decimal figures. *n. 813 A.D. Torkhede plates, (EI, III, p. 53; also IA, XXV, p. 345). The date Saka Sarhvat 735 is given in decimal figures. 12. 815 A.D. Buchkala inscription of Nagbhata, (EI, IX, p. 198). The date Sarhvat 872 is given in decimal figures. 13. 837 A>iD. Inscription of Bauka (Rajputana Museum, PLM, p. 127; EI, XVIII, jp. 87). The date Vikrama Sarhvat 894 is given in decimal figures. 14. 843 A.D. The inscriptions from Kanheri, No. 43 b., (IA, VIII, p. 133). The date Sarhvat 765 is given in decimal figures. 15. 851 A.D. The inscriptions from Kanheri, No. 15, (Ibid). The date 1 Sarhvat 775 is given in decimal figures. 16. 853 A.D. Pandukesvara Plates of Lalitasura- deva, (IA, XXV, p. 177). The date Sarhvat 21 of the King's reign is given in decimal figures. 17. 860 A.D. Ghatiyala Inscription of Kakkuka (EI, IX, p. 277). The date Vikra- ma Sarhvat 918 1 is given in decimal figures. 1 For correction of date see I A, XX, p. 421. 42 NUMERAL NOTATION 1 8. 862 A.D. Deogarh Jaina Inscription of Bhoja- deva, (EI, IV, p. 309). The dates Vikrama Sarhvat 919 and the corres- ponding Saka Sarhvat 784 are both given in decimal figures. 19. 870 A.D. Gwaliot inscription of the reign of Bhojadeva {Archaeological Survey of India, Report, 1903-4, plate 72). Al- though the date is not given, the slokas are numbered from 1 to 26 in decimal figures. 20. 876 A.D. Gwalior inscription of Allah, of the reign of Bhojadeva (EI, I, p. 159). The date Vikrama Sarhvat 933, as well as the numbers 270, 187 and 50 are given in decimal figures. 21. 877 A.D. The inscriptions from Kanheri, No. 43a, (I A, XIII? p. 133). The date Sarhvat 799 is. given in decimal figures. 22. 882 A.D. Pehava inscription (EI, I, p. 186). The date Sarhvat 276 (Sri Harsa Era) is given in decimal figures. 23. 893 A.D. Grant plate of Balavarmana, (EI} IX, p. 1). The date Vallabhi Sam- vat 5 74 is given in decimal figures. 24. 899 A.D. Grant plate of Avanivarmana, (EI, IX, p. 1). The date Vikrama Sam- vat 956 is given in decimal figures. 25. 905 A.D. The Ahar stone inscription (journ. United Provinces Hist. Soc, 1926, pp. 83 ff) contains several dates, written in decimal figures. 26. 910 A.D. Rastrakuta grant of Krishna II (EI, THE DECIMAL PLACE-VALUE SYSTEM 43 I, p. 53). The date is given in decimal figures. 27. 917 A.D. Sanskrit and old Canarese inscrip- tions, No. 170, (I A, XVI, p. 174). The date Sam vat 974 is given in decimal figures. The number 500 also occurs. 28. 930 A.D. Cambay plates of Govinda IV, (EI, VII, p. 26). The date Saka Sarhvat 852 is given in decimal figures. 29. 933 A.D. Sangli plates of Rastrakuta Govin- daraja IV, (IA, XII,' p. 249). The date Sarhvat 855 is given in decimal figures. 30. ■ 951 A.D. Sanskrit and old Canarese inscrip- tions, No. 135, (I A t XII, p. 257). The date Sarhvat 873 is given in decimal figures. 31. 953 A.D. Inscription of Yas*ovarmana, (£2, I, p. 122). The date Sarhvat ion is given in decimal figures. 32. 968 A.D. Siyadoni stone inscription (EI, I, p. 162). The inscription contains a large number of numerals expressed in decimal figures. 33. 972 A.D. Rastrakuta grant of Amoghavarsa, (IA, XII, p. 263). The date Saka 894 is given in decimal figures. Palaeographic evidences of the early use of the decimal place-value system of notation are found in the Hindu colonies of the Far East. 1 The most important ones among these are the three inscriptions of 1 G. Coedes, "A propos de l'origine des chiffres arabes," Bull. School of Oriental Studies (London), VI, 1931, pp. 323-8. 44 NUMERAL NOTATION Srivijaya, two found at Palembang in Sumatra, and the third in the island of Banka. These contain the dates 605, 606 and 608 of the Saka Era (corresponding respectively to A.D. 683, 684 and 686) written in numerical figures. Another instance giving the date 605 Saka is the inscription or Sambor in Cambodia. In an inscription at Po Nagar in Champa, occurs the date 735 Saka (=813 A.D.). Their Supposed Unreliability. The above list contains more than thirty undoubted epigraphic ins- tances of the use of the place-value notation in India. G.R.Kaye, 1 who believes in the theory of the non- Hindu origin of the place-value notation, states that all the early epigraphic evidences of its use in India are unreliable. On the basis of the existence of a few forged grant plates he asserts that in the ele- venth century "there occurred a specially great oppor- tunity to regain confiscated endowments and to acquire fresh ones" and thereby concludes that all early epi- graphic evidences must be unreliable. Such reasoning is obviously fallacious and needs no refutation. Most of the copper plates are legal documents recording gifts made by rich persons 01 kings to Brah- manas on religious occasions. The plates contain de- tails as to the occasion for making the gifts, the names of the donor and the donee, the description of the mov- able and immovable properties transferred by the gift, and the date of the gift which is always written out in full in words and very often in figures also. The forgeries may be of two kinds: (1) In the original documents, parts relating to either the names of the donor. or the donee, or the description of the immovable property may have been obliterated by being beaten out and new 1 "Notes on Indian Mathematics," J*4SB, (N. S., 1907), III, pp. 482-87. THE DECIMAL PLACE- VALUE SYSTEM 45 names or descriptions substituted. All such forgeries are easily detected, because of the uneven surface of the part of the plate that is tampered with and the difference in the writing. (2) An entirely new document may be forged. Such cases, though rare, are also easily detected, because there is obvious divergence as to the date recorded in the docu- ment, and that inferred on the basis of the forms of the characters used in the writing. Such for- geries are also marked by an obvious inferiority in execution, and inaccuracies in the statement of genea- logies and other historical facts. Epigraphists have so far found little difficulty in eliminating the spurious grant plates. It might be mentioned that the genuineness of the grant plates included in our list has not been questioned by any epigraphist. 1 Kaye, in his article quoted above, has given a list of eighteen inscriptions and grant plates and eliminates all but the last two as forgeries. The arguments he has employed and the assertions of facts that he has made are in most cases incorrect and misleading, so that his conclusions cannot be accepted. As an instance of his method, we quote his criticism about the Gurjara grant plate, No. 1 in our list. He writes: "Dr. Biihler quotes this Gurjara inscription of the Chedi year 346 or A.D. 594 as the earliest epigraphic instance of the use of the decimal notation in India, (i) An examination 1 If any of them is forged, the forgery is so good that it can- not be detected. The writing in such cases, if any, is so well forged as to be indistinguishable from that used in the period to which the plate is said to belong. Therefore, the evidence of these plates as tq the method of writing numbers, cannot be rejected, even if they be proved to be spurious at some future date — a con- tingency which is very unlikely to use. It may also be noted that the list contains several sfone inscriptions which cannot be spurious. 46 NUMERAL NOTATION of the plate (Ep. Ind., II., p. 20) suggests the pos- sibility that the figures were added some time after the plate was engraved. The date is engraved in words as well as in figures. It is 'three hundred years exceeded by forty-six/ The symbols are right at the end of the inscription from which they are marked off by a double bar in A most unusual manner, (ii) The figures are of the type of the period, but they were also in use much later, and in no other example are such symbols used with place-value, (iii) Also there are nine dates written in the old notation (Ep. Ind., V), e. g., there is another grant of the Gurjara of Bharoch in which the date Samvat 391 (i.e., A.D. 640) is given in the old notation. Again, there is no other Chedi date, at least before the eleventh century A.D., given in the modern (place-value) notation, (iv) There cannot be the remotest doubt as to the unsoundness of this particular piece of evidence of the early use of the modern system of notation in India." The following remarks will show to the reader that Kaye's criticism and his conclusion are unfounded and invalid: (;) An examination of the plate, (El, II, p. 19), will convince every one about its genuineness. The writing is bold' and clear, the numerical figures occur at the end, as they ought to be, immediately after the words 'three hundred years exceeded by forty-six.' They are separated from the written words by bars, just as they ought to be. There is absolutely nothing suspicious about this method of separation, as it is common custom in India to do so and occurs frequently. That it was the practice to write the date at the end of a document is well known. 1 In fact, the numeral 1 ,Many of the plates mentioned in our list contain the date at the end. THE DECIMAL PLACE-VALUE SYSTEM 47 figures of the date occasionally mark the end of the document. 1 The double vertical bar, 1 1 , is a sign of inter- punctuation. Although punctuation marks have been in use in India from the earliest known times, yet their use did not become either regular or compulsory till very recent times. 2 Different writers used the various marks differently.. In inscriptions, the double vertical bar has been found at the end of sentences, half verses, verses, larger prose sections and documents. In the Junar inscriptions it occurs after numerals and once after the name of the donor. 8 In manuscripts, the practice of separating numbers by vertical bars is com- mon. It is found in the Bakhshali Manuscript* and in several others. Thus the occurrence of the numerals at the end and the inter-punctuation mark of the double vertical bar cannot form valid grounds for suspecting the document. The suggestion that the figures were added some time after the plate was engraved is absurd, as there appears to us no reason why one should take the trouble to add the figures when the date was already written in words. («) Kaye admits that the figures are of the type of the period. His remark that they were in use much later is incorrect. The Tables III-V and XII show that the use of three horizontal bars to represent 3 is not 1 This is so in the Chargaon plates of Huviska (Arch. Survey Report, 1908-9, plate 56), in the Inscription of Rudradamana (J. A, VIII, p. 42) and in others. ? There are some copper plate grants which do not contain any punctuation marks; see Buhler, I.e., p. 90. 3 Buhler, i.e., p. 89. 4 E.g., I 5 I , 21 r ; I 2558 I , 2v; I 330 I , 17V; instances such as these: | 1 | 4 | 9 | 16 | , i6y; and | 2 | , '] 4 | , etc., 5V. are very common. Very often, isolated numbers are not separated. The double vertical bar also occurs before and after the words udd, siilram, etc. 48 NUMERAL NOTATION found after the eighth century. The figure for 4 used in our grant plate is not found after the sixth century, and the same is true for the figure for 6. The forms of the numerical signs alone fix the date of the writing to the sixth century and not later. (/»') The Chedi Samvat is one of the thirty-four eras, whose use has "been discovered in inscriptions and grant plates. The occurrence of nine dates in the Chedi Sam- vat, written in the old notation after this plate, does not prove the unsoundness of this particular piece of evi- dence, as Kaye would like us to conclude. It simply shows that in India too, the new system had to fight for supremacy over the older one just as in other countries. In Arabia the new system was introduced in the eighth century, but it did not come into common use until five or six hundred years later. In Europe we find that it was exceptional for common people to use the new system before the sixteenth century — a good witness to this fact being the popular almanacs. Calen- dars of 1 5 5 7-96 have generally Roman numerals, while Koebel's Calendar of 1578 gives the Hindu numerals as subordinate to the Roman. 1 We may, therefore, conclude that the Gurjara grant plate offers us a genuine instance of the use of the new system (with place-value) in India. Kaye's criticisms regarding the genuineness of some other plates included in our list (marked with asterisks) have been found to be baseless. Place of Invention of the New System. It has been already stated that the same numeral forms for the numbers 1 to 9, as were in use in India from the earliest known times, have been used in the new system of nota- tion with the place-value. Another noteworthy fact 1 Smith and Karpiftski, I.e., p. 133. THE DECIMAL PLACE-VALUE SYSTEM 49 regarding the new system is the arrangement of the atika (digits). It will be observed that the arrangement in the old system was that the bigger numbers were written to the left of the smaller ones. 1 This same arrangement continues in the new system with place- value, where the digits to the left, due to their place or position, have bigger values. The gradual change from the old system to the new one using the same numerical signs, is to be found in India alone, and this, in our opinion, is one of the ' strongest arguments in favour of the Hindu origin of the new system. The earliest epigraphic instance of the use of the new system is 594 A.D. No other, country in the world offers such an early instance of its use. Epigraphic evidence alone is, therefore, sufficient to assign a Hindu origin to the modern system of notation. Inventor Unknown. It is not known who the inventor of the new system was, and whether it was invented by some great scholar, or by a conference of sages or by gradual development due to the use of some form of the abacus. Likewise, it is not known to which place, city, district or seat of learning belongs the honour of the invention and its first use. Epigra- phic evidence cannot help us in this direction. For the system was used in inscriptions, a very long time after its invention, in fact, when it had become quite popular all over Northern India. Time of Invention. The grant plates were legal documents. They were written by professional writers. The existence of such writers is mentioned in the southern Buddhist canons and in the Epics. 2 They have 1 Showing thereby that the place assigned to a numeral de- pended upon its value. This has been incorrectly thought to be a sort of place-value system by some writers. 2 Biihler, I.e., p. 5. 4 JO NUMERAL NOTATION been called lekhaka, lipikara and later on divira, karana, kayastha, etc. According to Kalhana, 1 the Kings of Kashmir employed a special officer for drafting legal documents. He bore the title of pattopddhyaya, i.e., the teacher (charged with the preparation) of title deeds. The existence of manuals such as the Lekhapancasika, the Lek/japrakaJa, which give rules for drafting letters, land grants, treaties, and various kinds of bonds and bills of exchange, show beyond doubt that the writing of grant plates was a specialised art and that the style of writing those documents must always have been centuries behind the times, just as it is even to-day with respect to legal and state documents. The time of invention of the new system must, therefore, be placed several centuries before its first occurrence in a grant plate in the sixth century A.D. The exact period of invention may be roughly deduced from the history of the growth of numerical notations in other countries. According to Heath, 2 the Greek alphabetic notation was invented in the 7th century B.C., but it came into general use only in the second century A.D. Thus it took about eight hundred years to get popular. In Arabia the new notation was introduced in the 8th century A.D., but it came into common use about five or six hundred years later. The same was the case in Europe. The Arabs got the complete decimal arith- metic, including the method of performing the various operations, at a period when intellectual activity in Arabia was at its greatest height, but they could not make the decimal system common before about five or six hundred years had elapsed. 3 In legal documents 1 Kajatarangjnt, V, pp. 397k 2 Heath, History of Greek Mathematics, I, Oxford, 1921, p. 34. 3 The arithmetic written by Al-Kharki in the eleventh century does not use the decimal system, showing that at the time there were two schools amongst the Arab mathematicians, one favouring THE DECIMAL PLACE-VALUE SYSTEM J I and in recording historical dates, the Arabs even now use their old alphabetic notation. Epigraphic evidences show that the new system was quite common "in India in the eighth century and that the old system ceased to exist in Northern India by the middle of the tenth century. This would, therefore, place the invention of our system in the period bet- ween the first century B.C. and the third century A.D. . The exact date of the invention, however, would be nearer to the ist century B.C. or even earlier, because for a long time after its invention, the system must have been looked upon as a mere curiosity' and used simply for expressing large numbers. A still longer time must have elapsed before the method of perform- ing the operations of addition, subtraction, multiplica- tion, division and the extraction of roots, could be perfected. It would be only after the perfection of the methods of performing the operations that the system could be used by mathematicians. And then after this it would take about five hundred years, as in Arabia, to become popular. There should, therefore, be a gap of about eight centuries between the time of invention and its coming into popular use, just as was the case with the Greek alphabetic notation. There- fore, on epigraphic evidence alone, the invention of the place -value system must be assigned to the begin- ning of the Christian era, very probably the ist century B.C. This conclusion is supported by literary and other evidences which will be given hereafter. the Hindu numerals, while the other stuck to the old notation. See the article on "Hisab" by H. Suter.in the Encyclopaedia of Islam. 5Z NUMERAL NOTATION 9. PERSISTENCE OF THE OLD SYSTEM The occurrence of the old system of writing numbers, with no place-value, is found generally in inscriptions upto the seventh century A.D., after which it was gradually given up in favour of the new system with place-value. Occasional use of the old system, however, is to be met with in Nepal and in some South Indian inscriptions upto the beginning of the tenth century A.D., but after this period the old system seems to - have been forgotten, and completely gone out of use. In the seventh century the new system was in general use, but the old system seems to have been given preference in inscriptions. There are a number of grant plates of the eighth century A.D., in which the dates, although written in the old notation, are incorrectly inscribed, showing thereby that people had already forgotten the old system. In a grant plate of Siladitya VI, 1 dated the Gupta year 441 (c. 760 A.D.), the sign for 40, instead of the sign for 4, has been Sub- joined to the sign for 100 to denote 400, i.e., 4,000 has been incorrectly written for 400. There is another grant plate, dated the Garigeya year 183 {c. 753 A.D.), in which the figure 183 is wrongly written. 2 This plate is of special interest as it exhibits the use of the old and the new systems in ihe same document. 3 Another very inter- esting instance of the use of the old and the new systems in one and the same document is the Ahar stone 1 1 A, VI, p. 19, (plate). 2 EI, III, p. 133, (plate). In this the sign of 8 is written for 80 and that of 30 for 3. The number 20 has been written by placing a dot after 2. a For other instances showing admixture of both the old and the new systems, see Fleet Gupta Inscriptions, Corpus Inscriptiomwi Indicarnm, III, p. 292; also IA., XIV, p. ys\, where (800) (4) (9) -849. WORD NUMERALS 5 3 inscription. 1 The document records gifts made on several occasions ranging over thirty-seven years, the last entry corresponding to 905 A.D. In this inscription the old notation is used in the first six lines whilst in the follow- ing lines it has been discarded and the new place-value notation appearsT It is evident from the forms that the writer did not know the old system. For instance, 200 is written by adding the subscript 2 to the letter su (ico), instead of using a mdtrkd sign as in the old system. In the same way the sign for 10 is incorrect in so far as a small zero has been affixed to the usual sign for ten. The inscription shows that although the old system had gone out of use completely, yet people tried to use it in inscriptions, probably for the same reason that makes us use the Roman numerals in giving dates, in number- ing chapters of books, and in marking the hours on the face of a clock, even upto the present day. 10. WORD NUMERALS Explanation of the System. A system of expressing numbers by means of words arranged as in the place-value notation was developed and perfected in India *n the early centuries of the Christian era. In this system the numerals are expressed by narpes of things, beings or concepts, which, naturally or in accordance with the teaching of the Sdstras, connote numbers. Thus the number one may be denoted by anything that is markedly unique, e.g., the moon, the earth, etc.; the number two may be denoted by any pair, e.g., the eyes, the hands, the twins, etc.; ?nd similarly Others. The zero is denoted by words meaning void, sky, complete, etc. 1 C. D. Chatterjec, "The Ahar stone inscription," Jonrn. United Provinces Hist. Soc, 1926, pp. 83-119. 54 NUMERAL NOTATION The system is used in works on astronomy, mathe- matics and metrics, as -well as in the dates of inscrip- tions and in manuscripts. The ancient Hindu mathema- ticians and astronomers wrote their works in verse. Consequently they strongly felt the need for a convenient method of expressing the large numbers that occur so often in the astronomical works and in the statement of problems in mathematics. The word numerals were invented to fulfil this need and soon became very popular. Thev are used even upto the present day, whenever big numbers have to be expressed in Sanskrit verse. The words denoting the numbers from one to nine and zero, with the use of the principle of place-value, give us a very convenient method of expressing numbers by word chronograms. To take a concrete case, the number 1,2.30 mav be expressed in many ways: 1 . kba-guna-kara-ddi , z . kba-loka-karna-candra, 5. dkdsci-kdhi-netni-dbard, etc. It will be observed that the same number can be expressed in hundreds of ways by word chronograms. This property makes the word numerals specially suit- able for inclusion in metre. To secure still greater variety, the numbers beyond ten are also sometimes denoted by words. List of Words. The following is a list of words commonly used in this system to denote numbers: o is expressed by iunya, kha, gagam, ambara, dkdsa, abhra, vtyat, vyon/a, antarikja, nabhci, jaladbarapatba, piirna, randhra, visnnpada, ananta, etc. \j is expressed by ddi, sasi, indn, vidhu, catidra, ktild- dbara, himagn, s'//d///s'u, ksapdkpra, hin/di/isu, sitaraswi, prdleyawsu, soma, sasdrika, mrgchika, hiwakara, sudbdmsu, mjanikctra, s'as'adhara, u-e/a, abja, bbu, WORD NUMERALS 55 bhumi, ksiti, dhard, tirvard, go, vasundhard, prtlm, ksmd, dharani, vasudhd, ild, ku, mahi, rnpa, pitdwdha, ndyaka, tana y etc. 2 is expressed by yama, yamala, asvin, ndsatya, dasra, locana, netra, aksi, drsti, caksn, ainbaka, najana iksana, ^paksa, bdhu, kara, karna, kuca, ostha, gulpha, jdnu, jaiighd, dvaya, dvanda, jugala, yugma, ayana, kutumba, ravicandrau, nay a, 1 etc. 3 is expressed by rdma, guna, triguna, loka, trijagat, bhitvana, kdla, trikdla, trigata, trinetra, haramtra, sahodardh, agni, ana la , vafmi, pdraka, vaisvdnara, dahana, tapana, hutdsana, jvalana, sikhin, krsdnu, botr, pur a, ratna? (J aim), etc. 4. is expressed by veda, srnti,^ samtidra, sdgara, abdhi, atnbhodha, ambhodhijaladbi, iidadhi,jalanidhi, salildkara, visanidhi, vdridhi, payodhi, payonidhi, ambudhi, kendra* varna, dsrama, yuga, turya, krta, aya, dya, dis, bandhu, kostha, gati, kasdya, etc. 5 is expressed by banc, sara, sastra, sdyaka, isu, bhuta, parva, prdna^pavana* pdndava,artha, visaya, tuahdbhiita, tatva, bhdva, indriya, ratna, karamya* vrata, etc. 6 is expressed by rasa, anga, kdya, rtu, mdsdrdha, dar- s'ana, rdga, art, sdstra, tarka, kdraka, lekhya, dravya, 6 ik/jara, kumdravadana, sanmukha, etc. 7 is expressed by naga, aga, bhiibhrt, parvata, saila, acala, adri, giri, rsi, muni, yati, atri, vara, svara, 1 Method of comprehending things from particular stand- points — dravydrthika and parydydrthik.a. 2 Used by Mahavira only; others take it for five. 5 3 See SiSe, i. 27; SiSi, ganitddhydya, x. 2. Used also for 7 (See the quotations by Bhattotpala in his commentary on Brhat- samhitd, ch. ii). In AI-Biruni's list it is erroneously put for 9. 4 That which ought to be done; according to the Jainas — abimsd, snnrta l asteya, brabrnacarya l and aparigraba. r ' Used by Mahavira. J 6 NUMERAL NOTATION dhdtu, aha, turaga, vdji, haya, chandah, dhi, kalatra, tatva? dvrpa, patmaga," 1 bhaya* mdtrkd, vyasana, etc. 8 is expressed by vasu, ahi, ndga, gaja, danti, dvirada, diggaja, has tin, ibba, mdtanga, kunjara, dvipa, puskarin, sindhura, sarpa, taksa, siddhi, bhuti, anustubha, maii- gala, anika, karman* durita, tanu? dik," mada, 7 etc. 9 is expressed by arika, naiida, nidhi, graha, randhra, chidra,dvdra,go* upendra, kesava, tdrksyadhvaj, durgd, paddrtba* labdha, labdhi, etc. io is expressed by dis, dik, disd, dsd, angtdi, pankti, kakubb, rdvanasira, avatdra, karman, etc. ii is expressed by rudra, isvara, uirda, hara, isa, bhava, bharga, sulirt, mahadeva, aksaubini, etc, iz is expressed by ravi, stirya, ina, arka, mdrtanda, dyumani, bbdnu, dditya, divdkara, wdsa, rdsi, vyaya, etc. 13 is expressed by visvedevdh, visva, kdma, atijagati, agbosa, etc. 14 is expressed by manu, vidyd, indra, sakra, /oka, 10 etc. 15 is expressed by tithi, ghasra, dina, ahaii, paksa, 1 " 1 etc. 16 is expressed by nrpa, bhupa, bbupati, asti, kald, etc. 17 is expressed by atyasti, etc. 1 Used by Mahavira because the Jainas recognise seven tatias; used for five by others. 2 Used by Mahavira. 3 Used by Mahavira. * Used by Maharvira for 8 and by others for 1 o. 5 Used by Mahavira. 6 This word has been used for 8 as well as for 10. The use of dis or dik for 4 also occurs. 7 Used by Mahavira only. 8 This has been used for 1 also. Used by Mahavira only. 10 Also used for 3. 11 Also used for 2. WORD NUMERALS 57 1 8 is expressed by dbrti, etc. 19 is expressed by atidhrti, etc. 20 is expressed by nakha, krti, etc. 21 is expressed by utkrti, praktti, svarga, etc. 22 is expressed by krti, jdti (?), etc. 23 is expressed by vikrti. 24 is expressed by gdyatri, jina, arbat, siddha, etc. 25 is expressed by tatva? etc. 27 is expressed by naksatra, udu, bba, etc. 32 is expressed by danta, rada, etc. 33 is expressed by deva, amara, tridasa, sura, etc. 48 is expressed by jagati, etc. 49 is expressed by tana, etc. Word Numerals without Place-value. In the Veda we do not find the use of names of things to denote numbers, but we do find instances of numbers denoting things. For instance, in the Rgveda the number 'twelve' has been used to denote a year 2 and in the A.tharvaveda the number 'seven' has been used to denote a group of seVen things (the seven seas, etc.). 3 There are instances, however, of fractions having been denoted by word symbols, e.g., kald — T '^, kustha ■== *.,, sapba = -\. The earliest instances of a word being used to denote a whole number are found about 2,000 B.C., in the Satapatba Brdbmana* and Taittirtya Brdhmana? The 1 Generally used for 5 ; also for 7 by Mahavira. 2 "Deva hitim jugupurdvddasasya rtum narona praminantyete . .?..." (vii. 103, 1). ' 3 "Om ye trisapta pariyante . . . ." (i. 1, 1). 4 The word krta has been used for 4. " catusfomena krtena ayandm . . . ." (xiii. 3. z. 1). 5 "Ye vai catvdrah stomdh krtam tat. ..." (i. 5. 11. 1). 58 NUMERAL NOTATION Chdndogya Upanisad also contains several instances. In the Veddftga Jyotisa 1 (1,200 B.C.) words for numerals have been used at several places. The Sraitta-sfitras of Kdtjayana- and L.dtydyand* have the words gdyatri for 24 and jagati for 48. At- this early stage, however, the word symbols were nothing more than curiosities; their use to denote numbers was rare. Moreover, we find evidences of a certain indefiniteness in the numerical significance at- tached to certain words. For instance, in the same work, the A.itareya Brdhwana, the word virdt has been used to denote 10 at one place and 30 at another. The principle of place-value being unknown, the- word symbols could not be used to denote large numbers, which were usually denoted in terms of the numerical denominations or by breaking the number into parts. 4 The use of the word symbols without place-value is found in the Pitigala Cbandah-siitra composed before 200 B.C. The principle of place-value seems to have been applied to the word numerals between 200 B.C. and 300 A.D. Word Numerals with Place-value. The earliest instance of the use of the word numerals with place- value- in its current form is found in the slgnt-Purdtia, 5 1 riipa = 1, aya = 4, gima = yuga — iz, bbasavitiha = 27. See (YJ. 23, Aj. 31), {YJ. 13, Aj. 4), {AJ. 19), 07. 25) and {YJ. 20) respectively. 2 Weber's edition of Kdtjayana Sraufa Sutra, p. 1015. 3 ix. 4. 31. Y)asayutanamayutam sahasrdni ca vimsatih Kotyah sastisca sat caiva yo'smin rajan-mridbe batdb that is, 10 (10000) -f- 10000+20 (1000) -|- 60 (10,000,000) -f- 6 (10,000,000) — Alabdbbdrata, Striparva, xxvi. 9. 5 Anni-Vwdna, Barigabasi ed., Calcutta (1314 B.S.), chs. 122- 23, 131, 140, 141, 328-335. According to Paigiter, probably the giearest Pnr.inic scholar of modern times, "the purdnas cannot WORD NUMERALS 59 a work which belongs to the earliest centuries of the Christian era. Bhattotpala in his commentary on the Brbat-sambitd has given a quotation from the original Pulisa-siddhdnta 1 (c. 400) in which the word system is used. The number expressed in this quotation is kha (o) kha (o) asta (8) muni (7) rdma (3) asvi (2) netra (2) asta (8) sara (5) rdtripdh (1) = 1,582,237,800. There are in this work 2 several other quotations from the Pulisa-siddhdnta, which contain word . numerals. Later astronomical and mathematical manuals such as the Siirya-siddbdnta ,{c. 300), the Panea-siddhdntikd 3 (505), the Mahd- and Laghu-Bbdskariya 4 (522), the Brdbma-spbuta- siddhdnta 5 (628), the Trisatikd* (c. 750), and the Ganita- sdra-samgraha' (850), all make use of the word notation. 8 Word Numerals in. Inscriptions. The earliest epigraphic instances of the use of the word numerals are met with in two Sanskrit inscriptions 8 found in Cam- bodia which was a Hindu colony. They are dated 604 be later than the earliest centuries of the Christian era." (JKAS, 1 91 2, pp. 254-55). The Agni-Purdna is admitted by all scholars to be the earliest of the Purdnas. J Brhat-samhitd, ed. by S. Dvivedi, Benares, p. 163. ~ Ibid, pages 27, 29, 49, 5 1 , etc. We are, however, not sure whether those quotations are from the original work or from a later redaction of the same. 6 i. 8; viii. 1, etc. 4 See MBh, ch. 7 and LBb, ch. 1. 5 i. Ji-5 5, etc. 6 R. 6, Ex. 6, etc. 7 ii. ' 7, 9, etc. 8 In the face of the evidence adduced here, G. R. Kaye's assertion, {Indian Mathematics, Calcutta, 1915, p. 31) that the word numerals were introduced into India in the ninth century from the east, shows his ignorance of Indian mathematical works, or is a deliberate misrepresentation. u R. C. Mazumdar, Ancient Indian colonies in the jar east, — Campa, Vol. I, Lahore, 1927; see inscriptions Nos. 32, 39; also 40, 41, 43 and 44. 60 NUMERAL NOTATION A.D. and 625 A.D. Their next occurrence is found in a Sanskrit inscription of Java, belonging to the 8th century. 1 In India proper, although they were in use amongst the astronomers and mathematicians from the 3rd or 4th century A.D. onwards, it did not become the fashion to use them in inscriptions till a much later date. The earliest Hindu inscriptions using these numerals are dated 813 A.D. 2 and 842 A.D. 3 In the following century they are used in the plates issued by the Eastern Chalukya Amma II, in 943 A.D. 4 In later times the epigraphic instances become more frequent. The nota- tion is also found in several manuscripts in which dates are given in verse. 3 Origin and Early History. It should be noted that the arrangement of words, representing the numbers zero and one to nine, in a word chronogram is contrary to the arrangement that is followed when the same number is written with numerical signs. This fact has misled some scholars to think that the decimal notation and the word numerals were evolved at two different places. G. R. Kaye has gone so far as to suggest that the word numerals were imported into India from the east. This suggestion is incorrect for the simple reason that in no language other than Sanskrit do we find any early use of the word system. Moreover, in no country other than India do we find any trace of the use of a word system of numeration 'IA, XXI, p. 48. 2 The Kadab plates, I A, XII, p. 11; declared by Fleet to be suspicious (Kanarese Dynasties, Bombay Ga^atteer, I, ii, 399, note 7); cf. Buhler, I.e., p. 86, note 4. 3 The Dholpur Inscription, Zeitschrift der Deutschen Morgeniandis- chen Gesellschaft, XL, p. 42. * I A, VII, p. 18. 5 Buhler, I.e., p. 86, note 7. WORb NUMERALS 6 1 as far back as the fourth century A.D., at which period it "was in common use amongst the astronomers and mathematicians of India. During the earlier stages of the development of this system, we find that instead of the word symbols, the number names were used, being arranged from left to right just as the numerical signs. An instance of this is found in the Bakhshali Manuscript 1 (c. 200), where the number 265 3 296226447064994. ...8321 8 is expressed as Sadvimsasca (26) tripancdsa (5 3) ekonatrimsa (29) evacha Dvdsa [///'J(62) sadvimsa (26) catubcatvdrimsa (44) saptati (70) Catuhsasti (64) na\vanavati\ (99) ... msanantaram Trirasiti (83) ekavimsa (21) asta (8) ... pakam In the same manuscript, however, the contrary arrangement is used when the number 54 is expressed as catuh (4) panca {$). a Jinabhadra Gani (575) has used word symbols with the left to right arrangement to express numbers. 3 It seems, therefore, that in the beginning opinion was divided as to which method of arrangement should be followed in the word system. The extensive use of the word numerals by early mathematicians such as Pulisa, Varahamihira, Lalla and others appears to have set the fashion to write the word numerals with a right to left arrangement, which was generally followed by later writers. 1 Folio 58, recto. The dots indicate some missing figures. The problem apparently required the expression of a big number in numerical denominations. We do not find a problem of this type in any of the later works. Cf. B. Datta, "The Bakhshali Mathematics." BCMS, XXI, p. 21. '■* Folio 27, recto. 3 'brhat-ksetra-samdsa, i. 691!. 6z NUMERAL NOTATION No explanation as to why the right to left arrange- ment was preferred in the word system is to be found in any of the ancient works. The following explana- tion suggests itself to us, and we believe thac it is not far from the truth: The different words forming a number chronogram were to be so selected that the resulting word expression would fit in with the metre used. To facilitate the selection the number was first written down in numerical figures. The selection of the proper words would then, naturally, begin with the figure in the units place, and proceed to the left just as in arithmetical operations. This is in accordance with the rule "ankdndm vdmato gatih" i.e., 'the numerals proceed to the left,' which seems to have been very popular with the Indian mathematicians. The right to left arrange- ment is thus due to the desire of the mathematicians to look upon the process of formation of the word chrono- gram as a sort of arithmetical operation. Date of Invention. The use of the word numerals in the Agni-Purdna which was composed in the 4th century A.D. or earlier, shows that the word system of numerals must have become quite common in India at that time, the Purdnas being works meant for the common folk. That it was a well developed system in the fourth century is also shown by its extensive use in the Surya-siddhdnta and the Pulisa-siddhdnta. Its in- vention consequently must be placed at least two centuries earlier. This would give us the period, 100 A.D. to 200 A.D., as the time of its invention. This conclusion is supported by the epigraphic use of the word notation in 605 A.D., in Cambodia, which shows that by the end of the 6th century A.D., the knowledge of the system had spread over an area roughly of the size of Europe. It must be pointed out here that the decimal place- value notation and the word numerals were not invented ALPHABETIC NOTATIONS 63 at the same time. The decimal notation must have been in existence and in common use amongst the mathema- ticians long before the idea of applying the place-value principle to a system of word names could have been conceived. Thus we find that in the beginning (c. 200), the place-value principle, as is to be expected, was used with the number names. The word symbols were then substituted for the number names for the sake of metrical convenience. The right to left procedure was finally adopted because of the mathematicians' desire to look upon the formation of the word numeral as a sort of mathematical operation. The above considerations place the invention of the decimal place- value notation at a period, at least two or three centuries before the invention of the word system. The word notation, therefore, points to the 1 st century B.C. as the time of invention of the place- value notation. This conclusion agrees with that arrived on epigraphic evidence alone. 11. ALPHABETIC NOTATIONS The idea of using the letters of the alphabet to denote numbers can be traced back to Panini (c. 700 B.C.) who has used the vowels of the Sanskrit alphabet to denote numbers. 1 No definite evidence of the exten- sive use of an alphabetic notation is, however, found 1 In Panini's grammar there are a number of sutras (rules) which apply {o a certain number of sutras that follow and not to all. Such sutras are marked by signs according to Panini. Patarijali. commenting on sutra i. 3. 11, says that according to Katyayana (4th century B.C.) a letter (varna), denoting the number of sutras upto which a particular rule is to apply, is written over the sutra. Kaiyyata illustrates this remark by saying that the letter /' is written above Panini's sutra, v. 1. 30 to show that it applies only to the next two sutras. Thus according to Panini a = 1, ;' = 2, « — 3, 64 NUMERAL NOTATION upto the 5th century A.D. About this period a number of alphabetic notations were invented by different writers with the sole purpose of being used in verse to denote numbers. The word numerals gave big number chronograms, so that sometimes a whole verse or even more would be devoted to the word chronogram only. This feature of the word system was naturally looked upon with disfavour by some of the Indian astronomers ,vho considered brevity and conciseness to be the main attributes of a scientific composition. Thus the alpha- betic notations were invented to replace the word system in astronomical treatises. The various alpha- betic systems 1 are simple variations of the decimal place-value notation, using letters of the alphabet in the place of numerical figures. It must be noted here that the Hindu alphabetic systems, unlike those employed by the Greeks or the Arabs, were never used by the common people, or for the purpose of making calcula- tions; their knowledge was strictly confined to the learned and their use to the expression of numbers in verse. Alphabetic System of Aryabhata I. Aryabhata I (499) invented an alphabetic system of notation, which has been used by him in the Dasagitikd z for enumerating the numerical data of his descriptive astronomy. The 1 Some alphabetic systems used for the pagination of manus- cripts do not use the place-value principle. These systems were the invention of scribes who probably wanted to be pedantic and to show off their learning. Their use was confined to copyists of manuscripts. a The Dasagtikd as the name implies ought to contain ten stanzas, but actually there are thirteen. Of these the first is an invocation to the Gods, the second is the paribhdsd ("definition") given above and the thirteenth is of the nature of a colophon. These three stanzas are, therefore, not counted. Cf. W. E. Clark, "Hindu-Arabic Numerals," Indian Studies in Honour of Charles Rockwell Lanman, (Harvard Univ. Press), 19*9, p. 231. * ALPHABETIC NOTATIONS 6} rule is given in the Dasagitikd thus: Vargdksardni varge'varge' vargdksardni kdt nmau yah Khadvinavake svard nava varge'varge navdntyavarge vd The following translation gives the meaning of the rule as intended by the author: "The varga 1 letters beginning with k (are used only) in the varga* places, the avarga letters in the avarga* places, (thus) ya equals nmau {na plus ma); the nine vowels (are used to denote) the two nines of zeros of varga and avarga (places). The same (procedure) may be (repeated) after the end of the nine varga places." This rule has been discussed by Whish ? 4 Brock- khaus, 5 Kern, 8 Barth, 7 Rodet, 8 Kaye, u Fleet, 10 Datta, 11 Ganguly, 12 Das, 13 Lahiri" and Clark. 15 The translation of kha by "place" (Clark) or by "space" (Fleet) is incorrect. We do not find the word kha used in the sense of 'notational place' anywhere in Sanskrit literature. Its meanings are 'void', 'sky', etc., and it has been used for zero, in the mathematical and 1 Varga here means "classed," i.e., the classed letters of the alphabet. The first twenty-five letters of the alphabet are classed in groups of five, the remaining ones are unclassed. 2 Varga here means odd. 3 Avarga here means even. 4 Transaction; of the L,iterary Society of Madras, I, 1827, p. 54. 5 Zeitschrifte fiir die kunde des Morgen/andes, IV, p. 81. e JRAS, 1863, p. 380. 7 Oeuvres, III, p. 182. 8 J A, 1880, II, p. 440. JASB, 1907, p. 478; Indian Mathematics, Calcutta, 191 5, p. 30; The Bakhshi/i Manuscript, Calcutta, 1927, p. 81. 10 JARS', 191 1, p. 109. 11 Sabitya-Parisad-Patrikd, 1929, p. 22. 12 BCMS, 1926, p. 195. 13 IH<2, III, p. no. 14 History of the World (in Bengali), Vol. IV, p. 178. 15 Aryabhatiya of Aryabhata, Chicago, 1930, p. 2. . 66 NUMERAL NOTATION astronomical works. We thus replace "the two nines of places" in the translation given by Clark by "the two nines of zeros." Clark has given the following reason for not translating kha by zero: "That is equivalent to saying that each vowel adds two zeros to the numeri- cal value of the consonant. This, of course, will work from the vowel / on; but the vowel a does not add two zeros. It adds no zero or one zero depending on whether it is used with varga or avarga letters. It seems to me, therefore, more likely that a board divided into columns is implied rather than a symbol for zero, as Rodet thinks." The vowels do not add zeros. The explanation will not work for any of the vowels; for instance, i, according to this interpretation, would add two zeros to g but three zeros to j. What really is implied by kha is explained by the commentator Surya- deva as follows: "khdni sunyopalaksitdni, sankhydvinydsas- thdndni tesdm dvinavakam, khadvinavakam, tasmin khadvina- vake s'iinyopalaksitdstbdndstddasa (18) ityarthab-" that is-, "kha denotes zero; the places for putting (writing) the numbers are two nines {dvinavakam) , therefore, khadvinavake means the eighteen places denoted by zeros." It may be mentioned here that the Hindus denote the notational places by zeros. Bhaskara I (5 22), commenting on Ganitapdda, 2, which gives the names of ten notational places, says: ' "nydsaica sthdndndm 0000000000." i.e., "writing down the places we have 0000000000." Bhaskara I is more explicit in the interpretation of kha by zero, for in his comments on the above rule, he states: "khadvinavake svard nava varge: kha means zero (sunya). In two nines of zeros (kha), so khadvinavake', that is, in the eighteen (places) marked by zeros; " l 1 Commentary on the Dasagitikd by Bhaskata I, "khadvinavake svard nava varge khdni Mnydni, khdndm dvinavakam tasmin khadvinavake astddasa Jcinydksitesu " ALPHABETIC NOTATIONS 6j Thus kha must be translated by zero, although the kba (zero) here is equivalent to the 'notational places.' 1 What is implied here is certainly the symbol for the zero and not a board divided into columns. Clark finds great difficulty in translating navdntya- varge vd. The reading hau instead of vd suggested by Fleet is not acceptable. The translation given by us accords with the several commentaries (by Bhaskara I, Suryadeva, Paramesvara and Nilakantha) consulted by us. They all agree. Explanation. Arayabhata's rule gives the method of expressing the alphabetic chronogram in the decimal place-value notation, and vice versa. The notational places are indicated as follows: au o ai e I u avavavavavavavavav oooooo-ooooooooooo o where v stands for varga and a for avarga. It will be observed that the eighteen places are de- noted by zeros and they are divided into nine pairs, each pair consisting of a varga place and an avarga place, i.e., odd place and even place. 2 The varga letters k to m 3 .are used in varga places, i.e., odd places only, and denote the numbers i, 2, , 25 in succession. The 1 Nilakantha says : "khadvinavake, that is, there are eigh- teen places, the nine varga places and the nine avarga places " See Aryabhatiya, ed. by K. Sambasiva Sastri, Trivandrum, 1930, p. 6. 2 The later Indian treatises use the terms visama and sama for varga and avarga respectively. Varga is also used for a square number or the figure. 3 These are called varga or classified letters, because they are classified into groups of five each. 68 NUMERAL NOTATION avarga letters y to h are used in the avarga places, i.e., even places only, and denote the numbers 3, 4, ,10 successively. The first varga and avarga places together constitute the first varga-avarga pair, and so on. Nine such varga-avarga pairs are denoted by the nine vowels in succession. Thus the first varga-avarga pair, i.e., the units and the tens places are denoted by a; the second varga-avarga pair, i.e., the hundreds and thousands places by i; and so on. The vowels thus denote places — zeros according to the Indian usage of denoting the places — and have by themselves no numerical value. When attached to a 'letter-number' a vowel simply denotes the place that the number occupies in the decimal place-value notation. For instance, when the vowel a is attached toj", it means that the number 3 which y denotes is to be put in the first avarga place, i.e., the tens place. Thus ya is equal to 30. On the other hand when a is attached to one of the classed letters, it refers it to the first varga place, i.e., the units place. Thus ria is equal to 5 and ma is equal to 25 and rima 1 is equal to 30. Similarly yi denotes that the number 3 is to be put in the thousands place whilst gi would mean that the number 3 which g represents is to be put in the hundreds place (g being a varga lefter). Thus j-/=3,ooo, whilst ^'=300. It is possible that the zeros already written were rubbed out and the corresponding numerical figures as obtained from a given letter chronogram were substituted in their places. This would automatically give zeros in the vacant places. When this is not done and the numbers are written below the zeros indicating the places, then zeros have 1 When two consonants are together joined to a vowel, the numbers representing both are referred to the same varga-avarga pair. They are added together as in this case, nma = ria + ma = J '+ *J = 3°- ALPHABETIC NOTATIONS 69 to be written in the places that remain vacant. 1 The same procedure can be applied to express numbers occupying more than eighteen places, by letting the vowels with anusvara denote the next eighteen places, or by means of any other suitable device. One advantage of this notation is that it gives very brief chronograms. This advantage is, however, more than counterbalanced by two very serious defects. The first of these is that most of the letter chronograms formed according to this system are very difficult to pronounce. In fact, some of these 2 are so complicated that they cannot be pronounced at all. The second defect is that the system does not allow any great variety in the letter chronograms, as other systems do. Katapayadi System. In this system the con- sonants of the Sanskrit alphabet have been used in the place of the numbers 1-9 and zero to express num- bers. The conjoint vowels used in the formation of number chronograms, have no numerical significance. It gives brief chronograms, which are generally pleasant sounding words. Skilled writers have been able to coin chronograms which have connected meanings. It is superior to that of Aryabhata I, and also to the word system. Four variants of this system are known to have been used in India. It is probably due to this non- uniformity of notation that the system did not come into general use. 1 Some examples from the j ' ? u " '6o ;= cu, vu, ghu, thu, rthu, rthu, thu, ,rgha, rghu. 70 = cu, cu, thu, rthii, rghu, rmta. 80 = zs, oO. o. 0. 0. p u 90= &3.9.H.H ®- 100 == su, su, lu, a. 200 = su, a, lu, rghii.- 300 = sta, sua, nua, sa, su, surh, su. 400 = suo, sto, sta. 74 NUMERAL NOTATION It will be observed that to the same numeral there correspond various phonetical values. Very frequently the difference is slight and has been intentionally made, probably to distinguish the signs with numerical values from those with letter values. In some other cases there are very considerable variations, which (accord- ing to Bjihler) have been caused by misreadings of older signs or dialectic differences in pronunciation. The symbols are written on the margin of each leaf. Due to lack of spacer they are generally arranged one below the other in the Chinese fashion. This is so in the Bower manuscript which belongs to the sixth century A.D. In later manuscripts the pages are numbered both in the aksarapalli as well as in decimal figures. Sometimes these notations are mixed up as in the following: 1 la su su 33 = 3 ; ioo = o; 102 — o; o 2 su su su 131 = la; 150= P 209 = o 1 o rum The aksarapalli has been used in Jaina manuscripts upto the sixteenth century. After this period, the deci- mal figures are generally used. In Malabar, a system resembling the aksarapalli is in use upto the present day. 2 1 Cf. PLM, p. 108. 2 1 = na, 2 = nna, 3 = nya, 4 = skra, 5 = jhra, 6 = ha(ha), 7 = gra, 8 = pra, 9 = dre(?), 10 "= ma, . 20 = tha, 30 — la, 40 = pta, 50 = ba, 60 = tra, 70 = ru (tru), 80 = ca, 90 = na, 100 = na. (C/.JRAS, 1896, p. 790) THE ZERO SYMBOL 75 Other Letter Systems. {-A) A system of nota- tion in which are employed the sixteen vowels and thirty-four consonants of the Sanskrit alphabet is found in certain manuscripts from Southern India (Malabar and Andhra), Ceylon, Burma and Siam. The thirty-four consonants in order with the vowel a denote the numbers from one to thirty-four, then the same consonants with the vowel a denote the numbers thirty- five to sixty-eight and so on. 1 (B) • Another notation in which the sixteen vowels with the consonant k denote the numbers one to sixteen and with kh they denote the numbers seventeen to thirty-two, and so on, is found in certain Pali manus- cripts from Ceylon. 2 (C) In a Pali manuscript in the Vienna Imperial Library a similar notation is found with twelve vowels and thirty-four consonants. In this the twelve vowels 8 with k denote the numbers from one to twelve, with kh they denote the numbers from thirteen to twenty- four, and so on. These letter systems do not appear to have been in use in Northern India, at least after the third century A.D. They are probably the invention of scribes who copied manuscripts. 12. THE ZERO SYMBOL Earliest Use. The zero symbol was used in metrics by Pihgala (before 200 B.C.) in his Chandah-sutra. He gives the solution of the problem of finding the total number of arrangements of two things in n places, repetitions being allowed. The two things considered are r Burnell, South Indian Palaeography, London, 1878, p. 79. 2 Ibid. 3 The vowels r, r, I, /, are omitted. 76 NUMERAL NOTATION the two kinds of syllables "long" and "short", denoted by /and g respectively. To find the number of arrange- ments of long and short syllables in a metre containing « syllables, Pirigala gives the rule in short aphorisms : "(Place) two when halved;" 1 "when unity is sub- tracted then (place) zero;" 2 "multiply by two when zero;" 3 "square when halved."* The meaning of the above aphorisms will be clear from the calculations given below for the Gdyatri metre which contains 6 syllables. 5 Separately place Place the number Halve it, result 3 cannot be halved, therefore, subtract i, result 2 Halve it, result 1 1 cannot be halved, therefore, subtract 1, result The process ends. The calculation begins from the last number in column B. Taking unity double it at o, this gives 2; at 2 square this (2), the result is 2 2 ; then at zero double (2*), the result is 2"; ultimately, at 2 square this (2 s ), the result is 2", which gives the total number of ways A 6 3 B 1 Pingala Cbandah-sutra, ed. by Sri Sitanath, Calcutta, 1840, viii. 28. 2 Ibid, viii. 29. 3 Ibid, viii. 30. 4 Ibid, viii. 31. s For 7 syllables, the steps are : Subtract i 6 place o Double 2.2 n = 2 7 Halve 3 „ 2 Subtract i 2 „ o Square 2 B Double 2.'2 2 = 2 3 Halve i „ 2 Subtract i o o Square 2 2 Double i =2 giving 2 7 as the result. THE ZERO SYMBOL 77 in which two things can be arranged in 6 places. 1 It will be observed that two symbols are required in the above calculation to distinguish between two kinds of operations, viz., (i) that of halving and (2) that of the 'absence' of halving and subtraction of unity. These might have been denoted by any two marks arbitrarily chosen. 2 The question arises: why did Pingala select the symbols "two" and "zero"? The use of the symbol two can be easily explained as having been suggested by the process of halving- — division by the number two. The zero symbol was used probably because of its being associated, at the time, with the notion of 'absence' or 'subtraction.' The use of zero in either sense is found to have been common in Hindu mathematics from early times. The above reference to Pingala, however, shows that the Hindus possessed a symbol for zero {sunya), whatever it might have been, before 200 B.C. The Bakhshali Manuscript (c. 200) contains the use of zero in calculation. For instance, on folio 5 6 verso, we have : v ° 4 multiplied become 84 964 168 848320 141 12 The square of forty different places is 1 1600 [. On sub- tracting this from the number above (numerator), the ■ 1 . 846720 remainder is f ^ ' I 14112 factor, it becomes I 60 On removal of the common 1 This method of calculation is not peculiar to the 'Pingala Cbandab-sdtra. It is found in various other works on metrics as well as mathematics. The zero symbol has been similarly employed in this connection in later works also. Vide infra. 2 E.g., Prthudakasvami uses va (from varga, "square") and gu (from guna, "multiply"), while Mahavira uses the numerals 1 and o. Vide infra. 78 NUMERAL NOTATION There are a large number of passages of this kind in the work. It will be noticed that in such passages the sentences would be incomplete without the figures, so the figures must have been put there at the time of the original composition of the text, and cannot be suspected of being later interpolations. For an explicit reference to zero and an operation with it, we take the following instance from the work : 4 I visible 200 1 Adding 2 unity to zero 1 I 2 I 3 I " 8 In the Vanca-siddhdntikd (505) zero is mentioned at several places. The following is an instance: "In Aries the minutes are seven, in the last sign six; in Taurus, six (repeated) thrice; five (repeated) twice; four; four; in Gemini they are three, two, one, zero {sunyd) (each repeated) twice." 4 Zero is here conceived as a number of the same type as three, two or one. It cannot be correctly interpreted otherwise. Addition and subtraction of zero are also used in expressing numbers in this work for the sake of metrical convenience. For instance : "Thirty-six increased by two, three, nine, twelve, nine, three, zero (Jmyd) are the days." 6 Instances of the above type all occur in those 1 The zeros given here are represented in the manuscript by dots. The statement in modern symbols is equivalent to the equation, X -\- 2X -\- }*• + 4X = 200. 2 The Sanskrit word is jutam meaning literally "adding", but what is meant is "putting" unity for the unknown (zero). 3 BMs, r folio 22, verso. i PSi, vi. 12. s PSi, xviii. 35; other instances of this nature are in iii. 17; iv. 7; iv. 8; iv. 11; xviii. 44; xviii, 45; xviii. 48; xviii. 51. THE ZERO SYMBOL 79 sections of the Panca-siddhdntikd which deal with the teachings of Pulis"a. It seems, therefore, that such ex- pressions are quotations from the PuliJa-siddhdnta. As it is known that the word numerals were employed by PuliSa (c. 400), it can be safely concluded that he was conversant with the concept of the zero as a numeral. The writings of Jinabhadra Gani (529-589), a con- temporary of Varahamihira, offer conclusive evidence of the use of zero as a distinct numerical symbol. While mentioning large numbers containing several zeros, he often enumerates, obviously for the sake of abridgement, the number of zeros contained. For instance: 224,400,000,000 is mentioned as "twenty-two forty-four, eight zeros;" 1 3,200,400,000,000 as "thirty- two two zeros four eight zeros." 2 At another place in his work 407150 40715 241960 ' = 241960 „ J * J 483920 ^ y 48392 is described thus : "Two hundred thousand forty-one thousand nine hundred and sixty; removing (apavartand) the zeros, the numerator is four-zero-seven-one-five, and the deno- minator fo ur-eight- three-nine-two . " 3 It should be noted that the term apavartana means what is known in modern arithmetic as the reduction of a fraction to its lowest terms by removing the common factors from the numerator and the denominator. Hence the zero of Jinabhadra Gani is certainly not a mere concept of nothingness but is a specific numerical symbol used in arithmetical calculation. 1 Brhat-ksetra-samdsa, ed. with the commentary of Malayagiri, Bombay, i. 69. 2 Ibid, i. 71. Other such instances are in i. 90, 97, 102, 108, 113, 119, etc. 3 Ibid, i. 83. 82 NUMERAL NOTATION been given up long before. The quotation from Subandhu cannot, therefore, be taken as a definite proof of the use of the dot as a symbol for zero in his time. All that we can infer is that at some period before Subandhu, the dot was in use. We may go further and state that very probably, the earliest symbol for zero was a dot and not a small circle. The earliest epigraphical record of the use of zero is found in the Ragholi plates 1 of Jaivardhana II of the eighth century. The Gwalior inscriptions of the reign of Bhojadeva 2 also contain zero. The form' of the 'symbol in these inscriptions is the small circle. This is the form that has been in common use from quite early times, probably from before the eighth century. Other Uses of the Symbol. In the present elementary schools in India, the student is taught the names of the several notational glaces and is made to denote them by zeros arranged in a line. These, zeros are written as ooooooooo The teacher points out the first zero on the right and says 'units', then he proceeds to the next zero saying 'tens' and so on. The student repeats the names after the teacher. This practice of denoting the notational places by zeros can be traded back to the time of Bhaskara I, who, as already pointed out on page 66, in his commentary on the Aryabhatiya, Ganita-pdda,i y says : 'Writing down the places, we have ooooooooo o." In all works on arithmetic (^pdtiganitd) zero has 1 No. 4 in the list of inscriptions given before. 2 Nos. 19 and zo in the list. PLACE- VALUE IN HINDU LITERATURE 83 been used to denote the unknown. This use of zero can be traced back to the third century A.D. It is used for the unknown in the Bakhshali arithmetic. In algebra, however, letters or syllables have been always used for the unknown. It seems that zero for the unknown was employed in arithmetic, really to denote the absence of a quantity, and was not a symbol in the same sense as the -algebraic x (yd), for it does not appear in subsequent steps as the algebraic symbols do. This use of zero is mostly found in problems on proportion — the Rule of Five, Rule of Seven, etc. The Arabs also under Hindu influence used zero for the unknown in similar problems. Similar use of zero for the unknown quantity is found in Europe in a Latin manuscript of some lectures by Gottfried Wolack in the University of Erfurt in 1467-6 8. 1 The dot placed over a number has been used in Hindu Ganita to denote the negative. In this case it denotes the 'absence' of the positive sign. Similar use of the dot is found in Arabia and Europe obviously under Hindu influence. 2 13. THE PLACE- VALUE NOTATION IN HINDU LITERATURE Jaina Canonical Works. The earliest literary evidence of the use of the word "notational place" is furnished by the Anuyogadvdra-sutra? a work written before the Christian era. In this work the total number 1 Smith and Karpinski, I.e., pp. 53-54. 2 The occasional use by Al-Battani (929) of the Arabic negative Id, to indicate the absence of minutes (or second?), noted by Nallino (Verbandlungen des 5 congresses der Orientalisten, Berlin, 1882, Vol. II, p. 271), is similar to the use of the zero dot to denote the negative. 3 The passage has been already quoted in detail (vide supra p. 12). 84 NUMERAL NOTATION of human beings in the world is given by "a number which when expressed in terms of the denominations, koti-koti* etc., occupies twenty-nine places {sthand)." Reference to the "places of numeration" is found also in a contemporary work, the Vyavahdra-sutra.^ Puranas. The Puranas which are semi-religious and semi-historical works, also contain references to the notational places. These works were written for the purpose of spreading education on religious and historical matters amongst the common people. Refer- ence to the place-value notation in these works shows the desire of their authors to give prominence to the system. The A-gni-Purdna* says : "In case of multiples from the units place, the value of each place {sthand) is ten times the value of the preceding place." The Visnu-Purdna 3 has similarly : "O dvija, from one place to the next in succession, the places are multiples of ten. The eighteenth one of these (places) is called pardrdha." The Vdyu-Purdna* observes : "These are the eighteen places {sthand) of calcula- tion; the sages say that in this way the number of places can be hundreds." The above three works are the oldest among the Puranas and of these the -Agni and the Vdyu Puranas in their present form are certainly as old as the fourth century A.D. The Agni-Purdna is referred by some scholars to the first or second century A.D. 1 Ch. i.; cf. B. Datta, Srientia, JuJy, 193 1, p. 8. 2 The yigrti-Purana contains also the use of the word numerals with place -value {vide supra p. 58). 3 vi. 3 . * ci. lozf. PLACE- VALUE IN HINDU LITERATURE 8 J Works on Philosophy. The following simile has been used in Vydsa-Bhdsya 1 on the Yoga-sutra of Patanjali : "Thus the same stroke is termed one in the units place, ten in the tens place, and hundred in the hundreds place " 2 The same simile occurs in the Sdriraka-Bhdsya of Sankaracarya : "Just as, although the stroke is the same, yet by a change of place it acquires the values, one, ten, hundred, thousand, etc. . . ." s The first of the above works cannot be placed later than the sixth century whilst the second one not later than the eighth. The quotations prove conclu- sively that in the sixth century, the place-value notation was so well known that it could be used as an illustra- tion for a philosophical argument. Literary Works. A passage from the Vdsavadattd of Subandhu comparing the stars with zero dots has already been mentioned. Several other instances of the use of zero are found in later literature, but they need not be mentioned here. 4 1 111. 1 3 . 3 The translation is as given by J. H. Woods, The Yoga System of Patan/aii, p. 216. ' In a foot-note, it is remarked: "Contrary to Mr. G. R. Kaye's opinion, the following passages show that the place-value system of decimals was known as early as the sixth century A.D." The above passage is also noted by Sir P. C. Ray in his History of Hindu Chemistry, Vol. II, p. 117. 3 III. iii. i-j;cf.B. Datta, American Math. Monthly, XXXIII 1926, pp. 220-1. 4 E.g., the use of the Mnya-bindu in Naisadha-carita of Sriharsa (c. 12th century). Cf. B. Datta, Ibid, pp. 449-454. 86 NUMERAL NOTATION 14. DATE OF INVENTION OF THE PLACE-VALUE NOTATION We may now summarise the various evidences regarding the early use of the place-value notation in India : (1) The earliest palaeographic record of the use of the place-value system belongs to the close of the sixth century A.D. (2) The earliest use of the place-value principle with the word numerals belongs to the second or the third century A.D. It occurs in the A^gni-Purdna, the Bakhshali Manuscript and the Pulisa-siddhdnta. (3) The earliest use of the place- value principle with the letter numerals is found in the works of Bhas- kara I about the beginning of the sixth century A.D. (4) The earliest use of the place-value system in a mathematical work occurs in the Bakhshali Manuscript about 200 A.D. It occurs in the Aryabhatija composed in 499 A.D., and in all later works without exception. (5) References to the place- value system are found in literature from about 100 B.C. Three references rang- ing from the second to the fourth century A.D. are found' in the Puranas. (6) The use of a symbol for zero is found in Pihgala's Chandah-sutra as early as 200 B.C. The reader will observe that the literary and non- mathematical works give much earlier instances of the use of the place-value system than the mathematical works. This is exactly what one should expect. The system when invented must have for some time been used only for writing big numbers. A long time must have elapsed before the methods of performing arithmetical operations with them were invented. The system cannot be expected to occur in a mathematical DATE OF INVENTION OF PL ACE- VALUE 87 work before it is in a perfect form. Therefore, the evidences furnished by non-mathematical works should, in fact, be earlier than those of mathematical works. Mathematical works are not as permanent as religious or literary works. The study of a particular mathematical work is given up as soon as another better work comes into the field. In fact, a new mathematical work is composed with a view to removing the defects of and superseding the older ones. It is quite probable that works employing the place-value notation were written before Aryabhata I, but they were given up and are lost. It will be idle to expect to find copies of such works after a lapse of sixteen hundred years. In Europe and in Arabia it is still possible to find mss. copies of works using the old numerals or a mixture of the old numerals with the new place-value numerals, but in India absolutely no trace of any such work exists. In Europe the first definite traces of the place- value numerals are found in the tenth and eleventh centuries, but the numerals came into general use in mathematical text books in the seventeenth century. In India Aryabhata I (499), Bhaskara I (522), Lalla (c. 598), and Brahmagupta (628), all use the place -value numerals. There is no trace of any other system of notation in their works. Following the analogy of Europe, we may conclude, on the evidence furnished by Hindu mathematical works alone, that the place- value system might have been known in India about 200 B.C. As the literary evidence also takes us to that period, we may be certain that the place-value system was known in India about 200 B.C. Therefore we shall not be much in error, if we fix 200 B.C. as the probable date of invention of the place-value system and 88 NUMERAL NOTATION zero in India. It is possible that further evidence may force us to fix an earlier date. 15. HINDU NUMERALS IN ARABIA 1 The regular history of the Arabs begins after the flight of Mohammad from Mecca to Medina in A.D. 622. The spread of Islam succeeded in bringing to- gether the scattered tribes of the Arabian Peninsula and creating a powerful nation. The united Arabs, within a short space of time, conquered the whole of Northern Africa and the Spanish Peninsula, and extended their dominions in the east upto the western border of India. They easily put aside their former nomadic life, and adopted a higher civilisation. The foundations of Arabic literature and science were laid between 750-850 A.D. This was done chiefly with the aid of foreigners and with foreign material. The bulk of their narrative literature came to the Arabs in translation from Persian. Books- on r the science of war, the knowledge of weapons, the veterinary art, falconry, and the various methods of divination, and some books on medicine were translated from Sanskrit and Persian. They got the exact sciences from Greece and India. Before the time of Mohammad the Arabs did not possess a satisfactory numeral notation. The numer- ous computations connected with the financial adminis- tration of the conquered lands, however, made the use of a developed numeral notation indispensable. In some localities the numerals of the more civilised con- quered nations were used for a time. Thus in Syria, the Greek notation was retained, and in Egypt the 1 For details consult Cajori's History of Mathematics, and Smith and Karpinski's Hindu Arabic Numerals. HINDU NUMERALS IN ARABIA 89 Coptic. To this early period belongs the Edict of Khalif Walid (699) which forbade the use of the Greek lan- guage in public accounts, but made a special reservation in favour of Greek letters as numerical signs, on the ground that the Arabic language possessed no numerals of its own. 1 The Arabic letters gradually replaced the Greek ones in the alphabetic notation and the abjad notation came to be used. It is probable that the Arabs had come to know of the Hindu numerals from the writings of scholars like Sebokht, and aslo of their old ghobdr forms from other sources. But as their informants could not supply all the necessary informa- tion {e.g., the methods of performing the ordinary operations of arithmetic) these numerals had to wait for another century before they were adopted in some of their mathematical works. During the reign of the Khalif Al-Mansur (753- 774 A.D.) there came embassies from Sindh to Baghdad, and among them were scholars, who brought along with them several works on mathematics including the Brdhma-sphuta-siddhdnta and the Kbanda-khddyaka 6f Brahmagupta. With the 'help of those scholars, Al- fazari, perhaps also Yakub ibn Tarik, translated them into Arabic. Both works were largely used and exer- cised great influence on Arab mathematics. It was on that occasion that the Arabs first became acquainted with a scientific system of astronomy. It is believed by all writers on the subject that it was at that time that the Hindu numerals were first definitely introduced amongst the Arabs. It also seems that the Arabs at first adopted the ghobdr forms of the numerals, which they had already obtained (but without zero) from the 1 Theophanes (758-818 A.D.), "Chronographia;" Scriptores Historiae By^antinae, Vol. XXXIX, Bonnane, 1839, P- 575! q uot ed by Smith and Karpinski, I.e., p. 64, note. 90 NUMERAL NOTATION Alexandrians, or from the Syrians who were employed as translators by the Khalifs at Baghdad. Al-Khowarlzmi (825), one of the earliest writers on arithmetic among the Arabs, has used the ghobdr forms. 1 But not long afterwards, the Arabs realised that the ghobdr forms were not suited to their right-to-left script. Then there appears to have been made an attempt to use more convenient forms. But as people had got accustomed to the ghobdr forms, they did not like to give them up, and so we find a struggle 2 between the two forms, which continued for about two centuries (10th and nth) until at last the more convenient ones came into general use. The west Arabs on the other hand did not adopt the modified forms of the east Arabs, but continued to use the ghobdr forms, and were thus able to transmit them to awakening Europe. This, perhaps, explains in a better way the divergence in the forms of modern Arabic and modern European numerals, than any theory yet propounded. In a theory that was advanced by Woepcke, this divergence is explained by assuming that (1) about the second century after Christ, before zero had been invented, the Hindu numerals were brought to Alex- andria, whence they spread to Rome and also to west Africa; (2) that in the eighth century, after the notation in India had been already much modified and perfected 3 Smith and Karpinski, I.e., p. 98. 7 One document cited by Woepcke is of special interest since it shows the use of the ordinary Arabic forms alongside the ghobdr at an early date (970 A.D.). The title of the work is "Interest- ing and Beautiful Problems on Numbers" copied by Ahmed ibn Mohammed ibn Abdaljalil Abu Sa'id, al-Sijzi, (951-1024) from a work by a priest and physician, Nazif ibn Yumn, al-Qass (died 990). Sprenger also calls attention to this fact (in Zeit. d deutschen- morgenlandischen Gesselschaft, XLV, p. 367). AH ibn Ahmed Al-Nasavi (c. 1025) tells us that the symbolism of numbers was unsettled in his day (Smith and Karpinski, I.e., p. 98). HINDU NUMERALS IN ARABIA 9 1 by the invention of zero, the Arabs at Baghdad got it from the Hindus; (3) that the Arabs of the west borrowed the Columbus-egg, the zero, from those in the east but retained the old forms of the nine numerals, if for no other reason, simply to be contrary to their political enemies of the east; (4) that the old forms were remembered^ by the west Arabs to be of Hindu origin, and were hence called ghobdr numerals; (5) that, since the eighth century, the numerals in India underwent further changes and assumed the greatly modified forms of modern Devanagari numerals. Now, as to the fact that these figures might have been known in Alexandria in the second century A.D., there is not much doubt. But the question naturally arises: Why should the Alexandrians use and retain a knowledge of these numerals? As far. as we know, they did possess numeral notations of their own; why should they give preference to a foreign notation? These questions cannot be satisfactorily answered unless we assume that along with the nine symbols the principle of place-value and probably also the zero was com- municated to them. But as they were unprepared for the reception of this abstract conception, they adopted the nine numerals only and used them on the apices. These numerals were then transmitted by them to Rome and to west Africa. The second assumption that the Hindu numeral figures of the eighth century were adopted by the Arabs is not supported by fact. The figures that are found in the old Arabic manuscripts resemble either the ghobdr numerals or the modern Arabic more than the Hindu numerals of the eighth century. In fact, we have every reason to believe that the Arabs knew these ghobdr forms, perhaps without the principle of place -value and zero, long before they had -direct contact with India, and that they adopted zero only about 750 A.D. 92 NUMERAL NOTATION 16. HINDU NUMERALS IN EUROPE Boethius Question. It cannot be definitely said when and how the Hindu numerals reached Europe. Their earliest occurrence is found in a manuscript of the Geometry of Boethius {c. 500), said to belong to the tenth century. There are several other manus- cripts of this work and they all contain the numerals. Some of these contain the zero whilst the others do not. If these manuscripts (or the portions of them that contain the numerals) be regarded as genuine, it will- have to be acknowledged that the Hindu numerals had reached Southern Europe about the close of the fifth century. There are some who consider the passages dealing with the Hindu numerals in the Geometry of Boethius to be spurious. Their arguments can be summarised as below : (1) The passages in question have no connection, with the main theme of the work, which is geometry. The Hindu numerals have not been mentioned in the Arithmetic of Boethius. They have not been used by him anywhere else. Neither Boethius' contemporary Capella (c. 475), nor any of the numerous mediaeval writers who knew the works of Boethius makes any reference to the numerals. (2) The Hindu numeral notation was perfected in India much later than the fifth century, so that the numerals, even if they had been taken to Europe along the trade routes, had no cLirh to any superiority over the numerals of the west, and so could not have attracted the attention of Boethius. Of the above arguments, the second is against facts, for it is now established that the Hindu numeral nota- tion with zero was perfected and was in use in India during the earliest centuries of the Christian eta. The numerals could have, therefore, easily reached HINDU NUMERALS IN EUROPE 93 t Europe along the trade routes in the fifth century or even earlier. The first argument is purely speculative and throws doubt on the authenticity of the occurrence of the numerals in Boethius' Geometry. It does not prove anything. It seems to us unfair to question the genuineness of the occurrence of the numerals, when they are found in all manuscripts of the work that are in existence now. Their occurrence in the Geometry can be easily explained on the ground that Boethius' knowledge of those numerals was very meagre. He had obtained the forms from some source — from the JSfeo- Pythagoreans or direct from some merchant or wander- ing scholar — but did not know their use. He might have known their use in writing big numbers by the help of the principle of place- value and zero, but he certainly did not know how the elementary operations of arithmetic were to be performed with those numerals. Hence he could make no use of them in his arithmetic or any other work. The writings of Sebokht (c. 6 5 o) show that the fame of the numerals had reached the west long before they were definitely introduced there. The question of the introduction of the Hindu numerals through the agency of Boethius may, therefore, be regarded as an open one, until further investigations decide it one way or the other. Definite Evidence. The first writer to describe the ghobdr numerals in any scientific way in Christian Europe was Gerbert, a French monk. He was a distinguished scholar, held high ecclesiastical positions in Italy, and was elected to the Papal chair (999). He had also been to Spain for three years. It is not definitely known where he found these numerals. Some say that he obtained them from the Moors in Spain, while others assert that he got them from some other source, probably through the merchants. We find that Gerbert did not appreciate these numerals (and rightly, for there 94 NUMERAL NOTATION was neither zero nor the place-value), and that in his works, known as the ReguJa de abaco computi and the Libellus, he has used the Roman forms. We thus see that upto the time of Gerbert (died 1003) the principle of place-value was not known in Europe. As early as 711 A.D., the power of the Goths was shattered at the battle of Jarez de le Frontera, and immediately afterwards the Moors became masters of Spain, and remained so for five hundred years. The knowledge of the modern system of notation which was definitely introduced at Baghdad about the middle of the eighth century must have travelled to Spain and from there made its way into Europe. The schools estab- lished by the Moors at Cordova, Granada, and Toledo were famous seats of learning throughout the middle ages, and attracted students from all parts of Europe. Thus although Europe may not be directly indebted to. the Moors for its numerical symbols, it certainly is for that important principle which made the ordinary ghobdr forms superior to the Roman numerals. Several instances of the modern system of nota- tion are to be found in Europe in the twelfth century, but no definite attempt seems to have been made for popularizing it before the thirteenth century. Perhaps the most influential in spreading these numerals in Europe was Leonardo Fibonacci of Pisa. Leonardo's father was a commercial agent at Bugia, the modern Bougie, on the coast of Barbary. It had one of the best harbours, and at the close of the twelfth century was the centre of African commerce. Here Leonardo went to school to a Moorish master. On attaining manhood he started on a tour of the Mediter- ranean and visited Egypt, Syria, Greece, Italy and Provence, meeting with scholars and merchants and imbibing a knowledge of the various systems of numbers in use in the centres of trade. All these systems, he MISCELLANEOUS REFERENCES' 9 J however says, he counted as errors compared with that of the Hindus. 1 Returning to Pisa he wrote his Liber Abaci in 1202, rewriting it in 1228. 2 In this the Hindu numerals are explained and used in the usual compu- tations of business. At first Leonardo's book met with a cold reception from the public, because it was toe advanced for the merchants and too novel for the universities. However, as time went on people began to realise its importance, and then we find it occupying the highest place among the mathematical classics,, of the period. Among other writers whose treatises have helped the spread of the numerals may be mentioned Alexander de Villa Die {c. 1246) and John of Halifax (f- 1 « >- 5- i>* 1 '5 > ►s a rt 2 -f- •ft 1 u o a Q """ -* 1 " ^- jr 5- -**<» I *— i 1— 1 W 3 ^ =»■ 3" r £ *■ *- r 5-T J i-l pa < § 'a o «■ s s s | f u 3 ft !> a y ' 2 — 5 — s «s «• 5 TABLES 107 Ksatrapa & Andhra Inscrip- tions en ■*- T- 5 8.S 5V> 00 > =*» Q 9 *; e © 1 "2 A. 00 M C JJ- g x: 4 Is ^ 2 1 O B Q 4 rr->*| rC cV X \ 1 •a n X t ^ X i *-* ** 4 < ^ ►J < c .2 °u ■a L L * r ^ ^ ^: ^ ^ s$> x $» ^ t % 5 u j3 a, >-* u £ 1 cr 1 1 s s g « £ jaggayapeta Inscs. and Sivaskanda Varmana Plates r *\ \ A I (1 < Cr^JX. CS c- ^ c y^ r£- Cr- c- y-^ -3 £ 1 'S cq 1 *-* ^— < c 2 CO a '0 u « a. « u ■*-» C3 CO- 1 1 w j6 HI jK rC. d^ C~> • s Cl a M 10 J! t— 1 Ksatrapa & Andhra Inscriptions 1 It 1 •' 3^. n < 3 go u . H- 1 CO a ea C"D CO..JT 1 ^ c 1— 1 1 II III yt ic vXO ^ S* III JT- rC vSL- C- 5~ CV— z <*£ 1 1 i 1 — t D • O co »*o G *• •* «•» •» 4» ,^ r-» ^ 0- no NUMERAL NOTATION X* UL/ o rO J> 8 a s 1 ( ft l'( * li (o VI to VIII Cen- tury A. D. CO ■4-1 G 01 n o 2 > C 1 1 a :£> 1 1 I 7 > 9 133 3 G o ■*-» !> Pallava and Salankays Grants c c c 0^ CC c cc O*" >^ (J < > 2 < Vaka- taka Grants ^ ST 3 (_i 3 -3 g. / ►— < a o i, Parivraj Ucchakal nsctiptions *-* > a, a ( it nl j*. JL UJ\ ^ U ~ Fh O 1 c- lo n ** «^ K> ,j- tr» >o rv ^. o» TABLES III Vi 4-1 cu •c *&. tr fi? pg is is I Bower Manuscripts < f i if < fr t* ~ \< (u j6 *? 3i 9 «>\ /v i-i PQ v to viii Century A. D. Misc. Inscs. & Grants IX to X Century A.D. Pratihara Inscs. & Grants r*\. vn to vin Century (?) A.D. (jrants of the Ganga Dynasty J* »i •^ ••» * to ^ »n»* *» 112 NUMERAL NOTATION 4_* > < j— i Jaggaya- peta & Pallava Grants ih 3 -t-i o C '•3 u a « « ^ e 9) -5 •t-i t— * j— i W X ^» i i ^ 3- ^ e © c« 2 c «* <** 5 1—4 ^ 6 03 :* +- 1 CO to fc/ ?c w pa 3 Sri <*$ Hi, c a * 3 H tf y CD V 7< to > V 8 © fc* V3 b $PQ c . _ U Q w 3 w ro HU i 1^ a B^ > w o-s Q 5 «*> o ^ e & ♦? :rC BU < > z,s r C7 *2, P O e ■3 4-1 c . U 3 rt « w & •CO S _ "S 3 >>0 5? > SX) £ 1 C . 1 U B < > t^ P. S 3 & D *ta ^9 * 1— 1 1— 1 > > # 5 r$ "2 ^ O S 3 ■ « «-> - c u . U Q o 3 C CD o ) yo ^ <* ^ >-^t S © i ° » 6 o O . ^ -, *< m -J- *> ^ N Co <*N H4 NUMERAL NOTATION ■*-> a, •55 u W 3 q Jaina Manuscripts 0^ pt ^^^V^^k/3 DC ■J? Buddhist Manus- cripts from Nepal 2 1 3 u 8 V y ? 4} < J Miscellaneous Grants and Inscriptions c u 1— 1 H CO £ 9 20 TABLES "5 •5 <" O n ^ \S fst? ' > < S b 3 5 3 u ° O -a -3 c cu «•- rt <-> 4-J CO 3 i-< o fT" ^ ^ 7C7 3 r> * r- ^ rS o ml O £> o» £* rs T V— 1 r^ r n" t^ X r-t _" wi trt w "J m < 1-1 < l-H O *i B ■«.2* r b S u 3 . o ■ 3.2 p t fc 5 IE z g * l-l '■n C c.2 \*J i ■ ■■ u ffl 1— 1 " o .2* ^i in ^ ►— ( c -? 5 » M o o o ft « o r- . n6 NUMERAL NOTATION B* fc* El ■■< fcH u CO D a •— > a t# # bc & US XT to 5S 3 a v fir S^ CEO ^ -5 8 V— VIII Cent. 1 A.D. Misc. Inscrip- - tions £*fe * Q Q < & k 8 3 c £ 9 20 rt X 3 tV X a, U-l m < £ H HH O 60^ „ dQ 23 rt rt c°. 1 Gran theG Dyn r=> £ ^ Cl, 1 *>*r -4-1 H C M U >U CO ^ a c* it"Fc? D o o » S o a o o 5 i» «» o o «» ft ^» «Y m ■+ -A ^O K ^ •» TABLES "7 CO « Ul OJ 6 3 C 6 X, «rt u m a '-« CO u u M S 3 G -? -^ /Yj r^KJ i/ », *> «v «> t>. O >•«% t to < Sri i— i > to n C u-b-i^e^ Q $=> Su i . cv « -* V> *<, * * £ U8 NUMERAL NOTATION <*< 1 1 2 °<£ -> 1 00 < r- «r -< Q-T30 d, ^ a" C^ (P «"s >0 (/V o Bud- dhist Mss. f.yf^cfloPt-ZO Bakh- shali Mss. c n dh . *1 vgQ (U Ui en OJ U s s C- CT^CC^ Do ^ (joj c--) ^ ^ o>^ a Jajalla- deva Grants 1? Trilocana pala Grants ^-^ h- 1 h- 1 X a CO < 3-Q 2< rrt 5^ ? c/c^o"3) o Apara- Grants c-x C^ ^> ^ G^ o CO ■ Mula- raja Grants "^Nn > i' ; ^ ,,0 ^ e ^ «=» ^ 120 NUMERAL NOTATION TABLE XIV— Development of Ndgart Numerals 1 ^ »\ N ? ^ ? i. = ^ ^ * a. 5 a 5 ^ ^ 1 } A -f ¥ H * * V sr T ^h V ^ • •3L 6 € £ ^ ? 1 7 0> 8 1 1 5 3 Z. C 1 ? « ? ? V Q" c d TABLES 121 TABLE XV— 'Numeral Forms in Modern Hindu Scripts 3 J 8 Nagari &arada Tikarf n Gurumukhi 1 Kaithi \ Blngala MaithiU J Uriya C Gujarat! i Mar^thi i Telegu o Kanadl -i Malayalam ft Burmese P Siamese 9 Tibetan r * 3 it 4 £ n e J 3 K H * I kt T o j> Q, % H 4. & ^ r ° y & cr (»• r i* o Chapter II ARITHMETIC i. GENERAL SURVEY Terminology and Scope. Arithmetic forms the major part of the Hindu works on pdtiganita. The word pdtiganita is a compound formed from the words pdfi, meaning "board," and ganita > meaning "science of cal- culation;" hence it means the science of calculation which requires the use of writing material (the board). 1 It is believed that this term originated in a non-Sanskrit literature of India, a vernacular of Northern India. The oldest Sanskrit term for the board is phalaka or patta , not pdti. The word pdti seems to have entered into Sanskrit literature about the beginning of the seventh century A.D. 2 The carrying out of mathematical calcula- tions was sometimes called dh&li-karma ("dust-work"), because the figures were written on dust spread on a board or on the ground. Some later writers have used the term vyakta-ganita ("the science of calculation by the 'known'") for pdtiganita to distinguish it from algebra which was called avyakta-ganita ("the science? of calculation by the 'unknown' "). The terms. pdtiganita and dhuli-karma were translated into Arabic when. Sanskrit works were rendered into that language. The Arabic equivalents are ilm-hisdb-al-takht ("the science of 1 Paper being scarce, a wooden board was generally used for making calculations even upto the 19th century. 2 B. Datta, American Math. Monthly, XXXV, p. J26. 1 24 ARITHMETIC calculation on the board") and hisdb-al-ghobdr ("calcula- tion on dust") respectively. Bayley, Fleet and several others suspect that the origin of the term pdti in Hindu Mathematics lies in the use of the board as an abacus. This conjecture, however, is without foundation, as no trace of the use of any form of the abacus is found in India. According to Brahmagupta 1 there are twenty operations and eight determinations in pdtiganita. He says: "He who distinctly and severally knows the twenty logistics, addition, etc., and the eight determinations including (measurement by) shadow is a mathemati- cian." The twenty logistics, according to Prthudakasvami, are: (1) samkalita (addition), (2) vyavakalita or vyutkalita- (subtraction), (3) gunana (multiplication), (4) bhdgahdra (division), (5) varga (square), (6) varga-mula (square-root), (7) ghana (cube), (8) ghana-mula (cube-root), (9-13) panca jdti (the five rules of reduction relating to the five standard forms of fractions), (14) trairdsika (the rule of thrjee), (15) vyasta-trairdsika (the inverse rule of three), (16) pafcardsika (the rule of five), (17) sapta- rds'i/k.a (the rule' of seven), (18) navardsika (the rule of nine), (19) ekddasardsika (the rule of eleven), and (20) bbdnda-pntibhdnda (barter and exchange). The eight determinations are: (1) nrisraka (mixture), (2) sredhi (progression or series), (3) ksetra (plane figures), (4) 'ihdta (excavation), (5) clti (stock), (6) krdkacika (saw), (7) rdsi (mound), and (8) chdyd (shadow). Of the operations named above, the first eight have been considered to be fundamental by Mahavira and later writers. The operations of duplation (doubling) 1 BrSpSi, p. 172. GENERAL SURVE^ I2J and mediation (halving), which were considered funda- mental by the . Egyptians, the Greeks and some Arab and western scholars, do not occur in the Hindu mathematical treatises. These operations were essential for those* who did not know the place- value system of notation. They are not found in Hindu works, all of which use the place-value notation. Sources. The only works available which deal exclusively with patiganita are: the Bakbsbdli Manuscript (c. 200), the Trisatikd {c. 750), the Ganita-sdra-samgraha (r.. 850), the Ganita-tilaka (1039), the Lf/dvafi (11 50), the Ganita-kaumudi (1356), and the Pdti-sdra (1658). These works contain the twenty operations and the eight determinations mentioned above. Examples are also given to illustrate the use of the rules enunciated. Besides these there are a number of astronomical works, known as Siddhdnta, each of which contains a section dealing with mathematics. Aryabhata I (499) was the first to include a section on mathematics in his Siddhdnta, the Aryabhatiya. Brahmagupta (628) followed Aryabhata in this respect, and after him it became the general fashion to include a section on mathematics in a Siddhdnta work. 1 The earlier Siddhdnta works do not possess this feature. The Surya-siddhdnta (c. 300) does not contain a section on mathematics. The same is true of the Vdsistha, the Pitdmaha and the Romaka Siddhantas. Bhaskara I and Lalla, 2 although zealous followers of Aryabhata I, did not emulate him in in- cluding a section on mathematics in their astronomical works. 1 Amongst such works may be mentioned the Mahd-siddhdnta (950), the Siddhdnta-Iekhara (1036), the Siddbdnta-tattva-viveka (1658), etc. 2 It is stated by Bhaskara II that Lalla wrote a separate treatise on patiganita. 126 ARITHMETIC Exposition and Teaching. In India conciseness of composition, especially in scientific matters, was highly prized. The more compact and brief the composition, the greater was its value in the eyes of the learned. It is for this reason that the Indian treatises contain only a brief statement of the known formulas and results, sometimes so concisely expressed as to be hardly understandable. This compactness is more pronounced in the older works; for instance, the exposition in the Aryabbatiya is more compact than in the later works. This hankering after brevity, in early times, was due chiefly to the dearth of writing material, the fashion of the time and the method of instruction fol- lowed. The young student who wanted to learn pdtiganita was first made to commit to memory all the rules. Then he was made to apply the rules to the solution of problems (also committing the problems to memory). The calculations were made on a pdti on which dust was spread, the numbers being written on the dust with the tip of the fore-finger or by a wooden style,, the figures not required being rubbed out as the calculation proceeded. Sometimes a piece of chalk or soap-stone was used to write on the pdti. Along with each step in the process of calculation the sutra (rule) was repeated by the student, the teacher supervising and helping the student where he made mistakes. After the student had acquired sufficient proficiency in solving the problems contained in the text he was studying, the teacher set him other problems — a store of graded examples (probably constructed by himself or borrowed from other sources) being the stock-in-trade of every professional teacher. At this stage the student began to understand and appreciate the rationale of the easier rules. After this stage was reached the teacher gave proofs of the more difficult formula; to the pupil. GENERAL SURVEY 1 27 It will be observed that the method of teaching pursued was extremely defective in so far as it was in the first two stages purely mechanical. A student who did not complete all the three stages knew practi- cally nothing more than the mere mechanical applica- tion of a set of formulas committed to memory; and as he did not know the rationale of the formulae he was using, he was bound to commit mistakes in. their application. It may be mentioned that not many teachers themselves could guide a pupil through all the stages of the teaching, and the earnest student, if he had a genuine desire to learn, had to go to some seat of learning or to some celeberated scholar to complete his training. Mathematics is and has always been the most difficult subject to study, and as a knowledge of higher mathematics could not be turned to material gain there were very few who seriously undertook its study. In India, however, the religious practices of the Hindus required a certain amount of knowledge of astronomy and mathematics. Moreover, there have always been, from very early times, a class of people known as ganaka whose profession was fortune-telling. These people were astrologers, and in order to impress their clients with their learning, they used to have some knowledge of mathematics and astronomy. Thus it would appear that instruction in mathematics, upto a certain minimum standard, was available almost every- where in India. As always happens, some of the pupils got interested in mathematics for its own sake, and took pains to make a thorough study of the subject and to add to it by writing commentaries or independent treatises. Decay of Mathematics. All this was true when the times were normal. In abnormal times when there were 128 ARITHMETIC foreign invasions, internal warfares or bad government and consequent insecurity, the study of mathematics and, in fact, of all sciences and arts languished. Al-BIruni who visited north-western India after it had been in a very unsettled state due to recurrent Afghan invasions for the sake of plunder and loot complains that he could not find a pandit who would explain to him the principles of Indian mathematics. Although Al-Biruni's case was peculiar, for no respectable pandit would agree to help a foreigner, especially one belonging to the same class as the invaders and the despoilers of temples, yet we are quite sure that in the Punjab there were very few good scholars at that time. We, however, know of at least one very distinguished mathematician, Sripati, who probably lived in Kashmir at that time. It is certain, however, that after the 12th century very little original work was done in. India. Com- mentaries on older works were written and some new works brought out,, but none of these had sufficient merit as regards exposition or subject matter, so as to displace the works of Bhaskara II, which have held undisputed sway for nine centuries (as standard text books). The Fundamental Operations. The eight funda- mental operations of Hindu ganita are: (1) addition, (2) subtraction, (3) multiplication, (4) division, (5) square, (6) square-root, (7) cube and (8) cube-root. Most of these elementary processes have not been mentioned in the Siddhanta works. Aryabhata I gives the rules for finding the square- and cube-roots only, whilst Brahma- gupta gives the cube-root rule only. In the works on arithmetic (pdtiganita), the methods of addition and subtraction have not been mentioned at all or men- tioned very briefly. Names of several methods of multiplication have been mentioned, but the methods GENERAL SURVEY 1 29 themselves have been either very briefly described or not described at all. The modern method of division is briefly described in all the works and so are the methods of squaring, square-root, cubing and cube- root. Although very brief descriptions of these funda- mental operations are available, yet it is not difficult to reconstruct the actual procedure employed in perform- ing these operations in ancient India. These methods have been well-known and taught to children, practi- cally without any change, for the last fifteen hundred years or more. " They are still performed in the old fashion on a pdti ("board") by those who have obtained their primary training in the Sanskrit pathasala and not in the modern primary school. The details of these- methods are also available to us in the various com- mentaries, viz., the commentary of Prthudakasvami and the several commentaries on Bhaskara's ILildvati. As already mentioned, the calculations were per- formed on sand spread on the ground (dbtlli-karma 1 ) or on a pdti ("board"). Sometimes a piece of chalk or soap-stone {pdndu-lekha or svetavarnt) was used to write on the pdti. 2 As the figures written were big, so several lines of figures could not be contained on the board. Consequently, the practice of obliterating figures not required for subsequent work was common. Instances of this would be found in the detailed method of working (the operations) given hereafter. That all mathematical operations are variations of the two fundamental operations of addition and sub- traction was recognised by the Hindu mathematicians 1 Bhaskara II, SiSi, candragrahanddhikara, 4. 2 Bhaskara II: khatikdyd rekhd ucchddya..., i.e., "having drawn lines with a chalk,..," quoted by S. Dvivedi in his History of Mathe- matics (in Hindi), Benares, 1910, p,. 41. 130 ARITHMETIC from early times. Bhdskara I (c. 525) states: 1 "All arithmetical operations resolve into two cate- gories though usually considered to be four. 2 The two main categories are increase and decrease. Addi- tion is increase and subtraction is decrease. These two varieties of operations permeate the whole of mathema- tics (gamta). So previous teachers have said: 'Multipli- cation and evolution are particular kinds of addition; and division and involution of subtraction. Indeed every mathematical operation will be recognised to consist of increase and decrease.' Hence the whole of this science should be • known as consisting truly of these two only." 2. ADDITION Terminology. Aryabhata II (950) defines addition thus: "The making into one of several numbers is addition." 3 . The Hindu name for addition is samkalita (made together). Other equivalent terms commonly used .are samkalana (making together), misrana (mixing), sam- melana (mingling together), praksepana (throwing to- gether), samyojana (joining together), ekikarana (making into one), yu fed, yoga (addition) and abhydsa* etc. The word samkalita has been used by some writers in the general sense of the sum of a series. 5 The Operation. In all mathematical and astrono- mical works, a knowledge of the process of addition is 1 The quotation is from his commentary on the Aryabhatiya. 2 i.e., addition, subtraction, multiplication and division. 3 MS/\ p. 143. * This word has been used in the sense of addition in the Sulba only. It is used for multiplication in later 1 works. ~° E.g., Tris, p. 2; GSS, p. 17. ADDITION 131 taken for granted. Very brief mention of it is made in some later works of elementary character. Thus Bhaskara II says in the Lildvati: "Add the figures in the same places in the direct or the inverse order." 1 Direct Process. In the direct process of addition referred to above, the numbers to be added are written down, one below the other, just as at present, and a line is drawn at the bottom below which the sum is written. At first the sum of the numbers standing in the units place is written down, thus giving the first figure of the sum. The numbers in the tens place are then added together and their sum is added to the figure in the tens place of the partial sum standing below the line and the result, substituted in its place. Thus the figure in the tens place of the sum is obtained; and so on. An alternative method used was to write the biggest addend at the top, and to write the digits of the sum by rubbing out corresponding digits of this addend. 2 Inverse Process. In the inverse process, the num- bers standing in the last place (extreme left) are added together and the result is placed below this last place. The numbers in the next place are then added and so on. The numbers of the partial sum are corrected, if necessary, when the figures in the next vertical line are added. For instance, if 12 be the sum of the numbers in the last place, 12 is put below the bottom line, 2 being directly below the numbers added; then, if the 1 L,, p. 2; direct (krawa), i.e., beginning from the units place; inverse (ittkrama), i.e., beginning from the last place on the left. The commentator Gangadhara says: ankdndm vdmatogatirili vitarkena ekasthdnddi yojanam kramah ulkramaslu antyasfhdnddi yojanani, i.e., "According to the rule 'the numerals increase (in value) to- wards the left', the addition of units first is the direct method, the addition of figures in the last place first is the inverse method." 2 Dvivedi, History of Mathematics, Benares, 19 10, p. 60. 132 ARITHMETIC sum of the numbers in the next place is 13 (say), 3 is placed below the figures added and 1 is carried to the left. Thus the figure 2 of the partial sum 12 is rubbed out and substituted by 3. 1 The Arabs used to separate the places by vertical lines, but this was not done by the Hindus. 2 3. SUBTRACTION Terminology. Aryabhata II (950) gives the fol- lowing definition of subtraction: "The taking out (of some number) from the sarvadhana (total) is subtraction; what remains is called Jesa (remainder)." 3 The terms vyutkalita (made apart), vyutkalana (making apart), iodbana (clearing), pdtana (causing to fall), viyoga (separation), etc., have been used for subtraction. The terms sesa (residue) and antara (difference) have been used for the remainder. The minuend has been, called sarvadhana or viyojja and the subtrahend viyojaka. The Operation. Bhaskara II gives the method of subtraction thus: "Subtract the numbers according to their places in the direct or inverse order." 4 1 The Manoranjana explains the process of addition thus: Example. Add 2, 5, 32, 193, 18, 10 and 100. Sum of units 2,5,2,3,8,0,0 20 Sum of tens 3,9,1,1,0 14 Sum of hundreds i>o>o,i 2 Sum of sums 360 The horizontal process has been adopted by the commentator so that both the 'direct' and 'inverse' processes may be exhibited by a single illustration. It was never used in practice. 2 Cf. Taylor, Ulawati, Bombay, 18 16, Introduction, p. 14. 3 MSi, p. 143. * L, p. 2. SUBTRACTION 133 Direct Process. Suryadasa 1 explains the process of subtraction with reference to the example. 1000 — ' 360 thus: "Hence making the subtraction as directed, six cannot be subtracted from the zero standing in the tens place, so taking ten and subtracting six from it, the remainder (four) is placed above (six), and this ten is to be subtracted from the next place. For, as the places of unit, etc., are multiples of ten, so the figure of the subtrahend that cannot be subtracted from the corresponding figure of the minuend is subtracted from ten, the remainder is taken and this ten is deducted from the next place. In this way this ten is taken to the last place until it is exhausted with the last figure. In other words, numbers upto nine occupy one place, the differentiation of places begins from ten, so it is known 'how many tens there are in a given number' and, therefore, the number that cannot be subtracted from its own place is subtracted from the next ten, and the remainder taken." The above refers to the direct process, in which subtraction begins from the units place. Inverse Process. The inverse process is similar, the only difference being that it begins from the- last place of the minuend, and the previously obtained partial differences are corrected, if required. The process is suitable for working on a pdti m (board) where figures can be easily rubbed out and corrected. This process seems to have been in general use in India, and was considered to be simpler than the direct process. 2 1 In his commentary on the Lildvatl. ■ According to Garigadhara, the inverse process of working is easier in the case of subtraction, and the direct in the case of addition. / 134 ARITHMETIC 4 . MULTIPLICATION Terminology. The common Hindu name for multiplication is gunana. This term appears to be the oldest as it occurs in Vedic literature. The terms banana, vadha, ksaya, etc. which mean "killing" or "destroying" have been also used for multiplication. These terms came into use after the invention of the new method of multiplication with the decimal place-value numerals; for in the new method the figures of the multiplicand were successively rubbed out (destroyed) and in their places were written the figures of the product. 1 Synonyms of banana (killing) have been used by Arya- bhata I 2 (499), Brahmagupta (628), 'Sridhara (c. 750) and later writers. These terms appear also in the Bakhshall Manuscript. 3 The term abhydsa has been used both for addition and multiplication in the Sulba works (800 B.C.). This shows that at that early period, the process of multiplica- tion was made to depend on that of repeated addition. The use of the word parasparakrtam (making together) for multiplication in the Bakhshall Manuscript 4 is evi- dently a relic of olden times. This ancient terminology proves that the definition of multiplication was "a pro- cess of addition resting on repetition of the multiplicand as many times as is the number of the multiplicator." This definition occurs in the commentary of the Arya- bhatiya by Bhaskara I. The commentators of the Lildvati give the same explanation of the method of multipli- cation. 5 1 See the kppdta-sandhi method of multiplication, pp. 1 3 8ff. 2 ^A, ii. 19, 26, etc. 3 BMs, 65 verso. 4 BMs, 3 verso. 5 Colebrooke, Hindu Algebra, p. 133. MULTIPLICATION 1 3 5 The multiplicator was termed gunya and the multi- plier gunaka or gunakdra. The product was called gunana-phala (result of multiplication) or pratyutpanna (lit. "reproduced," hence in arithmetic "reproduced by multiplication"). The above terms occur in all known Hindu works. Methods of Multiplication. Aryabhata I does not mention the common methods of multiplication, probably because they were too elementary and too well- known to be included in a Siddhanta work. Brahma- gupta, however, in a supplement to the section on mathematics in his Siddhanta, gives the names of some methods with very brief descriptions of the processes: "The multiplicand repeated, as in gomutrikd, as often as there are digits 1 in the multiplier, is severally multiplied by them and (the results) added (according to places); this gives the product. Or the multiplicand is repeated as many times as there are component parts 2 in the multiplier." 3 ' "The multiplicand is multiplied by the sum or the difference of the multiplier and an assumed quantity and, from the result the product of the assumed quantity and the multiplicand is subtracted or added." 4 Thus Brahmagupta mentions four methods: (i) gomutrikd, (2) khanda, (3) bheda and (4) ista. The common and well-known method of kapdta-sandhi has been omitted by him. 1 khanda, translated as "integrant portions" by Colebrooke. 2 bheda, i.e., portions which added together make the whole, or aliquot parts which multiplied together make the entire quantity. J BrSpSi, p. 209; Colebrooke, I.e., p. 319. 4 BrSpSi, p. 209. Colebrooke {I.e., p. 320) thinks that this is a method to obtain the true product when the multiplier has been taken to be too great or too small by mistake. This view is incorrect. 1 36 ARITHMETIC Sridhara mentions four methods of multiplication: (1) kapdta-sandbi, (2) tastha, (3) rupa-vibhdga and (4) sthdna-vibbdga. Mahavira mentions the same four. Aryabhata II mentions only the common method of kapdta-sandbi. Bhaskara II, besides the above four, mentions Brahmagupta's method of ista-gunana. The five methods given by Bhaskara II were mentioned earlier by Sripati in the Siddbdnta-sekbara. Ganesa 1 (1545) mentions the gelosia method of multiplication under the name of kapdta-sandbi and adds that the intelligent can devise many more methods of multiplication. The method is also given in the Gatiita-maijjarL We have designated it as kapdta-sandbi (b). Seven 2 distinct modes of multiplication employed by the Hindus are given ,below. Some of these are as old as 200 A.D. These methods were transmitted • to Arabia in the eighth century and were thence com- municated to Europe, where they occur in the writings of mediaeval mathematicians. Door-junction Method. The Sanskrit term for the method is kapdta-sandbi. . Sridhara 3 describes it thus: "Placing the multiplicand below the multiplier as in kapdta-sandbi, 4 ' multiply successively, in the direct or inverse order, moving the multiplier each time. This method is called kapdta-sandbi." Aryabhata II 5 (950) gives the following without name: 1 Commentary on the Litdvati, MSS No. I. B. 6. in the Asiatic Soc. of Bengal, Calcutta, pp. 17, 18. In this work only two methods are given, (1) kapdta-sandhi and (2) kapdta-sandhi (b). 2 Or ten if we count also the sub-divisions under each head. 3 Tris, pp. 3f. * kapdta means "door" and sandhi means "junction"; hence kapdta-sandhi means "the junction of doors." 5 MSi, p. 143; the inverse method only has been given. MULTIPLICATION 1 37 "Place the first figure of the multiplier over the last figure of the multiplicand, and then multiply suc- cessively all the figures of the multiplier by each figure of the multiplicand." Sripati 1 (1039) gives the name kapdta-sandhi and states: "Placing the multiplicand below the multiplier as in the junction of two doors multiply successively (the figures of the multiplicand) by moving it (the multiplier) in the direct or inverse order." Mahavira refers to a method known as kapdta- sandhi, but does not give the details of the process. 2 Bhaskara II gives the method but not the name, while Narayana (1356) gives the method in almost the same words as Sridhara, and calls it kapdta-sandhi. The main features of the method are (/) the relative positions of the multiplicand and the multiplier and (//) the rubbing out of figures of the multiplicand and the substitution in their places of the figures of the product. 1 The method owes its name kapdta-sandhi to the first feature, and the later Hindu terms meaning "killing" or "destroying" for multiplication owe their origin to the second feature. The occurrence of the terms banana, vadha, etc., in the works of Aryabhata I and Brahmagupta, and in the Bakhshali Manuscript ihow beyond doubt that this method was known in India about 200 A.D. - The following illustrations 3 explain the two pro- cesses of multiplication according to the kapdta-sandhi plan: 1 SiSe, xiii. z; GT, 15. 2 GSS, p. 9. 3 The illustrations are based on the accounts given in the commentaries on the Ulavati, especially the Manoranjana which gives more details. 138 ARITHMETIC Direct Process: This method of working does not appear to have been popular. It has not been mentioned by writers after the nth century, Sripati (1039) being the last writer to mention it. Example. To multiply 135 by 12, The numbers are written down on the pati thus: 12 135 The first digit of the multiplicand (5) is taken and multiplied with the digits of the multiplier. Thus 5X2=10; o is written below 2, and 1 is to be carried over. 1 Then 5x1 = 5; adding 1 (carried over), we get 6. 5 which is no longer required is rubbed out and 6 written in its place. Thus we have 12 1360 The multiplier is then 'moved one place towards the left, and we have 12 1360 Now, 12 is multiplied by 3. The details are: 3 X 2=6; this 6 added to the figure 6 below 2 gives 12. 6 is rubbed out and 2 substituted in its place. 1 is carried ovef. Then 3X1 = 3; 3 plus 1 (carried over)=4. 3 is rubbed out and 4 substituted. After the multiplier 1 2 has been moved another place towards the left, the figures on the pati stand thus: 12 1420 Then, 1x2=2; 2+4=6; 4 is rubbed out and 6 substituted. 1X1 = 1, which is placed to the left of 6. 1 For this purpose it was probably noted in a separate portion of the pati by the beginner. MULTIPLICATION 1 39 As the operation has ended, 12 is rubbed out and the pdti has 1620 Thus the numbers 12 and 135 have been killed'' and a new number 1620 is born (pratyutpanna). 2 The reader will note that the position of the multiplier and its motion serve two important purposes, i>i%., (/') the last figure of the multiplier indicates the digit of the multiplicand by which multiplication is to be performed and, {it) the product is to be added to the number standing underneath the digit of the multiplier multiplied. Sometimes the product of a digit of the multipli- cand and the multiplier extends beyond the last place of the multiplier. In such cases, the last figure of the partial product is noted separately. The reader should note this fact in the case, 135X99, by performing the operation according to the above process. The beginner was liable to commit mistakes in such cases, (/') of not correctly taking into account the separately noted number, or (//') of rubbing out the digit of the multiplicand beyond the last digit of the multiplier. For these reasons, this process was not in general use and the inverse process was preferred. Inverse Method: There appear to have been two varieties of the inverse method. {a) In the first the numbers are written thus: 12 135 Multiplication begins with the last digit of the multipli- cand. Thus 1X2 = 2; 1 is rubbed out and 2 substi- 1 This explains the use of the term banana (killing) and its synonyms for multiplication. ■ Hence the product was termed pratyutpanna. 140 ARITHMETIC tuted; then 1x1 = 1, this is written to the left; 1 the multiplier 12 is moved to the next figure. The work on the pdti stands thus: 12 1235 Then, 3X2=6; 3 is rubbed out and 6 substituted; then 3X1=3 and 3 -J- 2= 5; 2 is rubbed out and 5 substituted in its place. The multiplier having been moved, the work on the pdti stands thus: 12 1565 Now, 5x2=10; 5 is rubbed out and o substituted in its place; then 5x1 = 5; 5 + 1=6; 6+6=12; 6 is rubbed out and 2 substituted, and 1 is carried over; then 1 + 5=6, 5 is rubbed out and 6 substituted in its place. The pdti has now, 1620 as the product {pratyutpannd). The figures to be carried over are noted down on a separate portion of the pdti and rubbed out after addition. ib) In the second the partial multiplications {i.e., the multiplications by the digits of the multiplicand) are carried out in the direct manner. These partial multiplications, however, seem to have been carried out in the inverse way, this being the general fashion. The following example will illustrate the method of working: Example. Multiply 324 by 753 The multiplier and the multiplicand are arranged thus: 753 ■ 324 'Or the alternative plan: 1X1=1 and then 1X2 = 2, thus giving 12 in the place of 1 in the multiplicand, etc. MULTIPLICATION 141 Multiplication begins with the last place of the multiplier. 3X7 gives 21; 1 is placed below the 7 of the multiplier and 2 to its left, thus: 753 21 324 Then 3X5 gives 15; 5 is placed below the 5 of the multiplier and 1 carried to the left; the 1 obtained in the previous step is rubbed out and (i + i)=2 is substituted, giving 753 225324 Then 3X3 gives 9; the 3 of the multiplicand is rubbed out and 9 substituted. The work on the pdti now stands thus: 753 225924 The multiplier is now moved one place to the right giving 753 225924 Then multiplying 7 by 2 we get 14. This 14 being set below the 7 gives 753 239924 Multiplying 5 by 2 and setting the result below it, we obtain 753 240924 Finally multiplying 3 by 2 and rubbing out 2, which is required no longer, and substituting 6 in its place, we get 753 240964 (0 753 243764 («') 75 J 243964 • ("/) 753 M3972 I42 ARITHMETIC The multiplier is 'then moved one step further giving 753 240964 Multiplying by 4 the digits of the multiplier 753, and setting the results as before we obtain multiplying 7X4 and setting the result; multiplying 5X4 and setting the result; multiplying 3X4 and setting the result. It may be again remarked that the position and motion of the multiplier play a very important part in the above process. The digits of the multiplier are also successively rubbed out in order to avoid confusion, thus 7 is rubbed out at stage (/), 5 at (//) and 3 at (/'//). The following variation of the above process is also found: 1 "Multiplicand 135, multiplier 12; the multiplier placed at the last place of the multiplicand gives J 3$ According to the rule 'the numerals progress to the left' the last figure of the multiplicand (the figure 1) is multiplied by 12. Then after moving (12) we get 12 1235 Again, the figure 3 next to the last of the multi- plicand being multiplied by the multiplier 12 gives 12 1265 3 1 Lildvatyuddbarana by Krparama Daivajna, Asiatic Society of Bengal, Calcutta, Ms. No. III. F. 1 10. A. MULTIPLICATION 1 43 Then after moving (12) we get 12 1265 3 Again, multiplying the first figure 5 of the multi- plicand with the multiplier 12, we get 12 1260 36 Then rubbing out the multiplier, the numbers 1260 36 being added according to places give 1620." Transmission to the West. The kapdta-sandhi method of multiplication was transmitted to the Arabs who learnt the decimal arithmetic from the Hindus. It occurs in the works of Al-Khowarizmi (825), Al-Nasavi 1 (c. 1025) Al-Hassar z (c. 1 175), Al-Kalasadi J (cr. 1475) and many others. The following illustration is taken from the work of Al-Nasavi who calls this method al-amal al-hindi and tdrik al-hindi ("the method of the Hindus"): Example. To multiply 324X753 43 309 nn 2t$$$2 .'. Product = 243972. 324 753 753 753 1 F. Woepeke, I (6), p. 407. 2 H. Suter, Bibl. Math., II (3), p. 16. 3 Ibid, p. 17. 144 ARITHMETIC In the above the arrangement of the multiplicand and multiplier is just the same as in the Hindu method. The multiplier is moved in the same way. As the work is performed on paper, the figures are crossed out instead of being rubbed out. It may be mentioned that in Europe, the method is found reproduced in the work of Maximus Planudes. Gelosia Method. The method known as the 'gelosia', 1 has been described in the Ganita-manjari (16th century) as the kapdta-sandhi method. It appears also in Ganesa's commentary on the" Lildvati. As the description of the kapdta-sandhi given by the older mathematicians is incomplete and sketchy, it is diffi- cult to say whether Ganesa is right in identifying the gelosia method with the kapdta-sandhi of older writers. In our opinion Ganesa's identification is incorrect. 2 We are at present unable to say definitely whether this method is a Hindu invention or was borrowed from the Arabs who are said to have used it in the 13 th century. 3 It occurs in some Arab works of the 14th century, and also in Europe about the same time. Ganesa was undoubtedly one of the best mathe- maticians of his time and the fact that he identified this method with the kapdta-sandhi which is the oldest known method shows that the gelosia method must have been in use in India from a long time before him. The only available description of the method runs as follows: "(Construct) as many compartments as there are places in the multiplicand and below these as many 1 We shall designate it as kapata-sandbi (b) method.' 2 Cf. the quotation from Sripati given before, p. 137. 3 Smith, History, II, p. 115. MULTIPLICATION 145 as there are places in the multiplier; the oblique lines in the first, in the one below, and in the other (com- partments) are produced. Multiply each place of the multiplicand, by the places of the multiplier (which are) one below the other and set the results in the com- partments. The sum taken obliquely on both sides of the oblique lines in the compartments gives the product. This is the kapdta-sandbi." 1 The following illustration is taken from Ganesa' s commentary on the L^ildvatr. To Multiply 135 by 12 I 3 5 / / / / I /~ 3 / 5 / 1 / / / 2 1 / 6 I / o| Cross Multiplication Method. This method has been mentioned by Sridhara, Mahavira, Sripati and some later writers as the tastha method. These writers, how- ever, do not explain the method. Sridhara simply states: "The next (method) in which (the multi- plier) is stationary is the tastha." 2 The. method is algebraic and has been compared to tiryak-gunana or vajrdbhydsa (cross multiplication) used in algebra. 3 It has been explained by Ganesa (c. 1545) thus: 1 Translated from the Ganita-manjari of Ganesa, son of Dhundhiraja. 2 TriJ, p. 3. 3 Colebrooke, I.e., p. 171, fn. 5. I46 ARITHMETIC "That method of multiplication in which the numbers stand in the same place, 1 is called tastba- gunana. It is as follows: after setting the multiplier under the multiplicand multiply unit by unit and note the result underneath. Then as in vajrdbhydsa multiply unit by ten and ten by unit, add together and set down the result in the line. Next multiply unit by hundred, hundred by unit and ten by ten, add together and set down the result as before; and so on with the rest of the digits. This being done, the line of results is the product." 2 This method was known to the Hindu scholars of the 8 th century, or earlier. The method seems to have travelled to Arabia and thence was transmitted to Europe, where it occurs in Pacioli's Suma 3 and is stated to be "more fantastic and ingenious than the others." GaneSa has also remarked that "this (method) is very fantastic and cannot be learnt by the dull without the traditional oral instructions." Multiplication by Separation of Places. This method of multiplication known as sthdna-khanda, is based on the separation of the digits of the multiplicand or of the multiplier. It has been mentioned in all the works from 628 A.D. onwards. Bhaskara II describes the method as follows: "Multiply separately by the places of figures and add together." 4 With reference to the example 135x12, Bhaskara II explains the method thus: 1 In contra-distinction to the method in which the multiplier moves from one place to another. 2 Ganesa's commentary on the Uldvati^ i, 4^6. 3 Smith (I.e., II, p. 112) quotes from this work. ') s i35 i35 1 2 270 '35 1620 Zigzag Method. The method is called gomutrikd.* It has been described by Brahmagupta. It is in all 1 In a manuscript used by Taylor, see his Uldwati, pp. 8-9. 2 This arrangement is found in the commentary of Gariga- dhara on the IMdvatT, in the library of the Asiatic Society of Bengal, Calcutta. 3 Found in Gangadhara, I.e. 4 The word gomfitrikd, means "similar to the course of cow's urine," hence "zigzag." Colebrooke's reading gosutrikd is in- correct. The method of multiplication of astronomical quantities is called gomiitrika even upto the present day by the pandits. I48 ARITHMETIC essentials the same as the sthdna-khanda method. The following illustration is based on the commentary of Prthudakasvami. "Example. To multiply 1223 by 235 The numbers are written thus : 2 1223 3 1223 5 1223 The first line of figures is then multiplied by 2, the process beginning at the units place, thus: 2 X 3=6; 3 is rubbed out and 6 substituted in its place, and so on. After all the horizontal lines have been multiplied by the corresponding numbers on the left in the vertical line, the numbers on the pdti stand thus: 2446 3669 6 1 1 5 287405 after being added together as in the present method. The sthdna-khanda and the gomutrikd methods resem- ble the modern plan of multiplication most closely. The sthdna-khanda method was employed when working on paper. Parts Multiplication Method. This method is mentioned in all the Hindu works from 628 A.D. onwards. Two methods come under this head: (/") The multiplier is broken up into two or more parts whose sum is equal to it. The multiplicand is then multiplied severally by these and the results added. 1 (//) The multiplier is broken up into two or more aliquot parts. The multiplicand is then multiplied by 1 Thus I2 Xi35 = (4+8)Xi35=(4X i35)+(8xi35)- MULTIPLICATION 149 one of these, the resulting product by the second and so on till all the parts are exhausted. The ultimate product is the result. 1 These methods are found among the Arabs and the Italians, having been obtained from the Hindus. They were known as the "Scapezzo" and "Repiego" methods respectively among the Italians. 2 Algebraic Method. This method was known as ista- gunana. Brahmagupta's description of the method has been already quoted. Bhaskara II explains it thus: "Multiply by the multiplicator diminished or in- creased by an assumed number, adding or subtracting (respectively) the product of the multiplicand and the assumed number." 3 This is of two kinds according as we (/) add or {it) subtract an assumed number. The assumed number is so chosen as to give two numbers "with which multiplication will be easier than with the original multiplier. The two ways are illustrated below: CO 135X i2=i3jx(i2+8)— 135X8 =2700 — 1080=1620 (it) 135 X 12=135 X (12—2)+ 135X2 = 1350+270=1620 This method was in use among the Arabs 4 and in Europe 5 , obviously under Hindu influence. 3 Thus 12X 135 = 3X 135 X4. 2 Smith, History, II, p. 117. 3 -L, p. 3- 4 E.g., Beha Eddin (c. 1600). See G. Enestrom, Bib/. Math., VII ( } ), p. 95 . 5 E.g., Widman (1489), Riese (1522), etc. See Smith, I.e., p. 120. I JO ARITHMETIC J. DIVISION Terminology. Division seems to have been regard- ed as the inverse of multiplication. The common Hindu names for the operation are bhdgahdra, bhdjana, harana, chedana, etc. All these terms literally mean "to break into parts," i.e., "to divide," excepting harana which denotes "to take away." This term shows the relation of division to subtraction. The dividend is termed bhdjj/a, hdrya, etc., the divisor bhdjaka, bhdgahara or simply hara, and the quotient labdhi "what is obtained" or labdha. The Operation. Division was considered to be a difficult and tedious operation by European scholars even as late as the ijth and 16th centuries; 1 but in India the operation was not considered to be difficult, as the most satisfactory method of performing it had been evolved at a very early period. In fact, no Hindu mathematician seems to have attached any great im- portance to this operation. Aryabhata I does not men- tion the method of division in his work. But as he has given the modern methods for extracting square- and cube-roots, which depend on division, 2 we conclude that the methdd of division was well-known in his time and was not described in the Aryabhatiya as it was considered to be too elementary. Most Siddhanta writers have followed Aryabhata in excluding the process of division from their works, e.g., Brahma- gupta (628), Sripati (1039), and some others. A method of division by removing common factors seems to have been employed in India before the inven- tion of the modern plan. This removal of common 1 Smith, I.e., p. 132. 2 He" has used the technical term labdha for the quotient. DIVISION 151 factors is mentioned in early Jaina works. r It has been mentioned by Mahavira who knew the modern method, probably because it was considered to be suitable in certain particular cases: "Putting down the dividend and below it the divisor, and then, having performed division by the method of removing common factors, give out the resulting (quotient)." 2 The modern method of division is not found in the Bakhshali Manuscript, although the name of the operation is found at several places. The absence of the method may be due to the mutilated form of the text, although it is quite possible that the method was not known at that early period (200 A.D.). The Method of Long Division. The modern method of division is explained in the works on pdtiganita, the -earliest of which, Sridhara's Trisatikd, gives the method as follows: 3 "Having removed the common factor, if any, from the divisor and the dividend, divide by the divisor (the digits of the dividend) one after another in the inverse* order." Mahavira says: 6 "The dividend should be divided by the divisor (which is) placed below it, in the inverse order, after having performed on them the operation of removing common factors." x Tatvdrthddhigama-sAtra, Bbdsya of Umasvati (/:. 160, ed. by H. R. Kapadia, Bombay, 1926, Part I, ii. 52, p. 225. 2 GSS, p. 11. The method would not give the quotient un- less the dividend be completely divisible by the divisor. 3 TriJ, p. 4. * Pratiloma. 5 GSS, p. 11; cf. Rangacarya's translation. 152 ARITHMETIC Aryabhata II gives more details of the process: 1 "Perform division having placed the divisor below the dividend; subtract from (the last digits of the divi- dend) the proper multiple of the divisior; this (the multiple) is the partial quotient, then moving the divisor divide what remains, and so on." Bhaskara II,' 2 Narayana 8 and others give the same method. The following example will serve to illustrate the Hindu method of performing the operation on a pdti: Example. Divide 1620 by 12. The divisor 12 is placed below the dividend thus : 1620 12 The process begins from the extreme left of the dividend, in this case the figure 16. This 16 is divided by 12. The quotient 1 is placed in a separate line, and 16 is rubbed out and the remainder 4 is substituted in its place. The subtraction is made by rubbing out figures successively as each figure of the product to be subtracted is obtained. Thus, the partial quotient r, being written, the procedure is 1620 1 12. line of quotients iX 1 = 1, so 1 of the dividend is rubbed out (as 1 — 1=0); then 1X2=2, so 4 is substituted in the place of 6 (as 6—2=4). The figures on the pa [tf are: 420 l 2 line of quotients 1 MSt, p. 144. 2 Bhaskara gives the process briefly as follows: "That number, by which the divisor being multiplied, balances the last digit of the dividend gives the (partial) quotient, and so on." (L, p. 3) 3 GK, i. 16. DIVISION 153 The divisor 12 is now moved one place to the right giving 420 12 line of quotients 42 is then divided by 12. The resulting quotient 3 is set in the "line of quotients," 42 is rubbed out and the remainder 6 substituted in its place. The figures now stand thus: 60 13 12 line of quotients Moving the divisor one place to the right, we have 60 12 On division being performed, as before the resulting quotient 5 is set in the "line of quotients" and 60 is rubbed out leaving no remainder. The line of quo- tients 1 has 135 which is the required result. The above process, when the figures are not obliterated and the successive steps are written down one below the other, becomes the modern method of long division. The method seems to have been invented in India about the 4th century A.D., if not earlier. It was trans- mitted to the Arabs, where it occurs in Arabic works from the 9th century onwards.' 2 From Arabia the method travelled to Europe where it came to be known as the galley {galea, batelld) method. 3 In this variation 1 The "line of quotients" was usually written above the divi- dend. 2 Al-Khowarizml (c. 825), AI-Nasavi (r. 1025); cf. Smith, I.e., pp. 138-139. 3 Also called the 'scratch method'. IJ4 ARITHMETIC of the method, the figures obtained at successive stages ate written and crossed out, for the work is carried out on paper (where the figures cannot be rubbed out). The method was very popular in Europe from the 15 th to the 1 8 th century. 1 The above example worked on the galley plan would be represented thus: 4 I /02O 1 m 1 II t 46 mo 13 n in it 46 vm 135 tm 1 Comparing the successive crossing out of the figures in I, II and III, with the rubbing out of figures in the corresponding steps according to the Hindu plan, it becomes quite clear that the galley method is exactly the same as the Hindu method. The crossing out of figures appears to be more cumbrous than the elegant Hindu plan of rubbing out. The Hindu plan of moving the divisor as the digits of the quotient were evolved, although not essential, was also copied and occurs in the works of such well-known Arab writers as Al-Khowarizmi (825), Al-Nasavi (c. 1025) and others. The mediaeval Latin writers called this feature the antirioratio. 1 For details see Smith, l.c, pp. 136-139. SQUARE IJ5 6. SQUARE Terminology. The Sanskrit term for square is varga or krti. The word varga literally means "rows" or "troops" (of similar things). But in mathematics it ordinarily denotes the square power and also the square , figure or its area. Thus Aryabhata I says: 1 "A square figure of four equal sides 2 and the (number representing its) area are called varga. The product of two equal quantities is also varga." How the word varga came to be used in that sense has been clearly indicated by Thibaut. He says: "The origin of the term is clearly to be sought for in the graphical representation of a square, which was divided in as many vargas or troops of small squares, as the side contained units of some measure. So the square drawn with a side of five padas length could be divided into five small vargas each containing five small squares, the side of which was one pada long." 3 This expla- nation of the origin of the term varga is confirmed by certain passages in the Sulba works. 4 The term krti literally means "doing," "making" or "action." It carries with it the idea of specific performance, probably the graphical representation. Both the terms varga and krti have been used in the mathematical treatises, but preference is given to the term varga. Later writers, while defining these terms in arithmetic, restrict its meaning. Thus Sridhara says: 5 *A, ii. 3. 2 The commentator Paramesvara remarks: "That four sided figure whose sides are equal and both of whose diagonals are also equal is called samacaturajra ("square")." 3 Thibaut, Sulba-sutras, p. 48. 4 ApSl, iii. 7; K$t, iii. 9; cf. B. Datta, American Math. Monthly, XXXIII, 193 1, p. 375. 5 Tw, p. 5. I 5 6 ARITHMETIC "The product of two equal numbers is varga." Prthudakasvami 1 , Mahavira 2 and others give similar definitions. The Operation. The occurrence of squaring as an elementary operation is characteristic of Hindu arith- metic. The method, however, is not simpler than direct multiplication. It was given prominence by the Hindu writers probably because the operation of square-root is the exact inverse of that of squaring. Although the method first occurs in the Brdhma-sphuta-siddhdnta, there is no doubt that it was known to Aryabhata I as he has given the square-root method. Brahmagupta gives the method 3 very concisely thus: "Combining the product, twice the digit in the less 4 (lowest) place into the several others (digits), with its {i.e., of the digit in the lowest place) square (repeatedly) gives the square." Sridhara (750) is more explicit: 5 "Having squared the last digit multiply the rest of the digits by twice the last; then move the rest of the digits. Continue the process of moving (the remain- 1 Cf. Colebrooke, I.e., p. 279. 2 GSS, p. 12. 3 The method is not mentioned in the chapter on Arithmetic, but seems to have been mentioned as an afterthought in the form of an appendix, (BrSpSi, p. 212). 1 Kdseriinam has been translated by Colebrooke as "the less portion." This translation is incorrect. He says that "the text is obscure" (p. 322, fn. 9), for according to his translation the rule becomes practically meaningless. The term rdhrunam must be translated by "the digit in the lowest place." Dvivedi agrees with the above interpretation (p. 212). The method taught here is "the direct method of squaring." s Tris, p. 5. The translation given by Kaye and Ramanu- jacharia is incorrect. (Bibl. Math., XIII, 1912-13). SQUARE I57 ing digits after each operation) to obtain the square." Mahavira 1 (850) gives more details: "Having squared the last (digit), multiply the rest of the digits by twice the last, (which is) moved for- ward (by one place). Then moving the remaining digits continue the same operation (process). This gives the square." Bhaskara II 2 writes: "Place the square of the last (digit) over itself; and then the products of twice the last (digit) and the others {i.e., the rest) over themselves respectively. Next, moving the number obtained by leaving the last digit (figure), repeat the procedure." He has remarked that the above process may be begun also with the units place. 3 The following is the method of working on the pdti, the process beginning from the last place, accord- ing to Sridhara, Mahavira, Bhaskara II and others: To square 125. The number is written down, 125 The last digit is 1. Its square is placed over itself. 1 125 Then twice the last digit 2X1 = 2; placing it below the rest of the figures (below 2 or below 5 according as the direct or inverse method of multiplication is used) 1 GSS, p. 12. 2 JL, p. 4- 3 -L, p- 5- I58 ARITHMETIC and rubbing out the last digit 1, the work on the pdti appears as 1 25 2 Performing multiplication by 2 (below) and placing the results over the respective figures, we get 150 25 One round of operation is completed. Next, moving the remaining digits, i.e., 25, we have 150 25 Now, the process is repeated, i.e., the square of the last digit (2) is placed over itself giving 154 2 5 Then, placing twice the last digit {i.e., 2X2=4) below the rest of the digits and then rubbing out 2, we have 154 5 4 Performing multiplication, 4X5 = 20, and placing it over the corresponding figure 5, {i.e., o over j and 2 carried to the left), the work on the pdti appears as 1560 5 Thus a second round of operations is completed. Then moving 5 we have 1560 5 SQUARE 159 Squaring 5 we get 25, and placing it over 5 (i.e., 5 over 5 and 2 carried to the left) we have 15625 5 As there are no 'remaining figures' the worlc ends. 5 being rubbed out, the pdti has 15625, the required square. According to Brahmagupta and also Bhaskara II, the work may begin from the lowest place (i.e., the units place). The following method is indicated by Brahmagupta: To square .125. The number is written down 125 The square of the digit in the least place, i.e., 5 2 =2 5 is set over it thus: t 25 125 Then, 2X5 = 10 is placed below the other digits, and five is rubbed out, thus: 25 12 10 Multiplying by 10 the rest of the digits, i.e., 12, and setting the product over them (the digits), we. have 1225 12 IQ Then rubbing out 10 which is not required and moving the rest of the digits, i.e., 12, we have 1225 12 l6o ARITHMETIC Thus one round of operations is completed. Again, as before, setting the square of 2 above it and 2X2=4 below 1, we have 1625 1 4 Multiplying the remaining digit 1 by 4, and setting the product above it, we have 5625 1 Then, moving the remaining digit 1, we obtain 5625 1 Thus the second round of operations is completed. Next setting the square of 1 above it the process is completed, for there are no remaining figures, and the result stands thus: 15625 Minor Methods of Squaring. The identity (/) t?={n— a)(n+a)+a°- has been mentioned by all Hindu mathematicians as affording a suitable method of squaring in some cases. For instance, ^ i5 2 =ioX 20+25=225. Brahmagupta says: "The product of the sum and the difference of the number (to be squared) and an assumed number plus the square of the assumed number give the square." 1 Sridhara (750) gives it thus: "The square is equal to the product of the sum and the difference of the given number and an assumed 1 BrSpSi, p. 212. SQUARE l6l quantity plus the square of the assumed quantity." 1 Mahavira, Bhaskara II, Narayana and others also give this identity. The formula («) (a+ by = a*-\-b 3 -\-zab, or its general form (a-\- b+t+ ....) 2 =.a t +b*+c*+ .... +zab + . • has been given as a method "of squaring. Thus Mahavira 2 says: "The sum of the squares * of the two or more portions* of the number together with their products each with the others multiplied by two gives the square." Bhaskara II 4 gives: "Twice the product of the two parts plus the square of those parts gives the square." The formula (tit) » 2 =i-t-34-5+ to n terms has been mentioned by Sridhara and Mahavira. Sridhara* says: "(The square of a number) is the sum of as many terms in the series of which one is the first term and two the common difference." 1 Tri/, p. 5. 2 GSS, p. .13. 3 The word sthdna has been used in the original. This word has been generally used in the sense of 'notational place.' Following the commentator, we have rendered it by "portion." As a given number, say, 125, can be broken into parts as J04-40+3J or as 100+20+5, an d as the rule applies to both, it is immaterial whether the word 'stbdna' is translated by 'place' or 'portion.' This rule appears to have been given as an explanation of the Hindu method of squaring used with the place-value numerals. for the reduction of the form of i-+ % of ( \ [+ c f b 3 ° f !>- 3 Tris, p. 11. 13 1 94 ARITHMETIC Reduce to a proper fraction: 3*+£ of 3 i+i- of ( 3 i + i- of 3 4)+4 + £ of j + f of (4+1 of 4). This was written as 3 i 2 i 2 I 4 I 3 i 6 i 4 Adding denominators to numerators of the lower fractions, and applying rule (/) to left-hand top com- partment to reduce it to a proper fraction, we get 7 2 i 2 5 4 4 3 7 6 5 4 Now performing multiplication as directed, i.e., mul- tiplying the denominator of the first fraction by all the lower denominators and the numerator by the sum of the numerators and denominators of the lower frac- tions, we get 4X-*X£=W-. and+X£X,|=-§£ i.e.. 245 48 20 24 have Then making denominators similar (savarnana), we 245 48 40 48 FRACTIONS J 95 performing the addition we have z^tf- or j-f-f- as the result. The rule for bhdgdpavdha is given in all the works on pdtiganita. It is the same as that for bhagdnubandha, except that "addition" or "increase" is replaced by "subtraction" or "decrease" in the enunciation of the rule for bhdgdpavdha. Lowest Common Multiple. Mahavira 1 was the first amongst the Indian mathematicians to speak of the lowest common multiple in order to shorten the process. He defines niruddha (L. C. M.) as follows: "The product of the common factors of the denominators and their resulting quotients is called niruddha'' The process of reducing fractions to equal deno- minators is thus described by him: 2 "The (new) numerators and denominators, obtained as products of multiplication of (each original) numera- tor and denominator by the (quotient of the) niruddha {i.e., L. C. M.) divided by the denominator give frac- tions with the same denominator." Bhaskara IP does not mention niruddha but observes that the process can be shortened. He says: "The numerator and denominator may be multi- plied by the intelligent calculator by the other deno- minator abridged by the common factor." The Eight Operations. Operations with frac- tions were known in India from very early times, the method of performing them being the same as now. *GSS, p. 3 3(5 6). *GSS,p. 33(56). 3 L, p. 6. 196 ARITHMETIC Although Aryabhata does not mention the elementary- operations, there is evidence to show that he knew the method of division by fraction by inverting it. All the operations are found in the Bakhshali Manuscript (c. 200). Addition and Subtraction. These operations were performed after the fractions were reduced to a common denominator. Thus Sridhara says: 1 "Reduce the fractions to a common denominator and then add the numerators. The denominator of a whole number is unity." Brahmagupta and Mahavira give the method under Bhdgajati. Mahavira differs from other writers in giving the methods of the summation of arithmetic and geo- metric series under the title of addition (samkalitd).* Later writers follow Sridhara. Multiplication. Brahmagupta says: 3 "The product of the numerators divided by the product of the denominators is the (result of) multipli- cation of two or more fractions." While all other writers give the rule in the same way as Brahmagupta, Mahavira refers to cross reduction in order to shorten the work: 4 "In the multiplication of fractions, the numerators are to be multiplied by the numerators and the deno- minators by denominators, after carrying out the process of cross reduction, 5 if that be possible." 1 TriJ, p. 7. *Cf. GSS, pp. 28 (22) ff. 3 BrSpSi, p. 173. * GSS, p. 25 (*)■ 8 Vajrdpavartana-vidhi, i.e., "cancellation crosswise," thus 2 r\ 71 6 v 1 * ^4 &x!= X =*=i- FRACTIONS *97 Division. Although the elementary operations are not mentioned in the A^ryabhatiya, the method of division by fraction is indicated under the Rule of Three. The . fxi Rule of Three states the result as J - 1 When these P quantities are fractional, we get an expression of the a c -rX-, form , for the evaluation of which Aryabhata I m ' J \ n states: "The multipliers and the divisor are multiplied by the denominators of each other." As will be explained later on (p. 204) the quantities are written as a b m n c d Transferring the denominators we have a m n b c d Performing multiplication, the result is — t-j- The above interpretation of a rather obscure line" in the A.ryabhatiya is based on the commentaries of Suryadeva and Bhaskara I. Thus Suryadeva says: 1 Where / = phala, i.e., "fruit," /= icchd, i.e., "demand or requisition" ff^=pramdna, i.e., "argument." 2 A, p. 43 . Previous writers seem to have been misled by the commentary of Paramesvara which is very vague; cf; Clark (p. 40) and P. C. Sengupta (p. 7.5). , 198 ARITHMETIC "Here by the word gunakdra is meant the multiplier and multiplicand, i.e., the phala and iccbd quantities that are multiplied together. By bhdgahdra is meant the pramdna quantity. The denominators of the phala and iccbd are taken to the pramdna. The denominator of the pramdna is taken with the pbala and iccbd. Then multi- plying these, i.e., (the numerators of) the pkala and iccbd and this denominator, and dividing by (the product of) the numbers standing with the pramdna, the result is the quotient of the fractions." Brahmagupta 1 gives the method of division as fol- lows: "The denominator and numerator of the divisor having been interchanged, the denominator of the dividend is multiplied by the (new) denominator and its numerator by the (new) numerator. Thus division of proper fractions is performed." Sridhara" adds the following to the method of multiplication: "Having interchanged the numerator and deno- minator of the divisor, the operation is the same as before." 3 Mahavira 4 explains the method thus: "After having made the numerator of the divisor 5 its denominator (and vice versa) the operation is the same as in multiplication." "Or, 6 when (the fractions constituting) the divisor 1 BrSpSi, p. 173. 2 TriJ, p. 8. 3 i.e., the same as that of multiplication. * GSS, p. 26 (8). 5 Mahavira uses the term pramdna-rdsi for divisor, showing thereby its connection with the 'rule of three.' G This is similar to the way in which Aryabhata I expresses the method. FRACTIONS [ 99 and dividend are multiplied by the denominators of each other and these products are without denomina- tors, (the operation) is as in the division of whole 1 numbers." Square and Square-root. Brahmagupta 2 says: "The square of the numerator of a proper fraction divided by the square of the denominator gives the square." "The square-root of- the numerator of a proper fraction divided by the square- root of the denominator gives the square-root." Other works contain the same rules. Cube and Cube-root. Sridhara 3 gives the rule as follows: "The cube of the numerator divided by the cube of the denominator gives the cube, and the cube-root of the numerator divided by the cube-root of the denominator gives the cube-root." Other works give the same rules. Unit Fractions. Mahavira has given a number of rules for expressing any fraction as the sum of a number of unit fractions. 4 These rules do not occur in any other work, probably because they were not considered important or useful. (i) To express i as the sum of a number («) of unit fractions. The rule for this is: 5 "When the sum of the different quantities having 1 The term for whole number is sakala. 2 BrSpSi, p. 174. 3 TV;/, p. 9. 4 There is no technical term for unit fraction. The term used is rupdmsaka-rdsi, i.e., "quantity with one as numerator." 5 GSS, p. 36 (75). 2O0 ARITHMETIC one for their numerator is i, the (required) denomina- tors are such as, beginning with i, are in -order multi- plied by 3, the first and the last being multiplied again by 2 and §." Algebraically the rule is 3 ' 3" 3* 3 n ~ 2 2-3" -2 (2) To express 1 as the' sum of an odd number of unit fractions. The rule for this is stated thus: 1 "When the sum of the quantities (fractions) having one for each of their numerators is one, the denomina- tors are such as, beginning with two, go on rising in value by one, each being further multiplied by that which is (immediately) next to it and then halved." Algebraically this is I = r + r-+ + 7 r r+ r 2.34 3.4.^ (2»-l).2«.£ 2Jl.\ (3) To express a unit fraction as the sum of a number of other fractions, the numerators being given? The rule for this is: "The denominator of the first (of the supposed or given numerators) is the denominator of the sum, that of the next is this combined with its numerator and so on; and then multiply (each denominator) by that which is next to it, the last being multiplied by its own numerator. (This gives the required denomina- tors)." iGSS,?. 36(77). 2 Each may be one. GSS, p. 36(78). FRACTIONS 20 1 Algebraically this gives : 1 _ a/ , a * n n (w+flj (»+') the two denominators are the factors 2 of the denominator of the sum, each multi- plied by their sum." Expressed algebraically the rules are: (0 - = — +— 3 w n p.n ~ p.n p^T a.b a(a+b) ' b(a+b) (6) To express any fraction as the sum of two other fractions whose numerators are given. The rule for this is: 4 "Either numerator multiplied by a chosen number, then combined with the other numerator, then divided by the numerator of the sum so as to leave no remainder, and then divided by the chosen number and multi- plied by the denominator of the sum gives rise, to one denominator. The denominator corresponding to the other (numerator), however, is this (denominator) multiplied by the chosen quantity," *GSS, p. 37(85). 2 hdra-hdra-labdha, lit. "the divisor and quotient by that divisor." 3 The integer^) is so chosen that n is divisible by (p — 1). * GSS, p. 38(87). THE RULE OF THREE 20$ Algebraically the rule is m a . b m X p ~m~ X p Xp A particular 1 case of this would be ma b n an-\-b an-\-b — x» m m provided that (an-\-b) is divisible by m. (7) To express a given fraction as the sum of an even number of fractions whose numerators are previously assigned. The rule for this is: 2 "After splitting up the sum into as many parts, having one for each of their numerators, as there are pairs (among the given numerators), these parts are taken as the sum of the pairs, and (then) the denominators are found according to the rule for finding two frac- tions equal to a given unit fraction." 12. THE RULE OF THREE Terminology. The Hindu name for the Rule of Three terms is trairasika ("three terms," hence "the rule of three terms"). The term trairdsika can be traced back to the beginning of the Christian era as it occurs in the 1 Evidently, the chosen number p must be a divisor of «, and such that is an integer. The solution given does not hold for any values of a and b, but only for such values as allow of an integer p to be so chosen as to satisfy the required conditions. 2 GSS, p. 38(89). 204 ARITHMETIC Bakhshali Manuscript, 1 in the A.ryabhatiya and in all other works on mathematics. About the origin of the name Bhaskara I (c. 525) remarks: 2 "Here three quantities are needed (in the statement and calculation) so the method is called trairdsika ("the rule of three terms")." A problem on the rule of three has the form: If p yields/, what will / yield? In the above, the three terms are p, f and /'. The Hindus called the term p, pramdna ("argument"), the term /, phala ("fruit") and the term /', iccbd ("requisi- tion"). These names are found in all the mathematical treatises. Sometimes they are referred to simply as the first, second and third respectively. Aryabhata II differs from other writers in giving the names mdna, vinimaya and iccbd respectively to the three terms. It has been pointed out by most of the writers that the first and third terms are similar, i.e., of the same deno- mination. The Method. Aryabhata I (499) gives the follow- ing rule for solving problems on the Rule of Three: "In the Rule of Three, the phala ("fruit"), being multiplied by the iccbd ("requisition") is divided by the pramdna ("argument"). The quotient is the fruit corresponding to the iccbd. The denominators of one being multiplied with the other give the multipliet {i.e., numerator) and the divisor (i.e., denominator)." 3 1 The term rail is used in the enumeration of topics of mathematics in the Sthdndtiga-siltra(c. 300 B.C.) (Sutra 747). There it probably refers to the Rules of Three, Five, Seven, etc. 2 In his commentary on the Aryabhatlya. 3 The above corresponds to dryd 26 and the first half of dryd 27 of the Ganitapdda of the Aryabhatiya; compare the working of Hxample I, where the interchange of denominators takes place. See also pp. 195/. THE RULE OF THREE 205 Brahmagupta gives the rule thus: "In the Rule of Three pramdna ("argument"), phala ("fruit") and icchd ("requisition") are the (given) terms; the first and the last- terms must be similar. The icchd multiplied by the phala and divided by the pramdna gives the fruit (of the demand)." 1 Sridhara states: "Of the three quantities, the pramdna ("argument") and icchd ("requisition") which are of the same deno- mination are the first and the last; the phala ("fruit") which is of a different denomination stands in the middle; the product of this -and the last is to be divided by the first." 2 Mahavira writes: "In the Rule of Three, the icchd ("requisition") and the pramdna ("argument") being similar, the result is the product of the phala and icchd divided by the pramdna"'* Aryabhata II introduces a slight variation in the terminology. He says: "The first term is called mdna, the middle term vinimaya and the last one icchd. The first and the last are of the same denomination. The last multiplied by the middle and divided by the first gives the result."* Bhaskara II, Narayana and others give the rule in the same form as Brahmagupta or Sridhara. The Hindu method of working the rule may be illustrated by the following examples taken from the Trisatikd'. 1 BrSpS/,p. 178. 2 TW?, p. 1 5 . *GSS,p. 5 8( 2 ). * MS:, p. 149. 2o6 ARITHMETIC Example I. 1 "If one pala and, one karsa of sandal wood are obtained for ten and a half pana, for how much will be obtained nine pala and one karsa?" Here i pala and i karsa=\\ pala, and 9 pala and 1 karsa —94- pala are the similar, quantities. The "fruit" io£ pana corresponding to the first quantity (i^ paid) is given, so that pramdna (argument) = 1^ phala (fruit) = io£ icchd (requisition) = 9^ The above quantities are placed in order as I I 4 10 1 2 9 1 4 Converting these into proper fractions we have 5 4 21 2 37 4 Multiplying the second and the last and dividing by the first, we have 21 2 5 4 37 4 2I V 3? TAT 5 Or transferring denominators 21 5 4 2 37 4 2I -4-37 5.2.4 pala 4 purdna, 13 pana, 2 kdkini and 16 vardtaka. In actual working the intermediate step SL) y 3 7 * 1 TriJ, p. 1 5 . THE RULE OF THREE 207 was not written. The denominators of the multipliers were transferred to the side of the divisor and that of the divisor to the multipliers, thus giving at once "•4-37 5.2.4 ' Example II. 1 "Out of twenty necklaces each of which contains eight pearls, how many necklaces, each containing six pearls, can be made?" Firstly, we have I 8 20 The result (performing the operation of the Rule of Three) is 160 pearls. Secondly, perform the operation of the Rule of Three on the following: If 6 pearls are contained in one necklace, how many necklaces will contain 160 pearls? Placing the numbers, we have 6 1 160 Result: necklaces z6, part of necklace Inverse Rule of Three. The Hindu name for the Inverse Rule of Three is vyasta-trairdsika (lit. "inverse rule of three terms"). After describing the method of the Rule of Three the Hindu writers remark that the operation should be reversed when the proportion is inverse. Thus Sridhara observes: "The method is to multiply the middle term by the first and to divide by the last, in case the proportion is different." 2 1 Tris, p. 17. 2 TriS, p. 18. 208 ARITHMETIC Mahavira says: "In the case of this (proportion) being inverse, the operation is reversed." 1 Bhaskara II writes: "In the inverse (proportion), the operation is re- versed." 2 He further observes: "Where with increase of the icehd (requisition) the phala decreases or with its decrease the phala in- creases, there the experts in calculation know the method to be the Inverse Rule of Three." 3 "Where the value of living beings is regulated by their age; and in the case of gold, where the weight and touch are compared; or when heaps are subdivided, 4 let the Inverse Rule of Three be used." 5 Hxample: Example II given under the Rule of Three above has been solved also by the application of* the Inverse Rule as follows: "Statement | 8 20 6 Result: necklaces 26 2 3 Here if the icehd, i.e., the number of pearls in a necklace, increases, the phala, i.e., the number of neck- laces, decreases, so that the Inverse Rule of Three is applied. Appreciation of the Rule of Three. The Rule of Three was highly appreciated by the Hindus because 1 GSS, p. 58(2). *L,p. 17. 3 -L,p. 17. * "When heaps oi grain, which have been meted with a small measure, are again meted with a larger one, the number decreases . . . . " (Com. of Suryadasa). *L, p. 18. THE RULE OF THREE ZO9 of its simplicity and its universal application to ordinary •problems. The method as evolved by the Hindus gives a ready rule which can be applied even by the "ignorant person" to solve problems involving proportion, with- out fear of committing errors. Varahamihira (505) writes: "If the sun performs one complete revolution in a year, how much does he accomplish in a given number of days? Does not even an ignorant person calculate the sun in such problems by simply scribbling with a piece of chalk?" 1 Bhaskara II has eulogised the method highly at several places in his work. His remarks are: "The Rule of Three is indeed, (the essence of) arithmetic." 2 "As Lord Sri Nirayana, who relieves the sufferings of birth and death, who is the sole primary cause of the creation of the universe, pervades this universe through His own manifestations as worlds, paradises, mountains, rivers, gods, men, demons, etc., so does the Rule of Three pervade the whole of the science of calculation. Whatever is computed whether in algebra or in, this (arithmetic) by means of multiplication and division may be comprehended by the sagacious learned as the Rule of Tjhree. What has been composed by the sages through the multifarious methods and operations such as miscellaneous rules, etc., teaching its easy variations, is simply with the object of increasing the comprehen- sion of the duller intellects like ourselves." 3 On another occasion Bhaskara II observes: 1 PSi, iv. 37. 2 L, p. 15. The same remark occurs in SiSi, Golddbydya, PraJnddbyaya, verse 3. s L,p. l 76'. 14 2 1 ARITHMETIC "Leaving squaring, square-root, cubing and cube- root, whatever is calculated is certainly variation of the Rule of Three, nothing else. For increasing the comprehension of duller intellects like ours, what has been written in various ways by the learned sages having loving hearts like that of the bird cakora, .has become arithmetic." 1 Proportion in the West. The history of the Hindu rules of proportion shows how much the West was indebted to India for its mathematics. The Rule of Three occurs in the treatises of the Arabs and mediaeval Latin writers, where the Hindu name 'Rule of Three' has been adopted. Although the Hindu names of the terms were discarded, the method of placing the terms in a line, and arranging them so that the first and last were similar, was adopted. Thus Digges (1572) remarked, 2 "Worke by the Rule ensueing Multiplie the last number by the seconde, and diuide the Product by the first number," ... "In the placing of the three numbers this must be observed, that the first and third be of one Denomination." The rule, as has been already stated, was perfected in India in the early centuries of the Christian era. It was transmitted to the Arabs probably in the eighth century and thence travelled to Europe, where it was held in very high esteem 3 and called the "Golden Rule." * Compound Proportion. The Hindu names for compound proportion are the Rule of Five, the Rule of 1 SiSi, Golddbyaya, Prafrtddhyaya, verse 4. 2 Quoted by Smith, I.e. p. 488. s The Arabs, too, held the method in very high esteem as is evidenced by Al-Birunt's writing a separate treatise, Fi raJikat al-hind ("On the raJika of the Hindus") dealing with the Hindu Rules of Three or more terms. Compare also India (I. 313) where an example of tyasta-trairdjika, ("the Inverse Rule of Three") is given. THE RULE OF THREE 211 Seven, the Rule of Nine, etc., according to the number of terms involved in the problems. These are some- times grouped under the general appellation of the "Rule of Odd Terms." The above technical terms as well as the rules were well-known in the time of Aryabhata I (499), although he mentions the Rule of Three only. That the distinction between the Rule of Three and Compound Proportion is more artificial than real was stressed by Bhaskara I (c. 525) in his com- mentary on the Aryabkatiya. He says: "Here Acarya Aryabhata has described the Rule of Three only. How the well-known Rules of Five, etc. are ' to be obtained? I say thus: The Acarya has described only the fundamentals of anupata (proportion). All others such as the Rule of Five, etc., follow from that fundamental rule of proportion. How? The Rule of Five, etc., consist of combinations of the Rule of Three .... In the Rule of Five there are two Rules of Three, in the Rule of Seven, three Rules of Three, and so on. This I shall point out in the examples." Remarks similar to the above concerning the Rules of Five, Seven, etc., have been made by the com- mentators of the Lilavati, especially by Ganesa and Suryadasa. 1 In problems on Compound Proportion, two sets of terms are given. The first set which is complete is called pramana paksa (argument side) and the second set in which one term is lacking is called the icchd paksa (requisition side). The Method. The rule relating to the solution of problems in compound proportion has been given by Brahmagupta as follows: "In the case of odd terms beginning with/ three 1 Noted by Colebrooke, J.c, p. 35, note. 212 ARITHMETIC. terms 1 upto eleven, the result is obtained by transposing the fruits of both sides, from one side to the other, and then dividing the product of the larger set of terms by the product of the smaller set. In all the fractions the transposition of denominators, in Hke manner, takes place on both sides." 2 Sridhara says: "Transpose the. two fruits from one side to the other, then having transposed the denominators (also in like manner) and multiplied the numbers (so obtained on each side), divide the side with the larger number of terms by the other (side)."* Mahavira* and Aryabhata II have given the rule in the same way as Sridhara. Bhaskara II has given it thus: "In the rules of five, seven, nine of more terms, after having taken the phala (fruit) and chid" from its 1 It should be observed that, as stated above, the Rule of Three is a particular case of the above Rule of Odd terms. Brahmagupta is the only Hindu writer to have included the Rule of Three also in the above rule. Some Arab writers have fol- lowed him in this respect by not writing the terms of the Rule of Three in a line, but arranging them in compartments, as for the other rules of odd terms. 2 BrSpSi, p. 178. 3 TriJ, p. 19. *GSS,.p. 62 (32). 5 MSi, p. 150, rules 26 and 27 (repeated with a slight varia- tion). 8 The commentators differ as regards the interpretation of this word. Some take it to mean. "divisor," i.e., "denominator," while others say that it means "the fruit of the other side." The rule is, however, correct with either interpretation. The first interpretation, however, brings Bhaskara's version in line with those of his predecessors. It may be mentioned here that Arya- bhata II repeats the rule twice. At first he does not direct the transposition of denominator, and at the second time he does so. THE RULE OF THREE 213 own side to the other, the product of the larger set of terms divided by the product of the smaller set, gives the result' (or produce sought)." 1 Illustration. We shall illustrate the Hindu method of working by solving the following example taken from the Ulavati: "If the interest of a hundred in one month be. five, what' will be the interest of 16 in 12 months? Also find the time knowing the interest and principal; and telT the principal knowing the time and interest." To find interest: The first set of terms (J>ramdna paksa) is: 100 niska, 1 month, 5 niska. (phala) The second set (icchd paksa) is: 16 niska, 12 months, ?c niska The terms are now written in compartments 2 as below: IOO 16 I 12 5 1 L, p. 18. 2 The terms of the same denomination are written in com- partments in the same horizontal line. 3 The figures are written in compartments in order to faci- litate the writing of fractions and also to denote the side which contains more terms after transposition of fruits. Sometimes, the compartment corresponding .to an absent term is left vacant as we find in a copy of MunisVara's Pdtisara (in the Government Sanskrit Library at Benares). When the terms are written in compartments, the symbol o to denote the unknown or absence of a term is unnecessary. In 'some commentaries on the ISlavati (Asiatic Society of Bengal manuscripts) we find the numbers written without compartments, but in such cases the symbol o is used to denote the absence of a term. After transposition, the side on which o occurs contains a smaller number of terms than the other. 214 ARITHMETIC In the above 5 (written lowest) is the "fruit" of the first side, and there is no "fruit" on' the second side. Interchanging the fruits we get IOO 16 I 12 5 The larger set of terms is on the second "side." The product of the numbers is 960. The product of the numbers on the side of the smaller set of terms is 100. Therefore, the required result is -?-$#= V"> written as | 4 g| or 9 niska, fraction \%\ To find Time: Here the sides are 100 niska, 1 month, 5 niska and 16 niska, x months, 4 ^ niska The terms are written as IOO 16 I 5 48 5 Transposing the fruits, i.e., transposing the numbers in the bottom compartment, we get IOO 16. I 48 5 5 Transposing the denominators we have IOO 16 I 48 5 5 THE RULE OF THREE 215 Here, the larger set of terms is on the first side and their product is 4800. The product of the numbers on the side of the smaller set is 400. Therefore, the result is 4800 400 = = 12 months. 400 To know the principal: The first side is 100 niska, 1 month, j niska The second side is x niska, 12 months, ^ niska This is written as I Oo I 12 5 48 5 After transposition of fruits (i.e., the terms in the bottom cells) we have IOO I 12 48 5 5 Transposing denominators we get IOO I 12 48 5 5 The product of the numbers in the larger set divided by the product of the numbers in the smaller set, gives *i6 ARITHMETIC I480O = 16 niska. Rule of Three as a Particular Case. According to Brahmagupta, the above method may be applied to the Rule of Three. Taking the first example solved under the Rule of Three, above, and placing the terms we have .21 2 6 5 4 37 4 Transposing the fruits, we have 21 2 37 4 5 4 Transposing denominators, we get 21 2 37 4 5 4 Therefore, the result is 2I ' 37 ' 4 as before. 2-5-4 If we consider the term corresponding to the un- known as the fruit, the. terms should be set as below: 1 Here, we consider f pala of sandal wood as the "fruit" of Qpana (money). The previous method forces us to consider %*- pana as the "fruit" or the middle term, because the "first" and "third" are directed to be alike. It will be observed that any of the terms may be considered to be the fruit in the alter- native method given here. THE RULE OF THREE 217 5 4. 37 4 21 2 Hence, as before, 1 the result is 5.4.2 The above method of working the Rule of Three is found among the Arabs,* although it does not seem to have been used in India after Brahmagupta. This points to the indebtedness of the Arabs to Brahma- gupta especially, for their knowledge of Hindu arith- metic. Written as above the method of working the Rule of Three appears to be the same as the method of proportion. In the same way the rule of other odd terms, when properly translated into modern symbolism, is nothing but the method of proportion. It has been stated by Smith 3 that the Hindu methods of solution "fail to recognize the relation between the Rule of Three and proportion." 1 This statement appears to have been made without sufficient justification, for the solutions have been evidendy obtained by the use of the ideas of proportionality and variation . The aim of the Hindu works is to give a method which can be readily used by common people. For this very reason, the cases in which the variation is inverse have been enumerated. Considered as a method which stimulated the student to think for himself, the method is certainly 1 The product of the numbers on the side of the larger set is divided by the product of the numbers on the side of the smaller set. o in this case is not a number. It is the symbol for the unknown or absence. 2 Thus Rabbi ben Ezra wrote 4 ? 6 „ for 47 : 7=63: x. See Smith, I.e., p. 4896 3 1.c, p. 488. 21 8 ARITHMETIC defective, but for practical purposes, it is, in our opinion, the best that could be devised. . 13. COMMERCIAL PROBLEMS Interest in Ancient India. The custom of taking interest is a very old one. In India it can be definitely traced back to the time of Panini (c. 700 B.C.) who in his Grammar lays down rules validating the use of the suffix &? to number names in case of "an interest, a rent, a profit, a tax or a bribe given." 1 The interest became due every month and the rate of interest was generally given per hundred, 12 although this was not always the case. The rate of interest varied in different localities and amongst different classes of people, but an interest of fifteen per cent per year seems to have been considered just. Thus in Kautilya's Arthasdstra, a work of the fourth century B.C., it is laid down: "an interest of a pana and a quarter per month per cent is just. Five pana per month per cent is commercial interest. Ten pana per. month per cent prevails in forests. Twenty pana per month per cent prevails among sea traders." 3 The Gotama Sutra states: "an interest of five mdsd per' twenty {kdrsdpana) is just." 4 Interest in Hindu Ganita. The ordinary pro- blems relating to the finding out of interest, principal or time etc., the other quantities being given, occur in the section dealing with the Rule of Five. The Hindu 1 Panini's Grammar, v. i. 22, 47, 49. 2 It has been pointed out by B. Datta that the idea of per cent first originated in India. See his article in the American Mathematical Monthly, XXXIV, p. 530. 3 Arthatestra, edited and translated into English by R. Sham- sastry, Mysore, III, ii, p. 214. * Gotama Sutra, xii. 26. Since 20 mdsd equal a kdrsdpana, the rate iff 1 5 per cent annually. COMMERCIAL PROBLEMS 21 £ works generally contain a section called milraka-vyava'- hdra ("calculations relating to mixed quantities") in which occur miscellaneous problems on interest. The contents of this section vary in different works, according to their size and scope. Thus the Aryabhatija contains only one rule relating to a problem on interest, whilst the Ganita-sdra-samgraha has a large number of such rules and problems. Problem involving a Quadratic Equation. Arya- bhata I (499) gives a rule for the solution of the following problem: The principal sum^>(=ioo) is lent for one month (interest unknown=.*). This unknown interest is then lent out for /(=six) months. After this period the original interest (x) plus the interest on this interest amounts to A (= sixteen). The rate-intere'st (x) on the principal (J>) is required. The above problem requires the solution of the quadratic equation tx*+px—Ap = o, u- u ■ -pl*±y/(j>lz)'+Apt which gives x — -. The negative value of the radical does not give a solution of the problem; so the result is VApt+(p/z)*-pfz x = - - ■ This is stated by Aryabhata I as follows: "Multiply the sum of the interest on the principal and the interest on this interest (A) by the time (/) and by the principal (p). Add to this result the square of half the principal {(p/z) 2 }. Take the square-root of this. Subtract half the principal (J>\z) and divide the 1 220 ARITHMETIC remainder by the time (/). The result will be the (unknown) interest (x) on the principal." 1 Brahmagupta (628) gives a more general rule. His problem is: The principal (p) is lent out for t x months and the unknown interest on this (=.*■) is lent out for t 2 months at the same rate and becomes A. To find x. This gives the quadratic whose solution is ^^^ The negative value of the .radical does not give a solution of the problem, so it is discarded. Brahmagupta states the -formula thus: "Multiply the principal (p) by its time (/,) and divide by the other time (A,) (placing the result) at two places.- Multiply the first of these by the mixture (A). Add to this the square of half the other. Take the square- root of this (sum). From the result subtract half the other. This will be the interest (x) on the principal." 2 Other Problems. Mahavira (8jo) gives two other types of problems on "mixture" requiring the solution of simultaneous equations. As an example of the first type may be mentioned the following: 3 "It has been ascertained that the interest for i£ months (/= rate-time) on 60 (c= rate-capital) is 2 J (/'= 1 A, p, 41. The Sanskrit terms are: m/t/a=piind.pal t pbala =interest. 2 BrSpSi, p. 183. This rule is also given by Mahavira, GSS, P- 7i (44)- "GJ-J-.p. 69(32). COMMERCIAL PROBLEMS 221 rate-interest). The interest (on the unknown capital P) .for. an unknown period (T) is 24 (=1), and 60 (=m =P-\-T) is the time combined with the capital lent out. What is the time (T) and what is the capital (P)?" The problem gives: p+r = « ... (2) P-T = z t V ^-^x 4 7 V„ 2 <■/ Hence P = £ (» ± V /w 2 - fL x 4/), and T = i (» ^/w 2 - ^-X 4I) The above result is stated by Mahavira thus: "From the square of the mixture (m) subtract the rate-capital (e) divided by the rate-interest (/') multiplied by the rate-time (/) and four times the given interest (4I). Then the operation of . sarikramana 1 is performed in relation to the square-root of this and the mixture 0*0" ■ The second type of problems may be illustrated by the following example: "The interest on 30 (P) is 5 (I) for an unknown 1 Given the numbers a and b, the process of sankramana is the a-\-b 1 a — b finding out of half their sum and difference i.e. and 2 GSS, p. 68(29). Jt should be noted that both the signs of the radical are used. 222 ARITHMETIC period (T), and at an unknown rate of interest (/) per ioo {c) per i£ month (/). The mixture (/»=/+ T) is 12^. Find /' and T." 1 The solution is given by T= 4 (^± V W =_^), P and consequently i / -r v , ctI.A\ Mahavira states the solution thus: "The rate-capital (^) multiplied by its time (/) and the interest (7) and the square of two (=4) is divided by the other capital (P). Then perform the operation of sankramana in relation to the square-root of the remainder (obtained as the result of subtracting the quotient so obtained) from the square of the mixture (m) and the mixture." 2 Miscellaneous Problems on Interest. Besides the problems given above various other interesting prob- lems are found in the Hindu works on pdtiganita. Thus Brahmagupta gives the solution of the following problem: Hxampk. In what time will a given sum s, the interest on which for / months is r, become k times itself? The rule for the solution of the above is: 3 "The given sum* multiplied by its time and divided 1 GSS, p. 69(34). 2 GSS, p. 69(33). 3 BrSpSi, p. 181. *-The Sanskrit term used is pramdna (argument). COMMERCIAL PROBLEMS 223 by the interest, 1 being multiplied by the factor 2 less one, is the time (required)." The Ganita-sdra-samgraha (850) contains a large number of problems relating to interest. Of these may .be mentioned the following: (1) "In this (problem), the (given) capitals are (e x =) 40, (c 2 =) 30, (c 3 —) 20 and fo=) 50; and the months are (/,=) 5, (/ 2 =) 4, (/•,=) 3 and (/ 4 =) 6 (respectively). The sum of the interests is («?=) 34. (Assuming' the rate of interest to be the same in each case, find the amounts of interest in each case)." 8 Here, if the rate of interest per month for 1 be r, then ^1 ^*2 -*3 Cjy Cj Z Cj 3*3 where x x , x 2y x 3y are ' the interests earned on the capitals c ly c 2y c 3y in t ly t 2y t 3 months res pectively. Therefore, X± X 2 ' . x 3 x l -\- x 2 -\- x 3 -\- ... * rJi c Ji c Js. ^i+< , ^2+^s+ • m or x, — ^ r , etc. This formula is given by Mahavira for the solution 1 .The Sanskrit term used is phala (fruit). 2 The Sanskrit term used is guna (multiple). *GSS,p. 70(38). 224 -ARITHMETIC of the above problem. 1 (2) "(Sums represented by) 10, 6, 3 and 15 are the (various given) amounts of interest, and 5, 4, 3 and 6 are the (corresponding) months (for which the interests have accrued); the sum of the (corresponding) capital amounts is seen to be 140. (Assuming the rate of interest to be the same in each case, find out these capital amounts)." 2 (3) "Here (in this problem) the (given) capital amounts are 40, 30, 20 and 50; and 10, 6, 3 and 15 are the (corresponding) amounts of interest; 18 is the quantity representing the mixed sum of the respective periods of time. (Find out these periods separately, assuming the rate of interest to be the same in each case)." 3 (4) "The interest on 80 for 3 months is unknown; 7^ is the mixed sum of that (unknown quantity taken as the) capital lent out and pf the interest thereon for i- year. What is the capital here and what the interest?"* (5) "The mixed sums (capital+ interest) are 50, 58 and 66, and the months (during which interests have accrued) are 5, 7 and 9 (respectively). Find out what 1 GSS, p. 70(37). The formula clearly shows that Mahavira -knew the algebraic identity a c e a-\- c-\- e-\- b d /-•■•- b + d+f+./ 2 GSS, p. 70(40). The solution is given by Rule 39 on the same page. 3 GSS, p. 70(43). The solution is given by Rule 42 on the ,$ame page. * GSS, p. 71(46). This is similar to Aryabhata's problem gir«n before (p. 217). COMMERCIAL PROBLEMS 225 the interest is (in each case, the capital being the same)?" 1 (6) "The mixed sums of the capital and periods of interest are 21, 23, and 25; here (in this problem) the amounts of interest are 6, 10 and 14. What is the common capital?" 2 (7) "Borrowing at the rate of 6 per cent and then lending out at the rate of 9 per cent, one obtains in the way of differential gain 81 at the end of 3 months. What is the capital (utilised here)?" 3 (8) "The monthly interest on 60 is exactly 5 . The capital lent out is 35; the (amount of the) instalment (to be paid) is 15 in (every) 3 months. What is the time of discharge of that debt?" 4 (9) "The mixed sum (of the capital amounts lent out) at the rates of 2, 6 and 4 pet cent per mensem is 4400. Here the capital amounts are such as have equal amounts of interest accruing after 2 months. What (are the capital amounts lent, and what is the equal interest)?" 6 (10) "A certain person gives once in 12 days an instalment of 2|, the rate of interest being 3 per cent (per mensem). What is the capital amount of the debt discharged in 10 months?"* (11) "The total capital represented by 8520 is in- vested (in parts) at the (respective) rates of 3, 5 and 8 per cent (per month). Theri, in this investment, in 5 1 GSS, p. 71 (48). The solution requires the use of the identity a c a — c 2 GSS, p. 72(52). 3 GSS, p. 72(55). *GSS, p. 73(59)- 5 GSS, p. 73(61). 6 GSS, p. 73(65). 15 226 ARITHMETIC months the capital amounts lent out are, on being diminished by the respective amounts of interest, (found to be) equal in value. (What are the respective amounts invested thus?)" 1 (12) "The total capital represented by 13740 is invested (in parts) at the (respective) rates of 2, 5 and 9 per cent (per month), then, in this investment, in 4 months the capital amounts lent out are, on being combined with the (respective) amounts of interest, (found to be) equal in value. (What are the respective amounts thus invested?)" 2 (13) "A certain man borrows a certain (unknown) sum of money at an interest of 5 per cent per month. He pays the debt in instalments, due every -| of a month. The instalments begin with 7 and increase in arithmeti- cal progression, with 7 as the common-difference. 60 is the maximum amount of instalment. He gives in the discharge of his debt the sum of a series in arithmetical progression consisting of -^2- terms. After the payment of each instalment, interest is charged only on that part of the principal which remains to be paid. What is the total payment corresponding to the sum of the series, what is the interest (which he paid), what is the time of the discharge of the debt, (and what is the principal sum borrowed)?" 3 Barter and Exchange. The Hindu name for barter is bbdnda-prati-bhdnda ("commodity for commodity"). All the Hindu works on pdtiganita contain problems relating to the exchange of commodities. It is pointed out in these works that problems on barter are cases of compound proportion, and can be solved by the 1 CSS, p. 74(67). 2 CSS, p. 74(67). 3 GSS, pp. 741", (72-73^). The text of the problem is verv obscure. The translation given here is after Rangacarya. COMMERCIAL PROBLEMS 227 Rule of Five, etc. A typical problem on barter is the following: "If three hundred mangoes be had in this market for one dramma, and thirty ripe pomegranates for a pana y say quickly, friend, how many (pomegranates) should be had in exchange for ten mangoes?" 1 Other Types of Commercial Problems. Of various other types of commercial problems found in the Hindu works may be mentioned (i) problems on part- nership and proportionate division, and (2) problems relating to the calculation of the fineness of gold. 2 Most of these problems are essentially of an algebraic character, but they are included in pdtiganita (arithmetic). The formulae giving the solution of each type of examples precede the examples. These formulae are too numerous to be mentioned. The following examples, however, will illustrate the nature and the scope of such problems: (1) A horse was purchased by (nine) dealers in partnership, whose contributions were one, etc., upto nine; and was sold by them for five less than five hundred. Tell me what was each man's share of the sale- proceed. (2) Four colleges, containing an equal number of pupils, were invited to partake of a sacrificial feast. A fifth, a half, a third and a quarter (of the total number of pupils in the college) came from the respective colleges to the feast; and added to one, two, three and four, they were found to amount to eighty-seven; or, with those deducted, they were sixty-seven. Find the actual number of the pupils that came from each college. 1 L, p. 20. 2 Such problems are found in the Uldvati, the Ganita-sdra- samgraha, the TriJatikd, etc. 228 ARITHMETIC (3) Three (unequal) jars of liquid butter, of water ,and of honey, contained thirty-two, sixty and twenty-four pala respectively: the whole was mixed together and the jars filled again. Tell me the quantity of butter, of water and of honey in each jar. 1 (4) According to an agreement three merchants carried out the operations of buying and selling. The capital of the first consisted of six purdna, that of the second of eight purdna, but that of the third was unknown. The profit obtained by these men was 96 purdna. In fact the profit obtained by him (the third person) on his unknown capital happened to be 40 purdna. What was the amount thrown by him into the transaction and what was the profit of each of the other two merchants? 2 (5) There were four merchants. Each of them obtained from the others half of what he had with him (at the time of the respective transfers of money). Then they all bec"ame possessed of equal amounts of money. What was the measure of money each had to start with? 3 (6) A great man possessing powers of magical charm and medicine saw a cock fight going on, and spoke separately in confidential language to both the owners of the cocks. He said to one, "If your bird wins, then you give the stake-money to me. If, however, your bird loses then I shall give you two-thirds of that stake-money." He went to the owner of the other cock and promised to give three-fourths (of his stake-money on similar conditions). In each case the gain to him could be only 12 (gold-pieces). Tell me, O ornament 1 This and the two previous examples are given by Prthudaka- svami to illustrate Rule 16 of the ganitddhydya of the Brdhma-sphufa- siddhdnta. 2 GSS, p. 94(223-5). 3 GSS, p. 99(267$). COMMERCIAL PROBLEMS 229 on the head of mathematicians, the money each of the cock-owners had staked. 1 (7) The mixed price of 9 citrons and 7 fragrant wood-apples is 107; again the mixed price of 7 citrons and 9 fragrant wood-apples is 101. O arithmetician, tell me quickly the price of a citron and of a wood- apple, having distinctly separated those prices.' 2 (8) Pigeons are sold at the rate of 5 for 3 {pana). sarasa birds at the rate of 7 for 5 {pana), swans at the rate of 9 for 7 {pana) and peacocks at the rate of 3 for 9 {pana). A certain man was told to bring at these rates 100 birds for 100 {pana) for the amusement of the king's son, and was sent to do so. What (amount) does he give for each (of the various kinds of birds that he buys)? 3 (9) There are 1 part (of gold) of 1 varna, 1 part of 2 varna, 1 part of 3 varna, 2 parts of 4 varna, 4 parts of 5 varna, 7 parts of 14 varna, and 8 parts of 15 varna. Throw- ing these into the fire, make them all into one (mass), and then (say) what the varna of the mixed gold is. This mixed gold is distributed among the owners of the fore- going parts. What does each of them get?* (10) Three pieces of gold, of 3 each in weight, and of 2, 3 and 4 varna (respectively), are added to (an unknown weight of) gold of 13 varna. The resulting varna comes to be 10. Tell me, O friend, the measure (of the unknown weight) of gold. 5 1 GSS, pp. 99-100(270^-2^). 'GSS, p. 8 4 (i 4 o4-2£). 3 GSS, P . 85(152-3). 4 GSS, p. 88 (170- 1-4). 5 GSS, p. 89(181). Similar examples occur in the TriJalika (p. 26) and the Laldvati (p. 25). 230 ARITHMETIC 14. MISCELLANEOUS PROBLEMS Regula Falsi. The rule of false position is found in all the Hindu works. 1 Bhaskara II gives prominence to the method and calls it ista-karma ("rule of supposition"). He describes the method thus: "Any number, assumed at pleasure, is treated as specified in the particular question, being multiplied and divided, increased or diminished by fractions (of itself); then the given quantity, being multiplied by the assumed number and divided by that (which has been found) yields the number sought. This is called the process of supposition." 2 Sridhara takes the assumed number to be one. 3 Mahavira gives a large variety of problems to which he applies the rule.* Ganesa in his commentary on the h.ildvati remarks, "In this method, multiplication, divi- sion, and fractions only are employed." The following examples will illustrate the nature of the problems solved by the rule of supposition: (1) Out of a heap of pure lotus flowers, a third, a fifth, a sixth were offered respectively to the gods Siva, Visnu and Surya and a quarter was presented to Bhavaru. The remaining six were given to the venerable preceptor. Tell quickly the number of lotuses. 6 (2) The third part of a necklace of pearls, broken in 1 The method originated in India and went to Europe through Arabia. There is a mediaeval MS., published by Libri in his Histoire, I, 304 and possibly due to Rabbi beri Ezra in which the method is attributed to the Hindus. For further details and references, see Smith, History, II, p. 437, foot-note 1. - L, p. 10. 3 See the rule on stambboddesa, Tris, p. 13. 4 These problems occur in chapters iii and iv of the Ganita- sdra-sariigraha. s L,p. ir. Cf. GSS.p. 48(7). MISCELLANEOUS PROBLEMS 23 I an amorous struggle, fell to the ground; its fifth part rested on the couch; the sixth part was saved by the wench; and the tenth part was taken by her lover: six pearls remained strung. Say, of how many pearls was the necklace composed? 1 (3) One-twelfth part of a pillar, as multiplied by ^ part thereof, was to be found under water; ^ of the remainder, as multiplied by -fa thereof, was found buried in the mire below; and 20 hasta of the pillar were found in the air (above the water). O friend, give out the length of the pillar.' 2 (4) A number of parrots descended on a paddy f eld, beautiful with crops bent down through the weight of ripe corn. Being scared away by men, all of them suddenly flew off. One-half of them went to the east, one-sixth went to the south-east; the difference between those that went to the east and those that went to the south-east, diminished by half of itself and again diminished by half of this (resulting difference), went to the south. The difference between those that went to the south and those that went to the south-east diminished by two-fifths of itself went to the south-west; the difference between those that went to the south and those that went to , the south-west, went to the west; the difference between those that went to the south-west and those that went to the west, together with three-sevenths of itself went to the north-west; the difference between those that went to the north-west and those that went to the west together with seven-eighths of itself, went to the north; the sum of those that went to the north-west and those that went to the north, diminished by two-thirds of itself went to the north-east; and 280 parrots were found to 1 Tris, p. 14, ' cf. GSS, p. 49 (17-22) for a similar example. 2 GSS, p. 55(60). Cf. Tris, p. 13. 23 2 ARITHMETIC remain in the sky above. How many were the parrots in all? 1 The Method of Inversion. The method of inversion called vilomagati ("working backwards") is found to have been commonly used in India from very early times. Thus Aryabhata I says: "In the method of inversion multipliers become divisors and divisors become multipliers, addition be- comes subtraction and subtraction becomes addition." 2 Brahmagupta's description is more complete. He says: "Beginning from the end, make the multiplier divisior, the divisor multiplier; (make) addition subtrac- tion and subtraction addition; (make) square square- root, and square-root square; this gives the required quantity." 3 The following examples will illustrate the nature of problems solved by the above method: (i) What is that quantity which when divided by 7, (then) multiplied by 3, (then) squared, (then) increased by 5, (then) divided by f, (then) halved, and then reduced to its square-root happens to be the number (2) The residue of degrees of the sun less three, being divided by seven, and the square-root of the quotient extracted, and the root less eight multiplied by nine, and to the product one being added, the amount is 1 GSS, pp. 4 8f(i2-i6). 2 A, Ganitapdda, 28. s BrSpSi, p. 301. The method occurs also in GSS, p. 102 (286); MSi, p. 149; .L, p. 9; etc. 4 GSS, p. 102 (287). Examples of this type are very common in Hindu arithmetic. They were also very common in Europe. Smith in his History, II, quotes two such problems from an American arithmetic of the 16th century. MISCELLANEOUS PROBLEMS 2} J a hundred. When does this take place on a Wednesday? 1 Problems on Mixture. The Hindu works on pdtiganita contain a chapter relating to problems on mixture (misraka-vyavahdra}. Miscellaneous problems on interest, problems on allegation, and various other types of problems, in which quantities are to be separated from their mixture, form the subject matter of misraka-vyavahdra. A chapter "on mixture" (De' mescolo) is found in early Italian works on arithmetic, evidently under Hindu influence. 2 Some of the problems of this chapter are deter- minate and some are indeterminate. A few relating to interest and allegation have already been given. 3 The following are some others: (i) In the interior of a forest, 3 heaps of pomegra- nates were divided (equally) among 7 travellers, leaving 1 fruit as remainder; 7 (of such heaps) were divided among 9, leaving a remainder of 3 (fruits), again 5 (of such heaps) were divided among 8, leaving 2 fruits as remainder. O mathematician, what is the numerical value of a heap?* (2) On a certain man bringing mango fruits home, his elder son took one fruit first and then half of what remained. The younger son did similarly with what was left. He further took half of what was left there- after; and the other took the other half. Find the number of fruits brought by the father? 5 1 Colebrooke, cba, p. 333 (18). 2 Smith, History \ II, p. 588, note 4. 3 See commercial problems, pp. 2i6ff; also problems on pro- portionate division (Jtraksepa-karana): TriJ, p. 26; GSS, p. 75(79i); Ms >> PP- 154-15 5- * GSS, p. 82 (128 J). Such problems are given under the rule of vallikd-kutttkdra by Mahavlra. "GSS, p. 82 (i } i£). 234 ARITHMETIC (3) A certain lay follower of Jainism went to a Jina temple with four gate-ways, and having taken (with him) fragrant flowers offered them in worship with devotion (at each gate). The flowers in his hand were doubled, trebled, quadrupled and quintupled (respectively in order) as he arrived at the gates (one after another). The number of flowers offered by him was sixty 1 at each gate. How many flowers were originally taken by him? (4) The first man has 16 azure-blue gems, the second has 10 emeralds, and the third has 8 diamonds. Each among them gives to each of the others 2 gems of the kind owned by himself; and then all three men come to be possessed of equal wealth. What are the prices of those azure-blue gems, emeralds and dia- monds? 2 (j) In what time will four fountains, being let loose together, fill a cistern, which they would severally fill in a day, in half a day, in a quarter and in a fifth part of a day? 3 Problems involving Solution of Quadratic Equations. The solution of the quadratic equation has been known in India from the time of Aryabhata I (499). Problems on interest requiring the solution of the quadratic equation have already been mentioned. Mahavira and Bhaskara II give many other problems. Mahavira divides these problems into two classes: (/) those that involve square-roots (muld) and («) those 1 GSS, p. 79 (11 2^-1 1 3^). The printed text has panca ("five"). According to it the answer is 43/12 which appears absurd. There are some other problems in the printed edition which give such absurd results. All those are, we presume, due to the defects of -the mss. consulted by the editor. So here we have made the emendation 'sixty.' *GSS, p. 87(165-166). 3 BrSpSi, p. 177 (com.); L, p. 23. MISCELLANEOUS PROBLEMS 235 that involve the square (vargd) of the unknown. The first type gives a single positive answer, while the second type has two answers corresponding to the two roots of the quadratic. Bhaskara II deals with the first type of problems only in his pdtiganita, the Ljldvati. The second type of problems, involving the square of the unknown has been treated by him in his Bijaganita (algebra). The following examples will illustrate the nature and scope of such problems: Problems involving the square-root: (1) One-fourth of a herd of camels was seen in the forest; twice the square-root of that had gone to mountain slopes; and three times five camels were found \o remain on the bank of a river. What was the numeri- cal measure of that herd of camels? 1 (2) Five and one-fourth times the square-root (of a herd) of elephants are sporting on a mountain slope; five-ninths of the remainder sport on the top of the mountain; five times the square-root of the remainder sport in a forest of lotuses; and there are six elephants then (left) on the bank of a river. How many are the elephants? 2 (3) In a garden beautified by groves of various kinds of trees, in a place free from all living animals, many 1 GSS, p. 51 (34)- The problem belongs to the type of the mila-jdti, and leads to an equation of the form x — {bx-\-c-\/x-\- a )= - The method of solution is given in GSS, p. 50 (33). 2 GSS, p. 52 (46). The problem is of the sesa-mil/a variety. It gives the equation x-^-V^-i (*-W3 -5 V*— ¥V^ -K*- VV5) = 6 - Mahavira reduces it by putting ^ = x — "J-y/x — & (•* V"V^)- to ^ — j-y'p'=6. In the general case a similar equation is again obtained-, which is again reduced, and so on till the equation is reduced to the form, x — b\f^ = d, from which x can be easily obtained. Z}6 ARITHMETIC ascetics were seated. Of them the number equivalent to the square-root of the whole collection were practis- ing yoga at the foot of a tree. One-tenth of the remainder, the square-root (of what remained after this), \ (of what remained after this), then the square- root (of what remained after this), \ (of what remained after this), the square-root (of what remained after this), \ (of what remained after this), the square-root (of what remained after this), \ (of what remained after this), the square-root (of what remained after this), \ (of what remained after this), the square-root (of what remained after this) — these parts consisted of those who were learned in the teaching of literature, in religious law, in logic, and in politics, as also of those who were versed in controversy, prosody, astronomy, magic, rhetoric and grammar, as well as of those who pos- sessed an intelligent knowledge of the twelve varieties of the anga-sdstra; and at last 12 ascetics were seen (to remain without being included among those mentioned before). O excellent ascetic, of what numerical value was this collection of ascetics? 1 (4) A single bee (out of a swarm of bees) was seen in the sky; \ of the remainder (of the swarm), and \ of the remainder (left thereafter) and again \ of the remainder (left thereafter) and a number of bees equal to the square-root of the numerical value of the swarm, were seen in lotuses; and two bees were on a mango tree. How many were there? 2 (5) Four times the square-root of half the number of a collection of boars went to a forest wherein tigers 1 GSS, p. 52 (42-45). The problem is of the same variety as the above one. The substitution will have to be made 6 times to reduce the resulting equation. 2 GSS, p. 5 3 (48). This problem is of the dviragra-tesa-mula variety. MISCELLANEOUS PROBLEMS 237 were at play; 8 times the square-root of ^ of the remain- der went to a mountain; and 9 times the square-root of 1 of the (next) remainder went to the bank of a river; and boars equivalent in (numerical) measure to 5 6 were seen to remain in the forest. Give the numerical measure of all those boars. 1 (6) The sum of two (quantities, which are respec- tively equivalent to the) square-root (of the numerical value) of a collection of swans and (the square-root of the same collection) as combined with 68, amounts to 34? How many swans there are in that collection? 2 (7) Partha (Arjuna), irritated in fight, shot a quiver of arrows to slay Kama. With half his arrows, he parried those of his antagonist, with four times the square-root of the quiver-full, he killed .his horses; with three he demolished the umbrella, standard and bow; and with one he cut off the head of his foe. How many were the arrows, which Arjuna let fly? 3 . (8) The square-root of half the number of a swarm of bees is gone to a shrub of jasmin; and so are eight- ninths of the whole swarm; a female is buzzing to one remaining male that is humming within a lotus, in which he is confined, having been allured to it by its 1 GSS, p. 54X56). This problem is of the amJa-mula variety, wherein fractional parts of square-roots are involved. The prob- lems give equations of the form x—a 1 y/J^—,a 2 y/h 2 (x—a 1 y/r^) — <* 3 \A>[(*— <*i\/£^0 — a^y/b^x— a x -\Jb x x) '—...=k. By repeated substitutions Mahavira reduces the equation to the form x — Ay/Bx — c = o. 2 GSS, p. 56 (68). This problem is of the mitla-miira variety, wherein the sum of square-roots is involved. It gives an equa- tion of the form y/x -\-\Jx±d = m. 3 L, p. 16. 238 ARITHMETIC fragrance at night. Say, lovely woman, what is the number of bees. 1 Problems involving the square of the unknown: (9) One-twelfth part of a pillar, as multiplied by -^ part thereof, was found under water; ^ of the remainder, as multiplied by ^ thereof, was found buried in the mire, and 20 hasta of the pillar were found in the air. O friend, give the measure of the length of the pillar. 2 (10) A number of elephants (equivalent to) -fa of the herd minus z, as multiplied by the same (^ of the herd minus 2), is found playing in a forest of sallaki trees. The remaining elephants of the herd equal in number to the square of 6 are moving on a mountain. How many are the elephants? 3 15. THE MATHEMATICS OF ZERO It has been shown that the 2ero was invented in India about the beginning of the Christian era to help the writing of numbers in the decimal scale. The Hindu mind did not rest satisfied till it evolved the complete arithmetic of zero. The Hindus included zero among the numbers (sankhya), and it was used 1 L, p. 16. 2 GSS, p. j j (60). The problem gives the equation v 12.30' 20.16 v 12.30 Also solved by regula falsi. Mahavira puts (x — x z )— ^, and then solves the quadratic 3 2 ^ 320 A " The roots of this are then used to get the values of x. ■ 3 Gxy, p. 55 (63). THE MATHEMATICS OF ZERO 239 in their arithmetic at the time when the original of the Bakhshali Manuscript was written, about the third century A.D. The operation of addition and subtrac- tion of zero are incidentally mentioned in the Panca- siddhdntikd of Varahamihira (505). The complete decimal arithmetic is found in the commentary of Bhaskara I {c. 525) on the A^ryabhattya. The results of operations by zero are found stated in the work of Brahmagupta (628) and in all later mathematical treatises. The treatment of zero in the arithmetic of the Hindus is different from that found in their algebra. In order, therefore, to bring out this difference clearly, we give separately the results found in pdtiganita (arithmetic) cind in bijaganita (algebra). Zero in Arithmetic. The Hindus in their arith- metic define zero as the result of the operation a — a = o This definition is found in Brahmagupta's work 1 and is repeated in all later works. It is directly used in the operation of subtraction. In carrying out arithmetical operations, the results of the operations of addition, subtraction and multiplication of zero and by zero are required. The Hindus did not recognise the opera- tion of division by zero as valid in arithmetic; but the division of zero by a number was recognised as valid.. Narayana in his pdtiganita (arithmetic) has clearly stated this distinction: "Here in pdtiganita, division by zero is not recog- nised, and therefore, it is not mentioned here. As it is of use in bijaganita (algebra), so I have mentioned division by zero in my Bijaganita." 2 1 BrSpSi, p. 309. Cf. B. Datta, BCMS, XVIII, pp. 165-176 for some other details regarding operations with 2ero. 2 GK, remark subjoined to i 30. 242 ARITHMETIC cipher. A number divided by zero is kha-hara (that number with zero as denominator). The product of (a number and) zero is zero, but it must be retained as a multiple of zero (kha-guna), if any further operations impend. Zero having become a multiplier (of a number), should zero afterwards become a divisor, the number must be understood to be unchanged. So likewise any number, to which zero is added, or from which it is subtracted (is unaltered)." 1 In the Bijaganita, the same results are given with the addition that if a quantity is subtracted from zero, its sign is reversed, while in the case of addition the sign remains the same. Zero as an Infinitesimal. It will be observed that Brahmagupta directs that the results of the opera- tions x -r- o and o -=- x should be written as f- and -Ir- respectively. It is not possible to tell exactly what he actually meant by these forms. It seems that he did not specify the actual value of these forms, because the value of the variable x is not known. Moreover, the zero seems to have been considered by him as an infinitesimal quantity which ultimately reduces to nought. If this surmise be correct, Brahmagupta is quite justified in stating the results as he has done. The idea of zero as an infinitesimal is more in evi- dence in the works ofBhaskara II. He says: "The product of (a number and) zero is zero, but the number must be retained as a multiple of zero {kha-guna), if any further operations impend." He further remarks that this operation is of great use in astronomical calculations. It will be shown in the section on Calculus, that Bhaskara II has actually used quantities which ultimately tend to zero, and has successfully evaluated the differen- tial coefficients of certain functions. He has, moreover, "L.p.8. THE MATHEMATICS OF ZERO 245 used the infinitesimal increment f'(x)§x of the function f(x), due to i change hx in x. The commentator Krsna proves the result oxa = o = aXoas follows: "The more the multiplicand is diminished, the smaller is the product; and, if it be reduced- in the ut- most degree, the product is so likewise: now the utmost diminution of a quantity is the same with the reduction of it to nothing; therefore, if the multiplicand be nought, the product is cipher. In like manner, as the multiplier decreases, so does the product; and, if the multiplier be nought, the product is so too." In the above zero is conceived of as the limit of a diminishing quantity. Infinity. The quotient of division by zero of a finite quantity has been called by Bhaskara II as kha- hara, which is synonymous with kha-cheda (the quantity with zero as denominator) of Brahmagupta. Regard- ing the value of the kha-hara, Bhaskara II remarks: "In this quantity consisting of that which has cipher for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God, at the period of the destruction or creation of worlds, though numerous orders of beings are absorbed or put forth." 1 From the above it is evident that Bhaskara II knew that — = 00 and 00 -f- k = 00 . 1 BBi, pp. 5-6. G. Thibaut (A.siranomie, Astrology und Ma- thematics,, Strasbourg, 1899, p. 72) thought that this passage was an interpolation. There appears no justification for considering this as an interpolation, as the passage occurs in the oldest known commentary and in all copies of the work so far found. Cf. Datta, I.e., p. 174. 244 ARITHMETIC Ganesa remarks that —is "an indefinite and unli- o mited or infinite quantity: since it cannot be determined how great it is. It is unaltered by the addition or subtraction of finite quantities: since in the preliminary operation of reducing both fractional expressions to a common denominator, preparatory to taking their sum or difference, both numerator and denominator of the finite quantity vanish." Krsna remarks: "As much as the divisor is diminished, so much is the quotient increased. If the divisor is reduced to the utmost, the quotient" is to the utmost increased. But, if it can be specified, that the amount of the quotient is so much, it has not been raised to the utmost: for a quantity greater than that can be assigned. The quo- tient, therefore, is indefinitely great, and is rightly termed infinite." Regarding the proof of — ± k ~ - Krsna makes the same remarks as Ganesa. He, however, goes a step further when he says that This is illustrated by him through the instance of the shadow of a gnomon, which at sun-rise and sun-set is infinite; and is equally so whatever height be given to the gnomon, and whatever number be taken for the radius. "... Thus, if the radius be 120; and the gnomon be 1, 2, 3 or 4; the expression deduced from the proportion, as sine of sun's altitude is to sine of zenith distance, so is gnomon to shadow, becomes -Hr 2 , ^-^, -^rp or - ± ^ a • Or, if the gnomon be, as it is usually framed, 12 fingers, and radius be taken THE MATHEMATICS OF ZERO 245 as 3438, 120, 100 or 90, the expression will be ^xa^s i^to > jj^oj) or li ^_ 0> which are aU aUke infinite _„ 1 ' Indeterminate Forms. Brahmagupta has made the incorrect statement that - = o o Bhaskara II has sought to correct this mistake of Brahmagupta. According to him L,im — — = a. e^>o s His language, however, in stating this result is defective, for he calls the infinitesimal s zero, not being in possession of a suitable technical term. That, in the above case, he actually meant by zero a small quantity tending to the limiting value zero, is abundantly clear from the use he makes of the result in his Astronomy. Taylor 2 and Bapu Deva Sastri 3 are also of this opinion. Bhaskara has given three illustrative examples. They are: (a-XcH — ) (/') Evaluate = 6*. o J From this he derives the result x= 14, which is correct if we consider o=e, a small quantity tending to zero. His other examples are: (") {£+*-9)*+(!+*-9)} ° = 9o giving x = 9; and 1 All the above passages are taken from the respective com- mentaries. They have been noted by Colebrooke, I.e. 2 Ulawatt, Bombay, 18 16, p. 29. 3 His Bi/a-ganita (in Hindi), Pt. I, Benares, 1875, p. 179 tt sq. 246 ARITHMETIC giving x=2* Bhaskara II's result a -Xo o is, however, not quite correct, as the form is truly 'indeterminate and may not always have the value a. His attempt, however, at such an early date to assign a meaning to the form — , and his partial solution of the problem are very creditable, seeing that in Europe mathematicians made similar mistakes upto the middle of the nineteenth century A. D. 2 - 1 The answers of this and the previous example are incorrect because o 2 has been taken to be equal to o. 2 Martin Ohm (1828) says: "If a is not zero, but b is zero, then the quotient ajb has no meaning" for the quotient "multi- plied by zero gives only zero and not a, as long as a is not zero." I^ehrbuch der niedern Analysis, Vol. I, Berlin, 1828, pp. no, 112. BIBLIOGRAPHY OF SANSKRIT MATHEMATICAL WORKS i. Apastamba SuJba Sutra by Apastamba (c. 400 B.C.). Edited with the commentary of Kapardisvami, Karavindasvami and of Sundararaja by D. Srinivasachar and V.S. Naras- imhachar, University of Mysore Sanskrit Series, 193 1; by A. Biirk, with German translation and note and comments in Zeitschrift der deutschen morgenldndischen Gesselschaft, LV, 1902, pp. 543-59 1 ; LVI > r 9°3> PP- 327-39 1 - 2. Arsa jyotisa. Edited by Sudhakara Dvivedi with his own commentary, Benares, 1906. 3. Aryabhatiya by Aryabhata I (499). Edited with the com- mentary, entitled Bhatadipika, of Paramesvara (1430) by H. Kern (Leiden, 1875); by Udai Narayan Singh, with explanatory notes in Hindi (Muzaflarpur, 1906); with the commentary, entitled Mahdbhdsya, of Nilakantha (1500) on the last three chapters by K. Sambasiva Sastri (Trivandrum, Part I, 1930; Part II, 193 1; Part III, in press). Transla- tions: by P. C. Sengupta ("The Aryabhatiyam," in the Journal of the Department of Letters in the University of Calcutta, XVI, 1927); by W. E. Clark {The Aryabhattya of Aryabhata, Chicago, 1930). The second chapter of the book has been translated also by L. Rodet ("Le5ons de calcul d'Aryabhata," in journal Asiatique, XIII (7), 1878; reprint, Paris, 1879) an d bj G- R. Kaye ("Notes on Indian Mathematics, No. 2 — Aryabhata," in Journal of the Asiatic Society of Bengal, IV, 1908). Other commentaries in MSS: (i) by Bhaskara I (522); (ii) BbataprakatikA by Suryadeva Yajva (12th century). 4. Atharvan Jyotisa. Edited by Bhagvad Datta, Lahore, 1 924. 5. Bakhshdli Manuscript — A Study in Mediaval Mathematics, Parts I, II and III edited by G. R. Kaye, Calcutta, 1927, 1933. 6. Baudhdyana Sulba Sutra by Baudhayana (c. 800 B.C.). Edited by G. Thibaut, with English "translation, cridcal notes and extracts from the commentary of Dvarakanatha Yajva, in 248 BIBLIOGRAPHY the Pandit (Old Series, IX and X, 1874-5; New Series, I, 1877). The text appears in the edition of the Baudhdyana Srauta Sutra (as its 30th chapter) by W. Caland (in 3 volumes, Calcutta, 1904, 1907, 191 3). 7." Bijaganita of Bhdskara II (11 50). Edited by Sudhakara Dvivedi (Benares) and revised by Muralidhara Jha (Benares, 1927); with the commentary, entitled Navdnkura, of Krsna (c. 1600) by D. Apte (Anandasrama Sanskrit Series, Poona, 1930). English translation by H. T. Colebrooke (Algebra with Arithmetic and Mensuration from the Sanscrit of Brahmegupta and Bhascara, London, 18 17). Other commentaries in MSS: (/') Bijaprabodha by Ramakrsna (c. 1648); («) by Suryadasa (born 1508). References to this treatise in our text are by pages of Muralidhara Jha's revised edition. 8. Bijaganita of Jnanaraja (1503). MS. 9. Bijaganita of Narayana (1350). MS. (incomplete). 10. Brdhma-sphuta-siddhdnta by Brahmagupta (628). Edited with explanatory notes by Sudhakara Dvivedi, Benares, 1902. Chapters xii and xiii of the work dealing respectively with arithmetic and algebra have been translated into English by H. T. Colebrooke (Algebra with Arithmetic and Mensura- tion etc.). Commentary by Prthudakasvami (860).. MS. (in- complete). 11. Brhajjdtaka of Varahamihira (505). Edited with the commen- tary of Bhattotpala (966) by Rasikmohan Chattopadhyaya (Calcutta, 1300, b.s.); by Sitaram Jha (Benares, 1^23). 12. Brhat-samhitd by Varahamihira (505). Edited by H. Kern (Calcutta, 1865); by Sudhakara Dvivedi, with the com- mentary of Bhattotpala (Vizianagram Sanskrit Series, 2 Vols., Benares, 1895). English translation by H. Kern (See his collected works). 13. Dhydnagrahopades'a of Brahmagupta (628). Edited by Sudha- kara Dvivedi and published as an appendix to his edition of the Brdhma-sphuta-siddhdnta. 14. Ganita-kaumudi of Narayana (1356). MS. 15. Ganita-manjari by Ganesa, son of Dhundhiraja (1558). MS. 16. Ganita-tilaka by Sripati (1039). Edited with the commentary of Sirhhatilaka Suri (c. 1275) by H. R. Kapadia (Gaikwad Sanskrit Series, Baroda, 1935). BIBLIOGRAPHY 249 17. Ganita-sdra-samgraha by Mahavira (850). Edited with English translation and notes by M. Rangacarya, Madras, 191 2. 18. Graha-ldghava of Ganesa Daivajfia (c. 1545)- Edited with the commentaries of Mallari, Visvanatha and his own by Sudha- kara Dvivedi, reprinted, Bombay, 1925. 19. Karana-kutuhala by Bhaskara II (1150)- Edited with the com- mentary of Sumatiharsa by Madhava Sastri, Bombay, 1901. 20. Karana-paddhati by Pathumana Somayaji (1733). MS. 21. Kdtydyana Sulba Sutra by Katyayana (c. 400 B.C.). Edited with explanatory notes by Vidyadhar Sharma, Benares, 1928. 22. Khanda-khddyaka by Brahmagupta (628). Edited with the commentary of Amaraja (c. 1250) by Babua Misra (Calcutta, 1925). English translation with notes and comments by Prabodh Chandra Sengupta (Calcutta, 1934). Other com- mentaries in MSS: (/) by Prthudakasvami (incomplete); (it) by Varuna; (//'/) by Bhattotpala. 23. Laghu-Bbdskanya by Bhaskara I (522). Commentary by Sarikara- narayana. MSS. 24. Laghu-mdnasa by Manjul'a (932). Commentaries by (i) Prthu- dakasvami (964); (ii) ParameSvara (143°)- MSS - 25. Ulavati by Bhaskara II (n 50). Edited with notes by Sudha- kara Dvivedi, Benares, 1910. Translations: (;) by H. T. Colebrooke (Algebra with Arithemtic and Mensuration etc.); (ii) by J. Taylor (Li/awati, Bombay, 1816). Colebrooke's translation has been re-edited with critical notes by Haran Chandra Banerji (Lildvati, 2nd ed., Calcutta). Commentaries in MSS: (/) Buddhivildsini by Ganesa (1 545); (ii) Ganitdmrtasdgari by Garigadhara (143 2 ); (."') Ga "~ itdmrtalahari by Ramakrsna (1339); (iv) Manoranjana by Raniakrsnadeva; (v) Ganitdmrtakilpi^d by Suryadasa (154O; (vi) Cintdmani by Laksmidasa (1500); (vii) NisrstadM by Munisvara (1608). References to Lf/dvati in our text are by the pages of Dvivedi's edition. 26. Mdnava Sulba Sutra by Manu, MS. English translation by N. K. Mazumdar (in Journal of the Department of letters in the University of Calcutta, VIII, 1922). 27. Mahd-Bhdskariya by Bhaskara I (522). Commentary by (i) Suryadeva (12th century); (ii) Paramesvara (1430). MSS. 18. Mahd-siddhdnta by Aryabhata II (950). Edited with explana- tory notes by Sudhakara Dvivedi, Benares, 1910. 25° BIBLIOGRAPHY 29. Panca-siddhdntikd by Varahamihira (505). Edited with a com- mentary in Sanskrit, translation into English and critical notes by G. Thibaut and Sudhakara Dvivedi, Benares, 1889. 30. Pdti-sdra by Munisvara (born 1603). MS. 31. Sadratnamdla by Saiikaravarman. MS. 32. Siddhdnta-sekhara by Sripati (1039). Edited by Babua Misra, Calcutta, Vol. I (193Z), containing chapters i-xii of the text, with the commentary of Makkibhatta (1377) on chapters i-iv and that of the editor on the rest. 33. Siddhdnta-siromani by Bhaskara II (11 50). Edited with the author's own gloss (Vdsandbhdsyd) by Bapu Deva Sastri, Benares); by Muralidhara Jha with the commentaries, Vdsandvdrttka of Nrsimha (1621) and Maria of Munisvara (1635), Vol. I (containing chap, i of the Ganitddhydyd) (Benares, 1917); by Girija Prasad Dvivedi, with original commentaries In Sanskrit and Hindi, Vols. I and II (Lucknow, 1911, 1926). English translation of the text only by Bapu Deva Sastri and Wilkinson (Calcutta, 1861). 34. Siddhdnta-tattva-viveka by Kamalakara.(i65 8). Edited with the Sesa-vdsand of the author, by Sudhakara Dvivedi, Benares, 1885. 3 5 . Sisya-dhi-vrddhida by Lalla (5 98). Edited by Sudhakara Dvivedi, Benares, 1886. Commentary by Mallikarjuna Suri (1179). MS. 36. Surya-siddhdnta. Edited by F. E. Hall and Bapu Deva Sastri with the commentary of Ratiganatha (Calcutta, 1859); kH Sudhakara Dvivedi with an original commentary in Sans- krit (Calcutta, 1909-11). Translation into English, with critical notes by E. Burgess and W. D. Whitney; into Bengali with critical notes by Vijnanananda Svami. 37. Tantra-samgraha by Nilakantha (1500). Commentary by an un- known writer. MS. 38. Trifatikd by Sridhara (750). Edited by Sudhakara Dvivedi, Benares, 1899. The principal rules in this text leaving the illustrative examples and their solutions, have been trans- lated into English by N. Ramanujachariar and G. R. Kaye and published with notes and comments ("Trisatika of Sridharacarya," Bib/. Math., XIII (3), 1912-13, pp. 2036".). Dvivedi's text is apparently incomplete. The manuscript in our collection, though not perfect, contains a few more rules and examples. INDEX Ababa, 12 Abbuda, 1 2 Abenragel, 99 Abhyisa, 130, 134 Abja, 13 Abu Sahl Ibn Tamim, 98 Addition, 150; direct process, 131; inverse process, 131; terminology, 130; the opera- tion, 130 Aghana, 175, 177, 178, 179, 180 Ahaha, 12 Akkhobhini, 12 Aksarapalli, 33, 34, 72, 74 Aksiti, 10 Al-Amuni Saraf-Eddin, 99 Al-Battani, 83 Al-Biruni, 39, 55, 98, 99, 100, 128, 210 Alexander de Villa Die, 95, 103 Al-Fazarl, 89 Al-Hassar, 143 Al-hindi, 143 Alibin Abil-Regal Abul-Hasan, 99 Alkalasadi, 99, 143 Al-Kharki, 50 Al-Khowarizmi, 90, 143, 153, 154 Al-Masudi, 97, 100 %. Al Nadim, 98 AINasavi, 90, 143, 153, 154, 174 Alphabetic Notations, 63 ; aksarapalli, 72; explanation, 67; katapayadi system, 69; other letter systems, 75; system of Aryabhata I, 64 Al-Qass, 90 Al-Sijzi, 90 Arh^a, 185, 186, 188 Ananta, 10 Aiikapalli, 33 Anta, 9 Antya, 10, 13, 163 Anuyogadvara-sutra, 11, 12, 83, 169 Apavartana, 189 Arbuda, - 9, 13 Archimedes, 11 Arithmetic, 125; exposition and teaching, 126; fundamental operations, 128; general sur- vey, 123; sources, 125; ter- minology and scope, 123 Arjuna, 10 Arkhand, 97 Arthasastra of Kautilya, 6, 7, 21, 187, 218 Aryabhata I, 12, 69, 71, 87, 125, 128, 134, 135, 137, 150, 155, 156, 162, 163, 173, 197, 198, 204, 211, 219, 232, 234 Aryabhata II, 71, 72, 130, 132, 136, 152, 172, 177, 184, 204, 205, 212, 240 Aryabhatiya, 65, 67, 69, 71, 80, 82, 86, 125, 126, 130, 134, 150, 170, 171, 175, 197, 204, 211, 217, 219, 239 Asankhyeya, 12 AsVaghosa, 2 2 5 2 INDEX Atata, 12 Athanasius Gamenale, 96 Atharvaveda, 18, 57 Avarga, 65, 66, 67, 69 Avesta, 101 Avicenna, 1 84 Ayuta, 9, 10, 12, 13 Bahula, 10 Bakhshali Manuscript, 47, 61, 65, 77» 8l > 86 > 134. i37> 151, 186, 192, 204, 239 Bapudeva Sastri, 245 Barnet, L.D., 6, 72 Barter and exchange, 226 Barth, 65 Bayley, 29, 30, 124 Beha Eddin, 100, 149, 184 BMga, 185, 186, 188, 190 Bhaga-bhiga, 191 Bhagahara, 124, 150, 198 Bhagajati, 196 Bhagamatr, 192 Bhaganubandha, 190, 192, 193, '195 Bhagapavaha, 191, 195 Bhagavati-sikra, 4, 7 Bhajaka, 150 Bhdjana, 150 Bhajya, 150, 177, 178 Bhanda-pratibhanda, 124, 226 Bhandarkar, 17, 19 Bhasa, 2 Bhaskara I, 66, 67, 80, 82, 86, 87, 125, 13°. J 34, i7°> *97> 204, 211, 239 Bhaskara II, 10, 13, 125, 128, 129, 131, 132, 136, 137, 146, i49» r 52, 157, !59. l6l > 162, 164, 167, 172, 184, 188, 195, 205, 208, 209,212, 230, 234, 23 5, 241, 242, 243, 245, 246 Bhattotpala, 5 5 Bhinna, 188 Bihari, 81 Bijadatta, 184 Bijaganita, 8, 235, 239, 241, 242, 245 Bindu, 12, 81 Bodhisattva, 10 Boethius, geometry, 93; question, 92 Bower manuscript, 74 Brahmagupta, 8, 87, 89, 124, 125, 128, 134, 135, 137, 147, i49> I 5°> x 56, i59> i6o > l6 3> 169, 170, 176, 188, 189, 196, 198, 199, 205, 211, 212, 217, 220, 222, 232, 239, 241, 242, 243. 245 Brahmana, Aitereya, 58; Panca- virhsa, 10; Satapatha, 57, 186; Taittiriya, 57 Brahma-sphuta-siddhanta, 8, 59, 89, 156, 228, 241 Brahmi numerals, 25; early occurrence and forms, 25; period of invention, 37; relation with letter forms, 33; resume, 37; theories about .their origin, 28 Brhat-ksetra-samasa, 79 Brhat-sarhhita, 55,59 Brnda, 163 Brockkhaus, 65 Buddha, 2, 36, 187 Buhler, 16, 17, 19, 21, 23, 24, 30, 33. 35. 37. 45> 47, 49> 6o > 74 Burnell, 30, 75 Caire, 97 Cajori, 88 Capella, 92 Caraka, 2 INDEX 253 Carta deVaux, 97, 100, 101, 102 Cataneo, 175 Catur, 1 3 Chandah-sutra, 58, 75, 76, 77, 86 Chatterjee, CD., 53 Chaya, 124 Checks on operations, 180 Chedana, 150 Chid, 212 Chuquet, 175 Citi, 124 Clark, W.E., 64, 65, 66, 67, 170, i7!> 175, 197 Coedis, G., 43 Colebrooke, 134, 13^5, 145, 147, 156, 164, 177, 211, 233, 245 Commercial Problems, 218; barter and exchange, 226; interest in ancient India, 218; interest in Hindu ganita, 218; other problems, 220; other types of, 227; pro- blem involving a quadratic equation, 219 Cowell, E.B., and Neil, R.A., 7 Cube, 162; minor methods of, 166; terminology, 162; the operation, 163" Cube-root, 175; terminology, 175; the operation, 175 Cullaniddesa, 4 Cunningham, 28, 29 D Dantidurga, 40 Darius, 23 Darius Hystaspes, 101 Dasa, 9, 12, 13 Dasagitika, 64, 65, 66, 71 Dasagunah sarhjM, 13 Dasagunottara sarhjna, 10 Dasa-koti, 1 3 Dasa-laksa, 13 Dasa-sahasra, 13 Datta, B., 7 , 61, 65, 81, 84, 85, 123. J 55, 17°. l8 4> 185, 218, 2 43 Deccke, W., 16 Decimal place-value system, 38; epigraphic instances, 40; forms, 39; important features, 38; inventor unknown, 49; place of invention . of the new system, 48; their sup- posed unreliability, 44; time of invention, 49 Delambre, 184 Devendravarmana, 40 Dharma-sutra, 17 Dhuli, 8; karma, 8, 123, 129 Digges, 210 Digha-Nikaya, 7 Division, 150; terminology, 150; the method of long, 151; the operation, 150 Divyavadana, 7 Djahiz, 97 Dramma, 227 Dvivedi, S., 59, 129, 131, 156, 168, 178, 183, 241 E Ekadasa-rasika, 124 Ekanna-catvarirhsat, 14 Ekanna-vimsati, 14 Ekikarana, 130 Ekona, 14 El-jowharee, 100 Elliot, 100 End, 1 01, see hend Enestrom, G., 149 Euting, J., 23 Fihrist, 98 Firdausi, 100 2H INDEX Firouzibadi, ioo, 101 Fleet, 52, 60, 65, 67, 124 Fliigel, G., 98 Fractions, 185; addition and sub- traction, 196; cube and cube- root, 199; division, 197; early use, 185; lowest common multiple, 195; multiplication, 196; reduction in combina- tions, 190; reduction to common denominator, 189; reduction to lowest terms, 189; square and square-root, 199; terminology, 188; the eight operations, 195; unit fractions, 199; writing of, 188 Frisius, Gemma, 175 Ganaka, 127 Ganesa, 136, 145, 146, 211, 230, 244 Gahgadhara, 131, 133, 147 Ganguli, 65 . Ganita, 4, y, 6, 7, 8, 83, 128, 130; avyakta, 123; vyakta, 123 Ganita-kaumudi, 125 Ganita-mafljari, 136, 144, 14; Ganitanuyoga, 4 Ganita-s&ra-samgraha, 80, 125, 219, 223, 227, 230, 240 Ganrta-tilaka, 125 Gerbert, 93, 94 Ghana, 8, 124, 162, 177, J 7 8 » J 79; P ada » J 75 Ginsburg, J., 95 Gomutrika, 135, 147, 148 Gotama-sutra, 218 Gunaka; 135 Gunakara, 135, 198 Gunana, 124, 134; ista, 136, 149; phala, 135; tastha, 146; tiryaka, 145; Gunya, 135 H Halifax, John of, 95 HalJ, F., 81 Halle, 104 Halsted, G.B., 38, 39 Hanana, 134, 139 Handasa, 102 Handasi, 10 1, 102 " Hara, 150 Harana, 150 Harya, 150 Heath, 50 Hemcandra, 12 Hend, 101 Hetuhila, 11 Hetvindriya, 11 Hind, 100, 101 Hindasa, 101, 102 Hindasi, 10 1 Hindi, 100, 101, 102 Hindisah, 98, 101 Hindu Numerals in Arabia, 88; Arabic reference, 96; definite evidence, 93; European re- ferences, 102; in Europe, 92; miscellaneous references, 95; Syrian reference, 95; the terms hindasa, etc., 101 Hisab-al-ghobar, 98, 124 Hunter, G.R., 19, 29 Ibn Albanna, 99 Ibn Hawkal, 100 Ibn Seedeh, 100 Ibn Tarik, 89 Ibn Wahshiya, 96 • Iccha, 198, 204-208; paksa, 211, 213 Idani, 186 Ilm-hisab-altakhta, 123 INDEX *55 Indraji, Bhagvanlal, 6, 26, 35-37 Infinity, 243 Inscription, Ahar Stone, 42, 5 2; at Po Nagar, 44; Belhari, 40; Buchkala — of Nagbhata, 41 ; Deogarh Jaina — of Bhojadeva, 42; Dholpur, 60; fromDhauli, 33; from Girnar, 33; from Kanheri, 41, 42; Ghatiyala — of Kakkuka, 41; Gurjara — 45; Gwalior — of Allah, 42; Gwalior — of the reign of Bhojadeva, 42, 82; Hathigumpha, 6; Hindu, 60; Junar, 47; Kanheri, 40; Kharosthl, 21; Ksatrapa, 34; Nanaghat, 25, 26; of Asoka, 16, 20, 21, 23, 28, 34; of Bauka, 41; of the Kusinas, 22, 34; of Rudradaman, 47; of the Parthians, 21; of the Sakas, 21; of Samanta Devadatta, 41; of Sambor, 44; of Srivijaya, 44; of Yasovar- mana, 43; Pehava, 42; Sanskrit and old Canarese, 43; Siyadoni stone, 43; stone, 45; word numerals in, 59 Isidorus of Seville, 102 Ista, 135 lstakarma, 230 Litakri, 100 Itarhi, 186 Jacobi, H., 7 Jadhr, 1 70 Jaivardhana II, 46, 82 Jaladhi, 1 3 Jarez de le Frontera, 94 Jati, 188 Jinabhadra Gani, 61, 79 John, 96 Jones, Sir W., 16 Jyotisa, 8 Kaccayana's Pali Grammar, 11 Kaiyyata, 63 Kakirti, 206 Kali,' 185, 188 Kalasavarnana, 8 Kalhana, 50 Kalidasa, 2 Kalpasutra, 6, 7 Kanda, 18 Karikara, 10 Kapadia, H.R., 80, 151 Kapatasandhi, 134, 135, 136, i37» i43» 144, 145 Karahu, 11 Karana, 5 o Karana-kutuhala, 184 Karani, 1 70 Karpinski, 97; see Smith Karsa, 206 Katapayadi system, 69; first variant, 70; second variant, 71; third variant, 72; fourth variant, 72 Kathana, 12 Katyayana, 58, 63 Kautilya, 2; Arthasastra of, see Arthasastra Kayastha, 5 o Kaye, G. R., 44, 45, 46, 47, 48, 59, 60, 65, 85^ 101,102, 156, 168, 170, 171, 175, 190 Keith, see Macdonell Kerala system, 72 Kern, 65 Kha-cheda, 243; -hara, 242, 243, 244; -guna, 242 Khalif al-Mansur, 89 Khalif Walid, edict of, 89 Khanda, 135 256 INDEX Khanda-khadyaka, 89 Kharavela, 6 Kharosthi numerals, 21; early occurrence, 21; forms and origin, 22; lipi, 21 Kharva, 13; maha, 13 Khata, 124 Kopp, 16 Koti, 10, 11, 12, 13 Koti- koti, 11, 12 Kotippakoti, 12 Krakacika, 124 Krama, 131 Krsna, 243, 244 Krti, 155, 169 Ksetraganita, 7, 8 Ksipra, 186 Ksiti, 13; maha, 13 Ksobha, 13; maha, 13 Ksobhya, 10 Ksoni, 13; maha,- 13 Kumuda, 12 Kustha, 185 Kuttaka, 8 Labdha, 150 Laghu-Bhaskariya, 59 Lahiri, 65 Lakkha, 11 Laksa, 11, 13 Lalitavistara, 10, 37, 187 Lalla, 61, 87, 125 Langdon, 25, 29 La Roche, 175 Latyayana, 58 Lekhapancasika, 5 o Lekhaprakasa, 50 Leonardo Fibonacci of Pisa, 94, 103 Lespius, 16 Lilavati, 125, 131, 132, 133, 134, i3 6 » 137. J44, M5> 146, 147, 2ii, 213, 227, 229, 230, 235, 241, 245; Ganesa's commentary, 144, 145, 146 Lowest Common Multiple, 195 M Macdonell and Keith, 9 Madhya, 9, 13 Maha-Bhaskariya, 59, 80 Mahabja, 13 MaMkathana, 12 Mahasaroja, 13 Mahavira, 5, 13, 55, 56, 57, 77, 8o, 136, 137, 145, 151, 156, 157, 161, 162, .164, 166, 167, 168, 172, 188, 191, 192, 195, 196, 198, 199, 205, 208, 212, 220, 222, 224, 23°. 2 33. 2 35, *37> 2 3 8 » 240, 241 Mahmud bin Qajid al-Amuni Saraf-Eddin, 99 Maitrayani Samhita, 9, 18, 185 Majjhima Nikaya, 4 Malayagiri, 79 Mana, 204, 205 * Manoranjana, 132, 1*37 Marre, A., 100, 184 Marshall, 19 Martin, Ohm, 246 Masa, 218 Mathematics, appreciation of, 3; decay of, 127; Hindus and, 3; in Hindu education, 6; of zero, 238; scope and develop- ment of Hindu, 7 Maximus Planudes, 104, 144, 184 Mazumdar, R.C.,59 Measures, see weights Megasthenes, 21, 37 Mihir Yast, 101 Milindapanho, 7 INDEX 257 Mille; 9 Miscellaneous problems, 230; problems involving solution of quadratic equations, 234; problems involving the square of the unkown, 238; problems involving the square- root, 23.5; problems on mixture, 233; regula falsi, 230; the method of inver- sion, 232 Misraka, 124; vyavahara, 219, 233 Misrana, 130 Mitra, Rajendra Lai, 10, 97, 187 Mohammad, 88 Mohammad Ben Musa, 102 Mohenjo-daro, 19; and Harappa, 19, 23, 29; and the Indus Valley civilization, 17; dis- coveries at, 1; finds of, 20 Montucla, J.F., 99 Mudra, 7 Mudrabala, 1 1 Muhurta, 186 Mukerjee, Sir Asutosh, 17 Mula, 169, 170, 220, 234; arhsa, 237; dviragra sesa, 236; ghana, 124, 175; jati, 235; misra, 237; sesa, 235; varga, . 124. i 6 9 Multiplication, 134; algeb- raic methods, 149; Brahma- gupta's method, ,136; by separation of places, 146; cross-multiplication method, 145; direct process, 1 3 8; door- junction method, 13^; gelosia method, 144; inverse method, 139; methods' of, 135: parts — method, 148; terminology, 134; transmission to the west, 143; zigzag method, 147 17 Munisvara, 213 Myriad, 9 N Nagabala, 10 Nagari script, 39 Nagarjuna, 2 Nahuta, 12 Naisadha-carita, 85 Nallino, 83 Narada, 4 Narayana, 13, 137, 152, 161, 162, 167, 168, 183, 184, 205, 240 Nau, F., 95 Nava, 13; — dasa, 15;— yirhsati, 15 Nava-rasika, 124 Neil, R.A., see Cowell Nikharva, 10, 13 Nilakantha, 67, 170 Ninnahuta, 12 Nirabbuda, 12 Niravadya, 1 1 Niruddha, 195 Niska, 213, 214, 215, 216 Niyuta, 9, 10, 12 Notation, abjad, 89; decimal place-value, 3 ; difference from other, 27; epigraphic instances of decimal place- value, 40; Greek alphabetic, 50; 'places of, 12; scale of, 9 Numeral, ghobar, 89/90, 91, 93, 94; Hieratic and Demotic, 28; Hindu, 38; Kharosthi and Semitic, 28 Numeral notation, 1; Brahmi, 25; earliest, 19; in spoken language, .13; Kharosthi, 21; terminology, 9 Numerical, Asokan — figures, 37; development of symbblism, 16 2 5 8 INDEX Nyarbuda, 9,10, 13 O Ojha, 17, 24 Oldenburg, 4 Operations, checks on, 180 Pacioli, 146 Pada, 155, 169, 170, 177 Padma, 13; maha, 13 Paduma, 1 2 Pana, 206, 216, 218, 227, 229 Panca, 13 Panca-rasika, 124 Panca-siddhantika, 59, 78, 79, 239 Panini, 2, 18, 33, 63, 218 Pankti, 173 Pannavand-sutra, 37 Papyrus Blacas, 24 Paramesvara, 67, 155, 197 Parardha, 9, 13 Parasparakrtarn, 134 Pargiter, 58 Parikarma, 8 Patala, 18 Patana, 132 Patanjali, 2, 63, 85 PatI, 8, 124, 126, 129, 138, i39> l 4°, Mi, 148, 152, 157. M 8 . J 59> I 73. !74, 177, 178, 180 Patiganita, 8, 82, 123, 126, 128, 151, 184, 187, 195, 222, 226, "7> 233= 2 35, 239» 2 4° Patlsara, 125, 213 Parta, 123 Pattopadhyaya, 50 Peurbach, 175 Phala, 198, 204, 205, 206, 208, 212, 216, 220, 223, 228 Phalaka, 123 Phoenician forms (of numerals), 24 Phoenician script, 17 Pingala, 75, 76, 77 Place-value, date of invention of the notation, 86; In Hindu literature, 83 ; in Jain cano- nical works, 83; in literary works, 85 ; in works on philosophy, 85; invention of system, 5 1 ; new notation, 5 3; principle, 39; the decimal notation, 40 ; the decimal system, 43 Plate, Cambay — of Govinda, 43; Chargaon— ^of Huviska, 47; Ciacole, 40; Daulatabad — of Sahkargana, 41 ; Dhiniki Copper, 40; Grant of Avani- varmana, 42; Grant of Balavarmana, 42; Gurjara Grant, 40, 45, 48; Kadab, 60; Pandukesvara — of Lalita- suradeva, 41; Ragholi, 40, 82; Sangli — of Rastrakuta Govindaraja,43;Torkhedi, 41 Prabhaga, 190 Prakespa karana, 233 Prakrta, 170 Praksepana, 130 Pramana, 198, 204, 205, 223 ; rasi, 198; paksa, 211, 213 Pratiloma, 1 5 1 Pratyutpanna, 135, 139, 140 Prayuta, 9, 12, 13 Princep, James, 33 Prthudakasvami, 77, 124, 129, 148, 156, 163, 228 Pulisa, 61, 79 Pundarika, 1 2 Purana, 2, 84, 86, 206, 228; Agni, 58, 59, 62, 84, 86 ; Vayu, 84; Visnu, 84 INDEX '59 Purva, 164 R Rabbi ben Ezra, 96, 103, 217, 230 Radix, 1 70 Rajatarahgirn, 50 Rajju, 8 Ramanuj acaria, 156, 168 Ramayana, 2 Rangacarya, 151, 226 R&si, 8, 124; ruparhsaka, 199 Ray, H.C., 17 Ray, Sir P.C., 85 Regula falsi, 230, 238 Reina'ud, 97, 98 Rgveda, 9, 15, 17, 18, 20, 57> l8 5 Rhys Davids, 7 Riese, 149 Rodet, 65, 66, 170, 175 Rosen, 102 Rule of Three, 203 ; apprecia- tion of, 208; as a particular case, 216; compound propor- tion, 210; illustration, 213 ; inverse, 207; proportion in the west, 210; terminology, 203; the method, 204, 211 Rupa, 6 Rupa-vibhaga, 136 Sachau, E.C., 98, 99 Sadgurusisya, 71 Sadratnamala, 70 Sahasra, 9, 12, 13 Sakala, 1 99 Salila, 10 Samacaturasra, 155 Samapta-lambha, 1 1 Samavayariga-sutra, 6, 37 Sarhkalana, 130 Sarhkalita, 124, 130, 196 Sarhkhya, 238 Sarhkhyana, 4, 6, 7 Sammelana, 130 Samudra, 9, 10 Sarhyojana, 130 Sanatkumara, 4 Satikaracarya, 85 Saiikha, 1 3 Sarikhyayana srauta sutra, 10 Sarikramana, 221, 222 Sariku, 1 3 Sapha, 185 Sapta, 1 3 Sapta-rasika, 124 Saritapati, 1 3 Sarvabala, 1 1 Sarvadhana, 132. Sarvajna, 1 1 Sarvanukramani, 71 Sastri, Madhava, 184 Sastri, Sambasiva, 67 Sata, 9, 12, 13; koti, 13 Satavahana, 25 Satottara ganana, 10 ; sarhjna, 10 Savarnana, 194 Script, Indian, 16; Nagari, 39; North Semitic, 17; Phoeni- cian, 17; South Semitic, 16, 17 Sebokht, 89, 93, 95, 96 Sefer ha-Misp'ar, 103 Sefer Yeslrah, 98 Senart, E., 26 Sengupta, 175, 197 Sesa, 132 Shamasastri, R., 6, 19, 187, 218 Siddhanta, 3, 125, 128, 135, 150; Brahma-sphuta, 8, 59, 89, 156, 228, 241; Maha, 125, 181, 183, 184, 240; Parasar, 3; Pitamaha, 3, 125; .Pulisa, 59, 62, 79, 86; Romaka, 125; Sekhara, 125, z6o INDEX 136; Surya, 3, 59, 62, 125; Tattva-viveka, 125; Vasistha, 3, 125 Siddhasena Gani, 80, 171 Siladitya VI, 52 • Silberberg, Moritz, 103 Sindhind, 97 Singh, A.N., 170, 171, 172, Sirsaprahelika., 12 Sitanath, Sri, 76 Skandasena, 188 Smith, 144, 146, 149, 150, 153, 154, 175, 210, 217. 23°. 2 32, 233; and Karpinski, n, 27, 30, 48, 8o, 83, 88; 89, 90, 95,103 Sodhana, 132 Sodhya, 177, 178 Sogandhika, 12 Square, 155; minor methods of squaring, 160; terminology, 155; the operation, 1 5 6 Square-root, 169; terminology, 169; the operation, 170 Srauta-sutra, 3S $redhi, 124 '13. 133; inverse process, 133; terminology, 1 3 2; the opera- tion, 132 Sulba 130, 134, 155, i 7 o, 185, 188 Sumatiharsa, 184 Sunya, 38, 54, 77;— bindu, 81, 8) Suryadasa, 133, 197, Z0 8, 211 Suryadeva, 67, 71 Susruta, 2 Suter, H., 51,99, 102, 143, 174 Sutra, 4 SVetavarni, 129 Sridhara, ■ 8 J 45, i55 161, 162, !72, 177, 192, 193, 134, 136, 156, 157, 160, 163, i6 7j 168, 190, 191, 198, 199, 188 196 205, 207, 212, .230, 240 Sridharacarya, see Sridhara Sriharsa, 85 Sripati, 128, • 136, 137, '138, '-144, 145, 150, 167, 172 Stambhoddesa, 230 Sthana, 12, 161, 166; — khanda, 146, 148 Sthananga-sutra, 8, 204 Subandhu, 81, 82, 85 Subtraction, 132; direct process, Taccheda, 241 Taittiriya Sarhhita, 9, 14, ij; Brabmana, see Brahmana Talkhis, 99 Tallaksana, 1 1 Tantric, 19 Tarik al-hindi, 143 Tastha, .136, 145; gunana, see gunana Tattvarthadhigama-sutra, 80, 151, 171 Taylor, 16 z % 132, 147, 245 Theon of Alexandria, 171 Theophanes, 89 Theory, Indraji's, 3 j Thibaut, 155 Titilambha, n Trairasika, 124, zoi, 204; vyasta, 124, 207, 210 Tri, 13; pada, 185 Triprasna, 5 Trisatika, 59, 125, iji, 187,205, 227, 229 Tropfke, 80 U Umasvati, 2, 80, iji, 189 una-virhsati, 15; trirhsat, 15 INDEX 261 Upanisad, Chandogya, 3, 58; Mundaka, 4$ Uppala, 12 Utkrama, 131 Utsanga, 10 Uttara, 163 Uttaradhyayana-svitra, 4 Vadava, 10 Vadha, 134 Vajasaneyi sarhhita, 9,-15 Vajrabhyasa, 145, 146 Vajrapavartana-vidhi, 196 Vallika-kuttikara, 233 Valmiki, 2 Varahamihira, 61, 79, 209, 239 Varataka, 206 Varga, 8, 65, 66, 67, 69, 124, 155, 156, 235; mula, see mula;— varga, 8 Varna, 63, 229 Vasavadatta, 81, 85 Vasistha, 1 7 Vedariga, 7, 19;— jyotisa, 7, 58 Vedisri, 25 Vibhutangama, 1 1 Vidya, apara, 4; Brahma, 4; naksatra, 4; para, 4; rasi, 4 Vikalpa, 8 Vinaya Pitaka,' 4, 7 Yinimaya, 204, 205 Visamjna-gati, 11 ViVaha, 10 Vivara, 10 Viyoga, 132 Viyojaka, 132 Viyojya, 132 Vrnda, 13 Vyasabhasya, 85 Vyavahara, 8; — sutra, 84 Vyavakalita, 1 24 Vyavasthana-prajfiapti, 1 1 Vyutkalana, 132 Vyutkalita, 124, 132 W Wahshiya, 97 Warner, A.G., and Warner, E., 100 Waschke, 104 Weber, 16, 17, 37, 58 Weights and Measures, 186 Whish, 65 Widman, 149 Woepcke, F., 90, 102, 143, 174, 184 Wolack, Gottfried, 83 Woods, J.H., 85 Word-numeral, 53; date of invention, 62; explanation of the system, 5 3; in inscrip- tions, 59; list of, 54; origin and early history, 60; without place-value, 57; with place- value, 58; Wright, W., 96 Writing in ancient India, 16 Y Yajurveda, 9, 10, 20 Yakub ibn Tarik, 89 Yavat tavat, 8 Yoga, 130, 236; — sutra, 85 Yojana, 171, 172 Yukti, 130 Zero, as an infinitesimal, 242, 243;- earliest use, 75; form of the symbol, 81; in algebra, 241; in arithmetic, 239; indeterminate forms, 245; other uses of the symbol, 82; the mathematics of, 238; the symbol, 75 HISTORY OF HINDU MATHEMATICS A SOURCE BOOK PART II AL GEBR A BY BIBHUTIBHUSAN DATTA AND AVADHESH NARAYAN SINGH COPY RIGHT, 193 J, BY AVADHESH NARAYAN SINGH ALL RIGHTS RESERVED (RV, x. 14. 25) To the Seers, our Ancestors, the first Path-makers PREFACE The present work forms Part II of our History of Hindu Mathematics and is devoted to the history of Algebra in India. It is intended to be a source book, and the subject is treated topicwise. Under each topic ' are collected together and set forth in chronological order translations of relevant Sanskrit texts as found in the Hindu mathematical works. This plan necessitates a certain amount of repetition. But it shows to the reader at a glance the improvements made from century to century To gather materials for the book we have examined - all the published mathematical treatises of the Hindus as well as most of the important manuscripts available in Indian libraries, a list of the most important of which has already been included in Part I. We have great pleasure in once more expressing our thanks to the autho- rities of the libraries at Madras, Bangalore, Trivandrum, Tripunithura, Baroda, Jammu, and Benares, and those of the India Office (London) and the Asiatic Society of Bengal for supplying transcripts of manuscripts or sending them to us for consultation. We are indebted also to Dr. R. P. Paranjpye, Vice-Chancellor of the Lucknow University, for help in securing for our use several manuscripts or their transcripts from the State libraries in India and the India Office. In translating Sanskrit texts we have tried to be as literal and faithful as possible without sacrificing the spirit of the original, in order to preserve which we have at a few places used literal translations of Sanskrit tech- nical terms instead of modern terminology. For in- stance, we have used the term 'pulveriser' for the equa- tion ax-\-by=j, and the term 'Square-nature 5 for the equation Nx 2 ^~c=j 2 . The use of symbols — letters of the alphabet to de- note unknowns — and equations are the foundations of the science of algebra. The Hindus were the first to make systematic use of the letters of the alphabet to denote unknowns. They were also the first to classify ■ and make a detailed study of equations. Thus they may be said to have given birth to the modern science of algebra. A portion of the subject matter of this book has been available to scholars through papers by various authors and through Colebrooke's Algebra' with Arithmetic and Mensuration from the Sanscrit of Brahmegupta and Bhas- cara, but about half of it is being presented here for the first time. For want of space it has not been possible to give a detailed comparison of the Hindu achievements in Algebra with those of other nations. For this the reader is referred to the general works on the history of mathematics by Cantor, Smith, and Tropfke, to Dixon's History of the Theory of Numbers and to Neugebauer's Mathematische Keilschrift-Texte. A study of this book along with the above standard works will reveal to the reader the remarkable progress in algebra made by the Hindus at an early date. It will also show that we are indebted to the Hindus for the technique and the fundamental results of algebra just as we owe to them the place-value notation and the elements of our arithmetic. We have pleasure in expressing our thanks to Mr. T. N. Singh and Mr. Ahmad AH for help in correcting proofs and to Mr. R. D. Misra for preparing the index to this volume. LUCKNOW BlBHUTIBHUSAN DATTA March, 1938 Avadhesh Narayan Singh CONTENTS CHAPTER III ALGEBRA Page i. GENERAL FEATURES I Name for Algebra — Algebra defined — Distinction from Arithmetic — Importance of Algebra — Scope of Al- gebra — Origin of Hindu Algebra. 2. TECHNICAL TERMS 9 Coefficient — Unknown Quantity — Power — Equation — Absolute Term. 3. SYMBOLS 12 Symbols of Operation — Origin of Minus Sign — Sym- bols for Powers and Roots — Symbols for Unknowns. 4. LAWS. OF SIGNS 20 Addition — Subtraction — Multiplication — Division — Evolution and Involution. 5. FUNDAMENTAL OPERATIONS 25 Nbmber of Operations — Addition and Subtraction — Multiplication — Division — Squaring — Square-root. 6 EQUATIONS 28 Forming Equations — Plan of writing Equations — Pre- paration of Equations — Classification of Equations. 7. LINEAR EQUATIONS IN ONE UNKNOWN 36 Early Solutions — Rule of False Position — Disappear- ance from later Algebra — Operation with an Optional Number — Solution of Linear Equations. 8. LINEAR EQUATIONS^WITH TWO UNKNOWNS 43 Rule of Concurrence — Linear Equations. contents Page 9. LINEAR EQUATIONS WITH SEVERAL UN- KNOWNS 47 A type of Linear Equations — Solution by False Posi- tion — Second Type — Third Type — Brahmagupta's Rule — Mahavira's Rule — Bhaskara's Rule. 10. QUADRATIC EQUATIONS , 59 Early Treatment— "Bakhshali Treatise — Aryabhata I — Brahmagupta's Rules — Sridhara's Rule — Mahavira — Aryabhata II— Sripati's Rules— BMskara IPs Rules- Elimination of the Middle Term — Two roots of the Quadratic — Known to Mahavira — Brahmagupta. 11. EQUATIONS OF HIGHER DEGREES 76 Cubic and Biquadratic — Higher Equations. 12. SIMULTANEOUS QUADRATIC EQUATIONS 8 1 Common Forms — Rule of Dissimilar Operations — Mahavira's Rules. 13. INDETERMINATE EQUATIONS OF THE FIRST DEGREE 87 General Survey — Its Importance — Three Varieties of Problems — Terminology — Origin of the name — Preliminary Operations — Solution of bx — ay=-±ic — Aryabhata I's Rule — Bhaskara I's Rules — Brahma- gupta's Rules — Mahavira's Rules — Aryabhata II — Sripati's Rule — Bhaskara II's Rules — Solution of by= ax±i — Constant Pulveriser — Bhaskara I's Rule — Brahmagupta's Rule — Bhaskara II's Rule — Solution of by-\-ax—±:C — Brahmagupta's Rule — Bhaskara II's Rule — Narayana — Illustrative Examples — Particular Cases. ''" 14. ONE LINEAR EQUATION IN MORE THAN TWO UNKNOWNS I2j 15. SIMULTANEOUS INDETERMINATE EQUA- TIONS OF THE FIRST DEGREE 127 Sripati's Rule — Bhaskara II's Rule — Special Rules — General Problem of Remainders — Conjunct Pulveriser — Generalised Conjunct Pulveriser — Alternative me- thod. CONTENTS Page 16. SOLUTION OF Nx*+i=y* 141 Square-nature — Origin of the Name — Technical Terms — Brahmagupta's Lemmas — Description by later writers — Principle of Composition — General Solution of the Square-nature — Another Lemma — Rational Solution — Sripati's Rational Solution — Illus- trative Examples — Solution in Positive Integers. 17. CYCLIC METHOD 161 Cyclic Method — Bhaskara's Lemma — Bhaskara's Rule — Narayana's Rule — Illustrative Examples. 18. SOLUTION OF Nx*±c=y* 173 Form Mn 2 x 2 ±:c=y 2 — Form a 2 x 2 ±c=y 2 — Form c—Nx 2 ==y s — Form Nx 2 — £?=)/*. 19. GENERAL INDETERMINATE EQUATIONS OF THE SECOND DEGREE : SINGLE EQUATIONS 1 8 1 Solution — Solution of ax 2 -\-bx-\-c==y 2t — Solution of ax 2J t-bx J rc=a'j iJ rl>'y J rc' — Solution of ax 2 -\-bj 2 -\-c= ^ — Solution of a 2 x 2j r by 2 -\-c=^ — Solution of ax 2 -\- bxy+cy 2 =?*. 20. RATIONAL TRIANGLES 204 Early Solutions — Later Rational Solutions — Integral Solutions — Mahavira's Definitions — Right Triangles having a Given Side — Right Triangles having a Given Hypotenuse — Problems involving Areas and Sides — Problems involving Sides but not Areas — Pairs of Rectangles — Isosceles Triangles with Integral Sides — Juxtaposition of Right Triangles — Isosceles Triangles with a Given Altitude — Pairs of Rational Isosceles Triangles — Rational Scalene Triangles — Triangles having a Given Area. 21. RATIONAL QUADRILATERALS . 228 Rational Isosceles Trape2iums — Pairs of Isosceles Trapeziums — Rational Trapeziums with Three Equal Sides — Rational Inscribed Quadrilaterals — Inscribed Quadrilaterals having a Given Area — Triangles and Quadrilaterals having a Given Circum-Diameter. contents Page zz. SINGLE INDETERMINATE EQUATIONS OF HIGHER DEGREES 245 Mahavlra's Rule — Bhaskara's Method — Narayana's Rule — Form ax in+2 -\-bx in ==y 2 — Equation ax^-\-bx 2 -\-c 23. LINEAR FUNCTIONS MADE SQUARES OR CUBES ' 250 Square-pulveriser — Cube-pulveriser — Equation bx±c =ay 2 — Equation bx-^c=ay n . 24. DOUBLE EQUATIONS OF THE FIRST DEGREE 258 25. DOUBLE EQUATIONS OF THE SECOND DEGREE 266 First Type — Second Type. 26. DOUBLE EQUATIONS OF HIGHER DEGREES 278 Double Equations in several Unknowns. 27. MULTIPLE EQUATIONS 283 28. SOLUTION OF mg=bx+cy-\-d 297 Bakhshali work— An Unknown Author's Rule — Brahmagupta's Rule — Mahavira's Rule — Sripati's Rule — Bhaskara II's Rule — Bhaskara's Proofs. INDEX 309 Chapter III ALGEBRA i. GENERAL FEATURES Name for Algebra. The Hindu name for the science of algebra is bijaganita. Bi/a means "element" " or "analysis" and ganita "the science of calculation." Thus bijaganita literally means "the science of calcula- tion with elements" or "the science of analytical calculation." The epithet dates at least as far back as the time of Prthudakasvami (860) who used it. Brahmagupta (628) calls algebra kuttaka-ganita, or simply kuttaka}- The term kuttaka, meaning "pulve- riser", refers to a branch of the science of algebra dealing particularly with the subject of indeterminate equations of the first degree. It is interesting to find that this subject was considered so important by the Hindus that the whole science of algebra was named after it in the beginning of the seventh century. Algebra is also called avyakta-ganita or "the science of calculation with unknowns" {avyakta— unknown) in contradistinc- tion to the name vyakta-ganita or "the science of calcu- lation with knowns" (iy«/te=known) for arithmetic »ncluding geometry and mensuration. Algebra Defined. Bhaskarall (1150) has defined algebra thus: "Analysis {bijci) is certainly the innate intellect assisted by the various symbols (varna), which, for the 1 See Bibhutibhusan Datta, "The scope and development of the Hindu Ganita," THQ, V, 1929, pp. 479-512; particularly pp. 4 8 9 f. Z ALGEBRA instruction of duller intellects, has been expounded by the ancient sages who enlighten mathematicians as the sun irradiates the lotus; that has now taken the name algebra (bijaganita)." 1 That algebraic analysis' requires keen intelligence and sagacity has been observed by him on more than one occasion. "Neither does analysis consist in symbols, nor are there different kinds of analyses; sagacity alone is ana- lysis, for wide is imagination." 2 "Analysis is certainly clear intelligence." 3 "Or intelligence alone is analysis." 4 In answer to the question, "if (unknown quantities) are to be discovered by intelligence alone what then is the need of analysis ?" he says: "Because intelligence is certainly the real analysis; symbols are its helps. The innate intelligence which has been expressed for the duller intellects by the ancient sages, who enlighten mathematicians as the sun irradi- ates the lotus, with the help of various symbols, has now obtained the name of algebra." 5 Thus, according to Bhaskara II, algebra may be de- fined as the science which treats of numbers expressed by means of symbols, and in which there is scope and pri- mary need for intelligent artifices and ingenious devices. Distinction from Arithmetic. What distinguishes algebra from arithmetic, according to the Hindus, will be found to some extent in their special names. Both deal with symbols. But in arithmetic the values of the sym- bols are vyakta, that is, known and definitely determinate, 1 BBi, p. 99. 2 BBi, p. 49; SiSi, Go/a, xiii. j. 3 L, p. 15; Sift, Go/a, xiii. 3. * BBi,p. 49. 5 BBi, p. 100. GENERAL FEATURES 3 while in algebra they are avyakta^ that is, unknown, indefinite. The relation between these two branches of ganita is considered by Bhaskara II to be this: "The science of calculation with unknowns is the source of the science of calculation with knowns." 1 He has put. it more explicitly and clearly thus: "Algebra is similar to arithmetic in respect of rules (of fundamental operations) but appears as if it were indeterminate. It is not indeterminate to the intelligent; it is certainly not sixfold, 2 but manifold." 3 The true distinction between arithmetic and- algebra, besides that of symbols employed, lies, in the opinion of Bhaskara II, in the demonstration of the rules. He remarks: "Mathematicians have declared algebra to be com- putation attended with demonstration: else there would be no distinction between arithmetic and algebra." 4 The truth of this dictum is evident in the treatment of the guna-harma in the Li/dvati and the madbyamdharana in the Btjaganita. Both are practically treatments of problems involving the quadratic equation. But whereas in the former are found simply the applica- tions of the well-known formulae for the solution of such equations, in the latter is described also the rationale of those formulas. Similarly we sometimes find included in treatises on arithmetic problems whose solutions require formulas demonstrated in books on algebra. The method of demonstration has been stated to be "always of two kinds: one geometrical {ksetragatd) and 1 BBi, p. i. 2 The reference is to the six fundamental operations recognised in algebra as well as to the six subjects of treatment which are essential to analysis. 3 L,p. 15. *BBi, p. 127. 4 ALGEBRA the other symbolical (rdsigata)." 1 We do not 'know who was the first in India to use geometrical methods for demonstrating algebraical rules. Bhaskara II (1150) ascribes it to "ancient teachers." 2 Importance of Algebra. The early Hindus regard- ed algebra as a science of great importance and utility. In the opening verses of his treatise 3 on algebra Brahma- gupta (628) observes: "Since questions can scarcely be known {i.e., solved) without algebra, therefore, I shall speak of algebra with examples. "By knowing the pulveriser, zero, negative and positive quantities, unknowns, elimination of the middle term, equations with one unknown, factum and the Square-nature, one becomes the learned professor [dcdryd) amongst the learned." 4 Similarly Bhaskara II writes: "What the learned calculators {sdmkhydh) describe as the originator of intelligence, being directed by a wise being {satpurusd) and which alone is the primal cause ibijd) of all knowns {vyakta), I venerate that Invisible God as well as that Science of Calculation with Unknowns. . . Since questions can scarcely be solved without the reason- ing of algebra — not at all by those of dull perceptions — I shall speak, therefore, of the operations of analysis." 5 1 BBi,p / ii5. z BBi, p. 127. 3 Forming chapter xviii of his Brdhma-sphuta-siddhdnta. 4 BrSpSi, xviii. 1-2. 5 In the first part of this passage every principal term has been used with a double significance. The term sdmkhydh (literally, "expert calculators") signifies the "Samkhya philosophers" in one sense, "mathematicians" in the other; safpurusa "the self- existent being of the Sarhkhya philosophy" or "a wise mathema- tician"; vyakta "manifested universe" or "the science of calculation with knowns." GENERAL FEATURES J Narayana (1350) remarks : "I adore that Brahma, also that science of calcula- tion with the unknown, which is the one invisible loot- cause of the visible and multiple-qualitied universe, also of multitudes of rules of the science of calculation with the known." 1 "As out of Him is derived this entire universe, visible and endless, so out of algebra follows the whole of arithmetic with its endless varieties (of rules). There- fore, I always make obeisance to Siva and also to {avyakta-) ganita (algebra)." 2 He adds : "People ask questions whose solutions are not to be found by arithmetic ; but their solutions can generally be found by algebra. Since less intelligent^ men do not succeed in solving questions by the rules of arithmetic, I shall speak of the lucid and easily intelli- gible rules of algebra." 3 Scope of Algebra. The science of algebra is broad- ly divided by the Hindus into two principal parts. Of these - the most important one deals with analysis (bija). The other part treats of the subjects which are essential for analysis. They are : the laws of signs, the arith- " metic of zero (and infinity), operations with unknowns, ' surds, the pulveriser (or the indeterminate equation of the first degree), and the Square-nature (or the so-called Pellian equation). To these some writers add con- currence and dissimilar operations, while others include them in arithmetic. 4 At the end of the first section of his treatise on algebra Bhaskara II is found to have 1 NBI, I, R. i. 2 NBi, II, R. i. 3 NBi, I, R. 5-6. * All writers, except Brahmagupta and Sripati, are of the latter opinion. 6 ALGEBRA observed as follows : "(The section of) this science of calculation which is essential for analysis has been briefly set forth. Next I shall propound analysis, which is the source of pleasure to the mathematician." 1 Analysis is stated by all to be of four kinds, for equations are classified into four varieties {vide infra). Thus each class of equations has its own method of analysis. Origin of Hindu Algebra. The .origin of Hindu algebra can be definitely traced back to the period of the Sulba (800-500 B.C.) and the Brdhmatia {c. 2000 B.C.). But it was then mostly geometrical. 2 The geometrical method of the transformation of a square into a rectangle having a given side, which is described in the important Sulba is obviously equivalent to the solution of a linear equation in one unknown, vi^., ax = c z . The quadratic equation has its counterpart in the cons- truction of a figure (an altar) similar to a given one but differing in area from it by a specified amount. The usual method of solving that problem was to increase the unit of measure of the linear dimensions of the figure. One of the most important altars of the obligatory Vedic sacrifices was called the Mahavedi (the Great Altar). It has been described to be of the form of an isosceles trapezium whose face is 24 units long, base 30 and altitude 36. If x be the enlarged unit of measure taken in increasing the size of the altar by m units of area, we must have 1 BBi, p. 4i- 2 Bibhutibhusan Datta, Tie Science of the Sulba, Calcutta, 19$ 2. or GENERAL FEATURES 972.x -2 = 972 -f- m. Therefore x ■V 1 + m 972' If m be put equal to 972 (« — i), so that the area of the enlarged altar is « times its original area, we get x =y«, some particular cases of which are described in the Su/ba. 1 The particular cases, when n = 14 or 14-f, are found as early as the Satapatha Brdhmana x {c. 2000 B.C.). The most ancient and primitive form of the "Fire- altar for the sacrifices to achieve special objects" was the Syenacit (or "the altar of the form of the falcon"). A B F Z M H P L F ' H' M' R Fig. Its body (ABCD) consists of four squares of one square purusa each; each of its wings (EFGH, E'F'G'H') is a rectangle of one purusa by one purusa and a prddesa (=1/10 of a purusa). This Fire-altar was enlarged in 1 .ftr, X. 2 3. 7ff. '8 ALGEBRA two ways: first, in which all the constituent parts were affected in the same proportion ; second, in which the breadth of the portions LFGM and L'F'G'M' of the wings were left unaffected. If x be the enlarged unit for enlargement in the first case we shall have to solve the quadratic equation zxXix+ z\xx(x + -)[ + x x{x + ^) = 7$+w, where m denotes the increment of the Fire-altar in .size. Therefore x 2 =H . In particular, when m = 94, we shall have x 2 = 13-^= 14 (approximately), which occurs in the Satapatha Brdhmana. In the second case of enlargement the equation for x will be 2XX2X+ z{x X(x-j- i)}+ x x(x+^jj) = 7^ + m, or jx 2 + \x = 7^ + m, which is a complete quadratic equation. The problem of altar construction gave rise also to certain indeterminate equations of the second degree such as, (1) X 2 +J 2 = % 2 , (2) x 2 + a 2 = z 2 ; and simultaneous indeterminate equations of the type ax + by + &(. + dw -= p, x -\-j -\- ^-\- h> ~ q. Further particulars about these equations will be given later on. TECHNICAL TERMS 9 2. TECHNICAL TERMS Coefficient. In Hindu algebra there is no syste- matic use of any special term for the coefficient. Ordinarily, the power of the unknown is mentioned when the reference is to the coefficient of that power. In explanation of similar use by Brahmagupta his commentator Prthudakasvami writes "the number (arika) which is the coefficient of the square of the unknown is called the 'square' and the number which forms the coefficient of the (simple) unknown is called 'the unknown quantity.' "* However, occasional use of a technical term is also met with. Brahmagupta once calls the coefficient samkbyd 2 (number) and on several other occasions gunaka, or gunakdra (multiplier). 3 Prthudakasvami (860) calls it arika* (number) or prakrtt (multiplier). These terms reappear in the works of Sripati (1039) 5 and Bhaskara II (1150). 6 The former also used rupa for the same purpose. 7 Unknown Quantity. The unknown quantity was called in the Sthdndnga-sittrcfi (before 300 B.C.) ydvat-tdvat (as many as or so much as, meaning an arbitrary quantity). In the so-called Bakhshali treatise, it was called jadrcchd, vdnchd or kdmika (any desired quantity). 9 This term was originally connected with the Rule of False Position. 10 Aryabhata I (499) 1 BrSpSi, xviii. 44 (Com)\ 2 BrSpSi, xviii. 63. 3 BrSpSi, xviii. 64, 69-71. * BrSpSi, xviii. 44 {Com). 6 SiSe, xiv. 33-5. e BBi, pp. 33-4. 7 SiSe, xiv. 19. 8 Sfitra 747; cf. Bibhutibhusan Datta, "The Jaina School of Mathematics," BCMS, XXI, 1929, pp. 11 5-145; particularly pp. 122-3. 9 BMs y Folios zz, verso; 23, recto & verso. 10 Bibhutibhusan Datta, "The Bakshshali Mathematics," BCMS, XXI, pp. 1-60; particularly pp. 26-8, 66. IO ALGEBRA calls the unknown quantity gulikd (shot). This term strongly leads one to suspect that the shot was probably then used to represent the unknown. From the begin- ning of the seventh century the Hindu algebraists are found to have more commonly employed the term avyakta (unknown). 1 Power. The oldest Hindu terms for the power of a quantity, known or unknown, are found in the XJttard- dhyayana-siitra {c. 300 B.C. or earlier). 2 In it the second power is called varga (square), the third power ghana (cube), the fourth power varga-varga (square-square), the sixth power ghana-varga (cube-square), and the twelfth power ghana-varga-varga (cube-square-square), using the multiplicative instead of the additive principle. In this work we do not find any method for indicat- ing odd powers higher than the third. In later times, the fifth power is called varga-ghana-ghdta (product of cube and square, ghdta= pro duct), the seventh power varga-varga-ghana-ghdta (product of square-square and cube) and so on. Brahmagupta's system of expressing powers higher than the fourth is scientifically better. He calls the fifth power panca-gata (literally, raised to the fifth), the sixth power sad-gata (raised to the sixth) ; similarly the term for any power is coined by adding the suffix gata to the name of the number indicating that power. 3 Bhaskara II has sometimes followed it consis- tently for the powers one and upwards. 4 ■ In the Anuyogadvdra-sutra* , a work written before the com- mencement of the Christian Era, we find certain interest- ing terms for higher powers, integral as well as fractional, particularly successive squares (varga) and square-roots (parga-milld). According to it the term prathama-varga 1 BrSpSi, xviii. 2, 41; SiSe, xiv. 1-2; BBi, pp. 7 ft. 2 Chapter xxx, io, 11. 3 BrSpSi, xviii. 41, 42. *BBi, p. 56. h Sutra 142. TECHNICAL TERMS II (first square) of a quantity, say a, means a 2 ; dviiiyavarga (second square) = (a 2 ) 2 = a A ; trtiya-varga (third square) = ((a 2 ) 2 ) 2 = a 8 ; and so on. In general, nth varga of a = a****** - t0 » terms = a 2 ". Similarly, prathama-varga-mula (first square-root) means y/a; dvitiya-varga-mtila (second square-root) = \J (yj a) — a Vi ; and, in general, «th varga-mula of a = a v%n . Again we find the term trtiya-varga-mfila-ghana (cube of the third square-root) for (a v2Z ) z = a 3 '*. The term varga for "square" has an interesting origin i-n a purely concrete concept. The Sanskrit word varga literally means "rows," or "troops" (of similar things). Its application as a mathematical term originated in the graphical representation of a square, which was divided into as many varga or troops of small squares, as the side contained units of some measure. 1 Equation. The equation is called by Brahma- gupta (628) sama-karana 2 or sami-karana z (making equal) or more simply samal (equation). Prthuda- kasvami (860) employs also the term sdmya* (equality or equation); and Sripati (103 9) sadrsi-karana* (making similar). Narayana (1350) uses the terms sami-karana, sdmya and samatva (equality). 7 An equation has always two paksa (side). This term occurs in the works of 1 G. Thibaut, Sulba-sutras, p. 48. Compare also Bibhutibhusan Datta, "On the origin of the Hindu terms for root," Amer. Math. Mori., XXXVIII, 1931, pp. 371-6. 2 BrSpSi, xvlii. 63. 3 Br SpSi, .xviii, subheading for the section on equations. 4 BrSpSi, xviii. 43. 6 BrSpSi, xii. 66 {Com). 8 SiSe, xiv. 1. 'NBi, II, R. 2-3. 12 ALGEBRA Sridhara (c. 750), Padmanabha 1 and others. 2 Absolute Term. In the Bakhshali treatise 3 the absolute term is called drsya (visible). In later Hindu algebras it has been replaced by a closely allied term rupcfi (appearance), though it continued to be employed in treatises on arithmetic. 5 Thus the true significance of the Hindu name for the absolute term in an algebraic equation is obvious. It represents the visible or known portion of the equation while its other part is prac- tically invisible or unknown. 3. SYMBOLS Symbols of Operation. There are no special symbols for the fundamental operations in the Bakh- shali, work. Any particular operation intended is ordinarily indicated by placing the tachygraphic abbre- viation, the initial syllable of a Sanskrit word of that import, after, occasionally before, the quantity affected. Thus the operation of addition is indicated by ju (an abbreviation from yuta, meaning added), subtraction by -f- which is very probably from ksa (abbreviated from ksaya, diminished), multiplication by gu (from gum or gunita, multiplied) and division by bhd (from bhdga or bhdjita, divided). Of these again, the most systemati- cally employed abbreviation is that for the operation of subtraction and next comes that of division. In the case of the other two operations the indicatory words 1 The algebras of Sridhara and Padmanabha are not available now. But the term occurs in quotations from them by Bhas- kara II (BBi, pp. 61, 67). 2 BrSpSi, xviii. ^(Com); SiSe, xiv. 14, 20; BBi, pp. 43-4. 3 BMs, Folio 23, verso; Folio 70, recto and verso (c). *BrSpSi, xviii. 43-4; SiSe. xiv, 14, 19; etc. 5 Tris, pp. 11, 12. SYMBOLS '3 are often written in full, or altogether omitted. In the latter case, the particular operations intended to be carried out are understood from the context. We take the following instances from the Bakhshali Manuscript : °5 x (i) 1 ~ } yu means — + — , and yu' means — -j- w ir 1 1 - ' -r 11 1 £1 I (ii) 2 3 3 3 3 3 3 3 ™g» iiiiiiii means 3X3X3X3x3x3 X3X10. (iii) 3 1 means .VI 1 3 1 2 2 5 i 2+ 3 7 4 9 (i + |) + {zx(i+|)-^} + {3^(i + |)-^j +'{^(1 + 1)-^}. Civ)* (v) B 1 1 1 1 bhd 1+1 1 1 2 3_ 4 +6 36 means 36 (i-iXi+ao-iXi+iy 40 M«2 l6o x 3 I I 1 2 160 means X13A. 40 " In later Hindu mathematics the symbol for subtrac- tion is a dot, occasionally a small circle, which is placed above the quantity, so that 7 or 7 means —7; other operations are represented by simple juxtaposition. 1 Folio 5 9,. recto. 2 Folio 47, recto. * Folio 25, verso. The beginning and end of this illustration are mutilated but the restoration is certain. ♦Folio 13, verso. 'Folio 42, recto. 14 ALGEBRA Bhaskara II (1150) says, "Those (known and unknown numbers) which are negative should be written with a dot (bindti) over them." 1 A similar remark occurs in the algebra of Narayana (1350). 2 Their silence about symbols of other fundamental operations proves their non-existence. Origin of Minus Sign. The origin of • or ° as the minus sign seems to be connected with the Hindu symbol for the zero, o. It is found tc have been used as the sign of emptiness or omission in the early Bakh- shali treatise as well as in th'e later treatises on arithmetic (vide infra). 3 It is placed over the number affected in order to distinguish it from its use in a purely numerical significance when it is placed beside the number. The origin of the Bakhshali minus sign '( + ) has been the subject of much conjecture. Thibaut suggested its possible connection with the supposed Diophantine negative sign ifr (reversed ty } tachy graphic abbreviation for leityis meaning wanting). Kaye believes it. The Greek sign for minus, however, is not ifr, but -j- . It is even doubtful if Diophantus did actually use it; or whether it is as old as the Bakhshali cross. 4 Hoernle 5 presumed the Bakhshali minus, sign to be the abbrevia- tion ka of the Sanskrit word kanita, or nu (or nu) of nyuna, both of which mean diminished and both of which abbreviations in the Brahmi characters would be denoted by a cross. Hoernle was right, thinks Datta, 6 so far as he sought for the origin of -J- in a tachygraphic abbre- viation of some Sanskrit word. But as neither the word kanita nor nyuna is found to have been used in the Bakhshali work in connection with the subtractive 1 BBi, p. 2. 2 NBi, I, R. 7. 3 p. 16. 4 Cf. Smith, History, II, p. 396. *1A, XVII, p. 34. 6 Datta, Bakh. Math., (BCMS, XXI), pp. 17-8. SYMBOLS I J operation, Datta finally rejects the theory of Hoernle and believes it to be the abbreviation ksa, from ksaya (decrease) which occurs several times, indeed, more than any other word indicative of subtraction. The sign for ksa, whether in the Brahmi characters or in Bakhshali characters, differs from the simple cross ( + ) only in having a little flourish at the lower end of the vertical line. The flourish seems to have been dropped subsequently for convenient simplification. Symbols for Powers and Roots. The symbols for powers and roots are abbreviations of Sanskrit words of those imports and are placed after the number affected. Thus the square is represented by va (from t>arga\ cube by gha (from gbana), the fourth power by va-va (from varga-varga), the fifth power by va-gha-ghd (from varga-ghana-ghdtd), the sixth power by gha-va (from ghana-varga), the seventh power by va-va-gha-ghd (from varga-varga-ghana-ghdtd) and so on. The product of two or more unknown quantities is indicated by writing bhd (from bhdvita > product) after the unknowns with or with- out interposed dots ; e.g., ydva-kdgha-bhd or ydvakdgbabbd means (jd) 2 (k£f. In the Bakhshali treatise the square- root of a quantity is indicated by writing after it mfi, which is an abbreviation for mMa (root). For instance, 1 and II I yu 5 i mtt 4 i II I 7+ i mu 2 I means V 1 1 -+- 5 = 4 means -\/ii — 7 = 2. In other treatises the symbol of the square-root is ka (from karani, root or surd) which is usually placed before the quantity affected. For example, 2 ka y ka 450 1 Folio 59, recto; compare also folio 67, verso. 2 J3J3/, p. 1 j. i6 ALGEBRA ka 75 ka 54 means V? + V45° + Vj] + V54- Symbols for Unknowns. In the Bakhshali treatise there is no specific symbol for the unknown. Consequently its place in an equation is left vacant and to indicate it vividly the sign of emptiness is put there. For instance, 1 10 I means x-\- ix-\-^x-\-4x- I 2 I 3 1 4 1 drsya zoo 1 :200. The use of the zero sign to mark a vacant place is found in the arithmetical treatises of later times when the Hindus had a well-developed system of symbols for the unknowns. Thus we find in the Trisatikd? of Sridhara (c. 750) the following statement of an arithme- tical progression whose first term {ddib) is 20, number of terms (gacchab) 7, sum (ganitam) 245 and whose com- mon difference (uttarab) is unknown: I ddib 20 [ u o \ gacchah 7 | ganitam 245 | This use of the zero sign in arithmetic was consi- dered necessary as algebraic symbols could not be used there. Lack of an efficient symbolism is bound to give rise to a certain amount of ambiguity in the re- presentation of an algebraic equation especially when it contains more than one known. For instance, in 3 o 5 ju wu o 11 1 sa 7+ wu o 1 1 1 which means V.v + 5 — s and V x — 7 — t, different unknowns have to be assumed at different vacant places. 1 BMs } Folio 2.z, verso. *BMs> Folio 59, recto. Tr/s, p. 29. SYMBOLS *7 To avoid such ambiguity, in one instance which contains as many as five unknowns, the abbreviations of ordinal numbers, such as pra (from prathama, first), dvi (from dvitfya, second), tr (from trtfya, third), ca (from caturtha, fourth) and path (from pancama, fifth), have been used to represent the unknowns; e.g.? 9 pra 7 dvi dvi 10 tr 10 tr 8 ca % ca ii pam 1 1 pam 9 pra yutam jatam pratyaika- (kramena) i6|i 7 |i8[i9)20 which means '""i( = 9) + x *( = 7) = l6 '> x 2( = 7) + - v 3( = IO ) = x 75 ■*»( = io) + x- 4 ( = 8) = 18 ; * 4 ( = 8) + x- B ( = 1 1) = 19; *6(=- IX ) + *i( = 9) = 2 °- Aryabhata I (499) very probably used coloured shots to represent unknowns. Brahmagupta (628) mentions varna as the symbols of unknowns. 2 As he has not at- tempted in any way to explain this method of symbolism, it appears that the method was already very familiar. Now, the Sanskrit word varna means "colour" as well as "letters of the alphabet," so that, in later times, the unknowns are generally represented by letters of the alphabet or by means of various colours such as kClaka (black), nilaka (blue), etc. Again in the latter case, for simplification, only initial letters of the names are generally written. Thus Bhaskara II (1150) observes, "Here (in algebra) the initial letters of (the names of) knowns and unknowns should be written for implying them." 3 It has been stated before that at one time the unknown quantity was called yavat-tavat (as many 1 Folio 27, verso. 2 BrSpSi, xviii. 2, 42, 51, etc. 3 BBi, p. 2; see also NB/\ I, R. 7. I 8 ALGEBRA as, so much as). In later times this name, or its abbreviation yd, is used for the unknown. According to the celebrated Sanskrit lexicographer Amarasirhha (c. 400 A.D.), ydvat-tdvat denotes measure or quantity (mdna). He had probably in view the use of that term" in Hindu algebra to denote "the measure of an unknown" {avyakta mdnd). In the case of more unknowns, it is usual to denote the first by ydvat-tdvat and the remaining ones by alphabets or colours. Prthudakasvami (860) says: "In an example in which there are two or more unknown quantities, colours such as ydvat-tdvat, etc., should be assumed for their values." 1 He has, indeed, used the colours kdlaka (black), nilaka (blue), pitaka (yellow) and haritaka (green). Sripati (1039) writes: "Ydvat-tdvat (so much as) and colours such as kdlaka (black), nilaka (blue), etc., should be assumed for the unknowns." 2 Bhaskara II (nyo) says: "Ydvat-tdvat (so much as), kdlaka (black), nilaka (blue), pita (yellow), lohita (red) and other colours have been taken by the venerable professors as notations for the measures of the unknowns, for the purpose of calculating with them." 3 "In those examples where occur two, three or more unknown quantities, colours such as ydvat-tdvat, etc., should be assumed for them. As assumed by the previous teachers, they are: ydvat-tdvat (so much as), kdlaka (black), nilaka (blue), pitaka (yellow), lohitaka (red), haritaka (green), svetaka (white), citraka 1 BrSpSi, xviii. 5 1 (Com). 2 SiSe, xiv. 2. " BBi, p. 7- SYMBOLS 19 (variegated), kapilaka (tawny), pi ngalaka (reddish-brown), dhumraka (smoke-coloured), pdtalaka (pink), savalaka (spotted), sydmalaka (blackish), mecaka (dark blue), etc. Or the letters of alphabets beginning with ka, should be taken as the measures of the unknowns in order to prevent confusion." 1 The same list with a few additional names of colours appears in the algebra of Narayana. 2 This writer has further added, "Or "the letters of alphabets {yarna) suth as ka> etc., or the series of flavours such as madhura (sweet), etc., or the names of dissimilar things with un- like initial letters, are assumed (to represent the unknowns)." Bhaskara II occasionally employs also the tachygra- phic abbreviation of the names of the unknown quantities themselves in order to represent them in an equation. For example, 3 in the following 5 ma 1 m 1 mu 1 va 1 md 7 ni 1 mu 1 va 1 md 1 ni 97 mu 1 va 1 md 1 ni 1 mu z va md stands for mdnikya (ruby), ff/ for (jndra-)nila (sapphire), mu for muktdphala (pearl) and va for (sad)vajra (diamond). H< has observed in this connection thus: "(The maxim), 'colours such as jdvat-tdvat, etc., should be assumed for the unknowns,' gives (only) one method of implying (them). Here, denoting them 1 BBi, pp. 7 6f. 2 JNBi, I, R. 17-8. These verses have been quoted by Mura- lidhara Jha in his edition of the Bijaganita of Bhaskara II (p. 7, footnote 5). 8 BBi, p. 50; compare also p. 28. 20 ALGEBRA by names, the equations may be formed by the intelli- gent (calculator)." It should be noted that ydvat-tdvat is not a varna (colqur or letter of alphabet). So in its inclusion in the lists of varna, as found enumerated in the Hindu algebras — though apparently anomalous — we find the persistence of an ancient symbol which was in vogue long before the introduction of colours to represent unknowns. To avoid the anomaly Muralidhara J ha 1 has suggested the emendation ydvakastdvat {ydvaka and also; ydvaka = red) in the place of ydvat-tdvat, as found in the available manuscripts. He thinks that being misled by the old practice, the expression ydvakastdvat was confused by copyists with ydvat-tdvat. In support of this theory it may be pointed out that ydvaka is found to have been sometimes used by Prthu Jakasvami to represent the unknown. 2 Bhaskara II has once used simply ydvat* Narayana used it on several occasions. The origin of the use of names of colours to represent unknowns in algebra is very pro- bably connected with the ancient use of differently coloured shots for the purpose. 4. LAWS OF SIGNS Addition. Brahmagupta (628) says: "The sum of two positive numbers is positive, of two negative numbers is negative; of a positive and a negative number is their difference." 4 Mahavira (8jo): "In the addition of a positive and a negative number 1 See the Preface to his edition of Bhiskara's Bijaganita. *BrSpSi, xii. 15 (Com); xii. 18 (Com). » BBi, p. jo. * BrSpSi, xviii. 30. LAWS OF SIGNS - 21 (the result) is (their) difference. The addition of two positive or two negative numbers (gives) as much posi- tive or negative numbers respectively." 1 Sripati (1039): "In the addition of two negative or two positive numbers the result is their sum; the addition of a posi- tive and a negative number is their difference." 2 "The sum of two positive (numbers) is positive; of two negative (numbers) is negative; of a positive and a negative is their difference and the sign of the difference is that of the greater; of two equal positive and negative (numbers) is zero." 3 Bhaskara II (11 50): "In the addition of two negative or two positive numbers the result is their sum; the sum of a positive and a negative number is their difference." 4 Narayana (1350): "In the addition of two positive or two negative numbers the result is their sum; but of a positive and a negative number, the result is their difference; subtract- ing the smaller number from the greater, the remainder becomes of the same kind as the latter." 6 Subtraction. Brahmagupta writes: "From the greater should be subtracted the smaller; (the final result is) positive, if positive from positive, and negative, if negative from negative. If, however, the greater is subtracted from the less, that difference is reversed (in sign), negative becomes positive and positive becomes negative. When positive is to be subtracted from negative or negative from positive, 1 GSS, i. 50-1.- * SiSe, xiv. 3. 3 Stfe, iii. 28. * BBi, p. 2. 5 NBi, I, R. 8. 22 ALGEBRA tlien they must be added together." 1 Mahavira: "A positive number to be subtracted, from another number becomes negative and a negative number to be subtracted becomes positive." 2 Sripati: "A positive (number) to be subtracted becomes negative, a negative becomes positive; (the subsequent operation is) addition as explained before." 3 Bhaskara II: "A positive (number) while being subtracted be- comes negative and a negative becomes positive; then addition as explained before." 4 Narayana: "Of the subtrahend affirmation becomes negation and negation affirmation; then addition as described before." 6 Multiplication. Brahmagupta says: "The product of a positive and a negative (number) is negative; of two negatives is positive; positive mul- tiplied by "positive is positive." 6 Mahavira: "In the multiplication of two negative or two positive numbers the result is positive; but it is negative tn the case of (the multiplication of) a positive and a negative number." 7 Sripati: "On multiplying two negative or two positive 1 BrSpSi, xviii. 31-z. * GSS, i. 51. 8 J7.fr, xiv. 3. *BBi,p. 3. 5 JVJB/', I, R. 9. *BrSpSt, xviii. 33. 7 GSS, i. 50. LAWS OF SIGNS 25 numbers (the product is) positive; in the multiplication of positive and negative (the result is) negative." 1 Bhaskara II: "The product of two" positive or two negative (numbers) is positive; the product of positive and nega- tive is negative." 2 The same rule is stated by Narayana. 3 Division. Brahmagupta states: "Positive divided by positive or negative divided by negative becomes positive. But positive divided by negative is negative and negative divided by posi- tive remains negative." 4 Mahavira: "In the division of two negative or two positive numbers the quotient is positive, but it is negative in the case of (the division of) positive and negative." 5 Sripati: "On dividing negative by negative or positive by positive, (the quotient) will be positive, (but it will be) otherwise in the case of positive and negative." 6 Bhaskara II simply observes: "In the case of divi- sion also, such are the rules {i.e., as in the case of multiplication)." 7 Similarly Narayana remarks, "What have been implied in the case of multiplication of positive and negative numbers will hold also in the case of division." 8 Evolution and Involution. Brahmagupta says: • "The square of a positive or a negative number is 1 SiSe, xiv. 4.' 2 BBi, p. 3. 3 NBi, I, R. 9. * BrSpSi, xviii. 34. . b GSS, i. 50. 6 SiSe, xiv. 4. 7 BBi, p. 3. s NBi I, R. 10. 24 ALGEBRA positive. . . . The (sign of the) root is the same as was that from which the square was derived." 1 As regards the latter portion of this rule the com- mentator Prthudakasvami (860) remarks, "The square- root should be taken either negative or positive, as will be most suitable for subsequent operations to be carried on." Mahavira: "The square of a positive or of a negative number is positive: their square-roots are positive and negative respectively. Since a negative number by its own nature is not a square, it has no square-root." 2 Sripati: "The square of a positive and a negative number is positive. It will become what it was in the case of the square-root. A negative number by itself is non- square, so its square-root is unreal; so the rule (for the square-root) should be applied in the case of a positive number." 3 Bhaskara II: "The square of a positive and a negative number is positive; the square-root of a positive number is positive as well as negative. Ther.e is no square-root of a nega- tive number, because it is non-square." 4 Narayana: "The square of a positive and a negative number is positive. The square-root of a positive number will be positive and also negative. It has been proved that a negative number, being non-square, has no square- root." 5 1 BrSpSi, xviii. 35. 2 GSS, i. 52. 3 Sife, xiv. j . * BBi, p. 4. 5 NB:, I, R. 10. FUNDAMENTAL OPERATIONS 2 J 5. FUNDAMENTAL OPERATIONS Number of Operations. The number of funda- mental operations in algebra is recognised by all Hindu algebraists to be six, vi^., addition, subtraction, multi- plication, division, squaring and the extraction of the square-root. So the cubing and the extraction of the cube-root which are included amongst the fundamental operations of arithmetic, are excluded from algebra. But the formula (a -f bf = a*+ $a*b + ^ + £ 3 , or (a + bf = a 3 + ^ab{a + b) + b 3 , is found to have been given, as stated before, in almost jll the Hindu treatises on arithmetic beginning with that of Brahmagupta (628). By applying it repeatedly, Maha- vira indicates how to find the cube of an algebraic ex- pression containing more than two terms; thus (a+b + c+d+...)* = cfi + ^a\b + c + d+....)+ ia(b + ,+ sa 7 + m& \ 1 1 The eqi lation x -J- zx + 3 X $x -J- 12 x 4X — 300 is statec as 4 1 . * 1 1 3 3 1 I2 4 1 1 | 1 1 1 1 1 driya 300. This plan of writing an equation was subsequently abandoned by the Hindus for a new one in which the two sides are written one below the other without any 1 NBi, II, R. 3 {Gloss). 3 Datta, Bakh. Math., (BCMS, XXI), p. 28. 3 Folio 59, recto. * Folio 23, verso. EQUATIONS 3 1 sign of equality. Further, in this new plan, the terms of similar denominations are usually written one below the other and even the terms of absent denominations on either side are expressly indicated by putting zeros as their coefficients. Reference to the new plan is found as early as the algebra of Brahmagupta (628). 1 Prthudaka- svami (860) represented the equation 2 io.v — 8 = .v 2 + 1 as follows: yd va o yd 10 ru 8 yd va 1 yd o ru 1 which means, writing x iotyd „v 2 .o + a-.io — 8 x 2 + x.o + 1 or ox- 2 + iox — 8 = x' 2 + ox -+- 1. If there be several unknowns, those of the same kind are written in the same column with zero coefficients, if necessary. Thus the equation 197.V — 1644J — £ = 6302 is represented by PrthudakasvamI thus: 3 yd 197 kd 1644 ni i ru o yd o kd o tii o ru 6302 which means, putting y for kd and £ for m, 197.* — 1644J — ^+o = ojc+qy+o^+ 6302. The following two instances are from the Bija- ganita of Bhaskara JI (1150): 4 (1) yd j kd 8 ni 7 ru 90 yd 7 kd 9 ni 6 ru 62 r BrSpSi, xvii. 43 (vide infra, p. 33). Compare also BBi, p. 127. 2 BrSpSi, xviii. 49 (Com). 3 BrSpSi, xviii. 54 (Com). *BBi, pp. 78, 101. 34 ALGEBRA been thus made..." 1 Sripati says: "From one (side) the square of the unknown and the unknown should be cleared by removing the known quantities; the known quantities (should be cleared) from the side opposite to that." 2 Similarly Bhaskara II: "Then the unknown on one side of it (the equation) should be subtracted from the unknown on the other side; so also the square and other powers of the un- known; the known quantities on the other side should be subtracted from the known quantities of another (i.e., the former) side." 3 Here we give a few illustrations. With reference to the equations from the commentary of Prthudaka- svami, stated on page 31, the author says: "Perfect clearance (samasodhand) being made in accordance with the' rule, (the equation) will be yd va 1 yd 10 ru 9 i.e., x 2 — iox = — 9. The following illustration is from the Bijaganita of Bhiskara II: 4 "Thus the two sides are jd va 4 yd 34 ru 72 yd va o yd o ru 90 On complete clearance (samasodhand), the residues of the two sides are 1 BrSpSi, xviii. 44 (Com). 2 SiSe, xvi. 17. »BB/,p. 44- «BB»,p. 6j. EQUATIONS 3 J ydva 4 yd 34 ru o jdva o yd o ru 18" /.*., 4V 2 — 34X =18. Classification of Equations. The earliest Hindu classification of equations seems to have been according to their degrees, such as simple (technically called ydvat- tdvat) i quadratic (yarga\ cubic (gbana) and biquadratic (yarga-vargd). Reference to it is found in a canonical work of circa 300 B. C. 1 But in the absence of further corroborative evidence, we cannot be sure of it. Brahma- gupta (628) has classified equations as: (1) equations in one unknown [eka-varna-samtkarand), (2) equations in several unknowns {aneka-varna-samikarand), and (3) equations involving products of unknowns (bhdvita). The first class is again divided into two subclasses, vi^., it) linear equations, and («) quadratic equations {avyakta- varga-samikarana). Here then we have the beginning of out present method of classifying equations according to their degrees. The method of classification adopted by Prthudakasvami (860) is slightly different. His four classes are: (1) linear equations with one unknown, (2) linear equations with more unknowns, (3) equations with one, two or more unknowns in their second and higher powers, and (4) equations involving products of un- knowns. As the method of solution of an equation of the third class is based upon the principle of the elimina- tion of the middle term, that class is called by the name madhyamdharam (from madhyama, "middle", dharana "elimination", hence meaning "elimination of the mid- dle term"). For the other classes, the old names given by Brahmagupta have been retained. This method of classification has been followed by subsequent writers. 1 Sthdndnga-sutra, Sutra 747. For further particulars see Datta, Jaina Math., (BCMS, XXI), pp. 11 9ft". J 6 ALGEBRA Bhaskara II distinguishes two types in the third class, vi%., (i) equations in one unknown in its second and higher powers and (ii) equations having two or more unknowns in their second and higher powers. Accord- ing to Krsna (i 5 80) equations are primarily of two classes: (1) equations in one unknown and (2) equations in two or more unknowns. The class (1), again, com- prises two subclasses: (/) simple equations and (//) quadratic and higher equations. The class (2) has three subclasses: (/) simultaneous linear equations, (ii) equa- tions involving the second and higher powers of un- knowns, and (Hi) equations involving products of un- knowns. He then observes that these five classes can be reduced to four by including the second subclasses of classes (1) and (2) into one class as madhyamdharana. 7. LINEAR EQUATIONS IN ONE UNKNOWN Early Solutions. As already stated, the geometrical solution of a linear equation in one unknown is found in the Sulba, the earliest of which is not later than 800 B.C. There is a reference in the Sthdndnga-sutra (c. 300 B.C.) to a linear equation by its name (ydvat-tdvaf) which is suggestive of the method of solution 1 followed at ■that time. We have, however, no further evidence about it. The earliest Hindu record of doubtless value of problems involving simple algebraic equations and of a method for their solution occurs in the Bakhshali treatise, which was written very probably about the beginning of the Christian Era. One problem is: 2 "The amount given to the first is not known. The second is given twice as much as the first ; the third 1 Datta, Jaifia Math., (BCMS, XXI), p. 122. 2 BMs, Folio 23, recto. LINEAR EQUATIONS IN ONE UNKNOWN 37 thrice as much as the second ; and the fourth foar times as much as the third. The total amount distributed is 132. What is the amount of the first ?" If x be the amount given to the first, then according to the probelm, x + zx -\- 6x + 2 ix + 9J = ioi. Or, in general, ax-\-by=m, bx + ay = n. Solution. "From the larger amount of price multiplied by the (corresponding) bigger number of things sub- tract the smaller amount of price multiplied by the (corresponding) smaller number of things. (The re- mainder) divided by the difference of the squares of the numbers of things will, be the price of each of the bigger number of things. The price of the other will be obtained by reversing the multipliers." 1 ^,, _ a m — bn _ an — bm Thus x — ^ 2 _ £2 ' J — a 2 _ £2 • Example. "A wizard having powers of mystic incantations and magical medicines seeing a cock-fight going on, spoke privately to both the owners of the cocks. To one he said, 'if your bird wins, then you give me your stake-money, but if you do not win, I shall give you two-thirds of that.' Going to the other, he promised in the same way to give three-fourths. From both of them his gain would be only 12 gold pieces. Tell me, O ornament of the first-rate mathematicians, the stake- money of each of the cock-owners." 2 i.e., x — ■ *j = l Z> y_2 x =lZ Or, in general, x — c a A 7 J=P> J~ T x=P> 1 GSS, vi. 1 3 9^. 2 GSS, vi. 270-2^. 46 ALGEBRA Solution'?- (c+ d)b - {a + b)c Pi y= '("+*) p The following example with its solution is taken from the Bijaganita of Bhaskara II : Hxample. "One says, 'Give me a hundred, friend, I shall then become twice as rich as you.' The other replies, 'If you give me ten, I shall be six times as rich as you.' Tell me what is the amount of their (res- pective) capitals ?" 2 The equations are x + 100 = z(j — 100), (1) y -J- 10 = 6(x — 10). (2) Bhaskara II indicates two methods of solving these equations. They are substantially as follows : First Method. 3 Assume x = 2% — 100, j = ^ + 100, so that equation (1) is identically satisfied. Substituting these values in the other equation, we get £ -J- 110= i2£ — 660; whence ^ =.70. Therefore, x = 40, j = 170. Second Method.* From equation (1), we get x = zy — 300, and from equation (2) x = rXj + 7°)- 1 GSS, vi. 268J-9J. i BBi, p. 41. 3 BBi, p. 46. 4 BBi, pp. 78f. LINEAR EQUATIONS WITH SEVERAL UNKNOWNS 47 Equating these two values of x, we have zy— 300^ \{j+ 70), or izjy — 1800 =jy -)- 70; whence j— 170. Substituting this value of y in any of the two expressions for x, we get x = 40. 1 1 is noteworthy that the second method of solution of the problem under consideration is described by Bhaskara II in the section of his algebra dealing with "linear equations with several unknowns," while the first method in that dealing with "linear equations in one unknown." In this latter connection he has observed that the solution of a problem containing two unknowns can sometimes be made by ingenious artifices to depend upon the solution of a simple linear equation. 9. LINEAR EQUATIONS WITH SEVERAL UNKNOWNS A Type of Linear Equations. The earliest Hindu treatment of systems of linear equations involving " several unknowns is found in the Bakhshall treatise. One problem in it runs as follows : "[Three persons possess a certain amount of riches each.] The riches of the first and the second taken together amount to 13 ; the riches of the second and the third taken together are 14 ; and the riches of the .fiist and the third mixed are known to be 15. Tell me the riches of each." 1 If x v x 2 , x 3 be the wealths of the three merchants respectively, then ■*i + *2 = *3> X 2 + *3 = x 4> *s + x i = T 5- ( J ) Another problem is 1 BMs, Folio 29, recto. The portions within f ] in this and the following illustration have been restored. 48 ALGEBRA "[Five persons possess a certain amount of riches each. The riches of the first] and the second mixed together amount to 16 ; the riches of the second and the third taken together are known to be 17 ; the riches of the third and the fourth taken together are known to be 18 ; the riches of the fourth and the fifth mixed together are 19 ; and the riches of the first and the fifth together amount to 20. Tell me what is the amount of each." 1 i.e., x 1 + x 2 = 16, x 2 + * 3 = 17, * 3 + x 4 == 18, *4 + X 5 = X 9> *5 + *1 = 2 °- ( 2 ) There are in the work a few other similar pro- blems. 2 Every one of them belongs to a system of linear equations of the type x l ~\~ x i = a \> x 2 ~H X 3 ~ a 2> '••> ' X n ~H X l = a n> W n being odd. Solution by False Position. A system of linear equations of this type is solved in the Bakhshali treatise substantially as follows : Assume an arbitrary value p for x x and then calculate the values of x 2 , x 3 , ... corresponding to it. Finally let the calculated value of x n + *i be equal to b (say). Then the true value of x x is obtained by the formula X l=P+U a n~ b )- In the particular case (1) the author 3 assumes the arbitrary value 5 for x-,; then are successively calculated the values x\ = 8, x' z = 6 and x\ + x\= 11. The correct values are, therefore, *i = 5 + (15 — 11)/ 2 = 7, x 2 = 6, x a = 8. 1 BMs, Folios 27 and 29, verso. 2 BMs, Folio 30, recto ; also see Kaye's Introduction, p. 40. 3 BMs, Folio, 29, recto. LINEAR EQUATIONS WITH SEVERAL UNKNOWNS 49 Rationale. By the process of elimination we get from equations (I) (a 2 — aj+fa— a z )+ +(*„-i— *»-a)+2*i = r stands for x x + x 2 + . . . + •*"„• But it will not be proper to say that equations of this, type have been treated in the Bakhshali treatise. 1 They have, however, been solved by Aryabhata (499) and Mahavlra (8 jo). The former says: "The (given) sums of certain (unknown) numbers, leaving out one number in succession, are added to- gether separately and divided by the number of terms less one; that (quotient) will be the value of the whole." 2 n i.e., sx=2 a T \{n — 1). r=l Mahavlra states the solution thus: "The stated amounts of- the commodities added to- gether should be divided by the number of men less 1 The example cited by Kaye (BMs, Introd., p. 40, Ex. vi) which conforms to this type of equations is based upon a mis- apprehension of the text. 2 A, ii. 29. 50 ALGEBRA one. The quotient will be the total value (of all the com- modities). Each of the stated amounts being subtract- ed from that, (the value) in the hands (of each will be found)." 1 In formulating his rule Mahavira had in view the following example: "Four merchants were each asked separately by the customs officer about the total value of their com- modities. The first merchant, leaving out his own invest- ment, stated it to be 22; the second stated it to be 23, the third 24 and the fourth 27; each of them deducted his own amount in the investment. O friend, tell me separately the value of (the share of) the commodity owned by each." 2 tr 111 22+234-24+27 Here x x + x 2 + ,v 3 + x 4 = ^ ] T =3^> 4—1 Therefore Xj = 10, a- 2 = 9, x 3 = 8, x 4 = 5. Narayana says: "The sum of the depleted amounts divided by the number of persons less one, is the total amount. On subtracting from it the stated amounts severally will be found the different amounts." 3 The above type of equations is supposed by some modern historians of mathematics 4 to be a modification of the type considered by the Greek Thymaridas and solved by his well known rule HLpanthema, namely, 5 * ■+ X l + - V 2 + + - V «-l = A x +*■, =a lt x + a- 2 = a 2t . . ., x + *■„_! = a n _ v 1 GSS, vi,. 159. 2 GSS, vi. 160-2. 3 GK, ii. 28. 4 Cantor, Vorksimgen iiber Geschhhte der Mathematik (referred to hereafter as Cantor, Geschhhte), I, p. 624; Kaye, Ind. Math., p. 13 ; JASB, 1908, p. 135. 5 Heath, Greek Math., I, p. 94. LINEAR EQUATIONS WITH SEVERAL UNKNOWNS J I The solution given is _ (m = /> 2 (2 x — x a — 2 'tf 2 .m)> x„ + 2 ^„ >m = £ n (2 x — A- n — 2 '*„,„), where 2 '#,.,„, denotes summation from m = i x.o m = n excluding «? = r. Therefore > 2 x + & + i)S '* ltfl , = (b x + i) (2 x — xj, 2^+(U i)S 'a z , m = (b 2 + i) (2 x — xa), Let k T = (i>, + i)2 '**» > r = i, 2, 5,. . : , «. 1 This is the name given by Mahavira to problems involving equations of type (III). *GSS t vi. 253J-5J. LINEAR EQUATIONS WITH SEVERAL UNKNOWNS 53 Then dividing the foregoing equations by i^+i, £ a + 1, .... , respectively, and adding together, we get k x k A k, X. Whence V = r -^ + Ml , k r + ^2 , , K + Mr-1 + i±^ f + ... + i i ±M,_ ( _ 2)4 | Mahavira describes the solution thus: i "The sum of the amounts begged by each person is multiplied by the multiple number relating to him as increased by unity. With- these (products), the amounts at hand are determined according to the rule Istaguna- ghna etc. 1 They are reduced to a common denominator, and then divided by the sum diminished, by unity of the multiple numbers 1 divided by themselves as increased by unity. (The quotients) should be known to be the amounts in the hands of the persons." 2 Problems of Oie above kind have been treated also by Narayana (1357). He says: ■* * 1 Ine reference is to rule vi. 241. 2 GSS, vi. 251^-2.52.^. J4 ALGEBRA "Multiply the sum of the monies received by each person by his multiple number plus unity. Then pro- ceed as in the method for "the purse of discord." Divide the multiple number related to each by the same as increased by unity. By the sum diminished by unity of these quotients, divide the results just obtained. The quotients will be the several amounts in their pos- session." 1 One particular case, where b 1 = b 2 = . . . . b n = i and c 1 = c 2 = . . . = c n =■■ c, was treated by Brahmagupta (628). He gave the rule: "The total value (of the unknown quantities) plus or minus the individual values (of the unknowns) multiplied by an optional number being severally (given), the sum (of the given quantities) divided by the number of unknowns increased or decreased by the mul- tiplier will be the total value; thence the rest (can be determined)." 2 2* ± cx 1 = a l3 S'a- ± cx 2 = a 2 , ,Sx±a tt = a n . Therefore ax =-■ '* + '*+ "' + '» • n ± c Hence x x - L ( ± a, T «i + «» + ■•• + ',. ); and so on. Brahmagupta's Rule. Brahmagupta (628) states th& following rule for solving linear equations involving several unknowns: "Removing the other unknowns from (the side of) the first unknown and dividing by the coefficient of the first unknown, the value of the first unknown (is obtain- ed). In the case of more (values of the first unknown), 1 GK, ii. 33f. 2 BrSpSi, xiii. 47. LINEAR EQUATIONS WITH SEVERAL UNKNOWNS 55 two and two (of them) should be considered after re- ducing them to common denominators. And (so on) repeatedly. If more unknowns remain (in the final equation), the method of the pulveriser (should be employed). (Then proceeding) reversely (the values of other unknowns can be found)." 1 Prthudakasvami (860) has explained it thus: ' "In an example in which there are two or more unknown quantities, colours such as ydvat-tdvat,. etc., should be assumed for their values. Upon them should be performed all operations conformably to the state- ment of the example and thus should be carefully framed '.wo or more sides and also equations. Equi-clearance should be made first, between two and two of them and so on to the last: from one side one unknown should be cleared, other unknowns reduced to a common denomi- nator and also the absolute numbers should be cleared from the side opposite. The residue of other unknowns being divided by the residual coefficient of the first un- known will give the value of the first unknown. If there be obtained several such values, then with two and two of them, equations should be formed after reduction to common denominators. Proceeding in this way to the end find out the value of one unknown. If that value be (in terms of) another unknown then the coefficients of those two will be reciprocally the values of the two unknowns. If, however, there be present more unknowns in that value, the method of the pulveriser should be employed. Arbitrary values may then be assumed for some of the unknowns." It will be noted that the above rule embraces the indeterminate as well as the determinate equations. In fact, all the examples given by Brahmagupta in illustra- 1 BrSpSi, xviii. 51. J 6 ALGEBRA tion of the rule are of indeterminate character. We shall mention some of them subsequently at their proper places. So far as the determinate simultaneous equations are concerned, Brahmagupta's method for solving them will be easily recognised to be the same as our present one. Mahavira's Rules. Certain interest problems treated by Mahavira lead to simple simultaneous equa- tions involving several unknowns. In these problems . certain capital a,mounts (c^ c 2 , c s ,. . . ) are stated to have been lent out at the same rate of interest (r) for different periods of time (f u t 2 , / 3 ,. . . ). If (i\, i 2 , i z ,. . . ) be the interests accrued on the several capitals, 'l = IOO ' 'a : _ rt 2 c 2 IOO ' '3 _ r h c z IOO > * • (i) If h + h + 4 + • . . . = I, C T and t r be known (for r : = 1. 2, • • ), we » have i > with similar values for i 2 , l 'z>- • • • (ii) Or, if c x + c 2 + c z + . . . = C, i r and t r be known (for r = i, 2,. . . ), we have and so on. -i — hlh + h(h + 5 (iii) Or, if we are given the sum of the periods /j + / 2 + ■■ T, c r and i r , then » h\ c \ + '2A2 + with similar values for f 2 , / 3 ,. . . . Mahavira has given separate rules for the. solution LINEAR EQUATIONS WITH SEVERAL UNKNOWNS J 7 of problems of each of the above three kinds. 1 Bhaskara's Rule. Bhaskara II has given practically the same rule as that of Brahmagupta for the solution of simultaneous linear equations involving several un- knowns. We take the following illustrations from his works. 'Example 1. "Eight rubies, ten emeralds and a hundred pearls which are in thy ear-ring were purchased by me for thee at an equal amount; the sum of the-price rates of the three sorts of gems is three less than the half of a hundred. Tell me, O dear auspicious lady, if thou be skilled in mathematics, the price of each." 2 If x,y, £ be the prices of a ruby, emerald and pearl respectively, then 8a- = icy = ioo£, x +J +Z = 47- Assuming the equal amount to be n>, says Bhaskara II, we shall get , x = »/8, y = rvjio, £ = a'/ioo. Substituting in the remaining equation, we easily get rv = 200. Therefore -v = 25, y = 20, £ = 2. Example 2. "Tell the three numbers which become equal when added with their half, one-fifth and one- ninth parts, and each of which, when diminished by those parts of the other two, leaves sixty as remainder." 3 Here we have the equations x -\- xjz =j + J//5 = z + */9. 0) y Z Z x x J r f \ X — — i- = y i = ? *- = 60. (2) S 9 9 2 2 5 1 GSS, vi. 37, 39, 42- 2 BBJ, p. 47. 3 BBi, p. 52. 60 ALGEBRA The geometrical solution of the simple quadratic equation 4& 2 — 4dh = — c 2 is found in the early canonical works of the Jainas (500- 300 B. C.) and also in the Tattvdrthddhigama-sutra of Umasvati (} 2 =64; 8jy = 24o; 8^=960; {z(S—s) + b} 2 4- 8JV£ = 1024; V1024 — 32; 324-8 = 40; 40 —- 8 = 5 = AT. For another problem 3 J - = 7, ^ = 5, .r = 5, b = y, then ' ^T+ysS? 6 The formula for determining the number of terms («) of an A.P. whose first term (a), common difference (b) and sum (s) ate known, is stated in the form V8&T 4-' (za — bf — (za — b) n = L — ^ f- * -. zb The working of the particular example in which s = 60. j + {za — by — {za — b) "~~ zb For the solution of the quadratic Brahmagupta uses also a third formula which is similar to the one now commonly used. Though it has not been expressly des- cribed in any rule, we find its application in a few 1 BrSpSi, xviii. 44. It will be noted that in this rule Brahma- gupta has employed the term madbya (middle) to imply the simple unknown as well as its coefficient. The original of the term is doubtless connected with the mode of writing the quadratic equation in the form ax 2 -f- bx-\- o = ox 2 + ox -f- c, so that there are three terms on each side of the equation. 2 BrSpSi, xviii. 45. 3 BrSpSi, xii. 18. 64 ALGEBRA instances. One of them is an interest problem: A certain sum (/>) is lent out for a period (/ x ); the interest accrued (x) is lent out again at this rate of interest for another period (/ 2 ) and the total amount is A. Find x. The equation for determining x is ~ a -^e a -f x = A. Ph Hence, we have x = \ !(Ph\ 2 1 A Ph Ph, which is exactly the form in which Bxahmagupta states the result. 1 There is a certain astronomical problem which in- volves the quadratic equation 2 (72 + a 2 )x 2 ^ 2^apx = H4(~ P 2 ), where a = agrd (the sine of the amplitude of the sun), b = palabhd (the equinoctial shadow of a gnomon 12 anguli long), JR. = radius, and x = konasaiiku (the sine of the altitude of the sun when his altitude is 45 °). Dividing out by (72 + <2 2 )> we have x 2 ^f zmx = n, where -._ **"P - - M4(R 2 /a - p*) 72 -j- a* 72 + * Therefore, we have x = y m 2 + tt ± m, as stated by Brahmagupta. This result is given also in the Sftryasiddhanta 2, (c. 300) and by Sripati (1039). 4 1 BrSpSi, xii. 15. Vide Part I, p. 220. 2 BrSpSi, iii. 54-55. 3 SuSi, iii. 30-1. * SiSe, iv. 74. QUADRATIC EQUATIONS 65 Sridhara's Rule. Sridhara (c. 750) expressly indicates his method of solving the quadratic equation. His treatise on algebra is now lost. But the relevant portion of it is preserved in quotations by Bhaskara II 1 and others. 2 Sridhara's method is : . "Multiply both the sides (of an equation) by a known quantity equal to four times the coefficient of the square of the unknown; add to both sides a known quantity equal to the square of the (original) coefficient of the unknown: then (extract) the root." 3 That is, to solve ax 2 + bx = ?> we have 4a 2 x 2 + 4abx — 4ac> or {iax + b) 2 = ^ac + b 2 . Therefore zax + b = ~\J a^c + b 2 . Hence Jf= V4^+^-^ _ za An application of this rule is found in Sridhara's Trisatikd, in connection with finding the number of terms of an A. P. 4 y/%bs+ (za — b) 2 — za+b i.e., n = L -^ r— ' ! — , zb l BBi, p. 61. 2 Jnanaraja (1503^ in his Brjaganita and Suryadasa (1J4 1 ) ' v n ' s commentary on Bhaskara's Bijaganita. 3 "Caturahatavargasamai rupaih paksadvayarh gunayet, Avyaktavargarupairyuktaii paksau tato mularn." This is the reading of Sridhara's rule as stated by Jnanaraja and Suryadasa and accepted also by Sudhakara Dvivedi. But according to the reading of Krsna (tr. 1580) and Ramakrsna (c. 1648), which has been accepted by Colebrooke, the second line of the verse should be "Purvavyaktasya krteh samarupani ksipet tayoreva" or "add to them known quantities equal to the square of the original coefficient of the unknown." 4 Tr/'s, R. 41. 66 ALGEBRA where a is the first term, b the common difference and s the sum of n terms. Mahavira. The only work of Mahavira (850) which is available now is the Ganita-sdra-samgraha. As it is admittedly devoted to arithmetic we cannot expect to find in it a rule for solving the quadratic. But there are in it several problems whose solutions presuppose a knowledge of the roots of the quadratic. One problem is as follows: "One-fourth of a herd of camels was seen in the forest; twice the square-root of the herd had gone to the mountain-slopes; three times five camels were on the bank of the river. What was the number of those camels P" 1 If x be the number of camels in the herd, then \x + i^/x + 1 5 = x. Or, in general, the equation to be solved is -r- x -\- cy/x -\- d — x, or ( 1 j~) x — ^V^* — d. Mahavira gives the following rule for the solution of this equation: "Half the coefficient of the root (of the unknown) and the absolute term should be divided by unity minus the fraction {i.e., the coefficient of the unknown). The square-root of the sum of the square of the coeffi- cient of the root (of the unknown) and the absolute term (treated as before) is added to the coefficient of the root (of the unknown treated as before). The sum squared is the (unknown) quantity in this mula type of problems." 2 1 GSS, iv. 34. 2 GSS, iv. J3. QUADRATIC EQUATIONS 67 I.e., X \i-a/l>^ V ^ 1 - a\b> ^ 1 - *//> I ' which shows that Mahavira employed the modern rule for finding the root of a quadratic. His solution for the interest problem treated by Brahmagupta is exactly the same as that of the latter. 1 We shall presently show that he knew that the quadratic has two roots, Aryabhata II. The formula for the number of terms («) of an A. P. whose first term (a), common difference (/;) and sum (s) are known is given by Aryabhata II (c. 950) as follows : 2 __ y/ibs + Q — bjzf — a + bjz n ~ b which shows that for solving the quadratic he followed the second method of Aryabhata I and Brahmagupta. Sripati's Rules. Sripati (1039) indicates two methods of solving the quadratic. There is a lacuna in our manuscript in the rule describing the first method, but it can be easily recognised to be the same as that of Sridhara. "Multiply by four times the coefficient of the square of the unknown and add the square of the coefficient of the unknown; (then extract) the square-root c'.ivided by twice the coefficient of the square of the unknown, is said to be (the value of) the'unknown." "Or multiplying by the coefficient of the square of the unknown and adding the square of half the coeffi- cient of the unknown, (extract) the square-root. Then (proceeding) as before, it is diminished by half the coefficient of the unknown and divided by the coefficient 1 GSS, vi. 44. 2 MSi, xv. 50. 68 ALGEBRA o£ the square of the unknown. This (quotient) is said to be (the value of) the unknown." 1 i.e., ■ ax 2 -\- bx = c, or a 2 x 2 + abx + {bfzf = ae + (bfz) 2 . Therefore ax -\- bjz = V ' ac -\- (b/z) 2 . Hence x = V^TW) 2 ~ &A. a Bhaskara IPs Rules. Bhaskara' II (1150) says: "When the square of the unknown, etc., remain, then, multiplying the two sides (of the equation) by some suitable quantities, other suitable quantities should be added to them so that the side containing the unknown becomes capable of yielding a root (pada-prada). The equation should then be formed again with the root of this side and the root of the known side. Thus the value of the unknown is obtained from that equation." 2 This rule has been further elucidated by the author in his gloss as follows : . "When after perfect clearance of the two sides, there remain on one side the square, etc., of the un- known and on the other side the absolute term only, then, both the sides should be multiplied or divided by some suitable optional quantity; some equal quantities should further be added to or subtracted from both the sides so that the unknown side will become capable of yielding a root. The root of that side must be equal to the root of the absolute terms on the other side\ For, by simultaneous equal additions, etc., to the two equal sides the equality remains. So forming an equa- tion again with these roots the value of the unknown is found." 3 1 Si$e, xiv. 1 7-8, 19. 2 BBi, p, 5 9. 3 BBi, p. 61. QUADRATIC EQUATIONS 69 It may be noted that in his treatise on arithmetic Bhaskara II has always followed the modern method of dividing by the coefficient of the square of the un- known. 1 Jnanaraja (1503) and Ganesa (1545) describe the same general methods for solving the quadratic as Bhaskara II. Elimination of the Middle Term. The method of solving the quadratic was known amongst the Hindu algebraists by the technical designation madhyamd- harana or "The Elimination of the Middle" (from madhyama = middle and dharana = removal, or destroy- ing, that is, elimination). The origin of the name will be easily found in the principle underlying the method. By it a quadratic equation which, in its general form, contains three terms and so has a middle term, is reduced to a pure quadratic equation or a simple equation involving only two terms and so having no middle term. Thus the middle term of the original quadratic is eliminated by the method generally adopted for its solution. And hence the name. Bhaskara II has observed, "It is also specially designated by the learned teachers as the madhyamdharana. For by it, the removal of one of the two 2 terms of the quadratic, the middle one, (takes place)." 3 The name is, however, employed also in an extended sense so as to embrace the methods for solving the cubic and the biquadratic, where also 1 L, pp. i 5 fF. 2 Referring to the two terms on the unknown side of the com- plete quadratic. Or the text varga-rdsdvekasya may be rendered as "of one out of the unknown quantity and its square." According to the commentators Suryadasa (1541) and Krsna (1580), it implies "of one between the two square terms," «'•<;., the square of the unknown and the square of the absolute number. 3J3£;,p. 59. 70 ALGEBRA certain terms are eliminated. It occurs as early as the works of Brahmagupta (628). 1 Two Roots of the Quadratic. The Hindus recog- nised early that the quadratic has generally two roots. In this connection Bhaskara II has quoted the following rule from an ancient writer of the name of Padmanabha whose treatise on algebra is not available now: "If (after extracting roots) the square-root of the absolute side (of the quadratic) be less than the negative absolute term on the other side, then taking it negative as well as positive, two values (of the unknown) are found." 2 Bhaskara points out with the help of a few specific illustrations that though these double roots of the quadratic are theoretically correct, they sometimes lead to incongruity and hence should not always be accepted. So he modifies the rule as follows: "If the square-root of the known side (of the quadratic) be less than the negative absolute term occurring in the square-root of the unknown side, then making it negative as well as positive, two values of the unknown should be determined. This is (to be done) occasionally." 3 Example 1 . "The eighth part of a troop of monkeys, squared, was skipping inside the forest, being delight- fully attached to it. Twelve were seen on the hill delighting in screaming and rescreaming. How many were they ?" 4 1 BrSpSi, xviii. 2. 2 "Vyaktapaksasya cenmulamanyapaksarnarupatah Alparh dhanarnagarh krtva dvividhotpadyate mitih" — BBi, p. 67. 3 BBi, p. 59; also compare the author's gloss on the same (P- 61). *BBi, p. 6y QUADRATIC EQUATIONS 71 Solution. "Here the troop of monkeys is x. The square of the eighth part of this together with 12, is equal to the troop. So the two sides are 1 ■%?x 2 + ox + 12 = ox 2 + x -f- o. Reducing these to a common denominator and then deleting the denominator, and also making clearance, the two sides become x 2 — 64X + o = ox 2 -fox — 768. Adding the square of 32 to both sides and (extracting) square-roots, we get x — 32 = ± (px + T 6). In this instance the absolute term on the known side is smaller than the negative absolute term on the side of the unknown; hence it is taken positive as well as negative; the two values of x are found to be 48, 16." 'Example 2. "The fifth part of a troop of monkeys, leaving out three, squared, has entered a cave; one is' seen to have climbed on the branch of a tree. Tell how many are they ?" 2 Solution. "In this the value of the troop is x ; its fifth part less three is \ x — 3; squared, fa x 2 — |x -f- 9; this added with the visible (number of monkeys), -fa x 2 — f x -f- 10, is equal to the troop. Reducing to a common denominator, then deleting the denominator and making clearance, the two sides become x 2 — 5 5X + o = ox 2 -f- ox — 256. Multiplying these by 4, adding the square of 55, and 1 We have here followed the modern practice of writing the two sides of an equation in a line with the sign of equality inter- posed, at the same time, preserving the other salient feature of the Hindu method of indicating the absent terms, if any, by putting zeros as their coefficients. 2 BBi, pp. 6sff. 72 ALGEBRA extracting roots, we get 2-x - — 5 5 = ±(«+ 45)- In this case also, as in the previous, two values are obtained: 50, 5. But, in this case, the second (value) should not be accepted as it is inapplicable. People have no faith in the known becoming negative." The implied significance of this last observation is this : If the troop consists of only 5 monkeys, its fifth part will be 1 monkey. How can we then leave out 3 monkeys ? Again, how can the remainder be said to have entered the cave ? It seems to have also a wider significance. Example 3. "The shadow of a gnomon of twelve fingers being diminished by a third part of the hypotenuse, becomes equal to fourteen fingers. O mathematician, tell it quickly." 1 Solution. "Here the shadow is (taken to be) x. This being diminished by a third part of the hypotenuse becomes equal to fourteen fingers. Hence conversely, fourteen, being subtracted from it, the remainder, a third of the hypotenuse, is x — 14. Thrice this, which is the hypotenuse, is 3X — 42. The square of it, yx 2 — 252JV+ 1764, is equal to the square of the hypotenuse, x 2 -f- 144. On making equi-clearance, the two sides become 8x 2 — 252X + o = ox 2 -\~'ox — 1620. Multiplying both these sides by 2 and adding the square of 63, the roots are 4 x — 63 = ± (ox + 47)- On forming an equation with these sides again, and (proceeding) as before, the values of x are 45/2, 9. 1 BBi, pp. 6 then the equation becomes Therefore y — 60 or -*/• Hence x — 9 \ / — = 60, whence ;*• == 150, 24. Again .*• — 9 *\ / 2 * _% 8" 7 _ " s "' whence x = |(6i ^ 31/385). Of the four values of x obtained above, only the value x = 150 can satisfy all the conditions of the problem; others are inapplicable. That will explain why Maha- vira has retained in his solution only the positive sign of the radical. (2) "Four times the square-root of the half of a col- lection of boars went into a forest where tigers were at play; twice the square-root of the tenth part of the remainder multiplied by 4 went to a mountain; 9 times the square-root of half the remainder went to the bank of a river; boars numbering seven times eight were seen in the forest. Tell their number." 2 1 CSS, iv. 54-5. 2 CSS, iv. 56. SIMULTANEOUS QUADRATIC EQUATIONS 8 1 If x be the total number of boats in the collection, + 9\AU* - 4V*7*) - 8V-^(jf-4Vx/2)} 4- 56 = x. Put y = x — 4V 'xjz ; then J — 8 V j/10 — 9V(j — 8 V j/io)/z = 56. Again put ^ =y — 8 V j/io ; then ^- 9 V^ = 56. Therefore £ — ( y v !— r — j X 3 = 128. Then y — sVjJio = 128; >- 1 / 8 4- y 64 4- 10.4.128x2 , ., whence y = { — v ) X^^ 160. Again x — 4\^xjz = 160; hence x = ( 4+Vit>+ 4.2.160 ^ x ^ Note that according to the problem the positive value of the radical has always to be taken. 12. SIMULTANEOUS QUADRATIC EQUATIONS Common Forms. Various problems involving .imultaneous quadratic equations of the following forms have been treated by Hindu writers : *~4z #-» -+£:;}...<*) x 2 -j-y 2 = c xy = b H<«) "IYjZIU") 82 ALGEBRA For the solution of (/) Aryabhata I (499) states the following rule : "The square-root of four times the product (of two quantities) added with the square of their difference, being added and diminished by their difference and halved gives the two multiplicands." 1 i.e., x = 4{V^ + 4£ + d), j = y^W+ljb - d). Brahmagupta (628) says : "The square-root of the sum of the square of the difference of the residues and two squared times the product of the residues, being added and subtracted by the difference of the residues, and halved (gives) the desired residues severally." 2 Narayana (1357) writes : "The square-root of the square of the difference of two quantities plus four times their product is their sum." 3 "The square of the difference of the quantities to- gether with twice their product is equal to the sum of their squares. The square-root of this result plus twice the product is the sum." 4 For the solution of (it) the following rule is given by Mahavira (850) : "Subtract four times the area (of a rectangle) from the square of the semi-perimeter; then by saiikramancP between the square-root of that (remainder) and the semi-perimeter, the base and the upright are obtained." 6 1 A, ii. 24. 2 BrSpSi, xviii. 99. 3 GK, i. 35. *GK,L 36. 6 Given a and b, the process of sankramana is the finding of half their sum and difference, i.e., and (see pp. 431). S GSS, vii. 129J. SIMULTANEOUS QUADRATIC EQUATIONS 83 ^ i.e., x = \{a + V <* 2 — 4t?), y = \{a — \?a 2 — 4b). Narayana says : "The square-root of the square of the sum minus four times the .product is the difference." 1 For {iii) Mahavlra gives the rule : "Add to and subtract twice the area (of a rectangle) from the square of the diagonal and extract the square- roots. By sarikramana between the greater and lesser of these (roots), the side and upright (are found)." 2 i.e., x = £(V* + *b + V* — zb), y = \{yj c -f- zb — VV— zb). For equations (iv) Aryabhata I writes : "From the square of the sum (of two quantities) subtract the sum of their squares. Half of the remain- der is their product." 3 The remaining operations will be similar to those for the equations (/'/); so that . x = \(a +\'zc — a 2 ), y = l(a — V ' ze — a 2 -). Brahmagupta says : "Subtract the square of the sum from twice the sum of the squares ; the square-root of the remainder being added to and subtracted from the sum and halved, (gives) the desired residues." 4 Mahavlra, 5 Bhaskara II 6 and Narayana 7 have also treated these equations. Narayana has given two other forms of simul- l GK, i. 35.- a GSS, vii. 127J. 3 A, ii. 23. « BrSpSi, xviii. 98. B GSS, vii. 125 £. fl L, p. 39. 'GK,i. i7 . 84 ALGEBRA taneous quadratic equations, namely, For the solution of (y) he gives the rule : "The square-root of twice the sum of the squares decreased by the square of the difference is equal to the sum."* - i.e., x -\-j = y zc — d 2 . Therefore .v = i( V zc - d* + d), J == K V 2C - d* ~ d). For («') Narayana writes : "Suppose the square of the product as the product (of two quantities) and the difference of the squares as their difference. From them by sankrama will be obtained the (square) quantities. Their square-roots severally will give the quantities (required)." 2 We have x 2 — J' 2 = m These are of the form (/). Therefore *2 = $(V/w* + 4b 2 -f m), jy a = i(V** a + 4^ 2 — »). Whence we get the values of x andjy. Rule of Dissimilar Operations. The process of solving the following two particular cases of simul- taneous quadratic equations was distinguished by most Hindu mathematicians by the special designation visama- kcirmcP (dissimilar operation) : 1 GK, i. 33. a GK, i. 34. 3 The name visama-karma originated obviously in contra- distinction to the name sankramana. This is evident from the term visama-sankramana used by Mahavira for visama-karma. SIMULTANEOUS QUADRATIC EQUATIONS * 8 J These equations are found to have been regarded by them as of fundamental importance. The solutions given are: for (0 * = Kt + *)' ^-Kt-»)» for (/,) x = ^ + ^), j,^^^-^). Thus Brahmagupta says: "The difference of the squares (of the unknowns) is divided by the difference (of the unknowns) and the quotient is increased and diminished by the difference and divided by two; (the results will be the two unknown quantities); (this is) dissimilar operation." 1 The same rule is restated by him on a different occasion in the course of solving a problem. "If then the difference of their squares, also the difference of them (are given): the difference of the squares is divided by the difference of them, and this (latter) is added to and subtracted from the quotient and then divided by two; (the results are) the residues; whence the number of elapsed days (can be found)." 2 Mahavtra states: "The sankramana of the divisor and the quotient of the two quantities is dissimilar (operation); so it is called by those who have reached the end of the ocean of mathematics." 3 Similar rules are given also by other writers. 4 1 BrSpSi, xviii. 36. z BrSpSi, xviii. 97. 3 GSS, vi. 2. 4 MSi, xv. 22; Si* \ GK, i. 32. 86 ALGEBRA Mahavira' s Rules. Mahavira (830) has treated certain problems involving the simultaneous quadratic equations : u -\- x = a, urw = ax, u -f- j — b. usn> = a j. Here r x a — u s j ~ b — //" Therefore u = - s Hence x=( )r, y =( )s, w =(-j -)ct. In the above equations x,y are the interests accrued on the principal u in the periods r, s respectively and w is the rate of interest per a. Mahavira states the result thus : "The difference of the mixed sums [a, b] multiplied by each other's periods [r, s], being divided by the difference of the periods, the quotient is known as the principal [//J." 1 Again, there are problems involving the equations: a -\- x = p, uxw = am, u + J = q- ttyw = an. Where x, j are the periods for which the principal u is lent out at the rate of interest w per a and ///, n are the respective interests. TT m x p — u Here — = — = J - . n y q — u Therefore // = — — . m — n 1 GSS, vi. 47. 2 ck, i. , ,Jma-karma originated ofc>VK name saiikramana. This is evident x. ana used by Mahavira for visama-karma. INDETERMINATE EQUATIONS OF THE FIRST DEGREE 87 Hence x = (^=^-) m, y = (P—^n, j^ = Mahavira gives the rule : "On the difference of the mixed sums multiplied by each other's interests, being divided by the difference of the interests, the quotient, the wise men say, is the principal." 1 13. INDETERMINATE EQUATIONS OF THE FIRST DEGREE General Survey. The earliest Hindu algebraist to give a treatment of the indeterminate equation of the first degree is Aryabhata I (born 476). He gave a method for finding the general solution in positive integers of the simple indeterminate equation by — ax = c for integral values of a, b, c and further indicated how to extend it to get positive integral solutions of simultan- eous indeterminate equations of the first degree. His disciple, Bhaskara 1 (522), showed that the same method might be applied to solve by — ax = — c and further that the solution of this equation would follow from that of by — ax = — 1. Brahmagupta and others simply adopted the methods of Aryabhata I and Bhas- kara I. About the middle of the tenth century of the Christian Era, Aryabhata II improved them by point- ing out how the operations can in certain cases be abridged . considerably. He also noticed the cases of failure of the methods for an equation of the form 1 GSS, vi. 51. 88 ALGEBRA by — ax — ^ c ■ These results reappear in the works of later writers. 1 Its Importance. It has been observed before that the subject of indeterminate analysis of the first degree was considered so important by the ancient Hindu algebraists that the whole science of algebra was once named after it. That high estimation of the subject continued undiminished amongst the later Hindu mathe- maticians. Aryabhata II enumerates it distinctively along with the sciences of arithmetic, algebra, and astronomy. 2 So did Bhaskara II and others. ■ As has been remarked by Ganesa, 3 the separate mention of the subject of indeterminate analysis of the first degree is designed to emphasize its difficulty and importance. On account of its special importance, the treatmentr of this subject has been included by Bhaskara II in his treatise of arithmetic also, though it belongs parti- cularly to algebra. 4 It is also noteworthy that there is a work exclusively devoted to the treatment of this subject. Such a special treatise is a very rare thing in the mathematical literature of the ancient Hindus. This work, entitled Kuttdkara-siromanif is by one Devaraja, a commentator of Aryabhata I. 1 For "India's Contribution to the Theory of Indeterminate Equations of the First Degree," see the comprehensive article of Professor Sarada Kanta Ganguly in Jotirn. Ind. Math. Soc, XIX, 1931, Notes and Questions, pp. 1 10-120, 129-142; see also XX, 1932, Notes and Questions. Compare also the Dissertation of D. M. Mehta on "Theory of simple continued tactions (with special reference to the history of Indian Mathematics)." - 2 AW, i. 1. ■■ 3 Vide his commentary on the Uldvati of Bhaskara IT 4 Bhaskara's treatment of the pulveriser in his liijaganiia is repeated nearly wo r d for word in his Uldvati. 5 There are four manuscript copies of this work in the Oriental Library, Mysore. INDETERMINATE EQUATIONS OF THE FIRST DEGREE 89 Three Varieties of Problems. Problems whose solutions led the ancient Hindus to the investigation of the simple indeterminate equation of the first degree were distinguished broadly into three varieties. The problem of one variety, is to find a number (N) which being divided by two given numbers (a, b) will leave two given remainders (R 1} R 2 ). Thus we have jy=«jc+R 1 = ^ + R 2 . Hence by — ax = K x — R 2 . Putting c = R t ~ R 2> we get by — ax = ±c the upper or lower sign being taken according as R t > or < R 2 . In a problem of the second kind we are required to find a number (x) such that its product with a given number (a) being increased or decreased by another given number (y) and then divided by a third given number ((3) will leave no remainder. In other words we shall have to solve ax + Y in positive integers. The third variety of problems similarly leads to equations of the form by -f- ax = ± c - Terminology. The subject of indeterminate analysis of the first degree is generally called by the 1 lindus kuttaka, kuttdkdra, kuttikdra or simply kutta. The names kuttdkdra and kutta occur as early as the Mabd-Bbdskarija of Bhaskara I (522). 1 In the cpmmen- tary of the Arjabbatiya by this writer we find the terms kuttaka and kuttdkdra. Brahmagupta has used kuttaka? kuttdkdra? and kutta? Mahavira, it appears, had a 1 MBh, i. 41, 49. 2 BrSpSi, xviii. 2, 11, etc. 3 BrSpSi, xviii. 6, 1 5 j etc. • i BrSpSi, xviii. 20, 25, etc. <)0 ALGEBRA preferential liking for the name kuttikdra. 1 In a problem of the first variety the quantities (a, U) are called "divisors" (bhdgahdra, bhdjak, cheda, etc.) and (Rj, R 2 ) "remainders" {agra, sesa, etc.), while in a problem of the second variety, p is ordinarily called the "divisor" and y the "interpolator" (ksepa, ksepaka, etc.); here a is called the "dividend" (bhdjya), the unknown quantity to be found (x) the "multiplier" {gunaka, gunakdra, etc.) and y the quotient (phald). The unknown (x) has been sometimes called by Mahavira as rdsi (number) implying "an unknown number." 2 Origin of the name. The Sanskrit words kutta, kuttaka, kuttdkdra and kuttikdra are all derived from the root kutt "to crush", "to grind," "to pulverise" and hence etymologically they mean the act or process of "breaking", "grinding", "pulverising" as well as an instrument for that, that is, "grinder", "pulveriser". Why the subject of the indeterminate analysis of the first degree came to be designated by the term kuttaka is a question which will be naturally asked. Ganesa (1545) says: "Kuttaka is a term for the multiplier, for multiplication is admittedly called by words import- ing 'injuring,' 'killing.' A certain given number being multiplied by another (unknown quantity), added or subtracted by a given interpolator and then divided by a given divisor leaves no remainder; that multiplier is the kuttaka: so it has been said by the ancients. This is a special technical term." 3 The same explanation as to the origin of the name kuttaka has been offered by Surya- dasa (1538), Krsna (e. 1580) and Rariganatha (1602). 4 1 GSS, vi. 79 i, etc. 2 GSS, vi. ujjff. 3 Vide his commentary on the Uldvati of Bhaskara II. 4 Vide the commentaries of Suryadasi on Uldvati and Bija- ganita, of Krsna on Bijaganita, and of Ranganatha on Siddhdnta- Jiromani. INDETERMINATE EQUATIONS OF THE FIRST DEGREE 91 But it is one-sided inasmuch as it has admittedly in view a problem of the second variety where we have indeed to find an unknown multiplier. But the rules of the earlier .algebraists such as Aryabhata I and Brahma- gupta were formulated with a view to the solution of a problem of the first variety. So the considerations which led those early writers to adopt the name kuttaka must have been different. Mahavira has once stated that, according to the learned, kuttikdra is another name for "the operation of praksepaka" (lit., throwing, scatter- ing, implying division into parts). 1 In fact, his writ- ing led his translator to interpret kuttikdra as "propor-. donate division", "a special kind of division or distribu- tion." 2 Bhaskara I, who had in view a problem of the second variety, once remarked, "the number is obtained by the operation of pulverising {kuttana), when it is desired to get the multiplier (gunakdra). . . . " 3 It will be presently shown that the Hindu method of solving the equation by — ax — ^ c is essentially based on a process of deriving from it successively other similar equations in which the values of the coefficients {a, b) become smaller and smaller. 4 Thus the process is indeed the same as that of breaking a whole thing into smaller pieces. In our opinion, it is this that led the ancient mathematicians to adopt the name kuttaka for the opera- tion. Preliminary Operations. It has been remarked by most of the writers that in order that an equation 1 "Praksepaka- karanamidarh kuttlkaro budhaissamuddis- tam" — GSS, vi. 79J. 2 Vide GSS (English translation), pp. 117, 300. 3 "Krta-kuttana-labdha-rasimesam Gunakararn samusanti " — MBh, i. 48. 4 It has been expressly stated by Suryadeva Yajva that the process must be continued "yavaddharabhajyayoralpata." 92 ALGEBRA of the form by — ax = ± c or by -\- ax = -4^ c may be solvable, the two numbers a and b must not have a common divisor; for, otherwise, the equation would be absurd, unless the number c had the same common divisor. So before the rules adumbrated hereafter can be applied, the numbers a, b, c must be made prime (drdha = firm, niccheda = having no divisor, nirapa- varta = irreducible) to each other. Thus Bhaskara I observes: "The dividend and divisor will become prime to each other on being divided by the residue of their mutual division. The operation of the pulveriser should be considered in relation to them." 1 Brahmagupta says: "Divide the multiplier and the divisor mutually and find the last residue; those quantities being divided by the residue will be prime to each other." 2 Aryabhata II has made the preliminary operations in successive stages. These will be described later on. 3 Sripati states: "The dividend, divisor and interpolator should be divided by their common divisor, if any, so that it may be possible to apply the method to be described." 4 "If the dividend and divisor have a common divisor, which is not a divisor of the interpolator then the problem would be absurd." 5 Bhaskara II writes: "As preparatory to the method of the pulveriser, 1 MBh, i. 41 . 2 BrSpSi, xviii. 9. 3 Vide infra, p. 104. 4 SiSe, xiv. 22. 5 Sife, xiv. 26.- solution of by — ax = ^ c 93 „the dividend, divisor and interpolator must be divided by a common divisor, if possible. If the number by which the dividend and divisor are divisible, does not divide the interpolator then the problem is absurd. The last residue of the mutual divi- sion of two numbers is their common divisor. The dividend and divisor, being divided by their common divisor, become prime to each other." 1 Rules similar to these have been given also by Nardyana, 2 Jnanaraja and Kamalakara. 3 So in our subsequent treatment of the Hindu methods for the solution in positive integers of the equation by ± ax — i c \ w e shall always take, unless otherwise stated, a, b prime to each other. Solution of by — ax = -^ c Aryabhata Fs Rule. The rule of Aryabhata I (499) 4 is rather obscure inasmuch as all the operations intend- ed to be carried out have not been described fully and clearly. So it has been misunderstood by many writers. 5 Following the interpretation of the rule by Bhaskara I (525), a direct disciple of Aryabhata I, Bibhutibhusan Datta has recently given the following translation: 6 1 L, p. 76; BBi, pp. 24f. a NBi, I, R. 5 3-4. 3 SiTVi, xiii. ij<)K. *^f, ii. 32-3. 5 L. Rodet, "Le$ons de calcul d'Aryabhatta," JA, XIII, 1878, pp. 303ff; G. R. Kaye, "Notes on Indian Mathematics. No. 2— Aryabhata," JASB, IV, 1908, pp. mff; BCMS, IV, p. 55; N. K. Mazumdar, "Aryyabhatta's rule in relation to Indeterminate Equations of the First Degree," BCMS, III, pp 11-9; P. C. Sen Gupta, "Aryabhatiyam," Jour. Dept. Let. Cal. Univ., XVI, 1927; reprint, p. zy.; S. K. Ganguly, BCMS, XIX, 1928, pp. i7ofl; W. E. Clark, ylryabhatiya^of Aryabhata, Chicago, 1930, pp. 42tF. 6 Bibhutibhusan Datta, "Elder Aryabhata's rule for the solution of indeterminate equations of the first degree," BCMS, XXIV, 1932, pp. 35-53. 94 ALGEBRA "Divide the divisor corresponding to the greater remainder by the divisor corresponding to the smaller remainder. The residue (and the divisor corresponding to the smaller remainder) being mutually divided, the last residue should be multiplied by such an optional integer that the product being added (in case the number of quotients of the mutual division is even) or subtracted (in case the number of quotients is odd) by the difference of the remainders (will be exactly divisible by the last but one remainder. Place the quotients of the mutual division successively one below the other, in a column; below them the optional multiplier and underneath it the quotient just obtained). Any number below (i.e., the penultimate) is multiplied by the one just above it and then added by that just below it. Divide the last number (obtained by doing so repeatedly 1 ) by the divisor corresponding to the smaller remainder; then multiply the residue by the divisor corresponding to the greater remainder and add the greater remainder. (The result will be) the number corresponding to the two divisors." He has further shown that it can be rendered also as follows: "Divide the divisor corresponding to the greater remainder by the divisor corresponding to the smaller remainder. The residue (and the divisor corresponding to the smaller remainder) being mutually divided (until the remainder becomes zero), the last quotient should be multiplied by ah optional integer and then added (in case the number of quotients of the mutual division is even) or subtracted (in case the number of quotients is odd) by the difference of the remainders. (Place the other quotients of the mutual division succes- 1 The process implied here is shown in detail in the working of the example on pages njf. solution of by — ax — -[- c 95 sively one below the other in a column; below them the result just obtained and underneath it the optional in- teger). Any number below (i.e., the penultimate) is multiplied by the one just above it and then added by that just below it. Divide the last number (obtained by doing so repeatedly) by the divisor corresponding to the smaller remainder; then multiply the residue by the divisor corresponding to the greater remainder and add the greater remainder. (The result will be) the number corresponding to the two divisors." Aryabhata's problem is : To find a number (N) which being divided by two given numbers (a, b) will leave two given remainders (R 15 R 2 ). x This gives: N = ax + K x = by + R 2 . Denoting as before by c the difference between R x and R 2 , we get (0 by = ax + c t if R, > R 2 , or (//) ax = by -j- c, if R 2 > R x the equation being so written as to keep c always posi- tive. Hence the problem' now reduces to making either ax + c by + c — J — or J- — ■ b a according as R t > R 2 or R 2 > R t , a positive integer. So Aryabhata says: "Divide the divisor corresponding to the greater remainder etc." 1 It has already been stated (p. 90) that in a problem of the first variety which gives an equation of the above form (and in which R x > R 2 ). a == divisor corresponding to greater remainder, b = divisor corresponding to lesser remainder, R x = greater remainder, R„ = lesser remainder. <)6 ALGEBRA Suppose &! > R 2 ; then the equation to be solved will be ax + c — by (I) a, b being prime to each other. Let b)a (q bq ~>\)b (ft £1 ft__ ^ 2 ) r i (ft r 2 ft /• 3 r m—\) r m-2 \3m~l ^m-nm-i 'm/ ' m— 1 Vy m ^m r ' m-ri Then, we get 1 a = bq + r is ^ = 'ift + r v r i = r aft + >"3, ^ = '* 3 ft + r «» 'm— 2 ' 'to— 12"*— 1 ~~ l~ 'm' Now, substituting the value of <2 in the given equa- tion (I), we get by — {bq + rjx -f- r. Therefore 1 When a < &, we shall have ^ = o, r x = a. solution of by — ax = ± c 97 where by x = r L x -f- ^- In other words, since a = bq + t\, on putting ; , = #*■ + J'i (0 the given equation (1) reduces to h J\ = r r< + '• (I- T ) Again, since b = r^ + r a , putting similarly „v — q Y y x + -Vj the equation (1. i) can be further reduced to Vi = r O\ — c ( L 2 ) and so on. Writing down the successive values and reduced equations in columns, we have CO (0 (3) (4) (5) (6) X = q x y + *-,, J'i = <72*i + J>'». - v i = q*y-i -f - v 2. .>2 = ?4*2 + J3. X 2 — ^5_)'3 "r ■*•'.■!« 'aJ'a = r 3*"i + f, 'Va — ''4J2 ^J r 4J'3 = ^^2 + e, r 5 x 3 = Va — <% (I.I) (1.2) (I- 3) (I- 4) (I- J) (1.6) fsii-ajn = '"an-l^n-1 + ^, (I- Z».— i) r 2«-l-'^7i — ^K^'ra f > (I. 2«) (2«— J'n-1 = #!>i-2*"*-l +J'„. (2«) *■„_! = ^n-lJ'n + -*n> (*» + J» = Wtt+l = 'Wl-*^ + «". (I- 2 « + Now the mutual division can be continued either (/) to the finish or (//) so as to get a certain number of quotients and then stopped. In either case the number of quotients found, neglecting the first one (q), as is usual with Aryabhata, may be even or odd. Case i. First suppose that the mutual division is continued until the zero remainder is obtained. Since a, b are prime to each other, the last but one remainder is unity. Subcase (J. i). Let the number of quotients be even. We then have r 2n J » r 2n + l ~ °> tfin = ^271-1* IOO ALGEBRA mutual division, one below the other, in the form of a chain. Now find by what number the last remainder should be multiplied, such that the product being sub- tracted by the (given) residue (of the revolution) will be exactly divisible (by the divisor corresponding to that remainder). Put down that optional number below the chain and then the (new) quotient underneath. Then multiply the optional number by that quantity which stands just above it and add to the product the (new) quotient (below). Proceed afterwards also in the same way. Divide the upper number (/>., the multi- plier) obtained by this process by the divisor and the lower one by the dividend; the remainders will respectively be the desired ahargana and the revolutions." 1 The equation contemplated in this rule is 2 ax — c . „. — r — = a positive integer. This fottn of the equation seems to have been chosen by Bhaskara I deliberately so as to supplement the form of Aryabhata I in which the interpolator is always made positive by necessary transposition. Further b is taken to be greater than a, as is evident from the following rule. So the first quotient of mutual division of a by b is always zero. This has not been taken into consideration. Also the number of quotients in the chain is taken to be even. 1 MBb, i. 42-4. The above rule has been, formulated with a view to its application in astronomy. 8 As already stated on p. 90, when the equation is stated in this second form a = dividend, b = divisor, c = interpolator, x = multiplier, y — quotient. solution of by — ax — ^ c ioi He further observes: "When the dividend is greater than the divisor, the operations should be made in the same way {i.e., according to the method of the pulveriser) after delet- ing the greatest multiple of the divisor (from the divi- dend). Multiply the (new) multiplier thus obtained by that multiple and add the (new) quotient; the /result will be the quotient here (required)." 1 That is to say, if in the equation ax ± c = by, !j = mb -\- a', we may neglect the portion mb of the divi- dend and proceed at once with the solution of a'x ± c = ky- Let .v = a, y = p be a solution of this equation. Then aTa ± c = b$ ; {mb + a')a ±e = b{ma + p), or tfa,± c = b{ma -\- p). Hence x = a, y = ma + p is a solution of the given equation. Brahmagupta's Rules. For the solution of Arya- ' bhata's problem Brahmagupta (628) gives the following rule: "What remains when the divisor corresponding to the greater remainder is divided by the divisor corres- ponding to the smaller remainder — that (and the latter divisor) are mutually divided and the quotients are severally set down one below the other. The last residue (of the reciprocal division after an even number of quotients has been obtained?) is multiplied by 1 Mhb, i. 47- 2 Compare the next rule: "Such is the process when the quotients (of mutual division) are even etc." 102 ALGEBRA such an optional integer that the product being added with the difference of the (given) remainders will be exactly divisible (by the divisor corresponding to that residue). That optional multiplier and then the (new) quotient just obtained should be set down (underneath the listed quotients). Now, proceeding from the lower- most number (in the column), the penultimate is multiplied by the number just above it and then added by the number just below it. The final value thus obtained (by repeating the above process) is divided by the divisor corresponding to the smaller remainder. The residue being multiplied by the divisor correspond- ing to the greater remainder and added to the greater remainder will be the number in view." 1 He further observes: "Such is the process when the quotients (of mutual division) are even in number. But if they be odd, what has been stated before as negative should be made positive or as positive should be made negative." 2 Regarding the direction for dividing the divisor corresponding to the greater remainder by the divisor corresponding to the smaller remainder, Prthudakasvami (860) observes that it is not absolute, rather optional; so that the process may be conducted in the same way by starting with the division of the divisor correspond- ing to the smaller remainder by the divisor correspond- ing to the greater remainder. But in this case of inver- sion of the process, he continues, the difference of the remainders must be made negative. That is to say, the equation by. = ax -f- c can be solved by transforming it first to the form ax = by — c, 1 BrSpSi, xviii. 3-5. 2 BrSpSi, xviii. 13. solution of by — ax = dc c io 3 so that we shall have to start with* the division of b by a. Mahavira' s Rules. Mahavira (850) formulates his rules with a view to the solution of ax -j- c in positive integers. He says: "Divide the coefficient of the unknown by the given divisor (mutually); reject the first quotient and then set down the other quotients of mutual division one below the other. When the residue has become sufficiently small, multiply it by an optional number such that the product, being combined with the inter- polator, which if positive must be made negative (and vice versa) in case (the number of quotients retained is) odd, will be exactly divisible (by the divisor correspond- ing to that residue). Place that optional number and the resulting quotient in order, under the chain of quo- tients. Now add the lowermost number to the product of the next two upper numbers. The number (finally obtained by this process) being divided by the given divisor, (the remainder will be the least value of the unknown)." 1 This method has been redescribed by Mahavira in a slightly modified form. Here he continues the mutual division until the remainder zero is obtained and further takes the optional multiplier to be zero. "With the dividend, divisor and remainder reduced (by their greatest common factor the operations should be performed). Reject the first quotient and set down the other quotients of mutual division (one below the other) and underneath them the zero 2 and the given remainder 1 GSS, vi. 115 \ (first portion). 2 VC'e have emended sagra of the printed text to khagra. 104 ALGEBRA (as reduced) in succession. The remainder, being multi- plied by positive or negative as the number of quotients is even or odd, should be added to the product of the next two upper numbers. The number (finally obtained by the repeated application of this process) whether posi- tive or negative, being divided by the divisor, the remainder will be (the least value of) the multiplier." 1 Aryabhata II. The details of the process adopted by Aryabhata II (950) in finding the general solution of {ax i c)\b = y in positive integers have been described by him thus: "Set down the dividend, interpolator and divisor as stated (in a problem): this is the, first operation. "Divide them by their greatest common divisor so as to make them without a common factor: this is the second operation. "Divide the dividend and interpolator by their greatest common divisor: the third operation. "Divide the interpolator and divisor by their * greatest common divisor: the fourth operation. "Divide the dividend and interpolator, then the interpolator (thus reduced) and divisor by their respec- tive different greatest common divisors: the fifth operation. "On forming the chain from these (reduced numbers), if the remainder becomes unity, then the object (of solving the problem) will be realised; but if the remainder in it be zero, the questioner does not know the method of the pulveriser. "Divide the (reduced) dividend and divisor reci- procally until the remainder becomes unity. (The quo- tients placed one below the other successively will form) 1 GSS, vi. 1 36 J (first portion). Our interpretation differs from those of Rangacharya and Ganguly. solution of by — ax — ^ c ioj the (auxiliary) chain. Note down whether the number of quotients is even or odd. Multiply by the ultimate the number just above it and then add unity. The chain formed on replacing the penultimate by this result is the corrected one. Multiply by the un-destroyed {i.e., corrected) penultimate the number just above it, then add the ultimate number; (now) destroy the ultimate. On proceeding thus (repeatedly) we shall finally obtain two numbers which are (techni- cally) called ktttta. I shall speak (later on) of those two quantities as obtained in the case of an odd number of quotients. If on dividing the dividend by the divisor once only the residue becomes unity, then the quotient is known to be the upper kutta and the remainder (i.e., unity) the lower kutta. "The upper and lower kutta thus obtained, being both multiplied by the interpolator of the given equation and then divided respectively by its dividend and divisor, the residues will be the quotient and multiplier respec- tively. "In the case of the third operation (having been performed before) multiply the upper kutta by the inter- polator" of the question and the lower kutta by the inter- polator as reduced by the greatest common divisor. The same should be done reversely in the case of the fourth operation. In the case of these two operations, the kutta nfter being multiplied as indicated should be divided respectively by the dividend and divisor stated by the questioner, the residues will be the quotient and multi- plier respectively. "In the fifth operation, multiply the upper kutta by the greatest common divisor of the dividend and the interpolator, and the lower one by the other (i.e., the greatest common divisor of the given divisor and the reduced interpolator). The products are the inter- Io6 ALGEBRA mediate quotient and 'multiplier. Multiply the divisor of the question by the intermediate quotient and also its dividend by the intermediate multiplier. Difference of these products is the required intermediate divider. The intermediate quotient and multiplier are multiplied by the interpolator of the question and then divided by the intermediate divider. The quotients thus obtained being divided respectively by the dividend and divisor of the question, the residues will be the quotient and multiplier (required). "The quotient and multiplier are obtained correctly by the process just described in the case of a positive interpolator when the chain is even and in the case of a negative interpolator if the chain is odd. In the case of an even chain and negative interpolator, also of an odd chain and positive interpolator, the quotient and multi- plier thus obtained are subtracted respectively from trie dividend and divisor made prime to each other and the residues give them correctly." 1 The rationale p£ these rules will be easily found to be as follows: t> (i) It will be noticed that to solve by = ax±c, ' (i) in positive integers, Aryabhata II first finds the solution of by — ax ± i- If x = a, y = p be a solution of this equation, we get b$ = aa.^ i, or &(49 = a(ca) -±- c. Therefore x = ca, y = c$ is a solution of (i). (it) Let a = a'g, c = c'g> then (i) reduces to bf = a'x ± c\ 1 MSi, xviii. 1-14. solution of by — ax = i c 107 where y r =y\g* Let x = a, y' = p be a solution of by' = #'.*• ^ T , so that we have b§ = a'a -j- 1. Hence £g/p = a 'gr'a ± c'g ; or b(e$) = a(c'a) ± c. Therefore x = /a, y = *rp is a solution of (1). (//'/') Let b=g'b', c=g'e"; then (1) reduces to £> = A** ± c", where x' = xjg*. If x' — a, y = p be a solution of we have £'p = ^o±i. Therefore ^V'P = age" a ± g'c", or %"p) = a(ea) ± c. Hence x = c a, y = {&}-'{&}*« 108 ALGEBRA Since gg" = a{g"a) ~ b(g$), we get ,( cr(gfi) ) _ C c{ g" n. Now, if the interpolator c is positive, it can be shown that (2) is not a solution. For, if it were, bq — c = x, an integer, = {m — n)b + p > b. But q < a, therefore, ^-'<*. solution of by — a x — ± c 109 which is absurd. Therefore, (1) must be the minimum solution in this case, not (2). Similarly, if the interpolator c is negative, it can be shown that (2) fe the minimum solution, not (1). Hence the following rule of Aryabhata II : "If the quotients (//?, ») obtained in the case of any proposed question be not equal, then the (derived) value for the multiplier should be accepted and that of the quotient rejected, if the interpolator is positive. On the other hand when the interpolator is negative, then the (derived) value for the quotient should be accepted and that for the multiplier rejected. How to obtain the quotient from the multiplier and the multiplier from the quotient correctly in all cases, I shall explain now. Multiply the (accepted) value of the multiplier by the dividend of the proposed question, add its interpolator and then divide by the divisor of the proposed question; the quotient is the corrected one. The product of the proposed divisor and the (accepted) quotient being added by the teverse of the interpolator and then divided by the dividend of the proposed question, the ■quotient is the (correct) multiplier." 1 He has further indicated how to get all positive integral solutions of the equation by = ax i c after having obtained the minimum solution. "The (minimum) quotient and multiplier being added respectively with the dividend and divisor as stated in the question or as reduced, after multiplying both by an optional number, give various other values." 2 That is to say, if x = a, jr — fi be the minimum solution, the general solution will be „v = bm -\- a, j>= am -f- (3. 1 MSi, xviii. 15-8. 2 MSf, xviii. 20. HO ALGEBRA Sripati's Rule. Sripati (1039) wr i tes : "Divide the dividend and divisor reciprocally until the residue is small. Set down the quotients one below the other in succession; then underneath them an optional number and below it the correspond- ing quotient, the optional number being determined- thus: (the number) by which the last residue must be multiplied such that the product being subtracted by the interpolator and then divided by the divisor (corres- ponding to that residue), leaves no remainder. It is to be so when the number of quotients is even; in the case of an odd number of quotients the interpolator, if negative, must be first made positive and conversely, if positive, must be made negative; so it has been taught by the learned in this (branch of analysis). Now multi- ply the term above the optional number by it (the optional number) and then add the quotient below. Proceeding upwards such operation should be per- formed again and again until two numbers are obtained. The first one being divided by the divisor, (the residue) will give (the least value of) the multiplier; similarly the second being divided by the dividend, will give (the least value) of the quotient." 1 Bhaskaia IPs Rules. Bhaskara II (11 50) des- cribes the method of the pulveriser thus: "Divide mutually the dividend and divisor made prime to each other until unity becomes the remainder in the dividend. Set down the quotients one under the other successively; beneath them the interpolator and then cipher at the bottom. Multiply by the penultimate the number just above it and add the 1 Sife, xiv. 22-25. This rule is the same as that of Bhaskara I and holds under the same conditions. (See pp. 991). solution of by — a x — ^ c i i-i ultimate; . then reject that ultimate. Do so repeatedly until only, a pair of numbers is left. The upper one of these being divided by the reduced dividend, the remain- der is the quotient; and the lower one being divided by the reduced divisor, the remainder is the multiplier. Such is precisely the process when the quotients (of mutual division) are even in number. But when they are odd, the quotient and multiplier so obtained must be subtracted from their respective abraders and the residues will be the true quotient and multiplier." 1 Bhaskara'II then shows how the process of solving a problem by the method of the pulveriser can some- times be abbreviated to a great extent. He says: "The "multiplier is found by the method of the pulveriser after reducing the additive and dividend by their common divisor. Or, if the additive (previously reduced or not) and the divisor be so reduced, the multiplier found (by the method) being multiplied by their common measure will be the true one. "Such is the process of finding the multiplier and quotient, when the interpolator is positive. On sub- tracting them from their ' respective abraders will be obtained the result for the subtractive interpolator." 2 Krsna (e. 1 5 80) gives the following rationale of these rules: We shall have to solve in positive integers by = ax ^ c. (1) (/) Suppose g is the greatest common measure of +a and c, so that a = a'g, c = c'g. Then by = a'gx ± c'g, or by' = a'x ± c' t C 1 - 1 ) where y' ~y/g. If x = a, y f — p be a solution of (1.1), 1 mi, pp. *5f; L, p. 77. 2 BBi, p. 26; L, pp. 78, 79. H4 ALGEBRA Then, forming the chain as directed in the rule, we get i i i 2 2 I 9° o By the rule, "Multiply by the penultimate the number just above it etc.," the two numbers obtained finally are 2430 and 1530. 1 Dividing these by 100 and 63 respectively, the remainders are 30 and 18. Hence x = i8,j= 30. Second Method. Reducing the dividend and the additive by their greatest common divisor (10), we have the statement: Dividend =10 Additive:=9 Since 10 Divisor = 63 63) 10 (o 10) 63 (6 60 T) 10 (3 9_ 1 1 Successive operations in the application of the rule are : 1 1 1 2 2 \ 90 90 1 1 1 1 1 1 1 1 \ 900 2 X 270 \ 90 91^ ~S> 630 X 270 9<^ \ 630 \ 9iS( ! 9^ j * ! % 1530 \ 9t^ 900 9ty \ 2430 1530 aty SOLUTION of by — «=±c 115 we get the chain o 6 3 9 o By the rule, "Multiply by the penultimate etc.," wc obtain finally the numbers 27 and 171. Dividing them respectively by 10 and 63, we get the residues 7 and 45. Since the number of quotients of the mutual division is odd, subtracting 7 and 45 from the corresponding abraders 10 and 63, we get 3 and 18. In this case we neg}cct 3. So x = 18; whence by the given equation j = 30. Or, multiplying the quotient 3 as obtained above by the greatest common divisor 10, we get the same result y = 30. Third Method. Reducing the divisor and the additive by their greatest common divisor (9), the statement is : Dividend = 100 A , ,. . tv ■ Additive = 10 Divisor = 7 Since 7) 100 (14 *)7(3 6_ 1 we get the chain 3 10 o By the rule, "Multiply by the penultimate etc.," wc obtain the two numbers 430 and 30. Dividing them by 100 and 7 respectively, the residues are 30 and 2. I16 ALGEBRA Multiplying the latter by the greatest common divisor 9, we get x = 18 and y = 30. Fourth Method. Dividing the divisor and the addi- tive by their common measure (9) and again the dividend and the reduced additive by their common measure (io), we have Dividend =10 Divisor = 7 Since 7) 10 (1 1_ 3)7(2 6 Additive we get the chain 1 2 I By the rule, "Multiply by the penultimate etc.," we have finally the numbers 3 and 2. Dividing them by 10 and 7 respectively, the residues are the same. Multiplying them respectively by the common measure 10 of the dividend and reduced additive, and 9 of the divisor and additive, we get as before x — 18 and y = 30. Adding to these minimum values (18, 30) of (x,y) optional multiples of the corresponding abraders (63, 100), we get the general solution of ioox + 90 = 6}j> in positive jntegers as x = 6}f» + 18, y = ioo« + 30, where m is any integer. Rules similar to those of Bhaskara II have been given by Narayana, 1 Jnanaraja and Kamalakara. 2 1 NBi, I, R. 5 5-60. *SiTVi, xiii. 183-190. solution of by = ax ^z i 117 Solution of by = ax i 1 Constant Pulveriser. Though the simple indeter- minate equation by = ax 4^ 1 is solved exactly in the same way as the equation by = a x ± c and is indeed a particular case of the latter, yet on account of its special use in astronomical calculations 1 it has received separate consideration at the hands of most of the Hindu algebra- ists. It may, however, be noted that the separate treat- ment was somewhat necessitated by the physical condi- tions of the problems involving the two types. In the case of by = ax ^ c the conditions are such that the value of either y or x, more particularly of the latter, has to be found and the rules for solution are formulated with tnat object. But in the case of the other (by = ax ^ 1) the physical conditions require the values of both y and x. The equation by = ax 4i 1 is generally called by the name of sthira-kuttaka or the "constant pulveriser" (from sthira, meaning constant, steady). Prthudakasvami (860) sometimes designates it also as drdba-kuttaka (from drdha = firm). But that name disappeared from later Hindu algebras because the word drdha was employed by later writers 2 as equivalent to niccheda . (having no divisor) or nirapavarta (irreducible). The origin of the name "constant pulveriser" has been explained by Prthudakasvami as being due to the fact that the inter- polator (^ 1) is here invariable. Ganesa 3 (1545) explains it in detail thus : In astronomical problems involving 1 Thus Bhaskara II observes, "This method of calculation is of great use in mathematical astronomy." (BBi, p. 31). He then points out how the solutions of various astronomical problems can be derived from the solution of the same indeter- minate equation. (BBi, p. 32; L, p. 81). 2 This special technical use of the word drdha occurs before Brahmagupta (628) in the works of Bhaskara I (522). 3 Vide his commentary on the l^ildvati of Bhaskara II. 1 1 8 ALGEBRA equations of the type by — ax = ^ c, the physical conditions are such that the dividend (a) and the divisor (b) are constant but the interpolator ( is found to be x==-"j,j=S. Multiplying these values by 5 and then abrading by 1 5 and 1 7 respectively, we get the required minimum solution x=i, j=6. Again a solution of 17.Y — 1 — =J 15 will be found to be x— 8, J>— 9. Multiplying these quantities by 5 and abrading by 15 and 17, we get the solution of ijx — 5 15 to be x=^io, y—ii. --.y Solution of by -\- ax = ± c An equation of the form by + ax = i c was gene- rally transformed by Hindu algebraists into the form by = — « ± f so that it appeared as a particular case of by = ax i c in which a was negative. Brahmagupta's Rule. Such an equation seems to 1 BBi, pp. 28, 31; L, pp. 77, 8r. soi-ution of by + ax = i <" 121 have been solved first by Brahmagupta (628). But his rule is rather obscure : "The reversal of the negative and positive should be made of the multiplier and inter- polator." 1 Prthudakasvami's explanation does not throw much light on it. He says, "If the multiplier be negative, it must be made positive; and the additive must be made negative: and then the method of the pulveriser should be employed." But he does not indicate how to derive the solution of the equation by = — ax + c (1) from that of the equation by — ax — c (2) The method, however, seems to have been this : Let x = a, y = |3 be the minimum solution of (2). Then we get £p = ca. — c or b(a — p) = — a(a — b) -f- c. Hence x = a — b, y — a — pis the minimum solution of (1). This has been expressly stated by Bhaskara II and others. Bhaskara IPs Rule. Bhaskara II says : "Those (the multiplier and quotient) obtained for a positive dividend being treated in the same manner give the results corresponding to a negative dividend." 2 The treatment alluded to in this rule is that of subtraction from the respective abraders. He has fur- ther elaborated it thus : "The multiplier and quotient should be deter- mined by taking the dividend, divisor and interpolator as positive. They will be the quantities for the additive interpolator. Subtracting them from their 1 BrSpSi, xviii. 13. ° 2 BBi, p. 26. 122 ALGEBRA respective abraders, the quantities for a negative inter- polator are found. If the dividend or divisor be nega- tive, the quotient should be stated as negative." 1 Narayana. Narayana (1350) says : "In the case of a negative dividend find the multi- plier and quotient as in the case of its being positive and then subtract them from their respective abraders. One of these results, either the smaller one or the greater one, should be made- negative and the other positive." 2 Illustrative Examples. Examples with solutions from Bhaskara II : 3 Example 1. \$y = — Gox ± 3. By the method described before we find that the minimum solution of i$y = 6ox+ 3 is x — 1 1 , _y = j 1 . Subtracting these values ftpm their respective abraders, namely 13 and 60, we get 2 and 9. Then by the maxim. "In the case of the dividend and divisor being of different signs, the results from the operation of division should be known to be so," making the quotient negative we get the solution of 1 3_y = — 6oa* + 3 as x = 2, y = — 9. Subtracting these values again from their respective abraders (13, 60), we get the solution of 1 $v = — 6o>r — 3 as x = 11, y= — j 1. Example 2. — iiy = \%x ^ 10. Proceeding as before we find the minimum solution of \\y = 1 Sx + 10 1 BBi, p. 29. 2 NBi, I, R. 63. *BBi, pp. 29, jo. * solution of by -\- ax =if n^ to be x = 8, y = 14. These will also be the values of x and y in the case of the negative divisor but the quotient for the reasons stated before should be made negative. So the solution of — xxy = \%x -f- 10 is x =■ 8, y = — 14. Subtracting these (i.e., their numerical values) from their respective abfaders, we get the solution of — 1 \y = i%x — 10 as x = 3, y == — 4. "Whether the divisor is positive or negative, the numercial values of the quotient and multiplier remain rhe same: when either the divisor or the dividend is negative, the quotient must always be known to be negative." The following example with its solution is from the algebra of Narayana : x 1J=— 3 *± 3- The solution of 1J = 3 * + 3 is x = 2, y = 9. Subtracting these values from the res- pective abraders, namely 7 and 30, and making one of the remainders negative, we get x = 5, y = — 21 and x — — 5, j* = 2i respectively as solutions of 1J= — }°*± 3- Particular Cases. The Hindus also round special types of general solutions of certain particular cases of the equation by -f- ax = c . For instance, we find in the Ganita-sdra-samgraha of Mahavira (850) problems of the following type : "The varna (or colours) of two pieces of gold weighing 16 and 10 are unknown, but the mixture of NBi, I, Ex. 29. 124 ALGEBRA them has the varna 4; what is the varna of each piece of gold P" 1 If x,y denote the required varna, then we shall have i(>x + loy = 4 X 2.6 ; or in general ax + by — c{a + b). Therefore a(x — c ) = &(f — y); whence x = c ± tn\a, y — c ^F /»/£, where w is an arbitrary integer. Hence the following rule of Mahavira : "Divide unity (severally) by the weights of the two ingots of gold. The resulting varna being set down at two places, increase or decrease it at one place and do reversely at the other place, by the unity divided by its own quantity of gold (the results will be the corres- ponding varna)"' 2 ' He has also remarked that "assuming an arbitrary value for one of the varna, the other can be found as before." 3 A variation of the above problem is found in the Lildvati of Bhaskara II : "On mixing up two ingots of gold of varna 16 and 10 is produced gold of varna 12 ; tell me, O friend, the weights of the original ingots." 4 That is to say,, we shall have to solve the equation i6x + loy = i2(x +.7); or in general ax + by = c(x -\-y). Hence x = m{c — b), y = m{a — e), where m is an arbitrary integer. 1 GSS, vi. 188. 2 GSS, vi. 187. 3 GSS, vi. 189. * L, p. 26. LINEAR EQUATION IN MORE THAN TWO UNKNOWNS 1 25 Hence the rule of Bhaskara II : "Subtract the resulting varna from the higher varna and diminish it by the lower varna; the remain- ders multiplied by an optional number will be the weights of gold of the lower and higher varna respec- tively." 1 In the above example c — b = z, a — c = 4. So that, taking m = 1, 2, or 1/2, Bhaskara II obtains the values of (x, j) as (2, 4), (4, 8) or (1, 2). He then observes that in the same way numerous other sets of values can be obtained. 14. ONE LINEAR EQUATION IN MORE THAN TWO UNKNOWNS To solve a linear equation involving more than two unknowns the usual Hindu method is to assume arbi- trary values for all the unknowns except two and then to apply the method of the pulveriser. Thus Brahma- gupta remarks, "The method of the pulveriser (should be employed), if there be present many unknowns (in an equation)." 2 Similar directions have been given by Bhaskara 11 and others. 3 One of the astronomical problems proposed by Brahmagupta 4 leads to the equation : 197.x - — 1644J — £ — 6302. Hence * = '644J>+ g + 6302 The commentator assumes ^ = 131. Then _ 1644?+ 6433 . 197 ' 1 .L, p. 25. 2 BrSpSi, xviii. 5.1. 3 BBL p. 76. * BrSpSi, xviii. 55. 126 ALGEBRA hence by the method of the pulveriser The following example with its solution is from the algebra of Bhaskara II : "The numbers of flawless rubies, sapphires, and pearls with one person are respectively j, 8 and j- f and O friend, another has 7, 9 and 6 respectively of the same gems. In addition they have coins to the extent of 90 and 62. They are thus equally rich. Tell quickly, O intelligent algebraist, the price of each gem." 1 If x, j, % represent the prices of a ruby, sapphire and pearl respectively, then by the question 5* + 8y + 73: + 90 = jx + 9j + 63; + 62. Therefore x = — •' > ' — . 2 Assume % = 1 ; then whence by the method of the pulveriser, we get x = 14 — m, y = zm -\- 1, where m is an arbitrary integer. Putting m = o, 1, 2, 3,... we get the values of (x,j, ^) as (14, 1, 1), (13^ 3, 1), (12, j, 1), (11, 7, 1), etc. Bhaskara II then observes, "By virtue of a variety of assumptions multiplicity of values may thus be obtained." Sometimes the values of most of the unknowns present in an equation are assumed arbitrarily or in terms of any one of them, so as to reduce the equation to a simple determinate one. Thus Bhaskara II says : "In case of two or more unknowns, x multiplied by 2 etc. (i.e., by arbitrary known numbers), or divided, 1 BBi, p. 77. SIMULTANEOUS INDETERMINATE EQUATIONS IZJ increased or decreased by them, or in some cases (simply) any known values may be assumed for the other unknowns according to one's own sagacity. Knowing these (the rest is an equation in one un- known)." 1 The above example has been solved again by Bhaskara II in accordance with this rule thus : 2 (i) Assume x == 33;, j = z%. Then the equation reduces to 3 8 ^-f 9° = 45 : <:4- 62. Therefore £ = 4. Hence x = 12, j = 8. (2) Or assume y = j, ^ = 3. Then the equation becomes ]X + IJI = JX + I2J. " Whence x — 13. 15. SIMULTANEOUS INDETERMINATE EQUATIONS OF THE FIRST DEGREE Ssrlpati's Rule. We have described before the rule of Brahmagupta for the solution of simultaneous equa- tions of the first degree. 3 In the latter portion of that rule there are hints for the solution of simultaneous indeterminate equations by the application of the method of the pulveriser. Similar rules have been given by later Hindu algebraists. Thus Sripati (1039) says : "Remove the first unknown from any one side of an equation leaving the rest, and remove the rest from the other side. Then find the value of the first by dividing the other side by its coefficient. If there be found thus several values (of the first unknown), the same (opera- 1 BBi, p. 44. * BBi, p. 46. s See pp. 54f. 128 ALGEBRA tions) should be made again (by equating two and two of those values) after reducing them to a common deno- minator. (Proceed thus repeatedly) until there results a single value for an unknown. Now apply the method of the pulveriser ; and from the values (determined in this way) the other unknowns will be found by pro- ceeding backwards. In the pulveriser the multiplier will be the value of the unknown associated with the dividend and the quotient, of that with the divisor." 1 Bhaskara IPs Rule. Bhaskara II (1150) writes : " Remove the first unknown from the second side of an equation and the others as well as the absolute number from the first side. Then on dividing the second side by the coefficient of the first unknown, its value will be obtained. If there be found in this way several values of the same unknown, from them, after reduction to a common denominator and then dropping it, values of another unknown should be determined. In the final stage of this process, the multiplier and quotient obtained by the method of the pulveriser will be the values of the unknowns associated with the dividend and the divisor (respectively). If there be several unknowns in the dividend, their values should be determined after assuming values of all but one arbitrari- ly. Substituting these values and proceeding reversely, the values of the other unknowns can be obtained. If on so doing there results a fractional value (at any stage), ♦-.he method of the pulveriser should be employed again. Then determining the (integral) values of the latter unknowns accordingly and substituting them, the values of the former unknowns should be found proceeding reversely again." 2 A similar rule has been given by JMnaraja. 1 SiSe, xiv. 15-6. 2 BBi, p. 76. SIMULTANEOUS INDETERMINATE EQUATIONS 1 29 Example from Bhaskara II: " (Four merchants), who have horses 5, 3, 6 and 8 respectively ; camels 2, 7, 4 and 1 ; whose mules are 8, 2, 1 and 3 ; and oxen 7, 1, 2 and 1 in number ; are all owners of equal wealth. Tell me instantly the price of a horse, etc." 1 If x, y, %, w denote respectively the prjces of a horse, a camel, a mule and an ox, and W be the total wealth of each merchant, we have 5* + V> 4 *Z 4 7^ = W (0 3* 4 iy + z z + * = w (*) 6* + 4JV 4 £ 4- w = W (3) %x+y+ iZ + »>=W (4) Then x = £(jj — 6^ — 6»), from (1) arid (2) = s(lj + Z — »)» from (2) and (3) = i(3JV — ^ 4- »), from (3) and (4) From the first and second values of x, we' get and from the second and third values, we have y = K*Z - J*0- Equating these two values of y and simplifying, 203; 4 i6iv = 243; — ijjy. Therefore v = ^ — . ^ 4 Take » = 4/ ; then 3 = 3i'» J = 76/, ^=85/. Special Rules. Bhaskara II observes that the physical conditions of problems may sometimes be such that the ordinary method of solving simultaneous in- 1 BBi, p. 79- 9 I JO ALGEBRA determinate equations of the first degree, which has been just explained, will fail to give the desired result. One • such problem has been described by him as follows : "Tell quickly, O algebraist, what number is that which multiplied by 23 and severally, divided by 60 and 80 leaves remainders whose snm is ioo." 1 Let the number be denoted by x ; the quotients by u, v ; and the remainders by s, t. Then we have 23*: — also Therefore 60 = *> 2$X — t 80 V s+t — IOO. 60a -+- s __ 8oy -f- t 23 23 T T 6ou 4- 8o# 4- s + t Hence x — ! ~ ■ — , 46 or x= 3Q« 4- 40^ + 50 For the solution of the above he observes : "Here, (although) there is more than one quotient («, v) in the dividend, the value of any should not be arbitrarily assumed ; for on so doing the process will fail." 2 "In a case like this," continues he, "the (given) sum of the remainders should be so broken up that each remainder will be less than the divisor corres- ponding to it and further that impossibility will not arise ; then must be applied the usual method." 1 In the present example we thus suppose s = 40, / — 60. Hence we have *6o# + 40 = 801* -f- 60 1 BBi, p. 91. » BBr, p. 91k SIMULTANEOUS INDETERMINATE EQUATIONS \$\ %OV -\- ZO dV -\- 1 or u = 7 = — - — ; 60 3 whence by the method of the pulveriser, we get v = 3»> -f- 2, « = 4JP + 3- Therefote x = — . 23 Again, applying the method of the pulveriser in order to obtain an integral value of x, we have w = 23 /w -f- 1, x = 240/w + 20. If we take s = 30, / = 70, we shall find, by proceed- ing in the same way, another value of x as 240/9 + 90. General Problem of Remainders. One type of simultaneous indeterminate equations of the first degree is furnished by the general problem of remainders, «!£., to find a number JV which being severally divided by a lt a % , a 3 , ..., a n leaves as remainders r v r 2 , r 3 , ..., r n respectively. In this case, we have the equations N = a x x x + r a = aje z + r 2 = ^3X3 + r 8 = ... = ***■« + >V The method of solution of these equations was known to Aryabhata I (499). For this purpose the term dviccheddgram occurring in his rule for the pulveriser must be explained in a different way so that the last line of the translations given before (pp. 94-5) will have to be replaced by the following: "(The result will be) the remainder corresponding to the product of the two divisors." 1 This explanation is, in fact, given by Bhaskara I, the direct disciple and earliest commentator of Aryabhata I. Such a rule is expressly stated by 1 See Bibhutibhusan Datta, BCMS, XXIV, 19} 2. t}Z ALGEBRA Brahmagupta. 1 The rationale of this method is simple : Starting •with the consideration of the first two divisors, we have N = a 1 x 1 + r x = a£c 2 + r 2 . By the - method described before we can find the minimum value a of x^_ satisfying this equation. Then the minimum value of N will be a x o. -j- r v Hence the general value of N will be given by AT = ai (aj + a) + r v = a^a 2 t + a x n -f r lt ' where t is an integer. Thus a 2 a + r x is the remainder left on dividing JV by a 1 a 2 , as stated by Aryabhata I and Brahmagupta. Now, taking into consideration the third condition, we have N = a x a 2 t + a-,0. + r t — agX 3 + r 3 , which can be solved in the same way as before. Pro- ceeding in this way successively we shall ultimately arrive at a value of N satisfying all the conditions. Prthudakasvami remarks: "Wherever the reduction of two divisors by a common measure is possible, there 'the product of the divisors' should be understood as equivalent to the product of the divisor corresponding to the greater remainder and quotient of the divisor corresponding to the smaller remainder as reduced (i.e., divided) by the common measure. 2 When one divisor is exacdy divisible by the other then the greater remainder is the (required) remainder and the divisor corresponding to 1 BrSpSi, xviii. 5 . 2 i.e., if p be the L.C.M. of a x and a 2 , the general value of N satisfying the above two conditions will be N = pt + a ia + /"i instead of N = a t a 2 t + a x a + r x . SIMULTANEOUS INDETERMINATE EQUATIONS 1 35 the greater remainder is taken as 'the product of the divisors.' (The truth of) this may be investigated by an intelligent mathematician by taking several symbols." 'Examples from Bhaska'ra I : (1) "Find that number which divided by 8 leaves 5 as remainder, divided by 9 leaves 4 as remainder and divided by 7 leaves 1 as remainder." 1 That is to say, we have to solve The solution is given substantially thus : The minimum value of N satisfying the first two conditions N = 8a- + 5 = 9 j + 4 is found by the method of the pulveriser to be 13. This is the remainder left on dividing the number by the product 8.9. Hence N = -jzt+ 13 = -jz+ i- Again, applying the same method we find the minimum number satisfying all the conditions to be 85. (2) "Tell me at once, O mathematician, that number which leaves unity as remainder when divided by any of the numbers from z to 6 but is exactlv divisible by 7." By the same method, says Bhaskara I (522), the number is found to be 721. By a different method Suryadeva Yajva obtains the number 301. It is in- teresting to find that this very problem was afterwards treated by Ibn-al-Haitam (c. 1000) and Leonardo Fibonacci cf Pisa (c. 1 202). 2 To solve a problem of this kind Bhaskara II adopts 1 See his commentary on A, ii. .3 2-3 . 2 L.E. Dickson, History of the theory of Numbers, Vol. II, referred to hereafter as Dickson, Numbers II, pp. 59, 60. 134 ALGEBRA two methods. One is identical with the method of Aryabhata I and the other follows from his general rule for the solution of simultaneous indeterminate equations of the first degree. They will be better understood from his applications to the solution 1 of the following problem which, as Prthudakasvami (860) observes, 2 was popular amongst the Hindus : To find a number N which leaves remainders 5, 4, 3, 2 when divided by 6, 5, 4, 3 respectively. /.*., N == 6x 4- 5 = 5jv 4- 4 = 4£ 4- 3 = 3^ 4- i. (1) We have ~ — V—i v _ 4? - * r _ 3^ ~ 1 6 ' J_ 5 ' ^~ 4 ' Now by the method of the pulveriser, we get from the last equation w = At + 3. r: = 3' + *, where / is an arbitrary integer. Substituting in the second equation, we get 5 To make this integral, we again apply the method of the pulveriser, so that t= JJ + 4, J=\2S+ 11. This value of jr makes x a whole number. Hence we have finally n> = 2.0s + 19, or b 2 n = b x m + (a x — a 2 ); supposing a x > a 2 . Solving this equation, we can find the value of m and hence of b x m -f- a x of x satisfying both the equations. The general value of x derived from this may be- equated to the value of x from the third equation and the resulting equation solved again, and so on. In illustration, of his rule Mahavira proposed several problems. One of these has already been given (Part I, p. 233). Here are two others : (1) "Five (heaps of fruits) added with two (fruits) were divided (equally) between nine travellers ; six (heaps) added with four (fruits) were divided amongst eight ; four (heaps) increased by one (fruit) were divided 1 SiSe, xiv. 28. SIMULTANEOUS INDETERMINATE EQUATIONS 1 39 amongst seven. Tell the number (of fruits in each heap)." 1 This gives the equations : 9Ji = 5* + 2 » 8 Ja = 6x- + 4, 7 j 3 = 4V + 1. (2) "The (dividends) are the sixteen numbers beginning with 3 j and increasing successively by three ; divisors are 32 and others successively increasing by 2 ; and 1 increasing by 3 gives the remainders positive and negative. What is the unknown multiplier ?" 2 This gives the equations : 3*Vi = 35* ± 1, 3472 = 3 8 *" ± 4, 36/3 = 4**" ± 7» ••• Alternative Method. In four palm-leaf manus- cript copies of the Lildvati of Bhaskara II Sarada Kahta Ganguly discovered a rule describing an alternative method for the solution of the generalised conjunct pulveriser. 3 There is also an illustrative example. The genuineness of this rule and example is accepted by him; but it has been questioned by A. A. Krishna- swami Ayyangar* who attributes them to some commen- tator of the work. His arguments are not convincing. 5 The chief points against the presumption, which have been noted also by Ganguly, are : (1) the rule and example in question have not been mentioned by the earlier commentators of the Uldvati and (2) they have not been so far traced in any manuscript of the Btjaganita, though the treatment of the pulveriser occurs nearly word for 1 GSS, vi. i2 9 £. a GSS, vi. 138J. 3 S. K. Ganguly, "Bhaskaricarya and simultaneous indeter- minate equations of the first degree," BCMS, XVII, 1926, pp. 89- 98. 4 A. A. Knshnaswami Ayyangar, "Bhaskara and samslishta Kuttaka," JIMS, XVIII, 1929. 6 For Ganguly's reply to Ayyangar's criticism se&JIMS, XIX, 193 1. 140 ALGEBRA word in the two works. Still we are in favour of accepting Ganguly's conclusion. 1 The rule in question is this : "If the divisors as well as the multipliers be different, find the value of the unknown answering to the first set of them. That value being multiplied by the second dividend and then added by the second interpolator will be the interpolator (of a new kuttakd); the product of the second dividend and first divisor will be the dividend there and the divisor will be the second divisor. The value of the unknown multiplier determined from the kiittaka thus formed being multiplied by the first divisor and added by the previous value of the unknown multi- plier will be the value (answering to the two divisors). The dividend (for the next step) has been stated to be equal to the product of the two divisors. So proceed in the same way with the third divisor. And so on with the others, if there be many." The rationale of this rule is as follows : Let a x be the least value of x satisfying the first equation of the system, vi^., Hence the general value is x = b x t -f- <*!> where / is any integer. Substituting this value in the second equation, we get b iJi = tf aV + ( a 2 a i ± r z)- If t = x be a solution of this equation, a value of x 1 Of the four manuscripts containing the rule and example in question two are from Pun, in Oriya characters, with the com- mentary of Sridhara Mahap&tra (1717) ; the other two, in Andhra characters and without any commentary, are preserved in the Oriental Libraries of Madras and Mysore. So these four manus- cript copies do not appear to have been drawn from the same source." This is a strong point in favour of the genuineness of the rule and example. SOLUTION OF N!v 2 + I =J 2 . 141 satisfying both the equations will be cu = b x x + « x as stated in the rule. Now the general value of / will be t = £ a /» -f- T » where m is an integer. Hence x = b x t + cij = ^ 2 «r + i^t + cij = ^iy^ + a 2 . Subs- tituting this value in the third equation we can find the least value of m and hence a value of x answering to the three equations. And so on for the other equations. The example runs thus : "Tell me that number ^which multiplied by 7 and then divided by 62, leaves the remainder 3. That number again when multiplied by 6 and divided by 101 leaves the remainder j ; and when multiplied by 8 and divided by 1 7 leaves the remainder 9. Also (give) at once the process of the pulveriser for (finding) the number with the remainders all positive." Symbolically, we have (0 6z Ji = 7* — 3> 1017, = 6x — 5, i 7 j 3 = 8x — 9; (2) 62^ == 7 x + 3, ioija = 6x + j, 17J3 = 8x + 9. 16. SOLUTION OF Nx* + 1 =j a Square-nature. The indeterminate quadratic equation Nx* ±, p. 100). Compare also "Tatra yavattavadvarge yo'nkah si prakrtih" (p. 107) ; "Istarh hrasvarh tasya vargah prakrtya ksunno..." (p. 33). * "Tatraikarh varnakrtim prakrtirh prakalpya..." (BBi, p. 106). Compare also "Sarupake varnakrti tu yatra tatrecchaikarh prakrtirh prakalpya..." (p. 105). SOLUTION OF Nx 2 -+ I =J' 2 I4J. employed the term prakrti to denote N only. 1 Brahma- gupta (628) uses the term gunaka (multiplier) for the same purpose. 2 This latter term, together with its variation guna, appears occasionally also in later works.* We presume that the name varga-prakrti origi- nated from the following consideration : The principle {prakrti) underlying the calculations in this branch of mathematics is to determine a number (or numbers) whose nature {prakrti) is such that its (or their) square (or squares, varga) or the simple number (or numbers) after certain specified operations will yield another number (or numbers) of the nature of a square. So the name is, indeed, very significant. This interpretation •seems to have been intended, at any rate, by the earlier writers who used the term in a wider sense. 4 It is perhaps noteworthy that we do not find in the works of Brabmagupta the use of the word prakrti either in the sense of N or of x 2 . Technical Terms. Of the various technical terms which are ordinarily used by the Hindu algebraists in connection with the Square-nature we have already dealt with the most notable one, prakrti, together with its synonyms. Others have been explained by Prthu- dakasvami (860) thus : "Here are stated for ordinary use the terms which 1 For instance, Prthudakasvami (860) -writes : "The multiplier (of the square of the unknown) is known as the prakrti ;" Sripati (1039) : "Krter-gunako prakrtirbhrs'oktah" (SiSe, xiv. 32) ; Kamalakara : "Guno yo rasi-vargasya saiva prakrtirucyate." 2 BrSpSi, xviii. 64. 3 For instance, Sripati employs the term gunaka (S/Se, xiv. 32); Bhaskara II and Narayana use guna (BBi, p. 42 ; NBi, I, R. 84). 4 For instance, Brahmagupta seems to have considered the scope of the subiect wide enough to include such equations as x +j = * 2 , x —j = v \ *y + 1 = a?, amongst others (cf. BrSpSi, xviii. 72). 144 ALGEBRA are well known to people. The number whose square, multiplied by an optional multiplier and then increased or decreased by another optional number, becomes capable of yielding a square-root, is designated by the term the lesser root {kanistha-pada) or the first root {ddya-muld). The root which results, after those operations have been performed, is called by the name the greater root (jyestha-pada) or the second root {anya-mtdd). If there be a number multiplying both these roots, it is called the augmenter (udvartakd) ; and, on the contrary, if there be a number dividing the roots, it is called the abridger (apavartaka)" 1 Bhaskara II (1150) writes : "An optionally chosen number is taken as the lesser root (brasva-mula). That number, positive or negative, which being added to or subtracted from its square multiplied by the prakrti ' (multiplier) gives a result yielding a square-root, is called the interpolator \ksepakd)'. And this (resulting) root is called the greater root (Jyestha-mula)." 2 Similar passages occur in the works of NSrayana, 3 Jfianarslja and Kamalakara. 4 The terms 'lesser root' and 'greater root' do not appear to be accurate and happy. For if x = m, y = n be a solution of the equation Nx 2 + c =y 2 , m will be less than n, if N and c are both positive. But if they are of opposite signs, the reverse will sometimes happen. 5 1 See Prthudakasvami's commentary on BrSpSi, xviii. 64. In the equation Nx* ± e =J Z , x = lesser root, y = greater root, H = multiplier, and e = interpolator. »BB/,p. 3 j. 8 NBi, I, R. 72. * SiTVi, xiii. 209. B For instance, take the following example from Bhaskara II (BBi, p. 4j) : SOLUTION OF Nx 2 + I =J 2 145 Therefore, in the latter case, where m > », it will be obviously ambiguous to call tn the lesser root and « the greater root, as was the practice in later Hindu algebra. This defect in the prevalent terminology was noticed by Krsna (1580). He explains it thus : "These terms are significant. Where the greater root is some- times smaller than the lesser root owing to the inter- polator being negative, there also it becomes greater than the lesser root after the application of the Principle of Composition." 1 The earlier terms, 'the first root' (adya-mtf/a) for the value of x and 'the second root' or 'the last root' {antya-muld) for the value of y, are quite free from ambiguity. Their use is found in the algebra of Brahmagupta (628). 2 The later terms appear in the works of his commentator Prthudakasvami (860). The interpolator is called by Brahmagupta ksepa, praksepa or praksepaka? Sripati occasionally employs the synonym ksipti* When negative, the interpolator is sometimes distinguished as 'the subtractive' (sodhakri). 13X 2 — 13 =.y. One solution of it is given by the author asw= 1,^ = 0; .so that here the lesser root is greater than the greater root. The same is the case in the solution x = z, y = 1 of his example (BBr, p. 43) — 5X 2 +21 =_>> 2 . Brahmagupta gives the example (BrSpSi, xviii. 77) 3X 2 — 800 =j 2 , •which has a solution (x = 2o,_y = 20) where the two roots are equal. 1 For example, by composition of the solution (1, o) of the equation 13X 2 — 13 =j 2 with the solution (f, ty) of the equation i}x 2 + 1 =j a , we obtain, after Bhaskara II, a new solution (V"> -tp) of the former, in which the greater root is greater than the lesser root. Similarly, by composition of the solution (2, 1) of the equation — 5X 2 +21 =j 2 with the solution (J, §) of the equation — 5x 2 -}- 1 = > 2 , we get a new solution (i, 4) of the former satisfying the same condition. 2 BrSpSi, xviii. 64, 66f. 3 BrSpSi, xviii. 65. 4 SiSe, xiv. 3 2. IO 146 ALGEBRA The positive interpolator is then called 'the additive.' 1 Brahmagupta's Lemmas. Before proceeding to the general solution of the Square -nature Brahmagupta has established two important lemmas. He says : "Of the square of an optional number multiplied by the gunaka and increased or decreased by another optional number, (extract) the square-root. (Proceed) twice. The product of the first roots multiplied by the gunaka together with the product of the second roots will give a (fresh) second root ; the sum of their cross- products will be a (fresh) first root. The (corresponding) interpolator will be equal to the product of the (previous) interpolators." 2 The rule is somewhat cryptic because the word dvidhd (twice) has been employed with double implica- tion. According to one, the earlier operations of finding roots are made on two optional numbers with two optional interpolators, and with the results thus obtained the subsequent operations of their composition are performed. According to the other implication ' of the word, the earlier operations are made with one optionally chosen number and one interpolator, and the subsequent ones are carried out after the repeated statement of those roots for the second time. It is also implied that in the composition of the quadratic roots their products may be added together or subtracted from each other. That is to say, if x = a, j = (3 be a solution of the equation Nx 2 + k =f, and x = a', j = f>' be a solution of Nx 2 + k' = j 2 , then, according to the above, 1 BrSpSr, xviii. 64-5. 2 BrSpSi, xviii. 64-5. SOLUTION OF Nx 2 + I = J 2 147 X = <*P' ± <*'p, J = PP' ± Naa' is a solution of the equation Nx 2 + kk' = j 2 . In other words, if IVa 2 + £ = p 2 , . Na' 2 4- k' = P' 2 , then N(«P' ± a'P) 2 + ^' = (PP' ± N a') 2 . (I) In particular, taking « = a', p = p', and k — /4', Brahmagupta finds from a solution x = a, y = p of the equation Nx 2 + >fe = y, a solution ,*• = 2ap, j = p 2 -f- IVa 2 of the equation Nx 2 + k 2 =y 2 . That is, if IVa 2 + £ = p 2 , then JV(2ap) 2 + k? = (p 2 + ATa 2 ) 2 . (II) This result will be hereafter called Brahmagupta' s Corollary. Description by Later Writers. Brahmagupta's Lemmas have been described by Bhaskara II (1150) thus : "Set down successively the lesser root, greater root and interpolator ; and below them should be set down in order the same or another (set of similar quantities). From them by the Principle of Composition can be obtained numerous roots. Therefore, the Principle of Composition will be explained here. (Find) the two cross-products of the two lesser and the two greater roots ; their sum is a lesser root. Add the product of the two lesser roots multiplied by the prakrti to the product of the two "greater roots ; the sum will be a greater root. In that (equation) the interpolator will be 148 ALGEBRA the product of the two previous interpolators. Again the difference of the two cross-products is a lesser root. Subtract the product of the two lesser roots multiplied by the prakrti from the product of the two greater roots; (the difference) will be a greater root. Here also, the interpolator is the product of the two (previous) inter- polators." 1 Statements similar to the above are found in the works of Narayana 2 (1350), Jnanaraja (1503) and Kamalakara 3 (1658). Principle of Composition. The above results are called by the technical name, Bbdvand (demonstration or proof, meaning anything demonstrated or proved, hence theorem, lemma ; the word also means composi- tion or combination). They are further distinguished as Samdsa Bbavand (Addition Lemma or Additive Composi- tion) and Antara Bbavand (Subtraction Lemma or Sub- tractive Composition). Again, when the Bbavand is made with two equal sets of roots and interpolators, it is called Tulya Bbdvand (Composition of Equals) and when with two unequal sets of values, Atulya Bbdvand (Compo- sition of Unequals). Krsna has observed that when it is desired to derive roots of a Square-nature, larger in value, one should have recourse to the Addition Lemma and for smaller roots .one should use the Subtraction Lemma. Brahmagupta's Lemmas were rediscovered and recognised as important by Euler in 1764 and by Lagrange in 1768. Proof. The proof of Brahmagupta's Lemmas has been given by Krsna substantially as follows : 1 BBi, p. 34. * NBi, I, R. 72-7 5 £. 3 SiTVi, xiii. 210-214. SOLUTION OF Nx 2 -f- I —J 2 1 49 We have Na 2 + k = P 2 , iVa' 2 + k' = p' 2 . Multiplying the first equation by P' 2 , we get s Na^' 2 -f ifep'a = pap'2 # Now, substituting the value of the factor p' 2 of the interpolator from the second equation, we get NaY* + k(Na'> + i') = pap'2, or lVa 2 p'^ + N£a'2 + &£' = p 2 p' 2 . Again, substituting the value of k from the first equa- tion in the second term of the left-hand side expression, we have 2V a 2|3'2 + jVa' 2 (p 2 - Na 2 ) + kk' = p 2 p' 2 , or iV(a2p'2 + a'-fpa) _|- kk' = P 2 P' 2 + N*a*cf*. Adding ^ 2NaPa'p' to both sides, we get iV(ap' ± a'p) 2 + &£' = ((3(3' -J- iVaa') 2 . Brahmagupta's Corollary follows at once from the above by putting a' = a, |3' = p and k' = k. General Solution of the Square-Nature. It is clear from Brahmagupta's Lemma (I) that when two solutions of the Square-nature, Nx 2 + 1 =jv 2 , are known, any number of other solutions can be found. For, if the two solutions be (a, b) and {a', b'), then two other solutions will be x = ab' ± a'b, j = bV ± Naa'. Again, composing this solution with the previous ones, we. shall get other solutions. Further, it follows from Brahmagupta's Corollary that if (a, b) be a solution of the equation, another solution of it is (zab, b 2 + Na 2 ). Hence, in order to obtain a set of solutions of the I 5 O ALGEBRA Square-nature it is necessary to obtain only one solu- tion of it. For, after having obtained that, an infinite number of other solutions can be found by the repeated application of the Principle of Composition. Thus Sripati (1039) observes : "There will be an infinite (set of two roots)." 1 Bhaskara 11 (11 50) remarks : "Here (i.e., in the solution of the Square-nature) the roots are infinite by virtue of (the infinitely repeated application of) the Principle of Composition as well as of (the infinite variety of) the optional values (of the first roots)." 2 Narayana (1350) writes, "By the Principle of Composition of equal as well as unequal sets of roots, (will be obtained) an infinite number of roots." 3 Modern historians of mathematics are incorrect in stating that Fermat (1657) was the first, to assert that the equation Nx 2 + 1 = y 2 , where JV is a non-square integer, has an unlimited number of solutions in inte- gers. 4 The existence of an infinite number of integral solutions was clearly mentioned by Hindu algebraists long before Fermat. \ Another Lemma. Brahmagupta says : "On dividing the two roots (of a Square-nature) by the square-root of its additive or subtractive, the roots for the interpolator unity (will be found)." 5 That is to say, if x = a, y — (3 be a solution cf the equation Nx 2 + k 2 =y\ then x = ajk, y — $jk is a solution of the equation Nx 2 + 1 ^y 2 . This rule has been restated in a different way thus : 1 SiSe, xiv. 3 3 . 2 "Ihanantyam bhavanabhistathestatah" — BB', p. 34. " NBi, I, R. 78. Compare also SiTVi, xiii. 217. 4 Smith, History, II, p. 453. 5 BrSpSi, xviii. 65. SOLUTION OF Nx 2 + I —J 2 151 "If the interpolator is that divided by a square then the roots will be those multiplied bv its square- root." 1 That is, suppose the Square-nature to be Nx 2 ±p 2 d=j 2 , so that its interpolator p 2 d is exactly divisible by the square p-. Then, putting therein u = x/p, v =-jfp, we derive the equation JVw 2 ± d = v\ whose interpolator is equal to that of the original Square-nature divided by p 2 . It is clear that the roots of the original equation are p times those of the derived equation. Bhaskara II writes : "If the interpolator (of a Square-nature) divided by the square of an optional number be the interpolator (of another Square-nature), then the two roots (of the former) divided by that optional number will be the roots (of the other). Or, if the interpolator be multi- plied, the roots should be multiplied." 2 The same rule has been stated in slightly different words by Narayana 3 and Kamalakara. 4 Jhanaraja simply observes: "If the interpolator (of a Square-nature) be divided by the square of an optional number then its roots will be divided by that optional number." Thus we have, in general, if x = a, y = p be r . solution of the equation Nx 2 ± k = j 2 , 1 BrSpSi, xviii. 70. 2 BBi, p. 54. 8 NBi, I, R. 76-764. « SiTVi, xiii. 215. I 5 2 ALGEBRA x = a\m, y — §\m is a solution of the equation Nx 2 ± k\m 2 =j 2 ; and x = #a, j/ = »p is a solution of the equation Nx 2 ± n 2 k =y 2 , where m, n are arbitrary rational numbers. By this Lemma, the solutions of the Square-natures (/') 6x 2 +12 =y 2 , (;;) 6x 2 + 15 =y\ and {Hi) 6x 2 +300 =y 2 , can be derived, as shown by Bhaskar'a II, 1 from those of 6x 2 -f- 3 =y 2 , since 12 = 2 2 .3, 75 = j 2 .3, and 300 = io 2 .3. How to solve this latter equation will be indicated later on. Rational Solution. In order to obtain a first solution of Nx 2 -\- 1 =j 2 the Hindus generally suggest the following tentative method : Take an arbitrary small rational number a, such that its square multiplied by the gunaka N and increased or diminished by a suitably chosen rational number k will be an exact square. In other words, we shall have to obtain empirically a relation of the form Na 2 ± k = p 2 , where a, k, p are rational numbers. This relation will be hereafter referred to as the Auxiliary 'Equation. Then by Brahmagupta's Corollary, we get from it the relation IV(2ap)2 + k i = (p 2 + jy a 2)2 3 or 1 BBi, p. 41. SOLUTION OF Nx 2 + I =J 2 153 Hence, one rational solution of the equation Nx 2 + i = j 2 is given by X -~F> y k — ' {A) Sripati's Rational Solution. Sripati (1039) nas shown how a rational solution o£ the Square-nature can be obtained more easily and direcdy without the intervention of an auxiliary equation. He says: "Unity is the lesser root. Its square multiplied by the prakrti is increased or decreased by the prakrti combined with an (optional) number whose square-root will be the greater root. From them will' be obtained two roots by the Principle of Composition." 1 If m* be a rational number optionally chosen, we have the identity N.i 2 + (w 2 — N) = m\ or iV.i 2 — (AT— /* 2 ) = m\ Then, applying Brahmagupta's Corollary to either, we get N(zm) 2 + (w 2 - JV) 2 = (w 2 + N) 2 ; N ( ™ ) 2 +1 =( m2+N )\ u m rrfl + N /m Hence x = —5 =r- T , y = —5 ==r Ti (Jt>) where //? is any rational number, is a solution of the equation Nx* + 1 =j 2 . The above solution reappears in the works of later Hindu algebraists. BMskara II says : 1 SiSe, xiv. 33. 154 ALGEBRA "Or divide twice an optional number by the differ- ence between the square of that optional number and the prakrti. This (quotient) will be the lesser root (of a Square-nature) when unity is the additive. From that (follows) the greater root." 1 Narayana states : "Twice an optional number divided by the difference between the square of that optional number and the gunaka will be the lesser root. From that with the additive unity determine the greater root." 2 Similar statements are found also in the works of Jnanaraja and Kamalakara. 3 If m be an optional number, it is stated that —5 5^ r m 2 <— N is a lesser root of Nx 2 + 1 —y 2 . Then, substituting that value of .y in the equation, we get / ^ 2 + N \ 2 ^m* ~ N> ' Hence the greater root is _ m 2 +N J ~ m 2 ~ N" The same solution will be obtained by assuming y = mx — 1. Krsna points out that it can also be found thus : 4 Nm 2 = (m 2 + N) 2 — (m 2 ~ Nf, identically. . ' . 4N/* 2 + (w 2 ~ N)» = (2) I IX 2 -f- I = J 2 . Solutions. "In the second example assume 1 as the lesser root. Multiplying its square by the prakrti, namely 11, subtracting 2 and then extracting the square-root, we get the greater root as 3. Hence the statement for / = — z iBB/, p. 35. 2 The abbreviations are : m = multiplier, / = lesser root, g = greater root and / = interpolator. In the original they are respectively pra, ka, jye, and kse, the initial syllables of the cor- responding Sanskrit terms. composition is 2 m = 11 /= 1 ") Let k = ± 2; then the auxiliary equation is No? ± 2 = p 2 . By Brahmagupta's Corollary, we have N(za$y + 4 = (p 2 + IVa 2 ) 2 , ,321 /\r a 2 . 2 N(aP) 2 +i^( P + , )■ or 2 Hence the required first solution is x=ap, y= >(p 2 + Aa 2 ). Since Na 2 = p 2 ^ 2, we have -J(p 2 -f- ATa 2 ) = P 2 ^ 1 = a whole number. (Hi) Now suppose £ = + 4 ; so that No 2 + 4 = p 2 . With an auxiliary equation like this the first integral solution of the equation Nx 2 + 1 = j 2 is if a is even; or x = -Jap, J = I(P 2 ~ 2); x- = ia(p 2 - 1), j/=4P(P?- 3 ); if p is odd. Thus Brahmagupta says : "In the case of 4 as additive the square of the second root diminished by 3, then halved and multiplied ■ by the second root will be the (required) second root ; The square of the second root diminished by unity and SOLUTION OF N.V 2 + I ~ J 2 I59 then divided by. 2 and multiplied by the first root will be the (required) first root (for the additive unity)." 1 The rationale of this solution is as follows : Since IVa 2 + 4 = p 2 , _ (1) 2 £t 2 we have N(-) + 1 = ( — ) . (2) Then, by Brahmagupta ? s Corollary, we get 7v(_^) 2 +I= (Pi + iV ^i) 2 . x 2 ' k 4 4 7 Substituting the value of N in the right-hand side ex- pression from (1), we have N(4) ! +I =(^i) S . (5) Composing (2) and (3), W-i( p '- , )} + '-{-f (»•-')}'• Hence .v = ^ i 2 or db 4 as interpolator. In order to derive integral roots corresponding to an equation with the additive unity from those of the equation with the interpolator ± 2 or i 4 tne Principle of Composition (should be applied)." 2 Suppose we have an equation of the form Na 2 + k = b\ (i) where a, b, k are simple integers, relatively prime, k being positive or negative. Then by Bhaskara's Lemma where m is an arbitrary integral number. In the above rule, m has been styled the indeterminate multiplier. Now, by means of the pulveriser, its value is determined so that — ~f— is a whole number. 1 The original text is cakravdlamidam jaguh. The commentator Krsna explains, "acarya etadganitarh cakiavalamiti jaguh" or "The learned professors call this method of calculation the Cakravdla." So Bh&skara II appears to have taken the Cyclic Method from earlier writers. But it is not found in any work anterior to him so far known. 2 BBi, pp. 36ff. 164 ALGEBRA Again, of the various such values, Bhaskara II chooses that one which will make \m 2 — N\ as small as possible. Let that value of m be n. Now let h an + b ~ k ' bn + Na k : « 2 - N The numbers 2 — 67 [ least. Substituting that in (3) we get 67.27 s — 2 — 221 2 . By the Principle of Composition of Equals, we get from this equation 67 (2.27.221) 2 + 4 = ("i 2 + 67. 2 7 3 )3, or 6 7 (ii 9 34) 2 + l 4 = (97684) 2 . Dividing out by 4, we have 67 (5967) a + 1 =(48842)*. 1 68 ALGEBRA Hence x = 5967, y = 48842 is a solution of (/'). («) 6i>r 2 + 1 = J a - Here we start with the auxiliary equation 61. i 2 -f 3 = 8 s . By the Lemma, we have 6i(^t- 8 )V m2 ~ Gl = ( 8 ^+ 61 ) 2 . (1) Now the solution of m-\- 8 ' — = an integer, is m = 3/ -f- 1. Putting / = 2, we get the value «? = 7 which makes \ m 2 — 61 | least. On substituting this value in (1), it becomes 61. 5 2 — 4= 39 s - Dividing out by 4, we get 6i(|) 2 -i=(- 3 2 9 -) 2 - (*) By the Principle of Composition of Equals, we have 6l(2.|.^-)2 + , = {( y_ )2 + 6l(|) 2 } 2 , or 6i(J-|^) 2 + 1 = (J-y-) 2 - (3) Combining (2) and (3), 61(3 805) 2 - 1- (2971 8) 2 . Composing this with itself, we get 61(2261 5 3980) 2 + 1 = (1766319049) 2 . Hence x = 226153980, ^=1766519049 is a solution of («). The following two examples have been cited by Narayana : (/«) 103X 2 4- 1 =J' 2 > (/V) 97X 2 4- 1 = y. CYCLIC METHOD 1 69 Their solutions are given substantially as follows : For {itt) we have the auxiliary equation 103 .i 2 — 3 = io 2 . By the Lemma, we get / m + io\ 2 , m 2 — 103 /iow+ 103 \ 2 103 ( ) H = ( )• v — 3 ' — 3 V — 3 ' ^ The general solution of m + 10 ! = an integer, is /» = _ 3/ _)_ 2. Putting / = — 3, we get m = 11. Then 103 -7 2 — 6 = 71 2 . Again, by the Lemma, IQ3 (7 «+ 7M 2 I « 2 — 103 _/ 7 I »+ iQ3-7 \ 2 _ The solution of 7* + 7i — a whole number, — 6 is n = — dt -\- 1 . Taking t = — 1 , we get 103 . zo 2 + 9 = 203 2 .. Next, we have /2o/> + 203 \ 2 ,p 2 — 103 /2.o$p + 103 . 20\ 2 \ 9 ' 9 ^ 9 ' ° Now, — *-— - = an integral number for p = 9/+ 2. When / = 1, p = 11. On taking this value we find 103.472+ 2 = 477 2 . 170 ALGEBRA Applying the Principle of Composition of Equals, we get 103(2.47.477)2 + 4 = (47 7 2 + 103. 4 7 2 ) 2 , or 103(44838)2 + 4 = (45 5056) 2 . Hence io3(224i9) 2 + 1 = (227528)^ which gives x = 22.419, j = 227528 as a solution of (///'). For the solution of (iv) the auxiliary equation is 97- l2 + 3 = iq2 - Therefore •m -j- io\ 2 w 2 — 97 _ / 10m + 97 < 971 3 The solution of m + 10 ) 4. m ~ 97 _ / 'Q»t 97 r an integer, 3 is m = 3/ + 2. Taking / = 3, w T e have w =11. Then 97.72+8 = 69.2 Next, we have 97 ( 7*+ 6 9 ) 2 | « 2 - 97 _ / ^+97-7 \ 2 The solution of 7/z + 69 - — 5 — - = an integer, 8 is n = 8/+ j. Taking t = 1, that is, « =13, we get 97.20 2 + 9 = 197 2 . Whence 91 / zo P+ T 9 7\ 2 j /> 2 — 97 ,_ /i9 7/> + 97-2Q \ 2 The solution of — ^ — <1 =-a whole number, CYCLIC METHOD 171 is p = 9/ + 5- Putting / = 1, we get ^> = 14. With this value of p we have 97-53 2 + 11 = 5" 2 - Whence /li£±J££\ 2 , ? 2 — 97 = / 5"? + 97-53 \ 2 ^ ^ n ' 11 ^ 11 ' The solution of 53^+ 522 . ^ ' ?Y — '- — = an integer, is q= 11/+ 8. The appropriate value of ^ is given by t = o. So, taking ^ = 8, we have 97- 86 2 - 3 = 8 4 7 2 - Next, we find / 86r+8 47 x | r 8 -97 = / 8 4 7r+97-8 = ^~' , , bn + Na and b x = ^ , we have k {a x ti — b x ) — a(n 2 — N), or A ( an _ h) = « 2 - N. a k Therefore — (a x n — b-^) is an integer. Since k and a have no common factor, a must divide a x ti — b x ; that is a,n — b A n % — N , -* * = — k-, = an integer. a k Hence b x also is a whole number. 1 1 Hankel's Proof: Hankel proves these two results thus : Since a-Je. = an -\- b and k = b 2 — Na z , we get a^b* — Na*) = an + b, or — {a, b — 1) = (» + Naa{). a Since a, b have no common factor, a must divide aj> — 1 ; that is, a.b — 1 - = an integer. solution of Nx 2 ^ c =J> 2 173 1 8. SOLUTION OF Nx 3 ± c =j a The general solution of the indeterminate quadratic equation Nx 2 ± c =f in positive integers was first given by Brahmagupta(628). He says : "From two roots (of a Square-nature) with any given additive or- subtractive, by making (combination) with the roots for the additive unity, other first and second roots (of the equation having) the given additive or subtractive (can be found)." 1 Eliminating n between ■ a x k = an -f- b, b x k = bn -f- Na, we get a x b — ab x = i. Hence b x — — = a whoJe number. now +-N = <+*- b r- N * a 2 _ a* k? — zbka! -I- k ~~ a 3 _ k{a^ k — i.ba l + i) ~~ a 3 k Therefore — -^ (a^k — iba x -f- i) is a whole number. Since a, k have no common factor, it follows that -* ^-* = — - k — = k x = an integer. Also k » 2 -N = a 1 *k-zba 1 +i • _ a^(b 3 — Na 3 ) - zba y + i — a 3 a 1 BrSpSi, xviii, 66. (^)-NV. 174 ALGEBRA Thus having known a single solution in positive integers of the equation Nx 2 ± c — j* 2 , says Brahma- gupta, an infinite number o' x other integral solutions can be obtained by making use of the integral solutions of Nx 2 -f- i = j 2 . If (p, q) be a solution of the former equation found empirically and if (a, (3) be an integral solution of the latter then, by the Principle of Com- position, x=p$±qa, j = tf±Npa will be a solution of the former. Repeating the opera- tions we can easily deduce as many solutions as we like. This method reappears in later Hindu algebras. Bhaskara II says : "In (a Square-nature) with the additive, or sub- tractive greater (than unity), one should find two roots by his own intelligence only ; then by their composition with the roots obtained for the additive unity an infinite number of roots (will be found)." 1 Narayana writes similarly : "When the additive or subtractive is greater than unity, two roots should be determined by one's own intelligence. Then, by combining them with the roots for the additive unity, an infinite number of roots can be obtained." 2 We take the following illustrative examples with solutions from Narayana : Example. "Tell me that square which being multi- plied by 13 and then increased or diminished by 17 or 8 becomes capable of yielding a square root." 3 * BBr, p. 42. . a NBi, I, R. 86. 8 NBi, I, Ex. 44. SOLUTION OF Nlv 2 ^ C —J 2 175 That is, solve (1) \$x 2 ± 17 =y, (2) i 3 ^±?=y. Solution. "In the first example it is stated that the multiplier' = 13 and interpclator = 17. "Now the "roots for the interpolator 3 are (1, 4), And for the interpolator 51, the roots are (1, 8). For the composition of these with the previous roots (1, 4} the statemc at will be m = 13 I ' = 1 g= 8 /' = 5 1 / = 1 ^ = 4 / = 3 So, by the iVddition Lemma, we get the roots corres- ponding to the interpolator 153 as (12,45). The rule says, 'If the interpolator (of a Square-nature) be divided by the square of an optional number etc' Now take the optional number to be 3, so that the interpolator may be reduced to 17. For 3 2 = 9 and 153/9 = 17. Therefore, dividing the roots just obtained by the optional number 3, we get the required roots (4, 15). "Applying the Subtraction Lemma and proceeding similarly we get the roots for the interpolator 17 as (4/3, 19/3)- "In the second example the statement is : multiplier = 13, interpolator = — 17. Proceeding as before we get (by the Addition Lemma) the roots (147, 530); and (by the Subtraction Lemma), the roots (3, io)." 1 Form Mn¥ rt c = y 2 . Brahmagupta says : "If the multiplier is that divided by a square, the first root is that divided by its root." 2 1 Our MS. does not contain the solution of the equations 2 BrSpSi, xviii. 70. I76 ALGEBRA That is to say, suppose the equation to be M f f-x*-±c=j\ (1) so that the multiplier (i.e., coefficient of x 2 ) is divisible by « 2 . Putting fix = u, we get Mu 2 ± c =j 2 . ■ (2) Then clearly the first root of (1) is equal to the first root of (2) divided by n. The corresponding second root will be the same for both the equations. The same rule is taught by Bhaskara II 1 and Nariyana. The latter says : "Divide the multiplier (of a Square-nature) by an arbitrary square number so that there is left no remainder. Take the quotient as the multiplier (of another Square-nature). The lesser root (of the reduced equation) divided by the square-root of the divisor will be the lesser, root (of the original equation)." 2 Form a 2 x 2 ± c = y 2 - F° r the solution of a Square-nature of this particular form, Brahmagupta gives the following rule : "If the multiplier be a square, the interpolator divided by an optional number and then increased and decreased by it, is halved. The former (of these results) is the second root ; and the other divided by the square- root of the multiplier is the first root." 3 Thus, it is stared that , _L/±f za ^ m J-H&T + *). 1 BBi, p. 42- 2 NBi, I, R. 84. 3 BrSpSi, xviii. 69. SOLUTION OF Nx 2 ^ C — J 2 1 77 where m is an arbitrary number, is a solution of the equation a z x 2 ^ c =jy 2 . The same solution has been given by Bhaskara II and Narayana. 1 Bhaskara's rule runs as follows : "The interpolator divided by an optional number is set down at two places ; the quotient is diminished (at one place) and increased (at the other) by that optional number and then halved. The former is again divided by the square-root of the multiplier. (The quotients) are respectively the lesser and greater roots." 2 The rationale of the above solution has been given by the commentators Suryadasa and Krsna substantially as follows : ± c = j 2 — a 2 x 2 = {y— ax){y + ax). Assume y — ax = m 7 m being an arbitrary rational number. Then y -f- ax = =!=-. Whence by the rule of concurrence, we get x = — ( — m), Fotm c — Nx 2 = y 2 . Though the equation of the form c — Nx 2 = j 2 has not been considered by any Hindu algebraist as deserving of special treatment, it occurs incidentally in examples. For instance, Bhaskara II has proposed the following problem : 1 NBi, I, R. 8j. » BBi, p. 42. 12 I78 ALGEBRA "What is that square which being multiplied by — 5 becomes, together with 21, a square ? Tell me, if you know, the method (of solving the Square-nature) when the multiplier is negative." 1 Thus it is required to solve — 5.V 2 + 21 =J 2 - C 1 ) Narayana has a similar example, «^., 2 — n* 2 + 60 =j % . (2) Two obvious solutions of (1) are (1, 4) and (2, 1). Composing them with the roots of — 5 x 2 + 1 =y, says Bhaskara II, an infinite number of roots of (1) can be derived. Form Nx 2 — k 2 = y 2 . Bhaskara II observes : "When unity is the subtractive the solution of the problem is impossible unless the multiplier is the sum of two squares." 3 Narayana writes : "In the case of unity as the subtractive, the multi- plier must be the sum of two squares. Otherwise, the solution is impossible." 4 Thus it has been said that a rational solution of Nx 2 — 1 = j 2 , and consequently of Nx 2 — & =y is not possible unless N is the sum of two squares. 1 BBi, p. 43. a NBi, I, Ex. 43. 8 "RupaSuddhau 1 khiloddistam vargayogo guno na cet" — BBi, p. 40. 'NBi, I, R. 83. SOLUTION OF Nx 2 ±C — J 2 J 79 For, if x = p/q, j = rjs be a possible solution of the equation, we have N(Plq) 2 -& = (rjs) 2 , or N = (qrjps) 2 -f (qkjp) 2 . Bhaskara II then goes on : "In case (the solution is) not impossible when unity is the subtractive, divide unity by the roots of the two squares and set down (the quotients) at two places. They are two lesser roots. Then find the correspond- ing greater roots at the two places. Or, when unity is the subtractive, the roots should be found as before." Thus, according to Bhaskara II, two rational solu- tions of Nx 2 — i =A where N = m 2 -f- n 2 , will be i 1 x = — m X i n n y — - — ■ > y m n will be So two rational solutions of ^ (/;/ 2 -f- tF) x 2 — k 2 =j 2 , x = y = m kn_ m x = y n km The following illustrative example of Bhaskara II 1 is also reproduced by Narayana : 2 13A- A 1 =-f. 1 BBi,p. 41. 2 NBi, I, Ex. 58. l8o ALGEBRA The former solves it substantially in the following ways : (i) Since 13 — z 2 4- 3 s two rational solutions are (1/2, 3/2) and (1/3, 2/3). (2) An obvious solution of 13* 2 — 4= J 2 is x = 1, j = 3. Then dividing out by 4, as shown before, we get a solution of the equation 13X 2 — 1 =y* as (1/2, 3/2). (3) Again, since an obvious solution of is x = i,j-= 2, we get, on dividing out by 9, a solution of our equation as (1/3, 2/3). (4) From these fractional roots, we may derive integral roots by the Cyclic Method. Since we have, by Bhaskara's Lemma, m being an indeterminate multiplier, , mjz-\- 3/2- \ 2 , rrfi— 13 = 7 3^/2+ i3/* \ 2 i3(^) 2 -f^£=^ =(^TT^)'- The suitable value of w which will make {m-\-^)\z an integer and |/# 2 — 13 J minimum is 3. So that we have i3-3 2 + 4 = ii 2 - From this again we get the relation I3 (3»+ IT ) 2 I "*— *5 = ( "* + i3-3 \ 2 GENERAL EQUATIONS OF SECOND DEGREE l8l l The appropriate value of the indeterminate multiplier in this case is « = 3. Substituting this value, we have i 3 .j2_ i = i8 2_ Hence an integral solution of our equation i$x 2 — 1 =j 2 is (5, 18). "In all cases like this an infinite number of roots can be derived by composition with the roots for the additive unity." 1 Nar^ana states the methods (2) and (3) only. 19. GENERAL INDETERMINATE EQUATIONS OF THE SECOND DEGREE: SINGLE EQUATIONS The earliest mention of the solution of the general indeterminate equation of the second degree is found in the Bijaganita of Bhaskara II (11 50). But there are good grounds to believe that he was not its first dis- coverer, for he is found to have taken from certain ancient authors a few illustrative examples the solutions of which presuppose a knowledge of the solution of such equations. 2 Neither those illustrations nor a treatment of equations of those types occurs in the algebra of Brahmagupta or in any other extant work anterior to Bhaskara II. Bhaskara II distinguishes two kinds of indeterminate equations : Sakrt saniikarana (Single Equations) and Asakrt samikarana (Multiple Equations). 3 Solution. For the solution of the general indeter- minate equation of the second degree, Bhaskara 1,1 (11 50) lays down the following rule: 1 «"Iha sarvatra padanam rupaksepapadabhyim bhavanayi'- naotyam" — BBi, p. 41. 2 Vide infra, pp. 2671". - 3 BBi, pp. 106, no. ' I 82 ALGEBRA "When the square, etc., of the unknown are present (in an equation), after the equi-clearance has been made, (find) the square-root of one side by the method des- cribed before for it, and the root of the other side by the method of the Square-nature. Then (apply) the method of (simple) equations to these roots. If (the other side) does not become a case for the Square-nature, then, putting it equal to the square of another unknown, the other side and so the value of the other (i.e., the new) unknown should be obtained in the same way as in the Square-nature ; and similarly the value of the first un- known. The intelligent should devise various artifices so that it may become a matter for (the application of) the Square -nature." 1 He has further elucidated the rule thus : "When, after the clearance of the two sides has been made, there remain the square, etc.,. of the unknown, then, by multiplying the two sides with a suitable number and by the help of other necessary operations as des- cribed before, the square-root of one side should be extracted. If there be present on the other side the square of the unknown with an absolute term, then the two roots of that side should be found by the method of the Square-nature. There the number associated with the square of the unknown is the prakrti ('multiplier'), and the absolute number is to be considered as the interpolator. What is obtained as the lesser root in this way will be the "value of the unknown associated with the multiplier (prakrti) ; the greater root is (again) the root of that square (formed on the first side). Hence making an equation of this with the square-root of the first side, the value of the unknown on the first side should be determined. '» BB:, p. 99. SOLUTION of ax* + bx + c =j i 183 "But if there be present on the second side the square of the unknown together with (the first power of) the unknown, or only the (simple) unknown with or without an absolute number, then it is not a case for the Square-nature. How then is the root to be found in that case ? So it has been said: 'If (the other side) does not become a case for the Square-nature etc' Then, putting it equal to the square of another unknown, the square-root of one side should be found in the way indicated before, and the two roots of the other side should then be determined by the method of the Square- nature. There again the lesser root is the value of the unknown associated with the prakrti and the greater root is equal to the square-root of that side of the equation. Forming proper equations with the roots, the values of the unknowns should be determined. "If, however, even after the second side has been so treated, it does not turn out to be a case for the Square-nature, then the intelligent (mathematicians) should devise by their own sagacity all such artifices as will make it a case for the method of the Square- nature and then determine the values of the un- knowns." 1 Having thus indicated in a general way the broad outlines of his method for the solution- of the general indeterminate equation of the second degree, Bhaskara II discusses the different types of equations severally, explaining the rules in every case in greater detail with the help of illustrative examples. (/') Solution of ax 2 -f- bx -f- c = y 2 For the general solution of the quadratic indeter- minate equation ax 2 -f- bx -f- c =j" 2 , (1) 1 BBi, p. 100. 1 84 ALGEBRA Bhaskara II gives the following particular rule : "On taking the square-root of one side, if there be on the second side only the square of the unknown together with an absolute number, in such cases, the greater and lesser roots should be determined by the method of the Square-nature. Of these two, the greater root is to be put equal to the square-root of the first side mentioned before, and thence the value of the first unknown should be determined. The lesser will be the value of the unknown associated with the prakrti. In this way, the method of the Square -nature should be applied to this case by the intelligent." 1 As an illustration of this rule Bhaskara II works out in detail the following example; "What number being doubled and added to six times its square, becomes capable of yielding a square- root ? O ye algebraist, tell it quickly." 2 Solution. "Here let the number be x. Doubled and together with six times its square, it becomes 6x 2 -f- zx. This is a square. On forming an equation with the square of j, the statement is 6x 2 -f- zx -f- oy 2 = ox 2 -f- ox + j 2 . On making equi-clearance in this the two sides are 6x 2 + zx and j 2 . "Then multiplying these two sides by 6 and superadding 1, the root of the first side, as described before, is 6x -j- 1 . "Now on the second side of the equation remains 6y 2 -\- 1. By the method of the Square-nature, its roots are : the lesser 2 and the greater 5, or the lesser 20 and the greater 49. Equating the greater root with the square-root of the first side, vi%., 6x -\- 1, the value of 1 BBi, pp. 100-1. 2 BBi, p. 101. SOLUTION of ax 2 -f bx' -\- c =j 2 185 x is found to be 2/3 or 8. The lesser root, 2 or 20, is the value of y, the unknown associated with the prakrti. In this way, by virtue of (the multiplicity of) the lesser and greater roots, many solutions can be obtained." 1 In other words the method described above is this : Completing the square on the left-hand side of the equation ax 2 -\- bx -\- c =y 2 , we have (ax + ±b) 2 = ay 2 + \{b 2 - 40c). Putting ^ = ax + \b, /& = \(b 2 — 4ac), we get af+k = Z 2 . (1 . 1) If j = / t ^ = m be found empirically to be a solution of this equation, another solution of it will be y = lq ±mp, z = wq± alp, where ap 2 -\- 1 = q 2 . Hence a solution of (1) is x = - — + ±(mg± alp), y = lq -j- mp. Now suppose x = r, when ^ = m ; that is, let m = ar--\- bjz. Substituting in the above expressions, we get the required solution of (1) as (1.2) ■*• = — (pq — b) + qr ± Ip, J = lq± iflpr + Ify); where ap 2 -+- 1 = q 2 and or 2 + br + c = I 2 . Thus having known one solution of ax 2 -\- bx -\- c =j 2 , an infinite number of other solutions can be 1 BBi, p. 101. I 86 ALGEBRA easily obtained by the method of Bhaskara II. The method is, indeed, a very simple and elegant one. It has been adopted by later Hindu algebraists. As the relevant portion of the algebra of Narayana (1350) is now lost, we cannot reproduce his description of the method. Jfianaraja (1503) says : "(Find) the square-root of the first side according to ' the method described before and, by the method of the Square-nature, the roots of the other side, where the coefficient of the square of the unknown is considered to be the prakrti and the interpolator is an absolute term. Then the greater root will be equal • to the previous square-root and the other (i.e., the lesser root) to the unknown associated with the prakrti." The above solution (1.2), but with the upper sign only, was rediscovered in 1733 by Eluer. 1 His method is indirect and cumbrous. Lagrange's (1767) method begins in the same way as that of Bhaskara II. by completing the square on the left-hand side of the equation. 2 (it) Solution oj ax 2 + bx + c = a'j z -\-b'y + c' Bhaskara II has treated the more general type of quadratic indeterminate equations : aX 2 _j_ fa _j_ c _ aJ 2 _j_ tfy _j_ ^ ( 2 ) His rule in this connection runs as follows : "If there be the square of the unknown together with the (simple) unknown and an absolute number, put- ting it equal to the square of another unknown its root (should be investigated). Then on the other side (find) 1 Leonard Euler, Opera Mathematka,vo\. II, 191 5, pp. 6-17; Compare also pp. 576-611. * Additions to Elements of Algebra by Leonard Euler, translated into English by John Hewlett, 5th edition, London, 1840, pp. 537ff- solution of ax 2 + bx + c = a'j 2 + b'j + *r ; where aa'p* -f 1 = ^ 2 , and ar 2 -\- br + <; = a's 2 - + ^j + /. The form (2.5) shows that having found empirically one solution of ax 2 + bx -f- c — a'j 2 -f- b'j + c' Bhaskara could find an infinite number of other solutions of it. Jnanaraja (1503) says : "If on the other side be present the square as well as the linear power of the unknown together with an absolute term, put it equal to the square of another unknown and then determine the lesser and greater roots. The lesser root will be equal to the first square- root and the greater to the second square-root." He gives with solution the following illustrative example : 3(x 2 +4*-)=j 2 + 4j, or ($x + <>) 2 == 3j>/ 2 -f- 1 zy -f- 36! Putting 3.V -j- 6 = %, where £ is the "first square- re ot" of Jnanaraja, we get Z*= 3J 2 + i^ 2 + 3 6 > or 3 £ 2 -= ( 3 j + 6) 2 + 72. Now put jy + 6 = iv, where n> is the "second square-root." Then 3^ 2 — 72 = TV 2 . Therefore, by the method of the Square-nature, ^ = 18, iv = 30. Whence x ==■ 4, y = 8, is a solution. 190 ALGEBRA (/»') Solution of ax 2 -f- by 2 -f- ^ = ^ 2 Bhaskara II followed several devices for the solution of the equation ak*+bf+c=& , (3) In every case his object was to transform the equation into the form of the Square-nature. He says : "In such cases, where squares of two unknowns with (or without) an absolute number are present, supposing either of them optionally as the prakrti, the rest (of the terms) should be considered as the interpolator. Then the roots should be investi- gated in the way described before. If there be more equations than one (the process will be especially help- ful)." 1 He then explains further : "Where on finding the square root of the first side, there remain on the other side squares of two unknowns with or without an absolute number, there consider the square of one of the unknowns as the prakrti ; the remainder will then be the interpolator. Then by the rule: 'An optionally chosen number is taken as the lesser root, etc./ 2 the unknown in the interpolator multiplied by one, etc., and added with one, etc., or not, according to one's own sagacity, should be assumed for the lesser root ; then determine the greater root." 3 There are thus indicated two artifices for solving the equation (3). They are : (i) Set x = my; so that equation (3) transforms into 1 BBi, pp. iosf. 2 The reference is to the rule for solving the Square-nature {vide supra p. 144) (BBi, p. 33). 3 BBi, p. 106. solution of ax 2 + by 2 -\- c = z 2 191 z 2 = i am2 + b )J 2 + c = « J 2 + c, where a = am 2 + b. Hence the required solution of ax 2 -f- by 2 -f- c = z 2 ls x = my = m(rq ± ps), y = rq±ps, Z = sq±apr; where s 2 = ar 2 -j- c and q 2 = ap 2 -\- 1. (//') Set x = my ± » ; then the equation reduces to ^ 2 '= a ^ 2 ^ zamny + y where a = aw 2 -f- £ and y = an 2 -j- *r. Completing the square on the right-hand side of this, we get CC£ 2 — p = w 2 , where w = ay ± amn and p = ya — a 2 m 2 n 2 = a{bn 2 -\- cm 2 ) -j- be. If z = s, w = r be a solution of this equation, another solution will be Z = sq ± #, ^ = r # ± «#; where ^ 2 = ap 2 + 1. Hence the solution of ax 2 + by 2 + e = £ 2 is x- = — (^ ± a -^ -F ^ w «) i »> j = -i- (rq ± asp =f amn), Z = sa ± r P\ where q 2 == ap 2 -j- 1, r 2 = as 2 — P, a = am 2 + £ and P = - Now, if y ~ /, w = r be a solution of »a = by 2 + <■, another solution of it will be J = /q±pr, w — rq±blp; where ap 2 -\- 1 = q 2 . Therefore, the solution of (3) will be y = /q±pr, a -\- m 2 . . , .. where ap 2 + 1 = q 2 and bl 2 + ± a, ^ = na> ± p. Substituting in the equation ax 2 -f- by* + c = Z 2 » we get {am 2 — n 2 )n> 2 ± zw (ama =f= »p) -j- by 2 + (c + aa 2 — p 2 ) = o. Putting \ — am 2 — n 2 , fi = ama =f »p, v = r -f- <7ct 2 — p 2 , this equation can be reduced to — Iby 2 + (n 2 — vJl) = * 2 , where # = Aa* ± n. Kamalakara gives also some other methods which are applicable only in particular cases. Case /'. Suppose that b and c are of different signs. 2 Two sub-cases arise: (i) Form ax 2 -\- by 2 ~ c = v*. First find «, v, says Kamalakara, such that au 2 — c = v 2 . Assuming x=\j — vy + », we have *x 2 + £j 2 — c = — ^ 2 j 2 + 2 \l?tuvy+ by 2 + (, we get ^-n> 2 =y 2 (c-— 2 ). Whence % — n> = X, where X is an arbitrary rational number. So . as stated in the rule. Therefore, X ~ zp\ l^ C 4 p' l) i *p 2 ' 1 Y. Mikamij The Development of Mathematics in China and Japan, Leip2ig, 191 3, p. 231. 8 BBi, p. 106. solution of ax 2 + bxy -f- cy 2 = £ 2 201 Now, if we suppose jv = W w > where -w, « are arbitrary integers, we get the solution of (5 . 1) as * = 8JL^i {« 8 (4^ 2 - ^ 2 ) - 4*- 2 pW ~ ^bmn), m Since the given equation is homogeneous, any multiple of these values of x,y, % will also be its solution. There- fore, multiplying by 8X/> 2 « 2 , we get the following solution of the equation p 2 x 2 -j- bxj 4- cy 2 = ^ 2 in in- tegers : x = m 2 {vp 2 — b 2 ) — 4^ 2 p 2 n 2 — ^bmn, ] y=%lmnp\ I (5.2) £ = w 2 (4#> 2 — b 2 ) + 4X a /> 2 « 2 , J where m y «, are arbitrary integers. In particular, putting a = b = c = 1, and \—p = 1 in (5 .2), we get x = 5m 2 — 4» (« + m), Z = $m 2 + 4« 2 , as the solution of the equation x 2 A- xj 4- jv 2 = ^ 2 . Dividing out by 8», the above solution can be put into the form 202 ALGEBRA j = m, as has been stated by Narayana : "An arbitrary number is the first. Its square less by its (square's) one-fourth, is divided by an optional number and then diminished by the latter and also by the first. Half the remainder is the second number. The sum of their squares together with their product is a square." 1 It is noteworthy that in practice Narayana approves of only integral solutions of his equation. For instance, he says : " 'Any arbitrary number .is' the first.' Suppose it to be 12. Then with the optional number unity, are obtained the numbers (12, 95/2). For integral values, they are doubled (24, 95). With the optional number 2, are obtained (12, 20). It being possible, these are reduced by the common factor 4 to (3, 5). In this way, owing to the varieties of the optional number, an infi- nite number of solutions can be obtained." 2 (/V) If neither a nor c be a square, the solution can be obtained thus : Multiplying both sides of the equation (5) by a and then completing a square on the left-hand side, the equation transforms into (ax + \bjf + (ac — ±P)f = a^. Putting. ax -f- \by = w and = ^(b 2 — /\ac\ we get w 2 = a^ 2 + Pj 2 . (5 • 3) l GK,i. 55. 2 See the example in illustration of the same. solution of ax 2 + bxy -+- cy 2 = % 2 203 The method of the solution of an equation of this form, according to Bhaskara II, has been described before. Assume rv = vy, % = uy ; so that the values of u, v will be given by v 2 = au 2 +$. (5.4) If « = m, v ^= n be a solution of (5 .4), another solution will be // = mq ±: pn, v = nq i amp; where ap 2 -f- 1 = q 2 . Therefore, a solution of (j) is x = -^KAm ± *™p) — b}, z=j(>»q±p")\ where ap 2 -{- \ — q 2 and am 2 -f- p = « 2 . Put » = #r -f- |-£ and y = — ; then we have t x = -~^{q(zar + b) ± zamp — b), s J = T ' Z = ^ zm< l ±P( zar + b ))- Multiplying by zat, we get the following solution of ax 2 -(- bxy -\- cy 2 = % 2 in integers : " x = s{ q (zar -f- b) ± 2^^> — £}, J = 2 then other rational solutions of it will be given by (/a, /p, li), where / is any rational number. This is clearly in evidence in the formula of Katyayana in which a is any quantity. It is now known that all rational solutions of x 2 -f- y 1 = ^ 2 can be obtained without duplication in this way. Later Rational Solutions. Brahmagupta (628) says: "The square of the optional (ista) side is divided and then diminished by an optional number; half the result is the upright, and that increased by the optional number gives the hypotenuse of a rectangle." 2 In other words, if m, n be any two rational numbers, then the sides of a right triangle will be The Sanskrit word ista can be interpreted as imply- ing "given" as well as "optional". With the former meaning the rule will state how to find rational right triangles having a given leg. Such is, in fact, the inter- pretation which has been given to a similar rule of Bhaskara II. 3 1 Datta, Sul&a, p. 179. 2 BrSpSi, xii. 35. 3 Vide infra p. 211 ; H. T. Colebrooke, Algtbra with Arith- metic and Mensuration from the Sanscrit of Brahmegupta and Bhascara, London, 1817, (referred to hereafter as, Colebrooke, Hindu Algebra), p. 61 footnote. RATIONAL TRIANGLES 207 A similar rule is given by Sripati (1039): "Any optional number is the side; the square of that divided and then diminished by an optional number and halved is the upright; that added with the previous divisor is the hypotenuse of a right-angled triangle. For, so it has been explained by the learned in the matter of the rules of geometry." 1 . Karavindasvami a commentator of the ^Apastamba Sulba, finds the solution /« 2 +2»\ /« 2 +2«+2\ m, ( ■ )m, ( ■ — ! — )m, V 2»-j-2 / V 2« + 2 ' by generalising a rule of the Sulba? Integral Solutions. Brahmagupta was the first to give a solution of the equation x 2 -f- j 2 = ^ a in integers. It is m 2 — » 2 , zmn, m z -f- « 2 , m, n being any two unequal integers. 3 Mahavira (850) says : "The difference of the squares (of two elements) is the upright, twice their product is the base and the sum of their squares is the diagonal of a generated rectangle." 4 He has re-stated it thus : "The product of the sum and difference of the elements is the upright. The sankramana^ of their squares gives the base and the diagonal. In the opera- tion of generating (geometrical figures), this is the process." 6 x SiSe, xiii. 41. 2 ApSl, 1.2 (Com.); also see Datta, Sulba, pp. 14-16. z BrSpSi, xii. 33 ; vide infra, p. 222. * GSS, vii. 90 J. 5 For the definition of this term see pp. 431". • GSS, vii. 93 J. 2o8 ALGEBRA Bhaskara II (115 o) writes: - "Twice the product of two optional numbers is the upright ; the difference of their squares is the side ; and the sum of their squares is the hypotenuse. (Each of these quantities is) rational (and integral)." 1 It has been stated before that the early Hindus recognised that fresh rational right triangles can be derived from a known one by multiplying or dividing its sides by any rational number. The same principle has been used by Mahavira and Bhaskara II in their treat- ment of the solution of rational triangles and quadri- laterals. Ganesa (1545) expressly states: "If the upright, base and hypotenuse of a rational right-angled triangle be multiplied by any arbitrary rational number, there will be produced another right- angled triangle with rational sides." Hence the most general solution of x 2 -f y 2 — ^ 2 in integers is (w 2 — n 2 )l, zwriJ, (w 2 -f- n 2 )/ where «?, n, I are integral numbers. Mahavira's Definitions. A triangle or a quadri- lateral whose sides, altitudes and other dimensions can be expressed in terms of rational numbers is called janya (meaning generated, formed or that which is generated or formed) by Mahavira. 2 Numbers which 2 GSS, introductory line to vii. 90J. The section of Mahavira's work devoted to the treatment of rational triangles and quadri- laterals bears the sub-title janya-vyavahdra {Janya operation) and it begins as "Hereafter we shall give out the janya operations in cal- culations relating to measurement of areas." Mahavira's treatment of the subject has been explained fully by Bibhutibhusan Datta in a paper entitled: "On Mahavira's solution of rational triangles and miadrilaterals," BCMS, XX, 1928-9, pp. 267-294. RATIONAL TRIANGLES 209 are employed in forming a particular figure are called its bija-samkhyd (element-numbers) or simply bija (element or seed). For instance, Mahavira has said: "Forming O friend ! the generated figure from the bija 2, 3," 1 "forming another from half the base of the figure (rectangle) from the bija z y 3," 2 etc. Thus, according to Mahavira, "forming a rectangle from the bija w, »" means taking a rectangle with the upright, base and diagonal as nfi — rfi, zmn, m 2 -\- n 2 respectively. It is noteworthy that Mahavira's mode of expression in this respect very closely resembles that of Diophantus who also says, "Forming now a right-angled triangle from 7, 4," meaning "taking a right-angled triangle with sides 7 2 — 4 2 , 2.7.4, 7 2 + 4 2 or 33, 56, 65. " 3 It should also be noted that Mahavira never speaks of "right- angled triangle." What Diophantus called "forming a right-angled triangle from m, n" Mahavira calls "form- ing a longish quadrilateral or rectangle from m, fi." Right Triangles having a Given Side. In the Sulba we find an attempt to find rational right triangles having a given side, that is, rational solutions of x 2 + a 2 = v*. In particular, we find mention of two such right triangles having a common side a, vz\., (a, 3^/4, 5^/4) and (a, ^ajiz, i^ajiz). A The principle underlying ythese solutions will be easily detected to be that of the reduction of the sides of any rational right triangle in the ratio of the given side to its corresponding 1 "Bije dve trini sakhe ksetre janye tu samsthapya" — GSS, vii. 9*5- 2 "He dvitribijakasya k$etrabhujardhena canyamutthapya" — GSS, vii. n 1 J. 3 Arithmetica, Book III, 19 ; T. L. -Heath, Diophantus of Alexan- dria, p. 167. ' 4 Datta, Sulba, p. 180. 14 2IO. ALGEBRA side. This principle of finding rational right triangles having a given side has been followed explicitly by Mahavira (850). 1 It has been stated before that one rule of Brahma- gupta 2 can be interpreted as giving rational solutions of x 2 -f- d 2 = ^ 2 as where n is any rational number. In fact, he has used this solution in finding rational isosceles triangles having a given altitude. 3 This solution has been expressly stated by Mahavira (850). He says : "The sankramana between any optional divisor of the square of the given upright or the base and the (respective) quotient gives the diagonal and the base (or upright)." 4 He has restated the solution thus : "The sankramana between any (rational) divisor of the upright and the quotient gives the elements ; or any (rational) divisor of half the side and the quotient are the elements." 5 The right triangles formed according to the first half of this rule are : fl 1 Vide infra, p. 213 2 Vide supra, p. 206. 3 Vide infra, p. 223 * GSS, vii. 97 J. 5 GSS, vii. 95 £. 8 The "elements" here are k(a/p -\- p) , \{a\p — p), where p is an optional number. RATIONAL TRIANGLES 211 and those according to the second half are i 1 a 2 o a 2 , 2 Bhaskara II gives two solutions one of which is the same as that of Brahmagupta. He says : "The side is given : from that multiplied by twice an optional number and divided by the square of that optional number minus unity, is obtained the upright ; this again multiplied by the optional number and diminished by the given side becomes the hypotenuse. This triangle is a right-angled triangle. "Or the side is given : its square divided by an optional number is put down at two places ; the optional number is subtracted (at one place) and added (at another) and then halved ; these results are the upright and the hypotenuse. Similarly from the given upright can be obtained the side and the hypotenuse." 2 That is to say, the two solutions are ma / zna \ and a, *(-£-_»), $(£- + „). Bhaskara II illustrates this by finding four right triangles having a side equal to 12, vj%., (.12, 35, 37), (12, 16, 20), (12, 9, 15) and (12, 5, 13).* j The rationale of the first solution has been given by Suryadasa (1538) thus : Starting with the rational right triangle n- — 1, 2/7, « 2 -j- i, he observes that if x, j, ^ 1 The "elements" here are q, ajiq, where q is an optional number. 2 L, p. 34. 3 L, pp. 341". 212 ALGEBRA be the corresponding sides of another right triangle, then x = -i = „ 2 5 T = k ( sa y)- « 2 — I 2« « 2 + I Hence x = £(« 2 — i), y = 2«/6, ^ = ^(« 2 -f- i). Therefore .v + % = zkn 2 = »y. If now we have x = a, "then a k = /r — i TT zna Hence j and « 2 - i' zna / zna \ The second rule has been demonstrated by Surya- dasa, Ganesa and Ranganatha thus : Since x 2 + a 2 ~ ^ 2 , we have a 2 = £ 2 — A -2 — (^ — a-) (^ -f- ,v). Assume ^ — x = n, where n is any rational num- ber ; then % + x = — . ^ =$(-!- + »), X = ^(^--«). Generalising the method of the Apastamba Sulba the commentators obtained the solution 1 /m 2 4- zm \ / m 2 + zm + z \ a -> ( ~, ) a > ( ! 1 ) a - v 2W -f- 2 ' v 2W -f- 2 ' 1 Datta, $idba, p. 16. RATIONAL TRIANGLES 21 3 Right Triangles having a Given Hypotenuse. For finding all rational right triangles having a given hypotenuse (c), that is, for rational solutions of x 2 + J 2 = ? 2 , Mahavira gives three rules. The first rule is : "The square-root of half the sum and difference of the diagonal and the square of an optional number are they (the elements)." 1 In other words, the required solution will be obtain- ed from the "elements" ■ v / (^ + ^ >2 )/ 2 ^d V(f~ P 2 )i 2 -> where p is any rational number. Hence the solution is p\ V' a - P\ '> The second rule is : "Or the square -root of the difference of the squares of the diagonal and of an optional number, and that optional number are the upright and the base." 2 That is, the solution is A V*-p 2 , c. These solutions are defective in the sense that y/c 1 — p* or \/r 2 — p 2 might not be rational unless p is suitably chosen. Mahavira's third rule is of greater importance. He says : "Each of the various figures (rectangles) that can be formed from the elements are put down ; by its diagonal is divided the given diagonal. The perpendi- cular, base and the diagonal (of this figure) multiplied by this quotient (give rise to the corresponding sides of the figure having the given hypotenuse)." 3 1 GSS, vii. 95 J. 2 GSS, vii. 97 J. 3 GSS, vii. 1 22 J. 214 ALGEBRA Thus having obtained the general solution of the rational right triangle, w'^., m % — » 2 , zmn, m 2 -J- # 2 > Mahavira reduces it in the ratio c/(m 2 + '« 2 ), so that all rational right triangles having a given hypotenuse c will be given by By way of illustration Mahavira finds four rec- tangles (39, 52), (25, 60), (33, 56) and (16, 63) having the same diagonal 65. 1 This method was later on rediscovered in Europe by Leonardo Fibonacci of Pisa (1202) and Vieta. It has been pointed out before that the origin of the method can be traced to the Sulba. Bhaskara II (11 50) says : "From the given hypotenuse multiplied by an optional number and doubled and then divided by the square of the optional number added to unity, is obtained the upright ; this is again multiplied by the optional number ; the difference between that (product) and the given hypotenuse is the side. "Or divide twice the hypotenuse by the square of an optional number added to unity. The hypotenuse minus the quotient is the upright and the quotient multiplied by the optional number is the side." 2 Thus, according to the above, the sides of a right- angled triangle whose hypotenuse is c are : zmc f zmc m / zmc \ tri 2 - -J- 1 ' \w 2 -f- 1 zmc zc or — ^— — , c 5— i — > c - 1 GSS, vii. 123-124J. a L, pp. 35, 36. RATIONAL TRIANGLES 21 J By way of illustration Bhaskara II finds two right triangles ( j i, 68) and (40, 75) having the same hypotenuse 85. 1 Suryadasa demonstrates the above substantially thus : If (*■» J> Z) be the Sides of the right triangle, we have — 2^ = -^— == — «rn = k (say), where m is any rational integer. Then x = A(m 2 — 1), y — zmk, % = k(m 2 + 1). Therefore a: + % = 2&!?? 2 = /Ky. Since % is given to be equal to tr, we have Hence J m 2 + 1 ' zmc m 2 + 1 ' and x = my — v = mi — s ) — c. - / ^ v m £ + 1 ' Problems Involving Areas and Sides. Mahavira proposes to find rational rectangles (or squares) in which the area will be numerically (samkhyayd) equal to any multiple or submultiple of a side, diagonal or perimeter, or of any linear combination of two or more of them. Expressed symbolically, the problem is to solve mx -f- ny + p% — rxy; j m, n, p, r being any rational numbers (r =fc o). For the solution of this problem he gives the following rule : 1 -L. PP- 3 jf- 21 6 ALGEBRA "Divide the sides (or their sum) of any generated square or other figure as multiplied by their respective given multiples by the area of that figure taken into its given multiple. The sides of that figure multiplied by this quotient will be the sides of the (required) square or other figure." 1 That is to say, starting with any rational solution of X > 2 + y> 2 = ^ (2) we shall have to calculate the value of mx' + ny' + p^ = jg, say. (3) Then the required solution of (1) will be obtained by reducing the values of x\y', ^' in the ratio of Q\rx'y' . Thus x = x'Q\rx'y' =Q\ry\ " y = / Q\rxJ = Q\rx' ,\ (4) Mahavira gives several illustrative examples some of which are very interesting: "In a rectangle the area is (numerically) equal to the perimeter ; in another rectangle the area is (numerically) equal to the diagonal. What are the sides (in each of these cases) ?" 2 Algebraically, we shall have to solve and x 2 +y 2 = ^2 i(x + y) = xy ;, } (x.x) Starting with the solution s 2 — t 2 , zst, s 2 + t 2 of (2) and putting m = n = z, p = o t r = i. in (4), we get 1 GSS, vii. 112J. a GSS, vii. H5J. RATIONAL TRIANGLES 217 the solution of (i . i) as 2(J 2 — t 2 ) + 4St l(S 2 — t 2 ) + 4St zst j 2 — / 8 {= zst(s*—t*) ) K ! + / 2 ). And putting m = the solution of (i . 2) - n as = 0, p = r = i, in (4), we have j 2 + t 2 zst ' J 2 j 2 - + ^ 2 — /*' 2 + Z 2 ) 2 2J"/(i 2 — z 2 )' Bhaskara II solves a problem similar to the second one above : Find a right triangle whose area equals the hypote- nuse. 1 He starts with the rational right triangle (3*, 4X, $x) ; then by the condition, area = hypotenuse, finds the value x = 5/6. So that a right triangle of the required type is (5/2, 10/3, 25/6). " He then observes : "In like manner, by virtue of various assumptions, other right triangles can also be found." 2 The general solution in this case is s 2 + fi 2(J 2 + t 2 ) (j 2 + t 2 ) 2 st ' s 2 — t 2 ' st(s* — t 2 ) Another example of Mahavira runs as follows : "(Find) a rectangle of which twice the diagonal, thrice the base, four times the upright, and twice the perimeter are together equal to the area (numerically)." 3 Problems Involving Sides but not Areas. Maha- vira also obtained right triangles whose sides multiplied iBS/.p. 56. 2 "Evamistavasadanye'pi" — BBi, p. 56. 3 GSS, vii. 1 17 J. 21 8 ALGEBRA by arbitrary rational numbers have a given sum. Algebraically, the problems require the solution of x 2 _f_y = 3* - j rx + {j + tz = A ; ) where r> s, t, A are known rational numbers. His method of solution is the same as that described above. Starting with the general solution of x' 2 +/ 2 = ^ 2 we are asked to calculate the value rx' + •[/ + *z' = P, say. Then, says Mahavira, the required solution is ,v = x'A/P, y =j'AIP, % = Z'AjP. One illustrative problem given by Mahavira is : "The perimeter of a rectangle is unity. Tell me quickly, after calculating, what are its base and upright." 1 Starting with the rectangle m 2 — « 2 , zmn, m 2 -\- « 2 , we have in this case P = z{m 2 — n 2 -f- zmri). Hence all rectangles having the same perimeter unity will be given by m 2 — n 2 mn z{m 2 — n 2 -j- zmiij m 2 — n 2 -f- zmn m, n being any rational numbers. The isoperimetric right triangles will be given by , m — ti\ np ( m 2 -\- n 2 } . . \ zm '^' m -\- n* \zm{m -\- ti)) ' where p is the given perimeter. Another example is : "(Find) a rectangle in which twice the diagonal, thrice the base, four times the upright and the perimeter together equal unity." 2 1 GSS, vii. 1 1 8$. " 2 GSS, vii. 1194. RATIONAL TRIANGLES 21 9 Pairs of Rectangles. Mahavira found "pairs of rectangles such that (/) their perimeters are equal but the area of one is double that of the other, or (it) their areas are equal but the perimeter of one is double that of the other, or (iit) the perimeter of one is double that of the ■ other and the area of the latter is double that of the former." These are particular cases of the following general problem contemplated in his rule : To find (x, y) and (//, v) representing the base and upright respectively of two rectangles which are related, such that 2M(X + j) = 2«(« + v), J , A) where m, #, p, q are known integers. His rule for the solution of this general problem is : "Divide the greater multiples of the area and the perimeter by the (respective) smaller ones. The square of the product of these ratios multiplied by an optional number is the upright of one rectangle. That diminished by unity will be its base, when the areas are equal. Other- wise, multiply the bigger ratio of the areas by that optional number and subtract unity ; three times the upright diminished by this (diiference) will be the base. The upright and base of the other rectangle should be obtained from its area and perimeter (thus determined) with the help of the rule, 'From the square of half the perimeter, etc.,' described before." 1 1 GSS, vii. 13 1 \- 133. The reference in the concluding line is to rule vii. 129J. 220 ALGEBRA In other words, to solve (A), assume j = j{(ratio of perimeters)(ratio of areas)} 2 , (i) and x =j — i, i£p = q, (2) or x — $[j — {/ (ratio of areas) — 1 }], if p ^ q, {tT) where s is an arbitrary number, and the ratios are to be so presented as always to remain greater than or equal to unity. Let m > «, q~^- p. ' Then we shall have to assume y x m 2 q 2 S l?pv 30 m 2 q 2 — s- (5) >'f+3), (4) t!*p 2 ' p Substituting these values in {A), we get , m 1 m 2 q'' U-\-V= UJ— ^t. ttfiq / m 2 q 2 a , \ uv = \s — ^-( s- ., ., s- 3 - -4- 1 ). p n l p v t^p* p ' Then Now, if the arbitrary multiplier s be chosen such that f - * 0) we have m, u V = -(4*3 2/72 nfq n - - tPp' 1 ^+»- From (4) and (6) we get m , m^q 2 1 5 sq ,„ , m 2 q 2 + 3), ismq 4»P' (6) (7) RATIONAL TRIANGLES 221 Substituting the value of J from (j) in (3) and (7) we have finally the solution of (A), when m > n, q ^ p, as m y X v = m /« , m*q N u = 4 — (4^5? — })■ (I) Mahavira has observed that 4 "when the areas are equal" we are to assume 1 J = J— 5-, *• w 1. 1 Bibhutibhusan Datta has shown that this restriction is not necessary. In fact, starting with the assumption J m 2 q 2 rfip 1 1 ; 'm>">q>P, and proceeding in the same way as above, he has obtained another solution of (A) in the form m J=> m 2 q rPp m*-q tflp v = q v ^ n*p f Datta finds two general solutions of (A), w°£. J = —ink + f > t?p* v ._ rm*q* /rm*q* _ rq f \ rmq Cn) (in) 222 ALGEBRA Isosceles Triangles with Integral sides. Brahma- gupta says : "The sum of the squares of two unequal numbers is the side ; their product multiplied by two is the altitude, and twice the difference of the squares of those two un- equal numbers is the base of an isosceles triangle." 1 Mahavira gives the following rule for obtaining an isosceles triangle from a single generated rectangle: "In the isosceles triangle (required), the two dia- gonals (of a generated rectangle 2 ) are the two sides, twice its side is the base, the upright is the altitude, and the area (of the generated rectangle) is the area." 3 Thus if m, n be two integers such that m ^= #, the sides of all rational isosceles triangles with integral sides are : (/) m 2 -f- n 2 , m 2 -f- n 2 , z(m 2 — « 2 ) ; or (//) m 2 -\- tr, m 2 -f- « 2 , 4/an. and rnPq 2 / rq \ rmqjrm 2 q 2 \ \ ~ tfip 2 \tp r np\ tn*p* ) ) (IV) where m~^ n, q^ p and r, t are any two integers. See Datta, "On Mahavira' s solution of rational triangles and quadrilaterals," BCMS, XX, 1928-9, pp. 267-294; particularly p. 285. 1 BrSpSi, xii. 33. 2 A rectangle generated from the numbers m and n has its sides equal to m 2 — n 2 and zmn and its diagonal equal to m 2 -f- n 2 . Cf. pp. 208-9. 3 GSS, vii. io8£. RATIONAL TRIANGLES 223 The altitude of the former is zmn and of the latter /# 2 — « 2 and the area in either case is the same, that is, zmnipfi — # 2 ). Juxtaposition of Right Triangles. It will be noticed that the device employed by Brahmagupta and Mahavira to find the above solutions is to juxtapose two rational right triangles — equal in this case — so as to. have a common leg. It is indeed a very powerful device. For, every rational triangle or quadrilateral may be formed by the juxtaposition of two or four rational •right triangles. So, in order to construct such rational figures, it suffices to know only the complete solution of .v 2 + j 2 = £ 2 in integers. The beginning of this principle is found as early as the Baudbayana Su/ba 1 (800 B.C.) wherein is described the formation of a kind of brick, called ubhayi (born of two), by the juxtaposition of the eighths of two suitable rectangular bricks of the same breadth (and thickness) but of different lengths. Isosceles Triangles with a Given. Altitude. Brahmagupta gives a rule to find all rational isosceles triangles having the same altitude. He says : "The (given) altitude is the producer (karani). Its square divided by an optional number is increased and diminished by that optional number. ' The smaller is the base and half the greater is the side." 2 That is to say, the sides and bases of rational isosceles triangles having the same altitude a are respectively, i(-+ m\ U~+ m) and (^-=^), where m is any rational number. 1 BSl, iii. 122 ; Compare Datta, Sulba, p. 45, where necessary figures are given. * ErSpSi, xviii. 37. 224 ALGEBRA In particular, let the given altitude be 8. Then taking m = 4 Prthudakasvami (860) obtains the rational isosceles triangle (10, 10, 12). Pairs of Rational Isosceles Triangles. Mahavira gives the following rule for finding two isosceles tri- angles whose perimeters, as also their areas, are related in given proportions : "Multiply the square of the ratio-numbers of the perimeters by the ratio-numbers of the areas mutually and then divide the larger product by the smaller. Multiply the quotient by 6 and 2 (severally) and then diminish the smaller by unity : again (find severally) the difference between the results, and twice the smaller one : these are the two sets of elements for the figures to be generated. From them the sides, etc., can be obtained in the way described before." 1 If (s 1} s 2 ) and (/\ v /S 2 ) denote the perimeters and areas of two rational isosceles triangles, such that x x : j 2 = m : «, A x : A 2 = P > q, (0 where the ratio-numbers ///, n, p, q are known integers, then the triangles will be obtained, says Mahavira, from the rectangles generated from ( 6 — £-, *■—£ 1 J and ( 4 — £- -f- 1, 4 — f 2 J, \ m*q wq ' x m-q m % q ' where rpp > m z q, when the dimensions of the first are multiplied by m and those of the second by n. The dimensions of the isosceles triangle formed from the first set of bija are : side = = m{(6 tflp \ 2 m'^q' + (* n*p nfiq 1 GSS, vii. 137. -)'}. RATIONAL TRIANGLES 225 base = 24/?/ — §-( 2 — '-- — 1 ), nrq x m z q ' altitude = ^{( 6 ^f-) 2 -(^ -0 2 }; and from the second set base =4„( 4 ^+,)(4^-z), altitude = „{( 4 ^-+i) 2 --( 4 ^-2) 2 j. ( v w 2 q ' x t?rq ' ) It can be easily verified that the perimeters and areas of the isosceles triangles thus obtained satisfy the conditions (i). In particular, putting m=n=p=q= i, we have two isosceles triangles of sides, bases and altitudes (29, 40, 21) and (37, 24, 5 j) which have equal perimeters (98) and- equal areas (420). This particular case was treated by Frans van Schooten the Younger (1657), J. H. Rabh (1697) and others. 1 - It is evident that multiplying the above values by »i' x q- we get pairs of isosceles triangles whose dimen- sions are integral. Rational Scalene Triangles. Brahmagupta says : "The square of an optional number is divided twice by two arbitrary numbers; the moieties of the sums of the quotients and (respective) optional numbers are the sides of a scalene triangle; the sum of the moieties of the differences is the base." 2 1 Dickson, Numbers, II, p. zoi. 2 BrSpSi, xii. 34. 15 226 ALGEBRA That is to say, the sides of a rational scalene triangle are m* j(f+>).*(-=^+ *).*(-=-->) + *(, i) where m, p, q are any rational numbers. The altitude (/»), area and segments of the base of this .triangle are all rational. Mahavira gives the rule : "Half the base of a derived rectangle is divided by any optional number. With'this divisor and the quo- tient is obtained another rectangle. The sum of the uprights (of these two rectangles) will be the base of the scalene triangle, the two diagonals, its sides and the base (of either rectangle) its altitude." 1 If m, n be any two rational numbers, the rational rectangle (AB'BH) H C " B Fig. 3 Fig. 4 formed from them is n fi — fp.^ zmn, m % -f- « 2 . If p, q be any two rational factors of ;/?«, that is, if mn — pq, the second rectangle (AC'CH) is p* - q\ zpq, f+f. 1 GSS, vii. 1 10J. RATIONAL TRIANGLES 227 Now, juxtaposing these two rectangles so that they do not overlap (Fig. 3), the sides of the rational scalene triangle are obtained as p 2 + q\ m 2 + * 2 , {{p 2 - q 2 ) + (>»* - * 2 )}, where mn — pq. Evidently the two rectangles can be juxtaposed so as to overlap (Fig. 4). So the general solution will be x The altitude of the rational scalene triangle thus obtained is zmn or zpq, its area pq(p 2 — q 2 ) -±_ mn(m 2 — n 2 ) and the segments of the base are p 2 — q 2 and m % — « 2 . In particular, putting m = 1 z, p = 6, q = 8 in Brahmagupta's general solution, Prthudakasvami derives a scalene triangle of which the sides (13, 15), base (14), altitude (12), area (84) and the segments of the base (y, 9) are all integral numbers. In order to get the above solutions of the rational scalene triangle the method employed was, it will be noticed, the juxtaposidon of two rational right triangles so as to have a common leg. In Europe, it is found to have been employed first by Bachet (162 1). The credit for the discovery of this method of finding rational scalene triangles should rightly go to Brahma- gupta (628), but not to Bachet as is supposed by Dickson. 1 Triangles having a Given Area. Mahavira proposes to find all triangles having the same given area A. His rules are : "Divide the square of four times the given area by three; The quotient is the square of the square of a side of the equilateral triangle." 2 1 Dickson, Numbers, II, p. 192. 2 GSS, vii. 154A. 228 ALGEBRA "Divide the given area by an optional number; the square-root of the sum of the squares of the quotient and the optional number is a side, of the isosceles tri- angle formed. Twice the optional number is its base and the area divided by the optional number is the altitude." 1 "The cube of the square-root of the sum of eight times the given area and the square of an optional number is divided by the product of the optional number and that square-root; the quotient is diminished by half the optional number which is the base (of the required triangle). The sankraniana between this remainder and the quotient of the square of the optional number divided by twice that square-root will give the two sides." 2 The last rule has been re-stated differently. 3 ' 21. RATIONAL QUADRILATERALS Rational Isosceles Trapeziums. Brahmagupta has shown how to obtain an isosceles trapezium whose sides, diagonals, altitude, segments and area are all rational numbers. He says : "The diagonals of the rectangle (generated) are the flank sides of an isosceles trapezium; the square of its side is divided by an optional number and then lessened by that optional number and divided by two; (the result) increased by the upright is the base and lessened by it is the face." 4 That is to say, y?e shall have (Fig. j) CD = i(^ -/,) + («• -A . 1 GSS, vii. 1 5 6£. 2 GSS, vii. 1584. 3 GSS, vii. 160J-161 J. « BrSpSi, xii. 36. RATIONAL QUADRILATERALS 229 AD = BC = m 2 +» 2 ; also £>H = m*- — n 2 , AH = zmn, area ABCD = mn(^^- - p). By choosing the values of m, n and^» suitably, the values of all the dimensions of the isosceles trapezium can be made integral. Thus, starting with the rectangle (5, 12, 13) and taking^ = 6, Prthudakasvami finds, by way of illustration, the isosceles trapezium whose flank sides = 13, base = 14, and face = 4. Its altitude (12), segments of base (5, 9), diagonals (15) and area (108) are also integers. Mahavira writes : "For an isosceles trapezium the sum of the per- pendicular of the first generated rectangle and the perpendicular of the second rectangle which is generated from any (rational) divisor of half the base of the first and the quotient, will be the base; their difference will be the face; the smaller of the diagonals (of the generated rectangles) will be the flank side; the smaller perpendi- cular will be the segment; the greater diagonal will be the diagonal (of the isosceles trapezium); the greater area will be the area and the base (of either rectangle) will be the altitude." 1 1 CSS, vii. 99 J. 230 ALGEBRA The first rectangle (AA'DH) generated from m, n is m 2 — n~, zwn, m 2 -f- n 2 . If p, q be any two rational factors of half the base of this rectangle, that is, if pq = w/i, the second rectangle (AB'CH) from these factors will be P 2 - q\ z P q, p* + q\ By judiciously juxtaposing these two rectangles, we shall obtain an isosceles trapezium of the type required (ABCD): D H / Fig. 5 Bo. CD = (p 2 - f) + (m 2 - n 2 ), AB = (J>Z— q 2 ) — (m 2 — « 2 ), AD = BC = m* + « 2 , if w 2 + « 2 < ^ 2 + ? 2 , DH = w 2 if w 2 « 2 < p 2_ ^C = I3D = p 2 + ? 2 , if p 2 + q 2 > m 2 + « 2 , AH = 2/v« = 2/^, area ABCD =.zpq (p 2 — q 2 ), if 2p^ (^> 2 — 2»»(w 2 — « 2 ). The necessity of the conditions w 2 -f- n % < p 2 + # 2 5 /// 2 — « 2 < p 2 — q 2 , etc., will be at once realised from a glance at Figs. 5 and 6. The above specifications of the dimensions of a rational isosceles trapezium will give Fig. 5 . But when the conditions are reversed so that RATIONAL QUADRILATERALS 231 M* + » 2 > p 2 + 4*, M i -f3 2 >p 2 -q\ ipqQP-q 1 ) = \{}?\r + r), (2) b'-a^^jr-r). (3) From (1) and (2), we get b'=(b + a)l2 + (^jr-r)jz, (4) a' = (b + a)lz-{h*lr-r)lz. (j) If a = 4, b = 14, c = 13, Z> = 12, taking r = 10, we shall have 1 a' = 34/5, £' = 56/5, / = 61/5. It has been stated above that, if m, 11, p, q are rational numbers such that m 2 ^ « 2 < p 2 ^ q 2 , we must have a = (p*— q 2 ) — {trfi - « 2 ), b = (p—q*) + {m 2 — « 2 ), <: = m 2 + » 2 , ^ = zmn = 2p^. 1 GW, vii. 174J. RATIONAL QUADRILATERALS 233 Substituting these values in (2), (4), (5) we get the dimensions of the equivalent isosceles trapezium as *' = (P 2 ~ T) ~ (4pYfr - r)/*, b' = (p 2 - f) + ( 4 pY \r - r)/2, / = (4PV/r+r)/2. If #/ 2 ^ « 2 > p 2 zt q 2 , the sides of the pair of isos celes trapeziuma equal in area and altitude will be a = (m 2 — » 2 ) — (p 2 — q 2 \ b = (w 2 — « 2 ) + (p 2 — q*), c =p 2 + q 2 ; a ' — (,,,1 {m 2 — « 2 ) — (4-w 2 « 2 /r — r)/2, £' = (»a — « 2 ) _j_ ( 4 mWj r — r)jz, c' = { 4 m 2 n 2 \r + r)/2. These two isosceles trapeziums will also have equal diagonals. Rational Trapeziums with Three Equal Sides. This problem is nearly the same as that of the rational isosceles trapezium with this difference that in this case one of the parallel sides also is equal to the slant sides. Brahmagupta states the following solution of the problem : "The square of the diagonal (of a generated rect- angle) gives three equal sides; the fourth (is obtained) by subtracting the square of the upright from thrice the square of the side (of that rectangle). If greater, it is the base; if less, it is the face." 1 The rectangle generated from m, n is given by w 2 — « 2 , zmn, m 2 -j- « 2 . 1 BrSpSi, xii. 3 7. 234 ALGEBRA If ABCD be a rational trapezium whose sides AB, BC, AD are equal, then AB = BC = AD = {m 2 + « 2 ) 2 , CD = ^{zmnf - (w 2 — « 2 ) 2 = i4w 2 » 2 - a* — «*, or CD = 3(«? 2 — « 2 ) 2 — {zmri) 2 = 3/v 4 -f- 3#* — iom 2 n 2 . In particular, putting w = 2, « = i, Prthudaka- svami deduces two rational trapeziums with three equal sides, viz., ( 2 5, *5> 25, 39) and (25, 25, 25, n). The first solution is also given by Mahavira who indicates the method for obtaining it. He says : "For a trapezium with three equal sides (proceed) as in the case of the isosceles trapezium with (the rect- angle formed from) the quotient of the area of a genera- ted rectangle divided by the square-root of its side multiplied by the difference of its elements and divisor; and that (formed) from the side and upright." 1 That is to say, from any rectangle (m 2 — n 2 , zmn, m 2 -f- n 2 ), calculate zmnim 2 — « 2 ) . / . » V zmn\m — n) Then from y ' zmn(m — n\ yj zmn{m + n) form the rectangle %m 2 n 2 , 4mn{m 2 —n v ), 4mn(m 2 -\-n 2 ). ' (1) Again from zmn, m 2 — » 2 form another rectangle 6/* 2 » 2 — w 4 - ««, 4*0 {m 2 - n 2 ), (m 2 -4- n 2 ) 2 (2) By the juxtaposition of the rectangles (1) and (2) we get Brahmagupta's rational trapezium with three 1 GSS, vii. 1 01 J. RATIONAL QUADRILATERALS 235 equal sider : CD = %m fi 1 + (6w 2 » 2 — nfl — «*) = I4^ 2 « 2 — m*> — ti\ AB = 8w 2 « 2 r- (6wV — w 4 — ««) = (/* 2 + « 2 ) 2 = ^D = BC, if nfi -\- n 2 < ^mn. The segment (CH), altitude (AH), diagonals (AC, BD) and area of this trapezium are also rational, for CH = 6w 2 « 2 — mt — «*, AH = 4mn(m 2 — » 2 ), AC = BD = 4?nn(m* + tfi), area ABCD = izm^tPim 1 — « 2 ). Rational Inscribed Quadrilaterals. Brahmagupta formulated a remarkable proposition : To rind all quadrilaterals which will be inscribable within circles arid whose sides, diagonals, perpendiculars, segments (of sides and diagonals by perpendiculars from vertices as also of diagonals by their intersection), areas, and also the diameters of the circumscribed circles will be expressible in integers. Such quadrilaterals are called Brahmagupta quadrilaterals. The solution given by Brahmagupta is as follows : "The uprights and bases of two right-angled tri- angles being reciprocally multiplied by the diagonals of the other will give the sides of a quadrilateral of uneqial sides : (of these) the greatest is the base, the least is the face, and the other two sides are the two flanks. " ] Taking Brahmagupta's integral solution, the sides of the two right triangles of reference are given by tii- — n~, z>?ni, w 2 -f- « 2 ; /> 2 - q\ zpq, f + f ; 1 lir^pSi, xil. 58. Z}6 ALGEBRA where ///, n, p, q are integers. Then 1 the sides of a Btahmagupta quadrilateral are zmnlp 2 - + q*), zpq(m* + /» 2 ). J ^ ; Mahayira says : "The base and the perpendicular (of the smaller and the larger derived rectangles of reference) multi- plied reciprocally by the longer and the shorter diagonals and (each again) by the shorter diagonal will be the sides, the face and the base (of the required quadrilateral). The uprights and bases are reciprocally multiplied and then added together; again the product of the uprights is added to the product of the bases; these two sums multiplied by the shorter diagonal will be the diagonals. (These sums) when multiplied respectively by the base and perpendicular of the smaller figure of reference will be the altitudes; and they when multiplied respectively by the perpendicular and the base will be the segments of the basev Other segments will be the difference of these and the base. Half the product of the diagonals (of the required figure) will be the area." 1 If the rectangle generated from m, n be smallet than that from^>, q, then, according to Mahavira, we obtain the rational inscribed quadrilateral of which the sides are (** 2 - fl 2 )^ + q*){m* + « 2 ), (p* - q*){m* + « 2 ) 2 , zmn^p* + q^ipP- + z? 2 ), zpq(m 2 + » 2 ) 2 ; whose diagonals are {zpq(t» 2 — » 2 ) -f zmn{p 2 — # 2 )}(^ 2 + » 2 ), {( W 2 _ „2^ p 2 _ qi) + A ,nnpq){m* + rfi) ; 1 GSS, vii. 103 J. RATIONAL QUADRILATERALS 237 whose altitudes are {zpq{m 2 — « 2 ) -f- zmn(p 2 — q 2 )}zmn, whose segments are {zpq{m 2 — « 2 ) + zmn(p 2 — q 2 ))(m 2 — « 2 ), {{m 2 — n 2 )(p 2 — q 2 ) + Ajnnpq)zmn ; and whose area is i{zpq(m 2 — n 2 ) + zmn(p 2 — q 2 )} {{m 2 — n 2 )(p 2 - f) + ^mnpq){t)i 2 -j- « 2 ) 2 . Sripati writes : "Of the two right triangles the sides and uprights are reciprocally multiplied by the hypotenuses; of the products the greatest is the base, the smallest is the face and the rest are the two flank sides of a quadrilateral with unequal sides." 1 Bhaskara II gives the rule : "The sides and uprights of two optional right triangles being multiplied by their reciprocal hypote- nuses become the sides : in this way has been derived a quadrilateral of unequal sides. There the two diagonals can be obtained from those two triangles. The product of the uprights, added with the product of the sides, gives one diagonal; the sum of the reciprocal products of the uprights and sides is the other." 2 Bhaskara II 3 illustrates by taking the right triangles (3, 4, 5) and (5, 12, 13) so that the resulting cyclic quadrilateral is (25, 39, 60, 52). The same example was 1 SiSe, xiii. 42. 2 L,, p. 5 1. a L, p. 52. 2 3 8 ALGEBRA given before by Mahlvira 1 and Prfhudakasvami. 2 This cyclic quadrilateral also appears in the Trisatika of Sridhara 3 and in the commentary of the Aryabbatiya by Bhaskara I (522). The diagonals of this, quad- rilateral are, states Bhaskara II, 56 (=3.12 + 4.5) and 63 (=4.12+ 3.5) (Fig. 7). He then observes: "If the figure be formed by changing the arrange- ment of the face and flank then the second diagonal will be equal to the product of the hypotenuses of the two right triangles (of reference), i.e., 65." (Fig. 8). Fig. 7 Fig. 8 By taking the right triangles (3, 4, 5) and (15, 8, 17) Bhaskara 11 gets another cyclic quadrilateral (68, 51, 40, 75), whose diagonals are (77, 85), altitude is 308/5, segments are 144/5 anc l 2 3 i /5j an d area is 3234. 4 (Fig. 9). With the sequence of the sides (68, 40, 51, 75) the 1 GSS, vii. 1 04 J. 3 Tris, Ex. 80. 2 BrSpSi, xii. 38 (Com.). * L, pp. 4 6ff. RATIONAL QUADRILATERALS 239 diagonals ate (77, 84) (Fig. 10), and with (68, 40, 75, 51) they are (84, 85). (Fig. 11). Fig. 9 Fig. 10 ^ -? <•- .-'' "/ ""•. --«* ,-<. 40 \t," " / Fig. 11 The deep significance of Brahmagupta's results has been demonstrated by Chasles 1 and Kummer. 2 1 M. ChasJes, Aperpi historique sur I'origine et development des methodes en geometrie, Paris, 1875, pp. 436^". 2 E. E. Kumjner, "Uber die Vierecke, deren Seiten und Dio- gonalen rational sind," Journ. fur Math., XXXVII, 1848, pp. 1-20. 240 ALGEBRA In fact, according to the sequence in which the quantities (A) are taken, there will be two varieties of Brahmagupta quadrilaterals having them as their sides :, (1) one in which the two diagonals intersect at right angles and (2) the other in which the diagonals intersect obliquely. The arrangement (A) gives a quadrilateral of the first variety. For the oblique variety, the sides are in the following order : zmn{f> + q 2 \ zpq(m 2 + n 2 ) ; j W or 2 - ? 2 )(w 2 + n% zmn{p 2 + q 2 \) ,~ (p* - n *)(f + f), zpq{m 2 + n 2 ). ) ^ Bhaskara II points out that the diagonals of the Brahmagupta quadrilateral are in the (A) variety, zpq(m 2 — « 2 ) + zmn(j> 2 — q 2 ), 4?vnpq-\- (p 2 — q 2 )(w 2 — r?"); in (5), zpq(m* - fi 2 ) -f zmn(p* - q 2 \ (p 2 + ^)(>w 2 + « 2 ); and in (C), Amnpq + (J) 2 - q 2 )(m 2 - « 2 ), (> 2 + q 2 )(m 2 + « 2 ). The diameter of the circumscribed circle in every case is (p 2 + q 2 )(m 2 + » 2 ). Ganesa (1545) shows that the quadrilateral is formed by the juxtaposition of four right triangles obtained by multiplying the sides of each of two rational right triangles by the upright and base of the other. He writes : "A quadrilateral is divided into four triangles by its intersecting diagonals. So by the juxtaposition of four triangles a quadrilateral will be formed: For that purpose the four triangles are assumed in this manner : Take two right triangles formed in the way indicated Compare also L. E. Dickson, "Rational Triangles and Quadril- aterals," Amer. Matb. Mors., XXVIII, 1921, pp. 244-250. RATIONAL QUADRILATERALS 241 before. If the upright, base and hypotenuse of a rational right triangle be multiplied by any arbitrary ■rational number, there will be produced another right triangle with rational sides. Hence on multiplying the sides of each of the two right triangles by an optional number equal to the base of the other and again by an optional nnmbex equal to the upright of the other, four right triangles will be obtained by the judicious juxta- position of which the quadrilateral will be formed." He then shows with the help of specific examples (see Figs. 12, 13 & 14) that we can obtain in this way Fig. 12 Fig. 13 Fig. 14 16 242 ALGEBRA from the same set of two rational right triangles two varieties of rational convex quadrilaterals': One in which the diagonals intersect each other perpendicularly; and the other in which they do so obliquely. Inscribed Quadrilaterals having a Given Area. Mahavira proposes to find .all rational inscribed rectangles having the same given area (A, say). He says : "The square-root of the exact area is a side of the square. The quotient of the area by an optional number, and that optional number will be the base and upright of the rectangle." 1 For finding all inscribed rational isosceles trapeziums having the same area A his rule is : "The given area multiplied by the square of an optional number is diminished by the area of a generated rectangle and then divided by the base of that rectangle ; the quotient divided by the optional number is the face ; the quotient added with twice the upright and divided by the Optional number gives the base ; the base (of the generated rectangle) divided by the optional number is the altitude ; and the diagonal divided by the optional number gives the two flank sides." 2 That is to say, if m 1 — » 2 , zmn, m 1 + « 2 be the upright, base and diagonal of a rectangle formed from m, n y the dimensions of the isosceles trapezium will be r _ S 2 A 2W«(» 2 » 2 ) — zmns ' base = — \ i £ + z(m* — « 2 ) s ( zmn v y > _ s 2 A -j- zmn{m 2 — « 2 ) imns " ' 1 GSS, vii. 146. 2 .GSS, vii. 148. RATIONAL QUADRILATERALS altitude s ' • j m 2 -\- n 2 side = ; s where s is an arbitrary rational number chosen such that s 2 A. > zmn(m 2 — « 2 ). For the construction of an inscribed trapezium of . three equal sides Mahavira gives the following rule : "The square of the given area is vdivided by the cube of an optional number and then increased by that optional number ; half the result gives the (equal) sides of a generated trapezium of three equal sides (having the given area) ; twice the optional number diminished by the side is the base ; and the given area divided by the optional number is the altitude." 1 In other words, the dimensions of an inscribed trape- zium of three equal sides having a given area A will be side =*(^pr + J )> : ; base = 2j- — -2-(^r" + s )l altitude = . To find inscribed quadrilaterals having a given area Mahavira gives the following rule : "Break up the square of the given area into any four arbitrary factors. Half the sum of these factors is diminished by them (severally). The remainders are,. the sides of an (inscribed) quadrilateral with unequal sides." 2 if 1 GSS, vii. 150. 2 GSS, vii. 1 5 2. This result follows from the fact that the area of a cyclic quadrilateral is V(r — a)(s — h)(s — c)(s — d). 244 ALGEBRA Triangles and Quadrilaterals having a Given Circum-Diameter. Mahavira proposes to find all rational triangles and quadrilaterals inscribable in a circle of given diameter. His solution is : "Divide the given diameter of the circle by the calculated diameter (of the circle circumscribing any generated figure of the required kind). The sides of that generated figure multiplied by the quotient will be the sides of the required figure." 1 In other words, we shall have to find first a rational triangle or cyclic quadrilateral ; then calculate the dia- meter of its circum-circle and divide the given diameter by it. .Dimensions of the optional figure multiplied by this quotient will give the dimensions of the required figure of the type. It has been found before (p. 227) that the sides of a rational triangle are proportional to m 2 + n\ p 2 + q\ (p 2 — q 2 ) ± (m* — tfi) and its altitude is proportional to imn (or zpq), m, n, p, q being any rational numbers- such that mn = pq. The diameter of the circle circumscribed about this triangle is proportional to (m 2 + n 2 )(p 2 + q 2 ) zmn Then the sides of a rational triangle inscribed in a circle of diameter D will be zmnD imnD jJp 2 — q 2 ) ± (^ 2 — « 2 ) p 2 + q 2 ' m 2 + « 2 ' 2 ' (m 2 + n 2 ) (p 2 + q 2 ) ' and its altitude will be 1 GSS, vii. 22 1 J. SINGLE INDETERMINATE EQUATIONS 245 The dimensions of a rational inscribed quadrilateral, as stated by Mahavira, have been noted before (p. 236). The diameter of its circum-circle is (p 2 + q 2 )^ 2 + « 2 ) 2 '. Then, according to Mahavira, the sides of a rational quadrilateral inscribed in a circle of diameter D, are D(-^- 2 ) } D(^^l D(-j^), D(£=X); v m 2 + rr ' ' v m 2 -J- rr ' * v p z -f- q 2 ' v p 2 + #y its diagonals are I {^(^- »*) + ^"(^-^ (y+^ + ^y {{m > _ W _ rt + 4OT ^ ) __^___. . and its area is x {(» a - « 2 )(/> 2 - ? 2 ) + 4«»^}; so that the sides, diagonals and area are all rational. 22. SINGLE INDETERMINATE EQUATIONS OF HIGHER DEGREES The Hindus do not seem to have paid much attention to the treatment of indeterminate equations of degrees higher than the second. Some interesting examples involving such equations are, however, found in the works of Mahavira (850), Bhaskara II (n 50) and Narayana (1350). Mahavira's Rule. One problem of Mahavira is as follows : Given the sum (s) of a series in A.P., to find its 246 ALGEBRA first term (a), common difference (b) and the number of terms (n). In other words, it is required to solve in rational numbers the equation {a+(^-)b)n = s, containing three unknowns a, b and n, and of the third degree. The following rule is given for its solution : "Here divide the sum by an optional factor of it ; that divisor is the number of terms. Subtract from the quotient another optional number ; the subtrahend is the first term. The. remainder divided by the half of the number of terms as diminished by unity is the common difference." 1 Bhaskara's Method. Bhaskara II proposes the problems : "Tell those four numbers which are unequal but have a common denominator, whose sum or the sum of whose cubes is equal to the sum of their squares." 2 If x, j, %, n> be the numbers, then (1) x +j + % + n> = x 2 + j 2 + £ a + » 2 , (2) x 3 +y» + £ 3 + w 3 = A- 2 '+y + z 2 + ^ 2 . Let the numbers be x, zx f $x, 4x, says Bhaskara II. That is, suppose j = zx, % = $x, w = 4X in the above 1 GSS, vii. 78. There are also other problems where instead of s, the given quantity is s -\- a, s -\- b, s + »ocs + a+b-\-n. (GSS, ii. 83 ; ef. also vi. 80). For such problems also the method of solution is the same as before, /'. = ^, t%, A. H. is a solution of (2). The following problem has been quoted by Bhas- kara II from an ancient author : "The square of the sum of two numbers added with the cube of their sum is equal to twice the sum of their cubes. Tell, O mathematician, (what are those two numbers)." 1 If x, j be the numbers, then by the statement of the question (* + yf + (* + j) 3 = 2C* 3 + j 3 )- - "Here, so that the operations may not become lengthy," says Bhaskara II, "assume the two numbers to be u + v and u — v." So on putting x ■= u-\- v, y — u — v, the equation reduces to 4# 3 + 4# 2 = i2/«> 2 , or 4# 2 + 4» = 1 zv z , or (zu + i) 2 = 122; 2 + 1. l BBi,p. 1 or. 248 AI^JEBRA Solving this equation by the method of the Square- nature we get values of u, v. Whence the values of (x,y) are found to be (j, i), (76, 20), etc. Narayana's Rule. Narayana gives the rule : "Divide the sum of the squares, the square of the sum and the product of any two optional numbers by the sum of their cubes and the cube of their sum, and then multiply by the two numbers (severally). (The results) will be the two numbers, the sum of whose cubes and the cube of whose sum will be equal to the sum of their squares, the square of the sum and the product of them." 1 That is to say, the solution of the equations (1) x» 4-ji 3 = x* +y, (4) (*• + j)* = *2 +y, (2) x* + J 3 = (x + yf, (5) (x +j/) 3 = (x 4- j) 2 , x — (3) X 3 +J 3 = XJ, are respectively {m 2 -f- n 2 )m __ {m 1 + n 2 )n . trfi 4- n 3 ' {m -\- tt) 2 m (6) (x 4- jf = xy, (1. 1) \ (4.i) J (2.1) (3-1) x y = X = y = m z -\-n z (m 4- nfn m z -f- « 3 : m 2 n m 3 -\- rP* mn 2 /w 3 + « 3 ' (5.i) 1 (m 2 4- n 2 ~)m (m -\- nf ' {m 2 4- n 2 )n m {m -+- rif . ' f (ffl| tif-m ! x = ty + n f ' x y (6.1) I = , (» + «) 2 * (/ (/%? 4- «; 3 m 2 n (m 4- «) 3 ' AT J /wr (w 4- ») 3 ' 1 GJC, i. 58. SINGLE INDETERMINATE EQUATIONS 249 where m, n are rational numbers. It will be noticed that the equation (2) can be reduced, by dividing out by x -\-j, to x 2 — xj -\-J 2 = x -\-y ; and similarly (5) can be reduced to With m = 1, n = 2 Narayana gives the following sets of particular values : (1 . 2) x,y = I, 1£ (4.2) x,j = &, \% (2 . 2) x,y =1,2 (5.2) x, y =4 4, (3.2) x,y= I, I (6.2) x,y = ^ l3 ^ r He then observes : "In this way one can find by his own intelligence two numbers for the case of difference, etc." Form ax 2 " + 2 -J- bx 2n = y 2 . For the solution of an equation of the form ax 2n+2 -\- bx 2n =j 2 , where n is an integer, Bhaskara II gives the following rule : "Removing a square factor from the second side, if possible, the two roots should be investigated in this case. Then multiply the greater root by the lesser. Or, if a biquadratic factor has been removed, the greater root should be multiplied by the square of the lesser root. The rest of the operations will then be as before." 1 Suppose ax 2 -\- b = £ 2 ; then the equation becomes y 2 = x 2n % 2 . y = X n ^. The method of solving ax 2 -f- b = £ 2 in positive integers has been described before. 1 BBJ, p. 102. 250 ALGEBRA Two examples of equations of this form occur in the Bijaganita of Bhaskara II : x (1) 5.x 4 — 100^* —j) 2 , (2) ix 6 + 49.x 4 = j 2 . It may be noted that the second equation appears in ■the course of solving another problem. Equation ax 4 + bx 2 + c = u 3 . One very special case of this form arises in the course of solving another problem. It is 2 (a + x 2 ) 2 + a 2 = « 3 , or x* -\- zax 2 + za 2 = u 3 . Let u = x 2 , supposes Bhaskara II, so that we get x 6 — x 4 = za 2 -\- zax 2 , or x 4 (zx 2 — 1) = {za + x 2 ) 2 , which can be solved by the method indicated before. Another equation is 3 In cases like this "the assumption should be always such," remarks Bhaskara II, "as will make it possible to remove (the cube of) the unknown." So assume u = mx ; then x = \m z . 23. LINEAR FUNCTIONS MADE SQUARES OR CUBES Square-pulveriser. The indeterminate equation of the type bx -\- c =y 2 1 BBi, pp. 103, 107. z BBi, p. 103 ; also vide infra, p. 280. 3 BBi, p. 50 ; also vide infra, p. 278. LINEAR FUNCTIONS toADE SQUARES OR CUBES 2 5. 1 is called varga-kuttaka or the "Square-pulveriser," 1 inas- much as, when expressed in the form r 2 — c - b x ' the problem reduces to finding a square (varga) which being diminished by c will be exactly divisible by b, which closely resembles the problem solved by the method of the pulveriser (kuttakd). For the solution in integers of an equation of this type, the method of the earlier writers appears to have been to assume suitable arbitrary values for j and then to solve the equation for .v. Brahmagupta gives the following problems : "The residue of the sun on Thursday is lessened and then multiplied by j, or by 10 Making this (result) an exact square, within a year, a person becomes a mathematician." 2 "The residue of any optional revolution lessened by 92 and then multiplied by 83 becomes together with unity a square. A person solving this within a year is a mathematician." 3 That is to say, we are to solve the equations : (i)- sx — 25 =y, (2) 10.V — 100 =j 2 , (3) 8 3 x— 7635 =y. Prthudakasvami solves them thus : (1 . 1) Supposej = 10 ; then x = 125. Or, put j = j ; then x = 10. (2 . 1) Suppose j = 10 ; then x = 20. (3 . 1) Assume j = 1 ; then x = 92. 1 BBi, p. 122. 2 BrSpSi, xviii 76. a BrSpSi, xviii. 79^ 2J2 ALGEBRA He then remarks that by virtue of the multiplicity of suppositions there will be an infinitude of solutions in every case. But no method has been given either by Brahmagupta or by his commentator Prthudakasvami to obtain the general solution. The above method is reproduced by Bhaskara II. 1 He has also given the following rule : "If a simple unknown be multiplied by the number which is the divisor of a square, etc., (on the other side) then, in order that its value may in such cases be integral, the square, etc., of another unknown should be put equal to (the other side). The rest (of the operations) will be as described before." 2 • His gloss on this rule runs as follows : "In those cases, such as the Square-pulveriser, etc., where on taking the toot of one side of the equation there remains on the other side a simple unknown multiplied by the number which was the divisor of the square, etc., the square, etc., of another unknown plus or minus; an absolute term should be assumed for (the value of this other side) in order that its value may be integral. The rest (of the operations) will be as taught before." Bhaskara has also quoted from an ancient author the following rule : "(Find) a number whose square is exactly divisible by the divisor, as also its product by twice the square- root of the absolute term. An unknown multiplied by that number and superadded by the square-root of the absolute term should be assumed (for the unknown on the other side). If the absolute term does not yield a square -root, then after dividing it by the divisor, the 1 Vide infra, p. 25 j f. 2 BBi, p. 1 20. LINEAR FUNCTIONS MADE SQUARES OR CUBES 253 lemainder should be increased by so many times the divisor as will make a square. If this is not possible, then the problem is not soluble." 1 Case i Let c be a square, equal to p 2 , say. Then we have to solve bx + p 2 =y 2 . The rule says, find^> such that p z = bq, zp$ = br. Then assume y = pu -f- P ; whence we get x = qu 2 + ru. Bhaskara II prefers the assumption j = bv + p, so that we have x = bv 2 + 2p^ . Czxtf ii. If = r 2 . Now assume y = bu ±r. Substituting this value in the equation bx + c = y 2 , we get bx -\- c = {bu i r) 2 = & 2 # 2 rb 2 ^ r # + ^ 2 > or Zw -j- c — r 2 = £ 2 # 2 ± 2^r«, or &*• + b{m — s) = £ 2 # 2 ^ 2#r#. at = £# 2 ± zru — (m — s). Example from Bhaskara II : 2 ■jx + 30 =J' 2 . On dividing 30 by 7 the remainder is found to be 2; we also know that 2 + 7-2 = 4 2 . Therefore, we 1 BBi, p. 121. 2 BB/, pp. 120, 121. 2 J4 ALGEBRA assume in accordance with the above rule J = 7" ± 4; whence we get x = ju 2 4_; 8« — z, which is the general solution. Cube-pulver iser . The indeterminate ' equation of the type bx -\- c = y 3 is called the ghana-kuttaka or the "Cube-pulveriser." 1 For its solution in integers Bhaskara II says : "A method similar to the above may be applied also in the case of a cube thus : (find) a number whose cube is exactly divisible by the divisor, as also its product by thrice the cube-root of the absolute term. An un- known multiplied by that number and superadded by the cube-root of the absolute term should be assumed. If there be no cube-root of the absolute term, then after dividing it by the divisor, so many times the divisor should be added to the remainder as will make a cube. The cube-root of that will be the root of the absolute number. If there cannot be found a cube, even by so doing, that problem will be insoluble." 2 Case i. Let c = p 3 . Then we shall have to find^> such that p z = bq , i)p§ = br. Now assume y = pv -\- p. Substituting in the equation bx -\- p 3 = y 3 we get bx + p 3 = {pv 4- p) 3 = ^ 3 + ipv^pv + P) 4- P 3 , or bx = bqv 3 -\- brv(pi> -\- p). x = qv 3 4- rv(pv -\- p). 1 BBi, p. 122. 2 BBi, p. 121. LINEAR FUNCTIONS MADE SQUARES OR CUBES 255 Or, if we assume y — bv + P, we shall have x = b*v z + tfv(bv + |3). Case ii. c ^ a cube. Suppose c — bm -\- n ; then find j such that » + sb = r 3 . Now assume y — bv -\- r, whence we get x = b 2 ^ + irv(bv + r) — (m — s), as the general solution. Example from Bhaskara II i 1 ja: + 6 =7 3 . Since 6=5.1 + 1 and' 1 -+- 43 . 5 = 6 3 , we assume y = ^v -\- 6. Therefore x == z$v z + 18^(5^ + 6) + 4 2 > is the general solution. Equation bx ± c = ay 2 . To solve an equation of the type ay* = bx zk.c* Bhaskara II says : "Where the first side of the equation yields a root on being multiplied or divided 2 (by a number), there also the divisor will be as stated in the problem but the abso- lute term will be as modified by the operations." 3 1 BBi, p. 122. 2 The printed text has hitva ksiptd (subtracting or adding). After collating with several copies Colebrooke accepted the reading batva ksip td (multiplying or adding). But we think that the correct reading should be hatva hftvd (multiplying or dividing) For in his gloss Bhaskara II has employed the terms gunito vibhakto va (multiplied or divided). Our emendation is further supported by the rationale of the rule. »BBi, p. 121. 2j6 ALGEBRA What is implied is this : Multiplying both sided of the given equation by a, we get cP-y*- = abx ^ ac, Put a = ay, v = ax. "Then the equation reduces" to ifi = bv ± ac, which can be solved in the way described before. We take the following illustrative example with its solution from Bhaskara II i 1 5j/ 2 + 3 = *6x. Multiplying by 5, and putting u= 5j, v= ^x, we get a 2 = i6v — 1 j. The solution of this is u = iw ^ 1, v = 4&> 2 ±: w -\- 1. Therefore, we have (1) 5 j = 8^ + 1, or (2) 5j = 8»-— 1. Now, solving by the method of the pulveriser, we get the solution of (1) as j=8/ + 5, »= j' + 3 ; and that of (2) as J=8/+3, 2y = 5/+ 2 ; where / is any rational number. Equation bx ^ c = ay n After describing the above methods Bhaskara II observes, ityagre'pi yojyamiti sesah or "the same method can be applied further on 1 $Bi, p. i2i. LINEAR FUNCTIONS MADE SQUARES OR CUBES 2J7 (i.e., to the cases of higher powers) " x Again at the end of the section he has added evam buddhimadbhiranyadapi yathasambhavam yojyam, i e , "similar devices should be applied by the intelligent to further cases as far as practicable." 2 What is implied is as follows : (i) To solve — =^=— =j. Put x = m% ± k. Then •- + nm?L (± £)-i + (± £)» ± r } = ~ | /W^" ± nm n -\ n - l k + . . . + nm^ (± /fe)" -1 1 + (±)- (££-') Now, if — =±=_ = a whole number, b X n -4- £* — ==— will be an integral number when (i) m = b or (2) b is a factor of m n , nm*- 1 k y etc. Or, in other words, knowing one integral solution of (1) an infinite number of others can be derived. (2) To solve t=£- —j. Multiplying by a n -\ we get cPx 11 -A- ca n ~ x ., =f ==j^- ] . 1 BBi, p. 121. s BBi, p. 122. 17 2j8 ALGEBRA Putting « = ax, v —ja"^ 1 , we have I which is similar to case (i). v, c z4. DOUBLE EQUATIONS OF THE FIRST DEGREE The earliest instance of the solution of the simulta- neous indeterminate quadratic equation of the type x ^ a = u 2 , ) x i b = v 2 , J is found in the Bakhshali treatise. The portion of th manuscript containing the rule is mutilated. The example given in illustration can, however, be restored as follows : "A certain number being added by five {becomes capable of yielding a square-root} ; the same number {being diminished by} seven becomes capable of yielding a square-root. What is that number is the question." 1 That is to say, we have to solve .V-v + 5 = », V-*" — 7 = v. The solution given is as follows : "The sum of the additive and subtractive is [12 its half I 6 j ; minus two [4J ; its half is \z\ ; squared | 4 |. 'Should be increased by the subtractive' ; {the subtrac- tive is} |j7_J; adding this we get | 11 1 . This is the number (required)." From this it is clear that the author gives the 1 BMs, Folio 59, recto. DOUBLE EQUATIONS OF FIRST DEGREE 2J9 solution of the equations x -\- a — u 2 , x — b = v 2 ; *-{*('-£-*— )}'+* where m is any integer. 1 Brahmagupta (628) gives the solution of the general case. He says : "The difference of the two numbers by the addition or subtraction of which another number becomes a square, is divided by an optional number and then increased or decreased by it. The square of half the result diminished or increased by the greater or smaller (of the given numbers) is the number (required)." 2 i.e., x = J K^- ± m ) \ ^ a > or x== l*(^i~ ^^j Tb > where m is an arbitrary integer. The rationale is very simple. Since a 2 = x i a -> v<1 = * ± b t we have " » 2 — v 2 = ^ a ^ b'. Therefore u — v = m, and // + v = — ■ — ', m where m is arbitrary. Hence K± a ^f b . % , . ,a — b . \ 1 In the above solution m is taken to be 2. 2 BrSpSi, xviiis 74. 260 ALGEBRA Since it is obviously immaterial whether u is taken as positive or negative, we have Similarly v — h( ~ • -£"*)• ra — b Therefore x = j £(- ± /s?) I =F a, ,a—b. or x where m is an arbitrary number. Brahmagupta gives another rule for the particular case : x -\- a = » 2 , x — b — v 2 . "The sum of the two numbers the addition and subtraction of which make another number (severally) a square, is divided by an optional number and then diminished by that optional number. The square of half the remainder increased by the subtractive number is the number (required)." 1 — {*<*£-*— )}'+'• Narayana (1357) says : "The sum of the two numbers by which another number is (severally) increased and decreased so as to make it a square is divided by an optional number and then diminished by it and halved ; the square of the result added with the subtrahend is the other number." 2 He further states : 1 BrSpSi, xviii. 73. 2 GK, i. 52. DOUBLE EQUATIONS OF FIRST DEGREE 26 1 "The difference of the two numbers by which another number is increased twice so as to make it a square (every time), is increased by unity and then halved. The square of the result diminished by the greater number is the other number." 1 t.e. t x=-- ( ) —a is a solution of x -f- a = u 2 , x -f- b = v 2 , a > b. "The difference of the two numbers by which another number is diminished twice so as to make it a square (every time), is decreased by unity and then halved. The result multiplied by itself and added with the greater number gives the other." 2 /a— b— i\ 2 . i.e., x — ( — r- ) -f- a is a solution of x — a = u 2 , x — b = v 2 , a > b. The general case ax+c = u 2 , ) ,, bx+d = v\) ' K } has been treated by Bhaskara II. He first lays down the rule : "In those cases where remains the (simple) unknown with an absolute number, there find its value by forming an equation with the square, etc., of another unknown plus an absolute number. Then proceed to the solution of the next equation comprising the simple unknown with an absolute number by substituting in it the root obtained before." 3 *GK,i.j}. a GX,i. 54. 3 BBi, pp. 1 17-8. 262 ALGEBRA He then proceeds to explain it further : "In those cases where on taking the square-root of the first side, there remains on the other side the (simple) unknown with or without an absolute number, find there the value of that unknown by forming an equation with the square of another unknown plus an absolute number. Having obtained the value of the unknown in this way and substituting that value (in the next equation) further operations should be proceeded with. If, however, on equating the root of the first with another unknown plus an absolute number, no further operations remain to be done, then the equation has to be made with the square, etc., of a known number." (/) Set u = mn> + a ; then substituting in the first equation, we get x — — {inhv 2 -\- zff/wo. -f- a 2 — c). Substituting this value of x in the next equation, we have b — (»V -f- lmwa. -\- a 2 — c) -\- d = l' 2 , (1.1) a which can be solved by the method of the Square-nature. {it) In the course of working out an example 1 Bhaskara II is found to have followed also a different procedure, which was subsequently adopted by Lag- range. 2 Eliminate x between the two equations. Then bu 2 + {ad — be) = av 2 , or abu 2 -\- k = n> 2 , (1.2) where n> = av, k = a 2 d — abc. 1 Vide infra, p. 265. a Addition to Euler's Algebra, p. 547. DOUBLE EQUATIONS OF FIRST DEGREE 263 If u = r, w = s be a solution of this transformed equation, another solution of it will be « = rq±ps, w = qs ± abrp ; where abp 2 -j- i = q 2 . Therefore, the general solution of (1) is * = -7^ ± P s f - ~> u = rq±ps, v = — (qs i abrp) ; where abp 2 + i = q 2 and abr 2 -j- «V — a^ = j -2 . Now, a rational solution of the equation abp 2 -j- 1 = q 2 is _ zt _ t 2 -\- ab P ' ~ t 2 — ab y q ~ t* — aW where t is any rational number. Therefore, the above general solution becomes where abr 2 + a 2 d — abc — s 2 . (iii) Suppose c and d to be squares, so that c = a 2 , d = p 2 . Then we shall have to solve ax + a 2 = » 2 , bx + P 2 = * 2 . 264 ALGEBRA The auxiliary equation abr 2 -f- a 2 cL — abc ~ s 2 in this case becomes abr 2 + ( + 1 J then from the first equation, x = 3_j' 2 + zy. Substituting this value in the other equation, we get 1 5j 2 + loy 4- 1 = v 2 , or (15J + j) 2 = ij^ 2 4- i'°- By the method of the Square-nature we have the solu- tions of this equation as v= 9 ) v= 71 J I5J4- 5 = 35)' *5J> + 5 = 2 -75)'"' Therefore j= 2, 18, ... Hence :*• = 16, 1008, ... (2) Or assume the unknown number to be *- = 4(**-i), l GK, i. 51. *BBi, p. 118. Z66 ALGEBRA so that the first condition of the problem (i.e., the first equation) is identically statisfied. Then by the second condition or (5«) 2 = 15P 2 -f 10. Now, by the method of the Square-nature, we get the values of (#, v) as (7, 9), (55, 71), etc. Therefore, the values of a- are, as before, 16, 1008, etc. The following example is by Narayana : "O expert in the art of the Square-nature, tell me the number which being severally multiplied by 4 and 7 and decreased by 3, becomes capable of yielding a square- root." 1 That is, solve : 4* — 3 = "\ ) ■jx — 3 = v 2 . ) Narayana says : "By the method indicated before the number is 1, 21, or 1057." 25. DOUBLE EQUATIONS OF THE SECOND DEGREE First Type. The double equations of the second degree considered by the Hindus are of two general types. The first of them is ax 2 -f ky 2 -\- c = u 2 , ) a'x 2 -\-b'y 2 +c' = v 2 .) Of these the more thoroughly treated particular cases are as follows : Case i. \ , (X' 1 GK, p. 40. + j 2 -f 1 = x 2 , —J 2 + 1 = v 2 1 DOUBLE EQUATIONS OF SECOND DEGREE 267 Case ii. f^+y-i-**, \x 2 — j 2 — 1 = v 2 . It should be noted that though the earliest treat- ment of these equations is now found in the algebra of Bhaskara II (1150), they have been admitted by him as being due to previous authors (ddyoddharanam). For the solution of (i) Bhaskara II assumes 1 so that both the equations are satisfied. Now, by the rBethod of the Square-nature, the solutions of the equa- tion 5^2 _ x = X 2 are ( If z)> (l7j 38 ) } ... Therefore) the solutions of (i) are x = 2} x = 38} j = z l> J==34 )' - Similarly, for the solution of (//'), he assumes which satisfy the equations. By the method of the Square-nature the values of ft, x) in the equation 5Z 2 +i=x* are (4, 9 ), (72, 161), etc. Hence the solutions of (//') are x = J = 9) x = 161) = *y j =144)' Bhaskara II further says that for the solution of equations of the forms (i) and (ii) a more general as- sumption will be where^>, m are such that P i ># 2 = a square, 1 BBi, p. 108. 268 ALGEBRA the upper sign being taken for Case /* and the lower sign for Case it. Both the equations are then identically- satisfied. Suppose _£ + /»* = J 2 , p— m* = P. Whence s ^= \l— + /r), where » is any rational number. Therefore Here he observes that m 2, should be so chosen that p will be an integer. Hence x 2 - i(^-h ^ 2 fi j 2 = trih& to the upper sign being taken for Case ;' and the lower sign for Case //'. ZW 2 Whence « = |( 1- «)^, ■im* n Or, we may proceed in a different way, says BMskara II Since {p 1 + q 2 ) ± *pq is always a square, we may assume x 2 = (/> 2 + # 2 )»/ 2 =F i, j2 _ 2.pqw 2 . DOUBLE EQUATIONS OF SECOND DEGREE 269 For a rational value of j, xpq must be a square. So we take p = ztn 2 , q~ n 2 . Hence we have the assumption X 2 = (4 ^4 + „4 )a ,2 q= Jt j j 2 = ^m 2 n 2 w 2 ; * j ^ ' the upper sign being taken for Case / and the lower sign for Case ii. Whence M = (2W 2 4~ n 2 )w, v — {zm 2 — n 2 )n>. It will be noticed that the equations (i) follow from (2) on putting w = \\zn. So we shall take the latter as our fundamental assumption for the solution of the equations (/') and («). Then, from the solutions of the subsidiary equations (4W 4 + w 4 )^ 2 T 1 = x 2 , by the method of the Square-nature, observes Bhaskara II, an infinite number of integral solutions of the original equations can be derived. 1 Now, one rational solution of (4W 4 + w 4 )^ 2 4- 1 = x 2 is zr TV = v (•4/8* + « 4 ) — r x = (4^ 4 + «*) + r* (4W* -+- n*) — r z ' Therefore, we have the general solution of x 2 -\-j 2 — 1 = v 2 \ x 2 —j 2 — 1 = u 2 ) 1 C£.BBi K p. no. M) 270 ALGEBRA (4^? 4 + w 4 ) + r 2 = 2r(2^ 2 + « 2 ) * — (4/K? 4 + O — r a> * (4^ + ffi) — r 2 ' _ 4%w _ zr{zm 2 — « 2 ) ■^ = (4« 4 + «*) — r 2 ' V ~ (4m 4 -+- « 4 j — r 2 ' where «?, «, r are rational numbers. For r = j/^, we get Genocchi's solution. 1 In particular, put m = it, n = 1, r = 8/ 2 — 1 in (^4). Then, we get the solution .8/ 2 — 1 v.2 64^ — 1 x 2/ V ' ' 8/ 2 ' 8/ 2 — I • ,/8/ 2 — i^ 2 J 2/ ' -^ 2/ ' (*> Putting m = ^, « = 1, r = 2/ 2 + 2/ -f- 1 in (^4), we have 2 1 ^ 2/ <*) Again, if we put «/ = /,#= 1, /• = 2/ 2 in (^4), we get j = 8/ 3 , v = 4 / 2 (2/ 2 — 1). J l ; These three solutions have been stated by Bhaskara II in his treatise on arithmetic. He says, 1 Now. Ann. Math., X, 18 ji, pp. 80-8 j ; also Dickson, Numbers., II, pp. 479- For a summary of important Hindu results in algebra see the article of A. N. Singh in the Arcbeon, 1936. 2 Here, and also in {c), we have overlooked the negative sign of x,_y, u and v. DOUBLE EQUATIONS OF SECOND DEGREE 271 "The square of an optional number is multiplied by 8, decreased by unity, halved and then divided by that optional number. The quotient is one number. Half its square plus unity is the other number. Again, unity divided by twice an optional number added with that optional number is the first number and unity is the second number. The sum and' difference of the squares of these two numbers minus unity will be (severally) squares." 1 ' "The biquadrate and the cube of an optional number is multiplied by 8, and the former product is again in- creased by unity. The results will be the two numbers (required)." 2 Narayana writes : "The cube of any optional number is the first number ; half the square of its square plus unity is the second. The sum and difference of the squares of these two numbers minus unity become squares." 3 That is, if m be an optional number, one solution of {it), according to Narayana, is x = 1- i, u = \m l -f- 2.) , j — m s , v = (m 2 — z) . It will be noticed that this solution follows easily from the solution (c?) of Bhaskara II, on putting t = mjz. This special solution was found later on by E. Clere (1850).* 1 L,p. 13. 2 L, p. 14. 3 GK, i. 46. * Notw. Ann. Math., IX, 1850, pp. 11 6-8; also Dickson, Numbers, II, p. 479 ; Singh, 1. c. 272 ALGEBRA Now, let us take into consideration the equation (4a* 4 + /^> 2 — 1 = x 2 . Its solutions are known to be w = x zm* ip =. x = zm 2 < zm £ From these, by the Principle of Composition, we get respectively two other solutions n> = x = i6m* + «* 32//?° -(- 6m 2 t£ w = x zm* _ « 6 + 3« 2 ^ 4 2«T Therefore, the general solutions of are x 2 -\-j 2 -\- 1 = » 2 , ) x—y 2 + \=v 2 ;) zm 2 x — zm 2 + « 2 -> X ri' > - zm 2^? 2 — n 2 («') X = -5(3 2#7 8 + 6«? 2 /2 4 ), * = ^(i6^ 4 + « 4 )(2^ 2 + n 2 ), v = 4( l6 ^ 4 + »*X^ 2 — « 2 ) ; (O DOUBLE EQUATIONS OF SECOND DEGREE 273 X = y zm n m ,2? it zm 2 -\- tfi 2 — « 2 V = zm zm' (O and x = — s(« 6 -|- 3« 2 ;^? 4 ), zm y = —l^ + >fi\ v = -^O 4 + «*)(z» a — » 2 ). (*") Putting « 1= 1 in (a') and {a"), we have the in- tegral solutions X — 2#? 2 , > H = 2/H7 2 + J 5 J/ = 2#f, 2-' = 2W 2 — I ; and x = zm\i6m z -\- 3), y = zm(\&m* + 1), # = (i6« 4 + i)(zm 2 + 1), y = (16JZ? 4 + i)(zm 2 — 1). Similarly, if we put m = i in (£') and (£"), we get (*'.i)i (a".i) and x = 1 « 2 , j = n, «=i(« 2 +2)J v = \{tfi-zy y ) x = *«*(»« +3), * = *(»* + 1) (« 2 + 2), j = »(«* + 1), v = £(** + 1) («2 _ 2 ). (C.i) :} ^"■ I > 1 This solution was given by Drummond (A.mer. Math. Mon., IX, 1902, p. 232). 18 274 ALGEBRA The solution (b f . i) is stated by Narayana thus : "Any optional number is the first and half its square is the second. The sum and difference of the squares of these two numbers with unity become capable of yielding a square-root." 1 Case iii. 'Form a x 2 + by 2 — u 2 , } a'x 2 + b'j 2 + c' = v 2 .) For the solution of double equations of this form Bhaskara II adopts the following method : The solution of the first equation is x = /;/;', u = try ; where am 2 + b = « 2 . Substituting in the second equation, we get {a'm 2 + b')y 2 + c' = v\ which can be solved by the method of the Square- nature. Example from Bhaskara II : 2 7* 2 + 8j 2 = * 2 j 7* 2 — 8j 2 + i=v 2 )' He solves it substantially as follows : In the first equation suppose x = zy ; then // = Gy. Putting x = 2)>, the second equation becomes 2oy 2 -f- i = v 2 . By the method of the Square-nature the values of y satisfying this equation are z, 36, etc. Hence the solu- tions of the given double equation are ,v = 4,72, ... ji= z, 36,... 1 CK,\. 45. *BBi, p. 119. DOUBLE EQUATIONS OF SECOND DEGREE 275 Case iv. Form a\x 2 ± y 2 ) + S= v 2 . ) Putting x 2 ± J 2 = Z Bhaskara II reduces the above equations to ■ a \+c' = v 2 ;) the method for the solution of which has been given before. Example with solution from Bhaskara II : x i(x 2 —j 2 ) + 3 = u 2 ) }(* 2 — J 2 )+ 3 =v 2 )' Set x 2 — j 2 = z ; then 2-Z + 3 = « 2 , 33: + 3 = v 2 . Eliminating % we g et 3» 2 = 2* 2 + 3, or (3#) 2 = 6^ 2 + 9. Whence y = 6, 60, . . . 3«= 15, 147,... Therefore # = 5, 49, ... Hence x 2 — j 2 = ^ = 11, 1199, ... Therefore, the required solutions are 11 " ->-*(t»--) where ^v is an arbitrary rational number. 1 BBi, p. 119. 276 ALGEBRA For m = 1, the values of (x,y) will be (6, 5), (600, 599),... For m = 11, we get the solution (60, 49), ... Case v. For the solution of the double equation of the general form ax 2 -\- by 2 + c = « 2 j a'x 2 + b'y 2 + c' = v 2 ) Bhaskara IPs hint 1 is : Find the values of x, u in the first equation in terms of y, and then substitute that value of x in the second equation so that it will be reduced to a Square-nature. He has, however, not given any illustrative example of this kind. Second Type. Another type of double equation of the second degree which has been treated is a 2 x 2 + bxy + cy 2 == u 2 , ~\ a'x 2 + b'xy + c'y 2 + d' = v 2 . ) The solution of the first equation has been given before to be v2 . b 2 v ) by za\ I v 4^2/ j za 2 where X is an arbitrary rational number. Putting k =y, we have x = ■->-((— — — 1 ) •*- = ay, za\ 4a* ' za* »= — (V — ^5 -!- 1 ); 2 \ 4(7" t where a = — [c— — :, — 1 ) — 9 - 2«v 4cr ' zir 1 K/V& jw/v a, pp. icjof.- DOUBLE EQUATIONS OF SECOND DEGREE 277 Substituting in the second equation, we get (y a 2 4. b'v+ c*)f + d' = v\ which can be solved by the method,of the Square-nature. This method is equally applicable if the unknown part in the second equation is of a different kind but still of the second degree. Bhaskara II gives the following illustrative example together with its solution :* x 2 -\- xy -f- J 2 == » 2 ) . (x-\-j)u+ 1 = W Multiplying the first equation by 36, we get (6x + iyf -\- zjy 2 = 36/f 2 . Whence 6x + 3jy = $(^2! *), X f X Gu=^Jf + l), where X is an arbitrary rational number. Taking X =y, we have 6x+ $3= iy, or x= £jy, and u = -I j. Substituting in the second equation, we get 56J 2 -+- 9 = yv 2 . By the method of the Square-nature the values of y are 6, 180, ... Hence the required values of (x,y) are (10, 6), (300, 180), ... 1 BBi, pp. loyf. 278 ALGEBRA 26. DOUBLE EQUATIONS OF HIGHER DEGREES There are a few problems which involve double equations of degrees higher than the second. The following examples are taken from Bhaskara II : Example 1. "The sum of the cubes (of two numbers) is a square and the sum of their squares is a cube. If you know them, then I shall admit that you are a great algebraist." 1 We have to solve the equations x 2 -\~ y 2 — " 3 x 3 -j-j 3 = *>) = p 2 .j The solution of this problem by Bhaskara II is as follows : "Here suppose the two numbers to be % 2 , z% 2 . The sum of their cubes is 9^. This is by itself a square and its square- ropt is 3£ 3 . "Now the sum of the squares of those two numbers is 5^ 4 . This must be a cube. Assuming it to be equal to the cube of an optional multiple of 5^ and removing the factor ^ 3 from both sides (we get £ = z$p 2 , where p is an optional number) ; so, as stated before, the numbers are (putting^ = 1) 625, 1250. The assump- tion should be always such as will make it possible to remove (the cube of) the unknown." 2 In general, assume x = w^ 2 , J = «^ 2 ; substituting in the second equation, we have a- 3 -f- j 3 = (w 3 -\- tJ 3 )^ 6 = i' 2 . If • /;/ 3 -f- « 3 = a square = p 2 , say, then v = pt^. 1 BBi, p. 56. 2 BBi, pp. j6f. DOUBLE EQUATIONS OF HIGHER DEGREES 279 Now, from the first equation, we have (m 2 + » 2 )^ = a 3 . Assume u = r^ ; then Z = r 2 m 2 -f- n 2 ' Hence we get mr 6 pr 6 x — f m i 1 «2N2> .y — (» a + « 2 ) 2 ' J (m 2 + « 2 ) 2 ' where r is any integer and m, n are such that ' m z + n 3 = a square. One obvious solution 1 of this equation is m = 1, n = z. Hence we get the solution x = , j = . 25 25 This particular solution has been given by Nariyana, who says : ' ' "The square of the cube of an optional number is the one and twice it is the other. These divided by 25 will be the two numbers, the sum of whose squares will 1 Now m a + « 3 can be made a square by putting » = U* + f)P, » = (J* + f)9> so that m 3 + tfi = (p a + q 3 ) 1 - Hence the solution of our equation •will be , A" 6 x = y = (Jfi + q*? (/>" + (64/25, 128/25); with 5> (625, 1250); with 1/2, (1/1600, 1/800); with 1/3, (1/18225, 2/18225). Thus by virtue of (the multiplicity of) the optional number many solutions can be found." Examp/e 2. "O most learned algebraist, tell me those various pairs of whole numbers whose difference is a square and the sum of whose squares is a cube." 2 That is to say, solve in positive integers j — x = u 2 , } j/2 _|_ x z ~ ?/1 3 < J Bhaskara Il's process of solving this problem is as follows : "Let the two numbers be x, j. Putting their differ- ence, j — x, equal to u 2 , we get the value of x as y — « 2 . Having thus found the value of x, the two numbers become y — u 2 , j. "The sum of their squares = zy 2 — zyu 2 -\- » 4 . This is a cube. Making it equal to » 6 and transposing we get tfi — « 4 = zy 2 — zyu 2 . Multiplying both sides by 2 and superadding « 4 , we get the square-root of the second side = zy — u 2 , and the first side = zu 6 — u*. Dividing out by » 4 (and putting w for zy/u 2 — i), we get zu 2 — 1= w 2 . By the method of the Square-nature the roots of this equation are *= 5, 29. , ••■ iv=~j, 41, ... 1 GK, i. 50. 2 BBi, p. 103. DOUBLE EQUATIONS OF HIGHER DEGREES 28 1 ''Then by the rule, 'Or, if a biquadratic factor has been removed, the greater root should be multiplied by the square of the lesser root,' 1 we get zy — 25 = 175, - ■ or zy — 841 = 34481. Therefore j = ioo, 17661, ... "Finding the respective values of the numbers, they are (75, 100), (16820, 17661), etc." Example 3. "Bring out quickly those two numbers of which the sum of the cube (of one) and the square (of the other) becomes a square and whose sum also is a square." 2 That is to say, solve {^+J* = «\ (1) 1 X -{-J = v 2 . (2) This problem has been solved by Bhaskara II in two ways, which are substantially as follows : First method. From (1) we get — !(£ + *). -»-*(t- 1 )- where X is an arbitrary number. Putting I = x, we get U = \{x 2 + x), J = \{x 2 — X). Substituting this value of j in (2), we get x 2 + x = zv 2 , or (zx+ i) 2 = 8^ 2 + 1. 1 The reference is to the rule on p. 249. 2 BBr, p. 107. 282 ALGEBRA By the method of the Square-nature we have v= 6 | *= 35) 2X+i = i7J' 2x-j-i = 99J'"' Whence the values of (x,y) are (8, 28), (49, n 76), ... Second Method. Assume x — zw 2 , j = jw 2 . Then x -f j = 9#' 2 = O^) 2 - So the equation (2) is satisfied. Now, substituting those values in (1) we get Zw 6 + 49^ = u 2 , or ^ 4 (8a^ 2 + 49) = u 2 . If 8^2 + 49 = 2J* then # = %ii> 2 . Now the values of w making S^ 2 -j- 49 a square are 2, 3, 7... Hence the required numbers (x, y) are (8, 28), (18, 63), (98, 343), .- Example 4. "What is that number which multiplied by three and added with unity becomes a cube ; the cube-root squared and multiplied by three becomes, together with unity, a square." 1 That is to say, solve = 2(^> 4 — ^ 2 ). Example 2. "If thou be expert in mathematics, tell me quickly those two numbers whose sum and difference are squares and whose product is a cube." 1 That is, solve x 4j_y = squares, j xj = a cube. J Bhaskara II says : "Here let the two numbers be 5^, 4.^. They are assumed such as will make their sum and difference both squares. Their product is 20^. This must be a cube. Putting it equal to the cube of an optional multiple 2 of io£ and removing the common factor ^ 3 from the sides as before, (we shall ultimately find) the numbers to be 10000, 12500." 1 BJB/, p. 56. Z G.K, i. 49. 286 ALGEBRA In general, let us assume, as directed by Bhaskara II, x = (m 2 -\- n 2 )^ 2 , j = zmn^, which will make x ^y squares. We have, therefore, only to make zmn{m 2 -f- w2 )^ 4 = a cube. Let zmn(m 2 -f- « 2 )^ 4 = p\ 3 . P 3 Then £ = Therefore x = y = zmn{m 2 -f- n 2 )' Q 2 + « 2 )/> 6 {2W»(//? 2 + « 2 )} 2 ' zmnp* {zmn{tti 2 + » 2 )} 2 ' where w, », ^> are arbitrary. This general solution has been explicitly stated by Narayana thus : • "The square of the cube of an optional number is divided by the square of the product of the two numbers stated above and then severally multiplied by those numbers. (Thus will be obtained) two numbers whose sum and difference are squares and whose product is a cube."* The two numbers stated above 2 are m 2 + « a and zntn whose sum and difference are squares. In particular, putting m = i, n = z,p = 10, Nara- yana finds x = 12500, y = 10000. With other values of m, n, p he obtains the values (3165/16, 625/4), (62500/117, 250000/507), (15625/1872, 15625/2028)-; and observes: "thus by virtue of (the multiplicity of) the optional numbers many values can be found." 1 GK, i. 49. 2 The reference is to rule i. 48. MULTIPLE EQUATIONS 287 Example 3. To find numbers such that each of them severally added to a given number becomes a square ; and so also the product of every contiguous pair increased by another given number. For instance, let it be required to find four numbers such that jT-a = ^ JZ + P = ii 2 , z -f a = r 2 , ^n> + |5 = X?. W -j- a — j- 2 , The method for the solution of a problem of this kind is indicated in the following rule quoted by Bhaskara II (11 50) from an earlier writer, whose name is not known : "As many multiple {guria) as the product-interpolator (yadha-ksepd) is of the number-interpolator (rasi-ksepa), with the square-root of that as the common difference are assumed certain numbers ; these squared and dimi- nished by the number-interpolator (severally) will be the unknowns." 1 ' In applying this method to solve a particular problem, to be stated presently, Bhaskara II observes by way of explanation : "In these cases, that which being added to an (unknown) number makes it a square is designated as the number-interpolator. The number-interpolator •multiplied by the square of the difference of the square- roots pertaining to the numbers, is equal to the product- interpolator. For the product of those two numbers added with the latter certainly becomes a square. The products of two and two contiguous of the square- roots pertaining . to the numbers diminished by the 1 BBi, p. 68. 288 ALGEBRA number-interpolator are the square-roots corresponding to the products of the numbers." 1 Since x=p 2 —a, j = q 2 —a, we get xj + P = 2 - «)(? 2 - «) + P = {p q - ay+{$-a(q-py). In order that xy -J- P may be a square, a sufficient condition is * ± Y- Thus, it is found that the square-roots p, q,_r\_ s form an A.P. whose common difference is y (= y/fi/a). Further, we have > x = p 2 — a, J={P± Y) 8 - «, K, = {P± 2 Y) 2 - a, V = (/> ± 3Y) 2 — a, as stated in the rule. These values of the unknowns, it will be easily found, satisfy all the conditions about their products. For *y+P = {p(P±y)-oL}\ J^+P={(P±Y)(p±2Y)-a} 2 , X* + P = {(P ± 2Y)(/> ± 3Y) - «} 2 . 1 BJB/, p. 67. MULTIPLE EQUATIONS 289 Thus we have i = (P± r)(P±2-r) — «, Z = (p±ZY)(p± 5 y)-a; as stated by Bhaskara II. It has been observed by him that the above principle is well known in mathematics. But we do not find it in the works anterior to him, which are available to us. It is noteworthy that 2 + 9*) + a 2 + P = I 2 as an indeterminate equation in q. Since we know one solution of it, namely q = p ± Y, £ = p{p ± Y) — <*, we can find an infinite number of other solutions by the method of the Square-nature. Now, suppose that another condition is imposed on the numbers, w'^., wx + 0' = ji2. *9 Z90 ALGEBRA On substituting the values of x and w this condition transforms into pi ± 6 Y /> 3 + ( 9 Y 2 + za)p 2 ± 6«ry^ + a* - 9 p + p' = & an indeterminate equation of the fourth degree in^>. In the following example and its solution from Bhaskara II we find the application of the above principle : Exaqipje. "What are those four numbers which together with 2 become capable of yielding square- roots ; also the products of two and two contiguous of which added by 18 yield square-roots ; and which are such that the square-root of the sum of all the roots added bv n becomes 13. Tell them to me, O algebraist friend.'* * Solution. "In this example, the product-interpolator is 9 times the number-interpolator. The square-root of 9 is 3. Hence the square-roots corresponding to the numbers will have the common difference 3. Let them be '*, x + h x + 6 > *+ 9- "Now the products of two and two contiguous of these minus the number-interpolator are the square- roots pertaining to the products of the numbers as increased by 18. So these square-roots are x 2 + $x — 2, x 2 + yx-\- 16, x 2 -f- 15^4- 5 2 -> "The sum of .these and the previous square-roots all together is )X 2 4-31x4- 84. This added with 1 1 1 BBi, p. 67. It will be noticed that by virtue of the last condition the problem becomes, in a way, determinate. MULTIP^LE EQUATIONS 29I becomes equal to 169. Hence 5X 2 -|- 3 ix -|- 95 = ox 2 -fox-f 169. "Multiplying both sides by 12, superadding 961, and then extracting square-roots, we get 6x -j- 3 1 = ox + 43 . .'. *■ = 2. "With the value thus obtained, we get the values of the square roots pertaining to the numbers to be 2, 5, 8, 11. Subtracting the number-interpolator from the squares of these, we have the (required) numbers as 2, 23,62, 119." Example 4. To find two numbers such that x — y -)- k — » 2 , - v + J -\- k. — v 2 , X 2 _ yl _{_ k' = S 2 , X* _j_ j,2 _j_ £' = / 2 . Bhaskara II says : "Assume first the value of the square-root pertain- ing to the difference (of the numbers wanted) to be any unknown with or without an absolute number. The root corresponding to the sum will be equal to the root pertaining to the difference together with the square-root of the quotient of the interpolator of the difference of the squares divided by the interpolator for the sum or difference of the numbers. The squares of these two less their interpolator are the sum and differ- ence of the numbers. From them the two numbers can be found by the rule of concurrence." 1 *BBi, pp. 1 1 iff. 292 ALGEBRA That is to say, if w is any rational number, we . assume u = w 4- a, where a is an absolute number which may be o. Then v = (w i a) + y/te'jte. Now x 2 — jy 2 + k' = (.*• — j)(x + j) 4- £' = (# 2 - i)(*< 2 - k) + # = # V — k(u 2 + z> 2 ) + k* + /£'. One sufficient condition that the right-hand side may be a square is k{v - uf = £', or v = « + \/ k'jM, which is stated in the rule. Therefore, x —j = {w ±: a) 2 — k, x -f-j/ = (22/ ± a + V F/£) 2 — & Hence x = ${(»> ± a) 2 + (»- ± a + Vi 7 ^) 2 — 2/6}, J = ${(» ± « + V^fe) 2 -(» + a) 2 }- Now, if y denotes ^jk'lk, we get x 2 +jv 2 = « 4 4- 2Y» 3 4- (3Y 2 — ikyfi 4- 2Y(Y 2 - k)u 4- i^ 2 + i(Y 2 - £) 2 . So it now remains to solve * 4 4- zytfi 4- (3 Y 2 — 2.k)tfi 4- 2y(y 2 — -£)# 4- \& 4- Ky 2 - &? + #' = ' 2 , which is an indeterminate equation in u. Applications. We take an illustrative example with its solution from Bhdskara II. "O thou of fine intelligence, state a pair of numbers, other than 7 and 6, whose sum and difference MUT/TIPLE EQUATIONS 293 (severally) added with 3 are squares ; the sum of their squares decreased by 4 and the difference of the squares increased by 12 are also squares ; half their product together with the smaller one is a cube ; again the sum of all the roots plus 2 is a square." 1 That is to say, if x >j, we have to solve X —J+ 3 = u\ X +J+3 — v\ X* ! -y + 12 = s*. •* 2 +y — 4 = t\ l x y+y=p z , This problem has been solved in two ways : First Method. As directed in the above rule, assume « = w — 1. Then x —y = {w — i) 2 — 3 = w 2 — 2#> — 2, X +J = {W — I + 2) 2 — 3 = W 2 + ZlV — 2. Therefore x = w 2 — 2, j = 2^. Now, we- find that * 2 — J 2 + I2 = (»^ 2 — 4) 2 , .v 2 -f- j/ 2 — 4 = a/ 4 , So all the equations except the last one are already satisfied. This remaining equation now reduces to zip 2 -f- 3»>— 2 = q 2 . Completing the square on the left-hand side Of this equation, we get (4^+ 3 ) 2 = 8^-f-25. 'M,p. 115. 294 ALGEBRA By the method of the Square-nature its solutions are q= 5 ) q= i75j 4^+3 = 15)' 4^'+3=495i , Therefore n> = 3, 123, ... Hence the values of (x,y) are (7, 6), (1 5 127, 246), ... Second Method. Or assume 1 x — y + 3 = o' 2 , then x + J + 3 = ^ 2 + A w + 4 = O*' + 2 ) 2 - Whence x = w 2 + 2^ — 1 , y = zw -\- z. Now, we find that •X" 2 —J 2 + I2 = (^ 2 + 23^ — 3) 2 , a -2 + J 2 — 4 = C^ 2 -\- 2.W -\- i) 2 , i*7 + J = (^ + J ) 3 - Then the remaining condition reduces to zw 2 -\- jo> -\- 3 = q*. Completing the square on the left-hand side, we get (4s; +7 ) 2 =8^+25. Whence by the method of the Square-nature, we get ,= ,}, ,-75]. 4^4-7=15)' 4»+l = 49iy Therefore w = 2, 122, ... Hence (v,j) = (7, 6), (15 127, 246), ... Another very interesting example which has been borrowed by Bhaskara 11 from an earlier writer is the following: 2 1 This is clearly equivalent to the supposition, u = w, v = w + 2. 2 The text is kasydpyuddharanam ("the example of some one"). This observation appears to indicate that this particular example ~ was borrowed by Bhaskara II from a secondary source ; its primary source was not known to him. MULTIPLE EQUATIONS 29 J "Tell me quickly, O sound algebraist, two numbers, excepting 6 and 8, which are such that the cube-root of half the sum of their product and the smaller one, the square-root of the sum of their squares, the square-roots of the sum and difference of them (each) increased by 2, and of the difference of their squares plus 8, -all being added together, will be capable of yielding a square- root." 1 ( That is to say, if x >J, we have to solve + V* +J.+ 2 + V-V — J + 2 = 4 2 . In every instance of this kind, remarks Bhaskara II, "the values of the two unknown numbers should be so assumed in terms of another unknown that all the stipulated conditions will be satisfied." In other words, the equation will have to be resolved into a number of other equations all of which have to be satisfied simul- taneously. Thus we shall have to solve X—J+2 = U\ ' X -\-J + 2 = V 2 , X 2—j2+ 8 = S 2 , X 2 + f = t\ u-\-v-\-s-\-t-\-p = q 2 . The last equation represents the original one. There- have been indicated several methods of solv- ing these equations. (/') Set x = a> 2 — 1, j = zip ; then we find that x — y + 2 = (» — i) 2 , x + j + 2 = (» + i) 2 , ' a .Bfi^p. j 10. Z<)6 ALGEBRA ■x- 2 ~ J 2 + 8 = (>» 2 - 3) 2 > So all the equations except the last one are identically satisfied. This last equation now becomes ZW 2 + $U> — 2 == ^ 2 . Completing the square on the left-hand side, we get (4^+3)2= 8^+ 2J . Solutions of this are 0=5} ?=3° ( ?= 175) 4^+3 = 15)' 4»+3 = 8jI' 4» + 3 = 495J'"' Therefore, we have the solutions of our problem as (*,j0= ( 8 > 6 )> (^77/4, 4i), (15 1*8, 246). ••• Or set ,* = •»+**, W j J = £? + 2; ,... x ("A- — W 2 — 2»", v ' | J — zn> — 2; f x = w 2 4- 4» 4- 3, v (jr= 2^4- 4. In conclusion Bhaskata II remarks : "Thus there may be a thousandfold artifices ; since they are hidden to the dull, a few of them have been indicated here out of compassion for them." 1 It will be noticed that in devising the various arti- fices noted above for the solution of the problem, Bhaskara II has been in each case guided by t he result that if u — w ± a, then, » = *±a4 yjkfjk. He has simply taken different values of a in the different cases. 1 BBi,p. no. solution of axy = bx + cy + d 297 28. SOLUTION OF axy = bx + cy+d Bakhshall Treatise. The earliest instance of a quadratic indeterminate equation of the type axy = bx -{- cy -\- d, in Hindu mathematics occurs in the Bakhshall Treatise (c. zoo). 1 The text is very mutilated. But the example that is preserved is xy = 3^+ 4 j=f= i, of which the solutions preserved are v— 3-4— 1 1 A __. x = \- 4 — 1 j, y = 1 + 3 =4; and ^-' + 4 = 5, ^ = t 4±J + 3 = lS . Hence, in general, the solutions of the equation xy = bx -\- cy -\- d, which appear to have been given are : x == m + c and m > a —t- — . If these conditions be re- m versed then x andy will have their values interchanged. The rationale of the above solutions can be easily shown to be as follows : axy = bx -f- cy -j- d, abx — acy = ad, c)(ay — b) = ad -f- be. a x — c = m y a rational number ; , ad + be b= ! . m or a 2 xy or (ax Suppose ax then ay 3 BrSpSi, xviii, , 60. solution of axy = bx + cy -\- d 299 Therefore x= — (m -f- c), ± , ad+bc y I a ad A- be a s m Or, we may put ay — b = m ; ad+ be then we shall have ax — e whence x = — ( (- c ), m 1 (ad-\- br m y = ^( m + *)• It will thus be found that the restrictive condition of adding the greater and lesser of the numbers m and (ad -\- bc)\ni to the lesser and greater of the- numbers b and c respectively as adumbrated in the above rule is quite unnecessary. Brahmagupta's Rule. Brahmagupta gives the following rule for the solution of a quadratic indeter- minate 'equation involving a factum : "With the exception of an optional unknown, assume arbitrary values for the rest of the unknowns, the product of which forms the factum. The sum of the products of these (assumed values) and the (respective) coefficients of the unknowns will be absolute quantities. The continued products of the assumed values and of the coefficient of the factum will be the coefficient of the optionally (left out) unknown. Thus the solution is effected without forming an equation of the factum. Why then was it done so?" 1 The reference in the latter portion of this- rule is to the method of the unknown writer. The principle 1 BrSpSJ, xviii. 62-3, vide supra, p. 297. 300 ALGEBRA underlying Brahmagupta's method is to reduce, like the Greek Diophantus (c. 27J), 1 the given indeterminate equation to a simple determinate one by assuming arbitrary values for all the unknowns except one. So it is undoubtedly inferior to the earlier method. Brahmagupta gives the following illustrative example : "On subtracting from the product of signs and degrees of the sun, three and four times (respectively) those quantities, ninety is obtained. Determining the sun within a year (one can pass as a proficient) mathe- matician." 2 If x denotes the signs and y the degrees of the sun, then the .equation is xy — ix — 4ji = 90. Thus this problem, as that of Bhaskara II (infra), appears to have some relation with that of the Bakhshili work. Prthudakasvami solves it in two ways. Firstly, he as- sumes the arbitrary number to be 17, then 1/ 90. 1 + 3. 4 , \ x = —{ 2 L - 2 - 5 + 4J = 10, 17 J = ~(i7 + 3) = 20. Secondly, he assumes arbitrarily jy = 20. On substituting this value in the above equation, it reduces to 20X" — 3X =170 ; whence x = 10. Mahavira's Rule. Mahavira (850) has not treated equations of this type. There are, however, two pro- blems in his Gapita-sdra-samgraha which involve similar 'equations. One of them is to find the increase or 1 Heath, Diophantus, pp. 192-4, 262. 2 BrSpSi, xviii. 61. solution of axy = bx -f- cy -f- d 301 decrease of two numbers (a, b) so that the product of the resulting numbers will be equal to another optionally given number (d). Thus we are to solve or ± x)(b ±y) = d, xy ^ {bx + ay) = d — ab. The rule given for solving this is : "The difference between the product of the given numbers and the optional number is put down at two places. It is divided (at one place) by one of the given numbers increased by unity and (at the other) by the optional number increased by the other given number. These will give in the reverse order the values of the quantities to be added or subtracted." 1 That is to say, d ~ ab ' d + b d ~ ab x = y a -f- 1 or x = j,= d '— ab T+T d ~ ab Thus the solutions given by Mahavira are much cramped. The other problem considered by him is to separate the capital, interest and time when their sum is given : If x be the capital invested and y the period of time in months, then the interest will be mxy, where m is the rate of interest per month. Then the problem is to solve mxy -f- x -4-y = p. Mahavira solves this equation by assuming arbitrary values for y. 2 1 CSS, vi. 284. *GSS, vi. 35. 3 OZ ALGEBRA Sripati's Rule. Sripati (1039) gives the following rule : "Remove the factums from one side, the (simple) unknowns and the absolute numbers from the other. The product of the coefficients of the unknowns being added to the product of the absolute quantity and the coefficient of the factum, (the sum) is divided by an optional number. The quotient and the divisor should be added arbitrarily to the greater or smaller of the coefficients of the unknowns. These divided by the coefficient of the factum will be the values of the un- knowns in the reverse order." 1 1 / ■ N i.e., x— — (/»-+- c) J a v m ' x a^ m j =, i-( w + b) where m is arbitrary. Bhaskara IPs Rule. Bhaskarall (1150) has given two rules for the solution of a quadratic indeterminate equation containing the product of the unknowns. His first method is the same as that of Brahmagupta: "Leaving one unknown quantity optionally chosen, the values of the other should be assumed arbitrarily according to convenience. The factum will thus be reduced and the required solution can then be obtained by the first method of analysis!" 2 Bhaskara's aim was to obtain integral solutions.- The above method is, however, not convenient for the purpose. He observes : "On assuming in this way an arbitrary known value for one of the unknowns, the integral values of the 1 SiSe, xiv. 20-1. 2 BBi, p. 123. ! solution of axy = bx-\- cy -\- d 303 two unknowns can be obtained with much difficulty." 1 So he describes a second method "by which they can be obtained with little difficulty." "Transposing the factum from one side chosen at pleasure, and the (simple) unknowns and the absolute number from the other side (of the equation), and then dividing both the sides by the coefficient of the factum, the product of the coefficients of the unknowns together with the absolute number is divided by an optional number. The optional number and that quotient should be increased or diminished by the coefficients of the unknowns at pleasure. They (results thus obtained) should be known as the values of the two unknowns reciprocally." 2 This rule has been elucidated by the author thus : "From one of the two equal sides the factum be- ing removed, and from the other the unknowns and the absolute number ; then dividing the two sides by the coefficients of the factum, the product of the coefficients of the unknowns on the other side added to the absolute number, is divided by an optional number. The optional number and the quo- tient being arbitrarily added to the coefficients of the unknowns, should be known as the values of the un- knowns in the reciprocal order. That is, the one to which the coefficient of the hdlaka (the second unknown) is added, will be the value of the ydvat-tdvat (the first unknown) and the one to which the coefficient of the ydvat-tdvat is added, will be the value of the kdlaka. But if, after that has been done, owing to the magnitude, the statements (of the problem) are not fulfilled, then 1 "Evamekasmin vyakte ra£au kalpit^ sati bahunayasenabhinnau raSi jnayete" — BBi, p. 124. 2 BBi, pp. iz4f. 3°4 ALGEBRA from the optional number and the quotient, the coeffi- cients of the unknowns should be subtracted, and (the remainders) will be the values of the unknowns in the reciprocal order." Thus Bhaskara's solutions are ±*' J = ^± n x ■■ ±n' J =■ — ± m where m' is any arbitrary number and «' = — 7 (-§ + — )• m y a 2 a The rationale of these solutions is as follows : axy = bx -\- cy -\- d, or or xy x - ^ a c d —J~ ~ a a ( x ■ c a )( -? - t) = t + ;£ = wV > sa y- be a* Then, either x = 4- fir a J- a or x J 7~±' b_ a ±m' whence the solutions. Bhaskara's Proofs. The same rationale of the above solutions has been given also by Bhaskara IT with the help of the following illustrative example. He observes that the proof "is twofold in every case : one geometrical (ksetragatd), the other algebraic (rdsigata)." 1 Example. "The c sum of two numbers multiplied by four and three, added by two is equal to the product BBi, p. 125. solution of axy = bx -\- cy -\- d 3°5 of those numbers. "Tell me, if thou knowest, those two numbers." 1 Solution. "Having performed the operations as stated, the sides are xy == 4X + 3 j + 2. The product of the coefficients of the unknowns plus the absolute term is 14. Dividing this by an optional number (say) unity, the optional number and the quotient are 1, 14. To these being arbitrarily added 4, 3, the coefficients of the unknowns, the values of (x,y) are (4, 18) or (17, 5). (Dividing) by (the optional number) 2, (other values will be) (5, 11) or (10, 6.)" 2 Geometrical Proof. "The second side of the equation is equal to the factum. But the factum is the area of an oblong quadrilateral of which the base and upright are the unknown quantities. Within this figure (Fig. 15) exist four x's, three j's and the absolute number 2. From this figure on taking off four x's and y minus four multiplied by its own coefficient, (i.e., 3), it becomes this (Fig. 16). * * .* T I 1 -r-n \ I I I y 1 4. LU Fig. 15 Fig. 16 The other side of the equation being so treated there- 1 BB/, pp. 123, 125. 20 2 BBi, p. 125. 306 ALGEBRA results 14. This must be the area of the figure remaining at the corner (see Fig. 16) within the rectangle represent- ing the factum, and is the product of its base and upright. But these are (still) to be known here. Therefore, assum- ing an optional number for the base, the upright will be obtained on dividing the area 14 by it. One of these, base and upright, being increased by 4, the coefficient of x will be the upright of the figure representing the factum, because when four x's were separated from the factum-figure, its upright was lessened by 4. Similarly the other being increased by 3, the coefficient of j, will be the base They are precisely the values of x andjy." 1 Algebraic Proof "This is also geometrical in origin. In this the values of the base and upright of the smaller rectangle within the rectangle whose base and upright are x andjy respectively, are assumed to be two other unknowns it and v. 2 One of them being increased by the coefficient of x will be the value of the upright of the outer figure and the other being increased by the coefficient of y will be taken to be the value of the base of the outer figure. Thus y = u +4, x = v + 3. Substituting these values of the unknowns x, j, on both sides of the equation, the upper side will be 3« + 4P + 26 and the factum side will be uv -f- 3// + 4^+12. On making perfect clearance between these sides, the lower side becomes uv and the upper side 14. This is the area of that inner rectangle and it is equal to the product of the coefficients of the unknowns plus the absolute number. How the values of the unknowns are to be thence deduced, has been already explained." 3 1 BBi, p, 126. 2 In the original text they are respectively nt (for nilakd) and pt {for pttakd). 8 BBi, p. 127. solution of axy = bx -f- cy + d 307 Bhaskara II further observes : "Thus the proof of the solution of the factum has been shown to be of two kinds. What has been said before — the- product of the coefficients of the unknowns together with the absolute number is equal to the area of another rectangle inside the rectangle representing the factum and lying at a corner — is sometimes other- wise. For, when the coefficients of the unknowns are negative, the factum-rectangle will be inside the other rectangle at one corner ; and when the coefficients of the unknowns are greater than the base and upright of the factum-rectangle, and are positive, the other will be outside the factum rectangle and at a corner, as (Figs. 17," 18). I I I r t 1 y i 1 t 1 1 v I . Fig. 17 Fig. 18 When it is so, the coefficients of the unknowns lessened by the optional number and the quotient, will be the values of x andj." 1 l BBi,p.iz-j. JIO INDEX atulya — 148 ; samasa — 148 ; tulya — 148 Bhavita 26, 35 Bija 1, 4, 5 Bijaganita 1, 2, 3, 65, 88, 90, 139 Bindu 14 Brahmagupta 1, 4, 5, 9, 10, 11, 17, 20, 21, 22, 23, 25, 26, 31, 33, 35, 40, 4 1 , 43. 44, 54, 55, 57, 6z , 6 3> 6 4, 67, 70, 74, 75, 82, 83, 85, 87, 89, 91, 92, 117, 121, i2j, 127, 132, 143, 145, 147, 157, 158, 160, 161, 173, 174, 175, 176, 181, 206, 207, 2IO, 211, 222, 223, 22J, 227, 2 33> 2 34, 23 5, 236, 239, 240, 251, 252, 259, 260, 264, 284, 300, 302 ; — 's Corollary 147, 149, 152; — 's Lemmas 146, 147, 148, 149 ; — 's Proof 148 Brahma na, 6 ; Satapatha — 7, 8 Brahma-sphuta-siddhanta 4, 32 Brahmi characters 14, 15 Brouncker 155 Cantor 50 Cakravala 162 Chasles 239 Clark, W. E. 93 Clere, E. 271 Coefficient 9 Colebrooke 33, 65, 141, 206, 255 Conjunct Pulveriser 135 ; gene- ralised — 137; alternative me- thod 139 Constant Pulveriser 117 Cube-pulveriser 254 Cyclic method 161 Datta, B. 1, 6, 9, 11, 14, 15, 30, 35, 3 6 > 59> 6 °, 93, 13 T > 2 °4> 205, 206, 207, 208, 209, 212, 221, 222, 223 Desboves 199 Devaraja 88 Dickson, L. E. 133, 199, 225, 227, 240, 270, 271 Diophantus 14, 209 Division 23, 27 Double equations ; — in several unknowns 283 ; — of higher degrees 278 ; — of the first degree 258; — of the second degree 266, first type 266, second type 276 Drsya 12 Drummond 273 Dvicchedagram 131 Dvidha 146 Dvivedi, Sudhakara 65 Elimination of the middle term Epanthema 50 Evolution and involution 23 Equations 11, 28; Classification of — 3 5 ; common forms of — 8 1 ; forming — 28 ; higher — 77 ; In- determinate^ — of the first de- gree 87, see pulveriser; Inde- terminate — of the second degree, see Square-nature ; linear — 44 ; linear — in one un- known 36 ; linear — with two unknowns 43 ; linear — with several unknowns 47, 125 ; — of highet degrees 76 ; plan of writing — 3 o ; preparation of INDEX 3 II — 33; Quadratic — 59; simul- taneous quadratic — 81 Euler 148, 166, 1 86, 262 Fermat 150, 166 Fibonacci, Leonardo 214 Frenicle 166 Gaccha 16 Ganesa 69, 88, 90, 117, 208, 212, 240 Gahgadhara 43 Ganguly, Sarda Kanta 31, 88, 93, 104, 139, 140 Ganita, 1, 3, 16 ; kuttaka — 1 ; avyakta — 1 ; vyakta — 1. Ganita-sara-sarhgraha 3 8, 66, 123, 300 General indeterminate equations of the second degree : single equations i8i General problem of remainders 131 General solution of Square- nature 149 Genocchi 270 Ghana 10, 15, 35; — varga 10, 15; — varga-varga 10 Ghata 10 Gulika 10 Gulikantara 40 Gunadhana 78 Gunaka 9 Gunakara 9 Guna-karma 3 H Haritaka 18 Heath 50 Hewlett, John 186 Hrasva mula 144 Hoernle 14, 15 . I Indeterminate equations 87, general survey 87, its import- ance 88, of the first degree 87, three varieties of problems 89, preliminary operations 91 ; simultaneous — of the first degree 127, Sripati's solution 127; single — of higher degrees 24$, Bhaskara's method 246, Mahavira's rule 245, Nari- yana's rule 248 Ibn-al-Haitam 133 Isosceles trapeziums 228 ; ratio- nal — 228 ; pairs of — 231 Isosceles triangles, — with a given altitude 223'; with integral sides 222 ; pairs of rational — 224 Istagunaghna 53 Ista-karma 39 Jha, Murlidhara 19, 20 Jnanaraja 28, 65, 69, 93, 116, 128, 144, 148, 1 j 1, 154, 186, 189, 193 J.yestha pada 144 Kfilaka 17 Kamalakara 93, 116, 141, 143, 144, 148, 151, 154, 193, 194, 195, 196, 197, 198 Kamika 9, 38 Kanistha pada 144 Karavindasvami 207 Katyayana 59, 204, 206 Kaye 14, 48, 49, 50, 61, 93 ;-297 Konasariku 64 JI2 INDEX Krsna 27, 36, 65, 69, 90, in, 118, 142, 148, 154, .163, 177 Krti-prakrti 141 Ksaya 12 Ksepaka 144 Ksetragata 3 Kummer 239 Kuttaka 1, 89, 91 ; citra — misra 52; drdha — 117; — ganita 1; origin of the name 90 ; — Siromani 88; sthira — 117 Kuttikara 89 Laghubhaskariya 284 Lagrange 148, 186 Laws of signs 20 Lemma 150; Brahmagupta's 146 ; Bhaskara's 162 LUavati 3, 88, 90, 117, 124, 139 Linear equations 36, 44; early solutions 36 ; in more than two unknowns 125 ; — in one unknown 36 ; rule of false position 37 ; solution of — 40 ; — with several unknowns 47 ; — with two unknowns 43 Lohita 18 M Madhura 19 Madhyamaharana 3, 35, 36, 69, 76 Maha-Bhaskariya 89 Mah&vedi 6 MaMvira 20, 22, 23, 24, 25, 38, .44, 49. 5°, 5i, 52, 53, 56, 66 > 6 7, 73» 74. 77, 7 8 , 79, 82, 83, '84, 85, 87, 89, 91, 137, 138, 207, 208, 209, 210, 213, 214, 215, 216, 21.7, 218, 219 221, 222, 223, 224, 226, 227, 229, 231, 234, 256, 238, 242, 243, 2 44, 245, 3 QI 5— ' s defini- tions 208 ; — 's rules 56, 86, 103, 300; rule of — 124. Matsunago 199 Mazumdar, N. K. 93 Mehta, D. M. 88 Mikami, Y. 200 Mula 15; varga — 10; dvitiya- varga — 11 ; n th varga — 11 ; prathama-varga — 1 1 ; tritiya- varga — ghana 11. Multiple equations 283 Multiplication 22, 26 N Narayana 5, 11, 14, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 4i, 43, J°> 52, 53, 8 2, 8 3> 84, 93, n6, 119, 122, 143, 144-, 148, 150, 151, 154, 164, 165, 168, 174, 176, 177, 178, 179, 181, 186, 202, 245, 248, 249, 260, 264, 266, 271, 274, 279, 284, 286 ; — 's rule 164, 248 Nilaka 17 Nyasa 30, 33 Nyuna 14 Operations, fundamental — 25 ; number of — 2 5 ; — with an op- tional number 39 Origin of minus sign 14 Padmanabha 12, 70 Pairs of rectangles 219 Paksa 11 Palabha 64 INDEX 313 Panca gata 10 Pellian equation 5 Pitaka 18 Power 10 Pradesa 7 Prakrti 9 Principle of Composition 145, 147, 148, J5° Prthudakasvami 1, 9, 11, 18, 20, 24, 29, 3-1, 32, 33, 34, 35. 4°, 41, 55. 75, I02 > IX 7> 12I » x 3 2 > 41, 55. 75, I02 > IX 7> 12I » x 3 2 > 134, l >)> x 43> '44, 145, 22 4, 227, 229, 234, 238, 251, 252, - J-T3 - J J *- ■ -- 227, 229, 234, 2 97, 2 9 8 > 3«° Purusa 7 Quadratic, two roots of 70, known to MaMvira 73. Quadrilaterals, rational — 228 ; Inscribed — having a given area 242 R Rahn, J. H. 225 Ramakrsna 65 Rangacarya 78, 104 Rariganath 90, 212 Rasigata 4 Rational, inscribed quadrilaterals 235 ; — isosceles trapeziums 228 ; — quadrilaterals 228 ; — right triangles having a given hypotenuse 213 ; — right tri- angles having a given side 209; — scalene triangles 225; — triangles 204, early solutions 204, integral solutions 206,. juxtaposition of — 223, later rational solutions 207 Rodet 5 1, 93 Rule, — of concurrence 43 ; — of false position 37, 38; dis- appearance from later algebra 38 Rupa 9, 12, 141 S Sadgata 10 SadrsTkaram n Sama 11 Sama-karana n Satmsodhana 33, 34 Samatva n Samikarana n, 28 ; aneka-varna — 33 ; avyaktavarga — 35 ; asakrt — 181 ; ekavarna — 35 ; sakrt — 181 Samsl. stakuttaka 136 Samya 11 'Sahkramana 43, 44, 82, 84 Satpurusa 4 Schooten, Fras van 225 Sen Gupta, P. C. 93 Siddhanta-sjromani 90 Siva 5 Singh, A. N. 270, 27 1 Smith 14, 32, 38, 150 Sodhana 33 Special rules 129 Squaring 27 Square pulveriser 250 Square-root 28 Sridhara 12, 16, 38, 65, 67, 238 ; — 's rule 65 Sridhara Mahapatra 140 Sripati j, 11, 18, 21, 22, 23, 24, 34, 40, 44, 64, 9 2 > 137, r 43, 145, 15°, r 55, 157, 2 o7, 2 37, 302; — 's rule 67, no, 127, 302 1 Sthananga-sutra 9, 35, 36 Subtraction 21 ; addition and — , 2 5 Sulba 6, 7, 36, 59, 204 Suryadasa 65, 69, 90, 162, 177, 2ii, 212, 215 3*4 INDEX Suryadeva Yajva 91 , 133 Suryasiddhanta 64 Symbols 12; — for powers and roots 15 ; — for unknowns 16 ; — of operation 12 Tattvarthadhigama-sutra 60 Technical terms 9, 143 Terminology 89 Thibaut, G. 11, 14 Thymaridas 5 o, 5 1 Trapeziums 228 ; rational — 228 ; rational — with three equal sides 133 Triangles, having a given area 227; — and quadrilaterals hav- ing a given area 244 TriSatika 16, 65, 238 U Umasvati 60 Unknown quantity