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A HISTORY OF
MATHEMATICAL NOTATIONS
Volume II
NOTATIONS MAINLY IN HIGHER MATHEMATICS
A HISTORY OF
MATHEMATICAL
NOTATIONS
By
FLORIAN CAJORI, Ph.D.
Professor of the History of Mathematics University of California
Volume II
NOTATIONS MAINLY IN HIGHER MATHEMATICS
The Open Court Publishing Company Chicago . Illinois
Copyright 1929 By
The Open Court Publishing Company Published March 1929 Second Printing September 1930 Third Printing January 1952
Paquin Printers
chicaco
PREFACE TO THE SECOND VOLUME
The larger part of this volume deals with the history of notations in higher mathematics. The manuscript for the parts comprising the two volumes of this History was completed in August, 1925, but since then occasional alterations and additions have been made whenever new material or new researches came to my notice.
Some parts of this History appeared as separate articles in scien¬ tific and educational journals, but later the articles were revised and enlarged.
I am indebted to Professor R. C. Archibald and to Professor L. C. Ivarpinski for aid in the arduous task of reading the proofs of this volume.
Florian Cajori
University of California
*
TABLE OF CONTENTS
Introduction to the Second Volume
PARAGRAPHS
I. Topical Survey of Symbols in Arithmetic and Algebra
(Advanced Part) . 388-510
Letters Representing Magnitudes . 388-94
Greek Period . 388
Middle Ages . 389
Renaissance . 390
Vieta in 1591 391
Descartes in 1637 . 392
Different Alphabets . 393
Astronomical Signs . 394
The Letters tv and e . 395-401
Early Signs for 3.1415 . 395
First Occurrence of Sign tv ... 396
Euler’s Use of tv . 397
Spread of Jones’s Notation . 398
Signs for the Base of Natural Logarithms . 399
The Letter e . 400
B. Peirce’s Signs for 3.141 .... and 2.718 . 401
The Evolution of the Dollar Mark . 402-5
Different Hypotheses . 402
Evidence in Manuscripts and Early Printed Books . . 403
Modern Dollar Mark in Print . 404
Conclusion . 405
Signs in the Theory of Numbers . 406-20
Divisors of Numbers, Residues . 407
Congruence of Numbers . 408
Prime and Relatively Prime Numbers . 409
Sums of Numbers . 410
Partition of Numbers . 411
Figurate Numbers . 412
Diophantine Expressions . 413
Number Fields . 414
Perfect Numbers . 415
Mersenne’s Numbers . 416
Fermat’s Numbers . 417
Cotes’s Numbers . 418
viii TABLE OF CONTENTS
PARAGRAPHS
Bernoulli’s Numbers . 419
Euler’s Numbers . 420
Signs for Infinity and Transfinite Numbers . 421
Signs for Continued Fractions and Infinite Series. . . . 422-38
Continued Fractions . 422
Tiered Fractions . 434
Infinite Series . 435
Signs in the Theory of Combinations . 439-58
Binomial Formula . 439
Product of Terms of Arithmetical Progression .... 440
Vandermonde’s Symbols . 441
Combinatorial School of Hindenburg . 443
Kramp on Combinatorial Notations . 445
Signs of Argand and Ampere . 446
Thomas Jarrett . 447
Factorial n . 448
Subfactorial N . 450
Continued Products . 451
Permutations and Combinations . 452
Substitutions . 453
Groups . 454
Invariants and Covariants . 456
Dual Arithmetic . 457
Chessboard Problem . 458
Determinant Notations . 459-68
Seventeenth Century . 459
Eighteenth Century . 460
Early Nineteenth Century . 461
Modern Notations . 462
Compressed Notations . 463
Jacobian . 464
Hessian . 465
Cubic Determinants . 466
Infinite Determinants . 467
Matrix Notations . 468
Signs for Logarithms . 469-82
Abbreviation for “Logarithm” . 469
Different Meanings of log x, lx, and Lx . 470
Power of a Logarithm . 472
Iterated Logarithms . 473
Marking the Characteristic . 474
Marking the Last Digit . 478
Sporadic Notations . 479
Complex Numbers . 480
TABLE OF CONTENTS ix
PARAGRAPHS
Exponentiation . 481
Dual Logarithms . 482
Signs of Theoretical Arithmetic . 483-94
Signs for “Greater” or “Less” . 483
Sporadic Symbols for “Greater” or “Less” . 484
Improvised Type . 485
Modern Modifications . 48G
Absolute Difference . 487
Other Meanings of ^ and ^ . 489
A Few Other Sporadic Symbols . 491
Signs for Absolute Value . 492
Zeroes of Different Origin . 493
General Combinations between Magnitudes or Numbers 494
Symbolism for Imaginaries and Vector Analysis .... 495-510 Symbols for the Square Root of Minus One .... 495
De Morgan’s Comments on V — 1 . 501
Notation for a Vector . 502
Length of a Vector . 504
Equality of Vectors . 505
Products of Vectors . 500
Certain Operators . 507
Rival Vector Systems . 508
Attempts at Unification . 509
Tensors . 510
II. Symbols in Modern Analysis . 511-700
Trigonometric Notations . 511-37
Origin of the Modern Symbols for Degrees, Minutes, and
Seconds . 511
Signs for Radians . 515
Marking Triangles . 516
Early Abbreviations of Trigonometric Lines . . . . 517
Great Britain during 1602-18 . 518
European Continent during 1622-32 519
Great Britain during 1624-57 . 520
Seventeenth-Century English and Continental Practices
are Independent . 521
England during 1657-1700 522
The Eighteenth Century . 524
Trigonometric Symbols of the Eighteenth Century . . 525
Trigonometric Sjunbols of the Nineteenth Century . . 526
Less Common Trigonometric Functions . 527
Quaternion Trigonometry . 528
Hyperbolic Functions . 529
Parabolic Functions . 531
TABLE OF CONTENTS
Inverse Trigonometric Functions .
Jonn Herschel’s Notation for Inverse Functions Martin Ohm’s Notation for Inverse Functions Persistance of Rival Notations for Inverse Functions
Inverse Hyperbolic Functions .
Powers of Trigonometric Functions
Survey of Mathematical Symbols Used by Leibniz
Introduction . .
Tables of Symbols ...
Remarks on Tables .
Differential and Integral Calculus
1. Introduction .
2. Symbols for Fluxions, Differentials, and Derivatives
а) Total Differentiation during the Seventeenth and
Eighteenth Centuries. Newton, Leibniz, Lan- den, Fontaine, Lagrange (1797), Pasquich, Griison, Arbogast, Kramp .
б) Criticisms of Eighteenth-Century Notations.
'Woodhouse, Lacroix, Lagrange ....
c) Total Differentiation during the Nineteenth
Century. Barlow, Mitchell, Herschel, Peacock, Babbage, Crelle, Cauchy (1823, 1829), M. Ohm' Cauchy and Moigno (1840), B. Peirce, Carr’ Peacock, Fourier .
d) Initial Differentials and Partial Derivatives.
Euler, Karsten, Fontaine, Monge, Condorcet, Legendre, Lagrange (1788), Lacroix, Da Cunha,’ L’Huilier, Lagrange (1797), Arbogast, Lagrange (1801), Crelle, Barlow, Cauchy, M. Ohm, W. R. Hamilton, W. Bolyai, Cauchy and Moigno, C. G. J. Jacobi, Hesse, B. Peirce, Strauch, Du- hamel, Carr, Meray, Muir, Mansion
3. Symbols for Integrals, Leibniz
4. Early Use of Leibnizian Notation in Great Britain.
5. Symbols for Fluents: Later Notations in Integral Cal¬
culus. Newton, Reyneau, Crelle, Euler, Fourier, Volterra, Peano, E. H. Moore, Cauchy’s Residual Calculus .
6. Calculus Notations in the United States .
7. Symbols for Passing to the Limit. L’Huilier, Weierstrass
Oliver, Riemann, Leathern, Dirichlet, Pringsheim Scheffer, Peano, W. H. Young .
8. The Sign . ]
9. Concluding Observations .
PARAGRAPHS
532
533
534
535
536
537 538-65
538 542
563-65
566-639
566
567
567-78
579
582
593
620
621
622
630
631
638
639
TABLE OF CONTENTS
xi
. # PARAGRAPHS
Finite Differences . 640 041
Early Notations . 640
Later Notations . 641
Symbols in Theory of Functions . 642-66
A. Symbols for Functions in General . 642
B. Symbols for Some Special Functions . 647
Symmetric Functions . 647
Gamma and Beta Functions . 649
Elliptic Functions . 651
Theta Functions . 656
Zeta Functions . 659
Power Series . 661
Laplace, Lame, and Bessel Functions . 662
Logarithm-Integral, Cosine-Integral, etc . 665
Symbols in Mathematical Logic . 667-99
Some Early Symbols . 667
The Sign for “Therefore” . 60S
The Sign for “Because” . 669
The Program of Leibniz . ' 670
Signs of
H. Lambert . 671
G. J. von Holland . 672
G. F. Castillon . 673
J. D. Gergonne . 674
Bolyai . 675
Bentham . 676
A. de Morgan . 677
G. Boole . 678
W. S. Jevons . 679
Macfarlane . 680
C. S. Peirce . 681
Ladd-Franklin and Mitchell . 682
R. G. Grassmann . 684
E. Schroeder . 685
J. H. MacColl . 686
G. Frege . 687
G. Peano . 688
A. N. Whitehead . 692
E. H. Moore . 693
Whitehead and Russell . 695
P. Poretsky . 696
L. Wittgenstein . 697
Remarks by Rignano and Jourdain . 698
A Question . . ‘ . 699
TABLE OF CONTENTS
xii
PARAGRAPHS
III. Symbols in Geometry (Advanced Part) . 700-711
1. Recent Geometry of Triangle and Circle, etc . 700
Geometrographie . 701
Signs for Polyhedra . 702
Geometry of Graphics . 703
2. Projective and Analytical Geometry . 704
Signs for Projectivity and Perspectivity .... 705
Signs for Harmonic and Anharmonic Ratios ... 706
Descriptive Geometry . 707
Analytical Geometry . 708
Plucker’s Equations . 709
The Twenty-seven Lines on a Cubic Surface . . . 730
The Pascal Hexagram . 711
IV. The Teachings of History . 712-50
A. The Teachings of History as Interpreted by Various
Writers. Individual Judgments . 712-25
Review of D. Andre . 712
Quotations from A. de Morgan . 713
J. W. L. Glaisher . 714
D. E. Smith . 715
A. Saverien ... ... 716
C. Maclaurin ....... 717
Ch. Babbage . 718
E. Mach . 719
B. Branford . 720
A. N. Whitehead . 721
PI. F. Baker . 722
IP. Burckhardt . 723
P. G. Tait . 724
O. S. Adams . 724
A British Committee . 725
B. Empirical Generalizations on the Growth of Mathe¬
matical Notations . 727-33
Forms of Symbols . 727
Invention of Symbols . 727
Nature of Symbols . 728
Potency of Symbols . 729
Selection and Spread of Symbols . 730
State of Flux . 731
Defects in Symbolism . 732
Individualism a Failure . 733
TABLE OF CONTENTS xiii
PARAGRAPHS
C. Co-operation in Some Other Fields of Scientific Endeavor 734
Electric Units . 734
Star Chart and Catalogue . 735
D. Group Action Attempted in Mathematics .... 736
In Vector Analysis . 737
In Potential and Elasticity . 738
In Actuarial Science . 739
E. Agreements To Be Reached by International Committees
the Only Hope for Uniformity of Notations . . . 740
Alphabetical Index
ILLUSTRATIONS
FIGURES
107. B. Peirce’s Signs for 3.141 .... and 2.718
108. From J. M. Peirce’s Tables, 1871 ....
109. Pillar Dollar of 1661 .
110. Forms That Are Not Dollar Symbols .
PARAGRAPHS
. . 400
. . 401
. . 402
. . 402
HI. Symbols for the Spanish Dollar or Peso Traced from Manu¬ script Letters, Contracts, and Account-Books . . . 403
112. The Modern Dollar Mark in the Making . 403
113. Dollar Marks in L’Hommedieu’s Diary, 1776 . 403
114. From Chauncey Lee’s American Accomptant, 1797. . . 404
115. From Chauncey Lee’s American Accomptant, Page 142 . . 404
116. Multiplication Table for Sexagesimal Fractions. . . . 513
117. Marking the Given and Required Parts of a Triangle, 1618 518
118. The Given and Required Parts of Another Triangle, 1618 518
119. Illustrating Girard’s Notation in Trigonometry . . . . 519
120. A Page of Isaac Newton’s Notebook Showing Trigonometric
Symbols . 522
121. Leibniz’ Figure in MSS Dated October 26, 1675 .... 570
122. Leibniz’ Figure in MSS Dated October 29, 1675 .... 570
123. From Arbogast’s Calcul des Derivations (1880), Page xxi . . 578
124. Manuscript of Leibniz, Dated October 29, 1675, in Which
His Sign of Integration First Appears . 620
125. G. Frege’s Notation as Found in His Grundgesetze (1893), Vol¬
ume I, Page 70 . 687
126. Bow’s Notation . 702
XV
INTRODUCTION TO THE SECOND VOLUME
It lias been the endeavor to present in the two volumes of this History a fairly complete list of the symbols of mathematics down to the beginning of the nineteenth century, and a fairly representative selection of the symbols occurring in recent literature in pure mathe¬ matics. That we have not succeeded in gathering all the symbols of modern mathematics is quite evident. Anyone hunting, for even an houi, in the jungle of modern mathematical literature is quite certain to bag symbolisms not mentioned in this History, The task of making a complete collection of signs occurring in mathematical writings from antiquity down to the present time transcends the endurance of a single investigator. If such a history were completed on the plan of the present work, it would greatly surpass this in volume. At the
present time the designing of new symbols is proceeding with a speed that is truly alarming.
Diversity of notation is bound unnecessarily to retard the spread of a knowledge of the new results that are being reached in mathe- matics. \\ hat is the remedy? It is hoped that the material here pre¬ sented will afford a strong induction, facilitating the passage from the ieaim of conjecture as to what constitutes a wise course of procedure to the realm of greater certainty. If the contemplation of the mistakes in past procedure will afford a more intense conviction of the need of some form of organized effort to secure uniformity, then this History wTill not have been written in vain.
xvu
ADDENDA
Page 28, lino 3, add the following: In the Commonplace Book of Samuel B. Beach, B.A., Yale, 1805, now kept in the Yale Uni¬ versity Library, there is given under the year 1804, “the annual expence about $700,” for the upkeep of the lighthouse in New Haven. The dollar mark occurs there in the conventional way now current. Prof. D. E. Smith found the symbol $ very nearly in the present form in DabolPs Schoolmaster’s Assistant, 4th edi¬ tion, 1/99, p. 20. Dr. J. M. Armstrong of St. Paul, Minn., writes that in the Medical Repository, New York (a quarterly publica¬ tion), Vol. Ill, No. 3, November and December, 1805, and January, 1806, p. 312, the $ is used as it is today.
Page 29, line 6, add the following: Since this volume was printed, important additional and confirmatory material appeared in our article “New Data on the Origin and Spread of the Dollar Mark ” in the Scientific Monthly, September, 1929, p. 212-216. Page 145, line 1, for Gioseppe read Giuseppe Moleti Page 323, lines 8 and 9, for in G. Cramer . . . found earlier read ^ Claude Rabuel s Commentaires sur la Geometric de M. Descartes , Lyons, 1730, and in G. Cramer . . . found also.
In the alphabetical index insert Mahnke, D., 542, 543, 563.
xyia
I
TOPICAL SURVEY OF SYMBOLS IN ARITHMETIC AND ALGEBRA (ADVANCED PART)
LETTERS REPRESENTING MAGNITUDES
388. Greek 'period. I he representation of general numbers by letters goes back to Greek antiquity. Aristotle uses frequently single capital letters, or two letters, for the designation of magnitude or number. For example, he says: “If A is what moves, B what is being moved, and r the distance over which it was moved, and A the time during which it was moved, then the same force A, in the same time could move the half of B twice as far as T, or also in half the time A exactly as far as r.”1 In other places2 he speaks of the “BT any force, “the time EZ.” In another place he explains how much time and trouble may be saved by a general symbolism.3
Euclid4 in his Elements represented general numbers by segments of lines, and these segments are marked by one letter,5 or by two letters placed at the ends of the segment,6 much the same way as in Aristotle. Euclid used the language of line and surface instead of numbers and their products. In printed editions of the Elements it became quite customary to render the subject more concrete by writing illustrative numerical values alongside the letters. For ex¬ ample, Clavius in 1612 writes (Book VII, Prop. 5, scholium) “A, | D f,;; and again (Book VIII, Prop. 4), “A, Q.B, 5.C, 4.D, 3 .” In Robeit Simson s translation of Euclid and in others, the order of the English Alphabet is substituted for that of the Greek, thus A B r A
E Z II 9, etc., in Euclid are A B C D E F G H , etc., in Simson and others.7
1 Aristotle Physics vii. 5.
2 Ibid. viii. 10.
Aristotle Analytica posteriora i. 5, p. 74 a 17. Reference taken from Gow, History of Greek Mathematics (Cambridge, 1884), p. 105, n. 3.
4 Euclid’s Elements , Book 7.
5 Euclid’s Elements , Book 7, Prop. 3 (ed. J. L. Heiberg), Vol. II (1884) p. 194-98.
6 Euclid’s Elements , Book 7, Prop. 1 (ed. Heiberg), Vol. II, p. 188-90.
7 A. de Morgan in Companion to the British Almanac , for 1849, p. 5.
1
o
A HISTORY OF MATHEMATICAL NOTATIONS
According to Pappus,1 it was Apollonius of Perga who, like Archi¬ medes, divided numbers into groups or myriads and spoke of double, triple myriads, and so on, and finally of the “k fold’ ’ myriad. This general expression of a myriad of as high an order as we may wish marks a decided advance in notation. Whether it was really due to Apollonius, or whether it was invented by Pappus, for the more ele¬ gant explanation of the Apollonian system, cannot now be determined. But Apollonius made use of general letters, in the manner observed in Euclid, as did also Pappus, to an even greater extent.2 The small Greek letters being used to represent numbers, Pappus employed the Greek capitals to represent general numbers.3 Thus, as Cantor says, “The possibility presents itself to distinguish as many general mag¬ nitudes as there are capital letters.”4
It is of some interest that Cicero,5 in his correspondence, used letters for the designation of quantities. We have already seen that Diophantus used Greek letters for marking different powers of the
unknown and that he had a special mark p° for given numbers. We have seen also a symbol ru for known quantities, and yd and other symbols for unknown quantities (Vol. I, § 106).
389. Middle Ages. — The Indian practice of using the initial letters of words as abbreviations for quantities was adopted by the Arabs of the West and again by the translators from the Arabic into Latin. As examples, of Latin words we cite radix , res, census, for the unknown and its square; the word dragma for absolute number.
In Leonardo of Pisa’s Liber abbaci (1202), 6 the general representa¬ tion of given numbers by small letters is not uncommon. He and other writers of the Middle Ages follow the practice of Euclid. He uses letters in establishing the correctness of the rules for proving operations by casting out the 9’s. The proof begins thus: “To show the foundation of this proof, let .a.b. and .b.g, be two given numbers which we wish to add, and let .a.g. be the number joint from them
Pappus Collectio, Book II (ed. Hultsch), Vol. I (1876), p. 2-29. See M. Cantor, Vorlesungen uber Geschichte der Mathematik , Vol. I (3d ed., 1907), p. 347.
2 See G. H. F. Nesselmann, Die Algebra der Griechen (Berlin, 1842), p. 128-30.
3 Pappus (ed. Hultsch), Vol. I (1876), p. 8.
4 M. Cantor, op. cit., Vol. I (3d ed., 1907), p. 455.
5 Epistolae ad atticum, Lib. II, epistola 3.
6 Scritti di Leonardo Pisano matematico ... da Bald. Boncompagni (Rome, 1857-62), Vol. I, p. 394, 397, 441. See G. Enestrom, Bibliotheca mathematica (3d ser.); Vol. XIII (1912-13), p. 181.
LETTERS REPRESENTING MAGNITUDES
3
. . . ; Observe the use of clots to bring into prominence letters oc¬ curring m the running text, a practice very common in manuscripts o that time. In another place Leonardo proposes a problem: a horses eat b oats in c days, d horses eat e oats in / days; if the same amount of feed is eaten in the two cases, then the first product M.bx. is equal to the second .d.e.f., etc.2 Still more frequent repre¬ sentation of numbers by letters occurs in Jordanus Nemorarius’ Arithmetica. Jordanus died 1237; the arithmetic was brought out in print m 149G and 1514 in Paris by Faber Stapulensis. Letters are used instead of special particular numbers.3 But Jordanus Nemorarius was not able to profit by this generality on account of the fact that he had no signs of operation— no sign of equality, no symbols for sub¬ traction, multiplication, or division. He marked addition by juxta¬ position. He represented the results of an operation upon two letters by a new letter.4 This procedure was adopted to such an extent that the letters became as much an impediment to rapid progress on a train of reasoning as the legs of a centipede are in a marathon race. Letters are used occasionally in the arithmetic (1321) of the Jewish writer Levi ben Gerson5 of Avignon who, like Nemorarius, has no signs of operation. Gerson uses letters in treating of permutations and combinations. A similar cumbrousness of procedure with letters is found in the printed editions (1483) of the Algorismus de integris of the Italian Prosdocimo de’ Beldomandi.6 Less extensive, but more skilful, use of letters is made in the thirteenth century by Meister Gernardus7 and in the fourteenth century by the Frenchman N. Oresme,8 who even prefixed numerical coefficients to the letters in passages like “A.a. excedunt. .3. a. in sesquitertia.”
390. Renaissance.— The employment of letters to represent the various powers of unknown quantities by Chuquet in his manuscript Triparty , by Pacioli, and by the early algebraists of the sixteenth
1 Scritti di Leonardo Pisano , Vol. I, p. 20, 11. 9-28.
2 Op. cit., Vol. I, p. 132, 133.
•! See M. Curtze in Zeilschrift f. Mathematik und PhysiJc, Vol. XXXVI 1189 1') histor.-liter. Abt., p. 1-3. v '
4 P. Treatlein in Abhandlungen zur Geschichte der Mathematik, Vol. II (1879), p. 132 33. *
‘hU™ Prf0n; ^ des Reclmers (trails. G. Lange; Frankfurt a. M„
19U9), J. Varlebach, Lewi ben Gerson als Mathematiker (Berlin, 1910).
6 M. Cantor, op. cit., Vol. II (2d ed., 1913), p. 206.
7 G. Lnestroin, Bibliotheca mathematica (3d ser.), Vol. XIV (1914), p. 99 £f.
8 N. Oresme, Algorismus proportionum (ed. M. Curtze; Berlin, 1868), p. 22.
4
A HISTORY OF MATHEMATICAL NOTATIONS
century has been explained elsewhere. The manuscript algebra of Adam Riese, found in 1855 in the Library at Marienberg, contains some use of letters to represent given general numbers.1 Parts of hi? Coss were written in 1524, other parts in the interval 1544-59. Gram mateus in his Rechenbuch of 1518 uses in one place2 letters as the terms of a proportion: “Wie sich hadt a zum b also hat sich c zum d. Auch wie sich hadt flzuc also had sich b zum d.” One finds isolated cases indicating the employment of letters for given numbers in other writ¬ ers, for instance, Chr. Rudolff (1525) who writes j/c+j/d, Cardan
(1570)3 who explains that R ~ is equal to j”, that is, that
During the close of the fifteenth and early part of the sixteenth cen¬ tury the development of symbols of operation in algebra proceeded rapidly, but quantities supposed to be given were, as a rule, repre¬ sented by actual numbers ; numerical coefficients were employed with few exceptions. A reader who goes over the explanations of quad¬ ratic and cubic equations in works of Tartaglia, Cardan, Stifel, is impressed by the fact. As yet literal coefficients, as we write them in ax2-\-bx = c and xz-Jrax = b, were absent from algebra. In consequence there could not be a general treatment of the solution of a cubic. In its place there was given a considerable number of special cases, illus¬ trated by equations having particular numerical coefficients appropri¬ ately chosen for each case. Thus, Cardan,4 on August 4, 1539, dis¬ cusses the irreducible case of the cubic, not by considering xs = ax-\-b,
when (^\ > (|) > by taking xz = 9z+ 10, where 27 >25.
391. Vieta in 1591. — The extremely important step of introducing the systematic use of letters to denote general quantities and general numbers as coefficients in equations is due to the great French alge¬ braist F. Vieta, in his work In artem analyticam isagoge (Tours, 1591). He uses capital letters which are primarify representatives of lines and surfaces as they were with the Greek geometricians, rather than pure numbers. Owing to this conception, he stresses the idea of homogeneity of the terms in an equation. However, he does not con¬ fine himself to three dimensions; the geometric limitation is aban-
1 Bruno Berlet, Adam Riese, sein Leben, seine Rechenbucher und seine Art zu rechnen. Die Cos von Adam Riese (Leipzig, 1892), p. 35-62.
2 Grammateus, Rechenbuch (1518), Bl. CIII; J. Tropfke, Geschichte der Ele¬ ment ar-M athematik, Vol. II (2d ed., 1921), p. 42.
3 J. Tropfke, op. cit., Vol. II (2d ed., 1921), p. 42.
4 M. Cantor, op. cit., Vol. II (2d ed., 1913), p. 489.
LETTERS REPRESENTING MAGNITUDES
5
doned, and he proceeds as high as ninth powers — solido-solido- solidum. The homogeneity is illustrated in expressions like 11 A planum +Z in B ,” the A is designated planum, a “ surface/’ so that the first term may be of the same dimension as is the second term, Z times B. If a letter B represents geometrically a length, the product of two B’s represents geometrically a square, the product of three B’s represents a cube.
Yieta uses capital vowels for the designation of unknown quanti¬ ties, and the consonants for the designation of known quantities. His own words are in translation: “As one needs, in order that one may be aided by a particular device, some unvarying, fixed and clear symbol, the given magnitudes shall be distinguished from the un¬ known magnitudes, perhaps in this way that one designate the required magnitudes with the letter A or with another vowel E, I, 0, U, Y , the given ones with the letters B, G, D or other consonants.’’1
392. Descartes in 1637. — A geometric interpretation different from that of Vieta was introduced by Rene Descartes in his La geometrie (1637). If b and c are lengths, then be is not interpreted as an area, but as a length, satisfying the proportion bc:b = c: 1.
Similarly, - is a line satisfying the proportion - : 1 = b : c. c c
With Descartes, if b represents a given number it is always a posi¬ tive number; a negative number would be marked —b. It was J. Hudde2 who first generalized this procedure and let a letter B stand for a number, positive or negative.
393. Different alphabets. — While the Greeks, of course, used Greek letters for the representation of magnitudes, the use of Latin letters became common during the Middle Ages.3 With the development of other scripts, their use in mathematics was sometimes invoked. In 1795 J. G. Prandel expressed himself on the use of Latin type in alge¬ braic language as follows:
“Why Latin and Greek letters are chosen for algebraic calcula¬ tion, while German letters are neglected, seems, in books composed in
1 Vieta, Isagoge (Tours, 1591), fol. 7.
2 J. Hudde, De reductione aequalionum (1657), published at the end of the first volume of F. Van Schooten’s second Latin edition of Rene Descartes’ Geometrie (Amsterdam, 1659), p. 439. See G. Enestrom in Encyclopedic des sden. math., Tom. I, Vol. II (1907), p. 1, n. 2; also Bibliotheca mathematica (3d ser.), Vol. IV (1903), p. 208: The Geometry of Descartes, by Smith & Latham, Open Court (Chicago, 1925), p. 301.
3 See, for instance, Gerbert in CEuvres de Gerbert, par A. Olleris (Paris, 1867), p. 429-45.
6
A HISTORY OF MATHEMATICAL NOTATIONS
our language, due to the fact that thereby algebraic quantities can be instantaneously distinguished from the intermixed writing. In Latin, French and English works on algebra the want of such a con¬ venience was met partly by the use of capital letters and partly by the use of italicized letters. After our German language received such development that German literature flourishes in other lands fully as well as the Latin, French and English, the proposal to use German letters in Latin or French books on algebra could not be recounted as a singular suggestion.”1
“The use of Greek letters in algebraic calculation, which has found wide acceptance among recent mathematicians, cannot in itself encumber the operations in the least. But the uncouthness of the Greek language, which is in part revealed in the shape of its alphabetic characters, gives to algebraic expressions a certain mystic appear¬ ance.”2
Charles Babbage3 at one time suggested the rule that all letters that denote quantity should be printed in italics, but all those which indicate operations should be printed in roman characters.
The detailed use of letters and of subscripts and superscripts of letters will be treated under the separate topics of algebra and geome¬ try.4 -
That even highly trained mathematicians may be attracted or repelled by the kind of symbols used is illustrated by the experience of Weierstrass who followed Sylvester’s papers on the theory of algebraic forms until Sylvester began to employ Hebrew characters which caused him to quit reading.5
394. Astronomical signs. — We insert here a brief reference to astro¬ nomical signs; they sometimes occur as mathematical symbols. The twelve zodiacal constellations are divisions of the strip of the celestial sphere, called the “zodiac”; they belong to great antiquity.6 The symbols representing these constellations are as follows:
1 Johann Georg Prandel’s Algebra (Miinchen, 1795), p. 4. 2 Ibid., p. 20.
3 Charles Babbage, art. “Notation, ” in Edinburgh Encyclopedia (Philadelphia, 1832).
4 Consult Vol. I, §§ 141, 148, 176, 188, 191, 198, 342, 343; Vol. II, 395-401, 443, 444, 561, 565, 681, 732.
5 E. Lampe in Naturwissenschaftliche Rundschau , Vol. Nil (1897), p. 361; quoted by R. E. Moritz, Memorabilia mathemaiica (1914), p. 180.
». 6 Arthur Berry, Short History of Astronomy (New York, 1910), p. 13, 14; W. W. Bryan, History of Astronomy (London, 1907), p. 3, 4; R. Wolf, Geschichte der -Aistronomie (Miinchen, 1877), p. 188-91; Gustave Schlegel, Uranographie chinoise, Vol. I (Leyden, 1875), Book V, “Des zodiaques et des planetes.”
LETTERS REPRESENTING MAGNITUDES
7
|
T |
Aries, the Ram |
Libra, the Balance |
|
|
3 |
Taurus, the Bull |
Scorpio, the Scorpion |
|
|
n |
Gemini, the Twins |
* |
Sagitarius, the Archer |
|
23 |
Cancer, the Crab |
V3 |
Capricornus, the Goat |
|
Q |
Leo, the Lion |
AW AW |
Aquarius, the Water-Bearer |
|
n |
Virgo, the Maid |
X |
Pisces, the Fishes |
|
The signs for the planets, sun, |
moon, etc., are as follows: |
||
|
O |
Sun |
n |
Jupiter |
|
( |
Moon |
b |
Saturn |
|
© |
Earth |
ft |
Ascending node |
|
S |
Mercury |
ts |
Descending node |
|
$ |
Venus |
6 |
Conjunction |
|
6 |
Mars |
8 |
Opposition |
According to Letronne,1 the signs for the five planets and the sun and moon occur in two manuscripts of the tenth century; these signs, except that for the moon, are not found in antiquity. The early forms of the signs differ somewhat from those given in printed books. The signs for ascending and descending nodes of the moon’s orbit occur in a Greek manuscript of the fourteenth century.2 Some forms Uear resemblance to the Hindu-Arabic numerals. Particularly those for Jupiter and Saturn look like the four and five, respectively. In the twelfth century there were marked variations in the forms of the Hindu-Arabic numerals and also in the forms of the signs for the planets, sun, and moon. It is believed by some3 that these astro¬ nomical signs and numeral signs (being used often by the same persons) mutually influenced each other, with regard to their forms, before the introduction of printing. Hence the resemblances.
Several of the astronomical signs appear as mathematical symbols. Apparently, the sign for Pisces was chosen by L. and T. Digges as bheir sign of equality, but they added an additional horizontal stroke, as a cross-line. Such strokes were applied by them also to their symbols for* powers -of the unknown (Vol. I, § 170). The sign for Taurus, placed horizontally, with the open end to the left, was, we believe, the sign of
1 Letronne, Revue archeologique (lstser.), Vol. Ill (Paris, 1846), p. 153, 253-63.
2 P. Tannery, Memoires scientifiques, Vol. IV. (1920), p. 356, 359. Tannery
gives facsimile reproductions. »
3 G. Horn-D’Arturo, “Numeri Arabici e simboli celesti,” P ubblicazioni dell’Ox-
■A
servatorio astronomico della R. Universita di Bologna , Vol. I (Roma, 1925), p. 187#
.% a ^
204.
8
A HISTORY OF MATHEMATICAL NOTATIONS
equality in the 1637 edition of Descartes’ Geometrie (§ 191). The sign for Aries, placed horizontally, serves with Kastner1 for “greater than” and “smaller than.” The sign for earth is employed extensively in the modern logical exposition of algebra (§ 494). Extensive use of astro¬ nomical signs occurs in Leibniz’ letters2 to Jacob Bernoulli; for in¬ stance, j" u {/ J dx, where each astronomical sign stands for a cer¬ tain analytic expression (§ 560). Kastner employed the signs for Sun, Moon, Mars, Venus, Jupiter, in the marking of equations, in the place of our modern Roman or Hindu-Arabic numerals.3 Cauchy sometimes let the sign for Taurus stand for certain algebraic expres¬ sions.4
THE LETTERS tt AND e
395. Early signs for 3.1415. ... * — John Wallis,5 in his Arith- metica injinitorum (1655), lets the square □ or, in some cases, the Hebrew letter “mem” which closely resembles a square, stand for 4/3,14149 . . . . ; he expresses □ as the ratio of continued products and also, as William Brounker had done before him, in the form of a continued fraction.
Perhaps the earliest use of a single letter to represent the ratio of the length of a circle to its diameter occurs in a work of J. Chr. Sturm,6 professor at the University of Altdorf in Bavaria, who in 1689 used the letter e in a statement, “si diameter alicuius circuli ponatur a, circumferentiam appellari posse ea (quaecumque enim inter eas fuerit ratio, illius nomen potest designari littera e).” Sturm’s letter failed of general adoption.
Before Sturm the ratio of the length of a circle to its diameter was represented in the fractional form by the use of two letters. Thus,
1 A. G. Kastner, Anfangsgriinde der Arithmetik , Geometrie .... (Gottingen, 1758), p. 89, 385.
2 C. I. Gerhardt, Leibnizens Mathematische S chr if ten, Vol. Ill (Halle, 1855),
p. 100.
3 A. G. Kastner, Anfangsgriinde der Analysis endlicher Grossen (Gottingen, 1760), p. 55, 269, 336, 358, 414, 417, and other places.
4 A. L. Cauchy, Comptes rendus, Yol. XXIV (1847) ; (Euvres completes (1st ser.), Vol. X, p. 282.
B John Wallis, Arithmetica infmitorum (Oxford, 1655), p. 175, 179, 182.
6 J. Christoph Sturm, Mathesis enucleata (Niirnberg, 1689), p. 81. This refer¬ ence is taken from A. Krazer’s note in Euleri opera omnia (1st ser.), Vol. VIII, p. 134.
LETTERS 7T AND e
9
William Oughtred1 designated the ratio (§185) by -. He does not
<5
define tt and 8 separately, but no doubt 7 r stood for periphery and 8 for diameter. The radius he represents by R. We quote from page 66 of the 1652 edition: “Si in circulo sit 7.22: : 8. tt: : 113.355: erit 8.ir::2R.P: periph. Et ir.8: : \P .R: semidiam. 8.ir::Rg. Circul. Et ir. 8. .±Pg. Circul. Oughtred’s notation was adopted by Isaac Bar- row2 and by David Gregory.3 John Wallis4 in 1685 represented by 7 r the periphery described by the center of gravity in a revolution. In 1698 De Moivre5 designated the ratio of the length of the circle to
the radius by - .
r
396. First occurrence of the sign 7 r. — The modern notation for 3.14159 .... was introduced in 1706. It was in that year that ^ illiam Jones6 made himself noted, without being aware that he was doing anything noteworthy, through his designation of the ratio of the length of the circle to its diameter by the letter 7 r. He took this step without ostentation. No lengthy introduction prepares the reader for the bringing upon the stage of mathematical history this distinguished visitor from the field of Greek letters. It simply came, unheralded, in the following prosaic statement (p. 263) :
“There are various other ways of finding the Lengths or Areas of particular Curve Lines , or Planes , which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Cir¬ cumference as 1 to ~ — -J^r— — &c =3 14159 Ac — v
5 2 3 9 3 53 2393 , 71.
This series (among others for the same purpose, and drawn from the same Principle) I received from the Excellent Analyst, and my much esteem’d Friend Mr. John Machin; and by means thereof, Van
1 W. Oughtred, Clavis mathematicae (1652), p. 66. This symbolism is given in the editions of this book of 1647, 1648, 1652, 1667, 1693, 1694. It is used also in the Appendix to the Clavis , on “Archimedis de Sphaera et Cylindro declaration This Appendix appeared in the editions of 1652, 1667, 1693.
2 W. Whewell, The Mathematical Works of Isaac Barrow (Cambridge, 1860), p. 380, Lecture XXIV.
3 David Gregory, Philosophical Transactions , Vol. XIX (London, 1697), p. 652 except that he writes -, p being the radius.
p
4 John Wallis, Treatise of Algebra (1685), “Additions and Emendations, ” p. 170.
6 De Moivre, Philosophical Transactions , Vol. XIX (1698), p. 56.
6 William Jones, Synopsis palmariorum matheseos (London, 1706), p. 263.
10
A HISTORY OF MATHEMATICAL NOTATIONS
Ceulen’s Number, or that in Art. 64.38. mav be Examin’d with all desirable Ease and Dispatch.” Then he writes “d = c-c 7r” and
“C = dX7T.”
This was not the first appearance of the letter 7 r in Jones’s book of 1706. But in earlier passages the meanings were different. On page 241 it was used in lettering a geometric figure where it represented a point. On page 243 one finds “Periphery (7 r),” as previously found in Wallis.
Nor did the appearance of t = 3.14159 .... on the stage attract general attention. Many mathematicians continued in the old way. In 1721 P. Varignon1 wrote the ratio S.7r, using for ratio the dot of Oughtred.
397. Eider’s use of 7 r. — In 1734 Euler2 employed p instead of 7 r
and g instead of . In a letter of April 16, 1738, from Stirling to Euler,
as well as in Euler’s reply, the letter p is used.3 But in 1736 he4 desig¬ nated that ratio by the sign 1 : ir and thus either consciously adopted the notation of Jones or independently fell upon it. Euler says: “Si
enim est m — J terminus respondens inuenitur - denotante 1 : 7 r ratio-
nem diametri ad peripheriam.” But the letter is not restricted to this use in his Mechanica, and the definition of 7 r is repeated when it is
taken for 3.1415 . He represented 3.1415 . . . . again by 7 r in
17375 (in a paper printed in 1744), in 1743, 6 in 1746, 7 and in 1748. 8 Euler and Goldbach used 7r = 3.1415 . . . . repeatedly in their corre¬ spondence in 1739. Johann Bernoulli used in 1739, in his correspond¬ ence with Euler, the letter c ( circumf erentia ), but in a letter of 1740
1 Pierre Varignon, Histoire de V Academic r. des sciences , annee 1721 (Paris, 1723), Memoires, p. 48.
2 Euler in “De summis serierum reciprocarum,” Comm. Acad. Petr., Vol. VII (1734-35), p. 123 ff. See von Braunmtihl, V orlesungen uber Geschichte der Trigo¬ nometric, Vol. II (Leipzig, 1903), p. 110.
3 Charles Tweedie’s James Stirling (Oxford, 1922), p. 179, 180, 185, 188.
4 L. Euler, Mechanica sive motus scientia analytice exposita, Vol. I (Petrograd, 1736), p. 119, 123; Vol. II, p. 70, 80.
5 L. Euler in Comm. Acad. Petr, ad annum 1737, IX (1744), p. 165. See A. von Braunmuhl, op. cit., Vol. II, p. 110. Euler says: “Posito ir pro peripheria circuli, cuius diameter est 1, ... . ”
6 L. Euler in Miscellanea Berolinensia, Vol. VII (1743), p. 10, 91, 136.
7 L. Euler in Histoire de V academie r. des sciences, et de belles lettres, annee 1745 (Berlin, 1746), p. 44.
8 L. Euler, op. cit., annee 1748 (Berlin, 1750), p. 84.
LETTERS 7r AND e
11
he began to use 7 r. Likewise, Nikolaus Bernoulli employed x in his letters to Euler of 1742.1 Particularly favorable for wider adoption was the appearance ol x for 3.1415 .... in Euler’s Introductio in
analysin infimtorum (1748). In most of his later publications, Euler clung to x as his designation of 3.1415 .
398. Spread of Jones’s notation. — In 1741, x = 3. 14159 .... is used in Sherwin’s Tables .2 Nevertheless, mathematicians in general were slow in following suit. In 1748 Diderot3 wrote, “Soit le rapport
du diametre a la circumference
— - ... J. A. Segner varied in his 0
piactice, in 1/51 he let x stand for the ratio, but in 1767 he5 repre¬ sented 3.14159 .... by <5:x, as did Oughtred more than a century earlier. Sa\s Segnei . Si ratio diametri ad peripheriam circuli, quam dedimus, vel alia verae satis propinqua, <5:x, et sit diameter circuli
data d, erit eiusdem circuli peripheria = ^.d” Again, later, he lets x
0
be “ dimidi um peripheriae” of the circle.6 Even more vacillating was Kastner, who in his Anfangsgriinde of 1758 lets 1 :P stand for the ratio of diameter to circumference,7 and x for the circumference. He uses T in this sense in his plane geometry, and the early part of his solid geometry. Then all of a sudden he writes (p. 323) the ratio in the 1 01 m 1 .x and continues this notation over nine consecutive pages. 4 uithei on ( p. 358 ) in his trigonometry he puts cos u = tv and sin u = p', he wHtes, on page 367, cos A = t, and on page 389, cos 4P = x. It cannot be said that in 1 / 58 Kastner had settled upon any one fixed use of the letter x. In 1760 his practice had not changed;8 he lets x be coefficient of the (n- f- l)th term of an equation; later he puts x equal
to the algebraic irrational V7 a ,then x = V — 1, then x is an angle AP M
1 See Paul H. von Fuss, Correspondance mathematique et physique de quelques ceMres geometres du XVIII siecle (1843). Also F. Rudio, Archimedes, Huygens, Lambert, Legendre (Leipzig, 1892), p. 53.
IL Sherwin s Alathematical I ables (3d 6d.,‘ revised by W illiam Gardiner London, 1741), p. 44.
^ Denys Diderot, M&moires sur differens sujets de malhematiques (Paris, 1748),
4 J. A. Segner, Histoire de Vacademie , annee 1751 (Berlin, 1753), p. 271.
6 J. A. de Segner, Cursus Mathematics, Pars I (2d ed.; Halae, 1767), p. 309.
6 Segner, op. cit., Pars IV (Halae, 1763), p. 3.
7 A. G. Kastner, Anfangsgriinde der Arithmetik, Geometrie und Trigonometric (Gottingen, 1758), p. 267, 268.
8 A. G. Kastner, Anfangsgriinde der Analysis endlicher Grossen (Gottingen, 1760), p. 107, 117, 211, 228, 254, 296, 326, 327, 413, 432.
12
A HISTORY OF MATHEMATICAL NOTATIONS
and 7 r = R, then 7r = tan MpH, then tv is a coefficient in the cubic z3+7r2+p = 0. After that tv — 3.14159 . . . . , then tv is a general ex¬ ponent of the variable x, and is =0 in a particular case and =3 in another, then again tv is the coefficient of a term in an equation, and an exponent of x. Evidently tv was still serving him in the role of a general-utility symbol. But in 1771, at last, Kastner1 regularly re¬ served tv = 3. 14159 .
Nicolas de Beguelin2 in 1751 adopted the notation tv = 3.14159 as did also Daniel Bernoulli3 in 1753, G. W. Krafft4 in 1753, Daviet de Foncenex5 in 1759.
Another noted German writer of textbooks of the eighteenth cen¬ tury, W. J. G. Karsten, uses tv in the first volume of his Lehrbegrif 6 to represent a polygon, and uses no letter for 3.14 . But in the sec¬
ond volume he is definite: “Wenn man hinfiihro ein fur allemahl die Zahl 3, 1415926 u.s.f. —tv setzt, . . . . so ist p = 2rTv = Tvd.” One finds tv for 3.14159 .... in publications of C. A. Vandermonde7 in 1770, and Laplace8 in 1782. About the middle of the eighteenth century the letter tv was used frequently by French mathematicians in mechanics and astronomy for other designations than 3.141 . . . . , but in the latter part of that century 3.141 . . . . came to be generally designated by tv. Unusual is the procedure of Wessel,9 who writes 7t = 360°, and of L. N. M. Carnot, who, in his Geometrie de position (1803), page 138, takes the radius to be unity and a-fourth of the length of the circle to be tv, so that “sin (x±a) = -f cos a.” Another unusual procedure is that of D. Lardner,10 who lets tv be the “approxi-
1 A. G. Kastner, Dissertationes mathematicae et physicae (Altenbvrgi, 1771), p. 41, 66, 67.
2 Beguelin, op. cit., annee 1751 (Berlin, 1753), p. 444.
3 Daniel Bernoulli, Histoire de Vacademie, ann6e 1753 (Berlin, 1755), p. 156.
4 Georg Wolffgang Krafft, Institutiones Geometriae Sublimioris (Tubingen, 1753), p. 122.
6 Daviet de Foncenex in Miscellanea philosophico-malhematica Taurinensis, Vol. I (1759), p. 130.
6 W. J. G. Karsten, Lehrbegrif der gesamten Mathematik. 1. Theil (Greifswald, 1767), p. 304, 412.
7 Vandermonde in Histoire de V Academie des Sciences, annee 1770 (Paris, 1773), p. 494.
8 Laplace in op. cit., ann£e 1782 (Paris, 1785), p. 15.
9 Caspar Wessel, Essai sur la representation analytique de la direction (Copen- hague, 1897), p. 15. This edition is a translation from the Danish (1799).
10 Dionysius Lardner, The First Six Books of the Elements of Euclid (London, 1828), p. 278.
LETTERS 7T AND e
13
mate ratio of the circumference of a circle to its diameter/’ but does not state which approximate value it represents. The Italian, Pietro Ferroni,1 in 1782 wrote the capital letters P for 3.14159 . . and n for 6.283 . . Perhaps the earliest elementary French schoolbook to con¬ tain 7 r in regular use was A. M. Legendre’s Elements de geometrie (1794), page 121.
399. Signs for the base of natural logarithms. — The need of a sym¬ bol to represent the base of the natural system of logarithms pre¬ sented itself early in the development of the calculus. Leibniz2 used the letter b in letters to Huygens of October 3/13, 1690, and January
27, 1691. In the latter he considers t= { -^v-- and writes =
) l — v2 l — v
“b estant une grandeur constante, dont le logarithme est 1, et le loga- rithme de 1 estant 0.” A reviewer3 of G. Cheyne’s Fluxionum methodus
inversa writes in 1703, “ j dx : x = lx et X* = av . (seu cum la = l)xlx = y,” thus suggesting the letter a.
400. The letter e. — The introduction of the letter e to represent the base of the natural system of logarithms is due to L. Euler. According to G. Enestrom, it occurs in a manuscript written in 1727 or 1728, but which was not published until 1862.4 Euler used e again in 1736 in his Mechanical Volume I, page 68, and in other places, as well as in articles of the years6 1747 and 1751. Daniel Bernoulli7 used e in this sense in 1760, J. A. Segner8 in 1763, Condorcet9 in 1771, Lambert10 in
1 Pietro Ferroni, Magnitudinum exponentialium .... theoria Florence (1782), p. 228, 252.
2 C. I. Gerhardt, Leibnizens Mathematische Schriften, Vol. II (Berlin, 1850), p. 53, 76.
3 Acta eruditorum (Leipzig, 1703), p. 451.
4 Euler’s art., “Meditatio in experimenta explosione tormentorum nuper in- stituta,” in the Opera posthuma (1862), Vol. II, p. 800-804. See G. Enestrom, Bibliotheca mathematica (3d ser.), Vol. XIV (1913-14), p. 81.
5 L. Euler, Mechanica sive motus scientia analytice exposita (St. Petersburg, 1736), p. 251, 256; also in Comm. Acad. Petr., Vol. VII (1740) p. 146.
6 L. Euler in Histoire de V Academie r. d. sciences et d. belles lettres de Berlin, ann6e 1745 (Berlin, 1746), p. 185; annee 1751 (Berlin, 1753), p. 270.
7 Daniel Bernoulli in Histoire de V Academie r. d. sciences, annee 1760 (Paris, 1766;, p. 12.
8 J. A. Segner, Cursus mathematici, Paris IV (Halae, 1763), p. 60.
9 N. O. de Condorcet, Histoire de V academie, ann6e 1771 (Paris, 1774), p. 283.
10 J. H. Lambert in Histoire de V Academie r. d. sciences et d. belles lettres, ann6e 1764 (Berlin, 1766), p. 188; ann6e 1764 (Berlin, 1766), p. 412.
14
A HISTORY OF MATHEMATICAL NOTATIONS
1764, J. A. Fas1 in 1775. On the other hand, D’Alembert2 in 1747 and in 1764 used the letter c for 2.718 as did also the astrono¬
mer Daniel Melandri3 of Upsala in 1787. The letter e for 2.718 is found in Abbe Sauri,4 in E. Bezout,5 in C. Kramp.6 In Italy, P. Frisi,7
NOTE ON TWO NEW SYMBOLS.
BY BI’N'J.VM I N PE I BO K,
Professor of Mathematics in Harvard College, Cambridge, Mass.
The symbols which are now used to denote the Neperian base and the ratio of the circumference of a circle to its diameter are, for many reasons, inconvenient ; and the close relation, between these two quantities ought to be indicated in their notation. 1 would propose the following characters, which I have used with suc¬ cess in my lectures : —
(t) to denote ratio of circumference to diameter,
(J) to denote Neperian base.
It will bo seen that the former symbol is a modification of the letter c ( circumference ), and thd latter of fi (base).
The connection of these quantities is shown by the equation,
Fig. 107. — B. Peirce’s signs for 3.141 .... and 2.718 ....
in 1782, and Pietro Ferroni,8 in the same year, used C for 2.718 . . . . , but Paoli9 adopted the e. A. de Morgan10 in 1842 used the epsilon e for 2.718 .... and E for el/ ~l.
1 J. A. Fas, Inleiding tot de Kennisse en het gebruyk der Oneindig Kleinen (Ley¬ den, 1775), p. 71.
2 D’Alembert in Ilistoire de V academie, annee 1747 (Berlin, 1748), p. 228; annee 1764 (Berlin, 1766), p. 412.
3 Daniel Melandri in Nova Acta Helvetica physico-mathematica, Vol. I (Basel, 1787), p. 102.
4 L’Abbe Sauri, Cours de mathematiques, Tome III (Paris, 1774), p. 35.
6 E. Bezout, Cours de mathematiques, Tome I (2d ed.; Paris, 1797), p. 124.
6 C. Kramp, Elements d' aritlimetique (Cologne, 1808), p. 28.
7 Paulii Frisii, Operum tomus primus (Mediolani, 1782), p. 195.
8 Pietro Ferroni, Magnitudinum exponentialium logarithmorum et Trigono- metriae sublimis theoria (Florence, 1782), p. 64.
9 Pietro Paoli, Elementi d’ algebra, Tomo I (Pisa, 1794), p. 216.
10 A. de Morgan, “On the Foundations in Algebra,” Transactions of the Cam¬ bridge Philos. Society, Vol. VII (Cambridge, 1842), p. 185.
DOLLAR MARK
15
The use of the letter c in place of e, found in the writings of a few French and Italian mathematicians, occurs again in the Analytic Mechanics of Benjamin Peirce.1
401. B. Peirce's signs for 3.11+1 .... and 2.718 .... * — An ex¬ traordinary innovation in notation for i r and e was suggested in 1859 by Benjamin Peirce. He made the state¬ ment2 shown in Figure 107.
His sons, Charles Saunders Peirce and James Mills Peirce, used this nota¬ tion in their articles; the latter placed the symbols shown in Figure 108 on the title-page of his Three and Four Place Tables (Boston, 1871). But Peirce’s other pupils, Joseph Winlock, Chauneey
Wright, and Truman Henry Safford, used the symbol ir in the first volume of the Mathematical Monthly.
Fig. 108.— From J. M. Peirce’s Tables (1871)
THE EVOLUTION OF THE DOLLAR MARK
402. Different hypotheses. — There are few mathematical symbols the origin of which has given rise to more unrestrained speculation and less real scientific study than has our dollar mark, $. About a dozen different theories have been advanced by men of imaginative minds, but not one of these would-be historians permitted himself to be hampered by the underlying facts. These speculators have dwelt with special fondness upon monogrammatic forms, some of which, it must be admitted, maintain considerable antecedent probability. Breathes there an American with soul so dead that he has not been thrilled with patriotic fervor over the “U.S. theory” which ascribes the origin of the $ mark to the superposition of the letters U and Sf This view of its origin is the more pleasing because it makes the sym¬ bol a strictly American product, without foreign parentage, apparently as much the result of a conscious effort or an act of invention as is the sewing machine or the cotton gin. If such were the case, surely some traces of the time and place of invention should be traceable; there ought to be the usual rival claimants. As a matter of fact, no one has ever advanced real evidence in the form of old manuscripts, or connected the symbol with a particular place or individual. Nor
1 Benjamin Peirce, Analytic Mechanics (New York, 1855), p. 52.
2 J. D. Runkle’s Mathematical Monthly , Vol. I, No. 5 (February, 1859), p. 167, 168, “Note on Two New Symbols.”
16
A HISTORY OF MATHEMATICAL NOTATIONS
have our own somewhat extensive researches yielded evidence in support of the “U.S. theory.” The theory that the $ is an entwined U and S, where U S may mean “United States” or one “Uncle Sam,” was quoted in 1876 from an old newspaper clipping in the Notes and Queries (London) ;l it is given in cyclopedic references. In the absence of even a trace of evidence from old manuscripts, this explanation must give way to others which, as we shall find, rest upon a strong basis of fact. Possibly these statements suffice for some minds. How¬ ever, knowing that traditional theories are dear to the heart of man, an additional coup de grace will not be superfluous. The earliest high official of the United States government to use the dollar mark was Robert Morris, the great financier of the Revolution. Letters in his own handwriting, as well as those penned by his secretary, which we have seen,2 give the dollar mark with only one downward stroke, thus, $. To assume that the symbol is made up of the letters U and S is to assert that Robert Morris and his secretary did not know what the real dollar symbol was; the letter U would demand two downward strokes, connected below. As a matter of fact, the “U.S. theory” has seldom been entertained seriously. Perhaps in derision of this fanciful view, another writer declares “surely the stars and stripes is the obvious explanation.”3
Minds influenced less by patriotic motives than by ecclesiastical and antiquarian predilections have contributed other explanations of our puzzle. Thus the monogrammatic form of I H S (often erroneous¬ ly interpreted as Jesus, Hominum Salvator) has been suggested.4 The combination of H S or I I S, which were abbreviations used by the Romans for a coin called sestertius, have been advocated.5 We should expect the supporters of these hypotheses to endeavor to establish an unbroken line of descent from symbols used at the time of Nero to the symbols used in the time of Washington. But sober genealogical inquiries of this sort were never made or, if made, they brought dis¬ aster to the hypotheses.
A suggestion worthy of serious attention is the Portuguese symbol, cifrao, used in designating “thousands,” as in 13$786 (Vol. I, § 94). Somehow this symbol is supposed to have received the new meaning of
1 Notes and Queries (5th ser.), Vol. VI (London, 1876), p. 386; Vol. VII, p. 98.
2 Letter of 1792, in Harper Memorial Library, University of Chicago; Robt. Morris’ Private Letter Book, in MSS Div. of Library of Congress.
3 Notes and Queries (5th ser.), Vol. VI, p. 434.
4 Standard Dictionary, art. “Dollar.”
5 M. Townsend, U.S., an Index, etc . (Boston, 1890), p. 420.
DOLLAR MARK
17
“dollar” and to have been transferred from its old position in thou¬ sands’ place to the new position in front of the number affected. The burden of proof that the two transformations actually took place lies with the advocates of this theory. But such a proof has never been attempted in print. I he present writer has examined many books and many manuscripts from which support might be expected for such a theory, if true, but nowhere has he found the slightest evidence. The names of monetary units used in Brazil at the beginning of the nineteenth century were reis, veintein, tuston, pataca , patacon. cruzado , and none of these was represented by the symbol $.
An interesting hypothesis is advanced by the noted historian, T. 1. Medina, of Santiago de Chile. He suggests that perhaps the dollar mark was derived from the stamp of the mint of Potosi in Bolivia, this stamp was the monogrammatic p and s. Against the validity of this explanation goes the fact that forms of p and s were used
as abbreviations of the peso before the time of the establishment of the mint at Potosi.
All the flights of fancy were eclipsed by those who carried the $ back to the Pillars of Hercules.” These pillars were strikingly im¬ pressed upon the ‘pillar dollar,” the Spanish silver coin widely used in the Spanish-American colonies of the seventeenth and eighteenth centuiies.1 dhe Pillars of Hercules” was the ancient name of the opposite promontories at the Straits of Gibraltar. The Mexican “globe dollar” of Charles III exhibited between the pillars two globes repre¬ senting the old ana new worlds as subject to Spain. A Spanish ban¬ ner or a scroll around the Pillars of Hercules was claimed to be the origin of the dollai mark." the theory supposes that the mark stamped on the coins wras copied into commercial documents. No em- bari assments were experienced from the fact that no manuscripts are knovm which show in writing the imitation of the pillars and scroll. On the contiaiy, the imaginative historian mounted his Pegasus and pranced into antiquity for revelations still more startling. “The device of the two pillars was stamped upon the coins” of the people who “built Tyre and Carthage”; the Hebrews had “traditions of the pillars of Jachin and Boaz in Solomon’s Temple,” “still further back in the remote ages we find the earliest known origin of the symbol in connection with the Deity. It was a type of reverence with the first people of the human race who worshipped the sun and the plains of
1 Notes and Queries (5th ser.), Vol. VII (London, Feb. 24, 1877); New Ameri¬ can Cyclopedia, Vol. VI (1859), art. “Dollar.”
2 M. Townsend, op. cit., p. 420.
18
A HISTORY OF MATHEMATICAL NOTATIONS
central Asia.” The author of this romance facetiously remarks, “from thence the descent of the symbol to our own time is obvious.”1 Strange to say, the ingenious author forgot to state that this connec¬ tion of the dollar mark with ancient deities accounts for the modern phrase, “the almighty dollar.”
Fig. 109. — “Pillar dollar” of 1661, showing the “Pillars of Hercules.” (From Century Dictionary , under “Pillar.”)
Most sober-minded thinkers have been inclined to connect the dollar symbol with the figure 8. We have seen four varieties of this theory. The Spanish dollars were, as a rule, equivalent to eight smaller monetary units, universally known in Spain as reales or reals. The “pillar dollar” shows an 8 between the two pillars. The Spanish dollar was often called a “piece of eight.” What guess could be more natural than that the 8 between two pillars suggested the abbreviation 8, which changed into $? So attractive is this explanation that those who advanced it did not consider it worth while to proceed to the prosaic task of finding out whether such symbols were actually em¬ ployed in financial accounts by merchants of English and Latin America. Other varieties of theorizing claimed a union of P and 8 (“piece of eight”)2 or of R and 8 (“eight reales ”)3 or of |8| (the vertical
1 American Historical Record, Vol. Ill (Philadelphia), p. 407-8; Baltimore American (June 3, 1874).
2 M. Townsend, op. cit., p. 420; Scribner1 s Magazine, Vol. XLII (1907), p. 515.
3 M. Townsend, op. cit., p. 420.
DOLLAR MARK
19
lines being marks of separation)1 or of 8/.2 The “PS theory” has been given in W ebster’s Unabridged Dictionary , not in its first edition, but in the editions since the fourth (1859) or fifth (1864). It is claimed that this widely accepted theory rests on manuscript evidence.3 One writer who examined old tobacco account-books in Virginia reproduces lithographically the fancifully shaped letter p used to represent the “piece of eight” in the early years. This part of his article is valuable. But when it comes to the substantiation of the theory that $ is a combination of P and 8, and that the $ had a purely local evolution in
Fig. 110. — Forms that are not dollar symbols
the tobacco districts of \ irginia, his facts do not bear out his theory. He quotes only one instance of manuscript evidence, and the reason¬ ing in connection with that involves evident confusion of thought.4 I o us the “PS theory” seemed at one time the most promising working hypothesis, but we were obliged to abandon it, because all evidence pointed in a different direction. We sent inquiries to recent advocates of this theory and to many writers of the present day on early Ameri¬ can and Spanish-American history, but failed to get the slightest manuscript evidence in its favor. None of the custodians of manu¬ script records was able to point out facts in support of this view. We ourselves found some evidence from which a superficial observer might draw wrong inferences. A few manuscripts, particularly one of the year 1696 from Mexico (Oaxaca), now kept in the Ayer Collec¬ tion of the Newberry Library in Chicago, give abbreviations for the Spanish word pesos (the Spanish name for Spanish dollars) which consist of the letter p with a mark over it that looks like a horizontal figure 8. This is shown in Figure 110. Is it an 8? Paleographic study goes against this conclusion ; the mark signifies os, the last two letters in pesos. This is evident from several considerations. The fact that in the same manuscript exactly the same symbol occurs in vezos , the contraction for vezinos, or “neighbors,” may suffice; an 8 is mean¬ ingless here.
1 Notes and Queries (5th ser.), Vol. VII (London), p. 317.
2 Scribner's Magazine, Vol. XLII (1907), p. 515.
3 American Historical Record, Vol. Ill, p. 271. 4 Ibid., Vol. Ill, p. 271.
20
A HISTORY OF MATHEMATICAL NOTATIONS
We have now described the various hypotheses.1 The reader may have been amused at the widely different conclusions reached. One author gives to the $ “a pedigree as long as chronology itself. ” Others allow it only about 125 years. One traces it back to the worshipers of the sun in Central Asia, another attributes it to a bookkeeper in a Virginia tobacco district. Nearly every one of the dozen theories seemed so simple to its advocate as to be self-evident.
403. Evidence in manuscripts and early printed books. — The his¬ tory of the dollar mark is difficult to trace. The vast majority of old documents give monetary names written out in full. This is the case also in printed books. Of nine Spanish commercial arithmetics of the seventeenth and eighteenth centuries, five gave no abbreviations whatever for the peso (also called piastre , peso de 8 reales, “piece of eight, ” “Spanish dollar”). In fact, some did not mention the peso at all. The reason for the omission of peso is that the part of Spain called Castile had monetary units called reales, ducados, maravedises, etc.; the word peso was used mainly in Spanish America and those towns of Spain that were in closest touch with the Spanish colonies. After the conquest of Mexico and Peru, early in the sixteenth century, Spanish-American mints, established in the various points in the Spanish possessions, poured forth the Spanish dollar in such pro¬ fusion that it became a universal coin, reaching before the close of the century even the Philippines and China. In the seventeenth century the Spanish “piece of eight” was known in Virginia, and much was done to promote the influx of Spanish money into that colony. The United States dollar, adopted in 1785, was avowedly modeled on the average weight of the Spanish-dollar coins in circulation. Thomas Jefferson speaks of the dollar as “a known coin, and most familiar of all to the minds of the people.”2 No United States dollars were actu¬ ally coined before the year 1794.3 We proceed to unfold our data and to show the evolution of the dollar mark by stages so easy and natural that the conclusion is irresistible. There are no important “missing links.” To enable the critical reader to verify our data, we give the sources of our evidence. No man’s ipse dixit is a law in the world of scientific research.
We begin with information extracted from early Spanish printed books, consisting of abbreviations used for peso or pesos.
1 Other possible lines of research on the origin of $ were suggested by Professor D. E. Smith in his Rara Arithmetica (1908), p. 470, 471, 491.
2 D. K. Watson, History of American Coinage (1899), p. 15.
3 Gordon, Congressional Currency, p. 118.
DOLLAR MARK
21
Ivan Vasquez de Serna1 . 1620 . Pes., pes de 8 real
Francisco Cassany2 . 1763 . p , also ps.
Benito Bails3 . 1790 . pe, seldom p
Manuel Antonio Valdes4 . 1808 . ps.
Here we have the printed abbreviations Pes., ps, pe, p. More interesting and convincing are the abbreviations found in manu¬ scripts which record commercial transactions. We can give only a small part of the number actually seen. In our selection we are not discriminating against symbols which might suggest a conclusion different from our own. As a matter of fact, such discrimination would be difficult to make, for the reason that all the abbreviations for the peso,, oi piece of eight, ” or piastre that we have examined point unmistakably to only one conclusion. We say this after having seen many hundreds of these symbols in manuscripts, antedating 1800, and written in Mexico, the Philippines, San Felipe de Puerto, New Orleans, and the colonies of the United States. It was a remarkable coincidence that all three names by which the Spanish dollar was best known, namely, the peso, piastre, and “piece of eight/’ began with the letter p and all three were pluralized by the use of the letter s. Hence p and ps admirably answered as abbreviations of any of these names. The symbols in Figure 111 show that the usual abbreviations was ps or p, the letter p taking sometimes a florescent form and the s in ps being as a rule raised above the p. The p and the s are often connected, showing that they were written in these instances by one uninterrupted motion of the pen. As seen in Figure 111, the same manuscript sometimes shows widely different shapes. The capital P is a rare occurrence. We have seen it used at the beginning of sentences and a few times written in ledgers at the top of columns of figuies.. In the sixteenth century the ps had above it a mark indicating the omission of part of the word, thus, p s. Sometimes the contrac¬ tion of the word pesos was pss. or pos. Not infrequently two or more different abbreviations are found in one and the same manuscript. The body of the text may contain the word written out in full, or
1 Ivan \ asquez de Serna, Reducciones de oro (Cadiz, 1620), p. 263 ff. (In the Hispanic Museum, New York City.)
2 Don Fr. Cassany, Arithmetica deseada (Madrid, 1763). (In the Library of Congress.)
3 Don Benito Bails, Arismelica (Madrid, 1790). (In the Library of the Ameri- can Philosophical Society, Philadelphia.)
. *Don- M* A- Valdes, Gazetas de Mexico (1808). (In the Newberry Library Chicago.) Jy
22
A HISTORY OF MATHEMATICAL NOTATIONS
Place of M8. Date of MS.
1 Spain abt. 1500
3 Mexico ( ?) 1601
/
o
\r
iC j
s
9
Mexico
Manila
Mexico
1644
1672
1718
S>
Date of MS. Place of MS.
1598 - Mexico City 2
1633 San Felipe de puerto 4
11 Chietla (Mexico) 1748
13
Mexico
1768
15 New Orleans 1778
17 Mexico City 1781
19 On the Mississippi 1787
21 Philadelphia 1792
23 “ Nouvelle Madrid ” 1794 (N. O.)
25 " Nouvelle Madrid " 1794 (PA O,)
27 New Orleans 1796
29 New Orleans 1798
^PJT 1649 Mexico City
1696
Mexico
6
8
1746 Mexico City 10
1768
1769
Manila
12
14
(1778) 1783 New Orleans 16
1786 New Orleans 18
1787 Mexico City 20
1793 “Nouvelle Madrid” 22
(N. O.)
1794 “ Nouvelle Madrid ” 24
(N. O.)
1794 “ Nouvelle Madrid ” 20 (N. O.)
1796 Philadelphia (?) 28
1799 Louisville (?) SO
Fig. 111. — Symbols for the Spanish dollar or peso , traced from MS letters, contracts, and account-books. No. 1: The historian, Dr. Cayetano Coll y Toste, of Porto Rico, says that this was the written symbol “during the time of Chris¬ topher Columbus.” Nos. 2, 3, 6, 9, 10, 11, 13, 14, 17, 20 are traced from MSS owned by W. W. Blake, Avenida 16 de Septiembre 13, Mexico City. Nos. 15, 16,
DOLLAR MARK
23
18, 19 are from the Draper Collection in Wis. Hist. Libr., Madison; Nos. 15, 16 in Clark MSS, Vol. XLVIII J, p. 37, 38; Nos. 18, 19 in Clark MSS, Vol. I, p. 136, 143. Nos. 4, 5, 7, 8, 12 are from the Ayer Collection, Newberry Libr., Chicago. No. 21 from letter of Robert Morris to the lion. Jeremiah Wadsworth, Esq., Hartford, Conn., in Harper Mem. Libr., University of Chit ago. Nos. 22, 23, 24, 25, 26, 27, 28, 29, 30 are from MSS in Chicago Hist. Soc. i,ibr. ; No. 22 in the Menard Collection, \ ol. LXI\ ; Nos. 23, 24 in the Menard Collection, Vol. LX, p. 187 ; Nos. 25, 30 in Autogr. Letters, Vol. LXI; No. 26 in the Menard Collection, \ ol. LXII; Nos. 27, 29 in the Menard Collection, Vol. LX11I; No. 28 in Autogr. Letters, Vol. LXX1, p. 76. The “N.O.” in the figure, following “Nouvelle Ma¬ drid,” should be “Mo.”
contracted to pss or pos, while the margin or the head of a column of figures may exhibit p.s or simply p. These were the abbreviations used by the Spanish-Americans from the sixteenth century down to about 1820 or 1830. The transition from the ps to our modern dollar mark was not made by the Spaniards ; it was made by the English-speaking people who came in contact with the Spaniards. At the time when Mexico achieved its independence (1821), the $ was not yet in vogue there. In a Mexican book of 1834 on statistics1 both the ps and the $ are used. Our $ was introduced into Hawaii by American missionaries in a translation of Warren Colburn’s Mental Arithmetic in 1835. 2
The transition from the florescent ps to our dollar mark is seen in Figure 112. Apparently it is a change introduced unconsciously, in the effort to simplify the complicated motion of the pen called for in the florescent ps. No manuscript on this point is so interesting and convincing as the two contemporaneous copies, made by the same hand, of a letter written in 1778 by Oliver Pollock, then “commercial agent of the United States at New Orleans.” Pollock rendered great service to the L’nited States, being to the west what Robert Morris was to the east. Pollock’s letter is addressed to George Roger Clark, who was then heading an expedition for the capture of the Illinois country. Both copies of that letter show the $ in the body of the let¬ ter, while in the summary of accounts, at the close, the $ and the florescent ps are both used. These documents show indeed “the mod¬ ern dollar mark in the making.” In the copy from which our photo¬ graph is taken, Figure 112, the 8613 dollars is indicated by the regular $, while in the other copy it is represented by the fancy p\ Carefully examining the two symbols in our photograph, we see that the ps is made by one continuous motion of the pen, in this order: Down on the left — up on the right — the loop of the p — the s above. On the other
1 J. A. Escudero, Nolicias esladislicas del Estado de Chihuahua (Mexico, 1834).
2 Copy in the Newberry Library, Chicago.
24
A HISTORY OF MATHEMATICAL NOTATIONS
hand, the $ symbol is made by two motions: One motion down and up for the p, the other motion the curve for the s, one symbol being superimposed upon the other.
Mr. Augustus H. Fiske, of Cambridge, Massachusetts, has pointed out to the present writer that the modern dollar mark occurs in a diary of Ezra FHommedieu for the year 1776. LTIommedieu was a native of Southold, Long Island, and a Yale graduate. He was a member of the New York Provincial Assembly, which, on July 10, 1776, styled itself the Convention of the Representatives of the State
Fig. 112. — The modern dollar mark in the making. (From copy of letter by Oliver Pollock at New Orleans to George Roger Clark, 1778, Wis. Hist. Libr., Madison, Draper Collection, Vol. XXXVIII J, p. 37.)
of New York. The first date in the diary is June 10, 1776; the last is December 5, 1776. Before August 21, 1776, most sums of money are expressed in pounds and shillings. When dollars are mentioned, the word “dollar” is written out in full. On August 21 occurs the first dollar symbol (see tracing 1 in Fig. 113). Under date of August 28 the treasurer is to advance $10 for removing military stores from New York (tracing 2). On October 2 a loan of $100,000 is obtained from the Continental Congress (tracing 3) ; on October 3 and 4 the same sum is referred to in a similar way (tracings 4 and 5). On October 4 the treasurer is to pay $6412§ bounty money to the rangers (tracing 6) . The $ signs now appear more frequently. Their shapes are shown in the remaining tracings. We see in this diary the gradual substitu-
DOLLAR MARK
25
tion of the conventional sign $ for the spelled word. The first eleven tracings have the S crossed by only one line; the last three have the double lines.
The origin of the dollar mark is simplicity itself. It is an evolution from ps. When the p was made by one long stroke only, as in Figure 111, Nos. 12, 14, 17, 20, then the mark took the form $, as used by Robert Morris (Fig. Ill, No. 21). Before 1800 the regular mark $ was seldom used. In all our researches we have encountered it in eight¬ eenth-century manuscripts not more than thirty or forty times. None
-f. *>. S. 4. -C
/ s / / /
% Z. f. JO.
/ / / J*
^ # / J?
Fig. 113. — Dollar marks in L’Hommedieu’s diary, 1776
of these antedates L’Hommedieu’s diary of 1776. But the dollar money was then very familiar. In 1778 theater prices in printed ad¬ vertisements in Philadelphia ran, “Box, one dollar.” An original manuscript document of 1780 gives thirty-four signatures of sub¬ scribers, headed by the signature of George Washington. The sub¬ scribers agree to pay the sum annexed to their respective names, “in the promotion of support of a dancing assembly to be held in Morris¬ town this present winter. The sums are given in dollars, but not one of the signers used the $ symbol; they wrote “Dollars,” or “Doll,” or “D.”1
1 American Historical and Literary Curiosities (Philadelphia, 1861), Plates 52, 22.
26
A HISTORY OF MATHEMATICAL NOTATIONS
It is interesting to observe that Spanish- Americans placed the ps after the numerals, thus 6 5ps, while the English colonists, being accustomed to write £ before the number of pounds, usually wrote the $ to the left of the numerals, thus $65. It follows after the numerals in some letters written by Joseph Montfort Street, at Prairie du Chien in Wisconsin and Rock Island, Illinois, in 1832 and 1836. 1 The dollar mark $ is usually written after the number, as in 85$, in many Latin- American arithmetics; for instance, in those of Gabriel Izquierdo,2 Florentino Garcia,3 Luis Monsante,4 Agustin cle La-Rosa Toro,5 and Maximo Vazquez.6 In the Argentine Republic the $ is still frequently written to the right of the numerals, like this, 65$.
404. Modern dollar mark in print. — It has been said by various writers, including myself, that the first appearance of the dollar mark in print is in Chauncey Lee’s American Accomptant , printed at Lansin- burgh, New York, in 1797. The statement is inaccurate. Lee’s sign for “dimes” more nearly resembles our dollar mark than does his sign for “dollars.” We may premise that Lee assumes in his arithmetic the attitude of a reformer. Impressed by the importance of decimal frac¬ tions, he declares that “vulgar fractions are a very unimportant, if not useless part of Arithmetic,”7 and in his Introduction he proceeds to propose a decimal system of weights and measures. In his table “Of Federal Money” (p. 56) he introduces, without comment, new signs for mills, cents, dimes, dollars, and eagles, as shown in Figures
1 See correspondence of Joseph Montfort Street in the Iowa State Historical Department, in one volume. On p. 31 is a letter from Street to the Hon. Lewis Cass, secretary of the Department of War, Oct. 4, 1832, in which occurs “800
on p. 57 is a letter from Street to David Lowry, Nov., 1836, in which one finds “27000 but also “$ 300,000.”
2 Gabriel Izquierdo, Tratado de aritmetica (Santiago, 1859), p. 219 ff.
3 Florentino Garcia, El Aritmetico Argentine), Quinta edicion (Buenos Aires,
1871) , p. 24 ff. It is noteworthy that the dollar mark in its modern form occurs in the first edition of this text, 1833. The author, Garcia y Coates, probably followed English texts, for the first edition contains the sign -h for division; the author was teaching in the “Academia espanola e inglesa” in Buenos Aires, and the general arrangement of the original text reminds one of arithmetics in the English lan¬ guage.
4 Luis Monsante, Lccciones de aritmetica demostrada , Setima edicion (Lima,
1872) , p. 119 ff.
6 Agustin de La-Rosa Toro, Aritmetica T eorico-P ractica, Tercera edition (Lima, 1872), p. 126.
6 Aritmetica practica ... por M. Maximo Vazquez ... Septima edicion (Lima, 1875), p. 113.
7 Ch. Lee, American Accomptant (1797), p. xiv.
DOLLAR MARK
27
114 and 115. It will be seen that his sign for “dollars” involves four strokes, the “dollar” being the fourth denomination of units. Two of the four strokes are curved, and they regularly inclose a space. They do not suggest the letter S in ps from which our dollar mark descended. The probability is that Lee was familiar with the dollar mark as it was found in handwritten business documents of his day, and that he modified it in the manner shown in his book, in order to arrive at a unified plan for constructing signs for mills, cents, dimes, and dollars. It appears, therefore, that. Lee’s publication marks a
! jo NO T A T I O N,
I '
Of Federal Money.
io Mills ( j) make i Ceru . i o Cents i Dime,
io Dimes - « i Dollar.
10 Dollars - i Eagle.*
C her after i flics.
//
it
E.
;V/
Q, What arc the names of the fever al foreign end f<*~
J , 7 j j . v/ O J ;
acral gold, Jilver and copper coins 5 circulating in the Uni - ! , ted States , and their value in bedtral Money ?
Fig. 114. — From Chauncey Lee’s American Accomptant (1797), showing the signs proposed by him for mills, cents, dimes, dollars, and eagles. (Courtesy of the Library of the University of Michigan.)
side excursion and does not constitute a part of the actual path of descent of our dollar mark. Erroneous, therefore, is the view of a writer1 claiming that Lee’s symbol for the dollar constitutes the primi¬ tive and true origin of the dollar mark.
After 1800 the symbol began to be used, in print, and also more frequently in writing. On September 29, 1802, William A. Washing¬ ton wrote a letter on the disposal of part of the land above the Poto¬ mac belonging to the estate of George Washington. In this letter there is mention of “$20,” “$30,” and “$40” per acre.2 In fact, the ledger
1 See Bankers Magazine , Vol. LXII (1908), p. 857.
2 Letter in Harper Memorial Library, University of Chicago.
28
A HISTORY OF MATHEMATICAL NOTATIONS
kept by George Washington himself, now preserved in the Omaha Public Library, contains the $ frequently. The earliest date in the ledger is January 1, 1799.
The dollar mark occurs a few times in Daniel Adams’ Scholar’s Arithmetic; or Federal Accountant, Keene, New Hampshire (4th ed., 1807) (p. 87, 88). The more common designation in this text is “Dolls.” or “D.” Adams gives also “D.,” “d.” “c.,” “m.” for dol¬ lars, dimes, cents, mills, respectively. With him, the dollar mark has the modern form except that the two strokes are not vertical on the page, but slanting like a solidus. The same form is found in an anonymous publication, uThe Columbian Arithmetician. By an Ameri¬ can” (Haverhill, Mass., 1811).
Examples'.
Confolldate c-qGq Mills into all the higher denomi-
^ //579>-4 ' '■ ' • —
ii CfQ. 6.4
-
. .* '
n n 6 A
7*9* °*T . • .. .... .. • •
. . .
Fig. 115. — From Chauncey Lee’s American Accornptant , p. 142
nations*
• • £4 ■ ■. , .., ■. > $
■ ■ ■
In newspapers the dollar mark rarely occurs during the first de- cennium of the nineteenth century. In the Boston Patriot of Sep¬ tember 1, 1810, one finds “$1 12,” (the cents being separated from the dollars by a blank space), but we have failed to find the mark in the numbers of the Columbian Centinel, published in Boston, for the period from August to December, 1801, and for August 12, 1809.
Samuel Webber’s Arithmetic (Cambridge, Mass., 1812) contains the dollar mark but not before page 125. In the first treatment of “Federal Money,” page 95, the abbreviations “E.,” “D.,” “<L,” “c.,” “m.” are used; in the second treatment, page 125, our dollar mark is introduced. The shape of the mark in this book is peculiar. The S is unusually broad and is heavy; the two slanting parallel lines are faint, and are close to each other, so short as hardly to pro¬ ject beyond the curves of the S, either above or below. Most of the business problems deal with English money.
In Jacob Willetts’ Scholars’ Arithmetic (2d ed., Poughkeepsie, 1817), the dollar mark appears in its modern form with the two strokes straight up and down. Willetts uses also the abbreviations “Dol.” and “D.”
THEORY OF NUMBERS
29
405. Conclusion. — It has been established that the $ is the lineal descendant of the Spanish abbreviation ps for pesos, that the change f i om the floiescent to $ was made about 1 / 75 by English- Americans who came into business relations with Spanish-Americans, and that the earliest printed $ dates back to the opening of the nineteenth century.
SIGNS IN THE THEORY OF NUMBERS
406. In the article on the theory of numbers in the Encyclopedic des sciences mathematiques , Tome I, Volume III (1906), page 3, which was originally written by P. Bachmann and later revised by E. Maillet, the great multiplicity and duplication of notations is de¬ plored in the following statement: “II n'y a malheureusement pas d’entente au sujet des notations relatives aux fonctions arithmetiques qui interviennent dans la theorie des nombres.” In the present article we cannot do more than enumerate what seem to be the more important symbols introduced; the preparation of an exhaustive list would seem very onerous and also of comparatively little additional value.
407. Divisors of numbers ; residues. — The notation fn for the sum of the divisors of n was introduced by Euler;1 when n is prime and k
an integer, he wrote (nk=l+n+n2+ .... = Barlow2
J n— 1
employed the sign to save repetition of the words “divisible by,” and the sign ^ to express “of the form of.” Cunningham3 lets “a(N) denote the sum of the sub-factors of N (including 1, but excluding N). It was found that, wTith most numbers, anN = 1, when the operation (<j) is repeated often enough.” Here c2(n) means a{a(n)}. Dickson writes s(n) in place of Cunningham's a(n); Dickson4 lets a{n) repre¬ sent the sum of the divisors (including 1 and n ) of n; he lets also crfc(n) represent the sum of the Tcth powers of the divisors of n.
The “symbol of Legendre” is
; it represents the residue, -f 1
1 L. Euler, “De numeris amicabilibus,” Opuscula varii argumenti, Vol. II (Ber¬ lin, 1750), p. 23-107 ; Commentationes arithmeticae, Vol. I (Petrograd, 1849), p. 102, 103; Opera omnia (1st ser.), Vol. VI, p. 21. See L. E. Dickson, History of the Theory of Numbers , Vol. I (1919), p. 42.
2 Peter Barlow, “Theory of Numbers” in Encyclopaedia Metropolitana, Pure Sciences, Vol. I (1845), p. 648.
3 Allan Cunningham, Proc. London Math. Soc., Vol. XXXV (1902-3), p. 40; L. E. Dickson, op. cit., Vol. I (1919), p. 48.
4 L. E. Dickson, op. cit., Vol. I, p. 53.
30
A HISTORY OF MATHEMATICAL NOTATIONS
c — 1
or —1, when N 2 is divided by c. In Legendre’s words: “Comme les
c — 1
quantites analogues a N 2 se recontreront frequemment dans le
/AT
cours de nos recherches, nous emploierons la caractere abrege I -
c — 1
pour exprimer le reste que donne N 2 divise par c ; reste qui, suivant ce qu’on vient de voir, ne peut etre que +1 ou — l.”1
Legendre’s notation was extended by Jacobi.2 If p—ff'f" . . . . , where /, f', are uneven prime numbers, then Jacobi defines
x
the symbol ( — ) by the equation
'x
X
X
J \/7 \f
X
7/
= + 1. In
Jacobi’s words: “1st nemlich, um diese Verailgemeinerung fur die quadratischen Reste anzudeuten, p irgend eine ungerade Zahl= jf'/" . . . . , wo /, /', f" ... . gleiche oder verschiedene Primzahlen bedeu- ten, so dehne ich die schone Legendre’sche Bezeichnung auf zusam-
x
mengesetzte Zahlen p in der Art aus, dass ich mit - , wenn x zu p
x
X
X
777
V
bezeichne.” Legendre’s
Primzahl ist, das Product ^
and Jacobi s symbols have been written3 also (N\c) and (x\p). A sign similar to that of Legendre was introduced by Dirichlet4 in connection
c — 1
with biquadratic residues; according as/c4 =+l or — 1 (mod. c), he
wrote
'k)
,C/4
+ 1 or —1, respectively. Dickson5 writes (k/c) 4.
Another symbol analogous to that of Legendre was introduced by Dirichlet6 in the treatment of complex numbers; he designated by k ~
the number +1, or —1, according as k is, or is not, quadratic
m
residue of m, such that one has p =
V
m
(mod. m), k and m being
1 A. M. Legendre, Essai sur la theorie des nombres (Paris, 1798), p. 186.
2 C. G. J. Jacobi, Crelle’s Journal, Vol. XXX (1846), p. 172; Werke, Vol. VI (Berlin, 1891), p. 262. Reprinted from Monatsberichten der Konigl. Akademie d. Wissenschaften zu Berlin vom Jahr 1837. See L. E. Dickson, op. cit., Vol. Ill, p. 84.
3 G. B. Mathews, Theory of Numbers (Cambridge, 1892), p. 33, 42.
4 G. L. Dirichlet, Abhand. d. K. Preussisch. Akad. d. Wissensch. (1833), p. 101- 21; Werke, Vol. I (1889), p. 230.
5 L. E. Dickson, op. cit., Vol. II, p. 370.
6 G. L. Dirichlet, Crelle’s Journal, Vol. XXIV (1842), p. 307; Werke, Vol. I (1889), p. 551.
THEORY OF NUMBERS
31
sign
in biquadratic residues of complex numbers to represent the
complex integers and p the norm of m. Eisenstein1 employed the n~ m
complex unit to which the power (mod. m) is congruent, where
ni is any primary prime number, n a primary two-term prime number different from m, p the norm of m.
Jacobi2 lets A(a) be the excess of the number of divisors of the form 4m+l, of x, over the number of divisors of the form 4??i+3, of x. His statement is “Sit B{x) numerus factorum ipsius x, qui formam 4m + 1 habent, C(*} numerus factorum, qui formam 4m+3 habent, facile patet, fore A(x) = Bix) — C(x) Glaisher3 represents this excess, for a number n, by E{n). Dickson4 lets Er{n) or Erin) be the excess of the sum of the rth powers of those divisors of n which (whose complementary divisors) are of the form 4?n+l over the sum of the rth powers of those divisors which (whose complementary divisors) are of the form 4m4-3. He also lets A'r(n) be the sum of the rth powers of those divisors of n whose complementary divisors are odd. Bach- mann5 lets t(ri), and Landau6 lets T(n), be the number of divisors of the positive integer n; Landau writes cf(n) = XT(n), C = Euler’s constant 0.57721 .... and R(x) = cT(x)-x log x-(2C-l)x, x>0. Dickson lets the sign T(n) = 7"(1) + T*(2)+ • . . . + where
STM is the number of divisors7 of n. Bachmann8 designates by co(n) the number of distinct prime divisors of n. Landau and Dickson9 designated by Og(T) a function whose quotient by g(T) remains nu¬ merically less than a fixed finite value for all sufficiently large values of T. Before them Bachmann10 used the form 0(n).
The constant C, referred to above as “Euler’s constant/’ was in-
1 G. Eisenstein, Crelle’s Journal, Vol. XXX (1846), p. 192.
2C. G. J. Jacobi, Werke, Vol. I (1881), p. 162, 163; Dickson, op. tit., Vol. I,
p. 281.
3 J. W. L. Glaisher, Proc. London Math. Soc., Vol. XV (1883), p. 104-12.
4 L. E. Dickson, op. tit., Vol. I, p. 296.
5 P. Bachmann, Encyklopadie d. math. Wissensch., Vol. I (Leipzig, 1898- 1904), p. 648.
6 Edmund Landau, Nachrichten von der K. Gesellschaft d. Wissensch. zu Gottingen, Math.-Phys. Klasse (1920), p. 13.
7 L. E. Dickson, op. tit., Vol. I, p. 279.
8 P. Bachmann, op. tit., p. 648.
0 L. E. Dickson, op. tit., Vol. I, p. 305; Landau, Handbuch der .... Prim- zahlen, Vol. I (1909), p. 31, 59.
10 P. Bachmann, Analytische Zahlentheorie (Leipzig, 1894), p. 401.
32
A HISTORY OF MATHEMATICAL NOTATIONS
troduced by Euler1 who represented it by the letter C. We mention it here, even though strictly it is not part of the theory of numbers. Mertens2 designated it by a German capital letter G. It is known also as “Mascheroni ’s constant. ”3 Mascheroni4 in 1790 designated it by the letter A. This designation has been retained by Ernst Pascal.5 Gauss5 wrote T0 = —0,5772156. W. Shanks7 adopted the designation “E. or Eul. constant.” The letter E for “Eulerian constant” was adopted by Glaisher8 in 1871 and by Adams9 in 1878. Unfortunately, E is in danger of being confused with E2to, sometimes used to designate Eulerian numbers.
In formulas for the number of different classes of quadratic forms of negative determinant, Kronecker10 introduced the following nota¬ tion which has been adopted by some later writers: n is any positive integer; m is any positive odd integer; r is any positive integer of the form 87c — 1 ; s is any positive integer of the form 87c+ 1 ; G(ri) is the number of non-equivalent classes of quadratic forms of determinant — n; Fin) is the number of classes of determinant — n, in which at least one of the two outer coefficients is odd; X{n) is the sum of all odd divisors of n; 4>(n) is the sum of all divisors of n; T(n) is the sum of the divisors of n which are >\/n, minus the sum of those which are <|/n; &(ri) is the sum of the divisors of n of the form 8k ±1, minus the sum of the divisors of the form 87c ± 3 ; T'(n) is the sum both of the divisors of the form 87c + 1 which are >j/n and of the divisors of the form 87c + 3 which are < j/n, minus the sum of both the divisors of the form 87c Hh 1 which are < j /n and of the divisors of the form 8k ±3 which are >j/n; <f>(n) is the number of divisors of n of the form 47c +1, minus the number of those of the form 47c — 1 ; \f(n) is the number of divisors of n of the form 3/c-j-l, minus the number of those of the form 37c — 1 ; <f>'(n) is half the number of solutions in integers of n = x2-\-CAy2;
1 L. Euler, Commentarii academiae Petropolitanae ad annum 1736, Tom. VIII, p. 14-16; M. Cantor, op. cit., Vol. Ill (2d ed.), p. 665; Vol. IV (1908), p. 277.
2 F. Mertens, Crelle’s Journal, Vol. LXXVII (1874), p. 290.
3 Louis Saalschutz, Bernoullische Zahlen (Berlin, 1893), p. 193.
4 L. Mascheroni, Adnotationes ad calculum integralem Euleri (Pavia, 1790-92), Vol. I, p. 11, 60. See also Euleri Opera omnia (1st ser.), Vol. XII, p. 431.
5 Ernst Pascal, Repertorium d. hoheren Mathematik, Vol. I (1900), p. 477.
6 C. F. Gauss, Werke, Vol. Ill (Gottingen, 1866), p. 154.
7 W. Shanks. Proc. Royl. Soc. of London, Vol. XV (1867), p. 431.
8 J. W. L. Glaisher, Proc. Roy. Soc. of London, Vol. XIX (1871), p. 515.
9 J. C. Adams, Proc. Roy. Soc. of London, Vol. XXVII (1878), p. 89.
10 L. Kronecker in Crelle’s Journal, Vol. LVII (1860), p. 248.
THEORY OF NUMBERS
33
t'(n) is half the number of solutions in integers of n = z2-f-3.64y2, in which positive, negative, and zero values of x and y are counted for both equations.
Some of the various new notations employed are indicated by the following quotation from Dickson: “Let x/c(x) be the sum of the /cth powers of odd divisors of x; Xk(x) that for the odd divisors >l/x; X'k'(x) the excess of the latter sum over the sum of the /cth powers of the odd divisors <j/x of x; Xk'(x) the excess of the sum of the /cth powers of the divisors 8s ± 1 >j/x of x over the sum of the /cth powers of the divisors 8s±3<i/x of x.”1
Lerch2 of Freiburg in Switzerland wrote in 1905, q(a) = (a*-1- 1) /Pt where p is an odd prime, and a any integer not a multiple of p. Simi¬ larly, Dickson3 lets the sign qu stand for the quotient (u^—D/v or for — l)/p.
Mobius4 defined the function bm to be zero if m is divisible by a square >1, but to be (— 1)* if m is a product of h distinct primes >1, while bi = l. Mertens5 writes pn, Dickson,6 M(n), for the bm of Mobius! this function is sometimes named after Mertens.
Dirichlet7 used the sign
, when n and s are integers and s^n.
to designate the largest integer contained in — . Mertens8 and later
authors wrote [x] for the largest integer ^x. Stolz and Gmeiner9
write ^ or [a :b]. Dirichlet10 denoted by N (a +bi) the norm a2+62 of
the complex number a+W, a symbolism used by H. J. S. Smith11 and others.
1 L. E. Dickson, op. tit., Vol. I, p. 305.
2 M. Lerch, Mathematische Annalen, Vol. LX (1905), p. 471.
3 L. E. Dickson, op. tit., Vol. I, p. 105, 109; see also Vol. II, p. 768.
, '.A- F* M5bius in Crelle’s Journal , Vol. IX (1832), p. Ill; Mobius, Werke Vol. IV, p. 598.
5 F. Mertens, Crelle’s Journal, Vol. LXXVII (1874), p. 289.
6 L. E. Dickson, op. tit., Vol. I, p. 441.
G. L. Dirichlet, Abhand. d. K. Preussisch. Akad. d. Wissensch. von 18A9 p. 69-83; Werke, Vol. II (1897), p. 52.
8 F. Mertens, op. tit., Vol. LXXVII (1874), p. 290.
9 Stolz und J. A. Gmeiner, Theoretische Arithmetik, Vol. I (2d ed.; Leipzig,
p. 20.
10 G. L. Dirichlet, Crelle’s Journal , Vol. XXIV (1S42), p. 295.
11 II. J. S. Smith, “Report on the Theory of Numbers/’ Report British Associa¬ tion (London, 1860), p. 254.
34
A HISTORY OF MATHEMATICAL NOTATIONS
The designation by n of a multiple of the integer n is indicated in the following quotation from a recent edition of an old text: aPour
exprimer un multiple d’un nombre nous mettrons un point audessus
• •
de ce nombre: ainsi ... 327^ signifie un multiple cle 327 ..., a,c ... signifie un multiple commun aux deux nombres a, et c.”1
408. Congruence of numbers. — The sign = to express congruence of integral numbers is due to C. F. Gauss (1801). His own words are: “Numerorum congruentiam hoc signo, = , in posterum denotabimus, mod ulurn ubi opus erit in clausulis adiungentes, —16 = 9 (mod. 5), — 7=15 (modoll).”2 Gauss adds in a footnote: “Hoc signum propter magnam analogiam quae inter aequalitatem atque congruentiam in- venitur adoptavimus. Ob eandem caussam ill. Le Gendre in com¬ ment. infra saepius laudanda ipsum aequalitatis signum pro con- gruentia retinuit, quod nos ne ambiguitas oriatur imitari dubitavi- mus.”
The objection which Gauss expressed to Legendre’s double use of the sign = is found also in Babbage3 who holds that Legendre vio¬ lated the doctrine of one notation for one thing by letting = mean: (1) ordinary equality, (2) that the two numbers between which the = is placed will leave the same remainder when each is divided by the same given number. Babbage refers to Peter Barlow as using in his Theory of Numbers (London, 1811) the symbol ff placed in a hori¬ zontal position (see § 407). Thus, says Babbage, Legendre used the symbolism an=— 1, Gauss the symbolism a”= — 1 (mod. p), and Barlow the symbolism an^>p. Babbage argues that “we ought not to multiply the number of mathematical symbols without necessity.” A more recent writer expresses appreciation of Gauss’s symbol: “The invention of the symbol = by Gauss affords a striking example of the advantage which may be derived from appropriate notation, and marks an epoch in the development of the science of arithmetic.”4
Among the earliest writers to adopt Gauss’s symbol was C. Kramp of Strassbourg; he says: “J’ai adopte de meme la signe de congruence, propose par cet auteur, et compose de trois traits paralleles, au lieu de
1 Claude-Gaspar Bachet, Problemes plaisants et delectables (3d ed., par A. Labosne; Paris, 1874), p. 13.
2 C. F. Gauss, Disquisitiones arithmeticae (Leipzig, 1801), art. 2; Werke, Vol. I (Gottingen, 1863), p. 10.
3 Charles Babbage, art. “Notation” in the Edinburgh Cyclopaedia (Phila¬ delphia, 1832).
4 G. B. Mathews, Theory of Numbers (Cambridge, 1S92), Part I, sec. 29.
THEORY OF NUMBERS
35
deux. Ce signe m’a paru essentiel pour toute cette partie de Tanalyse, qui admet les seules solutions en nombres entiers, tant positifs que negatifs.”1 The sign ^ is sometimes used for “incongruent.”2
409. Prime and relatively prime numbers. — Peano3 designates a piime by A v. Euler4 lets irT) stand for the number of positive integers not exceeding D which are relatively prime to D (“denotet character ?rZ) multitudinem istam numerorum ipso D minorum, et qui cum eo nullum habeant divisorem communem”). Writing n for D, Euler’s function ttD was designated <f>(n) by Gauss,5 and T(n) (totient of n) I)} S\ h ester.6 Gauss s notation has been widely used,’ it is found in Dedekind’s edition of Dirichlet’s Vorlesungen uber Zahlentheorie7 and in Wertheim’s Zahlentheorie .8 Jordan9 generalized Euler’s ttD function, and represented by [n, k] the number of different sets of k (equal or distinct) positive integers ^n, whose greatest common di¬ visor is prime to n. In place of Jordan’s [n, k] Story10 employed the symbol TkW, some other writers <fo(n), and Dickson11 Jk(n).
Meissel designates by the nth prime number,12 so that, for instance, ^(4) =5, and by rev (: reversio ) the function which is the opposite13 of xfy, so that urev rev (x)=x,” and by E rev (m) =
d{m) the number of primes in the natural series from 1 to m inclusive,14
:Cf. Kramp, “Notations,” Elemens d’ Arithmetique Universelle (Cologne, 1808).
See, for instance, L. E. Dickson, Algebras and Their Arithmetics (ChicaoD 1923), p. 38.
3 G. Peano, Formulaire mathematique, Tom. IV (1903), p. 68.
4 Acta Acad. Petrop ., 4 II (or 8), for the year 1755 (Petrograd, 1780), p. 18; Commentationes arithmeticae,Yo\. II (Petrograd, 1849), p. 127; L E Dickson ov cit., Vol. I, p. 61, 113.
6C. F. Gauss, Disquisitiones arithmeticae (Leipzig, 1801), No. 38. See also article by P. Baclimann and E. Maillet in Encyclopedic des sciences mathematiques Tome I, Vol. Ill (1906), p. 3.
6 J. J. Sylvester, Philosophical Magazine (5th ser.), Vol. XV (1883), p. 254.
7 P. G. L. Dirichlet, Vorlesungen uber Zahlentheorie, herausgegeben von R. Dedekind (3d ed.; Braunschweig, 1879), p. 19.
8 G. fV ertheim, AnfangsgriXnde der Zahlentheorie (Braunschweig, 1902), p. 42.
9 C. Jordan, Traite des substitutions (Paris, 1870), p. 95-97.
10 W. E. Story, Johns Hopkins University Circulars, Vol. I (1881), p. 132.
11 L. E. Dickson, op. cit., Vol. I, p. 147, 148.
12 E. Meissel, Crelle’s Journal, Vol. XLVIII, p. 310.
13 E. Meissel, ibid., p. 307.
14 E. Meissel, ibid., p. 313.
36
A HISTORY OF MATHEMATICAL NOTATIONS
which Dickson1 represents by 6(m). Landau2 writes t(x) for the num¬ ber of primes in the series 1, 2, .... , [x], where [x] is the largest integer ^ x . He also lets /(F) be a simpler function of x, such that t(x) -f(x)
7 r
(x)
->0 when x->oo . R. D. Carmichael3 uses H{y} to represent
the index of the highest power of the prime p dividing y, while Strids- berg4 uses Hm to denote the index of the highest power of the prime p which divides m!
410. Sums of numbers. — Leibniz5 used the long letter ^ not only
as a sign of integration in his calculus, but also as the sign for the sum of integers. For example, he marked the sum of triangular numbers thus :
“1+3+ 6+10+etc. — f x l+4+10+20+etc.= j j x l+5 + 15+25+etc.= Cx.”
This practice has been followed by some elementary writers; but certain modifications were introduced, as, for example, by De la
Caille6 who lets stand for the sum of certain numbers, j for the of their squares, f for the sum of their cubes, etc. Bachmann7
sum
lets
f(*0
%jm
stand for the
sum of the mth powers of the divisors of k, m
being any given odd number. A. Thacker8 of Cambridge used the notation <f>(z) = lw+2”-(- .... +2”, where z is an integer and n a positive integer. Dickson9 lets <j>k(n) be the sum of the kth powers of
1 L. E. Dickson, op. tit., Vol. I, p. 429.
2 Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen (Leipzig und Berlin), Vol. I (1909), p. 4.
3 R. D. Carmichael, Bull. Amer. Math. Soc., Vol. XV (1909), p. 217.
4 E. Stridsberg, Arkiv for Matematik, Astr., Fysik., Vol. VI (1911), No. 34. See L. E. Dickson, op. tit., Vol. I, p. 264.
5 G. W. Leibniz, “Historia et origo calculi differentialis” in Leibnizens Mathe- matische Schriften (ed. C. I. Gerhardt), Vol. V (1858), p. 397.
6 D. de la Caille, Ijectiones elementares mathematicae . ... in Latinum tra- ductae .... a C(arolo) S.(cherffer)e S. J. (Vienna, 1762), p. 107.
7 Encyklopadie d. Math. TFm., Vol. 1 (1900-1904), p. 638.
8 A. Thacker, Crclle’s Journal, Vol. XL (1850), p. 89.
9 L. E. Dickson, op. tit., Vol. I, p. 140.
THEORY OF NUMBERS
37
the integers and prime to n. Sylvester1 in 1866 wrote Sjti to express the sum of all the products of j distinct numbers chosen from 1> 2 i numbers. L. E. Dickson2 writes Sn>m for Sylvester’s Sj'i, and also Sn for ln+2n+ .... + (p — l)n. The sign Sn has been variously used for the sum of the nth powers of all the roots of an algebraic equation.3 The use of 2 for “sum” is given in § 407.
The sign F(a, N ) stands for a homogeneous symmetric poly¬ nomial, a and N being integers.4 Dickson5 introduces the symbol JE in the following quotations: “The separation of two sets of numbers by the symbol JL shall denote that they have the same sum of kth powers for k = 1, . . . . , n. Chr. Goldbach noted that a+/3+<5, a+7+<5, /3-f 7+ 8, <5jka + <5, /3+<5, 7 + 5, a+/3+7 + <5.”
411. Partition of numbers. — Euler’s “De partitione numerorum”6 contains no symbolism other than that of algebra. He starts out with the product (l+xaz)(l+xPz){\+x^z)(l+xsz){\ +xrz), etc., and indi¬ cates its product by
1 +Ps+Q2*+/te*+&z4+> etc.
and considers the term Nxnzm “whose coefficient indicates in what various ways the number n may be the sum of m different terms of the series a, /3, 7, 8, e, etc.”
Chrystal7 uses a quadripartite symbol to denote the number of partitions. “Thus, P(n\p\q) means the number of partitions of n into p parts, the greatest of which is q” ; “P(n\ + 1 >q) means the num¬ ber of partitions of n into any number of parts, no one of which is to exceed q; Pu{n\ >p\ + ), the number of partitions of n into p or any less number of unequal parts unrestricted in magnitude.” “P(n| + 1, 2, 22, 23, . . . .) the number of partitions of n into any number of parts, each part being a number in the series 1, 2, 22, 23, . . . .”
C. G. J. Jacobi represents the number of partitions, without repe¬ tition, of a number n, by N(n = 1 •xl+2-x2+3-x3), x^O. Sylvester,8
1 J. J. Sylvester, Griornale di matematiche, Vol. IV (Napoli, 1866), p. 344.
2 L. E. Dickson, op. cit., Vol. I, p. 95, 96.
3 See, for instance, M. W. Drobisch, Hohere numerische Gleichungen (Leipzig, 1834), p. 133, 134.
4 L. E. Dickson, op. cit., Vol. I, p. 84.
5 L. E. Dickson, op. cit., Vol. II, p. 705.
6 L. Euler, Introductio in analysin infinitorum, Vol. I, Editio Nova (Lugduni 1797), chap, xvi, p. 253.
7 G. Chrystal, Algebra , Part II (Edinburgh, 1889), p. 527, 528.
8 J. J. Sylvester, Collected Mathematical Papers, Vol. II (Cambridge, 1908), “On the Partitions of Numbers, ” p. 128.
38
A HISTORY OF MATHEMATICAL NOTATIONS
using the terms “denumerant” and “denuraeration,” writes GU and
n;
“HerscheFs symbol
G(U, V). “GU in its explicit form , ,
Cl/jOjC'j « • • • V y
rn . as a linear function of the nth powers of the rth roots of unity,
Tl*
.... will be replaced by — k” The coefficient1 of tn in the product of the series generated by ^ ^
,n » etc., is represented as — ; — — -
1 — ta 1 — tb ’ F a,b,c, .... If
the meaning of which is then extended and modified by Sylvester.2
“A system of equations3 in x, y, z, . . . u, may be denoted by S(x, y, z,
. . . . , u) or . . . . by S alone”; “RXS is the equation which expresses
that the coefficient cluster and primary of S balance about the axis
Ox.” “If S' is what S becomes when we write in S, fxf-g, or more
(f)X
generally $x, in place of x, we may denote S' symbolically by =S,”
x
where
4>x
= is an operative symbol
x
u
n — 3;
1, 2, 3, .
— expresses4 the r;
r-ary partibility of n when repetitions are allowed ;
n-
r+1;
1, 2, 3 .... Tj
the
same when repetitions are excluded.”
The sign p(n) is used to express the total number of ways in which a given positive integer n may be broken up into positive integral summands, counting as identical two partitions which are distin¬ guished only by the order of the summands.5
When the order of summands is taken into account, then the resulting “compositions” are often marked by c(n). G. H. Hardy and J. E. Littlewood6 use as a positive integer depending only on k, such that every positive integer is the sum of or fewer positive fcth powers; they also use G(k) to represent a number such that every sufficiently large number can be represented as the sum of not more than G(k) positive kth. powers. We have G2 = g-2 = 4; (r3^ 8, gr3 = 9.
412. Figurative numbers. — An old symbolism for square numbers is the geometric square □, for triangular numbers the triangle A, and so on. Hence the names “polygonal” numbers and “pyramidal” numbers and the general name “figurate” numbers. L. Euler and C.
1 Op. tit., p. 132. 2 Op. tit., p. 133. 3 Op. tit., p. 138. 4 Op. tit., p. 155.
6 A. J. Kempner, American Mathematical Monthly, Yol. XXX (1923), p. 357.
6 G. H. Hardy and J. E. Littlewood, Nachrichten von der K. Gesellschaft d. Wissensch. zu Gottingen, Math.-Phys. Klasse (1920), p. 34. See also A. J. Kempner, op. tit., p. 363, 364.
THEORY OF NUMBERS
39
Goldbach used the geometric signs □ and A in connection with num¬ ber theory in their correspondence/ about 1748. Occasionally they write also □ to denote the product of two unequal factors. On September 7, 1748, and again later, Goldbach2 lets □ be an even square and □ an odd square; he lets also 0 or 0 stand for a square which may be even or odd. Goldbach writes “40+ □ + □ + □ = 8m+5±2+ where, in case + is taken, 0 is an odd square, and in case — is taken 0 is an even square.
Legendre3 designates bv “pol. x ” a polygonal number of the order m+2, the side of the polygon being x, so that “pol. 2 = m+2, pol. 3 =
7YI
3m +3 and pol. x, of the order m+2, is —(x2— x)-\-x. For triangular
numbers, m = l; for square numbers, m = 2, etc.
In an arithmetical progression with the first term 1 and the com¬ mon difference m — 2, the sum of r terms is the r th m-gonal number,
designated by Pj.m) in the Encyclopedic des sciences mathemcdiqnes, Tome I, Volume 1, page 30, but is designated prm by L. E. Dickson,4 who lets P/ stand for the pyramidal number pm+p2m+ . . . . +prm» Dickson says further: “We shall often write Ar or A (r) for the rth triangular number r(r+l)/2, A or A' for any triangular number, □ for any square, 0, 0, or 0 for a sum of two, three, or four squares. The rth figurate number of the order n is the binomial coefficient rr_ /r+n— 1\ _ (r+n— l)(r+n— 2) . ... r „K Jn \ n ) 1.2 .... n
413. Diophantine expressions. — In the solution of ax+by = c and ax2+bx+c = □, L. Euler wrote (v, a ) for av-\- 1, (v, a, b) for (a6 + l)r+ b, etc., while C. F. Gauss6 employed the notations A = [a], B = [a, (3] = +4 + 1, C = [a, (3, y] = yB+A, D = [a, (3 , y, <5] = <5C+P , etc.
Gauss7 employed the notation ^/ for axx-\-a'x'x'-\-a"x"x"
1 P. H. Fuss, Correspondance mathematique et physique ... du XV I IB™* siecle (St. Peterbourg), Vol. I (1843), p. 451-63, 550, 602, 630.
2 P. H. Fuss, op. cit., Vol. I, p. 476, 492.
3 A. M. Legendre, Theorie des nombres (3d ed.), Vol. II (Paris, 1830), p. 340,
349.
4 L. E. Dickson, op. cit., Vol. II (1920), p. 1, 2.
5 L. E. Dickson, op. cit., Vol. II, p. 6, 7.
6 C. F. Gauss, Disquisitiones Arithmeticae (1801), § 27; Werke, Vol. I (1863), p. 20. See L. E. Dickson, op. cit., Vol. II, p. 49, 357.
7 C. F. Gauss, Disquisitiones Arithmeticae (1801), arts. 266-85; Werke, Vol. I (1863), p. 300. L. E. Dickson, op. cit., Vol. Ill, p. 206.
40 A HISTORY OF MATHEMATICAL NOTATIONS
-f-2 bx x +26 xx +26 xx . G. Eisenstein1 considered the cubic form
f={a, 6, c, d)=ctx3+3bx2y+3cxy2+dy 3
with integral coefficients and let A = b2 — ac, B = bc—ad, C = c2 — bd. He called Ax2JrBxyJrCy2 = F the determining form of the cubic form /. The sign (a, 6. c) is used by Dickson2 for the binary quadratic form with integral coefficients ax2+2bxy+cy2.
Kronecker denoted by K(D) the number of primitive classes of
discriminant D = b2-4ac. He put H(D)
) where h = 2sh', h! uneven, and
Legendre signs.3 Cresse4 explains the more current notation thus ; “A(D) denotes the number of properly primitive, and h'(D) the number of improperly primitive, classes of Gauss forms (ct, 6, c) of determinant D = b2-ac. Referring to Gauss’ forms, F(D), G(D), E(D), though printed in italics, will have the meaning which L. Kronecker assigned to them when printed in roman type. The class-number symbol H(D) is defined as G(D)-F(D). By K{D) or C(D) we denote the number of classes of primitive Kronecker forms of discriminant D = b2— 4ac.”
Val (co) is called the “valence of o>” and is a function of co in a field S; it was used by Dedekind5 in 1887. In the congruential theory of forms, L. E. Dickson uses E. H. Moore’s notation, GF[p»], to repre¬ sent6 the Galois field of order pn, the letter F being used here as in many other English articles to signify “field.”
414. Number fields. — The German designation for number field oi domain, namely, Zcihlkorper or simply Nor per , has given rise to the general notation7 Korper K, used by Dedekind. For special number
1 G. Eisenstein, Crelle’s Journal , Vol. XXVII (1844), p. 89; L. E. Dickson op. cit., Vol. Ill, p. 253.
2 L. E. Dickson, op. cit., Vol. Ill, p. 2.
3 L. Kronecker, Sitzungsberichte Akad. d. Wissensch. (Berlin 1885) Vol TT p. 768-80; L. E. Dickson, op. cit., Vol. Ill, p. 138.
4 G. H. Cresse in Dickson’s History, Vol. Ill, p. 93.
5R. Dedekind, CrellFs Journal, Vol. LXXXIII (1877), p. 275 See L E Dickson, op. cit., Vol. Ill, p. 125.
6 L. E. Dickson, op. cit., Vol. I, p. 249; Vol. Ill, p. 293.
7 See P. G. L. Dirichlet, Zahlentheorie , herausgegeben von R. Dedekind (3d ed.; Braunschweig, 1879), p. 465 n.
THEORY OF NUMBERS
41
fields Dedekind1 used Korper R for the field of rational numbers, Kd) per ,J for the field of all complex numbers o$ = x-\ -yi, where x and y aie ieal and rational. He lets A (co) stand for the norm (modulus) of co and its conjugate co', Korper ft signifies with him a finite field2 of degree n, but ft has been employed in the theory of equations as a moie general symbol for field. In 18/3, G. Cantor3 used (ft) to signify the Inbegriff of all numbers ft which are rational functions with integral coefficients of the given series of linearly independent real numbers
a'2>y • • • In his Was sind und was sollen die Zahlen (1888), Dedekind designates by A 3 S that a system A is a part of a system S; he indicates by $\(A, B, C . . . .) a system made up of systems A, B, C ... . and b y <&(A, B, C ... .) the system composed of all elements common to A, B, C . . . . ; he marks by 0(s) an Abbildung of a system S whose elements are s.
the introduction of the symbol GF[qn] to represent the Galois field of order qn is due to E. H. Moore4 of Chicago, in 1893.
415. Perfect numbers— Perfect numbers have been called also “Euclidean numbers,” and have been represented by the symbol Ep where Ep = 2p~1(2p- 1), when the second factor is prime. Some con¬ fusion is likely to arise between this and the E which sometimes is made to represent “Euler's constant,” or “Euler's numbers.” The notation Nprf ( nornbre parfait) was suggested by Nasso5 in 1900 and adopted by Peano.6 The notation Pm for a multiply perfect number n of multiplicity m (i.e., one, the sum of whose divisors, including n and 1, is the rath multiple of n) is used by Dickson.7
416. Mersenne numbers , named after Marin Mersenne, are marked Mq by Cunningham;8 Mersenne in 1644 asserted that numbers Mq = 2®-l, Q being prime and not greater than 257, are prime only when q= I.. 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257.
1 Op. tit., p. 435, 436.
2 Op. tit., p. 473.
3 G. Cantor, Crelle's Journal, Vol. LXXVII (1874), p. 261.
4E. H. Moore in Mathematical Papers Read at the International Congress, at Chicago, 1893 (New York, 1896), p. 211. See also E. Galois, (Euvres (ed. E.’ Picard; Paris, 1897), p. 15-23.
6 M. Nasso, Rivista di matematica (1900), p. 52.
6 G. Peano, Formulaire mathematique, Vol. IV (1903), p. 144.
7 L. E. Dickson, op. tit., Vol. I (1919), p. 33.
8 Allan Cunningham in Mathematical Questions and Solutions from the Educa¬ tional rimes , N.S., Vol. XIX (London, 1911), p. 81. See also L. E. Dickson, op. tit Vol. I (1919), p. 30.
42
A HISTORY OF MATHEMATICAL NOTATIONS
417. Fermat numbers, 22n-f-l, are represented by Fn in Dickson’s History}
418. Cotes’ s numbers are positive fractional numbers represented by W. W. Johnson2 by the sign nAr; the numbers occur in Cotes’s Harmonia Mensurarum as coefficients of a series. Cotes gave the values of the numbers for n— 1, 2, . . . . , 10. The values for n— 1 are embodied in the “trapezoidal rule,” and for n = 2 in the “parabolic rule.” A property of these numbers is that M.0+ VL i+ .... JrnAn= 1.
419. Bernoulli’s numbers are given in Jakob (James) Bernoulli’s
Ars conjedandi (1713), page 97. They occur in the formulas for the sums of the even powers of the first n integers, being the coefficients of n in the formulas. Bernoulli himself computed only the first five numbers: -TV> A-
We quote the first four formulas in Bernoulli:
“jn ^\nn-\~2n- Jnn^o^n2 + .
33 ±n5 +§n4 -^n ”
Bernoulli obtains by inspection the sum for nc. Then he states, “Literae capitales A, B, C , D etc., ordine denotant coefficientes ulti-
morum terminorum pro j nn, fn\ fn jn2, etc. nempe A ooj, B oo — 3V, C 00 D 00 — -fa.” (“The capital letters A, B, C, D, etc., de¬ note by their order the coefficients of the last terms for jnn, J"n4, / «6, f etc., namely, A=$, C=^, D= —fo.”)
Euler3 in 1755 used the German type of the capital letters A, B, C, D, . . . . , to represent the absolute values of the Bernoullian co¬ efficients, so that A = \, B = ^ 0, etc. Euler introduced the name “Bernoullian numbers.” In a paper of 1769, on the summation of series, Euler4 considers numbers representing the products of the Ber-
1 L. E. Dickson, op. ait., Vol. I, p. 375.
2 Roger Cotes, Harmonia Mensurarum (Cambridge, 1722), “De methodo differential! Newtoniana”; W. W. Johnson’s article on the Cotesian numbers is in the Quarterly Journal of Pure and Applied Mathematics (1914), p. 52-65.
3 L. Euler, I nstitutiones calculi differ entialis (1755), Vol. II, § 122, p. 420. See also Euleri Opera omnia (1st ser.), Vol. X, p. 321; Vol. XII, p. 431.
4 Nova Comment. Petrop. XIV (pro 1769), p. 129-67; M. Cantor, op. cit., Vol. IV (1908), p. 262.
THEORY OF NUMBERS
43
noullian numbers J, ^ etc., by the numbers 6, 10, 14, 18, etc.,
respectively, and marks these products by the German forms of the capital letters A, B, C, D, etc. In a paper for the year 1781, Euler1 designates the Bernoullian numbers in the same manner as he did in 1755. Eulers notation of 1755 was used by L. Mascheroni,2 and by Gauss.3 \ on Staudt4 employed the notation B ^ for the Tith number ; this notation occurs earlier in Klugel’s Worterbuch .5
According to G. Peano,6 it was Euler who designated the Bernoul¬ lian numbers by the signs B\, B3, B0, . . . . , but Peano does not give the leference. L. Saalschiitz7 says that formerly they were marked Bi, B3, B'0, . . . . , as, B%, H4, Bg, . . . . , had the value zero. Scherk8 and Grunert9 write the first, second, . . . . , nth Bernoullian numbers
1 3 2n — 1
thus: B, B, . . . . , B. De Morgan10 writes Bh Bs, B$. .... Binet11 adopts the signs B1} — B3, B5} —B7) .... Others, for instance, Pascal,12 represent these numbers by the even subscripts, B,, Bi} B6, ... . The notation B\, B2, B3, . , which is now in wide use, was em¬
ployed by Binet,13 Ohm,14 Raabe,15 Stern,16 Hermite,17 and Adams.18
I Acta Petrop. IV (1781), II, p. 45; M. Cantor, op. tit., Vol. IV, p. 277.
2L. Mascheroni, Adnotationes ad Calculum Integralum Euleri (Paris, 1790); Euleri Opera omnia (1st ser.), Vol. XII, p. 431.
3 C. F. Gauss, Werke, Vol. Ill (Gottingen, 18G6), p. 152.
4K. G. C. von Staudt, Crelle’s Journal, Vol. XXI (1840), p. 373.
0 G. S. Klugel’s Mathematisches Worterbuch, completed by J. A. Grunert, \ol. IV (Leipzig, 1823), “Summirung der Reihen,” p. 636.
c G. Peano, Formulaire mathematique, Tome IV (Turin, 1903), § 79, p.
248.
7 Louis Saalschiitz, Bernoullische Zahlen (Berlin, 1893), p. 4.
8 H. F. Scherk, Crelle’s Journal, Vol. IV (1829), p. 299.
9 J. A. Grunert, Supplement zu G. S. Klugel’s Worterbuch, Vol. I (Leipzig, 1833), “Bernoullische Zahlen.”
10 A. de Morgan, “Numbers of Bernoulli,” Penny Cyclopaedia.
II J. Binet, Comptes rendus, Vol. XXXII (Paris, 1851), p. 920.
12 Ernst Pascal, Repertorium d. hoheren Mathematik (ed. A. Schepp), Vol. I (Leipzig, 1900), p. 474.
13 J. Binet, Journal de Vecole polytechnique (Paris,) Vol. XVI, Cahier 27 (1839) p. 240.
14 Martin Ohm, Crelle’s Journal, Vol. XX (1840), p. 11.
15 J. L. Raabe, ibid., Vol. XLII (1851), p. 350-51.
16 M. A. Stern, ibid., Vol. LXXXIV (1878), p. 267.
17 C. Hermite, ibid., Vol. LXXXI (1876), p. 222.
J. C. Adams, Proc. Roy. Soc. of London, Vol. XXVII (1878), p. 88.
44
A HISTORY OF MATHEMATICAL NOTATIONS
420. Euler’s numbers,1 E2m, are coefficients occurring, as Ernst
x^m
Pascal states, in the series sec x= / < E^m 5 name was given
to these numbers by Scherk.2 Chrystal3 marks Euler’s numbers Eh E2, . . . thus Ei=l, E2 = 5, E3 = 61, E4=1385, etc.
SIGNS FOR INFINITY AND TRANSFINITE NUMBERS
421. The sign oo to signify infinite number was introduced by John Wallis in 1655 in his De sectionibus conicis in this manner: Esto enim oo nota numeri infinite (see also § 196). The conjecture5 has been made that Wallis, who was a classical scholar, adopted this sign from the late Roman symbol oo for 1,000. Nieuwentijt6 uses the letter m to represent quantitas infinita.
Wallis’ symbol for infinity came to be used at the beginning of the eighteenth century in the Calculus. Thus in the Acta eruditorum for 1708 (p. 344) one encounters the oddity “dy = oo, seu infinito.” The same publication7 contains “oo” potestas indeterminate numeri in- finiti” in a review of Cheyne’s Philosophical Principles of Natural Religion (London, 1705); Cheyne8 used this symbol freely. The symbol came to be used extensively, for instance, by Johann Ber¬ noulli9 in dealing with tautochrones. Sometimes, particularly in
1 L. Euler, Institutiones calculi differ entialis (1755), p. 542.
2 H. F. Scherk, Vier mathematische Abhandlungen (Berlin, 1825). The first Abhandlung concerns the Eulerian and Bernoullian numbers.
3 G. Chrystal, Algebra, Part II (Edinburgh, 1889), p. 318.
4 John Wallis, Opera mathematica, Vol. I (Oxford, 1695), p. 297, 405; De sec¬ tionibus conicis, Pars I, Prop. 1; also Arithmetica infinitorum, Prop. 91.
5 W. Wattenbach, Anleitung zur latein. Paldographie (2d ed., 1872), Appendix,
p. 41.
6 Bernhardi Nieuwentiitii, Analysis infinitorum seu curvilineorum proprie- tates ex polygonorum natura deductae (Amstelaedami, 1695). This reference is taken from H. Weissenborn, Die Principien der hoheren Analysis (Halle, 1856),
p. 124.
7 Acta eruditorum (Leipzig, 1710), p. 462.
8 See also George Cheyne, Philosophical Principles of Religion, Part II (Lon¬ don, 1716), p. 20, 21.
9 J. Bernoulli, Histoire de Vacademie r. d. scien., annee 1730 (Paris, 1732), Memoires, p. 78; Opera omnia, Vol. Ill (1742), p. 182, 183.
INFINITY AND TRANSFINITE NUMBERS
45
writings of Euler.1 and some later writers,2 the symbol is not a closed figure, but simply go . Another variation in form, due, apparently, to the exigencies of the composing-room, is seen in a book of Bangma,3 containing “Sec 270° = 0-0.” B. Fontenelle, in his Elements^de la geometrie de l inf ini (Paris, 1727), raises oo to fractional powers, as, for example (p. 43), ttI • go *, oo ■*, oo oo .” In modern geometry ex¬ pressions oo , go 2, . . . . , go w are used, where the exponents signify the number of dimensions of the space under consideration.
In the theory of functions as developed by Weierstrass4 and his followers, the symbol oo is used in more than one sense. First, it is used to represent an actual infinity. One says that a function fix),
when /(a) = 0, *s lnfmite x~ a, or for x = a, and one writes /(a) = oo .
In this case one does not use the -(-or — signs before oo ; that is, one does not write + oo or - oo . In writing /( oo) = b one means the same as
— b; in writing /(oo ) = oo one means the same as
y=0
in writing
y — O
Second, in considering the limit of a function
f(x) the +00 and — oo may arise as virtual infinities. For example, one
has x^™oof(x) = - oo , when, however large a positive constant P be
taken, one can take a positive number Q, such that f(x)<—P when x>Q.
In the theory of transfinite number, Georg Cantor5 represents by co an ordinal number of a rank next superior to any of the integers 1, 2, 3, ... . Previously,6 Cantor had used the sign oo , but he discarded
1 L. Euler in Histoire de Vacademie r. d. sciences et belles letlres, annee 1757 (Berlin, 1759), p. 311; annee 1761 (Berlin, 1768), p. 203. Euler, Institute calculi dijf. (1755), Vol. I, p. 511, 745; Euler in Novi Comment, acad. scient. imper. Petro- politanae, Vol. V, for the years 1754, 1755 (Petropoli, 1760), p. 209.
2 Matthias Hauser, Anfangsgriinde der Mathematik, 1. Thed (Wien 1778)
p. 122.
3 0. S. Bangma, Verhandeling over de ... . Driehoeks-meting (Amsterdam
1808), p. 26.
4 K. Weierstrass, Abh. Akad. (Berlin, 1876), Math., p. 12; Funktionenlehre (Berlin, 1886), p. 2; Werke, Vol. II, p. 78. See also A. Pringheim and J. Molk in Encyclopedic des scien. math., Tom. II, Vol. I, p. 20, 31, 32 (1909).
5 Georg Cantor, Grundlagen einer allgemcinen Mannichfaltigkeitslehre (Leipzig 1883), p. 33.
6 Georg Cantor, M alhematische Annalcn, Vol. XVII (1880), p. 357.
46
A HISTORY OF MATHEMATICAL NOTATIONS
that in 1882, in favor of co, “because the sign go is used frequently for the designation of undetermined infinities.”1 Cantor’s first ordinal of the second class of numbers (II) is therefore co; he2 marks the first ordinal of the third class (III) by 12; Bertrand Russell3 marks the first of the (p+2)th class by coy, so that cox is the same as 12.
Cantor designates by the sign aleph-zero JS0, which is the Hebrew letter aleph with a subscript zero attached, the smallest transfinite cardinal number,4 and by Si the next superior cardinal number. Peano,5 in 1895, indicated cardinal numbers by the abbreviation Nc.
Earlier than this the aleph had been used as a symbol for 60 in the Hebrew numeral notation, and it was used as a mathematical symbol for certain fundamental functions by H. Wronski,6 in his Philosophic des mathematiques (1811).
The cardinal number of the aggregate constituting all numbers in the linear continuum7 is marked c. The cardinal number of the aggregate F of all functions8 of a real variable (known to be greater than c) is marked f.
The different alephs of well-ordered transfinite aggregates are marked So Si .... S*, .... , Sw, . . . . , Sa, and Schoenflies9 writes the initial numbers of each class of ordinals, fi0, fii, . . . . , fi„, . . . . , 12a,, .... , 12^, so that fi0 = w and fii = fi, which latter is G. Cantor’s designation for the first number in his third class, and is sometimes10 written also an. G. Cantor11 marked a derived aggregate of p by //; Peano marked it Du in 189512 and 5u in 1903.
Other symbols used by G. Cantor13 in his theory of aggregates are
I Op. cit., Vol. XXI (1883), p. 577. 2 Op. cit., Vol. XXI, p. 582.
3 B. Russell, Principles of Mathematics, Vol. I (1903), p. 322.
4 G. Cantor, Mathematische Annalen, Vol. XLVI (1895), p. 492.
6 G. Peano, Formulaire de mathematiques, Vol. I (1895), p. 66; Vol. IV (1903),
p. 128.
6 Gergonne’s Annates de mathematiques (Nismes), Vol. Ill (1812, 1813), p. 53.
7 A. Schoenflies, Entwickelung der Mengenlehre und ihrer Anwendungen (1913), p. 54; E. V. Huntington, The Continuum (2d ed., 1917), p. 80.
8 A. Schoenflies, op. cit., p. 60. 9 A. Schoenflies, op. cit., p. 124.
10 E. V. Huntington, op. cit. (2d ed., 1917), p. 72.
II G. Cantor, Mathematische Annalen, Vol. V, p. 123.
12 G. Peano, Formulaire mathematique, Vol. I (1895), p. 69; Vol. IV (1903),
p. 121.
13 G. Cantor, Mathematische Annalen, Vol. XLVI (1895), p. 481-512; see also P. E. B. Jourdain’s translation of this article, and the one in Vol. XLIX (1897), p. 207-46, in a book, Contributions to the Founding of the Theory of Transfinite Numbers (Chicago and London, 1915).
47
INFINITY AND TRANSFINITE NUMBERS
where M suggests Menge (“aggregate”); (M, N, P, _ )
the uniting of the aggregates M, A , P, . . . . which have no common
elements into a single aggregate; M the general symbol for Mdchtigkeit ( powei ) or cardinal number” of M ; when two aggregates M and N are equi\ alcnt, Cantor writes J\I oo A . For two aggregates J\1 and N ,
the cardinal numbers are designated also by the signs a = Tf, b = N. The ‘ ‘covering-aggregate (. Belegungsmenge ) of N with M” is denoted by (N\M), and ab = (FlM).
If in a simply ordered aggregate M={m}, the element mx has a rank lower than m* in the given order of precedence, this relation is expressed m\Km^ m2 > m\. If the infinite aggregates are multiply ordered F . Riesz1 used the symbolism aj_b, a<ii b, a i>b to designate that in the fth order, a and b have equal rank, or a precedes b, or a follows b. If two ordered aggregates M and N are “similar,” they are marked by Cantor M^N; the “ordinal type” of M is marked M. If a is the ordinal type, the corresponding cardinal number is a. If in an ordered aggregate M all the relations of precedence of its ele¬ ments are inverted, the resulting ordered aggregate is denoted by *M and its ordinal type *a, provided a = M. Cantor2 designates a linear continuous series having both a first and a last element, a series of type d.
When two fundamental series {a,} and {a'v} of numbers of the first or second number-class are zusammengehorig (“coherent”)3 they are marked {a„}[|{a£}. The “e-numbers of the second number-class” are roots of the equation gj£ = £. The symbol E(y) represents the limit of the fundamental series {y„}, and is an e number.
G. Cantor4 in 1880 marked the least common multiple of aggre¬ gates M and N by m{M, N}, but E. Zermelo5 and A. Schoenflies6 adopted the sign @(M, N) suggested by the first letter in Summe (“sum”). The Durchschnitt, or all the elements common to M and N, are marked [M, N\ by Zermelo and N) by Cantor and Schoen¬
flies.7 Thes^ign Cv is used to represent the continuum8 of space of v
1 F. Riesz, Mathematische Annalen, Vol. LXI (1903), p. 407.
2 G. Cantor, Mathematische Annalen, Vol. XLVI (1895), p. 511.
3 G. Cantor, ibid., Vol. XLIX (1897), p. 222.
4 G. Cantor, ibid., Vol. XVII (1880), p. 355.
5E. Zermelo, Mathematische Annalen, Vol. LXV (1908), p. 265.
6 A. Schoenflies, Entwickelung der Mengenlehre und ihrer Anwendungen (Leip¬ zig und Berlin, 1913), p. 10.
7 A. Schoenflies, op. ait., p. 11.
8 A. Schoenflies, op. ait., p. 11, 56.
48
A HISTORY OF MATHEMATICAL NOTATIONS
dimensions. If Xa = 2xa are ordered aggregates, their product yields
A
an ordered product-aggregate which F. Hausdorff1 represents by nXa
a
and Schoenflies2 by *nXa; if such a product is a V ollprodukt (“com-
A
plete product”), Hausdorff (writing Ma for Xa) marks it {{JIM a))
a
a
and its type by ((n^a)). He designates the “maximal product” by T.
a
In the theory of the equivalence of aggregates E. Zermelo3 writes <f>^fM.N, to express the Abbildung of M upon N, where M and N are aggregates having no elements in common and $ is a subaggre¬ gate such that each element of M-\-N appears as element in one and only one element {m, n) of <f>.
Peano4 designates by a‘b the aggregate of all the couples formed by an object of a class a with an object of class 6, while the sign a;b designates the couple whose two elements are the classes a and b. J. Rey Pastor5 designates the existence of a one-to-one correspond¬ ence between infinite aggregates A and B, by the symbol A 7\ B.
SIGNS FOR CONTINUED FRACTIONS AND INFINITE SERIES
422. Continued fractions. — We previously mentioned (§ 118) a notation for continued fractions, due to al Hassar, about the begin¬ ning of the thirteenth century, who designates6 by f f § the ascending fraction
24
3+|
8
9
A Latin translation of the algebra of the famous Arabic algebraist and poet, Abu Kamil,7 contains a symbolism for ascending continued
1 F. Hausdorff, Mathematische Annalen , Vol. LXV (1908), p. 462.
2 A. Schoenflies, op. tit., p. 75-77.
3 E. Zermelo, Mathematische Annalen , Vol. LXV (1908), p. 267, 268.
4 G. Peano, Formulaire mathematique , Tome IV (1903), p. 125.
5 Revista matematica Hispano- Americana, Vol. I (Madrid, 1919), p. 28.
6 H. Suter in Bibliotheca Mathematica (3d ser.), Vol. II (1901), p. 28.
7 Paris MS 7377 A, described by L. C. Karpinski, Bibliotheca mathematica (3d ser.), Vol. XII (1911-12). Our reference is to p. 53. See also Vol. X (1909- 10), p. 18, 19, for H. Suter’s translation into German of a translation into Italian of a Hebrew edition of Abu Kamil’s book, On the Pentagon and Decagon , where f-| signifies f plus f of or §§.
CONTINUED FRACTIONS, INFINITE SERIES
fractions, which differs from that of al-Has§ar in proceeding from left to right. The fraction U stands for J pins ft; the fraction -ft or U stands for of + Abu Kamil himself presumably proceeded from light to left. I he Arabic notation is employed also by Leonardo1 of Pisa in his Liber abbaci of 1202 and 1228, and later still by the Arabic author al-Qualasadi.2 Al-Hassar, it will be observed, also used the fractional line for ordinary fractions, as did also Leonardo of Pisa.
423. Leonardo3 gives expressions like ft-ft 8, which must be
read from right to left and which stands in our notation for 8-f-— ^q1 ^
or 8++. He gives also as equal to IHKVtV; the value of these
ascending continued fractions is T2V¥.
Leonardo gi\es also two other notations, one of which he describes as follows: “Et si in uirga fuerint plures rupti, et ipsa uirga termina- uerit in circulo, tunc fractiones eius, aliter quam dictum sit denota- bunt, ut in hac -iHJO cuius uirge fractiones denotant, octo nonas unius integii, et sex septimas de octononis, et quatuor quintas sex septimarum de octo nonis et duas tertias quattuor quintarum sex septimarum de octo nonis unius integri.4 (“And if on the line there should be many fractions and the line itself terminated in a circle, then its fractions would denote other than what has been stated, as m this -MffO, the line of which denotes the fractions eight-ninths of a unit, and six-sevenths of eight-ninths, and four-fifths of six-sevenths of eight-ninths, and two-thirds of four-fifths of six-sevenths of eight- ninths of an integral unit. ) W e have here an ascending continued fraction
8+
9
48+
~7
192 -f 5
384
3
Leonardo’s third notation is described by him as follows: “Item
si uirgule protraherunter super uirgam in hunc modum =■- - ^
5 4 3 9’
denotant fractiones eius quinque nonas et tertiam et quartam et quintam unius none.”5 (“Also if a short line be drawn above the
1 Leonardo of Pisa, Scritti (pub. by B. Boncompagni), Vol. I (Rome, 1857), p. 24. See also Encyclopedic des scienc. math., Tome I, Vol. I (1904), p. 318, n. 330.
2 M. Cantor, Vorlesungen uber Geschichte der Mathematik, Vol. I (3d ed., 1907),
p. 813. ' ’
3 Leonardo of Pisa, op. cit., Vol. I, p. 85. * 4 Op. cit., Vol. I (1857), p. 24.
5 Op. cit., Vol. I (1857), p. 24, also p. 91, 92, 97. See G. Enestrom, Bibliotheca mathematica (3d ser.), Vol. XII (1911-12), p. 62.
50
A HISTORY OF MATHEMATICAL NOTATIONS
1115
fractional line in this manner ~ , they denote the following
fractions, five-ninths, and one-third, one-fourth and one-fifth of
one-ninth.”) The foregoing fraction signifies, therefore,
9
it is a complex fraction, but hardly of the “continued” type. Not to be confounded with these is Leonardo’s notation O f-f-f-J which does not represent a continued fraction either, but signifies the multiplica¬ tion1 of the fractions, thus j-.-f •-§-•-§, and resembles a mode of writing, f— f, for f.f, sometimes found in al-ITassar.2
424. Pietro Antonio Cataldi in 16063 discusses the ascending con-' tinued fraction as “vna quatita scritta, 6 proposta in forma di rotto di rotto” (“a quantity written or proposed in the form of a fraction of a fraction”). He explains an example and tells how mathematicians are accustomed to write it: “Sogliono i Pratici scriuere cosi f •-§-•$•-1 The meaning of this appears from a simpler example explained on the previous page (142). He says: “poniamo 3. quarti, & J. quarto, cioe, J.& or J. One has here a notation for an ascending continued fraction. Seven years later Cataldi introduced, for the first time, a notation for descending continued fractions, and he chose practically the symbolism that he had used for the ascending ones. In 1613 he explained: “Notisi, che no si potendo comodamete nella stampa formare i rotti, & rotti di rotti come andariano cioe cosi
2
8.&
2
8
come ci siamo sforzati di fare in questo, noi da qui inazi gli formaremo tutti a qsta similitudine
*-*i.*i.*i.
facendo vn punto all ’8. denominator de ciascum rotto, a significare, che il sequente rotto e rotto d’esso denominator.”4 (“Observe that
1 Leonardo of Pisa, op. cit., Vol. I, p. 24.
2 H. Suter, Bibliotheca mathematica (3d ser.), Vol. II, p. 27.
3 Pietro Antonio Cataldi, Seconda parte della pratica aritmetica (Bologna, 1606), p. 143.
4 Trattato del Modo Brevissimo di trouare la Radice quadra delli numeri, et Regole da approssirnarsi di continuo al vero nelle Radici de ’ numeri non quadrati ... Di Pietro Antonio Cataldi Lettore delle Scienze Mathematiche nello Studio di Bologna (in Bologna, 1613), p. 70, 71.
CONTINUED FRACTIONS, INFINITE SERIES 51
since in printing when proceeding hurriedly, one cannot conveniently f°im fi actions, and fractions of fractions, in this form
as here we are forcing ourselves to do it, we can easily denote all of them by adopting this imitation
placing a point after the denominator 8. in each fraction, to signify that the fi act ion following is a fraction of the denominator.”) But, in his simplified form of writing, as it appears further on in his book, the point is not placed after the numbers in the denominator, but is placed higher up, on _a level with the ” This is seen in Cataldi’s approximation to V18 (p. 71) :
“Di 18. la ft sia 4.& £.& -J.& j
6 vogliamo dire 4.& £.& £.& £.& f.& £.& i cioe 4.& •£.& |.& £.&
6 vogliamo dire 4.& f.& £.& £.& J che e 4 .& £.& £.& cioe 4.& £.& -J.&
6 vogliamo dire 4.& f .& £ che e 4 . & f •& cioe 4 . &
Except for the & in place of this notation conforms closely
with one of the modern notations.
425. We come next to Wallis’ representation of Brounker’s ex-
4
pression1 for -, namely,
7T
U
□ =
11 1
u
¥
42.
2
^ & c.,” and Wallis’
- 6
apC-d
7 S-&c. ”
I
i
1 John Wallis, Arithmetica infinilorum (Oxford, 1655), p. 182.
52
A HISTORY OF MATHEMATICAL NOTATIONS
The second expression is Wallis’ general representation1 of a continued fraction. The omission of the signs of addition makes this notation inferior to that of Cataldi who used the But Wallis adheres to his notation in his collected works2 of 1695, as well as in his letter3 to Leibniz of April 6, 1697. Leibniz,4 on the other hand, used an im¬ proved form,
a
a+l+-,
C +
1
d+
1
e+etc.”
in his letter to John Bernoulli of December 28, 1696.
A modern symbolism is found also in C. Huygens5 who expressed the ratio 2640858:77708431 in the form
u
29+i
+ A
+ |
+ *
+ i
+1 etc.
ff
The symbolism of Leibniz and Huygens came to be the regular form adopted by eighteenth-century writers and many writers of more recent date; as, for example, by J. A. Serret6 in his Higher Algebra. Among eighteenth-century writers we cite especially L. Euler and J. Lagrange.
426. L. Euler7 writes
“°+L-0
6+c+Netc.”
a
In some of his articles the continued fractions written in this nota¬ tion made extraordinary demands upon space, indeed as much as
1 Op. cit., p. 191.
2 John Wallis, Opera mathematical Vol. I (1695), p. 469, 474.
3 C. I. Gerhardt, Leibnizens Mathematische Schriften, Vol. IV (Halle, 1859),
p. 17.
4 Op. cit., Vol. Ill (Halle, 1855), p. 351, 352.
5 Christian Huygens, Descriptio automati planetarii (The Hague, 1698) ; Opuscula posthuma (Amsterdam, 1728), Vol. II, p. 174-79. See also S. Gunther, in Bullettino Boncompagni, Vol. VII (Rome, 1874), p. 247, 248.
6 J. A. Serret, Cours d’algebre superieure (Paris, 1854), p. 491; German edi¬ tion by G. Wertheim, Vol. I (2d ed.; Leipzig, 1878), p. 8.
7 L. Euler, Introductio in analysin infinitorum (1748); ed. 1922, Vol. I, p. 362.
CONTINUED FRACTIONS, INFINITE SERIES
53
lo cm. or four-fifths of a large page.1 In 17G2 he devised a new nota¬ tion in his Specimen Algorithmi Singularis ;2 a fraction
a+l+i
c
is represented by the symbol an infinite continued fraction by
(a, b, c, d, e etc.) ml . , /. .
(b c d e etc ) ' 'L*11S symiD0^sm oners superior advantages in
computation, as. for example: (a, b, c, d, e)=e(a, b, c, d) + (a, b, c), also (a, b, c, d, e) = (e, d, c, b, a).
427. E. Stone3 writes
„ 314159 100000
will be = 3
7+-
15-
15
1 + :
7+4 *
This extension of the fractional lines to the same limit on the right is found earlier in Leibniz (§ 562) and is sometimes encountered later, as, for instance, in the book on continued fractions by O. Perron (1913). 4
tu y the need of more compact nota¬ tions asserted itself and there was a return to Cataldi’s practice of placing all the partial fractions on the same level. Sir John W. Herschel5 writes the continued fraction
in the form
C\
GU +
C2
«2 +
C3
CI3 + . . . .
Cx
ax
Cl C2 Cz Cx
&1 + &2 + &3 + CLX 1
1 See L. Euler in Novi comment, academ. imper. Petropolitanae , Vol. V, for the years 1754, 1755 (Petropoli, 1700), p. 225, 226, 231.
2 N. Comm. Petr. IX, pro annis 1762 et 1763 (Petropoli, 1764), p. 53-69. See M. Cantor, op. cit., Vol. IV (1908), p. 155.
3 E. Stone, New Mathematical Dictionary (2d ed.; London, 1743), art. “Ratio.”
4 Oskar Perron, Lehre von den Kettenbriichen (Leipzig und Berlin, 1913), p. 3.
5J- F. W. Herschel, Collection of Examples of ... . Calculus of Finite Differences (Cambridge, 1820), p. 148.
54
A HISTORY OF MATHEMATICAL NOTATIONS
but states that this is “after the example” of Heinrich Burmann of Mannheim, a worker in C. F. Hindenburg’s school of combinatory analysis. The notation of Burmann and Herschel has been used in England by Hall and Knight,1 C. Smith,2 G. Chrystal,3 L. Ince,4 and is still widely used there. Chrystal (Part II, p. 402) also uses sym¬ bolisms, such as:
111111
1/13 = 3-
1+ 1+ 1-f- 1-j- 6+ 1 -f-
* *
where the indicate the beginning and end of the cycle of partial quotients.
429. Mobius5 says: “Der Raum-Ersparniss willen mogen nun die Kettenbriiche von der besagten Form, wie
durch (a), (a, b), (a, b, c), (a, b , c, d), u.s.w. ausgedriickt werden.” Mobius6 represents the function
_ 1 _
(a, .... e) (b, ... . e)(c, d, e){d, e)(e)
by the symbolism [a, b, c, d, e\. M. Stern7 designated the continued fraction
a+- &2
aid - h
a2 .
1 H. S. Hall and S. R. Knight, Elementary Algebra (ed. F. L. Sevenoak; New York, 1895), p. 369.
2 C. Smith, Elementary Algebra (ed. Irving Stringham; New York, 1895), p. 530.
3 G. Chrystal, Algebra, Part II (Edinburgh, 1889), p. 396, 397.
4 Linsay Ince in University of Edinburgh Mathematical Dept. (1914), Paper
No. 7, p. 2, 3.
6 A. F. Mobius, Crelle’s Journal, Vol. VI (1830), p. 217; Werke, Vol. IV (Leip¬ zig, 1887), p. 507.
6 Mobius, Crelle’s Journal , Vol. VI, p. 220.
7 M. Stern, Theorie der Kettenbriiche und ihre Anwendung (Berlin, 1834), p. 4, 22, 33; reprinted from Crelle’s Journal , Vol. X, XI.
CONTINUED FRACTIONS, INFINITE SERIES
55
by P (a, am) ; he lets a\, am stand symbolically for the denominator of the equivalent ordinary fraction, and a, am for its numerator, so that
F(a, am) = . Stern employs also the fuller form F(a, am)=F(a+
Q'l) &TTI
5i:ai+62:a2, etc.). For the special, infinite continued fraction F(1 : 1 + 1 : 2+9: 2+25 ;2, etc.) he suggests the form
x F[l:l + (2z+l)2:2] .
0 — co
430. J. H. T. Muller1 devised the notation
7 , o,\ ; :
o0 +V-+T-+ b\ 02
+
where each + sign may be replaced by a — sign when parts are nega¬ tive. Muller also wrote
&o+
a\
b/ ( 1 +2 +3 + .... +71)
Muller’s first symbolism found quite wide acceptance on the European continent, as is born out by statements of Chrystal2 and Perron.3 Per¬ ron remarks that sometimes the dots were omitted, and the continued fraction written in the form
6°+r+r+
Ol 02
+
a
n
'n
431. When all the numerators of the partial fractions are unity, G. Lejeune Dirichlet4 wrote down simply the denominators br , in the following form:
(&o, biy • • • • , bn — j, bn).
E. Heine5 6 writes the continued fraction
Mi
1 J. H. T. Muller, Lehrbuch der Mathematik. Ersler Theil, Die gesammte Arithmetik enthaltend (Halle, 1838), p. 384.
2 G. Chrystal, Algebra, Part II (Edinburgh, 1889), p. 397.
3 0. Perron, op. ciL, p. 3.
4 G. Lejeune Dirichlet, Abhandl. Akad. Berlin (1854), Math., p. 99; Werke,
Vol. II (Berlin, 1897), p. 141.
6 E. Heine, Theorie der Kugelfunctionen (Berlin, 1878), p. 261.
56
A HISTORY OF MATHEMATICAL NOTATIONS
in the form
|
Mi |
M2 . |
• • • Mti — 1 M71 |
|
Al |
X2 • |
. . . \n — 1 \n |
This notation is followed by Pascal1 in his Repertorium.
432. The modern notation, widely used on the European conti¬ nent,
5o±
All , 0-2 1 .
I&1 - |b> -
}
is due to Alfred Pringsheim,2 who represents this continued faction also by the symbol
n
771 + 1
he represents the continued fraction whose first term is bm and the denominator of the first partial fraction, 6m+i. Also in place of
he writes simply
av
bv
n
777 + 1
n
771 + 1
G. Peano3 represents the continued fraction
_1
Ul +
J.
a2-f etc. to an ,
by the symbolism Fc(ah 1 .... n).
In the continued fraction
a.4+ h _ +
^2 + ^3+ CLn ’
1 E. Pascal, Repertorium, der hoheren Mathematik (ed. P. Epstein and H. E. Timerding), Vol. I (2d ed., 1910), p. 444.
2 Alfred Pringsheim, Encyklopcidie der mathematischen Wissenschaften, 1. Band, 1. Theil (Leipzig, 1898-1904), p. 119. These notations occur also in the French edition, the Encyclopedie des scien. math., Tome I, Vol. I (1904), p. 283, 284.
3 G. Peano, Formulaire mathematique, Vol. IV (1903), p. 197.
CONTINUED FRACTIONS, INFINITE SERIES
57
the expression pn = dnPn-i+bnpn-2, where p0= 1, pi = ah is called by Chrystal1 a “continuant of the nth order” and is marked
Pu = k( F . M ,
\<2i, 0.2, ... . , anJ
a symbol attributed2 to Thomas Muir.
433. A notation for ascending continued fractions corresponding to Burmann and HerscheFs notation for descending ones is the fol¬ lowing,
61+62+63-f- ~{~bn
ai a2 az an ’
used, among others, by Weld.3 A. Pringsheim and J. Molk4 write an ascending continued fraction
I O'Tl
<h+?+"' F
TT(n I “2
in the form
434. Tiered fractions .— G. de Longchamps5 indicates by
Oi
a2
^n+l
tiered fractions ( fractions etagees) involving the numbers ah a2, . . . . , an+ 1, the bars between the numbers signifying division. If the bars be assigned different lengths, An has a definite arithmetical value. Thus,
0\ _ 0\0z0a . 0\
&2 02 ’ 02
Q3 a3 _ ai
o-a Oa a2aZ0A ‘
1 G. Chrystal, Algebra , Part II (1889), p. 4G6. 2 O. Perron, op. cit., p. 6.
3 Laenas G. Weld, Theory of Determinants (2d ed.; New York, 1896), p. 186.
1 A. Pringsheim and J. Molk, Encyclopedie des scien. math., Tome I Vol I (1904), p. 317.
J Gobierre de Longchamps, Giornale di matem. (lstser.), Vol. XV (1877), p. 299.
58
A HISTORY OF MATHEMATICAL NOTATIONS
An may have n\ different values, arising from the different relative lengths which may be assigned to the fractional lines.
435. Infinite series I — The early writers in the field of infinite series or infinite products indicated such expressions by writing down the first few terms or factors and appending “&c.” Thus Wallis in 1656 wrote: “1, 6, 12, 18, 24, &c.,”2 and again “lXfXfXj, &c.,”3 and similarly for infinite continued fractions.4 He uses the “&c.” also for finite expressions, as in “0+a3+63+c3 &c. cujus terminus ultimus l 3, numerus terminorum Z+ l”5 (whose last term [is] l 3, the number of terms Z+l).
Nicholas Mercator writes “ps = 1 — afi-aa — a3+a4, &c.,”6 and also
“a , ca ( cca , c3a
& deinceps continuando progressionem in infini¬
tum.”7 James Gregory, in the same year, says: “Si fuerint quantitates continue proportionales, A, B, C, D, E, F, &c. numero infinitae.”8 He places the sign + before “&c.” when the algebraic sum of the terms is expressed, as in “erit primus terminus +4 secundi +-§■ tertii +J quarti +4 quinti +&c. in infinitum = = spatio Hyperbolico S B KH.”9 In other passages he leaves off the “in infinitum.”10 Brounker11 in 1668 writes
“ rV2+3^4+5V6+7V8+9VIO ’ &C' in infinitum”
G. W. Leibniz12 gives the quadrature of the circle by the series — J+i, and ends with “+TV &c.” Cotes13 writes
“v*— &c.”
Wolff14 says “i+i+i+A+A und so weiter unendlich fort.”
1 See also § 408.
2 John Wallis, Arithmetica infinitorum (Oxford, 1655), p. 26.
3 Op. tit., p. 175. 4 Op. tit., p. 191. 5 Op. tit., p. 145.
6 Logarithmo-technia .... auctore Nicolao Mercatore ( London, 1668), Propo- sitio XVII, p. 32.
7 Op. tit., p. 25.
8 James Gregory, Exercitationes geometricae; in the part on Mercator’s quadra¬ ture of the hyperbola, p. 9.
9 James Gregory, op. tit., p. 11. 10 James Gregory, op. tit., p. 13.
11 L. Viscount Brounker, Philosophical Transactions (London), Vol. II, p. 645- 49; abridged edition by John Lowthrop, Vol. I (London, 1705), p. 10.
12 Philosoph. Transactions, abridged, Vol. I (London, 1705), p. 16. The prac¬ tice of Leibniz is exhibited also in articles in the Acta eruditorum (1691), p. 179; (1693), p. 178; (1694), p. 369; (1695), p. 312; (1701), p. 215.
13 Roger Cotes, Harmonia mensurarum (Cambrigiae, 1722), p. 6.
14 Christian Wolff, M athematisches Lexicon (Leipzig, 1716), p. 176.
CONTINUED FRACTIONS, INFINITE SERIES
59
43G. T. Watkins1 uses dots in place of “&c.” to represent the tail end of a series, but he does not write the + or the — after the last term written down. In a letter to Nikolaus I. Bernoulli, May 14, 1743, Euler2 writes l-\-x-\-x2-\-x3Jr .... +3°° , and also 1—3+5— 7+9- _ ±(2oo+l).
E. Waring3 writes series thus, “a+/3+Y+<5+e+r+etc.”
The indefinite continuance of terms is designated by Schultz4 in this manner, “1 +3 + 5+7+ .... ^,” the ~ being probably in¬ tended for Wallis’ oo . Owing no doubt to the lack of proper type in printing offices Wallis’ symbol was given often forms which stood also for “difference” or “similar.” Thus the sign co stands for “in¬ finity” in a publication of F. A. Prym5 in 1863.
The use of dots was resorted to in 1793 by Prandel6 when he writes
“□ Circ. =4 — -§-— — A — T88 . ” L’Huilier7 ends an infinite
series of positive terms with “+ . ”
L’Abbe de Gua8 writes a finite expression, marking the terms omitted, with dots and also with “&c.,” “3, 4, 5 .... &c n— m+2,” the commas indicating here multiplication. F. Nicole9 writes a pro¬ cession of factors, using dots, but omitting the “&c.” and also the bar: “nXn-lX . . . . n — 7.” The same course is pursued by Condorcet.10 On the other hand, C. A. Vandermonde11 used the “&c.” as in “a+6+c+& c.” Paulus Mako12 of Austria designates that the
series is infinite, by writing “5, bm, bm2y bmz, . fern00.” C. F.
Flindenburg13 uses dots between, say, the fourth term and the nth
1 Thomas Watkins, Philosophical Transactions , Vol. XXIX (1714—16), p. 144.
2 L. Euler, Opera posihuma, I (Petropoli, 1862), p. 538. See G. Enestrom in Bibliotheca mathematica (3d ser.), Vol. XI (1910-11), p. 272.
3 E. Waring, Meditationes algebraicae (Cantibrigiae; 3d ed., 1782), p. 75.
4 Johann Schultz, Versuch einer genauen Theorie des Unendlichen (Konigsberg, 1788), p. xxvii.
5 F. A. Prym, Theoria novafunctionum ultraellipticarum (Berlin, 1863), p. 1, 3, 4.
c Johann Georg Prandel, Kugldreyeckslehre u. hohere Mathematik (Miinchen, 1793), p. 37.
7 Simon l’Huilier, Principiorum Calculi Differentials et Integrals Expositio (Tubingae, 1795), p. 27.
8 De Gua in Histoire de Vacademie r. d. sciences , ann£e 1741 (Paris, 1744), p. 81.
9 F. Nicole, op. cit., annee 1743 (Paris, 1746), p. 226.
10 Le Marquis de Condorcet, op. cit., annee 1769 (Paris, 1772), p. 211.
11 C. A. Vandermonde, op. cit., annee 1771 (Paris, 1774), p. 370.
12 Compendiaria matheseos institutio .... Paulus Mako (3d ed.; Vienna, 1771),
p. 210.
13 C. F. Hindenburg, Infinitinomii dignitatum .... hisloria leges ac formulae .... auctore Carolo Friderico Hindenburg (Gottingen, 1779), p. 5, 6, 41.
60
A HISTORY OF MATHEMATICAL NOTATIONS
term, the + or the — sign being prefixed to the last or nth term of the polynomial. However, at times he uses “&c.” in place of the dots to designate the end of a polynomial, “cC-j-dD+eE+dkc.” E. G. Fischer1 writes a finite expression y = ax-\-bx2-\-cxs-{- . . . . -\-pxr and, in the case of an infinite series of positive terms, he ends with “+etc.”
437. M. Stern2 writes “1 — -£-+-£■— \ etc. =j.” Enno Heeren Dirk-
sen3 of Berlin indicates in 1829 by “-fete.” that the sum of the terms of an infinite series is intended, but a few years later4 he marks an infinite progression thus: “1, 2, 3, 4 in inf.” Martin Ohm5 says: “Jede unendliche Reihe (P) .... a-{-bx-\- cx2+dx3 .... in inf. kann durch das kombinatorische Aggregat S[Pa»xa] ausgedruckt werden, wenn man nur statt a nach und nach 0, 1,2, 3, 4, in inf. gesetzt denkt, und wenn Pa den Koeffizienten von xa vorstellt.” L. Seidel6 writes “f(x, n) fix , l)+/(x, 2) .... in inf.”; A. N. Whitehead7 in 1911 ends
with “+etc . ” A notation for infinite series now frequently
used is to place dots before and after the nth term, as in8 a0-f aiil/x) + ci 2 ( 1 / ~ f~ • • • • ~T ( 1 / xd) n -j- . . . .
There are recent examples of rather involved notations to indicate the number of each term in a finite series, like the following of 0. Stolz and J. A. Gmeiner.9
44
yOy
+8
1 Ernst Gottfried Fischer, Theorie der Dimensionszeichen (Halle, 1794), p. 27, 54.
2 M. Stern, Theorie der Kettenbritche (Berlin, 1834), p. 34. Reprint from Crelle’s Journal, Vol. X, XI.
3 Crelle’s Journal (Berlin, 1829), p. 170.
4 Abhandlungen der K. P. Akademie d. Wissenschaften (Berlin, 1832), Th. I, p. 77-107.
5 Martin Ohm, Versuch eines vollkommen consequenten Systems der Mathematik, Vol. II (Berlin, 1829), p. 264, 265.
6 Abhandlungen d. Math.-Phys. Classe d. k. Bayerischen Akademie der Wissen¬ schaften, Vol. V (Munchen, 1850), p. 384.
7 A. N. Whitehead, An Introduction to Mathematics (New York and London),
p. 212.
8 W. B. Ford, Studies on Divergent Series and Sum7nability (New York, 1916),
p. 28.
9 O. Stolz und J. A. Gmeiner, Theoretische Arithmetik (Leipzig), Vol. I (2d. ed., 1911), p. 104, 105, 109, 114, 121.
THEORY OF COMBINATIONS
61
438. The sign 2 for summation is due to L. Euler1 (1755), who says, “summam indicabimus signo 2.” This symbol was used by Lagrange,2 but otherwise received little attention during the eight¬ eenth century. A widely used notation for the sum of a series is the capital letter S; it is given, for instance, in G. S. Kliigel’s Worterbuch .3 The 2 to express “sum” occurs in 1829 in Fourier’s Theory of Heat,4 published in 1822, and in C. G. J. Jacobi’s5 elliptic functions of 1829.
n
Cauchy6 used three indices m, n, r, as in ^ r fr. Alfred Pringsheim7
m
00
marks the sum of an infinite series thus, f>vav.
o
Jahnke8 adopts in one place, as a substitute for 2 ai} the simpler sign a, which he borrows from geodesists, to designate the sum ai+«2+a3+ . Additional symbols for sum are given in § 410.
SIGNS IN THE THEORY OF COMBINATIONS
439. Binomial formula. — The binomial formula, as Newton wrote
m
it in his letter to Oldenburg of June 13, 1676, took this form, {P-\-PQ)n
= PnJr — AQ+-9--- BQ+m CQ-f-etc.,9 where A stands for the n 2n 3 n
m
first term Pn, B for the second term, and so on. Wallis,10 in his Algebra of 1693, gives Newton’s form of 1676 for the binomial formula. Leib-
1 L. Euler, Institutiones calculi differ entialis (St. Petersburg, 1755), Cap. I, § 26, p. 27.
2 J. Lagrange, GUuvres, Vol. Ill, p. 451.
3 G. S. Klugehs Mathematisches Worterbuch, completed by J. A. Grunert, Part V (Leipzig, 1831), “Umformung der Reihen,” p. 348; Part IV (1823), “Summirbare Reihe,” p. 577.
4 Joseph Fourier, La theorie analytique de la chaleur (Paris, 1822; Eng. trans. by A. Freeman, 1878), chap, iii, sec. 6, p. 208, and other places.
5 C. G. J. Jacobi, Fundamenta nova theoriae functionum elliyticarum (1829); Gesammelte Werke, Vol. I (Berlin, 1881), p. 94.
6 See G. Peano, Forrnulaire mathematique, Vol. IV (Turin, 1903), p. 132.
7 See, for instance, Encyklopadie der Math. Wissensch., Vol. I, Part I (Leipzig, 1898-1904), p. 77.
8 E. Jahnke, Archiv der Mathematik und Physik (3d ser.), Vol. XXV (1916), p. 317.
9 J. Collins, Commercium Epistolicum (London, 1712; ed. J. B. Biot and F. Lefort, Paris, 1856), p. 103.
10 John Wallis, De algebra tractatus (1693), p. 376.
62
A HISTORY OF MATHEMATICAL NOTATIONS
niz* 1 wrote in 1695,
u
m
-(5)
m
y+a = ym+jy
™ lz1 , , m-m— 1 m~2
a 1
1.2
y • a2, etc.,
>7
where the (5) simply marks the equation as being equation No. 5. In Newton’s Quadratura curvarum,2 1704, occurs the following pas¬ sage: “Quo tempore quantitas x fluendo evadit x-j-0, quantitas xn
evadet x-f-0|n; id est per methodum serierum infinitarum, xTO+n0xn_1
nn — n
00xw-2-l- etc.” In 1706 William Jones3 gives the form
a+x|
n— 0
--an-\-'\y an~1x-
n — 0. n—1
X " an~2x 2
, n— 0V n— lv ,n— 2 Q , . , d — j— X— 7>— X— g— an“3x3+etc.,
but Jones proceeds to develop the form given by Newton, except that Jones writes a-\-ag where Newton has P+PQ. In 1747 Jones4
adopted the abbreviations “n' = n
n—1
jt
; n — n •
/ n ^ . ^nr
; n =n
n
n— 3
— 4
niy=n"' • — Another notation due to Kramp will be noted further
m
/e a a r\ r-i , n . in , ^ I , VI _ . Hfl v m~U __ , UL^m-P
on (§ 445). Cotes5 wrote 1+£| =1 + n^+nX~2n'^+nX~2rT
X~3 ri~~ e^c-” Euler designated the binomial coefficient
n(n — 1) .... {n — p+1) . (n\ . ... . , ,
by - in a paper written m 1778 but not
1 *2 *3 .... p
published until 1806, 6 and by
m
in a paper of 1781, published in
1784.7 Rothe8 in 1820 denoted the pth binomial coefficient in the
1 Leibnizens M athematische Schriften (ed. C. I. Gerhardt), Vol. V (1858), p. 323.
2 Isaac Newton, Opera (ed. S. Horsley), Vol. I (London, 1779), p. 336.
3 William Jones, Synopsis Palmariorum Matheseos (London, 1706), p. 169, 170; the coefficients taking the form used by John Wallis in his Algebra of 1693, p. 358.
4 W. Jones, Philosophical Transactions for the Year 174-7 (London, 1748),
p. 563; abridged edition by John Martin, Vol. X (London, 1756), p. 17.
6 Roger Cotes, Harmonia mensurarum , the tract De methodo differ entiali Newtoniana (Cambridge, 1722), p. 30.
6 L. Euler in Nova acta acad. Petrop ., Vol. XV (1806), Math ., p. 33
7 L. Euler in Acta Acad . Petrop., Vol. V (1784), pars prior, § 18, p. 89. See Cantor, op. cit., Vol. IV, p. 206.
8 H. A. Rothe, Theorie der kombinatorischen Integrate (Ntirnberg, 1820). Taken from J. Tropfke, op. cit.., Vol. VI (2d ed., 1924), p. 44.
THEORY OF COMBINATIONS
63
expansion of (a+5)n by n , and Ohm1 in 1829 denoted the nth coeffi¬ cient in ( a-\-b)m by mn. The notation (”') which has become the more common was introduced in 1827 by von Ettingshausen,2 and was used in 1851 by Raabe.3 Stolz and Gmeiner4 employ for binomial coefficients (™) and also mr, thus following the symbolism of Itothe
and Ohm. Saalschutz5 lets (k)h stand for (J).
The quantic ax* A-Sbx-y -\-?>cxy2 dyz , in which the terms are affected with the binomial coefficients arising in the expansion of (x+y)\ is denoted by Cayley6 by the abbreviation (a, b, c, d)x, y). When the terms are not affected by bionomial coefficients, Cayley put
the arrow-head on the parenthesis, writing for instance (a, b, c, d\x, y) to denote ax3-\-bx2y-\-cxy2-\-dy*. Faa de Bruno7 designated by (( xv))F the coefficient of xv in the development of F, which is a function of x.
An imitation of Newton’s original mode of writing the binomial formula is seen in Stirling’s notation for series: “I designate the initial terms of a series by the initial letters of the alphabet, A, B, C, D, etc. A is the first, B the second, C the third, and sic porro. I denote any term of this kind by the letter T, and the remaining terms in their order of succession by the same letter affixed with the Roman numer¬ als I, II, III, IV, V, VI, VII, etc., for the sake of distinction. Thus, if T is the tenth, then T' is the eleventh, T" the twelfth, T'" the thirteenth, and so on. And whichever term of this kind is defined as T, the ones that follow are generally defined by T' T" T'" Tiv, etc. The distance of the term T from any given term I denote by the in¬ determinate quantity £.”8 Thus, following the notation of Newton (“more Newtoniana”), Stirling puts, in the series 1, \x, fa;2, T\a;3, etc.,
A = 1, B = %Ax, C = f Bx,
T' =
3+1
Tx.
1 Martin Ohm, op. cit., Vol. II (1829), p. 79.
2 Andreas von Ettingshausen, Vorlesungen uber hohere Mathematik, Vol. 1 (Vienna, 1827), p. 38. See E. Netto in Cantor, op. cit., Vol. IV, p. 206; J. Tropfke, op. cit., Vol. VI (1924), p. 44.
3 J. L. Raabe in Journal f. reine u. angewandte Mathematik, Vol. XLII (1851), p. 350. See Encyclopedic des sciences Math., Tome I, Vol. I (1904), p. 67, n. 20, 21.
4 O. Stolz and J. A. Gmeiner, Theoretische Arithmetik, Vol. I (Leipzig, 1902), p. 187.
5 Louis Saalschutz, Bernoullische Zahlen (Berlin, 1893), p. 3.
6 See G. Salmon, Modern Higher Algebra (3d ed.; Dublin, 1876), p. 92.
7 Fail de Bruno, Theorie des formes binaires (Turin, 1876), p. 161.
8 James Stirling, Methodus differ entialis (London, 1730), p. 3.
64
A HISTORY OF MATHEMATICAL NOTATIONS
440. Product of terms in an arithmetical progression. — Machin,1
in a paper on Kepler’s problem, adopts the notation — which he
says, “denotes by its Index m on the Right-hand, that it is a Compos¬ ite Quantity, consisting of so many Factors as there are Units in the Number m; and the Index a above, on the Left, denotes the common Difference of the Factors, decreasing in an Arithmetical Progression, if it be positive ; or increasing, if it be negative ; and so signifies, in the com¬ mon Notation, the composite Number or Quantity, n-f-a*n+a — a*
n-f-a — 2a»n+a— 3a» and so on.” Further on we shall encounter other notations for such a product, for instance, those of Kramp, Ampere, and Jarrett (§§ 445-47).
441. Vandermonde's symbols.— In 1770 C. A. Vandermonde, in an article on the resolution of equations,2 adopted the following con¬ tractions:
“(A) pour a+ &+c+&c (A2) pour a2+52-f-c2+& c (AB) pour ab-f-ac+&+6c+& c+
(A2B) pour a26+62a+c2a+&c+a2c+62c+c26+&c+&c+ .
Et en general par (AaB^Cy . . . . )r' indiquerai la somme de tous les termes differens qui resuteroient de celui-la, au moyen de toutes les substitutions possibles des petites lettres a, b, c, d , e, &c dans un ordre quel conque aux grandes, A, B, C, &c.” He writes also (AaB^CyD8Ee ....) = { a(3y5e ....}, or if several Greek letters are equal, he writes {an/3pyq .... }.
442. Laplace3 represented the resultant of three equations by the symbolism (1a2»68«c), the indices being placed to the left of a letter and above.
443. Combinatorial school of Hindenburg. — The notation used by members of the combinatorial school in Germany is often so prolix and involved that a complete account of their symbolism transcends the limits of our space. The generalization of the binomial theorem so as to involve any power of any polynomial, be the number of terms
1 John Machin, Philosophical Transactions (London), No. 147, p. 205 (Jan., etc., 1738); abridged by John Martyn, Vol. VIII, Part I, p. 78.
2 C. A. Vandermonde in Histoire de Vacademie r. d. sciences, annee 1771 (Paris, 1774), p. 370, 371.
3 P. S. Laplace, Histoire de Vacademie, r. d. sciences (Paris), annee 1772, p. 267, 294, 2. partie; Laplace, (Euvres , Vol. VIII, p. 365-406. See also Th. Muir, The Theory of Determinants in Historical Order of Development, Part I (2d ed., 1906), p. 30.
THEORY OF COMBINATIONS
65
in the pol\ nomial finite or infinite, was one of the problems considered. ( ail Friedrich Hindenburg, in a treatise of 1778, 1 represents the bi¬ nomial coefficients by capital German letters, thus
mm m(m-l) l)(m-2)
1 ’ * 1.2 ’ ^ - 1T2T3 - ’ -
This notation is followed by G. S. Klugel in his Worterbuch .2 Netto remarks that Hindenburg adheres to the practice which had been at least in part abandoned by Leibniz as inconvenient — the practice of using the alphabetical arrangement of the letters for the designation of order and for enumeration. Hindenburg employs the upright Latin capitals A, B, C to mark combinations, the slanting Latin capitals A, B, C for permutations. A superscript stroke placed to the left of a letter, 'A 'A, means a combination or permutation without repetitions, while a stroke placed to the right, A' A', means with repetitions. To mark coefficients that are not binomial, but poly¬ nomial, he uses s?nall German letters in place of the capital German letters given above.
In a book of 1779, Hindenburg uses small and capital letters in Roman, Greek, and German type. He employs superscripts placed not only before or after a letter, but also above the letter, the super¬ scripts either inclosed in a parenthesis or not. He3 calls mb, mc, mdf signa indefinita, used in the manner illustrated in the following ex¬ pressions:
Am = am
B'*=(A + b)m = Am+rnAm-lb + —'-™^AAm-‘tb2+&c = a”+»‘b .
1 • z
Cm=(B+c)m= .... =Bm-\-mc = amJrmbJrmc .
Further on (p. 17) Hindenburg puts a{a-\-b-\-c-\-d .... -j-co] = h4 , ai’A + 'B+'C+'D .... -f-'Q] = "A ,
a[',A + ''B+"C+,'D _ +"R] = '"A _ _ _
d["D _ + etc.,
nD = d{n~lD _ +W-T3] .
1 C. F. Hindenburg, Methodus nova et facilis serierum infinitarum, etc. (Got¬ tingen, 1778). Our knowledge of this publication is drawn from E. Netto in Cantor, op. cit., Vol. IV, p. 20.5-7.
2 G. S. Klugel, Mathematisches Worterbuch, 1. Theil (Leipzig, 1803), art “Binomial — Coefficientem”
3 C. F. Hindenburg, I nfinitmomii dignitalum .... Historia, leges ac formulae (Gottingen, 1779), p. 5, 17, 18, 20.
66
A HISTORY OF MATHEMATICAL NOTATIONS
He introduces (p. 21) f for summam partium (the parts chosen in a certain manner), which is to be distinguished from the sign of inte¬ gration J', but like f has vim transitivam, i.e., is distributive in
addition. Later still (p. 159), writing jB = 6+c+d+&c, C = c+d+e+
2 2
&c, . . . . , he puts B = bB-\-cC-\-dDJr&c, C = cC-\-dD-\-eE-\-& c, and
(2) (2)
so on; also (p. 161) B = bC-\-cD-\-dE . , C = cD-\-dE . ... ,
(3) (2) (2)
C = cD-\-dE . ... , and so on.
444. In 1796 and 1800 Hindenburg published collections of papers on combinatorial analysis. The first collection1 opens with an article by J. N. Tetens of Copenhagen who lets \n\ stand for the coefficient of the nth term of a polynomial, also lets the coefficient of the nth term ( terminus generalis) in the expansion of ( a-\-bx-\-cx2Jt- + \n\xn~l-\- -\-)m be indicated by T(a-\-bx-\- -\-\n\xn~l)m or simply by T(a- fi +|n|)w. In a footnote Hindenburg compares the symbols of Tetens with symbols of his own; thus, T(a+ -\-\n\)m = pm')£n, etc., where in Hindenburg’s notation of that time, pmJ^n is the nth coeffi¬ cient of the power pm.
In 1800 Hindenburg2 refers to Heinrich Burmann who advanced a purely combinatorial notation in his Developpment des fonctions com- binatoires. Hindenburg states that Burmann ’s short and expressive signs cannot be explained at this time, but states that the simplest monogrammata of Burmann from which the others are formed by additions and changes are r~i for series, L for combinations, 3 for discerption. In 1803 Hindenburg and Burmann brought out a joint publication, Ueber combinatorishe Analysis, and in 1807 Burmann pub¬ lished at Mannheim a Pangraphie , or system of universal notation. We have not seen these two publications. As we point out elsewhere (§ 428), J. W. F. ITerschel adopted a few of Burmann’s symbols.
445. Kramp on combinatorial notations. — Kramp3 of Cologne in 1808 expressed himself on matters of notation as follows: “If one designates by Q any polynomial, ordered according to the power of the variable x, such that axnJrbxn+rJrcxn+2r-ir, etc., the notation Qia,
CCfc 3, etc., will be very convenient for designating the coefficients of that polynomial, namely, a, b, c, etc. The coefficient Q ^ m is accord¬ ingly the one preceding the power of x whose exponent is n-\-(m— l)r.
1 Sammlung combinatorisch-analytischer Abhancllungen, herausgeg. von Carl Friedrich Hindenburg, Erste Sammlung (Leipzig, 1796), p. 6, 7.
2 Op. cit., Zweyte Sammlung (1800), p. xiii.
3 Christian Kramp, Elemens d’ arithmetique universelle (Cologne, 1808), “Notations.”
THEORY OF COMBINATIONS
67
The polynomial Q is therefore Q\lxn+Q\2.xn+r+Q'\fi>-xn+2r-\- , etc. "The power of the polynomial Qh, according to this same notation, becomes Q/iv5fcl .xnh+Qh'Jfc2-xnh+r+Qh^3-xnh+2r+, etc. And if by S one understands any other polynomial, ordered according to the powers of that same variable, such that N'fcl • .r/+£'fc2.£z+r+&'fc3 -xl+2r+ , etc., the product of these two polynomials QhS will be identical with (QAN)t-^+i+(Q^)1<2x^+z+-+(Q^)t3^n^+z+2r+, etc.
“Professor Hindenburg appears to be the first mathematician who has felt the indispensable need of this local notation in the present state of Analysis ; it is moreover generally adopted today by the mathe¬ maticians of his nation. Above all the polynomial coefficients 1, Qh\ 2, Qh ic3, etc. recur without ceasing in all that vast part of Analysis which has for its object the development in series of any function whether explicit or implicit .
“In the eighteenth chapter and in most of those which follow, I have conveniently employed the German letters a, b, c, b, etc., to denote the binomial factors of the power to the exponent n. One
Tl\Tl — 1 )
has therefore, a = n; b = — and so on.
I • A
“I employ the Latin letter D, placed before the polynomial func¬ tion Q, and I have named first derivative ( premiere derivee), what results from that function when one multiplies all its terms by their respective exponents and one then divides them by x. In the case that Q = a-\-bxfi-cx2-{-dx3Jre tc. one will therefore have, DQ = b-\-2cxJr 3dx2+4cx+3etc. One has in the same way for a second derivative (seconde derivee ), D2Q = 2c + Qdx +12 ex2 +20/Y3+ etc .
“The capital German letters are in general signs of functions. I employ particularly to this end the letters g and Thus g(x) designates any function whatever of the variable; ^(V) designates another.
“The derivatives ( derivees ) of Arbogast designate therefore the simple coefficients of the series; mine are veritable polynomial func¬ tions, developed in series, of which only the first terms are the deriva¬ tives of Arbogast. The notation of these last is identical with the local notation ^(a+x)'^ 2, S(a+a:)^3 etc. for the first
derivatives; and for the second, with (gX)^l, (gX)^2, (gX)^3 etc. always in the case that the exponents of the powers of x in the function X are the terms of the progression of the natural numbers 0, 1, 2, 3, etc .
“My researches on the calculus of derivatives go back to the year 1795. They appear for the first time in the work published in 1796
68
A HISTORY OF MATHEMATICAL NOTATIONS
by Professor Hindenburg, under the title Der polynomische Lehrsatz, where this mathematician did me the honor of joining my essay on combinatorial analysis with his and also with those of Messrs. Tetens; Pfaff and Kliigel .
“For the designation of the product whose factors form among themselves an arithmetical progression, such as a(a+r)(a+2r) . . . . (< a-\-nr—r ), I have retained the notation anlr already proposed in my analyse des refractions ; I have given it the nam efacultes. Arbogast substituted for it the choicer and more French designation of fac- torielles ”
Kliigel1 uses Kramp’s notation aw>r for a(a+r) .... (a+mr-r) and calls it a Facultat.
446. Signs of Argand and Ampere. — Independently of German writers, a few symbols were introduced by French writers. J. R. Argand,2 in treating a “Probleme de combinaisons,” designates by (m, n) the “ensemble de toutes les manieres de faire avec m choses n parts ... et par Z(m, n) le nombre de ces manieresV A few years
1 2 1 3 2
later A. M. Ampere3 lets [x] = x, [x\ = [x\(x-\-p) , [x\ = [x\(x-\-2p) . . . . ,
m+l m
[ x ] = [x](p+mp). This notation resembles that of Vandermonde; Ampere refers to the work of Vandermonde and Kramp. Ampere’s notation is used by Lentheric4 of Montpellier.
Crelle5 adopted the signs when m is an integer: (u, -\~x)m =
u(u-\-x)(u-\- 2x)(u-\-3x) .... (u+lm— l]x), ( u,-\-x)~y = Schellbach6 added to these symbols the following:
1
(• u — yx , ~{-x)y'
ao,-\-k a§ a ^ a 2k a%i$ .... ank — & , fn(x,+y)=f(,x) f(x+y) f(x+2y) . . • .f(x+ny-y),
(1 nn, + l) (1 af) (1 af) (1 af) .... (1 — an-f) ‘
Schellbach gives the symbolism nlas to mark the occurrence of n
♦
quantities a0, ah a2, , an- 1, where 5 takes successively the values
1 G. S. Kliigel, M athematisches Worterbuch , 1. Theil (Leipzig, 1803), art. ' ‘Facultat. ’ ’
2 J. R. Argand in J. D. Gergonne’s Annates de mathematiques pares et ap~ pliquees (Nismes), Vol. VI (1815 and 1816), p. 21.
2 Op. cit ., Vol. XV (1824-25), p. 370.
4 Op. cit., Vol. XVI (1825-26), p. 120.
5 A. L. Crelle, in Crelle’ s Journal, Vol. VII (1831), p. 270, 271.
6 Karl Heinrich Schellbach, Crelle’ s Journal, Vol. XII (1834), p. 74, 75. Schell¬ bach gave a discussion of mathematical notation in this article.
THEORY OF COMBINATIONS
69
0, 1, , n— 1. Accordingly, f(n\xs) stands for a function of x0,
Xi, , xn~i. He writes ( m , n \ a&) = the combinations, without repe¬ tition, of the elements a0 , ah .... , an-h taken m at a time. And [m,n\f aa] = the combinations with repetitions. He lets also n\aa = a0-{-ai+. . . . -fan-i, where a takes successively the values 0, 1, .... , n— 1. Also1
b — c r, -f 1\ 3
(fl-f 5) 3 — 4 j [3 — <j J 1 -f o' ! a-f 1 -f 5] [c t 4 — <j ! <5 — a]
1, +1 / '
and
similarly for (a-f 6)n, where the arrows indicate (p. 154) the direction of the combination.
447. Thomas Jarrett. — An extensive study of algebraic notations was made by Thomas Jarrett (1805-82), of Catharine Hall, at Cam¬ bridge in England. He2 published an article in 1830, but gave a much fuller treatment of this subject in an essay3 of 1831. He remarks that the demonstration of the legitimacy of the separation of the symbols of operation and quantity, with certain limitations, belongs to Servois. Jarrett refers to Arbogast, J. F. W. Herschel, Hindenburg, Lacroix, Laplace, Schweins, and Wronski. Jarrett points out that the following notations used by him are not original with him: First,
Ex(f){x) for </>fc-F Dx) is partly due to Arbogast; second, dr'u for - — is
dxn
due to Lacroix, although not used by him, being merely pointed out in a single line ( Calcul Diff., Vol. II, p. 527); third, ( u)x=a for the value assumed by u, when x is put equal to a, belongs to Schweins.
The principal symbols introduced by Jarrett are as follows:4
n
Smam, the sum of n terms, of which the rath is am (p. 1).
n; r
Sm dmj the rth term must be omitted.
r s s
S S am, n, the sum of r terms of which the rath is S am , n (p. 5).
m n m
m, n
Sr, +s(sar), the sum of every term that can be formed with the fol¬ lowing conditions: each term in the product of ra quantities in which r has the values of the successive
1 Schellbach, ov ■ cit., p. 154, 156.
2 Thomas Jarret in Transactions of the Cambridge Philosophical Society, Vol. Ill (Cambridge, 1830), p. 67.
3 An Essay on Algebraic Development containing the Principal Expansions in Common Algebra, in the Differential and Integral Calculus and in the Calculus of finite Differences. ... By the Rev. Thomas Jarrett, M.A., Fellow of Catharine Hall, and Professor of Arabic in the University of Cambridge (1831).
4 Thomas Jarrett, op. cit. (1831), “Index to the Symbols.”
70
A HISTORY OF MATHEMATICAL NOTATIONS
CO
nyr
nr
n
P a
J rrv^m j n, r
P a
natural numbers, while s has any m values such that their sum shall be n, zero being admissible as a value of s, and repetitions of the same value of that letter being allowed in the same term (p. 78).
cO
coefficient of xm in the development of (p. 79).
the sum of the series formed by giving to m every inte¬ gral value from n to r both inclusive; zero being also taken as a value if n is either zero or negative (p. 136).
the product of n factors, of which the rath is am (p. 12). the rth factor must be omitted.
a
= a(a+ra)(a+2m) .... (a+n — l*ra) (p. 15).
n, m
|a = a(a—l)(a — 2) .... (a — n+1).
n
a = a(a—l){a — 2) .... 2*1.
n
rn
{ am-\-bn {....{ c }....} denotes the result of the combination
m + l n + 1 n + 1
of the symbols {ai+&i{a2+52 . • • • {an+6n{ c }••••};
12 n n + 1 n + 1 1
the brackets being omitted after the expansion, if they are then without signification (p. 19).
n n m — 1 m
Theorem. {am-\-hm {....{ c = Smam>Pr 5r+c*F hr .
7n + l n+1 r
m
m, n P f CL ry
m, n; s Cr CL.
T)
the sum of every possible combination (without repe¬ tition of any letter in the same combination) that can be formed by taking ra at a time of n quantities of which the rth is ar.
as is to be everywhere omitted.
rn, n — m
Cr,s(cLr-bs), n quantities of which the rth is ar, and n others of which the sth is bs, every possible combination being formed of the first series, by taking them ra at a time, each com¬ bination thus formed being multiplied by n—m quanti¬ ties of the second series, so taken that in each of the com¬ binations the whole of the natural numbers from 1 to n shall appear as indices: thus (p. 22).
THEORY OF COMBINATIONS
71
273
Cr> «(<2r *frs) = CllCl2&3&4?>5_hfllfl3^2^4&5 4-<2lfl4^2^3^5_l_C[l^5^2^3^4T "
G2^3^1^4^5-)- fl2^4^1^3^5 4“ Ct2^5^1^3^4 4“ G3^4^1^2^5 4~
azaibibzbi+aiasbibibz .
(</>4-i/')nw means {(04-^)(</>4"W . . . . (04-^0 }w, (w round paren¬ theses) (p. 41).
n
{(& + &){ U means {(0l4-^l)(02 + ^2) .... (tn+'l'n)}'"' •
r r-fl
(■ u)x=a,<t>x=aM , denote respectively the values of u, and 4>x(u)t when x is put equal to a; this substitution, in the lat¬ ter case, not being made until after the operation indi¬ cated by </>x has been performed (p. 45).
Ex^u) means that in u, any function of x, x-\ -h is substituted for x.
Dx(u) means the excess of the new value of u above the original value.
EXt j/m) expresses either Ex*Ey{u) or Ey»Ex(u) (p. 58).
2m _ ^
fam-i = A j, where the right member is called the (2m — l)th
1
1
number of Bernoulli; £i = 2g=,083 (p. 89).
A turn through 180° of Jarrett’s sign for n-factorial yields "T, a symbol introduced by Milne1 in the treatment of annuities. He lets a stand for the expectation of life by an individual A, ta the probability of his surviving t years, and — the expectation of life after the ex¬ piration of t years. This sign was used similarly by Jones2 who lets a(m)— | n be the present value of £1 per annum, to be entered upon after n years, m being the present age.
448. Factorial “n.” — The frequency of the occurrence of 71- factorial in algebra and general analysis gives this expression sufficient importance to justify a separate treatment, even at the risk of some slight repetition of statements. In 1751 Euler3 represented the prod¬ uct 1.2.3 . m by the capital letter M. “Ce nombre de cas 1.2. 3.4.
. ... m etant pose pour abreger = M. ...” Probably this was not in¬ tended as a general representation of such products, but was intro¬ duced simply as a temporary expedient. The very special relation
1 Joshua Milne, Annuities and Assurances , Vol. I (London, 1815), p. 57, 58.
2 David Jones, Value of Annuities , Vol. I (London, 1843), p. 209.
3 L. Euler, “Calcul de la Probability dans le jeu de Recontre,” IUstoire de Vacademie r. d. sciences et des belles lettres de Berlin, annee 1751 (Berlin, 1753), p. 259, 265.
72
A HISTORY OF MATHEMATICAL NOTATIONS
m to M would go against the use of M as the product of, for example,
1.2.3 . r, or of 1.2. 3.4. 5. 6. A little-known suggestion came in
1774 from J. B. Basedow1 who used a star, thus 5* = 5.4. 3. 2.1. Other abbreviations were used in 1772 by A. T. Vandermonde: “Je repre-
n
sente par [p] le produit de n facteurs ... p- (p— 1) • (p — 2) • (p — 3) ... ou le produit de n termes consecutifs d’une suite dont les premieres differences sont 1, et les secondes differences sont zero.”2 He writes
n o —n
[p\ = P'(p~ 1) •(p — 2) •(? — 3) .... (p — n- fl); he finds [p] — 1, [p\
4
_ 1 _
(p+l)*(p+2).(p-f3).(p+4) .... (p+n)
= 11.9.7.5.
7
® 1.2. 3.4.5. ’
2. 2.4.4. 6. 6. . . .
1.3. 3. 5. 5. 7.
The special case
when p = n = a positive integer would yield the product n(n— 1) . . . . 3.2.1., but Vandermonde was operating with expressions in form more general than this (see also § 441).
A sign for n-factorial arises as a special case of a more general notation also in Christian Kramp of Strasbourg who in his Elemens d’ arithmetique universelle (1808) and in special articles3 lets am|r stand for a(a+r)(a+2r) .... [a+(m— l)r], and uses the special forms
a
m\0 = am a-m
■r —
(a+r)
mil —
m r ’
1.2.3
m ou a cette autre
forme plus simple ml” In 1808 Kramp said: aJe me sers de la nota¬ tion tres simple n\ pour designer le produit de nombres decroissans depuis n jusqu’a Funite, savoir n{n— l)(n — 2) .... 3.2.1. L’emploi continuel de Y analyse combinatoire que je fais dans la plupart de mes demonstrations, a rendu cette notation indispensable.”4 In a footnote to Kramp’s article, the editor, J. S. Gergonne, compares the notations of Vandermonde and Kramp. “Vandermonde fait a* (a— 1) . (a — 2) . . . . (a — m+1) = [a]w, d’ou il suit qu’en rapprochant les deux nota¬ tions, ou a [a]m= (a — m+l)w|1 = am|-1; . . . . , 1.2. 3. 4 .... m= lm'1
= m!” Kramp’s notation lw|1 found its way into Portugal where
Stockier5 used it in 1824.
1 Johann Bernhard Basedow, Bewiesene Grundsatze der reinen Mathematik, Vol. I (Leipzig, 1774), p. 259.
2 A. T. Vandermonde, Histoire de Vacademie r. d. sciences, annee 1772, Part I (Paris, 1775), Mem., p. 490, 491.
8 J. D. Gergonne, Annales de Mathematiques, Vol. Ill (1812 et 1813), p. 1.
4 C. Kramp, Elemens d’ arithmetique universelle (Cologne, 1808), “Notations.” See also p. 219.
6 Francisco de Borja Gar^ao Stockier, Methodo inverso dos Limites (Lisbon, 1824), p. 35.
THEORY OF COMBINATIONS
73
A new designation for n-factorial was introduced by Legendre. In 1808 he