Open Court Classics of Science and Philosophy,

iOMETRICAL LECTURES

OP

ISAAC BARROW

J. M, CHILD

CO

GEOMETRICAL LECTURES

ISAAC BARROW

Court Series of Classics of Science and

Thilo sophy, 3\(o. 3

THE GEOMETRICAL LECTURES

OF

ISAAC BARROW

. HI

TRANSLATED, WITH NOTES AND PROOFS, AND A

DISCUSSION ON THE ADVANCE MADE THEREIN

ON THE WORK OF HIS PREDECESSORS IN THE

INFINITESIMAL CALCULUS

BY

B.A. (CANTAB.), B.Sc. (LOND.)

546664

CHICAGO AND LONDON. THE OPEN COURT PUBLISHING COMPANY

1916

Copyright in Great Britain under the Ad of 191 1

33

LECTI ONES

Geometric^;

In quibus (praffertim)

GEHERALJA Cttr varum L iacaruw STMPTOUATA D E . C L A 7^ A T^ T II !(,

Audore IsAACoBARRow Collegii

S S. Trhiitatn in Acad. Cant ah. SociO, & Socictatif lie- ti<£ Sodale.

Oi <fvftt Aoj/r/ito; H< iruiv]<t TO. nxSHfufla, at tr& wVi^C) o%*( r«>riu- OITTI GfetJfft , a* rttrv -mufAttli it, yu pi <iow mt , fi>1/Si'» *'Ao &.'««> n$uw, OH 6)f «tr;t TO ^urt^i «oT»i a'uT&i/yi'jrf **ir. Plato de Repub. VII.

L ON D IN I,

Typis Cttlielmi Godkd , & proftant venales apud

J«h-tnitf>t Tt-wmsre, & OR.ivi.innm Pullejn Juniorcm.

UW. D C L X X.

Note the absence of the usual words " Habitae Cantabrigioe," which on the title-pages of his other works indicate that the latter were delivered as Lucasian Lectures. J. M. C.

PREFACE

ISAAC BARROW was Ike first inventor of the Infinitesimal Calculus ; Newton got the main idea of it from Barrow by personal communication ; and Leibniz also was in some measure indebted to Barrow's work, obtaining confirmation of his oivn original ideas, and suggestions for their further development, from the copy of JJ arrow's book (hat he purchased in 1673.

The above is the ultimate conclusion that I have arrived at, as the result of six months' close study of a single book, my first essay in historical research. By the "Infinitesimal Calculus," I intend "a complete set of standard forms for both the differential and integral sections of the subject, together with rules for their combination, such as for a product, a quotient, or a power of a function ; and also a recognition and demonstration of the fact that differentiation and integration are inverse operations."

The case of Newton is to my mind clear enough. Barrow was familiar with the paraboliforms, and tangents and areas connected with them, in from 1655 to 1660 at the very latest; hence he could at this time differentiate and inte- grate by his own method any rational positive power of a variable, and thus also a sum of such powers. He further developed it in the years 1662-3-4, and in the latter year probably had it fairly complete. In this year he com- municated to Newton the great secret of his geometrical constructions, as far as it is humanly possible to judge from a collection of tiny scraps of circumstantial evidence ; and it was probably this that set Newton to work on an attempt to express everything as a sum of powers of the variable. During the next year Newton began to "reflect on his method of fluxions," and actually did produce his Analysis per sfLquationes. This, though composed in 1666, was not published until 1711.

viii BARROW'S GEOMETRICAL

The case of Leibniz wants more argument that I am in a position at present to give, nor is this the place to give it. 1 hope to be able to submit this in another place at some future time. The striking points to my mind are that Leibniz bought a copy of Barrow's work in 1673, anc^ was a^^e "to communicate a candid account of his calculus to Newton " in 1677. In this connection, in the face of Leibniz' per- sistent denial that he received any assistance whatever from Barrow's book, we must bear well in mind Leibniz' twofold idea of the " calculus " :

(i) the freeing of the matter from geometry, (ii) the adoption of a convenient notation. Hence, be his denial a mere quibble or a candid statement without any thought of the idea of what the "calculus" really is, it is perfectly certain that on these two points at any rate he derived not the slightest assistance from Barrow's work ; for the first of them would be dead against Barrow's practice and instinct, and of the second Barrow had no knowledge whatever. These points have made the calculus the powerful instrument that it is, and for this the world has to thank Leibniz; but their inception does not mean the invention of the infinitesimal calculus. This, the epitome of the work of his predecessors, and its completion by his own discoveries until it formed a perfected method of dealing with the problems of tangents and areas for any curve in general, i.e. in modern phraseology, the differentiation and integration of any function whatever (such as were known in Barrow's time), must be ascribed to Barrow.

Lest the matter that follows may be considered rambling, and marred by repetitions and other defects, I give first some account of the circumstances that gave rise to this volume. First of all, I was asked by Mr P. E. B. Jourdain to write a short account of Barrow for the Monist ; the request being accompanied by a first edition copy of Barrow's Lectiones Opticce. et Geometries. At this time, I do not mind confessing, my only knowledge of Barrow's claim to fame was that he had been "Newton's tutor": a notoriety as unenviable as being known as " Mrs So-and-So's husband." For this article I read, as if for a review, the book that had been sent to me. My attention was arrested

PREFACE ix

by a theorem in which Barrow had rectified the cycloid, which I happened to know has usually been ascribed to Sir C. Wren. My interest thus aroused impelled me to make a laborious (for I am no classical scholar) translation of the whole of the geometrical lectures, to see what else I could find. The conclusions I arrived at were sent to the Monist for publica- tion ; but those who will read the article and this volume will find that in the article I had by no means reached the stage represented by this volume. Later, as I began to still further appreciate what these lectures really meant, I con- ceived the idea of publishing a full translation of the lectures together with a summary of the work of Barrow's more immediate predecessors, written in the same way from a personal translation of the originals, or at least of all those that I could obtain. On applying to the University Press, Cambridge, through my friend, the Rev. J. B. Lock, I was referred by Professor Hobson to the recent work of Professor Zeuthen. On communicating with Mr Jourdain, I was invited to elaborate my article for the Monist into a 2oo-page volume for the Open Court Series of Classics.

I can lay no claim to any great perspicacity in this dis- covery of mine, if I may call it so ; all that follows is due rather to the lack of it, and to the lucky accident that made me (when I could not follow the demonstration) turn one of Barrow's theorems into algebraical geometry. What I found induced me to treat a number of the theorems in the same way. As a result I came to the conclusion that Barrow had got the calculus; but I queried even then whether Barrow himself recognized the fact. Only on com- pleting my annotation of the last chapter of this volume, Lect. XII, App. Ill, did I come to the conclusion that is given as the opening sentence of this Preface ; for I then found that a batch of theorems (which I had on first reading noted as very interesting, but not of much service), on careful revision, turned out to be the few missing standard forms, necessary for completing the set for integration ; and that one of his problems was a practical rule for finding the area under any curve, such as would not yield to the theoretical rules he had given, under the guise of an "inverse-tangent" problem.

The reader will then understand that the conclusion is

x HARROW'S GEOMETRICAL LECTURES

the effect of a gradual accumulation of evidence (much a^ a detective picks up clues) on a mind previously blank as regards this matter, and therefore perfectly unbiased. This he will see reflected in the gradual transformation from tentative and imaginative suggestions in the Introduction to direct statements in the notes, which are inset in the text of the latter part of the translation. I have purposely refrained from altering the Introduction, which preserves the form of my article in the Monist, to accord with my final ideas, because I feel that with the gradual developrhent thus indicated I shall have a greater chance of carrying my readers with me to my own ultimate conclusion.

The order of writing has been (after the first full trans- lation had been made) :— Introduction, Sections I to VIII, excepting III; then the text with notes; then Sections III and IX of the Introduction ; and lastly some slight altera- tions in the whole and Section X.

In Section I, I have given a wholly inadequate account of the work of Barrow's immediate predecessors ; but I felt that this could be enlarged at any reader's pleasure, by reference to the standard historical authorities ; and that it was hardly any of my business, so long as I slightly expanded my Monist article to a sufficiency for the purpose of showing that the time was now ripe for the work of Barrow, Newton, and Leibniz. This, and the next section, have both been taken from the pages of the Encyclopedia Britannica ( Times edition).

The remainder of my argument simply expresses my own, as I have said, gradually formed opinion. I have purposely refrained from consulting any authorities other than the work cited above, the Bibliotheca Britannica (for the dates in Section III), and the Dictionary of National Biography (for Canon Overton's life of Barrow) ; but I must acknowledge the service rendered me by the dates and notes in Sotheran's Price Current of Literature. The translation too is entirely my own without any help from the translation by Stone or other assistance from a first edition of Barrow's work dated 1670.

As regards the text, with my translation beside me, I have to all intents rewritten Barrow's book; although throughout I have adhered fairly closely to Barrow's own

PREFACE xi

words. I have only retained those parts which seemed to me to be absolutely essential for the purpose in hand. Thus the reader will find the first few chapters very much abbreviated, not only in the matter of abridgment, but also in respect of proofs omitted, explanations cut down, and figures left out, whenever this was possible without breaking the continuity. This was necessary in order that room might be found for the critical notes on the theorems, the inclusion of proofs omitted by Barrow, which when given in Barrow's style, and afterwards translated into analysis, had an important bearing on the point as to how he found out the more difficult of his constructions ; and lastly for deductions therefrom that point steadily, one after the other, to the fact that Barrow was writing a calculus and knew that he was inventing a great thing. I can make no claim to any classical attainments, but I hope the transla- tion will be found correct in almost every particular. In the wording I have adhered to the order in which the original runs, because thereby the old-time flavour is not lost ; the most I have done is to alter a passage from the active to the passive or vice versa, and occasionally to change the punctuation.

I have used three distinct kinds of type : the most widely spaced type has been used for Barrow's own words ; only very occasionally have I inserted anything of my own in this, and then it will be found enclosed in heavy square brackets, that the reader will have no chance of confusing my explanations with the text ; the whole of the Introduc- tion, including Barrow's Prefaces, is in the closer type; this type is also used for my critical notes, which are generally given at the end of a lecture, but also sometimes occur at the end of other natural divisions of the work, when it was thought inadvisable to put off the explanation until the end of the lecture. It must be borne in mind that Barrow makes use of parentheses very frequently, so that the reader must understand that only remarks in heavy square brackets are mine, those in ordinary round brackets are Barrow's. The small type is used for footnotes only. In the notes I have not hesitated to use the Leibniz notation, because it will probably convey my meaning better ; but there was really no absolute necessity for this,

xii BARROW'S GEOMETRICAL LECTURES

Barrow's a and e, or its modern equivalent, /* and /&, would have done quite as well.

I cannot close this Preface without an acknowledgment of my great indebtedness to Mr Jourdain for frequent advice and help; I have had an unlimited call on his wide reading and great historical knowledge ; in fact, as Barrow says of Collins, I am hardly doing him justice in calling him my " Mersenne." All the same, I accept full responsibility for any opinions that may seem to be heretical or otherwise out of order. My thanks are also due to Mr Abbott, of Jesus College, Cambridge, for his kind assistance in looking up references that were inaccessible to me.

J. M. CHILD. DKKIIY, ENGLAND, Xmas, 1915.

P.S. Since this volume has been ready for press, I have consulted several authorities, and, through the kindness of Mr Walter Stott, I have had the opportunity of reading Stone's translation. The result I have set in an appendix at the end of the book. The reader will also find there a solution, by Barrow's methods, of a test question suggested by Mr Jourdain ; after examining this I doubt whether any reader will have room for doubt concerning the correctness of my main conclusion. I have also given two specimen pages of Barrow's text and a specimen of his folding plates of diagrams. Also, I have given an example of Barrow's graphical integration of a function ; for this I have chosen a function which he could not have integrated theoretically, namely, i/«/(i -^4), between the limits o and x ; when the upper limit has its maximum value, T, it is well known that the integral can be expressed in Gamma functions; this was used as a check on the accuracy of the method.

J. M. C.

TABLE OF CONTENTS

PAGE

INTRODUCTION

The work of Barrow's great predecessors .... I

Life of Barrow, and his connection with Newton . 6

The works of Barrow ...... .8

Estimate of Barrow's genius ... 9

The sources of Barrow's ideas .... . 12-

Mutual influence of Newton and Barrow .... i6-

Description of the book from which the translation has been

made . 20

The prefaces. ....... .25

How Barrow made his constructions ... . 28"

Analytical equivalents of Barrow's chief theorems ... 30

TRANSLATION—

LECTURE I. Generation of magnitudes. Modesof motion and the quantity of the motive force. Time as the independent variable. Time, as an aggregation of instants, compared with a line, as an aggregation of points 35

LECTURE II. Generation of magnitudes by "local move- ments." The simple motions of translation and rotation . 42

LECTURE III. Composite and concurrent motions. Com- position of rectilinear and parallel motions . . 47

LECTURE IV. Properties of curves arising from composition of motions. The gradient of the tangent. Generalization of a problem of Galileo. Case of infinite velocity . . 53

LECTURE V. Further properties of curves. Tangents. Curves

like the cycloid. Normals. Maximum and minimum lines 60

LECTURE VI. Lemmas ; determination of certain curves con- structed according to given conditions ; mostly hyperbolas . 69

LECTURE VII. Similar or analogous curves. Exponents or Indices. Arithmetical and Geometrical Progressions. Theorem analogous to the approximation to the Binomial Theorem for a Fractional Index. Asymptotes , . , 77

xiv BARROW'S GEOMETRICAL LECTURES

LECTURE VIII. Construction oftangentsby means of auxiliary curves of which the tangents are known. Differentiation of a sum or a difference. Analytical equivalents . . 90

LECTURE IX. —Tangents to curves formed by arithmetical and geometrical means. Paraboliforms. Curves of hyperbolic and elliptic form. Differentiation of a fractional power, products and quotients . . . . . . . 101

LECTURE X. Rigorous determination of ds/rfjc. Differentia- tion as the inverse of integration. Explanation of the "Differential Triangle" method; with examples. Differ- entiation of a trigonometrical function . . . .113

LECTURE XI. Change of the independent variable in inte- gration. Integration the inverse of differentiation. Differ- entiation of a quotient. Area and centre of gravity of a paraboliform. Limits for the arc of a circle and a hyperbola. Estimation of IT . . . . . . . . 125

LECTURE XII. General theorems on rectification. Standard forms for integration of circular functions by reduction to the quadrature of the hyperbola. Method of circumscribed and inscribed figures. Measurement of conical surfaces. Quadrature of the hyperbola. Differentiation and Integra- tion of a Logarithm and an Exponential. Further standard forms .1 55

LECTURE XIII. These theorems have not been inserted . 196

POSTSCRIPT

Extracts from Standard Authorities . . . . .198

APPENDIX

I. Solution of a test question by Barrow's method . . 207

II Graphical integration by Barrow's method . . . 211

III. Reduced facsimiles of Barrow's pages and figures . . 212

INDEX , ..... 216

INTRODUCTION

THE WORK OF BARROW'S GREAT PREDECESSORS

THE beginnings of the Infinitesimal Calculus, in its two main divisions, arose from determinations of areas and volumes, and the finding of tangents to plane curves. The ancients attacked the problems in a strictly geometrical manner, making use of the " method of exhaustions." In modern phraseology, they found "upper and lower limits," as closely equal as possible, between which the quantity to be determined must lie ; or, more strictly perhaps, they showed that, if the quantity could be approached from two " sides," on the one side it was always greater than a certain thing, and on the other it was always less ; hence it must be finally equal to this thing. This was the method by means of which Archimedes proved most of his discoveries. But there seems to have been some distrust of the method, for we find' in many cases that the discoveries are proved by a reductio ad absurdum, such as one is familiar with in Euclid. To Apollonius we are indebted for a great many of the pro- perties, and to Archimedes for the measurement, of the conic sections and the solids formed from them by their rotation about an axis.

The first great advance, after the ancients, came in the beginning of the seventeenth century. Galileo (1564-1642) would appear to have led the way, by the introduction of the theory of composition of motions into mechanics ; * he also was one of the first to use infinitesimals in geometry, and from the fact that he uses what is equivalent to "virtual velocities " it is to be inferred that the idea of time as the independent variable is due to him. Kepler (1571-1630) was the first to introduce the idea of infinity into geometry

* See Mach's Science of Mechanics for fuller details.

I

2 BARROW'S GEOMETRICAL LECTURES

and to note that the increment of a variable was evanescent for values of the variable in the immediate neighbourhood of a maximum or minimum ; in 1613, an abundant vintage drew his attention to the defective methods in use for estimating the cubical contents of vessels, and his essay on the subject (Nova Stereometria Dolioruni) entitles him to rank amongst those who made the discovery of the in- finitesimal calculus possible. In 1635, Cavalieri published a theory of "indivisibles," in which he considered a line as made up of an infinite number of points, a superficies as composed of a succession of lines, and a solid as a succession of superficies; thus laying the foundation for the "aggre- gations " of Barrow. Roberval seems to have been the first, or at the least an independent, inventor of the method ; but he lost credit for it, because he did not publish it, preferring to keep the method to himself for his own use ; this seems to have been quite a usual thing amongst learned men of that time, due perhaps to a certain professional jealousy. The method was severely criticized by contemporaries, especially by Guldin, but Pascal (1623-1662) showed that the method of indivisibles was as rigorous as the method of exhaustions, in fact that they were practically identical. In all probability the progress of mathematical thought is much indebted to this defence by Pascal. Since this method is exactly analogous to the ordinary method of integration, Cavalieri and Roberval have more than a little claim to be regarded as the inventors of at least the one branch of the calculus ; if it were not for the fact that they only applied it to special cases, and seem to have been unable to generalize it owing to cumbrous algebraical notation, or to have failed to perceive the inner meaning of the method when concealed under a geometrical form. Pascal himself applied the method with great success, but also to special cases only ; such as his work on the cycloid. The next step was of a more analytical nature; by the method of indivisibles, Wallis (1616-1703) reduced the determination of many areas and volumes to the calculation of the value of the series (o'"+ im+ 2™+ . . . «"')/(« + i)nm, i.e. the ratio of the mean of all the terms to the last term, for integral values of n ; and later he extended his method, by a theory of interpola- tion, to fractional values of n. Thus the idea of the Integral

INTRODUCTION 3

Calculus was in a fairly advanced stage in the days immedi- ately antecedent to Barrow.

What Cavalieri and Roberval did for the integral calculus, ^ Descartes (1596-1650) accomplished for the differential branch by his work on the application of algebra to geometry. Cartesian coordinates made possible the extension of in- vestigations on the drawing of tangents to special curves to the more general problem for curves of any kind. To this must be added the fact that he habitually^ used the index notation ; for this had a very great deal to do with the possibility of Newton's discovery of the general binomial expansion and of many other infinite series. Descartes failed, however, to make any very great progress on his own account in the matter of the drawing of tangents, owing to what I cannot help but call an unfortunate choice of a definition for a tangent to a curve in general. Euclid's circle-tangent definition being more or less hopeless in the general case, Descartes had the choice of three :

(1) a secant, of which the points of intersection with

the curve became coincident ;

(2) a prolongation of an element of the curve, which

was to be considered as composed of an infinite succession of infinitesimal straight lines ;

(3) the direction of the resultant motion, by which the

curve might have been described.

Descartes chose the first ; I have called this choice unfor- tunate, because I cannot see that it would have been possible for a Descartes to miss the differential triangle, and all its consequences, if he had chosen the second definition. His choice leads him to the following method of drawing; a tangent to a_curve in general. _ Describe a circle, whose centre is on the axis of x, to cut the curve ; keeping the centre fixed, diminish the radius until the points of section coincide ; thus, by the aid of the equation of the curve, the problem is reduced to finding the condition for equal roots of an equation.

For instance, \etyz = 4ax be the equation to a parabola, and (x - p)* +yz = r& the equation of the circle. Then we have (x - pf + 4ax r*. If this is a perfect square, x=p - 20, ; i.e. the subtangent is equal to 2a.

4 BARROW'S GEOMETRICAL LECTURES

The method, however, is only applicable to a small number of simple cases, owing to algebraical difficulties. In the face of this disability, it is hard to conjecture why Descartes did not make another choice of definition and use the second one given above ; for in his rule for the tangents to roulettes, he considers a curve as the ultimate form of a polygon. The third definition, if not originally due to Galileo, was a direct consequence of his conception of the composition of motions ; this definition was used by Roberval (1602-1675) and applied successfully to a dozen or so of the well-known curves; in it we have the germ of the method of "fluxions." Thus it is seen that Roberval occupies an almost unique position, in that he took a great part in the work preparatory to the invention Qiboth branches of the infinitesimal calculus ; a fact that seems to have escaped remark. Fermat (1590-1663) adopted Kepler's notion of the increment of the variable becoming evanes- cent near a maximum or minimum value, and upon it based his method of drawing tangents. Fermat's method of finding the maximum or minimum value of a function in- volved the differentiation of any explicit algebraic function, in the form that appears in any beginner's text-book of to- day (though Fermat does not seem to have the "function " idea) ; that is, the maximum or minimum values of f(x) are the roots of f'(x) = o, -where f(x) is the limiting value of [f(x + K) -f(x)\lh ; only Fermat uses the letter e or E instead of h. Now, if YYY is any curve, wholly con- cave (or convex) to a straight line AD, TZYZ a tangent to it at the point Y whose ordinate is NY, and the tangent meets AD in T; also, if ordinates NYZ are drawn on either side of NY, cutting the curve in Y and the tangent in Z; then it is plain that the ratio YN : NT is a maximum (or a mini- mum) when Y is the point of contact of the tangent.

Here then we have all the essentials for the calculus ; but only for explicit integral algebraic functions, needing the binomial expansion of Newton, or a general method of rationalization which did not impose too great algebraic difficulties, for their further development; also, on the

INTRODUCTION 5

authority of Poisson, Fermat is placed out of court, in that he also only applied his method to certain special cases. Follow- ing the lead of Roberval, Newton subsequently used the third definition of a tangent, and the idea of time as the ' independent variable, although this was only to insure that one at least of his working variables should increase uni- formly. This uniform increase of the independent variable would seem to have been usual for mathematicians of the period and to have persisted for some time ; for later we find with Leibniz and the Bernoullis that d(dy\dx) = (cPyldx^dx. Barrow also used time as the independent variable in order that, like Newton, he might insure that one of his variables, a moving point or line or superficies, should proceed uni- formly; it is to be noted, however, that this is only in the lectures that were added as an afterthought to the strictly geometrical lectures, and that later this idea becomes altogether subsidiary. Barrow, however, chose his own definition of a tangent, the second of those given above ; < and to this choice is due in great measure his advance over his predecessors. For his areas and volumes he followed the idea of Cavalieri and Roberval.

Thus we see that in the time of Barrow, Newton, and Leibniz the ground had been surveyed, and in many direc- tions levelled; all the material was at hand, and it only wanted the master mind to " finish the job." This was f possible in two directions, by geometry or by analysis; each method wanted a master mind of a totally different type, and the men were forthcoming. For geometry, Barrow: for analysis, Newton, and Leibniz with his in- spiration in the matter of the application of the simple and convenient notation of his calculus of finite differences to infinitesimals and to geometry. With all due honour to these three mathematical giants, however, I venture to assert that their discoveries would have been well-nigh impossible to them if they had lived a hundred years earlier; with the possible exception of Barrow, who, being a geometer, was more dependent on the ancients and less on the moderns of his time than were the two analysts, they would have been sadly hampered but for the preliminary work of Descartes and the others I have mentioned (and some I have not— such as Oughtred), but especially Descartes.

6 BARROW'S GEOMETRICAL LECTURES

II

LIFE OF BARROW, AND HIS CONNECTION WITH NEWTON

Isaac Barrow was born in 1630, the son of a linen-draper in London. He was first sent to the Charterhouse School, where inattention and a predilection for fighting created a bad impression ; his father was overheard to say (pray, according to one account) that " if it pleased God to take one of his children, he could best spare Isaac." Later, he seems to have turned over a new leaf, and in 1643 we nnd him entered at St Peter's College, Cambridge, and afterwards at Trinity. Having now become exceedingly studious, he made considerable progress in literature, natural philosophy, anatomy, botany, and chemistry the three last with a view to medicine as a profession, and later in chronology, geometry, and astronomy. He then proceeded on a sort of " Grand Tour " through France, Italy, to Smyrna, Con- stantinople, back to Venice, and then home through Germany and Holland. His stay in Constantinople had a great influence on his after life ; for he here studied the works of Chrysostom, and thus had his thoughts turned to divinity. But for this his great advance on the work of his pre- decessors in the matter of the infinitesimal calculus might have been developed to such an extent that the name of Barrow would have been inscribed on the roll of fame as at least the equal of his mighty pupil Newton.

Immediately on his return to England he was ordained, and a year later, at the age of thirty, he was appointed to the Greek professorship at Cambridge ; his inaugural lectures were on the subject of the Rhetoric of Aristotle, and this choice had also a distinct effect on his later mathematical work. In 1662, two years later, he was appointed Professor of Geometry in Gresham College ; and in the following year he was elected to the Lucasian Chair of Mathematics, just founded at Cambridge. This professorship he held for five years, and his office created the occasion for his Lectiones Mathematics, which were delivered in the years 1664-5-6 (Habitce Cantabrigice). These lectures were published, according to Prof. Benjamin Williamson (Encyc. Brit.

INTRODUCTION 7

(Times edition), Art. on Infinitesimal Calculus) in 1670; this, however, is wrong : they were not published until 1683, under the title of Lectiones Mathematics. What was published in 1670 was the Lectiones Optics et Geometries; the Lectiones Mathematics were philosophical lectures on the fundamentals of mathematics and did not have much bearing on the infinitesimal calculus. They were followed by the Lectiones Optics and lectures on the works of Archimedes, Apollonius, and Theodosius; in what order these were delivered in the schools of the University I have been unable to find out ; but the former were published in 1669, "Imprimatur" having been granted in March 1668, so that it was probable that they were the professorial lectures for 1667 ; thus the latter would have been delivered in 1668, though they were not published until 1675, and then probably by Collins. The great work, Lectiones Geometries, did not appear as a separate publication at first : as stated above, it was issued bound up with the second edition of the Lecliones Optics; and, judging from the fact that there does not, according to the above dates, appear to have been any time for their public delivery as Lucasian Lectures, since Imprimatur was granted for the combined edition in 1669; also from the fact that Barrow's Preface speaks of six out of the thirteen lectures as " matters left over from the Optics," which he was induced to complete to form a separate work ; also from the most conclusive fact of all, that on the title-page of the Lectiones Geometries there is no mention at all of the usual notice " Habitse Cantabrigiae " ; judging from these facts, I do not believe that the '•''Lectiones Geometries" were delivered as Lucasian Lectures. Should this be so, it would clear up a good many difficulties ; it would corroborate my suggestion that they were for a great part evolved during his professorship at Gresham College ; also it would make it almost certain that they would have been given as internal college lectures, and that Newton would have heard them in 1663-1664.

Now, it was in 1664 that Barrow first came into close personal contact with Newton ; for in that year, he examined Newton in Euclid, as one of the subjects for a mathematical scholarship at Trinity College, of which Newton had been a subsizar for three years ; and it was due

to Barrow's report that Newton was led to study the Elements more carefully and to form a better estimate of their value. The connection once started must have developed at a great pace, for not only does Barrow secure the succession of Newton to the Lucasian chair, when he relinquished it in 1669, but he commits the publication of his Lectiones Optica to the foster care of Newton and Collins. He himself had now determined to devote the rest of his life to divinity entirely ; in 1670 he was created a Doctor of Divinity, in 1672 he succeeded Dr Pearson as Master of Trinity, in 1675 ne was chosen Vice-Chancellor of the University; and in 1677 he died, and was buried in Westminster Abbey, where a monument, surmounted by his bust, was soon afterwards erected to his memory by his friends and admirers.

Ill THE WORKS OF BARROW

Barrow was a very voluminous writer. On inquiring of the Librarian of the Cambridge University Library whether he could supply me with a complete list of the works of Barrow in order of publication, I was informed that the complete list occupied four columns in the British Museum Catalogue \ This of course would include his theological works, the several different editions, and the translations of his Latin works. The following list of his mathematical works, such as are important for the matter in hand only, is taken from the Bibliotheca Britannica (by Robert Watt, Edinburgh, 1824) :

1. Euclid's Elements, Camb. 1655.

2. Euclid's Data, Camb. 1657.

3. Lectiones Opticortim Phenomenon, Lond. 1669.

4. Lectiones Optic&et Geometricce, Lond. 1670

(in 2 vols., 1674'; trans, Edmond Stone, 1735)-

5. Lectiones Mathematics, Lond. 1683. This list makes it absolutely certain that Williamson is

wrong in stating that the lectures in geometry were published under the title of "Mathematical Lectures." This, how- ever, is not of much consequence; the important point in

INTR OD UCTION 9

the list, assuming it to be perfectly correct as it stands, is that the lectures on Optics were first published separately in 1669. In ^e following year they were reissued in a revised form with the addition of the lectures on geometry.

The above books were all in Latin and have been translated by different people at one time or another.

IV ESTIMATE OF BARROWS GENIUS

The writer of the article on " Barrow, Isaac," in the ninth (Times) edition of the Encyclopedia Britannica, from which most of the details in Section II have been taken, remarks :

''By his English contemporaries Barrow was considered a mathematician second only to Newton. Continental writers do not place him so high, and their judgment is prob- ably the more correct one."

Founding my opinion on the Lectiones Geometric^ alone, I fail to see the reasonableness of the remark I have italicized. Of course, it was only natural that contemporary continental mathematicians should belittle Barrow, since they claimed for Fermat and Leibniz the invention of the infinitesimal calculus before Newton, and did not wish to have to con- sider in Barrow an even prior claimant. We see that his own countrymen placed him on a very high level ; and surely the only way to obtain a really adequate opinion of a scientist's worth is to accept the unbiased opinion that has been expressed by his contemporaries, who were aware of all the facts and conditions of the case ; or, failing that, to try to form an unbiased opinion for ourselves, in the position of his contemporaries. An obvious deduction may be drawn from the controversy between Newton and Hooke the opinion of Barrow's own countrymen would not be likely to err on the side of over-appreciation, unless his genius was great enough to outweigh the more or less natural jealousy that ever did and ever will exist amongst great men occupied on the same investigations. Most modern criticism of ancient writers is apt to fail, because it is in the hands of the experts ; perhaps to some degree this must be so, yet you would hardly allow a K.C- to be a fitting man for a jury.

io BARROW'S GEOMETRICAL LECTURES

Criticism by experts, unless they are themselves giants like unto the men whose works they criticize, compares, perhaps unconsciously, their discoveries with facts that are now common knowledge, instead of considering only and solely the advance made upon what was then common knowledge. Thus the skilled designers of the wonderful electric engines of to-day are but as pigmies compared with such giants as a Faraday.

Further, in the case of Barrow, there are several other things to be taken into account. We must consider his dis- position, his training, his changes of intention with regard to a career, the accident of "his connection with such a man as Newton, the circumstances brought about by the work of his immediate predecessors, and the ripeness of the time for his discoveries.

His disposition was pugnacious, though not without a touch of humour ; there are many indications in the Lectiones Geometries alone of an inclination to what I may call, for lack of a better term, a certain contributory laziness ; in this way he was somewhat like Fermat, with his usual " I have got a very beautiful proof of . . . : if you wish, I will send it to you ; but I dare say you will be able to find it for your- self"; many of Barrow's most ingenious theorems, one or two of his most far-reaching ones, are left without proof, though he states that they are easily deduced from what has gone before. He evidently knows the importance of his discoveries ; in one place he remarks that a certain set of theorems are a " mine of information, in which should any- one investigate and explore, he will find very many things of this kind. Let him do so who must, or if it pleases him." He omits the proof of a certain theorem, which he states has been very useful to him repeatedly ; and no wonder it has, for it turns out to be the equivalent to the differentia- tion of a quotient ; and yet he says, " It is sufficient for me to mention this, and indeed I intend to stop here for a while." It is not at all strange that the work of such a man should come to be underrated.

His pugnacity is shown in the main object pervading the whole of the Lectiones Geometricce he sets out with the one express intention of simplifying and generalizing the existing methods of drawing tangents to curves of all kinds and of

INTRODUCTION 11

finding areas and volumes ; there is distinct humour in his glee at " wiping the eye " of some other geometer, ancient or modern, whose solution of some particular problem he has not only generalized but simplified.

"Gregory St Vincent gave this, but (if I remember rightly) proved with wearisome prolixity."

" Hence it follows immediately that all curves of this kind are touched at any one point by one straight line only. . . . Euclid proved this as a special case for the circle, Apollonius for the conic sections, and other persons in the case of other curves."

His early training was promiscuous, and could have had no other effect than to have fostered an inclination to leave others to finish what he had begun. His Greek professor- ship and his study of Aristotle would tend to make him a confirmed geometrician, revelling in the "elegant solution" and more or less despising Cartesian analysis because of its then (frequently) cumbersome work, and using it only with certain qualms of doubt as to its absolute rigour. For instance, he almost apologizes for inserting, at the very end of Lecture X, which ends the part of the work devoted to the equivalent of the differential calculus, his " a and & " method the prototype of the " h and k " method of the ordinary text-books of to-day.

Another light is thrown on the matter of Cartesian geometry, or rather its applications, by Lecture VI ; in this, for the purpose of establishing lemmas to be used later, Barrow gives fairly lengthy proofs that

(i) my±xy = mx-/fi, (2) ±yx+gx my = mx'2/r

represent hyperbolas, instead of merely stating the fact on account of the factorizing of mx^jb + xy, »;x^jr±xy. The lengthiness of these proofs is to a great extent due to the fact that, although the appearance of the work is algebraical, the reasoning is almost purely geometrical. It is also to be noted that the index notation is rarely used, at least not till very late in the book in places where he could do nothing else, although Wallis had used even fractional indices a dozen years before. In a later lecture we have the truly terrifying equation (rrkk -