# Full text of "Who Was the First Inventor of the Calculus?"

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STOP Early Journal Content on JSTOR, Free to Anyone in the World This article is one of nearly 500,000 scholarly works digitized and made freely available to everyone in the world by JSTOR. Known as the Early Journal Content, this set of works include research articles, news, letters, and other writings published in more than 200 of the oldest leading academic journals. The works date from the mid-seventeenth to the early twentieth centuries. We encourage people to read and share the Early Journal Content openly and to tell others that this resource exists. People may post this content online or redistribute in any way for non-commercial purposes. Read more about Early Journal Content at http://about.jstor.org/participate-jstor/individuals/early- journal-content . JSTOR is a digital library of academic journals, books, and primary source objects. JSTOR helps people discover, use, and build upon a wide range of content through a powerful research and teaching platform, and preserves this content for future generations. JSTOR is part of ITHAKA, a not-for-profit organization that also includes Ithaka S+R and Portico. For more information about JSTOR, please contact support@jstor.org. 1919.] RECENT PUBLICATIONS. 15 then the coefficients of e^cc' -1 , e Px x*~ 2 , • • •, e Px in the result are respectively /03)-&2+(*~ ^f'VD-bi, (?) m'h+( t ~ 1 2 )f'(P)-b2 + ( t ~ 1 )/"08)-6i, /(/3).& 4 + (*~ 3 )/'(/3)-&3 + ('" 2 )/"(/3)-6 2 + (' ~ ^'"(fi-bu the law of formation being sufficiently obvious. Under the conditions above, for t = r + s, we have to set b\, - • •, b a = c\, •••, c s when /(/3) = f'ifi) = • • • = / (r_1) (/3) = while / (r) (/3) #= and equate these coefficients in order to those . of G(fi, s) of which the first r are zero and the last s in order are Xi, • • • , X s as in (3). Then the first r equations are identically satisfied and the last s equations suffice to determine Ci, • • •, c in terms of Xi, • • •, X 3 . In fact since / (r) (|8) 4= the (r + l)-st equation determines Ci in terms of Xi; for the same reason the next equation determines Ci in terms of X2 and c\, etc. Indeed to within a num- erical factor the determinant of the last s equations in c\, •••, c is [/ (r) (/3)] s 4= 0. If X is actually expressed in the form given in II the system (7) is immediately available, but as a rule other forms of X are preferable. RECENT PUBLICATIONS. WHO WAS THE FIRST INVENTOR OF THE CALCULUS? The Geometrical Lectures of Isaac Barrow. Translated by J. M. Child, Chicago and London, The Open Court Publishing Co., 1916. xiv + 218 pages. An English translation of so important a work as Isaac Barrow's Lectiones geometricm will be greatly welcomed. Few American mathematicians have had access to a translation into English by E. Stone, published in 1735; according to a statement made by W. Whewell in the preface to his Latin edition of The Mathe- matical Works of Isaac Barrow, Cambridge, 1860, Stone's translation "was so badly executed that it cannot be of use to any one." Few readers will object to Child's omission of certain parts of Barrow's geometrical lectures, parts which seem to be of little or no interest at the present time. Child's historical intro- duction and critical notes greatly assist in the deeper comprehension of Barrow's genial work. In fact, Child has aimed to do much more than simply to supply a translation. He has made a searching study of Barrow and has arrived at 16 RECENT PUBLICATIONS. [Jan., startling conclusions on the historical question relating to the first invention of the calculus. He places his conclusions in italics in the first sentence of his pre- face, as follows: "Isaac Babeow was the first inventor of the Infinitesimal Calculus; Newton got the main idea of it from Barrow by personal communication; and Leibniz was also in some measure indebted to Barrow's work, obtaining confirmation of his own original ideas, and suggestions for their further development, from the copy of Bar- row's book that he purchased in 1678." These claims are far-reaching. Either they are true and should be accepted or they are false and should be rejected. Before entering upon an examination of the evidence brought forth by Child, it may be of interest to review a similar claim set up for another man as inventor of the calculus who, like Barrow, was active before the time of Newton and Leib- niz. Such a claim has been made in favor of Pierre de Fermat (1601-1665) who died when Newton was just beginning his great career and when Leibniz had not yet brought out his first mathematical publication. Fermat was declared to be the first inventor of the calculus by Lagrange, Laplace, and apparently also by P. Tannery, than whom no more distinguished mathematical triumvirate can easily be found. Lagrange expressed himself as follows :* "One may regard Fermat as the first inventor of the new calculus. In his method De max- imis et minimis he equates the quantity of which one seeks the maximum or the minimum to the expression of the same quantity in which the unknown is increased by the indeterminate quantity. In this equation he causes the radicals and fractions, if any such there be, to disappear and after having crossed out the terms common to the two numbers, he divides all others by the indeter- minate quantity which occurs in them as a factor; then he takes this quantity zero and he has an equation which serves to determine the unknown sought. ... It is easy to see at first glance that the rule of the differential calculus which consists in equating to zero the differential of the ex- pression of which one seeks a maximum or a minimum, obtained by letting the unknown of that expression vary, gives the same result, because it is the same fundamentally and the terms one neglects as infinitely small in the differential calculus are those which are suppressed as zeroes in the procedure of Fermat. His method of tangents depends on the same principle. In the equation involving the abscissa and ordinate which he calls the specific property of the curve, he augments or diminishes the abscissa by an indeterminate quantity and he regards the new ordi- nate as belonging both to the curve and to the tangent; this furnishes him with an equation which he treats as that for a case of a maximum or a minimum. . . . Here again one sees the analogy of the method of Fermat with that of the differential calculus; for, the indeterminate quantity by which one augments the abscissa x corresponds to its differential dx, and the quantity ye/t, which is the corresponding augmentation 2 of y, corresponds to the differential dy. It is also remarkable that in the paper which contains the discovery of the differential calculus, printed in the Leipsic Acts of the month of October, 1684, under the title, Nova methodus pro maximis et minimis etc., Leibnitz calls dy a line which is to the arbitrary increment dx as the ordinate y is to the subtan- gent; this brings his analysis and that of Fermat nearer together. One sees therefore that the latter has opened the quarry by an idea that is very original, but somewhat obscure, which con- sists in introducing in the equation an indeterminate which should be zero by the nature of the question, but which is not made to vanish until after the entire equation has been divided by that same quantity. This idea" has become the germ of new calculi which have caused geometry and mechanics to make such progress, but one may say that it has brought also the obscurity of the principles of these calculi. And now that one has a quite clear idea of these principles, one sees 1 J. Lagrange, "Lecons sur le calcul des fonctions," legon dix-huiti&me, (Euvres de Lagrange, publiees par J. A. Serret, Tome X, p. 294. 2 Fermat lets e be the increment of x, and t the subtangent for the point x, y on the curve. 1919.] RECENT PUBLICATIONS. 17 that the indeterminate quantity which Fermat added to the unknown simply serves to form the derived function which must be zero in the case of a maximum or minimum, and which serves in general to determine the position of tangents of curves. But the geometers contemporary with Fermat did not seize the spirit of this new kind of calculus; they did not regard it but a special artifice, applicable simply to certain cases and subject to many difficulties, . . . moreover, this invention which appeared a little before the GgomStrie of Descartes remained sterile during nearly forty years. . . . Finally Barrow contrived to substitute for the quantities which were supposed to be zero according to Fermat quantities that were real but infinitely small, and he published in 1674 his method of tangents, which is nothing but a construction of the method of Fermat by means of the infinitely small triangle, formed by the increments of the abscissa e, the ordinate ey/t, and of the infinitely small arc of the curve regarded as a polygon. This contributed to the creation of the system of infinitesimals and of the differential calculus." Such is Lagrange's interpretation of the work of Fermat and of the place it should occupy in the history of the calculus. Even more positive is the dictum of Pierre Simon Laplace who, in his Essai philoso'phique sur le calcul des proba- bilities, speaks of Fermat in the following terms •} "This great geometrician [Fermat] expresses by the character E the increment of the abscissa; and considering only the first power of this increment, he determines exactly as we do by differ- ential calculus the subtangents of the curves, their points of inflection, the maxima and minima of their ordinates, and in general those of rational functions. We see likewise by his beautiful solution of the problem of the refraction of light inserted in the Collection of the Letters of Descartes that he knows how to extend his methods to irrational functions in freeing them from irrational- ities by the elevation of the roots to powers. Fermat should be regarded, then, as the true dis- coverer of Differential Calculus. Newton has since rendered this calculus more analytical in his Method of Fluxions, and simplified and generalized the processes by his beautiful theorem of the binomial. Finally, about the same time Leibnitz has enriched differential calculus by a nota- tion which, by indicating the passage from the finite to the infinitely small, adds to the advantage of expressing the general results of calculus, that of giving the first approximate values of the dif- ferences and of the sums of the quantities; this notation is adapted of itself to the calculus of par- tial differentials." P. Tannery, the noted historian of mathematics, expressed himself more re- cently as follows : 2 "Fermat is also honored with the invention of the differential calculus on account of his method of maxima and minima and of tangents, which, of the prior processes, is in reality the near- est to the algorithm of Leibniz; one could with equal justice, attribute to him the invention of the integral calculus; his treatise De aquationum localium transmutatione, etc., gives indeed the method of integration by parts as well as rules of integration, except the general powers of variables, their sines and powers thereof. However, it must be remarked that one does not find in his writings a single word on the main point, the relation between the two branches of the infinitesimal calculus." We proceed to quote two opinions, one English, the other French, relating to the attitude taken by Lagrange and Laplace. In the Edinburgh Review for Sep- tember, 1814, p. 324, we read (in an anonymous review now known to have been from the pen of John Playfair) : "To a passage of the latter [Laplace], however, we cannot but advert, and with much less satisfaction than we have generally felt in pointing out any of the remarks of this celebrated writer to the attention of our readers. 'II parait que Fermat, le veritable inventeur du calcul differentiel, 1 A Philosophical Essay on Probabilities, by Pierre Simon, Marquis de Laplace. Transl. by F. W. Truscott and F. L. Emory, New York, 1902, Part I, Chapter V, p. 46. The French orig- inal appeared in 1812. 2 P. Tannery, Article "Fermat" in La Grande Encyclopedic (Berthelot). 18 RECENT PUBLICATIONS. [Jan., a considers ce calcui comme une derivation de celui des differences finies,' etc. Against the affirma- tion that Febmat is the real inventor of the Differential Calculus, we must enter a strong and solemn protestation. The age in which that discovery was made, has been unanimous in ascrib- ing the honour of it either to Newton or Leibnitz; or, as seems to us much the fairest and most probable opinion, to both; that is, to each independently of the other, the priority in respect of time being somewhat on the side of the English mathematician. The writers of the history of the mathematical sciences have given their suffrages to the same effect;— 'Montucla, for instance, who has treated the subject with great impartiality, and Bossut, with no prejudices certainly in favour of the English philosopher. In the great controversy, to which this invention gave rise, all the claims were likely to be well considered; and the ultimate and fair decision, in which all sides seem to have acquiesced, is that which has just been mentioned. It ought to be on good grounds, that a decision, passed by such competent judges, and that has now been in force for a hundred years, should all at once be reversed. — Fermat . . . had certainly approached very near to the differential or fluxionary calculus, as his friend Roberval had also done. He considered the infinitely small quantities introduced in his method of drawing tangents, and of resolving maxima and minima, as derived from finite differences; and, as Laplace remarks, he has extended his method to a case, when the variable quantity is irrational. He was, therefore, very near to the method of fluxions; with the principle of it, he was perfectly acquainted; — and so at the same time were both Roberval and Wallis, though men much inferior to Fermat. The truth is, that the discovery of the new calculus was so gradually approximated, that more than one had come quite near it, and were perfectly acquainted with its principles, before any of the writings of Newton or Leibnitz were known. That which must give, in such a case, the right of being considered as the true inventor, is the extension of the principle to its full range; connecting with it a new calculus, and new analytical operations; the invention of a new algorithm with corre- sponding symbols. These last form the public acts, by which the invention becomes known to the world at large the judge by whiph the matter must be finally decided. Great, therefore, as is the merit of Fermat which no body can be more willing than ourselves to acknowledge; and near as he was to the greatest invention of modern times, we cannot admit that his property in it is to be put on a footing with that of Newton or of Leibnitz; — we should fear, that in doing so, we were violating one of the most sacred and august monuments that posterity ever raised in honour of the dead." Poisson says •} "As a magnitude approaches its maximum or its minimum, it varies less and less and its differential vanishes as it reaches one or the other of these extreme values. Starting from this principle, Fermat had the happy idea, for the determination of the maximum or minimum of a quantity, to assign to the variable upon which it depends, an infinitely small increment and to equate to zero the corresponding increment of that quantity previously reduced to the same order of magnitude as that of the variable. It is in this manner that he determined the path of light in passing from one medium into another upon the supposition conforming with the theory he had adopted, that the time of passage be a minimum. Lagrange considered him, for that reason, as the first inventor of the differential calculus, but this calculus consists of a set of rules suitable for finding immediately the differences of all functions, rather than of the use one makes of the infinitely small variations in the resolution of this or that species of problems; and from that point of view, the creation of the differential calculus does not go back beyond Leibnitz, the inventor of the algorithm and of the notation which have generally prevailed since the origin of the calculus and to which infinitesimal analysis is chiefly indebted for its progress. It should be observed, moreover, that the binomial formula which supplied Newton and Leibnitz the means of expressing very simply the differential of any power whatever, integral or fractional, positive or negative — that this formula was unknown to Fermat, that he could not differentiate the radi- cals which presented themselves in his problem, and that he replaced this operation by geometri- cal constructions and special devices, the avoidance of which is the special object of the differ- ential calculus." We have now quoted the views of Lagrange, Laplace, Poisson, Paul Tannery and an Edinburgh reviewer, on the invention of the calculus. We have quoted 1 Poisson "Memoire sur le calcui des Variations," Memoires de I'academie royale des sciences de I'institut de France, Tome XII, Paris, 1833, p. 223. 1919.] KECENT PUBLICATIONS. 19 the conclusion reached by Child in favor of Barrow. The practical question arises, which of these conflicting opinions is correct. It is easy to see that the answer to be given to this question hinges upon the conception we have as to what constitutes an invention of the calculus. Is it the creation of a method like that of Fermat? Is it the invention of a set of rules and a notation as de- manded by Poisson? Mr. Child endeavors to answer this question in his preface: "By the 'Infin- itesimal Calculus,' I intend ' a complete set of standard forms for both the differ- ential and integral sections of the subject, together with rules for their combi- nation, 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.'" Every one will admit that a set of rules for differentiation and integration is an essential part of the calculus. But does this include all the essentials to be met by a successful candidate for the honor of invention of the calculus? Could the calculus have fulfilled its mission had it not possessed a suitable notation both for differentiation and integration? We do not mean that the very symbols introduced by Leibniz or by Newton are essential parts of a differential and in- tegral calculus. But is not some sort of a suitable notation to designate the first differential or derivative as well as higher differentials or derivatives and to desig- nate integration a necessary part of the calculus— a sine qua non, without which the calculus could not render its service in the resolution of complicated prob- lems? We hold that by common and tacit agreement of all text book writers since the time of Newton and Leibniz such a notation is looked upon as an essen- tial of the calculus. We cannot recall a single author of a text on this subject since the time of Newton and Leibniz who has not recognized the need of a suit- able notation and who does not use one. On the strength of this unanimity of tacit testimony we must insist upon a suitable notation as a necessary require- ment to be met by any one for whom the invention of the calculus is claimed. In his geometrical lectures Barrow uses few symbols. He does not even use the ordinary symbols in trigonometry. He uses the letters a and e for the designa- tion of increments of variables, but this notation is altogether inadequate. Passing to another topic, we observe that most texts have invoked the aid of geometry in the development of the calculus. Geometrical figures help in the grasp of abstract relations. Nevertheless, the calculus has been largely analyt- ical. Algebraic, logarithmic and trigonometric symbols have been used habit- ually. We could not have rules for differentiation and integration, as ordinarily understood, unless the mathematical relations were expressed in analytical form. Now Barrow does not develop an analytical calculus. He establishes geometrical theorems which Child translates into analysis. In that way Child obtains a group of formulas for differentiation and integration, such as have been gathered also from other pre-Newtonian writers. 1 These formulas he sets down to Bar- 1 See H. G. Zeuthen, Geschichte der Mathematik im XVI. und XVII. Jahrhundert, deutsche Ausg. von R. Meyer, Leipzig, 1903, "Integrationen vor der Integralrechnung," pp. 248-300. 20 questions and discussions. [Jan., row's credit. For example Barrow proves the theorem (Lecture XII, Appendix 1,9): "Let EBK be an equilateral hyperbola (that is, one having equal axes), and let the axes be CED, CI; also let KI, KD be ordinates to these; let EVY be a curve such that, when any point R is taken at random on the hyperbola, and a straight line BVS is drawn parallel to DC, then SB CE, SV are in continued proportion; join CK; then the space CEYI will be double the hyperbolic sector KCE." The adjoining figure is our own; Child gives none. The determination of the area CEYI by calculus involves the Jo dy , y + Va 2 + y 2 - = log Va 2 + 2/- ,2 Hence Child claims that Barrow possessed this formula in geometrical garb. Accordingly, the geometrical process of integrating dy Jo Va 2 + ; would be to construct the hyperbola x 2 — y 2 = a 2 and from it the curve EVY, yielding CEYI as the area representing the definite integral. By the same argu- ment it may be claimed that when Dinostratus of old used the quadratrix in the quadrature of the circle, he worked out the part of the integral calculus contained in the formula I Va 2 — x 2 dx = f ira 2 . Dinostratus and Barrow were clever Jo men, but it seems to us that they did not create what by common agreement of mathematicians has been designated by the term differential and integral cal- culus. Two processes yielding equivalent results are not necessarily the same. It appears to us that what can be said of Barrow is that he worked out a set of geometric theorems suggesting to us constructions by which we can find lines, areas and volumes whose magnitudes are ordinarily found by the analytical processes of the calculus. But to say that Barrow invented a differential and integral calculus is to do violence to the habit of mathematical thought and ex- pression of over two centuries. The invention rightly belongs to Newton and Leibniz. Fi/okian Cajoei. University of California. QUESTIONS AND DISCUSSIONS. Edited by U. G. Mitchell, University of Kansas, Lawrence. DISCUSSIONS. The discussion given below is closely related to Professor Moritz's article "On the Construction of Certain Curves Given in Polar Coordinates" published