Skip to main content

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

See other formats


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 

Read more about Early Journal Content at 
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 


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. 



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 


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. 


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 

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). 


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. 


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 


"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 


dy , y + Va 2 + y 2 
- = log 

Va 2 + 2/- 


Hence Child claims that Barrow possessed this formula in geometrical garb. 
Accordingly, the geometrical process of integrating 



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 

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. 


Edited by U. G. Mitchell, University of Kansas, Lawrence. 

The discussion given below is closely related to Professor Moritz's article 
"On the Construction of Certain Curves Given in Polar Coordinates" published