Historical note

(Chapters I, II, and III)

(N.B. Roman numerals refer to the bibliography to be found at the end of this note.)

In 1604, at the apogee of his scientific career, Galileo believed he had demonstrated that in rectilinear motion where the velocity increases in proportion to the distance travelled the law of motion would be the very one ($x = ct^2$) he had discovered in the fall of heavy bodies (III, v. X, p. 115-116). Between 1695 and 1700 there is not a volume of the Acta Eruditorum published monthly at Leipzig in which there do not appear memoirs by Leibniz, the Bernoulli brothers, the marquis de l’Hôpital, treating, essentially in the notation which we still use, the most varied problems in differential calculus, integral calculus, and calculus of variations. Thus it is in the interval of almost exactly a century that the infinitesimal calculus, or as the English ended by calling it, the Calculus par excellence ("calculus") was forged; and nearly three centuries of constant use have not yet completely blunted this incomparable instrument.

The Greeks neither possessed nor imagined anything like it. No doubt, if they had known, they would have refused to use it, an algebraic calculus, that of the Babylonians, of which a part of their Geometry is possibly only a transcription, for it is strictly in the domain of geometrical invention that their most brilliant mathematical creativity appears, their method of treating problems which for us are a matter for the integral calculus. Eudoxus, in treating the volume of the cone and of the pyramid, gave the first models of this, which Euclid has transmitted more or less faithfully to us ((I), Book XII, prop. 7 and 10). But, above all, it is to these problems that almost all the work of Archimedes was devoted ((II) and (II bis)); and, by an odd chance, we may still read, in their original text, in the sonorous Dorian dialect in which they are so carefully composed, the greater part of his writings, and, up to the one recently rediscovered, where he expounds the "heuristic" procedures by which he was led to one of his most beautiful results ((II), v. II, p. 425-507). For this is one of the weaknesses of the "exhaustion" of Eudoxus: although an irreproachable method of proof (certain postulates having been agreed) it is not a method of discovery; its application necessarily rests on prior knowledge of the result to be proven; also, as Archimedes said, "of the results for which Eudoxus first found a proof, concerning the cone and of the pyramid ..., not a small part goes back to Democritus, who was the first to state them without proof" (loc. cit., p. 430). This circumstance makes a detailed analysis of the work of Archimedes particularly difficult, an analysis which, to tell the truth, seems not to have been undertaken by any modern historian; for in fact we do not know the extent of his awareness of the family connections which unite the various problems with which he dealt (connections which we would express by saying that the same integral reappears in many places in various geometrical aspects) and what importance he might have attributed to them. For example, let us consider the following problems, the first solved by Eudoxus, the others by Archimedes: the volume of the pyramid, the area of a segment of a parabola, the centre of gravity of a triangle, the area of the Archimedean spiral ($\rho = c \omega$ in polar coordinates); these all depend on the integral $\int x^2 dx$, and, without departing in any way from the spirit of the method of exhaustion, one can reduce them all to the evaluation of “Riemann sums” of the form $\sum_n a n^2$. This indeed is how Archimedes treated the spiral ((II), v. II, p. 1-121), using a lemma which amounts to writing

As for the centre of gravity of the triangle, he proved (by exhaustion, using a decomposition into parallel strips) that it lies on each of the medians, so at their point of concurrence ((II), v. II, p. 261-315). For the parabola he gave three procedures: the one, heuristic, designed only to “give some plausibility to the result”, reduces the problem to the centre of gravity of the triangle, by an argument from statics in the course of which he does not hesitate to consider the segment of the parabola as the sum of infinitely many line segments parallel to the axis ((II), v. II, p. 435-439); another method relies on a similar principle, but is drawn up in full rigour by exhaustion ((II), v. II, p. 261-315); a last proof, extraordinarily ingenious, but of lesser scope, gives the area sought as the sum of a geometric series, exploiting specific properties of the parabola. Nothing indicates a relation between these problems and the volume of the pyramid; it is even stated ((II), v. II, p. 8) that the problems regarding the spiral have “nothing in common” with certain others regarding the sphere and the paraboloid of revolution, of which Archimedes had occasion to speak in the same introduction, and among which one finds one (the volume of the paraboloid) which reduces to the integral $\int x dx$.

As one sees from these examples, the principle of exhaustion is, apart from special tricks, the following: by a decomposition into “Riemann sums” one obtains upper and lower bounds for the quantity under examination, bounds which one compares directly with the expression stated for this quantity, or else with the corresponding bounds for a similar problem which has already been solved. The comparison (which, in the absence of negative numbers, is necessarily done in two parts) is introduced by the ritual phrase: “if not, then, it would be either greater or smaller; suppose, if it were possible, that it were greater, etc.; suppose, if it were possible, that it were smaller, etc.”, whence the name “apagogic” or “by reduction to the absurd” (“ἀπαγωγὴ εἰσ ἀδύνατον”) which the scholars of the XVII$^{th}$ century gave to the method. It is in a similar form that the determination of the tangent to the spiral is set out by Archimedes ((II), v. II, p. 62-76), an isolated result, and the only one that we have to cite as an ancient source of “differential calculus” apart from the relatively easy determination of tangents to conics, and a few problems on maxima and minima. If, indeed, in what concerns "integration", an immense field of research was offered to the Greek mathematicians, not only by the theory of volumes, but also by statics and hydrostatics, they hardly had, lacking kinematics, occasion to tackle differentiation seriously. It is true that Archimedes gives a kinematic definition of his spiral; and, not knowing how he could have been led to knowledge of its tangent, one has the right to ask whether he had any concept of the composition of movements. But in this case would he not have applied so powerful a method to other problems of the same kind? It is more plausible that he must have utilised some heuristic procedure of passing to the limit that the results he knew about conics may have suggested; these, of course, are essentially simpler in nature, since one can construct the points of intersection of a line and a conic, and consequently determine the condition for these points to coincide. As to the definition of the tangent, this latter is conceived as a straight line which, in a neighbourhood of a certain point of the curve, has the curve entirely on one side; its existence is assumed, and it is also assumed that every curve is composed of convex arcs; under these conditions, to prove that a line is tangent to a curve it is necessary to prove certain inequalities, which is of course done with the most complete accuracy.

From the point of view of rigour, the methods of Archimedes leave nothing to be desired; and, even in the XVII$^{th}$ century, when the most scrupulous mathematicians wished to put a result judged to be particularly delicate entirely beyond doubt, it was an "apagogic" proof that they gave for it ((XI $a$) and (XII $a$)). As to their fruitfulness, Archimedes' œuvre is sufficient witness. But to have the right to see an "integral calculus" there, one would have to exhibit, through the multiplicity of geometric appearances, some sketch of a classification of the problems according to the nature of the underlying "integral". In the XVII$^{th}$ century, we shall see, the search for such a classification became, little by little, one of the principal concerns of the geometers; if one does not find a trace in Archimedes, is this not a sign that such speculations would have seemed exaggeratedly "abstract" to him, and that he has deliberately, on the contrary, on each occasion, kept as close as possible to the specific properties of the figure whose study he was pursuing? And must we not conclude that this wonderful œuvre, from which the integral calculus, according to its creators, is entirely drawn, is in a certain way the opposite of the integral calculus?

It is not with impunity, moreover, that one can, in mathematics, let a ditch appear between discovery and proof. In favourable times the mathematician, without abandoning rigour, has only to write down his ideas almost as he conceives them; sometimes he may even hope to do so by dint of a felicitous adjustment of the accepted language and notation. But often he has to resign himself to choosing between incorrect, though perhaps fruitful, modes of exposition, and correct methods which do not permit him to express his thought except by distorting it, and at the cost of a tiring effort. Neither way is free of danger. The Greeks followed the second, and it is there, perhaps, even more than the sterilising effect of the Roman conquest, that one must seek the reason for the surprising arrest of their mathematics almost immediately after its most brilliant flowering. It has been suggested, not implausibly, that the oral teaching of the successors of Archimedes and Appolonius must have contained many a new result without their believing it necessary to inflict on themselves the extraordinary effort required to publish in conformity with the received canons. In any case, these were not the scruples which stopped the mathematicians of the XVII$^{th}$ century, when, before the crowds of new problems which posed themselves, they searched assiduously in Archimedes’ writings for the means to overcome them.

While the great classics of Greek literature and philosophy were all printed in Italy by Aldus Manutius and his followers, and almost all of them before 1520, it was only in 1544 that the first edition of Archimedes, in Greek and Latin appeared $^4$, printed by Hervagius at Bâle, no previous Latin version having preceded it; and, far from the mathematicians of this time (absorbed as they were by their algebraic researches) immediately feeling the influence, one had to wait for Galileo and Kepler, both of them astronomers and physicists rather than mathematicians, for this influence to become manifest. From this moment, without break until about 1670, there is no name in the writings of the founders of the Infinitesimal Calculus which recurs so often as that of Archimedes. Many translated him and prepared commentaries on him; all, from Fermat to Barrow, cited him indiscriminately; all claimed to find both a model and a source of inspiration there.

It is true that these declarations, as we shall see, must not all be taken completely literally; here one finds one of the difficulties which hinder a correct interpretation of these writings. The historian must also take account of the organisation of the scientific world at this time, still very defective at the start of the XVII$^{th}$ century, while towards the end of that century, by the creation of learned societies and scientific journals, by the consolidation and development of the universities, it finished by strongly resembling what we know today. Lacking any periodical up to 1665, the mathematicians had no way of making their work known, other than by writing letters, or by printing a book, most often at their own expense, or at that of some Maecenas if they could find one. Editors and printers capable of work of this sort were rare, sometimes unreliable. After the long delays and the innumerable upsets which a publication of this sort involved, the author had often to face up to interminable controversies, provoked by adversaries who were not always in good faith, and sometimes pursued in a tone of surprising acerbity: for, in the general uncertainty on the very principles of the Infinitesimal Calculus, it was not difficult for anyone to find weak points, or at least obscure and contestable points, in the arguments of their rivals. One understands that in these circumstances many scholars who valued tranquility contented themselves with communicating their methods and results to a few chosen friends. Some, and above all some amateurs of science, such as Mersenne at Paris and later Collins at London, maintained a vast correspondence into every country, of which they communicated extracts here and there, not without mixing into these extracts errors of their own invention. Owners of “methods” which, for lack of concepts and general definitions, they were neither able to draft in the form of theorems nor even to formulate accurately, the mathematicians were reduced to

$^4$ Archimedis Opera quae quidem exstant omnia, nunc primus et gr. et lat. edita . . . Basileae, Jo. Hervagius, 1544, l vol. in-fol.

testing them on large numbers of particular cases, believing they could not measure their strength better than by hurling challenges at their colleagues, sometimes accompanied by the publication of their own results in coded language. The studious young travelled, more perhaps than nowadays; and the ideas of a scholar would sometimes spread more as a result of the travels of his pupils than by his own publications, but not without there being yet another cause of misunderstandings. Finally, as the same problems naturally occurred to many mathematicians, some of whom were very distinguished, but had only an imperfect knowledge of the results of the others, claims of priority could not fail to arise incessantly, and it was not unusual for accusations of plagiarism to be added.

Thus it is in the letters and private papers of the scholars of this time, almost as much or even more than in their publications proper, that the historian must seek his documents. But while those of Huygens, for example, have been preserved and made the object of an exemplary publication (XVI), those of Leibniz have not yet been published except in a defective and fragmentary manner, and many others are lost beyond recall. At least the most recent research, founded on the analysis of manuscripts, has put in evidence, in a manner which seems irrefutable, a point which partisan quarrels had almost obscured: it is that every time that one of the great mathematicians of this time has reported on his own work, on the evolution of his thought, on what has influenced him and what not, he has done so in an honest and sincere manner, in all good faith $^5$; these precious testimonies, of which we possess quite a large number, can therefore be used with full confidence, and the historian does not have to transform himself into an examining magistrate here. For the rest, most of the questions of priority were totally lacking in sense. It is true that Leibniz, when he adopted the notation $dx$ for the "differential", did not know that Newton had employed $\dot{x}$ for the "fluxion" for about ten years: but what if he had known? To take a more instructive example, who is the author of the theorem $\log x = \int dx/x$, and what is its date? The formula, as we have just written it, is due to Leibniz, for both terms are written in his notation. Leibniz himself, and Wallis, attribute it to Grégoire de Saint-Vincent. This latter, in his Opus Geometricum (IX) (it appeared in 1647, but was composed, he said, long before), proves only the equivalent of the following: if $f(a, b)$ denotes the area of the hyperbolic segment $a \leq x \leq b,\ 0 \leq y \leq A/x$, then the relation $b'/a' = (b/a)^n$ implies $f(a', b') = n\ f(a, b)$; to this his pupil and commentator Sarasa almost immediately added the remark $^6$ that the areas $f(a, b)$ can thus "take the place of logarithms". If he said no more on this, and if Grégoire himself said nothing, was it not because, for the majority of mathematicians of this time, logarithms were "aids to calculation" not yet naturalized with no right to be quoted in mathematics? It is true that Torricelli, in a letter of 1644 (VII bis) spoke of his research on a curve that we would write as $y = a e^{-x^2},\ x \geq 0$, adding that Napier (whom moreover he covered in praise) "pursued only practical

5 This applies for example to Torricelli (see (XII), v. VIII, p. 181-194) and to Leibniz (D Mahnke, Abh. Preuss. Akad. der Wiss., 1925 Nr. 1, Berlin, 1926). This is not to say, of course, that a mathematician cannot be subject to illusions on the originality of his ideas; but they are not the greatest who are the most inclined to deceive themselves in this respect. 6 Solutio problematis . . . Auctore P. Alfonso Antonio de Sarasa . . . Antverpiae, 1649.

arithmetic", while he himself "drew a speculation in geometry out of it"; and he left a manuscript, evidently prepared for publication, on this curve, though it remained unpublished until 1900 ((VII), v. I, p. 335-347). Descartes, furthermore, had met the same curve about 1639 in connection with "Debeaune's problem" and described it without mentioning logarithms ((X), v. II, p. 514-517). However it may have been, J. Gregory, in 1667, gave, without citing anyone at all ((XVII a), reproduced in (XVI bis), p. 407-462), a rule for calculating the areas of hyperbolic segments by means of (decimal) logarithms: this at once implies theoretical knowledge of the connection between the quadrature of the hyperbola and logarithms, and numerical knowledge of the connection between "natural" and "decimal" logarithms. Is it only at this last point that Huygens' claim applies, contesting directly the novelty of Gregory's result (XVI a)? This is no more clear to us than to their contemporaries; these in every case had the clear impression that the existence of a link between logarithms and the quadrature of the hyperbola was something known a long time, without being able to refer to anything other than epistolary allusions or even to the book of Grégoire de Saint-Vincent. In 1668, when Brouncker gave series for log 2 and for log(5/4) (with a meticulous proof of convergence, by comparison with a geometric series)(XIV), he presented them as expressions for corresponding segments of the hyperbolas, and added that the numerical values that he obtained were "in the same ratio as the logarithms" of 2 and of 5/4. But in the same year, with Mercator (XIII) (or more precisely with the exposition given immediately by Wallis of the work of Mercator (XV bis)), the language changed: since the segments of the hyperbola are proportional to the logarithms, and since it is known that logarithms are defined by their characteristic properties only up to a constant factor, nothing stops one from considering the segments of the hyperbola as logarithms, termed "natural" (in opposition to "artificial" or "decimal" logarithms) or hyperbolic; this last step taken (the series for log(1 + x), given by Mercator, contributed to this), the theorem $\log x = \int d.x/x$ was obtained, up to notation, or, rather, it even became the definition. What to conclude, if not that it was by almost imperceptible transitions that this discovery was made, and that a priority dispute in this matter strongly resembles a quarrel between the violin and the trombone over the exact moment when a certain motif appears in a symphony? And to tell the truth, although at the same time other mathematical creations, the arithmetic of Fermat, the kinematics of Newton, bear a strongly individual stamp, it is of the gradual and inevitable unrolling of a symphony, where the "Zeitgeist", at the same time composer and conductor, takes the baton, that the development of the infinitesimal calculus in the XVII$^{th}$ century reminds one: each executes his part with his own proper timbre, but none is the master of the themes which he makes heard, themes which are almost inextricably entangled by a scholarly counterpoint. It is thus under the form of a thematic analysis that the history has to be written; we shall be content here with a summary sketch, and shall not pretend to minute exactitude⁷. Here in any case are the principal themes which appear under superficial examination:

A) The theme of mathematical rigour, contrasting with that of the infinitely small, indivisibles or differentials. We have seen that both of these held an important place with Archimedes, the first in all of his œuvre, the second in his only treatise on Method, which the XVII$^{th}$ century did not know, so that if it had been transmitted and not reinvented, it could only have been by the philosophical tradition. The principle of the infinitely small appears, moreover, in two distinct forms, as it concerns “differentiation” or “integration”. As to this, suppose first that one is to calculate a plane area: one will divide it into infinitely many infinitely small parallel strips by means of infinitely many equidistant parallels; and each of these strips is a rectangle (even though the finite strips one would obtain using two parallels at a finite distance would not be rectangles). Similarly, a solid of revolution will be decomposed into infinitely many cylinders of the same infinitely small height, by planes perpendicular to the axis $^8$; similar modes of speech might be employed when decomposing an area into triangles by concurrent lines, or reasoning about the length of an arc of a curve as if it were a polygon with infinitely many sides, etc. It is certain that the rare mathematicians who possessed a firm command of Archimedes’ methods, such as Fermat, Pascal, Huygens, Barrow, would not, in each particular case, have found any difficulty in replacing the use of this language with rigorous proofs; also they often remark that this language is only a short-hand. “It would be easy”, says Fermat, “to give proofs after the manner of Archimedes; . . . , it is enough to have been warned once and for all in order to avoid continual repetitions . . . ” ((XI), v. I, p. 257); similarly Pascal: “thus the one of these methods does not differ from the other save in the manner of speaking” ((XII b), p. 352)⁹; and Barrow, with his sardonic conciseness: “longior discursus apagogicus adhiberi possit, sed quorsum?” (one could prolong this by an apagogic discourse, but to what benefit?) ((XVIII), p. 251). Fermat, it seems, was wary of advancing what he could not so justify, and so condemned himself to not stating any general result except by allusion or under the form of a “method”: Barrow, although very careful, was a little less scrupulous. As for the majority of their contemporaries, one can say at the very least that rigour was not their principal concern, and that the name of Archimedes was most often only a tent intended to cover merchandise doubtless of great price, but for which

⁷ In what follows, the attribution of a result to such an author, to such a date, only indicates that this result was known to him at that date (which as often as possible has been verified by examining the original texts); we do not mean to affirm absolutely that this author did not know it earlier, or that he did not receive it from someone else; even less do we want to say that the same result might not have been obtained independently by others, either earlier, or later. ⁸ See for example the exposition by Pascal in his “letter to M. de Carcavy” (XII b). One will note that, thanks to the prestige of an incomparable language, Pascal manages to create the illusion of perfect clarity, to the point where one of his modern editors goes into ecstasies over “the meticulousness and preciseness in the exactitude of the proof”! ⁹ But, in the Letter to Monsieur A. D. D. S.: “. . . without stopping, neither at the methods of movements, nor at that of the indivisibles, but following that of the ancients, so that the matter would henceforth be firm and beyond dispute” ((XII a), p. 256).

Archimedes would certainly not have assumed responsibility. All the more so when one deals with differentiation. If the curve to be rectified is assimilated to a polygon with infinitely many sides, it is an “infinitely small” arc of the curve that is assimilated to an “infinitely small” line segment, either the chord, or a segment of the tangent whose existence is assumed; or again it is an “infinitely small” interval of time that one considers, during which (while one is dealing only with velocity) the movement “is” uniform; yet bolder, Descartes, wanting to determine the tangent to the cycloid, which is not covered by his general rule, assimilates curves rolling on one another to polygons, to deduce that “in the infinitely small” the movement can be assimilated to a rotation about the point of contact ((X), v. II, p. 307-338). Here again, a Fermat, who bases his rules for tangents and for maxima and minima on such infinitesimal considerations, is able to justify them in each particular case ((XI b); cf. also (XI), v. II, passim, in particular p. 154-162, and the Supplément aux Œuvres (Gauthier-Villars, 1922), p. 72-86); Barrow gave exact proofs for many of his theorems after the manner of the ancients, starting from simple hypotheses of monotonicity and convexity. But it was no longer the moment to pour new wine into old skins. In all that, as we know today, it was the notion of a limit that was being elaborated; and, if one can extract from Pascal, from Newton, from others as well, statements that seem very close to our modern definitions, one need only put them back in their context to perceive the invincible obstacles that hindered rigorous exposition. When, from from the XVIII$^{th}$ century, some mathematicians concerned for clarity wished to put some order among the confused piles of their riches, such indications, in the writings of their predecessors, were precious to them; when d’Alembert, for example, explained that there is nothing in differentiation other than the notion of a limit, and defined this accurately (XXVI), one can believe that he was guided by Newton’s considerations on the “first and last reasons of vanishing quantities” (XX). But, so far as the XVII$^{th}$ century is concerned, it is very necessary to state that the way was not open to modern analysis except when Newton and Leibniz, turning their backs on the past, agreed provisionally to seek the justification of new methods, not in rigorous proofs, but in the fruitfulness and coherence of the results.

B) Kinematics. Archimedes, one has seen, had already given a kinematical definition of his spiral; and in the middle ages there developed (but, without evidence to the contrary, without infinitesimal considerations) a rudimentary theory of the variation of quantities as a function of time, and of their graphical representation, which one can perhaps trace back to Babylonian astronomy. But it was of the greatest importance for the mathematics of the XVII$^{th}$ century that, from the beginning, the problems of differentiation arose, not only regarding tangents, but also regarding velocities. Galileo, nevertheless, ((III) and (III bis)), investigating the law of the velocity in the fall of heavy bodies (after having obtained the law for distances $x = at^2$, from experiments on an inclined plane), does not proceed by differentiation: he makes various hypotheses about the velocity, first $v = dx/dt = cx$ ((III), v. VIII, p 203), then later $v = ct$ (id., p. 208), and seeks to recover the law of distances by reasoning, in a rather obscure way, about the graph of the velocity as a function of the time; Descartes (in 1618) argued similarly, about the law $v = ct$, but as a true mathematician and with as much clarity as the language of indivisibles permits $^{10}$ ((X), v. X, p. 75-78); in both of these, the graph of the velocity (in this instance a straight line) plays the principal rôle, and one may ask up to what point they were aware of the proportionality between the distances travelled and the areas contained between the axis of time and the curve of the velocities; but it is difficult to affirm anything on this point, even though Descartes’ language seems to imply an awareness of the fact in question (which certain historians would like to trace back to the middle ages $^{11}$), although Galileo makes no clear allusion to it. Barrow stated it explicitly in 1670 ((XVIII), p. 171); perhaps it was no novelty to anyone at this time, and Barrow did not present it as such; but, no more for this result than for any another, should one try to fix a date too precisely. As for the hypothesis $v = cx$, also envisaged by Galileo, he contented himself (loc. cit.) with proving that it is untenable (or, in modern language, that the equation $dx/dt = cx$ has no solution $\neq 0$ which vanishes for $t = 0$), by an obscure argument which Fermat later ((XI), v. II, p. 267-276) took the trouble to develop (and which comes close to saying that, $2x$ being a solution if $x$ is, $x \neq 0$ would be contrary to the physically evident uniqueness of the solution). But it is this same law $dx/dt = cx$ which, in 1614, served Napier to introduce his logarithms, for which he gave a kinematical definition (IV), and, in our notation, would be written as follows: if, on two straight lines, two moving bodies travel according to the laws $dx/dt = a,\ dy/dt = -ay/r,\ x_0 = 0,\ y_0 = r$, then one says that $x$ is the “logarithm” of $y$ (in modern notation, $x = r \log(r/y)$). We have seen that the solution curve of $dy/dx = c/x$ appeared in 1639 with Descartes, who described it kinematically ((X), v. II, p. 514-517); it is true that he classified all non-algebraic curves rather disdainfully as “mechanical”, and claimed to exclude them from geometry; but happily this taboo, against which Leibniz, much later, still believed he had to protest vigorously, was not observed by his contemporaries, nor by Descartes himself. The cycloid, and the logarithmic spiral, appeared, and were studied eagerly, and their study powerfully helped the interpenetration of geometric and kinematical methods. The principle of composition of movements, and more precisely of composition of velocities, was at the basis of the theory of movement of projectiles expounded by Galileo in the chef-d’œuvre of his old age, the Discorsi of 1638 (((III), v. VIII, p. 268-313), a theory which thus contains implicitly a new determination of the tangent to a parabola; if Galileo did not remark this expressly, Torricelli, on the other hand (VII), v. III, p. 103-159) insisted on this point, and founded on this very principle a general method for determining the tangents to curves for which a kinematical definition could be given. It is true that he had been forestalled in this, by several years, by Roberval (VIII a), who said he had been led to this method by the study of the cycloid; this same problem of the tangent to

$^{10}$ Decartes even adds an interesting geometrical argument by which he deduces the law $x = at^2$ from the hypothesis $dv/dt = c$. On the other hand it is ironic, ten years later, to see him muddle his notes, and copy for Mersenne’s use an incorrect argument on the same question, where the graph of the velocity as a function of time is confused with the graph as a function of the distance travelled ((X), v. I, p. 71). $^{11}$ H. Wielietner, Der “Tractatus de latitudinibus formarum” des Oresme, Bibl. Mat. (III), v. 13 (1912), p. 115-145.

the cycloid gave Fermat, besides, the opportunity to demonstrate the power of his method of differentiation ((XI b), p. 162-165), while Descartes, unable to apply his algebraic method here, invented the instantaneous centre of rotation for the occasion ((X), v. II, p. 307-338).

But, as the infinitesimal calculus developed, kinematics ceased to be a separate science. One sees more and more that in spite of Descartes the algebraic curves and functions have nothing, from the “local” point of view, that of the infinitesimal calculus, to distinguish them from other much more general ones; the functions and curves with a kinematical definition are functions and curves like the others, amenable to the same methods; and the variable “time” is merely a parameter whose temporal aspect is purely a matter of language. Thus, with Huygens, even when he dealt with mechanics, it was geometry that dominated (XVI b); and Leibniz did not give time any privileged rôle in his calculus. In contrast, Barrow devised, from the simultaneous variation of various quantities as a function of an independent universal variable conceived as a “time”, the foundation of an infinitesimal calculus with a geometric tendency. This idea, which must have come to him when he sought to recover the method of composition of movements, whose existence he knew only by hearsay, is expounded in detail, in very clear and very strong terms in the first three of his Lectiones Geometricae (XVIII); there, for example, he shows carefully that if a moving point has as projections on two rectangular axes AY, AZ, moving points one of which moves with a constant velocity $a$ and the other with a velocity $v$ that increases with time, then the trajectory has a tangent with gradient equal to $v/a$. and is concave towards the direction of increasing Z. In the rest of the Lectiones he pursued these ideas very far, and although he indulged the affectation of drafting them from beginning to end in a form as geometric and little algebraic as possible, one may see there, with Jakob Bernoulli ((XXIII), v. I, p. 431 and 453), the equivalent of a good part of the infinitesimal calculus of Newton and Leibniz. Exactly the same ideas served as the point of departure for Newton ((XIX c) and (XX)): his “fluentes” are various quantities, functions of a “time” which is only a universal parameter, and the “fluxions” are their derivatives with respect to “time”; the possibility that Newton permits himself of changing the parameter at need is equally present in the work of Barrow, though used less flexibly by him 12. The language of fluxions, adopted by Newton and imposed by his authority on the English mathematicians of the following century, thus represents the last outpost, for the period that we treat, of the kinematical methods whose true rôle was now over.

12 On the relations between Barrow and Newton, see Osmond, Isaac Barrow, His life and time, London, 1944. In a letter of 1663 (cf. St. P. Rigaud, Correspondence of scientific men ..., Oxford, 1841, vol. II, p. 32-33), Barrow speaks of his thoughts, already old, on the composition of movements, which had led him to a very general theorem on tangents (if it is that of Lect. Geom., Lect. X (XVIII), p. 247), which is so general that it actually contains as a particular case all that had been done up to then on the subject). On the other hand, Newton was the pupil of Barrow in 1664 and 1665, but said he had independently obtained his rule for deducing a relation between their “fluxions” from a relation between “fluentes”. It is likely that Newton had taken the general idea of quantities varying as a function of time, and of their velocities of variation, from Barrow’s teaching, concepts which his thoughts on dynamics (to which Barrow must have remained a total stranger) soon helped clarify.

C) Algebraic geometry. This is a parasitic theme, a stranger to our subject, which intrudes itself by the fact that Descartes, with his systematic attitude, claimed to make algebraic curves the exclusive object of geometry ((X), v. VI, p. 390); also, it was a method of algebraic geometry, and not like Fermat, a method from the differential calculus, that he gave for the determination of tangents. The results bequeathed by the ancients on the intersection of a line and a conic, the thoughts of Descartes himself on the intersection of two conics, and the problems which reduce to them, would have led him completely naturally to the idea of taking the coincidence of two intersections as the criterion for contact: we know today that in algebraic geometry this is the correct criterion, and of such great generality that it is independent of the concept of limit and of the nature of the “base field”. Descartes first applied it in a not very convenient way, trying to make two intersections of the curve under study and a circle having its centre on Ox coincide at a given point ((X), v. VI, p. 413-424); his pupils, van Schooten, Hudde, substituted a line for the circle and obtained the gradient of the tangent to the curve

in the form $-F'_x / F'_y$, the “derived polynomials” $F'_x, F'_y$ being defined by their formal rule of formation ((X bis), v. I, p. 147-344 and (XXII), p. 234-237); de Sluse also arrived at this result at about the same time ((XXII), p. 232-234). Of course the clear-cut distinctions which we note here, and which alone give a meaning to the controversy between Descartes and Fermat, could not have existed in any way in the minds of mathematicians of the XVII$^{th}$ century: we have mentioned them only to clarify one of the most curious episodes of the history which concerns us, and to remark the almost immediate complete eclipse of algebraic methods, for the time being absorbed by differential methods.

D) Classification of problems. This theme, we have seen, appears to be absent from Archimedes’ œuvre, and it was immaterial to him whether he solved a problem directly or reduced it to a problem already treated. In the XVII$^{th}$ century the problems of differentiation first appear under three distinct aspects: velocities, tangents, maxima and minima. As to these last, Kepler (V) made the observation (which one already finds in Oresme $^{13}$ and which had not escaped even the Babylonian astronomers) that the variation of a function is particularly slow in the neighbourhood of a maximum. Fermat, already before 1630 ((XI bis); cf. (XI), v. II, p. 71) inaugurated his infinitesimal method à propos these problems, which in modern language amounts to examining the first two terms (the constant term and the first order term) of the Taylor expansion, and to writing that the second vanishes at an extremum; he went on to extend his method to the determination of tangents, and even applied it to the search for points of inflection. If one takes account of what has been said above about kinematics, one sees that the unification of these three types of problem regarding the first derivative was accomplished quite early. As for problems regarding

$^{13}$ H. WIELEITNER, Der “Tractatus de latitudinibus formarum” des Oresme, Bibl. Mat. (III), v. 13 (1912), p. 115-145, in particular p. 141.

the second derivative, they do not appear until later, above all with the works of Huygens on the evolute of a curve (published in 1673 in his Horologium Oscillatorium (XVI b)); at this time, Newton, with his fluxions, was already in possession of all the analytical tools needed to solve such problems; and, despite all the geometric talent which Huygens expended on it (and of which later differential geometry would profit at its beginning), they did not serve for anything else, during the period which we treat, than to allow the new analysis to proclaim the power of its tools.

As to integration, this appeared with the Greeks as the calculation of areas, volumes, moments, as the calculation of the perimeter of the circle and the areas of spherical segments: to which the XVII$^{th}$ century added the rectification of curves, the calculation of the area of surfaces of revolution, and (with the work of Huygens on the compound pendulum (XVI $b$)) the calculation of moments of inertia. It was first a matter of recognising the connections between all these problems. For areas and volumes the first immense step was taken by Cavalieri, in his Geometry of Indivisibles (VI $a$). There he stated, and claimed to prove, more or less the following principle: if two plane areas are such that every line parallel to a given direction cuts them in segments which are in a constant ratio, then the areas are in the same ratio; an analogous principle is stated for the volumes cut by planes parallel to a fixed plane in areas whose measures are in a constant ratio. It is likely that these principles were suggested to Cavalieri by theorems such as those of Euclid (or rather of Eudoxus) on the ratio of the volumes of pyramids of the same height, and that before stating them in a general manner he had first verified their validity on a great number of examples taken from Archimedes. He “justified” them by employing a language on whose legitimacy one sees him question Galileo in a letter of 1621, although already in 1622 he used it without hesitation ((III), v. XIII, p. 81 and 86) and of which this is the essential part. Consider for example two areas

the one $0 \leq x \leq a,\ 0 \leq y \leq f(x)$, the other $0 \leq x \leq a,\ 0 \leq y \leq g(x)$;

the sums of the ordinates $\sum_{k=0}^{n-1} f(ka/n),\ \sum_{k=0}^{n-1} g(ka/n)$ are in a ratio which, for $n$ sufficiently large, is as close as one requires to the ratio of the two areas, and it would not even be difficult to demonstrate this by exhaustion when $f$ and $g$ are monotone; Cavalieri passed to the limit, made $n = \infty$, and spoke of “the sum of all the ordinates” of the first curve, which is in a ratio to the analogous sum for the second curve rigorously equal to the ratio of their areas; the same for volumes; this language was universally adopted later, even by authors, like Fermat, who had the clearest awareness of the precise facts which it covers. It is true that subsequently many mathematicians, such as Roberval (VIII $a$) and Pascal (XII $b$), preferred to see, in the ordinates of the curve of which one forms the “sum”, not line segments like Cavalieri, but rectangles of the same infinitely small width, which is no great advance from the point of view of rigour (whatever Roberval says), but perhaps keeps the imagination from going off the rails too easily. In any case, and since one deals only with ratios, the expression “sum of all the ordinates” of the curve $y = f(x)$ or, in short, “all the ordinates” of the curve, is, in the final analysis, as it also appears in the writings of Pascal, the exact equivalent of the Leibnizian $\int y\, dx$.

From the language adopted by Cavalieri, the principles stated above follow inevitably, and also immediately imply consequences which we shall state in modern notation, having understood that $\int f\, dx$ simply means the area contained between Ox and the curve $y = f(x)$. First, every plane area cut by every line $x =$ constant in segments the sum of whose lengths is $f(x)$, is equal to $\int f\, dx$; the same is true for every volume cut by each plane $x =$ constant in an area of measure $f(x)$. Further, $\int f\, dx$ depends linearly on $f$; we have $\int (f + g)\, dx = \int f\, dx + \int g\, dx$, $\int cf\, dx = c \int f\, dx$. In particular, all problems of areas and volumes are reduced to quadratures, that is to say, to the calculation of areas of the form $\int f\, dx$; and, what is perhaps novel and more important, one must consider as equivalent two problems that depend on the same quadature, and one has the tools to decide, in each case, if this is so. The Greek mathematicians never attained (or perhaps would never have agreed to attain) such a degree of “abstraction”. Thus ((VI), p. 133) Cavalieri “demonstrates” quite easily that two similar volumes are in the ratio of the cube of the ratio of similarity, whereas Archimedes stated this conclusion, for quadrics of revolution and segments of them, only at the end of his theory of these solids ((II), v. p. 258). But to reach this point it was necessary to throw Archimedean rigour overboard.

One thus had there the means of classifying problems, at least provisionally, according to the degree of real or apparent difficulty presented by the quadratures to which they reduce. This is where the algebra of the time served as a model: for in algebra too, and in the algebraic problems that arose in geometry, although the Greeks were interested only in the solutions, the algebraists of the XVI$^{th}$ and XVII$^{th}$ century had begun to turn their attention principally to the classification of problems according to the nature of the tools which might serve to solve them, thus anticipating the modern theory of algebraic extensions; and they had not only proceeded to a first classification of problems according to the degree of the equation on which they depend, but had already posed difficult questions on possibility: the possibility of solving every equation in radicals (in which many no longer believed), etc. (see the Historical Note to Book II, chap. V); they preoccupied themselves also with reducing all the problems of a given degree to a form of geometric type. Likewise in the Integral Calculus the principles of Cavalieri put him in a position to recognise immediately that many of the problems solved by Archimedes reduce to the quadratures $\int x^n\, dx$ for $n = 1, 2, 3$; and he devised an ingenious method of effecting this quadrature for as many values of $n$ as one wishes (the method amounts to observing that $\int_0^{2a} x^n\, dx = c_n a^{n+1}$ by homogeneity, and to writing

from which, on expanding, one obtains a recurrence relation for the $c_n$ ((VI a), p. 159 and (VI b), p. 269-273). But Fermat had already gone much further, showing first (before 1636) that $\int_0^a x^n\, dx = \frac{a^{n+1}}{n+1}$ for $n$ a positive integer ((XI), v. II, p. 83), by means of a formula for the sums of powers of the first N integers (a process copied from the quadrature of the spiral by Archimedes), then by extending this formula to all rational $n \neq -1$ ((XI), v. 1, p. 195-198); he did not draft a proof of this last result (communicated to Cavalieri in 1644) until much later, after reading the writings of Pascal on integration [^1] (XI c).

These results, joined to geometric considerations which took the place of change of variables and integration by parts, already allowed one to solve a large number of problems that reduce to elementary quadratures. Beyond that, one first encountered the quadrature of the circle and the hyperbola: since it is above all with “indefinite integrals” that one dealt at this time, the solutions of these problems, in modern terms, is furnished respectively by the inverse trigonometric functions and by the logarithm; these were given geometrically, and we have seen how the latter was introduced little-by-little into analysis. These quadratures formed the subject of numerous works, by Grégoire de St.-Vincent (IX), Huygens ((XVI c) and (XVI d)), Wallis (XV a), Gregory (XVII a); the first believed he had effected the quadrature of the circle, the last that he had proved the transcendence of $\pi$; they developed processes of indefinite approximation of the circular and logarithmic functions, some with a theoretical slant, others oriented towards numerical calculus, which would soon, with Newton ((XIX a) and (XIX b)), Mercator (XIII), J. Gregory (XVII bis), then Leibniz (XXII), come to general methods of series expansion. In every case, the conviction was born, little by little, of the “impossibility” of the quadratures in question, that is to say, of the nonalgebraic character of the functions that they defined; and at the same time, it became usual to consider that a problem was solved so far as its nature permitted, when it had been reduced to one of the “impossible” quadratures. This is the case, for example, of problems on the cycloid, solved by the trigonometric functions, and the rectification of the parabola, reduced to the quadrature of the hyperbola.

The problems of rectification, of which we have just cited two of the most famous, had a particular importance, since they formed a natural geometric transition between differentiation, which they presuppose, and integration under which they come; one can associate with them the problems on the area of surfaces of revolution. The ancients had treated only the case of the circle and of the sphere. These questions appeared only very late in the XVII$^{th}$ century; it seems that the difficulty, insurmountable at the time, of the rectification of the ellipse (considered the simplest curve after the circle) had discouraged effort. Kinematical methods gave some purchase on these problems, and allowed Roberval (VIII b) and Torricelli ((VII), v. III, p. 103-159), between 1640 and 1645, to obtain results on the arcs of spirals; but it is only in the years preceding 1660 that they became the order of the day; the cycloid was rectified by Wren in 1658 ((XV), v. I, p. 533-541); a little later the curve $y^3 = ax^2$ by various authors ((XV), v. I, p. 551-553; (X bis), p. 517-520; (XI d)), and several others too ((XI), v. I, p. 199; (XVI), v. II, p. 224) reduced the rectification of the parabola to the quadrature of the hyperbola (that is to say, to an algebraico-logarithmic function). This last example is the most important, for it is a particular case of the general principle that rectification of the curve $y = f(x)$ is nothing else than the quadrature of $y = \sqrt{1 + (f'(x))^2}$; and it is indeed from this principle that Heurat deduced it. It is no less interesting to follow the attempts of the aging Fermat, in his work on the curve $y^3 = ax^2$ (XI d); to the curve $y = f(x)$ of arc $s = g(x)$ he associated the curve $y = g(x)$, and determined the tangent to this from the tangent to the first (in modern language, he showed that their gradients $f'(x)$, $g'(x)$ are linked by the relation

one thinks oneself very near to Barrow, and one need only combine this result with that of Heurat (which is more or less what Gregory did in 1668 ((XVII bis), p. 488-491)) to obtain the relation between tangents and quadratures; but Fermat stated only that if, for two curves each referred to a system of rectangular axes, the tangents at points with same abscissa always have the same gradient, then the curves are equal, or, in other words, that the knowledge of $f'(x)$ determines $f(x)$ (up to a constant); and he justified this assertion only by an obscure argument of no probative value.

Less than ten years later, the Lectiones Geometricae of Barrow (XVIII) had appeared. From the outset (Lect. I), he states in principle that, in a rectilinear movement, the distances are proportional to the areas $\int_0^t v \, dt$ contained between the time axis and the graph of the velocity. One would expect him to deduce the link between the derivative conceived as the gradient of the tangent and the integral conceived as an area from this, and from his kinematical method already cited on the determination of tangents; but there is none of that, and he shows later, in a purely geometric manner ((XVIII), Lect. X, §11, p. 243) that, if two curves $y = f(x)$, $Y = F(x)$ are such that the ordinates Y are proportional to the areas $\int_a^x y \, dx$ (that is to say, if $cF(x) = \int_a^x f(x) \, dx$), then the tangent to $Y = F(x)$ cuts Ox at the point with abscissa $x - T$ determined by $y/Y = c/T$; the proof is, moreover, perfectly correct, starting from the explicit hypothesis that $f(x)$ is monotone: and it is stated that the sense of variation of $f(x)$ determines the sense of the concavity of $Y = F(x)$. But one may note that this theorem is somewhat lost among a crowd of others, of which many are very interesting; the unwarned reader is tempted only to see a means of solving by quadrature the problem $Y/T = f(x)/c$, that is to say, a particular problem of determining a curve from information about its tangent (or, as we would say, a differential equation of a particular kind); and all the more so since the applications which Barrow gave of it concern, before all, problems of the same kind (that is to say, differential equations integrable by "separation of variables"). The geometric language which Barrow imposed on himself is here a cause why the link between differentiation and integration, so clear where kinematics was concerned, was somewhat obscured.

On the other hand, various methods had taken shape, for reducing some problems of integration to others, and to "solving" them, or at least to reducing them to the already classified "impossible" problems. In its simplest geometrical form, integration by parts consists of writing the area bounded by O.x, O.y, and an arc of the monotone curve $y = f(x)$ joining a point $(a, 0)$ of Ox to a point $(0, b)$ of Oy, as $\int_0^a y\, dx = \int_0^b x\, dy$; and it is frequently used implicitly. The following generalization appears, already well hidden, in Pascal ((XII c), 287-288): $f(x)$ being as above, let $g(x)$ be a function $\geqslant 0$, and let $G(x) = \int_0^x g(x)\, dx$; then one has $\int_0^a y\, g(x)\, dx = \int_0^b G(x)\, dy$, which he proved ingeniously by evaluating the volume of the solid $0 \leqslant x \leqslant a,\ 0 \leqslant y \leqslant f(x),\ 0 \leqslant z \leqslant g(x)$ in two ways; the particular case $g(x) = x^n,\ G(x) = \frac{x^{n+1}}{n+1}$ plays an important rôle, both in Pascal (loc. cit., p. 289-291) and in Fermat ((XI), v. I, p. 271); the latter (whose work bears the significant title "Transmutation et émendation des équations des courbes, et ses applications variées à la comparaison des espaces curvilignes entre eux et avec les espaces rectilignes . . .") did not prove this, doubtless because he did not judge it useful to repeat what Pascal had just published. These theorems on "transmutation", where we would see a combination of integration by parts and of change of variables, take the place of this to a certain extent, for it was not introduced until much later; it is indeed contrary to the mode of thought of the time, still too geometric and too little analytic, to allow the use of variables others than those imposed by the figure, that is to say, one or other of the coordinates (or sometimes polar coordinates), or the arc length of the curve. It is thus that we find results in Pascal (XII d) which, in modern notation, can be written, putting $x = \cos t,\ y = \sin t$, and for particular functions $f(x)$:

and, in J. Gregory ((XVII bis), p. 489), for a curve $y = f(x)$ and its arc $s$, $\int y\, ds = \int z\, dx$, with $z = y \sqrt{1 + {y'}^2}$. It is only in 1669 that we see Barrow in possession of the general theorem on change of variables ((XVIII), p. 298-299); his statement, geometric as always, amounts to the following: let $x$ and $y$ be linked by a monotone relation, and let $p$ be the gradient of this relation at the point $(x, y)$; then, if the functions $f(x),\ g(y)$ are such that $f(x)/g(y) = p$ for every pair of corresponding values $(x, y)$, then the areas $\int f(x)\, dx,\ \int g(y)\, dy$, taken between corresponding limits, are equal; and conversely, if these areas are always equal ($f$ and $g$ being implicitly assumed to be of constant sign), then $p = f(x)/g(y)$; the converse naturally serves to apply the theorem to the solution of differential equations (by "separation of variables"). But Barrow inserted the theorem only in an appendix (Lect. XII, app. III, theor. IV), where, observing that many of his previous results are only particular cases, he excused himself as having discovered it too late to make more use of it.

Thus, around 1670, the situation was the following. One knew how to treat problems which reduce to the first derivative by uniform procedures, and Huygens had treated geometric questions which reduce to the second derivative. One knew how to reduce all problems of integration to quadratures; one had various techniques, of a geometric aspect, for reducing quadratures one to another, in the badly classified cases, and one was accustomed, from this point of view, to the handling of the trigonometric and logarithmic functions; one had become aware of the link between differentiation and integration; one had begun to tackle the “method of inverse tangents”, a name given at this time to problems which reduce to differential equations of the first order. The sensational discovery of the series $\log(1 + x) = - \sum_{n=1}^{\infty} (-x)^n / n$ by Mercator had just opened totally new perspectives on the application of series, and principally of power series, to problems that had been called “impossible”. On the other hand, the ranks of the mathematicians had been very much thinned: Barrow had resigned his professorial chair for that of a preacher; Huygens apart (who had almost all his mathematical œuvre behind him, having obtained already all the principal results of the Horologium Oscillatorium which he was setting himself to edit definitively), only Newton at Cambridge, and J. Gregory isolated at Aberdeen were active; and Leibniz would soon join them with neophyte ardour. All three, Newton already from 1665, J. Gregory from the publication of Mercator in 1668, Leibniz from about 1673, devoted themselves principally to the topic of the day, the study of power series. But, from the point of view of the classification of problems, the principal effect of the new methods seemed to be to obliterate all distinctions between them; and indeed Newton, more analyst than algebraist, did not hesitate to announce to Leibniz in 1676 (XXII), p. 224) that he knew how to solve all differential equations $^{15}$; to which Leibniz responded ((XXII), p. 248-249) that he, on the contrary, was concerned to obtain the solution in finite terms whenever he could “assuming the quadratures”, and also to know whether every quadrature could be reduced to those of the circle and hyperbola as was claimed in most of the cases already studied; in this connection he recalled that Gregory believed (with reason, as we know today) that the rectification of the ellipse and hyperbola were not reducible to the quadratures of the circle and hyperbola; and Leibniz asked up to what point the method of series, such as Newton employs, could give replies to these questions. Newton, for his part ((XXII), p. 209-211), declared himself in possession of criteria, which he did not indicate, for deciding, apparently by examining the series, on the “possibility” of certain quadratures (in finite terms), and gave a (very interesting) example of a series for the integral $\int x^a (1 + x)^b \, dx$.

One sees the immense progress made in less than ten years: the questions of classification were already stated in these letters in fully modern terms; if the one that Leibniz raised was solved in the XIX$^{th}$ century by the theory of abelian integrals, the other, on the possibility of reducing a given differential equation to quadratures, is still open despite important recent work. If this was so, it was that Newton and Leibniz, each on his own account, had already reduced the fundamental operations of the infinitesimal calculus to an algorithm; it sufficed to write, in the notation used by one or the other, a problem of quadrature or a differential equation, for its algebraic structure to appear immediately, disengaged from its geometric coating; the methods

15 In the course of the exchange of letters, which was not done directly between the interested parties, but officially through the secretary of the Royal Society as intermediary, Newton “staked his claim”, stating his method as follows: 5accdae 10effh 12i ...rrrssssstttuuu, an anagram in which one finds encapsulated the method of solution by a power series with undetermined coefficients ((XXII), p. 224).

of "transmutation" also can be written in simple analytic terms; the problems of classification are stated in a precise fashion. Mathematically speaking, the XVII$^{th}$ century had reached its end.

E) Interpolation and the calculus of differences. This theme (from which we do not separate the study of the binomial coefficients) appeared early and continued through the century, for reasons simultaneously theoretical and practical. One of the great tasks of the time was in fact the calculation of trigonometric, logarithmic, and nautical tables, made necessary by the rapid progress of geography, navigation, theoretical and practical astronomy, of physics, of celestial mechanics; and many of the most eminent mathematicians, from Kepler to Huygens and Newton, participated, either directly, or by theoretical research into the most efficient processes of approximation.

One of the first problems, in the use and even the preparation of tables, was that of interpolation; and as the precision of calculations increased, it was perceived in the XVII$^{th}$ century that the ancient procedure of linear interpolation loses its validity as the first differences (differences between successive values in the table) cease to be perceptibly constant; thus one sees Briggs, for example [^2], make use of differences of higher order, and even of rather high order, in the calculation of logarithms. Later, we see Newton ((XIX d) and (XX), Book III, Lemma 5) [^3] and J. Gregory ((XVII bis), p. 119-120), each on his own, pursue in parallel their research on interpolation and on power series; both arrived, moreover by different methods, on the one hand at the formula for interpolation by polynomials, called "Newtonian", and on the other at the binomial series ((XVII bis), p. 131; (XXII), p. 180) and at the principal power series expansions of classical analysis ((XVII bis); (XIX a and d) and (XXII), p. 179-192 and 203-225); it is hardly in doubt that these two lines of research reacted on one another, and were intimately linked also in the mind of Newton to the discovery of the principles of the infinitesimal calculus. Great concern for practical numerical work appears in Gregory as in Newton, in construction and usage of tables, in numerical calculation of series and integrals; in particular, although one does not find any careful proof of convergence, of the kind of that of Lord Brouncker cited above, both make constant mention of the convergence of their series from the practical point of view of their aptness for calculation. This is again how we see Newton, in reply to a question posed by Collins for practical purposes [^4] apply the so-called Euler-Maclaurin summation method to the approximate calculation of $\sum_{p=1}^{N} \frac{1}{n+p}$ for large values of N.

One also encounters early on the calculation of values of a function starting from the differences, employed as a practical procedure for integration, and even, might one say, of integration of a differential equation. Thus Wright, in 1599, having, with a view to nautical tables, to solve a problem which we would write as

proceeded by adding the values of sec $t$, by successive intervals of a second of arc $^{19}$ obtaining naturally, to all intents and purposes, a table of values of log tan $(\pi/4 + t/2)$; and this coincidence, observed since the calculation of the first tables of log tan $t$, remained unexplained until the integration of sec $t$ by Gregory in 1668 ((XVII c) and (XVII bis) p. 7 and 463).

But these questions also have a purely theoretical and even arithmetical aspect. Let us agree to denote by $\Delta' x_n$ the sequences of successive differences of a sequence $(x_n)_{n \in \mathbf{N}}$, defined by recurrence by means of $\Delta x_n = x_{n+1} - x_n$, and

and to denote by $S'$ the inverse operation of $\Delta$ and its iterates, thus putting $y_n = S.x_n$ if $y_0 = 0, \Delta y_n = x_n$, and $S'.x_n = S({S'}^{-1} x_n)$; one has $S' x_n = \sum_{p=0}^{n-r} \binom{n-p-1}{r-1} x_p$,

and, in particular, if $x_n = 1$ for all $n$, one has $S.x_n = n$, and the sequences $S^2 x_n$ and $S^3 x_n$ are those of the "triangular" and "pyramidal" numbers already studied by the Greek arithmeticians, and in general $S^r x_n = \binom{n}{r}$ for $n \geq r$ (and $S' x_n = 0$ for $n < r$); these sequences were introduced, from this point of view, certainly as early as the XVI$^{th}$ century; they appeared of their own accord also in the combinatorial problems, which, either on their own, or in connection with probabilities, played rather a large rôle in the mathematics of the XVII$^{th}$ century, for example, with Fermat and Pascal, then with Leibniz. They also appeared in the expression for the sum of the $m^{th}$ powers of the first N integers, whose calculation, as we have seen, underlay the integration of $\int x^m dx$ for $m$ an integer, by the first method of Fermat ((XI), v. II, p. 83). This was how Wallis proceeded in 1655 in his Arithmetica Infinitorum (XV a) not knowing the (unpublished) work of Fermat, and also, he said, in ignorance of the method of indivisibles except by reading Torricelli; it is true that Wallis, keen to finish, did not delay himself with meticulous research: once he had achieved the result for the first integral values of $m$ he supposed it true "by induction" for every integer $m$, passed correctly to the case $m = 1/n$ for integer $n$, then by an "induction" yet more summary than the first, to arbitrary rational $m$. But the interest and originality of his work is that he raised himself progressively from there to the study of the "Eulerian" integral $I(m, n) = \int_0^1 (1 - x^{1/m})^n dx$ (of which the value, for $m$ and $n > 0$ is $\Gamma(m+1)\Gamma(n+1)/\Gamma(m+n+1)$) and other similar integrals, drew up the table of values of $1/I(m, n)$ for integers $m$ and $n$, which is none other than that of the integers $\binom{m+n}{n}$, and, by methods almost identical to those

19 Ed. Wright, Table of Latitudes, 1599 (cf. Napier Memorial Volume, 1914, p. 97).

which one uses today in expounding theory of the $\Gamma$ function, ended at the infinite product for $\mathrm{I}(\frac{1}{2}, \frac{1}{2}) = \pi / 4 = (\Gamma(\frac{3}{2}))^2$; it was not difficult, besides, to render his method correct by integrations by parts and very simple changes of variables, and by the consideration of $\mathrm{I}(m, n)$ for all real values of $m$ and $n$, which he could hardly have thought of, but which the Newtonian analysis would soon make possible. In any case it was the "interpolation" effected by Wallis of the integers $\binom{m+n}{n}$ to non-integral values of $m$ (more precisely to values of the form $n = p/2$, with $p$ an odd integer) which served as a point of departure for Newton when he started out ((XXII), p. 204-206), leading him, first to study the particular case $(1 - x^2)^{p/2}$, to the binomial series, then to the introduction of $x^a$ (thus denoted) for every real $a$. and to the differentiation of $x^a$ by means of the binomial series; all this without a great effort to obtain proofs or even rigorous definitions; further, a remarkable innovation, was that from knowledge of the derivative of $x^a$ he deduced $\int x^a \, dx$ for $a \neq -1$ ((XIX a) and (XXII), p. 225). For the rest, although he was soon in possession of much more general methods for expansion in power series, such as the so-called Newton polygonal method (for algebraic functions) ((XXII), p. 221) and that of undetermined coefficients, he returned many times later, with a sort of predilection, to the binomial series and its generalizations; it was from there, for example, that he seems to have derived the expansion of $\int x^a (1 + x)^{\beta} \, dx$ mentioned above ((XXII), p. 209).

The evolution of these ideas on the continent, however, was very different, and much more abstract. Pascal had drawn close to Fermat in the study of the binomial coefficients (from which he formed and named the "arithmetic triangle") and their use in the calculus of probabilities and the calculus of differences; when he tackled integration he introduced the same ideas there. Like his predecessors, when he used the language of indivisibles, he conceived the integral $F(x) = \int_0^x f(x) \, dx$ as the value of the ratio of "the sum of all the ordinates of the curve"

to the "unit" $N = \sum_{0 \leq p < N} 1$ for $N = \infty$ ((XII b), p. 352-355) (or, when he abandoned this language for the correct language of exhaustion, as the limit of this ratio as $N$ increases indefinitely). But, having the problems of moments in view, he observed that, when dealing with discrete masses $y_i$ distributed at equidistant intervals, the calculation of the total mass comes back to the operation $S y_n$ defined above, and the calculation of the moment to the operation $S^2 y_n$; and, by analogy, he iterated the operation $\int$ to form what he called the "triangular sums of the ordinates", thus, in our language, the limits of the sums $N^{-2} S^2 (f(n/N))$, that is to say the integrals $F_2(x) = \int_0^x F(x) \, dx$; a new iteration gave him the "pyramidal sums" $F_3(x) = \int_0^1 F_2(x) \, dx$. the limits of $N^{-3} S^3 (f(n/N))$; the context makes clear that it was not for lack of originality either in his thought or his language that he stopped here, but only because he anticipated using only these, whose systematic employment is at the base of a good part of his results, and for which he immediately proved the properties which we would write as $F_2(x) = \int_0^x (x - u) f(u) du,\quad F_3(x) = \frac{1}{2} \int_0^x (x - u)^2 f(u) du$ ((XII b), p. 361-367); all this without writing a single formula, but in a language so transparent and so precise that one can immediately transcribe it into formulae as we have just done. With Pascal, as with his predecessors, but in a much clearer and systematic manner, the choice of the independent variable (which is always one of the coordinates, or else the arc of a curve) is implicit in the convention which fixes the equidistant (though “infinitely close”) points of subdivision of the interval of integration; these points, according to the case, are either on Ox or on Oy or on the arc of the curve, and Pascal takes care never to leave any ambiguity on this subject ((XII b), p. 368-369). When he has to change variable he does so by means of a principle which amounts to saying that the area $\int f(x) dx$ can be written $S(f(x_i) \Delta x_i)$ for every subdivision of the interval of integration into “infinitely small” intervals $\Delta x_i$, equal or not ((XII d), p. 61-68).

As one sees, we are already very close to Leibniz; and it was, one might say, a lucky accident that he, when he wished to be initiated into modern mathematics, had met Huygens, who immediately put the writings of Pascal into his hands ((XXII), p. 407-408); he was particularly prepared by his thoughts on combinatorial analysis, and we know that he made a profound study of it, as is reflected in his œuvre. In 1675, we see him transcribe the theorem of Pascal cited above, in the form $\mathrm{omn}(x \omega) = x.\mathrm{omn}\ \omega - \mathrm{omn}(\mathrm{omn}\ \omega)$, where $\mathrm{omn}\ \omega$ is an abbreviation for the integral of $\omega$ taken from 0 to $x$, for which Leibniz, several days later, substituted $\int \omega$ (the initials of “summa omnium $\omega$” ) at the same time as he introduced $d$ for the infinitely small “difference”, or as he would soon say, the differential ((XXII), p. 147-167). Conceiving these “differences” as quantities comparable among themselves though not with finite quantities, he nevertheless most often took, explicitly or not, the differential $dx$ of the independent variable $x$ as unity, $dx = 1$ (which amounts to identifying the differential $dy$ with the derivative $dy/dx$), and at first omitted it from his notation for the integral, which thus appeared as $\int y$ rather than as $\int y dx$; but he scarcely delayed introducing this latter, and kept to it systematically once he perceived its invariant character with respect to the choice of independent variable, which dispenses with having this choice constantly present in mind ^20 ; and he showed no little satisfaction when he returned to the study of Barrow, whom he had neglected until then, in noting that the general theorem on change of variable, of which Barrow makes so much, follows immediately from his own notation ((XXII), p. 412). Moreover in all this he kept very close to the calculus of differences, from which his differential calculus was deduced by a passage to the limit which of course he would have had to take great pains to justify rigorously; and in what followed he insisted deliberately on the fact that his principles apply equally well to one and to the other. He cited Pascal expressly, for example, when, in his correspondence with

^20 "J'avertis qu'on prenne garde de ne pas omettre dx ... faute fréquemment commise, et qui empêche d'aller de l'avant, du fait qu'on ôte par là à ces indivisibles, comme ici dx, leur généralité ... de laquelle naissent d'innombrables transfigurations et équipollences de figures." ((XXI b), p. 233).

Johann Bernoulli ((XXI), v. III, p. 156), referring to his first researches, he gave a formula from the calculus of differences which is a special case of that of Newton, and from it deduced by “passing to the limit” the formula $y = \sum_{1}^{\infty} (-1)^{n-1} \frac{d^n y}{dx^n} \frac{x^n}{n!}$ (where $y$ is a function vanishing for $x = 0$, and the $d^n y / dx^n$ are its derivatives for the value $x$ of the variable), a formula equivalent to a similar one which Bernoulli had just communicated to him ((XXI), v. III, p. 150) and (XXIV), v. I, p. 125-128), and which the latter proved by successive integrations by parts. This formula, as one sees, is very close to the Taylor series; and it was the same argument as Leibniz, by passage to the limit starting from the calculus of differences, that Taylor rediscovered in 1715 to obtain “his” series, $^{21}$ without however making great use of it.

F) One will already have seen, implicit in the evolution described above, the progressive algebraisation of the infinitesimal calculus, that is to say its reduction to an operational calculus endowed with a uniform system of notation of algebraic character. As Leibniz had indicated many times with perfect clarity ((XXI b) p. 230-233), it was a matter of doing for the new analysis what Viète had done for the theory of equations, and Descartes for geometry. To understand the need for this, one has only to read a few pages of Barrow; at no time can one cope without having under one’s eyes a sometimes complicated figure, described meticulously beforehand; he needs no fewer than 180 figures for the 100 pages (Lect. V-XII) which form the essence of his work.

There could hardly be a question of algebraisation, it was true, before some unity had appeared across the multiplicity of geometric appearances. However, Grégoire de St. Vincent (IX) had already introduced (under the name “ductus plani in planum”) a sort of law of composition which amounts to using systematically the integrals $\int_a^b f(x)g(x)\, dx$, considered as volumes of solids $a \leq x \leq b, 0 \leq y \leq f(x), 0 \leq z \leq g(x)$; but he was far from drawing the consequences which Pascal later deduced, as one has seen, from studying the same solid. Wallis in 1655, and Pascal in 1658, forged, each in his own way, languages of algebraic character, in which, without writing any formula, they drafted statements which one can transcribe immediately into formulae of the integral calculus as soon as one has understood the mechanism. The language of Pascal is particularly clear and precise; and, if one does not understand why he refused to use the algebraic notation, not only of Descartes, but even that of Viète, one can only admire the tour de force which he accomplished, and which his mastery of language alone made possible.

But let several years pass and all changes. Newton was the first to conceive the idea of replacing all operations of a geometric character, of contemporary infinitesimal

21 B. TAYLOR, Methodus Incrementorum directa et inversa, Lond., 1715. For the calculus of differences Taylor could naturally rely on the results of Newton, contained in a famous lemma of the Principia ((XX). Book III, lemma 5) and published more fully in 1711 (XIX d). As to the idea of passing to the limit, it seems to be typically Leibnizian; and one could hardly believe in the originality of Taylor on this point if one did not know many examples from all periods of disciples ignorant of all apart from the writings of their master and patron. Taylor cited neither Leibniz nor Bernoulli; but the Newton-Leibniz controversy was raging, Taylor was secretary of the Royal Society, and Sir Isaac was its all-powerful president.

analysis, by a single analytic operation, differentiation, and by the solution of the inverse problem; an operation which of course the method of power series allowed him to execute with extreme facility. Borrowing his language, we have seen that, with the fiction of a universal “temporal” parameter, he styles as “fluentes” the variable quantities as a function of this parameter, and their derivatives as “fluxions”. He does not seem to have attached a particular importance to notation, and his devotees later vaunted the absence of a systematic notation as an advantage; nevertheless, for his personal use, he soon adopted the habit of denoting the fluxion by a point, thus $dx/dt$ by $\dot{x}$. $d^2 x/dt^2$ by $\ddot{x}$, etc. As to integration, it seems that Newton, just like Barrow, had never envisaged it save as a problem (to find the fluent knowing the fluxion, to solve $\dot{x} = f(t)$), and not as an operation; also he has no name for the integral, nor, it seems, a standard notation (except sometimes a square, $\int \overline{f(t)}$ or $\square f(t)$ for $\int f(t)\,dt$). Was it because he was reluctant to give a name and a sign to what was not defined in a unique manner, but only up to an additive constant? For lack of a text one can only ask the question.

As much as Newton was empirical, correct, circumspect in his greatest boldnesses, so Leibniz was systematic, a generalizer, adventurous innovator and sometimes boastful. From his youth he had in his head the idea of a “characteristic” or universal symbolic language, which would be to the total of human thought what algebraic notation is to algebra, where every name or sign would be the key to all the qualities of the thing signified, and which one could not use correctly without as a result reasoning correctly. It is easy to treat such a project as chimerical; yet it is not an accident that its author was the very man who would soon recognise and isolate the fundamental concepts of the infinitesimal calculus, and endow it with its more or less definitive notation. We have already witnessed above its birth, and observed the care with which Leibniz, who seemed aware of his mission, modified them progressively to ensure the simplicity and above all the invariance that he sought (XXI $a$ and $b$). What is important to remark here is the clear concept of $\int$ and of $d$, of the integral and of the differential, as mutually inverse operators, from when he introduced them (knowing nothing yet of the ideas of Newton). It is true that in proceeding in this way he could not escape from the ambiguity inherent in the indefinite integral, which was the weak point of his system, over which he skates adroitly, as do his successors. But what strikes one, from the first appearance of the new symbols, is to see Leibniz immediately busy formulating their rules of use, asking if $d(xy) = dx\,dy$ ((XXII), p. 16-166), and answering himself in the negative, then coming progressively to the correct rule (XXI $a$), which he would later generalize to his famous formula for $d^n(xy)$ ((XXI), v. III, p. 175). Of course, at the moment that Leibniz was groping in this way, Newton had already known for ten years that $z = xy$ implies $\dot{z} = \dot{x}\,y + x\,\dot{y}$; but he never took the trouble to say so, not seeing in it anything other than a special case, not worthy of naming, of his rule for differentiating a relation $F(x,\ y,\ z) = 0$ among fluents. On the contrary, the principal concern of Leibniz was not to make his methods serve for the solution of such concrete problems, nor even to deduce them from rigorous and impregnable principles, but above all to set up an algorithm resting on the formal manipulation of a few simple rules. It was in this spirit that he improved algebraic notation by the use of parentheses, that he progressively adopted $\log x$ or $l_x$ for the logarithm $^{22}$, and that he insisted on the "exponential calculus", that is the systematic consideration of exponentials $a^x, x^x, x^y$, where the exponent is a variable. Above all, while Newton did not introduce fluxions of higher order except as strictly necessary in each concrete case, Leibniz early oriented himself towards the creation of an "operational calculus" by the iteration of $d$ and of $\int$; little-by-little becoming aware of the analogy between multiplication of numbers and the composition of operators in his calculus, he adopted, with happy audacity, the notation for exponents to write the iterates of $d$, thus writing $d^n$ for the $n^{th}$ iterate ((XXII), p. 595 and 601 $^{23}$, and (XXI), v. V, p. 221 and 378) and even $d^{-1},\ d^{-n}$ for $\int$ and its iterates ((XXI), v. III, p. 167); and he even tried to give a meaning to $d^\alpha$ for arbitrary real $\alpha$ ((XXI), v. II, p. 301-302, and v. III, p. 228).

This is not to say that Leibniz was not also interested in the applications of his calculus, well knowing (as Huygens often repeated to him ((XXII), p. 599)) that they are the touchstone; but he lacked the patience to go deeply into them, and above all he sought the opportunity to formulate new general rules. It was thus that in 1686 (XXI c) he treated the curvature of curves, and the osculating circle, to end up in 1692 (XXI d) with the general principles of the contact of plane curves $^{24}$; and in 1692 (XXI e) and 1694 (XXI f) he set out the bases for the theory of envelopes; concurrently with Johann Bernoulli, he effected in 1702 and 1703 the integration of rational fractions by decomposition into simple elements, but first in a formal manner and without fully appreciating the circumstances which accompany the presence of complex linear factors in the denominator (XXI, g and h). It was thus again that one August day in 1697, meditating in his carriage on questions of the calculus of variations, he had the idea for the rule for differentiation with respect to a parameter under the $\int$ sign, and enthusiastically sent it off immediately to Bernoulli ((XXI, v. III, p. 449-454). But when he got there, the fundamental principles of his calculus had been established a long time, and their use had begun to spread: the algebraisation of infinitesimal analysis was an accomplished fact.

G) The notion of function was introduced and clarified in many ways during the XVII$^{th}$ century. All kinematics rests on an intuitive, in some way experimental, idea of quantities that vary in time, that is, functions of time, and we have already seen how one thus comes to a function of a parameter, such as appears with Barrow, and, under

$^{22}$ But he had no symbol for the trigonometric functions, nor (lacking a symbol for $e$) for the "the number whose logarithm is $x$". $^{23}$ ... "c'est à peu près comme si, au lieu des racines et puissances, on vouloit toujours substituer des lettres, et au lieu de $xx$, ou $x^3$, prendre $m$, ou $n$, après avoir déclaré que ce doivent estre les puissances de la grandeur $x$. Jugés, Mons., combien cela embarasseroit. Il en est de mesme de $dx$ ou de $ddx$, et les differences ne sont pas moins des affections des grandeurs indéterminées dans leur lieux, que les puissances sont des affections d'une grandeur prise à part. Il me semble donc qu'il est plus naturel de les désigner en sorte qu'elles fassent connoistre immédiatement la grandeur dont elles sont les affections." $^{24}$ He first committed a singular error on that point, believing that the "kissing circle" (the osculating circle) has four points in common with the curve at the point of contact; he surrendered later on this matter, only with difficulty, to the objections of the Bernoulli brothers ((XXI), v. III, p. 187-188, 201-202 and 207).

the name fluent, with Newton. The notion of an "arbitrary curve" appeared often, but was rarely made precise; it could be that it was often thought of in a kinematic or in any case experimental form, and without judging it necessary for a curve to be susceptible of a geometric or analytic characterisation in order to serve as an object of argument; so it was, in particular (for reasons which we are better able to understand today) when it concerns integration, for example with Cavalieri, Pascal and Barrow; the latter, reasoning about the curve defined by $x = ct,\ y = f(t)$, with the hypothesis that $dy/dt$ should be increasing, even said expressly that "it does not matter" that $dy/dt$ increase "regularly according to some law, or even irregularly" ((XVIII), p. 191) that is, as we would say, be susceptible or not to an analytic definition. Unfortunately this clear and fruitful idea, which would, suitably clarified, reappear in the XIX\textsuperscript{th} century, could not then fight against the confusion created by Descartes, when the latter had, in the first place, banned from "geometry" all curves not susceptible of a precise analytic definition, and in the second place, restricted the admissible procedures of formation in such a definition to algebraic operations alone. It is true that, on this last point, he was not followed by the majority of his contemporaries; little-by-little, and often by very subtle detours, the various transcendental operations, the logarithm, the exponential, the trigonometric functions, quadratures, solution of differential equations, passage to the limit, summation of series, became established, though it is not easy in each case to mark the precise moment when the step forward was made; and, moreover, the first step forward was often followed by a step back. For the logarithm, for example, one must consider as important stages the appearance of the logarithmic curve ($y = a^x$ or $y = \log x$ according to the choice of axes), of the logarithmic spiral, the quadrature of the hyperbola, the series expansion of $\log(1 + x)$, and even the adoption of the symbol $\log x$ or $l x$. In what concerns the trigonometric functions, which in a certain sense reach back to antiquity, it is interesting to observe that the sinusoid did not first appear as defined by an equation $y = \sin x$, but with Roberval ((VIII a), p. 63-65), as a "companion of the cycloid" (so as a case of the curve

As for Leibniz, he seemed to hold to the Cartesian point of view, widened by the explicit adjunction of quadratures, and by the implicit adjunction of the other operations familiar to the analysis of his time: summation of power series, solution of differential equations. Likewise, Johann Bernoulli, when he wanted to consider an arbitrary function of $x$, introduced it as "a quantity formed in any way starting from $x$ and the constants" ((XXI), v. II, p. 150), sometimes specifying that it concerns a quantity formed "in an algebraic or transcendental manner" ((XXI), v. II, p. 324); and, in 1698, he agreed with Leibniz to call such a quantity a "function of $x$" ((XXI), v. III, p. 507-510 and p. 525-526) 26. Leibniz had already introduced the words "constant", "variable", "parameter", and refined, à propos envelopes, the notion of a family of curves depending on one or more parameters (XXI e). The questions of notation were clarified also in the correspondence with Johann Bernoulli: the latter readily writes X or $\xi$, for an arbitrary function of $x$ ((XXI), v. III, p. 531); Leibniz approved, but also proposed $x^{-|1|}$, $x^{-|2|}$, where we would write $f_1(x)$, $f_2(x)$; and he proposed, for the derivative $dz/dx$ of a function $z$ of $x$, the notation dx (in contrast to $dz$, which is the differential) while Bernoulli wrote $\Delta z$ ((XXI), v. III, p. 537 and 526).

Thus, with the century, the heroic epoch was over. The new calculus, with its notions and notation, was established, in the form which Leibniz had given it. The first disciples, Jakob and Johann Bernoulli, were rivals in discovery with the master, browsing through the rich expanses to which he had shown them the way. The first treatise on differential and integral calculus was written in 1691 and 1692 by Johann Bernoulli 27, for the benefit of a marquis who showed himself an apt pupil. It hardly mattered that Newton decided at last, in 1693, to publish a parsimoniously brief glimpse of his fluxions ((XV), v. II, p. 391-396); if his Principia provided food for thought for more than a century, on the terrain of the infinitesimal he had been caught up with, and on many points overtaken.

26 Until then, and already in a manuscript of 1673, Leibniz had employed this word as an abbreviation to denote a quantity "remplissant telle ou telle fonction" for a curve, for example the length of the tangent or of the normal (bounded by the curve and O.x), or even the subnormal, the subtangent, etc., thus, in sum, a function of a variable point on a curve, with a geometrico-differential definition. In the same manuscript of 1673 the curve is assumed to be defined by a relation between $x$ and $y$, "donnée par une équation", but Leibniz adds that "il n'importe pas que la courbe soit ou non géométrique" (that is to say, in our language, algebraic) (cf. D. Mainke, Abh. Preuss. Akad. der Wiss., 1925, Nr. 1, Berlin, 1926). 27 The part of this treatise on the integral calculus was published only in 1742 ((XXIV), v. III, p. 385-558); that which deals with the differential calculus was rediscovered and published only recently (XXIV bis); it is true that the Marquis de l'Hôpital had published it in French, slightly recast, under his own name, as Bernoulli remarked bitterly in his letters to Leibniz.

The weaknesses of the new system are nevertheless visible, at least to our eyes. Newton and Leibniz, abolishing at one blow a tradition of two millennia, accorded the primordial rôle to differentiation and reduced integration to be only its inverse; it would take all the XIX$^{th}$ century, and part of the XX$^{th}$, to reestablish a fair equilibrium, by putting integration at the basis of the general theory of functions of a real variable and of its modern generalizations (see the Historical Notes to the Book on Integration). This reversal of point of view was also responsible for the excessive, almost exclusive, rôle, given to the indefinite integral at the expense of the definite integral, already seen in Barrow, and above all in Newton and Leibniz: there also the XIX$^{th}$ century had to put things back in place. Finally, the characteristically Leibnizian tendency to formal manipulation of symbols continued to accentuate itself throughout the XVIII$^{th}$ century, well beyond what the resources of analysis at this time could justify. In particular, one must recognise that the Leibnizian notion of differential had, to tell the truth, no meaning; at the start of the XIX$^{th}$ century it fell into a disrepute from which it has been raised only little-by-little; and, if the employment of first differentials has been completely legitimised, the differentials of higher order, although so useful, have to this day not yet been truly rehabilitated.

However it may be, the history of the differential and integral calculus, starting from the end of the XVII$^{th}$ century, divides into two parts. The one relates to the applications of this calculus, always more rich, numerous and varied. To the differential geometry of plane curves, to differential equations, to power series, to the calculus of variations, referred to above, were added the differential geometry of skew curves, then of surfaces, multiple integrals, partial differential equations, trigonometric series, the study of numerous special functions, and many other types of problems, whose history will be expounded in the Books devoted to them. We treat here only the works which have contributed to fine-tune, deepen, and consolidate the very principles of the infinitesimal calculus, in what concerns functions of a real variable.

From this point of view, the great treatises of the middle of the XVIII$^{th}$ century offer few novelties. Maclaurin in Scotland $^{28}$, Euler on the continent (XXV $a$ and $b$), remained faithful to the traditions to which each was the heir. It is true that the first exerted himself to clarify the Newtonian concepts a little $^{29}$, while the second, pushing the Leibnizian formalism to its extreme, was content, like Leibniz and Taylor, to let the differential calculus rest on a very obscure passage to the limit starting from the calculus of differences, a calculus of which he gave a very careful exposition. But above all Euler completed the work of Leibniz in introducing the notation still in use today for $e$, $i$, and the trigonometric functions, and in spreading the notation $\pi$. On the other hand, and if most often he made no distinction between functions and analytic expressions, he insisted, à propos trigonometric series and the problem of vibrating strings, on the necessity of not restricting oneself to the functions so

$^{28}$ C. Maclaurin, A complete treatise of fluxions, Edinburgh, 1742. $^{29}$ They much needed, indeed, to be defended against the philosophico-theologico-humoristic attacks of the famous Bishop Berkeley. According to him, one who believed in fluxions could not find it difficult to have faith in the mysteries of religion: an argument ad hominem, lacking neither logic nor piquancy.

defined (and which he qualified as "continuous"), but to consider too, should the case arise, arbitrary, or "discontinuous", functions, given experimentally by one or more arcs of a curve ((XXV c), p. 74-91). Finally, although this goes a little outside our framework, it is impossible not to mention here his extension of the exponential function to the complex plane, whence he obtained his celebrated formulae linking the exponential with the trigonometric functions, as well as the definition of the logarithm of a complex number; there one finds elucidated definitively the famous analogy between the logarithm and the inverse circular functions, or, in the language of the XVII$^{th}$ century, between the quadratures of the circle and of the hyperbola, already observed by Grégoire de St. Vincent, made precise by Huygens and above all by Gregory, and which, with Leibniz and Bernoulli, appeared in the formal integration of

D'Alembert, meanwhile, the enemy of all mystique in mathematics as elsewhere, had, in remarkable articles (XXVI), defined with the greatest clarity the notions of limit and of derivative, and maintained forcefully that at bottom this is the whole "metaphysics" of the infinitesimal calculus. But this wise counsel did not have an immediate effect. The monumental work of Lagrange (XXVII) represents an attempt to found analysis on one of the most arguable of the Newtonian concepts, that which confuses the notions of an arbitrary function and of a function expandable in a power series, and to derive the notion of differentiation from that (by considering the coefficient of the term of first order in the series). Of course, a mathematician of the calibre of Lagrange could not fail to obtain important and useful results at that point, like, for example (and in a manner which was in reality independent of the point of departure just indicated) the general proof of the Taylor formula with remainder expressed as an integral, and its evaluation by the mean value theorem; besides, Lagrange's work was at the origin of Weierstrass' method in the theory of functions of a complex variable, as well as the modern algebraic theory of formal series. But, from the point of view of its immediate purpose, it represented a retreat rather than a progress.

With the teaching works of Cauchy, on the contrary (XXVIII), one finds oneself again on solid ground. He defined a function essentially as we do today, although in language still a little vague. The notion of limit, fixed once and for all, was taken as the point of departure; those of a continuous function (in the modern sense) and of the derivative are deduced immediately, as well as their principal elementary properties: and the existence of the derivative, instead of being an article of faith, becomes a question to study by the ordinary methods of analysis. Cauchy, to tell the truth, hardly interested himself in this; and on the other hand, if Bolzano, having come to the same principles, constructed an example of a continuous function not having a finite derivative at any point (XXIX), this example was not published, and the question was not settled publicly until Weierstrass, in a work of 1872 (and in his course from 1861) (XXXII).

As regards integration, the work of Cauchy represents a return to the sound traditions of antiquity and the first part of the xvii$^{th}$ century, but relying on still inadequate technical means. The definite integral, for a long time relegated to the second rank, again became the primordial notion, for which Cauchy adopted definitively the notation $\int_a^b f(x) \, dx$ proposed by Fourier (in place of the unwieldy $\int f(x) \, dx \left[ \begin{array}{c} x = b \\ x = a \end{array} \right]$ sometimes employed by Euler); and, to define it, Cauchy returned to the method of exhaustion, or as we would say, to “Riemann sums” (which it would be better to call Archimedes sums, or Eudoxus sums). It is true that the XVII$^{th}$ century never judged it apposite to subject the notion of area, which to it appeared at least as clear as that of an incommensurable real number, to critical examination; but the convergence of the “Riemann” sums towards the area under the curve, if only a monotone or piecewise monotone curve, was an idea familiar to all authors concerned with rigour in the XVII$^{th}$ century, such as Fermat, Pascal, Barrow; and J. Gregory, particularly well prepared by his thoughts on passage to the limit and his familiarity with an already very abstract form of the principle of “nested intervals”, had even drafted, it would appear, a careful proof, which remained unpublished ((XVII bis), p. 445-446), and could have served Cauchy almost without change, had he known it$^{30}$. Unfortunately for him, Cauchy claimed to prove the existence of the integral, that is the convergence of the “Riemann sums”, for an arbitrary continuous function; and his proof, though correct if based on the theorem on the uniform continuity of continuous functions on a closed interval, was denuded of all probative value for lack of this concept. Nor did Dirichlet seem to notice the difficulty when he composed his celebrated memoirs on trigonometric series, since he cited the theorem in question as “easy to prove” ((XXX), p. 136); it is true that, in the final analysis, he applied it only to bounded piecewise monotone functions; Riemann, more circumspect, mentions only these last, when it is a matter of using his necessary and sufficient condition for the convergence of the “Riemann sums” ((XXXI), p. 227-271). Once the theorem on uniform continuity had been established by Heine (cf. Gen. Top., II, Historical Note, p. 217), the question of course no longer offered any difficulty; and it was easily resolved by Darboux in 1875 in his memoir on the integration of discontinuous functions (XXXIII), a memoir where he happens to be in agreement on many points with the important researches of P. du Bois-Reymond, which appeared about the same time. As a result, one finds proved for the first time, but this time definitively, the linearity of the integral of continuous functions. On the other hand, the notion of uniform convergence of a sequence or a series, introduced by Seidel, among others, in 1848, and put to good use by Weierstrass (cf. Gen. Top., X, Historical Note, p. 347), had made it possible to give a solid basis, under conditions a little too restrictive, it is true, to the integration of series term-by-term, and differentiation under the $\int$ sign, while awaiting the modern theories of which we do not have to speak here, and which clarify these questions in a provisionally definitive manner.

We have thus reached the final stage of the classical infinitesimal calculus, that represented by the great Treatises on Analysis of the end of the XIX$^{th}$ century; from our point of view, that of Jordan (XXXIV) occupies the most eminent place among

$^{30}$ This at least is what the summary given by Turnbull from the manuscript would indicate.

them, in part for aesthetic reasons, but also because, if it constitutes an admirable exposition of the results of classical analysis, it announces modern analysis in many ways, and prepares the way for it. After Jordan came Lebesgue, and one enters into the subject of another Book of the present work.

(I) Euclidis Elementa, 5 vol., éd. J. L. Heiberg, Lipsiae (Teubner), 1883-88. (I bis) T. L. Heath, The thirteen books of Euclid’s elements ..., 3 vol., Cambridge, 1908. (II) Archimedis Opera Omnia, 3 vol., éd. J. L. Heiberg, 2e éd., Leipzig (Teubner), 1913-15. (II bis) Les Œvres complètes d’Archimède, trad. P. Ver Eecke, Paris-Bruxelles (Desclée-de Brouwer), 1921. (III) Galileo Galilei, Opere, Ristampa della Edizione Nazionale, 20 vol., Firenze (Barbera), 1929-39. (III bis) Discorsi e Dimonstrazioni Matematiche intorno à due nuoue scienze Attenti alla mecanica & i movimenti locali, del Signor Galileo Galilei Linceo, Filosofo e Matematico primario del Serenissimo Grand Duca di Toscana. In Leida, appresso gli Elsevirii, MDCXXXVIII. (IV) J. Neper, Mirifici logarithmorum canonis constructio, Lyon, 1620. (V) J. Kepler, Neue Stereometrie der Fäser, Ostwald’s Klassiker, n 165, Leipzig (Engelmann), 1908. (VI) B. Cavalieri: a) Geometria indivisibilibus continuorum quadam ratione promota, Bononiae, 1635 (2e éd., 1653); b) Exercitationes geometricae sex, Bononiae, 1647. (VII) E. Torricelli, Opere, 4 vol., éd. G. Loria et G. Vassura, Faenza (Montanari), 1919. (VII bis) E. Torricelli, in G. Loria, Bibl. Mat. (III), t. I, 1900, p. 78-79. (VIII) G. de Roberval, Ouvrages de Mathématique (Mémoires de l’Academie Royale des Sciences, t. III), Amsterdam, 1736: a) Observations sur la composition des mouvements, et sur le moyen de trouver les touchantes des lignes courbes, p. 3-67; b) Epistola ad Torricellium, p. 363-399. (IX) P. Gregorii a Sancto Vincentio, Opus Geometricum Quadraturae Circuli et Sectionum Coni ..., 2 vol., Antverpiae, 1647. (X) R. Descartes, Œuvres, éd. Ch. Adam et P. Tannery, 11 vol., Paris (L. Cerf), 1897-1909. (X bis) R. Descartes, Geometria, trad. latine de Fr. van Schooten, 2e éd., 2 vol., Amsterdam (Elzevir), 1659-61.

(XI) P. FERMAT, Œuvres, Paris (Gauthier-Villars), 1891-1912: a) De linearum curvarum cum lineis rectis comparatione ... Auctore M. P. E. A. S., t. I, p. 211-254 (trad. française, ibid., t. III, p.181-215); b) Methodus ad disquirendam maximam et minimam, t. I, p. 133-179 (trad. française, ibid., t. III, p. 121-156); c) De aequationum localium transmutatione et emendatione ad multimodam curvilineorum inter se vel cum rectilineis comparationem, cui annectitur proportionis geometricae in quadrandis infinitis parabolis et hyperbolis usus, t. I, p. 255-288 (trad. française, ibid., t. III, p. 216-240).

(XII) B. PASCAL, Œuvres, 14 vol., éd. Brunschvicg, Paris (Hachette), 1904-14: a) Lettre de A. Dettonville à Monsieur A. D. D. S. en lui envoyant la démonstration à la manière des anciens de l’égalité des lignes spirale et parabolique, t. VIII, p. 249-282; b) Lettre de Monsieur Dettonville à Monsieur de Carcavy, t. VIII, p. 334-384; c) Traité des trilignes rectangles et de leurs onglets, t. IX, p. 3-45: d) Traité des sinus du quart de cercle, t. IX, p. 60-76.

(XIII) N. MERCATOR, Logarithmotechnia ... cui nunc accedit vera quadratura hyperbolae ... Londini, 1668 (reproduced in F. MASÈRES, Scriptores, Logarithmici ..., vol. I, London, 1791, p. 167-196).

(XIV) LORD BROUNCKER, The Squaring of the Hyperbola by an infinite series of Rational Numbers, together with its Demonstration by the Right Honourable the Lord Viscount Brouncker, Phil. Trans., v. 3 (1668), p. 645-649 (reproduced in F. MASÈRES, Scriptores, Logarithmici ..., vol. I, London, 1791, p. 213-218).

(XV) J. WALLIS, Opera Mathematica, 3 vol., Oxoniae, 1693-95: a) Arithmeticca Infinitorum, t. I, p. 355-478.

(XV bis) J. WALLIS, LOGARITHMOTECHNIA NICOLAI MERCATORIS: discoursed in a letter written by Dr. J. Wallis to the Lord Viscount Brouncker, Phil. Trans., v. 3 (1668), p. 753-759 (reproduced in F. MASÈRES, Scriptores Logarithmici ..., vol. I, London, 1791, p. 219-226).

(XVI) CH. HUYGENS, Œuvres complètes, publiées par la Société Hollandaise des Sciences, 19 vol., La Haye (Martinus Nijhoff), 1888-1937: a) Examen de "Vera Circuli et Hyperboles Quadratura ...", t. VI, p. 228-230 (= Journal des Scavans, juillet 1668); b) Horologium Oscillatorium, t. XVIII; c) Theoremata de Quadratura Hyperboles, Ellipsis et Circuli ..., t. XI, p. 271-337; d) De Circuli Magnitudine Inventa ..., t. XII, p. 91-180.

(XVI bis) Christiani Hugenii, Zulichemii Philosophi, vere magni, Dum viveret Zelemii Toparchae, Opera Mechanica, Geometrica, Astronomia et Miscellanea, 4 tomes en 1 vol., Lugd. Batav., 1751.

(XVII) J GREGORY: a) Vera Circuli et Hyperbolae Quadratura ..., Pataviae, 1667; b) Geometriae Pars Universalis, Pataviae, 1668; c) Exercitationes Geometricae, London, 1668.

(XVII bis) James Gregory Tercentenary Memorial Volume, containing his correspondence with John Collins and his hitherto unpublished mathematical manuscripts . . . , ed. H. W. Turnbull, London (Bell and Sons), 1939.

(XVIII) I. Barrow, Mathematical Works, ed. W. Whewell, Cambridge, 1860.

(XVIII bis) Lectiones Geometricae: In quibus (praesertim) GENERALIA Curvarum Linearum SYMPTOMATA DECLARANTUR. Auctore Isaac Barrow . . . Londini . . . M.DC.LXX (= Mathematical Works, p. 156-320).

(XIX) I. Newton, Opuscula, t. I, Lausanne-Genève (M. M. Bousquet), 1744: a) De Analysi per aequationes numero terminorum infinitas, p. 3-26; b) Methodus fluxionum et serierum infinitarum, p. 29-199 (1st publication 1736); c) De Quadratura curvarum, p. 201-244 (1st publication 1706); d) Methodus differentialis, p. 271-282 (1st publication 1711); e) Commercium Epistolicum, p. 295-420 (1st publication 1712, 2nd ed. 1722).

(XX) I. Newton, Philosophiae Naturalis Principia Matematica, London, 1687 (new edition, Glasgow, 1871).

(XX bis) I. Newton, Mathematical principles of natural philosophy, and his System of the world, transl. into English by A. Motte in 1729, Univ. of California, 1946.

(XXI) G.W. Leibniz, Mathematische Schriften, éd. C. I. Gerhardt. 7 vol., Berlin-Halle (Asher-Schmidt), 1849-63: a) Nova Methodus pro Maximis et Minimis . . . , t. V, p. 220-226 (= Acta Erud. Lips., 1684); b) De Geometria recondita . . . , t. V, p. 226-233 (= Acta Erud. Lips., 1686); c) Meditatio nova de natura Anguli contactus . . . , t. VII, p. 326-328 (= Acta Erud. Lips., 1686); d) Generalia de Natura linearum . . . , t. VII, p. 331-337 (= Acta Erud. Lips., 1692); e) De linea ex lineis numero infinitis . . . , t. V, p. 266-269 (= Acta Erud. Lips., 1692); f) Nova Calculi differentialis applicatio . . . , t. V, p. 301-306 (= Acta Erud. Lips., 1694); g) Specimen novum Analyseos . . . , t. V, p. 350-361 (= Acta Erud. Lips., 1702); h) Continuatio Analyseos Quadraturarum Rationalium, t. V, p. 361-366 (= Acta Erud. Lips., 1703).

(XXII) Der Briefwechsel von Gottfried Wilhelm Leibniz mit Mathematikern, herausg. von C. I. Gerhardt, t. I, Berlin (Mayer und Müller), 1899.

(XXIII) Jakob Bernoulli, Opera, 2 vol., Genève (Cramer-Philibert), 1744.

(XXIV) Johann Bernouilli, Opera Omnia, 4 vol., Lausanne-Genève (M. M. Bousquet), 1742.

(XXIV bis) Johann Bernouilli: Die erste Integralrechnung, Ostwald’s Klassiker, n 194, Leipzig-Berlin, 1914; Die Differentialrechnung, Ostwald’s Klassiker, n 211, Leipzig, 1924;

(XXV) L. Euler, Opera Omnia: a) Institutiones calculi differentialis, (1) t. X, Berlin (Teubner), 1913; b) Institutiones calculi integralis, (1), t. XI-XIII, Berlin (Teubner), t. 1, 1913-14; c) Commentationes analyticae . . . (1), t. XXIII, Berlin-Zürich (Teubner-O. Füssli), 1938.

(XXVI) J. d’Alembert: a) Sur les principes métaphysiques du Calcul infinitésimal (Mélanges de littérature, d’histoire et de philosophie, nouv. éd., t. V, Amsterdam, 1768, p. 207-219); b) Encyclopédie, Paris, 1751-65, articles “Différentiel” et “Limite”. (XXVII) J.-L. Lagrange, Œuvres, Paris (Gauthier-Villars), 1867-1892: a) Théorie des fonctions analytiques, 2e éd., t. IX; b) Leçons sur le calcul des fonctions, 2e éd., t. X. (XXVIII) A.-L. Cauchy, Résumé des leçons données à l’École Royale Polytechnique sur le calcul infinitésimal, Paris, 1823 (= Œuvres complètes, (2), t. IV, Paris (Gauthier-Villars), 1899). (XXIX) B. Bolzano, Schriften, Bd. I: Funktionenlehre, Prag, 1930. (XXX) P. G. Lejeune-Dirichlet, Werke, t. I, Berlin (G. Reimer), 1889. (XXXI) B. Riemann, Gesammelte Mathematische Werke, 2e éd., Leipzig (Teubner), 1892. (XXXII) K. Weierstrass, Mathematische Werke, t. II, Berlin (Mayer und Müller), 1895. (XXXIII) G. Darboux, Mémoire sur les fonctions discontinues, Ann. E. N. S., (2) t. IV (1875), p. 57-112. (XXXIV) C. Jordan, Traité d’Analyse, 3e éd., 3 vol., Paris (Gauthier-Villars), 1909-15.

[^1]: It is remarkable that Fermat, so scrupulous, used the additivity of the integral, without a word to justify it, in the applications which he gave of his general results: did he base himself on the piecewise monotonicity, implicitly assumed, of the functions under study, from which it is indeed not difficult to justify the additivity by exhaustion? or was he already, despite himself, swept along by the language he employed? [^2]: H. Briggs, Arithmetica logarithmica, London, 1624, chap. XIII. [^3]: See also D.C. Fraser, Newton's Interpolation Formulas, Journ. Inst. Actuaries, v. 51 (1918), p. 77-106 and p. 211-232. and v. 58 (1927), p. 53-95 (articles reprinted as a booklet, London (undated)). [^4]: Cf. St. P. Rigaud, Correspondence of scientific men ... Oxford, 1841, v. II, p. 309-310.