17. Infinitesimal calculus.

In 1604, at the apogee of his scientific career, Galileo, believes he has proved that, in rectilinear movement where the speed grows proportional to the distance covered, the law of movement will be just the one ($x = ct^2$) that he had discovered in the falling of weights ([122 b], v. X, pp. 115-116). Between 1695 and 1700, there is no volume of Acta Eruditorum published monthly at Leipzig, where there do not appear memoirs from Leibniz, the Bernoulli brothers, the marquis de l’Hôpital, dealing with, to all intents and purposes with the notation of which we still make use, the most varied problems of differential calculus, integral calculus and the calculus of variations. It is thus almost exactly during the interval of a century that infinitesimal calculus was wrought or, as was eventually said by the English, Calculus at its best ("calculus"); and nearly three centuries of constant use have not completely blunted this incomparable instrument.

The Greeks neither had nor imagined anything like it. If they knew, no doubt, if only to refuse to use it, an algebraic calculus, that of the Babylonians, of which part of their Geometry is perhaps just a transcription, it is strictly in the domain of geometric invention that what is perhaps their most brilliant mathematical creation is registered, their method for treating problems that for us would tie in to the integral calculus. Eudoxus, treating the volume of the cone and the pyramid, gave the first models of it, which Euclid has more or less faithfully transmitted to us ([107], book XII, prop. 7 and 10). But above all, it is to these problems that most of the work of Archimedes was devoted [5 b and c]; and by a remarkable piece of fortune, we are in a position to read still in their original text, in the sonorous dorian dialect in which he had so carefully composed them, most of his writings, and up to the one, found again recently, where he sets out the "heuristic" procedures by which he was led to some of his most beautiful results ([5 b], v. II, pp. 425-507). For it is there that there is one of the weaknesses of the "exhaustion" of Eudoxus: an irreproachable method of proof (having assumed certain postulates), it is not a method of discovery; its application relies necessarily on advance knowledge of the result to be proved; also, says Archimedes, "of the results whose proofs Eudoxus was the first to discover, concerning the cone and the pyramid ..., not a small part is due to Democritus, who was the first to state them without proof" (loc. cit., p. 430). This circumstance makes it particularly difficult to analyse in detail the work of Archimedes, an analysis which, truth to tell, seems not to have been undertaken by any modern historian; for because of this fact we are ignorant to what point he was conscious of the lines of relationship which unite the different problems which he treats (links that we will express by saying that the same integral recurs in many places, under varied geometric 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 the parabola, the centre of gravity of the triangle, and the area of the spiral of Archimedes so-called ($\rho = c \omega$ in polar co-ordinates); they all depend on the integral $\int x^2 dx$, and, without in any way distancing ourselves from the spirit of the method of exhaustion, they can all be reduced to the calculation of "Riemann sums" of the form $\sum_n a n^2$. It is in this way in fact, that Archimedes deals with the spiral ([5 b], v. II, pp. 1-121), by means of a lemma that reduces to writing

As for the centre of gravity of the triangle, he proves (by exhaustion, by means of a decomposition into parallel strips) that it can be found on each median, hence at their meeting point ([5 b], v. II, pp. 261-315). For the parabola he gives three procedures: one, heuristic, meant only to "give some likelihood to the result", reduces the problem to the centre of gravity of the triangle, by an argument from statics during which he does not hesitate to consider the segment of the parabola as the sum to infinity of segments of straight lines parallel to the axis ([5 b], v. II, pp. 435-439); another method relies on an analogous principle, but is set out in full rigour by exhaustion ([5 b], v. II, pp. 261-315); a last proof, extraordinarily ingenious but of less scope, gives the area sought as a sum of a geometric series by means of particular properties of the parabola. Nothing indicates a relationship between these problems and the volume of the pyramid; it is even specified ([5 b], v. II, p. 8) that the problems relating to the spiral have "nothing in common" with certain others relative to the sphere and the paraboloid of revolution, of which Archimedes had the opportunity of speaking in the same introduction and amongst which there is to be found one (the volume of the paraboloid) which reduces to the integral $\int x dx$.

As can be seen from these examples, and apart from the use of particular contrivances, the principle of exhaustion is the following: by a decomposition into "Riemann sums", upper and lower bounds are obtained for the quantity under consideration, bounds that are compared directly with the expression given for the quantity, or else with corresponding bounds for an analogous problem which has already been solved. The comparison (which, because of the inability to use negative numbers is necessarily done in two parts) is introduced by the sacramental words: "if not, in fact, it would be either greater or less; assume, if possible, that it be greater, etc.; assume, if possible, that it be smaller, etc." whence the name of the method "apagogic" or "by reduction to the absurd" ("ἀπαγωγὴ εἰς ἀδύνατων") that was given to it by the scholars of the XVIIth century. It is in an analogous form that the determination of the tangent to the spiral is drawn up by Archimedes ([5 b], v. II, pp. 62-76), an isolated result, and the only one that we are able to quote as an ancient source for the "differential calculus" apart from the relatively easy determination of tangents to conics, and a few problems of maxima and minima. If in fact, in what concerns "integration", an immense field of research was offered to Greek mathematicians, not only by the theory of areas and volumes, but also by statics and hydrostatics, they hardly had, due to the lack of dynamics, the opportunity of seriously tackling differentiation. It is true that Archimedes gives a dynamic definition of his spiral; and, short of knowing how he could have been led to the knowledge of its tangent, we have the right to ask ourselves if he did not have some idea of the composition of movements. But in that case would he not have applied such a powerful method to other problems of the same kind? It is more likely that he must have used some heuristic procedure of passing to the limit that the results known to him concerning conics could have suggested; these, of course, are of an essentially simpler nature, since one can construct the points of intersection of a straight line and a conic, and in consequence determine the condition for coincidence of these points. As to the definition of the tangent, this is conceived of as a straight line which, in the neighbourhood of a certain point of the curve, leaves the curve entirely on the same side; the existence of it is assumed, and it is also assumed that every curve is made up of convex arcs; under these conditions, in order to show that a straight line is tangent to a curve, certain inequalities must be proved, which of course is done with the most complete accuracy.

From the point of view of rigour, the methods of Archimedes leave nothing to be desired; and, in the XVIIth century, still, when the most scrupulous mathematicians wish to put a result which is adjudged to be particularly delicate entirely beyond doubt, it is an "apagogic" proof that they give of it ([109], v. I, pp. 211-254; French translation, v. III, pp. 181-215 and [244], v. VIII, pp. 249-282). As to their fruitfulness, the work of Archimedes is sufficient testimony. But in order to have the right to see there an "integral calculus", it would be necessary to give in evidence, through the multiplicity of the geometric appearances, some sketch of the classification of the problems according to the nature of the underlying "integral". In the XVIIth century, as we will see, the search for such a classification becomes little by little one of the principal concerns of the geometers; if no trace of this is found in Archimedes, is it not a sign that such speculations would have appeared to him as exageratedly "abstract", and that he voluntarily, on the contrary, at each opportunity, kept as near as possible to the specific properties of the figure whose study he was pursuing? And must we not conclude that this admirable work, from which the integral calculus, as admitted by its creators, is entirely drawn, is in some sense the opposite of the integral calculus?

It is not with impunity, besides, that one can, in mathematics, allow a ditch to be dug between discovery and proof. In favourable times, the mathematician, without abandoning rigour, only has to write down his ideas almost as he conceives of them; sometimes even he can hope to do it so that it would be thus, at the cost of a felicitous change in the language and notations assumed. But often he must resign himself to choosing between incorrect and possibly fruitful methods of exposition, and correct methods that no longer allow him to express his thought except by distorting it and at the cost of a tiring effort. Neither one nor the other way is exempt from dangers. The Greeks followed the second, and it is perhaps there, even more than in the sterilising effect of the Roman conquest, that the reason for the surprising halt in their mathematics almost immediately after its most brilliant flowering must be sought. It has been suggested, without it being unlikely, that the oral teaching of the successors of Archimedes and Apollonius could have contained many a new result without their having thought it necessary to inflict on themselves the extraordinary effort required for a publication which conformed to the received canons. It is no longer such scruples in any case that stopped the mathematicians of the XVIIth century, when, faced with the new problems which were being stated in droves, they sought in assiduous study of the writings of Archimedes the means to bypass it.

While the great classics of the Greek literature and philosophy were all printed in Italy, by Alde Manuce and his followers, and almost all before 1520, it is only in 1544, and with Hervagius at Basle, that the earliest edition of Archimedes appears, both Greek and Latin [5 a], without any previous Latin publication having appeared to prepare its way; and, far from the mathematicians of the time (absorbed as they were by their algebraic researches) having immediately felt its influence, we must await Galileo and Kepler, both of them astronomers and physicists rather than mathematicians, for this influence to become manifest. From this moment, and unceasingly until about 1670, there is no name, in the writings of the founders of the infinitesimal Calculus, which recurs more often than that of Archimedes. Many translate and comment on him; all from Fermat to Barrow, quote him with envy; all declare finding there at the same time a model and a source of inspiration.

It is true that these declarations, as we will see, must not all be taken to the letter; there is to be found one of the difficulties that we come up against in trying to get a true interpretation of these writings. The historian must take account also of the organisation of the scientific world of the time, very defective still at the beginning of the XVIIth century, whereas at the end of the same century, by means of the creation of scholarly societies and scientific periodicals, by means of the consolidation and development of the universities, it ends up by resembling strongly what we know today. Deprived of all periodicals until 1655, mathematicians did not have the choice, in order to make their work known, of anything other than by way of letters, and the printing of a book, most often at their own cost, or at the cost of a patron if one could be found. The editors and printers capable of work of this sort were rare, often rather uncertain. After the long delays and the innumerable troubles that a publication of this kind implied, the author had most often to face up to interminable controversies, provoked by adversaries who were not always in good faith, and carried on sometimes in a surprising bitterness of tone: for, in the general uncertainty in which even the principles of the subject of the infinitesimal calculus was to be found, it was not difficult for each person to find weak points, or at least obscure and contestable, in the arguments of his rivals. It is understandable that under these conditions many scholars keen on peace contented themselves with communicating to a few chosen friends their methods and their results. Some, and especially certain science amateurs, such as Mersenne in Paris, and later Collins in London, kept up a vast correspondence in all countries, in which they communicated extracts here and there, not without mixing in with these extracts stupidities of their own vintage. Possessors of "methods" that, for lack of notions and general definitions, they could not draw up in the form of theorems nor even formulate with some accuracy, mathematicians were thus reduced to making the attempt by means of large numbers of particular cases, and believed that it was not possible to do better, in order to measure its power, than to throw out challenges to their colleagues, accompanied sometimes by publication of their own results in code. The studious youth journeyed, and more perhaps than today; and the ideas of such a scholar were spread sometimes better as a result of the journeys of his pupils that by his own publications, but not without having there another cause of misunderstandings. Finally, the same problems occurring necessarily to a large number of mathematicians, among whom many very distinguished ones, who had only an imperfect knowledge of the results the one of the others, claims for priority could not fail to arise ceaselessly, and it was not rare that accusations of plagiarism would be added on.

It is therefore in the letters and private papers of the scholars of the time, as much or even more than in their publications proper, that the historian must seek his documents. But, whereas those of Huygens for example have been kept and have been the object of exemplary publication [169 b], those of Leibniz have only been published in a fragmentary manner so far, and many others are lost without hope. At least the most recent research, based on the analysis of manuscripts, has brought out, in a way that seems irrefutable, a point that the partisan quarrels had somewhat obscured: it is that, each time that one of the great mathematicians of that time gave an account of his own work, of the evolution of his thought, of the influences that he came under and those that he did not come under, he has done it in an honest and sincere manner, and in all good faith;$^1$ these precious testimonies, of which we possess a fairly great number, can thus be used in all confidence, and the historian does not need in their case to transform himself into an examining magistrate. For the remainder, most of the questions of priority that have arisen are totally lacking in sense. It is true that Leibniz, when he adopted the notation $dx$ for the “differential”, was ignorant of the fact that Newton, for about ten years, had been using $\dot{x}$ for the “fluxion”: but what would it have mattered if he had known? To take a more instructive example, who is the author of the theorem $\log x = \int \frac{dx}{x}$, and what is the date thereof? The formula, as we have just written it, is from Leibniz since both one and the other members are written in his notation. Leibniz himself, and Wallis, attribute it to Grégoire de Saint-Vincent. This latter, in his Opus Geometricum [135] (appeared in 1647, but drawn up, he says, long before), proves only the equivalent of what follows: if $f(a, b)$ denotes the area of the hyperbolic segment $a \leq x \leq b, 0 \leq y \leq A/x$, the relation $b'/a' = (b/a)^n$ implies $f(a'/b') = n.f(a, b)$; to which his pupil and commentator Sarasa adds almost immediately [269] the remark that the areas $f(a, b)$ can therefore “take the place of logarithms”. If he says no more, and if Grégoire himself had said nothing about it, is it not because, for the majority of mathematicians of the time, logarithms were “calculation aids” with no right to be quoted in mathematics? It is true that Torricelli, in a letter of 1644 [208], speaks of his research on a curve that we would write as $y = ae^{-cx}, x \geq 0$, while adding that where Napier (whom he covers with praise besides) “was only after the arithmetical uses”, he himself “drew out of it a speculation on geometry”; and he had left a manuscript about this curve evidently prepared for publication, but which remained unpublished until 1900 ([310], v. I, pp. 335-347). Descartes had besides encountered the same curve already in 1639 in connection with the “problem of Debeaune”, and had described it without speaking of logarithms ([85 a], v. II, pp. 514-517). Whatever, J. Gregory, in 1667, gives, without quoting anybody whatever, ([136 a], reproduced in [169 a], pp. 407-462), a rule for calculating the areas of hyperbolic segments by means of (decimal) logarithms: which implies at the same time a theoretical knowledge of the link between the quadrature of the hyperbola and logarithms, and the numerical knowledge of the link between “natural” and “decimal” logarithms. Is it about this last point only that the claim of Huygens applies, who contests immediately the novelty of the result of Gregory ([169 b], v. VI, pp. 228-230)? It is that which is no clearer to us than to his contemporaries; these in any case had the clear impression that the existence of a link between logarithms and the quadrature of the hyperbola was something that had been known for a long time, without their being able on this to refer to anything other than epistolary allusions or to the book of Grégoire de Saint-

$^1$ This is applicable, for example to Torricelli (see [244], v. VIII, pp. 181-194) and to Leibniz [217]. That is not to say, of course, that a mathematician may not have a false idea about the originality of his ideas; but it is not the greatest who are most inclined to be at fault in this regard.

Vincent. In 1668, when Brouncker gives (with a polished proof of convergence, by comparison with a geometric series) series for log 2 and log (5/4) ([38]: reproduced in [218], v. I, pp. 213-218), he presents them as expressions of the corresponding segments of a hyperbola, and adds that the numerical values that he obtains are “in the same ratio as the logarithms” of 2 and of 5/4. But in the same year, with Mercator ([220], reproduced in [218], v. I, pp. 167-196) or more exactly in the exposition given immediately by Wallis of the work of Mercator ([326 b], reproduced in [218], v. I, pp. 219-226), the language changes: since the segments of a hyperbola are proportional to logarithms, and that it is well known that logarithms are not defined by their characteristic properties except up to a constant factor, nothing prevents considering the segments of a hyperbola as logarithms, qualified as “natural” (in opposition to the “artificial” or “decimal” logarithms), or hyperbolic; this last step taken (to which the series for log(1+x) contributes, given by Mercator), the theorem $\log x = \int \frac{dx}{x}$ is obtained, up to notation, or rather it became a definition. What to conclude, if not that it is by almost imperceptible transitions that this discovery was made, and that a dispute about priorities on this matter resembles strongly a quarrel between the violin and the trombone as to the exact moment when a particular motif appears in a symphony? And truth to tell, while at the same time other mathematical creations, the arithmetic of Fermat, the dynamics of Newton, carried a strongly individual cachet, it is very much of the gradual and inevitable development of a symphony, where the “Zeitgeist”, at the same time composer and conductor, holds the baton, that we are reminded by the development of the infinitesimal calculus in the XVIIth century: each undertakes his part with characteristic timbre, but no one is master of the themes that he is creating for the listener, themes that a scholarly counterpoint has almost inextricably entwined. It is thus under the form of a thematic analysis that the history of this must be written; we will be content here with a summary sketch, and which does not pretend to a meticulous exactitude.$^2$ Here in any case are the principal themes that a superficial examination brings out:

A) The theme of mathematical rigour, contrasting with that of the infinitely small, indivisibles or differentials. It has been seen that both hold an important place with Archimedes, the first in all his work, the second only in the treatise on his Method, that the XVIIth century did not know, so that, if it had been transmitted and not reinvented, it could only have been by philosophical tradition. The principle of the infinitely small appears besides in two distinct forms depending on whether it is a case of “differentiation” or of “integration”. As for this one, let it be first a case of calculating a plane

$^2$ In what follows, the attribution of a result to such and such an author, at such and such a date, indicates only that this result was known to him at that date (which has as often as possible been verified using the original texts); we do not mean to affirm absolutely that this author did not know of it earlier, or that he did not receive it from someone else; still much less do we wish to say that the same result was not possibly obtained independently by others, either earlier or later.

area: it will be divided into an infinity of parallel strips infinitely small, by means of an infinity of equidistant parallels; and each of these strips is a rectangle (even though none of the finite strips that are obtained by means of two parallels at a finite distance apart would be a rectangle). In the same way, a solid of revolution would be decomposed into an infinity of cylinders of the same infinitely small height, by planes perpendicular to the axis;$^3$ analogous ways of speaking could be used when it is a case of decomposing an area into triangles by means of concurrent straight lines, or to argue about the length of an arc of a curve as if it were the case of a polygon with an infinity of sides, etc.. It is certain that the rare mathematicians who possessed fully the ability to handle the methods of Archimedes, such as Fermat, Pascal, Huygens, Barrow, could, in each particular case, have found no difficulties in replacing the use of this language by rigorous proofs; also they often remark that this language is only there for conciseness. "It would be easy", says Fermat, "to give proofs in the manner of Archimedes ...; this being said, it is sufficient to have been warned once for all in order to avoid continual repetitions ..." ([109], v. I, p. 257); in the same way Pascal: "thus one of these methods only differs from another in the way of stating it" ([244], v. VIII, p. 352);$^4$ and Barrow, with his mocking conciseness: "longior discursus apagogicus adhiberi possit, sed quorsum?" (it would be possible to lengthen this by an apagogic discourse, but to what end?) ([16 b], p. 251). Fermat is very careful even, it would appear, not to bring forward anything that he could not justify thus, and condemns himself by that to announcing no general result except by allusion or in the form of a "method"; Barrow, although so careful, is somewhat less scrupulous. As for the majority of their contemporaries, it can be said at the very least that rigour is not their main concern, and that the name of Archimedes is most often only a dissembling meant to cloak merchandise of great price no doubt, but for which Archimedes would certainly not have assumed responsibility. All the more so is it the case where differentiation is concerned. If the curve, when it is a case of its rectification, is assimilated to a polygon with an infinity of sides, it is a case here of an "infinitely small" arc of the curve which is assimilated to an "infinitely small" segment of a straight line, either the chord, or a segment of the tangent whose existence is assumed; or again it is an "infinitely small" time interval that is being considered, during which (so long as it is a case only of the speed) the movement "is" uniform; still bolder, Descartes, wishing to determine the tangent

$^3$ See for example the exposition by Pascal, in the "letter to M. de Carcavy" ([244], v. VIII, pp. 334-384). It will be noted that, thanks to the prestige of an incomparable language, Pascal manages to create the illusion of perfect clarity, to the point that one of his modern editors is in ecstasy over "the minutiae and the accuracy in the exactitude of the proof"! $^4$ But, in the Letter to Monsieur A. D. D. S.: "...without stopping, nor for the methods of movement, nor for that of the indivisibles, but in following that of the ancients, so that the matter could henceforth be firm and beyond dispute" ([244], v. VIII, p. 256).

of a cycloid which does not come under his general rule, assimilates curves rolling one on the other to polygons, so as to conclude that "in the infinitely small" the movement can be assimilated to a rotation about a point of contact ([85 a], v. II, pp. 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 ([109], v. I, pp. 133-179; French translation, v. III, pp. 121-156); cf. also ([109], v. II, passim, in particular pp. 154-162, and Supplément aux Oeuvres pp. 72-86); Barrow gives precise proofs for a large part of his theorems, in the manner of the ancients, starting with simple hypotheses of monotonicity and convexity. But the time was already over for pouring new wine into old wineskins. In all that, as we know today, it was the notion of limits that was being developed; and, if it is possible to extract from Pascal, Newton and from others as well, statements that seem very close to our modern definitions, it is only necessary to put them back in their context to realise the irremovable obstacles that were in the way of a rigorous exposition. When starting in the XVIIIth century some mathematicians caught up by the need for clarity wished to put some order into the confused heap of their riches, such indications, encountered in the writings of their predecessors, were precious to them; when d'Alembert for example explains that there is nothing else in differentiation other than the notion of limit, and defines this with accuracy [75 b], it is possible to believe that he was guided by the considerations of Newton on the "first and last reasons of vanishing quantities" [233 b]. But, so long as it is a case of the XVIIth century, it must be realised that the way was not open for modern analysis until Newton and Leibniz, turning their back on the past, accept that they must seek provisionally the justification for their new methods, not in rigorous proofs, but in the fruitfulness and coherence of the results.

B) The dynamic. Already Archimedes, we have seen, gave a dynamic definition of his spiral; and in the Middle Ages a rudimentary theory of the variation of quantities as a function of time is developing (but proof to the contrary lacking, without infinitesimal considerations), and of their graphical representation, which must perhaps have had its origins in Babylonian astronomy. But it is of the greatest importance for the mathematics of the XVIIth century that, from the beginning, the problems of differentiation arose, not only in connection with tangents, but in connection with velocities. Galileo however ([122 a and b]), seeking the law of velocities in the falling of weights (after having obtained the law of distances $x = at^2$, by means of an inclined plane), does not proceed by using differentiation: he makes various hypotheses about the velocity, first $v = \frac{dx}{dt} = cx$ ([122 b], v. VIII, p. 203), then later $v = ct$ (id., p. 208), and seeks to find again the law of distances by reasoning, in a fairly obscure way, about the graph of the velocity as a function of time; Descartes (in 1618) argues in the same way, about the law $v = ct$, but as a true mathematician and with as much clarity as the language of indivisibles can muster$^5$ ([85 a], v. X, pp. 75-78); with both, the graph of velocity (in a straight line) plays the main role, and there is reason to ask to what point they were aware of the proportionality between the distances covered and the areas bounded by the axis of time and the curve of the velocity; but it is difficult to say anything on this point, although the language of Descartes seems to imply knowledge of the fact in question (that certain historians wish to put back to the Middle Ages [335]), whereas Galileo does not make any clear remark about it. Barrow states it explicitly in 1670 ([16 b], p. 171); perhaps at that time it was no longer a novelty for anyone, and Barrow does not present it as such; but, no more for this result as for any other, is it appropriate to give a date with too much accuracy. As for the hypothesis $v = cx$, considered also by Galileo, he is happy (loc. cit.) to prove that it is not sustainable (or, in modern language, that the equation $\frac{dx}{dt} = cx$ has no solution $\neq 0$ that is zero for $t = 0$), by an obscure argument that Fermat later ([109], v. II, pp. 267-276) takes the trouble of developing (and which amounts just about to saying that, $2x$ being a solution as well as $x, x \neq 0$ would be contrary to the obvious physical uniqueness of the solution). But this same law $\frac{dx}{dt} = cx$ which, in 1614, was used by Napier to introduce his logarithms, of which he gives a dynamic definition [230 a], which, in our notation, would be written as follows: if, on two straight lines, two bodies are in motion following the laws $\frac{dx}{dt} = a, \frac{dy}{dt} = -ay/r, x_0 = 0, y_0 = r$, then one says that $x$ is the "logarithm" of $y$ (in modern notation, we have $x = r \log(r/y)$). It has been seen that the curve which is a solution of $\frac{dy}{dx} = \frac{c}{x}$ appeared in 1639 with Descartes, who described it dynamically ([85 a], v. II, pp. 514-517); it is true that he qualified fairly disdainfully as "mechanics" all the non algebraic curves, and claimed to exclude them from geometry; but happily this taboo, against which Leibniz, much later, believes he still has to protest vigorously, was not observed by his contemporaries, nor by Descartes himself. The cycloid, the logarithmic spiral appear and are studied with eagerness, and their study is a powerful help in the interlacing of geometric and dynamic methods. The principle of composition of movements, and more precisely of velocities, was at the basis of the theory of movement of projectiles stated by Galileo in the chef-d'oeuvre of his old age, the Discorsi of 1638 ([122 b], v. VIII, pp. 268-313) a theory that contains thus implicitly a new determination of the tangent to the parabola; if Galileo does not expressly state it, Torricelli on the contrary ([310], v. III, pp. 103-159) insists on this point, and even bases on this same principle a general method of determination of tangents for curves which are capable of a dynamic definition. It is true that he was forestalled in this, by several years, by Roberval ([263], pp. 3-67) who says that he was led to

$^5$ Descartes even adds an interesting geometric reasoning by which he deduces the law $x = at^3$ from the hypothesis $\frac{dv}{dt} = ct$. By contrast, it is ironic that ten years later, we see him tied up in his notes, and copying out for the use of Mersenne an inexact argument about the same question, where the graph of velocity as a function of time is confused with the graph as a function of distance covered ([85 a], v. I, p. 71).

this method by the study of the cycloid; this same problem of the tangent to the cycloid gives besides the opportunity to Fermat of proving the power of his method of differentiation ([109], v. I, pp. 162-165), whereas Descartes, incapable of applying to it his algebraic method, invents for the occasion the instantaneous centre of rotation ([85 a], v. II, pp. 307-338).

But, as the infinitesimal calculus is developed, dynamics ceases to be a separate science. It is noticed more and more that in spite of Descartes, algebraic curves and functions do not have, from the “local” point of view which is that of the infinitesimal calculus, anything that distinguishes them from much more general functions; the functions and curves with a dynamic definition are functions and curves like the others, accessible to the same methods; and the variable “time” is no more than a parameter, whose temporal aspect is purely a matter of language. So with Huygens, even when it is a matter of mechanics, it is geometry that dominates ([169 b], v. XVIII); and Leibniz does not give to time any privileged place in his calculations. On the contrary Barrow conceives of making, of the simultaneous variation of several quantities as functions of a universal independent variable conceived of as “time”, the foundation of an infinitesimal calculus with a geometric tendency. This idea, which must have come to him as he sought to find again the method of composition of movements that he knew existed only by hear-say, is stated in detail, in very clear and very general terms, in the first three of his Lectiones Geometricae [16 b]; and for example he proves there with care that, if a moving point has as projections, on two rectangular axes $AY, AZ$, moving points such that one is moving with a constant speed $a$ and the other with a speed $v$ that grows with time, the trajectory has a tangent with slope equal to $v/a$, and is concave in the direction of increasing $Z$. In the following parts of the Lectiones, he pursues these ideas very far, and although he puts a sort of affectation in drawing them up almost end to end in a form that is as geometric and as sparingly algebraic as possible, it is permissible to see there, with Jakob Bernoulli ([19 a], v. I, pp. 431 and 453), the equivalent of a good part of the infinitesimal calculus of Newton and Leibniz. Exactly the same ideas form the point of departure for Newton ([233 a], v. I, pp. 201-244 and [233 b]): his “fluents” are the various quantities, functions of “time” which is only a universal parameter and the “fluxions” are the derivatives with respect to “time”; the possibility that Newton allows himself, of changing at need the parameter, is equally present in the method of Barrow, although used by him with less suppleness.$^6$ The language of fluxions, adopted by Newton and im-

$^6$ The question of the relations between Barrow and Newton is controversial; see [241] and [233 d]. In a letter of 1663 (cf. [262], vol. II, pp. 32-33), Barrow speaks of his thoughts, already old, on the composition of movements, which have brought him to a very general theorem on tangents (if it is that of the Lect. Geom., Lect. X ([16 b], p. 247), it is indeed so general that it has as a special case all that had been done until then on the subject). On the other hand, Newton could have been a pupil of Barrow in 1664 and 1665, but says that he obtained independently his rule for deducing, from a relation between “fluents”, a relation between their “fluxions”. It posed by his authority on English mathematicians of the following century, represents thus the last outpost, for the period that has our attention, of the dynamic methods whose true role was already finished.

C) Algebraic Geometry. This is a parasite theme, a stranger to our subject, which is introduced because Descartes, by his systematic approach, claimed to make of algebraic curves the exclusive object of geometry ([85 a], v. VI, p. 390); also it is a method of algebraic geometry, and not like Fermat a method from the differential calculus, that he gives for the determination of tangents. The results bequeathed by the ancients on the intersection of a straight line and a conic, the thoughts of Descartes himself on the intersection of two conics and the problems that reduce to that, were going to lead him quite naturally to the idea of taking as criterion for contact the coincidence of the two intersections: we know today that in algebraic geometry it 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 applies it first in a manner that is hardly convenient, in seeking to make two intersections of the curve under study and of a circle having its centre on $Ox$ coincide in a given point ([85 a], v. VI, pp. 413-424); his disciples, van Schooten, Hudde, substitute a straight line for the circle, and obtain in the form $-F'_x/F'_y$ the slope of the tangent to the curve $F(x, y) = 0$, the "derived polynomials" $F'_x, F'_y$ being defined by their formal rule of formation ([198 d], v. I, pp. 147-344 and [85 b], pp. 234-237); de Sluse also reaches this result at about the same time ([198 d], pp. 232-234). Of course the clear-cut distinctions that we draw out here, and which alone give meaning to the controversy between Descartes and Fermat, could in no way exist in the minds of mathematicians of the XVIIth century: we have only mentioned them in order to make clearer one of the more curious episodes in the history which concerns us, and to note almost immediately after the complete eclipse of algebraic methods, provisionally absorbed by differential methods.

D) Classification of problems. This theme, we have seen, seems absent from the work of Archimedes, to whom it appears to be a matter of indifference to solve a problem or to reduce it to a problem already treated. In the XVIIth century, the problems of differentiation appear at first under three distinct aspects: speed, tangents, maxima and minima. As for these last, Kepler [179 a and b] makes the observation (that is already found with Oresmus ([335], p. 141) and which had not even escaped the Babylonian astronomers) that the variation of a function is particularly slow in the neighbourhood of a maximum. Fermat, already before 1630 ([109], v. II, pp. 71 and 113-179), inaugurates in connection with such problems his infinitesimal method, which in modern language reduces in fact to seeking the first two terms (the constant term and the first order term) in the Taylor expansion, and to writ-

is possible that Newton had obtained from the teaching of Barrow the general idea of quantities varying as a function of time, and of their speed of variation, notions that his thoughts on dynamics (to which Barrow must have remained a complete stranger) soon contributed by making them more precise.

ing that at an extremum the second is zero; he goes from there to extend his method to the determination of tangents, and even applies it to the search for points of inflexion. If one takes account of what has been said above in connection with dynamics, it can be seen that the unification of the three types of problems relative to the first differential was accomplished at a fairly early stage. As for problems relative to the second differential, they only appear quite late, and especially with the works of Huygens on the evolute of a curve (published in 1673 in his Horologium Oscillatorium ([169 b], v. XVIII)); at this time, Newton, with his fluxions, was already in possession of all the analytical means necessary for solving such problems; and, in spite of all the geometric talent that Huygens expends on it (and of which later differential geometry would profit in its beginnings), they were hardly used for anything else, during the period with which we are concerned, than to allow the new analysis to become conscious of the power of its means.

As for integration, it had appeared to the Greeks as the calculation of areas, of volumes, of moments, as the calculation of the length of a circle and of areas of spherical segments; to which the XVIIth century adds the rectification of curves, the calculation of the area of surfaces of revolution, and (with the work of Huygens on the compound pendulum ([169 b], v. XVIII)) the calculation of moments of inertia. It was a case first of recognising the link between all these problems. For the areas and the volumes, this first and immense step forward is made by Cavalieri, in his Geometry of indivisibles [57 a]. He announces there, and claims to prove, the following principle more or less: if two plane areas are such that every parallel in a given direction intersects them in segments whose lengths are in a constant ratio, then these areas are in the same ratio; an analogous principle is stated for the volumes intersected by planes parallel to a fixed plane where the areas are in a constant ratio. It is likely that these principles were suggested to Cavalieri by theorems such as that of Euclid (or rather Eudoxus) on the ratio of volumes of pyramids with the same height, and that before stating them in a general form, he first verified their validity with a large number of examples taken from Archimedes. He "justifies" them by the use of a language, about whose legitimacy we see him asking Galileo in a letter of 1621, whereas in 1622 already he uses it without hesitation ([122 b], v. XIII, pp. 81 and 86) and of which here is the essential part. Let there be for example two areas, 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 one to the other in a ratio that, for $n$ sufficiently large, is arbitrarily near to the ratio of two areas, and it would even be not difficult to prove it by exhaustion for $f$ and $g$ monotonic; Cavalieri goes to the limit, makes $n = \infty$, and speaks of "the sum of all the ordinates" of the first curve, which is to the analogous sum for the second curve in a ratio rigorously equal to the ratio of the areas; in the same way for the volumes; and this language is afterwards universally adopted, even by authors, like Fermat, who have the clearest realisation of the precise facts that it covers. It is true that subsequently many mathematicians, such as Roberval ([163], pp. 3-67) and Pascal ([244], v. VIII, pp. 334-384), prefer to see, in the ordinates of the curve whose "sum" is being taken, not segments of a straight line like Cavalieri, but rectangles of the same infinitely small height, which is not great progress from the point of view of rigour (whatever is said by Roberval), but perhaps prevents the imagination from going off the rails too easily. In any case, and as it is a case only of ratios, the expression "the sum of all the ordinates" of the curve $y = f(x)$, or in short "all the ordinates" of the curve, is definitively, as it appears clearly for example in the writings of Pascal, the exact equivalent of the $\int y dx$ of Leibniz.

From the language adopted by Cavalieri, the principles stated higher follow inevitably, and imply immediately consequences that we will state in modern notation, it being understood that there $\int f dx$ will only mean the area bounded by $Ox$ and the curve $y = f(x)$. First of all, every plane area, intersected by each straight line $x =$ constant in segments the sum of whose lengths is $f(x)$, is equal to $\int f dx$; and it is the same with every volume intersected by each plane $x =$ constant in an area of value $f(x)$. What is more, $\int f dx$ depends linearly on $f$; we have $\int (f+g) dx = \int f dx + \int g dx$, $\int c f(x) dx = c \int f dx$. In particular all the problems of area and volume are reduced to quadratures, that is to say to calculations of area of the form $\int f dx$; and, what is perhaps even more novel and important, two problems depending on the same quadrature must be considered equivalent, and there is the means in each case to decide if it is so. The Greek mathematicians never attained (or perhaps never consented to attain) such a degree of "abstraction". Thus ([57 a], p. 133) Cavalieri "proves" without any trouble that two similar volumes are in a ratio equal to the cube of the ratio of similarity, whereas Archimedes only states this conclusion for quadrics of revolution and their segments, at the end of his theory of solids ([5 b], v. I, p. 258). But to get there it was necessary to throw Archimedean rigour overboard.

There was thus there the means of classifying problems, at least provisionally, following the degree of real or apparent difficulty that arose with the quadratures related to them. That is where the algebra of the time was of use as a model: for in algebra also, and in the algebraic problems that were arising in geometry, whereas the Greeks were only interested in the solutions, the algebraists of the XVIth and XVIIth centuries began to turn their attention mainly to the classification of problems according to the nature of the means employed for their solution, thus creating a prelude for 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 depended, but they had already stated for themselves difficult questions about possibilities: the possibility of solving every equation by radicals (in which some of them did not believe), etc.; they were preoccupied also with reducing to a type of geometric form all the problems of given degree. In the same way in the integral Calculus, the principles of Cavalieri put them in a position to recognise immediately that many of the problems solved by Archimedes reduce to quadratures $\int x^n dx$ for $n = 1, 2, 3$; and he invents an ingenious method for carrying out this quadrature for as many values of $n$ as desired (the method reduces to observing that one has $\int_0^{2a} x^n dx = c_n a^{n+1}$ by homogeneity, and to writing

whence, by expanding, a recurrence relation for the $c_n$ ([57 a], p. 159 and [57 b], pp. 269-273). But already Fermat had gone much farther, by proving first (before 1636) that $\int_0^a x^n dx = \frac{a^{n+1}}{n+1}$ for $n$ a positive integer ([109], v. II, p. 83), by means of a formula for the sums of powers of the first $N$ integers (a procedure imitated from the quadrature of the spiral of Archimedes), then in extending the same formula to all rational $n \neq 1$ ([109], v. I, pp. 195-198); from this last result (communicated to Cavalieri in 1644) he only draws up a proof much later, following his reading of the writings of Pascal on integration$^7$ ([109], v. I, pp. 255-288; French translation v. III, pp. 216-240).

These results joined to geometric considerations that take the place of change of variables and integration by parts, allow already the solution of a large number of problems that reduce to elementary quadratures. Beyond that, is met first the quadrature of the circle and that of the hyperbola: as it is especially "indefinite integrals" with which they were concerned at that time, the solution of these problems in modern terms, is supplied respectively by the inverse circular functions and by the logarithm; these were given geometrically, and we have seen how this last has been introduced little by little into analysis. These quadratures are the subject of much work, by Grégoire de Saint-Vincent [135], Huygens ([169 b], v. XI, pp. 271-337 and v. XII, pp. 91-180), Wallis (326 a], v. I, pp. 355-478), Gregory [136 a]; the first believes he has succeeded in the quadrature of the circle, the last believes he has proved the transcendence of $\pi$; with the ones and the others procedures of indefinite approximation are developed for the circular and logarithmic functions, the ones with a theoretical tendency, the others oriented towards numerical calculation, which is going to end soon, with Newton ([233 a], v. I, pp. 3-26 and 29-199), Mercator ([220], reproduced in [218], v. I, pp. 167-196), J. Gregory [136 d], then Leibniz [198 d], with general methods of expansion in series. In any case, the conviction comes to light little by little of the "impossibility" of the quadratures in question, that is to say of the non-algebraic character of the functions that they define; and at the same time, it becomes the custom

$^7$ It is remarkable that Fermat, so scrupulous, uses the additivity of the integral, without a word of justification, in the applications that he gives of his general results: does he base himself on the monotonicity by parts, implicitly assumed, of the functions that he studies, from which condition it is indeed not difficult to justify additivity by exhaustion? or is he already, in spite of himself, carried along by the language that he uses?

to consider a problem solved as far as its nature allows, when it has been reduced to one of the "impossible" quadratures. It is the case for example with the problems of the cycloid, solved by means of circular functions, and of the rectification of the parabola, reduced to the quadrature of the hyperbola.

The problems of rectification, of which we have just quoted two of the most famous, had a particular importance, as making up a natural geometric transition between differentiation, which they presuppose, and integration which they come under; problems on the area of surfaces of revolution can be associated with them. The ancients had only treated the case of the circle and the sphere. In the XVIIth century, these questions only appear very late; it appears that the difficulty, insurmountable for the time, of the rectification of the ellipse (considered to be the simplest curve after the circle) had discouraged effort. Dynamic methods give some access to these problems, which allows Roberval ([263], pp. 363-399) and Torricelli ([310], v. III, pp. 103-159), between 1640 and 1645, to obtain results on the arc of spirals; but it is only in the years that precede 1660 that they become the order of the day; the cycloid is rectified by Wren in 1658 ([326 a], v. I, pp. 533-541); soon after the curve $y^3 = ax^2$ is done by several authors ([326 a], v. I, pp. 551-553; [85 b] v. I, pp. 517-520; [109], v. I, pp. 211-255; French translation, v. III, pp. 181-215), and several authors also ([109], v. I, p. 199; [169 b], v. II, p. 334) reduce the rectification of the parabola to the quadrature of the hyperbola (that is to say to an algebraico-logarithmic function). This latter example is the most important, because it is a particular case of the general principle according to which the rectification of a curve $y = f(x)$ is none other that the quadrature of $y = \sqrt{1 + (f'(x))^2}$; and it is from precisely this principle that Heurat deduces it. It is no less interesting to follow the groping of Fermat in his old age, in his work on the curve $y^3 = ax^2$ ([109], v. I, pp. 211-255; French translation v. III, pp. 181-215); to the curve $y = f(x)$ of arc $s = g(x)$, he associates the curve $y = g(x)$, and determines the tangent to this one starting from the tangent to the first one (in modern language, he proves that their slopes $f'(x), g'(x)$ are linked by the relation $(g'(x))^2 = 1 + (f'(x))^2$); one can believe oneself very close to Barrow, and it would only be necessary to combine this result with that of Heurat (which is more or less what Gregory does in 1668 ([136 d], pp. 488-491)) to obtain the relation between tangents and quadratures; but Fermat states only that if, for two curves each with reference to a system of rectangular axes, the tangents at the points with the same abscissa always have the same slope, the curves are equal, or in other words that the knowledge of $f'(x)$ determines $f(x)$ (up to a constant); and he justifies this assertion only by an obscure argument without any convincing value.

Less than ten years later, the Lectiones Geometricae of Barrow [16 b] had appeared. From the beginning (Lect. I), he states in principle that, in a rectilinear movement, the spaces are proportional to the areas $\int_0^t vdt$ bounded by the axis of time and the curve of the velocity. One would believe that he is going to deduce from that, and from his dynamic method already quoted concerning the determination of tangents, the link between the differential conceived as the slope of the tangent, and the integral conceived as the area; but there is none of that, and he proves later, in a purely geometric way ([16 b], Lect. X, , 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 $c.F(x) = \int_a^x f(x) dx$, then the tangent to $Y = F(x)$ intersects $Ox$ at the point with abscissa $x - T$ determined by $y/Y = c/T$; the proof besides is perfectly precise, starting with the explicit hypothesis that $f(x)$ is monotonic, and it is said that the direction of variation of $f(x)$ determines the direction of concavity of $Y = F(x)$. But it will be noted that this theorem is somewhat lost amongst a crowd of others, many of which are very interesting; the reader who has not been warned is tempted to see there only a means for solving by quadrature the problem $Y/T = f(x)/c$, that is to say a certain problem of the determination of a curve starting from information on its tangent (or, as we would say, a differential equation of a particular type); and that all the more so as the applications of it given by Barrow are concerned above all with problems of the same kind (that is to say with differential equations integrable by "separation of variables"). The geometric language that Barrow imposes on himself is here a cause why the link between differentiation and integration, so clear as long as it was a case of dynamics, is somewhat obscured

On the other hand, various methods had taken shape, to reduce problems of integration the ones to the others, and to "solve" them or to reduce them to "impossible" problems that had already been classified. In its simplest geometric form, integration by parts consists in writing the area bounded by $Ox, Oy$ and an arc of the monotonic 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 in an implicit way. With Pascal ([244], v. IX, pp. 17-18) appears the following generalisation, already much better hidden: $f(x)$ being as above, let $g(x)$ be a function $\geq 0$, 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 proves ingeniously by evaluating in two ways the volume of the solid $0 \leq x \leq a, 0 \leq y \leq f(x), 0 \leq z \leq g(x)$; the particular case $g(x) = x^n, G(x) = \frac{x^{n+1}}{n+1}$ plays an important role, both with Pascal (loc. cit., pp. 19-21) and with Fermat ([109], v. I, p. 271); this latter (whose work carries the significant title of "Transmutation and emendation of equations of curves, and its varied applications to the comparison of curvilinear spaces between them and with rectilinear spaces ...") does not prove it, no doubt because he judges it useless to repeat what Pascal had just published. These theorems of "transmutation", where we would see a combination of integration by parts and changes of variables, take the place of this to some extent, which is only introduced very late; it is in effect contrary to the way of thinking, still too geometric and too little analytic, of the time, to allow the use of variables other than those imposed by the figure, that is to say one or other of the co-ordinates (or sometimes polar co-ordinates) then the arc of the curve. It is thus that we find with Pascal ([244], v. IX, pp. 60-76) results that, in modern notation, are written, putting $x = \cos t, y = \sin t$ and, for particular functions $f(x)$:

and, with J. Gregory ([136 d], 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 for change variables ([16 b], pp. 298-299); its statement, geometric as always, reduces to what follows: let $x$ and $y$ be linked by a monotonic relation, and let $p$ be the slope of the graph of this relation at the point $(x, y)$; then, if the functions $f(x), g(y)$ are such that one has $f(x)/g(y) = p$ for every corresponding pair of values $(x, y)$, 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 supposed to have constant sign), we have $p = f(x)/g(y)$; the converse is naturally used to apply the theorem to the solution of differential equations (by "separation of variables"). But the theorem is only inserted by Barrow in an appendix (Lect. XII, app. III, theor. IV), where, by making the observation that several of his preceding results are only particular cases of it, he excuses himself as having discovered it too late to make more use of it.

Thus, around 1670, the situation is as follows. It is known how to treat, by uniform procedures, the problems that arise with the first derivative, and Huygens has tackled geometric questions that arise with the second derivative. It is known how to reduce all problems of integration to quadratures; there are available various techniques, of a geometric aspect, to reduce quadratures the one to the others, in badly classified cases, and it has become customary, from this point of view, to handle circular and logarithmic functions; there is a consciousness of the link between differentiation and integration; the "inverse method of tangents" is beginning to be tackled, a name given at this time to problems that reduce to differential equations of the first order. The sensational discovery of the series $\log(1 + x) = - \sum_1^\infty (-x)^n / n$ by Mercator has just opened up quite new perspectives for the application of series, and mainly power series, to problems called "impossible". By contrast the ranks of mathematicians are significantly thinned: Barrow has left the chair of professor for that of preacher; Huygens apart (who has almost all his mathematical work behind him, having already obtained all the principal results of the Horologium Oscillatorium that he is setting himself to draw up definitively), only Newton at Cambridge, and J. Gregory at Aberdeen are active, whom Leibniz will join soon with the ardour of a neophyte. All three, Newton from 1665 already, J. Gregory from the publication of Mercator in 1668, Leibniz from about 1673, devote themselves principally to the subject in fashion, the study of power series. But, from the point of view of the classification of problems, the principal effect of the new methods seems to be to obliterate all distinctions between them; and in effect Newton, more of an analyst than an algebraist, does not hesitate to announce to Leibniz in 1676 ([198 d], p. 224) that he knows how to solve all differential equations,$^8$ to which Leibniz responds ([198 d], pp. 248-249) that on the contrary it is a case of obtaining the solution in a finite number of terms whenever it is possible "presupposing the quadratures", and also of knowing if every quadrature can be reduced to those of the circle and the hyperbola as had been noticed in the majority of cases studied so far; he brings out again in this connection that Gregory believed (with reason, as we know today) that the rectification of the ellipse and the hyperbola is irreducible to the quadratures of the circle and the hyperbola; and Leibniz asks to what point the method of series, such as Newton uses it, can give an answer to these questions. Newton, on his side ([198 d], pp. 209-211), declares that he is in possession of criteria, that he does not indicate, for deciding, apparently by the examination of series, about the "possibility" of certain quadratures (in finite terms), and gives an example of a (very interesting) series for the integral $\int x^\alpha (1 + x)^\beta dx$.

The immense progress that has been realised in ten years can be seen; questions of classification are stated in these letters in quite modern terms; if one of those that Leibniz raises has been solved in the XIXth century by the theory of abelian integrals, the other, about the possibility of reducing a given differential equation to quadratures, is still open despite important recent work. If it is thus, it is because already Newton and Leibniz, each on his own account, have reduced to an algorithm the fundamental operations of the infinitesimal calculus; it is sufficient to write, in the notations which one or the other uses, a problem on quadratures or differential equations, in order for its algebraic structure to appear immediately, freed from its geometric crust; the methods of "transmutation" also can be written in simple analytical terms; the problems of classification are stated in a precise way. Mathematically speaking, the XVIIth century has reached its end.

E) Interpolation and the calculus of differences. This theme (from which we will not separate the study of binomial coefficients) appears early and continues throughout the century, for reasons that are both theoretical and practical. One of the big tasks of the time is in fact the calculation of trigonometric, logarithmic, nautical tables made necessary by the rapid progress of geography, navigation, theoretical and practical astronomy, physics, celestial mechanics; and many of the most eminent mathematicians, from Kepler to Huygens and Newton, take part, either directly, or by means of theoretical research on the procedures for the most efficient approximations.

One of the first problems, in the use and even the construction of the tables, is that of interpolation; and, as the precision of the calculations grows, it is noticed in the XVIIth century that the ancient procedure of linear interpo-

$^8$ During this exchange of letters, which is not done directly between the interested parties, but officially through the intermediary of the secretary of the Royal Society, Newton "takes the date" by stating his method as follows: 5accdae 10effh 12i ... rrrssssstttuu, an anagram in which the method of solution by a power series with indeterminate coefficients is contained ([198 d], p. 224).

lation loses its validity as soon as the first differences (differences between the successive values which occur in the table) cease to be effectively constant; so Briggs for example is seen ([36], chap. 13) making use of differences of higher orders, and even of fairly high order, in the calculation of logarithms. Later, we see Newton ([233 a], v. I, pp. 271-282 and [233 b], book III, lemma 5; see also [114]) and J. Gregory ([136 d], pp. 119-120), each on his own account, pursuing parallel researches on interpolation and on power series; both one and the other end up, by methods which are different besides, on the one hand with the formula for interpolation by polynomials, called "of Newton", and on the other with the binomial series ([136 d], p. 131; [198 d], p. 180) and with the principal expansions in series of classical analysis ([136 d]; [233 a], v. I, pp. 3-26 and 271-282 and [198 d], pp. 179-192 and 203-225); it is hardly in doubt that these two lines of research reacted one on the other, and were intimately linked in the mind of Newton with the discovery of the principles of the infinitesimal calculus. With Gregory as with Newton a great concern for numerical practice, the construction and use of tables, the numerical calculation of series and integrals comes to light; in particular, although no polished proof of convergence is found in their work, in the way of that of Lord Brouncker quoted above, both constantly mention convergence of their series from the practical point of view of their aptitude for calculation. It is still thus that we see Newton, in reply to a question set by Collins to a practical end ([262], v. II, pp. 309-310), applying to the approximate calculation of $\sum_{p=1}^{N} \frac{1}{n+p}$, for large values of $N$, a particular case of the method of summation of Euler-Maclaurin so-called.

The calculation of the values of a function starting from their differences is also met early on, used as a practical procedure for integration, and even, it might be said, for the integration of a differential equation. Thus Wright, in 1599, having to solve, with nautical tables in view, a problem that we would denote by $\frac{dx}{dt} = \sec t = \frac{1}{\cos t}$, proceeds by adding the values of $\sec t$, by successive intervals of a second of arc ([338]; cf. [230 b], p. 97), naturally obtaining to all intents and purposes a table for the values of $\log \tan(\frac{\pi}{4} + \frac{t}{2})$; and this coincidence, observed from the calculations of the first tables of $\log \tan t$, remained unexplained until the integration of $\sec t$ by Gregory in 1668 ([136 c] and [136 d], pp. 7 and 463).

But these questions also have a purely theoretic and even arithmetic aspect. Let us agree to denote by $\Delta^r x_n$ the sequence of successive differences of a sequence $(x_n)_{n \in \mathbf{N}}$, defined by recurrence by means of $\Delta x_n = x_{n+1} - x_n, \Delta^r x_n = \Delta(\Delta^{r-1} x_n)$, and to denote by $S^r$ the inverse operation to $\Delta$ and its iterates, therefore stating $y_n = Sx_n$ if $y_0 = 0, \Delta y_n = x_n$ and $S^r x_n = S(S^{r-1} x_n)$; we have

in particular, if $x_n = 1$ for all $n$, we have $Sx_n = n$, the sequences $S^2 x_n$ and $S^3 x_n$ are those of the "triangular" and "pyramid" numbers already studied by Greek arithmeticians, and in general we have

these sequences were introduced, from this point of view, in any case from the XVIth century; they appear by themselves also in combinatorial problems, which, either on their own, or in connection with probabilities, played a fairly big role in the mathematics of the XVIIth century, for example with Fermat and Pascal, then with Leibniz. They also turn up in the expression for the sum of the $m$th powers of the first $N$ integers, whose calculation, as we have seen, is at the basis of the integration of $\int x^m dx$ for $m$ an integer, by the first method of Fermat ([109], v. II, p. 83). It is thus that Wallis proceeds in 1655 in his Arithmetica Infinitorum ([326 a], v. I, pp. 355-478), without knowing of the work (unpublished) of Fermat, and without knowing either, he says, of the method of indivisibles other than from the lecture of Torricelli; it is true that Wallis, keen to reach the end, does not slow himself down by meticulous research: once the result obtained for the first integer values of $m$, he states it to be true "by induction" for all integers $m$, goes on correctly from there to $m = 1/n$ for $n$ an integer, then, by an "induction" even more summary than the first, to $m$ an arbitrary rational. But what is interesting and original in his work, is that he progressively lifts himself from there to the study of the "Eulerian" integral $I(m, n) = \int_0^1 (1 - x^{1/m})^n dx$ (whose value, for $m$ and $n > 0$, is $\Gamma(m+1)\Gamma(n+1)/\Gamma(m+n+1)$) and other such, draws up, for $m$ and $n$ integers, the table of values of $1/I(m, n)$ which is none other than that of the integers $\binom{m+n}{n}$, and, by methods almost identical to those in use today for expounding the theory of the $\Gamma$ function, ends up with the infinite product for $I\left(\frac{1}{2}, \frac{1}{2}\right) = \frac{\pi}{4} = (\Gamma\left(\frac{3}{2}\right))^2$; it is not difficult besides to make the method correct by means of integration by parts and very simple changes of variables, and by considering $I(m, n)$ for all real values of $m$ and $n$, which he could hardly have thought of, but which the Newtonian analysis would soon make possible. It is in any case the "interpolation" carried out 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 provides a point of departure for Newton when he starts out ([198 d], pp. 204-206), bringing him, first to the study of a particular case $(1-x^2)^{p/2}$, to the binomial series, from there to the introduction of $x^a$ (denoted thus) for all real $a$, and to the differentiation of $x^a$ by means of the binomial series; all that without great effort for obtaining rigorous proofs or even definitions; what is more, a remarkable innovation, it is from the knowledge of the derivative of $x^a$ that he deduces $\int x^a dx$ for $a \neq -1$ ([233 a], v. I, pp. 3-26 and [198 d], p.

225). For the remainder, and although he was soon in possession of much more general methods for expansions in power series, such as the method of Newton’s polygon so-called (for algebraic functions) ([198 d], p. 221) and that of indeterminate coefficients, he returns often in what follows, with a kind of predilection, to the binomial series and to its generalisations; and it is from there, for example, that he seems to have drawn out the expansion of $\int x^\alpha (1 + x)^\beta dx$ which was referred to above ([198 d], p. 209).

The evolution of ideas on the continent, meanwhile, is very different, and much more abstract. Pascal had drawn close to Fermat in the study of binomial coefficients (from which he constructs what he calls the “arithmetic triangle”) and their use in the calculation of probabilities and in the calculation of differences; when he tackles integration, he introduces there the same ideas. Like his predecessors, when he uses the language of indivisibles, he conceives of 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 $N = \infty$ ([244], v. VIII, pp. 352-355) (or, when he abandons this language for the correct language of exhaustion, as a limit of this ratio for $N$ growing indefinitely). But, having problems about moments in view, he observes that, when it is a matter of discrete masses $y_i$ spread out at equidistant intervals, the calculation of the total mass reduces 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 iterates the operation $\int$ to form what he calls the “triangular sum of the ordinates”, thus, in our language, the limits of the sums $N^{-2} S^2 \left( f \left( \frac{n}{N} \right) \right)$, that is to say the integrals $F_2(x) = \int_0^x F(x)dx$; a new iteration gives him the “pyramidal sums” $F_3(x) = \int_0^x F_2(x)dx$, limits of $N^{-3} S^3 \left( f \left( \frac{n}{N} \right) \right)$; the context besides is sufficient to show that, if he stops there, it is not because of a lack of generality in his thought nor in his language, but solely because he only expects to use these, whose systematic employment is at the basis of a good part of his results, and of which he proves immediately the properties that we would write as

([244], v. VIII, pp. 361-367); all that without writing a single formula, but in a language so transparent and so precise that one can immediately transcribe it into formulae as has just been done. With Pascal as with his predecessors, but in a much clearer and more systematic way, the choice of the independent variable (which is always one of the co-ordinates, or the arc of the curve) is implicit in the convention that fixes the equidistant subdivision points (although “infinitely near”) of the intervals of integration; these points, respectively, are, either on $Ox$, or on $Oy$, or on the arc of the curve, and Pascal takes care never to allow any ambiguity about this to remain ([244], v. VIII, pp. 368-369). When he has to change variables, he does it by means of a principle which reduces to saying that the area $\int f(x)dx$ can be written as $S(f(x_i)\Delta x_i)$, for every subdivision of the interval of integration into "infinitely small" intervals $\Delta x_i$, equal or not ([244], v. IX, pp. 61-68).

As can be seen, we are already very near Leibniz; and it is, it might be said, a lucky break that this latter, when he wished to be initiated into modern mathematics, met Huygens, who immediately put Pascal's writings into his hands ([198 d], pp. 407-408); he was particularly well prepared for this by his thoughts on combinatorial analysis, and we know that he made a deep study of it, which is reflected in his work. In 1675, we see him transcribing Pascal's theorem above in the form omn$(x\omega) = x.$ omn$\omega$ - omn(omn $\omega$), where omn $\omega$ is an abbreviation for the integral of $\omega$ taken from 0 to $x$, for which Leibniz, a few days later, substitutes $\int \omega$ (initial of "summa omnium $\omega$") at the same time as he introduces $d$ for the infinitely small "difference", or, as he will soon say, the differential ([198 d], pp. 147-167). Conceiving of these "differences" as quantities which are comparable amongst themselves but not with finite quantities, he takes besides most often, 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 $\frac{dy}{dx}$), and at first omits it from his notation for the integral, which thus appears as $\int y$ rather than $\int y dx$; but he does not delay introducing this, and keeps to it systematically once he realises its invariant character with respect to the choice of independent variable, which dispenses with having this choice constantly in mind;$^9$ and he shows no little satisfaction, when he returns to the study of Barrow that he had neglected up till then, in noting that the general theorem of change of variables, of which Barrow makes such a big thing, follows on immediately from his own notation ([198 d], p. 412). In all this as it happens, he keeps very close to the calculus of differences, from which his differential calculus can be deduced by going to the limit which of course he would be hard put to justify rigorously; and in what followed he voluntarily insists on the fact that his principles apply without change to both one and the other. He quotes explicitly Pascal, for example, when in his correspondence with Johann Bernoulli ([198 a], v. III, p. 156), referring to his first researches, he gives a formula for the calculation of differences which is a particular case of that of Newton, and deduces from it by "passing to the limit" the formula $y = \sum_1^\infty (-1)^n \frac{d^n y}{dx^n} \cdot \frac{x^n}{n!}$ (where $y$ is a function which is zero for $x = 0$, and the $\frac{d^n y}{dx^n}$ are its derivatives for the value $x$ of the variable) a formula equivalent to a similar one that Bernoulli communicated to him ([198 a], v. III, p. 150 and [20 a], v. I, pp. 125-128), and that this latter proves by successive integration by parts. This formula, as can

$^9$ "I warn that care should be taken not to omit $dx$ ... a mistake commonly committed, and which prevents going forward, because by that is taken away from these indivisibles, such as $dx$ here, their generality ... from which are born innumerable transfigurations and equipollences of figures." ([198 a], v. V, p. 233).

be seen, is very close to Taylor series; and it is even the argument of Leibniz, going to the limit starting with the calculus of differences, that Taylor finds again in 1715 to obtain "his" series [305],$^{10}$ without besides making great use of it.

F) One would have already seen, implicit in the evolution described above, the progressive algebraisation of infinitesimal analysis, that is to say its reduction to an operational calculus furnished with a uniform system of notation of an algebraic character. As Leibniz had indicated several times with perfect clarity ([198 a], v. V, pp. 230-233), it was a case of doing for the new analysis what Viète had done for the theory of equations, and Descartes for geometry. In order to understand its necessity, just read a few pages of Barrow; at no time is it possible not to have under one's eyes a figure, sometimes complicated, described beforehand with meticulous care; no less than 180 figures are needed for the 100 pages (Lect. V-XII) which form the essential part of the work.

It could hardly have been a question of algebraisation, it is true, before some unity had appeared amongst the multiplicity of geometric appearances. Meanwhile Grégoire de St. Vincent [135] already introduces (under the name of "ductus plani in planum") a kind of law of composition which reduces to the systematic use of 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 is far from drawing out the consequences that Pascal later deduces, as has been seen, from study of the same solid. Wallis in 1655, and Pascal in 1658, forge, each for his own use, languages of algebraic character, in which, without writing any formula, they draw up statements that can immediately be transcribed into formulae of the integral calculus as soon as their mechanism has been understood. The language of Pascal is particularly clear and precise; and, if it can not be understood why he refused to allow himself the use of the algebraic notations, not only of Descartes, but even of Viète, one can only admire the tour de force that he accomplished, and which his mastery of language was the only means of his being capable of doing.

But just let a few years go by, and everything changes. Newton first conceives of the idea of replacing all the operations, of a geometric character, of contemporary infinitesimal analysis, by a unique analytic operation, differentiation, and by the solution of the inverse problem; an operation that it is well understood the method of power series allowed him to carry out

$^{10}$ For the calculus of differences, Taylor could naturally rely on the results of Newton, contained in a famous lemma of the Principia ([233 b], Book III, lemma 5) and published more fully in 1711 ([233 a], v. I, pp. 271-282). As for the idea of passing to the limit, it seems typically Leibnizian; and it would be difficult to believe in the originality of Taylor on this point if there were not known from all periods many examples of disciples ignorant of all except the writings of their master and patron. Taylor quotes 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.

with extreme facility. Borrowing his language, we have seen, from the fiction of a universal "temporal" parameter, he qualifies as "fluents" the quantities varying as a function of this parameter, and as "fluxions" their derivatives. He does not seem to have attached particular importance to the notations, and his devoted supporters later even boast that the absence of a systematic notation is an advantage; he nevertheless picks up the habit early on, for his personal use, of denoting fluxion by a point, thus $\frac{dx}{dt}$ by $\dot{x}$, $\frac{d^2 x}{dt^2}$ by $\ddot{x}$, etc.. As for integration, it appears that Newton, exactly like Barrow, never saw it except as a problem (find the fluent knowing the fluxion, thus solve $\dot{x} = f(t)$), and not as an operation; also he has no name for the integral, nor, it seems, standard notation (except sometimes a square $\boxed{f(t)}$ or $\Box f(t)$ for $\int f(t) dt$). Is it because he dislikes giving a name and a symbol to an object that is not defined in a unique way, but only up to an additive constant?) Lacking a text, the question can only be asked.

As much as Newton is empirical, concrete, circumspect in his greatest boldness, so much is Leibniz systematic, generalisor, adventurous and sometimes rash innovator. From his youth, he had in mind the idea of a "characteristic" or universal symbolic language, which would be to the whole of human thought what algebraic notation was to algebra, where each name or symbol was the key to all the qualities of the object symbolised, and that could not be employed correctly without at the same time reasoning correctly (see p. 6). It is easy to treat such a project as fanciful; it is not by chance though that its author was the man himself who was soon to recognise and isolate the fundamental concepts of the infinitesimal calculus, and endow it with more or less definitive notations. We have already assisted above at the birth of these, and observed the care with which Leibniz, who seems aware of his mission, modifies them progressively until he ensures the simplicity and above all the invariance that he is seeking ([198 a], v. V, pp. 220-226 and 226-233). What is important to note here, is, as soon as he introduces them (without knowing anything yet of the ideas of Newton), the clear conception of $\int$ and of $d$, of the integral and the differential, as inverse operators one of the other. It is true that in proceeding in this way he can not avoid the ambiguity inherent in the indefinite integral, which is the weak point of his system, over which he slides adroitly, like his successors. But what is striking, from the first appearance of the new symbols, is to see Leibniz immediately busy with the formulation of rules of use, asking himself if $d(xy) = dx dy$ ([198 d], pp. 165-166), and answering himself in the negative, to arrive progressively at the correct rule ([198 a], v. V, pp. 220-226), which he was later to generalise by his famous rule for $d^n(xy)$ ([198 a], v. III, p. 175). Of course, at the time at which Leibniz is groping thus, Newton knew for ten years already that $z = xy$ implies $\dot{z} = \dot{x} y + x \dot{y}$; but he never takes the trouble to say it, only seeing it as a particular case, not worthy of being named, of his rule for differentiating the relation $F(x, y, z) = 0$ between fluents. On the contrary, the principal concern of Leibniz is not to make his methods useful for the solution of such concrete problems, nor even to deduce them from rigorous and impregnable principles, but above all to put in train an algorithm relying on the formal handling of a few simple rules. It is in this spirit that he improves the algebraic notation by the use of parentheses, that he adopts progressively $\log x$ or $l x$ for the logarithm,$^{11}$ and that he insists on the "exponential calculus", that is to say, the systematic consideration of exponentials, $a^x, x^x, x^y$, where the exponent is a variable. Above all, while Newton does not introduce fluxions of a higher order except strictly in proportion to the necessity in each concrete case, Leibniz turns early towards the creation of an "operational calculus" by the iteration of $d$ and of $\int$; becoming aware little by little of the analogy between the multiplication of numbers and the composition of operators in his calculus, he adopts, by a fortunate boldness, the notation for exponents to write the iterates of $d$, writing thus $d^n$ for the $n$-th iterate ([198 d], pp. 595 and 601,$^{12}$ and [198 a], v. V, pp. 221 and 378) and even $d^{-1}, d^{-n}$ for $\int$ and its iterates ([198 a], v. III, p. 167); and he seeks even to give a meaning to $d^\alpha$ for $\alpha$ an arbitrary real ([198 a], v. II, pp. 301-302, and v. III, p. 228).

That is not to say that Leibniz is not also interested in the applications of his calculus, knowing well (as Huygens repeats to him often ([198 d], p. 599)) that they are its touch stone; but he lacks patience to go deeply into them, and seeks there above all the opportunity for new general rules. It is thus that in 1686 ([198 a], v. VII, pp. 326-329) he deals with the curvature of curves, and with the osculating circle, to end up in 1692 ([198 a], v. VII, pp. 331-337) with general principles on the contacts of plane curves;$^{13}$ in 1692 ([198 a], v. V, pp. 266-269) and in 1694 ([198 a], v. V, pp. 301-306) he sets down the basis for the theory of envelopes; concurrently with Johann Bernoulli, he effects in 1702 and 1703 the integration of rational fractions by decomposition into simple elements, but first of all in a formal manner and without fully realising the circumstances that accompany the presence of complex linear factors in the denominator ([198 a], v. V, pp. 350-366). It is thus again that one day in August 1697, meditating in a carriage on questions of the calculus of variations, he has the idea for the rule for differentiation with respect to a parameter under the sign $\int$; and, enthusiastic, sends it straightaway to Bernoulli ([198 a], v. III, pp. 449-454). But when he gets

$^{11}$ But he does not have any symbol for the trigonometric functions, nor (for lack of a symbol for $e$) for the "number whose logarithm is $x$".

$^{12}$ "...it is just about as if, instead of roots and powers, one would like to substitute letters, and instead of $xx$, or $x^3$, take $m$, or $n$, after having declared that they must be powers of the quantity $x$. Judge, Sirs, how that was embarrassing. It is the same with $dx$ or with $ddx$, and the differences are no less attachments of indeterminate quantities in their places, than the powers are attachments of a quantity set apart. It seems to me then that it is more natural to denote them in such a way that they make known immediately the quantity of which they are attachments."

$^{13}$ On that he commits first an unusual error, believing that the "circle that kisses" (the osculating circle) has at the point of contact four points in common with the curve; and he only surrenders with difficulty, later on, to the objections of the Bernoulli brothers on that point ([198 a], v. III, pp. 187-188, 201-202 and 207).

there, the fundamental principles of his calculus have been secured for a long time, and their use has started to spread: the algebraisation of infinitesimal analysis is an accomplished fact.

G) The notion of function is introduced and made precise in a multitude of ways during the XVIIth century. All dynamics is based on an intuitive, and in some way experimental, idea, of quantities varying with time, that is to say functions of time, and we have already seen how we end up thus at a function of one parameter, such as it appears with Barrow, and, under the name of fluent, with Newton. The notion of "arbitrary curve" appears often, but is rarely made precise; it may be that often it is thought of in a dynamic form or in any case experimental form, and that without it being judged necessary that a curve should be capable of a geometric or analytic characterisation in order to be of use as an object of reasoning; it is so, in particular (for reasons that we are better able to understand today) when it is a case of integration, for example with Cavalieri, Pascal and Barrow; this latter, reasoning about the curve defined by $x = ct, y = f(t)$, with the hypothesis that $\frac{dy}{dt}$ is increasing, even says expressly that "it matters not" that $\frac{dy}{dt}$ increases "regularly following an arbitrary law, or even irregularly" ([16 b], p. 191) that is to say, as we would say, were capable or not of an analytic definition. Unfortunately this clear and fruitful idea, which would, made appropriately precise, reappear in the XIXth century, could not fight against the confusion created by Descartes, when this latter had, in the first case, banned from "geometry" all curves not capable of a precise analytic definition, and in the second case restricted only to algebraic operations the admissible procedures of formation in such a definition. It is true that, on this last point, he is not followed by the majority of his contemporaries; little by little, and often by very subtle detours, various transcendental operations, the logarithm, the exponential, the trigonometric functions, quadratures, the solution of differential equations, passing to the limit, the summing of series, acquired the right of being quoted, without it being easy in each case to mark the precise moment when the step forward is being taken; and besides the first step forward is often followed by a step back. For the logarithm, for example, one must consider as important stages the apparition 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 expansion in series of $\log(1 + x)$, and even the adoption of the symbol $\log x$ or $lx$. In what concerns the trigonometric functions, and even though in a certain sense they go back to antiquity, it is interesting to note that the sinusoidal curve does not appear at first as being defined by an equation $y = \sin x$, but just, with Roberval ([263], pp. 63-65), as a "companion of the roulette" (in fact, it is a case of the curve

that is to say as an auxiliary curve whose definition is deduced from that of the cycloid. To meet the general notion of analytic expression, we must go to J. Gregory, who defined it in 1667 ([169 a], p. 413), as a quantity which is obtained starting from other quantities by a succession of algebraic operations "or of every other imaginable operation"; and he tries to make precise this notion in his preface ([169 a], pp. 408-409), by explaining the necessity of adjoining to the five operations of algebra$^{14}$ a sixth operation, which in none other in the end than passing to the limit. But these interesting thoughts are soon forgotten, submerged by the torrent of expansions in series discovered by Gregory himself, Newton, and others; and the prodigious success of this last method created an enduring confusion between functions that were susceptible to analytic definitions, and functions expandable in power series.

As for Leibniz, he seems to keep to the Cartesian point of view, expanded 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. In the same way Johann Bernoulli, when he wishes to consider an arbitrary function of $x$, introduces it as "a quantity made up in any way starting from $x$ and constants" ([198 a], v. III, p. 150), specifying sometimes that it is a matter of a quantity made up "in an algebraic or transcendental manner" ([198 a], v. III, p. 324): and, in 1698, he agrees with Leibniz to give to such a quantity the name "function of $x$" ([198 a], v. III, pp. 507-510 and 525-526).$^{15}$ Already Leibniz had introduced the words "constant", "variable", "parameter", and specified in connection with envelopes the notion of a family of curves depending on one or more parameters ([198 a], v. V, pp. 266-269). The questions about notation become precise also in correspondence with Johann Bernoulli: this latter readily writes $X$, or $\xi$, for an arbitrary function of $x$ ([198 a], v. III, p. 531); Leibniz approves, but also proposes $\bar{x}|^1_-$, $\bar{x}|^2_-$ where we would write $f_1(x), f_2(x)$; and he proposes, for the derivative $\frac{dz}{dx}$ of a function $z$ of $x$, the notation $dz$ (in contrast to $dz$ which is the differential) whereas Bernoulli writes $\Delta z$ ([198 a], v. III, pp. 537 and 526).

Thus, with this century, the heroic era comes to an end. The new calculus, with its notions and notations, is constituted, in the form that Leibniz gave it. The first disciples, Jakob and Johann Bernoulli, vie in discoveries with the master, exploring freely the rich territories to which he had shown the way. The first treatise on differential and integral calculus was written in 1691 and

$^{14}$ They consist of the four rational operations, and the extraction of roots of arbitrary order; J. Gregory never ceased to believe in the possibility of solving by radicals equations of all degrees.

$^{15}$ Until then, and already in a manuscript of 1673, Leibniz had used this word as an abbreviation to designate a quantity "fulfilling such or such a function" for a curve, for example the length of the tangent or the normal (limited to the curve and $Ox$), or the sub normal, the sub tangent, etc. ..., thus in conclusion a function of a point varying on the curve, with a geometrico-differential definition. In the same manuscript of 1673, the curve is assumed defined by a relation between $x$ and $y$, "given by an equation" (that is to say, in our language, algebraic) (cf. [217]).

1692 by Johann Bernoulli,$^{16}$ for the use of a marquis who shows himself a good pupil. It matters little besides that Newton decided finally, in 1693, to publish parcimoniously a short glimpse of his fluxions ([326 a], v. II, pp. 391-396); if his Principia have enough to supply the meditation of more than a century, on the ground of the infinitesimal calculus he has been joined, and on many points overtaken.

The weaknesses of the new system are besides visible, at least to our eyes. Newton and Leibniz, abolishing at a stroke a tradition two millennia old, gave the primary role to differentiation, and reduced integration to being only its inverse; it will take the whole of the XIXth century, and a part of the XXth century, to re-establish a fair equilibrium, by putting integration as the basis of the general theory of functions of a real variable, and of its modern generalisation (see p. 223). It is from this reversal of points of view also that follows the excessive, and almost exclusive, role, taking place already with Barrow, and especially starting with Newton and Leibniz, played by the indefinite integral at the cost of the definite integral: there also the XIXth century had to put matters back in their place. Finally the properly Leibnizian tendency to a formal handling of symbols was to go on being emphasised throughout the whole of the XVIIIth century, well beyond what could be allowed by the resources of the analysis of the time. In particular, it must be recognised that the Leibnizian notion of a differential has, truth to tell no meaning; at the beginning of the XIXth century, it fell into a disrepute from which it has only recovered little by little; and, if the use of first differentials ended up by being completely legitimised, the differentials of higher order, although so useful, have not yet been really rehabilitated to our day.

Whatever the case, the history of the differential and integral calculus, starting with the end of the XVIIth century, is divided into two parts. One is concerned with the applications of this calculus, always richer, more numerous and more varied. To the differential geometry of plane curves, to differential equations, to power series, to the calculus of variations, which have already been referred to above, have been joined the differential geometry of ill-shaped curves, then of surfaces, multiple integrals, partial differential equations, trigonometric series, the study of numerous special functions, and many other types of problems. We only should be concerned here with work which has contributed to clearing up, deepening and consolidating the very principles of the infinitesimal calculus, in what concerns functions of a real variable.

$^{16}$ The part of this treatise which is concerned with the integral calculus was published in 1742 only ([20 a], v. III, pp. 385-558 and [20 b]); that which is concerned with the differential calculus was only found again and published recently [20 c]; it is true that the marquis de l’Hôpital had published it in French, slightly modified, under his own name, which is remarked on with some bitterness by Bernoulli in his letters to Leibniz.

From this point of view, the great treatises of the middle of the XVII-Ith century offer few novelties. Maclaurin in England$^{17}$ [214], Euler on the continent ([108 a], (1), vv. X to XIII), remained faithful to the traditions of which each one was the inheritor. It is true that the first tries hard to clarify somewhat the Newtonian concepts$^{18}$ whereas the second, pushing Leibnizian formalism to an extreme, is content, like Leibniz and Taylor, to base the differential calculus on a very obscure passage to the limit starting with the calculus of differences, a calculus of which as it happens he gives a very polished exposition. But above all Euler finishes the work of Leibniz by introducing and ensuring the adoption of the notation still in use today for $e, i$ and the trigonometric functions, and spreading the notation $\pi$. On the other hand, and if most of the time he does not make any distinction between functions and analytic expressions, he insists, in connection with trigonometric series and the problem of vibrating strings, on the need not to limit oneself to the functions thus defined (and that he qualifies as "continuous") but to consider also, if the need arises, arbitrary, or "discontinuous", functions given experimentally by one or more arcs of a curve ([108 a], (1), v. XXIII, pp. 74-91). Finally, even though this is somewhat outside our framework, it is not possible not to mention here his extension of the exponential function to the complex domain, from which he obtains the famous formulae linking the exponential to the trigonometric functions, as well as the definition of the logarithm of complex numbers; from that is elucidated definitively the famous analogy between the logarithm and the inverse circular functions, or, in the language of the XVIIth century, between the quadratures of the circle and of the hyperbola, observed already by Grégoire de St. Vincent, made precise by Huygens and above all by J. Gregory, and which, with Leibniz and Bernoulli, appeared in the formal integration of

Meanwhile d'Alembert, the enemy of all mysticism in mathematics as elsewhere, had, in remarkable articles ([75 a], articles DIFFÉRENTIEL and LIMITE, and [75 b]), defined with the greatest clarity the notions of limit and of derivative, and maintained strongly that in the end that is all the "metaphysics" of the infinitesimal calculus. But this wise advice did not have any immediate effect. The monumental work of Lagrange ([191], vv. IX-X) represents an attempt to base analysis on one of the most arguable of the Newtonian concepts, that which confuses the notions of arbitrary functions and that of functions which can be expanded in a power series, and to draw from that (by consideration of the coefficient of the term of the first order

$^{17}$ Maclaurin was professor at Edinburgh in Scotland (translator's note). $^{18}$ They had great need, indeed, to be defended against the philosophico-theologico-humoristic attacks of the famous bishop Berkeley. According to him, those who believe in fluxions should not find it very difficult to have faith in the mysteries of religion: an argument ad hominem, which was not lacking in logic or in spice.

in the series) the notion of differentiation. Of course, a mathematician of the quality of Lagrange could not fail to obtain in this case important and useful results, as for example (and in a way which was in fact independent of the point of departure that we have just indicated) the general proof of the formula of Taylor with the expression for the remainder in the form of an integral, and its evaluation by the theorem of the mean; in any case the work of Lagrange is at the origin of the method of Weierstrass 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 objective, it represents a retreat rather than progress.

With the teaching works of Cauchy, on the contrary ([56 a], (2), v. IV) we find ourselves at last on solid ground. He defines a function essentially as we do it today, even though in a language which is still rather vague. The notion of limit, fixed once and for all, is taken as the point of departure; those of continuous function (in the modern meaning) and of derivative are immediately deduced, as well as their principal elementary properties; and the existence of the derivative, instead of being an act of faith, becomes a matter of study by the ordinary methods of analysis. Cauchy truth to tell, is hardly interested in this; and on the other hand, if Bolzano, having reached on his part the same principles, constructed an example of a continuous function not having a derivative at any point [27 a], this example was not published, and the question was only publicly settled by Weierstrass, in a work in 1872 ([329 a], v. II, pp. 71-74).

As far as integration is concerned, the work of Cauchy represents a return to the sane traditions of antiquity and the first part of the XVIIth century, but relying on technical means which are still insufficient. The definite integral, passed over into second place for too long, becomes again the primary notion, for which Cauchy ensures the definitive adoption of the notation $\int_a^b f(x) dx$ proposed by Fourier (instead of the awkward $\int f(x) dx \left[ \begin{array}{l} x = a \\ x = b \end{array} \right]$) sometimes used by Euler); and, in order to define it, Cauchy returns to the method of exhaustion, or as we would say, to "Riemann sums" (which would be better called sums of Archimedes, or sums of Eudoxus). It is true that the XVIIth century did not deem it necessary to submit the notion of area to a critical examination, an idea which had appeared to them to be at least as clear as that of incommensurable real number; but the convergence of "Riemann" sums to the area under the curve, so long as it is the case of a monotonic curve or a piecewise monotonic curve, was a familiar idea to all the authors concerned with rigour in the XVIIth century, such as Fermat, Pascal, Barrow; and J. Gregory, particularly well prepared by his thoughts on the passage to the limit and his familiarity with an already very abstract form of the principle of "nested intervals", had even drawn up, it would appear, a careful proof, which remained unpublished ([136 d], pp. 445-446), which could have been of use to Cauchy almost without any change if he had known it.$^{19}$ Unfortunately for him, Cauchy claimed to prove the existence of the integral, that is to say the convergence of the "Riemann sums", for an arbitrary continuous function; and his proof, which would have become correct if it were based on the theorem on uniform continuity of functions continuous in a closed interval, is stripped of all convincing value for lack of this notion. Dirichlet does not seem to have seen either the difficulty at the time when he drew up his famous memoirs on trigonometric series, since he quotes there the theorem in question as "easy to prove" ([92], v. I, p. 136); it is true that he only applies it definitively to bounded piecewise monotonic functions; Riemann, more circumspect, only mentions these latter when it is a matter of using his necessary and sufficient condition for the convergence of "Riemann sums" ([259 a], pp. 227-271). Once the theorem on uniform continuity had been established by Heine (cf. p. 145), the matter did not offer any more difficulties of course; and it is easily dealt with by Darboux in 1875 in his memoir on the integration of discontinuous functions [77], a memoir in which he meets on many points with the important researches of P. du Bois-Reymond, which appeared at about the same time. In the same way for the first time, but this time definitively, the linearity of the integral of continuous functions is proved. On the other hand, the notion of uniform convergence of a sequence or series, introduced among others by Seidel in 1848, and put into prominence in particular by Weierstrass (cf. p. 205) had allowed the construction of a solid basis, under conditions that were a bit too restrictive it is true, for the integration of series term by term, and for differentiation under the $\int$ sign, while awaiting modern theories of which we do not have to speak here, and which would clarify these questions in a manner which is provisionally definitive.

We have thus attained the final stage of the classical infinitesimal calculus, that which is represented by the great Treatises on Analysis of the end of the XIXth century; from the point of view that concerns us, that of Jordan [174 c] occupies amongst them a pre-eminent place, for aesthetic reasons on the one hand, but also because, if it constitutes an admirable setting out of the results of classical analysis, it announces in many ways modern analysis and prepares the way for it.

$^{19}$ It is at least what is indicated by the resumé given by Turnbull from the manuscript.