## La methode des fluxions [translated by Georges Louis Le Clerc].

Paris: De Bure l’aine, 1740.

First edition in French of Newton’s first exposition of his fluxional calculus, translated with a long and important preface by the celebrated naturalist Comte de Buffon. Originally written in 1671, in Latin, this was Newton’s first comprehensive presentation of his method of fluxions which, according to Hall ‘might have effected a mathematical revolution in its own day’ (*Philosophers at War*, pp. 65-6). It should properly be placed first in the great trilogy of Newton’s major works: *Fluxions*, *Principia* (1687) and *Opticks* (1704). Newton’s *Methodus fluxionum* remained unpublished until its English translation by John Colson in 1736. In it he presents a method of determining the magnitudes of finite quantities by the velocities of their generating motions. At its time of preparation, it was Newton’s fullest exposition of the fundamental problem of the calculus, in which he presented his successful general method. Newton prepared this treatise just before his death. The autograph manuscript, which survives in Cambridge University Library, was entrusted to Henry Pemberton after Newton’s death but he did not publish it. John Colson (1680-1760) based his translation on a copy of Newton’s original manuscript made by William Jones. Both Newton’s manuscript and Jones’s copy lack a title page and it is unknown what title, if any, Newton gave to the manuscript. The title ‘De methodus fluxionum’ originates with Colson. In the preface, Colson writes “I thought it highly injurious to the memory and reputation of our own nation, that so curious and useful a piece should be any longer suppressed.” Buffon translated Colson’s edition in 1737 and added his lengthy preface the following year. The most interesting part of the preface is that dealing with the conception of the infinite and the metaphysical errors to which it leads. This includes a discussion of Berkeley’s *The analyst* (1734) which, oddly, he criticizes although Berkeley’s conclusions are very similar to his own.

*Provenance*: Eugène Brand (signature on title dated 1890).

Newton wrote three accounts of the calculus. The composition of the first, a tract entitled ‘De analysi per aequationes numero terminorum infinitas,’ resulted from Newton’s reception from Isaac Barrow, in the early months of 1669, of a copy of Mercator’s *Logarithmotechnia*, a work which contained the series for log(1 + *x*). The work, in which Newton demonstrated his much more general methods of infinite series, was not published until 1711, when William Jones included it, along with a number of other tracts, in his *Analysis per quantitatum series*. In ‘De analysi,’ however, Newton “did not explicitly make use of the fluxionary notation or idea. Instead, he used the infinitely small, both geometrically and analytically, in a manner similar to that found in Barrow and Fermat, and extended its applicability by the use of the binomial theorem. … It will be noticed that although the work of Newton contains the essential procedures of the calculus, the justification of these is not clear from the explanation he gave. Newton did not point out by what right the terms involving powers of *o* were to be dropped out of the calculation, any more than Fermat or Barrow … His contribution was that of facilitating the operations, rather than of clarifying the conceptions. As Newton himself admitted in this work, his method is ‘shortly explained rather than accurately demonstrated’” (Boyer, *The Concept of Calculu*s, p.191).

It was first in ‘Methodus fluxionum’ that “Newton introduced his characteristic notation and conceptions. Here he regarded his variable quantities as generated by the continuous motion of points, lines, and planes, rather than as aggregates of infinitesimal elements, the view which had appeared in ‘De analysi’. … In the ‘Methodus fluxionum’ Newton stated clearly the fundamental problem of the calculus: the relation of quantities being given, to find the relation of the fluxions of these; and conversely” (*ibid*., pp. 192-3).

In Newton’s third exposition, *De quadratura*, which was composed some twenty years after ‘Methodus fluxionum’ and published as an appendix to the *Opticks*, “Newton sought to remove all traces of the infinitely small” (*ibid*.).

“It was often lamented that the world had had to wait so many years to see Newton’s masterpiece on fluxions. It is astonishing to realize that publication sixty years beforehand would have changed the history of the calculus and would have avoided for Newton any controversy over priority. In 1736 all the results contained in Newton's treatise were well known to mathematicians. However, it was too concise for a beginner, and Colson added almost 200 pages of explanatory notes. His commentary contributed to the establishment of a kinematical approach to the problem of foundations. In his explanatory notes Colson presents the ‘geometrical and Mechanical Elements of Fluxions’. He writes:

‘The foregoing Principles of the Doctrine of Fluxions being chiefly abstracted and Analytical. I shall here endeavour, after a general manner, to shew something analogous to them in Geometry and Mechanicks: by which they may become not only the object of the Understanding, and of the Imagination, (which will only prove their possible existence) but even of Sense too, by making them actually to exist in a visible and sensible form’.

“Colson was convinced that by using moving diagrams it is possible to exhibit ‘Fluxions and Fluents Geometrically and Mechanically … so as to make them the objects of Sense and ocular Demonstration'. The motivation for using the geometrical and mechanical elements of fluxions is clearly that of guaranteeing an ontological basis to the calculus; in fact:

‘Fluents, Fluxions, and their rectilinear Measures, will be sensibly and mechanically exhibited, and therefore must be allowed to have a place *in rerum natura’*.

“Colson’s approach to the calculus is representative of a whole generation of British mathematicians: his ‘sensibly exhibited rectilinear measures’ of fluxions are a naive anticipation of Maclaurin’s kinematic definitions of the basic concepts of the calculus” (Guicciardini, *The Development of Newtonian Calculus in Britain 1700-1800*, pp. 56-57).

“In his preface …, Colson noted:

‘The chief Principle, upon which the Method of Fluxions is here built, is… taken from the Rational Mechanicks; which is, That Mathematical Quantity, particularly Extension, may be conceived as generated by continued local Motion; and that all Quantities may be conceived as generated after a like manner. Consequently there must be comparative Velocities of increase and decrease during such generations, whose Relations are fixt and determinable, and may therefore … proposed to be found.’

“Thus, a line or a curve was seen as generated by a continuously moving point, a surface by the motion of a line and a solid by the motion of a surface. After defining fluxions, fluents and moments, Newton went on to show how, within this framework, significant results could be derived. Following an introduction in which it was shown how equations could be solved with the use of infinite series, seven major problems were considered:

- From the Following Quantities (fluents) given, to find their fluxions.
- From the given Fluxions to find the Flowing Quantities.
- To determine Maxima and Minima of Quantities.
- To draw Tangents to Curves.
- To find the Quantity of Curvature in any Curve.
- To find the Quality of Curvature in any Curve.
- To find any number of Curves that may be squared”

(Gjertsen, *Newton Handbook*, p. 158).

“Buffon did start his scientific career as a Newtonian. He agreed that science should search for nature’s laws and that those laws should be as simple and as universal as possible. Buffon’s strong stance in favor of an orthodox Newtonianism was most obvious during his academic polemics with Alexis Clairaut. Buffon also published translations of two English books: Stephen Hales’s *Vegetable Staticks* (1735) and Newton’s *Treatise on Fluxions* (1740). The young man who wrote the prefaces to these books praised the experimental spirit of the English. But to what extent did these texts in fact express Buffon’s supposed Newtonian position? …

“The case of the preface to Newton’s *Fluxions* (1740) was a different matter, since it appeared to be a sign of allegiance both to Newton and to mathematics (in the guise of the calculus). But in fact Buffon’s preface, while acknowledging the perfect clarity of Newton’s ideas, developed a metaphysical critique of the concept of the infinite that had been closely tied to the practice of geometry. Buffon asserted that our daily experience (by means of sensation) is restricted to the limited, the finite—and therefore that the arithmetical or geometrical infinite had no actual existence. The preface to the *Fluxions*, far from being a sign of Buffon’s loyalty to mathematical conceptions of science, instead stressed the lack of reality of mathematical ideas. Some of these strong statements would later be developed near the end of the ‘Premier discours’ of the *Histoire naturelle*” (Hoquet, pp. 39-41).

“In his preface, Buffon rewrote the history of the calculus – drawing inspiration largely from a book that Fontenelle had published in 1727, *Élémens de la géométrie de l’infini *– in which he sided strongly with Newton [against Leibniz]. He was rightly criticized for his lack of objectivity, and he became closely tied with English scholars whose point of view he blindly adopted. In France, furthermore, he became involved with Clairaut, Maupertuis and Voltaire in a battle in defense of Newton. His translation and preface must be viewed from his perspective – historical objectivity was not his main concern …

“The debate on infinity tells us something about Buffon’s intellectual temperament … At the end of the seventeenth century a lengthy evolution of ideas had led to the Newtonian conception of an infinite time and space and, therefore, an infinite universe … Calculus gave a new topicality to this philosophical debate, since it raised the question of whether the infinitely small quantities manipulated by the new calculus really existed. Leibniz did not believe so … In 1727 Fontenelle defended their real existence, and Buffon seemed at first to have accepted his argument. He now attacked Fontenelle without naming him …

“Buffon rejected Fontenelle’s conclusion, mainly because he did not differentiate between geometrical and metaphysical infinities. ‘The idea of infinity,’ he said, ‘is only an idea of absence, and has no concrete representation.’ Even ‘space, time, and duration are not real Infinities.’ Likewise, ‘there is no number that is at present Infinite or infinitely small, or smaller or bigger than an Infinity, etc.’ Because ‘Numbers are no more than representations, and never exist independently of the things they represent,’ they do not have a ‘real existence’, and things themselves cannot be infinite …

“The direct consequence of this philosophy was that mathematics does not teach us anything about reality. More precisely – and here Buffon distanced himself radically from Fontenelle – mathematics does not have its own reality. Fontenelle gives an intellectual reality to numbers and geometrical figures, independent of all physical and metaphysical reality. For Buffon, there was only physical reality. Thus, mathematics was only a tool, practical, even indispensable, but nothing more …

“The last argument in which Buffon intervened was the one that the idealistic philosopher Berkeley had provoked by attacking the metaphysical foundations of calculus …it is clear that Buffon addressed it only to defend his friend the English doctor and mathematician James Jurin. Regardless of what he said, Buffon certainly had not read Berkeley’s book [*The analyst*, 1734] attentively, otherwise he would have seen that Berkeley’s criticisms of the status of the infinitely small corresponded exactly to his own, although they were based on an extremely different metaphysics. As with Leibniz the fundamental philosophical differences prevented Buffon from recognising what they had in common. His attack on Berkeley was more satire than philosophical discussion. By intervening so lightly into a serious debate, Buffon exposed himself to criticism. The interesting thing about this episode is that it shows his friendship with James Jurin and suggest that it was Jurin who had advised him in the Leibniz-Newton controversy” (Roger, pp. 34-38).

Babson 173; Macclesfield 1533; Wallis 236. Hoquet, ‘History without Time. Buffon’s natural history as a nonmathematical physique, *I**sis* 101 (2010), pp. 30-61. Roger, *Buffo*n: *A Life in Natural History*, 1997.

4to (255 x 196 mm), pp. xxx, [4] (errata and privilege), 148, title printed in red and black, woodcut figures in text. Contemporary quarter-morocco and marbled boards, spine ruled and in gilt with red lettering-piece (a little rubbed, joints starting).

Item #5802

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Price:
$4,500.00
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