## Analysis per quantitatum series, fluxiones, ac differentias: cum enumeration linearum tertii ordinis.

London: Pearson, 1711.

First edition of the third of Newton’s great works on physics and mathematics, following *Principia *(1687) and *Opticks* (1704), and certainly the rarest of the three. This is a very fine copy in untouched contemporary English calf. This work contains ‘De Analysi per Aequationes Numero Terminorum Infinitas,’ written in 1669 and published here for the first time, containing Newton’s theory of infinite series; ‘Methodis differentialis,’ a treatise on interpolation written in 1676 and published here for the first time, the basis of the calculus of finite differences; two treatises, ‘De quadratura curvarum’ and ‘Enumeratio linearum tertii ordinis,’ first published in the *Opticks* but written in 1693 and 1695; the ‘Epistola prior’ and ‘Epistola posterior,’ first published in vol. III of John Wallis’ *Opera* (1699), a letter from Newton to Collins, written November 8, 1676, and one to Wallis dated August 27, 1692. Newton described *De analysi* “to Oldenburg as ‘a compendium of the method of these [infinite] series, in which I let it be known that, from straight lines given, the areas and lengths of all the curves and the surfaces and volumes of all the solids [formed] could be determined, and conversely with these [taken as] given the straight lines could be determined, and I illustrated the method there outlined by several series.’ Despite the use of the words ‘method of series’ rather than ‘method of fluxions’ (in the letter quoted Newton made no open reference of ‘fluxions’ at all), it is obvious from the inversion (lines to areas, areas to lines) that differentiation and integration, that is, the method of fluxions, is in question” (Hall, pp. 16-17). “Modern workers, Duncan Fraser noted in 1927, had only just struggled up to the level reached by Newton in 1676 [in ‘Methodus differentialis’]. Whiteside was equally impressed by the work, claiming that ‘During the years 1675-76 … Newton laid down the … modern elementary theory of interpolation by finite differences but … diffidently kept back his insights and discoveries therein for nearly forty years more’ (*Papers*, IV, pp. 7-8)” (Gjertsen, p. 357). Newton’s decision to allow the publication of the 1669 tract at this time was heavily influenced by the on-going priority dispute with Leibniz over the invention of calculus. “The work [*Analysis per quantitatum series* …] also contained a Preface drafted, no doubt, with Newton’s assistance. It contained no mention of Leibniz. It did, however, contain the claim that Newton had ‘Deduced the quadrature of the circle, hyperbola, and certain other curves by means of infinite series … and that he did so in 1665; then he devised a method of finding the same series by division and extraction of roots, which he made general the following year’” (*ibid*., p. 18). For good measure, the tract was included in its entirety the following year in *Commercium epistolicum*, the official report on the priority dispute, largely drafted by Newton himself.

“*De analysi*, the work which established Newton’s reputation outside the walls of Trinity College, was first heard of in a letter from Barrow to Collins dated 20 June 1669. ‘A friend of mine,’ Barrow wrote, ‘brought me the other day some papers, wherein he hath sett downe methods of calculating the dimension of magnitudes like that of Mr Mercator concerning the Hyperbola; but very Generall; as also of resolving equations’ (*Correspondence*, I, p. 13). The manuscript, with Newton’s permission, was sent to Collins on June 31. The author’s name was revealed to Collins on 20 August, when Barrow wrote that the author was ‘Mr Newton, a fellow of our College, and very young … but of an extraordinary genius and proficiency in these things’ (*ibid*., pp. 14-15).

“Not only was Collins the first outside Cambridge to see important work of Newton; he had also, although inadvertently, provoked the work. In the early months of 1669 he had sent Barrow a copy of Mercator’s *Logarithmotechnia* (1668), a work which contained the series for log(1 + *x*). Barrow was aware that Newton had worked out for himself a general method for infinite series some two years before. Mercator’s book warned Barrow, and through him Newton, that others were working along similar lines. Newton’s reaction was to write, probably in a few summer days of 1669, his treatise *De analysi* which showed, by its generality, how far ahead he was of all other rivals.

“Collins, like Barrow, had no difficulty in recognising the originality and power of Newton’s technique and, consequently, brought up the question of publication. An appendix to Barrow’s forthcoming optical lectures seemed a suitable place. Newton revealed, however, for the first time, his ability to frustrate even such skilled and persistent suitors as Collins. Immediate publication was rejected out of hand; thereafter Newton deployed a variety of excuses: a need to revise the work, a desire to add further material, the pressures of other business and, as a last resort when demands became too pressing, he simply failed to reply. As a result *De analysi* remained, with a good deal more of Newton’s early mathematical work, unpublished for half a century.

“Newton’s reluctance to publish did not prevent Collins from copying and distributing the work. One copy was found by Jones in 1709 and is now to be seen in the Royal Society. Another copy was sent to John Wallis, at some point passed to David Gregory, and is at present in the Gregory papers at St. Andrew’s. Others who heard from Collins of Newton’s work were James Gregory, de Sluse and, above all, Leibniz. In October 1676 Leibniz visited London, saw Collins, and was allowed to read *De analysi*. He took thirteen printed pages of notes, an event construed by Newton as undoubted evidence of Leibniz’s reliance upon the discoveries of others in his mathematical development.

“The work, Newton began, would present a general method ‘for measuring the quantity of curves by an infinite series of terms.’ To this end, three rules were formulated” (Gjertsen, pp. 149-150).

Rule 1 stated that the area under a curve (in modern terms, the integral) of the form *y = x ^{m/n}*, where

*m*and

*n*are positive whole numbers, is

*nx*

^{(m+ n)/n}/(

*m+n*).

Rule 2 stated that the area under a sum of different curves *y = X*_{1} + *X*_{2} is the sum of the areas under the individual curves *y = X*_{1} and *y = X*_{2} (the integral of a sum is the sum of the integrals).

Proofs were offered for both of these rules.

“In the third rule, which took up the bulk of the work, Newton considered cases where ‘the value of y, or any of its terms’ were so compounded as to require a reduction into more simple terms. This was done variously by division, by the extraction of roots, and by the resolution of affected equations [equations in which *y* is only implicitly defined in terms of *x*, such as by a polynomial equation involving both *x* and *y*]. Thus, if the curve was a hyperbola, and the equation was *y* = 1/(1 + *x*^{2}), Newton began by dividing 1 by 1 + *x*^{2} which yielded:

*y* = 1 – *x*^{2} + *x*^{4} - *x*^{6} + *x*^{8} - ….

Rule 2 was applied at this point and the area of the hyperbola was seen to be equal to:

*x* – 1/3 *x*^{3} + 1/5 *x*^{5} – 1/7 *x*^{7} +1/9 *x*^{9} - ….

The series, Newton noted, was an infinite one and therefore carried on indefinitely. What, then, of the area of the hyperbola? No matter, Newton somewhat complacently responded, as ‘a few of the initial terms are exact enough for any use.’

“There was more in *De analysi*, historians have noted, than the manipulation of infinite series. When at the conclusion of the paper Newton set out his proof of Rule 1, he revealed at the same time details of his method of fluxions. The proof required the use of ‘infinitely small’ areas (later to be called ‘moments’). It was, Boyer has noted, ‘the first time in the history of mathematics that an area was found through the inverse of what we call differentiation’, and thus made Newton ‘the effective inventor of the calculus’, for his ability ‘to exploit the inverse relationship between slope and area through his new infinite analysis.’

“Newton’s failure to establish his priority at this point by following the advice of Barrow and Collins would later involve him, and many others, in much distress and in considerable polemical effort” (Gjertsen, pp.151-152).

Born the son of a Welsh farmer, William Jones (1675-1749) earned his living teaching mathematics. One of his pupils later became the Earl of Macclesfield, who in due course took Jones upon his staff at Shirburn Castle, Oxfordshire. Jones was known to Newton through the publication of his *Synopsis palmariorum matheseos* (1706), a work which introduced the symbol *π* to denote the ratio of the circumference to the diameter of a circle.

“In 1708, [Jones] had obtained the papers of John Collins, including the original correspondence in which Barrow had discussed Newton’s mathematical work for the first time outside Cambridge. Also among Collins’ manuscripts were copies that had been made of a number of unpublished papers. One of these was an anonymous version of ‘De analysi’. From the correspondence now in his possession, Jones was quickly able to identify Newton as the author of this essay and he began to make preparations for its publication.

“Jones was perhaps fortunate in the moment of his acquisition of Collins’ archive. It is hard to imagine Newton wishing to collaborate on an edition of his juvenilia at any time before the middle of the first decade of the eighteenth century. Then, however, it started to become increasingly important to him to find clearly dated evidence of his work on infinitesimals during the 1660s. Only by doing this could he establish a significant interval between the moment of his own invention of the calculus and Leibniz’s discoveries. The prospect of access to the letters in which he had first described his mathematical activities must have seemed like a godsend. In particular, the correspondence with Collins about Newton’s planned additions to Mercator’s translation of Kinckhuysen [*Algebra Ofte Stel-konst*, 1661, an introduction to algebra] indicated that he had already reached beyond the mathematical competence of his Continental counterparts. Letters from Barrow and Collins testified to the extent of Newton’s abilities even before he had read Mercator’s *Logarithmotechnia *(1668), a book that had considerably extended contemporary knowledge of infinite series. It was true that Newton’s youthful letters expressed modesty and reservations about the nature of his own discoveries at this point. But the slightly later tract, ‘De analysi’, which Newton had planned to revise for publication in the early 1670s, suggested a more confident claim to the originality of his thinking. Moreover, as Newton almost certainly realised, Collins had allowed Leibniz sight of his copy of the manuscript when the young German natural philosopher visited London in October 1676.

“Newton communicated the autograph copy of ‘De analysi’ to Jones for use in the preparation of his edition. He also gave permission for Jones to include two other early mathematical papers, ‘Enumeratio linearum tertii ordines’ and ‘Methodus differentialis’ in his work. These dated in origin from the late 1660s and early 1670s, as notes in Newton’s ‘Waste Book’ and elsewhere indicate. They bore signs, however, of much more recent revision. This was even more true of the fourth essay that Jones edited, ‘De quadratura curvarum’, in which Newton’s full mastery of the dynamic nature of his calculus and of the peculiar notation that expressed it was made clear. Newton composed this work in the early 1690s, not in the 1660s, as he had hinted when he had published it as an appendix to the *Opticks* in 1704 …

“In his introduction to the edition, Jones quoted extensively from the correspondence that he had collected to prove Newton’s priority in the invention of the calculus. In about 1712, he placed many of the originals at Newton’s disposal. Some of these, together with both Collins’ copy and the autograph of ‘De analysi’, Newton later deposited in the Royal Society. Most of the earliest letters, however, entered the Macclesfield Collection [and are now in Cambridge University Library]. As a result of his efforts, Jones was elected a Fellow of the Royal Society in 1712” (*Footprints*, pp. 78-80).

“The [*Methodis differentialis*] began, as did so many of Newton’s mathematical papers, with a query. A certain John Smith working on a table of square and higher roots, at the suggestion of John Collins, sought Newton’s advice in 1675 on ways to reduce the immense computational labours involved. Newton advised Smith to pursue methods of interpolation and, more importantly, began to consider himself how such methods could be generalised. The *Methodis*, at its most general, sought to show how ‘Given some number of terms of any series whatever arranged at given intervals, to find any intermediate term you will with close approximation’ (*Papers *VIII, p. 251). Something of this work also emerged in *Principia* in Book III, lemma V, where it is demonstrated how ‘To find a curved line of the parabolic kind which shall pass through any given number of points.’

*De quadratura curvarum* was the last of Newton’s major treatises on calculus to be composed, in 1691, but the first to be published, as an appendix to the *Opticks* (1704). It was in this work that Newton developed the method of fluxions in terms of the ‘prime and ultimate ratios’, an early version of the theory of limits, first met with in *Principia*. “The work is significant in a number of ways. At the level of notation the manuscript of 1691-2 saw for the first time the use of Newton’s dotted fluxional notation … Also used was a capital Q to stand for the process of quadrature, rather than the summation sign ∫ adopted by Leibniz in his published work. On a more substantive issue, *De quadratura* contained the first published statement of the binomial theorem, discovered by Newton some forty years before.

“The text of *De quadratura*, in its published form, is in two parts. In the first part Newton, in the manner of *De analysi*, demonstrated how infinite series could be deployed to determine the quadrature and rectification of curves. In the second part he returned to the topic of fluxions, discussed at greater length in his then unpublished *De methodis* [eventually published as *The method of fluxions and infinite series* in 1736]” (Gjertsen, p. 579).

In the final treatise in this collection, *Enumeratio linearum tertii ordinis*, composed around 1695 and first published in the *Opticks*, Newton sought to classify cubic curves, in a manner analogous to the classification of quadratic curves (conics) into ellipses, parabolas and hyperbolas (and some degenerate cases). Newton identified 72 species of cubic curves, mostly classified in terms of the properties of their diameters and asymptotes. There are, in fact, 78 species: four were added by James Stirling in his *Lineae tertii ordinis Newtonianae* (1717), and the remaining two by François Nicole and Nicolas Bernoulli in the 1730s. Newton also stated a general theorem according to which all cubic curves can be obtained as ‘shadows’ cast by one particular species of cubic, in a manner analogous to that in which every conic section can be obtained as a shadow of a circle. This was proved by Nicole and Alexis-Claude Clairaut in 1731. “In some ways the *Enumeratio *is the most original of Newton’s mathematical work. It had no predecessors, met with no rivals claiming to have anticipated the results, or few even who acknowledged its results. No mention of the work appeared in the *Philosophical Transactions* before 1715, while Bernoulli, writing to Leibniz in 1714, commented ‘I have not yet been able to bring myself to examine this matter, since I do not willingly embroil myself with intricacies of that sort, utterly useless as they are indeed’ (*Papers*, VII, p. 572)” (Gjertsen, p. 187).

Babson 207; ESTC T18644; Gray 294 (incorrectly giving the date 1712); Horblit 66b; Wallis 293. Gjertsen, *Newton Handbook*, 1986. Hall, *Philosophers at War. **The Quarrel between Newton and Leibniz*, 1980. *Footprints of the Lion*: *Isaac Newton at Work*, 2001. Whiteside (ed.), *The Mathematical Papers of Isaac Newton*, 1967–81, vol. II, pp. 206–59, vol. III, pp. 3–19, & vol. VII, pp. 3–182.

4to (231 x 160 mm), pp. [14], 101, [1], with two folding plates. Woodcut initials, historiated intaglio head- and tail-pieces (some of the latter woodcuts), engraved allegorical vignette on title by Nutting incorporating a portrait of Newton as the source of light. The plates are etched tables on double leaves entitled: ‘Tabula curvarum simpliciorum quae cum ellipsi et hyperbola comparari possunt’ and ‘Residuum tabulae curvarum simpliciorum quae cum ellipsi et hyperbola comparari possunt’; both are signed: ‘Iohan. Senex sculpt.’ ‘Tractatus de quadratura curvarum’ and ‘Enumeratio linearum tertii ordinis’ have separate half-titles. The author’s name is given in the ‘Praefatio editoris’, which is signed W. Jones. Contemporary panelled calf, red lettering-piece on spine. A fine copy without any restoration. Rare in such good condition.

Item #4697

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Price:
$85,000.00
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