## Two Treatises of the Quadrature of Curves, and Analysis by Equations of an infinite Number of Terms, explained: Containing The Treatises themselves, translated into English, with a large Commentary; in which the Demonstrations are supplied where wanting, the Doctrine illustrated, and the whole accommodated to the Capacities of Beginners, for whom it is chiefly designed.

London: by James Bettenham, at the expense of the Society for the Encouragement of Learning, and sold by John Nourse … and John Whiston … booksellers to the said Society, 1745.

First edition, the superb Macclesfield copy, of the first English translations of two of Newton’s most important papers on the calculus and infinite series. The translation was by John Stewart and the tracts are accompanied by Stewart’s extensive commentary. This work contains English translations of two of Newton’s three published works on the calculus: ‘De Analysi per Aequationes Numero Terminorum Infinitas,’ written in 1669 and first published in 1711, containing Newton’s theory of infinite series; and ‘De quadratura curvarum’, first published in the *Opticks* (1704) but written in 1693, containing Newton’s mature presentation of his theory of fluxions (the third work, *The Method of Fluxions*, was composed in Latin in 1671 and first published in English in 1736). 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). ‘De quadratura curvarum’ was Newton’s first publication of his method of fluxions, or calculus, which he developed in terms of ‘prime and ultimate ratios’, an early version of the theory of limits; it includes the first published statement of the general binomial theorem and of ‘Taylor’s theorem’ on series expansions. The commentaries by the translator, John Stewart, are much longer than the treatises themselves. The Society for the Encouragement of Learning existed to enable authors to be rewarded for their labours. Unfortunately in this case they did not succeed: the book, of which 350 copies had been printed, was in large part remaindered two years later. For some reason there is a good deal of confusion in the literature as to what this volume actually contains. Several bookseller’s catalogues, auction records, and even Gjertsen (p. 554) incorrectly state that the ‘Two Treatises’ are those appended to the *Opticks* (one of those treatises was the ‘Ennumeratio linearum tertii ordinis,’ which is not translated here, and ‘De analysi’ was not appended to the *Opticks*), while Babson implies that only ‘De quadratura curvarum’ is translated in the present work, and Wallis describes the work as a translation of ‘De analysi’, with ESTC making the same mistake. Although uncommon, the work appears with somewhat greater frequency than one might have expected given the size of the print-run, with some 10 copies listed on ABPC/RBH in the last 40 years.

*Provenance*: the Earls of Macclesfield (embossed stamp on title and next leaf, armorial South Library bookplate on front paste-down); Kenneth Rapoport (bookplate); sold Sotheby’s, 14 April 2005, lot 1535 & Swann Galleries, 3 April 2014, lot 141, $27,500. It is probable that this was William Jones’s copy – although not signed by him, several other books in the Macclesfield library in very similar bindings were his.

“*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).

‘De quadratura’ was the first of Newton’s treatises on fluxions to be published, but the last to be composed, so that it represents his most mature view of the subject. It was prompted by a letter from David Gregory, on 7 November 1691, sending Newton “my method of squaring figures, published three years ago but now clarified by examples. If only I might be allowed to know your method too, which, as I have subsequently gathered, differs little from mine.” “By late December, Newton’s new tract was already far enough advanced to completion for him to tell Gregory at the beginning of a three-week visit he made to London that he planned to publish it … In the course of the same visit he likewise informed Fatio de Duillier more vaguely that his ‘treatise of curves would soon see the light.’ A month afterwards, having looked through the text of ‘De quadratura’ himself, Fatio could report to Christiaan Huygens that it compared with Leibniz’ letters and publications on ‘Calculus differentialis’ like a ‘finished original’ to a ‘lame and very imperfect copy,’ adding that ‘what he has on quadratures is infinitely more general than all that has been had before, and it is very simple and of wonderful use in all parts of geometry’ (Fatio to Huygens, 15 February 1692). Four weeks later, he added vividly if with exaggeration that ‘I have been chilled to see what Mr Newton has done, and have reproached him for rendering my own hard work useless and not wanting to leave anything for his friends who are come after him to do’ (*ibid*., 17 March). But by then, as Fatio sadly admitted in the same letter to Huygens, the heat of Newton’s interest in seeing his treatise into print was past and his attention had already strayed to other matters” (*Papers* VII, pp. 11-13).

“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. The notation was preserved in the published version of *De quadratura*. 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). But perhaps “Newton’s most important achievement in his ‘De quadratura’ [was] the first explicit enunciation of the Taylor expansion of a general function – Newton deduced the particular ‘Maclaurin’ form in his Corollary 3 (by successive differentiation, it would seem) and then passed to the general theorem in his Corollary 4” (*Papers *VII, pp. 18-19). The expansion was rediscovered by Brook Taylor in 1715.

Kown to his students as ‘John Triangles’, John Stewart (1708?-1766) was Colin Maclaurin’s successor in the Chair of Mathematics at the Marischal College, Aberdeen. He held this post for forty years, until he, his wife, and his eldest daughter died of ‘fever’ on the same day in 1766.

Babson 210; ESTC 1018643; Macclesfield 1535 (this copy); Wallis 303. Gjertsen, *Newton Handbook*, 1986. Hall, *Philosophers at War. **The Quarrel between Newton and Leibniz*, 1980. *Footprints of the Lion*: *Isaac Newton at Work*, 2001.

4to (265 x 195mm), pp. xxxii, 479, [5], last 2 leaves with errata etc., woodcut figures in text, tail-piece at end designed by William Kent and engraved by Virtue. Contemporary black morocco, gilt fillet on covers, spine gilt, red morocco lettering-piece, red edges.

Item #5396

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Price:
$55,000.00
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