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NEWTON'S ALGEBRA - THE FIRST EDITION AUTHORISED AND EDITED BY NEWTON. Second edition, but the first authorised and edited by Newton (probably with the assistance of John Machin - see below), of his treatise on algebra, or 'universal arithmetic,' his "most often read and republished mathematical work" (Whiteside, Papers V, p. xiv). "Included are 'Newton's identities' providing expressions for the sums of the ith powers of the roots of any polynomial equation, for any integer i [pp. 251-2], plus a rule providing an upper bound for the positive roots of a polynomial, and a generalization, to imaginary roots, of René Descartes' Rule of Signs [pp. 242-5]" (Parkinson, p. 138). About this last rule for determining the number of imaginary roots of a polynomial (which Newton offered without proof), Gjertsen (p. 35) notes: "Some idea of its originality . can be gathered from the fact that it was not until 1865 that the rule was derived in a rigorous manner by James Sylvester." The work is a printed version of lectures Newton prepared in the period 1672-83. Although the editor of the first edition, William Whiston, later claimed that he had Newton's permission to print the lectures, Newton was far from satisfied with the result, complaining that the titles and headings were not his and that it contained numerous mistakes. His real concern was that "an unfinished text composed so long before should now be presented to the world as though it represented his latest researches into the structure and applications of algebra" (Papers V, p. 11), and that Whiston had "too faithfully and impercipiently followed the parent manuscript, incorporating in his princeps edition its several inconsistencies and lapses into error without, in the main, even bringing them to the reader's notice . In his private library copy of the edition, Newton corrected many minor misprints, inserted more appropriate running heads ('Multiplicatio', 'Divisio', 'Extractio Radicum', 'De Forma Aequationis', 'Reductio Aequationum', 'Resolutio Quaestionum Arithmeticarum (Geometricarum)' and the like) and on the Arithmetica page 279 deleted an unwarranted half-title 'Aequationum Constructio linearis'; more radically, he mapped out a large-scale reordering of the sixty-one geometrical problems comprising its central portion, seeking to grade them into a more logical sequence and in increasing levels of difficulty, while in the concluding section on the 'curvilinear ' construction of equations he pared away all not directly needed flesh, reducing it to two skeletal conchoidal neuses, now denuded of their proof. That last savage act of butchery apart, all these improvements were incorporated in the Latin revise - future parent (and rightfully so) of all subsequent editions - which he himself brought to publication in 1722" (ibid., pp. 13-14). Babson notes that "This edition was the last issued during Newton's lifetime and is almost as rare as the first." In commerce, this edition is, in fact, much rarer than the first: ABPC/RBH list only two other copies of this edition since 1975, but ten of the first. "In fulfilment of his obligations as Lucasian Professor, Newton first lectured on algebra in 1672 and seems to have continued until 1683. Although the manuscript of the lectures in [Cambridge University Library] carries marginal dates from October 1673 to 1683, it should not be assumed that the lectures were ever delivered. There are no contemporary accounts of them and, apart from Cotes who made a transcript of them in 1702, they seem to have been totally ignored. Whiteside (Papers V, p. 5) believes that they were composed 'over a period of but a few months' during the winter of 1683-4" (Gjertsen, pp. 33-4). The Arithmetica "derives partly (in its discussion of the elemental algebraic operations and of the reduction and exact solution of equations) from Newton's earlier, unpublished 'Observations' on the introduction to Cartesian algebra presented by Gerard Kinckhuysen in his 1661.
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