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 of approximating to the roots of numerical equations. This is simply the method of Vieta improved. The same treatise contains "Newton's parallelogram," which enabled him, in an equation, $\scriptstyle{f(x,y)=0}$, to find a series in powers of x equal to the variable y. The great utility of this rule lay in its determining the form of the series; for, as soon as the law was known by which the exponents in the series vary, then the expansion could be effected by the method of indeterminate coefficients. The rule is still used in determining the infinite branches to curves, or their figure at multiple points. Newton gave no proof for it, nor any clue as to how he discovered it. The proof was supplied half a century later, by Kaestner and Cramer, independently.[37]

In 1704 was published, as an appendix to the Opticks, the Enumeratio linearum tertii ordinis, which contains theorems on the theory of curves. Newton divides cubics into seventy-two species, arranged in larger groups, for which his commentators have supplied the names "genera" and "classes," recognising fourteen of the former and seven (or four) of the latter. He overlooked six species demanded by his principles of classification, and afterwards added by Stirling, Murdoch, and Cramer. He enunciates the remarkable theorem that the five species which he names "divergent parabolas" give by their projection every cubic curve whatever. As a rule, the tract contains no proofs. It has been the subject of frequent conjecture how Newton deduced his results. Recently we have gotten at the facts, since much of the analysis used by Newton and a few additional theorems have been discovered among the Portsmouth papers. An account of the four holograph manuscripts on this subject has been published by W. W. Rouse Ball, in the Transactions of the London Mathematical Society (vol. xx., pp. 104–143). It is interesting to observe how Newton begins his research on the classification of cubic