NEWTON, ISAAC (b. Woolsthorpe, England, 25 December 1642; d. London, England, 20 March 1727), mathematics, dynamics, celestial mechanics, astronomy, optics, natural philosophy.

    of AB, corresponding to the moments a and b of A and B, respectively, is aB + bA. And, for the special case of A = B, the moment of A2 is determined as 2aA.

In order to extend the result from “area” to “content” or (“bulk”), from AB to ABC, Newton set AB = G and then used the prior result for AB twice, once for AB, and again for GC, so as to get the moment of ABC to be cAB + bCA + aBC; whence, by setting A = B = C, the moment of A3 is determined as 3aA2. And, in general, the moment of An is shown to be naAn-1 for n as a positive integer.

The result is readily extended to negative integral powers and even to all products AmBn, “whether the indices m and n of the powers be whole numbers or fractions, affirmative or negative.” Whiteside has pointed out that by using the decrements 1/2a, 1/2b and the increments 1/2a, 1/2b, rather than the increments a, b, “Newton ... deluded himself into believing” he had “contrived an approach which avoids the comparatively messy appeal to the limit-value of (A + a)/(B + b) - AB as the increments a, b vanish.” The result is what is now seen as a “celebrated nonsequitur.”135

In discussing lemma 2, Newton defined moments as the “momentary increments or decrements” of “variable and indetermined” quantities, which might be “products, quotients, roots, rectangles, squares, cubes, square and cubic sides, and the like.” He called these “quantities” genitae, because he conceived them to be “generated or produced in arithmetic by the multiplication, division, or extraction of the root of any terms whatsoever; in geometry by the finding of contents and sides, or of the extremes and means of proportionals.” So much is clear. But Newton warned his readers not “to look upon finite particles as such [moments],” for finite particles “are not moments, but the very quantities generated by the moments. We are to conceive them as the just nascent principles of finite magnitudes.” And, in fact, it is not “the magnitude of the moments, but their first proportion [which is to be regarded] as nascent.”

Boyer has called attention to the difficulty of conceiving “the limit of a ratio in determining the moment of AB.”136 The moment of AB is not really a product of two independent variables A and B, implying a problem in partial differentiation, but rather a product of two functions of the single independent variable time. Newton himself said, “It will be the same thing, if, instead of moments, we use either the velocities of the increments and decrements (which may also be called the motions, mutations, and fluxions of quantities), or any finite quantities proportional to those velocities.”

Newton thus shifted the conceptual base of his procedure from infinitely small quantities or moments—which are not finite, and clearly not zero—to the “first proportion,” or ratio of moments (rather than “the magnitude of the moments”) “as nascent.” This nascent ratio is generally not infinitesimal but finite, and Newton thus suggested that the ratio of finite quantities may be substituted for the ratio of infinitesimals, with the same result, using in fact the velocities of the increments or decrements instead of the moments, or “any finite quantities proportional to those velocities,” which are also the “fluxions of the quantities.” Boyer summarized this succinctly:

Newton thus offered in the Principia three modes of interpretation of the new analysis: that in terms of infinitesimals (used in his De analysi ...); that in terms of prime and ultimate ratios or limits (given particularly in De quadratura, and the view which he seems to have considered most rigorous); and that in terms of fluxions (given in his Methodus fluxionum, and one which appears to have appealed most strongly to his imagination).137

From the point of view of mathematics, proposition 10, book II, may particularly attract our attention. Here Newton boldly displayed his methods of using the terms of a converging series to solve problems and his method of second differences. Expansions are given with respect to “the indefinite quantity o,” but there are no references to (nor uses of) moments, as in the preceding lemma 2, and, of course, there is no use made of dotted or “pricked” letters.

The proposition is of particular interest for at least two reasons. First, its proof and exposition (or exemplification) are highly analytic and not geometric (or synthetic), as are most proofs in the Principia. Second, an error in the first edition and in the original printed pages of the second edition was discovered by Johann [I] Bernoulli and called to Newton's attention by Nikolaus [I] Bernoulli, who visited England in September or October 1712. As a result, Newton had Cotes reprint a whole signature and an additional leaf of the already printed text of the second edition; these pages thus appear as cancels in every copy of this edition of the Principia that has been recorded. The corrected proposition, analyzed by Whiteside, illustrates “the power of Newton's infinitesimal techniques in the Principia,” and may thus confute the opinion that “Newton did not (at least in principle, and in his own algorithm) know how ‘to formulate and resolve problems through the integration of differential equations.’”138

From at least 1712 onward, Newton attempted to impose upon the Principia a mode of composition that could lend support to his position in the priority