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

    densities of the planets (the numerical calculations in this proposition were much altered for the second edition). In proposition 9, Newton estimated the force of gravity within a planet and, in proposition 10, demonstrated the long-term stability of the solar system. A general “Hypothesis I” (in the second and third editions; “Hypothesis IV” in the first) holds the “centre of the system of the world” to be “immovable,” which center is given as the center of gravity of the solar system in proposition 11; the sun is in constant motion, but never “recedes” far from that center of gravity (proposition 12).

It is often asserted that Newton attained his results by neglecting the interplanetary attractions, and dealing exclusively with the mutual gravitational attractions of the planets and our sun. But this is not the case, since the most fully explored example of perturbation in the Principia is indeed that of the sun-earth-moon system. Thus Newton determined (proposition 25) the “forces with which the sun disturbs the motions of the moon,” and (proposition 26) the action of those forces in producing an inequality (“horary increment”) of the area described by the moon (although “in a circular orbit”).

The stated intention of proposition 29 is to “find the variation of the moon,” the inequality thus being sought being due “partly to the elliptic figure of the Moon's orbit, partly to the inequality of the moments of the area which the Moon by a radius drawn to the Earth describes.” (Newton dealt with this topic more fully in the second edition.) Then Newton studied the “horary motion of the nodes of the moon,” first (proposition 30) “in a circular orbit,” and then (proposition 31) “in an elliptic orbit.” In proposition 32, he found “the mean motion of the nodes,” and, in proposition 33, their “true motion.” (In the third edition, following proposition 33, Newton inserted two propositions and a scholium on the motion of the nodes, written by John Machin.) Propositions 34 and 35, on the inclination of the orbit of the moon to the ecliptic plane, are followed by a scholium, considerably expanded and rewritten for the second edition, in which Newton discussed yet other “inequalities” in the motion of the moon and developed the practical aspects of computing the elements of that body's motion and position.

Propositions 36 and 37 deal at length and in a quantitative fashion with the tide-producing forces of the sun and of the moon, yielding, in proposition 38, an explanation of the spheroidal shape of the moon and the reason that (librations apart) the same face of it is always visible. A series of three lemmas introduces the subject of precession and a fourth lemma (transformed into hypothesis 2 in the second and third editions) treats the precession of a ring. Proposition 39 represents an outstanding example of the high level of mathematical natural science that Newton reached in the Principia. In it he showed the manner in which the shape of the earth, in relation to the pull of the moon, acts on its axis of rotation so as to produce the observed precession, a presentation that he augmented and improved for the second edition. Newton here employed the result he had previously obtained (in propositions 20 and 21, book III) concerning the shape of the earth, and joined it to both the facts and theory of precession and yet another aspect of the perturbing force of the moon on the motion of the earth. He thus inaugurated a major aspect of celestial mechanics, the study of a three-body system.

Lemma 4, book III initiates a section on comets, proving that comets are “higher” than the moon, move through the solar system, and (corollary 1) shine by reflecting sunlight; their motion shows (corollary 3) that “the celestial spaces are void of resistance.” Comets move in conic sections (proposition 40) having the sun as a focus, according to the law of areas. Those comets that return move in elliptic orbits (corollary 1) and follow Kepler's third law, but (corollary 2) “their orbits will be so near to parabolas, that parabolas may be used for them without sensible error.”

Almost immediately following publication of the Principia, Halley, in a letter of 5 July 1687, urged Newton to go on with his work on lunar theory.166 Newton later remarked that his head so ached from studying this problem that it often “kept him awake” and “he would think of it no more.” But he also said that if he lived long enough for Halley to complete enough additional observations, he “would have another stroke at the moon.” In the 1690's Newton had depended on Flamsteed for observations of the moon, promising Flamsteed (in a letter of 16 February 1695) not to communicate any of his observations, “much less publish them, without your consent.” But Newton and Flamsteed disagreed on the value of theory, which Newton held to be useful as “a demonstration” of the “exactness” of observations, while Flamsteed believed that “theories do not command observations; but are to be tried by them,” since “theories are . . . only probable” (even “when they agree with exact and indubitable observations”). At about this same time Newton was drawing up a set of propositions on the motion of the moon for a proposed new edition of the Principia, for which he requested from Flamsteed such planetary observations “as tend to [be useful for] perfecting the theory of the planets,” to serve Newton in the preparation of a second edition of his book.