Dictionary of Scientific Biography

Dictionary of Scientific Biography

Electronic edition published by Cultural Heritage Langauge Technologies (with permission from Charles Scribners and Sons) and funded by the National Science Foundation International Digital Libraries Program. This text has been proofread to a low degree of accuracy. It was converted to electronic form using data entry.

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

illustrate the general principles of the motions of bodies, then again in book III, in application to the motions of planets and their satellites or of comets. While this mode of presentation makes the Principia more difficult for the reader, it does have the decided advantage of separating the Newtonian principles as they apply to the physical universe from the details of the mathematics from which they derive.

As an example of this separation, proposition 1 of book III states that the satellites of Jupiter are “continually drawn off from rectilinear motions, and are retained in their proper orbits” by forces that “tend to Jupiter's centre” and that these forces vary inversely as the square of their distances from that center. The proof given in this proposition is short and direct; the centripetal force itself follows from “Phen. I [of book III], and Prop. II or III, Book I.” The phenomenon cited is a statement, based upon “astronomical observations,” that a radius drawn from the center of Jupiter to any satellite sweeps out areas “proportional to the times of descriptions”; propositions 2 and 3 of book I prove by mathematics that under these circumstances the force about which such areas are described must be centripetal and proportional to the times. The inverse-square property of this force is derived from the second part of the phenomenon, which states that the distances from Jupiter's center are as the 3/2th power of their periods of revolution, and from corollary 6 to proposition 4 of book I, in which it is proved that centripetal force in uniform circular motion must be as the inverse square of the distance from the center.

Newton's practice of introducing a particular instance repeatedly, with what may seem to be only minor variations, may render the Principia difficult for the modern reader. But the main hurdle for any would-be student of the treatise lies elsewhere, in the essential mathematical difficulty of the main subject matter, celestial mechanics, however presented. A further obstacle is that Newton's mathematical vocabulary became archaic soon after the Principia was published, as dynamics in general and celestial mechanics in particular came to be written in the language of differentials and integrals still used today. The reader is thus required almost to translate for himself Newton's geometrical-limit mode of proof and statement into the characters of the analytic algorithms of the calculus. Even so, dynamics was taught directly from the Principia at Cambridge until well into the twentieth century.

In his “Mathematical Principles” Whiteside describes the Principia as “slipshod, its level of verbal fluency none too high, its arguments unnecessarily diffuse and repetitive, and its content on occasion markedly irrelevant to its professed theme: the theory of bodies moving under impressed forces.” This view is somewhat extreme. Nevertheless, the work might have been easier to read today had Newton chosen to rely to a greater extent on general algorithms.

The Principia is often described as if it were a “synthesis,” notably of Kepler's three laws of planetary motion and Galileo's laws of falling bodies and projectile motion; but in fact it denies the validity of both these sets of basic laws unless they be modified. For instance, Newton showed for the first time the dynamical significance of Kepler's so-called laws of planetary motion; but in so doing he proved that in the form originally stated by Kepler they apply exactly only to the highly artificial condition of a point mass moving about a mathematical center of force, unaffected by any other stationary or moving masses. In the real universe, these laws or planetary “hypotheses” are true only to the limits of ordinary observation, which may very well have been the reason that Newton called them “Hypotheses” in the first edition. Later, in the second and third editions, he referred to these relations as “Phaenomena,” by which it may be assumed that he now meant that they were not simply true as stated (that is, not strictly deducible from the definitions and axioms), but were rather valid only to the limit of (or within the limits of) observation, or were phenomenologically true. In other words, these statements were to be regarded as not necessarily true, but only contingently (phenomenologically) so.

In the Principia, Newton proved that Kepler's planetary hypotheses must be modified by at least two factors: (1) the mutual attraction of each of any pair of bodies, and (2) the perturbation of a moving body by any and all neighboring bodies. He also showed that the rate of free fall of bodies is not constant, as Galileo had supposed, but varies with distance from the center of the earth and with latitude along the surface of the earth.144 In a scholium at the end of section 2, book I, Newton further pointed out that it is only in a limiting case, not really achieved on earth, that projectiles (even in vacuo) move in Galilean parabolic trajectories, as Galileo himself knew full well. Thus, as Karl Popper has pointed out, although “Newton's dynamics achieved a unification of Galileo's terrestrial and Kepler's celestial physics,” it appears that “from a logical point of view, Newton's theory, strictly speaking, contradicts both Galileo's and Kepler's.”145

## The “Principia”: Definitions and Axioms.

The Principia opens with two preliminary presentations: the “Definitions” and the “Axioms, or Laws of

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