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

    Motion.” The first two entities defined are “quantity of matter,” or “mass,” and “quantity of motion.” The former is said to be the measure of matter proportional to bulk and density conjunctively. “Mass” is, in addition, given as being generally known by its weight, to which it is proportional at any given place, as shown by Newton's experiments with pendulums, of which the results are more exact than Galileo's for freely falling bodies. Newton's “quantity of motion” is the entity now known as momentum; it is said to be measured by the velocity and mass of a body, conjunctively.

Definition 3 introduces vis insita (probably best translated as “inherent force”), a concept of which the actual definition and explanation are both so difficult to understand that much scholarly debate has been expended on them.146 Newton wrote that the vis insita may be known by “a most significant name, vis inertiae.” But this “force” is not like the “impressed forces” of definition 4, which change the state of rest or uniform rectilinear motion of a body; the vis inertiae merely maintains any new state acquired by a body, and it may cause a body to “resist” any change in state.147

Newton then defined “centripetal force” (vis centripeta), a concept he had invented and named to complement the vis centrifuga of Christiaan Huygens.148 In definitions 6 through 8, Newton gave three “measures” of centripetal force, of which the most important for the purposes of the Principia is that one “proportional to the velocity which it generates in a given time” (for point masses, unit masses, or for comparing equal masses). There follows the famous scholium on space and time, in which Newton opted for concepts of absolute space and absolute time, although recognizing that both are usually reckoned by “sensible measures”; time, especially, is usually “relative, apparent, and common.” Newton's belief in absolute space led him to hold that absolute motion is sensible or detectable, notably in rotation, although contemporaries as different in their outlooks as Huygens and Berkeley demurred from this view.

The “Axioms” or “Laws of Motion” are three in number: the law of inertia, a form of what is today known as the second law, and finally the law that “To every action there is always opposed an equal and opposite reaction.” There is much puzzlement over the second law, which Newton stated as a proportionality between “change in motion” (in momentum) and “the motive force impressed” (a change “made in the direction ..., in which that force is impressed”); he did not specify “per unit time” or “in some given time.” The second law thus seems clearly to be stated for an impulse, but throughout the Principia (and, in a special case, in the antecedent definition 8), Newton used the law for continuous forces, including gravitation, taking account of time. For Newton, in fact, the concepts of impulse and continuous force were infinitesimally equivalent, and represented conditions of action “altogether and at once” or “by degrees and successively.”149 There are thus two conditions of “force” in the second law; accordingly, this Newtonian law may be written in the two forms f?d(mv) and f?d(mv)/dt, in which both concepts of force are taken account of by means of two different constants of proportionality. The two forms of the law can be considered equivalent through Newton's concept of a uniformly flowing time, which makes dt a kind of secondary constant, which can arbitrarily be absorbed in the constant of proportionality.

There may be some doubt as to whether or not Newton himself was unclear in his own mind about these matters. His use of such expressions as “vis impressa” shows an abiding influence of older physics, while his continued reference to a “vis” or a “force” needed to maintain bodies in a state of motion raises the question of whether such usage is one of a number of possibly misleading “artifacts left behind in the historical development of his [Newton's] dynamics.”150 It must be remembered, of course, that throughout the seventeenth and much of the eighteenth century the word “force” could be used in a number of ways. Most notably, it served to indicate the concept now called “momentum,” although it could also even mean energy. In Newton's time there were no categories of strict formalistic logic that required a unitary one-to-one correspondence between names and concepts, and neither Newton nor his contemporaries (or, for that matter, his successors) were always precise in making such distinctions.

The careful reader of books I-III should not be confused by such language, however, nor by the preliminary intrusion of such concepts. Even the idea of force as a measure of motion or of change of motion (or of change per se, or rate of change) is not troublesome in practice, once Newton's own formulation is accepted and the infinitesimal level of his discourse (which is not always explicitly stated) understood. In short, Newton's dynamical and mathematical elaboration of the three books of the Principia is free of the errors and ambiguities implicit in his less successful attempt to give a logically simple and coherent set of definitions and axioms for dynamics. (It is even possible that the definitions and axioms may represent an independent later exercise, since there are, for example, varying sets of definitions and axioms for the same system of dynamics.) One of