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Leibniz does not seem to be aware of the problem of the law of inertia for his system of dynamics together with metaphysics. Among our contemporary

physicists and philosophers of science, Julian Barbour is by far the most renowned for his consistent research on relational theories of dynamics. And he knows quite well the problem of inertia (the law and inertial mass as well). So let me begin with a quotation from his short paper:

Unlike his contemporaries Huygens and Leibniz, who both cheerfully used the law of inertia as the foundation of dynamics while stoutly maintaining the relativity of motion, Newton felt this problem so acutely that he could not conceive of any dynamics formulated without a rigid framework—- absolute space. (Barbour 1995a, 7)

This remark has a sting for Huygens and Leibniz, as well as for Newton. It is true that the law of inertia, in one form or another, was already used before them by such people as Galileo or Descartes. Galileo was using his version as regards motions on the earth, so that his “inertial motion” is along a circumference of the earth (hence not straight) , but experimentally, it worked well enough for Galileo’s purpose. For Huygens and Leibniz, an inertial motion must be uniform and on a straight line, whatever direction we may consider in the universe. But we may ask: what does uniform mean, and how can we know the path of that motion is straight? If we already have the criterion of uniformity, the uniformity of inertial motion makes sense, and if we may assume that our space is Euclidean then motion along a straight line is all right. But then, these two “ifs” must be shown to hold in the first place; just taking the law of inertia as “the foundation of dynamics” won’t do. To this question, add “the relativity of motion,” whether the motion is straight,

circular, or curvilinear. Then anything may be taken as a frame of reference, and the law of inertia must be supposed to be holding in it. Can we say this law is the same as the “law of inertia” holding in another reference frame? Newton must have thought this is absurd.

Barbour’s diagnosis may be briefly summarized as follows. An “inertial motion” may be called a “cosmic drift,” but we do not know whence it comes and what determines its course (Barbour 2001, 477). It stands alone, as it were, disengaged from the contingent world (ibid.), including the earth, the sun, and the stars. This makes a sharp contrast against the relativity of motion, which makes everything relative to something else; collisions, in particular, are determined only by relative velocities. Yet, in order to explain projectiles, collisions, or centrifugal forces, inertial

motions must be presupposed. This contrast and the necessity of collaboration of the two, inertia and relativity, is the source of enigma.

Since the reader may be puzzled by any explanation only in terms of ordinary words, let me supply the following Figure, for illustrating the implication of the relativity of motion, a simple case of circular motion and a generalized case. You may reconsider the status of the law of inertia, in view of this illustration.

Figure 16: Can Relativity be Generalized?

With Figure 16, the reader may see the difficulty of harmonizing the law of inertia with relativity. See the upper figure of circular motions. Geometrically, it may be all right to say that the two ways of describing are equivalent, but it is not all right if we assume the law of inertia. If B is rotating around A in an inertial system, a

centrifugal force arises for B, so that, physically, the two ways are not equivalent at all (this can be a version of Newton’s argument). Thus we have to distinguish physical relativity from mathematical relativity. And you may also notice that the radius of the revolution remains the same in either way. In order to emphasize this, I inserted several lines of distance in the lower, generalized case too. Relativity does not mean “anything goes” but presupposes something invariant. Thus, in order to assert the relativity of motion, Leibniz should have put forward more careful arguments. The conceptual apparatus necessary for a possible solution was not available to anyone, when Newton, Leibniz, and their contemporaries were writing. We need Gaussian coordinates and differential geometry, which Einstein utilized for

A A B B relativity of revolution relativity, generalized? A B B A

constructing his theory of gravity (it is usually called “general relativity,” but a caution is necessary for this word).

This enigma is a serious problem to be solved for Leibniz too. Was he ever aware of this problem? As far as I can see, within dynamics, he has not clarified the

foundation of the law of inertia; he merely said inertia is given as part of nature of body (See Part 1, Section 3). However, unlike Huygens, Leibniz prepared his own metaphysics Monadology, in addition to Analysis Situs, as the foundation of his dynamics. Thus, if we wish to look for his argument for, or clarification of, the foundation of the law of inertia, we have to examine his metaphysics and Initia Rerum again.