Definitions

The body of the text begins with a series of eight definitions and a long scholium that is an essay on absolute time, space, and motion. Only Definitions 3 and 5 have a counterpart in the original tract On Motion .

The following three definitions further refine Newton's view of a centripetal force by listing three specific quantities: the "absolute quantity," the "accelerative quantity," and the "motive quantity."

Principia : Definition 6. The absolute quantity of centripetal force is the measure of this force that is greater or less in proportion to the efficacy of the cause propagating it from the center through the surrounding regions .

Newton relates the absolute quantity of the force to "the center of force" itself, such as "the earth for the center of gravitational force." The more massive the earth, the greater the centripetal force that it will exert upon a unit mass.

Principia : Definition 7. The accelerative quantity of centripetal force is the measure of this force that is proportional to the velocity that it generates in a given time .

Newton relaStes the accelerative quantity of the force to "the place of the body," such as "the gravitating force is greater in valleys and less on the

peaks of high mountains." The more distant the center of force, the lesser the centripetal force that acts on a unit mass.

Principia : Definition 8. The motive quantity of centripetal force is the measure of this force proportional to the motion that it generates in a given time .

Newton relates the motive quantity of the force to "the body itself," such as the "propensity of the whole toward the center compounded of the propensities of all the parts." The motive quantity of a centripetal force is measured by the change in the quantity of motion (the change in the product of the velocity and quantity of matter) that it generates in a given time in a given mass. Newton stated that "for the sake of brevity, we may call these quantities of forces absolute, accelerative, and motive forces." It is important to note that the motive force depends upon the time required to generate the change in motion. When Newton gave his second law of motion, he stated that "a change in motion is proportional to the motive force impressed." If Law 2 implies that the change must be in a "given time," then Definition 8 confirms that implication. It has been argued, however, that Newton's concept of time as particulate permits quite a different reading of Law 2 (see the discussion on page 146).^[2]

Principia : Scholium. Thus far it has seemed best to explain in what sense less familiar words should be taken in what follows. For time, space, place, and motion I do not define since they are very well known to all. But I should say that common opinion conceives of these quantities only in relation to perceivable objects. And hence certain preconceptions arise, and in order for these to be removed, it is useful to separate them into absolute and relative, true and apparent, and mathematical and common [opinions ].

1. Absolute, true, and mathematical time, in itself and by its own nature, flows  . . .

2. Absolute space by its own nature and without reference to anything external  . . .

3. Place is the part of space which a body occupies, and it is either absolute or relative  . . .

4. Absolute motion is the translation of a body from an absolute place  . . .

After some ten pages of detailed discussion of these topics, Newton concludes the scholium with the following statement:

In the following pages, however, it will be set forth more fully how to determine true motions from their causes, effects, and apparent differences, and, conversely, how to determine from motions, whether true or apparent, their causes and effects. To this end I have composed the following treatise .

This long scholium has deservedly occupied the attention of many Newtonian scholars. It is not reproduced in its entirety here: in part because it already has been the object of much attention, and in part because the goal of my work is to detail Newton's analysis, which I believe undergoes relatively little change, even if the concepts employed in it undergo a gradual clarification. The topics of time and space, however, are so central to

all of Newton's analyses, and to the worldview of those who follow, that a few comments on its role are in order.^[3]

The key word is absolute : absolute time, absolute space, and absolute motion. The scholium represents Newton's attempt to explain visible relative motion by relating it to invisible absolute space and time. Attempts to understand the concepts of time and space can be found as early as the writings of Plato and Aristotle. For Plato, "time is a moving image of eternity," which refers to the eternal uniform circular motion of celestial bodies.^[4] Aristotle amended Plato's view by identifying time with "the numerical aspect" of the motion of these material bodies, rather than with the motion itself.^[5] Aristotle then defined space in terms of a material continuum and the place in such a space as "the outer limit of that contained."^[6] In contrast to this Aristotelian concept of motion as fixed to visible matter, Newton's concept of motion is fixed to invisible space. Newton's view of absolute space and time ultimately was rejected by physics in the twentieth century, but it provided a basis for physics in the eighteenth and nineteenth centuries.

Newton built upon Galileo's analysis of motion. In support of the Copernican system, Galileo had to overcome the common belief that if the earth moved, then falling bodies would be left behind. He did so by arguing that one cannot demonstrate absolute rectilinear motion.^[7] In Corollary 5 to the first law of motion (see the following), Newton states that the motion of bodies "are the same with respect to each other whether that space is at rest or moves uniformly in a straight line." Thus, it is impossible to construct an experiment to determine an absolute state of rest (though Newton will attempt to distinguish between a state of rotation and a state of rest in a discussion of the physics of a rotating bucket of water). Nevertheless, he postulates a space at absolute rest, perhaps with respect to the "fixed stars." Newton may have been motivated in part by his desire to refute Descartes's relativism of motion (and the complete denial of space by the Cartesian continuum of matter).^[8] In any event, Newton sets forth in this scholium the view of space and time that supports his analysis of motion and that became the accepted, "obvious" basis for the very successful science of the eighteenth and nineteenth centuries. Only after extended analysis of the invariance of the laws of physics in the twentieth century did Einstein and others have reason to challenge Newton's received view of absolute space and time.^[9] Julian Barbour neatly sums up the contribution of Newton's view to the development of dynamics:

The successes of the scholium speak for themselves. Within half a century or so Newtonian dynamics conquered the world. Men came to accept his concepts of absolute space and time—and they worked brilliantly. Descartes's confused notions were completely forgotten. You had to look quite hard to find the faults in the scholium .^[10]

It is interesting that the change that comes about in the twentieth century—which requires the invariance of the laws of motion to be primary to any preconceived notion of absolute space and time—gives rise to an Einsteinian concept of space and time that is closer to the Aristotelian concept of motion (as fixed to visible matter) than to the Newtonian concept of motion (as fixed to invisible space).