Ethics · Benedictus de Spinoza
PROPOSITIONS
Corollary.--Hence it follows, that a body in motion keeps in motion, until it is determined to a state of rest by some other body; and a body at rest remains so, until it is determined to a state of motion by some other body. This is indeed self--evident. For when I suppose, for instance, that a given body, A, is at rest, and do not take into consideration other bodies in motion, I cannot affirm anything concerning the body A, except that it is at rest. If it afterwards comes to pass that A is in motion, this cannot have resulted from its having been at rest, for no other consequence could have been involved than its remaining at rest. If, on the other hand, A be given in motion, we shall, so long as we only consider A, be unable to affirm anything concerning it, except that it is in motion. If A is subsequently found to be at rest, this rest cannot be the result of A's previous motion, for such motion can only have led to continued motion; the state of rest therefore must have resulted from something, which was not in A, namely, from an external cause determining A to a state of rest. Axiom I.--All modes, wherein one body is affected by another body, follow simultaneously from the nature of the body affected and the body affecting; so that one and the same body may be moved in different modes, according to the difference in the nature of the bodies moving it; on the other hand, different bodies may be moved in different modes by one and the same body.
Axiom II.--When a body in motion impinges on another body at rest, which it is unable to move, it recoils, in order to continue its motion, and the angle made by the line of motion in the recoil and the plane of the body at rest, whereon the moving body has impinged, will be equal to the angle formed by the line of motion of incidence and the same plane. So far we have been speaking only of the most simple bodies, which are only distinguished one from the other by motion and rest, quickness and slowness. We now pass on to compound bodies. Definition.--When any given bodies of the same or different magnitude are compelled by other bodies to remain in contact, or if they be moved at the same or different rates of speed, so that their mutual movements should preserve among themselves a certain fixed relation, we say that such bodies are in union, and that together they compose one body or individual, which is distinguished from other bodies by the fact of this union. Axiom III.--In proportion as the parts of an individual, or a compound body, are in contact over a greater or less superficies, they will with greater or less difficulty admit of being moved from their position; consequently the individual will, with greater or less difficulty, be brought to assume another form. Those bodies, whose parts are in contact over large superficies, are called hard; those, whose parts are in contact over small superficies, are called soft; those, whose parts are in motion among one another, are called fluid.
LEMMA IV. If from a body or individual, compounded of several bodies, certain bodies be separated, and if, at the same time, an equal number of other bodies of the same nature take their place, the individual will preserve its nature as before, without any change in its actuality (forma). Proof.--Bodies (Lemma i.) are not distinguished in respect of substance: that which constitutes the actuality (formam) of an individual consists (by the last Def.) in a union of bodies; but this union, although there is a continual change of bodies, will (by our hypothesis) be maintained; the individual, therefore, will retain its nature as before, both in respect of substance and in respect of mode. Q.E.D. LEMMA V. If the parts composing an individual become greater or less, but in such proportion, that they all preserve the same mutual relations of motion and rest, the individual will still preserve its original nature, and its actuality will not be changed. Proof.--The same as for the last Lemma. LEMMA VI. If certain bodies composing an individual be compelled to change the motion, which they have in one direction, for motion in another direction, but in such a manner, that they be able to continue their motions and their mutual communication in the same relations as before, the individual will retain its own nature without any change of its actuality.
Proof.--This proposition is self--evident, for the individual is supposed to retain all that, which, in its definition, we spoke of as its actual being. LEMMA VII. Furthermore, the individual thus composed preserves its nature, whether it be, as a whole, in motion or at rest, whether it be moved in this or that direction; so long as each part retains its motion, and preserves its communication with other parts as before. Proof.--This proposition is evident from the definition of an individual prefixed to Lemma iv. Note.--We thus see, how a composite individual may be affected in many different ways, and preserve its nature notwithstanding. Thus far we have conceived an individual as composed of bodies only distinguished one from the other in respect of motion and rest, speed and slowness; that is, of bodies of the most simple character. If, however, we now conceive another individual composed of several individuals of diverse natures, we shall find that the number of ways in which it can be affected, without losing its nature, will be greatly multiplied. Each of its parts would consist of several bodies, and therefore (by Lemma vi.) each part would admit, without change to its nature, of quicker or slower motion, and would consequently be able to transmit its motions more quickly or more slowly to the remaining parts. If we further conceive a third kind of individuals composed of individuals of this second kind, we shall find that they may be affected in a still greater number of ways without changing their actuality. We may easily proceed thus to infinity, and conceive the whole of nature as one individual, whose parts, that is, all bodies, vary in infinite ways, without any change in the individual as a whole. I should feel bound to explain and demonstrate this point at more length, if I were writing a special treatise on body. But I have already said that such is not my object; I have only touched on the question, because it enables me to prove easily that which I have in view.
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