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f GB will be greater than that of FB； and thus the weight of the finite body is greater than that of the infinite。 And； further； the weight of unequal masses will be the same； since the infinite and the finite cannot be equal。 It does not matter whether the weights are commensurable or not。 If （a） they are incommensurable the same reasoning holds。 For instance； suppose E multiplied by three is rather more than C： the weight of three masses of the full size of BD will be greater than C。 We thus arrive at the same impossibility as before。 Again （b） we may assume weights which are commensurate； for it makes no difference whether we begin with the weight or with the mass。 For example； assume the weight E to be commensurate with C； and take from the infinite mass a part BD of weight E。 Then let a mass BF be taken having the same proportion to BD which the two weights have to one another。 （For the mass being infinite you may subtract from it as much as you please。） These assumed bodies will be commensurate in mass and in weight alike。 Nor again does it make any difference to our demonstration whether the total mass has its weight equally or unequally distributed。 For it must always be Possible to take from the infinite mass a body of equal weight to BD by diminishing or increasing the size of the section to the necessary extent。 From what we have said； then； it is clear that the weight of the infinite body cannot be finite。 It must then be infinite。 We have therefore only to show this to be impossible in order to prove an infinite body impossible。 But the impossibility of infinite weight can be shown in the following way。 A given weight moves a given distance in a given time； a weight which is as great and more moves the same distance in a less time； the times being in inverse proportion to the weights。 For instance； if one weight is twice another； it will take half as long over a given movement。 Further； a finite weight traverses any finite distance in a finite time。 It necessarily follows from this that infinite weight； if there is such a thing； being； on the one hand； as great and more than as great as the finite； will move accordingly； but being； on the other hand； compelled to move in a time inversely proportionate to its greatness； cannot move at all。 The time should be less in proportion as the weight is greater。 But there is no proportion between the infinite and the finite： proportion can only hold between a less and a greater finite time。 And though you may say that the time of the movement can be continually diminished； yet there is no minimum。 Nor； if there were； would it help us。 For some finite body could have been found greater than the given finite in the same proportion which is supposed to hold between the infinite and the given finite； so that an infinite and a finite weight must have traversed an equal distance in equal time。 But that is impossible。 Again； whatever the time； so long as it is finite； in which the infinite performs the motion； a finite weight must necessarily move a certain finite distance in that same time。 Infinite weight is therefore impossible； and the same reasoning applies also to infinite lightness。 Bodies then of infinite weight and of infinite lightness are equally impossible。 That there is no infinite body may be shown； as we have shown it； by a detailed consideration of the various cases。 But it may also be shown universally； not only by such reasoning as we advanced in our discussion of principles （though in that passage we have already determined universally the sense in which the existence of an infinite is to be asserted or denied）； but also suitably to our present purpose in the following way。 That will lead us to a further question。 Even if the total mass is not infinite； it may yet be great enough to admit a plurality of universes。 The question might possibly be raised whether there is any obstacle to our believing that there are other universes composed on the pattern of our own； more than one； though stopping short of infinity。 First； however； let us treat of the infinite universally。
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Every body must necessarily be either finite or infinite； and if infinite； either of similar or of dissimilar parts。 If its parts are dissimilar； they must represent either a finite or an infinite number of kinds。 That the kinds cannot be infinite is evident； if our original presuppositions remain unchallenged。 For the primary movements being finite in number； the kinds of simple body are necessarily also finite； since the movement of a simple body is simple； and the simple movements are finite； and every natural body must always have its proper motion。 Now if the infinite body is to be composed of a finite number of kinds； then each of its parts must necessarily be infinite in quantity； that is to say； the water； fire； &c。； which compose it。 But this is impossible； because； as we have already shown； infinite weight and lightness do not exist。 Moreover it would be necessary also that their places should be infinite in extent； so that the movements too of all these bodies would be infinite。 But this is not possible； if we are to hold to the truth of our original presuppositions and to the view that neither that which moves downward； nor； by the same reasoning； that which moves upward； can prolong its movement to infinity。 For it is true in regard to quality； quantity； and place alike that any process of change is impossible which can have no end。 I mean that if it is impossible for a thing to have come to be white； or a cubit long； or in Egypt； it is also impossible for it to be in process of coming to be any of these。 It is thus impossible for a thing to be moving to a place at which in its motion it can never by any possibility arrive。 Again； suppose the body to exist in dispersion； it may be maintained none the less that the total of all these scattered particles； say； of fire； is infinite。 But body we saw to be that which has extension every way。 How can there be several dissimilar elements； each infinite？ Each would have to be infinitely extended every way。 It is no more conceivable； again； that the infinite should exist as a whole of similar parts。 For； in the first place； there is no other （straight） movement beyond those mentioned： we must therefore give it one of them。 And if so； we shall have to admit either infinite weight or infinite lightness。 Nor； secondly； could the body whose movement is circular be infinite； since it is impossible for the infinite to move in a circle。 This； indeed； would be as good as saying that the heavens are infinite； which we have shown to be impossible。 Moreover； in general； it is impossible that the infinite should move at all。 If it did； it would move either naturally or by constraint： and if by constraint； it possesses also a natural motion； that is to say； there is another place； infinite like itself； to which it will move。 But that is impossible。 That in general it is impossible for the infinite to be acted upon by the finite or to act upon it may be shown as follows。 （1。 The infinite cannot be acted upon by the finite。） Let A be an infinite； B a finite； C the time of a given movement produced by one in the other。 Suppose； then； that A was heated； or impelled； or modified in any way； or caused to undergo any sort of movement whatever； by in the time C。 Let D be less than B； and； assuming that a lesser agent moves a lesser patient in an equal time； call the quantity thus modified by D； E。 Then； as D is to B； so is E to some finite quantum。 We assume that the alteration of equal by equal takes equal time； and the alteration of less by less or of greater by greater takes the same time； if the quantity of the patient is such as to keep the proportion which obtains between the agents； greater and less。 If so； no movement can be caused in the infinite by any finite agent in any time whatever。 For a less agent will produce that movement in a less patient in an equal time； and the proportionate equivalent of that patient will be a finite quantity； since no proportion holds between finite and infinite。 （2。 The infinite cannot act upon the finite。） Nor； again； can the infinite produce a movement in the finite in any time whatever。 Let A be an infinite； B a finite； C the time of action。 In the time C； D will produce that motion in a patient less than B； say F。 Then take E； bearing the same proportion to D as the whole BF bears to F。 E will produce the motion in BF in the time C。 Thus the finite and infinite effect the same alteration in equal times。 But this is impossible； for the assumption is that the greater effects it in a shorter time。 It will be the same with any time that can be taken； so that there will no time in which the infinite can effect this movement。 And； as to infinite time； in that nothing can move another or be moved by it。 For such time has no limit； while the action and reaction have。 （3。 There is no interaction between infinites。） Nor can infinite be acted upon in any way by infinite。 Let A and B be infinites； CD being the time of the action A of upon B。 Now the whole B was modified in a certain time； and the part of this infinite； E； cannot be so modified in the same time； since we assume that a