on the heavens-第3节
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is that a principle is great rather in power than in extent； hence that which was small at the start turns out a giant at the end。 Now the conception of the infinite possesses this power of principles； and indeed in the sphere of quantity possesses it in a higher degree than any other conception； so that it is in no way absurd or unreasonable that the assumption that an infinite body exists should be of peculiar moment to our inquiry。 The infinite； then； we must now discuss； opening the whole matter from the beginning。 Every body is necessarily to be classed either as simple or as composite； the infinite body； therefore； will be either simple or composite。 But it is clear； further； that if the simple bodies are finite； the composite must also be finite； since that which is composed of bodies finite both in number and in magnitude is itself finite in respect of number and magnitude： its quantity is in fact the same as that of the bodies which compose it。 What remains for us to consider； then； is whether any of the simple bodies can be infinite in magnitude； or whether this is impossible。 Let us try the primary body first； and then go on to consider the others。 The body which moves in a circle must necessarily be finite in every respect； for the following reasons。 （1） If the body so moving is infinite； the radii drawn from the centre will be infinite。 But the space between infinite radii is infinite： and by the space between the radii I mean the area outside which no magnitude which is in contact with the two lines can be conceived as falling。 This； I say； will be infinite： first； because in the case of finite radii it is always finite； and secondly； because in it one can always go on to a width greater than any given width； thus the reasoning which forces us to believe in infinite number； because there is no maximum； applies also to the space between the radii。 Now the infinite cannot be traversed； and if the body is infinite the interval between the radii is necessarily infinite： circular motion therefore is an impossibility。 Yet our eyes tell us that the heavens revolve in a circle； and by argument also we have determined that there is something to which circular movement belongs。 （2） Again； if from a finite time a finite time be subtracted； what remains must be finite and have a beginning。 And if the time of a journey has a beginning； there must be a beginning also of the movement； and consequently also of the distance traversed。 This applies universally。 Take a line； ACE； infinite in one direction； E； and another line； BB； infinite in both directions。 Let ACE describe a circle； revolving upon C as centre。 In its movement it will cut BB continuously for a certain time。 This will be a finite time； since the total time is finite in which the heavens complete their circular orbit； and consequently the time subtracted from it； during which the one line in its motion cuts the other； is also finite。 Therefore there will be a point at which ACE began for the first time to cut BB。 This； however； is impossible。 The infinite； then； cannot revolve in a circle； nor could the world； if it were infinite。 （3） That the infinite cannot move may also be shown as follows。 Let A be a finite line moving past the finite line； B。 Of necessity A will pass clear of B and B of A at the same moment； for each overlaps the other to precisely the same extent。 Now if the two were both moving； and moving in contrary directions； they would pass clear of one another more rapidly； if one were still and the other moving past it； less rapidly； provided that the speed of the latter were the same in both cases。 This； however； is clear： that it is impossible to traverse an infinite line in a finite time。 Infinite time； then； would be required。 （This we demonstrated above in the discussion of movement。） And it makes no difference whether a finite is passing by an infinite or an infinite by a finite。 For when A is passing B； then B overlaps A and it makes no difference whether B is moved or unmoved； except that； if both move； they pass clear of one another more quickly。 It is； however； quite possible that a moving line should in certain cases pass one which is stationary quicker than it passes one moving in an opposite direction。 One has only to imagine the movement to be slow where both move and much faster where one is stationary。 To suppose one line stationary； then； makes no difficulty for our argument； since it is quite possible for A to pass B at a slower rate when both are moving than when only one is。 If； therefore； the time which the finite moving line takes to pass the other is infinite； then necessarily the time occupied by the motion of the infinite past the finite is also infinite。 For the infinite to move at all is thus absolutely impossible； since the very smallest movement conceivable must take an infinity of time。 Moreover the heavens certainly revolve； and they complete their circular orbit in a finite time； so that they pass round the whole extent of any line within their orbit； such as the finite line AB。 The revolving body； therefore； cannot be infinite。 （4） Again； as a line which has a limit cannot be infinite； or； if it is infinite； is so only in length； so a surface cannot be infinite in that respect in which it has a limit； or； indeed； if it is completely determinate； in any respect whatever。 Whether it be a square or a circle or a sphere； it cannot be infinite； any more than a foot…rule can。 There is then no such thing as an infinite sphere or square or circle； and where there is no circle there can be no circular movement； and similarly where there is no infinite at all there can be no infinite movement； and from this it follows that； an infinite circle being itself an impossibility； there can be no circular motion of an infinite body。 （5） Again； take a centre C； an infinite line； AB； another infinite line at right angles to it； E； and a moving radius； CD。 CD will never cease contact with E； but the position will always be something like CE； CD cutting E at F。 The infinite line； therefore； refuses to complete the circle。 （6） Again； if the heaven is infinite and moves in a circle； we shall have to admit that in a finite time it has traversed the infinite。 For suppose the fixed heaven infinite； and that which moves within it equal to it。 It results that when the infinite body has completed its revolution； it has traversed an infinite equal to itself in a finite time。 But that we know to be impossible。 （7） It can also be shown； conversely； that if the time of revolution is finite； the area traversed must also be finite； but the area traversed was equal to itself； therefore； it is itself finite。 We have now shown that the body which moves in a circle is not endless or infinite； but has its limit。
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Further； neither that which moves towards nor that which moves away from the centre can be infinite。 For the upward and downward motions are contraries and are therefore motions towards contrary places。 But if one of a pair of contraries is determinate； the other must be determinate also。 Now the centre is determined； for； from whatever point the body which sinks to the bottom starts its downward motion； it cannot go farther than the centre。 The centre； therefore； being determinate； the upper place must also be determinate。 But if these two places are determined and finite； the corresponding bodies must also be finite。 Further； if up and down are determinate； the intermediate place is also necessarily determinate。 For； if it is indeterminate； the movement within it will be infinite； and that we have already shown to be an impossibility。 The middle region then is determinate； and consequently any body which either is in it； or might be in it； is determinate。 But the bodies which move up and down may be in it； since the one moves naturally away from the centre and the other towards it。 From this alone it is clear that an infinite body is an impossibility； but there is a further point。 If there is no such thing as infinite weight； then it follows that none of these bodies can be infinite。 For the supposed infinite body would have to be infinite in weight。 （The same argument applies to lightness： for as the one supposition involves infinite weight； so the infinity of the body which rises to the surface involves infinite lightness。） This is proved as follows。 Assume the weight to be finite； and take an infinite body； AB； of the weight C。 Subtract from the infinite body a finite mass； BD； the weight of which shall be E。 E then is less than C； since it is the weight of a lesser mass。 Suppose then that the smaller goes into the greater a certain number of times； and take BF bearing the same proportion to BD which the greater weight bears to the smaller。 For you may subtract as much as you please from an infinite。 If now the masses are proportionate to the weights； and the lesser weight is that of the lesser mass； the greater must be that of the greater。 The weights； therefore； of the finite and of the infinite body are equal。 Again； if the weight of a greater body is greater than that of a less； the weight of GB will be greater than that of FB； and thus the weight of the finite body is greater than that of
