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king aither as equivalent to fire。 It is also clear from what has been said why the number of what we call simple bodies cannot be greater than it is。 The motion of a simple body must itself be simple； and we assert that there are only these two simple motions； the circular and the straight； the latter being subdivided into motion away from and motion towards the centre。
4
That there is no other form of motion opposed as contrary to the circular may be proved in various ways。 In the first place； there is an obvious tendency to oppose the straight line to the circular。 For concave and convex are a not only regarded as opposed to one another； but they are also coupled together and treated as a unity in opposition to the straight。 And so； if there is a contrary to circular motion； motion in a straight line must be recognized as having the best claim to that name。 But the two forms of rectilinear motion are opposed to one another by reason of their places； for up and down is a difference and a contrary opposition in place。 Secondly； it may be thought that the same reasoning which holds good of the rectilinear path applies also the circular； movement from A to B being opposed as contrary to movement from B to A。 But what is meant is still rectilinear motion。 For that is limited to a single path； while the circular paths which pass through the same two points are infinite in number。 Even if we are confined to the single semicircle and the opposition is between movement from C to D and from D to C along that semicircle； the case is no better。 For the motion is the same as that along the diameter； since we invariably regard the distance between two points as the length of the straight line which joins them。 It is no more satisfactory to construct a circle and treat motion 'along one semicircle as contrary to motion along the other。 For example； taking a complete circle； motion from E to F on the semicircle G may be opposed to motion from F to E on the semicircle H。 But even supposing these are contraries； it in no way follows that the reverse motions on the complete circumference contraries。 Nor again can motion along the circle from A to B be regarded as the contrary of motion from A to C： for the motion goes from the same point towards the same point； and contrary motion was distinguished as motion from a contrary to its contrary。 And even if the motion round a circle is the contrary of the reverse motion； one of the two would be ineffective： for both move to the same point； because that which moves in a circle； at whatever point it begins； must necessarily pass through all the contrary places alike。 （By contrarieties of place I mean up and down； back and front； and right and left； and the contrary oppositions of movements are determined by those of places。） One of the motions； then； would be ineffective； for if the two motions were of equal strength； there would be no movement either way； and if one of the two were preponderant； the other would be inoperative。 So that if both bodies were there； one of them； inasmuch as it would not be moving with its own movement； would be useless； in the sense in which a shoe is useless when it is not worn。 But God and nature create nothing that has not its use。
5
This being clear； we must go on to consider the questions which remain。 First； is there an infinite body； as the majority of the ancient philosophers thought； or is this an impossibility？ The decision of this question； either way； is not unimportant； but rather all…important； to our search for the truth。 It is this problem which has practically always been the source of the differences of those who have written about nature as a whole。 So it has been and so it must be； since the least initial deviation from the truth is multiplied later a thousandfold。 Admit； for instance； the existence of a minimum magnitude； and you will find that the minimum which you have introduced； small as it is； causes the greatest truths of mathematics to totter。 The reason 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 th