He then shows (by a long process of reasoning, not easy to follow) first, that there cannot be an infinite body with finite gravity; next, that there can be no infinite gravity. Accordingly there can be no infinite body at all, having gravity. At the end, he considers that this is established, (1) by the partial arguments (διὰ τῶν κατὰ μέρος) immediately preceding; (2) by the general reasonings in his other treatises respecting first principles, in which he explained the Infinite — in what sense it existed and did not exist; (3) by an argument about the Infinite, upon which he touches in the next chapter (ss. 4-13).
CH. 7. — Every body is of necessity either infinite or finite. If infinite, it is as a whole either of like constituents or of unlike. If the latter, either of a finite number of species, or of an infinite number. The last is impossible, if our fundamental assumptions are allowed to stand. For since the simple modes of motion are limited in number, the simple bodies must be alike limited; each simple mode of notion belonging to its own special simple body, and each natural body having always its own natural motion. But, if the Infinite be composed of a finite number of species, each of these constituent parts must be infinite; that is, water and fire must be infinite. Yet this too, is impossible; for we have seen that there cannot be either infinite levity or infinite gravity (the attributes of fire and water). Moreover, if these bodies be infinite, the places which they occupy, and the motions which they make, must also be infinite; but this also we have shown to be inadmissible, if our fundamental assumptions are admitted. The centripetal body cannot be carried to an infinite distance downward, nor the centrifugal body to an infinite distance upward. That which cannot come to pass, cannot be in course of coming to pass; thus, if a thing cannot come to be white, or a cubit long, or domiciled in Egypt, it cannot be in course of becoming white, or a cubit long, &c. It cannot be in course of being carried to a terminus which cannot be reached. It might be argued that fire, though discontinuous and dispersed, might still be infinite, in the sum total of its different masses. But body is that which is extended in every direction: how can there be many bodies unlike to each other, yet each of them infinite? Each of them, if infinite at all, ought to be infinite in every direction (ss. 1-5).
We thus see that the Infinite cannot consist of unlike constituents. But neither can it consist of constituents all similar. For, first, there are only three simple motions, and one of the three it must have; but we have shown that it cannot have either centripetal or centrifugal motion (i.e., that it cannot have either infinite gravity or infinite levity); nor can it again have circular motion, for the Infinite cannot be carried in a circle: this would amount to saying that the Heaven is infinite, which we have shown to be impossible. The Infinite indeed cannot be moved in any way at all; for, if moved, it must be moved either according to nature, or contrary to nature (violently), and, if its present motion be violent, it must have some other mode of motion which is natural to it. But, if it have any such, this assumes that there exists some other place belonging to it, into which it may be conveyed — an obvious impossibility (ss. 6, 7).
Farthermore, the Infinite cannot act in any way upon the Finite, nor be acted upon thereby (ss. 8-10). Nor can the Infinite be acted upon in any way by the Infinite (ss. 11-12).
If then every perceptible body possesses powers, as agent or patient or both, there can be no perceptible body which is infinite. But all bodies which are in any place are perceptible; therefore no body which is in any place can be infinite. There is no infinite body, indeed there can be no body at all, outside of the Heaven; for that which is outside of the Heaven is in a place. Even if perceivable only up to a certain point (μέχρι τινός), even if merely intelligible, it would still be in a place, and would therefore come under the foregoing argument — that there is no body outside of the Heaven (ss. 13, 14).
The foregoing reasoning may be summed up, in more general language (λογικώτερον), as follows:— The Infinite assumed as homogeneous cannot be moved in a circle, since the Infinite has no centre; nor in a straight line, since this would imply a second infinite place into which it must be moved according to nature, and a third infinite place into which it must be moved against nature, and since in either case the force which causes it to be moved must be infinite. But we have already argued, in treating of Motion (Phys. VIII. x.) that nothing finite can have infinite power, nothing infinite can have finite power; and, if that which is moved according to nature can also be moved contrary to nature, there must of necessity be two Infinites — Movens and Motum. Yet what can that be which causes the Infinite to move? If it cause itself to move, it must be animated (ἔμψυχον): but how can an infinite animated being (ζῷον) exist? And, if there be anything else which causes it to move, there must exist two Infinites, each distinguished from the other in form and power (ss. 15-17).
Again, even if we admit the doctrine of Leukippus and Demokritus — That the whole is not continuous, but discontinuous, atoms divided by intervening spaces — still the Infinite is inadmissible. For the nature and essence of these atoms is all the same, though they are different from each other in figure and arrangement; accordingly the motion of all must be the same: if one is heavy or centripetal, all must be so alike; if one is light or centrifugal, all must be so alike. But either of these motions would imply the existence of centre and periphery; which does not consist with an infinite whole. In the Infinite, there is neither centre nor periphery; no terminus prefixed either for upward or downward motion; no own place either for centripetal or centrifugal matter. Therefore in an infinite universe, there can be no motion at all (ss. 18, 19).
CH. 8. — There cannot be more than one Kosmos. All things both rest and are moved, either by violence, or according to nature. In that place to which it is carried by nature, it also rests by nature: in that place to which it is carried by violence, it rests by violence. If the current which we see towards the centre is by violence, the opposite current must be natural; if earth is carried by violence from thence hitherward, its natural current must be from hence thitherward; and, if being here it rests without violence, its current towards here must be a natural one. For there is one only which is natural. Now, if there be many Kosmi, they must be alike in their nature, and must be composed of the same bodies, having the same nature and powers — fire, earth, and the two intermediate elements: for, if the bodies here are not the same as those in other Kosmi — if the same names are given in an equivocal sense and do not connote the same specific attributes — the name Kosmos must be equivocal also, and there cannot be many true or real Kosmi, in the same sense. To the parts or elements of each Kosmos, therefore, the centripetal and centrifugal currents are natural; for the simple currents are limited in number, and each element is so named as to connote one of them specially; and, if the currents are the same, the elements must also be the same everywhere. If there were another Kosmos, the earth in that would tend towards the centre of our Kosmos, and the fire in that would tend towards the periphery of our Kosmos. But this is impossible; since in that case the earth in that Kosmos would run away from the centre of its own Kosmos, and the fire therein would run away from its own periphery. Either we must not admit the same nature in the simple elements of the numerous Kosmi; or, if we do admit it, we must recognize only one centre and one periphery. This difficulty prevents our recognizing more than one Kosmos (ss. 1-6).
It is unphilosophical to affirm that the nature of these simple elements becomes changed according as they are more or less distant from their own places. The difference is at best one of degree, not one of kind. That they are moved, we see plainly; there must therefore be some one current of motion natural to them. Accordingly every portion of the same element (or of elements the same in kind) must tend towards the same numerical place — towards this actual centre (πρὸς τόδε τι μέσον), or that actual periphery; and, if the tendency be towards one centre specie, but towards many centres numero, because particulars differ numero alone, and not specie, still the attribute will be alike in all, and will not be present in some portions, absent in others: I mean that, if the portions of this Kosmos are relative to each other, those in another Kosmos are in the like condition, and what is taken from this Kosmos will not be different from what is taken from the corresponding elements of any other Kosmos. Unless these assumptions can be overthrown, it is indisputably certain that there can be only one centre and one periphery; by consequence therefore, only one Kosmos and not more (ss. 7-10).
There are other reasons to show that there is a given terminus for the natural current both of fire and of earth. A thing moved, speaking generally, changes from something definite into something else definite; but there are different species of such change: the change called getting-well is from sickness to health; that called growth is from the little to great; that called local movement is from a terminus to another terminus, and local movements are specifically different from each other, according as the terminus a quo and the terminus ad quem is defined in each. The terminus is always a known and definite point: it is not accidental, nor dependent upon the arbitrium of the mover. Fire and earth therefore do not move on to infinity, but to definite points in opposite directions; and the local antithesis is between above and below: these are the two termini of the respective currents. Earth is carried with greater velocity, the nearer it approaches to the centre; fire is carried with greater velocity, the nearer it approaches to the periphery. This shows that its current does not stretch to infinity; for its velocity would then increase infinitely. Earth is not carried downward by the force of any thing else, nor fire upwards: not by any violence, nor by squeezing out (ἐκθλίψει), as some say. If this were so, a larger quantity of earth would move downward, and a larger quantity of fire upward, more slowly than a smaller. But the reverse is what occurs: the larger quantity of earth moves downward more rapidly than the smaller; if its motion had been caused by violence or by squeezing out, such motion would have slackened as it became more widely distant from the moving force (ss. 11-14).