C. To the Third Argument

412. As to the third principal argument of the question, when the argument is made that ‘an angel cannot be moved [continuously] because he is indisivible’ [n.301] -although one could easily reply that an angel occupies a divisible place and so, in respect of place, he is disposed as if he were divisible - or that, if he occupies existing as a point a point-place, he cannot be moved continuously so as always to have point-existence -yet, because there seems no reason to deny that an indivisible is moved (even if it were a per se existing indivisible of quantity), then one can concede that an angel, occupying a ‘where’-point, can, as always existing in a point, be continuously moved.

413. And what is here assumed about an indivisible can be proved in many ways: First that a sphere moved over a plane describes a line on the plane and yet only touches the plane at a point; therefore the point passes through the whole line, and yet not for this reason is the line that the point thus passes through composed of points. Therefore, by similarity, neither would this result follow if the point existed per se.

414. Multiple responses are made here:

That there is no spherical thing in nature but only in the intellect or imagination. -But this reply is nothing, because the heaven is simply spherical; and anyway, given that there were no simply spherical thing in nature, there would still be no contradiction on the part of sphere and plane that this thing move over that thing as a sphere over a plane (but there would be a contradiction if, from an indivisible moved over something, the result was that the thing moved over was indivisible).

415. A response in another way is that a natural sphere touches a plane at a line and not at a point. - But this seems impossible, because what is applied to a circular line (so as to touch the whole of it) is necessarily circular, because any circular part is circular in any part; but of a straight line no part is circular or curved.

416. Another response is that, because the point of the sphere [n.413] is moved per accidens, therefore there is no need that the space over which it moves be commensurate with it; but the sphere itself is moved per se, and it is divisible. - But against this is that, although a part in a whole is moved per accidens, yet it is always in a space equal to it, and it describes - in its passage - the whole space; indeed, if a whiteness (which is moved, when the extension is moved, more per accidens than any part or term of the extension) is compared to space according to the quantity it has per accidens , its accidental quantity would still be measured by space. Hence - as far as commensuration is concerned - it does not seem that ‘being moved per accidens’ takes away anything other than ‘being moved per se’.

417. Second [n.413], the line laid down by the sphere is not commensurate with the sphere (because then it would be a solid), and it is commensurate with something moved over it; therefore only with the point that is moved over it. If too the sphere is posited to be in a vacuum and only the line to be a plenum, and if per impossibile the sphere could be moved in a vacuum and the point could be moved over the line-plenum, the line-plenum would only be precisely described by the point. And so the conclusion intended follows from these considerations.

418. Further, take a solid cube and let it be moved. Its primary surface is always on something equal to it, and so always on a surface; or something corresponds to it in the magnitude placed underneath [sc. the magnitude over which the cube is moving], to wit a line - and thus, by always passing over something of the magnitude before something else of it, the cube passes over the whole magnitude; therefore the whole magnitude underneath is composed of a line, if their reasoning be valid [sc. those who say an indivisible cannot be moved continuously, n.412].The idea seems to be that if a cube is moved over a magnitude not continuously but indivisible by indivisible, then the surface of the cube in contact with the magnitude beneath will move over one line of the magnitude before another. So if we focus on just one line in the magnitude, we can consider the whole surface moving over that one line, which will thus be the magnitude the surface moves over. The magnitude moved over will then be composed of that one line. We can repeat this process for every subsequent line of the magnitude, and consequently the surface will always be moving over a magnitude composed of a line. Since this result is unacceptable, we must suppose that the cube moves continuously and not indivisible by indivisible. ³⁷

419. Further, let a first point be marked on a line over which another line is moving. This point on the line placed underneath describes the whole of the moved line, because just as any point of the moved line is always continuously at different points of the line underneath, so conversely any point of the line underneath is underneath different points of the moved line; and yet along with all of these points there stands a continuity of motion.

420. It can therefore be conceded (since the statement about motion per accidens [n.416] seems nothing but a subterfuge) that an indivisible could be per se moved if it existed per se, and still be moved continuously; nor from this does it follow that the magnitude passed over is composed of indivisibles.

421. However, because of what Aristotle means in the passages quoted [nn.302-304], one needs to understand that in local motion there is succession for two reasons, namely the divisibility of the movable and the divisibility of the space, and each of these causes, if it existed per se and precisely, would be a sufficient reason for succession; for any movable first passes over one part of the space before it passes over another, and so there would be succession on the part of the space when comparing the movable to the diverse parts of space; further any same part of the space goes by a first part of the movable before a second, and so there would be succession on the part of the movable when comparing it to any same part of the space. In like way too can it be said of the motion of alteration and perhaps of the motion of increase.

422. The philosopher denies therefore, and well denies, that an indivisible ‘as far as concerns itself’ can be moved or can move such that a continuity of motion on its part can be taken such that it is a movable possessing in itself the complete idea of continuous movable, because being continuously moved is not something it has in itself; yet moving or being continuously moved is not repugnant to an indivisible when taking the continuity of motion from something else [n.421].

423. And such, and nothing more, is what Aristotle’s reasons prove, as is plain by running through all of them:

For when Aristotle takes the principle that ‘everything that is moved is partly in the term from which and partly in the term to which’ [n.302], this principle is true if the movable is of the sort that, from its own idea, there is succession of motion; for such a movable is in both terms according to different parts of itself. However things are not so here [sc. in the case of an indivisible], but an indivisible is partly in one term and partly in the other according to the same part of itself - that is, it is in some intermediate stage, not by being at rest, but insofar as this intermediate stage is something of both terms, that is, insofar as it is that through which the indivisible tends from one term to the other; this is to say that it is under change and under something lying under change, and in this way the parts of motion are continuous. - But when the principle is taken that ‘the indivisible cannot be partly in one term and partly in the other because it does not have parts’ [n.302], this principle is true of the first sense of partly (and so I conclude and concede that the indivisible is not thus a movable), but it is false of the second sense of partly [sc. first and second in this paragraph: the first is that of a movable from whose own idea there is succession, and the second is that of an indivisible].

424. To the other argument [n.303], when it is said that ‘a movable passes through a space equal or less than itself before it passes through a greater space’, I reply by saying that ‘to pass through’ can be understood of a divisible passage or of an indivisible passage.

If for an indivisible one, the proposition is false if the understanding is that before passing through any greater space the movable universally passes indivisibly through some equal space; for then one would have to concede that there would be a first change in local motion; and not even those perverters (and not expositors), who say that Aristotle retracts [in Physics 8] what he said in Physics 6 [n.297], can reasonably say that he contradicts himself within Physics 6 itself. There is no need, then, that any successive passage, which is greater than the movable, be preceded by an indivisible passage.

But if ‘to pass through’ be understood of a divisible passage, then it can be understood of the whole, not by reason of the whole, but by reason of the part; and this not by comparing the part to a ‘where’ equal to it and the whole to a ‘where’ equal to it, because the continuous is that ‘whose motion is one and indivisible’, Metaphysics 5.6.1016a5-6; and in this way the part passes through a space corresponding to it at the same time as the whole movable passes through a whole space corresponding to it. But when understanding ‘to pass through’ with respect to some definite and determinate point in the space, the whole passes first through that point by reason of some part (and, in having passed through it, the whole has passed through something less than itself, speaking of a ‘where’ different from its own first total ‘where’), before it thus passes through a space equal or greater than itself; and this it does per accidens, insofar as the movable can have a ‘where’ less than its total ‘where’.

425. But if we speak of greater or lesser or equal ‘wheres’, according to which continuity of motion is immediately expected (and an infinite number of which ‘wheres’ are something in the first ‘where’), then simply the whole passes through a space greater than itself before it passes through one equal to itself. As such is the response to the issue at hand, saving what belongs to the per se idea of continuous motion and not what does not belong to the per se idea of it.

426. And if you object that, however it may be with Aristotle’s argument in itself [n.303], this point is always in a space equal to itself and so passes through the whole (so it is commensurate with the whole line underneath, and so this line underneath will be composed of points) - I say that it is ‘always’ in the sense that, in any indivisible, it is in a space equal to itself; but it is not ‘always’ in the sense of any part of time.

The same could be argued about the first surface of the cube solid [n.418], that although in any ‘now’ of time it is lying precisely on the line over which it is moved, yet in the intermediate time between two instants it is flowing over the continuous intermediate between the two extremes.

427. As to the last reason [n.304], I well concede that it is possible to take a time less than any given time, but from this does not follow that in that lesser time a lesser movable can be moved, save when speaking of a continuous movable that was, on its own part, the cause of the continuity of the motion.