Polygons

Within a finite extent is it possible to discover an infinite number of vacua?

Aristotle asked this in his Questions in Mechanics.

Let us draw a shape.

G is a center with an equiangular and equilateral polygon of any number of sides, say the hexagon ABCDEF.

Similar to this and concentric with it, describe another smaller one which we shall call HIKLMN.

Prolong the side AB, of the larger hexagon, indefinitely toward S.

In like manner, prolong the corresponding side HI of the smaller hexagon, in the same direction, so that the line HT is parallel to AS; and through the center draw the line GV parallel to the other 2.

This done, imagine the larger polygon to roll upon the line AS, carrying with it the smaller polygon.

If the point B, the end of the side AB, remains fixed at the beginning of the rotation, the point A will rise and the point C will fall describing the arc CQ until the side BC coincides with the line BQ, equal to BC.

But during this rotation the point I, on the smaller polygon, will rise above the line IT because IB is oblique to AS.

It will not again return to the line IT until the point C shall have reached the position Q.

The point I, having described the arc IO above the line HT, will reach the position (22) O at the same time the side IK assumes the position OP; but in the meantime the center G has traversed a path above GV and does not return to it until it has completed the arc GC.

This step having been taken, the larger polygon has been brought to rest with its side BC coinciding with the line BQ while the side IK of the smaller polygon has been made to coincide with the line OP, having passed over the portion IO without touching it; also the center G will have reached the position C after having traversed all its course above the parallel line GV.

Finally, the entire figure will assume a position similar to the first, so that if we continue the rotation and come to the next step, the side DC of the larger polygon will coincide with the portion QX and the side KL of the smaller polygon, having first skipped the arc PY, will fall on YZ, while the center still keeping above the line GV will return to it at R after having jumped the interval CR.

At the end of one complete rotation the larger polygon will have traced upon the line AS, without break, six lines together equal to its perimeter; the lesser polygon will likewise have imprinted six lines equal to its perimeter, but separated by the interposition of five arcs, whose chords represent the parts of HT not touched by the polygon: the center G never reaches the line GV except at six points.

From this it is clear that the space traversed by the smaller polygon is almost equal to that traversed by the larger, that is, the line HT approximates the line AS, differing from it only by the length of one chord of one of these arcs, provided we understand the line HT to include the five skipped arcs.

This case of hexagons is be applicable to all other polygons.

Whatever the number of sides, provided only they are similar, concentric, and rigidly connected, so that when the greater one rotates the lesser will also turn however small it may be.

The lines described by these two are nearly equal provided we include in the space traversed by the smaller one the intervals which are not touched by any part of the perimeter of this smaller polygon.

Let a large polygon of, say, one thousand sides make one complete rotation and thus lay off a line equal to its perimeter; at the same time the small one will pass over an approximately equal distance, made up of a thousand small portions, each equal to one of its sides, but interrupted by a thousand spaces which, in contrast with the portions that coincide with the sides of the polygon, we may call empty.

But now suppose that about any center, say A, we describe 2 concentric and rigidly connected circles; and suppose that from the points C and B, on their radii, there are drawn the tangents CE and BF and that through the center A the line AD is drawn parallel to them, then if the large circle makes one complete rotation along the line BF, equal not only to its circumference but also to the other two lines CE and AD, tell me what the smaller circle will do and also what the center will do.

As to the center it will certainly traverse and touch the entire line AD while the circumference of the smaller circle will have measured off by its points of contact the entire line CE, just as was done by the above mentioned polygons.

The only difference is that the line HT was not at every point in contact with the perimeter of the smaller polygon, but there were left untouched as many vacant spaces as there were spaces coinciding with the sides.

But here in the case of the circles the circumference of the smaller one never leaves the line CE, so that no part of the latter is left untouched, nor is there ever a time when some point on the circle is not in contact with the straight line.

How now can the smaller circle traverse a length greater than its circumference unless it go by jumps?

There are 2 reasons why this cannot happen.

1. There is no ground for thinking that one point of contact, such as that at C, rather than another, should slip over certain portions of the line CE.

But if such slidings along CE did occur they would be infinite in number since the points of contact (being mere points) are infinite in number: an infinite number of finite slips will however make an infinitely long line, while as a matter of fact the line CE is finite.

2. As the greater circle, in its rotation, changes its point of contact continuously the lesser circle must do the same because B is the only point from which a straight line can be drawn to A and pass through C.

Accordingly, the small circle must change its point of contact whenever the large one changes: no point of the small circle touches the straight line CE in more than one point.

But even in the rotation of the polygons, there was no point on the perimeter of the smaller which coincided with more than one point on the line traversed by that perimeter.

This is clear when you remember that the line IK is parallel to BC.

Therefore IK will remain above IP until BC coincides with BQ, and that IK will not lie on IP except at the very instant when BC occupies the position BQ.

At this instant the entire line IK coincides with OP and immediately afterwards rises above it.

In the case of polygons with 100,000 sides, the line traversed by the perimeter of the greater, i.e, the line laid down by its 100000 sides one after another, is equal to the line traced out by the 100000 sides of the smaller, provided we include the 100000 vacant spaces interspersed.

So in the case of the circles, polygons having an infinitude of sides, the line traversed by the continuously distributed [cantinuamente disposti] infinitude of sides is in the greater circle equal to the line laid down by the infinitude of sides in the smaller circle but with the exception that these latter alternate with empty spaces.

Since the sides are not finite in number, but infinite, so also are the intervening (25) empty spaces not finite but infinite. The line traversed by the larger circle consists then of an infinite number of points which completely fill it; while that which is traced by the smaller circle consists of an infinite number of points which leave empty spaces and only partly fill the line.

After dividing and resolving a line into a finite number of parts, that is, into a number which can be counted. it [72] is not possible to arrange them again into a greater length than that which they occupied when they formed a continuum [continuate] and were connected without the interposition of as many empty spaces.

But if we consider the line resolved into an infinite number of infinitely small and indivisible parts, we shall be able to conceive the line extended indefinitely by the interposition, not of a finite, but of an infinite number of infinitely small indivisible empty spaces.

This which has been said on simple lines also holds in the case of surfaces and solid bodies.

It is assumed that they are made up of an infinite, not a finite, number of atoms.

Such a body once divided into a finite number of parts it is impossible to reassemble them so as to occupy more space than before unless we interpose a finite number of empty spaces, that is to say, spaces free from the substance of which the solid is made.

But if we imagine the body, by some extreme and final analysis, resolved into its primary elements, infinite in number, then we shall be able to think of them as indefinitely extended in space, not by the interposition of a finite, but of an infinite number of empty spaces.

Thus one can easily imagine a small ball of gold expanded into a very large space without the introduction of a finite number of empty spaces, always provided the gold is made up of an infinite number of indivisible parts.