Expansion and Contraction

I will do both.

And first, just as, for the production of expansion, we employ the line described by the small circle during one rotation of the large one – a line greater than the circumference of the small circle – so, in order to explain contraction, we point out that, during each rotation of the smaller circle, the larger one describes a straight line which is shorter than its circumference.

For the better understanding of this we proceed to the consideration of what happens in the case of polygons.

Employing [94] a figure similar to the earlier one, construct the two hexagons, ABC and HIK, about the common center L, and let them roll along the parallel lines HOM and ABc.

Now holding the vertex I fixed, allow the smaller polygon to rotate until the side IK lies upon the parallel, during which motion the point K will describe the arc KM, and the side KI will coincide with IM. Let us see what, in the meantime, the side CB of the larger polygon has been doing.

Since the rotation is about the point I, the terminal point B, of the line IB, moving backwards, will describe the arc Bb underneath the parallel cA so that when the side KI coincides with the line MI, the side BC will coincide with bc, having advanced only through the distance Bc, but having retreated through a portion of the line BA which subtends the arc Bb.

If we allow the rotation of the smaller polygon to go on it will traverse and describe along its parallel a line equal to its perimeter; while the larger one will traverse and describe a line less than its perimeter by as many times the length bB as there (50) are sides less one; this line is approximately equal to that described by the smaller polygon exceeding it only by the distance bB.

Here now we see, without any difficulty, why the larger polygon, when carried by the smaller, does not measure off with its sides a line longer than that traversed by the smaller one; this is because a portion of each side is superposed upon its immediately preceding neighbor.

Let us next consider two circles, having a common center at A, and lying upon their respective parallels, the smaller being tangent to its parallel at the point B; the larger, at the point C. Here when the small circle commences to roll the point B Fig 9 does not remain at rest for a while so as to allow BC to move backward and carry with it the point C, as happened in the case of the polygons, where the point I remained fixed until the side KI coincided with MI and the line IB carried the terminal point B backward as far as b, so that the side BC fell upon bc, thus superposing upon the line BA, the portion Bb, and advancing by an amount Bc, equal to MI, that is, to one side of the smaller polygon. On account of these superpositions, which are the excesses of the sides of the larger over the smaller polygon, each net advance is equal to one side of the smaller polygon and, during one complete rotation, these amount to a straight line equal in length to the perimeter of the smaller polygon.

The number of sides in any polygon is comprised within a certain limit.

The number of sides in a circle is infinite.

The former are finite and divisible.

The latter infinite and indivisible.

In the case of the polygon, the vertices remain at rest during an interval of time which bears to the period of one complete rotation the same ratio which one side bears to the perimeter; likewise, in the case of the circles, the delay of each of the infinite number of vertices is merely instantaneous, because an instant is such a fraction of a finite interval as a point is of a line which contains an infinite number of points.

The retrogression of the sides of the larger polygon is not equal to the length of one of its sides but merely to the excess of such a side over one side of the smaller polygon, the net advance being equal to this smaller side; but in the circle, the point or side C, during the instantaneous rest of B, recedes by an amount equal to its excess over the side B, making a net progress equal to B itself.

In short the infinite number of indivisible sides of the greater circle with their infinite number of indivisible retrogressions, made during the infinite number of instantaneous delays of the infinite number of vertices of the smaller circle, together with the infinite number of progressions, equal to the infinite number of sides in the smaller circle – all these, I say, add up to a line equal to that described by the smaller circle, a line which contains an infinite number of infinitely small superpositions, thus bringing about a thickening or contraction without any overlapping or interpenetration of finite parts. This result could not be obtained in the case of a line divided [96] into finite parts such as is the perimeter of any polygon, which when laid out in a straight line cannot be shortened except by the overlapping and interpenetration of its sides.

This contraction of an infinite number of infinitely small parts without the interpenetration or overlapping of finite parts and the previously mentioned expansion of an infinite number of indivisible parts by the interposition of indivisible vacua is the most that can be said about the contraction and expansion of bodies.

This is unless we give up the impenetrability of matter and introduce empty spaces of finite size.

Remember that we are dealing with the infinite and the indivisible.

I cannot make the invisible visible.

You mentioned gold. We know that metal can be immensely expanded.

Take a rod of silver half a cubit long and 4 times as wide as one’s thumb.

This rod they cover with gold-leaf which is so thin that it almost floats in air.

• It has not more than eight or ten thicknesses.

Once gilded they begin to pull it, with great force, through the holes of a draw-plate.

Again and again it is made to pass through smaller and smaller holes, until, after very many passages, it is reduced to the fineness of a lady’s hair. Yet the surface remains gilded.

Imagine now how the substance of this gold has been expanded and to what fineness it has been reduced.

You are greatly mistaken. The surface increases directly as the square root of the length, a fact which I can demonstrate geometrically.

The original thick rod of silver and the wire drawn out to an enormous length are two cylinders of the same volume, since they are the same body of silver. If I determine the ratio between the surfaces of cylinders of the same volume, the problem will be solved.

The areas of cylinders of equal volumes, neglecting the bases, bear to each other a ratio which is the square root of the ratio of their lengths.

Take 2 cylinders of equal volume having the altitudes AB and CD, between which the line E is a mean proportional.

Then I claim that, omitting the bases of each cylinder, the surface of the cylinder AB is to that of the cylinder CD as the length AB (54) is to the line E, that is, as the square root of AB is to the square root of CD.

Cut off the cylinder AB at F so that the altitude AF is equal to CD.

Then since the bases of cylinders of equal volume bear to one another the inverse ratio of their heights, it follows that the area of the circular base of the cylinder CD will be to the area of the circular base of AB as the altitude BA is to DC: moreover, since circles are to one another as the squares of their diameters, the said squares will be to each other as BA is to CD.

But BA is to CD as the square of BA is to the square of E: and, therefore, these four squares will form a proportion; and likewise their sides; so the line AB is to E as the diameter of circle C is to the diameter of the circle A.

But the diameters are proportional to the circumferences and the circumferences are proportional to the areas of cylinders of equal height.

Hence the line AB is to E as the surface of the cylinder CD is to the surface of the cylinder AF.

Since the height AF is to AB as the surface of AF is to the surface of AB; and since the height AB is to the line E as the surface CD is to AF, it follows, ex oequali in proportione perturbata,* that the height AF is to E as the surface CD is to the surface AB, and convertendo, the surface of the cylinder AB is to the surface of the cylinder CD as the line E is to AF, i. e. , to CD, or as AB is to E which is the square root of the ratio of AB to CD. Q.E.D.;

If now we apply these results to the case in hand, and assume that the silver cylinder at the time of gilding had a length of only half a cubit and a thickness three or four times that of Fig 10.

one’s thumb when the wire has been reduced to the fineness of a hair and has been drawn out to a length of twenty thousand cubits (and perhaps more), the area of its surface will have been increased not less than two hundred times.

Consequently the ten leaves of gold which were laid on (55) have been extended over a surface 200 times greater, assuring us that the thickness of the gold which now covers the surface of so many cubits of wire cannot be greater than one twentieth that of an ordinary leaf of beaten gold.

Consider now what degree of fineness it must have and whether one could conceive it to happen in any other way than by enormous expansion of parts; consider also whether this experiment does not suggest that physical bodies [materie fisiche] are composed of infinitely small indivisible particles, a view which is supported by other more striking and conclusive examples.