Finite and Infinite

Between finite and infinite discrete quantities there is a third intermediate term which corresponds to every assigned number.

Are the finite parts of a continuum finite or infinite in number?

The best reply is that they are neither finite nor infinite but correspond to every assigned number.

Those parts should not be included within a limited number, for in that case they would not correspond to a number which is greater.

Nor can they be infinite in number since no assigned number is infinite; and thus at the pleasure of the questioner we may, to any given line, assign 100 finite parts, 1,000, 100,000, or any number we may please so long as it be not infinite.

I agree with the philosophers that the continuum contains as many finite parts as they please and I concede also that it contains them, either actually or potentially, as they may like.

But I must add that just as a line ten fathoms [canne] in length contains ten lines each of one fathom and forty lines each of one cubit [braccia] and eighty lines each of half a cubit, etc. , so it contains an infinite number of points; call them actual or potential, as you like, for as to this detail, Simplicio, I defer to your opinion and to your judgment.

The fact that something can be done only with effort or diligence or with great expenditure of time does not render it impossible.

You could not easily divide a line into 1,000 parts, and much less if the number of parts were 937 or any other large prime number.

But if I were to accomplish this division which you deem impossible as readily as another person would divide the line into forty parts would you then be more willing, in our discussion, to concede the possibility of such a division?

It is possible to divide a line into its infinitely small elements by following the same order which one employs in dividing the same line into 40, 60, or 100 parts.

This is done by dividing it into 2, 4, etc.

He who thinks that, by following this method, he can reach an infinite number of points is greatly mistaken.

If this process were followed to (37) eternity there would still remain finite parts which were undivided.

By such a method one is very far from reaching the goal of indivisibility; on the contrary he recedes from it and while he thinks that, by continuing this division and by multiplying the multitude of parts, he will approach infinity, he is, in my opinion, getting farther and farther away from it.

My reason is this.

In the preceding discussion we concluded that, in an infinite number, it is necessary that the squares and cubes should be as numerous as the totality of the natural numbers [tutti i numeri], because both of these are as numerous as their roots which constitute the totality of the natural numbers.

Next we saw that the larger the numbers taken the more sparsely distributed were the squares, and still more sparsely the cubes; therefore it is clear that the larger the numbers to which we pass the farther we recede from the infinite number; hence it follows [83] that, since this process carries us farther and farther from the end sought, if on turning back we shall find that any number can be said to be infinite, it must be unity.

Here indeed are satisfied all those conditions which are requisite for an infinite number; I mean that unity contains in itself as many squares as there are cubes and natural numbers [tutti i numeri].

Unity is at once a square, a cube, a square of a square and all the other powers [dignità]; nor is there any essential peculiarity in squares or cubes which does not belong to unity.

For example, the property of 2 square numbers that they have between them a mean proportional; take any square number you please as the first term and unity for the other, then you will always find a number which is a mean proportional.

Consider the 2 square numbers, 9 and 4.

Then 3 is the mean proportional between 9 and 1; while 2 is a mean proportional between 4 and 1; between 9 and 4 we have 6 as a mean proportional.

A property of cubes is that they must have between them two mean proportional numbers; take 8 and 27; between them lie 12 and 18; while (38) between 1 and 8 we have 2 and 4 intervening; and between 1 and 27 there lie 3 and 9.

Therefore, unity is the only infinite number.

These are some of the marvels which our imagination cannot grasp and which should warn us against the serious error of those who attempt to discuss the infinite by assigning to it the same properties which we employ for the finite, the natures of the two having nothing in common.

There is a vast change of character which a finite quantity would undergo in passing to infinity.

Draw the straight line AB.

Let the point C divide it into 2 unequal parts.

If pairs of lines be drawn, one from each of the terminal points A and B, and if the ratio between the lengths of these lines is the same as that between AC and CB, their points of intersection will all lie upon the circumference of one and the same circle.

Thus, for example, AL and BL drawn from A and B, meeting at the point L, bearing to one another the same ratio as AC to BC, and the pair AK and BK meeting at K also bearing to one another the same ratio.

Likewise, the pairs AI, BI, AH, BH, AG, BG, AF, BF, AE, BE, have their points of intersection L, K, I, H, G, F, E, all lying upon the circumference of one and the same circle.

Accordingly if we imagine the point C to move continuously in such a manner that the lines drawn from it to the fixed terminal points, A and B, always maintain the same ratio between their lengths as exists between the original parts, AC and CB, then the point C will, as I shall presently prove, describe a circle.

Fig 7

The circle thus described will increase in size without limit as the point C approaches the middle point which we may call O; but it will diminish in size as C approaches the end B.

So that the infinite number of points located in the line OB will, if the motion be as explained above, describe circles of every size, some smaller than the pupil of the eye of a flea, others larger than the celestial equator.

If we move any of the points lying between the two ends O and B they will all describe circles, those nearest O, immense circles;

But if we move the point O itself, and continue to move it according to the aforesaid law, namely, that the lines drawn from O to the terminal points, A and B, maintain the same ratio as the original lines AO and OB, what kind of a line will be produced? A circle will be drawn larger than the largest of the others, a circle which is therefore infinite.

But from the point O a straight line will also be drawn perpendicular to BA and extending to infinity without ever turning, as did the others, to join its last end with its first;

For the point C, with its limited motion, having described [85] the upper semi-circle, CHE, proceeds to describe the lower semicircle EMC, thus returning to the starting point.

But the point O having started to describe its circle, as did all the other points in the line AB, (for the points in the other portion OA describe their circles also, the largest being those nearest the point O) is unable to return to its starting point because the circle it describes, being the largest of all, is infinite; in fact, it describes an infinite straight line as circumference of its infinite circle.

There is a difference between a finite and an infinite circle since the latter changes character in such a manner that it loses not only its existence but also its possibility of existence.

There is no infinite circle.

Similarly, there is no infinite sphere, no infinite body, and no infinite surface of any shape.

What shall we say concerning this metamorphosis in the transition from finite to infinite?

Why should we feel greater repugnance, seeing that, in our search after the infinite among numbers we found it in unity?

Having broken up a solid into many parts, having reduced it to the finest of powder and having resolved it into its infinitely small indivisible atoms why may we not say that this solid has been reduced to a single continuum [un solo continuo] perhaps a fluid like water or mercury or even a liquefied metal?

Stones melt into glass. The glass itself under strong heat become more fluid than water.