Infinity and Indivisibility
Table of Contents
Give us this geometrical proof that there is always equality between these solids and between their bases.
IPC is a right angle. The square of the radius IC is equal to the sum of the squares on the 2 sides IP, PC.
But the radius IC is equal to AC and also to GP, while CP is equal to PH.
Hence, the square of the line GP is equal to the sum of the square (26s of IP and PH, or multiplying through by 4, we have the square of the diameter GN equal to the sum of the squares on IO and HL.
Since the areas of circles are to each other as the squares of their diameters, it follows that the area of the circle whose diameter is GN is equal to the sum of the areas of circles having diameters IO and HL, so that if we remove the common area of the circle having IO for diameter the remaining area of the circle GN will be equal to the area of the circle whose diameter is HL.
As for the other part, we leave its demonstration for the present, partly (30) because those who wish to follow it will find it in the twelfth proposition of the second book of De centro gravitatis solidorum by the Archimedes of our age, Luca Valerio,* who made use of it for a different object, and partly because, for our purpose, it suffices to have seen that the above-mentioned surfaces are always equal and that, as they keep on diminishing uniformly, they degenerate, the one into a single point, the other into the circumference of a circle larger than any assignable; in this fact lies our miracle. **
What about the other difficulty raised by Simplicio?
The infinity and indivisibility are in their very nature incomprehensible to us.
Imagine then what they are when combined. Yet if we wish to build up a line out of indivisible points, we must take an infinite number of them.
We are bound to understand both the infinite and the indivisible at the same time. Many ideas have passed through my mind concerning this subject, some of which, possibly the more important, I may not be able to recall on the spur of the moment; but in the course of our discussion it may happen that I shall awaken in you, and especially in Simplicio, objections and difficulties which in turn will bring to memory that which, without such stimulus, would have lain dormant in my mind.
Allow me therefore the customary liberty of introducing some of our human fancies, for indeed we may so call them in comparison with supernatural truth which furnishes the one true and safe recourse for decision in our discussions and which is an infallible guide in the dark and dubious paths of thought.
One of the main objections urged against this building up of continuous quantities out of indivisible quantities [continuo d’ indivisibili] is that the addition of one indivisible to another cannot produce a divisible, for if this were so it would render the indivisible divisible.
Thus if 2 indivisibles, say 2 points, can be united to form a quantity, say a divisible line, then an even more divisible line might be formed by the union of three, five, seven, or any other odd number of points.
Since however these lines can be cut into two equal parts, it becomes possible to cut the indivisible, which lies exactly in the middle of the line. In answer to this and other objections of the same type we reply that a divisible magnitude cannot be constructed out of two or ten or a hundred or a thousand indivisibles, but requires an infinite number of them.
Here a difficulty presents itself which appears to me insoluble.
One line can be greater than another. Each has an infinite number of points.
And so within the same class, we may have something greater than infinity, because the infinity of points in the long line is greater than the infinity of points in the short line.
This assigning to an infinite quantity a value greater than infinity is quite beyond my comprehension.
This is one of the difficulties which arise when we attempt, with our finite minds, to discuss the infinite, assigning to it those properties which we give to the finite and limited; but [78] this I think is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another.
To prove this I have in mind an argument which, for the sake of clearness, I shall put in the form of questions to Simplicio who raised this difficulty.
I take it for granted that you know which of the numbers are squares and which are not.
I am quite aware that a squared number is one which results from the multiplication of another number by itself; thus 4, 9, etc. , are squared numbers which come from multiplying 2, 3, etc. , by themselves.
The products are called squares and the factors are called sides or roots.
On the other hand, those numbers which do not consist of two equal factors are not squares.
Therefore if I assert that all numbers, including both squares and non-squares, are more than the squares alone, then I am speaking the truth.
How many squares there are? One might reply that there are as many as the corresponding number of roots, since every square has its own root and every root its own square, while no square has more than one root and no root more than one square.
But if I inquire how many roots there are, it cannot be denied that there are as many as there are numbers because every number is a root of some square.
Then there are as many squares as there are numbers because they are just as numerous as their roots, and all the numbers are roots.
Yet at the outset we said there are many more numbers than squares, since the larger portion of them are not squares.
Not only so, but the proportionate number of squares diminishes as we pass to larger numbers.
Thus up to 100 we have 10 squares, that is, the squares constitute 1/10 part of all the numbers; up to 10000, we find only 1/100 [79] part to be squares; and up to a million only 1/1000 part; on the other hand in an infinite number, if one could conceive of such a thing, he would be forced to admit that there are as many squares as there are numbers all taken together.
We can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and finally the attributes “equal,” greater," and “less,” are not applicable to infinite, (33) but only to finite, quantities.
When therefore Simplicio introduces several lines of different lengths and asks me how it is possible that the longer ones do not contain more points than the shorter, I answer him that one line does not contain more or less or just as many points as another, but that each line contains an infinite number.
Or if I had replied to him that the points in one line were equal in number to the squares; in another, greater than the totality of numbers; and in the little one, as many as the number of cubes, might I not, indeed, have satisfied him by thus placing more points in one line than in another and yet maintaining an infinite number in each?
So much for the first difficulty.