Squares decreases in a greater proportion, as we go on to bigger Numbers, for count to an Hundred you’ll find 10 Squares, viz. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, which is the same as to say the 10th Part are Squares; in Ten thousand only the 100th Part are Squares; in a Million only the 1000th: And yet in an infinite Number, if we can but comprehend it, we may say the Squares are as many as all the Numbers taken together.
“Sagr. What must be determin’d then in this Case?
“Salv. I see no other way, but by saying that all Numbers are infinite; Squares are Infinite, their Roots Infinite, and that the Number of Squares is not less than the Number of Numbers, nor this less than that: and then by concluding that the Attributes or Terms of Equality, Majority, and Minority, have no Place in Infinites, but are confin’d to terminate Quantities.”
The way in which the problem is expounded in the above discussion is worthy of Galileo, but the solution suggested is not the right one. It is actually the case that the number of square (finite) numbers is the same as the number of (finite) numbers. The fact that, so long as we confine ourselves to numbers less than some given finite number, the proportion of squares tends towards zero as the given finite number increases, does not contradict the fact that the number of all finite squares is the same as the number of all finite numbers. This is only an instance of the fact, now familiar to mathematicians, that the limit of a function as the variable approaches a given point may not be the same as its value when the variable actually reaches the given point. But although the infinite numbers which Galileo discusses are equal, Cantor has shown that what Simplicius could not conceive is true, namely, that there are an infinite number of different infinite numbers, and that the conception of
