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t 94 SCIENTIFIC METHOD IN PHILOSOPHY 



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 1 oooth : 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 mathe- 

 maticians, 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 



