THE POSITIVE THEORY OF INFINITY 193 



into my Head, which (that I may be the better under- 

 stood) I will propose by way of Interrogatories to 

 SimpliciuSy who started this Difficulty. To begin then : I 

 suppose you know which are square Numbers, and which 

 not ? 



" Simp. I know very well that a square Number is 

 that which arises from the Multiplication of any Number 

 into itself ; thus 4 and 9 are square Numbers, that arising 

 from 2, and this from 3, multiplied by themselves. 



" Salv. Very well ; And you also know, that as the 

 Products are call'd Squares, the Factors are call'd Roots : 

 And that the other Numbers, which proceed not from 

 Numbers multiplied into themselves, are not Squares. 

 Whence taking in all Numbers, both Squares and Not 

 Squares, if I should say, that the Not Squares are more 

 than the Squares, should I not be in the right ? 



" Simp. Most certainly. 



" Salv. If I go on with you then, and ask you, How 

 many squar'd Numbers there are ? you may truly answer, 

 That there are as many as are their proper Roots, since 

 every Square has its own Root, and every Root its own 

 Square, and since no Square has more than one Root, 

 nor any Root more than one Square. 



" Simp. Very true. 



" Salv. But now, if I should ask how many Roots 

 there are, you can't deny but there are as many as there 

 are Numbers, since there's no Number but what's the 

 Root to some Square. And this being granted, we may 

 likewise affirm, that there are as many square Numbers, 

 as there are Numbers ; for there are as many Squares as 

 there are Roots, and as many Roots as Numbers. And 

 yet in the Beginning of this, we said, there were many 

 more Numbers than Squares, the greater Part of 

 Numbers being not Squares : And tho' the Number of 



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