VOL. VI.] PHILOSOPHICAL TRANSACTIONS. 627 



he pleased, the demonstration had been just as good as now it is, without 

 changing one syllable: That is, it will equally prove the proportion of the semi- 

 diameter to the quadrantal arc, to be what you please : As any may presently 

 see, who does but read over his paper. 



In his second Paper, 



He pretends to confute a theorem, which has a long time passed for truth ; 

 and therefore no more concerns Dr. Wallis, than other men. And it is this, 

 the four sides of a square being divided into any number of equal parts, for ex- 

 ample, into 10; and straight lines drawn through the opposite points, which will 

 divide the square into 100 lesser squares : The received opinion (says he), and 

 which Dr. Wallis commonly uses, is, that the root of those 100, namely 10, 

 is the side of the whole square. Which to confute, he tells us, the root 10 is 

 a number of squares, whereof the whole contains 100; and therefore the root 

 of ] 00 squares is 10 of those squares, and not the side of any square ; because 

 the side of a square is not a superficies, but a line. 



For answer, I say that it is neither the opinion of Dr. Wallis, nor of any 

 other that I know, so far is it from being a received opinion, which Mr. 

 Hobbes insinuates as such, that 10 is the root of 100 squares : For surely a 

 bare number cannot be the side of a square figure : Nor yet, as Mr. Hobbes 

 would have it, that 10 squares is the root of 100 squares ; But that 10 lengths 

 is the root of 100 squares. It is true, that the number 10 is the root of the 

 number 100, but not of 100 squares: and that 10 squares is the root, not of 

 100 squares, but of 100 squared squares : Like as 10 dozen is the root, not of 

 100 dozen, but of 100 dozen dozen, or squares of a dozen. And as there you 

 must multiply, not only 10 into 10, but dozen into dozen, to have the square 

 of 10 dozen; so here 10 into 10, which makes 100, and length into length, 

 which makes a square, to obtain the square of 10 lengths, which is therefore 

 100 squares, and 10 lengths the root or side of it. But, says he, the root of 

 100 soldiers, i^ 10 soldiers. Ans. No such matter: For 100 soldiers is not 

 the product of 10 soldiers into 10 soldiers, but of 10 soldiers into the number 

 10 : And therefore neither 10, nor 10 soldiers the root of it. So 10 lengths 

 into the number 10, makes no square, but 100 lengths; but 10 lengths into 

 10 lengths makes, not 100 lengths, but 100 squares. 



So in all other proportions : As if the number of lengths in the square side 

 be 2; the number of squares in the plain will be twice two, because there will 

 be two rows of two in a row : If the number of lengths in the side be 3 ; the 

 number of squares in the plain, will be 3 times 3, or the square of 3 : If that 



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