VOL. VI.] PHILOSOPHICAL TRANSACTIONS. 620 



fraction -^. Again, if 14^ be the greatest root, what shall be the remaining 

 fraction? Doth he take the root of 200 to be more than 14vV by some further 

 remaining fraction? If so, he should liave told us what that fraction is; for -^ 

 it is not, this being part of his greatest root 14-V- But if we should allow, as 

 I think we must, that by the greatest root he means sometimes 14t-V, some- 

 times 14, that is, if we allow him to contradict himself, yet how comes he by 

 the fraction -fV ? For -^ is too much, the square of 14-jV being more then 200, 

 as by multiplying 14-^ into itself will appear; which destroys his whole design; 

 for 14, multiplied into 14-iV, will not make 200, but I98; contrary to his rule. 

 But further, it is so gross a mistake to make 200 the square of 14-^, that every 

 apprentice boy that can but multiply whole numbers and fractions, could have 

 informed him better, who would first have reduced the fraction to smaller 

 terms, putting 14-f for 14-rV, and then multiplying 142- into itself, would have 

 showed him, that the square of 14-^^, that is 14-^ multiplied into itself, is not 

 200, but 204VV- 



But the root of 200, is the said number \0^/1, \\h\ch is less than 14-fV, and 

 greater than 14-V* the square of that being somewhat more than 

 14^ 200; and of this somewhat less; but either of them within an unite 

 14^ of it. 



But this second proposition is, as I said, contradicted by his third, 



56 which makes the square of 14-V to be 100^'^, by what computation, 



14 we shall see by and by; and then finds fault, that this and the for- 



4 mer do not agree. But it is no wonder they should disagree, when 



4 both are false. The same square (says he) calculated geometrically, 



-^ consists (by Euclid 2, 4) of the same numeral great square 1 96, and 



of two rectangles under the greatest side 14 and the remainder of 



204^V the side, and further of the square of the less segment; which alto- 

 gether make 200^. He might have learned to reckon better; but 

 let us see how he makes it out. As by the operation itself (says he) appeared 

 thus: The side of the greater segment is 14-iV (this was but now the side of the 

 whole square, how^ comes it now to be but the side of the greater segment ?) 

 which multiplied into itself (says he) makes 200 : (no, but 204^ :) The pro- 

 duct of 14 the greatest segment into the two fractions -^v is 4, and that added 

 to 1 96 makes 200 : (if by two fractions -yV:. he mean, as he ought by his rule, 

 the fraction 4 twice taken, or the double of it, it will be not 4, but 8, and this 

 added to 196 makes 204; but all this he puts in his pocket, for it comes not 

 into account at all.) Lastly, the product of -jV into -yV? or 4- into -f is ^; 

 which with the first 200 makes 200V^ : But he forgets himself, for his lesser 

 segment was not -rV> but -jV ; he should therefore have said -jV into -^v, or 4. 



