380 3t EM ARES OS BARROW'S ECCJATJ. 



Remarks. This definition seems to be objectionable on this ground, 



that it defines a number A, to measure another number B y 

 by a third number C, when either C multiplied by A, or A 

 multiplied by C, produces the number 13. Now the possi- 

 bility, that Cx A can be equal to AxC forms the subject 

 of the 16th proposition of the very book to which this de- 

 finition is prefixed* To say the least, therefore, this de- 

 finition iS out of order: and as Euclid does not appear to 

 have made any use of it, till after the 16th proposition, so 

 certainly it ought not to have been given till the truth of the 

 proposition virtually implied in it had been demonstrated; 

 that is to say, till it had been proved, that C multiplied by 

 A is equal to A multiplied by C, to which proposition it 

 might have formed a corollary. 



Axiom 7. 



Axiom 7. u If one number, multiplying another, produce a third, 



the multiplier shall measure the product by the multiplied; 

 and the multiplied shall measure the same by the multi- 

 plier." 



Remarks. The first part of this axiom is admissible, since it only 



implies, that, if any number, A, be first multiplied by any 

 other number, B, and then divided by the same number, 

 B, the quotient will be A, — a truth which is evident from 

 the opposite effects of multiplication and division. The 

 latter part of this axiom appears to be objectionable, for it 

 does not, like the former part, first suppose an operation 

 to be performed upon a number A, and then the effect of 

 that operation to be done away, or withdrawn by another 

 operation of a directly opposite nature; for though by this 

 latter part it is required to multiply A by B as before, yet 

 it is not required afterward to divide by B, but by A : and 

 though it may be an obvious truth, that A first multiplied 

 by B. and then divided by B, will give A; yet it is by no 

 means so obvious, that A multiplied by B, and then di- 

 vided by A, will give B, for here the operations of multi- 

 plication and division are by different numbers. By the 

 former part of this axiom, if B be first multiplied by A, 

 and then divided by A, the result will be 13; and if the 

 latter part of it wtre self evident , namely, that A multi- 

 plied 



