382 



REMARKS ON B ARROW'S EUCLI&. 



demonstration of the same; from which it may be con- 

 cluded, that Mr. Legendre himself did not consider Euclid's 

 demonstration of this proposition as perfectly satisfactory. 

 Indeed it must be confessed, that in Euclid's demonstra- 

 tion, as given by Dr. Barrow at least, there is an air of 

 obscurity, which renders it difficult to be understood. For 

 the satisfaction of such of your readers as may not be in 

 possession of an edition of Euclid containing the 7th book, 

 it may be proper here to give both the enumeration and 

 demonstration of this proposition, as they are found in 

 Dr. Barrow. 



Proposition. 



a 



Fioposition 16. " If two numbers, A, B, mutually B 4 A3 



multiplying themselves produce any num- A3 B 4 



bers AB, BA; the numbers produced, AB 12 BA 12 

 AB and B A, shall be equal the one to the other." 



Euclid's de- 

 monstration. 



JReinarks. 



Demonstration. 



For because AB=AxB (a) therefore d 15 clef. 7 

 shall 1 be as often in A, as B in A B, (b) b 15 7 



and by consequence alternately 1 shall be as c 4 ax. 7 

 often in B as A in AB, But because BA=BxA, (a) 

 therefore shall 1 be as often in B, as A in B A ; therefore 

 as often as 1 is in A B, so often is 1 in B A ; and (c) so 

 AB=BA. W.W. D. 



With respect to this demonstration it must be observed, 

 that the attentive student meets with a difficulty in the very 

 beginning; for why does it follow, because AB=Ax B, 

 that 1 shall be as often in A as B in AB? That AB=Ax B 

 is an identical proposition, and implies no more than that 

 A multiplied by B is equal to A multiplied by B, from which 

 no inference can be drawn. The next step of the demon- 

 stration, namely, " And by consequence alternately 1 shall 

 be as often in B as A in AB, is deduced from the preceding 

 by virtue of the 15th proposition, which proves, that, if 

 1 be contained in B as often as D'is contained in E, then 1 

 is contained in D as often as B is contained in E. The de- 

 monstration proceeds with, u but because BA=BxA, 

 therefore shall 1 be as often in B, as A in BA." Now this 



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