REMARKS «» BARROW'S EUCLIB. g£) 



is objectionable upon the same principle as the first step of 

 the demonstration. The next step is in these words: 

 " Therefore as often as 1 is in AB, so often is 1 in BA." 

 But this does not appear to be the most natural and obvious 

 inference from what has been previously attempted to be 

 proved ; for, if it had been satisfactorily shown, that 1 is 

 contained as often in B as A in A B, and that 1 is contained 

 as often in B as A in BA, the natural inference it appears 

 would be, that A is contained in AB as often as A is con- 

 tained in BA, and so finally AB=BA. 



From the objections here stated the following demon* 

 atration is easily derived, which is submitted to the conside- 

 ration of the lovers of geometrical accuracy with the 

 greatest humility, as seeming to afford a more satisfactory 

 proof of the proposition thau the one above given. 



In this demonstration it may be proper to observe, that, 

 to avoid any ambiguity, the sign of multiplication, or x > 

 should be read by the words multiplied by. It has been 

 thought better also, instead of referring to the proposition, 

 definition, or axiom, on which any of the steps in the 

 process depend, to insert these at length. 



Demonstration. 

 Since by Axiom 5 u unity measures every number by New demo* - 

 the units that are in it, that is, by the same number," straUon - 

 therefore 1" measures A, A times; and since by the first part 

 of Axiom 7, "If one number multiplying another pro- 

 duces a third, the multiplier shall measure the product by 

 the multiplied ;" therefore B shall measure AxB, A times; 

 hence I shall be as often in A, as B in A x B : but by Pro- 

 position 15, if 1 measures A as often as B measures AxB, 

 then 1 shall measure B as often as A measures AxB, 

 or 1 shall be as often in B, as A in AxB: Again by 

 Axiom 5, as above quoted, 1 measures B, B times, and by 

 Axiom 7, A measures Bx A, B times, therefore 1 shall be 

 as often in Bas A in Bx A; but it was shown above, that 1 

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

 A is inBxA, so often is A in AxB: but by Axiom 4 

 u Those numbers, of which the same number, or equal 

 numbers, are the same parts, are equal amongst them- 

 selves ; M therefore B x A is equal to Ax B. W. "W. D. 



IX, Account 



