REMARKS ON BARROW'S EUCLID. ggj 



plied by B, and then divided by A, would give B also, it 



... BxA AxB . . 



would be — - — ~ — - — , or BxA=AxB: hence it ap- 

 A A v 



pears, that the latter part of this axiom virtually implies the 

 truth of the 10th proposition, and is therefore objection- 

 able on the same grounds as the 23d definition. 



Axiom 8. 

 " If one number measure another, that number by Axiom 8. 

 which it mcasureth shall measure the same by the units 

 that are in the number measuring, that is, by the number 

 itself that measures." 



A A 



This axiom implies, that if — - = C, then - T = B. Now Remarks. 



this is really more of a proposition than an axiom. By 

 the former part of the last axiom it may indeed be inferred, 



that, since ~-=C, Amustbe=Cx B; because — - — =C; 



but, as it has before been shown, it by no means follows because 



— -r — =C, that therefore - a, =B. This axiom therefore 



is objectionable upon the same grounds with the last, 



Axiom 9. 



u If a number measuring another, multiply that by Axiom 9. 

 which it measureth, or be multiplied by it, it produceth 

 the number which it measureth. 



This axiom implies, that, if a number A measures another Remarks, 

 number B by a third number C, then A multiplied by C, or 

 C multiplied by A, gives the same product B; that is to 

 say, this axiom implies the truth of the 16th proposition, 

 and is therefore objectionable on the grounds before stated. 



Proposition 16. 

 As there has been frequent occasion to refer to this pro- Proposition 16 

 position in the preceding remarks, it may not be improper " as eng^edtuo 

 to observe here, that it is one of ihose which has engaged mnny eminent 

 the attention of several eminent mathematicians of the pre- 

 sent day, and among others the celebrated Legendre, who, 

 in his u Essai sur la Theorie des Nombres" has given a 



demonstration 



mathematici- 

 ans. 



