392 ANTIENT METAPHYSICS. Book V. 



' one another *,' where the accident or quality of belpg equal to one 

 another, is praedicated of things equal to the fame thing. As there 

 is fomc dilpute about this axiom, before I come to inquire concerning 

 ic, I muft obferve, that, though the greater part of Euclid's axioms 

 relate to equality, yet he has given us no precife definition of it ; 

 which, however, was the more necefTary, that there is equality in 

 number^ as well as in itiagnitudc^ but very different one from another. 

 It is true, there is one of the axioms which lays it down, 'That 

 * things which are congruous, or fitting to one another t> are equal.* 

 But, fuppofing this to have been among the definitions, which it ought 

 to have been, it is not fufficiently precife; for congruous, ox fitting, are 

 terms of too indefinite a meaning for a mathematical definifion. The 

 true definition, therefore, of equality of magnitude^ with which only 

 geometry is concerned, is that given by Appollonius %, ' Magnitudes,' 

 fays he, ' are equal, which occupy the fame fpace.' And which de- 

 finition Dr Simpfon has very properly adopted. 



This being premlfed, the evidence of this firfl: axiom is eafiiy 

 explained ; for, let us fuppofe that the magnitudes are not equal to 

 one another, there would be a contradidion to the hypothefis, appa- 

 rent 



* I cannot help obferving a little inaccuracy In the exprelTion of this axiom, fuch as 

 is hardly ever to be found in Euclid, not, as far as I know, in any other inftance. It 

 fhould have been expreffed thus, * Things, which are each equal to the fiime, are equal 

 < to one another ;' for two or more things, taken together, (not feparately^) maybe 

 equal to a third thing, and yet not equal to one another. I am the more furprifed at 

 this inadvertency, and in the firfl: axiom too, that I obferve that, in his propofitions, 

 where he compares two things to other things, in point of equality, he exprefles, that 

 it is each to each, Uxn^ci Uxn^cA. See the fourth propoHtionof the firfl book. 



•f The Greek is, t« t!px^/^o^»vTx i-k uh>.n>-u, t<rx ctXXnXtii ic-n, which Dr. Simpfon has 

 thus tranflated, with more regard to the meaning tVan the words: * M.ignitudes, which 



* coincide with one another, that is, which exadly fill the fame fpace, are equal to 



* one another.' 



t See Proclus, in his Commentary upon the firil book of Euclid, page 70. 



