Cliap. VIIL ANTIENT METAPHYSICS. 445 



As to the axioms, they are all founded upon the definitions, being 

 felf-evident propofitions, refuliing from thofe definitions ; for, if it be 

 once admitted that magnitude has a certain ftandard by which it can 

 be meafured, viz. the fpace that it occupies, and that, therefore, one 

 magnitude may be faid to be equal or unequal to another ; from thefe 

 hypothefes, I fay, all the axioms neceflarily follow ; for axioms, as I 

 liave obferved, are only true, becaufe, to fuppofe the contrary, would 

 he a contradidion ; and the contradiction is to fome definition. For 

 example, the axiom, that if two magnitudes arc equal to a third, they 

 are equal to one another, is true ; becaufe, to fuppofe otherwife, would 

 be contradictory to the definition of equality ; for, if the one was great- 

 er than the other, they could not be both congruous to, or occupy the 

 fame place as a third magnitude*. And there is one of Kuclid's axioms 

 that implies this contradidion to a definition ftill more evidently, and 

 it is the I ith, which fays, that all right angles are equal to one ano- 

 ther ; for, to fuppofe the contrary, would be, at once, to deftroy the 

 definition of a right angle. And, as all the demonfi:rations are dedu- 

 ced from definitions, poftulates, and axioms, it follows, of necelTary 

 confequence, that, as Plato fays, the whole fcience of geometry is by^ 

 potheticaL 



The quefiion, then, is, what is to be done to make It a pofit'wc 

 fcience, and to give it a real foundation in the nature of things, not in 

 our ideas only ; for it is not fufficient to explain, as I have done, the 

 feveral hypothefes upon which Euclid proceeds, and to fliow that they 

 are perfedly intelligible, and fuch as are at leaft poffible to exlft ; but 

 it muft be further fhown, in order to make geometry a real fcience, 

 that they do actually exifl: ; for we are not to imagine that, becaufe 

 Euclid has defined the feveral figures which are the fubjeCt of his 

 demonfirations, therefore thofe fig)ires do really exift. A man may 

 define a Hippocentaiir or a Chimaa^a^ and,.from fuch definitions, may 

 draw confequences that are deflionftratively true ; but it will not from 



thence 



* See whatl have faid in explanation of this axiom, page 392. 



