Mr. T. S. Davies on Geometry and Geometers, 507 



end the notes on the 10. Prop. Page 380 of the English 4'" Edition 

 which he says ought to have preceded the 10*^. it can, it is true, be 

 done without the lO**^ but not so easily. 



" 10. In P. 271. at the bottom, he sayes ' the same objection oc- 

 ' curs again in the IS'** Prop, where it remains in its full force, for 

 ' though it be allowed that there are some equimultiples of C and E 

 ' and some of D and F such, that the multiple of C is greater than 

 ' the multiple of D but the multiple of E not greater than the multiple 

 ' of F ; yet it is not demonstrated nor in any sort shewn, that other 

 ' multiples of these quantities cannot be taken such that the very con- 

 ' trary shall hajjpen.' No indeed, for the very contrary may often 

 happen ; but this has not the least force against the demonstration 

 Euclid gives of the 13*** Prop, for since it is allowed, that such equi- 

 multiples can be taken as just now mentioned, the Demonstration of 

 the 13**^ remains firm and legitimate, but indeed what he sayes here 

 is so far from having the least appearance of an objection, that the 

 mildest thing [that] can be said of it is, that the author has been in 

 so great a hurry as not to have taken time to consider what he was 

 writing, 



"11. In P. 272. at the foot, he mentions the principle whereby the 

 difficulty he had before spoken of might be obviated, viz : ' that if a 

 ' magnitude of any kind be given, or propounded, there may (or can) 

 ' be another magnitude of the same kind which shall have to it any 

 ' ratio assigned,' or which is the same thing, that unto any three 

 magnitudes two of which are of the same kind, there can be found 

 a fourth proportional. This he sayes I will by no means admit of 

 (tho Euclid himself in 2. Prop. 12. has used it) and in P. 70 of his 

 Elements at the foot he had said that ' this kind of argumentation 

 ' is authorised and adopted by Euclid himself in his twelfth book' as 

 to which is to be observed that Euclid in the 12 Prop. 6 has shewn 

 that a fourth proportional can be found to any three straight lines, 

 from which what he assumes in the 2 Prop. 12 can be legitimately 

 deduced, as is shewn in the note at the foot of the page in the 4*° 

 Edition [and in all subsequent ones the same note is retained] at 

 that proposition, but his using here will not infer that he would 

 have used it before he had shewn the 12. Prop. 6. 



" 12. Near the end of P. 274 the author is for ' entirely rejecting 

 ' the lO'^'and 13"' Propp. of Book 5*^ and everything else founded on 

 ' the Definition of a greater and less ratio, as being of no other use in 

 ' the Elements than to open the way to these important theorems on 

 ' the alternation and equality of ratios' (by the last he means the 22 

 Prop. 5.) ' which may be better demonstrated without them, from the 

 • Definition of equal ratios alone, &c.' but tho the 14*** and 20"» 

 Propp. 5 on which the 16. 22. depend, can be shewn without the help 

 of the lO^'^and 13*** yet since these two last are frequently used both 

 by antient and modern geometers, they ought to have a place in the 

 Elements ; and since they are there, it was proper to demonstrate 

 the 14"> and 20''» by them as it is done without any construction 

 by taking equimultiples. 



" 13. As to what is said concerning a nature or idea (' see P. 273' 



