506 Mr. T. S. Davies on Geometry and Geometers. 



Yet whilst I avoid such extended discussions as I should be 

 entangled in by this subject, it is certainly due to Thomas 

 Simpson to state, that all geometers have not entertained, and 

 do not now entertain, that high estimate of Dr. Barrow's 

 treatment of ratio that Dr. Simson did, and which he here so 

 triumphantly quotes. It is satisfactory to be able to state my 

 own view in the felicitous language of a dispassionate judge, 

 and of one whose philosophical acumen in such matters needs 

 no praise of mine, Professor Powell. In a paper read before 

 the Ashmolean Society of Oxford in 1836, he thus expresses 

 himself: — " Dr. Barrow, in his celebrated Mathematical Lec- 

 tures for 1666, has treated the whole subject of ratios and 

 proportion in the most copious and elaborate manner, but, as 

 appears to me, with more learning than perspicuity; he is 

 extensively occupied in examining and refuting such objec- 

 tions as those just adverted to; and in doing so seems more 

 explicit and satisfactory, than in any attempts to elucidate di- 

 rectly the doctrine itself on real philosophical principles. In 

 point of fact, in the midst of his very extensive dissertations, 

 it is far from an easy matter to discover what is his own idea 

 of the nature of ratios ; and when it is developed, it is by no 

 means clear wherein it substantially differs from the views of 

 some of his opponents" (p. 14). 1 not only fully concur in 

 this view, but I believe that much of the praise bestowed upon 

 Dr. Barrow's Lectures on this subject, has been bestowed for 

 fashion's sake, and not from those writers having read and 

 unravelled that complicated series of discussions. 



" 9. In P. 379 of the 4'" English Edition it is said That in order 

 to prove what is affirmed in the Demonstration of 10. Prop. 5 in the 

 Greek Edition viz : that A cannot have a less ratio to C than B has 

 to it, it ought to have been shewn that if the ratio of A to C be 

 greater than that of B to C. and taking any equimultiples of A and B, 

 and any multiple of C, the multiple of A is ever greater than that 

 of C whenever the multiple of B is greater than that of C ; but this 

 is not done in the 10 Prop, but would easily follow from it and can- 

 not without it be easily demonstrated. The author of the notes on 

 the 8^° [4'° ? the 8vo was not published till after Simpson's death] 

 Edition sayes, that this point ought to have been cleared up by pro- 

 positions antecedent thereto and independent thereupon (I suppose 

 he means upon the lO''*. Proposition, else I do not understand him) 

 and adds that the 8'^^ Prop, seems the proper place for doing it. 

 then he gives an account of what is proved in Prop. 8. and next 

 gives a demonstration that the ratio of A to C cannot be less than 

 the ratio of B to C. but in the very first words of it, he supposes that 

 A is greater than B, which is the thing that is proved in the 10'^ 

 Prop, and therefore he has done it by the help of the 10**^, tho, a 

 little before, he had said it should be done by propositions indepen- 

 dent upon it ; his demonstration is exactly the same with that at the 



