Mr. T. S. Davies on Geometry and Geometers. 205 



a series of itself, I extract from this letter only what relates to 

 subjects mooted in the preceding one. 



" Your opinion about Kirkby's Ultimators I believe is very just ; 

 If Sir Isaac Newton had made use of the limits of ratios he had done 

 much better than he has done by bringing in Ultimate ratios which 

 I think is a very improper form, as in most cases there is no such 

 thing as an Ultimate ratio. 



-* -* * * 



" I was once thinking this Summer to have written for paper to 

 print the Sectio Determinata, which as you observed might have 

 helped the sale of the Loci plani, but all I have gotten done is to 

 revise the first of these books by the assistance of my good friend 

 Mr. William Trail who is Professor of Mathematics in the new Col- 

 lege of Aberdeen, but who has been staying here all this summer with 

 his uncle Dr. Trail Professor of Divinity, he has copied over for me 

 a few sheets, I believe but one or two at most, concerning the limits 

 of ratios from which the rules for finding fluxions are geometrically 

 deduced, and he also copied as short a paper concerning the elemen- 

 tary properties of Logarithms which I think have not yet been accu- 

 rately treated of ; if I find myself in tolerable health next Spring, for I 

 am afraid how I shall stand thro' the winter, I design to annex these 

 two small things to the Sectio Determinata and print them ; but if 

 God see fit to determine otherwise by my want of health, I design 

 to leave the care of printing them to Mr. Trail who is a worthy good 

 young man and an Excellent Mathematician for his age ; and I de- 

 sign to do the same with respect to the Porisms if I can get them 

 ready to be put into his hands." 



This letter is marked on the back with a memorandum in 

 Nourse's hand, "answered April 16, 1768;" and as Simson 

 lived only till the 1st of October, it is probable that he never 

 wrote in reply. 



The complaint which Simson makes of the ancient geo- 

 metry being "worn out," was then only too true in respect to 

 academic history : yet amongst non-academic mathematicians 

 in this country, of whom Dr. Simson seems to have known 

 little, nothing could be less accurate. I find that even in 

 Cambridge, a considerable degree of patronage was afforded 

 to Simson's Euclid ; for in the letters of Dr. Robert Smith, 

 there is notice of a remittance of 28/. 15s., dated April 4, 1756, 

 and another respecting 10/. 16s. on account of this work. We 

 ought also to recollect how very extensively the geometrical 

 writings of Thomas Simpson had been diffused ; and the pe- 

 riodicals devoted to mathematics (as the Lady's Diary, the 

 Gentleman's Diary, the Mathematician, and others) will show 

 how strongly geometry prevailed amongst non-academic men 

 in those days, and the eminent powers of investigation pos- 

 sessed by those geometers. It cannot be denied, however, 



