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



1st Feb. 1765, short but to the purpose : " Forget not to send 

 me Lawson's Translation of Apollonius concerning Tangen- 

 cies as he improperly names it." The work was in reality a 

 translation of Vieta's attempted restoration of that treatise of 

 Apollonius, and was most imperfectly executed. Many sub- 

 cases were omitted altogether; and the determinations in 

 general were very loosely given, even where given at all. Nor 

 are all the lemmas given by Pappus, as necessary to the solu- 

 tion of the problem employed by Vieta or his translator, — the 

 celebrated vii. 117 for instance. I may, however, refer to a 

 paper printed in the Mathematician, vol. iii. pp. 76-78, for 

 some further remarks on this subject. 



Meanwhile it is proper to keep in view that the restoration 

 of the solution given by Apollonius is not necessarily getting 

 the best solution. Indeed as regards this problem, it is cer- 

 tainly not getting the best. By taking different routes, several 

 foreign geometers have arrived at constructions of great sim- 

 plicity and beauty: for instance, Cauchy, Gaukier, Durrande, 

 Poncelet, Steiner and Gergonne. The last is very remarkable : 

 but there is a total absence of all attempts to fix the deter- 

 minations and discriminate the special sub-cases in all these 

 solutions. In short we have neither a complete restoration 

 nor a completed solution of this one of the Apollonian series 

 of problems. Yet it is worthy of both. 



The commentaries on Jack's edition of the Data, though 

 slight, are certainly correct and interesting; though there are 

 some geometers who still think that a perfect restoration of 

 Euclid's Data has not been accomplished, even by Simson 

 himself. 



" I thank you for what you write about Mr. Jack's Edition of the 

 Data which good or otherwise let me have a Copy by the first occa- 

 sion ; and in the mean time in your next, which pray let be as soon 

 as you can, let me have his translation of the first paragraph in page 

 512 of Gregory, viz. a part of Prop. 73. Also let me know if he 

 has put in the translation of the 2 d Demonstration of Prop. 24. which 

 is a mere paralogism." (Letter*, 10th Dec. 1756.) 



" I thank you for what you give me from Mr. Jack's book, though 

 he has by himself or from others found some thing amiss in these two 

 Propositions, yet he does not speak of them as if he knew wherein 

 the faults lie. I am just obliged to go out so cannot write particu- 

 larly about what he sayes ; only there is not the least occasion of 

 finding what he calls M, because that fourth proportional is already 

 expressly named by TK in the exposition which immediately follows 

 the Enunciation of the Proposition. * * * 



* In a postscript of the same letter he first expresses his intention to 

 " make an Edition of the Euclid in small [word missing, probably ' size'] 

 for the use of schools." 



