Mr. T. S. Davies's Notes on Geometry and Geometers. 203 



well worthy of transcription ; but the complexity of the requisite 

 diagrams renders it impossible, whilst the enunciations would be 

 unintelligible without them. Indeed one of the peculiarities of 

 Swale^s geometry is the complexity of his diagrams : he almost 

 invariably uses more lines than any other investigator ; but this 

 disadvantage is more than counterbalanced by the elegant sim- 

 plicity of his reasonings, and the vast number of collateral pro- 

 perties which he developes in his processes. In these respects 

 I know of no geometer who has so nearly equalled him as the 

 late Professor Da^nes, whose diagrams have frequently presented 

 such a similarity to those used by Mr. Swale as to lead some of 

 the friends of the latter to supjoose that Mr. Davies had access 

 to these J\ISS. long before he even knew of their existence. The 

 Swale MSS. were not seen by Professor Davies until " October 5, 

 1850," and he had only time to wi'ite the few notes respecting 

 them contained in No. VII. of this series of papei's before his 

 progress was arrested by the hand of death. 



" A Collection of Problems by the Compasses alone " occupies 

 pages 292-332 ; but since they are principally of an elementary 

 character, they need not be further particularized. His son 

 appears to have been very expert in such constructions ; for on 

 page 296, after giving an elegant determination of " the centre 

 of a given circle,^' he adds, " this method, which is more simple 

 and elegant than Mascheroni's, was discovered by J. H. Swale, 

 junior, 19th February 1829." The solution of isolated problems 

 by ordinary geometry is again resumed at page 332, and is con- 

 tinued throughout the remainder of the two volumes. In the 

 page just cited, a theorem occurs which appears to have been 

 " sent to Professor Leybourn, 6th Dec. 1830," but did not find 

 its way into the Repository ; it furnishes a ready proof of the 

 methods of finding the centre of a given circle already instanced 

 in No. VII. by Professor Davies, and has recently been published 

 as Question 378 of the Educational Times. 



The problems on " Tangencies," already mentioned, occur in 

 pp. 383-386 j the fifth and sixth cases heiug fi7-st constructed 

 from the principles of the poles of similitude, and uftenvards 

 reduced to Simpson's principle before stated. The last portion 

 of the volume appears to have been formed from an eai'lier ma- 

 nuscript, since u poiiion of its pages is occupied with the demon- 

 stration of several problems relating to poles of similitude appa- 

 rently deduced from Lawson's translation of Victa^'s Tangeneies, 

 the intei'veuing spaces being filled up with later speculations. 

 In volume III. a few isolated ])roblem8 from the Diary occur, 

 but the principal portion is occuj)ied with the extension and ap- 

 plication of the problems alreaily alluded to as jireparatory to 

 the solution of the problems on inclinations. He here treats 



