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



AL, given in position, in Q and R, so that the ratio, sum, differ- 

 ence, and rectangle of AQ and AR, may be respectively equal to 

 given quantities;" each successive variation unfolding new pro- 

 perties of the illustrative diagrams, and affording additional 

 proofs of the extensive powers and ingenuity of their author. A 

 question from the ^lathematician proposed " about 1750, by John 

 Turner, who had a school at Heath, near Wakefield, Yorkshire," 

 is considered on pp. 327-8 ; and a case of "ApoUonius on In- 

 clinations " appropriately closes the volume, which, from internal 

 evidence and the title affixed to the geometrical portion, most 

 probably contains the latest efforts of Mr. Swale's untiring mind. 



I have been the more particular in describing the contents of 

 these manuscript volumes, partly in consequence of their number 

 and extent, but mostly from the extreme improbability of even a 

 tithe of them being ever given to the public. Portions occur 

 here and there, like oases in a desert, which might be selected 

 for publication in a separate form did the taste for the ancient 

 geometry warrant such a ]u-oceeding ; l)ut since such is not likely 

 soon to be the case, the probability is that an immense mass of 

 Mr. Swale's speculations must ever I'cmain in an incomplete and 

 unprofitable condition, — a notable monument of misdirected 

 energy and useless expenditure of valuable time. His systematic 

 researches on tangencies, maxima and minima, the inscription of 

 poljrgons in circles and in each other, printed in his Apollonius, 

 afford convincing proofs of how much he was capable when his 

 extensive powers were directed to regular subjects of inquuy ; 

 for the elegant methods of research employed in these papers, 

 and the simplicity and beauty of the results obtained, must ever 

 command the admiration of geometers. His fertility of inven- 

 tion and originality of conception were inferior to those of no 

 contemporary geometer; and had he directed those energies to 

 systematic inquiries which he expended in the solution of some 

 thousands of isolated and comparatively unintei'csting questions, 

 he might have systematized scattered topics or originated new 

 theories, in which he would have rivalled Carnot in transversals, 

 Davies in spherics and porisms, or Chasles in anharmonic ratio, 

 and have secured for his owti name a permanent place in the 

 history of modern geometry. 



What will ultimately become of the MSS. is of course beyond 

 conjecture. That they will be almost religiously preserved by 

 his son during his life no one will doubt who is acquainted with 

 the profound veneration he entertains for the memory of a kind 

 and indulgent father ; but when we call to mind that a second 

 generation has deliberately burnt the ]\ISS. left by the Stewarts, 

 and that already much of Mr. Swale's correspondence, ik.c. has 

 been destroyed by an accidental fire, it may not be considered 



