Mr. T. S. Davies's Notes on Geometry and Geometers. 207 



from Simpson's Exercises, the Mathematical Repository, &c., 

 together with a variety of extracts in prose and verse agreeably 

 to the indications of the title-page ; but nothing appears to merit 

 particular notice unless it be a short discussion of the diflFerent 

 cases of the problem "to determine P in a lineMN of any order, 

 so that drawing the tangents PV, PT, to two given circles (A), 

 (B), they shall have a given ratio." The required point P is 

 veiy elegantly shown to lie in the intersection of the given line 

 with a ffiven circle, which Mr. Swale appropriately terms " the 

 circle of tangential ratio," and which obviously becomes the circle 

 of similitude when the given ratio is that of the radii of the two 

 given circles. The two MSS. numbered VIII. and IX. have 

 already been noticed by Professor Davies as volumes I. and II. 

 on the Mascheronian Geometry, and need not be further noticed. 

 No. X. is a short paper fully written out for insertion in the 

 third number of the " Apollonius ; " it contains fmr construe- 

 tions and demonstrations to the problem of having " a point P 

 and two parallel lines AQ, BR, given in position, to determine 

 the position of a line PQR, of section, making the rectangle, 

 sum of squares, or differences of squares, of the segments AQ, 

 BR, cut off from the hues given in position equal a given square 

 (y2) ;" which are designated by Mr. Swale as "diversified solu- 

 tions to the same problem ; or brief introductory Lessons for 

 young Geometricians." The paper is prefaced by a motto which 

 inculcates his favourite dogma, that "variety of method, orferti- 

 bty of resom-ce, is increased ^jotm-," and appears, with one excep- 

 tion, to be the only existing manuscript fully prepared for the 

 press. 



The eleventh and last volume in my possession is divided into 

 two parts; the first of which (pp. 5-87) is devoted to the solu- 

 tion of diophantine and other " algebraical inquiries " selected 

 from various authors, and the second part (pp. 281-338) to the 

 consideration of numerous original and selected problems under 

 the title of " Geometrical Amusements, to sooth an incurable 

 despondency." Pages 298-308 contain a discussion of the pro- 

 blem " to determme a point P, in AC, the side of a given triangle 

 ACB, such that drawing PQ perpendicular and PR parallel to 

 the base AB, the ratio, sum, difference, rectangle, sum of squares, 

 or difference of squares, of PQ and PR, may be respectively equal 

 to given quantities;" four different constructions and demon- 

 strations being given to each case. The problem partially con- 

 sidercd in MS. No. X., as extended to the cases of the ratio 

 sum, or difference of AQ and BR, occupies pp. 308-316, /o«r 

 different constructions, &c. being given to each of the six cases 

 as m the previous instance. In a similar manner he treats the 

 problem " to draw PQR, through a given point P, to meet AK 



