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



" Problem. — A point V, and two circles^ radii AT, BV, are given 

 in position ; to describe a circle through P, that shall touch the 

 given circles. 



" Construction I.— Draw the tangents PM, PN ; take AH : HB 

 = AIM : BN, and AI : IB = PM2 : PN^ ; to the circle centre I and 

 radius a fourth proportional to HB, HI, BN ; draw the tangent 

 PR, and let the direction BH meet the circles in K, L ; then 

 PL, PK, will meet them in T, V, the required points of contact. 



"Construction II.— Take AH : HB = AT : BV; inflect the tan- 

 gent HR (in the arc through H to centre P) to I and K ; to 

 which centres and radius IH, describe arcs intersecting in Q : 

 the circle through the points P, Q, to touch the circle AT, is the 

 one required." — {MS. pages 113-4.) 



The first of these constructions is identical with that given by 

 ]\I. Cauchy in the Correspondance sur I'Ecole Poly technique, 

 vol. i., a translation of which may be seen in Leybourn's 

 " Ladies^ Diaries," vol. iv. pp. 269, 270, wliose process has been 

 elegantly extended to the general case of tangencies by " Cen- 

 turion" in No. 1154 of the Mechanics' Magazine, and M'hich 

 again is almost identical with a construction given by Mr. Swale 

 himself in MS. vol. ii. p. 384. The second construction is de- 

 rived from a discussion of the tangencies contained in pp. 383-6 

 of the volume just cited, where the whole are most ingeniously 

 reduced on Simpson's principles (Select Exercises, Prob. 57), 

 to the subsidiary problem of determining '■' a point in a right 

 line given in position, such, that lines drawn thence to two given 

 points may have a given difference." The remainder of the 

 volume is occupied with the construction of numerous other 

 problems relating to the intersection of circles, tangents to them 

 from given points or in given ratios, many of which are equally 

 curious and interesting. ' A remarkably neat construction of the 

 problem, "to desci-ibe a circle that shall bisect the circumferences 

 of two given circles, centres A and B, and have a tangent from a 

 given point D of a given length P," is given in page 141 * ; and 

 also the construction of a fourth circle "to bisect the circum- 

 ferences of three given ones" in page 144, which has been pub- 

 lished as Quest. 343 of the "Educational Times." His objects 

 throughout appear to have been to extend and diversify Mas- 

 cheroni's methods, for he remarks at the foot of a construction 

 which closes the volume, " I find this is similar to Maschei'oni's," 

 and in these respects he has succeeded to a greater extent than 



* Construction. — Posite the diameter FG of the circle (A), and the ra- 

 dius BL of the circle (B) ])erpendicular to the direction AB, which meets 

 the circle FGL in P and Q ; inflect the line P (in the arc through D to 

 centre Pj to I and K ; to which centres and radius ID describe arcs inter- 

 secting in R : the circle PQR is the one required. 



