Mr. T. S. Davies's N^otes on Geometry and Geometers. 31 



of mathematical treatises which he was in the habit of taking 

 along with him when he set out on his annual excursions into 

 the country. The results of his study are the two manuscript 

 volumes already noticed, both of which contain numerous in- 

 teresting extensions of the use of the circle in geometrical con- 

 structions, and many examples of the highest ingenuity in its 

 application. No. I., as Mr. Davies terms it, or No. VIII. ac- 

 cording to Mr. Swale's enumeration, commences wath the division 

 and subdivision of lines, the di^^sion of arcs of circles, the drawing 

 of common tangents, and finding proportionals. He then proceeds 

 to the description of polygons, their inscription in circles and in 

 each other, the inscription and circumscription of circles in tri- 

 angles, &c., to many of which four or five different methods of 

 construction are given. "Fertility in resource is increased 

 power" was ever his favourite maxim, and throughout the whole 

 of his writings he has endeavoured fully to illustrate its truth. 

 No. II., or more correctly No. IX., is by far the most curious and 

 valuable. He commences with the problems of having " given 

 three or four tangential circles inscribed in a given circle, to de- 

 scribe another circle that shall touch the given one and any two 

 of the inscribed circles," and after having given elegant con- 

 structions to these, he proceeds to the construction of the various 

 cases of the Apollonian problem of tangencies, with the exception 

 of that where a tangent circle to three given circles is required to 

 be described, the enunciation only of which is given. Professor 

 Davies regrets this circumstance, owing to the " probability that 

 had Mr. Swale succeeded in this, it might have opened the road 

 to a new system of treatment of the general problem;" — but if 

 we are to be guided in our conjectures by what is already done 

 in the MS. with respect to the subsidiary problem of "de- 

 scribing through a given point, a circle which shall touch two 

 given circles," to which the case of a tangent circle to three given 

 ones may always be reduced, we may safely infer that ]\Ir. Swale 

 had obtained no clue to any essentially new process for the general 

 case left unconstructed. In his valuable paper " On Tangential 

 Circles" printed in the first number of the " A])ollonius," he 

 considers the cases when the three given circles " touch each 

 other," and when they are " unyhotv posited ;" both cases are 

 constructed solely from the properties of what arc now termed 

 the poles of similitude, the first agreeing in principle with the 

 construction given in Anderson's " Variorum Problematum 

 Practice," of which a translation accompanies the construction 

 in the " Apollonius," and the second reducing it by means of the 

 same properties to the subsidiary problem previously noticed, 

 The enunciation and construction of the subsidiary problem itself 

 are given in the manuscript, aa follows :— 



