206 Mr. T. T. Wilkinson's Additions to the late 



esting, however, as a fact in the history of this problem, that 

 Mr. Swale had been in possession of his method and its exten- 

 sion so many years before its publication. Pages 82, 83 contain 

 " an extract from one of [his] mathematical manuscript books " 

 relating to the premature death of his friend Mr. William Davis, 

 editor of the JNIathematical Companion ; and a fitting tribute is 

 paid to his memory in a notice of the event, which Mr. Swale 

 forwarded for insertion in the Leeds Mercury of the following 

 week. A few diophantine, dynamical, and other problems occur 

 in the remaining portions of this volume; but none of them 

 ap]Dear to possess much interest, if we except a dissertation on 

 the motion of a ball upon elliptical and triangular billiard tables 

 (apparently suggested by Questions 250 and 270 of the Mathe- 

 matical Repository), which determines the directions "of impul- 

 sion so that the ball may, for ever, pursue the same track " on 

 a triangular table, to be the sides of the triangle of minimum 

 perimeter inscribed in the given triangular table. 



The seventh volume is a bulky octavo, which seems from the 

 repetitions in the paging to have been made up of several smaller 

 manuscripts. It bears the same title as the preceding, and opens 

 with a series of " Lessons " for his sou, amongst which are no 

 fewer than twelve " original methods of dividing a given line in 

 extreme and mean ratio." They bear the date " 19th May 1833," 

 and would seem to have been satisfactory to their author, for he 

 adds, "we have now done justice to this Ancient Problem." 

 Many isolated solutions in this volume contain references to the 

 Geometrical Amusements, and were no doubt intended for 

 "Parts II. and III." of that valuable work; others appear to 

 have been copied from older slips relating to his friend Mr. 

 Nicholson, which, as is said in the Leeds Correspondent, are 

 "now brought forward as a sincere tribute of friendship and 

 respect for the memory of that ingenious Geometrician." In 

 page 82 he acknowledges the receipt of a letter from Mr. Ley- 

 bourn, dated "21st Mayl833," on which he remarks, "Leybourn 

 was one of my early scientific correspondents, ha\ing written to 

 him 38 years. I hope yet to spend a week with him at Bagshot. 

 At the same time I should greet Lowry, another old correspond- 

 ent, and Cunliffe. On such an occasion we should take our 

 harps down from the willows and once more tune them to the 

 cheering songs of science." 



The subject of extreme and mean ratio is again taken up at 

 page 171, and two other methods of division added to those 

 already noticed, after which the solution of some rather difficult 

 surd equations occur which had been sent to him for solution by 

 Mr. Harding. Much of the remaining portion of this manu- 

 script is occupied with solutions of geometrical problems selected 



