540 Mr. T. S. Davies on Geometry and Geometer's, 



dered by the event spoken of, destroyed the whole of them him- 

 self; and that these re-entries are those of some scraps which 

 accidentally escaped, and which, for some interest or other 

 attached to them, he thought worth preserving. 



The earliest in respect to date is a comprehensive development 

 and application of the geoaietry of the compasses, imder the 

 appropriate title of " Mascheronian Geometry." It was " begun 

 in Teb. 1829: revised, improved, and extended, Feb. 1830" (as 

 stated in a MS. note) ; and the latest date is April 23, 1830. 

 This treatise is in all respects a very remarkable one. From a 

 list of the books which appear to have constituted his entii'e 

 library at the time, it would seem that Mascheroni's own work 

 was not before him ; and a comparison of his solutions of the 

 several problems which are common to both with those of the 

 originator of the system, will prove that, even had the work been 

 in his hands, it could have afforded him but little assistance. 

 After the very simplest problems (and sometimes even in these) 

 the methods employed are generally different; INIr. Swale's 

 being more concise, direct and inventive than Mascheroni's. 

 ]\Iany of the problems which he constructs under these restricted 

 conditions are sufficiently difficult when only subjected to the 

 proper restrictions of the ancient geometiy ; and several of these 

 constructions, when divested of this restriction, constitute better 

 solutions of the same problems than any \\ith which I am other- 

 wise acquainted. Amongst them are constructions of the Apol- 

 lonian problem of tactions. The case of three circles being given, 

 is, however, left blank ; which is the more to be regretted from 

 the great interest which this case has excited amongst the 

 continental geometers, and 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. 



Unfortunately, this method has little to recommend it to the no- 

 tice of geometers besides its being difficult and curious. Had we, 

 indeed, any direct mode of geometrical analysis peculiarly adapted 

 to it, the case would be somewhat different, inasmuch as it might 

 at least be rendered suggestwe with respect to construction under 

 the ordinary conditions. I myself gave a good deal of attention 

 to the subject at one period of my life (indeed about the time 

 that Mr. Swale's MS. is dated) ; but I became convinced that 

 the pursuit was a useless one, inasmuch as it did not admit of 

 any independent analysis, but must consist of the adaptation of 

 methods otherwise previously known to the circumstances pi'e- 

 scribed for the compass-construction. This at least was the 

 only \ iew at which I arrived ; and I am inclined to think it was 

 the process by which Mr. Swale arrived at his constructions. 

 At the same time the ordinary constructions of the same pro- 



