Mr. T. S. Davies on Geometry and Geometers. 287 



The charge most commonly made against the science is, the 

 total absence of general methods of research, both as respects 

 construction and demonstration. It is alleged, that of two pro- 

 perties of a figure intimately related as to their subject matter, 

 the demonstration of the one furnishes no clue to the demon- 

 stration of the other ; and that the most elegant construction of 

 a problem fails to facilitate the construction of one nearly kin- 

 dred to it — often, indeed, of a converse problem. "All is iso- 

 lated," it is said ; " and it rather requires a certain kind of hap- 

 hazard dexterity of mind than the application of general methods 

 to make an able geometer. Tn the coordinate geometry, on the 

 contrary, we can always depend upon obtaining a solution, since 

 we can always reduce the conditions into the form of equations, 

 which only require the ordinary resources of algebraic transfor- 

 mation to complete the inquiry." 



No doubt there is a certain degree of truth in this, but there 

 is yet a greater degree of misapprehension. Still, the inference 

 being made from the general writings of geometers, and that too 

 from the survey which an unpractised mind is obliged to take, 

 even the misapprehension is pardonable. The brevity with which 

 geometers put down their steps (consisting only of what con- 

 structions they make in the individual case before them, and 

 the relations which successively result amongst the parts of the 

 figure), without the slightest reference as to why they adopted 

 their special method, tends veiy much to justify the opinion to 

 the mind of the uninitiated, that the ancient geometry is a sy- 

 stem of special expedients, each adapted to the individual case, 

 like the solution of an enigma, and the whole incapable of 

 reduction to any general principles. 



There is also another very plausible ground for the inference. 

 It cannot be denied that nearly all geometers, however much 

 they may add to the details of the science in the way of theorems 

 or problems, do yet pursue it merely as a technical system. 

 Their only ambition is " to discover new truths," to make mere 



to the Philosophical Magazine, there is nothing on the face of the ahove auto- 

 graph of Professor Davies to indicate with certainty, or to afford anything 

 like a conclusive inference, that he intended it to occupy its present posi- 

 tion. I am responsihlc for the title given to it. But, even if its original 

 destination be doubtful, it may with great propriety form part of this set 

 of articles. The manuscript now forwarded is a portion of a longer auto- 

 graph of Mr. Davies, which I have divided into two portions, thinking that 

 such a form would be more convenient for publication. When this has 

 appeared m print, I shall forward the other part for insertion in this ad- 

 mirable Journal. 



" Jambs Cockle. 

 "2 Pump Court, Temple, 

 December 20, 1851."] 



