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



stitute our first " Elements." In the more extended classes of 

 research, however, this becomes much more embarrassing ; and it 

 is to be regretted that no single work in which the different 

 classes of geometric research are intelligibly defined, can be 

 pointed out. With one more source of difficulty this formidable 

 list will be concluded ; though others, and those not of a minor 

 character, might have been added. 



The great object of the ancient geometers appears to have been 

 the solution of problems ; and hence the investigation of theorems 

 held no importance in their estimation, further than as they were 

 subsidiary to the demonstration of the constructions arrived at, 

 or in the analyses by which those constructions were obtained. 

 Instead, therefore, of investigating the properties of figures and 

 classing them according to any rule (good or bad), only those 

 were recorded that became subservient to some step or other in 

 the construction of a problem. This is strikingly manifested in 

 the seventh book of the Mathematical Collections of Pappus; 

 where we see given as isolated propositions many theorems 

 which form parts of the most beautiful and interesting classes of 

 research that have been yet discovered. That wonderful work 

 of M. Chasles (Aperqu Historique) bears witness to this in almost 

 every page, and it prevents the necessity of my adducing illus- 

 trative examples in this paper. 



It will probably be objected that the arbelon and some other 

 speculations mentioned by Pappus, as well as some of the minor 

 works of the ancients which have reached us, contravene this view 

 of the leading objects of the Greek Geometry. I know of none of 

 those ancient works, however, in which I cannot trace the ulti- 

 mate object to be the solution of some specific problem or class of 

 problems; and so far I see no force in such an objection. As 

 regards any of the sets of properties mentioned by Pappus, we 

 must recollect that he wrote and " collected" long after the period 

 when geometry could be said to " flourish" in the school of Plato 

 — long after the decadence of pure geometry amongst the Greeks. 

 The arbelon is itself, beyond being " pretty and curious," mere 

 geometrical trifling; just the kind of speculation that might be 

 supposed to be indulged in the age when the weak Proclus pre- 

 sided over that once illustrious school. Nothing of this kind 

 appears to have engaged the attention of geometers during the 

 period of Apollonius and Archimedes : even the various curves 

 that were devised by the ancients were not devised for the pur- 

 pose of investigating their properties, but of solving some in- 

 tractable problem by means of them. The conic sections come 

 the nearest to claiming an exemption from this general rule : but 

 though many properties are given by Apollonius, the immediate 

 application of which to constructive purposes might not readily 



