Mr. T. S. Davies on Geometry and Geometers. 525 



strike the mind, yet so many of them are subsidiary to the de- 

 monstration of properties which have that undoubted purpose, as 

 to require little concession on this point. Besides, my remarks 

 are more immediately made in reference to the propositions of 

 plane geometry ; and I think we may infer that if such classes of 

 properties had been investigated, the good taste and judgement 

 of Pappus would have led him to substitute them for the arbelon 

 at least*. 



* It is usually stated that the several treatises enumerated by Pappus in 

 the celebrated preface to his seventh book, " were written with a view of 

 facilitating the study of the geometrical analysis." High as is the authority 

 with which this opinion is enforced, I can only adopt it in a very modified 

 sense of the terms employed — and in a sense too, which its supporters do 

 not seem to include in their mode of understanding the statement. It is 

 only in the light of their forming exemplars of the geometrical analysis, 

 that I can view it as approaching to the fact ; although I should not, 

 perhaps, dispute the question if it were stated that these treatises are in 

 the main, solutions of the problems in which the analysis of other problems 

 often terminate. My principal objection to this latter view woidd be, that 

 though analyses do often terminate in one or other of these problems, they 

 as often do not ; and that even if they were found by experience to do 

 so still more frequently, there appears to be no reason why other classes of 

 problems may not present as much variety in respect to this circumstance 

 as those upon which the Greeks happened to spend their powers presented 

 of frequency. 



That Euclid's Data and his Porisms were subservient to analysis, and 

 intended to be so, there cannot exist a moment's doubt. Like his Elements 

 they are intended to be subsidiary, and appear to have no other object. 

 The treatises of Apollonius, on the contrary, can only be viewed as final and 

 complete, each in itself : the complete enumeration of the varieties of case 

 and circumstance, and the solution of each in succession, is the obvious end 

 of his undertaking — not the means of getting to something else beyond it. 

 Indeed, we may ask, to what purpose could these solutions have been ren- 

 dered subservient in the cultivation of analysis ? I cannot form the least 

 conjecture as to how they can be so employed. We are also compelled to 

 ask what could have been the nature of those problems which required such 

 an immense amount of preparation as these treatises would imply, even 

 supposing we could see how to apply them ? It is strange that no single 

 hint should have escaped the pen of Pappus on this topic, had there been 

 such wonderful problems or classes of problems. To me, therefore, every 

 one of the treatises of Apollonius appears to have nothing further to do 

 with analysis, than as far as analysis might have been employed in obtaining 

 the constructions ; even this being an assumption for which it might be 

 difficult to furnish convincing authority. Our views would be much more 

 in keeping at all events with the disputational character of tin- intercourse 

 of the geometers of those times, did we believe that the analysis was always 

 concealed, and only the construction and demonstration given. 



