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



that as a general rule, geometry has held a very subordinate 

 and precarious position in our universities since the commence- 

 ment of the preceding century ; whilst non-academic men, till 

 very recent times, have manifested extraordinary skill in such 

 researches. It is certainly a curious fact, that since our non- 

 academic men have devoted so much more of their attention 

 to analysis, (for I believe that Messrs. Lowry and Whitley 

 remain the only two representatives of the old English school 

 of geometers now left us,) the tide is setting, in one of our uni- 

 versities at least, in the direction of the ancient geometry. 



It would appear from the last of these letters, that the edi- 

 ting of the posthumous works of Simson was not a work of 

 considerable difficulty ; and indeed they are probably printed 

 nearly in the state that they were left by their distinguished 

 author. There is no doubt that with due attention there might 

 have been selected from his papers things much more worthy 

 of publication than those printed in the appendix, — the last 

 proposition, however, excepted. This one is important in re- 

 spect to a problem of great celebrity — that of the " tactions " 

 of Apollonius : but on this head I must refer to a paper in the 

 Mathematician, vol. iii. p. 77, where its importance is esta- 

 blished. 



There still exists (as I am informed by a friend to whom 

 I refer in the paper just quoted) in the library of the Escurial, 

 an Arabic translation of the Sectio Determinata. If it were 

 possible for another Halley to compare that translation with 

 the restoration of the work by Simson, it would be a sub- 

 ject of the greatest interest ; inasmuch as it would fully de- 

 cide the degree of confidence which we should place in any 

 " restorations " whatever, this being the one upon which Sim- 

 son appears to have bestowed more care than upon any of his 

 works except the Porisms. 



I think the conclusion of Kirkby's preface ought to disarm 

 criticism with respect to his " Doctrine of Ultimators." The 

 work itself, indeed, never appears to have attracted much at- 

 tention ; and it is now altogether forgotten, except by a few 

 collectors of mathematical curiosities. The remark of Simson, 

 however, is correct; for the equation which Kirkby brings 

 out (which is unexceptionable except as to the omission of the 

 sign ± from the radical) is that of the ellipse. As a general 

 rule, however, curves generated by means of ordinates to the 

 circle assume more convenient forms of equation when polar 

 co-ordinates are employed, though there occasionally occur 

 exceptions to it — of which the pyriformis is one. 



Royal Military Academy, 

 April 28, 1848. 



