20 The Three Sections, Tangentief!, 



tions; '^ and this is the more strange as the Section of Space 

 is by far the most useful of the three. 



The Determinate Section was solved by AVillebrord 

 Snel^ and since then by Dr. Robert Simscu^ William WaleSj 

 and Petro Giaunini. SnePs and Wales' solutions were re- 

 published at London in 1772^, by the Rev. John Lawson^ and 

 Giannini's at Parma in 1773. Dr. Simson's solution was 

 published in his Opera Reliqua in 1 776, at the private expense 

 of Earl Stanhope, and covers over 150 pages. 



However, though the lost writings of Apollonius occupied 

 the attention of Newton, Halley, Simson, Burrow, Huygens, 

 D'Omerique, Lalouere, and a host of other distinguished 

 geometers, it is a most remarkable fact that none of 

 them perceived the liaison of " The Three . Sections." 

 Indeed, it was only through the instrumentality of the 

 Homographic Theory, as systematised by M. Chasles, Pro- 

 fessor of Geometry to the Faculty of Sciences of Paris, that 

 this intimate connection was exposed, and analogous solutions 

 for the first time given. Chasles' solutions — extracted from 

 his correspondence with the late Professor Davies, of the 

 Royal Military Academy — dated 1848, were published in the 

 third volume of the Mathematician, and again in his recent 

 work entitled Traite de Geometric Superieure. 



These latter solutions are more in detail than those in the 

 ]Mathematician, and the following accompanying observations 

 of the author, who has been justly styled the Newton of 

 Geometry, are worthy of special attention. He says : — 



'"^Amongst the numerous questions to Avhich the homographic 

 theory can be most easily applied, are those which formed 

 the subject of the three Avorks of Apollonius, entitled the 

 Section of Ratio, the Section of Space, and the Determinate 

 Section. Each of these questions exacted a great number of 

 propositions. Pappus relates that there were 181 in the 

 Section of Ratio, 124 in the Section of Space, and 83 in the 

 Determinate Section. Tliese arose from the fact that the 

 solution to the general question was never given directly, as 

 the ancient geometers proceeded to first establish the most 

 simple cases, and then went step by step to the more general, 

 so that the solution of each case always depended on those 

 which preceded. IMoreover, each problem gave rise to as 

 many dififerent questions as there were varieties in the 

 different relative parts of the figure. 



In the two last centuries these problems have occupied the 

 attention of many eminent geometers, who endeavoured to 



