and Loci of Apollonius, ^c. 21 



restore the works of Apollonius ; bnt^ although they tried to 

 reduce the solution of each to as few propositions as possible, 

 it is yet the same long and tortuous method they have all 

 followed. For instance, J. Leslie gives four propositions to 

 the Section of Ratio, six to the Section of Space, and eight 

 to the Determinate Section, whilst, by my method, one 

 solution suffices for the three questions, considered in their 

 most general forms." 



Now, I have already recorded my opinion concerning the 

 peculiar method of investigation of the ancient geometers and 

 their modern imitators, namely, that it is attributable to the 

 want of precision and generality in the indicated operations, 

 and involved theorems ; but I Avill farther observe, in this 

 place, that the homographic theory must receive some 

 developments in Ihnits to the constants of the equations, 

 implicating the double points of divisions on the same straight 

 line, before it becomes thoroughly effective in its applica- 

 tions. And from the absence of such developments, Chasles' 

 solutions are necessarily defective. 



Take for instance his solution to the Section of Ratio, 

 which is as follows* : — 



" Draw AE parallel to NN, to cut MM in E ; draw AG 

 parallel to Ma\l to cut NN in G ; find I in MM such that 

 PI : RG :: m : 7i; bisect IE in O; in NN find H such 

 that PO : RH :: m : n; draw HA to cut MM in P; 

 from O as centre and radius = (OF'OE)^ describe a circle; 

 through either point C in which this circle cuts MM, draw 

 CA to cut NN in D : then will CAD be an answerable line." 

 And his only remarks in respect to the limits of the problem 

 are — " And if the segments OF and OE be not on the same 

 side of O, the two solutions will be imaginary." 



Here it is evident that when the given straight lines MM, 

 NN, are parallel, the method is not intelligibly applicable. And 

 it is but right to observe that this is the only case in which 

 the principal constniction given in Leslie^s Geometrical 

 Analysis (introducing the improvement of indicating opposite 

 directions by opposite signs) cannot be applied. However, the 

 general method of finding the double points of homographic 

 divisions svhicli is given in the Geometric Superieure, would, 

 if introduced, overcome this imperfection. But there is a 

 much more serious defect which cannot be rectified by the 

 " theory," such as it now exists, namely, the non-establish- 



* See the enunciation I gi\'e to this problem. 



