78 Improvements in Fundamental Ideas 



However, there are conflicting opinions as to the cause of 

 this peculiar piece-meal method adopted by Apollonius. 

 Mons. M. Chasles — after giving a general solution to the 

 Determinate Section, and finding expressions for the limits of 

 the involved ratio — makes the following remarks, bearing on 

 the subject : — " La recherche de ces expressions ofTrait surtout 

 des grandes difficultes qui ont exige toutes les resources du 

 Geometre. Car Apollonius ne connaissant pas les differentes 

 relations analytiques de ^involution dont nous avons fait 

 usage, c'est au moyen de figures, et par des considerations de 

 pure geometrie, differentes dans les trois cas, qu'il est par- 

 venu a la determination des valeurs en question." The 

 geometers of the English school, having unfortunately fol- 

 lowed in the footsteps of Apollonius, were unable to perceive 

 the cause which led to the necessity of having recourse to 

 supplementary figures. 



But, that I am right in attributing the method to the 

 want of precision and generality in the language employed 

 to express the elementary operations and theorems, will be 

 evident from the solutions which I have effected, independent 

 of involution and homographic theories, for the celebrated 

 problems of the Greek and French schools. 



These solutions I intend to present immediately to the 

 Institute, with critical and historical notes. 



DEFINITIONS. 



A and B representing points, it is to be remembered that 

 when we say ' line AB/ or simply AB, we mean the distance 

 between A and B understood as being described from A direct 

 to B ■ and that when we say ' line BA/ or ' BA/ we mean 

 the distance between these same points, as described from B 

 direct to A. 



The two directions along a straight line are said to be the 

 primitive directions of the line. 



To distinguish between the two directions along a straight 

 line, we call one of them ' left/ and the opposite one ' right/ 



The ' relative ; magnitude and distance of a finite straight 

 line BB, in respect to any other straight line CC, is the 

 magnitude and direction on this other line CC, measured 

 direct from the projection of its first point B to that of its 

 second point B. 



If a straight line AA be cut perpendicularly by another 

 BB, then, looking from the point of intersection along the 

 right direction on AA, and having the point of intersection 



