Improvements in Fundamental Ideas, Sfc. 77 



but, laying aside its intrinsic worth, in developing the fun- 

 damental ideas of the science, there are other reasons of great 

 importance why it should be introduced ; for by its means 

 only we are enabled, in some cases, to distinguish between 

 what is or what is not a perfect geometrical proposition. To 

 show this I need only instance the following enunciation : — 

 " Given two straight lines MM NN in position, and the 

 points PQ, one in each line; through two given points 

 BC to draw two straight lines BO CO, making the angle 

 BOC of a given magnitude 6, and such that EF, being 

 the respective points in which BO and CO cut MM and 

 NN, we shall have the segment PE to the segment QF 

 in a given ratio m : n ." 



Here, by not taking cognizance of the method of forma- 

 tion of the angular magnitude 0, and of the formation of 

 the segments as to direction on the given lines, and of the 

 sign of the ratio, we have in reality four distinct proposi- 

 tions under one enunciation. The result is that, to what 

 is meant by this enunciation, it is absolutely impossible to 

 give a direct general solution applicable to all particular 

 cases, from which the determination of the ' limits ' and 

 ' porismatic ' relations can be adduced. 



Now the Geometrie Superieure is almost entirely depen- 

 dent on homographic pencils and divisions, and their peculiar 

 developments — all founded on improved ideas. It has not 

 been found compatible with its principles to use the geo- 

 metrical truths transmitted to us by the Greeks, as many 

 of these truths have not sufficient generality and precision. 

 But it is possible to improve the ordinary geometry, and 

 give to it all the advantages claimed as the distinguishing 

 characteristic of modern theories : all that is required to 

 effect this is a skilful introduction of conceptions of motion 

 into its fundamental ideas and theorems. 



Whether a comprehensive motional philosophy (embracing 

 movements of rotation, translation, &c.) was ever used in the 

 writings of the Greeks, it is hard to determine : most pro- 

 bably it was the groundwork of Euclid's porismatic or second 

 elements. But, from the accounts of the method pursued 

 , by Apollonius in his books of solutions, it is evident this 

 great geometer did not make use of any elements established 

 on such principles : for, according to Pappus, the general 

 problems were divided into numerous particular cases, each 

 case receiving separate analysis, composition, and discussion, 

 " just," he says, " as variations in the figure might require." 



