72 ADDRESS TO THE BRITISH ASSOCIATION, 1881. 



and in the old systems a proposition was proved with reference to 

 a particular figure. This figure might, it is true, be drawn in any 

 manner within certain ranges of limitation : but if the limits were 

 exceeded, a new proof, and often a new enunciation, became neces- 

 sary. Gradually, however, it came to be perceived (e.g. by Carnot, in 

 his ' Geometrie de Position,') that some propositions were true even 

 when the quantities were reversed in direction. Hence followed a 

 recognition of the principle (of signs) that every line should be re- 

 garded as a directed line, and every angle as measured in a definite 

 direction. By means of this simple consideration, geometry has 

 acquired a power similar to that of algebra, viz. of changing the signs 

 of the quantities and transposing their positions, so as at once, and 

 without fresh demonstration, to give rise to new propositions. 



To take another instance. The properties of triangles, as esta- 

 blished by Euclid, have always been considered as legitimate elements 

 of proof; so that, when in any figure two triangles occur, their rela- 

 tions may be used as steps in a demonstration. But, within the 

 period of which I am speaking, other general geometrical relations, 

 e.g. those of a pencil of rays, or of their intersection with a straight 

 line, have been recognised as serving a similar purpose. With what 

 extensive results this generalisation has been attended, the Geo- 

 metric Superieure of the late M. Chasles, and all the superstructure 

 built on Anharmonic Ratio as a foundation, will be sufficient evi- 

 dence. 



Once more, the algebraical expression for a line or a plane involves 

 two sets of quantities, the one relating to the position of any point in 

 the line or plane, and the other relating to the position of the line or 

 plane in space. The former set alone were originally considered vari- 

 able, the latter constant. But as soon as it was seen that either set 

 might at pleasure be regarded as variable, there was opened out to 

 mathematicians the whole field of duality within geometry proper, 

 and the theory of correlative figures which is destined to occupy a 

 prominent position in the domain of mathematics. 



Not unconnected with this is the marvellous extension which the 

 transformation of geometrical figures has received very largely from 

 Cremona and the Italian School, and which in the hands of our 

 countrymen Hirst and the late Professor Clifford, has already brought 

 forth such abundant fruit. In this, it may be added, there lay 

 dormant, it is true, and long unnoticed the principle whereby 

 circular may be converted into rectilinear motion, and vice vcrsd 

 a problem which until the time of Peaucillier seemed so far from 



