ADDRESS. 39 



progress may perhaps be formed by considering, in one or two cases, 

 from what simple principles some of the great recent developments have 

 taken their origin. 



Consider, for instance, what is known as the principle of signs. In 

 geometry we are concerned with quantities such as lines and angles ; 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 necessary. Gradually, how- 

 ever, it came to be jierceived {e.g. by Carnot, in his ' Geometric 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 regarded as a directed line, and every 

 angle as measured in a definite direction. By means of this simple con- 

 sideration, 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 propo- 

 sitions. 



To take another instance. The properties of triangles, as established 

 by Euclid, have always been considered as legitimate elements of proof ; 

 so that, when in any figure two triangles occur, their relations may be 

 used as steps in a demonstration. But, within the period of which I am 

 speaking, other general geometi-ical 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 generalisa- 

 tion has been attended, the Geometric Superieure of the late M. Chasles, 

 and all the superstructure built on Anharmonic Ratio as a foundation, 

 will be sufficient evidence. 



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 variable, 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 country- 

 men 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 versa, — a problem which until the time of 

 Peaucillier seemed so far from solution, that one of the greatest mathe- 

 maticians of the day thought that he had proved its entire impossibility. 



