A HALF-CENTURY OF SCIENCE. 207 



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 man- 

 ner within certain ranges of limitation ; but if the limits were exceeded, 

 a new proof, and often a new enunciation, became necessary. Gradu- 

 ally, however, it came to be perceived (e. g., by Carnot, in his " Geo- 

 metrie 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 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 dem- 

 onstration, to give rise to new propositions. 



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

 lished 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, w T ithin the pe- 

 riod 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 recognized as serving a similar purpose. With what exten- 

 sive results this generalization 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 varia- 

 ble, 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 marvelous extension which the 

 transformation of geometrical figures has received very largely from 

 Cremona and the Italian school, and which in the hands of our coun- 

 trymen Hirst and the late Professor Clifford has already brought 

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

 mant, 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 mathematicians of the day thought that he had 

 proved its entire impossibility. In the hands of Sylvester, of Kempe, 

 and others, this principle has been developed into a general theory of 

 link-work, on which the last word has not yet been said. 



