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, however, it came to be perceived (e. g., by Carnot, in his "Géometrie 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 demonstration, to give rise to new propositions.
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 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 extensive results this generalization has been attended, the "Géometrie Supérieure" 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 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 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 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.