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 undiscovered planetary orb of speculation." In mathematics, as in all true sciences, no subject is considered in itself alone, but always as related to, or an outgrowth of, other things. The development of the notion of continuity plays a leading part in modern research. In geometry the principle of continuity, the idea of correspondence, and the theory of projection constitute the fundamental modern notions. Continuity asserts itself in a most striking way in relation to the circular points at infinity in a plane. In algebra the modern idea finds expression in the theory of linear transformations and invariants, and in the recognition of the value of homogeneity and symmetry.   SYNTHETIC GEOMETRY.

The conflict between geometry and analysis which arose near the close of the last century and the beginning of the present has now come to an end. Neither side has come out victorious. The greatest strength is found to lie, not in the suppression of either, but in the friendly rivalry between the two, and in the stimulating influence of the one upon the other. Lagrange prided himself that in his Mecanique Analytique he had succeeded in avoiding all figures; but since his time mechanics has received much help from geometry.

Modern synthetic geometry was created by several investigators about the same time. It seemed to be the outgrowth of a desire for general methods which should serve as threads of Ariadne to guide the student through the labyrinth of theorems, corollaries, porisms, and problems. Synthetic geometry was first cultivated by Monge, Carnot, and Poncelet in France; it then bore rich fruits at the hands of Möbius and Steiner in Germany and Switzerland, and was finally developed to still