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 -invariants, or differential-parameters, which have been investigated by Jacobi, C. Neumann, Sir James Cockle, Halphen, and elaborated into a general theory by Beltrami, S. Lie, and others. Beltrami showed also the connection between the measure of curvature and the geometric axioms.

Various researches have been brought under the head of "analysis situs." The subject was first investigated by Leibniz, and was later treated by Gauss, whose theory of knots (Verschlingungen) has been employed recently by J. B. Listing, 0. Simony, F. Dingeldey, and others in their "topologic studies." Tait was led to the study of knots by Sir William Thomson's theory of vortex atoms. In the hands of Riemann the analysis situs had for its object the determination of what remains unchanged under transformations brought about by a combination of infinitesimal distortions. In continuation of his work, Walter Dyck of Munich wrote on the analysis situs of three-dimensional spaces.

Of geometrical text-books not yet mentioned, reference should be made to Alfred Clebsch's Vorlesungen über Geometrie, edited by Ferdinand Lindemann, now of Munich; Frost's Solid Geometry; Durege's Ebene Curven dritter Ordnung.

The progress of algebra in recent times may be considered under three principal heads: the study of fundamental laws and the birth of new algebras, the growth of the theory of equations, and the development of what is called modern higher algebra.

We have already spoken of George Peacock and D. F. Gregory in connection with the fundamental laws of algebra. Much was done in this line by De Morgan.