In the preceding chapter we endeavoured to give a flash-light view of the rapid advance of synthetic geometry. In connection with hyperspace we also mentioned analytical treatises. Modern synthetic and modern analytical geometry have much in common, and may be grouped together under the common name "projective geometry." Each has
advantages over the other. The continual direct viewing of figures as existing in space adds exceptional charm to the study of the former, but the latter has the advantage in this, that a well-established routine in a certain degree may outrun thought itself, and thereby aid original research. While in Germany Steiner and von Staudt developed synthetic geometry,
Plücker laid the foundation of modern analytic geometry.
Julius Plücker (1801–1868) was born at Elberfeld, in Prussia. After studying at Bonn, Berlin, and Heidelberg, he spent a short time in Paris attending lectures of Monge and his pupils. Between 1826 and 1836 he held positions successively at Bonn, Berlin, and Halle. He then became professor of
physics at Bonn. Until 1846 his original researches were on geometry. In 1828 and in 1831 he published his Analytisch-Geometrische Entwicklungen in two volumes. Therein he adopted the abbreviated notation (used before him in a more restricted way by Bobillier), and avoided the tedious process
of algebraic elimination by a geometric consideration. In the
second volume the principle of duality is formulated analytically.
With him duality and homogeneity found expression
already in his system of co-ordinates. The homogenous or
tri-linear system used by him is much the same as the co-ordinates of Möbius. In the identity of analytical operation and geometric construction Plücker looked for the source of
his proofs. The System der Analytischen Geometrie, 1835, contains a complete classification of plane curves of the third order, based on the nature of the points at infinity. The
Theorie der Algebraischen Curven, 1839, contains, besides an enumeration of curves of the fourth order, the analytic relations between the ordinary singularities of plane curves known as "Plücker's equations," by which he was able to explain "Poncelet's paradox." The discovery of these relations
is, says Cayley, "the most important one beyond all
comparison in the entire subject of modern geometry." But in Germany Plücker's researches met with no favour. His method was declared to be unproductive as compared with the synthetic method of Steiner and Poncelet! His relations
with Jacobi were not altogether friendly. Steiner once