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 among the most curious of a series of models published by L. Brill in Darmstadt. It has been pointed out that if a fourth dimension existed, certain motions could take place which we hold to be impossible. Thus Newcomb showed the possibility of turning a closed material shell inside out by simple flexure without either stretching or tearing; Klein pointed out that knots could not be tied; Veronese showed that a body could be removed from a closed room without breaking the walls; C. S. Peirce proved that a body in four-fold space either rotates about two axes at once, or cannot rotate without losing one of its dimensions. 

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