Colloquium publications.

FORMS OF NON-EUCLIDEAN SPACE.

BY FREDERICK S. WOODS.
By a non-euclidean geometry we shall mean any system of
geometry which, while differing in essential particulars from that
of Euclid, is nevertheless in accord with the facts of experience
within the limits of the errors of observation. The space in which
such a geometry is valid is a non-euclidean space. It is clear that
the test of experience can be applied only within a restricted portion of space, so that non-euclidean spaces, while having essentially the same properties in. such a restricted region, may differ
widely when considered in their entirety. It is the purpose of the
present lectures to present especially those non-euclidean spaces,
investigated by Clifford, Klein and Killing, which have been
named by the last author the Clijford-Klein'sche _Rauinformen.*
For the sake of clearness it is necessary to begin with the
geometry of a restricted portion of space. Here the author has
followed the development of his own article on " Space of Constant Curvature,"t to which the reader is referred for references
to the literature and for fuller handling of some of the subject
matter of the first five paragraphs of these lectures.
The point of view adopted is that objective space presents certain phenomena of form, position and magnitude, which demand
explanation as do other physical phenomena. This explanation
the geometrician gives by the assumption of certain hypotheses,
*Clifford, W. K., "A Preliminary Sketch of Biquaternions," lMcathematical
Papers, No. XX.
Klein, F., "Zur Nicht-Euklidischen Geometrie," cMath. Annalen, vol. 37
(1890), p. 344. Lectures on IMathematics, Lecture XI, New York, 1894. " Zur
ersten Vertheilung der Lobatchewsky Preise," Math. Annalen, vol. 59 (1898),
especially pp. 591-592.
Killing, WV., "Ueber Clifford-Klein'sche Raumformen," Math. Annalen, vol.
39 (1891). EinfUhrung in die Grundlagen der Geometric, vol. 1, Chap. 4; Paderborn, 1893.
t Annals of Mathematics, ser. 2, vol. 3 (1902), p. 71.
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