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THE

AMERICAN
MATHEMATICAL
MONTHLY.

DEVOTED TO THE

SOLUTION OF PROBLEMS IN PURE AND APPLIED MATHEMATICS,
PAPERS ON MATHEMATICAL SUBJECTS, ETC.

EDITED BY

B. F. FINKEL, A. M., M. Sc.,

PROFESSOR OF MATHEMATICS AND PHYSICS IN DRURY COLLEGE,
SPRINGFIELD, MISSOURI.

-AND

LEONARD E. DICKSON, A. M., Ph. D.,

ASSISTANT PROFESSOR OF MATHEMATICS, THE UNIVERSITY OF CHICAGO,
CHICAGO, ILLINOIS.

SAUL EPSTEEN, Ph. D.,

ASSISTANT IN MATHEMATICS, THE UNIVERSITY OF CHICAGO.

OIXON BROS. PRINTERS, 288 EAST COMMERCIAL STREET, SPRINGFIELD, MO,

THE

AMERICAN

MATHEMATICAL MONTHLY.

VOL. XI.

Entered at the Post-office at Springfield, Missouri, as second-class matter.

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The past century was marked by two distinct yet closely related advances in elementary geometry; one concerned with the invention of new geometries, the other chiefly devoted to the revision of the foundations on which the older Euclidean geometry rests. Early in the century was invented the geometry of Lobatchevsky and about the middle of the century, that of Riemann. 2 The pursuit of this work with its extensions by Cayley, Klein, Beltrami, and numerous others, leads on the one hand to the relations of metric to projective geometry and to their applications to geometry and the theory of functions; on the other, to the study of differential forms, the theory of surfaces, and the theory of groups. The end of the century saw started, though by no means completed, a great movement to investigate the logical foundations of geometry and of math

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1. These lectures on Spherical Geometry constitute a short extract from the author's course entitled "Elementary Geometry" and offered annually at Yale University to members of the Senior class for the purpose of "training students to become effective teachers of Geometry in the secondary schools."'

2. We may here call attention to Manning's Non-Euclidean Geometry published by Ginn & Co. At present this seems to be the best book from which the student of meager mathematical development may learn the elements of Lobatchevskyan and Riemannian geometry, trigonometry, and analytic geometry. There is considerable criticism which might be advanced against the book but it is hoped that the readers of this series of lectures will be in a position to make their own criticisms.

8. The bibliography of modern investigations on noneuclidean geometry and allied topics is very long. G. B. Halsted has covered the ground up to about 1880 in a collection of references printed in The American Journal of Mathematics, Vol. 1, pp. 261-266, pp. 384-385, and Vol. 2, pp. 65-70. THE AMERICAN MATHEMATICAL MONTHLY has printed papers on noneuclidean geometry, chiefly by Halsted, as follows: Vol. 1, p. 70, 112, 149, 188, 222, 259, 301, 345, 378, 421; Vol. 2, p. 10, 42, 67, 70, 108, 137, 144, 181, 214, 256, 309, 846; Vol. 3, p. 1, 18, 35, 67, 109, 132, 237; Vol. 4, p. 10, 77, 101, 170, 200, 217, 269, 307; Vol. 5, p. 1, 67, 127, 290; Vol. 6, p. 59, 166, 219; Vol. 7, p. 123, 154, 247; Vol. 8, p. 81, 84, 161, 216; Vol. 9, p. 153. We may note further

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