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A. M., Ph. D., Ex-Fellow of Princeton College and Johns Hopkins University,

Professor of Mathematics in the University of Texas.

AUSTIN:
PUBLISHED BY THE UNIVERSITY OF TEXAS.

1891.

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TRANSLATOR'S PREFACE.

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Lobatschewsky was the first man ever to publish a non-Euclidian geom. etry.

Of the immortal essay now first appearing in English Gauss said, “The author has treated the matter with a master-hand and in the true geometer's spirit.

I think I ought to call your attention to this book, whose perusal can not fail to give you the most vivid pleasure."

Clifford says, “It is quite simple, merely Euclid without the vicious assumption, but the way things come out of one another is quite lovely.”

“What Vesalius was to Galen, what Copernicus was to Ptolemy, that was Lobatschewsky to Euclid."

Says Sylvester, “In Quaternions the example has been given of Al. gebra released from the yoke of the commutative principle of multiplication—an emancipation somewhat akin to Lobatschewsky's of Geometry from Euclid's noted empirical axiom."

Cayley says, “It is well known that Euclid's twelfth axiom, even in Playfair's form of it, has been considered as needing demonstration; and that Lobatchewsky constructed a perfectly consistent theory, wherein this axiom was assumed not to hold good, or say a system of non'Euclidian plane geometry. There is a like system of non-Euclidian solid geometry.”

GEORGE BRUCE HALSTED. 2407 San Marcos Street,

Austin, Texas. May 1, 1891.

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