After graduation, Halsted served as an instructor in mathematics at Princeton until beginning his post at the University of Texas at Austin in 1884.

From 1884 to 1903, Halsted was a member of the University of Texas at Austin Department of Pure and Applied Mathematics, eventually becoming its chair.

[1] In 1893 in Chicago, Halsted read a paper Some salient points in the history of non-Euclidean and hyper-spaces at the International Mathematical Congress held in connection with the World's Columbian Exposition.

In 1903, Halsted was fired from UT Austin after having published several articles that criticized the university for having passed over R. L. Moore, at that time a young and promising mathematician whom Halsted hoped to have as an assistant, for an instructor post in favor of a well-connected but less qualified candidate with roots in the area.

In a preface, Poincaré paid tribute to Halsted's inter-continental reach: He "has already taken the trouble to translate many European treatises and thus powerfully contributed to make the new continent understand the thought of the old.

There are four pages of index, 58 of text, and a lyrical preface: "Man imprisoned in a little body, with short-arm hands instead of wings, created for his guidance a mole geometry, tactile space, codified by Euclid in his immortal Elements, whose basal principle is congruence, measurement.

The angel of light in him created for the guidance of eye-life an independent system, a radiant geometry, a visual space, codified in 1847 by a new Euclid, by the Erlangen professor, George von Staudt, in his immortal Geometrie der Lage, published in the quaint and ancient Nurnberg of Albrecht Durer."

By developing concepts of eject and cut, the text relates abstraction to practice in perspective drawing or a picture plane (page 10).