The History of Mathematics: A Brief Course

NON-EUCLIDEAN GEOMETRY^345

Thus, ironically, the acceptance of the logical consistency of hyperbolic geom-

etry was accompanied by a nearly immediate rejection of any practical application

of it in astronomy or physics. That situation was to change only much later, with

the advent of relativity.

Lobachevskii was unaware of the work of Gauss, since Gauss kept it to him-

self and urged others to do likewise. Had Gauss been more talkative, Lobachevskii

would easily have found out about his work, since his teacher Johann Martin Chris-

tian Bartels (1769 1836) had been many years earlier a teacher of the 8-year-old

Gauss and had remained a friend of Gauss. As it was, however, although he contin-

ued to perfect his "imaginary geometry," as he called it, and wrote other mathemat-

ical papers, he made his career in administration, as rector of Kazan' University.

He at least won some recognition for his achievement during his lifetime, and his

writings were translated into French and German after his death and highly re-

garded.

Even though his imaginary geometry was not used directly to describe the

world, Lobachevskii found some uses for it in providing geometric interpretations

of formulas in analysis. In particular, his paper "Application of imaginary geometry

to certain integrals," which he published in 1836, was translated into German in

1904, with its misprints corrected (Liebmann, 1904)· Just as we can compute the

seemingly complicated integral

immediately by recognizing that it represents the area of a quadrant of a circle of

radius r, he could use the differential form for the element of area in rectangular

coordinates in the hyperbolic plane given by dS = (1/ siny') dxdy, where y' is the

angle of parallelism for the distance y (in our terms sin y' = sech y) to express certain

integrals as the non-Euclidean areas of simple figures. In polar coordinates the

corresponding element of area is dS = cot r' dr Üè = sinh r dr d9. Lobachevskii also

gave the elements of volume in rectangular and spherical coordinates and computed

49 integrals representing hyperbolic areas and volumes, including the volumes of

pyramids. These volumes turn out to involve some very complicated integrals

indeed. He proved, for example, that

Bolyai's fate. Janos Bolyai's career turned out less pleasantly than Lobachevskii's.

Even though he had the formula for the angle of parallelism in 1823, a time when

Lobachevskii was still hoping to vindicate the parallel postulate, he did not publish

it until 1831, five years after Lobachevskii's first publication. Even then, he had

only the limited space of an appendix to his father's textbook to explain himself.

His father sent the appendix to Gauss for comments, and for once Gauss became

quite loquacious, explaining that he had had the same ideas many years earlier, and

that none of these discoveries were new to him. He praised the genius of the young

Lobachevskii for discovering it, nevertheless. Bolyai the younger was not overjoyed

at this response. He suspected Gauss of trying to steal his ideas. According to Paul

Stackel (1862-1919), who wrote the story of the Bolyais, father and son (quoted

in Coolidge, 1940, p. 73), when Lobachevskii's work began to be known, Bolyai

immediately thought that Gauss was stealing his work and publishing it under the

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