The History of Mathematics: A Brief Course

NON-EUCLIDEAN GEOMETRY 343

(1784-1846) in 1829 that he had been thinking about the foundations of geome-

try off and on for nearly 40 years (in other words, from the age of 13 on), saying

that his investigations were "very extensive," but probably wouldn't be published,

since he feared the controversy that would result. Some time during the mid-1820s,

the time when he was writing and publishing his fundamental work on differential

geometry, Gauss wrote a note—which, typically, he never published—in which he

mentioned that revolving a tractrix about its asymptote produced a surface that

is the opposite of a sphere. This surface turns out to be a perfect local model of

the non-Euclidean geometry in which the angle sum of a triangle is less than two

right angles. It is now called a pseudosphere. This same surface was discussed a

decade later by Ferdinand Minding (1806-1885), who pointed out that some pairs

of points on this surface can be joined by more than one minimal path, just like

antipodal points on a sphere.

5.4. Lobachevskii and Janos Bolyai. From what has been said so far, it is clear

that the full light of day was finally dawning on the subject of the parallel postulate.

As more and more mathematicians worked over the problem and came to the same

conclusion, from which others gained insight little by little, all that remained was a

slight push to tip the balance from attempts to prove the parallel postulate to the

exploration of alternative hypotheses. The fact that this extra step was taken by

several people nearly simultaneously can be expressed poetically, as it was by Felix

Klein (1926, p. 57), who referred to "one of the remarkable laws of human history,

namely that the times themselves seem to hold the great thoughts and problems

and offer them to heads gifted with genius when they are ripe." But we need not be

quite so lyrical about a phenomenon that is entirely to be expected: When many

intelligent people who have received similar educations work on a problem, it is

quite likely that more than one of them will make the same discovery.

The credit for first putting forward hyperbolic geometry for serious consider-

ation must belong to Schweikart, since Gauss was too reticent to do so. However,

credit for the first full development of it, including its trigonometry, is due to the

Russian mathematician Nikolai Ivanovich Lobachevskii (1792-1856) and the Hun-

garian Janos Bolyai (1802-1860), son of Farkas Bolyai. Their approaches to the

subject are very similar. Both developed the geometry of the hyperbolic plane and

then extended it to three-dimensional space. In three-dimensional space they con-

sidered the entire set of directed lines parallel to a given directed line in a given

direction. Then they showed that a surface (now called a horosphere) that cuts

all of these lines at right angles has all the properties of a Euclidean plane. By

studying sections of this surface they were able to deduce the trigonometry of their

new geometry. In modern terms the triangle formulas fully justify Lambert's asser-

tion that this kind of geometry is that of a sphere of imaginary radius. Here, for

example, is the Pythagorean theorem for a right triangle of sides a, b, c in spherical

and hyperbolic geometry, derived by both Lobachevskii and Bolyai, but not in the

notation of hyperbolic functions. Since cos(ia;) = cosh(x) the hyperbolic formula

can be obtained from the spherical formula by replacing the radius r with ir, just

as Lambert stated.

Spherical geometry Hyperbolic geometry