The History of Mathematics: A Brief Course

5. NON-EUCLIDEAN GEOMETRY 341

clockwise about A to make angles that decreased to <9 0. He then—too hastily, as

we now know—drew the conclusion that èï would have the properties of both of the

sets of angles that it separated, that is, the line making this angle would intersect

BE and would also have a common perpendicular with it. In fact, it has neither

property. But Saccheri was determined to have both. As he described the situation,

the hypothesis of the acute angle implied the existence of two straight lines that

have a common perpendicular at the same point. In other words, there could be

two distinct lines perpendicular to the same line at a point, which is indeed a

contradiction. Unfortunately, the point involved was not a point of the plane, but

is infinitely distant, as Saccheri himself realized. But he apparently believed that

points and lines at infinity must obey the same axioms as those in the finite plane.

Once again, as in the case of Ptolemy, Thabit ibn-Qurra, ibn al-Haytham, and

others, Saccheri had developed a new kind of geometry, but resorted to procrustean

methods to reconcile it with the geometry he believed in.

5.2. Lambert and Legendre. The writings of the Swiss mathematician Johann

Heinrich Lambert (1728-1777) seem modern in many ways. For example, he proved

that ð is irrational (specifically, that tan ÷ and χ cannot both be rational numbers),

studied the problem of constructions with straightedge and a fixed compass, and

introduced the hyperbolic functions and their identities as they are known today,

including the notation sinh χ and cosh x. He wrote, but did not publish, a treatise

on parallel lines, in which he pointed out that the hypothesis of the obtuse angle

holds for great circles on a sphere and that the area of a spherical triangle is the

excess of its angle sum over ð times the square of the radius. He concluded that in a

sphere of imaginary radius ir, whose area would be negative, the area of a triangle

might be proportional to the excess of ð over the angle sum. What a sphere of

imaginary radius looks like took some time to discern, a full century, to be exact.

By coincidence, the hyperbolic functions that he studied turned out to be the

key to trigonometry in this imaginary world. Just as on the sphere there is a

natural unit of length (the radius of the sphere, for example), the same would be

true, as Lambert realized, on his imaginary sphere. Such a unit could be selected in

a number of ways. The angle èï mentioned above, for example, decreases steadily

as the length AB increases. Hence every length is associated with an acute angle,

and a natural unit of length might be the one associated with half of a right angle.

Or, it might be the length of the side of an equilateral triangle having a specified

angle. In any case, Lambert at least recognized that he had not proved the parallel

postulate. As he said, it was always possible to develop a proof of the postulate to

the point that only some small, seemingly obvious point remained unproved, but

that last point nearly always concealed an assumption equivalent to what was being

proved.

Some of Lambert's reasoning was recast in more precise form by Legendre, who

wrote a textbook of geometry used in many places during the nineteenth century,

including (in English translation) the United States. Legendre, like Lambert and

Saccheri, refuted the possibility that the angle sum of a triangle could be more than

two right angles and attempted to show that it could not be less. Since the defect of

a triangle—the difference between two right angles and its angle sum—is additive,

in the sense that if a triangle is cut into two smaller triangles, the defect of the

larger triangle is the sum of the defects of the two smaller ones, he saw correctly

that if one could repeatedly double a triangle, eventually the angle sum would have