A History of Mathematics From Mesopotamia to Modernity

202 A History ofMathematics

This means that you are adopting what Saccheri called the hypothesis of the acute angle. The
simple negation of postulate 5, though, is:

There exists a straight line, falling on two other straight lines, which makes the two interior angles on one side less

than two right angles, such that the two straight lines, produced indefinitely, never meet.

This is impossibly vague, and cannot be made a basis for any serious deductions. To have a workable
geometry, one would need—at least in the terms understood in the 1820s—to have rules for when
triangles were congruent, rules for angle sum, rules for areas of figures, and so on. In other words,
one would need measurement of a kind, and to construct a new geometry was in a certain way to
define a new way of measuring. This was to be made explicit by Riemann in the 1850s, but it was
not how Lobachevsky and Bolyai proceeded. Their expositions were surprisingly similar, each with
its advantages; Bolyai’s is perhaps the clearer, Lobachevsky’s the more complete.
It is easiest to start with Lobachevsky’s clarification of the vague statement above. Properly
analysed, he claimed, it must go as follows:

All straight lines in a plane which go out from a point can, with reference to a given straight line in the same plane, be

divided into two classes—intocuttingandnot-cutting.

Theboundary linesof the one and the other class of those lines will be calledparallelto the given line.

From the pointAlet fall upon the lineBCthe perpendicularAD, to which again draw the perpendicularAE(Fig. 9)...

In passing over from the cutting lines, asAF, to the not-cutting lines, asAG, we must come upon a lineAH, parallel

toDC, a boundary line, upon one side of which all linesAGare such as do not meet the lineDC, while upon the other

side every straight lineAFcuts the lineDC.

The angleHADbetween the parallelHAand the perpendicularADis called the parallel angle (angle of parallelism),

which we will here designate by

(p)forAD=p. (Lobachevsky, in Fauvel and Gray 16.B.3, pp. 524–5)

The above definition is hardly more sophisticated than the work of Lambert. Its essential importance
is that it changes the imprecise negation of postulate 5 into a precise statement about angles and
their relation to lengths; with everypis associated a
(p). It is, of course, crucial in constructing
a sensible geometry that
(p)depends only onp, but this follows from the ‘elementary’ results
which Lobachevsky gives at the outset. These include the standard rules for when two triangles

A D

F

K
 E C

H

GH

E
 K B

Fig. 9Lobachevsky’s diagram.