A History of Mathematics From Mesopotamia to Modernity

212 A History ofMathematics

C

B

Π(a)

a

Fig. 17Figure for Exercise 7.

∠D′A′F′=∠DAF. (This is where it is essential that the rules for congruent triangles work, but

Lobachevsky has assumed they do.) Hence angles at A′define cutting lines on B′C′if the same

angles at A define cutting lines on BC—and obviously vice versa. This is enough to conclude

that the angles of parallelism are the same for the two line-segments.

5. Ift=tan(x/ 2 ),
   2 t
   1 +t^2

=

2 tan(x/ 2 )

sec^2 (x/ 2 )

=2 sin(x/ 2 )cos(x/ 2 )

(writing tan as cos/sin, and sec as 1/cos), which is sinx. Hence, the second formula is

equivalent to:

sin(

(p))=

2 e(−Kp)

1 +e−^2 Kp

=

(

1

2

(eKp+e−Kp)

)− 1

=sech(Kp)

Aspincreases from 0 to∞, cosh(p/K)increases from 1 to∞, and its inverse sech decreases

from 1 to 0. Since by definition

(p)is between 0 andπ/2, the first equation implies that

(p)

decreases fromπ/ 2 (p= 0 )to 0(p→∞).

6. Asx→0, sinx/x→1. Hence, to first order in the lengthsa,b,c, the ‘sine formula’ can be
   replaced bya/sinA=b/sinB=c/sinC, the usual formula.


7. IfA =0, we are in the situation where the sidesb,care parallel in Lobachevky’s sense.
   We consider a triangle (cf. Fig. 17) withB=π/2 andC=
   (a). The formula gives 1=
   0 +sin(
   (a))cosh(Ka), using cos(π/ 2 )=0, sin(π/ 2 )=1. This can be transformed into the
   first version of the formula for
   (a).