A History of Mathematics From Mesopotamia to Modernity

Geometries andSpace 209

C

A

B

R

R

b O

c

a

Fig. 13Solving a spherical triangle.

Appendix B. The formulae of spherical and hyperbolic trigonometry

To state these, we need a convention, since the formulae will vary with the radius of the sphere.
The ancient convention was to work with a sphere of radius 60, but let us simply call the radius
R= 1 /K,Kbeing the ‘curvature’ of the sphere. So, the biggerKis, the more curved the sphere—
the smaller its radius.
Now for any arc on S of lengtha,Kais a number between 0 and 2π. And the two key formulae
in ‘solving’ a spherical triangle ABC as in Fig. 13, already mentioned in connexion with al-B ̄ir ̄un ̄i’s
work, are:

sin(Ka)

sin A

=

sin(Kb)

sin B

=

sin(Kc)

sin C

the analogue of the ‘sine formula’, and

cos(A)=−cos(B)cos(C)+sin(B)sin(C)cos(Ka)

one of two analogues of the ‘cosine formula’. The first of these goes over into the ordinary sine

formula whenKa,Kb,Kcare small (tend to zero).

The hyperbolic formulae are simply related to the spherical; one ‘replaces’ cos by cosh and sin by

isinh when dealing with lengths (but not with angles, since only the ordinary functions apply to

angles). They are, then,

sinh(Ka)

sin A

=

sinh(Kb)

sin B

=

sinh(Kc)

sin C

and

cos(A)=−cos(B)cos(C)+sin(B)sin(C)cosh(Ka)

Exercise 6.Prove the statement above, about the limit of the spherical sine formula for small values of

Ka,Kb,Kc.

Exercise 7.Use the second formula in the case when the angle A= 0 to deduce the formula for

(p).