Spherical trigonometry is not in vogue. A century ago, a Tripos student might resolve a half-dozen spherical triangles before breakfast. Today, even the basics of the subject are unknown to many students of mathematics. That is a pity, because there are many elegant and surprising results in spherical trigonometry. For example, two spherical triangles that are similar – having corresponding angles equal – have the same area. This contrasts sharply with the situation for plane geometry.
There is no denying the crucial importance of spherical trigonometry in astronomy and in the geosciences. A good memory is required to master all the fundamental results: the sine law, the cosine law for angles, the cosine law for sides, Gauss’s formulae and much more. But we can get a long way with a few simple and easily-remembered rules formulated by the inventor of logarithms.
Recently, when studying the analemmatic sundial on the East Pier in Dun Laoghaire, I had to derive the equation for a great circle. This involves the intersection of a plane and a sphere, an easy problem in three-dimensional Cartesian geometry. But when I saw the result,
tan φ = tan ε sin (λ – λ_0 )
it was clear that a more direct approach must be possible.
The formula for a great circle turns out to be one of Napier’s Rules. These rules are easy to state. Every spherical triangle has three angles and three sides. The sides are also expressed as angles, the angles they subtend at the centre of the sphere. For a sphere of unit radius, these angles (in radians) equal the lengths of the sides. Napier’s Rules apply to right-angled triangles. Omitting the right angle, we write the remaining five angles in order on a pie diagram (see figure below), but replace the three angles not adjacent to the right angle by their complements (their values subtracted from 90 degrees). If we select any three angles, we will always have a middle one and either two angles adjacent to it or two angles opposite to it. Then Napier’s Rules are:
SIne of mIddle angle = Product of tAngents of Adjacent angles
SIne of mIddle angle = Product of cOsines of Opposite angles.
With five choices for the middle angle and adjacent and opposite cases for each, there are ten rules in all. For example,
sin a = tan b tan B¯ = tan b cot B
As a mnemonic, note the correspondences of the first vowels in key words, indicated in red.
Napier’s Rules apply only to right triangles, but we can often handle a general spherical triangle by dividing or extending it. Suppose we want the great circle distance from Paris to Cairo, knowing the latitude and longitude of each. The meridians from the North Pole to these cities, together with the great circle between them, form a spherical triangle for which we know two sides and the included angle. We can apply the cosine law for sides to get the great circle distance. But what if we have forgotten the cosine law? We can drop a perpendicular from Paris to the meridian through Cairo and apply Napier’s Rules repeatedly to find the inter-city distance (it turns out to be about 3,200 km).
John Napier (1550 – 1617), formulator of the rules, is best remembered as the inventor of logarithms. Also out of vogue today, his tables of logs enabled Johannes Kepler to analyse Tycho Brahe’s observations and deduce the orbits of the planets. Napier also popularised the use of decimal fractions in arithmetic. But his work in mathematics was essentially recreational, for Napier was foremost a theologian. An ardent, even fanatical, protestant, he regarded his commentary on the Book of Revelations as his best work. In A Plaine Discovery of the Whole Revelation of St. John, he predicted that the apocalypse and the end of the world would occur in 1700.
Napier’s book on logarithms contained his “Rules of Circular Parts” of right spherical triangles. They are easily remembered and simple to apply. If you are ever marooned on a desert island and know the location, you can use them to work out how far you will have to swim home. I hope you make it.
See also a later post on Spherical Geometry: Pythagoras goes Global