Spherical trigonometry has all the qualities we expect of the best mathematics: it is beautiful, useful and fun. It played an enormously important role in science for thousands of years. It was crucial for astronomy, and essential for global navigation. Yet, it has fallen out of fashion, and is almost completely ignored in modern education.
A beautiful popular book, Heavenly Mathematics, has appeared recently (Van Brummelen, 2012). It traces the historical development of spherical trigonometry and describes its role in scientific progress, from ancient Greece, through India, the Islamic Enlightenment and the Renaissance, to more modern times.
Pythagoras on the Sphere
We have written on this blog about Napier’s Nifty Rules. These rules give ten relationships between the angles and sides of right-angled triangles on the sphere. (Recall that the sides of the triangle are expressed in terms of the angles they subtend at the centre; the radius of the sphere is taken to be unity).
One − and only one − of Napier’s Rules relates the three sides of the right-angled triangle. We take the right angle to be C, and the side opposite to it to be c. Then the rule is
cos c = cos a cos b
This beautifully simple equation may not appear familiar but, believe it or not, this is just the spherical form of Pythagoras’ Theorem!
Let us consider a small triangle, so all the sides a, b and c are small quantities. Then we may replace the cosine functions by their first few terms:
cos a ≈ (1 – ½a2 ) , cos b ≈ (1 – ½b2 ) , cos c ≈ (1 – ½c2 ) .
To second-order accuracy, we can write the equation cos c = cos a cos b as
c2 = a2 + b2 ,
the usual form of Pythagoras’ Theorem that we all know and love.
In the song from Gilbert and Sullivan’s The Pirates of Penzance, we learn from the Modern Major-General:
About binomial theorem I am teeming with a lot o’ news,
With many cheerful facts about the square of the hypotenuse.
And Danny Kaye sang, in the 1958 film Merry Andrew,
The square of the hypotenuse of a right triangle
is equal to the sum of the squares of the two adjacent sides.
The Cosine Law
For oblique spherical triangles (not having a right angle), the Cosine Law holds:
cos c = cos a cos b + sin a sin b cos C .
Note that this is the Pythagorean relationship with an “extra bit”. This is just the relationship that we need to calculate the distance between two places on Earth. Knowing the latitudes and longitudes, (λ1,φ1) and (λ2,φ2), we construct the spherical triangle comprised of two meridians from the pole to the points, and the great circle joining the points. Then we know a=½π −φ1, b=½π −φ2 and C= λ2−λ1 and we can immediately get c.
Once again, considering small triangles, we take all the sides a, b and c to be small. So, sin a ≈ a and sin b ≈ b. But we cannot assume that the angles are small, as their sum is greater than π. So, we leave cos C in exact form:
c2 = a2 + b2 – 2 a b cos C
This is just the Cosine Law of plane trigonometry; again, it is the Pythagorean relationship with an “extra bit”.
Glen Van Brummelen, 2012: Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry. Princeton Press.