Napier's Analogies
Learn Napier's analogies for solving oblique spherical triangles, with all four formulas, variables, and worked examples.
The Formula
tan(½(A + B)) = (sin(½(a + b)) / sin(½(a − b))) · cot(½C)
tan(½(a − b)) = (cos(½(A + B)) / cos(½(A − B))) · tan(½c)
tan(½(a + b)) = (sin(½(A + B)) / sin(½(A − B))) · tan(½c)
Napier's analogies (sometimes called Napier's rules of circular parts) are a set of four equations used to solve oblique spherical triangles. They were developed by John Napier, the Scottish mathematician also famous for inventing logarithms, and published posthumously in 1614. These formulas are the spherical geometry counterpart of Mollweide's equations in planar trigonometry.
A spherical triangle is formed by three great-circle arcs on the surface of a sphere. Unlike flat triangles, the angles of a spherical triangle always sum to more than 180 degrees. Spherical trigonometry was historically essential for celestial navigation, astronomy, and geodesy (measuring the Earth's surface), and remains important in modern satellite navigation and aerospace engineering.
Napier's analogies are particularly useful when you know two sides and the included angle (SAS), or two angles and the included side (ASA), of a spherical triangle and need to find the remaining parts. They provide a direct computational method that avoids the ambiguities that can arise with the spherical law of sines.
In the formulas, lowercase letters (a, b, c) represent the sides of the spherical triangle measured as angles (since sides of a spherical triangle are arcs, their lengths are expressed as the angle they subtend at the center of the sphere). Uppercase letters (A, B, C) represent the angles at each vertex. Side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C.
The first two analogies relate the angles A and B to the sides a, b and the angle C. The third and fourth relate the sides a and b to the angles A, B and the side c. Together, they allow complete solution of any oblique spherical triangle when sufficient information is given.
Variables
| Symbol | Meaning |
|---|---|
| A, B, C | Angles of the spherical triangle at vertices A, B, C |
| a, b, c | Sides opposite to angles A, B, C (measured as angles in degrees or radians) |
| tan | Tangent function |
| cot | Cotangent function (1/tan) |
| sin, cos | Sine and cosine functions |
Example 1
In a spherical triangle, a = 50°, b = 70°, and C = 60°. Find A − B using the first analogy.
tan(½(A − B)) = (cos(½(50 + 70)) / cos(½(50 − 70))) · cot(30°)
= (cos 60° / cos(−10°)) · cot 30°
= (0.5 / 0.9848) · 1.7321 = 0.5077 × 1.7321
tan(½(A − B)) = 0.8794
½(A − B) = arctan(0.8794) ≈ 41.3°, so A − B ≈ 82.6°
Example 2
Using the same triangle (a = 50°, b = 70°, C = 60°), find A + B using the second analogy.
tan(½(A + B)) = (sin(½(50 + 70)) / sin(½(50 − 70))) · cot(30°)
= (sin 60° / sin(−10°)) · 1.7321
= (0.8660 / (−0.1736)) · 1.7321 = −4.990 × 1.7321
tan(½(A + B)) = −8.642
½(A + B) = arctan(−8.642) ≈ −83.4° + 180° = 96.6°, so A + B ≈ 193.2° (valid, since spherical triangle angles sum to more than 180°)
When to Use It
Napier's analogies are used whenever spherical triangle calculations are needed.
- Celestial navigation: determining position from star observations
- Astronomy: computing angular distances between celestial objects
- Geodesy: calculating distances on the Earth's curved surface
- Satellite communication: determining look angles to satellites
- Great-circle route planning for aviation and maritime navigation
- Crystallography: analyzing angular relationships in crystal structures