# Delambre's Analogies

## Theorem

Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.

Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.

### Sine by Sine

- $\sin \dfrac c 2 \sin \dfrac {A - B} 2 = \cos \dfrac C 2 \sin \dfrac {a - b} 2$

### Sine by Cosine

- $\sin \dfrac c 2 \cos \dfrac {A - B} 2 = \sin \dfrac C 2 \sin \dfrac {a + b} 2$

### Cosine by Sine

- $\cos \dfrac c 2 \sin \dfrac {A + B} 2 = \cos \dfrac C 2 \cos \dfrac {a - b} 2$

### Cosine by Cosine

- $\cos \dfrac c 2 \cos \dfrac {A + B} 2 = \sin \dfrac C 2 \cos \dfrac {a + b} 2$

## Also known as

Delambre's Analogies are also known as Gauss's Formulas or Gauss's Formulae.

However, there are so many results and theorems named for Carl Friedrich Gauss that $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers to settle for Delambre.

The names of the individual formulas are not standard, but $\mathsf{Pr} \infty \mathsf{fWiki}$ needs some way to distinguish between them. Any advice on this matter is welcome.

## Also see

## Source of Name

This entry was named for Jean Baptiste Joseph Delambre.

## Historical Note

Delambre's Analogies, or Gauss's Formulas, were discovered by Jean Baptiste Joseph Delambre in $1807$ and published in $1809$.

Carl Friedrich Gauss subsequently discovered them independently of Delambre.

## Sources

- 1976: W.M. Smart:
*Textbook on Spherical Astronomy*(6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $16$.*Delambre's and Napier's analogies.* - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Delambre's analogies** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Gauss's formulae (Delambre's analogies)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Delambre's analogies** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Gauss's formulae (Delambre's analogies)**