Delambre's Analogies

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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.

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