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
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
, 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)