# Spherical Law of Sines/Historical Note/Mistake

## Source Work

2008: Ian Stewart: *Taming the Infinite*:

- Chapter $5$: Eternal Triangles
- Early trigonometry

## Mistake

*Georg Joachim Rhaeticus calculated sines for a circle of radius $10^{15}$ -- effectively, tables accurate to $15$ decimal places, but multiplying all numbers by $10^{15}$ to get integers -- for all multiples of one second of arc. He stated the law of sines for spherical triangles*- $\dfrac {\sin a} {\sin A} = \dfrac {\sin b} {\sin B} = \dfrac {\sin c} {\sin C}$

*and the law of cosines*- $\cos a = \cos b \cos c + \sin b \sin c \cos A$

*in his*De Triangulis*, written in $1462$-$3$ but not published until $1533$.*

## Correction

The tables published by Rhaeticus appeared in fact in his *Opus Palatinum de Triangulis*.

This was published in $1596$, some $20$ years after the death of its author.

The tables contained in it were computed in intervals of $10$ seconds of arc, not $1$ seconds of arc, and calculated to $10$ decimal places, not $15$.

The Spherical Law of Sines and Spherical Law of Cosines actually appeared in *De Triangulis Omnimodus* by Johannes Müller von Königsberg (Regiomontanus), not Georg Joachim Rhaeticus, who was not born until $1514$.

As for the suggestion that *De Triangulis Omnimodus* was not published until $1533$, this is suspect, as no corroboration for this can be found online.

It is accepted that there exists a $1533$ edition of this work, as this can be found everwhere you look. But while evidence of an actual $1464$ edition may be elusive, the fact that no online biographies mention the fact that its publication was delayed until $1533$, the suggestion is that this is false.

It is possible that Ian Stewart has conflated the two works in question: *De Triangulis Omnimodus* by Regiomontanus, and *Opus Palatinum de Triangulis* by Rhaeticus.

## Sources

- 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $5$: Eternal Triangles: Early trigonometry