# Spherical Law of Sines/Historical Note/Mistake

## Source Work

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.