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{\large The True History of the Law of Cosines.}
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In a nutshell, Al-Battani had the equipment, but did not formulate the law.
He works in astronomical, not geometric terms. It is true
that to go from one of his formulas to the law of cosines you only have to
write the length of the day in terms of the latitude and the
height of the Ecliptic; this he certainly could have done, but he did not.
Here are the details, as explained by Anton von Braunm\"{u}hl in his
{\em Vorlesungen \"{u}ber Geschichte der Trigonometrie}.\vs


This image is adapted from Braunm\"{u}hl's Fig. 15. It is
based on Al-Battani and presumably  shows
the calculation of the height $h = \overline{SC}$ of the sun at a given time
$t$.

As represented, the celestial sphere intersects the plane of
the figure in the great circle containing the poles $P,P'$
and the mid-day position $D$ of the Sun. The Sun rises
at $A$. The surface angle $\widehat{APD}$ = $t_0$
is $\frac{1}{2}$ the length of the day. The surface angle $\widehat{SPD}$ gives
the time $t$ of the observation, measured from noon. The
other quantities entering into the calculation are
the colatitude $\overline{H'P} = \varphi$ and the (complement of the)
 angle of the Earth's axis with the Ecliptic, measured by $\overline{ED} = 
\delta$. Al-Battani's formula is given in terms of the {\em sinvers} function,
equivalent to our cosine: $\sv(\alpha) = 1 - \cos(\alpha)$, and
reads as follows:

$$ \sv t = \sv t_0 -\frac{\displaystyle \sv t_0 \sin h}{\displaystyle
\sin(90^o - (\varphi - \delta))}.$$

This is described by Braunm\"{u}hl, footnote 1 on page 53. He goes on to
say that this is not quite the spherical law of cosines.
The statement that Al-Battani ``knew'' the law of cosines he quotes from
H. Hankel: ``Vor den trigonometrischen Fundamentals\"{a}tzen kennt
Al-Batt\'{a}ni au\ss er denen des Almagest bereits die Formel
$$ \cos a = \cos b \cos c + \sin b \sin c \cos\alpha$$
f\"{u}r schiefwinklige Dreiecke, die er daher nicht immer in
zwei rechtwinklige Dreiecke zerlegen mu\ss, und wei\ss~ dieselbe,
um eine Multiplikation zu ersparen, in die Form zu setzen
$$\sv \alpha = \frac{\displaystyle \cos(b-c)-\cos a}{\displaystyle 
\sin b \sin c}.''$$

Braunm\"{u}hl continues: Aber weder die erste, noch die zweite dieser
Formel kommt bei ihm irgendwo vor. Hankel's Irrtum scheint mir aus
Delambre's Hist. de l'Astr. du moyen age p. 20 zu stammen, denn dieser
f\"{u}hrt dort in die bei Al-Batt\'{a}ni vorkommende Gleichung

$$ \sv t = \sv t_0 -\frac{\displaystyle \sv t_0 \sin h}{\displaystyle
\sin(90^o - (\varphi - \delta))}$$

statt $\sv t_0$ den Wert $\frac{\displaystyle \cos(\varphi - \delta)}
{\displaystyle \cos \varphi \cos \delta}$ ein, wodurch sie dann allerdings in 
$\sv t = \frac{\displaystyle \cos(\varphi - \delta)-\cos h}{\displaystyle 
\cos \varphi \cos \delta}$ \"{u}bergeht. Aber siese Substitution hat Delambre,
nicht Al-Batt\'{a}ni ausgef\"{u}hrt, der \"{u}berhaupt nirgends mit
Gleichungen rechnet, soda\ss~  bei ihm von einem Zusammenhang  dieser
Formeln, oder besser Regeln, untereinander nirgends eine Spur vorhanden
ist.\vs

Braunm\"{u}hl also makes the important remark (page 54) that there is
no mention of the spherical triangle $PZS$ in Al-Batt\'{a}ni. (... ohne
irgendwelche Beziehung auf das sph\"{a}rische Dreieck $PZ\Sigma$.)\vs

In his section on Regiomontanus (pp 130-133) Braunm\"{u}hl gives evidence
that R. was well acquainted with Al-Batt\'{a}ni's work, and states that
``kein Zweifel mehr bestehen kann, da\ss~ er beim Studium des
Albategnius zu diesem Lehrsatze gefu\"{u}hrt wurde.'' But he also
says (still about the law of cosines) ``Dieses Verdienst mu\ss~ nach unserer
Ansicht Regiomontan ganz allein zugesprochen werden.''


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