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\title{Hipparchus Measures the Distance to the Moon}
 

\begin{document}
\maketitle
 
The date: March 14, 189 BCE. (Or maybe 190; sources disagree).
  The places:  Alexandria and the Hellespont.  The 
event: a solar eclipse.  The Hellespont is a narrow strait on the northwest coast 
of Turkey.  The eclipse was total on the coast of the Hellespont, but in Alexandria,
one-fifth of the Sun had remained visible during the eclipse.

Here's a map showing these two locations:
\medskip

\insertimage{AlexandriaMap.png}{0.5}
\medskip

The sun subtends an angle of about half a degree when viewed from Earth, and 
in saying that ``one-fifth of the Sun remained visible'' in Alexandria, apparently
what was meant was that the visible part of the sun subtended an angle of about 
one-fifth what the whole Sun subtended, thus one-tenth of a degree. 

In the diagram below, $H$ is the Hellespont, $A$ is Alexandria, and $M$ is the moon.
The two lines $MH$ and $MA$ show the path of light rays from the edge of the moon's shadow.
Since the Moon takes a month to go around the Earth, we can assume it stays in one place
during the short solar eclipse.  Similarly, the Earth's yearly revolution around the Sun 
(or the Sun's revolution around the Earth, in the geocentric system) does not matter.
The daily rotation (of the Earth or the Sun, it doesn't matter which for this purpose)
makes the shadow of the Moon move across the Sun from west to east.
Therefore only the difference in latitudes of $A$ and $H$ matter;  it was not 
necessary to observe the eclipse at the exact same time in both places (luckily, 
since there were no accurate clocks).   Those latitudes are $41^\circ$ for $H$ and 
$31^\circ$ for $A$ (and those were known to Hipparchus, at least within a degree).
Since the difference is only ten degrees, the straight-line distance $HA$ is approximately
the same as $10/360$ times the circumference of the earth.  

The sharp angle $AMH$  is the ``lunar parallax'',  which the eclipse showed was 0.1 degree.
The diagram below shows the geometry of the situation. 



\medskip


 
\insertimage{HipparchusEclipse.png}{0.7}
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 In the diagram, 

\begin{itemize}
\item $\delta$ is 
the declination of the Moon.  That is how far above the plane of the equator the moon appeared
that day; that was $-3^\circ$, which would have meant that the moon rose to a maximum height 
of $3^\circ$ less than the latitude.  So $\delta$ is known if you watch the moon and know your 
latitude.
\item $\zeta$ is the latitude of $H$ minus $\delta$.
\item $\zeta^\prime$ is a small fraction of a degree different from $\zeta$, since $OM$ and $HM$
are almost parallel because the Moon is pretty far away.  
\end{itemize}
\medskip

Every angle  in the diagram could be known before the eclipse, except the small angle
$AMH = 0.1^\circ$, which was measured during the eclipse.  That determines $M$ and hence 
the distance $D$, in terms of the radius $OH = OA$ of Earth.  
 
The angle $AMH$ to be measured is the ``lunar parallax'' for the two locations $A$ and $H$.

Hipparchus was perhaps the discoverer (or inventor?) of trigonometry.  He didn't invent 
the sine and cosine functions, but instead he used the ``chord'' function,  giving the length
of the chord of the unit circle that subtends a given angle.  He was able to solve 
the geometry problem to find $D$ as a multiple of the Earth's radius, and concluded
that the moon is 71 Earth radii away from Earth.  In a fashion befitting a modern scientist,
he reported his error bars:  he said the moon was between 35 and 41 Earth diameters away.
The modern value is about 60 radii, or 30 diameters.  

Today you can measure the distance 
to the moon by bouncing a laser beam off a mirror left there by astronauts, and timing the 
round-trip of the light.  We know the speed of light now very accurately, so to get the 
distance just requires a multiplication.

A few years later, Hipparchus made another attempt to measure the distance to the Moon, 
 using a lunar eclipse, and he got a larger answer; and his error bars from the two 
 calculations did not even overlap, which he honestly admitted.   The measurement that the eclipse was
 $4/5$ complete was probably not very accurate, modern calculations are said to show.
 We don't really know for certain the exact details of Hipparchus's calculations, though 
 several historians have published reconstructions;  only one fragment of his works has 
 survived.



In principle, the accuracy of such measurements could be improved by using a longer baseline.
That seems not to have been done until 1751, when 
the French astronomer Nicolas Louis de Lacaille at the Cape of Good Hope
measured the lunar parallax by comparing his observations with observations made
by other astronomers in Europe, and calculated that the lunar parallax that would 
be observed from opposite sides of the Earth (a baseline of 8000 miles) would
be about two degrees.  

\begin{thebibliography}{}
\bibitem{}
G. J. Toomer, Hipparchus on the distances of the sun and moon, {\em Archive for History of Exact Sciences}
{\bf 14} (1974), 126-�142.

\bibitem{}
Hirschfeld, Alan W., {\em Parallax:  The Race to Measure the Cosmos}, Freeman, New York (2001).

\end{thebibliography}
 

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