# Lunar distance measurement reference

While preparing trigonometric exercises for my students, I learned that, in 1771, French astronomers determined the distance of the Moon from the Earth by measuring the appropriate angles from both Berlin and Cape Town at the same time. I would like to know more about this such as who these astronomers were and where were the results published?

My Google-fu skills were obviously not black belt since I couldn't find anything. This could be because I searched with the date 1771 whereas I suppose a publication would come at a later date.

I can describe what they probably did, and why this measurement became possible in the end of 18th century, but to find the reference on this specific work, you give insufficient information. Names of these astronomers would help to find a reference.

Measuring a distance to the moon has a long history beginning at least from Hipparchus (2nd century BC) who obtained the right order of magnitude. It was improved by Ptolemy (2nd century AD). And was always improved since then, so the measurement that you mention is just one step in the long progress.

From your description, what they used is the most straightforward procedure, which leads to a very precise result, but the use of this procedure was technically impossible until the second half of 18th century.

If you have two observers A and B in Berlin and Paris, you can measure two angles at A and B of the triangle ABM, where M is a point on the Moon. Two angles determine the triangle up to similarity, so if you know the distance AB, you obtain the distance to the moon. The difference between these two angles is called parallax. There are two problems involved:

1. The distance AB. In 18th century this was determined with very high accuracy by triangulation. This is exactly when they started to make accurate maps based on land surveys using triangulation nets.

2. Clock synchronization in Berlin and Paris: the measurement had to be made simultaneously, since the Moon moves. This was a major scientific problem since XVI century. And the second half of 18th century is the time when it was finally solved. One of the solutions was the invention of an accurate mechanical clock which can be transported long distance while keeping exact time.

These two things made such a measurement possible.

All previous measurements were based on more primitive technology but on a much more sophisticated theory. Namely the theory which can predict the Moon motion, as seen from the center of the Earth. Then measuring the actual position of the Moon in the sky from a single place, you find the difference which is called parallax, and this gives you the distance. Interestingly, you can use the theory of Moon motion for time measurement, by a kind of reverse procedure.