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Math
, Philosophy

*We know that Gauss thought of making a test of the sum of the angles of
an enormous stellar triangle, and there are reports that he actually
carried out a similar test, on a terrestrial scale, by triangulating
three mountain tops in Germany. He was a professor at Gottingen, so it is
said that he chose a hill near the city and two mountain tops that could
be seen from the top of this hill. He had already done important work in
applying the theory of probability to errors of measurement, and this
would have provided an opportunity to make use of such procedures. The
first step would have been to measure the angles optically from each
summit, repeating the measurement many times. By taking the mean of
these observational results, under certain constraints, he could
determine the most probable size of each angle and, therefore, the most
probable value for their sum. From the dispersion of the results, he
could then calculate the probable error; that is, a certain interval
around the mean, such that the probablilty of the true value lying
within the interval was equal to the probability of it lying outside the
interval. It is said that Gauss did this and that he found the sum of
the three angles to be not exactly 180 degrees, but deviating by such a
small amount that it was within the interval of probable error. Such a
result would indicate either that space is Euclidean or, if
non-Euclidean, that its deviation is extremely small – less than the
probable error of the measurements.*

*Even if Gauss did not actually make such a test, as recent scholarship
has indicated, the legend itself is an important milestone in the
history of scientific methodology. Gauss was certainly the first to ask
the revolutionary question, what shall we find if we make an empirical
investigation of the geometrical structure of space? No one else had
thought of making such an investigation. Indeed, it was considered
preposterous, like trying to find by empirical means the product of
seven and eight.*

Taken from *An Introduction to the Philosophy of Science* by Rudolf
Carnap.