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.