The Determination of the Accuracy of Observations

When making the case for the so-called method of least squares, it is assumed that the probability of an error of observation Δ may be expressed by the formula (A)%% MathType!Translator!2!1!LaTeX.tdl!TeX -- LaTeX 2.09 and later! % MathType!MTEF!2!1!+- % feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca % WGObaabaWaaOaaaeaacqaHapaCaSqabaaaaOGaaiOlaiaadwgadaah % aaWcbeqaaiabgkHiTiaadIgacaWGObGaeyiLdqKaeyiLdqeaaaaa!4030!$$\frac{h}{{\sqrt\pi}}.{e^{ - hh\Delta \Delta }}$$ where π is the semi-perimeter of the unit circle, e is the base of natural logarithms, and h is also a constant that according to Section 178 of Theoria Motus Corporum Coelestium may be regarded as a measure of the accuracy of the observations. It is not at all necessary to know the value of h in order to apply the method of least squares to determine the most probable values of those quantities [parameters] on which the observations depend. Also, the ratio of the accuracy of the results to the accuracy of the observations does not depend on h. However, knowledge of its value is in itself interesting and instructive, and I will therefore show how we can arrive at such knowledge through the observations themselves.