Squares.

Squares. METHOD OF LEAST, an arithmetical process of great importance for combining observations, or sets of observations, so as to obtain the most probable value of a quantity which depends on these observations. It is in fact the scientific method of taking certain averages, and it finds its most constant use in astronomy and other physical sciences. The necessity for applying the method arises from the fact that, when the greatest precision of measurement is sought, repeated measurements of the same quantity do not agree. Thus, the altitude of a star at culmination, if carefully measured night after night by the same observer through the same instrument, will in general come out a little different in the different observations. All the measurements will, however, lie within a certain range of variation; and if all are equally trustworthy, the arithmetical mean will give the most probable value of the real altitude. The differences between this mean and the individual measurements on which it is founded are called the residuals. The important mathematical property of these residuals is that the sum of their squares is less than the sum of the squares of the differences between the individual measurements and any other single quantity that might be taken. Now, this principle of 'Least Squares' holds not only for the simple case just described, but also for more complicated cases in which one observed quantity ($y$) is to be expressed as an algebraic function of another or of several independently observed quantities ($x$). Here the object is to find the most probable values of the assumed constants or parameters which enter into the formula. When these values are calculated we can calculate in terms of them and the observed $x$'s a value of $y$ corresponding to each set of observations. Comparing the calculated $y$'s with the observed $y$'s, we get a set of residuals, the sum of whose squares is a minimum if the parameters have been calculated according to a particular process. It is this process which is described as the method of least squares. Its basis is found in the mathematical principles of Probability (q.v.). See Professor Merriman's Textbook on the Method of Least Squares (2d ed. 1885), or Chauvenet's smaller treatise (1879), and for elementary discussion any good treatise on practical astronomy and geodesy.