LEAST SQUARES, METIIOD OF. An applica tion of the theory of probabilities (q.v.) to the deduction of the most probable value from a number of observations. each of which is liable to certain aecidental errors. The methods by which this is done may be understood from a single example. Let it be found that a given bar has, at the temperatures of 20°. 40°, 50°, and 1i0° C., respectively, the lengths 1000.22, 1000.65. 1000.1)11. and 11)01.05 millimeters; and let it be required to a-certain the eoefficient of linear expansion. i.e., the amount of linear ex pansion per degree of temperature. If de notes the length of the bar at 0° C., c the co efficient of linear expansion, and //- the length of the bar at t° C., then t c = Substitut ing respectively '20 and 40 for t, and the corre sponding values of it, we get 20c = 1000.22 and + 40c = 1000.65. Solving these two equations for and e, we obtain: 1, = 999.79, and c = 0.215. But if these values of and o are then substituted in the equations correspond ing to t = 50 and t = 60, we find. respectively, 1000.87, = 1001.08. instead of the ex perimental figures 1000.90 and 1001.05. The difference between the 1000.87 and the 1000.90, 0.03, is called the residual of the third equa tion, while + 0.03 is obviously the residual of the fourth equation.
In the same way we might solve the first and fourth equations and obtain 999.80, c = 0.0208, in which case the residuals of the second and third equations would be + 0.02. ± 0.06. Other combinations of the given equations would give other residuals, and the smaller the resid uals the closer the probable approximation. It can be shown analytically and experimentally that in a series of observations affected by acci dental errors, errors whose law of recurrence is such that in the long run they are as often positive as negative, the number of errors of a given magnitude is a function of that magnitude.
This particular function is f (x) = where it is a constant for all observations of a series, and 7 and c have their usual meanings. The distribution of residuals follows
this law, which is represented graphically by the curve y = hid---c-h=". If x = 0, y = and therefore varies directly as 1r; hut as x becomes very large. at becomes very small. That is, the number of errors of very small magnitude is relatively large. and the number of errors of very large magnitude is small. It has further been found that the sum of the squares of the residuals. varies inversely as h. and hence when h is largest. Mr= is small est; in other words. that the most probable val ues of the unknowns are those which make 7x' a minimum. From this is derived the name 11 rthod of Least Squares.
For example. suppose a circumference. a-. bi sected by a diameter, is measured and found to be c, and the two seatieircuntferenecs are also measured and found to be s,. and we are re quired to find the most probable value of x. The residuals are c x, s, Hence, assuming only accidental error, f (x) = (c x)' + (8, ?,x = a min imum, or r (x) = 2 32 = 0, whence J` = ( 2C + 82), the most probable value.
The method of least squares is due to Legendre (1805), who introduced it in his Nouvelles no'lhodes pour la dOermina(ion des orbites des coin tcs. In ignorance of Legendre's contribu tion, however, au Irish-American writer, Admit, editor of The Analyst 11808), first deduced the law. He gave two proofs. the second being essentially the same as Herschel's (1850). Gauss gave the first proof which seems to have been known in Europe (the third after Ad rain's) in 1S09. To him is due much of the honor of placing the subject before the mathe matical world, both as to the theory and its applications.
For an introduction to the method of least squares, consult Comstock, Method of Least Squares (Boston, 1890) ; and for the history of the method, consult Merriman in the Transactions of the Connecticut Academy (1877, vol. iv., p. 151, with a complete bibliography).