LEAST SQUARES, METHOD OF. This is a method, which, since its first introduction, has been shown to be the method of finding the most probable truth, when a number of discordant observations have been made upon a phenomenon. The earliest attempt at any thing of the sort was made by Cotes, in a tract entitled' Estimatio Errorunt in mixta mathesi,' in which he very distinctly recommends a process which is Identical with that of the method of least squares. It is remarkable that Cotes proposes this theorem not merely as a mode of finding a convenient mean (as was done by Legendre and Gauss), but as giving positively the most probable result. Ho oven introduces the hypothesis of observations baring different weights (though not with perfect correctness), and comes as near as possible to the assertion afterwards proved by Laplace. It will be worth while to quote the postage, as follows :—' .Mihi vix quidquam ulterius desidemri vide atur postquam ostensum fuerit quit ration's Probabilitas maxima in his rebus hiberi posait, nil diverere observationes, in eundem finem institute paullulum diversas ab invicem conclusiones cxhibont. Id autem fist ad moclum sequentis exempli. Sit p locus objecti alicujus ex observatione prim& definitus, q, r, s, ejusdein objecti loca ex obser vationibum subsequentibun; sint insuper r, Q, tu, a, ponders reciproce proportionalia epatiis evagationum, per qure so diffundere possint Errores ex obeervationibus aingulis prodcuntes, qmeque demur ex dat6 errorurn dimptibns ; et ad puncta p, q, r, s, intelligantur ponders r, Q, n, a, et inveniatur oorum gravitatie centrum z : dico punctual fore locum objecti maxitne probabilem qui pro vero ejus loco tutis rime haberi potent.' Legendre, in his work on comets (1800, distinctly proposed the application of the method to any case, and Gauss afterwards stated he had been in the habit of using it since 1795. Finally, Laplace, In his Theory of Probabilities (1814), and we believe in a previous paper published in the ' Memoirs of the Academy of Sciences, showed that this method was in all cases the one which the principles of that theory pointed out as giving the result, which, from the observations, has the greatest weight of probability in its favour. The details and demonstration of this method may be found In the work of Laplace, cited In the Berlin • Antronomiaches Jahrbuch; for 1834 and the two following years, and in the treatise on Probabilities in the Encyclo paedia Metmpolitana.
The most simple case of this method has been in use as long as accurate observations have been made, under the name of taking an average or a mean. If three observations give 93, 94, and 98, then the mean of the three I. 95, and if this be assumed as true, it in also aantrued that the errors of the observations were 2, 1, and 3. Tho sum of the squares of these is 4 +1 + or 14, and this is the least passible sum which can be thus obtained. If for example, we assume anything but 95, say 951, the assumed errors are then 21, 11 and 2 V. the squares of which are 4.41, 1.21, and 8.41, the sum of which Is 14-03. more than 14. • But the more extended eases of the method of least squares are those in which the result I. not simply observed, but is to le deter mined by operation. upon the result., of observation. In all cases the nth is the same: namely, that result lea the greatest probability In its favour, the assumption of which makes the sum of the squares of the errors the least powille, provided that all the obeerestions are equally worthy of confidence. Without entering into further explana tion, we shall give the results of one case.
Suppose that A and a are to be determined by oisservation, the required result being Ai-a, or the solution of the equation a x=s. Suppose also, which is essential to the simple form of the method which we now give, that all the observations, both of A and a, are made under equally favourable circumstances. Say that four observa tion, are made of each ; those fur a being p, q, r, and s : those for being r, q, n, and a. If then all the observations were perfectly cor rect, each of the equations p x= r, q x= Q, r x = n, s x=s, would be identical wills a x= A. Supposing, however, that the observations are discordant, take what value of x we may, the several quantities p x r, q S—Q, r s will not be (as they should be) each equal to nothing. 'Whatever their value may be, the whole of each Value will be error : and the sum of the squares of the errors, or metnou 01 1 V.I.81. squares in now unworn/my uses rn astronomy, which is-perhaps the only science in which so delicate a test is abso lutely necessary.