SQUARES, METHOD OF LEAST, jr.) astronomy, the best mode hitherto discovered of obtaining the most correct result from a number of observations upon any phenomenon. These observations are assumed to differ slightly from each other, and to be all of equal value, that is, taken under equally favorable conditions, and with equal instruments. The ordinary and long-established mode of approximating to the truth in such cases is by finding the aritlunetic mean, and accepting.it as the correct result; but in all eases where the result required does not come directly from observation, but requires to be discovered by calculation, this simple and useful method is inapplicable, and that of "least squares," which gives more probable corrections, is adopted. Tire method is founded on a theorem Ivhich was first propounded by Legendre in 18(15, more for the sake of insuringuniformity among calculators than from any belief in its intrinsic value; but it was afterward thoroughly discussed and proved, by Gauss and Laplace, that "if the mean of it nurnber of distinct observations be so taken that the sum of the squares of its differences from the actual observations (generally designated errors) shall be a minimum, this mean will tic, under these circumstances, the correctest obtainable value." The process by which the mean thus obtained is shown to be the most trustworthy approximation is too long for insertion here; Indit may not be undesirable to give an example of the most ccmnion form of the method as occurring in astronomy. Let there
be a series of equations— X = x X, = + 2y + 5z, . X, = 4x + y 4z, X, = — x + 3y + 3z; where the unknown quantities are y, and z, connected by various (the more the better) equations with X, X,, etc., quantities which must be determined by actual observation. Suppose the values of the quantities thus found to be 3, 5, 21, and 14, then, since by hypothesis all these four observations are erroneous, the errors are 3 — X, 5 — X,, 21 — 14 — X., or, _ 3— y — 2z, 5 — 3.r — 2y — 5e, 21 — 4x — y — 4z, 14+ — 3y — 3z The squares of these four errors are now added together; and. to find the values of x, y, and z, which will render this sum (call it S) a ad/Mullin, we must differentiate S with respect to x„y, and z in turn, and putting each of these partial coefficients equal to zero• we obtain the three equations — 88 + 27x+ 8y +30z = 0; — 76+ gr + 15y + 25s = O. and — 1ST -j- 30.r + 25y + 54z = 0: from which the most trustworthy values of tr. y, nud z can be found by common alts lm.—For a full view of the NVI101• of this subje•i, see a lamer by Mr, Ellis in tire Cambridge Transactions, vol. Yid: