12. We may remark one striking evidence of the accuracy thus obtained by employing large masses of observations. In this way we may often detect inequalities much smaller than the errours by which they are encumbered and concealed. Thus the Diurnal Oscillations of the Barometer were discovered by the comparison of observations of many days, classified according to the hours of the day; and the result was a clear and incontestable proof of the existence of such oscillations although the differences which these oscillations produce at different hours of the day are far smaller than the casual changes, hitherto reduced to no law, which go on from hour to hour and from day to day. The effect of law, operating incessantly and steadily, makes itself more and more felt as we give it a longer range; while the effect of accident, followed out in the {215} same manner, is to annihilate itself, and to disappear altogether from the result. SECT. III.--_The Method of Least Squares._ 13. The Method of Least Squares is in fact a method of means, but with some peculiar characters. Its object is to determine the _best Mean_ of a number of observed quantities; or the _most probable Law_ derived from a number of observations, of which some, or all, are allowed to be more or less imperfect. And the method proceeds upon this supposition;--that all errours are not _equally_ probable, but that small errours are more probable than large ones. By reasoning mathematically upon this ground, we find that the best result is obtained (since we cannot obtain a result in which the errours vanish) by making, not the _Errours_ themselves, but the _Sum of their Squares_, of the _smallest_ possible amount. 14. An example may illustrate this. Let a quantity which is known to increase uniformly, (as the distance of a star from the meridian at successive instants,) be measured at equal intervals of time, and be found to be successively 4, 12, 14. It is plain, upon the face of these observations, that they are erroneous; for they ought to form an arithmetical progression, but they deviate widely from such a progression. But the question then occurs, what arithmetical progression do they _most probably_ represent: for we may assume several arithmetical progressions which more or less approach the observed series; as for instance, these three; 4, 9, 14; 6, 10, 14; 5, 10, 15. Now in order to see the claims of each of these to the truth, we may tabulate them thus. Sums of Sums of Squares Observation 4, 12, 14 Errours Errours. of Errours. Series (1) 4, 9, 14 0, 3, 0 3 9 " (2) 6, 10, 14 2, 2, 0 4 8 " (3) 5, 10, 15 1, 2, 1 4 6