11. In order to obtain very great accuracy, very large masses of observations are often employed by philosophers, and the accuracy of the result increases with the multitude of observations. The immense collections of astronomical observations which have in this manner been employed in order to form and correct the Tables of the celestial motions are perhaps the most signal instances of the attempts to obtain 214 accuracy by this accumulation of observations. Delambre’s Tables of the Sun are founded upon nearly 3000 observations; Burg’s Tables of the Moon upon above 4000.
But there are other instances hardly less remarkable. Mr. Lubbock’s first investigations of the laws of the tides of London[34], included above 13,000 observations, extending through nineteen years; it being considered that this large number was necessary to remove the effects of accidental causes[35]. And the attempts to discover the laws of change in the barometer have led to the performance of labours of equal amount: Laplace and Bouvard examined this question by means of observations made at the Observatory of Paris, four times every day for eight years.
[34] Phil. Trans. 1831.
[35] This period of nineteen years was also selected for a reason which is alluded to in a former [note]. It was thought that this period secured the inquirer from the errours which might be produced by the partial coincidence of the Arguments of different irregularities; for example, those due to the moon’s Parallax and to the moon’s Declination. It has since been found (Phil. Tr. 1838. On the Determination of the Laws of the Tides from Short Series of Observations), that with regard to Parallax at least, the Means of one year give sufficient accuracy.
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.
| Observation | 4, 12, 14 | Errours | Sums of Errours | Sums of Squares 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 |
Here, although the first series gives the sum of the 216 errours less than the others, the third series gives the sum of the squares of the errours least; and is therefore, by the proposition on which this Method depends, the most probable series of the three.