In like manner, if we try to reduce to law any meteorological changes, those of the Height of the Barometer for instance, we find that we can make little progress in the investigation, precisely because we do not know the Argument on which these changes depend. That there is a certain regular diurnal change of small amount, we know; but when we have abstracted this Inequality, (of which the Argument is the time of day,) we find far greater Changes left behind, from day to day and from hour to hour; and we express these in curves, but we cannot reduce them to Rule, because we cannot discover on what numerical quantity they depend. The assiduous study of barometrical observations, thrown into curves, may perhaps hereafter point out to us what are the relations of time and space by which these variations are determined; but in the mean time, this subject exemplifies to us our remark, that the method of curves is of comparatively small use, so long as we are in ignorance of the real Arguments of the Inequalities.
6. (II.) In the next place, I remark that a difficulty is thrown in the way of the Method of Curves by the Combination of several laws one with another. It will readily be seen that such a cause will produce a complexity in the curves which exhibit the succession of facts. If, for example, we take the case of the Tides, the Height of high water increases and diminishes with the Approach of the sun to, and its Recess from, the syzygies of the moon. Again, this Height increases and diminishes as the moon’s Parallax increases and diminishes; and again, the Height diminishes when the Declination increases, and vice versa; and all these Arguments of change, the Distance from Syzygy, the Parallax, the Declination, complete their circuit and 210 return into themselves in different periods. Hence the curve which represents the Height of high water has not any periodical interval in which it completes its changes and commences a new cycle. The sinuosity which would arise from each Inequality separately considered, interferes with, disguises, and conceals the others; and when we first cast our eyes on the curve of observation, it is very far from offering any obvious regularity in its form. And it is to be observed that we have not yet enumerated all the elements of this complexity: for there are changes of the tide depending upon the Parallax and Declination of the Sun as well as of the Moon. Again; besides these changes, of which the Arguments are obvious, there are others, as those depending upon the Barometer and the Wind, which follow no known regular law, and which constantly affect and disturb the results produced by other laws.
In the Tides, and in like manner in the motions of the Moon, we have very eminent examples of the way in which the discovery of laws may be rendered difficult by the number of laws which operate to affect the same quantity. In such cases, the Inequalities are generally picked out in succession, nearly in the order of their magnitudes. In this way there were successively collected, from the study of the Moon’s motions by a series of astronomers, those Inequalities which we term the Equation of the Center, the Evection, the Variation, and the Annual Equation. These Inequalities were not, in fact, obtained by the application of the Method of Curves; but the Method of Curves might have been applied to such a case with great advantage. The Method has been applied with great industry and with remarkable success to the investigation of the laws of the Tides; and by the use of it, a series of Inequalities both of the Times and of the Heights of high water has been detected, which explain all the main features of the observed facts. 211
Sect. II.—The Method of Means.
7. The Method of Curves, as we have endeavoured to explain above, frees us from the casual and extraneous irregularities which arise from the imperfection of observation; and thus lays bare the results of the laws which really operate, and enables us to proceed in search of those laws. But the Method of Curves is not the only one which effects such a purpose. The errours arising from detached observations may be got rid of, and the additional accuracy which multiplied observations give may be obtained, by operations upon the observed numbers, without expressing them by spaces. The process of curves assumes that the errours of observation balance each other;—that the accidental excesses and defects are nearly equal in amount;—that the true quantities which would have been observed if all accidental causes of irregularity were removed, are obtained, exactly or nearly, by selecting quantities, upon the whole, equally distant from the extremes of great and small, which our imperfect observations offer to us. But when, among a number of unequal quantities, we take a quantity equally distant from the greater and the smaller, this quantity is termed the Mean of the unequal quantities. Hence the correction of our observations by the method of curves consists in taking the Mean of the observations.
8. Now without employing curves, we may proceed arithmetically to take the Mean of all the observed numbers of each class. Thus, if we wished to know the Height of the spring tide at a given place, and if we found that four different spring tides were measured as being of the height of ten, thirteen, eleven, and fourteen feet, we should conclude that the true height of the tide was the Mean of these numbers,—namely, twelve feet; and we should suppose that the deviation from this height, in the individual cases, arose from the accidents of weather, the imperfections of observation, or the operation of other laws, besides the alternation of spring and neap tides. 212
This process of finding the Mean of an assemblage of observed numbers is much practised in discovering, and still more in confirming and correcting, laws of phenomena. We shall notice a few of its peculiarities.
9. The Method of Means requires a knowledge of the Argument of the changes which we would study; for the numbers must be arranged in certain Classes, before we find the Mean of each Class; and the principle on which this arrangement depends is the Argument. This knowledge of the Argument is more indispensably necessary in the Method of Means than in the Method of Curves; for when Curves are drawn, the eye often spontaneously detects the law of recurrence in their sinuosities; but when we have collections of Numbers, we must divide them into classes by a selection of our own. Thus, in order to discover the law which the heights of the tide follow, in the progress from spring to neap, we arrange the observed tides according to the day of the moon’s age; and we then take the mean of all those which thus happen at the same period of the Moon’s Revolution. In this manner we obtain the law which we seek; and the process is very nearly the same in all other applications of this Method of Means. In all cases, we begin by assuming the Classes of measures which we wish to compare, the Law which we could confirm or correct, the Formula of which we would determine the coefficients.
10. The Argument being thus assumed, the Method of Means is very efficacious in ridding our inquiry of errours and irregularities which would impede and perplex it. Irregularities which are altogether accidental, or at least accidental with reference to some law which we have under consideration, compensate each other in a very remarkable way, when we take the Means of many observations. If we have before us a collection of observed tides, some of them may be elevated, some depressed by the wind, some noted too high and some too low by the observer, some augmented and some diminished by uncontemplated changes in the moon’s distance or motion: but in the course of a year or two at the longest, all these causes of irregularity balance 213 each other; and the law of succession, which runs through the observations, comes out as precisely as if those disturbing influences did not exist. In any particular case, there appears to be no possible reason why the deviation should be in one way, or of one moderate amount, rather than another. But taking the mass of observations together, the deviations in opposite ways will be of equal amount, with a degree of exactness very striking. This is found to be the case in all inquiries where we have to deal with observed numbers upon a large scale. In the progress of the population of a country, for instance, what can appear more inconstant, in detail, than the causes which produce births and deaths? yet in each country, and even in each province of a country, the proportions of the whole numbers of births and deaths remain nearly constant. What can be more seemingly beyond the reach of rule than the occasions which produce letters that cannot find their destination? yet it appears that the number of ‘dead letters’ is nearly the same from year to year. And the same is the result when the deviations arise, not from mere accident, but from laws perfectly regular, though not contemplated in our investigation[33]. Thus the effects of the Moon’s Parallax upon the Tides, sometimes operating one way and sometimes another, according to certain rules, are quite eliminated by taking the Means of a long series of observations; the excesses and defects neutralizing each other, so far as concerns the effect upon any law of the tides which we would investigate.
[33] Provided the argument of the law which we neglect have no coincidence with the argument of the law which we would determine.