Thus when we have a series of good Observations, and know the argument upon which their change of magnitude depends, the Method of Curves enables us to ascertain, almost at a glance, the law of the change; and by further attention, may be made to give us a formula with great accuracy. The Method enables us to perceive, among our observations, an order, which without the method, is concealed in obscurity and perplexity. 206
3. But the Method of Curves not only enables us to obtain laws of nature from good Observations, but also, in a great degree, from observations which are very imperfect. For the imperfection of observations may in part be corrected by this consideration;—that though they may appear irregular, the correct facts which they imperfectly represent, are really regular. And the Method of Curves enables us to remedy this apparent irregularity, at least in part. For when Observations thus imperfect are laid down as Ordinates, and their extremities connected by a line, we obtain, not a smooth and flowing curve, such as we should have if the observations contained only the rigorous results of regular laws; but a broken and irregular line, full of sudden and capricious twistings, and bearing on its face marks of irregularities dependent, not upon law, but upon chance. Yet these irregular and abrupt deviations in the curve are, in most cases, but small in extent, when compared with those bendings which denote the effects of regular law. And this circumstance is one of the great grounds of advantage in the Method of Curves. For when the observations thus laid down present to the eye such a broken and irregular line, we can still see, often with great ease and certainty, what twistings of the line are probably due to the irregular errours of observation; and can at once reject these, by drawing a more regular curve, cutting off all such small and irregular sinuosities, leaving some to the right and some to the left; and then proceeding as if this regular curve, and not the irregular one, expressed the observations. In this manner, we suppose the errours of observation to balance each other; some of our corrected measures being too great and others too small, but with no great preponderance either way. We draw our main regular curve, not through the points given by our observations, but among them: drawing it, as has been said by one of the philosophers[30] who first systematically used this method, ‘with a bold but careful hand.’ 207 The regular curve which we thus obtain, thus freed from the casual errours of observation, is that in which we endeavour to discover the laws of change and succession.
[30] Sir J. Herschel, Ast. Soc. Trans. vol. v. p. 1.
4. By this method, thus getting rid at once, in a great measure, of errours of observation, we obtain data which are more true than the individual facts themselves. The philosopher’s business is to compare his hypotheses with facts, as we have often said. But if we make the comparison with separate special facts, we are liable to be perplexed or misled, to an unknown amount, by the errours of observation; which may cause the hypothetical and the observed result to agree, or to disagree, when otherwise they would not do so. If, however, we thus take the whole mass of the facts, and remove the errours of actual observation[31], by making the curve which expresses the supposed observation regular and smooth, we have the separate facts corrected by their general tendency. We are put in possession, as we have said, of something more true than any fact by itself is.
[31] Ib. vol. v. p. 4.
One of the most admirable examples of the use of this Method of Curves is found in Sir John Herschel’s Investigation of the Orbits of Double Stars[32]. The author there shows how far inferior the direct observations of the angle of position are, to the observations corrected by a curve in the manner above stated. ‘This curve once drawn,’ he says, ‘must represent, it is evident, the law of variation of the angle of position, with the time, not only for instants intermediate between the dates of observations, but even at the moments of observation themselves, much better than the individual raw observations can possibly (on an average) do. It is only requisite to try a case or two, to be satisfied that by substituting the curve for the points, we have made a nearer approach to nature, and in a great measure eliminated errours of observation.’ ‘In following the graphical process,’ he adds, ‘we have a conviction almost approaching to moral certainty that 208 we cannot be greatly misled.’ Again, having thus corrected the raw observations, he makes another use of the graphical method, by trying whether an ellipse can be drawn ‘if not through, at least among the points, so as to approach tolerably near them all; and thus approaching to the orbit which is the subject of investigation.’
[32] Ib.
5. The Obstacles which principally impede the application of the Method of Curves are (I.) our ignorance of the arguments of the changes, and (II.) the complication of several laws with one another.
(I.) If we do not know on what quantity those changes depend which we are studying, we may fail entirely in detecting the law of the changes, although we throw the observations into curves. For the true argument of the change should, in fact, be made the abscissa of the curve. If we were to express, by a series of ordinates, the hour of high water on successive days, we should not obtain, or should obtain very imperfectly, the law which these times follow; for the real argument of this change is not the solar hour, but the hour at which the moon passes the meridian. But if we are supposed to be aware that this is the argument, (which theory suggests and trial instantly confirms) we then do immediately obtain the primary Rules of the Time of High Water, by throwing a series of observations into a Curve, with the Hour of the Moon’s Transit for the abscissa.
In like manner, when we have obtained the first great or Semi-mensual Inequality of the tides, if we endeavour to discover the laws of other Inequalities by means of curves, we must take from theory the suggestion that the Arguments of such inequalities will probably be the parallax and the declination of the moon. This suggestion again is confirmed by trial; but if we were supposed to be entirely ignorant of the dependence of the changes of the tide on the Distance and Declination of the moon, the curves would exhibit unintelligible and seemingly capricious changes. For by the effect of the Inequality arising from the Parallax, the convexities of the curves which belong to the 209 spring tides, are in some years made alternately greater and less all the year through; while in other years they are made all nearly equal. This difference does not betray its origin, till we refer it to the Parallax; and the same difficulty in proceeding would arise if we were ignorant that the moon’s Declination is one of the Arguments of tidal changes.