2. For this purpose, I remark that the Colligation of ascertained Facts into general Propositions may be considered as containing three steps, which I shall term the Selection of the Idea, the Construction of the Conception, and the Determination of the Magnitudes. It will be recollected that by the word Idea, (or Fundamental Idea,) used in a peculiar sense, I mean certain wide and general fields of intelligible relation, such as Space, Number, Cause, Likeness; while by Conception I denote more special modifications of these ideas, as a circle, a square number, a uniform force, a like form of flower. Now in order to establish any law by reference to facts, we must select the true Idea and the true Conception. For example; when Hipparchus found[25] that the distance of the bright star Spica Virginis from the equinoxial point had increased by two degrees in about two hundred years, and desired to reduce this change to a law, he had first to assign, if possible, the idea on which it depended;—whether it was regulated for instance, by space, or by time; whether it was determined by the positions of other stars at each moment, or went on progressively with the lapse of ages. And when there was found reason to select time as the regulative idea of this change, it was then to be determined how the change went on with the time;—whether uniformly, or in some other manner: the conception, or the rule of the progression, was to be 188 rightly constructed. Finally, it being ascertained that the change did go on uniformly, the question then occurred what was its amount:—whether exactly a degree in a century, or more, or less, and how much: and thus the determination of the magnitude completed the discovery of the law of phenomena respecting this star.
[25] Hist. Ind. Sc. b. iii. c. iv. sect. 3.
3. Steps similar to these three may be discerned in all other discoveries of laws of nature. Thus, in investigating the laws of the motions of the sun, moon or planets, we find that these motions may be resolved, besides a uniform motion, into a series of partial motions, or Inequalities; and for each of these Inequalities, we have to learn upon what it directly depends, whether upon the progress of time only, or upon some configuration of the heavenly bodies in space; then, we have to ascertain its law; and finally, we have to determine what is its amount. In the case of such Inequalities, the fundamental element on which the Inequality depends, is called by mathematicians the Argument. And when the Inequality has been fully reduced to known rules, and expressed in the form of a Table, the Argument is the fundamental Series of Numbers which stands in the margin of the Table, and by means of which we refer to the other Numbers which express the Inequality. Thus, in order to obtain from a Solar Table the Inequality of the sun’s annual motion, the Argument is the Number which expresses the day of the year; the Inequalities for each day being (in the Table) ranged in a line corresponding to the days. Moreover, the Argument of an Inequality being assumed to be known, we must, in order to calculate the Table, that is, in order to exhibit the law of nature, know also the Law of the Inequality, and its Amount. And the investigation of these three things, the Argument, the Law, and the Amount of the Inequality, represents the three steps above described, the Selection of the Idea, the Construction of the Conception, and the Determination of the Magnitude.
4. In a great body of cases, mathematical language and calculation are used to express the connexion 189 between the general law and the special facts. And when this is done, the three steps above described may be spoken of as the Selection of the Independent Variable, the Construction of the Formula, and the Determination of the Coefficients. It may be worth our while to attend to an exemplification of this. Suppose then, that, in such observations as we have just spoken of, namely, the shifting of a star from its place in the heavens by an unknown law, astronomers had, at the end of three successive years, found that the star had removed by 3, by 8, and by 15 minutes from its original place. Suppose it to be ascertained also, by methods of which we shall hereafter treat, that this change depends upon the time; we must then take the time, (which we may denote by the symbol t,) for the independent variable. But though the star changes its place with the time, the change is not proportional to the time; for its motion which is only 3 minutes in the first year, is 5 minutes in the second year, and 7 in the third. But it is not difficult for a person a little versed in mathematics to perceive that the series 3, 8, 15, may be obtained by means of two terms, one of which is proportional to the time, and the other to the square of the time; that is, it is expressed by the formula at + btt. The question then occurs, what are the values of the coefficients a and b; and a little examination of the case shows us that a must be 2, and b, 1: so that the formula is 2t + tt. Indeed if we add together the series 2, 4, 6, which expresses a change proportional to the time, and 1, 4, 9, which is proportional to the square of the time, we obtain the series 3, 8, 15, which is the series of numbers given by observation. And thus the three steps which give us the Idea, the Conception, and the Magnitudes; or the Argument, the Law, and the Amount, of the change; give us the Independent Variable, the Formula, and the Coefficients, respectively.
We now proceed to offer some suggestions of methods by which each of these steps may be in some degree promoted. 190
Sect. II.—Of the Selection of the Fundamental Idea.
5. When we turn our thoughts upon any assemblage of facts, with a view of collecting from them some connexion or law, the most important step, and at the same time that in which rules can least aid us, is the Selection of the Idea by which they are to be collected. So long as this idea has not been detected, all seems to be hopeless confusion or insulated facts; when the connecting idea has been caught sight of, we constantly regard the facts with reference to their connexion, and wonder that it should be possible for any one to consider them in any other point of view.
Thus the different seasons, and the various aspects of the heavenly bodies, might at first appear to be direct manifestations from some superior power, which man could not even understand: but it was soon found that the ideas of time and space, of motion and recurrence, would give coherency to many of the phenomena. Yet this took place by successive steps. Eclipses, for a long period, seemed to follow no law; and being very remarkable events, continued to be deemed the indications of a supernatural will, after the common motions of the heavens were seen to be governed by relations of time and space. At length, however, the Chaldeans discovered that, after a period of eighteen years, similar sets of eclipses recur; and, thus selecting the idea of time, simply, as that to which these events were to be referred, they were able to reduce them to rule; and from that time, eclipses were recognized as parts of a regular order of things. We may, in the same manner, consider any other course of events, and may enquire by what idea they are bound together. For example, if we take the weather, years peculiarly wet or dry, hot and cold, productive and unproductive, follow each other in a manner which, at first sight at least, seems utterly lawless and irregular. Now can we in any way discover some rule and order in these occurrences? Is there, for example, in these events, as in eclipses, a certain cycle of years, after which like 191 seasons come round again? or does the weather depend upon the force of some extraneous body—for instance, the moon—and follow in some way her aspects? or would the most proper way of investigating this subject be to consider the effect of the moisture and heat of various tracts of the earth’s surface upon the ambient air? It is at our choice to try these and other modes of obtaining a science of the weather: that is, we may refer the phenomena to the idea of time, introducing the conception of a cycle;—or to the idea of external force, by the conception of the moon’s action;—or to the idea of mutual action, introducing the conceptions of thermotical and atmological agencies, operating between different regions of earth, water, and air.
6. It may be asked, How are we to decide in such alternatives? How are we to select the one right idea out of several conceivable ones? To which we can only reply, that this must be done by trying which will succeed. If there really exist a cycle of the weather, as well as of eclipses, this must be established by comparing the asserted cycle with a good register of the seasons, of sufficient extent. Or if the moon really influence the meteorological conditions of the air, the asserted influence must be compared with the observed facts, and so accepted or rejected. When Hipparchus had observed the increase of longitude of the stars, the idea of a motion of the celestial sphere suggested itself as the explanation of the change; but this thought was verified only by observing several stars. It was conceivable that each star should have an independent motion, governed by time only, or by other circumstances, instead of being regulated by its place in the sphere; and this possibility could be rejected by trial alone. In like manner, the original opinion of the composition of bodies supposed the compounds to derive their properties from the elements according to the law of likeness; but this opinion was overturned by a thousand facts; and thus the really applicable Idea of Chemical Composition was introduced in modern times. In what has already been said on the History of Ideas, we have seen how each science was in a state 192 of confusion and darkness till the right idea was introduced.
7. No general method of evolving such ideas can be given. Such events appear to result from a peculiar sagacity and felicity of mind;—never without labour, never without preparation;—yet with no constant dependence upon preparation, or upon labour, or even entirely upon personal endowments. Newton explained the colours which refraction produces, by referring each colour to a peculiar angle of refraction, thus introducing the right idea. But when the same philosopher tried to explain the colours produced by diffraction, he erred, by attempting to apply the same idea, (the course of a single ray,) instead of applying the truer idea, of the interference of two rays. Newton gave a wrong rule for the double refraction of Iceland spar, by making the refraction depend on the edges of the rhombohedron: Huyghens, more happy, introduced the idea of the axis of symmetry of the solid, and thus was able to give the true law of the phenomena.