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 2_t_ + _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.