Another thing to be observed is, that, in many cases, the result of a number of para .
cular facts, or the ,collective instance arising from them, can only be found out by geome try, which, therefore, becomes a necessary instrument in completing the work of induction. An example, which the science of optics furnishes, will make this clearer than any general description. When light passes from one transparent medium to another it is refracted, that is, it ceases to go on in a straight, line, and the angle which the incident ray makes with the superficies which bounds the two media, determines that which the refracted ray makes with the same superficies. Now, if we would learn any thing about the relation which these angles bear to one another, we must have recourse to experiment, and all that. experiment can do is, for any particular angle of incidence, to determihe the correspond ing angle of refraction. This may be done in innumerable eases ; but, with respect to the general rule which, in every possible case, determines the one of those angles from the other, or expresses the constant and invariable relation which subsists between them,— with respect to it, experiment gives no direct information. The methods of geometry must therefqre be called in to our assistance, which, when a constant though unknown re lation subsists between two angles, or two variable quantities of any kind, and when an in definite number of values of those quantities are given, furnishes infallible means of dis covering that unknown relation, either accurately, or at least by approximation. In this way it has been found, that, when the two media remain the same, the -cosines of the angles above mentioned have a constant ratio to one another. Thus it appears, that, after experiment has done its utmost, geometry must be applied before the business of induction can be completed. This can only happen when the experiments afford accurate mea sures of the quantities concerned, like the instantice radii, curriculi, &c. and this advantage of admitting generalization with so much certainty is one of their properties, of which it does not appear that even Bacon himself was aware.
Again, from the intimate connection which prevails among the principles of science, the sums of one investigation must often contribute to the success of another, in such a de gree as to make it unnecessary to employ the complete apparatus of induction froni the beginning. When certain leading principles have been once established, they serve, in
new investigations, to narrow the limits within which the thing sought for is contained, and enable the inquirer to arrive more speedily at the truth.
Thus, suppose that, after the nature of the reflection and refraction of light, and par ticularly of the colours produced by the latter, had been discovered by experiment, the cause of the rainbow were to be inquired into. It would, after a little consideration, ap pear probable, that the phenomenon to be explained depends on the reflection and refrac tion of light by the rain falling from a cloud opposite to the sun, Now, since the nature of reflection and refraction are supposed known, we have the principles previously ascertained which are likely to assist in the explanation of the rainbow. We have no occasion, there fore, to enter on the inquiry, as if the powers to be investigated were wholly unknown. It is the combination of them only which is unknown, and our business is to seek so to com bine them, that the result may correspond with the appearances. This last is precisely what Newton accomplished, when, by deducing from the known laws of refraction and re flection the breadth of the coloured arch, the diameter of the circle of which it is a part, and the relation of the latter to the place of the spectator and of the sun, he found all these to come out from his calculus, just as they observed in nature. Thus he proved the truth of his solution by the most clear and irresistible evidence.
The strict method of Bacon is therefore only necessary where the thing to be explained is new, and where we have no knowledge, or next to none, of the powers employed. This is but rarely the case, at least in some of the branches of Physics ; and, therefore, it occurs most commonly in actual investigation, that the inquirer finds himself limited, almost from the first outset, to two or three hypotheses, all other suppositions involving inconsistencies which cannot for a moment be admitted.. His business, therefore, is to compare the re sults of these hypotheses, and to what consequences may in any case arise from the one that would not arise from the other. If any such difference can be found, and if the matter is a subject of experiment, we have then an instantia crucis which must decide the question.