7. In constructing the formulæ by which laws of nature are expressed, although the first object is to assign the Law of the Phenomena, philosophers have, in almost all cases, not proceeded in a purely empirical manner, to connect the observed numbers by some expression of calculation, but have been guided, in the selection of their formula, by some _Hypothesis_ respecting the mode of connexion of the facts. Thus the formula of Dulong and Petit above given was suggested by the Theory of Exchanges; the first attempts at the resolution of the heavenly motions into circular functions were clothed in the hypothesis of Epicycles. And this was almost inevitable. 'We must confess,' says Copernicus[28\3], 'that the celestial motions are circular, or compounded of several circles, since their inequalities observe a fixed law, and recur in value at certain intervals, which could not be except they were circular: for a circle alone can make that quantity which has occurred recur again.' In like manner the first publication of the _Law of the Sines_, the true formula of optical refraction, was accompanied by Descartes with an hypothesis, in which an explanation of the law was pretended. In such cases, the mere comparison of observations may long fail in suggesting the true formulæ. The fringes of shadows and other diffracted colours were studied in vain by Newton, Grimaldi, Comparetti, the elder Herschel, and Mr. Brougham, so long as these inquirers attempted merely to trace the laws of the facts as they appeared in themselves; while Young, Fresnel, Fraunhofer, Schwerdt, and others, determined these laws in the most rigorous manner, when they applied to the observations the Hypothesis of Interferences. [Note 28\3: _De Rev._ l. i. c. iv.] 8. But with all the aid that Hypotheses and Calculation can afford, the construction of true formulæ, in {201} those cardinal discoveries by which the progress of science has mainly been caused, has been a matter of great labour and difficulty, and of good fortune added to sagacity. In the _History of Science_, we have seen how long and how hard Kepler laboured, before he converted the formula for the planetary motions, from an _epicyclical_ combination, to a simple _ellipse_. The same philosopher, labouring with equal zeal and perseverance to discover the formula of optical refraction, which now appears to us so simple, was utterly foiled. Malus sought in vain the formula determining the Angle at which a transparent surface polarizes light: Sir D. Brewster[29\3], with a happy sagacity, discovered the formula to be simply this, that the _index_ of refraction is the _tangent_ of the angle of polarization. [Note 29\3: _Hist. Ind. Sc._ b. ix. c. vi.]