In now taking leave of the system of epicycles let it be remarked that it has been held up to ridicule more than it deserves. On reading Airy’s account of epicycles, in the beautifully clear language of his _Six Lectures on Astronomy_, the impression is made that the jointed bars there spoken of for describing the circles were supposed to be real. This is no more the case than that the spheres of Eudoxus and Callippus were supposed to be real. Both were introduced only to illustrate the mathematical conception upon which the solar, planetary, and lunar tables were constructed. The epicycles represented nothing more nor less than the first terms in the Fourier series, which in the last century has become a basis of such calculations, both in astronomy and physics generally. [Illustration: “QUADRANS MURALIS SIVE TICHONICUS.” With portrait of Tycho Brahe, instruments, etc., painted on the wall; showing assistants using the sight, watching the clock, and recording. (From the author’s copy of the _Astronomiæ Instauratæ Mechanica_.)] FOOTNOTES: [1] For definition see p. 22. [2] _Ibid_. [3] For definition see p. 18. [4] For definition see p. 18. [5] In his great book Copernicus says: “The movement of the heavenly bodies is uniform, circular, perpetual, or else composed of circular movements.” In this he proclaimed himself a follower of Pythagoras (see p. 14), as also when he says: “The world is spherical because the sphere is, of all figures, the most perfect” (Delambre, _Ast. Mod. Hist_., pp. 86, 87). [6] Kepler tells us that Tycho Brahe was pleased with this device, and adapted it to his own system. [7] _Hist. Ast._, vol. i., p. 354. [8] _Hist. of Phys. Ast._, p. vii. [9] “Est enim Astronomi proprium, historiam motuum coelestium diligenti et artificiosa observatione colligere. Deinde causas earundem, seu hypotheses, cum veras assequi nulla ratione possit ... Neque enim necesse est, eas hypotheses esse veras, imo ne verisimiles quidem, sed sufficit hoc usum, si calculum observationibus congruentem exhibeant.” BOOK II. THE DYNAMICAL PERIOD 5. DISCOVERY OF THE TRUE SOLAR SYSTEM—TYCHO BRAHE—KEPLER. During the period of the intellectual and aesthetic revival, at the beginning of the sixteenth century, the “spirit of the age” was fostered by the invention of printing, by the downfall of the Byzantine Empire, and the scattering of Greek fugitives, carrying the treasures of literature through Western Europe, by the works of Raphael and Michael Angelo, by the Reformation, and by the extension of the known world through the voyages of Spaniards and Portuguese. During that period there came to the front the founder of accurate observational astronomy. Tycho Brahe, a Dane, born in 1546 of noble parents, was the most distinguished, diligent, and accurate observer of the heavens since the days of Hipparchus, 1,700 years before.