The two first books are almost strictly, as Kepler styles them, geometrical, relating in great measure to the inscription of regular polygons in a circle. The following passage is curious, presenting an analogous idea to that contained in one of the extracts already given from the Commentaries on Mars. "The heptagon, and all other polygons and stars beyond it, which have a prime number of sides, and all other figures derived from them, cannot be inscribed geometrically in a circle; although their sides have a necessary magnitude, it is equally a matter of necessity that we remain ignorant of it. This is a question of great importance, for on this account is it that the heptagon, and other figures of this kind, have not been employed by God in the adornment of the world, as the other intelligible figures are employed which have been already explained." Kepler then introduces the algebraical equation, on the solution of which this problem depends, and makes a remark which is curious at this period of the history of algebra—that the root of an equation which cannot be accurately found, may yet be found within any degree of approximation by an expert calculator. In conclusion he again remarks that "the side of the heptagon has no place among scientific existences, since its formal description is impossible, and therefore it cannot be known by the human mind, since the possibility of description precedes the possibility of knowledge; nor is it known even by the simple eternal act of an omniscient mind, because its nature belongs to things which cannot be known. And yet this scientific nonentity has some scientific properties, for if a heptagon were described in a circle, the proportion of its sides would have analogous proportions." The third book is a treatise on music, in the confined and ordinary sense in which we now use that word, and apparently a sober and rational one, at least as nearly so as Kepler could be trusted to write on a subject so dangerous to his discretion. All the extravagance of the work seems reserved for the fourth book, the title of which already conveys some notion of the nature of its contents. In this book he has collected the substance of the astrological opinions scattered through his other works. We shall content ourselves with merely citing his own words, without any attempt to explain the difference between the astrology which he believed, and that which he contemptuously rejected. The distinctive line seems very finely drawn, and as both one and the other are now discarded by all who enjoy the full use of their reasoning powers, it is not of much consequence that it should be accurately traced.