ON THE HARMONY OF FIGURES.

Lazarus Schöner’s Lazarus Schöner (1543–1607) was a German mathematician and follower of Ramus. opinion on the five solid figures.

...with harshness of speech, great recklessness, and in a manner most unworthy of such a man. Following this persuasion of Ramus, look, Schöner himself also believed that the regular bodies The five Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. are of no use; and not only this, but he also neglected or despised Proclus, following the judgment of Ramus. From Proclus, he could have learned the use of the five bodies both in Euclid’s Elements and in the fabric of the world. Indeed, the student was more fortunate than the Master, because Schöner joyfully accepted the use of the bodies revealed by me in the Fabric of the World Kepler refers here to his first major work, the Mysterium Cosmographicum (1596), where he proposed that the distances between the planets were determined by the five Platonic solids., which Ramus had rejected when it was emphasized by Proclus. For what does it matter if the Pythagoreans attributed these figures to the elements, and not, as I do, to the Spheres of the World? Ramus should have striven to remove this error of theirs regarding the true subject of the figures, as I have done; he should not have abolished this entire Philosophy with a single tyrannical word. What if the Pythagoreans taught this same thing as I do, but wove their opinion in veils of words? Does not the Copernican form of the World exist in Aristotle himself, though wrongly refuted by him under other names—while they called the Sun "Fire" and the Moon the "Counter-Earth"? Kepler suggests that ancient Pythagorean concepts like the "Central Fire" were actually hidden references to a sun-centered (heliocentric) universe like that of Copernicus.

The mystical interpretation of the Pythagoreans concerning the five figures.

For if the arrangement of the spheres was the same among the Pythagoreans as among Copernicus; if the five Bodies were known, and the necessity of their fivefold number; if they all constantly taught that the five bodies are the Archetypes of the parts of the World; how much is left for us but to believe that their opinion, read as a riddle by Aristotle, was refuted according to the literal sense of the words rather than their true meaning? While Aristotle reads "Earth," to which they gave the Cube, they perhaps meant Saturn, whose Sphere is moved away from Jupiter by the interposition of a Cube. Indeed, the common people ascribe rest to the Earth, but Saturn has been allotted the slowest motion, closest to rest; for this reason, even among the Hebrews, it obtained its name from "Rest" The Hebrew name for Saturn is Shabbatai, derived from the root sh-b-t, meaning to rest or cease, as in "Sabbath.". Likewise, Aristotle reads that the Octahedron was given to Air, when they perhaps meant Mercury, whose sphere is enclosed by an Octahedron; and Mercury is no less swift (being indeed the swiftest of all) than mobile Air is considered to be. By the word "Fire," perhaps Mars was suggested—which elsewhere is also named Pyroeis original: "Pyroeis," from the Greek for "fiery." from fire—and to it the Tetrahedron was given, perhaps because its sphere is enclosed by this figure. And under the veil of "Water," to which the Icosahedron was attributed, the star of Venus could be hidden (as its course is contained by the Icosahedron), because the humors In ancient medicine and astrology, humors were bodily fluids associated with planetary influences; Venus was linked to moisture. are subject to Venus, and she herself is said to have risen from the foam of the Sea, hence the name Aphrodite The Greek name for Venus, derived from aphros, meaning "foam.". Finally, the word "World" could have signified the Earth; and the Dodecahedron ascribed to the World, because the Earth’s course is contained by this figure, divided into twelve parts of longitude Referring to the twelve signs of the Zodiac., just as that figure is contained by twelve planes in its whole circuit. Therefore, the fact that the five figures were distributed in this way in the Mysteries of the Pythagoreans—not among the Elements as Aristotle believed, but among the Planets themselves—is most strongly confirmed by Proclus, who handed down this purpose of Geometry: that it teaches how the heaven received suitable figures for its specific parts.

Nor is there yet an end to the damage that Ramus has done us; look at Snell, the most skillful of modern Geometers, clearly supporting Ramus in his preface to the Problems of Ludolph van Ceulen Ludolph van Ceulen (1540–1610) was a Dutch mathematician famous for calculating pi to 35 decimal places.. First he says, Willebrord Snell’s Willebrord Snell (1580–1626) was a Dutch mathematician and astronomer. opinion on Binomials: that the division of ineffables into thirteen species is useless for practice. I concede this, if he recognizes no use except in common life, and if no con-

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