The Harmony of the World

ON DEMONSTRATION. BOOK I.

...of whether physical contemplations have a use for life. But why does he not follow Proclus, whom he cites, who recognizes some greater good in Geometry than the arts necessary for survival? If he had, the use of the tenth book would indeed have appeared in evaluating the types of figures. Snellius Willebrord Snellius (1580–1626), a Dutch mathematician famous for the law of refraction. cites authors and geometers who do not use the tenth book of Euclid. Truly, all of them treat either linear or solid problems, and deal with figures or quantities of a kind that do not have their end within themselves, but clearly tend toward other uses, and would not be sought out without them. But regular Figures are sought for their own sake as Archetypes; they have their perfection within themselves. They are among the subjects of planar Problems, despite the fact that a solid is also enclosed by planar faces; similarly, the material of the tenth book pertains primarily to planes. Why then should heterogeneous things be cited? Or why is a cargo esteemed as worthless because Codrus A stereotypical name for a poor man in Roman poetry. does not buy it to feed his belly, but Cleopatra buys it to adorn her ears? Is it merely a cross fixed for their wits? original: "Crux tantùm defixa est ingenijs?" Kepler is mocking those who find the study of irrational numbers to be a form of mental torture. Certainly it is for those who vex the Ineffables The "Ineffables" refers to incommensurable or irrational magnitudes, which cannot be expressed as simple ratios of whole numbers. with numbers—that is, by trying to speak them. But I treat these species not with numbers, nor through Algebra, but by the reasoning of the Mind; indeed, because I have no need of them to calculate the accounts of markets, but to explain the causes of things.

He Snellius/Ramus thinks these subtle matters should be separated from the Elementary Arrangement original: "στοιχειώσεı" (stoicheiōsei), referring to the systematic building up of geometric principles. and hidden away in Libraries. He acts entirely as a faithful disciple of Ramus, and spends his effort quite foolishly: Ramus took away the form from the Euclidean Building, he cast down the roof—the five [regular] bodies—and when these were removed, the whole structure was dissolved. The walls stand cracked, the arches threatening to ruin. Snellius therefore even takes away the Mortar, as if it has no use except for the solidity of a house joined together under the five figures. O happy understanding of a disciple! How skillfully he learned to understand Euclid from Ramus: namely, they think the Elements original: "Στοιχεῖα" (Stoicheia). are so named because there is found in Euclid an all-encompassing abundance of propositions, problems, and Theorems for every kind of Quantity and the arts occupied with them. Yet the book was called an Elementary Arrangement because of its form, because the following proposition always rests upon the preceding one, all the way to the last proposition of the last book (and partly the ninth book), which cannot do without any of the prior ones. They turn the Architect into a wood-cutter or a supplier of materials, thinking that Euclid wrote his book for the sake of serving all others, while he himself alone would have no house of his own. But enough of these matters in this place; I must return to the main point of the discourse.

The occasion for this Book I

For when I perceived that the true and genuine differences of geometric things (from which I must derive the causes of Harmonic Proportions) were generally ignored entirely; that Euclid, who had passed them down with great care, was being exploded and driven out by the quibbles of Ramus; and that, with the noise of the wanton disturbing him, he was heard by no one—or was telling the mysteries of Philosophy to the deaf; and that Proclus, who could have opened the mind of Euclid, unearthed hidden things, and made difficult concepts easy to grasp, was even a laughingstock and had not continued his Commentaries all the way to the tenth book: I saw that I absolutely had to do this. I decided that at the beginning, I would transcribe from the tenth book of Euclid those things which especially served my present purpose, and bring to light the series of matters in that book, with certain divisions cast between them...