The Harmony of the World

...what the use of physical contemplations is for life. But why does he Willebrord Snell (1580–1626), a Dutch mathematician mentioned earlier in the text. not follow Proclus, whom he cites, who acknowledges some greater good of Geometry than the arts necessary for life? If he had, then indeed the use of the tenth book would have appeared in assessing the types of figures. Snellius cites geometric authors who do not use Euclid’s tenth book. Truly, all those authors treat either linear problems or solid ones, and concern themselves with figures or quantities of such a kind that do not have their end within themselves, but clearly tend toward other uses, nor would they even be sought out without those uses.

But regular figures are sought for their own sake as Archetypes Archetypes: original models or patterns on which the structure of the world is based.; they have their perfection in themselves and are among the subjects of plane problems, notwithstanding the fact that a solid is also enclosed by plane faces; similarly, the matter of the tenth book belongs chiefly to planes. Why then should things of a different kind be cited? Or why is merchandise estimated as cheap because Codrus A character from Roman satire representing a poor man; here, a metaphor for someone concerned only with basic survival. does not buy it to feed his belly, but Cleopatra buys it to adorn her ears? Is such a torment fixed only for geniuses? Kepler likely refers to the "crux" or "cross" as a metaphor for a difficult intellectual puzzle. Indeed, it is for those who vex the Ineffable Ineffable: Kepler refers to irrational numbers (like the square root of 2) which cannot be expressed as a simple ratio of whole numbers, making them "unspeakable" in classical Greek mathematics. with numbers—that is, by trying to speak them out.

But I treat these types of figures not with numbers, not through Algebra, but by the reasoning of the Mind; truly because I have no need of them for calculating market accounts, but for explaining the causes of things. He Referring to Snellius or Ramus. thinks those subtleties should be separated from the Systematic Instruction original: "Στοιχειώσει" (Stoicheiosei), referring to the foundational arrangement of Euclid's Elements. and hidden away in libraries. He acts entirely as a faithful disciple of Ramus, and his effort is not misplaced: Ramus took the form away from the Euclidean building and pulled down the summit—the five bodies The five Platonic solids.; with these removed, the whole structure was dissolved, the walls stand cracked, and the arches threaten to ruin. Snellius therefore even takes away the cement Here, Euclid's tenth book, which provides the logical proofs for irrational magnitudes needed to construct the Platonic solids., since it has no use except for the solidity of the house joined together under the five figures.

O happy understanding of the disciple, how skillfully he learned to understand Euclid from Ramus! That is, they think they are called Elements original: "Στοιχεῖα" (Stoicheia) because there is found in Euclid an all-encompassing abundance of propositions, problems, and theorems for every kind of quantity and the arts concerned with them; whereas the book is called Elements original: "Στοιχειώσις" (Stoicheiosis) from its form, because the following proposition always relies on the preceding one, all the way to the last of the last book (and partly of the ninth), which cannot do without any of the prior ones. They turn the Architect into a forest-keeper or a timber-merchant, 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 saw that the true and genuine differences of geometrical things—from which I must derive the causes of Harmonic Proportions—are generally completely ignored; and that Euclid, who had handed them down with great care, was being driven out, oppressed by the quibbles of Ramus, and because of the disruptive noise of those behaving wantonly, he was heard by no one, or else was telling the mysteries of Philosophy to the deaf; and that Proclus Proclus (412–485 AD), a Neoplatonist philosopher whose commentary on Euclid was vital to Kepler’s thought., who could have opened the mind of Euclid, unearthed hidden things, and made difficult concepts easy to grasp, was also held in ridicule, and had not continued his commentaries as far as the tenth book; I saw that I absolutely had to do this: first, to transcribe from Euclid's tenth book those things which especially served my present purpose; and to bring to light the sequence of matters in that book, with certain divisions interspersed.

bring to light