Because the causes of harmonic proportions might be discovered by us from the divisions of a circle into the equal aliquot parts which are constructed geometrically and scientifically from the provable, regular, plane figures, I begin with what must be made known, I have brought forward at this time the conceptual differentiations of geometrical matters that, in as far as may be clear from what has been published, have not been known in the case of the solids. All the more so in that besides Euclid and his commentator Proclus, no one appeared among the ancients who indicated that he himself had knowledge of these specific differentiations of geometrical matters precisely. the distribution of problems into planes, solids and lines of Pappus of Alexandria, and of the ancients who he followed, was close enough to the quality of conception of one part of the geometrical subject arising to be explained. However, that treatment is both brief, and applied to practical questions. Pappus makes no mention of theory, but if we do not occupy our whole mind with the theory of this question, we will never be able to understand harmonic ratios. When Proclus Diadochus had published four books on Euclid's First, he wanted theoretical philosophy to be incorporated in the subject of mathematics by public profession. If he were to have left us his commentaries on the tenth book of Euclid, he would both have freed our geometers from stupid ignorance, and would have reduced my work in developing the differentiations of geometrical things in solid. For those distinctions of conceptual existences were known well enough by him, as is easily seen in the preface to his book, because he established that the principles of the whole essence of mathematics are the same as that principle which also pervades all existence, and that everything is caused by that, the finite and the infinite, the limited and the unlimited, the limit or circumscription in relation to a form, and the knowledge of the unlimited in relation to the matter of geometrical things.
Form and proportion are the characteristics of quantities, form of the particular, proportion of the combined. Form is completed by limits, a straight line by points, a plane surface by lines, a body is limited, circumscribed and formed by surfaces. Therefore, what has been made finite, circumscribed and formed, can also be comprehended by the mind. The infinite and the indeterminate cannot be constrained by any part of the knowledge which is given by definitions, and by no restraint of proofs. But, the figures have prior existence in the Archetype, then in the work. First they exist in the divine mind, then in created things, in the different modes of the subject, but nonetheless with the same form of its essence. Therefore, the formation with quantities, a certain mental or intellectual essence, creates the differentiations of their essence. That is much clearer when derived from proportions. When a form is completed by many limits, it is effected in such a way that the form would make use of proportions because of this plurality. But it cannot be possible to know what proportion would be at all without the action of the mind. And for that reason the person who gives limits to quantities in relation to the principle of essence, that person asserts that formed quantities have an intellectual essence. But there is no need for argument. Proclus' whole book should be read, it will be evident enough that he knew the intellectual differentiation of geometrical things in a provable way. And yet, when this had been confirmed, he did not go off on his own and assert it in isolation, but cried aloud so that the assertion could not be ignored, and so that he might even wake up sleepy minds. His language flows like a flooding river, layered thoughout with the most abundant and abstruse propositions of platonic philosophy, which is this, the argument of his whole book.
Truth has not freed our century from penetrating to such hidden matters. Proclus' book has been read by Pierre Ramee, but, in what concerns the heart of philosophy, it has been scorned and thrown aside, along with Euclid's tenth book. And, anyone who has written a commentary on Euclid, if such were to have been written in his defense, has been ridiculed and ordered to remain silent. The aroused wrath of the embittered censor has been turned on Euclid as on a criminal. The tenth book of Euclid, which, when read and understood may be able to unfold the mysteries of philosophy, has been doomed by savage sentence to not be read. Nothing more shameful was ever written by Ramee, I ask you to read his words from "The Study of Mathematics," Book 21:
"Stuff, he says, has been handed down in that tenth book, in such a way that I would never have found the same obscurity in human letters and arts. I say obscurity not to be understood, Euclid anticipates that (that could be clear to the illiterate and uneducated who only look at what is right in front of their eyes) but in order to investigate and search out what the purpose, and proposed use of the work might be, what the classes, types and differences of the subjects might be. I have never read anything more confused and involuted. Might not the Pythagorean superstition seem to have been drawn into this book as if into a pit, etc."
By God, Ramee, if you would not have believed that this book may be read with too much ease, you would never have slandered so much obscurity. You need more work. You need quiet. You need forethought. And, above all, you need attentiveness of mind. Then, you may understand the intent of the writer. With that, the good sort of mind will be lifted up to the point where, resolving to live at last in the light of truth, inspired, exulting with incredible pleasure, it perceives the whole world and all its different parts most exactly, as if from a very high place. But to you, you who act in this place as the advocate of ignorance, and of the common man seeking advantage from everything whether divine or human, I say to you, that these matters may be "unnatural sophistries," to you "Euclid will have been quickly and immoderately taken advantage of," to you, "this subtlety has no place in geometry." Let it be your lot to slander what you do not understand. For me, who hunts for the causes of things, no other path will lead to them apart from that which is in the tenth book of Euclid.
Lazarus Schoener followed Ramee in his geometry, he confessed that he was not able to see any use in the world for the five regular solids. Then he would have read the book I wrote, "The Secret of the Universe," in which I prove that the numbers and distances of the planets hae been chosen from the five regular solids. Now look what injuries professor Ramee inflicted on his student Schoener. First, once Ramee had read Aristotle, who refuted pythagorean philosophy on the properties of the elements derived from the five solids, he at once conceived in his soul contempt for the whole of the pythagorean philosophy, then, when he knows that Proclus was part of the pythagorean sect, he used to affirm to his student that he did not believe, what was most true, that the ultimate purpose of Euclid's book, towards which all the propositions of all the books together are brought back, is the five regular solids. This is the origin of Ramee's most confident conviction that the five solids ought to be removed from the end of the books of Euclid's Elements.
After the end of the book has been chopped off, like the shell of a levelled building, Euclid was left, a formless heap of propositions, against which, as if against some ghost, Ramee inveighs in all the 28 books of his "Study of Mathematics," speaking with a great harshness and a great rashness, most unbecoming to such a great man. Schoener followed this conviction of Ramee, and he himself believed that there is no use for the regular solids at all. But not completely, he neglected, or refused to follow Ramee's judgment on Proclus. He was able to learn the use of the five solids, both in Euclid's Elements, and in the making of the world, from Proclus. And the student was much happier than the professor because he accepted the use of the solids opened up by me in the making of the world which Ramee refused to impress upon him from Proclus. For what does it matter if the pythagoreans did attribute these figures to the elements, but not to the spheres of the world, as I do? Ramee would not have offered up one tyrannical word against this whole philosophy, he would have exerted himself to have removed this error of theirs in regard to the real subject of the figures, as I did.
What if the Pythagoreans did teach the same thing that I do, weaving their meaning into a cover of words? May not the Copernican form of the world have existed in Aristotle, may it not have been incorrectly refuted by him in other words when he would call the sun, fire, and the moon Antichthone? For if the same ordering of the orbits which was known to Copernicus was known to the Pythagoreans, if the five solids and the necessity for their five-fold number was known, if they continuously taught that the five solids are the Archetypes of the parts of the world, how little more would it be for us to believe that the thinking of the pythagoreans has been collected together secretly by Aristotle, but had alredy been refuted by the meaning of the words?
When Aristotle reads "earth," they were giving him "cube," because they may perhaps have understood Saturn whose orbit is distanced from Jupiter by an interposed cube. And the common sort attribute rest to earth, but Saturn moving the slowest of all has been marked out as the closest to rest, for which reason the planet was given the name "rest" by the Hebrews. In the same way Aristotle reads octahedron as given to "air," when they may, perhaps, understand Mercury, whose orbit is contained by an octahedron, and Mercury is no less fast (certainly the fastest of all the planets) than the air is mobile. Mars was perhaps worked in with the word "fire," the name for this planet is "pyrois," which is derived from fire, and a tetrahedron was given to it perhaps because the orbit of this planet has been enclosed by this figure. And "water" could have concealed the star of Venus to which the icosahedron was allotted (because the orbit is contained by an icosahedron) because fluids are subject to Venus, and she herself was said to have been born from the ocean spray, whence the name Aphrodite.
And lastly, the sound of the word "world" could mean "earth" and the dodecahedron be allotted to the world, because its orbit is contained by this figure, separated into twelve longitudinal parts so that this figure is contained by twelve planes for the whole orbit. Agreement has therefore been reached that in the mysteries of the Pythagoreans the five figures have not been distributed among the elements, as Aristotle believed, but rther, among the planets. This is a great confirmation of what Proclus handed down, among other things, as th purpose of geometry, and what he would tech, namely, how heaven would have accepted harmonious forms for its particular parts.
Although not this purpose, but injury is what Ramee inflicts on us, so Snell, the most skillful of today's geometers giving open support to Ramus, says first that "the very division of the unnameables (1) into thirteen different types is of no profitable use" in the preface to "The Problems of Ludolph of Coellen." I concede that, if he should know no use, if not in every day life, and if there would be no use for life in the investigations of physics.
But why does he not follow Proclus, who said he did not know any greater good of geometry than the arts which are necessary to life? But then the use of the tenth book would be clear from the evaluated types of the figures. Snell says that all those authors of geometrical works who do not use the tenth book of Euclid, deal with either the problems of lines, or solids, and with figures or such quantities which do not have their purpose within themselves but tend toward other uses, and that they may not be investigated separately from those other purposes.
But the regular figures are investigated for intrinsic reasons, they have their own perfection in themselves, and are included among the problems of planes, not withstanding the fact that a solid is enclosed by plane surfaces, and that the most important subject matter of the tenth book concerns planes. Why would anything different be mentioned? Or why are the goods, which Codrus does not buy to stuff his belly, but Cleopatra does to decorate her ears, thought so worthless in value? Has so much torment been fashioned by minds? The unnameables are offensive to those for whom this question must be defined by numbers. But I deal with these types by the reasoning of the mind, not by numbers, and not by algebra.
Because there is no work for me in reckoning up the balance sheets of merchants, but there is in developing the causes of things. It is a common opinion that these subtleties must be separated out from the Elements of geometry, and must be stuffed away in the archives. Ramee's altogether faithful student discusses that, and he does not perform an academic undertaking. Ramee took away the form from Euclid's construction, and overthrew the crown of the work, the five solids. The whole structure wa destroyed after these had been removed, the cracked walls remain standing, the jutting arches left in ruins, then Snell took away the cement as well, so there has been no use for the solidity of Euclid's house cemented together under the five figures.
What a happy comprehension by the student, how correctly he affirmed that he understood Euclid by reading Ramus. So they think of what was called the Elements, because an abundant variety of propositions, problems and theorems is discovered in Euclid for all the different kinds and quantities of the arts bound up with them, although the book "The Elements" might have been named from its form, because a subsequent proposition is always supported by a preceding one, right through to the last proposition of the last book (usually Book 9), what lack could there be of anything prior? They make a forest-ranger, or timber merchant out of the architect by thinking that Euclid obviously wrote his book so that it might supply all others, only he would have no dwelling of his own. This is more than enough of these matters for this point, now we must return to the main line of discussion.
Because I would understand the true and real differentiations of the geometrical matters by which the causes of harmonic proportions must be derived by me, I declare the following to be widely unknown: that Euclid who handed on these proportions carefully has been driven away overwhelmed by the mockery of Ramus, and confused by the babbling of the lewd, is heard by nobody, or else tells the mysteries of philosophy to the deaf. That Proclus who lays bare the mind of Euclid, digs up the hidden things, and may have been able to make easy again what is difficult to comprehend, was made an object of derision, and did not continue his commentary up to the tenth book. I saw that all this was to be done to me by me.
As I begin I would transcribe those things from Euclid's tenth book which may contribute in an especial way to my present undertaking. I would bring into the light the series of matters in that book, separated by certain definite divisions. I would show the reasons why some members of the divisions were omitted by Euclid, then, lastly the figures themselves must be discussed. I have been content to simply reference the propositions in those cases which were proven clearly by Euclid. There are many questions which have been proven by Euclid in a different way, now these must be reworked, or were separated, joined together again, or the order must be changed on account of the purpose that has been given me, namely the comparison of the figures which can be known and those which cannot. I have combined the series of definitions, propositions and theorems in numerical order, as I did in the "Dioptics," because of the ease of reference.
I have not been accurate in regard to the lemmasd, nor over anxious in regard to names, being more concerned with the constructions themselves. Certainly this is not yet geometry in philosophical terms, but in this part I do discuss the philosophy of geometry. Would that I were able to treat still more popularly of geometrical questions, provided the treatment were clearer and more palpable. But, I hope that readers equal to both will think about my work for the good, both because I teach geometry popularly, and because I was not able to overcome by my effort the obscurity of the subject matter. Finally I give this advice to any people who might be completely unfamiliar wityh mathematical questions, carried along by my exposition they should red only the propositions from number 30 to the end, and faithfully employing those propositions without proof, they should proceed to the other books, especially to the last. If such readers were to be terrified by the difficulty of the geometrical argument, they might deprive themselves of the most joyful fruit of contemplating the harmonies. Now, let us go to work with God.