De Divina
Proportione

To the most excellent Prince Ludovico Maria Sforza, Anglus, Duke of Milan, ornament of peace and of war: the epistle of Brother Luca of Borgo San Sepolcro, of the Order of Friars Minor, professor of Sacred Theology, on the Divine Proportion.

It being, most excellent Duke, the 8th day of February in the year of our salvation 1498, in the impregnable citadel of your illustrious city of Milan, the most worthy place of your customary residence: I stood in your presence at the laudable and scientific contest [a court disputation], attended by many most celebrated and learned men of every rank, religious as well as secular, in whom your magnificent court continually abounds. Of their number — besides the Most Reverend Lordships of bishops, protonotaries and abbots — there were, of our sacred seraphic Order, the reverend father and sublime theologian Master Gometio, with that most worthy 1v herald of Sacred Scripture, Brother Dominico surnamed Ponzone, and the reverend father Master Francesco Busti, at present appointed regent in our worthy convent of Milan. And of the seculars, first my particular patron, the illustrious Lord Galeazzo Sforza Sanseverino, Vicar, most valiant general of Your Highness, a captain in arms today second to none, and a diligent follower of our disciplines. And distinguished orators of the most illustrious powers, and supreme men of medicine and astronomy: the most illustrious and most acute student of Serapion and Avicenna, searcher of the higher bodies and interpreter of things to come, Ambrogio Rosa; the most learned healer of all ills, Luigi Marliano; the most diligent observer of medicine in its every part, Gabriele Pirovano; and one much admired and venerated by the aforesaid in all these matters, Nicolò Cusano, with Andrea Novarese, most expert in the same professions, and other eminent and most judicious doctors of both laws; and, of your most distinguished Magistracy, counsellors, secretaries, chancellors.

In the company of the most keen-sighted architects and engineers and assiduous inventors of new things, Leonardo da 2r Vinci, our Florentine compatriot, who in sculpture, casting and painting verifies his surname with every work [Vinci — "he conquers"]. This the admirable and stupendous equestrian statue makes plain — whose height from the nape to the level ground is twelve braccia, that is, thirty-six times the line .ab. shown here, and whose whole bronze mass rises to about 200,000 pounds, whereof the common ounce is the twelfth part of each — dedicated to the most holy, unconquered memory of your father, and wholly beyond envy of those statues of Phidias and Praxiteles on Monte Cavallo; and likewise the graceful image of the Ardent Desire of our Salvation [the Last Supper], painted by his own hand in that worthy and devout place of bodily and spiritual refection, the sacred temple of the Grazie — before which today Apelles, Myron, Polyclitus and the rest must yield. Nor is he sated with these: he now attends with every study to bring to its due end his inestimable work on local motion, on percussion and weights and on all forces, that is to say accidental weights — having already, with all diligence, brought to a worthy close his book on painting and human movements. And with him his brother-in-arts Giacomo 2v Andrea da Ferrara, a most careful follower of the works of Vitruvius, and in nothing thereby diminished in his singular military skill.

Your Highness, with your golden and mellifluous words, declared that man worthy of the greatest commendation before God and the world who, being endowed with some virtue, willingly communicates it to others; whence charity accrues to his neighbour, and to himself praise and honour, after the sacred saying: Quod et sine figmento didici et sine invidia libenter comunico — "what I learned without guile, I share gladly and without envy." The sense of those sweetest words fixed itself so firmly in my mind that nothing was ever more durably cut in marble. And though it was almost innate in me by nature to deal thus with everyone — above all in those faculties with which, among other men, it pleased the Most High in His immense kindness to endow me, namely the necessary sciences and most worthy mathematical disciplines — nevertheless, being already weary from laborious toils by day and by night, of body and of spirit alike (all of which will be plain to whoever diligently examines our great work compiled in these same disciplines and faculties, dedicated to the magnanimous 3r kinsman of Your Highness, Guidobaldo Duke of Urbino, together with the other works cited in its fifth distinction), I had already settled myself, like others, in some sunny corner to count out my years. But greatly stirred by those words of yours, I took breath again upon the deserted shore; and — as a seasoning to every other work of ours composed in these faculties, and as the summit and most delectable taste of all the aforesaid sciences and mathematical disciplines — I resolved to prepare this brief compendium and most useful treatise called On the Divine Proportion: for Your Highness, for the profit of your reverent subjects, and also for the honour and perfect adornment of your most worthy library, graced as it is with an innumerable multitude of volumes in every faculty and doctrine. And this book, together with all the material forms of the bodies contained in it, will give whoever visits that library no less cause for wonder than all the other volumes and worthy things laid up there, since these forms have until now been hidden from the living. In it we shall speak of high and sublime matters, which are in truth the touchstone and assay-cup of all the aforesaid sciences and disciplines, 3v and from which every other speculative operation — scientific, practical and mechanical — derives; without knowledge of them and their premise, nothing among human affairs can be well understood or well done, as shall be demonstrated. And therefore Your Ducal Highness, with shrewd intelligence, will exhort your household and your other reverent subjects to peruse it with delight, with utmost pleasure and with most useful fruit; for these are no old wives' fables nor other ridiculous and false pleasantries, nor yet lying and incredible poetic inventions that feed the ears with mere smoke. For although false things, according to the Philosopher, are useful to us for the knowledge of the true things that follow from them — as the reverse of the obverse, each the opposite of the other — yet true things will profit us the more, since from them nothing but truth proceeds. And among true things, as Aristotle and Averroes affirm, our mathematical sciences are the truest and stand in the first degree of certainty, and all the natural sciences follow after them. Let this suffice for introduction and argument to what follows here. With humility always, and with due reverence toward Your Ducal Highness, 4r to whom I supremely and continually commend myself. Quae felicissime ad vota valeat — may she fare most happily, according to her every wish.

Propter admirari ceperunt philosophari — through wondering, men began to philosophize. The authority just proposed, most excellent Duke, of the "master of those who know" [Aristotle], holds that knowing took its beginning from seeing, as the same master affirms in another place, saying: Quod nihil est in intellectu quin prius fuerit in sensu — that is, nothing is in the intellect which was not first in some way offered to the sense. And of our senses the wise conclude that sight is the most noble; whence, not undeservedly, it is said even among the common people that the eye is the first gate through which the intellect understands and tastes.

As is contained in that same place: the priests of Egypt, seeing the moon eclipse, stood long in wonder, and seeking the cause they found by true science that it comes about naturally through the interposition of the earth between the sun and the moon; with which they were satisfied. And from that time forward their successors, refining themselves hand 4v over hand by the light of the five intellectual windows, filled innumerable multitudes of volumes with their profound sciences for our profit; for just as one thought bursts forth from another, so from that one were many others afterward born. Turning this over within myself, I resolved to take up the pen for this most useful compendium chosen from the mathematical sciences, and together with it, with my own hand, materially to form their bodies in proper shape for the common profit, and to offer them, with the present compendium, to Your Ducal Highness. At their unaccustomed aspect — like a thing come down from heaven in our times — I do not doubt your graceful and keen intellect will take the greatest pleasure, above all when, by the aforesaid light, with no less searching than the ancient Egyptians spent on that eclipse, you find out the causes of these forms and their sweetest harmony, with the aid and support of the present treatise. Of this I am certain: if in the past she offered herself broad and ample to whoever was in part endowed with such sciences and disciplines, in the future she must show herself far more generous and most 5r ample; and that Your Highness will all the more, with every diligent care, exhort your dear household and reverent subjects and other well-wishers to their acquisition — since these mathematical sciences are the foundation and the ladder for arriving at the knowledge of every other science, standing as they do in the first degree of certainty, as the Philosopher affirms, saying: Mathematicae enim scientiae sunt in primo gradu certitudinis et naturales sequuntur eas [for the mathematical sciences are in the first degree of certainty, and the natural sciences follow them].

The sciences and mathematical disciplines stand, as has been said, in the first degree of certainty, and all the natural sciences follow them, and without knowledge of them it is impossible to understand any other science well. And in Wisdom also it is written: Quod omnia consistunt in numero, pondere et mensura — that is, everything deployed through the universe below and above is of necessity subject to number, weight and measure. And in these three things, says Aurelius Augustine in The City of God, the Supreme Craftsman is supremely praised, because in them "He made stand the things that were not." By your loving exhortation I understand that many, ignorant of the sweetest fruit of such utility, must wake from their torpor 5v and mental sleep, and give themselves wholly, with every study and solicitude, to inquiring into these things; and this will be the cause that in them the age renews itself in due season, and that in each study of whatever science men come more truly and swiftly to perfection. And beyond the fame and worthy commendation of Your Ducal Highness, it will breed in your excellent dominion no small ability in your dear household and beloved subjects, ever wholly ready for its defence — no less than the noble, ingenious geometer and most worthy architect Archimedes did for his own fatherland.

Archimedes — as is written — with his new and varied inventions of machines long kept the city of Syracuse safe against the onset and warlike advance of the Romans, until Marcus Marcellus openly undertook to storm it. And by daily experience it is not hidden from Your Ducal Highness — seeing that for many years your father's most holy memory was the author, preceptor and pattern of this to all Italy and to both Gauls, transalpine and cisalpine — that the defence of commonwealths great and small, by another 6r name called the art of war, cannot possibly be exercised with honour and profit without knowledge of Geometry, Arithmetic and Proportion. Nor can any worthy army, appointed to siege or to defence, ever be called fully provided unless there be found in it engineers, and a deviser of new machines specially ordained — as we said a little above of the great geometer Archimedes at Syracuse.

Consider generally all its engines of war — take whichever you will: bastions and other defences, bombards, stone-throwers, trebuchets, mangonels, ballistae, catapults, rams, testudos, siege-cats, with all the other innumerable machines, devices and instruments — they will always be found built and formed by force of numbers, of measure, and of their proportions. What else are strongholds, towers, ravelins, walls, outworks, moats, bridges, turrets, battlements, mantlets and the other fortifications of towns, cities and castles, but all geometry and proportion, levelled and set true with due levels and plumb-lines? For no other cause were the ancient 6v Romans so victorious — as Vegetius, Frontinus and other distinguished authors write — than for their great care and diligent preparation of engineers and other masters of works by land and by sea, whose sufficiency is impossible without the mathematical disciplines, that is Arithmetic, Geometry and Proportions. All of which the ancient histories of Livy, Dionysius, Pliny and others make clear and manifest; and from these Roberto Valturio, that most expert man of Rimini, drew everything that stands in his worthy work entitled De Instrumentis Bellicis, dedicated to the illustrious Lord Sigismondo Pandolfo. And with those same machines and instruments to the letter, as the Riminese sets them in his book, and with many more besides, the most happy memory of the near kinsman of Your Highness, Federico da Montefeltro, most illustrious Duke of Urbino, caused the whole stupendous edifice of his noble and admirable palace at Urbino to be adorned round about at its foot, in a frieze of living and beautiful stone, by the hands of most worthy stonecutters and sculptors, in ordered sequence. So, among other examples, one reads of Julius Caesar's ingenious bridge in his Commentaries; and 7r so to this day, in the worthy Umbrian city of Todi, in the church of San Fortunato, our sacred convent, there hang publicly a great multitude of very thick ropes of your father's most holy memory, which he duly disposed for a bridge over the Tiber on the way to his famous and achieved victory.

By no other means, again, did our most subtle Scotus attain to the great speculations of Sacred Theology than by knowledge of the mathematical disciplines, as appears through all his sacred works — above all if one considers well that question of his second book on the Sentences, where inquiring he asks whether the angel has its own determinate place of existence; in which he well shows that he had understood the whole sublime volume of our most keen-sighted Megarian philosopher Euclid. For no other reason, likewise, do all the texts of the prince "of those who know" — the Physics, the Metaphysics, the Posterior Analytics and the rest — show themselves difficult, than through ignorance of the said disciplines. For no other reason is there a dearth of good astronomers than for want of Arithmetic, Geometry, 7v Proportions and Proportionality; and of ten of them, nine govern their judgments by tables, almanacs and other things calculated by Ptolemy, Albumasar, Alì, Alfraganus, Geber, Alfonso, Bianchini, Prosdocimo and others — which, through the small care of the copyists, may be corrupted and vitiated; and consequently, trusting in them, they fall into the greatest and most evident errors, to no small harm and prejudice of those who rely upon them.

The supreme subtlety of all the municipal laws consists likewise — as men learned in them have several times explained to me — in judging of the alluvions and washings-round of waters in their excessive flooding; on which their eminent head, Bartolo da Sassoferrato, composed a particular treatise which he entitled Tiberina. In its proem he greatly extolled geometry together with arithmetic, affirming that he had learned them from a friar of ours named Guido, a professor of Sacred Theology; and in that treatise is contained the giving and the taking which the Tiber works at times by its flooding in those parts, above all from Perugia toward Deruta. 8r There, always with geometric figures rectilinear and curvilinear, part by part, he guided himself by citing our most keen-sighted philosopher Euclid, and concluded the matter with the greatest subtlety.

I do not speak of the sweet and suave harmony of music, nor of the supreme loveliness and intellectual comfort of perspective, nor of the disposition of architecture, with the description of the maritime and terrestrial universe and the doctrine of the courses and celestial aspects; for what has been said of them thus far appears clearly. I pass over, to spare the reader tedium, many other sciences practical and speculative, with all the mechanical arts necessary to human affairs, which without the suffrage of these can neither be acquired nor kept in due order. And therefore it is no cause for wonder that good mathematicians are few in our times: the rarity of good teachers is the reason, together with gluttony, sleep and idle featherbeds, and in part the weakness of the more recent wits. Whence among the wise it has magisterially passed into a common proverb: Aurum probatur igni et ingenium mathematicis — that is, fire proves the goodness of gold, 8v and the mathematical disciplines the rareness of the wit; which in substance means that the good wit, apt for mathematics, will be most apt for every science, these being of the greatest abstraction and subtlety, since they must always be considered apart from sensible matter. And truly they are, as the Tuscan proverb has it, the sciences that split the hair in mid-air.

For which cause the ancient and divine philosopher Plato, not undeservedly, denied entry to his most celebrated Gymnasium to those unversed in Geometry, when he set a motto at the top of his principal door, in great and legible letters, with these formal words, videlicet: Nemo huc geometriae expers ingrediatur — that is, let no one enter here who is not a good geometer. This he did because in her every other science lies hidden. Filled before him with her sweetest sweetness, Pythagoras, that most diligent contemplator of nature, for the invention of the right angle — as one reads of him, and Vitruvius recounts — made sacrifice 9r to the gods of a hundred oxen, with the greatest feasting and jubilation, as shall be told below.

And let this, for the present, suffice for the commendation of the mathematicians. Their number in this your illustrious city already begins, by the grace of Your Ducal Highness, to grow daily and not a little, through the assiduous public lecture upon them newly introduced by you, with the progress of the distinguished hearers according to the grace granted to me in these matters by the Most High: expounding to them clearly and with all diligence — so they judge — the sublime volume of the aforesaid Euclid in the sciences of Arithmetic and Geometry, Proportions and Proportionality; his ten books having already been brought to a most worthy end, always interposing our practice alongside his theory, for the greater utility and ampler understanding of them; and to the present dispatch of this work I devote the remainder of my time.

This word "mathematical," most excellent Duke, is Greek, derived from [the Greek mathéin, "to learn"], which in our tongue is as much as to say "teachable"; and for our purpose, by the mathematical sciences and 9v disciplines are understood Arithmetic, Geometry, Astrology, Music, Perspective, Architecture and Cosmography, and whatever other depends upon these. Nevertheless the wise commonly take the first four, that is Arithmetic, Geometry, Astronomy and Music, and the others are called subalternate, that is, dependent on these four. So hold Plato and Aristotle, and Isidore in his Etymologies, and Severinus Boethius in his Arithmetic. But our judgment, feeble and low though it be, constrains them to either three or five: that is, to Arithmetic, Geometry and Astronomy, excluding Music from their number for as many reasons as they exclude Perspective from the five; or else, adding Perspective to their four for as many reasons as they join Music to our three.

If they say that Music contents the hearing, one of the natural senses, Perspective contents the sight, which is by so much the more worthy as it is the first gate to the intellect. If they say that Music attends to sonorous number, and to the measure carried in the time of its prolations, Perspective attends to 10r natural number according to its every definition, and to the measure of the visual line. If the one refreshes the soul through harmony, the other delights it greatly through due distance and variety of colours; if the one considers its harmonic proportions, the other considers the arithmetical and the geometrical. And in brief, most excellent Duke — it is now many years that this contends in my head, and by no one has it been made clear to me why four rather than three or five; yet I esteem that so many wise men do not err, and by their sayings my ignorance is not uprooted. Alas, who is there who, seeing a graceful figure well disposed with its due lineaments, to which nothing seems wanting but breath, would not judge it a thing divine rather than human? And painting imitates nature as closely as can be told — which appears evidently to our eyes in that foretasted image of the Ardent Desire of our Salvation, in which it is not possible to imagine the apostles more intently alive at the sound of the voice of infallible truth when it said, Unus vestrum me traditurus est ["one of you shall betray me"]; where, with acts and gestures, one to another and another to one, 10v they seem to speak with living and afflicted wonder: so worthily did our Leonardo dispose it with his graceful hand.

So too one reads of Zeuxis in Pliny's De Picturis: being in contest of the same exercise with Parrhasius, and challenging one another with the brush, the one made a basket of grapes woven about with its vine-leaves, which, set in public, conquered the birds, who threw themselves upon it as upon true fruit. And the other made a veil. Then Zeuxis — Parrhasius having likewise set his work in public — believing it a veil that covered the work made for the contest, said to him: "Lift away the veil, and let everyone see yours as mine was seen." And so he remained conquered: for he had deceived the birds, irrational animals; but the other had deceived a rational animal — and a master.

If the great delight and supreme love I bear her — though I am unlearned in her — do not deceive me, there is universally no gentle spirit whom painting does not delight, seeing that she draws to herself both the one animal and the other, rational and irrational. Wherefore I shall still hold with this, if nothing further comes: that the principal ones are three, and the others subalternate; or five, if those men count Music among them — for by no means does it seem to me that 12r Perspective should be set behind, she being worthy of no less praise. And I am certain that, this being no article of faith, it will be tolerated in me. So much for what concerns the said name.

Next, for less trouble, in what follows it is to be noted: when there is alleged at times "the 1st of the first," "the 4th of the second," "the 10th of the fifth," "the 20th of the sixth," and so running on up to the fifteenth, by the first citation is always to be understood the number of the conclusions [propositions], and by the second the number of the books of our philosopher Euclid, whom we follow in everything as the archimandrite of these faculties. That is, to say "by the fifth of the first" means by the 5th conclusion of his first book; and so for the other partial books of his total book of the Elements and first principles of Arithmetic and Geometry. But when the authority we adduce is from another work of his, or from another author, we shall name that work and that author.

We shall proceed also by many various characters and ab11vbreviations which it is customary to use in such faculties — as indeed each faculty requires its own: so medicine uses hers for scruples, ounces, drams and handfuls; the silversmiths and jewellers theirs for grains, pennyweights and carats; the astrologers theirs for Jupiter, Mercury, Saturn, Sun, Moon and the rest; and the merchants, with like brevity, divers signs for lire, soldi, grossi and denari — and this only to avoid prolixity of writing and also of reading, since doing otherwise they would fill much paper with ink. So we too in the mathematical [sciences], for algebra — that is, the speculative practice — [use signs] denoting cosa, censo and cubo [the unknown, its square, and its cube] and the other terms, as is contained in our aforesaid work. Of their number we shall use some in this work as well, commonly set down with those that follow, videlicet: R., that is, root, one or more; R.R., that is, root of root, one or more; m̃, that is, "minus," in every quantity of whatever nature; p̃, that is, "plus," likewise in every quantity; q., that is, quantity or quantities; po., that is, power or powers; li., that is, line or lines; Geo., that is, geometry or geometric; Arith., that is, arithmetic or arithmetical; propor., that is, proportion or proportions; No., that is, number or numbers; □, that is, square, squares; Dra., that is, difference or differences;

p., that is, first (in all its genders); 2., that is, second; m.cato, that is, multiplied; m.care, that is, to multiply; s. propor. h. el m. e doi ex., that is: according to the proportion having the mean and two extremes.

Likewise these names — multiplication, product, rectangle — import one and the same thing. And again these — the square of a quantity, and the power of any quantity — are one and the same thing; for the power of the line stands in respect of its square by the last [proposition] of the first [book], and the utmost the line can do is its square. And these things must at times be observed in our process, so that there be no equivocation in the sense of the words. 12v

It seems to me, most excellent Duke, that the fitting title of our treatise must be On the Divine Proportion; and this for the many correspondences of likeness to God Himself which I find in our proportion, of which we treat in this our most useful discourse. Among others, we shall take four of them as sufficient to our purpose. The first is that it is one only and not more, and it is not possible to assign of it other species or differences: and unity is the supreme epithet of God Himself, according to the whole school of theology and of philosophy also. The second correspondence is that of the Holy Trinity: that is, just as in the divine nature one same substance subsists among three persons, Father, Son and Holy Spirit, so one same proportion of this sort must always be found among three terms; and never among more, nor among fewer, can it be found, as shall be said.

The third correspondence is that, just as God properly cannot be defined, nor made understood to us by words, so this our proportion can never be assigned by a number 13r we can understand, nor expressed by any rational quantity, but is always hidden and secret, and by the mathematicians called irrational. The fourth correspondence is that, just as God can never change, and is all in all, and all in every part, so our present proportion is one and the same in every quantity, continuous or discrete, great or small, and always invariable; in no way can it be changed, nor even otherwise apprehended by the intellect, as our process will demonstrate. A fifth correspondence may, not undeservedly, be added to the foregoing: that is, just as God confers being upon the Celestial Virtue, by other name called the Quintessence, and by means of it upon the other four simple bodies, that is the four elements Earth, Water, Air and Fire, and through these confers being on every other thing in nature — so this our sacred proportion gives formal being — according to the ancient Plato in his Timaeus — to heaven itself, by attributing to it the figure of the body called dodecahedron, otherwise the body of twelve pentagons, which, as shall be shown below, cannot 13v be formed without our proportion. And similarly he assigns to each of the other elements its own form, in no way coinciding one with another: to fire the pyramidal figure called tetrahedron, to earth the cubic figure called hexahedron [the cube], to air the figure called octahedron, and to water that called icosahedron.

And these forms and figures are by the wise all named regular bodies, as shall be said of each severally below; and then, by means of these, [being is given] to infinitely many other bodies called dependents. And these five regulars cannot be proportioned among themselves, nor be understood as circumscribable by the sphere, without our said proportion — all of which will appear below. And though many other correspondences might be adduced, let these suffice for the fitting denomination of the present compendium.

This our proportion, most excellent Duke, is of such prerogative and of excellence as worthy as could ever be told, in respect of its infinite power; seeing that without knowledge 14r of it, very many things most worthy of admiration, whether in philosophy or in any other science, could never come to light. This gift is certainly granted it by the invariable nature of the higher principles, as says the great philosopher Campanus, our most famous mathematician, upon the 10th of the 14th; above all seeing that she it is who, with a certain irrational symphony, accords among themselves so many diversities of solids — in size, in multitude of bases, and also in figures and forms — as will be understood in our process, when we set out the stupendous effects which, wrought upon a line divided according to her, are to be called not natural but truly divine.

When a straight line is divided according to the proportion having the mean and two extremes — for so, by another name, the wise call our foretasted proportion — then if to its greater part be added half of the whole line thus proportionally divided, it follows of necessity that the square of 14v their sum is always quintuple, that is, five times, the square of the said whole half.

Before we proceed further, it must be made clear how the said proportion is to be understood and placed among quantities, and how it is named by the most wise in their volumes. I say, then, that she is called proportio habens medium et duo extrema, that is, "the proportion having the mean and two extremes"; which is a proper attribute of every ternary, since whatever ternary be assigned will always have its mean with its two extremes, for the mean is never understood without them. And in this manner one is taught to divide a quantity in the 29th of the 6th, it having first been described, in the 3rd definition of the 6th, how such division is to be understood — though in his 2nd book, by the 11th, he shows how to divide the line with the same virtue and force, not otherwise naming proportion until the 5th book was passed. And by Campanus it is adduced among numbers in the 16th of the 9th. So much for its denomination. — How its mean and its extremes are understood. — Having understood how our proportion is called by its own particular 15r name, it remains to make clear how the said mean and extremes are to be understood in any quantity whatever, and how they must be conditioned, so that the said divine proportion may be found among them. For which it is to be known, as is assigned in the 5th, that among three terms of one same kind there are always of necessity two relations, or let us say proportions: one between the first term and the second, the other between the second and the third.

For example, let there be three quantities of the same kind — for otherwise no proportion is understood to exist among them. Let the first be .a., which shall be 9 in number; the second .b., 6; the third .c., 4. I say that among them are two proportions: one from .a. to .b., that is from 9 to 6, which among the common ones we called, in our work, sesquialter — this being when the greater term contains the lesser once and a half, since 9 contains 6 and also 3, which is half of 6; for this reason it is called sesquialter. But since we do not intend here to speak of proportions in general — having treated of them diffusely and in full, together with the proportionalities, in our 15v earlier-cited work — I do not care to enlarge upon them further here; let everything commonly said of them, with their definitions and divisions, be presupposed. Our present discourse shall be of this single one alone, since no one before has treated of her with so useful a process as this.

Now, returning to the purpose begun with the three quantities: there is again, from the second .b. to the third .c., that is from 6 to 4, another proportion, likewise sesquialter. Whether these be like or unlike we do not at present care: the sole intent is to make clear that among three terms of the same kind there must of necessity be found two proportions. I say likewise that our divine one observes the same conditions: among its three terms — the mean and the two extremes — it invariably contains two proportions, always of one same denomination. In the others, continuous or discontinuous, this can come about in infinitely various ways: at times among their three terms the proportion will be duple, at times 16r triple, et sic in ceteris, running through all the common species. But between the mean and the extremes of this one of ours, no variation is possible, as shall be said.

Whence deservedly came the fourth correspondence with the Supreme Craftsman; and in that she is counted among the other proportions without species or other difference, keeping the conditions of their definitions, we may liken her in this to Our Saviour, who came not to dissolve the Law but to fulfil it, and conversed with men, making himself subject and obedient to Mary and Joseph. So this our proportion, sent down from heaven, keeps company with the others in definition and conditions, and does not degrade them — rather she magnifies them the more amply, holding the principate of unity among all quantities indifferently and never changing; as of the great God says our Saint Severinus [Boethius], videlicet: Stabilisque manens dat cuncta moveri — "abiding stable, she gives all things their motion." For which cause it is to be known — so that she may be recognized among whatever quantities occur — that among her three terms she is always found invariably disposed in 16v continuous proportionality, in this manner: namely, that the product of the lesser extreme into the sum of the lesser and the mean is equal to the square of the mean; and consequently, by the 10th definition of the 5th, the said sum will of necessity be her greater extreme. And when three quantities of whatever kind are found so ordered, they are said to be according to the proportion having the mean and two extremes. And its greater extreme is always the sum of the lesser and the mean; so that we may say the said greater extreme is the whole quantity, divided into those two parts — the lesser extreme and the mean — under that condition. Whence it is to be noted that the said proportion cannot be rational: never can the lesser extreme, nor the mean, be denominated by any number while the greater extreme is rational; for they will always be irrational, as below shall openly be said. And in this she agrees with God in the third mode, ut supra.

These things well noted, we must know that to divide a quantity according to the 17r proportion having the mean and two extremes means to make of it two unequal parts, such that the product of the lesser into the whole undivided quantity is as much as the square of the greater part, as our philosopher declares by the 3rd definition of the 6th. And therefore, even should a problem never name dividing the said quantity according to the proportion having the mean and two extremes, but should merely say: make of it two parts so conditioned that the product of the one into the whole said quantity equals the square of the other part — whoever understands well, and is expert in the art, must reduce that problem to our said proportion, for otherwise it cannot be interpreted.

For example, should one say: "Make me of 10 two such parts that the one, multiplied by 10, makes as much as the other multiplied by itself" — this case and others like it, worked according to the instructions given by us in the speculative practice called algebra et almucabala, by other name the rule of the thing, set down in our earlier-cited work, will be found solved thus: the one part, the lesser, is 15 minus the root of 125; and the greater is the root of 125 minus 5. These parts, so described, are irrational, and in the art are called residues [apotomes], whose species 17v our philosopher, in the 79th of the 10th, assigns to be six. And commonly the said parts are pronounced thus — the lesser: "fifteen minus the root of one hundred twenty-five," such speech meaning: take the root of 125, which is a little more than 11, and subtract it from 15; there will remain a little more than 3, or say a little less than 4. And the greater is pronounced: "root of one hundred twenty-five minus five," meaning: take the root of 125, which is a little more than 11 as was said, and from it subtract 5; there would remain a little more than 6, or say a little less than 7, for the said greater part.

But such operations of multiplying, adding, subtracting and dividing residues, binomials and roots, and all other quantities rational and irrational, whole and broken, in every manner — these having been fully demonstrated in our aforesaid work, I do not care to repeat them here. Here we attend only to saying new things, not to reiterating those already said.

And every quantity thus divided, we shall always have three terms ordered in continuous proportionality: one will be the whole quantity so divided, that is, the greater extreme — here, in the proposed case, 10; the second is the greater part, that is, the mean, which is 18r the root of 125 minus 5; and the third, the lesser, is 15 minus the root of 125. Among these holds the same proportion — from the first to the second as from the second to the third, and conversely, from the third to the second as from the second to the first. And to multiply the lesser, 15 minus the root of 125, by the greater, which is 10, amounts to as much as multiplying the mean by itself, that is, the root of 125 minus 5: for the one product and the other is 150 minus the root of 12,500, just as our proportion requires. And for this, 10 is said to be divided according to the proportion having the mean and two extremes: its greater part is the root of 125 minus 5, and the lesser is 15 minus the root of 125 — each of which is of necessity irrational, as is proved by the 6th of the 13th, and again in the 11th of the 2nd and the 16th of the 9th. So much for knowledge of the quantity thus divided.

And because in our process it will often fall to us to name Roots, it seems well to make succinctly clear here what that imports — though in our own work it is spoken of diffusely, in every mode. I say, then, that the Root of a quantity is itself a quantity, which multiplied 18v by itself makes that quantity whose Root it is said to be; and that multiplication of it into itself is called the square of the said Root. As we say the Root of 9 is 3, of 16 is 4, of 25 is 5, and so in the others; and 9, 16 and 25 are called squares.

And hereby it is to be known that there are certain quantities which have no Root that can be exactly named by number. As 10 has no number which, multiplied into itself, makes precisely 10; and so 11, 12, 13 and others like them. Roots therefore are, and arise, of two sorts: the one called discrete, or let us say rational, being that which can be assigned exactly by number, as the Root of 9 is 3; the other is called surd, being that which cannot be given exactly by number, as we have said of the Root of 10 and others. And these by another name are called irrational — for all those quantities which cannot be exactly assigned by number are in the art called irrational, and those which can be so given are called rational. Let this suffice for our purpose concerning roots.

These things well noted, let us return to the first proposed 19r effect, and make it clear with evident examples. For its elucidation let the same case of 10, adduced in that place, be taken up again, without labouring further in other toilsome quantities; for what is said in this one always happens the same in every other. And by way of arithmetic, for the fuller knowledge of Your Highness, we shall go on following out all the rest — presupposing always that the scientific proofs of everything our process will contain are assigned geometrically, with all diligence, in the places we shall cite from our philosopher Euclid, according to the fitting exigency of the conclusions.

I say, then, that 10 being divided according to our proportion, its greater part is the root of 125 minus 5; upon which, by the said effect, place 5 — that is, half of the whole 10 — and it will make exactly the root of 125, since that "minus 5" comes to be restored and filled up by the "plus 5," half of 10. This sum — the root of 125 — multiplied by itself makes 125 for its square, which is five times the square of the half of 10 (the half being 5, and its square 25). Whence 125 is precisely quintuple of the said 25, square of the said half of 10, as was stated. And this 19v effect has place in every quantity of whatever nature, as the 1st of the 13th of our guide openly demonstrates.

If a quantity be divided into two parts, and upon one of them be placed a quantity such that the square of this sum is quintuple the square of the quantity added, it follows of necessity that the said added quantity is half of the first quantity divided into those two parts; that the part to which it was added is its greater part; and that the whole was divided into those parts according to our proportion.

For example, take 15 minus the root of 125, and the root of 125 minus 5, as the two integral parts of a quantity; and upon the one — the root of 125 minus 5 — place 5 as third quantity. The sum is the root of 125, whose square is 125; and the square of the added quantity is 25. Whence 125 is quintuple of 25, square of the added quantity. I say that the root of 25, that is 5, is half of the first quantity divided into those two parts; and that the part to which it was added is the greater part of the said first quantity, divided according to our proportion having the mean and two extremes — that is, of 10. And this is the converse of the preceding effect, as the 2nd of the 13th concludes geometrically. 20r

If a quantity be divided according to our proportion, and to its lesser part be added half of the greater, then the square of the sum will always be quintuple the square of the half of the said greater part.

For example, let 10 be the quantity divided according to our divine proportion, so that the one part — the greater — is the root of 125 minus 5, and the lesser is 15 minus the root of 125. I say: if to 15 minus the root of 125, the lesser, be added half of the root of 125 minus 5, the greater, then the sum of the lesser and that half, multiplied by itself, will be five times the square of the half of the greater. And so it appears: for half of the root of 125 minus 5 is the root of 31¼ minus 2½, which, joined with 15 minus the root of 125, the lesser, makes 12½ minus the root of 31¼; whence, multiplying 12½ minus the root of 31¼ by 12½ minus the root of 31¼, it makes 187½ minus the root of 19,531¼. And this is called the square of the sum.

Then square also the half of the said greater: that is, multiply the root of 31¼ minus 2½ by the root of 31¼ minus 2½; it will make 37½ minus the root of 781¼, and this is called the square of the half of the greater — which 20v is precisely the fifth part of the square of the sum. Consequently the said square of the sum is quintuple the square of the half of the said greater part of 10 thus divided. And this power, with the others, is much to be esteemed, as is all proved geometrically by the 3rd of the 13th of our author.

If a quantity be divided according to our divine proportion, and to the whole of that quantity be added its greater part, then the said sum and the said greater part will be the parts of another quantity divided in the same way; and the greater part of this second quantity [so divided will always be the whole first quantity].

For example, let the quantity divided according to our unique proportion be 10, whose greater part is the root of 125 minus 5, and the lesser 15 minus the root of 125. If then upon 10, the first quantity, be placed the root of 125 minus 5, its greater part, it will make a second quantity: the root of 125 plus 5. And this second quantity — the root of 125 plus 5 — I say is likewise divided according to our proportion into the said two parts, namely into the root of 125 minus 5, the greater part of the first, and into 10, which was the first quantity and is the greater part of this second one. And it appears thus: the product of the root of 125 minus 5 21r — which was the greater part of the first and is now the lesser of this second — into the whole of this second quantity, that is into the root of 125 plus 5, makes as much as the square of the mean, or say the greater part, of this second quantity, which is 10: for the one and the other make exactly 100, as the said proportion requires. And this power the 4th of the 13th also makes manifest to us geometrically.

If a quantity be divided according to our said proportion, the sum of the square of the lesser part with the square of the whole entire quantity will always be triple the square of the greater part.

For example, let 10 be the quantity divided as we have said, so that one part is 15 minus the root of 125, the lesser, and the other the root of 125 minus 5, the greater. I say that the square of 15 minus the root of 125, joined with the square of 10, the whole quantity, makes a sum which is triple — three times — the square of the greater part, the root of 125 minus 5. For the square of 15 minus the root of 125 is 350 minus the root of 112,500; and the square of 10 is 100; which, joined with 350 minus the root of 112,500, makes 450 minus the root of 112,500 for the said sum. And the square 21v of the root of 125 minus 5 is 150 minus the root of 12,500, which is the third part of the said sum, as appears: for 150 minus the root of 12,500, multiplied by 3, makes exactly 450 minus the root of 112,500. Therefore the said sum is triple the said square, just as we stated. Which effect the 5th of the 13th concludes geometrically.

No rational quantity can ever be divided according to our said proportion without each of its parts being irrational, called a residue.

For example, let 10 be the rational quantity to be divided according to the proportion having the mean and two extremes. I say that of necessity each of the parts must be a residue. The one will be 15 minus the root of 125, the lesser, and the other, greater, will be the root of 125 minus 5; whence each appears to be a residue, for so they are called in the art according to the 79th of the 10th. And this effect we have from the 6th of the 13th.

If the side of the equilateral hexagon be added to the side of the equilateral decagon, both being understood as described in one same 22r circle, their sum will always be a quantity divided according to our said proportion, and its greater part will be the side of the hexagon.

For example, let the side of an equilateral hexagon in the designated circle be the root of 125 minus 5, and let the side of the equilateral decagon in the same circle be 15 minus the root of 125; the diameter of that circle will be the root of 500 minus 10. I say that the sum of the root of 125 minus 5 with 15 minus the root of 125 — which is 10 — is divided according to our proportion; and its greater part is the root of 125 minus 5, and the lesser 15 minus the root of 125, as we have said many times in dividing 10. And this is made manifest by the 9th of the 13th, geometrically.

If a line be divided according to the proportion having the mean and two extremes, then of that circle in which the greater part is the side of the hexagon, the lesser part will always be the side of the decagon.

For example, if the divided line were 10, its greater part, the root of 125 minus 5, will always be the side of the hexagon of a circle whose diameter is the double of the root of 125 minus 5, that is, the root of 500 minus 10. I say that of that 22v same circle, 15 minus the root of 125, the lesser part, will be the side of the equilateral decagon placed in it. And of this converse Ptolemy makes much use in the 9th chapter of the first section of his Almagest, to demonstrate the quantity of the chords of the arcs of the circle; as likewise is openly demonstrated upon the aforesaid 9th of the 13th, geometrically.

If in a circle the equilateral pentagon be formed, and under two of its neighbouring angles two straight lines be subtended, drawn from the ends of its sides, then of necessity those lines will divide each other according to our proportion, and each of their greater parts will always be the side of the said pentagon.

For example, let the pentagon be .abcde., and from the extremes .c. and .a. let the chord .ac. be drawn, which subtends the angle .b.; and from the extremes .b. and .e. let the other chord .be. be drawn, which subtends the angle .a. I say that these two lines .ac. and .be. divide each other at the point .f. according to the proportion having the mean and two extremes, and the greater part of each is precisely the side of the said pentagon. Thus, of the line .ac. the greater part is .cf., 23r and the greater part of the line .be. is .ef.; and each of these is always equal to the side of the said pentagon. And by the mathematicians these two lines are called, by another name, chords of the pentagonal angle.

Thus, if each of the said chords were 10 — for they will be equal, their pentagon being equilateral in the circle — .cf. would be the root of 125 minus 5, and .af. 15 minus the root of 125; and the part .ef. would likewise be the root of 125 minus 5, and .bf. 15 minus the root of 125; and the side of the pentagon would likewise be the root of 125 minus 5. All of which the 11th of the 13th demonstrates geometrically, in fine fashion. And by this effect we can, from knowledge of the side, come to knowledge of all its chords and of all their parts; and so conversely, from knowledge of the chords we can come to knowledge of the side and of the parts of the said chords — working arithmetically and geometrically, as in our above-cited work we have taught how to handle them, with all diligence of binomials and other irrational lines, of which our philosopher treats in his 10th; and by lines he demonstrates it in the 11th of the second and the 29th of the sixth. So one comes easily to knowledge of the one and the other, in every 23v mode — a thing of the greatest utility in our scientific and speculative occasions.

If a quantity be divided according to the aforesaid proportion, then all the effects that can arise from it and from its parts arise identically — in relation, number, species and kind — from any other quantity so divided.

For example, let there be two lines so divided: the one .ab., divided at .c., with its greater part .ac.; the other .de., with its greater part .df.; and what we say of these two, we intend of infinitely many others, which can easily be assigned by way of arithmetic. Putting .ab. at 10, .ac. would be the root of 125 minus 5, and the other part 15 minus the root of 125; and putting .de. at 12, .df. would be the root of 180 minus 6, and the other part would be 18 minus the root of 180. I say that everything that can ever happen to one of the said lines — compared, multiplied, divided, and worked in every other way — the like always happens to the other: that is, from each to its greater part is the same proportion, and likewise from each to its lesser part is the same proportion; and so conversely, from 24r each of their parts to the wholes; and so with the product of the one into its parts and, conversely, to the said parts; and so it falls out in dividing and subtracting. Thus the proportion from 10 to its greater part, the root of 125 minus 5, is the very same as from 12 to its greater part, the root of 180 minus 6; and the proportion from the sum of 10 and the root of 125 minus 5 to the root of 125 minus 5 is the very same as from the sum of 12 and the root of 180 minus 6 to the root of 180 minus 6.

And so, briefly, to infinity: taken and turned about quomodocumque et qualitercumque — through permuted, converse, conjoined, disjoined, everted and equal proportionality — it will always agree in one same denomination and in the same effects intensively. Which thing without fail demonstrates the greatest harmony in all quantities so divided, as will appear below in the regular and dependent bodies. And all this, in substance, the second of the 14th concludes geometrically.

If the side of an equilateral hexagon be divided according to our divine proportion, its greater part will always of 24v necessity be the side of the decagon circumscribed by the same circle as the hexagon.

For example, if the side of the hexagon were 10, divided in the said manner, its greater part will be the root of 125 minus 5 — which I say is exactly the side of the decagon circumscribed by the same circle, whose diameter would come to be 20. And this is concluded by the 3rd of the 14th. Whence evidently, having the side of the one, the side of the other is easily found; and likewise, having the diameter of the circle, or its circumference, or its area, or any other part of it, by these we can always come to knowledge of the one and the other through the one — and so conversely, in every mode of circle, hexagon, decagon and triangle too, working arithmetically and geometrically. Which is a most useful thing, as was said above in the 9th effect concerning the pentagon, ideo etc.

If a quantity be divided according to our said proportion, then the root of the sum of the square of the whole quantity and the square of its greater part will always be, in 25r proportion to the root of the sum of the square of the said quantity and the square of its lesser part, as the side of the cube to the side of the triangle of the body of twenty bases [the icosahedron].

For example, let 10 be the quantity divided according to the proportion having the mean and two extremes, so that the one part, the greater, is, as we have said many times, the root of 125 minus 5, and the lesser 15 minus the root of 125. Now square — that is, multiply into itself — the quantity adduced, namely 10: it makes 100. Square also its greater part, the root of 125 minus 5, which multiplied into itself makes 150 minus the root of 12,500; and square also the lesser part, 15 minus the root of 125, which multiplied into itself makes 350 minus the root of 112,500. Now upon the square of the greater part — upon 150 minus the root of 12,500 — place the square of the whole quantity, that is of 10, which is 100: it makes 250 minus the root of 12,500. Place the same square of the said quantity — 100 again — upon the square of the lesser part, which we found to be 350 minus the root of 112,500: joined with 100 it makes 450 minus the root of 112,500. Now I say that the proportion of the root of the one sum — 250 minus the root of 12,500, made of the square 25v of the said quantity and of its greater part — to the root of the other sum, made of the square of the said quantity and of its lesser part, that is 450 minus the root of 112,500, is precisely as the proportion of the side of the cube to the side of the triangle of the body of twenty bases, when both the said bodies are circumscribed, or surrounded, by one same sphere.

These roots of sums are called lines potent upon the said sums: that is, the root of 250 minus root of 12,500 means a quantity whose power, or square, is precisely that sum; and likewise the root of 450 minus root of 112,500 means a quantity whose power, or let us say square, is precisely 450 minus the root of 112,500. Such roots are by the practitioners called, by another name, universal roots or bound roots, as appears in our earlier-alleged work in the 3rd treatise of its 8th distinction, beginning at folio 120 of the said volume. These quantities are of the subtlest scrutiny, and belong to the speculative practice, as appears diffusely in the said volume. And these, most excellent Prince, cannot be named by any lowlier 26r denominations; and this whole speculative effect is demonstrated by the 9th of the 14th, geometrically, with certain others adduced in that place by Campanus.

For its 13th effect it is no small wonder that without its suffrage the pentagon — the figure of five equal sides, adduced above in the 9th effect and to be adduced again below — can never be formed. And without the pentagon, as shall be said, it is not possible to form or even to imagine the body noblest above all the other regulars, called dodecahedron, that is, the body of twelve equilateral and equiangular pentagons, by other name called the body of twelve pentagonal bases; whose form, as shall be said, the divine Plato attributed to the Quintessence, that is, to Heaven, for most fitting reasons. Thus our philosopher, in the 4th book by the 10th, teaches us how to make a triangle of this condition: that each of its two angles standing upon the base be double the other. And this he did because, if we would know how to form the equilateral and also 26v equiangular pentagon, and inscribe and circumscribe it to the circle — that is, form it exactly within and without the circle — it was not possible unless he had first taught us to make the said triangle, as appears by the 11th and 12th of the said 4th.

And to make the said triangle it is of necessity required to divide a line according to our divine proportion, as by the said 10th of the 4th he shows us — although in that place he does not say that the line is divided under the said proportion and its conditions, not having yet given us notice of what proportion is, which he reserves for his 5th; for it is not his custom to bring into his demonstrations things that come later, of which notice has not yet been given, but he uses only the antecedents; and this order is observed through all his fifteen books. And therefore, for the purpose of the said triangle, he does not say to divide the line according to the proportion having the mean and two extremes, but says, according to the 11th of the 2nd, to make of it two parts such that the square of the one is equal to the product of the other part into the whole line. Which in effect means nothing else than to divide it 27r according to the said proportion, as appears by the 3rd definition of the 6th and by the 29th of the same. And we too said as much above, when it was declared how the mean and its extremes are to be understood, in treating its first effect.

It does not seem good to me, most excellent Duke, to extend myself at present into more of its infinite effects, for the paper would not suffice the ink to express them all. But these thirteen we have chosen from among the rest, out of reverence for the company of twelve and its most holy Head, Our Redeemer Jesus Christ; for since we have given the proportion the divine name, its effects should also end at the number of our salvation, of the twelve Articles and twelve Apostles together with Our Saviour. For which college I understand Your Ducal Highness to have a singular devotion, having caused it to be set forth by our aforesaid Leonardo with his graceful brush in the fore-mentioned most sacred place, the temple of the Grazie. Nevertheless, in the process that follows we shall not refrain from adducing others as occasion requires; seeing that, as shall be said, it would not 27v be possible to form or imagine the harmony and worthy correspondence among all the regular bodies and their dependents — to which end we have set forth those already given, that their sequel may be rendered the clearer.

Now, most excellent Duke, the virtue and power of our aforesaid proportion, with its singular effects — above all as we said above — manifests itself in the formation and composition of the bodies both regular and dependent. Of which, that they may be better grasped, we shall speak here following in order: first of the five essential ones, which by another name are called regulars, and then successively of a good number of their distinguished dependents.

But first it must be made clear why they are called regular bodies; secondly it must be proved that in nature it is not possible to form a sixth. They are called regular, then, because they are of equal sides and angles and bases, and each is exactly contained by the other, as shall be shown;

and they correspond to the five simple bodies in nature, that is, earth, water, air, 28r fire and quintessence — the Celestial Virtue which sustains all the others in their being. And just as these five simples are enough and sufficient in nature — to argue otherwise would be to make God superfluous, or deficient for the need of nature, which is absurd, since as the Philosopher affirms, God and Nature do not work in vain, that is, they neither fall short of the need nor exceed it — so likewise the forms of these five bodies, of which we must presently speak, are five ad decorem universi [for the adornment of the universe], and cannot be more, by what shall follow. And therefore, not undeservedly, as shall be said below, the ancient Plato in his Timaeus attributed the figures of the said regulars to the five simple bodies, as was said above in the fifth correspondence of the divine name attributed to our proportion. So much for their denomination.

It now behoves us to show how there cannot be more than five such bodies in nature — bodies, that is, whose bases are all equal among themselves, with equal solid and plane angles, and likewise equal sides. Which appears thus: for the 28v constitution of any solid angle the concourse of at least three surface angles is necessary, since of only two surface angles no solid angle can be completed. Now, since the three angles of any equilateral hexagon are equal to four right angles — and of the heptagon too, the figure of seven sides, and generally of every equilateral and equiangular figure of more sides, its three angles are always greater than four right angles, as evidently appears by the 32nd of the 1st — and every solid angle is less than four right angles, as the 21st of the 11th testifies: it is therefore impossible that three angles of the hexagon, or of the heptagon, or generally of any equilateral and equiangular figure of more sides, should form a solid angle. And hereby it is manifest that no solid figure, equilateral and of equal angles, can be formed of hexagonal surfaces, or of surfaces of more sides; for if the three angles of the equilateral and equiangular hexagon are greater than a solid angle, it follows that four or more will exceed the said solid angle much more.

But the three angles of the equilateral and 29r equiangular pentagon are manifestly less than four right angles, and its four are greater than four rights. Whence of the three angles of an equilateral and equiangular pentagon a solid angle can be formed; but of four of its angles, or more, no solid angle can possibly be formed. And therefore only one body is formed of equilateral and equiangular pentagons, which is called dodecahedron — otherwise, by the philosophers, the body of twelve pentagons — in which the angles of the pentagons, three by three, form and contain all the solid angles of the said body.

The same reasoning holds for quadrilateral figures of equal sides and angles, as was said of the pentagons: for every quadrilateral figure, if it be equilateral and also of equal angles, will by definition be a square, since all its angles will be right, as is shown by the 32nd of the 1st. Of three angles, then, of such a surface figure it is possible to form a solid angle; but of four of them, or more, it is impossible. Wherefore of such surface figures — quadrilateral, equilateral and of equal angles — one solid can be formed, which 29v we call the cube: a body contained by six square surfaces, having twelve sides and eight solid angles. And of equilateral triangles, six angles are equal to four rights, by the said 32nd of the 1st; therefore fewer than six are less than four rights, and more than six are greater than four rights; and so of six such triangular angles, or more, no solid angle can be formed — but of five, of four and of three it can. And since three angles of the equilateral triangle contain a solid angle, of equilateral triangles is formed the body of four triangular bases with equal sides, called tetrahedron; and when four such triangles concur, the body of eight bases is formed, called octahedron; and if five equilateral triangles contain a solid angle, then the body called icosahedron is formed, of twenty triangular bases and equal sides. Whence, both why the regular bodies are so many and such, and why also they are no more, is fully manifest by what we have said.

Having seen and understood what 30r the regular bodies are, and exactly how many, it follows now to tell how they are formed, so as to be exactly surrounded by a sphere; and further, what proportion and denomination they, or their sides, bear to the diameter of the sphere that exactly surrounds them — whereby one comes to knowledge of them all. First, therefore, we shall speak of the tetrahedron, the equilateral body of four triangular bases; and then of each of the others successively, following in order.

[Of the tetrahedron exactly in the sphere.] I say, then, that the said body is to be formed thus. First take the diameter of the sphere in which we intend to place it — let it be the line .ab. — and divide it at the point .c. so that the part .ac. is double the part .bc. Upon it describe the semicircle .adb.; draw the line .cd. perpendicular to the line .ab., and draw the lines .bd. and .da. Then make the circle .fgh. about the centre .e., whose semidiameter shall be equal to the line .cd. In this circle then make an equilateral triangle, as the 2nd of the 4th teaches; let this triangle be .fgh., and from the centre to its angles draw the lines .ef., .eg., .eh. Next, upon the centre .e. 30v raise the line .ek. perpendicular to the surface of the circle .fgh., as the 12th of the 11th teaches; set this perpendicular equal to the line .ac., and from the point .k. let fall the hypotenuses .kf., .kg., .kh. These things exactly observed, I say the pyramid of four triangular bases with equal sides is finished; and it will be exactly circumscribed by the sphere of that diameter .ab. And I say, as to the proportion between the diameter of the sphere and the side of the pyramid so built, that the square of the said diameter is sesquialter to the square of the side of the said pyramid — that is, the square of the diameter contains the square of the side of the pyramid once and a half, as 3 to 2 and 6 to 4. That is to say: if the square of the said diameter were 6, the square of the side of the pyramid would be 4. And so it is found proved in geometry.

It follows to demonstrate how the cube is formed, and what proportion holds between its side and the diameter of the sphere that exactly surrounds it. I say the cube is to be formed thus. First take the 31r diameter of the sphere in which we intend exactly to place it; let it be the line .ab., upon which I shall make the semicircle .adb. Then I shall divide the diameter at the point .c. just as I did in forming the preceding pyramid, so that the part .ac. is double the part .bc.; draw the line .cd. perpendicular to the line .ab., and draw also the lines .db. and .da. Then make a square all of whose sides are equal to the line .bd.; let that square be .efgh., and upon its four angles raise four lines perpendicular to the surface of the said square, as the 12th of the 11th teaches; and let each of these perpendiculars also be set equal to the line .bd.; let the four perpendiculars be .ek., .fl., .gm., .hn. These four perpendiculars will each be equidistant from the others, by the 6th of the said 11th; and the angles contained by them and by the sides of the square are right, by the definition of the line perpendicular to a surface. Then join the extremities of these perpendiculars, drawing the lines .kl., .lm., .mn., .nk. These things diligently and exactly observed, the cube we sought to form will be finished, 31v contained by six square surfaces, as is proved by the 34th of the 1st. The four surfaces that surround it — those whose opposite sides are the four perpendiculars — are all square.

That the base is square is manifest from our construction; and that the topmost surface, .klmn., is square as well is demonstrated again by the said 34th of the 1st and by the 10th of the 11th. So too, by the 4th of the said 11th, it is manifest that all the sides of the said cube stand orthogonally upon its two opposite surfaces. And this body will be exactly circumscribed by the sphere of the proposed diameter. Whence the said diameter will always be triple in power to the side of the said cube: that is, the square of the said diameter will be three times the square of the side of the cube. So, if the diameter were the root of 300, the side of the cube would have to be exactly 10. The knowledge of which is opportune in many necessary cases.

In the third place there comes to be built the body 32r of eight triangular bases called octahedron, likewise to be exactly surrounded by a proposed sphere, of which sphere only the diameter is known to us. It is done in this way. Take the diameter of the sphere — let it be the line .ab. — and divide it into equal parts at the point .c.; upon the whole line make the semicircle .adb., and draw .cd. perpendicular to the line .ab.; then join the point .d. with the extremities of the said diameter, that is with .a. and with .b. Then make a square all of whose sides are equal to the line .bd.; let this square be .efgh. In this square draw two diameters, the one .eg. and the other .fh., dividing each other at the point .k.; whence by the 4th of the 1st it is manifest that each of these diameters is equal to the line .ab., which was set as diameter of the sphere — seeing that the angle .d. is right, by the first part of the 30th of the 3rd. And each of the angles .e., .f., .g., .h. is right by the definition of the square; and it is manifest too that those two diameters .eg. and .fh. divide each other equally at the point .k., as appears by 32v the 5th and 32nd and 6th of the 1st, easily deducing. Now raise upon .k. the line .kl. perpendicular to the surface of the square, and set this perpendicular equal to half the diameter .eg. — or .fh.; then let fall the hypotenuses .le., .lf., .lg., .lh.; and all these hypotenuses, by the things said and presupposed, by means of the penultimate of the 1st repeated as many times as need be, will be equal among themselves and equal also to the sides of the square.

Thus far, then, we have a pyramid of four triangular bases with equal sides, set upon the said square; which pyramid is half of the body of eight bases that we intend. Then beneath the said square we shall make another pyramid like this one, in this way: we shall carry the said line .lk. onward, boring and penetrating the said square, to the point .m., so that the line .km., standing below the square, is equal to the line .lk. standing above it. And then I shall join the point .m. with all the angles of the square, drawing four other hypotenusal lines, .me., .mf., .mg., .mh.; and these too are proved to be 33r equal among themselves, and to the sides of the said square, by the penultimate of the 1st and the others adduced above, as was proved of the other hypotenuses above the square. And the aforesaid things being always diligently observed, the body of eight triangular bases with equal sides will be finished; and it will be exactly circumscribed by the sphere.

The proportion between the sphere and the said body is this: the square of the diameter of the sphere is exactly double the square of the side of the said body. That is, if the said diameter were 8, the side of the eight-based body would be the root of 32; their powers stand to each other in duple proportion — the square of the diameter is double the square of the side of the said body. And so we have both the construction, and the proportion with respect to the sphere.

To know how to make the body of twenty equilateral triangular bases, such that it be exactly surrounded — circumscribable — by a given sphere having a rational diameter. And the side of the said body will evidently be an irrational line, namely that which is called the minor line.

For example, let the diameter of the 33v given sphere again be .ab., set as rational either in length or in power only. Divide it at the point .c. so that .ac. is quadruple — four times — .cb., and upon it make the semicircle .adb.; draw .cd. perpendicular to .ab., and draw the line .db. Then, according to the quantity of the line .db., make the circle .efghk. about the centre .l., and inscribe in it an equilateral pentagon on the points noted, or say named; to whose angles, from the centre .l., draw the lines .le., .lf., .lg., .lh., .lk. And in the same circle make also an equilateral decagon. Divide, then, all the arcs whose chords are the sides of the pentagon into equal parts; and from the middle points to the extremities of all the sides of the inscribed pentagon let straight lines be drawn; and further, upon all the angles of the said pentagon raise the cathetus, or say perpendicular, as the 12th of the 11th teaches, each of which shall likewise be equal to the line .bd. And connect — or say join — the extremities of these five catheti with five corausti [a coraustus, in Pacioli's glossary at Chapter LXXI, is the straight line joining the tops of two raised lines]; and by the 6th of the 11th the five catheti so 34r raised will be equidistant among themselves; and since they are equal, the five corausti joining their extremities will also be, by the 33rd of the 1st, equal to the sides of the pentagon.

Let fall, then, from each summit of all the catheti, two by two, hypotenuses to the two adjacent angles of the inscribed decagon; and the extremities of these ten hypotenuses — which descend from the five extremities of the catheti to the five points which are the intermediate angles of the inscribed decagon — connect and join, forming another pentagon in the said circle, which will also be equilateral by the 23rd of the 3rd. And when you have done this, you will see that you have made ten triangles, whose sides are the ten hypotenuses, the five corausti, and the five sides of this inscribed pentagon. And that these triangles are equilateral you will grasp thus: since both the semidiameter of the described circle and each of the raised catheti are equal to the line .bd. by the hypothesis, each of the catheti will be, by the corollary of the 15th of the 4th, equal to the side of the equilateral hexagon made 34v in the circle whose diameter is equal to the line .bd.; and because, by the penultimate of the 1st, each of the ten hypotenuses is more potent than the cathetus by just as much as the side of the decagon has power [i.e., its square exceeds the cathetus's square by the square of the decagon's side], and further, by the 10th of the 13th, the side of the pentagon is more potent than the same by just as much as the same side of the decagon has power, each of these hypotenuses will be, by common science, equal to the side of the pentagon.

And of the corausti it has already been shown that they are equal to the sides of the pentagon. Whence all the sides of these ten triangles either are sides of the equilateral pentagon inscribed the second time in the circle, or are equal to them: the said triangles are therefore equilateral. Further, above the centre of the circle, which is the point .l., raise another cathetus equal to the first ones — let it be .lm. — and join its upper extremity, the point .m., with each extremity of the first ones by five corausti. By the 6th of the 11th this central cathetus, raised at the centre, will be equidistant from each of the angular catheti; and therefore, by the 33rd of the 1st, these five corausti will be 35r equal to the semidiameter of the circle and, by the corollary of the 15th of the 4th, each will be as the side of the hexagon. Then to the said central cathetus, at either end, add a line equal to the side of the decagon: above, add to it .mn.; and below, beneath the circle, add to it from the centre of the circle .lp. Then let fall from the point .n. five hypotenuses to the five upper angles of the ten triangles standing around the circuit, and from the point .p. another five to the other five lower angles. And these ten hypotenuses will be equal among themselves [and] to the sides of the inscribed pentagon, by the penultimate of the 1st and by the 10th of the 13th, just as was demonstrated before of the other ten.

You have, then, the body of twenty triangular and equilateral bases, all of whose sides are equal to the sides of the pentagon; and its diameter is the line .np. And of these twenty triangles, ten stand in the circuit upon the circle, five rise upward concurring at the point .n., and the other five concur below the circle at the point .p. And that this body called icosahedron, so formed, is exactly surrounded by the given sphere will be made manifest thus. Since the line .lm. 35v is equal to the side of the hexagon, and the line .mn. to the side of the decagon — both equilateral, and both circumscribed by the same circle .efg. — the whole .ln. will be, by the 9th of the 13th, divided according to the proportion having the mean and two extremes at the point .m., and its greater part will be the line .lm.

Divide, then, .lm. into equal parts at the point .q.; and .pq. will be, by common science, equal to .qn., since .pl. is set equal to the side of the decagon, just as .mn. is; whence .qn. is half of .np., just as .qm. is half of .ml. Since, then, the square of .nq. is, by the 3rd of the 13th, quintuple the square of .qm., the square of .pn. will also be, by the 15th of the 5th, quintuple the square of .lm.; for by the 4th of the 2nd the square of .pn. is quadruple the square of .qn., and the square of .lm. likewise quadruple the square of .qm., by the same. And quadruple to quadruple is as simple to simple, as the 15th of the 5th affirms. And the square of .ab. is quintuple the square of .bd., by the second part of the corollary of the 8th of the 6th and by the corollary of the 17th of the same; for .ab. is also quintuple .bc., since .ac. was quadruple 36r of the same. Since, then, .lm. is by the hypothesis equal to .bd., .ab. will be, by common science, equal to .np. Whence, if upon the line .np. a semicircle be made and carried round until it returns to the first place whence it began to move, the sphere made by its motion will be — by the definition of equal spheres — equal to the proposed sphere.

And because the line .lm. is the mean proportional between .ln. and .nm. — and therefore between .ln. and .lp. — each semidiameter of the circle will also be the mean proportional between .ln. and .lp., since .lm. is equal to the semidiameter of the circle. Whence the semicircle described upon .pn. will pass through all the points of the circumference of the circle .efg., and therefore also through all the angles of the fabricated solid that stand on that circumference. And because, by the same reasoning, all the corausti — which connect, or say join, the extremities of the angular catheti with the extremity of the central one — are mean proportionals between .pm. and .mn. (for each of them is equal to .lm.), it follows that the same semicircle passes 36v also through the other angles of the icosahedron figure so fabricated.

This body, then, is inscribable — or say placeable — in the sphere whose diameter is [.pn., and therefore also in the sphere whose diameter is] .ab. And the side of this solid figure I say is the minor line. For it is manifest that the line .bd. is rational in power, since its square is the sub-quintuple — say, the fifth part — of the square of the line .ab., which was set as rational either in length or in power only. Whence the semidiameter — and the semidiameters — of the circle .efg. are also rational in power, its semidiameter being equal to .bd. Therefore, by the 16th of the 13th, the side of the equilateral pentagon inscribed in this circle is the minor line; and further, as was shown in the process of this demonstration, the side of this figure is as much as the side of the pentagon. Therefore the side of this figure of twenty equilateral triangular bases is the minor line, just as was proposed.

To know how to make the body of twelve equilateral and equiangular pentagonal bases, such that the proposed sphere exactly 37r surrounds — or say circumscribes — it. And the side of the said body will manifestly be irrational, of the kind called residue.

Make a cube, according to the manner given, such that the assigned sphere exactly surrounds it; and let two surfaces of this cube be .ab. and .ac.; and let us now imagine that .ab. is the topmost surface, and the surface .ac. one of the lateral ones, and let the line .ad. be common to these two surfaces. Divide then, in the surface .ab., the two opposite sides into equal parts — namely .db. and the side opposite to it — and let the points of division be joined by the line .ef. The side .ad. too, and the one opposite it in the surface .ac., divide into equal parts, and let the points of division be joined by a straight line, of which let the half be .gh., and let [.h.] be the middle point of the line .ad. Likewise divide the line .ef. into equal parts at the point .k., and draw .hk.

Each, then, of the three lines .ek., .kf. and .gh. you will divide according to the proportion having the mean and two extremes, at the three points .l., .m., .q.; and let their greater parts be .lk., .km. and .gq., which are manifestly equal, since all the divided lines are equal — each of them being half the side of the cube. Then from 37v the two points .l. and .m. raise perpendiculars to the surface .ab., as the 12th of the 11th teaches, each of which you will set equal to the line .kl.; let them be .ln. and .mp. Likewise from the point .q. raise .qr. perpendicular to the surface .ac., which you will set equal to .gq. Then draw the lines .al., .an., .am., .ap., .dm., .dp., .dl., .dn., .ar., .aq., .dr., .dq. It is manifest, then, by the 5th of the 13th, that the two lines .ke. and .el. are, in power, triple the line .kl.; and therefore also the line .ln., since .kl. and .ln. are equal. And .ke. is equal to .ea.; therefore the two lines .ae. and .el. are in power triple the line .ln. Whence, by the penultimate of the 1st, .al. is in power triple .ln.; and therefore, by the same, .an. is in power quadruple .ln. And since every line is in power quadruple its half, it follows by common science that .an. is double .ln. in length. And because .lm. is double .lk., and .kl. and .ln. are equal, .an. will be equal to .lm., since their halves are equal. And because, by the 33rd of the 1st, .lm. is equal to .np., .an. will be equal to .np. And in the same way 38r you will prove the three lines .pd., .dr. and .ra. to be equal among themselves and to the two aforesaid.

We have then, by these five lines, the equilateral pentagon .anpdr. But perhaps you will say that it is not a pentagon, because perhaps it does not lie all in one same surface — which is necessary for it to be a pentagon. That it lies all in one same surface you will grasp thus. Let there spring — say issue — from the point .k. the line .ks., perpendicular to the surface .ab. and equal to .lk.; it will thereby be equal to each of the two .ln. and .mp. And since it is equidistant from each of them by the 6th of the 11th, and therefore lies with both in the same surface by the definition of equidistant lines, the point .s. must of necessity lie on the line .np. and divide it equally. Draw then the two lines .rh. and .hs.; whereby the two triangles .ksh. and .qrh. are constituted upon one angle, namely .khq. And the proportion of .kh. to .qr. is as that of .ks. to .qh.: for as .gh. is to .qr., so is .kh. to .qr., by the 7th of the 5th; and as .rq. is to .qh., so is .ks. to .qh., by the same. But .gh. is to .qr. as .qr. to .qh., since .qr. is equal to .gq. Therefore, by the 32nd of the 6th, the line 38v .rhs. is one line. Whence, by the 2nd of the 11th, the whole pentagon we dispute of lies in one same surface.

And I say further that it is equiangular, which will appear thus. Since .ek. is divided according to the proportion having the mean and two extremes, and .km. is equal to its greater part, the whole .em. too will be, by the 4th of the 13th, divided according to the proportion having the mean and two extremes, and its greater part will be the line .ek. And therefore, by the 5th of the 13th, the two lines .em. and .mk. — and therefore the two .em. and .mp., since .mp. is equal to .mk. — are in power triple the line .ek., and therefore also the line .ae., since .ae. is equal to .ek. Whence the three lines .ae., .em. and .mp. are in power quadruple the line .ae. It is clear too, by the penultimate of the 1st twice repeated, that the line .ap. is in power equal to the three lines .ae., .em. and .mp.; whence .ap. is in power quadruple the line .ae. And the side of the cube, being double the line .ae., is also in power quadruple of it, by the 4th of the 2nd. Therefore, by common science, .ap. is equal to the side of the cube; and since .ad. is one of the sides of the cube, .ap. will be equal to .ad., and therefore, by the 8th of the 1st, the angle .ard. is equal to the angle .anp.

In the 39r same way you will prove the angle .dnp. to be equal to the angle .dra., for you will prove the line .dn. to be in power quadruple the half of the side of the cube. Since then, by these things said, the pentagon is equilateral and has three equal angles, it will be equiangular by the 7th of the 13th. If, then, by this way and like reasoning we fabricate upon each of the other sides of the cube an equilateral and equiangular pentagon, there will be finished a solid contained by twelve equilateral and also equiangular pentagonal surfaces — since the cube has twelve sides.

It remains now to demonstrate that this solid is circumscribable — say, exactly surrounded — by the given sphere; which will appear thus. Draw from the line .sk. two surfaces dividing the cube: one dividing it along the line .hk., the other along the line .ef. By the 40th of the 11th, the common section of these two surfaces will divide the diameter of the cube, and conversely — say by converse — it will itself be divided equally by the said diameter. Let then their common section, carried to the diameter of the cube, be the line .ko., so that the point .o. is the centre of the cube; and 39v let the lines .oa., .on., .op., .od., .or. be drawn. It is clear that each of the two lines .oa. and .od. is a semidiameter of the cube, and therefore they are equal. And of the line .ok. it is clear, by the 40th of the 11th, that it is equal to .ek., that is, to half the side of the cube; and because .ks. is equal to .km., .os. will be divided at the point .k. according to the proportion having the mean and two extremes, and its greater part will be the line .ok., which is equal to .ek. Whence, by the 5th of the 13th, the two lines .os. and .sk. — and therefore also .os. and .sp., since .sp., to which this demonstration extends no further, is equal to .ks. — will be triple in power to the line .ok., and therefore to half the side of the cube. Whence, by the penultimate of the 1st, the line .op. is in power triple the half of the side of the cube. And by the corollary of the 14th of the 13th it is manifest that the semidiameter of the sphere is triple in power to half the side of the cube which is circumscribed — say surrounded — by the same sphere. Whence .op. is as much as the semidiameter of the sphere that exactly surrounds the proposed cube; and by the same reasoning, so are all the lines drawn from the point .o. to each of the angles of all the pentagons formed upon the sides of the 40r cube — that is, to all the angles proper to the pentagons, and not to those common to them and to the surfaces of the cube; proper, precisely, as are the three angles .n., .p., .r. in the pentagon formed.

And of those lines that come from the point .o. to all the angles of the pentagons which are common to the pentagons and to the surfaces of the cube — as, in the present pentagon, the two angles .a. and .d. — it is clear that they are equal to the semidiameter of the sphere that exactly surrounds the cube, since they are semidiameters of the cube by the 40th of the 11th. But the semidiameter of the cube is as the semidiameter of the sphere that exactly surrounds it, as appears by the reasoning of the 14th [of the 13th]. Therefore all the lines drawn from the point .o. to all the angles of the dodecahedron — that is, of the solid contained by twelve equilateral and equiangular pentagonal surfaces, for so it is called in Greek — are equal among themselves and to the semidiameter of the sphere. Whence if the semicircle drawn upon the whole diameter of the sphere — or of the cube — be carried round, it will pass through all its angles; whence 40v by the definition it is circumscribable — say surrounded — by the assigned sphere.

I say further that the side of this figure is an irrational line, namely that which is called residue, if the diameter of the sphere that exactly surrounds it be rational in length or in power; which appears thus. Since the diameter of the sphere is, by the 14th of the 13th, triple in power to the side of the cube, the side of the cube will be rational in power if the diameter of the sphere be rational in length or in power. And by the 11th of the 13th it is clear that the line .rp. divides the line .ad. — which is the side of the cube — according to the proportion having the mean and two extremes, and that its greater part is equal to the side of the pentagon. And because its greater part is a residue, by the 6th of the 13th, it is manifest that the side of the figure called dodecahedron is a residue. Which is what we wished to demonstrate.

The sides of the five aforesaid bodies, all exactly circumscribed by one same sphere — of which sphere only the diameter is proposed to us — to know how to find by means of the said diameter.

For example, let .ab. be the diameter of some sphere proposed to us, by which we must find the sides of the five aforesaid bodies, 41r all understood as placed in one same sphere, so that the sphere, touching one of the angles of each, touches them all — that is, exactly surrounds them all. Which we shall do thus. Divide this diameter at the point .c. so that .ac. is double .cb., and into equal parts at the point .d.; and upon it we shall make the semicircle .afb., to whose circumference let two lines be drawn perpendicular to the line .ab., namely .ce. and .df. And join .e. with .a. and with .b., and .f. with .b. It is manifest, then, by the demonstration of the 13th of the 13th, that .ae. is the side of the figure of four triangular and equilateral bases; by the demonstration of the 14th [of the same], that .eb. is the side of the cube; and by the demonstration of the 15th, that .fb. is the side of the figure of eight triangular and equilateral bases. Then let the line .ag. issue from the point .a., perpendicular to .ab. and also equal to the same .ab.; join .g. with .d., and let .h. be the point where .gd. cuts the circumference of the semicircle; and draw .hk. perpendicular to .ab. And because .ga. is double .ad., .hk. will be, by the 4th of the 6th, double .kd., since the two 41v triangles .gad. and .hkd. are equiangular by the 32nd of the 1st — the angle .a. of the greater being equal to the angle .k. of the lesser, each being right, and the angle .d. common to both.

Therefore, by the 4th of the 2nd, .hk. is quadruple in power to .kd.; therefore, by the penultimate of the 1st, .hd. is in power quintuple .kd. [and since .db. is equal to .hd. — for .d. is the centre of the semicircle — .db. too will be in power quintuple .kd.]. And since the whole .ab. is double the whole .bd., just as .ac., taken from the first, is double .cb., taken from the second, the remainder .bc. of the first will be, by the 19th of the 5th, double the remainder .cd. of the second; and therefore the whole .bd. is triple .dc. Therefore the square of .bd. is nonuple — nine times — the square of .cd.; and since it was only quintuple the square of .kd., the square of .dc. will be, by the second part of the 10th of the 5th, less than the square of .kd., and hence .dc. less than .kd. Let then .dm. be equal to .kd., and let .mn. rise to the circumference, perpendicular to .ab.; and join .n. with .b. Since then .dk. and .dm. are equal, the two lines .hk. and .mn. will be — by the definition of what it is for a line to be equidistant from the centre — equally distant, or say 42r remote, from the centre, and therefore equal to each other, by the second part of the 13th of the 3rd and by the second part of the 3rd of the same. Whence .mn. is equal to .km., since .hk. was equal to it. And because .ab. is double .bd., and .km. double .dk., and the square of .bd. quintuple the square of .dk., the square of .ab. will be, by the 15th of the 5th, likewise quintuple the square of .km. — for as the square of the double is to the square of the double, so is the square of the simple to the square of the simple. And by the demonstration of the 16th of the 13th it is manifest that the diameter of the sphere is quintuple in power to the side of the hexagon of the circle of the figure of twenty bases.

Therefore .km. is equal to the side of the hexagon of the circle of the figure of twenty bases, since the diameter of the sphere, .ab., is quintuple in power both to the side of the hexagon of the circle of that figure and to .km. And further, by the demonstration of the same, it is manifest that the diameter of the sphere is composed of the side of the hexagon and two sides of the decagon of the circle of the figure of twenty bases. Since then .km. is as the side of the 42v hexagon, and .ak. is equal to .mb. — for they are the residues, or say remainders, of equals when equals are taken away — .mb. will be as the side of the decagon. And since .mn. is as the side of the hexagon, being equal to .km., .nb. will be, by the penultimate of the 1st and by the 10th of the 13th, as the side of the pentagon of the circle of the figure of twenty bases. And because, by the demonstration of the 16th of the same, it appears that the side of the pentagon of the circle of the figure of twenty bases is the side of that same figure of twenty bases, it is clear that the line .nb. is the side of this figure.

Divide then .eb. — the side of the cube exactly surrounded by the proposed sphere — according to the proportion having the mean and two extremes, at the point .p., and let its greater part be .pb. It is clear, then, by the demonstration of the preceding, that .pb. is the side of the figure of twelve bases. Thus the sides of the five bodies premised — say set before us — are found by means of the diameter of the sphere alone proposed to us; and these sides are: .ae. of the pyramid of four bases; .eb., side of the cube; .fb., side of the eight bases; .nb., side of the twenty bases; and the line .pb., side of the twelve bases. And 43r which of these sides are greater than which among themselves appears thus. It is clear that .ae. is greater than .fb., since the arc .ae. is greater than the arc .fb.; and again .fb. is greater than .eb., and .eb. greater than .nb. And I say further that .nb. is greater than .pb.: for since .ac. is double .cb., the square of .ac. will be, by the 4th of the 2nd, quadruple the square of .cb.; and by the second part of the corollary of the 8th of the 6th, and by the corollary of the 17th of the same, it is clear that the square of .ab. is triple the square of .be. But by the 21st of the 6th, the square of .ab. is to the square of .be. as the square of .be. to the square of .cb., since the proportion of .ab. to .be. is as that of .be. to .bc., by the second part of the corollary of the 8th of the 6th. Whence, by the 11th of the 5th, the square of .be. is triple the square of .cb.; and since the square of .ac. is quadruple the same square, as has been shown, the square of .be. will be, by the first part of the 10th of the 5th, less than the square of .ac. And therefore the line .ac. is greater than the line .be., and .am. much greater still. And it is already manifest, by the 9th of the 13th, that if the line .am. be divided according to the proportion having the mean and two 43v extremes, its greater part will be the line .km., which is equal to .mn.; and again, when .be. is divided according to the same proportion, habens medium et duo extrema, its greater part is the line .pb. Since then the whole .am. is greater than the whole .be., .mn. — equal to the greater part of .am. — will be greater than .pb., which is the greater part of .eb. And this is manifest by the 2nd of the 14th book, which fortifies itself with firm demonstration without the help of any of those that follow. Therefore, by the 19th of the 1st, so much the more strongly is .nb. greater than .pb.

Whence it appears that the sides of the five aforesaid bodies exceed one another almost in the same order in which they follow one another. Only this exception stands: that order is not observed between the cube and the octahedron, the eight bases; for the side of the eight bases precedes the side of the cube, although the cube precedes the octahedron in fabric and formation, as appears in the 13th — and not without mystery. In formation the cube is set before the octahedron because, by the same division of the diameter 44r of the proposed sphere, are found both the side of the pyramid of four triangular bases and the side of the cube. Thus .ae., side of the pyramid, is greater than the sides of all the other bodies; after it, .fb., side of the eight bases, is greater than the sides of all the others that follow it; in the third place in greatness follows .eb., side of the cube; in the fourth place is .nb., side of the twenty bases, the icosahedron; and least of all is .pb., side of the dodecahedron, of the twelve pentagonal bases.

Having understood the sufficiency of the said five regular bodies, and shown the impossibility of there being more than five, together with the way of proceeding to infinity in their dependents, it follows that we must give a way to their proportions, between one and another and another and one, both as to capacity and content and as to their surfaces; and then of the inclusions of one in another and conversely; and first of their corporeal bulk.

The proportions of one to another will always be irrational, by reason of our proportion adduced above, which interposes itself in their compositions and formations 44v as has been said — except that, for the tetrahedron and the cube and the octahedron, through the very exactness of their proportions to the diameter of the sphere in which they are inscribed, it may at times perhaps be rational; but that of the icosahedron, and that of the dodecahedron, compared with whichever you please, can never be rational, for the reason said. And therefore it does not seem to me, most excellent Duke, that more should be said of it here, for it would be to swell the volume with infinite irrationalities, in which the intellect would sooner come to confusion than take pleasure — and on pleasure our study is ever intent. Let this much suffice: what has been said in our own work, in the particular treatise composed on the said bodies; to which, since it has been communicated in multitude to the world, recourse is easy. And by means of their dimensions set down in that place, according to the rareness of one's wits, one will always be able to draw from them great delight together with profit. And I say the same likewise of all their dependents, of which a good number are set down in that place.

True it is that, by the 10th of the 14th, the proportion of the dodecahedron to the icosahedron, 45r when both are made in the same sphere, is concluded to be exactly as that of all the surfaces of the one to all the surfaces of the other joined together. And the 16th of the same says that the octahedron is divisible into two pyramids of equal height, level with the semidiameter of the sphere in which it was fabricated; and their bases are square, and that superficial square is subduple to the square of the diameter of the sphere. Which knowledge much avails us for its measure; and by means of it many others can be reached.

Their surfaces, most excellent Duke, we may likewise say are proportional among themselves in the same manner as was said of their corporeal mass — that is, irrational, through the malice of the pentagonal figure that interposes itself in the dodecahedron. But those of the others can at times be rational, as those of the tetrahedron, cube and octahedron, being triangular and square, and known in proportion with the diameter of the sphere in which they are formed, as has been seen above. True it is that the 8th 45v of the 14th concludes: all the surfaces of the twelve pentagonal bases are to all the surfaces of the twenty triangular bases — of the dodecahedron to those of the icosahedron — as the side of the cube to the side of the triangle of the body of twenty bases, when all the said bodies are exactly contained, or circumscribed, by one same sphere. Wherefore it does not seem right to pass over in silence the marvellous correspondence between them in their bases: namely, that the bases of the dodecahedron and those of the icosahedron are each exactly circumscribed by one same circle, as the 5th of the said 14th shows — a thing worthy of note; and this, when they are fabricated in the same sphere.

And of all the surfaces of the tetrahedron to all the surfaces of the octahedron, the proportion is known by the 14th of the said 14th: one of the bases of the tetrahedron is once and a third one of the bases of the octahedron — in the sesquiterza proportion, which is when the greater contains the lesser once and a third, as 8 to 6 and 12 to 9. And the proportion of all the surfaces 46r of the octahedron joined together, to all those of the tetrahedron joined together, is sesquialter — once and a half — as if those of the octahedron were 6 and the others 4; which is when the greater contains the lesser once and a half; and this, when they are of one same sphere. And all those of the tetrahedron, joined with those of the octahedron, compose a surface called medial, as the 13th of the said 14th holds. And all the surfaces of the hexahedron, the cube, equal the double of the square of the diameter of the sphere that circumscribes it; and the perpendicular drawn from the centre of the sphere to each of the bases of the said cube is always equal to half the side of the said cube, by the last of the 14th. That is: if the said diameter were 4, all the said surfaces would be 32; and if the said perpendicular were 1, the side of the cube would be 2. Of which proportions and surfaces, having treated fully in our own work, let these stand as supplement, working diligently in every mode by algebra, together with those of the dependents.

It follows now to make clear how, among these five essential — that is, regular — bodies, the one is contained by the other; and which are, and which are not, and why. Speaking first of the tetrahedron, it is shown that it can in no way receive within itself any other than the octahedron, the body of eight triangular bases and six solid angles; for in it there are neither sides nor bases nor angles upon which the sides of the cube, or its angles or surfaces, might rest so as to touch equally, as their true inscription requires — as its material form demonstrates to the eye, and as is made manifest by true science in the 1st of the 15th. Nor yet either of the other two, the icosahedron and the dodecahedron.

When, therefore, we wish to inscribe or form the said octahedron in the said four-based body, the tetrahedron, we shall do it in this way. First we shall fabricate the said tetrahedron as we have taught above; which done, we shall divide each of its sides into equal parts, and join all their middle points with straight lines, 47r one with another and another with one. Which done, without doubt we shall have placed the said body exactly within it, so that its six solid angles will rest equally upon the six sides of the said tetrahedron. Which thing material experience will render open, and the 2nd of the 15th makes manifest.

The said tetrahedron will be placed in the cube in this way: first we shall make the cube according to the manners given above; then in each of its six square surfaces we shall draw the diagonal, or diameter; and the purpose will be concluded, as the 1st of the 15th demonstrates. For the said tetrahedron, as was said, has six sides corresponding to the number of the six surfaces of the cube, and these come to be their six diagonals drawn across its surfaces; and the four angles of the pyramid come to rest in four of the eight of the said cube. Which, again, the mistress of all things, holy experience, renders clear in their material forms.

And wishing to form the eight bases, the octahedron, 47v in the hexahedron, one must first have fabricated in the cube the equilateral triangular pyramid, whose sides, as was said, are the six diameters of its bases. Then, if we divide each of the said diameters into equal parts and join those middle points with straight lines one to another, without doubt the octahedron will be exactly formed in the proposed cube; and each of its solid angles will rest exactly in the bases of the said cube, by the 3rd of the 15th.

The hexahedron, or cube, will be made in the octahedron in this way: first we shall make the said octahedron according to the instructions given above in this work. It being so formed, find the centre of each of its triangular bases by the 5th of the 4th; these eight centres we shall then join one with another by twelve straight lines. And we shall have concluded our intent; and each of the solid angles of the cube will come to rest upon a base of the said octahedron, as the 4th of the 15th declares.

If we wish to form the equilateral triangular pyramid, 48r the tetrahedron, in the octahedron, we shall first make in it the cube, according to what was said above in the preceding; and then in the said cube the said tetrahedron will be made in the manner stated. And thus we shall likewise have placed the said tetrahedron in the given octahedron, as says the 5th of the 15th.

The icosahedron, as has been said, has twelve solid angles, each contained by five surface angles of five of its triangles. Therefore, to make the dodecahedron in it, one must first, as we have taught in this work, make the said icosahedron. And when it is duly disposed, find the centre of each of its triangular bases by the 5th of the 4th; these we shall then connect by thirty straight lines, all among themselves, in such a way that of necessity twelve pentagons will be formed, each opposite a solid angle of the said icosahedron. And each of the sides of the said pentagons is opposite, crosswise, to one of the sides of the said icosahedron; and just as in the said icosahedron there are twelve solid angles, so in the 48v dodecahedron there are twelve pentagons; and as in the one there are twenty triangular bases, so in the said dodecahedron there are twenty solid angles caused in the said bases by means of the said lines. And as in the one there are thirty sides, so in the dodecahedron there are thirty sides opposite to them, crosswise as has been said — all of which their form makes manifest, as also the 6th of the 15th concludes.

When one wishes to form the icosahedron in the dodecahedron, we shall first fabricate the latter according to the instruction given above in this work; and of its twelve pentagons that contain it we shall find the centre, as the 14th of the 4th teaches; and these we shall join among themselves with thirty lines, in such a way that in it will be caused twenty triangles and twelve solid angles, each contained by five surface angles of the said triangles. Their points will lie in the twelve centres of its twelve pentagons; and likewise its thirty lines set themselves crosswise against the thirty of the dodecahedron, just as was said of those against these; as appears also by the 7th of the said 15th.

The cube, again, we shall easily make in the said dodecahedron, seeing that the latter is formed upon the twelve sides of the cube, as is contained in the 17th of the 13th. For if to each of its twelve pentagons, according to the requirement of the said proposition, twelve chords be drawn, without doubt six equilateral quadrangular surfaces will be formed; and opposite each of them will stand two solid angles of the said dodecahedron, and in eight of its angles will be formed the eight of the inscribed cube — in such a way that upon each base of the cube there comes to remain almost the form of the roof-shaped body [the corpo seratile, Pacioli's "saw-horse" or ridge-roof solid: a triangular prism]. All of which is clear by the 8th of the 15th.

If in the dodecahedron the cube be first disposed, as was said in the preceding, then the octahedron will easily be formed in the said dodecahedron. For we shall divide into equal parts the six sides of the dodecahedron opposite the six surfaces of the cube — those sides, that is, which as it were make the ridge of the roof-shaped body, and which are exactly six. And those six middle points of theirs we shall connect by twelve straight lines, all among themselves, so that they will come to 49v cause six solid angles, each contained by four surface angles of the four triangles of the octahedron. And each touches one of the said six sides of the dodecahedron; and consequently what was sought is shown to be concluded, as is contained in the 9th of the 15th.

The tetrahedron, again, will be placed in the same dodecahedron if first the cube be formed in it, as has been said, and then in the said cube the tetrahedron be placed, as has also been shown. These things done, it will clearly appear that our purpose is concluded, in this way: since the solid angles of the cube rest in the solid angles of the dodecahedron, and the solid angles of the tetrahedron take their stand in those of the cube, it follows from first to last that the said tetrahedron is duly included in the proposed dodecahedron — which our experience, in the material models composed by us and offered into the hands of Your Highness, makes manifest, together with the scientific demonstration of the 10th of the said 15th.

The cube is formed in the icosahedron if first the dodecahedron be made in it, as we said before; and then in that dodecahedron the cube be made in the manner given. These things done, the intent will appear dispatched, by what was said before: for the solid angles of the dodecahedron all fall in the centres of the bases of the icosahedron, and the solid angles of the cube fall in the said solids of the dodecahedron. And consequently the intent is dispatched — as is also declared to us by the 11th of the 15th.

There is no doubt: if in the said icosahedron the cube be formed, as we taught above, and then in that cube the tetrahedron be fabricated, of necessity it too will come to be inscribed in the said icosahedron. For the solid angles of the pyramid of four triangular bases touch those of the cube, and those of the cube touch those of the icosahedron; it follows, from first to last, that those of the tetrahedron equally touch those of the icosahedron. And consequently our purpose is concluded, by the 12th of the 15th. And this is what concerns 50v their proposed inclusions.

Whence, most excellent Duke, by the things discoursed it is manifest that, the regular bodies being five, if each could duly be formed in each, as is presupposed, it would follow that each would receive four;

and consequently among them all there would come to be twenty inscriptions, that is, four times five. But because each does not receive each, as has been adduced, there are but twelve inscriptions. Namely: one only in the tetrahedron, that of the octahedron; and two in the cube, of the tetrahedron and of the octahedron; and two again in the octahedron, one of the cube and one of the tetrahedron. And three are those of the icosahedron — one of the dodecahedron, one of the cube, and the other of the tetrahedron; and four are those of the dodecahedron — one of the icosahedron, another of the cube, another of the octahedron, and the fourth of the tetrahedron. Which in all are twelve in number. For in the pyramid of four bases there are neither sides nor angles nor surfaces on which the angles of the three other regulars might rest, save only of the octahedron.

The cube, again, 51r can receive in itself only the pyramid and the octahedron; and the octahedron only the cube and the pyramid; and in none of these is it possible to place either of the other two, the icosahedron and the dodecahedron. And though the icosahedron gives lodging to three, to the octahedron alone has it denied it; and this comes about out of regard for the glorious sign that makes all the demons tremble — the Holy Cross. For its three lines, which cut one another squarely, drawn from one angle to the other diametrically, find no place in it where they might duly be drawn as the disposition of the said octahedron requires. But the dodecahedron, being among the others endowed with singular prerogative, has forbidden or denied lodging to none, being the receptacle of all. And for this also the ancient Plato, together with the other reasons adduced, attributed it to the Universe.

Above, as has been seen, most excellent Duke, we have demonstrated each of the said five regular bodies to be inscribable in the proposed sphere, and circumscribable by it. It remains now fittingly to demonstrate 51v how the said sphere may also be inscribed in each of them. Which we shall here set forth with evident clearness: that, vice versa, the sphere can be inscribed in each of them. And it will appear thus. From the centre of the sphere which circumscribes each of these bodies, to each and every base of each of them, let the perpendiculars issue, or be drawn; these will of necessity fall within the centres of the circles which exactly circumscribe the said bases; and since all the circles which exactly surround the said bases are equal, these perpendiculars will be equal. Whence, if according to the quantity of one of them we describe a circle about the centre of the sphere that circumscribes them, and turn its semicircle round until it returns to the place whence it began to move, then — since it must pass through all the extremities of all the perpendiculars — we shall establish, by the corollary of the 15th of the 3rd, that the sphere described by the motion of this semicircle touches, or exactly meets, all the bases of the assigned body at the meeting-points of the 52r perpendiculars; for the sphere can no more touch of the bases of the body than the semicircle touched as it moved. Whence it is manifest that we have inscribed the sphere in the assigned body, just as was proposed to do.

IIIIIIIIIIVVIThe plane tetrahedron, solid or hollow, is formed of six equal lines, which contain twelve surface angles and four solid; and they make among themselves four triangular bases, equilateral and equiangular. — Of the cut, or truncated. — The cut tetrahedron, or let us say truncated, solid plane or hollow, is contained by eighteen lines, which cause thirty-six surface angles and twelve solid. And eight bases surround it, of which four are hexagonal, of six equal sides, and the other four are triangular, likewise equilateral and also equiangular. But of the said eighteen lines, twelve are common to the triangular bases and to the hexagonal; which nonetheless all belong properly to those hexagons, because of 52v necessity those four hexagons, joined together by certain of their sides, cause those four triangles — as experience, in its own material form, renders clear to our eye. And it is born of the preceding body, its sides being uniformly cut at the third part. — [Of the elevated solid.] — The elevated tetrahedron, or let us say pointed, solid or hollow, has likewise eighteen lines, of which six are common. And it has thirty-six surface angles and eight solid, of which four are the cones of the surface pyramids, and four are common to the five pyramids — that is, to the interior one, which the eye cannot see but only the intellect apprehends, and to the other four exterior. Of which five pyramids the said body is composed, when they are among themselves equilateral, triangular and equiangular, as its own material form demonstrates to us. And the surfaces that clothe it — which are not properly to be called bases — are in all twelve in number, all triangular. And of this body the elevated-truncated cannot in any way be assigned, for the defect of the hexagons, which make no solid angles.

Plate 2 · Tav. I Tetracedron planus solidus">

Plate 3 · Tav. II Tetracedron planus vacuus">

Plate 4 · Tav. III Tetracedron abscisus solidus">

Plate 5 · Tav. IIII Tetracedron abscisus vacuus">

Plate 6 · Tav. V Tetracedron elevatus solidus">

Plate 7 · Tav. VI Tetracedron elevatus vacuus">

VIIVIIIVIIIIXIXIIXIIIXIIIIXVXVIThe hexahedron, or let us say cube, plane solid or hollow, has twelve lines — or sides, or ribs — and twenty-four surface angles and eight solid, and six bases or surfaces which contain it, all square, equilateral and also equiangular: like in form to the diabolical instrument otherwise called the die, or taxillus. — Of the cut, or truncated. — The cut hexahedron, or truncated plane, likewise solid or hollow, has twenty-four lines, which about it cause forty-eight surface angles, of which twenty-four are right and the others acute. And it has twelve solid angles, and is contained by fourteen surfaces or bases, namely six square and eight triangular. And all the said lines are common to the squares and to the trigons, because those six squares, joined together angularly, of necessity cause the eight triangles — just as the hexagons did in the truncated tetrahedron. And it is born of the cube uniformly cut at the half of each of its sides, as its own material form demonstrates to the eye. — Of the elevated. — For the constitution of the elevated hexahedron, solid or hollow, there concur of necessity 53v thirty-six lines, which, applied among themselves, cause seventy-two surface angles and six solid pyramidal ones, each contained by four surface angles. And it is clothed with twenty-four triangular surfaces, which properly are not to be called bases. And of those lines, twelve are common to all the surface triangles that contain and surround it. And the said body is composed of six exterior quadrilateral lateral pyramids, which all present themselves to the eye according to the situation of the body; and also of the interior cube on which the said pyramids rest, which the intellect alone imagines, for it is wholly hidden from the eye by the superposition of the said pyramids upon it. And of that cube the six square surfaces are the bases of the said six pyramids, which are all of the same height, and are hidden from the eye, and secretly surround the said cube. — [Of the truncated elevated solid.] — The truncated-elevated hexahedron, solid or hollow, has seventy-two lines, or sides, or ribs. These make one hundred forty-four surface angles, and of solid ones they make fourteen, all pyramidal: of which six are of quadrangular lateral pyramids 54r and eight of trilateral pyramids. And of the said lines, twenty-four are common to the trigonal and tetragonal pyramids. And it has forty-eight faces or surfaces surrounding it, all triangular; and this body so made is composed of the cut hexahedron, solid, within — perceptible to the intellect alone — and of fourteen pyramids, as has been said. And thrown upon a level space, it always comes to rest upon three pyramidal cones, or points, as its form demonstrates.

Plate 8 · Tav. VII Exacedron planus solidus">

Plate 9 · Tav. VIII Exacedron planus vacuus">

Plate 10 · Tav. VIIII Exacedron abscisus solidus">

Plate 11 · Tav. X Exacedron abscisus vacuus">

Plate 12 · Tav. XI Exacedron elevatus solidus">

Plate 13 · Tav. XII Exacedron elevatus vacuus">

Plate 14 · Tav. XIII Exacedron abscisus elevatus solidus">

Plate 15 · Tav. XIIII Exacedron abscisus elevatus vacuus">

XVIIXVIIIXVIIIIXXXXIXXIIXXIIIXXIIIIThe plane octahedron, solid or hollow, receives in itself twelve lines and twenty-four surface angles, and of solid ones it has six; and it is contained by eight triangular bases, equilateral and equally equiangular, as it presents itself to us in its own material form. — [Of the cut plane solid.] — The truncated, or cut, octahedron, plane solid or hollow, has thirty-six lines, which make seventy-two surface angles — forty-eight belonging to the hexagons and twenty-four to the squares. And it contains twenty-four solid angles, and has fourteen bases, of which eight are hexagonal, of six sides, and six are tetragonal, 54v that is square. But of the said lines, twenty-four are common — to the squares and to the hexagons. And those squares are formed by the hexagons when all eight, uniform, touch one another; of all which the eye, in its material form, makes the truth clearly known to the intellect. And of this one also it is not possible to form its elevated so that it present itself uniform, through the defect, likewise, of the hexagons, which, as was said of the truncated tetrahedron, cannot cause a solid angle. And it is formed from the preceding, uniformly cut at the third part of each of its sides. — [Of the elevated, solid or hollow.] — The elevated octahedron, solid or hollow, has thirty-six lines of equal length, and has seventy-two surface angles and eight solid pyramidal ones. And it is contained by twenty-four surfaces, [all] trigonal, equilateral and equiangular, which exactly surround it. But of those lines, twelve are common to all the triangles of the pyramids. And this body is composed of eight triangular lateral pyramids, equilateral and equiangular, of the same height, which all appear without; and also of the interior octahedron, perceptible by imagination alone to the 55r intellect; and the bases of that octahedron are the bases of the said eight pyramids, as its material form makes manifest to us.

Plate 16 · Tav. XV Octocedron planus solidus">

Plate 17 · Tav. XVI Octocedron planus vacuus">

Plate 18 · Tav. XVII Octocedron abscisus solidus">

Plate 19 · Tav. XVIII Octocedron abscisus vacuus">

Plate 20 · Tav. XVIIII Octocedron elevatus solidus">

Plate 21 · Tav. XX Octocedron elevatus vacuus">

XXVXXVIXXVIIXXVIIIXXIXThe plane icosahedron, solid or hollow, contains thirty lines or sides, all equal among themselves; and these cause in it sixty surface angles and twelve solid; and they also form in it twenty bases, all triangular, equilateral and equiangular. And each of the said solid angles is made, or contained, by five surface angles of the said triangular bases, as its material figure likewise demonstrates. — [Of the truncated plane solid.] — The truncated icosahedron, plane or solid, has ninety sides or lines, and it has one hundred eighty surface angles, of which one hundred twenty belong to the triangles concurring in its composition, and sixty to the pentagons which likewise enter into it, all of which are equilateral. And these lines form about the said body thirty-two bases, of which twenty are hexagonal, of six equal sides, and twelve are pentagonal, of five equal sides. 55v And each kind, in its degree, is equilateral and also equiangular among themselves — that is, all the hexagons have equal angles among themselves, and so the pentagons have equal angles among themselves; but the sides, of pentagons and hexagons alike, are all equal among themselves. Only in their angles do the pentagons and the hexagons differ. And this body so made is born of the preceding regular one, when each of its sides is uniformly cut at its third part. And from such cuts are caused twenty hexagons and twelve pentagons, as has been said, and [30] corporeal or solid angles. But of the said lines, sixty are common to the hexagons and pentagons, because from the twenty hexagons uniformly joined together, of necessity the twelve pentagons are caused. And of this one, again, the elevated cannot be given, for the defect of the said hexagon, as we said above of the truncated tetrahedron and the truncated octahedron. — Of the solid elevated. — The elevated icosahedron, solid or hollow, has in itself ninety lines, and has one hundred eighty surface angles and twenty solid pyramidal ones; and it has sixty bases or surfaces surrounding it, all 56r triangular, equilateral and also equiangular. But of the ninety lines, thirty are common to each of the surfaces of its twenty pyramids. And the said body is composed of twenty triangular lateral pyramids, equilateral and equiangular, of equal height, and of the entire interior icosahedron, perceptible by imagination alone to the intellect; and its bases are likewise the bases of the said twenty pyramids — all of which, again, its own material form makes plain.

Plate 22 · Tav. XXI Ycocedron planus solidus">

Plate 23 · Tav. XXII Ycocedron planus vacuus">

Plate 24 · Tav. XXIII Ycocedron abscisus solidus">

Plate 25 · Tav. XXIIII Ycocedron abscisus vacuus">

Plate 26 · Tav. XXV Ycocedron elevatus solidus">

Plate 27 · Tav. XXVI Ycocedron elevatus vacuus">

The plane dodecahedron, solid or hollow, has thirty equal lines or sides, which cause in it sixty surface angles. And it has twenty solid angles, and has twelve bases or surfaces which contain it; and these are all pentagonal, with sides and angles all equal among themselves, as appears in its form. — Of the truncated, or cut. — The cut, or truncated, dodecahedron, plane, solid or hollow, has sixty lines 56v all of equal length, and has one hundred twenty surface angles, and of solid ones thirty. But of the hundred twenty surface angles, sixty belong to the triangles and sixty to the pentagons; and those triangles are of necessity caused by the said pentagons when these are joined angularly among themselves — as was said in the causation of those of the truncated tetrahedron and octahedron, which were formed from hexagons and quadrangles and triangles; and so in those of the truncated icosahedron, from hexagons and pentagons, as the material figure demonstrates.

XXXIXXXIIXXXIIIXXXIIIIAnd each of the said solid angles is made and contained by four surface angles, of which two are of triangles and two of pentagons, concurring at one same point. And all its lines or sides are common to the triangles and to the pentagons; for the ones and the others, duly applied together, each is cause of the other — the triangles of the pentagons and the pentagons of the triangles. And just as the twelve equilateral pentagons, angularly joined, form in the said body twenty triangles, so too we may say that twenty equilateral triangles, angularly joined among themselves, 57r cause twelve pentagons likewise equilateral. And hereby it appears that all the said lines are common among them, as has been said. And the surfaces that surround this body are thirty-two, of which twelve are pentagonal, equilateral and equiangular, and twenty are triangular, likewise equilateral and equiangular — all, as we have said, reciprocally caused. So it appears in its material form; and this body derives from the preceding, uniformly cut at the half of each of its sides. — [Of the elevated solid.] — The elevated dodecahedron, solid or hollow, has ninety lines and one hundred eighty surface angles, and of solid ones, twelve elevated pyramidal pentagonal; and it has besides twenty corporeal hexagonal [angles]. And it has sixty surfaces, all triangular, equilateral and equiangular. But of the said ninety lines, thirty are common to the twelve bases of the pentagonal pyramids, whose bases likewise must be pentagonal; and these are the bases of the intrinsic regular dodecahedron which concurs in its composition, which the intellect comprehends by imagination alone. And these thirty common lines concur only in the causation of 57v the twenty depressed solid angles, which, as has been said, are hexagonal — six lines, that is, concurring in their formation. And the said body is formed of the aforesaid intrinsic regular dodecahedron and of twelve pentagonal lateral pyramids, equilateral, equiangular and of equal height; and their bases are the same bases of the intrinsic one, ut supra. — Of the cut elevated. — The truncated-elevated dodecahedron, solid or hollow, has sides or lines to the number of one hundred eighty, of which sixty are raised for the causation of the pentagonal pyramids, and sixty raised for the constitution of the triangular pyramids; the other sixty are the base-sides of each of the said pyramids, of the pentagonal and of the triangular alike. And this body so made is composed of the intrinsic cut plane dodecahedron, offered to the intellect by imagination alone, and of thirty-two pyramids, of which twelve are pentagonal, of heights equal among themselves, and the other twenty triangular, likewise of equal height. And the bases of these pyramids are the surfaces of the said truncated dodecahedron, referring 58r each to its own — the trigonal to the triangular pyramids and the pentagonal to the pentagonal. And falling on a plane, this body always comes to rest on six points, or pyramidal cones; of which cones one is of a pentagonal pyramid and the other five of triangular pyramids.

Which thing — that such points should stand on a level — seems absurd to the eye when the body hangs in air; and this, most excellent Duke, is of the greatest abstraction and profound science: whoever understands it will not, I know, let me lie. And its dimension is reached by the subtlest practice, above all of algebra et almucabala, known to few, and well demonstrated by us in our work, with ways of grasping it easily. And likewise that of the cut icosahedron, in which hexagons and pentagons interpose themselves, which make all the measures harsh.

Plate 28 · Tav. XXVII Duodecedron planus solidus">

Plate 29 · Tav. XXVIII Duodecedron planus vacuus">

Plate 30 · Tav. XXVIIII Duodecedron abscisus solidus">

Plate 31 · Tav. XXX Duodecedron abscisus vacuus">

Plate 32 · Tav. XXXI Duodecedron elevatus solidus">

Plate 33 · Tav. XXXII Duodecedron elevatus vacuus">

Plate 34 · Tav. XXXIII Duodecedron abscisus elevatus solidus">

Plate 35 · Tav. XXXIIII Duodecedron abscisus elevatus vacuus">

XXXVXXXVIXXXVIIXXXVIIIAnother body, most excellent Duke, much unlike those already told, is found, called of twenty-six bases, deriving from a most graceful beginning and origin. 58v Of its bases, eighteen are square, equilateral and rectangular, and eight are triangular, likewise equilateral and equiangular. And this body has forty-eight sides or lines, and has ninety-six surface angles, of which seventy-two are all right — those of its eighteen square bases — and twenty-four are acute — those of its eight equilateral triangles. And these ninety-six concur among themselves in the composition, in it, of twenty-four solid angles, each of which consists of one surface angle of a triangle and three right angles of three squares.

And of its forty-eight lines, twenty-four are common to the trigons and to the squares; for of those eighteen squares, joined together according to due opportunity, of necessity result those eight formed triangles, just as was said above of the other truncated bodies. And the origin of this body is from the hexahedron cut uniform in all its parts, as its material form likewise demonstrates to the eye. And its science is most useful, in many considerations, to whoever knows how to apply it well — above all in architecture. And this for the knowledge of its solid, plane and hollow. 59r — [Of the solid or hollow elevated.] — The twenty-six bases, solid or hollow, elevated, receives for its formation one hundred forty-four lines, which, applied among themselves according to fitting exigency, cause in it two hundred eighty-eight surface angles and twenty-six elevated pyramidal solid ones, of which eighteen are contained by four acute surface angles each, and eight by three acute. And the said body is composed of twenty-six lateral pyramids, of which eighteen are quadrangular and eight triangular — all of which can be discerned by the eye round about the outside — and of the preceding twenty-six bases, plane solid, within, comprehended by imagination only. And its twenty-six bases are equally the bases of the aforesaid twenty-six pyramids: the eighteen quadrangular [of the eighteen quadrangular lateral pyramids], and the eight triangular of the eight triangular pyramids. And in whatever way this body be thrown on a level space, it always comes to rest upon three points or pyramidal cones — of which the experience of its material model will further satisfy the eye.

Plate 36 · Tav. XXXV Vigintisex basium planus solidus">

Plate 37 · Tav. XXXVI Vigintisex basium planus vacuus">

Plate 38 · Tav. XXXVII Vigintisex basium elevatus solidus">

Plate 39 · Tav. XXXVIII Vigintisex basium elevatus vacuus">

Among these, fittingly, most excellent Duke, 59v is to be placed the body called of seventy-two bases, which our Megarian philosopher fully describes in the 14th of his 12th. This body, though it has its bases plane, sided, angular and unlike one another, is not to be said to have dependence or derivation from any of the regulars: it is formed and created only, as our philosopher demonstrates in the said place, by means of the duodecagonal figure, that of twelve equal sides. And of its aforesaid bases, forty-eight are quadrangular, inequilateral and inequiangular, having only their two opposite sides — those drawn toward the one and the other pole, or say cone — equal to each other; and its other twenty-four bases are triangular, likewise inequilateral. And of these, twelve stand about the one cone and twelve about the other, and each of them has two equal sides, namely those that tend to the point of the lower and of the upper pole.

Of this one too its elevated can always be formed, as has been done with the others; but through the unlikeness of its bases its science will be difficult, however much it might render no small loveliness to the eye. And there would be caused in it seventy-two pyramids 60r according to the number of its seventy-two bases, and the bases of those pyramids would be the same as its own, with the body itself imagined within. The form of this elevated I did not care to produce materially among these, so as to leave its part also to the reader, of whose wit I do not despair. And this seventy-two bases is much frequented by the architects in their dispositions of buildings, being a form well accommodated above all where tribunes or other vaults — or say heavens — are to be made. And though in such buildings just so many faces are not always taken exactly, still they govern themselves by its similitude, quartering and turning it in every way according to the place and site where they intend to set the building. To whose pattern very many are found disposed and fabricated in divers parts, as is manifest in the inestimable ancient temple of the Pantheon, today called by the Christians, in the capital of the world, the Rotonda — which was disposed with such diligent industry and observance of proportions that the light of a single little eye, left open in its summit, renders the whole of it splendid and luminous.

I pass over 60v many other famous and illustrious cities — Florence, Venice, Padua, Naples and Bologna — in which many buildings, sacred as well as profane, small or great, are made in the mirror of this body. Here too in your own Milan, in the worthy shrine of San Satiro, the ornate chapel is a part of this body split open and applied to the wall with a certain reservation of convexity, a rosette joined in each of its bases, which renders it adorned. And in your devout and most sacred temple of the Grazie, its tribune at the first altar, with the lateral ones, is likewise nothing but a part in the likeness of this, with those ornaments joined in its bases for greater loveliness.

And though many build and draw their forms at their own pleasure, having no more knowledge of Vitruvius than of any other architect, they use the art nonetheless without knowing it: as Aristotle says of the rough rustics that solegizant et nesciunt se solegizare — they syllogize and know not that they syllogize — so these utuntur arte et nesciunt se uti, use the art and know not that they use it. So too the tailor and the shoemaker use geometry and know not what it is; masons, carpenters, smiths and every artificer use measure and proportion and know it not; 61r for, as has been said at other times, everything consists in number, weight and measure.

But what shall we say of the modern buildings, ordered and disposed after their kind with various and divers models, which seem to the eye to render some prettiness while they are small, and then, built full-scale, do not bear the weight — and far from reaching a thousand years, fall to ruin before the thirtieth? And through their ill condition, busied more in remaking than in making, they cause great expense; calling themselves architects, though they never so much as saw the covers of the most excellent volume of our most worthy architect and great mathematician Vitruvius, who composed De Architectura, with supreme instruction for every structure. Whoever strays from it hoes in water and founds on sand, and quickly spoils the art; for, though named architects, they know not the difference between the point and the line — how then shall they know that of the angles, without which it is not possible to build well? This is made plain, as the aforesaid Vitruvius says, by the great jubilation and utmost gladness that Pythagoras had when, with certain science, he had found the true proportion of the two straight lines that contain 61v the right angle of the square: for which, making great sacrifice and feast to the gods, he immolated a hundred oxen. And this angle is of such excellence that it can never vary; and by another name the perfect geometers call it angulum iustitiae, the angle of justice, for without knowledge of it, it is not possible to tell good from ill in any operation of ours, nor without it can any certain measure ever be given in any way. Whence the modern cobblers of building, in their edifices, think they have done nothing unless, departing from the straight and due ancient norm, they interpose some incongruity of their follies — blaming those (for some such are yet found) who go reducing the art to the true and ancient manner. These are they who delight in our mathematical disciplines, imitating the true guide of all building in the works of the aforesaid Vitruvius. Departing from whom, one sees how our buildings stand, divine and profane alike: this one crooked, and that one twisted. And therefore most fitting is the device of Your Highness, and its effect — the hatchet that lops away all that is crooked.

And continuing what is already begun, Your Highness will shortly bring your own 62r Milan to a loveliness not less than Florence's, removing from it its abominable and inept stamp together with its authors. For in truth she understands these things better sleeping than they do watching with a thousand eyes — as the like was shown by your close kinsman, the most illustrious Duke of Urbino, in the admirable fabric of his worthy fore-cited palace. And this with the sufferance of those who might take ill what has been said thus far for their instruction; and for the said body let this suffice to the purpose.

Plate 40 · Tav. XXXVIIII Septuaginta duarum basium solidum">

Plate 41 · Tav. XL Septuaginta duarum basium vacuum">

It does not seem good to me, most excellent Duke, to extend myself further in the said bodies, seeing that their process tends to infinity through the continual and successive cutting away, hand by hand, of their solid angles, according to which their various forms come to multiply. And this each may pursue for himself, the way being opened by those already given; for it is ever said, Quod facile est inventis addere — it is not hard to add to things found: and so, taking away and adding more and less to the aforesaid, 62v it will be easy for every purpose. And we have followed this thus far only to show how from those five regulars the virtue always distils itself into the other dependents, in the similitude of the five simples which concur in the formation of every created compound. For which cause — as was hinted above — Plato was constrained to attribute the foretasted five regular forms to the five simple bodies, that is, to earth, air, water, fire and heaven, as appears diffusely in his Timaeus, where he treated of the nature of the universe.

And to the element of earth he attributed the cubic form, that of the hexahedron, since no figure has need of greater violence to be moved; and among all the elements, what is found more fixed, constant and firm than the earth? And that of the tetrahedron he gave to the element of fire, because, flying upward, it takes the pyramidal form — as our own fire makes plain to the eye, for we see it broad and uniform at the plane and below, and always narrowing as it rises, so that its flame ends its summit in a point, 63r just as does the cone of every pyramid. The form of the octahedron he attributed to air: for as air follows fire at a small movement, so this form, by its aptness for motion, follows the form of the pyramid. And the figure of twenty bases, the icosahedron, he deputed to water; for since it is surrounded by more bases than any of the others, it seemed to him to suit, within the sphere, rather the motion of the thing that descends pouring itself abroad than of that which ascends. And the form of the twelve pentagonal bases he attributed to heaven, as to that which is the receptacle of all things; and this dodecahedron is likewise the receptacle and lodging of all the other four regular bodies, as appears in their inscriptions one within another. And again, as Alcinous says upon the Timaeus of Plato: just as in heaven there are twelve signs in its zodiac, and each of them is divided into thirty equal parts, so that its whole annual revolution is 360, so this dodecahedron has in itself twelve pentagonal bases, each of which, resolved into five triangles by fixing the point in the middle, 63v and each of the said triangles into six scalenes, gives thirty triangles to each base — which among all the bases are 360, like the said zodiac.

And these forms are much commended by Chalcidius, that most celebrated philosopher, in expounding the said Timaeus; and so by Macrobius, Apuleius and very many others; for in truth they are worthy of every commendation, for the reasons adduced in their constructions, showing the sufficiency of the said five forms, like that of the five simple bodies, to admit in no way of more: and just as the number of the said simples cannot be increased in nature, so of these five regulars it is not possible to assign more that are equal in bases, sides and angles and that, placed in a sphere, one angle touching, all touch. For if in nature a sixth simple body could be assigned, the Supreme Craftsman would have proved deficient in His works, and would have to be judged wanting in prudence, as not having known from the beginning all that nature required. And moved by this certainly, and by nothing else, 64r I understand that Plato attributed these forms, as has been said, to each of the said simples — arguing as a most excellent geometer and most profound mathematician: seeing that beyond the five varied forms of these no other tending to the spherical, equal as said in sides, bases and angles, can be imagined or formed — as is shown in the penultimate of the 13th, and adduced by us in due place — he concluded, not undeservedly, that the said forms befall the five simples, and that from them every other form depends. And though these five alone are called regular, the sphere is not thereby excluded from being most regular above them all, and every other body from being derived from her, as from the most sublime cause of causes. And in her there is no variety, but uniformity throughout; and in every place she has her beginning and her end, her right and her left. Whence her form is caused we shall say next, here putting an end to the said dependents; and successively of all the other oblong bodies, those that are longer than they are broad.

XLIThe sphere has been defined by many as to what it is, above all by Dionysius, a worthy mathematician. Yet our author describes it with utmost brevity in his 11th, and that description is adduced by all who came after, where he says thus: the trace of the half circle makes the sphere. The SPHERE is that which contains the trace of the arc of the circumference of the half circle when, in whatever manner the semicircle be taken, the line of the diameter being held fast, the said arc is turned about until it returns to the place whence it began to move: that is, the semicircle being made upon whatever line you will, and that line held fast, the said semicircle is carried round through its whole revolution. The body so described is called a sphere, whose centre is the centre of the said semicircle so carried round. — Demonstration of the said definition. — Let the semicircle .c. be made upon the line .ab., with the point .e. made centre, and let its whole arc be the part of the circumference .adb. I say that, holding fast the said line .ab., diameter of the said semicircle, and carrying it round upon her, 65r beginning from the point .d., going toward the lower part and returning with its arc toward the upper to the said point .d. whence it first moved — or contrariwise, going toward the upper and returning toward the lower, still with the arc to the said point .d. — that round thing made by the said semicircle in its revolution is the said spherical body, the sphere: imagining, as one must, that the said semicircle, for example's sake, is half a material trencher; for otherwise it would form no body, since the arc alone, carried round, leaves no trace, being a line without breadth or depth. And let this be said for its knowledge and causation.

Plate 1 · Tav. XLI Sphera solida">

In this sphere, most excellent Duke, all five regular bodies are imagined in this way. First the tetrahedron: if upon its surface — its shell or vesture — four points be marked, or imagined, equidistant in every direction one from another, and these be joined by six straight lines, which of necessity will pass within the sphere, the aforesaid body will be exactly formed 65v in it. And whoever should draw the cut, in imagination, with a plane surface in every direction along the said straight lines, the said tetrahedron would remain exactly bare. As — so that by this the others may be better grasped — if the said sphere were a bombard-stone, and upon it the said four points were marked with equal distances: if a stonecutter, or carver, with his irons should chip and face it away, sparing the said four points, he would have made of that whole stone exactly the tetrahedron.

Likewise, if on the said spherical surface four [read: eight] points be marked equidistant among themselves, one from another and another from one, and these be joined with twelve straight lines, there will be placed in the said sphere, by imagination, the second regular body called hexahedron or cube — the figure of the diabolical instrument called the die. And these points likewise marked on a bombard-stone in the manner said, and joined by a stonecutter as above, he will have reduced the said ball to cubic form. And if on the said surface six points be noted, each at its 66r due distance as has been said, whoever joins — or say connects — them with twelve straight lines will have made exactly in the said sphere the third regular body, called octahedron; and the like done upon such a stone, the stonecutter will have made of a ball the body of eight triangular bases. And so, if twelve points be marked, these joined by thirty straight lines, he will likewise have placed in the said sphere the fourth body, called icosahedron; and in the same way the stonecutter will have reduced the stone to the body of twenty triangular bases. And if twenty points be noted in the manner said, joining them again with thirty straight lines, there will be formed in the said sphere the fifth and most noble regular body called dodecahedron, the body of twelve pentagonal bases; and so the stonecutter would have made the same form of the said ball. Thus with like imaginations all will be placed in the sphere in such a way that their angular points are situated on the spherical surface; and one of their angles touching the sphere, at once all touch; nor is it possible in any way that one touch without the others, when the said body is placed in a sphere.

And by this 66v infallible science Your Highness will at times be able, as we have been wont, to take sport with the said stonecutters, arguing their ignorance in this manner: ordering them to make of such stones some form with equal sides, faces and angles, which yet shall not be like any of the five regulars; for example, binding them to make a capital, or base, or cyma for some column, that shall be of four or of six equal faces in the manner said — and that of the four, they be not triangular, or of the six, not square; and likewise of eight or twenty faces and none triangular, or of twelve and none pentagonal: all which things are impossible. But they, like rash braggarts, will promise to move heaven and earth, maria et montes; for many are found who neither know, nor care to learn, against the moral precept that says: Ne pudeat quae nescieris te velle doceri — be not ashamed to wish to be taught what you know not. Such was the carpenter who, asked what he would do if no plane were to be found, replied he would make one with another; and the other joiner who said his square was too big for truing a small one — presupposing that right angles vary among themselves; and the one who, two equal little rods being set 67r before his eyes in the form of a tau, thus T, judged now the one, now the other the longer. And plenty of other such blockheads.

With one of these, at the time of the building of the palace of Count Girolamo of good memory in Rome, I was conversing in his presence; and as it happened, discoursing of the building — there being many worthy men of divers faculties in his company, among others the then-renowned painter Melozzo da Forlì — to give pleasure to speculation, Melozzo and I urged the Count to have a certain capital made in one of these forms, we not making the difficulty clear to the Count, but only that it would be a worthy thing. Assenting to this, the Count called the master to him and asked whether he knew how to make it. He answered that this was a small business and that he had made such things many times; whereat the Count doubted it were as worthy a thing as we commended it. We still affirming the same, and adding openly that he would not do it, for the impossibility adduced above, the Count called the said stonecutter back to him and asked him again 67v whether he would make it. Then, half scoffing, he smiled — in brief, for the yes and for the no he was ever ready to wager. The Count said to him: "If you do not do it, what will you lose?" And he, shrewdly, answered not badly: "My lord, as much more than I stand to gain as seems fit to Your Most Illustrious Lordship." And so they were agreed. He was assigned a term of twenty days — he asking but four — and it fell out that he spoiled many marbles and produced a zero on the abacus. In the end the Count held him bound only for the damage to the stones. He was left shamefaced; but he never rested until he would know the origin of the proposal, and learned it was the friar — so that he bore me thereafter no small rancour; and finding me he said: "Master, master, I forgive you not the injury done me, unless you teach me the way to make it." And I offered myself to him for what I was worth; and staying on in Rome for several days I was no churl to him, but opened to him these and other things pertinent to his art; and he, courteous, insisted that I carry away a worthy cape in his name. So I say that like occasions will at times serve Your Highness to make others aware of their errors, that they come not before you with so many braggings, as though despising every other man.

So 68r once did Hiero with the poet Simonides, as Cicero recounts in his De Natura Deorum. Simonides rashly bound himself, within the space of one day, to tell him exactly what God is, saying it was no such difficulty to know it as others claimed. When the said term was done, Hiero asked whether he had found it; he said not yet, and asked that he grant him a little more time. After which the like befell him again; and in brief, several terms interposed, he confessed he understood it less than before, and was left confounded with his rashness.

And this is as much as concerns their placing in the sphere.

It follows, most excellent Duke, for the full knowledge of this our treatise, that something be said, for their knowledge, of the oblong bodies — those that are longer, or taller, than they are broad, such as columns and their pyramids. Of both of these several sorts are found; and so we shall speak first of the columns and their origin, then of their pyramids.

XLIIColumns are of two makes, round and sided. 68v Just as among plane figures some are curvilinear — those contained by curved or bent lines — and others are called rectilinear — those contained by straight lines. The round column is a body contained between two equal circular bases, equidistant from each other. It is defined by our philosopher in the 11th thus: the round corporeal figure whose bases are two plane circles, equal at the extremities and in thickness — that is, height — is the trace of the rectangular parallelogram when, the side containing the right angle being held fast, the said surface is carried round until it returns to its place. And this figure is called the round column. Whence the round column and the sphere and the circle have one same centre.

For example, let the parallelogram be .abcd., that is, a quadrangular surface of equidistant sides [and right angles]. Hold fast the side .ab.; which so held, let the whole parallelogram be carried round until it returns to the place whence it began to move. The corporeal figure described by the motion of this parallelogram 69r is called the round column, whose bases are two circles: of the one the centre is the point .b., and the other is that which the line .da. makes in its motion or turning, and its centre is the point .a. And the axis of this column is the said line .ab., which stands still in the movement of the parallelogram. And if we imagine the parallelogram .abcd., when in its turning it arrives at the position .abef., as joined to the position whence it began to move, according to the continuation of the plane surface — so that the whole is one parallelogram .dcef. — and that we have drawn in it the diameter .de., that diameter .de. will also be the diameter of the column.

What is said — that the column and the sphere and the circle have one same centre — is to be understood when they have one same diameter. For example: we have said that .de. is the diameter of this column; therefore the sphere and the circle whose diameter is the line .de. must have one same centre with the centre of the proposed column. Let it be, then, that the line .de. divides the line .ab. at the 69v point .g.; and .g. will be the centre of the column, because it divides the axis of the column equally, and the diameter of the column equally too — which is proved by the 26th of the 1st, since the angles at .g. are equal by the 15th of the 1st, and the angles at .a. and at .b. are right by the hypothesis. And the line .ad. is also equal to the line .be.; whence .dg. is equal to .eg., and so .ag. equal to .gb. And since the angles .c. and .f. are right, if about the point .g., with the interval .dg., and upon the line .de., a circle be made, it will pass, by the converse of the first part of the 30th of the 3rd, through the points .c. and .f. Whence the point .g. is the centre of the circle whose diameter is the diameter of the column, and therefore of the sphere too; and hereby it is manifest that about every rectangular parallelogram the circle, and about every column the sphere, can be circumscribed. And so is clear what this theorem of our philosopher wished to set before us in the said definition of the round column — of which let this suffice. And 70r following on, we shall speak of the sided ones, as was promised.

Plate 58 · Tav. XLII Columna rotunda solida">

The other species or sort of columns are called sided. Of these the first is triangular: its bases, the upper and the lower, are two triangles equidistant from each other according to the height of the column, as in the one figured here, whose upper base is the triangle .abc. and the lower the triangle .def. And this figure, says our author, is called the corpo seratile [the "saw-horse" solid, a triangular prism], and is like the ridge of the roof of a house that has four faces or walls, whose roof sheds rain on two sides only, as the eye demonstrates. And the bases may be equilateral [or not equilateral]. And of such columns the three faces are always parallelograms, of four sides and rectangular; so that the said saw-horse body is contained by five surfaces, of which three are quadrangular and two triangular.

Plate 42 · Tav. XLIII Columna laterata triangula solida">

Plate 43 · Tav. XLIIII Columna laterata triangula vacua">

The second sort of the sided are the quadrilateral: those which have their two bases, in the manner said, 70v quadrangular; and four other surfaces surrounding them, likewise quadrilateral, equidistant from each other according to their opposition. And these likewise are at times equilateral, at times inequilateral, according to the disposition of their bases; for of plane rectilinear quadrilateral figures four sorts are assigned. The one called square, which has all its sides equal and its angles right, as here beside, the figure .A.; the other called tetragon longo, which has its opposite sides equal and its angles likewise right, but is longer than it is broad, as here beside, the figure .B. The third sort is called elmuaym, a figure equilateral but not rectangular, by other name called rhombus, as here the figure .C.; the fourth sort is called like-to-the-elmuaym, or by other name rhomboid, of which only the opposite sides are equal and equidistant from one another, and it has no right angles, as appears in the figure .D. All other four-sided figures besides these are called elmuariffe, that is, irregular, as are the figures marked .E. Now according to all these diversities of bases 71r the said quadrilateral columns may vary; but however that be, the equidistance between their bases must always be understood as their height. And these we may call regular after the likeness of their bases, and the others irregular, or elmuariffe.

Plate 44 · Tav. XLV Columna laterata quadrangula solida">

Plate 45 · Tav. XLVI Columna laterata quadrangula vacua">

In the third place are the pentagonal sided columns, those of five faces, as here the figure .AB., each face being tetragonal or quadrilateral. And the bases of such columns are always two pentagons, two rectilinear figures of five sides or angles — for in all rectilinear figures the number of the angles equals the number of the sides, and otherwise they cannot stand. And these too must be equilateral or inequilateral according as their bases permit, just as was said a little before of the quadrilateral sided ones. For some pentagons are equilateral and equiangular, and others inequilateral and consequently inequiangular; but every pentagon that has three angles equal among themselves, if it be equilateral, will of necessity be equiangular too, 71v as the 7th of the 13th demonstrates. This is said because a pentagon might have equal sides with two angles equal to each other, and yet not be wholly equiangular. And these two pentagons, upper and lower, are likewise to be understood in the said column with the equidistance of their height, whether the columns be equilateral or inequilateral, as you will.

For, most excellent Duke, the species of the sided columns can increase to infinity according to the varieties of the rectilinear figures of more and fewer sides; but for every sided column its two bases, the upper and the lower, must of necessity be two similar rectilinear figures — agreeing, that is, in the number of sides (not the one triangular and the other tetragonal), and equilateral and equiangular between themselves, for the uniformity of the columns — however much variety may otherwise be made in them, forming them now equilateral and now inequilateral. For which cause it does not seem good to me to extend myself further in them, but only to bring to memory that their denomination always derives from the bases: 72r that is, as the bases are, so are they named.

For example, if the bases are triangular, as above in the saw-horse body, they will be called triangular; if tetragonal or quadrilateral, they will be called quadrangular; if pentagonal, pentagonal; and if of six sides, they will be called hexagonal, et sic de singulis. But be the bases of whatever quality you will, the faces of each will always be tetragonal and rectangular. And of the one and the other, their material forms demonstrate to the eye what has been said, at the number set for them in their table; and also, below in this work, in plane figure in perspective, at the same number, as Your Highness will be able to see.

Plate 46 · Tav. XLVII Columna laterata pentagona solida">

Plate 47 · Tav. XLVIII Columna laterata pentagona vacua">

Plate 48 · Tav. XLVIIII Columna laterata exagona solida">

Plate 49 · Tav. L Columna laterata exagona vacua">

XLIIXLVIIIXLVIIIIFittingly now, it seems to me, the way to measure all sorts of columns should be set down. Though we have treated of this fully in our own work, yet I shall bring it in succinctly here, by way of a hint, for Your Highness; and first of all the round ones, for which let this be the general rule. First let one of its bases be measured, reducing it to a square according to 72v the approximate method found by the noble geometer Archimedes, set down in his volume under the rubric De quadratura circuli and adduced in our work with its demonstration — thus: find the diameter of the base and multiply it by itself, and of the product take the 11/14 — the eleven fourteenths; and these multiplied by the height of the column, this last product is the corporeal mass of the whole column.

For example, that it may be better grasped: let the round column be .abcd., whose height .ac. — or .bd. — is 10, and the diameters of its bases, the one .ab. and the other .cd., each 7. I say that, to square this and every like column, take one of the said diameters, whichever it be, .ab. or .cd. — it matters not, they being equal — namely 7; and this 7 must be multiplied by itself: it makes 49; and of this, I say, take the 11/14, which are 38½. And these, I say, be multiplied against the height, or length, of the whole column, that is against .bd. or .ac., which we set at 10: it makes 385; and so much shall we call the whole capacity, or corporeal content, of the whole said column. And this case means, most excellent Duke, 73r that if those numbers import braccia of whatever sort you will, there will be in it 385 little cubic blocks — like dice, one braccio in every direction: a braccio long, a braccio broad and a braccio high, as the figure here at the side demonstrates. And so, if the said numbers import feet, then as many feet as was said of the braccia; and if paces, paces, and palms, palms, et sic de singulis. And resolving the said column into cubes, one would make 385 of them; and let this suffice for the present intent. Nevertheless, for the squaring and dimension of the said circular bases many other ways are given, which all return to one, and which we have set out in order in our said work.

And the reason why one takes the said 11 of the 14 parts of the multiplication of the diameter by itself, in every circle, is because it has been found, with great approximation, by Archimedes, that the circle, in comparison with the square of its diameter, is as 11 to 14 — that is, if the square of the diameter were 14, the circle would be 11 — though not yet with precision by any sage; but it varies little, as here appears to the eye in the figure: 73v the circle is less than the said square by as much as the angles of the said square, which the circle loses of its space; and those angles are, of the whole square, the 3 of its 14 parts, and the 11 come to be comprised by the circular space — as appears in the square .abcd., whose sides equal the diameter of the circle, that is, the line .ef. which divides it through the middle, passing through the point .g. called the centre of the said circle, as at the beginning of his 1st our philosopher tells us. And this of the round ones.

The way being shown for the dimension of the round, there follows that of the sided. For which likewise let this be the general rule, and an exact one: always square one of its bases, whichever you will, and what that makes, multiply then by the height or length of the said column; and this last product is exactly its corporeal mass or capacity — be they of as many faces as you will, it never fails. As, for example, let the tetragonal sided column be .ab., 10 high, and its 74r bases each 6 in every direction. I say: first square one of the said bases — which, they being equilateral, is done by multiplying one of the sides by itself, 6 by 6 makes 36 — and this is exactly the space of the base. Now, I say, multiply this by the height or length of the whole said column, that is by 10: it makes 360. And so many braccia, or feet, exactly, will the said column square to, in the manner said above of the round. And so, if its bases were inequilateral or otherwise irregular, still let them always be squared according to the norms given by us in the said work, and the product multiplied by their height, and the thing sought will be had infallibly in every one. And for the dispatch of all the others this same rule is to be kept, be they trigonal or pentagonal or hexagonal or heptagonal, et sic de singulis: that is, according to the exigency of their bases, these must first be measured — if triangular, by the rule of triangles; if pentagonal, by the rules of pentagons; and if hexagonal, likewise. The rules for which forms and figures are assigned at length in our said work — to which, access being easy through its abundant printed multitude, by now spread through the world, I do not care to adduce them here again. And so we shall put an end to the said columns, and following on we shall speak of their pyramids. 74v

It follows in order, most excellent Duke, that we speak of the pyramids and their diversity; and first of those called round pyramids, and then successively of all the others. And for full knowledge we shall say, with our philosopher in his 11th, that the round pyramid is a solid figure, being the trace of a right-angled triangle when, one of the sides containing the right angle being held fast, it is carried round until it returns to the place whence it began to move. And if the fixed side be equal to the side carried round, the figure will be right-angled; if longer, acute-angled; if shorter, obtuse-angled. And the axis of the said figure is the fixed or held side, and its base will be a circle. And this is called the pyramid of the round column. For example, that this may be better grasped: let the triangle be .abc., whose angle .b. is right, and let the side that is held be .ab. Which held fast, let the said triangle be turned about 75r until it returns to the place whence it began to move. That corporeal figure, then, which is described or formed by the movement of this triangle, is called the round pyramid. Of which there are three differences or species: one is right-angled, another acute-angled, the third obtuse-angled.

And the first is formed when the side .ab. is equal to the side .bc.; and suppose the line .bc., when by the turning of the triangle it arrives at the position of the line .bd., so that the point .c. falls upon the point .d. and they become one same line — by which is understood that it then joins itself, in straightness, to the position from which it began to move; and this line will be as it were the line .bcd. And because, by the 32nd of the 1st and by the 5th of the same, the angle .cab. is half a right angle, the angle .cad. will be right; and therefore such a pyramid will be called a right-angled pyramid. But if the side .ab. be longer than the side .bc., it will be acute-angled: for then, by the 32nd of the 1st and by the 19th of the same, the angle .cab. will be less than half a right angle, and therefore the whole angle .cad. less than right, and acute; whence the said 75v pyramid is acute-angled. And if the side .ab. be less than the side .bc., the angle .cab. will be greater than half a right angle, by the 32nd of the 1st and the 19th of the same, and the whole .cad., which is double .cab., greater than right, and obtuse: then the pyramid is fittingly called obtuse-angled. And the axis of this pyramid is the line .ab.; and its base, the circle described by the line .bc. so carried round about the centre .b. And this pyramid is called the pyramid of the round column — of that column, namely, which the parallelogram arising from the two lines .ab. and .bc. would make, the side .ab. standing fixed, as was said above of the round column. And let this satisfy our purpose for the round pyramid and its differences; now let the others be spoken of.

Plate 59 Pyramis rotunda solida">

The sided pyramids, most excellent Duke, are of infinite sorts, even as the varieties of their columns whence they take their origin, as we shall presently conclude. But first let us set down the declaration of our philosopher, given in his 11th, where he says the sided pyramid is a corporeal figure 76r contained by surfaces which, all save one, are raised up to one opposite point. Whence it is to be noted that in every sided pyramid all the surfaces that surround it, except its base, rise to a point which is called the cone of the pyramid; and all these lateral surfaces are triangular, while most often their base is not triangular — as here in line drawing appears the triangular pyramid .A., whose cone is .B.; and the quadrilateral pyramid .D. with its cone .E.; and the pentagonal pyramid .F. and its cone .G. And so following, in all of them — and better in their own material form, at the numbers LI, LII, LIII, LIIII, LV, LVI, solid and hollow; and below in this work, in the plane by perspective, at the same numbers. And the derivation of these is from the sided columns of which we spoke above; and they are born in this way: fixing a point, actually or in imagination, in one of the bases of the sided column, and joining it by straight lines with each of the rectilinear angles of the other, opposite base of the said column. Then 76v the pyramid of the said column will be exactly formed, contained by as many triangular surfaces as there are lines, or sides, in the base of the said column. And the column and its pyramid will be denominated by the same numbers: if the sided column is trilateral or triangular, the pyramid too will be called trigonal or triangular; if the said column be quadrilateral, its pyramid will be called quadrilateral; if pentagonal, pentagonal, et sic de reliquis.

LIIIILVILVIIIWhence it is manifest that — as was said before of the said sided columns, that their species can multiply to infinity according to the diversity and variation of their rectilinear bases — the same, we say, must happen with their sided pyramids; since to every column, or cylinder, its pyramid answers, be it round or sided. And that point fixed in its base need not be situated exactly in the middle of the said base: so long as it does not leave the base, it matters not, for with the said lines drawn a pyramid is still caused; though the one whose lines are drawn exactly to the middle point is called an upright, level pyramid, and the others are 77r called leaning, or inclined. There are certain others called short, or truncated, pyramids: those that do not reach fully to the cone, but lack the summit, and are called cut, or docked. And of as many sorts are these as their entire ones, and so named, round or sided — as here in line drawing appear the truncated round .A., the short triangular .B., the cut quadrangular .C. And this seems to me sufficient for their knowledge. And following on, we shall next speak of their graceful measure.

Plate 50 · Tav. LI Pyramis laterata triangula solida">

Plate 51 · Tav. LII Pyramis laterata triangula vacua">

Plate 52 · Tav. LIII Pyramis laterata quadrangula solida">

Plate 53 · Tav. LIIII Pyramis laterata quadrangula vacua">

Plate 54 · Tav. LV Pyramis laterata pentagona solida">

Plate 55 · Tav. LVI Pyramis laterata pentagona vacua">

Plate 56 · Tav. LVII Pyramis laterata triangula inequilatera solida">

Plate 57 · Tav. LVIII Pyramis laterata triangula inequilatera vacua">

Plate 60 · Tav. LXI Pyramis laterata exagona vacua">

The just and precise quantity and measure, most excellent Duke, of every entire pyramid, round or sided, will be had from the quantity of their columns, in this way. First we shall find the area, or space, of the base of the pyramid we intend to measure, by way of the rules given above for finding the corporeal mass of all columns, round and sided; and having found it, we shall multiply it by the axis, that is, the height of the said pyramid: and what that makes will be the capacity of its whole column. And of this last multiplication we shall always 77v take the ⅓ — its third part; and just so much, exactly, is the corporeal quantity of the said pyramid; and it never fails.

For example, let the round pyramid be .abc., whose base is the circle .bc. — its diameter 7 — and its axis .ad., which is 10. I say: first square the base, as was done above with the round column — for, as has been said, the columns and the pyramids have the same bases and the same heights. We shall have for the surface of the base 38½; which, multiplied by the axis .ad., that is by 10, will make 385 for the capacity of its whole column. Now of this, I say, take the ⅓: there comes 128⅓, and this is the quantity of the said pyramid. Whence it is to be noted, as to the exactness adduced, that with the round ones the numbers must answer according to the proportion so far found between the diameter and the circumference, and, by the one said above, between 11 and 14 — which, as was said in that place, is not exact, but varies little, as found by Archimedes. Yet that takes nothing from what we have said: that the round pyramid in quantity is 78r exactly the ⅓ of its round column — though, again through the ignorance of the squaring of the circle, it cannot be exactly expressed by number. But its ⅓ it is; and the said column is its triple, three times its pyramid, as is proved by the 9th of the 12th. But all the other, sided ones can be exactly assigned by number, their bases being rectilinear. And just as was done with the round, the like is to be observed with all the sided ones; for of these it is proved, in the 8th of the 12th, that they are triple — three times — their pyramid. And let this be said for their sufficient dimension.

In the 6th of the 12th, most excellent Duke, our philosopher concludes that the saw-horse body — the first species of the sided columns, as was said above — is divisible into three equal pyramids, the base of each being triangular; and consequently the said body is triple each of them. And with this evidence every pyramid is shown to be 78v sub-triple to its cylinder, or column. And hence arises the rule given above, that of the quantity of the whole column one takes the third part; which in the rectilinear columns appears clearly, for all of them are resoluble into as many saw-horse bodies as their bases can be distinguished into triangles, and of so many they are always said to be composed, as is proved in the 8th of the 12th. Thus the quadrilateral column: its base, being quadrilateral, resolves into two triangles by drawing in it the diagonal line, from one opposite angle to the other; and upon these triangles are imagined — and actually made — two saw-horse bodies. And because each is triple its own pyramid, it follows that both together are triple both their pyramids. But the two saw-horse bodies are the whole quadrilateral column: therefore the two pyramids of the two saw-horse bodies are the third of the whole said column; and these two pyramids amount exactly to one total pyramid of the whole column, just as their two saw-horse bodies are the whole column, being its two equal and integral parts. So the rule given cannot 79r fail, for all the reasons adduced. And the same effect shows itself likewise in every other sided column — as also in their third species, called pentagonal, whose base is resoluble into three triangles; and by what has been said, the whole column into three saw-horse bodies, of which each is triple its pyramid, whereby all three are triple all their three pyramids; and these together amount to one pyramid of the whole column, just as their three saw-horse bodies remake the whole column. And so the same, running through all the others.

And the said resolution of bases into triangles is demonstrated in the 32nd of the 1st, where it is concluded that every polygonal figure — of more angles and sides — is always resoluble into as many triangles as it has angles, or sides, less two. For example, the quadrilateral has four angles and consequently four sides: it is resoluble into two triangles at the least — at its least resolution — as appears if a straight line be drawn in it from one of its opposite angles to the other, as here is seen in the figure of the tetragon .abcd., divided into the two triangles .abd. and .bcd. by 79v the line .bd., which in the art is called the diagonal line, and also diameter. And so the pentagonal resolves into three triangles at the least — by general rule, into two triangles fewer than its angles or sides. Which will appear if from any one of its angles two straight lines be drawn to the two opposite ones, as is done here in the pentagonal figure .abcde. described: the lines being drawn from its angle .a. to the two opposites .c. and .d., it is resolved into the three triangles .abc., .acd. and .ade. And each of the said lines, in the art, is called a chord of the pentagonal angle. And so the hexagonal resolve into four triangles, et sic in reliquis. So that we are much obliged, most excellent Duke, to the ancients, who with their vigils have enlightened our minds — above all to our Megarian Euclid, who gathered together in order from those who went before, and added of his own, in these most excellent disciplines and mathematical sciences, with so many diligent demonstrations: as appears through all his sublime volume, in which his genius shows itself not human but divine — above all in his 10th, 80r in which truly he exalted himself as high as is permitted to the human. Nor can I conceive how anything loftier could have been said of those most abstract, irrational lines, whose science is most profound above every other, in the judgment of those who know most of it. And of the entire pyramids, so far as concerns the purpose, here let there be an end.

For the short, or docked, pyramids, their measure is found by means of their entire ones, to which, as the imperfect to its perfect, they are reduced in this way. First we shall reduce the said short one to the entire, up to its cone, by the method given in our published work; and that entire one we shall measure by the ways aforesaid, and we shall have clear its whole capacity, which we shall keep. Then we shall take the measure of that little pyramid which was added to the docked one to make it entire — again by the ways given — and the quantity of this little pyramid we shall subtract from the quantity of the whole great one that we kept. The remainder of necessity comes to be, exactly, the quantity of the said 80v truncated pyramid. And of all ways this is the shortest and most secure; and be they round or sided, the same is observed.

It follows that we should speak of the dimension of the regular bodies and of their dependents. Of the said regulars I do not care to extend myself further here, having already composed a particular treatise on them, dedicated to the most illustrious kinsman of Your Ducal Highness, Guidobaldo Duke of Urbino, in our work dedicated to His Lordship; and recourse to it will be easy for the reader, it having come forth for the common profit, as was said before; and in this your illustrious city many copies of it are found. Their measure is by so much the more speculative as those bodies are more excellent and perfect than the others — matter certainly for the tragic buskin, and not for the fool. And in that place enough was said of it. But the way for the others that depend on them is like the one given for the short pyramids: one must reduce them to their perfect totals, and measure those with diligence by our rules given in the said place, and keep that quantity; and then measure separately, again by the rules of the pyramids, the supplement made to complete the whole; and what it makes, subtract from the quantity of its whole regular. The remainder will be exactly the quantity of the said dependent. 81r

When the said dependent is of the number of the truncated — as the truncated tetrahedron, which lacks the points with respect to its entire — these come to be all equal and uniform little pyramids; and so, one being measured, by it all the others are at once known, according to the number set upon their sides or bases or elsewhere, by which in practice one must always govern oneself; and these had, you will subtract them from its whole, as has been said. But if the said dependent be of the number of the elevated, then, to have its measure, there is added to its perfect the quantity of all its little pyramids, which of necessity come to be as many as the bases of its perfect. And so, briefly, more and less, in these one must guide oneself by the light of their perfects, adding to them and diminishing according to the occasions said. Whoever would govern himself otherwise would come into inextricable chaos. 81v And therefore let this be the fitting instruction concerning them — not that I distrust the rare wits and speculative intellects, ready for these and for every other faculty, which we have presupposed throughout our whole process: above all, supreme by excellence and antonomasia among all others, that of Your Ducal Highness — to whom in our discourse I do not mean to have spoken as to one ignorant, neither of these matters nor of any others in any way, seeing that Your Highness is endowed and adorned with every one of them alike. On which were I to enlarge, not paper only but life would not suffice. Sed quod patet expresse non est probare necesse — what is expressly plain needs no proof. When with your sole glance you heal and gladden every troubled sight, you are truly that sun which warms and lights the one pole and the other. And what more can be said of you today among mortals, save that you alone are the quiet and refreshment not of Italy only, but of all Christendom?

Splendid, ample, magnificent and magnanimous you show yourself to everyone; in you is mercy, in you is piety, 82r in you magnificence; in you is gathered whatever of goodness can be in a creature. Let Demosthenes yield, with Cicero and Quintilian, to your mouth — a fount that pours so broad a river of speech, nectar to the good and to the wicked a severe knife. Most observant of every religion, and of their temples not only restorer but assiduous builder; ever wholly given to the divine office by day and by night, with no less reverence than the professed themselves show with their most sacred prelates — as your most worthy devout chapel, deputed to the divine cult and adorned with most worthy singers, together with your other particular devotions, makes manifest. To every suppliant, above all the pious, you unbar your merciful ears without delay; and your benignity toward him who asks not only succours, but more often freely outruns the asking. For which things, not undeservedly, He to whom nothing is ever new has made you, singularly in our times, among all others in the whole universe, a partaker of His graces. Wherefore it is with no less fitness than Octavian, in his time, built in Rome the temple of Universal Peace, that Your Highness, in memory of so many graces, has constructed your most sacred temple of the Grazie in your illustrious city of Milan; and day by day you are never sated with adorning it in every way, and with succouring it in its every timely need. And this succinct discourse I pray the reader not to attribute to adulation, from which both by nature and by profession I am wholly alien; for did I otherwise, you, reader, would stand convicted of envy and spite toward His Highness no less than I of adulation, did you take no wonder at so many excellences and celestial gifts of his. 82v

Sed quod oculis vidimus testamur — we testify to what we have seen with our own eyes. And not I alone, but with my whole most sacred seraphic religion, with its chief and singular head and shepherd, our most reverend father Master Francesco Sansone da Brescia, its most worthy general, at our General Chapter of the present year, celebrated here in your illustrious city of Milan: at which was a very great number of most famous and celebrated men, doctors and bachelors in Sacred Theology and other sciences, from all the 83r world and of every nation quae sub celo est; in which, assiduously every day, cathedral and public disputations were held, always with the presence of the immense humanity of Your Ducal Highness and your devout condescension toward your servants, together with the most reverend Lordship of Monsignor your brother-in-law Ippolito, Cardinal of Este and most worthy Archbishop of Milan, and much other company of your most distinguished Magistracy. I pass over the plenty and affluent abundance in everything that flowed from the hands of Your Ducal Highness for the sustenance of so great a multitude, which sufficed not only for those then present, but for those who came after, for many months. For whose health and happy state the whole Minorite flock spreads its prayers with joined hands to the Most High — and particularly I, unworthy and wretched sinner, who continually commends himself devoutly to Your Ducal Highness.

Because where there is no order there is always confusion, therefore, for the fuller understanding of this our compendium — that all the proper figures set hereafter in perspective aspect may be found again, and the material ones too according to their public table — Your Highness will observe this method: namely, when you read above, in their chapters, of their creations and formations, you will look in that place, opposite, in the margin of the book, at the number marked in the ancient abacus figures — beginning from the first up to the 48th chapter, saying .I. .II. .III. .IIII. .V. and following to their end. And that very same number you will find below, where in this work the said bodies are all figured in order; and the same number will likewise be set in the margin in that place, .I. answering to .I., .II. to .II., .III. to .III., and so in all. And that figure will be of the said body made in the plane with all perfection of perspective, as our Leonardo da Vinci knows how. And these same numbers you will seek out also among the material forms of the said bodies, which hang with their names in 84r Greek and in Latin set on a label affixed above each, on its cord between two black ambers — each referring again, as said, to the number set in the margin where that body is treated. And so Your Highness will have their dispositions in the one way and the other; which deserved to be adorned not with base material — as poverty has forced upon me — but with precious metal and fine gems. But Your Highness will consider the affection and the spirit in your perpetual servant.

There are certain terms, most excellent Duke, introduced by the wise into the mathematical disciplines for the understanding of their parts, that in none of them there be equivocation; which to one not well versed in them would give trouble. They are often inserted above in this our compendium, as in reading you will have found; and so as not to stray from the ancients, we have observed them. 84v Of these it seems to me useful to give the reader notice here, succinctly. And first of the hypothesis.

[Of the hypothesis.] By the hypothesis is to be understood the presupposition, admitted and conceded between the parties — proposer and adversary — by means of which one intends to conclude; which being denied, no conclusion follows. And therefore it is not customary to admit it unless it be possible.

What the hypotenuse is in geometry. By the hypotenuse, in all rectilinear figures especially, is understood the line which lies opposite their greatest angle. But properly it has become customary to understand by it the side opposite the right angle in right-angled — or orthogonal — triangles, as they are called in the art; which triangles of necessity are always the half of the square figure, or of the long tetragon, that is, the rectangular figure of four sides longer than broad.

What the coraustus is among straight lines. By coraustus is understood a straight line which joins the extremities of two lines raised on high; and the corausti can be more or fewer, according to the number of the raised lines.

Of the cone, or pyramidal vertex. 85r The cone of the pyramid means the supreme point of the summit, where the lines that depart from its base concur.

Of the pentagonal chord. The pentagonal — or pentagonic — chord, or say the chord of the whole pentagonal angle, means a line drawn straight in the pentagonal figure, from any one of its angles to the one opposite it, as has been done many times.

The perpendicular. The perpendicular means a straight line raised, or set, upon another squarely — making, that is, one or more right angles about itself; and so also when it stands, in the manner said, set upon a plane surface. And it is commonly employed in triangles for their measure, as we said in its place in our said work.

The cathetus. Cathetus imports the same as the perpendicular; and by the vulgar, coarsely, in triangles it is commonly called the arrow of the triangle. It comes from the Greek word.

Of the diameter. Diameter, properly, is understood in the circle: a straight line 85v passing through its centre and touching the circumference with its extremities on either side, dividing the circle into two equal parts. But it is customary to speak of the diameter in squares as well; and therefore, to avoid equivocation, one says diameter of the circle and diameter of the square, to distinguish the one from the other.

Of the parallelogram. By parallelogram is understood a surface of equidistant sides. Properly such are the quadrilaterals — those four species which you had above in the 59th chapter, called square and long tetragon, rhombus and rhomboid, by other names elmuaym and like-the-elmuaym. And though every figure of an even number of sides has its opposite sides equidistant — the hexagon, octagon, decagon, duodecagon and the like — nevertheless those four are particularly to be understood.

What the diagonal line is. Diagonal principally means a straight line drawn from one angle to the opposite one in the long tetragon, dividing it into two equal parts — to distinguish it from that of the 86r square. And in the rhombus and rhomboid too it is customary so to call it.

Of the centre of the circle. Centre, properly, is said in the circle of that middle point in which, planting the immovable foot of the compasses, the other foot, turning, describes the circle with the line called circumference, or periphery. And from that point all lines drawn to the said circumference are equal among themselves. But it is used also, in the other rectilinear figures, to call centre the middle point of their surface — as in triangles, squares, pentagons, hexagons and the other equilateral and also equiangular figures; since from each of their angles the straight lines drawn to the said point will all likewise be equal among themselves.

The arrow. Arrow is the name of that straight line which moves from the middle point of the arc of any portion of the circle and falls squarely upon the middle of its chord. And it is called arrow in respect of the part of the circumference called the bow, in the similitude of the material bow, which also uses these three names: cord, bow and arrow.

Of many other terms. 86v Though very many other terms are in use, of which we have treated fully in our great work, I do not care to adduce them here; only these, necessary to the understanding of the present compendium, have I seen fit to adduce for Your Highness. And though this work be not concluded with so great a number of leaves, yet matter of no less substance, and the highest speculations, are treated in it. And truly, most excellent Duke, not lying to Your Highness, I say that the speculation of the mathematical disciplines cannot virtually extend itself higher, though at times the quantities fall out greater and lesser. And in these our Megarian philosopher concluded and ended his whole volume of Arithmetic, Geometry, Proportions and Proportionality, distinguished into fifteen partial books, as is clear to the understanding. And therefore no small grace and dignity will this work add to your aforesaid most worthy library, as we said before in your epistle, it being the only one composed of such order and matter, and known to none in all the universe thus far — save to Your Highness. And here, in your illustrious great city of 87r Milan, it was composed with no middling toils and long vigils, under the shadow of Your Highness and of your as-it-were son, my — undeservedly — particular and singular patron, the illustrious Lord Galeazzo Sanseverino of Aragon, second to none in arms and supreme lover of our disciplines, above all in the daily attendance of his assiduous lecture, tasting of them the most useful and sweet fruit.

And let there stand, for the conclusion of our process, the humble plea for pardon and due reverence of the perpetual servant of Your Highness, to whom he commends himself infinitely in every way. Quae iterum atque iterum ad vota felicissime valeat — may she again and again fare most happily, according to her every wish.

FINIS

The Bodies, to the Reader (tercet)

The sweet fruit, lovely and so much delighted in, once constrained philosophers to go seeking the cause of us — the fruit that feeds the intellect.

Distich: Querere de nobis fructus dulcissimus egit / Philosophos causam, mens ubi laeta manet. — A most sweet fruit drove the philosophers to seek our cause: there where the mind abides in joy.

87v FINIS

On the 13th day of December, in Milan, in our bountiful convent — the reverend father and professor of Sacred Theology, Master Francesco Mozanica, most worthy minister of the province, governing the whole of it — 1498, the Supreme Pontiff Alexander VI reigning, in the seventh year of his pontificate.

The text translated here is the Compendium de divina proportione that Luca Pacioli completed in Milan in December 1498 and presented to Ludovico Sforza, with sixty polyhedra painted by Leonardo da Vinci. Of the three manuscripts Pacioli had made, two survive: this one — given to the Biblioteca Ambrosiana in 1637 with the Arconati donation of Leonardo manuscripts — and a second in Geneva. The printed edition of 1509 added further treatises but reproduced the plates only as woodcuts. The Italian text followed here is the 1956 Fontes Ambrosiani transcription of the Ambrosiana codex.