CHAPTER II¶
Preface of the Reverend Father Magister Luca Pacioli of Borgo Sansepolcro, of the Order of Minorites and Professor of Sacred Theology, to the Compendium on the Divine Proportion drawn from the Mathematical Disciplines
Propter admirari coeperunt philosophari (“On account of wonder men began to philosophize,” Aristotle, Metaphysics I.2, 982b12). The authority of the “Master of those who know” [Aristotle; cf. Dante, Inferno IV.131] holds, Most Exalted Duke, that knowledge took its beginning from sight; just as he affirms in another place, saying: Quod nihil est in intellectu quin prius sit in sensu—that is, nothing is in the intellect that has not first offered itself in some way to the senses. And among our senses, the wise conclude that sight is the noblest; whence even among the common people it is not undeservedly said that the eye is the first portal through which the intellect understands and tastes.
As is recorded in that place, when the priests of Egypt beheld the moon eclipsed, they stood in great wonder; and seeking the cause, they discovered through true science that this occurs naturally through the interposition of the earth between the sun and the moon, wherewith they remained satisfied. From that time forward, as their successors progressively refined their inquiries with the light of the five intellectual windows [the five senses], they filled an innumerable multitude of volumes with their profound sciences for our utility, inasmuch as one thought bursts forth from another (l’uno pensier da l’altro scoppia; Dante, Inferno XXIII.10), so that many others were born therefrom.
Reflecting within myself upon this, I resolved to take up the pen for this most useful compendium chosen from the mathematical sciences, and together with it, with my own hands, to fashion materially in their proper forms the solid bodies belonging thereto for the common utility, and to offer them along with the present compendium to Your Ducal Highness. On account of their unwonted aspect—like something descended from heaven in our own days—I do not doubt that Your graceful and perspicacious intellect will take the greatest pleasure therein, especially when, by that same light and with no less investigation than the ancient Egyptians applied to the eclipse, You discover with the aid and support of the present treatise the causes and sweetest harmony of such forms.
Wherefore I am certain that if in the past Your Highness has shown Yourself generous and munificent toward those who possessed even a part of such sciences and disciplines, in the future You will show Yourself far more magnanimous and bountiful still, and—what is more—will exhort with all diligent care Your dear courtiers, reverent subjects, and other well-wishers to acquire them. For these mathematical sciences are the foundation and ladder (fondamento e scala) for arriving at knowledge of every other science, because they stand in the first degree of certainty, as the Philosopher affirms when he says: 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”).
Without knowledge of them, it is impossible to understand any other science well. Moreover, in the Book of Wisdom (11:21) it is written: Quod omnia consistunt in numero, pondere et mensura—that is, everything that unfolds throughout the lower and upper universe is necessarily subject to number, weight, and measure; and in these three things Aurelius Augustine in De Civitate Dei says that the Supreme Artificer is most highly praised, because in them fecit stare ea quae non erant (“He caused to stand fast the things that were not”).
Through Your loving exhortation, I perceive that many who are ignorant of such sweet fruit of utility will awaken from their torpor and mental slumber, and will devote themselves wholly to inquiring into it with every study and solicitude; and this will cause the age itself to be renewed in our time, and men to arrive with greater reality and swiftness at perfection in whatever science they study.
Furthermore, beyond renown and worthy commendation, it will increase no small probity in Your Ducal Highness’s exalted dominion among Your dear courtiers and beloved subjects, making them ever prepared for its defense—no less than the noble, ingenious geometer and most worthy architect Archimedes did for his own homeland. He, as is written, by his new and varied inventions of machines long preserved the city of Syracuse safe and unharmed against the assault and warlike might of the Romans, until Marcus Marcellus sought to storm it by stealth.
Indeed, by daily experience it is not hidden from Your Ducal Highness (even as for many years Your illustrious father’s memory [Francesco Sforza] was the author, preceptor, and standard thereof for all Italy and for both Transalpine and Cisalpine Gaul) that the defense of great and small commonwealths—otherwise called the military art—cannot be practiced egregiously with honor and profit without knowledge of Geometry, Arithmetic, and Proportion. Nor can any worthy army assigned to a siege or defense ever be said to be fully provided for unless there is found within it a dedicated engineer and deviser of new machines, such as we have just mentioned in the great geometer Archimedes at Syracuse.
If one examines carefully all artillery and siege engines in general—take whichever you will, such as bastions and other ramparts, bombards, briccole, trebuchets (trabocchi), mangonels, rompheae, ballistae, catapults, battering rams (arieti), and tortoises (testudines)—they will always be found to be fabricated and formed by the power of numbers, measure, and their proportions. What else are citadels (rocche), towers, ravelins, walls, outworks (antemuri), moats, bastioned keeps (turrioni), battlements (merli), mantlets, and other fortifications in towns, cities, and castles, if not pure geometry and proportions, poised and adjusted with due levels and plumb-lines (archipendoli)?
Through nothing else were the ancient Romans so victorious, as Vegetius, Frontinus, and other egregious authors write, than through their great care and diligent preparation of engineers and commanders by land and sea, whose sufficiency is impossible without the mathematical disciplines—namely Arithmetic, Geometry, and Proportions—as the ancient histories of Livy, Dionysius, Pliny, and others make clear and manifest. From these sources Roberto Valturio, the most skilled citizen of Rimini, drew all the machines he set forth in his worthy work entitled De re militari (De instrumentis bellicis), dedicated to the Illustrious Lord Sigismondo Pandolfo [Malatesta]. And the most blessed memory of Your Highness’s close kinsman Federico da Montefeltro, Most Illustrious Duke of Urbino, caused all those machines and instruments—letter for letter as the said author of Rimini sets them forth in his book, and many more besides—to be carved in order by the hands of most worthy stonecutters and sculptors in a frieze of living and beautiful stone all around the base of the stupendous edifice of his noble and admirable Palace in Urbino. Thus too one reads in the Commentaries of Julius Caesar concerning his ingenious bridge [across the Rhine]; and even to this day, in the worthy Umbrian city of Todi, in the church of San Fortunato—our sacred convent—there still hang publicly a great multitude of huge hempen cables which the illustrious memory of Your father [Francesco Sforza] duly arranged for a bridge across the Tiber to win his famous victory.
By no other means, moreover, did our most subtle Doctor [John Duns] Scotus arrive at the loftiest speculations of Sacred Theology than through knowledge of the mathematical disciplines, as appears throughout his sacred works—especially if one examines the question in the Second Book of his Sentences where he inquires whether an angel has a proper and determinate place (locus determinatus) for its existence, wherein he clearly shows that he understood the entire sublime volume of our most perspicacious Megarian philosopher Euclid.
For no other reason, likewise, do all the texts of the “Prince of those who know” [Aristotle]—the Physics, Metaphysics, Posterior Analytics, and the rest—appear difficult, save through ignorance of the aforesaid disciplines. For no other reason is there a scarcity of good astronomers, save for the lack of arithmetic, geometry, proportions, and proportionalities; so that nine out of ten govern their judgments by tables, almanacs (tacuini), and other calculations made by Ptolemy, Albumasar, Alfraganus, Geber [Jabir ibn Aflah], King Alfonso, Giovanni Bianchini, Prosdocimo [de’ Beldomandi], and others, which through the carelessness of copyists are often corrupted and vitiated; and consequently, trusting in them, they fall into the greatest and most manifest errors, to no small harm and prejudice of those who rely upon them.
The supreme subtlety of all municipal laws, too, consists—as experts therein have often explained to me—in adjudicating alluvial deposits and circumluvions of waters (alluvioni e circumluvioni de l’acque) caused by excessive flooding. On this subject their eminent chief, Bartolus of Sassoferrato, composed a special treatise entitled Tyberiadis [De fluminibus], in the preface of which he greatly extolled Geometry and Arithmetic, affirming that he had learned them from a friar of our Order named Brother Guido, Professor of Sacred Theology, in that treatise on what the Tiber sometimes gives and takes away through its flooding, especially in the region from Perugia toward Deruta—wherein, citing step by step our most perspicacious philosopher Euclid with rectilinear and curvilinear geometric figures, he guided his inquiry and brought it to a conclusion with the greatest subtlety.
I say nothing of the sweet and gentle harmony of music, nor of the supreme beauty and intellectual comfort of perspective, nor of the most skillful disposition of architecture, together with the description of the maritime and terrestrial universe [cosmography] and the doctrine of celestial bodies and aspects, because from what has already been said their dependence is clear. To spare the reader tedium, I pass over many other practical and speculative sciences along with all the mechanical arts necessary in human affairs, none of which can be acquired or maintained in due order without the aid of these disciplines.
Therefore one should not wonder if good mathematicians are few in our times: for the scarcity of good teachers is the cause thereof, together with gluttony, sleep, and idle feather-beds (la gola, il sonno e l’oziose piume; Petrarch, Canzoniere VII.1), and in part the weakness of more recent minds. Whence among the wise it is magisterially customary to say as a common proverb: Aurum probatur igni et ingenium mathematicis—that is, fire proves the goodness of gold, and the mathematical disciplines prove the excellence of the intellect. This means in substance that a keen intellect is most suited to mathematics, since these sciences are of the greatest abstraction and subtlety, having always to be considered apart from sensible matter; and truly they are, as the Tuscan proverb says, those that “split a hair in mid-air” (spaccano el pelo in l’aire).
For this reason the ancient and divine philosopher Plato not undeservedly denied entry into his celebrated Gymnasium [the Academy] to those unskilled in geometry, placing an inscription high above its main portal in large, legible letters with these formal words: Nemo huc geometriae expers ingrediatur—that is, let no one who is not a good geometer enter here. This he did because within geometry every other hidden science is found. Filled before him with its sweetest delight, that most diligent contemplator of nature, Pythagoras, upon discovering the property of the right angle ([I.47])—as is read of him and as Vitruvius recounts (De architectura IX.praef.6)—sacrificed a hundred oxen to the gods with the greatest festivity and jubilation, as will be told below.
Let this suffice at present in commendation of the mathematical sciences, the number of whose students in this Your illustrious city already begins to grow day by day through the grace of Your Ducal Highness, thanks to the assiduous public lectures recently instituted by You and the progress of the egregious auditors, as I expound to them clearly and with all diligence (in their judgment), according to the grace granted me therein by the Most High, the sublime volume of the aforesaid Euclid in the sciences of Arithmetic and Geometry, Proportions and Proportionalities. Having already brought its first ten Books to a most worthy conclusion—always interweaving our practical applications with his theory for their greater utility and fuller understanding—I now dedicate the remainder of my time to the present completion of this treatise.