Angelo Mazzocco, 
Spanish and Italian

Mark Peterson, 
Mathematics

Begun as a recovery and appropriation of classical texts in the area of the humanities (grammar, rhetoric, poetry, history, and moral philosophy), the Italian Renaissance eventually turned its attention to the recovery and examination of texts in the field of the mathematical sciences. Soon the works of such distinguished ancient mathematical scientists as Aristotle, Apollonius, Pappus, Proclus, Ptolomey, but especially Archimedes and Euclid became indispensable tools of the mathematicians of the Renaissance. Like their counterparts in the area of the humanities, the classical works in the field of mathematics were subjected to a rigorous textual criticism and were accorded much reverence. Indeed, the discipline of mathematics itself acquired much esteem and admiration among the mathematical scholars of the Renaissance. Hence aptitude for the mathematical sciences was regarded as a mark of true intelligence and the golden section came to be characterized as divine proportion.

Unlike the scholars of the humanities, who generally came out of the refined, classically-driven courts of Italy, those of the mathematical sciences usually emerged from the ordinary, practically-minded shops of the artisans of the peninsula. Consequently whereas the former were molded by a sophisticated educational training, the latter were formed by intense, persevering self-teaching. The higher social status of the literati together with the enormous prestige accorded to the humanities in the intellectual circles of Renaissance Italy led these scholars to argue that the humanities were the sole repositories of true and valuable knowledge. Such cultural supremacy on the part of the literati prompted the mathematicians to defend and validate their field of study. The mathematical works of Archimedes and Euclid, they maintained, were as useful as the rhetorical and philosophical treatises of Cicero, the poetic works of Virgil, or the historical writings of Livy. Moreover, mathematics was fundamental to every human activity and it contributed significantly to the well being of the individual and the state. Indeed, mathematics was essential to the painter, the architect, the sculptor, the cosmographer, the astronomer, the musician, the philosopher, the etymologist, and the tradesman. Mathematics aided the poet in his rhyme scheme, the general in his warfare, and the civil servant in his administrative transactions. Mathematics served as the lifeblood of the merchant class. Certainty and immutable truth are inherent characteristics of the mathematical science; consequently mathematics constituted the single most important component of the experimental method. Being influenced by the Neoplatonism that flourished in Florence in the second half of the fifteenth century, some mathematicians believed that number and proportions played a fundamental role in the structure of the universe; therefore, mathematics provided the key to the understanding of the eternal truths of God and humankind.

From its outset in the late-fourteenth century to its end in the late sixteenth century, the Italian Renaissance produced many distinguished mathematicians. However, by virtue of their influence and their overall representation of the mathematical trends of the period, the most important mathematicians of the early Italian Renaissance are Piero della Francesca (1410?-1492), Luca Pacioli (ca. 1445-1517), and Leonardo da Vinci (1452-1519). The three were linked intellectually. Piero was the teacher of Pacioli, a fellow townsman, and Pacioli influenced Leonardo's interests in geometry and proportions, among others, while they resided at the court of Lodovico Sforza in Milan. Piero was primarily a painter who nevertheless delved extensively into mathematics (he wrote a treatise on commercial mathematics) and used theoretical mathematics as an integral part of his painting, as can be seen in the intricate mathematical construction and symbolism of The Flagellation (link). Pacioli, as a painting of him demonstrates (link), was the Renaissance mathematician par excellence, being the foremost propagator of mathematical science and one of its most forceful defenders. Leonardo, an all-inclusive genius, made of mathematics the underlying principle of his entire intellectual enterprise. Moreover, mathematics, which Leonardo regarded as the sole repository of certainty, became the means with which he challenged the reasoning of the literati, a reasoning he considered highly fluid and therefore susceptible to interminable controversies. In fact, Leonardo emerged as a major advocate for the mathematical sciences in the dispute between the literati and the mathematicians alluded to earlier.

Given the reverence accorded to mathematics by the mathematicians of the Renaissance, and the centrality they attributed to it in the context of the culture of their time, as well as the interdisciplinary nature of the Renaissance, and the dispute it generated between the literati and the mathematicians about the cultural implications of the humanities and the mathematical sciences, we felt that an examination of the mathematical activities of Piero, Pacioli, and Leonardo and of the cultural milieu that produced them would address the concerns and goals expressed by the project on "Mathematics Across the Curriculum." Indeed, we believed that a study of the issues raised by the mathematicians and humanists of the Italian Renaissance, issues that, in varying degrees are very much part of the contemporary discourse on education, would alleviate math anxiety among students of the humanities and recognize the centrality of mathematics in the intellectual enterprise of our time. Moreover, a study of mathematical issues in the context of the rich culture of the Renaissance, a culture characterized by an interweaving of numerous disciplines, would prove that the mathematical discourse can serve as a tool for learning across the curriculum. We thus decided to include the material on Piero, Pacioli, and Leonardo in an all-encompassing course on the Italian Renaissance (see syllabus). The segment on mathematics was treated as just another facet of the Renaissance. Students were required to read selections from the three targeted authors as well as secondary sources on Renaissance mathematics (Link). They were also exposed to a CD-ROM on the scientific activities of Leonardo and a video on the painting techniques of Piero.

The presentation of the segment on mathematics lasted about a month and it consisted of two phases. Phase one explored the issues of Renaissance mathematics in the context of the Renaissance itself and in relationship to our own time: Why did mathematics become an important field of study in the Renaissance? What were the reasons for the polemic between the mathematicians and the humanists? How was this polemic resolved? Do the issues raised about mathematics in the Renaissance have any relevance for our own time? Do we still believe in the centrality of mathematics?

Phase two delved into the actual mathematics of the Renaissance. It was necessary to say something in class about the Platonic solids, since they are the subject of Piero della Francesca's Libellus de Quinque Corporibus Regularibus, and of Pacioli's Divina Proportione, and members of the class knew nothing about them. These five polyhedra: tetrahedron, octahedron, cube, icosahedron, and dodecahedron are the subject of Euclid's Book XIII. Among other things, it is proved there that they are the only convex polyhedra with all faces regular polygons congruent to each other, and all polyhedral vertices alike. (We had cardboard models of them to pass around.) These figures are also the basis for a speculative
quasi-mathematical theory of the elements in Plato's dialogue Timaeus. In this theory, atoms of the elements are imagined to"be" the Platonic solids, corresponding (in the order given above) to fire, air, earth, water, and the celestial material. In particular, the dodecahedron, corresponding to the celestial material, gets a special significance.

[Figure of polyhedra PSOLIDS.JPG can go here: I will soon provide an APPLET, which can replace the figure.]

The "poems" [LINK?] in Pacioli's introduction to Divina Proportione are incomprehensible without knowing this. Even knowing about Plato's idea, the "poems" seem very strange. We discussed them, and compared Pacioli's introduction, on the one hand, to Piero's dedication in his Libellus [LINK?], on the other. (Piero provides no other introduction to the Libellus). The difference in tone is revealing of two entirely different attitudes toward mathematics, one mystifying and deliberately arcane, the other workmanlike and serviceable. We discussed possible reasons for this difference.

The "divine proportion" (or golden ratio) was, according to Pacioli,inexpressibly marvellous. Even his Table of Contents [LINK?] seems to run out of superlatives only when they have been well and truly exhausted. Again, it was necessary here to fill in a small amount of mathematics for the class, really a definition: the "divine proportion" is, by definition, the way to divide a line segment A into two parts B, C (with B the longer one), such that A:B=B:C. Euclid calls this "the mean between two extremes," language Piero also uses. But for Pacioli it is "divine." The reason is that this is the division you must make in order to construct a regular pentagon, which, in turn, is the face of the dodecahedron, the divine "Platonic atom." It is on this basis that Pacioli celebrates the marvellous properties of this ratio.

The "divine ratio" really does have some interesting properties. If you start with a regular pentagon, you can connect each vertex to each other one, creating a 5-pointed star inside. The sides of the star cut each other in "divine" ratio. You will have constructed a "pentagram," which you may recognize as a figure associated with the occult (without knowing why it got this association). That this figure contains the divine ratio is the content of Chapter XVIII of Divina Proportione, "On its ninth effect, exceeding all the others." The series of "effects" begins in Chapter VII, "On the first effect of a line divided in our proportion," which says

"If a line [A] be divided [as above into B and C] in this proportion, which has a mean between two extremes (for this is another way which the learned have used to name the aforementioned proportion), and one adds the larger part [B] to half of the whole line [A], then it follows with certainty that the square of this sum is always five times as great as the square of the said half [of A]."

The class found this somewhat mystifying, but a couple of lines of algebra show that the statement is true (it follows from A:B=B:C). Pacioli was writing before algebra had a decent notation. This makes the "first effect" seem straightforward, but that was not how Pacioli understood it. In explaining what this means, he says, among other things,

"... so we can compare [this proportion] with Our Saviour, who came, not to dissolve the Law, but rather to fulfill it, and lived among men, in that He made Himself obedient and subject to Mary and Joseph. Just so does this proportion, sent from heaven, accomodate itself to the definition and conditions of others [i.e., ordinary proportions], and does not scorn them, but rather glorifies them ..."

Mathematics, for Pacioli, had a meaning it may never have had for anybody else. He was an influential teacher, though. (He strongly influenced Leonardo da Vinci.) How widespread were these exuberant sentiments? And what inspired others, like Piero, who did mathematics without this peculiar "occult" motivation? These are questions which we raised without having good answers available. Indeed, it may be that no one has reconstructed sufficiently the setting in which Renaissance mathematics was done.

We believe that we accomplished our goals in this course. In fact, following the presentation of the segment on Piero, Pacioli, and Leonardo, the students demonstrated a greater appreciation for mathematics and less hesitation in treating mathematical matters. One student observed that if today's teaching of mathematics partook of the excitement and freshness it enjoyed in the Renaissance, we might have more engineers and mathematicians. Another marveled at Pacioli's intense love for mathematics, which she compared with the platonic love that Petrarch, the father of the Italian Renaissance, had for Laura, the young Provencal woman who served as the source of inspiration for his poetry (see Evaluation Report).