History of the Renaissance: The Reciprocal Innovations in Mathematics and Art and Music
Certainly, the Renaissance was a period of innovation in forms of art, science (or proto-science), mathematics, philosophy, and literature. Considering these innovations, I was particularly interested in the reciprocal relationship between music, art, and mathematics/science in the Renaissance. Discussed are the innovations in mathematics and how they occurred and what impacts these innovations in mathematics had on art and music.
Perhaps one of the most important developments in Renaissance mathematics came from the development of humanist libraries in the fifteenth century. These libraries were very different from Medieval libraries. They were much larger and had many more classical works including many Greek codices. Before the advent of the Renaissance libraries usually only had translated Greek mathematical works by Euclid and occasionally Archimedes. These new libraries came to contain works by Euclid and Archimedes in Latin and Greek. Furthermore, these libraries also contained Greek texts from Apollonius, Diophantus, Hero, Pappus, and Proclus. There was a marked emphasis on locating, archiving, and studying works by Archimedes.
At the end of the fourteenth century, a group of scholars of Greek in Florence working with Emanuel Chrysoloras established the city as the main market for classical codices in Italy. Fra Ambrogio Traversari did extensive searches for codices in monasteries throughout Italy and Poggio Bracciolini did his searches in monasteries outside the Alps. Others reported to the Florentines, such as Francesco Filelfo and Giovanni Aurispa, who found codices in Byzantium. Filelfo found around fifty codices such as a Greek codex of Conica by Apollonius. At one point Filefo owned a text by Aratus, Geographia by Ptolemy, and the Mechanic by Aristotle. Interestingly, Filefo also made one of the earliest Renaissance references to the idea that the Earth revolves around the sun. Aurispa was able to find 238 codices when he traveled to the East in 1421-1423. For instance, he obtained a work by Athenaeus and another by Procus. He also found Mathematical Collections by Pappus. In the library of Antonio Corbinelli, a pupil of Chrysoloras, there are two Latin versions of works by Euclid, one Latin version of the Almagest by Ptolemy, and two works by Leonardo Fibonacci, and a Greek work by Euclid. Palla Strozzi owned a Greek Almagest, a Greek codex called Geometricha, and two works by Leonardo Fibbocci in Latin. He also had in his collection a Greek codex of Geographia by Ptolemy, a work that once translated achieved wide distribution in the Renaissance.
These conditions lead to a number of collections in the first part of the fifteenth century that would later be purchased by Cosimo de’ Medici and placed in the two great libraries. One of them, the Libreria Medicea Publica, was placed in the monastery of San Marco. The library in San Marco was primarily created from the libraries of one of the early students of Chrysoloras, Niccolò Niccoli, and Ser Filippo di Ser Ugolino Pieruzzi. Another Medici library, the Libreria Medicea Privata, was kept in the household of the family and generally only accessible by scholars associated with them. In 1456, the Liberia Medicea Privata had only one Greek codex, but by 1495 the library contained more than 460, including a copy of the Mechanica, a commentary on Ptolemy’s the Almagest by Theon of Alexandria, and two by Euclid. To grow the library, Lorenzo de’ Medici sent Janus Lascaris and Angelo Poliziano on searches for more manuscripts. From 1489 to 1492, Lascaris was able to find around 200 Greek manuscripts, such as Codex A by Archimedes and Pneumatica by Hero.
Works such as Perspectiva communis by John Peckham are also represented in San Marco. Paolo dal Pozzo Toscanelli, who although has no surviving mathematical works, was known by many like Vespasiano da Bisticci and Pico Della Mirandola, and Traversari to be talented in mathematics, astronomy, and humanist thought and was often a visitor of Cosimo de’ Medici. Later Toscanelli would even become acquainted with the architect Filippo Brunelleschi in what Toscanelli called, “The greatest association of my life.” Toscanelli initially taught Brunelleschi Euclid and geometry, but it is likely that they later influenced each other. Brunelleschi, for his part, was also acquainted with Pall Strozzi, Poggio, Traversari, and Cosimo de’ Medici as well as Niccoli and Leon Battista Alberti and knew the works of Vitruvius well.
Further demonstrating the learning of mathematics by humanists in the Renaissance is the Commentary by Lorenzo Ghiberti. Within the Commentary is a treatise on optics based on Alhazen’s and Euclid’s studies of optics. But more interesting to us is that Ghiberti says that the artist has to know geometry and the mathematical works from Archimedes, Apollonius, Ptolemy, and Alhazen. Another outstanding artist of the time, the architect Leon Battista Alberti, was praised for his their astronomical observations and his mathematics. Alberti wrote on statics and perspective as well.
In Rome, Nicholas V’s patronage built the Biblioteca Vaticana. Nicolas’ goal was to create the greatest library since Alexandria. For this goal, he hired copyists, translators, compilers, and people to search for manuscripts. In 1455, after Nicholas’ death, the library had 1209 codices with 414 in Greek. In 1484, the year of Sixtus IV’s death, the library had 3700 codices with 843 in Greek. But even before, in 1455, the Biblioteca Vatican had the largest collection of manuscripts in Europe.
Essentially this movement was created by a renewed interest in locating and translating manuscripts by Greek mathematicians, especially those by Archimedes. These developments resulted in a rebirth in mathematics and science (or at least proto-science).
Perhaps one of the most important works, as it relates to art, is the Book of Optics (known as Perspectiva or De Aspectibus in Latin) by Alhazen, which was probably translated in Spain. It influenced the “perspectivists” like the previously-mentioned John Peckham. Later it influenced people like the pioneering architect Brunelleschi. It also influenced Alberti, who wrote the first theoretical text on perspective in painting. To say the least, Alhazen’s work was nothing short of revolutionary, to the point that it would take five hundred years after his work for Kepler and Descartes. It is interesting to note that Alhazen’s work wasn’t even directly addressing images, but rather only on geometry and reflection and refraction of light.
In order to engage in his studies, Alhazen invented the first dark chamber. He was primarily observing the way light travels rather than paying attention to the images that are projected. As art historian Hans Belting notes, the geometry of rays that carry their single forms, as in a mosaic, from spots on the surfaces of visible objects to spots on the surface of the eye, is one thing, while the integral image that emerges in the brain is another. In the Renaissance, the opposite road was taken when pictorial perspective was introduced with the idea of constructing pictures that tell me how I see, or, to put it differently, which reproduce my perception mirror-like. The two cultures share the same mathematical theory but differ in its practical application and significance.
Basically, the concept of geometry in Arab optics is different than the Western representational concept of geometry in visual media, where the focus is on the individual gaze.
The philosopher and mathematician Biagio Pelacani criticized the perspectivists and Alhazen. but used Alhazen’s theories as a basis. He created a theory of mathematical space which was dependent on the individual’s perspective, something which Alhazen and others weren’t interested in. His ideas of space and spatial topology spurred new discussions of viewing and vanishing points, orthogonal, and the visual pyramid which created the idea of mathematical perspective that was necessary for the Renaissance innovation of perspective in art.
In the latter part of the Renaissance, new innovations on the study of sound helped create new musical innovations. Scientific thought was able to falsify premises used in the study of music and in the practice of music. It should be noted that music and the study of mechanics were much more closely related in the Renaissance than they are now, especially when considering that all the studies in the quadrivium of arithmetic, geometry, astronomy, and music are given equal weight. More interestingly, as historian Stillman Drake notes, is that the origins of experimental science seem to be in sixteenth-century music.
The Renaissance certainly already had innovations like the development of polyphony in Europe (or at least Western and Central Europe). In the time after 1450 and even more so after 1550, the practice of music changed fundamentally. These developments necessitated new developments in music theory, but mechanics still wasn’t able to fully analyze it yet. The predominant theories of harmony followed a sort of teleological and numerological approach from the Pythagorean and Ptolemaic theories of harmony. But by around 1563, the mathematician Giambattista Benedetti questioned this approach. Benedetti came to disagree with the theories on the harmony of prominent music theorist Gioseffo Zarlino, which were a modification of Pythagorean ratios. Benedetti noted that following Zarlino’s harmonic ratios strictly could result in a significant change in pitch in only a few bars. Benedetti, who is recognized as one of the most important precursors to Galileo, decided to relate pitch and harmony not to numbers but to frequencies of the vibrations from the source. He thought that consonance happened when the notes in the harmony concurred or recurred with each other frequently. It is not directly stated whether Benedetti’s assertion that string length is proportional to vibration frequency was based on experiment, though that he appealed to physical reality rather than the dominant teleological theory is notable.
Galileo Galilei’s father, Vincenzo Galilei also disagreed with Zarlino, his former teacher. In 1578, Galilei sent Zarlino a discourse arguing that approximately equal temperament is necessary. Zarlino, it appears, attempted to prevent the printing of Galilei’s discourse. Eventually, Galilei set up a series of scientific experiments to demonstrate the validity of his ideas.
From Drake: 'The ratios 2:1, 3:2, 4:3 will give octaves, fifths, and fourths for strings of like material and equal tension but of lengths in these ratios, or for columns of air of similar lengths. But if the lengths are equal and the tensions are varied, then the weights required to produce the tensions are as the squares of these numbers.' He found that the weight required for a string to have a given pitch is the inverse squares of the lengths. He never published his ideas but his son Galileo owned the manuscripts from his father. It is also pretty likely that Galileo was involved in his father’s experiments. Galilei’s experiments, which used strings and weights, also probably influenced Galileo’s interests in the pendulum. At around the same time, the Dutch mathematician Simon Stevin was advocating for an equal temperament based on the twelfth root of two, but never actually published his ideas. I would like to note, however, that the idea of a Western origin of equal temperament is questionable and probably came as an export from China.
Marin Mersenne, who was a friend of Descartes, published a work on the theory and practice of music in 1636-1637 called Harmonie Universelle, where he heavily emphasized the role of mechanics in music. Mersenne carried out falling body experiments which Galileo never fully described in his published books. This especially makes Drake’s thesis seem more valid.
The reciprocal influence between different regions, people, and fields of study is very well demonstrated by the Renaissance. Mathematics heavily influenced innovations in the arts in the Renaissance such as vanishing points and perspectives. But it could be equally said that it goes the other way; that is, that art incentivizes the study of mathematics. Mathematics also heavily influenced innovations in music, too. Even more interestingly, though, is that the study of music seems to have heavily influenced the development of modern science. I think that this demonstrates the value of the liberal arts and why the Renaissance really was a rebirth. It is also interesting to consider that the experimentation of Galilei and perhaps of Stevin and Benedetti seems to either be a consequence or cause of the shift into the Enlightenment and the scientific revolution. These developments in art and music were also a sign of attempting to further the Humanist idea while not being too focused on the writings of the ancients. I find it not surprising that these historical occurrences cross-pollinated each other. It is also interesting to note how the development of public (and private) libraries influenced the creation of new art, music, and mathematics.
Works Cited
- Belting, Hans. “Perspective: Arab Mathematics and Renaissance Western Art.” European Review, vol. 16, no. 2, 2008, pp. 183–190., doi:10.1017/S106279870800015X.
- Drake, Stillman. “Renaissance Music and Experimental Science.” Journal of the History of Ideas, vol. 31, no. 4, 1970, pp. 483–500. JSTOR, www.jstor.org/stable/2708256.
- Rose, Paul Lawrence. “Humanist Culture and Renaissance Mathematics: The Italian Libraries of the Quattrocento.” Studies in the Renaissance, vol. 20, 1973, pp. 46–105. JSTOR, www.jstor.org/stable/2857013.
