Renaissance (mathematical history)

Summary: The Renaissance’s resurgence in humanism also benefited mathematics and engineering.

The Renaissance or Rinascimento (both words mean “rebirth”) was a flourishing of philosophy, art, architecture, science, and high culture more generally beginning in fourteenth-century Europe. Renaissance thinkers thought of themselves as restoring the civilization of Greece and Rome after what they called “the Middle Ages.” The Renaissance saw the rise of humanism, hermeticism, Neoplatonism, and realist art involving optical perspective; the decline of feudalism; increased circulation of ideas due to printing; the Protestant Reformation; a strong interest in classical literature and history; a strengthened interest in science and mathematics and their applications; and world exploration.

Early Renaissance (c. 1300–1450)

The Renaissance can be traced back to the thirteenth-century writings of Dante Alighieri, Francesco Petrarca, and Brunetto Latini and the paintings of Giotto di Bodone. Such work was sponsored by bankers, merchants, and industrialists who rose to great wealth and influence, displacing the Church and landed nobility as primary sponsors of high culture.

Starting in the mid-fourteenth century, humanist scholars searched libraries to recover the lost texts of classical Rome. Many edited texts went to print, increasing their accessibility at (relatively) low cost. After approximately 50 years, attention turned to recovering the Greek heritage, which—though mostly lost in the West—had continued on in Byzantium. Many Greek scholars migrated west at this time, bringing their expertise and manuscripts to Venice, in particular. The recovery and translation of Plato’s works, along with several tracts in neoplatonism and hermeticism, fueled an interest in applying simple numerical ratios and geometric regularity in fields as diverse as art and architecture, cosmology, alchemy, and musical tuning. The intentions included occult efforts to replicate cosmic structures, invoking astral influences at the human scale. More visceral results were achieved by composers, such as Josquin des Prez, who brought polyphonic techniques to Italy from the Low Countries, laying foundations for important Italian composers (such as Giovanni Pierluigi di Palestrina) toward the end of the sixteenth century.

Renaissance (c. 1450–1500)

The Renaissance spread north from Tuscany and across the Alps during the second half of the fifteenth century. Political philosophy, exemplified by Niccolò Machiavelli’sPrince and Discourses on Livy, attempted a rational analysis of political structures contextualized by cultural difference and the practicalities of everyday life. Vernacular languages came to be used for scholarly writing, making texts more widely readable as did printing, which advanced rapidly with the establishment of fine publishing houses in the Veneto. Examples include the Aldine Press, where italic typefaces were invented and Erhard Ratdolt’s press, which pioneered the printing of mathematical diagrams when producing the first edition of Euclid’s Elements in 1482.

The mid-Renaissance was centered on the Republic of Florence, largely sponsored by a powerful banking family, the Medici. The ideals of this period are expressed in Florentine architecture, such as Filippo Brunelleschi’s Church of San Lorenzo, which has a legible geometric regularity, bright and even light, openness, and a delicately balanced stillness. Ideals in painting included realism based on optical theory. Artists could occupy the leading edge of mathematical research; Piero della Francesca, for example, produced treatises on perspective theory in addition to painting with perspective techniques. Sculpture also developed a scholarly foundation through both historical study of the classical texts that had survived and hands-on dissection of fresh cadavers.

High Renaissance (c. 1500)

The High Renaissance lasted only briefly before transforming into Mannerism. It was focused on Rome, owing to the patronage of Pope Julius II. Art gained a level of dynamism best known through the works of Rafaello Sanzio (Raphael) and Michelangelo Buonarotti in Rome, and Tiziano Vecelli (Titian) and Giorgione in Venice. Leonardo da Vinci’s Last Supper, Raphael’s School of Athens, and Michelangelo’s ceiling in the Sistine Chapel were painted during the High Renaissance.

Further north, the Renaissance adapted to local cultures and circumstances. In Germany, for example, goldsmiths crafted clocks, automata, and mathematical and astronomical instruments for their patrons. Reformation printers published a wide range of medieval texts alongside Lutheran tracts, largely shedding the refined typography of Venice in favor of speed and quantity. Gothic elements remained strong in the art and architecture of England, the Netherlands, and Scandinavia and Renaissance influences reached those countries only after they had become Mannerist. Because of Protestantism, secular authorities replaced the Catholic Church as the primary sponsor of cultural works.

Renaissance Science and Mathematics

Renaissance scholars initially reacted against Scholastic natural philosophy by turning to Neoplatonism, taking an often mystical and magical approach to nature, often with practical goals. This shift can be seen in the intertwining of alchemy and astrology, for example, and in the wide range of applications described in Giambattista della Porta’s 1558 book Natural Magic. The title reflects a distinction drawn between natural magic, which invoked empirical knowledge of nature to achieve results; in contrast to spiritual magic, which regulated astral influence using amulets and talismans; and demonic magic, which invoked supernatural beings.

The Church’s need for calendrical reform led Nicolaus Copernicus to develop heliocentric astronomy as an improvement upon the Hellenistic methods maintained and developed throughout the Middle Ages. Astronomy was favored also in Protestant territories owing to the educational reformer Philip Melanchthon arguing that it was an ideal way to learn about divine creation.

Artillery motivated studies in ballistics, leading to stellated polygonal designs for fortresses, such as Naarden in the Netherlands and the Kronborg in Denmark. Aristotelianism, however, still provided qualitative theory for ballistics and other practical endeavors, such as hydraulic engineering.

The development of machines and engineering techniques inspired efforts to classify and theorize about them, as shown by the published “theaters of machines” by Jacques Besson and Agostino Ramelli.

The influences of exploration can be dated at least as far back as 1488, when Bartholomeo Dias found a connection between the Atlantic and the Indian Ocean that led to trade routes established beginning in 1498 with Vasco da Gama’s arrival in Calicut, six years after Christopher Columbus found the West Indies. Such journeys motivated developments in navigation and shipbuilding as well as an outward-looking attitude. Trade expanded, especially in Spain, Portugal, and—as the new knowledge spread north—the Netherlands. Descriptions and specimens brought back from foreign regions caused disputes and reforms in biological taxonomy that were eventually settled in the eighteenth century by Charles Linnaeus.

Progressive rational problem-solving, combined with the growth of theoretical method and a growing preference for naturalistic rather than occult explanations, provided many elements needed for the eventual emergence of modern empirical science.

Mathematics was boosted early by the ascendance of merchants and bankers who needed computational methods to manage money and later to solve problems in navigation and cartography. Some advanced material was assimilated from Arabic sources, such as geometric methods and high-precision trigonometric tables. Solving polynomial equations became a display of virtuosity; the quadratic had been solved in antiquity, now Girolamo Cardano and other mathematicians developed solutions for cubics and higher order problems. As algebra developed, many algebraic symbols were invented and evolved into the forms used today. Hindu-Arabic numerals replaced Roman numerals but the calculation of the products, ratios, and square roots of large numbers in astronomy and navigation was still onerous and error-prone. These operations were facilitated by conversion into addition and subtraction problems using prosthaphaeresis (based on trigonometric transforms), and later through the invention of logarithms.

Bibliography

Field, J. V. The Invention of Infinity: Mathematics and Art in the Renaissance. Oxford, England: Oxford University Press, 1997.

Goulding, Robert. Defending Hypatia: Ramus, Savile, and the Renaissance Rediscovery of Mathematical History. New York: Springer, 2010.

Hall, Marie Boas. The Scientific Renaissance, 1450–1630. New York: Dover Publications, 1994.

Hay, Cynthia. Mathematics from Manuscript to Print 1300–1600. Oxford, England: Oxford University Press, 1988.