Geometry

Discoveries in Renaissance geometry were most significant for architecture, art, optics, cartography, navigation (see the Mercator projection in chapter 9), and surveying. In 1400, the basic text for geometry in universities were the first few books of Euclid’s Elements. Pertaining mostly to triangles and circles, Euclid’s text had been used in schools since antiquity. Other Greek texts on geometry, however, had been lost. Renaissance humanists produced new editions of these works as they were discovered, often adding lengthy commentaries. An important Greek text on conics was published in Latin in 1566, and the works of Archimedes appeared in print in 1588. Meanwhile, in the abacus schools (see chapter 11) where technical subjects were taught, solid geometry was beginning to be developed. The three-dimensional technique required for surveying also applied to architecture and painting, and Luca Pacioli and Daniele Barbaro (1513-70) contributed significant publications. Pacioli taught mathematics throughout much of northern Italy, discussing his ideas with colleagues in several cities. Leonardo de Vinci (1452-1519), who was interested in geometry, illustrated Pacioli’s 1509 publication on proportion. Bar-baro’s profusely illustrated La practica della perspettiva (The practice of perspective, 1568, 1569) included information on polyhedra. More important, he expounded the intellectual value of studying such shapes, because “by the secret intelligence of their forms we ascend to the highest speculations concerning the nature of things” (Kemp 1990, p. 76).

No one studied forms more closely than artists. Several 15th-century artists contributed to the development of linear perspective (see chapter 3), notably Leon Battista Alberti (1404-72) and Filippo Brunelleschi (1377-1446). Although not published during the Renaissance, De prospettiva pingendi (On perspective in painting) by the painter Piero della Francesca (c. 1420-92) circulated in manuscript copies after circa 1474. Written from a practical point of view, this treatise explained through geometric diagrams the techniques for drawing threedimensional objects in a two-dimensional space. Commencing with Euclidean principles, Piero described additional proportional relationships, paying particular attention to the sides and diagonals of geometric shapes. Albrecht Durer published Underweysung der Messung (Treatise on measurement, in four sections) in 1525. Because he was famous as an artist, and because this was the first printed treatise in German on geometry, the book was quite popular. Besides illustrating theoretical points, Durer applied geometric polyhedra to several artisanal topics, such as typography. The first section of the book, on lines, illustrates how to create sections of cones; the second treats both the theory and the applications of polygons; the third discusses the attributes of solid forms; the fourth pertains to polyhedra. This last book shows solids in two-dimensional shapes, flattened out as templates.

The problem of expressing three-dimensional shapes on a two-dimensional surface also concerned

Handbook to Life in Renaissance Europe

Navigators and cartographers (see chapter 9). The Mercator projection, though useful, was not to scale, and shapes of the continents were somewhat distorted. Mercator’s system, based on the concept of cylindrical projection, allowed mariners to set their course in a straight line, but measurement of distance was problematic. After gaining practical experience in navigation, the English mathematician Edward Wright (1558-1615) used trigonometry to calculate latitudinal distances. His 1599 treatise, Certain Errors in Navigation Detected and Corrected, interpreted and improved Mercator’s system.