Overview: Mathematics 1450-1699

Background

In the years after 1450 European mathematics flourished, yet during the previous 850 years almost all mathematical developments occurred in other parts of the world. In Medieval Europe the religious and philosophical works of the ancient Greeks and Romans were still preserved, but many mathematical texts had been lost. Yet while European mathematics suffered through the Dark Ages, elsewhere in the world mathematical innovation continued. In India mathematics was well advanced by a.d. 700, especially in trigonometry. The use of the zero helped improve the Hindu numerical system, which was the basis for the numbers we use today. Chinese mathematicians developed a decimal system and discovered solutions for a number of cubic and quadratic equations. The Arab regions of the Middle East were geographically fortunate, having easy access to Babylonian, Greek, and Indian mathematics. Arab mathematicians assembled, and then developed, a wide range of knowledge. However, while these cultures made fundamental discoveries in mathematics, they all reached a peak of activity, then faded. For various reasons the continuation of mathematical research was not given support or encouragement in these societies.

Europeans rediscovered many of the ancient Greek texts through contact with Arab mathematics. However, these works suffered somewhat from the multiple translation process, from Greek to Arabic to Latin, in which mistakes were often compounded. Innovations were often resisted, such as Gerbert's (c. 945-1003) and Leonardo of Pisa's (c. 1170-1240) separate attempts to popularize Arabic-Hindu numerals. Not until the sixteenth century were they widely accepted.

The Printing Press Aids the Development of Mathematics

By the middle of the fifteenth century translators such as Regiomontanus (1436-1476) had recovered most of the ancient Greek sources from Arabic copies. However, the books were few and extremely expensive, as they had to be painstakingly handwritten. The development of movable type and papermaking after 1450 resulted in an information revolution in Europe. Books became easy and cheap to mass-produce, and language and type were slowly standardized.

One of the earliest mathematical texts printed was Luca Pacioli's (1445-1517) Summa de Arithmetica (1494). It was difficult to read, and many commentators have noted that there were superior alternatives, such as Nicolas Chuquet's (c. 1450-1500) Triparty (1484), which contained practical exercises in geometry and commerce and was clearly written. However, it remained in manuscript form, and so Pacioli's printed book reached a much wider audience.

Most texts early in the period relied heavily on ancient sources, such as Rafael Bombelli's (1526-1572) influential treatise on algebra (1572). This fascination with ancient texts also led to the revival of ancient mystical themes within mathematics. Scholars like John Dee (1527-1608) became interested in the occult power of numbers and symbols, assigning numerical values to letters in words to expose hidden meanings. These magical elements led some scientists, such as Francis Bacon (1561-1626) and Giordano Bruno (1548-1600), to ridicule or downplay the role of mathematics. Yet the popularity of numerology and astrology gave rise to many mathematical and scientific innovations, such as precise computations of the position and motion of the stars and planets.

Slowly European mathematics caught up with Indian, Chinese, and Arabic mathematical knowledge, but often information was not shared. Scipione del Ferro (1465-1525) discovered a method of solving some cubic equations, which his pupil, Antonio Maria Fior, kept secret in order to win mathematical contests. However, Fior was utterly defeated in 1535 by Niccolò Fontana, better known as Tartaglia (1500-1557), who figured out the method. Girolamo Cardano (1501-1576) heard of Tartaglia's victory and persuaded him to allow one of the cubic solutions to be printed. However, Cardano also worked out several more cases, and Tartaglia became furious when those too were published. Secretive findings in mathematics and science were common, but many thinkers eventually began to see the benefits of sharing information. A number of important figures, such as Fr. Marin Mersenne (1588-1648), acted as intermediaries between many mathematicians, corresponding widely and spreading new ideas.

Mathematical innovations came from many sources. The symbols + and - were originally used by German bookkeepers, as they allowed easy cocomputation of deficiencies and excesses in stock. A Welsh doctor, Robert Recorde (1510-1558), created the = sign, and François Viète (1540-1603) introduced the use of letters to represent numbers in algebraic equations. Such developments combined to give mathematics its modern look and allowed easy communication across language barriers.

Albert Girard (1595-1632) was one of a number of mathematicians who expanded the field of algebra, writing an influential text, Invention nouvelle en l'algebre (1629). Girard was a military engineer, and like others of this profession he found mathematics increasingly useful. The development of cannons led to the study of the flight of cannonballs. Cannons also made older fortifications obsolete, as tall, thin, straight walls were easily toppled, and so new, geometrically elaborate fortifications were constructed to withstand the impact of artillery.

John Napier Develops Logarithms

The development of logarithms by John Napier (1550-1617) had practical applications in many areas, from astronomy to bookkeeping. Starting from ideas suggested by Chuquet, Napier produced many tables of computed logarithms, numbers that reduced multiplication and division problems to simple addition and subtraction. Henry Briggs (1561-1630), a keen astronomer, expanded and helped popularize Napier's works. Napier is also remembered for his system of rods to make calculations easier, and many versions of such devices were constructed, eventually producing the slide rule, a device that remained popular until the invention of the electronic calculator.

The development of European painting produced a new interest in geometry. To create an illusion of depth in a two-dimensional painting, the art of perspective was revived. Gerard Desargues (1591-1661), an architect and engineer, studied the figures created when a plane intersects a cone, and introduced the concept of projective geometry. Geometrical studies also helped improve mapmaking and navigation.

The popularity of gambling led Blaise Pascal (1623-1662) and Pierre de Fermat (1601-1675) to calculate probabilities of winning at dice games. Both men also contributed to a wide range of other mathematical topics. Fermat left behind one of the most puzzling mathematical challenges, his Last Theorem, which remained unsolved until the final decade of the twentieth century.

The Invention of Calculus

Many fields of mathematics were developed in this period. Number theory, induction, new methods for calculating π, decimal fractions, and a host of other important work helped make the years 1450-1699 one of the most productive periods of mathematical research. However, it was the invention of calculus that was the crowning achievement of the time. A number of seemingly independent developments combined to give rise to this new field. René Descartes (1596-1650) introduced algebra into geometry, and along with others, such as Fermat, developed the field of analytic geometry, which describes curves in the form of equations. The study of curves led to a number of new techniques being developed to analyze their slope and rate of change. Descartes developed a method for finding the perpendicular to the tangent of a curve, while Fermat worked on the problem of finding the maxima and minima of variables. Bonaventura Cavalieri (1598-1647), a Jesuit priest, developed a method for calculating the area or volume of a geometric figure, the method of indivisibles. Then Isaac Newton (1642-1727) and Gottfried Leibniz (1646-1716) independently put all the pieces together and developed methods for differentiation and integration, the beginnings of calculus.

Newton is also important for the way he viewed mathematics. His writings popularized the use of mathematical analysis in scientific subjects, such as physics and optics. He helped make mathematics the new language of science. Mathematics had also been readily adopted in areas as diverse as commerce, mapmaking, navigation, astronomy, astrology, and even gambling. The continuation of mathematical development in Europe owed much to the practical applications the field offered. Financial support and patronage from wealthy individuals and governments enabled many mathematicians to dedicate their lives to their studies.

Looking ahead to the 1700s

Mathematics in the following century would be dominated by calculus. Newton and Leibniz's creation was to occupy the minds of the best and brightest mathematical minds of the eighteenth century, Leonhard Euler (1707-1783), Joseph Lagrange (1736-1813), and Pierre Simon Laplace (1749-1827) to name a few. Yet, despite the work of these great mathematicians, by the end of the eighteenth century calculus was a field in trouble. While calculus worked, it seemed to have no logical foundation. It would be the mathematicians of the nineteenth century who had to deal with such objections. The eighteenth century also saw important work in other mathematical fields. Advances in analytic geometry, differential geometry, and algebra would all play important roles in the development of modern mathematics.

DAVID TULLOCH