Mathematics

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Mathematics

The field of mathematics made great strides during the Renaissance. Social and economic changes, along with the recovery of ancient Greek texts, helped promote new discoveries in mathematics. These advances, in turn, affected fields such as accounting, art, navigation, and astronomy. The boom in mathematical awareness set the stage for the scientific revolution of the 1600s.

Roots of Renaissance Mathematics. Like many other areas of culture, mathematics benefited from the renewed interest in classical* knowledge during the Renaissance. For example, humanists* of the period explored the ideas of the Greek philosopher Plato and his followers. Their writings promoted the view that mathematics was the key to understanding nature. Other cultural traditions also influenced mathematics during the Renaissance. The modern numbering system, for example, had originated in the Hindu-Arabic world.

The Rise of Algebra. In the early 1300s, mathematicians in Italy wrote a series of practical texts on the Islamic technique known as algebra. These books, intended for use by merchants and accountants, taught the Hindu-Arabic system of numerals. They also introduced new symbols and abbreviations for use in algebra. The texts used algebra to solve problems related to commerce, banking, and weights and measures. These works stimulated an interest in algebra in other countries, including France, England, and Germany. The first German algebra text appeared in 1525, and the royal physician Robert Recorde published the first English algebra in 1557. This book, The Whetstone of Witte, introduced the equal sign. Other now-familiar symbols that first appeared during the Renaissance included +, -, ÷ , <, and >.

The most famous Renaissance algebra was Girolamo Cardano's Great Art, published in 1545. It was the first European text to make major advances beyond earlier Islamic works. Cardano provided solutions to several problems that had frustrated European mathematicians for years. Perhaps more importantly, he inspired a wide range of new problems and topics for exploration. In addition to Great Art, Cardano also published the first treatise* on applying mathematics to games of chance.

In 1591, François Viète of France began to use letters to represent entire classes of numbers in algebraic equations. This change enabled mathematicians to solve equations in general terms. For example, instead of solving the equation "x2 + 5x + 6 = 0," they could write out the general equation "ax2 + bx + c = 0" and solve it for x in terms of a, b, and c. This meant that mathematicians could use the general solution to solve all problems of the same type. Viète's invention made algebra both more abstract and more powerful. Viète also introduced the "let x = the unknown" approach to solving word problems, which is now standard in algebra.

Trigonometry and Geometry. Advances in geometry and trigonometry (a branch of mathematics dealing with the properties of triangles) paved the way for new methods in navigation, mapmaking, and art. In the early 1500s, maps of the seas showed meridians—imagi-nary circles on the Earth's surface that pass through the North and South Poles—as straight lines an equal distance apart. However, the flat maps of the time did not account for the fact that the Earth is round. A sailor who used a meridian on such a map as a guideline for plotting a straight course would travel in a spiral that ended up at the North or South Pole.

Gerard Mercator, a maker of mathematical instruments in the Flemish* city of Louvain, created a map that solved this problem. Mercator's map projected the Earth's surface onto the surface of a cylinder. This kind of projection involves, in effect, wrapping a flat sheet of paper around a sphere, then tracing a straight line out from every point on the sphere's surface to a matching point on the paper. The unrolled sheet then becomes a flat map.

Mercator's map successfully represented a constant course as a straight line. The map worked well if one stayed on the same latitude—that is, the same distance north or south of the equator. However, it did not account for the fact that the distances between meridians grow shorter as one moves farther from the equator. In 1599, the English sailor and mathematician Edward Wright solved the problem by working out the trigonometric theory behind Mercator's projection. His work enabled navigators to use Mecator's map to calculate distances between points anywhere on Earth.

Renaissance artists used geometry to create perspective—the illusion of three-dimensional space on the flat surface of a drawing or painting. The rediscovery of a work on optics by the ancient Greek mathematician Euclid led several artists, including the Italian Piero della Francesca, to do original work in geometry. In his treatise On Perspective in Painting, Piero explained how to show objects in three dimensions from any point on the plane of the picture. Albrecht DÜrer, one of the best-known artists in Europe, wrote the first geometric text in German.

The field of astronomy also benefited from work in mathematics. Johann Müller of Königsberg, better known as Regiomontanus, wrote the first complete European work on trigonometry in 1464. His book, On Triangles of All Kinds, influenced Georg Joachim Rheticus, a student of the Polish astronomer Nicolaus Copernicus. Copernicus drew on Rheticus's knowledge of this subject to work out the details of his new idea that the Earth revolved around the Sun, rather than the other way around.

As astronomy and navigation became more complex, so did the calculations needed to chart the stars and plot courses at sea. In the early 1600s, Scotland's John Napier developed the system of logarithms to make calculations in trigonometry easier. A logarithm is the power to which a base, such as 10, must be raised to produce a given number. It is possible to find the product of any two numbers by adding their logarithms. The slide rule, a simple computer for adding logarithms, put fast calculation within the reach of everyone who worked with numbers.

Mathematics and the Future of Europe. Mathematicians in the Middle Ages had begun to work on problems whose answers did not have definite limits. A number of Renaissance scholars continued and refined this work, resulting in major advances in mathematics and science.

For example, the German astronomer Johannes Kepler became curious about how wine merchants estimated the volumes of the barrels they used to store wine. He invented a system that involved thinking of the area of a curved surface as the sum of an infinite number of straight lines. This technique gave rise to the field of analytic geometry, created by French mathematicians Pierre de Fermat and René Descartes in the early 1600s. In time, these new methods led to the development of calculus, which revolutionized science and mathematics in Europe.

Along with its practical benefits, mathematics had a broader cultural meaning. It taught European thinkers the ideas of symbolic reasoning and the power of abstract ideas. It also showed that mathematical analysis could solve many types of real-world problems. The advances in mathematics during the Renaissance reflected a general sense of intellectual power and progress.

* classical: in the tradition of ancient Greece and Rome
* humanist: Renaissance expert in the humanities (the languages, literature, history, and speech and writing techniques of ancient Greece and Rome)

* treatise: long, detailed essay

* Flemish: relating to Flanders, a region along the coasts of present-day Belgium, France, and the Netherlands

Top Secret Equations

Italian mathematicians sometimes took part in public contests to win money, honor, and teaching positions. In 1535, two mathematicians competed to solve a series of problems called cubics. Girolamo Cardano later persuaded the winner of this contest to tell him his method, which he promised not to reveal. However, Cardano then published the solution in his book Great Art. The outraged scholar responded by publishing the text of the oath Cardano had broken. This story reveals how Renaissance society often encouraged scholars to keep their findings secret—an idea quite foreign to modern mathematicians.

Renaissance: An Encyclopedia for Students