History of mathematics VI

Medieval European mathematics (c. 500–1400)

Medieval European interest in mathematics was driven by concerns quite different from those of modern mathematicians. One driving element was the belief that mathematics provided the key to understanding the created order of nature, frequently justified by Plato's Timaeus and the apocryphal biblical passage (in the Book of Wisdom) that God had ordered all things in measure, and number, and weight^[44].

[edit] Early Middle Ages (c. 500–1100)

Boethius provided a place for mathematics in the curriculum when he coined the term quadrivium to describe the study of arithmetic, geometry, astronomy, and music. He wrote De institutione arithmetica, a free translation from the Greek of Nicomachus's Introduction to Arithmetic; De institutione musica, also derived from Greek sources; and a series of excerpts from Euclid's Elements. His works were theoretical, rather than practical, and were the basis of mathematical study until the recovery of Greek and Arabic mathematical works.^[45][46]

[edit] Rebirth of mathematics in Europe (1100–1400)

In the 12th century, European scholars traveled to Spain and Sicily seeking scientific Arabic texts, including al-Khwarizmi's The Compendious Book on Calculation by Completion and Balancing, translated into Latin by Robert of Chester, and the complete text of Euclid's Elements, translated in various versions by Adelard of Bath, Herman of Carinthia, and Gerard of Cremona.^[47][48]

These new sources sparked a renewal of mathematics. Fibonacci, writing in the Liber Abaci, in 1202 and updated in 1254, produced the first significant mathematics in Europe since the time of Eratosthenes, a gap of more than a thousand years. The work introduced Hindu-Arabic numerals to Europe, and discussed many other mathematical problems.

The fourteenth century saw the development of new mathematical concepts to investigate a wide range of problems.^[49] One important contribution was development of mathematics of local motion.

Thomas Bradwardine proposed that speed (V) increases in arithmetic proportion as the ratio of force (F) to resistance (R) increases in geometric proportion. Bradwardine expressed this by a series of specific examples, but although the logarithm had not yet been conceived, we can express his conclusion anachronistically by writing: V = log (F/R).^[50] Bradwardine's analysis is an example of transferring a mathematical technique used by al-Kindi and Arnald of Villanova to quantify the nature of compound medicines to a different physical problem.^[51]

One of the 14th-century Oxford Calculators, William Heytesbury, lacking differential calculus and the concept of limits, proposed to measure instantaneous speed "by the path that would be described by [a body] if... it were moved uniformly at the same degree of speed with which it is moved in that given instant".^[52]

Heytesbury and others mathematically determined the distance covered by a body undergoing uniformly accelerated motion (today solved by integration), stating that "a moving body uniformly acquiring or losing that increment [of speed] will traverse in some given time a [distance] completely equal to that which it would traverse if it were moving continuously through the same time with the mean degree [of speed]".^[53]

Nicole Oresme at the University of Paris and the Italian Giovanni di Casali independently provided graphical demonstrations of this relationship, asserting that the area under the line depicting the constant acceleration, represented the total distance traveled.^[54] In a later mathematical commentary on Euclid's Elements, Oresme made a more detailed general analysis in which he demonstrated that a body will acquire in each successive increment of time an increment of any quality that increases as the odd numbers. Since Euclid had demonstrated the sum of the odd numbers are the square numbers, the total quality acquired by the body increases as the square of the time.^[55]

[edit] Early modern European mathematics (c. 1400–1600)

In Europe at the dawn of the Renaissance, mathematics was still limited by the cumbersome notation using Roman numerals and expressing relationships using words, rather than symbols: there was no plus sign, no equal sign, and no use of x as an unknown.^{[citation needed]}

In 16th century European mathematicians began to make advances without precedent anywhere in the world, so far as is known today. The first of these was the general solution of cubic equations, generally credited to Scipione del Ferro c. 1510, but first published by Johannes Petreius in Nuremberg in Gerolamo Cardano's Ars magna, which also included the solution of the general quartic equation from Cardano's student Lodovico Ferrari.

From this point on, mathematical developments came swiftly, contributing to and benefiting from contemporary advances in the physical sciences. This progress was greatly aided by advances in printing. The earliest mathematical books printed were Peurbach's Theoricae nova planetarum (1472), followed by a book on commercial arithmetic, the Treviso Arithmetic (1478), and then the first extant book on mathematics, Euclid's Elements, printed and published by Ratdolt in 1482.

Driven by the demands of navigation and the growing need for accurate maps of large areas, trigonometry grew to be a major branch of mathematics. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595. Regiomontanus's table of sines and cosines was published in 1533.^[56]

By century's end, thanks to Regiomontanus (1436–76) and Simon Stevin (1548–1620), among others, mathematics was written using Hindu-Arabic numerals and in a form not too different from the notation used today.

[edit] 17th century

The 17th century saw an unprecedented explosion of mathematical and scientific ideas across Europe. Galileo, an Italian, observed the moons of Jupiter in orbit about that planet, using a telescope based on a toy imported from Holland. Tycho Brahe, a Dane, had gathered an enormous quantity of mathematical data describing the positions of the planets in the sky. His student, Johannes Kepler, a German, began to work with this data. In part because he wanted to help Kepler in his calculations, John Napier, in Scotland, was the first to investigate natural logarithms. Kepler succeeded in formulating mathematical laws of planetary motion. The analytic geometry developed by René Descartes (1596–1650), a French mathematician and philosopher, allowed those orbits to be plotted on a graph, in Cartesian coordinates.

Building on earlier work by many predecessors, Isaac Newton, an Englishman, discovered the laws of physics explaining Kepler's Laws, and brought together the concepts now known as calculus. Independently, Gottfried Wilhelm Leibniz, in Germany, developed calculus and much of the calculus notation still in use today. Science and mathematics had become an international endeavor, which would soon spread over the entire world.^[57]

In addition to the application of mathematics to the studies of the heavens, applied mathematics began to expand into new areas, with the correspondence of Pierre de Fermat and Blaise Pascal. Pascal and Fermat set the groundwork for the investigations of probability theory and the corresponding rules of combinatorics in their discussions over a game of gambling. Pascal, with his wager, attempted to use the newly developing probability theory to argue for a life devoted to religion, on the grounds that even if the probability of success was small, the rewards were infinite. In some sense, this foreshadowed the development of utility theory in the 18th–19th century.