The European Mathematical Awakening

A Journey Through the History of Mathematics from 1000 to 1800

Frank J. Swetz

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486310275
• Topic: Mathématiques
• Subtopic: Histoire et philosophie des mathématiques

Absorbing and entertaining, these thirty-two articles by distinguished educators offer a reader-friendly introduction to the history of mathematics. The newly corrected and updated essays cover eight centuries of discoveries, ranging from the medieval practice of finger calculus to the pioneering work of Leonhard Euler.
Fascinating topics include the geometry behind the windows of Gothic churches, the development of complex numbers, the evolution of algebraic symbolism, and mathematical considerations on the trajectory of a cannon ball. Profiles of historic figures include Leonardo Fibonacci, Johannes Kepler, Isaac Newton, Galileo, the Bernoulli family, and other well- and less-known personalities, including mathematicians of the French Revolution and women in mathematics. Suitable for readers with no background in mathematics, this volume offers an excellent guide for high school students and college undergraduates as well as anyone with an interest in the history of mathematics.

Perspective: The European Mathematical Awakening

By the beginnings of the 11th century, Europe had survived a series of cataclysmic events: barbarian invasions; plagues and crop failures, to emerge from a period of intellectual and political stagnation known as the “Dark Ages.” Advances in agriculture and animal husbandry provided a better and larger food supply. Improved nutrition stimulated population increase. Land reclamation projects were undertaken and new towns founded. The harnessing of wind and water power made many of life’s daily chores easier. Opportunities in villages and towns attracted freemen, craftsmen and artisans. Road systems were improved. Trade and commerce increased. Cities grew and prospered. In particular, the Italian maritime city states of Pisa, Venice and Genoa benefited from the weakening Islamic domination of the Mediterranean to become trade entrepôts for the goods and commodities of the Levant. Merchants from these cities established trading houses abroad, conducting business at the sources of their imports. Interacting with local merchants, they learned their customs and habits and brought much of this new knowledge back to Europe. Civic authority and structure were resurrected and strengthened throughout Europe, giving rise to regional and local identities, beginnings of a sense of nationalism. Trade guilds provided a united voice for some skilled working classes. These new institutions of identity and wealth gave rise to political power, which would challenge and modify existing sources of authority. Since the fall of the Roman Empire, the dominant encompassing political, as well as spiritual, authority in Europe was the Catholic Church. The Church was primarily concerned with otherworldly matters, and initially viewed inherited Greek scientific and mathematical theories as suspect pagan knowledge. However, the study of mathematics was formally sanctioned by St. Augustine (ca.400) as worthy of Christian involvement; still, the Catholic Church’s actual interest with mathematics was minimal, limited primarily to the determination of the Church calendar based on a correct dating of Easter. Handbooks called computi were written to assist in this task. A few churchmen pursued mathematics for its intrinsic and classical values. One such scholar was the French monk, Gerbert of Aurillac (ca. 950–1003), who eventually became Pope Sylvester II in the year 999. Gerbert sought out existing Arabic sources of Euclid’s Elements to compile a practical geometry text, employed little known Hindu Arabic numerals and improved counting table computing techniques. The 11th century also witnessed a revitalization and reform of the Church’s monastic movement, which resulted in a widespread establishment of new monastic centers. These centers also helped to foster a sense of regional coherence and also contributed to an intellectual revival by establishing libraries and schools. Monks labored in scriptoria to copy and preserve extant antique works. Cathedral and monastic schools became available for the education of youth.

The new sources of political power began to assert themselves and challenge the overriding power of the Catholic Church. The church in promoting other worldliness, discouraged intellectual curiosity of the physical world and disdained the accumulation and manipulation of wealth. The rising climates of commercial expansionism and humanistic inquiry found themselves in direct conflict with these beliefs. Scholars now began to take a closer look at the world around them and tried to understand the physical forces that controlled nature and human existence. Mathematics became a primary tool in these quests of understanding.

European merchants in their travels abroad were avid observers of foreign practices and customs that would improve their profits margins. Such practices would be brought back to Europe and adapted or refined to suit the local commercial milieu. Leonardo of Pisa (Fibonacci), (ca.1175-1250), a member of a prominent Pisan merchant family, worked in their trade colony in Bougie on the coast of North Africa. Leonardo studied mathematics under the tutelage of Arab instructors. He learned a new set of numerals, said to have originated in India. Accompanying these numerals were computing schemes, algorisms that could be carried out with pencil and paper; freeing problem solvers from the labors of a computing table. Leonardo published his findings on the Hindu Arabic numerals for a European reading audience in Liber Abaci (1202). Eventually, this new system of arithmetic became popular with the European merchant community replacing the use of the cumbersome counting table and the figura imperiale, Roman numerals. Soon, special teachers called reckoning masters: in Italian, maestri d’abbaco; in German, Rechenmeister, taught this new form of computation to the merchant community and paying students. Adam Riese (1492-1559) became a well respect member of this movement. Books and manuscripts called abaci and practicae appeared promoting the new arithmetic. The advent of printing with movable type greatly helped to disseminate this new knowledge. The first printed European arithmetic book appeared in 1478 in Treviso, a small commercial town, outside of Venice. It is called simply the “Treviso Arithmetic.” Now, written or printed calculations allowed for retrospection, analysis and the possible perception of patterns and structure in mathematics. Also at this time, the classics of Greek scholarship, preserved in Islamic libraries were reintroduced into Europe, translated into local languages, read and studied. Printed copies of Euclid’s Elements were particularly in demand. Soon, this new instruction on arithmetic moved from the limited tutelage of the reckoning masters to the quadrivium of the monastic schools and to the newly founded universities or guilds of scholars. The use of numbers and calculation became available to a wide segment of the population and with this knowledge came an increased awareness of the usefulness and power of mathematics; power to make a livelihood and power to better understand the world.

The new numerals assisted in communication involving mathematics. Repeated phrases or operations printed in arithmetic manuals lent themselves to standardization, and abbreviation. Now mathematical problems that had for so long been expressed in sometime obscure rhetorical form, could be understood as a form of relationships between numbers. Soon there were special symbols adopted to represent the basic operations of addition and subtraction, multiplication, division and “equals”. Often, in written or printed statement, one word or phrase was repeated many times. For example, in Italian the word for “unknown” or “thing” was “cosa”; on a single typeset page of arithmetic book, cosa could appear a dozen times during an explanation. Such repetition, prompted the mathematician Luca Pacioli (ca. 1445-1509) to use the abbreviation “co” for cosa in writing his Summa de arithmetica, geometrica, proportioni et proportionalita published in 1494. As a result of such innovations, rhetorical algebra evolved into symbolic algebra. Pacioli’s Summa was believed to be the compendium of 15th century mathematical knowledge. In his closing comments, the author noted that obtaining an algebraic solution for the cubic equation was impossible. Within forty years of this statement, solution techniques for the cubic equation were developed and perfected by such mathematicians as Scipione del Ferro, Nicolo Tartaglia, Girolamo Cardano and Rafael Bombelli. In the work done to obtain the solution process for cubic equations, a foundation was laid for a development of a theory of equations. Algebraic manipulation now encompassed the use of imaginary numbers. Improved astronomical techniques and measurement prompted a maturing of trigonometry and highlighted urgency for improved computational efficiency and mathematical accuracy. In part to meet these needs, a variety of concrete computing devices were invented and computational capacities were strengthened by Simon Stevin’s systematization of decimal fractions (1585) and the appearance of John Napier’s logarithms (1614). New scientific theories were proposed and mathematically investigated by a series of individuals. Nicolo Tartaglia (1499-1557) and Galileo Galilei (1564-1642) explored the forces acting on a cannonball in motion. René Descartes (1596-1650) sought to unravel the workings of a rainbow and the forces propelling the planets through space. Greek number mysticism was replaced by a more mathematically based theory of numbers. Communities of natural scientist/mathematicians working in unison or consecutively, sought to unravel the mysteries of nature. National academies and societies were founded to facilitate cooperation and the exchange of scientific information. The British Royal Society opened its doors in 1660 and the French Academy in 1666. The trajectories of the planets were explored by Nicolo Copernicus (1473-1543), Johannes Keple...