# Overview: Mathematics 1700-1799

# Overview: Mathematics 1700-1799

Following on the resounding successes of the preceding century, eighteenth-century mathematics not only continued to break important new ground, but also paused to consolidate the gains made by Isaac Newton (1642-1727) and Gottfried Leibniz (1646-1716) through their invention of the calculus. Leonhard Euler (1707-1783), one of the greatest mathematicians of all time, helped to standardize and formalize mathematical notation, while also making important contributions to virtually every branch of the discipline. At the same time, mathematics education was revolutionized, resulting in a more standard approach to teaching mathematics to students at all levels and adding to the mathematical sophistication of the population who attended school. Add to these the development of fundamental tools, such as probability and statistical theory, imaginary numbers, and the continued development of algebraic tools, and it is apparent that the eighteenth century was a time of great mathematical progress, setting the stage for virtually all subsequent mathematical advances that were to follow in the next two centuries.

*Eighteenth-Century Society*

The eighteenth century was a century of war. As it opened Russia was at war with the Swedes, Saxony had invaded Livonia, European powers were fighting to determine who would become the King of Spain, and fighting in Europe, Asia, and the Americas continued through most of the century. The eighteenth century was also a century of the arts, with Bach, Handel, Haydn, Mozart, and other great composers writing some of the most famous and beautiful music ever heard. And the eighteenth century was a century of revolution. The philosophical principles of the Enlightenment inspired the American colonies to rebel against their mother country, while Russian, English, French, and Dutch citizens also asserted their rights to self-government at one time or another.

In and around the political upheavals, the Industrial Revolution was taking place. As the growth of steam power began to replace humans and animals for many laborious tasks, the mechanization of industry and society began to take hold. People began to worry about machines taking the place of humans, concerned that they may be without work because of this. At the same time, engineers continued to develop evermore sophisticated and elaborate machines, designed to do an increasing number of jobs. As these events took shape, the role of science in general, and mathematics in particular, became increasingly important to governments, the educated population, and society in general.

*Eighteenth-Century Mathematics*

At the dawn of the eighteenth century mathematics was in the process of taking its place as a formal discipline. This paralleled the emergence of science in general as a field in its own right, worthy of full-time pursuit at a professional level by highly trained practitioners. This recognition owed much to the successes of Newton, Leibniz, the Bernoulli family, and others who showed that science and mathematics were interwoven and that success in these fields could be a source of national pride. In fact, Newton is often considered the first scientist to win popular recognition during his lifetime, even though his discoveries were not widely understood by the great majority of the population. It is entirely possible that this very complexity also helped convince the general population of the legitimacy of science as a career, simply because so much study was necessary to understand the fundamental tools, such as the calculus.

One result of this recognition was the publication of mathematics textbooks. In previous centuries, many mathematicians viewed their craft as proprietary, hiding their discoveries from each other and from other scientists. With the growing importance of mathematics, however, it became apparent that this was not only unnecessary, but counterproductive. Some of the first mathematics textbooks were written in the eighteenth century, making it possible to teach advanced mathematics, in addition to basic arithmetic, in schools. This, in turn, helped turn out a more mathematically sophisticated population, at least among those who attended university classes.

Following these successes, it should be no surprise that scientists and mathematicians became increasingly well known throughout Europe. At times, in fact, mathematicians were almost placed on exhibit by royal courts. Euler was recruited by both Frederick the Great in Berlin and by Catherine the Great of Russia. Members of the Bernoulli family were also recruited by both of these leaders, and Jean d'Alembert (1717-1783) was influential in French mathematics as well as French politics.

At the same time, nations were becoming increasingly dependent on science and engineering, both of which required the development and use of increasingly sophisticated mathematical tools. Newton and Leibniz had taken the first steps to show that mathematics could reveal deep truths about the universe in which we live; Joseph Lagrange (1736-1813), Pierre Laplace (1749-1827), the Bernoulli family, Carl Gauss (1777-1855), and others followed this example in revealing the fundamental connections between mathematics and the physical world. And, as the Industrial Revolution progressed, it became obvious that the new machines upon which a nation's industry, economy, and military strength came to depend required engineers to be trained in the new mathematical tools. As the machines became more complex, they pushed the technology of the day to its limits, requiring mathematically trained engineers to carefully design improvements rather than simply building something that seemed like it should work. By century's end even the navy, often the bastion of traditional thinking, was training its captains to be "scientific sailors," using mathematics and physics to help plot courses, determine a ship's position, shoot cannons, and determine how to arrange the forces on a ship to maximize its sailing characteristics.

Leaving all social and technological impacts aside, the field of mathematics itself made giant strides during the eighteenth century. In one sense, much of eighteenth-century mathematics consisted of consolidating the advances in calculus made in the last years of the previous century. In particular, French mathematicians of the eighteenth century made impressive strides in understanding the calculus. Differential equations, infinite series, the wave equation, and the calculus of variations were all introduced during the this time—and all proved important to other fields as well as mathematics.

Outside of France, most mathematicians worked in the shadows cast by Euler and the members of the Bernoulli family. Both Euler and the Bernoullis contributed in so many areas that even a simple summary could fill an entire book. Indeed, Euler and the Bernoulli family contributed to, originated, or redefined virtually every major branch of mathematics at that time, and much of their work continues to be important today. Euler alone authored or co-authored over 500 books and scientific papers during his life, much of it after going blind in both eyes.

Finally, mathematicians began to recognize the importance of a number of mathematical concepts that are today recognized as vital in many fields. For example, they recognized that π is a number that will never terminate or repeat itself, forever ending hopes of "squaring the circle." Another step forward was the recognition that negative numbers can have square roots, and that they can be manipulated in much the same way as real numbers. This gave rise to the field of complex analysis, which today forms the mathematical foundation of much of the field of electronics. And, in the area of mathematical notation, Euler helped to standardize the symbols used to represent various concepts, making it easier for mathematicians to frame their thoughts and to explain these thoughts to each other in an unambiguous manner.

The eighteenth century is said to have come after the "Century of Genius" and before the "Golden Century" of mathematics. This is true to the extent that no discoveries were made to rival those of Newton and Leibniz, while the sheer volume of landmark mathematical work was far short of what was to come. However, eighteenth-century mathematics formed a crucial link between these two centuries. It was a time during which mathematicians began to develop the conceptual tools to take full advantage of the discoveries made during the previous century, and to construct a solid foundation upon which to build during the Golden Age of the nineteenth century.

**P. ANDREW KARAM**

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# Overview: Mathematics 1700-1799

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**Overview: Mathematics 1700-1799**