Mathematics and the Industrial Revolution

Summary: New energy sources, management styles, and more intensely divided labor revolutionized manufacturing and technology.

The term “Industrial Revolution” refers to the great social transformation, beginning in the mid-eighteenth century, during which manufacturing replaced agriculture as the center of productive activity. This transition had profound implications for economic and political institutions and international relations, as well as for the landscape and environment, family, education, and culture. Its two main dimensions were technological innovation and the social organization of production. The Industrial Revolution was facilitated by the increased use of realistic perspectives in painting and drawing that flourished in the Renaissance, as well as by the invention of the printing press in the fifteenth century, which spurred intellectual growth in many fields, including mathematics. These developments allowed for better visual representation and distribution of mathematical ideas and inventions to a much broader audience than the older master-apprentice models.

Characteristics

Some historians question the use of the term “revolution,” since these developments indisputably occurred incrementally over a period of a century or more. Nonetheless, their cumulative impact dramatically changed virtually every aspect of life, first in Great Britain and eventually worldwide. New technologies both drew on existing mathematics and prompted its further development. New institutions of intellectual life also fostered the emergence of increasingly abstract mathematics.

The key technological feature of the Industrial Revolution was the application of new sources of power: first the steam engine (late eighteenth century), and later electricity and the internal combustion engine (late nineteenth century). As the Industrial Revolution spread in the late twentieth century, nuclear energy and emerging “green energy” sources have been developed. A crucial problem of the early Industrial Revolution was the means of transmitting power from the steam engine to the machines used in production itself. This problem gave rise to the mathematical theory of linkages.

Equally important to the Industrial Revolution was the large-scale organization of labor. In England, the Enclosure Acts (1760–1845) forced small farmers into urban areas, while vagrancy laws, poor laws, and workhouses (places where those who were not able to support themselves could seek shelter and employment) instilled labor discipline. A large labor pool was thus created for the new factories. Market competition impelled factory owners to use the cheapest possible labor—children as young as 5 as well as adult women and men—and to maximize profits by extending the working day to 14 hours or more per day, seven days per week.

The vastly larger scale of production made possible by mechanization and the steam engine created a qualitatively distinct industrial organization of labor. It intensified the division of labor, de-skilling some jobs and creating new forms of specialization.

The Industrial Revolution therefore meant profound changes in work, residence patterns, family relations, and urban life. This in turn sparked interest in social statistics. Edwin Chadwick (1800–1890) and Friedrich Engels (1820–1895) pioneered the use of quantitative measures to describe social problems. Belgian mathematician Adolphe Quetelet applied the statistical techniques previously used in astronomy to social problems, further developing them and helping to institutionalize the discipline of statistics.

James Watt and the Steam Engine

James Watt (1736–1819), the grandson of a mathematics teacher, possessed the combination of manual dexterity and an aptitude for mathematics. He trained as a maker of mathematical instruments, securing a position at the University of Glasgow, a major center of the British Industrial Revolution, where he first encountered the inventive yet inefficient Newcomen steam engine. While the Newcomen engine served to pump water from coal mines, Watt’s improvements turned the steam engine into a practical means of supplying power to factories and of transporting manufactured goods to market.

James Watt’s parallel motion mechanism (1804), in particular, allowed the force of an engine to act in both push and pull directions, converting rotary motion to linear motion. This provided an empirical, though imprecise, solution to the geometrical problem of constructing a straight line without tracing a straight line. In Euclidean geometry, it is axiomatic that a straight line can be produced, but—in contrast to the circle—no method existed to do so.

Following Watt, a spatial linkage that traced exact straight lines was created by mathematician Pierre-Frederic Sarrus in 1853 and proved geometrically by Charles-Nicolas Peaucellier in 1864. The mathematical theory of linkages was further developed by Pafnuty Chebyshev, James Joseph Sylvester, Alfred Kempe, and Arthur Cayley.

Mathematics and the Industrial Revolution

The late eighteenth and early nineteenth centuries were extremely fruitful in the development of modern mathematics. However, the connections between this work and the Industrial Revolution are mainly indirect.

A notable exception was Charles Babbage (1791–1891) and his work on some of the earliest computing machines. Numerical tables used in applied mathematics were calculated by hand and often contained many errors. Babbage sought to replace these human “computers” with machines, as so many manufacturing jobs were being mechanized. He began work on his first “difference engine” in 1822, moved on to a programmable “analytical engine,” and continued experimenting with steam-powered computing machines for much of the rest of his life. Ada Lovelace, generally credited as the first computer programmer, created a program that could have run on Babbage’s machine, had it been built.

Some technical problems that arose in connection with the Industrial Revolution proved amenable to solution via abstract mathematics developed in other contexts. For example, analysis of electrical circuits, waves, and oscillations is simplified by using complex numbers, originally explored in relation to the solution of algebraic equations.

In France, the École Polytechnique, founded by mathematicians Lazare Carnot and Gaspard Monge in 1794 to train military engineers, supplied technical training and expertise for emerging French industries. Its faculty, students, and examiners included many of the most influential French mathematicians of the nineteenth century, and its textbooks, such as the calculus texts of Adrien-Marie Legendre and Sylvestre-Francois Lecroix, influenced mathematics instruction internationally.

Bibliography

Musson, A. E., and Eric Robinson. Science and Technology in the Industrial Revolution. New York: Gordon & Breach, 1989.

Sangwin, Christopher. “Revisiting James Watt’s Linkage with Implicit Functions and Modern Techniques.” Mathematics Magazine 81, no. 2 (2008).

Weightman, Gavin. The Industrial Revolutionaries: The Making of the Modern World 1776–1914. New York: Grove Press, 2010.