Development of the Calculus

Locale England and continental Europe

Date 1615-1696

Sir Isaac Newton and Gottfried Wilhelm Leibniz, working independently and building on the work of predecessors, created the calculus, a new branch of mathematics.

Key Figures

• Sir Isaac Newton (1642-1727), English natural philosopher and mathematician
• Gottfried Wilhelm Leibniz (1646-1716), German philosopher and mathematician
• John Wallis (1616-1703), English mathematician
• Blaise Pascal (1623-1662), French mathematician and physicist
• Pierre de Fermat (1601-1665), French mathematician
• Bonaventura Cavalieri (1598-1647), Italian Jesuit priest and mathematician
• Johannes Kepler (1571-1630), German astronomer and mathematician
• Isaac Barrow (1630-1677), English mathematician and theologian

Summary of Event

No person, even a genius, creates in a vacuum. Sir Isaac Newton recognized this when he stated, “If I have seen further than other men, it is because I stood on the shoulders of giants.” Even though the creation of the calculus has been associated more closely with Newton than with any other mathematician, his work depended on the contributions of others. That seventeenth century mathematicians had been primed for the calculus is evidenced by its independent discovery by Gottfried Wilhelm Leibniz, who also used the contributions of his mathematical precursors.

It is no accident that the calculus originated during the Scientific Revolution, since the calculus provided scientists with efficacious ways of solving such problems as centers of gravity, instantaneous velocities, and projectile trajectories. The calculus was also different from such previous disciplines as geometry and algebra, since it involved a new operation by which, for example, a circle’s area could be calculated by means of the limit of the areas of inscribed polygons, as their number of sides increased indefinitely.

During the first half of the seventeenth century, several mathematicians devised new methods for determining areas and volumes. For example, Johannes Kepler, stimulated by the problem of discovering the optimum proportions for a wine cask, published in 1615 a treatise in which he used immeasurably minute quantities, or “infinitesimals,” in his volumetric calculations. Some scholars have seen Kepler’s achievement as the inspiration for all later work in the calculus.

In 1635, Bonaventura Cavalieri published a book that challenged Kepler’s in popularity, and some mathematicians attribute the development of the calculus to its appearance. Cavalieri used extremely small segments called “indivisibles” to devise theorems about areas and volumes. Although he compared these indivisibles to pages in a book, he made use of an infinite number of them in solving various problems. Blaise Pascal also used indivisibles in calculating the areas under various curves, but in his method he neglected “differences” of higher order, which some scholars see as the basic principle of the differential calculus. Building on the work of Pascal, his friend Pierre de Fermat formulated an ingenious method for determining the maximum and minimum values of curves.

In England, John Wallis, who had studied the mathematical methods of Fermat and others, attempted to arithmetize the geometric treatment of areas of volumes in his Arithmetica infinitorum (1655; arithmetic of infinitesimals), but Isaac Barrow was critical of Wallis’s work. Barrow favored a geometric approach in determining tangents to curves, and his advocacy of geometry influenced Newton. In studying problems of tangents and quadratures (constructing squares equal in area to a surface), Barrow recognized the basic inverse relationship between differentiations and integrations, but he never generalized his method. This became the principal mathematical achievement of his pupil, Isaac Newton.

According to Newton’s personal testimony, he began the steps that led to his invention of the calculus while he attended Barrow’s lectures at Cambridge University. He was also studying Wallis’s work and the analytic geometry of René Descartes. Because of an outbreak of the bubonic plague, Newton returned to his home in Lincolnshire where he derived a method of using infinite series to express the area of a circle. He also devised a differentiation method based not on ultimately vanishing quantities but on the “fluxion” of a variable. For example, Newton was able to determine instantaneous speeds through fluxions of distance with respect to time. Fluxions, Newton’s name for the calculus, were descriptive of the rates of flow of variable quantities.

During the next four years, Newton made his methods more general through the use of infinite series, and he circulated his discoveries among his friends in a work titled De analysi per aquationes numero terminorum infinitas (1669; on analysis by means of equations with an infinite series of terms), which was not formally published until 1711 (in Analysis per quantitatum series, fluxiones, ad differentias: Cum enumeratione linearum tertii ordinis ). Not only did Newton describe his general method for finding the instantaneous rate of change of one variable with respect to another but he also showed that an area under a curve representing a changing variable could be obtained by reversing the procedure of finding a rate of change. This notion of summations being obtained by reversing differentiation came to be called the fundamental theorem of the calculus. Though Newton’s predecessors had been groping toward it, he was able to understand and use it as a general mathematical truth, which he described fully in Methodus fluxionum et serierum infinitarum (The Method of Fluxions and Infinite Series , 1736), written in 1671 but not published until after Newton’s death. In this work, he called a variable quantity a “fluent” and its rate of change the “fluxion,” and he symbolized the fluxion by a dot over the letter representing the variable.

Newton’s third work on the calculus, Tractatus de quadratura curvarum (treatise on the quadrature of curves), was completed in 1676 but published in 1711 (also part of Analysis per quantitatum series). He began this treatise by stating that, instead of using infinitesimals, he generated lines by the motion of points, angles by the rotation of sides, areas by the motion of lines, and solids by the motion of surfaces. This work exhibits Newton’s mastery of the increasingly sophisticated and powerful methods he had developed. Though he made sparse use of fluxions in his greatest work, Philosophiae naturalis principia mathematica (1687; The Mathematical Principles of Natural Philosophy, 1729, best known as the Principia ), he did offer three ways to interpret his new analysis: using infinitesimals (as he did in De analysi), limits (as he did in De quadratura), and fluxions (as he did in Methodus fluxionum). The Principia was Newton’s last great work as a mathematician.

The chief work on the calculus that rivaled Newton’s work was that of Leibniz. Leibniz’s early mathematical interests were arithmetic and geometry, but in studying the problem of constructing tangents to curves he used a “differential triangle” (used earlier by Pascal, Fermat, and Barrow) to arrive at solutions. He recognized that the ratio of the differences in the horizontal and vertical coordinates of a curve was needed to determine tangents, and by making these differences infinitely small, he could solve these and other problems. He also realized that the operations of summation and determining differences were mutually inverse. In a 1675 manuscript on his “inverse method of tangents,” Leibniz used an integral sign for the sum and “dx” for the difference. With his new notation he was able to show that integration is the inverse of differentiation. Like Newton, Leibniz did not circulate his ideas immediately, but, in 1682, he published his discoveries in a new journal, Acta eruditorum (proceedings of the learned). In this and later articles he presented his methods of determining integrals of algebraic functions and solving differential equations.

Significance

Both Newton and Leibniz saw the calculus as a general method of solving important mathematical problems. Though their methods were essentially equivalent, Newton, the geometer, and Leibniz, the algebraist, developed and justified their discoveries with different arguments. They both reduced such problems as tangents, rates, maxima and minima, and areas to operations of differentiation and integration, but Newton used infinitely small increments in determining fluxions whereas Leibniz dealt with differentials. Scholars have attributed this contrast to Newton’s concern with the physics of motion and Leibniz’s concern with the ultimate constituents of matter.

Initially, no priority debate existed between Newton and Leibniz, both of whom recognized the basic equivalence of their methods. Controversy began when some of Newton’s disciples questioned Leibniz’s originality, with a few going so far as to accuse Leibniz of plagiarism (since Leibniz had seen Newton’s De analysi on a visit to London in 1676). Nationalism played a part in the controversey as well. The English and the Germans desired the glory of the calculus’s discovery for their respective countries. Though the controversy generated many hurt feelings and some unethical behavior on both sides in the seventeenth century, scholars now agree that Newton and Leibniz discovered the calculus independently.

The significance of this priority controversy was not a question of victor and vanquished but the divisions it created between British and Continental mathematicians. The English continued to use Newton’s cumbersome fluxional notation, whereas Continental mathematicians, using Leibniz’s superior formalism, were able to systematize, extend, and make a powerful mathematical discipline of the calculus. Consequently, for the next century, British mathematicians fell behind the mathematicians of Germany, France, and Italy, who were able to develop the calculus into a powerful tool capable of helping mathematicians, physicists, and chemists solve a wide variety of important problems.

Bibliography

Berlinski, David. A Tour of the Calculus. New York: Vintage Books, 1997. This book was written for general readers “who wish to understand the calculus as an achievement in human thought.” Though the author’s emphasis is on the concepts and formalism of the calculus, he also makes use of historical analysis, with many references to Newton and Leibniz. Includes an index.

Boyer, Carl B. The History of the Calculus and Its Conceptual Development. New York: Dover, 1959. This reprint of a classic work makes available for a wide variety of readers an authoritative and comprehensive treatment of the entire history of the calculus. Includes an extensive bibliography and an index.

Cohen, I. Bernard, and George E. Smith, eds. The Cambridge Companion to Newton. New York: Cambridge University Press, 2002. Several prominent scholars examine Newton’s principal achievements. A. Rupert Hall’s “Newton Versus Leibniz” is particularly relevant. Includes a thematic bibliography and an index.