The Emergence of the Calculus

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The Emergence of the Calculus

Overview

The calculus describes a set of powerful analytical techniques, including differentiation and integration, that utilize the concept of a limit in the mathematical description of the properties of functions, especially curves. The formal development of the calculus in the later half of the seventeenth century, primarily through the independent work of English physicist and mathematician Sir Isaac Newton (1642-1727) and German mathematician Gottfried Wilhelm Leibniz (1646-1716), was the crowning mathematical achievement of the Scientific Revolution. The subsequent advancement of the calculus profoundly influenced the course and scope of mathematical and scientific inquiry.

Background

Important mathematical developments that laid the foundation for the calculus of Newton and Leibniz can be traced back to mathematical techniques first advanced in ancient Greece and Rome. In addition to existing methods to determine the tangent to a circle, the Greek mathematician and inventor Archimedes (c. 290-c. 211 b.c.) developed a technique to determine the tangent to a spiral, an important component of his water screw.

The majority of other ancient fundamental advances ultimately related to the calculus were concerned with techniques that allowed the determination of areas under curves (principally the area and volume of curved shapes). In addition to their mathematical utility, these advancements both reflected and challenged prevailing philosophical notions regarding the concept of infinitely divisible time and space. Two centuries before the work of Archimedes, Greek philosopher and mathematician Zeno of Elea (c. 495-c. 430 b.c.) constructed a set of paradoxes that were fundamentally important in the development of mathematics, logic, and scientific thought. Zeno's paradoxes reflected the idea that space and time could be infinitely subdivided into smaller and smaller portions, and these paradoxes remained mathematically unsolvable in practical terms until the concepts of continuity and limits were introduced.

Archimedes also built upon the work of Greek astronomer, philosopher, and mathematician Eudoxus of Cnidus (c. 400-c. 350 b.c.). Eudoxus developed a method of exhaustion that could be used to calculate the area and volume under curves and of solids (e.g. the cone and pyramid) that relied on the concept that time, space, and matter could be divided into infinitesimally small portions. Moreover, the method of exhaustion pointed the way toward a primitive geometric form of what in calculus terms would become known as integration.

Although other advances by classical mathematicians also set the intellectual stage for the ultimate development of calculus during the Scientific Revolution, it is apparent that ancient Greek mathematicians failed to find a common link between problems dedicated to finding the area under curves and to the problems requiring the determination of a tangent. That these process are actually the inverse of each other became the fundamental theorem of the calculus eventually developed by Newton and Leibniz.

During the Medieval Age philosophers and mathematicians continued to ponder questions relating to the movement of objects. These inquiries led to early efforts to plot functions relating such variables as time and velocity. In particular, the work of French Roman Catholic bishop Nicholas Oresme (c. 1325-1382) proved an important milestone in the development of kinematics (the study of motion) and geometry, especially Oresme's proof of the Merton theorem, which allowed for the calculation of the distance traveled by an object when uniformly accelerated (e.g. by acceleration due to gravity). Oresme's proof established that the sum of the distance traversed (i.e. the area under the velocity curve) by a body with variable velocity was the same as that traversed by a body with a uniform velocity equal to the middle instant of whatever period was measured. In other words, the area under the curve was a sum of all distances covered by a series of instantaneous velocities. This work was to prove indispensable to the quantification of parabolic motion by Italian astronomer and physicist Galileo Galilei (1564-1642) and later influenced Newton's development of differentiation techniques.

During the Renaissance in Western Europe, a rediscovery of ancient Greek and Roman mathematics spurred the increased use of mathematical symbols, especially to denote algebraic processes. The rise in symbolism also allowed the development and increased application of the techniques of analytical geometry principally advanced by French philosopher and mathematician René Descartes (1596-1650) and French mathematician Pierre de Fermat (1601-1665). Beyond the practical utility of establishing that algebraic equations corresponded to curves, the work of Descartes and Fermat laid the geometrical basis for calculus. In fact, Fermat's methodologies included concepts related to, and to the determination of, minimums and maximums for functions that are mirrored in modern mathematical methodology (e.g. setting the derivative of a function to zero). Both Newton and Leibniz were to rely heavily on the use of Cartesian algebra in the development of their respective calculus techniques.

Although many of the fundamental elements for the calculus were in place, the recognition of the fundamental theorem relating differentiation and integration as inverse processes continued to elude mathematicians and scientists. Part of the difficulty related to a lingering philosophical resistance toward the philosophical ramifications of the limit and the infinitesimal.

Accordingly, in one sense the genius of Newton and Leibniz lay in their ability to put aside the philosophical and theological ramifications of the utilization of the infinitesimal to develop a very practical branch of mathematics. Neither Newton or Leibniz gave serious address to the deeper philosophical issues regarding limits and infinitesimals in their publications on technique. In this regard Newton and Leibniz worked in the spirit of empiricism that grew during throughout the Scientific Revolution.

Impact

Although largely carried over into the eighteenth century, and affecting more the elaboration of the calculus rather than the initial development of the techniques, the acrimonious controversy surrounding whether Newton or Leibniz deserved credit for the development of the calculus was grounded in the actions of both men during the late seventeenth century. There is clear and unambiguous historical documentation that establishes that Newton's unpublished formulations of the techniques of calculus came two decades prior to Leibniz's preemptory publications in 1684 and 1686. Although their correspondence (mostly through a third party) makes Leibniz's path to calculus less clear, scholars generally conclude that Leibniz independently developed his own set of the techniques. Although the mathematical outcomes were identical, the differences in symbolism and nomenclature used by Newton and Leibniz evidence independent development. The dispute regarding credit for the calculus quickly evolved into a feud that drew in supporters along blatantly nationalistic lines that subsequently divided English mathematicians who relied on Newton's "fluxions" from mathematicians in Europe who followed the conventions established by Leibniz. In particular, the publications and symbolism of Leibniz greatly influenced the mathematical work of two brothers, Swiss mathematicians Jakob Bernoulli (1654-1750) and Johann Bernoulli (1667-1748).

Working separately, the Bernoulli brothers widely applied the calculus. Johann Bernoulli was the first to apply the term integral, and he spread Leibniz-based methodologies and nomenclature among influential French mathematicians. Jakob Bernoulli incorporated the calculus into his work regarding probability and statistics. In addition to the greater utility and translatability of Leibniz's notational systems, Bernoulli's spread of Leibniz's notation is one of the major reasons modern calculus much more closely resembles the original notations set forth by Leibniz than those of Newton.

In modern mathematical texts, Newton is often cited as the inventor of the differential calculus, and Leibniz is given credit for the development of integration. Both men, however, developed techniques for differentiation and integration. Accordingly, any awarding of credit for the development of the respective techniques more properly recognizes the varying mathematical and philosophical emphasis exhibited by Newton and Leibniz.

Following Leibniz's publication, Newton published his own work in his Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), Opticks, and in John Wallis's works. Newton's writing carefully reflected ancient Greek philosophical ideas. In fact, the concept of the limit in Principia is defined as a "ratio of evanescent quantities."

The publications of Newton and Leibniz emphasized the utilitarian aspects of calculus. Nevertheless, the respective development of nomenclature and techniques by Newton and Leibniz also mirrored their own philosophical leanings. Newton developed the calculus as a practical tool by which to attack problems regarding the effects of gravity and as an accurate calculator of planetary motion. Accordingly, Newton emphasized analysis, and his mathematical methods attempted to describe the effects of forces on motion in terms of infinitesimal changes with respect to time. Leibniz's calculus sought to derive integral methods by which discrete infinitesimal units could be summed to yield the area of a larger shape. Thus, he derived inspiration from the idea that incorporeal entities were the driving basis of existence and change in the larger world experienced by mankind.

Although philosophical debates regarding the underpinning of the calculus simmered, the first texts in calculus were able to appear before the end of the seventeenth century. Despite the fact that modern scholars now credit much of the content of the text to Johann Bernoulli, the first textbook in calculus was published by French mathematician Guillaume François Antoine l'Hospital (1661-1704). L'Hospital's Analyse des Infiniment Petits four l'intelligence des lignes courbes first appeared in 1696 and helped bring the calculus into wider use throughout continental Europe.

Although the philosophical debate on the logical consistency of the calculus would gain importance in the eighteenth century, it is a telling note of the intellectual climate of the time that within a few decades the calculus was quickly embraced and applied to a wide range of practical problems in physics, astronomy, and mathematics. Why the calculus worked, however, remained a vexing question that would eventually open calculus to attack on philosophical and theological grounds. This school of critics—eventually to be led in eighteenth-century England by the Anglican Bishop George Berkeley (1685-1753)—argued that the fundamental theorems of calculus derived from logical fallacies and that the great accuracy of calculus actually resulted from the mutual cancellation of fundamental reasoning errors.

Within a century the attacks upon calculus, because they resulted in an increased rigor in mathematical analysis, ultimately proved beneficial to the development of modern mathematics. The practical genius of Newton and Leibniz, grounded in their respective recognition of the fundamental theorem of calculus (that differentiation and integration are inverse processes), endured to provide a powerful analytical tool that fueled Enlightenment Age inquiries into the natural world and offered the mathematical basis for the development of modern science.

K. LEE LERNER