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The Elaboration of the Calculus

Overview

Many of the most important and influential advances in mathematics during the eighteenth century involved the elaboration of the calculus, a branch of mathematical analysis that describes properties of functions (curves) associated with a limit process. Although the evolution of the techniques included in the calculus spanned the history of mathematics, calculus was formally developed during the last decades of the seventeenth century by English mathematician and physicist Sir Isaac Newton (1643-1727) and, independently, by German mathematician Gottfried Wilhelm von Leibniz (1646-1716). Although the logical underpinnings of calculus were hotly debated, the techniques of calculus were immediately applied to a variety of problems in physics, astronomy, and engineering. By the end of the eighteenth century, calculus had proved a powerful tool that allowed mathematicians and scientists to construct accurate mathematical models of physical phenomena ranging from orbital mechanics to particle dynamics.

Background

Although it is clear that Newton made his discoveries regarding calculus years before Leibniz, most historians of mathematics assert that Leibniz independently developed the techniques, symbolism, and nomenclature reflected in his preemptory publications of the calculus in 1684 and 1686. The controversy regarding credit for the origin of calculus quickly became more than a simple dispute between mathematicians. Supporters of Newton and Leibniz often argued along bitter and blatantly nationalistic lines and the feud itself had a profound influence on the subsequent development of calculus and other branches of mathematical analysis in England and in Continental Europe.

The first texts in calculus actually appeared in the last years of the seventeenth century. 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 both improved and made wide application of calculus. Johann Bernoulli was the first to apply the term integral to a subset of calculus techniques allowing the determination of areas and volumes under a curve. During his travels Johann Bernoulli sparked intense interest in calculus among French mathematicians, and his influence was critical to the widespread use of Leibniz-based methodologies and nomenclature.

Impact

Physicists and mathematicians seized upon the new set of analytical techniques comprising the calculus. Advancements in methodologies usually found quick application and, correspondingly, fruitful results fueled further research and advancements.

Although the philosophical foundations of calculus remained in dispute, the arguments proved no hindrance to the application of calculus to problems of physics. The Bernoulli brothers, for example, quickly recognized the power of the calculus as a set of tools to be applied to a number of problems. Jakob Bernoulli's distribution theorem and theorems of probability and statistics, ultimately of great importance to the development of physics, incorporated calculus techniques. Johann Bernoulli's sons, Nikolaus Bernoulli (1695-1726), Daniel Bernoulli (1700-1782), and Johann Bernoulli II (1710-1790), all made contributions to the calculus. In particular, Johann Bernoulli II used calculus methodologies to develop important formulae regarding the properties of fluids and hydrodynamics.

The application of calculus to probability theory resulted in probability integrals. The refinement made immediate and significant contributions to the advancement of probability theory based on the late seventeenth-century work of French mathematician (and Huguenot) Abraham De Moivre (1667-1754).

English mathematician Brook Taylor (1685-1731) developed what became known as the Taylor expansion theorem and the Taylor series. Taylor's work was subsequently used by Swiss mathematician, Leonard Euler (1707-1783) in the extension of differential calculus and by French mathematician Joseph Louis Lagrange (1736-1813) in the development of his theory of functions.

Scottish mathematician Colin Maclaurin (1698-1746) advanced the expansion of a special case of a Taylor expansion (where x = 0). More importantly, in the face of developing criticism from Irish Bishop George Berkeley (1685-1753) regarding the logic of calculus, Maclaurin set out an important and influential defense of Newtonian fluxions and geometric analysis in his 1742 publication Treatise on fluxions.

The application of the calculus to many areas of math and science was most profoundly influenced by the work of Euler, a student of Johann Bernoulli. Euler was one of the most dedicated and productive mathematicians of the eighteenth century. Based on earlier work done by Newton and Jakob Bernoulli, in 1744 Euler developed an extension of calculus dealing with maxima and minima of definite integrals termed the calculus of variation (variational calculus). Among other applications, variational calculus techniques allow the determination of the shortest distance between two points on curved surfaces.

Euler also dramatically advanced the principle of least action formulated in 1746 by Pierre Louis Moreau de Maupertuis (1698-1759). In general, the principle asserts an economy in nature (i.e., an avoidance in natural systems of unnecessary expenditures of energy). Accordingly, Euler asserted that natural motions must always be such that they make the calculation of a minimum possible (i.e., nature always points the way to a minimum). The principle of least action quickly became an influential scientific and philosophical principle destined to find expression in later centuries in various laws and principles, including LeChatelier's principle regarding equilibrium reactions. The principle profoundly influenced nineteenth century studies of thermodynamics.

On the heels of an influential publication covering algebra, trigonometry and geometry (including the geometry of curved surfaces) Euler's 1755 publication, Institutiones Calculi Differentialis, influenced the teaching of calculus for more than two centuries. Euler followed with three volumes published from 1768 to 1770, titled Institutiones Calculi Integralis, which presented Euler's work on differential equations. Differential equations contain derivatives or differentials of a function. Partial differential equations (PDE) contain partial derivatives of a function of more than one variable. Ordinary differential equations (ODE) contain no partial derivatives. The wave equation, for example, is a second-order differential equation important in the description of many physical phenomena including pressure waves (e.g., water and sound waves). Euler and French mathematician Jean Le Rond d'Alembert (1717-1783) offered different perspectives regarding whether solutions to the wave equation should be, as argued by d'Alembert, continuous (i.e. derived from a single equation) or, as asserted by Euler, discontinuous (having functions formed from many curves). The refinement of the wave equation was of great value to nineteenth century scientists investigating the properties of electricity and magnetism that resulted in Scottish physicist James Clerk Maxwell's (1831-1879) development of equations that accurately described the electro-magnetic wave. The disagreement between Euler and d'Alembert over the wave equation reflected the type of philosophical arguments and distinct views regarding the philosophical relationship of calculus to physical phenomena that developed during the eighteenth century.

Although both Newton and Leibniz developed techniques of differentiation and integration, the Newtonian tradition emphasized differentiation and the reduction to the infinitesimal. In contrast, the Leibniz tradition emphasized integration as a summation of infinitesimals. A third view of the calculus, mostly reflected in the work and writings of French mathematician Joseph Louis Lagrange (1736-1813) was more abstractly algebraic and depended upon the concept of the infinite series (i.e., a sum of an infinite sequence of terms). Converging series, for example, have sums that tend to a limit as the number of terms increases. The differences regarding a grand design for calculus were not trivial. According to the Newtonian view, calculus derived from analysis of the dynamics of bodies (e.g., kinematics, velocities, and accelerations). Just as the properties of a velocity curve relate distance to time, in accord with the Newtonian view, the elaborations of calculus advanced applications where changing properties or states could accurately be related to one another (e.g., in defining planetary orbits, etc.). Calculus derived from the Newtonian tradition allowed the analysis of phenomena by artificially breaking properties associated with the phenomena into increasingly smaller parts. In the Leibniz tradition, calculus allowed accurate explanation of phenomena as the summed interaction of naturally very small components.

Lagrange's analytic treatment of mechanics in his 1788 publication, Analytical Mechanics (containing the Lagrange dynamics equations) placed important emphasis on the development of differential equations. Lagrange's work also profoundly influenced the work of another French mathematician, Pierre-Simon Laplace (1749-1827) who, near the end of the eighteenth century, began important and innovative work in celestial mechanics.

Despite the great success of eighteenth-century mathematicians in developing the techniques of calculus and in the application of those techniques to an increasingly wide variety of problems, a philosophical void remained with regard to the logical underpinnings of calculus. That calculus worked was apparent, but why it worked remained a question that eluded mathematicians. More importantly, the philosophical void in the logic of calculus open calculus to critical attacks. Most prominent among the critics of calculus in England was the influential Anglican Bishop, George Berkeley. The culmination of Berkeley's attacks on Newton's reasonings was formulated in a 1734 work titled, The Analyst: or a discourse addressed to an infidel mathematician. Berkeley, worried about the growing intellectual dominance and reliance upon mathematics and science to provide the most accurate depictions of the natural world, attempted to argue that apparent utilitarian accuracy of calculus was intellectually misleading. In particular, Berkeley argued that the theorems of calculus were derived from logical fallacies. Berkeley contended that the apparent accuracy of calculus resulted from the mutual cancellation of fundamental errors in reasoning.

Scholars took Berkeley's criticisms seriously and set out to vigorously support the logical foundations of calculus with well-reasoned rebuttals, including attempts to incorporate the rigorous mathematical arguments of the Greeks into the calculus. D'Alembert published two influential articles titled Limite and Différentielle published in the Encyclopédie that offered a strong rebuttal to Berkeley's arguments and defended the concept of differentiation and infinitesimals by discussing the notion of a limit. Regardless, the debates over the logic of calculus resulted in the introduction of new standards of rigor in mathematical analysis that laid the foundation for a subsequent rise in pure mathematics in the nineteenth century.

K. LEE LERNER

Further Reading

Boyer, Carl. The History of the Calculus and Its Conceptual Development. 2nd ed. New York: Dover, 1959.

Boyer, Carl. A History of Mathematics, New York: John Wiley and Sons, 1991.

Edwards, C. H. The Historical Development of the Calculus. Springer Press, 1979.

Hall, Rupert. Philosophers at War: The Quarrel Between Newton and Leibniz. Cambridge: Cambridge University Press, 1980.

Kuhn, Thomas S. The Structure of Scientific Revolutions. Chicago: University of Chicago Press, 1970.


CREDIT FOR CALCULUS

During the later half of the seventeenth century, a set of analytical mathematical techniques eventually known as the calculus was developed by English mathematician and physicist Sir Isaac Newton. At the same time German mathematician Gottfried Wilhelm von Leibniz developed the calculus along different philosophical lines. Although the two approaches attacked similar problems they differed greatly in symbolism and nomenclature. In 1684 and 1686 Leibniz published his work, and a year later Newton's version appeared in Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), a book destined to dominate the intellectual and scientific landscape for the next two centuries.

A feud over credit for calculus was on, and the dispute quickly became more than a scholarly tussle. Supporters of Newton or Leibniz often argued along bitter and sometimes nationalistic lines. Newton's scientific stature, especially within the influential British Royal Society, and the fact that Newton and Leibniz were in communication during the development of the calculus eventually resulted in a charge of plagiarism against Leibniz by members of the Royal Society. Supporters of Leibniz subsequently leveled similar charges against Newton. Leibniz petitioned the Royal Society for redress, but Newton hand-picked the investigating committee and prepared reports dealing with the controversy for committee members to sign. Leibniz's mathematical work became subsumed into the dispute over the invention of the calculus. Before the dispute was resolved, however, Leibniz died. Newton's anger at Leibniz remained unabated by the grave. In many of Newton's papers he continued to specifically set out mathematical and personal criticisms of Leibniz.

The feud over credit for calculus adversely affected communications regarding the development of calculus between English and continental European mathematicians. English mathematicians used Newton's "fluxion" notations exclusively when doing calculus. In contrast, European—especially Swiss and French—mathematicians used only Leibniz's dy/dx notation.

Although the notations and nomenclature used in modern calculus most directly trace back to the work of Leibniz, Newton has received the most credit for the development of the calculus in textbooks. Modern historians of science generally conclude that the feud between Newton and Leibniz was essentially groundless. A modern analysis of the notes of Newton and Leibniz clearly established that Newton secretly developed calculus some years before Leibniz published his version but that Leibniz independently developed the calculus so often credited exclusively to Newton.

K. LEE LERNER


The Elaboration of the Calculus

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