Euler and the Concept of Function

Date 1748

The introduction of algebraic expressions for curves helped mathematicians to analyze geometrical figures. Leonhard Euler’s 1748 work marked a change in perspective by putting the function first and the curve second. By inverting the order of the expression, Euler reconceived the very subject matter of mathematics.

Locale Berlin, Prussia (now in Germany)

Key Figures

• Leonhard Euler (1707-1783), Swiss mathematician and physicist
• Johann I Bernoulli (1667-1748), Swiss mathematician
• Gottfried Wilhelm Leibniz (1646-1716), German mathematician and philosopher
• Joseph-Louis Lagrange (1736-1813), French mathematician

Summary of Event

The traditional objects of mathematical study have been numbers and geometrical shapes. In the seventeenth century, René Descartes juxtaposed algebra and geometry, allowing algebra to be used to study geometrical objects. In Descartes’s conception, it was still the geometrical object that was primary; the algebraic expression was simply a translation of the shape into equations, a new way to represent the object. With the work of Leonhard Euler, however, a change began that led mathematicians to think of the expression representing a curve as the primary object of study. The geometrical object itself came to be seen as a mere illustration of equations, rather than the object the equations sought to describe. This shift shaped mathematical practice and attitudes for more than two centuries.

From ancient times to the seventeenth century, the objects of study of mathematics were clearly recognized as numbers and figures. In other words, mathematicians sought to understand and to identify the properties of numbers (arithmetic) and shapes (geometry). The purpose of algebra was simply to solve arithmetic problems. It was analogous to the use of illustrations to solve geometrical problems, even when it was recognized that the geometrical object itself could not be drawn on a piece of paper. Even arithmetic was sometimes considered to be a branch of geometry, as expressed in Euclid’s Elementa (c. 300 b.c.e.; The Elements of Geometrie of the Most Auncient Philosopher Euclide of Megara, 1570; commonly known as Elements).

In the seventeenth century, René Descartes introduced an approach to geometry that was to transform the subject. A geometrical figure could be represented by an algebraic equation, and study of the algebraic equation and its properties enabled a mathematician to arrive at conclusions about the geometrical object. For example, if one had the equation for a given circle and one found that plugging the value 0 in for x produced an equation with no solutions for y, then one could conclude that the circle did not cross the y-axis. This kind of property was dependent on the coordinate representation of the circle, but others dealt with intrinsic properties of the figure.

Even after Descartes’s innovation, however, the geometrical circle was taken by mathematicians to be prior to any algebraic representation of it. After all, even to write down an algebraic equation for a geometrical figure required selecting a coordinate system, and geometers should not have to resort to that choice in order to be able to make conclusions about shapes and figures. (That is, it should not be necessary to know where a shape is in order to know things about the unchanging properties of the shape.) The algebraic equation was a tool for arguing about the geometrical object but was only ancillary (and was dispensable).

With the advent of the calculus in the late seventeenth century, however, techniques were introduced that could apply directly to the equation for a geometric shape. For example, using the calculus, a mathematician could find the area of a parabola without using the geometric definition of the curve. Mathematicians like Pierre de Fermat earlier in the seventeenth century had been able to find expressions for the area of a given parabola directly by geometric means, but it was much harder to see how to extend those techniques to curves in general. The calculus offered a relatively easy way to extend results to general categories.

A function is a formula for obtaining values for one quantity from those for another quantity (for example, finding the distance traveled by an object given its duration of travel). By the seventeenth century, it was recognized that some functions (like the values of a polynomial) could be obtained by algebraic means and some (like the sine function) could not. While algebraically obtainable functions supported the view that numbers were the primary object of mathematics, the other kind of function not only resisted this view but also made it difficult to understand precisely what a given function did to an input value in order to produce the output value. Johann I Bernoulli offered a characterization of a quantity obtained “in any manner” from another quantity as the basis for his idea of function, which was a term subsequently employed by Gottfried Wilhelm Leibniz.

It was Euler’s Introductio in analysin infinitorum (1748; Introduction to Analysis of the Infinite, 1988-1990) that put the issue of the definition of functions squarely before the mathematical audience at large. Euler had already attained quite a reputation for his mathematical ingenuity in solving problems. Benefiting, moreover, from the standing of Johann I Bernoulli, his mathematical guide at the University of Basel, he had rapidly become part of the research community. Euler was to become the most prolific mathematician of all time, as measured by published papers, and he moved back and forth between positions in St. Petersburg and Berlin. His influence on mathematical ideas and even notation was unmatched.

In the opening chapter of his Introduction to Analysis of the Infinite, Euler analyzes functions of various kinds. He defines a function generally as an “analytic expression,” which puts an emphasis on its ability to produce a resulting quantity by plugging an original quantity into an equational expression. There is no need to represent the quantity to be obtained as a variable on one side of the equation, so he allows for functions that are defined implicitly.

Euler then defines algebraic and transcendental functions, the latter being the kind that cannot be expressed algebraically (in which term he includes polynomials and ratios of polynomials). The discussion even takes on the difference between singly valued functions and multiply valued functions. The latter category encompasses functions in which, for example, the variable to be obtained only appears as a square, in which case one input can produce two separate outputs, one positive and one negative.

Over the course of his work, Euler tackles all sorts of interesting problems about infinite series, infinite products, and continued fractions. Many of those results were put to use by Euler himself, as well as his successors. His treatment of the idea of function at the beginning of his text, however, has perhaps been Euler’s most important mathematical legacy. The algebraic expression has become the central subject for mathematical investigation, and the geometrical object that it describes has been demoted to second place.

Significance

Euler’s characterization of a function as an “analytic expression” was one that even he found restrictive in the course of his later mathematical career. By the end of the eighteenth century, the mathematician Joseph-Louis Lagrange had characterized a function as a combination of operations. This more inclusive idea of a function could apply to situations beyond those in which numbers were understood as quantities known in advance. In addition, one could take other sorts of mathematical objects (permutations, for example) and regard those as the ingredients to be plugged into the recipe to which the function corresponded.

The drive toward abstraction that characterized mathematics through most of the twentieth century can be seen as an outcome of Euler’s treatment of function as the central idea of mathematics. The branch of mathematics called “category theory” starts with the idea of a function as primary and is not so concerned with the sort of object on which the function acts. Even the philosophy of mathematics has been influenced by the view that the nature of mathematical objects is not so important as the kind of functions that act on them.

Geometrical objects have not been entirely neglected over the centuries since Euler wrote. It remains the case, however, that students asked to demonstrate something about geometrical objects will almost always turn to the algebraic representation. In the nineteenth century, the characterization of branches of geometry was according to the kind of function that left the geometric objects unchanged. The heritage of the idea of function has cut across all branches of mathematics and continues to affect our views of its subject matter.

Bibliography

Bottazzini, Umberto. The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass. New York: Springer-Verlag, 1986. Starts with a chapter on Euler’s role in the formulation of functions.

Dunham, William. Euler: The Master of Us All. Washington, D.C.: Mathematical Association of America, 1999. The discussion of logarithms looks at the distinctive nature of transcendental functions before and after Euler.

James, Ioan. Remarkable Mathematicians: From Euler to von Neumann. New York: Cambridge University Press, 2002. Some biographical background with more attention on Euler’s other work.

Kline, Morris. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1972. Appeared at a time when discussion of the concept of function was a lively pedagogical issue and pursues the subject in detail through the period before Euler.

Lützen, Jesper. “Euler’s Vision of a General Partial Differential Calculus for a Generalized Kind of Function.” In Sherlock Holmes in Babylon, edited by Marlow Anderson, Victor Katz, and Robin Wilson. Washington, D.C.: Mathematical Association of America, 2004. Indicates the slow rate at which changes in the concept of function crept through the mathematical community.