Legendre Introduces Polynomials

Legendre found, in the course of trying to solve a differential equation, a family of polynomials that satisfied the same kind of properties that ordinary polynomials did. This suggested the use of those polynomials to represent all functions that had certain features, and similar families have been studied by mathematicians and physicists ever since.

Summary of Event

The introduction of differential equations by the founders of calculus Calculus led to the formulation of problems about the natural world in mathematical terms. For the earliest workers in the field, there was a limited repertoire of possible solutions to such equations. When none of those solutions seemed to work for equations that were needed to analyze changes in the physical world, new kinds of functions had to be introduced. One of the central groups of equations lacking solutions was orthogonal polynomials, and Legendre’s efforts to solve a problem about the solids of revolution brought a particular family of such polynomials to the fore. [kw]Legendre Introduces Polynomials (1784)
[kw]Polynomials, Legendre Introduces (1784)
[g]France;1784: Legendre Introduces Polynomials[2530]
[c]Mathematics;1784: Legendre Introduces Polynomials[2530]
[c]Science and technology;1784: Legendre Introduces Polynomials[2530]
Legendre, Adrien-Marie
Laplace, Pierre-Simon
Euler, Leonhard
Fourier, Joseph

Polynomials had been known in mathematics for centuries by the time that Legendre was undertaking his investigations. They are expressions involving combinations of whole-number powers of the variable, and the solution of general equations involving polynomials of the third and fourth degree had been part of Renaissance mathematics in the sixteenth century. One of the central features of polynomials that made them crucial in algebra Algebra was the principle of undetermined coefficients. Undetermined coefficients principle This principle states that if two polynomials are equal for all values of the variable, then the coefficients of like powers of the variable have to be equal. This was taken for granted in the seventeenth century but investigated more rigorously in the eighteenth century.

In view of the centrality of polynomials in algebra, they played an important role in the new field of calculus as well. The field of differential equations Differential equations involves taking a mathematical statement about the rate at which a quantity is changing and trying to figure out an expression for the original quantity. If it was possible to get the original quantity as a simple polynomial, then the solver could use everything that was known about polynomials to analyze the solution. The earliest differential equations, however, already involved functions Function in mathematics that were more complicated than polynomials, like the trigonometric and exponential functions. It seemed as though the background from polynomials was not going to be useful in analyzing such solutions.

Leonhard Euler made an immense contribution to understanding the analysis of such solutions by treating even complicated functions as a kind of polynomial with no limit to the highest power of the term. Such a polynomial of “infinite” degree is called an infinite series, and Euler was a master of manipulating infinite series Infinite series (mathematics) for many kinds of functions. Once Euler could demonstrate that a function could be represented uniquely as an infinite series, he was able to put information about polynomials to use in talking about complicated functions, although subsequent generations have sometimes found a lack of rigor in his treatment. Nevertheless, his intuition sufficed to get remarkable formulae connecting the solutions of differential equations.

Adrien-Marie Legendre managed to carry the work of Euler further with the help of his colleague Pierre-Simon Laplace. Both men were interested in the question of how to simplify the problem of gravitational attraction by a body that was spread over space, and the work of the founders of calculus, Sir Isaac Newton and Gottfried Wilhelm Leibniz, indicated that the attraction could be expressed as a differential equation. Solving such differential equations was quite difficult, especially if it was not clear what form the solution was going to take. It was clear that the result was not going to be a simple function, but the problem facing Legendre was to figure out some kind of expression.

Legendre came up with the idea of representing the solution of the differential equation in which he was interested as a series involving powers of the cosine of the angle made at the center of the solid he was studying by two lines connecting the center with the surface. Each of the coefficients of the series would be a polynomial, and from that he could obtain an expression that could be evaluated. If it were possible to determine properties of the polynomials in question, then the solutions for a whole family of differential equations could be evaluated.

In his 1784 paper on celestial mechanics, Legendre generated a number of results about the polynomials that he had derived in the course of working on the solution to the differential equation. In particular, he could derive properties of the polynomials without having to write down their explicit forms (which could be quite complicated). He could figure out how the polynomials interacted with one another. Most important, he was able to show that functions of certain kinds could only be represented in one way as expressions involving his polynomials.

This combination knowledge—of both how the Legendre polynomials (as they came to be known) interacted with one another and how the representation of certain kinds of functions was unique in series involving those polynomials—led to the study of similar classes of polynomials called “orthogonal.” Orthogonal polynomials The evaluation of the expressions that arose in Legendre’s paper required the help of Laplace, and Legendre polynomials are sometimes also called “Laplace coefficients.” Laplace coefficients Legendre did not himself develop the study of such polynomials in detail, as he continued to move about in branches of mathematics like geometry and number theory in addition to differential equations. Nevertheless, the use of orthogonal polynomials as a kind of series offered a solution technique for differential equations that would attract engineers as well as mathematicians and physicists. Even when it might be hard to justify the application of techniques on rigorous grounds, the ability to compute a solution as needed enabled defects in rigor to be overlooked.


One of the major subjects for study in physics in the early nineteenth century was the behavior of waves. Regular trigonometric functions Trigonometric functions like the sine function had simple graphs, but observation found plenty of more complicated curves. Trying to analyze them in terms of ordinary trigonometric functions did not seem helpful, but they also did not fit in with standard polynomials.

The mathematician Joseph Fourier recognized that the waves could be analyzed by using a series of trigonometric functions, using the same kind of approach that Legendre had with his polynomials. These Fourier series enabled mathematical physicists to represent the waves uniquely, and the coefficients could be calculated on the basis of the experimental data. Without Legendre’s study of the earlier kind of orthogonal polynomials, Fourier’s results (which were still regarded with suspicion by members of the mathematical community with a concern for rigor) would have been even harder to swallow.

The importance of orthogonal series continued to be demonstrated in the twentieth century. One way of interpreting the results of quantum mechanics is in terms of a certain kind of infinite-dimensional space. While this is clearly beyond what Legendre would have envisaged, the notion that one could still be using the properties of polynomials even in such a remote setting was a guide for those who sought to analyze mathematically the behavior of waves in nature.

Further Reading

  • Dunham, William. The Calculus Gallery: Masterpieces from Newton to Lebesgue. Princeton, N.J.: Princeton University Press, 2005. A specialist on Euler ties Legendre’s work into Euler’s.
  • Freud, Geza. Orthogonal Polynomials. Oxford, England: Pergamon Press, 1971. Not much history but an exposition of the ideas underlying Legendre’s creation.
  • Itard, Jean. “Adrien-Marie Legendre.” In Dictionary of Scientific Biography, edited by Charles C. Gillispie. Vol. 8. New York: Charles Scribner’s Sons, 1970. Survey of Legendre’s mathematical work.
  • James, Ioan. Remarkable Mathematicians: From Euler to Von Neumann. New York: Cambridge University Press, 2002. Looks forward to Legendre’s influence on Fourier.
  • Kline, Morris. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1972. Most detailed analysis of the text in which Legendre introduces his polynomials and the problem they were intended to solve.

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