By the early eighteenth century, various approaches had been tried to solve several mathematical problems that had been known since antiquity. Johann I Bernoulli organized much of the earlier material and produced an account that could be followed by a wide range of mathematical readers.
The calculus of variations was a method in applied calculus that required a full command of cutting-edge developments in mathematics in the early eighteenth century. It was therefore used as arena in which the leading mathematicians of the day could display their superior skills and knowledge. The calculus of variations was applicable to a wide range of problems, but in its earliest days there did not seem to be any common thread connecting the arguments and the solutions to these various problems. With the 1718 publication of Swiss mathematician Johann I Bernoulli’s essay on the calculus in the Histoire de l’Académie Royale des Sciences: Avec les Memoires de Mathematique et de Phisique pour la même année (history of the royal academy of sciences: with the memoirs of mathematics and physics for the same year), the first glimmerings of a general theory emerged, a theory that was to play an important role in the natural sciences as well as in mathematics over the next two centuries.
The new field of the calculus was particularly applied to the study of the cycloid, a curve that is traced out by a point on the circumference of a circle as it rolls along a flat surface. The Dutch mathematician Christiaan Huygens
Before the calculus, a curve was usually expressed in the form of an explicit equation connecting two variables. After the development of the calculus by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the latter half of the seventeenth century, it was recognized that curves could best be described by the variation of the points on the curve over time. Such a description of a curve took the form of a differential equation—that is, an equation specifying the derivatives of the positions of the points on the curve. Mathematicians had learned, by study of the cycloid, to recognize its unique differential equation.
Newton had posed the problem in Philosophiae Naturalis Principia Mathematica (1687; The Mathematical Principles of Natural Philosophy, 1729; best known as the Principia) of which physical shape would encounter the least resistance from air when rotated. To solve this problem, he made some simplifying assumptions (which were to be characteristic of the field in its later developments). His solution in the Principia involved geometry, but he was aware of a way to enunciate and solve the problem using the calculus.
What served as the jumping-off point for the calculus of variations was not Newton’s problem, however, but a question posed by Johann I Bernoulli in a 1696 issue of Acta Eruditorum, a journal published in Leipzig for scholars in mathematics and other areas. Bernoulli asked what path an object traveling between two points in space should follow in order to make the journey in the least possible time. Bernoulli declared that the curve that solved the problem would be called the brachistochrone, from the Greek for “least time.” The problem had originally been posed by the Italian physicist Galileo in the early seventeenth century, but he had mistakenly thought that the answer was a circular arc.
Bernoulli’s challenge attracted the attention of many distinguished mathematicians, including Newton, Leibniz, and Bernoulli’s elder brother Jakob I Bernoulli. The results of the challenge were included in an issue of the Acta Eruditorum for the following year. The various contributors had taken different paths to the solution, but they all succeeded in identifying the shape of the brachistochrone as a cycloid. This came as quite a surprise to the Bernoullis, who had not thought that the curve they had learned about in Huygens’s work would appear in this rather remote setting.
Newton had sent in his submission anonymously, but no one had any doubts about the origins of that solution. It was not, however, Newton’s solution that was to be the most influential, but that of Jakob Bernoulli. He approached the problem by allowing a point on the curve to vary (which accounts for the name of the resulting discipline, the calculus of variations) and derived a differential equation as a result. The differential equation was that of the cycloid, and its familiarity may explain why so many solutions were forthcoming. Jakob Bernoulli’s methodology, however, could be generalized to solve problems whose solutions were less familiar. Recognizing this, he pursued the method that he had developed to deal with his brother’s challenge in a subsequent paper of 1701.
There were two other kinds of problems to which the calculus of variations could be applied. One was that of finding a geodesic, or the shortest path between two points on a surface. On a flat surface, or plane, it is easy to see that the geodesic is always a straight line; however, for two points located on a sphere or an ellipsoid, for example, the geodesic is less obvious. Moreover, since eighteenth century mathematicians and physicists were aware that the surface of the Earth is ellipsoid in shape, they were particularly interested in finding a method to determine the geodesic of an ellipsoid. Such an equation would have obvious benefits for navigation.
The other kind of problem that could be solved with the calculus of variations was an “isoperimetric” problem: Given a specific perimeter, what figure with that perimeter will have the greatest area? A story from classical antiquity had served as a model for the tricks that could be concealed in apparently straightforward situations, but the mathematicians of the seventeenth century were looking for a systematic way of showing, for instance, that the rectangle of given perimeter with the greatest area was always a square.
Jakob Bernoulli was able to make headway on all of these problems—brachistochrone, geodesic, and isoperimeter—with the methods he introduced in his 1701 paper, also published in the Acta Eruditorum. He entitled his paper “Analysis of a Great Isoperimetric Problem,” and he observed that he was calling the problem great not because it was important in itself but because it led to methods that would be applicable to many other problems as well. Jakob was able to draw upon his success in solving the problem raised by his brother about the brachistochrone to attract Johann’s attention to his more ambitious paper.
Despite his wealth of ideas, however, Jakob Bernoulli’s exposition left a good deal to be desired. His presentation verged on the pedantic, and even those with an interest in the subject found his article difficult going. After his death in 1705, his brother Johann took up the task of providing an exposition of the bases for the calculus of variations. Johann was also trying to promote his own ideas, some of which differed significantly from those of his brother. Finally, in 1718, Johann published a paper in Histoire de l’Académie Royale des Sciences presenting the calculus of variations as a systematic discipline with important applications to a range of problems. Johann’s 1718 paper proved to be the source from which the next generation of mathematicians was to draw in further developing and applying the calculus.
Johann I Bernoulli’s publication served as a guide to the mathematicians of the later eighteenth century, who gave their own formulations to some of the principles he had put forward. In particular, Bernoulli himself was the teacher of Leonhard Euler,
The Bernoullis were not entirely forgotten, however, even after Euler’s magisterial supersession of their earlier treatment. The twentieth century mathematician Constantin Carathéodory revisited Johann Bernoulli’s ideas and acknowledged their importance to his own work in the calculus of variations. The field had continued to develop in the nineteenth century, and many of the problems to which it was applied by Carathéodory and his contemporaries would have been unfamiliar to Bernoulli. Nevertheless, the idea of looking at a possible solution and trying to vary a point on it to see what sort of differential equation would result remained the source of inspiration in the calculus of variations for centuries.
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Bliss, Gilbert Ames. Calculus of Variations. LaSalle, Ill.: Open Court, 1925. An account designed for college students that starts with several chapters on the problems that led to the Bernoullis’ work. -
Chabert, Jean-Luc. “The Brachistochrone Problem.” In History of Mathematics: Histories of Problems. Paris: Ellipses, 1997. Looks at the creation of the field of calculus of variations as a response to the specific problem raised by Johann Bernoulli. -
Goldstine, Herman H. A History of the Calculus of Variations from the Seventeenth Through the Nineteenth Century. New York: Springer-Verlag, 1980. Supplies more contemporary mathematical reconstructions of the original arguments. -
Kline, Morris. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1972. Chapter 24 is devoted to the calculus of variations, one of the areas in which Kline himself worked. -
Van Brunt, Bruce. The Calculus of Variations. New York: Springer, 2004. A modern treatment of the subject, starting from the work of the Bernoullis.
De Moivre Describes the Bell-Shaped Curve
Bernoulli Proposes the Kinetic Theory of Gases
D’Alembert Develops His Axioms of Motion
Agnesi Publishes Analytical Institutions
Euler Develops the Concept of Function
Bayes Advances Probability Theory
Legendre Introduces Polynomials
Jean le Rond d’Alembert; Leonhard Euler; Joseph-Louis Lagrange; Gaspard Monge.