JACOB (1654–1705)

By the time Jacob began his duties at the University he had already mastered Descartes' Geometry and had begun his work on Probability that would not be published in its entirety until after his death, in the book entitled Ars Conjectandi (Basel, 1713). This pioneering work included the formulation of the "Law of Large Numbers" as well as the introduction and generation of the Bernoulli numbers. Influenced by the writings of Wallis and Barrow on Mathematics, and in Physics by the work of Archimedes, Huygens, and Mariotte, he now directed his attention to the questions in Mechanics and Optics raised in the writings of Huygens and others. From there he was shortly coming to grips with infinitesimals. Following his investigations into the application of the calculus to elasticity, which he tested in experiments designed and performed by himself, he became sceptical as to the validity of a linear stress-strain law: these results together with some queries on points of Calculus he sent directly in his first communication to Leibniz (1687).

In 1687 he was appointed Professor of Mathematics in Basel—a position that would remain in the family for more than a century. But more important than the appointment was the attention he paid to the work of Leibniz in the Acta Eruditorum. Because he received no reply to the enquiring letter on points of the calculus written to Leibniz (Leibniz was travelling so that almost three years elapsed before the letter reached him), Jacob embarked on the establishment, development and application of the calculus—with the assistance of his younger brother Johann who was by now rapidly learning from him.

In the ensuing years he obtained penetrating insights into Infinite Series (1689), formulated the Law of Large Numbers, and gave a detailed study (1690) of the isochrone—the curve along which a particle under the action of gravity moves so that its height is a linear function of time. Along with this, he investigated a range of problems in geometry, including the logarithmic spiral, and was the first to exhibit an elliptic integral (1691). The latter integral arises in the course of his analysis of the Elastica and, following a discussion of its properties, the integral is computed numerically.

The tautochrone, or problem of constant descent, is defined as the curve along which a particle under gravity will fall to reach its nadir in a fixed time irrespective of the point of release. This problem, whose solution is the cycloid, had been identified by Huygens in his Horologium Oscillatorium (1673) was posed by Leibniz (1687) with solutions provided by Huygens (1687) and Leibniz (1689). In his analysis of the problem (1689–90) Jacob showed himself a master of the differential and integral calculus and formulated the problem of the catenary as a related problem. In this he showed his early awareness that such questions had opened a new area to be explored.

At the same time he was involved with the Geometry of Geodetic Curves, where he discovered the "Theorema Aureum"—his formula for the radius of curvature which he published in 1694: this appears to be the first non-trivial and successful application of derivatives of order higher than the first. But above all else he took pleasure in the logarithmic spiral—spira mirabilis—being its own evolute. He requested that it be engraved on his tombstone, with the motto "Eadem Mutata Resurgo": it is unfortunate that when this request was being complied with, the curve actually engraved was (mistakenly) the Archimedean spiral.

His pioneering work on series, begun in 1689, evolved over the years and took the form of a series of theses assigned for defence to his students as follows:

Although it was evident during those years that he found his greatest satisfaction in mathematical investigations, nevertheless he never turned away from the most practical questions in physics—and even in banking—and, frequently, through them was led to fruitful mathematical results. Among many others he addressed such issues as

(1) The best shape for a boat-sail

(2) The best design for a watch-spring

(3) The most efficient design for a drawbridge.



From his encounter with the question of compound interest and his expertise with the binomial theorem he derived the exponential function in series form with the help of those numbers that still bear his name. At the same time he was labouring to put the Theory of Differential Equations on a satisfactory basis. One of the more difficult and absorbing problems that occupied him was that of the Elastica, which he had formulated in its most general mathematical form in the years 1691 to 1694 and whose analysis he continued to pursue through the following decade to his last work in 1705.

In the course of this work we have for the first time a physical problem formulated in the form of a variational principle (1701), as distinct from extremum problems, several of which had been addressed over the previous decade. Previous to this, when his brother Johann issued a challenge for the solution of the problem of the Brachistochrone in 1696, Jacob had promptly solved it. As anticipated by Leibniz, there were only four others who could solve it, namely Johann, Leibniz, Newton and l'Hospital—the latter with help from Johann. But it was Jacob alone who recognised that a new field was opened by such questions, namely the Calculus of Variations, which he created and developed in the following years.

In the various branches of eighteenth century Mathematics, Geometry of Curves, Probability, Series, Differential Equations, Calculus of Variations and above all Mechanics (with but one exception—Number Theory), Jacob was a pioneer. This pioneering spirit is highlighted in a major work of his appearing in 1703, on the "center of oscillation" motivated by the problem of the pendulum. The method used which he terms the "principle of the lever" is in fact the equilibrium of moments. Implicit in the paper also is a version of what later came to be known as the "Principle of d'Alembert" whereby the law for static equilibrium may be extended to describe dynamic equilibrium through the addition of the force of inertia as measured by the reversed mass acceleration. As demonstrated by Truesdell this combination is the first formulation of the Law of Moment of Momentum. The idea is already in evidence in incorrect form in an earlier paper of Jacob which appeared in 1686. Thus can Truesdell assert that this law as an independent law of Mechanics is due to Jacob I Bernoulli, first proposed as a generalization of the (static) Principle of Moments, in print in 1686, and thus predating the publication of Newton's Laws (1687). The intermediate work of 1691, done under the stimulus and inspiration of the Marquis de l'Hospital (who has primacy in giving the correct form of the inertial term) gives the valid but not yet fully general form.

Jacob Bernoulli died unexpectedly at the height of his powers in 1705. By temperament, he had no inclination to pursue general comprehensive physical theories, but rather by mastering the essentials of each problem that came his way, he made his radical and extensive contributions, no matter what may have been the initial source of the problem. The example that best illustrates this gift of Jacob is the structuring he made on a problem as simple (in concept) as the Elastica. In this work he was so far ahead of the general theory (of the nineteenth century) that only rarely does one find in the standard works a recognition of his extraordinary achievement. However, it is noteworthy that in his recent book "Mathematical Models for Elastic Structures" P. Villaggio, in contemplating a list of outstanding results produced over the last three centuries, begins the list with Jacob's (1705) equation for the elastica.

Jacob was also the first of the Bernoulli family to earn his living as a professional mathematician. In comparison with that of some of his prolific contemporaries, the quantity of his correspondence is relatively small; however, only in this century has there been a realization of the riches of its content.

Ó Mathúna, 1999