DANIEL (1700–1782)

Immediately he was a candidate for two chairs in Basel—those of Anatomy and Logic: the choice was made by lot in each case and Daniel lost in both cases. This ill-luck was to follow him for quite a while. Ostensibly to pursue his Medical Studies he went in 1723 to Italy where he cultivated the practical side under Michelotti in Venezia. He was the victim of a bout of illness during a stay in Bologna.

Meantime he had not neglected that subject nearest his heart, as in 1724 in Venezia he published Exercitationes Quaedam Mathematicae—a treatise addressing four main problem areas, namely the game of pharaon and recurrent series, the flow of water from a vessel, the Riccati differential equation, and the Lunulae. Herein was already evident the range of his interests, Probability, Hydromechanics, Calculus and Geometry—a range that strengthened and widened with time.

This work attracted wide attention. The same year he was awarded the prize of the Paris Academy—for the first time: eventually he was to be awarded the prize on ten occasions. The work for which he was awarded this prize was that on hourglasses (St. 8). Besides, he was now invited, with his brother Nicolaus, to the St. Petersburg Academy. They both accepted and arrived in the Russian capital in 1725. But the harsh climate never agreed with him and in his years there he was on three occasions an unsuccessful candidate for positions in Basel. Moreover the death of Nicolaus in 1726 had dampened whatever enthusiasm he had for the atmosphere of the Academy which was so conducive to his work. At length in 1732 he was appointed to the Chair of Anatomy and Botany in Basel. In 1733 in the company of his younger brother Johann II, he returned home making a sojourn through Germany, the Netherlands and France: in every city in which they stopped, the learned elite came to meet and welcome them.

In spite of his unhappiness while there, those were nevertheless fruitful years which Daniel spent in St. Petersburg where he had the company of Euler from 1727. It was there he laid out his work-program in Hydromechanics, Probability and the Theory of Oscillations—areas he continued to explore following his return to Basel. The investigation of the Theory of Oscillations was to be a life-long project. Ten years after his return, in 1743, he managed to exchange his responsibility in Botany for Physiology—a subject nearer his heart. And at length at the age of fifty (1750), he was appointed to the Chair of Physics—the position he had always wished for. He stayed in that position for the remainder of his career until his retirement in 1776. During that quarter-century that he lectured in Physics, his fame spread for the excellence of his lectures and the exemplary experiments with which he illustrated them. Present at these lectures were many "mature students" and "auditors" who came to observe what they knew was the best show in town: these included the three Counts Teleki, two of whom kept diaries wherein there are frequent references to Daniel and his lectures. Daniel was demonstrating that as an Experimental Physicist he was without a peer: he was finally in a position that suited him.

In the earlier years (1733–50) in spite of his other responsibilities, he continued his work in areas begun in St. Petersburg. In 1734 he had ready his book Hydrodynamica though four years would elapse before the publication of a revised version in Strasbourg (1738). This work was a feat in its time wherein it is possible to follow Bernoulli's train of thought from when he first addressed such problems in his Exercitationes ten years earlier. No matter how unsatisfactory may be some of the derivations, there are found here the concepts which ever since have been part of the subject; namely, energy, pressure, and in the case of the "elastic fluid" namely, the gas, there is a version of the Kinetic Theory of Gases giving insight into the Townley-Boyle law, as well as a form of the "Equation of State" and, of course, Bernoulli's Equation, though not in the local form in which we know it from the subsequent formulation of Euler. Besides that, there is discussion of the utilization of such insights to improve the technology in such areas as sail-boats, mill-wheels, design of oars, as well as the Archimedian screw and other features of the machinery of the time. In his discussion of technical applications (Section 9), the notion of power is introduced and discussed: it is of particular interest that it appears that Daniel Bernoulli was the pioneer who introduced this crucial concept to technology. This book was a stimulus for the later important work by Euler.

Although it was against his will that he pursued his course in Medicine, he soon recognised that this profession presented problems having a physical basis, particularly in the area of physiology. In his basic thesis, De Respiratione, there is a comprehensive account of the mechanics of respiration. In 1728 in the first volume of the Commentarii Academiae Petropolitanae (CP I) he published a novel though invalid mechanical theory of muscle contraction, independent of the theory hitherto accepted. In the same volume of the Commentarii there is a treatise by him giving a clear picture of the shape and location where the optic nerve enters the bulbus—explaining the blind spot, and confirming an hypothesis of Mariotte: also in this volume is the fundamental paper (St. 9) on the Principles of Mechanics—essentially an axiomatic/geometric investigation of the law of composition and resolution of forces (the parallelogram law).

In 1737, on the occasion of the graduation of two of his students, he delivered a celebratory speech in Latin on work on estimating the heart's capacity for mechanical work—the first attempt at such a measurement, whose accuracy was later recognised as remarkable. (There is an echo of this in the experiments performed by Haldane and Bohr at the end of the nineteenth century). This interesting work was re-issued (together with a German translation) in Basel in 1941. Although his future work would be in other areas, he never lost sight of the insights physics could yield in the study of physiology.

In his work on strictly mathematical topics, again the beginning was made in the Exercitationes, particularly in the solutions obtained for various cases of Riccati's Equation. From the same basis he developed a topic arising from the problem of the pharaon, namely recurrent series; in CP III, 1728 (1732), he illustrates how these can be used to obtain approximations to the roots of algebraic equations. In his work on series, he did not give due attention to convergence, so much a concern of his cousin, Nicolaus I: in this he did not differ much from many of his contemporaries. However there is inherent in his work the early illustration of the concept of summability which would receive precise formulation only much later from Cesàro and others.

Starting in 1726 for more than twenty years there came from his hand a series of dissertations dealing with various aspects of Mechanics: these include the investigation of the motion of a system of mass-points under mutual attraction, motion subject to frictional forces, and the rotation of rigid bodies. Though in all of these topics, Euler was to make a far more significant contribution, very often Bernoulli was the trailblazer who formulated the questions and drew the attention of Euler to the subject through sustained correspondence that persisted after Bernoulli's return to Basel.

In the area of Probability, his most notable work is Specimen theoriae novae de mensura sortis appearing in CP V 1731, (1738). This work, done in St. Petersburg, proposes a novel evaluation of capital gains and by proposes a mathematical formulation of a new theory of value in political economy. Many years later, similar ideas occur in his treatment of a problem in Probability arising in Medicine—the death-risk from smallpox (1760). On this basis he strongly advocated widespread inoculation, estimating it would lead to an extension of three years in the average life expectancy. Always inclined to have recourse to the Calculus in determining the influence of rate-changes, he applied the same approach to the theory of Errors: in this he reflects a closer affinity to the modern perspective than to that of his contemporaries. It appears he was the first to suspect a universal law of error—although the one proposed by him was invalid. Many years later he was to propose replacing mean estimation by an alternative which is essentially the maximum likelihood estimate.

Perhaps the most important and most satisfying of his labours was the sustained investigation of oscillatory systems where his mathematical ability and his physical insight complemented each other perfectly. Starting with CP III, 1728 (1732), there came a series of dissertations that won him recognition as master. He would subsequently yield primacy as a mathematician to Euler, but for many years there were, besides d'Alembert, only the two of them laboring in the field. But as an experimental physicist, Daniel had no need to yield to anyone and it was from him that came the stimulus that set Euler to work in this field where he reaped such a bountiful harvest.

At first addressing the shape of a perfectly flexible thread subject to certain force conditions, he found access to a source for such a varied series of curves as the velaria, the lintearia, the catenary as well as several others. At the same time he determined the shape of an elastic bar under the action of an external force as well as its own weight: in this work he proposed the linearised form of the moment-curvature relation in use since. On his departure from St. Petersburg he left one of his most significant papers for publication which, due to progressive delay in the printing of the Commentarii—a delay lengthening by the year—did not appear for several years, in CP VI, 1732/3 (1739). The stimulus for this work came from an account of the work done by his father Johann I. He addressed the case of a hanging rope to which are attached several body-weights: his analysis showed that the number of simple oscillations equalled the number of bodies—the number of "degrees of freedom". Then, considering a free-hanging rope of uniform density and by showing that the frequencies are the zeros of a function that would later be recognised as a Bessel function (this was the first appearance of the latter) he showed that such a system has an infinite set of frequencies, each of which can be put in tune with that of a simple pendulum.

On returning to Basel the correspondence with Euler on these issues continued: following the insights provided by Bernoulli, Euler, with his unique strength in mathematical analysis, soon provided a surfeit of results on various aspects of oscillation systems. Then Bernoulli addressed the problem of the elastic bar subject to various end-conditions: he verified these results by experiment, for which Bernoulli with his sharp ear for tone detection, was now establishing his mastery. In this, as pointed out by Truesdell, he was the first in this subject to use theory to facilitate experiment.

In 1738 he had proposed to Euler, whom he acknowledged as the expert in mathematical analysis, the exploration of the general variational problem, , where R denotes the radius of curvature of a bent bar. In 1742 in another communication he was requesting Euler to direct his attention to the variational problem, , as describing the equilibrium (minimization of elastic energy) in an elastic bar—the variational form of the elastica under a linearized stress-strain law. At least partly from this stimulation Euler produced the rich body of results incorporated as an Additamentum to his 1744 Volume (Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, Lausanne/Geneva, 1744), which Truesdell has characterized as "a timeless masterpiece in our subject". However, by the late forties, the wave equation firstly posed by d'Alembert, was being fully and intensively analysed by Euler. With its novel notation for partial derivatives now a necessity, it was no longer easy for Bernoulli to keep in step with the advances being made by Euler. After publishing some further items on wave vibration of strings under various loading conditions, it appears he decided to leave the field to Euler henceforth.

It was then that he directed his attention to a problem for which he had no peer, namely the oscillations in organ pipes, in which he provided comprehensive insights, using only the most basic mathematics. Also came a few treatises on the vibration of non-uniform strings. In all this work, which was motivated by the desire to understand musical sounds, there is widespread though implicit use of a principle which we now recognise as the Principle of Superposition. That concept, which would be expanded by Fourier into such a fruitful field of mathematical investigation over half a century later, was clear to Bernoulli through his physical insight that a musical note can be analysed into its harmonic components, since it is a superposition of an infinite series of its natural frequencies. One can only conjecture what might have developed if his physical curiosity had been matched by a comparable mathematical curiosity. For it must be acknowledged that Daniel Bernoulli never wrote down a partial differential equation, thus foreclosing any possible development in that direction. The relevance of these concepts to the wave mechanics of the twentieth century need not be elaborated here.

The treatises submitted for the Prize of the Paris Academy deal chiefly with such topics as marine navigation, the motion of the tides, astronomy, magnetism, and oceanography. He was awarded the prize on ten occasions. It was clear that any problem in technology could attract his attention, from which some form of solution would result. For the year 1757, the topic on offer was how to dampen the rolling and pitching motions of a ship on a turbulent sea. The essay of Euler was confined to the free oscillations: but Bernoulli addressed the crucial issue of forced oscillations. The judgments arrived at by Bernoulli in that prize-winning essay remained the norm for a hundred years thereafter.

The most notable of these treatises is that for the year 1740, Traité sur le flux et le reflux de la mer. It is a massive work (St. 33: pp. 327–438 of WDB-3) second only to the volume Hydrodynamica; it is also the most elegantly written—and the most frequently printed of all his works. At the instigation of Pope Benedict XIV (Prospero Lambertini), the Principia of Newton were reissued in Geneva in 1742 under the joint editorship of Jacquier and Le Seur: included as Appendices are the treatise of Bernoulli as well as those of two of the three with whom he shared the Prize, namely Euler and MacLaurin; significantly the fourth treatise, that of Cavallini, a Cartesian, is not included. This volume was one of the crucial factors in ensuring the ultimate acceptance of Newton's Principia on the Continent. This treatise of Bernoulli shows his versatile resourcefulness at its best.

His contemporary fame was largely due to the excellence of his semi-public lectures: accounts of these have survived—they are described in the diaries of the Hungarian Counts Teleki. And we are still benefitting from his insights into so many physical problems. It appears, as was later revealed by Fritz Burckhardt, that he was the first to guess the inverse square law of Coulomb but was unable to prove it experimentally. Euler has noted that in his work on Magnetism, Daniel was the first to construct an apparatus for measuring the inclination of the earth's field—done in collaboration with Dietrich, the craftsman who invented the horse-shoe magnet.

In summary, it could be said that if he had the desire and had taken the opportunity to formulate explicitly and pursue the mathematical concepts that were suggested to him by his physical insights, it is possible that his mathematical reputation might have ranked with those of his famous predecessors.

Ó Mathúna, 1999