**Balthasar van der Pol**was known as Balth. His father, also named Balthazar van der Pol, was a rich tea merchant with broad cultural interests and he provided his son with ample opportunity to develop his many talents. His mother was Gerhardina Clasina Steffens. Balth attended the HBS (hogere burgerschool) in Utrecht and graduated from the gymnasium in 1911. He then entered the University of Utrecht where he studied physics and mathematics until 1916 when he graduated with a degree in Physics for which he was awarded the highest distinction.

In 1916 van der Pol went to study with John Ambrose Fleming, an English electrical engineer and physicist who was Pender Professor at University College London. Fleming was the first professor of Electrical Engineering at University College, and is well known for inventing a diode, the first thermionic valve, in 1904. On 2 June 1917 van der Pol married Pietronetta Posthuma in London; they had a son and two daughters. After a year working with Fleming, van der Pol remained in England, but went to Cambridge to work with John Joseph Thomson at the Cavendish Laboratory. Thomson had announced his discovery of the electron in 1897, a discovery which came out of his study of cathode rays. He was President of the Royal Society during the years that van der Pol worked with him. During his time at the Cavendish Laboratory between 1917 and 1919, van der Pol met and became friends with Edward Appleton who, thirty years later in 1947, was awarded the Nobel Prize of Physics for his contributions to understanding the ionosphere. Appleton explained the work on which van der Pol was engaged during these two years in the Cavendish Laboratory [4]:-

Although most of the work for van der Pol's doctoral thesis was undertaken at Cambridge, his thesis supervisor was Willem Henri Julius, director of the Physics Laboratory of the University of Utrecht, described by Einstein as 'one of the most original exponents of solar physics'. On returning to the Netherlands in 1919, van der Pol was appointed to the Teylers Museum in Haarlem named after Pieter Teyler van der Hulst, a wealthy 18Van der Pol's radio work was begun at Cambridge and was under two heads, experimental and theoretical. We can say that Heaviside had 'invented' the ionosphere to explain Marconi's remarkable transmission by radio over the Atlantic in1901. In his theoretical work van der Pol gave a quantitative proof that, if one neglected the influence of a reflecting layer, experiment and theory were in flagrant disagreement in the case of long-distance propagation. To do this van der Pol familiarised himself with the theoretical work on the diffraction of radio waves round a conducting earth and used this to make a direct comparison between(a)signal strength predicted and(b)signal strength received, in a practical case of radio transmission. The paper in question is in the Phil. Mag. Sept.1919and is worth consulting. ... In this work van der Pol showed great experimental skill. He made his own apparatus, as was customary in those days of J J Thomson. His triode oscillator gave him waves of about3metres, which was about as short as anyone else had produced in those days. All this work on the dielectric constant of ionised air formed the basis of his Doctor's thesis. It was round about this same period that van der Pol and I collaborated in the study of non-linear phenomena using triode circuits to cheek our theories. Together we worked at(a)oscillation hysteresis and(b)forced vibrations in a non-linear system. Van der Pol's work on relaxation oscillations was, of course, entirely his own.

^{th}century cloth merchant. An Amsterdam banker of Scottish descent, Teyler had bequeathed his fortune for the advancement of art and science. Hendrik Lorentz had been appointed director of research at Teylers in 1910 and under his leadership Teylers Museum conducted research in optics, electromagnetism, radio waves and atomic physics. It was an ideal environment for van der Pol who completed his doctoral thesis

*De invloed van een geioniseerd gas op het voortschrijden van electromagnetische golven en toepassingen daarvan op het gebied der draadlooze telegraphie en bij metingen van glimlichtontladingen*Ⓣ and submitted it to the University of Utrecht. He was awarded the degree of doctor of science (with distinction) by the University of Utrecht on 27 April 1920.

In 1922 van der Pol left Teylers Museum to take up an appointment as Head Physicist at Philips Physical Laboratory in Eindhoven. The company was established in Eindhoven in 1891 making carbon-filament lamps. The physics research laboratory to which van der Pol was appointed had been founded in 1914. By 1922 it was involved in many diverse projects from X-ray radiation to radio reception. Van der Pol later became Director of Scientific Radio Research. In 1927 he was made a Knight of the Order of Oranje Nassau for establishing the first radio-telephonic communication between the Netherlands and the Dutch East Indies. He continued to work at Philips until 1949 but during part of this time, from 1938 to 1949, he held the Chair of Theoretical Electricity at the Technical University of Delft.

Let us look at some of van der Pol's scientific contributions. Morris Kline writes:-

H Bremmer in [2] lists van der Pol's main contributions under the headings: propagation of radio waves; non-linear circuits: relaxation oscillations; transient phenomena, and operational calculus. Let us look first at the last of these topics on which Bremmer and van der Pol collaborated in writing the classic textThe scientific work of Balthasar van der Pol covered pure mathematics, applied mathematics, radio, and electrical engineering. ... even in mathematics, his papers covered number theory, special functions, operational calculus and nonlinear differential equations. In this last field he was a pioneer.

*Operational Calculus: Based on the Two-Sided Laplace Integral*. Arthur Erdélyi, reviewing the first edition published in 1950, writes:-

This 1950 book was not the first joint work of van der Pol and Bremmer on the operational calculus, for example they published two papers with the titleThis book is intended as a treatise on the application of the operational calculus in its modern form to mathematics, physics, and engineering. The authors have adopted a rather attractive and unique mode of presentation to cater to the needs of applied mathematicians and engineers. They seem to feel that on the one hand modern applications of the operational calculus require theorems under rather general conditions, and on the other hand the readers they have in mind are not interested in, or not able to follow, proofs under such general conditions. ... The book is very well written, and may be read with profit by both mathematicians and engineers. The mathematical treatment is more advanced than in most other books devoted primarily to applications, and on this account the work is not suitable as a text-book except for a graduate course for engineers or physicists of unusually strong mathematical background ...

*Modern operational calculus based on the two-sided Laplace integral*in 1948. A second edition of their text was published in 1955, then a third edition in 1987.

Of course, to most mathematicians the name of van der Pol is associated with the differential equation which now bears his name. This equation first appeared in his article *On relaxation oscillation* published in the *Philosophical Magazine* in 1926. Details are given in several of the references (see for example [4], [9], [10] and [11]). Besterenko in [10] writes:-

We mentioned above that van der Pol was interested in number theory. Robert Rankin surveys his work in this area in [4]. We give an extract:-We consider the first attempts at solving nonlinear problems of the theory of oscillations. The first observation of the inapplicability of the linear theory to problems of this type was made in radio-engineering by van der Pol. We explain the history of the development of the equation carrying his name, and also the origins of the method of finding the first approximation to the solution of this equation(the method of slowly varying coefficients).

As another example of his work on number theory we mention the paperOf van der Pol's papers on the theory of numbers[An electro-mechanical investigation of the Riemann Zeta function in the critical strip(1947)]is perhaps the best known. In it he combined his knowledge of radio technology and number theory to advantage. In order to investigate the behaviour of the Riemann zeta-functionζ(s)on the lineRes=^{1}/_{2}he derives a formula ... the saw-tooth[part of which]was cut on the circumference of a paper disk and a beam of light was projected past the teeth on to a photocell. The electric current so produced eventually yielded a record, rather like an anemometer trace, of the modulus ofζ(^{1}/_{2}+it), from which the first73zeros could be read off with decreasing accuracy for increasing values of t. The branch of number theory, however, which lay closest to his heart was the theory and applications of theta-functions. His published work on this subject is contained in four papers; mention should also be made of his highly individual 'Lectures on a modern unified approach to elliptic functions and elliptic integrals'(mimeographed notes)given at Cornell University in1958.

*The primes in*

*k*(

*p*) (1951). D H Lehmer writes:-

Mary Cartwright gives a full list in [4] of the honours which were give to van der Pol:-This paper presents, in graphical representation, a list of primes in the quadratic field defined by a primitive cube root of unity. The diagram is based on regular hexagons and gives all primes whose norms do not exceed10000. The symmetry of the diagram produces a strikingly beautiful effect.

Mary Cartwright, who knew van der Pol personally, also writes [4]:-From1945to1946he was in addition[to his other pasts]President of the Temporary University at Eindhoven which was founded to replace other Netherlands universities in occupied territories, and for his work as president he was made a Knight of the Order of the Netherlands Lion in1946. ... He was an outstanding figure in the various societies established to promote the study of radio. He became a member and then fellow and life member of the Institute of Radio Engineers(U.S.A.)in1920; he was Vice-President of it in1934, and was awarded its Medal of Honour in1935for contributions to circuit theory. He was a Founder-member of the Nederlandsch Radiogenootschap, for a time its President, and later an honorary member. From1934to1952he was Vice-President of the Union Radio Scientifique Internationale(U.R.S.I.)and from1957to1959he represented it on the Executive Board of the International Council of Scientific Unions. He travelled widely in connection with his work, attending plenary sessions of the U.R.S.I. or telecommunications' conferences nearly every year in various foreign countries. He was an honorary member of the Institute of Radio Engineers of Australia. Many foreign academies and universities also honoured him; the Danish Academy of Technical Sciences awarded him the Valdemar Poulsen Gold Medal in1953for outstanding contributions to radiotechnics and particularly for research in the field of electro-magnetic oscillations and wave-propagation and for international scientific cooperation and organization of technical questions relating to radio communication; the Technical University of Warsaw gave him an honorary degree in1956, and the University of Geneva in1959, and the French Academy of Sciences made him a Corresponding Member in1957.

In fact van der Pol corresponded with Nikolai Mitrofanovich Krylov about the theory of nonlinear oscillations; a letter sent by van der Pol to Krylov is published in [9].Van der Pol did much to popularize his subject; he was an engaging lecturer, and often took the opportunity of bringing together phenomena over a wide field of science which could be elucidated by a single mathematical relation such as the equation for relaxation oscillations. His summary['The non-linear theory of electric oscillations'(1934)]of work on non-linear oscillations up to1934was a quite masterly account of the theory up to that time. It gives many references, including some to Russian work ...

After his retirement in 1949 he became Director of the Comité Consultatif International des Radiocommunications in Geneva. He continued in this position until 1956 after which he settled to Wassenaar. However, he held a temporary professorship at the University of California, Berkeley, for the year 1957 and then was the Victor Emanuel Professorship at Cornell in Ithaca, New York, in 1958.

**Article by:** *J J O'Connor* and *E F Robertson*

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