Leonhard Euler

Ferdinand Rudio, professor at the Federal Polytechnic in Zurich gave a talk at the Town Hall in Zurich on the 6th December 1883. We give a translation by Stefanie Eminger.


Ladies and Gentlemen!1

A few weeks ago we witnessed the centenary of the day when mathematics lost one of its most outstanding representatives, a man whose name no mathematician will mention without a feeling of utmost admiration even today: Leonhard Euler from Basel. Under these circumstances and at this meeting in particular I will surely need no further justification for giving you a broad overview over the life and works of this distinguished mathematician.

The human intellect, ladies and gentlemen, manifests itself in the most diverse shapes and forms, but despite this diversity few are blessed with the ability to set themselves apart from their fellow human beings, and fewer still have the privilege of leaving lasting marks of their lives. Thus, when we know of a man, characterized by outstanding intelligence or an unusual artistic aptitude, and see that he employs his talents for the benefit and enjoyment of his fellow human beings, then, although we will not be able to follow all of his ways, we will have sympathy for him from a human point of view alone because we see a bit of human perfection embodied in him, to which we aspire ourselves, though often in vain.

And just imagine to what extent our interest is awakened when we talk about a man whose tremendous genius exceeded the ordinary to the extent that he left a mark of his intellect on an entire science for a whole century and beyond.

Leonhard Euler was such a man.

Lately an academic celebration commemorating Euler took place in his native town. The purpose of this celebration was to attest to the fact that we fully appreciate the great legacy that Leonhard Euler left us. With this in mind, this evening may be seen as a commemoration, which we dedicate, in grateful acknowledgement and admiration, to the Manes of a scientist whom you may regard as one of the greatest prides of your home country Switzerland.

Leonhard Euler2 was born on 15 April 1707 in Basel. His father, Paul Euler, was a preacher at St Jakob; his mother, Margarethe, came from the Brucker family. Euler spent his first childhood years not in Basel, but in the nearby village Riehen, whereto his father had been appointed as preacher already in 1708. The humble rural conditions of Leonhard Euler's upbringing surely contributed to his simple, modest attitude, as well as his impartiality, which he managed to preserve up to old age. There is an amusing anecdote from when he was four years old. Living in the countryside, the young Leonhard naturally had many opportunities to see how hens hatch eggs and thus produce their young. This natural process must have strongly impressed the young boy: one day he went missing, but after a long search he was eventually found in the henhouse, sat on top of a large pile of eggs that he had collected. To the bewildered question what on earth he was doing there he replied, with childlike earnestness, that he wanted to hatch chickens.

Leonhard's first teacher was his father, who prepared him for entry to the secondary schools in Basel. Note that Paul Euler liked to study mathematics when he was young; he had been regarded as a talented student of the great Basel mathematician Jakob Bernoulli. Thus it is not surprising that mathematics had a special place in his teaching. However, he did this not because he wanted to make a mathematician of his son; in fact, for him it was a matter of course that Leonhard would become a preacher, too, some day and possibly even his successor in Riehen. He appreciated mathematics as a useful training of the mind and because he saw it as the basis of any solid scientific education.

Leonhard Euler was a man of many talents, which meant that it was not too difficult for him to obey his father's wish. Thus, when he matriculated at the University of Basel later on, he did join the faculty of theology and studied oriental languages with great enthusiasm. But Euler's readiness of mind and incredible memory allowed him to engage in deep mathematical speculations as well and in particular to attend lectures by Johann Bernoulli, the brother of Jakob Bernoulli. Soon he attracted the attention of his teacher to such an extent that Europe's most famous mathematician at the time did not consider it below him to grace this youth of barely sixteen years with a more personal relationship.

Fortunately, around about that time Leonhard succeeded in finally getting his father's permission to fully devote his studies to his favourite subject, mathematics. Paul Euler had understood that his son was not born to live the contemplative life of a humble country preacher, but that he was destined to take the lead in mathematics some day, as a worthy successor of the great Bernoullis.

After Euler had gained all common academic qualifications, he competed for a prize that the Paris Academy awarded for the best paper on the rigging of ships at the young age of nineteen. Admittedly, he was awarded only second place, but the young mathematician, who had never left Basel and hence had never seen a big ship, had the satisfaction of having been defeated only by a nautical engineer, who had been considered an authority both in theory and in praxis for many years.

Around about the same time, in spring 1727, Euler applied for the vacant professorship of physics in Basel. At the time there was a curious custom at the University of Basel: the successful applicant was chosen from among the approved candidates by lot. The outcome was to Euler's disadvantage and at the same time influenced the whole of his living conditions.

Two years previously, the two brothers Daniel and Nikolaus Bernoulli, sons of Johann Bernoulli and friends of Euler, had been appointed to posts at the St Petersburg Academy then founded by Catherine I of Russia. Before they had left Basel, they had promised their young friend that if at all possible they would get him a position at the St Petersburg Academy, too. Now Euler received a message saying that a suitable vacancy had opened up, as long as he would be happy to lecture on physiology rather than mathematics. "Come to St Petersburg as soon as you can and show the Academy that, although I have told them many good things about you, I have not told them everything by far. I would argue that by your appointment I render a much greater service to our academy than to yourself", Daniel Bernoulli wrote to the then nineteen-year-old Euler.

The prospect of getting a job at such a big academy was too tempting for Euler to be put off by the condition mentioned above. He had a remarkable knowledge of the sciences as well and had already written a theory of sound. After having familiarized himself with anatomy and physiology with great enthusiasm and success, he left his fatherland that very year, 1727, at the young age of twenty. He never returned to his home country.

Upon his arrival in St Petersburg he was appointed assistant at the mathematical institute at the Academy straightaway. Strangely enough, nobody ever mentioned physiology anymore. Euler had the privilege of working alongside his friend Daniel Bernoulli for six years. The lively rivalry that developed between the two great mathematicians and that persisted until Daniel Bernoulli's death in 1782 was very important to mathematics. At this point I would like to emphasise that their friendship was never tainted by jealousy; a factor that adds a particular zest to reading their extensive correspondence, published by Fuss.

Daniel Bernoulli returned to Basel in 1733 since he could not tolerate the climate in St Petersburg, to which his brother Nikolaus had fallen victim already a few years previously. Although Euler was a mere 26 years old then, his scientific importance was already so widely recognised that nobody had any concerns about appointing him Daniel Bernoulli's successor.

In 1735 Euler gave a truly startling demonstration of the astonishing effortlessness with which he solved even the most complicated problems, but unfortunately it had a most catastrophic outcome for him. The Academy had been commissioned to make some astronomical calculations that had to be carried out in the shortest amount of time possible. All the other mathematicians at the Academy said that they would need several months for these calculations. Euler completed them in the space of three days. But what a sacrifice he made to science! The -- one would almost say superhuman -- strains to which he subjected himself during those three days led to a dangerous illness, which in turn resulted in the loss of his right eye. For any other person, the loss of such an important organ would have been a very good reason to take it easy, but Euler's industriousness increased rather than decreased because of the misfortune that befell him.

Meanwhile, the political circumstances in Russia had become unbearable for any intelligent human being, and surely I may add that this was the case for a Swiss in particular. You are all too familiar with the favouritism that spread under the successors of Peter the Great -- some of them incompetent and some of them despotic -- that I do not have to go into more detail about these unfortunate conditions, which are summarised in this sentence: "The Russian constitution is despotic, albeit alleviated by assassination".

Thus it is understandable how happy Euler was to accept the brilliant offer that Friedrich the Great made him in 1741.

It is well known that Friedrich I, the grandfather of Friedrich the Great, set up the Berlin Academy of Sciences in 1700; its first president was Leibniz. Under his successor, Friedrich Wilhelm I, whose well-known preference for soldiers did not allow for a proper understanding of scientific ambitions, the Academy decayed. When Friedrich the Great ascended to the Prussian throne in 1740 he did so with the ambition of putting his country in a respectable position, not just with regards to politics, but also to social issues and science.

To that end he first attempted to resurrect the Berlin Academy by appointing the most distinguished scholars in Europe.

Ladies and gentlemen, it must make you proud indeed to learn that among all the mathematicians alive then, the 34-year old Leonhard Euler from Basel was regarded as the most worthy one to head the series of many distinguished names that have since adorned this famous institute. Euler came to Berlin in 1741 and was appointed director of the Academy's mathematical institute immediately. He held this post for 25 years, until 1766; and he was, alongside Voltaire, undoubtedly the most prominent representative of the select circle that gathered around Friedrich the Great at the time.

The following anecdote gives us an impression of the pressure under which Euler must have lived during his last months in St Petersburg: The Queen Mother once wondered about Euler's conspicuous reticence, which he had no reason for whatsoever as she had always treated him most kindly. Euler did not fail to give the necessary explanation. "I come from a country where one gets hanged when one speaks", he replied.

Ladies and gentlemen, I have arrived at a period in Euler's life where it would be suitable to have a look at the scientific works of this marvellous man.

Now, you could not possibly expect me to cover the progress that is linked to Leonhard Euler in great detail. The nature of this lecture bans me from doing this, but Euler's immense productivity, possibly unparalleled in the history of all sciences, does so even more.

If I wanted to quickly read out only the titles of his works, I would need more time than you would want to grant me, as the index of all of his works alone fills more than 60 printed pages. This index lists more than 800 scientific publications, among them many that fill thick volumes. If one were to publish a complete edition of his works, which, I'm sorry to say, we do not have and might never have, then this edition would comprise 40 stately quarto volumes. After having returned to St Petersburg later on, Euler claimed on several occasions that he would be able to write so many mathematical papers that they would last the Memoirs of the Academy for 20 years after his death. And he did more than he had promised: His papers adorned the Memoirs of the St Petersburg Academy until 1823, i.e. 40 years after his death, and the papers that had been left in the archive were published in 1830. Furthermore, when his works were compiled in 1843, i.e. 60 years after his death, and it was believed that his mammoth legacy had finally been conquered, all of a sudden more than 50 further unpublished papers were found, which had been missed all the same.

Ladies and gentlemen, you are amazed already having just heard this dry listing; how much more would you marvel if I were able to acquaint you with the content of Euler's papers, too. But you will rightfully ask me to give you a broad overview over the area where Euler has achieved so many great things. I believe that I will accomplish this most easily by starting with a few introductory remarks on the relationship between mathematics and the sciences.

The task of the sciences is to find the laws that govern the physical world, i.e. to fathom the dependencies among the individual phenomena. Whether these connections are logically necessary, i.e. not possible in any other way, is an idle question. To us, the -- admittedly empirical -- certainty with which we can deduce the appearance of one phenomenon from the appearance of another phenomenon suffices and must suffice.

In modern sciences, people aim to consider all phenomena in terms of motion: the nature of sound consists in the vibrations of a sounding body and sound is transmitted to us by means of oscillations through the air around us; the nature of light consists in the oscillations of this extremely fine, weightless matter that is called aether and permeates all bodies, according to the wave theory of light established by Huygens and Euler and developed by Fresnel and Thomas Young; the nature of heat consists in a more or less intensive motion of the smallest particles of the heated body, according to the principles of thermodynamics. I deliberately mention these examples because the name Euler is linked to all of them, as we will see shortly.

Ultimately, the laws that govern these phenomena of motion are expressed in terms of numbers. May I use this opportunity to point out to you how accurate a feeling the Pythagoreans had developed more than 2000 years ago, in considering numbers to be the ultimate underlying principle of all being. Our modern point of view differs from the Pythagorean one only insofar as we have actually conducted experiments on a range of natural phenomena and have shown that ultimately, their nature can indeed be expressed in terms of numerical ratios.

Please allow me to pick some of the above examples and use them to derive the terms that one should know in order to be able to get an impression of Euler's research areas.

First of all, imagine that you are on top of a high tower and drop a stone. The laws that govern the motion of the falling stone have first been laid down by Galilei and can been summarised as follows: If you measure the time that the stone needs in order to cover a certain distance, it will cover the exact same distance in the same amount of time, no matter how often you repeat the experiment. If the time is doubled, the stone will cover a distance that is 2×2 or four times as long; if it's tripled, the distance is 3×3 or nine times as long; if the time is ten times as long, then the distance covered is 10×10 or 100 times as long, etc. Thus, if you have measured that the stone covers a distance of 5 metres in one second, you are now able to calculate the distance that the stone will cover in an arbitrary amount of time. For example, in 4 seconds it will cover a distance that is 4×4, i.e. 16 times as long as the distance it covers in one second, i.e. in 4 seconds it will cover a distance of 16×5 or 80 metres. Of course, you have to count the seconds from the point onwards when you drop the stone.

You can see that the distance that an object covers in free fall in a given time can be deduced mathematically. We say that the distance covered is dependent on the length of time or that it is a function of time, and since 2×2 is called the square of 2, 3×3 the square of 3, and 10×10 the square of 10, we say that the distance increases proportionally to the squares of the given lengths of time.

Let us move on to a second example: to Kepler's laws. Kepler discovered that the Earth orbits the Sun in an ellipse, with the Sun being one of its foci. In more popular terms, the Earth rotates around the Sun on a circle-like line and the Sun is located almost at the centre of this circle. Now imagine a line drawn from the Sun to the Earth. We will call this line a radius vector. As the Earth moves around the Sun, this radius vector moves around the Sun and sweeps out a portion of the circle-like area during a given interval of time. We call such a portion a sector. Kepler's second law now states that the Earth orbits the Sun in such a way that the radius vector sweeps out equal sectors during equal intervals of time. Thus, if you have found by observation how large a sector the radius vector sweeps out in, say, one hour, then the sector swept out in two hours will be twice as large, the one swept out in three hours will be three times as large, the one swept out in ten hours will be ten times as large, etc. We say that the sector is dependent on time, or a function of time. Moreover, the sector grows proportionally to time.

Let us move on to the third example. Imagine a point of light and, at some distance from it, a sheet of white paper. Then the point of light illuminates the sheet to a certain degree. The brightness of the sheet increases the closer the sheet is to the point of light, and decreases the further away it is from the point. Now if you measure the sheet's brightness at a given distance, then the brightness will decrease by a factor of 2×2 or 4 as the distance is doubled, by a factor of 3×3 or 9 if the distance is three times as big, by a factor of 10×10 or 100 if the distance is ten times as big, etc. We say that the brightness is dependent on distance, or that it is a function of distance. In particular, we say that the brightness decreases proportionally as the squares of distance increase.

Having heard these examples, you will now understand what we mean by saying that the aim of the sciences is to express the interdependencies of individual phenomena in terms of mathematical functions, in that we consider a function to be the dependency of two quantities expressed in terms of numbers. Since the natural phenomena are dependent on each other in the most varied ways, there are infinitely many mathematical functions -- but please do not believe that these dependencies and hence the corresponding functions are always as simple as in the examples I mentioned above. There are some highly complex cases around. We will now refer to mathematics as the language with which natural phenomena can be described most simply and at the same time most thoroughly. As an example, it would not be possible to describe the motions of Earth around the Sun in a more simple and comprehensive manner than by Kepler's laws.

We have now arrived right at the centre of Euler's research area. It is one of Euler's main achievements to have studied the myriad of functions that were either offered to him directly by nature or that his ingenuity first had to derive, in extenso for the first time: he investigated their properties and identified the source of these properties, and he grouped them together according to common features. Furthermore, he ascribed functions that had been regarded as distinct, such as the so-called trigonometric and exponential functions for example, to each other. He dedicated two particular major works to these investigations: his Introduction to Infinitesimal Calculus and his Manual for Differential and Integral Calculus. Even today, more than a hundred years later, these books are still the most readable of all textbooks on higher analysis. Although many books have been written on this topic since, almost all of them are more or less variations of the area studied by Euler.

But I cannot move on from reviewing Euler's mathematical work without having considered an important factor. I have said that mathematics is a language in which natural phenomena can be described in the simplest and most comprehensive manner. With this in mind, you will understand how important it is to express mathematical thoughts themselves as concisely and clearly as possible. In this respect, Euler's work was epoch-making. We can be safe to say that the whole form of modern mathematical thinking has been created by Euler. If you read any author immediately before Euler, it is very difficult indeed to understand his terminology, as he has not yet learned how to let the formulas speak for themselves. This art was not taught until Euler came along.

But Euler was not just a great mathematician, he was also a great physicist and astronomer. He wrote several rather major works on the motion of celestial bodies and was the first to write on analytic mechanics. He might have been the only one in his century to have a correct idea of the nature of heat: he taught that there is no special heat matter, but that the nature of heat consists in the motion of the smallest particles of the heated body. Furthermore, and in opposition even to an authority like Newton, he supported the theory first expressed by Huygens that the nature of light does not consist in a special light matter, but in the oscillations of the aether filling the universe.

Please allow me to point out a special achievement of Euler's in optics. You will all have noticed at some point that, when you look through cut glass, i.e. a glass prism or a glass lens, the objects you see appear to have not only contorted shapes, but also coloured fringes. These coloured fringes are caused by the varied refrangibility of the individual colours that are the components of colourless light. But when using optical instruments, such chromatic fringes are highly distracting -- you can see this for yourselves by using a bad lorgnette, for example. Euler discovered that these chromatic distortions are absent in the human eye, which, at the end of the day, is an optical instrument, too. The reason for their absence is that light passes through several substances of different refractivity inside the eye, so that the various chromatic distortions cancel out. Inspired by this discovery, he calculated which combinations of lenses one has to use in order to construct achromatic instruments, i.e. instruments free from those chromatic distortions. When Euler published his results, he was attacked most severely from all directions; in particular by the English physicist Dollond, who referred to Newton's explanation that achromatic instruments were considered impossible. But Euler was so certain that his calculations were correct, that he did not budge until his opponent Dollond himself constructed the first achromatic telescope using a combination of flint and crown glasses in 1758. In doing so, he validated Euler's results most splendidly. Dollond is generally referred to as the creator of this seminal invention for optical instruments in physics textbooks, but it would only be an act of justice if Euler were mentioned alongside Dollond, as indeed he is the intellectual creator of this important invention.

Although Euler was without a doubt the most important mathematician of the last century (and perhaps of all centuries), he still found time to delve into studying a range of purely practical problems. I will only mention that we owe a comprehensive treatment of artillery sciences to him, in which he developed a complete theory of the motion of thrown objects. Furthermore, he made a range of valuable contributions to shipbuilding by developing the theory of floating bodies and deducing which shapes ships need to have in order to be as manoeuvrable as possible whilst also being as stable as possible. These works caused greatest sensation at the time and have been translated into almost all European languages.

I have now reached an area in this short overview over Euler's merits that might be of particular interest to you: his popular works.

I will only talk about one of them, Euler's Letters to a German Princess. These letters are addressed to a niece of Friedrich the Great and are the continuation of the lessons that she received from Euler. He covers the most important aspects of astronomy, of mathematical and physical geography, of physics, and of philosophy in 234 mostly very short letters, using such a clear, lucid language -- I would almost go as far as to say pleasant -- that the letters may still be considered an exemplary popular presentation. Time bars me from looking at them in more detail so I have to limit myself to pointing out those letters to you. However, I would count myself lucky if this evening would at least result in these Letters to a German Princess by Euler attracting the interest that they so highly deserve among a wider circle of readers.

Euler left Berlin in 1766, i.e. in the 60th year of his life, and returned to St Petersburg. The motivations for this change were in part some differences with the Berlin Academy, but in particular the splendid offers made by Empress Catherine II and which Euler, who had a very big family, did not dare reject. Having barely arrived in St Petersburg, he was taken severely ill. Although he recovered in time, he lost also his second eye entirely. Thus, Euler was completely blind during the last 17 years of his life. However, now Euler's unusual mental abilities truly showed themselves: henceforth, the totally blind old man developed an almost frenetic activity; almost half of his entire production dates to the years when Euler was bereft of a scholar's most valuable organ.

Shortly afterwards, he was hit by another misfortune, which, given the circumstances, must have affected him particularly severely. His house, a gift from the Empress, fell prey to a big fire. His library and part of his manuscripts were burnt. He himself would have died in the fire had it not been for a man called Grimm, who was from Basel but lived in St Petersburg. He saw the danger that his famous compatriot was in, entered the burning house at the risk of his own life and carried the blind old man out of the flames across his shoulders.

We truly have to admire the exceptional calmness and serenity of mind that Euler must have possessed to be able to return to his scientific research again and again after such severe strokes of fortune. Admittedly, his incredible imagination and a downright phenomenal memory certainly helped him in this. Euler belonged to those mathematicians who had the entirety of their science at their command at every moment.

The subsequent details might offer an indication of his mental abilities. In a sleepless night, Euler, aged 75, calculated the first six powers of the first 20 numbers and recited them forwards and backwards for several days. In his old age he still knew the entire Aeneid by heart; in fact, he could state the first and last verse on every page of the edition that he used in his youth.

Euler possessed what we today call a general education to a very high degree. He had a thorough knowledge of classic Antiquity, of history and of literature. He knew more about medicine and the sciences than most people; we have heard that he was appointed a physiologist at the St Petersburg Academy at the mere age of twenty. He dedicated his leisure time to the musical arts, but he revealed himself as a mathematician at the piano, too, he even wrote a theory of musical arts.

Euler was an excellent person, not something that you can say about every great man. He was unusually kind-hearted and almost naively pious. You might also be interested to learn that Euler never stopped being a Swiss, for although he lived in Berlin for 25 years and in St Petersburg for 31 years, he always used the genuine Basel vernacular with all its peculiarities, often to the amusement of those around him.

Euler's death was worthy of a great scholar; he passed away whilst being engaged in research. Even on 18 September 1783 he studied the motion of balloons, which had just started to emerge, with the same vivid interest with which he approached every new invention. He had mastered a complicated calculation and talked about it with a friend, when he suddenly fell back and his quill fell out of his hand: Euler had ceased to calculate and to live.

A century has passed since then, a century full of progress in mathematics. But as great as the brilliant discoveries by Lagrange, Gauss, Jacobi, are, we are still under the dominant influence of this formidable person that was Euler. We do not read his works out of historical interest or to learn what people thought about this or that difficult problem last century. We recognize him as our teacher, to whose guidance we still submit ourselves today, full of the humility and admiration that his intellectual superiority inspires in us.

However, I want to conclude this remembrance of Euler by looking at a different angle. The century that separates us from Euler is abundant, perhaps over-abundant in technological progress. But for all that, it is an undisputed fact that this progress is very closely linked to the development of mathematics, even if this connection is not always as obvious as in the case of Euler inventing the achromatic telescope. Thus, Euler's contribution to the great achievements that humanity takes both pride and delight in today is not to be underestimated, and hence his name deserves to be known and recognized even by those who have no interest in mathematics.

At the St Petersburg cemetery, a mighty block made from Finnish granite, with the inscription "Leonardo Eulero Academia Petropolitana", reminds the wanderer that he is at the same place where the mortal remains of this outstanding mathematician are buried. It is possible that, in thousands and thousands of years, the stone will have been removed due to events of some kind, that its inscription will be weathered and its significance will have been forgotten. But the name Leonhard Euler will live on as a symbol of intellectual perfection for as long as there is civilization, for he himself has raised himself a monument that is greater, more sublime and more imperishable than any man-made structure: his immortal works.

1 This lecture, first published in the well-known collection by Benno Schwabe, has been out of print for many years now. Although demand never ceased, I would never have considered to re-publish it, had it not been for the fact that just now, as we are about to launch a fundraising campaign in aid of an edition of Euler's works, people expressed the wish that the call for donations would be accompanied by a short biography of Euler aimed at the general public. Thus, the lecture that I gave more than a quarter of a century ago may set off into the world again. And if it helps contribute to Euler's works finally being reborn in an edition worthy of the eminent mathematician, then the reprint is not uncalled-for after all. And at the same time a wish will come true, a wish that I had already hinted at then (see p. 462), but at the time I did not dare hope that I would witness its implementation in my lifetime.

2 For the biographical part, I used the commemorative speeches by Condorcet and Fuss, the Correspondance mathématique published by Fuss, as well as the Biographies in Switzerland's Cultural History by Prof. R Wolf.


JOC/EFR April 2013

The URL of this page is:
http://www-history.mcs.st-andrews.ac.uk/Extras/Rudio_Euler.html