Math Tutorials and More
by George

In this section we will look at some of the areas in which Leonhard Euler made significant contributions. However, Euler's interests were so broad and the number of his publications so large that it is impossible to cover them all. Hopefully, what I have presented will give you some idea of the importance of his scientific work.

Dioptrics

Different colors, having different wavelengths, are refracted differently by a lens. The image produced by a spherical lens has fringes of rainbow like color. Euler investigated the cause of this distortion. In Memoires of the Berlin Academy in 1747, 1752, and 1753 he showed how to eliminate chromatic distortion of images by lenses. He showed that it was necessary to use multiple lenses with different refractive properties. The motivation for his work came from looking at the human eye that doesn't have this problem.

Newton had tried to do this using two lenses separated by a water layer, but did not succeed. Newton claimed that it was not possible to correct for chromatic distortion in lenses and he began looking at reflective type telescopes using mirrors. Euler's theory predicted that correction should be possible using lenses of the type Newton used. The Englishman Dolland performed experiments to test Euler's theory. He actually believed going in to the experiment that Newton was correct, but his results confirmed Euler's theory. In 1757 Dolland was able to construct a lens system made of two types of glass that was free of chromatic aberration. Such a lens is called an achromatic lens. This is a good example of where a theoretical discovery has led to a practical application.

Euler's three large volumes entitled Dioptrica (1761–1771) provide a complete theory of the way light waves act in lenses. The first volume presents the general theory while the second and third volumes examine manufacturing processes for producing eye glasses, telescopes and microscopes.

Naval Science

Euler was very interested in the construction of ships and in navigation. His major work on this topic was Scientia Navalis published in 1749. It consisted of two large volumes. Some of the topics covered were

1. the equilibrium of ships;
2. the stability of equilibrium;
3. the oscillations of ships;
4. inclination under the influence of arbitrary forces;
5. the effect of rudders;
6. the effect of oars;
7. the construction of rowed ships;
8. the resistance of water to moved bodies;
9. the force exerted by the wind on a sail;
10. masting of sailing ships;
11. a ship on a skew course.

Later Euler was worried that the above work could not be understood by ordinary seamen. Therefore, in 1773 he published a work that could be understood by laymen. It was entitled Complete theory of construction and piloting of ships as to be applied to those who navigate.

Lunar Motion

Calculating the orbit of the moon is much more difficult than calculating the orbit of the earth around the sun. In calculating the earth's orbit it is only necessary to consider the gravitational attraction between the earth and the sun. To first order the attraction of the other planets and the moon are negligible. However, for the moon, both the earth's attraction and the sun's attraction are important. Euler published a number of articles on this subject. Two of the longer publications were Theory of the motion of the moon which exhibits all its irregularities, 1753 (355 pages) and Theory of lunar motion, 1772 (791 pages). These publications contain results that involved a tremendous amount of calculation. The Theory of lunar motion has one table that is 144 pages long. Euler won the first prize of the Paris Academy of Sciences in 1770 and 1772 with papers on Lunar motion. Tobias Mayer used Euler's formulas to calculate Lunar tables for use in the calculation of Longitude. The Board of Longitude awarded 500 pounds to Mayer and 300 pounds to Euler for their contribution to navigation.

Analysis and Mechanics

I have lumped analysis and mechanics together since analysis is the key to understanding mechanics and mechanics provided most of the problems in analysis that math­ematicians in the eighteenth century addressed. Over one half of the pages Euler published were either expressly devoted to mechanics or involved closely related topics [2]. Newton and Leibnitz invented calculus in the seventeenth century, but it was Euler in the eighteenth century that developed it to the point where it was useful in solving physical problems. He showed that Newton's laws and other laws of physics could be formulated in terms of differential equations. He also investigated techniques for solving these differential equations. In particular, he pioneered the use of power series expansions.

Euler was the first to publish a paper on partial differential equations. He also published the first textbook on calculus that could in any sense be considered complete. Historian Carl Boyer calls it the most important textbook of modern times. Here is a quote from The Foremost Textbook of Modern Times (1950) [4].

The most influential mathematics textbook of ancient times is easily named, for the Elements of Euclid has set the pattern in elementary geometry ever since. The most effective textbook of the medieval age is less easily designated; but a good case can be made out for the Al-jabr of Al-Khwarizmi, from which algebra arose and took its name. Is it possible to indicate a modern textbook of comparable influence and prestige? Some would mention the Géométrie of Descartes or the Principia of Newton or the Disquisitions of Gauss; but in pedagogical significance these classics fell short of a work by Euler titled Introductio in analysin infinitorum. Here in effect Euler accomplished for analysis what Euclid and Al-Khwarizmi had done for synthetic geometry and elementary algebra respectively. Coordinate geometry, the function concept, and the calculus had arisen by the seventeenth century; yet it was the Introductio which in 1748 fashioned these into the third member of the triumvirate — comprising geometry, algebra, and analysis.

… Euler avoided the phrase analytic geometry, probably to obviate confusion with the older Platonic usage; yet the second volume of the Introductio has been referred to, appropriately, as the first textbook on the subject. It contains the earliest systematic graphical study of functions of one and two independent variables, including the recognition of the quadrics as constituting a single family of surfaces. The Introductio was first also in the algorithmic treatment of logarithms as exponents and in the analytic treatment of the trigonometric functions as numerical ratios.

The Introductio does not boast an impressive number of editions, yet its influence was pervasive. In originality and in the richness of its scope it ranks among the greatest of textbooks; but it is outstanding also for clarity of exposition. Published two hundred and two years ago, it nevertheless possesses a remarkable modernity of terminology and notation, as well as of viewpoint. Imitation is indeed the sincerest form of flattery.

Euler did fundamental work dealing with the gravitational interaction of point masses, the motion of rigid bodies, and with elastic fluids and solids. His publications in these areas are too numerous to list.

Hydrodynamics and Hydraulics

Euler published the first general treatise on hydrodynamics. It consisted of three parts. The first paper entitled Principles of the motion of fluids was published in 1761. The second entitled Section two of Principles of Fluid Motion was first presented to the Saint Petersburg Academy in 1766. The third entitled The third chapter on the linear motion of fluids, especially of water was also presented in 1766. Euler's treatise on hydrodynamics contained the general equations for the motion of an ideal fluid (called Euler's equation), the conservation of mass (Equation of Continuity), and the conservation of fluid energy. It would be another 100 years before another treatise on hydrodynamics was published.

In hydraulics, Euler was the first to provide a complete theory of fluid driven turbines. His most in-depth treatment of the subject is contained in the paper Complete theory of machines activated by water, 1756. His treatment is so complete that an engineer today could use it to design a turbine.

Insurance

Euler wrote several fundamental papers in actuarial science that is the foundation of the insurance industry. One of these is entitled General investigations on the mortality and the multiplication of the human race, 1767. In it he considers such topics as

1. A certain number of men, of whom all are the same age, being given, to find how many of them are probably yet alive after a certain number of years.
2. To find the probability that a man of a certain age be still alive after a certain number of years.
3. One demands that probability that a man of a certain age will die in the course of a given year.
4. To find the term in which a man of a given age is able to hope to survive, of the kind that it is equally probable that he die before this term as after.
5. To determine the life annuity that it is just to pay to a man of any age all the years, until his death, for a sum which will have been advanced first.
6. When the interested parties are some infants newly born and when the payment of the life annuities must begin only when they will have attained a certain age, to determine the amount of these life annuities.

The Calculus of Variations

In calculus it is shown that a smooth curve has zero slope at places where it has a maximum or a minimum. The slope of the function defining a curve is called the derivative. Thus, the derivative of a smooth function is zero at points where it achieves an extreme value. The calculus of variations began as a generalization of this concept. Here the maximum or minimum is over a set of functions rather than over a set of points. For example, Fermat (1601–1665) studied optics problems involving more than one medium with different light speeds. He found that light follows the path requiring the least travel time. Here the travel time is minimized over all possible paths. A generalization of the derivative, called the variation, was defined for problems of this type that is zero for the desired optimum. It turns out that virtually all the laws of Physics can be expressed in variational form, i.e., in a form in which some scalar quantity has zero variation at the desired solution. Although the calculus of variations started out looking at maximums and minimums, the condition that the variation is zero over a class of function may not correspond to a maximum or minimum.

The calculus of variations played a big role in the development of Einstein’s general theory of relativity. Leonhard Euler was responsible for much of the early development in this field. Historians tell us that the motivation for the early development of the calculus of variations came from a belief in a God who designed the physical world in an efficient and optimal manner. Euler began by considering geodesic problems, i.e., finding the curve joining two points on a surface having minimum length. In 1728 he wrote a paper entitled On finding the equation of geodesic curves. Later, in 1744, he published a more general work entitled A method for discovering curved lines that enjoy a maximum or minimum property, or the solution of the isoperimetric problem taken in the widest sense. One of the things he considered in this paper was the minimization of integrals of the form

\begin{equation*} I=\int_a^b f(x,y,y')\,dx. \end{equation*}

He showed that a necessary condition for a minimum was the Euler equation

\begin{equation*} \frac{\partial f}{\partial y}-\frac{d}{dx}\left(\frac{\partial f}{\partial y}\right)=0. \end{equation*}

Euler also applied variational methods to problems in mechanics. The application of variational methods to mechanics began with the principle of least action. The credit for this principle is usually given to the French math­ematician Maupertuis (1744), but was probably discovered earlier by Euler. It appeared in Euler's 1744 paper cited above. In the form developed by Euler, the action was the time integral of twice the kinetic energy (then known as the living force) over the path. The principle states that the actual path minimizes the action. It is now known that the principle in this form only holds for conservative systems, i.e., systems for which the sum of the kinetic energy and the potential energy is a constant.

Topological problems

Euler was very interested in geometric problems in which the concept of distance was not involved. He used the term geometry of position to describe such problems. Today we would include such problems under topology. An example of this type of problem is the Königsberg bridge problem. In the eighteenth century Königsberg was a city in Germany. It is now called Kaliningrad and is located in Russia. Figure 6 shows what present day Kaliningrad looks like. The Pregel river passed through Königsberg forming two islands as is shown in Figure 7a. There were seven bridges crossing the river. The problem was to see if there was a path a person could take that crossed each bridge exactly once and returned to the original starting point. Euler solved this problem in 1736, showing that it was not possible. It is often incorrectly stated that Euler solved this problem using graph theory [5]. It is true that today this problem is usually solved by making a graph as shown in Figure 7b.

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