Euler : Some Papers

translated and annotated by
Ian Bruce et al.

     Euler's early papers mathematical papers show the influence of Johan Bernoulli, his mentor. E001 and E003 are concerned with the isochrone and brachistochrone problems. E002 is a dissertation on sound, which was presented to the University at Basel in a vain attempt to gain the vacant physics chair; E005 is concerned with curves that intersect orthogonally, a much longer paper, and this is perhaps Euler's first mature paper; all of course show his mathematical brilliance, although he has not yet made the transition to completely analytical methods. E006 is concerned with the involute of the circle, formed by unwinding a string along the tangent, as a correcting mechanism for the period of a chronometer, developed originally by Huygens in the Horologia. E007 is an attempt to explain atmospheric phenomena in terms of air vesicles, fine matter, and centrifugal force. E008 is about the general solution of heavy planar curves under various loadings, catenaries, sails, etc. E009 is a classic early paper on the shortest curve joining two points on a surface. E010 is the start of Euler's love affair with the exponential function, related to easing the pain of solving differential equations. The papers presented here in pdf format are taken from the appropriate volumes of Euler's works. E011 is a later paper, and relies on previous work not yet covered in this series of translations. E012 is a masterful work, in which Euler first establishes a surprisingly simple geometric condition for tautochronic curves, and then shows how to generate such curves, both analytic and algebraic, starting from the familiar cycloid ; E013 extends the analysis to a resistive medium where the resistance is in proportion to the square of the speed. E014 is an elementary treatment of finding the pole star from three measurements on a star over time. Both the translation and the Latin version are presented, the latter usually at the end. These are amongst the most popular of the translations presented on this site. I wish to thank the members of the Euler Archive based at Dartmouth College for providing most of the original pdf files.


The (erroneous) construction of tautochronous curves in media with different forms of resistance. This is Euler's first paper E001; the math in the translation has been clarified (Feb. '07) to account for Euler's mistake.

Link to E001 by clicking here.

You can do likewise for the other papers; use the browser 'Back' arrow to return to this screen.

A short survey in two chapters of the state of the theory of sound at the time and of sources, with special reference to the flute. Note an error in Wikipedia that I have corrected (March 2010) : this paper is not Euler's Ph.D. thesis – he never did a doctorate – but was written in support of his application for the position of Physics Professor at Basle University at the time, in which he was unsuccessful. This error appears all over the web with the message that I have translated it!

A method for finding algebraic reciprocal trajectories is presented, now complete.

Euler's essay on the location, height, and number of the masts on ships to maximize the speed.

 Presented to the French Academy of  Science in 1727 and published the following year.

The Solution of the Problem of Reciprocal Trajectories. This problem had been solved initially by the editor of the 3rd edition of Newton's Principia, Pemberton, and the solution send in code to Johan Bernoulli as a challenge. This was a source of great embarrassment to Bernoulli, though of course he came up with his own solution; hence Euler's interest in presenting a general solution.

A blend of analysis, dynamics, geometry, and algebra is used in this interesting paper to derive the curve necessary to provide s.h.m. for large angles for a cylinder which forms the timing mechanism for a marine chronometer, modifying an original design of Sully. The associated correcting curve is the involute of the circle. Both analytic and algebraic solutions are given.

What amounts to an equation of state is established for air, even though temperature enters in an odd way. However, it is based on a hypothetical model due to J. Bernoulli, involving spherical vesicles containing fine matter with a permeable membrane inflated by centrifugal force. Euler's ability shines through as he analyses this model, but he is unable to fit it to experimental data in a convincing manner, which is not really very surprising. However, a read of this paper will convince you of the progress made in the last 300 years in understanding the structure of matter.

Another blend of analysis, statics, geometry, and algebra is used here in this interesting paper, in which the Calculus has definitely 'Come of Age', as it were. Heavy curves associated with chains, ropes, sails, strings, etc, are analysed both with and without elasticity, after a general formula has been found from statics principles.

In which Euler determines a general differential equation by means of which points on surfaces are joined by curves of the shortest length. The method is applied to general cylinders, conical surfaces, and surfaces of revolution.

In which Euler presents a method for reducing the order of certain second order differential equation to first order by means of changes of variables using exponential functions, for which one variable has a constant rate of change; and these reductions can also be established directly from carefully chosen exponential transformations.

In which Euler presents further methods for transforming differential equations; however, it dates from 1735, and lay some years in the future from his current work, and quotes from a paper E031 on Riccati differential equations.

In which Count Ricatti presents his thoughts on solving second order differential equations by using some cunning transformations that were to inspire Euler. Originally published in the Act. Er. of 1724.

In which Euler presents methods for producing tautochrone curves including cycloids and other curves, after establishing an interesting basic geometrical principle involving a differential of order zero from the height of the curve.

E013 In which Euler presents a method for producing tautochrone curves in a resistive medium. The method compares the motions of both cylinders and spheres on a cycloid in a vacuum with those on the curves sought. In the analysis, the series expansion for the exponential function is introduced, as well as the procedure for evaluating a constant of integration.

In which Euler presents a method for checking the elevation of the pole star and the declination of a fixed star from 3 measurements over a period of time.

A translation of Euler's Mechanica Vol. I is available here.

A translation of Euler's Mechanica Vol. II is available here.

A fascinating paper in which Euler explores transcendental progressions in which the general terms are infinite products related to quotients of factorials - relating initially to the Wallis product for pi, and in which integrals are found for the general terms; a derivation of the 'half-derivative' is given finally as a final consequence. This paper is considered as an application of the next paper presented E020. [An error in the original date of publication has been corrected ; Dec. '08]

Another fascinating paper in which Euler explores the summation of transcendental progressions in which the general terms are generally derived from the familiar g.p.; the beginnings of Euler's fascination with the sum of the inverse squares of integers is made here.

Here Euler returns to enlarge on an earlier paper, E006, but using the later approach of E012, and investigates a possible timing mechanism for a clock, in which two weights unwind from curves attached to a pulley. The physics is interesting, as he shows how the vis viva method is used with the rotational motion of the pulley. The latter part of the paper is purely mathematical, and relates to finding appropriate pairs of isochronous curves : one may be given and the other found ; or both are the same. A number of quoted results come from the Mechanica without reference.


Here Euler investigates the straight line collision of two bodies, and determines the law of conservation of linear momentum, this is not done by applying Newton's Laws directly, and Newton does not even get mentioned, but by making use of the artifact of a massless connecting spring during the collision to replace the compression, and by regarding the speed of each body to be derived from the distance fallen by the same mass; this in the modern view is thus by making use of the conservation of  mechanical energy. The important thing to note, is that the collision is not analysed in terms of time as the independent variable, but rather by the distance between the masses, making the calculation time independent. This approach was used  successfully in other situations by Hermann in his Phoronomia, where the so-called moment of the speed, vdv is put equal to the moment of the force Fdx or adx, for unit mass.

This completes with E019 and E020 a trilogy of papers on the summation of series. A synthetic method is demonstrated whereby many kinds of series can be summed using integration and differentiation to reduce a sum to a geometric series. This paper culminates on summing hypergeometric series; a truly fascinating work.

These two papers show Euler's early involvement in providing answers to two of the questions posed Fermat; firstly, the discrediting of Fermat's formula for primes 22^n+1, and secondly, the establishment of Fermat's Little Theorem. In addition, a letter from Fermat to Wallis is given, in which the formula for primes is first mentioned.

Here Euler sets out a method for finding a number which gives certain known remainders when divided by given divisors : now known as the Chinese Remainder Theorem, and tackled by modular arithmetic. Euler finally applies his method to the vexing question of the birth date of Jesus, which I do not think I have understood properly.

This later paper develops Euler's answers to some of the problems arising from the two papers E26 & E54 ; it is of interest as it show how he has developed systematic methods for dealing with such problems; although he admits to not having found all the answers; at any rate, Euler now has developed a more respectful outlook on Fermat's work, and one may presume Fermat is now recognised by Euler for the outstanding mathematician he was.

This is a most interesting paper, and shows the extent that the genius of Euler had reached at this time; he applies himself to a new method of solving first order differential equations, as applied to the general Ricatti equation : though it is hard to classify his method. Certainly series expansions of variables are made as power series, integrations are performed as a summation of special series technique, and results are obtained in general as integrals. A small flaw arises when Euler is absent-minded about an integration. I have pointed it out, but left the subsequent work as it is, as the method is the same, just the wrong coefficients. People have pointed out this paper as seminal in the development of integral transforms, and it is certainly heading in this direction, but it has its origins fixed in traditional methods. Well worth a read , I think. A few mistakes in the original have been amended from the corrected version in the Opera Omnia as noted in the text, since this paper was first posted. The origins of this paper are as an extension of E30, in which the arc length of an ellipse is required. It is referred to near the end of Ch. II of the Mechanica as an alternative way of solving a tautochrone problem proposed to Euler by Daniel Bernoulli.

This is one of Euler's most celebrated papers, in which he demonstrates formulas such as Pi ^2= sum of the inverse squares of the positive integers, and many more, on equating the sine expansion of a circular arc to the infinite product of the simple factors of the associated multiple arcs.

This is an extensive paper that develops a method for finding a family of curves arising from the constant of integration of dz = Pdx, which is treated as the second variable; the rudiments of partial differentiation are presented, and there is an extensive survey of homogeneous functions centred around what is now know as Euler's Theorem for such functions. The origins of this paper would seem to be Proposition 15 of Vol. 2 of the Mechanica, relating to families of tautochronous curves, where an integration relying on Euler's Theorem is required.

This is an equally extensive paper that continues the development of methods for finding a family of curves arising from the constant of integration of dz = Pdx, which is treated as the second variable. A method is developed for finding the modular equation for the first order equation that is extended to cover a number of cases; this in turn is extended to second and higher orders. The method involves finding suitable functions to integrate, starting from a part of the modular equation that is integrable, so that the whole equation is of this form. This paper is noteworthy in addition as it seems to be the first in which the function notation, albeit in a slightly different form from the modern meaning, is introduce. I have not been able to check all the equations at this stage.

E071  Dissertation on Continued Fractions.

Here Euler produces an extended work on continued fractions, and indicates the use of calculations of this kind in finding successive approximations to such irrational numbers as the square roots of numbers, pi, e, etc. There is too much in this paper to be listed briefly here.

E248 The Archimedes Screw.
This is a long paper full of formulas, in which Euler tries to establish a working kinematic model for the movement of water up or down the spiral : he treats the type of the machine in which a narrow spiral is wound on the outside of a cylinder. There is a lack of physical reasoning on the basic mode of operation, and the problem is tackled in a kinematic manner, where methods are given for establishing how the water will flow under various conditions, according to the vis viva idea. I would be very happy for someone to spend a day or two going through it, as I probably have not got all the details correct ; Euler appears to have started the paper in the 1730's while D. Bernoulli was at St. Petersburg, left it and returned to it many years later, as there are some inconsistencies present. It is a mathematical tour de force.

E 278
The Mechanics of Solid or Rigid Bodies Vol. I. & Vol. II .This work is now completely translated.


E 279

Euler attempts to find whole number for which the general quadratic ax2+bx+c becomes a perfect square; this is done by guessing such a number  and finding an indefinite number of other like numbers.


An Introduction to Natural Science, ...... This philosophical work from Euler's Opera Postuma (E842) translated from German by Dr. E. Hirsch. Click on
the above link to access the available chapters. This now includes also a translation of E81, Thoughts on the Elements of Bodies.....


Here Euler gives us an introduction to continued fractions, in which the genius of Euler is evident almost from the first line; this was to be built on by Euler and later mathematicians, and was given a new beginning when the connection between matrices and continued fractions came into being.


Here Euler gives us an update, inspired by Lagrange, of his understanding of the propagation of sound in one dimension. Some notes have been added for you to clarify matters at times. No attempt is made, of course, to improve on the theory presented.


Here Euler continues the task he has set himself in the last paper, and extends his calculations into two and three dimensions, essentially deriving the wave equation in differential form for waves propagating in two and three dimensions; this is a paper that laid the mathematical foundations for many later investigations by others into physical phenomena. Of course, hardly any of the prerequisite experimental knowledge was available at the time ; yet the result is a testimonial to Euler's mathematical genius.


Here Euler tidies up some loose ends from the previous paper; he recalls that Ricatti's method can be used to integrate the 3 dimensional equation he has derived for the propagation of sound; finally, he writes down what the solution should be essentially in terms of travelling waves, and works backwards to derive the wave equation. As the reader may note, a number of physically interesting ideas are not investigated; for example, Euler almost makes a start to Fourier analysis by decomposing a pulse into the different wavelengths that fit into an interval……


Here Euler presents his method for extracting square roots by continued fractions, and then uses this to obtain solutions to his so-called Pell problems whereby quadratic formulas are given values resulting in perfect squares.


Spherical Trigonometry all derived briefly and clearly from first principles.


Concerning the Summation of Infinite Series........ : E736
In which Euler sets out extensions of his celebrated papers E020 & E041; however, he adopts a rather cavalier attitude to the convergence of some of the series, which actually diverge.

A Remarkable Problem... : E407
In which Euler prepares a 4 by 4 square of numbers in which the rows and columns have remarkable properties. Translated by Johan Sten.

On Euler Angles : E478
In which Euler sets out the general formulae for the translation and rotation of any rigid body. Translated by Johan Sten.

In which Euler presents some results relating to spherical triangles, complementing some work by the Finnish astronomer Lexell. Now with imbedded diagrams. Translated by Johan Sten.

Lexell's Paper on Spherical Triangles
In which a young associate of Euler, A. J. Lexell sets out some new results on spherical geometry. Translated by Johan Sten.

Lexell's Paper on the Motion of a Rigid Body
In which Lexell continues Euler's work in the translation of a rigid body. Translated by Johan Sten.

Ian Bruce. Aug. 6th , 2020 latest revision. Copyright : I reserve the right to publish this translated work in book form. However, if you are a student, teacher, or just someone with an interest, you can copy part or all of the work for legitimate personal or educational uses. Please feel free to contact me if you wish, especially if you have any relevant comments or concerns about this work.