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

CONTENTS

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.

E002

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

E003

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

E004

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

Presented to the

E005

The Solution of the Problem of Reciprocal Trajectories. This problem had been
solved initially by the editor of the 3rd edition of

E006

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.

E007

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.

E008

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.

E009

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.

E010

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.

E011

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.

Ricatti

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.

E012

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.

E014

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.

E015

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

E016

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

E019

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]

E020

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.

E021

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.

E022

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.

E025

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.

E026&E054

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 2^{2^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.

e036tr

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.

E134

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.

E031

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.

E041

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.

E044

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.

E045

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

**E
278 **

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

Euler attempts to find whole
number for which the general quadratic *ax ^{2}+bx+c*
becomes a perfect square; this is done by guessing such a number and finding an indefinite number of other
like numbers.

**ContentsE842
**

**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..... **

E281

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.

E305

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.

E306

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.

E307

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……

E323

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.

E524

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.

E698

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. 6^{th} , 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.