**E****ULER'S
**

Ian Bruce

This marks the completion of part of a large project that will
take a year or two to complete : yet I feel that someone should do it in its
entirety, since Euler's calculus works are interconnected in so many ways, as
one might expect, and Euler had a habit of returning to earlier ideas and
making improvements. John D. Blanton has already translated Euler's *Introduction to Analysis* and approx. one
third of Euler's monumental *Foundations
of Differential Calculus* : this is not really much help to me, as I would
have to refer readers to texts that might not be available to them, and even
initially, I have had to delve into Ch.18 of Part II of the latter book to
obtain explanations of the formulae used in the first chapter of the
integration, which was our main concern here initially. I have decided to start
with the integration, as it shows the uses of calculus, and above all it is
very interesting and probably quite unlike any calculus text you will have read
already. Euler's abilities seemed to know no end, and in these texts well
ordered formulas march from page to page according to some grand design. I hope
that people will come with me on this great journey : along the way, if you are
unhappy with something which you think I have got wrong, please let me know and
I will fix the problem a.s.a.p. There are of course, things that Euler got
wrong, such as the convergence or not of infinite series; these are put in
place as Euler left them, perhaps with a note of the difficulty. The other
works mentioned are to follow in a piecemeal manner alongside the integration
volumes, at least initially on this web page. These works are available in the
public domain on the Euler Archive website and from Google Books ; I have made the corrections suggested from
time to time by the editors in the *Opera
Omnia* edition, to all of whom I express my thanks. The work is divided as
in the first edition and in the Opera Omnia into 3 volumes. All the chapters
presented here are in the books of Euler's original treatise, which corresponds
to Series I volumes 11,12 & 13 of the *O.O.*
edition. I have done away with the sections and parts of sections as an
irrelevance, and just call these as shown below, which keeps my computer much
happier when listing files.

The supplements of the
posthumous volume IV are attached here following Vol. III.

**The
Investigation of Functions of Two Variables from a Relation of the
Differentials of the First Order.**

Click
here for the 1^{st} chapter
**: ***Concerning the nature of differential equations, from which functions
of two variables are determined in general.* Euler establishes a criterion
for the total derivative of functions of two variables using the idea of a
multiplier, which is quite ingenious,
and which can be shown to be equivalent to a well-known vector identity.
I have used the word *valid* in the to
indicate such functions, rather than *real*
or *actual*, as against absurd, which
Euler uses. He then shows how this criterion can be applied to several
differential equations to show that they are in fact integrable, other than by
using an integrating factor ; this includes a treatment of the normal
distribution function.

Click
here for the 2^{nd} chapter
**: ***Concerning the resolution of equations in which either differential
formula is given by some finite quantity.* Euler establishes the solution to
a number of equations in which *p* = (*dz/dx*) is the partial derivative first
of a constant, then a function of *x*,
then a function of *x* and *y*, and finally a function of *x, y,* and *z*. These solutions are found always by initially assuming that y is
fixed, an integrating factor is found for the remaining equation, and then the
complete solution is found in two ways that must agree. The historian of
mathematics will be interested to know that the arbitrary form of the function *f* : *x
* related initially to the shape of a
stretched string, as described here.

Click
here for the 3^{rd} chapter
**: ***Concerning the resolution of equations in both differential formulas
are given in terms of each other in some manner.* Euler establishes the
solution of some differential equations in which there is an easy relation
between the two derivatives *p *and *q*. Perhaps they are equal, depend on
each other in a linear manner, or there is some other simple relation between *p* and* q*, etc.

Click
here for the 4^{th} chapter
**: ***Concerning the resolution of equations in which a relation is proposed
between the two differential formulas and a single third variable quantity.*
This chapter sees a move towards the generalization of solutions of the first
order d.e. considered. Initially a solution is established from a simple
relation, and then it is shown that on integrating by parts another solution
also is present. Several examples are treated, and eventually it is shown that
any given integration is one of four possible integrations, all of which must
be equivalent. A simple theory of functions is used to show how this comes
about; later Euler establishes the conditions necessary for a particular
relation to give rise to the required first order d.e.

Click
here for the 5^{th} chapter
**: ***Concerning the resolution of equations in which a relation is given between
the two differential formulas (dz/dx) , (dz/dy) and any two of the three
variables x, y, and z.* This chapter is a continuation of the methods
introduced in ch.4 above. A very neat way is found of introducing integrating
factors into the solution of the equations considered, which gradually increase
in complexity. All in all a most enjoyable chapter, and one to be recommended
for students of differential equations.

Click
here for the 6^{th} chapter
**: ***Concerning the resolution of equations in which some relation is given
between the two differential formulas (dz/dx) , (dz/dy) and all three variables
x, y, and z.* This chapter completes the work of this section, in which
extensive use is made of the above theoretical developments, and ends with a
formula for function of function differentiation.

**Volume III, Section II**.

**The
Investigation of Functions of Two Variables from a Relation of the
Differentials of the Second Order***.*

Click
here for the 1^{st} chapter
**: ***Concerning Differential Formulas of the Second Order in General.*

This
is a short chapter but in it there is much that is still to be found in
calculus books, for here the chain rule connected with the differentiation of
functions of functions is introduced. Much frustration is evident from the bulk
of the formulas produced as Euler transforms second order equations between
sets of variables *x, y* and *t, u. * A lead is given to the Jacobi determinants of
a later date that resolved this difficulty.
Happy reading!

Click
here for the 2^{nd} chapter
: *In
which a Single Formula of the Second Order Differential is given in terms of
some other remaining quantities.*

This
is a reasonably straight-forwards chapter in which techniques employed before
are given new territory ; essentially the equation *d ^{2}z/dx^{2}* ,

Click
here for the 3^{rd} chapter
: *In
which two or all of the Formulas of the Second Order are given in terms of some other remaining
quantities.*

This
is a long but interesting chapter similar to the two above, but applied to more
complex differential equations; at first an equation resembling that of a
vibrating string is investigated, and the general solution found. Subsequently
more complex equations are transformed and by assuming certain parts vanishing
due to the form of transformation introduced, general solutions are found
eventually. Examples are provided of course.

Click
here for the 4^{th} chapter
: *A
Special Method of Integrating Equations of this kind in another way.*

This
is also a long but very interesting chapter wherein Euler develops the solution
of general second order equations in two variables, with non-zero first order
terms, in terms of series that may be finite or infinite; the coefficients
include arbitrary functions of *x* and *y* in addition, leading to majestic
formulas which are examined in cases of interest – especially the case of
vibrating strings where the line density changes, and equations dealing with
the propagation of sound. Euler himself seems to have been impressed with his
efforts.

Click
here for the 5^{th} chapter
: *A
Special Method of Integrating Equations of this kind in another way.*

This
is also a very long but very interesting chapter wherein Euler develops a
transformation, initially of the first order differentials, whereby the
solution of a second differential equation
in terms of the unknown z can be
found from the solution of a similar equation in the variable *v*, related by the form *z = (Mdv/dx) + Nv*. The main part of the
chapter is taken up finding appropriate values for *s = M/N. *The method is extended to forms involving the second
degree. This is the last chapter in this section.

**Volume III, Section III**.

**The
Investigation of Functions of Two Variables from a Relation of the
Differentials of the Third of Higher Order***.*

Click
here for the 1^{st} chapter
**: ***Concerning the Resolution of the Simplest Differential Formulas
involving a Differential Formula.*

This
is a short chapter but in it there are some interesting developments. Thus, the
distinction is made between repeated integrals and integrals over two or more
dimensions. The use of arbitrary functions takes the place of constants in
these general integrations considered.

Click
here for the 2^{nd} chapter
**: ***Concerning the Integration of Higher Order Differential Equations by Reduction
to Lower Orders.*

This
is a another short chapter in which Euler explores relatively easily solved
differential equations by using a transformation based on *z*(*x, y*)* = e ^{ax}v*(

Click
here for the 3^{rd} chapter
**: ***Concerning the Integration of Homogeneous Equations where the
Individual Terms contain Differential Formulas of the same Order.*

This is another short chapter, as Euler reaches
the bounds of his knowledge. Here initially he investigates the solution of the
equation *A*(*ddz/dx ^{2}*) +

**Volume III, Section IV**.

**The
Investigation of Functions of Three Variables from a Given Relation of the
Differentials***.*

Click
here for the 1^{st} chapter
**: ***Concerning Differential Formulas Involving Functions of Three
Variables.*

This is another short chapter, forming a basis for
the integration of functions of three variables: differential formulas of
higher orders are established at first; however, it soon becomes apparent that
this is a far more difficult task, due to the introduction of arbitrary
functions of the other variables in the integration.

Click
here for the 2^{nd} chapter
**: ***Concerning the Finding of Functions of Three Variables from the Value
of a Certain Differential Formula.*

In this chapter, arbitrary functions of the integration
are introduced for the variables not integrated over at the time, and the work
relates back to extending results already established for one or two variables.

Click
here for the 3^{rd} chapter
**: ***Concerning the Resolution of Differential Equations of the First Order.*

In this chapter some procedures are put in place
for the integration of such equations in general, which are then applied to
certain cases in which a simple relation exists between the first order partial
derivatives. This is the penultimate chapter.

Click
here for the 4^{th} chapter
**: ***Concerning the Resolution of Homogeneous Differential Equations.*

In this chapter, Euler applies his skills to the
solution of homogeneous differential equations in two and three dimensions,
especially those of orders one, two, and three. By making linear substitution
of the coordinate *x, y*, and *z*, he is able to derive algebraic
equations to be solved in a straight forwards manner, and also to reduce the
number of variables to two. The chapter ends with some hints as to his current
research on the theory of fluids, in which such integrations find a place. This
is the final chapter of the original work; an appendix on the Calculus of
Variations follows. A later posthumous edition published in 1793 ran to four
volumes, where additions to most chapters were put in place in volume IV.

**Volume III, APPENDIX**.

**THE CALCULUS OF VARIATIONS**

Click here for the 1^{st} chapter **: ***Concerning the Calculus of
Variations in General.*

In this chapter there are no formulas : Euler
devotes considerable space to explaining the nature of variations, which are to
be applied both to the differential as well as to the integral calculus in the
chapters that follow.

Click here for the 2^{nd} chapter **: ***Concerning the Variations
of Differential Formulas involving two Variables.*

In this chapter Euler sets out the ground work for
his treatment, where a single order of the variation (delta) δ is
considered. The differential operator *d*
selects neighbouring points on a line or surface, and may run to any order we
please, while (delta) δ selects
points on neighbouring lines or surfaces of the same kind; it is shown
initially that *d* and delta commute,
and the variations of *p, q, r*, etc. ,
defined already can be investigated in terms of differentials.

Click here for the 3^{rd} chapter **: ***Concerning the Variations
of Simple Integral Formulas involving two Variables.*

In this chapter the groundwork is established for
calculating the variation of integrals involving two variables. This follows on
from above, and is extended to some intriguing results regarding the
integrability of integrals.

Click here for the 4^{th} chapter **: ***Concerning the Variations
of Complicated Integral Formulas involving two Variables.*

In this chapter the method produced above is
applied to more complicated cases, where one or more additional intermediate
integrals are inserted in evaluating the variation of the integral *Vdx, *so that *V* becomes a function of an intermediate variable also expressed by
an integral.

Click here for the 5^{th} chapter **: ***Concerning the Variations
of Integral Formulas involving three Variables and including two relations.*

In this chapter the method of the previous chapters
is extended further to include two independent functions *y* and *z* of *x*, and the variations thereof; this is
further extended to the case of an extra variable *v* defined via an integral. The final two chapters follow the same
sort of procedures.

Click here for the 6^{th} chapter **: ***Concerning the Variations
of Integral Formulas involving three Variables the relation of which is
contained by a single relation.*

Click here for the 7^{th} chapter **: ***Concerning the Variations
of Integral Formulas involving three Variables in which one is considered as a
function of the other two.*

**Volume III, APPENDIX**.

**SUPPLEMENT ON THE INTEGRATION OF CERTAIN SPECIAL DIFFERENTIAL
EQUATIONS**

Click here for the supplement.

In this final supplement, Euler expands on his
method of integrating by the use of multipliers, which he extends to wider and
wider classed of functions of two variables; one can see the tentative origins
of transforms introduced as Euler grapples with this problem, in his attempt to
reduce integrals to known forms for which he can evaluate the complete
integral.

**Volume IV, Supplements (E660)**.

Sup.1 This supplement is concerned with removing the
irrationality from integrands, which subsequently can be integrated either by
logarithms or the arcs of circles.

Sup.2 This supplement investigates several series expansions
leading to term by term integrations of integrands ; series summation is
undertaken is a new way, and extended to infinite products.

Sup.3a
: The
evaluation of a certain integral formula*, *with the integration extending from zero to one is
considered.This supplement investigates the investigation of a particular class
of functions, which give rise to factorial sums, and which depend on inverse
circular functions, logarithmic functions, and repeated integrations. This is a
long and rather complex supplement, which I have split up into 3 parts, of
which this is the first.

Sup.3b
Concerning the value of another certain
integral formula , in the case where after the integration there is put the end
limit one. Here three different methods are shown for evaluating certain
functions, which lead to some interesting results.

Sup.3c A
number of integral expansions are set out here, and these are related to a
general theorem, which is set out without proof, but for which all the special
cases considered thereafter are correct. This supplement has its origins in the
work by Wallis on infinite products involving pi and the associated areas of
circles. A good place to look for some
references is :

T*he Early History of the Factorial Function*

Author(s): Jacques
Dutka

Archive for History of Exact Sciences, Vol.
43, No. 3 (1991), pp. 225-249; and

*The Early History
of the Hypergeometric Function*

Author(s): Jacques
Dutka

Source: Archive for History of Exact
Sciences, Vol. 31, No. 1 (1984), pp.
15-34

E321 This is a rather difficult paper, and involves a number of concepts not handled directly, but referred to from previous publications not detailed explicitly, that I have illustrated by way of notes, which should not be considered as proofs, but as indications of how to go about proving these assertions. The topic is the investigation of a class of integrals, related to the modern Beta function, and which can only be evaluated as infinite products, some of which can be given in terms of circular quadratures, and other simpler ones that can actually be integrated.

Sup.4a This is an interesting supplement, in which a number Euler's special techniques on integration become apparent; the use of the combinatorial function with negative indices is especially noteworthy, as well as his use of imaginary exponential expansions in terms of cos and sine.

Sup.4b This is another interesting supplement,
in which Euler starts with essentially the formula for the cosine rule for the
square of a side of a triangle; systematically he shows how to integrate such
formulas both in the denominator and numerator, with a factor cos*i**f*
included. Such integrations produce intersesting finite expansions in terms of
squared binomial terms and sines or cosines of angles; eventually, a rigorous
proof is produced for such expansions. Thus, as in music, we might call these
variations on a theme. The orthogonal nature of the cosine function becomes
apparent, and it appears as if Euler is knocking on the door, as it were, of
Fourier Analysis. E672/3/4 form a large part of this supplement.

Sup.5a This is another interesting supplement, this time a number of issues are handled, leading to new methods of integration, at this time, and finally calculating the sums of infinte series of inverse powers. The text is that of E464, though it does rely on E463 and E59, which I am thinking of translating before proceeding further.

E59 This is an early paper, full of interesting derivations, too numerous to describe in detail; essentially a way is shown for producing the values of infinite products related to sines, cosines, and tangents. The sums of infinite series are represented by integrals, relating to the trig. functions. These again are related to infinite products ; a major final part looks at an application of previous work (E122) which compares infinite products with the ratio of integrals.

E463 This paper supplies a lot of the answers to E464, and again is full of interesting derivations, essentially showing a connection between infinite series expansions, integrals, and formulas relating to the sine and tangent functions.

Sup.5b This is the second question addressed
in this long chapter, this time the integral formulas arising from a particular
integral, where the integrand is *x ^{p-1}dx/(1-x^{n})^{1-q/n}*
, which is investigated between the limits 0 and 1 for ascending powers of

Sup.5c This is the third question addressed in this long chapter, in this case the integrals are evaluated from 0 to infinity; Euler makes use of results from his famous theorem relating to de Moivre's formula.

Sup.5d Two questions are addressed in this
long chapter; these are E588 dealing with the integrand * x ^{m-1}dx/(1+x^{k})^{n}*,
while the other E589 handles

Sup.5e This is the final part of this
supplement: *A Method of finding Integral Formulas which in certain cases maintain a
given Ratio amongst themselves; where likewise a Method is treated of summing
Continued Fractions*.
E595 in the *O.O*. ; this leads Euler back to dealing with
continued fractions, where he extends
the work of Brouncker and Wallis. Interesting.

Ian Bruce. Feb. 16^{th}, 2017,
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 by
clicking on my name, especially if you have any relevant comments or concerns.