**EULER'S **

Ian Bruce

This is the start 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.

Click here for some ** introductory
material**, in which Euler defines integration as the inverse process of
differentiation.

Click here for
the 1^{st} Chapter :* **Concerning** the integration
of rational differential formulas*. A large part of Ch.18 from Part II
of Euler's Differential Calculus is presented here for the reader's
convenience, in order that the
derivations of formulas used in the reduction of rational functions can be
understood. This is now available below in its entirety.

Click here for
the 2^{nd} Chapter :* *** Concerning
the integration of irrational differential formulas**. Euler finds ways
of transforming irrational functions into rational functions which can then be
integrated. He makes extensive use of differentiation by parts to reduce the
power of the variable in the integrand.

Click here for
the 3^{rd} Chapter :* *** Concerning the integration of differential formulas by infinite series**.
Particular simple cases involving inverse trigonometric functions and
logarithms are presented first. Following which a more general form of
differential expression is integrated, applicable to numerous cases, which
gives rise to an iterative expression for the coefficients of successive powers
of the independent variable. Finally, series are presented for the sine and
cosine of an angle by this method. Here Euler lapses in his discussion of
convergence of infinite series; part of the trouble seems to be the lack of an
analytic method of approaching a limit, with which he has no difficulty in the
geometric situations we have looked at previously, as in his

Click here for
the 4^{th} Chapter **: **

Click here for
the 5^{th} Chapter **: **

Click here for
the 6^{th} Chapter **: **

Click here for
the 7^{th} Chapter **: **

Click here for
the 8^{th} Chapter **: **

Click here for
the 9^{th} Chapter **: Concerning the development of integrals as
infinite products.** The integral

Click here for
the 1^{st} Chapter **: Concerning the separation of variables.**
The focus now moves from evaluating integrals treated above to the solution of
first order differential equations. You should find most of the material in
this chapter to be straightforward. Euler finds to his chagrin that there is to
be no magic bullet arising from the separation of the variables approach, and
he presents an assortment of methods depending on special transformations for
particular families of first order differential equations; he obviously spent a
great deal of time examining such cases and this chapter is a testimony to
these trials.

Click here for
the 2^{nd} Chapter **: Concerning the integration of differential
equations by the aid of multipliers.** Euler now sets out his new method,
which involves finding a suitable multiplier which allows a differential
equation to become an exact differential and so be integrated. This chapter
relies to some extend on Ch. 7 of Part I of the Differential Calculus, a small
relevant part of which has been included here. Euler refers to such
differential equations as integral by themselves; examples are chosen for which
an integrating factor can be found, and he produces a number of examples
already treated by the separation of variables technique, to try to find some
common characteristic that enables such equations to be integrated without
first separating the variables. This task is to be continued in the next
chapter.

Click here for
the 3^{rd} Chapter **: Concerning the investigation of differential
equations which are rendered integrable by multipliers of a given form.**
Euler moves away from homogeneous equations and establishes the integration
factors for a number of general first order differential equations. The
technique is to produce a complete or exact differential, and this is shown in
several ways. For example, the d. e. may consist of two parts, and each part is
provided with its own general integrating factor : a common factor can then be
chosen from the two on giving introduced variables particular values. A general
method of analyzing integrating factors in terms of consecutive powers equated
to zero is presented. There is much material and food for thought in this
Chapter.

Click here for
the 4^{th} Chapter **: Concerning the particular integration of
differential equations.**

Euler declares that while the complete integral includes an unspecified constant: the particular integrals to be defined and investigated here may relate to the existence of solutions where the values of the added constant is zero or infinity, and in which cases the solution, perhaps found by inspection, degenerates into an asymptotic line, in which no added constant is apparent. Other situations to be shown arise in which an asymptotic line is evident as a solution, while some solutions may not be valid. A number of situations are examined for certain differential equations, and rules are set out for the evaluation of particular integrals.

Click here for
the 5^{th} Chapter **: Concerning the comparison of transcendental
quantities contained in an integral of the
form Pdx/sq.root(quadratic in x).**

This is a most
interesting chapter, in which Euler cheats a little and writes down a
biquadratic equation, from which he derives a general differential equation for
such transcendental functions. From the general form established, he is able
after some effort, to derive results amongst other things, relating to the
inverse sine, cosine, and the log. function as special cases of the general
known integral. More general differential equations of the form discussed are
gradually introduced. It is interesting to note the use of F: *x*
which would later be written as F(*x*),
for the notation for a function.

Click here for
the 6^{th} Chapter **: On the comparison of transcendental
quantities contained in the form
Pdx/sq.root(quadratic in x^{2}).**

This is a continuation of the previous chapter, in which the mathematics is more elaborate, and on which Euler clearly spent some time. It seems best to quote the lad himself at this point, as he put it far better than I, in the following Scholium :

" § 611. Now here
the use of this method, which we have arrived at by working backwards from a
finite equation to a differential equation, is clearly evident. For since the
integration of the formula *dx /sqrt(A +
Cx ^{2} +Ex^{4}*) cannot be produced either from logarithms
or the arcs of circles, it is certainly a wonder that such a differential
equation thus can be integrated algebraically; which equations indeed in the
preceding chapter have been treated with the help of this method, and also which are able to be elicited by the
ordinary method, as the individual differential formulas can be expressed
either by logarithms or circular arcs, the comparison of which is then reduced
to an algebraic equation. Now since here by such an integration clearly no
treatment can be found, clearly no other
method is apparent, by which the same integral, that we have shown here, can be
investigated. Whereby we shall set out
this argument more carefully. "

Thus, Euler sets to work on the penultimate chapter of this section, which is a wonder of Eulerian trickery, relying on the symmetric biquadratic formula announced in the previous chapter, but now extended to higher powers.

Click here for
the 7^{th} Chapter **: Concerning the approximate integration of
differential equations.**

In this final chapter of this part, a number of techniques are examined for the approximation of a first order differential equation; this is in addition to that elaborated on above in Section I, CH. 7.

**Volume I, Section III.**

Click here for
the single chapter **: Concerning the resolution of more
complicated differential equations.**

In this single chapter which marks the final section of Part
I of Book I, use is made of a new variable, *p
= dy/dx*, in solving some more difficult first order differential
equations.

**EULER'S **

Ian Bruce

Click here for
the 18^{th} Chapter of Part II :*
*** Concerning the resolution of
rational functions of the form into partial fractions**. This is a most
extensive investigation, in which amongst other things of interest, use is made of De Moivre's Theorem
in the reduction of powers of quadratic terms to simple terms.

Ian
Bruce. June 29^{th},
2010 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.