**EULER'S **

Ian Bruce

This is the second part of a large project
that is now complete : 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 so I have felt the
need to provide my own independent translation of this work. These works are
available in the public domain on the Euler Archive website and from Google
Books, as well as being obtainable from libraries.

Click here for the
Chapter 1 :* *** Preface and finite differences**.
Euler makes use of his great organizational skills to present a masterful
outline of finite differences. Prior to this he expands on the troublesome
times had by mathematicians in coming to grips with calculus, the use or misuse
of infinitesimal quantities; he does not of course use the modern concept of
limit, although it can be seen in his writings here.

Click here for the
Chapter 2:* *** Concerning the use of differences in the
teaching of series**. Several methods are set out for finding the sums of
finite series, initially those which end with a constant set of differences,
for which a summatory series can be set up, the differences of the terms of
which is the original series. Euler moves on to several other techniques which
are of interest, some of which are well known.

Click here for
the Chapter 3:* *** Concerning infinity and the infinitely small**.

This is a most interesting
chapter in which Euler grapples with the concept of infinity and with
differentials. These involve grey areas of mathematics, where quantities are
considered that cannot be described by numbers. The infinitesimals occupy such
a grey area, being neither quite just zero, as Euler maintains, but yet they
are too small to be measured on any scale. Thus, to say *dy* and *dx* are zero, but
only to be used in a ratio which need not be zero, as Euler maintains, cannot be faulted, but there is a characteristic
of direction associated with differentials not considered here by Euler, that is not the case for quantities that are
just plain zero, such as a single point on a line; thus we must consider at
least 2 points for the differentials *dx*,
etc. which are closer together than any finite quantity. The final discussion on divergent series is
also of interest.

Click here for the
Chapter 4:* *** Concerning the nature of the differentials
of each order**.

This is a sort of housekeeping chapter, in which everything is put in place for the great feast that lies in wait of the reader in the following chapters : here Euler is concerned that common fallacious manners of thinking are avoided, and a clear understanding of differentials is put in place.

Click here for the
Chapter 5:* *** Concerning the differentiation of algebraic
functions involving one variable**.

This chapter deals in a thorough
manner with the differentiation of algebraic functions of *x*; the familiar rules for products and quotients are set out in
several ways, short cuts are noted, and results leading to general formulas
investigated; in all a most rewarding chapter, especially set out for the
beginner from a higher viewpoint than usual in modern equivalent texts.

Click here for the
Chapter 6:* *** Concerning the differentiation of
transcending functions**.

This chapter deals in a thorough manner with the differentiation of the transcending functions know at the time, starting with the hyperbolic log, the exponential function, and the arcsine, arccos, and arctan of a circular arc; special functions arising from these such as the versed sine are also treated; finally, the sine, cos, and tan are differentiated with numerous examples. Recall that this was, after all, a calculus text for the beginner.

Click here for the
Chapter 7:* *** Concerning the differentiation of functions
involving two or more variables**.

This chapter deals at some length with the idea of the total derivative, although this term is not used here, and use is made of homogeneous functions as an intermediate step, and at last the partial derivatives w.r.t. the different variables produces the same quantity on repeated differentiation. This chapter is definitely designed for the instruction of the uninitiated.

Click here for the
Chapter 8:* *** Concerning the differentiation of formulas
of higher differentials**.

This chapter is very interesting, as it shows how Euler thought about matters that were on the boundary of what was known, and what was unknown at the time; the subject is differentials of higher order, some that clearly appear in some form from time to time in various problems. Some higher order differentials lead to fixed or definite forms with some meaning, others seem to be pointless. Order is the criterion as always, and Euler shows that the presence of the second order differential must be eliminated either by making the first order constant, or by introducing a scheme whereby the second order differentials cancel each other out, and one imagines that at least some of these quantities will appear in future chapters, and of course elsewhere.

Click here for the
Chapter 9:* *** Concerning differential equations**.

A number of issues are covered in this final chapter of Part I. We may note in general the rule governing the total derivative in three dimensions, leading from a finite algebraic equation to a first order differential equation in three dimensions. Real or actual and imaginary or absurd differential equations can be distinguished by this criterion. Thus the way is paved for integration, where a differential equation may be tested for solubility.

Non calculus methods involving the use of simple transformations are used to establish dual series with the same sum; these are not always converging in the modern sense, but are very interesting, and lead to some useful results, such as the logarithmic integral. If well as opening a window into this field of study, Euler's work stimulated other mathematicians of the day to do further research into the nature of convergence of series.

Euler goes to town in
this chapter as he develops new series by differentiation of existing series,
taken mainly from his previous work, the *Introductio,
*which already exists in translation, and to which the reader is here
referred as needed. Innumerable new sums of series, both finite and infinite,
arise under Euler's magic touch, and one can only gaze in wonder at his
accomplishments.

Euler notes the added complexity of finite differences from which differentials can be found; he considers the possibility of proceeding in the opposite direction, and derives essentially what is know as the Taylor expansion for a finite interval; from this expansion the finite differences of any order are apparent; he then digresses to consider iterative formulas of sums of powers that arise, and finally applies the series expansion backwards to find the value of a function at the origin.

Euler now applies the Taylor series type formulas found above to find new values of functions from known ones ; this is done initially for algebraic functions, then roots of numbers are found, leading on to extrapolation and interpolation of the inverse trig. functions, and finally the sine, cosine, etc. functions are shown. The use of such techniques in generating tables of functions is evident.

Euler now applies his
analytical methods to the sums of powers of integers and to the sums of powers
of the reciprocals of integers. This is a most amazing chapter ; Euler
constructs methods for performing these tasks, which eventually encompass and
extend the Bernoulli numbers, which had much humbler beginnings. Definitely a
chapter for the mathematical *aficionado*,
I think.

This chapter follows on from the last with a number of very interesting applications; for example the sums of the inverse powers of integers are handled using the alternative series expansion developed; other series are investigated including series from which an expansion of pi can be made.

This chapter again augments the above work by examining other kinds of series that may be summed, but using the same general method of comparing the sums for a jump in the index of one step, and using a Taylor expansion either backwards or forwards to arrive at a formula for the increase in the sum. A number of cases are introduced; any difficulties encountered by the modern reader eventuate from the lack of appropriate function notation, subscripts, etc.

This chapter provides an application of differentiation : an fraction with algebraic numerator and or denominator is equated to an unknown series, and the terms of this devised series may be found by continued differentiation; the process is continued for the transcendental functions known at the time, and contains many admirable series expansions, most of which are well-known.

This chapter provides a fascinating insight into Euler's methods for finding the approximate roots of polynomial equations, and certain transcendental equations. Whether these methods are now all know to the expert in these things I am not altogether sure; at any rate, well worth a look.

This chapter provides an excellent introduction the finding of maxima and minima of rational functions, as well as some examples of transcending functions.

This chapter continues the work on maxima and minima, extending it to the case of functions having more than one value, for which the above traditional method may fail, at a branch point of the function. The traditional methods are then developed slowly for functions of two variables, and include numerous examples.

This is an interesting application of differentiation to finding bounds for the real roots of polynomials of the second, third, and fourth degrees, based on the calculation of max. and min. A few extra equations of higher order are also analyzed in this manner.

This is a chapter on the theory of equations; in it Euler explores the possibility of determining the number of real and imaginary roots from the properties of the coefficients of the polynomial, and finding the bounds within which such roots may lie. The work appears to be an ongoing project, arising from the earlier efforts of Harriot, Descartes, and Newton, the rule of signs, etc., and involving some inventions of his own. Unfortunately it falls short of Sturm's Theorem, which followed on from the earlier discoveries.

In this chapter cases are
concerned that do not conform to the ordinary rule of integration, and involve
the total derivative running to higher orders, and also perhaps which involve
ambiguous values. A careful evaluation is made of the maxima and minima for
such cases, shown also by a number of examples. The argument starts with
algebraic functions, and then is extended to transcendental functions, for
which extra forms of differentials are introduced. One can see why the
mathematicians of the 19^{th} century went to such trouble to establish
a notation for cases such as enumerated here with a perhaps less than
completely satisfactory notation, though it is still manageable.

In this chapter cases are concerned that may be considered a bugbear in other chapters; namely whenever the ratio zero divided by zero appears, and which Euler often writes down without a care. A careful review is made of such cases, and the related ones where infinitudes occur; methods are presented for dealing with these, involving the complete derivative or as we would say in most cases , a Taylor expansion is made, or it follows from L'Hôpital's rule, etc.; both algebraic and transcending cases are considered. Many examples are given; in all a very useful chapter for the beginner.

Euler continues with his *pot pourri* of problems that have been
bothering him. In this chapter he sets out a class of functions that does not
fit into any previous scheme. These take the form of generalised sums or
products ending (or beginning)with some real number rather than an integer, and
the intention is to show how to find the complete derivatives of such functions
when they are extended to infinity, where the terms of some order of difference
eventually are equal; thus, diverging series are included in this
derivation. The examples chosen arise
from a generalised harmonic function, or sums of inverse powers of the integers
of some order, and from a generalised *n*!

Clearly, it is to gain a greater understanding of these problems that this chapter has been inserted; it is unlikely that a beginner would be able to understand it, then or even now.

This content of this chapter follows on from the previous one; Euler shows how to interpolate infinite series and products according to the previous schemes established; there is most interest in sums involving inverse powers of integers, and in interpolating binomial coefficients. Usually the half indices are interpolated as examples.

Click here for Chapter 18 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. Feb. 2,
2011 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
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