EULER'S
INSTITUTIONUM CALCULI INTEGRALIS
Translated
and annotated by
Ian Bruce
Introduction.
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.
Volume I,
Section I. (E342)
Click here for some introductory
material, in which Euler defines integration as the inverse process of
differentiation.
Click here for the 1st 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 2nd 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 3rd 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 Mechanica.
Click here for the 4th Chapter : Concerning the integration of differential formulas involving logarithmic and exponential functions. Particular simple cases involving logarithmic functions are presented first; the work involves integration by parts, which can be performed in two ways if needed. Progressively more difficult differentials are tackled, which often can be integrated by an infinite series expansion. A new kind of transcendental function arises here. Those who delight in such things can see the exponential function set out as we know it, and various integrations performed, including the derivation of some very cute series, as Euler himself notes in so many words.
Click here for the 5th Chapter : Concerning the integration of differential formulas involving angles or the sines of angles. Again, particular simple cases involving sines or powers of sines and another function in a product are integrated in two ways by the product rule for integrals. This leads to the listing of numerous integrals, on continuing the partial integrations until simple integrals are arrived at; the chapter culminates with the sine and cosine function being linked to an exponential function of the angle ; the case where such an exponent disappears on summing to infinite is considered.
Click
here for the 6th Chapter :
Concerning the development of integrals in series
progressing according to multiple angles of the sine or cosine. This chapter considers differential
expressions such as d(phi)/(1+ncos (phi)) which can be readily expanded in a power
series of cosines, which then is changed into equivalent series of cosines of
multiple angles, which then can be integrated at once. Much labour is involved
in creating the coefficients of the cosines of the multiple angles. This
chapter is thus heavy in formulas; recursive relations of the second order are
considered; means of evaluating the coefficients from infinite sums are
considered; all in all a rather heady chapter, some parts of which I have just
presented, and leave for the enthusiast to ponder over.
Click here for the 7th Chapter : A general method by which integrals can be found approximately. This chapter starts by considering the integral as the sum of infinitesimal strips of width dx, from which Euler forms upper and lower sums or bounds on the integral, for a dissection of the domain of integration into sections. This lead to an improved method involving successive integration by parts, applied to each of the sections, and leading to a form of the Taylor expansion, where the derivatives of the integrand are evaluated at the upper ends of the intervals. This method is applied to a number of examples, including the log function. Various cases where the integral diverges are considered, and where the divergence may be removed by transforming the integrand.
Click here for the 8th Chapter : Concerning the value of integrals on taking certain cases only. This chapter starts by considering the integral xmdx/sqrt(1-x2) for various values of m. The even powers depend on the quadrature of the unit circle while the odd powers are algebraic. Products of the two kinds are considered, and the integrands are expanded as infinite series in certain ways. These integrals lead to more complex forms such as xmdx/cu.rt((1-x3)2) and xmdx/cu.rt((1-x3), and again products are formed and series expansions made. Integrals that are the forerunners of the Betta and Gamma functions are considered, while the final masterful stroke is to consider the integration of xm-1dx/ (1+x2) , which will be shown in the following chapter from infinite products rather than from infinite sums.
Click here for the 9th Chapter : Concerning the development of integrals as infinite products. The integral dx/sqrt(1-x2) is first expanded as an infinite product between the limits 0 and 1, relying on the general method established earlier, in contrast to using repeated integration to reduce the power of the variable in the integrand considered in Ch. 8. Euler proceeds to investigate a wide class of integral of this form, relating these to the Wallis product, etc. Eventually he devises a shorthand way of writing such infinite products or their integrals, and investigates their properties on this basis. One might presume that this was the first extensive investigation of infinite products. This chapter ends the First Section of Book I.
Volume I,
Section II.
Click here for the 1st 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 2nd 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 3rd 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 4th 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 5th 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 6th Chapter :
On the comparison of transcendental
quantities contained in the form
Pdx/sq.root(quadratic in x2).
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 + Cx2 +Ex4) 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 7th 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.
Volume
II, Section I. (E366)
The resolution of differential equations of the second
order only.
Click here for the 1st chapter : Concerning the integration of simple differential formulas of the second order. Use is made of the variables, p = dy/dx and q = dp/dx in solving some more difficult first order differential equations. At this stage the exclusive use of the constant differential dx, which can be seen in the earlier work of Euler via Newton is abandoned, so that ddx need not be zero, and there are now four variables available in solving second order equations : p, q, x, and y. Euler admits that this is a more powerful method than the separation of variables in finding solutions to such equations, where some differential quantity is kept constant.
Click here for
the 2nd chapter : Concerning second order differential
equations in which one of the variables is absent.
Further use is made of the variables, p = dy/dx and q = dp/dx in solving some second order differential equations. The idea of solving such equations in a step–like manner is introduced; most of the equations tackled have some other significance, such as relating to the radius of curvature of some curve, etc.
Click here for the 3rd chapter : Concerning homogeneous second order differential equations, and those which can be reduced to that form. Further use is made of the variables, p = dy/dx, q = dp/dx, p = ux, and q = v/x in solving some homogeneous second order differential equations. In these examples a finite equation is obtained between some of the variables, as x disappears. Euler displays his brilliance in finding integrating factors for these equations, to one of which I have added a note (§807 Scholium) ; others I have left to the intrepid investigators of this work.
Click here for the 4th chapter : Concerning second order differential equations in which the other variable y has a single dimension. A careful exposition is made of equations of the form y'' +Py' + Qy = X, where P, Q, and X are functions of x, written of course in the Euler manner ddy + Pdy + ...etc. A lot of familiar material is uncovered here, perhaps in an unusual manner : for example, we see the origin of the particular integral and complementary function for integrals of this kind.
Click here for the 5th chapter : Concerning the integration by factors of second order differential equations in which the other variable y has a single dimension. Now equations of the form y'' +Py' + Qy = X, where P, Q, and X are functions of x, are considered that can be solved completely. The use of multipliers is used in conjunction with the formation of total differentials, applied in succession solving such equations for particular forms of P and Q.
Click here for the 6th chapter : Concerning the integration of other second order differential equations by putting in place suitable multipliers. This is a harder chapter to master, and more has been written by way of notes by me, though some parts have been left for you to discover for yourself. The methods used are clear enough, but one wonders at the insights and originality of parts of the work. The use of more complicated integrating factors is considered in depth for various kinds of second order differential equations. How much of this material is available or even hinted at in current texts I would not know; it seems to be heading towards integral transforms, where the integral of the transformed equation can be evaluated, and then the inverse transform effected : but this latter operation is not attempted here.
Click here for
the 7th chapter : On the resolution of the second order
differential equation ddy +a x^n ydx^2 = 0 by infinite series. This chapter
is rather labour intensive as regards the number of formulas to be typed out;
however, modern computing makes even this task easier. The relatively easy task
of setting up an infinite series for the integral chosen is accomplished; after
which considerable attention is paid to series that end abruptly due to the
introduction of a zero term in the iteration, thus providing algebraic
solutions. Euler had evidently spent a great deal of time investigating such
series solutions of integrals, and again one wonders at his remarkable
industry. Recall that this book was meant as a teaching manual for integration,
and this task it performed admirably, though no thought was given to
convergence, a charge often laid.
Click here for the 8th chapter : Concerning the resolution of other second order differential equation by infinite series. This chapter is also rather labour intensive as regards the number of formulas to be typed out; here a more general second order differential equation is set up and integrated by a series expansion. The emphasis is now on degenerate cases, which arise when the roots of the indicial equation are equal or imaginary, and the ln function is introduced as a multiplier of one of the series; there is a desire to obtain the complete integral for these more trying cases.
Click here for
the 9th chapter : Concerning the resolution of other second
order differential equation of the form
Lddy + Mdxdy +Nydx2=0. This is a most interesting chapter, in which other second order equations are transformed in various ways into other like equations that may or may not be integrable. It builds on the previous chapter to some extent, and ends with some remarks on double integrals, or the solving of such differential equations essentially by double integrals, a process which was evidently still under development at this time.
Click here for the 10th chapter : On the construction of second order differential equations from the quadrature of curves. In this chapter there is a move into functions of two variables. The idea is to take an integral of some function V, treat it as a function of two variables x and u, and to form a differential equation of the form Lddy + Mdydu +Nydu2=0 from this integral by differentiating within the integral. This is set equal to a chosen function U, which is itself differentiated w.r.t. x. enabling the coefficients L, M, and N to be determined. Thus, the differential equation becomes equal to a function U with the limits w.r.t. x taken, so that U is a function of u only. Essentially the work proceeds backwards from a solution to the responsible differential equation. These details are sketched here briefly, and you need to read the chapter to find out what is going on in a more coherent manner. A number of examples of the procedure are put in place, and the work was clearly one of Euler's ongoing projects.
Click here for the 11th chapter : On the construction of second order differential equations sought from the resolution of these by infinite series. This chapter follows on from the previous one : more degrees of freedom are introduced by introducing a series with two–fold coefficients, enabling a more general differential equation to be tackled, that has been met before. An integral is established finally for the differential equation, the bounds of which both give zero for the dummy variable, an artifice that enables integration by parts to be carried out without the introduction of extra terms. The variable x in the original d.e. is treated as a constant in the integration.
Click here for the 12th chapter : Concerning the integration of second order differential equations by approximations. This chapter is not about what you might think from the title, and does not offer much in the way of the approximate evaluation of integrals numerically, even if they are of second order, apart from advocating the use of very small intervals, and eventually a more involved way that allows the second derivative to change in the initial interval is set out for use, and giving rise to quadratic quadrature over each interval. If anything, the chapter sets the stage for an iterative program of some kind, and thus is of a general nature, while what to do in case of diverging quantities is given the most thought. This marks the end of Section I.
Volume
II, Section II.
The
resolution of differential equations of the third or higher orders which
involve only two variables.
Click here for the 1st chapter : Concerning the integration of simple differential formulas of the third or higher orders.
Euler derives some very pretty results for the integration of these simple higher order derivatives, but as he points out, the selection is limited to only a few choice kinds. Thus the chapter is rather short.
Click here for the 2nd chapter : Concerning the integration of differential formulas of this form Ay +By' +Cy'' .... +Ny(n) on considering dx constant. This is the most beautiful of chapters in this book to date, and one which must have given Euler a great deal of joy ; there is only one thing I suggest you do, and that is to read it.
Click here for the 3rd chapter : Concerning the integration of differential formulas of this form Ay +By' +Cy'' .... +Ny(n) = X, on considering dx constant. This is clearly a continuation of the previous chapter, where the method is applied to solving y for some function of X, using the exponential function with its associated algebraic equation. Serious difficulties arise when the algebraic equation has multiple roots, and the method of partial integration is used; however, Euler tries to get round this difficulty with an arithmetical theorem, which is not successful, but at least provides a foundation for the case of unequal roots, and the subsequent work of Cauchy on complex integration is required to solve this difficulty. This is a long chapter, and I have labored over the translation for a week; it is not an easy document to translate or read; but I think that it has been well worth the effort.
Click here for the 4th chapter : The application of the method of integration treated in the last chapter to examples. The examples are restricted to forms of X above for which the algebraic equation has well-known roots. Much light is shed on the methods promulgated in the previous chapter, and this chapter should be read in conjunction with the preceding two chapters. Euler takes the occasion to extend X to infinity in a Taylor expansion at some stages.
Click here for the 5th chapter : Concerning the integration of differential formulas of this form X=Ay +Bxy' +Cx2y'' + etc. This is a chapter devoted to the solution of one kind of differential equation, where the integrating factor is simply xdx. Simple solutions are considered initially for distinct real roots, which progress up to order five. Most of the concern as we proceed is about real repeated roots, which have diverging parts that are shown to cancel in pairs, and complex conjugate pairs, which are easier to handle, and the general form of the solution is gradually evolved by examining these special cases, after which terms are picked out for parts of the general integral. This is the end of Euler's original Book One.
Volume III,
Section I. (E385)
The
Investigation of Functions of Two Variables from a Relation of the
Differentials of the First Order.
Click here for the 1st 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 2nd 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 3rd 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 4th 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 generalisation 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 5th 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 6th 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 1st 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 2nd 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 d2z/dx2 , d2z/dxdy , or d2z/dy2 = some function of dz/dx and other possible functions of x and y are integrated in general, with the customary examples.
Click here for
the 3rd 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 4th 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 5th 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.
EULER'S
INSTITUTIONUM CALCULI DIFFERENTIALIS
Translated
and annotated by
Ian Bruce
Click here for the 18th 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 9th,
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.