Translated and annotated by
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.

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.









Translated and annotated by
Ian Bruce

Click here for the 18th Chapter of Part II : Concerning the resolution of rational functions of the forminto 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 29th, 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.