EULER'S

INSTITUTIONUM CALCULI INTEGRALIS

VOLS III & IV

Translated and annotated by
Ian Bruce

Introduction.


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.

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 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 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.

 

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 1st 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 2nd 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) = eaxv(x, y).

 

Click here for the 3rd 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/dx2) + B(ddz/dxdy) + C (ddz/dy2) =0 , where he finds a primitive first order form and an algebraic equation. The method is extended to more general cases.

Volume III, Section IV.

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

 

Click here for the 1st 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 2nd 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 3rd 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 4th 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 1st 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 2nd 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 3rd 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 4th 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 5th 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 6th chapter : Concerning the Variations of Integral Formulas involving three Variables the relation of which is contained by a single relation.

Click here for the 7th 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 :

The 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 cosif 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 xp-1dx/(1-xn)1-q/n , which is investigated between the limits 0 and 1 for ascending powers of n, and the asssociated powers of p and q, as far as n=10. This work is E640.

 

 

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 xm-1dx/(1+xk)n, while the other E589 handles xm-1dx/(1-2xkcosq+x2k).

 

 

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.

 

Sup.6 : This is E391 in the O.O. , in which Euler considers double and repeated integrals and their uses in finding volumes and areas; the celebrated problem of V. Viviani's curve formed by the intersection of a sphere and cylinder is also considered, with the associated areas.

 

Sup.7 : This is E581 in the O.O. , in which Euler considers a special class of integrals. This is rather an interesting but challenging development.

 

 

Sup.8a : This is E506 in the O.O. , in which Euler reconsiders a special class of integrals, involving the integration of the functions dx/root X=dy/root Y, for algebraic functions X and Y, in comparison with Lagrange's determinations. This is rather a long paper, but easy to follow.

 


Ian Bruce. March 20th , 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.