This was the last book published by Euler in his lifetime in 1783, and consists of a number of papers presented to the St. Petersburg Academy of Sciences. It should be observed that such works were not written to educate aspiring mathematicians, as were his calculus and analytical books, but rather took the form of what we would now call research papers. Clearly he felt they were the most worthy of his attention at that point of his life, which had not been published separately, or were latter-day thoughts on earlier papers. Several other books are useful as references for these works ; these include :
A Source Book of Mathematics, edited by D.J.Struik, Harvard;
History of the Theory of Numbers, Vol. 1 & 2 especially, L.E.Dickson. Dover;
Number Theory….., Andre Weil, Birkhauser;
Biscuits of Number Theory, edited by Benjamin & Brown, MAA.
Click here for the 1st Section [E550]: On series in which the products from two contiguous terms constitute a given progression.
This is rather a long paper, in which two methods are introduced for finding the general term of a special kind of series of the form a, ab, bc, cd, de, etc. , where the series or rather sequence has the terms designated by A, B, C, D, etc., where the first term a may not be known initially. This gives rise to solutions in terms of infinite products that may be solved either by integrals [E223] , or by continued fractions , originally investigated in E122. Most of the work consists of setting up products of quadratic equations of various kinds which can be recast as continued fractions, using the comparison of coefficients to show how this is done in these cases.
Click here for the E19 :
Click here for the E122 : Concerning transcendental progressions, or of which the general terms are unable to be given algebraically.
This again is rather a long paper, and depends on E19, on which it relies heavily, especially on the theorem in section 16 of that, where essentially infinite products are displayed as products of integrals. This work is a continuation of that, and is difficult at times due to the lack of decent notations.
Click here for the 2nd Section [E551] : Various method of enquiring into the nature of series.
This is rather quite a short paper and quite delightful; it sets out to form the sum of the series to be found from middle maximum terms of expansions of the trinomial (1+x+x2)n with n, starting from an inductive proof which is fallacious; then the genius who was Leonhardt Euler takes over ……
Click here for the 3rd Section [E552]: Observations regarding the division of squares by prime numbers.
This is also rather a short paper, in which Euler inducts the reader into the charm of remainders or residues of squares divided by prime numbers, which laid the foundations for further work, especially in Gauss's masterful treatment in his Disquisitiones Arithmeticae. See e.g. D. J. Struik's A Source Book in Mathematics, 1200-1800 (1969, Harvard University Press), pp. 40-46.
Click here for the 4th Section [E553]: Analytical observations regarding continued fractions.
This is a longer paper, and one on which it appears Euler spent some time; in it he considers two related kinds of continued fractions, and establishes transformations between the two.
Click here for the 5th Section [E554]: A more accurate inquiry concerning the remainders left….
This is a longish paper, and one on which it appears Euler also spent some time; in it he considers amongst other things the remainders or residues left on dividing the squares of the natural numbers by prime numbers of certain elementary forms.
Click here for the 6th Section [E555]: Concerning the outstanding use of the method of interpolation in the theory of series. This is another longish paper, and one on which it appears Euler also spent considerable time; in it he considers amongst other things approximations for π, the sums of numerous series, and his continued fascination with a class of integrals arising in Vol. IV of the Integral Calculus.
Click here for the 7th Section [E556]: Regarding the criteria of the equation fxx+gyy = hzz, whether or not that may admit a resolution. This is an interesting paper, but perhaps confusing at times; the basic idea is to find an algorithm for expressing the sum of integral multiples of two squared prime numbers as an integral multiple of another prime squared. Euler confesses that he has not been able to give a general proof for this, but presents a large number of possible techniques for special cases; in a sense perhaps it indicates how he reasoned towards general solutions to problems, by considering special cases. It may well be that there is no general method for doing this….
Click here for the 8th Section [E557]: Regarding certain outstanding properties arising from the divisors of powers. This paper has its origins in the previous work by Euler in proving that Fermat's formula for primes was not true in general, F5 or 232+1 has the factors 641 and 6700417; here he finds simple formulas for the numbers that must be taken from numbers of the form 4an+1 to produce primes, applicable to the square a2. This is a long paper with an extension, with much labour having been expended by Euler in its development.
Click here for the 9th Section [E558]: Some progression is sought beginning from unity, so that just as many of its terms may be required to be added to the minimum, so that all the intervening numbers may be produced. This paper has its origins in well known theorem of Fermat regarding polygonal or figurative numbers of any order. Beguelin, at the Berlin Academy, had tried unsuccessfully, to prove Fermat's theorem, and now Euler was trying to do the same. Interesting, but have your copy of Dickson handy!
Click here for the 10th Section [E559]: Some aids in the solution of Pell's equation. This paper has its origins in a former work of Euler, and shows how to reduce certain equations to the Pell form, as well as elaborating on known solutions.
Click here for the 11th Section [E560]: Some small analytical works. Six problems are set out here with their solutions, the last of which is an extension of a previous one.
Click here for the 12th Section [E561]: Various observations regarding angles in geometric progression. Euler investigates trigonometric ratios in infinite products with the angles in g.p.
Click here for the 13th Section [E562]: Sines and cosines of multiple angles : an extension of a previous paper.
Click here for the 1st Section [E586]: Considerations on a theorem of Fermat regarding the resolution of numbers into polygonal numbers. Euler had a great penchant to prove Fermat's Theorem above; however, he was unable to supply such a proof , which turned out to be very difficult for the trigonal case, which he considers here.
Click here for the 2nd Section [E587]: Observations on some theorems of the most illustrious Lagrange. Euler has began the study of integrals involving two variables in an earlier paper; Lagrange had received this enthusiastically but found a paradox, which Euler resolves here to his satisfaction, and considers further developments.
Click here for the 3rd and 4th Sections [E588&9]: Investigation of the integral formulas….. Here Euler examines two classes of integrals, which he solves with a great deal of dexterity. I have put these together, each followed by its Latin original.
Click here for the 5th Section [E590]: Certain Analytical Theorems of which the demonstration is now desired. Euler examines a number of situations which he finds perplexing; these are concerned mainly with situations involving complex numbers, and it has not occurred to him to move into the complex plane. Finally, he considers the rectification of curves consisting of circular arcs, and concludes that such curves can be circles only. However, following on his later work on rectifying curves, he finds that this is not the case, which he presents in E783, translated here
Click here for the E783 : Concerning algebraic curves all of which may be measured by circular arcs.
Click here for the 6th Section [E591]: Concerning a relation being established between three or more quantities. Euler examines how a number of disparate numbers may be linked together by an extension of the method of the least common denominator; thus, for example, using this method, he looks for a connection between pi and ln2. He is largely unsuccessful in his quests, though the method itself is of interest.
Click here for the 7th Section [E592]: Concerning the resolution of transcending fractions into an infinite number of simple fractions. Euler was fascinated always with his infinite product proof for pi, and here he tackles the inverse problem, of adding an infinitude of simple fractions involving pi to give some finite sum, involving trigonometric functions. The problems are tackled in two ways, one involving essentially a limiting process, and the other straight differentiation of a given series.
Click here for the 8th Section [E593]: Concerning the transformation of series into continued fractions where likewise this theory has been made considerably enlarged. Another of Euler's fascinations was the study of continued fractions, which he presents here from basic principles to several ways of accomplishing the required continued fraction from a variety of series.
Click here for the 9th Section [E594]: A method of resolving integrals…. involving continued fractions. This Commentary can be found also in Supp. 5e of Vol. IV of the Integral Calculus.
Click here for the 10th Section [E595]: The summation of continued fractions of which the indices form an arithmetical progression ….. Euler adds another facet to his work on the Ricatti differential equation as he understand it, and provided a solution involving continued fractions.
Click here for the 11th Section [E596]: The summation of series from prime numbers of the form…. Euler continues his studies into the sum of the signed inverses of prime numbers and extends this to higher powers, using Leibnitz's familiar expansion for π/4. This work provided a stimulus for further investigations into such sums ; however, he leaves this work in a exasperated state of mind, as he is unable to make further progress. An interesting read.
Click here for the 12th Section [E597]: The summation of series of reciprocal numbers powers by a new and easier method. Euler continues his studies into the sum of the signed inverses of prime numbers and extends this to higher powers further, making use of known integrals involving the sine and tangent functions. This work provides a much easier method investigation such sums. Another interesting read.
Click here for the 13th Section [E598]: Concerning a significant advance in the science of numbers. As you may well be aware, there was an ongoing intellectual battle of sorts going on between Euler and Lagrange: the latter would devour what Euler had written in his own rather quaint expositary manner, polish things up and add his own deductions, and hey presto, another paper, but without the detailed explanations…. That is the case here, where Euler in turn contemplates the latest Lagrange emissive on factors prime to the quadratic form pp + nqq, on which he himself had worked on for many years. This is a longish paper, running to 30 odd pages; the mathematics is elementary, but the understanding is not quite so elementary, and the treatment is mainly intuitive.
Click here for the 14th & 15th Sections [E599&E600]. E599: The solution of a question relating to the calculation of the probability of how much two spouses should pay, so that after the death of each, a certain sum of money may be paid to their heirs. The solution of a number of situations where both spouses are still alive after so many years, on or other have passed away, or both are analysed, using mortality tables as part of the calculations.
E600: The solution of more difficult questions in the calculus of probability. An analysis of the probabile results of repeated lottery draws with 95 tickets is presented. The paper is presented in an unusual way in that the demonstrations of the theorems is not given until the end.
Ian Bruce. Dec. 3rd , 2017 latest revision. Copyright : I reserve the right to publish this translated work in book form. You are not given permission to sell all or part of this translation as an e-book. 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. See note on the index page. Please feel free to contact me if you wish by clicking on my name at the end of the index page, especially if you have any relevant comments or concerns.