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. A general but very brief summary of this work can be found in Ronald S. Calinger's Leonhard Euler, pp. 529-30. 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.
Click here for the 1st Chapter [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 Chapter [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 Chapter [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 Chapter [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 Chapter [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 Chapter [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 Chapter [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 Chapter [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
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