**EULER'S **

Ian Bruce

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.

**VOLUME I**

Click here for
the 1^{st} 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 2^{nd} 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*+*x*^{2})^{n} with *n*,
starting from an inductive proof which is fallacious; then the genius who was
Leonhardt Euler takes over ……

Click
here for the 3^{rd} 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 4^{th} 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 5^{th} 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 6^{th} 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 7^{th} 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 8^{th} 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, F

Ian
Bruce. August. 8^{th}
, 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
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