**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, 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.

**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

Click here for the 9^{th} Chapter [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 10^{th} Chapter [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 11^{th} Chapter [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 12^{th} Chapter [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 13^{th} Chapter [E562]: ** Sines and cosines of multiple angles **: an extension of a previous paper.

**VOLUME II**

Click here for
the 1^{st} Chapter [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 2^{nd} Chapter [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 3^{rd}
and 4^{th} Chapters [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 5^{th}
Chapter [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 6^{th}
Chapter [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 7^{th}
Chapter [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 8^{th}
Chapter [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 9^{th}
Chapter [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 10^{th}
Chapter [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 11^{th} Chapter
[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 12^{th} Chapter
[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.

Ian
Bruce. Nov. 9^{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 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.