St. Petersburg: Academia Imperialis Scientiarum, 1783-85.
First edition of this collection of 28 previously unpublished papers by Euler (listed below), of which the most important is ‘Observationes circa divisionem quadratorum per numeros primos’, which gives the first clear statement of the ‘law of quadratic reciprocity,’ called the ‘golden theorem’ by Gauss who gave its first proof in his Disquisitiones Arithmeticae (1801). The first volume was published in the year of Euler’s death, most of the papers having been presented to the St. Petersburg Academy a decade or more earlier.
The majority of the papers in these volumes deal with topics in number theory (3, 5, 7-11, 14, 19). Euler had studied questions related to quadratic reciprocity for decades, starting with a letter to Christian Goldbach dated 28 August, 1742. He gradually accumulated numerical evidence, but did not feel able to formulate the ‘law’ until he did so in (3). He continued to work on topics related to quadratic reciprocity – paper (5) is an example – but a proof eluded him, as it did Legendre later.
Apart from the papers dealing with quadratic reciprocity, the most interesting of the number-theoretic papers is perhaps ‘Nova subsidia pro resolutione formulae axx + 1 = yy,’ which gives a new method of solving Pell’s equation x2– dy2 = 1. Euler had developed a method of solution using continued fractions in 1765 (‘De usu novi algorithmi in problemate Pelliano solvendo’) but noted that for some values of d this method leads to very tedious calculations (he mentions the case d = 61, for which the smallest solution is x = 1766319049 and y = 226153980). The new method, which generates solutions of Pell’s equation from solutions of closely related equations, is much more efficient, although it applies only to certain values of d. It has been repeatedly rediscovered over the centuries, and has recently been generalized to apply to any value of d.
Another group of papers deals with finite and infinite series (1, 2, 20, 24, 25). Euler had famously solved the problem of exactly summing the zeta function series ς (n) = 1/1n + 1/2n + 1/3n + 1/4n + … in the case n = 2, which had defeated the Bernoullis, and in paper (25) he calculates ς (2n) up to n = 17. In ‘De summa seriei ex numeris primis formatae 1/3 - 1/5 + 1/7 + 1/11 - 1/13 ... ubi numeri primi formae 4n - 1 habent signum positivum, formae autem 4n + 1 signum negativum,’ Euler notes that the sum of the reciprocals of the primes diverges, as does the harmonic series 1 + (1/2) + (1/3) + (1/4) + …. His derivations start with the ‘Leibniz’ series, 1 - (1/3) + (1/5) - (1/7) + (1/9) - ... = /4.
In ‘De eximio usu methodi interpolationum in serierum doctrina’ Euler presents his discovery of the ‘Lagrange interpolation formula,’ a fundamental technique in numerical analysis. This was published by Lagrange in 1795, and earlier by Waring in 1779 (‘Problems concerning interpolations,’ Phil. Trans., Vol. 69, pp. 59-67), but Euler’s paper was presented to the Academy on May 18, 1772.
The last two papers discuss topics in probability, a subject not normally associated with Euler. ‘Solutio quaestionis ad calculum probabilitatis pertinentis’ treats a problem in annuities: How much should be paid by a couple, so that a certain sum of money can be paid to the heir after the death of the other? In ‘Solutio quarundam quaestionum difficiliorum in calculo probabilium,’ Euler studies the ‘Genoese lottery,’ a game of chance similar to today’s lotteries in which numbered balls are placed in a large wheel, five or six are drawn at random, and players attempt to guess the numbers. Euler became interested in such lotteries after he was asked by King Frederick II for his analysis of a proposal for a state lottery involving the drawing of five numbers from 1 to 90.
Other papers deal with definite integrals (15-17), infinite products (12, 13), and several with one of Euler’s favourite topics, continued fractions (4, 21-23).
The works included in these volumes are as follows:
1. De seriebus, in quibus producta ex minis terminis contiguis datam constituunt progressionem
2. Varia artificia in serierum indolem inquirendi
3. Observationes circa divisionem quadratorum per numeros primos
4. Observationes analyticae
5. Disquitio accuratior circa residua ex divisione quadratorum altiorumque potestatum per numeros primos relicta
6. De eximio usu methodi interpolationum in serierum doctrina
7. De criteriis aequationis fxx + gyy = hxx, utrum ea resolutionem admittat necne?
8. De quibusdam eximiis proprietatibus circa divisores potestatum occurrentibus
9. Proposita quacunque protressione ab unitate incipiente, quaeritur quot eius terminos ad minimum addi oporteat, ut omnes numeri producantur
10. Nova subsidia pro resolutione formulae axx + 1 = yy
11. Miscellanea analytica
12. Variae observationes circa angulos in progressione geometrica progredientes
13. Quomodo sinus et cosinus angulorum multiplorum per producta exprimi queant
14. Considerationes super theoremate Fermatiano de resolutione numerorum in numeros polygonales
15. Observation in aliquot theoremata illustrissimi de la Grange
16. Investigatio formulae integralis (xm-1 dx)/(1+xk)n casu, quo post intagrationem statuitur x =
17. Investigatio valoris integralis (xm-1 dx)/(1-2xkcos +x2k) a termino x = 0 ad x = extensi
18. Theoremata quaedam analytica, quorum demonstratio adhuc desideratur
19. De relatione inter ternas pluresve quantitates instituenda
20. De resolutione fractionum transcendentium in infinitas fractiones simplices
21. De transformatione serierum in fractiones continuas, ubi simul haec theoria non mediocriter amplificatur
22. Methodus inveniendi formulas integrales, quae certis casibus datam inter se teneant rationem, ubi sumul methodus traditur fractiones continuas summandi
23. Summatio fractionis continuae cuius indices progressionem arithmeticam constituunt dum numeratores omnes sunt unitates ubi simul resolutio aequationis Riccatianae per huiusmodi fractiones docetur
24. De summa seriei ex numeris primis formatae 1/3 - 1/5 + 1/7 + 1/11 - 1/13 ... ubi numeri primi formae 4n-1 habent signum positivum, formae autem 4n+1 signum negativum
25. De seriebus potestatum reciprocis methodo nova et facillima summandis
26. De insigni promotione scientiae numerorum
27. Solutio quaestionis ad calculum probabilitatis pertinentis
28. Solutio quarundam quaestionum difficiliorum in calculo probabilium
Two volumes, large 4to (257 x 205 mm), pp. [iv], 364; [iv], 346, with two folding engraved plates. Bound in two fine uniform calf bindings with gilt armorial stamp of Collége Royal de Henri IV to the front boards, some light wear to spines and capitals and a 5 cm crack to the front hinge to second volume (still strong though), in all a very fine and unrestored set.