# Euler and Lagrange

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## Abstract

The invention of the calculus and its early exploitation was certainly due to Newton and Leibniz, but the invention of classical analysis was very largely due to Leonhard Euler, of Basel, who worked mainly in St. Petersburg. He was the student of Johann or Jean Bernoulli the First, and maintained a justifiable lifelong admiration and affection for the family.

## Keywords

Summation Formula Bernoulli Number Generate Fraction Euler Polynomial Indicial Equation
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## Reference

- 1.Nörlund [1924], pp. 480–481. In fact the whole bibliography from the eighteenth century onwards is very complete. It is on pp. 464–531 of Nörlund.Google Scholar
- 2.Euler 1, VIII [1748], pp. 133–134. This value of IT, according to the editors, Euler got from Th. F. deLagny [1719], p. 135. Recall that de Moivre, Maclaurin and Stirling used c for 27r. It was William Jones, whom we mentioned on p. 72 who first used the symbol Tr. Cf. Jones [1706], p. 243.Google Scholar
- 3.Euler 1, VIII [1748], pp. 122–128.Google Scholar
- 4.Euler 1, XVII 11749], pp. 195–232. Cf. p. 196n for references to the papers by Bernoulli and Leibniz.Google Scholar
- 5.Euler 1, XVII [1749], p. 210.Google Scholar
- 6.Euler 1, XVII [1749], pp. 167–169. Euler was preceded in the study of trinomial factors by Newton. It is perhaps worth remarking that in his discussion of trigonometric functions Euler, following de Moivre [1730], p. 1, notes that (cos
*z*± √−1 • sin*z*)^{n}= cos*nz*± √−1 • sin*nz*.Google Scholar - 9.These results are essentially Newtonian but were in the cases n = 1, 2, 3, 4 first derived by Albert Girard (1590–1633), of Lorraine (Girard [1629]). (Girard was also the first to write cube and fourth roots as
^{−3}√,^{−4}√.) Newton’s derivation was “an unproved generalization of his computations on the coefficients (expressed as symmetric functions of the roots) of an equation of the eighth degree. No rigorous demonstration of the rule was published till the contents of Maclaurin’s letter to Stanhope on July 8, 1743 systematically establishing it was printed by the Publisher’ (Patrick Murdoch?) in Part II, Chapter XII of Maclaurin’s posthumous*A Treatise of Algebra*..., pp. 286–296.” Cf. Newton,*Papers*, Vol. V, p. 361n.Google Scholar - 10.Euler 1, VIII [1748], pp. 177–181. In this place he goes up to
*n*= 26. Later he took up the general case as we shall see below. The case*n*= 2 just given is, of course, one of the most celebrated expansions. In Section 3.1 below we see the case*n*= 1; cf. p. 131.Google Scholar - 11.Euler 1, X [1755]. Cf. also, Euler 1, XV [1738].Google Scholar
- 12.Euler 1, X [1755], pp. 321ff. This is Chapter V.Google Scholar
- 13.Euler 1, X [1755], p. 321.Google Scholar
- 14.This result was first discovered by Euler. Cf., e.g., Nörlund [1924], pp. 23–26. The form of R„ has been discussed by a number of people, including Darboux and Hermite (cf. Nörlund [1924], p. 34.) The first 20 Eulerian numbers are tabulated by Nörlund [1924], p. 458. Boole’s result appears in Boole [1880]. The so-called Boole polynomials are discussed at length by Jordan [1939], pp. 317ff. He also gives another form of a summation formula of Boole. It involves differences instead of derivatives and the Boole instead of the Euler polynomials. These Boole polynomials may be defined with the help of the relationGoogle Scholar
- 15.Attached to Euler 1, XII [1768/69], pp. 415–542, are commentaries written by an Italian humanist and mathematician, Lorenzo Mascheroni (1750–1800), who first taught in a gymnasium in Ticino and then became professor of mathematics in Pavia (cf. Mascheroni [1790] and [1792]). On p. 431 of Euler 1, XII [1768/69], Mascheroni gives Euler’s constant as 0.577215 664901 532860 61811 090082 39 (cf. also p. 442); later Gauss recalculated the constant and in 1878 J. C. Adams (Adams I [1878]) gave its value to 263 places. Mascheroni’s value was wrong in the 20th, 21st, and 22nd places. It is interesting to see how Mascheroni expressed definite integrals. He wrote, e.g., He used this notation throughout whereas Euler wrote, e.g., the integral and then added the words
*integration ab x*= 0*ad x*= 1.Google Scholar - 16.Euler 1, X [1755], p. 350. Note that Euler used a bold-faced capital S lying on its side to denote infinity Apparently it was Wallis who introduced this symbol, as we noted on p. 79.Google Scholar
- 19.Euler 1, X [1755], p. 384. Thus, e.g., in the expression for 1 − 2
^{3}+ … the trinomial with the in front is to be discarded, leaving for the infinite sum −1/8. It was not until Gauss investigated the hypergeometric series that we find a true discussion of convergence. Later Cauchy, in his*Analyse Algébrique*(Paris, 1821) also gave a rigorous treatment of series convergence; it should be noted though that Leibniz did give a criterium for the convergence of alternating series. The interested reader may consult F. Cajori [1919], pp. 373–377.Google Scholar - 20.Euler 1, X [1755], p. 419. Kowalewski points out that K should be 2404879675441.Google Scholar
- 21.Raabe [1848]. Cf. also, [1851]. In this paper he relates the Bernoulli and Euler numbers to the tangent and secant coefficients. Cf. also, Glaisher [1914].Google Scholar
- 22.Nörlund [1924], p. 98. Euler’s first letter on the Gamma function appears in P.-H. Fuss [1843]; the relevant letter is dated October 13, 1729 and contains the relation (3.10) below. There are a number of others on the same topic. (There is an interesting discussion of the relation between Wallis’s and Euler’s inspiration for this sort of interpolation in N. Bourbaki [1960], pp. 226–227.)Google Scholar
- 23.Euler 1, X [1755], p. 640ff. Cf. Chap. XVII.Google Scholar
- 24.Whittaker, W W, p. 237. The letter is in Fuss [1843].Google Scholar
- 25.Euler 1, XII [1768/69], pp. 269–270. This is a volume of his
*Institutiones Calculi Integralis*See also the footnote on p. 270 for further references. This function is also discussed in a number of papers contained in Vol. XVII starting on p. 233. He used the notation (g) for the integralCf. Euler 1, XVII [1762/65], pp. 268ff. (Here again the integral is to be understood as being evaluated at the limits*x*= 0 and*x*= 1, and n,*p*,*q*are positive integers.)Google Scholar - 26.Euler 1, XVII [1765/65], pp. 268–315. This is made possible by the previous paper, pp. 233–267. Cf. also, pp. 316ff. Note that
*B*,(*p*,*q*) = 1*B*(*n*,*n*).Google Scholar - 28.Brown [1896], p. 241. For the most part, Brown’s work, as improved in certain important respects by W. J. Eckert, is still the standard for calculating lunar ephemerides. (A certain emendation was also made by G. M. Clemence.) The interested reader may wish to consult Brouwer [1961] or
*Ephemeris*[1952/59].Google Scholar - 29.Cf. Euler 2, XXIII [1753], pp. 64–336.Google Scholar
- 30.Cf. also, Tisserand [1889], Tome III, 1894, pp. 65–75, for the first theory, and pp. 76–88 for the second, and Brown [1896], pp. 239–241.Google Scholar
- 31.Bernoulli, D. [1728].Google Scholar
- 32.de Moivre [1722], pp. 162–178 and [1730], pp. 4ff. It is fascinating to read on p. 178 of the first reference a remark by de Moivre that while his 1722 paper was in press (
*Dum superiores paginae praelo subjiciebantur*,…) he accidentally discovered in the*Acta Leips*. 1702/03 that Leibniz had used similar methods for treating algebraic fractions.Google Scholar - 35.Lagrange I [1759], p. 28.Google Scholar
- 36.Lagrange I [1759], p. 33. Note that he has reversed the usual order of the indices.Google Scholar
- 37.Lagrange I [1759], p. 35.Google Scholar
- 38.Lagrange I [1762], pp. 493–498. The study of this equation is central to Lagrange’s work on “Fourier” series. It is in his famous method of “variation of parameters,” which he may have evolved from a study of Euler’s method for analyzing the perturbations in the orbits of the moon and planets caused by third bodies. It was used by Euler to study the famous “three-body problem” of astronomy. Cf. Euler 2, I [1736/42]. Lagrange hailed this as the first great application of analysis to dynamics.Google Scholar
- 39.Lagrange I [1762], pp. 473ff.Google Scholar
- 40.The first major problem solved by Daniel Bernoulli was the solution of this equation. This had also been studied by Jacques Bernoulli. Riccati (1676–1754) himself was a member of a Venetian family that produced a number of well-known scientists. His works were published posthumously (Riccati [1758].)Google Scholar
- 41.Lagrange I [1762], pp. 488–490.Google Scholar
- 42.Lagrange I [1762], pp. 514–516.Google Scholar
- 45.Presumably it was Daniel Bernoulli to whom he referred. (Cajori [1919], p. 242. Cajori gives here a brief discussion of the topic.)Google Scholar
- 54.This is a special case of what is known as Herschel’s theorem. (Milne-Thomson [1933], p. 32. Cf. also, Boole [1880].)Google Scholar
- 58.Lagrange VI [1772], pp. 604–606. Cf. also, Mayer [1770]. The use of these tables is explained on pp. 97–99 of that work.Google Scholar
- 60.Lagrange VII [1778], p. 553. He also said on p. 547 that his method was free of theGoogle Scholar
- 62.Lagrange VII [1778], p. 548. Here Lagrange has given for the homogeneous case the now well-known method of “Gaussian” elimination for solving linear systems. We say more about this later on p. 216.Google Scholar
- 63.Lagrange VII [1778], pp. 549–550.Google Scholar
- 71.Clairaut [1759]; see especially, pp. 544–564.Google Scholar
- 72.These matters appear in three long papers. Cf. Lagrange I [1759’], I [1760], and I [1761]. They occupy pp. 39–332 of Vol. I.Google Scholar
- 73.Lagrange I [1759’], pp. 111–112. He found this value,
*m*+ 1/2 − 1/2, using l’Hôpital’s rule.Google Scholar

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© Springer-Verlag, New York, Inc. 1977