59. Schlesinger conjectured that these are definite integrals taken between 0 and ∞. Indeed [53], [54], and [59] are all readily expressible in terms of beta and gamma functions, as he pointed out.
60. The division of the lemniscatic arc into n equal pieces depends on an equation of degree n2 (unlike the division of the circular arc); n2 – n of the roots are complex. Gauss inferred that the lemniscatic functions are doubly periodic, which is the real breakthrough.
61. A slip of the pen: for mm + 6mn + nn read m4 – 6m2n2 + n4,
where the sums are taken over all m, n not both zero. Each Sk can be written as a sum of terms
and Sk depends on
.
The series sk are nowadays called Eisenstein sk series, and were introduced by Eisenstein [1847]. They play a crucial role in the Weierstrassian theory of elliptic functions. The connection between the sk and the periods is explained in Weil [1976].
62. By ‘geometrically’ Gauss meant by ruler and compass. The hint Gauss dropped about this in D.A. §335 greatly inspired Abel in his work on elliptic functions (see Ore [1971, p. 16]). It also makes clear that Gauss was considering the ‘arithmetic’ and ‘analytic’ aspects of elliptic functions together almost from the start.
63. So, if sin lemn
and cos lemn
, then
M(2 ϕ) = 2M(ϕ) N(ϕ) μ(ϕ) ν(ϕ), N(2ϕ) = M(ϕ)4 + N(ϕ)4.
Gauss was working towards a representation of elliptic functions as quotients (of theta functions).
64. This relates to the considerations in the D.A. §147–150 on how the divisors of x2 – α depend on α, when α = 4n ± 1, 4n ± 2.
65, 66. The method depended on the theory of Lagrange resolvents, see also no. 37.
67. The proof appears in the D.A. §358.
69. D.A. § 42.
70. Klein and Schlesinger pointed out that the last lines contain slips of the pen and should read, in part.
– (a + b – d – e)(a – c – d + f ) + (a – c – d + f )2
= – (a + b – d – e)(b + c – e – f ) + (b + c – e – f )2
71. See no. 73, 71, 73, 75–78 are again on cyclotomy.
72. This refers to Gauss’s interest in the foundations of Euclidean geometry. In a letter to W. Bolyai, 6 March 1832 [Gauss, Werke VIII, 224] Gauss indicated that the usual definition of a plane presumed too much.
73. Bachmann pointed out that for Gauss to have made an error in no. 71 it is most likely that n should be taken as composite and periods interpreted in the sense of Kummer [1856] and Fuchs [1863].
74. The e periods with f terms may he transformed into 2e periods with f /2 terms and the members of the new periods are determined from the old ones by a quadratic equation. This concerns the relationship between Gauss sums of order n and 2n, nowadays treated by the Hasse-Davenport theorem, see [Berndt , Evans, 1981, 122].
78. For Aug. 1 read Aug. 31 (no. 77).
80. Gauss received his doctorate with the first of his four proofs of the fundamental theorem of algebra [Werke III, 1 –30].
81. The proof is by means of similar triangles inscribed in a semicircle with hypotenuse as radius, given in full in Werke X.l, 524–525.
82. Schlesinger pointed out that for √.3x, read ∛x. The series is the Bessel function
c.f Watson [1962].
83. Schlesinger’s lengthy analysis fails to suggest what function l might be, but it is neither log(l +x) nor the function
which Gauss elsewhere denoted l.
84. Klein and Bachmann pointed out that for classes read genera, in conformity with D.A. §287.
85. Nothing in the Nachlass suggests what the proof might have been.
86. Schlesinger conjectured, on the basis of a letter from Gauss to Hindenburg (8 October 1799, Werke X.l, p. 429) that the generalization was inspired by Laplace’s proof of Lagrange’s theorem.
87. The series (a2 + b2 – 2ab cos θ)n = ∑ an cos nθ was studied by Ivory [1798]. If x denotes the eccentricity of an ellipse with major axis 1, then 1 – x2 cos2ϕ = a2 + b2 – 2ab cos 2ϕ where
so x is the ratio of the geometric and arithmetic means of a and b. If n = ½, then
which yields Gauss’s series when x = 1, a = ½ = b.
88. Klein and Schlesinger noted that Gauss, in a letter to Olbers B4 January 1812 = Werke VIII, p. 140), subsequently dated his first investigations into the method of least squares by means of this entry. Laplace’s error method was described in his [1793].
91. a, b These entries are imperfectly presented in the diary but they show Gauss was accumulating numerical evidence leading to 92.
92. Gauss refers to his discovery of the Fourier series expansions of his functions P(ϕ) and Q(ϕ) [Werke III, 465], which play the role of theta-functions in his theory of elliptic functions.
P(x) := 1 + 2x + 2x4 + ∙∙∙ + 2xn² + ∙∙∙
Q(x) := 1 – 2x + 2x4 + ∙∙∙ + (-1)n 2xn² + ∙∙∙
R(x) := 21/4 + 29/4 + 225/4+ ∙∙∙ + 2n²/4 + ∙∙∙
and Gauss observed, amongst “One hundred theorems on the new transcendents”, that the arithmetico-geometric mean of P(x)2 and Q(x)2 is always 1. These last observations date from 1818, see Geppert [1927]. Klein’s and Schlesinger’s lengthy comments are particularly worthy of note here, and on [95].
94. Gauss applied his theory of elliptic functions to show that the attraction exerted by a planet on an arbitrary point is equal to the attraction exerted by a distribution of mass along the orbit whose density is proportional to the time the planet spends traversing each part of the orbit. Gauss [1818].
95. Gauss sought trigonometric expansions of log P and log Q, and on the basis of numerical calculations conjectured a relation between the periods of lemniscatic integrals and the arithmetico-geometric mean (see no. 98), as Klein and Schlesinger pointed out.
97. Gauss was much interested in the parallax of the moon.
98. As Klein and Schlesinger pointed out this entry may represent a conclusion or a conjecture. Euler had shown [1768, § 334] that
Stirling [1730–57] had calculated the value of ϖ to 17 decimal places, and had also taken 1 and √2 as specific values in computations for the rectification of the ellipse. A letter from Gauss to Pfaff (24 November 1799 = Werke X.l, 232) makes it clear that Gauss did not yet have a proof that M [√2, 1) = π/ϖ.
99. Stäckel connected this with Gauss’s work on the area of a triangle, a topic of central importance for the study of Euclid’s parallel postulate. Gauss wrote to Bessel on the matter on 16 December 1799, see Werke, VIII, 159.
100–102. Klein and Schlesinger conjectured that these entries refer to Gauss’s discovery that the reciprocal of the arithmetico-geometric mean is a solution of a linear differential equation. In fact the arithmetico-geometric mean satisfies Legendre’s equation and its inverse function is Gauss’s modular function. See Geppert pp. 40–42.
104. A variant of Dirichlet’s test, valid provided ϕ ≠ 0,
105, 106. These signify Gauss’s realisation of the importance of the arithmetico-geometric mean for the general elliptic integral, and not just for the lemniscatic case.
107. The first to give arithmetical rules for determining Easter was Lambert [1776]. Gauss was familiar with much of Lambert’s work but one cannot be certain of an influence in this respect.
108. Only Gauss’s later treatments [Werke III, 401, 473] have survived; they date from 1825 and 1827.
109. The multiplicity follows from allowing the arithmetico-geometric mean to become a complex function.
110. Elliptic transcendents means elliptic integrals of the first kind.
112. Klein and Schlesinger connected this with the calculation of powers of e.
113. The problem is, given M between 0 and 1, written as a continued fraction
,
what is the probability P(n,x) that the fraction
lies between 0 and x. Gauss discussed this problem with Laplace in a
letter dated 1812. He found that P(1,x) = Ψ(x) – Ψ(0), where Ψ(x) = d/dx log Π(x) and Π(x) is Gauss’s factorial function, Π(x) = Γ(x + 1), and
See [Khinchin §15].
114, 115. See Werke II, 285, 286.
116. See D.A. §§365, 366, where this assertion is repeated, but again without a proof. Loewy in his commentary supplied a proof using only the techniques available to Gauss, i.e, avoiding Galois theory.
118. See no. 123, For a good discussion of Gauss sums see [Berndt and Evans, 1981].
119. Gauss tackled the problem of locating Ceres, observed for the first time (by Piazzi) on January 1801 and lost 42 days later when it went too near to the sun. His successful solution was published in Zach’s Comm. Liter, for December 1801 = Werke, VI, 1874, 199–204.
120–122. The theory of planetary orbits, their observation, and the treatment of observational errors was congenial to Gauss, being useful, as it seemed to him, and offering much scope for his calculating prowess.
125. See Theoria Motus §§ 88–97.
126. See Theoria Motus, II, §§ 115–163.
127. See Theoria Motus §§ 33ff.
129. See Theoria Motus, II, §§ 164–171.
130–133. Gauss’s Theoria Residuorum Biquadraticorum, Commentatio prima of 1825 = Werke II, 65–92.
134. The sixth proof of the golden theorem.
135. Art. 358 of the D.A. as rederived in Disquistionum circa aequationes puras ulterior evolutio, Werke II, 243.
137. This was taken up in a long and interesting paper by Eisenstein [1844].
138. The quantity ε is not a cube root of unity, but a rational root of the congruence ε2 + ε + 1=0 (mod p) where p is a prime of the form 3n + 1.
146. Gauss vwote ‘=’ for ‘≡’. This, the final entry, has become the most famous. Bachmann commented that the connection between lemniscatic functions with the theory of biquadratic residues remained to be cleared up. A. Weil pointed out [1974, 106] that the substitution z = y (1 – x2) reduces 1 = x2 + y2 + x2y2 to z2 = 1 – x4, which makes the connection with lemniscatic functions clear. When taken mod p it is also a question about biquadratic residues, and the investigation of ax4 – by4 ≡ 1 (mod p) is connected with higher Gauss sums. Weil remarks that it was while studying Gauss’s two papers on biquadratic residues that he was led eventually to formulate the Weil conjectures. It is interesting to see that Gauss, by counting infinity, is already thinking of the curve projectively, when the curve is z2y2 = y4 – x4. Had one written z̃2 = z, the projective curve would have been z̃4 = y4 – x4. Fermat knew, and indeed published a proof [1659], that there are no solutions in integers to y2 = z4 + x4 and thus z̃4 = y4 – x4, so this curve is truly momentous in its significance for mathematics.
Annotated Bibliography
Cyclotomy, number theory, algebra
Gauss’s claim to have provided the first rigorous proof of the fundamental theorem of algebra, in 1799, has been found excessive by recent historians (see Gilain [1991] and Baltus [1998]). On the one hand, Gauss’s own argument makes claims about algebraic curves that could not be proved until Ostrowski rescued them in 1927. On the other, it is easy enough, using modern techniques, to rescue one of d’Alembert’s proofs of the fundamental theorem of algebra. Gauss claimed that d’Alembert’s proof showed only that, if the roots of a polynomial equation exist, then they must be of the form a + b √-1, but not that such roots exist. An argument, familiar in embryo to Cauchy and made explicit by Kronecker, shows that one may always adjoin numbers of this form to a field containing the coefficients of a given polynomial so as to factorise the polynomial. Gauss’s estimate of the originality of his early work is therefore doubly in question.
On the other hand, none would dispute the power and originality of Gauss’s work on the theory of numbers. Dunnington chose to skirt this achievement, and it would take many pages to navigate through it. Better, then, to cite a few works that illuminate the topic from an accessible modern standpoint that Gauss would also recognise: Berndt and Evans’ paper [1981], the book by Berndt , Evans, and Williams [1998], and the book by Ireland and Rosen [1982]. Karsten Johnsen has essayed several difficult passages in Gauss’s diary, and three of his papers are listed in the bibliography. Readers will also enjoy Waterhouse’s papers, one of which, [1994], is a reconstruction of the interaction between Germain and Gauss that led him to produce a surprisingly large counterexample to one of her remarks.
Geometry and topology
Particular mention should be made of Epple’s charming explanation [1999] of a passage in Gauss’s Nachlass, where Gauss introduced the linking number of two curves in the context of electro-magnetic theory. As Epple points out, this is but one illustration of the roots knot theory has in nineteenth century physics (see also Nash [1999]).
Statistics
Dunnington’s treatment is unashamedly pro-Gauss, and in this he has the support of Sprott among the historians of statistics, and the distinguished statistician R.A. Fisher (as quoted in the opening pages of Sprott [1978]). Gauss’s priority in the discovery of the method of least squares is generally agreed, and if his claim that he discovered it in 1794, at the age of 17, is hard to find good evidence for it does not mean that it is wrong. Dutka [1996] showed that Gauss was certainly using the method only a few years later, in 1799.
However, as Stigler argued (see especially his book [1986], pp. MI-MS) there was also a vigorous French tradition, exemplified by Legendre and Laplace, and Legendre never conceded priority to Gauss.
Stigler went on to argue that Gauss’s first published defence of it, in 1809, was deeply flawed, because he combined both a non sequitur and vicious circle in the same argument, and was subsequently rejected by Gauss himself. Therefore, the result was better derived fi-om the Central Limit Theorem by Laplace, and indeed by Gauss himself in 1821, and Gauss’s appreciation of its generality may be due to his reading of Legendre [1805]. However, Waterhouse [1990] looked again at Gauss’s argument, and found that it was neither circular nor vitiated by a gap in its reasoning, although it was true that Gauss did shift his position in response to Laplace [1811], when he discussed what estimate of the ‘losses’ due to error was the most suitable (expected square error or expected absolute error). Gauss, like his contemporaries, took the arithmetic mean of several equally good observations to be the right estimate of the true value, and deduced that the corresponding distribution of errors would then be that described by the normal distribution. He gave this distribution the same ‘axiomatic’ status as the ubiquitous assumption that the arithmetic mean was the right estimate to take, because the normal distribution followed, as a mathematical fact, from the assumption about the arithmetic mean.
Stigler also claims that the fame of Gauss’s method is due to its absorption by Laplace, but this downplays its use within the German astronomical community. As Darrigol ([2000], p. 44) notes, the Germans were much more committed to error analysis than the French in this period, and astronomy seems to have been the science that set the highest standards of accuracy in the period, as much of Gauss’s own work illustrates.
Telegraph and magnetism
The accounts given in Garland [1979] and Darrigol [2000] (pp. 50–54) are much more thorough mathematically than Dunnington’s. Two pictures, reproduced from Garland’s account (Figures 6a and b, pp. 16–17), give a very clear impression of just how good Gauss’s survey of terrestrial magnetism actually was. Darrigol also argues that it was Gauss’s insistence on the value of absolute units that drove Weber’s originality, which “implied new kinds of instruments with simple geometry and high sensitivity, and required the verification of the laws according to which these instruments were analysed.” (Darrigol [2000], p. 74).
Baltus, C. 1998 Lagrange and the fundamental theorem of algebra, Proceedings of the Canadian Society for the History and Philosophy of Mathematics 11, 85–96.
Berndt, B.C. and Evans, R.J. 1981 The determination of Gauss sums, Bulletin of the American
Mathematical Society 5,2, 107–130.
Berndt, B.C, Evans, R.J. and Williams, K.S. 1998 Gauss and Jacobi Sums, Wiley-Interscience, New York.
Biermann, K.R. 1963 Zwei ungeklärte Schlüsselworte von C.F. Gauss, Monatsberichte der Deutschen Akademie der Wissenschaften zu Berlin 5, 241–244.
Biermann, K.-R. 1970 Carl Friedrich Gauss in Autographenkatalogen. Schriften zur Geschichte der Naturwissenschaften, Technik und Medizin 7,1, 60–65.
Biermann, K-R 1971 Zu Dirichlets geplantem Nachruf auf Gauss, NTM Schriften zur Geschichte der Naturwissenschaften Technik und Medizin 8 (1), 9–12.
Biermann, K.-R. 1983 C. F. Gauss als Mathematik- und Astronomiehistoriker, Historia Mathematica 10,4, 422–434, A survey, written by Gauss, of the previous 100 years.
Biermann, K-R. 1990 (ed.) Carl Friedrich Gauss, Der “Furst der Mathematiker” in Briefen und Gesprächen, Leipzig, Urania-Verlag.
Breitenberger, E. 1984 Gauss’s Geodesy and the Axiom of Parallels, Archive for History of Exact Sciences 29, 273–289.
Breitenberger, E. 1993 Gauss und Listing: Topologie und Freundschaft, Mitteilungen der Gauss-Gesellschaft, Göttingen 30, 3–56.
Bühler, W.K. 1981 Gauss: A Biographical Study. New York, Springer-Verlag, (bibliography lists volumes of correspondence between Gauss and others, up to 1981).
Collison, M.J. 1977 The origins of the cubic and biquadratic reciprocity laws. Archive for History of Exact Sciences 17,1, 63–69.
Cox, D.A. 1984 The arithmetic-geometric mean of Gauss, Enseignement Mathematique (2) 30, 275–330.
Carl Friedrich Gauss, Titan of Science_A Study of His Life and Work Page 55