Zimmermann, Paul. “Gauss’ Zulassungsgesuch zur Promotion,” Braunschweigisches Magazin, No. 15 (July 16, 1899), 113–117.
————. “Zum Gedächtniss an C F. Gauss,” Braunschweigisches Magazin, No. 16 (July 30, 1899). Supplement to Br. Anz, No. 209, pp. 124–127.
————. “C. F. Gauss’ Briefe an seine Tochter Minna und deren Gatten H. A. Ewald,” Braunschweigisches Magazin, Dec, 1915 (Wolfenbüttel), pp. 133–141.
————. “Neue kleine Beiträge zu C. F. Gauss’ Leben und Wirken,” Grimme Natalis Monatsschrift, Heft XI (1921). 12 pages.
“Zur Erinnerung an C. F. Gauss,” Göttinger Zeitung, No. 16 (Apr. 15, 1877), 121–124,
“Zwei Briefe von Gauss,” Im neuen Reich, Jan, 1877.
An Introduction to Gauss’s Mathematical Diary
by Jeremy Gray
On March 1796, when still 18, Gauss opened a mathematical diary that he maintained as a private record of his discoveries for the next several years. It begins with his celebrated discovery that the regular 17-gon can be constructed by ruler and compass alone. Forty-nine entries were made in that year, most of them to do with number theory and algebra, as were the bulk of the thirty-three entries of 1797, although the lemniscatic integral (to be defined below) also makes its appearance. Gauss’s interests then shift away from number theory to probability only to return in 1800 to the elaboration of a general theory of elliptic integrals. By the end of 1801 the demands of astronomy take over, and the final entries lack the enthusiasm and profusion of the first, but by 1805, when the entries resume. Gauss was 28, an established mathematician and astronomer, and, by October, a married man. He no longer needed the diary.
In any case, his use for the diary seems to have been psychological reassurance. Discoveries may have been recorded to establish independence, even priority, or simply to capture the flood of ideas sweeping over the young man’s mind. The bulk of the entries reveal a marvellous eye for the significant in mathematics, but (as Klein noted in his introductory essay) some are hopelessly vague. Few record how a problem was solved, and the lack of even a clue to the solution is sometimes so dramatic that on one occasion Gauss himself was unable to reconstruct the discovery later (see [141]). Gauss was plainly excited to see that he could compare himself favourably with mathematicians of earlier generations, notably Euler and Legendre (Schlesinger’s essay on him makes this very clear, see Gauss, Werke X.2). But above all the diary is testimony to the rapidity with which ideas came to Gauss at the start of his career, so that they seem almost to crowd one another out (which was Klein’s explanation of why several of Gauss’s best ideas were never published at all).
Number theory
Gauss’s discovery of the constructibility of the regular 17-gon derives from an already considerable grasp of the algebraic nature of the problem. He thought of the 17 vertices as the 17 roots of z17 – 1 = 0, of which one is trivially z = 1, and the other 16 satisfy z16 + z15 + ∙ ∙ ∙ + z + 1 = 0. Gauss’s crucial observation, which he described in detail in the Disquisitiones §354, is that since 17 is a prime and 16 = 24 the calculation of the roots reduces to the successive solution of 4 quadratic equations. Solving quadratics can be done geometrically by ruler and compass, so the 17-gon is constructible geometrically, and Gauss more or less showed how it can be done by writing down the 4 quadratic equations explicitly. He also gave the complex coordinates of each point to ten decimal places, showing that he did regard the plane of complex numbers geometrically in advance of Argand and Wessel. But what is more remarkable is that Gauss was in sight of a far-reaching theory of equations of the form zn – 1 = 0. We can see some of the steps he took in the direction of the theory of cyclotomy, as it is called, in later entries in the diary, and it is not surprising that on making this discovery Gauss decided to study mathematics and not philology.
The second entry in the diary is equally dramatic for it concerns quadratic reciprocity, a topic somewhat slighted by Dunnington. One says that an integer a is a (quadratic) residue mod p if the congruence x2 ≡ a (mod p) has a solution (where p is prime). The quadratic reciprocity theorem asserts that, given two odd primes p and q then p is a residue mod q if and only if q is a residue mod p unless both are of the form 4n + 3, in which case p is a residue mod q if and only if q is not a residue mod p. There are auxiliary statements concerning the prime 2 and the number –1: 2 is a residue mod p if and only if p is of the form 8k ± 1, and –1 is a residue mod p if and only if p is of the form 4k – 1. Because a product ab is a residue mod p if and only if the factors are either both residues or both not residues, it is easy to determine whether any given number is a residue modulo a prime.
Dunnington noted that whatever the situation concerning prior discovery of the quadratic reciprocity theorem, the difficult task was to come up with a proof, and that Gauss was the first to do. His first proof, however, is generally agreed to be unpleasant, and Gauss speedily found a second (see entry [16]). This invoked the theory of binary quadratic forms (which are expressions of the form Ax2 + 2Bxy + Cy2).
The central problems in this theory are to find what integers can be written in this way for specified integers A, B, and C, and when two given quadratic forms represent the same set of integers. Gauss called two binary quadratic forms Ax2 + 2Bxy + Cy2 and A1x12 + 2B1x1y1 + C1y12 properly equivalent if one is obtained from the other by writing
where a, b, c, d are integers with ad–bc = 1. He then showed that proper equivalence classes of forms with the same square-free discriminant (B2 – AC) obey a composition law (in modern terminology, they form a finite commutative group). The idea of using composition of forms to analyse families of quadratic forms is profound; Edwards calls it Gauss’s great contribution to the theory. Put in modern language, the crucial observation is that there is a subgroup consisting of the squares in this group. Gauss, following Legendre, called the cosets of this subgroup the genera. He distinguished the genera by their ‘total character’ (D.A. §231) which is indeed their character in the modern sense of group representation theory, whence the name.
Gauss gave several illustrative examples in the Disquisitiones Arithmeticae (§230, 231). For example, the form 10x2 + 6xy + 17y2 has discriminant 32 – 10 ∙ 17 = –161 = –1 ∙ 7 – 23. The numbers which this form can represent must be non-residues mod 7 and non-residues mod 23. The odd numbers this form represents must also be congruent to 1 mod 4. These three pieces of information make up the total character of the class defined by the form 10x2 + 6xy + 17y2, and one sees that a system of plus and minus 1’s has been associated to it according as the numbers it represents are or are not residues modulo the primes that divide the discriminant (that is how it ‘discriminates’). Around 1880 Weber and Dedekind began to interpret the Gaussian total character as a homomorphism from the group of forms under consideration to the subgroup {±1} of the nonzero complex numbers.
The fact that half the assignable characters cannot be ascribed to a quadratic form in which A, 2B, and C have no common divisor (D.A. §261) is the basis of the second proof of quadratic reciprocity (D.A. §262). The connection can be made because the theorem in §261 was established by Gauss independently of the law of quadratic reciprocity, so an argument by contradiction based upon the theorem allows Gauss to deduce the law: if the law was false forms can easily be written down for all assignable characters.
How much of this theory was apparent to Gauss on June 27, 1796 is not clear. Some of an extant preliminary version of the Disquisitiones Arithmeticae, called the Analysis Residuorum, appears in Gauss, Werke, II, 199–240, and more was found in Berlin by U. Merzbach (see [Merzbach, 1981]). Unfortunately, the relevant chapter V is still missing, but we do know Gauss wrote it at least four times in 1797 and 1798, because he said so in a letter to W. Bolyai (quoted in [Merzbach, 1981] p. 175). One supposes that in 1796 Gauss saw how the proof would go, if not all the details.
In mid-May 1801 Gauss found a fifth way to prove the quadratic reciprocity theorem, when he s
aw how it was connected with what are now called Gauss sums
The proof itself continued to elude him until August 30th 1805 (when he wrote to Olbers that ‘As lightning strikes the puzzle was solved’). It depends on showing that if
then
It is fairly easy to show that the sum is ±√n or ±i√n in each case; the difficulty lies in determining the sign, and it is this problem that occupied Gauss until 1805. For a good recent survey of Gauss sums which explains this and other matters, see Berndt and Evans [1981].
Among the final entries in the diary are some, [130–138], that show Gauss was preoccupied in 1807 with extending the theory to cubic and biquadratic residues, and incidentally thereby discovered a sixth proof of the quadratic reciprocity theorem. When Gauss published his papers on biquadratic residues, in 1825 and 1831 (Werke, II, 65–92 and 93–148), he observed that the theorems were best stated in terms of imaginary numbers, or, as we would call them, Gaussian integers. By then Jacobi had discovered the cubic reciprocity law, and in the 1840s Eisenstein also found it, together with a proof and connections with the theory of elliptic functions. A brief account of these developments and the unfortunate priority dispute they provoked is given in Collison [1977] and Weil [1974, 1976].
Gauss’s legacy to the next generation of German mathematicians was not just the remarkable theorems about numbers that he discovered. It was also the deep and often surprising inter-cormections he found, and the profundity of the proofs involved. Even when expositors like Dirichlet made Gauss’s theory more comprehensible, by casting out binary forms in favour of quadratic integers, they accepted and endorsed its importance, and as a result the Disquisitiones Arithmeticae started a tradition, which continues to the present day, which finds the theory of numbers to be one of the most profound and important branches of mathematics. Early nineteenth century neo-humanists such as Alexander von Humboldt, who valued learning for its own sake and trusted applications to follow naturally, had no trouble sharing Gauss’s taste in this matter. Neither, perhaps, does the large modern public who thrilled to the final resolution of Fermat’s Last Theorem by Andrew Wiles in 1996.
Elliptic function theory
As early as September 9, 1796, another major theme emerged in the diary, that of elliptic functions, which may be thought of as a far-reaching generalisation of the familiar trigonometric functions. Gauss worked on and off on the theory of functions of a complex variable and in particular elliptic functions for the next 35 years, but he published very little of what he found. Instead he left a profusion of notes and drafts, which were collected in the Gauss Werke, where they have been thoroughly analysed by Schlesinger and others.
In September 1796 Gauss considered the so-called lemniscatic integral
which gives the arc length of the lemniscate. In January 1797 he read Euler’s posthumous paper [1786] on elliptic integrals, and learned that if
then AB = π/4. Gauss now began ‘to examine thoroughly the lemniscate’. By analogy with the integral for arcsin, he regarded the integral above as defining x as a function of z, and he eventually called this function sl for sinus lemniscaticus. He pursued the analogy with the trigonometric functions, defining the analogue of the cosine function, which he denoted cl(x), and noting early on that the equation for sl(3x) in terms of sl(x) is of degree 9, whereas the equation for sin(x) as a function of sin(x) is only a cubic equation (sin(3x) = 3sin x – 4sin3 x). On March 19 he noted in his diary ‘Why dividing the lemniscate into n parts leads to an equation of degree n2. The reason is that the roots are complex. He thereupon regarded sl and cl as functions of a complex variable.
He had already seen that, as a real function, sl was a periodic function with period 2ω = 4A. It followed from the addition law for sl and cl that the complex function sl had two distinct periods, 2ω and 2iω, and so, when m and n are integers, sl(x + (m + in) 2ω) = sl(x), the first occurrence of the Gaussian integers, m + in. This observation enabled Gauss to write down all the points where sl or cl were zero or infinite, and so to write them as quotients of two infinite series. Gauss wrote sl(x) = M(x)/N(x) and did one of his provocative little sums. He calculated N(ω) to 5 decimal places and log N(ω) to 4, and noted that this seemed to be π/2. He wrote ‘Log hyp this number = 1,5708 = l/2π of the circle?’ [Werke, X.l, 158]. In his diary (March 29, 1797, nr. 63) he checked this coincidence to 6 decimal places, and commented that this ‘is most remarkable and a proof of this property promises the most serious increase in analysis’.
In July 1798 an improved representation of M and N led Gauss back to the calculation of ω, and he commented in his diary (July, nr. 92): ‘we have found out the most elegant things exceeding all expectations and that by methods which open up to us a whole new field ahead.’ Entry was delayed, however, by a year, and the path unlocked by the arithmetico-geometric mean. This is defined for two real numbers a and b as follows: set a0 = a and b0 = b, and recursively
Then it is easily seen that the two sequences (an) and (bj converge to the same value, called the arithmetico-geometric mean of a and b, which may be written M(a, b) here. On May 30, 1799 Gauss wrote in his diary (nr. 98): ‘We have found that the arithmetico-geometric mean between 1 and 2 is π/ω to 11 places, which thing being proved a new field in analysis will certainly be opened up.’ The proof was still eluding Gauss even in November 1799, as a letter from Pfaff to Gauss makes clear (Werke, X.l, 232).
In early 1800 Gauss could show that
This reminded Gauss of similar formulae for the arc-length of an ellipse as a function of its eccentricity, and so he could connect his ideas with existing algorithms for computing such things, known to Lagrange (for elliptical arc-length) and Legendre (for elliptic integrals in their own right). The connection with the arithmetico-geometric mean function was, however, original with Gauss, and was to prove most instructive. Gauss soon found this striking power series:
From it, he deduced that the function 1/M(1 + x, 1 – x) √n
Y″ x(x2 – 1) + Y′ (3x2 – 1) + xY = 0 ,
and noted that another independent integral of this differential equation is 1/M(1,x). An ingenious series of calculations (described in Cox, [1984], a masterly account of Gauss’s work, well worth considering on several counts) led Gauss to the result he had long sought: M(1, √2 ) = π/ω̑.
More results followed until on May 6 he believed he had led the theory of elliptic integrals ‘to the summit of universality’ (Diary entry nr. 105) only to find by May 22 that the theory was ‘greatly increased and unified’, and was becoming ‘most beautifully bound together and increased infinitely’ (nr. 106). Gauss had begun to study the general elliptic function with a real modulus k:
As before, Gauss regarded this integral as defining x as a complex function of z, and, as other authors were to later, he allied this approach to one that developed elliptic functions (complex functions having one real and one imaginary period) directly as quotients of entire functions, without reference to an integral. The natural conjecture is that these two approaches describe the same objects. To show that a function defined via an elliptic integral is indeed a quotient was, for Gauss, a straightforward generalisation of the lemniscatic case. It was much harder to show that every doubly-periodic function defined as a quotient arises as the inverse of a suitable elliptic integral, but using his theory of the arithmetico-geometric mean Gauss was able to establish this.
The whole question becomes much harder when doubly-periodic functions are admitted having two complex periods (to avoid trivialities, the quotient of the periods must not be real). Now the arithmetico-geometric mean becomes complex, and complex numbers have two square roots, so there is a choice to be made at every stage of the procedure. However, on 3 June 1800 Gauss wrote that the connection between the infinitely many means has been completely cleared up (Diary entry nr. 109). Two days later he remarked “We have now immediately applied our theory to elliptic transcendents” (nr. 110), which confirms that Gauss generalised his theory of ellipti
c functions by making the arithmetico-geometric mean into a complex function. Unfortunately, nothing survives from 1800 to indicate what Gauss’s complete solution was. Later work by Gauss, in the 1820s, shows that Gauss did not start from elliptic integrals with a complex modulus k but worked intensively with power series in complex variables.
Gauss was clear from an early stage in his work on function theory about the importance of the complex domain. As he wrote in a famous letter to Bessel (December 18, 1811, in Werke, X.l, 366, but not quoted in Dunnington) he asked of anyone who wished to introduce a new function into analysis to explain:
if he would apply it only to real quantities, and imaginary values of the argument appear so to speak only as an offshoot, or if he agrees with my principle that the imaginaries must enjoy equal rights in the domain of quantities with the reals. Practical utility is not at issue here, but for me analysis is an independent science that through the neglect of any fictitious quantity loses exceptionally in beauty and roundness and in a moment all truths, that otherwise would be true in general, have necessarily to be encumbered with the most burdensome restrictions.
Gauss was confident that a viable theory of a complex variable required only the representation of complex numbers as points in the plane. In his Disquisitiones Arithmeticae of 1801 he had illustrated his discoveries in number theory with that representation, for example in his theory of the ruler and compass construction constructibilty of the regular 17-gon. So he had in mind a geometrical, rather than a formal or purely algebraic theory of functions of a complex variable.
Carl Friedrich Gauss, Titan of Science_A Study of His Life and Work Page 51