[94] Edited the completed theory of comets.
Göttingen July.
[95] New things in the field of analysis opened up to us, namely investigation of a function etc.
October.
1799
[96] We have begun to consider higher forms.
Brunswick February 14.
[97] We have discovered new exact formulas for parallax.
Brunswick April 8.
[98] We have proved that the arithmetico-geometric mean between 1 and √2 is π/ϖ to 11 places, which thing being proved a new field in analysis will certainly be opened up.
Brunswick May 30
[99] We have made exceptional progress in the principles of Geometry.
Brunswick September.
[100] We have discovered many new things about the value of the arithmetico-geometric mean.
Brunswick November.
[101] We had already discovered long ago that the arithmetico-geometric mean was representable just as a quotient of two transcendental functions: now we have discovered the second of these functions is reducible to integral quantities.
Helmstadt December 14.
[102] The arithmetico-geometric mean itself is an integral quantity. Proved.
December 23.
1800
[103] Succeeded in determining the reduced forms in the theory of ternary forms.
February 13.
[104] The series a cos A + a' cos(A + ϕ) + a" cos(A + 2ϕ) + etc. leads to a limit if a, a', a", etc. form a progression without change of sign which converges continually to 0. Proved.
Brunswick April 27.
[105] We have led the theory of transcendental quantities:
to the summit of universality.
Brunswick May 6.
[106] Succeeded in making a great increase in this theory, Brunswick May 22, by which at once all the preceding, not just the theory of the arithmetico-geometric means, is most beautifully bound together and increased infinitely.
[May 22.]
[107] On about these days (May 16) we most elegantly resolved the problem of the chronology of the Easter Feast. Published in Zach’s Comm Liter August 1800 p. 121, 223.
[May 16.]
[108] Succeeded in reducing the numerator and denominator of the lemniscatic sine (interpreted most universally) to integral quantities, at once derived from true principles the development in infinite series of all functions of lemniscates which can be thought of; a most beautiful invention which is not inferior to any of the above. Moreover in these days we have discovered the principles according to which the arithmetico-geometric series must he interpolated, so that it is now possible to exhibit terms in a given progression corresponding to any rational index by algebraic equations.
Late May or June 2, 3.
[109] Between two given numbers there are always infinitely many means both arithmetico-geometric and harmonico-geometric the mutual cormection of which we have fortunately completely cleared up.
Brunswick June 3.
[110] We have now immediately applied our theory to elliptic transcendents.
June 5.
[111] Finished the rectification of the ellipse in three different ways.
June 10.
[112] We have invented a totally new numerico-exponential calculus.
June 12.
[113] We have solved the problems in the calculus of probabilities concerning continued fractions which at one time were attacked in vain.
October 25.
[114] November 30. This has been a good day, in which it has been given to us to determine the number of classes of binary forms by three methods, thus
1) by infinite products
2) by infinite sums
3) by finite sums of cotangents or logarithms of sines.
Brunswick.
[115] Dec 3. We have discovered a fourth method and the most simple of all for negative determinants which is derived only from the number of numbers ρ, ρ', etc. if Ax + ρ, Ax + ρ', etc. are linear forms of divisors of forms □ + D.
The same.
1801
[116] Proved that it is impossible to reduce the division of the circle to an equation of lower degree than our theory suggests.
Brunswick April 6 [1801]
[117] These days we have learned to determine the Jewish Easter by a new method.
April.
[118] A method for proving the fifth fundamental theorem has been found by means of a most elegant theorem in the division of the circle, thus
According as a ≡ 0 1 2 3 (mod 4) substituting for n all numbers from 0 to (a – 1).
Brunswick mid-May.
[119] A new, most simple and expeditious method for investigating the elements of the orbit of celestial bodies.
Brunswick mid-September
[120] We are attacking the theory of the motion of the moon.
August.
[121] We have discovered many new formulae most useful in theoretical astronomy.
Month of October.
1805
[122] In the following years 1802, 1803, 1804 astronomical work took up the greatest part of my free time, first of all carrying out the calculations concerning the theory of new planets. From this it happened that in those years this catalogue was neglected. And those days in which it was possible to make some increase to mathematics are forgotten.
[123] The proof of the most charming theorem recorded above. May 1801, which we have sought to prove for 4 years and more with every effort, at last perfected. Commentationes recentiores, 1.
1805 August 30.
[124] We have further worked out the theory of interpolation.
1805 November
1806
[125] We have discovered a new and most perfect method for determining the elements of the orbit of a body moving about the sun from two heliocentric positions.
1806 January.
[126] We have carried forward to the highest degree of perfection a method for determining the orbit of a planet from three geocenttic positions.
1806 May.
[127] A new method for reducing the ellipse and hyperbola to a parabola.
1806 April.
[128] At about the same time we finished off the resolution of the function
into four factors.
1807
[129] A new method for determining the orbit of a planet from four geocentric positions of which the last two are incomplete.
1807 January 21.
[130] Began the theory of cubic and biquadratic residues.
1807 February 15.
[131] Further worked out and completed. Proofs thereto are still wanted.
[132] The proof of this theory discovered by most elegant methods so that it is totally perfect and nothing further is desired. By this residues and quadratic non-residues are illustrated exceptionally well at the same time.
February 22.
[133] Theorems, which attach to the preceding theory, most valuable additions, provided with an elegant proof (namely for which primitive roots b itself must be taken positive and which negative, aa + 27bb = 4p; aa + 4bb = p).
February 24.
[134] We have discovered a totally new proof of the fundamental theorem based on totally elementary principles.
May 6.
1808
[135] The theory of division into three periods (Art 358) reduced to far simpler principles.
1808 May 10.
[136] The equation X – 1 = 0 which contains all primitive roots of the equation xn – 1 = 0 cannot be decomposed into factors with rational coefficients. Proved for composite values of n.
June 12.
[137] I have attacked the theory of cubic forms, solutions of the equation x3 + ny3 + n2y3 – 3nxyz = 1.
December 23.
1809
[138] The theorems for the cubic residu
e 3 proved with elegant special methods by considering the values of
where the three always have the values α, αε, αεε, with the exception of two which give ε, εε but these are
with product ≡ ⅓.
1809 January 6.
[139] Series pertaining to the arithmetico-geometric mean further evolved.
1809 June 20.
[140] We have finished the division into five by the arithmetico-geometric mean.
1809 June 29.
1812
[141] The preceding catalogue interrupted by unjust fate a second time resumed the beginning of the year 1812. In the month November 1811 we succeeded in giving a purely analytical proof of the fundamental theorem in the study of equations but because nothing of this was put to paper an essential part of it was completely forgotten. This, sought for unsuccessfully for a rather long time, we however have happily rediscovered.
1812 February 29.
[142] We have discovered an absolutely new theory of the attraction of an elliptical spheroid on points outside the body.
Seeberg 1812 September 26.
[143] We also solved the remaining part of the same theory by new and remarkably simple methods.
October 15 Göttingen.
1813
[144] The foundation of the general theory of biquadratic residues which we have sought for with utmost effort for almost seven years but always unsuccessfully at last happily discovered the same day on which our son is born.
1813 October 23 Göttingen.
[145] This is the most subtle of all that we have ever accomplished at any time. It is scarcely worthwhile to intermingle it with mention of certain simplifications pertaining to the calculation of parabolic orbits.
1814
[146] I have made by induction the most important observation that connects the theory of biquadratic residues most elegantly with the lemniscatic functions. Suppose a + bi is a prime number, a – 1 + bi divisible by 2 + 2i, the number of all solutions to the congruence
1 ≡ xx + yy + xxyy (mod a + bi)
Including x = ∞, y = ±i, x = ±1, y = ∞ is = (a – 1)2 + bb.
1814 July 9.
Commentary on Gauss’s Mathematical Diary
by Jeremy Gray
1. See the introduction.
2. This assertion is trivial, and is regarded as a slip of the pen. Klein and Bachmann conjectured plausibly that Gauss meant to assert that for every prime p ≥ 5, there is a prime q, q < p, such that p is not a quadratic residue mod q. This theorem (for p = 4n+ 1) is described by Gauss in a hand-written note to his own copy of the Disquisitiones Arithmeticae (§130) as having been discovered on April 8, 1796, and is an essential step on the road to Gauss’s first proof of the theorem of quadratic reciprocity.
3. Johnsen [1968] observed that if one defines aj := cos 2jπ/p, 1 ≤ j ≤ (p – l)/2, as the cosines of the submultiples of the angles, and sets r := cos 2π/p + i sin 2π/p, then Gauss called d1 := ∑(k/p)=1 rk and d1 := ∑(k/p)=−1 rk the periods (D.A. § 343), and the entry refers to the way the aj are linearly independent over the rational numbers, ℚ, but may become linearly dependent over ℚ(d1). This has implications for the irreducibility of the cyclotomic polynomials.
4. Generalised quadratic reciprocity, see Disquisitiones Arithmeticae [D.A.] § 133.
5. Related to Goldbach’s conjecture.
7. Both the series and the continued fractions diverge. Euler had begun the study of these transformations in his [1754/55]. See also no. 58.
8. If G(x) is a polynomial of degree at most n – 1, and G(x)(l + a1x + a2x2 + ∙∙∙ + anxn) is developed as a power series in x, say s0 + s1x + s2x2 + ∙∙∙, then sn+k = a1sn+k-1 + a2sn+k-2 + ∙∙∙ + ansk, k = 0, 1, 2, . . .. De Moivre called a1, .., ak the scale (Index, or Scala) of the series [De Moivre. 1730, 22].
9. Presumably Gauss was approaching the prime number theorem which he later said, in a letter to Encke (Werke, II, 444), that he had suspected when reading Lambert’s tables in 1792 or 1793.,
11. If an integer N is written as a product of primes, N =
, then
12. Period is defined in D.A. §46 as follows. If p is a prime, not dividing N, and d is the exponent of N (the least integer such that Nd = 1 (mod p), then the period is the set of powers 1, N, .., Nd-1 (mod p). Bachmann argued that Gauss meant to write ((n + l) n – n ) for
and this expression counts the number (= Summa) of periods.
14. Gauss claimed, correctly, that ∑n≤x d(n) is asymptotically equal to π2x/6 where d(n) denotes the number of divisors of n.
15. In Gauss’s theory of binary quadratic forms, D.A. §287, a divisor of x2 + Ay2 is a prime p such that a multiple, pm, is equal to x2 + Ay2 for some relatively prime integers x, y, and also a quadratic form px2 + 2qxy + ry2 with q2 – pr = –A.
16. The second proof of the ‘golden’ theorem. See his marginal note to the D.A. §262, where this date is also confirmed, printed in Werke, 1, 476.
17. The symbol □ denotes a square number, the forms considered are undoubtedly binary quadratic forms. See D.A. §§279, 280.
17a. Struck out in the diary and scarcely legible; omitted from the Werke edition. If
so x2, y2, and z2 are in continued proportion, then
m4 + m2n2 + n4 = (m2 + mn + n2)( m2 – mn + n2)
which is never prime unless m = n= 1, when m2 + mn + n2 = 3 and m2 – mn + n2 = 1.
18. Every number is a sum of three triangular numbers (the triangular numbers are
Fermat conjectured that every number is the sum of 3 triangular numbers, 4 squares, 5 pentagonal numbers and so on, a result first proved by Cauchy in 1815.
19. Euler investigated [1752/53] when a number of the form 4n + 1 is prime as part of a general investigation in which he showed that all such primes are a sum of two squares, unlike those of the form 4n +3.
20. See no. 8.
21. Euler [1748, II, §§92, 93] gave a simple proof of a theorem of Apollonius [Conics, III, §§17, 19, 22]: if AB and A'B' are two chords of a conic meeting at O, then the ratio
is a constant independent of the position of O.
22. Obscure.
23. The details are taken up in the Analysis Residuorum, Werke, II, 230–234. Gauss had seen a connection between quadratic and cyclotomic questions, which he then investigated until he achieved success on September 2nd, when it yielded two more proofs of the golden theorem.
25. There is no evidence as to what the matter was, except that, like nos. 22, 23, 26, 27 it is underlined in red and so may refer to the same family of ideas.
26. See Werke II, 224; (a) is a polynomial in x with a root x = a, (ap) is a polynomial whose roots are the pth powers of the roots of (a), p is a prime.
28. The Exercitiones were published in Werke X.l 138–143, with notes by Schlesinger.
29. The series satisfies the differential equation d ny/dxn = y with the initial conditions that, at x = 0, y = 1,
30. Further proofs of the golden theorem. The ‘small points’ are taken up in no. 68. See also Gauss’s Analysis Residuorum, Werke II, 230–234.
31. If A(n) denotes the number of fractions a/b in lowest terms such that b ≤ n, then A(n) =
where ϕ is Euler’s ϕ function. If B(n) denotes the number of fractions a/b with 1 ≤ a ≤ b ≤ n, then B(n) = ½n(n + 1). Gauss asserted that
a result first published by Dirichlet in 1849.
32. The first appearance in the diary of the inversion of elliptic integrals. Gauss, as was customary in his day, used x for both the variable of integration and tacitly, its upper limit.
33. Here A stands for the first term of the series (i.e, z), B the second (i.e,
C the third (i.e,
), etc. This formalism was introduced by Newton in his study of the binomial series [1676 = Correspondence II, 130–132].
34. The Tschimhausen transformation replaces x in an equation xn + axn-1 + bxn-2 . . . = 0 by y,
with a view to simplifying it until it becomes solvable.
37. The Lagrange resolvent of a polynomial equation. Gauss hoped to use it to solve arbitrary polynomial equations. Loewy noted that by 1797 Gauss had become convinced that this was impossible, a view he implied in his dissertation and in the D.A. §359.
38. Gauss wants to reduce the study of xn – 1 = 0 to the study of the equations
, where
.
39. Here m denotes the number of solutions x3 – y3 ≡ 1 (mod p).
40. The irreducibility of the cyclotomic polynomial with prime exponent.
43. Schlesinger [Gauss, Werke, X.l, part 2, 291], Biennann [1963] and Schuhmann [1976] have all offered conjectures. In the absence of other evidence none of these can be conclusive.
44. Loewy conjectured the Lagrange interpolation formula is meant.
45. Also considered by Euler, and subsequently, by Riemann. The expression is equal to (1 – 21-ω) ζ(ω) where ζ is the Riemann Zeta function.
47. Differentiation with arbitrary indices, presumably, but no other trace remains.
48. Parabolic curves are those of the form y = a0xn + a1xn-1 + ∙∙∙ + an.
50, 51. Elliptic integrals; ‘elastic’ struck out in original.
52. Euler’s criterion (strictly, Newton’s) concerns integrals of the form ∫ xm (a + bxn)μ/ν dx. See Newton Mathematical Papers 1670–1673, III, p. 375.
55. If p – 1 = abc.., then the construction of a regular p-gon depends on the ability to divide a given angle into a, b, c, . . . equal parts, and hence on the cyclotomic equations xa – 1 = 0, xb – 1 = 0, . . . .
56. See D.A. § 145.
57. If a2 + b2 + c2 – bc – ac – ab = α, then (2a – b – c)2 + 3 (b – c)2 = 4α. So every odd divisor of the former divides x2 + 3y2. Conversely, if p divides the latter, then there is an odd integer A such that A2 ≡ –3 (mod p) and, if B = l, A2 + 3B2 = 0 (mod p), so if a = 0, b = (A + B)/2, c = (A – B)/2, then p divides a2 + b2 + c2 – bc – ac – ab.
Carl Friedrich Gauss, Titan of Science_A Study of His Life and Work Page 54