The most thorough analysis of the question was made by Breitenberger [1984]. He confronts the question: ‘if von Waltershausen was not simply confused in some way, what was he saying?’ and he gives it an elegant answer. Surveying Hanover threw up many triangles and many numbers (a figure of a million is sometimes mentioned). Conclusions were drawn (and maps made) on numbers which are the result of many calculations, and at every stage discrepancies between real and expected results lay within expected error bounds (Gauss analysed the errors quite carefully). Not only was Euclidean geometry never called into question, because the errors were only what was to be expected, each calculation amounted to a tacit defence of Euclidean geometry. But the measurements of the BHI triangle were not fed into such a mill. They show that, within experimental error, space is described by Euclidean geometry. To be sure “as a single instance it proves very little, but it has been designed so as to be transparent, and hence it will drive a point home” (Breitenberger [1984], p. 288). Newton dropped an apple in conversation to similar purpose and effect. The myth, Breitenberger concludes, is that the BHI triangle was surveyed as part of a deliberate test of Euclidean geometry. But it did incidentally show that Euclidean geometry is true to within the limits of the best observational error of the time. Put that way, the gap between Scholz and Breitenberger may be quite small.
Gauss’s Mathematical Diary
Between 1796 and 1814 Gauss kept an informal record of his mathematical discoveries. After his death it remained in the possession of the Gauss family until, in 1898, a grandson of Gauss gave it to Stäckel for use in the preparation of the later volumes of Gauss’s Werke. It was first published in the Mathematische Annalen (LVII), 1903, with a brief introduction and some notes by Klein, the editor-in-chief of the Werke. It was then re-published in the Werke (X.l, 1917, 485–574), with extensive notes supplied by the team of editors, the diary itself being reprinted in facsimile form as a supplement.
1796
[1] The principles upon which the division of the circle depend, and geometrical divisibility of the same into seventeen parts, etc.
[1796] March 30 Brunswick.
[2] Furnished with a proof that in case of prime numbers not all numbers below them can be quadratic residues.
April 8 ibid.
[3] The formulae for the cosines of submultiples of angles of a circumference will admit no more general expression except into two periods. [Or, see Johnsen [1986], pp. 168–169: The formulae for the cosines of submultiples of angles of a circumference will admit a more general expression only in terms of both of the two periods.]
April 12 ibid.
[4] An extension of the rules for residues to residues and magnitudes which are not prime.
April 29 Göttingen.
[5] Numbers which can be divided variously into two primes.
May 14 Göttingen.
[6] The coefficients of equations are given easily as sums of powers of the roots.
May 23 Göttingen.
[7] The transformation of the series 1 – 2 + 8 – 64 + ∙∙∙ into the continued fraction
and others.
May 24 Göttingen.
[8] The simple scale in series which are recurrent in various ways is a similar function of the second order of the composite of the scales.
May 26.
[9] A comparison of the infinities contained in prime and compound numbers.
May 31 Göttingen.
[10] A scale where the terms of the series are products or even arbitrary functions of the terms of arbitrarily many series.
June 3 Göttingen.
[11] A formula for the sum of factors of an arbitrary compound number: general term
June 5 Göttingen.
[12] The sum of the periods when all numbers less than a [certain] modulus are taken as elements: general term [(n + 1) a – na] an-1.
June 5 Göttingen.
[13] Laws of distributions.
June 19 Göttingen.
[14] The sum to infinity of factors = π2/6 ∙ sum of the numbers.
June 20 Göttingen.
[15] I have begun to think of the multiplicative combination (of the forms of divisors of quadratic forms).
June 22 Göttingen.
[16] A new proof of the golden theorem all at once, from scratch, different, and not a little elegant.
June 27.
[17] Any partition of a number a into three □ gives a form separable into three □.
July 3.
[17a] The sum of three squares in continued proportion can never be a prime: a clear new example which seems to agree with this. Be bold!
July 9.
[18] EUREKA, number = ∆ + ∆ + ∆.
July 10 Göttingen.
[19] Euler’s determination of the forms in which composite numbers are contained more than once.
[July Göttingen.]
[20] The principles for compounding scales of series recurrent in various ways.
July 16 Göttingen.
[21] Euler’s method for demonstrating the relation between rectangles under line segments which cut each other in conic sections applied to all curves.
July 31 Göttingen.
[22]
can always be solved.
August 3 Göttingen.
[23] I have seen exactly how the rationale for the golden theorem ought to be examined more thoroughly and preparing for this I am ready to extend my endeavours beyond the quadratic equations. The discovery of formulae which are always divisible by primes:
(numerical).
August 13 ibid.
[24] On the way developed
August 14.
[25] Right now at the intellectual summit of the matter. It remains to furnish the details.
August 16 Göttingen.
[26] (ap) = (a) mod p, a the root of an equation which is irrational in any way whatever.
[August] 18.
[27] If P, Q are algebraic functions of an indeterminate quantity which are incommensurable. One is given tP + uQ= 1 in algebra as in number theory.
[August] 19 Göttingen.
[28] The sums of powers of the roots of a given equation are expressed by a very simple law in terms of the coefficients of the equation (with other geometric matters in the Exercitiones).
[August] 21 Göttingen.
[29] The summation of the infinite series
, etc.
[same day August 21.]
[30] Certain small points aside, I have happily attained the goal namely if pn ≡ l (mod π) then xπ – l is composed of factors not exceeding degree n and therefore a sum of conditionally solvable equations; from this I have deduced two proofs of the golden theorem.
September 2 Göttingen.
[31] The number of different fractions whose denominators do not exceed a certain bound compared to the number of all fractions whose numerators or denominators are different and less then the same bound when taken to infinity is 6:π2.
September 6
[32] If
is denoted Π:x = z and x = Φ:z then
September 9.
[33] If
then
[34] An easy method for obtaining an equation in y from an equation in x, if given xn + axn-1 + bxn-2 ∙∙∙ = y.
September 14.
[35] To convert fractions whose denominator contains irrational quantities (of any kind?) into others freed of this inconvenience.
September 16.
[36] The coefficients of the auxiliary equation for the elimination are determined from the roots of the given equation.
Same day.
[37] A new method by means of which it will be possible to investigate, and perhaps try to invent, the universal solution of equations. Namely by transforming into another whose roots are αρ′ + βρ″ + γρ‴ + ∙∙∙ where
etc. and the number n denotes the degree of the equation.
&n
bsp; September 17
[38] It seems to me the roots of an equations xn – 1 [= 0] can be obtained from equations having common roots, so that principally one ought to solve such equations as have rational coefficients.
September 29 Brunswick.
[39] The equation of the third degree is this:
where 3n + 1 = p and m is the number of cubic residues omitting similarities. From this it follows that if n = 3k then m + 1 = 3l, if n = 3k ± 1 then m = 3l. Or z3 – 3pz + pp – 8p –9pm = 0. By these means m is completely determined, m + 1 is always □+3□.
October 1 Brunswick.
[40] It is not possible to produce zero as a sum of integer multiples of the roots of the equation xp – 1 = 0.
⊙ October 9 Brunswick.
[41] Obtained certain things concerning the multipliers of equations for the elimination of certain terms, which promise brightly.
⊙ October 16 Brunswick.
[42] Detected a law: and when it is proved a system will have been led to perfection.
October 18 Brunswick.
[43] Conquered GEGAN.
October 21 Brunswick.
[44] An elegant interpolation formula.
November 25 Göttingen
[45] I have begun to convert the expression
into a power series in which ω increases.
November 26 Göttingen
[46] Trigonometric formulae expressed in series.
By December.
[47] Most general differentiations.
December 23.
[48] A parabolic curve is capable of quadrature, given arbitrarily many points on it.
December 26.
[49] I have discovered a true proof of a theorem of Lagrange.
December 27.
1797
[50]
1797 January 7.
[51] I have begun to examine thoroughly the [elastic] lemniscatic curve which depends on
January 8.
[52] I have spontaneously discovered the ground for Euler’s criterion.
January 10.
[53] I have invented a way to reduce the complete integral to quadratures of the circle.
January 12.
[54] An easy method for determining
[55] I have found a distinguished supplement to the description of polygons. Namely, if a, b, c, d, . . . are the prime factors of the prime number p diminished by one, then to describe a polygon of p sides nothing other is required than that:
1) the indefinite arc is divided into parts a, b, c, d, . . .
2) and that polygons of a, b, c, d, . . . sides be described.
January 19 Göttingen
[56] Theorems on the residues –1, ±2 proved by similar methods to the rest.
February 4 Göttingen.
[57] Forms aa + bb + cc – bc – ac – ab, pertaining to the divisors, coincide with this: aa + 3bb.
February 6.
[58] An amplification of the penultimate proposition on page 1, namely
l – a + a3 – a6 + a10 ∙∙∙ =
From this, all series where the exponents form a series of the second order are easily transformed.
February 16.
[59] Established a comparison between integrals of the form
and
[60] Why dividing the lemniscate into n parts leads to an equation of degree n2.
March 19.
[61] On powers of the integral
depends
[1797 March.]
[62] The lemniscate is divisible geometrically into five parts.
March 21.
[63] Amongst many other properties of the lemniscatic curve I have observed:
The numerator of the decomposed sine of the doubled arc is = 2 ∙ Numerator denominator sine × numerator denominator cosine of the simple arc.
In fact the denominator = (numerator sine)4 + (denominator sine)4. Now if the denominator for the arc πl is θ, then the denominator of the sine of the arc kπl = θkk.
Now θ = 4,810480, the hyperbolic logarithm of which number is = 1,570796, i.e, π/2 which is most remarkable and a proof of which property promises the most serious increase in analysis.
March 29.
[64] I have discovered even more elegant proofs for the connection of divisors of the form □ – α, +1 with –1, ±2 .
June 17 Göttingen.
[65] I have perfected a second deduction of the theorem on polygons.
July 17 Göttingen.
[66] It can be shown by both methods that only pure equations need be solved.
[67] We have given a proof of what on October 1st was discovered by induction.
July 20.
[68] We have overcome the singular case of a solution of the congruence xn – 1 = 0 (evidently when the auxiliary congruence has equal roots), which troubled us for so long, with most happy success, from solutions to congruences when the modulus is a power of a prime number.
July 21
[69] If xμ+ν + a xμ+ν-1 + b xμ+ν-2 + ∙∙∙ + n (A)
is divided by
xμ + α xμ-1 + β xμ-2 + ∙∙∙ + m (B)
and all the coefficients a, b, c, etc. in (A) are integer numbers and indeed all the coefficients in (B) are rational then these will also be integers and the ultimate m a divisor of the ultimate n.
July 23.
[70] Perhaps all products of (a + bρ + cρ2 + dρ3 + ∙∙∙), where ρ denotes all primitive roots of the equation xn = 1 can be reduced to the form (x – ρy)(x – ρ2y) ∙∙∙. For example
(a + bρ + cρ2) × (a + bρ2 + cρ) = (a – b)2 + (a – b)(c – a) + (c – a)2
(a + bρ + cρ2 + dρ3) × (a + bρ2 + cρ2 + dρ) = (a – c)2 + (b – d)2
(a + b + cρ2 + dρ3 + eρ4 + fρ5) × = (a + b – d – e)2
– (a + b – d – e) (a – c – d – f ) + (a – c – d – f )2
= (a + b – d – e)2 + (a + b – d – e) (b + c – e – f ) + (b + c – e – f )2
Seen February 4.
This is false. For it would follow from this that two numbers of the form (x – ρy) have a product of the same form, which is easily refuted.
[July.]
[71] Demonstrated that the several periods of the roots of the equation xn = 1 cannot have the same sum.
July 27 Göttingen.
[72] I have demonstrated the possibility of the plane.
July 28 Göttingen.
[73] What we wrote on July 27 involved an error: but happily we have now worked out the thing more successfully since we can prove that none of the periods can be a rational number.
August 1.
[74] How one should attach the signs in doubling the number of periods.
[August.]
[75] I have found the number of prime functions by a most simple analysis.
August 26.
[76] Theorem: If 1 + ax + bxx + etc. + mxμ is a prime function with respect to the modulus p, then
is divisible by this function with respect to this modulus, etc. etc.
August 30.
[77] Showed, and the way to much more is laid open by the introduction of multiple moduli.
August 31.
[78] August 1 more generally adapted to any moduli.
September 4
[79] I have uncovered principles by which the resolution of congruences according to multiple moduli is reduced to congruences with respect to a linear modulus.
September 9.
[80] Proved by a valid method that equations have imaginary roots.
Brunswick October.
Published in my own dissertation in the month of August 1799.
[81] New proof of Pythagoras’ theorem.
Brunswick October 16.
[82] Considered the sum of the series
and showed that = 0 , if
1798
[83] Setting l(1 + x
) = ϕ'(x); l(1 + ϕ'(x)) = ϕ"(x); l(1 + ϕ"(x)) = ϕ'"(x) etc, then
Brunswick April.
[84] Classes are given in any order, and from this the representability of numbers as three squares is reduced to solid theory.
Brunswick April 1.
[85] We have found a genuine proof of the composition of forces.
Göttingen May.
[86] The theorem of Lagrange on the transformation of functions extended to functions of any variables.
Göttingen May.
[87] The series
connected to the general theory of series of sines and cosines of angles increasing arithmetically.
June.
[88] The calculus of probabilities defended against Laplace.
Göttingen June 17.
[89] So solved the problem of elimination that nothing extra could be desired.
Göttingen June.
[90] Various rather elegant results concerning the attraction of a sphere.
June or July.
[91a]
July.
[91b] arc sin lemn sin ϕ – arc sin lemn cos ϕ =
arc sin lemn sin [ϕ] = 0,95500598 sin [ϕ]
– 0,0430495 sin 3[ϕ]
+ 0,0018605 sin 5[ϕ]
– 0,0000803 sin 7[ϕ]
sin2 lemn [ϕ] = 0,4569472 =
– ․ ․ ․ cos 2[ϕ]
arc sin lemn sin ϕ =
sin5 [ϕ] = 0,4775031 sin [ϕ] + 0,03 ․ ․ ․ [sin 3ϕ] ․ ․ ․
[92] On the lemniscate, 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.
Göttingen July
[93] Solution of a problem in ballistics.
Göttingen July.
Carl Friedrich Gauss, Titan of Science_A Study of His Life and Work Page 53