Euler&Gauss

The Disquisitiones and Clock Arithmetic

In 1801, at 24, Gauss published the Disquisitiones Arithmeticae — and number theory went from a collection of clever puzzles to a systematic science. Its opening move looks almost too simple: a new notation for remainders.

Two numbers are congruent modulo m if they leave the same remainder on division by m — if they land on the same position of an m-hour clock. The genius is in the ≡ sign itself: Gauss chose a symbol that looks like equality because congruences behave like equations. You can add them, multiply them, and (carefully) divide them. Whole theories — quadratic reciprocity, which Gauss proved at 19 and loved enough to prove seven more times, calling it his theorema aureum, the golden theorem — flow from this one piece of good notation.

Euler had proven individual results about remainders; Gauss built them a home. Every hash table, every checksum, every cryptographic protocol on Earth now speaks Gauss's language of congruence.

% helper: Octave's built-in is powmod; alias for portability
if ~exist('powermod','file')
  function r = powermod(a,e,n), r = powmod(a,e,n); endfunction
endif

% --- Congruence basics --------------------------------------
mod(1777, 7)                     % Gauss's birth year on a 7-clock
mod(-3, 12)                      % Octave gives 9: always in 0..m-1

% --- A multiplication table mod 7 (a field!) ----------------
m = 7;
[K, J] = meshgrid(0:m-1);
mod(K .* J, m)                   % every nonzero row is a permutation

% --- Quadratic residues: which numbers are squares mod p? ---
p = 13;
QR = unique(mod((1:p-1).^2, p))  % -> 1 3 4 9 10 12 (exactly (p-1)/2)

% --- Euler's criterion links the two men directly -----------
% a is a square mod p  <=>  a^((p-1)/2) ≡ 1 (mod p)
a = 10;
powermod(a, (p-1)/2, p)          % 1 -> yes, 10 is a QR mod 13
a = 5;
powermod(a, (p-1)/2, p)          % p-1 (i.e. -1) -> not a square

The Seventeen-Sided Polygon

On the morning of March 30, 1796, a nineteen-year-old still deciding between philology and mathematics woke up knowing something no human had known in the 2,000 years since Euclid: the regular 17-gon can be drawn with compass and straightedge alone. He chose mathematics that day, and started a diary.

The Greeks could construct regular polygons with 3, 4, 5, 6, 8, 10, 12, 15 sides — but 7, 9, 11, 13, 14 defeated them, and everyone assumed 17 was equally hopeless. Gauss saw the problem wasn't geometry at all. The corners of an n-gon are the complex roots of zⁿ = 1 — points e^2πik/n on Euler's unit circle. Compass and straightedge can only take square roots. So the question becomes pure algebra: can cos(2π/17) be built from nested square roots? Gauss proved it can — because 17 = 2⁴ + 1 is a Fermat prime, and 16 halves down to 1 in exactly four steps: 16 → 8 → 4 → 2 → 1, each step a square root.

He was so proud of it that he asked for a 17-gon on his tombstone. (The stonemason refused — with that many sides it would just look like a circle — and carved a 17-pointed star on a memorial in Brunswick instead.)

Finding a Lost Planet: Ceres and Least Squares

On New Year's Day 1801, the astronomer Piazzi spotted a faint new "planet" between Mars and Jupiter — then lost it in the sun's glare after tracking it across just 3° of sky. Europe's astronomers despaired of ever finding it again. A 24-year-old with no telescope announced where it would reappear. In December, it was exactly there.

Gauss's weapon was a question nobody had answered properly: when your measurements disagree — and they always disagree — what is the best estimate they collectively point to? His answer: choose the parameters that minimize the sum of the squared errors.

Why squares? Gauss showed that if measurement errors follow his bell-shaped curve (next section), least squares gives the most probable parameters — the method and the distribution justify each other. Squaring also punishes large errors heavily, keeps everything differentiable, and reduces the whole problem to solving a linear system (by, naturally, Gaussian elimination). Two centuries later, least squares is still the beating heart of statistics, GPS positioning, machine learning's loss functions, and every trend line ever fitted.

% Noisy observations of a hidden line y = 2.5x - 1 -------------
rng(1801)  % Octave ≥7: use rng; older: rand("seed",1801);              % the year of Ceres
x = linspace(0, 10, 25).';
y = 2.5*x - 1 + randn(size(x))*1.5;

% --- Way 1: the normal equations, exactly as Gauss did -------
A = [x, ones(size(x))];          % design matrix
beta1 = (A.'*A) \ (A.'*y)         % solve A'A β = A'y

% --- Way 2: Octave's backslash (QR under the hood) -----------
beta2 = A \ y

% --- Way 3: polyfit -------------------------------------------
beta3 = polyfit(x, y, 1)

% --- Judge the fit --------------------------------------------
yhat = A*beta1;
SSres = sum((y - yhat).^2);
SStot = sum((y - mean(y)).^2);
R2 = 1 - SSres/SStot
plot(x, y, 'o', x, yhat, '-', 'linewidth', 2);
title(sprintf('Least squares: y = %.2fx + %.2f  (R^2 = %.3f)', ...
      beta1(1), beta1(2), R2));

The Bell Curve: A Portrait of Error Itself

Every measurement lies a little. Gauss asked what the lies look like in aggregate — and derived the one curve consistent with reasonable assumptions about error: symmetric, concentrated, with the arithmetic mean as the best estimate. The result is the most famous shape in science.

Look closely at the formula: e and π — Euler's constants — are right there in the portrait of randomness. The mean μ slides the bell; the standard deviation σ sets its width; the √(2π) (from the Gaussian integral ∫e^−x²dx = √π) makes the total probability exactly 1.

The reason this curve is everywhere — heights, exam scores, noise in electronics, thermal motion — is the central limit theorem: add up many small independent influences, whatever their individual shapes, and their sum drifts inevitably toward the Gaussian. It is less a law of nature than a law of aggregation.

% Sums of UNIFORM randomness become GAUSSIAN -------------------
N = 20000;                        % experiments
k = 12;                           % dice per experiment
sums = sum(rand(k, N));           % each column: sum of k uniforms

mu  = k/2;  sigma = sqrt(k/12);   % theory for uniform sums
z = (sums - mu) / sigma;          % standardize

histogram(z, 60, 'Normalization','pdf');  hold on;  % was hist(z,60,1) – deprecated         % normalized histogram
t = linspace(-4, 4, 200);
plot(t, exp(-t.^2/2)/sqrt(2*pi), 'linewidth', 2);
title('12 uniform dice, summed: the bell emerges');

% The 68-95-99.7 rule, checked empirically ---------------------
for s = 1:3
  printf('within %d sigma: %.2f%% (theory %.2f%%)\n', s, ...
         100*mean(abs(z) < s), 100*erf(s/sqrt(2)));
end

% Bonus: the Gaussian integral that tames sqrt(2*pi) -----------
quadgk(@(x) exp(-x.^2), -10, 10)  % sqrt(pi)   % -> 1.7725 = sqrt(pi)

Counting the Primes: A Teenager's Conjecture

At fifteen, Gauss was given a table of logarithms with a list of primes bound in the back. He began tallying primes in blocks of a thousand — a lifelong hobby; he eventually counted into the millions — and noticed that their density near a number x thins out like 1/ln x.

Here π(x) counts the primes up to x. Gauss's refined guess, the logarithmic integral Li(x), is astonishingly accurate: at x = 10⁹ it errs by 0.0003%. He never published the conjecture — pauca sed matura — mentioning it only in an 1849 letter. The proof took a century and required, fittingly, the machinery Euler had seeded: Riemann carried the Euler product into the complex plane, and the primes' rhythm turned out to be written in the zeros of the zeta function. The two protagonists of this guide meet inside the deepest unsolved problem in mathematics, the Riemann Hypothesis.

% --- Sieve of Eratosthenes, vectorized -----------------------
function pr = sieve(N)
  isp = true(1, N); isp(1) = false;
  for k = 2:floor(sqrt(N))
    if isp(k), isp(k*k:k:N) = false; end
  end
  pr = find(isp);
endfunction

N = 1e6;
pr = sieve(N);
x  = logspace(2, 6, 40);
pix = arrayfun(@(v) sum(pr <= v), x);   % π(x) by counting

approx1 = x ./ log(x);                    % Gauss, age 15
approx2 = arrayfun(@(v) quadgk(@(t) 1./log(t), 2, v)  % quad is deprecated → quadgk, x);  % Li(x)

% Relative errors of the two approximations -------------------
semilogx(x, 100*(approx1 - pix)./pix, '-o', ...
         x, 100*(approx2 - pix)./pix, '-s');
legend('x/ln x', 'Li(x)'); grid on;
ylabel('error (%)');
title('Gauss vs the primes: Li(x) is eerily good');

printf('pi(10^6) = %d;  Li: %.0f;  x/ln x: %.0f\n', ...
       pix(end), approx2(end), approx1(end));

The Remarkable Theorem: Curvature from Within

In the 1820s Gauss spent summers doing something unglamorous: surveying the Kingdom of Hanover, hauling instruments up hills. Out of that muddy fieldwork came the deepest idea of his career — that a surface's curvature can be measured entirely from inside it.

Gauss defined the curvature K of a surface at a point as the product of its two principal curvatures — how sharply it bends in its most- and least-curved directions. A sphere of radius R has K = 1/R² everywhere; a flat sheet has K = 0; a saddle has K < 0. The shock is his Theorema Egregium ("remarkable theorem"): K is unchanged by any bending that doesn't stretch the surface. Creatures living on the surface, measuring only distances along it, can compute K without ever peeking at the third dimension.

Everyday consequences first: a flat sheet (K = 0) can roll into a cylinder (still K = 0) but can never wrap a sphere (K > 0) without wrinkling or tearing — which is why every flat map of Earth lies, and why a slice of pizza droops until you fold it lengthwise, forcing its zero curvature to spend itself on stiffness. Deeper consequence: if curvature is intrinsic, then a space needs no outside to be curved in. Gauss's student Riemann generalized this to any number of dimensions; Einstein then proposed that gravity is the intrinsic curvature of four-dimensional spacetime. The line from Hanover's muddy survey triangles to general relativity is direct.

His private notebooks show he had worked out non-Euclidean geometry and suppressed it, fearing — as he wrote to Bessel — the "clamor of the Boeotians." When Bolyai and Lobachevsky published, priority passed to bolder hands. Pauca sed matura cut both ways.