The cubic was solved three times before it was published once, under an oath, after a notebook in Bologna was held to have released it. The formula then forced complex numbers into existence, and its failure at degree five became the theory of groups.
Scipione del Ferro, professor at Bologna, solved the depressed cubic $x^{3}+px=q$ around 1515 and told almost no one. A solution was worth more unpublished than published: university posts were defended by public challenge, and a method nobody else had was a method nobody else could answer. He passed it on his deathbed to his student Antonio Maria Fiore.
Fiore challenged Niccolò Tartaglia in 1535 with thirty problems, all of that one type. Tartaglia, who had rediscovered the method himself, answered all thirty in two hours and Fiore answered none. In 1539 Gerolamo Cardano got the method out of Tartaglia in the form of a poem, under an oath not to publish it.
The cubic formula has a defect that the quadratic does not. When a cubic has three distinct real roots — the case any picture makes look easiest — the formula reaches them only through square roots of negative quantities. This is the casus irreducibilis, and it is not a failure of the particular formula: no expression by real radicals can reach those roots in general. The negative square roots are not an artefact to be cleverly avoided. They are load-bearing.
Rafael Bombelli took this seriously in 1572, worked with the quantities anyway, and watched the imaginary parts cancel to leave the real roots standing. Complex numbers entered mathematics not from solving $x^{2}+1=0$ — which can simply be declared to have no solution — but because a formula for real answers to a real question refused to work without them.
The sixteenth century solved the cubic and the quartic and then stopped, and the stopping was the discovery. Three hundred years of failure at the quintic was not incompetence; it was evidence, and Abel and Galois turned it into a theorem about which equations yield to radicals at all. A negative result about formulas became a structural theory of symmetry.
— 1545 to 1832, in one sentenceRuffini in 1799 and Abel in 1824 proved no general solution by radicals exists beyond degree four. Évariste Galois, dead at twenty in 1832, answered the sharper question: given a particular polynomial, a finite group attached to it decides whether radicals suffice, and solvability of the group is solvability by radicals. The question stopped being find the formula and became describe the symmetry — the same relocation Klein would make for geometry forty years later.
book4/ladder-polynomials.html studies the n-bonacci family
$x^{n}=x^{n-1}+\cdots+x+1$, whose roots govern the growth rates the corpus uses throughout.
Those polynomials have Galois groups, and for $n \ge 5$ they are generically the full
symmetric group, which is not solvable — so the dominant root has no radical expression,
and the numerical value is not a shortcut around an algebraic fact but a consequence of one.
That is the honest link from this chapter to the corpus, and it is a computation rather than a
resemblance: the group is finite, nameable, and checkable for each $n$.
The del Ferro–Tartaglia–Cardano episode is also the series’ best available case study in attribution, which the corpus has had reason to think about. A result discovered three times, withheld twice, published once under an oath that a notebook was held to have released, and named after the man who wrote it down. Every party behaved defensibly by the norms of the day and the outcome still satisfies nobody. That is closer to how credit actually settles than any rule about it.
| del Ferro c.1515 | The depressed cubic $x^{3}+px=q$, solved and unpublished; the notebook seen at Bologna in 1543. |
| Tartaglia 1539 | The method, given to Cardano in verse, under oath. |
| Cardano 1545 | Artis Magnae, sive de Regulis Algebraicis — the cubic and, by Ferrari, the quartic; del Ferro, Tartaglia and Ferrari credited in the text. |
| Bombelli 1572 | L'Algebra — arithmetic with square roots of negatives, and the casus irreducibilis carried through to real roots. |
| Ruffini 1799; Abel 1824 | No general solution by radicals above degree four. |
| Galois 1832 | Solvability by radicals is solvability of the associated group. |
| Verification | book7/ch-cardano-verify.py — 5 blocks. The casus irreducibilis carried through on x³ − 7x + 6, imaginary parts cancelling to 1e-16, and the n-bonacci roots rising to 2. Ends with an [HONESTY] block naming what is evidence rather than proof. |
| Internal | book4/ladder-polynomials (the n-bonacci family and its Galois groups) · ch-felix-klein (the same relocation, for geometry) |