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In "Mathematical Thought from Ancient to Modern Times" Morris Kline claims (on page 209) that Leonardo da Pisa (Fibonacci) "showed that the roots of $x^3+2x^2+10x=20$ are not constructible with straightedge and compass". I'd be interested to know how this was proved because the tools we nowadays use to show that certain straightedge and compass constructions are not possible weren't available at that time. Can someone point me to a textbook or an article where Fibonacci's proof is explained?

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    $\begingroup$ Kline messed up, Fibonacci gave no such proof, see McTutor's biography:"Fibonacci proves that the root of the equation is neither an integer nor a fraction, nor the square root of a fraction." That is not enough to prove non-constructibility, $\sqrt{2+\sqrt{2}}$ is none of the above, but constructible. First non-constructibility proofs were given by Wantzel in 1837, see Suzuki, A Brief History of Impossibility. $\endgroup$
    – Conifold
    Commented Oct 20, 2022 at 10:58

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In the Flos (Flos Leonardi Bigolli Pisani super solutionibus quarundam questionibus ad numerum et ad geometriam, vel ad utrumque pertinentium), Fibonacci reinterprets in algebraic form the geometric theory of incommensurables in Book X of Euclid's Element and studies a third-degree equation namely:

Altera uero questio a predicto magistro Iohanne proposita fuit, vt inueniretur quidam cubus numerus, qui cum suis duobus quadratis et decem radicibus in unum collectis essent uiginti: super hoc meditando putaui huius questionis solutionem egredi ex his que continentur in X lib. Euclidis; et ob hoc super ipso X Euclidis accuratius studui, adeo quod sui teoremata ipsius memorie commendaui, et ipsarum intellectum comprehendi.

i.e.

Another question was proposed by the aforesaid teacher John (i.e. Iohannes Panormitanus, or John from Palermo, one of the scholar in the circle of Frederick II), namely, to find a certain cube number, which with its two squares and ten roots collected into one would be twenty. And for this reason I studied the book X of Euclid more carefully, so much so that I committed his theorems to his memory, and grasped their understanding.

and, so, in modern notation, to solve $$x^3+2x^2+10x=20.$$

Then Fibonacci shows that the positive root is not rational, nor is it a quadratic irrational (Ostendam rursus impossibile esse quod numerus .ab. sit radix alicuius numeri ratiocinati...). If the root were a quadratic irrational, say $x =\sqrt{n}$, in the equality $$x(x^2+10) = 20-2x^2$$ the term on the left would be irrational and the term on the right rational. So, unable to solve the equation exactly, he numerically calculates an approximation to 9 correct decimal places:

Et quia hec questio solui non potuit in aliquo suprascriptorum, studui solutionem eius ad propinquitatem reducere. Et inueni unam ex .X. radicibus nominatis, scilicet numerum .ab., secundum propinquitatem, esse unum et minuta .XXII. et secunda .VII. et tertia .XLII, et quarta .XXXIII. et quinta .IIII. et sexta .XL.

in modern notation:

$$x\approx 1 + \frac{22}{60} + \frac{7}{60^2}+\frac{42}{60^3}+\frac{33}{60^4}+ \frac{4}{60^5}+\frac{40}{60^6} $$

So actually Fibonacci does not prove non-constructibility of the roots, but that they are not rational nor of the form $\sqrt{n}$ for some rational $n$.

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    $\begingroup$ Isn't it interesting to compare the modern "$x^3+2x^2+10x=20$" with Fibonacci's "to find a certain cube number, which with its two squares and ten roots collected into one would be twenty". $\endgroup$ Commented Oct 21, 2022 at 6:33

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