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E. T. Bell wrote in Men of Mathematics:

Though the idea behind it all is childishly simple, yet the method of analytic geometry is so powerful that very ordinary boys of seventeen can use it to prove results which would have baffled the greatest of the Greek geometers -- Euclid, Archimedes, and Apollonius.

I don't necessarily believe everything in the book, but this passage sounds plausible enough to make me wonder.

Are there any good examples of an open problem from antiquity which seemed inaccessible, but was later easily solved by converting it to the Cartesian plane?

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    $\begingroup$ @Asvin I think those impossibility proofs involved a lot more than just analytic geometry. Like, Field Theory, and transcendence of $\pi$. $\endgroup$ – Gerry Myerson Nov 4 '19 at 5:57
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    $\begingroup$ @Asvin I wish to specifically disqualify those examples. No Galois theory. No calculus, for that matter. $\endgroup$ – Kim Nov 4 '19 at 6:09
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    $\begingroup$ How do you precisely know that a problem "seemed inaccessible" to the old geometers? $\endgroup$ – Francesco Polizzi Nov 4 '19 at 6:29
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    $\begingroup$ I think a less speculative version of this questions would be: Are there elementary geometric statements whose proof with analytic geometry is much easier to find than without it. $\endgroup$ – Lennart Meier Nov 4 '19 at 8:35
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    $\begingroup$ Archimedes knew how to trisect an angle using a "neusis construction" with a marked ruler. en.wikipedia.org/wiki/Angle_trisection#With_a_marked_ruler. The ancient Greeks didn't entirely ignore practical solutions to problems! $\endgroup$ – alephzero Nov 4 '19 at 22:51
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A classic problem in this category is Alhazen's billiard problem. I reproduce a quote from 100 Great Problems of Elementary Mathematics. The problem could not be solved using compass and ruler because its solution requires taking a cube root (see references at MathWorld).

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    $\begingroup$ So there is Galois theory involved on the solution, against the OP's requirements $\endgroup$ – Francesco Polizzi Nov 4 '19 at 7:47
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    $\begingroup$ @FrancescoPolizzi --- the solution given by Jack Elkin in Mathematics Teacher seems to require only elementary algebra. $\endgroup$ – Carlo Beenakker Nov 4 '19 at 8:15
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    $\begingroup$ Archimedes could have computed this to any desired accuracy. Pappus would have said: we see how to construct this by neusis, but not by plane methods like straightedge and compass. If Bell's comment suggests there are examples where the Greeks would have understood an answer, but were baffled in being unable to find it -- then this example does not support that claim. $\endgroup$ – Matt F. Nov 4 '19 at 23:32
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This paper contains a very readable account of Descartes invention of analytic geometry and describes some questions that it can solve reasonably easily compared to methods familiar to the ancient greeks.

Even defining conics sections in the plane is quite clumsy using the directrix, focus and eccentricity. The analytic form of a second degree equation is easier to remember and clearly superior when computing actual points of intersection for example.

Descartes' Theorem is one that is tough to prove or perhaps even discover without analytic geometry. Descartes himself used this as an example of the power of his method in his correspondence with Princess Elizabeth of Bohemia:

Bos, Erik-Jan, Princess Elizabeth of Bohemia and Descartes’ letters (1650-1665), Hist. Math. 37, No. 3, 485-502 (2010). ZBL1200.01012.

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  • $\begingroup$ So, you are answering a different version of the question? I mean, not the one with "being inaccessible", since Apollonius already knew a lot of things about conic sections. $\endgroup$ – Francesco Polizzi Nov 5 '19 at 13:25
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    $\begingroup$ @FrancescoPolizzi I interpreted "seemed inaccessible" to mean very difficult or even impossible if approached using classical greek mathematics. I agree that he greeks knew a huge amount about the properties of conic sections but they didn't have a systematic way to approach basic questions such as intersection or tangency. $\endgroup$ – Ivan Meir Nov 5 '19 at 13:43
  • $\begingroup$ I understand your interpretation, but the question has the tag "History of Maths". And, from the historical point of view, "seemed inaccessible" has a precise meaning: the perception of inaccessibility must be documented. Otherwise, as I said before, this is speculation. $\endgroup$ – Francesco Polizzi Nov 6 '19 at 6:26
  • $\begingroup$ We can really say that Descartes' Theorem would have "baffled" Apollonius? I cannot say this for sure, after all the Greeks were able to prove related statements like Pappus' theorem. What if Apollonius would have thought enough about Descartes' theorem? Nobody can know. $\endgroup$ – Francesco Polizzi Nov 6 '19 at 6:30
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    $\begingroup$ @FrancescoPolizzi I understand you make a very fair point, it's just that for this type of question I think it's worth allowing a looser interpretation and therefore a broader set of possible answers. I wouldn't say that Pappus' theorem is particularly related to Descartes' Theorem. Pappus' theorem is about points on lines and collinearity so very classical in nature. It is a fundamental proposition that is almost axiomatic, being equivalent to commutativity of multiplication. $\endgroup$ – Ivan Meir Nov 6 '19 at 9:25
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I read this excerpt of Men of Mathematics as a private joke. The very ordinary boy is Gauss of course. The number seventeen is a reference to the heptadecagon that Gauss constructed at age nineteen. This is the first progress since the Greeks concerning compass and straight-edge construction of regular polygons. It makes use of cartesian geometry, arithmetics, trigonometry and requires the solving of a degree seventeenth polynomial equation. Here is the solution from the Disquisitiones Arithmeticae.

\begin{align} 16\,\cos\frac{2\pi}{17} = & -1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+ \\ & 2\sqrt{17+3\sqrt{17}- \sqrt{34-2\sqrt{17}}- 2\sqrt{34+2\sqrt{17}}}\\ = & -1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+ \\ & 2\sqrt{17+3\sqrt{17}- \sqrt{170+38\sqrt{17}}}. \end{align}

This would certainly have baffled the Greek. I understand childish in Bell's excerpt as refering to Gauss as a child prodigy.

Another mathematician commonly endowed with supernatural powers as a child is Pascal, who supposedly rediscovered all the Euclid axioms at age eleven (debunked by his sister). His treaty Essai sur les coniques written at age sixteen contains his famous hexagrammum mysticum theorem, which generalises a previous result by Pappus. This a result that would have certainly baffled the Greeks. Pascal used projective geometry to reduce the proof to the case of a circle. Nowadays there are many short elegant proofs of Pascal theorem using analytic geometry.

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    $\begingroup$ The polynomial equation in question is of degree 16 (given by the 17th cyclotomic polynomial). $\endgroup$ – Achim Krause Nov 25 '19 at 9:23
  • $\begingroup$ @Krause Indeed, thanks. I was thinking to the equation $X^{17}-1 = 0$. $\endgroup$ – coudy Nov 25 '19 at 9:40
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Provocative answer (but not too much): checking the Fifth Postulate is a triviality in the Cartesian plane.

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    $\begingroup$ That doesn't really answer the question. The cartesian plane is a model for the complete set of euclidean postulates, so naturally you can not use it to decide whether or not one of the postulates is independent of the others. (I am sure you are aware of this yourself, but that doesn't change anything.) $\endgroup$ – RP_ Nov 4 '19 at 10:03
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    $\begingroup$ As mathematicians, I feel we should not force our own standards of rigour upon others (mathematicians or otherwise). A differential equation can be ill-posed, a question can not be. The question has already received one answer that seems sensible to me; for me that is enough to conclude that the question has merit. $\endgroup$ – RP_ Nov 4 '19 at 10:18
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    $\begingroup$ "A differential equation can be ill-posed, a question can not be" So there exist no speculative questions? Interesting point of view. $\endgroup$ – Francesco Polizzi Nov 4 '19 at 10:25
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    $\begingroup$ I simply require the asker of the question to adopt the standard of precision required by the MO community. And, in my opinion, saying that a problem from the antiquity "seemed inaccessible" is merely speculative, unless there are strong evidences of the contrary (like the famous three classical problems). $\endgroup$ – Francesco Polizzi Nov 4 '19 at 11:04
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    $\begingroup$ You are not convincing me that you are not being overly critical here. The OP is quoting E.T. Bell as the direct inspiration for this question. At the very minimum, this question can be meaningfully construed as "Are there any examples to back up Bell's claim, or is this an unwarranted exaggeration?" $\endgroup$ – RP_ Nov 4 '19 at 11:21
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I come across what looks like a good fit in Brianchon, Solution de plusieurs problèmes de géométrie, J. École Polytechnique 4, nº 10 (1810) 1–15, page 5:

Pappus reports also that the Greek geometers had tried in vain to solve this more general problem.

« Given a circle and three poles, arranged in arbitrary manner, inscribe in this circle a triangle whose sides, extended if necessary, each go through one of the given poles. »

With the help of analysis applied to geometry, the moderns easily overcame the difficulty, and this once famous question now amounts to very little; Lagrange has given a beautiful analytic solution (Mémoire de Berlin, 1776)

(etc.; according to Senapati (2019) Pappus had solved the case where the 3 points are aligned, in Mathematicae collectiones, Book 7, Prop. 117.)

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