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edited Feb 9, 2015 at 0:16

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Disclaimer: The following perhaps isn't an answer to your question as stated, so my apologies if this answer is useless to you. However, you're asking for how to treat this problem "honestly", and I think that adding the right kind of historical perspective falls under the heading of honesty.

Anyway, I think it is important to observe here that the ancient Greeks themselves did not limit their solutions to plane constructions. As can be read in Sir Thomas L. Heath's A History of Greek Mathematics, Vol. 1, pp. 246-9, Archytas proposed a solution to the problem where he intersected three surfaces of revolution in Euclidean $3$-space (a cone, a cylinder, and a torus) to obtain a point whose coordinates generate the field extension $\mathbb{Q}(\sqrt[3]{2})$.

It is somewhat misleading, I think, to keep referring to these problems as "the three famous unsolved problems of Greek mathematics", because the Greeks in fact solved them many times over, Archytas' solution being only one solution out of a multitude. Moreover, they even recognized that the solution could not be achieved by plane methods, in a way: Pappus has it that the Greeks classified construction problems as "plane", "solid", and "linear", according to the methods with which the problem could be solved. Of course, they never tried to make this very precise, let alone tried to prove it, but then they weren't trying to do the impossible either.

Disclaimer: The following perhaps isn't an answer to your question as stated, so my apologies if this answer is useless to you. However, you're asking for how to treat this problem "honestly", and I think that adding the right kind of historical perspective falls under the heading of honesty.

Anyway, I think it is important to observe here that the ancient Greeks themselves did not limit their solutions to plane constructions. As can be read in Sir Thomas L. Heath's A History of Greek Mathematics, Vol. 1, pp. 246-9, Archytas proposed a solution to the problem where he intersected three surfaces of revolution in Euclidean $3$-space (a cone, a cylinder, and a torus) to obtain a point whose coordinates generate the field extension $\mathbb{Q}(\sqrt[3]{2})$.

It is somewhat misleading, I think, to keep referring to these problems as "the three famous unsolved problems of Greek mathematics", because the Greeks in fact solved them many times over, Archytas' solution being only one solution out of a multitude. Moreover, they even recognized that the solution could not be achieved by plane methods, in a way: Pappus has it that the Greeks classified construction problems as "plane", "solid", and "linear", according to the methods with which the problem could be solved. Of course, they never tried to make this very precise, let alone tried to prove it, but then they weren't trying to do the impossible either.

Disclaimer: The following perhaps isn't an answer to your question as stated, so my apologies if this answer is useless to you. However, you're asking for how to treat this problem "honestly", and I think that adding the right kind of historical perspective falls under the heading of honesty.

Anyway, I think it is important to observe here that the ancient Greeks themselves did not limit their solutions to plane constructions. As can be read in Sir Thomas L. Heath's A History of Greek Mathematics, Vol. 1, pp. 246-9, Archytas proposed a solution to the problem where he intersected three surfaces of revolution in Euclidean $3$-space (a cone, a cylinder, and a torus) to obtain a point whose coordinates generate the field extension $\mathbb{Q}(\sqrt[3]{2})$.

It is somewhat misleading, I think, to keep referring to these problems as "the three famous unsolved problems of Greek mathematics", because the Greeks in fact solved them many times over, Archytas' solution being only one out of a multitude. Moreover, they even recognized that the solution could not be achieved by plane methods, in a way: Pappus has it that the Greeks classified construction problems as "plane", "solid", and "linear", according to the methods with which the problem could be solved. Of course, they never tried to make this very precise, let alone tried to prove it, but then they weren't trying to do the impossible either.

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created Feb 9, 2015 at 0:10

4.7k
19
43
67

Disclaimer: The following perhaps isn't an answer to your question as stated, so my apologies if this answer is useless to you. However, you're asking for how to treat this problem "honestly", and I think that adding the right kind of historical perspective falls under the heading of honesty.

Anyway, I think it is important to observe here that the ancient Greeks themselves did not limit their solutions to plane constructions. As can be read in Sir Thomas L. Heath's A History of Greek Mathematics, Vol. 1, pp. 246-9, Archytas proposed a solution to the problem where he intersected three surfaces of revolution in Euclidean $3$-space (a cone, a cylinder, and a torus) to obtain a point whose coordinates generate the field extension $\mathbb{Q}(\sqrt[3]{2})$.

It is somewhat misleading, I think, to keep referring to these problems as "the three famous unsolved problems of Greek mathematics", because the Greeks in fact solved them many times over, Archytas' solution being only one solution out of a multitude. Moreover, they even recognized that the solution could not be achieved by plane methods, in a way: Pappus has it that the Greeks classified construction problems as "plane", "solid", and "linear", according to the methods with which the problem could be solved. Of course, they never tried to make this very precise, let alone tried to prove it, but then they weren't trying to do the impossible either.