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February 03, 2008

TMA 1.2

"Consider the problem of doubling the cube. Explain what it is, and why it was important in the development of mathematics during Greek times."

Cube duplication in Greek mathematics

The problem of cube duplication is that of constructing a cube with exactly double the volume of a given cube. It is equivalent to the problem of doubling the volume of an arbitrary solid figure whilst maintaining its proportions. An early result, attributed to Hippocrates of Chios [1], was that the problem could be solved if it were possible to construct two 'mean proportionals' for two given lengths[2]. Given the side of the original cube, a line double this, and two mean proportionals between, the first proportional is the side for the doubled cube. It was in this 'reduced' form that the problem was subsequently studied.

A traditional account of the problem's origin is given by Theon[3]:

when the god announced to the Delians by oracle that to get rid of a plague they must construct an altar double the existing one, their craftsmen fell into a great perplexity in trying to find how a solid could be made double of another solid, and they went to ask Plato about it.
Of course, the problem must have been know previously if Hippocrates had worked on it a century earlier. However, a number of geometers associated with Plato did work on the problem.[4]

Even without this story the problem is intelligible in the context of the Greek geometric research tradition. It is analogous to the planar problem of doubling a square, the solution of which appears in one of Plato's Socratic dialogs[5]. A letter, supposedly by Eratosthenes and quoted by Eutocius[6], indicates that the solution of the problem of two mean proportionals would allow the cubature of many other solid figures. This is analogous to the relationship between the quadrature of rectilinear, planar figures and the finding of single mean proportionals.[7] The letter also states that a solution allows for arbitrary scaling, not just doubling.

This letter also suggests a practical significance. Scaling solid figures correctly is useful in building, and also in the construction of war engines. The cubature of solid figures helps in the manipulation of measures of volume. This slant is reinforced by a criticism that earlier researchers had produced theoretical solutions but not practical constructions. (The solution given in the letter is the design of a mechanical device for finding mean proportionals.)

Proclus credits Hippocrates with being the first to use reduction as a technique in difficult constructions. It was later used in other areas.[8] A reduction is, in a sense, an uncompleted analysis. It has been suggested[9] that this reduction was via a generalisation of the problem to the cubature of a square-based cuboid, and was analogous to a lemma concerning cylinders in a proposition of Archimedes.[10]

Eutocius' commentary on this proposition is an important source on Greek research in this area. He gives a collection of solutions for the reduced problem, using a variety of construction means: no line-and-circle solutions[11]. The mechanical flavour of the solutions sits ill with Plato's notion of geometry. One solution due to Menaechmus, a young contemporary of Plato, involved the intersection of a parabola and hyperbola: curves produced by planar sections of a cone.[12]

The Greek interest in conic sections appears to start with Menaechmus' solution. Conics was an important topic in later Greek mathematics, reaching a state of considerable sophistication with Apollonius around a century later.

The importance of conics is indicated by a three-fold classification of geometric problems given by Pappus[13]: plane (soluble by line and circle), solid (needing conic sections), linear (needing more exotic curves). He says of "ancient geometers"[14]:

they were as yet unfamiliar with the conic sections and were baffled for that reason

The significance of the problem is two-fold. One is the use of reduction, which allow problems to be investigated and understood in relation to other problems, even if they were left unsolved. The other is that this problem[15] pushed Greek geometry beyond the confines of line-and-circle constructions: specifically, it initiated the study of conics, which became an important topic in its own right.

Footnotes

[1] By Proclus. See SB2.F2
[2] Which is to say, two lines of intermediate lengths such that the following ratios are all equal: between the first given line and the first proportional, between the second proportional and the second given line, and between the proportionals themselves.
[3] Theon credits Eratosthenes as his source. See SB2.F1
[4] The version of the story by Eutocius mentions Archytas, Eudoxus, and Menaechmus
[5] In Meno; see SB2.E1
[6] See SB2.F3
[7] For more details see Saito (1995)
[8] e.g. Archimedes reduction of the quadrature of the circle to the rectification of its circumference
[9] Saito (1995)
[10] Proposition II.1 of On the sphere and the circle; see SB4.A6.
[11] Which are now know would be impossible
[12] Other given solutions included a construction involving intersecting half-cylinders from Archytus, a neusis construction from Heron, and a use by Nicomedes of a curve called a conchoid. See Thomas (1991).
[13] See SB5.B4.
[14] Strictly speaking he is referring to another problem— angle trisection—but the point applies to any problem that is ”by its nature a solid problem“. Pappus also criticises the use of neusis constructions for solid problem: these are only necessary for linear problems.
[15] Together with the other 'classic' problems: circle quadrature and angle trisection

Additional References

Thoughts

My main recollection from this was that I had great difficulty keeping within the word limit. I had the idea that quotations and footnotes shouldn't count. Discounting quotations meant that if I could find some primary source saying something I needed said then I could just use that. I discounted footnotes mainly so that I could demonstrate that I knew what 'mean proportionals' are without wasting words on it. (I tried to come up with a compact definition, but couldn't devise anything really suitable.) Understandably, my tutor didn't hold with my use of footnotes.

At the time I thought that there ought to be more to say about Pappus' classification of geometric problems, but I now can't quite remember what. Part of it might have been that it was never clear to me quite why solutions involving conic sections were considered more acceptable than neusis constructions. It also might have been to do with an analogy with Descartes' later classification of curves (wherein he used a tenuous notion of a curve's 'simplicity' in order to exclude from geometry figures for which his method did not work).

Posted by robin2 at 07:31 PM | Comments (0)