Omar Khayyam on Cubics

One of the accomplishments of the Persian mathematician Omar Khayyam 
was to give geometrical constructions for the roots of a cubic as the 
intersections of two conics.  Of course, this approach had been used 
earlier by Menaechmus and others to solve certain special cubics 
(notably in relation to the problem of "duplicating the cube"), but 
Khayyam generalized it to cover essentially all cubics (albeit with 
many individual cases so as to avoid negative numbers).

It's usually said that Khayyam erroneously believed the cubic could 
not be solved algebraically, but I think we need to be careful about 
assuming that Khayyam was referring to the modern idea of what 
constitutes an "algebraic" solution.  One of his most famous quotes
     " attention should be paid to the fact that
      algebra and geometry are different in appearance.
      Algebras are geometric facts which are proved."

This is usually cited as evidence of how Khayyam contributed to
reconciling the two fields of geometry and algebra, that had been
so assiduously separated by the Greeks, and thereby casting 
Khayyam as a forerunner of Descartes.  There is certainly truth 
in this view, because Khayyam was definitely far more inclined 
than the Greeks to treat his geometrical line segments as numerical 
quantities rather than strictly as spatial magnitudes.  In fact, 
he developed a numerical version of Euclid's (Eudoxus') theory of 
proportion that comes very close to Dedekind's definition of 
irrational numbers.

However, I think it's worth noting that he also said with regard
to cubic equations

    "This cannot be solved by plane geometry, since it
     has a cube in it.  For the solution we need conic

Here we might credit Khayyam with anticipating the eventual proof
of the unsolvability of the Delian problem (duplicating the cube)
by straight-edge and compass, but it seems to me this comment may
also shed some light on his statement that the cubic cannot be
"solved algebraically".  

Remember that, to Khayyam, "algebras are geometric facts which 
are PROVED", and he was still strongly influenced by the Greek 
insistence on straight-edge and compass constructions as the 
only valid "proofs" in a certain strict formal sense.  This is 
evidenced by the three ancient problems of squaring the circle, 
trisecting the angle, and duplicating the cube, each of which 
was known to be easily done by various geometrical methods, but 
those methods were not strictly in conformity with Euclidean 
construction, and so were regarded as logically inferior, i.e., 
they were "mechanical constructions" (similar to what we 
might view as "plausibility arguments") and didn't constitute 
demonstrations from the only existent system of strict logical 

Thus, it seems conceivable that when Khayyam said the cubic cannot 
be "solved algebraically" he was using his definition of "an algebra" 
as a geometric fact that is PROVED, and he was adhering to the Greek 
notion that the only theoretical proof of a geometric fact is on the 
basis of Euclid's axiomatic system (i.e., straight-edge and compass).
Of course, with this interpretation his statement was perfectly 
correct, and in fact was simply another way of expressing his 
assertion that the Delian problem cannot be solved by straight-
edge and compass.

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