322 11. POST-EUCLIDEAN GEOMETRY
proportion (see Problem 11.3). Working with the sides of the squares, it would
then be legitimate to multiply means and extremes—that is, to form rectangles on
the sides—since the appropriate theorems were proved in Book 6 of Euclid. He
could have said that the triangle ABT equals the rectangle on Ó and EH, which in
turn equals the rectangle on a and â. The assertion that the triangle ABT is the
rectangle on a and â is precisely Heron's theorem. What he has done up to this
point would not have offended a logical Euclidean purist. Why did he not finish
the proof in this way?
The most likely explanation is that the proof came from Archimedes, as many
scholars believe, and that Heron was aiming at numerical results. Another possible
explanation is that our reconstruction of what Heron could have done lacks the
symmetry of the process described by Heron, since á and â do not contain the
sides in symmetric form. Whatever the reason, his summing up of the argument
leaves no doubt that he was willing to accept the product of two areas as a product
of numbers.
1.4. Pappus. Book 4 of Pappus' Synagoge contains a famous generalization of
the Pythagorean theorem: Given a triangle ABT and any parallelograms BTZH
and ΑΒÄΕ constructed on two sides, it is possible to construct (with straightedge
and compass) a parallelogram ATMh. on the third side equal in area to the sum of
the other two (see Fig. 3).
The isoperimetric problem. In Book 5 Pappus states almost verbatim the argument
that Thcon of Alexandria, quoting Zenodorus, gave for the proof of the isoperimet-
ric inequality. Pappus embroiders the theorem with a beautiful literary device,
however. He speaks poetically of the divine mission of the bees to bring from
heaven the wonderful nectar known as honey and says that in keeping with this
mission they must make their honeycombs without any cracks through which honey
could be lost. Being endowed with a divine sense of symmetry as well, the bees
had to choose among the regular shapes that could fulfill this condition, that is,
triangles, squares, and hexagons. They chose the hexagon because a hexagonal
