"Euclid's *Elements* ... to the modern reader is incredibly dull.
There are no examples; there is no motivation; there are no witty remarks;
there is no calculation. There are simply definitions, axioms, theorems,
and proofs....
If one reads Book I from the beginning, one never has any idea what will
come next. It is only when one gets to the end of the book, where Euclid
proves the Pythagorean theorem, that one realizes that Book I's basic
purpose is to lead to the proof of that result....
As usual, Euclid does not show how he arrived at the [pentagon]
construction ..."

It seems like Prof. Katz has been reading one of my favorite books, "Proofs and Refutations: The Logic of Mathematical Discovery", by I. Lakatos (Cambridge UP, 1976). (That would be a good choice for a book review, come to think of it.) Lakatos sees Euclid as the source of the Original Sin of mathematicians, our careful and proud presentation of ideas in the most logical order, which leaves our students spending most of their time asking, "Why are we doing this?"

It seems worthwhile to note (from Heath, Vol. I, pp. 221-223)
an even earlier reference to the method. Antiphon, a contemporary of
Socrates, discussed the squaring of the circle this way: Inscribe a
square in the circle; on each side of the square, as base, put an
isosceles triangle with its vertex on the nearby arc of the circle; on
each side of this octagon put an isosceles triangle; etc. According to
the commentary of Simplicius, "Antiphon thought that in this way the area
[of the circle] would be used up.... And, as we can make a square equal to
any polygon ... we shall be in a position to make a square equal to a
circle." Aristotle mentioned Antiphon's argument only in order to denounce
it as a well-known fallacy, and of course it is a fallacy within the
Euclidean ruler-and-compass framework. But Antiphon was clearly ahead of
his time! Heath says, "But the objection [that the whole area will never
be used up] is really no more than verbal; Euclid [presumably following
Eudoxus] uses exactly the same construction in XII.2, only he expresses
the
conclusion in a different way, saying that, if the process be continued
far enough, the small segments left over will be together less than any
assigned area. Antiphon in effect said the same thing, which again we
express by saying that the circle is the *limit* of such an inscribed
polygon when the number of its sides is indefinitely increased." How close
Antiphon, Eudoxus, and Euclid came, not only to Archimedes, but to
Leibniz, Cauchy, Riemann, and Weierstrass!

- He showed that Euclid's 4th postulate, "All right angles are congruent to each other", can be proved from the previous three postulates. ("Right angle" is defined as "angle congruent to its supplement".)
- He replaced Euclid's cumbersome statement of his 5th postulate (see Allen, "Euclid", p. 8; change the last "to" to "two") by the conceptually clearer statement familiar to us today: "For any line L and any point P not on L, there is a unique line M through P that is parallel to L" [i.e., does not intersect L (even if the lines are indefinitely extended)]. This postulate was reintroduced by John Playfair in 1795 and is called the Proclus-Playfair version of the parallel postulate. Spoiler alert: Stillwell Chap. 18.

*a/x = x/y = y/b*.

Why were the Greeks so interested in these problems? Here is my guess:
In the first case, let *b=2a*. Then you quickly see that
*x= sqrt(2) a*.
So, if we can solve the mean proportion problem, we can construct a
line segment that is sqrt(2) times another segment. Square roots are
called "constructible numbers" because Euclidean constructions yield
them via the Pythagorean theorem. Now consider the problem of two numbers
in mean proportion and consider the case where *b = sqrt(2) a*
(not a big deal, in view of the previous result).
You should find that
*y ^{3} = 2a^{3}*. Voila! We have duplicated the cube!
Of course, the end of the story is that irrational cube roots are