312 10. EUCLIDEAN GEOMETRY
c
D
A Β
FIGURE 24. Diagonal and side of a square.
of incommensurables, is an attempt to bring into sharper focus the theorems already
proved and to test the underlying assumptions of a theory—to rigorize. Are these
kinds of activity complementary, opposed, or simply unrelated to each other?
10.10. Hippocrates' quadrature of a lune used the fact that the areas of circles
are proportional to the squares on their radii. Could Hippocrates have known this
fact? Could he have proved it?
10.11. Plato apparently refers to the famous 3-4-5 right triangle in the Republic,
546c. Proclus alludes to this passage in a discussion of right triangles with commen-
surable sides. We can formulate the recipes that Proclus attributes to Pythagoras
and Plato respectively as
(2n + l)^2 + (2n^2 + 2n)^2 = (2n^2 + 2n+ l)^2
and
(2n)^2 + (ç^2 - l)^2 = (ç^2 + l)^2.
Considering that Euclid's treatise is regarded as a compendium of Pythagorean
mathematics, why is this topic not discussed? In which book of the Elements
would it belong?
10.12. Proposition 14 of Book 2 of Euclid shows how to construct a square equal in
area to a rectangle. Since this construction is logically equivalent to constructing the
mean proportional between two line segments, why does Euclid wait until Book 6,
Proposition 13 to give the construction of the mean proportional?
10.13. Show that the problem of squaring the circle is equivalent to the problem of
squaring one segment of a circle when the central angle subtended by the segment
is known. (Knowing a central angle means having two line segments whose ratio is
the same as the ratio of the angle to a full revolution.)
10.14. Referring to Fig. 18, show that all the right triangles in the figure formed
by connecting B' with C, C with K, and K' with L are similar. Write down a
string of equal ratios (of their legs). Then add all the numerators and denominators
to deduce the equation
(BB' + <%"+·· + KK' + LM) : AM = A'Β : ΒΑ.
