298 10. EUCLIDEAN GEOMETRY
part. After it is established that four lines are proportional when the rectangle
on the means equals the rectangle on the extremes (Proposition 16, Book 6), it
becomes possible to convert this construction into the construction of the Section
(Proposition 30, Book 6).
Books 3 and 4 take up topics familiar from high-school geometry: circles, tan-
gents and secants, and inscribed and circumscribed polygons. In particular, Book 4
shows how to inscribe a regular pentagon in a circle (Proposition 11) and how to
circumscribe a regular pentagon about a circle (Proposition 12), then reverses the
figures and shows how to get the circles given the pentagon (Propositions 13 and 14).
After the easy construction of a regular hexagon (Proposition 15), Euclid finishes
off Book 4 with the construction of a regular pentakaidecagon (15-sided polygon,
Proposition 16).
Book 5 contains the Eudoxan theory of geometric proportion, in particular
the construction of the mean proportional between two lines (Proposition 13). In
Book 6 this theory is applied to solve the problems of application with defect and
excess. A special case of the latter, in which it is required to construct a rectangle
on a given line having area equal to the square on the line and with a square excess
is the very famous Section (Proposition 30). Euclid phrases the problem as follows:
to divide a line into mean and extreme ratio. This means to find a point on the
line so that the whole line is to one part as that part is to the second part. The
Pythagorean theorem is then generalized to cover not merely the squares on the
sides of a right triangle, but any similar polygons on those sides (Proposition 31).
The book finishes with the well-known statement that central and inscribed angles
in a circle arc proportional to the arcs they subtend.
Books 7 9 were discussed in Chapter 7. They are devoted to Pythagorean
number theory. Here, since irrationals cannot occur, the notion of proportion is
redefined to eliminate the need for the Eudoxan technique.
Book 10 occupies fully one-fourth of the entire length of the Elements. For its
sheer bulk, one would be inclined to consider it the most important of all the 13
books, yet its 115 propositions are among the least studied of all, principally because
of their technical nature. The irrationals constructed in this book by taking square
roots are needed in the theory developed in Book 13 for inscribing regular solids in a
sphere (that is, finding the lengths of their sides knowing the radius of the sphere).
The book begins with the operating principle of the method of exhaustion, also
known as the principle of Archimedes. The way to demonstrate incommensurability
through the Euclidean algorithm then follows as Proposition 2: //, when the smaller
of two given quantities is continually subtracted from the larger, that which is left
never divides evenly the one before it, the quantities are incommensurable. We
used this method of showing that the side and diagonal of a regular pentagon are
incommensurable in Chapter 8.
Book 11 contains the basic parts of the solid geometry of planes, parallelepipeds,
and pyramids. The theory of proportion for these solid figures is developed in
Book 12, where one finds neatly tucked away the theorem that circles are propor-
tional to the squares on their diameters (Proposition 2), which we quoted above.
Book 12 continues the development of solid geometry by establishing the usual
proportions and volume relations for solid figures; for example, a triangular prism
can be divided by planes into three pyramids, all having the same volume (Propo-
sition 7), a cone has one-third the volume of a cylinder on the same base, similar