- THE EARLIEST GREEK GEOMETRY 291
ΕÇFIGURE 14. Proof that circles are proportional to the squares on
their diameters.Let ΑΒÃΑ and ΕÆÇÈ be two circles with diameters Β A and ÈÆ, and suppose
that the circles are not proportional to the squares on their diameters. Let the ratio
Β A^2 : ÈÆ^2 be the same as ΑΒÃΑ : Ó, where Ó is an area larger or smaller than
ΕÆÇÈ. Suppose first that Ó is smaller than the circle ΕÆÇÈ. Draw the square
ΕÆÇÈ inscribed in the circle ΕÆÇÈ. Since this square is half of the circumscribed
square with sides perpendicular and parallel to the diameter ÈÆ, and the circle is
smaller than the circumscribed square, the inscribed square is more than half of
the circle. Now bisect each of the arcs EZ, ZH, ÇÈ, and È.Å at points Ê, Α, M,
and N, and join the polygon ΕÊÆΑÇÈÍΕ. As shown above, doing so produces
a larger polygon, and the excess of the circle over this polygon is less than half of
its excess over the inscribed square. If this process is continued enough times, the
excess of the circle over the polygon will eventually be less than its excess over Ó,
and therefore the polygon will be larger than Ó. For definiteness, Euclid assumes
that this polygon is the one reached at the first doubling: ΕÊÆΑÇÈÍΕ. In the
first circle ΑΒÃΑ, inscribe a polygon ΑÎΒÏÃÚΙΑÑ similar to ΕÊÆΑÇÈÍΕ. Now
the square on BA is to the square on ÆÈ as ΑÎΒÏÃÐΑÑ is to ΕÊÆΑÇÈÍΕ.
But also the square on BA is to the square on ÆÈ as the circle ΑΒÃΑ is to Ó. It
follows that ΑÎΒÏÃÚΙΑÑ is to ΕÊÆΑÇÈÍΕ as the circle ΑΒÃΑ is to Ó. Since
the circle ΑΒÃΑ is larger than ΑÎΒÏÃÚΙΑÑ, it follows that Ó must be larger than
ΕÊÆΑÇÈÍΕ, But by construction, it is smaller, which is impossible. A similar
argument shows that it is impossible for Ó to be larger than ΕÆÇÈ.
A look ahead. Ratios as defined by Euclid are always between two magnitudes of
the same type. He never considered what we call density, for example, which is the
ratio of a mass to a volume. Being always between two magnitudes of the same
type, ratios are "dimensionless" in our terms, and could be used as numbers, if only
they could be added and multiplied. However, the Greeks obviously did not think
of operations on ratios as being the same thing they could do with numbers. In
terms of adding, Euclid does say (Book 6, Proposition 24) that if two proportions
have the same second and fourth terms, then their first terms and third terms can