For instance, Dicaearchus says that the distance from
the Peloponnese to the Pillars is ten thousand stades
and still further to the head of the Adriatic; and from
the Peloponnesus to the Sicilian straits three thousand;
and therefore the remainder, from the Straits to the Pillars,
is seven thousand stades. I say nothing about the three
thousand stades, whether they are right or wrong; but
the seven thousand cannot be made out, whether you measure
along the coast or straight across the sea. The coast route is
a kind of obtuse angle, contained by two lines resting on the
straits and the pillars respectively; so that we have a triangle,
of which the apex is Narbo, and the base the straight line
representing the course by the open sea; of the two sides of
the triangle which contain the obtuse angle, that which extends
from the straits to Narbo is more than eleven thousand two
hundred stades, the other from Narbo to the Pillars is a little
under eight thousand. The longest distance from Europe to
Libya across the Tuscan sea is allowed to be not more than
three thousand stades, that by the Sardinian sea is somewhat less; but let us call it three thousand stades. Now
suppose a perpendicular let down through the gulf of Narbo
to the base of the triangle, that is to the straight seacourse, measuring two thousand stades; it requires only a
schoolboy's geometry to prove that the coasting voyage is
longer than the direct sea voyage by nearly five hundred
stades.

Polybius proves his point by the demonstration of the proposition "The square of hypotenuse of a right-angled-triangle is equal to the squares of the sides containing the right angle."

By applying this principle AD = 7745.9 . . and DC = 11019.9 . . . and the whole AC = 18765.8: whereas AB + BC (i.e. the coasting voyage) = 19200 stades (a difference of 434.2 stades, not 500). Add to this the 3000 from the Peloponnese to the Straits, the total coast voyage is 22,200 stades, as against Dicaearchus's 10,000.

And when the three thousand stades from theIn talking such nonsense he might well be regarded as having gone beyond even Antiphanes of Berga, and, in fact, to have left no folly for his successors to commit. . . .