From the Beginning to Plato

Figure 8.1
GREEK ARITHMETIC, GEOMETRY AND HARMONICS 253
on what is traditionally called the axiom of Archimedes or Archimedean
condition. Euclid purports to prove a form of the ‘axiom’ in:
X.1. Two unequal magnitudes being set out, if from the greater there be
subtracted a magnitude greater than its half, and from that which is left a
magnitude greater than its half, and if this process be repeated continually,
there will be left some magnitude which will be less than the lesser
magnitude set out.
Most of the results proved in Book XII concern solids, but the first and simplest,
XII.2, establishes that circles C, C′ are to one another as the squares sq(d) and sq
(d′) on their diameters d and d′ (XII.2). To prove this Euclid first shows (XII. 1)
that if P and P′ are similar polygons inscribed in C and C′, then P:P′ :: sq(d):sq(d
′). He then argues indirectly by assuming that C is not to C′ as sq(d) is to sq(d′),
but that for some plane figure C* which is, say, smaller than C′, C:C* :: sq(d):sq
(d′). He inscribes in C′ successively larger polygons P 1′, P 2′,… (see Figure 8.1)
until a P n′ of greater area than C* is reached, and then inscribes a similar polygon
P n in C. By XII.1, P n:P n′ :: sq(d):sq(d 1 ), so that C:C* :: P n:P n′, and C:P n :: C*:P n′.
But this is impossible since C is greater than P n and C* is less than P n′.
In book XIII Euclid shows in XIII.13–17 how to construct each of the five
‘cosmic figures’ or regular solids, the triangular pyramid contained by four
equilateral triangles, the octahedron contained by eight such triangles, the
icosahedron contained by twenty, the cube contained by six squares, and the
dodecahedron contained by twelve regular pentagons. In XIII. 13–17 Euclid also
characterizes the relationship between the edge e of the solid and the diameter d
of the circumscribed sphere. For triangular pyramid, cube, and octahedron the
results are simply stated; for example, for the triangular pyramid the square on d
is 1½ times the square on e. However for the other two solids Euclid uses
materials from Book X, taking the diameter d to be a ‘rational’ straight line, and
showing that the edge of the dodecahedron is an apotome, and that the
icosahedron is a line which he calls minor and defines in X.76. These