- THE EARLIEST GREEK GEOMETRY^287
Chapter 3, where he laments the lack of public support for research into solid ge-
ometry.) There is a long-standing legend that Plato's Academy bore the following
sign above its entrance:^13
ÁÃÅÙÌÅÔÑÇÔÏÓ ÌÇÄÅÉÓ ÅÉÓÉÔÙ
("Let no ungeometrical person enter.") If Plato was indeed more concerned with
geometry than with arithmetic, there is an obvious explanation for his preference:
The imperfections of the real world relate entirely to geometry, not at all to arith-
metic. For example, it is sometimes asserted that there are no examples of exact
equality in the real world. But in fact, there are many. Those who make the as-
sertion always have in mind continuous magnitudes, such as lengths or weights, in
other words, geometrical concepts. Where arithmetic is concerned, exact equality
is easy to achieve. If I have $11,328.75 in the bank, and my neighbor has $11,328.75
in the bank, the two of us have exactly the same amount of money. Our bank ac-
counts are interchangeable for all monetary purposes. But Plato's love for geometry
should not be overemphasized. In his ideal curriculum, described in the Republic,
arithmetic is still regarded as the primary subject.
1.8. Eudoxan geometry. To see why the discovery of incommensurables created
a problem for the Pythagoreans, consider the following conjectured early proof of a
fundamental result in the theory of proportion: the proposition that two triangles
having equal altitudes have areas proportional to their bases. This assertion is half
of Proposition 1 of Book 6 of Euclid's Elements. Let ABC and AC ¼ in Fig. 12 be
two triangles having the same altitude. Euclid draws them as having a common
side, but that is only for convenience. This positioning causes no loss in generality
because of the proposition that any two triangles of equal altitude and equal base
have equal areas, proved as Proposition 38 of Book 1.
Suppose that the ratio of the bases BC : CD is 2 : 3, that is, 3BC = 2CD.
Extend BD leftward to Ç so that BC = BG - GH, producing triangle AHC,
which is three times triangle ABC. Then extend CD rightward to Ê so that
CD = DK, yielding triangle ACK equal to two times triangle ACD. But then,
since GC = 3BC = 2CD = CK, triangles AGC and ACK are equal. Since
AGC = 3ABC and ACK = 2ACD, it follows that ABC : ACD = 2:3. We, like
Euclid, have no way of actually drawing an unspecified number of copies of a line,
and so we are forced to illustrate the argument using specific numbers (2 and 3 in
the present case), while expecting the reader to understand that the argument is
completely general.
An alternative proof could be achieved by finding a common measure of BC
and CD, namely \BC = \CD. Then, dividing the two bases into parts of this
length, one would have divided ABC into two triangles, ACD into three triangles,
and all five of the smaller triangles would be equal. But both of these arguments
fail if no integers m and ç can be found such that mBC = nCD, or (equivalently)
no common measure of BC and CD exists. This proof needs to be shored up, but
how is that to be done?(^13) These words are the earliest version of the legend, which Fowler (1998, pp. 200 201) found
could not be traced back earlier than a scholium attributed to the fourth-century orator Sopatros.
The commonest source cited for this legend is the twelfth-century Byzantine Johannes Tzetzes,
in whose Chiliades, VIII, 975, one finds ÌçäåÉò ÜçåùìÝôñçôïò åßóßôù ìïí ôçí óôå^çí. "Let no
ungeometrical person enter my house."