Greeks and‘Origins’ 55
Mirror
BD
A
C
Fig. 6Idea of proof by reflection of ‘Thales’ theorem’.
Sop= 2 r,say,andp^2 = 4 r^2. Rewriting, 4r^2 = 2 q^2 ,soq^2 = 2 r^2. Henceq^2 is even, and so
qis; but this contradicts the supposition thatpandqhave no common factor.
- This is vaguely formulated, and rather hard to make precise. It is clear from the proposition
that the area of the parallelogram is equal to the area of the (unique)rectanglewith the
same base and height. To go further, and talk of ‘multiplying base by height’, you have
to say what kind of numbers base and height are. The easy case is where the two ‘have a
common measure’ (see Exercise 2); if one isp.L and the otherq.L, then it is not hard to
show that the rectangle ispqtimes the square of side L. To go further, you need the general
theory of ratios. In theory, this would allow you to show that a rectangle of sides
√
√ 2 and
3 was equal to one of base
√
6 and height 1; however, it is difficult, to say the least.^14
- There are two questions involved; one is what is an acceptable simple proof, and the other is
what proof might have been used by someone at a time when organized deductive geometry
did not exist. Euclid’s proof (I.6) is quite complicated, involving extra construction lines;
it depends on I.4, that if triangles ABC and DEF have AB=DE, AC=DF, and the angles
A and D equal, then they are congruent. In theory one could apply this to show that
if ABC was isosceles (AB= AC), then it is ‘congruent to itself ’,ABC= ACB. In
practice it would be more natural to use properties of reflection; for example, that ABC is
unchanged by reflection in a mirror which bisects angle A (Fig. 6). However, any of these
ideas (congruence, reflection,...) are at the basic starting points of geometry, and one
wonders what Thales and his contemporaries, if the story is true, would have considered
a proof.
6&7. The property usually taken to define the ‘regular’ solids is that their faces are all regular
polygons and all of the same kind (e.g. all squares); and that in a given solid S the same
number of polygons meet at each point, or vertex as it is usually called. If we write (p,q)to
denote the solid whose faces arep-sided, withqat each vertex, then we have:
- Dedekind claimed in the nineteenth century (see Chapter 9) that he was the first person to have proved that
√
2 ·
√
3 =
√
6,
and this is commonly accepted.
