A History of Mathematics- From Mesopotamia to Modernity

Greeks and‘Origins’ 53

A

BC

DE

Fig. 4Constructing a regular pentagon. The angles of the pentagon are 108◦, so angles like ADE are 36◦. Hence the triangle ABC
has angles A= 36 ◦,B=C= 72 ◦. (Greek astronomers used degree measurements of angle, even if Euclid did not, so this
description is not too modernized.)

A

E

BC

x

x

x

Fig. 5The ‘extreme and mean section’ picture. CE bisects angle C, which is twice angle A. So the triangles ABC, AEC, ECB are all
isosceles, so the lines markedxare all equal. The statement that EB : EA=EA : AB now follows by using similar triangles (CBE, ABC)
and the various identifications we have.

2. To construct the triangle, you need to divide the side AB at E so that the ratio of BE to AE is
   equal to the ratio of AE to the whole of AB (see Fig. 5) This is called (by Euclid) ‘dividing the
   line in extreme and mean ratio’. More fancifully in the Middle Ages it acquired the name of
   the ‘golden section’ or ‘golden ratio’. If we want to use algebra: suppose AB has length 1, and
   AE=xso BE= 1 −x, then
   1 −x
   x

=

x

1

; x^2 = 1 −x

which we would now solve to givex=^12 (

√

5–1)=0.618.... This is ‘obviously’ irrational

to us (perhaps!), because of the

√

5 in it, but as usual we are not sure whether, or how, the

Pythagoreans discovered the fact. Again, Euclid’s proof is rather late in book X.^13

13. Even this is simplifying. Euclid proves in book X that what he calls an ‘apotome’ is irrational, and in book XIII that the golden
    ratio is an apotome.