A History of Mathematics From Mesopotamia to Modernity

Greeks,Practical andTheoretical 63

Fig. 3Regular hexagon circumscribed about a circle.

B

C

uu

A


A–A

Fig. 4Picture for Exercise 4.

Exercise 3.Show that the perimeter of the circumscribed hexagon is 4

√

3 r.

Exercise 4.Rule 2 above is clearly Pythagoras’s theorem. But where does rule 1 come from?

3. Heron or Hero

Three centuries after Archimedes (probably in the first centuryce) a very different mathematician
left a number of works which were both accessible and popular. This was Heron, or Hero of
Alexandria. (Because of translation problems, you may find either name used; I shall keep to the
more usual ‘Heron’ in what follows.) His works are not easy to find, except in small extracts, but
they are numerous and quite astonishingly diverse, dealing with theory and practice sometimes
separately and sometimes together. That he was not despised, despite his practical bent and what
some historians have seen as weak mathematical attainments, is shown by Pappus’s description of
his work—or that of his ‘school’, which in turn suggests influence.

The mechanicians of Heron’s school say that mechanics can be divided into a theoretical and a manual part;

the theoretical part is composed of geometry, arithmetic, astronomy and physics, the manual of work in metals,

architecture, carpentering and painting and anything involving skill with the hands.

...the ancients also describe as mechanicians the wonder-workers, of whom some work by means of pneumatics,

as Heron in hisPneumatica, some by using strings and ropes, thinking to imitate the movements of living things, as