Dual transitions in the works of Hero of Alexandria

Y. B. Karasik,
Thoughts Guiding Systems Corp.,
Ottawa, Canada.
e-mail:karasik@sympatico.ca

Geometrization of arithmetics (and algebra) by Greeks gave way to arithmetization/algebraization of geometry since the times of Descartes. There were attempts to change it with putting forward the ideas of synthetic geometry in XIX century, but it could not turn the tide.

Hero's works were exception of the rule. Unlike other Greeks that used to give geometric definitions to arithmetics operations (e.g. to multiplication), he switched to giving arithmetic definitions to geometric quantities. For instance, irrational numbers initially were in geometry only.

In this regard this achievement, as his many other inventions (e.g. steam turbine), was ahead of time.

Hero is best remembered for his famous Hero's formula. It is akin to Green's formula. Both express characteristics of the entire object through characteristics of its boundary.