In the years Julius Caesar began his bid for power in the burgeoning Roman Empire, Geminus, a stoic Greek philosopher from the island of Rhodes, wrote the Theory of Mathematics.  Geminus' work "dealt with the logical subdivisions of the mathematical sciences, discussing the philosophical principles of their classification, distinguishing carefully between such terms as ‘hypothesis' and ‘theorem,' ‘postulate' and ‘axiom.'" In particular, it appears that much of the content in Theory of Mathematics focused on geometry and geometric solids.

Much of Geminus' work is predicated upon Euclid's Elements, and he published his work to defend the science of mathematics against the Epicureans and Sceptics, who were enemies of the sciences.  He attempted to distinguish between "the things which must be taken for granted but are incapable of proof and the things which must not be assumed but are matter for demonstration."  To do so, he critically examined the definitions, postulates, and axioms posited by Euclid and discussed the legitimacy of each of Euclid's claims.  His efforts required that he make a clear distinction between a postulate and an axiom and that he clearly define fundamental concepts such as "line," "surface," and "angle," often providing additional classifications for clarity.  For example, he distinguishes between "composite" and "incomposite" lines, and he further classifies "incomposite" lines as "simple" and "mixed."

Geminus also devoted part of the Theory of Mathematics to proving Euclid's fifth postulate, the parallel-postulate.  To prove the parallel-postulate, Geminus constructed a series of parallelograms using parallel lines.  However, his proof was not theoretically sound, for it never " proved that ‘the geometrical locus of points equidistant from a straight line is a straight line,'" which requires another postulate.  Regardless, the Theory of Mathematics was an invaluable source for later mathematicians and for math historians.

Besides his work in mathematics, Geminus was also an astronomer, and he published several astronomy texts, including one titled Introduction to Astronomy that he based largely on Hipparchus' work.  Geminus is believed to have died shortly after 70 B.C.

Author: Tony Brinsko

References:
Biography in Dictionary of Scientific Biography (New York 1970-1980).

T L Heath, A History of Greek Mathematicians II (Oxford, 1921), 222-234.

http://www-history.mcs.st-and.ac.uk:/history/Mathematicians/Geminus.html