In Greek geometry, there is an arithmetic of magnitudes in which, in
terms of numbers, only integers are involved. This theory of measure is
limited to exact measure. Operations on magnitudes cannot be actually
numerically calculated, except if those magnitudes are exactly measured
by a certain unit. The theory of proportions does not have access to
such operations. It cannot be seen as an "arithmetic" of ratios. Even if
Euclidean geometry is done in a highly theoretical context, its axioms
are essentially semantic. This is contrary to Mahoney's second
characteristic. This cannot be said of the theory of proportions, which
is less semantic. Only synthetic proofs are considered rigorous in Greek
geometry. Arithmetic reasoning is also synthetic, going from the known
to the unknown. Finally, analysis is an approach to geometrical problems
that has some algebraic characteristics and involves a method for
solving problems that is different from the arithmetical approach. 3.
GEOMETRIC PROOFS OF ALGEBRAIC RULES Until the second half of the 19th
century, Euclid's Elements was considered a model of a mathematical
theory. This may be one reason why geometry was used by algebraists as a
tool to demonstrate the accuracy of rules otherwise given as numerical
algorithms. It may also be that geometry was one way to represent
general reasoning without involving specific magnitudes. To go a bit
deeper into this, here are three geometric proofs of algebraic rules,
the frrst by Al-Khwarizmi, the other two by Cardano.