Ancient mathematics was mostly applied arithmetic. But there are other
facits.
Number -- ordinal vs cardinal
Base -- binary, ternary, decimal, sexigesimal, etc
Arithmetic -- addition, multiplication, etc
Geometry -- areas and volumes
Number Theory -- pythagorean triples, primes, etc
Algebra -- solution of equations
Number and Base.
The very way numbers are represented has a
profound impact on the mathematics of a civilization.
Number and types of symbols
Positional and non-positional representation
Positive numbers vs negative numbers
Arithmetic had simple and pragmatic origins. Many different
systems were developed. Our own system evolved more than 2,000 years before
it took it's present form.
Geometry.
Born of practical needs to compute areas and volumes
and eventually make angular measurements, geometry eventually became the
theoretical model for all sciences to emulate.
Ideas of similarity of figures
Formulas for areas and volumes (some incorrect)
quadrilaterals and circles
parallelopipeds and pyramids
Classification of curves and shapes
Regular solids
Full axiomatic development
Method of exhaustion (the first limit)
Number Theory.
As with most ancient mathematics, number theory
had its origins with construction of shapes, but it rapidly assumed a life of
its own.
figurate numbers
Pythagorean triples
Characterization of numbers into primes, amicable, abundant, deficient
Theorems about numbers, tables of results
rational numbers - incommensurable numbers, (The square root of two
took many years to assimilate though its geometric origins are the most
basic.)
Algebra.
The first problems of algebra were practical, dividing inheritances, computing logistic needs, allocation of resources, determining volumes and areas, etc.
Solution of linear equations
Solution of linear systems
Solution of quadratics -- algebraically vs geometrically
Iterative methods
Positive vs negative solutions
Other issues of ancient mathematics:
Level of rigor -- from examples to formulas to theorems
Dynamically vs statically defined curves
Rhetorical vs symbolic expressions
Applied vs pure mathematics
Angular measurements -- astronomy
Quadratures of shapes -- the need for a limiting-like process.
