It seems to me that in transition from the algebra of number systems to
abstract algebra, the principle characteristic of algebra is completely
transformed.
Algebra appears first as a feature of numerical rather than geometric
mathematics, but ends as a feature of abstract mathematics (concerned with
classes of structures, e.g. groups, rings, categories) rather than of concrete
mathematics (concerned with particular structures, such as the natural numbers
or the reals)..

It is of interest methodologically as one line in the development of methods which bear upon the formalisation and mechanisation
of mathematics.

Other historical threads which contribute to this broader topic include:

- the axiomatic method
- the formalisation of mathematics
- formal methods in computer science and information engineering
- the automation of reason