Algebra up to, let us say, 1830, was a fruitful but not very rigorously founded enterprise, in which algebraic laws known
to be true for the natural numbers were applied to a much larger number system, elements of which were controversial.
At this time there was still controversy even over the use of negative numbers,
but also, particularly of complex numbers.
The status of transcendental numbers and of the number system as a whole was not well defined.

George Peacock, a professor of mathematics at Cambridge, attempted to put together a rationale for the algebra of the time
which is illuminating.
He divided algebra into two areas:

- arithmetical algebra
- symbolic algebra

The first deals with symbols representing the positive integers and is on solid ground, subject to the constraint that only
operations yielding positive integers are allowed.
"symbolic algebra" adopts the rules of "arithmetical algebra" but removes the restriction to positive integers.
All results deduced in arithmetical algebra which are general in form, though restricted in the application to positive integer
values, are deemed to be valid in symbolic algebra without the restriction on the range of the variables.
This is known as the "principle of permanence of form".