What's the exact meaning of this sentence from George Peacock? I am reading the book "A history of abstract algebra" by Israel Kleiner. The following sentence is said by George Peacock. I am not a native English speaker. So could someone translate it into plain English?

In symbolical algebra, the rules determine the meaning of the
  operations... we might call them arbitrary assumptions, inasmuch as
  they are arbitrarily imposed upon a science of symbols and their
  combinations, which might be adapted to any other assumed system of
  consistent rule.

 A: The context makes the meaning clear. In 1830, George Peacock wrote a book called "A Treatise on Algebra". In it, he attempted to justify things like $(-x)(-y) = xy$ (note that negative numbers were still far from universally accepted then, much less the meaning of multiplying them), by arguing as follows: that $(a-b)(c-d) = ac + bd - ad - bc$ is a law of "arithmetical algebra" whenever $a > b$ and $c > d$, so it is a law of symbolical algebra (always holding, whatever the values of $a, b, c, d$). With $a = 0$ and $d = 0$, you get $(-b)(-d) = bd$ for all values of $b$ and $d$.
The significance was looking at operations like $(a-b)(c-d)$ in an axiomatic sort of way, in terms of rules (like that the distributive law should hold). Rather than looking at such expressions only as place-holders for what happens when dealing with positive integers (which is what he meant by "arithmetical algebra"), he was looking at them more abstractly, letting $a, b, c, d$ be truly and purely just symbols, not necessarily positive integers or even integers.
Under certain circumstances (certain values of $a$, $b$, $c$, etc.), it may be possible to interpret a result of symbolical algebra, or give meaning to it. Under other circumstances, it may not be, but still the result holds in symbolical algebra. For instance, he says that a product of three symbols, like $abc$, can be interpreted as volume, in the case when they are all positive numbers. Something like $abcd$ has no interpretation (he says), because there are only three dimensions, but still we can work with expressions like that in symbolical algebra, as symbolical algebra is just a system of rules for operating on symbols. We don't have to worry at every step about the meaning of the operations; the meaning is just whatever the rules say.
Thus, going over (the quoted part of) his quote line by line:

In symbolical algebra, the rules determine the meaning of the operations...

In symbolical algebra, the meaning of operations (like $(-b)(-d)$  or $(a-b)(c-d)$) is determined by the rules (like the distributive law).

we might call them arbitrary assumptions, 

You can take these rules to be arbitrary assumptions.

inasmuch as they are arbitrarily imposed upon a science of symbols and their combinations, 

The symbolical algebra can be thought of as "a science of symbols and their combinations", with rules arbitrarily attached to it. For instance, Boolean algebra was another "science of symbols and their combinations", with different arbitrary rules attached: it had rules like $x + x = x$, for instance.

which might be adapted to any other assumed system of consistent rules.

You could assume different rules, and as long as they were consistent, you would still have a symbolical algebra. The Boolean algebra above is an example.
A: Here's an attempt.  I do not find the text particularly clear myself.
"In symbolic algebra, the rules (for manipulating the symbols) determine the meaning (of the symbols).  The rules are freely chosen.  If different rules were chosen, the same symbols could be used with different meanings."  
If there is something specific you're wondering about, I (or someone else) could attempt to elucidate further.  
