a(b + c) = ab + ac
and
(b + c)a = ba + ca
(6).
To these laws, which have been investigated by Sir William Hamilton and by Hermann Grassmann, the former has given special names. He calls the laws expressed in
| (1) and (3) the commutative law for addition; (5) the commutative law for multiplication; (2) and (4) the associative laws for addition; (6) the distributive law. |
§ 23. Having proved that these six laws hold, we can at once prove every one of the above propositions in their algebraical form.
The first is proved geometrically, it being one of the fundamental laws. The next two propositions are only special cases of the first. Of the others we shall prove one, viz. the fourth:—
(a + b)² = (a + b)(a + b) = (a + b)a + (a + b)b
by (6).