Division.-From the commutative law for multiplication, which shows that 3 X 4d.

When we are familiar with the treatment of quantities by equations, we may ignore the units and deal solely with numbers; and (ii.) (a) and (ii.) (b) may then, by the commutative law for multiplication, be regarded as identical.

May be regarded as resulting from the commutative law for addition and subtraction.

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This is due to the fact that there are really two kinds of subtraction, respectively involving counting forwards (complementary addition) and counting backwards (ordinary subtraction); and it suggests that it may be wise not to use the one symbol - to represent the result of both operations until the commutative law for addition has been fully grasped.

Rotation will bring a point from C to C and that the second will bring it back to C; the result is therefore equivalent to a rotation about OC. We note also that if the given rotations had been effected in the inverse order, the axis of the resultant rotation would have been OC, so that finite rotations do not obey the commutative law.

To transpose a term which is not the last term on either side we must first use the commutative law, which involves an algebraical transformation.

This is, of course, on the usual assumption that the sign of a product is changed when that of any one of its factors is changed, - which merely means that-1 is commutative with all other quantities.