A Treatise of Algebra.

1 3

SECT. IV.

Of Multiplication.

B EFORE I come to lay down Rules for multiply-ing Quantities one by another, it will be necessary,in order to give the Learner a clear Idea of the Reasonand Certainty of such Rules, to premise the two follow-ing Lemmas.

Lemma i.

When several Quantities are to be multiply'd continuallytogether , the -Product toilI come cut the fame > multiplythem according to what Order you will.

Thus 2x3x4 is = 3x4x2 = 24; and 5x7x9= 5x9x7 = 315.

Lemma 2.

If any Number of Quantities be multiply’d continuallytogether , and any other Number of Quantities be also mul-tiply'd continually together , and then the two Products oneinto another , the Quantity thence arising will be equal tothe Quantity which would arise by multiplying all the pro-posed Quantities continually together:

Thus 3x7 (or 21J multiply*?! by 5x9 (or 45) willbe equal 103x7x5x9 = 945 ; and 2x3x4 (or 24)multiply’d by 5x6x7 (or 210) will be equal to 2 x 3 X4 x 5 x 6> 7 = 5040 ; and so of any other.

The following Demonstrations, of these two Lemmas,may perhaps seem a little difficult to a Beginner; if so,I would advise him to pass them over, and proceed im-mediately to the Rules; since the Examples alreadygiven, with other particular Cases, of the fame kind,which he may try at his Leisure, will abundantly satisfyhim in the Truth of what has been laid down abovein general Terms.

Note. In the ensuing Demonstration, those Pro-ducts are said to be of the same Class, which have thefame Letter or Quantity for their last Factor, and those1 of