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31. Hitherto we have considered only positive
numbers; and there can be no doubt, but that the products
which we have seen arise are positive also: viz. + a by +
b must necessarily give + ab. But we must separately
examine what the multiplication of + a by - b, and of - a
by - b, will produce.
32. Let us begin by multiplying - a by 3 or + 3. Now,
since - a may be considered as a debt, it is evident that if
we take that debt three times, it must thus become three
times greater, and consequently the required product is -
3a. So if we multiply - a by + b , we shall obtain - ba,
or, which is the same thing, - ab. Hence we conclude,
that if a positive quantity be multiplied by a negative
quantity, the product will be negative; and it may be laid
Of the multiplication of algebraic quantities.
And first, how to find the sign of the product in
multiplication, from those of the multiplicator and
multiplicand given.
5. Before we can proceed to the multiplication of
algebraic quantities, we are to take notice, that if the
signs of the multiplicator and multiplicand be both alike,
that is, both affirmative, or both negative, the product
will be affirmative, otherwise it will be negative: thus +4
multiplied into +3, or - 4 by -3 produces in either case
+12 : but - 4 multiplied into +3, or +4 into -3 produces
in either case -12.
If the reader expects a demonstration of this rule, he
must first be advertised of two things : first, that numbers
are said to be in arithmetic progression, when they
increase or decrease with equal differences, as 0, 2, 4, 6;
or 6, 4, 2, 0; also as 3, 0, -3; 4, 0, -4; 12, 0, -12; or
-12,0, 12: whence it follows, that three terms are the
fewest that can form an arithmetical progression; and that
of these, if the two first terms be known, the third will
easily be had: thus if the two first terms be 4 and 2, the
next will be 0; if the two first be 12 and 0, the next will
be -12; if the two first be -12 and 0, the next will be
+12, &c.
2dly, If a set of numbers in arithmetical progression,
as 3, 2 and 1, be successively multiplied into one
132
down as a rule, that + by + makes + or plus; and that, on
the contrary, + by - , or - by +, gives - , or minus.
33. It remains to resolve the case in which - is
multiplied by - ; or, for example, - a by - b. It is evident,
at first sight, with regard to the letters, that the product
will be ab; but it is doubtful whether the sign + , or the
sign -, is to be placed before it; all we know is, that it
must be one or the other of these signs. Now, I say that it
cannot be the sign - : for - a by + b gives - ab, and - a
by - b cannot produce the same result as - a by + b; but
must produce a contrary result, that is to say + ab;
consequently, we have the following rule: - multiplied by
- produces + , that is the same as + multiplied by + .
The next extract is from Saunderson's Elements of Algebra, which ran to five
editions between 1740 and 1792.
common multiplicator, as 4, or if a single number, as 4,
be successively multiplied into a set of numbers in
arithmetic progression, as 3, 2 and 1, the products 12, 8
and 4, in either case, will be in arithmetical progression.
This being allowed, (which is in a manner self-
evident,) the rule to be demonstrated resolves itself into
four cases:
1st, That +4 multiplied into +3 produces +12.
2dly, That -4 multiplied into +3 produces -12.
3dly, That +4 multiplied into -3 produces -12.
And lastly, that -4 multiplied into -3 produces +12.
These cases are generally expressed in short thus: first +
into + gives + ; secondly - into + gives - ; thirdly + into -
gives - ; fourthly - into - gives + .
Case 1st. That +4 multiplied into +3 produces +12,
is self-evident, and needs no demonstration; or if it
wanted one, it might receive it from the first paragraph of
the 3d article: for to multiply +4 by +3 is the same thing
as to add 4 + 4 + 4 into one sum; but 4 + 4 + 4 added into
one sum give 12, therefore + 4 multiplied into +3, gives
+12.
Case 2d. And from the second paragraph of the 3d art.
it might in like manner be demonstrated, that - 4
multiplied into +3 produces -12: but I shall here
demonstrate it another way, thus: multiply the terms of
this arithmetical progression 4, 0, -4, into +3, and the