to both. If the quantity turn out to be additive, it answers the former; if subtractive, the latter question. It is true this result originated in the absurdity of the statement of the question; we say absurdity in this case, because no consideration was required to convince us that one of the true conditions was reversed. But similar questions will arise, in which this term will not be justly applicable,-questions wherein it is impossible that any previous reflection short of the actual solution of the problem could satisfy us as to the correctness of our hypothesis. In addition to this, a large branch of analysis actually owes its existence to the circumstance of the introduction of negative quantities. What then is to be understood by such quantities? A careful examination of the problem before us will give the

answer.

It was required to determine in how many years the event would happen. Let us suppose our point of reckoning (the origin of the measure of time) to be the date of the birth of the son. Our anticipation of a positive solution was equivalent to the expectation that a certain number of years would have to be added to the first assigned age of the person. But as the result is negative, we reasonably conclude that our expectation was incorrect, and that we ought to subtract the number instead. By doing this we obtain the correct result. Let us see, then, what modification we have made in the assumed meaning of the symbols + and -. As long as they connected together quantities which admitted of combination by addition and subtraction, these symbols were made to signify addition and subtraction, and nothing more. We must, therefore, if we retain the rules deduced from this restriction of their nature, attach to these symbols a signification which of necessity carries with it the ideas implied in addition and subtraction. Thus if + 4 signifies that 4 years are to be added to the age of an individual, — 4 occurring under the same circumstances must signify that 4 years are to be taken from his age. We need not, then, trouble ourselves to enter into a discussion of the different ways in which + and

CHAPTER II.

FUNDAMENTAL OPERATIONS-ADDITION, SUBTRACTION,

MULTIPLICATION, AND DIVISION.

29. THE rules of addition, subtraction, etc., differ in nowise from the corresponding rules in arithmetic. Yet, in order to familiarize the student with processes which will have frequent occurrence as he proceeds, it has been thought expedient to devote some space to the discussion of examples, which may serve to habituate him to computation.

Previous, however, to entering on the consideration of the separate rules, it may be desirable to make a few general remarks.

The AXIOM on which the operations of arithmetic depend is this, that total operations are performed by means of corresponding partial ones in any order. This axiom is applied to algebra, with the additional law

LAW 1. Quantities affected with the signs + and are in no way influenced by the quantities to which they are united by these signs: thus, a connected with b in the expression b + a has the same force as when it is connected with any other quantity, c.

The second statement in the axiom of arithmetic is sometimes amplified into the two following laws :—

LAW 2. THE DISTRIBUTIVE LAW.-Additions and subtractions may be performed in any order.

LAW 3. THE COMMUTATIVE LAW. Multiplications and divisions may be performed in any order.

The annexed question in common addition has for its object the determination of the sum of three numbers. But the process employed consists in breaking each of the num- 147 bers into three parts and then uniting the corresponding 293 parts of each, units with units, tens with tens, and so 614 In the same way the multiplication of 256 by 78

on.

is performed by multiplying every part of the one number by every part of the other, and then adding together the results. It is perfectly immaterial whether the operation commence with multiplying by 8 or by 70. The adoption of a certain order of proceeding may afford facilities, but it is not an essential element in the work.

ADDITION.

30. The rules for addition are:

1. Like quantities with the same sign are combined by adding together their coefficients.

2. Like quantities with different signs are combined by subtracting their coefficients, and retaining the sign of that quantity which has the greater coefficient.

Three or more quantities of different signs are combined, either by collecting into one term all the quantities which have the positive, and all those which have the negative sign respectively, and proceeding as before, or by performing the operation on two terms, then on the result and a third term, and so on.

These rules are too obvious to require a formal demonstration; we shall therefore intersperse our examples with a few observations, in preference to attempting any such.

As we have

Ex. 1. To add together 15x+a and 18x+b. before stated, 15x added to 18x will give 33x; if x, for instance, represent 7, the sum of 15 times 7 and 18 times 7 is manifestly 33 times 7. Now, the result must consist of a, b,

C

and a certain number of x's; but as a, b, and x, are unlike quantities, all we can do is to represent the addition by the sign; we, therefore, write the result 33x+a+b.

2. To add 12a-x to 11a - 2x. Here, if we first take no notice of the x, our result will be 23a, but from the first part x is to be subtracted, and from the second 2x; therefore, on the whole, 3x is to be subtracted from 23a, and the result is 23a-3x.

3. To add 12a - 2x to 12x-4α. The sum of the first term in each is 12a+12x, but from one of these we must subtract 2x and from the other 4a.

Now, it requires no consideration to shew, that any diminution of one of the quantities to be added, must cause an exactly equal diminution in the whole result; consequently, our process requires only that we subtract 4a and 2x successively from 12a and 12x; therefore, 8a+ 10x is the sum sought.

4. To add 12a-15x to 6x-7a. By adding the first terms in each we get 12a + 6x; from which, by virtue of the second, we must subtract 7a and 15x; if we perform the first subtraction, it remains that from 5a + 6x is to be subtracted 15x.

Now, if our operation consisted merely in subtracting 6x, the result would be 5a; but having done this, there still remains 9x to be subtracted, which operation can only be indicated, so that the final result is written 5a - 9.x.

5. To 12a + 7x+8y add 4a - 9x-7y. The result is 16a-2x+y. Observe that 9x exceeds 7x, and thus the sign of 2x in the result is

6. To 7a+7x + 2y add 3y - 17x - 18a. The result is 5y-10x-11a.

In this example, we transposed the order of the quantities, writing y first in the final result. This, however, is quite unnecessary, as will appear from the following reasoning: Our definition of is that it signifies increased by, and that of diminished by. It would, then, appear rather strange to commence operations by writing-11a, since the phrase "diminished

by 11a" wants some preceding expression to make it intelligible. But if we supply the phrase "a person's property," or something of the kind, the difficulty will be removed. Nor is it even requisite to supply so much as this phrase in the example last given; it will quite suffice if we understand that the capital which is to be diminished shall follow some time in the course of the statement. If we adopt the more general definitions of + and given in Art. 27, even this consideration will be unnecessary.

1. 2a+3a + 7a + 11a + a.

2. a2 - a3 - 3a - 2a + 2a3 + 7a.

3. a-b+ a2 - b2 — ab + 7b + 8a+ 3b2 — a2 + ab.

-

4. 3x2y + 3xу2 + 3x3 + 4y3 − x3y — xy3 — x3 — 2y3.

5. (ax)2 + (bx)2 – (ab)2 — a3b + 2ax2 + 2a3b + 2ab2 + 2(ab)2 +2(bx)2 - a2b.

6. 5xу3 + 2x22 + (xy)3 + (xz)3 + 3xy3 — 3(xz)3 — (xy)3 + (x2)3