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and of their

which are thence derived, extending their application to all the meanvalues of the symbols and adopting also as the subject matter of ing of the operations our operations or of our reasonings, whatever quantities or forms of themselves, symbolical expression may result from this extension: but, inas- results. much as in many cases, the operations required to be performed are impossible, and their results inexplicable, in their ordinary sense, it follows that the meaning of the operations performed, as well as of the results obtained under such circumstances, must be derived from the assumed rules, and not from their definitions or assumed meanings, as in Arithmetical Algebra.

We will endeavour to illustrate this important and fundamental distinction between these two sciences in the case of the two operations which form the subject of this Chapter.

different

subtraction

552. The rule of subtraction, derived rigorously from the Consideradefinition or assumed meaning of that operation in Arithmetical tion of three Algebra, directs us to change the signs of the terms of the cases of subtrahend and to write them, when so changed, in the same in the tranline with the terms of the minuend, incorporating, by a proper Arithmetirule, like terms into one. In the application of this rule, three cal to Symcases will present themselves, which it may be proper to con- Algebra. sider separately.

sition from

bolical

1st. Where the subtrahend is obviously less than the minuend. When the If the minuend be a + c and the subtrahend be a, then a + c subtrahend is less than - a + c = c. If the minuend be 3a + 76 and the subtrahend be the minuend. 2a + 6b, then 3a + 7b − (2a + 6b) = 3a + 7 b − 2 a − 6b = a + b.

In these examples, the results follow necessarily from the definition of subtraction in its ordinary usage, merely supposing that the symbols represent magnitudes of the same kind.

Examples.

2nd. Where the subtrahend may or may not be less than When the the minuend, according to the relation of the values of the subtrahend symbols involved.

If the minuend be 3a + 4b and the subtrahend 2a + 5b, then

3a + 4b - (2a +5b)=3a+4b−2 a − 5b = a - b.

may be less than the minuend, but not necessarily

80.

If a be greater than b, this is a result of Arithmetical Al- Examples. gebra, but it ceases to be so when a is less than b: as long therefore as the relation of the values of a and b remains undetermined, it is uncertain whether it is a result of Arithmetical or of Symbolical Algebra: it is one of an infinite number of cases in which these two sciences may be said to inosculate with each other.

If the minuend be 3a-4b and the subtrahend be 2a – b, then 3a-46-(2a-b)=8a-4b-2a+b=a-3b. Unless a be greater than 36 this is an example of Symbolical Algebra only. Where the subtrahend is obviously greater than the

When the 3rd. subtrahend minuend. is greater

than the minuend. Examples.

bra, it is in

what order

the terms

succeed

If the minuend be a and the subtrahend a + c, then

a − (a + c) = a − a — c — — c.

If the minuend be 2a+3b and the subtrahend 3a + 46: then

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If the minuend be 2a+3b+4c and the subtrahend be 4a+7b+10c: then

2a+3b+4c-(4a+7b+10c)

= 2a+3b+4c-4a-7b-10c

=-2a-4b-6c = − (2a + 4b+6c).

In these examples, the results are obtained, by the application of the rule for the removal of the brackets when preceded by the sign minus and for the incorporation of like terms, under circumstances which are not recognized in Arithmetical Algebra, inasmuch as they are not necessary consequences of the definition of the operation of subtraction, though they are necessary results of the unlimited application of the rule for performing it.

In Symbo- 553. Again, in Arithmetical Algebra, it has been shewn lical Alge- to be indifferent in what order, terms connected by the signs different in plus and minus, succeed each other, so long as a positive term occupies the first place (Art. 22) and the several operations indicated are possible. Thus a+b is equivalent to b+a: a-b +c is equivalent to a+c-b or c+a-b or c-b+a*. The same rule is transferred to Symbolical Algebra, without any restriction with respect to the values or signs of the symbols involved, in virtue of the assumptions made in Art. 546. Thus

each other.

Art. 21 and note. Thus if b was greater than a, but less than a+c, the operation, or rather succession of operations, indicated in the expression a−b+c would be impossible, but would cease to be so in the expression a+c-b: it is however convenient, even in Arithmetical Algebra, to consider an expression as representing any value which an interchange of its terms would render it capable of expressing in virtue of this convention, we might consider -b+a, as equivalent to a-b, and recognise its use in Arithmetical Algebra, if a was greater than b.

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a-b is equivalent, in this latter science, to b + a, for all values of a and b:

3a + 5a is equivalent to 5a-3a: a−b+c

is equivalent to a + c

b or c+a-b or c-b+a or b+a+c

or − b+c+a, and similarly in all other cases.

not common to

certained

554. Inasmuch as the results of symbolical addition and The meansubtraction are obtained from an assumed rule of operation, and results of ing of those not from the definition of the operation itself, it will follow symbolical algebra, that their meaning, when capable of being interpreted, must which are be dependent upon the conditions which they are required to satisfy but as the rules for performing these operations and arithmetical algebra, the results obtained are or may be made identical in those must be astwo sciences in all cases which are equally within their province, it is allowable to assume that the operations and their tation. results, within those limits, possess precisely the same meaning: it is only when the results of these rules are not common to Arithmetical Algebra, that it will be found necessary to resort to an interpretation of their meaning, upon principles which we shall proceed to establish and to exemplify in the case of the operations which are under consideration.

by interpre

for the in

555. The addition of a symbol preceded by a negative Conditions sign is equivalent to the subtraction of the same symbol pre- terpretation ceded by a positive sign and conversely.

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It appears, therefore, that in the case of negative symbols, the operation of addition is no longer associated with the fundamental idea of increase, nor that of subtraction with that of decrease: and thus a change of sign from plus to minus, in the symbol operated upon, is equivalent to a change of operation from addition to subtraction and conversely.

of positive and negative symbols.

556. The signs plus and minus, when prefixed to symbols The signs denoting quantities of the same kind, cannot denote modifica- used inde

-

Thus if a be greater than b, the symbolical result b+a is convertible into a-b, which is, under this form, a result of arithmetical algebra.

+ and

pendently, can symbolize convertible af

† These formulæ express the rule for the concurrence of like and unlike signs fections of in symbolical addition and subtraction, which' is as follows: "when two unlike magnitude only. signs come together, they are replaced by the single sign; and when two like signs come together, they are replaced by the single sign +."

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Possible and impossible quan

tities.

Interpretation of the

- in the

case of

tions of magnitude, but only such affections or qualities of the magnitudes represented, as are convertible by the operations of addition and subtraction: it is on this account that a can admit of no interpretation, as compared with a or +a, when a denotes an abstract number, to which no qualities are attributed.

557. Quantities and their symbols are said to be real or possible, when they can be shewn to correspond to real or possible existences: in all other cases, they are said to be unreal, impossible or imaginary. It will follow, therefore, that when positive symbols represent real quantities, the same symbols with a negative sign will be said to be impossible or imaginary, whenever they are not capable of an interpretation, which is consistent with the conditions they are required to satisfy. It remains to shew that there exist large classes of magnitudes which possess qualities which can be correctly symbolized by the signs + and -, and that consequently the terms negative and impossible + are not coextensive in their application.

558. Our first example of the existence of qualities of signs + and magnitudes which can be thus symbolized will be in expressing the opposite directions of lines in geometry, and which will symbols re- be found to constitute one of the most extensive applications presenting lines in of Symbolical Algebra: the discussion of the following problem geometry. will form the most simple introduction to this most important theory.

Problem.

Its solution by the principles of

Arithmeti

cal Alge

bra.

"A traveller moves southwards for a miles, and then returns northwards for 6 miles: what is his final distance from the point of departure ?"

Let A be the point of departure and AB the distance, expressed in magnitude by the symbol a, to

A

N

S

B

which he travels southwards: let BC be the distance, expressed in magnitude by the symbol b, through which he returns north

1

For if a and b continue to denote magnitudes of the same kind, they may be replaced by the ordinary symbols of Arithmetical Algebra, such as c and d, when c+da+(-b) = a−b is always greater than c-d or a-(-b) or a+b, results which are contradictory to each other.

† So numerous are the cases in which negative quantities admit of a consistent interpretation, that the term impossible has never been applied to them: it has been uniformly applied however to a second class of symbolical quantities, though not, as will be hereafter shewn, with perfect propriety.

wards: his final distance AC, when he stops, from the point of departure, is the excess of AB above AC, and is correctly represented by a -b: or if we suppose c to represent the distance AC, then we have

a-b= c.

As long as AB is greater than BC, or a is greater than b, the traveller continues on the same side of the point of departure A, and the solution of the problem is strictly within the limits of Arithmetical Algebra: but if we suppose the traveller to return farther northwards than he went, in the first instance, southwards, or AB to be less

than BC, then his distance C

B

AC to the north of the point of departure A, is not capable of being represented in Arithmetical Algebra by a-b, since a is less than b, and the operation thus indicated is impossible: but inasmuch as, in this case, AC the final distance of the traveller to the north of A is the excess of BC above AB, or of b above a, it will be correctly represented in Arithmetical Algebra by b-a: and if b = a+c, we shall have

b- a = c.

tinct cases

of

this problem when solved by the principles of

There are therefore two distinct cases of this problem, when Two dissolved by the principles of Arithmetical Algebra, according as the traveller stops on the south or on the north of the point of departure: and it will be observed that the solution obtained in each case expresses the absolute magnitude of the final distance only, and not its quality or affection, whether south or cal Alge

north.

Arithmeti

bra.

trical solu

In the geometrical solution of the problem, the distances are Its geomeexpressed and exhibited to the eye, both in quality and mag-trical nitude: and there is no such interruption of continuity in passing from south to north of the point of departure, or, in other words, through the zero point, as occurs in the solution of the problem by the principles of Arithmetical Algebra.

by the principles of

Algebra.

Let us next consider the solution of this problem by the prin- Its solution ciples of Symbolical Algebra. Denoting AB by a and BC by b, when AB is greater than Symbolical BC, the distance AC from the point of departure is denoted by a-b: and inasmuch as the operation denoted by - is assumed to be possible for all relations of value of a and b (Art. 546), we may suppose b equal to a + c as well as to a-c, or to be greater as well as less than a: in one case we get, when bac,

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