95. The words greater and less are often used in Algebra in an extended sense. We say that a is greater than b or that b is less than a if b is a positive quantity. This is consistent with ordinary language when a and b are themselves both positive, and it is found convenient to extend the meaning of the words greater and less so that this definition may also hold when a or b is negative, or when both are negative. Thus, for example, in algebraical language 1 is greater than 2 and 3. 2 is greater than

96. Before leaving this part of the subject we may make a few general remarks. The subject of Algebra has been divided by some modern writers into two parts, which they have called Arithmetical Algebra and Symbolical Algebra. In Arithmetical Algebra symbols are used to denote the numbers and the operations which occur in Arithmetic. Here, as shewn in the preceding chapters of the present work, we begin by defining our symbols, and then arrive at certain results, as for example, at the result (a+b) (a - b) = a - b3. In Symbolical Algebra we assume that the rules of Arithmetical Algebra hold universally, and then determine what must be denoted by the symbols and the operations, in order to ensure this result. Thus we may consider, that in the present chapter we have been examining what meanings must be given to the symbols to make the results of the previous chapters hold universally. And we have thus been led to the theory of negative quantities, and to an extension of the meaning of the words addition, subtraction, multiplication and division.

97. In some of the older. works on Algebra, scarcely any reference is made to the extensions of meaning which we have given to some simple arithmetical terms. In such works the proofs and investigations are only valid so long as the symbols have purely arithmetical meanings; and the proofs and investigations are really assumed without demonstration to hold when the symbols have not purely arithmetical meanings. In recent works, as in the present, an attempt is made to establish the proofs completely. It must not however be denied that this branch of

the subject presents considerable difficulty to the beginner, and it will probably only be after repeated examination of the subject that the student will obtain a conviction of the universal truth of the fundamental theorems.

The student is recommended to proceed onwards as far as the chapter on equations; he will there see some further remarks on negative quantities, and he may afterwards read the present chapter again. It would be inconsistent with the plan of this work to enter very largely on this branch of Algebra; but the present chapter may furnish an outline which the student can fill up by his future reading and reflection.

We shall require in the course of the work certain propositions which are obvious axioms in Arithmetic, and which are also true when we give to the terms and symbols their extended meanings.

98. If equal quantities be added to equal quantities, the sums will be equal.

99. If equal quantities be taken from equal quantities, the remainders will be equal.

Thus, for example, if A=pB + C, then by taking C from these equal quantities we have A-C = pB.

100. If equal quantities be multiplied by the same or equal quantities, the products will be equal.

Thus too if ab then a"-b" and "/a/b.

=

101. If equal quantities be divided by the same or equal quantities, the quotients will be equal.

102. If the same quantity be added to and subtracted from another, the value of the latter will not be altered.

103. If a quantity be both multiplied and divided by another, its value will not be altered.

104. It is important to draw the attention of the reader to the fact, that these propositions are still true whether the quantities spoken of are positive or negative, and when the terms addition, subtraction, multiplication, and division have their extended meanings. For example, if a=b, and c=d, then ac-bd; this is obvious if all the letters denote positive quantities. Suppose however that c is a negative quantity, so that we may represent it by -y; then d must be a negative quantity, and if we denote it by -8, we have y=8; therefore aybd; therefore - ay=-b8; and thus ac= = bd.

1. Shew that x2+ y2+4x2 + 2xy + 8xz and 4 (x + 2)2 become identical when x and y each = ɑ.

2. If a = 1, b

=

2

3'

x=7 and y=8, find the value of

5 (a − b) 3/ {(a + x) y3} − b √ {(a + x) y} + a.

5

3. If a = b x=5 and y=2'

7'

=

(10a +20b){(x-b) y} - 3a /{y2 (x —b)}+5b.

5. Substitute y+3 for x in x-x3 + 2x2 - 3 and arrange

result.

6. Prove that

the

{(a − b)2 + (b − c)2 + (c — a)3}3 = 2 {(a — b)* + (b − c)* + (c − a)1}.

7. If 28=a+b+c, shew that

2 (8-a) (8-b) (8-c) + a (s—b) (s—c) + b (s—c) (8-a)

8. Prove that

+ c (s− a) (s—b) = abc.

(a + b + c)* − (b + c)* − (c + a)* − (a + b)*+ a* + b* + c*= 12abc (a+b+c).

-

( s − a ̧ )2 + ( s − a ) 3 + ... + (s — a„ )2 = a, ̧3 + a ̧3 +.... +a„3.

10. If 2s = a+b+c and 2o3 = a2 + b2 + c3, shew that

(∞3 — a3) (σ3 — b3) + (∞3 − b3) (σ3 — c3) + (o3 — c3) (o3 — a2)

=

= 4s (s — a) (8 — b) (8 — c).

VI. GREATEST COMMON MEASURE.

105. In Arithmetic the greatest common measure of two or more whole numbers is the greatest number which will divide each of them without remainder. The term is also used in Algebra, and its meaning in this subject will be understood from the following definition of the greatest common measure of two or more Algebraical expressions. Let two or more Algebraical expressions be arranged according to descending powers of some common letter; then the factor of highest dimensions in that letter which divides each of these expressions without remainder is called their greatest

common measure.

106. The term greatest common measure is not very appropriate in Algebra, because the words greater and less are seldom applicable to Algebraical expressions in which specific numerical values have not been assigned to the various letters which occur. It would be better to speak of the highest common divisor or of the highest common measure; but in conformity with established usage we retain the term greatest common measure. The letters G. C. M. will often be used for shortness instead of this term.

When one expression divides two or more expressions without remainders we shall say that it is a common measure of them, or more briefly, that it is a measure of them.

107. The following is the rule for finding the G. C. M. of two Algebraical expressions:

Let A and B denote the two expressions; let them be arranged according to descending powers of some common letter, and suppose the index of the highest power of that letter in A not less than the index of the highest power of that letter in B. Divide A by B; then make the remainder a divisor and B the dividend. Again, make the new remainder a divisor and the preceding divisor the dividend. Proceed in this way until there is no remainder; then the last divisor is the G. C. M. required.

108. Example: find the G. C. M. of

x2-6x+8 and 4x3- 21x2 + 15x + 20. x2-6x+8) 4x-21x+15x+20 (4x+3 4x3-24x2+32x

109. The truth of the rule given in Art. 107 depends upon the following principles :

(1) If P divide A, then it will divide mA. For since P divides A, we may suppose A = aP, then m4=maP, thus P divides mA.

For

(2) If P divide A and B, then it will divide mA ±nB. since P divides A and B, we may suppose A= aP, and B=bP,

then ma±nB = (ma±nb) P; thus P divides mA ±nB.

We can now prove the rule given in Art. 107.