and y =

65 or 4 xy

(3).

CC + y = d.........

(3.), and 2 way = VÕ or 4 xy = b............... (4.) From (3) and (4) we easily find x } (a + Na” – b),

} (a - Nã

Na2 - 6). Hence, from (1), the square root required isV} (a + Nav - ) + 1} (a - Na - b).

b NOTE.-It is evident that, unless (a? 6) is a perfect square, our result is more complicated than the original expression, and therefore the above method fails in that case.

Ex. 1. Find the square root of 14 + 615.
Let W14 + 6.75 W X + N Y..........

.... (1.) Squaring, then, 14 + 65 = x + y + 3xy.

+
Hence, equating the rational and irrational parts-

2C + Y
14.......

.. (2.),

180.......... From (2) and (3) we easily find x = 9, y = 5.

5. Hence the square root required is 79 + 15 or 3 + 15. Ex. 2. Find the square root of 39 + 1496.

Let 39 + 7 1496 dx + dý. Squaring, &c., we have, a + y = 39;

1496. From these equations we easily find a =

17. Hence, the square root required is 122 + 1 17.

21. The square roots of quantities of this kind may often be found by inspection.

Ex. 1. Find the square root of 19 + 8 13.

We shall throw this expression into the form a2 + 2 ab + bo, which we know is a perfect square.

Dividing the irrational term by 2, we have 4 W3. Now all we have to do is to break this up into two such factors that the sum of their squares shall be 19. The factors are evidently 4 and 13.

and 4 xy

22, y

Thus, we have 19 + 8N3 = (4)2 + 2 (4) N3 + (13)2

= (4 + 232 The

square root is therefore 4 + 13. Ex. 2. Find the square root of 29 + 1215. We have 29 + 12 V5 = (3)2 + 2 (3)2 15 + (2 5)

(3 + 2 55)? The square root is therefore 3 + 2 15.

+

15.

.

+

+
1

*I* (c + a)2 (23 a)
(2

3

3

فروت» .17

+ 9

m

Find the value of 16. V12 + 148 – 2/3, 156 + 7189. Va*b*a* +

laxc N 6

b 18. 927 " +673 8a+m+9 + 3 264 a"

Multiply19. a + Nab + b by Nã - Nb, až + b} by Vă - . 20. (x + y)} by (x + y)}, a + b Vd by a? - ab Vã + b'd.

+ a

ca 21. cd

by Na

a (a + b) VÕ3. 22. a: + b3 + c+ dby aš

l 63 c. dt. Divide23. a* + xy + y by x + ały + y. 24. 23 - y by wł + y$.

až Rationalize the denominators of

5 4

a

с

d

+

+

3
4

1
26.
2 + 13' 372
3 273' 75

V3 3

2

1/3 + 12 27. 1 + 12 + 73' 72 + 3 + 5 13 - 2

V2 1

6 28.

x 7' 32 + 13° x + ałyk + y Find the square roots of — 29. 11 + 4 27, 8 + 2 Vī5, 30 - 10 15. 30. 8 + 2 12, 9 - 6 w2, 20 – 10 13:

a

CHAPTER V.

RATIO AND PROPORTION.

a

a + m

CL

Ratio. 22. The student is referred to Chapter II. of the Arithmetic section of this work for definitions and observations . which need not be repeated here.

23. A ratio of greater inequality is diminished, and a ratio of less inequality is increased, by increasing the terms of the ratio by the same quantity. Let a:b or be the ratio, and let each of its terms be 6

a + m increased by m. It will then become

b + me Now, b + m

as (a + m) b = (b + m) d,

7 or, as ab + bm + ab + am; or, as om z am, or as 6 z d. Hence the ratio is increased when b sid, that is, wheni

b it is a ratio of less inequality; and is diminished when b < a, that is, when it is a ratio of greater inequality.

Cor. It may be shown in the same way that

A ratio of greater inequality is increased, and a ratio of less inequality is diminished, by diminishing the terms of the ratio by the same quantity.

24. When the difference between the antecedent and consequent is small compared with either, the ratio of the higher powers of the terms is found by doubling, trebling, &c., their difference.

at a Let a t X : or

be the ratio, where x is small compared with a. (a + x)" a + 2 ani + 2

2 x Then

1 aa a + 2 x

nearly. a

a

a

+

nearly =

a

330 =lt

=

(a + c)3 ai + 3 asx + 3 ax + 2013 a

a a + 3 C

nearly; and so on.
Ex. (1002)2 : (1000) = 1004 : 1000 nearly.

2
(1002) : (1000)3 = 1006 : 1000 nearly.

a

=

Proportion. 25. Proportion, as has been already said, is the relation of equality expressed between ratios. Thus, the expression a : b = c:d,

, or a:b::c:d,

a

с

or

с

a

с

(1.) Since

às then

a or

=

63

с

a

с

or

a

b ď is called a proportion. 26. The following results are easily obtained :

6

6
then o x
Х

Х
6

6
à

'
.. a:0::6:d (alternando).

6 d (2.) 1 - = 1;

d

c' .. b:a:: d:c (invertendo). Also, by Art. 64, page 214, we have (3.) a + 6:6:: c + d:d (componendo). (4.) a -6:6:: 0 - d:d (dividendo).

b (5.) a - b:a:: c- d:c (convertendo).

dic (6.) a + b:a b:: C + d :C d (componendo and

dividendo). 27. If a:b::c:d and e:f::g:h, we may compound the proportions. Thus we have

9
(1), and = :(2).
6

h

cg (1) x (2), then,

dh' or ae : bf :: cg : dh.