Surds, or irrational quantities, are such as have no exact root; but are usually denoted by means of the sign of radicality (/) placed before them. Thus 5, or 52 is a surd, being the square root of 5.

$/7, or 1, cube root of 7, is a surd.

, or y2 is a surd, at least, when a2 is not a cabe number.

The most useful problems relating to surds, are the following. PROB. I. To Reduce a Rational Quantity to the Form of a Surd.

Rule. Raise the quantity to a power equivalent to that denoted by the index of the surd; then over this new quantity place the radical ɛign, and it will be of the form required.

Thus, 1. To reduce 3 to the form of the square

root.

First, 3×3=32=9; whence 9 is the answer. :. To reduce 2x3 to the form of the cube root. First, 2x2x2x2x2x2=(2x2)3=8x6;

whence 3/8x or (86) is the answer.

A rational quantity may be put into the form of a surd, by reducing its index to the form of a fraction of the same value.

Thus, a = a2=√√/a2

a2b=e31b3=3/a°b3

PROB. II. To Reduce Quantities of different Indices, to other equivalent ones, that shall have a common Index. Rule 1. Divide the indices of the quantities by the given index, and the quotients will be the new indices for those quantities.

2. Over the said quantities, with their new indices, place the given index, and they will make the equivalent quantities required.

3. A common index may also be found by reducing the indices of the quantities to a common denominator, and involving each of them to the power denoted by its numerator.

1. Reduce 15 and 9 to equivalent quantities having the common index §.

4÷÷4=4×14= the 1st index.
14=1x=4= the 2d index.
(91) 2

Therefore (152)2 and

quired.

are the quantities re

5-3 8+2√15.

=

2

=

=

PROB. IV. To add Surd Quantities together. Rule. 1. Reduce such quantities as have unlike indices to other equivalent ones, having a common index.

2. Bring all fractions to a common denominator, and reduce the quantities to their simplest terms, as in the last problem.

3. Then, if the surd part be the same in them all, annex it to the sum of the rational parts, with the sign of multiplication, and it will give the total sum required.

But if the surd part be not the same in all the quantities, they can only be added by the signs +

and

=3

1. It is required to add 27 and /48 together. First, √27 √9X3 = 31/3; and 48 = /16X3=4/3; whence, 3√3 + 4√/3 = (3 + 4) 3=7/3=sum required.

2. Add together 2 3/a36 and 3 3/64bx®. 23/a3b=23/a3 × 3/b = 2 a 3/b. 33/64bx6=33/64 xo × 3/6 = 24x3 3/b. Therefore, 2 a3/b+24x3 3/6= (2a + 24x3) 3b, the sum.

PROB. V. To find the Difference of Surd Quantities.

Rule. Prepare the quantities as in the last rule, and the difference of the rational parts annexed to the common surd, will give the difference of the surds required.

-.

But if the quantities have no common surd, they can only be subtracted by means of the sign 1. Required to find the difference of 448 and √112.

First, 448=√64x7=8√√7; and √112= √16X74√7.

Whence 8/7-4/7=4/7 the difference re

== the 1st index. the ad index.

PROB. III. To reduce Surds to their most simple

Terms.

Rule. 1. Divide the given surd into factors, one of which at least is a complete power; set the root of this factor before the other factor or factors, with the radical sign between them.

2. When the surd is a fraction whose denominator is not a complete power, multiply both numerator and denominator by such a number as will make the denominator a complete power. Then proceed as above.

1. Reduce 3/108 to its simplest terms. V/108/27X4=2/27X3/4=3×3/4-3/4

2. Find the difference of 8 3/36 and 37a6b. 8 3/236-8 a3/b, and 3/ab=a2 3/b. Therefore the difference is (8 a D a2) 3/b.

PROB. VI. To Multiply Surd Quantities together.

Rule. Reduce the surds to the same index; next multiply the rational quantities together, and the surds together; then the one product annexed to the other will give the whole product required; which may be reduced to its most simple terms by Problem III.

1. Find the product of 3/4 and Here × ×?×?=IVII-IV=M×]=IVH =x}x 15= V15.

Ex. 2. Given 6+z=' √+z+z2, to find z. First, by squaring both sides of the equation, we have, &2 + 2bz + z2 = b2 + z √ a2 + z2

Then by striking the 6 from both sides, results, 2bz+22=% √ a2 + z2.

Dividing by z gives, 2b + z = √ aa + z2.
Squaring both sides, 462+4x+2a == «a + 23.
Striking the 22 from both sides, and transposing
40 gives, 4&z=qa— 4i2.

40

-b.

a2 - 469 02 And, dividing by 4b, z=— 46 Of reducing double, triple, &c. Equations, containing two, three, or more unknown Quan

ties.

FOB. L. To Exterminate trvo unknoron Quantities; or, to Reduce the two Simple Equations containing them, to a Single one.

Rule 1. 1. Observe which of the unknown quantities is the least involved, and find its value in each of the equations, by the methods already

explained.

2. Let the two values, thus found, be made equal to each other, and there will arise a new equation with only one unknown quantity in it, whose value may be found as before.

Rate II. 1. Consider which of the unknown quantities you would fir exterminate, and let its value be found in that equation where it is least involved.

2. Substitute the value, thus found, for its equal in the other equation, and there will arise a new equation, with only one unknown quantity, whose value may be found as before.

Rule II. 1. Let the given equations be multiplied, or divided, by such numbers or quantities as will make the term which contains one of the unknown quantities the same in both equations. 2. Then, by adding or subtracting the equations according as the case may require, there wili arise a new equation, with only one unknown quantity,

as before.

Paos. II. To Exterminate three Unknoron Quantities; er, to Reduce the three Simple Equations, containing

them, to a Single one.

Rule. 1. Let x, y, and z, be the three unknown quantities, to be exterminated.

2. Find the value of x from each of the three given equations

3. Compare the first value of x with the second, and an equation will arise involving only y and z. 4. In like manner, compare the first value of x with the third, and another equation will arise involving only y and z.

5. Find the values of y and z from these two equations, according to the former rules; and x, and Z, will be exterminated as required.

واقع

Nate. Much in the same manner may any number of unknown quantities be exterminated. But there are often shorter methods for performing

practice.

to find x, y, z.

(1), x+y=a (2). x+x=b

(3) y+x=c

Their sum is (4). 2x+2y+2x=a+b+c.
Its half is (5). x + y + z = 4a+13+ $c.
From this take equa. (3). and there remains x
$a+$6-$c.

From (5) take (2), leaves ya— £b + {c.
From (5) take (1), leaves z=— -4a+1b + Sc.

QUEST. I. The paving of a square at 28. a yard, costs as much as the inclosing it at 5s. a yard: required the side of the square?

Let r-side of the square sought, Then 4r=yards of inclosure, And yards of pavement; Hence 41 × 5201 =price of inclosing, And x2 × 2=2rprice of paving, Therefore 22-20, and 10-length of the side But 22-20x by the question, required.

2. A market woman bought in a certain number of eggs at two a penny, and as maay at three a penny, and sold them all out again at the rate of five for two-pence, and by so doing lost fourpence; what number of eggs had she? Let x number of eggs of each sort, . Then will price of the first sort, And 4x price of the second sort;

=

But 5 2 2 (the whole number of eggs): Ar;
Whence r price of both sorts, at 5 for 2 pence,
And 4+4 by the question;
That is x+4x-3x=8;

Or

Or

3x+2x-x=24; 15a+101-24r=120,

Or r=120=number of eggs of each sort required.

3. A person has two horses, and a saddle worth gol. now if the saddle be put on the back of the first horse, it will make his value double that of the second; but if it be put on the back of the second, it will make his value triple that of the

first, what is the value of each horse?

Let the first horse be denoted by a, and the second by y. Then x + 50=2y.

And y+50=3r.
From the first equation, x-2y-co.
This value substituted for x in the second,
gives y + 50=3(2y50)=6y—150:
Consequently 6y+y=50+150;
that is 5y=200, or y=40.
Hence x-2y-50=80—50=30.

SECT. VI. QUADRATIC EQUATIONS. A Simple Quadratic Equation, is that which involves the square of the unknown quantity only,

An affected or adfected quadratic equation, is that which involves the square of the unknowa

quantity in one term, and the first power in another term.

Thus, ab, is a simple quadratic equation; And ar2 + bx=c, is an affected quadratic equa

tion.

The rule for a simple quadratic equation has been given already.

Rule. 1. Transpose all the terms involving the unknown quantity to one side, and the known terms to the other; and so that the term containing the square of the unknown quantity may be positive.

2. If the square of the unknown quantity is multiplied by any coefficient, all the terms of the equation are to be divided by it, so that the coefficient of the square of the unknown quantity may be 1.

3. Add to both sides the square of half the coefficient of the unknown quantity, and the side of the equation involving the unknown quantity will be a complete square.

two roots become equal; but if is less than 6, the quantity under the radical sign becomes negative, and the two roots are impossible.

nitudes abstractly considered, where a contrariety
4. If the equation express the relation of mag-
cannot be supposed to take place, the negative
roots cannot be of use, or rather there are no
is unintelligible, and therefore the square root of
such roots; for then a negative quantity by itself
a positive quantity must be positive only.
Ex. 1.
Here, x2—fx=423—20}=223 by transposition;
Given x2-x+20=42}, to find z.
And 12-4x=44} by multiplying by 2.
Then x2-x+441+1=441 by completing the
square;

Hence x-44-63 by evolution;
Therefore x=6+1=7 the answer.
Given ax2 + bx=e, to find x.

2.

First, x2+

I.

square;

And

x +

tion;

b

20

=

a

= + 4a3

Therefore r±√G

4ac + b2 4.x2

=

by completing the

When the terms of a quadratic of this form a12-bx=-c, are so related that b-c=a, then will the two roots of the equation be unity and

a

Thus, if 4x2-7x=-3, then 1=1 or = ÷

If 7x-23=-16, then x=1, or x=1. And so of others. The demonstration of this useful property is left for the learner's ingenuity.

In many cases the roots of quadratic equations may be more readily obtained by a table of logarithmic sines and tangents, than by any other method: the precepts for this purpose are these:

1. If the relation of three assumed lines a, b, and r be such, that ar-r2=62; then it will be, as a: b radius: the sine of an angle. And as radius: the tangent (or cotangent) of half that angle: : b:

: 2.

2. If the relation of three lines a, b, and 1, be such that aar=62; then it will be, asa: b:: radius: the tangent of an angle. And as radius: the tangent or cotangent of half this angle (ac cording as the sign of ar is positive or negative)

:: b: x.