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3. Required the fifth root of x3—5x1y+10x3y2— 10x2y3+5xy1——y3.

x5—5x1y+10x3y2—10x2y3+5xy1—y3 (x—y

x5

5x1)-5x1y

x-5x1y+10x3y2-10x2y3+5xy'—y3—(x—y)3

EXAMPLES FOR PRACTICE.

1. Required the cube root of x3+6x2y+12xy2+8y3.

Ans. x+2y.

2. Required the fourth root of y-4y3+6y3—4y+1.

Ans. y-1.

3. Required the fifth root of x2+5x1+10x2+10x2 +5 +1.

Ans, x+1.

4. Required the fourth root of x-4xy3+6x3yŝ —4x3y+y+

Ans. x-y

RATIOS.

RATIO is the relation which one quantity bears to another in respect of magnitude, the comparison being made by considering what multiple, part, or parts, one quantity is of the other. The first quantity being called the antecedent, and the latter the consequent. Thus, the ratio of 2 to 1 is double ratio, that of 3 to 1 triple, &c.

The two quantities that are compared are called the terms of the ratio, as 6 and 2; the first of these, 6, being called the antecedent, and the latter, 2, the consequent. Also, the index, or exponent of the ratio, is the quotient of the two terms, so the index of the ratio of 6 to 2 is 6-2 or 3, which is called triple ratio.

When two terms of a ratio are equal, the ratio is that of equality; as of 3 to 3, whose index or quotient is 1, denoting the single or equal ratio. But when the terms are not equal, as of 6 to 2, it is a ratio of inequality.

When the antecedent is the greater term, as in 6 to 2, it is said to be a ratio of greater inequality; but when the antecedent is the less term, as in the ratio of 2 to 6, it is said to be a ratio of less equality.

COMPOUND RATIO.

COMPOUND RATIO is that which is made up of two or more other ratios, by multiplying the exponents together, and so producing the compound ratio of the product of all the antecedents to the product of all the consequents. Thus, the compound ratio of 5 to 3, and 7 to 4, is the ratio of 35 to 12.

If a ratio be compounded of two equal ratios, it is called the duplicate ratio; if of three equal ratios, the triplicate ratio; if of four equal ratios, the quadruplicate, &c. The simple ratio of 3 to 2 are thus :the duplicate ratio 9 to 4, the triplicate ratio 27 to 8, the quadruplicate ratio 81 to 16, &c.

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Ratio is sometimes confounded with proportion, but very improperly, as being quite different things; for proportion is the similitude, or equality, or identity of two ratios. So the ratio of 6 to 2 is the same as that of 3 to 1, and the ratio of 15 to 5 is that of 3 to 1 also; therefore the ratio of 6 to 2 is similar or equal, or the same with that of 15 to 5, which constitutes proportion, and is thus expressed, as 6 is to 2, so is 15 to 5, or thus, 6:2::15; 5. So that ratio exists between two terms, but proportion between two ratios, or four terms.

PROPORTIONAL THEOREMS,

THE demonstrations of which may be found in the fifth book of Euclid.

1. Four quantities are proportionals, when the first is the same multiple of the second, that the third is of the fourth. Thus, if a, b, c, and d, be the four proportionals, they are placed thus, a bed, and are read, a is to b as c to d; therefore =

a

b

с

d

2. When four quantities are proportionals, the product of the means is equal to the product of the extremes; that is, if a: b: cd, then will ad = bc. Also, if a b bc, then will ac = b2. Whence it follows, that if the product of two quantities be equal to the product of two other quantities, the four quantities will be proportionals, thus, ad be, and this is turned into a proportion by making the terms of one product the extremes, and the terms of the other product the means.

3. If four quantities be proportionals, the sum of the first and second is to the second, as the sum of the third and fourth is to the fourth. Thus, if a:b::c:d, then will a+b;b::c+d;d.

4. If four quantities be proportionals, the difference between the first and second is to the second, as the difference between the third and fourth is to the fourth. Thus, ifa:b::c:d; then will a-b:b::c-d:d.

5. If four quantities be proportionals, the sum of the first and second is to their difference, as the sum of the third and fourth is to their difference. Thus, if a:b::c:d, then will a+b;a-b::c+d;c-d.

6. If four quantities be proportionals, the difference between the first and second is to their sum, as the difference between the third and fourth is to their sum; that is, a -ba+b::c―d: c+d.

7. If four quantities be proportionals, they are proportionals when taken alternately. Thus, if a:b::c:d, then will bad: c.

8. If a number of quantities be proportionals, the antecedent is to its consequent, as the sum of all the antecedents is to the sum of all the consequents. Thus, if ab::c:d::x:y::r:s, then will a:b::a+c+ x+r:b+d+y+s.

9. If four quantities be proportionals, the first and second may be multiplied or divided by any quantity, as well as the third and fourth, and the resulting quantities will retain the same ratio. Thus, if a t

abcd, then will

r r

:: cxdx.

10. When four quantities are proportionals, the first and third may be multiplied or divided by any quantity, as well as the second and fourth, and the resulting quantities will be proportionals. Thus, if

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11. If the corresponding terms of two or more ranks of proportionals be multiplied together, their products will be proportionals.

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Then will aei bfj:: cgk :: dhl.

12. When four quantities are proportionals, the same roots or powers of these quantities will be proportionals. Thus, if a bed,

Then will ab*::*: dor/a:√/b :: No√d

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REDUCTION OF EQUATIONS.

DEFINITIONS.

1. When two quantities are connected together by the sign of equality, the whole is called an equation. This equation is a proposition, and denotes the equality of two quantities or things expressed algebraically.

2. A simple equation is one which, when reduced into its simplest form, contains only the first power of the unknown quantity. Thus x+5=9 is an equation, expressing, that if 5 be added to x, the sum will be equal to 9. In this case, a+5 being equal to 9, the value of x must be 4.

Also, if x-7=0, this denotes, that if 7 be subtracted from a, nothing will remain; therefore, the positive part of the expression, which is x, is equal to the negative part 7.

Known quantities are represented by numbers, or the leading letters of the alphabet, and the unknown quantities by x, y, and z, or any other letter.

3. The resolution of equations is finding some quantity or quantities, from the given quantities, that will answer the conditions of the question.

4. Five axioms deduced from Euclid's Elements.

1. If equal quantities be added to equal quanties, their sums will be equal.

2. If equal quantities be subtracted from equal quantities, their remainders will be equal.— Cor. Hence, if the same quantity be added to, and subtracted from any given quantity, its value will not be altered.

3. If equal quantities be multiplied by equal quantities, their products will be equal.

4. If equal quantities be divided by equal quantities, their quotients will be equal.

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