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Seek how often the divisor may be had in the dividend, and place the result in the quotient. Multiply the triple square by the last quotient figure, and write the product under the dividend; multiply the triple quotient by the square of the last quotient figure, and place this product under the last; under these write the cube of the last quotient figure, and call their sum the subtrahend. Subtract the subtrahend from the dividend, and to the remainder bring down the next period for a new dividend, with which proceed as before; and so on, till the whole is finished.

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4. What is the cube root of 9. What is the cube root of 303464448?

Ans. 672.

436036824287? Ans. 7583.

272. Solids of the same form are in proportion to one another as the cubes of their similar sides or diameters.

1. If a bullet, weighing 72 | 3×3×3-27 and 6×6×6=216 lbs. be 8 inches in diameter,

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Then 27 :: 4 :: 216.

Ans..32 lbs. 3. If a ball of silver 12 inchin diameter be worth $600, what is the worth of another ball, the diameter of which is 15 inches ?

Ans. $1171.87+.

EXTRACTION OF ROOTS IN GENERAL.

ANALYSIS.

273. The roots of most of the powers may be found by repeated extractions of the square and cube root. Thus the 4th root is the square root of the square root; the sixth root is the square root of the cube root, the 8th root is the square root of the 4th root, the 9th root is the cube root of the cube root, &c. The roots of high powers are most easily found by logarithms. If the logarithm of a number be divided by the index of its root, the quotient will be the logarithm of the root. The root of any power may likewise be found by the following

RULE.

274. Prepare the given number for extraction by pointing off from the place of units according to the required root. Find the first figure of the root by trial, subtract its power from the first period, and to the remainder bring down the first figure in the next period, and call these the dividend. Involve the root already found to the next inferior power to that which is given, and multiply it by the number denoting the given power for a divisor. Find how many times the divisor may be had in the dividend, and the quotient will be another figure of the root. Involve the whole root to the given power; subtract it from the given number as before, bring down the first figure of the next period to the remainder for a new dividend, to which find a new divisor, and so on till the whole is finished.

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Between two numbers to find two mean proportionals. RULE.-Divide the greater by the less, and extract the cube root of the quotient; multiply the lesser number by this root, and the product will be the lesser mean; multiply this mean by the same root, and the product will be the greater mean.

EXAMPLE.-What are the two mean proportionals between 6 and 162?

And

162÷6-27 and 27—3; then 6X3-18, the lesser. 18x3-54, the greater. Proof, 6: 18:54: 162.

REVIEW.

1. If the length of a line, or any number be multiplied by itself, what will the product be (253)? What is this operation called? What is the length of the line, or the given number, called?

2. What is a cube (61)? What is meant by cubing a number (254) ? Why is it called cubing? By what other name is the operation called? What is the given number called?

3. What is meant by the biquadrate, or 4th power of a number? What is the form of a biquadrate?

4. What is a sursolid? What its form? What is the squared cube? What its form? What are the successive forms of the higher powers (258)?

5. What is the raising of powers called? How would you denote the power of a number? What is the small figure which denotes the power called? How would you raise a number to a given power?

6. What is Evolution? What is meant by the root of a number? What relation have Evolution and Involution to each other?

7. How may the root of a number be denoted? Which method is preferable? Why (262)?

8. Has every number a root? Can the root of all numbers be expressed? What are those called which cannot be fully expressed?

9. What is the greatest number of figures there can be in the continued product of a given number of factors? What the least? What is the inference? How, then, can you ascertain the number of figures of which any root will consist?

10. What does extracting the square root mean? What is the rule? Of what is the square of a number consisting of tens and units made up (266)? Why do you subtract the square of the highest figure in the root from the left hand period? Why double the root for a divisor? In dividing, why omit the right hand figure of the dividend? Why place the quotient figure in the divisor? What is the method of proof?

11. When there is a remainder, how may decimals be obtained in the root? How find the root of a Vulgar Fraction? What propor tion have circles to one another? When two sides of a right angled triangle are given, how would you find the other side? What is the proposition on which this depends (68)? What is meant by a mean proportional between two numbers ? How is it found?

12. What does extracting the cube root mean? What is the rule ? Why do you multiply the square of the quotient by 300? Why the quotient by 30? Why do you multiply the triple square by the last quotient figure ? Why the triple quotient by the square of the last quotient figure? Why do you add to these the cube of the last quotient figure? With what may this rule be illustrated? Explain the

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SECTION IX.

MISCELLANEOUS RULES.

1. Arithmetical Progression.

275. When numbers increase by a common excess, or decrease by a common difference, they are said to be in Arithmet ical Progression. When the numbers increase, as 2, 4, 6, 8, &c., they form an ascending series, and when they decrease, as 8, 6, 4, 2, &c., they form a descending series. The numbers which form the series are called its terms. The first and last term are called the extremes, and the others the means.

276. If I buy 5 lemons, giving for the first, 3 cents, for the second, 5, for the third, 7, and so on with a common difference of 2 cents; what do I give for the last lemon?

Here the common difference, 2, is evidently added to the price of the first lemon, in order to find the price of the last, as many times, less 1 (3+2 +2+2+2 11 Ans.), as the whole number of lemons. Hence,

I. The first term, the number of terms, and the common difference given to find the last term.

RULE. Multiply the number of terms less 1, by the common difference, and to the product add the first term.

2. If I buy 60 yards of cloth, and give for the first yard 5 cents, for the next 8 cents, for the next, 11, and so on, increasing by the common difference, 3 cents, to the last, what do I give for the last yard? 59X3177, and 177+5 182 cts. Ans.

3. If the first term of a series be 8, the number of terms 21, and the common difference 5, what is the last term?

20X5+8=108 Ans.

4. If the first term be 4, the difference 12, and the number of terms 18, what is the last term? Ans. 208.

277. If I buy 5 lemons, whose prices are in arithmetical progression, the first costing 3 cents, and the last 11 cents, what is the common difference in the prices?

Here 11-3-8, and 5-1-4; 8 then is the amount of 4 equal differences, and 4)8(-2, the common difference. Hence,

II. The first term, the last term, and the number of terms given trind the common difference.

RULE.-Divide the difference of the extremes by the number of terms, less 1, and the quotient will be the common difference.

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278. If I give 3 cents for the first lemon, and 11 cents for the last, and the common difference in the prices be 2 cents, how many did I buy?

The difference of the extremes divided by the number of terms, less 1, gives the common difference (277); consequently the difference of the extremes divided by the common difference, must give the number of terms, less 1 (11-3-8, and 8-2-4, and 4+1) 5 Ans. Hence,

III. The first term, the last term, and the common difference given to find the number of terms.

RULE.-Divide the difference of the extremes by the common difference, and the quotient, increased by 1, will be the answer.

2. If the first term of a series be 8, the last 108, and the common difference 5, what is the number of terms?

108-8-5-20, and 20+1

21 Ans.

3. A man on a journey travelled the first day 5 miles, the last day 35 miles, and increased his travel each day by 3 miles; how many days did he travel?

Ans. 11.

279. If I buy 5 lemons, whose prices are in arithmetical progression, giving for the first 3 cents, and for the last 11 cents, what do I give for

the whole ?

The mean, or average price of the lemons will obviously be half way between 3 and 11 cents the difference between 3 and 11 added to 3 is (11-3-2-) 7, and 7, the mean price, multiplied by 5, the number of lemons, equals (7X5) 35 cents, the answer. Therefore,

IV. The first and last term, and the number of terms given to find the sum of the series.

RULE.-Multiply half the sum of the extremes by the number of terms, and the product will be the sum of the series.

2. How many times does a common clock strike in 12 ours?

1+12 2x12-78 Ans.

3. Thirteen persons gave presents to a poor man in arithmetical progression; the first gave 2 cents, the last 26 cents; what did they all give? Ans. $1,82.

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