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2X1X28

2300+2x30=1260)4167

27

1200X3-3600

65X3X3 540

3x3x3

4167

12167(23 root but the value of 8 is 8000, and the 2 is 20, that is, 8000 feet of the stone will make a pile measuring 20 feet on each side, and (12167– 8000) 4167 feet remain to be added to this pile in such a manner as to continue it in the form of a cube. Now it is obvious that the addition must be made upon 3 sides; and each side being 20 feet square, the surface upon which the additions must be made will be (20×20×3=2×2×300) 1200 feet, but when these additions are made, there will evidently be three deficiencies along the lines where these additions come together, 20 feet long, or (20×3=2×30) 60 feet, which must be filled in order to continue the pile in a cubic form. Thus the points upon which the additions are to be made, are (1200+60) 1260 feet and 4167 feet, the quantity to be added divided by 1260, the quotient is (4167-1260) 3, which is the thickness of the additions, or the other figure of the root. Now if we multiply the surface of the three sides by the thickness of the additions, the product (1200×3), 3600 feet, is the quantity of stone required for those additions. Then to find how much it takes to fill the deficiencies along the line where these additions come together, since the thickness of the additions upon the sides is 3 feet, the additions here will be 3 feet square, and 60 feet long, and the quantity of stone added will be (60×3×3=) 540 feet. But after these additions there will be a deficiency of a cubical form, at the corner, between the ends of the last mentioned additions, the three dimensions of which will be just equal to the thickness of the other additions, or 3 feet, and cubing 3 feet we find (3×3×3=) 27 feet of stone required to fill this corner, and the pile is now in a cubic form, measuring 23 feet on every side, and adding the quantities of the additions upon the sides, the edges, and at the corner together, we find them to amount to (3600+540+27) 4167 feet, just equal to the quantity remaining of the 12167, after taking out 8000. To il lustrate the foregoing operation, make a cubic block of a convenient size to represent the greatest cube in the left hand period. Make 3 other square blocks, each equal to the side of the cube, and of an indefinite thickness, to represent the additions upon the three sides, then 3 other blocks, each equal in length to the sides of the cube, and their other dimensions equal to the thickness of the square blocks, to represent the additions along the edges of the cube, and a small cubic block with its dimensions, each equal to the thickness of the square blocks, to fill the space at the corner. These, placed together in the manner described in the above operation, will render the reason of each step in the process perfectly clear. The process may be summed up in the following

RULE.

271. 1. Having distinguished the given number into periods, of three figures each, find the greatest cube in the left hand period, and place its root in the quotient. Subtract the cube from the left hand period, and to the remainder bring down the next period for a dividend. Multiply the square of the quotient by 300, calling it the triple square, and the quotient by 30, calling it the triple quotient, and the sum of these call the divisor.

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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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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,

what is the diameter of a bul-
let weighing 9 lbs. ?
72:83:9: 64 Ans. 4 in,
2. A bullet 3 inches in di-
ameter weighs 4 lbs. what is
the weight of a bullet 6 inches
in diameter?

es

Then 27: 4 :: 216.

Ans. 32 lbs. 3. If a ball of silver 12 inch

in diameter be worth $600, what is the worth of another ball, the diametor 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.

QUESTIONS FOR PRACTICE.

1. What is the cube root of 2. What is the fourth root 48228544? of 19987173376 ?

33-27

48228544 (364

Ans. 376.

3. What is the of 191102976?

82×3=27)212 dividend.

363-46656

362X3-3708) 15725 2d div❜d.

3643 48228544

sixth root Ans. 24.

4. What is the seventh root of 3404825447? Ans. 23.

5. What is the fifth root of 307682821106715625?

Ans. 3145.

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?

1626-27 and 27-3; then 6×318, the lesser. 18x354, the greater. Proof, 6 18:54

And

162.

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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

117

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 fig. ure 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 What is the method of

2. What is a cube (61)? What is meant by cubing a number (251)? 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 biquad-divisor? rate, or 4th power of a number? proof? 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 he 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?

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

process.

13. What proportion have solids to one another? How can you find the roots of higher powers (273) ? State the general rule.

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 Arithmetical 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 tern 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 (3T2 +2+2+2=11 Ans.), as the whole number of lemons. Heuce,

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? 59X3-177, 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 5-14; 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 to find 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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