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12167(23 root but the value of 8 is 8000, and the 21x*x?8.
2 is 20, that is, 8000 feel of the
stone will make a pile measuring 20 3***0072X30=1260)4167 feet on each side, and (12167–
8000–) 4167 feet remain to be add1200X3_3600
ed to this pile in such a manner as 69X3X3= 510
to continue it in the form of a cube. 3X3X3 27
Now it is ubvious that the addition
must be made upon 3 sides; and 4167 each side being 20 feet square, the
surface upon which the additions must be made will be (20X20X32X2X300=) 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 (20X32X30=) 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 thickuess 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 (1200X3=), 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 to gether, 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 (60X3X33) 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 fiad (3X3X3=) 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 (3600T5407273) 4167 feet, just
qual 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 my resent the greatest cube in the left hand period. Make 3 others blocks, each equal 10 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 sqnare blocks, to represent the additicos along the edges of the cube, and a small cubic block with its dimensions, each equal to the thickuess of the square blocks, to fill the space at the corner. These, placed together in the manner deseribed in the above operation, will render the reason of each step in the process perfectly clear. The process may bo summed up in Wie toowing
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, call ing it the triple quotient, and the sum of these call the divisor,
115 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,
QUESTIONS FOR PRACTICE. 2. What is the cube root of 5. What is the cube root of 181 5848?
The decimals are obtained by IX!X300—3001815848(122
annexing ciphers to the remainder, 1x 30 30
as in the square root, with this difDivisor 330 815 divid. ference, thai 3 instead of 2 are an
nexed each time. .
6. What is the cube root of 23= 8
27054036008 ? Ans. 3002.
17. What is the cube foot of
23 = 8 18. What is the cube root of
subtra. 878-18 3. What is the cube root of 10648? .
Ans. 22. Ans. 4. What is the cube mot 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 sidles or diameters.
1. If a bullet, weighing 72 | 3X3X3—27 and 6X6X65 216 Hbs. be 8 inches in diameter, Then 27:4:: 216. what is the diameter of a bul- |
Ans. 32 lbs. let weighing 9 lbs. ?
3. If a ball of silver 12 inch 72:83 ::9: 64 Ans. 4 in, es in diameter be worth
2. A bullet 3 inches in di- | $600, what is the worth of ameter weighs 4 lbe. what is another ball, the diametor of the weight of a bullet 6 inches i which is 15 inches ? in diameter ?
EXTRACTION OF ROOTS IN GENERAL.
ANALYSIS. 273. The roots of most of the powers may be found by repeated oxtractions of the square and cube root. Thus the 4th root is the square root uf 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, ihe 9th root is the cube roof of the cube ront, &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 hy 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?
Ans. 376. 33–27
| 3. What is the sixth root 82X3–27) 212 dividend of 191102976? Avg. 24.
4. What is the seventh root 363=46656
of 3404825447 ? Ans. 23. 363 X3=3708 ) 15725 2d div'd. 5. What is the fifth root of
1 307682821106715625 ? 3643=48228544
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 legger number by this roots 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 8 and 162?
1625627 and 27–3; then 6Xat18, the tesser. And 18X3-54, the greater.
Proof, 6 : 18 ; : 54 : 162
1. If the length of a line, or any | 10. What does extracting the Dumber he multiplied by itself, what square root mean? What is the wilt the product be (253)? What rule? Of what is the square of a is this operation called ?' What is number consisting of tens and units the length of the line, or the given made up (266)? Why do you subnumber, called ?
tract the 'square of the highest fig, 2. What is a cube (61)? What | ure in the root from the left hand is meant by cubing a number (251) ? | period? Why double the root for Why is it called cubing? By whai la divisor? In dividing, why omit other name is the operation calledt ? | the right hand figure of the dividend ? What is the given number called? Why place the quotient figure in the
3. What is meant by the biquad-vivisor? What is the method of rate, or 4th power of a number ? | proof? What is the forın of a biquadrate ? 11. When there is a remainder,
4. What is a sursolidl ? What its ! how may decimals be obtained in form? What is the squared cube ? the root? How tind the root of a What its form? Whai are the suc- | Vulgar Fraction? What proporcessive forms of the higher powers tion have circles to one another ? (253) ?
| When two sides of a right angled 5. What is the raising of powers | triangle are given, how would you called ? How would you denote the find the other side ? What is the power of a number ? What is the proposition on which this depends, small figure which denotes the power 1 (68)? What is meant by a mean called 3* How would you raise a I proportional hetween two numbers ? number to a given power ?
How is it found ? .. 6. What is Evolution? What is 1 12. What does extracting the cube meant by the root of a number? | root mean? What is the rule? What relation have Evolution and | Why do you multiply the square of Involution to each other ?
the quotient by 300? Why the 7. How may the root of a number quotient by 30? Why do you mulbe denoted ? 'Which method is prel: | tiply the Iriple square by the last erable ? Why (62) ?
quotient figure ? 'Why the triple 3. Has every number a root ? quotient by the sa
he last Can the root of all numbers he ex- ' quotient figure ? Why do you add pressed ? What are those called | to these ihe cube of the last quowhich cannot be fully expressed? tient figure? With what may this
9. What is the greatest number of rule be illustrated ? Explain the figures there can be ir the continued process. prodluct of a given number of fac 13. What proportion have solids iors ? What the least? Whal is to one another?' How can you find the inference ? How, then, can you the roots of higher powers (273) ? ascertain the 'nuinber of ugures of State the general sula, which any root will consist 7
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 cominon 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+=ll Ans.), as the whole number of lemons. Hence,
I. The first lerm, the number of terms, and the common difference given to find the last term.
RULE. Multiply the number of terms less I, by the common difference, and to the product add the first term.
2. If I buy 60 yards of cloth, 1 3. If the first term of a seand give for the first yard 5 ries be 8, the number of terms cents, for the next 8 cents, for 21, and the common difference the next, 11, and so on, in- | 5, what is the last term? creasing by the common differ
2 0X5+8=108 Ans. ence, 3 cents, to the last, what 4. If the first term be 4, the do I give for the last yard ? | difference 12, and the number
59X3=177, and 177+5= of terms 18, what is the last 182 cts. Ans.
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 coinmon difference in the prices ?
Here 11-38, and 5~134; 8 then is the amount of 4 equal differences, and 4)8(=2, the common difference. Hence,
II. The first term, the last terin, and the number of terms given to find the common difference.
ROLE.—Divide the difference of the extremes by the number a terms, less 1, and the quotient will be the common difference,