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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.
QUESTIONS FOR PRACTICE. 2. What is the cube root of 5. What is the cube root of 181 5848?
The decimals are obtained by IX1X300=300|1815848 ( 122
annexing ciphers to the remainder, 1x 305 301 Divisor 330
as in the square root, with this dif815 divid. ference, thai 3 instead of 2 are an
nexed each time.
6. What is the cube root of 23= 8
27054036008 ? Ans. 3002.
7. What is the cube root of
13 Ans. 360X22= 1440 23 =
8. What is the cube root of
3. What is the cube root of
* .666666+1 Ans.
9. What is the cube root of 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 3X3X3–27 and 6X6X62216 Ibs. 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 inch72:83 ::9: 64 Ans. 4 in.
in diameter be worth 2. A bullet 3 inches in di- | $600, what is the worth of ameter weighs 4 lbs. what is another ball, the diameter of the weight of a bullet 6 inches | 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 extractions of the square and cube root. Thus the 4th rooi 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, ihe Ith root is the cube root of the cube &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? 48228544 (364
Ans. 376. 33—27
3. What is the sixth root
of 191102976? Ans. 24. 32X3-27 )212 dividend.
4. What is the seventh root 363-46656
of 3404825447 ? Ans. 23. 362X3=3708 ) 15725 2d div'd. 5. What is the fifth root of
307682821106715625 ? 3643=48228544
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?
162:6–27 and 327=3; then 6X3–18, the lesser. And 18X3=54, the greater.
Proof, 6 : 18 : : 54 : 102
1. If the length of a line, or any 10. What does extracting the number be multiplied by itself, what square root mean? What is the will 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 (254) ? | period? Why double the root for Why is it called cubing? By what à divisor? In dividing, why omit other name is the operation called ? 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- divisor? What is the method of rate, or 4th power of a number ? | proof? What is the form of a biquadrate ? 11. When there is a remainder,
4. What is a sursolid ? What its how may decimals be obtained in form? What is the squared cube ? the root? How find the root of a What its form? What are the suc- Vulgar Fraction ? cessive forms of the higlier powers
tion have circles to one another ?
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 (68)? What is meant by a mean called ? How would you raise a proportional between two numbers ? number to a given power
How is it sound ? 6. What is Evolution? What is 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 pref- tiply, the triple square by the last erable? Why (262)?
quotient figure ? "Why the triple 8. Has every number a root ? quotient by the square of the 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 quo: which 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 in the continued process. product of a given number of fac- 13. What proportion have solids tors ? What the least? What is to one another? How can you find the inference ? How, then, can you
the roots of higher powers (273) ? ascertain the number of figures of Siate the general rule. which any root will consist ?
MISCELLANEOUS RULES. 1. Arittzctical 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 scries, 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 terın 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, ?, 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, 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- 20X5+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.
277. If I buy 5 lemons, whose prices are in arithmetical progression, the first costing 3 cents, and the last Il cents, what is the common 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 terın, 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, 2. If the first term of a se- 3. A man has 12 sons whose ries be 8, the last 108, and ages are in arithmetical prothe number of terms 21, what gression ; the youngest is 2 is the common difference ? years old, and the oldest 35 ; 108-8-21-1=5 Ans.
what is the common difference
in their ages
Ans. 3 yrs.
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 exiremes divided by the coinmon diíterence, inust give the number of terms, less 1 (11—3=8, and 872–4, and 4+1=) 5 Aus. 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 se- 3. A man on a journey travries be 8, the last 108, and the elled the first day 5 miles, the common difference 5, what is last day 35 miles, and increasthe number of terms ?
ed his travel each day by 3 108—3—5—20, and 20+1= miles; how many days did he 21 Ans.
- 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=1 the difference between 3 and 11 added to 3 is (1143--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 niany times does a 3. Thirteen persons gave common clock strike in 12 presents to a poor man hours ?
arithmetical progression; the 1+12=2X12–78 Ans. first gave 2 cents, the last 26
cents; what did they all give ?