6. Subtract the subtrahend from the dividend, and to the re: mainder bring down the next period for a new dividend, with which proceed as before ; and so on till the whole is finished. Examples. 1. What is the cube root of 3. What is the cube root of 10648 ? 41.063625? Ans. 3.45 2x2x300=1200 triple square. 4. What is the cube root of 2 ? 2X30= 60 trip. quot. Ans. 1.25 + 10648(22 The decimals are obtained by an2x2x2=8 nexing ciphers to the remainder, as in the square root, with this differ ence, that 3 ipstead of 2 are annex. 1260)2648 dividend. ed each time. Application. similar sides or diameters. 3X3 X3=27 and 6x6x6=210 8x8x8=512 lb. Thus 27 : 4 :: 216 Ans. 32 lbs. 9 3. If a ball of silver 12 inches 72)4608(64 the cube root of which i in diameter be worth $600, what 432 is 4, the Answer. is the worth of another ball, the diameter of which is 15 inches? 288 Ans. $1171.87+ 288 cwt. qr: 4. If a cable 12 inches round 6. There is a cistern which require an anchor of 18 cwt. what contains 8204 solid inches, I demust be the weight of an anchor | mand the side of a cubical box for a 15 inch cable ? which shall contain the same cwt. Ib. quantity. Ans. 14.12+ in. 123 : 18 :: 153 : 35 0 17. Ans. 7. A person wanted a cylin5. The diameter of a legal drick vessel of 3 feet deep, that Winchester bushel is 182 inches, shall hold twice as much as anoand its depth 8 inches ; what ther of 23 inches deep, and 46 inust the diameter of that bushel | inches in diameter; what must be whose depth is 74 inches ? be the diameter of the required Ans. 19.10671. vessel ? Ans. 57.37 in. Between two given numbers to find two mean proportionals. RULE.—Divide the greater by the less, and extract the cube root of the quotient. Multiply the least given number by the root for the lesser, and this product by the same root for the greater of the two numbers sought. Examples. 1. What are the two mean pro 2. What are the two mean proportionals between 2 and 163 portionals between 6 and 162 ? 1642=8 and 8f=2 Ans. 18 and 54: thus 2x2=4 the lesser, and 4x2=8 the greater. Proof. 2 : 4 ::8: 16. 3. TO EXTRACT THE ROOT OF ANY POWER Rule 2. 1. Prepare the given number for extraction by pointing off froin the place of units according the required root. 2. 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. 3. 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. 4. Find how many times the divisor may be had in the dividend, and the quotient will be another figure of the root. 5. 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. Examples. 1. What is the cube root of 2. What is the fourth root of 18228544? 19987173376 ? Ans. 376. QUESTIONS. 1. What is the cube root of a num 10. When there is a remainder, how ber? how do you proceed to find de2. What is a cube? cimal places in the root ? 3. What is its root? 11. What proportion have solids of 4. What is the first step in the rule the same form to one another? for extracting the root? 12. How are two mean proportion5. What is the second step? als between two given num6. What the third ? bers found ? 7. What the fourth ? 13. What is the rule for extracting 8. What the fifth ? the roots of any powers ? 9. What the sixth ? NOTES. 1. Why do you multiply the square of | 4. Why do you multiply the triple the quotient by 300 ? quotient by the square of the last 2. Why the quotient by 30 ? quotient figure ? What is found by multiplying the 5. Why do you add to these the cube triple square by the last quotient of the last quotient figure ? Forene? 127 Arithmetical Progression. A rank of numbers is in Arithmetical Progression, when they increase by common excess, or decrease by a common difference. When the numbers increase, they form an ascending series, and when they decrease, a descending series. Thus, 1, 2, 3, 4, &c. and 3, 6, 9, 12, &c. are ascending series, and 10, 9, 8, 7, &c. and 20, 16, 12, 8, &c. descending series. The terms of the progression are the numbers which form the series. The first term and last term are called the extreme. If any three of the five following things be given, the other two are readily found, viz. the first term, the last term, the number of terms, the common difference, and the sum of all the terms. Problem I. The first term, the last term, and the number of terms given, to find the sum of all the terms. Rule. *-Multiply the sum of the extremes by the number of terms, and half the product will be the answer. Examples. 1. The first term of an arith 2. How many times does a metical progression is 1, the last common clock strike in 12 hours ? term 21, and the number of terms Ans. 78 times. 11 ; what is the sum of the series? 21 last term. 22 22 22 3. Thirteen persons gave their donations to a poor man, in arithmetical progression, the first gave 2 cents, and the last 26; what did the poor man receive ? Ans. $1.82 2)242 121 Ans. * Suppose another series of the same kind with the given one, to be placed under it in an inverse order; then will the sum of every two corresponding terms be the same as that of the first and last; consequently, any one of these gums, multiplied by the number of terms, will give the whole sum of the two series, and half this sum will evidently be the sum of the given series ; thus, 2. 4 6 8 10 given series. - 10 8 6 4 2 the same inverted. 12+12 4:12-412-7:12-12 X 560 and 69=30=2+4+6+8610 Problem 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 coinmon difference. Examples. 1. The extremes are 2 and 2. A man has 12 sons, whose 53, and the number of terms 18; ages are in arithmetical progreswhat is the common difference ? SIOD, the youngest is 2 years old, 53 and the oldest 35 ; what is the 2 common difference in their ages ? 18 Ans. 3 years. Problem III. The first term, the last term, and common difference given, to find the number of terms. Rule.t--Divide the difference of the extremes by the common difference, and the quotient, increased by 1, is the number of terms required. Examples. 1. The extremes are 2 and 53, 2. A man on a journey, travand the common difference 3 ; elled the first day 5 miles; the what is the number of terms ? last day 35 miles, and increased 53-2=51 and 3)51(17 17 his travel each day by 3 miles; 3 Ans. 11 days. QUESTIONS. 1. When is a rank of numbers in 5. What are the terms of a progresArithmetical Progression ? sion ? 2. What is meant by an ascending 6. What is the first problem ? series? 7. What the rule ?, 3. What by a descending ? 8. What the second ? &c. 4. What are the extremes ? * The difference of the first and last terms evidently shows the increase of the first term, by all the subsequent additions, till it becomes equal to the last ; and as the number of those additions is evidently one less than the number of terms, and the increase by every addition equal, it is plain that the total increase, divided by the number of additions, will give the difference at every one separately; whence the rule is manifest. t By Problem II. the difference of the extremee, divided by the number of terms, less 1, gives the common difference ; consequently, the same divided by the common difference, must give the number of terms less 1 ; hence this quofrent, increased by 1, must be the answer to the questioni |