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. Solids of the same form are in proportion to one another as the cubes of their similar sides or diameters. 4. If a cable 12 inches round require an anchor of 18 cwt. what 6. There is a cistern which contains 8204 solid inches, I de must be the weight of an anchormand the side of a cubical box for a 15 inch cable ? which shall contain the same quantity. Ans. 14.12+ in. cwt. cwt. qr. lb. 123 18:153: 35 0 17 Ans. 5. The diameter of a legal Winchester bushel is 18 inches, and its depth 8 inches; what must the diameter of that bushel be whose depth is 71⁄2 inches? Ans. 19.10671. 7. A person wanted a cylindrick vessel of 3 feet deep, that shall hold twice as much as another of 23 inches deep, and 46 inches in diameter; what must be the diameter of the required 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 proportionals between 2 and 16? 162-8 and 8-2 2. What are the two mean proportionals between 6 and 162? Ans. 18 and 54: 3. TO EXTRACT THE ROOT OF ANY POWER. Rule 2. 1. Prepare the given number for extraction by pointing off from 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. 1. Why do you multiply the square of | 4. Why do you multiply_the_triple the quotient by 300 ? 2. Why the quotient by 30? What is found by multiplying the 5. triple square by the last quotient fore? quotient by the square of the last quotient figure? Why do you add to these the cube of the last quotient figure ? SECTION II. 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. *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 sums, 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, 24. 4 6 8 10 given series. 12+12+12+12+12-12 X 560 and 60-30—2+4+6+8+10 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 common 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?sion, the youngest is 2 years old," and the oldest 35; what is the common difference in their ages? Ans. 3 years. 53 18 1 17 17)51(3 Ans. 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. 2. 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 days. QUESTIONS. 1. When is a rank of numbers in Arithmetical Progression? 2. What is meant by an ascending series? 3. What by a descending? 4. What are the extremes ? 5. What are the terms of a progres- 6. What is the first problem? 8. What the second? &c. *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. + By Problem 11. the difference of the extremes, divided by the number of terms, less 1, gives the common difference; consequently, the same divided by hence this quothe common difference, must give the number of terms less 1; frent, increased by 1, must be the answer to the question. |