the cube of the same figure, and take the sum for the subtrahend; subtract the subtrahend from the dividend, and to the remainder bring down the next period for a dividend, with which proceed as with the preceding, so continuing until the whole is completed. NOTE-1. Decimals must be pointed from left to right, by placing a point over the third, sixth, &c., figures in that direction. 2. If the divisor is not contained by the dividend, place a cipher in the quotient, and annex two ciphers to the divisor, and bring down the next period for a dividend, and use the divisor, as thus increased, for finding the next quotient figure. 3. If there is a remainder after finding the integer of the root, annex a period of three ciphers thereto, and proceed for the decimal of the root as if seeking for the integer, annexing a period of three ciphers to each remainder until the decimal is carried to as many places of figures as desired. EXAMPLE.-Required the cube root of 47421875.6324. 47421875.632400 (361.959+. Ans. 92 X 36195 X 30 93 87953850 7293537298622079 64218902921 EXAMPLE. -- Required the cube root of 32768. Also, of 8489664. General Rule for extracting the roots of all powers, or for finding any proposed root of a given number. 1. Point off the given number into periods of as many figures each, counting from right to left, as correspond with the denomination of the root required; that is, if the cube root be required, into periods of three figures, if the fourth root, into periods of four figures, &c. 2. Find the first figure of the root by inspection or trial, and place it at the right of the number, in the form of a quotient; raise this quotient figure to a power corresponding with the denomination of the root sought, and subtract that power from the left hand period, and to the remainder bring down the first figure of the next period, for a dividend. 3. Raise the root thus far found (the quotient figure) to a power next inferior in denomination to that of the root required, multiply this power by the number or index figure of the root required, and take the product for a divisor; find the number of times the divisor is contained in the dividend, and place the result (except that the remainder is large, diminished by one or two units) in the quotient, for the second figure of the root. 4. Raise the root thus far found (now consisting of two figures) to a power corresponding in denomination with the root required, and subtract that power from the two left hand periods, and to the remainder bring down the first figure of the third period, for a dividend; find a new divisor, as before, and so proceed until the whole root is extracted. - EXAMPLE. Required the fifth root of 45435424. 45435424(34. Ans. 35=243. 34 X 5) 2113 345: EXAMPLE.-Required the fifth root of 432040.0354. 432040.03540 (13.4+. Ans. 15=1 14 X 5) 33 135 371293 134 X 5) 607470 13.45 43204003424 116 For instructions touching special cases, see NOTES relative to the extraction of the square root, and to the extraction of the cube root. A series of three or more numbers, increasing or decreasing by equal differences, is called an arithmetical progression. If the numbers progressively increase, the series is called an ascending arithmetical progression; and if they progressively decrease, the series is called a descending arithmetical progression. The numbers forming the series are called the terms of the progression, of which the first and the last are called the extremes, and the others the means. The difference between the consecutive terms, or that quantity by which the numbers respectively increase upon each other, or decrease from each other, is called the common difference. Thus, 3, 5, 7, 9, 11, &c., is an ascending arithmetical progression, and 11, 9, 7, 5, 3, is a descending arithmetical progression. In these progressions, in both instances, 11 and 3 are the extremes, of which 11 is the greater extreme, and 3 is the less extreme. The numbers between these, (9, 7, 5,) are the means. In every arithmetical progression, the sum of the extremes is equal to the sum of any two means that are equally distant from the extremes; and is, therefore, equal to twice the middle term, when the series consists of an odd number of terms. Thus, in the foregoing series, 3+11=5+9=7X2. The greater extreme, the less extreme, the number of terms, the common difference, and the sum of the terms, are called the five properties of an arithmetical progression, of which, any three being given, the other two may be found. Let s represent the sum of the terms. "6 E 66 the greater extreme. the common difference. the number of terms. The extremes of an arithmetical progression and the number of terms being given, to find the sum of the terms. EXAMPLE. -What is the sum of all the even numbers from 2 to inclusive? 100, EXAMPLE. - How many times does the hammer of a common clock strike in 12 hours? Έ (1+12) × 12÷278 times. Ans. (+1)+E+ sum of the terms. (EX 2-n-1 × d) × † n = sum of the terms. (2 e + n − 1 × d) × }; n = sum of the terms. -- The greater extreme, the common difference, and the number of terms of an arithmetical progression being given, to find the less extreme. EXAMPLE. A man travelled 18 days, and every day 3 miles farther than on the preceding; on the last day he travelled 56 miles; how many miles did he travel the first day? 56(18-1X 3) = 5 miles. Ans. ~ (EX2+ d)2-sXdX8+d less extreme, when 2 = (2E+d)-8sd is equal to, or greater than d. (2 E + d)2 - 8s d d less extreme, when { E + d)2 — 8 3 d s = The extremes of an arithmetical progression and the common difference being given, to find the number of terms. E—e÷d+1= number of terms. EXAMPLE. -As a heavy body, falling freely through space, descends 16 feet in the first second of its descent, 483 feet in the next second, 80 in the third second, and so on; how many seeonds had that body been falling, that descended 305 feet in the last second of its descent? 305-16289 ÷ 327=9+1=10 seconds. Ans. No (2ed)2+8s d—d—e÷d+1=number of terms. 2 2s+E+(2 E + d)ɛ — 8 s d +d number of terms when 2 ➡ (2E+ d)2 — 8 s d is equal to, or greater than d. 2 s÷E+(2 E + d)2 — 8 s dd number of terms when |