3. For greater exactness, call the root last found the assumed root, and repeat the operation. Or, Let N represent the number to be extracted, r the nearest rout, to be found by repeated trials. being the same as the irrational formula of Dr. Halley. The last theorem is the same as Birks' rule. Xr the cube root of N ; that is, n+2r3: 2N+r3 :: r : N; being the same as the rational formula of Dr. Halley. As these algebraical theorems or rules, are all exactly the same in principle, the learner may use that which he conceives to be most convenient. But in the application of any one of them, the operation will, in general, be shorter if you find the root by Rule I. to three places of figures, instead of finding it by repeated trials. Prop. 2. To extract the cube root of a vulgar fraction. Rule. Reduce the fraction to its lowest terms, then extract the cube root of the numerator for a new numerator, and the cube root of the denominator for a new denominator; but, if the terms will not extract even, multiply the numerator by the square of the denominator, and the cube root of the product, divided by the denominator will give the root required.—Or reduce the fraction to a decimal, and then extract the root. Examples to Prop. 1, Rule 1. (1.) Extract the cube root of 48627·125. (2.) Required the cube root of 122615327232. (6.) What is the cube root of 254358061056000 ? (10.) Required the cube root of 517.375475. (11.) Extract the cube root of 20874107909304, (12.) Extract the cube root of 1551328-215978515625. Examples to Rule 2. (13.) Extract the cube root of 98003·449 to 6 places of decimals. 98 003 1419 (4.61 assumed root.. 4X4×4=64 2x300 4800 divisor. 34003 dividend. 33336 subtrahend. 636181 subtrahend. Secondly. The cube of 46·1=97972∙181, the cube of the assumed root. 293947-811: 293979·979 :: 46·1: 46·104937, &c. the root sought. By cubing 46-104937, and repeating the latter part of the operation, the root may be obtained, truly, to nearly 20 figures. (14.) What is the cube root of 7154-10916753 ? (15.) Extract the cube root of 8302348000000 to four places of decimals. (16.) Extract the cube root of 2 to eleven places of decimals, (17.) Extract the cube root of 0001357 to ten places of decimals. (18.) Extract the cube root of 13.6' to nine places of decimals. (19.) Extract the cube root of 92398647506217 to four places of decimals. First Examples to Prop. 2. (20.) Extract the cube root of 243. ; the cube root of 27 is 3, and that of 64 is 4, there fore the cube root of is 2. (21.) Extract the cube root of 3. 2x3 squared-18, the cube root of which (according to the preceding examples) is 2.620741, &c. This, divided by 3, gives 87358, &c. for the cube root of; or 3=66666, &c. the cube root of which is 87358, &c. as before. (22.) What is the cube root of 17? CLASS II. 373 The six following questions depend upon the 33rd proposition of the XIth book of Euclid, and the 12th and 18th propositions of the XIIth book; or, the 9th, 19th, and 21st propositions of the Xth book of Keith's Geometry; where it is demonstrated, that all solid bodies are in proportion to each other as the cubes of their similar sides, diameters, lines, &c. (27.) If the diameter of a globe be 1 inch, its solidity will be 5236 inch; what will be the solidity of a globe of 15 inches diameter ? (28.) The solid content of a block of marble is 31185 inches; what will be the side of a cubical piece of equal solidity? (29.) A malster agreed with a carpenter to make him a cubical bin, to hold 60 quarters of barley; what will be the internal length of one of its sides, 2150.42 cubic inches being a Winchester bushel ? (30.) If a stone, 20 inches long, 15 inches broad, and 8 inches thick, weigh 2171b., what will be the length, breadth, and thickness, of a similar stone that weighs 9000lb. ? (31.) Admit the length of a ship's keel to be 125 feet, the breadth of the mid-ship beam 25 feet, and the depth of the hold 15 feet; required the dimensions of two other ships, of a similar construction, the one to carry 3 times, the other, the burthen of that given above? (32.) A sugar-loaf, in the form of a cone, the perpendicular height whereof is 20 inches, is to be divided into 3 equal parts; what will be the perpendicular height of each part? (33.) It is required to find two mean proportionals between 4 and 108; or, which is the same thing, there are four numbers in geometrical progression, the first term is 4 and the last 108, what are the two middle terms? (34.) There are seven numbers in geometrical progression, the first is 9, and the seventh 36864, what are all the intermediate terms; or, which is the same thing, find five mean proportionals between 9 and 36864. TO EXTRACT ANY ROOT OF A POWER. Rule I. Point the root into periods as the question requires. Find the nearest root to the first period, and subtract its power therefrom; to the remainder bring down the first figure in the next period for a dividend. Involve the root to the next lower power than the given one, and multiply it by the index of the given power for a divisor, the quotient is the next figure in the root. Then involve the whole root as before, and subtract. Repeat the ope ration till all the figures are brought down. 2. When the index of the power to be extracted is a composite number, the work may be performed more concisely than by this general rule. Indeed, rules of this kind will never be made use of, except by those who have not acquired such a knowledge of the mathematics as will enable them to make use of better methods. Thus, the square root of the square root the biquadrate, or fourth rect, for 4×1. The cube root of the square root, or the square root of the cube root the sixth root, for. The square root of the fourth root= the eighth root, for x=1, or xx. The cnbe root of the cube root the ninth root, for x}=}, &c. 89 Rule II. If N be any given power whatever, whose root is sought, n the index of the power, r the nearest rational root; or rn the nearest rational power to N, whether greater or less. Then will (1.) Extract the 5th root of 307682821106715625. By Rule I. 307 8628211067 15625|(3145 root. 35=3×3×3×3×3=243 subtrahend. 314x5=405) 648 first dividend, 315-28629151 subtrahend. 31+x5=4617605)21391311 second dividend. 314)53052447761824 subtrahend. 314+5=48605856080)243804492431 third dividend. 31455-307682821106715625 subtrahend, |