2. An army consisting of 87616 men, is to be arranged in the form of a square; how many will a side contain? Ans. 296 men. 3. What will the paling of a square garden consisting of 2209 square yards cost, at 3s. 6d. per yard? Ans. 321. 18s. 4. Required the length of a ladder, the foot of which being pitched 12 feet from the wall, its top will reach a window 18 feet from the ground? Ans. 21 feet, 7.59969, &c. inches. 5. A rope 120 fathoms long is extended from the top of a cliff, to a boat moored at SO fathoms from its base; required the perpendicular height of the cliff? Ans. 89.4427, &c. fathoms. 6. Required a mean proportional between 36 and 2401*? Ans. 294. 7. A rectangular field has its sides equal to 210 and 300 yards respectively; what length must the side of a square be to contain an equal area? Ans. 250.9979, &c. yards. 8. The diagonal (or straight line joining the opposite corners) of a chess-board, measures 30 inches; required the length of the side? Ans. 21.213203, &c. inches. 9. A wall is supported 13 feet from the ground by a shoar 16 feet long; how far is the foot of the shoar distant from the base of the building? Ans. 9 feet, 3.9285, &c. inches. 10. A gentleman has a table 5 feet wide, and 14 feet long, and wants three others to be made, each square, and all together of equal dimensions with the former; required the side of each? Ans. 4 feet, 9.9654, &c. inches. EXTRACTION OF THE CUBE-ROOT. 279. The following table contains the first nine whole num * The length of the ladder, the perpendicular distance of its top from the ground, and the distance of its foot from the wall, together form a right angled triangle; and it is demonstrated in the 47th proposition of the first book of Euclid's Elements, that the square of the longest side of such triangle is equal to the sum of the squares of the two remaining sides; wherefore in the present instance, 12 + 182 = 21.633, &c. feet, the length required. 2 y From the foregoing note it follows, that the difference of the squares of the longest and of either of the remaining sides, is equal to the square of the other side; whence 1201 -80289.44, &c. fathoms. 2 The square root of the product of any two numbers is a mean proportional between them. bers which are exact cubes, together with their cube roots, where each root is placed exactly under the number of which it is the root, whereby the cube root of any cube number within its limits may be found. ༤. 8. CUBES ...... 27. 64. 125. 216, 343. 512, 729. CUBE-ROOTS. 1. 2. 3. 4. 5. 6. 7. 8. 9. 280. To extract the cube root, consisting of several figures, from any number. RULE I. Put a point over the units' place, and also one over every third figure, counting from the units; whereby the given number will be divided into periods of three figures each, except the left hand period, which may be either one, two, or three figures. II. Find by the table the greatest cube in the left hand period, set the said cube under that period, and its root in the quotient. III. Subtract the cube from the period above it, and to the remainder bring down the next period for a dividend. IV. Multiply the square of the root by 300, and place the product to the left of the dividend for a divisor. V. Find how often the divisor is contained in the dividend, and place the number in the quotient for the next figure of the root. VI. Multiply the divisor by the last figure of the root. Multiply all the figures in the root, except the last, by 30, and that product by the square of the last. Cube the last figure of the root. Add these three together, and call their sum the subtrahend. VII. Subtract the subtrahend from the dividend, and bring down the next period for a new dividend. VIII. Find a new divisor by proceeding as before, viz. multiplying the square of the whole of the root found by 300. Divide. Find a new subtrahend as before, and proceed in this manner until the work is finished. IX. Decimals must likewise be pointed over every third figure from the units' place, and if there are not decimals enough to complete the right hand period, the deficiency must be supplied by ciphers. If the given number consists of whole and as numbers and decimals, the root will consist of as many places of whole numbers, as there are periods of whole numbers; many decimals, as there are periods of decimals. If there is a remainder after all the figures are brought down, the work may be continued, by bringing down periods of ciphers *. fore put 2 in the quotient. I multiply the divisor by this quotient figure, and place the product 5400 under the dividend; I multiply all the figures in the root, (viz. 3.) except the last, by 30, and that product by 22 the square of the last, which gives 360. I then cube the last figure, viz. 2, which gives 8: these three I add together for the subtrahend, which being the same as the dividend, and all the periods being brought down, the work is finished. a When the root of a decimal only, having several places of ciphers on its left, is to be extracted, for every complete period of ciphers, observe to prefix a cipher to the root. This rule is proved by involving the root found to the cube, and adding in the remainder (if any) to the last line of the work; the sum will be equal to the given number, if the operation is right. 6. What is the cube root of 14886936 ? Root 246. 7. What is the cube root of 43614208? 8. Extract the cube root of 128.024064. 9. Required the cube root of 1879080.904. 10. Extract the cube root of 1.006012008. Root 352. Root 5.04. 11. Required the cube root of 27407.028375. 12. What is the cube root of .0001357 Root 123.4. Root 30.15. Root .05138, &c. 13. What is the cube root of 2? Root 1.2599, &c. 281. To extract the cube root of a vulgar fraction, or mixed number. RULE. If the terms of the fraction be both rational, extract the root of the numerator for a numerator, and of the denominator for a denominator (Art. 280.); but if they are not both rational, reduce the fraction to a decimal, (Art. 233.) and extract the root of the latter". (Art. 280.) For a mixed number, reduce the fractional part to a decimal, b The former part of the rule is the most convenient, when it can be applied; the latter part is general, and applies equally, whether the terms of the given fraction be rational or irrational. The operations are proved by involution, as in the preceding rule. (Art. 233.) to which prefix the whole number, and extract the root as before. 1 2 2 Thus = .5, then√.5 = .7938, &c. the root. 16. Required the cube root of 34. 282. PROMISCUOUS EXAMPLES FOR PRACTICE. 1. What is the length, breadth, and height, of a cubical room, which contains 2197 cubic feet of air? Ans. 13 feet. 2. Required the side of a cubical box, which will hold 2744 cubic inches of flour. Ans. 14 inches. 3. A cubical box holds 9261 cubic inches of corn, how many square feet of deal are there in it? Ans. 18 feet, 54 inches. 4. A cubical cistern contains 125 cubic feet of water, what is the value of the lead, at 2d. per lb. allowing 44 lb. to every square foot, and 12 square feet for the rim ? Ans. 61. 15s. 6d. .5. 5. The solid content of the earth is estimated at 265404598080 cubic miles; required the side of a cube containing an equal quantity of matter, of the same density? Ans. 6426.4 miles nearly. 6. Required two mean proportionals between 2 and 54°? Ans. 6 and 18. To find two mean proportionals, divide the greater extreme by the less, and extract the cube root of the quotient; then multiply the less extreme by |