figure being necessary to preserve the 529 [ 23 form of the square, by filling the corner, we place it at the right of the divisor, in place of the cipher which is always un83 ) 129 derstood there, and then multiply the 123 whole divisor by the last" figure of the 2323=2529 proof. root. As we may conceive every root to be made up of tens and units, the above reasoning may be applied to any number whatever, and may be giren in the following general RULE. 267. Distinguish the given numbers into periods ; find the joot of the greatest square number in the left hand period, and place the root in the manner of a quotient in division, and this will be the highest figure in the root required. Subtract the equare of the root already found from the left band period, and to the remainder bring down the next period for a dividend. Double the root already found for a divisor; seek how many tines the divisor is contained in the dividend, (excepting the right hand figure and place the result for the next figure in the root, and also on the right of the divisor. Multiply the divisor by the figure in the root last found; subtract the pro: Duct from the dividend, and to the rernainder bring down tbe next period for a new dividend. Double the root now found for a divisor, and proceed as before to find the next figure of the root, and so on, till all the periods are broug bt down. QUESTIONS FOR PRACTICE. 1. What is the equare root 1 6. What is the square of 529 ? | root of ? Ans. .64549. 2. What is the square root Reduce to a decimal and of2? Ans. 1.41426. then extract the root, (130). The decimals are found hy an. 7. What is the square nexing pairs of ciphers continually so the remainiler for a new divideod. root of a? Ans. In this way a surd root may be ob sained to any assigned degree of | 8. What is the square exactness. root of 19 Ans. 14. 3. What is the square root of 9. An army of 567009 inea o 182.25 ? Ans. 13.5. i are drawn up in a solid body, 4. What is the square root ! in form of a square; what is of .0003272481 ? Ans. .01809. the number of men in raok Hence ibe root of a decimal is and file? Aps. 753. Ereater than its powers. 10. What is the length of 5. What is the square root of 5499025 ? A08. 2345. - 1 261. SQUARE ROOT. 167 the side of a square, which ameter of a circle 4 times as shall contain an acre, or 160 large? Ans. 24. rods ? Ans. 12.649 + rods. i Circles are to one another as the 11. The area of a circle is squares of their diameteis; there. 234.09 rods; what is the fore square the given diameter, multiply or divide it by the given length of the side of a square proportion, as the required diamoof equal area? Ans. 15.3 rods." Iter is to be greater or less than the 12. The area of a triangle given diameter, and the square is 45244 feet; what is the root of the product, or quotient, will be the diameter required. length of the side of an equal square ? Ans. 212 feet. 14. The diameter of a cir. cle is 121 feet; what is the 13. The diameter of a circle | diameter of a circle one half is 12 joobes; what is the di- ! - as large? Ans. 85.5+feet. 268. Having two sides of a right angled triangle given to find the other side : RULE.-Square the two given sides, and if they are the two pides which include the right angle, that is, the two shortest sides, add them together, and the square root of the sum 'will be the length of the longest side ; if not, the two shortest; 'subtract the square of the less from that of the greater, and the square root of the remainder will be the length of the side required. (See demonstration, Part I. Art. 68.) - QUESTIONS FOR PRACTICE. 1. In the right angled | If A B be 45 inches, and triangle, A B C, the side A C A C 36 inches, what is the is 36 inches, and the side B C length of BCS 27 inches; what is the length i A B2 =45 x 45 =2025 of the side A B? AC2-36 X 36–1296 /B B C2 = 729 BC=4729=27 inches. If A B=45, B C=27in. what is the length of A C. AC2=36 X 36–1296 | AB2=452, B C2272, A B C2-27 x 27€ 729 ; C? and AC-V 1296-36in, 2. Suppose a man travel east and one on the other side of 40 miles, (from A to C) and the street 21 feet from the then turn and travel north 30 ground; what is the width of miles ; (from C to B) how far the street ? is be from the place (A) where Ans. 56.64+ feet. he started? Ans. 50 miles. 5. A line 81 feet long, will 3. A ladder ,48 feet long exactly reach from the top of will just reach from the oppo- a fort, on the opposite bank of site side of a ditch, known to a river, known to be 69 feet be 35 feet wide, to the top of broad; the height of the wall a fort; what is the height of is required. - Ans. 42.425 feet. the fort ? Aps. 32.8-f feet. 4. A ladder 40 feet long, 6. Two ships sail froin the with the foot planted in the | same port, one goes due east same place, will just reach a | 150 miles, the other due north window on one side of the 252 miles; how far are they street 33 feet from the ground, as asunder? Ans. 293.26 miles. 269. To find a mean proportional between two numbers. RULE.--Multiply the two given pumbers together, and the square root of the product will be the mean proportional songht. QUESTIONS FOR PRACTICE. 1. What is the mean propor- | portional between 49 and 64? tional between 4 and 36 ? Ans. 56. 36x24=]44 and v144=12 Ans 3. What is the mean proThen 4:12 :: 12 : 36. portional between 16 and 64? 2. What is the mean pro- | Ans. 32 EXTRACTION OF THE CUBE ROOT. ANALYSIS. - 270. To extract the cube root of a given pumter, is to find a number which multiplied hy its square will produce the given number, or it is to find the length of the side of a cube of which the given number expresses the content. 1. I have 12167 solid feet of stone, which I wish to lay up in a cubical pile ; what will be the length of the sides ? or, in other words, what is the cube root of 12167? , By distinguishing 12167 into periods we find the root will consist of two figures. (265) Since the cube of tens (264) can contain no significant figures less than thousands, the cube of the tenis in the root must be found n the left hand period. The greatest cube in 12 is 8, whose root is 2, 271. CUBE ROOT. 189 but the value of 8 in 8000and the 2 js 20, 12167 (23.root. that is, 8000 feet of the stone will make a 2X2 X 2=8 pile measuring 20 feet on each side, and (12167-8000=) 4167 feet remain to be 2*300*2*4167 added to this pile in such a manner as to 30=1260 continue it in the form of a cube. Now 1200 x3=3600 it is obvious that the addition must be 60 X 3x3=540 made upon 3 sides; and each side being 3x3x3=27 20 feet square, the surface upon which the additions must be made will be (20 4167 20 m3=2X2 X300=) 1200 feet, but when these additions are made, there will evi. dently be three deficiencies along the lines where these additions come together, (20 feet long, or 20 >3=Ž 30 ) 60 feet, which must be filled in order to continue the pile in a cubic form. Thus the points upon which the additions are to be made are 0200 +60=) 1260 feet and 4167 feet, the quantity to be added divided by 1260, ihe quotient is (4167- 1260=) 3, which is the thickness of the additions, or the other figure of the root. Now if we multiply the surface of the three sides by the thickness of the additions, the product, (120043 =) 3600 feet, is the quantity of stone required for those additions. Then to find how much it takes to fill the deficiencies along the line wbere these additions come together, since the thickness of the additions upon the sides is 3 feet, the additions here will be 3 feet square, and 60 feet, and the quantity of stone added will be (60X3>3=) 540 feet. But after these additions there will be a deficiency of a cubical form, at the corner, between the ends of the last mentioned additions, the three dimensions of wbich will be just equal to the thickness of the other additions, or 3 feet, and cubing 3 feet we find (3X3X3X3=) 27 feet of stone required to fill this corner, and the pile is now in a cubic form measuring 23 feet on every side, and adding the quantities of the additions upon ihe sides, the edges, and at the corner together, we find them to amount to (36004540 + 273) 4167 feet, just equal to the quantity remaining of the 12167 after taking out 8000. To illustrate the foregoing operation, make a cubic block of a convenient size to represent the greatest cube in the left hand period. Make 3 other square blocks, each equal to the side of the cube, and of an indefinite thickness, to represent the additions upon the three sides, then 3 other blocks, each equal in length to the sides of the cube. and their other dimensions equal to the thickness of the square blocks, to represent the additions along the edges of the cube, and a small cubic block with its dimensions each, equal to the thickness of ths square blocks, to fill the space at the corner. These placed together in the manner des. cribed in the above operation, will render the reason of each step in the process perfectly clear. The process may be summed up in the following RULE. 271. 1. Having distinguished the given number into periods of three figures each, find the greatest cube in the left hand period, and place its root in the quotient. Subtract the cube from the left hand period, and bring down the next period for a dividend. Multiply the square of the quotient by 300, calling it the triple square, and the quotient by 30, callirg it the triple quotient, apú tte-sum of these call the divisor. Seek how often the divisor may be had in the dividend, and place the result in the quocient. Multiply the triple square by the last quotient figure, and write the product under the dividend ; multiply the triple quotient by the square of the last quotient figure, and place this product under the last; under these write the cube of the last quotient figure, and call their sum the subtrahend. Subtract the subtrahend from the dividend, and to the remainder bring down the next period for a new dividend, with which proceed as before; apd so on till the whole is finished. QUESTIONS FOR PRACTICE. 2. What is the cube root of 5. What is the cube root of 1815748? 2 ? Ans. 1.25+ 1x1x300=300|i815748(122 The decimals are obtained by annexing ciphers to the remainder, 17305 301 as in the square root, with this dif. Divisor 330 815 divid. { ference, that 3 instead of 2 are an. nexed each time. 300x2=600 6. What is the cube root of 30 x 22=120 | 27054036008? Aps. 3002. 7. What is the cube root 728 sub, of 18? 122 , 300+12430 43560)87848 0135 1 Ans. 43200 2=86400 * 360*22= 1440! 8. What is the cube root 23= 8 of ? subtra. 87848 3. What is the cube root of 10648? Ans. 22. 4. What is the cube root of 303464448? Aps. 672. 8 272. Solids of the same form are in proportion to one another as the cubes of their similar sides or diameters. J. If a bullet weighing 72 | es in diameter? lbs. be 8 inches in diameter, 3x3x3=28 & 6*6*6=216 what is the diameter of a bul b. Thus 27 : 4 :: 216. let weighing 9 lbs. ? Ans. 32 lbs. 72; 83: :9: 464 Ans. 4 in. 3. If a ball of silver 12 in. 2. A bullet 3 inches in dia. | ches in diameter be worth meter weighs 4 lbs, what is $600, what is the worth of the weight of a bullet 6 inch- another ball, the diameter of which is 15 inches? Ans, $1171.87+ |