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 he 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. the fort? Ans. 32.8+ feet. Ans. 42.42 6 feet. 4. A ladder 40 feet long, 6. Two ships sail from 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, I asunder? Ans. 293.26 miles. 269. To find a mean proportional between two numbers. RULE.—Multiply the two given numbers together, and the square root of the product will be the mean proportional sought. QUESTIONS FOR PRACTICE. 1. What is the mean proportional between 4 and 36 ? 36X4=144 and ✓144=12 Ans. Then 4 : 12 : : 12 : 36. 2. What is the mean proportional between 49 and 64 ? Ans. 56. 3. What is the mean proportional between 16 and 64? Ans. 32. EXTRACTION OF THE CUBE ROOT. ANALYSIS. 270. To extract the cube root of a given number, is to find a number which, multiplied by 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 tens in the root must be found in the left hand period. The greatest cube in 12 is 8, whose root is %, 12167(23 root but the value of 8 is 8000, and the 23—2X2X2_8 2 js 20, that is, 8000 feet of the stone will make a pile measuring 20 22 X 300+2X30=1260 )4167 feet on each side, and (12167– 8000) 4167 feet remain to be add1200X3=3600 ed to this pile in such a manner as 60X3X3= 540 to continue it in the form of a cube. 3X3X3 27 Now it is obvious that the addition must be made upon 3 sides; and 4167 each side being 20 feet square, the surface upon which the additions must be made will be (20X20X3–2X2X300=) 1200 feet, but when these additions are made, there will evidently be three deficiencies along the lines where these additions come together, 20 feet long, or (20X32X30=) 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 (1200+60=) 1260 feet and 4167 feet, the quantity to be added divided by 1260, the quotient is (4167- 1260–) 3, which is the thickuess 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 (1200X3=), 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 where 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 long, and the quantity of stone added will be (60X3X3=) 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 which will be just equal to the thickness of the other additions, or 3 feet, and cubing 3 feet we find (3X3X3=) 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 the sides, the edges, and at the corner together, we find them to amount to (36007540427=) 4167 feet, just cqual 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 the square blocks, to fill the space at the corner. These, placed together in the manner described 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 to the remainder 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, call. ing it the triple quotient, and the sum of these call the divisor. Seek how often the divisor may be had in the dividend, and place the result in the quotient. 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 ; and so on, till the whole is finished. QUESTIONS FOR PRACTICE. 2. What is the cube root of 5. What is the cube root of 181 5848? 2? Ans. 1.25+ 1XIX 300—300|1815848 ( 122 The decimals are obtained by 1x 30 — 301 annexing ciphers to the remainder, Divisor 330 as in the square root, with this dif815 divid. ference, thai 3 instead of 2 are an nexed each time. 6. What is the cube root of 23 - 8 27054036008 ? Ans. 3002. 7. What is the cube root of 728 sub. 122 X 3007-12X30 43560)87848 3138? 43200X2_86400 l=f Ans. 5130 27 360X22 1440 23 8 8. What is the cube root of subtra. 87848 3. What is the cube root of 10648? Ans. 22. 4. What is the cube root of 303464448? Ans. 672. .66666671 —.873+ Ans. 9. What is the cube root of 436036824287 ? Ans 583. 272. Solids of the same form are in proportion to one another as the cubes of their similar sides or diameters. 1. If a bullet, weighing 72 3X3X3=27 and 6X6X63216 lbs. be 8 inches in diameter, Then 27 : 4 :: 216. what is the diameter of a bul Ans. 32 lbs. let weighing 9 lbs. ? 3. If a ball of silver 12 inch72:83::9: 64 Ans. 4 in. in diameter be worth 2. A bullet 3 inches in di- | $600, what is the worth of ameter weighs 4 lbs. what is another ball, the diameter of the weight of a bullet 6 inches ( which is 15 inches ? in diameter? Ans. $1171.87+ es EXTRACTION OF ROOTS IN GENERAL. ANALYSIS. 273. The roots of most of the powers may be found by repeated extractions of the square and cube root. Thus the 4th root is the square root of the square root : the sixth root is the square root of the cube root, the 8th root is the square root of the 4th root, the 9th root is the cube root of the cube root, &c. The roots of high powers are most easily found by logarithms. If the logarithm of a number be divided by the index of its root, the quotient will be the logarithm of the root. The root of any power may likewise be found by the following RULE. 274. Prepare the given number for extraction by pointing off from the place of units, according to the required root. 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. 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. Find how many times the divisor may be had in the dividend, and the quotient will be another figure of the root. 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. QUESTIONS FOR PRÁCTICE. 1. What is the cube root of 2. What is the fourth root 48228544? of 19987173376 ? 48228544 (364 Ans. 376. 33–27 3. What is the sixth root of 191102976? Ans. 24. 32X3=27 )212 dividend. 4. What is the seventh root 363=46656 of 3404825447 ? Ans. 23. 362X3=3708 ) 15725 2d div'd. 5. What is the fifth root of 307682821106715625 ? 3643=48228544 Ans. 3145. Between two numbers to find two mean proportionals. Rule.- Divide the greater by the less, and extract the cube root of the quotient ; multiply the lesser number by this root, and the product will be the lesser mean; multiply this mean by the same root, and the product will be the greater mean. EXAMPLE.—What are the two mean proportionals between 6 and 162? 162:6–27 and 27=3; then 6X3=18, the lesser. And 18x3=54, the greater. Proof, 6 : 18 : : 54 : 162. What propor (258) ? REVIEW. 1. If the length of a line, or any 10. What does extracting the number be multiplied by itself, what square root mean? What is the will the product be (253) ? What rule? Of what is the square of a is this operation called ?' What is number consisting of tens and units the length of the line, or the given made up (266) ? Why do you subnumber, called ? tract the square of the highest fig. 2. What is a cube (61)? What ure in the root from the left hand is meant by cubing a number (254) ? period? Why double the root for Why is it called cubing? By what a divisor? In dividing, why, omit other name is the operation called ? the right hand figure of the dividend ? What is the given number called ? Why place the quotient figure in the 3. What is meant by the biquad. divisor ? What is the method of rate, or 4th power of a number ? proof? What is the form of a biquadrate ? 11. When there is a remainder, 4. What is a sursolid ? What its how may decimals be obtained in form? What is the squared cube ? the root? How find the root of a What its form? What are the suc- Vulgar Fraction ? cessive forms of the higher powers tion have circles to one another ? When two sides of a right angled 5. What is the raising of powers triangle are given, how would you called ? How would you denote the find the other side ? What is the power of a number ? What is the proposition on which this depends small figure which denotes the power (68)? What is meant by a mean called ? How would you raise a proportional between two numbers ? number to a given power ? How is it found ? 6. What is Evolution ? What is 12. What does extracting the cube meant by the root of a number? root mean? What is the rule ? What relation have Evolution and Why do you multiply the square of Involution to each other? the quotient by 300? Why the 7. How may the root of a number quotient by 30 s Why do you mulbe denoted ? Which method is pref- tiply the triple square by the last erable? Why (262) ? quotient figure ? Why the triple 8. Has every number a root ? quotient by the square of the last Can the root of all numbers he ex- quotient figure? Why do you add pressed? What are those called to these the cube of the last quowhich cannot be fully expressed ? tient figure ? With what may this 9. What is the greatest number of rule be illustrated ? Explain the figures there can be in the continued process. product of a given number of fac- 13. What proportion have solids tors What the least? What is to one another?' How can you find the inference ? How, then, can you the roots of higher powers (273) ? ascertain the number of úgures of State the general rule. which any root will consist ? |