12167(3 root but the value of 8 is 8000, and the lex?X? 2 is 20, that is, 8000 feet of the stone will make a pile measuring 20 3°**0+%X30=1260)4167 feet ou each side, and (12167– 8000) 4167 feel remain 10 bc add. 1200X 33600 ed to this pile in such a manner as 69X3X3= 510 to continue it in the form of a cube. 3X3X3 27 Now it is ubvious 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 (20X20X32X2X300=) 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 (20X32X303) 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 4161 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 to gether, 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 fiad (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 ingether, we find them to amount to (36007-5407273) 4167 feet, just equal to the quantity remaining of the 12167, after taking out 8000. To il. \ustrate 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 10 the side of the cube, and of au indefinite thickness, to represent the additions upon the three sides, theu 3 other blocks, each equal in length to the sides of the cube, and their other dimensions equal to the thickuess of the square blocks, to represent the additicns along the edges of the cube, and a small cubic block with its dimeusions, 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 bo summed up in Wie lwowing 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. Seck 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 squarc 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+ IX1X300_300i815848 (122 The decimals are obtained by 1x 30 301 annexing ciphers to the remainder, Divisor 330 as in the square root, with this dit 815 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 foot of 728 sub. 128x300+12X30 43560 ) 87848 538? 43200X2=86400 1}= Ans. 360X22= 1440 23 8 8. What is the cube root of subira. 878-18 3. What is the cube root of .6666667 10648? Ans. 22, Ans. 4. What is the cube root of 9. What is the cube root of 303464448 ? Ans. 672 436036824287 ? Ans. 7583. =.873+ 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 6X6X6_216 Hos. 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 inch 72:83 ::9: 64 Ans. 4 in, in diameter be worth 2. A bullet 3 inches in di- į $600, what is the worth of ameter weighs 4 lbs. what is another ball, the diametor 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 oxtractions 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 roof 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 hy 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 PRACTICE. 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? Abs. 24. 82X3–27 )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 ? 3643548228544 Ans. 3145. Between two numbers to find two, mean proportionals. ROLE.—Divide the greater by the less, and extract the cube root of the quotient; multiply the lesser number by this rook, 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 8 and 162? 1625-6227 and 2–3 ; then 6Xt18, the tesser. And 18X3354, the greater. Proof, 6 : 18 ; : 54 : 162 771 REVIEW, 117 What propor: REVIEW. 1. If the length of a line, or any 10. What does extracting the Dumbes he 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 inade 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 (251) ? | period? Why double the root for Why is it called cubing? By what a divisor? In dividing, why, omit other name is the operation calleet ? 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 biquado "ivisor? What is the method of rate, or 41h power of a number ? | proof? What is the forin of a biquadrate ? 11. When there is a remainder, 4. What is a sursolie? What its how may decimals be obtained in form? What is the squared cube ? the root? How tind the root of a What its form? What are the suc- Vulgar Fraction ? ceesive forms of the higher powers tion have çircles to one another ? (252) ? 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 hetween 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 1 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 num!ser quotient by 30 ? Why do you mulbe denoted ? 'Which method is preti tiply the iriple square by the last erable ? Why (262) ? quotient figure ? "Why the triple 8. Hlas 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 in these the cube of the last quowhich cannot be fully expressert? tient figure? With what may this 9. What is the greatest number of rule be illustrated ? Explain the figures there can he ir: the continued process. procluct of a given number of fac- 13. What proportion have solids Lors? 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 nuinber of ugures of State the general rule. which any rool will consist ? SECTION IX. MISCELLANEOUS RULES. 1. Arithmetical Progression. 275. When numbers increase by a common excess, or decrease by a common difference, they are said to be in Arithmetical Progression. When the numbers increase, as 2, 4, 6, 8, &c., they form an ascending series, and when they decrease, as 8, 6, 4, 2, &c., they form a descending series. The numbers which form the series are called its terms. The first and last tenn are called the extremes, and the others the means. 276. If I buy 5 lemons, giving for the first, 3 cents, for the second, 5, for the third, 7, and so on with a common difference of 2 cents; what do I give for the last lemon ? Here the cominon difference, 2, is evidently added to the price of the first lemon, in order to find the price of the last, as many times, less 1 (3+2 +2+2+2=11 Ans.), as the whole number of lemons. Hence, I. The first lerm, the number of terms, and the common difference given to find the last term. RULE. Multiply the number of terms less 1, by the common difference, and to the product add the first term. 2. If I buy 60 yards of cloth, 3. If the first term of a seand give for the first yard 5 ries be 8, the number of terms cents, for the next 8 cents, for 21, and the common difference the next, ll, and so on, in- 5, what is the last term ? creasing by the common differ- 20x5+8=108 Ans. ence, cents, to the last, what 4. If the first term be 4, the do I give for the last yard ? difference 12, and the number 59X3=177, and 177+5= of terms 18, what is the last 182 cts. Ans. tcrm? Ans, 208. 277. If I buy 5 lemons, whose prices are in arithmetical progression, the first costing 3 cents, and the last 1l cents, what is the coinmon difference in the prices ? Here 11-338, and 5m134; 8 then is the amount of 4 equal differences, and 4)8(=2, the common difference. Hence, II. The first term, the last terin, and the number of terms given to find the common difference. ROLE.—Divide the difference of the extremes by the number of terms, less 1, and the quotient will be the common difference, |