3. If & gallon cost £3, what 5. A lends B $48 for of will f tun cost? a year; how much must B of as of 1 of f=hto tun. lend A of a year to bal. Σστε, ::: 5. Ans. £140. ance the favor? 4. If my horse and chaise Ans. $86.40. be worth $175, and the value 6. A person owning of a of my horse be that of my farm, sells of his share for chaise, what is the value of £171; what is the whole each? farm worth? Ans. £380. 1:176 ::: $105 horse. 1:175 : $70 chaise. MISCELLANEOUS. For miscellaneous exercises, let 5. Two thirds and of a the pupil review Section IV. Part I. and also the following articles: 51, person's money amounted to 5%, 55, 56, 57, 58, and 59. $760; how much had he ? Ans. $600. 1. In an orchard I the trees bear apples, & peaches, & 6. A man spent ț of his plums, 30 pears, 15 cherries, life in England, 4 in Scotland, and 5 quinces; what is the and the remaining 20 years, in whole number of trees ? the United States : to what +++++=+it+= age did he arrive ? t; then 50=1 and 1=50 Ans. 48 years. X 12=600, Ans. 2. One half, 4 of a school, 7. A pole is in the mud, and 10 scholars, make up the + in the water, and 12 feet school : how many scholars out of the water; what is its are there? Ans. 60. length ? Ans. 70 feet. 3. There is an army, to 8. There is a fish whose which if you add }, }, and head is 1 foot long, his tail itself, and take away 5000, as long as his head and half the sum total will be 100000; the length of his body, and what is the number of the bis body as long as his head whole army? Ans. 50400 men. and tail both; what is the leagth of the fish ? 4. Triple, the half, and the Ans, 8 feet fourth of a certain number we equal to 104; what is that 9. What number is that number? whose 6th part exceeds its Ans. 27H 8th part by 20? Ans. 480. VULGAR FRACTIONS. 105 10. What sum of money / son's legacies £257 38. 4d. : is that whose 3d part, 4th part what was the widow's share ! and 5th part are $94 ? Ans. £035 1033d. Ans. $120. 15. A man died, leaving 11. If to my age there added be his wife in expectation of an One half, one 311, and 3 times three, Six score and len their sum will be; heir, and in his will ordered, Wbat is my age ? pray show it me. that if it were a son, of the Ans. 66 years. 12. Seven eighths of a cer. estate should be his, and the remainder the mother's; but tain number exceeds four if a daughter, the mother fifths, by 6; what is that should have , and the daughnumber? ter }; but it happened that 13. What number is that she had both, a son and a from which if you take of 3, daughter, in consequence of and to the remainder add which the mother's share was of t, the sum will be 10 ? $2000 less than it would have Ans. 10/2015 been if there had been only a 14. A father gave is of his daughter; what would have estate to one of his sons, and been the mother's portion, had to of the residue to another, there been only a son ? and the surplus to his relict Ans. $1750 for life; the difference in the ܪ REVIEW, 1. What are fractions ? Or how 6. When are fractions said to have many kinds are fractions? In what a conmou denominator ? do they differ? 7. What is the common multiple 2. How is a vulgar fraction ex- of two or more numbers 1-the least pressed? What is denoted by the common multiple ?-a prime num. denominator (22)? By the numera- her ?--he aliquot parts of a num. tor? ber?-a perfect number? Explain. 3. What is a decimal fraction ? 8. What is denoted by a vulgar How is it expressed ? How is it fraction (129)? How is an impropread ? How may it be put into the er fraction changed to a whole or a mixed number (216) ?-a whole or 4. What is a proper fraction ?.-. mixed number to au improper frac an inproper fraction? What are tion ? the terms of a fraction ? What is a 9. How is a fraction multiplied by compound fraction ?ma mixed mm- a whole number (219) ---divided by ber a whole lumber? 5. What is ineant by a common 10. How would you multiply divisor of two numbers !--by the whole number by a fraction (222) i greatest coinmou divisor i ma fraction by a fraction ! 11. How would you divide a a common denominator (239)-O whole nuinber by a fraction (225) ? | the lcast coinmon denominator ? a fraction by a fraction ? 16. How are fractions of a higher 12. How may you enlarge the denomination changed to a lower terms of a fraction (229) ? How di- denomination (213)?-into integers minish them ? of a lower ?--a lower denoinination 13. How would you find the great. to a higlier ?-into integers of a est cominon divisor of two numbers ? higher ? How reduce a fraction to its lowest 17. Is any preparation necessary terms ? in oriler to add fractions (249)? 14. How would you find a com- why must they have the same demon multiple of two nuinbers (236) ? ' uominator ? How are they added ? -ihe leasi cominon multiple ? How is subtraction of fractions per15. How are fractions brought to forined? How the rule of three ? SECTION VIII. POWERS AND ROOTS. 1. Anvolution. A ANALYSIS. * 253. Let A represent a line 3 feet long; if this length A de multiplied by itself, the product (3X33), 9 feet, is ihe area of the square, B, which ineasures 3 feet on every side. B Hence, if a líne, or a number, be multiplied by itself, it is said to be squured, or because it is used twice as a factor, it is said to be raised to the second power; and the lins which makes the sides of the square is called the first power; the root of the square, or ils squure root. Thus, the square root of B9, is A—3. 254. Again, if the square, B, be multiplied by B its roul, A, the product (9x3=), 27 feet, is the volume, or content, of the cube, A CE, which measures 3 fcet oirevery side. Hence, if a line or a aumber be multiplied iwice into itself, it is said to F be cubed, or because it is employed 3 times as a factor (3X3X3_-27), it is said to be raised to the third power, and the line or number which show's the dimensions of the cube, is called its cube ront. Thus the cube root of A C E27, is AS3. 255. Again, if the cube, D, be multiplied by its root, A, the product (27X3=), 81 feet, is the content of a parallelopipedon, À CE, whose D jength is 9 feet, and other dimensions, 3 feet each way, equal to 3 cubes, AC E, placed end to end.' Hence, if a given number be multiplied 3 times into itself, or employed four times as a factor (3X3X3X3=81), it is raised to the fourth power, or biquadrute, of which the giveu number is called the Lurth root. :56 09. INVOLUTION. 107. 260. Again, if the biquadrate, D, be multiplied by its rook, A, the duct, (, E laid down in a square form, and called the sursolid, or fifth power, of whicb A is the fifth root. 257. Again, if the sursolid, or fifth power, be multiplied by its root, A, the product (243X3=), 729, is the content of a cube equal to 27 cubes, A CE, and is called a squared cube, or sixth power, of which A is the sixth root. 258. From what precedes, it appears that the form of a root, or firs' power, is a line, the second power, a square, the third power, a cube, the fourth power, a parallelopipedon, the fifth power, a plank, or square solid, and the sixth power, a cube, and proceeding to the higher powers, it will be seen that the forms of the 3d, 4th and 5th powers are continually repeated ;, that is, the 3d, 6th, 9th, &c. powers will be cubes, the 41b, 7th, 10th, &c. parallelopipedons, and the 5th, 8th, 11th, &c. planks. The raising of power of numbers is called INVOLUTION. 259. The number which denotes the power to which another is to be raised, is called the index, or exponent of the power. To denote the second power of 3, we should write 32, to denote the 3d power of 5, we should write 5s, and others in like manner, and to raise the number to the power required, multiply it into itself continually as many times, less one, as are denoted by the index of the power, thus : 3-3 53, first power of 3, the root. 3233X3 -9, second power, or square of 3. 3333X3X3 =27, third power, or cube of 3. 34=3X3X3X3=81, fourth power, or biquadrate of 3. power of 45 ? QUESTIONS FOR PRACTICE. 1. What is the fifth power 2. What is the second of 6? 6 Ans. 2025. 6 3. What is the square of 0.25 feet (121)? 36 20 ponera Ans. 0.0625 m 6 4. What is the square of 216 3d power. 3 inch? Ans. fin. 6 5. What is the cube of lf, or 1.5? 1296 4th power. Ans. 2=3], or 3.373. 6 6. How much is 44.7 62 ? Ans. 7776 Sth power. 83? 75 ? 114 ? 1010? 260. The powers of the nine digits, from the first to the sixtb inclusive, are exhibited in the following TABLE. Roots, or 1st powers, 111 21 31 51 61 71 81 Squares, or 2d powers, 111 41 9! 161 25 36 491 64) 81 Cubes, or 3d powers, 111 81 271 641 125 2161 343 121 729 Biquadrates, or 4th p. 11126] 81| 2561 625) 1296| 21011 40961 656) Sursolids, or 5th powers, 11/32243|1024| 31251 7776] 16807) 327681 59049 Square cubes, or 6th p. 11,641729, 4096|15625146656|117649|262144)5314414 2. Evolution. ANALYSIS. 261. The method of ascertaining, er extracting the roots of numbers, or powers, is called Evolution. The root of a number, or power, is a numo ber, which, multiplied by itself continually, a certain number of times, will produce that power, and is named from the denomination of the power, as the square rool, cube root, or 2d root, 3d root, &c. Thus 27 is the cube or Sd power of 3, and hence 3 is called the cube, or 3d, root of 27. 262. The square root of a quantity may be denoted by this character ✓ called the raštical sign, placed before it, and the other roots by the same sign, with the index of the root placed over it, or by fractional indices placed on the right hand Thus, VI, or 94, denotes the square root of 9, 32, or 27t, denotes the cube root of 27, and 16, or 167, denotes the 4th root of 16. The latter method of denoting roots is preferable, inasmuch as by it we are able to denote roots azd powers at the same time. Thus , 8% signifies that 8 is raised to the second power, and the cube root of that power extracted or that the cube root of 8 is extracted, and this root raised to the second power ; that is, the numerator of the index denotes the power, and the denominator the root of the number over which it stands. 263. Although every number must have a root, the roots of but very few numbers can be fully expressed by figures. We can, however, by the belp of decimals approximate the rnois of all sufficiently near for all prac tical purposes. Such roots as cannot be fully expressed by figures are denominated surds, or irrational uumbers. 284. The least possible root, which is a whole number, in l. The aguaro of is (1x1=) 1, which laas one figure less than the number |