49. How much wood in a pile 96 ft. 6 in. long, 4 feet wide, if of the length is 3 st. 3 in. high, and i, 3 ft. 10 in. and the remaining part 4 ft. ? Ans. 11 C. 42 ft. 2 in. 2". 50. At 98. 7 d. for 3 lb. of coffee, what will of 2 lb. come to ? Ans. 4 s. 31 d. 51. At 13} cents for of a pound of cinnamon, what will 4 Ib. 5 oz. come to! Ans. $2,30. 52. A has ļof a yard of broadcloth for which he gave at the rate of $8} per yard. He gives the broadcloth and 50 cents for 1 yards of cassimere. What did the cassimere cost him per yard ? Ans. $2,663. 53. A gentleman was 6 days travelling from Boston to Washington city. He paid 6 cents per mile for stage fare, and $1,75 a day for other expenses. His whole expense was $37,75; what is the distance between the two places ? Ans. 436 miles. 54. A gentleman left his estate to be divided between his widow, two sons and a daughter as follows: the widow's share was to equal the elder son's share plus $525; the elder son's to equal the younger son's plus the daughter's ; the younger son's to equal : of of the whole estate; and the daughter's share to equal of the younger son's share. What was the share of each, and the amount of the whole estate ? See Rule, page 175. Widow's share, $1575. Elder son's 66 $1050. Daughter's $315. $3675. 55. At 6 o'clock the minute and hour hands of a elock point in opposite directions; how long before they will first point in the same direction. Ans. 32 minutes. SECTION IX. INVOLUTION. DEFINITIONS. A Root is a number from which any power of that number may be obtained by being multiplied into itself a certain number of times. A Power is the last product of a root multiplied into itself a given number of times. [1f 5 be multiplied by itself, 5 is the root and 25 the 2d power.] An Index is a small figure or figures set to the right, and a little above the root, to denote what power is required. [Thus, 21 signifies the 2d power of 21.] A Line is length without breadth or thickness. change its direction, or it is A B the shortest distance between two points; as A B. An Angle is the opening of two lines when they meet in the same point, or it is the space included between them when they proceed from the same point, as C D E. When an angle is read, the letter standing at the point where the lines meet, is read in the middle, as CDE, or EDC. A Perpendicular Line is one which forms a right angle with another line. [If the angle A CB be equal to the angle A CD, the line A C is perpendicular to the line B D, and the angles A C B B -D and A C D are right an. gles.] Parallel Lines have the same perpendicular dis tance between them in ba every part, as a b and d a cd. с A Triangle is a figure having three sides. If one of the angles in a triangle be a right angle, the triangle is called a right angled triangle. FG His a triangle. H .4 Parallelogram is a figure of four sides, having the opposite sides parallel. If I K IK be parallel to ML, and K L to I M, the figure I K L M is a parallelogram. When the angles in a parallelogram are right ones, it is called a rectan. gle. A Square is a figure having 4 equal sides, and 4 right angles. If O P be equal P to, or as long as each of the other sides, and the angles Q O RQ is a square. Involution is the method of finding a required power from a given root. Remark. The 2d power of a number is generally called the square, and the third power, the cube of that number. RULE, For finding any power of a given root. Multiply the given root into itself as many times as the index has units wanting one, the last product will be the power required. Note.-The reason for multiplying once less than the number of units in the index, is because the root itself, or given number, is called the first power, and the first product must be the 2d, and the 2d product, the third power, * &c. 1. What is the square, or 2d power, of 8 ? Here 8 X 8=64 the square of 8. 2. What is the cube, or 3d power, of 8 ? 8x8=64, and 64 X8=512 the answer. 3. The square of 15 is how much? Ans. 225. 4. What number is equal to 163 ? Ans. 4096. 5. The 204 is how much? Ans. 160000. 6. What is the square of 1 ? Ans. 1. 7. The 33 is how much ? Ans. 27. 8. What number is equal to 72 ? Ans. 49. 9. 93 is equal to what number? Ans. 729. 10. The 55 is how much ? Ans. 3125. 11. What is the sum of 14, 23, 32 ? Ans. 18. 12. Required the sum of 25,4", 5, and 502. Ans. 2913. 13. What is the sum of 1o, 2o, 3o, 4o, 5o, 6o, 7o, 83, 92'? Ans. 285. 14. What is 13+23+33 +43 +53 +63+73 +8% Ans. 2025. 15. What is the difference between 373 and 454 ? Ans. 4'049'972. 16. What is the prosluct of 132 multiplied by 133 ? Ans. 371 293. 17. What is the 5th power of 13? Ans. 371'293. 18. What is the product of the 23 into the 28 ? Ans. 2048. +93? * The power of a number is so called because it is what the root is just capable of producing, when multiplied into itself. 19. What is the 11th power of 2? Ans. 2048. In the 16th question, we multipled the third power of 13 by the square of 13; in the 17th question, we raised 13 to the 5th power, and found the same number as when we multiplied the 3d power of 13 by the 2d power of 13. In the 18th question we multiplied the 3d power of 2 by the 8th power of 2, and obtained the same number, as when we raised 2 to the 11th power in the 19th question. By examining these four questions, it will be seen that multiplying two powers of the same number together, we shall obtain that power of the same number, or root, whose index is equal the sum of the indices of those powers multiplied into themselves. In the 16th question, the sum of the indices is 5, and in the 17th question, the given index is 5 ;-in both these questions we obtain the 5th power of 13. The sum of the indices in the 18th question, is 11, a number equal to the given index in the 19th, and in both questions we find the 11th power of 2. This principle will hold true in all cases, for mul. tiplying one power by another, we increase the power multiplied as much faster than we should were we to multiply by the root itself, as the multiplier is greater than the root. If the 4th power of 4 be required, we may obtain it by multiplying 4 three times into itself, as in the first operation below, or by multiplying the 2d power of 4, which is 16, by 16, as in the ad operation. Operation 1. Operation 2. 1 6 1 6 |