(30) Divide a straight line into two parts, so that one of the parts shall be three times the length of the other part. *(31) If the straight line bisecting the angles at the base of an isosceles triangle be produced to meet, show that they will contain an angle equal to an exterior angle at the base of the triangle. * (32) Divide a right-angled triangle into two isosceles triangles. (33) Trisect a right angle. Let A B C be a right angle. In BC take any point D. On BD describe an equilateral triangle B E D. Bisect the angle BED by the straight line B F; then the angle A B C is trisected by the lines B E, B F. We leave the demonstration as an exercise for the pupil. Note. An exercise may often be worked out in more than one way. For example, No. 29 may be demonstrated thus, (The pupil can draw the figure for himself) : Let A B C be a triangle; let D, E, be the middle points of A B and A C. Join D E, D C, E B. Then the triangle D E B is equal to the triangle A E D (I. 38), and the triangle EDC is also equal to the triangle AED (I. 38); therefore the triangle DEB is equal to the triangle ED C. But the equal triangles DE B, EDC, are upon the same base D E, therefore they are between the same parallels (I. 39); therefore D E is parallel to B C. ARITHMETIC. (MALES.) INVOLUTION. Involution is the operation by means of which a number is raised to a certain power. The first power of a certain number is the number itself. The second power (or square) of a certain number is the number multiplied by itself. The third power (or cube) of a certain number is the continued product of that number taken three times. The fourth power of a certain number is the continued product of that number taken four times. Similarly, any higher power of a certain number is the continued product of that number taken as many times as are indicated by the number denoting the power. Thus the 1st power of 6 = 61 = 6. 4th 6=64 = 6 x 6 x 6 x6 = 1296. And so on, for the fifth, sixth, and higher powers of 6. Observe that the power to which a number is required to be raised is denoted by a figure (or figures) placed over the number to the right hand. Such figure (or figures) is called the index, or exponent. Thus, 54 fourth power of 5 = = 5 × 5 × 5 × 5 = 625. The first, second, third, or any higher power of I is 1, because if I be multiplied by itself any number of times, the product is 1. Thus, 16 = IXI XIX I XIXI = I. Ι To Find any Power of a Given Number. The method of proceeding will be clear from the following examples : (1) Find the square of 29. The square is the second power. square of 29=292 = 29 X 29841. (2) Find the cube of 16. The cube is the third power. .. cube of 16 = 163 = 16 × 16 × 16 = 4096. (3) Find the fourth power of 9. Fourth power of 9 = 91= 9 × 9 × 9 × 9=6561. (4) Find the cube of. = Cube of (})3 = } > } > } = (5) Find the fourth power of §. (5)1 = 5 × 5 × 8 × 8: = (6) Find the square of 23. 625 1296 (2)2= 2 × 210 x 10 = 361541. (7) Find the cube of 3*7. (37)3 =37 X 3:7 × 37 = 50·653. (8) Find the fourth power of 17. (1·7)=(13)'= (16 X 16 X 16 X 16)= 65536 = 9·9856+ 6561 Since 383 × 3 × 3 × 3 × 3 × 3 × 3 × 3 =(3 × 3 × 3 × 3 × 3) × (3 × 3 × 3) It follows that the eighth power of a number may be found by multiplying the fifth power by the third power (the cube). Similarly, it may be found by multiplying the second power (the square) by the sixth power (2 + 6 = 8); by multiplying the fourth power by the fourth power (4 + 4 = 8); and so on. In the same way, since 2 + 2 + 2 + 2 = = 8, the eighth power may be found by multiplying the product of the square taken four times; or by finding the fourth power of 32; so that 38: = (32)4, where 2 X 4 = 8. (9) Find the ninth power of 5. (We here work this example by three different methods): (a) 5° = 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 = 1953125. (6) 5o = 54+5=5a × 55 = 625 × 3125 = 1953125. (c) 5o = 53X3 = (53)3= (125)3 = 1953125. So the fourth power of a number may be found by squaring the square of the number, since 2 × 2 = 4. Exercise 1. (1) Find the squares of 5; 45; 61; 76; 90; 100; 109; 187. (2) Find the cubes of 1; 11; 27; 39; 73; 110; 143; 156; 1000. (3) Find the fourth powers of 6; 19; 86; 200. (4) Find the seventh powers of 4; 7; 9. EVOLUTION. The number which has been raised to a certain power is called the root of that power. Thus, since 9 is the second power or square of 3, then 3 is the second or square root of 9. 3 is the root; 9 is the square or second power of 3. Similarly, 2 is the cube root of 8, because 8 is the cube of 2; 4 is the fourth root of 256, because 256 is the fourth power of 4. Evolution is the operation by means of which the root of a certain number is found. Thus, if 3721 be given as the square of some number, by evolution we discover that number of which 3721 is the square, or the square root of 3721. Evolution in Arithmetic is generally limited to finding the square and cube roots of numbers. The reasons for the rules employed are given in works on Algebra, and must be taken for granted by the arithmetical student. To Find the Square Root of a Given Number. Rule.-Divide the number into periods of two figures each by placing a dot over the units figure, and over every alternate figure to the left in integers, and to the right in decimals. Each period is supposed to end in a figure having a dot over it. A cipher may be added to the decimals if required. Now take the first period to the left, which may consist of either one or two figures, according as it has a dot over the first figure or not. Find the greatest square number in that first period, place its root in the quotient on the right as in long division, and set the square number itself under the first period. Subtract the square number from the first period, and to the remainder annex the second period for the next dividend. The figure in the quotient is the first figure of the root required: double it, and set the product to the left of the dividend just obtained as a divisor. Find how many times this divisor is contained in the number obtained by omitting the last figure to the right in the dividend, place the result in the quotient for the second figure of the root, and also annex it to the right of the divisor. Multiply the divisor thus completed by the second figure of the quotient, place the product below the dividend, and subtract it from it. (If the product be greater than the dividend, a lower number must be taken as the second figure of the quotient and the units figure of the divisor.) To the remainder bring down the figures of the third period to form a new dividend; find the first figures of a new divisor by doubling the figures in the quotient, divide all the figures in the dividend except the last by the figures in the new divisor, and place the result in the quotient as the third figure of the root, and also to the right of the new divisor in order to complete it. Then divide as before, and continue the operation above explained as long as any periods remain to be brought down. If there should be a remainder after all the periods in the number have been dealt with, the operation may be continued decimally by annexing two ciphers for each additional period till a sufficiently accurate result has been arrived at. |