4. Find the product of 999.75 and 75.85. Ans. 75831.667. 5. Find the product of 85, .097, and .125. Ans. 1.03062. Let DIVISION BY LOGARITHMS. 16. Proposition. The logarithm of the quotient of two numbers is equal to the logarithm of the dividend minus the logarithm of the divisor. ((1) b2=m; then, by def., log m = x. {8 (2) b'n; then, by def., log ny. 18. Examples. 1. Divide 73.125 by .125. 17. Rule. 1. Find the logarithms of the numbers, subtract the logarithm of the divisor from the logarithm of the dividend, and the remainder will be the logarithm of the quotient. 2. Find the number corresponding which will be the quotient. m n 2. Divide 7.5 by .000025. 3. Divide 87.9 by .0345. 5. Divide 85734 by 12.7523. -y. quotient = 585. Ans. 300000. Ans. 2547.824. Ans. 44.171. Ans. 6723. ARITHMETICAL COMPLEMENT. 19. Definition. The arithmetical complement of a logarithm is the result obtained by subtracting that logarithm from 10. Thus, denoting the logarithm by l., and its arithmetical complement by a. c. l., we shall have the formula, 10-l. a. c. l. The arithmetical complement of a logarithm is most readily found by commencing at the left of the logarithm, and subtracting each digit from 9 till we come to the last numeral digit, which must be subtracted from 10. Thus, to find the a. c. of 3.47540, we say: 3 from 9, 6; 4 from 9, 5; 7 from 9, 2; 5 from 9, 4; 4 from 10, 6; O from 0, 0. a. c. of 3.475406.52460. 20. Proposition. The difference of two logarithms is equal to the minuend, plus the arithmetical complement of the subtrahend, minus 10. For, -1(10)—10. It is convenient to use the a. c. in division when either the dividend or the divisor is the indicated product of two or more factors. Thus, let it be required to find x in the proportion: 37.5 678.5: 27.56: x; .* . x 678.5 27.56 .. log log 678.5+ log 27.56+ a. c. log 37.5 - 10. x = log 678.5 2.83155 a. c. log 37.5= 8.42597 log x = 2.69780 ・・・ x=498.656. 21. Examples. 1. Given 125.5 : .0756 :: x: .0034532, to find x. 2. Given 843 : x :: 732.534 : .759, to find x. 3. Given x .034 :: .784 : .00489, to find x. 32.015 .874 .000216 × 90257 Ans. 5.7325. , to find x. = 4. Given x= 5. Given .753 X 12.234 87.5 X 3.7547 :: 56.5x, to find x. Ans. 2014.96. · Ans. .87346. INVOLUTION BY LOGARITHMS. 22. Proposition. The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power. Let 24. Examples. Ans. 5.451125. Ans. 1.4353. 23. Rule. 1. Find the logarithm of the number and multiply it by the exponent of the power, and the product will be the logarithm of the power. 2. Find the number corresponding which will be the power. 1. Find the cube of .034. (1) log .034 2.53148 (1) X3 (2) log .0343-5.59444 ...0343.000039305. 2. Find the square of 25.7. Ans. 660.47. 3. Find the fourth power of .75. 4. Find the cube of 8.07. 5. Find the fifth power of .9. √ (1) EVOLUTION BY LOGARITHMS. 25. Proposition. The logarithm of any root of a number is equal to the logarithm of the number divided by the index of the root. Let (1) b*=n; then, by def., log n= x. b=n; then, by def., log √n= log n =(2) ... log in= r 26. Rule. 1. Find the logarithm of the number, divide it by the index of the root, and the quotient will be the logarithm of the root. 2. Find the number corresponding which will be the root. Ans. .3164. Ans. 525.55. Ans. .59047. 27. Examples. 1. Extract the square root of .75. (1) log .75 1.87506 (1)÷ 2 = (2) log V .75=1.93753 ... V .75 .86602. Scholium. 1.875062 (2+1.87506) 21.93753. ÷ 2. Extract the cube root of 91125. Ans. 45. 3. Find the value of V5. Ans. .89443. 4. Extract the fifth = root of .075. 3 5. Find the value of 37.5 X (.78)2 X r Ans. .676317. TRIGONOMETRY. 28. Definition and Classification. Trigonometry is that branch of Mathematics which treats of the solution of triangles. Trigonometry is divided into two branches - Plane and Spherical. PLANE TRIGONOMETRY. 29. Definition. Plane Trigonometry is that branch of Trigonometry which treats of the solution of plane triangles. 30. Parts of a Triangle. Every triangle has six parts-three sides and three angles. If three parts are given, one being a side, the remaining parts can be computed. If the three angles only are given, the triangle is indeterminate, since an infinite number of similar triangles will satisfy the conditions. 31. Sexagesimal Division of Angles and Ares. The horizontal diameter, O P, called the primary diameter, and the vertical diameter, O' P', called the secondary diameter, divide the circumference into four equal parts, called quadrants. O O' is the first quadrant, O' P the second, P P' the third, and P' O the fourth. P P' |