(Log. of -75.46, though incorrect, is used for the sake of brevity.) 6. Find the product of -0.017, 25, and 165.4. 7. Find the product of —14, —7.643, and −0.004. Ans. -.428008. DIVISION BY LOGARITHMS. 11. RULE. From the Logarithm of the dividend subtract the Logarithm of the divisor, and the remainder will be the Logarithm of the quotient. E. g. 1. Divide 78.46 by 0.00147. Log. of 78.46 66 Quotient, 53374.1. 1.894648 2. Divide 0.0014 by 756. Log. of 0.0014 2.878522 Log. 6.267606 Negative numbers can be divided in the same manner as positive, taking care to prefix to the quotient the proper sign, according to the rules of Algebra. 3. Divide .7478 by 0.00456. Ans. 163.99+ INVOLUTION BY LOGARITHMS. 12. RULE. Multiply the Logarithm of the number by the exponent of the power required. 1. Find the 15th power of 1.17. Ans. 10.538. Log. of 1.17 0.068186 Log. 1.022790 H Ans. 4.401765+. 3. Find the 4th power of 0.0176. Ans. 0.000000095951+. 5. Find the 3d power of -.017. Ans. -0.000004913. Ans. 7529536. L EVOLUTION BY LOGARITHMS. 13. RULE. Divide the Logarithm of the number by the exponent of the root required. Negative numbers are evolved in the same manner, taking care to prefix to the root the proper sign, according to the rules of Algebra. For the sake of convenience, where the characteristic of a Logarithm is negative, and not divisible by the index of the root, we can increase the negative characteristic so as to make it divisible, providing we prefix an equal positive number to the decimal part of the Logarithm. g. 1. Find the 5th root of 0.0173. Log. of 0.0173 is 2.238046, which is equal to 5+3.238046, and dividing this by 5 gives 1.647609, which is the Log. of 0.4442. 14. Instead of subtracting one Logarithm from another, it is sometimes more convenient to add what it lacks of 10, which difference is called the complement, and from For the sum reject 10. The result is evidently the same. xy=x+(10-y)-10. The complement is easiest found by beginning at the left of the Logarithm of the number, and subtracting each figure from 9, except the last significant figure, which must be subtracted from 10. In proportion, therefore, we have the following rule: Add the complement of the Logarithm of the first term to the Logarithms of the second and third terms, and from the sum reject 10. E. g. 1. Find a fourth proportional to 14, 175, and 7486. Complement of Log. of Ans. 93575. 2.243038 2. Given the first three terms of a proportion, 416, 584, and 256, to find the fourth. 3. Find the value of 179 × 4968 489. Ans. 359.38+. Ans. 1818.552+. Ans. 4.7776+ 331.9 (√2.04 V1.203) Ans. 0.000197055. 7. In a system whose base is 4, what is the Logarithm of 4? of 16? of 64? of 2? of 8? of 1? of? of? of } ? of 0 ? 8. Solve the equation 125* = 25. CHAPTER II. TRIGONOMETRIC FUNCTIONS. DEFINITIONS. 15. Trigonometry is that branch of mathematics which treats of methods of computing angles and triangles. 16. Plane Trigonometry treats of methods of computing plane angles and triangles. 17. The circumference of every circle is divided into 360 equal parts, called degrees (°), each degree into 60 equal parts, called minutes ('), and each minute into 60 equal parts, called seconds ("). 18. As angles at the centre vary as their arcs, or arcs as their corresponding angles, the measure of an angle is the arc included between its sides and described from its vertex as a centre (Geom., III. 14). 19. As the sum of all the angles about the point C is equal to four right angles, one right angle, A C B, would be measured by one quarter of the circumference, or 90° (Geom., III. 15). 20. The Complement of an arc or angle is 90° minus this arc or angle. Thus, the arc AD is the complement of D B, and the angle A CD of DC B. When an arc or angle is greater than 90°, its complement is negative. 21. The Supplement of an arc or angle is 180° minus this arc or angle. Thus, the arc EAD is the supplement E I A B of D B, and the angle ECD of DC B. When an arc or angle is greater than 180, its supplement is negative. 22. The Sine of an arc or angle is the line drawn from one end of the arc, perpendicular to the diameter passing through the other end; or it is half the chord of double the arc. Thus, DF is the Sine of the arc D B, or of the angle DC B. · 23. The Versed Sine of an arc or angle is that part of the diameter which E H I K Thus, BF is the is between the foot of the sine and the arc. 24. The Cosine of an arc or angle is the sine of the complement of the arc or angle, or the radius minus the versed sine of the arc or angle. Thus, HD CF is the Cosine of the arc BD, or of the angle BCD. (Co in Cosine, &c., stands for complement.) = 25. The Tangent of an arc or angle (in Trigonometry) is the line touching one extremity of the arc, and terminated by a line drawn from the centre through the other extremity. Thus, BI is the Tangent of the arc BD, or of the angle BCD. 26. The Cotangent of an arc or angle is the tangent of the complement of the arc or angle. Thus, A K is the Cotangent of BD, or of the angle B C D. 27. The Secant of an arc or angle (in Trigonometry) is the line drawn from the centre through one end of the arc, and terminated by the tangent to the other end. Thus, CI is the Secant of BD, or of the angle BCD. 28. The Cosecant of an arc or angle is the secant of the complement of the arc or angle. Thus, C K is the Cosecant of BD, or of the angle BCD. *Those who prefer the Analytical Method will turn from this point to Chapter IV. |