2. To raise .09163 to the 4th power. 4 3. To find the cube of 3.07146. Log. 3.07146 = 0.487345 3 Power, 28.9758 = 1.462035 4. To raise 1.0045 to the 45th power. 1950 45 9750 7800 Power, 1.22391 = 0.087750 EVOLUTION BY LOGARITHMS. 23. When it is required to extract roots by means of logarithms, find, in the table, the logarithm of the number whose root is to be extracted; divide this logarithm by the index of the root : the quotient is the logarithm of the root sought. If the logarithm of the number whose root is to be found, have a negative characteristic, and this be not divisible by the index of the root, it must be increased by so many negative units as will make it divisible by the index of the root, and carry the units borrowed, as so many tens to the decimal part of the logarithm. The reason of which is obvious. For, if the characteristic be not divisible by the index, the remainder, which is negative, cannot be carried to the decimal part of the logarithm, which is positive. But, by adding the negative units, and as many positive units, the value of the logarithm is not changed, and the difficulty is avoided. 1. To find the square mot of 365. Log. 365=230 1233 2. To find the 10th root of 2. Loc. ? = 0.301031) Divided by 10 = 001303 3. To find the square mot of 0.093. Log. 0.033 = 2.9083 Divdet by 2, gres 1.45+241; 4. To find the cube root of 0.00048. Log. 0.00048 = 2.6SP1 Divided by 3. gives 2.893747 1 CHAPTER II. PLANE TRIGONOMETRY. 24. PLANE TRIGONOMETRY is that branch of Mathematics which treats of the methods of finding, by calculation, the unknown sides and angles of a plane triangle, from those sides and angles that are given. 25. For the purposes of trigonometrical calculations, the circumference of the circle is supposed to be divided into 360 equal parts, called degrees; each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds. 26. As the circumference of a circle may be regarded as a proper measure of angles, having their vertices at the centre, the four right angles, which can be formed about the same point, are measured by 360 degrees; two right angles, by 180 degrees ; one right angle, by 90 degrees; and an angle less than a right angle, by an arc less than 90 degrees. 27. Degrees, minutes, and seconds are usually designated by the respective characters, ° ' ". Thus, 16° 12' 15" is read, 16 degrees, 12 minutes, and 15 seconds. 28. The complement of an angle is what remains after subtracting the angle from 90°. The sum of an angle and its complement, is equal to 90°. 29. The supplement of an angle is what remains after subtracting the angle from 180°. The sum of an angle and its supplement, is equal to 180°. 30. The sine of an angle is the perpendicular let fall from one extremity of the arc which measures it, on the diameter passing through the other extremity. Thus, BD (Pl. I. Fig. 1) is the sine of the angle AOC, or of the arc AB. 31. The main of an angle, or arr, is the part of the drameIrr interempted between the foot of the side and centre. OD is the cuts.De AB The of an are is the be which touches it at Ont' runan, and as and brain arun through the other CHIRON and to cut Gr AC s te tangent of The ARR T: RVA er et me du fait centre tornak , and by het tok Ceria. (the XA IR cos. A sin. A R? Sin. A : cos. A :: R: cot. A=R cos. A COS. A sin. A Also, OAC and OTE being similiar, OA : AC :: ET : OE, or, R2 =tang. A cot. A. We see, from these relations, that the cosine of an arc can be found, when radius and the sine are given; and if radius, the sine, and cosine be known, that the secant, cosecant, tangent, and cotangent are determined from them. 38. We see, in examining the figure, that if the point B coincides with the point A, or the arc AB=0, the sin. AB becomes =0, the tang. AB becomes =0, the sec. AB=R, the cos. AB =R; the cot. AB, and cosec. AB, become infinite, since OC and ET are then parallel. When the arc AB=45', sin. A= cos. A, tang. A=cot. A, sec. A= =cosec. A. When the arc AB becomes AE, or equal to 90°, then the sin. A, or sin. 90°= R; cos. A, or cos. 90o=0; tang. A and sec. A become infinite, as the lines AC and OC do not intersect. 39. If the arc ABEF, greater than 90%, be considered, FH is its sine (30); OH, its cosine (31); AQ, its tangent (32); and OQ, its secant (33). But FH is the sine of the arc GF, the supplement of ABF (29), and OH is its cosine (31); hence, the sine of an arc is equal to the sine of its supplement ; and the cosine of an angle, to the cosine of its supplement. Furthermore, AQ is the tangent of the arc AEF (32), and OQ is its secant (33); GL is the tangent, and OL the secant, of the supplemental arc GF. But, as AQ is equal to GL., and OQ to OL, it follows, that the tangent of an arc or angle is equal to its supplement; and the secant of an arc, to the secant of its supplement.* * These relations are between the values of the trigonometrical lines ; the algebraic signs which they have in the different quadrants, are not considered. |