kla B Prob. 5; then draw two others at the same distance, across the former, as N and O; by the crossing of these lines will be made a figure ABCD, of four sides; extend the compasses at pleasure, and setting one foot in D, describe the arch cde; with the seme extent, set one foot in B, and describe the arch fgh; then set one foot in C, and contract them so as to reach the point e, and describe the arch Im; with the same extent, and one foot in A, describe the arch ik, and the oval will be completed. In the same manner, with a greater or less extent of the compasses, may a greater or less oval be made by the same four sided figure ABCD.. A MA D g N OF PLAIN TRIGONOMETRY. RIGHT AND OBLIQUE ANGLED. PLAIN Trigonometry is that science, by which we measure the sides and angles of plain triangles. SECTION I. Of Rectangular Trigonometry. In a right triangle, the longest side is usually called the hypothenuse, the next longest, the base, and the shortest the perpendicular. Logarithmick sines, tangents, and secants, are called the tabular sides of a triangle, and are the sines, &c. of the opposite angles. The length of the sides are called the natural sides. All the three angles of a triangle are equal to two right angles, or 180°. The proportion ought to be made between sides and sides; and between angles and angles. When a side is required, any side (whether known or not) may be made radius; but when an angle is required, then a known side only, must be made radius. Note. A side is said to be made radius, when one foot of the dividers is set in one end of the side, and such a circle described, of which the side is the semidiameter: Also, that when the hypothenuse is radius, it is the sign of the right angle, or 90°, and the base and perpendicular, usually called the legs, become sines of their opposite angles: but when one of the legs is made radius, the other becomes the tangent of the opposite angle, and the hypothenuse, the secant of the same angle. Tangent's radius is 45°. When a side is to be found, the two first terms of the proportion must be tabular sides, and the last a real one: but when an angle is to be found, the two first terms must be real sides, and the third, a tabular one. The given parts, whether sides or angles, are marked with and the part required, with O. Angles are measured by the arch of a circle. The periphery of every * To work on the fcale with a fecant, you must take the fines backward, that is se fines for 10 fecants, &c. every circle, whether great or small, is divided into 360 degrees, each degree into 60 minutes, every minute into 60 seconds, and so on, to thirds, fourths, &c. L H D E - Any portion of the periphery of a circle, as ECF, is called an arch, and a line drawn from the ends of an arch, as, EIF, is called the chord of the arch. Half the chord of any arch, as EI, is called the sine of the arch EC, and IC it called the versed sine of the same arch EC: So, also, EG is the sine of the arch ED. A line drawn perpendicular to the diameter of a circle, so as to touch the circle and not cut it, is called a tangent, as CH, which is the tangent of the arch EC, because the line BH, drawn from the centre B, through E, called the secant, meets it in the point H. A B C I The complement of an arch is the remainder, after the arch is takfrom 90°; thus KD is the complement of the arch ALK, taken from the arch AD. The cosine or sine complement of an arch is the sine of the complement of that arch, as ED is the complement of EC. PROBLEM I. The Angles and one of the Legs given, to find the Hypothenuse and other Leg. EXAMPLE. In the triangle ABC, right angled at B, suppose the leg AB, 86 equal parts (as feet, yards, miles, &c.) the angle A=33°, 40, and the angle C-56°, 20': Required the length of the hypothenuse AC, and the leg BC? Geometrically. Draw AB equal to 86, from any line of equal parts, then upon the point B, erect the perpen- same line of equal parts that AB was taken from. C B By Calculation. By making the hypothenuse radius, the leg's will become the sines of the opposite angles; and as natural sides are required, the proportions must begin with tabular sides: therefore, for the hypothenuse, As the sine of C Is to radius So is the side AB To the side AC 56°,20′ 9.92027 90,00 10. 86 1-934.50 Here, I add the logarithms of the 2d and 3d terms, and from their sum subtract the first, and the remainder is the logarithm. 11-93450 of the side sought, which 9-92027 gives 103.3. The same must be done in all the folFor 103.3 2.01423 (lowing cases. For the Leg BC. As the sine of the angle C 56°,20′ 9.92027 So is the side AB To the side BC It might have been as easily found by the following proportion : As R: S,A :: AC: BC. PROBLEM II. The Angles and Hypothenuse given, to find the Legs. EXAMPLE. In the triangle ABC, suppose the hypothenuse AC= 146, the angle A= 36°,25', and the angle C = 53°,35': Required the legs AB and BC? Geometrically. Draw the line AB at pleasure, and make the angle A = 36°,25′; then take AC = 146 from any line of equal parts; lastly, from the point C let fall the perpendicular CB on the line AB: so the triangle is constructed, and AB and BC may be measured from the line of equal parts. B By Calculation. Making AC radius, the legs become sines, as before, and as the angles are given to find the sides, we must begin the proportion with angles, or tabular sides. As we had before found AB, the proportion might have been, As S.C: S,A :: AB : BC. PROBLEM III. and IV. The two Legs given, to find the Angles and Hypoth enuse. EXAMPLE. In the triangle ABC, suppose the leg AB=94, and BC 56: Required the angles and hypothenuse. = Geometrically. Draw AB 94 from any line of equal parts, then, from the point B raise BC perpendicular to AB, and take BC from the former line of equal parts = 56; lastly, join the points A and C with the straight line AC, so the triangle is constructed. AC may be found by taking it in your dividers and applying it to the line of equal parts; and C the angles may be measured by the 6th Geometrical Problem. By By Calculation. 1st. For the angle A; supposing the base AB the radius, then the hypothenuse becomes secant of the angle A, and the perpendicular BC, the tangent of the angle A and as an angle is required, we must begin the analogy with a natural side. As AB Is to BC 94 1.97313 56 1.74819 So is tangent's radius 45°,00 10. 30°,47′ 9.77506 The perpendicular might have been made radius, and then the proportion would have been, as BC: AB:: tan. rad. : tang. of C. Now, as we have found the angle A, and as the angles A and C, taken together, are equal to 90°, therefore from 90°,00 Take the angle A = 30°,47 And we have the angle C = 59°,13′ 2d. For the Hypothenuse. The base still being radius, we have this analogy for finding the hypothenuse: as T. R: Sec. A: AB: AC. But this may be done without the help of secants: for, having found the angles, we may now make the hypothenuse radius; and as a natural side is required, we must begin the proportion with a tabular side; therefore, As the sine of C 59°,13' 9.93405 Is to radius So is AB To AÇ 90,00 10. 94 1.97318 109.4 2.03908 Or the analogy might have been, as S. C: R:: BC: AC. PROBLEM V. and VI. The Hypothenuse and one of the Legs given, to find the Angles and other Leg. EXAMPLE. In the triangle ABC, suppose the leg AB-83, and the hypothenuse AC 126: Required the angles A and C, and the leg BC? = Geometrically. Draw A B = 83 from any line of equal parts; and from the point B, raise the perpendicular BC of any length, then take the length of AC 126 from the same line of equal parts, and setting one foot of the dividers in A, with the other cross the perpendicular BC in C; lastly, join AC, so the triangle will be constructed, and the angles may be measured as directed in Problem 3d and 4th." By Calculation. First, for the angle C; and as an angle is required, we must begin with a side, making the hypothenuse radius. As AC 126 2-10037 And we have the angle A 48°,4S' For the side BC. As a side is now required, we must begin with an angle; therefore, As radius 90°,00' 10. SECTION II. Of oblique angular Trigonometry. In any triangle, the sides are proportional to the sines of the opposite angles. When two angles of any triangle are given, their sum, being subtracted from 180°, leaves the third angle; and when one angle is given, that being subtracted from 180°, leaves the sum of the two unknown angles. When any angle exceeds 90°, subtract it from 180°, and work with the remainder. When the given and required parts, viz. sides and angles are opposite. As the sine of any angle is to the sine of any other angle; so is the side opposite to the first angle, to the side opposite to the other angle. Or, as one side is to any other side; so is the sine of the angle opposite to the first side, to the angle opposite to the other side. When any tavo sides with the angle included between them are given. RULE 2. As the sum of any two sides is to their difference; so is the tangent of the half sum of the two opposite angles, to the tangent of half the difference of those two opposite angles; which half difference being added to the half sum, gives the greater of the two angles, and, being subtracted from the half sum, leaves the less of the two unknown angles. When the three sides are given, to find the Angles. RULE 3. As the base of any triangle (or sum of the segments of the base) is to the sum of the other two sides: so is the difference of those sides, to the difference of the two segments of the base, made by letting fall a perpendicular to the base from the angle opposite to it; half of which difference, being added to half the sum of the two segments, gives the longest, and being subtracted, leaves the shortest. The learner being now somewhat acquainted with the common method of working by logarithms, it will be proper to shew how to perform those proportions without subtracting the first number from the sum of the second and third, which is done by setting down the arithmetical complement of the first term instead of the logarithm. This may be readily done thus; subtract the first figure of the logarithm from 10, and set down the remainder: then subtract each of the |