4 tions, in which a fourth term is to be found from three given terms, by multiplying the fecond and third together, and dividing the product by the first, in working with the natural numbers, whether they be fides, or fines, tangents, or fecants of angles. Or, in working with the logarithms, add the log. of the 2d and 3d terms together, and from the fum fubtract the log. of the ift term; then the number anfwering to the remainder will be the 4th term required. To work a ftating inftrumentally, as fuppofe by the log. lines on one fide of the two-foot fcales.-Extend the compaffes from the firft term to the fecond, or third, which happens to be of the fame kind with it; then that extent will reach from the other term to the fourth, taking both extents towards the fame fide. Note. For the fides of triangles you use the line of numbers (marked Num.) and for the angles, the lines of fines or tangents (marked Sin. and Tan.) according as the proportion refpects fines or tangents. If the extent upon the tangents reach beyond the line, fet it fo far back as it reaches over. In a triangle there must be given three parts, one of which, at leaft, must be a fide; because the fame angles are common to an infinite number of triangles. In plane trigonometry there are three cafes or varieties only, viz. 1. When two of the three parts given, are a fide and its oppofite angle. 2. When there are given two fides and their included angle. 3. When the three fides are given. A Table 80 8 I 7920 660 63360 5280 1760 320 Note alfo, An inch is fuppofed equal to 3 barley corns in length, 4 inches-a hand, 6 feet, or 2 yards—a fathom, 3 miles a league, 60 nautical or geographical miles-a degree, or 69 ftatute miles nearly, 360 degrees, or 25000 miles nearly— is the circumference of the earth. PROBLEM I. Given Three Parts, fuch, that an Angle and its Oppofite Side are Two of them, to find the rest. In any plane triangle, the fides are proportional to the fines of their oppofite angles. That is, As one fide Is to another fide. :: So is fin. angle opp. the former : Note * DEMONSTRATION. Let ABC be any triangle: in AB affume any point D, take CE = AD, and upon AC demit the perpendiculars DF, EG, BH; then will DF and EG be the fines of the angles A, C, to the general radius AD or CE. Now, from fimilar triangles, we shall AB: BH :: AD: DF CB: BH AD (CE): EG have { B E D Λ F II G C and hence, of equality, AB : BC :; EG : DF. Note 1. To find an angle, begin the proportion with a fide oppofite a given angle; and to find a fide, begin with an angle oppofite a given fide. 2. An angle found by this rule is always ambiguous, except it be a right angle, or except that the magnitude of the given angle prevent the ambiguity; because the fine anfwers to two angles which are the fupplements of each other: and accordingly the conftruction gives two triangles with the fame given parts; and when there is no reftriction or limitation included in the propofition, either of them may be taken. The degrees, in the table, answering to the fine, is the acute angle; and if the angle be obtufe, take thofe degrees from 180°, and the remainder will be the obtufe angle. When the given angle is obtufe, or right, there can be no ambiguity; for then neither of the other angles can be obtufe, and the conftruction will produce but one triangle. 1. Draw the line AB fcale of equal parts. 345 from fome convenient 37° 20. 3. With the center в and radius 232, taken from the fame scale of equal parts, cross Ac in c. 4. Draw BC, and the triangle is conftructed. Then Then the angles в and c, measured by the scale of chords, and the fide Ac, measured by the fcale of equal parts, will be found to be as follows: viz. fubtr. 152 56 or 101 44 fub. from 180 00 180 00 leaves 21 04 or 78 16 the ▲ B. In the first proportion, Extend from 232 to 345 upon the line of numbers; that extent will reach, upon the fines, from 37° to 64° the angle c. In the fecond proportion, Extend from 37° to 27° or 78° upon the fines; that extent will reach, upon the numbers, from 232 to 174 or 374, for the fide AC. |
