Remark, that to find a side, any side may be made radius : then say, as the name of the side given is to the name of the side required ; so is the side given to the side required. But to find an angle, one of the given sides must be made ra dius : then, as the side made radius is to the other side ; so is the name of the first side (which is always radius) to the name of the second side ; which fourth propor tional must be found among the sines, or tangents, &c. to be determined by the side made radius : against it is the re quired anglf:. In a right-angled triangle you ,must always have two sides, or the angles and one side given, to find the rest.
Axiom II. In all plane triangles, the sides are in direct proportion to the sines of their opposite angles. Thus, " if two angles and one side be given, to find ei tiler of the other legs." In fig. 22, the angle B C D is 101° 25', the angle C B D is 42', and the given kg B C is equal to 76. of the scale assumed : to find the sides CD and BD.
To find DC As the sine angle D 101° 25' . 9.99132 Is to the side BC 76 . . . . 1.88081 So is sine angle B 44° 42' . . 9.84720 11.72801 9.99132 -- To the side DC 54.53 . . 1.73669 The foregoing is worked by logarithms, thus : add the logarithm of the second and third terms together, then deduct the logarithm of the first term, and the re mainder is the logarithm of the fourth term, or number sought. When an an gle is greater than the sine, tangent, and secant of the supplement, (i. e. of the number of degrees wanting of are to be used.
" Two sides, and an angle opposite to one of them, being given, to find the other opposite angle, and the third side, fig. 23." The side B C 10.6, D B 65 miles, and the angle B C D 31° 49' given, to find the angle B D C obtuse, and the side CD.
• To find D. As the sine B D 65 . • . • 1.8129; Is to the angle C 31°49' . . • 9.72198 So is the side B C 106 . . . 2.02531
11.74729 1.81291 To sine angle D 43' . . 9.93438 To find 1)13 As sine angle C 49' . . • 9 72198 Is to the sine B D 65 . . 1.81291 So is sine angle B 27.28 . . 9.66392 11.47683 9.72198 To the sine D C 56.88 . . . 1.75485 • Axiom HI. In• every plane triangle it will be as the sum of any two sides is to - their difference ; sb is the tangent of half the sum of the angles opposite there, to the tangent of half their difference. Which half difference, being added to half the sum of the angles, gives the greater ; but if subtracted, the remainder will be the lesser angle.
" Two sides, and their contained angle given, to find either of the other angles, and die third side, fig 24." The side B C 109, B D 76 leagues, and the angle C B D 101° 30' being given, to find the angle B D C, or B C D, and the side C D.
' Axiom IV. In any plane triangle, as the base, or greater side, is to the sum of the ether two sides ; so is the difference of the sides to the difference of the seg ments of the base, made by a perpendi cular let fall from the angle opposite to the base : and if half the difference of the segments be added to half their sum, it will give the greater segment ; but if sub tracted, the remainder will be the lesser segment. The triangle being thus cut, becomes two right angled triangles ; the hypothenuses and bases of which are given to find the angles by Axiom I.
I-laving divided the right-angled trian gle into two right-angled triangles, the bypotbenuses and bases of which are given, to find the angles by Cunter. 1. The extent from 105 to 1.b will reach from 35 to 45 on the line of sines. 2. The extent from 85 to 75, on the line of num bers, will reach from radius to 61° 56', the angle B DA on the line of sines. 3. T,he extent from 50 to 30, on the line of , numbers, will reach from radius to angle A D C 36° 53', on the line of sines.