practice, may be set down with nearly the same facility as the logarithms: we may thus practically dispense with introducing the tabular radius. Applying the formula in (51), which is the one commonly used when only one angle is required, since the numbers composing it generally lie near each other in the table, we have, to find C in the preceding example, . Example II. Given the two sides of a triangle 375 and 814, and the included angle=74° 47', to find the other angles and the third side. Supposing a 814, b=375, we have a+b=1189, therefore A=78° 23′ 30′′, and B=26° 49′ 30′′. To find the side c we have the proportion sin B sin C::b:c : log sin 26° 49′ 30′′ co ar 0 3455665 The side c may be found independently of A or B, by (55.); and, owing to the nature of the quantities concerned, with greater facility: for this purpose we have the equations log cos 37° 23' 30".... 9.9000955 log c........ ..... 2.9040998 ...c-801·86. The slight difference observed in the values of log c is owing to the imperfection of the tables. Example III. Suppose the two angles A and B to be 30o, and 42° 14', and the contained side c 27; required the other parts. By the equation A+B+C, we have C =107° 46', and by (48.) the sides a and b are found to be 14.4386 and 19 4099 respectively. Example IV. Given the angle A=23° 27', the side a 115 08, and the side c=75, to find the rest. By (48.) sin C csin A, the value of C must a be taken less than (vid. art. 58.) The angle B 2 will now be found from the equation A+B+CT, and the side from (48). Performing the calculations, we find, b C-15° 1' 54", B=141° 31′ 6′′, b=179.948. If, in this example, c had been greater than a, the angle C might have been taken either greater or less than (58). But when the case is applied to the solution of a problem, the nature of the inquiry will remove all ambiguity. (60.) In the determination of the heights and distances of objects, horizontal angles are observed by the theodolite; and vertical angles either by the theodolite furnished with a vertical arch, or by a quadrant having a plummet suspended from the centre, and furnished with open or telescopic sights. Lines upon the ground may be measured by a Gunter's chain, or a common measuring tape: but a better method consists in measuring upon a stretched rope in the direction of the line, by means of a seasoned deal rod twenty feet long. At the end of each rod, a pin may be stuck in the rope during the replacement of the rod. When the rope has to be drawn forward, the position of the end of the last rod must be accurately noted, and the measurement may be carried on as before. To take the angle of elevation or depression of an object, B or B' (fig. 10.) by means of a common quadrant, the eye at E or A respectively, must be directed through the sights, to the object; the plumbline will then mark along the arch graduated from D to E, the angle D A P, which is evidently the measure of the angle of elevation B A R, or of depression HA B'. (61.) When the necessary lines and angles have been measured, others must be calculated from these, so as to lead to the end in view; but as this depends upon the particular problem under consideration, no general rules can be given. The student will therefore have to depend upon his own re sources. Example I. Standing upon a horizontal plain, |