become sines, tangents, cosines, or the like, and write upon them the proper designations accordingly. Then say,

As the name of the given side is to the given side,

So is the name of the required side to the required side. 4. To find an angle. Assume one side to be radius, and mark the remaining sides as before. Then say,

As the side made radius is to radius,

So is the other given side to the name of that side;

Which determines the opposite angle.

5. Applying this method to the rightangled triangle ABC, and calling the hypothenuse a radius, we shall have,

c = a sin. C÷R; hence sin. C = Rc÷a.
b a cos. CR; hence cos. C

Rba

Then, assuming the side b to be radius, we shall have,

cb tang. C÷R; hence tang. C

= Rcb.

If radius be called 1, the natural sines and cosines will be used in the application of these formulas; they are often more convenient than logarithms in railroad practice, especially when the numbers which measure the sides of the triangle are either less than 12, or are resolvable into factors less than 12. 6. The simpler relations between these natural functions are as follows:

:

1

ADJUSTMENT AND USE

OF

INSTRUMENTS.

IX.-XV.

1

ADJUSTMENT AND USE

OF

INSTRUMENTS.

IX.

GENERAL REMARKS ON ADJUSTMENTS.

1. Care should be taken in all instrumental adjustments, where screws work in pairs, to loosen one before tightening its opposite.

2. Remember that the eye-piece inverts the image of the cross-hairs, and that consequently any movement of it, by weans of the small capstan head screws on the outside of the telescope-barrel, should be in the direction which would seem to increase the error requiring correction.

3. Before beginning the adjustments, screw the object-glass close home, and make a pin-scratch across its rim and the end of the tube, by which to mark its proper place; draw out the eye-piece until the cross-hairs are exactly in focus; that is to say, until no movement of the eye shall appear to displace them, and bring the object to be observed clearly into view.

4. Never permit the glasses to be rubbed with a gritty fabric. To remove the dust from them, use a soft, clean handkerchief, and change often the part applied.