818

THEORY AND PRACTICE OF ENGINEERING.

Book II.

To find the height of the upper part of a tower,as that from O to P. First find the height fromthe ground at S to O, then measure the distance tothe station R. Elevate then the alhidade until youcatch the point P ; remark where it stands on the*side A B, as at Z ; let R be distant 38 feet from S,and the height SO, for example, be 34 feet, and thenumber of parts on the instrument from A to Z40. Then as 40 : 38 :: 100 : 95, to which add theheight of the square, and we obtain the wholeheight of the tower ; from this sum subtract theheight SO, and the remainder will be the heightfrom O to P,

It is a well-known property of a right-angled tri-angle, that if the ratio of any pair of its sides beknown, the angles and ratios of the other sides maybe found ; this is indeed the principle upon whichtrigonometry is formed ; as there are three pairs ofsides in a right-angled triangle, differently related toeither of its acute angles, so there are three ratioswhich will determine the angle.

Let w be the angle, y the opposite side, .and x thecontaining side, and v the hypotenuse; the anglew may be indifferently determined by any of the

three numbers, ^ The first - is the sine of the

r x x r

angle w, the second - is the tangent, and the third

p

Fig. 1003.

Heights of Objects which arenot accessible may also be taken,for instance placing the geo-metric square at the station V,so that its centre is level withthe point R: turn its planeobliquely in such a manner thatby boning along the side D A,the point T is seen, and by boningalong the side D C, a piquetcan be placed in the visual ray, asat X, measured at a certaindistance from the station V.

Then remove the square to thepiquet X, and turn it in such amanner that by boning alongits side D A, we see in the linethe piquet V, and by the sightsof the alhidade the point T.

Remark the number of divi-sions from C to Y, where thealhidade stands, and state therule of three sum by placingfor a first term the number ofdivisions from C to Y ; in thesecond term the distance in feetfrom the two stations V and X ;and lastly 100 for the third term,being the side A D : the quotientgives the line VT. For thesecond operation, place thesquare at V, so that its centreshall be exactly in the sameplace as before, and its plane per-pendicular to the horizon, and its side DC parallel to it. Then elevate the alhidade untilthe point T is seen through its sights, and remark where the alhidade stands on theside C B, as at the point Z.

Fig. 1004.