Chat. VIII.

GEOMETRY.

825

To measure with the sector inaccessible heights , as that of the line A B. Place two piquetsor stations, as at C D. in a right line with the inaccessible line A B : measure the distancebetween the stations C and D, which is 72 feet.

Place the sector at these stations, mounted onits foot, to form and measure the acute angleBCD 29°, and the obtuse angle C D B 130°, withthe adjacent side DC, 72 feet. To find thesides BC and B D, follow the previous rule, andthe unknown angle A B C will be 21°, the sideB C 153 feet, and B D 97. To get the height G Band afterwards the whole height A B, observethe triangle D B G, in which the angles G D B and B DG are required : we shall have the first,

G D B , by subtracting from 180° the angle C D B,

130° ; the remainder 50° will be the value of theangle G D B , and if from 90° we subtract this50, we shall have 40° for the angle D BG. In the triangle DBG, find the two anglesG D B 50°, and DBG 40°, with the adjacent side D B 97 feet. The length G B is known,by the method before cited for measuring distances by means of a triangle, where two anglesand the adjacent sides are known, and it will be found that the height G B is 74 feet,to which add 5 feet for the height of the sector, it will be found that the height A B is 79feet.

Of the Astrolabe and its application. — This instrument is of wood or brass, andconsists of a round plane, a foot or 18 inches in diameter, having two faces, one ofwhich is sunk and called the sea, the other

Fig. 1023.

Fig. 1024.

the altimetric scale ; it is an ancient invention,and is said to have been in use as earlyas the days of Ptolemy . The sunk faceof the astrolabe contains several lines orbands of metal; the first has engraved on itseveral stars, and on the others are circlesof azimuth, &c. for different elevations of thepole, serving to examine and ascertain themotions of the stars. On the other face,which is used by geometricians, and whichcomes more under our notice, are severalgraduated circles, serving to take the heights ofvarious objects above the horizon. Thesecircles are divided into four equal parts bythe two diameters AB and CD, of whichA B, descending from the ring of the in-strument, represents vertical lines, and thediameter C D, which cuts A B at a right an-gle in the point E, represents the horizon orthe level of sea or land. Below the horizontal line C D is an oblong figure F G H I, whichis called the altimetric scale, and is composed of two geometric squares, each havingtwo of their sides divided into twelve equal parts. We count from the horizontal lineC D, or from the points F and G, the divisions on the sides FI and G H, on which versedshadow is written, and those on the other two sides which compose the long side I H, onwhich right shadow is written, from right to left of the vertical line A B.

At the centre of the astrolabe is an alhidade fitted with sights, and at the vertical pointA of the instrument a ring passes through a movable socket, which serves to suspend theastrolabe in a vertical position.

Measuring distances with the astrolabe , when onlyone of the objects is accessible : as the distancefrom the station M to the inaccessible point N,its edge being level with the ground at M. Plantthe staff MO, about 5 feet in length, perpendicularat the station M, and divide it into twelve equalparts, without comprising the portion which isplanted in the earth. Then, holding the astrolabeby its ring, raise or lower it until in the samevisual ray, when looking from the top of the staffO, through the sights of the alhidade, the point Nis discovered. Observe then at what point of thealtimetric scale the alhidade is standing, as at thethird point of the side GH, versed shadow. Then

Fig. 1025.