Chap. VIII.

GEOMETRY.

823

Fig. 1017.

The length of the building from I to K maybe similarly found, by placing the sector at L,and then measuring the angle ILK, and thelength LI and LK.

To measure distances with the sector by forminga triangle, two angles of which as well as theadjacent side are known: as to find the distanceA B, from the base line A C : construct and mea-sure the angle CAB, 84° from the station A,and the angle A C B, 47°, and the base line134 feet : then to obtain the other angle, ABC,add together the two known angles, and sub-tract them from 180°, the value of three anglesof any triangle, and the remainder, 49°, is thevalue of the angle ABC. To obtain thelength of A B, take in the compasses thelength of the side AC, 134 feet, from the centreof the sector along the side of equal parts; thenturning the face of the sector, place one limbof the compasses open to 134 at the 98 pointon the line of chords, which is twice 49, thevalue of the inaccessible angle ABC; thenleaving the sector open, take from the same lineof chords the value of twice the angle oppositethe side A B, which will be 94°; this measuredon the sector from the .centre along the lineof equal parts gives 129 feet for the side A B.

The side CB may also be found by takingtwice the angle CAB, 168°, instead of twicethe angle A C B, and following out the samemethod.

When angles are greater than 124*, its doublenot being contained on the line of chords, wemust, to find this double, subtract 124.} from180, the remainder, 55}, doubled will give 111for the double required.

To measure distances with a sector when theobjects are inaccessible, as that of the sideA B: select two stations at pleasure, as Cand D, 42 feet apart, or any other distance.

From the station C, form aryi measure withthe sector the angle D C B, which in this casewe will suppose to be 40}°, and then the angleD C A, 76}°: from the station D form andmeasure the angle C D A, 76}° and the angleCDB, 124}°.

These three angles, with the distance betweenthe stations C and D, form several triangleswhich may be done by the preceding rules,and it will be found that the side CA is 91feet long, and the side D A also the same di-mension. The same rules apply to the othertriangles, and the sides CB and DB may befound; and lastly the triangle DAB may befound, the side DA 91 feet, DB 108 feet, andthe angle A D B 48°; then following theprinciple which has been previously explained,viz. the method of measuring by the sector thedistances formed by a triangle of which we knowthe length of two sides, and the angle comprisedby them, it will be seen that the length A Bis 84 feet.

To obtain heights or distances by means oftrigonometry, it is only necessary that thelength of one line should be ascertained by

measurement, the magnitude of the angles being taken by observation; the sector, as wellas the other instruments, are graduated with great precision, but all observations require

3 g 4

Fig. 1018.

Fig. 1019.