822

THEORY AND PRACTICE OF ENGINEERING.

Book II.

-Vb

Fig. 1012.

Fig. 1013.

To draw any angle on a right line. — From the point or extremity of the given linedescribe an arc of any radius, keeping the compasses open ;apply it to the sector, opening it until the points fall upon the /

two 60° of the line of chords : then take the opening on the /

same line of chords of the degrees of the proposed angle:place one foot of the compasses where the arc touches thegiven line, and let the other fall on the arc through thispoint, and that of the extremity of the given line whence thearc is described; draw a right line, and it will be the requiredangle. As, if it is required to construct at the point A, onthe line A B, an angle of 56°, describe from the point A,the arc C D, of any radius : then with the same opening ofthe compasses carried to the two 60° on the sector, andplaced on the line of chords, which must be done byopening the limbs of the sector: then take the distance be-tween the two 56° on the same line, which will be equalto. the required angle, and apply it to the arc, when it willtouch the point E : then draw a right line through this pointfrom that of A, and the angle BAE will be one of 56°.

By the same means the opening of any rectilineal angle maybe measured, and its degrees ascertained : great care mustbe taken always to plumb dow r n the centres of the piquets, asthat of C E is found to be at D.

When the sector is used for surveying, the larger it is made, the lessliable it will be to error, and too much care cannot be observed in placingthe piquet, the centre of which, if planted obliquely in the ground as shownat C, will produce considerable error.

An angle in geometry denotes the inclination of one straight line toanother, and in this simple acceptation must be less than two right angles;but in trigonometry the term angle has a more extended signification. LetA B be a fixed line and A a given point in it, and suppose A E to revolvein one plane about A ; then the whole angular space described by AE inits revolution about A is called an angle, which may therefore in this casebe of any magnitude ; or if with the centre A and any radius, we describe acircular arc, subtending any angle A C D, this arc cannot, according to thegeometrical definition of an angle be greater than the semi-circumference of the circle ; butaccording to the trigonometrical definition, the subtending arc may be of any magnitude,consisting of any number of circumferences, or any portion of a circumference.

To form and measure angles by the sector : asfrom the point A to form and measure the angleB A C. Place the sector at the station A, the faceof the chords being uppermost, and bone by itssights one of the objects as B, and by the other limbthe object C, and the angle B A C will be formed.

To measure this angle, take in the compasses thedistance between the two 60° on the line of chords,and putting one foot in the centre of the sector,let the other fall on line of chords which will be at65°, which indicates the angle.

To measure distances by the sector, by forminga triangle of which the two sides are known as w-ellas the comprised angle : as to find the distancebetween B C, when the two sides D B and D C,with the comprised angle B D C, are known.

Place the sector at D, in order to form and mea-sure the angle BDC, which we will suppose tobe 83° : then measure the length of its sides, D Band BC, the first of which is 80 and the second75 feet; this being done, remove the sector from itsstand, and turn the under side uppermost, keepingit open at the angle BDC, 83°. Then place onefoot of the compasses on the line of equal parts atthe figure 80, the number of feet contained in theside DB, and open the other limb to 75 thenumber contained in D C: this opening of thecompasses, measured from the centre of the scalealong the line of equal parts, will give 108 feetfor the distance from B to C. Fig. 1016 .

Fig. 1014.

Fig. 1015.