824

THEORY AND PRACTICE OF ENGINEERING.

Book II.

correction by other methods, which do not depend upon the accuracy of the instrumentnor the mechanical skill; one of the most important is the principle of repetition, by whichany error in the graduation, and in reading off the number of degrees, to which a singleobservation is liable, is divided among many repeated observations, so that by a sufficientnumber of repetitions the required angle can be accurately obtained, or nearly so : anothermethod for correcting erroneous graduation and reading off, consists in taking the mean ofseveral readings upon different parts of the instrument.

To find by the sector the three angles of a triangle , when the threesides are known. Take in the compasses from the centre ofthe line of equal parts the length of the side opposite to the angleto be ascertained, and carry this opening to the same equalparts, and open the sector in such a manner that each foot of thecompasses answers to the two other known sides. Keep the sectorat this opening, and turn it over ; take in the compasses thedistance between the 60° on the line of chords, which being mea-sured from the centre of the same line will give the requiredangle.

In the triangle ABC the three sides of which are known,viz. A B 10 feet, B C 7, and C A 13 feet, it is required to know theangle A 15 C. The side C A opposite the required angle being13, take in the compasses the distance between the centre ofthe sector and the 13 point in the line of equal parts. Applythis length to the sector, opening and shutting it until each foot ofthe compasses (open to the 13 parts) answers precisely to 10 feetof A 15 and the 7 of 15 C : keep the sector at this opening, andturn it over, taking in the compasses the distance between thetwo G0° on the line of chords; place one foot in the centre of the sector, and the other legfalling on the same line of chords will indicate 98° 31' for the angle ABC. By thesame process it will be found that the angle 15 C A is 49° 15’ and CAB 32° 14'.

To measure with the sector heights whose feet are ac-cessible _The height of the line A 15, whose foot B is

accessible, is found by taking a station at any distance,as at F, 40 feet, and planting the sector there, mountedon its foot, with its line of chords or visual rayparallel with the horizon. Then lift the other limbof the sector, until the top on the point B is discoveredthrough its sights : this will give 23° for the angleDC 15, and as the visual ray cuts the line A B in aright angle at the point D, a triangle will be formedof which two angles and the adjacent sides are known,viz. the angle D C B, 23°, the angle C D B, 90°, withthe adjacent side DC 40 feet. The angle CDBwill be found by taking the known angle D C 15 23°from 90°; the remainder, 67°, will be the unknownangle C B D. To have the side D B, take in the com-passes from the centre of the sector, and along theline of equal parts, the value of the known side DC,

40 feet, and carry this to twice the angle C B D,

67 = 134 on the line of chords. Take in the com-passes on the same line twice the angle which is op-posite to the required side B D, which will be 46, theangle D C B being 23, and keeping this opening,turn the sector to measure it on the line of equalparts, which will give 16 for the number of feet fromD to B, to w’hich sum the height of the sector ECmust be added for the whole height of the line A B.

In like manner, by placing the sector at B, the height F G may be ascertained.

The sector is still much employed in France , where it is called the compass of proportion:its principle depends upon the fourth proposition of Euclid’s sixth book, in which it is shownthat equiangular triangles have their homologous sides proportionals. The scales now putupon sectors are divided into single and double; the former has a line with inches dividedinto eighths or tenths, a second into decimals containing 100 parts, a third into chords, on afourth are sines, on a fifth tangents, on a sixth rhumbs, on a seventh and eighth latitudesand hours, &c. &e. The double scale contains a line of lines, a line of chords, another ofsines, a fourth of tangents to 45°, a fifth of secants, a sixth of tangents above 45°, and aseventh of polygons.

Thg single scales may be used when the sector is either open or shut, blit the doublerequire it to be opened.

Fig. 1021

Fig. 1022.

Fig. 1020.