9

With the line of lines we operate on any given line that

will come within the opening of the sector ; and with the

line of chords we can work with any radius of similar ex-

tent. This last is constructed by making the lateral dis-

tance of the chord of 60 degrees, which is radius, equal in

length to the line of lines. All the intermediate degrees

between 1 and 60, are then set off laterally from the centre,

on both rulers, by taking on the line of lines a measure

equal to twice the natural sine of half the angle. Thus, for

the chord of 80 degrees, refer to SherwiiTs Tables, or to

others of equal authority, and find the natural sine of 15 de-

grees, which is 2588190, when radius is 10,000,000, and

the double of this sine is 5176380. Now the line of lines

as radius is equal to 100, in place of 10,000,000, and the

measure of the double sine must therefore be taken from

it in two places of figures, instead of seven. We see at

once that the length of the chord is between 51 and 52 on

the line of lines. Take in the compasses, as nearly as

possible, 51, and three-fourths, and this measured from the

centre on the line of chords, will give the chord of 80 de-

grees. The young draughtsman ought to exercise himself

and test his sector by this and similar operations ; for it is

equally important that he understand the structure of his

scale, and be able to ascertain that the various lines of his

sector have a due relation to each other.

The line of chords is principally used to protract and

measure angles.

To protract or lay down any angle less than 60 degrees,

say an angle of 30 degrees (Fig. 17.) Open the sector at

pleasure, and with the trans-

verse distance 60.60 in the

compasses, as a radius, describe

an arc of a circle ; take the

transverse distance of 30 de-

grees, and set it upon the arc;

then draw right lines from

the centre to the points on the

arc, and the required angle is formed. When it is de-

sired to measure any angle of not more than 60 degrees,

take in the compasses the transverse distance of 60.60 at

any opening of the sector, and with this radius describe,

from the angular point, an arc across the given angle ; take

the measure of the arc included in the given angle, in the

compasses, and apply this transversely to the line of chords,

and the similar divisions on which the points of the com-

passes fall will express the true measure of the angle.

To protract an angle of more than 60 degrees, take as

radius the transverse distance of 60.60 at any convenient

opening of the sector, and describe an arc as in the former

case; then take the transverse distance of one-half or one-

third of the given number of degrees, and set off twice or

three times on the arc, as the case may be ; afterwards

form the angle by right lines from the centre to the two

outermost points of measure on the arc. Thus, if the angle

is to contain 100 degrees, having described the arc, set off

50 degrees twice, and thus obtain the required measure.

Any angle of more than 60 degrees is measured in por-

tions, in like manner

To lay down an angle of less than 10 degrees, it is more

convenient to set off radius on the arc, and form an angle

of 60 degrees, and then deduct the difference between this

angle and the one required. Thus, suppose the angle to

be 7 degrees. Protract one of 60 degrees, and set off the

complement 53 degrees ; then the remainder of the arc will

contain 7 degrees, the required angle. The reason for thus

laying down small angles is, that the divisions of the sec-

toral line of chords are not so readily distinguished when

they approach within 10 degrees of the centre of the in-

strument. To measure any small angle, protract an angle

of 60 degrees, that shall include it, then take the comple-

ment to 60 degrees in the compasses, and this applied

transversely to the sector will show the measure of the

supplement, which being deducted from 60 degrees, the

remainder will express the measure of the given angle.

The line of chords is distinguished by the letter C on

each leg of the sector

The Line of Polygons.—This line is placed near the inner

edges of the sector, and marked POL on both scales. Its

use is to divide the circumference of a circle into a number

of equal parts, and to determine the sides of regular figures

that can be inscribed within the circle or described about

it. It is constructed by setting off, from a line of chords,

lateral distances equal to the chords of the central angles

of the square, pentagon, hexagon, heptagon, octagon, nona-

gon, decagon, undecagon, and duodecagon,—figures of 4,

5, 6, 7, 8, 9, 10, 11, and 12 sides respectively. The cen-

tral angle is found by dividing 360 degrees by the num-

ber of sides in the figure; and the length of the chord is to

be measured on a line equal in length to the sectoral line

of chords, but graduated to the full quadrant, or 90 de-

grees ; the reason of which is, that the chords of the cen-

tral angles of the square and pentagon, the one 90, and

the other 72 degrees, could not otherwise be contained in

the length of the sector.

To inscribe a regular polygon, say a figure of five equal

(Fi/j. is.) sides, in any given circle,

make the radius of the circle

; a transverse distance to 6, 6,

! the chord of 60 degrees, and

: the transverse of 5, 5, will

/ then give the side of the pen-

tagon. This set off 5 times

on the circumference of the

circle, and the points connected by chord lines, will com-

plete the figure. (Fig. 18.)

To construct an octagon on any given line, make the

line a transverse to 8.8 on the line of polygons (8 being

the number of equal sides in the figure); with the sector

thus set, take the transverse of 6.6 for a radius, and from

each termination of the given line describe arcs intersect-

ing each other. From the point of intersection, and with

the same radius, describe a circle passing exactly through

the terminations of the given line ; which thus becomes

one side of the required octagon, and is to be set off eight

times round the circumference of the circle to complete the

figure. To describe a regular polygon about any circle

13

(Fig. 17.)