7

satisfactory if accurately constructed ; but there can be no

question, that one with a vernier applied to the first sub-

divisions, would give minute measures with much greater

certainty, and no case of instruments ought now to leave

the maker without having this addition on one face of

the plain scale.

The diagonals may be safely applied to a scale where

only one subdivision is required. Thus, if seven lines be

ruled, inclosing six equal spaces, and the length be divided

into primaries, as A B, B C, &c., the first primary AB, may

be subdivided into twelfths by two diagonals running

from 6, the middle of A B, to 12 and 0. We have here a

{Fig 13.)

A 6 B C

7/\5

8/ \4

_ 9/ \3

10/ \2

11/ V

12 O 1 2 5 4

very convenient scale of feet and inches. From C to 6, is

1 foot 6 inches; and from C, on the several parallels to

the various intersections of the diagonals, we obtain 1 foot

and any number of inches from 1 to 12. All of which is

evident from the figure.

On the face of the plain scale that carries the diagonal

one, there is usually a line of inches and tenths, and

underneath it, a decimal scale. These can be used sepa-

rately, and in conjunction ; and in the latter case the

primaries of the decimal scale being taken as feet, the

subdivisions of the upper line are inches.

Line of Chords. — This is usually introduced on the

plain scale. It is an unequally divided scale, giving the

length of the chord of an arc, from 1 degree to 90 degrees.

The quadrant, or quarter of a {Fig. u;.

circle, A C, contained between the

two radii at right angles, BA and

B C, has its extremities joined by

the line A C, to which the measures

of the chords are to be transferred.

The quadrant is divided accurately

into ten equal parts ; then from C

as a centre, each division is transferred by an arc to the

line A C, and the chords of every 10 degrees obtained.

These primary divisions can be subdivided into tenths of

1 degree each, by division of the corresponding arcs. This

is rather an illustration of the construction, than a true

method of performing it. A line of chords can be laid

down accurately only from the tabular sines, delicately

set off by the beam compasses. In using this scale, it is

to be remembered that the chord of 60 degrees is equal to

radius. Therefore to lay down an angle of any number

of degrees, draw an indefinite straight line; take in the

compasses the chord of 60 degrees, and from one termina-

tion of the line, as a centre, describe an arc of sufficient

extent; then take from the scale the chord of the required

angle, and set it off on the arc; lastly, draw another line

from the centre cutting the arc in the measure of the chord.

To ascertain the degrees of an angle, extend the angular

lines if necessary, that they may be at least equal to the

chord of 60 degrees; with this chord in the compasses

describe an arc from the angular point; then take the

extent of the arc and apply it to the scale, which will

show the number of degrees contained in the angle.

The Plain Protractor.—The plain scale is sometimes

made of greater width, in order to contain all the preced-

ing lines, and also a protractor for setting off and measur-

ing angles. The most eligible form for this instrument is

the circle or half circle, which construction will presently

come before us. It will suffice for the present to say, that

the plain scale protractor is a portion of a semicircle, having

radii drawn from its centre to every degree of its circum-

ference. If, therefore, the centre on the lower side is

made to coincide with a given point, an angle of any

number of degrees may be measured or set off around its

edges.

A small roller is sometimes inserted in a slot to make

the plain scale serve the purpose of a parallel ruler, but

considerable care is necessary in thus applying it, lest the

roller slide or shift at either extremity.

Double Scales.

Each of the scales we have described has a fixed measure

that cannot be varied ; but Ave come now to speak of those

double scales in which we can assume a measure at con-

venience, and subdivide lines of any length, measure chords

and angles to any radius, &c.

The Sector.—This instrument consists of two fiat rulers,

united by a central joint, and opening like a pair of com-

passes. It carries several plain scales on its faces, but its

most important lines are in pairs, running accurately to

the central joint, and making various angles according to

the opening of the sector. The principle on which the

double scales are constructed, is contained in the 4th Prop,

of the 6tli Book of Euclid, which demonstrates that “ the

sides about the equal angles of equiangular triangles are

proportionals, &c.” Noav

let A C I (Fig: 15) be a

sector, or in other words,

an arc of a circle contained

between two radii; and let

C A, C I, be a pair of sec-

toral lines, or a double

scale. Draw the chord A I,

and also the lines B IT, D G,

E F, parallel to A I. Then shall C E, C D, C B, C A, be

proportional to E F, D G, B H, and A I respectively.

That is, as C A : A I : : C B : B IT, &c. Hence at every

opening of the sector, the transverse distances from one

{Fig. 15.)