13

The length of a logarithmic tangent is measured off

from the commencement of the line at the left hand, by

extending the compasses to the degree or minute required.

We give two examples of the application of the scale to

the solution of questions in trigonometry. 1. The base of

a right-angled triangle is 25, and the perpendicular 15,

what is the angle opposite to the perpendicular ? Here,

if the base is considered radius, the perpendicular will be

the tangent of the angle opposite to it; therefore,

As 25 :15:: Radius: Tangent.

Extend the compasses from 15 to 25 on the line of num-

bers, and this opening will reach backwards from 45

degrees on the line of tangents to 31 degrees, the angle

required. 2. The base of a right-angled triangle is 20,

and the angle opposite to the perpendicular 50 degrees,

what is the perpendicular ?

As Radius:Tan.50°::20 :Perpendicular sought.

Extend the compasses from 45 degrees to 50 on the line

of tangents, and apply them, thus opened, from 20 towards

the right hand, to 23J-, the perpendicular. This example

shows the method of working when the angle exceeds 45

degrees. The extent taken from the tangents is only from

45 to 40, the complement of 50 degrees ; and we therefore

apply it from 20 towards the right hand to obtain the

length of the perpendicular; but had the angle been 40

degrees, the extent would have been applied from 20 to-

wards the left hand, to 16|, which would, in that case,

have been the perpendicular.

We have now gone systematically through the sector,

which contains a great deal of what may be termed me-

chanical mathematics, and offers much that is valuable to

the draughtsman in the way of suggestion for the con-

struction and management of scales.

Protractors.

We have already referred to the protractor on the plain

scale. The semicircle (Fig. 21), though different in form,

is the same in prin-

ciple. It is a half

circle of brass, or

other metal, having

a double graduation

on its circular edge.

The degrees run both

ways to 180; so that

any angle, from 1 to

90 degrees may be set off on either side. Each graduation

marks an angle and its supplement; thus, 10, 20, 30, coin-

cide with 170, 160, 150; and are the supplements of each

other. An angle is protracted or measured by this instru-

ment with great facility. To protract an angle, draw a

line, and lay the straight edge of the protractor upon it,

with its centre on the point where the angle is to be formed;

the required number of degrees is next marked off close to

the circular edge ; the instrument is then laid aside, and

a, line drawn from the angular point, to the one which

measures the extent of the angle. Thus in the figure, B

is the centre, or angular point, D the measure of the angle,

and B D the line by which it is formed. The converse

operation of measuring an angle is equally simple ; the

angular point and the centre of the protractor are made

to coincide, and the straight edge of the instrument is laid

exactly upon one line of the angle, when the other will

intersect the circular edge, and indicate the number of

degrees. The plain scale protractor is used in the same

manner; but it is by no means so convenient an instru-

ment as the semicircle. Either of them may be employed

occasionally to raise short perpendiculars. For this pur-

pose, make the centre and the graduation of 90 degrees

coincide with the line upon which the perpendicular is to

be raised.

Parallel Ruler.

This is a well-known instrument, consisting of two

rulers connected by slides, moving on pivots, and so ad-

justed, that at every opening of the instrument, the rulers

and the slides form a parallelogram. In use, its edge is

made to coincide exactly with the line to which others are

to be drawn parallel; the lower ruler is then held firmly

down, and the upper one raised to any required distance,

when a line drawn along its edge will be parallel to that

from which it started (Fig. 22). There are several methods

Ufy. 22.)

L_z_p._I

of uniting the rulers ; but we are not aware that any one

has very decided advantages over the others. The ordi-

nary form, as shown in the figure, is perhaps the simplest,

and, therefore, the best. The straight edge of the

{Fig. 23.) >

drawing-board and the T square, 9-re the surest

A means of all for drawing parallels and perpendi-

frf culars ; and the parallel ruler will never be used

' |!!! when these can be employed.

i

Drawing Pens.

The drawing pen differs from the pen-leg of

the compasses only in its having a long straight

handle, the top of which usually unscrews and

forms a tracer or pin, to set off angles by the edge

of the protractor (Fig. 23). The dotting pen is a

similar modification of another leg of the mov-

able compasses. The use of both is to draw

straight, continuous, or dotted lines in ink. A

place is usually provided in the drawing case for

a thin pencil, to rule in straight lines, that may

afterwards either be obliterated or made per-

manent by the ink pen.

Pricker.

This is a simple instrument, consisting of a

fine needle-point firmly fixed into the end of a wooden