Armengaud, Jacques Eugène; Leblanc, César Nicolas [Hrsg.]; Armengaud, Jacques Eugène [Hrsg.]; Armengaud, Charles [Hrsg.]
The engineer and machinist's drawing-book: a complete course of instruction for the practical engineer: comprising linear drawing - projections - eccentric curves - the various forms of gearing - reciprocating machinery - sketching and drawing from the machine - projection of shadows - tinting and colouring - and perspective. Illustrated by numerous engravings on wood and steel. Including select details, and complete machines. Forming a progressive series of lessons in drawing, and examples of approved construction — Glasgow, 1855

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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.


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.)


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.


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.


This is a simple instrument, consisting of a
fine needle-point firmly fixed into the end of a wooden
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