DOI Seite / Zitierlink:
https://doi.org/10.11588/diglit.25888#0023
CONSTRUCTION AND USE OF DRAWING INSTRUMENTS.
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.)
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.)