DOI Seite / Zitierlink:
https://doi.org/10.11588/diglit.25888#0027
CONSTRUCTION AND USE OF DRAWING INSTRUMENTS.
11
upper tangents at one-fourth from the centre. It is thus
seen that we can command a great variation of radius for
the lower tangents, but are restricted, in the upper tan-
gents, to such a radius as can be taken between the legs
of the sector at only one-fourth from the centre. These
lines are constructed on the same principle as the secants
and sines. We take the measure of the tabular tangents
from 1 to 45 degrees and set off the lower line; and one-
fourth of the tabular tangents from 45 to 75 degrees, to
graduate the upper line. Thus, the tabular tangent of
30 degrees is 57J when radius is 100 ; take therefore 57f
from the line of hues, and transfer it from the centre to
the line of lower tangents on each leg, as the measure of
30 degrees. Again, the tabular tangent of 65 degrees is
about 214 when radius is 100 ; take therefore the fourth
part of 214, or 53j, and transfer it from the centre to the
line of upper tangents on each leg. To illustrate the use
of the two lines take the foll^ving questions :—What is
the tangent of 30 degrees when radius is 40 ? Take 40
from the line of lines and make it a transverse to 45.45 at
the end of the sector, then the transverse of 30 degrees
will measure 23^ on the line of lines. The converse ope-
ration gives radius when the length of the tangent is
known. What is the tangent of 70 degrees when radius
is 20 ? Make 20 a transverse to 45.45 at one-fourth from
the centre, and the transverse of 70.70 will measure 55k
The annexed diagram {Fig. 20) shows the mechanical
construction of the secants and tan-
gents. The line C D is a line of
tangents to 75 degrees, and it is
formed by drawing lines from B, the
centre, through the graduations of
the quadrant, to meet the tangential
line C D, standing at right angles to
B C. A B is a line of secants, formed
by transfer with the compasses of
the radial lines from the centre B.
It is therefore seen that the radius,
tangent, and secant, are the base,
perpendicular, and hypothenuse of a
right-angled triangle. A line of sines
would be formed by graduating the
radius 10, B, with lines drawn
through the degrees of the quadrant,
and parallel to B C.
The lines of sines and tangents are
frequently of use to the draughts-
man in the determination of a number of points through
which an eccentric curve can be drawn. We give two
examples of the use of the sectoral lines in the solution of
questions in Trigonometry.
1. A right-angled triangle has base 12, perpendicular
16, required the hypothenuse. Set the sector at right
angles, by making the lateral distance of 5 on the line of
lines a transverse to 3 and 4. Then take the transverse
of 12 on one leg to 16 on the other, and this, measured on
the line of lines, will give 20 for the hypothenuse.
2. A right-angled triangle has perpendicular 30, and
the angle opposite thereto 37 degrees, required the hypo-
thenuse. Take 30 from the line of lines, and make it
a transverse to 37 degrees on the line of sines, then the
transverse of 90.90, will measure 50 on the line of lines,
the length of the hypothenuse.
Logarithmic Lines.
The Line of Numbers.—This line, commonly called
Gunter s Line, and marked N on the sector, is divided
into spaces forming a geometrical series, and is simply a
table of logarithms expressed by relative measures of
length. It is constructed in the following manner:—The
entire line is divided into two equal parts, and each of
these parts into nine unequal primary divisions, corre-
sponding to the logarithms they are to represent. These
primaries are to be the measures of the numbers 2 to 9,
whose logarithms may be considered 30, 47, 60, 70, 78,
84, 90, and 95, as will be seen on reference to the ordinary
tables. To make a scale of equal parts for setting off
these quantities, take one half of the line in the compasses
and make it a transverse to 10.10 on the line of lines.
Then take successively the transverse distances of 30.30 ;
47.47; &c., and set them off from the commencement on
the first half of the line of numbers for the primary
divisions 2, 3, &c. These same spaces may next be trans-
ferred to the second half of the line for its primary divi-
sions. Thus we have obtained the logarithms of 20.30.
40.50.60.70.80.90.100 on the first half; and those of 100,
200, &lc., on the second half. Now, for the subdivision
of the space between 1 and 2, we must set off from the
commencement of the line, in succession, the logarithms
of 11, 12, &c., to 19 ; and for that between 2 and 3, the
logs, of 21, 22, &c., to 29 ; and thus proceed till we come
to the space between 6 and 7, which is too short to admit
the decimal divisions. Graduate therefore this last space,
and all onward to 10, into two-tenths; and consequently
take the logs, of 62, 64, 66, 68; 72, 74, 76, 78; &c. All
these subdivisions set off from the commencement on the
first half of the line, may also be set off from 1 in the
middle, in the second half. Thus, we have found on the
one half, the logs, of tens and units, and on the second half
those of hundreds and tens. But there is a farther sub-
division of the space between 1 and 2 on the second half,
which is graduated to twenty places; and this halving of
the first subdivisions is effected by setting off successively
from 1, the logs, of 105, 115, 125, &c. This done, the line
is constructed.
In using this line any value may be attached to the
primary divisions, merely observing their relative propor-
tion to each other. Thus, if the primaries on the first
half are units, and their subdivisions tenths, those on the
second half will be tens and their subdivisions units.
Whatever is the value of a primary or subdivision on the
first half, the corresponding primary and subdivision on
the second half will have ten times that value. We
illustrate the use of the line by a few examples. Take off
the measures of the numbers 896, 1150, 2050. For the
first, place one foot of the compasses at the beginning of
{Fig. 20.)
11
upper tangents at one-fourth from the centre. It is thus
seen that we can command a great variation of radius for
the lower tangents, but are restricted, in the upper tan-
gents, to such a radius as can be taken between the legs
of the sector at only one-fourth from the centre. These
lines are constructed on the same principle as the secants
and sines. We take the measure of the tabular tangents
from 1 to 45 degrees and set off the lower line; and one-
fourth of the tabular tangents from 45 to 75 degrees, to
graduate the upper line. Thus, the tabular tangent of
30 degrees is 57J when radius is 100 ; take therefore 57f
from the line of hues, and transfer it from the centre to
the line of lower tangents on each leg, as the measure of
30 degrees. Again, the tabular tangent of 65 degrees is
about 214 when radius is 100 ; take therefore the fourth
part of 214, or 53j, and transfer it from the centre to the
line of upper tangents on each leg. To illustrate the use
of the two lines take the foll^ving questions :—What is
the tangent of 30 degrees when radius is 40 ? Take 40
from the line of lines and make it a transverse to 45.45 at
the end of the sector, then the transverse of 30 degrees
will measure 23^ on the line of lines. The converse ope-
ration gives radius when the length of the tangent is
known. What is the tangent of 70 degrees when radius
is 20 ? Make 20 a transverse to 45.45 at one-fourth from
the centre, and the transverse of 70.70 will measure 55k
The annexed diagram {Fig. 20) shows the mechanical
construction of the secants and tan-
gents. The line C D is a line of
tangents to 75 degrees, and it is
formed by drawing lines from B, the
centre, through the graduations of
the quadrant, to meet the tangential
line C D, standing at right angles to
B C. A B is a line of secants, formed
by transfer with the compasses of
the radial lines from the centre B.
It is therefore seen that the radius,
tangent, and secant, are the base,
perpendicular, and hypothenuse of a
right-angled triangle. A line of sines
would be formed by graduating the
radius 10, B, with lines drawn
through the degrees of the quadrant,
and parallel to B C.
The lines of sines and tangents are
frequently of use to the draughts-
man in the determination of a number of points through
which an eccentric curve can be drawn. We give two
examples of the use of the sectoral lines in the solution of
questions in Trigonometry.
1. A right-angled triangle has base 12, perpendicular
16, required the hypothenuse. Set the sector at right
angles, by making the lateral distance of 5 on the line of
lines a transverse to 3 and 4. Then take the transverse
of 12 on one leg to 16 on the other, and this, measured on
the line of lines, will give 20 for the hypothenuse.
2. A right-angled triangle has perpendicular 30, and
the angle opposite thereto 37 degrees, required the hypo-
thenuse. Take 30 from the line of lines, and make it
a transverse to 37 degrees on the line of sines, then the
transverse of 90.90, will measure 50 on the line of lines,
the length of the hypothenuse.
Logarithmic Lines.
The Line of Numbers.—This line, commonly called
Gunter s Line, and marked N on the sector, is divided
into spaces forming a geometrical series, and is simply a
table of logarithms expressed by relative measures of
length. It is constructed in the following manner:—The
entire line is divided into two equal parts, and each of
these parts into nine unequal primary divisions, corre-
sponding to the logarithms they are to represent. These
primaries are to be the measures of the numbers 2 to 9,
whose logarithms may be considered 30, 47, 60, 70, 78,
84, 90, and 95, as will be seen on reference to the ordinary
tables. To make a scale of equal parts for setting off
these quantities, take one half of the line in the compasses
and make it a transverse to 10.10 on the line of lines.
Then take successively the transverse distances of 30.30 ;
47.47; &c., and set them off from the commencement on
the first half of the line of numbers for the primary
divisions 2, 3, &c. These same spaces may next be trans-
ferred to the second half of the line for its primary divi-
sions. Thus we have obtained the logarithms of 20.30.
40.50.60.70.80.90.100 on the first half; and those of 100,
200, &lc., on the second half. Now, for the subdivision
of the space between 1 and 2, we must set off from the
commencement of the line, in succession, the logarithms
of 11, 12, &c., to 19 ; and for that between 2 and 3, the
logs, of 21, 22, &c., to 29 ; and thus proceed till we come
to the space between 6 and 7, which is too short to admit
the decimal divisions. Graduate therefore this last space,
and all onward to 10, into two-tenths; and consequently
take the logs, of 62, 64, 66, 68; 72, 74, 76, 78; &c. All
these subdivisions set off from the commencement on the
first half of the line, may also be set off from 1 in the
middle, in the second half. Thus, we have found on the
one half, the logs, of tens and units, and on the second half
those of hundreds and tens. But there is a farther sub-
division of the space between 1 and 2 on the second half,
which is graduated to twenty places; and this halving of
the first subdivisions is effected by setting off successively
from 1, the logs, of 105, 115, 125, &c. This done, the line
is constructed.
In using this line any value may be attached to the
primary divisions, merely observing their relative propor-
tion to each other. Thus, if the primaries on the first
half are units, and their subdivisions tenths, those on the
second half will be tens and their subdivisions units.
Whatever is the value of a primary or subdivision on the
first half, the corresponding primary and subdivision on
the second half will have ten times that value. We
illustrate the use of the line by a few examples. Take off
the measures of the numbers 896, 1150, 2050. For the
first, place one foot of the compasses at the beginning of
{Fig. 20.)