0.5

1 cm

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.)