0.5

1 cm

12

ENGINEER AND MACHINIST’S DRAWING-BOOK.

the line on the left hand, and extend the other over eight

primaries and four subdivisions, and nearly to the end of

the fifth subdivision. For the second, extend over the

ten primaries of the first half and three subdivisions of

the second half. For the third, extend over the first ten

primaries, and one primary of the second half, and half a

subdivision beyond. To multiply, say, 135 by 48: take

the extent from 1 on the left hand to 48 in the first

interval; and apply it to 135 in the second interval,

when it will reach to 648, or 6480. To divide 6480 by

135 : extend backwards from 135 to 1 on the left hand,

and this will measure back from 6480 to 48. To find a

fourth proportional to the numbers 3, 8, and 15: take the

extent from 3 to 8 in the first interval, and this will

reach from 15 to 40 in the second interval, for 3:8:: 15:40.

The Line of Sines.—This line gives the sines of angles

to SO degrees in a geometric series; their logarithms

being expressed by relative spaces. It is constructed by

laying down the logarithms of the sines from the same

scale of equal parts by which the line of numbers was

measured. Its two intervals are not, however, of equal

length ; and hence we cannot set off the primaries in both

from the same measure. We therefore require a scale of

equal parts of twice the length, or one the whole length

of the line of numbers, to enable us to set off all the sines

from the commencement of the scale on the left hand.

The simplest way is to make the length of the line of

numbers, a transverse to 10.10 on the sectoral line of

lines, and take the transverse measures of half the loga-

rithms. Now the logs, of the primaries, 1, 2, 3, &c. to 10,

in the first interval, and of 20, 30, &c. to 90, in the second

interval, are these: 242, 543, 719, 843, 940, 1019, 1085,

1143, 1194, 1239; 1534, 1699, 1808, 1884, 1937, 1972,

1993, 2000. Take, therefore, the halves of these logs,

tranversely from the line of lines, and lay them down suc-

cessively, on the line of sines, from the beginning of the

scale. The subdivision of the primaries into minutes and

degrees is proceeded with in the same manner. The

degrees in the first interval are divided into six spaces,

each being 10 minutes; but between 10 and 20, in the

second interval, there are 20 subdivisions, each represent-

ing 30 minutes, or half a degree ; between 20 and 30, and

30 and 40, there are 10 subdivisions, each being 1 degree ;

between 40 and 50, 50 and 60, 60 and 70, there are 5

graduations, each 2 degrees ; and the space between 70

and 90 admits only of one subdivision to divide it for 80

and 90 degrees. The commencement of the scale is inter-

fered with by the sectoral line of sines, so that the measure

of 50 minutes is the least sine that can be laid down on

the logarithmic line. With this sine, whose log. is 162, we

commence the subdivisions, and then proceed to the sines

of 1° 10', 1° 20', &c.; 2° 10', 2° 20', &c., until the graduation

is completed.

There is another method of construction, by which the

sines are measured off from the termination of the line on

the right hand. For this purpose the logs, of the arith-

metical complements of the sines are taken, that is to say,

the difference between them and radius. Thus the log.

of the sine of 30 degrees, taking all the places of figures,

is 9.6989700, and this deducted from radius, or 10.0000000,

leaves a remainder of .3010300. If, therefore, 301, or its

half by the proposed scale, be laid from the end of the line

of sines at the right hand, it will reach the graduation of

30 degrees. There is yet another method : in place of the

arithmetical complement of the sine, take the secant of the

complementary angle, viz., 60 degrees, and set off from the

right hand as in the former case. We mention these vari-

ous modes of construction to call the young mechanician’s

attention to the relation between different angles, and as

suggestions for more scientific inquiry concerning them.

The manner of taking off a logarithmic sine from the

scale is obvious : one foot of the compasses is placed at the

commencement, and the other extended to the required

degree or minutes. The use of the line in conjunction

with the line of numbers may be illustrated by one ex-

ample. The base of a right-angled triangle is 30, and the

angle opposite to it 30 degrees, what is the liypothenuse ?

Now, Sine of Angle: 30 :: Radius: liypothenuse.

Set one foot of the compasses on 30 degrees, and extend

the other to 30 on the line of numbers; and with this

opening, set one foot on 90 degrees of the line of sines, and

the other foot will reach to 60 on the line of numbers—

the liypothenuse required.

The Line of Tangents.—This line gives the logarithmic

measures of the tangents to 45 degrees, and thence back-

wards to 88° 30'. The tangent of 45 degrees being equal

to radius, or the line of numbers, the graduation cannot

be extended beyond this angle; but the upper tangents

are obtained by reckoning backwards, 40 for 50, 30 for

60, 20 for 70, &c.; and this method of obtaining the longer

tangents is compensated by a peculiarity of operation

when the line is wrought in conjunction with the line of

numbers. This scale is constructed by measuring off,

successively, from the commencement at the left hand, the

logarithms of the primaries and subdivisions as required.

Thus, the first interval has for its primaries the degrees

from 1 to 10 ; and, in the second, every primary is 10

degrees, except the last, which is only 5. The subdivisions

in the first interval are ten minutes; and in the second,

between 10 and 20 and 20 and 30, they are 30 minutes, or

half a degree; and from 30 to 45, 1 degree each. Make

a scale of equal parts, as for the sines, by applying the

length of the line of numbers to 10.10 on the line of lines;

then take the transverses of half the logarithmic tangents

found in the tables. Thus, the logs, of 1, 2, 3, 4, 5, 6, 7,

8, 9, 10; 20, 30, 40, 45 degrees, are 242, 543, 719, 844,

941, 1021, 1089, 1148, 1199, 1246; 1561, 1761, 1924,

2000. Take therefore the halves of these numbers from

the scale, and transfer them to the line for the primaries,

The subdivisions are commenced at 1° 30', whose log. is

418 ; and then continued 1° 40', 1° 50', 23 10', &c., until

completed. As in the case of the sines, the line of tan-

gents may likewise be constructed by laying down the

arithmetical complements of the tangents backwards, from

45 degrees to the commencement of the scale.

ENGINEER AND MACHINIST’S DRAWING-BOOK.

the line on the left hand, and extend the other over eight

primaries and four subdivisions, and nearly to the end of

the fifth subdivision. For the second, extend over the

ten primaries of the first half and three subdivisions of

the second half. For the third, extend over the first ten

primaries, and one primary of the second half, and half a

subdivision beyond. To multiply, say, 135 by 48: take

the extent from 1 on the left hand to 48 in the first

interval; and apply it to 135 in the second interval,

when it will reach to 648, or 6480. To divide 6480 by

135 : extend backwards from 135 to 1 on the left hand,

and this will measure back from 6480 to 48. To find a

fourth proportional to the numbers 3, 8, and 15: take the

extent from 3 to 8 in the first interval, and this will

reach from 15 to 40 in the second interval, for 3:8:: 15:40.

The Line of Sines.—This line gives the sines of angles

to SO degrees in a geometric series; their logarithms

being expressed by relative spaces. It is constructed by

laying down the logarithms of the sines from the same

scale of equal parts by which the line of numbers was

measured. Its two intervals are not, however, of equal

length ; and hence we cannot set off the primaries in both

from the same measure. We therefore require a scale of

equal parts of twice the length, or one the whole length

of the line of numbers, to enable us to set off all the sines

from the commencement of the scale on the left hand.

The simplest way is to make the length of the line of

numbers, a transverse to 10.10 on the sectoral line of

lines, and take the transverse measures of half the loga-

rithms. Now the logs, of the primaries, 1, 2, 3, &c. to 10,

in the first interval, and of 20, 30, &c. to 90, in the second

interval, are these: 242, 543, 719, 843, 940, 1019, 1085,

1143, 1194, 1239; 1534, 1699, 1808, 1884, 1937, 1972,

1993, 2000. Take, therefore, the halves of these logs,

tranversely from the line of lines, and lay them down suc-

cessively, on the line of sines, from the beginning of the

scale. The subdivision of the primaries into minutes and

degrees is proceeded with in the same manner. The

degrees in the first interval are divided into six spaces,

each being 10 minutes; but between 10 and 20, in the

second interval, there are 20 subdivisions, each represent-

ing 30 minutes, or half a degree ; between 20 and 30, and

30 and 40, there are 10 subdivisions, each being 1 degree ;

between 40 and 50, 50 and 60, 60 and 70, there are 5

graduations, each 2 degrees ; and the space between 70

and 90 admits only of one subdivision to divide it for 80

and 90 degrees. The commencement of the scale is inter-

fered with by the sectoral line of sines, so that the measure

of 50 minutes is the least sine that can be laid down on

the logarithmic line. With this sine, whose log. is 162, we

commence the subdivisions, and then proceed to the sines

of 1° 10', 1° 20', &c.; 2° 10', 2° 20', &c., until the graduation

is completed.

There is another method of construction, by which the

sines are measured off from the termination of the line on

the right hand. For this purpose the logs, of the arith-

metical complements of the sines are taken, that is to say,

the difference between them and radius. Thus the log.

of the sine of 30 degrees, taking all the places of figures,

is 9.6989700, and this deducted from radius, or 10.0000000,

leaves a remainder of .3010300. If, therefore, 301, or its

half by the proposed scale, be laid from the end of the line

of sines at the right hand, it will reach the graduation of

30 degrees. There is yet another method : in place of the

arithmetical complement of the sine, take the secant of the

complementary angle, viz., 60 degrees, and set off from the

right hand as in the former case. We mention these vari-

ous modes of construction to call the young mechanician’s

attention to the relation between different angles, and as

suggestions for more scientific inquiry concerning them.

The manner of taking off a logarithmic sine from the

scale is obvious : one foot of the compasses is placed at the

commencement, and the other extended to the required

degree or minutes. The use of the line in conjunction

with the line of numbers may be illustrated by one ex-

ample. The base of a right-angled triangle is 30, and the

angle opposite to it 30 degrees, what is the liypothenuse ?

Now, Sine of Angle: 30 :: Radius: liypothenuse.

Set one foot of the compasses on 30 degrees, and extend

the other to 30 on the line of numbers; and with this

opening, set one foot on 90 degrees of the line of sines, and

the other foot will reach to 60 on the line of numbers—

the liypothenuse required.

The Line of Tangents.—This line gives the logarithmic

measures of the tangents to 45 degrees, and thence back-

wards to 88° 30'. The tangent of 45 degrees being equal

to radius, or the line of numbers, the graduation cannot

be extended beyond this angle; but the upper tangents

are obtained by reckoning backwards, 40 for 50, 30 for

60, 20 for 70, &c.; and this method of obtaining the longer

tangents is compensated by a peculiarity of operation

when the line is wrought in conjunction with the line of

numbers. This scale is constructed by measuring off,

successively, from the commencement at the left hand, the

logarithms of the primaries and subdivisions as required.

Thus, the first interval has for its primaries the degrees

from 1 to 10 ; and, in the second, every primary is 10

degrees, except the last, which is only 5. The subdivisions

in the first interval are ten minutes; and in the second,

between 10 and 20 and 20 and 30, they are 30 minutes, or

half a degree; and from 30 to 45, 1 degree each. Make

a scale of equal parts, as for the sines, by applying the

length of the line of numbers to 10.10 on the line of lines;

then take the transverses of half the logarithmic tangents

found in the tables. Thus, the logs, of 1, 2, 3, 4, 5, 6, 7,

8, 9, 10; 20, 30, 40, 45 degrees, are 242, 543, 719, 844,

941, 1021, 1089, 1148, 1199, 1246; 1561, 1761, 1924,

2000. Take therefore the halves of these numbers from

the scale, and transfer them to the line for the primaries,

The subdivisions are commenced at 1° 30', whose log. is

418 ; and then continued 1° 40', 1° 50', 23 10', &c., until

completed. As in the case of the sines, the line of tan-

gents may likewise be constructed by laying down the

arithmetical complements of the tangents backwards, from

45 degrees to the commencement of the scale.