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.