DOI Page / Citation link:
https://doi.org/10.11588/diglit.20658#0023
OF DRAWING.
9
scales or lines represent degrees. The line of tangents is divided into half degrees, or 30 minutes,
as is also the line of chords ; the line of sines is also divided into half degrees from the center to the
sixtieth degree ; but from thence to the seventieth degree it is divided in whole degrees; from 70 to
80 into two degrees, and from 80 to 90 the divisions must be estimated by the eye. The lesser
line of tangents is divided to every two degrees, from 45 (where it begins) to 50; but from 50 to
60 it is divided to each degree, and from 60 to the end into half degrees. In the line of secants,
the first 10 degrees are to be estimated by the eye ; from 20 to .50 it is divided into every two
degrees ; from 50 to 60 to every degree ; and from 60 to the end, to every half degree.
To multiply numbers by the line of lines. Make the lateral distance of one of the factors the pa-
rallel distance of 10; then the parallel distance of the other factor will be the product. Thus if it
were required to multiply 5 by 8, extend the compasses from the center of the sector to 5 on the
primary divisions, and open the sectortill this distance becomes equal to the parallel distance of 10
from 10 ; then the parallel distance of 8 from 8, measured with the compasses from the center, will
extend to 4, which is to be reckoned 40, as there is no extent over. The sector remaining open the
same width, the parallel distance of 7 from 7, will extend from the center to 35; that of 9 to 45, Sec.
To divide by the line of lines. Make the lateral distance of the dividend the parallel distance
of the divisor; then the parallel distance of 10 is the quotient. Thus to divide 40 by 8, make the
lateral distance of 40 (viz. 4 of the primary divisions) the parallel distance of 8; then the parallel
distance of 10, taken in the compasses, will extend from the center to 5, the quotient.
To find a fourth proportional number by the line of lines. Make the lateral distance of the se-
cond number the parallel distance of the first number ; and the parallel distance of the third num-
ber is the fourth number required. Suppose it were required to find a number, bearing the same
proportion to 9 as 2 does to 6; take the lateral distance of 2, and make it the parallel distance of
6; then the parallel distance of 9, extended from the center, will reach to 3, the proportional
required.
In any calculations on the line of lines, if the number to be made a parallel distance be too
great for the sector, it is to be divided by any number which will divide it without a remainder,
and such quotient take instead of the dividend ; but in this case the answer is to be multiplied
by the number by which the other number was divided. Thus, if it were required to find the
fourth proportional to 4, 8, and 6, the second term 8 must be divided by 2 (its lateral distance
being too great to be made the parallel distance of 4); and the lateral distance of 4, the quotient,
is to be taken in its stead, and made the parallel distance of the first term 4 ; then the parallel
distance of the third term 6, will reach from the center to 6 (the half of 12). In that manner,
if a number be too small to be made the parallel distance, it may be multiplied by any number,
and the answer is to be divided by the same number.
Any numbers consisting of units and tens (or hundreds, tens, and units, if the units be 5), may
be divided, multiplied, or proportionals to them found, in the same manner; counting the larger
divisions of the line tens, and the second division units, for units and tens: and the larger divi-
sions hundreds, the second tens, and the third divisions fives.
To protract angles by the line of chords.—Case I. When the given degrees are under 60.—
Describe an arch with any radius ; make the same radius the transverse distance of 60 from the
line of chords. Take the transverse distance of the given number of degrees (suppose 25) on the
chord line, and lay them on the described arch from the radius ; then a line drawn from the cen-
ter to the arch, where the compasses intersect it, will form the required angle with the radius.
d Case
9
scales or lines represent degrees. The line of tangents is divided into half degrees, or 30 minutes,
as is also the line of chords ; the line of sines is also divided into half degrees from the center to the
sixtieth degree ; but from thence to the seventieth degree it is divided in whole degrees; from 70 to
80 into two degrees, and from 80 to 90 the divisions must be estimated by the eye. The lesser
line of tangents is divided to every two degrees, from 45 (where it begins) to 50; but from 50 to
60 it is divided to each degree, and from 60 to the end into half degrees. In the line of secants,
the first 10 degrees are to be estimated by the eye ; from 20 to .50 it is divided into every two
degrees ; from 50 to 60 to every degree ; and from 60 to the end, to every half degree.
To multiply numbers by the line of lines. Make the lateral distance of one of the factors the pa-
rallel distance of 10; then the parallel distance of the other factor will be the product. Thus if it
were required to multiply 5 by 8, extend the compasses from the center of the sector to 5 on the
primary divisions, and open the sectortill this distance becomes equal to the parallel distance of 10
from 10 ; then the parallel distance of 8 from 8, measured with the compasses from the center, will
extend to 4, which is to be reckoned 40, as there is no extent over. The sector remaining open the
same width, the parallel distance of 7 from 7, will extend from the center to 35; that of 9 to 45, Sec.
To divide by the line of lines. Make the lateral distance of the dividend the parallel distance
of the divisor; then the parallel distance of 10 is the quotient. Thus to divide 40 by 8, make the
lateral distance of 40 (viz. 4 of the primary divisions) the parallel distance of 8; then the parallel
distance of 10, taken in the compasses, will extend from the center to 5, the quotient.
To find a fourth proportional number by the line of lines. Make the lateral distance of the se-
cond number the parallel distance of the first number ; and the parallel distance of the third num-
ber is the fourth number required. Suppose it were required to find a number, bearing the same
proportion to 9 as 2 does to 6; take the lateral distance of 2, and make it the parallel distance of
6; then the parallel distance of 9, extended from the center, will reach to 3, the proportional
required.
In any calculations on the line of lines, if the number to be made a parallel distance be too
great for the sector, it is to be divided by any number which will divide it without a remainder,
and such quotient take instead of the dividend ; but in this case the answer is to be multiplied
by the number by which the other number was divided. Thus, if it were required to find the
fourth proportional to 4, 8, and 6, the second term 8 must be divided by 2 (its lateral distance
being too great to be made the parallel distance of 4); and the lateral distance of 4, the quotient,
is to be taken in its stead, and made the parallel distance of the first term 4 ; then the parallel
distance of the third term 6, will reach from the center to 6 (the half of 12). In that manner,
if a number be too small to be made the parallel distance, it may be multiplied by any number,
and the answer is to be divided by the same number.
Any numbers consisting of units and tens (or hundreds, tens, and units, if the units be 5), may
be divided, multiplied, or proportionals to them found, in the same manner; counting the larger
divisions of the line tens, and the second division units, for units and tens: and the larger divi-
sions hundreds, the second tens, and the third divisions fives.
To protract angles by the line of chords.—Case I. When the given degrees are under 60.—
Describe an arch with any radius ; make the same radius the transverse distance of 60 from the
line of chords. Take the transverse distance of the given number of degrees (suppose 25) on the
chord line, and lay them on the described arch from the radius ; then a line drawn from the cen-
ter to the arch, where the compasses intersect it, will form the required angle with the radius.
d Case