MAT 
equal parts : it will fall on 40, which is in the 
same proportion to 15 that 8 is to 3. And 
this is demonstrable by common arithmetic 
for 3 being | of 8, and 15 being | of 40, the 
solution given by this scale must be correct. 
This depends entirely on the mathematical 
axiom ; viz. that “ parallel lines under the 
same angle are to each other in proportion 
to their respective distances from the angu- 
lar point.” 
“ To set off an angle by a line of chords 
of 60“ only,” (fig. 2.) Open the sector to 
any extent at pleasure, and with the dis- 
tance between 60 and, 60 describe a seg- 
ment at least equal to the space you think 
the angle will occupy. On the same line 
of cords take on your compasses, the num- 
ber of degrees you intend the angle to be, 
say 27, and applying one leg to the com- 
mencement of your segment, (which we 
suppose to be a given point on a given line) 
measure the same space on the segment. 
The two points thus ascertained on the 
segment will show an angle of 27 degrees ; 
which will be better seen by drawing lines 
from them respectively to the centre where 
the segment was described. When the 
angle is to be more than 60 degrees, ano- 
ther operation on a second line, made at 60 
degrees, will give the angle required ; thus 
you may make an angle of 60 degrees in 
the intended direction ; and if the whole 
angle to be made amount to 73, you may 
add a second angle of 13. But the neatest 
and shortest way is to draw a perpendicular 
to the given line, on the point whence the 
segment arises, and from that to make an 
angle equal to the complement : thus, if the 
angle is to be 73, from the base line, you 
should make an angle equal to 17, which 
added to 73 complete 90 degrees, and thus 
obtain the desired angle by inversion. 
“ A line being given, to find the sine of a 
segment whose radius shall be the hypothe- 
nuse of a triangle (at any given angle), 
formed by that line, as a base, and by the 
sine as a perpendicular thereto,” (fig. 3.) 
Here we have one of the most important, 
yet simple, operations in mathematics ; viz. 
the ascertaining a sine upon an undescribed 
segment. Let the base line, A B, be 174, 
and the given angle be 42 ; make the angle 
at one end, B, of the base, and at the other, 
A, raise a perpendicular which is to become 
the sine, when intercepted by the hypothe- 
nuse C B. Take 174 from the line of equal 
parts on your compasses, and open your 
sector until the distance between 48 and 
48 on tlie lines of sines corresponds there- 
MAT 
with. Now measure the distance between 
42 and 42 on the lines of sines, and their re- 
sult, 162, will be the length of the sine to a 
segment, of which the hypothenuse of the 
triangle is radius, and whose versed sine will 
be found by continuing the base line until it 
meets the segment: the base line in this 
case will be equal to the co sine; since a 
perpendicular raised at the angular point 
parallel to the sine, A C, would, if the seg- 
ment were continued thereto, complete the 
quadrant of a circle. 
But if, instead of taking the hypothenuse 
for a radius, we take only the length of the 
base line ; and from the same point as be- 
fore, draw'^a segment, A D, from the end of 
the base to the hypothenuse ; then, instead 
of being a sine, the line whose length wj? 
have just ascertained to be 162 will be a 
tangent, and comes under the next ex- 
ample. 
“ To ascertain the length of a tangent 
under a given angle, on a given line.” Take 
the distance 174 (equal to the radius), from 
the line of equal parts, and open your sec- 
tor, so that it may be the distance be- 
tween 45 and 45 on the lines of tangents. 
Then take the distance from 42 to 42 on 
the same lines, and it will be found equal to 
162 on the line of equal parts. Hence we 
see that the tangent of a segment made on 
the base as a radius is the line of a segment 
made on the hypothenuse as a radius ; the 
angle in both instances being the same, and 
not exceeding 45°. 
“ To find the length of the secant in the 
same figure.” Take the length of the base, 
as before, from the line of equal parts, and 
spread the sector until that measure reaches 
from 0 to 0 (that is from the very beginning) 
of the lines of secants; measure the dis- 
tance from 42 to 42 on the lines of secants; 
it will reach to 238 on the line of equal 
parts, and give that for the length of the 
hypothenuse, which is in this case cohsi- 
dered as a secant. 
Besides the lines already described, there 
are some that require the sector to be com- 
pletely unfolded, so as to be all in one line. 
These are the artificial lines of numbers, 
sines, and tangents, taken from Gunter’s, 
tables, whicli depend on logarithms for the 
solution of their operations ; as will be seen 
under the head of Navigation, in which 
the properties of Gunter’s scale are illus- 
trated. 
MATHEMATICS, originally signified 
any discipline or learning ; but, at present, 
denotes that science which teaches, or com 
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