26 
INSTRUMENTS, 
The Scales commonly put on the best Sectors, ate 
Sinjrle 
("Inches, each Inch divided into 8 and 10 parts. 
I Decimals, containing 100 parts. 
I Chords, 
line 
of 
Sines, 
Tangents, 
Rhumbs, 
Latitude, 
Hours, 
Longitude, 
Inclin. Merid. 
. the T Numbers, 
Loga- f Sines, 
rithms C Versed Sine3, 
of j Tangents, 
marked 
"Cho. 
Sin. 
Tang. 
Rum. 
l.at. 
lion, 
j Lon. 
In. Me. 
Num. 
Sin. 
V. Sin. 
^Tan. 
Double 
^ 4 y 
5 
6 
L 7 J 
a 
line 
of 
"Lines, or of equal parts, 
Chords, 
Sines, 
Tangents to 45° 
Secants, 
Tangents above 45° 
Polygons, 
> marked 
Lin. 
Cho. 
Sin. 
Tan. 
See. 
Tan. 
Pol. 
'The scales of lines, chords, sines, tangents, 
rhumbs, latitudes, hours, longitude, inch 
inerfcl. may be used, whether the instrument 
is shut or open, each of these scales being 
contained on one of the legs only. The 
scales of inches, decimals, log. numbers, log. 
sines, log. versed sines, and log. tangents, are 
to be used with the sector quite opened, part 
of each scale lying on both legs. 
'1 he double scales of lines, chords, sines, 
and lower tangents, or tangents under 45 de- 
grees, are all of the same radius or length ; 
they begin at the centre of the instrument, 
and are terminated near the other extremity 
oi each leg; viz. the lines at the division 10, 
the chords at 60, the sines at 90, and the 
tangents at 45 ; the remainder of tire tan- 
gents, or those above 45 degrees, are on other 
scales beginning at | of the length of the 
former, counted from the centre, wdiere they 
are marked with 45, and run to about 76°. 
* The secants also begin at the same dis- 
tance from the centre, where they are marked 
with 0, and are from thence continued to as 
many degrees as the length of the sector will 
allow, which is about 75 degrees. 
Each double scale, one being on each leg 
and proceeding from the centre, make an 
angle ; and in an equal angular position are 
all the double scales, whether of lines, or of 
chords, or of sines, or of tangents to 45 de- 
grees. 
And the angles made by the scales of upper 
tangents, and of secants, are also equal ; and 
sometimes these angles are made equal to 
those made by the other double scales. 
The scales of polygons are put near the 
inner edge of the legs : their beginning is not 
so far removed from the centre, as the 60 on 
the chords is: where these scales begin, they 
• are marked with '4, and from thence are fi- 
gured backwards, or towards the centre, to 
12. 
From this disposition of the double scales, 
it is plain, that those angles which were equal 
to each other, while the legs of the sector 
were close, will still continue to be equal, 
although the sector be opened to any distance 
it will admit of. 
We shall now' illustrate the nature of this 
instrument by examples. 
Iz?t CL, CL, (fig. 13) be the two lines of 
lines upon the sector, opened to an angle 
JLCL; join the divisions 4 and 4, 7 and' 7, 
10 and 10, by the dotted lines a, h, c, d, I. L. 
Then by the nature of similar triangles, it is 
C L to C h, as LL to ab ; and CL to C d, 
as L L to c d ; and therefore a b is the same 
part of LL as C b is of C L. Consequent- 
ly, if LL be 10, then a b will be 4, and c d 
will be 7 of the same parts. 
And hence, though the lateral scale C L 
be fixed, yet a parallel scale LL, is obtain- 
able at pleasure ; and therefore though the 
lateral radius is of a determined length in the 
lines of chords, sines, tangents and secants, 
yet the parallel radius may be had of any size 
you w r ant, by means of the sector, as far as 
its length will admit; and all the parallel 
sines See. peculiar to it; as will be evident 
by the following examples in each pair of 
lines. 
Ex. I. In the lines of equal parts. Hav- 
ing 3 numbers given, 4, 7, 16, to find out a 
4th proportional. To do this, take the late- 
ral extent of 16 in the line C L, and apply 
it parallel-wise, from 4 to 4, by a proper 
opening of the sector; then take the parallel 
distance from 7 to 7 in your compasses, and 
applying one foot in C, the other will fall on 
28 in the line of lines C L, and is the number 
required; for 4: 7:: 16: 28. 
Ex. 2. In. the lines of chords. Suppose 
it required to lay off an angle A C B, (fig. 4) 
equal to 35 degrees ; then with any conveni- 
ent opening of the sector, take the extent 
from 60 to 60, and with it (as radius) on the 
point C describe the arch A D indefinitely ; 
then in the same opening of the sector take 
the parallel distance from 35 degrees to 35 
degrees, and set it from A to B in the arch 
A 1) and draw A B, and it makes the angle 
at C required. 
Ex. 3. In the lines of sines. The lines 
of sines, tangents, and secants, are used in 
conjunction w'ith the lines of lines in the so- 
lution of all the cases of plain trigonometry ; 
thus let there be given in the triangle A B C, 
(fig. 1 4) the side A B = 230 ; and the angle 
AB C = 36° 30' ; to find the side A C. 
Here the angle at C is 53° 30'. Then take 
the lateral distance 230, from the line of 
lines, and make it a parallel from 53° 30' to 
53° 30' in the line of sines; then the parallel 
distance between 36° 30' in the same lines, 
will reach laterally from the centre to 170, 
19 in the line of lines for the side A C .re- 
quired. 
Ex. 4. In the lines of tangents. If instead! 
of making the side B C radius (as before) 
y ou make A B radius ; then A C w hich before 
was a sine, is now the tangent of the angle 
B ; and therefore to find it, you use the lines 
of tangents, thus: 
Take the lateral distance 230 from the fine 
of liif. s, and make it a parallel distance on 
the tangent radius, viz. from 45° to 45°, then 
the parallel tangent from 36° 30', to 36° 30', 
will measure laterally on the line of line* 
170, 19, as before, for the side A C. 
Lx. 5. In the lines of secants. In the 
same triangle, in the base A B, and the an- 
gles at B and C given, as before, to find the 
side or hypothenuse B C. Here B C is the 
secant of the angle B. 
Take the lateral distance 230 from the line 
of lines, and make it a parallel distance on the 
tangent radius or beginnings of the lines of 
secants ; then the parallel secant of60°3')' will 
measure laterally on the line of lines 287, 12, 
for the length of B C as required. 
Lx. 6. In the lines of sines and tangents 
conjointly. In the solution of spherical trian- 
gles, you use the line of sines and tangents 
only, as in the following example. In the sphe- 
rical triangle A BC(fig. 15) right-angled at A, 
there are given the side AB = 36° 1 5', and the 
adjacent angle B = 42° 34', to find the side 
A C. The analogy is radius r sine of A B : : 
tangent of B : tangent of A C ; therefore 
make the lateral sine of 36° 15' a parallel at 
radius, or between 90 and 90 ; then the pa- 
rallel tangent of 42° 34' will give the lateral 
tangent of 28° 30' for the side A C. 
Lx. 7. In the lines of polygons . It has 
been observed that the chord of 60 degrees 
is equal to radius ; and 60° is the sixth part of 
360° ; therefore such a chord is the side of 
a hexagon, inscribed in a circle : so that in 
the line of polygons, if you make the pa- 
rallel distance between 6 and 6, the radius 
of a circle, as A C (fig. 16), then if you take 
(lie parallel distance between 5 and 5, and 
place it from A to B, the line A B will be the 
side of a pentagon A B D E F, inscribed in 
the circle; in the same manner may any 
other polygon, from 4 to 12 sides, be inscrib- 
ed in a circle, or upon any given line A B. 
Lx. 8. Of Gunter's lines. We have now 
shewn the use of all that are properly called 
sectoral lines, or that are to be used sector- 
wise ; but there is another set of lines usually 
put upon the sector, that will in a more ready 
and simple manner give the answers to the 
questions in the above examples, and these 
are called artificial lines of numbers, sines, 
and tangents: because the)’ are only the 
logarithms of the natural numbers, sines, and 
tangents, laid upon lines of scales, which 
method was first invented by Mr. Edmund 
Gunter, and is the reason why they have ever 
since been called Gunter’s lines, or the Gun- 
ter. 
Logarithms are only the ratios of numbers, 
and the ratios of all proportional numbers 
are equal. Now all questions in multipli- 
cation, division, the rule of three, and the 
analogies of plain and spherical trigonometry, 
are all stated in proportional numbers or 
terms; therefore, if iiUhe compasse* you take 
the extent (or ratio) between the first and 
second terms, that will always be equal to 
the extent (or ratio) between the third and 
fourth terms ; and consequently, if with the 
