901 
SECTOR. 
The great advantage of the sector above the common 
•scales, &c., is, that it is made so as to fit all radiuses, and all 
scales. By the lines of chords, sines, &c., on the sector, we 
have lines of chords, sines, &c., to any radius betwixt the 
length and breadth of the sector when open. 
The sector is founded on the fourth proposition of the 
sixth book of Euclid ; where it is demonstrated, that similar 
triangles have their homologous sides proportional. 
The instrument consists of two rulers, or legs, of brass or 
ivory, or any other matter, representing the radii, moveable 
round an axis or joint, the middle of which expresses the 
centre; whence several scales are drawn on the faces of the 
rulers. It is usually put up in the small cases of mathe¬ 
matical instruments that are in every body’s hands. 
The scales commonly put upon the best sectors, are— 
- / 
bB / 
i) 
i\ 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
\14 
1 
2 
3 
4 
5 
6 
7 
/ Inches, each inch divided into 8 and 10 parts. 
Decimals, containing 100 parts. 
Chords, 
Sines, 
Tangents, 
=« Rhumbs, 
o, / Latitude, 
;5 ^ Hours, 
Longitude, 
Inclin. Merid. 
the f N umbers, 
Loga- I Sines, 
ritlims | Versed sines, 
\ of (Tangents, 
/ Lines, or of equal parts, 
\ Chords, 
1 Sines, 
/ Tangents to 45°,. 
/Cho. 
Sin. 
Tang. 
Rum. 
Lat. 
/ Hou. 
c3 \ Lon. 
2 J In. Mer. 
I Num. 
I Sin. 
V. Sin. 
\Tan. 
) ( Cho. 
*3 ) Sin - 
> < Tan. 
I) Sec. 
Tan. 
VPol. 
j Secants, 
/ Tangents to above 45°, 
V Polygons, 
In describing the use of the sector, the terms lateral 
distance and transverse distance, often occur. By the 
former is meant the distance taken with the compasses on 
one of the scales only, beginning at the centre of the sector; 
and by the latter, the distance taken between any two cor¬ 
responding divisions of the scales of the same name, the legs 
of the sector being in an angular position: but in taking 
these transverse distances, it is to be observed, that each of 
the several scales hath three parallel lines, across which the 
divisions of the scale are marked, and that the points of the 
compasses must be always set on the inside line, or that line 
next the inner edge of the leg, which is the only line, in 
•each scale, which runs to the centre. 
For the use of the logarithmic scale of numbers, see 
Gunter’s Line. 
To use the Line of Lines on the Sector. — 1. To divide 
•a given line into any number of equal parts; e. g. 9. Make 
the length of the given line, or some known part of it, a 
transverse distauce to 9 and 9: then will the transverse 
distance of 1 and 1 be the ^th parth of it; or such a sub- 
multiple of the Jth part, as was taken of the given line: 
or the Jth part will be the difference between the given line 
and the transverse distance of 8 and 8. 
Hence, 2. To make a scale of a given length, to contaiu 
a given number of equal parts; e. g. Let the scale to the map 
of a survey be 6 inches long, and contain 140 poles, and let 
it he required to open the sector, so that a corresponding 
scale may be taken from the line of lines. Make the trans¬ 
verse distance 7 and 7 (or 70 and 70, viz. '|°) equal to 3 
inches (= jj); and this position of the line of "lines will pro¬ 
duce the given scale. 
3. To divide a given line (e. g. of 5 inches) into any 
assigned proportion, as of 4 to 5. Make 5 inches, the 
length of the given line, a Iransverse distance to 9 and 9, 
the sum of the proposed parts; and the transverse distances 
of the assigned numbers, 4 and 5, will be the parts required. 
4. To two given lines, viz., 2 and 6, to find a third pro¬ 
portional. Take between the compasses the lateral distance 
Yol. XXII. No. 1549. 
of the second tehh, viz., 6; set one point on the division 
expressing the first term, viz., 2 on one leg, and open the 
legs of the sector till the other point will fall on the corre¬ 
sponding division on the other leg: keeping the legs of the 
sector in this position, take the transverse distance of the 
second term, viz., 6, and this distance is the third term re¬ 
quired, which distance, measured laterally from the centre, 
will give 18, the number required: for 2 : 6 :: 6 : 18. 
Otherwise, take the distance 2 laterally, and apply it trans¬ 
versely to 6 and 6, the sector being properly opened : then 
the transverse distance at 2 and 2, being taken with the 
compasses, and applied laterally from the centre of the sec¬ 
tor on the scale of lines, will give the third term, when the 
proportion is decreasing; for 6 : 2 :: 2 : §. If the legs of 
the sector will not open so far as to let the lateral distance of 
the second term fall between the divisions expressing the first 
term, then take f, §, L t , or any aliquot part of the second 
term, that will conveniently fall within the opening of the 
sector, and make such part the transverse distance of the 
first term: then, if the transverse distance of the second term 
be multiplied by the denominator of the part taken of the 
second term, the product will give the third term. 
5. To three given lines, viz., 3, 7, and 10, to find a fourth 
proportional. Open the legs of the sector, till the trans¬ 
verse distance of the first term, 3, be equal to the lateral 
distance of the second term, 7, or to some part of it; then 
will the transverse distance of the third term, 10, give the 
fourth term, 23g, required; or such a submultiple of it, as 
was taken of the second term; for 3 : 7 :: 10 : 23J. 
Otherwise, set the lateral dis'ance, 7, transversely from 10 
to 10, opening the sector accordingly; and the transverse 
distance, at 3 and 3, applied laterally, will give 2^; for 
10 : 7 :: 3 : 2*. 
6. To diminish a line of four inches, in the proportion of 
8 to 7. Open the sector till the transverse distance of 
8 and 8 be equal to the lateral distance of 7: mark the 
point, where four inches, as a lateral distance, taken from 
the centre, reaches; and the transverse distance taken at 
that point will be the line required. If the line should be 
too long for the legs of the sector, take f, A, or a, &c. part 
of the given line for the lateral distance, and the correspond¬ 
ing transverse distance, taken twice, thrice, or four times, &c. } 
will be the line required. 
7. To open the sector, so that the two scales of lines shall 
make a right angle. Take the lateral distance from the cen¬ 
tre to the division marked 5, between the points of the com¬ 
passes, and set one foot in the division marked 4, on one of 
the scales of lines; and open the legs of the sector till the 
other foot falls on the division marked 3, on the other scale 
of lines, and then will those scales stand at right angles to 
one another; for the lines 3, 4, 5, or any of their multiples, 
constitute a right-angle triangle. 
8. To turn right lines given, e. g. 40 and 90, to find a 
mean proportional. Set the two scales of lines at right 
angles; find the half sum of the given lines, viz., 65, and 
the half difference, viz., 25, and take with the compasses the 
lateral distance of the half sum, G5, and apply one foot to 
the half difference, 25, the other foot transversely will reach 
to 60, the mean proportional required; for 40 : 60 :: 60 : 90. 
To use the Scale of Chords on the Sector. —1. To open 
the sector so that the two scales of chords may make an 
angle of any number of degrees, e. g. 40. Take the distance 
from the joint to 40, the number of degrees proposed on the 
scale of chords; open the sector till the transverse distance 
from 60 to 60, on each leg, be equal to the aforesaid lateral 
distance of 40: then do the scales of chords make the angle 
required. 
2. The sector being opened, to find the degrees of its 
aperture. Take the extent from 60 to 60, and lay it off on 
the scale of chords from the centre: the number, where it 
terminates, shews the degrees of its opening. By applying 
sights on the scales of chords, the sector may be used to 
take angles, as a surveying instrument. 
3. To protract or lay down an angle of any given number 
of degrees. 1. Let the number of degrees be less than 60, 
10 U viz., 
