902 
SEC 
viz., 46. At any opening of the sector, take the transverse 
distance of 60 and 60 on the chords; and with this open¬ 
ing describe an arc: take the transverse distance of the 
given number of degrees, 46, and lay this distance on the 
arc described, marking its extremities; from the centre of 
the arc, through these extremities, draw two lines, and they 
will contain the angle required. 2. When the degrees given 
are more than 60, viz., 148; describe the arc as before; take 
the transverse distance of f or § of the given degrees, 148, 
e. g. | = 491 degrees: lay this distance on the arc thrice: 
and from the centre draw two lines to the extremities of the 
arc thus determined, and they will contain the required angle. 
N. B. If the radius of the arc or circle is to be of a given 
length, then make the transverse distance of 60 and 60, equal 
to that assigned length. 
4. To find the degrees which a given angle contains. 
About the vertex describe an arc, and open the sector till the 
distance from 60 to 60, on each leg, be equal to the radius 
of the circle; then taking the chord of the arc between the 
compasses, and carrying it on the legs of the sector, see 
what equal number, on each leg, the points of the' compasses 
fall on: this is the quantity of degrees the given angle con¬ 
tains. 
5. To take an arc, of any quantity, from off the circum¬ 
ference of a circle. Open the sector till the distance from 
60 to 60 be equal to the radius of the given circle; then 
take the extent of the chord of the number of degrees, on each 
leg of the sector, and lay it off on the circumference of the 
given circle. By this use, may any regular polygon be in¬ 
scribed in a given circle, as well as by the line of polygons: 
e. g. in a circle whose diameter is given to describe a regular 
polygon of 24 sides. Make the given diameter a transverse 
distance from 60 to 60 on the scales of chords; divide 360 
by 24, and take the transverse distance of 15 and 15, the 
quotient, and this will be the chord of the twenty-fourth 
part of the circumference. In order to prevent errors, where 
the distance is to be repeated several times, it will be best to 
proceed thus: with the chord of 60 degrees divide the cir¬ 
cumference into six equal parts; in every division of 60 
degrees lay down, first, the chord of 15 degrees, and next 
the chord of 30 degrees, and then the chord of 45 degrees, 
beginning always at the same point. Thus the error in 
taking distances, will not be multiplied into any of the divi¬ 
sions following the first. 
To use the Line of Polygons on the Sector. 1. In a 
given circle to inscribe a regular polygon, e. g. an octagon. 
Open the legs of the sector, till the transverse distance of 6 
and 6 be equal to the given diameter, then will the transverse 
distance of 8 and 8 be the side of an octagon, which may 
be inscribed in the given circle. In like manner may any 
other polygon, the number of whose sides does not exceed 
12, be inscribed in a given circle. 
2. On a given line to describe a regular polygon, e. g. a 
pentagon. Make the given line a transverse distance to 5 
and 5: at that opening of the sector, take the transverse 
distance of 6 and 6; and with this radius, on the extremi¬ 
ties of the line, as centres, describe arcs intersecting each 
other; and on the point of intersection, as a centre, with 
the same radius, describe a circumference passing through 
the extremities of the given line; and in this circle may the 
pentagon, whose side is given, be inscribed. By a like pro¬ 
cess may any other polygon, of not more than 12 sides, be 
described on a given line. 
3. On a right line, to describe an isosceles triangle, hav¬ 
ing the angles at the base double that at the vertex. Open 
the sector till the ends of the given line fall on 10 and 10 on 
each leg: then takejhe distance from 6 to 4; this will be the 
length of the two equal sides of the triangle. 
To use the Scales of Sines, Tangents, and Secants on 
the Sector. By the several lines disposed on the sector, we 
have scales to several radiuses: so that, 1, having a length, 
or radius, given, not exceeding the length of the sector 
when opened, we find the chord, sine, &c., thereto: e. g. 
suppose the chord, sine, or tangent, or 10 degrees to a 
radius of three inches required. Make three inches the 
TOE. 
aperture, or transverse distance, between 60 and 60 on the 
scales of chords of the two legs; then will the same extent 
reach from 45 to 45 on the scale of tangents, and from 90 to 
90 on the scale of sines on the other side: so that to what¬ 
ever radius the line of chords is set, to the same are all the 
others set. In this disposition, therefore, if the aperture, or 
transverse distance, between 10 and 10, on the scales of 
chords, be taken with the compasses, it will give the chord 
of 10 degrees; if the transverse distance of 10 and 10 be in 
like manner taken, on the scales of sines, it will be the sine 
of 10 degrees: lastly, if the transverse distance of 10 and 
10 be in like manner taken on the scales of tangents, it gives 
the tangent of 10 degrees to the same radius. 
2. If the chord, or tangent, of 70 degrees were required, 
for the chord, the transverse distance of half the arc, viz., 
35, must be taken, as before; which distance, being re¬ 
peated twice, gives the chord of 70 degrees. To find the 
tangent of 70 degrees, to the same radius, the scale of upper 
tangents must be used, the other only reaching to 45: mak¬ 
ing, therefore, three inches the transverse distance between 
45 and 45 at the beginning of that scale; the extent between 
70 and 70 degrees, on the same, will be the tangent of 70 
degrees to three inches radius. 
3. To find the secant of an arc, make the given radius the 
transverse distance between 0 and 0 on the line of secants; 
then will the transverse distance of 10 and 10, or 70 and 70, 
on the said lines, give the secant of 10 degrees, or 70 degrees. 
The scales of upper tangents and secants do not run quite 
to 76 degrees; but those of a greater number of degrees may 
be found by the sector in the following manner. Thus, the 
tangent of any number of degrees may be taken from the 
sector at once; if the radius of the circle can be made a 
transverse distance to the complement of those degrees on 
the lower tangent. £. g. To find the tangent of 78 degree's 
to a radius of two inches. Make two inches a transverse 
distance of 12 degrees on the lower tangents; than the trans¬ 
verse distance of 45 degrees will be the tangent of 78 degrees. 
In like manner the secant of any number of degrees may be 
taken from the sines, if the radius of the circle can be made 
a transverse distance to the cosine of those degrees. Thus, 
making two inches a transverse distance to the sine of 12 
degrees, then the transverse distance of 90 and 90 will be 
the secant of 78 degrees. Hence it will be easy to find the 
degrees answering to a given line, expressing the length of 
a tangent or secant, which is too long to be measured on 
those scales, when the sector is set to the given radius. 
Thus, for a tangent, make the given line a transverse dis¬ 
tance to 45 and 45 on the lower tangents; then take the 
given radius, and apply it to the lower tangents: and the 
degrees, where it becomes a transverse distance, give the 
cotangent of the degrees answering to the given line. And 
for a secant, make the given line a transverse distance to 90 
and 90 on the sines: then the degrees answering to the given 
radius, applied as a transverse distance on the sines, will 
be the cosine of the degrees answering to the given secant 
line. 
4. If the converse of any of these things were required, 
that is, if the radius be required, to which a given line is 
the sine, tangent, or secant; it is but making the given line, 
if a chord, the transverse distance on the line of chords, 
between 10 and 10, and then the sector will stand at the 
radius required; that is, the aperture between 60 and 60, 
on the said line, is the radius. 
If the given line were a sine, tangent, or secant, it is but 
making it the transverse distance of the given number of 
degrees; then will the distance of 90 and 90 on the sines, of 
45 and 45 on the lower tangents near the end of the sector, 
and of 45 and 45 on the upper tangents towards the centre 
of the sector, and of 0 and 0 on the secants, be the radius. 
5. If the radius, and any line representing a sine, tangent 
or secant, be given, the degrees corresponding to that line 
may be found by setting the sector to the given radius, ac¬ 
cording as a sine, tangent, or secant, is concerned; taking 
the given line between the compasses, applying the two 
feet transversely to the scale concerned, and sliding the feet 
along 
