Description of the SECTOR

Description of the SECTOR. This instrument consists of two equal legs or rules of brass, &c. riveted together, but so as to move easily on the rivet: on the faces of the instrument are placed several lines; the principal of which are the line of equal parts, line of chords, line of sines, line of tangents, line of secants, and line of polygons.

The line of equal parts, called also the line of lines, marked L, is a line divided into 100 equal parts, and where the length of the leg will allow it, each of these is subdivided into halves and quarters. It is found on each leg, on the same side, and the divisions numbered 1, 2, 3, 4, 5, &c. to 10, which is near the extremity of each leg. Note, in practice, 1 represents either 1, 10, 100, 1000, 10000, &c. as occasion requires; in which case, 2 represents 2, 20, 200, 2000, 20000, &c. and so of the rest. The line of chords, marked C on each leg, is divided after the usual manner, and numbered 10, 20, 30, &c. to 60. The line of sines, denoted on each leg by the letter S, is a line of natural sines, numbered 10, 20, 30, &c. to 90. The line of tangents, denoted on each leg by the letter T, is a line of natural tangents, numbered 10, 20, 30, &c. to 45. Besides which, there is another little line of tangents on each leg, commencing at 45°, and extending to 75°, denoted by the letter $\backslash pi$. Line of secants, denoted on each leg by the letter $\backslash sec$, is a line of natural secants, numbered 10, 20, 30, &c. to 75, not commencing at the centre of the instrument, but at some distance therefrom. The line of polygons, denoted by the letter P on each leg, is numbered 4, 6, 5, &c. to 12, which falls considerably short of the centre of the instrument.

Besides these lines, which are essential to the sector, there are others placed near the outward edges on both sides, and parallel thereto; which are in all respects the same as those on Gunter's scale, and used after the same manner. Such are the lines of artificial sines marked S, of artificial tangents marked T, and Gunter's line of numbers marked N; these lines do not extend to the end of the instrument. There are sometimes other lines placed, to fill the vacant spaces, as the lines of hours, latitudes, and inclination of meridians, which are used the same as on the common scales.

The lines found by the sector are of two kinds, lateral and parallel; the first are such as are found by the sides of the sector, as AB, AC, (fig. 6.) the latter such as go across from one leg to the other, as DE, BC. Note, the lines are not placed in the same order on all sectors, but they may be easily found by the above directions.

Use of the Line of Equal Parts on the Sector. 1. To divide a given line into any number of equal parts, suppose seven. Take the given line in your compasses; and setting one foot in a division of equal parts, that may be divided by seven, for example, 70,

whose seventh part is 10, open the sector till the other point fall exactly on 70, in the same line on the other leg. In this disposition, applying one point of the compasses to 10, in the same line, shut them till the other fall in 10, in the same line, on the other leg, and this opening will be the seventh part of the given line. Note, if the line to be divided be too long to be applied to the legs of the sector, divide only one half or one fourth by 7, and the double or quadruple thereof will be the seventh part of the whole.

2. To measure the lines of the perimeter of a polygon, one of which contains a given number of equal parts. Take the given line in your compasses, and set it parallel, upon the line of equal parts, to the number on each leg expressing its length. The sector remaining thus, set off the length of each of the other lines parallel to the former, and the numbers each of them falls on will express their lengths.

3. A right line being given, and the number of parts it contains, suppose 120, to take from it a shorter line, containing any number of the same parts, suppose 25. Take the given line in your compasses, open the sector till the two feet fall on 120 on each leg; then will the distance between 25 on one leg, and the same number on the other, give the line required.

4. To multiply by the line of equal parts on the sector. Take the lateral distance from the centre of the line to the given multiplier; open the sector till you fit that lateral distance to the parallel of 1 and 1, or 10 and 10, and keep the sector in that disposition; then take in the compasses the parallel distance of the multiplicand, which distance, measured laterally on the same line, will give the product required. Thus, suppose it were required to find the product of 8, multiplied by 4: take the lateral distance from the centre of the line to 4 in your compasses, i. e. place one foot of the compasses in the beginning of the divisions, and extend the other along the line to 4. Open the sector till you fit this lateral distance to the parallel of 1 and 1, or 10 and 10. Then take the parallel distance of 8, the multiplicand; i. e. extend the compasses from 8, in this line, on one leg, to 8 in the same line on the other; and that extent, measured laterally, will give the product required.

5. To divide by the line of equal parts on the sector. Extend the compasses laterally from the beginning of the line to 1, and open the sector till you fit that extent to the parallel of the divisor; then take the parallel distance of the dividend, which extent, measured in a lateral direction, will give the quotient required. Thus, suppose it was required to divide 36 by 4; extend the compasses laterally, the beginning of the line to 1, and fit to that extent the parallel of 4, the divisor; then extend the compasses parallel, from 36 on one leg to 36 on the other, and that extent, measured laterally, will give 9, the quotient required.

Use of the Sine of Chords on the Sector. 1. To open the sector so as the two lines of chords may make an angle or number of degrees, suppose 40. Take the distance from the joint to 40, the number of the degrees proposed, on the line of chords; open the sector till the distance from 60 to 60, on each leg, be equal to the given distance of 40; then will the two

Sector. lines on the sector form an angle of 40 degrees, as was 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 line of chords from the centre; the number whereon it terminates will show the degrees, &c. required.

3. To lay off any number of degrees upon the circumference of a circle. Open the sector till the distance between 60 and 60 be equal to the radius of the given circle; then take the parallel 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. Hence any regular polygon may be easily inscribed in a given circle.

Use of the Line of Polygons on the Sector. 1. To inscribe a regular polygon in a given circle. Take the semidiameter of the given circle in the compasses, and adjust it to the number 6, on the line of polygons, on each leg of the sector: then, the sector remaining thus opened, take the distance of the two equal numbers, expressing the number of sides the polygon is to have; e. gr. the distance from 5 to 5 for a pentagon, from 7 to 7 for a heptagon, &c. These distances carried about the circumference of the circle, will divide it into so many equal parts.

2. To describe a regular polygon, e. gr. a pentagon, on a given right line. Take the length of the line in the compasses, and apply it to the extent of the number 5, 5, on the lines of polygons. The sector thus opened, upon the same lines take the extent from 6 to 6; this will be the semi-diameter of the circle the polygon is to be inscribed in. If then, with this distance, from the ends of the given line, you describe two arches of a circle, their intersection will be the centre of the circle.

3. On a right line, to describe an isosceles triangle, having 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 take the distance from 6 to 6. This will be the length of the two equal sides of the triangle.

Use of the Lines of Sines, Tangents, and Secants, on the Sector. By the several lines disposed on the sector, we have scales to several radii; so that having a length or radius given, not exceeding the length of the sector when opened, we find the chord, sine, &c. thereto. e. gr. Suppose the chord, sine, or tangent, of 10 degrees, to a radius of 3 inches required; make 3 inches the aperture, between 60 and 60, on the lines of chords of the two legs; then will the same extent reach from 45 to 45 on the line of tangents, and from 90 to 90 on the line of the sines on the other side; so that to whatever radius the line of chords is set, to the same are all the others set. In this disposition, therefore, if the aperture between 10 and 10, on the lines of chords, be taken with the compasses, it will give the chord of 10 degrees. If the aperture of 10 and 10 be in like manner taken on the lines of sines, it will be the sine of 10 degrees. Lastly, if the aperture of 10 and 10 be in like manner taken on the lines of tangents, it gives the tangent of 10 degrees.

If the chord, or tangent, of 70 degrees were required; for the chord, the aperture of half the arch, viz. 35, must be taken, as before; which distance,

repeated twice, gives the chord of 70 degrees. To find the tangent of 70 degrees to the same radius, the small line of tangents must be used, the other only reaching to 45: making, therefore, 3 inches the aperture between 45 and 45 on the small line; the extent between 70 and 70 degrees on the same, will be the tangent of 70 degrees to 3 inches radius.

To find the secant of an arch, make the given radius the aperture between 0 and 0 on the line of secants: then will the aperture of 10 and 10, or 70 and 70, on the said lines, give the tangent of 10°, or 70°.

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 aperture 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 aperture of the given number of degrees; then will the distance of 90 and 90 on the sines, of 45 and 45 on the tangents, of 0 and 0 on the secants, be the radius.

Astronomical Sector. See Astronomical Sector. SECULAR, something which is temporal; in which sense the word stands opposed to ecclesiastical: thus we say, secular power, secular jurisdiction, &c.

Secular is more peculiarly used for a person who lives at liberty in the world, not shut up in a monastery, nor bound by vows, or subjected to the particular rules of any religious community; in which sense it stands opposed to regular. The Romish clergy are divided into secular and regular.