SECTOR

, of a Circle, is a portion of the circle comprehended between two radii and their included arc. Thus, the mixt triangle ABC, contained between the two radii AC and BC, and the arc AB, is a Sector of the circle.

The Sector of a circle, as ABC, is equal to a triangle, whose base is the arc AB, and its altitude the radius AC or BC. And therefore the radius being drawn into the arc, half the product gives the area.

Similar Sectors, are those which have equal angles included between their radii. These are to each other as the squares of their bounding arcs, or as their whole circles.

Sector also denotes a mathematical instrument, which is of great use in geometry, trigonometry, surveying, &c, in measuring and laying down and finding proportional quantities of the same kind: as between lines and lines, surfaces and surfaces, &c: whence the French call it the compass of proportion. |

The great advantage of the Sector above the common scales, &c, is, that it is contrived so as to suit all radii, and all scales. By the lines of chords, sines, &c, on the Sector, we have lines of chords, sines, &c, to any radius between the length and breadth of the Sector when open.

The Sector is founded on the 4th proposition of the 6th book of Euclid; where it is demonstrated, that similar triangles have their like sides proportional. An idea of the theory of its construction may be conceived thus. Let the lines AB, AC represent the legs of the Sector; and AD, AE, two equal sections from the centre: then if the points BC and DE be connected, the lines BC and DE will be parallel; therefore the triangles ABC, ADE will be similar, and consequently the sides AB, BC, AD, DE proportional, that is, as AB : BC :: AD : DE; so that if AD be the half, 3d, or 4th part of AB, then DE will be a half, 3d, or 4th part of BC: and the same holds of all the rest. Hence, if DE be the chord, sine or tangent, of any arc, or of any number of degrees, to the radius AD, then BC will be the same to the radius AB.

The Sector, it is supposed, was the invention of Guido Baldo or Ubaldo, about the year 1568. The first printed account of it was in 1584, by Gaspar Mordente at Antwerp, who indeed says that his brother Fabricius Mordente invented it, in the year 1554. It was next treated of by Daniel Speckle, at Strasburgh, in 1589; after that by Dr. Thomas Hood, at London, in 1598: and afterwards by many other writers on practical geometry, in all the nations of Europe.

Description of the Sector. This instrument consists of two rules or legs, the longer the better, made of box, or ivory, or brass, &c, representing the radii, moveable round an axis or joint, the middle of which represents the centre; from whence several scales are drawn on the faces. See the fig. 1, plate xxvi.

The scales usually set upon Sectors, may be distinguished into single and double. The single scales are such as are set upon plane scales: the double scales are those which proceed from the centre; each of these being laid twice on the same face of the instrument, viz. once on each leg. From these scales, dimensions or distances are to be taken, when the legs of the instrument are set in an angular position.

The scales set upon the best Sectors are

The manner in which these scales are disposed on the Sector, is best seen in the figure.

The scales of lines, chords, sines, tangents, rhumbs, latitudes, hours, longitude, incl. merid. may be used, with the instrument either 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 open, with the two rulers or legs stretched out in the same direction, part of each scale lying on both legs.

The double scales of lines, chords, sines, and lower tangents, or tangents under 45°, are all of the same radius or length; they begin at the centre of the instrument, and are terminated near the other extremity of 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 the tangents, or those above 45°, are on other scales beginning at 1/4 of the length of the former, counted from the centre, where they are marked with 45, and run to about 76 degrees.

The secants also begin at the same distance from the centre, where they are marked with 10, and are from thence continued to as many degrees as the length of the Sector will allow, which is about 75°.

The angles made by the double scales of lines, or chords, of sines, and of tangents to 45 degrees, are always equal. And the angles made by the scales of upper tangents, and of secants, are also equal.

The scales of polygons are set near the inner edge of the legs; and where these scales begin, they are marked with 4, and from thence are figured backwards, or towards the centre, to 12.

From this disposition of the double scales, it is plain, that those angles that are 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.

The scale of inches is laid close to the edge of the Sector, and sometimes on the edge; it contains as many inches as the instrument will receive when opened; each inch being usually divided into 8, and also into 10 equal parts. The decimal scale lies next to this: it is of the length of the Sector when opened, and is divided into 10 equal parts, or primary divisions, and each of these into 10 other equal parts; so that the whole is divided into 100 equal parts: and by this decimal scale, all the other scales, that are taken from tables, may be laid down. The scales of chords, rhumbs, sines, tangents, hours, &c, are such as are described under Plane Scale.

The scale of logarithmic or artificial numbers, called Gunter's scale, or Gunter's line, is a scale expressing the logarithms of common numbers, taken in their natural order.

The construction of the double scale will be evident by inspecting the instrument. As to the scale of poly- | gons, it usually comprehends the sides of the polygons from 6 to 12 sides inclusive: the divisions are laid down by taking the lengths of the chords of the angles at the centre of each polygon, and laying them down from the centre of the instrument. When the polygons of 4 and 5 sides are also introduced, this line is constructed from a scale of chords, where the length of 90° is equal to that of 60° of the double scale of chords on the Sector.

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 corresponding divisions of the scales of the same name, the legs of the Sector being in an angular position.

Of the Line of Lines. This is useful, to divide a given line into any number of equal parts, or in any proportion, or to make scales of equal parts, or to find 3d and 4th proportionals, or mean proportionals, or to increase or decrease a given line in any proportion. Ex. 1. To divide a given line into any number of equal parts, as suppose 9: make the length of the given line a transverse distance to 9 and 9, the number of parts proposed; then will the transverse distance of 1 and 1 be one of the equal parts, or the 9th part of the whole; and the transverse distance of 2 and 2 will be 2 of the equal parts, or 2/9 of the whole line; and so on. 2. Again, to divide a given line into any number of parts that shall be in any assigned proportion, as suppose three parts, in the proportion of 2, 3, and 4. Make the given line a transverse distance to 9, the sum of the proposed numbers 2, 3, 4; then the transverse distances of these numbers severally will be the parts required.

Of the Scale of Chords. 1. To open the Sector to any angle, as suppose 50 degrees: Take the distance from the joint to 50 on the chords, the number of degrees proposed; then open the Sector till the transverse distance from 60 to 60, on each leg, be equal to the said lateral distance of 50; so shall the scale of chords make the proposed angle of 50 degrees.—By the converse of this operation, may be known the angle the Sector is opened to; viz, taking the transverse distance of 60, and applying it laterally from the joint.

2. To protract or lay down an angle of any given number of degrees. At any opening of the Sector, take the transverse distance of 60°, with which extent describe an arc; then take the transverse distance of the number of degrees proposed, and apply it to that arc; and through the extremities of this distance on the arc draw two lines from the centre, and they will form the angle as proposed. When the angle exceeds 60°, lay it off at twice or thrice.—By the converse operation any angle may be measured; viz, With any radius describe an arc from the angular point; set that radius transversely from 60 to 60; then take the distance of the intercepted arc and apply it transversely to the chords, which will shew the degrees in the given angle.

Of the Line of Polygons. 1. In a given circle to in- scribe a regular polygon, for example an octagon. Open the legs of the Sector till the transverse distance from 6 to 6 be equal to the radius of the circle; then will the transverse distance of 8 and 8 be the side of the inscribed octagon. 2. Upon a line given to describe a regular polygon. Make the given line a transverse dis. to 5 and 5; and at that opening of the Sector take the transverse distance of 6 and 6; with which as a radius, from the extremities of the given line describe arcs to intersect each other, which intersection will be the centre of a circle in which the proposed polygon may be inscribed; then from that centre describe the said circle through the extremities of the given line, and apply this line continually round the circumference, for the several angular points of the polygon.—3. On a given right line as a base, to describe an isosceles triangle, having the angles at the base double the angle at the vertex. Open the Sector till the length of the given line fall transversely on 10 and 10 on each leg; then take the transverse distance to 6 and 6, and it will be the length of each of the equal sides of the triangle.

Of the Sines, Tangents, and Secants. By the several lines disposed on the sector, we have scales of several radii. So that, 1. Having a length or radius given, not exceeding the length of the Sector when opened, we can find the chord, sine, &c, to the same: for ex. suppose the chord, sine, or tangentof 20 degrees to a radius of 3 inches be required. Make 3 inches the opening or transverse distance to 60 and 60 on the chords; then will the same extent reach from 45 to 45 on the tangents, and from 90 to 90 on the sines; so that to whatever radius the line of chords is set, to the same are all the others set also. In this disposition therefore, if the transverse distance between 20 and 20 on the chords be taken with the compasses, it will give the chord of 20 degrees; and if the transverse of 20 and 20 be in like manner taken on the sines, it will be the sine of 20 degrees; and lastly, if the transverse distance of 20 and 20 be taken on the tangents, it will be the tangent of 20 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 taken 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 under one only reaching to 45: making therefore 3 inches the transverse distance to 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 3 inches radius.— 3. To find the secant of an arc; make the given radius the transverse distance between 0 and 0 on the secants; then will the transverse distance of 20 and 20, or 70 and 70, give the secant of 20 or 70 degrees.— 4. 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, according as a sine, tangent, or secant is concerned; then taking the given line between the compasses, and applying the two feet transversely to the proper scale, and sliding the feet along till they both rest on like divisions on both legs; then the divisions will shew the degrees and parts corresponding to the given line. |

By means of the double scales, which are the parts more peculiar to the Sector, all proportions are worked by the property of similar triangles, making the sides proportional to the bases, that is, on the Sector, the lateral distances proportional to the transverse ones; thus, taking the distance of the first term, and applying it to the 2d, then the distance of the 3d term, properly applied, will give the 4th term: observing that the sides of triangles are taken off the line of numbers laterally, and the angles are taken transversely, off the sines or tangents or secants, according to the nature of the proportion. For example, in a plane triangle ABC, given two sides and an angle opposite to one of them, to find the rest; viz, given AB = 56, AC = 64, and [angle]B = 46° 30′, to find BC and the angles A and C. In this case, the sides are proportional to the sines of their opposite angles; hence these proportions, as AC (64) : sin. [angle]B (46° 30′) :: AB (56) : sin. [angle]C, and as sin. B : AC :: sin. A : BC.

Therefore, to work these proportions by the Sector, take the lateral distance of 64 = AC from the lines, and open the Sector to make this a transverse distance of 46° 30′ = [angle]B, on the sines; then take the lateral distance of 56 = AB on the lines, and apply it transversely on the sines, which will give 39° 24′ = [angle]C. Hence, the sum of the angles B and C, which is 85° 54′, taken from 180°, leaves 94° 6′ = [angle]A. Then, to work the 2d proportion, the Sector being set at the same opening as before, take the transverse distance of 94° 6′ = [angle]A, on the sines, or, which is the same thing, the transverse distance of its supplement 85° 54′; then this applied laterally to the lines, gives 88 = the side BC sought.

For the complete history of the Sector, with its more ample and particular construction and uses, fee Robertson's Treatise of such Mathematical Instruments, as are usually put into a Portable Case, the Introduction.

Sector of a Sphere, is the solid generated by the revolution of the Sector of a circle about one of its radii; the other radius describing the surface of a cone, and the circular arc a circular portion of the surface of the sphere of the same radius. So that the spherical Sector consists of a right cone, and of a segment of the sphere having the same common base with the cone. And hence the solid content of it will be found by multiplying the base or spherical surface by the radius of the sphere, and taking a 3d part of the product.

Sector of an ellipse, or of an hyperbola, &c, is a part resembling the circular Sector, being contained by three lines, two of which are radii, or lines drawn from the centre of the figure to the curve, and the intercepted arc or part of that curve.

Astronomical Sector, an instrument invented by Mr. George Graham, for finding the difference in right ascension and declination between two objects, whose distance is too great to be observed through a sixed telescope, by means of a micrometer. This instrument (fig. 2, pl. 26,) consists of a brass plate, called the Sector, formed like a T, having the shank CD, as a radius, about 2 1/2 feet long, and 2 inches broad at the end D, and an inch and a half at C; and the cross-piece AB, as an arch, about 6 inches long, and one and a half broad; upon which, with a radius of 30 inches, is described an arch of 10 degrees, each degree being divided in as many parts as are convenient. Round a small cylinder C, containing the centre of this arch, and fixed in the shank, moves a plate of brass, to which is fixed a telescope CE, having its line of collimation parallel to the plane of the Sector, and passing over the centre C of the arch AB, and the index of a Vernier's dividing plate, whose length, being equal to 16 quarters of a degree, is divided into 15 equal parts, fixed to the eye end of the telescope, and made to slide along the arch; which motion is performed by a long screw, G, at the back of the arch, communicating with the Vernier through a slit cut in the brass, parallel to the divided arch. Round the centre F of a circular brass plate abc, of 5 inches diameter, moves a brass cross KLMN, having the opposite ends O and P of one bar turned up perpendicularly about 3 inches, to serve as supporters to the Sector, and screwed to the back of its radius; so that the plane of the Sector is parallel to the plane of the circular plate, and can revolve round the centre of that plate in this parallel position. A square iron axis HIF, 18 inches long, is screwed flat to the back of the circular plate along one of its diameters, so that the axis is parallel to the plane of the Sector. The whole instrument is supported on a proper pedestal, so that the said axis shall be parallel to the earth's axis, and proper contrivances are annexed to fix it in any position. The instrument, thus supported, can revolve round its axis HI, parallel to the earth's axis, with a motion like that of the stars, the plane of the Sector being always parallel to the plane of some hour circle, and consequently every point of the telescope describing a parallel of declination; and if the Sector be turned round the joint F of the circular plate, its graduated arch may be brought parallel to an hour-circle; and consequently any two stars, whose difference of declination does not exceed the degrees in that arch, will pass over it.

To observe their passage, direct the telescope to the preceding star, and fix the plane of the Sector a little to the westward of it; move the telescope by the screw G, and observe at the transit of each over the cross wires the time shewn by the clock, and also the division upon the arch AB, shewn by the index; then is the difference of the arches the difference of the declination; and that of the times shews the difference of the right ascension of those stars. For a more particular description of this instrument, see Smith's Optics, book iii, chap. 9.

SECULAR Year, the same with Jubilee.