Sector

(Use of the scale of chords on the), 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 GO, 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 S) B/C! T/ Gt Be to 60, and Jay 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 scale of chords, the sector may be used to take angles, as a surveying in. | Strumeut. 3. ‘To protract or lay down an angle, of any given number of degrees, 1. Let the number of degrees be le than 60, viz. 46. At any opening of the sector, take the transverse dis. tance of 60 and 60 on the chords; and with | this opening describe an arc: take the trang. verse distance of the given number of degrees | 46, and lay this distance on the are 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 are as before; take the) transverse distance of 4 or 3 of the given des grees, 148, €. g. 4404 degrees: lay this dise tance on the are thrice; and from the centre | draw two lines to the extremities of the are | thus determined, and they will contain the re« quired angle, N. 1. If the radius of the are or circle is to be of a given length; then make | the transverse distance of 60 and 60 equal to that a igned length. 4. To find the degrecs 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 are between the compa es, and carrying it on the legs of the sector, see what equal number, on. each leg, the points of the compa es fall on: this is the quantity of degrees the given angle | contains, 5. To take an arc, of any quantity, from off the circumference 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 circume | ference of the given circle. By this use, may any regular polygon be inscribed in a given circle, as well as by the line of polygons: e. g. | ina 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. 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 circumference into 6 equal parts; in eve division of 60 degrees lay down, first the chell 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 ang of the divisions following the first.. SecTor (Use of the line of polygons on the), 1. Ina 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 radius, 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 ina given circle. 2. Ona 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 extremities of the line, as centres, describe ares, intersecting each’ other; and on the point of intersection, asa centre, with the same radius, describe a circumference pa ing through the extremities of the given line; and in this circle may the pentagon, whose side is given, be inscribed. By a like proce may any other polygon, of not more than 12 sides, be diese ised on a given line. 3. Ona right line, to describe an isos- -celes 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 wil] be the length of the two equal sides of the triangle. Sector (Use of the scales of sines, tangents, and secants, on they. By the several lines disposed on the sector we have scales. to several radiuses: so that lt. having a length, or radius, given, not exceeding the length of the sector when opened, we find the chord, sine, . “thereto: e. g. suppose the chord, sine, or tangent, of 10 degrees to a radius of 3 inches required. Make 3 inches the aperture, or transyerse 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 g0 to gO on the scale of 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, or transverse distance, between 10 and 10, on the scales of chords, be taken with the compa es, it will give the chord of 10 degrees; if the transverse distance of 10 and 10 be in like manner taken, on the -seales 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 repeated twice, gives the chord of 70 degrees, Yo find the tangent of 70 degrees, to the same radius, the scale of upper tangents must be used, the other only reaching to 45: making, therefore, 3 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 t0 3 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 sectcr at once; if the radius of the circle can be made a transverse distance to the complement of those degrees on the lower tangent. E.g, To find the tangent of 78 degrees to a radius of 2 inches. Make 2 inches a transverse distance to 12 degrees on the lower tangents: then the transverse distance of 45 de- ‘grees will be the tangent of 78 degrees, 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 2 inches a transverse distance to the sine of 12 degrees, then the transverse distance of g0 and gO will be the secant of 78 degrees. Hence it will be easy tu find the degrees answering to a given line, expre ing 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, fora tangent, make the given Jine a transverse distance 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 transvetse 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 nnmber of degrees: then will the distance of 90 and gO 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, aiid of O and 0 on the secants, be the radius. 5. If the radius, and any line representing a sine, tangent, or secant, be given, the degrecs corresponding to that line may be found by setting the sector to the given radius, according as a sine, tangent, or secant, is concerned; taking the given line between the compa es, applying the two feet transversely to the scale concerned, and sliding the feet along till they both rest on like divisions on both legs: and the divisions will shew the degrees and parts corresponding to the given line. For the method of determining the degrees answering to any tangent, or secant, that can- not be thus measured, see above 6. To find the length of a versed sine to a given number of degrees, and a given, radius. _ Make the transverse distance of 90 aud 90 in the sines equal to the given radius; take ‘the transverse distance of the sine complement of the given degrees; if the given degrees are le than 90, the difference; but if greater, the sum of the sine complement and radius gives the versed sine. 7. To open the legs of the sector so that the corresponding double scales of lines, chords, sines, tangents, may make, each of them, a right angle. On the lines, make the lateral distance 10, and a distance between 8 on one leg, and 6 on the other leg; on the sines, make the lateral distance 90 a transverse distance from 45 to 45, or from 40 to 50, or from 30 to 60, or from the sine of any degrees to their complement: or, on the sines, make the lateral distance of 45 a transverse distance between 30 and 30. 575 576