SEC
U4]
SEC
the Points of the Compares fall on : This is the Quantity of Degrees, the given Angle contains.
5. To take an Arch, of any Quantity, from off the Circumference of a Circle. Open the Setter, till the Di- ftance from 60 to 60 be equal to the Radius of the given Circle : Then take the Extent of the Chord, of the dumber of Degrees, on each Leg of the Sector, and lay it off, on the Circumference of the given Circle. By this Ufe may any regular Polygon he inicribed in a given Circle, as well as by the Line of Polygons.
Ufe of the Line of Polygons en the Sector.
i°, To iufribe a regular 'Polygon, in a given Circle. Take the Semi-diameter of a given Circle, in the Com- pafTes, and adjuft it to rhe Number 6, on the Line of Polygons, on each Leg of the Settor : Then the Setter remaining thus opened, take the Diftance of the two equal Numbers, exprclfmg the Number of Sides the Polygons is to have. B. gr. The Diftance from 5 to 5 for a'Pentagon, from 7 to 7 for a Heptagon, £•>£. Thefe Diltances carried about the Circumference of the Circle, will divide it into ib many equal Parts,
2 . To defcribe a regular "Polygon, e. gr. a Pentagon on a given right Line : Take the Length of the Line in the Compares, and apply it to the Extent of the Num- ber 5, 5, on the Lines of Polygons. The Setter thus opened, upon the fame Lines take the Extent, from 6 to 6, this will be the Semi. diameter of the Circle the Po. lygon is to be infenbed in. It, then, with this Diftance, from the Ends of the given Line, you defcribe two Arches of a Circle, their Interferon will be the Center ot the Circle.
3 . On a right Line, to defcribe an Ifofceles Triangle : Having the Angles at the Bale, double thofe at the Ver- tex : Open the Setter, till the Ends of the given Line fall on 10 and \o on each L .eg ; Then take the Diftance from 6 to 6. This will be the Length of the two equal iides of the Triangle.
Ufe cf the Lines of Sines > Tar.gcnts, and Secants ) on the Sector.
By the feveral Lines difpofed on the Settor, we have Sea es to feveral Radius's, So that having aLength, or Ra- dius given, not exceeding the Length of the Settor when opened ; we find the Chord, Sine, i§c. thereto. E.gr. Sup- pofe the Chord, Sine, or Tangent, of 10 Degrees, to a Radius of 3 Inches, required: Make 3 Inches the Aper- ture, between 60 and 60, on the Lines ot Chords of the two Legs 5 then will the fame Extent reach from 45, to 45 on the Line of Tangents, and from 90 to 90 on rhe Line of Sines on the other Side 5 fo that to whatever Radius the Line ot" Chords is let, to the fame are all the other let. In this Dilpofition therefore, if the Aper- ture between 10 and 10, on the Lines of Chords, be taken with the CompafTes, it will give the Chord ot 10 De- grees. If the Aperture ot 10 and 10 be in like manner taken on the Lines of Sines, it will be the Sine of 10 Degrees. Laftly, 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 re- quired 5 for the Chord, the Aperture of half the Arch, viz. 35, mull be taken as before 5 which Diftance, re- peated twice, gives the Chord of 70 . To find the Tan- gent of 70 to the fame Radius, the fmall Line of Tan- gents mull be ufed, the other only reaching to 45 : Making therefore, 3 Inches the Aperture between 45 and 45 on the Imall Line ; the Extent between 70 and 70 Degrees, on the fame, will be the Tangent of 70°. to 3 Inches Radius.
To find the Secant of an Arch, make the given Ra- dius the Aperture between o and o on the Line of Secants: Then will the Aperture of io and 10, or 70 and 70, on the faid Lines, give the Tangent of io° or 70 .
If the Converfe of any of thefe Things were required ; that is, if the Radius be required, to which a given Line, is the Sine, Tangent, or Secant, 'tis but making the given Line, if a Chord, the Aperture on the Line of Chords, between 10 and 10, and then the Settor will {land at the Radius required ; that is, the Aperture be- tween 60 and 60, °n the faid Line, is the Radius. If the given Line were a Sine, Tangent or Secant, 'tis but making it the Aperture of the given Number of Degrees, then will' the Diftance of 90 and 90 on the Sines, of 45 and 45 on the Tangents^ of and on the Secants, be the Radius.
Ufe of the S
ECTOR in
Tigommetry.
\°. The Safe and Perpendicular of a rettangled Triangle being given, to find the Hypothenufe .- Suppole the Bafe A C (Plate Trigonom. Fig. 2.) 40 Miles, and the Perpendicular " A B 30 j open the Setter till the two Lines of Lines make a right Angle : Then for the Bafe, take 40 Parts on the Line of Lines on one Legj and for the Perpen- dicular 30 on the fame Line on the other Leg: Then the Extent from 40 on the one, to 30 on the other, taken in the CompafTes, will be the Length of the Hypothenufe which Line will be found 50 Miles.
2 Q - The 'Perpendicular A B of a right-angled Triangle ABC, being given 30, and the Angle B C A 37°, to find the Hypothenufe B C: Take the given Side A B, and fet it over, on each Side, upon the Sine of the given Angle ACBj then the Parallel Diftance of Radius, or of 90 and 90, will be the Hypothenufe BC5 which will meafure 50 on the Line of Sines.
3 . The Hypothenufe and Safe being given, to find the 'Perpendicular : Open the Setter, till the two Lines of Lines be at right Angles 5 then lay off the given Bale on one of thofe Lines from the Centre : Take the Hypo- thenufe in your CompafTes, and fetting one Foot in the Point of the given Bafe, let the other fallon the Line of Lines, on the other Leg : The Diftance from the Centre to the Point where the CompafTes fall, will be the Length of the Perpendicular.
4 . The Hypothenufe being given, and the Angle ACB, to find the 'Perpendicular : Make the given Hypothenufe a Parallel Radius 5 i. e. make it the Extent from 90 to 90 on the Lines of Sines ; then will the, Parallel Sine of the 'Angle A C B be the Length of the Side A B.
5 tf . The Safe and 'Perpendicular A B given, to find the Angle B C A : Lay offthe Bafe A-C on both Sides the Setter, from the Centre, and note its Extent: Thea take the given Perpendicular, and to it open the Settor, in the Terms of the Bafe , the Parallel Radius will be the Tangent of B C A.
6 9 - In any right-angled Triangle, two Sides being given, it'ith the included Angle, to hid the third Side: Suppofe the Side A C 20, the Side B C 30, and the in- cluded Angle ACB no° ; Open the Setter, till the two Lines of Lines make an Angle equal to the given Angle, viz. no . Lay off" the given Sides of the Tri- angle from the Center of the Settor, on each of the Lines of Lines j the Extent between their Extremes is the Length of the Side A B fought, viz. 412.
7°. The Angles CAB and ACB given, and the Side C B, to find the Safe A B : Take the given Side C B, and turn it into the Parallel Sine of its oppofite Angle C A Bj and the Parallel Sine of the Angle ACB will be the Length of the Bafe A B.
8°. The three Angles of a Triangle being given, to find the Proportion of the Sides ; Take the lateral Sines of the feveral Angles, and meafure them in the Line of Lines j the Numbers anfwering thereto, give the Propor- tion of rhe Sides.
9 . The three Sides beinggiven, to find the Angle ACB: Lay the Sides AC, C B, along rhe Line of Sines, from the Centre, and fet over the Side AB in theirTerms: So is the Setter opened, in thefe Lines, to the Quantity of the Angle ACB.
10"- The Hypothenufe A C {Fig. 3. ) of a right-angled Spherical ABC %iven, e. gr. 43 ° ; and the Angle CAB 20°. to find the Side C B. The Rule is: As Radius is to the Sine of the given Hypothenufe 4; , fo is the Sine of the given Angle 20 to the Sine of the Perpendicular C B. Take, then, 20° from the Centre, along rhe Line of Sines, in your CompafTes, and fet the Extent, from 90 to 90, on the two Legs, and the Pa- rallel Sine of 430, the given Hypothenufe, will, when meafured from the Centre on the Line of Sines, give 130 g ' the Side required.
1 1°. The Perpendicular B C, and the Hppothenufi A G given, to find the Safe AB: As the Sine Complement of the Perpendicular B C is to Radius, fo is the Sine Complement of the Hypothenufe, to the Sine Complement of the Bafe- — Therefore, make the Radius a Parallel Sine of the given Perpendicular, e.gr. 76 30'; Then the Parallel Sine of the Complement of the Hypothenufe, e. gr. 47 meafured along the Line of Sines, will be found 49° 25', the Complement of the Bale requited 1 s Confequently the Bafe itfelf will be 40° 35'.
^Particular Ufes of the Sector in Geometry, &c.
i°. To make a regular Polygmt, whofe Area fiall be of any given Magnitude: Let the Figure required be a Polygon, whofe fuperficial Area is 125 Feet: Extract the Square Root of ' of 125, it will be found c. Make a Squaw-,
whofe
