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The great Advantage of the Se&or above the common Scales, &>c. is, that 'tis made to fit all Radius'sand Scales. For, by the Lines of Chords, Sines, £f?c. on the Se&or, we have Lines of Chords, Sines, Z$c. to any Radius betwixt the Length and Breadth of the Se&or when opened.
The Se&or is founded on the Fourth Proportion of the £thBookofi;/c//W; where 'tis demonftrated, That iimilar Triangles have their homologous Sides proportional : An Idea of its Foundation may be conceived thus ; Let the Lines AB, AC (Fig. 14.) reprefenr the Leg of the Se&or-, and AD, AE two equal Sections from the Centre. If now the Points C B and D E be connected, the Lines CB and DE will be Parallel 5 therefore the Triangles ADE, ACli Similar, and confequently the Sides AD, pE, AB and BC proportional: That is, as AD :DE: AB : BC $ whence if AD be the Half* Third or Fourth Part of A B ; DE will be a Half, Third or Fourth Part of CB, and the fame holds of all the reft. If. therefore, AD be the Chord, Sine, or Tangent of any Number of Degrees to the Radius AB ; DE will be the fame to the Radius BC.
Defcrifition of the Sector.
The Se&or confifts of two equal Rules or Legs of Brafs or otherMatter,rivetted together jbut lb as to moveeafy on the Rivet. See its Figure, Tab. Geometry Fig,i5, On the Faces oi the Instrument are placed feveral Lines^the principal are the Line ot equal Parts, Line of Chords, Line of Sines, Line of Tangents, Line of Secants, and Line of Polygons.
The Line of Equal 'Parts, called aUb Line of Lines, marked L, is a Line divided into 100 equal Parts ; and, where the Length of the Leg will allow ir, each is fub- divided into Halves, and Quarters. Tis found on each Leg, on the lame Side 5 and the Divisions numbred 1, 2> 3, 4, 1£>c. to 10, which is near the Extremity of each Leg. Note, in Practice, 1 is taken for 10, or 100, or 1000, or 10000, $$C. as Occafion requires $ in which Cales 2 is taken tor 20 or 200, or 2000, &c. and fo of the reft. %%e Line of Chords, marked C on each Leg, is divided after the ufual Manner,and numbred 10, 20, 30, &c. to 60. See Chord. TbeZ/?/e ofSines i denoted on each Leg by the Letter S, is a Line of Natural Sines, numbred 10, 20, 30, &c. to 90. See Sines. Line of tangents, denoted on each Leg by the Letter T, is a Line of natural Tangents, numbred 10, 20, 30, SS>c to 45 : befides which, is another little Line of Tangents on each Leg, commencing at 45 and extending to 75 denoted by the Letter t. See Tangent. Line of Stcants, denoted on each Leg by the Letter S, is a Line of natural Secants, number'd 10, 20,30, f$e. to 75, and commencing, not from the Centre of the lnftrument, but at Two Inches Diftance therefrom. See Secant. Line of 'Polygons, denoted by the Letter P on each Leg, is number'd 4, 5, 6, £S?c. to 12, which falls 3 Inches ihort of the Centre ot the Iniirument. See Polygon.
Befide thefe Lines, which are eiTential to the Se&or ; there are others placed near the outward Edges on both Faces, and parallel thereto, which are, in all refpe&s, the fame as in Giwter's Scale, and ufed after the fame Manner. Such are the Line of Artificial Sines marked S $ The Line of Artificial "tangents 5 A Line of 12 Inches, mark'd In; and Gunter's Line of 'Numbers, mark'd N. For the Ufes of all which, See Gunter's Scale. There are fome times, other Lines placed,, to fill the vacant Spaces, as the Lines of Hours, Latitudes and Inflations of Meridians-, which are ufed the lame as on the common Scales. See Scale.
The Lines found by the Se&or are of two Kinds, late- ral and parallel. The Firft are fuch as are found by the Sides of the Se&or as A B, A C. The latter, fuch as go flcrofsfrom one Leg to the other, as ~D E, C B. Note, the Order of the Lines in the newer Se&ors, is different from what it is in the old ones ; for the fame Line is not now put at the fame Diftance from the Edge on both Legs 5 but •he Line of Chords, e.gr. is innermoft upon the one, and the Line of Tangents on the other. The Advantage here- of is, that when the lnftrument is fet to a Radius for the Chords, ir ierves alfo for the Sines and Tangents without ftirring it; for the Parallel betwixt 60 and 60 of the Chords, go and 90 of the Sines, and 45 and 45 of the Tangents, are all equal.
C 43 D
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Vfe of the Line of Equal Tarts on the Sector.
1° To 'Divide a given Line Xmto any Number of equal ™ans, e.gr. Seven : Take the given Line in your Com- pafles, and fetting one Foot in a Divifion of the Line of ec lual Parts, that may ealily be divided by 7, e. gr. 70 , Wofc Seventh Part is 10 : Open the Sector till the other joint fall exactly on 70 in the fame Line, in the other !^ e g- In this Difpofition, applying one Point of the ^nvpaffes to 10 in the fame Line ; (hut them till the ether fall ; n , ;„ trie fame Line of the other Leg. This
Aperture will be the .thwart of the given Line. Note, if \L 7,u'\ % divided be too long to be applied to the Legs of the Setter only div.de one Half, or one Fourth by 7, and the Double or Quadruple thereof will be the 7 th Part of the whole. '
2° To meafure the Lines of the Perimeter of a Pohgon one of which contains a given Number of equal Parrs - lake the given Line in your Compafles, and let it over upon the Line of equal Parts, to the Number of Parts on each Side exprefltng its Length. The Settor remaining thus, fet off the Length of each of the other Lines paral lei to the former ; and the Numbers each of them falls on will exprefs their Lengths.
3° ji Right Line being given, aid the Number of Tarts it contains, e.gr. 120, to take from it a lefs Line, containing any humbtr of the fame 'Parts, e.gr. 2* : Taking the given Line m your Compafles, open the Setter till the two Feet fall on 120 on each Leg ; then, the Diftance from 2* to 25 gives the Lrne required.
4" To find a third 'Proportional to T-vo given Lines, and a. Fourth to Three. For rhe Firft, take the Length of the firft given Line in your Compafles, and lay it off on the Line of equal Parts from the Centre, to find the Number where it terminates: Then open the Settor, till the Length of the Second Line be included in the Aperture of the Extreme of the firft : The Setter remaining thus, lay oft the Length of the Second Line on one of the Legs from the Centre ; and note the Nomber where it ter- minates ; the Diftance between that Number on the Two Legs, gives the Third Proportional. In the Second Cafe, take the Second Line in your Compafles, and opening the Setter, apply this Extent to the Ends of the Firft laid off from the Centre on both Legs. ' The Seclor thus open'd, lay oft the Third Line from the Centre, and the Extent between the Number whereon it terminates on both Legs, is the Fourth Proportional.
5 To divide a Line in any given Proportion, e gr. into Two Parts, which mall be to each other as 40 to 70: Add the Two Numbers together, their Sum is no. Then, between your Compafles take the Line propofed, fuppofe 165, and open the Settor till this Diftance
which
reach from no, to n on both Legs. The Settor thu open'd, take the Extent from 40 to 46, as alfo from 70 to 70 i the firft will give 60, the laft 105, which will be the Parts propofed, for 40 : 70 : 60 : 105.
6° To open the Settor, fo as the Two Lines of equal Parts may make a Right Angle : Find Three Numbers that may exprefs the Sides of a Right-angled Triangle, as 3, 4 and 5, or their Equimultiples, as 60, 80 and 100. Take, then, in your Compafles the Diftance from the Centre to 100 ; and open the Settor, till, one Point fet 1 on So, ' the '
.,, 1° **&* a R '& h ' Urn equal to the Circumference of a. Circle: The Diameter of a Circle being to the Circum- ference, nearly as 50 to 157, take the Diameter in your Compafles, and fet it over on the Legs of the Setter, from 50 to jo. The Settor thus open'd, take the Diftance from "57 «° 157 ln your Compafles. This will be your Cir- cumference required.
Ufe of the Line of Chords at the Sector.
1°. To open the Sector, fo as the two Lines of Chords may make an Angle of any Number of 'Degrees, - — --•
...~ .u xuu , a.iu u^cii uie sector, tin, one roint let up- 80, the other full upon 60 in the other Leg ; rhen do Two Lines of equal Parrs, include a Right Anole.
Take the Diftance from i
40:
the Joint to 40, the Number of Degtees propofed, on the Line of Chords; open the Settor, till the Diftance from 60 to 60, on each Leg, be equal to the forefaid Diftance of 40 ; then does the Line of Chords make the Angle required.
2°. The Settor being opened, to find the degrees of its Aperture : Take the Extent from Co to 60, and lay it off on rhe Line of Chords, from the Centte : The Num- ber, whereon it terminates, fhews the Degrees of its
Opening By applying Sighrs on the Line of Chords,
the Settor may be ufed to take Angles, as a Surveying lnftrument. J 5
3°. To make an Angle of any given Number of Degrees, with a given Line: On the given Line delcribe a Circu. lar Arch, rhe Centte whereof is the Point whereon the Angle is to be made. Set off the Radius from 60 to 60 ; and the Settor remaining thus, take the Diftance of the two Numbers on each'Leg, exprefling the propofed De- grees, and lay it from the Line upon the Arch defcribed. Laftly, drawing a Line from the Centre, thtough the End of the Arch, it will make the Angle propofed.
4 . To find the Degrees a given Angle contains : About the Vertex delcribe an Arch, and open the Settor, till the Diftance from 60 to to on each Leg, be equal to' the Radius of the Circle: Then taking the Chord of the Arch between the Compafles, and carrying it on the Legs of the Settor, fee what equal Number, on each Leg,
the
