The PREFACE.

quite opened; or leffer than the Distances from 90 to go of the Sines, 60 and 60 of Chords, or 45 and 45 of Tangents, when the Sector is quite fut: And fo by this Inftrument the Sphere may readily be projected, upon the Plan of any Circle whofe Radius not exceeds the afore-prefcribed Limit. Whence the Excellency of this Inftrument above the Plain-Scale manifeftly appears; for that is a Scale but to one Radius; and therefore by it, it is not easy to project the Sphere, unless upon a Circle whofe Radius is of a given Length.

In the fourth Chapter, you have the chief particular Ufes of the Lines of Lines, and the Lines of Polygons.

In the fifth Chapter, is fhewn the Manner of working Proportions with Lines and Sines, Sines and Tangents, Lines and Tangents, &c. As alfo the excellent Ufe of the Sector in drawing the Hour Lines upon an horizontal and erect South Plan.

Lastly, In the fixth Chapter is contained, the Ufe of the Lines of Numbers, artificial Sines and Tangents. What is faid in this Chapter, I am of opinion, is fufficient to fhew Perfons, indifferently Skilled in Arithmetick and Geometry, the Manner of using thofe Lines.

Thus, courteous Reader, have I given you the Contents of what is to be read in the following little Tract, hoping you will candidly receive the fame. And fo I remain your's, &c. .

E.-S.

THE

THE

Defcription, Nature, and General Use

OF THE

Sector and Plain-Scale, &c.

CHA P. I.

Concerning Definitions of Chords, Sines, Tangents, &c. and the Manner of Projecting them, and putting them on Rulers, commonly called Plain-Scales.

B

ECAUSE of the fix Parts of a Triangle, viz. the three Sides and the three Angles, any three of them being given, independent of one another, the other three are thereby limited, and confequently fome way or other to be found. And fince the Angles of a Triangle, that is, the Arcs of equal Circles de

scribed

fcribed about, the angular Points, and comprehended between the Sides (which are proportional to the Angles, from Prop. 33. lib. 3. Euclid) are not proportional to its Sides, the Ancients devifed certain right Lines, appertaining to a Circle, which come in competition, with Angles or Arcs. These right Lines are thus defined:

A Chord of an Arc or Angle is a right Line drawn from one End of an Arc to the Fig.1. other; as the Line AB is the Chord of the Arc AB: it is alfo the Chord of the Arc AFB, because it is common to both Arcs of the Circle.

Whence it is manifeft. (per Prop. 15. lib. 3. Euclid) that the greatest Chord that can be drawn in a Circle is its Diameter, or the Chord of 180 Degrees, or a Semicircle: Whence all Chords of Arcs, greater than a Semicircle, are leffer than the Diameter,

A right Sine of an Arc is a right Line drawn from one End of that Arc, perpendicular to a Diameter drawn to the other End of it, and belongs to both Arcs of a Semicircle; as BC is the right Sine of the Arc AB, and alfo of the Arc FB. Hence it is manifeft, from the aforecited Propofition of Euclid, that the greatest Sine is equal to the Semidiameter GE, or the Sine of 90 Degrees: Therefore all Sines of Arcs greater than 90 Degrees, or a Quadrant, are leffer than the Semidiameter, or Radius,

The verfed Sine of an Arc is that Portion of the Diameter included between the right Sine of the faid Arc, and the Arc itfelf. As AC is the

verfed

verfed Sine of the Arc AB, and FC the verfed Sine of the Arc FGB; hence the greatest verfed Sine is the Diameter AF, viz. the verfed Sine of 180 Degrees, or a Semicircle.

A Tangent of an Arc is a right Line touching the Circumference of a Circle, and (from Prop. 16. lib. 3. Euclid) is at right Angles to the Diameter drawn through the Point of Contact, and limited by the Secant of the fame Arc; as AD is the Tangent of the Arc AB; whence there can be no Tangent of 90 Degrees, or a Quadrant, for then the Tangent will be infinitely long,

The Secant of an Arc is a right Line drawn from the Center of a Circle, through one End of the Arc it is the Secant of, till it meets the Tangent, raised at right Angles to a Diameter drawn to the other End of the Arc; as ED is the Secant of the Arc AB; whence there can be no Secant of 90 Degrees, or a Quadrant, because then the Secant will be infinitely long.

The Half, or Semitangent of an Arc, is that Portion, next to the Center, of a right Line drawn from the Center of the Circle parallel to the Tangent of the Arc, cut off by the Chord of the Complement of the Arc to 180 Degrees, as EH is the half Tangent of the Arc AB. But because (from Prop. 20. lib. 3. Euclid) the Angle BFA is half of the Angle BEA, the Side FE equal to the Side EA, and the Angles EAD, FEH right ones, HE will be equal to the Tangent of half the Arc BA; that is, the Semitangent of any Arc is but the Tangent of half that

Arc

Arc; whence the greatest Semitangent, or that of 180 Degrees, is infinite.

The Circumference of every Circle is fuppofed to be divided into 360 equal Parts, called Degrees; and each Degree into 60 equal Parts, called Minutes, &c. This Number was chofen by Geometricians for the Divifion of a Circle, because it may be divided into a greater Number of Parts without any Remainder, than any leffer Number than 360.

The Complement of an Arc, or Angle, is what it wants of a Quadrant, or 90 Degrees; or of a Semicircle, or 180 Degrees; or laftly, of a whole Circle, or 360 Degrees. As 20 Degrees is the Complement of 70 Degrees to a Quadrant, because 20 Degrees is the Remainder of 70 Degrees fubtracted from 90 Degrees; likewife 50 Degrees is the Complement of 130 Degrees to 180 Degrees, and 70 Degrees the Complement of 290 Degrees.

How to Project the Plain-Scale.

Firft draw a Circle, ABDC, which Fig.2. cross at right Angles with the Diameters, AD, CB; then continue out AD to G, and upon the Point B raise BF perpendicular to CB. Now draw the Chord AB, and divide the Quadrant AB into 9 equal Parts, fetting the Figures 10, 20, 30, &c. to 90, to them; each of which 9 Parts again fubdivide into 10 equal Parts, and then the Quadrant will be divided into 90 Degrees. Now fetting one. Foot of your Compaffes in the Point A, trans

fer