Note, Befides the Characters already Explain'd in Book I Pages 2, 3 and 4. these following are added, Viz. Angles. 328 PARTI. Of TRIGONOMETRY. Definitions. 1. A Circle is fuppofed to be divided into 360 equal Parts, called Degrees, and each Degree into 60 equal Parts,called Minutes, and each Minute into 60 equal Parts called Seconds, &c. Any Portion of whofe circumference is called an Arc, and is measured by the number of Degrees it contains. 2. A Chord is a Right-line joyning the extremities of an Arch, as AC is the Chord of the Arches ABC, ADC. 3. A Right Sine is a Rightline drawn from one End of N. B. A great part of fcrib'd out of Heyn's this Part I. was tran Trig. an Arc, perpendicular to that Diameter paffing through the other End; or it is half the Chord of twice the Arc. So A E is the Right-fine of the Arcs A B, A D. And here it is Evident, that the Sine of 90 Degrees, (which is equal to the Radius or Semidiameter of the Circle) is the greateft of all Sines, the Sine of an Arc greater than a Quadrant, being less than the Radius. 4. A Verfed Sine is the Segment of the Diameter intercepted between the Arc and its Right-fine, thus; E B is the Versed Sine of the Arc A B, and ED of the Arc A D. 5. A Tangent of an Arc is a Right-line drawn perpendicu lar to the end of a Diameter paffing through one end of an Arc, and its Length is limited by a Right-line drawn from the Center through the other End of the Arc, and this Line is called the Serant; thus, B M is the Tangent, and FM the Secant of the Arcs A B, A D. 6. The 6. The Difference of an Arc from a Quadrant, whether it be greater or lefs, is called its Compliment; fo GA is the Compliment of the Arcs AB, ADHA is the Sine of that Compliment, or Co-Sine; & the Tangent of that Compliment, or Co-tangent; FI the Secant of that Compliment, or Co-fecant. ond 7. The Difference of an Arc from a Semi-circle is called its Supplement. That part of the Radius which is betwixt the Center and Right-Sine, is equal to the Co-fine; thus FE is HA. A 9. If an Arc be greater or less than a Quadrant, the Sum or Difference of the Radius and Co-fine is equal to the Verfed Sine. AXE In a Triangle are Six Parts, vig. Three Sides and three Angles; any three of which being given (except the three Angles of a Plane Triangle) the other three may be found, either Mechanically by the help of a Scale of Equal Parts and Line of Chords, or by an Arithmetick Calculation (if fuppofing the Radius divided into any Number of Equal Parts, we know how many of thofe Equal Parts are in the Sine, Tangent, or Secant of any Are propofed): The Art of inferring which is called Triganometry, and is either Plain or Spherical. The Table of Natural Signs, Tangents, &c. may be thus made. X=AY+(V. A E the Sine of an Arc being given to N. B. The Radius is always fuppos'd to be given OFA g? A Eg:+FEg: (By 47.1. Excl. FAQ:AEG: FE4: (By Tranfp.) √ F A q: A E q: FE (By Evolution.) That is, Rg — S q : — Σ. 2. AE the Sine of an Arc being given, to find BN the Sine of half the Arc. ft of this) and confe FE the Co-fine, is known (by the quently EB; then √AEq: + EBq: AB (by 47. 1. Eucl. Elem.) And 4B = BN (by Third Definition.) i. e. t √ S q : + U q := S& Arc. U u 3. BN Part I. 3. BN the Sine of an Arc being given to find AE the Sine of twice that Arc... FN the Co-fine is (or may be made) known by the 1ft; and the AFBN and ABE, have the NCE Land the B common to both, confequently the NFB is = Z BAE (by 32. 1. Eucl. Elem.) Therefore the As FBN and ABE are fimilar; therefore (by 4. 6, Eucl. El.) FB..FN::ABA E. i. e." 4. BO and XP, the Sines of two Arcs BD and XB, being given to find Xp, the Sine of the Sum of the Arcs. From the Point Plet fall P VIDF and P.YX 9. Then the Co-fines F O and FP are known (by the 1ft.) And the ASFBO and FPV are fimilar; therefore (by 4. 6. Eucl. Elem.) FBFP :: BO...PV. Again, the XuP and Fu as being each, and the Eucl. El.) Wherefore they are B Y VOD have the Ls XPu and equal, PuX=LouF (by 15.1, Similar. But the As Fou and FOB are also (manifeftly) Similar; therefore the 4. PXu is 4 BFQ; and confequently the As PXY and BFO are Similar: Wherefore, FB FO:: XP. XY; But Y(PV)+YX=Xp= FOXXP+FPXBO FB 3 That is, the Sine of one Arc into the others Co-fme the Sine of the other Arc into the Co-fine of the first Arc, and this Sum divided by Radius, is S, of the Sum of the two Arcs; and if the Sines of the fame two Arcs were given, and it were required to find the Sine of their Difference, it would be FOXX P R FPXBQ =S Difference of the two Arcs. Let |