Chap. II. Containing the Dorine of Plane-Triangles. T Rigonometry, is that part of Geometry, which treats more particularly about the measuring of Triangles, wherein having three Things given, either all Sides, or Sides and Angles, or Angles only in Spherics; a 4th (Side or Angle) may be found; and is either Plane or Spheric; It's the former we begin with. Section I. Of Things necessary to be understood, relating to Plane Trigonometry. 1. A Triangle, is any Three corner'd Figure; it confifteth of Six Things, Three Sides and Three Angles, and is either Plane or Spheric. 2. A plane Triangle is projected on a Plane or Flat Superficies, and therefore its Sides are Right-lines; but the Sides of a Spheric Triangle, are Arcs of the Sphere, of which more in the Chapter of Spheric Trigonometry. 3. An Angle is the meeting of any two Lines, making a Corner; and is either a Right-angle, containing juft 90 deg. or Oblique, more than 90 deg. called an Obtuse-Angle, or less than 90 deg. called an Acute-angle. 4. When two Lines crofing one another, make the Angles. on every Side equal, then thofe Angles are Right, and the Lines are perpendicular to each other. 5. A Degree, is the 360 Part of the Periphery of any Circle; the half 180 is a Semicircle and the 4th Part, 90, is called a Quadrant; Alfo a Degree containing 60 Min. and a Min. 60 Seconds, &c. as in Prob. IX. Defin. 1. of Geometry, in Page 15. 6. By Complement of any Number of Degrees, understand what thofe Degrees want of 90. 7. A Triangle, is either Right-angled, having one Rightangle; or Oblique, having no Right-angle. 8. In a Right-Angle-Triangle, the Side oppofite to the Rightangle, is called the Hypothenufe, and the two Sides containing the Right-angle, are called Legs. 9. In all Plane Triangles, the Sum of the three Angles is 180 Degrees. 10. The Angles (of a Plane-Triangle) being given, the Proportion of the Sides only can be determined; therefore, 1 11. In a Right-Angle-Triangle, two things given (one of them a Side) are fufficient to find a Third: But, 12. In Oblique-Triangles three things, and one of them a Side, must be given to find a Fourth. 13. Three Letters fignify an Angle; as BAC fignify the Angle A; And two Letters a Side, AB fignify the Side AB, &c. 14. Given Things, whether Sides or Angles, are marked with a Dafb, thus ('.) and required Things; with a Cypher, thus (°.) after any Number fignifies {Degrees, 25d. is 25 Degrees, and S. S. c. T. T. c. Sec. Sec. c. Co. Ar. 31m. is 31 Minutes. [Sine. } Sine Complement, or Co-Sine. thus Tangent-Complement, or Co-Tan. ftands for Secant. fo is; As thus, 24:36. That is, As 2 is to 4, fo is 3 to 6. 16. There are feven Cafes in Right-Angle-Triangles, and fix in Oblique. Their Solutions follow; but firft of Right-AngleTriangles. Section II. The firft Axiom, and the feven Cafes of Plane Right- Axiom 1.TN all Plane Right-Angle-Triangles, if one of the J. If the Hypothenufe be Radius, each Leg is the Sine of its oppofite Angle, See Plate 2. Fig. 1. marked for the first Axiom. 2. If one Leg be Radius, the Hypothenufe is a Secant, and the other Leg is a Tangent of the Angle oppofite to this Leg. See Plate 2. Fig. 2. titled for the first Axiom. Note 1. To find a Side, any Side may be Radius, faying thus; As one given Side, is to the Word on it; C 2 Obferve Obferve to begin with the Side made Radius. And what Proportion the Side (made Radius) hath to Radius; the fame hath the other Sides, to the Sines, Tangents, or Secants by them reprefented; and the contrary: And, These two Notes (to the diligent Reader) are fufficient to frame any Proportion by the firft Axiom, making any Side of a. Right-Angled-Triangle the Radius. Problem I. Cafe 1. The Angles, and Hypothenufe given; to find either of the Legs. Examp. In the Right-Angle-Triangle ABC, Plate z. Fig. 1. The This Triangle is made by Prob. 10. of Geometry, in Page 17.. If you make the Hypothenuse AC Radius, the Proportion (by Axiom 1. and Note 1.) is thus, As Radius, is to the Hypothenufe AC; fo is the Sine of the Angle BAC, to the Leg BC. Or thus briefly; Radius. Hypoth. AC: S. BAC... Leg BC.. S. god. 121 Leag. 54d. 30m. To work Proportions by-Logarithms, obferve this General Rule: Add the Logarithm of the Second and Third Terms together; and from that Sum fubtract the Logarithm of the firft Term, the Remainder is the Logarithm of the fourth. Term, or Number fought. As in the foregoing Proportion, the Sum of the Logarithms of the fecond and third Terms added together is 11.993471, from which it is easy to fubtract the Logarithm of the firit Term (being Radius) by cancelling (cutting off, or leaving out) the first Figure to the Left-hand, and then it is 1.993471, which brings forth 98.51; that is, Leagues 98.51 Parts of 100 for the Leg 'BC, the fourth Term, or Side required. .. By Gunter's Scale, thus, :: S. BAC Radius AC Sine 54d. 3om. (on the fame Sines) to Leagues 98.5 Tenths, (on the Line of Numbers) for the Leg BC, nearly as above. Obferve the like in all that follows, except in thofe Proportions wherein is the Word Secant, which is wrought only by the Logarithms. The three feveral Proportions, making each Side Radius, to find the Leg BC, axe these which follow. Radius ..S. BAC Sec. BAC.. T. BAC Sec. ACB Radius } :: Hypothenufe AC.. Leg BC. Likewife to find the Leg AB, they are thefe following: Radius ..S. ACB Sec. BAC.. Radius Hypothenufe AC.. Leg AB Note; When Radius is not the firft Term in the Proportion, then take the Complement-Arithmetic of the Logarithm of the firft Term; (which how to find, is fhew'd in Chap. 1. Sect. 2. Proportion 9. of the Ufe of the Table of Sines, Tangents and Secants, in Page 300.) This Comp. Arith. or Co. Ar. add to the Logar. of the Second and Third Terms, and from the Characteristic of their Sum fubtract 10 or 20, the remaining Figures is the Logarithm of the Fourth Term fought; as may be seen in the following Proportion in Problem II. Prob. II. Cafe 2 and 3. The Angles and one Leg given; to find the Hypothenufe, and the other Leg. Example. In the Right-Angle-Triangle ABC. Plate 2. Fig. 2. The {Angle ACB 35d, 30m. } given: {HYP. AC BC 98 Leagues Hyp. req. This Triangle is made by Prob. 12. of Geometry, P. 17 & 18. 1. To find the Hypothenufe AC, make it Radius, and the Proportion (by Axiom 1. and Note 1.) is thus, As the Sine of the Angle BAC, is to the Leg BC; so is Radius, to the Hypothenufe AC. Or thus, S. BAC Leg BC :: Radius. Hypothenufe AC, 54d. 30m. 98 Leg. By Gunter's Scale, thus ; S. BAC. Leg BC.: Radius. Hypothenufe AC. That is, S. 54°. 30.98 L.:: S. 9od... Leagues 120.4 Tenths, That is, the Extent from Sine of 54d. 30m. (on the Line of Sines) to 98 Leagues, (on the Line of Numbers) will reach from Sine of god. to Leag. 120.4 Tenths, the Hypothenufe required. The three feveral Proportions, making each Side Radius, to find the Hypothen use, are these following. S. BAC Radius } T. BAC. Sec. BAC:: Leg BC. Hypothenufe AC. And to find the Leg AB, they are these ; T. BAC.. Radius:: Leg BC.. Leg AB. Note; In working by Gunter, when a Tangent is mentioned, the Radius then is the Tangent of 45d. as in the two laft Proportions, it is, As T. BAC.. Radius: Leg BC. Leg AB. Which is, T. 54d. 30m... T.45d.:: 98 Leag... Leag. 69.9 Tenths. That is, the Extent from 54d. 30m. to 45d. (on the Line of Tangents) will reach from 98 Leagues, to Leagues 69.9 Tenths on the Line of Numbers, for the Leg AB. As Rad. .. T. ACB Leg BC Leg AB. Which is, T. 45d... T. 35° 30':: 98 Leag... Leag. 69.9 Tenths. That is, the Extent from T. 45d. to T. 35d. 30m. will reach from 98 Leagues, to Leagues 69.9 Tenths, as before. Prob. III. Cafe 4 and 5. The Hypathenufe and one Leg given, to find the Angles and the other Leg. Example. In the Right-Angle-Triangle ABC, Plate 2. Fig. 3. The {LYP: AB 125} Leagues given { Hyp. AC ACB or BAC 1 BBC req. This Triangle is made by Prob. 11. of Geometry, Page 17. 1. To find the Angles. If you make the Hypothenufe AC Radius, the Proportion (by Axiom 1. and Note 2.) is thus ; As the Hypothenufe AC, is to Radius; fo is the Leg AB, to the Sine of the Angle ACB. Or thus, Hypot. AC. Rad.:: Leg ABS. ACB. 121 Leag. 9od.:: 69 Leag... S. 34d. 46m. Which Subtract from Remainder is the Angle BAC god. com. 55d. 14m. by the 9th of |