And here it may not be improper to hint, that no Perfon, though accustomed to construct Plans of Gentlemens Eftates by the Theodolite, &c. fhould attempt to survey a County, or the Sea-Coasts or Channels of any Kingdom, till he has made himself Master of these Problems, with as much of Navigation or Aftronomy, at leaft, as teaches to find the Latitudes and Longitudes of Places; which done, for his Encouragement, we will venture to affirm there will not arife any Difficulty, in Practice, which he may not easily furmount. The Writers on this subject are numerous; as almost all Authors, who have treated of Surveying, Navigation, &c. have written preparatory Treatifes on Plane Trigonometry: Some of them are, Pitifcus, in the Year 1614; Gunter, 1623; Briggs, 1633: Newton, 1658; Sturmy, 1667; Norwood, 1631; Seller, 1669; Dechales, 1674; Phillipps, 1678; Sir Ja nas Moore, 1681; Atkinson, 1686; Leybourn, 1690; M. Bouger, 1694; Leybourn, 1704; Sherwin, 1705; Harris, 1706; Hodgson, 1706; Ozanam, 1712; Laurence, 1717; Wilson, 1720; Hawney, 1725; Keil, 1728; Wolfius, 1732; Kelly, 1734; Martin, 1736; Hauxley, 1743; Simpfon, 1748; Muller, 1748; Emerfon, 1749; Barrow, 1750; Webster, 1751, M. Bouger, 1753; Holliday, 1756; Martin, 1759; Maferes, 1760; Mountaine, 1763; Robertfon, 1764; Patoun, 1765; Simfon, 1767; Hutton, 1770; Payne, 1772. A CHAP. I. Of Definitions, and the Nature of Sines, Tangents, &c. CHAP. 2. Of the Conftruction of Scales P. I CHAP. 3. Of the Solution of the common Cafes of right-angled plane Triangles BOOK Of Oblique-Triangles 10 II. 22 Of DEFINITIONS, and the Nature of SINES, TAN I. T GENTS, &c. RIGONOMETRY is the Science or Doctrine of the Properties of Triangles. 2. A Triangle is a Figure contained under three Lines. 3. Trigonometry may be either Plane or Spherical Plane Trigonometry is, when the three Lines which form the Triangle are Right Lines. It is called Plane, because these Lines are in the fame Plane, or flat Surface. By fome Authors it is called Plain, because the common Principles of Plane are Something easier to be understood than thofe of Spherical Trigonometry. 4. Spherical Trigonometry treats of the Properties of Triangles, formed, or conceived to be formed, on the Superficies of a Globe or Sphere; which are therefore called Spherical Triangles. But this must be the Subject of fome future Effay. B of Of the Chord, Sine, Tangent, &c. of an Arc. 5 Let ADB be a Semicircle, C the Center, the Pl. I. Fig. Arcs AD and DB each '90 Degrees, or a Quarter of I, 2. a Circle. First. Affume any Point, F, in the Arc BD; join B, F ; then the Line BF is called the Chord of the Arc BF. 6. Secondly. From the Point F, let fall the Line FE, perpendicular to AB; or (which is the fame) draw FE parallel to DC; then is the Line FE called the Sine, and EB the Verfed-Sine, of the Arc BF. 7. Thirdly. At the Point B, on AC, raife the Perpendicular BG; join the Points, C, F, by the Line CF; which produce, to interfect BG in G; then is BG called the Tangent, and CG the Secant, of the Arc BF. 8. Fourthly. Any Arc, BF, lefs than a Quarter of a Circle, being fubtracted from 90 Degrees, or the Quarter of the Circle DB, the remaining Arc FD is called the Complement of the Arc BF. 9. Fifthly. From the Point, F, draw FI, perpendicular to CD; which is easiest done by drawing it parallel to AB: Alfo, from the Point D, DH parallel to AB, (or, which is the fame, perpendicular to DC,) till it interfects CG in H: Then is IF the Sine-Complement, DH the Tangent-Complement, and CH the Secant-Complement, of the Arc FB; or (which is the fame Thing) the Sine, Tangent, and Secant, of the Arc DF. 10. Inftead of Sine-Complement, Tangent-Complement, and Secant-Complement, it is very common to write Co-fine, Co-tangent, and Co-fecant. 11. By the Conftruction it appears that CE is always equal to IF, the Cofine of the Arc BF. 12. Hence The Reference, 217. g. at this Mark in the Margin, fignifies, Article 217, Elements of Geometry: And, in general, when g. is added to a Reference, it denotes the Elements of Geometry; otherwife, the Article in this Effay. |