10. The oppofite Angles made by the croffing of two Lines are equal, as the Angle DBE is equal to the Angle ABC. 11. An Angle is either Right or Oblique. 12. A right Angle is that whofe Measure is 90 degrees, as FBH. 13. An oblique Angle is either Acute, or Obtufe. 14. An acute Angle is that whofe Measure is lefs than 90 degrees; as the Angle ABC, or the Angle ACB. 15. An obtufe Angle is that whofe Measure is more than 90 degrees, as the Angle DBC. 16. All Angles coming (or meeting) together in one point upon a right Line; all of them taken together are equal to two right Angles, as the Angles DBH, HBC, and CBA meeting all together in the point B, upon the Line DF, are equal to two right Angles DBH, HBF. 17. A Triangle hath fome of its Sides equal, or elfe they are all unequal. 18. A Triangle of fome unequal Sides, is either Equicrural, or Equilateral. 19. An Equicrural (called alfo an Ifofceles) Triangle, is equiangled at the Bafe. And if a Perpendicular be let fall from the meeting of the equal Sides, it cuts the Bafe and the Angle oppofite thereunto into two equal parts; as in the Equicrural Triangle DLC, the Perpendicular DA cuts the Base CL into two equal parts in A, and also bissecteth the Angle CDL. 20. An Equilateral Triangle is that whofe Sides are all equal, and whofe Angles contain each 60 degrees. 21. A Triangle is Right-angled, or Oblique-angled. 22. A Right-angled Triangle is that which hath one right Angle, as the Triangle CBA Right-angled at A. (See the former Figure.) 23. An Oblique-angled Triangle is that which hath all its Angles Oblique, as the Triangle BDC. 24. An oblique-angled Triangle is either acute angled, or obtufe angled. 25. An acute angled Triangle is that which hath all its Angles acute, as the Triangle BDE. 26. An obtufe angled Triangle is that which hath one Angle obtufe, as the Triangle CDB, obtuse angled at B. 27. All equiangled (or like) Triangles have their Sides about the equal Angles proportional, (Eucl. Lib.6. Pr. 4.) E 1 B C DA F Let ABC and DEF be two Triangles equiangled, fo as the Angles at Band D, at A and E, and alfo at C and F,be equal one to another, each to its correfpondent; then I fay it will be, As AB: BC:: DE: DF. As AB: AC:: DE: EF. As AC: CB:: EF: DF. 28. If any one side of a plain Triangle be continued, the outward Angle (made by that Continuation) is equal C to the two inward and oppofite Angles in the fame Triangle, (Eucl. Lib. 1. Pr. 32.) As in the Triangle ABC, the Side CB, being continued to D. I fay, the Angle ABD is equal to both the Angles, at A and C. B 29. The three Angles of every plain Triangle, are together equal to two right Angles, (Eucl. Lib. 1. Pr. 32.) And from hence will follow thefe Corollaries. 1. That there can be but one right, or one obtufe Angle in any plain Triangle.. 2. And if one Angle be either right or obtufe, the other two shall be acute. 3. That the third Angle of any plain Triangle, is the Complement of the other two, to two right Angles. 4. That if two Triangles be equiangled in any two of their Angles, they are wholly equiangled. CHA P. II. Concerning Sines, Tangents, and Secants, by which the Angles of Right-lined, and Sides and Angles of Spherical Triangles, are measured. "T HE Menfuration of Triangles is by knowing any three of the fix Parts (Vide 1. of Chap. 1.) whether Sides or Angles, or both, to find out any of the other parts of the Triangle whether Sides or Angles. 2. And this is performed by the Rule of Proportion (commonly, and defervedly called the Golden Rule of Arithmetick) which teacheth of four Numbers proportioned one to another, any three of them being given, to find out the fourth. 3. So that for the Measuring (which is also called the Refolving) of Triangles, there must be certain Proportions of all the parts of a Triangle one to another, and thofe Proportions explain'd in Numbers. But the certain knowledge of the Proportions that all the parts of a Triangle have one to another, is not yet known; for that every crooked Line in a Triangle (as the Measures of the Angles in the Right-lined, and the Measures of Sides and Angles in a Spherical Triangle are) must be reduced to a Right Line, a Proportion for which was never yet B 3 found, found. But crooked Lines are nearly reduced to right Lines by the definition of the quantity which right Lines applied to a Circle have in refpect of the Ra dius (or Semidiameter) of the Circle. 4. Right Lines applied to a Circle are Subtenfes (or Chords) Sines, Tangents, and Secants. 5. A Subtenfe (or Chord) is a right Line infcribed in a Circle, dividing the whole Circle into two parts; and fo fubtendeth both the Segments. 6. A Subtenfe paffing thro the Center of a Circle, divides it into two equal parts, and fubtends both Segments or Semicircles; and this Subtenfe is the greatest. 7. A Subtenfe not the greateft, is that which divideth the Circle into two unequal parts; and fo on the one fide fubtends an Arch lefs than a Semicircle, and on the other fide an Arch more than a Semicircle as in the following Q E B Figure, AB paffeth thro the Center, and divides it into two equal parts; but CD does not pafs thro the Center, therefore divides the Circle into two unequal parts, or Segments, CQD the greater, and CPD the leffer. 8. A Sine is either right or versed. 9. A right Sine is one half of the Subtenfe of the double Arch: As the right Sine of PD, or DBQ, is FD, it being the half of CD the Subtenfe of CPD, equal to twice PD. So the right Sine of DB, or DPCA, is the right Line DH, the half of the Subtenfe DG. Hence it follows, that the right Sine of an Arch, more or lefs than a Quadrant, and lefs lefs than a Semicircle, is one and the fame. Hence it is alfo, that whenfoever the right Sine is called the Sine of the Complement, or Cofine, it is understood only the Sine of the Complement of an Arch lefs than a Quadrant: As the right Sine of the Complement PD is the right Sine of DB, that is, the Line DH. 10. Every right Sine is perpendicular to the Diameter drawn from one extreme of the Arch given; as the right Sine FD is perpendicular to PQ, and DH is perpendicular to AB. 11. The right Sine of the Complement is equal to the Segment of the Diameter intercepted between the right Sine and the Center of the Circle; as the Cofine of PD is the Segment of the Diameter FO, equal to HD. 12. The verfed Sine is the Segment of the Diameter intercepted between the right Sine and the Circumference; as the verfed Sine of the Arch PD is the Segment of the Diameter PF. 13. Of verfed Sines fome are greater and some are leffer. 14. A greater verfed Sine is the verfed Sine of an Arch greater than a Quadrant, and a leffer verfed Sine is the verfed Sine of an Arch leffer than a Quadrant. As the verfed Sine of DBQ is FQ, the verfed Sine of an Arch greater; and the verfed Sine of PD (as before) is PF, the verfed Sine of an Arch lefs. 15. A Tangent is a right Line drawn perpendicularly on the extremity of the Diameter of the Circle, and paffing by the other end of the Arch; as BE is the Tangent of BL. 16. A Secant is a right Line drawn from the Center of the Circle, by the end of the Arch, till it meet the Tangent; as the Secant of BL is the right Line OE. 17. The Tables of Sines, Tangents, and Secants, are extended no further than 90 Degrees (or a Qua B 4 drant.) |