and also less than it, as being comprehended thereby, which is absurd; therefore the Sphere A B C to a Sphere less than the Sphere D EF, hath not a triplicate Proportion of that which B C has to E F. After the same manner it is demonstrated, that the Sphere D E F to a Sphere less than A BC, has not a triplicate Proportion of that which E F has to B C: I say, moreover, that the Sphere A B C to a Sphere greater than D E F, hath not a triplicate Proportion of that which B C bas to EF: For, if it be possible, let it have to the Sphere L M N greater than DEF; then (by Inversion) the Sphere L M N to the Sphere ABC, shall have a triplicate Proportion of that which the Diameter E F has to the Diameter B C. But as the Sphere L M N is to the Sphere A B C, so is the Sphere D E F to some Sphere less than A B C, because the Sphere L M N is greater than DEF. Therefore the Sphere D E F to a Sphere less than ABC, hath a triplicate Proportion of that which EF has to BC, which is absurd, as has been before proved. Therefore the Sphere ABC to a Sphere greater than DEF, has not a triplicate Proportion of that which B C has to E F. But it has also been demonstrated, that the Sphere A B C to a Sphere less than D E F, has not a triplicate Proportion of that which B C has to EF: Threfore, the Sphere ABC to the Sphere DEF, bas a triplicate Proportion of that which B C has to EF; which was to be demona Itrated. THE THE EL EM ENTS Of PLANE and SPHERICAL TRIGONOMETRY. DEFINITION S. HE Business of Trigonometry is, to find the or the Ratios of the Sides, when the Angles are given, and to find Sides and Angles, when Sides and Angles are given : In order to which it is necessary, 'that not only the Peripheries of Circles, but also certain Right Lines in and about Circles, be supposed divided into some determined Number of Parts. And so the ancient Mathematicians thought fit io divide the Periphery of a Circle into 360 Parts, which they call Degrees; and every Degree into 60 Minutes; and every Minute into 60 Seconds; and, again, every Second into 60 Thirds; and so on. And every Angle is said to be of such a Number of Degrers and Minutes, as there are in the Arc measuring that Angle. There are some that would have a Degree divided into centefimal Parts, rather than sexagesimal ones ; and perhaps it would be more useful to divide, not only a Degree, but even the whale Circle, into a duplicate Ratio ; which Division may some Time or other gain Place. Now, if a Circle contains 360 Degrees, a Quadrant thereof, which is the Measure of a Right Angle, will be 90 of those Parts : And if it contains 100 Parts, a Quadrant will be 25 of these Parts. Tbe Complements of an Arc is the Difference thereof from a Quadrant. A Chord or Subtense, is a Right Line drawn from one End of the Arc to the other, The Right Sine of any Arc, which is also commonly called only a Sine, is a Right Line drawn, from one End of an Arc, perpendicular to the Radius drawn through the other End of the said Arc; and is, therefore, the Semisubtense of double the Arc; viz. DE="DO, and the Arc 'D Oʻis double of the Arc D B. Hence, the Sine of an Arc of 30 Degrees is equal to one half of the Radius. For (by Corol. 15. El. 4.) the side of an Hexagon inscribed in a Circle, that is, the Subtense of 60 Degrees, is equal to the Radius. A Sine divides the Radius into two Segments CE, E B: one of which CE, which is intercepted between the Centre and the Right Sine, is the Sine of the Complement of the Arc D B to a Quadrant (for Ć E=F D, which is the Sine of the Are DH), and is called the Cofire: The other Segment B E, which is intercepted between the Right Sine and the Periphery, is called a Versed Sine, and sometimes a Sagitta. And if the Right Line C G be produced from the Centre C, thro' one End D of the Arc, until it meets the Right Line BG, which is perpendicular to the Diameter drawn through the other End B of the Arc; then CG is called the Secant, and BG the Tangent, of the Are DB. The Cosecant and Cotangent of an Arc ate the Secant and Tangent of that Arc which is the Complement of the former Arc to a Quadrant. Note, As the Chord of an Arc, and of its Complement to a Circle, is the fume i jo, likewise, are the Sine, Tangent, and Secont, of an Arca the same as the Sine, Tangent, and Sciant, of its Complement to a Semicircle. I be Sinus Totus is the greatest Sine, or the Sine of go Degrees, which is equal to the Radius of the Circle. A Trigonometrical Canon is a Table, which, brginning frem ane Minute, ar derly exprefes the Lengths inat every Sine, Tangent, and Secant have, in respect of the Radius, which is supposed Unity; and is conceived to be divided inta 10,000,000 or more decimal Parts. And so the Sine, Tangent, or Secant, of an Arc, may be had by by Help of this Table ; and, contra' iwise, a Sine, Tane gent or Secant, being given te mroy find the Arc it expreses. Take notice, That in the following Trail, R. fignifies the Radius, S. a Sine, Cor. a Cofine, r. a Tangent, and Cot. a Cotangent; als A C q fignifies the Square of the Right Line AC; and the Marks or Charalters t, -, =, :: :, and V, are, severally, used zo signify Addition, Subtraclion, Equality, Proportionality, and the Extraction of the Square Root : Again, when a Line is drawn over the Sum or Difference of two Qxantities, then that Sum or Difference is to be considered as one Quantity. The CONSTRUCTIONS of the Trigonometrical Canon. THEOREM. given, the other side is also given. FOR (by 47 of the first Element) ACq=A Bq + BC and AC9-B C q=A Bq and interchangeably A C9-A Bq=BC 9. Whence, by the Extraction of the Square Root, there is given AC = VAB 9+BCq; and A B=VAC-DCq; and BC=VAC-A B q. PROPOSITION II. PROBLEM. The Sine DE of the Arc B D, and the Radius CD, being given, to find the Cosine D F. T! HE Radius C D, and the Sine D Е, being given in the Right-angled Triangle C D E, there will be given (by the last Prop.) ✓ CD-DEF=(CE=) DF. PRO T 4 |