3. To find the Angle C. The Analogy sBC:R::SBA:s C. In Numbers 70545 : 100009 :: 59130 : 83819 ; Now 83819 is the Natural Sine of 569 57'=C. Cafe 3. Given . as BA = 36 15 1. To find the Hypothenuse B C. The Analogy R:05 B : :AB :ctB C. In Numbers 100000 : 73649 :: 136382 : 100291 ; Now 100291 is the Tangent of the Complement of 448 52' = B.C. 2. To find the other Leg A C. The Analogy R:SBA:: B : 1 AC. In Numbers 100000 : 59130 : :91847 : 54295 : fo is 54295 the Natural Tangent of 28% 30'=AC. 3. To find the other Angle C. The Analogy R :05BA :: 5 B :cs C. In Numbers 100000 : 80644 :: 67644 : 54537 5 fo fhall 54537 be the Co-fine of 560 57' = C. 1. To find the Hypothenuse B C. The Analogy s B : s AC :: R:s BC. In Numbers 67644 : 47715 : : 100000 : 70545 ; then is 70545 the Sine of 44° 52' = B'C. 2. To find the other Leg B A. The Analogy i B.: TAC :: RisB A. In Numbers 91847 : 54295 : : 100000 : 59130 ; thus is 59130 the Sine of 36° 15' = BA. 3. To find the other Angle C. The Analogy csCA:R::cśB:s C. In Numbers 87881 : 100000 :: 73649 : 83819 ; so is 83819 the Natural Sine of 56° 57' = C. Case 5. Given and the Leg The Leg' :: 6. 6 BA = 36 15 AC = 28 30 1. To find the Hypothenuse B C. The Analogy R :0.8CA :: CB A:45 BC. In Numbers 100000 ': 87881 :: 80644 : 70875 ; Thus shall 70875 be the Co-sine of 44° 52' = B.C. 2. To find the Angle B. The Analogy sBA:R::1CA : t B. In Numbers 59130 : 100000 :: 54295 : 91847 ; foʻis 9184the Tangent of the Angle B = 42° 34'. VOL. II. Q 2 3. TO To find the Angle C. 3. The Analogy sCA:R ::BÁ : 10. In Numbers 47715 : 100000 :: 73323 : 153692'; i thus shall 153692 be the Tangent of 56-57 C. : Cafe 6. Given both the Angles { B = 42 34 ; 1. To find the Hypothenuse. B C. The Analogy 4C : CtB :: R:6 BC; In Numbers 153692 :: 108876 : 100000 70857 ; thus you have 70857 the Co-line of 44° 52' = BC. 2. To find the Leg A B. The Analogy s B : CSC ::R: csBA. In Numbers 67644 : 54537 :: 100000 : 80644 ; fo that 80644 is the Co-sine of 36% 151 = BA. 3. To find the Leg A C. The Analogy sĆ :-65B :: ROD'S AC; In · Numbers 83819 : 736492 :: 100000 # 87881"; and thus we have 87881 the. Co-fine of 289 30'AC. Thus we have the Manner of working all the Cases of a Right-angled Spherical Triangle by Natural Sines and Tangents. C HA P. C H A P. XI. The third Method of solving Right-angled Spherical Triangles, by the Lord Napier's Five Circular Parts. HAT these Five Circular Parts are, I have already said in the Definitions ; and I have also demonstrated in Theorem 40, and 41, their Properties, wherein the Use and Excellency of this Invention of that noble Lord doch wholly consist ; and that is, by shewing at once the Proportion for resolving any Case of a Right-angled Spherical Triangle, by one Universal Proposition or Catholic Canon ; viz. The Radius and Sine of the Middle Part, are re ciprocally proportional to the Tangents of the Extream Parts conjunet, and to the Co-fines of the Exa treams disjunet. Now in order to set forth the Advantage of this Universal Canon, there is contrived an Instrument which shews by Inspection the Order of the Terms, in any Proportion or Analogy, for the Solution of aný Cafe, by having the several Parts of this Canon, and 'the Five Circular Parts, fo aptly placed thereon, as to correspond to each other; and thereby conftitute the Analogy at once. But before I come to describe this Instrument, and shew its use ; I must desire the young Learner to consider well the following Particulars. 1. That if the Extreme Parts be conjunct, then the Analogy must be performed by Sines and Tangents together. 2. But if the Extremes be Disjunct, then the Analogy will consist of Sines only. 3. If the Side or Angle fought chance to be the Middle Part, then muft Radius begin the Analogy ; or be the First Term. 4. But if one of the Extreams be the Part sought, then the other Extream must be the first Term of the Analogy 5. That in using the Instrument, if it chance that Co-fine and Complement should correspond, then you must understand the Sine of the Part thereby. For the Co-sine of the Complement of an Arch, is the Sine itself of that Arch. The Instrument now to be described (a Scheme of which I have here delineated) may be made of Paper, Pastboard, Wood, Brass, Silver,' &c. and confifteth of two Parts, of which the leffer and inmost moveth on the other, by some kind of Rivet, on the Center, in order to be turned round as Occasion requires. On this inmost moveable Piece is described a Right-angled Spherical Triangle ABC, whose five Circular Parts are, by five straight Lines referred to the outmost fixed Piece, which is divided into two Circular Margins, both which are divided into five equal Parts. |