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II. If the Extreams (in any Proportion) be Diftinct from the Mid. Fig.XXI. dle Part, the Proportion will be performed by Sines only: But if the Extreams be Conjunct to the Middle Part, it must be performed by Sines and Tangents jointly.
The Demonftration of the Univerfal Propofition is obvious Demonftration_of_the_Univerfal enough: For, where the Extream Parts are Disjunct, the Proportions differ nothing from the common ones: And in the Extreams Conjunct, where it is commonly faid,
As Radius, to the Tangent,
We here fay,
As the Co-Tangent, to the Radius.
And likewife Inverfly and Contrarily, which is plainly the fame thing: Because,
The Radius is a Mean Proportional, between the Tangent of an Arch, and the Tangent Complement of the fame Arch. Note, That when a Complement in any Proportion doth chance to concur with a Complement in the Circular Parts, you must then (always) take the Sine, it felf, or the Tangent it felf; inftead of the Co-fine, or Co-Tangent, in the Circular Parts: Because the Co-fine of the Co-fine, is the Sine it felf, and the Co-tangent of the Co-tangent, is the Tangent it self,
As is the Sixth Cafe following; where C B and C A are given, and the Angle at B is required: Here CA is the Middle Part, and C B, and B, are the Extreams Disjunct: Wherefore (by the fecond Part of the foregoing Corollary) the Proportion will be performed by Sines only. And (by this laft) because the Two Extreams, CB, and B, fall upon Complements in the Circular Parts; therefore, inftead of Co-fine B C, and Co-fine B, you muft fay Sine B C, and Sine B.
These Things premifed, we will exemplifie in the Solution of
CHA P. IV.
The Analogies, or Proportions, for the Solution of the Several Cales of Right-angled Spherical Triangles, by the Univerfal Propofition.
OR the Performance hereof, I fhall make use of this Rightangled Spherical Triangle A B C, Right-angled at A, the Quantities of whofe Sides and Angles are adfixed to their refpective Circular Parts in the Diagram noted with B, in Fig. XXI. And alfo in this Table, both in Sexagenary Degrees and Minutes; and in Decimal Parts alfo.
And in every Cafe I fhall diftinguifh the Two Given Terms, (be fides the Right Angle, which is, always the third) by marking the Sides or Angles Given, by a fhort Stroak, (1), and the Term Required, I fhall mark with (o.) All which are to be seen in Figure XXI.
The XVI Cafes of Right-angled Spherical Triangle, Refolved,
The Hypotenuse BC, and the Angle at C, given; To find
To Sine C, 56 Deg. 52 Min.
To Sine B A, 50 Deg. 10 Min.
9.9229334 9.9623978 19.8853312
CASE II: The Side Adjacent A C, Extream Conjunct.
As Co-tangent B C, 23 Deg. 30 Min.
To Radius, go Deg."
So Co-fine C, 33 Deg. 8 Min.
To Tangent C A, 51 Deg, 30 Min...
CASE III. The other Angle B, Extream Conjunct.
The Hypotenufe B C, and Side A-C, given, to find
CASE IV: The Oppofite Angle at B, Ex. Disj.
As Sine B C, 66 Deg. 30 Min.
So Sine, CA, 51 Deg. 30 Min.
To Radius, 90 Deg.
To Sine B, 58 Deg. 35 Min.
As Radius, 90 Deg.
To Tangent CA, 51 Deg. 30 Min.
CASE V. The Adjacent Angle C, Middle Part.
So Co-tangent BC, 23 Deg. 30 Min.
To Co-fine C, 33 Deg. 8 Min..
CASE VI. The other Side A B, Extream Disjunct.
Fig.XXI. The Side A C, and the Angle oppofite there to B, being Given ;
CASE VIII. The other Angle at C, Extream Disjunct.
As Cc-fine C A, 38 Deg. 30 Min.
To Radius, 90 Deg.
So Co-fine B, 31 Deg. 25 Min.
CASE IX. The Hypotenufe B, C, Extream Disjunct.
As Sine B, 58 Deg. 35 Min.
To Radius, 90 Deg.
So Sine CA, 51 Deg. 30 Min.
To Sine B C, 66 Deg. 30 Min.
The Side CA, and the Angle C, adjacent thereto, given; To find
CASE XI. The other Angle B: Middle Part.
CASE XII. The Hypotenuse BC, Extream Conjunct.
As Tangent CA, 51 Deg. 30 Min.
10.0993948 Cafe XII.
CASE XIII. Either Angle, as C: Extream Conjunct.
CASE XIV. The Hypotenufe C B: Middle Part.
To Co-fine € B, 23 Deg. 30 Min.
The Two Angles B and C, given; To find
CASE XV. Either of the Sides, as AC: Extream Disjunct. Cafe XV.