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Fig.XXI. Note, Thefe are the Propertions anfwerable to the Univerfal Propofition, yet may (many of them) be varied; fo that the Radius may be brought into the firft Place, and that by the latter part of the foregoing Corollary: Which faysRadius is a Mean Proportional between the Tangent of an Arch, and the Tangent Complement of the fame Arch. -So that (in the XIIIth CASE) where it is faid,

As the Tangent BA, Is to Radius:

So is Sine C A, To Co-tangent C.

It is all one, as if you fhould fay,

As Radius, 90 Deg.

To the Co-tangent B A, 39 Deg. 50 Min.
So the Sine ofCA, 51 Deg. 30 Min.

To the Co-tangent of C, 33. Deg. 3 Min.


9.9222466 9 8935444 19.8157910

The like Courfe may be taken in the Second, Third, Tenth and Twelfth CASES.

And thus you have the whole Doctrine of the Dimenfion of Right-angled Spherical Triangles, performed by Help of this one Catholick Propofition.


Some Pranotions concerning Oblique-angled, Spherical
Triangles, in order to the Solution of them.


N Oblique-angled Spherical Triangles there are XII Cafes, Ten of which may be refolved by the Univerfal Propofition; but then the Oblique Triangle must be reduced into Two Right-angled Triangles by help of a Perpendicular let fall, fometimes within, fometimes without, the Triangle: And to know whether it fall within or without, the fubfequent Rules are to be observed.

RULE I. If the Angles at the Bafe of the Triangle be both of the fame Affection, that is, both Acute or Obtufe, the Perpendicular let fall from the Vertical Angle shall fall within: "But if of different Affections, without.


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As in the Oblique angled Triangle ABC, whofe Angles at B Fig. and Care both Acute, the Perpendicular AD fhall. fall within XXII. the Triangle: For, if it fall not within, it must be the fame with one of the Sides, or elfe it muft fall without the Triangle: If it be the fame with either of the Sides, then the angle at B or C mutt be a Right Angle, which is contrary to the Propofition: If it fall without the Triangle, as fpofe at E, then the Angle A E B fhall be a Right Angle: But the Angle AB E is Obtufe, for it is the Complement of the Acute Angle ABC, and therefore the Side A E is greater than a Quadrant:, And the Angle ACE be- Fig. ing Acute, A E thall be alfo lefs than a Quadrant: But, that XXII. the fame Side Thould be both More and Lefs than a Quadrant, is abfurd: And therefore, in this Cafe, the Perpendicular thall fall within the Triangle.

But, In the Triangle A E B, Obtufe-angled at B, and Acute at E, the Perpendicular A D fhall fall without the Triangle upon! the Side EB, contrived: Or, if otherwife, it must be the fame with one of the Sides, or fall within the Triangle: It cannot be the fame with either of the Sides, for then the Angle at B or E fhould be a Right Angle: And it cannot fall within the Triangle, because then the Angles at B and E muft either be both Obtufe, or both Acute, as hath been already proved. If therefore the Angles at the Bafe be of different Affections, the Perpendicular tha fall without; as was to be proved.

However this Perpendicular falleth, it must be always oppofite to a known Angle, and for better Direction herein, take this General Rule.

RULE II. Let your Perpendicular fall from the End of a Side given, and adjacent to an Angle given.

As in this Triangle A B C, if there were given the Side A B, and the Angle at A; by the former, and this, Rule, the Perpen dicular mult fall from B upon the Side. A C.

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But if there were given the Side AC, and the Angle at A, the Fig. Perpendicular must C, upon the Side AB, continued XXIII. to E,And -And to know whether the Side upon which the Perpen all from dicular fhall fall, muft be continued or not, is no more than to ask whether the Perpendicular muft fall within or without the Triangle. But, if the former Directions be not fufficient, the Calculation will determine it. For,






RULE III. If the Ark found at the firft Operation (whether
be of a Side or an Angle) be more than the Arch given, the
Perpendicular shall fall without; if lefs, within the Triangle.

And this will plainly appear in the Solution of the fol lowing Cafes.

General Rules to be obferved in the Second Operation of the Solution of Oblique angled Spherical Triangles, when they are reduced into Two Right-angled Triangles.

After the first Operation, whereby either the Segments of the Bafe, or the Angle at the Cathetus, or Perpendicular, is found; a diligent Care being had to the Addition or Substraction of them: The fecond Operation will be performed by one of the Four following Rules.

RULE I. The Sines of the Complement of the Hypotenuses, to the Sines of the Complement of the Bafes, are in direct Proportion. So,

csBA: csDA::cs BC:cs DC. RULE II. The Sines of the Bafes, to the Tangents of the Angles at the Bafe, are in Reciprocal Proportion: So,

SBA DA::ct BetD::t B: D.

RULE III. The Sines of the Complement of the Angles at the
Bafe; to the Sines of the Complement of the Angles at the
Cathetus (or Perpendicular) are in dire Proportion: So,

SBCA: DCA::csB: cs B: cs D.

RULE IV. The Tangents of the Hypotenufes, to the Sines of the Complement of the Angles at the Catbetus (or Perpendicular) are in Reciprocal Proportion: So,

cs BCA:cs DCA::ct BC: ct DC :: DC: 1 BC.


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