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In any plane Triangle ABC, it will be, as the Line CE bisočiing the vertical Angle, is to the Base AB, so is the Secant of half the vertical Angle ACB, to the Tangent of an Angle; and, as the Tangent of half this Angle is to Radius, so is the Sine of half the vertical Angle, to the Sine of either Angle, which the bisoffing Line makes with the Base.

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Prop. xviii.
In any plane Triangle ABC, it will be, as the Per-

pendicular is to the Sum of the two Sides, so is the

Tangent of half the Angle at the Vertex, to the Tangent of an Angle; and, as Radius is to the Tangent of half this Angle, so is the Sum of the two Sides, to the Base of the Triangle.

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In any plane Triangle ABC, it will be, as the Perpendicular is to the Difference of the two Sides, so is the Co-tangent of half the vertical Angle, to the Tangent of an Angle; and, as Radius is to the Co-tangent of half this Angle, so is the Difference of the Sides to the Base of the Triangle.

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The Hypothemuse AC, and the Sum, or Difference, of the Legs, AB, BC, of a right-angled spherical Triangle ABC, being given, to determine the Triangle.

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Hence, if the two Legs be supposed equal to each other (or the given Difference = 0), then will the Co-fine of the Double of each, be equal to twice the Co-fine of the Hypothenuse minus the Radius.

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AC+AB, Tang. Acrae, that is, As the Co2 tang. of half the given Leg, is to its Tangent; so is the Co-tang. of half the Sum of the Hypothenus; and the other Leg, to the Tangent of half their Difference.

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The Angle at the Base and the Sum, or Difference, of the Hypothemuse and Base, of a right-angled spherical Triangle being given, to determine the Triangle,

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