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In any right-angled spherical Triangle (ABC) it will be, as Radius is to the Sine of either Angle, so is
Tangent of either Angle to the Co-langent of the other
For (CEF being as in the last) it will be, as Radius: Sine CE :: Tang. C: Tang. EF (by Theorem 4.) that is, Radius: Co-fine AC :: Tang. C : Cotang. A. Q. E. D.
As the Sum of the Simes of two unequal Arches is to their Difference, so is the Tangent of Half the Sum of those Arches to the Tangent of Half their Difference: And, as the Sum of the Cosines is to their Difference, so is the Co-langent of Half obe Sum of the Arches to the Tangent of Half the Difference of the fame Arches.
, In any spherical Triangle ABC it will be, as the Co-langent of Half the Sum of the two Sides is to the Tangent of Half their Difference, so is the Co-tangent of Half the Base to the Tangent of the Distance (DE) of the Perpendicular from the Middle of the Base.
AC-BC; Tang. DE, and it is proved, in p. 11. 2
that the Tangents of any two Arches are, inversely,
as their Co-tangents; it follows, therefore, that
Tang. AE : Tang. actic :: Tang. AC — BC
: Tang, DE; or, that the Tangent of Half the Base, is to the Tangent of Half the Sum of the Sides, as the Tangent of Half the Difference of the Sides, to the - Tangent
Cor. to Theor. 2.) there- .