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But, it will be proper to take Notice here (once for all) that, if in these, or any other Theorems, the Tangent, Secant, Co-sine, Co-tangent, &c. of an Arch greater than 90 Degrees be concerned; then, instead thereof, the Tangent, Secant, Cofine &c. of an Arch, as much below 90 Degrees, is to be taken, with a negative Sign; according to the Observation in Page 3.

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PR op. IV.
As the Sum of the Tangents of any two Angles
BAC, BAD, is to their Difference, so is the Sine of
the Sum of those Angles, to the Sine of their Dif.

ference.

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Hence it also appears, that the Base (CD) of a plane Triangle, is to (Cd) the Difference of its two Segments (made by letting fall a Perpendicular), as the Sine of the Angle (CAD) at the Vertex, to the Sine of the Difference of the Angles at the Base.

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In any plane Triangle ABC, it will be, as the Sum of the two Sides plus the Base, is to the Sum of the two Sides minus the Base, so is the Co-tangent of half either Angle at the Base, to the Tangent of half the other Angle at the Base, 2

In - In AC, produced, take CD = BC, and letBD be drawn; Then (by Theor. 5. p. 6.) it will be, A D + AB: AD – AB :: Tang. AbbreCos A 2 z

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In any plane Triangle ABC, it will be, as the Base plus the Difference of the two Sides, is to the Base minus the same Difference, so is the Tangent of half the greater Angle at the Base, to the Tangent of half the lesser.

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PR o P. VII,

As the Base of any plane Triangle ABC, is to the Sum of the two Sides, so is the Sine of half the vertical Angle, to the Cosine of half the Difference of the Angles at the Base.

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As the Base of any plane Triangle ABC, is to the Difference of the two Sides, so is the Co-one of half

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As the Difference of the two Sides AC, BC, of a plane Triangle, is to the Difference of the Segments o the Base AQ, BQ (made by letting fall a Perpendi. cular from the Vertex), so is the Sine of half the vertical Angle, to the Co-sine of half the Difference of the Angles at the Base.

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PR op. X,
As the Sum of the two Sides of a plane Triangle, is

to the Difference of the Segments of the Base (see the

preceding Figure), so is the Co-sine of half the vertical Angle, to the Sine of half the Difference of the Angles of the Base, - For,

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