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But, it will be proper to take Notice here (once for all) that, if in thefe, or any other Theorems, the Tangent, Secant, Co-fine, Co-tangent, &c. of an Arch greater than 90 Degrees be concerned then, inftead 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 Obfervation in Page 3.

D

A

O B

Thus, for Inftance, let BA be an Arch greater than 90°, and let the Tangent of the Sum of AB and AC be required; fuppofing T to reprefent

the Tangent of AD (the Supplement of AB) and t the Tangent of AC: Then, by writing -Tinstead of T, in the first of the foregoing Theorems, we

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fhall have Tang. of BC=

•T+t.

=

-T+t

1-tx-T ! +T

and therefore Tang. DC (- Tang. BC) =

T-t

1+T

which is the very Theorem demonftrated in the 2d Cafe.

PROP. IV.

As the Sum of the Tangents of any two Angles BAC, BAD, is to their Difference, fo is the Sine of the Sum of thofe Angles, to the Sine of their Difference.

Let

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that Ad=AD and d AB DAB, and, confequently, that CAd is the Difference of the two Angles BAC and BAD.

Now, by reafon of the fimilar Triangles CDE and CdF, it will be, CD (CB + BD) : Cd (CB BD):: DE: dF; but DE and dF are Sines of DAE and dAF to the equal Radii AD and Ad: Whence the Truth of the Propofition is manifeft.

COROLLARY.

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.

PROP. V.

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

In

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ABC), that is, AC + BC + AB : AC

2

2

+ BC -AB:: Co-tang. A: Tang. ABC. 2, E. D.

PROP. VI.

In any plane Triangle ABC, it will be, as the Bafe plus the Difference of the two Sides, is to the Bafe minus the fame Difference, fo is the Tangent of half the greater Angle at the Bafe, to the Tangent of half the leffer.

A

In the leffer Side CA, produced, take CD = CB, fo that AD may be the Difference of the two Sides; and let BD be B drawn: Then it is manifeft that the Angle CBD will be equal to D: But

(by Theor. 5. p. 6.) A B+ AD: ABAD::

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D+DBA

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2

CAB
2

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by 10. 1.

1.) Tang.

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Q, E.D.

PROP.

PROP. VII.

As the Bafe of any plane Triangle ABC, is to the Sum of the two Sides, fo is the Sine of half the vertical Angle, to the Co-fine of half the Difference of the Angles at the Bafe.

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D = DBC (by 6. 1.) and, confequently,

D

E

F

B

the ver

tical Angle ACB =

D+CBD

(by 10. 1.)

D.

2

the Sum of the

Moreover, feeing DCB is Angles A and CBA, at the Base (by 10. 1.) it is evident that BCF (or DCF) is equal to half that Sum; and, therefore, as ECF is the Excefs of the greater ABC (= BCE (by 8. 1.) above the half Sum (BCF), it must, manifeftly, be equal to half the Difference of the fame Angles A and CBA.

But (by Theor. 3.) AB: AD (AC + BC) :: Sine D(ACB): Sine ABD = Sine CED (by Cor. 1. to 8. 1.) Sine FEC Co-fine ECF. 2. E. D.

PROP. VIII.

As the Bafe of any plane Triangle ABC, is to the Difference of the two Sides, fo is the Co-fine of balf the vertical Angle, to the Sine of half the Difference of the Angles at the Bafe.

In

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In the greater Side CA let there be taken CD= CB, and let BD be drawn, and likewife CE, perpendicular to BD. It is manifeft, because CDCB, that CDB and CBD

=

are equal to one another, and that each of them is alfo equal to half the Sum of the Angles CBA and A at the Base (by Cor. 3. to 10. 1.); therefore ABD, being the Excess of the greater CBA above the half Sum, must confequently be equal to half the Difference of the fame Angles.

But (by Theor. 3.) AB: AD (AC-BC) :: Sine D (Co-fine DCE, or 1⁄2 C): Sine ABD. Q.E.D..

PROP. IX.

As the Difference of the two Sides AC, BC, of a plane Triangle, is to the Difference of the Segments of the Bafe AQ, BQ (made by letting fall a Perpendicular from the Vertex), fo is the Sine of half the vertical Angle, to the Co-fine of half the Difference of the Angles at the Bafe.

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As the Sum of the two Sides of a plane Triangle, is to the Difference of the Segments of the Bafe (fee the preceding Figure), fo is the Co-fine of half the vertical Angle, to the Sine of half the Difference of the Angles at the Bafe.

For,

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