Axiom II.

In any Triangle the Sides are proportional to the Sines of the oppofite Angles.

Demonftration,

Produce the leffer Side of the Triangle ABC, to wit AB to F, making AF BC: Let fall the Perpendiculars BD, FE, upon the Side CA produc'd if Need be; then will FE be the Sine of the Angle A, and BD the Sine of the Angle C, to the Radius BC

AF.

Now the Triangles ABD and AFE, having the A common to them both, and the DE to a Right-angle, are fimilar; wherefore (by 4. 6 Eucl. Elem.) AF (BC): AB::FE: BD; viz. Si. A: Si. C. Q. E. D. Otherwife thus; by Ax. I. AB: R:: BD: Si. A, and BC: R:: BD: Si. C; therefore ABX Si. A(RX BD) = BC x Si. C; wherefore AB: BC:: Si. C: Si. A. 2. ED.

Axiom III.

The Sum of the Legs of any Angle of a Plane Triangle is to their Difference, as the Tangent of half the Sum of the Angles oppofite to thofe Legs is to the Tangent of half their Difference. Demonftration.

In the Triangle ABC produce CB, the leffer Leg of the Angle B, till BD becomes BA, the greater Leg, and then bifect CD in E; join AD and bifect it alfo in F; draw BF, which (by 8. 1 Eucl. El.) will be

perpen. to AD; and

1

2

D

draw EF, which (by 2. 6 Eucl. Elem.) will be parallel to AC. Then will the Angle ABF FBD ABD, which external Angle ABD is (by 32. 1 Eucl. Elem.) BAC+ C, that is to the Sum of the oppofite Angles required. Draw then BG parallel to CA, fo will the Angle GBA be (by 29. 1 Eucl. Elem.) equal to its Alternate one BAC; and if from half the

Sum

Sum of the oppofite Angles you take the leffer Angle, i. e. If from ABF you take the GBA, there will remain GBF half the Difference of the oppofite Angles: And fo alfo, if from CE half the Sum of the Legs, you take CB the leffer Leg, there will remain BE equal to half the Difference of the Legs. And then fince the AABF is Right-angled, if BF be made Radius, AF will be the Tangent of ABF (i. e. the Tangent of half the Sum of the oppofite Angles); and in the little ▲ GBF, FG will be the Tangent of the GBF (i.e. the Tangent of half the Difference of the oppofite Angles): But the Segments of the Legs of any Triangle cut by Lines parallel to the Bafe, being (by Schol. to 2. 6 Eucl. El.) proportional; EC:EB :: FA: FG; that is in Words, half the Sum of the Legs is to half their Difference, as the Tangent of half the Sum of the oppofite Angles is to the Tangent of half their Difference: But Wholes are as their Halves; wherefore the Sum of the Legs is to their Difference, as the Tangent of half the Sum of the Angles oppofite is to the Tangent of half their Difference. 2. E. D.

Axiom IV.

The Bafe, or greatest Side of any Plane Triangle is to the Sum of the Legs, as the Difference of the Legs is to the Difference of the Segments of the Bafe made by a Perpendicular let fall from the Angle oppofite to the Base.

Demonftration.

From the B, on the Base AC, of the A ABC, let fall the Per

Z

B

D

pendicular BD; on B, as a Center, with the greater Leg BC, as a Radius, defcribe the Circle BxCyZ; and produce AB to x and y, and CA to Z. Then, by the 35. 3 Eucl. Elem. A yx Axis = AC × AZ; viz.: BC-BA: x: BC+BA: AC x: DC-DA: therefore AC: BC+ BA:: BC-BA: DC-DA. Q. E. D. Otherwise, let the Difference of the Squares of the Sides BC and AB be taken and divided by the Bafe AC, the Quotient shall be the Difference of the Segments of the Bafe aforefaid: Or, fquare all the 3 Sides, and deduct the Square of one of the less Sides out of the Sum of the other two Squares, divide half the Remainder by the longest Side, the Quotient is the Alternate Segment of the Bafe. The Proportions for the Solution of the fix Cafes of Plane oblique Triangles. [See the last Fig.]

N. B. 1, If the given Angle be Obtufe, the other 2 Angles then are each Acute, 2dly, If the Side oppofite to the given Angle is longer than the Side opposite to the Angle fought, then is the Angle fought Acute; but if fhorter, then is the faid Angle doubtful, and may be either Acute or Obtufe, because both the Sine and its Complement to two Right Angles are the fame: Wherefore to be certain, of what Quality the Argie oppofite to the greateft Side is. Take the Sum and Difference of the greateft Side and Middle (or leaft) and their Logarithms, if the half of them be equal to the Logarithm of the third Side, the Angle oppofite to the greatest Side is a Right Angle ; but if the Logarithm of the third Side be greater than the half it is Acute, if lefs, it is Obtufe. Or, without Logarithms, multiply the faid Sum by the Difference abovefaid, and extract the Square Root,

Equal to
Greater than
Lefs than

AB: BC:: Si. C: Si. A.

{

Right Obrufe Acute

Hence, by Subtraction, the B will be 2
known.

Si. A: Si. B:: BC: AC,

2

BC AC and C

3

First, find the Angles by the lait ;
then Si. C: Si. B:: AB: AC.

AC: BC+BA :: BC-BA: DC—DA :
AC+

2

DC — ¦ DA = DC.
DC-DA:= DA.

32

5

From half the Sum of the Sides fubduct each particular Side, and let the Sum of the Logarithm of the half Sum and Difference of the Side fubtending the enquired Angle be fubducted from the Sum of the Lg. of the other Difference and the dourled Radius, half the Remainder fhall be the Log, of the Tangent of half the enquired Angle.

Agreeable to this Axiom in Gellibrand's Trig. Britannica, p. 46. As the Rectangle of balf the Sum of the Sides and the Difference between that balf Sum and the Side oppofite to the Angle required, is to the Rectangle of the other two Remainders; fo is the Square of Radius to the Square of the Tangent of balf the Angle fubt. Ex Angulis latera, vel ex lateribus Angules & mixtim in Triangulis tam planis quam Sphæricis affequi, Summa Gloria Mathematici eft: Sic enim Coelum & Terras & Maria felici & admirando calculo Menfurat. Fran. Vieta.

THE