L 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 (ie. 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 Base made by a Perpendicular let fall from the Angle oppofite to the Bafe. Demonstration. From the B, on the Bafe AC, of the AABC, let fall the Per ㄠˋ 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 × : DC-DA: therefore AC: BC+ BA:: BC-BA: DC-DA. Q. E. D. Otherwife, let the Difference of the Squares of the Sides BC and AB be taken and divided by the Bafe AC, the Quotient fhall be the Difference of the Segments of the Base aforefaid: Or, fquare all the 3 Sides, and deduct the Square of one of the lefs Sides out of the Sum of the other two Squares, divide half the Remainder by the longeft 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 laft Fig.] N. B. f, 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 oppofite 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 Angle oppofite to the greateft Side is. Take the Sum and Difference of the greatest 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 greateft 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 abovesaid, and extract the Square Root, Equal to AB: BC:: Si. C: Si. A. { BC AC Hence, by Subtraction, the B will be Right 2 } 2 Then AB: AD:: R: Cofi. A. And CB DC:: R: Cofi. C. And 180°-A—¿C=ZB. Or more readily at one Operation. From half the Sum of the Sides fabduct 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 Log. of the other Difference and the doubled Radius, half the Remainder thail be the Leg. of the Tangent of half the enquired Angle. Agreeable to this Axiom in Gellibrand's Trig. Britannica, p. 46. As the Rectangle of half 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 tavo Remainders ; fo is the Square of Radius to the Square of the Tangent of half the Angle fought. Ex Angulis latera, vel ex lateribus Angulis & mixtim in Triangulis tam planis quam Sphæricis affequi, Summa Gloria Mathematici eft: Sic enim Cœlum & Terras & Maria felici admirando calculo Menfurat. Fran. Vieta, THE |