equal to the exterior and oppofite angle FBH, (22. 3.); wherefore their halves BGD, HBK will be equal to one another; the right angled triangles BGD, HBK will therefore be equiangular, and GD will be to BD, as BH to HK, and the rectangle contained by GD, HK will be equal to the rectangle DBH or BFC: but fince AH is to HK, as AD to DG, the rectangle HAD (22. 6.) will be to the rectangle contained by HK, DG, or the rectangle BFC, (as the fquare of AD is to the square of DG, that is) as the square of the radius to the fquare of the tangent of the angle DAG, that is, the half of BAC: but HA is half the perimeter of the triangle ABC, and AD is the excefs of the fame above HD, that is, above the base BC; but BF or CL is the excess of HA or AL above the fide AC, and FC, or HB, is the excefs of the fame HA above the fide AB; therefore the rectangle contained by half the perimeter, and the excefs of the fame above the base, viz. the rectangle HAD, is to the rectangle contained by the straight lines by which the half of the perimeter exceeds the other two fides, that is, the rectangle BFC, as the fquare of the radius is to the fquare of the tangent of half the angle BAC oppofite to the base. Q.E. D.

PROP. VII. FIG. 12. 13.

Na plain triangle, the bafe is to the fum of the

fides, as the difference of the fides is to the fum or difference of the fegments of the bafe made by the perpendicular upon it from the vertex, according as the fquare of the greater fide is greater or less than the fum of the fquares of the leffer fide and the base.

Let ABC be a plane triangle; if from A the vertex be drawn a ftraight line AD perpendicular upon the bafe BC, the bafe BC will be to the fum of the fides BA, AC, as the difference of the fame fides is to the fum or difference of the fegments CD, BD, according as the fquare of AC the greater fide is greater or lefs than the sum of the squares of the leffer fide AB, and the base BC.

About A as a center, with AC the greater fide for a distance, let a circle be defcribed meeting AB produced in E, F, and CB in G: it is manifeft that FB is the fum, and BE the difference of the fides; and fince AD is perpendicular to GC, GD, CD

will be equal; confequently GB will be equal to the fum or difference of the fegments CD, BD, according as the perpendicular AD meets the bafe, or the bafe produced; that is, (by Conv. 12. and 13. 2.) according as the fquare of AC is greater or lefs than the fum of the fquares of AB, EC: but (by 35. 3.) the rectangle CBG is equal to the rectangle EBF: that is, (16. 6.) BC is to BF, as BE is to BG; that is, the bafe is to the fum of the fides, as the difference of the fides is to the fum or difference of the segments of the base made by the perpendicular from the vertex, according as the fquare of the greater fide is greater or less than the sum of the squares of the leffer fide and the base. Q. E. D.

T

PROP. VIII. PROB. FIG. 14.

HE fum and difference of two magnitudes being given, to find them.

Half the given fum added to half the given difference, will be the greater, and half the difference fubtracted from half the fum,

I will be the lefs.

For, let AB be the given fum, AC the greater, and BC the lefs. Let AD be half the given fum; and to AD, DB, which are equa, let DC be added, then AC will be equal to BD, and DC together; that is, to BC, and twice DC; confequently twice DC is the difference, and DC half that difference; but AC the greater is equal to AD, DC; that is, to half the fum added to half the difference, and BC the less is equal to the excess of BD, half the fum above DC half the difference. Q. E. D.

SCHOLIUM.

Of the fix parts of a plane triangle (the three fides and three angles) any three being given, to find the other three is the bufinefs of plane trigonometry; and the feveral cafes of that problem may be refolved by means of the preceding propofitions, as in the two following, with the tables annexed. In these, the solution is expreffed by a fourth proportional to three given lines; but if the given parts be expreffed by numbers from trigonometrical tables, it may be obtained arithmetically by the common Rule of Three.

Note. In the Tables the following abbreviations are ufed. R, is put for the Radius; T, for Tangent; and S, for Sine. Degrees, minutes, feconds, &c. are written in this manner; 30° 25′ 13", &c. which fignifies 30 degrees, 25 minutes, ¥3 feconds, &c.

SOLUTION of the CASES of Right Angled TRIANGLES.

N

IN

GENERAL PROPOSITION.

a right angled triangle, of the three fides and three angles, any two being given befides the right angle, the other three may be found, except when the two acute angles are given, in which cafe the ratios of the fides are only given, being the fame with the ratios of the fines of the angles oppofite to them.

It is manifeft from 47. 1. that of the two fides and hypothenuse any two be given the third may also be found. It is alfo manifest from 32. 1. that if one of the acute angles of a rightangled triangle be given, the other is also given, for it is the complement of the former to a right angle.

If two angles of any triangle be given, the third is also given, being the supplement of the two given angles to two right angles. Fig, 15. The other cafes may be refolved by help of the preceding propofitions, as in the following table.

SOLUTION of the CASES of Oblique-Angled
TRIANGLES.

GENERAL PROPOSITION.

IN an oblique-angled triangle, of the three fides and

three angles, any three being given, the other three may be found, except when the three angles are given; in which cafe the ratios of the fides are only given, being the fame with the ratios of the fines of the angles oppofite to them.

|P-BC :: Rq: Tq, C, and hence C is known. (5.) Otherwise.

Let AD be perpendicular to BC. 1. If ABq be less than ACq+CBq. Fig. 16. BC: BA+ AC :: BA AC: BD-DC, and BC the fum of BD, DC is given; therefore each of them is given. (7.)

2 If ABq be greater than ACq+CBq. Fig. 17. BC: BA+AC:: BA-AC: BD +DC; and BC the difference of BD, DC is given, therefore each of them is given. (7.)

And CA: CD:: R: COS, C. (1.) and C being found, A and B are found by cafe 2. or 3.