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The folution of the cafes of oblique plane triangles..

Cafe.

Given

Sought

Solution.

The angles Either of As fine C: AB:: fine A: I and one fide the other BC (by Theor. III.)

2

3

4

AB

Two fides

fides BC

The other As AB: fin. C:: BC: fin. A AB, BC and angles A (byTheor. III.) which added to an ang. C op. and ABC C, and the fum fubtracted from to one of 'em. 180gives theother angle ABC. Two fides The other Let the angle ABC he found, AB, BC and fide AC by the preceding cafe, and an opp. anthen it will be, fin. C: AB:: gle C fin. ABC: AC (by Theor. III.)

Two fides The other As fum of AB and AC: their AC, AB and angles Cd tang. of half the fum of the included and ABC ABC and C: tang of half their angle A

diff. (by Theor.V.) which added to, and fubtracted from, the half fum, gives the two angles.

Two fides The other Let the angles be found by 5 AC, AB and fide BC the laft cafe, and then BC. the incl A.

6

by cafe 1.

All the three An angle, Let fall a perp. BD opp. to the fides. fuppofe Areq. angle: then (by Theor. IV.

as AC: fum of AB and BC: their dif. :dift. DG of the perp. from the middle of the bafe; whence, AD being alfo known, the angle A will be found by Cafe 2. of right-angles.

Note,

Note, The 2d and 3d cafes are ambiguous, or admit of two different answers each, when the fide. AB oppofite the given angle C (see fig. 2.) is lefs than the given fide BC, adjacent to it (except the angle found is exactly a right one): for then another right-line Ba, equal to BA, may be drawn from B to a point in the base, somewhere between C and the perpendicular BD, and therefore the angle found by the proportion AB (aB): fin. C. :: BC: fin. A (or of CaB,) may, it is evident, be either the acute angle A, or the obtufe one CaB (which is its fupplement), the fines of both being exactly the fame.

Having laid down, the method of refolving the different cafes of plane triangles, by a table of fines and tangents; I fhall here fhew the manner of conftructing fuch a table (as the foundation upon which the whole doctrine is grounded); in order to which, it will be requifite to premife the following propofitions.

PROPOSITION I.

The fine of an arch being given, to find its cofine, verfed fine, tangent, co-tangent, fecant, and co-fecant.

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will be known. Then because of the fimilar triangles CFE, CAT, and CDH, it will be (by 14. 4.)

1. CF FE:: CA: CT; whence the tangent is known.

2. CF: CE (CA) :: CA: CT; whence the fecant is known.

3. EF CF: CD: DH; whence the co-tangent is known.

4. EF: EC (CD) :: CD: CH; whence the co-fecant is known.

Hence it appears,

1. That the tangent is a fourth proportional to the co-fine, the fine, and radius.

2. That the fecant is a third-proportional to the co-fine and radius.

3. That the co-tangent is a fourth proportional to the fine, co-fine, and radius.

4. And that the co-fecant is a third proportional to the fine and radius.

5. It appears moreover (because AT: AC:: CD (AC): DH), that the rectangle of the tangent and co-tangent is equal to the fquare of the radius (by 10. 4.): whence it likewife follows, that the tangent of half a right angle is equal to the radius; and that the co-tangents of any two dif ferent arches (represented by P and Q) are to one another, inversely as the tangents of the fame arches: for, fince tang. P x co-tang. P= =fqu. rad. tang. Qx co-tang. Q; therefore will co

tang.

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tang. P: co-tang. Q: tang. Qtang. Pot as co-tang. P: tang. Q: co-tang. Q: tang. P(by 10. 4.)

PROP. II.

If there be three equidifferent arches AB, AC, AD, it will be, as radius is to the co-fine of their common difference BC, or CD, fo is the fine CF, of the mean, to half the fum of the fines BE4DG, of the two extremes: and, as radius to the fine of the common difference, fo is the co-fine FO of the mean, to half the difference of the fines of the two ex

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AE F G

H

meeting DG in H and v.

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2. For let BD be drawn, interfecting the radius OC in m; alfo draw mn parallel to CF, meeting AO in n; and BH and mv, parallel to AO,

Then, the arches BC and CD being equal to each other (by hypothefis), OC is not only perpendicular to the chord BD, but alfo bifects it (by 13.) and therefore Bm (or Dm) will be the fine of BC (or DC), and Om its co-fine: moreover mn, being an arithmetical mean between the fines BE, DG of the two extremes (because Bm Dm) is therefore equal to half their fum, and Dv equal to half their difference. But, because of the fimilar triangles OCF, Omn and Dvm,

It will be

OC: Om:: CF : mn
OC Dm :: FO: Dv

} Q. E. D.

} 2

COROL

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2

20m x CF

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Hence, if the mean arch AC be fuppofed that. of 60°; then OF being the co-fine of 60°, fine 30° chord of 60°OC, it is manifeft that DG-BE will, in' this cafe, be barely Dm; and confequently DG = Dm+ BE. From whence, and the preceding corollary, we have these two useful theorems.

1. If the fine of the mean, of three equidifferent arches (fuppofing radius unity) be multiplied by twice the co-fine of the common difference, and the fine of either extreme be fubtracted from the product, the remainder will be the fine of the other extreme.

2. The fine of any arch, above 60 degrees, is equal to the fine of another arch, as much below 60°, together with the fine of its excefs above 60o.

*Note,

Om x CF
OC

taken in a geometrical fenfe, de

notes a fourth-proportional to OC, Om and CF; but, arithmetically, it fignifies the quantity arifing by dividing the product of the measures of Om and CF by that of OC. Underfand the like of others.

PROP.

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