Hence the Angles A, E, G, being found by the foregoing Cafe, will be equal to the Sides DB, Complement of D C, and the Side B C, in the original Triangle D B C.

I could have fhewn fome other Ways to work fome of the foregoing Cafes; but their Demonftrations being very large and intricate, and the Operations thereby very tedious and troublefome; and having already given the best and eafieft numerical Methods for their Solution; I think 'twill be much better to omit the aforefaid perplexed Methods, and inftead thereof to fhew the Method of folving them by Projection, which is truly artful and ingenious, and oftentimes the most expedient and ufeful Way we can take; and therefore is what the young Trigonometer fhould be exceeding well acquainted with. The Manner of which, is the Subject of the following Chapter.

CHA P. XX.

The Method of folving the Six Cafes of Oblique Spherical Triangles by Projec

tion.

TH

HE Quantities of the Sides and Angles of the Triangle to be projected, let be fuppofed as follows, viz.

This is done by Prob. 5, of the Stereographic Projection; thus; Lay a Ruler on the Angular Point C, and on the Poles a and b of the two Oblique Circles DCG and B CF, and it will cut the Primitive in VOL. II, D d

two

two Points c, and d; then c d measured on the Chords will be found 76°, whofe Complement to 180° is 104, the Angle required.

2. To measure the Side D C.

This is done by the Reverse of Problem 6, by reducing it to the Primitive, and measuring it on the Chords; for it will be there found to be 24° 04'.

3. To measure the Side B D.

This is done on the Lines of Chords, and is 42° 09'.

be projected, as was required.

1. To measure the Angle D.

Lay a Ruler from the Angle D to the two Poles a and b, and it will cut the Primitive in d and c'; then dc measured on the Chords will be found 468 18' = D.

2. To measure the Side B D.

This is done by laying a Ruler on a and D, which will cut the Primitive in e; then Be measured on the Chords will be found 42° 09′ = B.D.

3. To measure the Side D C.

This, in the fame manner, reduced to the Primitive, is Cf 24° 04' on the Line of Chords.

Note, As the Parallel cuts the Oblique Circle in two Points, fo the Angle C will be obtufe or acute according as D C G paffeth through this or the other Point of Interfection; and ought to be foreknown, otherwife two Triangles may be made.

1. To measure the Angle D.

Take KL in your Compaffes, and measure it on a Scale of Half-tangents from 90 downward, and you'll find it to be 46° 18'.

2. To measure the Angle C..

This is done as in Cafe 1, and is found to be 104° Co'.

3. To meafure the Side B C.

This is done by reducing it to the Primitive, and meafuring it on a Line of Chords, and it will be found 30° col.