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Hence the Angles 1, E, G, being found by the foregoing Case, will be equal to the Sides D B, Complement of D C, and the Side B C, in the original Triangle D BC.
I could have shewn some other Ways to work some of the foregoing Cases; but their Demonstrations being very large and intricate, and the Operations thereby very tedious and troublesome ; and having already given the best and eafiest numerical Methods for their Solution; I think 'twill be much better to omit the aforesaid perplexed Methods, and instead thereof to thew the Method of solving them by Projection, which is truly artful and ingenious, and oftentimes the most expedient and useful Way we can take ; and therefore is what the young Trigonometer should be exceeding well acquainted with. The Manner of which, is the Subject of the following Chapter.
CHA P. XX.
The Method of solving the Six Cases of Ob
lique Spherical Triangles by Projection.
HE Quantities of the Sides and Angles of the Triangle to be projected, let be fupposed as follows, viz.
Let the Sides
B D = 42 09
36 08, Given The Angle
D= 46 18, And Opposite Side
BC= 30 00.
The Projection. Describe the Primitive Circle BIFH;
B draw the Oblique Circle BCF,
D to make the Angle
DBC = 36° 08'; then draw the Parallel of 600 ACE,
I to cut the Oblique
Circle in through that Point,
draw the Oblique
Circle - DCG, to make the Angle
ta BDC = 46° 18'; so will the Triangle
BDC be made or projected, as was required.
1. To measure the Angle C. 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.
two Points c, and d ; then cd measured on the Chords will be found 76', whose 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 429 09'.
B = 36 08, the Angle
C = 104 00, and the included Side B C = 30 oo.
que Circle BDF,
from - B to C;
B.CD = 104° ;
gle B C D
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 ine; then Be measured on the Chords will be found 429 og' = B.D.
3. To measure the Side D C. This, in the same manner, reduced to the Primitive, is Cf = 24° 04' on the Line of Chords.
B F, HI,
B to D; H
65° 56' ACE; then draw the Obli
que Circle BCE to make DBC=
369 081; VOL. II.
D d 2
this will interfect the Parallel in
C; through the Points
C and G, draw the Oblique Circle
DCG; so shall the Triangle
BDC, be made as required.
Note, As the Parallel cuts the Oblique Circle in two Points, so the Angle C will be obtuse or acute according as D C G pafseth through this or the other Point of Intersection ; and ought to be foreknown, other. wile two Triangles may be made.
1. To measure the Angle D. Take K L in your Compasses, and measure it on a Scale of Half-tangents from go downward, and you'll find it to be 46° 18'.
2. To measure the Angle C. This is done as in Case 1, and is found to be 104° col.
3. To measure the Side B C. This is done by reducing it to the Primitive, and measuring it on a Line of Chords, and it will be found 300 col.