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3. Rules for the Twelve Cases of Oblique Angled Spherical Triangles.

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Rule.

Draw the perpendicular CD from C the unknown angle, not required, on AB.

R: cos A tan AC: tan AD, less or greater than 90°, as A and AC are of the same or of different affection. BD AB AD.

sin BD sin AD: tan A: tan B, of the same or of different affection, as AD is less or greater than AB.

cos AD: cos BD:: cos AC: cos BC, less or greater than 90°, as A and BD are of the same or of different affection.

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sin BC sin AC : sin A: sin B. The affection of B
is doubtful; unless it can be determined by this rule,
that according as AC+BC is greater or less than 180°,
A+B is also greater or less than 180o.

R: cos AC: tan A: cot ACD, less or greater than 90°,
as AC and A are of the same or of different affection.
tan BC: tan AC cos ACD: cos BCD greater than
90o, if one or all of the 3 terms ACD, AC, BC, are
greater than 90°; otherwise less than 90°.
ACD + BCD=ACB, doubtful. If ACD+BCD ex-
ceed 180°, take their difference; if BCD is greater
than ACD, take their sum, for the angle ACB.

R: cos A tan AC: tan AD, less or greater than 90°,
as A and AC are of the same or of different affection.
cos AC cos BC: cos AD: cos BD, greater than 90°,|
if one or all of the 3 terms AD, AC, BC, are greater
than 90°; otherwise less than 90o. AD+BD AB,|
doubtful. If AD+BD exceed 180°, take their dif
ference; if BD is greater than AD, take their sum for
AB.

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Rule.

sin B: sin A :: sin AC: sin BC, doubtful; unless it can be determined by this rule, that according as A+B is greater or less than 180°, AC+BC is also greater or less than 180°.

R: cos A tan AC: tan AD, less or greater than 90o, as A and AC are of the same or of different affection. tan B: tan A: sin AD: sin BD, doubtful. AB= AD+BD. If AD+BD be greater than 180°, take their difference; if BD is greater than AD, take their sum for AB.

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12.

A side

AC.

cos AC=

sin AC. sin AB

̄ sin ( ¦ S—A C). sin (§ S—A B). rad3

sin S. sin (S-BC)

Let S denote the sum of the three angles; then will
cos (S—A). cos (S-C). rad2
sin A. sin C

ad2].

4. Rules for the first Ten Cases of Oblique Angled Spherical Triangles, expressed in general terms.

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James Walker, Printer,

6. James's Court, Lawnmarket,

Edinburgh.

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