CASE IV._Philosophical transactions, Vol. L. Part II. For the year 1758. Giving some account of the present undertakings, studies, and labours, of the ingenious, in many considerable parts of the world.

_When the part required stands opposite to a part, which is likewise

unknown_: Having from the _data_ of Case I. found a fourth part, let

the sines of the given sides be S, _s_; those of the given angles Σ,

σ; and the sines of half the unknown parts _a_ and _b_; and we shall

have, as before, S_sa_² + _d_² - _b_² = 0; and if the equation of the

supplements be (Σσα² + δ²) - β² = 0; then, because α² = 1 - _b_² = 1 -

(S_sa_² + _d_²), and β² = 1 - _a_², substituting these values in the

second equation, we get

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