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north and weft, and then meets with the first ship at C; required the course of each ship?

A TABLE of Proportions, for folving the several Cases of right angled Triangles. Fig. 27. and 28.

Cafes. Given. Sought.|

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AB, BC A & C AB: BC: Rad. Tan. A AB whofe complement is C.

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AC, AB A & C CA:AB:: Rad.: Sine CAC whofe complement is A.

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The folution of the ft Cafe admits of no variation. In the ad Cafe, if AB be made radius, the proportion may be, as AB: AC:: Rad.: Secant A, whofe comp. is C. In the 3d Cafe, if AC be made radius, the folution may be as Sine C: Sine A:: AB: BC; and, if AB be made radius, as Rad.: Secant A::AB: AC. The folution of the 4th Cafe may be, as Sec. A:R:: AC: AB; and, as Sec. C: R:: AC: CB, each fide being radius.

These proportions are true, but unneceffary; for, fince every cafe may be folved by a right application of fines or tangents, there is no reason for introducing fecants.

CONSTRUCTION of the feveral Cafes of Right angled Triangles.

CASE 1. Make the right angle ABC; take the given lengths of AB and BC from a scale of equal parts, and fet them from B to A and C, and join AC. Measure the angle at A by a line of chords, or protractor.

CASE 2. Make the right angle ABC; take the given length of AB from a scale of equal parts, and set it from B to A; take AC from the fame scale, and setting one foot of the compaffes in A, with the other cross the line BC in C, and join AC; measure the angle ACB by a line of chords.

CASE 3. Make the angle at A of the given quantity, by a line of chords or protractor; take the given length of AB from a scale, and fet it from A to B ; and from B raise the perpendicular BC. Measure BC and AC, by taking them severally in the compasses, and applying the extents to the scale from which AB was ta

ken.

CASE 4. Make the angle CAB of the given quantity; take the length of AC from a scale, and fet it from A to C; and from C let fall the perpendicular BC. Measure the fides AB and BC on the fcale from which AC was taken.

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A TABLE of PROPORTIONS for folving the several Cafes of Oblique angled Triangles. Fig. 31. 34.

and 36.

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AB: BE: Rad. : Sine BAE,
whofe complement is B.
AC: CE :: Rad Sine CAE,
whofe complement is C.

And the fum of BAE and CAE
is the angle BAC.

This cafe may be folved by the following Theorem : In any plain triangle, the rectangle contained by half the fum of the fides, and the excefs of that half fum above the fide oppofite to the angle fought, is to the rectangle contained by the differences between the half sum of the fides, and the two fides containing the angle fought; as the fquare of the radius is to the fquare of the tangent of half the angle fought.

EXAMPLE I. of Cafe 4. refumed. Fig. 36.

1. To find the angle BAC.

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