Required the logarithmic cosine, co-tangent, and co

secant of 35° 44′ 24′′.

10.142730

266

24

1064

532

6,0)638,4

106 prop. parts.

In a similar manner may the natural cosine, &c. be found.

To find the degrees, minutes, or degrees, minutes, and seconds, corresponding to any given logarithm, sine, tangent, &c.

Look out for the nearest logarithm to the given one, and the degrees answering to it will be found at the top, if the name be there, and the minutes in the left hand column; but if the name be at the bottom of the table, the degrees must be found there, and the minutes in the right hand column.

41

SECTION III.

C

From the definitions, it appears, 1st. that in any right angled triangle, if the hypothenuse AC be made radius of a circle, the perpendicular BC is the sine of the angle A, and that AB is its cosine.

A B

But the sine of either of the acute angles of a right angled triangle is the cosine of the other, and the contrary. Hence it follows that BC is the cosine of the angle C, and AB its sine: that is, when the hypothenuse is made radius, the base and perpendicular become the sines of their opposite angles, or the sine and cosine of the same angle.

2dly. If the base AB be made the radius of the circle, BC is obviously the tangent of the angle A, and AC is the secant thereof. But as the tangent of one angle is the cotangent of the other, likewise the secant of one angle is the cosecant of the other: hence it follows that BC is the cotangent of the angle C, and AC its cosecant.

B

C

A

3dly. If BC be made radius, it is evident that AB will be the tangent of the angle C, and AC its secant. But, as the tangent and secant of the angle Care the cotangent and cosecant of the angle A; hence, when the perpendicular CB is the radius of the A

B

circle, the base AB and hypothenuse AC will be the tangent and secant of the angle C, and the cotangent and cosecant of the angle A.

Principles for the Solution of Rectilineal Triangles.

Before we enter upon the investigation of this subject, it is necessary to observe, that the sines, tangents, secants, &c. of all angles are calculated to the logarithmical radius of ten thousand millions, and arranged in tables for the convenience of calculation.

Now, as no triangle can be formed, but another may be formed similar to it, it is manifest that by means of the tables any part of a triangle may be discovered, by having any three other parts given (the three angles excepted.)

1. In any right-angled triangle, the radius is to the

sine of one of the acute angles, as the hypothenuse is to the side opposite this angle.

Let ABC be the proposed triangle, right-angled at B; from A as a centre, with the radius AD (=AE), which let us suppose to be equal to the radius of the tables, describe the arc DE, which is the measure of the angle A; from E let fall the perpendicular EF, which is the sine of the angle A. Then the triangles