EXAMPLE II.

Required, the tangent of 54 degrees 30 minutes.

Look at the foot of the page (because the proposed degrees are more than 45.) for 54 degrees, and in the right hand column for 30 minutes ; then in the column marked tangent at its bottom, and on the same line with the 30 minutes, in the side column, we find 10.14673, which is the log-tangent required.

The reverse of this, viz. The logarithm of a sine, tangent, or secant, being given, to find the arc belonging to it, is performed by only looking in the proper column, for the nearest logarithm to that proposed, and the degrees and minutes answering thereto, are those required.

We will now shew how any sine, tangent, or secant may be had, though the figures in the tables were defaced, mis-printed, or obliterated.

PROB. I.

To find the tangent which is defaced, by the sine and co-sine.

The co-sine taken from the sine added to 90, or radius, which is 10.00000, the remainder is the tangent, (by part 1. theo. 24.)

EXAMPLE.

4. Suppose the tangent of 41°. 20' was defaced, but the sine and co-sine of it visible.

From the sine of 41°. 20'+10.00000,

or radius,

Take the co-sine of 41°. 20′

19.81983

9.87557

The rem. is the tan. of 41°. 20′ req. viz. 9 94426

2. To find a sine which is mis-printed, by help of the co-sine and tangent.

From the sum of the tangent and co-sine, take 10.00000, or radius, or (which is the same thing) cut off the first figure in the index, the remainder is the sine required (by part 2. theo. 24.)

EXAMPLES.

Suppose the sine of 46°. 50'. was defaced, but the

tangent and co-sine visible.

To the tangent of 46°. 50'

Add the co-sine of 46°, 50'

10.02781

9.83513

The co-tangent and co-sine of any arc, may be had by the same method; the complement of any degree, being only its residue from 90, or a quadrant, as before observed (by theo. 24. part 3 and 4.)

3. To find a tangent by the help of a co-tangent only.

From twice the radius, which is 20.00000, take the co-tangent, the remainder is the tangent (by theo. 24. part 5.)

EXAMPLE.

Required the tangent of 29°. 50′ being defaced, as also the sine and co-sine defaced, by the co-tangent only.

From twice the radius,

Take the co-tangent of 29°. 50'

20.00000

10.24148

9.75852

The rem. is the tang. of 29°. 50′ req.

4. To find the secant by the help of a co-sine; which may be found of great use when a table of sines and tangents can only be had.

From twice the radius, which is 20.00000, take the Co-sine, and the remainder will be the secant, (by theo. 24. part 6.)

EXAMPLE.

Required, the secant of 57°. 20' by the help of the co-sine only:

From the double radius,

Take the co-sine of 57°. 20'

20.00000

9.73219

The rem. is the secant of 57°. 20' req. 10.26781

5. To find a secant by the help of the sine and tangent.

From the tangent added to radius, take the sine, the remainder will be the secant (by theo. 24. part 7.)

EXAMPLE.

Required, the secant of 57°. 20' by help of the sine and tangent.

From the tan. of 57°. 20'+10.00000

the radius,

Take the sine of 57°. 20'

20.19303

9.92522

The rem. is the secant of 57°. 20' req. 10.26781.

The secants in these tables might have been omitted, because all proportions in which they are concerned, may be wrought by sines and tangents only, as shall be shewn in the several cases of plane trigonometry; and are here only inserted that all the various

SECT. II.

PLANE TRIGONOMETRY.

DEFINITIONS.

PLANE TRIGONOMETRY is the art, whereby, having given any three parts of a plane triangle, except the three angles, the rest are determined. For this purpose it is requisite that not only the periphery of circles, but also certain right lines, in, and about circles, be supposed to be divided into some assigned number of equal parts.

2. The periphery of every circle is supposed to be divided into 360 equal parts, called degrees; and each degree into 60 equal parts, called minutes; and each minute, into 60 equal parts, called seconds; &c.

3. Any part of the periphery, is calledian arch; and is the measure of the angle at the centre, which it subtends.

4. The difference of any arch from 90°. or a quadrant, is called the complement of that arch, and its difference from 180°, or a semi-circle, is called its supplement.

5. A Chord, or Subtense of an arch, is a right line joining its extremities.