If AC be made the radius, the sines of the angle A and C, described by the radius AC, will be proportional to the sines of the like arcs, or angles in the circle, that the tables now mentioned were calculated for. So if BC was required, having the angles and AB given, it will be,

Fig. 1,

As S.C: AB :: S.A: BC.

That is, as the sine of the angle C in the tables, is to the length of AB; (or sine of the angle C, in a circle whose radius is AC;) so is the sine of the angle A in the tables, to the length of BC, (or sine of the same angle, in the circle, whose radius is AC.)

In like manner the tangents and secants represented by making either leg the radius, will be proportional to the tangents and secants of a like are, as the radius of the given arc is to 10.000000, the radius of the tables aforesaid.

Hence it is plain, that if the name of each side of the triangle be placed thereon, a proportion will arise to answer the same end as before: thus if AC be made the radius, let the word radius be written thereon; and as BC and AB, are the sines of their opposite angles; upon the first let S.A, or sine of the angle A, and on the other let S.C, or sine of the angle C, be written. Then, When a side is required, it may be obtained by this proportion, viz.

As the name of the side given is to the side given,

So is the name of the side required

to the side required.

Thus, if the angles A and C, and the hypothe nuse AC were given, to find the sides; the proportion will be

That is, as radius is to AC, so is the sine of the angle A, to BC. And,

2. R: AC: S.C: AB.

That is, as radius is to AC, so is the sine of the angle C to AB.

When an angle is required, we use this

tion, viz.

As the side that is made the radius,

is to radius,

So is the other given side,

to its name.'

propor

Thus, if the legs were given to find the angle A, and if AB be made the radius, it will be

Fig. 2.

AB: R :: BC : T.A.

That is, as AB, is to radius, so is BC, to the tangent of the angle A.

After the same manner, the sides or angles of all right angled plane triangles may be found, from their proper data.

We here, in plate 4, give all the proportion requisite for the solution of the six cases in right-angled trigonometry; making every side possible the radius.

In the following triangles this mak-in an angle denotes it to be known, or the quantity of degrees it contains to be given; and this mark' on a side, denotes its length to be given in feet, yards, perches, or miles, &c. and this mark", either in an angle or on a side, denotes the angle or side to be required.

From these proportions it may be observed; that to find a side, when the angles and one side are given, any side may be made the radius; and

to find an angle, one of the given sides must be made the radius. So that in the 1st, 2d, and 3d cases, any side as well required as given may be made the radius, and in the first statings of the 4th, 5th, and 6th cases, a given side only is made the radius.

RIGHT ANGLED TRIANGLES.

CASE I

The angles and hypothenuse given, to find the base and perpen: dicular.

PL. 5. Fig. 4.

In the right angled triangle ABC, suppose the angle A = 46°. 30. and consequently the angle C =43°. 30. (by cor. 2. theo. 5.); and AC 250 parts, (as feet, yards, miles, &c.) required the sides AB and BC.

1st. BY CONSTRUCTION.

Make an angle of 46°. 30', in blank lines, (by prob. 16. geom.) as CAB; lay 250, which is the given hypothenuse, from a scale of equal parts, from A to C; from C, let fall the perpendicular (BC, by prob. 7. geom.) and that will constitute the triangle ABC. Measure the lines BC, and AB, from the same scale of equal parts that AC was taken from; and you have the answer.

2d. BY CALCULATION.

1. Making AC the radius, the required sides are found by these propositions, as in plate 4,

case 1.

R: AC:: S.A: BC.
R: AC: SC : AB.

If from the sum of the second and third logs. that of the first be taken, the number will be the log. of the fourth; the number answering to which will be the thing required; but when the first log. is radius, or 10.000000, reject the first figure of the sum of the other two logs. (which is the same thing as to subtract 10.000000;) and that will be the log. of the thing required.

2. Making AB the radius.

Secant A: AC:: R: AB.
Secant A: AC:: T.A: BC.

That is, As the secant of A=46° 30'

10.162188

*For finding the logarithmic Sine, Co-sine, &e. of any number of degrees and minutes in the table; also the degrees, minutes, &c. of any logarithmis Sine, Co-sine, &c. the reader is referred to Table 2, at the end of this Trea

tise.

That is, as the secant of C=43° 30′

10.139438

So is the tangent of C =43° 30′

9.977250

12.375190

to AB,

Or, having found one side, the other may be obtained by cor. 2. theo. 14. sect. 4.

3d. By Gunter's scale.

The first and third terms in the foregoing proportions, being of a like nature, and those of the second and fourth being also like to each other; and the proportions being direct ones, it follows; that if the third term be greater or less than the first, the fourth term will be also greater or less than the second; therefore the extent in your