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.

In the following triangles this mark-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 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.

RECTANGULAR TRIGONOMETRY.

CASE. I.

The angles and hypothenuse gi

ven, to find the base and perpendicular. fig. 64.

13

fig. 64.

B

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 legs AB and BC.

Geometrically.

Make an angle of 46°. 30'. in blank lines (by prob. 16. sect. 1.) as CAB; lay 250, which is the given by. pothenuse, from a scale of equal parts, from A to C; from C, let fall the perpendicular BC (by prob. 7. sect. 1.) 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 an

swer.

By Calculation.

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

If from the sum of the second and third logs. that of the first be taken, the remainder 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 10.00000, reject the first figure of the sum of the other two logs. (which is the same thing as to subtract 10.00000) and that will be the log. of the thing required.

Or having found one leg, the other may be obtained by cor. 2. theo. 14 sect. 1.

By Gunter's Scale.

On this scale there are lines of numbers, sines, and tangents, as well as lines of sine and tangent rhombs, versed sines, meridional parts, and equal parts; but the three first lines are sufficient for our present purpose.

The divisions on these respective lines, are the logarithms of numbers, sines and tangents, taken from a scale of equal parts, and applied on the lines of the scale.

The first and third terms in the foregoing proportions, being of a like nature, and those of the second

portions 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 compasses, from the first to the third term, will reach from the second to the fourth.

Thus to extend the first of the foregoing proportions.

1. Extend from 90° to 46° 30′, on the line of sines; that distance will reach from 250 on the line of numbers, to 181, for BC.

2. Extend from 90° to 43° 30' on the line of sines; that distance will reach from 250 on the line of numbers, to 172, for AB.

If the first extent be from a greater, to a less number; when you apply one point of the compasses to the second term, the other must be turned to a less; and the contrary.

By def. 8. sect. 1. The sine of 90° is equal to the radius; and the tangent of 45° is also equal to the radius; because if one angle of a right-angled triangle be 45°, the other will be also 45°; and thence (by the lemma preceding theo. 7. sect. 1.) the tangent of 45° is equal to the radius: for this reason the line of numbers of 10.00000, the sine 90°, and tangent of 45° being all equal, terminate at the same end of the scale; where there are small brass centres, usually placed to preserve the scale.

It was said before, that the tangents ended at 45°; but because the logarithms of tangents more than 45°, must pass off the scale; such distances therefore as