Demonstr. AB being the given leg, in the right-angled triangle ABC; with the centre A, and any assumed radius AD, describe an arc DE, and draw DF perpendicular to AB, or parallel to BC. Now it is evident, from the definitions, that DF is the tangent, and AF the secant, of the arc DE, or of the angle A which is measured by that are, to the radius AD. Then, because of the parallels BC, DF, it will be as AD: AB:: DF: BC:: AF: AC, which is the same as the theorem is in words.

D

B

Make AB 162 equal parts, and the angle A = 53° 7′ 48'; then raise the perpendicular BC, meeting AC in C. So shall AC measure 270, and BC 216.

Extend the compasses from 45° to 53°, on the tangents. Then that extent will reach from 162 to 216 on the line of numbers.

Note. There is sometimes given another method for right-angled triangles, which is this:

ABC being such a triangle, make one leg AB radius, that is, with centre A, and distance AB, describe an arc BF. Then it is evident that the other leg BC represents the tangent, and the hypothenuse AC the secant, of the arc BF, or of the angle A.

In like manner, if the leg BC be made radius; then the other leg AB will represent the tangent, and the hypothenuse AC the secant, of the arc Be or angle C.

But if the hypothenuse be made radius; then each leg

F

B D

will represent the sine of its opposite angle; namely, the leg AB the sine of the arc AE or angle C, and the leg BC the sine of the arc CD or angle A.

And then the general rule for all these cases, is this, namely, that the sides of the triangle bear to each other the same proportion as the parts which they represent.

And this is called, Making every side radius.

OF HEIGHTS AND DISTANCES, &c.

By the mensuration and protraction of lines and angles, are determined the lengths, heights, depths, and distances of bodies or objects.

Accessible lines are measured by applying to them some certain measure a aumber of times, as an inch, or foot, or yard. But inaccessible lines must be measured by taking angles, or by some such method, drawn from the principles of geometry.

When instruments are used for taking the magnitude of the angles in degrees, the lines are then calculated by trigonometry: in the other methods, the lines are calculated from the principle of similar triangles, without regard to the measure of the angles.

Angles of elevation, or of depression, are usually taken either with a theodolite, or with a quadrant, divided into degrees and minutes, and furnished with a plummet suspended from the centre, and two sides fixed on one of the radii, or else with telescopic sights.

To take an angle of altitude and depression with the quadrant.

Let A be any object, as the sun, moon, or a star, or the top of a tower, or hill, or other emi nence and let it be required to find the measure of the angle ABC, which a line drawn from the object makes with the horizontal line BC.

D

Fix the centre of the quadrant in the angular point, and move it round there as a centre, till with one eye at D, the other being shut, you perceive the object A through the sights: then will the arc GH of the quadrant, cut off by the plumb line BH, be the measure of the angle ABC as required.

The angle ABC of depression of any object A, is taken in the same manner; except that here the eye is applied to the centre, and the measure of the angle is the arc GH, on the other side of the plumb line.

B

G

H

The following examples are to be constructed and calculated by the foregoing methods, treated of in Trigonometry.

EXAMPLE L

Having measured a distance of 200 feet, in a direct horizontal line, from the bottom of a steeple, the angle of elevation of its top, taken at that distance, was found to be 47° 30′ from hence it is required to find the height of the steeple.

Construction.

Draw an indefinite line, upon which set off AC = 200 equal parts, for the measured distance. Erect the indefinite perpendicular AB; and draw CB so as to make the angle C = 47° 30′, the angle of elevation; and it is done. Then AB, measured on the scale of equal parts, is nearly 2184

What was the perpendicular height of a cloud, or of a balloon, when its angles of elevation were 35o and 64°, as taken by two observers, at the same time, both on the same side of it, and in the same vertical plane; their distance as under being half a mile or 880 yards. And what was its distance from the said two observers?

Construction.

Draw an indefinite ground line, upon which set off the given distance AB = 60; then A and B are the places of the observers. Make the angle A = 35°, and the angle B = 64°; and the intersection of the lines at C will be the place of the balloon; from whence the perpendicular CD, being let fall, will be its perpendicular height. Then, by measurement, are found the distances and height nearly as follows; viz. AC 1631, BC 1041, DC 936.

Or, having found the sine, the cosine will be found from it, by the property of the right-angled triangle CBF, viz. the cosine CF = √ CB2 — BF2, or c = VT — s2.

There are also other methods of constructing the canon of sines and cosines, which, for brevity's sake, are here omitted.

PROBLEM II.

To compute the tangents and secants.

The sines and cosines being known, or found by the foregoing problem; the tangents and secants will be easily found, from the principle of similar triangles, in the following manner :—

In the first figure, where, of the arc AB, BF is the sine, CF or BK the cosine, AH the tangent, CH the secant, DL the cotangent, and CL the cosecart, the radius being CA, or CB, or CD; the three similar triangles CFB, CAH, CDL, give the following proportions:

1st, CF: FB:: CA: AH; whence the tangent is known, being a fourth proportional to the cosine, sine, and radius.

2d, CF: CB:: CA: CH; whence the secant is known, being a third proportional to the cosine and radius.

3d, BF: FC:: CD: DL; whence the cotangent is known, being a fourth proportional to the sine, cosine, and radius.

4th, BF: BC:: CD: CL; whence the cosecant is known, being a third proportional to the sine and radius.

Having given an idea of the calculation of sines, tangents, and secants, we may now proceed to resolve the several cases of Trigonometry; previous to which, however, it may be proper to add a few preparatory notes and observations, as below.

Note 1.-There are usually three methods of resolving triangles, or the ca of trigonometry; namely, Geometrical Construction, Arithmetical Computa and Instrumental Operation.

In the First Method. The triangle is constructed by making the parts d' the given magnitudes, namely, the sides from a scale of equal parts, and angles from a scale of chords, or by some other instrument. Then, me the unknown parts, by the same scales or instruments, the solution will be tained near the truth.

In the Second Method. Having stated the terms of the proportion accorg to the proper rule or theorem, resolve it like any other proportion, in which a fourth term is to be found from three given terms, by multiplying the see and third together, and dividing the product by the first, in working with natural numbers; or, in working with the logarithms, add the logs, of the cond and third terms together, and from the sum take the log. of the first t then the natural number answering to the remainder is the fourth term sh In the Third Method.-Or Instrumentally, as suppose by the log. lines at one side of the common two foot scales; Extend the compasses from the is term, to the second or third, which happens to be of the same kind we : then that extent will reach from the other term to the fourth term, as regar taking both extents towards the same end of the scale.

Note 2.-In every triangle, or case in trigonometry, there must be given. parts, to find the other three. And, of the three parts that are given, or

them at least must be a side; because the same angles are common to an infitate number of triangles.

Note 3.-All the cases in trigonometry may be comprised in three varieties only; viz.

1st, When a side and its opposite angle are given.

2d, When two sides and the contained angle are given.

3d, When the three sides are given.

For there cannot possibly be more than these three varieties of cases; for each of which it will therefore be proper to give a separate theorem, as follows:

THEOREM I.

When a side and its opposite angle are two of the given parts.

Then the sides of the triangle have the same proportion to each other, as the sines of their opposite angles have.

That is, As any one side,

Is to the sine of its opposite angle;

So is any other side,

To the sine of its opposite angle.

D

Demonstr.-For, let ABC be the proposed triangle, having AB the greatest side, and BC the least. Take AD = BC, considering it as a radius; and let fall the perpendiculars DE, CF, which will evidently be the sines of the angles A and B, to the radius AD or BC. But the triangles ADE, ACF, are equiangular, and therefore AC : CF :: AD or BC : DE; that is, AC is to the sine of its opposite angle B, as BC to the sine of its opposite angle A.

A

E F B

Note 1.-In practice, to find an angle, begin the proportion with a side opposite a given angle. And to find a side, begin with an angle opposite a given side.

Note 2— An angle found by this rule is ambiguous, or uncertain whether it be acute or obtuse, unless it be a right angle, or unless its magnitude be such as to prevent the ambiguity; because the sine answers to two angles, which are supplements to each other; and accordingly the geometrical construction forms two triangles with the same parts that are given, as in the example below; and when there is no restriction or limitation included in the question, either of hem may be taken. The degrees in the table, answering to the sine, is the rute angle; but if the angle be obtuse, subtract those degrees from 180o, and be remainder will be the obtuse angle. When a given angle is obtuse, or a ight one, there can be no ambiguity; for then neither of the other angles can e obtuse, and the geometrical construction will form only one triangle.