PART III.

TRIGONOMETRY.

Is that branch of mathematical science which is employed in calculating the sides and angles of a Triangle; and its doctrines are founded on the mutual proportions which subsist between the sides and angles of that figure. These proportions are discovered by first finding the relations between the radius of a circle, and certain other lines drawn in and about the circle, called chords, sines, tangents, and secants; which lines are either expressed in numbers, and arranged in tables, called the tables of sines, tangents, &c.; or are actually laid down in lines upon scales, sectors, &c. The parts, then, of a supposed triangle being ready calculated in the tables of Sines and Tangents, the parts of other similar angled triangles are easily found by the rule of Proportion; for the lines in all triangles naving equal angles, are in direct proportion to each other.

The circumference of every circle is divided (or supposed to be divided) into 360 degrees; every degree into 60 minutes, &c. And there are various right lines so related to these degrees, or parts of a circle, as to be called their sines, tangents, &c. as in fig. 6. The radius is supposed to be 100000, and

* Trigonometry not being essential to the mere practical Surveyor, may be passed over by the student whose time for study is limited.

all the other lines are exactly calculated to every degree and minute, after this standard, in the tables.

By inspecting the figure, it will be perceived that a chord of an arc is a right line which divides a circle into two unequal parts; and is common to both those parts as AB is a chord, as well to the part AEDB, as it is to that part of the circle included between A

A right sine of an arc is a line drawn from one end of that arc, perpendicularly to the radius; as ab is the sine of the arc Aa. A versed sine is that part of the radius which is contained between the right sine and the circumference, as bA. The cosine is da.

A tangent of an arc is a right line drawn touching one extremity of the arc, and terminating at the secant of that arc; as AF is the tangent of the arc Aa. Bc is the co-tangent of the same arc.

A secant of an arc is a right line drawn from the centre through the extremity of the arc, and produced until it meets the tangent; as CF. The co-secant is Cc.*

These lines, with a line of equal parts, are commonly laid down parallel to one another, on wood or brass rules; and constitute what is commonly called (from

* The complement of an arc is what it wants of a quadrant, or 90°; the supplement of an arc is what it wants of a semicircle, or 180°.

The sine, tangent, and secant, of the complement of an arc, are called the co-sine, co-tangent, and co-secant of that arc.

Some peculiar properties of the lines in and about a circle are as follows:

1. The square of the diameter is equal to the sum of the squares of the chord of an arc, and of the chord of its supplement to a semi-circle.

2. The square of the radius is equal to the sum of the squares of the sine and co-sine.

3. The sum of the co-sine and versed-sine is equal to the radius.

4. The sine is a mean proportional between the versed sine of double the arc, and the sum of the co-sine and radius.

5. Radius is to the sine as twice the co-sine is to the sine of twice the arc or, as the secant is to the tangent.

6. As the co-sine is to the sine, so is radius to the tangent. 7. Radius is a mean proportional between the tangent and the co-tangent. And also between the secant and co-sine.

*

its inventor) Gunter's plain scale (fig. 7.); and is used in constructing triangles and other plane figures.

On the same scales are other lines of sines, tangents, numbers, &c., constructed from the logarithms of numbers; which lines are used in finding (by proportion) the angles and sides of triangles.+

Plane triangles are composed of six parts; three angles and three sides; and in all solutions, three of these parts (one of which must be a side) are always given, to find any or either of the others.

There are three methods of resolving triangles; viz. geometrical construction, instrumental operation, and arithmetical computation. Each of which I shall explain; but the latter particularly recommend.

In the first method, the triangle is constructed by 'drawing and laying down the given sides, from a scale of equal parts; and the given angles, from the line of chords or protractor: then the unknown parts are measured by the same scales, and so become known.

In the second method, by the logarithmic lines on Gunter's scale, extend the compasses from the first term to the second or third, whichever happens to be

* Mr. Gunter was a celebrated Professor of Astronomy at Gresham College.

For further particulars of the construction of these and other lines on plain scales, the reader is referred to BION on Mathematical Instruments, Book I.; or more especially to HUTTON'S Mathematical and Philosophical Dictionary, under the article Scale.

of the same kind as the first; then, that extent laid off from the other term (whether second or third) will reach to the fourth, or answer. In these logarithmic lines, the 10 in the line of numbers, 90 on the sines, and 45 on the tangents, are set perpendicular to each other.

In the third method, the terms of the proportion must be stated according to rule; which terms consist partly of the numbers which express the given length of sides, and partly of the logarithmic sines or tangents of the given angles, taken from the tables; in which case the logarithms of the second and third terms are to be added together; and, from their sum, the first must be subtracted, as has been heretofore directed in logarithmical arithmetic or else, when radius is not concerned in the analogy, by taking the arithmetical complement of the first term, and adding it to the logarithms of the second and third terms, the natural number of which aggregate logarithm is the fourth term of the proportion.

There are other methods of finding the proportions which the sides and angles of triangles bear to each other; but as they consist of long and tedious multiplications and divisions, I shall not waste the learner's time in explaining them.

TRIGONOMETRY has three cases; viz.

Case I.

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

Case II.

When two sides and their included angle are given.

Case III.

When the three sides are given.