Therefore upon the same principle can any rate for a parish or county be calculated.

ON PLANE TRIGONOMETRY.

Plane trigonometry is the art of measuring and computing the sides of plane triangles, or of such whose sides are right lines.

As this work is not intended to teach the elements of mathematics, it will be sufficient to point out a few of the principles, and give the rules of plane trigonometry for those cases that occur in surveying. In most of these cases it is required to find lines or angles, whose actual admeasurement is difficult or impracticable; they are discovered by the relation they bear to other given lines or angles, a calculation being instituted for that purpose; and as the comparison of one right line with another right line is more convenient and easy than the comparison of a right line to a curve, it has been found advantageous to measure the quantities of angles, not by the arc itself, which is described on the angular point, but by certain lines described about that arc.

The circumference of every circle is supposed to be divided into 360 equal parts, called degrees; each degree into 60 minutes; each minute into 60 seconds; and so on.

The sine of three angles of every triangle, or two right angles, are equal to 180 degrees.

The sum of two angles in any triangle, taken from 180 degrees, leaves the third angle.

In a right angled plane triangle, if either acute angle be

taken from 90 degrees, the remainder will give the other acute angle.

When the sine of an obtuse angle is required, subtract such obtuse angle from 180 degrees, and take the sine of the remainder, or supplement.

If two sides of a triangle are equal, a line bisecting the contained angle will be perpendicular to the remaining side, and divide it equally.

Before the required side of a triangle can be found by calculation, its opposite angle must first be given, or found.

The required part of a triangle must be the last term of four proportionals, written in order under one another, whereof the three first are given or known.

In four proportional quantities, either of them may be made the last term; thus, let A B C D be proportional quantities: As first to second, so is third to fourth, A: B :: C: D As second to first, so is fourth to third, B: A :: D: C As third to fourth, so is first to second, C: D: A: B As fourth is to third, so is second to first, D: C:: B: A Against the three first terms of every proposition or stating must be written their respective values taken from the proper tables.

If the value of the first term be taken from the sum of the second and third, the remainder will be the value of the fourth term or thing required, because the addition and subtraction of logarithms correspond with the multiplication and division of natural numbers.

If to the complement of the first value be added the second and third values, the sum rejecting the borrowed index will be the tabular number expressing the thing required. This method is generally used when radius is not one of the proportionals.

The complement of any logarithm, sine, or tangent, in the common table, is its difference from the radius 10.000.000, or its double, 20.000.000.

The complement of an arc is what it wants of 90 degrees.

L

The supplement of an arc is what it wants of 180 degrees.

A sine or right sine of an arc is a line drawn from one extremity of the arc perpendicular to the diameter, as B F, or the supplement to the arc B D E. (Fig. 1, Plate 28.)

The versed sine of an arc is that part of the diameter intercepted between the arc and its sine; as A F is the versed sine of the arc A B.

The tangent of an arc is a line perpendicular to the diameter touching the circle, as A H.

A secant is a line drawn from the centre C through any point of the circumference until it intersects the tangent as at H; then CH is the secant of the arc A B; also EI is the tangent and C I the secant of the supplemental arc BDE; and this latter tangent and secant are equal to the former, but are accounted negative, as being drawn in an opposite direction to the former.

The co-sine, co-tangent, co-secant of an arc, are the sine, tangent, and secant of the complement of that arc, the co. being only a contraction of the word complement. Thus, the arcs A B and B D being the complements of each other, the sine, tangent, or secant of one of these is the co-sine, co-tangent, or co-secant of the others; so B F the sine of AB, is the co-sine of B D; and B K the sine of B D, is the co-sine of A B. In like manner, A H the tangent of A B, is the co-tangent of BD; and DL the tangent of D B, is the co-tangent of A B. Also CH the secant of A B, is the co-secant of B D; and C L the secant of B D, is the co-secant of A B.

Corollary. Hence several remarkable properties easily follow from their definitions; as

1st. That an arc and its supplement have the same sine, tangent, and secant; but the two latter, the tangent and secant, are counted negative when the arc is greater than a quadrant, or 90 degrees.

2nd. When the arc is zero, or nothing, the sine and tangent are nothing, but the secant is the radius C A.

3rd. Of any are A B, the versed sine A F and co-sine B K

are equal to the radius C A-the radius, tangent, and secant forming a right angled triangle CAH; so also do the radius, co-tangent, and co-secant, another right angled triangle C D L. All these right angled triangles are similar to each other.

The sine, tangent, or secant of an angle, is the sine, tangent, or secant of the arc by which the angle is measured, or of the degrees, &c., in the same arc or angle.

The method of constructing the scale of chords, sines, tangents, and secants, usually engraved on instruments for practice, is shown by Fig. 1, Plate 28, called a Trigonometrical Canon.

A trigonometrical canon is a table showing the length of the sine, tangent, and secant to every degree and minute of the quadrant, with respect to the radius which is expressed by unity, or 1, with any number of cyphers. The logarithms of these sines, tangents, and secants are all ranged in the tables; and these are most commonly used, as they perform the calculations by only addition and subtraction, instead of the multiplication and division, by the natural sines, &c., according to the nature of the logarithms.

Having given an idea of the calculations and use of sines, tangents, and secants, we now proceed to resolve the several cases of trigonometry; previous to which it will be proper to add a few preparatory notes and observations.

Note 1. There are three methods of resolving trianglesnamely, geometrical construction, instrumental operation, and arithmetical computation.

In the first method, the triangle is constructed by laying down the sides by a scale of equal parts, and the angles from the scale of chords or protractor. Then measuring the unknown parts by the same scale for the lengths of the sides, and the angles by the scale of chords or protractor.

The second method, by logarithmic lines, commonly called Gunter's scales. These scales are to be perpendicularly over each other, as thus: the 10 on the line of numbers, 90 on the sines, and 45 on the tangents. (See Sector, Plate 41, Part V.)

In working proportions with these lines, attention must be paid to the terms, whether arithmetical or trigonometrical, that the first and third term may be of the same name, and the second and fourth of the same name. To work a proportion, take the extent on its proper line from the first term to the third with the compasses, and applying one point of the compasses to the second, the other applied to the right or the left, according as the fourth term is to be more or less than the second, will reach to the fourth.

In the third method, the terms must be stated according to rule; which terms consist of the given lengths of the sides, and of the sines or tangents of the given angles taken from the logarithmic tables; in which case the second and third terms are added together, and from this sum the first must be subtracted, excepting when radius is not concerned in the analogy, by taking the arithmetical complement of the first term, and adding to it the logarithms of the second and third terms, the natural number of which aggregate logarithm is the fourth term of the proposition.

Note 2. A triangle consists of six parts-viz. three sides and three angles; and in every case in trigonometry there must be given three parts to find the other three. Also of the three parts that are given, one of them must be a side, because with the same angles the sides may be greater or less in proportion. Note 3. All cases in trigonometrical surveying are comprised in three varieties-viz. :

1st. When two angles and a side are given.

2nd. When two sides and the included angle are given. 3rd. When three sides are given.

Problem 60.

Case 1. The following proportion is to be used when two angles of a triangle and a side opposite to one of them are given to find the other side:

Rule. As the sine of the angle opposite the given side is to