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9. In order to explain the use of the instrument here described, suppose it be required to draw at the point a in the line AB a line making with AB an angle of 22°.
Place the protractor so that its centre shall be upon the point a, and its straight
edge or diameter upon the line AB.
Then mark the
paper at the point c against the 22d division of the protractor, and a line joining c and A will form with AB the angle required.
10. We are now prepared to construct triangles when three parts are given, the angles in degrees and the sides in feet, yards, or other linear units.
In order to show the practical utility of trigonometry at the same time that we explain the solution of a triangle, let us take the following problem in the calculation of distances to inaccessible objects.
Suppose a fort situated upon an island, and a light-house upon the main shore, and let the distance from the light-house to the nearest salient of the fort be required.
Measure a line along the shore
of any length at pleasure, say 500 yards, beginning at the light-house. Then if two lines be imagined to be drawn from the extremities of the line just measured, to the salient of the fort, a large triangle will be formed having its two longest sides resting upon the sea. If now the angles which these two sides form with the first side, which we will call the base, could be de
termined by observation upon the shore, there would be known in this triangle a side and the two adjacent angles, which would be sufficient data to construct the triangle on a small scale, and to obtain the length of the required side extending from the light-house to the salient of the fort.
A somewhat rude instrument for the purpose of observing such angles as those alluded to above, might be easily made.
Let there be a circle, or flat circular ring
of wood gra
duated to degrees, and having a tin tube moveable upon a pivot at the centre of the circle ; the tube being closed at one end except a very small orifice, and having two threads crossing at right angles in the centre of the other end, so that in looking through the tube with the eye at the small orifice, the line of sight may coincide with the axis. Let this apparatus be mounted upon a three legged stand called a tripod, so that the plane of the circle shall be horizontal : then, by placing the instrument thus formed at the light-house, in the example above, and sighting with the tube, first to a staff at the other extremity of the base, and then to the salient of the fort, keeping the circle stationary, the number of degrees passed over upon its circumference by the tin tube will indicate the angle of the triangle at the light-house. This angle we shall suppose to be 1054. The angle at the other extremity of the base might be found in the same manner, and suppose it 470.*
To construct the triangle with these data, draw on paper a line AB, and make it equal in length to five hundred divisions of some scale of equal parts.f Then draw an indefinite line AC, making with AB an angle of 10510. Also
А B lay off in a similar manner at the point B an
* The instrument here described is of course very rude. It was deemed not advisable to encumber the work with a detailed description of more accurate instruments, which belongs properly to a treatise on surveying.
+ This may be done conveniently by taking 50 divisions, and considering each division as equal to ten.
angle of 47°, and the two lines AC and BC will meet at c. Take the line AC in the dividers and apply them to the scale. The number of equal parts upon the scale between the feet of the dividers, will show the number of yards from the lighthouse to the fort. This number is 791.
If the angle at c were required, it might be measured by applying to it the protractor; or it is equal to 180°—(a+b.) The side B c if among the sought parts might also be measured from the scale.
11. The instrument described above may be rendered suitable for application to the determination of heights. If a round bar be made to project horizontally from the top of the tripod, so that the graduated circular frame can be suspended by the socket at its centre in a vertical position, it will then serve to measure angles in a vertical plane.*
To show the use of the instrument thus prepared, take the following problem.
Required the height of a tower which stands upon horizontal ground, and the base of which is accessible.
tance the base line; at the extremity of the base line place the instrument arranged for taking vertical angles; suspend a plumb line from the centre of the circle, and the point 90° distant from that in which the plumb line cuts the circumference will be the point through which a horizontal radius would pass. Then sight with the tube to the top of the tower; the number of degrees between the tube and the horizontal radius just mentioned, will be the measure of the angle included between a line drawn to the top of the tower and the base line; let this number be 30°. Constructing a right
* A vertical plane is one perpendicular to the surface of the earth.
angled triangle upon paper, having its base 200 and angle at the base 30°, the perpendicular of this triangle will be the height of the tower. The height of the instrument must be added to the result found.
N. B. The sides found will always be expressed in units of the same kind as the base.
12. It is evident that when any three parts of a triangle, one of which is a side, are given, the other three may be discovered by a process similar to those just exhibited.
This kind of solution is said to be by construction.
The accuracy of the results must depend upon the niceness of the instruments, and the care with which the construction is made.
A degree of accuracy so uncertain and so variable, is quite inadequate for many purposes to which Trigonometry is applied.
A method of calculating the required from the given parts of a triangle, which should produce always the same results from the same data, and be either perfectly, or so nearly exact, as to leave an error of no importance, however great the dimensions employed, would be evidently a desideratum. Such a method we have, and it is that which it will be the object of the residue of the present treatise to unfold.
To give the student a general view of what is before him, it will be well to state that a number of equations will be found, each containing four quantities, which quantities will be general expressions for the measures of parts of a triangle. The equation will express the true relation between these parts. By making one of these parts the unknown quantity and resolving the equation with respect to it, its value will be expressed in terms of the other three. If now these three were given, the value of the fourth would be known the moment the values of the three given were substituted for their general representatives.
It is plain that as many such general equations will be required, as there can be formed essentially different combinations of four out of the six parts of a triangle.
Equations like those here alluded to are called formulæ be
cause each is a general form, under which a multitude of particular examples are included.
As these general forms require of necessity the use of algebraic symbols and processes, and as algebra, from its power and application to decompose combinations of quantity so as to extricate their elements, is often called analysis, the subject upon which we are now about to enter is called
13. The sides and angles of a triangle are not quantities of a similar kind, and therefore do not admit of direct comparison. Since angles are expressed in degrees, and sides in units of length, one of the first principles of equations, namely, that the members and terms should express quantities of the same kind, would be violated by the introduction of angles and sides together, without some modification of one or both.
The expedient which has been invented to accommodate these heterogeneous quantities to each other, is that of employing straight lines, so related to the arcs which measure the angles of a triangle, as to depend upon these arcs for their length, in such a manner that when the arcs are known, these straight lines may be known also; and vice versa. The chords of arcs are plainly lines of this description, and chords were at one time used for the purpose of which we here speak; but a more convenient kind of lines, of which
, there are three principal sorts termed sines, tangents and secants, of an arc or angle called, when spoken of collectively, trigonometrical lines, were introduced by the Arabs, and are now in general use. These lines being straight and expressed, as they will be found to be, in linear dimensions, like the sides of a triangle, they may be employed with the latter in equations or formulæ; and when, by the resolution of an equation of this description, one of these trigonometrical lines is found in terms of one or more sides of the triangle, the angle to which the trigonometrical line belongs may also be supposed to be known. How the former is known from the latter will be hereafter explained. Let it be taken for granted here that