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 formula 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

ANALYTICAL TRIGONOMETRY.

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

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 formula 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

ANALYTICAL TRIGONOMETRY.

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 whic the trigonometrical line belongs may also be supposed to known. How the former is known from the latter will hereafter explained. Let it be taken for gran

the knowledge of a trigonometrical line is equivalent to the knowledge of its arc or angle, and vice versa.

The trigonometrical lines are sometimes called trigonometrical functions of an arc or angle; the reason for which will be understood if we first explain the signification of the word function as employed in mathematics.

One quantity is said to be a function of another, when the former depends in any way upon the latter for its value.

It is said to be an increasing function when it increases as the quantity upon which it depends increases; and a decreasing function when it diminishes as the other increases.

The quantity upon which a function depends is called its variable, because this is supposed to change its value at pleasure, the function changing to correspond.

Now a trigonometrical line depends upon the magnitude of its arc for its length; it is therefore properly termed a function of the arc; and by way of distinction a trigonometrical function.

Of these trigonometrical lines, we now proceed to explain the nature and properties.

THE SINE.

14. The sine of an arc is a perpendicular let fall from one extremity of the arc upon the diameter drawn through the other extremity.

ference have, it thus appears, the same sine. Two such arcs are