(3.) Prop. To express the magnitude of an angle in terms of the subtending arc and its radius.

It is proved (see Geom. III. § 2. Prop. 33. and Schol. Prop. 34.), that the ratio which the circumference of any circle bears to its radius, is a constant quantity.

Let its numerical value be denoted by 2 .* The circumf. = 2. rad.

even

(4.) This equation may be made to assume a more simple form; but before we proceed with it, we will make a few remarks on the manner in which quantity may be estimated and expressed numerically (on which this simplification will depend), though it might perhaps be allowed us to assume that the student, in his progress through algebra, before entering on this elementary part of trigonometry, had acquired sufficient knowledge on this subject for our purpose. We think it better, however, to direct his attention particularly to it, because clear and distinct notions on this point are so essential for the right numerical interpretation, not only of trigonometrical formulæ, but of those also which result from the application of algebra to every other science; it being more particularly our object to guard him against erroneous conceptions respecting the numerical estimates of angles.

We may observe, then, that our estimate of quantity is either absolute or relative-absolute when formed without reference to another quantity of the same nature, and relative when formed by means of such reference. Our estimate is ordinarily of the latter kind; for if we would form an accurate notion, for example, of the length of a line, the extent of a sur

* =3.14159..... See Geom. B. III. § 4. Prop. 34. Schol. Also art. 66 of this work.

face, or the content of a solid, we usually consider how many yards, feet, or inches there may be in the line, how many square yards, square feet, or square inches in the surface, or how many cubic yards, cubic feet, or cubic inches in the solid content: thus forming a relative estimate of the quantity proposed, by referring it to some other fixed quantity of the same kind as a standard with which to compare it, and which may be assumed as we please. In the relative estimate of length, this standard may be an inch, a foot, a yard, or any other established length to which it may be convenient to refer, and the same will hold with respect to quantities of any other kind.

This manner of estimating quantity relatively not only assists us in forming ourselves an accurate idea of the absolute magnitude or quantity of anything, but becomes essential if we would convey that idea to the minds of others, which is effected by expressing the relation which exists between the proposed quantity and some other of the same kind generally recognized as the standard to which such quantities shall be referred. This relation is most easily and most accurately expressed by numbers : for if we represent the standard quantity by the number one, or unity, the quantity which is five times as large will be exactly represented by the number 5; that which is seven times and a half as large, by 7, or (in the decimal notation) by 7·5, and so on, whatever be the relation to be expressed. Any quantity thus arbitrarily chosen to be represented by unity is called the unit of that quantity; and the number expressing the relation which the proposed quantity bears to the standard or unit is called the numerical value of that quantity. It will manifestly depend, not only on the absolute value of the quantity to be expressed, but also on that of the quantity selected as the unit. Thus, if a foot be taken for the unit of length, the numerical value of a line two yards long will be 6; but if an inch be taken for the unit, the numerical value of this line will be 72. Conversely, if we assume a given number for the numerical value of a given quantity, the absolute quantity which must be represented by unity is easily determined. Thus, if 36 be assumed for the numerical value of a line a yard in length, a line whose length is one inch must be the linear unit; if the yard be represented numerically by 18, a line two inches long must be the linear unit, and so on. All this produces no uncertainty or ambiguity in our formulæ and results, provided we carefully bear in mind, in the interpretation of them, the suppositions which may have been made respecting the units of the quantities involved in our investigations. No such assumptions are necessarily made in deducing algebraical results, because the operations of algebra are altogether independent of them: if, however, they have been made, they will affect the form of the resulting equations, and must therefore be carefully borne in mind in the interpretation of such equations, or in deducing from them those numerical results which are almost always necessary in problems of practical importance. (5.) The equation of art. (3), to which we may now return, may exemplify the observations of the last paragraph. This equation is

and expresses the relation which exists generally between the angle, the subtending arc, and its radius, no assumption having been made respecting the magnitude of the angle or of the line which shall be taken respectively for the angular and linear units. Suppose, now, we denote a

right angle by unity, or make that angle our angular unit, and denote also by N' the number of such units or parts of a unit contained in the angle of which the general symbol is A. Then, putting also for its numerical value,

Again, if we take half a right angle for our angular unit, and N" to denote the number of such units or parts of the unit contained in A, we have

N' and N" are the numerical values of the angle whose absolute magnitude is denoted by A. The latter is evidently twice as large as the former; but considering the magnitude of the angular unit in each case, it is manifest that each of these numerical values indicates an angle of the same absolute magnitude.

We may also observe that the numerical value of the angle is independent of the linear unit, since the ratio

a

depends only on the absolute

values of a and r, and not at all on their numerical values. It is manifest, for instance, that, whether a and r be estimated in feet or inches, the

a

numerical fraction which expresses the value of the ratio will, when

reduced to its lowest terms, be necessarily the same.

(6.) The above assumptions as to the magnitude of the angular unit do not, however, lead to convenient equations expressing the relation between the numerical value of the angle and that of the ratio since,

a

a

r

as we have seen, the numerical multipliers of in those equations are

r

not whole numbers. By following an inverse process, however, to those above, we may assume the value of this coefficient, and determine the corresponding magnitude of the angular unit. Thus, suppose we wished the equation to be

which shows that the angular unit, in order that the above equation may be true, must be rather less than the third of a right angle.

(7.) The simplest relation, however, between the quantities in question is

which shows that the angular unit must in this case be rather less than two-thirds of a right angle. Also we may observe that N, the numerical value of the angle, becomes unity when a r, which shows that, in order that the equation 2rt <= 3.14159.... may be true, the unit of angle must be that angle, the subtending arc of which is equal in length to the radius.

We have thought it better to denote the numerical values of the angle by symbols N, N', &c., that the student may not confound them with the absolute value denoted by A. This distinction, however, is frequently not attended to, and the equation

In such cases it must be carefully recollected that A represents the numerical value of the angle, on the supposition just mentioned as to the angular unit.

(8.) The value of an angle expressed by the last equation is generally and often tacitly assumed in investigations involving relations between the angle itself and any of its trigonometrical functions to be hereafter described, and therefore should be carefully borne in mind by the student. At the same time, this is not the most convenient method to adopt when it is simply our object to express numerically the value of any proposed angle, independent of any formulæ or investigations such as those just alluded to. For this purpose the following is much more convenient.

Suppose the circumference of the circle A PB (fig. Art. 3) completed, and divided into 360 equal portions called degrees *; each degree into 60 equal parts called minutes; each minute into 60 equal parts called seconds ; &c. The magnitude of the angle ACP is expressed by the number of degrees, minutes, &c. contained in the subtending arc A P. A right angle will contain 90 of these degrees.

This measure of the angle is quite independent of the length of the subtending arc or its radius; for if, with any other radius C A', we describe another circle about C, it is manifest that the lengths of the degrees in each circumference will have the same ratio as the arcs A P, A' P', and therefore that the number of degrees, &c. in the one will equal that in the other.

If we take an angle of one degree for the angular unit, 90 will be the numerical value of a right angle, 180, of an angle equal to two right angles, 2.5 will be that of an angle of 2° 30', 3·75 of an angle of 3° 45', and so on.

*Degrees are denoted by the symbol (°), minutes by ('), seconds by ("), thirds by (""), &c. Thus 5 degrees 10 minutes and 15 seconds are written 5° 10′ 15′′.

(9.) Having given the numerical value (N°) of a proposed angle when the angular unit is one degree, to find the numerical value (N) when the angular unit is the angle subtended by an arc whose length equals the radius.

The numerical value of an angle equal to the sum of two right angles is 180 on the former supposition, and on the latter. Hence, since the numerical value of any proposed angle increases in the same ratio as that in which the angular unit is diminished, it is manifest that

which gives a rule for passing from the numerical value in one case to that in another.

If an angle of one minute be taken for the angular unit, and N' denote the numerical value in this case, or the number of minutes in the angle, we shall have

Similarly, if N" denote the number of seconds in the angle, and an angle of one second be taken for the angular unit,

(10.) If, in the equation Art. (3), we put 180 for the numerical value of an angle equal to two right angles, and, as before, N° for the corresponding numerical value of A, we have

Ex. 1. To find the length of an arc subtending an angle of one degree. Since N°1

Ex. 2. To find the length of the arc subtending an angle of 6° 12′ 36′′. Here N° 6.21

Ex. 3. To find the number of degrees, minutes, &c. in an angle subtended by an arc whose length equals that of the radius.

In this case a = r

Or, expressing the decimals in minutes, seconds, &c.

N° 57° 17′ 44" 48"