is a number which is the same for all circles. This number cannot be found exactly, but we can approximate to it as closely as we please. A first approximation is 3, a nearer one is 355 The number or 3.14159, 22 יך and a still nearer one is 113' 355 113 is the numerical value of the ratio correct to five decimal places, and is close enough for all practical purposes. The ratio we are considering is one of the most important numbers in mathematics, and is always denoted by the Greek letter π. Our assumption, then, in this Article is, that in every circle circumference =π, where diameter represents a number the correct value of which to five decimal places is 3.14159. If r be the radius of a circle, 2r is the diameter, and there Ex. 1. The radius of a circle is 6 furlongs. length of the circumference in yards? Here, › the radius= 6 furlongs = 6 × 220 yards; therefore circumference = 2 × π × 6 × 220 yards What is the = 2 × 3.14159 × 6 × 220 yards, putting = 3.14159, = Ex. 2. If the quarter of the circumference of a circle is 3 feet 4 inches, what is the length of the radius in inches? therefore circumference = 3 feet 4 inches = 40 inches; Hence, if r denote the radius of the circle in inches, 5. Theoretical Unit of Angle-the Radian. The radian is the angle subtended at the centre of any circle by an arc equal to the radius. Let ABCDE be any circle, radius r, and let a length AB equal to the length of the radius r be measured from A along the circumference. Then the angle AOB subtended at the centre by the arc AB is the radian. To justify this definition, we must show that it always gives the same angle whatever be the radius of the circle. This is easily proved if we make the assumption of the preceding Article. For AD and CE being diameters at right angles, the arc AC is circumference or Ꭰ C B E A ×2, and the angle AOC is a right angle. But by Euclid VI. 33, angle AOB: angle AOC :: arc AB : arc AC; Since represents a definite number, the radian is a definite angle, which is independent of the radius of the circle. As we cannot find the exact value of π, we cannot find the exact number of degrees in the radian, but taking #=3·14159, we get radian 2 right angles π 180° 3.14159 = 57°2958, to four decimal places. Roughly speaking, the radian is an angle a little less than the angle of an equilateral triangle which is an angle of 60°. The radian is also called the unit of circular measure. 6. Radian or Circular Measure of an Angle. Since the radian is a definite angle, it may be taken as a unit of angle, and then the number expressing the magnitude of any other angle is the number of times that angle contains the radian. This is called the radian or circular measure of the angle. Since the radian = 57° 2958, very nearly, if we multiply the radian measure of an angle by 57-2958, the product will be, very approximately, the number of degrees in the angle. Conversely, if an angle be given in degrees-minutes and seconds being expressed as the decimal of one degree-the radian measure will be the quotient obtained by dividing the number of degrees by 57.2958. Ex. 1. The radian measure of an angle is 1·23; express the angle in degree measure. The angle = 1.23 radians = 1.23 × 57° 2958, very nearly, = 70°4738, to four decimal places. 4738 degrees. 60 Hence the angle contains 70 degrees and 4738 of one degree. We can convert this decimal of a degree into minutes and seconds by multiplying by 60 for minutes, and again by 60 for seconds. The result is 28' 25" 68. Hence the angle in degrees, minutes, and seconds, is 70° 28′ 25′′ 68. 28 428 minutes. 60 60)30 .5 Ex. 2. Find the radian measure of the angle 120° 19′ 30′′. We first convert 19′ 30′′ to the decimal of a degree. Dividing 30 by 60, we find that 30" is 5 of a minute; then dividing 19.5 by 60, we find that 19′ 30′′ is 325 of a degree. Hence the angle expressed in degrees is 120°325; therefore the radian measure of the angle 60)19.5 *325 or angle AOP angle AOB :: arc AP : arc AB, A or, the number of radians in the angle AOP may be found by dividing the number expressing the length of the arc AP by the number expressing the length of the radius. Hence, if the symbol denote the radian measure of AOP, =, and therefore 1= re. Ex. 1. Find the angle, in degree measure, subtended at the centre of a circle whose radius is 100 inches by an arc whose length is 9 inches. therefore the degree measure = '09 × 57°•2958 (Art. 6) =5°1566 =5° 9′ 23′′ 76. Ex. 2. Find the length of the arc which subtends the angle 63° 35′ 54′′ at the centre of a circle whose radius is 10 feet. . 63° 35′ 54′′ 63°.5983. Hence the radian measure of the angle Putting = 111, r = 10, in the formula l=re, we find 7 = 10 × 1·11 = 11.1 feet. 8. Radian Measure of an Angle in terms of π. In Art. 6 we gave two approximate rules for converting from degree measure to radian measure, and conversely. We shall now show how the radian measure of an angle may be expressed exactly in terms of π. Since (Art. 5) there are ☛ radians in 2 right angles or 180°, the radian measure of any angle will bear to π the same ratio that the number of degrees in the angle bears to 180. Thus π π π the radian measure of 90° is of 60° is of 45° is of 30° is π 6' 2' 4' π of 1° is &c. Generally, the radian measure of D degrees 180' Dπ 180 is D times the radian measure of 1° or Hence, if be the radian measure and D the degree measure of the same angle, two formulæ connecting the degree measure and the radian measure in terms of T. If we put in these formulæ the approximate value 3.14159 for π, we are led again to the two approximate rules of Art. 6. The method of expressing the radian measure of an angle in terms of is generally adopted when we wish more to specify a definite angle than to express the number of radians in it. Thus, if we wish to express that a certain angle is equal to the angle of an equilateral triangle, we say that the angle is equal to 60° or to radians. We might for shortness say π 3 π (as is often done) that the angle is equal to the word radians 3' being omitted, as the presence of the letter π indicates that Ex. 1. Express, in terms of π, the radian measures of the angles 135°, 120°, 15°, 10′′. By formula (1) of this Article, the radian measures are respectively |