which shews that the size of the angle depends on arc since BAX is a fixed angle. radius 4. There are 7 diameters in the circumference of a circle 2π radii or 2πr. T 31416, or 22 355 or 7' 113 5. The complement of an angle is its defect from one П , 2 right angle. Subtract the angle from 90°, 100° or to find its complement. 6. The supplement of an angle is its defect from two right angles. Subtract the angle from 180o, 200o or π, to find its supplement. 7. Rule for reduction of the measures of angles, The following general examples in angles will suffice to give confidence in the use of this rule. Obs. As the circular measure represents a pure number, a ratio, it is always placed without any mark of denomination after it, as an angle these numbers are the circular measure. 3 4 or 21; where This is a pure convention, somewhat analogous to our expression "That child is just eight," or "That man is worth a million," meaning, of course, years and pounds sterling. The other measures must always have their marks added. PAPERS ON ANGLES. I. 1. What is the Circular Measure of an Angle? Give the circular measure of the angle subtended by a circular watch-spring 3 in. long and radius 14 in. 2. Find the number of degrees in the angles 10o, 30°, 25o, and the number of grades in 45°, 60°, 150°. 1 3. Find the circular measure of of a right angle; 8 2 of 4 right angles; one grade; one degree. 3 4. Add together 33°. 45′. 21′′; 2o. 22''; 3′. 21'; 5. What angle in degrees, minutes and seconds is meant by 0=2 ? II. 1. Find the angle whose circular measure is 1, and deduce the number of degrees in 3·05. 2. What number of grades, &c. is made by an arc of 5 chains at the centre of a railway curve, radius 80 chains? 3. Change to circular measure; 3°.14.25"; 185.21.5"; 105°.5′′. 4. Through what angle does a steersman move his wheel when he passes n out of the m spokes? 5. What convenience is there in the French system of angle measurement? III. 1. Prove that an angle may be measured by the fraction and find the arc which will arc radius subtend one degree at the centre of a circle, radius 1 foot. and 2. Change to grades the angles; 13°. 12′; 406°.3′′; 3. Find the angle in circular measure made by the hands of a clock at 5". 15', a quarter to 8, 3.30, 6.5. 5. Find the circumference of a circle where an 'angle of 33° is subtended by an arc of 4 inches. CHAPTER III. THE TRIGONOMETRICAL RATIOS. 17 THERE remains to be considered, the Trigonometrical measurement of angles, that is, the measurement of angles by triangles. The inclination of a slope on a railway to the horizon is measured in this way. A railway is said to rise 1 in 40 when for every 40 yards of the railway it rises 1 yard. This is an indication of the angle made by the railway and the horizon, and is denoted by saying that the sine of the angle of inclination is 1 40 The sine of an angle shews its size. If an angle it represents a slope of 1 in 2, a known 1 slope and angle. If the sine be it represents a slope 4 of 1 in 4. A mistake must here be guarded against. 1 The angle whose sine is is not double the angle 18 19 whose sine is a figure. 1 , as may easily be seen by drawing AMP is a right-angled triangle, and so APM is PM the complement of PAM. And it will be seen that AP is the sine of PAM; while M ment APM. This is called the cosine of A, and represents the rate of progress A along the horizontal line compared with the progress up the hill. If for every 40 up the hill AP, 13 is gone along the horizon AM, the cosine of PAM or 13 cos A is said to be 40° The values of sine A and cosine A may be made familiar thus: I go 50 yds. straight up a hill-side AP. I rise PM or 30 yds. This shews that the slope is 3 in 5 and the sine of its inclination is map 3 5° Also I go 40 yds. along a horizontal line on the This shews me that the cosine of the inclination |