if we omit the decimals after the first two places. 2. If n yards be taken as the unit of length, what will be the numerical measure of m feet? The numerical measure of any magnitude is the ratio which it bears to the magnitude which is taken as the unit of measure m ment. Now the ratio of m feet or yards to n yards is 3 which is therefore the numerical measure of m feet required. m 3п 3. If one French minute be taken as the unit of angular magnitude, what is the measure of an angle of one English minute? 50 Hence (being the ratio of the given angle which is to be 27 measured to the angle which is taken as the unit of angular measurement) is the measure required. 4. What must be the unit angle if the sum of the measures of a degree and a grade is r'? Let x be the number of degrees in the unit angle, .. the unit is an angle which contains 1.9 degrees. 5. If the measure of an angle be equal to the sum of the number of degrees, and half the number of grades in it, what is the unit of angular measure? Let x be the number of degrees in the angle. The measure of the angle is therefore represented by the Let y be the number of degrees in the unit. Then xxy=the number of degrees in the angle, 14 =x, 9 14 9 Or the unit of measurement is ths of a degree. 14 6. The measures of the three angles of a triangle expressed respectively A in degrees, B in grades, and C in circular measure, are numerically equal to one another; find A. Let x, y, and z be the number of degrees in the three angles. Then since they are the angles of a triangle, Substituting in (1) the values of y and z in terms of x obtained from (2), we have ... x 9 180 П IOT+9T+1800 .*. x= x= = 180, 800) = 1800π IOT 19π + 1800' 180, which is the number of degrees in the angle A. 9. Find the number of grades in 53° 4′ 21′′ and in 27° 19′ 1′′. Ans. 588 96 944 and 30° 35' 21` ·6... 10. Find the number of degrees in 278 19` 1“ and in 53o 4`21“. Ans. 24° 28′ 15" 924 and 47° 44′ 16′′ ·404. 11. Find the number of degrees and minutes in 255 of a right angle. Ans. 22° 57'. 12. The angles of a quadrilateral inscribed in a circle, taken in order, when multiplied by 1, 2, 2, 3 respectively, are in arithmetical progression; find their values. 13. How many sides has a polygon which contains as many grades in all its angles together as there are degrees in all the angles of a dodecagon? Ans. II. 14. The numbers of the sides of two regular polygons are as 2 : 3, and the number of grades in an angle of one equals the number of degrees in an angle of the other. Find how many sides they each have. Ans. Eight and twelve. 15. The interior angles of an irregular polygon are in arithmetical progression; the least angle is 120° and the common difference 5o. Find the number of sides. Ans. 16 or 9 sides. 16. The number of grades in an angle of a regular polygon is to the number of degrees in an angle of another as 5: 3. Find the number of sides in each, shewing that there are only three different solutions. 17. Find the circular measure of 10" of 36° 15′ 22′′ and of π grades. Ans. '000015π, •6328 nearly, π2 200 18. Find the number of degrees, minutes and seconds in the angles of which the circular measures are +1 and II 21 Ans. 237° 17′44′′ nearly, and 30° 0′ 43′′*45. |