of a grade, that is, 37 of a grade; and 30 cen tesimal seconds are 30 of a grade, that is, '003 of a (100)2 (110)th grade. Hence 5o 37' 30" may be written as 5o373; and since a grade is 100 of a right angle, 5373 may be written as '05373 of a right angle. Notwithstanding this great advantage of the centesimal method, the sexagesimal method has been retained in practical calculations, because the latter had become thoroughly established by long use in mathematical works, and especially in mathematical tables, before the former was proposed; and such works and tables would have been rendered almost useless by the change in the method of estimating angles. The centesimal method is not practically used even in France. 8. Although the centesimal method is not used in practical calculations it is customary to give an account of the method in works on Trigonometry; and it is shewn how to compare the numbers which measure the same angle in the English and French methods. This we shall explain in the next three Articles. 9. To compare the number of degrees in any angle with the number of grades in the same angle. Let D be the number of degrees in any given angle, G the number of grades in the same angle. Then, since there are 90 degrees in a right angle, Ꭰ 90 expresses the ratio of the given angle to a right angle; and, since there G are 100 grades in a right angle, also expresses the 100 ratio of the given angle to a right angle. The formula D=G· 1G gives the following rule: 10 From the number of grades in any angle subtract onetenth of that number; the remainder is the number of degrees in the angle. The formula G=D+D gives the following rule: To the number of degrees in any angle add one-ninth of that number; the sum is the number of grades in the angle. 10. To compare the number of English minutes in any angle with the number of French minutes in the same angle. m Let m be the number of English minutes in any angle, μ the number of French minutes in the same angle. Then, since there are 90 × 60 English minutes in a right angle, expresses the ratio of the given angle to a right 90 × 60 angle; and since there are 100 × 100 French minutes in a also expresses the ratio of the given μ right angle 100 x 100 angle to a right angle. 11. Similarly, if s be the number of English seconds in any angle, and σ the number of French seconds in the same angle EXAMPLES. I. Express the following six angles in the French mode: Express the following six angles in the English mode: 17. The angle of an equilateral triangle. 18. The angle at the vertex of the isosceles triangle described in Euclid IV. 10. 19. The sum of two angles is 30 grades, and their difference is 9 degrees: find each angle. 20. The difference of the two acute angles of a rightangled triangle is 20 grades: find the angles in degrees. 21. Find the number of English minutes in a grade. 22. Find the number of English seconds in a French minute. 23. Find the number of French minutes in a degree. 24. Find the number of French seconds in an English minute. 25. Find the ratio of an angle of 1° 25′ to an angle of 1o 25'. II. Trigonometrical Ratios. 12. There are certain quantities connected with an angle which are called the Trigonometrical Ratios of the angle; in the present Chapter we shall define these Trigonometrical Ratios, and demonstrate some of their most important properties, confining ourselves to angles less than a right angle. It will be seen as we proceed with the book that the whole subject rests on the definitions and properties contained in the present Chapter. 13. Let BAC be any acute angle; take any point in either of the containing straight lines, and from it draw a perpendicular to the other straight line: let P be the point in AC, and PM perpendicular to AB. We shall use the letter A to denote the angle BAC. C P M B The following are the definitions of the Trigonometrical Ratios of the angle A: When the cosine of A is subtracted from unity the remainder is called the versed sine of A. When the sine of A is subtracted from unity the remainder is called the coversed sine of A. But the term versed sine is not often used, and the term coversed sine is scarcely ever used. 14. The words sine, cosine, &c. are usually abbreviated in writing and printing; thus the above definitions may be expressed as follows: 15. The sine, cosine, tangent, cotangent, secant, cosecant, versed sine, and coversed sine of an angle are called the Trigonometrical Ratios of the angle: it will be seen from the definitions that the term ratio is appropriate, because each of the quantities defined is the ratio of one length to another, that is, each of the quantities is some arithmetical number or fraction. The Trigonometrical Ratios have been sometimes called Trigonometrical Functions, and sometimes Goniometrical Ratios or Functions. 16. The excess of a right angle over any angle is called the complement of that angle. Thus if A be the number of degrees in any angle, 90-A is the number of degrees in the complement of the angle. This affords another method of defining some of the Trigonometrical Ratios: after defining, as in Art. 14, the sine, tangent, and secant of an angle we may say: the cosine of an angle is the sine of the complement of that angle; the cotangent of an angle is the tangent of the complement of that angle; the cosecant of an angle is the secant of the complement of that angle. |