Fig. 3. 8. An obtuse angle is greater than a right angle; as A DE, Fig. 3. 9. An acute angle is less than a right angle; as E DB, Fig. 3. Note. When three lettors are used to express an angle, the middle letter denotes the angular point. Fig. 4. A E 10. A circle is a round figure bounded by a single line, in every part equally distant from some point, which is called the centre. Fig. 4. 11. The circumference or periphery of a BI circle is the bounding line; as A DEB, Fig. 4. 12. The radius of a circle is a line drawn from the centre to the circumference; as CB, Fig. 4. Therefore all radii of the same circle are equal. 13. The diameter of a circle is a right line drawn from one side of the circumference to the other, passing through the centre; and it divides the circle into two equal parts, called semicircles; as AB or DE, Fig. 5. 14. The circumference of every circle A is supposed to be divided into 360 equal parts, called degrees; and each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds ; and these into thirds, &c. Note. Since all circles are divided into the same num ber of degrees, a degree is not to be accounted a quantity of any determinate length, as so many inches or feet, &c. but is always to be reckoned as being the 360th part of the circumference of any circle, without regarding the size of the circle. 15. An arc of a circle is any part of the circumference; as BF or F D, Fig. 5; and is said to be an arc of as many Fig. 6. H 16. A chord is a right line drawn from one end of an arc to another, and is the measure of the arc; as HG is the chord of A the arc HIG, Fig. 6. D B NOTE. The chord of an arc of 60 degrees is equal in length to the radius of the circle of which the arc is a part. 17. The segment of a circle is a part of a circle cut off by a chord; thus, the space comprehended between the arc HIG and the chord H G is called a segment. Fig. 6. 18. A sector of a circle is a space contained between two radii, and an arc less than a semicircle; as BCD, or AC D, Fig. 6. 19. The sine of an arc is a line drawn Fig. 7. from one end of the arc, perpendicular to the radius or diameter drawn through K the other end : or, it is half the chord of DI double the arc; thus, HL is the sine of H the arc H B, Fig. 7. 20. The sines on the same diameter incrcase in length till they come to the cen-A HB C L tre, and so become the radius, after which they diminish. Hence it is plain that the sine of 90 degrees is the greatest possible sine, and is equal to the radius. 21. The versed sine of an arc is that part of the diameter or radius which is between the sine and the circumference; thus LB is the versed sine of the arc HB, Fig. 7. 22. The tangent of an arc is a right line touching the circumference, and drawn perpendicular to the diameter; and is terminated by a line drawn from the centre through the other end of the arc; thus, B K is the tangent of the arc BH, Fig. 7. NOTE. The tangent of an arc of 45 degrees is equal in length to the radius of the circle of which the arc is a of 23. The secant of an arc is a line drawn from the centre through one end of the arc till it meets the tangent; thus, CK is the secant of the arc B H, Fig. 7. 24. The complement of an arc is what the arc wants of 90 degrees, or a quadrant: thus, H D is the complement of the arc B H, Fig. 7. 25. The supplement of an arc is what the arc wants of 180 degrees, or a semicircle; thus A DH is the supplement of the arc B H, Fig. 7. Note. It will be seen by reference to Fig. 7, that the sine any arc is the same as that of its supplement. So, likewise, the tangent and secant of any arc are used also for its supplement. 26. The sine, tangent, or secant, of the complement of any arc, is called the co-sine, co-tangent, or co-secant of the arc; thus, F H is the sine, D I the tangent, and C I the secant of the arc DH; or they are the co-sine, co-tangent, and co-secant of the arc B H, Fig. 7. [The terms sine, tangent, and secant, are abbreviated thus : sin., tan., So, likewise, co-sine, co-tangent, and cosecant, are written co-sin., co-tan., and co-sec.] 27. The measure of an angle is the arc of a circle contained between the two lines which form the angle, the angular point being the centre; thus, the angle HCB, Fig. 7, is measured by the arc BH; and is said to contain as many degrees as the arc does. Note. An angle is esteemed greater or less, according to the opening of the lines which form it, or as the arc intercepted by those lines contains more or fewer degrees. Hence it may be observed, that the size of an angle does not depend at all upon the length of the including lines; for all arcs described on the same point, and intercepted by the same right lines, contain exactly the same num. ber of degrees, whether the radius be longer or shorter. 28. The sine, tangent, or secant of an arc, is also the sine, tangent, or secant of the angle whose measure the arc is. Fig. 8. 29. Parallel lines are such as are equal A B ly distant from each other; as A B and C D, and sec. Fig. 9. C 30. A triangle is a figure bounded by three lines; as A B C, Fig. 9. 31. An equilateral triangle has its three sides equal in length to each other. Fig. 9. 32. An isosceles triangle has two of its sides equal. Fig. 10. Fig. 11. 33. A scalene triangle has three unequal sides. Fig. 11. Fig. 12. 34. A right angled triangle has one right angle. Fig. 12. Fig. 13. 35. An obtuse angled triangle has one obtuse angle. Fig. 13. 36. An acute angled triangle has all its angles acute. Fig. 9, or 10. 37. Acute and obtuse angled triangles, are called oblique angled triangles, or simply oblique triangles; in which the lower side is generally called the base, and the other two, legs. 38. In a right angled triangle, the longest side is called the hypothenuse, and the other two, legs, or base, and perpen PART II. GEOMETRICAL PROBLEMS. Fig. 22, PROBLEM I. To draw a line paral C D lel to another line at any given distance ; as at the point D, to make a line parallel A to the line A B. Fig. 22. -В With the dividers take the nearest distance between the point D and the given line A B; with that distance set one foot of the dividers any where on the line A B, as at E, and draw the arc C; through the point D draw a line so as just to touch the top of the arc C. A more convenient way to draw parallel lines is with a parallel rule. [The parallel, rules, however, found in cases of mathematical instruments, are often inaccurate.] Fig. 23. E PROBLEM II. To bisect a given line ; or, to find the middle of it. Fig. 23. A B D Open the dividers to any convenient distance, more than half the given line A B, and with one foot in A, describe an arc above and below the line, as at C and D; with the same distance, and one foot in B, describe arcs to cross the former; lay a rule from C to D, and where the rule crosses the line, as at E, will be the middle. Fig. 24. H PROBLEM III. To erect a perpendicular from the end, or any part of a given line. Fig. 24. |