Fig. 9. с 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. AU Fig. 10. 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 perpenNote. The three angles of every triangle being added together will, amount to 180 degrees; consequently the two acute angles of a right angled triangle amount to 90 degrees, the right angle being also 90. Fig. 14. A 39. The perpendicular height of a triangle is a line drawn from one of the angles perpendicular to its opposite side; thus, the dotted line A D, Fig. 14, is the perpendicular height of the triangle A B C. В. D Note. This perpendicular may be drawn from either of the angles; and whether it falls within the triangle, or on one of the lines continued beyond the triangle, is immaterial. Fig. 15. 40. A square is a figure bounded by four equal sides, and containing four right angles. Fig. 15. Fig. 16. 41. A parallelogram, or oblong square, is a figure bounded by four sides, the opposite ones being equal, and the angles right.* Fig. 16. Fig. 17. А 42. A rhombus is a figure bounded by four equal sides, but has its angles oblique. Fig. 17. B * Any four sided figure, having its opposite sides parallel, is a parallelogram; bat, in this book, the term is understood as it is here ex Fig. 18. 43. A rhomboid is a figure bounded by four sides, the opposite ones being equal, but the angles oblique. Fig. 18. B 44. The perpendicular height of a rhombus or rhomboi. des, is a line drawn from one of the angles to its opposite side; thus, the dotted lines A B, Figs. 17 and 18, represent the perpendicular heights of those figures. Fig. 19. 45. A trapezoid is a figúre bounded by four sides, two of which are parallel, though of unequal lengths. Fig. 19 and Fig. 20. Note. Fig. 19 is sometimes called a right angled trapezium. Fig. 21. B 46. A trapezium is a figure bounded by four unequal sides. Fig. 21. 47. A diagonal is a line drawn between two opposite angles; as the line A B, Fig 21. A 48. Figures which consist of more than four sides are called polygons; if the sides are equal to each other, they are called regular polygons, and are sometimes named from the number of their sides, as pentagon, or hexagon, a figure of five or six sides, &c.; if the sides are unequal, they are called PART II. GEOMETRICAL PROBLEMS. Fig. 22. PROBLEM I. To draw a line paral 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. B F 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. * PROBLEM II. To bisect a given Aline ; or, to find the middle of it. Fig. 23. -B 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. PROBLEM III. To erect a perpendicular from the end, or any part of a given line. Fig. 24. Open the dividers to any convenient distance, as from D to A, and with one foot on the point D, from which the perpendicular is to be erected, describe an arc, as A EG; set off the same distance A D, from A to E, and from E to G; upon E and G describe two arcs to intersect each other at H; draw a line from H to D, and one line will be perpendicular to the other. Note. There are other methods of erecting a perpendicular, but this is the most simple. Fig. 25. PROBLEM IV. From a given point, as at C, to drop a perpendicular A on a given line A B, Fig. 25. 1 B D With one foot of the dividers in C, describe an arc to cut the given line in two places, as at F and G; upon F and G describe two arcs to intersect each other below the line, as at D; lay a rule from C to D, and draw a line from C to the given line. Perpendiculars may be more readily raised and let fall, by a small square made of brass, ivory, or wood. Fig. 26. C G PROBLEM V. To make an angle at E, equal to a given angle ABC, Fig. 26. H Н E Open the dividers to any convenient distance, and with one foot in B, describe the arc FG; with the same distance, and one foot in E, describe an arc from H; measure the arc FG, and lay off the same distance on the arc from H to I; draw a line through I to E, and the angles will be equal. Fig. 27. PROBLEM VI. To make an acute angle equal to a given number of degrees, suppose 36. Fig. 27. A 360 |