MENSURATION We may consider that which has no dimension, that which has one dimension, that which has two dimensions, or that which has three dimensions. Draw points and lines situated like the above, and from each point draw a perpendicular to the nearest line. TWO DIMENSIONS That which has two dimensions is a surface. If any two points of a surface are connected by a straight line, that line will lie wholly on the surface, a plane surface, or plane; or it will not lie wholly on the surface, a curved surface. If straight lines inclose a surface, the figure is a polygon. The least number of straight lines which can inclose a plane is three, a triangle, (1). The three lines may be equal, an equilateral triangle, (2); two of them may be equal, an isosceles triangle, (3); or no two of them may be equal, a scalene triangle, (4). A triangle may have one right angle, a right-angled triangle, (6); one obtuse angle, an obtuse-angled triangle, (7); or three acute angles, an acute-angled triangle, (8). The next number of straight lines which can inclose a plane is four, a quadrilateral, (9). The quadrilateral may have both pairs of its opposite sides parallel, a parallelogram, (10); one pair parallel, a trapezoid, (11); or neither pair parallel, a trapezium, (12). A parallelogram may have its angles right angles, a rectangle, (13); or not right angles, a rhomboid, (14). The rectangle may have its sides all equal, a square, (15); the rhomboid may have its sides all equal, a rhombus, (16). Five sides may inclose a surface, pentagon; six sides, hexagon; seven sides, heptagon; eight sides, octagon; nine sides, nonagon; ten sides, decagon, .... A polygon may have its sides and angles equal, a regular polygon; or its sides and angles not equal, an irregular polygon. Hence we may have regular and irregular pentagons, regular and irregular hexagons . . .... A regular polygon of an infinite number of sides is a circle. Regular hexagon. Irregular hexagon. Circle. Define: 23. A surface. 24. A polygon. 25. A triangle. 26. An equilateral triangle. 27. An isosceles triangle. 28. A scalene triangle. 29. A right-angled triangle. 30. An acute-angled triangle. 31. A regular polygon. 32. A regular pentagon. 33. A regular hexagon. 34. A regular heptagon. 36. A circle. 37. Beginning with "plane surface," (see Note) define: parallelogram; rectangle; rhomboid; rhombus; square. 38. Beginning with quadrilateral, (see Note) define: parallelogram; rectangle; square; rhombus. NOTE. A definition may begin with different terms, e.g.: A square is a plane surface bounded by two pairs of opposite sides, having each pair parallel, having its angles all right angles, and having its sides all equal. Or, A square is a quadrilateral having each pair of its opposite sides parallel, hav ing its angles all right angles, and having its sides all equal. Or, A square is a parallelogram having its angles all right angles, and having its sides all equal. Or, A square is a rectangle, having its sides all equal. That definition is the best which is the shortest, provided it begins with a term which is understood by the person for whom the definition is given. |
