GEOMETRY. BOOK I. DEFINITIONS. GEOMETRY is a science which treats of the properties and relations of quantities or magnitudes having extension. 1. A Point is that which has position, but no magnitude nor dimensions; neither length, breadth, nor thick ness. 2. A Line is length without breadth or thick ness. 3. A Surface or Superficies is an extension or a figure of two dimensions, length and breadth; but without thickness. 4. A Body or Solid is a figure of three dimensions, namely, length, breadth, and depth, or thickness. 5. Lines are either Right or Curved, or Mixed of these two. 6. A Right Line or Straight Line lies all in the same direction, between its extremities, and is the shortest distance between two points. From one point to another only one straight line can be drawn. When a Line is mentioned simply, it means a Right Line. 7. A Curve continually changes its direction between its extreme points. 8. Lines are either Parallel, Oblique, Perpendicular, or Tangential. 9. Parallel Lines are always at the same perpendicular distance; and they never meet, though ever so far produced. 10. Oblique Lines change their distance, and would meet if produced on the side of the least distance. A 11. One line is Perpendicular to another when it inclines not more on the one side than the other, or when the angles on both sides of it are equal. 12. A Line or Circle is Tangential, or is a Tangent to a Circle or other curve, when it touches it without cutting, although both are produced. 13. An Angle is the inclination or opening of two lines having different directions and meeting in a point. 8 14. Angles are Right or Oblique, Acute or Obtuse. 15. A Right Angle is that which is made by one line perpendicular to another. Or, when the angles on each side are equal to one another, they are right angles. 16. An Oblique Angle is that which is made by two oblique lines, and is either less or greater than a right angle. 17. An Acute Angle is less than a right angle. 18. An Obtuse Angle is greater than a right angle. 19. Superficies are either Plane or Curved. 20. A Plane Superficies, or a Plane, is that with which a right line may every way coincide. Or, if the line touch the plane in two points, it will touch it in every point. But if not, it is curved. 21. Plane figures are bounded either by right lines or curves. 22. Plane figures that are bounded by right lines have names according to the number of their sides or of their angles; for they have as many sides as angles; the least number being three. 23. A figure of three sides and angles is called a Triangle; and it receives particular denominations from the relations of its sides and angles. 24. An Equilateral Triangle is that whose three sides are all equal. 25. An Isosceles Triangle is that which has two sides equal. 26. A Scalene Triangle is that whose three sides are all unequal. 27. A Right-angled Triangle is that which has one right angle. 28. Other Triangles are Oblique-angled, and are either obtuse or acute. 29. An Obtuse-angled Triangle has one obtuse angle. 30. An Acute-angled Triangle has all its three angles acute. 31. A figure of Four sides and angles is called a Quadrangle, or a Quadrilateral. 32. A Parallelogram is a quadrilateral which has both its pairs of opposite sides parallel. And it takes the following particular names, viz., Rectangle, Square, Rhombus Rhomboid. 33. A Rectangle is a parallelogram having a right angle. 34. A Square is an equilateral rectangle, having its length and breadth equal, or all its sides equal, and all its angles equal. 35. A Rhomboid is an oblique-angled parallelogram. 36. A Rhombus is an equilateral rhomboid, having all its sides equal, but its angles oblique: 37. A Trapezium is a quadrilateral which has not its opposite sides parallel. 38. A Trapezoid has only one pair of opposite sides parallel. 39. A Diagonal is a line joining any two opposite angles of a quadrilateral. 40. Plane figures that have more than four sides are, in general, called Polygons; and they receive other particular names, according to the number of their sides or angles. Thus, 41. A Pentagon is a polygon of five sides; a Hexagon, of six sides; a Heptagon, seven; an Octagon, eight; a Non |