GEOMETRY. 



tiling-. 13. A figure is contained under 

 one or more bounds. 14. A circle is a 

 pluiu figure, contained in one line, called 

 the circumference, every where equally 

 distant from a certain point within it. 

 15. That eq'ui-distant point within the 

 circle is called its centre. 16. A line 

 passing from one side to the other of a 

 circle, and through its centre, is the 

 greatest line it can contain, and is called 

 its diameter. 17. The diameter divides 

 the circle into two equal and similar parts, 

 called semi-circles. 18. When a line 

 shorter than the diameter is drawn from 

 one point to another on the circumfer- 

 ence of a circle, it is called a chord. 

 19. The part of the circle, so cut off or 

 divided by sucli line or chord, is called 

 an arc or segment. 20. Figures con- 

 tained under right lines are called right- 

 lined figures. 21. A figure having three 

 sides is called a triangle. 22. If all the 

 sides of a triangle are of the same length, 

 it is called an equilateral triangle. 23. 

 If all the sides and angles are unequal, 

 it is called a scalene triangle. 24. If two 

 of the sides are of equal length, it is call- 

 ed an isosceles, or equi-crural triangle. 

 25. If containing a right angle, it is call- 

 ed a right-angled triangle. 26. The long 

 side subtending, and opposite to, the 

 right angle, is called the hypothenuse. 

 27. When the two shortest sides of a 

 triangle stand at a greater angle than 90 

 degrees, the figure is said to be " obtuse;" 

 and when all the angles are acute, it is 

 called an acute-angled triangle. 28. When 

 two lines preserve an equal distance from 

 each other in every part, they are said 

 to be parallel. 29. Parallel lines may 

 be either straight or curved, but can ne- 

 ver meet. 30. A figure having four 

 equal sides, and all the angles equal, is 

 a square. 31. But if its opposite angles 

 only be equal respectively, the figure 

 will then be a rhombus, or lozenge. 

 32. When all the sides of a figure are 

 right lines, and that the opposite sides 

 are parallel and equal, it is called a paral- 

 lelogram. 33. If the opposite sides are 

 equal, the others being unequal, the fi- 

 gure is called a rhomboides. 34. Four- 

 sided figures unequal in all respects, are 

 called trapesia. 35. Figures having more 

 than four sides are called polygons, and 

 are thus distinguished: with five sides, 

 it is called a pentagon; with six, an 

 hexagon ; with seven, an heptagon ; with 

 eight, an octagon ; with nine, an ennea- 

 gon ; with ten, a decagon ; with eleven, 

 an endecagon; with twelve, a dodeca- 



gon. 36. A solid has length, brearlthT 

 and thickness. 37. A pyramid is a solid 

 standing on a base, of any number of 

 sides, all of which converge from the 

 base to the same point or summit. 38. 

 When standing on a triangular base, it 

 is called a triangular pyramid ; on four, 

 a square pyramid ; on five, a pentagonal ; 

 and thus in conformity with the figure 

 of its base. 39. Every side of a pyramid 

 is a triangle. 40. A cone is found by 

 the revolution of a triangle on its apex, 

 or summit, and a point situated in the 

 centre, of its base ; therefore a cone (like 

 a sugar-loaf) has a base, but no sides. 

 41. A prism is a figure contained under 

 planes, whereof the two opposite are 

 equal, similar, and parallel ; and all the 

 sides parallelograms. 42. A sphere is a 

 solid figure, generated by the revolution 

 of a circle on its diameter, which is then 

 called the axis. 43. A cube is a solid 

 formed of six equal and mutually parallel 

 sides, all of which are squares. 44. A 

 tetrahedron is a solid contained under 

 four equal, equilateral triangles. 45. A 

 dodecahedron is a solid contained under 

 twelve equal, equilateral, and equiangu- 

 lar pentagons. 46. An icosahedron is a 

 solid contained under twenty equal, equi- 

 lateral triangles. 47. A parallelopipedon 

 is a figure considered under six quadrila- 

 teral figures or planes, whereof those 

 opposite are respectively parallel. 48. 

 Figures, or bodies, are said to be equal, 

 when their bulks are the same ; and si- 

 milar, when they are alike in form, 

 though not equal. 49. Therefore simi- 

 lar figures or bodies are to each other in 

 proportion to their respective areas -or 

 bulks. 50. The line or space on which 

 a figure stands is called its base ; its al- 

 titude is determined by a line drawn 

 parallel to its base, and touching its 

 vertex, or highest part. 51. A right- 

 lined figure is said to be inscribed within 

 another, when all its projecting angles 

 are touched thereby. 52 The figure sur- 

 rounding or enveloping another is said to 

 be described around, or on it. 53. When 

 a line touches a circle, and proceeds with- 

 out cutting it, such line is called a tan- 

 gent. 54. Any portion less than a semi- 

 circle, taken out from a circle by two 

 lines, or radii, proceeding from the cejitre, 

 is called a sector. 



Certain AXIOMS are likewise proper to 

 be carried in mind ; viz. I. That things 

 equal to one and the same thing are equal 

 to one another. 2. If to equal things (or 

 numbers) we add equal things, (or num- 



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