DEFINITIONS.
1. A point 'lath neither parts nor mag nitude. 2. A line has length, without breadth. 3. The ends, or bounds, of a line are points. 4. A right line lies evenly between two points. 5. A superficies or plane has only length and breadth. 6. Planes are bounded by lines. 7. A plain superficies lies evenly and level between its lines. 8. A plain angle is formed by the meeting of two right lines. 9. When an angle measures 90 degrees, it is call ed a right angle. 10. When less than 90 degrees, it is said to be an acute angle, 11. When more than 90 degrees, it is called an obtuse angle. 12. A term, or bound, implies the extreme of any thing. 13. A figure is contained under one or more bounds. 14. A circle is a plain figure, contained in one line, called the circumference, every where equally distant from a certain point within it. 15. That equidistant 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 f 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 such 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 fines, 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.
gm), 36. A solid has length, breadth, 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 centre, is called a sector.
Certain Axmars are likewise proper to be carried in mind ; viz. 1. 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.
hers) the whole will be equal. 3. If from equal things we take equal things, the remainder will be equal, and the reverse in respect to unequal things. 4. The whole is greater than any of its parts. 5. Two right lines do not contain a space. 6. All the angles within a circle cannot amount to more nor less than 360 degrees, nor in a semicircle to more nor less than 180 degrees. 7. The value, or measure, of an angle is not affected or changed by the lines where by it is formed being either lengthened or shortened. 8. Two lines standing at an angle of 90 degrees from each other will not be affected by any change of po sition of the entire figure in which they meet, but will still be mutually perpen dicular.
After thus much preparation, we may conclude the student to be ready to proceed in the solution of problems, which we shall study to exhibit in the most simple, as well as in a progressive manner.