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19. A chord is a right line joining the extremities of an
20. A segment is any part of a circle bounded by an arc and its chord,
21. A semicircle is half the circle, or a segment cut off by a diameter.
The half circumference is sometimes called the Semicircle.
22. A sector is any part of a circle which is bounded by an arc, and two radii drawn to its extremities.
23. A quadrant, or quarter of a circle, is a sector having a quarter of the circumference for its arc, and its two radii are perpendicular to each other. A quarter of the circumference is sometimes called a quadrant.
24. An angle is the inclination or opening of two lines, having different directions, and meeting in a point.
According to circumstances, they are distinguished into right or oblique ; and the oblique are again distinguished into acute or obtuse.
25. When one line standing on another line makes the adjacent angles equal to one another, each of them is called a right angle, and the straight lines are said to be perpendicular to one another.
26. An oblique angle is that which is made by two oblique lines; and is either less or greater than a right angle.
Of oblique angles, that which is less than a right angle, is called acute ; and that which is greater than a right angle, is called an obtuse angle.
27. Plane figures are bounded either by right lines or curves.
28. 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.
29. A figure of three sides (and consequently three angles) is called a triangle : and it receives particular denominations from the relations of its sides and angles.
When defined according to its sides, it is equilateral, isosceles, or scalene.
30. An equilateral triangle is that whose three sides are all equal.
31. An isosceles triangle is that which has two equal sides
32. A scalene triangle is that whose three sides are all unequal.
When defined according to its angles, it is either right- angled, obtuse-angled, or acute-angled.
33. A right-angled triangle is that which has one right angle.
All other triangles are oblique-angled, and are either obtuse or acute.
34. An obtuse-angled triangle has one obtuse angle.
36. A figure of four sides and angles is called a quadrangle, or a quadrilateral, or a trapezium.
37. 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.
38. A rectangle is a parallelogram, having a right angle.
39. A square is an equilateral rectangle; having its length and breadth equal, or all its sides equal, and all its angles equal
40. A rhomboid is an oblique-angled parallelogram.
41. A rhombus is an equilateral rhomboid ; having all its sides equal, but its angles oblique.
42. A trapezium is a quadrilateral which has not its opposite sides parallel.
43. A trapezoid has only one pair of opposite sides parallel.
44. A diagonal is a line joining any two opposite angles of a quadrilateral, or of any other right lined figure.
45. 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,
46. A pentagon is a polygon of five sides; a hexagon, of six sides; a heptagon, seven ; an octagon, eight; a nonagon, nine; a decagon, ten; an undecagon, eleven; and a dodecagon, or duodecagon, twelve sides.
47. A regular polygon has all its sides and all its angles equal. If they are not both equal, the polygon is irregular.
48. An equilateral triangle is also a regular figure of three sides, and the square is one of four : the former being also called a trigon, and the latter a tetragon.
49. Any figure is equilateral when all its sides are equal : and it is equiangular, when all its angles are equal. When both these are equal, it is called a regular figure.
50. By the distance of a point from a line is meant the shortest line that can be drawn from the point to the line. It is shown that this is the perpendicular. See th. 21.
51. When two or more lines are considered in relation to one another, they take different names, either parallel, oblique, perpendicular, or tangential.
52. Parallel lines are always at the same distance; and they never meet, though ever so far produced.
53. Oblique lines change their distance, and would meet, if produced on the side of the least distance.
54. 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.
55. 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.
56. The height or altitude of a figure is a perpendicular let fall from an angle, or its vertex, to the opposite side, called the base.
57. In a right-angled triangle, the side opposite the right angle is called the hypothenuse; and the other two sides are called the legs, and sometimes the base and perpendicular.
58. When an angle is denoted by three letters, of which one stands at the angular point, and the other two on the two sides, that which stands at the angular point is read in the middle : thus BAD signifies the angle contained by the lines BA and AD, and so of the other angles DAE and EAC.
59. For the purpose of calculation the circumference of every circle is supposed to be divided into 360 equal parts called degrees ; and each degree into 60 minutes, each minute into 60 seconds, and so on. Hence a semicircle contains 180 degrees, and a quadrant 90 degrees.
60. The measure of an angle, is an arc of any circle contained between the two lines which form that angle, the angular point being the centre; and it is estimated by the number of degrees contained in that arc, it being shown at prop. 4, that such a mode of admeasurement is consistent with the principles and truths of geometry.
61. Lines, or chords, are said to be equi-distant from the centre of a circle, when perpendiculars drawn to them from the centre are equal.
62. And the right line on which the greater perpendicular falls, is said to be farther from the centre.
63. An angle in a segment is that which is contained by two lines, drawn from any point in the arc of the segment, to the two extremities of that arc.
64. An angle on a segment, or an arc, is that which is contained by two lines, drawn from any point in the opposite or supplemental part of the circumference, to the extremities of the arc, and containing the arc between them.
65. An angle at the circumference, is that whose angular point or summit is any where in the circumference: and an angle at the centre, is that whose angular point is at the centre.
66. A right-lined figure is inscribed in a circle, or the circle circumscribes it, when all the angular points of the figure are in the circumference of the circle.
67. A right-lined figure circumscribes a circle, or the circle is inscribed in it, when all the sides of the figure touch the circumference of the circle.
68. One right-lined figure is inscribed in another, or the latter circumscribes the former, when all the angular points of the former are placed in the sides of the latter.
69. A secant is a line that cuts a circle, lying partly within, and partly without it.
70. Two triangles, or other right-lined figures, are said to be mutually equilateral, when all the sides of the one are equal to the corresponding sides of the other, each to each : and they are said to be mutually equiangular, when the angles of the one are respectively equal to those of the other.
71. Identical figures, are such as are both mutually equilateral and equiangular; or that have all the sides and all the angles of the one, respectively equal to all the sides and all the angles of the other, each to each ; so that if the one figure were applied to, or laid upon the other, all the sides of the one would exactly fall upon and cover all the sides of the other; the two becoming as it were but one and the same figure.
72. Similar figures, are those that have all the angles of the one equal to all the angles of the other, each to each, and the sides about the equal angles proportional.
73. The perimeter of a figure, is the sum of all its sides taken together.
74. The first part of geometry has respect to figures traced upon a plain superficies, or in which only two dimensions are concerned. It is called, therefore, the geometry of two dimensions ; and often also plane geometry, though general usage has restricted the term, plane geometry, to a particular class of such figures as may be traced upon a plane. These figures are entirely composed of straight lines and circles, which are therefore called geometrical lines ; whereas lines constructed any other way are called mechanical curves, or lines of the higher orders.
AXIOMS. Axioms are those fundamental truths, which, from their simplicity, are evident to
every mind, and are essential in a body of science as the foundation of a system of reasoning. 1. Things which are equal to the same thing are equal to each other. 2. When equals are added to equals, the wholes are equal. 3. When equals are taken from equals, the remainders are equal. 4. When equals are added to unequals, the wholes are unequal. 5. When equals are taken from unequals, the remainders are unequal.
6. Things which are double of the same thing, or equal things, are equal to each other.
7. Things which are halves of the same thing, are equal.
8. Every whole is equal to all its parts taken together, and greater than any of them.
9. Things which coincide, or fill the same space, are identical, or mutually equal in all their parts.
10. All right angles are equal to one another.
If two triangles have two sides and the included angle in the one, equal to two sides
and the included angle in the other, the triangles will be identical, or equal in all respects.
In the two triangles ABC, DEF, if the side AC be equal to the side DF, and the side BC equal to the side EF, and the angle C equal to the angle F; then will the two triangles be identical, or equal in all respects.
For conceive the triangle ABC to be applied to, or placed on, the triangle DEF, in such a manner that the point C may coincide with the point F, and the side AC with the side DF, which is equal to it.
Then, since the angle F is equal to the angle C (hyp.), the side BC will fall on
* In the complete process by which a theorem is enunciated and established, the following parts are to be always found.
I. The general enunciation, already explained, (see def. 6.) and which comprises
2. The predicate or affirmation made respecting the hypothetical figure. These are alike to be found in the general and particular enunciations.
II. Sometimes a preliminary construction is required to connect the hypothetical with the predicated parts of the figure.
III. One or more syllogisms, by which the necessary dependence of the predicate upon the hypothesis is established.
The syllogism is composed of three propositions,
2. The minor premise :—a proposition which agrees in subject with this axiom or theorem, and in predicate with the enunciated theorem.
3. The conclusion, which is the theorem in question, and inferred from the premises.