19. A Curvilineal Angle is that whose legs are curved lines. 20. A Mixtilineal Angle is that which is contained by a right line and a curve. When an angle is mentioned simply, it means a rectilineal angle. 21*. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which. stands on the other is called a perpendicular to it. Beginners are apt to suppose that a perpendicular must be a vertical, or plumb line, which it need not be. 22*. An Obtuse Angle is that which is greater than a right angle. 23*. An Acute Angle is that which is less than a right angle. The length of the legs, or the lines forming any angle, does not affect the quantity of that angle, since the inclination remains unaltered, whatever be the length of the legs. 24*. A Term or Boundary is the extremity of any thing. 25*. A Figure is that which is enclosed by one or more boundaries. 26. A Plane figure is that which is described upon a plane. 27*. A Circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another. 28*. And this point is called the Centre of the circle. 29. A line drawn from the centre to the circumference of a circle, is called a Radius. 30*. A Diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. 31*. A Semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter. 32. An Arc of a circle is any part of the circumference. 33. A Chord is a right line joining the extremities of an arc. 34. A Sector is a part of a circle contained by an arc, and two radii drawn to its extremities. 35. A Segment is a part of a whole; thus, if a right line be cut into two or more parts, each part is a segment of the whole line. 36*. A Segment of a circle is the figure contained by a straight line, and the circumference it cuts off. By means of the circle, a standard of measurement for angles has been obtained. Among English mathematicians, the circumference of a circle is supposed to be divided into 360 equal parts, called degrees; an angle which is formed by two radii will therefore contain a certain number of those parts, and is said to be of so many degrees. A right angle thus contains 90 degrees, and half a right angle 45 degrees, written 90°. 45°. 37. A Quadrant is a sector, which contains 90 degrees, or whose radii are at right angles to each other, and, consequently, is a quarter of a circle. 38. The Complement of an arc or angle, is the difference between it and an arc or angle of 90 degrees; thus, the complement of an arc or angle of 30 degrees, is an arc or angle which contains 60 degrees; 20 degrees are the complement of 70 degrees, and so on. 39. The Supplement of an arc or angle is the difference between it and 180 degrees; thus the supplement of 40 degrees is an arc or angle of 140 degrees. 40*. Rectilineal figures are those which are contained by straight lines. 41*. Trilateral figures, or triangles, by three straight lines. 42*. Quadilateral, by four straight lines. 43*. Multilateral figures, or polygons, by more than four straight lines. Note.-The sides of a trilateral figure must of necessity form three angles; those of a quadrilateral, four; and of a multilateral, as many angles as there are sides to the polygon. The three angles of any triangle are together equal to two right angles, or 180 degrees. 44*. Of three-sided figures, an Equilateral Triangle is that which has three equal sides. Its three angles are also equal to one another, and each contains 60 degrees. 45*. An Isosceles Triangle is that which has only two sides equal. Two of its angles are equal, namely, those opposite to the equal sides. 46*. A Scalene Triangle is that which has three unequal sides. All its angles are unequal. 47*. A Right-angled Triangle is that which has a right angle. It may be either isosceles or scalene, without its properties as a right-angled triangle being affected. The side opposite the right angle is called the hypothenuse, one of the other sides is called the base, and the remaining side the perpendicular. 48. Any triangle which has not a right angle is called an Oblique-angled Triangle. 49*. An Obtuse-angled Triangle is that which has an obtuse angle. The two remaining angles will necessarily be acute. 50*. An Acute-angled Triangle is that which has three acute angles. 51*. Of four-sided figures, a Square is that which has all its sides equal, and all its angles right angles. 52*. An Oblong is that which has all its angles right angles, but has not all its sides equal. Its opposite sides are necessarily equal. 53*. A Rhombus is that which has all its sides equal, but its angles are not right angles. Its opposite angles are necessarily equal. 54*. A Rhomboid is that which has its opposite sides equal to one another, but all its sides are not equal, nor its angles right angles. Its opposite angles are necessarily equal. 55*. Parallel straight lines are such as are in the same plane, and which being produced ever so far both ways, do not meet. 9 ク 56*. A Parallelogram is a four-sided figure, of which the opposite sides are parallel; and the diameter is the straight line joining two of its opposite angles. Vide Prop. XXXIV. Book I. of Euclid. 57*. A Rectangle is a right-angled parallelogram. 58*. All other four-sided figures besides these are called Trapeziums. Modern Geometers have distinguished all the other figures referred to here by the two following names : 59. A Trapezium is a quadrilateral which has not either pair of its opposite sides parallel. 60. A Trapezoid is a quadrilateral which has only one pair of its opposite sides parallel. 61. A Diagonal is a right line drawn from one vertex to another of a rectilineal figure: such figure may, by diagonals, be divided into as many triangles, minus two, as it has sides. Curved Lines are parallel when in the same plane, and kept at a given radiating distance throughout. 62. Parallel Planes are such as are at any given perpendicular distance throughout. The polished surfaces of perfect plate glass are parallel planes. 63. Oblique Planes are such as are, in relation to each other, neither parallel nor at right angles. Polygons are named after the number of their sides. A Trigon has three sides; a Tetragon, four sides; a Pentagon, five; a Hexagon, six; a Heptagon, seven; an Octagon, eight; a Nonagon, nine; a Decagon, ten; an Undecagon, eleven; and a Dodecagon, or Duodecagon, twelve sides. etc. Others are named, as polygons of thirteen or fourteen sides, 64. A Regular polygon has all its sides, and all its angles equal; if this is not the case, the polygon is Irregular. 65. The angle at the Circumference of a polygon is that which is contained by any two sides. 66. The angle at the Centre is that contained by two radii, drawn from the centre of the polygon to the extremities of one side. 67. The Exterior angle is that which is contained by one side and its adjacent side produced. 68. If two sides of a polygon contain an angle, whose vertex projects towards the interior of the figure, it is |