24. A spherical triangle is any portion of the superficies of a sphere, contained within three great circles cutting each other.

XXIV. Of three sided figures, an equi

lateral triangle is that which has three

equal sides.

Its three angles are also equal to one another, and each contains 60 degrees.

The three angles of any triangle, are together equal to two right angles or 180 degrees.

XXV. An isosceles triangle is that which

has only two sides equal.

A

Two of its angles are equal, namely, those opposite to the equal sides.

XXVI. A scalene triangle is that which has three unequal sides.

All its angles are unequal.

XXVII. 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.

25. Any triangle which has not a right angle, is called an oblique angled triangle.

XXVIII. An obtuse-angled triangle is

that which has an obtuse angle.

The two remaining angles will necessarily be acute.

XXIX. An acute angled triangle is that which has three acute angles.

XXX. Of four-sided figures, a square is that which has all its sides equal, and all its angles right angles.

XXXI. An oblong is that which has all its angles right angles, but has not all its sides equal.

Its opposite sides are necessarily equal.

XXXII. A rhombus is that which has all its sides equal, but its angles are not right angles.

Its opposite angles are necessarily equal. XXXIII. 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.

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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.

The four preceding figures are parallelograms.

26. 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.

27. A Rectangle is a right-angled parallelogram.

XXXIV. 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 :

28. A trapezium is a quadrilateral which has not either pair of its opposite sides parallel.

B

Only two of its angles can be equal, or they may all be unequal.

29. A Trapezoid is a quadrilateral which has only one pair of its opposite sides parallel.

It may have each pair of angles at the extremities of each of its parallel sides equal, or all its angles may be unequal. XXXV. Parallel straight lines are such

as are in the same plane, and which, being produced ever so far both ways, do not meet.

Curved lines are also parallel, when in the same plane, and kept at a given radiating distance throughout.

30. Parallel Planes are such as are at any given perpendicular distance throughout.

The polished surfaces of perfect plate glass are parallel planes.

31. Oblique Planes are such as are, in relation to each other, neither parallel, nor at right angles.

32. A Polygon is a figure of many angles, and is contained by as many sides as angles. Every figure bounded by straight lines is a polygon. 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.

Others are named according to the number of their sides; as polygons of 13, or 14 sides, &c.

33. A Regular polygon has all its sides, and all its angles equal: if they are not both equal, the polygon is Irregular.

34. The angle at the Circumference of a polygon, is that which is contained by any two sides.

35. The angle at the Centre, is that contained by two radii, drawn from the centre of the polygon to the extremities of one side.

36. The Exterior angle, is that which is contained by one side and its adjacent side produced.

37. If two sides of a polygon contain an angle, whose vertex projects towards the interior of the figure, it is called a Re-entering angle. Those whose vertices project outwards, are Salient angles.

A Solid is that which hath length, breadth, and thickness. Vide Def. I. Book XI. of Euclid.

Surfaces, are the extremities of solids; lines, the extremities of surfaces; and points, the extremities of lines.

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38. The Periphery of a figure, is its circumference, when the figure is curvilineal.

39. The Perimeter of a figure, is the sum of all its sides; whether curved, rectilineal, or mixed.

40. The Area of a figure, denotes its superficial content.

41. The Base of a figure or drawing, is its lowest line, and is usually horizontal; but in theoretical geometry, a right line in any direction may be called a base, to establish upon it the demonstration of a truth.

42. A Subtense is a right line extended under an arc or angle; thus a chord is the subtense of an arc: each of the three angles of a triangle also may be said to be subtended by the side opposite to it.

The Altitude of any figure is the straight line drawn from its vertex perpendicular to the base. Vide Def. IV. Book VI. of Euclid.

43. The Vertex of an angle, is its angular point; that is, the point where the legs of the angle meet. That of a figure, the uppermost angular point above the base.

44. When two lines cross each other, the opposite angles they make are called Vertical angles; but if a line be drawn upon, and any where between the extremities of another line, the angles thus made are called Adjacent angles. Vide Def. X.

45. Concentric figures are such as have the same centre.

46. Eccentric, or Non-concentric figures, are such as have different centres; hence if one circle be within another, the circumferences not being parallel, they are called eccentric circles.

47. One right-lined figure is said to be Circumscribed about another, or the latter is Inscribed in the former; when all the vertices of the inner figure, touch the sides of the

outer one.

48. A right-lined figure is Inscribed in a circle, or the circle Circumscribes it, when all the vertices of the former, touch the circumference of the circle.

49. A right-lined figure Circumscribes a circle, or the circle is Inscribed in it, when all the sides of the former are tangents to the circle.

50. If one figure be circumscribed about another, it is also said to be Described about it.

51. If a triangle be inscribed in a circle, either of its angles is In one segment, and stands On the other segment by which the said angle is subtended.

52. If two radii be drawn in a circle, the angle contained by them is called the angle at the Centre of the circle; and if from the points where these radii touch the circumference, two lines be drawn to any other point in the circumference, the angle contained by them is called the angle at the Circumference; and this angle contains exactly half the number of degrees of the angle at the centre.

53. The Sum of lines or planes, is the quantity produced by addition. A line 3 inches long is equal to the sum of two other lines of 1 inch and 2 inches long; also a triangle whose area is 3 square inches, is equal to the sum of two other triangles, whose areas are 1 square inch and 2 square inches.

54. The Product of two lines is a rectangle, having one of the given lines for its base, and the other for its height.

55. A Multiple of a line or figure, is another line or figure that is exactly 2, 3, 4, or any other number of times as large as the first line or figure.

56. A Measure of a line or figure, is any line or figure, which, being applied to the first, would divide it into any number of parts, each equal to the said measure.

57. To Transform a figure, is to change it into another figure of the same superficial or solid content; thus, if a square be made equal in area to a given triangle, the first is said to be a Transformation of the second.