27. Other triangles are oblique-angled, and are either acute or obtuse.

28. An obtuse-angled triangle has one ob

tuse angle.

29. An acute-angled triangle has its three angles acute. 30. A figure of four sides and angles is called a quadrangle, or a quadrilateral.

31. 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, and rhomboid. 32. A rectangle is a parallelogram, having

a right angle.

33. A square is an equilateral rectangle, having its length and breadth equal.

34. A rhomboid is an oblique-angled parallelogram, whose opposite sides are equal.

35. A rhombus is a parallelogram, having all its sides equal, but its angles oblique.

36. A trapezium is a quadrilateral, which has neither two of its opposite sides parallel.

37. A trapezoid has only one pair of its opposite sides parallel.

38. A diagonal is a line joining any two opposite angles of a quadrilateral.

39. 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,

40. A pentagon is a polygon of five sides; a hexagon, of six sides; a heptagon, of seven; an octagon, of eight; a nonagon, of nine; a decagon, of ten; an undecagon, of eleven; and a dodecagon, of twelve sides.

41. A regular polygon has all its sides and all its angles equal. If they are not both equal, the polygon is irregular.

42. 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.

43. 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 a regular figure.

44. A circle is a plane figure, bounded by a curve line, called the circumference, which is everywhere equidistant from a certain point, called its centre. The circumference itself is often called a circle, and also the periphery.

45. The radius of a circle is a line drawn from the centre to the circumference, as A B.

A

H

D

B

E

F

G

46. The diameter of a circle is a line drawn through the centre and terminating at the circumference on both sides, as A C.

47. An arc of a circle is any part of the circumference, as A D.

48. A chord is a right line joining the extremities of an arc, as EF.

49. A segment is any part of a circle bounded by an arc and its chord, as E F G.

50. A semicircle is half the circle, or segment cut off by a diameter. The half circumference is sometimes called the semicircle, as A G C.

51. A sector is any part of the circle bounded by an arc and two radii drawn to its extremities, as A B H.

52. A quadrant, or quarter of a circle, is a sector, having a quarter of its circumference for its arc, and its two radii are perpendicular to each other. A quarter of the circumference is sometimes called a quadrant, as

A B is the base, BC the perpendicular, and AC the hypothe

nuse.

55. 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, the angle contained by the lines BA and AD is called the angle BAD, or DA B.

D

B

56. 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.

NOTE.

GEOMETRICAL PROBLEMS.

PROBLEM I.

To divide a line, A B, into two equal parts.

It would be useful for the pupil to be furnished with a pair of dividers and a rule, and to be required to draw the diagrams here given.

Set one foot of the dividers in A, and, opening them beyond the middle of the line, describe arches above and below the line; with the same extent of the dividers, set one foot in the point B, and describe arches crossing the former; draw a line from the intersection above the line

A

-B

to the intersection below the line, and it will divide the line AB into two equal parts.

PROBLEM II.

To erect a perpendicular on the point C, in a given line.

Set one foot of the dividers in the given point C, extend the other foot to any distance at pleasure, as to D, and with that extent make the marks D and E. With the dividers, one foot in D, at any extent above half the distance D E, describe an arch above the line, and with the same extent, and one foot in E, de

D

-E

scribe an arch crossing the former; draw a line from the intersection of the arches to the given point C, which will be perpendicular to the given line in the point C.

PROBLEM III.

To erect a perpendicular upon the end of a line.

D

E

Set one foot of the dividers in the given point B, open them to any convenient distance, and describe the arch CD E; set one foot in C, and with the same extent cross the arch at D; with the same extent cross the arch again from D to E; then with one foot of the dividers in D, and, with any extent above the half of D E, describe an arch a; take the dividers from D, and, keeping them at the same extent, with one foot in E, intersect the former arch a in a; from thence draw a line to the point B, which will be a perpendicular to AB.

PROBLEM IV.

A

C

B

From a given point, a, to let fall a perpendicular to a given line

А В. dividers in

Set one foot of the the point a, extend the other so as to reach beyond the line AB, and describe an arch to cut the line AB in C and D; put one foot of the dividers in C, and, with any extent above half CD, describe an arch b; keeping the dividers at the same extent, put one foot in D, and intersect the arch b in b; through

which intersection, and the point a, draw a E, the perpendicular required.

PROBLEM V.

To draw a line parallel to a given line A B.

Set one foot of the dividers in E any part of the line, as at c; extend the dividers at pleasure, unless a distance be assigned, and describe an arch b; with the same

F

A.

B

C

e

extent in some other part of the line A B, as at e, describe the arch a; lay a rule to the extremities of the arches, and draw the line EF, which will be parallel to the line A B.

PROBLEM VI.

To make a triangle whose sides shall be equal to three given lines, any two of which are longer than the third.

Let ABC be the three given lines; draw a line, A B, at pleasure; take the line C in the dividers, set one foot in A, and with the other make a mark at B; then take the given line B in the dividers, and, setting one foot in A, draw the arch C; then take the line A in the dividers, and, setting one foot in B, intersect the arch C in C; lastly, draw the lines A C and BC, and the triangle will be completed.

PROBLEM VII.

A

A

B
C

D

B

To make a square whose sides shall be equal to a given line. Let A be the given line; draw a line, A B, equal to the given line; from B raise a perpendicular to C, equal to AB; with the same extent, set one foot in C, and describe the arch D; also, with the same extent, set one foot in A, and intersect the arch D; lastly, draw the line AD and CD, and the square will be completed.

A

B

In like manner may a parallelogram be constructed, only attending to the difference between the length and breadth.

PROBLEM VIII.

To describe a circle, which shall pass through any three given points, not in a straight line.

Let the three given points be A B C, through which the circle is to pass. Join the points A B and BC with right lines, and bisect these lines; the point D, where the bisecting lines cross each other, will be the centre of the circle required. Therefore, place one foot of the dividers in D, extending the other to either of the given points, and

B

A