Sol. Let M, N, P and Q be the given sides, and O the angle included by M and N. Draw AB=M. At A make an angle CAB = 0, and take AC=N. From the point Cas a centre, with a radius equal to P, and from the point B as a centre, with a radius = Q, describe 2 arcs which will intersect at D. Draw DC and DB. ABCD is the quadrilateral required.

CONSTRUCTION OF POLYGONS IN GENERAL.

87. To construct a hexagon which shall be equal to a given hexagon ABCDEF, (fig. 76.)

Sol. Divide the hexagon by diagonals into 4 triangles. Construct 4 triangles equal to those of the given hexagon and placed together in a similar order. The entire polygon GHMLKI thus constructed will be equal to ABCDEF.

In a similar manner polygons may be constructed, which shall be equal to a given polygon, whatever may be the number of its sides.

88. To construct a regular polygon in and about a

circle.

Sol. Suppose the required polygon is an octagon. With the compasses divide the circumference of the circle into 8 equal parts. Connect the 8 division-points by chords; and at the same points draw tangents to the circle. In this manner one regular octagon will be constructed within, and another without the circle.

The problem can be solved by another mode. Construct a square in a circle. Bisect each of the sides of this square. Draw radii through these division points. Connect the ends of these radii by chords with the two nearest vertices of the square. The required octagon will thus be constructed.

In a similar manner, by means of an inscribed equilateral triangle, we may construct a regular 6, 12, 24, &c. sided polygon in a circle, and by drawing tangents at the points where the vertices of the angles of such figures touch the circumference, we may construct a polygon of an equal number of sides about a circle.

CONSTRUCTION OF CIRCLES.

89. To describe a circle about a triangle.

Sol. (Fig. 63. 1.) Bisect 2 sides of the triangle, and at each division-point erect perpendiculars, which will intersect each other at O. From O as a centre, with a radius equal to the distance from the point O to the vertex of one of the angles of the triangle, describe a circle. The circumference of this circle will pass through the vertices of all the angles of the triangle; it will therefore be the circle required.

Remark. If the triangle is right-angled, the centre of the circle will be in the middle of the hypothenuse; if the triangle is acute-angled, this centre will be within the triangle, and if it be obtuse-angled, it will be without the triangle.

90. To describe a circle in a given triangle ABC, (fig. 63. 2.)

Sol. Bisect the angles A and B by straight lines, which will intersect each other at O. From the point O let fall perpendiculars upon the 3 sides of the triangle. From O as a centre, with a radius equal to either of these perpendiculars, describe a circle. The circumference of this circle will touch the 3 sides of the triangle. It will therefore be the circle required.

91. To describe a circle in and about a given square. Sol. Draw 2 diagonals in the given square, and from the point where they intersect each other as a centre, with a radius equal to half a diagonal, describe a circle. The circumference of this circle will pass through the vertices of all the angles of the square, and thus we have a circle described about a square.

Again, from the point of intersection of the diagonals let fall a perpendicular upon one of the sides of the square; then, from the same point as a centre, with a radius equal to this perpendicular, describe a circle. It will be a circle inscribed in square.

92. To describe a circle in and about a regular polygon.

Sol. Bisect 2 adjacent sides of the polygon, and at the division points erect perpendiculars. From the point where these perpendiculars intersect each other as a centre, with a radius equal to one of the perpendiculars, describe a circle; it will be an inscribed circle.

Again, from this centre draw a line to the vertex of one of the angles of the polygon; and then with a radius equal to this line, describe a circle; it will be a circle circumscribed about the polygon.

CONSTRUCTION OF THE SKELETONS OF SOLIDS.

93. We have before made rude diagrams of the skeletons of the solid bodies. We are now prepared to

construct them more accurately with the aid of instruments. The solid the skeleton of which is to be conConstruct the

structed should be placed before us.

skeleton

1. Of the cube, by placing together 6 equal squares, as shown in fig. 5.

2. Of the triangular, quadrangular, pentagonal, and polygonal prisms, as shown in figs. 1, 2, 3, 4.

3. Of the cylinder, fig. 6. The upper and lower sides of the rectangle must each be of the same length as the circumference of each circle.

4. Of the triangular pyramid, fig. 77.

5. Of the polygonal pyramids, figs. 7, 8, 9, and 10. 6. Of the cone, fig. 11. The curved side of the triangle must be of equal length with the circumference of the circle.

7. Of the regular solids, as shown in figs. 77, 78, 79, 80, 5.

PART THIRD.

COMPARISON AND MENSURATION.

I. POINTS.

94. A point has no length, breadth, or thickness; it has in fact no extension; a point is not the smallest particle of a line. As a point has no extension it cannot be measured; one point is as large as another, or rather neither has any magnitude. The representation of a point on paper or on the board has a magnitude, else it would not be visible; but that which is represented has none. A point has only a position. Where a definite line, whether straight or curved, ends, there is a point. If two lines intersect, there is at the intersection a point, which lies in both lines. Place two points together, and the position of the one will not vary from the position of the other; they will have the same position, and will coincide.

95. If a point be moved, the path which it describes in moving will be a line. If the point moves forward in the same direction, it describes a straight line; if the direction be changed every moment, it describes a curved line; if the direction be changed only once, it describes one line, composed of two straight lines joined together, or a