EXPLANATION.-1. The right line upon which a rectilineal or other figure is supposed to stand, is called its base.

2. Such right line is not always a horizontal line.

3. One of the sides of a triangle may sometimes be used as a base.

EXERCISE.-Example V, in various sizes.

QUESTIONS Should be asked by the teacher upon the subjects contained in this chapter.

CHAPTER X.

RECTILINEAL FIGURES.

Polygons.

Quadrilaterals and

DEFINITION." Quadrilateral figures are those which are contained by four straight lines."

Multilateral figures or polygons are those which are contained by more than four straight lines.

Contained may be explained as bounded.

EXAMPLE I. To draw a SQUARE and

its diagonal.

DIAGRAM, No. XV.

INSTRUCTION.-1. Draw a horizontal line;

2, And at each end thereof, let fall a perpendicular of the same length as the horizontal,-beginning on the left side.

3. Join the extremities of the perpendiculars.

EXPLANATION.-If this is done correctly, the four sides of the figure will be equal. The figure will then be called a square.

For the DIAGONAL.-Within the figure, draw a line slanting from the top of the right side of the square to the bottom of the left.

This line, which is called the diameter or the diagonal will bisect the figure;—that is, it will divide it into two equal triangles.

EXERCISE.-The figure in various sizes. The same with respect to Examples II. and III.

DEFINITION.-1. "A square is a quadrilateral or four-sided figure, having all its sides equal, and all its angles right-angles."

2. "The diameter or diagonal of a four-sided figure is the straight line joining two of its opposite angles."

D

EXAMPLE II.-To subdivide a square.
DIAGRAM, No. XVI.

INSTRUCTION.-1. Describe a square, as directed

in the last exercise.

2. Draw a diameter or diagonal from left to right. 3. Draw another diagonal from right to left.

4. Through the point where these diagonals cross, or cut one another, draw a straight line, parallel with either of the horizontal sizes, to cut the perpendicular sides of the square.

5. Again, through the same point, draw a straight line, parallel with either of the perpendicular sides of the square, to cut the horizontals.

EXPLANATION.-These lines, if correctly drawn, will divide the figure into four small squares. The diagonal will divide it into four isoceles triangles, and will also bisect the small squares.

Many QUESTIONS may be asked on the angles and triangles within this square. To vary the exercise, the oblong may be introduced.

EXAMPLE III.

INSTRUCTION.-1. Draw a horizontal line.

2. Let fall a perpendicular from each end thereof,making the length of the perpendicular less than that of the horizontal.

3. Join the ends of the horizontal.

An oblong will be formed.

Diagonals may be introduced at discretion.

DEFINITION.-1. “An oblong is that which has all its angles right-angles, but has not all its sides equal."

2. "A parallelogram is a four-sided figure, of which the opposite sides are parallel: and the diameter or diagonal is the straight line joining two of its opposite angles."

INSTRUCTION.-1. Draw a horizontal line.

2. At each extremity, draw a line of the same length as the horizontal, slanting towards the left. Care must be taken that the lines slant equally in the same direction; that is, that they are parallel.

3. Join the extremities of the slanting lines.

NOTE.-The figure must have the four sides equal, although they are made to slant.

4. Draw a diagonal from the upper angle on the left of the figure to the lower angle on the right.

DEFINITION.- "A rhombus has all its sides equal, but its angles are not right-angles.

EXAMPLE V.-To draw a

rhombus. Fig. 2.

DIAGRAM, No. XVIII.

INSTRUCTION.-1. Draw a horizontal line.

2. From each end thereof, draw a line of the same length as the horizontal, slanting towards the right;— beginning with the left side,-and making the slanting line parallel.

3. Join the opposite angles of the slanting sides.

A rhombus will be formed.

EXERCISE.-1. The present figure in various

sizes.

2. One or two examples of Fig. 1 may be introduced with this figure.

3. Contrast the angles of the rhombus in this exercise with those in the last, and shew wherein they differ.

EXAMPLE V.-To draw a

rhomboid.

DIAGRAM, No. XIX.

This figure is drawn as the rhombus-except that the horizontal parallels are to be made longer than the slanting sides.