EXAMPLE II:-To draw a circle touching the extremities of lines which cross one another at right angles.

DIAGRAM, No. XXIII.

INSTRUCTION.-1. Draw a horizontal line.

2. Upon the middle of this line, erect a perpendicular, equal in height to half the horizontal.

3. Continue the perpendicular in a downward direction, through the horizontal, until its length below the horizontal equals its height above it.

4. The horizontal and perpendicular, by thus cutting one another, have the appearance of four lines, diverging from one point. These lines are all equal in length, but are in opposite directions.

5. Join the four extremities of these lines by means of circular curves;* commencing at the highest point of the perpendicular; first drawing the curves on the left side of the figure, and afterwards those on the right.

6. The exterior figure thus completed will be a circle.

7. The point where the lines cut one another will De the centre of the circle.

* Dots made at different distances will be found of use at the outset, to guide the pupils in making the curve. Here also it should be explained to the pupils that every part of the curve should be of the same distance from the point where the lines cross.

EXERCISE.-Let the pupils draw any of the parts of a circle which are described in the following defini

tions.

DEFINITIONS, which may be illustrated by the diagrams; and upon which suitable questions may be asked.

1. "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."

"And this point is called the centre of the circle."

(A pair of dividers or compasses will be useful in explaining this definition.)

2. "A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference."

3. "A semicircle is the figure contained by a diameter, and the part of the circumference cut off by the diameter;" or in other words, it is half a circle.

4. "A quadrant is the quarter of a circle."

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5. "A radius of a circle is a right line drawn from the centre to the circumference."

6. "An arc of a circle is any part of a circumference."

7. "A chord is a right line joining the extremities

of an arc."

8. "A segment is any part of a circle bounded by an arc and its chord."

9. "A sector is any part of a circle which is bounded by an arc, and two radii drawn to its extremities."

EXAMPLE III.-To draw a circle, without the use of lines at right angles.

DIAGRAM, NO, XXIV.

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INSTRUCTION 1.-At the point where two imaginary lines would cross one another, as in the last exercise, make a dot, for the centre of the circle.

2. Also dot the four extremities of the same imaginary lines.

3. Join the dots by circular curves, as in the preceding exercise.

4. Rub out the centre dot.

¶ A circle will be formed.

EXERCISE. This figure in various sizes.

EXAMPLE IV.-To draw a circle within a square, or to describe a square

about a circle.

DIAGRAM, No. XXV.

To draw a circle within a square.

INSTRUCTION.-1. Draw a square.

2. Draw a diagonal to the square, parallel to the base; so that its opposite vertical sides shall be cut exactly in the middle points.

3. Draw another diagonal, parallel to the sides of the square, and equidistant from them.

4. Join the extremities of the perpendicular and horizontal lines with circular curves, as in Example II.

¶ A circle is drawn within a square.

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To describe a square about a circle.

INSTRUCTION.-1. Draw a circle, touching the extremities of two lines, which cross one another in the middle at right angles;-as in Example II.

2. Draw a perpendicular line on each side of the circle; making it touch the circumference at those points where the horizontal diameter cuts it.

3. These lines must be parallel to the perpendicular within the circle.

4. Draw two other straight lines, parallel with the horizontal diameter;-taking care that these lines touch the circle at those points where the perpendicular diameter cuts it.

5. Rub out such parts of the exterior lines as are beyond the angles at which they meet.

¶ A square is described about a circle.

DEFINITION.

"A straight line is said to touch a circle, when it meets the circle, and being produced does not cut it."

A line in such position is sometimes called a Tangent.

EXAMPLE V.-To illustrate the rela

tion of angles to a circle.

DIAGRAM, No. XXVI.

INSTRUCTION.-1. Draw a figure in every respect similar to that which is described in Chapter X. Example VII.-Diagram, No. XXI.