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expeditious of which is, holding the pencil between the thumb and fore finger, pressing the nail of the fore finger hard upon the slate or paper, and then turning the slate round. But as this exercises the judgement but little, and the eye still less, it should only be allowed when despatch is required.

A radius is a right line drawn from the centre of a circle to any part of its circumference. (fig. 1.)

A diameter of a circle is a right line drawn from one side of the circumference through the centre to the opposite side. (fig. 2.)

An arc of a circle is any portion of its circumference. Thus in figure 3, that part of the circumference between

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A and B is an arc and a right line drawn from one end of an arc to the other is called a cord. In fig. 3, the cord is represented by the dotted line.

The monitor will mark a centre, and draw a radius for the two following questions.

2. Make a circle round a given centre. (fig. 1.)

3. Describe a circle round a given radius. (fig. 1.) 4. Cut a circle by two perpendicular diameters.(fig.4.) 5. Cut a circle into eight equal parts. (fig. 4.) To do this, cut the circle into four parts, as in propsition 5, and then halve the quarters,

[blocks in formation]

6. Describe three concentrick circles. (fig. 5.) In fig. 5, all the circles have the same centre.

7. Describe three concentrick circles equidistant from each other.

8. Draw two concentrick circles, the diameter of one being three times that of the other.

9. Draw an arc of a circle, and mark its centre with a dot.

The centre of an arc, is the centre of the circle of which the arc is a part.

10. Draw an arc of a given radius. (fig. 6.)

It is easier to describe a whole circle than merely an arc of it. By placing the dividers on the centre, the monitor will easily test the correctness of the arc.

11. Cut an arc into two equal parts. (fig. 6.)

12. Cut an are into three equal parts. (fig. 7.)

13. Cut an arc into three smaller ones, and draw the cord of each.

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14. Describe a circle, and draw a tangent to it. (fig. 8.)

Tangent comes from a latin word, which means to touch. A tangent is a right line which touches a circle, but does not cut any of it off. If a right line be drawn from the centre of the circle or arc to the point of contact (that is, the point where the tangent touches the circle) the two right lines, (that is, the radius and the tangent) will be perpendicular to each other. (fig. 9.)

The monitor may test this with his quadrant of pasteboard; or, marking two places on the tangent at equal distances from the point of contact, he may see with his dividers if these points are at equal distances from the centre.

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15. Draw four tangents to a circle, forming a quadrilateral or four sided figure.

This figure need not form a perfect square, as in fig

ure 10.

16. Circumscribe or surround a circle with a square. (fig. 10.)

When the four tangents make right angles with each other, the figure is a square. In other cases, any direction may be given to two of the tangents.

In figure 10, we say the circle is circumscribed by the square, or the circle inscribed in a square.

17. Inscribe a square in a circle. (fig. 11.)

When a polygon has all the points of its angles touching a circle, it is said to be inscribed in a circle, and the circle circumscribes the polygon.

18. Double an arc of a circle. (fig. 6.)

This is more difficult than Prop. 11. First draw an arc and mark its centre, then prolong the arc to two, three, &c. times its former size.

19. Draw a tangent to a circle from a given point outside. (fig. 8.)

20. Draw two tangents to a circle from a given point. (fig. 12.)

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Observe that in drawing a tangent to a circle in problem 14, any part of the circle may be taken, but when a tangent is drawn from a given point, it can hit but two points as in fig. 12.

21. Cut a circle into six equal parts, (or, in other words, inscribe a regular hexagon in a circle.) (fig.13.)

The radius of any circle is equal to one side of the hexagon to be inscribed in it. The monitor, therefore,

may measure the radius with his dividers, and then apply them to each side of the hexagon. In other words, the cord of an arc, which is the sixth part of a circle, is equal to a radius or half diameter, (usually called a semi-diameter.)

22. Cut a circle into three equal parts, and inscribe an equilateral triangle. (fig. 13.)

After the hexagon is correctly drawn by problem 21st, it is easy to inscribe the triangle required in this, by drawing a cord between two points of the hexagon.

If cords be then drawn between the three remaining points of the hexagon, another triangle will be formed, whose base will be opposite the base of the other triangle, forming a beautiful figure resembling a star.

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23. Make two unequal circles tangent outside. (fig. 14.)

Unequal circles are circles of unequal size only.

24. Make two unequal circles tangent inside. (fig.15.)

25. The centres and the point of contact being given, perform problems 23 and 24.

The monitor will mark the centres, &c. When the circles touch either within or without, the point of contact and the two centres will be in a right line, and these may be tested with a rule.

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