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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.

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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A regular polygon has all its sides equal, and all its angles of an equal opening. When such a polygon is inscribed in a circle, the sides are cords of equal arcs, and the points cut the circle into equal parts.

26. Inscribe a regular octagon in a circle. (fig.16.)

Draw two diameters perpendicular to each other, then divide each quarter of the circle into halves by other diameters; then draw arcs from diameter to diameter.

27. Inscribe a regular pentagon in a circle. (fig. 17.)

It is difficult by the eye alone to divide the circumference into five equal parts, and the object of this problem is to exercise the pupils.

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28. Make a triangle, and circumscribe a circle. (fig. 18.)

First make a triangle, and then the object is to describe a circle which shall cut each of its three points. To do this, raise a perpendicular on the middle of one

of the sides, and then raise another on another side. These perpendiculars will cross each other, and the point of section (that is, the point where they cut each other) will be the centre of the circle required.

. In the figure, the dotted lines show the perpendiculars and centre.

29. Make a circle, and draw a tangent triangle. (fig. 19.)

Three tangents to a circle are easily made, but the monitor may increase the difficulty by giving directions to the tangent sides. Thus, let two sides be at right angles, obtuse or acute; let the triangle be equilateral, &c.

30. Draw a regular pentagon, and circumscribe it with a circle. (fig. 17.)

31. Draw a regular hexagon, and circumscribe it with a circle. (fig. 13.)

32. Draw a regular octagon, and circumscribe it with a circle. (fig. 16.)

In the former problems, the circle was made first, now the polygon.

33. Inscribe a circle in a triangle. (fig. 19.)

To find the centre of the circle, draw a line from the middle of either side of the triangle to the point opposite, then do the same by another side and its opposite point; the place where these two lines cross each other, will be the centre of the circle to be inscribed. See the dotted lines in fig. 19.

34. Make an arc which shall pass through two given points. (fig. 20.)