33. Draw obliques equidistant (that is, equally distant) from a perpendicular.

Draw first a horizontal, raise a perpendicular on its centre, and then draw a line from the top of the perpendicular to each end of the horizontal. The figure will then be an isoceles triangle, as in fig. 11.

34. Make a scalene triangle. (fig. 13.)

As it is not difficult to make a triangle of unequal sides, it will be well for the monitor to prescribe the length of one or more of them. Thus, he may say : "Make a scalene triangle, of which the three sides shall measure two inches, one inch, and a half inch."

35. Make an equilateral triangle. (fig. 20.)

After the pupil makes the figure exactly, let the length of the sides be given, as one, two, three, &c. inches. Then require the point to be under the base, turned to the right, &c.

36. From a given point draw a perpendicular.

First draw a right line, then make the proposed point, and lastly draw the perpendicular.

37. Raise a perpendicular on the end of a right line. (22)

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38. Make a Rhomb or Lozenge. (fig. 21.)

The four sides are equal as in the square, but the angles are not right angles. To draw this figure, make a right line, cross it with a perpendicular, like the dotted lines in the figure, and then draw the sides.

If the Rhomb or Lozenge have all the angles equal, the figure is merely a square placed obliquely, as in fig. 22.

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39. Cut a rectangle into halves. (fig. 23.)

This will make two angles, whose exactness may be tested by an eighth part of a circle of pasteboard, the rectangle being quarter of a circle, as was stated under Prop. 24.

40. Cut an acute angle into two equal parts. (fig.24) 41. Double an angle.

then make another Suppose the lower then by making the

Make an angle of any size, and of the same size by the side of it. angle of fig. 24 to be made first, upper right line, the angle will be doubled.

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42. Triple an angle. (fig. 25.)

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43. Cut an angle into three equal parts. (fig. 25.) 44. Cut an angle into six equal parts. (fig. 26.) These three propositions need no explanation.

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SECOND CLASS.

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1. Make two angles of perpendicular sides.

After having made one angle, the pupil will draw a perpendicular to one of the sides, and then a perpendicular to the other side, until the perpendiculars cross each other.

One of the angles is acute, and the other obtuse; and if you lengthen one of the perpendiculars, a new angle will be formed exactly like the angle first made, as the dotted continuation of the perpendicular in fig. 1 shows.

2. Make two triangles of perpendicular sides. (fig.2.) Make one triangle, and then draw a perpendicular to each side, until the perpendiculars touch and form angles. Each side of each triangle must be perpendicular to some side of the other triangle.

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3. Make a trapezium. (fig. 3.)

A Trapezium has four sides, of which, two, called the bases, are parallel. In the figure, these are the upper and lower sides. The height is a perpendicular from base to base. As this figure is easily made, the length of the bases and the height may be given: thus, "Make a trapezium whose height shall be one inch, and whose bases shall be an inch and a half, and two inches."

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4. Make a six sided polygon of unequal sides. (fig.4.) The word polygon means many-angled. To make a polygon, the best method is, first to place dots at the angles, and then draw right lines from dot to dot.

5. Make a five sided polygon of unequal sides. (fig.5.) 6. Make two polygons of unequal but parallel sides. (figs. 5 and 6.)

7. Make a six sided polygon of equal sides.

8. Make a five sided polygon of equal sides.

A polygon of equal sides is called a regular polygon. 9. From one point of a polygon draw diagonals, and then draw a parallel polygon. (fig. 7.)

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After drawing a polygon, from either of the points carry right lines to all the rest, thus making several triangles. These lines are called diagonals. Then you have only to draw parallels to the several sides from diagonal to diagonal.

To vary the exercise, let the pupil draw a polygon outside of that first drawn. He will then only have to lengthen the diagonals.

10. Draw a polygon, and from a central point draw diagonals, then draw a parallel polygon within and outside of the first drawn. (fig. 8.)

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11. Make a triangular pyramid. (fig. 9.) First draw the triangle which forms the base, then place a dot for the point or apex, and draw right lines from the point to each angle of the base.

12. Draw a quadrangular (or four angled) pyramid, (fig. 10,) then cut it by a plane parallel to its base.

The process is the same as in the preceding figure. The plane, or parallel to the base, must be the last thing done. The height of a pyramid is a perpendicular dropped from the apex or summit to the base. The pupil must be careful to distinguish the front lines from the back lines of the figure.

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13. Make a six sided pyramid. (fig. 11.)

14. Make a five sided pyramid. (fig. 12.)

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15. Make a five sided pyramid, and cut it by a plane parallel to the base. (fig. 12.)