NOTE. In order to make the curves on the opposite sides of the perpendicular correspond exactly with one another, the pupils must be directed to see that the lengths of the horizontal lines which the curves cut off are equal on the opposite sides of the perpendicular.

EXERCISE.-1. The figure as in the Diagram,
2. The same figure, upon one line only as its base.

EXAMPLE V. To draw a spiral.

DIAGRAM, No. XXXIII.

INSTRUCTION.-1. Draw a horizontal and dicular, cutting one another at right angles.

perpen

2. Divide each of the upper and lower portions of the perpendicular and that portion of the horizontal which lies to the left of the perpendicular into three equal parts; marking the points at which they are so divided.

3. On the remaining portion of the horizontal, mark off, in the first place, a length equal to one and a half of the divisions in the other lines; and then let the remainder of the line be divided as all the others.

4. Commencing with the upper portion of the perpendicular, but at the lowest mark, and in the direction of the left side, draw a circular curve, joining

the point of commencement with the corresponding mark in the horizontal line.

5. Continue this curve to meet the first mark in the lower portion of the perpendicular.

6. Commencing at the second mark in the upper portion of the perpendicular, draw a corresponding curve to the second mark in the lower portion.

7. From the highest point in the perpendicular to the furthest point in the horizontal, draw another corresponding curve.

¶ One side of the spiral will be formed.

8. On the right of the perpendicular, commencing at the second mark in the upper portion, draw a circular curve, cutting through the mark in the horizontal line, and proceeding to the first mark in the lower portion of the perpendicular.

9. In a similar manner, draw the remaining curves on the right side.

The spiral will be complete.

EXERCISE.-1. The spiral as in the Example.

2. The spiral with a greater number of curves.

EXAMPLE VI.-To draw curves on

opposite sides of a slanting line.

DIAGRAM, No. XXXIV.

INSTRUCTION.-1. Draw a slanting line, and describe the curves as usual.

2. In this figure, the upper curve, which ought to be drawn first, should be divided at its highest point, but the remaining parts of the curve must be drawn throughout in a downward direction.

EXERCISES.-1. Similar curves

on the opposite sides of successive slanting lines; as in the diagram.

2. The same; changing the direction of the slanting line from left to right.

3. The same kind of curves, commenced on the under side of successive slanting lines.

XXXV.

DIAGRAM, No.

4. Corresponding curves on the opposite sides of successive parallels. DIAGRAM, No. XXXVI.

5. Corresponding curves in opposite directions within successive parallels. DIAGRAM, No. XXXVII.

INSTRUCTION.-1. Draw a line for the base, sloping a little, from the left towards the right.

2. Upon this line, draw a triangle, having the left side rather longer than the right.

¶ The front of the pyramid will be described.

The side of the figure.

3. Draw the base of another triangle, sloping rather more than that of the first; and let it meet the base of the front of the figure at the right extremity.

4. The right side of the triangle already constructed, will serve as the side of the new triangle.

5. From the apex of the triangle forming the front of the figure, draw a line to the extremity of the new base.

¶ The side of the pyramid is described.

OBSERVE.-1. The figure now described represents the solid four-sided or quadrangular pyramid, shewing only two of its sides.

To draw the same kind of pyramid complete,— that is, to represent the two sides and the base which are here necessarily concealed, draw dotted lines, as in the Example, to mark out the remaining parts of the figure.

The following DEFINITIONS will be best illustrated by means of SOLID FORMS.

*

DEFINITIONS.-1. "A SOLID is that which has length, breadth, and thickness."

2. "That which bounds a solid is a superficis."

3. "A pyramid is a solid whose sides are all triangles, meeting in a point at the centre, and the base any plane figure whatever."+

EXPLANATION.-1. In familiar language the superfices of any solid may be described as its surface; and the plane superfices, as its smooth or level surface.

2. A solid is sometimes described as that which excludes everything else from the place it occupies.

* Boxes of suitable forms may be obtained at almost any School Depository; but the best adapted, and the most useful, although not the cheapest, are sold by W. Parker, West Strand, and adapted to Mr. Butler Williams's Exercises. There are others, also very useful, sold by Taylor and Walton, Gower-street, London.

† Sabine's Practical Mathematician.