In this section a few of the more useful projections only are given, it being presumed that the reader is already acquainted with the elementary principles of the subject.

The following projections will prove useful, as leading up to an understanding of the system employed in drawing a solid body in any position and projecting its surfaces horizontally and vertically, or at any angle or on to any plane as may be required. In the setting out of cylindrical or other curved surfaces a knowledge of this important section is most essential, and a careful study of the methods employed will greatly assist the student in grasping the system of projecting the shapes of regular or irregular stones forming the component parts of Vaults, Domes, etc.

## Note

In all the following problems the line forming the junction of the vertical and horizontal planes is called the Line of Intersection, and is abridged in the description to "L. of I."

Problem - To project the elevation of a solid triangular prism one face of which forms an angle of 30 degrees with the vertical plane. Vertical height = e1f1

Let abcd (Fig. 82) be the plan of the prism, at an angle of 30degrees from the L. of I., and e1f1 the height above the horizontal plane, while eg is the plan of the apex line.

Through e, b, c, g, and d project lines to the line of intersection, those from e and g being prolonged to the proposed height of the prism at f1 and g2 respectively. Make g1g2 = e1j1 and join f1g2 Join f1b1, g2cl, and g2d1. The figure blflg2dl will be the required elevation.

Fig. 82.

The dotted line connecting f1a will be the remaining arris, in this case not visible.

## Problem

To project a pyramid having a square base, one of its sides being at an angle of 30 degrees with the L. of I. Height e1f(see Fig. 83).

Fig. 83.

Let abcd be the plan, with e as apex.

Project lines of construction from points b, c, and dto the L. of I.; and from e to a point f1 above the L. of I., making the height equal to the given height e1f.

Join f1 to b1 c1 and d1 thus completing the elevation of the required pyramid. A dotted line connecting f1 and a will give the position of the remaining edge.

## Problem

To project a right cylinder having a horizontal axis at an angle of 30 degrees from the L. of I., and at a given distance above the horizontal plane.

In this case the plan will form a rectangle, as abed (Fig. 84), 30 degrees from the L. of I.

Let abcd be the plan of the cylinder, placed at an angle of 30 degrees with the L. of I. Bisect bc and ad in e and f, and join ef. Then ef will be the plan of the axis of the cylinder.

Draw a line gh parallel to L. of I., and at the required height above it. Through b, e, c, d,f, and a project lines at right angles to the L. of I., intersecting gh in b1 e1 c1 d1f1 and a1 respectively. Make ejand f1k each equal to be. Join jk. Bisect je1 and kf in land m, and join these points. Produce Im both ways, meeting the vertical lines bb1 cc2, and dd1 in b2, c2, and d2 respectively. Make an ellipse jc2e1b2 with je1 as major axis, and b2c2 as minor axis; and a semi-ellipse kd2f1 (the other half, not visible, is shown dotted), thus completing the elevation of the cylinder.

## Problem

To find the plan of a cone whose axis is at 30 degrees from the horizontal and parallel with the vertical plane.

Fig. 84.

Let abc (Fig. 85) be the elevation of the cone, the axis ad forming an angle of 30 degrees with the line of intersection. (The elevation of the cone in this instance forms an isosceles triangle.)

Fig. 85.

Project a, b, d, and c on and beyond the L. of I. to the horizontal plane, making d1d2 - bc. Bisect d1d2 in e, and draw b1ea1 parallel to the L. of I. Make ef x b1e; and with d1d2, as major axis, and b1f as minor axis, construct the semi-ellipse d1fd2b1 Draw a1d1 and a1d2 tangentially to the ellipse, thus completing the plan. The hidden portion of the circular base of the cone is shown dotted.

It is as well to remember the following axioms -

The plan of a circle lying parallel to the horizontal will be a circle; its elevation will be a straight line. If inclined to the horizontal its plan will be an ellipse.

If vertical, it will be a straight line on plan.

A cylinder, if its axis is parallel with the horizontal, will always be a rectangle on plan; its elevation will be a rectangle if it be parallel with the vertical; if at an angle, the ends will form ellipses.

A pyramid standing on its base will be a right-lined figure on plan and a pyramid in elevation.

A cone with base parallel to the horizontal will be a circle on plan, in elevation a triangle.

A sphere is a circle from all points of view.