understood by reference to Fig. 80a, b, which represent the planes of projection in position with a solid standing in the HP of which we require to draw a front elevation and two end elevations. The projection of the front elevation F E needs no explanation. To obtain the elevation of end A, we suppose a vertical plane, M, placed on the other side of the solid, either touching the end B or removed from it, perpendicular to the lines of sight, and, therefore, in this case perpendicular to the V P. The view looking on the end A is then projected on this plane in the usual way, after which the plane is supposed to be turned upon its vertical trace, V T, as a hinge in the directior of the arrow away from the solid until it coincides with the VP, and the view from end A becomes projected upon the ordinary V P of projection on the right-hand side of the front elevation, as at A. For the elevation looking on end B a vertical plane, N, is placed on the further side of the solid, and then projected as before into the V P giving the view from end B upon the lefthand side of the front elevation. Notice that in each case all

points in the projection travel in arcs of circles having the V T of the planes as centre as the planes turn from their original position into the V P.

The plan and three elevations thus found are shown in Fig. 806 as they would be drawn upon the paper, and by the help of Fig. 80a should present no difficulty. With these two drawings before him the student should very carefully verify the following

statements:

(a) The end elevation looking upon the right-hand end of the solid is shown upon the left hand of the front elevation.

(b) The end elevation looking upon the left-hand end of the solid is shown upon the right hand of the front elevation.

(c) In an end elevation, that part of the drawing nearest to the front elevation represents the back portion of the solid, and that part of the drawing furthest from the front elevation represents the front portion of the solid.

These three conditions should be adhered to in all kinds of mechanical drawing whatsoever. It is remarkable what different customs prevail with regard to the position of the end elevation; many draughtsmen use either position without any regard to uniform methods, and as a result considerable confusion prevails. At a later stage the student may find it apparently most convenient to put an end elevation next the end which it represents, rather than at the opposite end; but it is certainly not accurate. The front elevation shows only height and length; the end elevation only height and width, and the plan only length and width. Hence we see that the three dimensions of a solid are shown in any two of the three views, and that if any two views are given the third can be obtained from them. For example, in Fig. 806, lines are drawn from the front elevation to the end elevation, to give its dimensions in a vertical direction, while its dimensions in the other directions, as a' b' and c' d', are obtained from the plan, and are equal respectively to the sizes marked a b and c d. The drawing of a view from others is a very important part of practical projection, and the student should notice that, although it may be desirable to work one or two problems by drawing the arcs marked 1, 2, 3, 4 in order to obtain an end elevation, it is better and quicker to adopt the method of taking the distance with the dividers direct from the plan.

Since the drawing of additional elevations requires the use of other vertical planes of projection, we see that other ground lines will be obtained where these planes intersect the HP. But this simply amounts to drawing a new ground line, and then obtaining a new elevation above this XY in the usual way, knowing that the heights above the ground line are the same as in the first elevation. This method is often referred to as an alteration of the ground line. By exactly similar methods we may obtain additional plans from the first elevation, for we may suppose other horizontal planes to be placed in different positions relatively to the first elevation making new ground lines with the V P. In order to distinguish the different ground lines, they are marked as X1 Y1, X2 Y2, X3 Y3, &c.

PROBLEM L. (Fig. 81).-Given a plan and elevation of a solid to obtain a second elevation on a given ground line X1 Y1 and a second plan on a given ground line X2 Y2.

The plan and elevation of a simple solid are shown at P and E,

it is required to draw a second elevation on X1 Y1 and a second plan on X2 Y2.

To obtain an elevation on X1 Y1 draw projectors through each point in the plan P perpendicular to X1 Y1, and mark off distances along each from X1 Y1 equal to the height of the point above the H P—that is, equal to the distance of the point above X Y in the first elevation E. The construction for one end is shown in the figure, the distance e2 d2 being equal to e1 d1, and f2 a2 equal to fla1, and so on for each point.

Some of the lines in this elevation will be dotted, and as this generally follows in the projection of inclined solids, it is convenient to always draw the parts of the solid shown by full lines before those parts shown by dotted lines-that is, first draw those parts nearest the point of observation, or furthest from the X Y.

To obtain a plan on X2 Y2, draw projectors perpendicular to X2 Y2 through each point of the first elevation Ê, and make the distance of each point in front of X2 Y2 equal to its distance in front of X Y, as, for example, the distance n d3 equals the distance e1 d, and n h3 = e1 h.

The construction of the last problem shows how the projection of solids in difficult positions may be simplified, for they can first be drawn in a simple position, and then by a suitable alteration of the ground line, they can be projected as required. For example, if we required an elevation of the block in the last problem, with its long edges inclined at 30° to the V P, we should first draw a plan and elevation, as at P and E, and then obtain a second elevation on a ground line X' Y', drawn at an

angle of 30° to the long edges of the solid in the plan P-that is, the angle would be 30°. But care must be taken to arrange the first plan and elevation so that they fulfil at least one of the required conditions; as, for example, suppose we require the plan and elevation of a hexagonal prism with its axis inclined 60° to the HP, and one face parallel to the V P, we should first draw a plan and elevation, with a base in the HP, and one edge of the base parallel to the V P-that is, to the X Y-thus giving a face parallel to V P, and satisfying one condition of the problem. The problem would then be completed by drawing a new ground line X2 Y2, making an angle of 60° with the axis of the solid in the elevation, and from this elevation obtain the required plan. If the solid had been first drawn without an edge of the base, parallel to the V P, no alteration of the XY would have satisfied the conditions.

By an extension of this principle we can often obtain the plans or elevations of inclined solids in a simpler and quicker manner than by the method described in Problems xlviii. and xlix. An example is given in the following problem, from which it will be seen that the construction results in the required plan and elevation being obtained in a better, and in the more usual, position on the paper, than with the method of Problem 1.

PROBLEM LI. (Fig. 82).—To draw the plan and elevation of a solid made up of a square prism and a cylindrical block, when the axis of the block is inclined to the ground, and the faces of the prism make equal angles with the vertical plane.

Draw the XY and a line s't' inclined to it at the required inclination, 0, of the axis to the ground. At any part of the line draw a square A B C D equal to the base of the prism, having a diagonal A C on the line, and from the centre of the square draw a circle equal in diameter to the circular block. The square and circle then represent a view of the solid looking from above it in the direction of the axis. Draw the elevation of the solid as shown, commencing with the base of the circular block.

(A little difficulty will be found in making e' f' equal to f'g', but this can be avoided by drawing the line at any convenient position on the axis, and then drawing another X Y to pass through the point e'.)

The two drawings now made may represent the plan and elevation of the solid when the ground line is X' Y. As the solid is not stated to be a given distance in front of the vertical plane, we may draw a line st to represent the plan of the axis at any convenient distance in front of the X Y. Then, to obtain the plan, proceed as follows:- -Draw a projector from a'c' to beyond the plan of the axis, the plan of a and c must be on this

line, and a distance apart equal to the distance AC in the first view drawn, or what is better still for general application, make

the distance of a and c on
either side of the plan of the
axis, equal to the distance
8' A or 8' C. The other points
in the plan of the square
prism can be obtained in the
same way.
To obtain the
plan of the circular block, we
must first stripe it as in
Problem xlix., Fig. 80,
therefore divide the circle
centre s' into a convenient
number of equal parts, and
mark as shown 1, 2, 3, 4,
5, . . . 8. Draw the eleva-
tion of the stripes on the
elevation of the block, and
mark. Then to obtain the
plan, draw projectors
through the elevation of the
points, as 2' 4', and, as
before, mark off a distance

along the projector from the point where it cuts the axis plan equal to the distance t 2 or t4 in the first view drawn-that is, p 2 and p 4 equal t 2 or t 4. The completion of the plan needs no further description.

EXAMPLES.

EX. 1.-Draw the trace of a plane parallel to and 24′′ above the HP, and determine the projections of a point in this plane 31" from the ground line. (S. & A. E., 1886.)

EX. 2.-Two points ab on the ground line are 21" apart. A point P is 2" from a, and 27" from 6, and 11′′ from the V P. Obtain its projections. (S. & A. Adv., 1886.)

EX. 3. Draw the plan of a square prism, height 23", side of base 11′′, a diagonal being vertical. (S. & A. Adv., 1891.)

(By diagonal is meant a diagonal of the solid; first draw in a simple position and then alter the X Y.)

EX. 4.-Draw the plan of a cube of 28" edge with a diagonal of the solid vertical.