Measure a length AD = half the sum of the sides along one of the lines containing the angle.

At D draw a straight line at right angles to AD to meet AXY in Y, this gives the point Y.

With centre Y and distance YD draw the escribed circle.

With centre A and distance AF make another circular arc, this can be done, for the length of AF is given.

Draw a common tangent to these two circles between them. This tangent is the required base BC (66).

(2) To find the triangle ABC when its altitude, sum of sides and base are given.

Draw the straight line AD

=

half the sum of the sides, from

D measure the base DE, this gives the point E.

Next take the proportion AE : BC :: AF : 2YD, we obtain the length of YD, draw YD at right angles to AD at D of the length thus found, then AC is a tangent to the circle, with centre Y and distance YD, drawn from the given point A.

As before make a circle centre A and distance AF, and draw BC a common tangent to the two circles.

WHEN a person stands before a window and looks upon objects without, the different points upon these objects are seen by the eye along straight lines converging to a point in the eye. Now if these straight lines, called projectors, intersect the plane of the window in certain points we can, by representing these intersections by marks made on the window, obtain an exact representation of the objects as seen by the eye. This vertical plane of the window is called the picture plane. The line representing the intersection of this plane with the horizontal plane under the feet of the observer (the ground plane) is the ground line, sometimes called the intersecting line or axis, denoted by XY.

When objects are far off they appear smaller than when nearer the picture plane, sets of parallel lines appear each to converge to a point, and lines drawn on the picture are not proportional to their actual lengths.

This method of representing objects is called perspective projection.

But now suppose the observer to move further away from the picture plane, along a straight line perpendicular to it, and yet to have the power of clearly seeing the objects beyond it, the distances of which from the picture are small as compared with the distance of the eye from it.

All projectors will now be perpendicular to the picture plane, or nearly so; the lines on the picture when parallel to

one another will be proportional to the actual lengths of the lines in the object, and the heights of all points above the ground line may be accurately measured.

The projectors are taken as perpendicular to the vertical plane.

This is called Orthographic Projection.

The picture so obtained is called an elevation of the object on the Vertical Plane.

By taking another ground line we obtain another elevation. The heights of all points are still the same as in the first elevation but their positions with respect to one another are changed.

The elevation of a solid determines the heights of all points on it above the ground plane, and its breadth measured parallel to XY.

It is clear however that an elevation of an object is not all that is required.

Suppose however the eye of the observer were placed up above the object, and a view of it obtained at a considerable distance. Now we can find the lengths and breadths but not the heights of the objects. This observation is called the plan of the objects. Its outline is made by drawing perpendiculars from every point of the object to the horizontal ground plane. These perpendiculars are also called projectors.

Take a rectangular solid ABCGEDHF, for example a brick hung up as represented in fig. 163.

The plan of this solid is abhf and its elevation a'f'ég'. A and B have the same elevation, namely a', H and F have the same elevation f', and so on; while A and G have the same plan, namely a, and so on for other pairs of points.

But A is distinguished from B or G for A has not the same plan as B, or the same elevation as G.

It is found more convenient not to represent the plan and elevation as in the first figure but to revolve the vertical plane (or V.P.) backwards in the direction of the arrow till it is in the same plane with the horizontal plane (or H.P.).

The drawing is then shewn as in the second figure (163).