The Elements of Perspective / arranged for the use of schools and intended to be read in connection with the first three books of Euclid

by John Ruskin

Fig. 50.

As the inclined plane is given, one of its steepest lines must be given, or may be ascertained.

Let A B, [Fig. 50.], be a portion of a steepest line in the [p62] ]given plane, and V the vanishing-point of its relative horizontal.

Through V draw the vertical G F upwards and downwards.

From A set off any portion of the relative horizontal A C, and on A C describe a semicircle in a vertical plane, A D C, cutting A B in E.

Join E C, and produce it to cut G F in F.

Then F is the vanishing-point required.

For, because A E C is an angle in a semicircle, it is a right angle; and therefore the line E F is at right angles to the line A B; and similarly all lines drawn to F, and therefore parallel to E F, are at right angles with any line which cuts them, drawn to the vanishing-point of A B.

And because the semicircle A D C is in a vertical plane, and its diameter A C is at right angles to the horizontal lines traversing the surface of the inclined plane, the line E C, being in this semicircle, is also at right angles to such traversing lines. And therefore the line E C, being at right angles to the steepest lines in the plane, and to the horizontal lines in it, is perpendicular to its surface.