[p 24 ] PROBLEM V. TO DRAW A HORIZONTAL LINE, GIVEN IN POSITION AND MAGNITUDE, BY MEANS OF ITS SIGHT-MAGNITUDE AND DIVIDING-POINTS .

[Geometric diagram]
Fig.16.

Let AB (Fig.16.) be the given line.

Find the position of the point A in a.

Find the vanishing-point V, and most convenient dividing-point M, of the line AB.

Join aV.

Through a draw a horizontal line ab' and make ab' equal to the sight-magnitude of AB. Join b'M, cutting aV in b.

Then ab is the line required.

[p25]
COROLLARY I.

[Geometric diagram]
Fig.17.

Supposing it were now required to draw a line AC (Fig.17.) twice as long as AB, it is evident that the sight-magnitude ac' must be twice as long as the sight-magnitude ab'; we have, therefore, merely to continue the horizontal line ab', make b'c' equal to ab', join cM', cutting aV in c, and ac will be the line required. Similarly, if we have to draw a line AD, three times the length of AB, ad' must be three times the length of ab', and, joining d'M, ad will be the line required.

The student will observe that the nearer the portions cut off, bc, cd, etc., approach the point V, the smaller they become; and, whatever lengths may be added to the line AD, and successively cut off from aV, the line aV will never be cut off entirely, but the portions cut off will become infinitely small, and apparently “vanish” as they approach the point V; hence this point is called the “vanishing” point.

[p26]
COROLLARY II.

It is evident that if the line AD had been given originally, and we had been required to draw it, and divide it into three equal parts, we should have had only to divide its sight-magnitude, ad', into the three equal parts, ab', b'c', and c'd', and then, drawing to M from b' and c', the line ad would have been divided as required in b and c. And supposing the original line AD be divided irregularly into any number of parts, if the line ad' be divided into a similar number in the same proportions (by the construction given in AppendixI.), and, from these points of division, lines are drawn to M, they will divide the line ad in true perspective into a similar number of proportionate parts.

The horizontal line drawn through a, on which the sight-magnitudes are measured, is called the “Measuring-line.”

And the line ad, when properly divided in b and c, or any other required points, is said to be divided “IN PERSPECTIVE RATIO” to the divisions of the original line AD.

If the line aV is above the sight-line instead of beneath it, the measuring-line is to be drawn above also: and the lines b'M, c'M, etc., drawn down to the dividing-point. Turn Fig.17. upside down, and it will show the construction.