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APPLICATION OF THE VANISHING SCALE TO DRAWING FIGURES AT AN ANGLE WHEN THEIR VANISHING POINTS ARE INACCESSIBLE OR OUTSIDE THE PICTURE

This is exemplified in the drawing of a fence (Fig. 84). Form scale _aS_, _bS_, in accordance with the height of the fence or wall to be depicted. Let _ao_ represent the direction or angle at which it is placed, draw _od_ to meet the scale at _d_, at _d_ raise vertical _dc_, which gives the height of the fence at _oo_. Draw lines _bo_, _eo_, _ao_, &c., and it will be found that all these lines if produced will meet at the same point on the horizon. To divide the fence into s.p.a.ces, divide base line _af_ as required and proceed as already shown.

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THE REDUCED DISTANCE. HOW TO PROCEED WHEN THE POINT OF DISTANCE IS INACCESSIBLE

It has already been shown that too near a point of distance is objectionable on account of the distortion and disproportion resulting from it. At the same time, the long distance-point must be some way out of the picture and therefore inconvenient. The object of the reduced distance is to bring that point within the picture.

[Ill.u.s.tration: Fig. 85.]

In Fig. 85 we have made the distance nearly twice the length of the base of the picture, and consequently a long way out of it. Draw _Sa_, _Sb_, and from _a_ draw _aD_ to point of distance, which cuts _Sb_ at _o_, and determines the depth of the square _acob_. But we can find that same point if we take half the base and draw a line from base to distance. But even this distance-point does not come inside the picture, so we take a fourth of the base and a fourth of the distance and draw a line from base to distance. We shall find that it pa.s.ses precisely through the same point _o_ as the other lines _aD_, &c. We are thus able to find the required point _o_ without going outside the picture.

Of course we could in the same way take an 8th or even a 16th distance, but the great use of this reduced distance, in addition to the above, is that it enables us to measure any depth into the picture with the greatest ease.

It will be seen in the next figure that without having to extend the base, as is usually done, we can multiply that base to any amount by making use of these reduced distances on the horizontal line. This is quite a new method of proceeding, and it will be seen is mathematically correct.

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HOW TO DRAW A LONG Pa.s.sAGE OR CLOISTER BY MEANS OF THE REDUCED DISTANCE

[Ill.u.s.tration: Fig. 86.]

In Fig. 86 we have divided the base of the first square into four equal parts, which may represent so many feet, so that A4 and _Bd_ being the retreating sides of the square each represents 4 feet. But we found point D by drawing 3D from base to distance, and by proceeding in the same way from each division, _A_, 1, 2, 3, we mark off on _SB_ four s.p.a.ces each equal to 4 feet, in all 16 feet, so that by taking the whole base and the distance we find point _O_, which is distant four times the length of the base _AB_. We can multiply this distance to any amount by drawing other diagonals to 8th distance, &c. The same rule applies to this corridor (Fig. 87 and Fig. 88).

[Ill.u.s.tration: Fig. 87.]

[Ill.u.s.tration: Fig. 88.]

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HOW TO FORM A VANISHING SCALE THAT SHALL GIVE THE HEIGHT, DEPTH, AND DISTANCE OF ANY OBJECT IN THE PICTURE

If we make our scale to vanish to the point of sight, as in Fig. 89, we can make _SB_, the lower line thereof, a measuring line for distances.

Let us first of all divide the base _AB_ into eight parts, each part representing 5 feet. From each division draw lines to 8th distance; by their intersections with _SB_ we obtain measurements of 40, 80, 120, 160, &c., feet. Now divide the side of the picture _BE_ in the same manner as the base, which gives us the height of 40 feet. From the side _BE_ draw lines 5S, 15S, &c., to point of sight, and from each division on the base line also draw lines 5S, 10S, 15S, &c., to point of sight, and from each division on _SB_, such as 40, 80, &c., draw horizontals parallel to base. We thus obtain squares 40 feet wide, beginning at base _AB_ and reaching as far as required. Note how the height of the flagstaff, which is 140 feet high and 280 feet distant, is obtained. So also any buildings or other objects can be measured, such as those shown on the left of the picture.

[Ill.u.s.tration: Fig. 89.]

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MEASURING SCALE ON GROUND

A simple and very old method of drawing buildings, &c., and giving them their right width and height is by means of squares of a given size, drawn on the ground.

[Ill.u.s.tration: Fig. 90.]

In the above sketch (Fig. 90) the squares on the ground represent 3 feet each way, or one square yard. Taking this as our standard measure, we find the door on the left is 10 feet high, that the archway at the end is 21 feet high and 12 feet wide, and so on.

[Ill.u.s.tration: Fig. 91. Natural Perspective.]

[Ill.u.s.tration: Fig. 92. Honfleur.]

Fig. 91 is a sketch made at Sandwich, Kent, and shows a somewhat similar subject to Fig. 84, but the irregularity and freedom of the perspective gives it a charm far beyond the rigid precision of the other, while it conforms to its main laws. This sketch, however, is the real artist's perspective, or what we might term natural perspective.

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APPLICATION OF THE REDUCED DISTANCE AND THE VANISHING SCALE TO DRAWING A LIGHTHOUSE, &C.

[Above ill.u.s.tration: Perspective of a lighthouse 135 feet high at 800 feet distance.]

[Ill.u.s.tration: Fig. 93. Key to Fig. 92, Honfleur.]

In the drawing of Honfleur (Fig. 92) we divide the base _AB_ as in the previous figure, but the s.p.a.ces measure 5 feet instead of 3 feet: so that taking the 8th distance, the divisions on the vanishing line _BS_ measure 40 feet each, and at point _O_ we have 400 feet of distance, but we require 800. So we again reduce the distance to a 16th. We thus multiply the base by 16. Now let us take a base of 50 feet at _f_ and draw line _fD_ to 16th distance; if we multiply 50 feet by 16 we obtain the 800 feet required.

The height of the lighthouse is found by means of the vanishing scale, which is 15 feet below and 15 feet above the horizon, or 30 feet from the sea-level. At _L_ we raise a vertical _LM_, which shows the position of the lighthouse. Then on that vertical measure the height required as shown in the figure.

The 800 feet could be obtained at once by drawing line _fD_, or 50 feet, to 16th distance. The other measurements obtained by 8th distance serve for nearer buildings.

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