The Theory and Practice of Perspective Part 5

So likewise all diagonals drawn to the point of distance, which are contained between these parallels, such as _Ad_, _af_, &c., must be equal. For all straight lines which meet at any point on the horizon are perspectively parallel to each other, just as two geometrical parallels crossing two others at any angle, as at Fig. 49. Note also (Fig. 48) that all squares formed between the two vanishing lines _AS_, _BS_, and by the aid of these diagonals, are also equal, and further, that any number of squares such as are shown in this figure (Fig. 50), formed in the same way and having equal bases, are also equal; and the nine squares contained in the square _abcd_ being equal, they divide each side of the larger square into three equal parts.

[Ill.u.s.tration: Fig. 48.]

[Ill.u.s.tration: Fig. 49.]

From this we learn how we can measure any number of given lengths, either equal or unequal, on a vanishing or retreating line which is at right angles to the base; and also how we can measure any width or number of widths on a line such as _dc_, that is, parallel to the base of the picture, however remote it may be from that base.

[Ill.u.s.tration: Fig. 50.]

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GEOMETRICAL AND PERSPECTIVE FIGURES CONTRASTED

As at first there may be a little difficulty in realizing the resemblance between geometrical and perspective figures, and also about certain expressions we make use of, such as horizontals, perpendiculars, parallels, &c., which look quite different in perspective, I will here make a note of them and also place side by side the two views of the same figures.

[Ill.u.s.tration: Fig. 51 A. The geometrical view.]

[Ill.u.s.tration: Fig. 51 B. The perspective view.]

[Ill.u.s.tration: Fig. 51 C. A geometrical square.]

[Ill.u.s.tration: Fig. 51 D. A perspective square.]

[Ill.u.s.tration: Fig. 51 E. Geometrical parallels.]

[Ill.u.s.tration: Fig. 51 F. Perspective parallels.]

[Ill.u.s.tration: Fig. 51 G. Geometrical perpendicular.]

[Ill.u.s.tration: Fig. 51 H. Perspective perpendicular.]

[Ill.u.s.tration: Fig. 51 I. Geometrical equal lines.]

[Ill.u.s.tration: Fig. 51 J. Perspective equal lines.]

[Ill.u.s.tration: Fig. 51 K. A geometrical circle.]

[Ill.u.s.tration: Fig. 51 L. A perspective circle.]

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OF CERTAIN TERMS MADE USE OF IN PERSPECTIVE

Of course when we speak of +Perpendiculars+ we do not mean verticals only, but straight lines at right angles to other lines in any position.

Also in speaking of +lines+ a right or +straight line+ is to be understood; or when we speak of +horizontals+ we mean all straight lines that are parallel to the perspective plane, such as those on Fig. 52, no matter what direction they take so long as they are level. They are not to be confused with the horizon or horizontal-line.

[Ill.u.s.tration: Fig. 52. Horizontals.]

There are one or two other terms used in perspective which are not satisfactory because they are confusing, such as vanishing lines and vanishing points. The French term, _fuyante_ or _lignes fuyantes_, or going-away lines, is more expressive; and _point de fuite_, instead of vanishing point, is much better. I have occasionally called the former retreating lines, but the simple meaning is, lines that are not parallel to the picture plane; but a vanishing line implies a line that disappears, and a vanishing point implies a point that gradually goes out of sight. Still, it is difficult to alter terms that custom has endorsed. All we can do is to use as few of them as possible.

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HOW TO MEASURE VANISHING OR RECEDING LINES

Divide a vanishing line which is at right angles to the picture plane into any number of given measurements. Let _SA_ be the given line. From _A_ measure off on the base line the divisions required, say five of 1 foot each; from each division draw diagonals to point of distance _D_, and where these intersect the line _AC_ the corresponding divisions will be found. Note that as lines _AB_ and _AC_ are two sides of the same square they are necessarily equal, and so also are the divisions on _AC_ equal to those on _AB_.

[Ill.u.s.tration: Fig. 53.]

The line _AB_ being the base of the picture, it is at the same time a perspective line and a geometrical one, so that we can use it as a scale for measuring given lengths thereon, but should there not be enough room on it to measure the required number we draw a second line, _DC_, which we divide in the same proportion and proceed to divide _cf_. This geometrical figure gives, as it were, a bird's-eye view or ground-plan of the above.

[Ill.u.s.tration: Fig. 54.]

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HOW TO PLACE SQUARES IN GIVEN POSITIONS

Draw squares of given dimensions at given distances from the base line to the right or left of the vertical line, which pa.s.ses through the point of sight.

[Ill.u.s.tration: Fig. 55.]

Let _ab_ (Fig. 55) represent the base line of the picture divided into a certain number of feet; _HD_ the horizon, _VO_ the vertical. It is required to draw a square 3 feet wide, 2 feet to the right of the vertical, and 1 foot from the base.

First measure from _V_, 2 feet to _e_, which gives the distance from the vertical. Second, from _e_ measure 3 feet to _b_, which gives the width of the square; from _e_ and _b_ draw _eS_, _bS_, to point of sight. From either _e_ or _b_ measure 1 foot to the left, to _f_ or _f_. Draw _fD_ to point of distance, which intersects _eS_ at _P_, and gives the required distance from base. Draw _Pg_ and _B_ parallel to the base, and we have the required square.

Square _A_ to the left of the vertical is 2 feet wide, 1 foot from the vertical and 2 feet from the base, and is worked out in the same way.

_Note._--It is necessary to know how to work to scale, especially in architectural drawing, where it is indispensable, but in working out our propositions and figures it is not always desirable. A given length indicated by a line is generally sufficient for our requirements. To work out every problem to scale is not only tedious and mechanical, but wastes time, and also takes the mind of the student away from the reasoning out of the subject.

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