An example will be necessary to make this problem clear to the general
student.
The nearest corner of a piece of pattern on the carpet is 4½ feet
beneath the eye, 2 feet to our right and 3½ feet in direct distance
from us. We intend to make a drawing of the pattern which shall be
seen properly when held 1½ foot from the eye. It is required to fix
the position of the corner of the piece of pattern.
[Illustration: Fig. 51.]
Let _AB_, Fig. 51., be our sheet of paper, some 3 feet wide. Make _ST_
equal to 1½ foot. Draw the line of sight through _S_. Produce _TS_,
and make _DS_ equal to 2 feet, therefore _TD_ equal to 3½ feet. Draw
_DC_, equal to 2 feet; _CP_, equal to 4 feet. Join _TC_ (cutting the
sight-line in _Q_) and _TP_.
Let fall the vertical _QP′_, then _P′_ is the point required.
If the lines, as in the figure, fall outside of your sheet of paper,
in order to draw them, it is necessary to attach other sheets of paper
to its edges. This is inconvenient, but must be done at first that
you may see your way clearly; and sometimes afterwards, though there
are expedients for doing without such extension in fast sketching.
It is evident, however, that no extension of surface could be of any
use to us, if the distance _TD_, instead of being 3½ feet, were 100
feet, or a mile, as it might easily be in a landscape.
It is necessary, therefore, to obtain some other means of
construction; to do which we must examine the principle of the
problem.
In the analysis of Fig. 2., in the introductory remarks, I used the
word “height” only of the tower, _QP_, because it was only to its
vertical height that the law deduced from the figure could be applied.
For suppose it had been a pyramid, as _OQP_, Fig. 52., then the image
of its side, _QP_, being, like every other magnitude, limited on the
glass _AB_ by the lines coming from its extremities, would appear only
of the length _Q′S_; and it is not true that _Q′S_ is to _QP_ as _TS_
is to _TP_. But if we let fall a vertical _QD_ from _Q_, so as to get
the vertical height of the pyramid, then it is true that _Q′S_ is to
_QD_ as _TS_ is to _TD_.
[Illustration: Fig. 52.]
Supposing this figure represented, not a pyramid, but a triangle on
the ground, and that _QD_ and _QP_ are horizontal lines, expressing
lateral distance from the line _TD_, still the rule would be false for
_QP_ and true for _QD_. And, similarly, it is true for all lines which
are parallel, like _QD_, to the plane of the picture _AB_, and false
for all lines which are inclined to it at an angle.
Hence generally. Let _PQ_ (Fig. 2. in Introduction, p. 6) be any
magnitude _parallel to the plane of the picture_; and _P′Q′_ its image
on the picture.
Then always the formula is true which you learned in the Introduction:
_P′Q′_ is to _PQ_ as _ST_ is to _DT_.
Now the magnitude _P_ dash _Q_ dash in this formula I call the
“SIGHT-MAGNITUDE” of the line _PQ_. The student must fix this term,
and the meaning of it, well in his mind. The “sight-magnitude” of a
line is the magnitude which bears to the real line the same proportion
that the distance of the picture bears to the distance of the object.
Thus, if a tower be a hundred feet high, and a hundred yards off; and
the picture, or piece of glass, is one yard from the spectator,
between him and the tower; the distance of picture being then to
distance of tower as 1 to 100, the sight-magnitude of the tower’s
height will be as 1 to 100; that is to say, one foot. If the tower is
two hundred yards distant, the sight-magnitude of its height will be
half a foot, and so on.
But farther. It is constantly necessary, in perspective operations,
to measure the other dimensions of objects by the sight-magnitude of
their vertical lines. Thus, if the tower, which is a hundred feet
high, is square, and twenty-five feet broad on each side; if the
sight-magnitude of the height is one foot, the measurement of the
side, reduced to the same scale, will be the hundredth part of
twenty-five feet, or three inches: and, accordingly, I use in this
treatise the term “sight-magnitude” indiscriminately for all lines
reduced in the same proportion as the vertical lines of the object. If
I tell you to find the “sight-magnitude” of any line, I mean, always,
find the magnitude which bears to that line the proportion of _ST_ to
_DT_; or, in simpler terms, reduce the line to the scale which you
have fixed by the first determination of the length _ST_.
Therefore, you must learn to draw quickly to scale before you do
anything else; for all the measurements of your object must be
reduced to the scale fixed by _ST_ before you can use them in your
diagram. If the object is fifty feet from you, and your paper one
foot, all the lines of the object must be reduced to a scale of one
fiftieth before you can use them; if the object is two thousand feet
from you, and your paper one foot, all your lines must be reduced to
the scale of one two-thousandth before you can use them, and so on.
Only in ultimate practice, the reduction never need be tiresome, for,
in the case of large distances, accuracy is never required. If a
building is three or four miles distant, a hairbreadth of accidental
variation in a touch makes a difference of ten or twenty feet in
height or breadth, if estimated by accurate perspective law. Hence it
is never attempted to apply measurements with precision at such
distances. Measurements are only required within distances of, at the
most, two or three hundred feet. Thus it may be necessary to represent
a cathedral nave precisely as seen from a spot seventy feet in front
of a given pillar; but we shall hardly be required to draw a cathedral
three miles distant precisely as seen from seventy feet in advance of
a given milestone. Of course, if such a thing be required, it can be
done; only the reductions are somewhat long and complicated: in
ordinary cases it is easy to assume the distance _ST_ so as to get at
the reduced dimensions in a moment. Thus, let the pillar of the nave,
in the case supposed, be 42 feet high, and we are required to stand
70 feet from it: assume _ST_ to be equal to 5 feet. Then, as 5 is to
70 so will the sight-magnitude required be to 42; that is to say, the
sight-magnitude of the pillar’s height will be 3 feet. If we make _ST_
equal to 2½ feet, the pillar’s height will be 1½ foot, and so on.
And for fine divisions into irregular parts which cannot be measured,
the ninth and tenth problems of the sixth book of Euclid will serve
you: the following construction is, however, I think, more practically
convenient:—
The line _AB_ (Fig. 53.) is divided by given points, _a_, _b_, _c_,
into a given number of irregularly unequal parts; it is required to
divide any other line, _CD_, into an equal number of parts, bearing
to each other the same proportions as the parts of _AB_, and arranged
in the same order.
Draw the two lines parallel to each other, as in the figure.
Join _AC_ and _BD_, and produce the lines _AC_, _BD_, till they meet
in _P_.
Join _aP_, _bP_, _cP_, cutting _cD_ in _f_, _g_, _h_.
Then the line _CD_ is divided as required, in _f_, _g_, _h_.
In the figure the lines _AB_ and _CD_ are accidentally perpendicular
to _AP_. There is no need for their being so.
[Illustration: Fig. 53.]
Now, to return to our first problem.
The construction given in the figure is only the quickest mathematical
way of obtaining, on the picture, the sight-magnitudes of _DC_ and
_PC_, which are both magnitudes parallel with the picture plane. But
if these magnitudes are too great to be thus put on the paper, you
have only to obtain the reduction by scale. Thus, if _TS_ be one foot,
_TD_ eighty feet, _DC_ forty feet, and _CP_ ninety feet, the distance
_QS_ must be made equal to one eightieth of _DC_, or half a foot; and
the distance _QP′_, one eightieth of _CP_, or one eightieth of ninety
feet; that is to say, nine eighths of a foot, or thirteen and a half
inches. The lines _CT_ and _PT_ are thus _practically_ useless, it
being only necessary to measure _QS_ and _QP_, on your paper, of the
due sight-magnitudes. But the mathematical construction, given in
Problem I., is the basis of all succeeding problems, and, if it is
once thoroughly understood and practiced (it can only be thoroughly
understood by practice), all the other problems will follow easily.
Lastly. Observe that any perspective operation whatever may be
performed with reduced dimensions of every line employed, so as to
bring it conveniently within the limits of your paper. When the
required figure is thus constructed on a small scale, you have only to
enlarge it accurately in the same proportion in which you reduced the
lines of construction, and you will have the figure constructed in
perspective on the scale required for use.