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*** START OF THE PROJECT GUTENBERG EBOOK 60816 ***

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  |   ∠ angle, ∶ ratio, ∷ proportion, ∴ therefore, ′ prime, and        |
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  +----------------------------------+
  |                                  |
  |         Library Edition          |
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  |                                  |
  |                                  |
  |        THE COMPLETE WORKS        |
  |                OF                |
  |           JOHN RUSKIN            |
  |                                  |
  |                                  |
  |     ELEMENTS OF DRAWING AND      |
  |           PERSPECTIVE            |
  |          THE TWO PATHS           |
  |          UNTO THIS LAST          |
  |         MUNERA PULVERIS          |
  |        SESAME AND LILIES         |
  |        ETHICS OF THE DUST        |
  |                                  |
  |                                  |
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  |   NATIONAL LIBRARY ASSOCIATION   |
  |   NEW YORK             CHICAGO   |
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           THE ELEMENTS OF PERSPECTIVE

         ARRANGED FOR THE USE OF SCHOOLS

  AND INTENDED TO BE READ IN CONNECTION WITH THE
           FIRST THREE BOOKS OF EUCLID.




                CONTENTS.

                                                              PAGE
  Preface                                                       ix

  Introduction                                                   1

                PROBLEM I.
  To fix the Position of a given Point                          10

                PROBLEM II.
  To draw a Right Line between two given Points                 13

                PROBLEM III.
  To find the Vanishing-Point of a given Horizontal Line        17

                PROBLEM IV.
  To find the Dividing-Points of a given Horizontal Line        23

                PROBLEM V.
  To draw a Horizontal Line, given in Position and Magnitude,
    by means of its Sight-Magnitude and Dividing-Points         24

                PROBLEM VI.
  To draw any Triangle, given in Position and Magnitude, in a
    Horizontal Plane                                            27

                PROBLEM VII.
  To draw any Rectilinear Quadrilateral Figure, given in
    Position and Magnitude, in a Horizontal Plane               29

                PROBLEM VIII.
  To draw a Square, given in Position and Magnitude, in a
    Horizontal Plane                                            31

                PROBLEM IX.
  To draw a Square Pillar, given in Position and Magnitude,
    its Base and Top being in Horizontal Planes                 34

                PROBLEM X.
  To draw a Pyramid, given in Position and Magnitude, on a
    Square Base in a Horizontal Plane                           36

                PROBLEM XI.
  To draw any Curve in a Horizontal or Vertical Plane           38

                PROBLEM XII.
  To divide a Circle drawn in Perspective into any given
    Number of Equal Parts                                       42

                PROBLEM XIII.
  To draw a Square, given in Magnitude, within a larger
    Square given in Position and Magnitude; the Sides of the
    two Squares being Parallel                                  45

                PROBLEM XIV.
  To draw a Truncated Circular Cone, given in Position and
    Magnitude, the Truncations being in Horizontal Planes,
    and the Axis of the Cone vertical                           47

                PROBLEM XV.
  To draw an Inclined Line, given in Position and Magnitude     50

                PROBLEM XVI.
  To find the Vanishing-Point of a given Inclined Line          53

                PROBLEM XVII.
  To find the Dividing-Points of a given Inclined Line          55

                PROBLEM XVIII.
  To find the Sight-Line of an Inclined Plane in which Two
    Lines are given in Position                                 57

                PROBLEM XIX.
  To find the Vanishing-Point of Steepest Lines in an Inclined
    Plane whose Sight-Line is given                             59

                PROBLEM XX.
  To find the Vanishing-Point of Lines perpendicular to the
    Surface of a given Inclined Plane                           61


                APPENDIX.

                I.
  Practice and Observations on the preceding Problems           69

                II.
  Demonstrations which could not conveniently be included in
    the Text                                                    99




PREFACE.


For some time back I have felt the want, among Students of Drawing, of a
written code of accurate Perspective Law; the modes of construction in
common use being various, and, for some problems, insufficient. It would
have been desirable to draw up such a code in popular language, so as to
do away with the most repulsive difficulties of the subject; but finding
this popularization would be impossible, without elaborate figures and
long explanations, such as I had no leisure to prepare, I have arranged
the necessary rules in a short mathematical form, which any schoolboy
may read through in a few days, after he has mastered the first three
and the sixth books of Euclid.

Some awkward compromises have been admitted between the first-attempted
popular explanation, and the severer arrangement, involving irregular
lettering and redundant phraseology; but I cannot for the present do
more, and leave the book therefore to its trial, hoping that, if it be
found by masters of schools to answer its purpose, I may hereafter bring
it into better form.[1]

An account of practical methods, sufficient for general purposes of
sketching, might indeed have been set down in much less space: but if
the student reads the following pages carefully, he will not only find
himself able, on occasion, to solve perspective problems of a complexity
greater than the ordinary rules will reach, but obtain a clue to many
important laws of pictorial effect, no less than of outline. The subject
thus examined becomes, at least to my mind, very curious and
interesting; but, for students who are unable or unwilling to take it up
in this abstract form, I believe good help will be soon furnished, in a
series of illustrations of practical perspective now in preparation by
Mr. Le Vengeur. I have not seen this essay in an advanced state, but the
illustrations shown to me were very clear and good; and, as the author
has devoted much thought to their arrangement, I hope that his work will
be precisely what is wanted by the general learner.

Students wishing to pursue the subject into its more extended branches
will find, I believe, Cloquet’s treatise the best hitherto published.[2]


  [1] Some irregularities of arrangement have been admitted merely for
      the sake of convenient reference; the eighth problem, for
      instance, ought to have been given as a case of the seventh, but
      is separately enunciated on account of its importance.

      Several constructions, which ought to have been given as problems,
      are on the contrary given as corollaries, in order to keep the
      more directly connected problems in closer sequence; thus the
      construction of rectangles and polygons in vertical planes would
      appear by the Table of Contents to have been omitted, being given
      in the corollary to Problem IX.

  [2] Nouveau Traité Élémentaire de Perspective. Bachelier, 1823.




THE ELEMENTS OF PERSPECTIVE.




INTRODUCTION.


When you begin to read this book, sit down very near the window, and
shut the window. I hope the view out of it is pretty; but, whatever the
view may be, we shall find enough in it for an illustration of the first
principles of perspective (or, literally, of “looking through”).

Every pane of your window may be considered, if you choose, as a glass
picture; and what you see through it, as painted on its surface.

And if, holding your head still, you extend your hand to the glass, you
may, with a brush full of any thick color, trace, roughly, the lines of
the landscape on the glass.

But, to do this, you must hold your head very still. Not only you must
not move it sideways, nor up and down, but it must not even move
backwards or forwards; for, if you move your head forwards, you will see
_more_ of the landscape through the pane; and, if you move it backwards,
you will see _less_: or considering the pane of glass as a picture, when
you hold your head near it, the objects are painted small, and a great
many of them go into a little space; but, when you hold your head some
distance back, the objects are painted larger upon the pane, and fewer
of them go into the field of it.

But, besides holding your head still, you must, when you try to trace
the picture on the glass, shut one of your eyes. If you do not, the
point of the brush appears double; and, on farther experiment, you will
observe that each of your eyes sees the object in a different place on
the glass, so that the tracing which is true to the sight of the right
eye is a couple of inches (or more, according to your distance from the
pane,) to the left of that which is true to the sight of the left.

Thus, it is only possible to draw what you see through the window
rightly on the surface of the glass, by fixing one eye at a given point,
and neither moving it to the right nor left, nor up nor down, nor
backwards nor forwards. Every picture drawn in true perspective may be
considered as an upright piece of glass,[3] on which the objects seen
through it have been thus drawn. Perspective can, therefore, only be
quite right, by being calculated for one fixed position of the eye of
the observer; nor will it ever appear _deceptively_ right unless seen
precisely from the point it is calculated for. Custom, however, enables
us to feel the rightness of the work on using both our eyes, and to be
satisfied with it, even when we stand at some distance from the point it
is designed for.

Supposing that, instead of a window, an unbroken plate of crystal
extended itself to the right and left of you, and high in front, and
that you had a brush as long as you wanted (a mile long, suppose), and
could paint with such a brush, then the clouds high up, nearly over your
head, and the landscape far away to the right and left, might be traced,
and painted, on this enormous crystal field.[4] But if the field were so
vast (suppose a mile high and a mile wide), certainly, after the picture
was done, you would not stand as near to it, to see it, as you are now
sitting near to your window. In order to trace the upper clouds through
your great glass, you would have had to stretch your neck quite back,
and nobody likes to bend their neck back to see the top of a picture. So
you would walk a long way back to see the great picture—a quarter of a
mile, perhaps,—and then all the perspective would be wrong, and would
look quite distorted, and you would discover that you ought to have
painted it from the greater distance, if you meant to look at it from
that distance. Thus, the distance at which you intend the observer to
stand from a picture, and for which you calculate the perspective,
ought to regulate to a certain degree the size of the picture. If you
place the point of observation near the canvas, you should not make the
picture very large: _vice versâ_, if you place the point of observation
far from the canvas, you should not make it very small; the fixing,
therefore, of this point of observation determines, as a matter of
convenience, within certain limits, the size of your picture. But it
does not determine this size by any perspective law; and it is a mistake
made by many writers on perspective, to connect some of their rules
definitely with the size of the picture. For, suppose that you had what
you now see through your window painted actually upon its surface, it
would be quite optional to cut out any piece you chose, with the piece
of the landscape that was painted on it. You might have only half a
pane, with a single tree; or a whole pane, with two trees and a cottage;
or two panes, with the whole farmyard and pond; or four panes, with
farmyard, pond, and foreground. And any of these pieces, if the
landscape upon them were, as a scene, pleasantly composed, would be
agreeable pictures, though of quite different sizes; and yet they would
be all calculated for the same distance of observation.

In the following treatise, therefore, I keep the size of the picture
entirely undetermined. I consider the field of canvas as wholly
unlimited, and on that condition determine the perspective laws. After
we know how to apply those laws without limitation, we shall see what
limitations of the size of the picture their results may render
advisable.

But although the size of the _picture_ is thus independent of the
observer’s distance, the size of the _object represented_ in the picture
is not. On the contrary, that size is fixed by absolute mathematical
law; that is to say, supposing you have to draw a tower a hundred feet
high, and a quarter of a mile distant from you, the height which you
ought to give that tower on your paper depends, with mathematical
precision, on the distance at which you intend your paper to be placed.
So, also, do all the rules for drawing the form of the tower, whatever
it may be.

Hence, the first thing to be done in beginning a drawing is to fix, at
your choice, this distance of observation, or the distance at which you
mean to stand from your paper. After that is determined, all is
determined, except only the ultimate size of your picture, which you may
make greater, or less, not by altering the size of the things
represented, but by _taking in more, or fewer_ of them. So, then, before
proceeding to apply any practical perspective rule, we must always have
our distance of observation marked, and the most convenient way of
marking it is the following:

  [Illustration: Fig. 1. PLACING OF THE SIGHT-POINT, SIGHT-LINE,
   STATION-POINT, AND STATION-LINE.]


I. THE SIGHT-POINT.—Let _ABCD_, Fig. 1., be your sheet of paper, the
larger the better, though perhaps we may cut out of it at last only a
small piece for our picture, such as the dotted circle _NOPQ_. This
circle is not intended to limit either the size or shape of our picture:
you may ultimately have it round or oval, horizontal or upright, small
or large, as you choose. I only dot the line to give you an idea of
whereabouts you will probably like to have it; and, as the operations of
perspective are more conveniently performed upon paper underneath the
picture than above it, I put this conjectural circle at the top of the
paper, about the middle of it, leaving plenty of paper on both sides and
at the bottom. Now, as an observer generally stands near the middle of a
picture to look at it, we had better at first, and for simplicity’s
sake, fix the point of observation opposite the middle of our
conjectural picture. So take the point _S_, the center of the circle
_NOPQ_;—or, which will be simpler for you in your own work, take the
point _S_ at random near the top of your paper, and strike the circle
_NOPQ_ round it, any size you like. Then the point _S_ is to represent
the point _opposite_ which you wish the observer of your picture to
place his eye, in looking at it. Call this point the “Sight-Point.”


II. THE SIGHT-LINE.—Through the Sight-point, _S_, draw a horizontal
line, _GH_, right across your paper from side to side, and call this
line the “Sight-Line.”

This line is of great practical use, representing the level of the eye
of the observer all through the picture. You will find hereafter that if
there is a horizon to be represented in your picture, as of distant sea
or plain, this line defines it.


III. THE STATION-LINE.—From _S_ let fall a perpendicular line, _SR_, to
the bottom of the paper, and call this line the “Station-Line.”

This represents the line on which the observer stands, at a greater or
less distance from the picture; and it ought to be _imagined_ as drawn
right out from the paper at the point s. Hold your paper upright in
front of you, and hold your pencil horizontally, with its point against
the point _S_, as if you wanted to run it through the paper there, and
the pencil will represent the direction in which the line _SR_ ought to
be drawn. But as all the measurements which we have to set upon this
line, and operations which we have to perform with it, are just the same
when it is drawn on the paper itself, below _S_, as they would be if it
were represented by a wire in the position of the leveled pencil, and as
they are much more easily performed when it is drawn on the paper, it is
always in practice, so drawn.


IV. THE STATION-POINT.—On this line, mark the distance _ST_ at your
pleasure, for the distance at which you wish your picture to be seen,
and call the point T the “Station-Point.”

  [Illustration: Fig. 2.]

In practice, it is generally advisable to make the distance _ST_ about
as great as the diameter of your intended picture; and it should, for
the most part, be more rather than less; but, as I have just stated,
this is quite arbitrary. However, in this figure, as an approximation to
a generally advisable distance, I make the distance _ST_ equal to the
diameter of the circle _NOPQ_. Now, having fixed this distance, _ST_,
all the dimensions of the objects in our picture are fixed likewise, and
for this reason:—

Let the upright line _AB_, Fig. 2., represent a pane of glass placed
where our picture is to be placed; but seen at the side of it,
edgeways; let _S_ be the Sight-point; _ST_ the Station-line, which, in
this figure, observe, is in its true position, drawn out from the paper,
not down upon it; and _T_ the Station-point.

Suppose the Station-line _ST_ to be continued, or in mathematical
language “produced,” through _S_, far beyond the pane of glass, and let
_PQ_ be a tower or other upright object situated on or above this line.

Now the _apparent_ height of the tower _PQ_ is measured by the angle
_QTP_, between the rays of light which come from the top and bottom of
it to the eye of the observer. But the _actual_ height of the _image_ of
the tower on the pane of glass _AB_, between us and it, is the distance
_P′Q′_ between the points where the rays traverse the glass.

Evidently, the farther from the point _T_ we place the glass, making
_ST_ longer, the larger will be the image; and the nearer we place it to
_T_, the smaller the image, and that in a fixed ratio. Let the distance
_DT_ be the direct distance from the Station-point to the foot of the
object. Then, if we place the glass _AB_ at one-third of that whole
distance, _P′Q′_ will be one-third of the real height of the object; if
we place the glass at two-thirds of the distance, as at _EF_, _P″Q″_
(the height of the image at that point) will be two-thirds the height[5]
of the object, and so on. Therefore the mathematical law is that _P′Q′_
will be to _PQ_ as _ST_ to _DT_. I put this ratio clearly by itself that
you may remember it:

  _P′Q′_ ∶ _PQ_ ∷ _ST_ ∶ _DT_

or in words:

  _P_ dash _Q_ dash is to _PQ_ as _ST_ to _DT_

In which formula, recollect that _P′Q′_ is the height of the appearance
of the object on the picture; _PQ_ the height of the object itself; _S_
the Sight-point; _T_ the Station-point; _D_ a point at the direct
distance of the object; though the object is seldom placed actually on
the line _TS_ produced, and may be far to the right or left of it, the
formula is still the same.

For let _S_, Fig. 3., be the Sight-point, and _AB_ the glass—here seen
looking _down_ on its _upper edge_, not sideways;—then if the tower
(represented now, as on a map, by the dark square), instead of being at
_D_ on the line _ST_ produced, be at _E_, to the right (or left) of the
spectator, still the apparent height of the tower on _AB_ will be as
_S′T_ to _ET_, which is the same ratio as that of _ST_ to _DT_.

  [Illustration: Fig. 3.]

Now in many perspective problems, the position of an object is more
conveniently expressed by the two measurements _DT_ and _DE_, than by
the single oblique measurement _ET_.

I shall call _DT_ the “direct distance” of the object at _E_, and _DE_
its “lateral distance.” It is rather a license to call _DT_ its “direct”
distance, for _ET_ is the more direct of the two; but there is no other
term which would not cause confusion.

Lastly, in order to complete our knowledge of the position of an object,
the vertical height of some point in it, above or below the eye, must be
given; that is to say, either _DP_ or _DQ_ in Fig. 2.[6]: this I shall
call the “vertical distance” of the point given. In all perspective
problems these three distances, and the dimensions of the object, must
be stated, otherwise the problem is imperfectly given. It ought not to
be required of us merely to draw _a_ room or _a_ church in perspective;
but to draw _this_ room from _this_ corner, and _that_ church on _that_
spot, in perspective. For want of knowing how to base their drawings on
the measurement and place of the object, I have known practiced students
represent a parish church, certainly in true perspective, but with a
nave about two miles and a half long.

It is true that in drawing landscapes from nature the sizes and
distances of the objects cannot be accurately known. When, however, we
know how to draw them rightly, if their size were given, we have only to
_assume a rational approximation_ to their size, and the resulting
drawing will be true enough for all intents and purposes. It does not in
the least matter that we represent a distant cottage as eighteen feet
long, when it is in reality only seventeen; but it matters much that we
do not represent it as eighty feet long, as we easily might if we had
not been accustomed to draw from measurement. Therefore, in all the
following problems the measurement of the object is given.

The student must observe, however, that in order to bring the diagrams
into convenient compass, the measurements assumed are generally very
different from any likely to occur in practice. Thus, in Fig. 3., the
distance _DS_ would be probably in practice half a mile or a mile, and
the distance _TS_, from the eye of the observer to the paper, only two
or three feet. The mathematical law is however precisely the same,
whatever the proportions; and I use such proportions as are best
calculated to make the diagram clear.

Now, therefore, the conditions of a perspective problem are the
following:

  The Sight-line _GH_ given, Fig. 1.;
  The Sight-point _S_ given;
  The Station-point _T_ given; and
  The three distances of the object,[7] direct, lateral, and vertical,
    with its dimensions, given.

The size of the picture, conjecturally limited by the dotted circle, is
to be determined afterwards at our pleasure. On these conditions I
proceed at once to construction.


  [3] If the glass were not upright, but sloping, the objects might
      still be drawn through it, but their perspective would then be
      different. Perspective, as commonly taught, is always calculated
      for a vertical plane of picture.

  [4] Supposing it to have no thickness; otherwise the images would be
      distorted by refraction.

  [5] I say “height” instead of “magnitude,” for a reason stated in
      Appendix I., to which you will soon be referred. Read on here at
      present.

  [6] _P_ and _Q_ being points indicative of the place of the tower’s
      base and top. In this figure both are above the sight-line; if the
      tower were below the spectator both would be below it, and
      therefore measured below _D_.

  [7] More accurately, “the three distances of any point, either in the
      object itself, or indicative of its distance.”




PROBLEM I.

TO FIX THE POSITION OF A GIVEN POINT.[8]


Let _P_, Fig. 4., be the given point.

  [Illustration: Fig. 4.]

Let its direct distance be _DT_; its lateral distance to the left, _DC_;
and vertical distance _beneath_ the eye of the observer, _CP_.

[Let _GH_ be the Sight-line, _S_ the Sight-point, and _T_ the
Station-point.][9]

It is required to fix on the plane of the picture the position of the
point P.

Arrange the three distances of the object on your paper, as in
Fig. 4.[10]

Join _CT_, cutting _GH_ in _Q_.

From _Q_ let fall the vertical line _QP′_.

Join _PT_, cutting _QP_ in _P′_.

_P′_ is the point required.

If the point _P_ is _above_ the eye of the observer instead of below it,
_CP_ is to be measured upwards from _C_, and _QP′_ drawn upwards from
_Q_. The construction will be as in Fig. 5.

  [Illustration: Fig. 5.]

And if the point _P_ is to the right instead of the left of the
observer, _DC_ is to be measured to the right instead of the left.

The figures 4. and 5., looked at in a mirror, will show the construction
of each, on that supposition.

Now read very carefully the examples and notes to this problem in
Appendix I. (page 69). I have put them in the Appendix in order to keep
the sequence of following problems more clearly traceable here in the
text; but you must read the first Appendix before going on.


   [8] More accurately, “To fix on the plane of the picture the apparent
       position of a point given in actual position.” In the headings of
       all the following problems the words “on the plane of the
       picture” are to be understood after the words “to draw.” The
       plane of the picture means a surface extended indefinitely in the
       direction of the picture.

   [9] The sentence within brackets will not be repeated in succeeding
       statements of problems. It is always to be understood.

  [10] In order to be able to do this, you must assume the distances to
       be small; as in the case of some object on the table: how large
       distances are to be treated you will see presently; the
       mathematical principle, being the same for all, is best
       illustrated first on a small scale. Suppose, for instance, _P_ to
       be the corner of a book on the table, seven inches below the eye,
       five inches to the left of it, and a foot and a half in advance
       of it, and that you mean to hold your finished drawing at six
       inches from the eye; then _TS_ will be six inches, _TD_ a foot
       and a half, _DC_ five inches, and _CP_ seven.




PROBLEM II.

TO DRAW A RIGHT LINE BETWEEN TWO GIVEN POINTS.


  [Illustration: Fig. 6.]

Let _AB_, Fig. 6., be the given right line, joining the given points _A_
and _B_.

Let the direct, lateral, and vertical distances of the point _A_ be
_TD_, _DC_, and _CA_.

Let the direct, lateral, and vertical distances of the point _B_ be
_TD′_, _DC′_, and _C′B_.

Then, by Problem I., the position of the point _A_ on the plane of the
picture is _a_.

And similarly, the position of the point _B_ on the plane of the picture
is _b_.

Join _ab_.

Then _ab_ is the line required.


COROLLARY I.

If the line _AB_ is in a plane parallel to that of the picture, one end
of the line _AB_ must be at the same direct distance from the eye of the
observer as the other.

Therefore, in that case, _DT_ is equal to _D′T_.

Then the construction will be as in Fig. 7.; and the student will find
experimentally that _ab_ is now parallel to _AB_.[11]

  [Illustration: Fig. 7.]

And that _ab_ is to _AB_ as _TS_ is to _TD_.

Therefore, to draw any line in a plane parallel to that of the picture,
we have only to fix the position of one of its extremities, _a_ or _b_,
and then to draw from _a_ or _b_ a line parallel to the given line,
bearing the proportion to it that _TS_ bears to _TD_.


COROLLARY II.

If the line _AB_ is in a horizontal plane, the vertical distance of one
of its extremities must be the same as that of the other.

Therefore, in that case, _AC_ equals _BC′_ (Fig. 6.).

And the construction is as in Fig. 8.

  [Illustration: Fig. 8.]

In Fig. 8. produce _ab_ to the sight-line, cutting the sight-line in
_V_; the point _V_, thus determined, is called the VANISHING-POINT of
the line _AB_.

Join _TV_. Then the student will find experimentally that _TV_ is
parallel to _AB_.[12]


COROLLARY III.

If the line _AB_ produced would pass through some point beneath or above
the station-point, _CD_ is to _DT_ as _C′D′_ is to _D′T_; in which case
the point _c_ coincides with the point _c′_, and the line _ab_ is
vertical.

Therefore every vertical line in a picture is, or may be, the
perspective representation of a horizontal one which, produced, would
pass beneath the feet or above the head of the spectator.[13]


  [11] For by the construction _AT_ ∶ _aT_ ∷ _BT_ ∶ _bT_; and therefore
       the two triangles _ABT_, _abT_, (having a common angle _ATB_,)
       are similar.

  [12] The demonstration is in Appendix II. Article I.

  [13] The reflection in water of any luminous point or isolated object
       (such as the sun or moon) is therefore, in perspective, a
       vertical line; since such reflection, if produced, would pass
       under the feet of the spectator. Many artists (Claude among the
       rest) knowing something of optics, but nothing of perspective,
       have been led occasionally to draw such reflections towards a
       point at the center of the base of the picture.




PROBLEM III.

TO FIND THE VANISHING-POINT OF A GIVEN HORIZONTAL LINE.


  [Illustration: Fig. 9.]

Let _AB_, Fig. 9., be the given line.

From _T_, the station-point, draw _TV_ parallel to _AB_, cutting the
sight-line in _V_.

_V_ is the Vanishing-point required.[14]


COROLLARY I.

As, if the point _b_ is first found, _V_ may be determined by it, so, if
the point _V_ is first found, _b_ may be determined by it. For let _AB_,
Fig. 10., be the given line, constructed upon the paper as in Fig. 8.;
and let it be required to draw the line _ab_ without using the point
_C′_.

  [Illustration: Fig. 10.]

Find the position of the point _A_ in _a_. (Problem I.)

Find the vanishing-point of _AB_ in _V_. (Problem III.)

Join _aV_.

Join _BT_, cutting _aV_ in _b_.

Then _ab_ is the line required.[15]


COROLLARY II.

We have hitherto proceeded on the supposition that the given line was
small enough, and near enough, to be actually drawn on our paper of its
real size; as in the example given in Appendix I. We may, however, now
deduce a construction available under all circumstances, whatever may be
the distance and length of the line given.

  [Illustration: Fig. 11.]

From Fig. 8. remove, for the sake of clearness, the lines _C′D′_, _bV_,
and _TV_; and, taking the figure as here in Fig. 11., draw from _a_, the
line _aR_ parallel to _AB_, cutting _BT_ in _R_.

  Then _aR_ is to _AB_ as _aT_ is to _AT_.
       ----       ---- as _cT_ is to _CT_.
       ----       ---- as _TS_ is to _TD_.

That is to say, _aR_ is the sight-magnitude of _AB_.[16]

  [Illustration: Fig. 12.]

Therefore, when the position of the point _A_ is fixed in _a_, as in
Fig. 12., and _aV_ is drawn to the vanishing-point; if we draw a line
_aR_ from _a_, parallel to _AB_, and make _aR_ equal to the
sight-magnitude of _AB_, and then join _RT_, the line _RT_ will cut _aV_
in _b_.

So that, in order to determine the length of _ab_, we need not draw the
long and distant line _AB_, but only _aR_ parallel to it, and of its
sight-magnitude; which is a great gain, for the line _AB_ may be two
miles long, and the line _aR_ perhaps only two inches.


COROLLARY III.

In Fig. 12., altering its proportions a little for the sake of
clearness, and putting it as here in Fig. 13., draw a horizontal line
_aR′_ and make _aR′_ equal to _aR_.

Through the points _R_ and _b_ draw _R′M_, cutting the sight-line in
_M_. Join _TV_. Now the reader will find experimentally that _VM_ is
equal to _VT_.[17]

  [Illustration: Fig. 13.]

Hence it follows that, if from the vanishing-point _V_ we lay off on
the sight-line a distance, _VM_, equal to _VT_; then draw through _a_ a
horizontal line _aR′_, make _aR′_ equal to the sight-magnitude of _AB_,
and join _R′M_; the line _R′M_ will cut _aV_ in _b_. And this is in
practice generally the most convenient way of obtaining the length of
_ab_.


COROLLARY IV.

Removing from the preceding figure the unnecessary lines, and retaining
only _R′M_ and _aV_, as in Fig. 14., produce the line _aR′_ to the other
side of _a_, and make _aX_ equal to _aR′_.

Join _Xb_, and produce _Xb_ to cut the line of sight in _N_.

  [Illustration: Fig. 14.]

Then as _XR′_ is parallel to _MN_, and _aR′_ is equal to _aX_, _VN_
must, by similar triangles, be equal to _VM_ (equal to _VT_ in
Fig. 13.).

Therefore, on whichever side of _V_ we measure the distance _VT_, so as
to obtain either the point _M_, or the point _N_, if we measure the
sight-magnitude _aR′_ or _aX_ on the opposite side of the line _aV_, the
line joining _R′M_ or _XN_ will equally cut _aV_ in _b_.

The points _M_ and _N_ are called the “DIVIDING-POINTS” of the original
line _AB_ (Fig. 12.), and we resume the results of these corollaries in
the following three problems.


  [14] The student will observe, in practice, that, his paper lying flat
       on the table, he has only to draw the line _TV_ on its horizontal
       surface, parallel to the given horizontal line _AB_. In theory,
       the paper should be vertical, but the station-line _ST_
       horizontal (see its definition above, page 5); in which case
       _TV_, being drawn parallel to _AB_, will be horizontal also, and
       still cut the sight-line in _V_.

       The construction will be seen to be founded on the second
       Corollary of the preceding problem.

       It is evident that if any other line, as _MN_ in Fig. 9.,
       parallel to _AB_, occurs in the picture, the line _TV_, drawn
       from _T_, parallel to _MN_, to find the vanishing-point of _MN_,
       will coincide with the line drawn from _T_, parallel to _AB_, to
       find the vanishing-point of _AB_.

       Therefore _AB_ and _MN_ will have the same vanishing-point.

       Therefore all parallel horizontal lines have the same
       vanishing-point.

       It will be shown hereafter that all parallel _inclined_ lines
       also have the same vanishing-point; the student may here accept
       the general conclusion—“_All parallel lines have the same
       vanishing-point._”

       It is also evident that if _AB_ is parallel to the plane of the
       picture, _TV_ must be drawn parallel to _GH_, and will therefore
       never cut _GH_. The line _AB_ has in that case no
       vanishing-point: it is to be drawn by the construction given in
       Fig. 7.

       It is also evident that if _AB_ is at right angles with the plane
       of the picture, _TV_ will coincide with _TS_, and the
       vanishing-point of _AB_ will be the sight-point.

  [15] I spare the student the formality of the _reductio ad absurdum_,
       which would be necessary to prove this.

  [16] For definition of Sight-Magnitude, see Appendix I. It ought to
       have been read before the student comes to this problem; but I
       refer to it in case it has not.

  [17] The demonstration is in Appendix II. Article II. p. 101.




PROBLEM IV.

TO FIND THE DIVIDING-POINTS OF A GIVEN HORIZONTAL LINE.


  [Illustration: Fig. 15.]

Let the horizontal line _AB_ (Fig. 15.) be given in position and
magnitude. It is required to find its dividing-points.

Find the vanishing-point _V_ of the line _AB_.

With center _V_ and distance _VT_, describe circle cutting the
sight-line in _M_ and _N_.

Then _M_ and _N_ are the dividing-points required.

In general, only one dividing-point is needed for use with any
vanishing-point, namely, the one nearest _S_ (in this case the point
_M_). But its opposite _N_, or both, may be needed under certain
circumstances.




PROBLEM V.

TO DRAW A HORIZONTAL LINE, GIVEN IN POSITION AND MAGNITUDE, BY MEANS
OF ITS SIGHT-MAGNITUDE AND DIVIDING-POINTS.


  [Illustration: 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.


COROLLARY I.

  [Illustration: 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.


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 Appendix I.), 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.




PROBLEM VI.

TO DRAW ANY TRIANGLE, GIVEN IN POSITION AND MAGNITUDE, IN A HORIZONTAL
PLANE.


  [Illustration: Fig. 18.]

Let _ABC_ (Fig. 18.) be the triangle.

As it is given in position and magnitude, one of its sides, at least,
must be given in position and magnitude, and the directions of the two
other sides.

Let _AB_ be the side given in position and magnitude.

Then _AB_ is a horizontal line, in a given position, and of a given
length.

Draw the line _AB_. (Problem V.)

Let _ab_ be the line so drawn.

Find _V_ and _V′_, the vanishing-points respectively of the lines _AC_
and _BC_. (Problem III.)

From _a_ draw _aV_, and from _b_, draw _bV′_, cutting each other in
_c_.

Then _abc_ is the triangle required.

If _AC_ is the line originally given, _ac_ is the line which must be
first drawn, and the line _V′b_ must be drawn from _V′_ to _c_ and
produced to cut _ab_ in _b_. Similarly, if _BC_ is given, _Vc_ must be
drawn to _c_ and produced, and _ab_ from its vanishing-point to _b_,
and produced to cut _ac_ in _a_.




PROBLEM VII.

TO DRAW ANY RECTILINEAR QUADRILATERAL FIGURE, GIVEN IN POSITION AND
MAGNITUDE, IN A HORIZONTAL PLANE.


  [Illustration: Fig. 19.]

Let _ABCD_ (Fig. 19.) be the given figure.

Join any two of its opposite angles by the line _BC_.

Draw first the triangle _ABC_. (Problem VI.)

And then, from the base _BC_, the two lines _BD_, _CD_, to their
vanishing-points, which will complete the figure. It is unnecessary to
give a diagram of the construction, which is merely that of Fig. 18.
duplicated; another triangle being drawn on the line _AC_ or _BC_.


COROLLARY.

It is evident that by this application of Problem VI. any given
rectilinear figure whatever in a horizontal plane may be drawn, since
any such figure may be divided into a number of triangles, and the
triangles then drawn in succession.

More convenient methods may, however, be generally found, according
to the form of the figure required, by the use of succeeding problems;
and for the quadrilateral figure which occurs most frequently in
practice, namely, the square, the following construction is more
convenient than that used in the present problem.




PROBLEM VIII.

TO DRAW A SQUARE, GIVEN IN POSITION AND MAGNITUDE, IN A HORIZONTAL
PLANE.


  [Illustration: Fig. 20.]

Let _ABCD_, Fig. 20., be the square.

As it is given in position and magnitude, the position and magnitude
of all its sides are given.

Fix the position of the point _A_ in _a_.

Find _V_, the vanishing-point of _AB_; and _M_, the dividing-point of
_AB_, nearest _S_.

Find _V′_, the vanishing-point of _AC_; and _N_, the dividing-point of
_AC_, nearest _S_.

Draw the measuring-line through _a_, and make _ab′_, _ac′_, each equal
to the sight-magnitude of _AB_.

(For since _ABCD_ is a square, _AC_ is equal to _AB_.)

Draw _aV′_ and _c′N_, cutting each other in _c_.

Draw _aV_, and _b′M_, cutting each other in _b_.

Then _ac_, _ab_, are the two nearest sides of the square.

Now, clearing the figure of superfluous lines, we have _ab_, _ac_,
drawn in position, as in Fig. 21.

  [Illustration: Fig. 21.]

And because _ABCD_ is a square, _CD_ (Fig. 20.) is parallel to _AB_.

And all parallel lines have the same vanishing-point. (Note to
Problem III.)

Therefore, _V_ is the vanishing-point of _CD_.

Similarly, _V′_ is the vanishing-point of _BD_.

Therefore, from _b_ and _c_ (Fig. 22.) draw _bV′_, _cV_, cutting each
other in _d_.

Then _abcd_ is the square required.


COROLLARY I.

It is obvious that any rectangle in a horizontal plane may be drawn by
this problem, merely making _ab′_, on the measuring-line, Fig. 20.,
equal to the sight-magnitude of one of its sides, and _ac′_ the
sight-magnitude of the other.


COROLLARY II.

Let _abcd_, Fig. 22., be any square drawn in perspective. Draw the
diagonals _ad_ and _bc_, cutting each other in _C_. Then _C_ is the
center of the square. Through _C_, draw _ef_ to the vanishing-point of
_ab_, and _gh_ to the vanishing-point of _ac_, and these lines will
bisect the sides of the square, so that _ag_ is the perspective
representation of half the side _ab_; _ae_ is half _ac_; _ch_ is half
_cd_; and _bf_ is half _bd_.

  [Illustration: Fig. 22.]


COROLLARY III.

Since _ABCD_, Fig. 20., is a square, _BAC_ is a right angle; and as
_TV_ is parallel to _AB_, and _TV′_ to _AC_, _V′TV_ must be a right
angle also.

As the ground plan of most buildings is rectangular, it constantly
happens in practice that their angles (as the corners of ordinary
houses) throw the lines to the vanishing-points thus at right angles;
and so that this law is observed, and _VTV′_ is kept a right angle, it
does not matter in general practice whether the vanishing-points are
thrown a little more or a little less to the right or left of _S_: but
it matters much that the relation of the vanishing-points should be
accurate. Their position with respect to _S_ merely causes the
spectator to see a little more or less on one side or other of the
house, which may be a matter of chance or choice; but their
rectangular relation determines the rectangular shape of the building,
which is an essential point.




PROBLEM IX.

TO DRAW A SQUARE PILLAR, GIVEN IN POSITION AND MAGNITUDE, ITS BASE AND
TOP BEING IN HORIZONTAL PLANES.


Let _AH_, Fig. 23., be the square pillar.

Then, as it is given in position and magnitude, the position and
magnitude of the square it stands upon must be given (that is, the
line _AB_ or _AC_ in position), and the height of its side _AE_.

  [Illustration: Fig. 23.]  [Illustration: Fig. 24.]

Find the sight-magnitudes of _AB_ and _AE_. Draw the two sides _ab_,
_ac_, of the square of the base, by Problem VIII., as in Fig. 24. From
the points _a_, _b_, and _c_, raise vertical lines _ae_, _cf_, _bg_.

Make _ae_ equal to the sight-magnitude of _AE_.

Now because the top and base of the pillar are in horizontal planes,
the square of its top, _FG_, is parallel to the square of its base,
_BC_.

Therefore the line _EF_ is parallel to _AC_, and _EG_ to _AB_.

Therefore _EF_ has the same vanishing-point as _AC_, and _EG_ the same
vanishing-point as _AB_.

From _e_ draw _ef_ to the vanishing-point of _ac_, cutting _cf_ in
_f_.

Similarly draw _eg_ to the vanishing-point of _ab_, cutting _bg_ in
_g_.

Complete the square _gf_ in _h_, by drawing _gh_ to the
vanishing-point of _ef_, and _fh_ to the vanishing-point of _eg_,
cutting each other in _h_. Then _aghf_ is the square pillar required.


COROLLARY.

It is obvious that if _AE_ is equal to _AC_, the whole figure will be
a cube, and each side, _aefc_ and _aegb_, will be a square in a given
vertical plane. And by making _AB_ or _AC_ longer or shorter in any
given proportion, any form of rectangle may be given to either of the
sides of the pillar. No other rule is therefore needed for drawing
squares or rectangles in vertical planes.

Also any triangle may be thus drawn in a vertical plane, by inclosing
it in a rectangle and determining, in perspective ratio, on the sides
of the rectangle, the points of their contact with the angles of the
triangle.

And if any triangle, then any polygon.

A less complicated construction will, however, be given hereafter.[18]


  [18] See page 96 (note), after you have read Problem XVI.




PROBLEM X.

TO DRAW A PYRAMID, GIVEN IN POSITION AND MAGNITUDE, ON A SQUARE BASE
IN A HORIZONTAL PLANE.


  [Illustration: Fig. 25.]

Let _AB_, Fig. 25., be the four-sided pyramid. As it is given in
position and magnitude, the square base on which it stands must be
given in position and magnitude, and its vertical height, _CD_.[19]

  [Illustration: Fig. 26.]

Draw a square pillar, _ABGE_, Fig. 26., on the square base of the
pyramid, and make the height of the pillar _AF_ equal to the vertical
height of the pyramid _CD_ (Problem IX.). Draw the diagonals _GF_,
_HI_, on the top of the square pillar, cutting each other in _C_.
Therefore _C_ is the center of the square _FGHI_. (Prob. VIII.
Cor. II.)

  [Illustration: Fig. 27.]

Join _CE_, _CA_, _CB_.

Then _ABCE_ is the pyramid required. If the base of the pyramid is
above the eye, as when a square spire is seen on the top of a
church-tower, the construction will be as in Fig. 27.


  [19] If, instead of the vertical height, the length of _AD_ is
       given, the vertical must be deduced from it. See the Exercises
       on this Problem in the Appendix, p. 79.




PROBLEM XI.

TO DRAW ANY CURVE IN A HORIZONTAL OR VERTICAL PLANE.


  [Illustration: Fig. 28.]

Let _AB_, Fig. 28., be the curve.

Inclose it in a rectangle, _CDEF_.

Fix the position of the point _C_ or _D_, and draw the rectangle.
(Problem VIII. Coroll. I.)[20]

Let _CDEF_, Fig. 29., be the rectangle so drawn.

  [Illustration: Fig. 29.]

If an extremity of the curve, as _A_, is in a side of the rectangle,
divide the side _CE_, Fig. 29., so that _AC_ shall be (in perspective
ratio) to _AE_ as _AC_ is to _AE_ in Fig. 28. (Prob. V. Cor. II.)

Similarly determine the points of contact of the curve and rectangle
_e_, _f_, _g_.

If an extremity of the curve, as _B_, is not in a side of the
rectangle, let fall the perpendiculars _Ba_, _Bb_ on the rectangle
sides. Determine the correspondent points _a_ and _b_ in Fig. 29., as
you have already determined _A_, _B_, _e_, and _f_.

From _b_, Fig. 29., draw _bB_ parallel to _CD_,[21] and from _a_ draw
_aB_ to the vanishing-point of _DF_, cutting each other in _B_. Then
_B_ is the extremity of the curve.

Determine any other important point in the curve, as _P_, in the same
way, by letting fall _Pq_ and _Pr_ on the rectangle’s sides.

Any number of points in the curve may be thus determined, and the
curve drawn through the series; in most cases, three or four will be
enough. Practically, complicated curves may be better drawn in
perspective by an experienced eye than by rule, as the fixing of the
various points in haste involves too many chances of error; but it is
well to draw a good many by rule first, in order to give the eye its
experience.[22]


COROLLARY.

If the curve required be a circle, Fig. 30., the rectangle which
incloses it will become a square, and the curve will have four points
of contact, _ABCD_, in the middle of the sides of the square.

  [Illustration: Fig. 30.]

Draw the square, and as a square may be drawn about a circle in any
position, draw it with its nearest side, _EG_, parallel to the
sight-line.

Let _EF_, Fig. 31., be the square so drawn.

Draw its diagonals _EF_, _GH_; and through the center of the square
(determined by their intersection) draw _AB_ to the vanishing-point of
_GF_, and _CD_ parallel to _EG_. Then the points _ABCD_ are the four
points of the circle’s contact.

  [Illustration: Fig. 31.]

On _EG_ describe a half square, _EL_; draw the semicircle _KAL_; and
from its center, _R_, the diagonals _RE_, _RG_, cutting the circle in
_x_, _y_.

From the points _x_ _y_, where the circle cuts the diagonals, raise
perpendiculars, _Px_, _Qy_, to _EG_.

From _P_ and _Q_ draw _PP′_, _QQ′_, to the vanishing-point of _GF_,
cutting the diagonals in _m_, _n_, and _o_, _p_.

Then _m_, _n_, _o_, _p_ are four other points in the circle.

Through these eight points the circle may be drawn by the hand
accurately enough for general purposes; but any number of points
required may, of course, be determined, as in Problem XI.

The distance _EP_ is approximately one-seventh of _EG_, and may be
assumed to be so in quick practice, as the error involved is not
greater than would be incurred in the hasty operation of drawing the
circle and diagonals.

It may frequently happen that, in consequence of associated
constructions, it may be inconvenient to draw _EG_ parallel to the
sight-line, the square being perhaps first constructed in some oblique
direction. In such cases, _QG_ and _EP_ must be determined in
perspective ratio by the dividing-point, the line _EG_ being used as a
measuring-line.

  [_Obs._ In drawing Fig. 31. the station-point has been taken much
  nearer the paper than is usually advisable, in order to show the
  character of the curve in a very distinct form.

  If the student turns the book so that _EG_ may be vertical,
  Fig. 31. will represent the construction for drawing a circle in a
  vertical plane, the sight-line being then of course parallel to
  _GL_; and the semicircles _ADB_, _ACB_, on each side of the
  diameter _AB_, will represent ordinary semicircular arches seen in
  perspective. In that case, if the book be held so that the line
  _EH_ is the top of the square, the upper semicircle will represent
  a semicircular arch, _above_ the eye, drawn in perspective. But if
  the book be held so that the line _GF_ is the top of the square,
  the upper semicircle will represent a semicircular arch, _below_
  the eye, drawn in perspective.

  If the book be turned upside down, the figure will represent a
  circle drawn on the ceiling, or any other horizontal plane above
  the eye; and the construction is, of course, accurate in every
  case.]


  [20] Or if the curve is in a vertical plane, Coroll. to Problem IX.
       As a rectangle may be drawn in any position round any given
       curve, its position with respect to the curve will in either
       case be regulated by convenience. See the Exercises on this
       Problem, in the Appendix, p. 85.

  [21] Or to its vanishing-point, if _CD_ has one.

  [22] Of course, by dividing the original rectangle into any number
       of equal rectangles, and dividing the perspective rectangle
       similarly, the curve may be approximately drawn without any
       trouble; but, when accuracy is required, the points should be
       fixed, as in the problem.




PROBLEM XII.

TO DIVIDE A CIRCLE DRAWN IN PERSPECTIVE INTO ANY GIVEN NUMBER OF EQUAL
PARTS.


Let _AB_, Fig. 32., be the circle drawn in perspective. It is required
to divide it into a given number of equal parts; in this case, 20.

Let _KAL_ be the semicircle used in the construction. Divide the
semicircle _KAL_ into half the number of parts required; in this case,
10.

Produce the line _EG_ laterally, as far as may be necessary.

From _O_, the center of the semicircle _KAL_, draw radii through the
points of division of the semicircle, _p_, _q_, _r_, etc., and produce
them to cut the line _EG_ in _P_, _Q_, _R_, etc.

From the points _PQR_ draw the lines _PP′_, _QQ′_, _RR′_, etc.,
through the center of the circle _AB_, each cutting the circle in two
points of its circumference.

Then these points divide the perspective circle as required.

If from each of the points _p_, _q_, _r_, a vertical were raised to
the line _EG_, as in Fig. 31., and from the point where it cut _EG_ a
line were drawn to the vanishing-point, as _QQ′_ in Fig. 31., this
line would also determine two of the points of division.

  [Illustration: Fig. 32.]

If it is required to divide a circle into any number of given
_un_equal parts (as in the points _A_, _B_, and _C_, Fig. 33.), the
shortest way is thus to raise vertical lines from _A_ and _B_ to the
side of the perspective square _XY_, and then draw to the
vanishing-point, cutting the perspective circle in _a_ and _b_, the
points required. Only notice that if any point, as _A_, is on the
nearer side of the circle _ABC_, its representative point, _a_, must
be on the nearer side of the circle _abc_; and if the point _B_ is on
the farther side of the circle _ABC_, _b_ must be on the farther side
of _abc_. If any point, as _C_, is so much in the lateral arc of the
circle as not to be easily determinable by the vertical line, draw the
horizontal _CP_, find the correspondent _p_ in the side of the
perspective square, and draw _pc_ parallel to _XY_, cutting the
perspective circle in _c_.

  [Illustration: Fig. 33.]


COROLLARY.

It is obvious that if the points _P′_, _Q′_, _R_, etc., by which the
circle is divided in Fig. 32., be joined by right lines, the resulting
figure will be a regular equilateral figure of twenty sides inscribed
in the circle. And if the circle be divided into given unequal parts,
and the points of division joined by right lines, the resulting figure
will be an irregular polygon inscribed in the circle with sides of
given length.

Thus any polygon, regular or irregular, inscribed in a circle, may be
inscribed in position in a perspective circle.




PROBLEM XIII.

TO DRAW A SQUARE, GIVEN IN MAGNITUDE, WITHIN A LARGER SQUARE GIVEN IN
POSITION AND MAGNITUDE; THE SIDES OF THE TWO SQUARES BEING PARALLEL.


  [Illustration: Fig. 34.]

Let _AB_, Fig. 34., be the sight-magnitude of the side of the smaller
square, and _AC_ that of the side of the larger square.

Draw the larger square. Let _DEFG_ be the square so drawn.

Join _EG_ and _DF_.

On either _DE_ or _DG_ set off, in perspective ratio, _DH_ equal to
one half of _BC_. Through _H_ draw _HK_ to the vanishing-point of
_DE_, cutting _DF_ in _I_ and _EG_ in _K_. Through _I_ and _K_ draw
_IM_, _KL_, to vanishing-point of _DG_, cutting _DF_ in _L_ and _EG_
in _M_. Join _LM_.

Then _IKLM_ is the smaller square, inscribed as required.[23]


COROLLARY.

  [Illustration: Fig. 36.]

If, instead of one square within another, it be required to draw one
circle within another, the dimensions of both being given, inclose
each circle in a square. Draw the squares first, and then the circles
within, as in Fig. 36.


  [23] [Illustration: Fig. 35.] If either of the sides of the greater
       square is parallel to the plane of the picture, as _DG_ in
       Fig. 35., _DG_ of course must be equal to _AC_, and _DH_ equal
       to _BC_/2, and the construction is as in Fig. 35.




PROBLEM XIV.

TO DRAW A TRUNCATED CIRCULAR CONE, GIVEN IN POSITION AND MAGNITUDE,
THE TRUNCATIONS BEING IN HORIZONTAL PLANES, AND THE AXIS OF THE CONE
VERTICAL.


Let _ABCD_, Fig. 37., be the portion of the cone required.

  [Illustration: Fig. 37.]

As it is given in magnitude, its diameters must be given at the base
and summit, _AB_ and _CD_; and its vertical height, _CE_.[24]

And as it is given in position, the center of its base must be given.

  [Illustration: Fig. 38.]

Draw in position, about this center,[25] the square pillar _afd_,
Fig. 38., making its height, _bg_, equal to _CE_; and its side, _ab_,
equal to _AB_.

In the square of its base, _abcd_, inscribe a circle, which therefore
is of the diameter of the base of the cone, _AB_.

In the square of its top, _efgh_, inscribe concentrically a circle
whose diameter shall equal _CD_. (Coroll. Prob. XIII.)

Join the extremities of the circles by the right lines _kl_, _nm_.
Then _klnm_ is the portion of cone required.


COROLLARY I.

If similar polygons be inscribed in similar positions in the circles
_kn_ and _lm_ (Coroll. Prob. XII.), and the corresponding angles of
the polygons joined by right lines, the resulting figure will be a
portion of a polygonal pyramid. (The dotted lines in Fig. 38.,
connecting the extremities of two diameters and one diagonal in the
respective circles, occupy the position of the three nearest angles of
a regular octagonal pyramid, having its angles set on the diagonals
and diameters of the square _ad_, inclosing its base.)

If the cone or polygonal pyramid is not truncated, its apex will be
the center of the upper square, as in Fig. 26.


COROLLARY II.

If equal circles, or equal and similar polygons, be inscribed in the
upper and lower squares in Fig. 38., the resulting figure will be a
vertical cylinder, or a vertical polygonal pillar, of given height and
diameter, drawn in position.


COROLLARY III.

If the circles in Fig. 38., instead of being inscribed in the squares
_bc_ and _fg_, be inscribed in the sides of the solid figure _be_ and
_df_, those sides being made square, and the line _bd_ of any given
length, the resulting figure will be, according to the constructions
employed, a cone, polygonal pyramid, cylinder, or polygonal pillar,
drawn in position about a horizontal axis parallel to _bd_.

Similarly, if the circles are drawn in the sides _gd_ and _ec_, the
resulting figures will be described about a horizontal axis parallel
to _ab_.


  [24] Or if the length of its side, _AC_, is given instead, take
       _ae_, Fig. 37., equal to half the excess of _AB_ over _CD_;
       from the point _e_ raise the perpendicular _ce_. With center
       _a_, and distance _AC_, describe a circle cutting _ce_ in _c_.
       Then _ce_ is the vertical height of the portion of cone
       required, or _CE_.

  [25] The direction of the side of the square will of course be
       regulated by convenience.




PROBLEM XV.

TO DRAW AN INCLINED LINE, GIVEN IN POSITION AND MAGNITUDE.


We have hitherto been examining the conditions of horizontal and
vertical lines only, or of curves inclosed in rectangles.

  [Illustration: Fig. 39.]  [Illustration: Fig. 40.]

We must, in conclusion, investigate the perspective of inclined lines,
beginning with a single one given in position. For the sake of
completeness of system, I give in Appendix II. Article III. the
development of this problem from the second. But, in practice, the
position of an inclined line may be most conveniently defined by
considering it as the diagonal of a rectangle, as _AB_ in Fig. 39.,
and I shall therefore, though at some sacrifice of system, examine it
here under that condition.

If the sides of the rectangle _AC_ and _AD_ are given, the slope of
the line _AB_ is determined; and then its position will depend on that
of the rectangle. If, as in Fig. 39., the rectangle is parallel to the
picture plane, the line _AB_ must be so also. If, as in Fig. 40., the
rectangle is inclined to the picture plane, the line _AB_ will be so
also. So that, to fix the position of _AB_, the line _AC_ must be
given in position and magnitude, and the height _AD_.

  [Illustration: Fig. 41.]

If these are given, and it is only required to draw the single line
_AB_ in perspective, the construction is entirely simple; thus:—

Draw the line _AC_ by Problem I.

Let _AC_, Fig. 41., be the line so drawn. From _a_ and _c_ raise the
vertical lines _ad_, _cb_. Make _ad_ equal to the sight-magnitude of
_AD_. From _d_ draw _db_ to the vanishing-point of _ac_, cutting _bc_
in _b_.

Join _ab_. Then _ab_ is the inclined line required.

  [Illustration: Fig. 42.]

If the line is inclined in the opposite direction, as _DC_ in
Fig. 42., we have only to join _dc_ instead of _ab_ in Fig. 41., and
_dc_ will be the line required.

I shall hereafter call the line _AC_, when used to define the position
of an inclined line _AB_ (Fig. 40.), the “relative horizontal” of the
line _AB_.


OBSERVATION.

  [Illustration: Fig. 43.]

In general, inclined lines are most needed for gable roofs, in which,
when the conditions are properly stated, the vertical height of the
gable, _XY_, Fig. 43., is given, and the base line, _AC_, in position.
When these are given, draw _AC_; raise vertical _AD_; make _AD_ equal
to sight-magnitude of _XY_; complete the perspective-rectangle _ADBC_;
join _AB_ and _DC_ (as by dotted lines in figure); and through the
intersection of the dotted lines draw vertical _XY_, cutting _DB_ in
_Y_. Join _AY_, _CY_; and these lines are the sides of the gable. If
the length of the roof _AA′_ is also given, draw in perspective the
complete parallelopiped _A′D′BC_, and from _Y_ draw _YY′_ to the
vanishing-point of _AA′_, cutting _D′B′_ in _Y′_. Join _A′Y_, and you
have the slope of the farther side of the roof.

  [Illustration: Fig. 44.]

The construction above the eye is as in Fig. 44.; the roof is reversed
in direction merely to familiarize the student with the different
aspects of its lines.




PROBLEM XVI.

TO FIND THE VANISHING-POINT OF A GIVEN INCLINED LINE.


If, in Fig. 43. or Fig. 44., the lines _AY_ and _A′Y′_ be produced,
the student will find that they meet.

Let _P_, Fig. 45., be the point at which they meet.

From _P_ let fall the vertical _PV_ on the sight-line, cutting the
sight-line in _V_.

Then the student will find experimentally that _V_ is the
vanishing-point of the line _AC_.[26]

Complete the rectangle of the base _AC′_, by drawing _A′C′_ to _V_,
and _CC′_ to the vanishing-point of _AA′_.

Join _Y′C′_.

Now if _YC_ and _Y′C′_ be produced downwards, the student will find
that they meet.

Let them be produced, and meet in _P′_.

Produce _PV_, and it will be found to pass through the point _P′_.

Therefore if _AY_ (or _CY_), Fig. 45., be any inclined line drawn in
perspective by Problem XV., and _AC_ the relative horizontal (_AC_ in
Figs. 39, 40.), also drawn in perspective.

Through _V_, the vanishing-point of _AV_, draw the vertical _PP′_
upwards and downwards.

Produce _AY_ (or _CY_), cutting _PP′_ in _P_ (or _P′_).

Then _P_ is the vanishing-point of _AY_ (or _P′_ of _CY_).

  [Illustration: Fig. 45.]

The student will observe that, in order to find the point _P_ by this
method, it is necessary first to draw a portion of the given inclined
line by Problem XV. Practically, it is always necessary to do so, and,
therefore, I give the problem in this form.

Theoretically, as will be shown in the analysis of the problem, the
point _P_ should be found by drawing a line from the station-point
parallel to the given inclined line: but there is no practical means
of drawing such a line; so that in whatever terms the problem may be
given, a portion of the inclined line (_AY_ or _CY_) must always be
drawn in perspective before P can be found.


  [26] The demonstration is in Appendix II. Article III.




PROBLEM XVII.

TO FIND THE DIVIDING-POINTS OF A GIVEN INCLINED LINE.


  [Illustration: Fig. 46.]

Let _P_, Fig. 46., be the vanishing-point of the inclined line, and
_V_ the vanishing-point of the relative horizontal.

Find the dividing-points of the relative horizontal, _D_ and _D′_.

Through _P_ draw the horizontal line _XY_.

With center _P_ and distance _DP_ describe the two arcs _DX_ and
_D′Y_, cutting the line _XY_ in _X_ and _Y_.

Then _X_ and _Y_ are the dividing-points of the inclined line.[27]

_Obs._ The dividing-points found by the above rule, used with the
ordinary measuring-line, will lay off distances on the retiring
inclined line, as the ordinary dividing-points lay them off on the
retiring horizontal line.

Another dividing-point, peculiar in its application, is sometimes
useful, and is to be found as follows:—

  [Illustration: Fig. 47.]

Let _AB_, Fig. 47., be the given inclined line drawn in perspective,
and _Ac_ the relative horizontal.

Find the vanishing-points, _V_ and _E_, of _Ac_ and _AB_; _D_, the
dividing-point of _Ac_; and the sight-magnitude of _Ac_ on the
measuring-line, or _AC_.

From _D_ erect the perpendicular _DF_.

Join _CB_, and produce it to cut _DE_ in _F_. Join _EF_.

Then, by similar triangles, _DF_ is equal to _EV_, and _EF_ is
parallel to _DV_.

Hence it follows that if from _D_, the dividing-point of _Ac_, we
raise a perpendicular and make _DF_ equal to _EV_, a line _CF_, drawn
from any point _C_ on the measuring-line to _F_, will mark the
distance _AB_ on the inclined line, _AB_ being the portion of the
given inclined line which forms the diagonal of the vertical rectangle
of which _AC_ is the base.


  [27] The demonstration is in Appendix II., p. 104.




PROBLEM XVIII.

TO FIND THE SIGHT-LINE OF AN INCLINED PLANE IN WHICH TWO LINES ARE
GIVEN IN POSITION.[28]


As in order to fix the position of a line two points in it must be
given, so in order to fix the position of a plane, two lines in it
must be given.

  [Illustration: Fig. 48]

Let the two lines be _AB_ and _CD_, Fig. 48.

As they are given in position, the relative horizontals _AE_ and _CF_
must be given.

Then by Problem XVI. the vanishing-point of _AB_ is _V_, and of _CD_,
_V′_.

Join _VV′_ and produce it to cut the sight-line in _X_.

Then _VX_ is the sight-line of the inclined plane.

Like the horizontal sight-line, it is of indefinite length; and may be
produced in either direction as occasion requires, crossing the
horizontal line of sight, if the plane continues downward in that
direction.

_X_ is the vanishing-point of all horizontal lines in the inclined
plane.


  [28] Read the Article on this problem in the Appendix, p. 97, before
       investigating the problem itself.




PROBLEM XIX.

TO FIND THE VANISHING-POINT OF STEEPEST LINES IN AN INCLINED PLANE
WHOSE SIGHT-LINE IS GIVEN.


  [Illustration: Fig. 49.]

Let _VX_, Fig. 49., be the given sight-line.

Produce it to cut the horizontal sight-line in _X_.

Therefore _X_ is the vanishing-point of horizontal lines in the given
inclined plane. (Problem XVIII.)

Join _TX_, and draw _TY_ at right angles to _TX_.

Therefore _Y_ is the rectangular vanishing-point corresponding to
_X_.[29]

From _Y_ erect the vertical _YP_, cutting the sight-line of the
inclined plane in _P_.

Then _P_ is the vanishing-point of steepest lines in the plane.

All lines drawn to it, as _QP_, _RP_, _NP_, etc., are the steepest
possible in the plane; and all lines drawn to _X_, as _QX_, _OX_,
etc., are horizontal, and at right angles to the lines _PQ_, _PR_,
etc.


  [29] That is to say, the vanishing-point of horizontal lines drawn at
       right angles to the lines whose vanishing-point is _X_.




PROBLEM XX.

TO FIND THE VANISHING-POINT OF LINES PERPENDICULAR TO THE SURFACE OF A
GIVEN INCLINED PLANE.


  [Illustration: Fig. 50.]

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

Let _AB_, Fig. 50., be a portion of a steepest line in the given plane,
and _V_ the vanishing-point of its relative horizontal.

Through _V_ draw the vertical _GF_ upwards and downwards.

From _A_ set off any portion of the relative horizontal _AC_, and on
_AC_ describe a semicircle in a vertical plane, _ADC_, cutting _AB_ in
_E_.

Join _EC_, and produce it to cut _GF_ in _F_.

Then _F_ is the vanishing-point required.

For, because _AEC_ is an angle in a semicircle, it is a right angle;
and therefore the line _EF_ is at right angles to the line _AB_; and
similarly all lines drawn to _F_, and therefore parallel to _EF_, are
at right angles with any line which cuts them, drawn to the
vanishing-point of _AB_.

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

       *       *       *       *       *

The preceding series of constructions, with the examples in the first
Article of the Appendix, put it in the power of the student to draw
any form, however complicated,[30] which does not involve intersection
of curved surfaces. I shall not proceed to the analysis of any of
these more complex problems, as they are entirely useless in the
ordinary practice of artists. For a few words only I must ask the
reader’s further patience, respecting the general placing and scale of
the picture.

As the horizontal sight-line is drawn through the sight-point, and the
sight-point is opposite the eye, the sight-line is always on a level
with the eye. Above and below the sight-line, the eye comprehends, as
it is raised or depressed while the head is held upright, about an
equal space; and, on each side of the sight-point, about the same
space is easily seen without turning the head; so that if a picture
represented the true field of easy vision, it ought to be circular,
and have the sight-point in its center. But because some parts of any
given view are usually more interesting than others, either the
uninteresting parts are left out, or somewhat more than would
generally be seen of the interesting parts is included, by moving the
field of the picture a little upwards or downwards, so as to throw the
sight-point low or high. The operation will be understood in a moment
by cutting an aperture in a piece of pasteboard, and moving it up and
down in front of the eye, without moving the eye. It will be seen to
embrace sometimes the low, sometimes the high objects, without
altering their perspective, only the eye will be opposite the lower
part of the aperture when it sees the higher objects, and _vice
versâ_.

There is no reason, in the laws of perspective, why the picture should
not be moved to the right or left of the sight-point, as well as up or
down. But there is this practical reason. The moment the spectator
sees the horizon in a picture high, he tries to hold his head high,
that is, in its right place. When he sees the horizon in a picture
low, he similarly tries to put his head low. But, if the sight-point
is thrown to the left hand or right hand, he does not understand that
he is to step a little to the right or left; and if he places himself,
as usual, in the middle, all the perspective is distorted. Hence it is
generally unadvisable to remove the sight-point laterally, from the
center of the picture. The Dutch painters, however, fearlessly take
the license of placing it to the right or left; and often with good
effect.

The rectilinear limitation of the sides, top, and base of the picture
is of course quite arbitrary, as the space of a landscape would be
which was seen through a window; less or more being seen at the
spectator’s pleasure, as he retires or advances.

The distance of the station-point is not so arbitrary. In ordinary
cases it should not be less than the intended greatest dimension
(height or breadth) of the picture. In most works by the great masters
it is more; they not only calculate on their pictures being seen at
considerable distances, but they like breadth of mass in buildings,
and dislike the sharp angles which always result from station-points
at short distances.[31]

Whenever perspective, done by true rule, looks wrong, it is always
because the station-point is too near. Determine, in the outset, at
what distance the spectator is likely to examine the work, and never
use a station-point within a less distance.

There is yet another and a very important reason, not only for care in
placing the station-point, but for that accurate calculation of
distance and observance of measurement which have been insisted on
throughout this work. All drawings of objects on a reduced scale are,
if rightly executed, drawings of the appearance of the object at the
distance which in true perspective reduces it to that scale. They are
not _small_ drawings of the object seen near, but drawings the _real
size_ of the object seen far off. Thus if you draw a mountain in a
landscape, three inches high, you do not reduce all the features of
the near mountain so as to come into three inches of paper. You could
not do that. All that you can do is to give the appearance of the
mountain, when it is so far off that three inches of paper would
really hide it from you. It is precisely the same in drawing any other
object. A face can no more be reduced in scale than a mountain can. It
is infinitely delicate already; it can only be quite rightly rendered
on its own scale, or at least on the slightly diminished scale which
would be fixed by placing the plate of glass, supposed to represent
the field of the picture, close to the figures. Correggio and Raphael
were both fond of this slightly subdued magnitude of figure. Colossal
painting, in which Correggio excelled all others, is usually the
enlargement of a small picture (as a colossal sculpture is of a small
statue), in order to permit the subject of it to be discerned at a
distance. The treatment of colossal (as distinguished from ordinary)
paintings will depend therefore, in general, on the principles of
optics more than on those of perspective, though, occasionally,
portions may be represented as if they were the projection of near
objects on a plane behind them. In all points the subject is one of
great difficulty and subtlety; and its examination does not fall
within the compass of this essay.

Lastly, it will follow from these considerations, and the conclusion
is one of great practical importance, that, though pictures may be
enlarged, they cannot be reduced, in copying them. All attempts to
engrave pictures completely on a reduced scale are, for this reason,
nugatory. The best that can be done is to give the aspect of the
picture at the distance which reduces it in perspective to the size
required; or, in other words, to make a drawing of the distant effect
of the picture. Good painting, like nature’s own work, is infinite,
and unreduceable.

I wish this book had less tendency towards the infinite and
unreduceable. It has so far exceeded the limits I hoped to give it,
that I doubt not the reader will pardon an abruptness of conclusion,
and be thankful, as I am myself, to get to an end on any terms.


  [30] As in algebraic science, much depends, in complicated
       perspective, on the student’s ready invention of expedients,
       and on his quick sight of the shortest way in which the
       solution may be accomplished, when there are several ways.

  [31] The greatest masters are also fond of parallel perspective,
       that is to say, of having one side of their buildings fronting
       them full, and therefore parallel to the picture plane, while
       the other side vanishes to the sight-point. This is almost
       always done in figure backgrounds, securing simple and balanced
       lines.




APPENDIX.


I.

PRACTICE AND OBSERVATIONS.


II.

DEMONSTRATIONS.




I.

PRACTICE AND OBSERVATIONS ON THE PRECEDING PROBLEMS.




PROBLEM I.


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.




PROBLEM IX.


The drawing of most buildings occurring in ordinary practice will
resolve itself into applications of this problem. In general, any
house, or block of houses, presents itself under the main conditions
assumed here in Fig. 54. There will be an angle or corner somewhere
near the spectator, as _AB_; and the level of the eye will usually be
above the base of the building, of which, therefore, the horizontal
upper lines will slope down to the vanishing-points, and the base
lines rise to them. The following practical directions will, however,
meet nearly all cases:—

  [Illustration: Fig. 54.]

Let _AB_, Fig. 54., be any important vertical line in the block of
buildings; if it is the side of a street, you may fix upon such a line
at the division between two houses. If its real height, distance,
etc., are given, you will proceed with the accurate construction of
the problem; but usually you will neither know, nor care, exactly how
high the building is, or how far off. In such case draw the line _AB_,
as nearly as you can guess, about the part of the picture it ought to
occupy, and on such a scale as you choose. Divide it into any
convenient number of equal parts, according to the height you presume
it to be. If you suppose it to be twenty feet high, you may divide it
into twenty parts, and let each part stand for a foot; if thirty feet
high, you may divide it into ten parts, and let each part stand for
three feet; if seventy feet high, into fourteen parts, and let each
part stand for five feet; and so on, avoiding thus very minute
divisions till you come to details. Then observe how high your eye
reaches upon this vertical line; suppose, for instance, that it is
thirty feet high and divided into ten parts, and you are standing so
as to raise your head to about six feet above its base, then the
sight-line may be drawn, as in the figure, through the second division
from the ground. If you are standing above the house, draw the
sight-line above _B_; if below the house, below _A_; at such height or
depth as you suppose may be accurate (a yard or two more or less
matters little at ordinary distances, while at great distances
perspective rules become nearly useless, the eye serving you better
than the necessarily imperfect calculation). Then fix your sight-point
and station-point, the latter with proper reference to the scale of
the line _AB_. As you cannot, in all probability, ascertain the exact
direction of the line _AV_ or _BV_, draw the slope _BV_ as it appears
to you, cutting the sight-line in _V_. Thus having fixed one
vanishing-point, the other, and the dividing-points, must be
accurately found by rule; for, as before stated, whether your entire
group of points (vanishing and dividing) falls a little more or less
to the right or left of _S_ does not signify, but the relation of the
points to each other _does_ signify. Then draw the measuring-line
_BG_, either through _A_ or _B_, choosing always the steeper slope of
the two; divide the measuring-line into parts of the same length as
those used on _AB_, and let them stand for the same magnitudes. Thus,
suppose there are two rows of windows in the house front, each window
six feet high by three wide, and separated by intervals of three feet,
both between window and window and between tier and tier; each of the
divisions here standing for three feet, the lines drawn from _BG_ to
the dividing-point _D_ fix the lateral dimensions, and the divisions
on _AB_ the vertical ones. For other magnitudes it would be necessary
to subdivide the parts on the measuring-line, or on _AB_, as required.
The lines which regulate the inner sides or returns of the windows
(_a_, _b_, _c_, etc.) of course are drawn to the vanishing-point of
_BF_ (the other side of the house), if _FBV_ represents a right angle;
if not, their own vanishing-point must be found separately for these
returns. But see Practice on Problem XI.

  [Illustration: Fig. 55.]

Interior angles, such as _EBC_, Fig. 55. (suppose the corner of a
room), are to be treated in the same way, each side of the room having
its measurements separately carried to it from the measuring-line. It
may sometimes happen in such cases that we have to carry the
measurement _up_ from the corner _B_, and that the sight-magnitudes
are given us from the length of the line _AB_. For instance, suppose
the room is eighteen feet high, and therefore _AB_ is eighteen feet;
and we have to lay off lengths of six feet on the top of the room
wall, _BC_. Find _D_, the dividing-point of _BC_. Draw a
measuring-line, _BF_, from _B_; and another, _gC_, anywhere above. On
_BF_ lay off _BG_ equal to one third of _AB_, or six feet; and draw
from _D_, through _G_ and _B_, the lines _Gg_, _Bb_, to the upper
measuring-line. Then _gb_ is six feet on that measuring-line. Make
_bc_, _ch_, etc., equal to _bg_; and draw _ce_, _hf_, etc., to _D_,
cutting _BC_ in _e_ and _f_, which mark the required lengths of six
feet each at the top of the wall.




PROBLEM X.


This is one of the most important foundational problems in
perspective, and it is necessary that the student should entirely
familiarize himself with its conditions.

In order to do so, he must first observe these general relations of
magnitude in any pyramid on a square base.

Let _AGH′_, Fig. 56., be any pyramid on a square base.

  [Illustration: Fig. 56.]

The best terms in which its magnitude can be given, are the length of
one side of its base, _AH_, and its vertical altitude (_CD_ in
Fig. 25.); for, knowing these, we know all the other magnitudes. But
these are not the terms in which its size will be usually
ascertainable. Generally, we shall have given us, and be able to
ascertain by measurement, one side of its base _AH_, and either _AG_
the length of one of the lines of its angles, or _BG_ (or _B′G_) the
length of a line drawn from its vertex, _G_, to the middle of the side
of its base. In measuring a real pyramid, _AG_ will usually be the
line most easily found; but in many architectural problems _BG_ is
given, or is most easily ascertainable.

Observe therefore this general construction.

  [Illustration: Fig. 57.]

Let _ABDE_, Fig. 57., be the square base of any pyramid.

Draw its diagonals, _AE_, _BD_, cutting each other in its center, _C_.

Bisect any side, _AB_, in _F_.

From _F_ erect vertical _FG_.

Produce _FB_ to _H_, and make _FH_ equal to _AC_.

Now if the vertical altitude of the pyramid (_CD_ in Fig. 25.) be
given, make _FG_ equal to this vertical altitude.

Join _GB_ and _GH_.

Then _GB_ and _GH_ are the true magnitudes of _GB_ and _GH_ in
Fig. 56.

If _GB_ is given, and not the vertical altitude, with center _B_, and
distance _GB_, describe circle cutting _FG_ in _G_, and _FG_ is the
vertical altitude.

If _GH_ is given, describe the circle from _H_, with distance _GH_,
and it will similarly cut _FG_ in _G_.

It is especially necessary for the student to examine this
construction thoroughly, because in many complicated forms of
ornaments, capitals of columns, etc., the lines _BG_ and _GH_ become
the limits or bases of curves, which are elongated on the longer (or
angle) profile _GH_, and shortened on the shorter (or lateral) profile
_BG_. We will take a simple instance, but must previously note another
construction.

It is often necessary, when pyramids are the roots of some ornamental
form, to divide them horizontally at a given vertical height. The
shortest way of doing so is in general the following.

  [Illustration: Fig. 58.]

Let _AEC_, Fig. 58., be any pyramid on a square base _ABC_, and _ADC_
the square pillar used in its construction.

Then by construction (Problem X.) _BD_ and _AF_ are both of the
vertical height of the pyramid.

Of the diagonals, _FE_, _DE_, choose the shortest (in this case _DE_),
and produce it to cut the sight-line in _V_.

Therefore _V_ is the vanishing-point of _DE_.

Divide _DB_, as may be required, into the sight-magnitudes of the
given vertical heights at which the pyramid is to be divided.

  [Illustration: Fig. 59.]  [Illustration: Fig. 60.]

From the points of division, 1, 2, 3, etc., draw to the
vanishing-point _V_. The lines so drawn cut the angle line of the
pyramid, _BE_, at the required elevations. Thus, in the figure, it is
required to draw a horizontal black band on the pyramid at three
fifths of its height, and in breadth one twentieth of its height. The
line _BD_ is divided into five parts, of which three are counted from
_B_ upwards. Then the line drawn to _V_ marks the base of the black
band. Then one fourth of one of the five parts is measured, which
similarly gives the breadth of the band. The terminal lines of the
band are then drawn on the sides of the pyramid parallel to _AB_ (or
to its vanishing-point if it has one), and to the vanishing-point of
_BC_.

If it happens that the vanishing-points of the diagonals are awkwardly
placed for use, bisect the nearest base line of the pyramid in _B_, as
in Fig. 59.

Erect the vertical _DB_ and join _GB_ and _DG_ (_G_ being the apex of
pyramid).

Find the vanishing-point of _DG_, and use _DB_ for division, carrying
the measurements to the line _GB_.

In Fig. 59., if we join _AD_ and _DC_, _ADC_ is the vertical profile
of the whole pyramid, and _BDC_ of the half pyramid, corresponding to
_FGB_ in Fig. 57.

  [Illustration: Fig. 61.]

We may now proceed to an architectural example.

Let _AH_, Fig. 60., be the vertical profile of the capital of a
pillar, _AB_ the semi-diameter of its head or abacus, and _FD_ the
semi-diameter of its shaft.

Let the shaft be circular, and the abacus square, down to the level
_E_.

Join _BD_, _EF_, and produce them to meet in _G_.

Therefore _ECG_ is the semi-profile of a reversed pyramid containing
the capital.

Construct this pyramid, with the square of the abacus, in the required
perspective, as in Fig. 61.; making _AE_ equal to _AE_ in Fig. 60.,
and _AK_, the side of the square, equal to twice _AB_ in Fig. 60. Make
_EG_ equal to _CG_, and _ED_ equal to _CD_. Draw _DF_ to the
vanishing-point of the diagonal _DV_ (the figure is too small to
include this vanishing-point), and _F_ is the level of the point _F_
in Fig. 60., on the side of the pyramid.

Draw _Fm_, _Fn_, to the vanishing-points of _AH_ and _AK_. Then _Fn_
and _Fm_ are horizontal lines across the pyramid at the level _F_,
forming at that level two sides of a square.

  [Illustration: Fig. 62.]

Complete the square, and within it inscribe a circle, as in Fig. 62.,
which is left unlettered that its construction may be clear. At the
extremities of this draw vertical lines, which will be the sides of
the shaft in its right place. It will be found to be somewhat smaller
in diameter than the entire shaft in Fig. 60., because at the center
of the square it is more distant than the nearest edge of the square
abacus. The curves of the capital may then be drawn approximately by
the eye. They are not quite accurate in Fig. 62., there being a
subtlety in their junction with the shaft which could not be shown on
so small a scale without confusing the student; the curve on the left
springing from a point a little way round the circle behind the shaft,
and that on the right from a point on this side of the circle a little
way within the edge of the shaft. But for their more accurate
construction see Notes on Problem XIV.




PROBLEM XI.


It is seldom that any complicated curve, except occasionally a spiral,
needs to be drawn in perspective; but the student will do well to
practice for some time any fantastic shapes which he can find drawn on
flat surfaces, as on wall-papers, carpets, etc., in order to accustom
himself to the strange and great changes which perspective causes in
them.

  [Illustration: Fig. 63.]

The curves most required in architectural drawing, after the circle,
are those of pointed arches; in which, however, all that will be
generally needed is to fix the apex, and two points in the sides. Thus
if we have to draw a range of pointed arches, such as _APB_, Fig. 63.,
draw the measured arch to its sight-magnitude first neatly in a
rectangle, _ABCD_; then draw the diagonals _AD_ and _BC_; where they
cut the curve draw a horizontal line (as at the level _E_ in the
figure), and carry it along the range to the vanishing-point, fixing
the points where the arches cut their diagonals all along. If the arch
is cusped, a line should be drawn, at _F_ to mark the height of the
cusps, and verticals raised at _G_ and _H_, to determine the interval
between them. Any other points may be similarly determined, but these
will usually be enough. Figure 63. shows the perspective construction
of a square niche of good Veronese Gothic, with an uncusped arch of
similar size and curve beyond.

  [Illustration: Fig. 64.]

In Fig. 64. the more distant arch only is lettered, as the
construction of the nearest explains itself more clearly to the eye
without letters. The more distant arch shows the general construction
for all arches seen underneath, as of bridges, cathedral aisles, etc.
The rectangle _ABCD_ is first drawn to contain the outside arch; then
the depth of the arch, _Aa_, is determined by the measuring-line, and
the rectangle, _abcd_, drawn for the inner arch.

_Aa_, _Bb_, etc., go to one vanishing-point; _AB_, _ab_, etc., to the
opposite one.

In the nearer arch another narrow rectangle is drawn to determine the
cusp. The parts which would actually come into sight are slightly
shaded.




PROBLEM XIV.


Several exercises will be required on this important problem.

I. It is required to draw a circular flat-bottomed dish narrower at
the bottom than the top; the vertical depth being given, and the
diameter at the top and bottom.

  [Illustration: Fig. 65.]

Let _ab_, Fig. 65., be the diameter of the bottom, _ac_ the diameter
of the top, and _ad_ its vertical depth.

Take _AD_ in position equal to _ac_.

On _AD_ draw the square _ABCD_, and inscribe in it a circle.

Therefore, the circle so inscribed has the diameter of the top of the
dish.

From _A_ and _D_ let fall verticals, _AE_, _DH_, each equal to _ad_.

Join _EH_, and describe square _EFGH_, which accordingly will be equal
to the square _ABCD_, and be at the depth _ad_ beneath it.

Within the square _EFGH_ describe a square _IK_, whose diameter shall
be equal to _ab_.

Describe a circle within the square _IK_. Therefore the circle so
inscribed has its diameter equal to _ab_; and it is in the center of
the square _EFGH_, which is vertically beneath the square _ABCD_.

Therefore the circle in the square _IK_ represents the bottom of the
dish.

Now the two circles thus drawn will either intersect one another, or
they will not.

If they intersect one another, as in the figure, and they are below
the eye, part of the bottom of the dish is seen within it.

  [Illustration: Fig. 66.]

To avoid confusion, let us take then two intersecting circles without
the inclosing squares, as in Fig. 66.

Draw right lines, _ab_, _cd_, touching both circles externally. Then
the parts of these lines which connect the circles are the sides of
the dish. They are drawn in Fig. 65. without any prolongations, but
the best way to construct them is as in Fig. 66.

If the circles do not intersect each other, the smaller must either be
within the larger or not within it.

If within the larger, the whole of the bottom of the dish is seen from
above, Fig. 67. _a_.

  [Illustration: Fig. 67.]

If the smaller circle is not within the larger, none of the bottom is
seen inside the dish, _b_.

If the circles are above instead of beneath the eye, the bottom of the
dish is seen beneath it, _c_.

If one circle is above and another beneath the eye, neither the bottom
nor top of the dish is seen, _d_. Unless the object be very large, the
circles in this case will have little apparent curvature.

II. The preceding problem is simple, because the lines of the profile
of the object (_ab_ and _cd_, Fig. 66.) are straight. But if these
lines of profile are curved, the problem becomes much more complex:
once mastered, however, it leaves no farther difficulty in
perspective.

Let it be required to draw a flattish circular cup or vase, with a
given curve of profile.

The basis of construction is given in Fig. 68., half of it only being
drawn, in order that the eye may seize its lines easily.

  [Illustration: Fig. 68.]

Two squares (of the required size) are first drawn, one above the
other, with a given vertical interval, _AC_, between them, and each is
divided into eight parts by its diameters and diagonals. In these
squares two circles are drawn; which are, therefore, of equal size,
and one above the other. Two smaller circles, also of equal size, are
drawn within these larger circles in the construction of the present
problem; more may be necessary in some, none at all in others.

It will be seen that the portions of the diagonals and diameters of
squares which are cut off between the circles represent radiating
planes, occupying the position of the spokes of a wheel.

Now let the line _AEB_, Fig. 69., be the profile of the vase or cup to
be drawn.

Inclose it in the rectangle _CD_, and if any portion of it is not
curved, as _AE_, cut off the curved portion by the vertical line _EF_,
so as to include it in the smaller rectangle _FD_.

Draw the rectangle _ACBD_ in position, and upon it construct two
squares, as they are constructed on the rectangle _ACD_ in Fig. 68.;
and complete the construction of Fig. 68., making the radius of its
large outer circles equal to _AD_, and of its small inner circles
equal to _AE_.

The planes which occupy the position of the wheel spokes will then
each represent a rectangle of the size of _FD_. The construction is
shown by the dotted lines in Fig. 69.; _c_ being the center of the
uppermost circle.

  [Illustration: Fig. 69.]

Within each of the smaller rectangles between the circles, draw the
curve _EB_ in perspective, as in Fig. 69.

Draw the curve _xy_, touching and inclosing the curves in the
rectangles, and meeting the upper circle at _y_.[32]

Then _xy_ is the contour of the surface of the cup, and the upper
circle is its lip.

If the line _xy_ is long, it may be necessary to draw other rectangles
between the eight principal ones; and, if the curve of profile _AB_ is
complex or retorted, there may be several lines corresponding to _XY_,
inclosing the successive waves of the profile; and the outer curve
will then be an undulating or broken one.

  [Illustration: Fig. 70.]

III. All branched ornamentation, forms of flowers, capitals of
columns, machicolations of round towers, and other such arrangements
of radiating curve, are resolvable by this problem, using more or
fewer interior circles according to the conditions of the curves.
Fig. 70. is an example of the construction of a circular group of
eight trefoils with curved stems. One outer or limiting circle is
drawn within the square _EDCF_, and the extremities of the trefoils
touch it at the extremities of its diagonals and diameters. A smaller
circle is at the vertical distance _BC_ below the larger, and _A_ is
the angle of the square within which the smaller circle is drawn; but
the square is not given, to avoid confusion. The stems of the trefoils
form drooping curves, arranged on the diagonals and diameters of the
smaller circle, which are dotted. But no perspective laws will do work
of this intricate kind so well as the hand and eye of a painter.

IV. There is one common construction, however, in which, singularly,
the hand and eye of the painter almost always fail, and that is the
fillet of any ordinary capital or base of a circular pillar (or any
similar form). It is rarely necessary in practice to draw such minor
details in perspective; yet the perspective laws which regulate them
should be understood, else the eye does not see their contours rightly
until it is very highly cultivated.

  [Illustration: Fig. 71.]

Fig. 71. will show the law with sufficient clearness; it represents
the perspective construction of a fillet whose profile is a
semicircle, such as _FH_ in Fig. 60., seen above the eye. Only half
the pillar with half the fillet is drawn, to avoid confusion.

_Q_ is the center of the shaft.

_PQ_ the thickness of the fillet, sight-magnitude at the shaft’s
center.

Round _P_ a horizontal semicircle is drawn on the diameter of the
shaft _ab_.

Round _Q_ another horizontal semicircle is drawn on diameter _cd_.

These two semicircles are the upper and lower edges of the fillet.

Then diagonals and diameters are drawn as in Fig. 68., and, at their
extremities, semicircles in perspective, as in Fig. 69.

The letters _A_, _B_, _C_, _D_, and _E_, indicate the upper and
exterior angles of the rectangles in which these semicircles are to be
drawn; but the inner vertical line is not dotted in the rectangle at
_C_, as it would have confused itself with other lines.

Then the visible contour of the fillet is the line which incloses and
touches[33] all the semicircles. It disappears behind the shaft at the
point _H_, but I have drawn it through to the opposite extremity of
the diameter at _d_.

Turned upside down the figure shows the construction of a basic
fillet.

The capital of a Greek Doric pillar should be drawn frequently for
exercise on this fourteenth problem, the curve of its echinus being
exquisitely subtle, while the general contour is simple.


  [32] This point coincides in the figure with the extremity of the
       horizontal diameter, but only accidentally.

  [33] The engraving is a little inaccurate; the inclosing line
       should touch the dotted semicircles at _A_ and _B_. The student
       should draw it on a large scale.




PROBLEM XVI.


It is often possible to shorten other perspective operations
considerably, by finding the vanishing-points of the inclined lines of
the object. Thus, in drawing the gabled roof in Fig. 43., if the gable
_AYC_ be drawn in perspective, and the vanishing-point of _AY_
determined, it is not necessary to draw the two sides of the
rectangle, _A′D′_ and _D′B′_, in order to determine the point _Y′_;
but merely to draw _YY′_ to the vanishing-point of _AA′_ and _A′Y′_ to
the vanishing-point of _AY_, meeting in _Y′_, the point required.

Again, if there be a series of gables, or other figures produced by
parallel inclined lines, and retiring to the point _V_, as in
Fig. 72.,[34] it is not necessary to draw each separately, but merely
to determine their breadths on the line _AV_, and draw the slopes of
each to their vanishing-points, as shown in Fig. 72. Or if the gables
are equal in height, and a line be drawn from _Y_ to _V_, the
construction resolves itself into a zigzag drawn alternately to _P_
and _Q_, between the lines _YV_ and _AV_.

The student must be very cautious, in finding the vanishing-points of
inclined lines, to notice their relations to the horizontals beneath
them, else he may easily mistake the horizontal to which they belong.

Thus, let _ABCD_, Fig. 73., be a rectangular inclined plane, and let
it be required to find the vanishing-point of its diagonal _BD_.

Find _V_, the vanishing-point of _AD_ and _BC_.

Draw _AE_ to the opposite vanishing-point, so that _DAE_ may represent
a right angle.

Let fall from _B_ the vertical _BE_, cutting _AE_ in _E_.

Join _ED_, and produce it to cut the sight-line in _V′_.

  [Illustration: Fig. 72.]

Then, since the point _E_ is vertically under the point _B_, the
horizontal line _ED_ is vertically under the inclined line _BD_.

  [Illustration: Fig. 73.]

So that if we now let fall the vertical _V′P_ from _V′_, and produce
_BD_ to cut _V′P_ in _P_, the point _P_ will be the vanishing-point of
_BD_, and of all lines parallel to it.[35]


  [34] The diagram is inaccurately cut. _YV_ should be a right line.

  [35] The student may perhaps understand this construction better
       by completing the rectangle _ADFE_, drawing _DF_ to the
       vanishing-point of _AE_, and _EF_ to _V_. The whole figure,
       _BF_, may then be conceived as representing half the gable roof
       of a house, _AF_ the rectangle of its base, and _AC_ the
       rectangle of its sloping side.

       In nearly all picturesque buildings, especially on the
       Continent, the slopes of gables are much varied (frequently
       unequal on the two sides), and the vanishing-points of their
       inclined lines become very important, if accuracy is required
       in the intersections of tiling, sides of dormer windows, etc.

       Obviously, also, irregular triangles and polygons in vertical
       planes may be more easily constructed by finding the
       vanishing-points of their sides, than by the construction given
       in the corollary to Problem IX.; and if such triangles or
       polygons have others concentrically inscribed within them, as
       often in Byzantine mosaics, etc., the use of the
       vanishing-points will become essential.




PROBLEM XVIII.


Before examining the last three problems it is necessary that you
should understand accurately what is meant by the position of an
inclined plane.

Cut a piece of strong white pasteboard into any irregular shape, and
dip it in a sloped position into water. However you hold it, the edge
of the water, of course, will always draw a horizontal line across its
surface. The direction of this horizontal line is the direction of the
inclined plane. (In beds of rock geologists call it their “strike.”)

  [Illustration: Fig. 74.]

Next, draw a semicircle on the piece of pasteboard; draw its diameter,
_AB_, Fig. 74., and a vertical line from its center, _CD_; and draw
some other lines, _CE_, _CF_, etc., from the center to any points in
the circumference.

Now dip the piece of pasteboard again into water, and, holding it at
any inclination and in any direction you choose, bring the surface of
the water to the line _AB_. Then the line _CD_ will be the most
steeply inclined of all the lines drawn to the circumference of the
circle; _GC_ and _HC_ will be less steep; and _EC_ and _FC_ less steep
still. The nearer the lines to _CD_, the steeper they will be; and the
nearer to _AB_, the more nearly horizontal.

When, therefore, the line _AB_ is horizontal (or marks the water
surface), its direction is the direction of the inclined plane, and
the inclination of the line _DC_ is the inclination of the inclined
plane. In beds of rock geologists call the inclination of the line
_DC_ their “dip.”

To fix the position of an inclined plane, therefore, is to determine
the direction of any two lines in the plane, _AB_ and _CD_, of which
one shall be horizontal and the other at right angles to it. Then any
lines drawn in the inclined plane, parallel to _AB_, will be
horizontal; and lines drawn parallel to _CD_ will be as steep as _CD_,
and are spoken of in the text as the “steepest lines” in the plane.

But farther, whatever the direction of a plane may be, if it be
extended indefinitely, it will be terminated, to the eye of the
observer, by a boundary line, which, in a horizontal plane, is
horizontal (coinciding nearly with the visible horizon);—in a vertical
plane, is vertical;—and, in an inclined plane, is inclined.

This line is properly, in each case, called the “sight-line” of such
plane; but it is only properly called the “horizon” in the case of a
horizontal plane: and I have preferred using always the term
“sight-line,” not only because more comprehensive, but more accurate;
for though the curvature of the earth’s surface is so slight that
practically its visible limit always coincides with the sight-line of
a horizontal plane, it does not mathematically coincide with it, and
the two lines ought not to be considered as theoretically identical,
though they are so in practice.

It is evident that all vanishing-points of lines in any plane must be
found on its sight-line, and, therefore, that the sight-line of any
plane may be found by joining any two of such vanishing-points. Hence
the construction of Problem XVIII.




II.

DEMONSTRATIONS WHICH COULD NOT CONVENIENTLY BE INCLUDED IN THE TEXT.




I.

THE SECOND COROLLARY, PROBLEM II.


In Fig. 8. omit the lines _CD_, _C′D′_, and _DS_; and, as here in
Fig. 75., from _a_ draw _ad_ parallel to _AB_, cutting _BT_ in _d_;
and from _d_ draw _de_ parallel to _BC′_.

  [Illustration: Fig. 75.]

  Now as _ad_ is parallel to _AB_—
    _AC_ ∶ _ac_ ∷ _BC′_ ∶ _de_;
  but _AC_ is equal to _BC′_—
    ∴ _ac_ = _de_.

  Now because the triangles _acV_, _bc′V_, are similar—
    _ac_ ∶ _bc′_ ∷ _aV_ ∶ _bV_;
  and because the triangles _deT_, _bc′T_ are similar—
    _de_ ∶ _bc′_ ∷ _dT_ ∶ _bT_.

  But _ac_ is equal to _de_—
    ∴ _aV_ ∶ _bV_ ∷ _dT_ ∶ _bT_;
  ∴ the two triangles _abd_, _bTV_, are similar, and their angles
  are alternate;
    ∴ _TV_ is parallel to _ad_.

  But _ad_ is parallel to _AB_—
    ∴ _TV_ is parallel to _AB_.




II.

THE THIRD COROLLARY, PROBLEM III.


In Fig. 13., since _aR_ is by construction parallel to _AB_ in
Fig. 12., and _TV_ is by construction in Problem III. also parallel to
_AB_—

    ∴ _aR_ is parallel to _TV_,
    ∴ _abR_ and _TbV_ are alternate triangles,
    ∴ _aR_ ∶ _TV_ ∷ _ab_ ∶ _bV_.

  Again, by the construction of Fig. 13., _aR′_ is parallel to _MV_—
    ∴ _abR′_ and _MbV_ are alternate triangles,
    ∴ _aR′_ ∶ _MV_ ∷ _ab_ ∶ _bV_.

  And it has just been shown that also
    _aR_ ∶ _TV_ ∷ _ab_ ∶ _bV_—
    ∴ _aR′_ ∶ _MV_ ∷ _aR_ ∶ _TV_.

  But by construction, _aR′_ = _aR_—
    ∴ _MV_ = _TV_.




III.

ANALYSIS OF PROBLEM XV.


We proceed to take up the general condition of the second problem,
before left unexamined, namely, that in which the vertical distances
_BC′_ and _AC_ (Fig. 6. page 13), as well as the direct distances _TD_
and _TD′_ are unequal.

In Fig. 6., here repeated (Fig. 76.), produce _C′B_ downwards, and
make _C′E_ equal to _CA_.

  [Illustration: Fig. 76.]

Join _AE_.

Then, by the second Corollary of Problem II., _AE_ is a horizontal
line.

Draw _TV_ parallel to _AE_, cutting the sight-line in _V_.

  ∴ _V_ is the vanishing-point of _AE_.

Complete the constructions of Problem II. and its second Corollary.

Then by Problem II. _ab_ is the line _AB_ drawn in perspective; and by
its Corollary _ae_ is the line _AE_ drawn in perspective.


From _V_ erect perpendicular _VP_, and produce _ab_ to cut it in _P_.

Join _TP_, and from _e_ draw _ef_ parallel to _AE_, and cutting _AT_
in _f_.

Now in triangles _EBT_ and _AET_, as _eb_ is parallel to _EB_ and _ef_
to _AE_;—_eb_ ∶ _ef_ ∷ _EB_ ∶ _AE_.

But _TV_ is also parallel to _AE_ and _PV_ to _eb_.

Therefore also in the triangles _aPV_ and _aVT_,

  _eb_ ∶ _ef_ ∷ _PV_ ∶ _VT_.

Therefore _PV_ ∶ _VT_ ∷ _EB_ ∶ _AE_.

And, by construction, angle _TPV_ = ∠ _AEB_.

Therefore the triangles _TVP_, _AEB_, are similar; and _TP_ is
parallel to _AB_.

Now the construction in this problem is entirely general for any
inclined line _AB_, and a horizontal line _AE_ in the same vertical
plane with it.

So that if we find the vanishing-point of _AE_ in _V_, and from _V_
erect a vertical _VP_, and from _T_ draw _TP_ parallel to _AB_,
cutting _VP_ in _P_, _P_ will be the vanishing-point of _AB_, and (by
the same proof as that given at page 17) of all lines parallel to it.

  [Illustration: Fig. 77.]

Next, to find the dividing-point of the inclined line.

I remove some unnecessary lines from the last figure and repeat it
here, Fig. 77., adding the measuring-line _aM_, that the student may
observe its position with respect to the other lines before I remove
any more of them.

Now if the line _AB_ in this diagram represented the length of the
line _AB_ in reality (as _AB_ _does_ in Figs. 10. and 11.), we should
only have to proceed to modify Corollary III. of Problem II. to this
new construction. We shall see presently that _AB_ does not represent
the actual length of the inclined line _AB_ in nature, nevertheless we
shall first proceed as if it did, and modify our result afterwards.

In Fig. 77. draw _ad_ parallel to _AB_, cutting _BT_ in _d_.

Therefore _ad_ is the sight-magnitude of _AB_, as _aR_ is of _AB_ in
Fig. 11.

  [Illustration: Fig. 78.]

Remove again from the figure all lines except _PV_, _VT_, _PT_, _ab_,
_ad_, and the measuring-line.

Set off on the measuring-line _am_ equal to _ad_.

Draw _PQ_ parallel to _am_, and through _b_ draw _mQ_, cutting _PQ_ in
_Q_.

Then, by the proof already given in page 20, _PQ_ = _PT_.

Therefore if _P_ is the vanishing-point of an inclined line _AB_, and
_QP_ is a horizontal line drawn through it, make _PQ_ equal to _PT_,
and _am_ on the measuring-line equal to the sight-magnitude of the
line _AB_ _in the diagram_, and the line joining _mQ_ will cut _aP_ in
_b_.


We have now, therefore, to consider what relation the length of the
line _AB_ in this diagram, Fig. 77., has to the length of the line
_AB_ in reality.

Now the line _AE_ in Fig. 77. represents the length of _AE_ in
reality.

But the angle _AEB_, Fig. 77., and the corresponding angle in all the
constructions of the earlier problems, is in reality a right angle,
though in the diagram necessarily represented as obtuse.

  [Illustration: Fig. 79.]

Therefore, if from _E_ we draw _EC_, as in Fig. 79., at right angles
to _AE_, make _EC_ = _EB_, and join _AC_, _AC_ will be the real length
of the line _AB_.

Now, therefore, if instead of _am_ in Fig. 78., we take the real
length of _AB_, that real length will be to _am_ as _AC_ to _AB_ in
Fig. 79.

And then, if the line drawn to the measuring-line _PQ_ is still to cut
_aP_ in _b_, it is evident that the line _PQ_ must be shortened in the
same ratio that _am_ was shortened; and the true dividing-point will
be _Q′_ in Fig. 80., fixed so that _Q′P′_ shall be to _QP_ as _am′_ is
to _am_; _am′_ representing the real length of _AB_.

But _am′_is therefore to _am_ as _AC_ is to _AB_ in Fig. 79.

Therefore _PQ′_ must be to _PQ_ as _AC_ is to _AB_.

But _PQ_ equals _PT_ (Fig. 78.); and _PV_ is to _VT_ (in Fig. 78.) as
_BE_ is to _AE_ (Fig. 79.).

Hence we have only to substitute _PV_ for _EC_, and _VT_ for _AE_, in
Fig. 79., and the resulting diagonal _AC_ will be the required length
of _PQ′_.

  [Illustration: Fig. 80.]

It will be seen that the construction given in the text (Fig. 46.) is
the simplest means of obtaining this magnitude, for _VD_ in Fig. 46.
(or _VM_ in Fig. 15.) = _VT_ by construction in Problem IV. It should,
however, be observed, that the distance _PQ′_ or _PX_, in Fig. 46.,
may be laid on the sight-line of the inclined plane itself, if the
measuring-line be drawn parallel to that sight-line. And thus any form
may be drawn on an inclined plane as conveniently as on a horizontal
one, with the single exception of the radiation of the verticals,
which have a vanishing-point, as shown in Problem XX.


THE END.



Transcriber’s Note

A handful of unequivocal typographical errors has been corrected.

For increased clarity, a few diagrams have been shifted from their
original position in the text.





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