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.