IMPROVEMENTS IN ARCHITECTURE, IN ITS FORMS AND COMBINATIONS, AND ALSO IN THE SUPERFICIAL FIGURES WHICH MAY BE EMPLOYED; ALSO FOR AN IMPROVEMENT OR IMPROVEMENTS IN THE SUR- FACES OF BUILDINGS
Year: 1838

page_text: TO ALL TO WHOM THESE PRESENTS SHALL COME, I, JOHN
ROBERTSON, junior, of Great Charlotte Street, Buckingham Gate, in the
County of Middlesex, Gentleman, send greeting.
WHEREAS Her present most Excellent Majesty Queen Victoria, by Her
Letters Patent under the Great Seal of Great Britain, bearing date at West-
minster, the Eighteenth day of July, in the second year of Her reign, did,
for Herself, Her heirs and successors, give and grant unto me, the said
John Robertson, Her especial licence, full power, sole privilege and authority,
that I, the said John Robertson, my exors, admors, and assigns, or such
others as I, the said John Robertson, my exors, admors, or assigns, should
at any time agree with, and no others, from time to time and at all times.
during the term of years therein expressed, should and lawfully might
make, use, exercise, and vend, within England, Wales, and the Town of
Berwick-upon-Tweed, my Invention of “IMPROVEMENTS IN ARCHITECTURE, IN
ITS FORMS AND COMBINATIONS, AND ALSO IN THE SUPERFICIAL FIGURES WHICH
MAY BE EMPLOYED; ALSO FOR AN IMPROVEMENT OR IMPROVEMENTS IN THE SUR-
FACES OF BUILDINGS;" in which said Letters Patent is contained a proviso,
that I, the said John Robertson, shall cause a particular description of
the nature of my said Invention, and in what manner the same is to be
performed, to be inrolled in Her said Majesty's High Court of Chancery
within six calendar months next and immediately after the date of the said
in part recited Letters Patent, as in and by the same, reference being there-
unto had, will more fully and at large appear.

page_text: NOW KNOW YE, that in compliance with the said proviso, I, the said
John Robertson, do hereby declare the nature of my Invention, and the
manner in which the same is to. be performed, are fully described and
ascertained in and by the following statement thereof, reference being had
to the Drawings hereunto annexed, and to the figures and letters marked
thereon (that is to say):-
First, with regard to the improvements of architecture, in its forms and
combinations, and also in the superficial figures which may be employed.
These form a new system of architecture upon principles which I consider to
be the natural ones, of relational construction, and that it is only according to
these that such construction can be completed in relief or with forms bounded
by curved lines. The process of formation may be generally characterised as
that of modifying the properties of unity of form for relations, and combining
the modified forms or parts. A sphere, which is unity of form, is modified for
six relations, and is thus rendered the primary capital; a circle is modified for
four relations, and generally the forms are bounded with modifications of
circular arcs of 45°, such as may be called either relational or proportional arcs,
convex and concave, which are in some cases elongated, and in others fore-
shortened or contracted. With these the greatest simplification of the con-
struction may be effected, they are at the same time fitted for forming varieties
and complications. These proportional arcs may be formed of greater or less
depth, as compared with the breadth of the sections of forms which they bound,
and so the whole construction may be made with greater or less fullness or
relief. I shall afterwards state my reasons for preferring a certain degree of
fullness with which I have formed the figures for illustrating the description,
with very few exceptions. It will be easy to diverge from this to make the
differences of extension, accompanying a change of comparative depth of the
arcs, sufficiently understood with the proportions thus made to correspond.
I consider the construction to be proper, though not having the greatest fullness
or relief. As far as rectilenear boundaries are used in the construction, the pro-
portions are ruled by those of the curvilinear forms. I consider this system
to be also a relational or proper one to the eye, likewise to possess an economy
of strength as compared with other architecture. I proceed to the description
of the formation referring to the annexed Drawings, and to the letters and
figures of reference marked thereon. The Drawings are not intended to
present models, but illustrations of the description.
I have first to explain the nature of proportional arcs. This may be seen
in the following mechanical method by which their formation may be approxi-
mated-In Figure 1, a circle is divided into arcs of 45°, and diameters

page_text: drawn terminating in the points of division A, A, B, B, D, E, D, F. From
one of the points of division D, a line D, G, C, is drawn equal and parallel to the
radius of the circle A, C, and from the point with the distance C, G, D, the
arc D, M, L, A, is described, which is thus the arc D, H, A, in an inverted or
concave position to the breadth of the figure, and from this a concave pro-
portional arc is to be formed, as from the arc D, M, L,. A, on the other side of
the figure, forming a convex boundary to it, a convex proportional arc. is to le
formed. By using the same letters, the same demonstration will serve in the
cases of both the arcs. Divide each of them into any number of lesser arcs
A, L; L, M, M, D, all equal to one another. To render the approximations
to proportional. arcs indefinitely close, the operation would require to be
performed with an indefinitely great number of divisions; but the three taken
will suffice for shewing the nature of the operation. From the centre. with
which each arc is described C, draw radii to the points of division.C, L, C, M.
Then, in the case of each arc, from the centre C, with the distance C, A, having
described the arc A, L, instead of proceeding with that centre and distance,
find the distance of L from the nearest point in B, B; measure off upon
L, C, L, o, equal to that distance, and from o draw o, m, parallel to C, M;
then from o as a centre, with the distance o, L,, describe the arc L, m, until
it arrives at o, m, in m; find the distance of m from the nearest point in B,.B,
and measure off upon m; o, m, n, equal to that distance; draw n, d, parallel to
C, D; then from.n as: a centre, with the distance n, m, describe. the arc m, d,
until it arrives at n, d, in d. A, L, m, d, are certain approximations to pro-
portional arcs, convex and concave, and the greater number of divisions of the
circular arcs this operation is performed with, the nearer will be the approxi-
tions obtained. I shall call the line B, B, the diameter of formation of each
of the proportional arcs; and the line A, A, I call the base, to which B, B,
is perpendicular. To form proportional arcs of any greater or less degree of
depth as compared with the breadth of form bounded, the above operation is to
be varied as follows:-In Figure 2, circular. arcs of 45°, convex and concave,
H, K, are on one side of the figure placed upon the line of the base,, which is
extended, H, A, A, at a greater, and on the other side at a less distance from
the diameter of formation B, B, than that of the length of a radius
at the base, as in last examples; at the same time they are not altered
in inclination to B, B, nor in magnitude; let each of them be divided into any
number of parts all equal to one another, and from. the centre of each G,
draw radii G, H, G, L, G, M, G, K, to the points of division. By using the
same letters, the same demonstration serves in the cases of all the four arcs.
Then in each, from the centre G, with the distance. G, H, having described the

page_text: arc H, L, instead of proceeding with that centre and distance, find the distance
of L from the nearest point in B, B; measure off upon L, G, L, o, bearing
the same proportion to that distance that G, HI, does to C, H, that is, that
a radius of the arc bears to its distance at the base from the diameter of
formation; draw o, m, parallel to G, M; then from o as a centre, with the
distance o, L, describe the arc L, m, until it meets o, m, in m; find the
distance of m from the nearest point in B, B, and measure off upon m, o,
m, n, bearing, as before, the same proportion to that distance that H, G,
does to C, H; draw n,. . k, parallel to G, K; then from n as a centre, with
the distance n, m, describe the arc m, k, until it arrives at n, k, in k. H, L,
m, k, are certain approximations to the proportional arcs intended, and the
greater number of divisions of the circular arcs this operation is performed with,
the nearer will be the approximations obtained. The distances in a proportional
arc, or in forming it, as may be observed above, are always to be measured to its
diameter of formation in the direction perpendicular to that line, or in other
words, measuring from each point in the proportional arc to the nearest in the
diameter of formation, not to any fixed point or centre, and the diameter of
formation may be considered as of indefinite extension. The depth of a porpor-
tional arc is the difference between its greatest and least distance from the
cliame ter of formation. The breadth, or what I shall rather distinguish as the
depth of form bounded, is that between a proportional arc and its diameter of
By
formation; the form may be different on the other side of that line.
placing the circular arc at a greater comparative distance from the diameter of
formation, the proportional are derived from it is of less comparative depth
If the
that is, as compared with the depth of form bounded, and vice versa.
comparison were made between the depths of the proportional arcs and that of
the circular one, the reverse of this would in some degree be the case, but it is
as compared with the depth of form bounded that I distinguish proportional arcs
'as of greater or less comparative depth, or of deeper or shallower formation.
The nature then of proportional arcs is, first, such that considering the circular
arc of 45° to be made up of an indefinitely great number, or as great a number
as we need suppose, of minute parts, all equal to one another, these, which are
minute arcs, but which, for some purposes in the demonstration we may also
treat as chords, are, in modifying the circular are into a proportional one, to
retain the same inclinations and conjunctions as in the circular arc, but are,
say, commencing from the base, to vary in magnitude, considered in reference to
the length of radius with which each is described, or regarding them as chords
are to vary in length, as their distance from the diameter of formation varies.
The magnitude of curvature, therefore, as I believe I may term it, meaning in

page_text: reference to the length of radius with which at each point it is described,
being once taken, say, at the base, of a certain proportion to the depth of
form bounded at that point, takes the same proportion throughout at every other
point, and this proportional curvature may also be carried continually through
any number of stages or gradations of proportional arcs, each derived from a
circular one of 45°. Thus, returning to Figure 1, A, c, A, f, are convex
proportional arcs, derived from A, E, A, F; their heads are joined by the line
e, f, which is parallel to A, A. Then upon e, f, as the new base, and with the
extended diameter of formation B, B, b, the proportional arcs, convex and
concave, f, t, e, r, are derived from the circular arcs of 45°, f, T, e, v, R; these
circular arcs having the same magnitude as compared with their distance at the
base from the diameter of formation as those upon the former base A, A, and
with the same magnatude of curvature that A, c, E, f, leave off at e and f,
e, r, f, t, commence, and thus we may proceed through further gradations of
convex or concave arcs, or with alternations of them. And in a corresponding
manner, successions may be made of proportional arcs of any other comparative
depth.
ares;
I have next to explain the elongating and foreshortning of proportional
their extension is to be increased and decreased equably, and in
directions perpendicular to their bases. To consider this further, in Figure 3,
a circle is divided into arcs of 45°, and diameters drawn through the
points of division; A, d, are proportional arcs, convex and concave, derived
from the circular ares A, D ; A, Q, D, B, B, being the diameter of forma-
tion. Then let these proportional arcs be divided proportionally into any
number of parts. By "divided proportionally" is meant that the divisions
shall bear the same proportions to one another as do their distances from
the diameter of formation, being, therefore, such as have been modified from
equal divisions of the circular ares. Let three proportional divisions so
taken in each arc, A, Q, Q, S, S, d, represent in the present demonstration -
and indefinitely great number, then drawing chords to them, their chords
may be regarded as being of the same proportions to one another as are their
distances from the diameter of formation. Upon the chords, let triangles
be formed, each having one of its other sides perpendicular to the base A, A, .
and the other parallel with it. Then if the arcs A, d, be equally elongated or
foreshortened in the direction perpendicular to A, A, and with them the
triangles upon the chords. of their divisions, the sides of these triangles which
are perpendicular to A, A, will continue to bear the same proportions to one.
another as before, the sides parallel with A, A, will not be altered in length,
the chords will not have the same proportions to one another as before; and it..

page_text: is only as considering the progress of the arcs in each of the two directions,
perpendicular and parallel to the base, separately, that the same proportions
remain when the arcs are elongated or foreshortened. If a circular arc were
first elongated or foreshortened and then modified; the effect would be the
same as by first modifying and then elongating or foreshortening. I do not speak
of proportional ares being elongated or foreshortened parallel to the base or in
their depth, because the whole depth of form will then properly be elongated or
foreshortened in the same degree, and the change upon the proportions of the
figure may be considered as that of comparative extension, perpendicular to
the base. For the sake of distinguishing between the two directions in
reference to a proportional arc, I shall call that only extension which is
measured perpendicularly to the base; that which measured parallel to the
base, I call depth of the arc or depth of form; as a proportional arc is
increased or decreased in extension, so is each of its proportional divisions
to be in the same proportion. The corresponding points of division will thus
in all convex proportional arcs of the same depth of formation, be always
at the same proportions of their extension, they will likewise be always at
the same proportions of their depth; so also with the concave proportional
But the convex and concave will not agree in this neither, though
they should be measured from different ends.
arcs.
For it is here to be observed
that the concave proportional are will not be merely an inversion of the
convex one. The progress of the concave arc is earlier towards the diameter
of formation, as compared with its progress from the base, thereby the diminu-
tion of the magnitude of curvature is earlier as compared with the pro-
gress from the base, the whole amount of which will therefore also be
less in the first formation, and will afterwards have to be supplied withi
greater extension compared with what it at first possesses, to make its ex-
tension keep pace with that of the corresponding convex arc. They will both
have the same depth; for the depth might be calculated as supposing the
progress to be along the base.
. I next proceed to shew how relational circles, or circles modified for four
relations, are to be completed with proportional arcs. In Figure 3, in the
same manner as the convex circular arcs A, D, A, D, A, E, A, F, may be
modified into convex proportional arcs, having A, A, for their base, and B, B,
for their diameter of formation to the convex circular arcs, B, D, B, D,
B, E, B, F, may be modified into convex proportional ones having B, B
for their base, and A, A, for their diameter of formation. Then all these
proportional arcs being formed, they are to be elongated until they arrive
at the produced diameters G, L, H, K, at the points G, H, L, K, thus com-

page_text: pleting a relational circle of certain fullness, or of which the proportional
arcs forming it are of certain comparative depth. With proportional arcs of
any other comparative depth, the relational figure may be in a similar manner
enclosed by interchanging, as above, the diameters of formation and bases, and
giving the arcs such degree of extension as is required to make them meet'
and terminate upon the diagonal lines G, L, H, K. The figure may likewise be
inclosed with concave proportional arcs, which with a due degree of extension will
meet and terminate on the diagonal lines in the same point as the corresponding
convex ones. As in Figure 3, A, a, L, B, y, L, meet in the point L, I shall,
for distinction, call the figure, when it is thus enclosed on all sides, a concave
relational circle. In the same manner as there are gradations of the arcs, we
shall also have gradations of similar figures bounded by them. In forming
the various figures when the proportional arcs are said to be joined or con-
nected, it is to be understood that they have the same diameter of formation
and are in the same plane; they may be joined either at their bases or heads,
or the base of one to the head of another. When they are said to meet or
cross, that is, 'cut their diameters of formation, one perpendicular to one another
and cut being in the same plane. But the arcs meeting or crossing will only
be in the same plane with one another when bounding the same plane section,
or being upon a plane figure. In configurations upon a plane surface all the
diameters of formation will be in two directions parallel with and perpendicular
to one another; in solid formations they will have three directions, and be
parallel with and perpendicular to one another; such is the case in regular
formations. Two convex proportional arcs joined at their bases having equal
extension, I call a proportional quadrant; such in Figure 3 is G, B, H; two
concave proportional arcs joined at their other extremities, their heads, and
having equal extension, I call a concave proportional quadrant. A relational
semicircle is that division which has one of the diameters of formation for its
base:
I shall proceed to the consideration of solid forms, without further treating
separately of plane figures, as the sections of the solids and their front faces,
and the proportional divisions or patterns upon them, when geometrically
drawn, as is the case in the following figures, are also proper representations of
plane figures according to this system. But I have first to state the degree
of depth of the arcs with which I have formed nearly the whole of the figures,
and my reasons for preferring that proportion. I suppose the construction to
have the greatest fullness when the relational sphere is the nearest modification
that can be made for the required relations from a proper sphere; but I am
unable to determine by calculation with what depth of the ares this will be

page_text: the case, and not only upon this most difficult point, but also upon others,
must refer for greater certainty and accuracy to the data upon which calcu-
lations may be made. When a proportional arc is formed from a circular
one in its original position in the circle, or its distance at the base from the
diameter of formation being a radius, the depth of the proportional arc will
be, as nearly as I can calculate it, 4th and 5th of the distance at the base;
but in modifying the circular arc in its original position, the modification is all
thrown towards the head of the proportional arc, and a reverting effect upon
the magnitude of curvature at the base is not allowed for, taking the medium
between the distance of the circular arc of 45° at its base and at its other ex-
tremity from the diameter perpendicular to its base, then as a radius is to the
depth of the circular arc, so is that medium to 4th the distance at the base.
There are certain coincidences, which will appear afterwards, in favor of
making the depth of the proportional ares the diminution of the depth of
form during a gradation exactly 4th of the depth of form at the base; diminu-
tions of th and th appear also upon this ground to be proper ones when a
flatter formation is intended. In the following illustrations, I always use the
proportion of th the depth of form at the base as that of the depth of the
proportional arcs, except in one or two examples where I specify otherwise;
I have, of course, been greatly ruled in this by the appearance of the figures,
and consider the greatest fullness of formation to be at last very closely
approximated with the proportion preferred.
I now proceed to describe the formation of solids. Figure 4 is an elevation,
of which Figure 5 is the plan. The same letters denote corresponding parts
in the elevation and plan, also in the subsequent Figures to Figure 10, what
would be corresponding parts but for alterations which are to be described.
The plan presents relational circles placed regularly within one another, or
being upon the same diameters of formation A, A, B, B, and having D, D,
their diagonal lines, C, their common centre. The outer relational circle,
which is the plan at the base of the elevation, is one gradation greater than
the inner, which is the plan at the top, and is parallel with that of the base.
The arcs of vertical extension bounding the elevation have all a common
diameter of formation which rises perpendicularly from the plane of the base,
in which plane the bases of these arcs are lines in the centre C in the plan.
They rise from points of proportional divisions in the relational circle at the
base, and arrive at corresponding points in that at the top of the elevation.
The arcs A, C, and B, F, in the elevation are equal to and have the same
extension at A, D, in the plan, dividing the arcs of vertical extension pro-
portionally, each into the same number of divisions as in the horizontal ones;

page_text: then through their corresponding points of division, relational circles bounding
the form horizontally will pass, having upon the plan the same diameters of
formation as have those of the base and top of the elevation; and all the arcs,
horizontal and vertical, will be cut by the cross directional arcs into. corre-
sponding proportional divisions. Passing to Figure 5, the elevation is in it
altered, by the arcs D, G, bounding the diagonal sections being foreshortened
or lessened in extension th. The arcs F, G, E, G, unite with their former
curvature round the form, the same curvature downwards, and thus their heads
arrive at the lowered points G, and the effect is that the arcs E, G, F, G,
become the same as the arcs D, G, now are, and bound like them diagonal
sections of the form when it is completed. The other arcs of vertical exten-
sion, except A, E, B, F, are also diminished in extension, so that they may
continue to terminate in the arcs E, G, F, G, which they will do in the points
of proportional division, as before, and the other arcs of. horizontal extension,
except B, D, A, D, are to unite with their former curvature round the form,
so much curvature downwards as to follow the points of proportional division
of the arcs of vertical extension, and they will with this still be proportional
arcs and proportionally divided. The arcs of horizontal extension will now all
have their diameters of formation brought down to the base of the elevation,
being the lines A, A, B, B, of the plan.
Proceeding to Figure 6, the last figure is doubled, the arcs of vertical
extension forming proportional quadrants, and a face is added above and
below as follows:-The distance from E to E is equal on all sides of the.
figure, and the proportional quadrants E, V, E, are formed equal to and.
in the same plane with E, A, E; then proportional arcs may be drawn
from E, V, E, to G, F, G, the same as they are drawn from E, A, E,..
to G, D, G, and the cross directional arcs may also be drawn. The sur-
faces of these upper and lower faces are thus defined, and a proportional.
sphere or sphere, modified for six relations, completed. It has six faces,.
and, represented geometrically, as viewed in front of each, of its outline.
presents a relational circle, and the outline of the front face an inner one.
of the next gradation less than the outer, and placed regularly within it.
There are three diameters of formation of its arcs, one passing through it
from V to V, another from A to. A, and another through B, perpendicular.
to one another. Upon these sections, bounded by its proportional quadrants,
cut some upon one, some upon another, and all cut in the point where the.
diameters of formation cut, which I call the centre of the relational sphere..
Its arcs have very different degrees of extension; the front arcs have the
greatest, and the diagonal ones the least. In Figure 8, a diagonal section

page_text: upon the lines E, G, B, G, E, or V, G, D, G, V, is represented, and presents
the depth as compared with the extension of the proportional quadrant G, D, G,
bounding it. Considering the front face of Figure 6 to be a plane figure, the
construction is still proper, and the proportional quadrants have the due
proportion of depth to their distances from the diameters of formation
D, D, F, F.
Figure 9 may be termed a concave relational sphere. Its description is the
same as that of the convex one, changing convex into concave proportional
arcs. Its sections upon the lines A, A, V, V, are the same as that of which
the outline is presented by that of the figure; Figure 10 is its diagonal section.
In Figure 11, at L, the gradation of a pyramida coping, crossed with
intermediate arcs, is an alteration of so much of a relational sphere as
is presented in Figure 5. It covers the upper face of a relational sphere
used as a capital, and the arcs of its base therefore unite with their former
curvature round the form, the same curvature downwards, and the other
arcs of horizontal extension take in due degrees the curvature downwards,
following the points of proportional division of the arcs of vertical extension,
which, except the front ones, are elongated downwards to terminate in
the arcs of the base. The arcs remain proportional arcs, and, as before,
proportionally divided. Those of horizontal extension take the due proportion
of inclined depth to that of the inclined sections of the depth of form which
they bound. Again, in the short shaft at M, instead of the curvature of the
terminal arcs at the top being downward, as in the last example, it is upward,
in the same proportion, following those of the face of the capital which is
covered by it. The alteration of extension of the arcs of vertical extension
takes place accordingly, and following the points of proportional division of
these, the arcs of horizontal extension have the sections they bound of more
or less inclination upward or downward, and become proportioned accordingly;
the diameters of formation remain in the same plane, and upon them, the
sections bounded cut this plane, and their depth or breadth, whichever it may
be called, is measured to where they cut it again. The crossed part of the
cupola coping at A is an alternation of the form described at L, the arcs of
vertical extension being concave instead of convex arcs. In passing through
the proportional divisions of these, the convex arcs of horizontal extension take
the convex curvature downwards, agreeing with that of the boundaries at the
top and bottom of the gradation. Again, at O the crossed half of what I call
an architrave in this system, is an alteration of the form described at N, and
proceeds from the side face. instead of the upper face of the capital; all of its
arcs which are now of horizontal extension, one elongated on account of the

page_text: distance of the capitals, and towards the diagonal ones are so in a greater
degree, to bring all their heads into the plane of a direct transverse section.
At P is what I shall call an inter-base, beneath which is an inter-sub-base.
The inter-base has the same formation as the half of the form at o, but that
its arcs of horizontal extension are less extended. Qis the half of an inter-
columniation; its arcs of horizontal extension are concave ones, having the
same direct depth as the corresponding arcs of the architrave above. Its arcs
of vertical extention have the same depth which corresponding ones defining
the architrave above will have, and, but for the differences of vertical
extension, an intercolumniation would be the same as half the front face,
dividing it horizontally, of the architrave above. In Figure 12, the corrc-
sponding parts might be correspondingly described, only substituting concave
arcs wherever there are convex ones in the parts in Figure 11. The different
parts in each of these two Figures are regularly connected. Figure 13 will
shew the nature of the plans. The Figure marked R presents the halves of
traverse sections through the middle of the architraves, supposing the inter-
columniations to be filled up, and supposing also the architraves as well as the
capitals to have copings; such coping is represented in Figure 14. The
outlines of the capitals, shafts, bases, and so forth, and the lines where these
are joined by the architraves, intercolumniations, inter-bases, and so forth, are
likewise shewn upon the transverse plane; these latter have the like relation
to one another as the former. The bases are the halves of relational spheres,
the plans of which and the sub-bases are relational circles. From the bases
lines are drawn perpendicularly down, bounding the sub-bases to such length
as is required for inscribing the other half of the relational sphere of the base.
Where there are not more arcs required for describing the figures, only the
outlines, and those where diagonal sections would be made, or which separate
the faces, are drawn; as much as is seen of the convex relational spheres in
the present Figure 11 are so represented. The ascending arcs of the two
Figures are in connexion throughout, as are also those of horizonal extension,
or they would be completely so if the intercolumniations were filled in; one
arc joining another is either of the same or of the next gradaion to it; the
shafts are at present at what I consider their shortest extension, their front
ares having the same extension as those of the relational sphere. In the
next degree the front arcs will have the proportion of extension which the
representations of the diagonal ones have at present, and so on. In the
same manner the concave relational quadrants bounding the inter-bases,
as seen in the outer lines of the planes, are at present at their least

page_text: extension, those of the architraves at the next, and so on. I shall afterwards.
have to speak of gradations, of long extension.
With regard to elongating, I shall turn to Figure 15, which represents
an architectural wall, having the same plan as if it were columns and
intercolumniations. The shafts between which the curtain of the wall extends
are elongations of the halves of relational spheres, that is to say, the
sides are elongated, the upper face remains the same; its arcs are not of
vertical extension; the curtain of the wall is in the same manner the elon-
gation of the sides of an inter-base, the upper face remaining the same.
As the bounding lines at the top and bottom are not altered, the inclinations
of the arcs of horizontal extension will be the same as before the forms were
elongated. S is a transverse section through the middle of the curtain of
the wall, shewing the inclinations of the arcs of horizontal extension and
the lines of the sections they bound; to the plane of their diameters of
formation, which of these arcs are horizontal in it, and of those of vertical
extension are vertical. If the form were elongated all ways without being
separated, it would contain a dead space or solidity, bounded by planes; the
depth of the proportional formation would only extend to these. There.
might be constructions of this nature, especially for assimilating this system to
Gothic architecture, by having clustered columns; but I am not prepared to
go into such complications. On part of Figure 14, not only the lines of the arcs
of horizontal and vertical extension are drawn, but also others diagonally to
these; from these may be picked out various lines of patterns, of which certain
are shewn; these may be in connexion over the whole formation. There appears
the facility of making in relief with this system the regular configurations or
patterns, the corresponding ones to which may be made with right lines on a
plane surface. The geometrical representations of the front faces of the solids,
of this system, and of patterns upon them, are also duly proportioned for plane
figures.
I have next to refer to another description of capitals, those of double
curvature. In Figure 16, at A is one of these capitals and half of an archi-
trave, also of double curvature, attached to it. The under part of the capital
and architrave are the under half of the primary capital or relational sphere
and of the architrave belonging to it, as already described. Then concave
arcs, their heads joining the bases of the convex ones, continue the formation
outwards another gradation. There is the same proportion of extension given
to the arcs of vertical extension of this gradation as to the corresponding ones
of the last. The upper face of the capital is the same as that of a relational

page_text: sphere, except being flatter or having greater extension compared with it
height. This arises from the inequality of height of the diagonal and front
arcs taking place when they are at their largest, and not as in the relational
sphere, when they are at their smallest; the inequality therefore affords a less
height compared with the breadth. The upper face of the architrave is in like
manner flatter; each of the four faces of the capital, represented as viewed 'in
front, presents the like outline as the whole capital would do, but of the next
less gradation. The transverse section through the middle of the architrave pre-
sents the next less gradation again of the same figure. At B is the base and
inter base corresponding with this capital and architrave; they are merely in-
versions of them, but would be of the next larger gradation, and they have sub-
bases; that of the base I have made of the full proportion of height to depth of
the face of a relational sphere, not of the less proportion of height of the upper
face of these capitals, though I am doubtful which should be preferred, and in
the same manner as to the height of the entablature, upon which the base is
placed. This additional formation at C is constructed by carrying out a
gradation further, an inclined plane continuing the planes of sections upon the
upper arcs of the side faces of the capital and architrave. In the same manner
a gallery may be formed upon a column with a primary capital, as represented
in Figure 18. Figure 17 presents the construction of B and C upon the
transverse plane. Figure 19 represents a section through the middle of the
architraves, at the same time making a section of the roof or ceiling which
springs from them as they do from the shafts. Figure 20 represents the plan
of a ceiling or part of one; the architraves may be carried the whole way
across the apartment at once, and with them the ceiling between them, or they
may cross one another, there being where they cross suspended capitals, from
which the architraves are also to spring, as if they were underpropped. Such
appear the most direct varieties, but there may be many others; flat roofs may
have cornices, as represented in Figure 21, or may rest upon the half capital,
and there may also be cornices of lessen formation; architraves might be carried
up to form a sloping roof, though not quite regularly, according to this system;
I suppose such would rather be executed squaring off the formation of such
roof, making it plane within and without. I prefer what I believe are called
pavillion roofs, sloping in at the ends as well as sides, though gables and
tympanies may also be formed. The right lines drawn are to shew the different
heights at which I consider floors would best be laid; the highest is such as
would square in the relational sphere of the base, for this the shaft should
have a corresponding height; the next is at the bottom of the front lines of the
shafts, and there may be others lower.

page_text: I have now to state the principles upon which I consider the long
extensions of the shafts and architraves are to be regulated. Fig ure 22
is an arch consisting of two relational semi-circles, one of the next gra-
dation less than the other, and placed regularly within one another. Going
from this to squareness of the contained area and of form, except in the
relief upon the thickness of the walls, the shafts without bases or capitals
containing between them that area; take the proportional diminution used
in the present construction upon their thickness at the base, not upon
the breadth of the whole area, as is done by. the side arcs: of the arch. The
total height of the shafts is a third, proportional to their diminution and breadth.
At the base, meaning between their front arcs, that is, the diminution. being
th the height is 4; if th, the height would be 6, and so on; but the
height of the front arcs of the shaft is less than this by the diminution; I
consider the distance at which these shafts, the 'outer ones, stand to be the
completest one of a long extension, but of this there are also gradations; shafts
rising from these again will have the gradation less of height and thickness to
the extension between them, and the contrary of shafts formed to receive
those of the figure upon them, consequently of the next gradation larger. The
shafts of columns in the lowest tiér, in Figure 23, are of that size, and the
outer ones at that distance from one another. The height of their front arcs
is the same as in the shafts in Figure 22; this I consider the lowest long
height, and both the long distances between the outer shafts and the short
distance between them and the intermediate shafts I consider to be the shortest
of their respective regulations. With the bases at the nearness this gives them,
the concave relational arcs of the inter-bases connecting them are of their least
extension, or that with which they stand in the concave relational circle.
This coincidence between the short and long distances does not take place but
with the diminution during a gradation of exactly th; it also takes place with
a diminution of exactly th, if we make the facade hexastyle; and with a
diminution of th, if octastyle. In the tier of columns above these an incre-
ment is made to the proportion of height by squaring them in to the breadth
of a shaft at the base and placing the capital so squared in above the shaft so
squared in.
In the tier above these again the squaring in is to the breadth of a base of
a shaft: I do not, however, mean this to be done generally in succession
above one another, but choosing one of these heights makes successions with
it. With capitals of double curvature, I consider we may make increment of
height again by squaring off to the breadth of their bases; by this squaring
off we relieve the capitals and bases from the shafts. There are also grada-

page_text: tions of extension of another kind, not only by small increments; they are
obtained by taking always the diagonal: arcs as they stand upon the plane of
delineation or the breadth of the faces for the breadth of the whole forms, and
retaining the heights, as. before. Figure 24 is an example of construction
with a diminution of 3rd, instead of th; and Figure. 25. is an example
with diminution of th. With a diminution of 3rd; the extension of the: arc
in the relational circle, is twice, its depth; with a diminution of th, it is thrice
its depth; of th, it is five times its depth, and so proportionally. Windows;
except of a more elaborate, or arched description, I consider should be merely
cut out of the intercolumniation, generally oblong squares with flat sides; the
columns and architraves are their frames. I. consider the proportional divisions
of the surface, will afford proportions in placing them, taking double the number
of proportional divisions in the arcs of vertical extension that we do, in those of
horizontal extension, unless when we want to bring the windows to the ground,
or rather, make doors, then taking out, the same number of; the, proportional
divisions from top and bottom and each of the sides, instead of leaving the
sides curved, square then off and make them rectilinear. With right-lined
openings the appearance of a front will be firmer besides the convenience;
arch of this system, such as is shewn in Figure 22, I consider to be of due
formation for the purpose of an arch; as it departs from a flat arch and curves
inwards and downwards, its magnitude of curvature becomes lessened; as it: on
the other hand approximates. to. being, a flat arch, its proportions become
equalized; supposing it to be cut into proportional divisions, and these.to.be
the wedges of the construction, as, they; curve down to the base they become
lessened in breadth; as they are taken in, the line of a flat, arch they approxi
mate to becoming of, equal breadth. What arching there is, is directly sub-
sidiary to the support of a flat arch, and, at least for, the support, of a flatness
above, I consider the arch to be right. It appears well calculated to resist
lateral pressure; it has not the heaviness towards the diagonal lines which a
circular arch has; admits of being extended or heightened; also seems proper
for forming pointed arches by, bringing together its side, arcs. I consider
that complete vaulting may be well formed with such arches, but am not pre-
pared to describe such.
the
I have also to specify the improvement I propose in the surfaces of
buildings; it is to lay them over with paper of lead, or very thin lamina
of that metal, to give plaster especially a skin, prevent discoloration, assist
in hiding cracks and imperfections, and preserve the surface. I have been
led to the consideration of this by the other object of my Patent, it being
very desirable to have an apparantly amalgamated surface, not shewing lines

page_text: of division of the material. The difficulty of making such curvilinear forms
may probably make it desirable to use moulds, properly so called; first,
building up the structure roughly, and then building up the mould and filling
up a certain depth of surface, as we proceed, with plaster. At all events, con-
sidering a plastic material most adapted for them, it is desirable to give a good
surface. I have found the lead to cling indifferently well upon paint, having
been laid on it when wet, but I consider the best material for fixing it to be
gold size, such as usually used for large work or work in the open air; withi
this I have found it cling well. I mean it for interiors as well as exteriors of
buildings, and in all sorts of architectures; the lead, of course, may be either
washed or painted over.
In witness whereof, I, the said John Robertson, have hereunto set my
hand and seal, this Eighteenth day of January, in the second year of
the reign of our Sovereign Lady Victoria, by the grace of God, of the
United Kingdom of Great Britain and Ireland, Queen, Defender of the
Faith, and in the year of our Lord One thousand eight hundred and