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Class of 1898
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COURSE IN
DESCRIPTIVE GEOMETRY AND STEREOTOMY.
By S. EDWARD WARREN, C. E.
PROFESSOR OF DESCRIPTIVE GEOMETRY, ETC. ; FORMERLY IN THE RENSSELAER POLYTECHNIC
INSTITUTE, TROY.
The following works, published successively since i860, have been well received by all the scien-
tific and educational periodicals, and are in use in most of the Engineering and Scientific Schools of
the country, and the elementary ones in many of the Higher Preparatory Schools.
The Author, by his long-continued engagements in teaching, has enjoyed facilities for the prep-
aration of his works which entitle them to a favorable consideration.
I.— ELEMENTARY WORKS.
These are designed and composed with great care ; primarily for the use of all higher Public and
Private Schools, in training students for subsequent professional study in the Engineering and Sci-
entific Schools ; then, provisionally, for the use of the latter institutions, until preparatory training
shall, as is very desirable, more generally include their use ; and, finally, for the self-instruction of
Teachers, Artisans, Builders, etc.
1. ELEMENTARY FREEHAND
GEO ME TRICA L DRA WING. A Series
of Progressive Exercises on Regular Lines and
Forms ; including Systematic Instruction in
Lettering. A training of the eye and hand for
all who are learning to draw. i2mo, cloth,
many cuts, 75 cents.
2. ELEMENTARY PLANE PROB-
LEMS. On the Point, Straight Line, and
Circle. Division I. — Preliminary or Instru-
mental Problems. Division II. — Geometrical
Problems. i2mo, cloth, $1.25.
3. DRAFTING INSTRUMENTS
AND OPERATIONS. Division I. — In-
struments and Materials. Division II. — Use
of Drafting Instruments, and Representation
of Stone, Wood, Iron, etc. Division III. —
Practical Exercises on Objects of Two Dimen-
sions (Pavements, Masonry Fronts, etc.) Di-
vision IV. — Elementary ^Esthetics of Geo-
metrical Drawing. One volume, i2mo, cloth,
51.25.
Volumes 1 and 3 bound together, $1.75.
4. ELEMENTARY PROJECTION
DRA WING. Third edition, revised and en-
larged. In five divisions. I. — Projections of
Solids and Intersections. II. — Wood, Stone,
. and Metal Details. III. — Elementary Shad-
ows and Shading. IV. — Isometrical and Cab-
inet Projections (Mechanical Perspective).
V. — Elementary Structures. This and the
last volume are especially valuable to all Me-
chanical Artisans. i2mo, cloth, $1.50.
5. ELEMENTARY LINEAR PER-
SPECTIVE OF FORMS AND SHAD-
OWS. With many Practical Examples. This
volume is complete in itself, and differs from
many other elementary works in clearly dem-
onstrating the principles on which tlie prac-
tical rules 0/ perspective are based, without
including such complex problems as are usu-
ally found in higher works on perspective. It
is designed especially for Young Ladies' Semi-
naries, Artists, Decorator', and Schools of De-
sign, as well as for the institutions above men-
tioned. One volume, i2mo, cloth, $1.00.
II. -HIGHER WORKS.
These are designed principally for Schools of Engineering and Architecture, and for the members
generally of those professions ; and the first three also for all Colleges which give a General Scien-
tific Course, preparatory to the fully Professional Study of Engineering, etc.
I. DESCRIPTIVE GEOMETRY.
Adapted to Colleges and Liberal Education,
as well as to Technical Schools and Technical
Education . Part I. — Surfaces of Revolution.
The Point, Line, and Plane, Developable Sur-
faces, Cylinders and Cones, and the Conic Sec-
tions, Warped Surfaces, the Hyperboloid,
Double-Curved Surfaces, the Sphere, Ellip-
soid, Torus, etc , etc. Complete in itself by
giving, as may be preferred, Warped Surfaces
in immediate connection with their applica-
tions. One volume, 8vo, twenty-four folding
plates and wood-cuts, cloth, $4.00.
II. GENERAL PROBLEMS OF
SHADES AND SHADOWS. A wide
range of problems ; gives variety without rep-
etition, and a thorough discussion of the prin-
ciples of Shading. One volume, 8vo, with
numerous plates, cloth, $3.50.
III. HIGHER LINEAR PER-
SPECTIVE. Containing a concise sum-
mary of various methods of perspective con-
struction ; a full set of standard problems ; and
a careful discussion of special higher ones.
With numerous plates. 8vo, cloth, $4.00.
IT. ELEMENTS OF MACHINE
CONSTRUCTION AND DRAWING.
On a new plan, and enriched by many stand-
ard and novel examples of the best present
practice. Text 8vo. Plates 4to. $7.50.
V. STONE CUTTING. Cloth, 8vo, ten
plates, §2.50.
STERNOTOMY.
PROBLEMS
STONE CUTTING.
IN FOUR CLASSES.
I. — PLANE-SIDED STRUCTURES.
II. —STRUCTURES CONTAINING DEVELOPABLE SURFACES.
III. — STRUCTURES CONTAINING WARPED SURFACES.
IV. — STRUCTURES CONTAINING DOUBLE-CURVED SUR-
FACES.
FOR STUDENTS OF ENGINEERING AND ARCHITECTURE.
S. EDWARD WARREN C. E.
PROFESSOR IN THE MASSACHUSETTS NORMAL ART SCHOOL, ETC., AND FORMERLY IN
THE RENSSELAER POLYTECHNIC INSTITUTE.
NEW YORK:
JOHN WILEY AND SON,
15 Astor Place.
1875.
Entered, according to Act of Congress, in the year 1875, by
S. Edward Warken, C. E.,
In the Office of the Librarian of Congress, at Washington.
RIVERSIDE, CAMBRIDGE.
8TEREOTYPED AND PRINTED BY
H. O. HOUGHTON AND COMPANY.
TO ALL THOSE GRADUATES OF THE
KENSSELAER POLYTECHNIC INSTITUTE,
IN MANY SUCCESSIVE TEARS J WHO DOUBTLESS STILL RETAIN A CLEAR REC-
OLLECTION, AND, I HOPE, A PLEASANT REMEMBRANCE
OP THE "OBLIQUE ARCH," AND THE
" COMPOUND WING-WALL,"
AS DRAWN BY THEM UNDER MY INSTRUCTION ;
Cijts little Walnme,
WHICH CONTAINS IMPROVED ILLUSTRATIONS OF BOTH, WITH MANY OTHER
PROBLEMS, IS MOST KINDLY AND VERY RESPECTFULLY DED-
ICATED BY THEIR FORMER TEACHER AND
LASTING FRIEND
THE AUTHOR.
PKEFAOE.
This manual has been composed with the idea of represent-
ing, essentially, every class of structures, and every principal
variety of surface, so as to make it most widely useful to the
student in solving any other problems which he might meet.
The student needs, desires, and appreciates explicit detailed
information in due abundance, not to prevent him from think-
ing for himself, but to train him to do so by examples fully
explained. On this principle, and supported by the best author-
ities, I have discussed the few problems which could be admit-
ted within the proposed limits, so thoroughly as to satisfy, I
trust, all who enjoy the most — indeed, the only universally
— available help, viz., a printed text.
I have attached scales and dimensions to the problems, which
teachers and students may use or not, according as they prefer
to work as if drawing actual structures for practical purposes,
or to study the purely geometrical principles and operations
involved. In either case, the figures should be made, generally,
from two to three times as large as those of the plates in this
volume, carefully following the text, and under frequent inter-
rogation by the teacher, in doing so. I should add that this
work presupposes a fair acquaintance with descriptive geometry,
though many of its problems could be understood after the
study of my " Elementary Projection Drawing." It is, how-
ever, complete in itself in regard to several collateral topics
required for use in it, and not as conveniently found else-
where.
I must here acknowledge my indebtedness to Leroy for
suggestions of practical problems, and to Adhemak, in per-
VI PREFACE.
fecting the treatment of the oblique arch. It seemed better to
take examples essentially like actual structures, rather than
imaginary ones, merely for the sake of greater apparent origi-
nality. But I have in every case made such changes, in vari-
ous details of design and treatment, including, especially, the
novel feature of numerous examples for practice, as to make
my volume as much as possible a new contribution as well as a
text-book. Moreover, problems VIII. and XV., and the sys-
tematic arrangement presented in the general table, will not
be found elsewhere.
Newton, Mass., May, 1875.
NOTE TO TEACHERS.
Two complete successive courses, elementary and higher, can be
made up from this work, as follows : —
Each problem to be drawn in Plan, Elevation, Section, Details, in
isometric or oblique projection, and Developments.
I. Elementary Course. — a. Plane-sided structures. — 1. The but-
tressed walls (Prob. II.). 2. The plate band (Prob. III.).
3. A plane-sided wing-wall (Arts. 14-16).
b. Involving developable surfaces. — [See Arts. 17-28, and Probs.
IV., V., and XVII. (the bracket), with their examples.] ■ —
1. The segmental — 2. The full centred — 3. The sloping
front — 4. The skew front — 5. The cylindrical-faced —
6. The rampant — 7. The conical recessed, arch. 8. The
trumpet bracket.
c. Involving warped surfaces. — 1. Circular stairs around a central
post. [See Arts. 119-125, and 126.] 2. The warped-faced
wall (59).
d. Involving double-curved surfaces. — 1. The Niche (Prob. XVII.)
2. The dome only (Prob. XIX.).
II. Higher Course. — Any selection from the above, as introductory,
for those who have not previously taken the foregoing elementary
course, together with any of the other problems of each of the four
classes noted in the table of contents.
V11L
STONE-CUTTING.
GEOMETRICAL CLASSIFICATION.
The Characteristic Surface.
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CONTENTS.
Preface . v
Note to Teachers vii
STEREOTOMY : Stone-cutting : First Principles .... 1
CLASS I.
Plane-Sided Structures.
PROBLEM I. — To form plane surfaces of stone, making any angle with
each other 4
PROBLEM II. — A sloping wall, and truncated pyramidal buttress . 5
PROBLEM III. — The recessed flat arch, or plate-band .... 8
Plane-sided Wing-Walls 10
CLASS II.
Structures containing Developable Surfaces.
Arches. — Definitions 12
Classification 13
Preliminary constructions 14
§ Conic Sections ...... 14
1°. — To construct a circle by points, having given its radius . . 14
_2°. — To construct an arc of a circle by points, knowing its chord
and versed-sine, or rise. 15
3°. — To construct an ellipse by points on given axes. Also, nor-
mals to it (two methods) 15
4°. — To construct the arc of a parabola ; on a given segment of the
axis, and a chord which is perpendicular to the axis. Also,
normals to it . . . . . . . . 16
5°. — To construct an arc of a hyperbola, on a given chord, and seg-
ment of the axis, perpendicular to the chord . . . 17
§§ Poli/central Arch Curves . . . . .18
Three-Centred Ovals . . . . . . . . . 18
1°. — To construct the general case of the semi-oval of three centres 18
2°. — First special case. The semi-oval of three centres when the
lesser arc is 60° 18
3°. — Second special case. The ratio — to be a minimum . . 19
X CONTENTS.
Five-Centred Ovals 20
4°. — To construct a five-centred semi-oval, which shall conform as
nearly as possible to a semi-ellipse, on the same axes . 20
5°. — To construct the five-centred oval, by a method applicable to
an oval having any number of centres . . . .21
Illustrations.
PEOBLEM IV. — A three-centred arch in a circular wall . . . .22
PEOBLEM V. — A semi-cylindrical arch, connecting a larger similar gal-
lery, perpendicular to it, on the same springing plane ; with an en-
closure which terminates the arch by a sloping skew face . . .25
Groined, and Cloistered Arches ....... 29
Theorem I. — Having two cylinders of revolution, whose axes intersect ,
the projection of their intersection, upon the plane of their axes, is a
hyperbola 30
PEOBLEM VI. — The oblique groined arch' 32
PEOBLEM VII. — The groined and cloistered, or elbow arch . . .35
Conical, or Trumpet Arches ........ 37
PEOBLEM VIII. — A trumpet in the angle between two retaining walls . 37
PEOBLEM IX. — A trumpet arched door, on a corner .... 40
PEOBLEM X. — An arched oblique descent 44
CLASS III.
Structures containing Warped Surfaces.
PEOBLEM XL — The recessed Marseilles gate . . . . . 49
The Oblique Arch.
Preliminary topics. — Elementary mechanics of the arch . . . .53
The resulting standard, or essentially perfect design for an
oblique arch ......... 55
PEOBLEM XII. — The partial, and trial construction of the orthogonal, or
equilibrated arch 56
The Helix . . . . . . . . . . .61
Theorem II. — The projection of the helix on a plane parallel to its axis
is a sinusoid .62
The Helicoid .......... 62
PEOBLEM XIII. — A segmental oblique arch, on the helicoidal system . 63
I. The Projections. (Arts. 81-104) . . 63
II. The Directing Instruments . . . . . . . . 73
III. The Application 76
Useful Numerical Data ......... 79
Modifications of the Orthogonal and Helicoidal Systems . . .81
Wing- Walls . . ... . .83
PEOBLEM XIV. — The compound, or piano-conical wing-wall . . 85
The Conoid 92
PEOBLEM XV— The conoidal wing-wall 94
CONTENTS. XI
Stairs ...... 97
PKOBLEM XVI. — Winding stairs on an irregular ground plan . . 99
Other forms of stairs .........
CLASS IV.
Structures containing Double-Curved Surfaces.
PEOBLEM XVII. — A trumpet bracket, with basin and niche . . .103
Theorem III. — The conic section whose principal vertex and point of
contact with a known tangent are given, will be a parabola, ellipse,
or hyperbola, according as the given vertex bisects the subtangent,
or makes its greater segment within or without the curve . .106
PEOBLEM XVIII. — The hooded portal ...... 107
PEOBLEM XIX. — An oblique lunette in a spherical dome . .110
Pendentives ........ .114
Spirals ......... .115
PEOBLEM XX. — The annular and radiant groined arch . 120
STEREOTOMY.
STONE-CUTTING.
FIRST PRINCIPLES.
1. Stereotomy is that application of Descriptive G-eometry
which, comprehensively defined, treats of the cutting or shap-
ing of forms, whether material or immaterial, so as to suit cer-
tain given conditions.
2. Stereotomy, thus defined, embraces, either by etymology,
or established usage, the following subjects : —
1°. Shades and Shadows, or the cutting of the volume of space
from which an opaque body excludes the light, by any given sur-
fac%, on which the shadow of the body is thus said to fall.
2°. Perspective, or the cutting of the cone, of which the ap-
parent limit of a given body is the base, and the eye the ver-
tex, by any given plane, whose intersection with this cone is
called the perspective of the given body.
3°. Dialing, or the cutting of metal plates so that their shad-
ows upon a given surface shall mark the hours of the day.
4°. Cinematics, or the shaping of mechanical forms, so that
by their mutual action they shall produce certain motions.
5°. Structural articulations, or the shaping of the articula-
tions of wood and iron framings of every kind, with reference
to convenience of construction and use.
6°. Carpentry, or the cutting of wooden pieces, so that when
united they shall form a self-supporting whole.
7°. Stone-cutting, or the cutting of stone pieces of prescribed
form, from the rough block, so that when combined in an as-
signed order, they shall form a given or predetermined whole.
Of these, the last two are the most obviously characteristic ;
that is, most clearly illustrative of the definition (1).
3. Stone-cutting as a science embraces three distinct parts: —
1°. The construction, on large scales in practice, of the pro-
jections of at least so much of a proposed structure as will per-
mit —
i
2 STEREOTOMY.
2°. The derivation therefrom of the directing instruments,
used by the workman as guides in cutting the rough block to
its intended form by the chisel and mallet.
3°. The rules for the application of these directing instru-
ments in the proper order and manner.
4. The first two of the three parts just mentioned consist
of operations of applied descriptive geometry. The number of
directing instruments, and the mode of their application, will
depend considerably on the ingenuity of the designer.
5. Practical stone-cutting, or the actual formation of the
finished stone, belongs to the student, only so far as it may, in
the absence of models, serve him in gaining familiarity with
those complex masonry forms which cannot be readily imag-
ined from drawings alone.
In such modelling, the intended pieces would be wrought in
plaster, by the aid of their wooden, or paper directors, derived
from the drawings.
6. Slopes are variously expressed. 1°. In PL L, Fig. 1, the
Tk
slope H"k, for example, may be expressed by the ratio, ;/ , =
Tk 5
the tangent of the angle TH"k. Here ^7577 = T ' rea ^' a s ^°P e
of five to one.
2°. By degrees. Slopes of 30°, 40°, etc., make these angles
with the horizontal plane.
3°. A batter of 1 inch to 1 foot, etc., means a horizontal de-
parture of one inch from a vertical direction, for each foot of
altitude.
4°. Nearly level slopes, as of railways, are described as a rise
of 1 in 100, etc., 40 feet to the mile, etc.
5°. Once more ; a slope of 45° being naturally described as
that of 1 to 1, every other slope may be described by naming
its horizontal component distance first, and by taking its least
component as the unit. Thus a slope of 4 horizontal, to 1 ver-
tical, may be described as a slope of 4 to 1. But one of 1
horizontal, to 7 vertical, for instance, as a slope of 1 to 7 ; or a
batter of 1 in 7.
The nature of the case, or a reference to the figure, will
show, in each problem, the meaning of any expression of slope
that may be used.
STONE-CUTTING. 3
Directing instruments.
7. The directing instruments (3) used in stone-cutting are
of three kinds, bevels, templets, and patterns.
Bevels, as the common steel square, give the relative posi-
tions of required lines, or surfaces of a stone, by showing the
angles between them. In the former case they are plane bev-
els ; in the latter, diedral bevels.
Templets give the forms of required edges, or other distin-
guishing lines of a surface.
Patterns show the forms of plane or of developable surfaces.
In the former case, they may be made of any stiff, thin mate-
rial. In the latter, they must be flexible.
These instruments will be designated by numbers in the sub-
sequent problems, and in every case, No. 1 will be a straight-
edge, and No. 2 the square.
Notation.
8. For the sake of brevity, the horizontal and vertical planes
of projection will, on account of the frequent reference which
must be made to them, be denoted, respectively, as the planes
H and V.
The usual rules for inking visible and invisible, given or
auxiliary lines and planes (Des. Geom. 45), will be followed,
unless in particular cases greater clearness may result from
disregarding them.
When important lines are hidden by viewing a structure, as
usual, vertically downwards from above it, its horizontal pro-
jection may be inked as if the object were seen by looking at it
vertically upward from below it.
The greater complexity of some of the figures will make it
convenient to adopt the rule of distinguishing invisible lines
of the structure from the lines of construction, by dotting the
former and marking the latter in short dashes ; a distinction
not shown, however, on the plates of this volume.
9. In order to secure a brief, yet comprehensive exhibition of
the elements of stone-cutting, that is, one representing every
important class of structures, and form of surface, they may
be classified as in the General Table, the frontispiece.
CLASS I.
Plane-Sided Structures.
Problem I.
To form plane surfaces of stone, making any given angle with
each other.
This fundamental problem, being of constant occurrence, is
here separately explained, in order to avoid repetition.
Kg. l-
Fig. 2.
10. First. Fig. 1. represents the first steps in forming a
plane upon a wholly unwrought block. Having two straight-
edges, AB and CD, of equal width, ledges, as m n and p q, are
cut on opposite edges of the stone, until the tops of the
straight-edges placed on them, as shown, are found by sight-
ing, as from E, to be in the same plane.
The portion of rough stone between m n and p q is then cut
away until found, by frequent test with the straight-edge, ap-
plied transversely, as at FG, to be wrought down to the plane
of m n and p q.
Second. Having prepared one plane surface, as shown in
Fig. 2, a second plane face, perpendicular to the former, may
be formed, as shown, by cutting two or more channels in any
direction on the required face, until one arm, BC, of a square
will fit any of them, while the other arm, AB, coincides with
STONE-CUTTING. 5
the given face, and in a direction perpendicular to the common
edge of the two surfaces. The intermediate rough stone is then
cut down to the plane of these channels by applying the straight-
edge transversely to them. For any other than a right angle,
use a bevel giving such angle.
11. The principle of the first operation is, that if two lines,
m n and p q, are in the same plane, all lines, as FG, which in-
tersect them, are in that plane also.
That of the second operation is, that if two planes are per-
pendicular to each other, any line, as AB, in one of them, and
perpendicular to their intersection, will be perpendicular to all
lines, BC, etc., drawn through its foot and in the other plane.
Problem II.
A sloping wall and truncated pyramidal buttress.
I. The Projections (3) . — These, PI. I., Fig. 1, are made partly
from given linear dimensions, and partly from given slopes of
the inclined faces of the buttress and wall.
The plan and front elevation can be made wholly from data
of the kinds just mentioned ; but an end elevation is added,
as a check upon errors, and as showing a different method of
operation.
Let the wall be 8 ft. in height, and 3 ft. 6 in. thick at
its base, and with a slope of 1 to 6 on its front. Let the ver-
tical height of the buttress at c" be 7 ft. ; the slope of its front
1 to 5, that of its sides 1 to 4, and that of its top 4^ to 1 (6).
Then —
1°. Construct the end elevation by making G"m = <t of G"E",
and A"C" parallel to E"m ; also E"N = 10 ft., ON = 2 ft., and
Wh parallel to OE" ;' and C"L = 4 ft. 6 in., LM = 1 ft., and
a"c" parallel to MC", having made c"e 1 ft. below C"E".
These operations will give the required slopes seen in the
end view.
2°. The construction of the plan and front elevation of the wall
is obvious on inspection, all the heights in elevation being de-
termined by projecting across from the end view, and all the
widths in plan, by transferring the horizontal distances, E"C",
etc., from the end view to any convenient line of reference, as
e x h^ perpendicular to the ground line.
6 STEREOTOMY.
3°. The projections of the buttress. — The dimensions of its
base are, JK = 6 ft. ; HI = 4 ft. ; and A^U = 2 ft. 6 in. With
these draw the base, whence, by means of the end view and
the given slopes, all its lines can be found.
Thus, ah and cd, drawn indefinitely at first, are at distances,
e x b x and e x d^ from EF respectively equal to a"f and c"e.
The slope of the sides being 1 to 4, draw ij parallel to HJ,
and at a distance, gi, from it, equal to one fourth of G''E".
Then Qj and ij, being in the plane of the top of the wall, Jj is
the intersection of the face of the wall with the left side of the
buttress ; and c, its intersection with d x d, is a back upper cor-
ner of the buttress. Drawing the horizontal a"f, and making
e l n l ,=fn", draw n x n parallel to AB, and from n, draw na par-
allel to JH, and na will meet ah at a. Then draw ac and aH,
which will complete the left side of the buttress. Its right side
can be laid off from the axis of symmetry UV.
The vertical projection of the buttress is made by simply
projecting the points of the plan up to the traces H'K', a'b',
and c'd' of the horizontal planes in which they lie, and then
joining the points as shown.
In constructions like that of Jj, the pairs of parallels, A J
and HJ ; C/ and ij, which determine the required line, should
be as far apart as possible.
II. The Directing Instruments. — These are, besides Nos. 1
and 2 (7), No. 3, a pattern of the base of the upper stone of
the buttress, and seen in its true size in plan : No. 4, a bevel
containing the diedral angle, shown on the end elevation, be-
tween the base and the back of the buttress : No. 5, the like
angle between the base and front of any buttress stone : No. 6,
a bevel giving the angle between the base and either of the
sides of the buttress.
No. 6 is found by revolving any line of greatest declivity, as
R/i, perpendicular to JH, until parallel to V, as at RA" — Wh' n ;
when it will show the true slope of the side of the buttress.
No. 9 gives the angle between the top and back of the but-
tress.
If patterns are used as checks upon the operation of the bev-
els, they may be found as follows.
No. 7 is a pattern of the top of the buttress, and shown in
its true form by revolution about cd — c'd', till parallel to V.
This is done by making c'"V = a"c" and drawing a'"b" r paral-
STONE-CUTTING. 7
lei to a'b' and limited by the vertical projections, oV" and b'b'",
perpendicular to c'd', of the arcs described by ad and bb' in re-
volving about cd — c'd'.
No. 8 is a pattern of the right side of the upper abutment
stone, and is shown in its true form by revolving it about PQ
— P'Q' till horizontal. Then d", the revolved position of dd',
may be found by describing an arc with P as a centre and a
radius equal to d'P'", shown by the construction to be the true
length of Pd — P'd', and noting where it intersects dd" per-
pendicular to the axis PQ. The point b" being similarly found,
PQb"d" is the required pattern. These patterns would con-
veniently be open wooden frames.
III. The Application. — We will here illustrate this topic by
the manner of working the top stone of the buttress. This,
being the most irregular stone in the problem, the full explana-
tion of the manner of forming it will enable the student to de-
vise means for working any of the others.
Having selected a rough block in which the finished stone
might be inscribed, first bring the intended base of the stone to
a plane by Prob. I. Next scribe the edges of the base by pat-
tern No. 3.
Having thus the bottom edges of all the lateral faces of the
stone, these may be wrought from the base, and in their true
relative position, by the bevels Nos. 4, 5, and 6. By marking
the distances sc" and ra" on bevels Nos. 4 and 5 respectively,
the edges ab — a'b' and cd — c'd' will be determined on the
stone, which can then be finished by the use of No. 9 and the
straight edge.
For increased accuracy, patterns Nos. 7 and 8, and others
similarly found for the front and rear faces of the stone, can be
used in determining the edges of the lateral faces more posi-
tively than by the natural intersection of these faces as found
by the bevels and straight edge.
Examples. — 1°. Construct patterns 10 and 11, of the front and rear faces of
the top buttress stone.
Ex. 2°. Consmict a bevel, No. 12, giving the angle between the front, and an
adjacent lateral face.
Ex. 3°. Show the construction and use of the guides necessary in working the
stone below the top one of the buttress.
Ex. 4°. Do. for the top stone, supposing ijg top surfaces to be a pyramid hav-
ing its vertex at vv'.
Ex. 5°. Construct, on a larger scale than in Fig. 5, a plan and front elevation
b STEREOTOMY.
of the wall whose end elevation is there shown, with an isometrical figure of the
stone NFEDCg 1 . [The inclined surfaces, aspq, help to prevent sliding in the direc-
tion of the arrow, from the pressure of materials at the back, AN, of the wall.]
Problem III.
The recessed flat arch, or plate-band.
12. An arch is an assemblage of blocks, mutually support-
ing, by means of radiating joints between them, and side sup-
ports to confine them laterally. When the arched surface,
usually cylindrical, is plane, the structure is called a Plate-
band.
I. The Projections. — PL I., Figs. 2, 3, and 4. The con-
struction of these will, after a brief description, be sufficiently
obvious on inspection of the figures, since, on all of them, the
same letter indicates the same point.
Tt Jj — T / T"J 7 U / , Fig. 2, is the rectangular door-way, or
opening through a wall. The door, not shown, folds against
the vertical plane surface, U'V'J'Q'T"S"S'T', which is in the
plane qs. The recess which contains the door when shut, is
bounded by the surfaces just pointed out, and by the three
surfaces, Qq — V'Q'; Q^Ss— Q'S" ; and Ss — S'S", all of
which are perpendicular to the plane V-
The surfaces PQ — V'W'P'Q', PQRS — PQ'R"S", and
E,S — R/S'R/'S", collectively, form the convergent sides, or
jambs of the door- way.
The joints, AT)', etc., divide T"J' into equal parts, and
radiate from a point O, found by making OT"J' an equilateral
triangle. The lapping of the end stones, as Y, over the upper
jamb stones, as at G'F', is designed to give greater security.
Fig. 3 is an oblique projection of the stone A'D'E'F', as seen
in looking at it obliquely upward so as to see its front, right-
hand, and under surfaces.* The identity of its several sur-
faces with the same ones as seen in Fig. 2, is shown by the
lettering.
Fig. 4 is an isometrical drawing of the upper jamb-stone
G'X'N', showing its front, left-hand, and under surfaces.
II. The Directing Instruments. — Of these the only ones re-
quiring any further graphical operations for their construction
* See my Elementary Projection Drawing : Div. IV. Chap. V., for an
explanation of this species of projection.
STONE-CUTTING. 9
are the patterns of the radiating joints. One of these is shown
at l'p'p n f" ! f"k'"k", by revolving the joint, vertically projected
in f'p', about that line, considered as in the plane RX, as an
axis ; whence p'p"=Rr ; k'k"=cu ; k'k" f =cA, etc.
The true form of the joint on J'G' may be similarly found.
III. The Application. — Let it be required to work the stone
A'D'F', Figs. 2 and 3. There is no one precise order of oper-
ations which is indispensable ; but we should naturally work
first, either the top surface, EeD<i, as the simplest, or the back
deaj, or joint on A'D', as the largest and most important.
Supposing then that we have a block in which the stone can
be inscribed, as shown at fhmo; bring the intended joint sur-
face on A'D' to an indefinite plane by Prob. I., and by the
pattern, No. 3, mark its edges. Next, the back and front can
both be wrought square with this surface by No. 2, and their
edges marked by their patterns A'D'E'F'G'J' (No. 6), and
C'D'E'F'G'H' (No. 7). The top and bottom surfaces can
then be wrought, like all the lateral surfaces, either square
with the back, by No. 2 ; or, from the joint surface A'D' by
the bevels Nos. 4 and 5.
The surfaces on E'F' and G'F' are square with each other
and with the front and back. A pattern, No. 8 (not shown),
of the joint on J'G', and a bevel, No. 9, set to the angle QPX,
between the vertical and jamb surfaces, will suffice for the ac-
curate completion of the stone.
13. The stones A'D'F' and G'F'N' cannot be more clearly
shown than in Figs. 3 and 4, in which all their most irregular
portions are made visible. But the student cannot here be-
come too familiar with the methods of representation there
given, on account of their special usefulness in the drawing of
complicated masonry. Hence the nature of a. part of the fol-
lowing examples.
Ex. 1° Make an isometric drawing of the stone Y, as inscribed in a prism
containing its top and bottom faces.
Ex. 2°. Make the same drawing of Y as inscribed in the prism fhmo.
Ex. 3°. Make an oblique projection of the upper left-hand jamb-stone.
Ex. 4°. Draw the instruments, and describe their use, for working the upper
jamb stones as G'F'N'.
Ex. 5°. Devise and describe any other convenient manner of working the stone
Y, than that given in the text.
Ex. 6°. Make the top of the door-way semi-octagonal, by sloping the under
sides of Y and A'D'G' at 45° to the vertical sides, on Tt and J/, of the opening.
10 STEREOTOMY.
Plane Sided Wing-Walls.
14. Wing-walls, are those which flank the approach to any
passage-way, or area, so as to prevent obstructing it.
15. PI. L, Figs. 6-9, illustrate in outline the leading forms
of plane sided wing-walls ; rectilinear, prismatic, and pyram-
idal.
Fig. 6, a rectilinear wing- wall, AB — A'B', is such as might
be used to retain a roadway ascending behind it, and bending,
as shown by the arrows, to cross a bridge, one of whose abut-
ments is CC. Such a construction is often seen where a rail-
way or canal is crossed at right angles by a road which, on one
or both sides of the crossing, is parallel to the railway or canal.
Fig. 7 represents wing-walls, as abc — b'a'c', which, with the
arch-wall, cd — c'd', form three sides of a vertical hollow quad-
rangular prism, in which the tops of the wing- walls are trun-
cated by an oblique plane, a'c'd'g'. Had there been a slope or
batter to the arch and wing-walls, the inner surfaces, collec-
tively, would have formed part of a vertical quadrangular pyra-
mid. Either construction would serve in case of a railway
tunnel occupying the line of a city street.
Fig. 8 represents wing-walls, which, with the connecting
arch-wall, DF, form half of a vertical hollow hexagonal prism,
wholly truncated by an oblique plane, AEG — A'E'G' ; and
flanking the inlet to a pipe culvert, mn — m'n', by which the
water of some gulley or small ravine may be led through an
embankment.
Fig. 9 differs from Fig. 8, in that the top of the arch-wall,
dstg — d'g', is horizontal, and the inner faces of the walls form
a part of a vertical inverted pyramid.
16. A coping is a layer of thin stones, laid on the top of a
wall, and made broader than the thickness of the wall, which
they cover, for the purpose of sheltering the joints from the
weather, while their adaptation to this end and the effect of
their shadows, add to the beauty of the whole.
Had the arch-wall, Fig. 9, been truncated by the horizontal
plane P'Q', it might have received a coping, whose front edge
would have been, in plan, at pq, formed by projecting r' at r.
Lines between, and parallel to be and fk, would represent the
front of the copings of the wing- walls.
STONE-CUTTING. 11
Problems I. — III. will enable the student to work the fol-
lowing —
Examples. — 1°. Draw PI. I., Fig. 7, to scale from measurements, with a section
on the plane nNn', and describe the cutting of a stone at the corner dd', which, to
break joints with courses below, shall extend from kf in the arch- wall, cd, to pj
in the wing-wall, ee'.
Ex. 2°. Draw PI. I., Fig. 8, in like manner, making the vertical joint surfaces,
perpendicular to FD and FG, and describe, with an isometrical or oblique projec-
tion, the working of the top stones at one of the angles D or F.
Ex. 3°. In Ex. 2°, let there be a coping, and let the top be truncated as at P'Q',
Fig. 9.
Ex. 4°. Treat PI. I, Fig. 9, as described in Ex. 2°.
Ex. 5°. In Ex. 4, let there be a coping. Also add square piers at the foot of
the wing-walls, each side of which in plan shall be equal to ac, and whose
height shall be a'a" .
CLASS II.
Structures containing Developable Surfaces.
17. Walls having either cylindrical or conical faces, will be
treated incidentally, in connection with other constructions of
which they form a part.
Arches (12) include as particular cases, portals or arched
openings through walls ; and combined, or groined, and clois-
tered arches. Vaults include, more comprehensively, domes,
and other over-arched areas.
Only cylindrical arches will now. be considered, and under
the heads of, —
1°. Definitions. III . Preliminary Constructions.
11°. Classification. IV°. Practical Illustrations.
1°. — Arches — Definitions.
18. Arch Masses. — The supporting walls of an arch are
called its abutments when backed by earth, etc., and piers,
when exposed on all sides. The superincumbent masonry,
supported by the arch, is called the spandril, or backing.
Each row of blocks extending through the length of the
arch is a course. Each block in the course is an arch-stone,
or voussoir. The extreme voussoirs of each course are the
quoins, or ring-stones. The middle ring-stone is the key-stone.
19. Arch /Surfaces. — The end surfaces of an arch are its
faces, a'd' — aQDd, PI. III., Fig. 27. (In certain cases, where
the elevation is more important, or naturally made first, the
accented letters are attached to the plan (8) ). The inner
cylindrical surface, ABD, is the intrados. The outer one,
whether cylindrical, Fig. 29, or plane-sided, Fig. 27, is the
extrados. The top of the abutment, if level, is called the
springing plane ; if inclined radially, the skew-back.
The longer sides, mm'rr', Fig. 26, of a voussoir, are its beds,
and are continuous from stone to stone. Its ends, are its heads,
and these terminate, or break, in the coursing surfaces. Its in-
ner side is its soffit ; the outer, its back.
STONE-CUTTING. 13
20. Arch Lines. — The intersections of the faces with the
intrados and extrados are the face lines of the arch, and of
those surfaces. The intersections of the springing plane, or
the skew-back, with the same surfaces, are the springing lines.
The axis joins the centres of the face lines, and is parallel to
the springing lines in an arch of uniform cross section, and is
also straight in a cylindrical, and a conical arch. The edges,
parallel to the a±is, of the courses are the coursing joints ;
the transverse edges of the voussoirs are the heading joints ;
the edges lying in the thickness of the arch are the radial
joints, and the joints in the face are the face joints. The
span is the perpendicular distance between the springing lines.
The rise is the greatest height of the intrados above the span.
21. Arch Points. — The highest point of any right section of
an arch is its crown. The intersection of the axis with the
face is the face centre. When the face joints radiate from any
other point than the face centre, such point is called a, focus.
11°. — Arches — Classification.
22. Arches may be classified according to either, 1°, the form
of the intrados ; 2°, the relation of the axis to other parts ; or
3°, the forms of the face lines.
The two former are the most important grounds of division,
since they give rise to more radical differences of design. The
latter occasions only the many minor modifications of form.
23. Arches, divided according to the forms of their intra-
dos, are plane, developable, ivarped, or double-curved.
Plane arches have been illustrated in Prob. III.
Developable arches are those in which the intrados is near-
ly always cylindrical.
Warped and double-curved arches will be treated in con-
nection with other problems involving like surfaces.
24. Arches, divided according to the direction of the axis,
relative to other parts, are right, oblique, descending, and ram-
pant.
A right arch is one in which the axis is horizontal, and the
planes of the faces at least so nearly perpendicular to it that
the coursing joints can all be parallel to the axis. The face
may be oblique to H, when it has a batter ; or oblique to a
vertical plane through the axis, when it is a skew-face. It may
be oblique to both of these planes at once.
14 STEREOTOMY.
25. The oblique arch, properly so called, is one in which the
coursing joints, in order to be nearly or quite perpendicular
to the face, are not parallel to the axis, the latter being hori-
zontal, and oblique to the horizontal lines of the face.
26. A descending arch is one whose axis is inclined to a
horizontal plane.
A rampant arch is one in which one springing line is higher
than the other. The descending arch is also "sometimes called
rampant. The two cases may then be distinguished as longi-
tudinally rampant, when one end of the axis is higher than the
other, and transversely rampant, when one springing line is
higher than the other.
27. Arches, divided according to the form of the face line,
may be circular, elliptic, parabolic, etc., according to the form
of the right section of the intrados. They are also depressed,
PL II., Figs. 13, 15, or super-elevated, Fig. 14, according as
the rise (20) is less or more than half of the span. They are
also pointed, if, as in Fig. 14, the right section is composed of
intersecting arcs.
When the right section of an arch is a semicircle, semi-
ellipse, etc., the arch is said to be full-centred. When this
curve is less than a half one, the arch is called segmental.
28. When the right section is compound, having three, five,
or many centres, the arch is said to be three-centred, etc., or
poli/central, PL III., Figs. 11, 21.
111°. — Arches — Preliminary Constructions.
§ Conic Sections.
1°. — To construct a circle by points, having given its
radius.
29. Let Oe, PL II., Fig. 10, be the radius of the required
circle, and cd = 20e the diameter perpendicular to Oe. Draw
ep and cp, which, with Oe and Oe, will form a square, and
hence be tangent at c and e. Divide Oe and ep into the same
number of equal parts, numbered as in the figure. Then lines
radiating from c and d, through the like points on ep and Oe,
respectively, will meet at points v, r, etc., of the circle whose
radius is Oc. For Ofd and cgp are equal right triangles, and
the angle Odf is moreover common to the triangles Ofd and
STONE-CUTTING. 15
One. Then by subtraction, the angles ned and Ofd are equal,
which makes the triangles Ofd and den, similar, and hence den
right angled at n. Hence n is a point of the semicircle whose
diameter is cd.
2°. — To construct an arc of a circle by points, knowing
its chord and versed-sine or rise.
30. Let 2cb, PI. II., Fig. 15, be the given chord, and ab the
given rise of the required arc. Draw ac, and ee perpendicular
to it, and limited by the tangent ae at a ; also en parallel to
ab. Then divide cb, ae, and en, each into the same number
of equal parts, and number them as in the figure. Then lines
joining like points on cb and ae, will meet those radiating from
a to the points on en in points of the required arc.
Parallels to Oe and Oe, Fig. 10, from any point of the arc
ee, will show the correctness of this construction, and of points
as k on the arc ae, Fig. 15, produced.
Making cc' radial and equal to aa' we may likewise, as seen
in the figure, construct the concentric arc a'c r of the extrados
of a segmental arch (27).
31. The radius of the segmental arch, given by its span and
rise, may be desired, and may be found thus. PL II., Fig.
15.
Let C, intersection of abC and rC, the bisecting perpendicu-
lar to ae, be the centre of the arc ac. Then let R =■ Ca ; s =■
be and v = ab. Now ab : ae : : ar : aC.
Whence aO = ac X , ar .
ab
But ar — h ae ; hence
aC =
ac 2 s 2 -\- v 2
2ab~~ 2v '
3°. — To construct an ellipse by points on given axes. Also,
normals to it.
32. Let AB and CD be the given axes, PI. II., Fig. 16.
Then as the projection of a circle seen obliquely is an ellipse
(Des. Geom., Theor. VI.), the curve may be found as at a, b,
c, as in Fig. 10, as is evident by comparing Figs. 10 and 16.
In an elliptic arch the face joints should be normal to the
elliptic face line of the intrados.
33. Construction of normal Face Joints. First Method. —
16 STEREOTOMY
Let N be a point at which such a joint is to be drawn. With
C or D as a centre, and a radius equal to Ah, describe arcs
intersecting AB at F and F, which are the foci of the ellipse.
Then the normal Nm, at N, bisects the angle FNF.
34. Second Method. — The normal at any point being
perpendicular to the tangent at the same point, to draw the
normal at T, for example ; describe an arc as At, with cen-
tre h, and radius Ah, and produce dT, perpendicular to AB, to
meet this arc at t. This arc may represent the circle, which,
when revolved about AB till oblique to the paper, is projected
in the given ellipse. Then draw the tangent tK to the arc,
and as K, being in the axis AB, remains fixed, KT is the tan-
gent to the ellipse at T. Then Tk, perpendicular to KT, at
T, is the required normal at T.
4°. — To construct the arc of a parabola; on a given seg-
ment of the axis, and the chord which is perpendicular
to the axis. Also, normals to it.
35. On comparing Figs. 10, 16, and 20, PI. II., and knowing
from geometry (Des. Geom., Arts. 206-209), that the several
conic sections have certain general properties in common, their
construction may be put in general terms applicable to all cases,
thus. Points of any conic section are found at the intersection
of sets of lines which radiate from the two vertices of an, axis
of the curve ; those from one vertex radiating to points of equal
division on the common perpendicular to the chord and to
the tangent at that vertex ; and those from the other vertex, to
like points on the semi-chord, in the manner shown in the fig-
ures. Thus, PL II., Fig. 20, let AU be a semi-chord, and BU
a segment of the axis of a required parabola. Draw the tan-
gent BV, limited by AV, parallel to BU. Divide AV and
AU into the same number of equal parts, and number like
points, reckoned from A, with like numbers. Then radials
from B to the points on AV, will meet radials from the oppo-
site vertex (B') of the axis BU to like points on AU, in points,
k, c, N, etc., of the parabola. But note that as the axis is of
infinite length, the radials from its infinitely distant extremity,
B', will be parallel to BU as shown at Nm, etc.
The figure further shows a depressed gothic arch, each half
composed of parabolic arcs, as AB and DE.
36. Construction of the Normal Joints. — Let such a joint
STONE-CUTTING. 17
be constructed at N. Draw NN' perpendicular to the axis
BU, make BT = BN', and TN will be the tangent at N by
the property that the subtangent, N'T, is bisected by the
vertex, B, of the curve. Then Nw, the normal, is perpendicular
to the tangent NT.
In the parabola, the sub-normal N'w is constant ; hence, to
construct any other normal, as at e, we have only to draw ee'
perpendicular to BU, make e'h' = N'w, and h'e will be the nor-
mal at e.
37. PI. II., Fig. 13, represents a depressed gothic arch
formed of circular arcs, in imitation of Fig. 20. The con-
struction (Draft. Insts. & Oper. p. 91) may be obvious on in-
spection. The joints in each arc radiate to the centre of that
arc. Fig. 14 represents a pointed or lancet gothic arch, where
the radii, as ca and cb, are greater than the span, de.
5. —To construct an arc of a hyperbola, on a given chord,
and segment of the axis perpendicular to the chord.
38. This form of the conic section is seldom needed by the
mason. Yet to complete the series of similar constructions it
is here given. Let act,', PI. II., Fig. 18, be the chord, and Ve
the segment of the axis. Make a't' and at equal and parallel
to Ve, and divide at and ae similarly, as in the previous ex-
amples. Then, WV being that axis of the hyperbola which
is perpendicular to the given chord aa\ radials from V to the
points on at will meet those from W, to the like numbered
points on ae, in points as 5, c, d, of the required hyperbola.
39. This construction is essentially the perspective of that
of Fig. 10 on a plane so placed as to cut the visual cone whose
base is the circle, in a hyperbola ; while that of Fig. 20 is
similarly the perspective of Fig. 10 upon a plane cutting the
visual cone whose base is the circle, in a parabola.
40. In Figs. 16, 18, and 20, we may substitute any chord
and its conjugate diameter (that is, the one which bisects the
given chord and its parallel tangent) for the axis and a chord
perpendicular to it ; since such constructions would merely be
the projections, or perspectives of Fig. 10 on planes not paral-
lel to either of the given diameters, Oe and Oe.
2
18 STEREOTOMY.
§§ Polycentral Arch- Curves.
41. The practical convenience of circular, above that of
other curved work, with other reasons which will appear, has
led to the frequent adoption of artificial, or compound curves,
composed of circular arcs of different radii, in place of the true
conic sections ; especially in case of separate elliptic arches,
i. e. when not combined as in groined arches. And as there
will be found to be a choice in the relative lengths of these
radii, they should not be chosen arbitrarily, but so as to give the
best proportions to the intended oval, as will next be shown.
Three-Centred Ovals.
1°. To construct the general case of the semi-oval of three
centres.
42. In PL II., Fig. 17, let ab and bg be the given semi-axes
of the proposed curve. Assume ac at pleasure, but less than
bg; make ge = ac, and draw ce ; bisect ce by the perpendicular
to it, fd, meeting gb produced at d. Then will c, d, and a
point <? 1? taken on ah produced so that bc x = be, be the three
centres of the required oval.
43. From this beginning, the following investigation leads to
useful special cases. Let
ab = a ; bg = b ; ac = r ; dg = R.
Then from the triangle bed we have,
(cdy = (bdy + (b c y.
That is (R — r) 2 = (R — 5) 2 + (a— r) 2 (X)
whence ^ a 2 -\- V — 2ar , Q .
K - 2(6 — r) W
equations which obviously allow an infinite number of solutions
for the same given axes ; as the construction shows, where
ac, = r, was assumed at pleasure, only less than bg ; and thence
R was found.
2°. First special case. The semi-oval of three centres,
■when the lesser arc is 60°.
44. Let the arc am, PI. II., Fig. 11, = 60° ; whence bed =
cdc ! = 60°, and the arc mn=. 60°. Now put r = a — x, where
bc= x, then R = md = ac' = a -J- x.
STONE-CUTTING. 19
Substituting these values of R and r in (1) Art. 43, we find
ar — (a — o)x ■== - — •£-!—
whence, neglecting the negative value of x,
a — b - a — b .—
* = -g— + — g— V 3, (4)
which is constructed as follows :
Take bf = a — 5, bh — — ^ — , and on fh describe a semi-
circle ; in which (bhy = bh X bf. Then describe the arc he
from h as a centre, which will give be = x. Then make be' = be
and ede' an equilateral triangle on ec ! , which will give the three
centres required.
For, as bf= 2 bh; (bh) 2 = bhx 2bh = 2(bhy
and (hh) 2 = (bhy + (bhy = 3(bhy
or, (hk) = (»)vr= ^^VT;
then be = bh -\- hh = — 1 „ — \/ 73] = x ; and
r — a — x, and R = a -f- a; as required.
45. The curve in Fig. 11 has the advantage, in application
to bridge arches, that when b is > f a, it affords a greater in-
terior capacity for the flow of water than does an ellipse on the
same axes. For the radius of curvature at a, Fig. 17, of the
b 2
ellipse on ab and bg as semi-axes, is r x = - , but
(a — b . a — b .-$■ \
— 2 r — 2~ V 3 )
Now when b = § a,
a y
but r = (l — 1+ y 8> i a = .545 a
= (l _l±VI)^,
R
3°. Second special case. The ratio - to be a minimum.
r
46. From Eq. (2) Art. 43, — = - + h \~ 2< T
v ' r 2r (b — r)
Differentiating by the rule for fractions (a and b, constants)
dividing by dr, and placing the result = 0, we have
7 1 R
£ =__jr= ar 2 — (a 2 -f P) r + \b (a 2 + b 2 ) = ;
20 STEREOTOMY.
which solved with respect to r gives, after reducing, and neg-
lecting that value of r which makes r > b,
r = ygqrg / vv+F-(«_-S)\ (6)
Equating this with (3) and reducing,
R= V^+V / V?+g + («--ft) \ (6)
The direct reduction being somewhat tedious, note the sym-
metry of Eqs. (2) and (3), where (3) is obtained by substitut-
ing a for b and R for r in (2) ; and it will be obvious that (6)
is obtained from (5) by a like substitution.
47. The construction, which is very simple, is shown in PL
II., Fig. 19. Draw the chord ae, make ef = a — b, bisect af
by the perpendicular gd, meeting eb produced, and ab, at d and
c, the required centres for the half curve aoe. For, the similar
triangles age, aeb, and gde give
' ^gi + b 2 — (g — b) \
and R=de = ~ X eg = ^^ (v ff + ^+ (« -- *) ,)
Five-Centred Ovals.
4°. To construct a five-centred semi-oval, which shall con-
form as nearly as possible to a semi-ellipse on the same
axes.
48. Five-centred ovals are preferable to three centred ones,
when ~— < -j ; and are generally most pleasing when, as here
required, they most nearly resemble an ellipse, described on
the same axes. In PI. II., Fig. 21, let fg = ^ ag. It is a prop-
erty of the ellipse that its radius of curvature, at the extremity
of the minor axis, is a third proportional to the semi-minor and
semi-major axes. Hence make fe = 2ga and e is one of the
five centres. Again, the radius of curvature at the extremity
of the major axis, is a third proportional to the semi-major and
semi-minor axes, hence, make ca = c'b = \fg<> and c and c' will
be two other centres. Now, since the radius of curvature of an
ellipse is changing continually, a radius may be found which
shall be a mean proportional between the radii already found,
and such a radius is also a mean proportional between the
semi-axes, for/e : ga: : gf : ae; . • . fe X ac = ga X gf .
■ ae =
de =
ae
ab
ae
~be~
X
X
ag-.
eg
II II
2 + 6'
a
=
z 2 + b*
b
STONE-CUTTING. 21
that is, R X r = a X b, and calling
the intermediate radius r x , we have by hypothesis,
r* = R . r. Hence r* = a . b .
that is, a : r x : : r x : 6.
Hence make gj=gf, describe a semicircle on aj ; draw #&,
and we have gk, a mean proportional between ag, = a, and
#/, == 5, and hence equal to the radius r x . Now make if=gk ;
draw the arc did' ; make ah = #& ; draw the arc hd, and simi-
larly draw h'd' ; then ^ and <#'are the remaining centres; for/g
may be regarded as containing all the lesser radii found in
moving from f to a about e as a centre ; i. e., at some point
of the motion of ef about e, it becomes = d m. Likewise ab
may be regarded as containing all the radii greater than ac,
found in moving ab about c as centre, i. e., at some point of
the motion of ab about c, ac becomes = d m, hence d must be
at the intersection of arcs di and dh, drawn with e and c re-
spectively as centres, ah and fi each being equal to r x . A
general construction of the five-centred oval is given in my
" Drafting Instruments and Operations," p. 92.
5°. To construct the five-centred oval, by a method ap-
plicable to an oval having- any number of centres.
49. See PI. II., Fig. 12, where, as before, put ao = a r and
bo = b. The problem is indeterminate when the extreme radii
R and r are both chosen arbitrarily, for calling r x =■ the inter-
mediate radius, we have ec' = r x — r, and e"c' = H — r x ; mak-
ing but two equations for the three unknown quantities cc',
c"c', and r L . Note that as cc' -f- c'c" = R — r, the centre c'
will be on an ellipse whose foci are c and c", and whose major
axis is R — r.
In order to render the problem determinate, put
1°, oc" == 3 oc, i. e.
R — b = 3 {a — r), then, 2°, ec" = | oc", and
3°, cd == J co.
The problem can now be solved geometrically, and without
assuming any of the required radii, thus. Assume ag ; make
og" = 2>og ; bisect og" at e'; join g and e', and with ^asa centre,
and radius ag, describe the arc aG. Then take dg=-\ og ;
draw dg", which will cut ge' in a point g', which is the centre
of the arc GV ; and g" is the centre of the arc VC. In
general, however, it is useless to draw the arcs as these will
not be, except by accident, the ones required.
22 STEREOTOMY.
But the polygons, ogg'g", and occ'c", having their sides made
proportional, will always be similar, hence if we put,
oc =x; og = p; cc'-\-c'e" = z;
oc" = y ; of = q ; gg'+g'g" = s,
we shall have
x V x z i ii
- — -: _ = -• and 2-4-a — z=y-\-b,
from which after eliminating z,
p + q — s * p+q — s
or, when as was assumed, q= 3 p, then y = Sx, and
IV°. — Arches — Illustrations.
Problem IV.
A three-centred arch in a circular wall.
I. The Projections. In PL III., Fig. 22, a'a"d'd" is the
plan of a segment of the wall, 4 feet 7 inches thick, of a
circular room, which has a radius, Cb ( = 65") of 25 feet 8
inches. This wall is pierced by a horizontal cylinder, whose
axis C"C — C, intersects the vertical line, at b, that of the
room, at b, C, and whose right section is ABD ; forming an
arch, whose span, AD, is 17 feet 4 inches ; rise, CB, is 6 feet
2 inches ; interior height, GB, is 17 feet 2 inches; and height
to top of keystone is 20 feet 2 inches.
The face line, ABD, is found by (47) the joints radiate to
the centres O, c' and c", and are adjusted to the horizontal
joints of eleven equal courses of stone, each 22 inches thick.
From these numerical data, and general descriptions, the
drawings can be made (better on at least a scale of ? l) and,
in the plan, should show all the coursing joints (20) of one
half of the intrados, as e'" is shown at e 4 e 5 .
II. The Directing Instruments. In illustration of the
derivation of these from the projections of the arch, we shall
consider some one stone. Let it be mnpqr. As usual, the
instruments required will be of two kinds : patterns to deter-
STONE-CUTTING. Z6
mine the forms of the faces of the stone ; and bevels to deter-
mine their relative positions (7).
1°. The pattern, No. 3, of the top, pq, will be the figure
P'P"Q'Q", seen in its real size in the plan.
2°. That of the side on qr, No. 4, will be simply a rectangle,
of width qr, and length Q Q".
3°. That of the radial bed on mr, No. 7, requires the con-
struction of the true form of the elliptical face-joint, mr — M'Q'.
As this construction is the same for all such joints, it is here
given for them all. (See Des. Geometry, Part I, Problem
LXXXIX., 2°.) The circle of radius CK = bb" = bO, is the
horizontal, and the tangents to it, as KA, are the vertical pro-
jection of the vertical cylinder of revolution, from which the
face-joints, mr, np, are cut, as at e'e, e"g, etc., by planes c'f
c"h, etc., perpendicular to \/, and oblique to the axis of the
cylinder. These planes will therefore cut the cylinder in
ellipses whose semi-transverse axes are c'f c"h, etc. ; and
whose semi-conjugate axes are each equal to the radius, CK,
of the cylinder.
These ellipses, being thus known by their axes, each may be
shown first by revolving it into, or parallel to a plane of pro-
jection ; or, second, all may be shown in one figure constructed
on a common conjugate axis.
First Method. The plane of np, may be -revolved to the
right about the axis ; sb'" — np, parallel to the vertical plane,
when, after revolution, 6"'N' will appear at riN'" ; kM.' a>tjj",
and sP' afcpp" ; all perpendicular to np. Then p"j"W will
be the true form of the elliptic arc, N'P' — np, forming a face
joint. Others may be similarly found.
Second Method. Again, in Fig. 25, lay off from K', on K'H
produced, a distance = KC, from Fig. 22, giving a point O.
Then on a perpendicular to K'O, at O, lay off, from O, dis-
tances to the left, equal to the semi-transverse axes, CK (of the
circular right section in the plane CK) c'f, c"h, etc., and con-
struct quarter ellipses on K'O and these several semi-transverse
axes. Finally, since CD, c'e', c"e", etc., are the perpendicular
distances of the inner extremities, D, e', e", etc., of the joints
from the conjugate axes, perpendicular to the paper at C, c', c",
etc., lay off these distances from O on the semi-transverse
axes in Fig. 25, giving points from which draw ordinates, par-
allel to K'O, which will give D, e', e", e'", etc., corresponding
24 " STEEEOTOMY.
to D, e', e" e'", etc., in Fig. 22. The elements of the intrados
being parallel to the conjugate axes, at C, c', etc., of the ellip-
ses, Fig. 22 ; aT)c, de'f, ge"i, etc., Fig. 25, are bevels giving
the true forms and positions of the face-joints Dd, e'e, e"g,
etc., Fig. 22, relative to the elements of the intrados.
Returning to the joint np ; pp" , jj", and nW", being revolved
positions of lines parallel to the axis of the arch, if we lay off,
from p", j" and N", on these lines produced, distances equal
to N'N", M'M" and P'P", we shall have the true form of the
face-joint, np — N"P", and of the pattern No. 5, of the radial
bed WWM'W!. The pattern of the radial bed at mr may-
be similarly found.
4°. The soffit mn. The patterns of this, and of all the other
like surfaces, are shown on the development, Fig. 28, of the
entire intrados, ABD, of the arch. The length of this devel-
opment, = 2AC, Fig. 28, = 2AmnB, Fig. 22. Projecting over
GI from M'M", and HJ from N'N", and proceeding likewise
for the other stones, we get GHIJ as the pattern No. 6, of the
surface mn — M'M"N'N", for example.
5°. The patterns, Nos. 8 and 9, of the two faces of the stones
will be found at once by developing the concave, and the con-
vex cylindrical faces, A'C'D', and A"C"D", of the arch.
6°. Having now the patterns of all the faces of the stone mn
pq, their relative position may be determined by the square as
shown at S ; a bevel No. 10, set to the angle npq, and an arch
square, as VY, No. 11.
7°. Both plane beds being first finished, the templet vw, No.
13, might replace No. 11.
III. Application. This, so far as not already evident from
the description of the directing instruments, would be as fol-
lows.
Taking the stone just considered, the radial bed on np,
would first be wrought by No. 1, the straight edge, and the
pattern No. 6. Thence the top could be wrought by No. 10,
and the pattern, No. 3 ; and the soffit, by No. 11, and the pat-
tern, No. 6. Also No. 2, held in planes parallel to the face qr,
will give elements of the cylindrical faces of the stone in their
true relation to the top pq. Or, a frame TT', No. 12, whose
parallel bars TT and T'T' are curved to the radius of the face,
and held in planes parallel to the top, pq, would give circular
lines of the face, between which No. 1, held perpendicularly to
STONE-CUTTING. 25
TT, would test the proper cutting away of the intermediate
rough stone.
Examples. — 1°. Substitute for the circular wall, shown in the plan, a straight
wall, with a sloping face, as shown in Fig. 24, and make the isometrical drawing
of any one of the stones ; as indicated in Fig. 26 when the batter is on the front.
Ex. 2°. Make the isometric drawing, Fig. 23, of a stone from the cylindrical
wall.
Ex. 3°. Construct any of the arches already described, on Pis. II. and III. on a
large scale ; with an isometrical drawing of any one of the stones ; also a develop-
ment of all the faces of one stone into the plane of the paper.
Ex. 4°. Do. for PI. III., Fig. 27.
Ex. 5°. Do. for PI. III., Fig. 29, supplying a plan, and batter at one end.
Ex. 6°. In Prob. IV. let the face of the arch be in a vertical plane, but oblique to
the axis of the arch.
Ex. 7°. Let the arch, either segmental or full centred, be in a recess, with diver-
gent sides, as in the plate band, Prob. III. ; and therefore conical above the spring-
ing lines.
Pkoblem V.
A semi-cylindrical arch, connecting a larger similar gallery,
perpendicular to it, on the same springing plane ; with an in-
closure which terminates the arch by a sloping skew-face.
I. The projections. PI. IV., Figs. 30-32.
1°. Let the following be the given dimensions.
Skew (24) of the oblique wall PQC 4 , = 18°.
Batter " " » " " = T ^.
Radius, O'A', of intrados of arch = 3'.
" O'a' " extrados " " = 4' : 6".
" O'K, " intrados of gallery = 6' : 6".
Least thickness of wall = 2 ! : 3".
Greatest " " " = 5' : 4".
Let the springing plane be taken as the plane H> an d let the
given thicknesses of the wall be in it ; and let the plane V he
perpendicular to the axis, 0"0 — O', of the required arch.
Let there be five voussoirs, dividing the section A'E'B', of the
intrados equally, and let them be completed by horizontal and
vertical planes, as CD' and G'C, through the outer extrem-
ities, D' and G', of the radial beds.
Then, with a scale of not less than £%, = 2' to 1", in order to
be more easily accurate, the given dimensions can be drawn, as
shown, where JR is the horizontal trace of the vertical side of
the wall.
JR — KR' is one springing line of the gallery, all of whose
26 STEEEOTOMY.
elements are therefore parallel to JR, in front of the vertical
plane JR, and above the plane H 5 of the springing lines, JR —
KR' ; and AA" — A', and BB" — B', of the arch.
2°. Declivity of the plane face of the arch. — PQ, at 18° with
JR, is the horizontal trace of the plane of this face. Its bat-
ter, ^\ (3 to 10), is perpendicular to PQ. Hence, assume hh,
perpendicular to PQ, revolve it about a vertical axis at b, to
5L", parallel to V 5 make b'p' = 10, from any convenient scale
of equal parts, and p'L' = 3, from the same scale ; and \Jb' will
be the revolved position of the line of declivity, 5L, showing
the real slope of the plane arch face.
From this batter we find next, for convenience in projecting
points, the slope, taken in vertical planes parallel to the axis,
0"0 — O'. Thus LI, parallel to PQ, is the horizontal projec-
tion of a horizontal line in the plane face of the arch. Note I,
its intersection with the vertical plane bb", revolve I to I", pro-
ject it to F, onp'U, and lib' is the vertical projection of lb,
and is the declivity of the plane face of the arch, in the vertical
plane bb".
3°. Horizontal projection of the plane face. — Draw horizon-
tals, through all the points of the face, as G'Cj through C and D' ;
produce them to meet Vb', as at C 2 ; then, for instance, project
C 2 at C 3 , and revolve it to C 4 ; then C 4 C, parallel to PQ, will
be the horizontal projection of C'C^ and will intersect the pro-
jecting lines, from C and D', at C and D, the horizontal pro-
jections of C and D'. Find other points similarly, or —
Otherwise : project k' at k" and revolve it to k, when kC
will coincide with C 4 C, whence, as before, etc.
Again : as C 1 C 2 is the true distance of C and D' in front of
the vertical plane on PQ, and in the direction of the axis 0"0
■ — O', make iD —jC == C 1 C 2 , and we have C and D as before.
The horizontal projections of the radial joints all meet at O.
The semi-elliptic face line, AEB, has AB and 20S for a pair
of conjugate diameters ; hence it is tangent to A A" and BB" at
A and B, and at S has a tangent parallel to AB.
50. The last of the three constructions of CD, etc., just given,
namely, by the method of compass transference of knotvn dis-
tances is advantageous, in avoiding numerous lines of con-
struction, as all from C 2 to C ; yet for the same reason, disad-
vantageous, in not preserving upon the paper such traces of the
construction as would enable any one to recall it from the
drawing alone.
STONE-CUTTING. 27
4°. Horizontal projection of the cylindrical face. — JKJ' is a
profile plane, which contains a semicircular right section of the
cylindrical gallery. Revolving this plane about a vertical axis
at J, the centre of that section will appear as at O x by making
KO x equal to the given internal radius of the gallery ; and Kc 2 ,
with O x as a centre, will be the revolved position of the section.
Thence the horizontal projection of any points of the cylindri-
cal face of the arch can be found as before.
Thus, produce D'C to c 1? and then either make cC" = c x c^
the true distance of C"C in front of the vertical plane on JR ;
or, by showing the counter revolution, etc., project c 2 on JR, at
c 3 , not shown, counter revolve c s to c 4 , not shown, on KJ pro-
duced, whence project it by a line parallel to JR, till it meets
CC", giving C". In like manner all points of the cylindrical
face may be found.
The radial joints D"E", etc., of this face are arcs of ellipses,
being sections of the cylindrical intrados of the gallery by the
planes OO'D', etc., which cut it obliquely.
Opposite joints, symmetrical with 00", as D"E" and d x e x .
form parts of one ellipse, D"0"c?!, in horizontal projection,
since d x e x — d\e\ is exactly over that part of the ellipse D"0"
— D'O' whose vertical projection is on D'O' produced.
All the lines of the cylindrical face are invisible, and hence
dotted, except such top and lateral edges, as CD" and C"G".
II. The directing Instruments. — We may either show to-
gether the patterns of like faces of all the stones, or the pat-
terns of all the faces of one stone.
Adopting the latter method as clearer, while illustrating all
the operations required by the former, let the stone C'D'F' be
chosen for detailed representation.
Development of the stone C'D'F'. — A right section of this
stone is the polygon CD'E'F'G', which will develop in a straight
line as AiB x , Fig. 31.
Then, supposing the top face to be the plane of develop-
ment, and that the faces to the right of D', around to G', are
developed to the right of the edge D"D — D', while the face
C'G' revolves to the left about the edge C"C — C into the
plane CD' ; we shall make, in Fig. 31, A x c ; cd ; de ; ef; and
fB 1 equal respectively to G'C ; CD' ; D'E' ; E'F, and F'G' in
Fig. 30. At these points draw lines perpendicular to A^,
28 STEREOTOMY.
for the indefinite developments of those edges of the voussoir
which are parallel to the axis, 0"0 — O'.
To develope the edges of the plane end of the stone. — Lay off
the perpendicular distances of their extremities from some
plane of right section, as JR. Thus A x g x = cG ; cC x = cC,
etc., the second term of each equality being taken from the
plan in Fig. 30. The curved edge, EF — E'F', is developed
by taking one or more intermediate points, as HH7, and mak-
ing eA, and 7iHj, Fig. 31, respectively equal to E'H 7 , and H^,
in Fig. 30. Also, for additional accuracy, make ^0 2 = D'0',
and OA = 0"0, then O x , Fig. 31, will be a point of D^ pro-
duced. Likewise, make B^ and o 2 o x , Fig. 31, == G'O', and
0"0, Fig. 30, and o x will be a point on G^ produced.
To develope the edges of the cylindrical face of the same vous-
soir. — As before in Figs. 31 and 30, respectively, A x g 2 = eG" ;
cC 2 = cC" ; Nrc = Nw ; R 2 h = W\, etc. Also, as C"G" —
C'G' is an arc of the right section of the gallery, C 2 g 2 , Fig. 31,
is drawn with a radius == O x K, Fig. 30, and laid off on the
perpendicular at the middle of the chord C 2 g 2 . The elliptic arc
D"E"0" is developed in its real size at D 2 E 2 2 ; and as every
edge of the cylindrical face is curved, except C 2 D 2 , they will be
curved in the development, and must be there found by inter-
mediate points, as, at Mm, = Mm, Fig. 30.
To develope the plane face of the voussoir. — Revolve it about
CD — CD' into the plane of the top. The true lengths of G#,
F/, and Ee, Fig. 30, found at k'r, k's, and k't, on the line of
declivity L'5', will then be laid off on ^G, / X F, and ^E, per-
pendicular to C^Dj, Fig. 31 ; where g, f x and e x are transferred,
as shown, from CD, Fig. 30, to its equal, C X T> X , Fig. 31. The
true size of the radius EO — E'O' is found by revolving it
about OO" into the plane H, at e'"0. That of FO — F'O' is,
likewise, /"'0 ; then arcs from E and F, Fig. 31, as centres,
with these radii, respectively, will give 3 , the development
of the centre of the ellipse of which the face line, EF, is a part,
and the point to which GF and D X E radiate.
To develope the cylindrical face of the voussoir. — Here Fig.
32, F/, for example, is the true length, taken from c 2 / 2 , Fig. 30,
of the circular arc whose vertical projection is F'f. Other
points, being similarly found and joined, give the pattern, which
must be flexible, of the cylindrical face. When flat, all its
edges are curved except CD and GG. The developed faces of
STONE CUTTING.
the voussoir are the patterns of those faces,
voussoirs would be similarly found.
29
Those of the other
III. Application. — Select a stone whose right section will
circumscribe C'D'F'G', and whose length shall not be less than
Ee'', and work the top by No. 1, the straight edge (7) and No.
3. Make the left side square with the top by No. 2, and finish
it by its pattern. No. 4, with the pattern of the bed at D'E',
will complete that face. The arch square, No. 5, with the pat-
tern of the intrados, will give the surface EE"FF" ; and in like
manner the whole convex surface can be wrought. All the
edges of both ends of the stone being thus given, their patterns,
with the straight edge, will serve to complete them.
Otherwise : the plane and cylindrical ends can be wrought
next after the top, by bevels Nos. 6 and 7, respectively.
This problem includes the following simpler cases, added as
examples.
Examples. — 1°. Let both ends be plane; one, vertical on PQ ; the other ver-
tical on JR.
Ex. 2°. Let the cylindrical end be replaced by a plane one, having JR. for its
horizontal trace, and a batter of ^.
Ex. 3°. Construct a rampant (ascending) arch covering a straight flight of
steps.
Grroined, and Cloistered Arches.
51. Both of these kinds of arches are compound, being
formed by the intersection of two single arches, each of which
Fig. 3.
30 STEREOTOMY.
is usually cylindrical. They are distinguished from each other
as follows : —
In the groined arch, Fig. 3, that part of each cylinder is real
which is exterior to the other. Thus EF, LM, eF, and KH are
real portions of elements.
In the cloistered arch, that part of each cylinder is real which
is within the other. Thus in Fig. 3, HF and FM would be the
real portions of the elements.
The groined arch therefore naturally covers the quadrangular
space at which two arched open passages, ABC and abc, inter-
sect ; while the cloistered arch forms the doubly arched cover,
or quadrangular dome, of a quadrangular enclosed room, or
cell.
Theorem I.
Saving two cylinders of revolution, whose axes intersect, the pro-
jection of their intersection, upon the plane of their axes, is a
hyperbola.
See PI. IV., Fig. 30, where 2 — O' is the axis of the cyl-
inder A'S'B', and a parallel to JR through 2 , where 0"0 2 =
O x K, is the axis of the cylinder all of whose elements are par-
allel to the one JR.
A"S"B" is the horizontal projection of the intersection of
these cylinders, that is, upon the plane of their axes, and it
will be shown to be a hyperbola by proving that (0 2 h"y —
(R"h"y = (0 2 S") 2 ; where H" is any point on A"S"B" and
H"h" a perpendicular (ordinate) to the axis 2 0.
Suppose (0 2 h»y — (H" A") 2 = (0 2 S") 2 . (1)
Now (H"A") 2 = (R'h"')* = S'h'" X sh"> ; where. S'« = A'B'.
And S'h'" X sh'" = S'h'" X (S'O' + O'A"')
= S'h'" X (S 2 s 2 + H 2 A 2 ).
But (S 2 s 2 y= (OA) 2 — (o^) 2 .
And (H s A s ) 8 = (0 1 S a ) 2 — (Oxh 2 y.
Whence, since Oi« 2 = 2 S", and 1 h 2 = 2 h"
(s 2 s 2 y — (H 2 h 2 y=(0 2 h»y— (0 2 s»y.
But also (S 2 s 2 ) 2 — (H 2 A 2 ) 2 = (O'H') 2 — (O'A'") 2 = (K'h">y
= (E"h"y.
Hence (0 2 A") 2 — (0 2 S") 2 = (H"A") S .
Or, (0 2 h«y — (H"h"y = (o.s"y.
Thus equation (1) is proved ; and, calling 2 h" = x ; and
STONE-CUTTING. 31
H"h" = y ; and 2 S" = a ; we have x % — y 2 = a 2 ; which is
the equation of an equilateral hyperbola, that is, one whose
axes are equal. Thus the theorem is proved.
52. An important consequence of the last theorem is, that
when the cylinders become equal, the vertices of the curve, of
which S" is one, unite at 2 . Now when the two vertices of a
hyperbola coincide, the curve reduces to the special case of two
intersecting straight lines.
But the actual intersections of two cylinders (whose axes are
not parallel) are curves. Hence if any projection of these
curves is straight, they are plane curves, and hence ellipses.
53. A little consideration of the properties of hyperbolas
will sufficiently show, what there is not room here to strictly
prove : 1st, that a change in the angle between the axes of the
cylinders would only cause the curve A"S"B" to become a hy-
perbola referred to the new axes as conjugate diameters ;
and 2d, that the substitution of elliptic for circular cylinders
would only yield a general, in place of the equilateral form of
the hyperbola. The conclusion of (52) is therefore true of all
cylinders whose diameters, measured in a plane perpendicular
to that of their axes, are equal.
54. Cylinders, situated as just described, will have a pair
of common tangent planes parallel to that of their axes. This
fact, added to the last two articles, affords several statements of
what is really the same proposition, each statement being ap-
propriate to certain given conditions. Thus —
1°. If two ellipses intersect in a common semi-axis, of the
same kind for each, their other axes being in the same plane P
(thus if two ellipses whose transverse axes bisect each other
in H have a common vertical semi-conjugate axis), lines join-
ing points, which are on the two ellipses, and which are at
equal distances from P, will be parallel, and will therefore form
a pair of cylinders, of which these ellipses will be the intersec-
tions.
2°. If two cylinders intersect in one plane curve, as an
ellipse, there will be a second branch of the intersection, which
will also be plane.
3°. If two cylinders whose axes are in the same plane P,
also have two common tangent planes, parallel to P, they will
intersect in two plane curves, which will cross each other at
32 STEREOTOMY.
the intersections of the elements of contact of the tangent
planes.
Problem VI.
The oblique groined arch.
55. Design. — Suppose that, in the collecting system of cer-
tain water-works, supplied by several ponds, two conduits, each
covered by semi-cylindrical arches of nine feet span, unite at
an angle of 67° : 30' and discharge into one of twelve feet span,
covered by a semi-elliptic arch, having the same rise and spring-
ing plane as the former ones.
56. We may note in passing that, supposing the water to be
four feet deep in each of the nine-foot conduits, the sum of
their water sections is 72 square feet. Then, in the large con-
duit, if the water be but four feet deep, this conduit should be
18 feet wide. Or, if but 12 feet wide, as in the problem, its
floor should be sunk so that the water in it should be six feet
deep ; or else its declivity should be increased to give such a
velocity to the water in it that its section of 48 square feet
would transmit as much water per minute as passes the sec-
tions of 36 square feet each, of the two nine-foot conduits.
I. The Projections. — Three planes of projection are used:
the horizontal plane, H? containing the springing lines, Hf and
YN, YM and Y"M", of the two smaller conduits, and HH' X and
H"H 2 , of the larger one ; a vertical plane V? whose ground line
is P'Q', and which is perpendicular to the axis OX — X' of one
of the smaller conduits ; and a vertical plane Vn whose ground
line is H1H2, and which is perpendicular to the axis Om — 0{
of the larger conduit.
The lines in the horizontal plane can be first laid down from
the given dimensions and axes, OX and Om, of the arches ;
giving the four springing points H, Y, Y", H", of the groin.
Then OY and OY" are the horizontal projections of the quar-
ter ellipses (52), in which the arch whose axis is OR inter-
sects the right hand half of the one whose axis is OX. Like-
wise OH and OH" are the projections of the intersections of
the arch whose axis is Om with the left hand half of the one
whose axis is OX.
Next make the elevation on P'Q', making the spandril stones
40" thick, the thickness O'l' at the crown 3 ft., and the radius,
STONE-CUTTING. 33
q'V, of the extrados, 9 ft. The radial joints from X' divide the
intrados into five equal parts.
The elevation on H^H^ is now made as follows : —
1°. The face line Hi O" W 2 . — Any point, as a', is the ver-
tical projection of the element A" Am, or of any point of it.
Hence a' is horizontally projected upon the groin curves at A
and A". The projections of these points on Vi will then be
at a[ and a", at heights above H^Hg equal to that of a' above
PQ.
2°. Other points of the elevation on Vi* — As the extrados
is seldom a finished surface, we need not construct the groin
curves of the extrados, but may proceed as follows, to find ex-
tremities of radial joint lines ; these joints being normal to the
intrados in both elevations.
To find f\ for example. Draw the tangent g'V perpendic-
ular to X'y, project V at I in the vertical plane OH of the groin,
when Gl will be the horizontal projection of g'V ', considered as
the tangent to the groin HO at Gg'. Then projecting G at
g[ and I at l[, gives g[l{ as the new vertical projection of this
tangent, and mg{, perpendicular to it, as the like projection of
the joint in the face of the main arch, corresponding to X.'g' on
that of the arch H'O'Y'. Finally f[ is the intersection of the
joint mg[ with e\f{ parallel to the ground line HiH^, and at a
height from it eiP" = e'P'.
In like manner, other points may be found, as may be seen
at/i, extremity of an auxiliary joint X'/' — m l j'i.
3°. The curve b"I'"ei, right section of the extrados of the
elliptic arch is not, as might be supposed, an ellipse, derived
from b'j' as 0"Hi was from OH ; since the two conditions of
normal joints, and equal heights of like joints on both eleva-
tions prevent this.
That is, i'i, for example, determined from i' by projecting
i' upon DO as the straight horizontal projection of the outer
groin, and thence to i[, as g[ was found from g' would not co-
incide with i' x of the figure, found on the given normal mg x and
at the same height as i' ; since each determination is complete
in itself and hence independent of the other.
Points, as B, C,etc, of the horizontal projection of the outer
groin, are at the intersection of projecting lines from b' and b{ ;
c' and c[, etc. ; and are not in a straight line with O ; thus
showing that the outer groin is not, in space, a plane curve.
3
34 STEREOTOMY.
4°. The Projections of a Stone. — The stones of a groined
arch in jointed masonry are partly in each arch. Hence, tak-
ing the most irregular stone as an example, it is that whose sec-
tion in the front elevation is a'b'c'd'e'f'g'. The side elevation
of the end in the elliptic arch is a[b[d' 1 e / ig[ i and its plan is lim-
ited by the figure AadDd^.
II. The Directing Instruments. — Passing by Nos. 1 and 2,
the straight-edge and square, which are of constant use (7),
there are the following : —
No. 3, = the pattern a'b'd 1 . . . . g' of the end in the vertical
plane at ad.
No. 4, = the pattern, dcCcid^ of the plane portion of the top.
No. 5 = the pattern Ffdd^ of the plane portion of the
under side.
Fig. 34 shows, together, the patterns, Nos. 6, 7, 8, and 9, of
the principal lateral faces within the circular arch. There,
ab = a'b' ; bc= the arc b'c' ; ag = the arc a'g' ; and fg =f'g'.
Also Cc, Bb, Aa F/ = Cc, Bb etc., in the plan,
Fig. 33. Thus No. 6 is the pattern of the plane radial bed
ABab — a'b' ; No. 7, of the cylindrical extrados BCbc — b'c' ;
No. 8, of the cylindrical intrados, AGag — a'g' ; and No. 9, of
the plane radial bed, FG/# — /'#'•
Detailed description of the analogous patterns, Nos. 10, 11,
12, and 13, Fig. 35, of the corresponding lateral faces of that
part of the stone lying in the elliptic arch, is unnecessary, as
the construction of Fig. 35 is entirely similar to that of Fig. 34.
Thus, a t p^ Fig. 35 = a\p{ from the elliptical elevation;
and Aai Pp x , etc., Fig. 35, = Aa 1 Pp l5 etc., from the plan ;
hence AGa^i is the pattern of the soffit AGa^ — a\g\.
As a further aid to accuracy, there should be one or more
right section bevels, as Nos. 14 and 15, to test the relative po-
sition of the lateral faces. No. 16 is the pattern of the end in
the plane a^.
III. Application. — Choosing a block of the thickness hk,
and in the plan of which Aadd x a x can be inscribed, first work
the end in the plane ad, and complete it by the pattern No. 3.
From this, by means of No. 2, the square, determine the direc-
tion of all the lateral faces, including the upper and under
ones ; and the back, whose horizontal projection is Dd and
Dd x ; and finish them by their patterns, Nos. 4-9, and the arch-
square, No. 14.
STONE-CUTTING. 35
This done, the three edges, Cid x , fxd v and d 1 — e' x d[ of the end
in the plane a x d x will be known, whence this end can be wrought
square with the top and bottom plane surfaces, and completed
by its pattern, No. 16. Thence the remaining lateral surfaces
within the elliptic arch can be directed by the square, and com-
pleted by their patterns, Nos. 10-13, and No. 15.
Examples. — 1°. Let the cylinders (having a common springing plane, and
equal heights in every case) be equal.
Ex. 2°. Let them be at right angles.
Ex. 3°. Construct the patterns for the key-stone (which will extend in one piece
from 0, on all the cylinders).
Ex. 4°. The arches being at right angles, let the circular cylinder be real be-
tween OH and OH", and the elliptic one, between OH and OY, forming a clois-
tered arch.
Ex. 5°. In Ex. 4°, construct the patterns for the groin stone corresponding to
the one described in the groined arch (Aa will be real from A forward, and
A«i will be real from A to the right).
Ex. 6°. In Exs. 2° and 4°, let the extrados be finished as in PI. IV., Fig. 30.
Problem VII.
The groined and cloistered, or elbow arch.
57. This problem is designed to illustrate very compactly,
that is, without repetition of similar parts, first, the difference
between a groined and a cloistered arch ; and, second, the mode
of proceeding when it is determined that the intersection of the
two extrados, that is, the outer, as well as the inner groin curve,
shall be a vertical ellipse.
I. The Projections. — A gallery, PL VI., Fig. 47, whose width
A'B' is 8 ft., intersects one of 10 ft. in width, each of them end-
ing at its intersection, ADB, with the other. These galleries
are covered by arches of equal rise, CD' = CD", and whose
axes, CD and C"D, are in the same plane. They therefore in-
tersect in a vertical ellipse (52), whose horizontal projection
is ADB. The radius, E'Ci, of the given extrados is 5' : 3".
The portion A'C'DCA" thus forms one quarter of a right-
angled groined arch ; while B'C'DC'B" forms one quarter of
a right-angled cloistered arch. The construction will mostly
be obvious on inspection, by comparison with PL IV., Fig. 33.
A stone of the cloister. — MPRr — M'P' is its circular intrados ;
OPRQ— O'P'isa radial joint; NOQ? — N'O' is its circular
extrados; MN^r — M'N' is another radial joint, and MO —
36 STEBEOTOMY.
M'N'O'P' is its vertical plane end. The like surfaces in the
elliptic portion are obvious by reading the drawing.
The proposed elliptical outer groin may coincide with the
inner one AB, in plan. That is, both may be in the same
vertical plane. Then, taking the joint at a'c' for illustration,
d will be horizontally projected at n instead of at c ; and n will
thence be projected at n" , at the same height as c' ; likewise
F, at/, on BA produced, and at/" ; and so on for other points.
Then Wn"f n will be an ellipse, it being the right section of
the cylinder whose oblique section is the vertical" ellipse T)nf.
Producing the normal joint a"c" to the new extrados Wn"f",
we find its new outer extremity o" ; and the other vertical pro-
jection, o'o'", at the same height as o", of the element of the
new extrados, containing o". But o'"o' must be limited by the
plane, CV, of the joint a'c', as at o', whose horizontal projection
is o, on the horizontal projection, os, of the element at o".
Considering then the stone below the joint a'c', a"c", its
radial surfaces are si"ao ; oan, where on — o'c' — o"n" is an
elliptic arc ; and ui'an.
This design is more complex than the usual one, but gives a
greater increase of radial thickness toward the springing of the
elliptic arch. Indeed, unless the radius CjE' exceeds a certain
limit, relative to CD' and D'E', the joint, or normal e"F" will
be less than D"E", which is quite undesirable, relative to sta-
bility. The joints of the intrados, being the most important,
are inked full ; that is, as if seen from below ; as it is often
convenient to do (8).
The outer groin may be elliptical, found, like the inner one,
from given extrados, if the consequent deviation of the de-
rived face joints from a normal direction be only very slight.
II. The Directing Instruments. — These can be constructed
as in the last problem.
III. Application. — See also the last problem.
Examples. — 1°. Let both arches be circular.
2°. Let both be elliptical.
3°. Let their axes intersect at any other than a right angle.
4°. Turn the figure right for left, and ink the plan as seen from above.
5°. Construct the patterns for the key-stone of the groin ; and of any other
stone, both in the groined and in the cloistered angle.
6°. Assume an elliptic extrados of suitable proportions on the elliptic arch, and
find the corresponding curve joining the corresponding extremities of the normal
joints of the circular arch.
STONE-CUTTING. 37
Conical or Trumpet Arches.
58. When two walls, whether having vertical or sloping
faces, meet, so as to enclose any angle, it is sometimes desira-
ble to connect them at or near the top by self-supporting
masonry, which may, for example, afford additional area for
standing or passage. The original ground room enclosed by
the supposed angle at which the walls meet is not encroached
upon by the arched connection of the walls, since the latter is
self-supporting from the walls.
Such arch work, projecting from one or more wall faces, or
piercing them, and generally with a conical intrados, has been
called a, trumpet.
Peoblem VIII.
A trumpet in the angle between tioo retaining walls.
I. The Projections. — 1°. Description. Let AC and BC, PI.
V., Fig. 36, be the horizontal traces of two walls, enclosing a
right angle ; and whose faces, CAtZF, and CBeF, have the same
slope or batter of 2 to 5.
Let FDE be the level top of a quadrantal platform, with a
radius of 6' : 10" ; and forming a quarter of the base of an in-
verted oblique cone, whose axis coincides with FC — F'C, the
line of intersection of the faces of the walls ; and a portion of
whose convex surface forms the front face, ADEB J — A'D'E'
B'J', partly broken away on the right, of the platform.
This platform is then supported by a trumpet, whose sur-
face, AJBC — A' J'B'C, is a segment of a cone of revolution
whose axis HC — C is perpendicular to the plane V, and in
the plane H. From this general description, we proceed to
the details of the construction.
2°. Outlines of the walls and platform. — Having laid down
the traces, AC and BC, of the walls, CC, the horizontal pro-
jection of their intersection will bisect ACB, because the walls
have the same slope. Next draw Cm, and on it lay off from
C, Jive from any scale of equal parts ; and Ix, parallel to BC =
two from the same scale ; then, by the given conditions, Cx
will be the section of the face of the wall AdC, made by the
vertical plane, on BC, and revolved about the trace BC, into
38 STEREOTOMY.
H> Now lay off Cf= the height of the platform, and Cm, that
of the wall, draw fW and mW, parallel to Ix, project M' at
C, and F" at F, on CC, and draw C'd and C'e for the hori-
zontal projections of the upper edges of the walls, and FD and
FE, respectively parallel to them, and of the given dimensions
for the intersections of the platform with the faces of the walls.
3°. Conical front of the platform, and its vertex. — Sup-
posing it required for good appearance that the conical front
face of the platform shall intersect the walls in their lines
of declivity, draw DA and EB, perpendicular to AC and BC,
as such intersections ; then AKB, with C as its centre, will be
the horizontal trace of this conical front : and as DA and EB
are two of its elements, their intersection, V, will be the hori-
zontal projection of its vertex.
The projections of VD and VE on the vertical plane BCM,
are BCM and B ; which after revolution, as in (2°) will appear
at M'C and EB produced, which meet at the point indicated as
V". Then, making CV = BV", we have V the vertical pro-
jection of the vertex of the conical front of the platform.
4°. Outlines and joints of the trumpet. — Let the vertical
semicircle, AHB — A'H'B', on the chord, AB, of the horizon-
tal trace of the inverted cone, be the directrix of the cone of
revolution forming the trumpet. Dividing A'H'B' conveniently
into equal parts, here three, project i' and f, the points of di-
vision, at i and j, giving Ci — Ci', etc., as elements, and joints,
of the trumpet.
To find the face line of the trumpet, find where its elements
meet the surface of the cone VV. Since the vertex. CC of the
trumpet cone is in the axis, VC — VC, of the other one, this
is easily done by assuming any element, as Ci — Ci', of the
trumpet and finding its trace, and that of the axis, VF — VF',
upon the plane D'E' of the upper base of the platform. These
traces are N'N and F'F, giving FN as the trace of the plane of
these lines upon the top of the platform. FN cuts the circum-
ference, EID, of the platform at aa', giving aV — a'V for the
element of the cone VV in the same plane with Ci — Ci 1 , and
hence intersecting the latter at bb', a point of the required
face line.
The highest point. — This, JJ 7 , is found by revolving the ele-
ments VI and CH of the two cones, and in their common ver-
tical meridian plane VC, about any convenient vertical axis, as
STONE-CUTTING. 39
the vertical trace, C'F', of that plane, till they fall in, or par-
allel to the plane V^ Each point revolved moves in a hori-
zontal arc, at its proper height, as can be read from the figure,
giving V"K"I" and C"H" as the respective revolved elements.
These intersect at J" ; which, by counter revolution, gives JJ',
the required highest point.
To give a neater design to the top of the platform, the planes
of the joints of the trumpet radiate from the axis of the cone
VV', giving the radial lines as aC for the joints of the platform.
But to prevent the stones from coming to an edge along that
axis, they abut against a conical faced stone whose upper base
is gkh, of any convenient assumed radius, and whose intersec-
tion ucv — u'c'v' with the trumpet cone may be found as
A5B — AW J' was.
This stone may be built into the walls, as along the planes
Qgu and hvV, to any extent desired for stability. Also the
side stones, as A'D'a'b', of the trumpet may be likewise built
into the wall to avoid a thin edge along AC — A'C. The
small component of the reactions of the side stones tending to
thrust the central one forward would be sufficiently resisted by
the adhesion of the cement. Otherwise : the intermediate
stones, one or more, could easily be supported by forming the
stone ghuv, as indicated in Fig. 37, with a conical step as s.
II. The Directing Instruments, — These, taking the central
stone for illustration, will consist of patterns of its four lateral
faces, with a few bevels.
After Nos. 1 and 2, there will be No. 3, the pattern, aknp,
of the top, No. 4, that of the two equal radial beds, and No. 5,
that of the conical intrados ; which are constructed as follows :
No. 4 shows the full size, and real form of the radial bed npqr,
which is supposed to be revolved about its horizontal upper
edge np, till horizontal ; when r, its lowest point, will be found
at a distance s^, Fig. 42, from np, equal to the hypothenuse
of a right angled triangle, of which rs, perpendicular to np, and
the vertical distance of r' below n'p', are the other sides. Also,
WjSj = ns. Finding tyx in like manner, we have the pattern
No. 4, which will serve for both of the radial beds of this
stone.
No. 5 is the development of the conical intrados, made by
describing the arc iyj x = i'j', and with the radius C"H' ; , = the
slant height to the circular directrix AB — A'H'B', and laying
40 STEREOTOMY.
off the true lengths of the elements as CA = O'b"' = the true
length of Cb — Ob' revolved first at Ob" into the vertical plane
COT' and thence to O'b"'.
Bevels, Nos.6 and 7, set to the angles s^qi and s 1 n 1 r 1 ^ and
held in the plane CC'F', will be useful in giving the positions
of the two ends. The latter surfaces, being parts of oblique
cones, can be developed only by the usual construction, in which
the intersection of such a cone with a sphere whose , centre is
the vertex (VV) of the cone is found ; the development of
such intersection being a circle. But these developments are
unnecessary.
Let rs now be considered as the horizontal trace of a vertical
plane, perpendicular to the edge np, and cutting the adjacent
radial bed in sr, and the top in a line also horizontally pro-
jected in sr. By revolving the former sr about the latter till
horizontal, the true size of the diedral angle between the top,
and the radial bed npr, will be found. A bevel, No. 8, set to
this angle will be useful.
III. Application. — Having chosen a suitable block, in which
the finished stone could be inscribed, first form the top, by No.
1, and by its pattern No. 3 ; next, the radial beds, directing
their position by No. 8, and their form by No. 4 ; next the con-
ical intrados, of which No. 5 is the pattern. All the edges of
the two ends will thus be known, and as their radial edges are
elements of the cones to which they belong, it is only necessary
to transfer to ap and bq, from the drawings, points where ele-
ments meet those lines, and bring the front end apbq to its
proper conical form by cutting away the stone till No. 1 will
apply to it, at the corresponding points of division of ap and
bg. Hence it is, that, as already said, the tediously found de-
velopments of these conical ends may be dispensed with.
Problem IX.
A trumpet arched door on a comer.
I. The Projections. 1°. Outlines of the Plan. — PI. V.,
Fig. 40. The projections are here arranged partly with a view
to the greatest compactness. Two walls of an enclosed space,
and of the thickness, AC '== BD = 4' : 9", meet at right angles.
STONE-CUTTING. 41
From a, the equal distances aA and «B, each = 8' : 6" are set
off as the external limits of the trumpet as seen in plan.
If CD, the width of the door were also given, it would deter-
mine the angle A/B at the vertex / of the conical surface of
the trumpet. We here suppose A/B = 90°, which, with the
previous data, makes CD = 5' : 3|" very nearly. A vertical
semicircle, A.5B, which, revolved about AB till horizontal, gives
AEFB, is taken as the base, or linear directrix of the trumpet.
Then CdD is a vertical semicircular section of the trumpet, and
also of a second conical zone CEFD whose vertex is e, and
which serves to widen the approach, EFH, to the door. To
avoid crowding the figure, CD is shown as a single line, which
it might, in fact, be, in case of an opening having no gate, or
of a thin iron gate ; but in case of a gate of considerable thick-
ness, the edge at CdD should be cut away giving a narrow cyl-
indrical band, fitted to the gate top ; or there should be a gate
recess as in PL VI., Fig. 45.
2°. The Elevation in general, and plans of the elements. —
The foregoing being the main features as seen in plan, the ele-
vations are shown on two vertical planes ; one having aB, and
the other, A"B" for its ground line. Of these, only the former
is necessary for the purposes of the mason, showing as it does
the true sizes of the lines in the external face of one of the
walls ; while the other, on the vertical plane at A"B" is only
useful as helping to give an idea of the structure, as seen in
looking directly through the doorway, in the direction of.
The vertical planes Aa and Ba, being parallel to the respec-
tive opposite elements, B/ and A/, cut the trumpet cone AB/
in equal parabolas ; hence, to avoid a too great inequality in
the sizes of the arch stones as seen in the exterior of the walls,
divide the semicircle A/B, or its equal A"b"F", into conven-
ient unequal parts, the largest, W'l" = Bl ! being laid off from
the springing at B" of the arch. Project V, or I" ; h', or h" ;
etc., at I, h, etc., and through I, h, etc., dxa,wflJc, fg, etc., hori-
zontal projections of elements of the trumpet. These at n, i,
etc., pass to the conical zone CEFD, as at nm, and thence to
the cylindrical band EGHF in parallel elements, as mo.
3°. To find the parabola, Aa, in its own plane.
1st. Without the elevation on A"W. — The trumpet, being
a cone of revolution, and its axis af horizontal, its elements fa,
fg, etc., revolved about its axis,/r, and towards B, will come
42 STEREOTOMY.
to coincide with the extreme element /B produced, as at fa x ,
fg v etc. ; where aa\, gg v etc., perpendicular to fx, are the hor-
izontal projections of the arcs described by a, g, etc. Then a',
g', etc., vertical projections of a, g, etc., extremities of elements
of the trumpet, are at the intersections of the perpendiculars
aa', gg' , etc., to the ground line aB, with the arcs a Y a', gig', etc.,
all having x for their centre, and which being in the vertical
plane on «B, are seen in their true size. (The arc kjc', being
confused with B&', is not shown.)
2d. With the use of the elevation on A"B". — Having h", for
example, vertical projection of h, draw f'h", the vertical pro-
jection of the element fh, and project g upon it, at g". Then
g r is at a height gg', equal to that of g" above A"B" ; and in
like manner other points of Ba', except a', can be found. As
before, aa' = aa x .
4°. Determination of the radial beds. — These, if the face
joints, as g'V, were made normal to the parabolas, of which
Ba' is one, would be determined by these joints, with the ele-
ment joints fg, etc., and hence could not also contain the axis
fx of the trumpet, since g'V, etc., if normal to Ba', do not in-
tersect that axis. But if these radial beds do not contain the
axis fx, which is also the axis of the cone cEF, and of the cyl-
inder EGHF, their planes cannot cut the two latter surfaces in
elements, and the stones of the trumpet would properly termi-
nate in a vertical plane on CD, and be succeeded by others,
radiating from the axis fex, and covering the surfaces named
between CD and GH.
We therefore choose beds radiating from the common axis
fex and extending from Aa and Ba to LH. The top edges of
these beds, in the horizontal surfaces as I'K' — F'K" will then
be parallel to fx ; and the face joints g'V — g"V, etc., will
radiate from the point x, f" in the plane Ba.
II. The Directing Instruments. — These, besides Nos. 1 and
2, (7) will consist of patterns of the surfaces of the stones,
with such other bevels besides No. 2, as may be considered use-
ful as checks.
Taking the stone gkinmor — g'VK'J'k' — I"K"J"n"i"r"m",
No. 3, the pattern of its back, is I"K"3"r"m", which is in the
vertical plane LH.
No. 4, is the pattern of its front, VK'J'Jc'g'.
Nos. 5 and 6, are the patterns of the radial beds on f"J rr ,
STONE-CUTTING. 43
and on /"I". These are both shown as revolved about the axis
fx of the trumpet, till they become horizontal. Thus HI 2 =
I"r" ; gj.! = g'l' ; Dgi shows the true length of ig — i"g" ; etc.
No. 7, the pattern of the top, is a trapezoid of altitude I"K"
and bases equal to JiJ 2 and LI 2 .
No. 8, is a pattern of the conical intrados, gink, found, if
flexible, as in previous similar constructions by developing the
cone whose vertex is /. But as it is only the elements of the
intrados that must be found, it is enough to develop the pyra-
mid whose edges coincide with these elements. Hence in Fig.
41, the chords BZ and Ih are equal to the chords Wl" and l"h",
Fig. 40 and nikg is the pattern required.
Flexible patterns 9 and 10 of inm, and srom can obviously
be found. The vertical surface on K'J', forming No. 11, is
simply a rectangle, = K'J' X JiJ2-
Nos, 12, 13, and 14, will be bevels set to the respective angles
K"l"g", between the top and a radial bed ; K"J"k" ; and HFD,
between a vertical side as J'K' and the front, and taken in a
horizontal plane.
Fig. 44, is an oblique projection of the stone just described,
made intelligible by means of the like letters at like points.
III. The Application. — First, work the back by No. 1, and
mark its form by No. 3. Second, all the lateral faces adjacent
to the back are made square with it by No. 2. Also, the top
and the vertical side on J'K' are at right angles. Third, the
forms of the faces just mentioned can then be marked by their
patterns (5-7), 10, and 11 ; and the bevels, 12 and 13, can be
used as checks on their position. Fourth, make the front
square with the top, or at the angle HFD with the vertical side,
holding No. 14 perpendicular to the edge J'K'.
Fifth, mark the elements, gi and nk by No. 9, whence inm
can be wrought by No. 1, placed upon corresponding points of
division of in and ms into equal parts.
The upper joints of the trumpet stones being parallel to fx,
there will be some three-cornered stones adjacent to them in
the wall, where the joints are parallel to AC and BD.
Examples. — 1°. Let the walls include any other than a right angle.
2°. Let their exterior be a vertical tangent cylinder from A to B.
3°. Let the cone be other than right angled at its vertex f.
4°. Let FH be increased till LLj shall be long enough to embrace the back ends
of all the trumpet stones.
44 STEREOTOMY.
5°. Let LLi be a, sloping wall.
6°. Let there be a batter to the exterior faces Aa and Ba, of the wall.
7°. Let there be no opening at CD, and describe with an oblique or isometric
projection, the stone GECDFH — f, necessary to fill the opening, and admit the
extension of the trumpet surface to its vertex f.
PftOBLEM X.
An arched oblique descent.
I. The Projections. — Various conditions may give occasion
for a structure of this kind. Thus it might lead from a side
walk to an underground railway ; or from a hydraulic canal to
a turbine wheel pit ; or it might cover an arched stairway lead-
ing to an arched gallery.
1°. The perpendicular projections. — ABCD, PL V, Fig. 43, is
the horizontal projection of the section in the springing plane.
A vertical plane on AC here makes an angle of 28° with a ver-
tical plane on A x A perpendicular to AB. The line CD — D'
is one springing line of the intrados of a semi-cylindrical gallery,
of radius MD = 11' : 6" ; the perpendicular length, AA 15
of the horizontal projection is 7 ft. ; and AB and EF are re-
spectively 15 ft. and 9 ft.
Two principal vertical planes of projection are used ; V, that
of the head, on AB, and one Vi whose ground line is BD. The
line AB is at a height, BB 1? = 3' : 3" above CD, and hence,
strictly, the diameter AB of the vertical projection, AI'B, of the
head should be a line A'B' (not shown) parallel to AB and
3' : 3" above it. But to condense the figure, and because this
position of AI'B is not essential, the vertical plane, V, of the
head is revolved about its trace on the springing plane B X DC,
instead of about its horizontal trace.
2°. The oblique projections. — The projection of the arch
upon the vertical plane, Vn on BD, might be made in the usual
way, by projecting lines perpendicular to that plane. But, as
may be seen by trial, the result would be a much more com-
plicated, but no more useful figure than the present projection
on BD ; which is an oblique projection, formed by projecting
lines parallel to AB.
Thus, since the plane V is vertical, By perpendicular to BD,
is its trace on the vertical plane Vn and is also the oblique pro-
jection of the head, on Vi- Likewise, the quadrant DV, being
STONE-CUTTING. 45
the revolved semi-right section of the gallery reached by the
descent, M X D is the oblique projection of its horizontal radius,
MD, and similarly S, T, etc., are obliquely projected on Vi a ^
Si, T\, etc. Then making S^ = Ss ; T^ = Tt, etc., Dt^ is
an arc of the section of the gallery by the plane Vi 5 and D/ 2 ,
where j"j 2 is parallel to BJD, is the oblique projection of the
cylindrical face of the descent. That is, ^f'j^D is the oblique
projection of the outlines of the descent.
Laying off the heights of the several points of the semicir-
cular face above AB, from B x on B lc /", and drawing lines par-
allel to BiD and limited by D/ 2 , through the points so found,
we find the elements and edges parallel to them, of the descend-
ing arch. Thus, BJ)" = BB' by drawing Wb parallel to BB 2
and the arc bb" with centre Bi ; then b"i 2 is parallel to BiD, and
all the other parallels to BiD are similarly found, as may be
seen by the figure.
3°. The right section. — This is in any plane perpendicular
to the elements of the arch. To condense the figure (though
at the expense of confusing it somewhat, having first sought to
make it on the largest scale possible), assume XY, perpendicu-
lar to BD, and XI 1? perpendicular to B t D, as the traces of such
a plane. Then as usual, choose auxiliary planes parallel to
the axis of the cylinder ; here, vertical planes, parallel to Vi«
Each of these will contain an element of the arch ; and a line
of the plane YXI X , whose horizontal trace will be on XY and
whose vertical projection will be parallel to XIj. Thus the
plane I'PI", cuts from the arch the element at P" — P I", and
from the cutting plane YX1\ the line whose horizontal trace is
p" (intersection of the horizontal traces XY, and Pp") and
whose vertical projection is pp\ parallel to Xl\ through p the
projection of p" on XD. Having found, as above, b"i 2 also p'q',
by making B^' = B^ = BQ = PP', as shown ; p', and i', the
intersections of pp' with q'p' and b"i 2 are two points of the ob-
lique projection of the right section. Then making p"P" = pp'
and p"I" = pi 1 , as shown by revolving p' and i 1 to p x and I 2 ,
about p as a centre, and projecting p^ and I 2 by the lines p^P"
and LJ" to P" and I" on PP", we have the position of the points
p' and i' when revolved about XY into the horizontal plane.
All other points, both of the oblique projection and the re-
volved position of the right section, are similarly found, as is
sufficiently indicated by the lettering of other points.
46 STEREOTOMY.
Making Xe = Xe^, and drawing e Y to Y, the intersection of
XY with DC, the horizontal trace of the springing plane ; eY
is the revolved position of the intersection of the plane, YXI,
of right section, with the springing plane BiDY.
4°. Special Constructions. — Ye is made to pass through F,
in order to compare FO'E and FO'"E more nearly, though this
is not necessary. Ye is thus placed by first assuming YXI X at
pleasure, and finding the corresponding position of eY ; when,
if this position does not contain F, draw a parallel to it that
will, viz., eY as on the figure, which will meet DC at that posi-
tion of Y whence the corresponding desired position of YX can
be drawn.
The tangent to FO'"E, parallel to AB. Considering AB for
a moment as a line in the revolved plane of right section and
parallel to such a tangent, its horizontal trace would be y ; and
its vertical trace B 2 would be found by making XB 2 = XB.
Then making the height Bb 3 = B 2 6 2 , we get b 3 y, its projection
on V, and the parallel tangent at K' gives K'K two projections
of the point of contact of the required tangent from which K"
the point of contact on the revolved right section, of a tangent
parallel to CD is found as in (3°).
II. The directing Instruments. — Taking the springing stone,
FBB'I'R', these will be as follows, besides Nos. 1 and 2. No.
3, the top, is a parallelogram of width B"I", and length b"i 2 .
Then with centre B, and radius b"i 2 cut DM at i s , and Bz 3 will
be the position of b 2 i 2 — B', after revolving till horizontal, about
AB as an axis, since i 2 is in that right section of the gallery
whose horizontal projection is MD. Then ij$ and BP will be
two sides of the parallelogram, No. 3.
No. 4, the pattern of the vertical side of the stone, is of per-
pendicular width = eB", bottom length = BJD, and top length
= b"i 2 . One end is the vertical line BB', the other the arc
Dii of DV (2°) corresponding to Di 2 .
No. 5 is the pattern of the right section eB"I"R"F, used in
one method of working the stone.
No. 6, = FB'BFR', is the pattern of the plane end.
No. 7, the pattern of the opposite cylindrical end, dif-
fers from No. 6, in that BB' would be replaced by the develop-
ment of the arc Di 4 ; and RR', by that part of DV, correspond-
ing to Dn 2 , while the developed joint RT would be curved, as
STONE-CUTTING. 47
found by means of an intermediate point u, as in Prob. V.,
etc.
The remaining patterns require the construction of other de-
velopments. Making FOT'E", Fig. 43 (taken on CD, only
to bring the figure within the plate), equal to the right section
FR"N"E, Fig. 42 ; note that the real distances, estimated on
elements, from the right section, e'o'f^ to the heads of the arch,
are seen on the oblique projection. Then make F"F, Fig. 43,
=/iBi, Fig. 42, and in like manner passing from one figure to
the other, make R"R = p'q< ; 0"0 = o'o'» ; N"N = w'N l5 and
E"E = e'B 1 ; and FRNE will be the development of the face
line, FO'E, of the plane end of the arch.
Next, make FH, Fig. 43, = B X D, Fig. 42 (i. e. F'H=/ 1 D),
and likewise, RRi = q'r 2 ; NN X = N^, and EG = BjD ; and
HRjNiG, will be the development of the face line of the cylin-
drical end of the descent. Then No. 8 = FRR X H, the devel-
oped intrados of the stone considered.
Finally, the centre, 00^ of the plane end, to which its
joints radiate, is at the distance O^ from the centre Oi(0")
of the right section ; hence in Figs. 43 and 42, respectively,
make 1 2 , at this distance, 1 B 1 from the right section F"E" ;
make R0 2 = R'O ; or r0 2 = R"0", and 2 RI = OR'I'. Then,
in the two figures, make Il x = b"i 2 , and O 2 o 2 = BJD, and
IiRiOg will be the developed joint on the cylindrical end, cor-
responding to R'I'O, Fig. 42, on the plane end. Hence No. 9,
= RIIjRj, is the pattern of the radial bed on RT.
No. 10, = FBHD, Fig. 43, and similarly found, is the pat-
tern of the springing surface whose horizontal projection is
FBHD, Fig. 42.
Besides these patterns, bevels, Nos. 11 and 12, set to the
angles B'T'O" and i 2 b"B , respectively, will be useful in one
method of working the stone. Also No. 13, giving the angle,
BB"I", between the side and top of the stone, in a plane of
right section.
III. Application. — 1°. The method by squaring. — Choose a
stone on which a right section can be formed, exterior to the
finished stone, by No. 1, and No. 5 = eB"FR"F. Next, make
all the lateral surfaces square with this right section, by Nos. 1
and 2, and mark their edges by their patterns, Nos. 3, 4, 8, 9, and
10. This operation will give all the edges of both ends, which
48 STEREOTOMY.
can thus be formed by cutting away the rough stone on the ends
down to them, applying No. 1, in a direction parallel to AB on
both ends. This method is simple and accurate, but wasteful
of the stone between the actual plane end and the exterior
provisional right section, and of the labor of making the plane
surface of this right section. It may, however, be employed in
all cases, like many of the preceding, where the actual ends are
curved, or oblique to the right section.
2°. The method by oblique angled bevels. — Choosing a block
in which the finished stone can be inscribed, work first the ver-
tical side, that being the largest, and mark its edges by No. 4.
Second, work the top square with the former, if the arm of the
square in the top be guided by a small plane bevel, laid in the
top, and set to the angle %BF. Otherwise, use the level, No. 13,
held perpendicularly to the top edge Bi (BD — b"i^). Thence,
finish the top, by No. 3. Likewise work the under side, and
the radial bed on BT — B/'F, the latter by No. 11, held per-
pendicular to the top edge, 11^ Fig. 43. Next proceed with
the plane end, using No. 12 to give its position relative to the
top. From the finished plane end, the lengths at all points
being known from the side elevation, the remaining sides and
the cylindrical end can be easily and accurately wrought.
Examples. — 1°. Make the side elevation in perpendicular projection.
2°. Let the arch ascend from the plane end to the gallery.
3°. Construct the indicated pattern, No. 7.
4°. Let the descent be direct, BD perpendicular to AB.
5°. Construct the patterns for the key-stone.
6°. To avoid confusion, take X to the right of B.
CLASS III.
Structures containing Warped Surfaces.
59. Warped faced wing walls. — Suppose that the inner
faces, as bm — b'm', PL I., Fig. 7, instead of being vertical, were
sloping, but in such a way that the lowest lines of the fronts
of the walls should be, as seen in plan, parallel to bm and np.
Thus let them be as at h'h". The rate of slope at mh, where
the wall is highest, would then be less than at bh", where the
wall is lowest. The face of the wall would therefore be a
warped surface, and would be a portion of a hyperbolic para-
boloid ; generated either by hh", moving on bh" and mh so as
to remain horizontal ; or by bh", moving on bm and hh", and
parallel to the vertical plane on mn.
Example. — Construct the case just described in a large figure, with an aux-
iliary elevation showing the face of one of the wing-walls ; and take the joints to
coincide with positions of hh" and bh".
Problem XI.
The recessed Marseilles Crate.
I. The Projections. — 1°. The problem is this. Given a
straight wall in which is a recess with diverging sides, and in
the recess a round topped portal, closed by gates of like
form ; it is required to cover the top of the recess by a surface
which shall be agreeable, easily constructed, and practicable in
not interfering with the turning of the gate. Thus, having a
vertical straight wall, PI. VI., Fig. 45, bounded in thickness
by the parallel planes AB and C"D ; and in which is the pas-
sage EF, having a semicircular top, E'G'F', and covered by
gates of like form, fitted to the recess EFHI — H'E'G'G"F'P;
it is required to cover the diverging recess or embrasure re-
maining between the vertical planes HI and AB, in the man-
ner enunciated.
It will be agreeable that AH should be not less than HG,
the width of the gate; and that the front top edge, AB —
50 STEREOTOMY.
A'K'B', of the recess should be arched, in which case the
vertical height, G"K', and the radius, AO — A'O", should be
so adjusted that A' and B' shall not be lower than G", the
highest point of the gate.
So much being fixed, let the axis, OY — O", of the arch, and
the face lines, H'G'T, of the gate recess, and A'K'B', of the
embrasure, be the three given directrices of a warped surface,
generated by the motion of a straight line upon them. (Des.
Geom. 251.)
One of these directrices, OY — O", being straight, any de-
sired elements of the proposed warped surface may readily be
found by noting the points in which any plane containing
OY — O" cuts the other two directrices. Thus 00" A' is a
plane containing O Y — O", the point AA' of the front face line,
and cutting HI — H'G'T at L'L, giving ALY— A'L'O" for an
element of the warped surface.
But the limited directrices limit this warped surface by
the elements AL — A'L' and BM — B'M', so as to still leave
undetermined the surfaces projected in ALH and BMI.
On extending the warped surface just formed, by producing
the directrix A'K'B', it will generally intersect the sides, AH
and BI, of the recess in curves, which would prevent the full
opening of the gates.
We therefore proceed to complete the proposed top of the
recess by means of warped surfaces having the two direc-
trices, OY — O", and H'G"I', in common with the preceding
warped surface, and for a new third directrix a curve through
II' and BB', so formed as not to interfere with the full open-
ing of the gate. This third directrix is conveniently shown,
first, in its real form, by revolving the face, BI, to a position
parallel to the plane V, when BB' will appear at GB'".
2°. Determining conditions of the new third directrix. These
are: —
1st. That it should enclose I'G", and be tangent to it at I'.
2d. That it should also contain the point B'".
$>d. That the new warped surface directed by it should be
tangent to the preceding one along the common element, MB
— M'B', in order to avoid any visible edge of transition, or
break, in passing this common limit of the two surfaces.
The last condition will be fulfilled if the two warped sur-
faces be made to touch each other at any three points of their
STONE-CUTTING. 51
common element, YMB — 0"M'B'. But this they evidently
do at the two points, YO", and MM', since there the linear
directrices are the same for both surfaces.
Let BB' be the third point of YMB — 0"M'B', at which the
two warped surfaces shall be tangent. For this purpose, they
must there have a common tangent plane. Such a plane will
be determined by two tangent lines at BB', of which the most
convenient are YMB — 0"M'B', which is tangent to itself ;
and B'T', the tangent, at BB', to the directrix A'K'B'. Now
M'N', parallel to B'T', is the trace of this tangent plane on the
plane HI ; and N', where it meets the intersection, I — I'N',
of the planes HI and IB, is one point of its trace on the plane
IB. But BB' is another point of the same trace, which is,
therefore, in revolved position, N'B'".
The third directrix of the new warped surface is therefore, as
seen in the revolved position, a curve which shall be tangent
at I' to I'G", and at B'" to N'B'".
3°. Choice of curves. — Preferring a natural, to an artificial
curve for the directrix now determined, we may attempt an
ellipse having either I'O" produced for its transverse axis; or a
line from I', parallel to N'B'" for a diameter. But in either
case, unless its radius of curvature at I' be not less than 0"P,
it will intersect I'G", and thus be impracticable. Hence the
choice must generally lie between a curve of two centres com-
posed of a part of I'G", and an arc, tangent to it and to N'B"
at B'" ; or a tangent line to I'G" from B'", with the portion of
I'B'" from I' to the point of contact.
Preferring the former, draw B'"P, perpendicular to B'N',
and equal to I'O" ; draw 0"P, and a perpendicular to it at its
middle point will meet B"'P produced at the centre of the
required arc. But this centre will generally be quite remote,
and too acutely determined for accuracy; hence proceed as
follows. The contact, </'", of the two arcs may be found by
an application of the problem : To draw a line through a given
point, which shall pass through the intersection of two given
lines. Thus, construct any triangle, as 0"P1, of which O"
shall be one vertex ; the two others being on B'"P and li, the
perpendicular at the middle of 0"P. Then draw 2, 3, parallel
to OP ; 3, 4, parallel to PI ; and 4, 2, parallel to 0"1, will
meet 3, 2, in a point, 2, of the required radius, 0"2, which lim-
its the arc ~B'"g'" at g'". Having thus found g'", we can, when,
52 STEREOTOMY.
as in practice, making the drawings on a very large scale, find
the arc W"g'" by points as in (30).
4°. Test for Interference. The next step is to ascertain
whether the surface, generated by the gate-top in opening, in-
terferes with the top of the recess. The former surface is, for
the left-hand gate, for example, a portion of the annular torus
generated by the revolution of the circle of radius 0"H' around
H — H'H" as an axis. In such a torus, the gorge or interior
opening reduces to nothing, and the portion used in the prob-
lem is generated by the quadrant H'G". The proposed test is
made by taking horizontal planes, as Q'a', and finding whether
their intersections with the two surfaces intersect within or
without the jamb AH. Thus the plane Q'a' cuts the gate
torus in the horizontal circle of radius HQ, and the top of the
recess in the curve a5Q, found by projecting a', 5', etc., upon
the horizontal projections, 7c, LA, etc., of the elements which
contain them. When the circles as Qc?, described by points, as
QQ' of the gate, everywhere intersect the curves, like Qba, cut
from the recess roof by the respective planes of these circles,
outside of AH, as at d, no interference exists. But when, as
at n, the curve L/c and circle \me intersect within the jambs,
there is an interference. In the latter case, one or more of four
means may be used to remedy the difficulty : —
1st. To increase the radius K'O', estimated from K/.
2d. To raise the point K', without increasing the radius
K'O'.
2>d. If the interference is very slight, a portion of the recess
roof may be hewn out to coincide with the torus generated by
H'G", without disfiguring the surface by abrupt or too obvious
changes of form.
4th. The radius of the gate-top may be slightly diminished.
II. The Directing Instruments. The construction of these
will be best illustrated by showing patterns of all the devel-
opable surfaces of the most irregular one of the voussoirs, viz.,
that between the radial joints q'o' and r'u'.
Besides the straight edge, and square, and the rectangular
patterns of the top, and of the vertical side, at u'v', of this
stone, Nos. (1-4), there are, No. 5, a pattern of the back end,
q'o'v'u'r' ; No. 6, that of the front end, shown at Wm'o'v'u't' ;
No. 7, that of the radial joint at r'u', which is shown by re-
STONE-CUTTING. 53
volving it into the horizontal plane CjB", after supposing the
back, CD, of the wall to coincide with the vertical plane.
Then E'r x = rr" ; c x g x = the perpendicular distance of g from
CD ; T 1 t 1 = CC", etc. No. 8 shows, in like manner, the joint
in the plane 00" o ; and No. 9 == B^ij, Fig. 46, the true
size of the surface \g — B'g't', forming a part of the jamb IB.
Bevels may also be provided, set to as many of the diedral
angles, as m'o'v', m'q'r', etc., as may seem best. Nos. 10, 11,
etc.
III. Application. Form the plane rectangular top, to the
dimensions, o'v' and CC of pattern No. 3 ; then the three
vertical plane surfaces, viz., the two ends, and the side on wV,
square with each other and with the top, and shaped by their
patterns, 4, 5, and 6. Work the radial beds square with the
back, just completed by No. 5, checking their positions by the
right section bevels, 10, 11, etc. ; and scoring their edges by
patterns 7 and 8. The portion, Wr-^sJ^ determines the cylin-
drical surfaces on r'q' and p's', and the annular plane portion,
p'q'r's'.
Every edge of the warped portion of the stone being now
determined, this surface can be wrought by the straight edge,
No. 1, held in the direction of elements of the surface, and
these will be found by transferring their extremities as k' and
A', M' and B', from the drawings to the stone.
THE OBLIQUE ARCH.
60. This, the most extended of all the problems in Stone
Cutting, is usually made the subject of a separate treatise ; for
which its many and marked varieties, as well as its complexity,
make it sufficient. Yet, by the full and careful exhibition of
all the essential features of its usual form, the student can be
prepared both to design and superintend the construction of
an oblique arch as commonly built, and to read the works in
which the subject is treated more elaborately.
Preliminary Topics.
Elementary Mechanics of the Arch.
61. Let ABC be an ordinary semi-cylindrical arch, of which
we will first consider only one half, as ABTa, Fig. 4. Let G
54
STEEEOTOMY.
represent the centre of gravity of this half, and GH its weight,
acting vertically downward. The half ABT, as a whole, and
-£■
Fig. 4.
thus actuated by its unresisted weight, would fall, by turning
about a as a centre. As it would prevent the use of the arch
to oppose GH by props underneath, it is counteracted by a hori-
zontal force acting, as at T, at any point of BT, and this hor-
izontal force consists in the reaction of the other half, BTC, of
the arch and its immovable backing. This understood, as a
force may be considered as acting at any point on its own
proper line of direction, the pressure at T, and the weight con-
centrated at G, may be considered as both acting at g ; the for-
mer at gt, the latter at gh, whence gR, would be their resultant.
Now it is necessary for the stability of the half arch that gH
should intersect the base of the arch between A and a, to pre-
vent rotation about one or the other of those points ; and what
is thus evident for the half arch as a whole, is true of the sep-
arate stones of which it is composed. That is, by considering
the successive segments from BT to the successive radial joints
of the arch, in the same manner as just explained for the
whole, we should find a series of forces like gR, one for each
segment, and whose intersections would form a polygon, called
the line of resistance, which should lie wholly within the arch
in order to secure its stability.
62. Passing to the oblique arch (25), Fig. 5, it is evident
from the foregoing explanations that the portions, ABC and
ADEC, of the left half, are only more or less imperfectly sup-
ported by the opposite half. Hence, in discussing the oblique
arch relative to its stability, it is usual to consider it as divided
STONE-CUTTING.
55
by planes parallel to a face, as GH, into an indefinite number
of laminse ; each of which will be a right arch of a span equal
to ab, the oblique span of the given oblique arch.
[c
Fig. 5.
That is, the " thrust" " lines of pressure" or " lines of re-
sistance " in an oblique arch, are assumed to act in planes
parallel to its faces.
The resulting standard or essentially perfect design for an
Oblique Arch.
63. The conclusion of the last topic affects the form and
disposition of the joints of an oblique arch, and thence their
graphical construction, in the following manner : —
When ttvo surfaces are pressed together, the pressure at each
point should act in the direction of the normal to the surfaces
at that point. Else it can be resolved into two components :
one, normal to the surfaces ; and one parallel to them, which
will tend to produce slipping.
Thus, if the surfaces of two bodies meet in the plane whose
trace upon the paper is AB, Fig. 6, and are acted upon by a
force producing a pressure at any point, p, which may be rep-
56
STEEEOTOMY.
resented by OF=pT i 1 , this pressure may be decomposed at
any point as p of its line of direction into the normal com-
s
A
V
St
A
V
\v
Fig. 6.
...
*i Pi
ponent pN 1 = ON ; and the parallel component pS 1 = OS ;
which last will tend to produce slipping in the direction pS v
64. In applying this principle to the design of the joints
seen on the intrados (19) of an oblique arch, the " transverse"
"heading" or "broken" joints are made in planes parallel to
the faces, and thus represent on the arch itself the direction of
its thrust. The " longitudinal" " coursing" or " continuous "
joints are then made so that each shall intersect at right angles
all the transverse joints which it meets.
65. On account of the rectangular intersections of the joints,
arches thus designed are often described as forming the orthog-
onal system. On account of the equilibrium of the pressures
thus acting in them, they are also often called equilibrated
arches.
The coursing joint is also often called the trajectory.
Problem XII.
The partial, and trial construction of the orthogonal or equili-
brated arch.
66. Construction of a coursing joint. — 1st. In vertical
projection. Let the plane V? PI- "VI., Fig. 48, be considered
as parallel to the faces ABC and DEF of the arch. These
semicircles, with the equal ones having any convenient number
of equidistant points <?, o l5 o 2 , etc., on DC, as centres, represent
the vertical projections of sections of the arch parallel to its
transverse joints (64). ~Now,jirst, a line is normal to a curve
at a point, when it is perpendicular to the tangent at that
point ; and, second, as we learn from descriptive geometry, if
STONE-CUTTING. 57
one side only of a right angle be parallel to a plane, the projec-
tion of the angle on that plane will be a right angle. Hence
in the figure, as the tangents to the circles ABC, etc., are
parallel to V? the vertical projection of the normal curve to all
these circles will be perpendicular to the projections of their
tangents at its intersections with these circles. Thus, if a Q ac x
represent an arc of the curve, it will be perpendicular at a to
the tangent aT, and therefore tangent at a to the radius ao.
Hence we have the following construction : —
Let a be one of the points of division of one face, ABC, of
the arch, through which a coursing joint is to pass. Draw the
radius ao ; from a\, its intersection with circle o u the radius a^ ;
from %, intersection of a x o x with circle o 2 , draw the radius a 2 o 2 ;
from a 3 , similarly found, the radius a 3 o 3 , etc. ; and the line
aa x a 2 . . . . « n , thus found, will nearly coincide with the required
curve ; only that all the points except a are evidently a little
too low. It is an advantage, however, that the errors are wholly
on one side of the truth.
Now draw aoj, meeting circle o x at e 1 ; thence o x o^ likewise
giving c 2 ; thence c 2 o 3 , etc., and again, ac x e^ c n will be
nearly the required curve, but all its points besides a will be a
little too high. How now shall we approximate more closely
to the correct curve ?
67. At this point a general observation upon the relative
exactness of constructions made by plotting to scale the results
given by analysis from numerical data, or by geometrical con-
struction from graphical data, may be useful. The finest lines,
as perfect both in length and direction as can be laid down on
a large scale, and with the nicest care, on firm and smooth
paper, will give results equal, in nearly or quite every case, for
all practical purposes, to those of analysis.
The perfect length and location of lines are analogous to
absence of errors of calculation ; and fineness and large scale of
construction are analogous to the carrying out of numerical
computations to a large number of decimal places.
Besides, in all cases where drawings are necessary as guides
in the execution of a work, the results of calculation are ex-
posed to instrumental errors in plotting them to scale.
68. Returning now to the means of approximating to the
true curve, any one or more of the following methods may be
employed.
58 STEREOTOMY.
1°. Increase to any desired extent the number of circles be-
tween the faces, ABC and DEF, of the arch ; since this will
evidently cause the curves aa 5 and ac 5 , which are on opposite
sides of the true curve, to approach each other ; that is, will
cause either of them to approach the true curve.
2°. Bisect the spaces a ± c^ a 2 c 2 , etc., and the curve through
the points thus formed will sensibly coincide with the true
curve ; especially in a figure drawn on a large scale,- and with
numerous auxiliary circles.
3°. Bisect the distances oo x ; o 2 o 2 , etc., between the centres
of the successive circles ABC, etc., at the points 1, 2, 3, etc.,
and draw al, which will evidently divide a l c l at some point as
%, not shown. Then draw %, 2 ; meeting a 2 c 2 , at a point n^,
etc., and a % n 2 , etc, will very nearly coincide with the true
curve.
4°. Suppose the point w x , just described, to be a point of the
true curve. Then, observing that the curvature of the required
curve rapidly increases as it ascends, tangents to it at a and %
would meet nearer to a than to n. Hence assuming t as one
point of a required curve, draw to 5 , and take r on the part ts
and a little nearer t than to s ; and draw ro±, and on the portion
t^i of this line take r x , nearer the middle of t^ than r was to
the middle of ts ; thence draw r x o 3 , and on the portion t 2 s 2 of
this line take r 2 nearer the middle of t 2 s 2 than r x was to the
middle of t\%\ ; and so on ; and tlfa, etc., will be very exactly
the true curve, tangent at t, t x , t 2 , etc., to io 5 , ro 4 , r x o%, etc.
Example. — Work out each of the above four methods with the circles ABC,
etc., drawn to a radius of at least three inches.
69. Horizontal projection of a coursing joint. — Assum-
ing merely for illustration that ac 5 is the true vertical pro-
jection of such a joint, simply project its points upon the
horizontal projections of corresponding circles, as a at a' ; C\ at
c[; c 2 at c 2 , etc. ; where the accents are given to the horizontal
projection, as is sometimes done, when the vertical projection is
first made, or of most service.
70. Development of a coursing joint. — Let the cylin-
der A'C'D'F' be first made tangent to the horizontal plane
along the element, CF — C'F', and then rolled out upon that
plane. All its elements will then be parallel to C'F' in de-
velopment, and at distances from it equal to the true lengths
STONE-CUTTING. 59
of the arcs of right section between them and C'F'. Hence
make m"n' equal to the true length of the elliptic arc m'n' — •
mn of right section, as it would appear projected upon a
plane perpendicular to the axis 00 5 — oo y (This construction
is not shown, for want of room). Likewise v'a$ equals the
true length of the elliptic arc (arc of right section, whatever it
may be, in any case) v'a'o — va ; and q'a" equals the true
length of a'q' — aq; etc. for any sufficient number of points.
Then p'm"a'o; C'a" ; q'c'{, etc., are the developments, all
alike, of the equal semicircles, a' p' — a p ; OC' — oC ; etc. ;
and a§a v c'{c'i, etc., is the development of the coursing joint
a' c 5 — a c 5 ; and it crosses the developed semicircles at right
angles.
71. Identity of form of the coursing joints. — Through any
point, as a, Fig. 48, draw an element meeting the other circles
at points as 5, homologous with a. It is perfectly evident from
(70) that, if we construct the trajectory passing through b, as
in (6Q~), it will be of the same form as that through a. That is,
all the trajectories have parallel tangents at points on the same
element, and hence are all alike.
Likewise, the horizontal projections, and the developments
of like portions of the trajectories are identical in form. Hence
having any projection or development of any trajectory, the
portion of any other one included between the same elements
of the cylinder would be drawn simply by a card-board pattern
of this initial one.
72. Convergence of the coursing joints. — It will be imme-
diately evident on constructing any other coursing joint, as the
one through m'm for example, that these joints converge, rap-
idly at first ; as they approach CF — C'F' in the direction from
C to F', and DA — D'A' in the direction from D' to A' when
produced indefinitely in both of these directions.
The result of this convergence is, that no two stones in the
same course are alike ; though stones in like positions in differ-
ent courses are so. This greatly increases the number of
necessary patterns, difficulty of execution, and the expense of
constructing an equilibrated arch.
73. Creneration of the joint surfaces. — We know that 1°, the
normal to a surface at any point isperpendicular to the tangent
plane at that point ; 2°, that when a line is perpendicular to a
plane, the projections of the line are perpendicular to the
60 STEREOTOMY.
traces of the plane. Now aT is the vertical trace of the tan-
gent plane to the intrados along the element ab, at aa! ; its
horizontal trace would be a line through T, parallel to C'F'.
Hence the normal to the intrados at aa', for example, is oa —
a'a". If then an arm de, normal, at c 2 , be fixed to a rod
OxO which coincides with the axis 00 5 , and if the rod and arm
move together, the rod moving so as to coincide with the axis,
and the arm, so that the point c 2 of the arm, shall continue on
the semicircle 2 u, the point d, considered as one point of the
extrados, will generate the transverse 'joint of the extrados, cor-
responding to the semicircle 2 u of the intrados ; and the por-
tion c 2 d will generate one of the transverse joint or heading
surfaces ; which is thus everywhere normal to the intrados.
Likewise, the rod OxO continuing to coincide with the axis
as it moves, with the arm, let the pair move so that the point
c 2 of the arm shall trace the coursing joint a'c 5 , the point d
will then trace the extradosal trajectory corresponding to a'cs
of the intrados, and the line c\d will generate the normal joint
surface having d'c 5 for its directrix.
74. Nature of the joint surfaces. — These are evidently both
warped, but of a variable twist ; the former, the transverse
one, is so by reason of the variable relative velocity of the axial
and the rotary motions of the frame O x ed, which is imme-
diately evident on constructing any three of its positions ; the
latter, the coursing joint surface, is variably warped by reason
of the variable curvature of the coursing joints.
75. Summary. — Without detailed drawings, or further in-
vestigation, five grave practical objections to the equilibrated
arch are already apparent.
1°. The inequality of the stones in each course, resulting in
the evils already noted (72).
2°. The variable twist of the normal joint surfaces, which
still further complicates their execution.
3°. The non-parallelism of opposite faces of the same stone,
whereby the actions and reactions upon those faces are not in
the same line, and therefore in the aggregate yield resultant
couples, tending to produce rotation of the arch about a verti-
cal axis.
4°. The convergence of the coursing joints, while like points
of each are not, as we have seen, in the same section parallel
to the face, also makes the stones in the face of unequal width,
STONE-CUTTING. 61
unless they are arbitrarily made equal by breaking joints with
those behind them.
5°. The same convergence, finally, renders it impossible to
build equilibrated arches of common brick, as is often de-
sirable.
For these reasons, we shall here dismiss the equilibrated
arch, only referring the reader, who wishes to become familiar
with its details, to the works of Bashforth, Grraeff, and Ad-
HEMAR ; and shall seek an approximation, which, as nearly as
possible, retains the merits, but avoids the defects of the equili-
brated arch.
We will begin by seeking a simpler form of warped surface,
normal to the cylindrical intrados, for the heads and beds of
the voussoirs.
The Helix.
i
76. Let o— 0', 12', PL VII., Fig. 50, be a fixed axis, and
let 0,0' be the initial position of a generating point which has
these two simultaneous motions ; one of revolution around that
axis, and one of translation, parallel to it. By the first motion
alone, the point, 00', would generate the circle 0,3,6,9,12 ; by
the second alone, the straight line, — - 0'12' ; but, by both
motions combined, it would generate the curve called a helix.
The height 0'12' corresponding to a full revolution, 0,3,9,12, is
called the pitch of the helix.
The usual case is, that each of these motions is uniform,
giving rise to the common helix ; or simply the helix, as com-
monly understood.
77. Three elementary properties immediately follow from the
definition just given.
1°. The helix will lie on the convex surface of a cylinder of
revolution whose axis, o — 0'12', is that of the helix; and whose
radius is the perpendicular distance Qo of the generatrix 00'
from the axis.
2°. Since the two component motions of the generatrix, one
around the cylinder at right angles to its elements, and the
other, parallel to them, are each uniform, the path of this point,
i. e. the helix, must cross all the elements at a constant angle.
3°. It now follows that when the convex surface of the cyl
inder is developed, the development of the helix ivill be straight ..
for the elements of the cylinder will be parallel in development
62 STEREOTOMY.
and a line which crosses parallels at a constant angle must be
straight.
It thus appears that the joints, CD 15 KQ 1? etc., BJV, C5,
etc., Fig. 49, primarily chosen as convenient and sufficient sub-
stitutes for the theoretic ones of the orthogonal system, are
helices.
78. The construction of the helix also follows immediately
from its definition (76), as is shown in PL VIL, Fig. 50.
For let the distance 0'12' on the axis, be the height attained
by the generatrix while also making one revolution around the
axis from 0,0' to 12,12'. This revolution is evidently indicated
in horizontal projection by the circle of radius o0. Then, as
both motions are uniform, divide this circle, and the correspond-
ing height (pitch) 0'12', into the same number of equal parts,
and draw horizontal lines through the points of division on the
axis, and number them, as shown, to correspond with the num-
bers of their horizontal projections, oO, ol, etc. Then project
up 1, 2, 3, etc., from the plan upon the lines similarly num-
bered in elevation, at 1', 2', 3', etc., and the curve 0', 1', 2', 3',
etc., will be the vertical projection of the helix.
Theorem II.
The projection of the helix on a plane parallel to its axis is a
sinusoid.
A sinusoid is a curve in which if the abscissas, 0'b', 0'g', etc.,
are equal to, or proportional to the arcs of a given circle, the
corresponding ordinates W, 2 ! g', etc., are equal to the sines of
those arcs. Now Vb' = the sine of the arc 01 in the plan ;
2'g', = that of the arc 02 in the plan, etc. ; and the spaces 0'5',
0'g', etc., are proportional to those arcs. Hence, the curve 0',
1', 2', 3', 12' is a sinusoid.
The Helicoid.
79. Let now all the points of the horizontal line o0 — 0',
PI. VIL, Fig. 50, have the same two simultaneous motions that
have just been given to its outer point, 00'. Every point of the
line oO — 0' thus describes a helix, having the same pitch, 0'12\
and whose horizontal projection would be a circle centred at o.
All the consecutive helices, from the axis 0'12', to 0', 1', 2',
STONE-CUTTING. G3
3' 12', and so on outward without limit, thus described,
will constitute the surface called a right helicoid. The portion
within the cylinder of radius oO, is shown by the shaded area
of Fig. 50.
80. The surface just denned is called a helicoid, because its
generatrix, the line, oO — 0', moves upon a given helix, as
0', 1', 2', 3' 12', as a directrix ; and a right helicoid
because this generatrix is perpendicular to the axis o — 0'12'.
The surface is the same as that of the screw surfaces of a
square threaded screw, or of the plastering on the underside
of circular stairs.
The right helicoid is evidently normal to the convex sur-
face of the right cylinder having the same axis, since all its
elements are perpendicular to the axis of that cylinder.
Thus we see that the joint surfaces of the voussoirs (18) nor-
mal to the cylindrical intrados of the oblique arch are right
helicoids.
Problem XIII.
A segmental oblique arch, on the helicoidal system.
I. 81. The Projections. Primary outlines. — Let ABCD,
PL VII., Fig. 49, be the plan of the intrados of a segmental
(27) arch having a square span, AG, of 13 ft. ; an angle of
skew, CAB, of 54° ; and a perpendicular length, yy x , of 14
feet, between the vertical planes, AB and CD, of the faces.
Let EF — O" be the projections of the axis of the arch,
taken perpendicular to the plane V, and let the arc A'E'B' of
120°, centred at O", indicate the extent of the segment of the
cylinder of revolution, which forms the intrados.
Then AB^C is evidently (see PI. III., Fig. 28, etc., the
development of the intrados ; where CC X = 15'. 708 = the
length of the right section Cc 3 — A'E'B' ; and the curves AB X
and CT> 1 are the developments of the face lines, AB — A'E'B',
and CD — A'E'B' ; as will be presently explained more in
detail.
This being understood, and the curve CDj representing, at
each point, the direction of the thrust of the arch at that
point (62) the straight line CD X symmetrical with the curve, so
nearly coincides with it as to sufficiently replace it as a proper
direction for the developed transverse joints of the intrados ;
64 STEREOTOMY.
but, being straight, is evidently the development of a helical
arc (77, 3°).
This settled, next divide the straight line CDj into the odd
number of equal parts, chosen as the number of voussoirs, in
the width of the arch, in this case nine. Then, as the trans-
verse and coursing joints should be nearly or quite at right
angles to each other, let the developed coursing joints, B^v,
etc., be perpendicular to CD 1? or as nearly so as the length
DjBj, or width CCi, if unalterable, will allow. Such coursing
joints, being evidently both parallel and equidistant, the arch
may be built of brick if desired, (75, 5°) and, if of stone, all
the voussoirs in all the courses will be of equal width (75,
1°, 3°, 4°.)
Returning to Fig. 50, and comparing Figs. 49 and 50, the
axis of the arch replaces the axis o — 0', 12', and is perpendic-
ular to V- Hence, A'E'B' is the vertical projection of the
right cylindrical surface of the intrados, and of all the helices
traced upon it ; and the horizontal projections of these helices
will be similar, in form and construction, to the vertical projec-
tion of the helix in Fig. 50.
82. Horizontal projections of transverse helical joints on the
intrados. — KQ X , parallel to CD X , being the development of
one of these joints (77, 3°), and K& 4 that of the right section
at K, shows that Q x & 4 is the partial pitch due to the arc A'E'B'
of the circular projection of the same helical joint. By counter
development, Q : ^ 4 returns to Qk s . Hence, if we divide A'E'B'
and Q& 3 , each into the same number of equal parts, numbered
similarly from B' and Q as zero points, the points of the hori-
zontal projection, KPQ, of the helix considered, will be a,t the
intersection of the projecting lines from 1', 2', 3', etc., on
A'E'B', with parallels to Kk 3 , through the corresponding points
on Q& 3 , as shown in the figure. Thus, M 2 and B t are projected
from 2' and 3' upon the second and third parallels from Q.
83. Horizontal projections of intrados al helical coursing joints.
— In like manner, RAT, one of the parallels drawn through
points of equal division on CD X is the development of so much of
coursing joint as lies on the given segment of the cylinder taken
to form the arch. Tr 5 is, therefore, its proportion of the pitch
(76) of a coursing joint, and A'E'B' is its vertical projection.
Then, as before, divide Tr 5 and A'E'B' into the same number of
equal parts, number similarly from B' and r 5 as zero points,
STONE-CUTTING. 65
and the intersections of projecting lines from 1', 2', 3', etc.,
with the parallels to Rr fi through the like numbers on Tr s , will
be points R . . . R 2 , R 3 , etc. of the coursing helix RST.
84. Construction of horizontal projections of helices from their
developments. — The above constructions, having been inserted
to render them more intelligible by their analogy with Fig. 50,
it will now be shown that the sinusoid (Theor. II.) projection
of the helix can be found from its circular projection and devel-
opment. For the same parallels that divide Q^ 4 = Q& 3 , the
pitch of KPQ — A'E'B' into equal parts, divide the develop-
ment, KQ 15 of the same helix into the same number of equal
parts ; and the like is true of ^R{£. Hence, in cases like the pres-
ent, where the developments of the helices of the intrados are
given, their horizontal projections, as QPK, are most naturally
found as the intersections of the projecting lines from 1', 2',
3', etc., on A'E'B', with perpendiculars to the axis EF, from the
developments of the same points, 4 , M 3 , B 4 , etc., thus giving
respectively 3 , M 2 , B 3 , etc., on QPK.
All the other intradosal helices are found in a precisely
similar manner, as is now sufficiently evident by inspection.
85. The development of the face lines of the intrados. — 1st
Method. Parallels to AC through the ^oints of division 1, 2,
3, etc., on AB X , are the developments of the elements whose
vertical projections are the vertical projections 1', 2', 3', etc.,
of the same points of division. The horizontal projections of
the extremities of these elements are found by projecting 1', 2',
.... 6', 7', etc., upon both face lines, AB and CD, as at 6
and 7 on AB, and 7 on CD. Thence, in being developed,
they pass in planes 6 — z x ; 7 — z 2 , etc., perpendicular to the
axis EF, to z v z 2 , etc., of AB M and from 7, etc., on CD, to vh,
etc., on CDi, upon the developed elements, 7 vn, etc.
86. Construction of the developed face-line as a sinusoid.
(Theor. II.) If the equal divisions of A'E'B' be projected upon
AB, as at 6, 7, etc., these points will divide AB in a certain
manner. If the latter points be thence projected upon GiB x ,
projection of AB upon G^, the lines AB and G^ will be
divided similarly ; hence, if B^m^ be an arc similar to A'E'B',
and on the chord B^, it will be divided equally by the paral-
lels to GGd through the points B .... 6, 7, etc., on AB.
Hence, we may regard the arc B^Gi, and the straight line
B : A, as respectively the circular projection and the develop-
5
66 STEEEOTOMY.
ment of a helix, points of whose sinusoid projection, BjA, are
found as the other like curves, KPQ, have been, from like
data ; viz., at the intersections of parallels to GG X , from m x . . .
m 6 , m T , etc. with parallels tb G^ through like points, 1, 2, 3,
.... 6, 7, etc., on the straight line B X A.
87. Useful limits of the arc of the intrados. — These are evi-
dent from the development while considering the helical joints.
Were the arch extended to a 3 a 4 — A A 4 , so as to become full
centred (27), CDi would be extended each way, as shown at
DiD 2 , where Did 2 = A'A , and D 2 d 2 = a 3 v ; and the angle at
D 2 is 90°. Hence, we see that parallel coursing joints, quite
nearly perpendicular to the face line between C and D u as is
desirable, — become more and more oblique to it as we approach
D 2 on DiD 2 . Moreover, as can readily be imagined with a
given case in view, the greater the obliquity, that is the more
acute the angle CAB, the less should be the segment taken to
form the intrados.
88. The extrados. — While the outer surface of the actual
arch would be left rough, yet it is convenient to represent an
ideal extrados, or extrados of construction, to aid in forming
the voussoirs.
Such an ideal extrados is a cylindrical surface having the
same axis as the intrados, and terminated by the same vertical
planes of the faces. Its projections are, therefore, AiB 2 C 1 D 2 ,
and the arc A 2 F'B 2 concentric with A'E'B'. Then A 2 C 2 and
B 2 D 2 are the outer springing lines.
The intrados and extrados being thus concentric cylinders,
the point, as qB' 2 , in which the generatrix, Qq — B'B 2 , per-
pendicular to the axis EF — O" (79), of any of the helicoids
intersects the extrados, will generate helices upon the extra-
dos, whose developments and horizontal projections will be
found in the same manner as has now been explained for the
intrados.
89. Development of the extrados. — As the helicoidal sur-
faces of the voussoirs are right helicoids (79), their elements
are perpendicular to the axis EF — O" ; hence Kk and Qq,
perpendicular to EF, are such elements, and the outer helix
(helix on the extrados) corresponding to the inner helix
(helix on the intrados) KPQ will extend from k to q. Hence,
making w 9 a 2 = A 2 FB 2 = 20'. 94 (77, 3°) a 2 c u parallel to B 2 D 2
is the developed position of A 2 c ; and, carrying k across to this
STONE-CUTTING. 67
line at & 1? gives qk x as the development of the helix from q
to Jc.
The other developed transverse helices, as ba x and dc x , are
parallel to qk x , and are similarly divided, to give the developed
extradosal coursing helices bd 4 , and the parallels to it, corre-
sponding to BiIV, and its parallels on the intrados.
90. Contrast between the intrados and extrados. — Here we
meet with two points of difference. First — As the pitch,
Q& 3 , is the same for both of the helices, KQ X and qk^ while
the latter is longer, the angles o v qo and o x oq are greater than
the corresponding angles 4 QxOi and QiOj0 4 of the intrados.
Hence qo Y o and its equals at all the intersections of helices on
the extrados, are less than Q1O4O1, and all like angles on the
intrados.
Second — As the outer helix, corresponding to an inner one
from C to D, connects c with d, while the outer face line is
C 2 D 2 , corresponding to the inner one CD, the developed extra-
dosal face line, D 2 c 2 , and extreme helix, dc u do not terminate
at the same points, as they do on the intrados where the like
lines are CvnDx and C7D 2 .
91. An interesting consequence of tl~ Urst of the preceding
differences is, that to equalize the angles qo x o, and Q1O4O1, so
that the helicoidal surfaces should be normal to each other
somewhere between the intrados and the extrados, the initial
coursing joint, B^v, should be drawn to a point, IV, on the
side towards D 1? next to x, the foot of the perpendicular from
1$! to CDi, whenever x does not coincide with one of the points
of division of CD^
92. The construction of the developed face lines. — This may
be either as at V", etc., by projection from 1", etc., to 1, etc.,
on CD, and transference thence, by perpendiculars, to BD, to
elements through w 2 , etc. ; or as at B 2 a 2 by the sinusoid projec-
tion of a helix, from B 2 w 5 w 9 , and ba u the latter substituted,
without affecting the result, for a straight line, B 2 a 2 , as its cir-
cular projection and development. Either method is obvious
on inspection, in connection with the like constructions of the
developed face lines of the intrados.
The construction of the horizontal projections JcPq, etc., and
rSt, etc., of extradosal helices, is now obvious by inspection.
Thus m 2 is the intersection of the projecting line m'm 2 with
m' 3 m 2 perpendicular to EF.
68 STEREOTOMY.
93. Convenient checks upon the accuracy of the horizontal
projections and developments of the helices and face lines occur
as follows: First — The necessary intersection of correspond-
ing inner and outer helices on EF, in horizontal projection, as
at L, S, P, and N, because the elements of the right helicoids
containing such helices are vertical at these points. Second —
The developed positions as L x and l ly or N x and n x of such
points are in the same line, LjL^, or N^Nn^ perpendicular to
EF, and on E^ and e x f x , the developments respectively of
EF — E' and EF — F. Third — As helices from C to D and
from c to d necessarily cross EF at F, the middle point of CD
(as in Fig. 50 the helix crosses 0il2' at its middle point, 6'),
their developments, CD^ and dc x , cross ; the first, CviiDx and
EiFx at F x , the middle point of CvuDi ; and the other, dc ± and
^i/i a t/i, the middle point of dc v
94. Completion of the abutments. — These being alike, the
construction is shown only on one. The back of the abutment
is properly stepped by vertical planes, parallel and perpen-
dicular to that of the face (which represents the direction of
the thrust (62) of the arch), as shown at oZ u and mZ x . Then,
having regard to symmetry, and to the protection of the cor-
ners of the abutment, make mL = oZ l5 and draw BZ ; and, for
the opposite end of the abutment, make the angle D 2 D2= ZBB 2 ,
and Dz = BZ. These last details being, however, unessential,
may be varied at pleasure.
95. The face joints. — These are the intersections of the
planes of the faces with the coursing helicoids.
Now, referring to Fig. 50, as all the elements intersect the
axis, and are perpendicular to it, any plane containing the axis
or a perpendicular to it at any point of a helix, will contain an
element of the helicoid.
Comparing with Fig. 49, the plane, AB, of the face for ex-
ample, does not contain S, P, or L ; hence, it does not contain
elements of the helicoids SRr, K&P, or MraLI, which it there-
fore intersects in curves. It intersects the coursing helicoid
SRr in the face joint R"V".
96. The direct construction of the face joints will then be, to
project the points as R'" (derived from R"", extremity of a
coursing joint on the developed inner face line) at R' ; and r"',
likewise derived from rj u on the developed outer face line, at
r' ; when RV, though straight, would very nearly be the
STONE-CUTTING. 69
proper face joint. And so we might operate to find all the face
joints.
97. Representation by tangents. — The face joints are so
nearly straight, especially in a segmental arch, that their tan-
gents, at their inner extremities, may often be sufficient to
represent them. Now we know from descriptive geometry:
first, that the tangent line at a given point, t, of the intersec-
tion of any plane, P, with a surface, S, is the intersection of
the plane P with the tangent plane to the surface S, at the
point t ; second, that the tangent plane to a helicoid, at a given
point, is determined by the element through that point and the
tangent to the helix through that point.
98. The following construction depends on the principles just
given. Let the tangent at R'"R' be constructed. First — AB
is the plane of the curve, and its trace on the plane, EE„ of
right section is E — 0"F'. Second — The tangent plane at
R"'R', to the helicoid, RrS, is determined by the element
R'"L 2 — R'O", and the tangent, at R'"R', to the helix, RS —
B'R'E. Now the tangent to a helix at a given point lies in
the plane which is tangent to the cylinder containing the helix,
along the element containing the given point, and it makes the
same angle with that element that the helix does. But as the
latter angle is constant for all the elements, the development
of the helix, upon the tangent plane must coincide with its
tangent in that plane.
Hence the tangent at R' is the vertical projection, and
R""R", coinciding with SjRj, is the development of the tangent
line at R'"R' ; and R"W" is the projection, upon EE^ of the
portion R""R" of this tangent. Hence make R'W' = R"W",
and W will be one point of the vertical trace, on the plane
EEi, of the tangent plane to the helicoid RrS, at R'"R'. But
the element, R'"L 2 — -R'O", is parallel to the vertical plane
EEi, hence WU', parallel to R'O" is the vertical trace of the
auxiliary tangent plane. This meets E — 0"F', the like trace
of the plane of the curve, at U', which is therefore one point of
the required tangent line. The given point of contact R'"R' is
another; hence ER" — U'R' is the tangent, at R"'R', to the
face joint through that point.
99. Focus of like Tangents to Face Joints. — Draw R'X,
perpendicular to the horizontal 0"X ; and 0"Uj perpendicular
to U'W and hence to 0"R'. The angles at U' and R' in the
70 STEREOTOMY.
triangles O^UjU' and R'XO" thus formed are therefore equal,
and those at Ux and X are equal, being right angles. Hence
these triangles are similar ; whence we have : —
0"U' cos. U'CUi = O"^ (1)
Also, 0"U 1 = R'W (2)
Remembering that R'W'=R"W"; and calling the pitch,
= 3TV 5 , of the entire helix, 3RST, = j^, we have
9-n-f>"R'
R'W = h . W"W". (3)
From the triangle R"'EE 2 ; R'"E 2 = EE 2 tang. R'"EE 2 (4)
But, EE 2 = 0"X. (5)
andO"X=0"R'cos.R'0"X; or,0"X=0"R'cos. U'O"^ (6)
Now cancelling all the terms which are common to the two
columns of left hand and right hand members in these six
equations, and multiplying together the remaining terms in
like columns, we have :
0"U'=27T.O"R'. tan. E'"EE 2 ,„.
h ( 7 >
Now in this equation, every term in the right hand member is
constant, hence 0"U' is constant.
We thus find the interesting property that the tangents to
all the face-joints at their inner extremities, meet at a common
point on the vertical line through the centre E,O u of the face
of the arch.
The point U' is called the focus, and the distance 0"U', the
eccentricity of the arch ; or, more strictly, of the intrados ;
for it is evident that a similar construction exists for the extra-
dos, or for all the points in any cylinder, concentric with the
intrados.
100. Condensed construction of Foci. — U 7 being indepen-
dent of the position of R' on A'E'B', take A a 3 , the point in
the horizontal plane of the axis, and draw through it a devel-
oped helix as a^u meeting the plane of right section FF X (cor-
responding to EEj) at w, then, projecting a 3 on FF 2 at o", gives
o"u = 0"V.
Likewise, for the extrados, take A 4 <x 4 , corresponding to A a 3
in the intrados, and draw a 4 v parallel to the developed extra-
dosal helices, 6wi, etc., and project a 4 A 4 on FF : at 2 , then 2 v
STONE-CUTTING. 7 1
will be the eccentricity to lay off from O", giving U" as the
focus for the extrados ; so that, for example, r'U" is the
vertical projection of the tangent at r'"r' to the face-joint
R/V" — RV.
101. Foci for any Cylinder concentric with the Intrados. —
If, in a given arch, we take such a new cylinder, only the
terms 0"U' and 0"R/ will change ; hence, as is readily obvi-
ous, calling R , Ro> etc., the successive values of the radius
0"R', and calling the consequent positions of the focus, U', U",
etc. we have,
K _ TO 2 - ctc
0"U' ~ 0"U" ~~
Then putting ~^. f = c, we find 0"U" = ( -^
whence 0"U" is found simply as a third proportional. Thus
having drawn U'A , and A T, perpendicular to it, gives 0"T
(on 0"F produced) = c ; since R* = (O"A ) 2 = 0"U" X 0"T.
Then c being constant, draw for example TA 4 , and A 4 U" per-
pendicular to it, and U" will be the foeir of A 4 F'B 2 , which is
chosen for illustration, to save additional lines. For,
(0"A 4 ) 2 =0"U"xO"T
(0"U") = off - = ~- as above -
In this simple way we finally find the focus for any cylinder,
and thence as many tangents as we please to each face joint in
making an exact working drawing on a large scale. In the
figure, then, the face joint RV will be a curve, tangent to U'R'
at R' and to U'V at r'.
102. Curves of the Face-joints. — To get an idea of these,
see Fig. 50 again, where PQP', PiQiPl (PiQi not shown) and
P2Q2P2 are three parallel planes cutting the right helicoid
shown by the shaded area of the figure.
Auxiliary horizontal planes will intersect both the helicoid
and any one of the given planes in straight lines, whose inter-
section, in each of the auxiliary planes, will be a point of the
required curve.
Thus, taking the plane PQP', the horizontal plane O'Q cuts
from the helicoid the element o0 — 0', and from the plane,
the line PQ. These being parallel, meet only at infinity ;
72 STEREOTOMY.
hence PQ is an asymptote to the curve. The plane b'V cuts
from the helicoid the element ol — b'V, and from the plane,
the perpendicular to V at c', which meets the element at c'e.
Likewise, project d' at d, etc. Then at o' the intermediate plane
o'a' cuts from the helicoid the element oa — o'a', and from the
plane, the line oQ — o' which meet at oo', showing that the
horizontal projection of the curve passes through the centre of
the circle 0-3-6-12. Next, the plane 6'q' cuts the helicoid
and the given plane in parallels at 6' and q' respectively, which
only meet i ' infinity, hence qr is another asymptote. The
branch, cdoq, meets this asymptote at infinity towards q ; while
the new branch projected from s' , t',u f . . . . e', f at s,t,u ....
e,/ meets the same asymptote at infinity towards r, and the
asymptote /&, in the plane, 12'/, at infinity towards/.
Thus we see that the entire intersection consists of two
branches ; and that when the given plane, PQP', as here placed,
cuts the helix twice on the same side of the axis, as at h' and v\
the curve is shaped as at s t e, and does not pass through o ; but
that when it thus cuts the helix but once, as at the point a little
below c', the curve does pass through o, in the horizontal pro-
jection.
103. Applying this construction to the arch ; AB represents
the plane PQP' of Fig. 50, and each coursing helicoid, in suc-
cession, will represent the helicoid of Fig. 50. Taking the
helicoid, RrS, for illustration, the intersection of AB with suc-
cessive elements parallel to Rr would, when projected upon the
vertical projections, BgB' j 1"1', etc., of these elements, be the
vertical projections of points of the indefinite face joint R'r' ;
and a parallel to EF, at s 2 , the intersection of the plane AB
with the plane S«i, would be an asymptote to this curve which
would resemble cdoq in Fig. 50.
104. Other projections. — In finished drawings for exhibition
or other purposes, elevations on planes parallel to the face, or
to the axis, may be desired. These are easily made, as shown
in Fig. 51, a fragment of the vertical section through the axis.
A"E"B" is the right section of the intrados, and the parallels
to A"B" through its points of equal division 1, 2, 3, etc., are the
vertical projections of the elements at 1', 2', 3', etc., on the new
plane.
The portion, RS, of the inner helix, RST, is then projected ;
R, at Rj ; R 2 , at K 2 ; S, at Si, etc. Other helices could be
STONE-CUTTING. 73
similarly projected. The projections of the half face lines, EB
and FD, would be elliptical arcs. 1
II. TJie directing Instruments. — These are —
No. 1. The straight edge, applicable to any ruled surface,
in the direction of its straight elements.
No. 2. The mason's square, applied wherever two lines, or
two surfaces are to be perpendicular to each other.
No. 3. The pattern, MmZjoO, of the bed of a top stone of
the abutment ; also called an impost stone, or springer.
No. 4. The modification of No. 3, applicable to the bed,
DdZiZ, of the springer at the obtuse corner, D, of the abut-
ment.
No. 5. The second modification of No. 3, for marking the
bed, MmZB, of the springer at the acute corner, B, of the abut-
ment.
No. 6. The bevel, A' 2 ,A'T, for marking the joints li, K&,
etc., of the skew-back (19) in the plane, TJ f O"A' v
No. 7. The internal impost arch square, D"B'l', for marking
right sections of the intrados of the springers, in their proper
positions relative to the face of the abutment.
No. 8. The external impost arch square, 18A^', which de-
termines right sections of the extrados, in case they are wrought
of a cylindrical form, in their true relation to the level tops, as
jYi, of the springers.
No. 9. The flexible pattern, OiC^Qx, of the intrados, II 2 J,
of a springer.
No. 10. The corresponding flexible pattern, i'iYi#i, of the
extrados, ii 2 j, of a springer.
Nos. 11 and 12. The modifications of No. 10, which, put
together, equal No. 10, and which apply to the partial extra-
dosal surfaces of the two end springers ; which exist in conse-
quence of the different points, a x and a 2 , at which the developed
face line, B 2 a 2 and helix ba Y , terminate.
No. 13. The twisting frame, 20, 21, 22 — 20', 21', 22'. This
consists of three rulers, lying in three planes of right section,
and, as shown by the drawing, coinciding with three elements,
1 While writing these pages, an article on skew arches, by E. W. Hyde, C. E.,
has appeared in Van Nostrand's Magazine, Feb.-April, 1875 ; which may be
read with much interest by those who have acquired a sufficient knowledge of De-
scriptive Geometry, of its applications to the problem as exemplified in the mainly
graphical construction which I have here given ; and of higher mathematics.
74 STEREOTOMY.
20 — 20' ; 21 — 21', and 22 — 22', of a coursing helicoid, C c 7 S 2 .
Their perpendicular distances apart in a direction parallel to
the axis, are given in plan. Their angles with the horizontal
plane are given in the elevation. From these data they can
be rigidly framed together ; and can then be used to deter-
mine elements, in their true relative position upon a helicoidal
side of a voussoir ; having first notched upon their edges their
intersections with the helix C S 2 . In order that No. 13 may be
shifted along to determine successive elements, the perpendicu-
lar from 20 to 22 should be less than the length of the side of
the stone taken in the direction of the axis, and hence called
its axial length.
No. 14. The soffit frame. — This, used in case the intrados
of a stone is wrought first, consists of three parallel pieces, 23,
24, 25, framed together in planes of right section and giving
arcs of right section of the intrados, as RgSPr'", of a stone.
Their circular edges can be notched, by the aid of the draw-
ings, so as to show their intersections with the inner helical
edges, as R 3 S, of the voussoir.
No. 15. The arch square, e 3 3'e 4 , used in giving the cylindri-
cal intrados of a stone from its helicoidal side when the latter
is wrought first.
No. 16. A small, plane bevel, of the angle between a cours-
ing joint and an arc of right section ; and held against the
curved arm of No. 15, in the intrados, in order to guide No.
15 in a plane of right section.
No. 17. The helix templet, 5 4 c# 4 c 4 , whose curved edge, though
circular, will sensibly coincide with an inner helical arc, as
B B 3 , when placed against the helicoidal side of a stone. Mak-
ing the perpendicular from J 4 to 5 4 <? 4 , equal to that from 7', for
example, to the chord of the two divisions, as 6'-8', the eleva-
tion of an arc equal to 5 4 <? 4 , we have three points, 5 4 <# 4 and c 4 ,
to determine the required arc. No. 17 must be applied to give
a helical edge as B B 3 , before No. 15 can be applied, in succes-
sion to- No. 13.
No. 18. A flexible pattern, M!M 4 M 3 B 4 , of the development
of the intrados of a stone, gives the three remaining inner heli-
cal edges of that intrados, as B 3 M 2 , M 2 M, and MB , after hav-
ing found one, B B 3 , by No. 17. Or, after beginning with No.
14, it will, by the aid of No. 16, give all these edges.
No. 19. Fig. 52, is a bevel frame for working the helicoidal
STONE-CUTTING. 75
side of a stone from its cylindrical intrados, after having fin-
ished the latter by Nos. 1, 14, and 18.
No. 20. The radial joint bevel, Wi<? 4 w 2 , is used jointly with
No. 17, as indicated by the figure, to locate the corner edges as
B 3 6 3 of a stone upon the indefinite helicoidal side.
No. 21. The flexible pattern, p^f s ^sfsi which replaces No. 18,
for a voussoir which appears in the face of the arch. There
must be a pattern of this kind for every voussoir of one end of
the arch.
No. 22. This is a bevel varying for each stone in the face of
the arch, and giving the angle between the tangent to the face
joint, and the tangent to an inner coursing helix, at their inter-
section on the face-line.
We find this bevel for the point R'"R' as follows : The tan-
gent, R'U 7 , to the face-joint, meets the horizontal plane, A'B',
at y' 2 y 2 . The tangent, R'Y 2 , to the coursing helix through
R'"R' pierces the same plane at Y 2 Y 2 , where Y 2 is thus found.
RaWi, on the developed intrados, is the length of the vertical
projection of the development of the portion, R'"^ — R'B',
of this helix ; hence making Ww' = R^, and projecting w'
upon RiWiiv at w, gives R'"w, the horizontal projection of R'w',
the tangent to the helix at R'"R'. Then projecting Y 2 upon
R'"w, gives the trace Y 2 of this tangent upon A'B'. Hence
Y 2 y 2 is the like trace of the plane of the required angle, Y 2 R'"t/ 2
— Y 2 R'i/ 2 . To find this angle we have only to revolve R'"R'
about Y 2 y 2 as an axis, into the plane A'B' at a point R 6 , not
shown ; when Y 2 R 6 y 2 will be the required angle.
No. 22 is sometimes obtained mechanically, while building
the arch, as follows, Fig. 7. Let ILF be a portion of the upper
4 "
vk/ |taw /v y*fi
J?
-<
r s *i/ L^#-^-c
^ /tM^H
£
Fig. 7.
surface of the centring, coinciding therefore with the intrados of
the arch, and on which the inner helices of both kinds, and the
face-lines, are carefully chalked by means of thin flexible rules
bent over the centring, and through points transferred upon
chalked elements from the drawings.
76 STEREOTOMY.
Let FL be the face-line, and IH an inner coursing joint ;
and let r, r x , r 2 be three arcs of right section. Then set up and
fasten together three arch squares (No. 15) in the three paral-
lel planes of r, r x , and r 2 ; and tack on a strip oh, whose upper
edge shall coincide with a coursing helix, by being equidistant
from the angles of the squares, where it crosses their radial
arms. Then pass a plumb-line, pb, along oh till, as at n, the
bob rests on the face-line, when no will be a face-joint, and Icn
will be the required angle.
No. 23 is a pattern of the face of a face-stone, as FS 1 B", one
end of which is in a face of the arch. It would be found, for
each stone, on an elevation parallel to the face.
III. The Application. — This division of the problem will be
sufficiently illustrated by the working of two of the principal
stones ; viz., one of the intermediate springers, and one of the
voussoirs.
The springer, IYJI 2 e 2 — A'l^'A^Y'lO.
This, being the most irregular stone, is further represented
in oblique projection, Fig. 53. As in all other cases of irregu-
lar bodies, the stone is conceived to be inscribed in a rectangu-
lar prism, 14, 15, 16, 17 — 10, 11, 12, 13, from whose corners
the points of the given body may be located by rectangular co-
ordinates, which will therefore be seen in their true size in the
oblique projection.
Merely adding that like points are lettered or numbered with
like letters or numbers, the figure will mostly explain itself.
Thus, ii s , Fig. 53, = ii s from the plan ; i 3 k = ii 2 from the
plan ; and ki 2 = 26i' from the elevation. Similarly for all
other points, horizontal distances are taken from the plan, and
vertical ones from the elevation.
• 1°. Bring its bed, ODZyo, Fig. 53, to a plane in the usual
way, and mark its form by No. 3.
2°. On the edges given by No. 3, as directrices, work the
lateral vertical faces, DZil, "LyiY, etc., Fig. 53, square with
the bed by No. 2 ; and thence the level portion, Yji, of the
top.
3°. With No. 6, mark the joints, J/ and li, or test them.
4°. With No. 7, form channels of right section in the cylin-
drical face, II 2 J ; bring it to a cylindrical form by No. 1 applied
in the direction of the elements, and mark its edges by No. 9.
STONE CUTTING. 7T
5°. With No. 8, No. 1, and No. 10, similarly complete the
cylindrical triangle, 0$, of the extrados.
6°. Work the helicoidal cheeks, Iii 2 I 2 and ijIJ, which re-
ceive respectively a side and an end of adjoining voussoirs, sim-
ply by dividing the helical arcs, II 2 and ii 2 and JI 2 and ji 2 , each
into the same number of equal parts, and applying No. 1 upon
the corresponding points of division, as shown in Fig. 53.
A voussoir. — As we face the vertical plane, one of these
stones is, in detail, as follows : Mm^Bo is its front end or head,
forming a part of the transverse helicoid MmJjIi. Its opposite
and equal back end or head, is M 2 m 2 5 3 B 3 . Its right side or bed,
MwM 2 m 2 , is a part of the coursing helicoidal surface, beginning
at Mm, Its left hand bed is the equal surface Bo^B^g. Then
MB M 2 B 3 is its cylindrical intrados, whose development is
M]M 4 M 3 B 4 , and whose vertical projection is B'3'. Its similar
extrados is mb x m^ z , whose development is mm' s b 5 b G , and vertical
projection B 2 'e 4 .
To form this stone from the rough block, choose a block in
which the finished stone could be inscribed, then —
1°. By No. 13, form one of its indefinite helicoidal beds, as
MwM 2 m 2 . In doing this, first cut three elements on which the
edges of No. 13 will apply ; then placing rule 20 on the second
element, and rule 21 on the third element, rule 22 will be in a
position to determine a fourth element of the same helicoid.
In the same way any number of other elements may be
found, and the stone between them cut away by No. 1.
2°. By No. 17 and 20, placed together as shown on the plate,
mark the helical inner edge MM 2 , in its proper position relative
to the determining elements marked by No. 13.
3°. By No. 15, having its straight arm applied to the ele-
ments given by No. 13, and its curved arm guided in a plane
of right section by No. 16, applied on a small temporary plane
area cut on the intrados, cut any desired number of arcs of
right section in the intrados, MB M 2 B 3 ; and finish the intrados
as a cylindrical surface, by No. 1, placed in the direction of
the elements, as shown in the plan.
4°. By No. 18, placed in the cylindrical intrados, first
wrought, and with one of its edges coinciding with MM 2 , mark
the three remaining edges of the intrados. These edges will
serve as directrices of the three remaining helicoidal surfaces
of the stone.
78 STEKEOTOMY.
5°. By No. 15, or, more exactly, by No. 19, Fig. 52, which
consists of Nos. 1 and 15, very accurately made and firmly
braced together, work the helicoidal bed, BoB^A ; and mark
its corner edges, B 3 5 3 and B^, by Nos. 17 and 20 together.
6°. Having now finished all the faces of the stone, except
the extrados, which is to be left unwrought, and the ends, the
latter can be wrought by No. 2, the square, from the intrados ;
one arm being held in the direction of the elements of the in-
trados, and the other applied to corresponding points of division
of B 3 M 2 and 5 3 m 2 , transferred from the drawings.
In working a face-stone, as R 3 R"V', use No. 22 instead of
No. 20, No. 21 instead of No. 18, and No. 23 in working the
face.
Systems of Oblique Arch Stone-cutting.
105. Adhemae, with great elaborateness of graphical illus-
tration and refinement of detail, devotes no less than eighty-
three pages and seven large plates, to several different methods
of cutting the stones of an oblique arch, arranged under two
systems.
1st. The method by squaring, in which a rectangular prism
of stone is first formed, in which the finished voussoir can be
inscribed, and from whose sides, or edges, those of the required
voussoir can be located with the square, by measurement.
2d. The method by bevels. This method is illustrated by
the use of No. 2 (6 — 9), 15, and 19; a system which might
be extended by framing Nos. 13 and 14 together, or by adding
to No. 19 a second arch square, whose straight arm should be
to the left of EE, giving the relative position of an element,
and a right section of the intrados, and an element in each
helicoidal bed, all in one frame, or compound bevel.
106. Choice of circumscribing block. This point, which
may perplex a beginner, working only by the first of the above
methods, may be settled in three ways.
First Method. The sides of the circumscribing rectangular
prism may all be either parallel or perpendicular to the plane
of right section. Thus, for the voussoir R 3 SR 6 /* 6 , the length of
the prism (the prism not shown, to avoid confusion), would be
the perpendicular between the radial elements at R 3 and R 6 r 8 ;
while its width and thickness would be shown in elevation by a
rectangle circumscribing 5'5i2'w', since the vertical projections
of r 7 and r 6 are 5[ and m'.
STONE-CUTTING. 79
This method is the most elementary, requiring, besides the
square and straight edge, only Nos. 18 and 23 ; but it is obvi-
ously very wasteful of labor and material.
Second Method. Taking the same voussoir as before, pro-
ject all its corners upon a plane tangent to the extrados along
the element mid-way between r 6 and r 8 . Then the least rect-
angular prism will be that, one of whose sides shall be in this
tangent plane, and which shall include within it all the corners
of the stone. By this method there will be very little waste
of material in any case. And if this prism be first wrought,
the voussoir can be simply extracted from it mostly by squar-
ing from the edges of the prism, as before.
If, to save labor, the voussoir be wrought directly from the
rough block, the latter will be chosen similar to the prism here
described, but larger, and the method of cutting will be that
above described in detail.
The price of labor and material will determine whether it
will be best to cut the finished prism, or to risk an occasional
spoiled voussoir mis-cut directly from the rough block, or from
a block which by an error of choice may be too small.
Third Method. Here the faces of the circumscribing prism
are all parallel or perpendicular to the plane of the face of the
arch. This method is particularly adapted to a voussoir one
end of which is in the face of the arch, and can therefore be
immediately marked upon the face of the finished provisional
circumscribing prism.
107. Adaptation to a cut stone spandril. In this case, the
tops of the face-stone are finished with vertical and horizontal
surfaces, as in PI. IV., Fig. 30, and extending through the
thickness of the surmounting parapet. The horizontal sur-
faces are plane, and the vertical ones are cylindrical, having
the extradosal coursing helices for their directrices. The cut-
ting of the face-voussoirs thus designed, offers no special diffi-
culty.
Useful Numerical Data.
108. By some writers the problem of the oblique arch has
been treated almost wholly by the equations of its lines, and
of their projections; from which all its points could be located
by merely plotting to scale the results of computations, made
by substituting numerical data in the formulas. 1
l Bashforth, London, 1855. Buck, London, 1839. Graef, Paris, 1853.
80 STEREOTOMY.
But such treatment is less adapted for general use than the
graphical one, as most thoroughly exhibited by Adhemar ; and
which is all-sufficient, mainly on account of the simplicity and
similarity of the principal lines, on which all the points depend.
Nevertheless, a few simple formulas, such as all can use, are
useful as checks upon purely graphical constructions, especially
when the latter are of half, or whole size, upon a platform, as
they should be in practice.
109. Such formulas are the following, with the numerical
data, and results, used in the present example, substituted : —
The half span 0"'B' = S = 6.5 ft.
O'"B'O" = a = 30°.
Then 0'"0" = S. tan a = 3.75 ft.
and 0"B' = E = VS 2 + 0'"0" 2 = 7.5 ft. (7.504 ft.)
Again, let the angle of skew, E AC, =
then EAG = 90° — 0,
whence BG = 2S. tan (90° — 0) (1)
and AB = h/AG 2 +BG 2 = */l3 2 + (2S. tan (90 — Of
oo
or, AB = 2S. sec. (90 — 0) = 2S. cosec 6 = -~ .
v J sin
Also CC, = A'E'B' = I = R X .0174533 X 120° (2)
= 15.708 ft.;
where A'E'B' = 120°,
and .0174533 is the length of 1° where R= 1°.
Likewise, putting 0"Ba = R u
n d a 2 = AjF'B; = l 1 = R l x .0174533 X 120°, (3)
where Rx = 7.5 + 2.5 = 10, whence n s a 2 = 20.94 ft.
Further ; let C 2 y 4 , perpendicular to AB, = L = 14 ft.
Then, as A 2 C 2 ^ = (90 — 0.)
AC = BA = A 2 C 2 = L. sec (90 — 0) = L. cosec = ^
And, see (1) and (2), AB : = CD X = ^AGf+B^l (4)
Now suppose CD X to be divided into n equal parts of which
Diiv are m parts. Then D,iv = — CDi.
or, in this example, see (4), D,iv=|. CD X .
Let ABxGx = B 1 D 1 C = ft then ^ = tan (3. Or, as by (1)
B 1 G 1 = BG = 2S. cot0
5T -
—
.
2r
28
|
-
-
-
-
—
|
-}c-
Jfc^f
n
24
82 STEREOTOMY.
4°. Avoiding the acute diedral angles, between the face and
the intrados, near the acute angles, as B, of the abutments, by-
terminating the cylindrical intrados by a plane parallel, and
near to AB ; through r 5 for example, and then completing the
intrados between this plane and AB by a conoid, whose direc-
trices should be the ellipse in the plane r 5 , and a vertical line on
BD, as at Q, and equal to 0"'E' ; and whose plane director
should be horizontal.
5°. The substitution of brick for stone ; either wholly, or ex-
cept in the face ring of stones.
6°. The substitution of a circular for an elliptic face. This
will make the right section an arc of an ellipse whose longer
axis will be vertical. 1
111. Among much more important hinds of modification, are
the following : —
1°. Convergent arches. — When an arch is quite long, the
central portion of it may properly, and with great economy,
be built as a right arch, either of brick or stone. Then, near
the ends the coursing joints may gradually bend till perpendic-
ular to the face line at their intersections with it.
2°. Hart's System. 2 — In this system, the transverse joint
surfaces are planes, parallel to the face ; while each coursing
joint surface is a conoid, generated by a line moving parallel
to the face as a plane director, and upon the axis, and a helical
coursing joint as directrices.
3°. Cylindrical coursing joint surfaces. — This very inter-
esting modification of the orthogonal or equilibrated system, is
strongly advocated by Adhemar, s first, as dispensing with all
cut surfaces except plane and cylindrical ones, so related as to
be the ones most easily wrought with economy of material, la-
bor, and graphical construction ; second, as wholly avoiding those
components of pressure which act towards the faces, and tend
to produce dislocation.
The transverse joint surfaces in this system are planes par-
allel to the face.
The coursing joint surfaces are the horizontal cylinders, per-
1 Such arches, on the helicoidal system, are the subject of a work by Praly,
Paris, 1853.
2 John Hart, London, 1848.
3 Only regretting that prescribed limits imperatively forbid adequate illustra-
tion of this system ; the reader is referred to the ample exhibition of it given by
Adhemar, Paris, 1861.
STONE-CUTTING. 53
pendicular to the plane of the face, which project the trajec-
tories, or intradosal coursing joints of the orthogonal arch,
Prob. XII., upon the plane of the face.
4°. The Cow's Horn. — The form of oblique arch whose in-
trados is a portion of the warped surface so named, may be
substituted, in case of small arches, as oblique door-ways.
See PI. VII, Fig. 54.
The Cow's Horn (Come de Vache) is a warped surface, of
the kind having three given directrices. In its usual form,
it is generated by a straight line, AC — A'C, moving so as
always to rest upon two equal and parallel semicircles, AB —
A'N'P/, and CD — C'N'D', whose diameters are in the same
plane ; and upon a straight line, 00" — O', perpendicular to
the planes of these semicircles, at the point of symmetry O'.
This surface is the intrados of the arch. The semicircles
are its face lines. The semicircle on L/M' is the ideal extrados
of construction. The face-joints, as a'c' and m'p', are in pairs,
symmetrical with O'M' ; the coursing joints, ab — a'6', etc., are
elements of the warped surface ; and the available height of
the passage is O'N'. This height diminishes with the increas-
ing obliquity of the arch, till, when A' and D' coincide, the
arch degenerates into two cones, tangent to each other on 00"
— O' and the passage becomes closed. This is an objection to
this form of arch.
Examples. — 1°. Make a full construction, with patterns, etc., for the Cow's
Horn.
2°. Do. of a segmental, helicoidal arch, ?e/i!-handed, that of Prob. XIII. being
ngfa-handed, of 12 ft. span, 55° skew, 120° arc of right section and 2 ft. radial
thickness, and 12' : 6" perpendicular between faces.
3°. Do. of a full centred, helicoidal arch, with 60° angle or skew.
4°. Construct an arch on the cvlindric system.
5°. On Hart's system.
6°. A segmental, helicoidal, left-handed arch of 14 ft. span ; perpendicular
length 12 ft., 48° skew; 135° arc of right section; radial thickness 2 ft., 2 in., and
with the face voussoirs adapted to a cut stone spandril wall. Add several oblique
and isometric projections of springers and voussoirs.
Wing- Walls. x
112. Prismatic and pyramidal wing-walls have been suf-
ficiently illustrated in outline on PI. I., Figs. 6-9. The vertical,
quarter-cylindrical • one, with an oblique plane top as in PI. L,
Fig. 8, is described in my " Elementary Projection Drawing."
The top of the straight wall, which contains the face of the
84 STEREOTOMY.
arch, or passage, and which is flanked by the wing-wall, is some-
times horizontal : while either wing-wall is a segment of a
cylinder, oat off obliquely, as in PL L, Fig. 9. The top of either
wing-wall is then a portion of a right helicoid, having a vertical
axis ; that is, if such a wing-wall were a quarter-cylinder, its
coping would be one quarter of a revolution of a square-threaded
screw of stone.
All the foregoing forms are in use in the wing-walls of small
arches, culverts, and mounded cemetery vaults.
There yet remain, however, two forms of wing-wall, adapted
to larger structures, and which will now be described.
113. PI. VIII., Fig. 55, is a sketch, in plan and elevation, of
the crossing of two lines of communication, at different levels,
and at an acute angle. The lower one, rr, passes, by an oblique
arch, flanked by wing-walls, AC and BCx, through the embank-
ment, MM, which supports the upper line. The axis, parallel
to rn, of the arch, makes an angle, Cnr, with the direction,
CCi, of the embankment through which the arch runs.
This understood, it is determined that the wing-walls shall
extend a given distance in front of Cd, without being as far
apart at A and B as they would be if wholly curved. They
are, therefore, in plan, made partly straight and partly curved,
with a quite small radius, as Oo.
The straight wall, containing one face of the arch, lies between
the vertical planes CO and Cft^ Then, if the face of the wall
slopes, that of the curved portion of each wing-wall will be a
segment of a cone having a vertical axis ; and the faces, both of
the arch-wall and of the straight part of the wing-wall, will
be tangent planes to this cone. All parts of the face of the
wall will thus have the same batter.
From the complete plan, it will be seen that if the straight
parts of the two wing-walls are to be parallel to the centre line
of the arch, and equidistant from it, the conical portions must
be unequal segments, and of unequal radii, the larger segment,
adjacent to C x , having the less radius.
STONE-CUTTING. 85
Problem XIV.
The compound, or piano-conical wing-wall.
I. The Projections. 1°. Preliminaries. — PL VIII., Fig. 56,
represents the plan and elevation of a wing-wall, similar to that
part of Fig. 55 which is to the left of CO ; but with the ad-
dition of a pier at the foot of the wall, and a coping ; also, the
plane V? instead of being taken as in Fig. 55, parallel to CC X ,
is parallel to a plane, as OC, perpendicular to CCV
Let the vertical line at O be the axis of the inverted cone of
revolution, a part of whose convex surface is the face of the
conical part of the wall.
The circle with radius, OA, of four feet, is the horizontal
trace of this cone, and the arc AG, of 60°, is that of the conical
portion of the wall. The tangents to AG at A and G are re-
spectively the horizontal traces of the planes of the faces of the
arch- wall, and of the straight portion of the wing-wall ; which
are respectively tangent to the conical part, along the elements
whose horizontal projections are VB and VGT. Let the height
of the wall under the coping be 6 ft. 4 in. = 76 ins. and the
batter, £. The horizontal, AB, of the batter is, then, 19 ins. ;
and, making BC, the thickness of the wall at top, 2 ft., we
reach, at C, the back of the wall.
2°. Outlines of the elevation. — We must now turn to the
elevation, and make D'C = 6 ft. 4 ins. ; draw C'B' parallel to
the ground line RD', and project A at A' and B at B'. Then
A'B'C'D' is the section of the wing- wall at its junction with
the arch- wall, in the vertical plane OC. Hence V, the vertical
projection of the vertex of the conical portion of the face of
the wall, is found by producing the element BA — B 7 A', parallel
to Vi till it meets the axis O'O.
The section of the coping, in the vertical plane OC, is
C'E'F'H' ; 8 inches thick by 2 ft. 3 inches wide.
Next, suppose the plane slope, QC, of the embankment,
behind the wing- wall, to be one of \\ to 1, and to contain the
back top edge, QC — C, of the arch-wall. Accordingly, ma-ke
D'Q = | D'C, then, as the plane of the slope of the em-
bankment is perpendicular to the plane OC, Figs. 55 and 56, its
horizontal trace, as AB, Fig. 55, PQ, Fig. 56, is parallel to CO x .
Hence PQC, Fig. 56, is the plane of this slope.
86 STEREOTOMY.
3°. Conditions for the design of the top of the wall. — Let
these be three, as follows : —
1st. Its front and back edges shall be plane curves ; sections
of the front and back surfaces of the wall.
2c?. It shall be of uniform width, as seen in horizontal pro-
jection.
3d. Its straight elements shall be horizontal.
Let the plane SRB', parallel to PQC, be that of the front
top edge, whose vertical projection is therefore in RB'. To
find its horizontal projection, aTB, proceed as in Des. Geom.,
Prob. LVII. Thus, projecting G at G', gives VGT— V'G'T'
as one of the limiting elements of the conical face of the wall,
and T', its intersection with RB', is the vertical projection of a
point of the required edge, whose horizontal projection, T, is
therefore the projection of T' upon VG produced. In like
manner, any other points can be found. The front upper edge,
TB-T'B', of the conical part, is an elliptical arc, whose horizon-
tal projection, TB, is also elliptical, but in this case so slightly
so, that it may be represented, without sensible error, by a
circular arc whose centre, k, on OA, may be found by trial.
The front edge, aT, of the straight portion of the wall, is
tangent at TT' to TB — T'B', it being the intersection of the
tangent plane, GS, to the conical part, along the element GT,
with the plane, SRB', of the top edge. Hence (Des. Geom.
172) S is a point of this tangent, which is therefore projected
in ST— R'T'.
Supposing the wall to be terminated at a\ at a height,/*/"',
of 20 ins. from the ground ; project a! upon ST at a, make
af = BC = 2 ft., by condition 2° ; project/ upon the horizontal
a'f at/', and /'C will be the vertical projection of the back top
edge of the wall, by condition 1°. By condition 2°, draw CK
with centre k (in general, a curve at a constant normal distance
from BT), and limited by &K, and K/ will be parallel to aT.
Now, by condition 3°, draw T'h' parallel to the ground line,
till it meets f'C at h' ; project h' at h upon the curve KC, and
Th — T'h' will be one element of the top of the wall. Again :
project K at K", draw K'7', project V (which is on a'B') at I on
aT ; and Kl — K"l' will be another element of the top of the wall ;
which, as can now be seen, consists of three different warped
surfaces, as follows.
By condition 3°, all have the plane H as their common plane
director. Then, —
STONE-CUTTING. 87
1 S £. K/ — K"/' and al-a'l', both being straight, are directrices
of aflK — a'fl'K", which is thus a hyperbolic paraboloid (Des.
Geom. 278).
2c?. KA — K"h' being curved, while IT — I'T' is straight, these
are the directrices of a small conoidal portion, KhTl — K"h'T'V.
(See 114, p. 92.)
3d. TABC — T'h'WO is a warped surface without specific
name, of the kind having two directrices, TB — T'B' and
Ck — Oh' and a plane director, which is H«
These surfaces are evidently tangent to each other along
KZ— K»V and Th—Vh ! .
4°. The pier. — This is rectangular in plan,/c, being one side,
and perpendiculars to it at e and/, 3 ins., longer than ef, being
the adjacent sides ; so that the coping stone, odg, of the pier
shall be a square of 2 ft. 6 ins., overhanging on three sides. The
top of this stone is finished as a square pyramid 8 ins. high.
The pier having no batter, project e at e' ; and the vertical
at e' is a corner of the pier, and e'a' is the intersection of its
back with the face of the wall. Project d in the vertical at d',
for the corner of the pier ; and c in the vertical, c'c", for the
junction of the pier coping with the front of the wall coping ;
both being vertical surfaces. Then c'F', and the parallel through
H', are the upper and lower front edges of the wall coping.
Coping edges. — FLc — c"YL', being the intersection of the
warped top surface of the wall, produced, by a vertical plane and
cylinder, whose combined horizontal trace is FLc, is not strictly
a plane curve ; but it is so near to the plane curve, BTa — Wa',
that it may be considered as one without sensible error ; espe-
cially in practical cases, where OA would be several times 4
ft., and hence the changes of form of all the curved surfaces
much less quick. F'LV would be strictly constructed by pro-
jecting points of FLc upon the vertical projections of the same
horizontal elements, Kl — K"l', etc., on which they were taken.
The other edges of the coping are FLc — FV and CKf—E'f.
5°. The wall joints. — In the face of the wall, the coursing
joints of the conical part are horizontal circles, as b'r' ; right
sections of this portion. Those of the straight part are parallel
to Ge. The heading joints are elements of the conic portion,
and parallel to GT on the straight portion.
The bed surfaces. — To avoid acute-angled edges of stones by
making adjacent surfaces mutually perpendicular, the bed sur-
88 ■ STEREOTOMY.
faces, that is, those extending from AG, br, etc., to the back
of the wall, may be formed in two ways.
First. If two lines, as V'B' and v'j'j", perpendicular to each
other, as at j' y revolve about the common vertical axis V — W,
which they both intersect, V'B' will generate the conical front
of the wall; v'j'j" will generate a conic bed surface, whose
vertex is "W, and everywhere normal to the conical front of
the wall, and jj' will describe the horizontal circle,, jp — j'p',
intersection of these two cones, as desired for a coursing joint.
With like results, revolve b'b", and the other parallels to v'j",
about the same axis, V — W.
Second. If VV be made the centre of a series of spheres, of
radii V'A', Y'b', Vj', etc., each of them will be normal to the
conical face of the wall, at their intersections with it, since the
elements of the wall are radii of the spheres ; also they will
intersect the conic face in horizontal circles, as before.
Preferring the former system as simpler, let the beds of the
stones of the conic wall be conic surfaces generated by v'j'j",
etc., and let the ends of the stones be in vertical planes, VB,
VI, etc., radiating from the axis V-VV of the conical wall.
Thus, sqrpJJ and s'q'r'p'I'J'JJ'W are the complete projec-
tions of one stone of the conical wall.
The beds of the stones in the straight wall are planes, tan-
gent, as on r'W and p'J', to the conical beds in the conical
wall. Their heads are in vertical planes parallel to VGT.
The coping joints. — These simply divide the top of the
coping suitably, so as to break joints with the upper stones
of the wall, and are in vertical places through the horizontal
.straight elements KL — K'L', MN — M'N', etc., of the top of
the coping.
Thus, MN — M'N'P'Q', LK — L'K'K"L", etc., are vertical
joint surfaces of the coping.
6°. Location and limitation of the face joints. — It is desirable
that a coursing joint should coincide with the to£ of the pier,
but not always convenient to make a suitable common divisor
of the heights of the pier and wall ; hence arises the problem :
to arrange the stones in courses of diminishing thickness up-
wards, and so that a coursing joint shall coincide with the top
of the pier. Now parallel lines will be divided proportionally
by lines which meet at a point, P ; hence, if the latter lines
divide a given line, A, equally, they will divide a line, B, not
parallel to A, unequally.
l'l. IV
5
J^^
3
^1
UB
SoitZ.
SuJO.
SoJL
s. ir«
a..
1>,
STONE-CUTTING. 89
Hence, choosing in this example five courses of stones, as-
sume any point as a' on the level a'b' of the pier, drop a per-
pendicular B'B" from B', and draw a'B' and a'B". This done,
adjust any convenient edge scale of equal parts, by trial, till
the of the scale being placed on a'B", the point 1 of the scale
shall fall on a'b', and 5 of the scale on a'B'. Holding the scale
in this position, prick off the other points of division from to
5 upon the paper, and draw lines from a' through the points so
found, to meet B'B", as at y, etc. Then, through y, etc.,
draw the horizontal joints.
To avoid acute intersections of the coursing joints, q'p', etc.,
with the top edge, a'B', of the wall, the former must, as shown,
terminate on convenient heading joints, as b'a' on u'x', q'p' on
G'T', etc.
II. The Directing Instruments. These, besides Nos. 1 and
2, as in all other problems, are as follows : —
No. 3, a bevel, set to the diedral angle made by the slant
height of the pyramid, odg, with the vertical surfaces of the
pier coping. This angle is here shown by revolving the verti-
cal plane of this bevel till horizontal, as at o u f Y a.
No. 4 is a pattern of each face, as odg, of the pyramid, found
by revolving it about dg till horizontal, giving o'"dg, where
o'"f 1 = oy 1 .
No. 5 is the twisting frame for working the top of the coping
of the straight wall. It consists of two rulers framed together
so as to be in parallel planes, and so as to give the relative
position of the edges, as Zc — Z'c' and Yf — Y'f", of the front
and back of this coping. These rulers are shown in their real
relative position, in the separate figure (No. 5), by making c3
and c2 horizontal and equal respectively to cZ and/Y on the
plan ; and then 3z and 2Y, perpendicular to <?3 and <?2, and
respectively equal to 3Z' and 2Y' on the elevation. The per-
pendicular distance of the rule cYY" behind czz" is af; so that
the former applies to the edge Yf — Y'f" of the stone, while
the latter applies to Zc — Z'c'.
A similar frame can be made for the next stone, YL, of the
coping.
No. 6 is an elliptical templet fitted to the front top edge of
the conic wall. It is found simply by revolving the elliptic
arc, BT — B'T', around its transverse axis, BO — B'a', till
90 STEREOTOMY.
parallel to the plane V, when ordinates, as N"M K to this axis,
will appear as at w/"N 2 , perpendicular to a'W, and in their
real size, since N"N X is horizontal, and therefore seen in its
real size.
No. 7, not shown, is similar to No. 6, and similarly found,
and shows the real form of the curves, as FL a — F'Li, of
the front of the coping, assumed as plane curves because so
near to B'T' (4°). Hence, the ordinates for finding' No. 7 will
be laid off on perpendiculars to FV, and equal to their hori-
zontal projections, which are perpendiculars from FL : to BO.
No. 8, the dip bevel, shows the inclination of the elements,
b'b", etc., of the conical beds of the stones to the horizontal
plane.
No. 9, the coursing joint templet, is one of a set, one for
each coursing joint being evidently required by the increasing
radii of the latter. Its acting edge, ism, is cut to the circular
arc of a coursing joint.
No. 10 is flexible, and gives the relative position of a radial
joint, as j"j', and a coursing joint, as j'p' ; and hence, when
laid flat, gives the angle between an element and an arc of the
developed right section of a bed cone.
There is thus one for each bed, applicable to the two conic
surfaces, top of one stone and bottom of the next above it,
which unite on that bed.
Thus, in the one shown, the curved edge of radius v'j' is an
arc of the development of the horizontal circular section, jp —
j'p', of the bed cone whose vertex is v', while its straight
edge coincides with 0/ — v'j', produced.
No. 11 shows in like manner the relative position of a head-
ing and a coursing face joint, as s'q' and p'qf ; hence, when laid
flat, its curved edge, in the example shown, is drawn with the
radius V'j', slant height of the face cone to p'j', as a base or
right section, while its straight arm coincides with the ele-
ment V'j'.
No. 12 is a twisting frame for the curved portion of the
coping. It consists of three or more rulers, framed together,
each in a vertical plane, and so that their acting edges have
the relative position of three or more elements, as at X7 — 7' ;
L^ — LiKi, and LK — L/K', of the top of the coping.
No. 13, used in connection with No. 8, as indicated, gives
the vertical joints, as L'L", of the front of the coping.
STONE-CUTTING. 91
No. 14, a bevel between the bed and back of a stone, serves
to keep these surfaces in their proper relative position, so far
as the back, being exposed, is wrought.
III. The Application. This, so far as not already sufficiently
understood from the description of the directing instruments,
can be sufficiently illustrated in the working of a stone from
the body of the conical wall, and of the top of the wall as a
whole.
Take for illustration the stone rspqIJ — r's'p'q'XJ'J' ; but as
it is obscurely represented in the general projections, see Fig.
57, an oblique projection of it, which explains itself by letter-
ing like points with the same letters as in Fig. 56.
As in all other cases, points, as ii, not in the edges or faces
of the auxiliary circumscribing rectangular prism r x ur z — r 2 TPI",
are found by one or more ordinates, perpendicular to those
faces, and to each other ; and hence seen in their real size in
Fig. 57, where all parallels to r t r s , r±p", and r x u, are shown in
their real size.
Thus, in the two figures r x i 2 = r x i 2 ; i 2 i x = i&i (plan) and {$
= «!*' (elevation).
Having then chosen a rough block capable of containing
the finished stone, proceed as follows : —
1°. Work a portion as mm x nn x of its upper surface to a plane,
and on this temporary plane portion (sometimes called a sur-
face of operation) mark the circular front edge, pq, by No. 9.
2°. By No. 8, having its straight arm in the temporary
plane, mpnq, and radial to the curve, as shown, form any num-
ber of elements of the conical top, pJIq, of the stone ; and
gauge the thicknesses, as ql, etc., on these elements, Fig. 56,
which will give the back edge, IJ, of the stone.
3°. By No. 10, placed as shown, mark the radial edges, ql
and pJ, of the ends of the stone.
4°. Work the conical front, square with the top, by No. 2,
the square, as shown, placing its arms to coincide with elements
of both surfaces, and mark the vertical joints, sq and rp, by
No. 11, placed as shown.
5°. The bottom edge, rs, is equidistant from pq, measured
on elements, or it may be marked by one of the set No. 9, or
by one of No. 11, or yet again by No. 15 (not shown, but
readily found) the development of the entire front face, rspq —
92 STEREOTOMY.
r's'p'q' considered as on the convex surface of the cone, VV,
of the face of the conical wall.
6°. The ends are wrought simply by the straight-edge, from
their edges, already found, as directrices. The bottom bed is
square with the front, and also tested by the suitable form of
the set, No. 10.
7°. The top of the wall. — Having finished the body of the
wall, except the top, its front edge, BTa — B'a', can be imme-
diately located by measuring on the heading joints, from any
one coursing joint, as a'b f , or from a firm platform built at any
suitable level, as a horizontal reference plane. Having thus
the edge, Wa', as a directrix, any number of level chisel lines
can be cut in the top of the wall, by the aid of a spirit-level,
till the top of the wall is finished.
Then, having wrought the top of the coping, by the forms of
Nos. 5 and 12, suited to each stone, gauge the coping every-
where of a uniform vertical thickness, which may be done by
forms of No. 13, suited to different positions along FLc. The
under surface may then be wrought, either by suitable forms of
Nos. 5 and 12, or even by No. 1 only.
When material and labor are cheap enough, the somewhat
difficult stones of the coping of the conical wall might be
wrought by the method of squaring (105) from a completely
finished circumscribing rectangular prism.
Examples. — 1. Construct a front elevation of the wing- wall.
2. Construct an isometric drawing of a stone from the body of the conic wall.
3. Construct an isometric view, and an oblique projection of a coping stone, to
illustrate the cutting of it by squaring.
4. Work out the entire problem for the opposite wing- wall, CiB, Fig. 55.
5. Work out the problem of the vertical quarter cylindrical wing-wall with a
right helicoidal top (whose elements will therefore be horizontal).
The Conoid.
114. The conoid is a warped surface, which is generated by
a straight line which moves upon a straight line and a curve
as directrices, and always parallel to a given plane, called its
plane-director. ,
In its simplest form, partly shown in PI. VIII., Fig. 58,
which is an oblique projection, the directrices are a straight
line, OF, and a circle as that on AC, both perpendicular to the
plane director, RCO, the plane, FOQ, being also perpendicular
to that of the circle. The plane RCO is H> and RBC is V-
STONE-CUTTING. 93
115. Elliptic sections. — Every plane section parallel to
ABC is an ellipse. Let atb indicate such a section. Since
te = dq, and TE = DQ, we have from this, and the triangle
OAQ, TE : AQ : : te : aq, which expresses a property of two
ellipses having an axis in each, equal. Hence atb is an ellipse.
116. Tangents. — If a line moves upon three fixed lines, it
can have but one position at each point of any one of these
lines. But a hyperbolic paraboloid consists of two sets of ele-
ments, all those of each set parallel to one plane, and each
element of each set intersecting all ; and hence, any three of
the other set. Hence, if we take any three lines, as RK, rJc,
and OF, tangent at points on the same element, as TH, of the
conoid, and parallel to one plane, \f, they will be elements of
one generation of a hyperbolic paraboloid, whose other genera-
tion is formed by moving TH, upon these tangents. TH being
parallel to H, will remain so, and hence RrO is one of its posi-
tions, and KF another. That is KQ = FO. Thus RTKFHO
is a hyperbolic paraboloid, tangent to the conoid along TH,
and having H and V for the plane directors of its two genera-
tions.
117. Normal surface. — But as we could, in the last article,
have taken any other tangent at T, and parallels to it at all
points of TH, there may be an indefinite number of tangent hy-
perbolic paraboloids along TH. Of these, one will contain all
those tangents which are perpendicular to TH. Now, let all
these latter tangents be revolved 90°, about TH as an axis, and
they will all become normals to the conoid on TH. But as they
do not thus change their position relative to each other, they will
still form a hyperbolic paraboloid, which is thus the normal sur-
face along a given element.
118. General conclusion. — No distinctive property of the co-
noid, but only those of the hyperbolic paraboloid, having been
employed in this demonstration, this shows that the result is gen-
eral ; viz., that the normal surface to any warped surface at a
given element, is a hyperbolic paraboloid.
94 STEREOTOMY.
Problem XV.
The eonoidal wing-wall.
I. The Projections. — This novel form of wing-wall ia
founded in the idea, that, at the foot, FC, of the wall, where
the pressure of earth from behind is slight, no batter would be
necessary to give increased stability by increased thickness at
the base ; while, on the other hand, at AB — A'B', where the
height is greatest, the need of a slope or batter to the front
would also be greatest.
1°. To fulfill these conditions, let the face of the wall be a
quarter of a right conoid having the quadrant EF — E'F' for
its curved direction ; the perpendicular, OF — O', to the plane
V, for its straight directrix (O' being the intersection of FO
with B'E', produced, where B'E' has a slope of 3 to 10), and the
plane V f° r its plane director.
Hence, divide EAF, a quadrant of 9 ft. radius, into any
convenient number of parts, and OE, Oig, OJi, etc., drawn
through the points of division, and parallel to the ground-line
D'F', will be the horizontal projection of elements of the eo-
noidal face of the wall.
2°. The vertical projections of these elements will be found
by projecting E at E', g at g', etc., and drawing E'O' with a
batter, Wb', of 3 to 10, then g'O', etc. But, in this figure, O'
is thus made inaccessible ; hence proceed as follows : —
Draw A'B' at the intended height, 10 ft., of the wall, and
produce it as the vertical trace, A'e', of a horizontal plane, in
which is the quadrant Bag, whose centre is O. Divide this
quadrant in the same manner as EF, as at a, c, etc., project a,
c, etc., at a', c', etc., and as a'g', c'h', etc., necessarily pass
through O' (being identical with the vertical projections of the
elements of a cone whose vertex is 00', and base E7&F) they
are the vertical projections of the element of the conoid.
3°. The front top edge, BXF — B'C, is assumed to be the
intersection of the eonoidal front-face of the wall with a plane,
parallel to the straight directrix OF, and having a slope of 3
to 2. . BF is then found by projecting down B', y', X', etc.,
intersections of B'C with given elements, upon the horizontal
projections of the same elements, as at B, y, X, etc.
The back of the wall, made of the given top thickness AB,
STONE-CUTTING. 95
may then be made concentric with BF, as seen in plan, by-
making it tangent to any sufficient number of arcs, of radius
AB, and with their centres on BXF.
The top edge of the bach is here assumed to be the intersec-
tion of the vertical cylindrical surface of the back, with a plane
perpendicular to V? and whose vertical trace is A'C
The top surface of the ivall will thus naturally be a warped
surface, having BF — B'F ; and AC — A'C for directrices,
and H for its plane director.
The face-joints. — These shall be the elements just found,
for the heading joints ; while the coursing joints shall be equi-
distant horizontal sections, niF — m'T', etc., which (115) are
ellipses.
The joint surfaces. — With the quick curvatures arising from
small dimensions and large batter, as in the present example,
these surfaces should be normal to the face of the wall along
the lines just fixed upon as the joints of the face.
The coursing surfaces, or beds, will thus be warped surfaces,
which, for the joint mW — m'T', for example, will be generated
by a line mA — m'B", normal to EB — E'B', and moving
upon m¥ — m'T' as a directrix, so as to continue normal to the
face of the wall. Now, since there can be but one tangent
plane at any one point of a surface, and but one perpendicular
to a plane at any one point, there can be but one normal line
to a surface at any one point ; hence, the warped surface thus
generated is determinate.
The heading surfaces will be the normal hyperbolic para-
boloids along the elements of the conoid (117).
5°. Construction of the joint surfaces. — If two lines are
perpendicular to each other, and one of them be parallel to a
plane of projection, their projections on that plane will be
perpendicular to each other. Now the elements, as lu — l'u',
of the conoid, are parallel to V \ hence, the vertical projections,
r's', n'q', etc., of the normals to the conoid, along lu — l'u',
will be perpendicular to l'u'. And the like is true for the
other elements. Again, the normals at points of the ellipses
mF, etc., are perpendicular to the tangents to those ellipses at
the same points ; hence, rs, nq, etc., oQ, etc., are perpendicular
to the tangents at r, n, o, etc., respectively, to the ellipses
HrF, mriF, etc.
Projecting A, Q, R, etc., upon the vertical projections,
96 STEREOTOMY.
ra'B", o'Q ! , Jc'R', etc., normals at m', o', Jc', etc., we have
B"Q'R' . . . . T, tangent to m'T at C,T', for the vertical
projection of a coursing joint on the back of the wall; and
mAFC — B"m'T'S'R' as the normal coursing surface contain-
ing the ellipse m¥ — m'F ! .
Likewise, projecting p, q, s, etc., upon the perpendiculars to
Vv! at I', n', r ! , etc., we find p'q't', a heading joint, upon the
back of the wall.
6°. The tangents at o, w, r, etc., to the ellipses w?F, etc., and
to which the normals oQ, etc., are made perpendicular, may be
drawn in various ways, as most convenient for each point.
1st. That at zz', to the ellipse toF, is here drawn by the
method of bisecting the angle MsN included by lines from z to
the foci, one of which, /, is shown, and both of which are at
the intersections of an arc with radius Om, and centre F, with
the transverse axis mO ; produced to find the other focus f v
2d. That at n, to the same ellipse, is drawn by the method
of revolution ; the quadrant, EF, being the projection of mF
after a certain revolution about OF. Then n appears at I, and
IG is the revolved position of the required tangent, which, by
counter-revolution, appears at Gn.
3d. The tangent, J>, is likewise found from Jl, where the
revolution of the ellipse, HF, take place about HO, till it ap-
pears as a circle of radius OH.
4th. If adjacent figures were not in the way, so that other
quadrants of each ellipse could be shown, we might proceed as
follows, by the method of conjugate diameters. Thus, at o,
for example, draw Oo and any chord, parallel to it, and the
tangent at o to moF would then be parallel to the line from O
to the middle point of this chord, for such line would be the
diameter conjugate to Oo.
7°. Approximate joint surfaces. — With the considerably
larger dimensions, and less declivity of face, which would be
generally found in practice, the part from X to F would be
nearly a vertical plane, and from BE to X, nearly a conical,
or even an almost vertical cylindrical surface. Hence, in such
a case the coursing surfaces could be safely horizontal planes ;
and the heading surfaces could be planes. That through the
element lu — l'u', for example, might be a plane, determined
by the element together with the horizontal element, at K', of
the top surface.
J'l V
STONE-CUTTING. 97
II. The Directing Instruments and their Application. As
every surface, except the back, which would generally be left
rough, of all the stones between UU' and gg' is warped, no
patterns can be used. It would therefore be best to begin, at
least, by working some one surface by the method of squaring
(105) from the sides of a circumscribing prism, so far finished
as to admit of the application of this method.
Having the front, for example, thus wrought, beds could be
made square with it by keeping one arm of the square (No. 2)
on an element of the front, and the other on an element of the
bed.
With the often admissible approximate plane joints, already
described, the operations would be much easier, as patterns of
all the faces except the conoidal front could be easily found
and applied.
These general guiding observations, added to previous exam-
ples, will enable the student to construct whatever instruments
may be necessary.
Examples. = 1°. Construct the front elevation of this wall, and an isometric,
or oblique projection of one of its stones.
2°. Construct the wall when OE is less than OF.
STAIRS.
119. Stairs vary in form ; first, as a ivhole, depending on
the form of the space which they occupy ; second, in detail,
that is, in the form and arrangement of the separate steps.
Certain practical conditions and geometrical principles are,
however, common to all cases ; hence, a single example of a
general case, fully explained, will serve as a standard, from
which variations may be made to any particular forms.
120. General Geometrical Principles. — All stairs may be
divided into two principal kinds.
1st. Straight stairs ; in which the height, or rise, and the
width, or tread, are each uniform, on each and all of the steps.
2d. Winding stairs ; in which the rise remains uniform,
while the tread is variable at different points of each step.
In this sense, winding stairs which consist only of successive
short flights of straight stairs running in different directions,
are not included.
7
98 STEREOTOMY.
121. Winding stairs are, again, of two species : —
First, those which wind around a single vertical axis from
which the edges of the steps radiate.
Second, those which radiate around no single central axis
from which the steps radiate.
In the former, the tread is uniform on a line of ascent taken
at any given distance from the axis. In the latter, there is
but one such line, and it is taken at that distance from the
hand-rail which any one would naturally choose in passing
up or down the stairs, and may be called the line of passage.
Stairs mostly straight are often partly winding, at one or
both ends, and will then be classed under one or the other of
the varieties just indicated.
122. In winding stairs of the first species, the natural line
of passage upon them is obviously a common or circular helix,
as in PL VII., Fig. 50 ; where, if a horizontal and a vertical
plane be passed through each element, they would evidently
intersect each other so as to form the steps of such stairs as
would wind around the axis of a cylindrical pit.
In winding stairs of the second species, the horizontal projec-
tion of the line of passage will not be a circle as in PI. VII. ,
Fig. 50, but some other curve, as LP, Fig. 8, better conformed
to the ground area covered by the stairs.
123. Now let LP, Fig. 8, be the horizontal trace of a verti-
cal cylinder, on which a point, m, moves so that the horizontal
and vertical components of its motion are equal, that is, so that
if w*%=w 1 w*2, etc., the heights of on x above m; of m 2 above m u
etc., will be equal. The point m will thus generate a helix, h,
but of a more general kind than the circular helix, PL VII.,
Fig. 50.
124. Also if a horizontal straight line taken as a generatrix
G, move upon this new helix LP, in the same manner as in
PL VII., Fig. 50, the resulting surface will still be a helicoid, H,
but of a more general form than the usual particular form shown
in that figure.
But as the plan, LP, of h is no longer a circle, while the line
G continues perpendicular to it as seen in plan, Fig. 8, the lines
g, g u g 2 , etc., horizontal projections of successive positions of G,
will not pass through any one point, as at o in PL VII., Fig.
50, but will intersect each other, as in Fig. 8, so that when
g, g\,g<i, g n , and thence their intersections, p, q, r, etc.,
STONE-CUTTING. 99
become consecutive, p, q, r, r„, will form a curve to
which g, g v g 2 , g n will be tangent.
125. We thus reach the following important conclusions;
foundation of the design of stairs of every form.
Fig. 8.
1°. The curve pqr is the horizontal projection of a vertical
cylinder, C, which replaces the vertical straight line (axis) o —
0'12' in PI. VII., Fig. 50.
2°. The curve LP is the horizontal projection of a general
form of a helix, h, (123).
3°. The straight line g, then moves upon the helix A, and
remains horizontal and tangent to the cylinder C. It thus gen-
erates a general form of helicoidal surface, H ; such as forms
the under surface of the stairs, and such as will contain the
similar radial edges of all the steps.
Problem XVI.
Winding stairs on an irregular ground plan.
I. The Projections. Let a landing at AdE, PI. VIII. , Fig.
60, provided for a door in the wall ES, be connected with a
floor three feet higher at CH, by a flight of five steps, counting
the upper level. And let these steps be built three inches into
the wall, whose base is the segment HGFE of an irregular
polygon.
100 STEREOTOMY.
Let the line of passage (121) be the arc CD of the ellipse,
whose semi-axes, OA, and OB, are determined by convenience.
The tread of each step, measured from D on AC, is 12 ins.
giving I, J, K, C. At these points draw normals to the ellipse,
AC, as Dd, etc., which will be edges of steps. On these nor-
mals lay off 18 ins. inward from CD, to locate the inner end,
cd, of the steps, or the circumference of the well, as it is
called.
For firmer support, let each step extend as at Kg, or Jk
( J 2 &i) and K^, 7 ins. under the next upper one ; and then
normals to CD at k and g, will be the under edges of the step
JK, and, see k 2 and g 2 , in the general under surface of the steps.
This under surface of the steps is a right helicoid, H, gen-
erated by Qkq moving upon a helix whose horizontal pro-
jection is CD, normal to it, and parallel to H as its plane
director.
Normal joints. — These, perpendicular on Qq, etc., to the
helicoid, H, should (by 118) be hyperbolic paraboloids. But
it is a sufficient approximation to make them normal planes at
some suitable mean point. They are here made normal at g,
k, etc., points on the helix CD, for two principal reasons. 1st.
Only the helix CD will be straight in development, it alone
having equal arcs for equal ascents (123). Hence, normals at
its points can be more easily drawn. 2d. All the curvatures
are quicker within than without CD ; hence the warped, and
plane normal surfaces, will differ less if the normals be nearer
ed than to the wall.
The development. — Proceeding as just indicated, make C 2 D 2
= CD, and C^ = the total rise = 3 ft. and C^wiH be the
development of the helix over CD, and which cuts the front
edges of the steps.
Next, after dividing C^ into four equal parts, make J X J 2 =
\ 0^2 ; JiK! = JK ; J 2 &! = Jk ; Ki<fr — Kg ; and g^g 2 and ~k x
h 2 , each, for example equal to ^ JiJ 2 , when g 2 s x parallel to CJ)^
will be the development of that helix, projected in CD, which
is in the helicoid H.
Finally, at g 2 , k 2 , etc., draw g 2 fi, hh, etc., perpendicular to
C 1 D 1 , and they will be lines of the required normal planes to
H, at g, k, etc. Do the same for the other steps, and the figure
EMijfeJa will be the complete development of the section of the
steps made by the vertical cylinder whose base is CD.
STONE-CUTTING. 101
Then, to complete the plan, make gf = Teh = g\f x , etc., and
the lines as Mhm, parallel to Qkq, etc., will be the traces of
the normal planes to H upon the treads.
II. The directing instruments, besides Nos. 1 and 2, are
these : —
No. 8, the pattern, Mhrnl, of the top of a step.
No. 4, the pattern, M'L'B/, of the wall end of a step. This
is found by projecting the outer end, QL, of a step upon a par-
allel plane, as shown ; making all the .vertical distances equal
to the like ones on the development ; Q'Qi = g x g 2 ; P'Pi =
KiK. (greater than JiJ 2 , since K^fr is greater than k'k{).
No. 5, the pattern, n^bi, of the inner end of a step. This,
for the step EcZrR, is the development of its inner end. Then
r x n x ; r x \ ; r x d x , etc., equal rn, rl, rd, etc., on the plan ; and the
heights, h} 2 , etc., equal those on the delopment KjKg, etc.
No. 6 is a normal joint bevel, giving the constant angle
III. Application. — Having chosen a sufficient block, bring
its intended top to a plane, and mark its form by No. 3.
Work the rise and the end, square with the top, using Nos.
4 and 5 to give the forms of the ends. Work the normal
joints by No. 6, and the helicoidal under side by No. 1, ap-
plied on points transferred from the drawing, where elements
would meet ql and QL, for the step I J, for example.
Other Forms of Stairs.
126. Other stairs are, for want of space, merely suggested by
the steps, illustrated in Figs. 61, 62, which are both adapted to
circular stairs ; that is, those placed in a cylindrical case. They
are contrasted in the manner of support. In Fig. 61, which
shows a plan, elevation of the back edges, and a development
of the cylindrical outer end BD, the central open cylinder, or
well, is filled by a core, composed of the cylindrical wings, or
ears, O, solid with the inner end of each step. The core is
sometimes larger, and then solid, and with the steps indented
into it, as at the outer ends in Fig. 60.
Fig. 62 is an oblique projection of one step of circular stair
with an open well ; and the steps are supported by an ear CE
102
STEREOTOMY.
at the outer end whose whole thickness, Fe, is indented into the
wall, giving them a wide horizontal support.
Examples. — 1°. Observing that the curves, CDA and XY, PI. VIII., Fig. 60,
have the relation of involute and evolute to each other, represent, with the pat-
terns, stairs in which XY shall be assumed.
2°. Stairs in which cd shall be assumed.
3°. Construct stairs whose steps shall be formed as indicated in Fig. 62.
4°. Construct circular stairs in a cylindrical case, with a central post formed of
steps like that of Fig. 61.
5°. In a flight of five, or more, steps against one reach of wall, as GF, Fig. 60,
construct the intersection of that wall with the helicoid, H, of the under surface of
the stairs.
CLASS IV.
Structures containing Double-Curved Surfaces.
Problem: XVII.
A trumpet bracket with basin and niche .
I. The Projections. These are a plan, RrE ; a front eleva-
tion A'B'E" ; and a sectional side elevation, 0"'R"E"".
1°. The outlines of the bracket. Rrr x PI. IX. Fig. 63, is a
wall of the general thickness QK ; but, where the bracket is
attached, of the additional thickness, r x s. The front of the
bracket is composed of two cylindrical surfaces ; one vertical,
with the radius OA ; the other horizontal, with its elements,
AB, na, etc., parallel to the ground line A'B'.
The form of the latter cylinder is made to depend on the
given curve, AEB — A'E'B', of their intersection, where A'E'B'
is a semicircle. This curve would be the intersection of the
vertical cylinder, with a cylinder of revolution whose vertical
projection would be A'E'B', as shown more clearly in the aux-
iliary Fig. 64, where AEB — A'E'B' is the intersection of the
vertical cylinder, CD — CD', with the cylinder, UT — A'E'B',
which is perpendicular to the vertical plane. By Theorem I.,
the projection, E'"0"', of the intersection AEB — A'E'B', on a
plane, as 0"'X, parallel to the plane, EF, of the axes of the cyl-
inders of revolution, is a hyperbola. The vertical cylinder is
then cut away to this hyperbolic profile, E'"0'" ; so that the
face of the bracket within A'E'B' is a cylinder parallel to the
ground line and with a hyperbolic right section, or base O'E' —
0"'E'".
The top of the bracket is a plane annular surface, between
AEB — A"B" and the circular edge, of radius OH, of the hem-
ispherical basin, H'G'I' — J'G"J".
2°. The joints of the bracket. — Having found 0"'E'", as may
be seen by inspection, divide A'E'B' into equal parts, here five,
and draw the radial joints, as A'O'; c'O', etc., limited, to avoid
thin edges, by a semi-cylindrical stone of radius UT — O'T', =
R'V". Then —
104 STEREOTOMY.
To find intermediate points in the joints on the vertical cylin-
der. — Assume b' as such a point. Its horizontal projection is
b, and auxiliary projection J 1? which by revolution appears at b",
intersection of b 2 b" and b'b".
To find intermediate points in the joints on the horizontal cyl-
inder. — Here it is the horizontal projection of the point that
needs construction. Assuming d', for example, project it at
d", on 0"'E', thence to sX and revolve upon Xm; whence pro-
ject upon d'd at d. Or, by the method of transference, make
d x d = d"'d" (50).
By the same method, applied to b", make b'"b" = ob, to find
b". Also, make h"e'" = hh ! ", to find h". In similar ways the
points of all the joints of the bracket can be found ; as at
e'"h'{ = h'"h x , in finding A l5 a point of the basin joint, h x gk
— h'g'—Kfh'i'.
The horizontal projections, as h^gk, of the basin joints, above
gg', on the vertical semicircle, H'GT, of the basin, are found
by horizontal circles, as that with radius j'V, each of which will
contain four points of the two basin joints, of which points i$
is one.
3°. The Niche. — The niche is a vertical semi-cylinder, CFD
— C'C'D'D", covered by the quarter sphere included between
the horizontal semicircle, CFD — CD", and the vertical one,
CD — C"E"D". The joints of the spherical part are circular,
and in planes which radiate from the diameter, OF — O" —
0""F" ; and are limited by the cylindrical stone, 0"L/P".
Intermediate points of these joints, as N', are readily found
on the plan and side elevation, in various ways. Thus, MM X
— M'N'Mi — M"N" is a vertical circle through W, which point
is thence projected upon MM^ at N, and upon M /r N" at N".
This circle might have been made horizontal through N' ; or
the plane of the joint might have been revolved about OK —
0"K' as an axis, when the circular joint would have fallen on
K'E" and W at n". Then make r^N = n"W, which will give
N, as before.
Varied constructions are useful, in case some one of them
does not conveniently apply to certain points. Points, as K',
being projected at K, and K", and L/ at L, and L", KNL and
K"N"L"are the other two projections of the circular joiDt K'L\
II. The Directing Instruments. — These will consist of pat-
PIT!
I
STONE-CUTTING. 105
terns of the plane and developable faces of all the stones, with
a sufficient number of bevels to insure the correct relative po-
sitions of all the surfaces.
After the numerous preceding examples of developing the
convex surfaces of stones of irregular shape, the construction of
the required patterns can be made from the following general
directions.
Take the stone, A" A' VW, of the bracket, and L'K'p' of the
niche, as being the most irregular ones. Let each extend to the
vertical plane back, Br — R/'Q", of the wall. Develop the
entire convex surface, both lateral and end faces, of each, as in
previous examples. The following, including the necessary
bevels, will thus be found.
No. 1, the straight edge.
No. 2, the square. Then for the bracket stone alone —
No. 3, the pattern of the back, = A"h'Y'v't'.
No. 4, that of the top, = A'"A 2 A 3 HA 1 /iA.
No. 5, that of the radial joint, hiVuh 2 h 3 gh l — Y'h'.
No. 6, that of the opposite radial joint, AA'"v 1 «».
No. 7, that of the surface, Yvv x u.
No. 8, the development of the vertical cylindrical surface,
AM — A"h'i't'.
No. 9, the development of the portion, nivY — n'i'v'Y' — i"f",
of the horizontal hyperbolic cylinder of the front.
Nos. 10 and 11, as shown on the figure.
The corresponding guides for the niche stone can readily be
found.
To avoid the too acute angle at h x in the bracket stone, H'G'I'
might have been an arc of 120° or less ; or the portion of the
bracket above E' might have been a single stone thick enough
to contain the basin.
III. The Application. This is, in the main, sufficiently ob-
vious, from the description of the patterns, and the previous
essentially similar cases. The back of both the bracket and
the niche stone may properly be wrought first, since all the lat-
eral faces are perpendicular to it. The order and manner of
using the remaining guides may be left to the workman.
Examples. — 1°. Construct the niche alone.
2°. Construct the bracket alone, and without the basin.
106
STEREOTOMY.
Theorem III.
The conic section whose principal vertex and point of contact
with a known tangent are given, will be a parabola, ellipse, or
hyperbola ; according as the given vertex bisects the sub-
tangent, or makes its greater segment without, or within, the
curve.
In both the ellipse and the hyperbola, referred to their cen-
tres and axes, the subtangent is a fourth proportional to the
abscissa of contact, and those segments of the transverse axis
which meet on the ordinate of contact. That is, in Figs. 9
and 10, CO : Oa : : OA : OT ;
whence, by division, CO : Oa — CO : : OA : OT— OA ;
or CO : CA : : OA : TA
Fig. 9,
C T
I
Fig. 10.
Now, always, in the ellipse, CA>CO
.-. TA>OA
But in the hyperbola, CA<CO
.-. TA<OA.
In the parabola, AO = AT.
STONE-CUTTING. 10T
Problem XVII.
The hooded portal.
The Projections. — PL IX., Fig. 65. This construction is
known in France as the recessed gate of St. Anthony, it being
that gate near to the Bastile, which led to the suburb called St.
Anthony. The figure represents a broken plan and front ele-
vation, and a vertical section through the axis of the portal.
Its design is to give to a portal, closed by rectangular gates,
something of the grander effect of a semicircular topped portal,
closed by gates of like form, as in PI. VI., Fig. 45.
Let AcFmVO be half of a horizontal section of the portal,
below its top, V'F'. Then, as in Problem XI., the entire
passage embraces three parts, of which one half of each is as
follows : the portal proper, EVmF ; the gate recess, ~Ec"ce ;
and the converging embrasure, OecA, flanked by the jambs,
one of which is Ac.
The tops of the two former parts are horizontal planes. The
latter is covered by the very peculiar double-curved surface,
characteristic of the structure, and generated by a variable
semi-ellipse ; which, starting from the semicircle of radius O A,
as its initial position, moves so that its centre shall remain on
Oe — O', one vertex upon Ac, and another upon a fixed ellipse,
C"p"e", whose axes are O — O'C and Oe — O'. Hence, this
elliptical generatrix varies from the semicircle of radius OA to
the straight line 2ec. Thus, when this ellipse has reached the
plane ap, its semi-axes will be ap and O'p', = qp". Likewise,
bf and O'f are the semi-axes of another position of the movable
ellipse.
We may note in passing that, beginning at the vertical
plane OA, the horizontal axis will diminish more rapidly than
the vertical one, until we reach the point of contact of that
tangent to c"p"e", which makes the same angle with 0"D"
that Ac does with OA. Hence, as O'A'C is a circle, a few
positions of the movable ellipse near it will have their longer
semi-axes vertical.
The joints, as L'J, in the vertical plane portion exterior to
the recess," are straight, and radial to the semicircle of ra-
dius O'A'.
The face joints, as L'M 7 , within the recess, are best made
108 STEREOTOMY.
normal to both of the limiting positions of the variable ellip-
tic generatrix. This result may be obtained by making these
joints, as seen in front elevation, as circular arcs, tangent, as at
I/, to the radial joints, L' J, etc., just described, and with their
centres on O'A'. But such joints will divide 20'c' unequally.
If the latter result be thought undesirable, 20V, the top line
of the gate recess, may be divided equally, as in the figure, and
the joints may be either conic sections or curves of two centres,
tangent, as at L', to the radial joints, and, as at M', to vertical
lines.
The construction is illustrated in the joint K'G', which
(Theor. III.) is elliptical, since the greater segment, K'O', of
the subtangent, O'o, is exterior to the curve. Bisecting the
chord G'K', and joining its middle point, n', with d ! , the inter-
section of the tangents at G' and K', we have, by a property of
the ellipse, d'n' as a diameter ; which therefore meets the axis,
O'A' (and the opposite symmetrical diameter), at X, the centre
of the curve ; whence the shorter axis can be found from the
property of the subnormal, oS, expressed by the proportion,
Xo : oS : : a 2 : 5 2 ,
where a is the semi-transverse axis, XK' ; and 5, which equals
the semi-conjugate axis, = AB, Fig. 66, is found by the usual
construction from elementary geometry.
Examining the joints M'L' and P'Q', we find M'0'>* i'h',
but P'O' < PV. Hence, by Theor. III., the joint %1'V
should be elliptical, and P'Q' hyperbolic ; but the differt *ces,
M'O' — M'A' and P'O' — PV, are so small, that they are here
made with sufficient accuracy as circular arcs, whose centres
are on O'X.
The horizontal projections of the joints are found by pro-
jecting down their intersections with the contours of the sur-
face, made by the vertical planes, pa and /A, as is fully shown
for the joint K'I'H'G' ; whose horizontal projection is KIHG.
II. The Directing Instruments. — Most of these can be
sufficiently indicated by a description of the most irregular
stone of the structure; that whose vertical projection is
RYy F'K'G', and which is more clearly exhibited in the oblique
projection, Fig. 67, like points having like letters in both
figures.
The many surfaces of this stone are : —
STONE-CUTTING. 109
1°. The vertical rectangular plane side, Yz/UY'.
2°. The vertical plane back, Y'Uk.
3°. The vertical plane front, GRY?/Z.
4°. The horizontal plane base, ~UkK"c"zZy.
5°, 6°. The horizontal plane top, RYR'Y' ; and small hori-
zontal plane surface, Kcc'K'.
7°, 8°, 9°, 10°. The four minor vertical plane faces, K'K"&,
K'K' W, czc", and AZz, respectively, in the portal, gate recess,
and jamb.
11°. The oblique plane surface, GRG'R'.
12°. The elliptic cylindrical surface, G'k'KRG. Fig. 68, is
the development of the like surface on M'L', joined with the
plane portion on L/J.
13°. The double-curved surface, AcKHG.
The last surface being non-developable, no pattern of it can
be made, but templets fitted to any of its vertical sections,
parallel to C"e", or to f'b', or to its horizontal sections, can be
made.
These templets, with patterns, easily made, of the other sur-
faces, and the square and straight edge, will be ample guides in
working this stone.
III. Application. — First form the surface, YY'Uy, it being
the largest and simplest ; next the back ; and then the base
and front, and all the other plane surfaces, each of which is
square with one or more of the others.
The cylindrical surface, GG'Kk', may then be wrought
square with the back upon G'k', as a given edge, or directrix,
previously found by the pattern of the back. Or it may be
wrought by templets fitted to the profile, R'G'k'. Its edges
may then be scored on the stone by a pattern corresponding
to that of the cylindrical joint on M'L/, shown in Fig. 68.
These operations will give all the bounding edges of the
one remaining surface, which is double-curved. After approx-
imately hewing out this portion of each of the stones, they can
be accurately put in place, since all the other surfaces of each
will have been previously completed. The total double-curved
surface of the recess can then be wrought at once, by means of
the templets, last described in the list of guiding instruments.
Examples. — 1°. Construct the figure with two centred joints in the front
elevation.
110 STEEEOTOMY.
2°. Make an isometrical or an oblique projection showing the under side of
the stone shown in Fig. 67.
3°. Make like projections of the stone M'JTR.
Problem XIX.
An oblique lunette in a spherical dome.
I. The Projections. — A lunette is formed by the intersection
of two arched spaces, both of stone, and of unequal heights, so
that the groin curves will be of double curvature.
1°. Arrangement of projections. — PL X., Fig. 69. These
are a plan, and two elevations, on two vertical planes, V an d Vu
at right angles to each other, and whose ground lines are respec-
tively O'X and 0"X. As in all similar cases, the projections
of any point on V and on Vi> will then be at equal heights
above O'X and 0"X.
Given parts and dimensions. — In the plan, the circles, OA
of 11 ft. radius, and OH of 13' : 6" radius, are the horizontal
traces of the interior and exterior surfaces of the dome. The
former is a hemisphere ; the latter, partly cylindrical, as indi-
cated in the section shown on the plane 0"X, is there gener-
ated by H'H", 6': 10" high. The radius, D"x, of the extrados,
is 16 ft., where # is 3 ft. below the centre, O", of the intrados.
The elevation on O'X shows a right section of the arch, its
inner radius 4 ft., its outer one 8 ft., its thickness at the
crown 1' : 6" ; and the perpendicular distance of its axis, o x o',
from the diameter, HO, 6' : 3".
From these data all the remaining constructions are made.
2°. The groin. — Any horizontal plane will cut a horizontal
circle from the sphere, and two elements from the arch, which
will meet that circle in two points of the groin. Thus, the
plane, a' l m(m 1 v'), cuts from the sphere the circle of radius
Oa x (= vy) and from the arch the two elements, of which one
at a[, being projected on H, intersects circle Oa x at a x , as shown,
and thence gives its side elevation a x on vy. Other points
being found in the same manner, give the groin curve* aca 5 —
a'c'a' 5 — a"c"a's.
3°. The horizontal projection of the groin is an arc of a
parabola. — To prove this, refer the intrados of the sphere and
cylinder to the three rectangular coordinate axes : OH l5 as the
axis of X ; OH, as the axis of Y ; and the vertical at O, as
STONE-CUTTING. Ill
the axis of Z. Then, neglecting the usual negative sign of or-
dinates to the left of the origin, O, as not relating to the form
of the line sought, we have for the point a^f, for example,
(OA) 2 + (ha,) 2 + (A'O 2 = R 2
where R = the radius, OA, of the sphere.
That is z 2 + # 2 -f-z 2 = R 2 . (1)
And as the like is' true for every point of the sphere, (1) is
called the equation of the sphere, referred to its centre.
Again, (V A') 2 + (h'a'^f = (Vai) 2 = r 2 .
That is, calling O'o'—a,
(a — xy + z 2 = r*. (2)
and as the like is true for every point of the cylinder, this is
called the equation of the cylinder for the given axes of
reference.
Now that points, as a x a' v may be common to both surfaces,
and hence be points of their intersection, the x, y, and z of (1)
and (2) must be the same. That is, (1) and (2) will both be
true at once for the same point, so that we can substitute any
term in one for the like term in the other.
Then, from (1) , y 1 — R 2 — (z 2 + z 2 )
and from (2), (x -f- 2 2 ) = (r 2 -J- 2a# — a 2 )
whence, y 2 = — lax -f- (R 2 — r 2 -f- a ) (3)
which, since z is eliminated, is the equation of the curve aa 5 , in
the plane XY. Also, the term in the parenthesis is constant,
being made up of constants, and as a is a part of it, it may be
written 2wa, and (3) then becomes
y 2 = — 2ax -f- 2na = — 2a(x — ri) (4).
If now we shift the origin O to the left, on the axis of X, so
as to make x = x-\-n i
(4) will become, y 2 = — 2ax (5).
Restoring now the neglected sign of x, we finally have
y 2 = 2ax (6)
the usual form of the equation of a parabola lying, as a 5 a does,
to the right of its vertex taken as the origin. The curve aa 5 is,
therefore the arc of a parabola, of which HO is the axis.
4°. Joint-lines and surfaces of the sphere. — The coursing
joints on that part of the sphere which is independent of
the lunette, are horizontal circles, IQ — I"Q", GR — G"R",
112 STEREOTOMY.
etc., found by dividing the meridian of radius 0"A" into an odd
number of equal parts, — here eleven. The broken joints,
RQ — R"Q", etc., are arcs of meridians.
The beds of the dome voussoirs are the conical surfaces, as
P"Q"I" J", having the centre, 00", of the sphere for a com-
mon vertex, and intersecting the spherical surfaces in the
horizontal circles, as I"Q" and J"P'\
5°. Radial joint-surfaces of the lunette. — These are wholly
plane, and their edges are the intersections of these planes
with the several surfaces of the dome and arch.
Divide the arc a'c'a\ so that a[ shall he lower than E", the
corresponding first one from A", of the eleven equal divis-
ions, on the dome section ; here, into five equal parts. The
reason for this will soon appear.
The lunette joints in the spherical intrados. — -lb 2 is the trace
on V of the plane, R%, of the horizontal circle GR — G"R".
This plane cuts the plane of the joint o'd 2 in a horizontal line
at b' 2 , which, by projection, gives b 2 , and thence b 2 . Hence,
a 2 b 2 — a' 2 b 2 — a 2 b'2 is one oi these joints, showing that a x must
be lower than E", in order that there should be such a joint.
The others are found in the same way.
6°. The lunette joints in the conical beds of the dome. — One
of these is the intersection of the plane o'd 2 , with the conical
bed, C"G"R"R'". To find it, draw qd' 2 , at the height of C" qi ,
to give qd 2 , the trace of the horizontal plane of Cd 2 — C"q v
upon V» As before, this plane cuts from the plane o'd 2 a per-
pendicular to V at d' 2 , which, in horizontal projection, gives d 2 ,
on the horizontal projection, Cd 2 (C being projected from C"),
of the circle considered ; and thence d 2 . The joint sought is
evidently a hyperbola, it being the intersection of the ' plane
o x o ] d' 2 with the cone whose axis is the vertical at O, and whose
slant is that of G"0". Hence, make p'O'o' = G"0"A", and
Op — O'p' is that element whose intersection with the plane,
0]p'd 2 , is the vertex, p'p, of this hyperbola; whose horizontal
projection pb 2 d 2 can now be more accurately drawn than with-
out the aid of the vertex p. Finally, make O"/?"— the height
of p', and p"b 2 d 2 is the vertical projection of the same hyper-
bolic joint, of which only b 2 d? — b' 2 d 2 — b 2 d 2 is real.
Any other hyperbolic joints are found in the same way.
Other lunette lines. — These are, for the same joint plane
o'd' 2 , the circular arc d 2 e — d 2 e' — d 2 e", on the spherical extrados •
pi.vir
D r'-4*_
STONE-CUTTING. 113
ef 2 — e', on the horizontal ledge, generated by D"H" ; f 2 g 2 —
e'g' 2 on the cylindrical back of the dome ; g 2 u x — 9i-> on the
extrados of the arch; u x u — g 2 d 2 , a radial edge in the arch;
and ua 2 — d 2 , on the intrados of the arch.
' The large diameter of the arch, as compared with the radius
OA x , carries the point tt't" nearly out of the quadrant, OAA x ,
unless, as shown at ft", it be taken lower than the correspond-
ing point bb\.
II. The directing Instruments. — These, besides Nos. 1 and
2, are patterns of all the plane, cylindrical, and conical surfaces
of voussoirs, with certain bevels, as follows, taking for illustra-
tions the stone between o'd' 2 and o'g{ of the lunette, and the
stone R"Q"S"T", of the dome.
No. 3 shows the real form, a\g']a' 2 g 2 , of the plane end of the
lunette stone, which is in the plane V.
Nos. 4, 5, and 6, Fig. 70, are patterns of the intrados (No.
5) of the same stone, and of the two radial plane joints when
folded into the paper. Their construction is obvious, since like
points have like letters with Fig. 69, and are found by ordi-
nates from the vertical plane end, No. 3, in the plane V-
Useful bevels (not shown), would be No. 7, giving the angle
h"WW; and Nos. 8 and 9, giving the positions of the plane
beds on d 2 d 2 and d^g\, relative to the intrados d x d 2 .
No. 10, the pattern of that plane end, MNW, of this stone,
which is in the dome, is G"C"D"H"&"B"E".
From MN to a 2 b 2 is a spherical zone.
From N5 to b x d x is a conical zone, No. 11.
From 5W in the plane Wk" to d x f x , is a horizontal plane
surface, No. 12.
From the vertical line, W — k"H", extends the vertical cyl-
indrical back, No. 13, of the dome, intersected by those sur-
faces of the lunette which are parallel to its axis o'o^
The three remaining surfaces of the part of the stone in the
dome, are the plane annular portion, No. 14, generated by
D"H"; the spherical portion generated by D"C", and a conical
portion, No. 15, generated by C"G"; all starting from the
plane OW, and all limited at their intersections with that
portion of the stone which is in the arch.
Patterns of these surfaces, so far as developable, may readily
114 STEREOTOMY.
be made ; also bevels, conveniently giving the position of their
horizontal edges, relative to the end in the plane OW.
Thus this very irregular stone has thirteen faces, plane,
cylindrical, conical, and spherical.
To gain as full an idea of it as drawings alone can give, com-
plete its projection on V ; and make two or more isometrical,
or oblique projections of it.
For the proposed stone of the dome, the pattern, No. 16,
C"G"I"J", of its vertical plane end will be needed ; and those
of its conical beds, as P"Q"V"S", Nos. 17 and 18.
No. 17, for example, Fig. 71, is the development of
R"R" / U' / T", found by describing the arcs from O, with radii
equal to 0"G" and 6"C", and by making R"T" = RT from
the plan.
Finally, bevels like No. 19, will be useful, giving the rela-
tive positions of elements of the conical beds, and great circles
of either the intrados or extrados of the dome. And a tem-
plet, No. 20, should be cut to an arc of a great circle of the
spherical intrados. \
III. Application. — For a stone as irregular as that of the
lunette, the method by squaring (105) is preferable, if not in-
dispensable. Then form a right prism, the pattern of whose
base shall be the horizontal projection, ^^WM, of this
stone ; and upon whose rear and lateral faces the two plane
heads can be marked by Nos. 3 and 16.
Next, the intrados and plane joints of the arch portion of the
stone can readily be made square with the back by No. 2, and
formed by Nos. 4, 5, and 6.
The plane, W/^5, is readily made ; square with the end
on MW, and marked by No. 12 ; the cylindrical back, square
with the last surface, and marked by No. 13 ; and the spher-
ical surface, MNa 1 aJ> 2 , by Nos. 19 and 20.
Pendentives.
127. In connection with domes, the related subject of square
areas, covered by spherical surfaces, may be noticed ; though
detailed figures must be omitted for want of room. PI. X., Fig.
72, shows a skeleton sketch of such a design, which is some-
times adopted on account of the stately appearance of a dome-
like ceiling. Here let ABCD be the half of a square floor, of
STONE-CUTTING. 115
which the circumscribed circle, of radius OB, is the base of a
hemisphere. The four walls of the room will then be bounded
by vertical small semicircles, as BC — A'Q'D' ; the ceiling
F'H'E', within the circle of radius OA will be a spherical seg-
ment ; and the four areas like ABI will be covered by spherical
gores, shown more clearly at A"B"I", in the elevation made on
a vertical plane, whose ground line is mq, perpendicular to the
diagonal BO.
128. Two joint-systems. — The joints of the spherical surface
may then be either (a) horizontal small circles, and vertical
meridians; or (F) vertical small circles, and meridians, all
having BO — B" for a common diameter ; the beds bounded by
the small circle joints being conical in both cases.
Examples. — 1°. Make figure 69, on a scale of ^L, or larger, and with the arch
smaller in proportion.
2°. The same with the two elevations side by side.
3°. The same, with the axis of the arch coinciding with a horizontal diameter
of the dome.
4°. Complete the projection of the dome on V-
5. Construct the dome with pendentives, Eig. 72, in detail on a large scale, and
by each joint-system.
SPIRALS.
129. A few observations on the spirals found in the next
problem are here added, as they may not be conveniently acces-
sible elsewhere.
A SPIRAL is a plane curve, generated by a point which has
two simultaneous motions, or, more precisely, whose actual
motion can be revolved into two components ; one, a rotary mo-
tion, around a central point called the pole ; the other, a radial
motion, outward from the pole.
130. Illustrations. The spiral of Archimedes. — This is the
simplest of all the spirals ; since each of the component motions
Fig. 11.
116 STEREOTOMY.
of the generatrix is uniform. Thus in Fig. 11, let O be the
pole, and OA, the initial line, so-called, on which the successive
equal increments of the radial movement are laid off. Then
divide any circle, having O for its centre, into equal parts, as at
1, 2, 3, etc., and make 0^ = 06; 0<?j = Oc; 0^ = A, etc.,
and Oa^tf! tangent to OA at O, will be a spiral of
Archimedes. The distance of any point of the curve from the
pole is called its radius vector.
In this example the circle is divided in 16 equal parts ; hence,
a circle of a radius, which we will call 0A X , comprising 16 of the
parts of OA, from O, would be divided into the same number
of parts as OA^ As OA and any fractional part of the circle
of radius OA may be divided into the same number of equal
parts there may be an infinite variety of spirals of Archimedes.
Let the — th part of circle OA be divided into the same
n A
number of parts as the radius OA. Then, calling the radius
vector = r ; OA = a, and the arc, as A3, corresponding to any
radius vector, as Oc, = 0, we have, directly from the definition,
r :
r.
\a
n
a6
.2ira;
.2-rra
m
.2tt
whence r = = CI)
which is the general equation of the spiral in the form most
convenient for use in drawing tangents to it by the method of
resultants.
131. Tangent to the spiral of Archimedes. Differentiating (1)
-r = - 2tt ; where -r-, or is the ratio of the rotary and
dr n dr n *
the radial components of the motion of the generatrix, the
former being referred to the circumference of the circle whose
radius is a. The application will be better understood by an
example.
As we may generally make m = 1, write at once -r = — .
Then let it be required to construct the tangent at 5|, in Fig.
12. Here, n = 4, since i of the circle OA is divided into the
same number of parts as are found on the line OA. Then lay
off on Fbi produced, b Y n = 4, on any convenient scale ; and
2p = 2tt, by the same scale, on the tangent at 2 to the circle O A ;
reduce the rotary component as estimated with the radius P2,
STONE-CUTTING.
117
to its actual value, byS, parallel to 2p, at b v by drawing the
radius Pp. Then, b x t, the diagonal of the parallelogram on the
components, bin and b^s, is the required tangent at b x .
n 1-
Fig. 12.
If n = 1, PcZ, and the circle of radius Fd will be divided into
the same number of equal parts, -5- = -j-' and 2p would be
4 times 2p, or ^w would have been called 1 instead of 4.
132. The subnormal method. — Draw PQ perpendicular to
P5 1? and limited at Q by the normal ^Q. Then PQ is the sub-
normal. Now the triangles PQ5i and b^it are similar, and give,
whence PQ = b,n X -r-*-
But if b x n is made constant for each point, *2p will be so also ;
and hence, as we see from the figure, b y s will vary as PZ^ ; that
Vb
is, the ratio ~ will be constant. Thus PQ is constant. Hence,
having any one tangent, any other can be found as follows.
Take, for example, the point e x . Draw a perpendicular, PQx,
not shown, to its radius vector Pg 2 ; at P, and equal to PQ.
Then Q^ will be the normal at e u where the tangent will then
be perpendicular to Q^j.
133. The tangentoid spiral. — This is one of a series of
curves known as the trigonometrical spirals, in each of which
the radius vector, r, is some trigonometrical function of the
angle, (9, between it and the fixed initial line.
118
STEREOTOMY.
The tangentoid spiral is that in which the increments of the
radius vectors are equal, or proportional, to the increments of
the vectorial angle, 6.
a-*
»/
.- :: i, - -
a, "■
-P- b a /
-
a-a
p
Fig. 13.
Thus, let P, Fig. 13, be the pole, and Pa = a, the initial line.
Divide the circle Pa at pleasure as at 1, 2, etc., then on P&, for
example, make r == PB = ah = Pa X the tangent of BPa,
and B will be a point of a tangentoid spiral, whose equation,
simply expressing the construction, is, r = a. tan 6 ; or, if a = 1,
then r = tan 6 ; that is, r is equal to the tangent of 0. But if,
having drawn either a-J) x , or a 2 b 2 parallel to ah, we should make
PB X (not shown), = a-}> x ; or, PB 2 = a 2 b 2 , we should find new
forms of the spiral ; where if Pa x == a x and Pa 2 = a 2 , then-
equations would be r = a v tan 6 ; and r = a 2 . tan 0, and r
would be proportional to tan 0.
134. Initial line a secant. — In Fig. 13, the initial line, Pa,
is evidently tangent to the spiral at the pole, P. This is not
always so.
K
\.
a
c h
/ \
\ a.
E^
~p r \^
«<j/'
/'' ':
,-;/^H
*K
/.--—
c i /' !
/
/
rkn,
i ,-"'
d \i 4
In,/
\
M A' /
/ /k
/Ii
iK\
A\\
/ /*'/
Fig. 14.
STONE-CUTTING. 119
Thus, Fig. 14, let Ea and EP include any angle whatever,
and let them be divided proportionally by parallels as Aa, Bb
etc., including PL. Then arcs from A, B, etc., will meet the
corresponding radials, Pa, P5, etc., in points a x ,b t , etc., of a
curve which will still be a tangentoid spiral. For the incre-
ments, CD, CB, etc., of the radius vector, r, estimated from the
circle Cc l5 where Pc x c is perpendicular to Ea, are proportional
to the increments, cd, cb, etc., of the tangents of 6, where 8 is
estimated from Pc.
Evidently P is the pole, and PL the tangent at P.
Then, in order to write r = tan 6 ; make ML = Pc : ; and
take 6 = MPL. Then ML = tan 6 ; and MP = r = 1.
Also the tangent increment Mm = the radial increment
cji = CA, where Pro is the radius vector drawn through the
point a v Likewise, Mn = CE, etc.
dv
135. The tangent line. — From r = tan 6, -75-= sec 2 6 =
sec 2
l 2 '
Now at «!, for example, Pra = sec 8, and PM = 1. Then
a third proportional (x) to PM, and Pm, will be the longer
side of a rectangle ; equal to (Pm) 2 , and whose other side equals
PM. ThatisPMxz=(Pra) 2 - Butthe figures VMXx and PM 2
(= l 2 ) having the common altitude PM, are to each other as
x and PM. That is, Jq = -^~ = (m2) = ^^ = _ .
Hence, make ^H = the 3d proportional, x, for the radial
component of the motion of the generatrix ; and M& = MP = 1
for the rotatory component, referred to the line on which tan 6 is
estimated, and which will be reduced by the line Vk to its
value, M l5 for the circle of radius Ya v Then, making HK =
hk u K«j is the required tangent at a x .
We now close with the following, which, besides its interest
as a structure, embraces, as appropriate for a final problem,
representatives of all the four classes of surfaces which form
the main divisions of descriptive geometry, and of its applica-
tion in this volume.
120 STEEEOTOMY.
Problem XX.
The annular and radiant groined arch.
136. Supposed conditions. — Suppose that a building for a
private library or cabinet, or, if of suitable dimensions, for a
locomotive engine bouse, is to consist of a gallery, annular in
plan, and inclosing a central circular area. Also let the
gallery be divided into seven compartments by arches, whose
straight elements shall radiate from a vertical line at the centre
of the central area.
I. The Projections. 1°. — Let a vertical line at O, Plate X.,
Fig. 73, be the axis of four concentric vertical cylinders gener-
ated by the revolution about this axis of vertical lines at A,
a, 5, and B, where OA is 27 ft. 6 in. ; OB, 12 ft. 6 in. ; AB
15 feet ; and ah 9 feet, making the thickness of the walls Aa
and B5, 3 feet.
Let the circular gallery between the walls be covered by an
arch, whose intrados is the half annular torus generated by the
revolution of the vertical semicircle aob, of 4 ft. 6 in. radius,
shown at ac 2 b on the plane V- Let this gallery be divided into
seven equal compartments, one of which is bounded by the arc
AD, of 24.68 ft., and the radials, OA and OD.
This arc, AD, may be accurately determined by laying off
any suitable fractional part, as — of it, n times. Or, from a
table of sides of inscribed regular polygons, we can find the
length (here 23.86 ft.) of the chord equal to one side of the
regular polygon (here a heptagon), indicated by the number
of compartments.
Setting off each way from A, D, etc., 3 ft. on the outer cir-
cumference, as at AE and AF, and drawing the radii, as OE
and OF, we have the piers, as shown at Ee/F, and HhH ; and
the area GFIJ which is to be covered by the radiant arch. This
arch is here represented as closed at its ends by twelve inch
walls ; which, with the piers, form alcoves.
2°. The intrados of the radiant arch will, in any case, natur-
ally be a right conoid (114), extending from OF to OG, and
whose springing plane will be the same as that of the annular
arch; viz., the horizontal plane containing the diameter ab.
The conoid will then be generated by the straight line OF,
Pi.vnr
STONE-CUTTING. 121
moving so as to be parallel to the springing plane as a plane
director, while moving upon the vertical line at O, and some
curve o£ a height equal to oc 2 , and included symmetrically be-
tween OF and OG.
3°. Two systems. — At this point, there is a choice between
the two methods, one or the other of which would be most
naturally chosen.
First. The curved directrix may be the ellipse whose trans-
verse axis is the chord of any arc, as oo Y , or GF, included be-
tween OG and OF, and whose semi- conjugate axis is a verti-
cal, equal to oc 2 at the middle point of such chord.
Second. The curved directrix, may, instead, be the curve of
double curvature, formed by wrapping upon some of the vertical
cylinders, as DGA, a semi-ellipse whose transverse axis is
the true length of the corresponding arc, as GF, and whose
semi-conjugate axis equals oc 2 . as before.
4°. Adopting the second system. — KGiK^ is the curved
directrix of the conoidal intrados ; where KG^ tangent to
GKF at K, equals the arc KG, and KK X perpendicular to KGi
equals oc 2 . The method of concentric circles on the given
axes, is adopted, as shown, for constructing this eclipse, on ac-
count of its convenience in affording, at the same time, an easy
construction of any desired tangents, as also shown.
5°. The groin curves. — These are found by auxiliary hori-
zontal planes, taken for convenience though the inner extremi-
ties a 1? etc., of the radial joints of the annular arch, where ac 2 b
is divided here into five equal parts. Each plane will then
cut two horizontal circles, centred at O, from the torus, and
two elements from the conoid, whose intersections with the
circles will be four points of the groin curves. Thus g, f, e, /,
are the points determined by the springing plane ; c l the apex
of the groin, is the intersection of the circle Oo with the
element KO. Then, for example, making b", at the height,
b'{x 2 , = b x q ; and Kx x , = Kx 2 = KM, the two circles with
radii, Op and Oq, will intersect the corresponding elements
Ox x and OM, at the four points, I, Z x , m, and m x of the groin
curves.
6°. Nature of the horizontal projections of the groin curves. —
Each of those just found is an arc of a spiral of Archimedes
(130), for, from the properties of ellipses having an axis in
each equal (oc 2 = KK X ) we have,
122 STEREOTOMY.
oa 3 : op :: KX 2 : Kx 2 ;
or, by the substitution of equal terms,
n 2 n : m 2 m :: KN : KM.
That is, the increments, n 2 n and m 2 m, of the radius vectors,
are proportional to the corresponding increments, KN and KM
of the arc which marks their angular movement.
The pole and the initial line. — O, the given intersection of
the radius vectors, is the pole ; and, knowing the character of
the curve, find a fourth proportional to^ (= «6) GF (= 2G X K)
and JO (== JO) and lay it off from G to the left, or from F to
the right, on the circle OK, and points will be found, where
the radii to O will be the initial lines of the spirals, fc x j and
gc x i, respectively ; and hence, tangent to them at O.
The portions of these spirals beyond ^ and/", and within i
and j, as at jCO, being projections of no actual lines of the
structure, are called parasites.
7°. The tangentoid system. — Turning to this (3°) for a mo-
ment, for comparison, suppose that the vertical ellipse, of trans-
verse axis, gf, and vertical semi-conjugate axis at o 2 , = oe 2 , had
been chosen as the curved directrix of the conoid. Then, as o x
and o 2 are at equal heights, lines parallel to o x o 2 , as y x y 2 , would
give elements ; circular, with radius Oy x , through y x on the
torus, and straight at y 2 on the conoid, which would meet as
at y, a point of the curve of intersection of the torus and the
new conoid; a curve whose horizontal projection, gyc x , etc.,
would be an arc of a tangentoid spiral (133). For evidently
the increments, as o x y x , y x g, etc., of the radius vectors, are pro-
portional to the increments, as o 2 y 2 , y 2 g, etc., of the tangents of
the angles made by these radius vectors with Oo 2 as an initial
line (134).
8°. Section and joints of the annular arch. — Let the figure
AujVxWfrB be the section of the annular arch. By revolution
about the vertical axis at O, this figure, being in the vertical
plane on OA, will generate the volume of masonry, covering
the annular arch ; and its radial lines, as a x v x , will generate the
coursing surfaces, or beds, of the voussoirs. These beds will
obviously be conical surfaces, as at Wna^v, all having the ver-
tical line at O for their common axis. The heads, or transverse
joints, as at YZ, are in vertical planes through O.
9°. The joints of the conoidal arch. — The bed surfaces
STONE-CUTTING. 123
should, strictly, be normal to the conoidal inti-ados, along the
elements, as MM l5 which are the coursing joints. They will
therefore be hyperbolic -paraboloids (117).
But at the highest element, the conoidal surface is develop-
able along one element, KO, since a horizontal tangent plane
will evidently there be tangent all along that element. Hence,
the normal surface on KO will also be a plane, and the normal
surfaces on elements near KO will therefore be very nearly
plane. Hence, the bed surface on NN X may properly be plane.
10°. Construction of the warped bed on MM X . — Assume at
V2 a vertical plane of projection, L/T', perpendicular to MM 1 ;
on which MMj is therefore projected in the point m! , at the
height \Jm' = pa x = b x x 2 on Vi- Drawing the tangent b'{T', as
shown, make MT, tangent at M, equal to x 2 T", and by project-
ing T at T 7 , m'T will be its vertical projection. Then (116)
TO will be an element of one generation, and the horizontal
trace of the tangent hyperbolic paraboloid, generated by the
motion, parallel to H, of MO, upon MT — m'T and O — L'M'.
Hence, mt, m x u, MiU, and Ss, are elements of the other gener-
ation of the same tangent surface ; and m't' , m'u', etc., are their
vertical projections.
Taking, now, a horizontal plane, M's' x , at a height, L'M',
equal to v x q x ; perpendiculars m'W, mV, etc., to m'T, m't', etc.,
and limited by M.'s{, will be those elements of the normal hy-
perbolic paraboloid, which are the lines MT — m'T, etc., re-
volved 90° about MMx — m' as an axis (117). Hence, project
R' on MT at R ; r' on mt at r, etc., and MmM^rR will be
the horizontal projection of the required normal bed on MM L .
The actual limits of this bed are at its intersections mp 2 Q,
and w?iQi, with the corresponding conical beds, Qq^m, and
QiW?^, of the annular arch. These are found as follows : As-
same any intermediate point P on the joint a x v x , maker'V at the
height from L'T', equal to Fp x , and project r" at r'" on MT;
2' at z, on mt ; s" at s'", etc., and r'"zs'" will be the intersection
of the normal joint on MM 1? with the plane r"s" ; limited at p 2
by p\p 2 , the corresponding horizontal section of the torus.
Therefore, MRQp 2 m is a definite normal joint. M. x m l Q^k l is
another.
11°. The outer edge, RR l5 of the tvarped bed on MM y ' is a hy-
perbola. — This we know from the properties of the normal sur-
face, since any plane which cuts the intersection of the two
124 STEREOTOMY.
plane directors cuts the surface in a hyperbola ; and as the tan-
gent hyperbolic paraboloid on MM X , revolves 90° on MM X to
become the normal one, its former plane director, of the gener-
ation MM l5 which is H? becomes a vertical plane, parallel to
MMj. The other plane director is perpendicular to MM X . The
intersection of the two is thence a vertical line, which accord-
ingly cuts the horizontal plane M'si, containing the curve RR l5
which is therefore a hyperbola.
Otherwise ; by direct demonstration. The triangles T'L'm'
and m'M'R', and other like pairs, are similar, and give
L'T' : Wm r :: L'm' : M'R'
Also, L't' : Wm' : : L'm' : Mr'
whence L'T' X M'B/= L't' X M¥.
That is, MT X MR = mt X mr.
Substituting for MT and mt, the proportionals to them, MO
and mO,we have
MO X MR = raO X mr ;
which may be written, xy = x'y' ;
if the curve be referred to OM, and a perpendicular, 00 2 , to it
at O, as axes.
At the point equidistant from the axes,
x' = y', and putting x'y' = k 2 , we have xy = Je 2 ; which is the
equation of the hyperbola, referred to its asymptotes.
At the point | l5 for example, x= OS, y=. S?i.
The mean proportional to these is Sl=&; whence 2, the
vertex, is found, as shown, on the transverse axis, 02, of the
curve, which bisects the angle, SOO x , between the asymptotes.
This being here a right angle, the hyperbola is of the form
called equilateral.
12°. Construction of an approximate plane-normal joint, on
NNj. — The point N is symmetrical with Xj, hence the tangent
at X x is symmetrical with that at N. Then draw the tangent
b'ik, Make Vi tangent to the cylinder DKA at X 15 then Xja:
(== X 2 K) will be the horizontal projection of this tangent.
But b%k being straight, and oblique to the elements, as Kk of
the cylinder, will, when wrapped upon the cylinder DKA, be
transformed into a helix, whose projection, and that of its tan-
gent, on Vi, will in the vicinity of k, be sensibly b'^k produced.
Hence, -make arc Xjz" =X 1 a;, project x" on b'{h produced, and
x"~k x , will be the true height at which the tangents at X^^)
and at N, will pierce the plane OK.
STONE-CUTTING. 125
This found, take x'v' for the ground line of a new vertical
plane V3 ; project N upon it at the height a 3 n' (= a 3 a 2 ), and x
at ki at the height x"k x ; when n' Tc x will be the element at Nw/ of
the tangent hyperbolic paraboloid on NN X ; and n'V perpendic-
ular to it, an element of the normal hyperbolic paraboloid on
NN l5 where v'Y' =vY. Hence, projecting V at N 2 (on the
tangent at N) N 2 N", parallel to NN l5 will be the trace, on the
horizontal plane V'V" of the top of the annular arch, of the
tangent plane at ~Nn' to the normal hyperbolic paraboloid ;
that is, of the approximate plane joint.
In fact, two such planes should have been similarly made ;
one tangent at the middle point of Nw, for the bed NN 2 N'w ;
and one at the middle point of N^j, for the bed N^N".
II. The Directing Instruments. — After the previous thorough
working up of all the lines and surfaces composing the required
structure, these can now be summarily described.
Besides Nos. 1 and 2 (7), there should be, for the stone
"S^'Wna^qi, for example, —
No. 3. The pattern l$"''N'vq l , of the horizontal plane top.
No. 4. The pattern of the cylindrical back, which, to the
the right of Qt, is temporary. In No. 4, q 3 Q = q^N ,N ; J 4 6 ==
n{V', both in length and direction; 5 4 4 = X 3 5 2 '; 4 5, and 5b" s
are respectively equal to the like spaces on the plan ; and
q s q i = V^i- No. 5 is a pattern of the end, W 1 v 1 a 1 a 2 , in the arch.
No. 6 is the development of the conical bed, Qq^pm. The
vertex of this cone is found after revolution, at O', the inter-
section of 00' perpendicular to OA, with v x a x produced. Hence
with O' as a centre draw a 1 m s =pm, and v 1 Q 1 = ^ 1 Q ; likewise
VP l =pip 2 , and Q ll v l a l m 3 will be the required pattern.
No. 7 is the pattern found in the same way as was No. 6, of
the conical bed Wva 3 n.
No. 8, which could exist only for a plane bed, is the pattern
N 5 N 4 4w, of the plane joint N'"N'w4 ; hence N 4 4 = Y'n'.
Nos. 9 and 10, bevels giving respectively the angles V 1 V« 2 ,
and Ya 2 a^ will be useful in determining the relative position of
the top, and the conical bed Wva z n ; and of this bed and the
annular intrados ; both 9 and 10, being held in meridian planes
of the torus.
No. 11, set to the angle n'Y ! n", will, in like manner, serve
to fix the proper position of N'N'"4w, relative to the plane top.
126 STEBEOTOMY.
III. Application. — Choosing a sufficient block, bring it pro-
visionally to the form of a prism, whose base is na 3 q 2 4:, except
that the vertical cylindrical face on na need not be wrought.
Then, mark the intended top by No. 3, and the back by No.
4. The lines thus given, with the form of the end given by No.
5, will guide the cutting out of the portion Q,tqiq 2 .
The lower conical bed can then be directed from the end,
and from the vertical surface on V x Vi, and shaped by No. 6.
The upper conical bed will then be determined by No. 9 and
No. 7.
The remaining work is obvious enough, the construction of
No. 4 showing the ordinates from the base of the provisional
prism in the plane 45 3 to the edges w4 and na z .
137. St. Gales' Screw. — This term is applied to circular
stairs with solid central core, or column, where the surface seen
overhead in ascending the stairs is a double-curved helicoidal
surface, such as would be generated by the section, AaV'BJ, of
an annular arch, PI. X., Fig. 73, were it to move so that every
point of it should describe a helix about the vertical axis at O ;
the course of stones generated by ~Qbb x v z being solid with the
core. The coursing surfaces of such an arch would be oblique
helicoids, generated by a x v^ etc. ; and the method of cutting
of the stone may be sufficiently understood from the cutting of
the voussoirs of the oblique arch.
Examples. — 1°. Make the full construction of Fig. 73 on the tangentoid system.
2°. Make drawings of a key-stone, extending both ways from a in each arch.
3°. Also of an inner pier groin stone, as Mi/niQi.
4°. Construct the plane joints indicated in 12°.
5°. Make the necessary illustrative drawings of the double-curved arched cover-
ing of circular stairs, known as the St. Giles' screw (137). The coursing joints are
oblique helicoids, generated by radii of the semicircle ac^b, Fig. 73. The heading
joints are in planes perpendicular to the helix generated by the highest point, &i,
of the same semicircle. The coursing joints thus generated are not quite normal
to the double-curved arch surface, but are nearly enough so. To be perfectly so,
they should be generated by normals to that intrados, that is, by lines, not only
perpendicular to the tangents to the semicircle, as at its points of division, etc.,
but also to the tangents to the helices at the same points. The construction would
be too laborious, and thence more apt to be inexact.
PI. IX
M| - v ' t: t
if !
V
B' Y'
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