V
APPLETONS
CYCLOPAEDIA OF DRAWING,
DESIGNED AS
E KK A T A.
In " Kule and Examples," pp. 223 and 224, rule should be :
" Multiply one-half the weight of the rafter and the weight distributed
on it by one-half the span, and divide the product by the pitch."
In example, weight, instead of " 8,500 Ibs.," should be " 4,250 Ibs.,"
and result " 0.708 square inches."
P. 224, instead of "40x31x10=12,400 Ibs.," should be "40x35x
10=14,000," and corresponding changes in rest of calculations.
NEW AND ENLARGED ED
YOKK:
D. APPLETON AND COMPANY,
90, 92 & 94 GRAND STREET.
1869.
APPLETONS'
CYCLOPEDIA OF DRAWING,
DESIGNED AS
A TEXT-BOOK
MECHANIC, ARCHITECT, ENGINEER, AND SURVEYOR,
COMPRISING
GEOMETRICAL PROJECTION, MECHANICAL, ARCHITECTURAL, AND TOPOGRAPHICAL
DRAWING, PERSPECTIVE AND ISOMETRY.
EDITED BY
W. E. WORTHED.
NEW AND ENLARGED EDITION,
NEW YOKK:
D. APPLETON AND COMPANY,
90, 92 & 94 GRAND STREET.
1869.
ENTERED, according to Act of Congres?, in the year 1857, by
D. APPLETOX & CO.,
In the Clerk's Office of the District Court of the United States for the Southern
District of New York.
ENTERED, according to Act of Congress, in the year 186S, by
D. APPLETON & CO.,
In the Clerk's Office of the District Court of the United States for the Southern
District of New York.
PEEFAOE.
AT the suggestion of the publishers, this work was undertaken to
form one of their series of Dictionaries and Cyclopaedias. In this view,
it has been the intention to make it a complete course of instruction
and book of reference to the mechanic, architect and engineer. It
has not, therefore, been confined to the explanation and illustration
of the methods of projection, and the delineation of objects which
might serve as copies to the draughtsman, matters of essential impor-
tance for the correct and intelligible representation of every form ; but
it contains the means of determining the amount and direction of
strains to which different parts of a machine or structure may be sub-
jected, and the rules for disposing and proportioning of the material
employed, to the safe and permanent resistance of those strains, with
practical applications of the same. Thus while it supplies numerous
illustrations in every department for the mere copyist, it also affords
suggestions and aids to the mechanic in the execution of new designs.
And although the arranging and properly proportioning alone of mate-
rial in a suitable direction and adequately to the resistance of the strains
to which it might be exposed, would produce a structure sufficient in
point of strength for the purposes for which it is intended, yet as in
many cases the disposition of the material may be applied not only
practically, but also artistically, and adapted to the reception of orna-
ment, under the head of Architectural Drawing, the general charac-
teristics of various styles have been treated of, and illustrated, with
brief remarks on proportion and the application of color.
Within the last few years, both here and abroad, a number of works
have been published on " Practical Drawing," but no one work has illus-
trated all departments of the sitbject. In the mechanical, the works of
M. Le Brun and M. M. Arrnengaud are the standard which have been
made the basis of two English works, " The Practical Draughtsman's
Book of Industrial Design" and the "Engineer's and Machinist's Draw-
ing Book." From the latter of these works we have drawn most of
our chapters on Geometrical and Mechanical Drawing, and Shades and
Shadows. In neither the French nor the English works has the science
of architectural construction and drawing been adequately illustrated,
nor has Topographical Drawing been treated of. In these two
departments a varied selection has been made from the best authorities.
In the Architectural, Ferguson and Garbett have been the most con-
sulted; in the Topographical, Williams, Gillespie, Smith, and Frome.
The work will be found quite fully illustrated, and the drawings and
engravings have been carefully executed, mostly under the supervision of
Mr. H. Grassau.
Like most cyclopaedias, this work claims for its articles but little of
novelty or originality ; the intention of the compiler was, to collect within
moderate compass as much valuable matter as possible, in practical Draw-
ing and Design ; and to this purpose he brings the experience of series of
years in each of the departments treated. Practically, he has had means
of knowing the necessities of the trade and of the profession, and trusts
that the selection now made will be found useful for the purposes for
which it was intended. W.
TABLE OF CONTENTS.
GEOMETRICAL DEFINITIONS AND TECHNICALI-
TIES, 1-6.
DRAWINO INSTRUMENTS. Description and
use; pencil; ruler; triangle; T square;
parallel ruler ; sweeps and variable curves,
compasses, or dividers; drawing pens;
dotting point ; drawing pins ; scales ; pro-
tractor ; vernier scales ; scale of rhumbs,
longitude, chords, sines, tangents ; the sec-
tor; Marquois's scales; triangular com-
passes ; wholes and halves ; beam com-
passes; portable compasses; screw di-
viders ; circular protractor ; pentograph ;
camera lucida ; drawing table and draw-
ing board, 7-35.
Drawing paper ; tracing paper ; mouth
glue; mounting paper and drawings;
varnishing; management of instruments,
35^4.
GEOMETRICAL PROBLEMS. Drawing of lines,
division of lines, perpendiculars to, par-
allels; construction of angles; division of
angles; description of arcs and circles;
connection of straight lines by arcs, and
arcs with arcs ; compound curves, 42-56.
On circles and rectilinear figures; triangles,
squares, rectangles, parallelograms; in-
scribed and described circles ; pentagons ;
hexagons, octagons, polygons; table of
polygonal angles, 56-63.
On the use of the T square, and triangle in
construction of preceding problems; di-
vision of lines, 63-05.
Simple application of regular figures, 65.
Problems on proportional lines and equiv-
alent figures, 66.
On the ellipse, parabola, hyperbola, cycloid,
epicycloid, involute, and spiral, 68-79.
| GEOMETRICAL PROJECTION. Of the point,
line, solids; plans, elevations, sections,
80-82.
Shade lines in outline drawings, 83.
Projections of simple bodies; hexagonal
pyramid ; prism, 83-88.
Construction of conic sections, 90.
Penetrations or intersections of solids; of
cylinders, cones, and prisms; cylinders,
prisms, spheres, and cones, 92-99.
Of the helix, 99.
Development of surfaces, cylinder, cone,
sphere, 102.
MECHANICS. The mechanical powers; the
lever, wheel, and axle, pulley, inclined
plain, wedge, screw, 105-110.
Forces, parallel, inclined, parallelogram of,
composition and resolution of, centre of
gravity, 110-114.
Friction and limiting angle of resistance, ex-
periments by M. Moriri, 115-117.
Equilibrium of the polygon of rods or cords,
application to framing, 117-119.
Mechanical properties of materials; tables
of strength of woods, of metals; resist-
ance to compression, bricks, granite, cast-
iron pillars ; tensile strength ; transverse
strength, beams, girders ; detrusion ; ten-
sion, 120-129.
Mechanical work or effect, of animals, of
water, of steam ; the indicator ; effect of
expansion; table of pressure, temperature
and volumes of steam; table of weights
and evaporative power of different fuels ;
determination of water, fuel, and size of
boiler to produce a given power, 130-136.
DBA WING OF MACHINERY. Shafting, sec-
tions of wooden, cast and wrought iron ;
TABLE OF CONTEXTS.
table of diameters of journals; tranverse
strain; water-wheel shafts; section of
water-wheel ; crank shaft of steam engine ;
table of diameters of journals for torsional
strain ; line shafts, 137-142.
Bearings or supports for journals; steps;
suspension bearing of turbine, step or
guide for same; pillow-block standard;
side, sprawl, yoke hangers; couplings,
face, sleeve, screw, clamp; horned, slide,
or clutch, bayonet, and friction cone ; pul-
leys, plate, plain and curved arms, faced
coupling; drums, wooden; cone; belts,
table of strain on, strength of; fast and
loose pulleys ; oblique shafts, 142-156.
Gearing; spur, bevel-wheels ; internal gear,
rack and pinions; trundle gear; trans-
mission of motion ; size of gear ; pitch ;
table of pitch, diameter and number of
teeth ; form and proportions of teeth ; by
scale ; table of stress at pitch circle, thick-
ness of teeth and pitch ; table of pitch ;
thickness, length, and breadth of teeth and
velority ; fundamental principle of toothed
wheels; epicycloidal teeth; the trundle,
templates, involute teeth, 156-175.
Projections of a spur-wheel; of a bevel-
wheel ; of a skew bevel ; of a pinion driv-
ing a rack ; of a rack driving a pinion ; of
a wheel and tangent, or endless screw;
internal spur-wheel driving a pinion; an
internal driven by a pinion ; eccentrics,
175-189.
Drawing of screws; triangular threaded
screw and nut ; square threaded screw and
nut ; table of diameters of bolts and nuts
and threads per inch, 189-191.
Hooks, form and proportions of, 192.
Frames of cam-punch, and shear ; of plan-
ing machine; jack-screw; hydraulic-press,
action of same ; frames of American
marine engines ; working beams, Ameri- j
can and English with details ; cranks, pro- !
portion of eyes ; connecting rods with
details, 192-198.
Location of machines ; example of weaving j
rooms, 189-200.
Machines; marine engines, and locomotives,
in skeleton drawings ; cataract of a Cor- j
nish engine ; details of 48 stop gate j
Brooklyn "Water-works ; sections of a
locomotive boiler ; elevation and sections ;
of engine of Golden Gate ; elevation and
sections of a Lowell turbine, with rules
for describing curves and proportioning
turbines, 201-208.
AECUITECTUKAL DRAWINGS. Foundations,
walls, bond of and thickness of; extract
from London and Liverpool building acts ;
mortar; arches, 209-216.
Framing, beams, flooring, bridging, girders;
size of joists ; stirrup irons ; floors ;
trussed beams; fire-proof floors; parti-
tions ; roofs, pitch, form for various span,
size and proportions of parts; table of
same; joints; varieties of roofs, hipped,
gambrel or Mansard, circular : eaves ; iron
roofs, details of one ; Crystal Palace
girders, cast and wrought iron; princi-
ples of bracing; use of counters; truss
by tension rod; system of suspension
truss ; a completely braced frame ; bridge
trusses, 216-234.
Size and proportion of rooms ; dining rooms ;
parlors, drawing, and bed rooms, pantries;
passages, height of stories; details of parts;
stairs ; doors ; windows ; bases and surbase- ;
cornices; fireplaces; privies; water closets
and outhouses; cess pools; wood and coal
sheds, 234-245.
Drawing, applications of, to the laying out of
house; plans, elevations, and section of a
house ; plans of familiar forms of houses,
245-250.
Mouldings, Greek and Roman; orders of-
architecture, with examples ; Tuscan,
Doric, Ionic, Corinthian, and Composite ;
Arcades ; Romanesque and Gothic mould-
ings, jamb, ai'ch; capitals, string courses;
cornices ; arches, semicircular, segmen-
tal, stilted, horse-shoe, pointed ' ogee,
Tudor, and foiled ; domes and vaults ;
Byzantine, Roman, Gothic ; buttresses ;
towers; pinnacles; spires. Windows, Ro-
manesque, Byzantine, Norman, Gothic;
doorways ; the Renaissance, Florentine,
Venetian ; ornament, Greek, Roman, By-
zantine, Romanesque, Saracenic, Gothic,
Renaissance ; balustrades, 250-279.
Elevations of Houses, city and country;
details of windows ; verge-boards ; chim-
ney-tops ; balcony ; stables ; city tene-
ment house ; stores and warehouses ;
School-houses and furniture ; Lecture
TABLE OF CONTENTS.
rooms, Churches, Theatres, Legislative
Halls ; transmission of sound ; space re-
quired for seats; size of pews; require-
ments of churches; examples from city
practice ; requirements of theatres ; di- j
mensions of several ; New York Crystal j
Palace, 280-299.
Material for building ; appropriate color ; i
ventilation and warming ; air required for j
respiration, lighting and heating; circu-
lation; table of grains of moisture in 1
cubic foot of air ; methods of heating fire-
places, stoves x hot-air furnaces, steam and
hot water, circulation; ventilators, 299-
306.
Drainage; sewer pipes; privy vaults ; light-
ing; water supply; wells and water pipes,
306-308.
Principles of Design ; Extracts from Fergu-
son's Hand-Book of Architecture, the
Encyclopaedia Britannica; the two great
principles of art, 308-312.
Shading and Shadows. Diffusion of light ;
direct and cast shadows; problems for de-
termining the outline of shadows, by a
straight line upon plain and curved sur-
faces, by a circle, by a hexagonal pyramid ;
the limit of shade and shadow of a cylin-
der; reversed cone ; a prism; the shadows
cast in the interior of a cylinder, of a hemi-
sphere, of a niche ; the line of shade in a
sphere, and its shadow on a plane; line
of shade on the surface of a ring ; the out-
lines of shadows cast on surfaces of screws
and nuts, triangular and .square, threaded,
313-328.
Manipulation of shades and shadows; me-
thods of tinting surfaces' in the light, in
the shade ; shading by flat tints ; by
softened tints ; elaboration of shading and
shadows; depth of shadows ; examples of
finished shading, 328-338.
Finished coloring ; color of materials ; prep-
aration of tints ; body color ; manipula-
tion, washing, or sponging; color for
wrought iron, brass, copper ; intensity of
shades and shadoAvs ; margin of light ; ad-
vantage of washing ; conventional tints for
materials, 339-348.
TOPOGRAPHICAL DRAWING. Conventional
signs and representation of features of a
country; distinctive marks for edifices,
for metals ; methods of representing hills,
vertical and horizontal slopes by a scale
of shade ; contours, 339-354.
Plotting ; rough sketch ; choice of scale, and
scales prescribed by different commis-
sions ; lines of survey ; variation of needle ;
survey by compass and plot ; balancing of
error; plotting by latitudes and depar-
tures ; plotting of offsets by scale ; plot of
railway curves ; table of degrees of curva-
ture, radii, and central ordinates ; railway
plot and profile ; the two combined ; pro-
file and cross-section paper ; regulations
of the English Parliament for railway
plans ; geographical sections ; hydro-
graphic and marine surveys ; rough draft ;
transferring ; tracing ; photography ; copy-
ing glass ; transfer paper ; reduction and
enlargement of plans, 355-369.
Finishing plan ; direction of light ; boundary
lilies ; lettering ; examples of alphabets ;
construction of letters mechanically ;
spacing of letters; lines of lettering;
titles, 369-377.
Tinted topographical drawing ; conventional
tints ; colors used by French military en-
gineers ; imitation of conventional signs ;
representations of hills, woods, rivers, by
the brush ; effect of oblique light ; prep-
aration of paper for tinted drawing ;
application of tints, lettering, flourishes,
378-382.
Map of portion of the city of London, show-
ing drainings, contours, gas and water
mains, and occupancy of buildings; a
larger portion, showing effect of contour
lines, 382-284.
PERSPECTIVE DRAWING. Direction of lumin-
ous rays ; angle at which objects can be
seen; linear and aerial perspective; the
planes of a picture ; point of sight ; par-
allel and angular perspective ; to draw a
square and cube in parallel perspective ;
construction of a scale ; to determine the
position of any point in the ground plane ;
to draw an octagon, a circle, a pyramid,
a cone, in parallel perspective ; to draw a
square, cube, octagonal pillar, circular
pillar, octagonal pyramid, cone, elevation
of a building, arched bridge, in angular
perspective; to draw the interior of a
room, a flight of stairs, and to find the
viii
TABLE OF CONTENTS.
reflections of objects in water; the per-
spective projection of shadows ; the most
agreeable angle of vision for a represen-
tation in perspective, 385-404.
ISOMETRICAL DsAwiNG. Principle of iso-
metrical representation ; projection of a
cube and its general application ; collec-
tion of cubes and sections of cubes ; pro-
jection of curved lines; division of the
circumference of a circle ; projection of a
bevel wheel, of a pillar block, of a cul-
vert, section of a boiler, bridge truss;
horizontal section or plan of school-house ;
portion of a roof truss ; perspective draw-
ings, in which the point of sight is above
the plane of the picture, 415-424.
ENGINEERING DRAWING. Tredgold's defini-
tion of civil engineering; foundations;
piles ; rule to find the weight .which a
pile iwill bear ; weight of ram ; size of
piles ; sheet piling ; hollow cast-iron piles,
how driven ; Harlem bridge ; coffer dams ;
foundation of Susquehanna bridge ; sec-
tion of river-wall, Thames embankment,
extracts from specifications for ; crib pier
for quarantine establishment for the port
of New York, extracts from specifications
for same, 415-424.
Dams across the Connecticut River at Hoi-
yoke, Mass. ; across Merrimack, at Lowell,
Mass. ; across Mohawk, at Cohoes, K Y. ;
across Croton Eiver, N. Y. ; gauging
of streams; rain-fall and evaporation;
head-gates at Cohoes Dam ; hoisting ap-
paratus for same, 424-431.
Canals, dimensions of; section of Erie; of
Northern, at Lowell ; wall of same ; locks
of canals ; details of Chemung and Erie,
enlarged ; extracts from specifications of
New York State canals ; ponds or reser-
voirs ; flumes ; headgates of flume at Hoi-
yoke; section of conduit of Brooklyn
City Water-works ; of Croton Aqueduct ;
pipes across Harlem Eiver ; Croton new
reservoir : extracts from specification for
same ; water mains ; dimension of Brook-
lyn Water- works pipes ; formulas for
discharge of pipes ; extracts from speci-
fication for Brooklyn pipes, weights of;
sewers, dimensions of ; man-holes; catch-
basins, 431-448.
Gas supply ; weight of pipes ; discharge of,
449.
Roads and streets; division and width of,
here and in Paris ; pave ; carriage-way ;
Belgian, wooden, and asphalt pave ; roads,
McAdam, Telford, Central Park; grades
of; table of inclinations, feet per mile,
and angles; railways; gauge, width of
cut and embankment ; resistance of car-
riages on roads, 450-455.
Bridges ; piers ; trestles ; arches, table of
dimensions of several ; rule for depth of
key ; height of spandrel ; thrust of arch ;
resistance of abutment ; skew arch ;
frame bridge ; tension and compression of
chords ; specification for Howe's truss ;
Cubitt's cast-iron girders; wrought-iron
truss across Connecticut River ; suspen-
sion bridges, dimensions of several, 455-
466.
Steam engines ; horse-power ; boilers, evap-
oration of; rate of combustion ; work-
ing-pressure ; strength of shell and flues ;
joints, stays; locomotive fire-box; chim-
neys ; size of flues ; stationary engine,
details of, stuffing-box, piston ; founda-
tion for engine, 466-474.
PROJECTIONS OF THE GLOBE. Globular and
stereographic projections of hemisphere ;
construction of maps by development ;
table of miles to degrees of longitude ;
Mercator's chart, construction of, 475-
481.
! Specifications, form of, 482.
j APPENDIX. Extracts from Building Act,
'city of New York, 483-485.
Francis's tables on size of shafting ; power
required to drive cotton and woollen
' mill, 486-488.
Profile and cross-section paper ; application
of profiles to flow of water, resistance
and movement of trains, 489, 490.
INDEX, 491-496.
LIST OF PLATES.
PAGE
PLATES I., II. Projections of a regular Hexagonal Pyramid, .... 86
PLATES III., IV. Projections of Prisms, 88
PLATE V. Conic Sections, 90
PLATES VI., VII. Penetrations of Cylinders, 92
PLATES VIII., IX. Penetrations of Cylinders, Cones, and Spheres, ... 94
PLATES X., XI. Penetrations of Cylinders, Prisms, Spheres, and Cones, . 98
PLATE XII. The Spiral and Helix, 100
PLATE XIII. Development of the Surface of Intersected Cylinders and Cones, 102
DRAWING OF MACHINERY.
PLATE XIV. Drawings of Water-wheel Shafts, .... 139
PLATE XV. " of a Standard, 146
PLATE XVI. " of a Sprawl and bracket Hanger, ...... 148
PLATES XVII., XVIII. Drawings of Spur Wheels, 176
PLATES XIX., XX. Oblique Projections of a Spur Wheel, 178
PLATES XXI., XXII. Projections of a Bevel Wheel, 180
PLATES XXIII., XXIV. Rack Gear and Pinion, Worm and Wheel, . . .183
PLATES XXV., XXVI. Internal Gearing, 184
PLATE XXVII. Projections of Eccentrics, 185
DRAWING OF SCREWS.
PLATES XXVIH., XXIX. Projections of a Triangular-threaded Screw and Nut, 189
FRAMES.
PL ATE XXX. Iron Frames of Tools, 192
PLATE XXXI. Elevation of the Frames of American Marine Engines, . . 194
PLATE XXXII. Working-Beams and Cranks, 195
PLATE XXXIII. Steam-engine Connecting Rods, and Details, . . . 197
LIST OF PLATES.
LOCATION OP MACHINES.
PAGE
PLATES XXXIV., XXXV. Plans of the Location of Machines, .... 200
PLATES XXXVI., XXXVII., XXXVIII. Elevation, Section, and Plan of the
48" Stop-gate in use at the Nassau Water-works, Brooklyn, L. I., . . 202
PLATE XXXIX. Sections of a Locomotive Boiler, 204
PLATE XL. Front Elevation of one of the Engines of the Golden Gate, . 204
PLATE XLI. Side Elevation of the same, 204
PLATE XLLT. Vertical Section through the Centre of a Turbine Wheel, . . 207
PLATE XLIII. Plan of the Turbine Wheel and Wheel-pit, 207
PLATE XLIV. Plan of the Wheel, Guides, and Garniture, .... 208
ARCHITECTURAL DRAWING.
PLATE XLV. Elevations and Details of Framed Roofs, 220
PLATE XL VI. Iron Roof and Trussed Girders, 231
PLATES XLVII.-LI. Plans and Elevations of a House, 248
PLATE LII. Example of the Tuscan Order, 254
PLATE LILT. " of the Doric Order, . . 254
PLATE LIV. " of the Ionic Order, . 255
PLATE LV. " of the Corinthian Order, 256
PLATE LVI. Roman Arches and Entablatures, Gothic and Byzantine Columns, 260
PLATE LVII. Buttresses, Towers, and Spires, .268
PLATE LVIII. Windows, 270
PLATE LIX. Doorways, 271
PLATE LX. Ornaments and Brackets, 274
PLATE LXI. Roman and Saracenic Ornament, 274
PLATE LXII. Ornamental Mouldings, 276
PLATE LXIII. Ornaments of the Renaissance, 278
PLATE LXIV. Front Elevation of a High-Stoop House, New York City, . 280
PLATE LXV. Elevation of a House, from " Holly's Country Seats," . . .282
PLATE LXVI. Plan and Elevation of a Farm-house, in the English Rural Style, 281
PLATE LXVII. Elevation and Plan of a plain Timber Cottage Villa, . . .281
PLATE LXVIII. A Villa, Rural Gothic Style, 282
PLATE LXIX. A Villa in the Italian Style, 284
PLATE LXX. A Tenement House, 284
PLATE LXXI. Elevation of a Store Front, 286
PLATE LXXII. Fa9ade of two Stores erected on Broadway .... 285
PLATE LXXIII. Elevation of Store Front, executed in Cast Iron, . . . 288
PLATE LXXIV. Plan and Elevation of a District School-house, ... 288
PLATE LXXV. Design for a Church in the English Decorated Gothic Style, . 294
PLATE LXXVI. Front Elevation of the Roman Catholic Cathedral, Fifth Ave-
nue, New York City, 294
PLATE LXXVII. Byzantine Church, Park Avenue, 295
PLATE LXXVIII. Interior Perspective View of the New York Crystal Palace, 297
LIST OF PLATES.
SHADING AND SHADOWS.
PAGE
PLATE LXXIX. Forms of Shadows, 314
PLATE LXXX. Outlines of Shadows cast upon two Planes of Projection, . 318
PLATE LXXXI. Outlines of Shadows cast upon the Interior of a hollow Cyl-
inder and Ring, ............ 325
PLATE LXXXII. Outlines of Shadows cast upon the Surfaces of Screws and
Nuts, both Triangular and Square-threaded, 327
METHODS OF TINTING.
PLATE LXXXIII. Shading by Flat Tints, ...... K .. 330
PLATE LXXXIV. Shading by Softened Tints, ...... 332
ELABORATION OF SHADING AND SHADOWS.
PLATES LXXXV., LXXXVI. Examples in Lithography of Shades and Shadows
of different Solids, 336
PLATE LXXXVII. Effects of Light, Shade, and Shadow, on Screws, . . 338
FINISHED COLORING.
PLATES LXXXVIII. and LXXXIX. Illustrations in Chromo-Lithography from
Colored Drawings, 348
TOPOGRAPHICAL DRAWING.
PLATE XC. Examples of Topographical Drawings, 349
PLATE XCI. Meridians and Borders, 370
PLATE XCII. Mechanical Method of Constructing Letters and Figures, . 374
PLATE XCIII. Example of Titles illustrating the Form of Letters, . . .374
PLATE XCIV. Map of the Harbor and City of New Haven, .... 382
PLATE XCV. Examples of topographical Drawing, 382
PLATE XCVI. " " " " in colors, ... 382
PLATE XCVII. Geological Map, from Blake's " Survey of California," . . 382
FRONTISPIECE. r
PLATE XCVHI. Photograph from a Drawing of the Engine and Boiler of the
Steamer Pacific, 388
PERSPECTIVE DRAWING.
PLATE XCIX. A Square and Cube in Parallel Perspective, 390
PLATE C. in Angular Perspective, .... 396
PLATE CI. Projection of an Octagonal Pillar, Cylinder, Pyramid, and Cone, in
Angular Perspective, 398
PLATE CII. The Elevation of a Building in Angular Perspective, . . . 400
PLATE CHI. An Arched Bridge in Angular Perspective, and Interior of a Room, 400
x ii LIST OF PLATES.
PAGE
PLATE CIV. A Flight of Stairs, and Reflections in Water, .... 402
PLATE CV. Perspective Projection of Shadows, 402
ISOMETRICAL DRAWING.
PLATE CVI. Sections of Cubes, 408
PLATE CVII. Bevel Wheel and Pillow Block, 409
PLATE CVIII. Projection of a Culvert, such as were built beneath Croton
Aqueduct, 409
PLATE CIX. Elevation and Section in Isometry of the District School-house
given in Plate LXXIX., 412
ENGINEERING DRAWING.
PLATE CX. Transverse Section of the River-wall, Thames Embankment, . 421
PLATE CXI. Isometrical View of the Overflow and Outlet of the Victoria and
Regent Street Sewers in the Thames Embankment, 423
PLATE CXII. Section of the Dam across the Connecticut River, at Holyokc,
Mass., 430
PLATES CXHI. and CXIV. Drawings in Plan and Detail of the Headgates, and
the Machinery for hoisting them, at the Cohoes Company Dam, . . 430
PLATE CXV. Elevation and Section of the Headgates of a Flume, . . 438
PLATE CXVI. Sections of the Fire-box of a Locomotive, 470
PLATE CXVII. Elevations and Section Tubular Boiler, . ... 471
PLATE CXVIII. Sections of Chimneys, 472
PLATE CXIX. Elevation and Details of a Stationary Engine, . . . 473
PLATE CXX. Plan and Elevations of Foundation of a Stationary Engine, . 474
CYCLOPAEDIA OF DRAWING.
GEOMETRICAL DEFINITIONS AND TECHNICALITIES.
A point is mere position without magnitude, as the intersection of two
lines, or the centre of a circle.
Lines are measured by length merely, and may be straight or curved.
Straight lines are generally designated by letters or figures at their ex-
tremities, as the line A B, the line 1 2. Curved lines, by
additional intermediate letters or figures, as the curved line ABC.
A given point or given line expresses a point or line of fixed position
or dimension.
Surfaces or superficies are measured by length and breadth only. They
may be plane or curved.
Solids are measured by length, breadth, and thickness. The extremi-
ties of lines are points, the boundaries of surfaces are lines, and the boun-
daries of solids are surfaces.
Parallel lines are lines in the same plane
which are equally distant from each other
Fig. 1.
at every part (fig. 1).
Horizontal lines are such as are parallel to the horizon, or level.
Vertical lines are such as are parallel to the position of a plumb-line
suspended freely in a still atmosphere.
Inclined lines occupy an intermediate between horizontal and vertical
lines. Also two lines which converge towards each other, and if produced,
would meet or intersect, are said to incline to each other.
1
GEOMETRICAL DEFINITIONS AND TECHNICALITIES.
An angle is the opening between two straight lines which meet one
another. "When several angles are at
one point B, any one of them is expressed
by three letters, of which the letter that
is at the vertex of the angle, that is, at
the point in which the straight lines that
contain the angle meet one another, is
put between the other two letters : Thus
the angle which is contained by the
straight lines, AB, CB, is named the angle ABC, or CBA ; but if there be
only one angle at a point, it may be expressed by a letter placed at that
point ; as the angle at E.'
When a straight line standing on another straight line makes the adja-
cent angles equal to one another, each of the angles is called a right angle /
and the straight lines are said to be perpendicular to each other (fig. 3).
An obtuse angle is that which is greater than a right angle (fig. 4).
Fig. 3.
Fig. 4
Fig. 5.
Fig. 6.
An acute angle is that which is less than a right angle (fig. 5).
A triangle is a flat surface bounded by three straight lines ; when the
three sides are equal, the triangle is equilateral ; when only two of its
sides are equal, isosceles; when none equal, scaline; when one of the
angles is a right angle, the triangle is right angled, and then the longest
side, or that opposite the right angle is called the hypothenuse. The
upper extremity of the triangle is called the apex, the bottom line the
lose, and the two other including lines the sides.
A Quadrilateral figure is a surface bounded by four straight lines.
Fig. 7. Fig. 8. Fig. 9. Fig. 10. .
When the opposite sides are parallel, it is a parallelogram ; if its angles
GEOMETRICAL DEFINITIONS AND TECHNICALITIES.
8
are right angles, it is a rectangle (fig. 7) ; if the sides are also equal, it is a
square (fig. 8) ; if all the sides are equal, but the angles not right angles,
it is a rhombus (fig. 9). A trapezium has only two of its sides parallel (fig.
10). A diagonal is a straight line joining two opposite angles of a figure.
Plane figures of more than four sides are called polygons. When the
Fig. 12. Fig. 13.
Fig. 11.
Fig. 14.
Fig. 15.
Pentagon, five sides. Hexagon, six sides. Heptagon, seven sides. Octagon, eight sides.
sides are equal, they are regular polygons ; of which figs. 11-14 are ex-
amples, annexed to which are their respective designations.
A circle is a plane figure contained by one line,
which is called the circumference, and is such that all
straight lines, drawn from a certain point within the
figure to the circumference, are equal to one another.
And this point is called the centre of the circle.
The term circle is very generally used for the cir-
cumference, and will be found to be employed in this
work with this twofold meaning.
Any straight line drawn from the centre and terminating in the cir-
cumference is termed a radius / if drawn through the centre, and termi-
nated at each end by the circumference, it is termed a diameter.
An arc of a circle is any part of the circumference.
A sector of a circle is the space enclosed by two radii and the inter-
cepted arc. When the radii are at right angles, the space is called a quad-
rant or one-fourth of a circle. Half a circle is called a semicircle.
A chord is a straight line joining -the
extremities of an arc, as a 1). The space
cut off' by the chord is termed a segment.
A tangent to a circle or other curve
is a straight line which touches it at only
one point, as c d touching the circle at
only e.
Circles are concentric when described Fig. is.
from the same centres. Eccentric when described from different centres.
Triangular or other figures with a greater number of sides are inscribed
GEOMETRICAL DEFINITIONS AND TECHNICALITIES.
rig. 17.
Fig. 18.
in, or circumscribed ~by a circle, when the vertices of all their angles are in
the circumference (fig. 17).
A circle is inscribed in a
straight-sided figure, when it
is tangent to all the sides (fig.
18).
All regular polygons may
be inscribed in circles, and
circles may be inscribed in
polygons; hence the facility
With which polygons may be constructed.
For the measurement of angles, the circumference of a circle is divided
into 360 equal arcs, called degrees ,
which are again subdivided into min-
utes ' and seconds " ; 60 minutes to a
degree, and 60 seconds to a minute,
the vertex of the angle being placed
at the centre of the circle, the angle
is measured by the arc enclosed be^-
tween the sides. Thus the angle
DCB is measured by the arc DB ; the
line DH, a line drawn from one ex-
tremity of the arc perpendicular to
the radius passing through the other
extremity is called the sine of the
angle, GD is the cosine, HB the versed
sine, AB the tangent, FE the cotangent, AC the secant, and CE the cosecant.
An ellipse is an oval-shaped curve from any point P in which, if straight
lines be drawn to two fixed points
FF ; , their sum will be always the
same. FF' are the foci, the line
passing through the foci is 'called
the transverse axis, the line CD
\S perpendicular to the centre of
this line the conjugate axis.
A parabola is a curve in
which any point P is equally dis-
tant from a certain fixed point
F and a straight line KK' ; thus,
GEOMETRICAL, DEFINITIONS AND TECHNICALITIES. 5
PF. is always equal to PD=. F is called the focus, and the line KK' the
directrix (fig. 21).
K
Fig. 21.
Fig. 22.
An hyperbola is a curve from any point P in which, if two straight lines
be drawn to two fixed points FF' the foci, their difference shall always be
the same (fig. 22).
A cycloid is the curve described by a point P in the circumference of
Fig. 23.
Fig. 24.
a circle which rolls along an extended straight line until it has completed
a revolution.
If the circle be rolled on the circumference of another circle, the curve
then described by the point P is called an epicycloid (fig. 24).
Epicycloids are external or internal, according as the rolling or gener-
ating circle revolves on the outside or inside of the fundamental circle.
The internal epicycloid is sometimes called a hypocycloid.
OF SOLIDS.
A prism is a solid of which the ends are equal, similar, and parallel
straight-sided figures, and of which the other sides are parallelograms.
When all the sides are squares, it is called a cube (fig. 25).
A pyramid is a solid having a straight-sided base, and triangular sides
terminating in one point or vertex (fig. 26).
Prisms and pyramids are distinguished as triangular, quadrangular,
GEOMETRICAL DEFINITIONS AND TECHNICALITIES.
pentagonal, hexagonal, &c., according as the base has three, four, five, six
sides, <fec.
Fig. 25.
Fig. 26.
Fig. 27.
A sphere or globe (fig. 27), is a solid bounded by a uniformly curved
surface, every point of which is equally distant from the centre, a point
within the sphere. A line passing through the centre, and terminating
both ways at the surface, is a diameter.
Fig. 29. Fig. 30.
A cylinder is a round solid of uniform thickness, of which the ends are
equal and parallel circles (fig. 28).
A cone is a round solid, with a circle for its base, and tapering uni-
formly to a point at the top (fig. 29).
When a solid is cut through transversely by a plane parallel to the
base, the part cut off is a segment, and the part remaining is afrustrum ol
the solid. The latter term is usually limited to pyramids and cones.
Fig. 81.
Fig. 82.
Fig. 33.
Fig. 34.
The tetrahedron, bounded by four equilateral triangles (fig. 30).
The fexahedron, or cube, bounded by six squares (fig. 31).
The octahedron, bounded by eight equilateral triangles (fig. 32).
The dodecahedron, bounded by twelve pentagons (fig. 33).
The icosahedron, bounded by twenty equilateral triangles (fig. 34).
Regular solids may be circumscribed by spheres, and spheres may bo
inscribed in regular solids.
DRAWING INSTRUMENTS.
DRAWING INSTRUMENTS.
Lead pencil. Pencils are of various qualities, distinguished by letter
marks, of which the most common in use by draftsmen are HH and HUH.
The pencil used for drawing straight lines should be sharpened to a chisel
edge ; for making dots and marking points (.), the pencil should have a
round sharp point. Pencil lines intended to be made permanent in ink,
should be drawn quite delicately. The pencil should not be held tightly ;
a slight hold without slackness, inclined a little to the side toward which
the line is drawn. Never extend the line beyond what is necessary, and
avoid as much as possible the use of rubber, as it roughs the paper, mak-
ing it difficult to trace a smooth line in ink, and readier to receive and
retain dust.
The common ruler or straight edge. Rulers should be of close-grained,
thoroughly seasoned wood, such as mahogany, maple, pear, &c. They
should be about \ of an inch thick, bevelled a little on one edge, and from
1 to 2i inches wide, according to their length. Every draftsman should
have at least two rulers, the shortest from 9 inches to a foot long, and the
other as long as he may require in his drawing. As the accuracy of a
drawing depends greatly on the straightness of the lines, the bevelled edge
of the ruler should be perfectly straight. To test this, place a sheet of
paper on a perfectly smooth board ; insert two very fine needles in an up-
right position through the paper into the board, distant from each other
nearly the length of the ruler to be tested ; bring the edge of the ruler
against these needles, and draw a line from one needle to the other ; re-
verse the ruler, bringing the same edge on the opposite side and against
the needles, and again draw a line. If the two lines coincide, the edge is
straight ; but if they disagree, the ruler is inaccurate. When one ruler
has been tested, the other can be examined by placing their edges against
the correct one, and holding them between the eye and the light.
Triangles are made of the same kinds of wood as the ruler, and some-
what thinner, and of various sizes. They should be right-angled, with acute
angles of 45, or of 60 and 30. The most convenient size for general use
measures from 3 to 6 inches on the side. A larger size from 8 to 10 inches
DRAWING INSTRUMENTS.
o
Fig. 85.
long on the side is convenient for making, drawings to a large scale. Cir-
cular openings are made in the body of the triangle for the insertion of the
end of the finger to give facility in sliding the triangle on the paper. Tri-
angles are sometimes made as large as 15 to 18 inches on the side ; but in
this case they are framed in three pieces of about 1
wide, leaving the centre of the triangle open. The
value of the triangle in drawing perpendicular
lines depends on the accuracy of the right angle.
To test this (fig. 36), draw a line with an accurate
ruler on paper. Place the right angle of the tri-
angle near the centre of this line, and make one
of the adjacent sides to coincide with the line ; now
draw a line along the other, adjacent side, which,
if the angle is strictly a right angle, will be per-
pendicular to the first line. Turn the triangle on
this perpendicular side, bringing it into
the position ABC 7 ; if now the sides of
the triangle agree with the line BC' and
AB, the angle is a right angle, and the
sides straight. The straightness of the
hypothenuse or longest side can be tested
like a common ruler.
The triangle is used for the drawing of lines parallel or perpendicular
to each other. Thus (fig. 37), if it were required to draw lines parallel
and perpendicular to c d, place
one side of the triangle so as to
coincide accurately with the given
linec<?/ keeping the triangle in
this position with the right hand,
bring the edge of the ruler against
the hypothenuse of the triangle ; if
now the ruler be held securely by
the left hand, the triangle may be
slid along the edge of the ruler,
and any line drawn along the upper side of the triangle will be parallel to
the line c d, and the lines drawn along the other side of the triangle will
be perpendicular to this same line ; in this way a rectangle may be drawn
through three given points without moving the position of the ruler.
It is evident that for the drawing of parallel lines merely, either side
may be brought in contact with the ruler ; but the longer the side in con-
S
Fig. 86.
DRAWING INSTRUMENTS. 9
tact, the more accurately may the parallelism be preserved in sliding the
triangle.
The T square is a thin "straight edge" or ruler, , fitted at one
end with a stock, 5, applied transversely at right angles. The stock
being so formed as to fit and slide against one edge of the drawhig board,
the blade reaches over the surface, and presents an edge of its own at right
angles to that of the
board, by which par-
allel straight lines
may be drawn upon
the paper. To suit
a 41-inch board, the
blade should meas- Fi s- 88 -
ure 40 inches long clear of the stock, or one inch shorter than the board,
to remove risk of injury by overhanging at the end ; it should be 2
inches broad by -^ inch thick, as this section makes it sufficiently stiff
laterally and vertically. The tip of the blade may be secured from
splitting by binding it with a thin strip inserted in a saw-cut. The
stock should be 14 inches long, to give sufficient bearing on the edge of
the board, 2 inches broad and inch thick, in two equal thicknesses
glued together. With a blade and stock of these sizes, a well propor-
tioned T square may be made, and the stock will be heavy enough to act
as a balance to the blade, and to relieve the operation of handling the
square. The blade should be sunk flush into the upper half of the stock
on the inside, and very exactly fitted. It should be inserted full breadth,
as shown in the figure ; notching and dovetailing is a mistake, as it weak-
ens the blade, and adds nothing to the security. The lower half of the
stock should be only If inches broad, to leave a |-inch check or lap, by
which the upper half rests firmly on the board and secures the blade lying
flatly on the paper.
For the smaller sizes of board, reduce the proportions of both the blade
and the stock.
Fig. 39
One half of the stock, c (fig. 39), is in some cases made loose, to turn
upon a brass swivel to any angle with the blade a, and to be clenched by
10
DRAWING INSTRUMENTS.
a screAved nut and washer. The loose stock is useful for drawing parallel
lines obliquely to the edges of the board, such as the threads of screws,
oblique columns, and connecting-rods of steam-engines. A square of this
sort should be rather as an addition to the fixed square, and used only
when the "bevel edge is required, as it is not so handy as the other.
The edges of the blade should be very slightly rounded, as the pen will
thereby work the more freely. It is a mistake to chamfer the edges, that
is, to plane them down to a very thin edge, as is sometimes done with the
object of insuring a correct position of the lines ; for the edge is easily
damaged, and the pen is liable to catch the edge, and to leave ink upon it.
A small hole should be made in the blade near the end, by which the
square may be hung up.
In many drawing cases will be found the parallel ruler (fig. 40), consist-
ing of two rulers connected
by two bars moving on
1 pivots, and so adjusted that
the rulers, as they open,
form the sides of a paral-
lelogram. The edge of one
of the rulers being retained
in a position coinciding
with, or parallel to a given line ; the other ruler may be moved, and lines
drawn along its edge must also be parallel to the given line. This instru-
ment is only useful in drawing small parallels, and in accuracy and con-
venience does not compare with the triangle and ruler or J square.
Fig. 40.
SWEEPS AND VARIABLE CURVES.
For drawing circular arcs of large radius, beyond the range of the or-
dinary compasses, thin slips of wood, termed sweeps, are usefully employed,
of which one or both edges
are cut to the required circle.
For curves which are not
circular, but variously ellip-
tic or otherwise, "universal
sweeps," made of thin wood,
of variable curvature, are
veiy serviceable. The two
examples have been found
from experience to meet almost all the requirements of ordinary drawing
Fig. 42. One fourth full size.
DRAWING INSTRUMENTS. 11
practice. "Whatever be the nature of the curye, some portion of the uni-
versal sweep will be found to coincide with
its commencement, and it can be continued
throughout its extent by applying successively
such parts of the sweep as are suitable, taking
care, however, that the continuity is not in- Fig 42 Or ,^ rth full siZ( T
jured by unskilful junction.
ISTo varnish of any description should be applied to any of the wooden
instruments used in drawing, as the best varnish will retain dust, and soil
the paper. Use the wood in its natural state, keeping it carefully wiped.
Various other materials besides wood have been used, as steel for the blades
of the T square and the ruler ; the objection is the liability to soil the
paper. Glass is frequently used for the ruler and the triangle, and retains
its correctness of edge and angle, but it is too heavy, and liable of course
to fracture.
THE C.OMP ASSES OR DIVIDERS.
The best compasses are constructed with joints of two different metals,
as steel and brass, whereby the wear is more equal, and the motion of the
legs uniform and steady, and not subject to sudden jerks in opening or
shutting. This motion will occasionally require some adjustment to render
it uniformly smooth, and to move stiffer or easier at pleasure, but so that
they may keep steadily any position that may be given to them. This ad-
justment is performed by the application of a turnscrew to the axis of the
joint. In the common compasses, a simple screw forms the axis, which
may be turned with a screwdriver ; but in the' best made instruments, a
steel pin passes through the joints, having at one end a head of brass
riveted fast upon it, and on the other end a similar plate or nut is screwed,
on a diameter of which are drilled two small holes for the application of a
key (fig. 43). The points of a well made instrument should be
of steel so tempered, as neither to be easily bent or blunted ; not
too fine and tapering, and yet meeting closely when the com-
passes are shut.
Instruction for using dividers, which are applied only to
measure and transfer distances and dimensions, may appear Fig. 43.
superfluous; but there are a few simple directions which may save the
young draughtsman much perplexity .and loss of time. It is, TO course, de-
sirable to work the compasses in such a.manner that, when the dimension
12 DRAWING INSTRUMENTS.
is taken, it may suffer no disturbance in its transfer from the scale to the
drawing. In order to this, the instrument is to be held by the head or
joint, the forefinger resting on the top of the joint, and the thumb and
second finger on either side. When held in this way, there is no pressure
except on the head and centre, and the dimension between the points can-
not be altered ; but if the instrument be clumsily seized by a thumb on
one leg, and two fingers on the other, the pressure, in the act of transfer-
ence, must inevitably contract, in some small degree, the opening of the
compasses ; and if the dimension has to be set off several times, the proba-
bility is, that no two transfers will be exactly the same. And whilst it is all
important to keep the dimension exact, it is also desirable to manipulate in
such a way, when setting off the same dimension a number of times, that
the point of position be never lost. Persons unaccustomed to the use of
compasses, are very apt to turn them over and over in the same direction,
when laying down a number of equal measures, and this necessitates a fre-
quent change of the finger and thumb, which direct the movement of the
instrument ; the consequence is, either that the fixed leg is driven deep
into the drawing, or it loses position. Now, if the movement be alternately
above and below the line on which the distances are being set off, the coin-
passes can be worked with great freedom and delicacy, and
without any liability to shifting. If a straight line is drawn, and
semicircles be described alternately above and below the line, it
will show the path of the traversing foot. If the two movements
are tried, the superiority of the one recommended will at once be
discovered. The forefinger rests gently on the head ; and the
thumb and second finger, without changing from side to side,
direct the movement for setting off any number of times that
may be required.
The hair com/passes (fig. 44) are constructed in the same man-
ner as the common compasses. The only difference consists in a
contrivance, whereby the lower or point half of one shank can
be moved a very small quantity either .to wards or from the other
point, so that when the compasses are opened nearly to the re-
quired extent, by the help of the screw 5 the points may be set
with great precision, which cannot be done so well by the motion
of the joints alone.
Compasses with movable points (fig. 45) are a pair of compasses
^Jtorhich the point half of one of the legs is movable, to admit
7i s- of adapting singly a pen, a pencil, or a dotting point. The pen
point is used for drawing circles or arcs with ink. The pencil point is a
DRAWING INSTRUMENTS.
13
Fig. 45.
The annexed engraving
a tube adapted to hold a piece of lead pencil for describing circles or arcs,
and the dotting point consists of two blades, between which revolves a
small wheel, with numerous points round its
circumference, resembling the rowel of a spur.
The space between the blades being supplied
with Indian ink, as the compasses describe a
circle or arc, each point, as the wheel revolves,
will pass through the ink, and transfer it to
the paper beneath, making equidistant dots in
the circle which the compasses describe.
The movable points have a joint in them,
just under that part which locks into the
shank of the compasses, by which the part be-
low the joint may be set perpendicular to the
plane on which the lines are described, when
the compasses are open.
An additional piece, called a lengthening
bar, is frequently applied to these compasses,
to enable them to strike larger circles, or mea-
sure greater extents than they otherwise could,
represents this instrument and its appendages.
A, the compasses, with a movable point at B ; C and D, the joints to
set each point perpendicular to the paper ; E, the pencil point ; F, the pen
point ; G, the lengthening bar.
Bow compasses. These are a small pair, either having a
point for ink or pencil, used to describe small arcs or circles,
which they do more conveniently than large compasses. Fig.
46 is adapted for describing arcs of a radius intermediate between
those described by the above-named compasses, and
those capable of being produced by the bows repre-
sented by fig. 47. In fig. 46, the legs can be opened a
considerable width by the joint, whilst in fig. 47, the
opening is limited, the two blades or legs being formed
out of one solid piece of steel, and tempered so as to
form a spring at the upper part ; the spring of the two
blades is then kept in obedience by an adjusting screw
D, by which the two points may be set to any required
degree of minuteness, and very small circles may be
described with precision. F1 s- 46 - Fi s- 4T -
The pen bows (figs. 48, 49) are similar in their construction to the pen-
DRAWING INSTRUMENTS.
Fig. 4a Fig. 49.
cil bows. In fig. 49 there is a second joint A, by which, when the instru-
ment is open for use, the pen may be set perpendicular, or nearly so, to
the paper, which is essential in the use of the draw-
ing pen.
Similar to fig. 48 in their construction are the
spring dividers (fig. 50), particularly useful for re-
peating divisions of a small but equal extent, a
practice that has acquired the name of stepping.
The drawing pen (fig. 51) is used for drawing
straight lines. It consists of two blades with steel
points fixed to a handle ; and they are so bent, that a
sufficient cavity is left between them for the ink, when
the ends of the steel points meet close together, or
nearly so. The blades are set with the points more
or less open by means of a millheaded screw, so as
to draw lines of any required fineness or thickness.
Fig. 50. One of the blades is framed with a joint, so that by
taking out the screw, the blades may be completely opened, and the points
effectively cleaned after use. The ink is to be put between the blades by
a common pen, and in using the pen it should be
slightly inclined in the direction of the line to be
drawn, and care should be taken that both points
touch the paper ; and these observations equally apply
to the pen points of the compasses before described.
The drawing pen should be kept close to the ruler
or straight edge, and in the same direction during
the whole operation of drawing the line. Care
must be taken in holding the straight edge firmly
with the left hand, that it does not change its posi-
tion.
For drawing close parallel lines in mechanical
and architectural drawings, or to represent canals
or roads, a double pen (fig. 52) is frequently used,
with an adjusting screw to set the pen to any re-
quired small distance. This is usually called the
road pen. The best pricking point is a fine needle
held in a pair of forceps (fig. 53). It is used to mark
the intersection of lines, or to set off divisions from
the plotting scale and protractor. This point may
Fig. 51. Fig. 52. rig. 53. also be used to P rick through a drawing upon an
X
DRAWING- ESTSTRUMEIsrTS. 15
intended copy, or, the needle being reversed, the eye end forms a good
tracing point.
For filling up the broad lines of borders, a goose quill is often used
with a short nib and no slit (fig. 5-i). In drawing with this pen, incline
Fig. 54.
the drawing-board so that the ink will follow the pen, which prevents blots
or the accumulation of too much ink at any one point.
The dotting point (fig. 55) resembles a drawing pen, except that the
points are not so sharp. On the back blade, as seen in the engraving, is a
pivot, on which may be placed a dotting wheel, a, resembling the
rowel of a spur ; the screw 5 is for opening the blades to remove the \ jc
wheel for cleaning after use, or replacing it with one of another
character of dot. The cap c, at the upper end of the instrument, is
a box containing a variety of dotting wheels, each producing a dif-
ferent shaped dot. These are used as distinguishing marks for dif-
ferent classes of boundaries on maps ; for instance, one kind of dot
distinguishes county boundaries, another kind town boundaries, a
third kind distinguishes that which is both a county and a town
boundary, &c., &c. In using this instrument, the ink must be in-
serted between the blades above the dotting wheel, so that, as the
wheel revolves, the points shall pass through the ink, each carrying
with it a drop, and marking the paper as it passes. It sometimes
happens that the wheel will revolve many times before it begins to
deposit its ink on the drawing, thereby leaving the first part of the
line altogether blank, and in attempting to go over it again, the first
made dots are liable to get blotted. This evil may be mostly reme-
died by placing a piece of blank paper over the drawing to the very point
the dotted line is to commence at, then begin with drawing the wheel over
the blank paper first, so that by the time it will have arrived at the proper
point of commencement, the ink may be expected to flow over the points
of the wheel, and make the dotted line perfect as required.
Drawing pins (fig. 56) are used to hold paper down upon a
drawing or other board in any required position, and in most
cases answer better than heavy weights, which are frequently
used for that purpose, as the board may be shifted from place Fig ' 56t
DRAWING INSTRUMENTS.
to place without moving the paper. They consist of a brass head, with
a steel point at right angles to its plane. A represents it as seen edgewise,
and B as seen from above.
SCALES.
Fig. 57 represents the usual scale to be found in the common boxes of
drawing instruments. It contains, on its
two sides, simply divided scales, a diagonal
scale and a protractor. The simply divided
scales consist of a series of equal divisions of
an inch, which are numbered 1, 2, 3, &c.,
beginning from the second division on the
left hand.
It will be seen (figure 58) that the dif-
ferent scales are marked 30, 35, 40, &c., and
that the upper part of the left division in
eat;h is subdivided into twelve equal parts,
and that the lower part of the same division
is subdivided into ten equal parts. If now
these last subdivisions or tenths be consid-
ered as units, one mile, or one chain, or one
foot, then each primary division will repre-
sent ten units, ten miles, ten chains, or ten
feet, and the scale is said to be 30, 35, 40 (ac-
cording to the scale selected) miles, chains,
or feet to the inch. Tims, suppose that it
were required on a scale of 30 feet to the
inch, to lay off 47 feet. On the scale marked
30, place one point of the compasses or di-
viders at 4, and bring the other point to the
7th lower subdivisions, counting from the
right, and we have the distance required.
Each of the primary divisions may be re-
garded as unit, one foot for instance ; then
the upper subdivisions are twelfths of a foot
or inches, and the lower subdivisions tenths
of an inch,
rig. ST. In fig. 57, the scales are marked at the
DRAWING INSTUrMK.NTS.
17
left, 1 in. i, , i, but the divisions and subdivisions are as above. In this
fig., the primary divisions are one inch, f , , and J of an inch. These scales
are more generally used for drawings of machinery and of architecture,
ki i, vriojqrrrEi
i IT
i I .1 II I
LA.
17
17
Fig. 58.
while fig. 58 are for topographical drawings. The application of these
scales are similar to those already described. When the primary divisions
are considered inches, then the drawings will be each full, f , , or J size,
according to the scale adopted.
On the selection of the scale. In all working architectural and me-
chanical drawings, use as large a scale as possible ; neither depend, even
in that case, that the mechanics employed in the construction will measure
correctly, but write in the dimensions as far as practicable. For architec-
tural plans, the scale of \ an inch to the foot is one of very general use,
and convenient for the mechanic, as the common two-foot rule carried by
all mechanics is subdivided into iths, |ths, and sometimes sixteenths of an
inch, and the distances on a drawing to this scale can therefore be easily
measured by them. This fact should not be lost sight of in working draw-
ings. When the dimensions are not written, make use of such scales that
the distances may be measured by the division of the common two-foot
rule ; thus, in a scale of \ or \ full size, 6 inches or 3 inches represent
one foot ; in a scale of an inch to the foot or twelfth full size, each \ an
inch represents 6 inches, \ 3 inches ; but when \ or -j- 1 ^ an inch to the foot,
or any similar scale, is adopted, it is evident that these divisions cannot
be taken by the two-foot rule. The scale should be written on every
drawing, or the scale itself should be drawn on the margin. In topographi-
cal and geodesic drawings the latter is essential, as the scale adopted fre-
quently has to be drawn for the specific purpose, and the paper itself con-
tracts or expands with every atmospheric change, and the measurements
will therefore not agree at all times with a detached scale ; and moreover,
a drawing laid down from such detached scale, of wood or ivory, will not
be uniform throughout, for on a damp day the measurements will be too
short, and on a dry day too long. Mr. Holtzapffel has sought to remedy
this inconvenience by the introduction of paper scales ; but all kinds of paper
do not contract and expand equally, and the , error is therefore only par-
tially corrected by his ingenious substitution of one material for another.
18
DRAWING INSTRUMENTS.
5FFP*
B
fffl
m
Hi
m~~~
3
rj_
dT
rW-
J)
1
Fig. 59.
Diagonal scales. The simply divided scales give only two denomina-
tions, primaries and tenths, or twelfths ; but more minute subdivision is
attained by the diagonal scale, which consists of a number of primary
divisions, one of which is divided into tenths, and subdivided into hun-
dredths by diagonal
lines (fig. 59). This
scale is constructed in
the following manner :
Eleven parallel lines
are ruled, enclosing ten
equal spaces ; the length is set off into equal primary divisions, as DE. El,
&c. ; the first DE is subdivided, and diagonals are then drawn from the sub-
divisions between A and B, to those between D and E, as shown in the dia-
gram. Hence it is evident that at every parallel we get an additional tenth
of the subdivisions, or a hundredth of the primaries, and can therefore obtain
a measurement with great exactness to three places of figures. To take a
measurement of (say) 168, we place one foot of the dividers on the primary
1, and carry it down to the ninth parallel, and then extend the other foot
to the intersection of the diagonal, which falls from the subdivision 6, with
the parallel that measures the eight-hundredth part (fig. 60). The pri-
maries may of course be considered as yards, feet, or inches ; and the sub-
n \ 3 , 2 divisions as tenths and
J hundredths of these
respective denomina-
tions.
The diagonals may
be applied to a scale
where only one sub-
division is required.
Thus, if seven lines be
(fig. 61) ruled, enclos-
ing six equal spaces, and the length be divided into primaries, as AB, BC,
&c., the first primary AB may be subdivided into twelfths by two diagonals
A. G s (T running from 6, the mid-
dle of AB, to 12 and 0.
We have here a very con-
venient scale of feet and
inches. From C to 6 is
1 foot 6 inches ; and from
Fig. 60.
7/V
-4- v
/ \x
JO/ \2
/. \,
' US
0/2
DRAWING INSTRUMENTS.
1!)
C on the several parallels to the various intersections of the diagonals, we
obtain 1 foot and any number of inches from 1 to 12.
Plotting scales and rulers are scales of equal parts, with the divisions
placed on a fiducial edge, by which' any length may be pricked off on to the
paper without using the compasses, whose points, by frequent use, destroy
the fineness of the graduation.
On the scale (fig. 57) in common boxes of drawing instruments, the
edge of one side is divided as a protractor, for the laying out of angles.
The instrument, when by itself, consists of a semicircle of thin metal or
horn (fig. 62), whose circumference is divided into 180 equal parts or de-
grees (180). In the larger protractors each of these divisions is sub-
divided.
Application of the protractor. To lay off a given angle from a given
point on a straight line, let the straight line a J) of the protractor coincide
with the given line, and the point c with the given point ; now mark on
the paper against the division on the periphery, coinciding with the angle
required ; remove the protractor, and draw a line through the given point
and the mark.
The instruments already described are those to be found in the usual
cases of drawing instruments, and are sufficient for all the ordinary pur-
pose of draughtsmen ; but there are others adapted to special purpose, or of
careful and elaborate workmanship, which are useful where great accuracy
and finish are required, and of some of which descriptions will be given.
Vernier scales are preferred by some to the diagonal scale already
described. To construct a vernier scale by which a number to three
places may be taken, divide all the primary divisions into tenths, and
20 DRAWING INSTRUMENTS.
number these subdivisions 1, 2, 3, from left to right. Take off now
with the compasses eleven of these subdivisions, set the extent off back-
wards from the end of the first primary division, and it will reach beyond
the beginning of this division, or zero point, a distance equal to one of the
subdivisions. Now divide the extent thus set off into ten equal parts,
marking the divisions on the opposite side of the divided line to the strokes
marking the primary divisions and the subdivisions, and number them 1,
2, 3, &c., backwards from right to left. Then, since the extent of eleven
subdivisions has been divided into ten equal parts, so that these ten parts
exceed by one subdivision the extent of ten subdivisions, each one of these
equal parts, or, as it may be called, one division of the vernier scale, ex-
ceeds one of the subdivisions by a tenth part of a subdivision, or a hun-
dredth part of a primary division ; thus, if the subdivision be considered
10, then frorii to the first division of the vernier will be 11 ; to the sec-
ond, 22 ; to the third, 33 ; to the fourth, M; to the fifth, 55 ; and so on,
66, 77, 88, 99.
10
1 1 1
8 1
rn i i 1 i i i i
i i i i i i TT i
i i i i 1 i i i i
Ann I
1 l 1 1 (
*Pg_J Si * 2
Fig. 63.
To take off the number 253 from this scale, place one point of the di-
viders at the third division of the vernier ; if the other point be brought to
the primary division 2, the distance embraced by the dividers will be 233,
and the dividers must be extended to the second subdivision of tenths to
the right of 2. If the number were 213, then the dividers would have to
be closed to the second subdivision of tenths to the left of 2. To take off
the number 59 from the scale, place one point of the dividers at the ninth
division of the vernier ; if the other point be extended to the mark, the
dividers will embrace 99, and must therefore be closed to the fourth subdi-
vision to the left of 0.
These numbers, thus taken, may be 253, 25-3, 2*53 ; 213, 21*3, 2*13 ; 59,
5*9, .59, according as the primary divisions are taken as hundreds, tens, or
units.
The construction of this scale is similar to that of the verniers of theod-
olites and surveying instruments ; but, in its application to drawing, is not
as simple as the diagonal scales, figs. 59, 61.
On some of the plain scales in the instrument boxes will be found divi-
sions marked as in fig. 64. Many of the divisions here laid down have no
application to drawing, according to the scope of this work ; a brief ex-
planation and application will therefore only be given. Under definitions
DRAWING INSTRUMENTS.
and technicalities, the signification of the terms chords, tangents, sines, and
secants, has been defined. The chord of 60 is equal to radius, or half
Rim
Laii
CIw
Siit,
Tan
S\)
Fig. 65.
Fig. 64
the diameter. The line of chords is used to set off an angle, or to measure
an angle already laid down.
To set off an angle. An angle of 35 for instance : open the compasses
to the extent of 60 on the scale of chords, setting one point at A on the
line A B, describe with the other point an arc ; again with the compasses
open to the extent of 35 on the scale, setting one point on B, describe an
arc, cutting the arc B C ; through this intersection and the point A,
draw the line A C, and we have the angle CAB, 35.
To measure the angle contained by the straight lines A B and A C al-
ready laid down. , Open the compasses to the extent
of 60 on the line of chords, as before, and with this
radius describe the arc B C, cutting A B and A C,
produced, if necessary, in the points B and C ; then,
extending the compasses from B to 0, place one
point of the compasses on the beginning or zero
point, of the line of chords, and the other point will extend to the number
upon this line, indicating the degrees in the angle BAG.
The lines of sines, secants, tangents, and semitangents are principally
used for the several projections, or perspective representations, of the circles
of the sphere, by means of which maps are constructed.
The line of rhumbs is a scale of the chords of the angles of deviation
from the meridian denoted by the several points and quarter points of the
compass, enabling the navigator, without computation, to lay down or
measure a ship's course upon a chart.
The line of longitudes shows the number of equatorial miles in a de-
gree of longitude on the parallels of latitude indicated by the degrees on
the corresponding points of the line of chords. Example. A ship in lati-
tude 60 K. sailing E. Y9 miles, required the difference of longitude be-
tween the beginning and end of her course. Opposite 60 on the line of
chords stands 30 on the line of longitudes, which is, therefore, the number
of equatorial miles m a degree of longitude at that latitude. Hence, ||=
2 38', the required difference of longitude.
DRAWING INSTRUMENTS.
The sector (fig. 66) consists of two flat rulers united by a central joint,
and opening like a pair of compasses. It carries several plain scales on its
faces, but its most important lines are in pairs,
running accurately to the central joint.
Plain scales on the sector. On the outer edge
of the sector is usually given a decimal scale from
1 to 100 ; and in connection with it, on one of the
sides, a scale of inches and tenths. These are
identical with the lines on the plain scale, previ-
ously mentioned, but the latter are more commo-
diously placed for use. On the other side we have
logarithmic lines of numbers, sines, and tangents.
Sectoral double scales. These are respectively
named the Lines of Lines, Chords, Secants, Sines,
and Tangents. These scales have one line on each
ruler, and the two lines converge accurately in the
central joint of the sector.
The principle on which the double scales are
constructed is, that similar triangles have their
like sides proportional. Let the lines A B, A C,
represent the legs of the sector, and
A D, A E, two equal sections from
the centre ; then, if the points B C
and D E be connected, the lines B C
and D E will be parallel ; therefore,
the triangles A B C, A D E, wiU be
similar, and consequently the sides
A B, B C, A D, D E, proportional,
that is, as A B : B C : : A D : D E ;
so that if A D be the half, third, or
fourth part of A B, then D E will
be a half, third, or fourth part of
B C ; and the same holds of all the
rest. Hence, if D E be the chord, sine, or tangent
Fig. ee. of any arc, or of any number of degrees to the
radius A D, then B C will be the same to the radius A B. Thus at every
opening of the sector, the transverse distances D E and C B from one ruler
to another, are proportional to the lateral distances, measured on the lines
A B, A C. It is to be observed, that all measures are to be taken from
the inner lines, since these only run accurately to the centre.
DRAWING INSTRUMENTS. 23
The line of lines, marked L on each leg of the sector. This is a line
of 10 primaries, each subdivided into tenths, thus making 100 divisions.
Its use is, to divide a given line into any number of equal parts ; to give
accurate scale measures for the construction of a drawing ; to form any
required scale ; to divide a given line in any assigned proportion ; and to
find third, fourth, and middle proportionals to given right lines.
To divide a given line into eight equal parts. Take the line in the com-
passes, and open the sector so as to apply it transversely to 8 and 8, then
the transverse from 1 to 1 will be the eighth part of the line.
To form any required scale of equal parts. Take one inch in the com-
passes, and open the sector, till this extent becomes a transverse distance
at the division indicating the number of parts in an inch of the required
scale.
Example. To adjust the sector as a scale of one inch to four chains.
Make one inch the transverse distance of 4 and 4 ; then the transverse
distances of the other corresponding divisions and subdivisions will repre-
sent the number of chains and links indicated by these divisions : thus, the
transverse distance from 3 to 3 will represent three chains.
To construct a scale of feet and inches in such a manner, that an ex-
tent of three inches shall represent twenty inches. Make three inches a
transverse distance between 10 and 10, and the transverse distance of 6 and
6 will represent 12 inches. Set off this extent, divide it into 12 equal parts,
each of these divisions will represent an inch. Place the figure at the
right, and set off again the extent of the whole twelve parts, from to 1,
1 to 2, &c., to represent the feet.
Proportion. Two lines being given, to find a third proportional.
Example. The given lines = 2 and 6, a third proportional required.
Take between the compasses the lateral distance of the second term 6 on any
convenient scale, and open the sector until this distance becomes the trans-
verse distance to the first term 2 ; then the transverse distance of the second
term 6, measured upon the same scale as the former, will equal 18, the
third proportional required.
Example. to find a fourth proportional to the numbers 2, 6, and 10.
Take the lateral distance of the second term 6, from any convenient
scale of equal parts, and open the sector until that quantity, or any aliquot
part thereof, becomes the transverse distance of the first term 2, then the
transverse distance of the third term 10, taken from the same scale of equal
parts, will give 30, the fourth proportional required.
Line of Chords, marked C on each leg of the sector. The double scales
of chords upon the sector are more useful than the single line of chords de-
24 DRAWING INSTRUMENTS.
scribed on the plane scale ; for on the sector, the radius with which the arc
is to be described may be of any length less than the transverse distance
of 60 and 60 when the legs are opened as far as the instrument will admit
of. But with the chords on the plane scale, the arc described must be
always of the same radius.
To protract an angle BAG, which shall contain a given number of de-
grees, suppose 36.
Make the transverse distance of 60 and 60 equal to the length of the
radius of the circle, and with that opening de-
scribe the arc B C.
Take the transverse distance of the given de-
grees 36, and lay this distance on the arc from
the point B to C.
From the centre A of the arc, draw A C,
rig. 68. A B, and these two lines will contain the angle
required.
To protract an angle of more than 60, divide the required angle by 2
or 3, and set off as above twice or thrice the arc.
From what has been said about the protracting of an angle to contain
a given number of degrees, it will easily be seen how to find the degrees
(or measure) of an angle already laid down.
Line of Polygons. The line of polygons is chiefly useful for the ready
division of the circumference of a circle into any number of equal parts
from 4 to 12 ; that is, as a ready means to inscribe regular polygons of any
given number of sides, from 4 to 12, within a given circle. To do which,
set off the radius of the given circle (which is always equal to the side of
an inscribed hexagon) as the transverse distance of 6 and 6 upon the line
of polygons. Then the transverse distance of 4 and 4 will be the side of a
square ; the transverse of 5 and 5 the side of a pentagon.
If it be required to form a polygon, upon a given right line set off the
extent of the given line, as a transverse distance between the points upon
the line of polygons, answering to the number of sides of which the poly-
gon is to consist, as for a pentagon between 5 and 5, or for an octagon be-
tween 8 and 8 ; then the transverse distance between 6 and 6 will be the
radius of a circle, whose circumference would be divided by the given line
into the number of sides required.
All regular polygons, whose number of sides will exactly divide 360
(the number of degrees into which all circles are supposed to be divided)
without a remainder, may likewise be set off upon the circumference of a
circle by the line of chords. Thus, take the radius of the circle between
DRAWING INSTRUMENTS. 25
the compasses, and open the sector till that extent becomes the transverse
distance between 60 and 60 upon the line of chords ; then having divided
360 by the required number of sides, the transverse distance between the
numbers of the quotient will be the side of the polygon required. Thus,
for an octagon, take the distance between 45 and 45 ; and for a polygon
of 36 sides, take the distance between 10 and 10, &c.
Lines of sines, tangents and secants. Given, the radius of a circle,
required the sine and tangent of 28 30' to that radius
Open the sector, so that the transverse distance of 90 and 90 on the
sines, or of 45 and 45 on the tangents, may be equal to the given radius ;
then will the transverse distance of 28 30', taken from the sines, be the
length of that sine to the given radius ; or if taken from the tangents, will
be the length of that tangent to the given radius.
But if the secant of 28 30' was required
Make the given radius a transverse distance of and 0, at the begin-
ning of the line of secants, and then take the transverse distance of the de-
grees wanted, viz., 28 30'.
A tangent greater than 45 degrees (suppose 60) is found thus :
Make the given radius a transverse distance to 45, and 45 at the begin-
ning of the scale of upper tangents, and then the required degrees (60) may
be taken from the scale.
Given the length of the sine, tangent, or secant of any degrees, to find
the length of the radius to that sine, tangent, or secant.
Make the given length a transverse distance to its given degrees on its
respective scale. Then
If a sine, ) ( 90 and 90 on the sines ) will be
If a tangent under 45, ( the transverse ) 45 and 45 on the tangents ( the ra-
If a tangent above 45, f distance of J 45 and 45 on the upper tangents f dius
If a secant, ( and on the secants ) sought.
To find the length of a versed sine to a given number of degrees, and a,
given radius.
Make the transverse distance ' of 90 and 90 on the sine equal to the
given radius. Take the transverse distance of the complement of the sine
of the given number of degrees. If the given number of- degrees is less
than 90, subtract the complement of the sine from the radius, the remain-
der will be the versed sine.
If the given number of degrees are more than 90, add the complement
of the sine to the radius, and the sum will be the versed sine.
To open the legs of a sector, so that the corresponding double scales of
lines, chords, sines, tangents, may make each a right angle.
4 6 DRAWING INSTRUMENTS.
On the line of lines, make the lateral distance 10, a transverse distance
between 8 on one leg and 6 on the other leg.
On the line of sines, make the lateral distance 90, a transverse distance
from 45 to 45, or from 40 to 50, or from 30 to 60, or from the sine of any
degrees to their complement.
On the line of tangents, make the lateral distance of 45 a transverse
distance between 30 and 30.
Marquois*s scales (fig. 69). These scales consist of a right-angled tri-
angle, of which the hypothenuse or longest side is three times the length
of the shortest, and a rectangular rule. Our figure, which is drawn one-
third the actual size of the instruments from which it is taken, repre-
sents the triangle and a rule, as being used to draw a series of paral-
lel lines. The rule is one foot long, and has, parallel to each of its
edges, two scales, one placed close to the edge, and the other immediately
Fig. 69.
within this, the outer being termed the artificial, and the inner the natural
scale. The divisions upon the outer scale are three times the length of
those upon the inner scale, so as to bear the same proportion to each other
that the longest side of the triangle bears to the shortest. In the artificial
scales, the zero point is placed in the middle of the edge of the rule, and
the primary divisions are numbered both ways from this point to the two
ends of the rule, and are every one subdivided into ten equal parts, each
of which is, consequently, three times the length of a subdivision of the
corresponding natural scale.
The triangle has a short line drawn perpendicular to the hypothenuse
near the middle of it, to serve as an index or pointer ; and the longest of
the other two sides has a sloped edge.
To draw a line parallel to a given line, at a given distance from it. 1.
Having applied the given distance to the one of the natural scales which
is found to measure it most conveniently, place the triangle with its sloped
DRAWING INSTRUMENTS. 27
edge coincident with the given line, or rather at such small distance from
it, that the pen or pencil passes directly over it when drawn along this
edge. 2. Set the rule closely against the hypothenuse, making the zero
point of the corresponding artificial scale coincide with the index upon the
triangle. 3. Move the triangle along the rule, to the left or right accord-
ing as the required line is to be above or below the given line, until the
index coincides with the division or subdivision corresponding to the num-
ber of divisions or subdivisions of the natural scale, which measures the
given distance ; and the line drawn along the sloped edge in its new posi-
tion will be the line required.
The natural scale may be used advantageously in setting off the dis-
tances in a drawing, and the corresponding artificial scale in drawing
parallels at required distances.
The advantages of Marquois's scales are : 1st, that the sight is greatly
assisted by the divisions on the artificial scale being so much larger than
those of the natural scale to which the drawing is constructed ; 2d, that
any error in the setting of the index produces an error of but
one-third the amount in the drawing.
If the triangle be accurately constructed, these scales may
be advantageously used for dividing lines with accuracy and
despatch.
Triangular compasses. Fig. 70 represents this instrument
closed up. That which appears in the fig. as one limb A, con-
sists of a pair of compasses of the ordinary construction. The
single point limb B has a compass joint at <z, by which its point
may be opened at right angles to the plane of the pair of com-
passes A, when the three points will form a triangle. The com-
pass joint a is firmly attached to the centre of the compasses A,
which, by means of a nut and screw 5, may be turned round
without moving the limbs, of which it is the centre. The double
motion thus given to the point limb B (both at right angles to,
and parallel to the plane of the. compasses A), partakes of the
nature of a universal joint, and enables the three points of the
instrument to be placed at the angular points of any shaped tri-
angle whatever. This instrument is chiefly useful in transfer-
ring of points from one paper to another. The two points of
the compasses A being set upon such points of the drawing as have been
already copied, the third, B, is brought upon any other point ; then, by
applying the points A to the corresponding points on the copy, the point
B will establish the other and new point on the copy.
28 DRAWING INSTRUMENTS.
Wholes and halves. For copying and reducing drawing to half size,
compasses called wholes arid halves are used (fig. 71), in which the longer
legs being twice the length of the shorter, when the former are opened to
any given line, the shorter ones will be #'
opened to the half of that line. By their
means then, all the lines of a drawing may
be reduced to one-half, or enlarged to
double their length. These compasses are
also useful for dividing lines by continual
bisections.
The proportional compasses (fig. Y2) are
somewhat similar in their construction to
wholes and halves, but of more varied ap-
plication. The principle is the same, with
this difference, that the screw-joint C
passes through slides moving in the slots
of the bars, and admits of the centre being
rig. 71. adjusted for various relative proportions
between the openings A B and D E. Different sets of num-
bers are engraved on the outer faces of the bare, and by these
the required proportions are obtained. The instrument must Fi(r 72
be closed for adjustment, and the nut C loosened ; the slide is then moved
in the groove, until a mark across it, named the index, coincides with the
number required ; which done, the nut is tightened again.
The scales usually engraved on these compasses are named Lines,
Circles, Planes, and Solids.
The scale of lines is numbered from 1 to 10, and the index of the slide
being brought to any one of these divisions, the distance D E will measure
A B in that proportion. Thus, if the index be set to 4, D E will be con-
tained four times in A B.
The line of circles extends from 1 to 20, and the index being set to
(say) 10, D E will be the tenth part of the circumference of the circle,
whose radius is A B.
The line of planes, or squares, determines the proportion of similar
areas. Thus, if the index is placed at 3, and the side of any one square be
taken by A B from a scale of equal parts, D E will be the side of another
square of one-third the area. And if any number be brought to the index,
and the same number be taken by A B from a scale of equal parts, D E
will be the square root of that number. And in this latter case, D E will
also be a mean proportional between any two numbers, whose product is
equal to A B,
DRAWING INSTRUMENTS. 29
The line of solids expresses the proportion between cubes and spheres.
Thus, if the index be set at 2, and the diameter of a sphere, or the side of
a cube, be taken from a scale of equal parts by A B, then will D E be a
diameter or side of a sphere or cube of half the solidity. And if the slide
be set to (say) 8, and the same number be taken from a scale of equal
parts, then will D E measure 2 on the same scale, or the cube root of 8.
Beam compasses (fig. 73). When it is required to set off with accuracy
distances of considerable extent, or describe arcs of over a foot radius, the
beam compass is used. This instrument consists of a beam, A A, of any
length required, generally made of well-seasoned mahogany ; upon its
face is inlaid throughout its whole length a slip of holly or boxwood, a a,
upon which are engraved the divisions or scale, either feet and decimals or
inches and decimals, or whatever particular scale may be required ; but
ordinary beam compasses are constructed with a plain beam, with no scale
whatever. Two brass boxes, B and C, are adapted to the beam ; the latter
may be moved, by sliding, to any part of its length, and fixed in position
by tightening the clamp screw E. Connected with the brass boxes are
the two points of the instrument G and H, which may have any extent of
opening by 'sliding the box C along the beam, the other box, B, being
firmly fixed at one extremity. The object to be attained ia the use of this
instrument, is the nice adjustment of the points G H to any definite dis-
tance apart ; this is accomplished by two vernier of reading plates 5 c, each
fixed at the side of an opening in the brass boxes to which they are attached,
and afford the means of minutely subdividing the principal divisions a a
on the beam, which appear through those openings. D is a clamp screw
for a similar purpose as the screw E, namely, to fix the box B, and prevent
motion in the point it carries after adjustment to position. F is a slow
motion screw, by which the point G may be moved any very minute quan-
tity for perfecting the setting of the instrument, after it has been set as
nearly as possible by the hand alone.
The method of setting the instrument for use may be understood from
the above description of its parts, and also by the following explanation of
the method of examining and correcting the adjustment of the vernier J,
30
DRAWING INSTRUMENTS.
which will occasionally get deranged ; this verification must be done by
means of a detached scale. Tims, suppose, for example, that our beam
compass is divided to feet, inches, and tenths, and subdivided by the ver-
nier to hundredths, &c. First set the zero division of the vernier to the
zero of the principal divisions on the beam, by means of the slow motion
screw F. This must be done very nicely. Then slide the box C, with its
point G, till the zero on the vernier c exactly coincides with any principal
division on the beam, as twelve inches or six inches, &c., which must also
be done very accurately ; then apply the points to a similar detached
scale, and if the adjustment is perfect, the interval of the points G H will
measure on it the distance to which they were set on the beam. If they
do not by ever so small a quantity, it should be corrected by turning the
screw F till the points exactly measure that quantity on the detached scale ;
then, by loosening the little screws which hold the vernier I in its place,
the position of the vernier may be gradually changed, till its zero coincides
with the zero on the beam, and then tightening the screws again, the ad-
justment will be complete.
Pat-table or turn-in compasses (fig. 74) comprise in themselves a port-
able case of drawing instruments, consisting of a large pair of compasses
with movable points, which are
also so contrived, that one forms
in itself a small pencil bow, the
other a pen bow ; and when the
whole instrument is put together
and folded up, they occupy but a
space three inches long, and may
be carried in the pocket without
being an incumbrance.
Fig. 74: represents the instru-
ment when all its parts are to-
gether. The principal legs of the
instrument are F and G, movable
as usual by a joint at A. The
lower joints, B and C, afford the
means of setting the point limbs D and E perpendicular to the paper.
Each of the point limbs may be removed from the legs F and G, and
by means of their joints B and C, form perfect instruments, the one a pen
bow represented at H, and the other a pencil bow, shown at I K ; the point
limbs of these lesser instruments are all adapted to slide into the principal
legs F and G of the larger one, which are made hollow for their reception.
Fig. 74.
DRAWING INSTRUMENTS.
31
It may easily be seen from the engraving, that by reversing either of
the points in the principal instrument, it may be supplied with a pen or
pencil as may be required, leaving the other fine or plain point E or D to
act as a centre.
Mr. Brunei has introduced what are called Tubular Compasses, in
which the upper part of the legs lengthens out like the slide of a telescope,
thus giving greater extent of radius when required. The movable legs are
double, having points at one end, and a pencil or pen at the other ; and
they move on pivots, so that the pen or pencil can be instantly substituted
for the points, or vice versa, and that with the certainty of a perfect adjust-
ment. The design is very ingenious, and offers many conveniences, but
the instrument is too delicate for ordinary hands. Without extreme care
it is soon disarranged.
Large, screw dividers (fig. T5) are used for accurately dividing lines
into a definite number of equal parts, or for setting off equal distances.
A is the centre about which the legs A
and A B open or shut. B and C are
joints, by which the point limbs may be
set perpendicular as usual ; the extent or
opening between the points is regulated
by a screw passing through a socket
F, and terminated at the other extremity
by a milled head E, by which the screw
is turned round. Between this milled
head and the nearest point limb is fixed
what is called a micrometer head, deci-
mally divided round its outer or cylindri-
cal edge. One turn of the screw carries
the micrometer head completely round ;
therefore, when part of a turn only is
given to the screw, the divisions on the head show what fraction of a turn
has been given, and if it be known what number of turns or-threads of the
screw are equal to one inch, the points of these compasses may be thus set
to any small definite measure of length with the utmost precision. The
index or zero for reading the fraction of a turn of the screw is marked on
the point limb below J3. Thus this instrument may be considered as a
beam compass of small dimensions and minute accuracy.
The circular protractor (fig. 76) is one of the best kind of protractors.
It is a complete circle, A A, connected with its centre by four radii, aaaa,
The centre is left open, and surrounded by a concentric ring or collar, 5,
Fig. T5.
32 DRAWING INSTRUMENTS.
which carries two radial bars, c c. To the extremity of one bar is a pinion,
d, working in a toothed rack quite round the outer circumference of the
protractor. To the opposite extremity of the other bar, <?, is fixed a vernier,
which subdivides the primary divisions on the protractor to single minutes,
and by estimation to 30 seconds. This vernier is carried round the pro-
tractor by turning the pinion d. Upon each radial bar c c, is placed a
branch e e, carrying at their extremities a fine steel pricker, whose points
are kept above the surface of the paper by springs placed under their sup-
ports, which give way when the branches are pressed downwards, and
allow the points to make the necessary punctures in the paper. The
branches e e are attached to the bars c c, with a joint which admits of their
being folded backwards over the instrument when not in use, and for pack-
ing in its case. The centre of the instrument is represented by the inter-
section of two lines drawn at right angles to each other on a piece of plate
glass, which enables the person using it to place it, so that the centre, or
intersection of the cross lines, may coincide with any given point on the
plan. If the instrument is in correct order, a line connecting the fine
pricking points with each other would pass through the centre of the in-
strument, as denoted by the before-mentioned intersection of the cross
lines upon the glass. In using this instrument, the vernier should first be
set to zero (or the division marked 360) on the divided limb, and then
Fig. 76.
placed on the paper, so that the two fine steel points may be on the given
line (from whence other and angular lines are to be drawn), and the centre
of the instrument coincides with the given angular point on such line. This
done, press the protractor gently down, which will fix it in position by
means of very fine points on the under side. It is now ready to lay off the
DRAWING INSTRUMENTS.
33
given angle, or any number of angles that may be required, which is done
by turning the pinion d till the opposite vernier reads the required angle.
Then press downwards the branches e e, which will cause the points to
make punctures in the paper at opposite sides of the circle ; which being
afterwards connected, the line will pass through the given angular point,
if the instrument was first correctly set. In this manner, at one setting of
the instrument, a great number of angles may be laid off from the same
point.
It is not essential that the centre be over the given point, when applied
to the given line-, provided the pricking points exactly fall upon the line,
for the inclined line may be transferred to pass through the given angular
point by a parallel ruler.
The pentagraph (fig. 77) is used for the copying of drawings either on
Fig. 77.
the same scale, on a reduced scale, or on an enlarged scale, as may be re-
quired. It is represented (fig. 77) as in the act of copying a plan H, upon
3
34 DRAWING INSTRUMENTS.
a reduced scale h. The pentagraph consists of four rulers. A, B, C, and D,
made of stout brass. The two longer rulers, A and B, are connected to-
gether by, and have a motion round a centre, shown at the upper part of
the engraving. The two shorter rulers are, in like manner, connected with
each other, and with the longer rulers. The whole instrument is supported
by small pillars resting upon ivory castors, a a a, &c., which have a motion
in all directions. The rulers A and C have each an equal number of simi-
lar divisions, marked , , &c. ; and likewise a sliding index, E and F,
which can be fixed to any divisions on the ruler by a milled-headed clamp
screw shown in the engraving. The sliding indexes, E and F, have each
of them a tube adapted to slide on a pin, rising from a heavy circular
weight called the fulcrum, which acts as a centre for the whole instrument
to turn upon when in use, or to receive a sliding holder with a pencil or a
tracing point, as may be required.
To explain the method of using the instrument, the engraving repre-
sents the instrument in the act of reducing a plan to a scale of one half the
original. For this purpose the tracing point is fixed in a socket at G, over
the original drawing H. The pencil is placed in a similar tube or socket at
F, over the paper, to receive the copy ; and the fulcrum is fixed to that at E,
the scale being one half the original. The sliding indices were first clamped
at those divisions on the rulers marked |. The instrument being thus set for
use, if correct, the three points, E, F, and G, will be in one straight line,
as shown by the dotted line in the figure. This will invariably be the case
at whatever division the indices may be set to. Now, if the tracing point
G be passed delicately and steadily over every line of the plan H, a true
copy, but of one half the scale of the original, will be marked by the pencil
at F on the paper h beneath it. The fine thread represented as passing
from the pencil quite round the further extremity of the instrument to the
tracer at G, is to enable the draftsman at the tracing point to raise the
pencil from the paper, whilst he passes the tracer from one part of the ori-
ginal to another, and prevents false lines being made on the copy. Like-
wise, it may be noticed, that the pencil holder F is represented as sur-
mounted by a cup, which is for the purpose of putting some small shot in,
to press the pencil heavier upon the paper, whenever such expedient may
be found necessary.
If the object had been to enlarge the drawing to double its scale, then
the tracer must have been placed at F, and the pencil at G. And if a copy
be required, retaining the scale of the original, then the slides E and F
must be placed at the divisions marked 1. The fulcrum must take the
middle station, and the pencil and tracer those on the exterior rules A and
B of the instrument.
DRAWING INSTRUMENTS. 35
The camera lucida is sometimes used for copying and reducing topo-
graphical drawings. A description of the use of this instrument will be
found under the head of topographical drawing.
The drawing table and drawing board. The usual size of the drawing
table should be from 5 to 6 feet long, and 3 feet wide, of li or 2 inch white
pine plank well seasoned, without any knots, closely joined, glued, dowelled,
and clamped. It should be fixed on a strong firm frame and legs, and of
such a height that the draughtsman, as he stands up, may not have to
stoop to his work. The table is usually provided with a shallow drawer to
hold paper or drawings. Drawing tables are made portable, by having
two horses for their supports, and a movable drawing board for the top ;
this board is made similar to the top of the drawing table, but of inch
boards, and barred at the ends. Various woods are used for the purposes,
but white pine is by far the cheapest and best. Drawing boards should be
made truly rectangular, and with perfectly straight sides for the use of the
T square. Two sizes are sufficient for common purposes, 41 x 30 inches
to carry double elephant paper with a margin, and 31 X 24 inches for im-
perial and smaller sizes. Boards smaller than this are too light and un-
steady in handling.
Small boards are occasionally made, as loose panels fitting into a frame,
flush on the drawing surface, with buttons on the back to secure it in
position. The panel is mostly of white pine, with a hard-wood frame.
DRAWING- PAPER.
Drawing paper, properly so called, is made to certain standard sizes as
follows :
Demy . v ... 20 inches by 15 i inches.
Medium, ' . .' 22 " ITi "
Eoyal, . ... 24 " 19 \ "
Super Eoyal, ^_'. . 2Ti " 19i "
Imperial, . . \ 30 " 22 "
Elephant, . ... 28 " 23 "
Columbier, 1 ". 35 " 23i "
Atlas, . . ... 34 26 "
Double Elephant, . 40 27 "
Antiquarian, . . .53 " 31 "
Emperor, ... 68 " 48 "
Of these, Double Elephant is the largest in common use by engineers, and
it is the most generally useful size of sheet. Demy and Imperial are the
36 DRAWING INSTRUMENTS.
only other sizes worth providing for a drawing establishment. Whatman's
white paper is the quality most usually employed for finished drawings ;
it will bear wetting and stretching without injury, and when so treated,
receives color readily. For ordinary working drawings, where damp-
stretching is dispensed with, cartridge paper, of a coarser, harder, and
tougher quality, is preferable. It bears the use of indiarubber better, re-
ceives ink on the original undamped surface more freely, shows a fully
better line, and as it does not absorb very rapidly, tinting lies better and
more evenly upon it. For delicate small-scale line-drawing, the thick
blue paper, such as is used for ledgers, &c., imperial size, answers exceed-
ingly well; but it does not bear damp-stretching without injury, and
should be merely pinned or waxed down to the board. "With good man-
agement, there is no ground to fear the shifting of the paper. Good letter
paper receives light drawing very well ; of course it does not bear much
fatigue.
Large sheets, destined for rough usage and frequent reference, should
be mounted on linen, previously damped, with a free application of paste.
Tracing paper is a preparation of tissue paper, transparent and quali-
fied to receive ink lines and tinting without spreading. When placed
over a drawing already executed, the drawing is distinctly visible through
the paper, and may be copied or traced directly by the ink-instruments ;
thus an accurate copy may be made with great expedition. Tracings may
be folded and stowed away very conveniently ; but, for good service, they
should be mounted on cloth, or on paper and cloth, with paste.
Tracing paper may be prepared from thick tissue paper, by sponging
over one surface with a mixture of one part raw linseed oil and five spirits
of turpentine ; five gills of turpentine and one of oil will go over from
forty to fifty sheets of paper.
Tracing cloth is a similar preparation of linen, and is preferable for its
toughness and durability.
Mouth Glue, for the sticking of the edges of drawing paper to the
board, is made of glue and sugar or molasses ; it melts at the temperature
of the mouth, and is convenient for the draughtsman.
Drawing paper may be fixed down on the drawing board by the pins
at the corners, by weights, or by gluing the edges. The first is sufficient
when no shading or coloring is to be applied, and if the sheet is not to be
a very long time on the board ; and it has the advantage of preserving
the paper in its natural state. For shaded or tinted drawings, the paper
must be damped and glued at the edges, as the partial wetting of paper,
loose or fixed at the corners merelyj by the water colors distorts the surface.
DRAWING INSTRUMENTS. 37
Damp-stretching is done as follows : The edges of the paper should
first be cut straight, and, as near as possible, at right angles with each
other ; also the sheet should be so much larger than the intended drawing
and its margin, as to admit of being afterwards cut from the board, leaving
the border by which it is attached thereto by glue or paste, as we shall
next explain.
The paper must first be thoroughly and equally damped with a sponge
and clean water, on the opposite side from that on which the drawing is to
be made. When the paper absorbs the water, which may be seen by the
wetted side becoming dim, as its surface is viewed slantwise against the light,
it is to be laid on the drawing board with the wetted side downwards, and
placed so that its edges may be nearly parallel with those of the board ;
otherwise, in using a T square, an inconvenience may be experienced.
This done, lay a straight flat ruler on the paper, with its edge parallel to,
and about half an inch from one of its edges. The ruler must now be held
firm, while the said projecting half inch of paper be turned up along its
edge ; then, a piece of solid or mouth glue, having its edge partially dis-
solved by holding it in boiling or warm water for a few seconds, must be
passed once or twice along the turned up edge of the paper, after which,
by sliding the ruler over the glued border, it will be again laid flat, and
the rule being pressed down upon it, that edge of the paper will adhere to
the board. If sufficient glue has been applied, the ruler may be removed
directly, and the edge finally rubbed down by an ivory book-knife, or by
the bows of a common key, by rubbing on a slip of paper placed on the
drawing paper, so that the surface of the latter may not be soiled, which
will then firmly cement the paper to the board. This done, another but
adjoining edge of the paper must be acted upon in like manner, and then
the remaining edges in succession ; we say the adjoining edges, because
we have occasionally observed, that when the opposite and parallel edges
have been laid down first, without continuing the process progressively
round the board, a greater degree of care is required to prevent undula-
tions in the paper as it dries.
Sometimes strong paste is used instead of glue ; but as this takes a
longer time to set, it is usual to wet the paper also on the upper surface to
within an inch of the paste mark, care being taken not to rub or injure the
surface in the process. The wetting of the paper in either case is done for
the purpose of expanding it ; and the edges being fixed to the board in its
enlarged state, act as stretchers upon the paper, while it contracts in dry-
ing, which it should be allowed to do gradually. All creases or undula-
38 DRAWING INSTRUMENTS.
tions by this means disappear from the surface, and it forms a smooth
plane to receive the drawing.
To remove the paper after the drawing is finished, cut off inside the
pasted edge, and remove the edge by warm water and the knife.
With panelled boards, the panel is taken out, and the frame inverted ;
the paper, being first damped on the back with a sponge slightly charged
with water, is applied equally over the opening to leave equal margins,
and is pressed and secured into its seat by the panel and bars.
MOUNTING PAPER AND DRAWINGS, VARNISHING, ETC.
In mounting paper upon canvas, the latter should be well stretched
upon a smooth flat surface, being damped for that purpose, and its edges
glued down, as was recommended in stretching drawing paper. Then with
a brush spread strong paste upon the canvas, beating it in till the grain of
the canvas be all filled up ; for this, when dry, will prevent the canvas
from shrinking when subsequently removed ; then, having cut the edges
of the paper straight, paste one side of every sheet, and lay them upon the
canvas sheet by sheet, overlapping each 6ther a small quantity. If the
drawing paper is strong, it is best to let every sheet lie five or six minutes
after the paste is put on it, for as the paste soaks in, the paper will stretch,
and may be better spread smooth upon the canvas ; whereas, if it be laid
on before the paste has moistened the paper, it will stretch afterwards and
rise in blisters when laid upon the canvas. The paper should not be cut
off from its extended position till thoroughly dry, which should not be
hastened, but left in a diy room to do so gradually, if time permit ; if not,
it may be exposed to the sun, unless in the winter season, when the help
of a fire is necessary, provided it is not placed too near a scorching heat.
In joining two sheets of paper together by overlapping, it is necessary,
in order to make a neat joint, to feather edge each sheet ; this is done by
carefully cutting with a knife half way through the paper near the edges,
and on the sides which are to overlap each other ; then strip off a feather-
edged slip from each, which, if done dexterously, will form a very neat
and efficient joint when put together.
For mounting and varnishing drawings or prints, stretch a piece of linen
on a frame, to which give a coat of isinglass or common size, paste the back
of drawing, which leave to soak, and then lay it on the linen. When dry,
give it at least four coats of well made isinglass size, allowing it to dry be-
DRAWING INSTRUMENTS. 39
tween each coat. Take Canada balsam diluted with the best oil of turpen-
tine, and with a clean brush give it a full flowing coat.
MANAGEMENT OF THE INSTRUMENTS.
In constructing preparatory pencil-drawings, it is advisable, as a rule
of general application, to make no more lines upon the paper than are ne-
cessary to the completion of the drawing in ink ; and also to make these
lines just so dark as is consistent with the distinctness of the work. With
respect to the first idea, it is of frequent application : in the case, for
example, of the teeth of spur wheels, where, in many instances, all that is
necessary to the drawing of their end view in ink are three circles, one
of them for the pitch line, and the two others for the tops and bottoms
of the teeth ; and again, to draw the face view of the teeth, that is, in the
edge view of the wheel, we have only to mark off by dividers the positions
of the lines which compose the teeth, and draw four pencil lines for the
two sides, and the top and bottom of the elevation. And here we may
remark the inconvenience of that arbitrary rule, by which it is by some in-
sisted that the pupil should lay down in pencil every line that is to be
drawn, before finishing it in ink. It is often beneficial to ink in one part
of a drawing, before touching other parts at all ; it prevents confusion,
makes the first part of easy reference, and allows of its being better done,
as the surface of the paper inevitably contracts dust, and becomes other-
wise soiled in the course of time, and therefore the sooner it is done with
the better.
Circles and circular arcs should, in general, be inked in before straight
lines, as the latter may be more readily drawn to join the former, than the
former the latter. When a number of circles are to be described from one
centre, the smaller should be inked first, while the centre is in better con-
dition. When a centre is required to bear some fatigue, it should be pro-
tected with a thickness of stout card glued or pasted over it, to receive the
compass-leg.
Indiarubber is the ordinary medium for cleaning a drawing, and for cor-
recting errors in the pencil. For slight work it is quite suitable ; that sub-
stance, however, operates to destroy the surface of the paper ; and by re-
peated application, it so ruffles the surface, and imparts an unctuosity to it,
as to spoil it for fine drawing, especially if ink shading or coloring is to be ap-
plied. It is much better to leave trivial errors alone, if corrections by the
pencil may be made alongside without confusion ; as it is, in such a case,
time enough to clear away superfluous lines when the inking is finished.
40 DRAWING INSTRUMENTS.
For cleaning a drawing, a piece of bread two days old is preferable to
indiarubber, as it cleans the surface well and does not injure it. "When ink
lines to any considerable extent have to be erased, a small piece of damped
soft sponge may be rubbed over them till they disappear. As, however,
this process is apt to discolor the paper, the sponge must be passed through
clean water, and applied again to take up the straggling ink. For ordi-
nary small erasures of ink lines, a sharp rounded pen-blade applied lightly
and rapidly does well, and the surface may be smoothed down by the
thumb-nail. In ordinary working drawings, a line may readily be taken
out by damping it with a hair pencil, and quickly applying the indiarub-
ber ; and to smooth the surface so roughened, a light application of the
knife is expedient. In drawings intended to be highly finished, particular
pains should be taken to avoid the necessity for corrections, as every thing
of this kind detracts from the appearance.
In using the square, the more convenient way is to draw the lines off
the left edge with the right hand, holding the stock steadily but not very
tightly against the edge of the board with the left hand. The convenience
of the left edge for drawing by is obvious, as we are able to use the arms
more freely, and we see exactly what we are doing.
To draw lines in ink with the least amount of trouble to himself, the
mechanical draughtsman ought to take the greater amount of trouble with
his tools. If they be well made, and of good stuff originally, they ought
to last through three generations of draughtsmen ; their working parts
should be carefully preserved from injury, they should be kept well set,
and, above all, scrupulously clean. The setting of instruments is a matter
of some nicety, for which purpose a small oil-stone is convenient. To dress
up the tips of the blades of the pen or of the bows, as they are usually
worn unequally by the customary usage, they may be screwed up into con-
tact in the first place, and passed along the stone, turning upon the point
in a directly perpendicular plane, till they acquire an identical profile.
Being next unscrewed and examined to ascertain the parts of unequal
thickness round the nib, the blades are laid separately upon their backs on
the stone, and rubbed down at the points, till they be brought up to an
edge of uniform fineness. It is well to screw them together again, and to
pass them over the stone once or twice more, to bring up any fault ; to re-
touch them also on the outer and inner side of each blade, to remove barbs
or frasing ; and, finally, to draw them across the palm of the hand.
The China ink, which is commonly used for line-drawing, ought to be
rubbed down in water to a certain degree, avoiding the sloppy aspect of
light lining in drawings, and making the ink just so thick as to run freely
DRAWING INSTRUMENTS. 41
from the pen. This medium degree may be judged of after a little practice
by the appearance of the ink on the pallet. The best quality of ink has a
soft feel when wetted and smoothed; free from grit or sediment, and
musky. The rubbing of China ink in water tends to crack and break away
the surface at the point ; this may be prevented by shifting at intervals the
position of the stick in the hand while being rubbed, and thus rounding the
surface. ]S"or is it advisable, for the same reason, to bear very hard, as
the mixture is otherwise more evenly made, and the enamel of the pallet
is less rapidly worn off. When the ink, on being nibbed down, is likely
to be for some time required, a considerable quantity of it should be pre-
pared, as , the water continually vaporises ; it will thus continue for a
longer time in a condition fit for application. The pen should be levelled
in the ink, to take up a sufficient charge ; and to induce the ink to enter
the pen freely, the blades should be lightly breathed upon before immer-
sion. After each application of ink, the outsides of the blades should be
cleaned, to prevent any deposit of ink upon the edge of the squares.
To keep the blades of his inkers clean, is the first duty of a draughtsman
who is to make a good piece of work. Pieces of blotting or unsized paper
and cotton velvet, washleather, or even the sleeve of a coat, should always
be at hand while a drawing is being inked. "When a small piece of blot-
ting paper is folded twice so as to present a corner, it may usefully be
passed between the blades of the pen now and then, as the ink is liable to
deposit at the point and obstruct the passage, particularly in fine lining ;
and for this purpose the pen must be unscrewed to admit the paper. But
this process' may be delayed by drawing the point of the pen over a piece
of velvet, or even over the surface of thick blotting paper ; either method
clears the point for a time. As soon as any obstruction takes place, the
pen should be immediately cleaned, as the trouble thus taken will always
improve and expedite the work. If the pen should be laid down for a short
time with the ink in it, it should be unscrewed to keep the points apart,
and so prevent deposit ; and when done with altogether for the occasion,
it ought to be thoroughly cleaned at the nibs. This will preserve its edges
and prevent rusting.
For the designing of machinery, it is very convenient to have some
scale of reference by which to proportion the parts ; for this purpose, a
vertical and horizontal scale may be drawn on the walls of the room.
42 GEOMETRICAL PROBLEMS.
GEOMETEICAL PEOBLEMS.
ON STRAIGHT LINES.
IT is desirable that the beginner should construct the following prob-
lems, not copying them, but adopting some scale, which will give him the
use of the scale, and imprint the problems on his memory.
PROBLEM I. To draw a straight line through given points.
Let A and B (fig. 78) be two given points, represented by the intersec-
tion of two lines, or pricked
I 4 . 5 : | into the surface. Surround
the points by small circles,
Fig. T8.
when advisable for assisting
to define their locality, as thus ; place the straight edge at or so near
the points, that the point of the pen or pencil may pass through them, and
draw the line firmly and steadily.
Lines in drawing are divided into several classes, asfutt, broken, dotted,
and broken and dotted, &c. ; these again are divided into fine, medium, and
heavy, according to the breadth of the line (fig. 79).
The lines of a problem which are either given or are to be found, and
Full. Broken. Dotted. Broken and dotted.
Fig. 79.
the outlines of an object that can be seen from the point of view in which
it is represented should be full, and either fine, medium, or heavy, accord-
ing to the particular efifect that the draughtsman wishes to give. The por-
tions of the outline that are hidden from view, but which are requisite to
give a complete idea of the object, should be dotted or broken.
GEOMETRICAL PROBLEMS. 43
The other lines are used for conventional purposes by the draughtsman,
as boundaries of parishes or estates, or to show a change in position of an
object, &c., &c.
PKOB. II. To set off a gi/ven distance along a straight line C D,froin
a given point A on it (fig. 80).
Take off the given distance from the scale of equal parts with the
dividers. Set one foot of the
dividers on A, and bring the c? . A* % s p
other foot upon the line, and Fig. so.
mark the point B, either by
pricking with the foot of the dividers, or by a small dot with the sharp
point of a lead pencil.
When the distance as at A to be set off is too small to be taken off
from the scale with accuracy, set off any convenient distance A 5 greater
than the given distance ; set off from ~b towards A, the length by which
A b is greater than the given distance ; the part A a will be the required
distance.
To set off a number of distances on a straight line, set off their succes-
sive sums. Thus, to set off successively the distances = 10, 15, 20, set off
as above 10, 25, and 45, always starting from the same point. The object
of performing the operation in this manner is to avoid carrying forward
any inaccuracy that might be made were the respective distances set off
separately, one after the other. If the distances to be set off are equal, it
will be more accurate to set off a distance equal either to the whole aggre-
gate, or such a number of them as can be contained by the compasses, and
then dividing the line into the required parts.
PKOB. III. To divide a given line into two equal parts (fig. 81).
Open the dividers to as near as possible half the given line, place one
point of the dividers on the end of the line, bring the other point to the
line, and turn on this point ; if now the point of the dividers coincide with
Fig. 81.
the other end of the line, we have the division required ; but should the point
of the dividers fall within or without the end of the line, divide this deficit
or excess by the eye into two equal parts, and contract or open the dividers
44 GEOMETRICAL PROBLEMS.
to this point, and apply them again as at first ; perform the operation till
the revolution of the compasses coincides with the given line. Thus (fig.
81), suppose it were required to divide the line A B into two equal parts,
and the distance A C' was the first guess or opening of the dividers ; turn-
ing on the point C', the point of the dividers that was at A falls on the
point D beyond B, keeping the point of the dividers still on the point C',
open them till they embrace the distance C' J, 5 being at or near as can
be judged by the eye the centre of D B ; begin again from the point A with
the distance C' 5 contained in the dividers, and apply the distance as at
first, dividing the deficit or excess of the two revolutions, till the point of
the dividers that was at A falls by revolution on B. The eye, by practice,
becomes so accustomed to this means of division, that a plan may be re-
duced to half scale as quickly and with as little chance of mistake as by the
proportional compasses.
To divide a line into any number of. equal parts. If the number is
divisible by two, bisect the line, or divide it into two equal parts as above,
and continue this as long as the number remaining is divisible by two ; but
when the number is uneven, measure the given line on the scale, divide
numerically the length thus found by the number of parts into which it
is to be divided, and take on the scale as accurately as possible the quotient
thus obtained ; apply this length successively on the line, and if the last
distance set off does not agree with the extremity of the line ; thus if, as in
fig. 81, when the line is to be divided into two parts, the repeated length
exceeds the line, divide this excess by the eye into as many parts as the
given line is to be divided, and close the dividers so as to include a
length less by one of these parts. If the point D should fall inside
\j,/ B, divide the deficit as before by
the number of parts, but open
\ the dividers by one of these
parts.
The above problems may be
constructed geometrically as fol-
lows : To Insect or divide into
two equal parts a given line A B.
/ From A and B, with any radius
greater than half of A B, describe
Fig - 82< arcs intersecting each other above
and below the given line, the line C D connecting these intersections will
bisect A B, and also be perpendicular to it.
GEOMETRICAL PROBLEMS.
45
PROB. IY. To divide a given line into a given number of equal parts.
Let A B be the distance
to be divided, for example,
into four equal parts ; draw
the line A C, making an
acute angle with A B ; on
A C lay off any four equal
distances, each as near as
may be to 1 of A B, con-
nect the last division 4 with
B, and through the other
points 1, 2, 3, draw lines
parallel to 4 B, the inter-
sections of these lines 5,c,<7,
with the line A B, will divide it into four equal parts.
PROB. Y. To draw a perpendicular to a straight line, from, a, point
without it.
\st Method (fig. 84). From the point A, with a^ sufficient radius, cut
the given line at F and G, and from these points describe arcs cutting at E ;
through E draw A E, which will be the perpendicular required. If there be
no room below the line, the intersection may be taken above ; that is, be-
tween the line and the given point. This mode is not, however, likely to
be as exact in practice as the one given.
Fig. 83.
Fig. 84
Fig. 85.
2<# Method (fig. 85). From any two points B and C, at some distance
apart, in the given line, and with radii B A, C A, respectively, describe arcs
GEOMETRICAL PROBLEMS.
cutting at A and D. Draw the perpendicular required, A D. This method
is useful where the given point is opposite the end of the line, or nearly so.
PROB. YI. To draw a perpendicular to a straight line from, a given
point A in that line.
1st Method (fig. 86). With any radius, from the given point A, in the
given line B C, cut the line at B and C ; with a greater radius describe arcs
from B and C, cutting each other at D, and draw D A the perpendicular.
E/'
\F
A
Fig. 66.
Fig.8T.
2d Method (fig. 87). From any centre F, above B C, describe a circle
passing through the given point A, and cutting the given line at D ; draw
D F, and produce it to cut the circle at E ; and draw A E the perpendicular.
This method is useful when the point A is at or near one end ; and in prac-
tice, it is expedient in the first place to strike out a preliminary arc, of any
convenient radius, from the point A, as any point in that arc may be
chosen for the centre F, with the certainty that the arc from this centre
will pass through A, without the delay of adjusting the point of the com-
pass to it. This expedient is of general use where an arc is to be passed
through a given point, and particularly if the point of the pencil be round
or misshapen, and therefore uncertain.
3d Method (fig. 88). From A describe an arc E C, and from E, with the
Fig. 69.
<*ame radius, the arc A C, cutting the other at C ; through C draw a line E C D,
and set off C D equal to C E, and through D draw A D the perpendicular
GEOMETRICAL PROBLEMS. 47
required. This method, like the previous one, is useful when the point A
is at one end.
th Method (fig. 89). From the given point A, set off a distance A E
equal to three parts, by any scale ; and on the centres A and E, with radii
of four and five parts respectively, describe arcs intersecting at C. Draw
A C for the perpendicular required. This method is most useful on very large
scales, where straight edges are inapplicable, as in laying down perpendi-
culars or right angles on the ground ; as in laying out the corners of houses,
beams and girders may be set square with the sides of the houses, columns
and the like may be set perpendicularly by the same method. The
numbers 3, 4, 5, are, it is to be observed, taken to measure respectively
the base, the perpendicular, and the slant side of the triangle AEG. Any
multiples of these numbers may be used with equal propriety, as 6, 8, 10,
or 9, 12, 15, whether feet, yards, or any other measure of length.
PROB. YII. To draw a straight line parallel to a given line, at a given
distance apart (fig. 90).
From the centres A, B, in the given line, ""'
with the given distance as radius, describe
arcs C, D ; and draw the parallel line C D
touching the arcs. The method of drawing
tangents will be afterwards shown ; mean- -^ "
time, in all ordinary cases, the line C D may
be drawn by simply applying a straight edge by the eye.
PROB. YIII. To draw a parallel through a given point.
1st Method (fig. 91). With a radius equal to the distance of the given
point C from the given line A B, describe the arc D from B, taken con-
siderably distant from C ; draw the parallel through G to touch the arc D.
A/ In
/f
Fig. 91. Fig. 92.
%d Method (fig. 92). From A, the given point, describe the arc F D,
cutting the given line at F ; from F, with the same radius, describe the
arc E A ; and set off F D equal to E A. Draw the parallel through
the points A, D.
GEOMETRICAL PROBLEMS.
PROB. IX. To construct an angle
equal to a given angle (fig. 93).
Thus, on the line a 5 to construct an an-
gle which shall be equal to the given angle
CAB. With the dividers describe the arc
C B ; from the point #, with the same ra-
dius describe c 1> ; with the dividers mea-
sure the length of the arc C B, and on c &
lay off this distance ; through c draw c a,
and we have the required angle or open-
ing c a 5, equal to the given angle CAB.
PKOB.X. From a point A of a given
line D E, to draw a line making an angle
of 60 with the given line (fig. 94).
Take any convenient distance in the
dividers, and from A describe the arc B C. From B, with the same dis-
tance, describe an arc, and mark the point C where the arcs cross. Draw
the line A, C. This line will make with the given one the required
angle of 60.
Fig. 93.
^
Fig. 94
Fig. 95.
PKOB. XI. From a point 'Bon a given line DE, to draw a line making
an angle of 45 with it (fig. 95).
Set off any distance B a, along D E, from B. Construct a perpendicu-
lar to D E at <z, and set off on this perpendicular a c equal to a B ; draw
through B c a line, w r hich will make with D E the required angle of 45.
PEOB. XTT. To divide a given
angle, as B A C (fig. 96), into two
Fig. 96.
From the point A, or vertex of
the angle, with any radius describe
an arc b c ; from 5 and c, the inter-
sections of the arc with the sides of
the angle, with any radius greater
GEOMETRICAL PROBLEMS.
than half the arc I c, describe two arcs intersecting each other, as at D ;
through A and D draw a line which will bisect or divide into two equal
parts the angle BAG.
PKOB. XIII.: To bisect the angle contained between two lines, as AB and
CD (fig. 97), when the intersecting point or vertex of the angle is not on the
drawing.
Set off a point 5 at any convenient distance from A B, and through this
point draw a parallel to
A B ; at the same distance
from C D draw a paral-
lel ; extend these parallels
till they intersect at c j bi-
sect the angle 5 c d by c a,
which will also bisect the
angle contained between
the lines A B and C D.
PKOB. XIV. Through two given points, as B and C (fig. 98), to describe
an arc of a circle with a given radius.
From B and 0, with an opening of the dividers equal to the given
radius, describe two arcs crossing at A ; from A, as a centre, with the same
radius describe an arc which will be the one required. It is to be observed,
that there are two points A, one above and one below the line B C, from
which, as centres, arcs can be described with the given radius, and passing
through B and 0.
Fig. 9T.
Fig. 93.
Fig. 99.
PEOB. XY. To find the centre of a given circle, or of an arc of a circle.
Of a circle (fig. 99). Draw the chord A B, bisect it by the perpendic-
ular C D, whose extremities lie in the circumference, and bisect C D for
the centre G of the circle.
Of an arc, or of a circumference (fig. 100). Select the points A, B, C, in
the circumference well apart ; with one radius describe arcs from these
50
GEOMETRIC! AT. PKOBLEMS.
three points, cutting each other ; and draw the two lines D E, F G, through
their intersections : the point O, where they cut, is the centre of the circle
or arc.
Fig. 100.
PEOB. XYI. To describe a circle passing through three given points.
Join the given points A, B, C (fig. 100), and proceed as in last problem
to find the centre O, from which the circle may be described.
This problem is of utility : in striking out the circular arches of bridges
upon centering, when the span and rise are given ; describing shallow
pans, or dished covers of vessels ; or finding the diameter of a fly-wheel, or
any other object of large diameter, when only a part of the circumference
is accessible.
PEOB. XVII. To describe a circle passing through three given points,
when the centre is not available.
~Lst Method (fig. 101). From the extreme points A B, as centres, de-
scribe arcs A H, B G. Through the third point C draw A E and B F,
cutting the arcs. Divide A F and B E into any number of equal parts,
and set off a" series of equal parts of the same length on the upper portions
of the arcs beyond the points E F. Draw straight lines B L, B M, &c., to
the divisions in A F ; and A I, AK, &c., to the divisions in E G : the suc-
cessive intersections ~N, O, &c., of these lines, are points in the circle re-
quired, between the given points A and C, which may be filled in accord-
ingly. Similarly, the remaining part of the curve B C may be described.
2d Method (fig. 102).
Let A, D, B, be the given
points ; draw AB, AD, DB,
and /, parallel to A B. Di-
vide D A into a number of
equal parts, 1, 2, 3, &c.,
GEOMETRICAL PROBLEMS.
51
and from D describe arcs through these points to meet ef. Divide the arc
A e into the same number of equal parts, and draw straight lines from D to the
points of division. The intersections of these lines successively with the
arcs 1, 2, 3, &c., are points in the circle, which may be filled in as before.
Note. The second method is not perfectly true, but sufficiently so for
arcs less than one-fourth of a circle.
To describe the arc mechanically with three strips of board forming a
triangle. Insert two stiff pins or nails at A and B ; place the strips as
shown in fig. 103,
one against the pins
at A and B, and
having D at their
intersection; fasten
the two strongly to-
gether at this point and at the base of the triangle by the third strip ; plac-
ing the pencil at D, and keeping the edges against A and B, moving the
triangle to the right and left, the pencil will describe the circle.
PROB. XVIII. To draw a tangent to a circle from a given point in the
circumference.
\st Method. Through the given point A (fig. 104) draw the radial line
A C, and the perpendicular F G for the tangent required.
Fig. 103.
Fig. 104 Fig. 105.
%d Method. From A (fig. 105) set off equal segments, A B, A D ;
join B D, and draw A E parallel to it, for the tangent. This method is
useful when the centre is inaccessible.
PROB. XIX. To draw tangents to a circle from a point without it (fig.
106).
Draw A C from the given point A to the centre of the circle, bisect it
at D, from D describe an arc through C, cutting the circle at E and F.
Draw A E, A F for the required tangents.
To construct within the sides of an angle a circle tangent to these sides,
at a given distance from the vertex. In fig. 107, to describe a circle or arc
tangent at a and 5, equally distant from the vertex A ; draw perpendicu-
52
GEOMETRICAL PROBLEMS.
lars to A C at a, and to A B at I ; the intersection of these will be the
centre of the required circle.
Fig. 106.
Fig. 10T.
In the same fig., to find the centre, the radius being given, and not the
points a and 5. Draw parallels to A C and A B at a distance equal to the
given radius, and their intersection will be the centre required.
PROB. XX. To describe a circle from a given point to touch a given
circle (figs. 108, 109).
D E being the given circle, and B the point, draw from B to the centre C,
and produce it, if necessary, to cut the circle at A, and with B A as radius
describe the circle F G, touching the given circle. The operation is the
same whether the point B be within or without the circle.
Fig. 108.
Fig. 109.
It will be remarked that, in all cases of circles tangential to each other,
their centres and their points of contact must lie in the same straight line.
PROB. XXL
To draw tangents
to two given circles.
1st Method (fig.
110). Draw the
straight line ABC
through the cen-
Fig.no. tres of the two
GEOMETRICAL PROBLEMS.
53
given circles ; from the centres A, B, draw parallel radii A D, B E, in
the same direction ; join D E, and produce it to meet the centre line at C,
and from C draw tangents to one of the circles by Prob. XIX. Those tan-
gents will touch both circles, as required.
2d Method (fig. 111). Draw A B, and in the larger circle draw any
radius A II, on which set off II G equal to the radius of the smaller circle ;
Fig. 111.
on A describe a circle with the radius A G, and draw tangents B I, B K, to
this circle from the other centre B ; from A and B draw perpendiculars to
these tangents, and join C, D, and E, F, for the required tangents.
Note. The second method is useful when the diameters of the circles
are nearly equal.
PROB. XXII. Between two inclined lines to draw a series of circles
touching these lines and touching each other (fig. 112).
Bisect the inclination of the given lines A B, D, by the line K O ;
this is the centre line of the circles to be inscribed. From a point P in this
line draw the perpendicular P B
to the line A B, and from P de-
scribe the circle B D touching the
given lines and cutting the centre
line at E ; from E draw E F per-
pendicular to the centre line, cut-
ting A B at F, and from F describe
an arc E G, cutting A B at G ; Fig . n 2 .
draw G II parallel to B P, giving H the centre of the second touching
circle, described with the radius H E or H G. By a similar process the
third circle I N is determined. And so on.
Inversely, the largest circle may be described first, and the smaller
ones in succession.
Note. This problem is of frequent use in scroll work.
PROB. XXIII. Between two inclined lines to draw a circular segment
to jill up the angle, and touching the lines (fig. 113).
GEOMETRICAL PROBLEMS.
Let A B, D E, be the inclined lines ; bisect the inclination by the line
F C, and draw the perpendicular A F D to define the limit within which
the circle is to be drawn. Bisect the angles A and D by lines cutting at
C, and from C with radius C F, draw the arc H F G as required.
Fig. 114
PROB. XXIY. To -fill up the angle of a straight line and a circle, with
a circular arc of a given radius (fig. 114).
In the given circle A D draw a radius C B and produce it, set off B E
equal to the radius of the required arc, and on the centre C with the radius
C E, describe the arc E F. Draw G F parallel to the given line H I, at
the distance G H equal to the radius of the required arc, and cutting the
arc E F at F. Then F is the required centre ; draw the perpendicular F I,
and the radius F C cutting the circle at A, and with the radius F A or F I
describe the arc A I as required.
PROB. XXY. To fill up the angle of a straight line and a circle, with
a circular arc to join the circle at a given point (fig. 115).
In the given circle draw the radius A and produce it ; at A draw a
tangent meeting the given line at
D ; bisect the angle A D E so formed
with a line cutting the radius at F ;
and on the centre F describe the
arc A E as required.
PROB. XXVL To describe a
circular arc joining two circles, and
to touch one of them at a given
point (fig. 116).
Let A B and F G be the given
circles, to be joined by an arc
touching one of them at F. Draw the radius E F, and produce it both
ways ; set off F H equal to the radius A C of the other circle ; join C H
\
GEOMETRICAL PROBLEMS.
55
and bisect it with the perpendicular L I cutting E F at I ; on the centre I
with radius I F describe the arc F as required.
Fig. nr.
PEOB. XXTRTofindthe arc which shall
be tangent to a given point A on a straight line,
and pass through a given point outside the line
(fig. 117).
Erect at A a perpendicular to the given line ; connect C A, and bisect
it by a perpendicular ; the intersection of the two perpendiculars at a will
be the centre of the required arc.
PKOB. XXVIII. To connect two parallel lines ty a reversed curve com-
posed of two arcs of equal radius, and tangent to the lines at given points,
as at A and B (fig. 118).
Join A B, and divide it into two equal parts at C ; bisect C A and C B
by perpendiculars ; at A and B erect perpendiculars to the given lines, and
the intersections a and b will be the centres of the arcs composing the re-
quired curve.
Fig. 118.
Fig. 119.
PROB. XXIX. To join two given points, as A and B (fig. 119), in two
given parallel lines by a reversed curve of two equal arcs, whose centres lie
in the parallels.
Join A B, and divide it in equal parts at C, as above. Bisect also A C
and B C by perpendiculars ; the intersections a and b of the parallel lines
by these perpendiculars will be the centre of the required arcs.
PROB. XXX. On a given line, as A B (fig. 120), to construct a com-
pound curve of three arcs of circles, the radii of the two side ones being
56
GEOMETRICAL PROBLEMS.
equal and of a given length, and their centres in the given line j the central
arc to pass through a given point, as C, on the perpendicular bisecting the
given-line, and tangent to the other two arcs.
Draw the perpendicular CD; lay off A a, B b, and C c, eacli equal to
the given radius of the
side arcs ; join a c;
bisect a c by a per-
pendicular ; the inter-
section of this line
with the perpendicu-
lar C D will be the
required centre of the
central arc. Through
a and b draw the lines
D e and D e' ; from a
and b with the given radius, equal to a A, b B, describe the arcs A e and
B e', from D as a centre, with a radius equal to C D, and consequently by
construction D e and D e', describe the arc e C e', and we have the com-
pound curve required.
For the -construction of compound curves of five arcs, see construction
of ellipses, page 72.
0, X
\ \
/
\ \
/
\ \
/
\\
/
li
/
Fig. 120.
PROBLEMS ON CIRCLES AND RECTILINEAR FIGURES.
PROB. XXXI. To construct a triangle upon a given straight line or
base, the length of the two sides being given.
First, an equilateral triangle (fig. 121). On the ends A B of the given
line, with A'B as radius, describe arcs cutting at C, and draw A C, B C ;
then A B C is the triangle required.
Fig. 121. Fig. 122.
Second, when the sides are unequal (fig. 122). Let A D be the base,
and B and C the two sides. On either end, as A, of the base-line, with the
GEOMETRICAL PROBLEMS.
57
line B as radius, describe an arc ; and on D, with C as radius, cut the arc
at E. Draw A E, E D, then A E D is the triangle as required.
This construction is used also to find the position of a point, when its
distances are given from two other given points, whether joined by a line
or not.
PKOB. XXXII. To construct a square or a rectangle upon a given
straight line.
First, a square (fig. 123). Let A B be the given line ; on A and B as
centres, with the radius A B, describe arcs
cutting at C ; on C, with the same radius,
describe arcs cutting the others at D and
E ; and on D and E, cut these at F G.
Draw A F, B G, cutting the arcs at H, I ;
and join H I to form the square as re-
quired.
Second, a square or rectangle (fig. 124).
To the base E F draw perpendiculars E H,
F G, equal to the sides, and join G H to complete the rectangle.
When the centre lines of the square or rectangle are given, the figure
may be described as follows :
c.
it e K it L
Fig. 123.
Fig. 124
Let A B and C D (fig. 125) be the centre lines, perpendicular to each
other, and E the middle point of the figure ; set off E F, E G, equal each
to the half length of the rectangle, and E H, E J, each equal to half the
height. On the centres H, J, with a radius equal to the half length, de-
scribe arcs on both sides ; and on F, G, with a radius of half the height,
cut these arcs at K, L, M, N. Join the four intersections so formed, to
complete the rectangle.
PKOB. XXXIII. To construct a parallelogram, of which the sides and
one of the angles are given (fig. 126).
Let A and B be the lengths of the two sides, and C the angle ; draw a
straight line, and set off D E equal to A ; from D draw D F equal to B,
and forming an angle with D E equal to C ; from E with D F as radius,
58
GEOMETRICAL PROBLEMS.
describe an arc, and from F with D E as radius, cut the arc at G, and
draw F G and E G, to complete the parallelogram. Or, the remaining
sides may be drawn parallel to D E and D F, cutting at G, and the figure
is thus completed.
Fig. 126.
Fig. 127.
PROB. XXXIY. To describe a circle about a triangle (fig. 127).
Bisect two of the sides A B, A C, of the triangle at E, F ; from these
points draw perpendiculars cutting at K. From the centre K, with K A
as radius, describe the circle A B C, as required.
PROB. XXXV. To inscribe a circle in a triangle (fig. 128).
Bisect two of the angles A, C, of the triangle A B C, by lines cutting
at D ; from D draw a perpendicular D E to any side, as A C ; and with
D E as radius, from the centre D, describe the circle required.
"When the triangle is equilateral, the centre of the circle is more easily
found by bisecting two of the sides, and drawing perpendiculars, as in the
previous problem. Or, draw a perpendicular from one of the angles to
the opposite side, and from the side set off one-third of the length of the
perpendicular.
E
Fig. 128.
Fig. 129.
PROB. XXXYI. To inscribe a square in a circle ; and to describe a
circle about a square (fig. 129).
GEOMETRICAL PROBLEMS.
59
To inscribe the square. Draw two diameters A B, C D, at right angles,
and join the points A, B, C, D, to form the square as required.
To describe the circle. Draw the diagonals A B, C D, of the given
square, cutting at E ; on E as a centre, with E A as radius, describe the
circle as required.
In the same way, a circle may be described about a rectangle.
PROB. XXX YIL -To inscribe a circle in a square ; and to describe a
square about a circle (fig. 130).
To inscribe the circle. Draw the diagonals A B, CD, of the given
square, cutting at E ; draw the perpendicular E F to one of the sides, and
with the radius E F, on the centre E, describe the circle.
To describe the square. Draw two diameters A B, C D, at right angles,
and produce them ; bisect the angle D E B at the centre by the diameter
F G, and through F and G draw perpendiculars A C, B D, and join the
points A, D, and B, C, where they cut the diagonals, to complete the square.
PROB. XXXYIII. To inscribe a pentagon in a circle (fig. 131).
Draw two diameters A C, B D, at right angles ; bisect A O at E, and
Fig. 131.
from E with radius E B, cut A C at F ; from B, with radius B F, cut the
circumference at G H, and with the same radius step round the circle to I
and K ; join the points so found to form the pentagon.
PROB. XXXIX. To construct a regular hexagon upon a given straight
line (fig 132).
From A and B, with a radius equal to the given line, describe arcs
cutting at g / from ^, with the radius g A, describe a circle ; with the same
radius set off from A the arcs A G, G F, and from B the arcs B D, D E.
Join the points so found to form the hexagon.
PROB. XL. -To inscribe a regular hexagon in a circle (fig. 133).
Draw a diameter A B, from A and B as centres, with the radius of the
circle A C, cut the circumference at D, E, F, G ; draw straight lines A D,
D E, &c., to form the hexagon.
60
GEOMETEICAL PROBLEMS.
The points of contact, D, E, &c., may also be found by setting off the
radius six times upon the circumference.
Fig. 134.
PEOB. XLI. To describe a regular hexagon about a circle (fig. 134).
Draw a diameter A B of the given circle ; with the radius A D from A
as a centre, cut the circumference at C ; join A C, and bisect it with the
radius D E ; through E draw F G parallel to A C, cutting the diameter at
F, and with the radius D F describe the circle F II. Within this circle
describe a regular hexagon by the preceding problem ; the figure will touch
the given circle as required.
PEOB. XLIE. To construct a regular octagon upon a given straight line
(fig. 135).
Produce the given line A B both ways, and draw perpendiculars A E,
B F ; bisect the external angles at A and B by the lines A H, B C, which
make equal to A B ; draw C D and H G parallel to A E and equal to A B ;
and from the centres G, D, with the radius A B, cut the perpendiculars at
E,F, and draw E F to complete the octagon.
, E
A
Fig. 136.
PBOB. XLIII. To convert a square into a regular octagon (fig. 136).
Draw the diagonals of the square cutting at e' } from the corners A, B, C, D,
with A e as radius, describe arcs cutting the sides at g h, &c. ; join the
points so found to complete the octagon.
GEOMETRICAL PROBLEMS.
Gl
PKOB. XLIV. To inscribe, a regular octagon in a, circle (fig. 137").
Draw two diameters A 0, B D, at right angles ; bisect the arcs A B,
B C, &c., at e,fj &c. ; and join A e, e B, &c., for the inscribed figure.
r 7f
Fig. 1ST.
Fig. 133.
PEOB. XLV. To describe a regular octagon about a circle (fig. 138).
Describe a square about the given circle A B ; draw perpendiculars
h k, &c., to the diagonals, touching the circle. The octagon so formed is
the figure required.
Or, to find the points h, k, &c., cut the sides from the corners of the
square, as in Prob. XLIII.
PKOB. XLYI. To inscribe a circle within a regular polygon.
When the polygon has an even number of sides, as in fig. 139, bisect
two opposite sides at A and B, draw A B, and bisect it at C by a diagonal
D E drawn between opposite angles ; with the radius C A describe the
circle as required.
"When the number of sides is odd, as in fig. 140, bisect two of the sides
at A and B, and draw lines A E, B D, to the opposite angles, intersecting
at ; from C with C A as radius, describe the circle as required.
PKOB. XL VII. To describe a circle without a regular polygon.
When the number of sides is even, draw two diagonals from opposite
angles, like D E (fig. 139), to intersect at C ; and from C with C D as
radius, describe the circle required.
62 GEOMETRICAL, PROBLEMS.
When the number of sides is odd, find the centre C (fig. 140) as in last
problem, and with C D as radius describe the circle.
The foregoing selection of problems on regular figures is the most use-
ful in mechanical practice on that subject. Several other regular figures
may be constructed from them by bisection of the arcs of the circumscrib-
ing circles. In this way a decagon, or ten-sided polygon, may be formed
from the pentagon by the bisection of the arcs in Prob. XXXVIIL, fig.
131. Inversely, an equilateral triangle may be inscribed by joining the
alternate points of division found for a hexagon.
The constructions for inscribing regular polygons in circles are suitable
also for dividing the circumference of a circle into a number of equal
parts. To supply a means of dividing the circumference into any number
of parts, including cases not provided for in the foregoing problems, the
annexed table of angles relating to polygons, expressed in degrees, will be
found of general utility. In this table, the angle at the centre is found by
TABLE or POLYGONAL ANGLES.
Number of Sides of Regular
Polygon ; or number of
equal parts of the circum-
Angle at
Centre.
Number of Sides of Regular
Polygon.
Angle at
Centre.
ference.
No.
Degrees.
No.
Degrees.
3
120
12
30
4
90
13
27 rV
5
72
14
25|
6
60
15
24
7
51.3
16
224-
8
45 7
17
21A
9
40
18
20
10
36
19
18H
11
32 T T
20
18
dividing 360, the number of degrees in a circle, by the number of sides
in the polygon ; and by setting off round the centre of the circle a suc-
cession of angles by means of the protractor, equal to the angle in the
table due to a given number of sides : the radii so drawn will divide the
circumference into the same number of parts. The triangles thus formed
are termed the elementary triangles of the polygon.
PROB. XLYIII. To inscribe any regular polygon in a given circle ; or to
divide the circumference into a given number of equal parts, by means of
the angle at the centre (fig. 141).
GEOMETRICAL PROBLEMS.
63
Suppose the circle is to contain a hexagon, or is to be divided at the
circumference into six equal parts. Find the angle
at the centre for a hexagon, or 60 ; draw any ra-
dius B C, and set off by a protractor or otherwise
the angle at the centre C B D, equal to 60 ; then
the interval C D is one side of the figure, or seg-
ment of the circumference ; and the remaining
points of division may be found either by stepping
along the circumference with the distance C D in
the dividers, or by setting off the remaining five
angles of 60 each round the centre.
Fig. 141.
THE USE OF TIIE T SQUARE AND TRIANGLE IN THE CONSTRUCTION OF SOME
OF THE FOREGOING PROBLEMS.
From the description of the T square it may be seen, that by sliding
the stock along two contiguous edges of the board, the left hand and
bottom edges, any number of parallel and perpendicular lines may be
drawn. In so far, therefore, the T square supersedes the application of all
the problems for drawing parallels and perpendiculars, coinciding in
direction with the edges of the board ; for the square need only be set
with its edge coincident with the points through which the line is to be
drawn, and the pen or pencil drawn along the edge will describe the line
required. When the perpendiculars or upright lines are of short length,
the triangle and ruler are used. For this purpose, the triangle or set
square of 60 is preferable to that of 45, as it is longer and lighter.
When the lines to be drawn do not coincide in direction with the edges
of the board, the square may be adjusted with its bevel stock to the
obliquity required, and the lines may be drawn as before. This is proba-
bly the best plan when the oblique lines are numerous or extensive. In
most cases, however, oblique lines are only occasional, and when their
position is given, they may be drawn with a straight-edge. When the
oblique parallels and perpendiculars are short, as in oblique framing, short
rods or bars, bolt-heads, and the like, the com-
bined use of the straight-edge and triangle is
expedient. Square figures may be described
on a given centre, at one setting of the /
straight-edge, as in the drawing of the head Fig. 142.
of a square nut n (fig. ,142). From the given centre, with a radius equal
64
GEOMETRICAL. PROBLEMS.
to half the side of the square, describe a circle, and with the aid of the tri-
angle draw lines tangent to four sides.
To draw an octagon, apply the set-square of 45 to the corners, after
completing a square figure, and draw tangents to the inscribed circle, as,
for example, the line h k (fig. 138).
To draw an equilateral triangle upon a given line A B (fig. 143), it is
only necessary to apply the slant edge of the set-square of 60 to each end
Fig. 143.
Fig. 144.
Fig. 145.
of the base, with the short side b c applied to the square-blade, and to
draw the two sides A 0, B C. If the given side A B be upright (fig. 144),
apply the long side a l> to the straight-edge, and draw as before.
To draw a regular hexagon about a circle, with two of its sides parallel
to the lower edge of the board : draw the centre line A B (fig. 145), and
the upper and lower sides D E, F G, touching the circle, and apply the
triangle of 60 touching the circle for the four remaining sides, as shown
in the figure.
When the hexagons are to be inscribed in the circle, first draw the
centre line A B (fig. 133) as a diameter, and from the ends A, B, with the
set-square, draw four sides cutting the circle
at D, E, F, G, and join D E, F G.
The triangles of 45 and 60 are useful in
setting out the centre-lines of wheels with 3,
4, 6, 8, &c., arms, by drawing lines through
the centre of the wheel. To set out 12 spokes
in a wheel (fig. 146) : Draw two diameters,
A B, CD, parallel to the two edges of the
board ; in the quadrant A C, draw radii E a,
E J, with the long and the short sides of the
triangle against the square-blade. These will
divide the quadrant equally ; and the same
construction being employed for the other quarters of the circle, 12 centre
Fig. 146.
GEOMETRICAL PROBLEMS.
65
lines, equally distant, will be described. Should the triangle be large
enough to embrace the whole circle at once, the opposite quadrants A C
and B D may be divided with the same setting of the triangle.
A short method of dividing a line or surface into a number of equal
parts is illustrated by fig. 147 ; and it is convenient where an ordinary
rule does not evenly
measure the dimen-
sion. Suppose the
width A C is to be di-
vided into seven equal
parts, and that it mea-
sures 7| inches ; an or-
dinary inch rule, it is
plain, does not afford the subdivisions when applied directly ; but if 14
inches of length, or double the number of parts, be applied obliquely across
the space between the parallels A B, C D, so as to measure it exactly,
then point off two-inch intervals on the edge of the rule, and in this way
7 equal subdivisions will be effected, through which parallels may be
drawn.
SIMPLE APPLICATIONS OF REGULAR FIGURES.
PROB. XLIX. To cover a surface with equilateral triangles, hexagons,
or lozenges.
Describe an equilateral triangle
ABC, and produce the sides in-
definitely. Set off from one angle
A, equal intervals at a, 5, a', V,
&c., as required; and through
these points draw parallels to each
of the sides of the triangle. The
area will be covered with triangles
as required.
For hexagons, or equilateral ^ <L b c d e B f
triangles and hexagons on the Fig.ua
same surface, or lozenges, group the triangles.
PROB. L. To cover a surface with octagons and squares.
Draw two straight lines A B, A C, at right angles ; set off equal inter-
5
66
GEOMETRICAL PROBLEMS.
vals Ad,de, &c., on each line, equal to the breadth of the octagon to be
described, and through these points draw parallels to the given lines, to
Fig. 149.
form squares. "Within these squares construct octagons, by Probs. XLTTT.
or XLIY., and finish as in the figure.
PROBLEMS ON PROPORTIONAL LINES AND EQUIVALENT FIGURES.
PROB. LI. To divide a given straight line into two parts proportional
to two given lines.
Let A B (fig. 150) be the line to be divided ; draw the straight line
A D at any angle with A B, and set off A E, E D, equal to the other two
given lines. Join D B, and draw C E parallel to it ; this line divides A B
at C in the required ratio.
PROB. LII. To divide a straight line into any number of parts of given
proportions ; or similarly to a given straight line.
Let A B (fig. 151) be the line to be divided. Draw B G at any angle
AC JJ
Fig. 150.
H I A
Fig. 151.
with it, and set off by any convenient scale, B C, C D, &c., to G, respec-
tively, equal to the given divisions. Join A G, and from the points of
GEOMETRICAL PROBLEMS.
67
division of B G draw parallels to A G, cutting it at H, I, &c. The paral-
lels so drawn will divide A B as required.
PROB. LIU. To find a fourth proportional to three given lines.
Draw two lines I K, I N (fig. 152), at any angle, and set off I M, I N,
equal to the two first of the given lines, and set off I L equal to the third.
Join L M, and draw N K parallel to it. Then I K is a fourth proportional
as required.
The two first lines may "be set off successively on the same line, as from
I to M, and from M to N, and the third from I to L ; then L K will be
the fourth line required.
PROB. LIY. Tojmd a mean proportional between two given lines (fig.
153).
Let a I and I c be the given lines. Set off, on a straight line, A B, B C,
equal to the given lines ; bisect A C at D, and with D A as a radius describe
the semicircle AE C ; draw B E perpendicular to A C, meeting the circle
at E. Then B E is the mean proportional required.
^,_^_,.
/
E, ,
\
\
1
C D I
Fig. 153.
)
.4
T)
Fig. 154
PROB. LY. To construct a triangle equal in area to a gvven
Bisect the base B C (fig. 154) of the rectangle at D, and draw the per-
pendicular D A equal to twice the height, D E, of the rectangle. Draw
B A, A C ; the triangle A B C is equal in area to the rectangle B G.
PROB. LYI. To construct a square equal to a gwen rectangle (fig. 155).
Let A B D be the rectangle; ^ Jf G
produce A B, and set off B E equal
to the side B C of the rectangle ; bi-
sect A E at K, and describe a semi-
circle on A E ; draw the perpendicu-
lar B H, cutting the circle at H, and
on B H describe the square B G
required. Fig. 155.
PROB. LYII. To construct a triangle equivalent to any regular polygon.
Find the radius of the circle inscribed in the polygon. Set off on a
GEOMETRICAL PKOBLEMS.
right line a distance equal to half the sum of the sides of the polygon.
This distance will be the base of the equivalent triangle, and the radius of
the inscribed circle its perpendicular or altitude.
PKOBLEMS ON THE ELLIPSE, THE PAKABOLA, THE HYPEBBOLA, THE CYCLOID,
AND THE EPICYCLOID.
PEOB. LVlii- To describe an ellipse, the length and breadth, or the two
axes
1st Method (fig. 156). Bisect the transverse axis A B at C, and through
,/ V
C draw the perpendicular D E, making D and C E each equal to half
the conjugate diameter. On D as a centre, with C A as radius, describe
arcs cutting at F, F', for the foci. Divide A C into a number of parts at
the points 1, 2, 3, &c. "With radius A 1 on F and F' as centres, describe
arcs ; and with radius B 1, on the same centres, describe arcs inter-
secting the others as shown. Repeat the operation for the other divisions
of the transverse axis. The series of intersections thus found will be
points in the curve, and they may be as numerously found as is desir-
able ; after which a curve traced through them will form the complete
ellipse.
2d Method (fig. 15T). The two axes, A B, D E, being given. On A B
and D E as diameters from the same centre C, describe circles F G, H I ;
take a convenient number of points, a, 5, &c., in the semi-circumference
A F B, and draw radii cutting the innner circle at a! o', &c. ; from , 5,
&c., draw perpendiculars to A B, and from a', V, &c., draw parallels to
GEOMETRICAL PROBLEMS.
A B, cutting the respective perpendiculars at n, o, &c. The points of in-
tersection so found are points in the curve.
3d Method (fig.
158). Along the
straight edge of a slip
of stiff paper mark off
a distance a c equal
to A C, half the trans-
verse axis ; and from
the same point, a dis-
tance a l> equal' to
C D, half the conju-
gate axis. Place the
slip so as to bring the
point 5 on the line
A B of the transverse
axis, and the point c
on the line D E ; and
set off on the drawing
the position of the point
a. Always keeping the
point T) on the transverse
axis, and the point c on
the conjugate axis, any
required number of points
may be found.
4th Method (fig. 159).
--By the above method
large curves may be de-
scribed continuously by
means of a bar m, k, with
steel points m, Z, Jc, riveted
into brass slides, adjusted to
the length of the semi-axes,
and fixed with set-screws.
A rectangular cross E G,
with guiding slots, is placed
to coincide with the two
axes of the ellipse A C and B H ; by sliding the points &, Z, in the slots,
70
GEOMETRICAL PROBLEMS.
and carrying round the point m, the curve may be completely described.
If desirable, of course, a pen or pencil may be fixed at in.
5th Method (fig. 160). Given the two axes A B, C D ; on the centre
C, with A E as radius, describe
an arc cutting A B at F and G,
the foci ; fix a couple of pins
into the transverse axis at F
and G, and loop on a thread
or cord upon them, equal in
length, when looped on, to A
B, so as, when stretched, as per
dot-line FOG, just to reach
the extremity, C, of the conju-
gate axis. Place a pencil or
draw-point inside the cord, as
at H, and guiding the pencil
in this way, keeping the cord equally on tension, pass round the two
points F, G, and describe the curve as required.
This method is employed in setting off elliptical garden-plots, walks,
&c.
PEOB. LTX. To draw a tangent to an ellipse through a given point
in the curve (fig. 161).
From the given point T draw straight lines to the foci F, F' ; produce
F T beyond the curve to <?, and bisect the exterior angle c T F' by the line
T d. This line T d is the tangent required.
PROB. LX. To draw a tangent to an ellipse from a given point with-
out the curve (fig. 162).
From the given point T as centre, with a radius equal to its distance
from the nearest focus F, describe an arc ; from the other focus F', with
the transverse axis as radius, cut the arc at K, L, and draw K F', L F',
GEOMETRICAL PROBLEMS.
71
cutting the curve at M, 1ST ; then the lines T M, T K, are tangents to the
curve.
PROB. LXL To describe an
, by means of circular
arcs.
First, with arcs of two radii (fig. 163). Take the difference of the trans-
verse and conjugate axes, and set it off from the centre O to a and c, on
O A and O C ; draw a c, and set off half a c to d; draw d i parallel to a c,
set off O e equal to O d, join e i, and draw em, dm, parallels to d i, i e. On
centre ra, with radius m C, describe an arc through 0, and from centre i
describe an arc through D ; on centre d, e, also, describe arcs through A
and B. The four arcs thus described form approximately an ellipse. This
method does not apply satisfactorily when the conjugate axis is less than
two-thirds of the transverse axis.
Second, with arcs of three radii (fig. 164). On the transverse axis A B,
draw the rectangle B G, equal in height to O C, half the conjugate -axis.
Draw G D perpendicular to A C ; set off O K equal to O C, and on A K
as a diameter, describe the semicircle A N K ; draw a radius parallel to
O C, intersecting the semicircle at 1ST and the line G E at P ; extend O C
GEOMETRICAL PKOBLEMS.
to L and to D ; set off O M equal to P IS", and on D as a centre, with a
radius D M, describe an arc ; from A and B as centres, with a radius O L,
Fig. 164
intersect this arc at a and I. The points H, #, D, 5, IF, are the centres of
the arcs required ; produce the lines a H, D a, D 5, 5 H', and the spaces
enclosed determine the lengths of each arc.
This process works well for nearly all proportions of ellipses. It is
employed in striking out vaults and stone bridges.
The Parabola.
The parabola may be denned as an ellipse whose transverse axis is in-
finite ; its characteristic is that every point in the curve is equally distant
from the directrix E N and the focus F (fig. 165).
PBOB. LXII. To construct a parabola when the focus and directrix are
given.
1 Method (fig. 165). Through the focus F draw the axis A B perpen-
dicular to the directrix E N", and bisect A F at e, then e is the vertex of the
curve. Through a series of points C, D, E, on the directrix, draw parallels
to A B ; connect these points C, D, E, with the focus F, and bisect by
perpendiculars the lines F C, F D, F E. The intersections of these per-
pendiculars with the parallels will give points in the curve C' D' E', through
which trace the parabola.
2d Method (fig. 166). Place a straight edge to the directrix E N, and
GEOMETRICAL PROBLEMS.
73
apply to it a square LEG; fasten at G one end of a cord, equal in length
to E G ; fix the other end to the focus F ; slide the square steadily along
Fig. 165.
Fig. 166.
the straight edge, holding the cord taut against the edge of the square by
a pencil D, and it will describe the curve.
PEOB. LXIH. To construct a parabola, when the vertex A, the axis A B,
and a point M of the curve are given (fig. 167).
Construct the rectangle A B M C ; -divide M C into any number ol
equal parts, four for instance ; di-
vide A C in like manner ; con-
nect A 1, A 2, A 3 ; through 1'
2' 3', draw parallels to the axis.
The intersections I, II, III, of
these lines are points in the re-
quired curve.
PKOB. LXIY. To draw a tangent to a given point II of the parabola
(fig. 167).
From the given point II let fall a perpendicular on the axis at ~b / ex-
tend the axis to the left of A ; make A a equal to A ~b ; draw a II, and
it is the tangent required.
The lines perpendicular to the tangent are called normals. To find
tJie normal to any point I, having the tangent to any other point II. Draw
the normal II c / from I let fall a perpendicular I d on the axis A B ; lay
off d e equal to 5 c / connect I e, and we have the normal required. The
tangent may be drawn at I by a perpendicular to the normal I e.
Fig. 167.
74: GEOilETEICAI, PEOBLEMS.
The Hyperbola.
An hyperbola is a curve from any point P in -which, if two straight
lines be drawn to two fixed points, F F' the foci, their difference shall
always be the same.
PBOB. LXY. To describe an hyperbola (fig. 168).
From one of the foci F, with an assumed radius, describe an arc, and
from the other focus F', with another radius exceeding the former by the
given difference, describe two small arcs, cutting the first as at P and^>.
Let this operation be repeated with two new radii, taking care that the
second shall exceed the first by the same difference as before, and two new
points will be determined ; and this determination of points in the curve
may thus be continued till its track is obvious. By making use of the
same radii, but transposing, that is, describing with the greater about F,
and the less about F', we have another series of points equally belonging
to the hyperbola, and answering the definition ; so that the hyperbola con-
sists of two separate branches.
Fig. 168. Fig. 169.
The curve may be described mechanically (fig. 169). By fixing a ruler
to one focus F', so that it may be turned round on this point, connect the
extremity of the ruler R to the other focus F by a cord shorter than the
whole length F R of the ruler by the given difference ; then a pencil P
keeping this cord always stretched, and at the same time pressing against
the edge of the ruler, will, as the ruler revolves around F', describe an
hyperbola, of which F F' are the foci, and the differences of distances from
these points to every point in the curve will be the same.
PBOB. LXVI. To draw a tangent to any point P of 'an hyperbola (fig.
170).
GEOMETRICAL PROBLEMS.
75
, and from P let fall a
Fig. no.
On F' P lay off P p equal to F P ; conn
perpendicular on this line F j?, and
it will be the tangent required.
The three curves, the ellipse,
the parabola, and the hyperbola,
are called conic sections, as they
are formed by the intersections of a
plane with the surface of a cone
(plate Y).
If the cone be cut through both
its sides by a plane not parallel to
the base, the section is an ellipse ;
if the intersecting plane be parallel to the side of the cone, the section is a
parabola ; if the plane have such a position, that when produced it meets
the opposite cone, the section is a hyperbola. The opposite cone is a
reversed cone formed on the apex of the other by the continuation of its
sides.
The Cycloid.
The cycloid is the curve described by a point in the circumference of a
circle rolling on a straight line.
PEOB. LXYIL To describe a cycloid (fig. 171).
Draw the straight line A B as the base ; describe the generating circle
tangent to the centre of this line, and through the centre draw the line
E E parallel to the base ; let fall a perpendicular from upon the base ;
divide the semicircumference into any number of equal parts, for instance
six ; lay off on A B and C E distances 1', V 2' . . ., equal to the divisions
of the circumference ; draw the chords D 1, D 2 . . ., from the points 1', 2', 3' ...
on the line C E, with radii equal to the generating circle, describe arcs ;
76
GEOMETRICAL PROBLEMS.
from the points 1', 2', 3', 4', 5' on the line B A, and with radii equal suc-
cessively to the chords D 1, D 2, D 3, D 4, D 5, describe arcs cutting the
preceding, and the intersections will be points of the curve required.
. 2d Method (fig. 172). Let
9' be the base line, 4 9 the
half of the generating circle ;
divide the half circle into any
number of equal parts, say 9,
and draw the chord 01, 02,
3, &e. ; lay off on the base
1', V 2', 2' 3' . . . . , equal re-
spectively to the length of one
of the divisions of the half
circle 1 ; draw through the
points 1', 2', 3' lines par-
a\ allel to the chords 1, 2,
Wl 03....; the intersections I,
II, in .... of these lines are centres of the arcs a, a 5, b c . . . . , of which
the cycloid is composed.
The Epicycloid.
The epicycloid is formed by a point in the circumference of a circle
revolving either externally or internally on the circumference of another
circle as a base.
PKOB. LXVIIL To describe an epicycloid.
Let us in the first place take the exterior curve. Divide the circum-
ference A B D (fig. 173) into a series of equal parts 1, 2, 3 . . . ., beginning
from the point A ; set off in the same manner, upon the circle A M, A N,
the divisions 1', 2', 3' equal to the divisions of the circumference A B D.
Then, as the circle A B D rolls upon the circle A M A N, the points 1, 2, 3
will coincide successively with the points 1', 2', 3' ; and, drawing radii
from the point O through the points 1', 2', 3', and also describing arcs of
circles from the centre O, through the points 1, 2, 3, . . . ., they will inter-
sect each other successively at the points c, d, e Take now the dis-
tance 1 to c, and set it off on the same arc from the point of intersection, *,
of the radius A C ; in like manner, set off the distance 2 to d, from I to A 1 ,
and the distance 3 to e to A s , and so' on. Then the points A 1 , A a , A 1 ,
will be so many points in the epicycloid ; and their frequency may be in-
GEOMETRICAL PROBLEMS.
77
creased at pleasure by shortening the divisions of the circular arcs. Thus
the form of the curve may be determined to any amount of accuracy, and
completed by tracing a line through the points found.
As the distances 1 to c, which are near the commencement of the
curve, must be very short, it may, in some instances, be more convenient
to set off the whole distance i to 1 from <?, and in the same way the distance
b to 2 from d to A a , and so on. In this manner the form of the curve is
the more likely to be accurately defined.
2d Method. To find the points in the curve, find the positions of the
centre of the rolling circle corresponding to the points of contact 1', 2', 3',
&c., which may be readily done by producing the radii from the centre O,
through the points I/, 2', 3', to cut the circle B C. From these centres
describe arcs of a circle with the radius of C A, cutting the corresponding
arcs described from the centre 0, and passing through the points A 1 , A 2 ,
A 3 , .... as before.
When the moving circle A B D is made to roll on the interior of the
circumference A M, A 1ST, as shown (fig. 174), the curve described by the
point A is called an interior epicycloid. It may be constructed in the
78
GEOMETRICAL PROBLEMS.
same way as in the preceding case, as may be easily understood, the same
figures and letters of reference being used in both figures.
Fig. 174
The Involute.
The involute is a curve traced by the extremity of a flexible line un-
winding from the circumference of a circle.
PROB. LXIX. To describe an involute.
Divide the circumference of the given circle (fig. 175) into any number
GEOMETRICAL PROBLEMS.
79
of equal parts, as 0, 1, 2, 3, 4, ; at each of these points draw tangents to
the given circle ; on the first of these lay off the distance 1 I/, equal to the
arc 1 ; on the second lay off 2 2', equal to twice the arc 1 or the arc
2 : establish in a similar way the points 3', 4', 5', as far as may be
requisite, which are points in the curve required.
It may be remarked, that in all the problems in which curves have
been determined by the position of points, that the more numerous the
points thus fixed, the more accurately can the curve be drawn.
The involute curve may be described mechanically in several ways.
Thus, let A (fig. 176) be the cen-
tre of a wheel for which the form
of involute teeth is to be found.
Let m n a be a thread lapped
round its circumference, having
a loop-hole at its extremity a;
in this fix a pin, with which de-
scribe the curve or involute a 5
A, by unwinding the thread
gradually from the circumfer- rig. ITS.
ence, and this curve will be the proper form for the teeth of a wheel of the
given diameter.
The Spiral.
The spiral is the involute of a circle produced beyond a single revolution.
PKOB. LXX. To describe a spiral (figs. 5 and 6, plate XII).
Divide the circumference of the primary into any number of equal
parts, say not less than eight. To these points of division e,f, i, &c., draw
tangents, and from these points draw a succession of circular arcs ; thus,
from e as a centre, with the radius e g, equal to the arc a e reduced to a
straight linej describe the arc a g Kj from /", with the radius &/", describe
the arc g h from i the next arc, and so on. Continue the use of the centres
successively and repeatedly to the extent of the revolutions -required.
Thus the point a in the fig. is used as a centre for three arcs, bl,em,dn.
80
GEOilETEICAL PROJECTION.
GEOMETKICAL PKOJECTIOK
AKCHTTECTUKAL and mechanical drawing is generally the delineation of
bodies by geometrical or orthographic projection ; the representation, on a
sheet of paper which has only two dimensions length and breadth, of solids
which have three, length, breadth, and thickness.
Since the surfaces of all bodies may be considered as composed of
points, the first step is to represent the position in space of a point, by re-
ferring it planes whose position is established. The projection of a point
upon a plane is the foot of the perpendicular let fall from the point on the
plane. If, therefore, on two planes not parallel to each other, whose posi-
tions are known, we have the projections of a point, the position of this
point is completely determined by erecting perpendiculars from each plane
at the projected points : their intersection will be the point.
If from every point of an indefinite straight line A B (fig. 177), placed
in any manner in space, perpendiculars be let
fall on a plane L M N O, whose position is given,
then all the points in which these perpendiculars
meet the plane will form another indefinite straight
line a I : this line is called the projection of the
line AB on this plane. Since two points, are
sufficient to determine a straight line, it is only
Fig. 177. necessary to project two points of the line, and
the straight line drawn through the two projected points will be the pro-
jection of the given line. The projection of a straight line, itself perpen-
dicular to the plane, is the point in which this perpendicular meets the
plane.
If the projections a b and a' V of a straight line on the two planes
L M K O and L M P Q (fig. 178) are known, this line A B is determined ;
for if, through one of its projections a 5, we suppose a plane drawn perpen-
GEOMETRICAL PROJECTION.
81
dicularly to LMNO, and if through a' V another plane be drawn perpen-
dicular to L M P Q, the intersection of the two planes will be the line A B.
To delineate a solid, as the form of a machine for instance, it must be
referred to three series of dimensions, each of them at right angles to the
plane of the other. g
Fig. ITS.
Fig. 1T9.
Thus, let a~b G (fig. 179) be a parallelepiped in an upright position, of
which the plane a I is horizontal, and the planes a c and c I vertical. Let
d 0, df, and d g, be the boundary planes of a cubical space in which the
body a T> G is placed ; the sides of the body being parallel to those planes,
each to each, let the figure of the parallelepiped be projected on these
planes ; for this purpose draw parallel lines from the angles of the body
perpendicular to the planes, as indicated by the dotted lines ; then upon the
plane d e we shall have a! &', the projection of the surface a I : this is called
the plan of the object. Upon the plane dfwe have a' c', the projection
of the surface a c, the front elevation / and upon the plane dg, the projec-
tion V c' of the surface 5 c, the side elevation. Here, then, we have three
distinct views of the regular solid a 1) c delineated on plane surfaces, which
convey an accurate and sufficient idea of its form. Indeed, any two of
these representations are sufficient as a description of the object. From
the two figures a' c', V c', for example, the third figure a' V may be com-
pounded, by merely drawing the vertical lines c' k, V i, and a! k, c' I, to
meet the plane d e, and by producing them horizontally till they meet and
GEOMETRICAL PKOJECTTON.
m, a.
form the figure a' V . Similarly, the figure V c' may be deduced from the
other two by the aid of the lines h, *, from a' ', and the lines m, n, from
a' c'.
It is in this way that a third view of any piece of machinery is to be
found from two given views ; and in many cases two elevations, or one
elevation and a plan, may afford a sufficiently complete idea of the con-
struction of a machine. In other cases, many parts may be concealed by
others in which they are enclosed ; this suggests the occasional necessity
of views of the interior, in which the machine is supposed to be cut across
by planes, vertically or horizontally, so as properly to reveal its structure.
Such views are termed sections, and, with reference to the planes of section,
are denominated vertical and horizontal sections. To all such drawings
is given the general title of geometrical drawings, as distinguished from
perspective drawings.
By the aid of drawing instruments, measurements are transferable from
one position to another ; and there
is no necessity for erecting three
such planes as are supposed in fig.
179, upon which to execute draw-
ings of a machine. In practice,
the drawings are done upon one
common surface, and we may readi-
ly suppose the plane d g moved'
back into the position d g', and d e
also moved to d e', both of these
positions being in the plane of df.
This being done, we have the three
views depicted on one plane surface
(fig. 180). In this figure, the same
letters of reference are employed
as in fig. 179 ; d I and d m are the
ground and vertical lines. It is
Fig - m evident that the positions of the
same points in a' G' and a' V are in the same perpendicular from the ground
line : that, in short, the position of a point in the plane may be found by
applying the edge of the square to the same point as represented in the
elevation. The same remark is applicable as between the two elevations.
Hence the method of drawing several views of one machine upon the same
surface of paper in strict agreement with each other.
GEOMETRICAL PROJECTION.
83
OF SHADE LINES.
In outline drawings, or drawings which consist simply of the lines em-
ployed to indicate the form of the object represented, the roundness, the
flatness, or the obliquity of individual surfaces, is not indicated by the
lines, although it may generally be inferred from the relation of different
views of the same part. The direct significance of an outline drawing may,
however, be considerably increased, by strengthening those lines which in-
dicate the contours of surfaces resting in the shadow ; and this distinction
also improves the general appearance of the drawing. The strong lines, to
produce the best effect, ought to be laid upon the sharp edges at the sum-
mits of salient angles ; but bounding lines for curve surfaces should be
drawn finely, and should be but slightly, if at all, strengthened on the
Fig. 181.
Fig. 183.
Fig. 185.
Fig. 182.
Fig. 184.
Fig. 186.
shade side. This distinction assists in contrasting flat and curve surfaces.
To understand and apply the shade lines, however, we must know the
direction in which the light is supposed to fall upon the object, and thence
the locality of the shadows.
It is necessary for the explicitness of the drawing, that firstly, the light
be supposed to fall upon the object in parallel lines, that all the parts may
84 GEOMETRICAL PKOJECTION.
be shade-lined according to one uniform rule ; secondly, that the light
should be supposed to fall upon the object obliquely, as in this way both
the horizontal and vertical lines may be relieved by shading. To distribute
the shadows equally, the light is supposed to fall in directions forming an
angle of 45 with both the horizontal and the vertical planes of projection.
In general, the light should fall, as it were, from towards the upper left-
hand corner of the sheet of paper, supposing it square, making also an
angle of 45 with the surface.
, To illustrate what has been stated, let a I c d and a' I 1 ef (figs. 181 and
182) represent the elevation and plan of a solid rectangular body, N O
being the ground line. Let the direction of the light in both views be
represented in projection by the arrows A B ; these lines form the angle
45 with the line !N" O, and by drawing the parallels at J, d, a', e', so as to
embrace the extreme contour, we may readily perceive the way in which
the light falls upon the body : it falls upon three faces, namely, the two
vertical faces a'f,fe, and the top a' V ef. Consequently, the intersec-
tions or lines at which these planes meet ought to be lightly drawn, namely,
al,ad; a'fandfe. Again, the lateral planes represented by I c, cd,
V e', and a! V, are obviously in the shade, as no light falls upon them
directly ; and these lines are strengthened to express the distinction.
In figs. 183 and 184, the portion of the exterior from I by c to d is in
the shade, while the rest is light ; and the inverse is the case with the inner
edges. A peculiarity, however, occurs at d, for here the edges, inner and
outer, are parallel to the direction of the light. It is plain that the surfaces
which come up to these edges will be in a medium shade, and that the
lines at d should be of medium thickness.
Figs. 185 and 186 represent a hollow cylinder in projection. In the
plan, two lines, a, c, drawn parallel to the direction of the light, and touch-
ing the exterior of the cylinder, define the semicircular outline a V c, which
is thrown in the shade, and ought to be strengthened. The outlines a and
c are, like the edges at d (fig. 183), parallel to the light, and the contour
on each side gradually recedes and advances to the light. The thickness
of the line should, therefore, be rather gradually reduced at the points a, c.
In the elevation, the base-line df should be shaded, and b d is often half-
shaded, as it lies in a curve surface ; more generally full-shaded.
If, again, the cylinder be hollow, presenting in plan the interior contour
circle e h, then the semicircle e g li expresses the shady side of the interior,
the light striking directly upon the oppposite semicircle.
These examples illustrate every case of shade-lining that occurs in out-
line drawings. The effect is enhanced by proportioning the thickness of
GEOMETRICAL PROJECTION.
85
the lines to the depth of the surfaces to which they belong, below the
original surfaces from which the shadows
arise.
In the later French system of shading,
the light is supposed, in plan, to strike to-
wards the right hand upper corner, falling,
as it were, in front of the objects; but in
elevation, towards the right hand and foot of
the sheet (figs. 188, 18T).
It will be observed in the illustrations of
this work, that in the tinted drawings, the
shadow is thrown according to the French
system ; that is, the light is supposed to fall
on the drawing over the left shoulder at an
angle of 45. But in outline drawings, on
account of its greater simplicity, the more rig. m
usual system of throwing the shade line one way, both in plan and eleva-
tion, is adopted.
PROJECTIONS OF SIMPLE BODIES.
Projections of a regular hexagonal pyramid (PI. I, II). It is evident
that two distinct geometrical views are necessary to convey a complete
idea of the form of the object : an elevation to represent the sides of the
body, and to express its height ; and a plan of the upper surface, to ex-
press the form horizontally.
It is to be observed that this body has an imaginary axis or centre-line,
about which the same parts are equally distant ; this is an essential charac-
teristic of all symmetrical figures.
Draw a horizontal straight line L T through the centre of the sheet ;
this line will represent the ground line. Then draw a perpendicular Z 71
to the ground line. For the sake of preserving the symmetry of the draw-
ing, the centres of the lower range of figures are all in the same straight
line M N, drawn parallel to the ground line.
Figs. 1, 2. In delineating the pyramid, it is necessary, in the first
place, to construct the plan. The point S', where the line Z 71 intersects
the line M N", is to be taken as the centre of the figure, and from this
point, with a radius equal to the side of the hexagon which forms the base
of the pyramid, describe a circle, cutting M N in A' and D'. From these
86 GEOMETRICAL PROJECTION.
points with the same radius, draw four arcs of circles, cutting the primary
circle in four points. These six points being joined by straight lines, will
form the figure A' B' 0' D' E' F', which is the base of the pyramid ; and
the lines A' D', B' E', and C' F', will represent the projections of its edges
fore-shortened as they would appear in the plan. If this operation has
been correctly performed, the opposite sides of the hexagon should be
parallel to each other and to one of the diagonals ; this should be tested
by the application of the square or other instrument proper for the pur-
pose.
By the help of the plan obtained as above described, the vertical pro-
jection of the pyramid may be easily constructed. Since its base rests
upon the horizontal plane, it must be projected vertically upon the ground
line ; therefore, from each of the angles at A', B', C', and D', raise per-
pendiculars to that line. The points of intersection, A, B, C, and D, are
the true positions of all the angles of the base ; and it only remains to
determine the height of the pyramid, which is to be set off from the point
G to S, and to draw S A, S B, S C, and S D, which are the only edges of
the pyramid visible in the elevation. Of these it is to be remarked that
S A and S D alone, being parallel to the vertical plane, are seen in their
true length ; and moreover, that from the assumed position of the solid
under examination, the points F' and E' being situated in the lines B B'
and C C', the lines S B and S C are each the projections of two edges of
the pyramid.
Figs. 3 and 4. To construct the projections of the same pyramid, hav-
ing its base set in an inclined position, but with its edges S A and S D still
parallel to the vertical plane.
It is evident, that with the exception of the inclination, the vertical
projection of this solid is precisely the same as in the preceding example,
and it is only necessary to copy fig. 1. For this purpose, after having
fixed the position of the point D upon the ground line, draw through this
point a straight line D A, making with L T an angle equal to the desired
inclination of the base of the pyramid. Then set off the distance D A, fig.
1, from D to A, fig. 3 ; erect a perpendicular on the centre, and set off
G S equal to the height of the pyramid. Transfer also from fig. 1 the dis-
tance B G and C G to the corresponding points in fig. 3, and complete the
figure by drawing the straight lines A S, B S, C S, and D S.
In constructing the plan of the pyramid in this position, it is to be re-
marked, that since the edges S A and S D are still parallel to the vertical
plane, and the point D remains unaltered, the projection of the point A
will still be in the line M X. Its position at A' (fig. 4) is determined by
GEOMETRICAL PROJECTION. 87
the intersection of the perpendicular A A' with that line. The remaining
points B', C', &c., in the projection of the base, are found in a similar
manner, by the intersections of perpendiculars let fall from the correspond-
ing points in the elevation, with lines drawn parallel to M N", at a distance
(set off at 0,_p,) equal to the width of the base. By joining all the con-
tiguous points, we obtain the figure A' B' C' D' E' F', representing the
horizontal projection of the base, two of its sides, however, being dotted,
as they must be supposed to be concealed by the body of the pyramid.
The vertex S having been similarly projected to S', and joined by straight
lines to the several angles of the base, the projection of the solid is com-
pleted. -'., ;
Figs. 5 and 6. To find the horizontal projection of a transverse section
of the same pyramid, made by a plane perpendicular to the vertical, but
inclined at an angle to the horizontal plane of projection ; and let oil the
sides of the base be at an angle with the ground line.
Having drawn the vertical S S', the centre line of the figures, its point
of intersection with the line M JN" is the centre of the plan. Since none of
the sides of the base are to be parallel with the ground line, draw a diam-
eter A' D' making the required angle with that line, and from the points
A' and D' proceed to set out the angular points of the hexagon as in fig. 2.
Then, in order to obtain the projections of the edges of the pyramid, join
the angular points which are diametrically opposite ; and, following the
method pointed out in reference to fig. 1, project the figure thus obtained
upon the vertical plane, as shown at fig. 5.
ISTow, if the cutting plane be represented by the line a d in the eleva-
tion, it is obvious that it will expose, as the section of the pyramid, a poly-
gon whose angular points being the intersections of the various edges with
the cutting plane, will be projected in perpendiculars drawn from the
points where it meets these edges respectively. If, therefore, from the
points a, /, b, &c., we let fall the perpendiculars a a', ff, b b', &c., and
join their contiguous points of intersection with the lines A' D', F' C', B' E',
&c., we shall form a six-sided figure, which will represent the section re-
quired. The edges F S and E S being concealed in the elevation, but
necessary for the construction of the plan, have been expressed in dotted
lines, as also the portion of the pyramid situated above the cutting plane,
which, though supposed to be removed, is necessary in order to draw the
lines representing the edges. "VVe have here introduced the ordinary
method of expressing sections in purely line-drawings, by filling up the
spaces comprised within their outlines with a quantity of parallel straight
lines drawn at equal distances.
88 GEOMETRICAL PROJECTION.
Figs. 7 and 8. To find the horizontal projection of the transverse sec-
tion of a regular five-sided pyramid, cut by a plane perpendicular to the
vertical, but inclined to the horizontal plane ; and let one edge of the pyra-
mid he in a plane perpendicular to both planes of projection.
The plan of the pyramid is constructed by describing from the centre
S' a circle circumscribing the base, and from B' dividing the circumference
into five equal parts, and joining the contiguous points of division by
straight lines to form the polygon A' B' C' D' E', each of whose angles,
being joined to the centre S', shows the projections of the edges of the
pyramid. Then, following the method above explained, we obtain the
elevation and the horizontal projection of the section made by the plane
a c. But that method will not suffice for the determination of the point V,
because the perpendicular let fall from the corresponding point 5, in the
elevation, coincides with the projection of the edge B S. Let the pyramid
be supposed to be turned a quarter of a revolution round its axis ; the line
B' S' will then have assumed the position S' J 4 . Project the point V to 5 s ,
and join S V. Then, since the required point must also be conceived to
have described a quarter of a circle in a plane parallel to the horizontal
plane, and that its new position must be in the line S 5 s , it is obvious that
its vertical projection is the point J 4 , the intersection of a horizontal line
drawn through 5, with that line. The distance 5 4 , then, being transferred
from S' to V , determines the position of the latter point in the plan ; or,
following a more methodical process, by projecting the point J 4 to V 1 , and
describing a circle from the centre S' passing through V ; its intersection
with B' S' is the point sought.
PROJECTIONS OF A PRISM.
Plate III., figs. 1 and 2. Required to represent in plan and elevation
a regular six-sided prism in an upright position.
Lay down the ground line and centre line, and describe the hexagon as
already directed. Project the plan thus delineated by perpendiculars to
the ground line from each of its angular points ; and since the prism is
upright, these angular points themselves represent the horizontal projec-
tions of all its edges, and their elevations coincide with the perpendiculars
A' G, B' H, &c. Set off from G to A the height of the prism, and through
A draw A D parallel to the ground line. This will be the vertical projec-
tion of the upper surface. The edges being all parallel to the vertical
plane, are, of course, seen in their actual length.
Figs. 3 and 4. To form the projections of the, same prism, supposing it
GEOMETRICAL PROJECTION. 89
to have been moved round the point G, in a plane parallel to the vertical
plane.
Copy the elevation (fig. 1) on an inclined base. Now, by letting fall
perpendiculars from all the angles in the elevation, and joining the con-
tiguous points of intersection with the horizontal lines appropriate to these
points respectively, we obtain the polygon A' B' C' D 7 E 7 F' as the projec-
tion of the upper surface, and G' H' I 7 K' I/M 7 as that of the base of the
prism. Finally, it will be observed that all the edges are represented, in
the horizontal projection, by equal straight lines, as D' K 7 , A 7 G', &c., and
that the sides A' B', G 7 H', &c., remain still parallel to each other, which
will afford the means of verifying the accuracy of the drawings.
Figs. 5 and 6. Required the projections of the same prism set into a
position inclined to both planes of projection.
Assuming that the inclination of the prism upon the horizontal plane
is the same as in the preceding figures for the sake of simplifying the opera-
tion, the first process is to copy fig. 4, which may be done by drawing a
centre line X X, so as to form the required angle of the prism with the
vertical plane ; then, having set off upon this line a distance equal to
A 7 K', fig. 4, transfer the distances A 7 G' and D 7 K 7 also to fig. 6 ; and in
order to find the remaining angular points, make A 7 a equal to the corre-
sponding distance in fig. 4, and through a draw B 7 F 7 perpendicular to the
centre line, and transfer the distances a B 7 , a F 7 . Through the points B 7
and F 7 , draw straight lines parallel to A 7 K 7 , and join A 7 B 7 , A 7 F 7 ; and
since we have already seen that all the other sides must be parallel to these,
the figure is completed by drawing through the points G 7 , D 7 , and K 7 ,
straight lines parallel to A 7 B 7 and A 7 F 7 respectively.
Now, since the prism has been supposed to have preserved its former
inclination to the horizontal plane, it is obvious that every point in it, such
as A, has, in assuming its new position, simply moved in a horizontal
plane, and will, therefore, be in the line A A parallel to the ground line,
and since the same point has been projected to A 7 , fig. 6, it will also be in
the/ perpendicular A! A ; the point of intersection A, fig. 5, is, therefore,
its projection in the elevation. The remaining angular points in this view
are all determined in the same manner by the aid of figs. 3 and 6 ; and
having joined the contiguous points, and the corresponding angles of the
upper and lower surface, we obtain the complete vertical projection of the
prism in its doubly-inclined position.
90 GEOMETRICAL PROJECTION.
CONSTRUCTION OF THE CONIC SECTIONS.
Plate V. The plan of the cone (fig. 2) is simply a circle, described
from the centre S' of a diameter equal to that of the base. Its elevation
(fig. 1) is an isosceles triangle, obtained by drawing tangents A' A, B' B,
perpendicular to and intersecting the ground line ; then set off upon the
centre line the height C S, and join S A, S B. These lines are called the
exterior generatrices of the cone.
Figs. 1 and 2. Given tlie projections of a, cone, and the direction of a
plane X X, cutting it perpendicularly to the vertical, and obliquely to the
horizontal plane ; required to find, first, the horizontal projection of this
section and, secondly, the outline of the ellipse thus formed.
Through the vertex of the cone draw a line S E to any point within the
base A B ; let fall a perpendicular from E, cutting the circumference of
the base in E', and join E' S' ; then another perpendicular let fall from e
will intersect E 7 S' in a point e' ', which will be the horizontal projection
of a point in the curve required ; and so on for any required number of
points.
The exterior generatrices A S and B S being both projected upon the
line A' B', the extreme limits of the curve sought will be at the points a'
and y on that line, which are the projections of the points of intersection
a and 5 of the cutting plane with the outlines of the cone. And since the
line a' V will obviously divide the curve symmetrically into two equal
parts, the points /', g' , h', &c., will be readily obtained by setting off
above that line, and on their respective perpendiculars, the distances d' d?,
e' e\ &c. A sufficient number of points having thus been determined, the
curve drawn through them (which will be found to be an ellipse) will be
the outline of the section required.
This curve may be obtained by another method, depending on the prin-
ciple that all sections of a cone by planes parallel to the base are circles.
Thus, let the line F G represent a cutting plane ; the section which it
makes with the cone will be denoted on the horizontal projection by a
circle drawn from the centre S', with a radius equal to half the line F G ;
and by projecting the point of intersection II of the horizontal and oblique
planes by a perpendicular H H', and noting where this line cuts the circle
above referred to, we obtain two points IT and I' in the curve required.
By a similar construction, as exemplified in the drawings, any number of
additional points may be found.
As the projection obtained by the preceding methods exhibits the sec-
GEOMETRICAL PROJECTION. 91
tion as fore-shortened, and not in its true dimensions, we shall now proceed
to the consideration of the second question proposed. Let the cutting
plane X X be conceived to turn upon the point b, so as to coincide with
the vertical line b &, and (to avoid confusion of lines) let b k be transferred
to a' b', which will represent, as before, the extreme limits of the curve
required. Now, taking any point, such as d, it is obvious that in this new
position of the cutting plane, it will be represented by d*, and if the cutting
plane were turned upon a! V as an axis till it is parallel to the vertical
plane, the point which had been projected at d* would then have described
round a' V an arc of a circle, whose radius is the distance d? d? (fig. 2).
This distance, therefore, being set off at d' and f on each side of a' b' ',
gives two points in the curve sought. By a similar mode of operation any
number of points may be obtained, through which, if a curve be drawn, it
will be an ellipse of the true form and dimensions of the section.
Figs. 3 and 4. To find the horizontal projection and actual oiiiline of
the section of a cone, made by a plane Y Y parallel to one side or generatrix,
and perpendicular to the vertical plane.
Determine by the second method laid down in the preceding problem
any number of points, as F', G', J', K', &c., in the curve representing the
horizontal projection of the section specified. The horizontal plane pass-
ing through M gives only one point M' (which is the vertex of the curve
sought), because the circle which denotes the section that it makes with
the cone is a tangent to the given plane.
In order to determine the actual outline of this curve, suppose the
plane Y Y to turn as upon a pivot at M, until it has assumed the position
M B, and transfer M B parallel to itself to M' B'. The point F will thus
have first described the arc F E till it reaches the point E, which is then
projected to E 2 ; suppose the given plane, now represented by M' B', to
turn upon that line as an axis, until it assumes a position parallel to the
vertical plane, the point E 2 , which is distant from the axis M' B' by the
distance F' S' (fig. 4), will now be projected to F' (fig. 3). The same dis-
tance F' S' set off on the other side of the axis H' B' gives another point
G' in the curve required, which is the parabola.
Figs. 7, 8, 9. To draw the vertical projection of the sections of two op-
posite cones made by a plane parallel to their axis.
Let C E D and C B A be the two cones, and X X the position of the
cutting plane (fig. 7). Project in plan either of the cones, as in fig. 8 ;
from its centre, with a radius equal to L H, describe a circle, and draw
the tangent b a ; ba will be the horizontal projection of the cutting plane.
Draw the line II' M' (fig. 9) parallel to the cutting plane ; H', M' corre-
92 GEOMETRICAL PROJECTION.
spending in position to the intersections H, M (fig. 7"), of the plane with the
cones. From H' and M' lay off distances equal to L K, K I, and the length of
the cone, and through these points draw perpendiculars, as/' <?', d' c', V a',
&c., which must be made equ^l to the chords/ 0, dc^ba (fig. 8), made by
the cutting plane a 5, with circles whose radii are G K, IF, and the radius
of the base of the cone. Through the points a', c', e', H',/', d', V, draw
the curve, and we have the projection required. A similar construction
will give the sectional projection of the opposite cone at W. The curve
thus found is the hyperbola.
PENETRATIONS OR INTERSECTIONS OF SOLIDS.
On examining the minor details of most machines, we find numerous
examples of cylindrical and other forms, fitted to, and even appearing to
pass through each other in a great variety of ways. The examples grouped
in plates VI. to XI. are selected with the view of exhibiting those cases
which are of most frequent occurrence, and of elucidating general principles.
PENETRATIONS OF CYLINDERS.
Plate VI. Figs. 1 and 2 represent the projections of two cylinders of
unequal diameters meeting each other at right angles ; one of which is
denoted by the rectangle A B E D in the vertical, and by the circle
A! H' B' in the horizontal projections ; while the other, which is supposed
to be horizontal, is indicated in the former by the circle L P I !N", and in
the latter by the figure L' I' K' M'. From the position of these two solids
it is evident that the curves formed by their junction will be projected in
the circles A' H' B' and L P I !N" ; and further, that such would also be
the case even although their axes did not intersect each other.
But if the position of these bodies be changed into that represented at
figs. 3 and 4, the lines of their intersection will assume in the vertical pro-
jection a totally different aspect, and may be accurately determined by
the following construction.
Through any point taken upon the plan (fig. 4) draw a horizontal line
a' I', which is to be considered as indicating a plane cutting both cylinders
parallel to their axes ; this plane would cut the vertical cylinder in lines
drawn perpendicularly through the points c' and d'. To find the vertical
projection of its intersection with the other cylinder, conceive its base I' L',
GEOMETRICAL PROJECTION. 93
after being transferred to P L 3 , to be turned over parallel to the horizontal
plane ; this is expressed by simply drawing a circle of the diameter I 2 L 2 ;
and producing the line a' b' to a 2 ; then set off the distance a* e' on each
side of the axis I K, and draw straight lines through these points parallel
to it. These lines a &, g A, denote the intersection of the plane a! b' with
the horizontal cylinder, and therefore the points c, d, m, o, where they cut
the perpendiculars c c', d d', are points in the curve required. By laying
down other planes similar to a! I', and operating as before, any number of
points may be obtained. The vertices i and k of the curves are obviously
projected directly ; and their extreme points are determined by the inter-
sections of the outlines of both cylinders. When the cylinders are of
unequal diameters, as in the present case, the curves of penetration are
hyperbolas.
Figs. 5 and 6. When the diameters of the cylinders are equal, and
when they cut each other at right angles, the curves of penetration are pro-
jected vertically in straight lines perpendicular to each other, as in fig. 5,
where the projections of some of the points are indicated in elevation and
plan by the same letters of reference.
Figs. 7 and 8. To delineate the intersections of two cylinders of equal
diameters at right angles, when one of the cylinders is inclined to the ver-
tical plane.
Supposing the two preceding figures to have been drawn, the projec-
tion c of any point such as c' may be ascertained by observing that it must
be situated in the perpendicular c' <?, and that since the distance of this
point (projected at c in fig. 5) from the horizontal plane remains unaltered,
it must also be in the horizontal line c c. Upon these principles all the
points indicated by literal references in fig. 7 are determined ; the curves
of penetration resulting therefrom intersecting each other at two points
projected upon the axial line L K, of which that marked q alone is seen.
The ends of the horizontal cylinder are represented by ellipses, the con-
struction of which will also be obvious on referring to the figures ; and
they do not require further consideration here.
PENETRATIONS OF CYLINDERS, CONES, AND SPHERES.
PL YIIL, figs. 1 and 2. To find the curves resulting from the inter-
section of two cylinders of unequal diameters, meeting at any angle.
For the sake of simplicity, suppose the axes of both cylinders to be
parallel to the vertical plane, and let A B E D and N O Q P be their pro-
94; GEOMETRICAL PROJECTION.
jections upon that plane. In constructing, in the first place, their horizon-
tal projection, observe that the upper end A B of the larger cylinder is
represented by an ellipse A' K' B' M', which may easily be drawn by the
help of the major axis K' M' equal to the diameter of the cylinder, and of
the minor A' B', the projection of the diameter. The visible portion of
the base of the cylinder being similarly represented by the semi-ellipse
I/ D' H', its entire outline will be completed by drawing tangents I/ M'
and IF E7. The upper extremity P K of the smaller cylinder will also be
projected in the ellipse^' i' W.
~R ow, suppose a plane, as a' g' (fig. 2), to pass through both cylinders
parallel to their axes ; it will cut the surface of the larger cylinder in two
straight lines passing through the points f and g' on the upper end of the
cylinder ; these lines will be represented in the elevation, by projecting
the points y and g' to/*, g j and drawing af and c g parallel to the axis.
The plane af g' will in like manner cut the smaller cylinder in two straight
lines, which will be represented in the vertical projection by d h and e i,
and the intersections of these lines with af and c g will give four points
I, k, ra, and n, in the curves of penetration. Of these points one only, that
marked ?, is visible in the plan, where it is denoted by I'.
Fig. 1. To find the curves of penetration in the elevation without the
aid of the plan.
Let the bases D E and Q O of both cylinders be conceived to be turned
over into the vertical plane after being transferred to any convenient dis-
tance, as D 2 E 2 and Q 2 O 2 , from the principal figure ; they will then be
represented by the circles D 2 H 2 E 2 and Q 2 G' O 2 . Now draw a*c* paral-
lel to D E, and at any suitable distance from the centre I ; this line will
represent the intersection of the base of the cylinder with a plane parallel
to the axes of both, as before. The intersection of this plane with the
base of the smaller cylinder will be found by setting off from E, a distance
Rj?, equal to I o, and drawing through the point p a straight line parallel
to Q O. It is obvious that the intersection of the supposed plane with the
convex surfaces of the cylinders will be represented by the lines af, c g,
and d A, e i, drawn parallel to the axes of the respective cylinders through
the points where the chords a* c z and d 2 e z cut the circles of their bases ;
and that, consequently, the intersections of these lines indicate points in
the curves sought. These points may be multiplied indefinitely by con-
ceiving other planes to pass through the cylinders, and operating as
before.
Eigs. 3 and 4. To find the curves of penetration of a cone and,
GEOMETRICAL PROJECTION. 95
Let D S be the axis of the cone, A' I/ B' the circle of its base, and the
triangle A B S its projection on the vertical plane ; and let C, C', be the
projections of the centre, and the circles E' K7 F' and E G F those of the
circumferences of the sphere.
This problem, like most others similar to it, can be solved only by the
aid of imaginary intersecting planes. Let a b (fig. 3) represent the pro-
jection of a horizontal plane ; it will cut the sphere in a circle whose diam-
eter is a T), and which is to be drawn from the centre C' in the plan.
Its intersection with the cone is also a circle described from the centre S'
with the diameter c d; the points e' and/', where these two circles cut
each other, are the horizontal projections of two points in the lower curve,
which is evidently entirely hidden by the sphere. The points referred to
are projected vertically upon the line a 5 at e and f. The upper curve,
which is seen in both projections, is obtained by a similar process ; but it
is to be observed that the horizontal cutting planes must be taken in such
positions as to pass through both solids in circles which shall intersect each
other. For our guidance in this respect it will be necessary, first, to de-
termine the vertices m and n of the curves of penetration.
For this purpose, conceive a vertical plane passing through the axis of
the cone and the centre of the sphere ; its horizontal projection will be the
straight line G' L' joining the centres of the two bodies. Let us also make
the supposition that this plane is turned upon the line C C' as on an axis,
until it becomes parallel to the vertical plane ; the points S' and L' will
now have assumed the positions S 2 and L 2 , and consequently the axis of
the cone will be projected vertically in the line D' S 3 , and its side in S 3 L 3 ,
cutting the sphere at the points p and r. Conceive the solids to have
resumed their original relative positions, it is clear that the vertices or
adjacent limiting points of the curves of penetration must be in the hori-
zontal lines p o and r q, drawn through the points determined as above ;
their exact positions on these lines may be ascertained by projecting ver-
tically the points ra' and n' t where the arcs described by the points p and
r, in restoring the cone to its first position, intersect the line S L.
It is of importance further, to ascertain the points at which the curves
of penetration meet the outlines A S and S B of the cone. The plane
which passes through these lines being projected horizontally in A' B', will
cut the sphere in a circle whose diameter is i'j'j this circle, described in the
elevation from the centre C, will cut the sides A S and S B in four points
at which the curves of penetration are tangents to the outlines of the cone.
Figs. 5 and 6. To find the lines of penetration of a cylinder and a
cylindrical ring or torus.
96 GEOMETRICAL PROJECTION.
Let the circles A' E' B', F' G' K/, represent the horizontal, and the
figure A C B D the vertical projection of the torus, and let the circle
H'/'L', and the rectangle H I M L be the analogous projections of the
cylinder, which passes perpendicularly through it. Conceive, as before,
a plane a ~b (fig. 5), to pass horizontally through both solids ; it will ob-
viously cut the cylinder in a circle which will be projected in the base
IT f I/ itself, and the ring in two other circles, of which one only, part
of which is represented by the arc f b 3 b', will intersect the cylinder at
the points f and J 3 , which being projected vertically to fig. 5, will give
two points/ 1 and 5* in the upper curve of penetration.
Another horizontal plane, taken at the same distance below the centre
line A B as that marked a b is above it, will evidently cut the ring in
circles coinciding with those already obtained ; consequently the points/'
and b 3 indicate points in the lower as well as in the upper curves of pene-
tration, and are projected vertically at d and e. Thus, by laying down
two planes at equal distances on each side of A B, by one operation four
points in the curves required are determined.
To determine the vertices m and n, following the method explained in
the preceding problem, draw a plane O ri ', passing through the axis of the
cylinder and the centre of the ring, and conceive this plane to be moved
round the point O as on a hinge, until it has assumed the position O B',
parallel to the vertical plane ; the point n', representing the extreme
outline of the cylinder in plan, will now be at r', and being projected ver-
tically, that outline will cut the ring in two points p and r, which would
be the limits of the curves of penetration in the supposed relative position
of the two solids ; and by drawing the two horizontal lines r n and p m,
and projecting the point n' vertically, the intersections of these lines, the
two points, m and n, are the vertices of the curves in the actual position of
the penetrating bodies.
The points at which the curves are tangents to the outlines II I and L M
of the cylinder, may readily be found by describing arcs of circles from
the centre O through the points H' and L', which represent these lines in
the plan, and then proceeding, as above, to project the points thus obtained
upon the elevation. Lastly, to determine the points, as j, z, &c., where
the curves are tangents to the horizontal outlines of the ring, draw a circle
P' B' j' with a radius equal to that of the centre line of the ring, namely,
P D ; the points of intersection d and j' are the horizontal projections of
the points sought.
Required to represent the sections which would be made in the ring now
before us, by two planes, one of which, N' T', is parallel to the vertical
GEOMETRICAL PROJECTION. 97
plane, while the other, T" E', is perpendicular to both planes of projec-
tion.
The section made by the last-named plane must obviously have its ver-
tical projection in the line C D, which indicates the position of the plane ;
but the former will be represented in its actual form and dimensions in
the elevation. To determine its outlines, let two horizontal planes g q and
i k, equidistant from the centre line A B, be supposed to cut the ring ;
their lines of intersection with it will have their horizontal projections in
the two circles g' o' and h f q' which cut the given plane N' T' in o' and q f .
These points being projected vertically to o, q, k, &c., give four points in
the curve required. The line N' T' cutting the circle A' E' B' at 1ST', the
projection N" of this point is the extreme limit of the curve.
The circle P' s' f, the centre line of the rim of the torus, is cut by the
planes N' T' at the point s', which being projected vertically upon the
lines D P and C I, determines s and I, the points of contact of the curve
with the horizontal outlines of the ring. Finally, the points t and u are
obtained by drawing from the centre O a circle T' v f tangent to the given
plane, and projecting the 'point of intersection v f to the points v and a?,
which are then to be replaced upon C D by drawing the horizontals v t
and x u.
PENETRATIONS OF CYLINDERS, PRISMS, SPHERES, AND CONES.
Plate X., figs. 1 and 2. Required to delineate the lines of penetra-
tion of a sphere and a regular hexagonal prism whose axis passes through
the centre of the sphere.
The centres of the circles forming the two projections of the sphere are,
according to the terms of the problem, upon the axis C C' of the upright
prism, which is projected horizontally in the regular hexagon D' E' F' G'
IF I'. Hence it follows, that as all the lateral faces of the prism are equi-
distant from the centre of the sphere, their lines of intersection with it will
necessarily be circles of equal diameters. Now, the perpendicular face
represented by the line E' F' in the plan, will meet the surface of the
sphere in two circular arcs E F and L M (fig. 1), described from the centre
C, with a radius equal to c r V or a' c'. And the intersections of the two
oblique faces D'E' and F' G' will obviously be each projected in two arcs
of an ellipse whose major axis dg is equal to the diameter of the circle a c o,
and the minor axis is the vertical projection of that diameter, as represented
at e' f (fig. 2). But as it is necessary to draw small portions only of these
curves, the following method may be employed.
7
98 GEOMETRICAL PROJECTION.
Draw D G ; through the points E, F, divide the portions E F and F G
respectively into the same number of equal parts, and drawing perpen-
diculars through the points of division, set off from F G the distances from
the corresponding points in E F to the circular arc E C F, as points in the
elliptical arc required. The remaining elliptical arcs should be traced by
the same method.
Figs. 3 and 4. Required to draw the lines of penetration of a cylinder
and a sphere, the centre of the sphere leing without the axis of the cylinder.
Let the circle D' E' I/ be the projection of the base of the given cylin-
der, the elevation of which is shown at fig. 3, and let A B be the diameter
of the given sphere. If a plane, as c' d', be drawn parallel to the vertical
plane, it will evidently cut the cylinder in two straight lines G G', H IT,
parallel to the axis, and projected vertically from the points G' and IF.
This plane will also cut the sphere in a circle whose diameter is equal to
c' d', and which is to be described from the centre C with a radius of half
that line ; its intersection with the lines G G' and H H' will give so many
points in the curves sought, viz., G, H, I, K.
The planes a' V and e' f , which are tangents to the cylinder, furnish
only two points respectively in the curves ; of these points E and F alone
are visible, the other two, L and M, being concealed by the solid ; there-
fore, the planes drawn for the construction of the curves must be all taken
between a' V and e'f. The plane which passes through the axis of the
cylinder cuts the sphere in a circle whose projection upon the vertical
plane will meet at the points D, N, and g, h, the outlines of the cylinder,
to which the curves of penetration are tangents.
Figs. 5 and 6. To find the lines of penetration of a truncated cone
and a prism.
The . straight line C D is the axis of a truncated cone, which is repre-
sented in the plan by two circles described from the centre C' ; and the
horizontal lines M N and M' N' are the projections of the axis of a prism
of which the base is square, and the faces respectively parallel and per-
pendicular to the planes of projection.
In laying down the plan of this solid, it is supposed to be inverted, in
order that the smaller end of the cone, and the lines of intersection of the
lower surface F G of the prism may be exhibited. According to this
arrangement, the letters A' and B' (fig. 6) ought, strictly speaking, to be
marked at the points I' and H/, and conversely ; but as it is quite obvious
that the part above M' J\ f is exactly symmetrical with that below it, the
distribution of the letters of reference adopted in our figures can lead to no
confusion.
GEOMETRICAL PROJECTION. 99
The intersection of the plane F G with the cone is projected horizon-
tally in a circle described from the centre C', with the diameter F' G'.
The arcs I' F' A' and H' G' B' are the only parts of this circle which re-
quire to be drawn.
Figs. 7 and 8. To describe the curves formed ly the intersection of a
cylinder with the frustum of a cone, the axes of the two solids cutting each
Bother at right angles.
The axes of the solids and their projections are laid down in the fignres
precisely as in the preceding example. The intersections of the outlines
of the cone in the elevation with those of the cylinder, furnish, obviously,
four points in the curves of penetration ; these points are all projected
horizontally upon the line A' B'. Now, suppose a plane, as a o (fig. 7),
to pass horizontally through both solids ; its intersection with the cone
will be a circle of the diameter c d, while the cylinder will be cut in two
parallel straight lines, represented in the elevation by a 5, and whose hori-
zontal projection may be determined in the following manner : Conceive
a vertical plane f g, cutting the cylinder at right angles to its axis, and let
the circle g ef thereby formed be described from the intersection of the
axes of the two solids ; the line^' h will now represent, in this position of
the section, the distance of one of the lines sought from the axis of the
cylinder. Now set off this distance on both sides of the point A', and
through the points Jc and a' thus obtained, draw straight lines parallel to
A' B' ; the intersections of these lines with the circle drawn from the
centre C' of the diameter c d will give four points m' ', p', n, and 0, which
being projected vertically upon a 5, determine two points in and^> in the
curves required.
In order to obtain the vertices or adjacent limiting points of the curves,
draw from the vertex of the cone a straight line t e, touching the circle g ef,
and let a horizontal plane be supposed to pass through the point of con-
tact e. Proceed according to the method given above to determine the
intersections of this plane with each of the solids in question, the four
points *', /, , and s, which being projected vertically upon the line e r,
determine the vertices i and r required.
OF THE HELIX.
Plate XII. The Helix is the curve described upon the surface of a
cylinder by a point revolving round it, and at the same time moving
parallel to its axis by a certain invariable distance during each revolution.
This distance is called the pitch of the screw.
100 GEOMETEICAL PROJECTION.
Figs. 1 and 2. Required to construct the helical curve described by the
point A upon a cylinder projected horizontally in the circle A! C' F ; , the
pitch being represented by the line A.' A 3 .
Divide the pitch A' A 3 into any number of equal parts, say eight ; and
through each point of division, 1, 2, 3, &c., draw straight lines parallel to
the ground line. Then divide the circumference A' C' F' into the same
number of parts ; the points of division B', C', E', F', &c., will be the
horizontal projections of the different positions of the given point during
its motion round the cylinder. Thus, when the point is at B' in the plan,
its vertical projection will be the point of intersection B of the perpendicu-
lar drawn through B' and the horizontal drawn through the first point of
division. Also, when the point arrives at C' in the plan, its vertical pro-
jection is the point C, where the perpendicular drawn from C' cuts the
horizontal passing through the second point of division, and so on for all
the remaining points. The curve A B C F A 3 drawn through all the
points thus obtained, is the helix required.
Figs. 1 and 2. To draw the vertical elevation of the solid contained
between two helical surfaces and two concentric cylinders.
A helical surface is generated by the revolution of a straight line
round the axis of a cylinder ; its outer end moving in a helix, and the line
itself forming with the axis a constant and invariable angle.
Let A' C' F 7 and K' M' O' represent the concentric bases of the cylin-
ders, whose common axis S T is vertical ; the curve of the exterior helix
A C F A 3 is the first to be drawn according to tHe method above shown.
Then having set off from A to A 2 the thickness of the required solid, draw
through A 2 another helix equal and similar to the former. Now construct,
as above, another helix, K C O, of the same pitch as the last, but on the-
interior icylinder ; as also another, K 2 C 2 O 2 , equal and parallel to the
former. The lines A' K', B' I/, C' M', &c., represent the horizontal pro-
j ections of the various positions of the generating straight line, which, in
the present example, has been supposed to be horizontal ; and these lines
are projected vertically at A K, B L, &c.
It will be observed^ that in the position A K the generating line is
projected in its actual length, and that at the position C' M' its vertical
projection is the point C. The same remark applies to the generatrix of
the second helix. The parts of both curves which are visible in the ele-
vation may be easily determined by inspection.
Figs. 3 and 4. To determine the vertical projection of the solid formed
by a sphere moving in a helical curve.
Let A' C' E' be the base of a cylinder, upon which the centre point C'
GEOMETRICAL PROJECTION. 101
of a sphere whose radius is a! C' describes a helix, which is projected on
the vertical plane in the curve A C E J. After determining as above the
various points A, B, C, D , in this curve, draw from each of these
points as centres, circles with the radius a! C' ; the circumferences of these
circles will denote the various positions of the sphere during its motion
round the cylinder; and if lines be drawn touching these circles, the
curves thereby formed will constitute the figure required. One of these
curves will disappear at O, which is its point of contact with the circle
described from the point E, the intersection of the helix with the perpen-
dicular E E' ; it will again reappear at the point I when it becomes a tan-
gent to the circle described from the point J in the prolongation of the
line A A 7 . The exterior and interior circles (fig. 4) represent the horizon-
tal projection of the solid in question.
The conical helix differs from the cylindrical one in that it is described
on the surface of a cone instead of on that of a cylinder ; but the construc-
tion differs but slightly from the one described. By following out the same
principles, helices may be represented as lying upon spheres or any other
surfaces of revolution.
In the arts are to be found numerous practical applications of the heli-
cal curve, as wood and machine screws, geers, and staircases, the construc-
tion of which will be still farther explained under their appropriate heads.
102 THE DEVELOPMENT OF SUEFACES.
THE DEVELOPMENT OF SURFACES.
THE development of the surface of a solid is the drawing or unrolling
on a plane the form of its covering ; and if that form be cut out of paper,
it would exactly fit and cover the surface of the solid. Frequently in
practice, the form of the surface of a solid is found by applying paper, or
thin sheet brass directly to the solid, and cutting it to fit. Tin and copper-
smiths, boiler-makers, &c., are continually required to form from sheet
metal forms analogous to solids ; to execute which they should be able to
construct geometrically the development of the surface of which they are
to make the form.
The development of the surface of a plain cylinder is evidently but a
plane sheet, of which the circumference is one dimension whilst its length
is the other.
THE DEVELOPMENT OF THE STIRFACE OF INTERSECTED CYLINDEES.
Plate XIIL, figs. 1 and 2. To draw the surface of a cylinder formed
~by the intersection of another equal cylinder, as the knee of a stove pipe.
Let A B C D be the elevation of the pipe or cylinder. Above A B
describe the semicircle A' 4' B' of the same diameter as the pipe ; divide this
semicircle into any number of equal parts, eight for instance ; through these
points I 7 , 2', 3', &c., draw lines parallel to side A C of pipe, and cutting
the line C D of the intersection of the two cylinders. Lay off A!' B" equal to
the semicircle A' 4' B', and divided into the same number of equal parts ;
through these points of division erect perpendiculars to A" B", and on these
perpendiculars lay off the distances A" C", V I", 2" 2", 3" 3"-, and so on,
corresponding to A C, 1 1, 2 2, 3 3, &c., in preceding figure. Through the
points C", 1", 2", D", then draw connecting lines, and we have the
developed surface required. It is to be remarked, that this gives but one
half of the surface of the pipe, the other being exactly similar to it.
THE DEVELOPMENT OF SURFACES. 103
Figs. 3 and 4. To develop, the surface of a cylinder intersected by an-
other cylinder, as in the formation of a "[pipe.
The construction is similar to the preceding, and as the same letters
and figures are preserved relatively, the demonstration will be easily
understood from the foregoing.
The development of the surface of a right cone (figs. 5 and 6). From C'
(fig. 6) as a centre, with a radius C' A' equal to the inclined side A C of
the cone (fig. 5), describe an arc of a circle A' B' A" ; on this arc lay off
the distance A' B' A" equal to the circumference of the base of the cone ;
connect A' C' and C' A*, and A! B' A" C' is the developed surface re-
quired.
To develop the surface of afrustrum of a cone, D A B E (fig. 5).
On fig. 6 develop the cut-off cone C D E as in preceding construction,
and we have A' B' A" D" E 7 D as the developed surface of the right
frustrum.
To develop the surfqce of a frustrum of a cone, when the cutting plane
a 1) (fig. 5) is inclined to the hase.
On A B the base describe the semicircle A 3' B ; divide the semicircle
into any number of equal parts, six for instance ; from each point of divi-
sion 1', 2', 3', 4', 5', let fall perpendiculars to the base ; at 1, 2, 3, 4, 5, con-
nect each of these last points with the apex C. Divide now the arc A' B'
(fig. 6) into six equal parts, or the arc A' B' A" into twelve ; each of these
parts by the construction is equal to the arc A 1', V 2' (fig. 5) ; connect
these points of division with the point C' ; on C' A' (fig. 6) take C' a' equal
to C a of fig. 5, a being the point at which the plane cuts the inclined side
of the cone ; in the same way on C' B', lay off C' V equal to h.
It is evident that all the lines connecting the apex C with the base,
included within the two inclined sides, are represented as less than their
actual length in fig. 5, and must be projected on the inclined sides to de-
termine their absolute dimensions; project, therefore, the points 1", 2 // , 3",
4", 5", at which the cutting plane intersects the lines C 1, C 2, C 3, - C 4,
C 5, by drawing parallels to the base through these points to the inclined
side C B'. On fig. 6 lay off C' V" ', 0' 2"", &c., equal to C V", C 2,"', &e.
(fig. 5) ; connect the points a', \"" , 2"", V, a", and we have the
developed surface a' A! B' A" a" V required.
To develop the surface of a sphere or hall (figs. 189, 190).
It is evident that the surface cannot be accurately represented on a
plane surface. It is done approximately by a number of gores. Let CAB
(fig. 189) be the eighth of a hemisphere ; on C D describe the quarter
circle D A c / divide the arc into any number of equal parts, six for in-
104:
THE DEVELOPMENT OF SURFACES.
stance ; from the points of division 1, 2, 3, ... let fall perpendiculars on
C D, and from the intersections with this line describe arcs V 1", 2' 2", 3' 3",
cutting the line C B at 1", 2", 3", ; on the straight line C' D' (fig.
190), lay off C' D' equal to the arc D A c, with as many equal divi-
Fig. 190.
Fig. 1S9.
sions ; then from either side of this line lay off 1'" 1"", 2'" 2"" : . . . D' B'
equal to the arcs 1' 1", 2' 2", D B (fig. 189). Connect the points C',
1"", 2"", and C' A' B' is the developed surface.
It is to "be remarked, that in the preceding demonstrations, the forms are
described to cover the surface only ; in construc-
tion, allowance is to be made for lap by the addi-
tion of margins on each side as necessary. It
is found difficult in the formation of hemispherical
ends of boilers, to bring all the gores together at
the apex ; it is usual, therefore, to make them, as
shown (fig. 191), by cutting short the gores, and
FI. i9L surmounting the centre with cap piece.
MECHANICS.
105
MECHANICS.
THE profession of an architectural or mechanical draughtsman should
embrace not merely the mere copying of examples which may be furnished
him, but also the designing of new edifices and machines, in which he may
draw from the results of his own experience ; from good models, by col-
lating suitable parts from divers designs ; or by the rules of mechanics,
proportioning' the parts according to the magnitude and direction of the
strains to which they are to be subject, arid the materials of which they are
to be composed; introducing as much of ornament as the subject may
require.
THE MECHANICAL POWEES.
The simple machines or mechanical
powers which enter as elements into the
composition of all machinery are the
lever, the pulley r , and the inclined plane,
to which some add the toggle-joint and
the hydrostatic press. The lever em-
braces the wheel and axle, and the in-
clined plane the wedge and screw.
It is usual to regard the lever as of
three kinds, distinguished by the relative
position of power P, weight W, and ful-
crum F. In the first the fulcrum is be-
tween the power and the weight ; in the
2d, the weight is between the power and
the fulcrum ; and in the 3d, the power is
between the weight and the fulcrum.
This division is rather nominal than real;
all applications of the lever may be re-
solved under one general rule : that the
IST CLASS.
W F
CLASS.
Q
106 MECHANICS.
intermediate weight, pressure, or tension, is equal to the sum of the outside
ones, and the comparative size of these last two depends on their position
in reference to the first. Let the intermediate pressure be exerted through
the means of a spring balance c, Fig.
192, it will mark the sum of the
weights a and ft.
x _, Calling x the distance from a to
\l\ |_fj C, and y from c to ft/ knowing any
FIG. 192. three of the four, #, ft, x and y, the
fourth may be found ; for a multiplied by x is equal to ft multiplied by y
(ax=ly). Thus, if a be 6 Ibs., x 5 ft., and y 10 ft., then 6 multiplied by
5=30, is equal to 10 multiplied by ft/ therefore f,or 3 Ibs., is equal to ft,
and the load at c is 6-f 3=9 Ibs.
As c divides the lever proportionally to the weights, and as it is the
sum of the weights, knowing c, x, and y, a and ft may be calculated. Thus,
if the load at c be 100 Ibs., and x 7 ft., and y 3 ft., then one of the weights
will be T 7 o of c, or 70 Ibs., and the other T 3 o, or 30 Ibs, ; and, as the heavier
weight is at the shortest end of the lever, ft will be 70 Ibs., and a 30 Ibs.
a is to ft as 3 to 7, but x is to y as 7 to 3 ; therefore, the weights a and ft
are to each other as the opposite ends of the lever, as ytox; or the point
c divides the lever in the inverse proportion of the weights.
If either a or ft be known, and the lengths of the lever x and y, the
load at c may be calculated directly, c is equal to either weight, a or ft,
multiplied by the whole length of the lever, divided by y or a?, the part of
the lever opposite the known weight, c = + . It will be readily
understood that a, ft, or c may be considered at option, either fulcrum,
weight, or power, according to the requirements of the mechanism.
The Wheel and Axle. If a weight, P, be suspended from the periphery
of a wheel, fig. 193, whilst another weight, "W, is suspended on the oppo-
site side of a barrel or axle attached to the wheel, the principle of action is
the same as that of the lever. . P multiplied by its length of lever or radius
ca of the wheel is equal to "W multiplied by its length of lever or radius
of the axle cb / the axis c is the fulcrum. If a movement downward be
communicated to P, as shown by the dotted line, a rotary motion is given
to the wheel and axle ; the cord of P is unwound whilst that of W is
wound up, but P is still suspended from a and W from ft / the leverage, or
distance from the fulcrum, of each is the same as at first. The wheel and
axle is a lever of continuous and uniform action. Since the wheel has a
larger circumference than the axle, by their revolution more cord will be
MECHANICS.
107
unwound from the former than is wound up on the latter, P will descend
faster thanW is raised, in the proportion of the circumference of the
wheel to that of the axle, or of their radii ca to
,.-. **"" "**<
cl. When P has reached the position if] W
Fio. 193.
will have reached I W \ . If c a be 4 times c 5, then /
P will have moved 4 times the distance that ~W \
has. The movement is directly as the length of "
the levers, or the radii of the points of suspension.
It will be perceived, therefore, to move a large
weight by the means of a smaller one, that the
smaller must move through the most space, and
that the spaces described are as the opposite ends
of the lever, or inversely as the weights.
It is the fundamental principle of the action of
all mechanical powers, that whatever is " gained
in power," as it is said, is lost in space travelled ;
that, if a weight is to be raised a certain number
of feet, the force exerted to do this must always
be equal to the product of the weight by the
height to which it is to be raised ; thus, if 200 Ibs. are to be raised 50 ft.,
the force exerted to do this must be equal to a weight, which, if multiplied
by its fall, will be equal to the product 200 x 50, or 10,000 ; and it is
immaterial whether the force be a weight of 10,000 Ibs. falling one ft., or
1 Ib. 10,000 ft.
It is now common to refer all forces exerted to a unit of Ibs. ft. ; that
is, 1 Ib. falling 1 ft. ; and the effect to the same unit of Ibs. ft;, 1 Ib.
raised 1 ft. Thus, in the example above, the force exerted or power is
10,000 Ibs. ft. falling ; the effect 10,000 Ibs. ft. raised. In practice, the
Ibs. ft. of force exerted must always be more than the Ibs. ft. of effect pro-
duced ; that is, there must be some excess of the former to produce move-
ment, and to overcome resistance and friction of parts.
The measure of any force, as represented by falling weight, is termed
the absolute power of that force ; the resulting force, or useful effect for the
purposes for which it is applied, is called the effective power.
The Pulley. The single fixed pulley, fig. 194, consists of a single
grooved wheel movable on a pin or axis ; called fixed, because the strap,
through which the pin passes, is attached to some fixed object. A rope
passes over the wheel in the groove ; on one side the force is exerted, and
on the other the weight is attached and raised. It may be considered a
108
wheel and axle of equal diameters, or as a lever in
which the two sides are equal, the pin being the ful-
crum. P, the force exerted, must therefore be equal to
the weight, "W, raised ; and, if movement takes place,
"W will rise as much as P descends.
The fixed pulley is used for its convenience in the
application of the force ; it may be easier to pull down
than up, for instance ; but the Ibs. of force must be
FIO. iw. equal to the lbs< of e ff ect T he tension on the rope is
equal to either the force or weight.
Fig. 195 is a combination of a fixed pulley A, and movable pulley B.
The simplest way to arrive at the principle of this combination is to con-
sider its action. Let P be pulled down, say 2 feet ;
the length of rope drawn to this side of the pulley
must be furnished from the opposite side. On that
side there is a loop, in which the movable pulley
with the weight W attached is suspended. Each
side of this loop, 2 and 3, must go to make up the
2 ft. for the side or end 1. Cords 2 and 3 will
therefore furnish each 1 ft. As these cords are
shortened 1 ft. the weight W is raised 1 ft., and as
the movement of "W is but 1 ft. for the 2 ft. of P, W must be twice that of
P, because the 2 Ibs. ft. of P must equal Ibs. ft. of W.
In the combination of pulleys, Fig. 196. Let P be pulled, say 3 ft.,
,,/,.','.-///. then this length of rope drawn from the opposite
side of the pulley is distributed over the 3 cords 2,
. 3, 4, and the weight W is raised 1 ft. ; conse-
quently, the weight W is 3 times that of P. The
cord 1 supports P, the cords 2, 3, 4, the weight "W,
or 3 times P ; consequently, the tension on every
cord is alike. The same rope passing freely round
pulleys must have the same tension throughout ; so
that, to determine the relation of W to P, count the
number of cords which sustain the weight. Thus
in Fig. 197 the weight is sustained by 4 cords ; con-
sequently it is 4 times the tension of the cord, or 4
times the force P. In order not to confuse the
cords, the pulleys are represented as in the figures ; but, in construction,
the pulleys, or sheaves, are usually of the same diameter, and those in con-
nection, as A and B, and C and D, run on the same pin.
FIG. 195.
FIO. 196.
MECHANICS.
109
The Inclined Plane. To support a weight by means of a single fixed
pulley, the force must be equal to the weight. Suppose the weight,
instead of hanging freely, to rest upon an inclined plane 5 d, Fig. 198 ; if
motion ensue, to raise the weight "W the height a 5, the rope transferred
from the weight side of the pulley will be equal to bd, and P will have
consequently fallen this amount; thus, if Id be 6 ft., and al 1 ft, whilst
W is raised 1 ft. P has descended 6 ft., and as Ibs. ft. of power must equal
Ibs. ft. of effect, P will be of W ; and, by reference to the figure, P is to
W as a 5 is to fid, or as the height of the incline is to its length. If the
FIG. 198. FIG. 199.
end of the plane d be raised, till it becomes horizontal, the whole weight
would rest on the plane, and no force would be necessary at P to keep it
in position ; if the plane be revolved on 5, till it becomes perpendicular,
then the weight is not supported by the plane at all, but it is wholly
dependent on the force P, and is equal to it. Between the limits, there-
fore, of a level and a perpendicular plane, to support a given weight W,
the force P varies from nothing to an equality with the weight.
The construction, Fig. 199, illustrates the principle of the wedge, which
is but a movable inclined plane ; if the wedge be drawn forward by the
weight P, and the weight W be kept from
sliding laterally, the fall of P, a distance
equal to a d, will raise the weight W a
height eft. P will therefore be to W, as cb
is to a d. For example, if the length of the
wedge ad be 10 ft., and the back cb 2 feet,
then P will be to W, as 2 to 10, or } of it.
Let the inclined plane aid, Fig. 198, be
bent round, and attached to the drum A,
Fig. 200, to which motion of revolution on F IO> 200.
its axis is given, by the unwinding of the turns of a cord from around its
periphery, through the action of a weight P suspended from a cord passing
over a pulley. If the weight W be retained in its vertical position, by the
revolution of the drum it will be forced up the incline, and when the cord
has unwound one-half turn from the drum, and consequently the weight P
110
MECHANICS.
descended a distance ce equal to one-half the circumference of the drum,
the weight W has been raised to the height a I by
the half revolution of the plane ; P must therefore
be to W as a b is to one-half the circumference.
Extend the inclined plane so as to encircle the
;pdrum, Fig. 201. The figure illustrates the
mechanism of the screw, which may be considered
as formed by wrapping a fillet-band or thread
around a cylinder at a uniform inclination to the axis. In practice, the
screw or nut, as the case may be, is moved by means of a force applied at
the extremity of a lever, a complete revolution raises the weight the dis-
tance from the top of one thread to the top of the one above, or the pitch.
If the force be always exerted at right angles to the lever, Fig. 202, the
lever may be considered the radius of a wheel, at the
circumference of which the force is applied. Thus, if
the lever be 3 ft. long, the diameter of the circle
would be 6 ft., and the circumference 6 x 3.1416 or
18 T Vo ft-, if the pitch be 1 inch, or T V of a foot, then
the force would be to the weight as T V is to 18.85.;
and if the force be 1 lb., the weight would be
226.20 Ibs.
PARALLEL FORCES. If two horses be harnessed to a load, the effect is
to draw a load equal to the sum of their forces exerted in a line opposed
to the resistance of the load at the point where the whiffle-tree is attached
to the load. The forces, both of exertion and resistance, act in parallel
lines, and the resultant of the two forces, which is their sum, and counter-
balances the third, must be applied at a point intermediate between,
and distant from each of them, inversely as the forces exerted. The
composition of two parallel forces acting in the same direction is to be
solved as an example of a lever, with an intermediate fulcrum. The
forces are represented by the weights, and the point of application of the
resultant by the fulcrum, which is acted on, as if there were but a single
force, equal to the sum of the two, opposed to it.
The resultant, of any number of parallel forces, acting in one direction,
is equal to their sum, acting in the same direction at some intermediate
point ; that is, the effect of all these forces is just the same, as if there were
but one force, equal to their sum, acting at tliis point, and is balanced by
an equal force acting in the opposite direction. This central point may be
determined by finding the resultant, i. e., the sum, and the point of appli-
MECHAOTCS.
Ill
cation for any two of the forces, and then of other two, the resultants thus
determined being again added together like simple forces.
As parallel forces can be added together ; reciprocally, they can also be
divided. The single force, acting intermediately, may be resolved into
forces acting at the ends of a lever, whose sum, whatever their number,
will be equal to the central force.
Inclined Forces. When two men of equal strength pull directly opposite
to each other, the resultant is nothing. Let a third take hold of the centre
of the rope, and pull at right angles to
the rope; he will make an angle in the
rope and the two others are now pulling in
directions inclined to each other. The less
the force exerted at the centre, the less the
flexure in the rope ; but when it becomes
equal to the other two, the two, to balance it, must pull directly against it,
bringing the ends of the rope together, and acting as parallel forces. Be-
tween the smallest force and the largest, that can be exerted at the centre
and maintain a balance or equilibrium, the ends of the rope assume all
varieties of angles, which angles bear definite relations to the forces.
Kepresent these forces by weights, as in Fig. 203 ; let P and "W be the
extreme forces, acting over pulleys, and tending to draw the rope straight,
which the weight C prevents: to
find what must be the weight of
to balance the others. Above the
rope draw the lines cp and cw,
parallel to the rope, from c lay off
on cp parts of an inch, equal to
number of pounds or units of
weight in P; that is, if P be 5
Ibs., and W 6 Ibs., lay off cp f , T \,
or any fractions of an inch that
may be the most convenient, say f ; and c w of an inch ; draw ^? a parallel
to cw, and wa parallel to cp, connect ac / if the length ac be measured
in Stlis of an inch, or whatever fractional parts may be adopted for the
other weights, it will represent the weight of C in Ibs. (in this case about
5 parts, or Ibs.), and the direction in which C acts ; the work of the
weights P and W is to support C, and this would be done by a force equal
to C, 5| Ibs., acting in a* line c a directly above it. Therefore, the force
opposed to the direction of 0, and equal to it, will be the resultant of the
two forces, P and W acting at an angle to each other.
FIG. 203.
112
MECHANICS.
As two forces may
be compounded, so con-
versely one force may be
resolved into two; thus,
let the weight P, Fig.
204, be supported by two
inclined rafters CA and
C B. Each resists a part
of the force exerted by
the weight P. To find
the force exerted against
the abutments A and B,
in the direction of CA
and CB, draw cA.' par-
allel to C A, cB' to C B,
and cd, a continuation of
the line C P, the direction
in which the weight P acts ; lay off c d from a scale of equal parts, a
length which will represent the number of Ibs., or whatever unit of weight
there may be in the weight P; draw da parallel to cB', and d b parallel
to cA'j ca, measured on the scale of equal parts adopted, will represent
the Ibs. or units of weight exerted against A in the direction of C A, and
c I the Ibs. or units of weight exerted against B in the direction of C B.
This method of finding the resultant of two forces, or the components
of one force, is called the parallelogram of forces. If two sides of a paral-
lelogram represent two forces in magnitude and direction, the resultant of
these two forces will be represented in magnitude and direction by the
diagonal of the parallelogram, and conversely.
The sum of ac and cb is greater than cd j that is, the weight P exerts
a greater force in the direction of the lines C A and C B, against A and
B, than its own weight ; but the down pressure upon A and B is only equal
to the weight of P and of the rafters which support it, which last, in the
present consideration, is neglected. Resolve cb, the force acting on B in
the direction of c B', into gl) or c e the downward pressure, and c g or e 5
the horizontal thrust on the abutment B, and c a into cf and fa. To
decompose a force, form a triangle with the direction of the other forces,
upon the line representing the magnitude and direction of the given force.
ce represents the weight on B, cforde the weight on A ; cd, or ce + de,
the whole weight P ; therefore, the weight upon the two abutments A and
B is equal to the whole weight of P.
MECHANICS.
113
Connect A' B', and extend the line c d to E. c e is to e d as A' E is to
E B/ and as the weight is distributed on the abutments A and B in propor-
tion to ed and ce, it will be also as B'E is to A'E ; or the whole construc-
tion may be considered a lever, in which the weight is suspended at E and
distributed between the supports inversely as its distance from them, af
and e ft represent the horizontal thrust on the abutments A and B, but as
af and b e are equal, the force tending to separate them, or to tear asunder
A' B' if a tie-rod, would be represented by either of them.
In the application of forces to a lever we have considered them parallel
and perpendicular to the lever ; if they are inclined, they may be resolved
into forces acting perpendicular to and in the direction of the lever, or
they may simply be referred to the fulcrum, as the axis of a wheel and axle,
of which the perpendiculars let fall from the fulcrum upon the direction of
the forces are the radii. Thus, if the levers, Fig. 205, be acted on by
forces of which the direction is shown by the arrows, the leverage, as it is
called, or effective length of arm at which they act, are the perpendiculars
Fa and F J, on the directions of the forces.
Centre of Gravity. The action of weight, or its tendency
downward to the earth, is called gravity. Let a mass P, Fig.
206, be suspended by a cord, each particle of the mass is acted
upon by gravity, like an innumerable number of threads pulling
it downward, and all these parallel forces may be resolved into
a single force opposed to the direction of the string, and equal
the sum of all the forces or the weight.
Suspend the mass C, Fig. 207, by a cord from P ; the line
Fio. 206.
FIG. 208.
114
MECHANICS.
PM will represent the direction of the resultant through the mass. Sus-
pend the same mass again from Q, the resultant will now take the direc-
tion Q K, and the two resultants have one point C, their intersection in
common ; this point is called the centre of gravity ; all the weight may be
supposed concentrated at this point.
A body placed on a horizontal plane will fall over, unless the vertical
line passing through its centre of gravity fall within its base ; thus, in Fig.
208, the body will stand firmly, whilst in Fig. 209 it will fall over. Per-
sons carrying loads, Fig. 210, adjust their position in-
sensibly so that the vertical line from the common cen-
tre of gravity, ^, of their own bodies G, and of their
load H, should fall within the area bounded by their feet.
The body is thrown to the side opposite the load.
When bodies are of symmetrical forms and homo-
geneous, the determination of the centre of gravity is
FIG. 210. the finding the centre of the figure. The centre of
gravity of a triangle is in a line drawn from its summit to the middle of
the opposite side, and at of its length from the base.
The centre of gravity of any quadrilateral may be found by dividing it
into two triangles, and finding the centre of gravity of these triangles, con-
necting them by a straight line, and dividing this line into parts inversely
proportional to the surfaces of the two triangles, the point of division
being the centre of gravity of the figure. In the same
way any polygon may be subdivided into triangles
of which the centres of gravity may be found, and re-
solved by connections with each other.
The centre of gravity of the triangular pyramid,
Fig. 211, is in the straight line A E, connecting the apex
A with the centre of gravity of the base triangle BCD,
FIO. 211. and distant of the length of the line A E from E.
The centre of gravity of solids, which may be divided into symmetrical
figures and pyramids, as for all practical purposes most may be, can be
found by determining the centre of gravity of each of the solids of which
it is compounded, and then compounding them, observing that each centre
of gravity represents the solid contents" of its own mass or masses of which
it may be composed. The centre of gravity of bodies enclpsed by more or
less regular contours, as a ship for instance, is determined by dividing it
into parallel and equidistant sections, finding the centre of gravity of each,
and compounding them into a single one.
MECHANICS. 115
FRICTION, AND THE LIMITING ANGLE OF RESISTANCE.
Suppose a mass A (fig. 212) be pressed upon another, B, by means of
a force acting in a direction perpendicular to the common surface of the
two bodies, and let a second force Q act also upon it in a direction paral-
lel to this surface. Then, since the forces Band Q act in directions per-
pendicular to each other, they manifestly cannot counteract one another,
and it would be expected that the body should move in the direction of
the second force. This, however, is not always the case ; except the force
Q exceed a certain limit, no motion ensues. Some new force F, therefore,
has been produced in the system, counteracting the force Q ; that force is
called friction. It acts always in a direction parallel to the surfaces in
contact, and is always for surfaces of the same nature the same fraction,
or part of the force P by which these are pressed together, whatever be the
amount of that force, or whatever the extent of surfaces in contact. This
fraction is called the coefficient of friction. Whilst it is thus the same for
the same surfaces, whatever- be the extent of the surfaces, or the force with
which they are pressed together, it is different for different surfaces.
Construct the parallelogram of forces P P" Q M (fig. 213). P" M rep-
resent the resultant of the two forces P and Q. The actual friction is
always a certain given fraction of P acting parallel to the impressed sur-
face. Take M Q' equal to this given fraction of P M, complete the paral-
lelogram, and draw the diagonal P' M. Since, then, M Q' represents the
friction of the body upon the plane, or the force called into action by P M,
which opposes the motion of the body ; since, moreover, Q M represents
^ F
ft' Q C M A
Fig. 212. Fig. 213. Fig. 214.
the force tending to produce motion, it follows that the body will or will
not move, according as Q M is greater or less than M Q', or as the angle
P M P" is greater or less than P' M P. The angle P' M P is called the
limiting angle of resistance. It depends upon the coefficient of friction,
and is therefore the same for the same surfaces, whatever be the actual
amount of the impressed force P.
Hence it appears, that force impressed upon the surface of a solid body
at rest, by the intervention of another solid body, will be destroyed, pro-
116
vided the angle which the direction of that force makes with the perpen-
dicular to the surface does not exceed a certain angle called the limiting
angle of the resistance at that surface, and this is true, however great the
force may be. Also, that if the direction of the impressed force lie with-
out this angle, it cannot be sustained by the resistance of the surfaces in
contact, and that this is true, however small the force may be.
Suppose a heavy mass (fig. 214), whose centre of gravity is G, to be
placed on an inclined plane A B. The whole pressure of the mass may
be supposed to act in the direction of the vertical line G M, and this pres-
sure will be just destroyed by the resistance of the surface of the plane
when the angle G' P Q, which G P makes with the perpendicular P Q, is
equal to the limiting angle of resistance. A mass of any substance will,
therefore, just be sustained on an inclined plane, without slipping, when
the inclination of the plane is equal to the limiting angle of the resistance
of the surfaces in contact ; that is, when the angle B A C is equal to the
angle G P Q.
EXPERIMENTS ON FRICTION, BY M. MORIN.
WITHOUT UNGUENTS.
UNCTUOUS SURFACES.
PRICTION OF MO-
FRICTION OF QUI-
FRICTION OP MO-
FRICTION OF QUI-
SURFACES OF CONTACT.
TION.
ESCENCE.
TION.
ESCENCE.
Co-efflci-
Limiting
Co-effici-
Limiting
Co-effici-
Limiting
Co-effici-
Limiting
ent of fric-
angle of
ent of fric-
angle of
ent of fric-
angle of
ent of fric
angle of
tion.
resistance.
tion.
resistance.
tion.
resistance.
tion.
resistance.
Oak upon oak, fibres parallel to the
motion. .
0.4T8
25 33'
0.625
32 1'
0.108
6 10'
6.890
21 a 19'
Oak npon oak, fibres of the moving
body perpendicular to the motion,
0.324
1T.53
0.540
28.23
0.143
8^9'
0.314
17=26'
Oak upon elm, fibres parallel, .
^0.246
iaso
0.376
20.87
0.136
7.45
Elm upon elm, " " .
0.140
7.59
Wrought Iron npon oak, . . . .
0.619
3147'
0.619
81 47'
" " " wrought iron, .
0.13S
7.52
0.137
7.49
0.177
10.3
u cjjgfc a
0.194
1<P59'
0.194
1059
0.11S
6.44
brass, . /-
0.1T2
9.46
0.160
9.6
Cast iron on elm, ....
0.195
11.3
0.125
7.S
" " cast iron, . .
0.152
8.39
0.162
9.13
0.144
8.12
" " wrought iron, ,.
0.143
8.9
brass, . . . .
0.14T
8.22
0.132
7.32
Brass upon cast iron, . . .
0.21T
12.15
0.107
6.7
" " wrought iron, . .
0.1C1
9.9
" " brass, ." ,. . .
0.201
11.22
0.154
7.38
0.164
9.19
Leather oxhide, well tanned, on oak,
0.296
16.30
'.'** " on cast iron, wetted,
0.229
12.54
2.67
1457
" belts on oaken drums, .
0.27
0.47
" " " cast iron pulleys, .
0.28
Common building stones upon the
same, . .
0.88 to
0.65
20.49
83.2
0.65
0.75
83.2
36.53
MECHANICS. 117
From his experiments M. Morin found, that the friction of two surfaces
which had been considerable time in contact was not only different in its
amount, but in its nature, from the friction of surfaces in continuous
motion ; especially in this, that this friction of quiescence is subject to
causes of variation and uncertainty, from which the friction of motion is
exempt; but that the slightest jar or shock, the most imperceptible move-
ment of the surfaces of contact, was sufficient to change the friction from
the quiescent state into that which accompanies motion. Hence, as every
machine or structure of whatever kind may be considered as subject to
such shock or imperceptible motion, all questions of construction depend-
ing upon the state of friction should be referred to that which accompanies
continuous motion. The friction of two surfaces outside the limits of abra-
sion is independent of the extent of superficies, and when in motion, of
the velocity of the motion also.
There are three states in respect to friction into which the surfaces of
bodies in contact may be made to pass ; one, a state in which no unguent
is present ; second, a state in which the surfaces are unctuous, but inti-
mately in contact ; the third, a state in which the surfaces are separated
by an entire stratum of the interposed unguent. From experiments on
this last class Morin deduces, " that with the unguents olive oil and lard
interposed in a continuous stratum between them, surfaces of wood on
metal, metal on wood, wood on wood, and metal on metal, when in motion
have all of them very nearly the same coefficient of friction, the value of
that coefficient being in all cases included between 0.07 and 0.08, and the
limiting angle of resistance between 4 and 4 35'. For tallow as an'
unguent, the coefficient is the same as the above, except in case of metals
upon metals, in which case the coefficient was found to be 0.10."
ON THE EQUILIBRIUM OF THE POLYGON OF RODS OR COEDS.
If we take all the forces excepting those which act upon the extremi-
ties of the polygon, and find the direction of their resultant, then the two
extreme sides of the polygon being produced will meet this direction in
the same point. Thus, in the polygon represented loaded with the weights
P l , P 2 , P 3 , if we find the vertical R T passing through the centre of grav-
ity of these weights, and produce P A and P 6 B, these will meet R T in
the same point T (fig. 215).
Similarly in the funicular curve or catenary (fig. 216), if we draw tan-
gents at the points of suspension A and B, these, being in the direction of
118
MECHANICS.
the forces sustaining the curve at these points, will meet when produced
in the vertical line G T, passing through the centre of gravity G of the
curve. Let G T represent the weight of the cord A B ; draw G M and
G Is" parallel to A T and B T ; K T will represent the tension at A, and
Fig. 216.
M T the tension at B. Such a curved line is more liable to rupture,
therefore, at the upper point of suspension, and in construction, when pos-
sible, should be of greater dimensions.
If a polygon of rods be reversed, that is, placed upright instead of sus-
pended, the position in which it will stand is that which it will assume for
itself when loaded with the same weights, and suspended. Hence, to de-
termine the positions in which any number of beams should be arranged
in a polygon, so as to support one another, the timbers of a gambrel roof
for instance; let a cord be taken, and distances be measured along it,
equal respectively in length to the sides of the polygon ; let weights be
attached to these, equal each to one half the sum of the weights of the two
adjacent sides. Then the two ends of the string being held at a distance
apart equal to the length of the base of the polygon, the form which the
string will assume when hanging freely will be that in which the beams
should be arranged.
To the continual equilibrium of an upright framework it is essential
that its joints should be stiffened. IvTow this cannot be brought about by
any peculiarity in the joint itself, for the different parts of such a joint,
being situated exceedingly near to the centre about which each rod tends
to move, are, on the principle of the lever, readily crashed by the action
of a force, however slight, acting at the extremity of the rod. It is, there-
fore, requisite that each joint should be stiffened by subsidiary framing.
And out of the necessity for this strengthening arises the greater economy
of the suspended than the upright polygon or framing. In the suspended
polygon or curve, the only precaution necessary is, that the parts should
MECHANICS.
119
not tear asunder. In its uptight position, their flexibility as well as the
chance of their compression, must be guarded against.
The methods of giving rigidity to a system of rods are various. They
all of them, however, resolve themselves directly or indirectly into the
arrangement of the component rods in triangles. Of all simple geometri-
cal figures, the triangle is the only one which cannot alter its form with-
out at the same time altering the dimensions of its sides, and which cannot,
therefore, yield, except by separating at its angles, or tearipg its sides
asunder. Hence, therefore, a triangle whose joints cannot separate, and
whose sides are of sufficient strength, is perfectly rigid ; and this can be
asserted of no other plane figure whatever. Thus a parallelogram may
have sides of infinite strength, and no force may be sufficient to tear its
joints asunder, and yet may it be made to alter its form by the action of
the slightest force impressed upon it. And this is true in a greater or less
degree of all other four-sided figures and polygons. It is for these reasons
that in all framing, care is taken to combine all the parts, as far as pos-
sible, in triangles ; which being once done, we know that the rigidity of
the system may be insured by giving the requisite strength to the timbers
and joints.
The framing of a gate presents a very simple illustration of this prin-
ciple. The outline of the form of the gate is that of a rectangular paral-
lelogram. If, as in the accompanying figure (fig. 217), the parts which
compose it had been arranged in directions parallel to its sides only, so
Fig. 217.
Fig. 213.
that the whole frame should have been composed of elementary parallelo-
grams, each component parallelogram, and, therefore, the whole frame of
the gate, would readily have altered its form.
A bar placed diagonally across the gate remedies the evil, converting
the elementary parts of the gate from parallelograms into triangles, and
thus giving perfect rigidity to the frame (fig. 218).
Further illustrations of the principles of framing, together with the
120
MECHANICS.
equilibrium of solid bodies in contact, as in the construction of retaining
walls and arches, will be found under the head of Architectural Drawing.
THE MECHANICAL PROPERTIES OF MATERIALS.
To proportion properly the parts of a machine or edifice, the draughts-
man should understand not only the kind of strain, the direction, and
amount to which its different parts may be subjected, but also the nature
and properties of the materials of which it may be composed, and their
capability of resisting uninjured the required stress. Many experiments
have been made on the strength of materials, and in the tables beneath
will be seen great differences in select specimens of the same materials, and
of small dimensions, showing the necessity of practical knowledge in selec-
tion, and proper allowance against contingencies.
The forces to which materials in constructions are subjected are com-
pression, tension, flexure ', and torsion.
STRENGTH OF WOODS. From the results of Experiments of Capt. T. J.
Rodman, of the IT. S. Ordnance Department.
HATEBIAI&
Specific
gravity.
Weight per
cubic foot
Resistance per
To crushing.
square inch.
To tension.
Transverse
resistance.
s lw
Pounds.
Pounds.
Pounds.
Pounds.
Ash, ....
.51 to .73
32 to 46
4,500 to 8,800
11 to 24,000
500 to 900
Birch, .
.70
44
8,000
15,000
700
Bass . .
48 to 50
50 to 32
4 600 to 5 200
12 to 15 000
650
Box, . . . ' .'
.90
56
10,500'
23,000
Beech, ....
.67 to .73
42 to 46
5,800 to 6,900
15 to 18,000
750
Cedar, red, . . .
.38
23
6,000
10,000
100
Chestnut, . .
.47 to .54
29 to 34
5,100 to 5,600
12 to 13,000
350
Cherry, . . . .
.58
36
6,100
12,000
450
Cypress, . . ...
.55
34
8,500
17,000
350
Dogwood, . . .
.86
54
7,400
23,000
550
Elm, ....
72t6.77
45 to 48
6,200 to 6,600
15,000
700
Fir, yellow, . . .
.55 to .63
34 to 39
7,400 to 9,200
14 to 17,000
400 to 600
" red and white, .
.46
29
6,600 to 7,000
13 to 15,000
250 to 350
Gum, black, .
.61
38
6,700
16,000
500
Hickory, ....
.82 to .95
51 to 60
5,500 to 9,900 18 to 35,000
900
" red, . . .
.72 to .87
45 to 54
7,700 to 10,900
13 to 27,000
900 to 950
" white,
.90 to. 99
56 to 62
8,900 to 11,200
36 to 40,000
950 to 1,100
Hemlock,
.45
28
6,800
16,000
400
Hackmatack, . . . ' >
.59
36
450
Lignumvitse, . . .
1.26
79
9,800
16,000
900
Locust, . . .
.83
52
9,100
27,000
800
Mahogany St. Domingo, .
.76
47
7,400
12,000
550
Maple, ....
.6810.73
42 to 45
7,700 to 8,600
22 to 23,000
650
Oak, white, . . ,; .
.63 to .88
39 to 55
4,700 to 9,100 ! 12 to 21,000
500 to 950
" yellow, . . P
.71
44
6,300
25,000
600
" live,
1. to 1.1
62 to 69
6,500 to 7,200
16,000
450 to 550
Pine, pitch,
1.1
69
8,900
11,000
" white, . . tf
.36 to .46
23 to 29
5,000 to 5,800
11 to 12,000
350
" yellow, . . .
.53 to .67
33 to 42
7,800 to 8,400
12 to 19,000
500 to 650
Poplar .
43 to 50
27 to 31
5 700 to 6 600
8 to 15 000
300 to 450
Redwood, Cal., . .
.39
24
6,100 '
11,000
250
Spruce, . . .
.44
27
5,100 to 6,800
11 to 14,000
350
Teak, ....
.96
60
10,800
31,000
1,000
Walnut, black,
.53 to .65
33 to 41
5,800 to 7,500
16 to 18,000
450 to 650
MECHANICS.
121
STRENGTH OF METALS.
MATERIALS.
Specific .
Weight per
Resistance
per square inch.
gravity.
cubic foot.
To crtishing.
To tension.
Pounds.
Pounds.
Pounds.
Brass, cast,
8.40
523
10,000
18,000
" wire,
49,000
Bronze, .
8.70
542
42,000
Copper, cast,
8.61
537
22,000
wire,
61,000
Gold, cast,
19.26
1,200
20,000
Lead, sheet,
11.41
711
3,300
Platinum wire
22.07
1,376
56,000
Silver, cast,
10.48
653
40,000
Tin, cast, Banca,
7.22
450
3,700
" wire,
7,000
Zinc, cast,
7.22
450
2,900
" sheet,
16,000
Iron, cast, -
7 to 7.3
436 to 456
13 to 25,000
" wrought,
7.6 to 7.8
474 to 487
20 to 110,000
" cable, .
54, to 75,000
" wire,
86 to 113,000
Steel, cast, forged bar,
7.78
485
85 to 145,000
soft,' .
65 to 105,000
" plate, .
41,000
Homogeneous metal,
85 to 100,OOQ
Resistance to Compression. Parts of structures are usually subjected
to a compressive force in the direction of their length, and when this
dimension is less than a certain proportion (depending on the nature of
the material) to its diameter or shortest side, rupture takes place from the
absolute crushing of the parts. In the actual practice of construction,
materials cannot with safety be subjected to any pressure approaching
their ultimate strength. They are liable to various occasional and acci-
dental pressures, and to others of a permanent kind, resulting from settle-
ment, and other causes, of which no previous account can be taken, for
which allowance must nevertheless be made. Navier, from existing struc-
tures, deduces the rule that wood and stone should not be subjected to a
strain over one-tenth of that which breaks them, and iron to not over one-
fourth.
Bricks have been experimented on which have withstood a compressive
force of 13,000 Ibs. to the square inch, whilst others have failed under a
pressure of less than 500 Ibs. ; and among the building stones the range
would be even greater. The granite of Eockfort, Mass., as tested by Capt.
Eodman, crushed under 15,300 Ibs. to the inch. These experiments were
probably made on small cubes, and are of little use to the architect ; in
construction brick is seldom subjected to a pressure of 100 Ibs. per square
inch ; in fact, few structures have failed from an absolute crushing of either
122 MECHANICS.
brick or stone, but unequal settlements of the foundation or compression
of the mortar in the joints may change the strain upon the material from
a compressive to a transverse one and cause destructive cracks. The
thickness given for walls under the head of Architectural Drawing, will
be a sufficiently practical guide for the dimensions of such works, if
ordinary care be used in the selection of materials and construction.
Wood and iron, when used to resist a compressive force, are generally
of such lengths in comparison with their sides or diameter, that rupture
takes place partly from compression, partly from flexure. It has been
found, that if the length of a circular post exceed eight times its diameter,
the tendency under pressure will be to bend ; and the longer the pillar,
the other dimensions remaining the same, the more this tendency develops
itself.
For cast-iron posts, the circular form is usually adopted, as being the
strongest form for the same amount of material, the hollow pillar being in
this respect preferable to the solid. Mr. Hodgkinson, from his experi-
ments on cast-iron pillars, deduces the following rules :
1st. In all long pillars of the same dimensions, the resistance to frac-
ture by flexure is three times greater when the ends of the pillar are flat
and firmly bedded, than when they are rounded and capable of moving.
This shows the importance of having the ends of pillars turned square, and
of having the ends of braces square and not rounded, as has been proposed
and adopted by some architects. .
2 d. The strength of a pillar with one end round and the other flat, is
the arithmetical mean between that of a pillar of the same dimensions,
with both ends rounded and both ends flat.
Sd. A long uniform pillar, with its ends firmly fixed, whether by discs
or otherwise, has the same power to resist breaking as a pillar of the same
diameter and half the length, with the, ends rounded.
4th. Some little additional strength is given to a pillar by enlarging its
diameter at the middle part ; but this increase is not over one-seventh of
the breaking weight.
6th. In cast iron pillars of the same length, the strength is as the 3.6
power of the diameter nearly.
6th. In cast iron pillars of the same diameter, the strength is inversely
proportioned to the 1.7 power of the length.
The breaking weight of solid cylindrical cast iron pillars, with their
d 3 ' 6
ends flat and incapable of motion, is in tons 4A x -j-^, I being the length
in feet, d the diameter in inches. In hollow pillars the same rule ap-
MECHANICS.
123
plies, but for d we use D 36 d 3 - 6 , D being the external and d the internal
diameter. For pillars with ends movable and rounded, one-third of the
above formula will be the breaking weight.
TABLE I. Diameters to the 3.6 power.
Inches.
Inches.
Inches.
2.
12.125
5.25
391.36
8.
1782.9
2.5
27.076
5.5
462.71
8.25
1991.7
3.
52.196
5.75
543.01
8.5
2217.7
3.25
69.628
6.
632.91
8.75
2461.7
3.5
90.917
6.25
733.11
9.
2724.4
3.75
116.55
6.5
844.28
9.5
3309.8
4.
147.03
6.75
967.15
10.
3981.1
4.25
182.89
7.
1102.4
10.5
4745.5
4.5
224.68
7.25
1250,9
11.
5610.7
4.75
272.96
7.5
1413.3
11.5
6584.3
5.
328.32
- 7.75
1590.3
12.
7674.5
TABLE II. Lengths to the 1.7 power.
Inches.
Inches.
Inches.
1.
1.
9.
41.900
17.
123.53
2.
3.249
10.
50.119
18.
136.13
3.
6.473
11.
58.934
19.
149.24
4.
10.556
12.
68.329
20.
162.84
5.
15.426
13.
78.289
21.
176.92
6.
21.031
14.
88.801
22.
191.48
7.
27.332
15.
99.851
23.
206.51
8.
34.297
16.
. 111.43
24.
222.
Example. To find the breaking weight of a hollow cast iron pillar
with square ends, whose outside diameter is 6 inches ; inside or core, 4 ;
and length, 16 feet.
Then from Table I.: Against 6. is 632.91; against 4.5, 224.68.
632.91 224.68 = 408.23 ; 408.23 x 44 = 17962.12; dividing this by
1 7969 1 2
111.43, the number against 16 in Table II, we have -[-^3, = 162 tons
very near of 2,240 Ibs. each as the weight which would break this column.
The above formulas apply only to long pillars ; that is, those whose
length is at least 25 to 30 times their diameter, and the result arrived at is
124: MECHAOTCS.
their ultimate strength ; but the permanent load should not exceed one-
fourth of the breaking weight. Short cylindrical pillars may be loaded
perfectly safe with 10 tons to the square inch of area of base. It is, of
course, important that the columns should be straight with a square base,
so that the direction of the strain should be through the axis.
Tensile Strength. Woods are seldom used to resist tensile strains, but
when so used, the size should be very much larger than that merely re-
quired to resist the strain, say at least ten times. Wrought iron is the
material the most generally employed to resist a tensile force; and its
elastic power in that capacity, or the load with which it may safely be
trusted, is ten tons per square inch for best iron, and for ordinary iron,
10,000 Ibs.
Transverse Strength of Materials. The strength of a square or rectan-
gular beam to resist lateral pressure, acting in a direction perpendicular to
its length, is as the breadth and the square of the depth ; and inversely as
the length, or the distance from or between points of support. Thus a
beam twice the breadth of another, other proportions being alike, has twice
the strength ; or twice the depth, four times the strength ; but twice the
length, only half the strength.
The general formula is W j , in which W is the breaking weight ;
{/
S, a number determined by experiment on different materials (see table, p.
120) ; b, the breadth, and d, the depth in inches ; and I, the length in feet.
To find the breaking weight of a beam supported at the ends and
loaded in the centre : Rule Multiply the constant S, for the material from
the above table, by the breadth and square of the depth in inches, and
divide the product by the length in feet.
Example. What is the ultimate strength of a beam of white pine, 20
feet long, 8 inches wide, and 14 inches deep ?
S. 400 x 8 x 14' = 627.200. 62 ^ = 31.360 Ibs.
zo
From the above formula we can determine either the breadth, depth, or
length, the other quantities being known: 5 = ^-T a , and d /-.
To determine the depth; the weight, breadth, and length being
known : Multiply the length in feet by the weight in pounds ; also the
tabular number S by the breadth in inches. Divide the first product
by the last, and the square root of the quotient will be the required
depth.
To find the depth of an oak timber 15 feet long and 6 inchel wide, to
MECHANICS. 125
support a weight of 10,000 Ibs. at its centre. Multiply the given weight
by 10, and establish the depth on this basis; thus, to support 10,000
securely and permanently , find a beam whose ultimate strength is equal to
100,000 Ibs.
15 x 100,000 = 1,500,000. S 700 x 6 = 4,200. ^gff = 357.1
1/357.1 = say 19 inches, the depth required.
When the load is not on the middle of the beam : Divide four times
the product of the distance of the weight in feet from each bearing, by
the whole distance between the points of support, and the quotient is the
equivalent length of the beam loaded in the middle.
Suppose a beam 30 feet in length, with a load placed 9 feet from one
end ; required the equivalent length.
30-9 = 21. 21 X J t X * = 25.2 feet.
o()
When the load is distributed over the whole length of the beam, it will
bear double the load which it would support in the middle ; therefore,
in calculations for the strength of a beam with distributed load, use double
the tabular number S ; if the ends of this beam are firmly fixed, use three
times S. If loaded at the middle with ends firmly fixed, use 1^ times S.
"When a beam is fixed at one end, and the weight is placed on the
other (fig. 218), use only one-fourth of the tabular number ; if the load is
distributed on a like beam, use one-half of S.
Ex. To find the depth of a white pine beam 10 inches wide, project-
ing 5 feet from a wall, and capable of supporting with safety 2,000 Ibs.
Call the "Breaking weight 6 X 2.000 = 12.000 Ibs.
4-00
5 X 12.000 = 60.000. =-x 1<T- 1.000. 60. 4/60 = 7.74 in.
i l.UUU
It is only necessary that the dimensions thus obtained be preserved at
the points of greatest strain, that is, at A B (fig. 218). i
When the beam has two points of support, and the load I p
is intermediate, the point of suspension of the weight is
tike point of greatest strain, and the beam may be reduced (^\
towards the points of support without breaking it. Fig. 213.
The forms of beams which afford equal strength throughout are para-
bolic (figs. 219, 220, 221), of which the axis A B and the vertex A are
given, and the points M determined by calculations. Figs. 220, 221 are
oftener used when the force is applied on alternate sides of A B.
If a beam be subjected to a transverse strain, one side is compressed,
while the other side is extended ; and therefore, where extension terminates
126
MECHANICS.
and compression begins, there is a lamina or surface which is neither ex-
tended nor compressed, called the neutral surface or neutral axis. As the
strains are proportional to the distance from this axis, the material of
which the beam is composed should be concentrated as much as possible
Fig. 219.
at the outer surface. Acting on these principles, Mr. Hodgkinson has
determined the most economical form for cast-iron beams or girders, of
which the section is given (fig. 222) ; it has been found, that the strength
of cast-iron to resist compression is about six times that to resist exten-
sion ; the top web is therefore made only one-sixth the area of the lower
one. The depth of the beam is generally about T V of its length, the deeper
of course the stronger ; the thickness of the stem or the upright part
should be from 1 an inch to 1| inches, according to the size of the beam.
The rule for finding the ultimate strength of beams of the above section
4
Fig. 222.
is : Multiply the sectional area of the bottom flange in square inches by
the depth of the beam in inches, and divide the product by the distance
between the supports in feet, and 2.16 times the quotient will be the break-
ing weight in tons (2240 Ibs.) As has already been shown above, the sec-
tion thus determined need only be that of the greatest strain, and can be
reduced towards the points of support, either by reducing the width of
the flanges to a parabolic form (fig. 222), or by reducing the thickness
of the bottom flange ; the reduction of the girder in depth is not in general
as economical or convenient.
For railway structures subject to an impulsive force, the upper flange
MECHANICS.
127
should be | of the lower one. For wrought iron beams, as this material
affords less resistance to compression than tension, the top flange is generally
made larger than the lower one in the.proportion of 5 to 3. The following
may be taken for the formula to determine the strength of solfd wrought iron
beams : W = j , in which Tfis the breaking weight, A the area of
v
the section, d the depth, I the distance between the supports. C is a con-
stant determined by experiment for each form of beam ; for the beam
shown in section (fig. 223) C is found to be about 40.000 Ibs.
a = 1. in. x 2.75 = 2.75 sq. in.
5 = 8. " x 0.380 = 3.04 "
c = 0.42 " x 4.3 = 1.806 "
d = 9.42
A = 7.596
Fig. 223.
For wrought iron girders of large span, the box form is generally
adopted.
Experiments on the transverse strength of rectangular tubes of wrought iron,
supported at each end, and the weight laid on the middle.
Distance be-
tween the
supports.
Breaking
"Weight of tubes weights, exclu-
between the sup- sive of the
ports. weights of the
tubes.
External
depth of the
tubes.
|
External
breadth of
tubes.
Thickness of the
plates of the
tubes.
Feet.
Tons.
Inches.
Inches.
Inches.
30.0
42.62 cwt.
57.5
24
16
.525
7.5
72.36 Ibs.
4,454
6
4
.1325
30.0
23.09 cwt.
22.84
24
16
.272
7.5
35.53 Ibs.
1.409
6
4
.065
3.75
9.65 Ibs.
1.1
3
2
.061
3.75
4.34 Ibs.
.3
3
2
.03
45.0
130.36 cwt.
114.76
86
24
.75
3.75
9.65 Ibs.
1.1
3
2
.061
30.0
39 cwt
54.3
24
16
.50
A.%
Fig. 224.
In several of these experiments, .the tubes gave way by
the metal at the top becoming wrinkled.
In similar tubes, the strength, and consequently the
breaking weight, is proportioned to (1.9) power of the lineal
dimensions.
Approximate formula for rivetted tubes :
'"" \ B D 3 1) d 3 1 the breaking weight in tons.
128 MECHAIHCS.
In which,
A C = D, and a c = d in inches ; 1) = length in feet.
A B = B, and a I = I "
If the thickness of the metal be equal to t inches completely round
the section, then I = B 2 t, and d = D 2 t.
What is the breaking weight of a rectangular tube 40 feet long, depth
2 feet 6 inches, thickness of plate \ inch, and breadth 18 inches ?
( 18 x 30 3 17.5 x 29.5 3
-, 486000 440167 = 22.96 tons.
loUU (
It is found of iron beams and tubes, that they may be safely reduced
in strength from the middle towards the extremities in the ratio indicated
by theory.
It must be observed, that in the formula given for determining the
strength of material, the force exerted is supposed to be dead weight or
pressure, and that no consideration is paid to impulsive force, except such
slight shocks as are incident to all structures. It is impossible to give
rules .to calculate the strength necessary to resist active forces, varying in
intensity and frequency ; we can only give instances of practical structures
which have been found sufficient, as mere data on which to form judg-
ment.* It must be remarked, that where rigidity is required, stiffness of
beams, unlike their ultimate strength, is directly as the breadth, and the
cube of their depth, and inversely as the cube of their length.
Detrusion. The resistance to detrusion, or the force necessary to shear
across any material, is called into play at the joints, and in the bolts of
framings of timber and iron. The resistance of spruce to detrusion in the
direction of its fibre, is about 600 Ibs. per square inch ; of cast iron to de-
trusion, about 73,000 Ibs. per square inch ; of wrought iron, 45 to 50,000
Ibs.
Torsion. When two forces act in opposite directions upon a body, tend-
ing to turn its extremities in different directions, or twist them, it is said to
be subjected to torsion. Thus the main shaft of a steam-engine, at one end
of which the power acts through a crank ; which at the other is transmitted
through a gear or pulley ; the resistance which the load presents on the
one hand, and on the other, the power applied to the crank, represent two
forces, subjecting the shaft to the action of torsion.
When the torsion exceeds a certain limit, depending on the material
* For the girders of railway bridges, they should be of such dimensions as to bear a
strain of two tons per foot in length.
MECHANICS. 129
and its form, the fibres arc torn asunder, and the axles twisted off. For
the determination of the size of axles subject to a twisting force, we de-
duce from "Weisbacli the following rule, allowing a five-fold security or
strength above the absolute breaking twist : Multiply the weight in pounds
by the leverage in inches, and divide the product by C determined for
different forms and material, and the cube root of the product will be half
3 l-p-^
the thickness of the axle, expressed by formula r = \f 79-.
L/
Value of C for wrought and cast iron, circular section, 12,600 Ibs.
" square " 15,000
" " wood, circular " 1,260 "
" " square 1,500 "
In the square section, the rule gives as the cube root of the product one-
half the side of the square.
Example. The shaft of a turbine exerts, through a toothed wheel of 30
inch pitch or 15 inch radius, a force of 2,500 Ibs., what must be the diam-
eter of the shaft ?
OKAA -.- OAA 3 /37500 3 7375 1 ., . ,
2500 x ID = 3 < oOO. r = * = ^ = 1.44 inches.
1.44 x 2 = 2.88, say 3 inches, diameter required.
The length of the axle subjected to torsion does not affect the actual
amount of pressure required to produce rupture, but only the angle of tor-
sion which precedes rupture, and therefore the space through which the
pressure must be made to act. The ultimate strength of a long shaft to re-
sist torsion may be sufficient, but its elasticity will be found to be too much.
To determine the degrees of the angle of torsion of a given shaft, mul-
tiply the load in pounds by the leverage, and also by the length of the
shaft, between the points of the applied forces ; and divide the product by
the fourth power of the half diameter or half square, as the case may be,
multiplied by a constant determined by experiment, and the quotient is
P a I
the number of degrees in the angle, that is, ang. = ~ 4 .
Value of C.
Circular section. Square section.
Wood, . . . . . 3500 5800
Cast iron, . . . 160000 280000
Steel and wrought iron, . 280000 470000
Example. If the distance of the toothed wheel from the water wheel
in former experiment be 60 inches, what is the angle of torsion ?
9
130 MECHANICS.
37.500 x 60 375 x 6 375 x 6
160.000 r* 160 x 1.44 4 160 x 4.28 4.28
An angle too considerable for practice ; it should not exceed one degree.
To calculate the size of the shaft, so that the angle is ^ , the formula be-
4 I p a i
comes, r = \J -^ - ; then in the above example,
6 x o-
37.500 x 60 _ 375 x 60 _ 375 x 6 _
~80~~ ~S~ ~~ 281 ' 25
V281.25 = V16.77 = 4.1.
4.1 x 2 = 8.2, diameter of shaft.
MECHANICAL WORK OK EFFECT.
To work, considered in the abstract, is to overcome, during any certain
period of time, a continuously replaced resistance, or series of resistances.
Mechanical work is the effect of the simple action of a force upon a
resistance which is directly opposed to it, and which it continuously de-
stroys, giving motion in that direction to the point of application of the
resistance. It follows from this definition, that the mechanical work or
effect of any motor is the product of two indispensable quantities or terms :
First^ The effort, or pressure exerted.
Second, The space passed through in a given time, or the velocity.
The amount of mechanical work increases directly as the increase of
either of these terms, and in the proportion compounded of the two when
both increase. If, f OK example, the pressure exerted be equal to 4 Ibs., and
the velocity one foot per second, the amount of work will be expressed by
4x1 = 4. If the velocity be double, the work becomes 4 x 2 = 8, or
double also ; and if, with the velocity double, or 2 feet per second, the
pressure be doubled as well that is, raised to 8 Ibs. the work will be
8 x 2 = 16 Ibs. ft.
The unit of mechanical effect adopted in England and this country is
the horse power, which is equal to 33,000 Ibs. weight or pressure, raised or
moved through a space of 1 foot in a minute of time. The corresponding
unit employed in France is the kilogramme tre, which is equal to a kilo-
gramme raised one metre high in a second. The horse power is repre-
sented by 75 kilogrammetres ; that is, 75 kilog. raised 1 metre high per
second. When we speak of small amounts of mechanical effect, it is
generally said that they are equal to so many pounds raised so many feet
MECHANICS. 131
liigli in some given time, as a minute for example. The time must always
be expressed or understood. It is impossible tp express or state intelligibly
an amount of mechanical effect, without indicating all the three terms
pressure, distance, and time.
The motors generally employed in manufactures and industrial arts are
of two kinds living, as men and animals ; and inanimate, as water and
steam.
AVhat may be termed the amount of a day's work, producible by men
'and animals, is the product of the force exerted, multiplied into the distance
or space passed over, and the time during which the action is sustained.
There will, however, in all cases be a certain proportion of effort, in rela-
tion to the velocity and duration which will yield the largest possible pro-
duct or day's work for any one individual, and this product may be termed
the maximum effect. In other words, a man will produce a greater me-
chanical effect by exerting a certain effort at a certain velocity, than he
will by exerting -a greater effort at a less velocity, or a less effort at a
greater velocity, and the proportion of effort and velocity which will yield
the maximum effect is different in different individuals.
In the manner and means in which the strength of men and animals
is applied, there are three circumstances which demand attention :
1st. The power, when the strength of the animal is exerted against a
resistance that is at rest.
2d. The power, when the stationary resistance is overcome, and the
animal is in motion. And,
3d. The power, when the animal has attained the highest amount of
its speed.
In the first case, the animal exerts not only its muscular force or
strength, but at the same time a very considerable portion of its weight or
gravity. The power, therefore, from these causes must be the greatest
possible. In the second case, some portion of the power of the animal is
withdrawn to maintain its own progressive motion ; consequently the
amount of useful labor varies with the variations of speed. In the third
case, the power of the animal is wholly expended in maintaining its loco-
motion ; it therefore can carry no weight.
The following table exhibits the average amount of mechanical effect
produced by men and animals in different applications ; the animal work-
ing with a mean velocity and effort during an average day's work, thereby
producing the maximum effect.
132
MECHANICS.
Nature of the work.
Effort exerted.
Velocily per
second.
Effect per
second.
Duration.
Mechanical effect
per day.
Founds.
Feet.
Hoars.
Man -working at a lever, as in pump-
tog,
10.5
3.5
37.45
8
1.07S.560
" at a crank. Length of crank 16
to 18 inches ; height of axis of
shaft, 36 to 39 inches, .
17.
2.4
40.3
8
1.175.040
" tread-mill at level of axis, .
123.
0.43
61.44
S
1.769.472
" " " at angle of 24= from
the vertical, ....
25.|
2.25
57.75
8
1.663.200
" at a vertical capstan, .
25 i
1.9
43.45
8
1.395.360
Horse at a whim gin not less than 20
feet radius
153.
2.9
443.7
s
12.77S.560
Draught bv traces, according to
Geretner :
.Weight.
Man 150
30.
2.5
75.
8
2.160.000
Horse, 600
120.
4.
430.
8
13.S24.000
Mule, 500
100.
3.5
350.
3
10.080.000
Ox 600
120.
2.5
300.
8
8.640.000
Gerstner also gives the following formula to calculate the effect of
change of velocity.
K and c representing the effort and velocity per second, as given in the
Table, v the assumed velocity, and P the resulting effect.
Example. Suppose a horse to travel at the rate of 6 feet per second.
"What effort will he exert, and what will be the mechanical effect per
second ?
From the table we have c = 4. ; K = 120. ; v is assumed at 6 ; then
/
P = (2 120 = 60. = effort.
60 x 6 = 360 ft. Ibs. effect per second.
It is evident that this formula will not apply to extreme values of 'P or
v / yet it may be considered sufficiently near for most practical purposes.
not veiy different from the mean, and is illustrative of the ill effects re-
sulting from increase of velocity.
Water power. "Water acts as a moving power, or moves machines
either by its weight, by pressure, or by impact ; and is applied through
various forms of wheels. The mechanical effect inherent in water is the
product of its weight into the height from which it falls ; but there are
many losses incurred in its application to machinery, so that only a portion
of the mechanical effect becomes available ; that is, the efficiency of any
water wheel is represented by a certain per-centage of the absolute effect
of the water.
MECHANICS. 133
Example, The quantity of water supplied to the mills at Lowell is
3,596 cubic feet per second ; the net fall is 33 feet ; the absolute dynami-
cal effect of this water is :
3596 x 33 X 62.33 = 7.396.576 Ibs. ft. per second.
62.33 being the weight of a cubic foot of water, at 60 Fahr. ; on an average
it may be assumed, that a useful effect is derived equal to two-thirds of
the total power of the water expended ; two-thirds of 7.396.576, divided
by 550, gives 8965.5 horse power as absolutely available. 550 Ibs. 1 foot
high per second represent a horse power, being equal to 33,000 Ibs. 1 foot
high per minute.
/Steam is the elastic fluid into which water is converted by a continuous
application of heat. It is used to produce mechanical action almost in-
variably by means of a piston movable in a cylinder. The horse power
of a steam-engine is computed, by multiplying the area of the piston in
square inches by the effective pressure in Ibs. on each square inch of piston,
and the product by velocity in feet through which the piston moves per
minute, dividing this last product by 33,000.
The area of the piston is found by squaring the diameter, and multiply-
ing the square by 0,7854.
Example. Let the diameter of the piston be 18 inches, the effective
pressure 45 Ibs. per square inch, and the speed 300 feet per minute, what
will be the horse power of the engine ?
18 X 18 x 0,7854 = 254.46 square inches, area of piston.
254.46 X 45 X 300
33,000 =104. horse power.
To determine the effective pressure on the piston, recourse must be had
to an indicator, and take the mean pressure, as shown on the diagram ; the
pressure on the boiler is readily known, but the steam in its passage to the
cylinder is subject to various losses, as of wire-drawing, condensation, &c.,
so that frequently the pressure on the piston does not exceed two-thirds of
that on the boiler. The boilers of most of our stationary engines are sub-
jected to pressures of from 50 to 75 Ibs. per square inch ; the smaller en-
gines, say less than 10 horse power, are generally worked with full steam ;
effective pressure from 30 to 60 Ibs. Larger ones are generally worked
expansively, cutting off at from one-half to one-sixth stroke. The principle
of working steam expansively is as follows : If a cubic foot of air of the
atmospheric density be compressed into the compass of half a cubic foot,
its elasticity will be increased from 15 Ibs. on the square inch to 30 Ibs. ;
if the volume be enlarged to two cubic feet, the pressure will be one half,
134
MECHANICS.
or 7 1 Ibs. The same law holds in all other proportions for gases and va-
pors, provided their temperature is unchanged.
Tims, let E (fig. 225) be a cy-
linder, J the piston ; let the cylin-
der be supposed to be divided in
the direction of its length into any
number of equal parts, say twenty,
and let the diameter of the piston
represent the initial pressure of the
steam, which we call 1. If now
the piston descend through 5 of
the divisions, and the valve be then
shut, the pressure at each subse-
quent position of the piston may
be calculated by the law above
given, and represented as shown
in the figure. If the squares above
the point, when the steam was cut
off, be counted, they will be found
to amount to 50, those below to
about 68 ; so that while, by an
expenditure of a quarter of cylinder full of steam, we get an amount of
power represented by 50, we get 68 without any further expenditure, by
merely permitting expansion. Practically, for large cylinders, it may be
stated :
Cutting off at
^ stroke,
Saves of fuel
41 per cent.
58 "
68 "
Gains in effect
70 per cent.
70 X 2 = 140 per cent.
70 x 3 = 210
Mean pressure at different densities, and rate of expansion.
pressure.
EXPANSION BY EIGHTHS.
i
J
1
1
i
10
9.896
9.C37
9.1 87
8.465
7.417
5.965
3.S4S
15
14844
14456
13.781
12.697
11.126
8.947
5.773
'20
19.792
19.275
18.375
16.930
14.835
11.930
7.697
25
24.740
24.093
22.963
21.162
18,548
14.912
9.621
80
29.6SS
28.912
27.562
25.395
22.252
17.895
11.546
85
84.636
33.731
33.156
29.627
25.961
20.877
13.470
40
39.585
8S.550
86.750
83.860
29.670
23.860
15.395
45
44.533
43.368
41.343
88.099
33.378
26.842
17.319
50
49.4S1
4S.1S7
45.937
42.325
37.067
29.825
19.243
MECHANICS.
135
Water converted into steam under the pressure of the atmosphere, i. <?.,
15 pounds per square inch, expands to 1700 times its volume ; under double
the pressure, or 30 pounds, the volume would be one-half; and this pro-
portion would be strictly accurate but for the fact that the temperatures at
which water boils in these cases are different.
In the following table are given the total pressure of steam in pounds
per square inch, the corresponding temperature, and the number of cubic
inches of steam which would be produced by one cubic inch of water.
Total pressure
in pounds
per square
inch.
Corresponding
temperature.
Cubic inches of ;
steam produced by
a cubic inch of
Total pressure
in pounds
per square
inch.
Correspond ing
temperature.
Cubic (inches of
steam produced by
a cubic inch of
water.
14
209.1
1778
54
288.1
516
15
212.8
1669
55
289.3
503
20
22S.5
1281
56
290.5
500
25
241.0
1044
57
291.7
492
30
251.6
883
53
292.9
484
35
260.9
761
59
294.2
477
40
269.1
679
60
295.6
470
45
276.4
610
61
296.9
463
46
2TT.8
598
62
298.1
456
47
2T9.2
586
63
299.2
449
4S
280.5
575
64
300.3
443
49
231.9
564
65
301.3
437
50
2S3.2
554
66
302.4
431
51
284.4
544
67
803.4
425
52
285.7
534
68
804.4
419
53
286.9
525
69
305.4
414
It must be remarked, that in non-condensing engines, the effective pres-
sure is the excess above the pressure of the atmosphere.
TaUe showing the weights, evaporative powers per weight, and ~bulk and
character of fuels, from the report of Prof. "Walter K. Johnson, 1844.
Designation of fuel.
Specific
gravity.
Weight
per cubic
foot.
Water
evaporst'd
by one Ib.
of fuel.
Designation of fuel.
Specific
gravity.
Weight
per cubic
foot.
Water
evaporat'd
byonelb.
of fuel.
1UTUMINOUS.
Ibs.
Ibs.
ANTHRACITE.
Ibs.
Ibs.
Cumberland, maximum
1.313
82.09
10.7
Peach Mountain,
1.464
91.5
10.11
" minimum
1.337
83.28
9.44
Beaver Meadow, No. 5,
1.554
96.9
9.8S
Blossburgh,
1.324
82.73
972
Lackawana,
1.421
88. S
9.79
Newcastle, .
1.257
78.54
S.66
Beaver Meadow, No. 3, '
1.610
100.6
9.21
IMctou,
1.318
82.S3
8.41
Lehigh,
1.500
99.3
S.93
Pittsburgh, .
1252
78.37
8.20
Sydney,
Liverpool, .
Clover Hill,
1.333
1.262
1.2S5
83.66
78.89
80.36
7.99
7.84
7. 07
Natural Virginia,
Cumberland,
1.323
82.70
8.47
8.99
Cannelton, la.
1.273
79/4
7.34
WOOD.
Scotch,
1.519
94.95
G.95
Dry Pine "Wood,
21.01
469
136 MECHANICS.
The above table exhibits the ultimate effects. As a safe estimate im-
practical values, a deduction (for the coals) of T W should be made.
From these two tables it is easy to calculate the amount of fuel which
must be expended to produce a given power.
Example. To find the consumption of water and fuel required by a
high pressure engine, 12 inch cylinder, 4 feet stroke, the effective pressure
on the piston being 40 Ibs., and the number of double strokes 35 per min.
Area of piston = 12 x 12 x 0.7854 = 113.09.
Telocity of piston = 35x8 = 280 ft. per min.
Then, 113.09 X 280 X 12 = 379.982 cubic inches of steam used during 1
minute, or 379.982 X 60 = 22.798.920 cub. in. consumption per hour. Look-
ing in the first table against the pressure 55, that is 15, or atmosphere
added to 40 given above, we find 508 ; dividing, therefore, 22.798.920 by
508, we have 44880, the number of cubic inches of water used per hour ;
44880
~T~~28 ~ 26 Cll bi c f eet nearly.
Multiplying this by the weight of a cubic foot of water,
26 x 62.33 = 1620.58 Ibs.
Taking the safe estimate for the anthracites of the evaporation of 8 Ibs. of
water by 1 Ib. of fuel, we have,
= 202.5 Ibs. the consumption of coal per hour.
o
On an average of boilers, 1 square foot of grate surface is allowed for
the consumption of 14 Ibs. of coal per hour, from 15 to 25 square feet of
heating surface, and one-sixth of a square foot of flue at the base of the
chimney. Continuing the previous example, we have
202.5
"--Tr = 14.5 square feet of fire grate.
14.5 x 20 = 290 square feet of heating surface.
^- = 2.42 square feet of flue.
The horse power of the above engine would be
113,09 X 280 X 40
= 38.6 horse power.
A portion of which power would be consumed in the driving of the engine
itself, leaving about 35 horse power as effective on the first shaft.
DBA WING OF MACHINERY.
137
DRAWING OF MACHINERY.
HAVING tlms laid down the principles of geometrical projection, and
the rules by which to proportion parts, according to the stress to which
they may be subjected, we now proceed to the practical application ot
the principles and rules in the drawing of machinery.
Wood, cast and wrought iron.
SHAFTING.
Shafts are made of wood, cast and wrought iron. Fig. 226 repre-
sents the sections of the usual forms. Wooden shafts are mostly of an
octagonal or polyhedral form, and are seldom used but as shafts for water-
wheels, but are not equal
to those of cast iron ex-
cept in cheapness, and are
seldom adopted when the
latter can be readily ob-
tained. Cast iron is used
for the shafts of water-
wheels, and the heavier
kinds of mill- work, when rig. 220.
the strain is rather transverse than torsional. The most economical form
for cast iron shafts is the tubular, but the more usual are the feathered
shafts, that is, with a circular or square centre, and ribs running longi-
tudinally. Wrought iron shafts are used for the main and counter shafts
of mills, and for heavy shafts subject to torsional or to unequal stress and
shocks, and is by far the best material for shafts. The more usual and the
best form is the circular.
Shafts are termed first, second, and third movers ; the first are the first
recipients of power, as the jack-shaft from a water-wheel, or the fly- wheel
shaft of engines ; the second are the next in succession, distributing the
138
DRAWING OF MACHINERY.
power, as the main shafts of mills ; and, third, the counters or shafts trans-
mitting the power to the machines. The strain upon a shaft may be trans-
verse, torsional, or both. In all breast, overshot, or undershot water-
wheels, the jack-geer may be so placed that there will be no torsional
strain on the shaft of the wheel ; in many other shafts, no strain will be
transmitted through the journal. In these cases, the size of the journal
may be estimated from the transverse strain or weight to which it is sub-
jected. The following table is taken from the Practical Draughtsman,
calculated on this formula, D = YM> x .1938, D being the diameter in
inches, and w the weight to be sustained in Ibs.
Table of the diameters of the journals of water-wheel and other shafts for
heavy work.
1
DIAM. OF JOCIt
iAL IN IXCHES.
CMt iron.
Wrought iron.
Cut iron.
Wrought iron.
1099.0
2
1.7
100156
9
7.7
2146.7
2*
2.1
117793
9i
S.I
3709.5
3
2.5
1373SS
10
8.6
5890.5
^
3.0
15S604
10i
9.0
SS05.6
4
34
1S2S64
11
9.4
12019.5
<:*
8.8
20S950
1H
9.9
17175.5
5
43
237296
12
10.3
22S53.0
t*
4.7
26S012
12i
10.7
29676.0
6
5.1
311666
13
11.2
S7730.0
Ci
5.G
33S026
13*
11.6
43S73.0
7
6.0
376993
14
12.0
58915.7
7i
6.4
41SS45
1J|
12.5
70353.0
's
6.9
4636S5
15
12.9
&4373.0
64
7.3
The length of the journal should be from once to twice the diameter.
The size of the shaft at the point at which the load is applied may be
determined from previous rules ; but for all shafts less than three feet be-
tween bearings, the size as calculated for the journal need only be enlarged
enough to cut the key-seat.
PL XIV. Figs. 1, 2, 3, represent different views of a wooden water-
wheel shaft. Fig. 1 shows at one end the side external elevation of the
shaft, furnished with its iron ferules or collars and its gudgeon ; at the
other end, the shaft is shown in sections, giving the ferules in section,
but showing the central spindle with its feathers in an external elevation.
Generally, in longitudinal sections of objects enclosing one or more pieces,
DRAWING OF MACHINERY.
139
the innermost or central piece is not sectioned unless it lias some internal
peculiarity, the object of a section being to show and explain peculiarities,
and being therefore unnecessary when the object is solid ; on this account,
bolts, nuts, and solid cylindrical shafts are seldom drawn in section. Fig.
2 is a cross or transverse section through the centre of the shaft, to show
the outward octagonal form. Fig. 3 is an end view of the shaft, showing
the tit-ting of the spindle B and its feathers into the end of the shaft, and
the binding of the whole by ferules or hoops a a. The spindles B, which
are let into the ends, are cast with four feathers or wings c. The tail-piece
J is by many millwrights omitted. The ends of the beam are bored for
the spindle, and grooved to receive the feathers ; the casting is then driven
into its place, hooped with hot ferules, and after this hard- wood wedges are
driven in on each side of the feathers, and iron spikes are sometimes driven
into the end of the wood.
Figs. 4, 5, 6, represent different views of a cast iron shaft of a water-wheel.
Fig. 4 is an elevation of the shaft, with one half in section to show the form
of the core ; fig. 5, an end elevation ; fig. 6, a section on the line c c across the
centre. The body is cylindrical and hollow, and is cast with four feathers
c c, disposed at right angles to each other, and of an external parabolic out-
line. Kear the extremities of these feathers four projections are cast, for the
attachment of the bosses of the water-wheel. These projections are made
with facets, so as to form the corners of a circumscribing square, as shown
in fig. 5, and they are planed to receive the keys by which they are fixed
to the naves which are grooved to receive them. The shaft is cast in one
entire piece, and the journals are turned.
In all line drawings, the portions of an object represented in section
are shaded with diagonal lines, as in figs. 1, 2, 4, and 6.
Fig. 227 represents the sec-
tion of a portion of a water-wheel,
with a cast iron shaft, in use in
this country, in which stiffness is
given to the wheel by wooden
trusses, and a tensional strain is
given to the centre of the shaft.
These shafts are cast circular in
two lengths connected at the cen-
tre, with circular bosses on which
the naves of the wheel are keyed.
When the load upon a shaft Fig. 227.
is not central between the bearings, the size of the journals should be pro-
140
DRAWING OF MACHINERY.
portioned to the weight it will be required to support, which will be in-
versely as their distance from the centre of pressure.
Fig. 228 represents the fly-wheel shaft of a stationary engine. The
parts of least diameter are the journals ; their length is 1^ times the diam-
Fig. 22S.
,9 "5-
Fig. 229.
eter ; the centre of the shaft is enlarged to receive the hub of the fly-wheel,
and for convenience in driving the keys. Shafts of this form are mostly
of wrought iron, the reduction being made by steps, as a convenience in
swedging. Fig. 229 is a plan of the crank, from the wheel side.
The torsional strain on a shaft is as the power transmitted through it.
It is evident, power being weight multiplied by velocity, that the greater
the velocity of the shaft, the less the strain to transmit the same amount
of power ; and it is the modem practice to drive the shafts at high veloci-
ties, and reduce the weight of the geering. In first movers, the strain is
often compound ; and when the journals bear but little transverse strain,
the determination of their size must depend entirely on their capacity to
resist torsion. The formula given in the Practical Draughtsman for de-
termining the proper diameter is :
C being for cast iron, 1st movers, 419 ; 2cl, 206 ; 3d, 106.
wr'ght" " " 249; " 134; " 67.6.
Which formula is simplified and tabellated, so that it is only necessary
to divide the number or revolutions of the shaft by the horse power, and
find the diameter corresponding to the quotient in the table.
DRAWING OF MACHINERY. 141
Table of diameters for shaft journals, calculated with reference to torsional
strain*
JOURNALS OP CAST-IRON SHAFTS.
JOURNALS OF WROUGHT-IRON SHAFTS.
Diameter in
inches.
First movers.
Second movers.
Third movers.
First movers.
Second mover.
Third movers.
H
124.133
61.037
81.408
73.778
39.704
20.030
2
52.875
25.750
13.250
31.125
16.750
8.450
2*
26.S1G
13.190
6.790
15.372
S.576
4.327
8
15.519
7.630
3.922 .
9.222
4.963
2.504
*l
9.7T3
4.305
2.475
5.808
3.123
1.577
4
6.547
3.219
1.656
3.S91
2.094
1.563
4*
4.593
2.266
1.163
2.782
1.475
.742
5 *
3.852
1.64S
.343
1.992
1.072
.541
6*
2.519
1.239
.637
1.497
.806
.406
6
1.940
.954
.491
1.153
.620
.313
H
1.526
.750
.336
.906
.433
.246
i
1.222
.601
.309
.726
.391
.197
T*
1.002
.493
.253
.595
.325
.162
8
.833
.402
.207
487.
.261
.133
Si
.632
.835
.173
.405
.213
.110
9
.575
.232
.145
.341
.134
.093
9i
.439
.240
.124
.290
.156
.079
10
.419
.206
.106
.249
.134
.063
10i
.362
.173
.092
.215
.116
.058
11
.314
.155
.079
.137
.101
.051
1H
.275
.185
.069
.163
.089
.044
12
.242
.119
.061
.144
.078
.039
t
.214
.105
.054
.127
.063
.034
13
.191
.094
.049
.114
.061
.031
181
.170
.034
.043
.101
.054
.027
14
.153
.075
.033
.091
.049
.024
14*
.137
.067
.035
.082
.044
.022
15
.124
.061
.081
.074
.039
.020
1
2
3
4
5
6
7
Example, "What must be the diameter of the journal of a wrought iron
first mover, transmitting 30 horse power, and making 50 revolutions per
minute ?
1.667 in the table is intermediate between 1.992 and 1.497, corresponding
to 5 and 5|, and should be about 5f inches.
It is the common practice to make w r rought iron 2d and 3d movers of
an uniform diameter, without reduction at the journal ; the shaft is pre-
vented from sliding endways by collars keyed on. The usual length of
main shafts is from 7 to 10 feet between bearings ; and that they may run
142 DRAWING OF MACHINERY.
smooth, and not spring intermediately, it is desirable that they should
never be less than 2 inches diameter, and that the pulleys or geers through
which the power is transmitted to the next mover or to the machine should
be as near as possible to the bearing.
Tig. 230 represents a line of shafting. A is an upright shaft ; a a,
bevel-geers ; 5 5, bearings for the shaft ; c, coupling or connection of
the several pieces of shafting. These shafts are intended to be of wrought
iron. No reduction is made for the journal, no bosses for pulleys or geers.
As the power is distributed from this line of shafting, the torsional strain
diminishes with the distance from the bevel-geers or first movers, and the
diameter of each piece of shafting may be reduced consecutively, if neces-
sary ; but uniformity will generally be found to be of more importance
than a small saving of iron. The drawing given is of a scale large enough
to order shafting by, but the dimension, should be written in. It is often
usual in the order to the machinists merely to give the lengths of the shafts
and diameters as thus :
.3 .S .2
v HN a f f- y ft ft HN v A ft H-* v
X Q^ O 11. X > i.ln Qq X ' J.L. <jq X
The x marks represent the bearings ; the joints or couplings are generally
made near the bearings, and it is also usual to bring the pulleys as near
the bearings as possible. It frequently happens, therefore, that the coup-
ling and pulley are needed at the same point ; to remedy this, as the posi-
tion of the pulley depends on the machine which it is required to drive, it
frequently cannot be moved without considerable inconvenience or loss of
room ; the shaft will have, therefore, to be lengthened or shortened, to
change position of coupling ; or better, the coupling and pulley may be
made together.
BEARINGS OR SUPPORTS FOR THE JOURNALS OF SHAFTS.
for upright shafts. Footstep or step for an upright shaft. Fig. 231
represents an elevation ; fig. 232, a plan of the step. It consists of a foun-
dation or bed-plate A, a box B, and a cap or socket C. The plate A is
firmly fastened to the base on which it rests ; in the case of heavy shafts,
DRAWING OF MACHINERY.
143
often to a base of granite. The box B is placed on A, the bearing surface
being accurately bevelled, and fitted either by planing or chipping and
Fig. 232.
filing ; 5, 5, 5, are what are commonly called chipping-pieces, which are the
bearing surfaces of the bottom of B. A and B are held together by two
Jjjt
screws ; the holes for these are cut oblong in the one plate at right angles
144
DRAWING OF MACHINERY.
to those of the other ; this admits of the movement of the box in two direc-
tions to adjust nicely the lateral position of the shaft, after which, by
means of the screws the two plates are clamped firmly to each other. C,
the cup or bushing, which should be made of brass, slips into a socket in
B. Frequently circular plates of steel are dropped into the bottom of
this cup for the step of the shaft. The cup C, in case of its sticking to the
shaft, will revolve with the shaft in the box B ; if plates are used, these
also admit of movement in the cup.
Fig. 233 represents the elevation of a bearing for an upright shaft, in
which the shaft is held laterally by a box and bracket above the step. The
step B is made larger than the shaft, so as to reduce the amount of wear
incident to a heavy shaft. The end of the shaft, and the cup containing
oil, are shown in dotted line. The bed-plate A rests on pillars, between
which is placed a pillow-block or bearing for horizontal shaft.
Figs. 234, 235, represent the elevation and vertical section of the sus-
pension bearing used by Mr. Boyden for the support of the shaft of his tur-
bine wheels. It having been found difficult to supply oil to the step of such
wheels, it was thought preferable by him to suspend the entire weight of
wheel and shaft, where it could be easily attended to. The shaft (see sec-
tion) is cut into necks, which rest on corresponding projections cast in the
Fig. 284.
box 5 / the spaces in the box are made somewhat larger than the necks of
the shaft, to admit of Babbitting, as it is termed, the box ; that is, the shaft
being placed in its position in the box, Babbitt, or some other soft metal
melted, is poured in round the shaft, and in this way accurate bearing sur-
faces are obtained ; projections or holes are made in the box to hold the
metal in its position. The box is suspended by lugs 5, on gimbals c, simi-
lar to those used for mariners' compasses, which give a flexible bearing,
so that the necks may not be strained by a slight sway of the shaft. The
DRAWING OF MACHINERY.
screws e e support the gimbals, consequently the shaft and wheel ; by these
screws the wheel can be raised or lowered, so as to adjust its position accu-
rately ; beneath the box will be seen a movable collar, to adjust the lateral
position of shafts.
Figs. 236, 237 are the plan and elevation for the step, or rather guide
(as it bears no weight), of the foot of the shaft of these same turbines.
The plate A is firmly bolted to the floor of the wheel-pit ; the cushions, C,
holdino- the shaft, are either wooden or cast iron, and admit of lateral ad-
1
Fig. 236.
Fig. 237.
justment by the three sets of set-screws. "Wooden steps are often used to
support the shafts of the smaller horizontal wheels beneath the surface of
the water ; the fibres of the wood are placed vertically, and afford a very
excellent bearing surface. AVhen cast iron or steel is used for the step, it
is usual to encase the box, and supply oil by leading a pipe, sufficiently high
above the surface of the water, to force the oil down.
For long upright shafts, it is very usual to suspend the upper portion
by a suspension box, and to run the lower on a step, connecting the two
portions by a loose sleeve or expansion coupling, to prevent the unequal
mashing of the bevel wheels, incident to an alteration of the length of shaft
by variations of temperature. The suspension is frequently made by a
single collar at the top of the shaft.
When a horizontal shaft is supported from beneath, its bearing is usu-
ally called a pillow or plumber-block, or standard; if suspended, the sup-
ports are called hangers.
Figs. 238, 239 are the elevation and plan of a pillow-block. It consists
of a base plate A, the body of the block B, and the box C. The plate, as
in the step, is bolted securely to its base ; the surface on which the block
B rests being horizontal. A and B are connected by bolts passing through
oblong holes, so as to adjust the position in either direction laterally. The
10
140
DRAWING OF MACHINERY.
box or bush C is of brass, in two parts or halves, extending through the
block, and forming a collar by which it is retained in its place. The cap of
Fig. 239.
the block is retained by the screws o o o ; in the figure there are two screws
on one side and one on the other ; often four are used, two on each side,
but most frequently but one on each side.
PROJECTIONS OF A STANDAKD.
PI. XV. The standard is simply a modification of the pillow-block,
being employed for the support of horizontal shafts at a considerable dis-
tance above the foundation-plate. Fig. 1 is a front elevation ; fig. 2, a
plan ; and fig. 3, an end elevation of a standard. Like the pillow-block,
the plate A is fastened to the foundation itself, and the upper surface is
placed perfectly level in both directions. On these bearing surfaces a a a
the body of the standard rests, and can be adjusted in position horizontally,
and then clamped by screws to the foundation-plate, or keyed at the ends.
Fig. 4 is a plan of the upper part of the standard with the cover off, show-
ing the form of the box, with a babitted -bearing surface.
Whilst drawing the front elevation, mark off on figs. 3 and 4 the out-
DRAWING- OF MACHINERY.
147
lines of all such parts as arc immediately transferable by the help of the
square and compasses, from one figure to the other. The outline e k and
fl are arcs, whose centres lie in ef produced, and pass through the points
e, k and/ 1 , Z. To find the projection of this arc upon the plan (fig. 2), draw
through any points m and n, taken at pleasure upon the arc c k (fig. 3), the
horizontals m m, n n, and through m and n (fig. 1) draw m m and n n paral-
lel to C D, then set off the distances o m and^ n (fig. 3) to the corresponding
points on the lower side of centre line M X (fig. 2) : thus the curve emnk
will be determined. By a similar method the curve c m, n' will be ob-
tained, as also the projections of all such arcs as are denoted by rq (fig. 3).
To draw on fig. 3 the s t h (fig. 2), which is the line of penetration of
two cylinders, a similar construction to the preceding may be adopted.
But to avoid drawing too many lines on the figures, this projection is con-
structed (see fig. 5) on another part of the sheet, in which s t h' represent
the plan of the curve s t h (fig. 2), and h v* t the elevation, as at fig. 1.
Divide h v* t into any number of equal parts ; let fall perpendiculars h h'
v 2 v f . . . from the points of division, and horizontal and parallel lines li h,
v 2 v . . . ; lay off on each side from the half chords made on the semicircle,
and we have the curve h v t v s, which may easily be transferred to its
position in fig. 3.
It will be observed, that one side of the elevation (fig. 1) is represented
as broken ; this is often done in drawing, when the sides are uniform, and
economy of space on the paper is required.
Fig. 240. Fig. 241.
Suspended bearings or hangers for horizontal shafts are divided into
two general classes side hangers (figs. 240, 241), and sprawl hangers
148 DRAWING OF MACHINERY.
(pi. XVL, fig. 1) ; the figures will sufficiently explain the distinction. The
side hanger is the more convenient when it is required to remove the shaft,
and when the strain is in one direction, against the upright part ; they are
generally used for the smaller shafts, but sprawl hangers affording a more
firm support in both directions, are used as supports for all the heavier
shafts. Hangers are bolted to the floor timbers, or to strips placed to sus-
tain them, the centres of the boxes being placed accurately in line, both
horizontally and laterally.
PI. XYI. Fig. 1 represents the elevation of a sprawl hanger ; fig. 2,
the plan looking from above, with cover of box off ; fig. 3, a section on
the line A B, fig. 1.
Fig. 4 represents the elevation of a bracket, or the support of a shaft
bolted to an upright ; the box is movable, and is adjusted laterally by the
set-screws. Fig. 5 is a front elevation of the back plate cast on the post ;
it will be seen that the holes are oblong, to admit of the vertical adjust-
ment of the bracket. Fig. 6 is a side elevation of the box fig. 7, a sec-
tion lengthways, showing aperture for grease, and the points which retain
the babbit-metal lining in its place ; fig. 8 is a plan of the bottom half of
the box ; fig. 9, plan of the top.
Fig. 2-42 represents different views of what may be called a yoke-
hanger. Fig. 1 is a front and fig. 2 a side elevation ; fig. 3 a plan of the
hanger, looking up ; and fig. 4 a plan of the yoke, looking down upon it.
A is the plate which is fastened to the beam, E is the yoke, and B the stem
of the yoke, cut with a thread so as to admit of a vertical adjustment ; the
box D of the shaft C is supported by two pointed set-screws passing through
the jaws of the yoke ; this affords a very flexible bearing, and a chance for
lateral adjustment.
Couplings are the connections of shafts, and are varied in their con-
struction and proportions often according to the mere whim of the me-
chanic making them.
The Face Coupling (fig. 243) is the one in most general use for the con-
necting of wrought iron shafts ; it consists of two plates or discs with long
strong hubs, through the centre of which holes are accurately drilled to
fit the shaft ; one-half is now drawn on to the shaft, and tightly keyed ;
the plates are faced square with the shaft, and the two faces are brought
together by bolts. The number and size of the bolts depend upon the size
of the shaft, never less than 4 for shafts less than 3 inches diameter, and
more as the diameter increases ; the size of the bolts varies from to 1 \ in.
in diameter. The figure shows a usual proportion of parts for shafts of
from 2 to 5 inches diameter ; for larger than these, the proportion of the
diameter of the disc to that of the shaft is too large.
DRAWING OF MACHINERY.
149
Fig. 244 is a rigid sleeve coupling for a cast iron shaft ; it consists of a
solid 1mb or ring of cast iron hooped with wrought iron ; the shafts are
made with bosses, the coupling is slipped on to one of the shafts, the ends
Fig. 242.
of the two are then brought together ; the coupling is now slipped back
over the joint, and firmly keyed. This is an extremely rigid connection.
Fig. 243.
Fig. 244.
Fig. 245 is a screw coupling, a very neat and excellent rigid coupling,
for the connecting of wrought iron, more especially the lighter kinds. It
will be observed that this coupling admits of rotation but in one direction,
150
DRAWING OF MACHINERY.
the one tending to bring the ends of the shafts towards each other, the
reverse motion tends to unscrew and throw them apart, and uncouple
them.
Fig. 246 is a clamp coupling for a square shaft.
In many cases it occurs that rigid couplings, such as we have given,
are objectionable ; they necessarily imply that, to run with the least strain
Fig. 245.
Fig. 246.
possible, the bearings should be in accurate line ; any displacement involves
the springing of the shaft, and if considerably moved, fracture of shaft or
coupling. Wherever, then, from any cause the allignment cannot be very
nearly accurate, some coupling that, admits of lateral movement should be
adopted. The simplest of these is the box or sleeve coupling (tig. 247),
sliding over the end of two square shafts, keyed to neither, but often held
I
Fig. 243.
in place by a pin passing through the coupling into one of the shafts. For
round shafts, the loose sleeve coupling is a pipe or hub, generally 4 to 6
times the diameter of the shaft in length, sliding on keys fixed on either
shaft.
Fig. 248 represents a horned coupling. The two parts of the coupling
are counterparts of each other, each firmly keyed to its respective shaft,
but not fastened to each other ; the horns of the one slip into the spaces
DRAWING OF MACHINERY.
151
of the other ; if the faces of the horns are accurately fitted, it affords an
excellent coupling, and is not perfectly rigid.
It often happens that some portion of a shaft or machine is required to
be stopped whilst the rest of the machinery continues in motion. It is
evident that, if one half of a horned coupling be not keyed to the shaft, but
permitted to slide lengthways on the key, the key being fixed in the
shaft, forming in this case what is more usually called a feather, by slid-
ing back the half till the horns are entirely out of the spaces of the other
half, communication of motion will cease from one shaft to the other.
Couplings are made on this principle, called slide or clutch couplings.
As usually the motion is required but in one direction, the more general
form of this coupling is given in fig. 249. A represents the half of the
coupling that is keyed to the shaft, B the sliding half, c the handle or lever
which communicates the sliding movement ; the upper end of the lever
terminates in a fork, enclosing the hub of the coupling, and fastened by
two bolts or pins to a collar c' round the neck of the hub ; 1) is a box
or bearing for the shaft A ; to support B the end of its shaft extends
a slight distance into the coupling A. It will be observed that the
horns are ratchet-shaped by this form motion can be transmitted but in
one direction ; but should it be necessary to reverse the motion, it is ne-
cessary that the horns of the coupling be square. Shafts cannot be
Fig. 250. Fig. 251.
engaged with this form of coupling while the shaft is in
rapid motion, without great shock and injury to the ma-
chinery. To obviate this, other forms of coupling are re-
quisite ; one of these is represented (fig. 250). On the shaft
B is fixed a drum or pulley, which is embraced by a friction
band as tightly as may be found necessary ; this band consists of two straps
of iron, clamped together by bolts, leaving ends projecting on either side ;
the portion of the coupling on the shaft A is the common form of bayonet
152 DRAWING OF MACHINERY.
clutch ; the part c c is fixed to the shaft, and affords a guide to the prongs
or bayonets 1} 5, as they slide in and out. Slipping these prongs forward,
they are thrown into geer with the ears of the friction band ; the shaft A
being in motion, the band slips round on its pulley till the friction becomes
equal to the resistance, and the pulley gradually attains the motion of the
clutch.
But of all slide couplings to engage and disengage with the least shock,
and at any speed, the friction cone coupling (fig. 251) is by for the best.
It consists of an exterior and interior cone, a, ~b ; a is fastened to the shaft
A, whilst 1} slides in the usual way on the feathery of the shaft B ; press-
ing 5 forward, its exterior surface is brought in contact with the interior
conical surface of a; this should be done gradually; the surfaces of the
two cones slip on each other till the friction overcomes the resistance, and
motion is transmitted comparatively gradually, and without danger to the
machinery. It must be observed, that the longer the taper of the cones,
the more difficult the disengagement ; but the more blunt the cones, the
more difficult to keep the surfaces in contact. From the table given, page
116, it will be seen that the limiting angle of resistance for surfaces of cast
iron upon cast iron is 8 39', and this angle with the line of shaft will give
a very good angle for the surfaces of the cones of this material. When
thrown into geer, the handle of the lever or skipper is slipped into a notch,
that it may not be thrown out by accident.
Pulleys are used for the transmission of motion from one shaft to
another by the means qf belts ; by them every change of velocity may be
effected. The speed of the two shafts will be to each other in the inverse
ratio of the diameter of their pulleys. Tims, if the driving shaft make 100
revolutions per minute, and the driving pulley be 18 inches in diameter,
whilst the driven pulley is 12 inches, then,
12 : 18 :: 100 : 150;
that is, the driven shaft will make 150 revolutions per minute. Where
there is a succession of shafts and pulleys, to find the velocity of the last
driven shaft : Multiply together all the diameters of the driving pulleys
by the speed of the first shaft, and divide the product by the product of
the diameters of all the driven pulleys.
Pulleys are made of cast iron and of every diameter, from 2 in. up to 20
ft. The number of arms vary according to the diameter ; for less than 8 in.
diameter the plate pulley is preferable (fig. 252) ; that is, the rim is attached
to the hub by a plate ; for pulleys of larger diameters, those witli arms
are used, never less than 4 in number. The arms are made either straight
DRAWING OF MACHINERY.
153
(fig. 253), or curved (fig. 254). When large pulleys are cast entire, it is
better that the arms should be curved to admit of contraction in
for the smaller it is unimportant.
Fig. 252.
Fiz. 253.
Fig. 254.
Fig. 255 represents a portion of the elevation of a pulley sufficient to
show the proportion of the several parts, and fig. 256 a section of the same.
Fig. 255.
Fig. 250.
The parts may be compared proportionately with the diameter of shaft ;
thus the thickness of the hub is about \ the diameter of the shaft, this pro-
portion is also used for the hubs of couplings ; the width of the arms from
| to full diameter ; the thickness half the width ; the thickness of the rim
from i to ~ the diameter ; the length of hub the
same as the width of face.
Fig. 257 represents a faced coupling pulley,
an expedient sometimes adopted when a joint oc-
curs where a pulley is also required, the two are
then combined ; the pulley is cast in halves two plate pulleys, with plates
at the side instead of central, faced and bolted together.
154
DB AWING OF MACHINERY.
TTooden pulleys are commonly called drums ; these are now but sel-
dom used except for pulleys of very wide face. Fig. 258 represents one
form of construction in elevation and longitudinal section. It consists of
Fig. 258.
two cast iron pulleys A A, or spiders, with narrow rims ; they are keyed
on to the shaft at the required distance from each other, and plank or lag-
ging is bolted on the rims to form the face of the drum ; the heads of the
bolts are sunk beneath the surface of the lagging, and the face is turned.
Fig. 259 represents a wooden pulley which may be termed a wooden
plate pulley. The plate consists of sectors of inch boards firmly glued and
nailed together, the joints of the boards being always broken. The face
is then formed in a similar way, by nailing and gluing arcs of board one
to another to the required width of face ; these last should be of clear
stuff. The whole is retained on the shaft by an iron hub, cast with a plate
on one side, and another separate plate sliding on to the hub ; the hub is
placed in the centre of the pulley, the two plates are brought in contact
Fig. 259.
Fig. 260.
with the sides of the pulley, and bolted through ; the face of the pulley is
now turned in the lathe. A similar arrangement of hub is used for the
hanging of grindstones.
Cone pulleys arc used to change the speed of the driven shaft. Fig.
260 represents a cone pulley with its hangers ; on the machine there is a
DRAWING OF MACHINERY.
155
similar set, but with ends reversed ; that is, the large end of the hanging
or driving-pulley connects with the small end of the pulley on the machine.
At this time the maximum of velocity is attained on the driven shaft ; but
if the belt is at the opposite end, small pulley on to a large one, the speed
is the minimum, the speed of the shafts being in the inverse ratio of the
diameters of their pulleys. By this arrangement speed may be varied
within any required limit. It is not necessary that the two pulleys should
be counterparts of each other, but only that such proportions should be
preserved, that the belt may be tight on whatever set it is placed.
The wddth of the face of the pulley depends upon the width of the belt
necessary to transmit the power ; it should exceed by about half an inch
on each side the width of the belt for the ordinary sizes. To determine the
width of the belt, determine first as near as possible the power required to
be transmitted. The strain on the belt is determined by dividing the
power to be transmitted by the velocity ; thus, if a belt moving at a velo-
city of 1500 feet per inin. be required to transmit 5 horse power ; that is,
33000 x 5 = 165000 Ibs. ft. ; then = 110 Ibs., the strain on the belt
JL)Ou
to convey the power. In addition to this strain, it must be remarked, that
the belt is stretched on the pulleys, so that it does not slip while con-
veying the power. The strain given above may be considered approxi-
mately as the difference of tension between the two sides. Morin gives the
folio wins: Table to determine the strain on each side of the belt.
Portion of the
embraced by the
belt.
VALUE OF K.
New belts on
wooden drums.
Ordinnr
On wooden drums.
y belts
On iron pulleys.
Wet belts on iron
pulleys.
0.20
1.S7
l.SO
1.42
1.61
0.30
2.57
2.43
1.C9
2.05
0.40
3.51
3.2G
2.02
2.6;)
0.50
4.S1
4.8S
2.41
3.30
0.60
6.59
5.S8
2.87
4.19
O.TO
9.00
T.90
8.43
5.32
O.SO
12.34
10.62
4. 09
6.75
0.90
10.90
14.27
4.S7
8.57
1.00
1
23.14
19.16
5.S1
10.S9
Application of the table. Find in the table the value of K according
to the given circumstances ; from this number subtract unit or one, and
divide the strain on the belt to convey the power by this remainder, and
the quotient will be the minimum tension or that on the slack side. Add
156 DRAWING OF MACHINERY.
to this quotient 10 per cent, for friction clue to shafting, or other causes.
The tension on the leading or tight belt will be the above product added
to the strain, as given by the power required to be conveyed.
Applying this to the example above of a strain on the belt of 110 Ibs.
with the ordinary belt embracing 1 or 0.50 of the circumference, the value
of K in the table is 2,41 ; subtract 1., = 1.41 ; 110 divided by 1.41 = 78
Ibs.;
78 + 10 per cent, or 78 + 7.8 = 85.8 x ;the tension on the slack belt.
85.8 + 110 = 195.8, the tension on the tight belt.
Good belting of an ordinary thickness of T 3 g of an inch should sustain
a strain of 50 Ibs. per inch of width without risk, and without serious wear
for a considerable time. Therefore, in the example above, the belt mov-
ing at- a velocity of 1500 feet per minute, required to transmit a power of
195 8
five horses, should be ' , or very nearly 4 inches in width.
For the engaging and disengaging of a machine, that is, for putting into
or out of motion, the arrangement of a fast-and-loose pulley is adopted as
simpler and better than the clutches before given. It consists merely of two
pulleys in juxtaposition on the same axis, one fast, the other loose, so that
the belt which transmits the motion may be shifted from one to the other.
The face of the driving pulley, that is, the one on the driving shaft, ought
to be equal in width to that of both the fast and loose pulleys. By making
the face of the pulleys slightly convex, the belt is prevented from slipping
off, as the tendency of a belt is always to the larger diameter.
When the belt is shifted, whilst in motion, to a new position on a drum
or pulley, or from fast to loose pulley, or vice versa, the lateral pressure
must be applied on the advancing side of the .belt, on the side on which
the belt is approaching the pulley, and not on the side on which it is run-
ning off. It is only necessary that a belt, to maintain its position, should
have its advancing side in the plane of rotation of that section of the pulley
on which it is required to remain, without regard to the retiring side. On
this principle, motion may be conveyed by belts to shafts oblique to each
other. Let A and B (fig. 261) be two shafts at right angles to each other,
A vertical, B horizontal, so that the
line run perpendicular to the direction
of one axis is also perpendicular to the
other, and let it be required to connect
F , B 261 them by pulleys and a belt, that their
direction of motion may be as shown by the arrows and their velocities
DRAWING OF MACHINERY. 157
as 3 of A to 2 of B. On A describe the circumference of the pulley pro-
posed on that shaft ; to this circumference draw a tangent a Z> parallel to
m n, this line will be the projection of the edge of the belt as it leaves A,
and the centre of the belt as it approaches B ; consequently, lay off the
pulley b on each side of this line, and of a diameter proportional to the
velocity required. To fix the position of the pulley on A, let fig. 262 be
another view taken at right angles to fig. 261, and let the axis B have the
direction of motion indicated by the arrow, then the circle of the pulley
being described, and a tangent a' V drawn to it perpendicular to the
axis B as before determined, the position of the pulley on the shaft A is
likewise
Fig. 262.
The positions of the two pulleys are thus fixed in such a way, that the
belt is always delivered by the pulley it is receding from, into the plane of
rotation of the pulley towards which it is approaching. If the motion be
reversed, the belt will run off ; thus (fig. 1 263), if the motion of the shaft A
is reversed, the pulley B must be placed in the position shown by the
dotted lines.
It is not an essential condition that the shafts should be at right angles
to each other to have motion transferred by a belt. They may be placed
at any angle to each other, provided the shafts lie in parallel planes, so
that the perpendicular drawn to one axis is perpendicular to the other. If
otherwise, recourse must be had to guide-pulleys, which should be con-
siderably convex on their face.
Geering. The term geering, in general sense, is applied to all arrange-
ments for the transmission of power ; it is also used in a particular sense,
as toothed geering.
Toothed geering may be divided into two great classes spur and level
wheels. In the former, the axes of the driving and driven wheels are
parallel to each other ; in the latter they may be situated at any angle : if
of equal size and at right angles, they are called mitre-geei's.
Spur wheels, strictly so called, consist of wheels of which the teeth are
disposed at the outer periphery of the wheel (PI. XVIIL), in direction of
radii from their centres.
158 DRAWING OF MACHINERY.
Internal geering, in which the teeth are disposed in the interior peri-
phery of the wheel, in direction of radii from their centres (plate XXVI.)
Hack geer and pinion are employed to convert a rotatory into a recti-
linear motion, or vice versa. In this arrangement the pinion is a spur-
wheel, acting on teeth placed along a straight bar (plate XXIV., fig. 1.)
Bevel-geering, strictly so called, consists of toothed wheels formed to
work together in different planes, their teeth being disposed at an angle to
the plane of their faces (plate XXII.)
Trundle-geer or wheel is constructed by inserting the extremities of a
certain number of cylindrical pieces, called staves^ into equi-distant holes
formed near the circumferences of two parallel plates. The trundle or lan-
tern is in mill-work made of wood, and is very useful when iron geers can-
not be easily got or repaired. The trundle may be used either with a spur
wheel to transmit motion to parallel shafts, or with face or crown wheels,
to transmit motion to shafts at right angles to each other. Face or crown
wheels are such as have their teeth perpendicular to the plane of their faces.
The sides of the teeth should be radialj the outer edges cornered, and in-
serted in a single plate or disc, instead of two as the trundle.
On the transmission of motion. The velocity of rotation of a driven
wheel depends on its relative diameter to, and the velocity of the driving
wheel with which, it is connected. Thus, if the diameter of the driven
wheel be one-half that of the driver, then the driven wheel must make two
revolutions for one of the driver. The driver is often called a leader, the
driven a follower. Hence, to obtain the diameters of two wheels having
the distances apart : Divide the distances between their centres into parts,
inversely proportional to the number of revolutions which the wheels are
to make in the same unit of time. Tims, let A and B (fig. 264) be 'the
given centres, the ratio of the.ir velocities being respectively two and three ;
if the line joining the centres A and B be divided into 2 + 3 = 5 equal
parts, that is, into as many equal parts as there are units in the terms of
the given ratio, the radius of the wlieel upon A will contain three of these
parts, and the radius of the pinion on B will contain the remaining two
parts.
In determining the size of a pair of bevels, we are not, however, limited
to any particular diameters as when the axes are parallel ; the wheels may
be made of any convenient sizes, and the teeth consequently of any breadth,
according to the stress they are intended to bear. The question is the mode
of determining the relative sizes of the pair ; and this resolves itself into a
division of the angle included between the two axes inversely as the ratio of
their angular velocities. Let B and C (fig. 265) be the position of the two
DRAWING OF MACHINEBY.
159
given axes, and let them be prolonged till they meet in a point A. Further,
let it be required that C make seven revolutions while. B makes four. From
N,
Fig. 2C4.
Fig. 2C5.
any points D and E in the lines A B, A C, and perpendicular to them,
draw D d and E e of lengths (from a scale of equal parts) inversely as the
number of revolutions which the axes are severally required to make in
the same unit of time. Thus, the angular velocity of axis B being 4 (fig.
265), and that of the axis C being 7, the line D d must be dra\vn = 7, and
the line E e = 4. Then through d and e parallel with the axes A B and
A C draw d c and e c till they meet in c. A straight line drawn from A
through c will then, make the required division of the angle BAG, and
define the line of contact of the two cones, by means of which the two roll-
ing frusta may be projected at any convenient distance from A.
Otherwise, haA'ing determined the relative perimeters, diameters, or
radii, of the pair, then the lines D d and E e are to each other directly as
these quantities.
The point c may also be found more directly thus : From A towards C
in the axis A C, set off from a scale as many equal parts (A/*) as there are
units in the number (7) expressing the velocity of that axis ; from the point
f draw f c parallel to A B, and set off from the same scale as many parts
(f c} as there are units in the number (4) expressing the velocity of the
axis A B ; then a line drawn from A through c, as before, w r ill divide the
angle as required.
The case in which the axes are neither parallel nor intersecting admits
of solution by means of a pair of bevels upon an intermediate axis, so
situated as to meet the others in any convenient points.
160 DRAWDfG OF MACHINERY.
When the contiguity of the shafts is such as to permit of their being
connected by a single pair, skewed bevels are sometimes employed.
When the axes are at right angles to each other, and do not intersect,
the wheel and screw may be employed to connect them. The velocity of
motion is in this arrangement immediately deduced from that of the screw,
its number of threads, and the number of teeth in its geering wheel. Tims,
if it be required to transmit the motion of one shaft to another contiguous,
and at right angles to it the angular motions being as 20 to 1 ; then, if
the screw be a single-threaded one, the wheel must have 20 teeth ; but if
double-threaded, the number of teeth will be increased to 40, for 2 teeth
will be passed at every revolution. If the velocities be as 2 to 1, the con-
dition is, that the screw have half as many threads upon its barrel as there
are teeth on the wheel ; and if 1 to 1, the wheel and screw lose their dis-
tinctive characters : both become many-threaded screws under the form
of wheels. Wheels of this sort may often be applied with peculiar advan-
tage, especially in light geering ; and when so applied, it is not essentially
necessary that the axes be at right angles to each other any more than it
is in bevel-geer.
If the screw have few threads compared with the number of teeth of
the wheel, it must always assume the position of driver on account of
the obliquity of the thread to the axis ; and in this respect its action is
analogous to that of a travelling rack, moving endwise one tooth, whilst
the screw makes one revolution on its axis.
On the pitch of wlieels. The primary object aimed at in the construc-
tion of toothed-geer is the uniform transmission of the power, supposing
that to be constant and equal. This implies that the one wheel ought to
conduct the other, as if they simply touched in the plane, passing through
both their centres. This plane is denoted by the line A B in fig. 264.
When this line which is usually denominated the line of centres is
divided into two parts, A c and B c, proportional to the number of teeth
formed upon the perimeters of the pinion and wheel, these two parts
are proportioned or primitive radii of the pair ; and a circle being de-
scribed from each centre passing through the common point c, limits what
is called the pitch line or circle; that is, a circle described from the centre
A, and another from the centre B, through the same point, are called, the
first, \\\Q pitch circle or pitch line of the pinion, and the other of the wheel.
They are also sometimes called the primitive and proportional circles. If
the pitch circle be divided into as many equal parts as there are teeth
to be given to the wheel, the length of one of these parts is termed the
pitch of the teeth. One of these arcs comprehends a complete tooth and
DRAWING OF MACHINERY.
161
space, meaning by space the hollow opening between two contiguous
teeth. In bevel and conical wheels, the pitch circle is the base of the
frustum.
fiules. I. To find the pitch of the teeth of a wheel, the diameter and
number of teeth being given, divide the diameter D (in inches) by the
number of teeth N, and multiply the quo-
tient by 3.1416 : the product is the pitch
in inches or parts of an inch.
II. To find the diameter of a wheel
the number of teeth and pitch being
given, divide the pitch by 3.1416, and
multiply the quotient by the number of
teeth.
III. To find the number of teeth, the
diameter and pitch being given, divide
3.1416 by the pitch, and multiply the re-
sult by the diameter in inches.
In ordinary geering, the pitches most
commonly in use range from 1 inch to 4
inches, increasing up to two inches by
eighths, and beyond by fourths of an
inch. Below inch the pitches decrease
by eighths down to { inch.
The rules given above may be greatly
simplified by the use of the annexed
table, which will be found very conven-
ient when the diameter D is to be deter-
mined, the pitch P and number of teeth
N" being given ; and conversely, when
the diameter and pitch are given, to find
the number of teeth.
Ex. 1. Given a wheel of 88 teeth,
1
p
D= x N.
TT
7T
N = x D
P
Pitch in inches
?
a
f
inch.
the ilium, in inches,
multiply the number
of teeth by the tabu-
lar number answer-
ing to the given
pitch.
the number of teeth,
multiply the given
diameter in inches
by the tabular num-
T answering to the
given pitch.
Values of P.
Values of
Values of
p
e
6
1.9095
.5236
e
5
1.5915
,6233
_
H
1.4270
.6931
4
1.2732
.7854
8i
1.1141
.8976
t
3
.9547
1.0472
4
2}
.8754
1.1333
D-
Bf
.7953
1.2566
7
2*
.7135
1.3963
a
2
.6366
1.5708
e
is
.5937
1.6755
u
.5570
1,7952
r
if
.5141
1.9264
f
H
.4774
2.0944
I
If
.4377
2.2S4S
11
.8979
2.5132
I*
.3563
2.7926
-
1
.3133
3.1416
1
i
.2785
S.5904
1
i
.2337
4.1 8SS
i
.1939
5.0266
i
.1592
6.2332
f.
.1194
8.3776
i
.0796
12.5664
2i-inch pitch, to find the diameter of the pitch circle. Here the tabular
number in the second column answering to the given pitch is .7958, which
multiplied by 88 gives 70.03 for the diameter required.
2. Given a wheel 33 inches diameter, If -inch pitch, to find the number
of teeth. The corresponding factor is 1.7952, which multiplied by 33,
gives 59.242 for the number of teeth, that is, 59| teeth nearly. Now 59
would here be the nearest whole number, but as a wheel of 60 teeth may
11
162
DRAWING OF MACHESTERY.
be preferred for convenience of calculation of speeds, we may adopt that
number, and find the diameter corresponding. The factor in the second
column answering to If pitch is .557, and this multiplied by 60 gives 33.4
inches as the diameter which the wheel ought to have.
Another mode of sizing wheels in. relation to their pitches, diameters,
and number of teeth, is adopted in some machine shops, by dividing the
diameter of the pitch circle into as many equal parts as there are teeth to
be given to the wheel. To illustrate this by an arithmetical example, let
it be assumed that a wheel of 20 inches diameter is required to have 40
teeth ; then the diametral pitch,
20
777
40
that is, the diameter being divided into equal parts corresponding in num-
ber to the number of teeth in the circumference of the wheel, the length
of each. of these parts is | an inch, consequently m 2 ; and according to
the phraseology of the workshop, the wheel is said to be one of two pitch.
In this mode of sizing wheels, a few determined values are given to
m, as 20, 16, 14, 12, 10, 9, 8, T, 6, 5, 4, 3, 2, 1, which includes a variety
of pitches from } inch up to 3 inches, according to the following table,
which shows the value of the circular pitches corresponding to the assigned
values of m.
Values of m.
1
2
3
4
5
1
7
8
9
10
12
14
16 j 20
Corresponding circular pitch in deci- 1
mals of an inch, J
3.142
1.571
1.047
.785
.025
524
.449
m
.849
314
2C2
.224
196 .157
As it is convenient to express all the dimensions in terms of the same
unit, and the pitch being an appropriate quantity, it is nearly universally
Fig. 266.
adopted as the term of comparison. The following are the proportions
adopted by different workshops, some preferring one and some the other
(fig. 266).
DRAWING OF MACHINERY. 163
A C = Pitch of teeth, = 1 pitch, or = 15 parts.
ac Depth to pitch line, P P, = T 3 r " " = o* "
A.a + 'ac = Working depth of tooth, = T V . " " = 11 "
Cc A a = Bottom clearance, = TO " " = 1 "
C a = Whole depth to root, ' = T V u " = 12 "
C b = Tliickness of tooth, = T \ " = 7 "
A 5 = Width of space, = -A " " = 8 "
In practice, these proportions are often laid off in lines for the conveni-
ence of the workmen in the pattern shop, so that for any given pitch the
other dimensions may at once be determined by means of the compasses.
In figs. 267 and 268, two diagrams of that sort are given. Fig. 267 con-
tains the proportions last enumerated, in which the pitch is supposed to be
divided into 15 equal parts ; and fig. 268 is constructed nearly according
to the proportions first given, but embraces the recognised principle, that
the relative amount of clearance ought to vary inversely as the pitch,
wheels of small pitch requiring more clearance relatively than those in
which the pitch is greater.
The construction of these scales is very simple. Thus, in fig. 267, let
A B be divided into 15 equal parts, and draw B C perpendicular to it ; and
again divide B C into a determinate number of parts from B, actual meas-
ures of the pitches for which the scale is intended to be used ; that is, B a
= i inch ; B 5 = 1 inch ; B c = 2 inches, and so on, and join a and A,
5 and A, c and A, and so on. To complete the scale, draw 15 parallels to
B C from the points numbered in the line A B, and also the two parallels
T and II equidistant from the parallels on each side of them.
The scale is thus ready for use. To get from it the several proportions
for a given pitch, say of three inches = B J, let the compasses be extended
from the intersection of the parallel marked T, with the line A B, to the
point where it intersects the line Ad; this will be the part of the tooth
from the pitch line to the point, and equivalent to 5| parts of the pitch,
(viz., of B d) ; similarly, the compasses being extended from the intersec-
tion of the parallel U with the line A B to its point of intersection of the
line A d, will give the part of the length of the tooth from the pitch line
to the root, and equivalent to 6 parts of the pitch. For the whole 'length
of the tooth (if wanted in one measurement), set the compasses to the point
where the parallel marked 12 meets the line A B, and extend -to its point
of intersection of the line A d at *, the length is 12 parts of the pitch B d;
the working depth is in like manner found from the parallel marked 11,
the thickness from that marked 7, and the width of space from that
marked 8.
164
DEAWING OF MACHINERY.
The proportions for any other given pitch comprised in the scale are
found in precisely the same way ; and if the scale -be well constructed, they
may be measured off with the utmost accuracy and readiness. To save
confusion, it is, however, better in practice to insert in the diagram only
Proportion scales for geering.
those parallels, namely, T, U, 12, 11, 8, 7, which are required ; the others
are not requisite, and by inattention may lead to error.
The description of the scale as here given supposes that the lateral
DRAWING OF MACHINERY. 165
clearance is constantly l-15th of the pitcli ; but as it is commonly desirable
that this should vary slightly with the pitch, relatively increasing as the
pitch decreases, two other lines, in n and p q, have been introduced into
the scale, to enable such modification to be adopted should it be required.
These lines are drawn at such angles as to give a clearance at 6 inches
pitcli of l-18th, which is increased at |-inch pitch to l-10th. From these
lines the thickness and space are to be taken, instead of using the lines
marked T and 8, setting the compasses in the points of intersection with
the pitch lines, and extending perpendicularly to the line A B ; in other
words, the shortest distance from the point of intersection with the pitch
line to the line A B is the required measure of the space when the
line p q is taken, and of the thickness of tooth when the line m n is
taken.
Fig. 268 is more complete than the one described ; the principle of its
construction is in cifect the same, but its use is more extended, the diam-
eter of the wheel being found from it simultaneously with the length and
thickness of tooth, width of space, and clearances. The scale is adapted to
wheels of all the pitches, from \ inch up to 3 inches. The mode of con-
struction is this : having drawn the line A D of any convenient length,
raise the perpendicular C B to it, also of any convenient length. On the
line A D lay off the greatest pitcli of the scale from C to A ; then from
C towards D lay off seven times the pitch once or twice, according to the
sizes of wheels to which the scale is intended to be applied. In the scale
given, double of seA r en times the pitch is laid off, namely, 42 inches ; then
each of these great divisions being subdivided into 11 equal parts, one of
these parts will be equal to four teeth upon the radius of the wheel, so that
the whole line C D will be divided into 88 radial pitches. Next on the
line C B set off the pitches which may be required in the scale, and^
through these points draw the 24 parallels to A D, terminating in the lines
A B and D B. Then each parallel measured from the line B C to its
point of termination in B D is the radius of a wheel of 88 teeth of the par-
ticular pitch marked against it on the line A B. They also express the
radii of wheels having less than 88 teeth when measured only to the
corresponding point in the line joining B, and the divisional on C D,
against which the number of teeth is marked. Thus, the radius of a
wheel of 52 teeth and If-inch pitch is r s = 15 7-16th inches very nearly.
(The true answer by the table, page 161, is 30.8724 -f- 2 = 15.4362
inches).
The scale may also be used when the number of teeth exceeds 88 ; for
example, to find the radius of a wheel having 100 teeth. Thus, having
166 DRAWING OF MACHINERY.
found the radius answering to 88 teeth, upon the same parallel take off the
measure answering to the difference 100 88 = 12 teeth ; and the two
measures together will be the radius required.
To adapt the scale to odd numbers of teeth, the first division on the
right of C is divided into single radial pitches, so that the radius of any
wheel may be measured off without having recourse to calculation of any
kind. Tims, for example, if the wheel is intended to contain 50 teeth, the
parallel, answering to the particular pitch, comprehended between 52 B
and 2 B, will give the radius required, that is, a radius answering to
52 2 = 50 teeth ; and any other number of teeth, when not marked
against the base, may be found in the same way, observing that it is more
convenient to subtract than to add in this use of the scale.
For the proportions of the te^th, set off C a = h ( '-tenths of the pitch,
then will A a = 2>-tenths of the pitch, which corresponds to the depth from
the point of the tooth to the pitch line. Again, set off C 5 = 1 -fifteenth*
of the 3-inch pitch, and ^-elevenths on the parallel against the 1-inch pitch ;
this will be the thickness of the tooth, allowing from a fifteenth for clear-
ance on the largest pitch to a tenth on those from f-inch and under ; and
A 5 will be the width of space, including the clearance. Lines being
drawn from those points to B complete the diagram, which will be found
to contain all the proportions enumerated in the preceding table.
To use the scale, lay off the addendum of the tooth ; that is, the length
beyond the pitch line, equal to A a = T \ pitch, and the same length
marked off within the pitch line will give the whole working depth of the
tooth, namely, 6-10ths pitch. Then with the measure C a = T \ pitch in
the compasses, mark off the whole length of the tooth, and this will allow
l-10th at bottom for clearance. Again, set off the thickness of tooth =
C 5, and the space = A 5, which will contain the clearance for the par-
ticular pitch, varying from l-15th to fully l-10th on the small pitches. It
is hardly necessary to observe, that these measurements must be taken
upon the parallel corresponding to the particular pitch under considera-
tion at the time.
The amount of bottom clearance is here presumed to be uniformly
l-10th of the pitch ; but if it be thought advisable to make this vary, as in
the case of the lateral clearance, it will then be necessary to insert a third
line c B in the scale, and so related to a B that the space a c shall be
throughout equal to the depth of tooth from the pitch circle to the root,
and giving any bottom clearance that may be desired.
In relation to the strength of wheels, M. Morin gives it as a rule, that
when the velocity of the pitch circle does not exceed five feet per second,
DRAWING OF MACHINERY.
167
the breadth of the tooth measured parallel to the axis ought to be equal
to four times the thickness ; but when the velocity is higher, the breadth
ought to be equal to five thicknesses, the teeth being constantly greased.
If the teeth be constantly wet, he recommends the breadth to be made
equal to six thicknesses at all velocities.
With respect to the thickness of the tooth, it is plain that it must be
dependent on the pressure which the tooth is required to sustain, and upon
the nature of the material of which it is formed. We subjoin the follow-
ing table for calculating the strength of cast iron wheels.
For teeth of wood add 50 per cent, to
the thickness as given by Table.
When cast iron and wooden teeth
work together, their action upon each
other tends, in consequence of the elas-
ticity of the wood, to maintain a more
uniform distribution of the strain, and
being at first commonly more accurately
dressed, to prevent abrasion of wood by
the iron, they work with much less fric-
tion, are less liable to shocks, and nearly
exempt from accident by hard particles
coming between the teeth.
The best practice, when a mortise and
iron wheel are to work together, is to
make both of the same pitch, and in the first instance, of the same thick-
ness of tooth the pitch being of course calculated for the wooden teeth ;
afterwards, to dress down the teeth of the iron wheel by the chipping-iwl,
or the geer-cutting machine and file, to the exact form and thickness, as
given by the Table ; that is, to a thickness in relation to the thickness of
the wooden teeth, which shall have the ratio of 25 to 38. The following
table may be useful in determining the relation of the dimensions of the
teeth of wheels of the given pitches, and the power which they are capable
of transmitting safely at the various speeds named.
To find the power which a wheel is capable of transmitting for other
velocities than those in the table : For 6 feet per second, double the result
given for 3 feet ; for 8 feet, double the result at 4 feet, and so on ; and for
lower velocities than those given, divide the tabular number by the ratio
which they bear to those enumerated. Thus, for 2^ feet velocity, take
half the result at 5 feet, and so of other velocities.
Stress in Ibs.
5
Thickness of
Actual pitches to which
circle.
lb>.
inches.
inches.
400
0.50
H to H
800
O.T1
1} " H
1200
O.ST
H " 2
1600
1.00
2 2J
2000
1.12
2 i 2 j
2400
1.22
2* " 2f
2SOO
1.32
2* " 2}
3200
1.41
2i " 3
3600
1.50
8* " 3}
4000
1.5S
8t " 8f
4400
1.66
Sf " 3}
4800
1.T3
8* " 34
5200
l.SO
8J " 3}
5600
1.8T
3} " 4
6000
1.94
4 " 41
168
DRAWING OF MACHINERY.
When a wheel and pinion, which differ very much in size, work
together, the teeth of the latter, on account of their unequal thickness, are
capable of sustaining much less pressure than the teeth of the wheel : they
are in effect, if not in fact, much reduced in thickness ; and in applying
Pilch.
Thickness
Length
Least
breadth of
Velocity of the wheel at the pitch circle.
of teeth.
of teeth.
teeth.
Three feet
Four feet
Fire feet
Seven feet
Eleven feet
per second.
per second.
per second.
per second.
per second.
Inches.
Inches.
Inches.
Inches.
H. P.
B. P.
H. P.
H P
B P
6
2.9
42
8.4
43.2
57.6
72.
100.2
158.4
5*
2.6
3.S5
7.7
3G.3
48.4
60.5
847
133.1
4
1.9
2.S
5.6
19.
25.5
32.
45.
70.5
3*
1.6
2.45
4.9
1475
19.5
24.5
3425
54.
3
1.4
2.1
4.2
11.
14.5
18.
25.
39.5
2*
1.2
1.T5
3.5
7.5
10.
12.5
17.5
27.5
2
0.95
1.4
2.8
4.75
6.25
8.
11.
17.25
u
O.S3
1.225
2.45
3.5
5.
6.25
8.5
13.5
4
o.7i
1.05
2.1
2.75
3.5
45
6.25
10.
1}
0.59
0.875
1.75
2.
2.5
3.125
42
6.8
l*
0.53
O.TS75
1.575
1.5
2.25
2.5
3.5
5.5
i
0.43
0.7
1.4
1.2
1.6
2.
2.S
4.4
1
0.41
0.6125
1.225
1.
1.4
1.75
2.5
3.S
i
0.36
0.525
1.05
.7
.9
1.125
1.5
2.5
1
0.33
0.4375
0.675
.5
.625
.75
1.
1.7
i
0.24
0.35
0.7
.3
.4
.5
.7
1.1
1
0.18
0.2625
0.525
.2
.25
.3
.4
.6
i
0.12
0.175
0.85
.075
.1
.125
.175
.275
rules to the calculation of the strength of wheels, the difference of size of
the pair ought not to be overlooked, unless, as is indeed very common in
practice, the deficiency of strength be made up to the pinion by a flange
cast on one or both sides of the rim, of the same depth as the teeth, and
binding these together like the staves of a trundle. In this case the
pinion is commonly the stronger wheel of the pair.
In the construction of wheels, the problem which presents itself relative
to the shape of the teeth is this, that the surfaces of mutual contact shall
be so formed, that the wheels shall be made to turn by the intervention of
the teeth, precisely as they would by the friction of their circumferences.
Fundamental principle. In order that two circles A and B (fig. 269)
may be made to revolve by the contact of the surfaces of the curves in m
and n n of their teeth precisely as they, would by the friction of their cir-
cumferences, it is necessary and sufficient that a line drawn from the point
of contact t of the teeth to the point of contact c of the circumferences
(pitch circles), should, in every position of the point , be perpendicular to
the surfaces of contact at that point ; that is, in the language of mathe-
maticians, that the straight line be a normal to both the curves m m and
DRAWING OF MACHINERY.
169
n n. The principle here announced exhibits a special application of one
particular property of that curve known to mathematicians as the epicy-
cloid (see page 76).
C\l/0s>
Of epicycloidal teeth. The simplest illustration of the action of epicy-
cloidal teeth is when they are employed to drive a trundle, as represented
in fig. 270. Let it be assumed that the staves of the trundle have no sen-
sible thickness ; that the distance of their centres apart, that is their pitch,
and also their distance from the centre of the trundle, that is their pitch
circle, are known. The pitch circles of the trundle and wheel being then
drawn from their respective centres B and A, set off the pitches upon these
circumferences, corresponding to the number of teeth in the wheel and
number of staves in the trundle ; let five pins a 5 c, &c.,' be fixed into the
pitch circle of the trundle to represent the staves, and let a series of epicy-
cloidal arcs be traced with a describing circle, equal in diameter to the
radius of the pitch circle of the trundle, and meeting in the points "klm n,
&c., alternately from right and left. If, now, motion be given to the wheel
in the direction of the arrow, then the curved face in r will press against
the pin 5, and move it in the same direction ; but as the motion continues,
the pin will slide upwards till it reaches m, when the tooth and pin will
quit contact. Before this happens, the next pin a will have come into
contact with the face a I of the next tooth, which repeating the same
action, will bring the succeeding pair into contact ; and so on continually.
To allow of the required thickness of staves, it is sufficient to diminish
the size of the teeth of the wheel by a quantity equal to the radius of the
staves (sometimes increased by a certain fraction of the pitch for clear-
170
DRAWING OF MACHINERY.
ance), by drawing within the primary epicycloids at the required distance
another series of curves parallel to these. In practice, a portion must be
cut from the points of the teeth, and also a space must be cut out within
the pitch circle of the driver, to allow the staves to pass ; but no particular
form is requisite, the condition to be attended to is simply to allow of suffi-
cient space for the staves to pass without contact.
The action of a wheel and trundle being understood, it is easy to com-
prehend that of the teeth of a pair of wheels of the ordinary construction.
Let A. and B of fig. 271 be respectively the centres ot a wheel and pinion
of which the teeth are intended to be
of the epicycloidal form, and A c and
B c their primitive radii. To lay off
the teeth of this pair, having deter-
mined the pitch and number of teeth
in the wheel and pinion, let the pitch
lines be divided into as many equal
parts, setting out from the point of
contact c, as there are teeth in them
respectively. Let the thickness of
the teeth be next set off, taking c a
for the thickness of a tooth of the
wheel, and c l> for that of a tooth of
the pinion. Upon the radii A c and
B c as diameters describe two circles,
having also their point of contact at
Fi s- 271 - c and their centres at X and Y. Xow
let the circle Y be made to roll upon the pitch line of the wheel, and a
point in its circumference at c will describe the epicycloidal arc c m, and
this curve determines the form of the point of the tooth of the wheel. In
the same way describe the epicycloidal arc c n, by making the circle X to
roll upon the pitch circle of the pinion, and this curve will determine the
form of the part of the tooth of the pinion beyond the pitch line.
The curve c m of the tooth of the wheel is constantly in contact with the
radius B c, and its point of contact is at the same time situated in the circum-
ference of the circle X ; the contact will therefore cease when the extrem-
ity m of the tooth becomes the point of contact ; and this occurs when the
point m has arrived at the circumference of Y. If, then, an arc of a circle be
described from the centre A with the radius A m, its pointy of intersection
with the circumference of Y is that at which the tooth ought to cease to act,
DRAWING- OF MACHINERY. 171
to secure uniformity of motion, and at the same instant that another new
tooth advances into geer. The determination of the point/ limits also the
useful length of the flank; for if from B as a centre, with radius B/, an
arc be described, the part cf of the radius c B is that which is in contact
with the tooth till it arrives at the position B/, and consequently this is
the useful length of the flank of the pinion. In the same manner it may
be determined, that the useful length of the flank of the tooth of the wheel
c a is the portion c <j.
To find the form of the portions of the teeth within their respective
pitch circles, it is usually considered enough that sufficient space for play
be allowed whilst the tooth remains strong enough for its work. As every
tooth moves between two flanks, and touches only one of them, the space
may be bounded literally by radial lines prolonged towards the centre of
the wheel. The bottom of the space being sufficiently removed from the
pitch circumference to allow the tops of the teeth of the pinion to pass
round without touching them, may be described by arcs of a circle drawn
from the centre. In practice, and especially when the teeth of the wheels
are small, it is not usually considered necessary to apply strictly the form
of the epicycloid to find curves of the teeth, but to define them approxi-
mately.
The forms of the teeth are also occasionally described by arcs, of which
the radii are equal to the pitch, and with the centres taken upon the pitch
lines. When the diameters of the wheels are not very unequal, and the
teeth not thick, in many cases the curves of the teeth are described by arcs
drawn from the centres of the adjacent teeth upon the pitch line. This
gives a radius equal to the pitch, plus half the thickness of a tooth. In
like manner, the sides of the teeth also are sometimes described by arcs
from the centres of the adjacent teeth, giving a radius equal to the pitch,
minus half the thickness of a tooth. This form of the tooth may be defec-
tive, in the case of a very small pinion having to transmit great pressure,
as the extremities of the teeth may be too much reduced. In this case,
the curves or faces of the teeth may be described with radii equal to three-
fourths of the pitch ; and if this be not sufficient curvature, radii equal to
some smaller fraction of the pitch may be used. When, on the contrary,
the pitch is large, and the pressure comparatively small, the teeth may be
too short ; this will be remedied by employing arcs, of which the radius is
one and a half or twice the pitch. In practice, the ordinary mode in
which epicycloidal teeth are set out is by the mechanical method of form-
ing the pattern teeth by templets (figs. 272, 273, 274, 275).
Having determined the pitch of the teeth and the radius of the pitch
172
DRAWING OF MACHINERY.
circle, describe on a thin slip of wood say 1 in. thick an arc of the pitch
circle, and on another similar slip an arc of a circle equal to the diameter
Fig. 272.
of the wheel to the points of the teeth. The slips are then cut to the cir-
cumferences of the circular arcs described upon them. These two pieces
so prepared are fastened together by screws, as in the fig. 272 ; the piece
Fig. 273.
A, whose edge s f is an arc of the pitch circle, is fixed upon B, whose
edge is an arc of the extreme circumference of the wheel, the space * s be-
tween those edges being in breadth equal to the length of the teeth from
Fig. 274.
Fig. 275.
the pitch circle to the points. This done, a like templet is prepared for
\\\Q pinion (fig. 273).
The pair of templets being thus prepared, two tracing points m m are
DRAWING OF MACHINERY. 173
inserted obliquely, and from behind into the piece D of the pinion tem-
plet. One of these points passes out at the edge of the piece C, and the
other at the edge of the piece D, and the templets are then placed upon
each other, as shown in figs. 274 and 275 annexed, so that the circumfer-
ence of the piece C, that is the pitch circumference of the pinion, shall
meet the circumference of the piece A, that is the pitch circumference of
the wheel. If in this position the templets be made to roll upon each
other through a certain arc, pressing them at the same time slightly
together, the tracing points will mark two epicycloidal curves upon the
pieces A and B, as v and r; of these two curves, that marked -y, which is
traced on the face of the piece A, will be the curve of the lower portion or
flank, and that marked r will be the curve of the upper portion, or face of
a tooth of the wheel. If now the thickness of the tooth be marked off on
the edge of the piece C, that is, on the pitch circle, and the corresponding
tracing point be made to coincide with that point, the curves of the oppo-
site side of the tooth will be formed by making the templets roll together
in the contrary direction. A complete outline of a tooth of the wheel is
thus described, to which a pattern tooth may be cut and used to shape the
teeth by in making the wheel pattern. Instead of forming pattern teeth,
many prefer to lay off the teeth by circular arcs coinciding approximately
with the epicycloidal arcs found by the templets.
The preceding mode of obtaining the curves of the teeth of a pair of
wheels is faulty in as far as it gives a form of teeth smaller at the root than
at the pitch circles, and also in the circumstance that a pair of wheels,
formed in the manner described, will only work correctly with each other,
and not with wheels of any other numbers, although of the same pitch.
To obviate this, it is necessary to employ a modification of the ordinary
practice.
Let a thin slip of wood be provided, and let an arc of the pitch circle
be struck upon it ; divide the slip into two portions through the line of
this arc with a fine saw ; one part, A, will have a concave, and the other,
B, a corresponding convex circular edge. Describe an arc dd of the pitch
circle upon a second board C D, upon which the pattern tooth is to be
drawn. Fix the piece B upon the board, so that its circular edge may
accurately coincide with the circumference of the arc d d. A portion of
a circular plate D is next provided, of the same radius which it is proposed
to give to the generating circle : this plate has a fine tracing point nip
inserted into it, and projecting slightly from its under surface, and accu-
rately coinciding with its circumference. Having set off the ^hickness of
the tooth a c upon the pitch circle d d, so that twice this width increased
174
DRAWING OF MACHINEET.
Fig. 276.
by the clearance which it is desired the teeth should have, may be equal
to the pitch, the generating circle D is made to roll upon the convex edge
of B ; meantime the point at p will
trace upon the board the curve of
the faces of the tooth, having caused
the point to coincide successively
with the two points a and c, and
the circle to roll from right to left,
and vice versa.
Let the piece B be now re-
moved, and the piece A applied
and fixed, so that its concave
edge may accurately coincide with
the circular arc d d; then, with
the same circular plate D pressed
against the concave edge of A, and
made to roll upon it, the point at p, which is made as before to coincide
successively with the points a and c, will trace upon the surface of the
board C D the two hypocycloidal arcs a 5, c 5, which form the flanks of
the tooth. The complete tooth, thus formed, will work correctly with the
teeth similarly described iipon any other wheel, provided the pitch of the
teeth be the same, and the same generating circle D be used to strike the
curves upon the two wheels. In this manner the general forms of the
teeth of the pair are determined ; and it only remains to cut them off at.
such lengths that they shall come into contact in the act of passing through
the line of centres.
It has already been observed, that having found the epicycloidal curves
of the teeth by means of the templets, a common method of proceeding is
to find, by trial, a centre and small radius, by which the arc of a circle
can be described that will coincide nearly with the templet-traced curve.
A more commodious and certain method of determining the centre and
radius of the approximate arc has been supplied by Prof. "Willis in the
construction of his Odontograpli, manufactured by Messrs. Holtrapfel &
Co., London, for description of which see Appletojis* Dictionary of Me-
chanics, pp. 819, 820.
Involute teeth. Involute teeth have the disadvantage of being, when
in contact, too much inclined to the radius, by which an undue pressure is
transferred to their axes. Their mutual friction is thereby little affected,
but that of the axes is increased, and their journals are more speedily
worn. But they have at the same time the advantage of working with
DRAWING OF MACHINERY.
175
more accuracy under derangement and incorrectness of fitting, and any
pairs of them will work truly together in sets within certain limits, how-
ever different in diameters, the pitch being the same.
To describe this curve for the teeth of a pair, of which the radii of the
pitch circles and pitch of the tooth are determined, we may employ the
mode illustrated by fig. 277. Let A and B be the centres of the pair, and
e 1) be their pitch lines ; join A and B by a right line passing through c;
from this last point draw c d, c d, perpendicular to the radials B d, A d,
and cutting them in d and d; this line d d is then a common normal to
the teeth in contact, and the perpendiculars A d, B d, are the radii of the
involute circles which form the acting faces of the teeth.
A.'
Fig. 277.
u
Fig. 278.
Having thus elucidated the principles of the operation of toothed geer-
ing, and the form and proportions to be given to their parts, we proceed
now to give complete drawings of different toothed wheels ; premising,
that except in case of working drawings to full size, arcs of circles are
almost invariably used by the draughtsman to describe the forms of the
teeth ; and that in practice, too little attention is paid to the construction
of epicycloidal and involute teeth. In many drawings details are unneces-
sary, and spur geers are represented by pulleys and bevel geers as in fig.
278, the pitch being written in in figures.
PROJECTIONS OF A SPUR WHEEL.
PL XVII. To draw side elevation (fig. 1), an edge view (fig. 2), and a
vertical section (fig. 3), of a spur wheel with 34 teeth, and a pitch of two
inches :
Determine the radius of the pitch circle from table, page 161 ; against
2 in. we find in the second column .6366 ; .6366 X 54 = 34.38, the diam-
eter. Draw the central line A C B and the perpendicular D E ; on C as a
centre, with a radius 17.19, describe the pitch circle, and divide it into 54
176 DRAWING OF MACHINERY.
equal parts. To' effect this division, without fraying by repeated trials
that part of the paper on which the teeth are to be represented, describe
from the same centre C', with any convenient radius, a circle a I c d; with
the same radius divide its circumference into six equal parts, and sub-
divide each sixth into nine equal parts, and draw radii to the centre C' ;
these radii will cut the pitch circle at the required number of points.
Divide the pitch (2 in.) into 15 equal parts ; mark off beyond the pitch
circle a distance equal to 5i of these parts, and within it a distance equal
to 6 parts (see page 163), and from the centre C describe circles passing
through these points ; these circles are projections of the cylinders bound-
ing the points of the teeth and the roots of the spaces respectively.
In forming the outlines of the teeth, the radii which, by their intersec-
tions with the pitch circle, divide it into the required number of parts,
may be taken as the centre lines of each tooth. The thickness of the tooth,
measured on the pitch circle, is T 7 j of the pitch, and the width of the
space is equal to T 8 j. These distances being set off, take in the com-
passes the length of the pitch, and from the centre y describe a circular
arc h ij and from the centre^', with the same radius, describe another arc
Ti If, touching the former ; these arcs being terminated at the circles bound-
ing the points of the teeth and the bottoms of the spaces respectively, form
the curve of one side of a tooth. The other side is formed in a similar
manner, by drawing from the centre I the arc in n, and from the centre p
the arc m 0, and so on for all the rest of the teeth.
The teeth having been thus completed, we proceed to the delineation
of the rim, arms, and eye of the wheel. The thickness of the rim is usu-
ally made equal to that of the teeth, namely, y 7 ^ of the pitch, which dis-
tance is accordingly set off on a radius within the circle of the bottoms of
the spaces, and a circle is described from the centre C through the point
q thus obtained. AVithin the rim, a strengthening feather q r, in depth
about of the thickness of the rim, is generally formed, as shown in
the plate. The eye, or central aperture for the reception of the shaft, is
then drawn to the specified diameter, as also the circle representing the
thickness of metal round the eye, which is usually made equal to the pitch
of the wheel.
To draw the arms, from the centre C, with the radius C u equal to the
pitch, describe a circle ; draw all the radii, as C L, which are to form the
centre lines of the arms, and set off the distance L v, equal to f pitch, on
each side of these radii at the inner circumference of the rim, and through
all the points thus obtained draw tangents to the circle passing through
u. The contiguous arms are rounded off into each other by arcs of circles,
DRAWING OF MACHINERY. 177
whoso centres are obtained by the following construction : Taking, for
example, the arc M P Q, it is obvious that its centre is situated in the
straight line E which divides equally the interval between two contigu-
ous arms. Having fixed the point P (which should be at the same distance
from t as the breadth of the feather at the back of the rim) draw through
it a perpendicular R P to the line C E ; the question now becomes simply
a geometrical problem, to draw a circle touching the three straight lines
M X, P R, and S Q. Divide the angle P R M into two equal parts by
the straight line R O, which cuts C E in the point O, the centre of the
circle required ; its radius is the line O M perpendicular to M 1ST. If, now,
a cirple be drawn from the centre C, Avith the radius C O, its intersection
with the radii bisecting all the intervals between the arms will give the
remaining centres, such as O', of the arcs required ; and the circle pass-
ing similarly through M, marks all the points of contact M Q M', &c. To
draw the small arcs terminating the extremities of the arms, set off upon
the line C E, within the point r, the required radius of the arcs, and from
the centre C with a radius C w describe a circle ; the distance r w being
then transferred to the extremities of the arms at the points where they
are cut by the circle, as at S a?, will give the centres of the arcs required.
Draw the central web of the arm by lines parallel to their radii, making
the thickness about f inch for wheel of about this size.
Having thus completed elevation, the construction of the edge view
and vertical section becomes comparatively simple. Draw the perpendicu-
lars F G and II I (figs. 2, 3) as central lines in the representations ; set off on
each side of these lines half the breadth .of the teeth, and draw parallels ;
project the teeth of fig. 1 upon fig. 2, by drawing through all the visible
angular points straight lines parallel to A B, and terminated at either ex-
tremity by the verticals representing the outlines of the breadth of the
wheel ; project in like manner the circles of the hub ; lay off half length
on each side of F G, and draw parallels to it. The section (fig. 3) is sup-
posed to be made on the line D E of the elevation ; project, as in fig. 2,
those portions which will be visible in this section, and shade those parts
which are in section. The arms are made tapering in width, and somewhat
less than the face of the wheel.
Since the two projections (figs. 1 and 3) are not sufficient to exhibit fully
the true form, a cross section of one of them is given at fig. 4 ; this section
is supposed to be made by a plane passing through X X' and Y Y'. The
points y, 2, in fig. 1, and corresponding lines in fig. 3, represent the edges
of key-seat.
12
ITS DRAWING OF MACHINERY.
OBLIQUE PROJECTION OF A SPUR WHEEL.
Plate XIX. In drawing a spur wheel or other object in an oblique
position with respect to the vertical plane of projection, it- is necessary, in
the first place, to lay down the elevation and plan as if it were parallel to
that plane, as represented at figs. 1 and 2. Then transfer the plan to fig.
4, giving it the same inclination with the ground line which the wheel
ought to have in relation to the vertical plane ; and assuming that the
horizontal line A B represents the axis of the wheel, both in the parallel
and oblique positions, the centre of its front face in the latter position will
be determined by the intersection of a perpendicular raised from the point
C' (fig. 4) with that axis. Now it is obvious, that if we take any point, as
a in fig. 1, the projection of that point on fig. 3 must be in the line a #,
parallel to A B ; and further, this point being projected at a' (fig. 4), it
must be in the perpendicular a' a; therefore the intersection of these two
lines is the point required. Thus all the remaining points J, c 1 , cl, &c., may
be obtained by the intersections of the perpendiculars raised from the
points &', c', d', &c. (fig. 4), respectively, with the horizontals drawn
through the corresponding points in fig. 1. It will also be observed, that
since the points e and/, in the further face of the wheel, have their projec-
tions in a and 5 (fig. 1), their oblique projections will be situated in the
lines a a and 5 5, but they are also at e and// consequently, the lines e a
and/ b are the oblique projections of the edges a' e' and It' '/'. We have
now to remark, that all the circles which, in the rectangular elevation
(fig. 1), have been employed in the construction of this w^heel, are pro-
jected in the oblique view into ellipses, the length and position of whose
axes may be determined without any difiiculty ; lor since the plane F' G',
in which these circles are situated, is vertical, the major axes of all the
ellipses in question will obviously be perpendicular to the line A B, and
equal to the diameters of the circles of which they are respectively the pro-
jections ; and the minor axes, representing the horizontal diameters, will
all coincide with the line A B. Thus, to obtain the ellipse into which the
pitch circle is projected, it is only necessary to set off upon the vertical
D E (fig. 3), above and below the point C, the radius of the pitch-circle,
whose horizontal diameter ij being at i'j' (fig. 4) is projected to ij (fig. 3) ;
and thus having obtained the major and minor axes, the ellipse in question
may easily be constructed. The intersection of the horizontal lines gg, h k,
&c., with this circle gives the thickness of the teeth at the pitch line ; and
by projecting in the same manner the circles bounding the extremities and
DRAWING OF MACHINERY. 179
roots of the teeth, these points in each individual tooth may be determined
by a similar process. But since, in cases where strict accuracy is required,
a greater number of points is necessary for the construction of the curva-
ture of the teeth, two additional circles m n and o p may be drawn on fig.
1, and projected to fig. 3, and the points of their intersection with the
curves of the teeth projected to fig. 3, where the corresponding points are
indicated by the same letters.
It is almost unnecessary to observe, that the instructions we have given
for the drawing of the anterior face F' G' of the wheel are equally appli-
cable to the posterior H' I', which is parallel to it, and in all respects the
same ; the common centre of all the circles in it being at O' (fig. 4), is pro-
jected to O in fig. 3. Hence, it will be easy to construct the ellipses
representing these circles in the oblique projection, and consequently to
determine the points e, /, ', &c., in the curvature of the teeth ; observing,
that as their centre lines converge to C in the front face, they all tend to
O in the remoter surface, which is, however, for the most part concealed
by the former.
It would be superfluous to enter into any details regarding the con-
struction of the oblique view of the rim, eye, and arms which are drawn
upon precisely similar principles to those we have already so fully ex-
plained.
PROJECTIONS OF A BEVIL WHEEL.
Plate XXI. Fig. 1 is a face view, fig. 2 an edge view, and fig. 3 a
vertical transverse section. We have explained (page 159) the determina-
tion of the division of the angle of inclination of the axes of a pair of bevil
wheels ; their size and proportion are to be determined by the rules given
for spur-wheels ; thus, consider the base of the cone A B (figs. 2, 3) as the
diameter of the pitch circle of a spur-wheel, and proportion the pitch, form,
and breadth of teeth, according to the stress to which they are to be subjected.
Having determined and laid down, according to the required condi-
tions, the axis O S of the primitive cone, the diameter A B of its base, the
angle A S O which the side of the cone makes with the axis, and the
straight lines A o, D o', perpendicular to A S, and representing the sides
of two cones, between which the breadth of the wheel (or length of the
teeth) is comprised, the first operation is to divide the primitive circle, de-
scribed with the radius A C, into a number of equal parts corresponding
to the number of teeth or pitch of the wheel. Then upon the section (fig.
3), draw with the radius o A or o B, supposed to move parallel to itself,
180 DRAWING OF MACHINERY.
outside the figure a small portion of a circle, upon which construct the
outlines of a tooth M, and of the rim of the wheel, with the same propor-
tions and after the same manner as we have explained in reference to
spur-wheels ; set off from A and B the points a, d, andy, denoting respec-
tively the distances from the pitch line to the points and roots of the teeth,
and to the. inside of the rim, and join these points to the vertex S of the
primitive cone, terminating the lines of junction at the lines D </, E o' ';
the figure a 1) c d will represent the lateral form of a tooth, and the figure
c df c, a section of the rim of the wheel, by the aid of which the face view
(fig. 1) may easily be constructed.
The points a, 5, <?, c?, and <?, having been projected upon the vertical
diameter A 7 B r , describe from the centre C' a series of circles passing
through the points thus obtained, and draw any radius, as C' L, passing
through the centre of a tooth. On either side of the point L set off the
distances L k, L /, making up the thickness of the tooth M at the point,
and indicate, in like manner, upon the circles passing through the points
B 7 and d', its thickness at the pitch line and root ; then draw radii through
the points i, I, &, g, m, &c., terminating them respectively at the circles
forming the projections of the corresponding parts at the inner extremity
of the teeth ; these radial lines will represent the rectilinear edges of all
the teeth. The curvilinear outlines may be delineated by arcs of circles,
tangents to the radii g C' and i C', and passing through the points obtained
by the intersections of the radii and the various concentric circles. The
radii of these circular arcs may in general, as in the case of spur wheels,
be taken equal to the pitch, and their centres upon the interior and exte-
rior pitch-circles ; thus the points g and i, n and <?, for example, are the
centres foi: the arcs passing through the corresponding points in the next
adjacent teeth, and vice versa.
The drawing of the teeth in the edge view (fig. 2), and of such portions
of them as are visible in the section (fig. 3), is sufficiently explained by in-
spection of the lines of projection which we have partially introduced
into the plate for this purpose. We have only to remark, that in the con-
struction of these views, every point in the principal figure from which
they are derived is situated upon the projection of the circle drawn from
the centre C', and passing through that point. Thus the points g and ',
for example, situated upon the exterior pitch-circle, will be determined in
fig. 2 by the intersection of their lines of projection with the base A B of
the primitive cone ; and the points k and I will be upon the straight line
passing through a a (fig. 3), and so on. Farther, as the lateral edges of
air the teeth in fig. 1 are radii of circles drawn from the centre C', so in
DRAWING OF MACHINERY.
181
fig. 2 they are represented by lines drawn through the various points
found as above for the outer extremities of the teeth, and converging
towards the common apex S ; while the centre-lines of the exterior and
interior extremities themselves all tend to the points o and o' respectively.
This circumstance will suggest a mode of materially simplifying the opera-
tion of drawing the edge view of the teeth when the wheels are small, or
executed to a small scale ; and in all cases it affords a means of testing
the accuracy of the operations, if the method of projecting numerous points
be adopted.
Skew-bevels. When the axes of wheels are inclined to each other, and
yet do not meet in direction, and it is proposed to connect them by a
single pair of bevels, the teeth must be inclined to the base of the frusta to
Fig. 279.
allow them to come into contact. Set off a e equal to the shortest distance
between the axes, (called the eccentricity^) and divide it in c, so that a c
is to e c as the mean radius of the frustum to the mean radius of that with
which it is to work ; draw c in d perpendicular to a e. The line c m d gives
182 DRAWING OF MACHINERY.
the direction of the teeth ; and if from the centre a, with radius a c, a
circle be described, the direction of any tooth of the wheel will be a tan-
gent to it, as at c. Draw the line d e perpendicular to c m d, and with a
radius d e equal to c e describe a circle ; the direction of the teeth of the
second wheel will be tangents to this last, as at d.
SYSTEM COMPOSED OF A PINION DRIVING A RACK.
PL XXIII., fig. 1. The pitch line M N of the rack, and the primitive
circle A B D of the pinion being laid down touching one another, divide
the latter into twice the number of equal parts that it is to have of teeth,
and set off the common distance of these parts upon the line M K, as many
times as may be required ; this marks the thickness of the teeth and width
of the spaces in the rack. Perpendiculars drawn through all these points
to the solid part of the rack will represent the flanks of the teeth upon
which those of the pinion are to be developed in succession. The curvature
of these latter should be an involute A c of the circle A B D. The teeth
might be cut off at the point of contact d upon the line M "N, for at this
position the tooth A begins its action upon that of the rack E ; but it is
better to allow a little more length ; in other words, to describe the circle
bounding the points of the teeth with a radius somewhat greater than C d.
"With regard to the form of the spaces in the rack, all that is required
is to set off from M Is", as at the point e, a distance slightly greater than
the difference A a of the radius of the pitch circle, and that of the circle
limiting the points of the teeth, and through this point to draw a straight
line F G parallel to M K From this line the flanks of all the teeth of the
rack spring, and their points are terminated by a portion of a cycloid A 5,
which, however, may in most instances be replaced by an arc of a circle.
The depth of the spaces in the pinion obviously depends upon the height
of this curved portion of the teeth ; their outline is formed by a circle
drawn from the centre C, with a radius a little less than the distance from
this point to the straight line, bounding the upper surface of the teeth of
the rack.
SYSTEM COMPOSED OF A RACK DRIVING A PINION.
In this case the construction is in all respects identical with that of the
preceding example, with this exception, that the form proper to be given
to the teeth of the rack is a cycloid generated by a point A in the circum-
DRAWING OF MACHINERY. 183
ference of the circle A E C, in rolling on the line M 1ST. The curvature
of the teeth of the pinion is an involute as before.
SYSTEM COMPOSED OF A WHEEL AND TANGENT, OB ENDLESS SCREW.
Fig. 2. In the construction of this variety of geering, we must first fix
upon the number of teeth in the wheel, and the distance of its centre from
the axis of the screw. Then conceive a plane passing through the axis
E F of the screw, parallel to the face of the wheel, and let C be. the centre
of its primitive circle. If now a perpendicular C G be drawn from C upon
E F, and C A be taken as the radius of the pitch circle B A D of the
wheel, the difference A G will represent the radius of a cylinder, which
may be termed the primitive cylinder of the screw ; and a line M N drawn
through A, parallel to E F, will be a generatrix of that cylinder, which
will serve the purpose of determining the form of the teeth.
The section having been made through the axis, the question obviously
resolves itself into the case of a rack driving a pinion ; consequently the
curve of the teeth, or rather thread, of the screw should be simply a cycloid
generated by a point in the circle A E C, described upon A C as a diam-
eter, and rolling upon the straight line M 1ST. It is to be remarked, fur-
ther, that the outlines of the teeth are helical surfaces described about the
cylinder forming the screw, with the pitch A 5 equal to the distance, mea-
sured upon the primitive scale, between the corresponding points of two
contiguous teeth. These curves have been drawn on our figure, but being
for the most part concealed, they are expressed by dotted lines. The teeth
of the wheel are not, as in ordinary kinds of geering, set perpendicularly
to the plane of its face, but at an angle, and with surfaces corresponding
to the inclination and helical form of the thread of the screw. In some
instances, the points of the teeth and bottoms of the spaces are formed of a
concave outline adapted to the convexity of the screw, in order to present
as much bearing surface as possible to its action. In this kind of geering,
for obvious reasons, it is invariably the screw that imparts the motion.
Fig. 3 represents an edge elevation of the wheel, projected as in pre-
vious examples.
SYSTEM COMPOSED OF AN INTERNAL SPUR-WHEEL DRIVING A PINION.
PI. XXY., fig. 1. The form of the teeth of the driving wheel is in this
instance determined by the epicycloid described by a point in the circle
A E C, rolling on the concave circumference of the primitive circle M A N".
184 . DRAWING OF MACHINERY.
The points of the teeth are to be cut off by a circle drawn from the centre
of the internal wheel, and passing through the point E, which is indi-
cated, as before, by the contact of the curve with the flank of the driven
tooth.
The wheel being supposed to be invariably the driver, the curved por-
tion of the teeth of the pinion may be very small. This curvature is a
part of an epicycloid generated by a point in the circle MAX rolling
upon BAD.
SYSTEM COMPOSED OF AX INTERNAL WHEEL DRIVEN BY A PINION.
Fig. 2. This problem involves a circumstance which has not hitherto
come under consideration, and which demands, consequently, a different
mode of treatment from that employed in the preceding cases. The epicy-
cloidal curve A #, generated by a point in the circle having the diameter
A O, the radius of the circle MAX, and which rolls upon the circle
BAD, cannot be developed upon the flank A &, the line described by
the same point in the same circle in rolling upon the concave circumfer-
ence MAX; and for this obvious reason, that that curve is situated with-
out the circle BAD, while the flank, on the contrary, is within it. It
becomes necessary, therefore, in order that the pinion may drive the wheel
uniformly according to the required conditions, to form the teeth so that
they shall act always iipon one single point in those of the wheel. This
may be most advantageously effected by taking for the curvature of the
teeth of the pinion the epicycloid A d described by the point A in the
circle M A X, rolling over the circle BAD. It will be observed that, as
in the preceding examples, the tooth E of the pinion begins its action upon
the tooth F of the wheel at the point of contact of their respective primi-
tive circles, and that it is unnecessary that it should be continued beyond
the point , because the succeeding tooth II will then have been brought
into action upon G ; consequently the teeth of the wheel might be bounded
by a circle passing through the point c. It is, however, one of the prac-
tical advantages which this species of geering has over wheels working ex-
ternally, that the surfaces of contact of the wheel and pinion admit of
being more easily increased ; and by making the teeth somewhat longer
than simple necessity demands, the strain may be diffused over two or
more teeth at the same time. The flanks of the teeth of the wheel are
formed by radii drawn to the centre O, and their points are rounded off
to enable them to enter freely into the spaces of the pinion.
DRAWING OF MACHINERY. 185
PROJECTIONS OF ECCENTRICS.
The term eccentric is applied in general to all such curves as are com-
posed of points situated at unequal distances from a central point or axis.
The ellipse, the curve called the heart, and even the circle itself, when
supposed to be fixed upon an axis which does not pass through its centre,
are examples of eccentric curves.
The object of such curves, which are of frequent occurrence in machin-
ery, is to convert a rotatory into an alternating rectilinear motion ; and
their forms admit of an infinite variety, according to the nature of the
motion desired to be imparted. Examples of their application occur in
many arrangements of pumps, presses, valves of steam-engines, spinning
and weaving machines, &c.
Fig. 1, pi. XXYII. To draw the eccentrical symmetrical curve called
the heart, which is such as, when revolving with a uniform motion on its
axis, to communicate to a movable point A,a uniform rectilinear motion of
ascent and descent.
Let C be the axis or centre of rotation upon which the eccentric is
fixed, ami which is supposed to revolve uniformly ; and let A A' be the
distance which the point A is required to traverse during a half revolution
of the eccentric. From the centre C, with radii respectively equal to C A
and C A', describe two circles ; divide the greatest into any number of
equal parts (say 16), and draw through these points of division the radii
C 1, C 2, C 3, &c. Then divide the line A A' into the same number of
equal parts as are contained in the semicircle (that is, into 8 in the
example now before us), and through all the points 1', 2', 3', <fec., draw
circles concentric with the former ; the points of their intersection B, D, E,
&c., with the respective radii C 1, C 2, C 3, &c., are points in the curve
required, its vertex being at the point 8.
It will now be obvious that when the axis, in its angular motion, shall
have passed through one division, in other words, when the radius C 1
coincides with C A 7 , the point A, being urged upwards by the curvature
of the revolving body on which it rests, will have taken the position indi-
cated by 1' ; and further, when the succeeding radius C 2 shall have
assumed the same position, the point A will have been raised to 2', and so
on till it arrives at A', after a half revolution of the eccentric. The re-
maining half A G F 8 of the eccentric, being exactly symmetrical with the
other, will enable the point A to descend in precisely the same manner as
it is elevated. It is thus manifest that this curve is fitted to impress a uni-
186 DRAWING OF MACHINERY.
form motion upon the point A itself, but in practice a small friction
roller is usually interposed between the surface of the eccentric and the
piece which is to be actuated by it. Accordingly, the point A is to be
taken as the centre of this roller, and the curve whose construction we
have just explained is replaced by another similar to, and equidistant from
it, which is drawn tangentially to arcs of circles described from the various
points in the primary curve with the radius of the roller. This second
curve is manifestly endowed with the same properties as the other ; for,
supposing the point e, for example, to coincide with A, if we cause the
axis to revolve through a distance equal to one of the divisions the point
fj which is the intersection of the curve with the circle whose radius is
C 1', will then obviously have assumed the position V ; at the next por-
tion of the revolution, the point y (which is such that the angle f C g is
equal to e C f] will have arrived at 2', and so on. Thus it is plain that
the point a will be elevated and depressed uniformly by means of the
second curve, in the same manner as .that denoted by A is actuated by
tlis first,
It is obvious that the movable point a must, in actual working, be held
in contact with the surface of the eccentric ; this is generally accomplished
by the action of a weight or of a spring ; but in forms similar to fig. 1, in
which all the diameters, as A B, B F, D G, &c., are equal, two frictions
connected and placed diametrically opposite each other may be used,
which will be thus alternately and similarly impelled ; in many cases an
eccentric groove is cut, and the friction roll or point a is made to slide in
this groove.
Fig. 2. To draw a double eccentric curve, which shall impart a uni-
form molion of ascent and descent to the point A, traversing an arc of a
circle A A'.
First, divide the given arc A A' into any number of equal parts (8 in
the present example), and from the common centre, or axis C of the eccen-
tric, describe circles passing through each of the points of division 1', 2', 3',
&c. Divide also the circle passing through O, the centre of the arc A A',
into twice the number of equal parts ; then taking up in the compasses the
length A O, and placing one of the points at the division marked 1, de-
scribe an arc of a circle, which will cut at B the circle drawn with the
radius C I/; from the next point of division 2, mark off, in the same man-
ner, the point D in the circle whose radius is C 2', and so on. The points
B, D, E, &c., thus obtained, are points in the curve required, which, sup-
posing the eccentric to revolve uniformly, will possess the property of
communicating to the point A a uniform motion of ascent and de-
DE AWING OF MACHINERY. 187
scent along the arc A A'. This admits of easy demonstration. The angle
B C 1' is half of 2' C D, and consequently, when the point B has arrived
at 1', the radius C D, then coinciding with C B, will have passed through
an angle equal to I' C B, and again, at the next point in the revolution,
will coincide with C 2'. Therefore the portion B D of the curve will
impel the given point through the arc V 2', in the same time and with the
same velocity, as the part A B will have raised it from A to 1'. By a
similar process of reasoning it will be manifest, that the angle 1' C B being
just one-third of 3' C I, the point A will also traverse the space 2' 3' with
a uniform motion.
By a glance at the figure it will be seen that this curve is not symmetri-
cal ; in other words, that the part A F E is not equal or similar to A D E.
This may be accounted for by observing, that the arc 1) 1', for instance, is
equal to V B, and consequently the point ~b (which is determined by the
intersection of the circle passing through V with- the arc described from
the centre 15) cannot be situated in the same position in relation to A as
the point B, since the radius C A does not pass through 1' ; the same re-
mark applies to all the other arcs, d 2', &c. It is not the less certain,
however, that the part A F E of the eccentric will cause the given point
to descend through the arc A' A in the same uniform manner as it had
been elevated by the part A D E.
In the two preceding examples of eccentrics it has been shown, that
the point A moves through equal spaces in equal times, both in ascending
and descending. In some cases, however, this is by no means desirable ;
thus, if the eccentric is destined to give motion to a mass of matter which
offers considerable resistance, such a form would give rise to injurious and
destructive shocks. In such cases, it is necessary so to regulate the curva-
ture of the eccentric, that the point A shall move at the beginning and
end of its stroke with diminished velocity ; and that for this purpose,
the space A A should be unequally divided, as in the example which
comes next under notice.
Fig. 3. To draw a double and symmetrical eccentric curve, suck as to
cause the point A to move in a straight line, and with an unequal motion ;
the velocity of ascent being accelerated in a given ratio from the starting
point to the vertex of the curve, and the velocity of descent being retarded
in the same ratio.
Upon A A' as a diameter describe a semicircle, and divide it into any
number of equal parts ; draw from each point of division 1', 2', 3', &c.,
perpendiculars upon C A'; and through the points of intersection I 2 , 2 2 , 3 s ,
&c., draw circles having for their common centre the point C, which is to
188 DRAWING OF MACHINERY.
be joined, as before, to all the points of division on the circle (A' 48.)
The points of intersection of the concentric circles with the radii C 1, C 2,
C 3, &c., are points in the curve required.
Fig. 4. To construct a double and symmetrical eccentric, which shall
produce a uniform rectilinear motion, with periods of rest at the points
nearest to, and farthest from, the axis of rotation.
The lines in the figure above referred to indicate sufficiently plainly,
without the aid of further description, the construction of the curve in
question, which is simply a modification of the eccentric represented at
Fig. 1. In the present example, the eccentric is adapted to allow the
movable point A to remain in a state of rest during the first quarter of a
revolution B D ; then, during the second quarter, to cause it to traverse,
with a uniform motion, a given straight line A A', by means of the curve
D G ; again, during the next quarter E F G, to render it stationary at the
elevation of the point A! ; and finally, to allow it to subside along the
curve B E, with the same uniform motion as it was elevated, to its original
position, after having performed the entire revolution.
Fig. 5 represents an edge view of this eccentric, and fig. 6 a vertical
section of it.
Figs. 7, 8, and 9, a Circular Eccentric. These figures represent a
model of a variety of the circular eccentric, which is the contrivance
usually adopted in steam-engines for giving motion to the valves regu-
lating the action of the steam upon the piston. The circular eccentric
is simply a species of disc or pulley fixed upon the crank-shaft, or other
rotating axis of an engine, in such a manner that the centre or axis
of the shaft shall be at a given distance from the centre of the pulley. A
ring or hoop, either formed entirely of, or lined with brass or gun metal,
for the purpose of diminishing friction, is accurately fitted within project-
ing ledges on the outer circumference of the eccentric, so that the latter
may revolve freely within it ; this ring is connected by an inflexible rod
with a system of levers, by which the valve is moved. It is evident, that
as the shaft to which the eccentric is fixed revolves, an alternating recti-
linear motion will be impressed upon the rod, its amount being determined
by the eccentricity, or distance between the centre of the shaft and that
of the exterior circle. The throw of the eccentric is twice'the eccentricity
C E ; or it may be expressed as the diameter of the circle described by
the point E. The nature of the alternating motion generated by the cir-
cular eccentric is identical with that of the crank, which might in many
cases be advantageously substituted for it.
Fig. 8 is the edge view, fig. 9 the section of the eccentric, in this par-
DRAWING OF MACHINERY. 1$9
ticular example, formed in a single piece, and which can be applied only
when the shaft to which it is to be attached is straight and uninterrupted
by cranks, <fec. The mode of representing the arm in fig. 9, which is a
section on the line D F, is not strictly accurate, but is a license frequently
practised in similar cases, and which is attended with obvious advantage.
In many machines, the eccentric is used for the raising of a weight a
certain height and then letting it fall, as in the case of ore stampers, cloth
beetles, trip hammers, and the valve rod of some steam engines. In these
cases the eccentric may be considered as merely a single long tooth geer,
in which commonly, on account of the uniformity of action, the wiping or
rubbing surface is an involute curve, the boss of the eccentric being the
generating circle.
In practice, the term eccentric is generally confined to the circular
eccentric ; all others, with exception of that last described, or wypers,
being called cams.
DRAWING OF SCREWS.
The screw is a cylindrical piece of wood or metal, in the surface of
which one or more helical grooves are formed. The thread of the screw
is the solid portion left between the grooves ; and the pitch of the screw.
is the distance, measured on a line parallel to the axis of the cylinder, be-
tween the two contiguous centres of the same thread.
Projections of a, triangular-threaded screw and nut, pi. XXVIII., fig. 1.
Having drawn the ground line A B, and the centre lines C C' of the
figures, from O as a centre, with a radius equal to that of the exterior
cylinders, describe the semicircle a 3 6 ; describe in like manner the semi-
circle 1) c e with the radius of the interior cylinder. Now draw the per-
pendiculars a a" and 6 6", T) V and e c", which will represent the ver-
tical projections of the exterior and interior cylinders. Then divide the
semicircle a 3 6 first described into any number of equal parts, say 6, and
through each point draw radii, which will divide the interior semicircle
similarly. On the line a' a" set off the length of the pitch as many times
as may be required ; and through the points of division draw straight
lines parallel to the ground line A B. Then divide each distance or pitch
into twice the number of equal parts that the semicircles have been divided
into, and following instructions already laid down (page 100), construct
the helix a' 3' 6 both in the screw and nut.
Having obtained the point V by the intersection of the horizontal line
passing through the middle division of a! a with the perpendicular b V , de-
190 DRAWING OF MACHINERY.
scribe the helix V c' e ', which will represent the bottom of the groove.
The apparent outlines of the screw and its nut will then be completed by
drawing the lines V a!, a' I', etc., to the curves of the helices ; these
are not, strictly speaking, straight lines, but their deviation from the
straight line is, in most instances, so small as to be imperceptible, and it
is therefore unnecessary to complicate the drawing by introducing the
method of determining them with rigorous exactness.
When a long series of threads have to be delineated, they should be
drawn mechanically by means of a mould or templet, constructed in the
following manner : Take a small slip of thin wood or pasteboard, and
draw upon it the helix a' 3' G to the same scale as the drawing, and pare
the slip carefully and accurately to this line. By applying this templet
upon fig. 1, so that the points a! and 6 on the plate shall coincide with
#'and-6 on the drawing, the curve a' 3' 6 can be drawn mechanically,
and so on for the remaining curves of the outer helix. The same templet
may be employed to draw the corresponding curves in the screw-nut by
simply inverting it ; but for the interior helix a separate one must be cut,
its outlines being laid off in the same manner.
Projections of a square-threaded screw and nut (fig. 2). The depth of
the thread is equal to its thickness, and this latter to the depth of the
groove. The construction is similar to the preceding, and will be readily
understood from the drawing, the same letters and figures marking rela-
tive parts. The parts of the curve concealed from view are shown in
dotted lines.
It will be observed, that the heads and nuts of the screws are repre-
sented as broken, which is done for economy of space.
It is s.eldom necessary to delineate so exactly the outlines of screws,
as they are generally drawn to a much smaller scale. Fig. 3 shows the
simplest form by which a screw may be represented. Figs. 4, 5, 6, repre-
sent a triangular-threaded, a square-threaded screw, and a serpentine, in
which the helical curves are replaced by straight lines, and these forms
will be found sufficiently exact and graphic for most of the cases occur-
ring in practice.
Screws may have two, three, or even a greater number of threads, ac-
cording to the velocity which their action may be required to produce.
A double-threaded screw is one in which the pitch of any individual helix
includes two threads ; a three-threaded screw, one in which it embraces
three threads, and so on.
Size and proportion of bolts. The diameter of the bolt depends, of
course, on the strain to which it is to be subjected ; but since the tensile
DRAWING OF MACHINERY.
191
strength of common bolts is reduced at least one quarter by the cutting of
the thread, as a safe rule the tension ought not to exceed 4 tons on the
square inch of section. It will be found economical often, where the bolt
is long, to cut the thread on a larger wire, and weld the piece to a rod of
the interior diameter of the screw. The section of the thread most ap-
proved of for strength and easy motion of the nut is the equilateral tri-
angle, thus A , the bevelled .sides being equal between themselves and to
the base.
Diameter of
bolts.
Threads in
inch.
Short diameter
of nut.
Diameter of
bolts.
Threads in
inch.
Short diameter
of nut
Inches.
Inches.
Inches.
Inches.
1
12
I
If
5
l
5
11
. irV
11
**
8|
i
10
!A
2
*l
iV
7
?
8?
IT"*
2J
4
8|
1
8
If
21
4
4
M
n
2
II
8*
4}
1*
7
21
2*
8}
*A
l
6
V*
2|
3
4?
H
51
tt
2?
3
*H
If
5
sj
3
3
*i
The thickness of the nut should be equal to the diameter of the bolt.
The head of the bolt is usually square ; the nut may be of the same form,
but as often is six-paned or six square. When the head of the bolt is in-
Fig. 280.
Fig. 281.
tended to \>Q flush or even with the surface of the piece into which the bolt
is inserted, the inside of the head is made conical like the common wood-
192 DRAWING OF MACHINERY.
screw, or pyramidal, and it is then said to be countersunk. "When bolts
are employed in wood, washers are usually placed beneath the nut and
head, to give a more extended bearing surface.
Figs. 280 and 281 represent two wrought iron hooks, in which the
material is distributed according to the strain to which the parts may be
subjected. The following are the proportions on which fig. 280 is con-
structed : Assuming the neck of the hook as the modulus or 1, the diam-
eter of journals of the traverse are 1.1 ; width of traverse at centre, 2 ;
distance from the centre of the hook to the centre of the traverse, 7.5 ;
interior circle of the hook, 3.4 ; greatest thickness of the hook, 2.8. As-
suming (fig. 281) the diameter of the wire of the chain as 1 : interior circle
of hook is 3.2, and greatest thickness of hook, 3.5.
/
FRAMES.
Plate XXX. represents the application of iron in the frames of tools.
Fig. 1 represents the cam-punch and shear; in this case, the force
exerted whilst the machine is in the operation of punching or shearing,
tends to open the jaws a aj and the tendency increases with the depth of
the jaw, the strain obviously being the greatest at the inmost part of the
jaw. The frame consists of a plate of cast iron, with two webs around its
edges; the front web being subjected to a tensile strain, should be in the
area of its section about six times that of the rear web which is subjected
to a cornpressive force.
Fig. 2 is the side frame of a planing machine. The force here exerted
is horizontal against the cutter, which can be raised or lowered at pleasure,
according to the magnitude of the work to be planed ; the upright has,
therefore, to be braced, which is done in a curved form for beauty of out-
line.
Fig. 3 is a common jack-screw, in which the pressure is vertical ; the
base is made extended to give it stability.
Fig. 4: is a plan of the top plate, and fig. 5 the elevation of a hydraulic
press. The top and bottom plates and platen are cast iron, the four rods
are wrought iron ; the strain upon the rods is tensile, and it is only neces-
sary to give them such a size as to resist securely the power which maybe
required on the press. The plates are beams, supported at tiie four cor-
ners ; subjected to a breaking strain, it will be evident that the bottom
DRAWING OF MACHINERY. 193
plate is the strongest, as in this case the bottom of the plate being sub-
jected to a tensile strain, is a flange or platen, and affords more material
remote from the neutral axis than the ribs of the upper plate which are
subjected to the tensile strain. The movable platen is braced by triangu-
lar wings or flanges radial from the piston. In this particular case the
cylinder is cast iron hooped with wrought iron ; it is very common to
make the whole cylinder wrought iron.
Principle of the action of the hydrostatic press. Let A B C D (fig. 282)
epresent a vertical section of a cylindrical vessel filled with
an incompressible non-elastic fluid, as water for instance ; let
E and F be two pistons of different magnitudes connected with
the cylinder, and fitting closely their respective orifices ; now,
whatever pressure be exerted by the piston F on the fluid Fi s- 2S2<
in the cylinder, it will be repeated on the piston E as many times as the
area of the small piston is contained in the large piston ; that is, if the
area of F was 1 square inch, and the pressure exerted 10 Ibs., and the pis-
ton E 100 square inches, then the pressure on E would be 10 x 100, or
1000 Ibs. F corresponds to the plunger of the force-pump, E to the piston
or ram of the press. The thickness of metal of the cylinder, if of cast-iron,
should not be less than one-half the diameter of the ram. Adopting this
as the rule, to find the entire pressure in tons which -a cylinder can sustain,
the diameter of the ram being given :
Multiply the square of the diameter in inches by 3, and the product
will be the pressure in tons.
Or, the pressure in tons being given :
Divide the given pressure in tons by 3, and the square root of the quo-
tient will be the diameter of the piston in inches. Tims, the diameter of
the piston being 10 inches, the thickness of metal 5 inches, the pressure
might be 10 x 10 x 3 = 300 tons. ,
Figs. 6 & 7 represent a housing for rolls. The screw a presses down
upon the top of the box of the journal, and the effect is a tensile strain on
the sides of the frame ; but it must be remarked, that frames of this sort
are subject to percussive and intermittent strain vastly exceeding the mere
tensile strain, and proper allowance is to be made for this ; and it is much
better to depend in part on mass or on dead weight of material to resist
such strains than upon cohesive strength merely.
PL XXXI. represents the elevation of the frames of three classes of
American marine engines.
Figs. 1 and 2 represent the frame-work of the New World. It is com-
posed of four pieces of heavy pine timber dd, which are formed into two*
13
194 DRAWING OF MACHINERY.
triangles, and inclined slightly laterally to each Other (fig. 2) ; their lower
ends rest on. the keelsons, and upon their upper extremities are placed the
pillow-block c of the working beam. They are solidly fastened together
and to the boat by numerous horizontal and diagonal timbers, which are
secured by wooden knees and keys, and are heavily bolted. The two front
legs are bolted to flanges cast on the sides of the condenser, and the other
end of the framing is attached to a large mass of timbers, which support
the shaft pillow-block &. The framing is further steadied by two addi-
tional timbers, and rods running from the beam pillow-blocks outside the
shaft to the keelsons of the boat ; a represents the guides, which are bolted
at the bottom to the cylinder flange, and retained in their vertical position
by wrought iron braces connected with the framing. The entire fastening
of the engine and its framing is so disposed as to reduce all the strains to
direct ones of extension or compression on the fibres of the iron and wood
employed in the construction. The height of the frame is 46 feet, width
at bottom 31 feet.
Fig. 3 represents the side elevation of the frame of the side lever ocean
steamer Pacific. In this frame the two large hollow pillow-blocks which
sustain the shaft on each side of the cranks are supported by four wrought
iron columns G G on the forward extremity of the bed-plate I, the centre
of the shaft being 23 feet^above the keelson. The pillow-blocks thus sup-
ported are connected by two strong inclined braces D to the cylinder, by
means of solid facings cast with it on each side of the steam opening. The
columns are connected by horizontal braces A A, composed of hollow tubes,
through which bolts pass, and the frames of the two engines are connected
at the same points by similar tubes, and also two diagonal horizontal braces
cast together. Similar braces C C are used to connect each extremity of
the pillow-blocks, and the two engine frames are connected by a horizontal
wrought iron cross. To resist the tendency of the engines, in the rolling
of the ship, to press the outer bearings, there are in a vertical transverse
plan three wrought iron cross or diagonal braces F between the pillow-
blocks and bed-plates. Four cross braces II and J connect the extremities
of the cylinder and the frame. The cylinders are also connected by a
horizontal tubular brace. It will be thus seen that this frame is a sys-
tem of bracing and cross-bracing, in which the material is most economi-
cally disposed to resist the various strains.
The bed-plate consists of a single casting, 32 feet long and 9 feet broad,
which is securely fastened to the keelsons and ship's bottom ; the diameter
of the cylinder is 9p inches, and the stroke 9 feet.
Fig. 4 represents the side view of the frame of the inclined engines of
DRAWING OF MACHINERY. 195
the war steamer Susquehanna. The cylinder A is rested between two tri-
angular frames, on the inclination of the longest side, and is securely bolted
to each frame. The two frames are connected together with braces similar
to those of the Pacific, and the whole is securely bolted on to the keelsons
and bottom of the ship. In this illustration, the main pieces of the frame
are made of boiler iron, constructed like box girders ; but in smaller en-
gines, it is usual to make these parts of wood. The diameters of the cylin-
ders are 5 feet 10 inches, the length of stroke 10 feet.
PL XXXII. Fig. 1 represents the working-beam of the New World.
It is composed of a skeleton frame of cast iron, round which a wrought
iron strap A is fixed. This strap is forged in one piece, and its extreme
ends are formed into large eyes, which are bored out to receive the end
journals. The skeleton frame is a single casting, and contains the eyes for
the main centre and air-pump journal ; the centre hub is strengthened by
wrought iron hoops a a, which are shrunk upon it. At the points of con-
tact of the strap and skeleton, key-beds are prepared, into which the keys
are carefully fitted and tightly driven ; the keys are afterwards riveted
over at both ends, which retains them in their places, as well as the strap
on the skeleton frame. The strap is also secured to the frame by straps
5 5 and keys. The skeleton frame is still further braced by wrought iron
straps C C, which tie the middle of the long arms of the cross to the ex-
tremities of the shorter ones. This form of beam is that usually adopted
for the engines of eastern American river boats ; the proportions are some-
what varied, but the form is identical. The following are the dimensions
of our illustration: From centre to centre of end journals, 26 feet; this
is somewhat less than the usual proportion to length of stroke, being but
slightly less than double the stroke ; length of centre hub, 26 inches ;
diameter of main centre eye, 15 f ; of eye for air-pump journal, 6| ; of end
journal, 8} inches.
Fig. 3 represents a side elevation ; fig. 4, a plan ; and fig. 5 a section of
a cast iron working beam of an English stationary engine. It will be per-
ceived that the outline of the beam is a parabola, it being in effect a beam
supported at the centre and loaded at the extremities.
From the following table of practical examples from " Architecture of
Machinery," we would assume as a safe rule for land engines, that the
depth at centre should be the diameter of the cylinder, and the length of
beam three times the length of stroke. Hence we can construct the out-
line, having for the vertex the extremity of the beam and the point B in
the curve at the centre. The sectional area may be estimated from rules
already given, knowing the load at the extremity, that is ; the pressure on
196
DRAWING OF MACHINERY.
the piston, the weight of the same and its connections, and also the force
required to drive the" air-pump, estimated at the extremity of the lever.
As an engine is subject to shocks, the load should be estimated as six
times the absolute load. " Five per cent, of the nominal power of the
engine may be considered the maximum of power required to drive the
air-pump." Ed. Tredgold.
Diameter of
cylinder.
Length of stroke.
Description of
work.
Length of beam
from centre.
Depth at centre.
Sectional area.
inches.
ft. in.
ft. in.
inches.
square inches.
4TJ
s
Rolling,
12 4
43
240
401
7
Pumping,
10 4
36
162
39*
6 9
Blowing
9 6
85*
96i
864
6 3
Boiling,
9 3
80
60
84
5
Mill work,
8
25
50
W
4
"
6 10
22J
50
42
4
Marine,
6 3
23
133
42
4 2
G 6
2T
216
32
3
11
5
22
132
1
Figs. 6 and 7 represent a side and a front elevation of a crank, such
as is usually adopted on the engines of American river boats. The main
body of the crank is of cast iron, with two horns
a a projecting from the central hub, and the
whole is bound with a strap of wrought iron.
It is evident that this form of crank gives the
greatest amount of strength with the least ma-
terial, and belongs to the same class of construc-
tion as the working beam (fig. 1). The eye of
the crank is usually made one-fourth the diam-
eter of the cylinder. The table from Eedten-
bacher here inserted gives the relative sizes of
central and end eyes of cranks, depending on
the proportion between the length of crank and
the diameter of central eye. The first column
ni.VMETEK OP EYE, BEING UNITS.
For wrought
iron shaft.
Cast iron
shafts.
2
0.85
0.62
3
0.69
0.51
4
0.60
0.44
5
0.54
0.39
6
0.49
0.36
1
-. 0.45
0.33
8
0.42
0.31
9
0.40
0.23
10
0.3S
0.23
11
0.36
0.26
12
0.34
0.25
13
0.33
0.24
exhibits the number of times the diameter of eye is contained in the length
of crank ; the other columns exhibit the diameter of crank-pin.
From this table may be determined for any crank the diameter of
either eye, one being known, and the length of the crank.
Figs. 8, 9, a side view and front elevation of a wrought iron crank and
their practical proportions ; the eye for the crank-pin is a slightly conical
hole, and the pin is made of a corresponding taper.
PI. XXXIII. represents steam-engine connecting-rods and their details.
Figs. 1, 2, represent the front and side elevation of a cast iron connect-
DRAWING OF MACHINERY. 197
ing-rod. It is a bar, strengthened throughout the greater part of its length
by four ribs or feathers, whose outlines .in the direction of their length are
parabolic curves. Its upper extremity is formed into two projecting arms,
upon each of which a close wrought iron strap A is fixed by means of a
key or cotter c. These straps are provided and formed for the reception
of the brass bushes a and J, which are accurately fitted to the journals or
bearings of the cross head.
The lower end of the connecting-rod is made of a form suitable for the
reception of the brasses, and other adjusting mechanism necessary for the
purpose of acting freely, but without play, upon the pin of the crank.
In the present example, the end of the crank-pin is concealed by a slight
brass cover or disc, fixed to the connecting-rod by two small, screw pins,
which serves to protect the working surfaces from dust, and imparts an
elegant finish to the whole.
In fig. 3 the ends of the connecting-rod are represented upon a scale
of double the magnitude of the preceding figures. One of the upper links
or straps, with its adjusting apparatus, is supposed to be cut by a vertical
plane passing through the axes, so as to expose the interior arrangement.
This section exhibits distinctly the mode of fixing the links upon the arms
of the connecting-rod by means of the cotters <?, c, and projecting discs e, e,
cast upon the arms ; as also the contrivance for retaining the brasses a and
1) in their places. Fig. 5, which is a vertical section, shows the corre-
sponding provisions for the lower end of the rod ; a small oblique hole for
the introduction of oil will be observed in the upper brass m y while the
lower n is formed with a spherical projection entering a concave recess in
the cast iron, for the purpose of preventing its displacement by the friction
of the crank-pin, which is regulated and adjusted by the cotter d.
Fig. 6 is a horizontal section on the line a J, showing the form of the
body and feathers of the rod.
Fig. 4 represents the end of a connecting-rod, in which the arrange-
ment for tightening the brasses consists of a gib b and cotter a. The small
end of the cotter is made with a screw, which passing through a lug on
the gib, is fitted with a nut, by means of which the cotter is adjusted and
retained in any position required.
Figs. Y and 8 represent the side and front elevation of a wrought iron
connecting-rod, such as are generally used on American river boat engines.
The extremities are fitted with brasses, straps, gibs, and cotters, similar
to those already described. The peculiarity over the general English con-
struction is the economy of material, and the means adopted to give the
required stiffness. It consists of a double truss brace a a of round iron,
198 DRAWING OF MACHINERY.
which is fastened by bolts to the rod near each end ; struts 5 J, cut with a
screw, and furnished with nuts pass through the centre of the brace, by
which means the braces are tightened.
The length of connecting-rods, as recommended "by English mechanics,
is three times that of the stroke ; in this country shorter connecting-rods
are used, twice the length of stroke being not an unusual proportion. The
connecting-rod at its smallest part near the extremities is of the same diam-
eter as the piston-rod ; the boss in the centre is from 1 to 2 inches more.
ON THE LOCATION OF MACHINES.
In the arrangement of a manufactory or workshop, it is of the utmost
importance to know how to place the machinery, both as to economy of
space and also of working. "Where a new building is to be constructed for
a specific purpose of manufacture, it will be found the best to arrange the
necessaiy machines as they should be, and then build the edifice to suit
them. For defining the position of a machine, we merely need in outline
the space it occupies in plan and elevation, and the position of the driven
pulley or geer, and of the operative. To illustrate this subject, we have
selected a two- story weaving room, of which fig. 283 is an elevation and
plates XXXIY, and XXXV. plans.
In this example the building is rectangular, of a width and length to
accommodate the required machinery. The illustration is confined to a
few rooms in one angle, the rest being but a repeat of the same. The tim-
bering and planking are the same as adopted at all our large manufacturing
places. Beams 14 to 16 inches deep, and of little less width, placed from
8 to 9 feet apart from centre to centre, and floored with 3 to 4 inch
plank dowelled or matched, with top floors and bottom sheathing. The
form of construction being fixed, and the size of the building being deter-
mined for the number of looms, knowing the space they require for the
machines and the alley ways ; lay down the outlines of the building, and
dot in, or draw in red or blue, the position and width of beams. This last
is of importance, as it will be observed (fig. 283), that no driving-pulley
can come beneath the beam, and also that this is the position for the
hanger. Lay off now the width of the alleys and of the machines. The
first alley, or nearest the wall, is a back alley ; that is, where the operative
does not stand, and so on alternate alleys. Draw the lines of shafting cen-
tral to the alleys, as in this position the belts are least in the way. One
operative usually tends four looms ; they are therefore generally arranged'
in sets of four, two on each side of the alley, being placed as close to each
DRAWING OF MACHINERY.
199
other as possible, say one inch between the lathes, a small cross alley being
left between them and the next set. Lay off now the required alley at the
end of the room, and space oif the length of two rows of looms with alleys
at the end of alternate looms, and mark the position of the pulleys. It
1 i 1 1 J 1 1 j j J 1 I
will be observed that looms are generally rights and lefts, so that the
pulleys of both looms come in the space where there is no alley. Should
the pulley come beneath a beam, the loom must be either moved to avoid
2.00 DRAWING OF MACHINERY.
it, or the pulley may be shifted to the opposite end of the loom. Parallel
with the pulleys on the looms draw the driving-pulleys on the shafts, that
is, ~k parallel with &, I with i, / with /, and so on. Proceed now to draw
the third and fourth row of looms, since the second and third rows are
driven from the same shaft ; if they are placed on the same line, it will be
impossible to drive both from the same end, and as this is important, we
move the third row the width of the pulley 5, and for the sake of unifor-
mity, the fourth row also. Lay off now the length of looms and position
of pulleys as before, and parallel with the pulleys the driving-pulleys on
the shaft, that is c against c, f against f, and so on. Having in this way
plotted in all the looms, every alternate set being on a line with the third
and fourth row, we proceed now to lay down the position of the looms in
the floor above ; and since for economy of shafting it is usual to drive
from the lines in the lower rooms, to avoid errors, interference of belts and
pulleys, it is usual to plot the upper room on the same paper or board
as the lower room, using either two different colored inks, or drawing the
machines in one room in deep and in the other in light line, as shown in
plate XXXY. If the width of the rooms are the same, the lateral lines
of looms and alleys are the same, and it is only necessary, therefore,
to fix the end lines. Now, as the first loom in the oiiter row of looms,
in the lower room, occupies for its belt the position k on the shaft,
the loom in the upper room must be moved either one way or the
other to avoid this ; thus the position i of the pulley on the loom must
be made parallel to the pulley i on the shaft, so in the other looms a to ,
e to e, d to d, and 5 to &.
Besides the plan, it is often necessary, and always conveniet, to draw a
sectional' elevation (as in fig. 283), of the rooms, with the relative positions
of the driving pulleys and those on the machines, to determine suitably
the length of the belts, and also to see that their position is in every way
the most convenient possible. For instance, in the figure, one of the lower
belts should have been a cross belt, and one of the upper ones straight :
now had the belts to the second row of looms in the upper story, been
drawn as they should have been, straight, the belt would have interfered
a little with the alley, and it would have been better to have moved the
driving shaft a trifle towards the wall.
From this illustration of the location of machines, knowing all the re-
quirements, in a similar way any machinery may be arranged with economy
of spaces, materials, power, and attendance. These two last items are of
the more importance as they involve a daily expense, where the others are
almost entirely the first outlay.
DRAWING OF MACHINERY.
201
Fig. 2S4
MAC H IN E S.
Thus far the illustrations have been almost entirely confined to geomet-
rical projections and the delineation of parts of machinery. We now
proceed to give a few representations of complete machines, taken from
actual constructions, that may serve not only as copies for the draughts-
man, but as examples for the engineer.
In many cases, where the mere
working of the machine is to be
shown, a few lines will be sufficient
for the illustration, as in fig. 284,
which is a skeleton drawing of
Messrs. Maudsley and Field's direct
acting marine engine, in which A
A 1 are the steam cylinders, L the T
piece connecting the two pistons, E
the crank, F the wheel shaft, and
H the air pump.
Fig. 285 is a more finished draw-
ing of the same machine, in which
the engine frame and the T plate,
with its connections and guides,
are shown. It is very common in drawings to express piston rods, working
beams, and cranks thus, by
single lines, with circles
for their pins or pivots;
as their forms are well
established," and their full
delineation would add noth-
ing to the information to be
conveyed to the mechanic,
but might cover up and
confuse the drawing of
more important parts of the
machine.
Figs. 286 and 287 are
mere outline drawings of
two English locomotives,
and yet sufficient to express
Fig. 235.
-202
DRAWING OF MACHINERY.
Fig. 2S6.
the form of the engines, and the arrangement and comparative size of
drivers and other wheels.
"Working drawings are complete illustrations of a machine, either as a
whole or in detail, sufficient to enable the mechanic to construct it. They
should be of a large enough scale, that all the parts may be readily meas-
ured, or with the dimen-
sions in figures this last
is of importance, even
when the scale is mea-
surable. To the me-
chanic, it saves time and
one source of error, but
throws more responsi-
bility on the draughts-
man. Working draw-
~" ings should be almost
entirely in line, with shading only sufficient to distinguish circular from
flat parts. As many views should be given in plans, elevations, sections,
and detailed parts, as may fully explain the whole construction and work-
ing. In the designing of the machine, where there are moving parts, it
may often be necessary
to draw these parts in
different positions, to be
sure that they do not in-
terfere with some other
part of the machine.
If these lines are neces-
sary to the mechanic,
they are left in; one
position being shown in
full line, the other or others in dotted, or light, or red lines.
Figs. 288 and 289 are sections, at right angles to each other, of the cata-
ract of a Cornish Pumping Engine. To make these complete working
drawings, there should be a plan in addition ; but if it be explained that
all the parts are circular, except the lever e' f and weight A, it could be
readily constructed, especially if the dimensions were figured. In explana-
tion of the working of the cataract, it may be briefly said to consist of a
cast-iron cistern G, partly filled with water, in which is a pump a', the
plunger V of which is connected with a lever ef, the curved end of which
is depressed by a tappet on one of the plug rods C, and the plunger thereby
Fig. 2S7.
DEAWESTG OF MACHINERY.
203
raised, water flowing in freely through the valve c' . As the tappet leaves
the lever, the plunger is forced back to its original position by the weight
A, and with speed dependent on the escape of water from beneath the
plunger through the opening left round the regulating plug d, the descent
of the plunger detaches catches by which the different valves of the steam
cylinder are opened.
Fig. 2SS.
Plates XXXVI., XXXYIL, and XXXVIII. are elevations, sections,
and plans of the 48" stop gate in use at the Nassau Water Works, Brook-
lyn, L. I., forming a complete set of working drawings. It will be ob-
served that Plates XXXYI. and XXXVII., which are views at right
angles to each other, are both elevations and sections, one-half of each
being in elevation and one-half in section a method in use to economize
the number of drawings, when the two halves are complete duplicates.
Fig. 1, PI. XXXVIII., is a plan, and fig. 2 the horizontal section.
204: DRAWING OF MACHINERY.
Plate XXXIX. fig. 1, is a longitudinal section of a locomotive boiler,
and fig. 2 an interior view of the smoke box.
The steam space C at the fire end of the boiler is half globe shaped,
and surmounted by a dome. The object of the dome is to carry the steam
as high as possible above the water line before its introduction into the
steam pipe p, in order that the water held in suspension near the surface of
the water, may not be carried over into the cylinders. The steam pipe trav-
erses the length of the boiler, and in the smoke box branches off to each
cylinder ; I is the regulating or throttle valve, worked by the handle which
passes out through a stuffing box in the end of the boiler ; j is the fire box.
surrounded on all sides except at bottom with a water space ; the top or
crown sheet of the fire box is strengthened by pieces of iron, and the flat
sides are securely bolted together. The tube sheet is sufficiently stayed
by the tubes themselves ; this sheet is often made of copper, as are the
side sheets, from 6 to 7 inches below, to about the same height above the
coal line, in locomotive boilers burning anthracite or bituminous coal ; -7?
are the fire tubes varying in different boilers from li to 3 inches in diam-
eter and from T to 14 feet in length; the longer the tube the larger the
diameter. II II are the cylinders, a? a? the steam chests ; uu the exhaust
pipes, which are connected together, and pass up into the centre of the
smoke stack or chimney. The exhaust furnishes by its blast draught to
the chimney, and if the outlet be contracted, the greater the force of the
issuing steam, and the stronger the draught ; but of course the greater the
back pressure in the cylinders.
Plate XL. is the front elevation, and Plate XLI. is the side ele-
vation and section through air-pump of one of the oscillating engines of the
Golden Gate, in which a is the main shaft, 5 crank-pin, c cylinder, d trun-
nions on which the cylinder oscillates to accommodate itself to the motion
of the crank. 0, stuffing-box on the cylinder head. This is made as long
as practicable, to give as much bearing as possible for oscillating the
cylinder, ff, belt-passage connecting the trunnion with g g, side pipe.
h h, valve-stems connecting with the balance puppet- valves, in i i valve-
chests. The lower valve on the right or steam side is concealed by j,
air-pump. The air-pump bucket is provided with India-rubber valves, and
is worked by A*, crank on the intermediate shaft. I Z, condenser. There
are two condensers and two air-pumps, they are located between the cylin-
ders and inclined towards each other, one only being represented.
The passage ff, together with the side pipes, valve-chests, and appur-
tenances, are fixed to the cylinder, and oscillate with it, the steam being
received through one trunnion, and allowed to escape to the condenser
DRAWING OF MACHINERY. 205
through the opposite one. m is an injection cock, admitting the water
upon a scattering plate in the condenser.
The valves are worked by the toes o o in the usual manner. The rock-
shafts pp receive motion partly from the movement of the cylinder, and
partly from the eccentric. Levers are permanently attached to the trip-
shafts q <?, the ends of which work in a slotted piece curved to the centre
of the trunnion. This piece is guided, as represented in the engraving, by
vertical rods sliding in bushes attached to the fixed framing, and is con-
nected by a rod to the starting-lever r / all the levers for working by hand
being so balanced, that the engineer with one hand can work the engine
up to the usual speed.
The cut-off valve is placed outside the trunnion, and is a balance pup-
pet-valve, worked by the ordinary cam motion, and so arranged as to act
either as cut-off or throttle, or both, the levers being placed within reach
of the engineer when working the engine.
Plate XL1I. is a vertical section through the centre of a TURBINE
"WHEEL, and the axis of the supply pipe.* Plate XLIII. is a plan of the
Turbine and wheelpit. Fig. 1, Plate XLIY., is a plan of the whole wheel,
the guides and garniture. This Turbine was constructed for the Tremont
Manufacturing. Co. at Lowell, by Mr. James B. Francis, and contains
most of Mr. Boyden's improvements. Its expenditure of water under 13
feet head and fall, is about 139 cubic feet per second, and its ratio of useful
effect to the power expended, about 79 per cent.
B, the surface of the water in the wheelpit, represented at the lowest
height at which the turbine is intended to operate. C, the masonry of the
wheelpit. D, the floor of the wheelpit. To resist the great upward pres-
sure which takes place when the wheelpit is kept dry by pumps, three cast-
iron beams are placed across the pit, the ends extending about a foot under
the walls on each side ; on these are laid thick planks, which are firmly
secured to the cast-iron beams by bolts. To protect the thick planking
from being worn out by the constant action of the water, they are covered
with a flooring of one inch boards. E, the wrought-iron supply pipe.
This is constructed of plate iron three-eighths of an inch thick, riveted
together. The supply pipe is furnished with the man hole and ventilating
pipe G-, and the leak box H, to catch the leakage of the head gate, when-
ever it is closed for repairs of the wheel.
The lower end of the supply pipe is formed by the cast-iron curbs III.
The curbs are supported from the wheelpit floor by four columns, resting
* By permission of the author we take the following plates and description from the standard
work, " Lowell Hydraulic Experiments."
206 DRAWING OF MACHINERY.
on the cast-iron beam, O ; the beams X', rest immediately upon the col-
umns, and the curb upon the beams, the latter projecting over the columns
far enough for that purpose. The beams ~N' also act as braces from the
wheelpit wall to the curb, and are strongly bolted at each end.
K, the disc. This is of cast-iron, and is turned smooth on the upper
surface, and also on its circumference. It is suspended from the upper
curb I, by means of the disc pipes 31 M. The disc carries on its upper
surface, thirty-three guides (fig. 1, Plate XLIV.,) for the purpose of giving
the water entering the wheel, proper direction. They are made of Rus-
sian plate iron, one-tenth of an inch in thickness, secured to the disc by
tenons, riveted on the under side. The upper corners of the guides, near
the wheel, are connected by the garniture L, which is intended to diminish
the contraction of the streams entering the wheel, when the regulating
gate is fully raised. The garniture is composed of thirty-three pieces of
cast-iron, carefully fitted to fill the spaces between the guides ; they are
strongly riveted to the guides and to each other.
The upper flange of the disc pipe is furnished with adjusting screws,
by which the weight is supported upon the upper curb. The escape of
water between the upper curb and the upper flange of the disc pipe, is
prevented by a band of leather on the outside, which is retained in its
place by the wrought-iron ring. The top of the disc pipe, just below
the upper flange, has two wings, fitting into recesses in the top of the curb,
to prevent the disc from rotating in the opposite direction to the wheel.
R, H, the regulating gate. [Represented, Plate XLII., as fully raised.
The gate is of cast-iron ; the upper part of the cylinder is stiffened by a
rib, to which are attached three brackets, S, S. To these brackets are at-
tached wrought-iron rods, by which the gate is raised or lowered. To one
of the rods is attached the rack Y. The other two rods are attached by
means of links, to the levers T T. The other ends of these levers carry
geered arch heads, into which, and into the rack Y, work three pinions,
"W, of equal pitch and size, fastened to the same shaft, so arranged that by
the revolution of the pinion shaft, the gate is moved up or down, equally
on all sides. The shaft on which the pinions are fastened, is driven by the
worm wheel X ; this is driven by the worm a, either by the governor Y,
or the hand wheel Z. The shaft on which the worm a is fastened, is fur-
nished with movable couplings, which, when the speed gate is at any inter-
mediate points between its highest aud lowest positions, are retained in
place by spiral springs ; in either of the extreme positions, the couplings
are separated by means of a lever moved by pins in the rack Y ; by this
means, both the regulator and hand wheel are prevented from moving the
DRAWING OF MACHINERY. 207
gate in one direction, when the gate has attained either extreme position.
If, however, the regulator or hand wheel should be moved in the opposite
direction, the couplings would catch, and the gate would be moved. The
weight of the gate is counterbalanced by weights attached to the levers
T T, and by the intervention of a lever to the rack V.
l> &, the wheel, consists of a central plate of cast-iron, and two crowns,
c c, of the same material to which the buckets are attached. The buckets
are forty-four in number, made of Eussian plate iron, T of an inch in
thickness, and are secured to the crowns by grooves cut in the crowns of
the exact form of the buckets, and by tenons entered into the mortises
in both crowns, and riveted on the opposite sides.
dd, the vertical shaft, of wrought-iron, runs upon a series of collars,
resting upon corresponding projections in the suspension box e'. The part
of the shaft on which the collars are placed, is made separate from the
main shaft, and is pinned to it at/", by means of a socket in the top of the
main shaft, which receives a corresponding part of the collar piece. The
collars are made of cast steel ; they are separately screwed on, and keyed
to a wrought-iron spindle.
The suspension box is made in two parts, to admit of its being taken
off and put on the shaft ; it is lined with Babbit rrfetal. It is found that
bearings thus lined will carry from fifty to a hundred pounds to the square
inch, with every appearance of durability.
f'f, the upper and lower bearings are of cast-iron, lined with Babbit
metal, adjustable horizontally by means of screws. The suspension box <?',
rests upon the gimbal g. The gimbal itself is supported on the frame h h
by adjusting screws, which give the means of raising and lowering the
suspension box, and with it, the vertical shaft and wheel. The lower end
of the shaft is fitted with a cast-steel pin, i. This is retained in its place
by the step, which is made in three parts, and lined with case-hardened
wrought-iron.
The weight of the wheel, upright shaft, and bevel geer, is supported
by means of the suspension box e' on the frame &, which rests upon the
long beams m, reaching across the wheelpit, and supported at the ends by
the masonry, and also at intermediate points by the braces n n.
Mr. Francis deduces the following rules for proportioning turbines :
The sum of the shortest distances between the buckets, should be equal to the diameter
of the wheel.
The height of the orifices at the circumference of the wheel, should be equal to one-
tenth of the diameter of the wheel.
The width of the crowns should be four times the shortest distance between the buckets.
208 DRAWING OF MACHINERY.
The sum of the shortest distances between the curved guides, taken near the wheel,
should be equal to the interior diameter of the wheel.
The number of buckets is, to a certain extent, arbitrary. As a guide in practice, to
be controlled by particular circumstances, and limited to diameters of not less than two
feet, the number of buckets should be three times the diameter in feet, plus thirty. The
Tremont Turbine is 8^ feet in diameter, and according to the proposed rule, should have
fifty-five buckets instead of forty-four. The number of the guides is also to a certain ex-
tent arbitrary ; the practice at Lowell has been, usually, to have from a half to three-
fourths of the number of buckets.
As turbines are generally used, a velocity of the interior circumference of the wheel,
of about fifty-six per cent, of that due to the fall acting upon the wheel, appears most
suitable.
To lay out the curve of the buckets.
Referring to Plate XLIV., fig. 2, the number of buckets, j\7J having been determined
by the preceding rules, set off the arc g i = -^-=- . Let o> = g A, the shortest distance
between the buckets : t the thickness of the metal forming the buckets. Make the arc
g Te = 5a>. Draw the radius Ok, intersecting the interior circumference of the wheel at I;
the point I will be the inner extremity of the bucket. Draw the directrix Z TO tangent to
the inner circumference of the wheel. Draw the arc o n, with the radius o> -f t, from i, as
a centre ; the other directrix gp, must be found by trial, the required conditions being,
that, when the 'line m Us revolved round to the position g t, the point TO being constantly
on the directrix gp, and another point at the distance mg = rs, from the extremity of the
line describing the bucket, being constantly on the directrix m I, the curve described shall
just touch the arc no. A convenient line for a first approximation, may be drawn by
making the angle g p = 11. After determining the directrix according to the preceding
method, if the angle Ogp should be greater than 12, or less than 10, the length of the
arc g'Tc should be changed, to bring the angle within these limits.
The trace adopted for the corresponding guides is as follows : The number n having
been determined, divide the circle in which the extremities of the guides are found, into n
equal parts, v w, w a-, &c. Put a>' for the width between two adjoining guides, and t' for
the thickness of the metal forming the guides. AYe have by rule, a>' = . With w' as
a centre, and the radius <a' + t\ draw the arc yz; and with a as a centre, and the radius
2(' + '), draw the arc a' &'. Through v draw the portion of a circle v c', touching the
arcs y z and a! V; this will be the curve for the essential part of the guide. The remainder
of the guide, c' d\ should be drawn tangent to the curve c' v ; a convenient radius is one
that would cause the curve c' d\ if continued, to pass through the centre 0.
ARCHITECTURAL DRAWING. 209
AKCHITECTUKAL DKAWING.
THE art of architecture consists in the designing of a building, so as to
be most suitable and convenient for the purposes for which it is intended ;
in selecting and disposing of the materials of which it is composed, so as to
withstand securely and permanently the strains and wear to which they
may be subjected ; and arranging the parts so as to produce the most artis-
tic effect consistent with the use of the building and its location, and ap-
plying to it such appropriate ornament as may express the purpose, and
harmonize with the construction.
As an art it consists in the convenient and appropriate combination of
established forms and ornamentations, and the adaptation of old and recog-
nized styles to present requirements. Success in its practice should there-
fore depend not only on the talent of the architect, but on a thorough study
of the best masters of the science, and an extended acquaintance with the
effects of practical examples and the principles of construction.
In domestic architecture, by far the most extensive branch of the pro-
fession, most persons can give some idea of the kind of building which they
wish to have constructed, and perhaps express by line the general arrange-
ment of rooms ; but it is left to the architect to settle the style of building
appropriate to the position, to determine the dimensions and positions of
rooms and passages, thickness of walls and partitions, arrangements for
drainage, heating, and ventilating in fact all the details which are to be
drawn or specified for the construction. It is, therefore, indispensable that
he should understand what are the best proportions of parts, and what are
the necessities of construction.
FOUNDATIONS.
In preparing for the foundation of any building, there are two sources
of failure to be guarded against inequality of settlement, and lateral escape
210 ARCHITECTURAL DRAWING.
of the supporting material. It is not so much an unyielding as a uni-
formly yielding foundation that is required. When the character of the
ground is not known, it is, therefore, important that, previous to the com-
mencement of the work, soundings should be taken to ascertain the nature
of the soil and the lay of the strata, to determine the kind of foundation ;
and the more important and weighty the superstructure, the more careful
and deeper the examination.
Natural foundations. The best foundation is a natural one, such as a
stratum of rock or compact gravel. If circumstances prevent the work
being commenced from the same level throughout, the ground must be
carefully benched out, i. e., cut into horizontal steps, so that the courses
may all be perfectly level. It must also be borne in mind, that all work
will settle more or less, according to the perfection of the joints, and
therefore in these cases it is best to bring up the foundations to a uniform
level with large blocks of stone or with concrete, before commencing the
superstructure, which would otherwise settle most over .the deepest parts,
on account of the greater number of mortar joints, and thus cause unsightly
fractures. Foundations in soil should be excavated to a depth below the
action of frost.
Artificial foundations. "Where the ground in its natural state is too
soft to bear the weight of the proposed structure, recourse must be had to
artificial means of support, and, in doing this, whatever mode of construc-
tion be adopted, the principle must always be that of extending the bear-
ing surface as much as possible. There are many ways of doing this as
by a thick layer of concrete or beton, or by layers of planking, or by a net-
work of timber, or by increasing width of wall, or these different methods
may be' combined. The weight may also be distributed over the entire
area of the foundation by inverted arches. The use of timber is objection-
able where it cannot be kept constantly wet, as alternations of dryness
and moisture soon cause it to rot, and for such localities concrete is to be
preferred.
To prevent the lateral escape of the supporting material, when build-
ing in running sand or soft clay, which would ooze out from below the
work and allow the superstructure to sink, in addition to protecting the
surface with planking, concrete, or timber, it is often necessary to enclose
the whole area of the foundation with piles of timber or plank driven close
together ; this is called sheet-piling.
Where there is a hard stratum below the soft ground, but at too great
a depth to allow of the solid work being brought up from it without greater
expense than the circumstances of the case will allow, it is usual to drive
ARCHITECTURAL DRAWING.
211
down wooden piles, often shod with iron, until their bottoms are firmly
fixed in the hard ground. The upper ends of the piles are then cut off
level, and covered with a platform of timber, on which the work is built
in the usual way. The piles are generally of about 1 foot diameter, and
are driven at distances of from 2 to 3 feet from centre to centre.
Where a firm foundation is required to be formed in a situation where
no firm bottom can be found within an available depth, piles are driven,
to consolidate the mass, a few feet apart over the whole area of the foun-
dation, which is surrounded by a row of sheet-piling to prevent the escape
of the soil ; the space between the pile-heads is then filled to the depth of
several feet with stones or concrete ; and the whole is covered with a timber
platform on which to commence the solid work.
WALLS.
"Walls of permanent structures are almost exclusively composed of
either stone or brick, or both, and are included in one general term as
masonry.
Fig. 1 represents the front of a wall called the face ; fig. 2, a section ;
and fig. 3, the view of rear, or the "backing. The interior of the wall is
1
1
I
II 1
p
ii
nzz
ii
J-i
II
ii
1;
II
ll
I
Ji
li
](_
1; II
1
II
I!
_J
D II
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d
ll
p ( |i
1
1 -
^L_
II
ll
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_P
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t
II
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I
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ir
J
IEZ
cur
Fig. 1.
Fig. 2.
JLJL
Fig. 3.
called the filling. The term course is applied to each horizontal layer of
stone or brick ; if all the stones in a layer are of equal thickness, it is
termed regular coursing, footings are the lower projecting courses (figs.
1 and 2) resting on the foundation, usually not less than double the width
of the wall above in walls of buildings ; but for other walls, the width de-
pends on the nature of the foundation. Hock foundations need no extra
width of wall. String courses, or belting, are upper courses projecting
212 AECeiTECTUEAL DKAWIXG.
slightly beyond the face of the wall. Coping is the top courses, usually
got out in considerable lengths in comparison with the stones in the rest
of the work.
The beds of a stone or brick are the surfaces on which they rest ; the
build is the upper bed on which the stone above is placed ; the inter-
stices between the stones are termed joints. /Stretchers are stones or brick
which have their length disposed lengthways of the wall ; headers have
their length crossways. Quoins are the corners of a wall. Bond is the
lapping of the stones or brick on each other in the ^ construction, so as to-
tie the separate pieces together. Three classes of bond are shown in the
face (fig. 1) ; the lowest six courses consist of alternate courses of headers
and stretchers, the next six courses above have alternate headers and
stretchers in the same course, and in the remaining courses a header occurs
at every third stone ; this is the most usual bond. Headers should not be
placed one above the other in alternate courses.
Figs. 4 and 5 represent brick bonds ; fig. 4, the old English bond, and
,. >l f t i I 1
1 j
i L CI~~]
1
k^ l I i ' , 1 *
' K
lilt
1
Fig. 4. Fig. 5.
fig. 5, the Flemish bond. The most common bond in this country is to
lay a certain number of stretching courses, and then a heading course.
The fire-law of New York requires brick work to be built with headers
every five courses, but every seventh course a heading course is more com-
monly used. In all masonry, no vertical joint should extend through two
courses, but the vertical joints should be as near as possible midway be-
tween those below ; in other words, break joint with them.
Walls are composed of stones laid either with or without mortar. The
latter is called dry masonry ; rough wall is dry work of rough stones ; if
laid in mortar, it is called rubble work, but frequently this term is made to
include all rough work. Cut stone is called ashler / if only cut on beds
and joints, it is called rock or quarry faced ashler.
On the thickness of walls. Retaining-^dhs are such as sustain a lateral
pressure from an embankment or head of water (figs. 6 and 7). The width
of a retaining-wall depends upon the height of the embankment which it
may have to sustain, and the kind of earth of which it is composed (the
steeper the natural slope at which the earth would stand, the less the thrust
AKCHTTECTTJKAL DRAWING.
213
Fig. 6.
against the wall), and the comparative weight of the earth and of the ma-
sonry. The formula given by Morin for ordinary earths and masonry is
I = 0.285 h + h' ; that is, to
find the breadth of a wall
laid in mortar, multiply the
whole height of the embank-
ment above the footing by
285
1000
thickness one-fourth more.
Most retaining or brick
walls have an inclination or
batter to the face, sometimes
also the same in the back, but offsets (fig. 6) are more common. The usual
batter is from 1 to 3 inches horizontal for each foot vertical. To determine
the thickness of a wall having a batter, " determine the width by the rule
above, and make this width at one-ninth of the height above the base."
In most large cities there are Building Acts in force, to which all con-
structions within their limits must conform. Copious extracts from the
New York law may be found in the Appendix.
Figs. 8, 9, 10, 11, p. 214, represent the thickness of external brick walls
to the first, second, third, and fourth-rate buildings, as provided by the
Building Act for the city of London. Figs. 12, 13, 14, 15, show the same with
respect to party-walls. The figures 1, If, 2, 2f, represent the number of
lengths of brick in the wall. The following table gives the rate in Liver-
pool, not differing, constructively, materially from that of London.
HEIGHTS AKD WIDTHS OF BUILDINGS.
First-rate dwelling-house.
Second-rate dwelling-house.
Third-rate dwelling-house.
Fonrth-rate dwelling-house.
Exceeding forty -four feet
in height, or twenty -seven
feet front.
Not exceeding forty-four
feet in height, or twenty-
seven feet front
Not exceeding thirty-six
feet in height, or twenty-
one feet front.
Not exceeding thirty-two
feet in height, or fifteen
feet front.
" Every brewery, distillery, manufactory, or warehouse, of whatever
height or extent of frontage, is considered to be a first-rate building, the
external walls of which are in their respective stories to be 2f , 2, and 1
bricks in thickness, and the party-walls of 2 and 1| bricks."
Walls built of rubble should also be somewhat thicker than those of
brick on an average at least one quarter thicker, but depending on the
character of the stone. The common form of cut- work is to make the face
ashler, and back up with rubble or brick.
ARCHITECTURAL DRAWING.
Fig. 12
Fig. 8
Fig. 10 Fig. 9
Fig. 11
Fig. 15
X
rnLirnrj
dJL~l
a an an 11
I U\ I 2 JL 2
CZl OUEZJLlJLJrillUJ
81
ARCHITECTURAL DRAWING.
215
Mortar is a mixture of lime or cement, or both, with water and sand.
In the preparation of mortar, the materials should be well selected ; the
sand sharp and clean, the proportions properly preserved, and the whole
intimately mixed. As a general rule, the lime or cement should be suf-
ficiently fine to cover all the grains of sand, with the thinnest possible stra-
tum. Practically, about three or four cubic feet of sand are added to one
cubic foot of half liquid lime, for fat limes ; lean lime may not bear more
than half this sand. Cement is generally mixed with sand in proportions of
1 to 3, but in situations where a quick set is necessary, in equal proportions.
Arches. Arches are of various shapes, as,
.i^ ^~
\
/
\ /
/
V
Elliptical,
Segmental,
Circular.
Pointed,
The outer surface of the arch is called the extrados or back of the arch,
the inner or concave surface the intmdos or the soffit; the joints of all
arches should be perpendicular to the surface of the soffit. The stones
are called arch stones or voussoirs. The first course on each side are
termed springers, which rest on the imposts or abutments. In case of a
segmental arch, the course beneath the springers are called skew-backs.
The extreme width between springers is called the span of the arch, and
the versed sine of the curve of the soffit the rise of the arch. The highest
portion of the arch is called the crown, and the centre course of voussoirs
the key-course. The side portions of arches between the springing and the
crown, are termed haunches or flanks. All arches should be well sustained
by backing on the haunches, called spandrel-backing. The line of inter-
section of arches cutting across each other transversely is called a groin,
and the arches themselves groined arches.
That the voussoirs of an arch may resist crushing, they must have a
certain depth proportioned to the pressure of the arch ; and as this in-
creases from the curve towards the springing, the depth of the voussoirs
should likewise increase from the crown to the springing. Peronnet has
given as a rule for the depth at the crown the formula d = .07 r + 1 foot,
in which formula r is the greatest radius of curvature of the in trades.
This formula is applicable to arches less than fifty feet radius ; but beyond
this it gives greater dimensions than in ordinary practice.
216
ARCHITECTURAL DRAWING.
FRAMING.
Framing is the art of arranging beams for the various purposes to
which they are applied in structures. Timber and iron are the only ma-
terials in common use for frames.
"Wooden beams are usually represented longitudinally both in eleva-
tion and in section by their outlines merely, or in end view by rectangles
with diagonals from opposite corners (fig. 18), and in end section by the
usual diagonal lines in
one direction across the
face. Sometimes in
more finished drawings,
or when a distinction is to be marked between different materials, the
grain of the wood is represented as in figs. 19 and 20, a side and end ele-
vation of a beam.
Flooring. The timbers Avhich support the flooring-boards and ceiling
of a room are called the naked flooring.
The simplest form of flooring, and the one usually adopted in the con-
struction of city houses and stores is represented in plan and section, fig.
21. It consists of a single series of beams or deep joists, reaching from
wall to wall. As a lateral brace between each set of beams, a system of
Fig. 13.
Fig. 19.
Fig. 20.
11
Fig. 21.
bridging is adopted, of which the best is the herring-bone bridging, formed
ARCHITECTURAL DRAWING.
217
of short pieces of joists about 2x3, crossing each other, and nailed securely
to the top and bottoms of the several beams, represented by a and 5 in
fig. 21, and wherever a flue occurs, or a stairway or well-hole prevents one
or more joists from resting on the wall, a header, H, is framed across the
space into the outer beams or trimmer-beams T T, and the beams cut off
or tail-beams are framed into the trimmer.
Whenever the distances between the walls exceed the length that can
safely be given to joists in one piece, an intermediate beam or girder,
running longitudinally, is introduced, into
which the joists are framed (fig. 22). Very
often the joists are merely notched on to
beams. Flooring is still further varied, by
framing with girders longitudinally ; beams
crossways, and framed into or resting on the
girders ; and joists framed into the beams,
running the same direction as the girders. rig. 22.
It is evident, that when the joists are not flush or level with the bottom
of the beams or girders, either that in the finish the beams will show,
or that ceiling-joists or furrings will have to be introduced.
On the size of joists. The following dimensions, taken in part from
the Liverpool Building Act, may be considered as safe sizes for ordinary
constructions, the distances from centre to centre being one foot.
Joists in floors, clear bearing
Exceeding 7 feet, and not exceeding 10 feet, to be not less than 6x2 inches.
" 10 " " " 12 " " " 6 x 2J- "
, " 12 " " 14^ " " " 7 x 2J- "
" 14J- " " " 16 " " " 8 x 2i "
" 1G " " " 18 " " 9 x 2| "
" 18 " " " 20 " " 10 x 2 "
" 20 " " " 22 " " 11 x 3 "
22 " " 24 " " " 12 x 3 "
It is to be observed that lumber is seldom sawed to dimensions of frac-
tions of an inch ; we must therefore adopt a width of an integral inch, and
proportion the distances from centre to centre, according to the increase or
decrease of width given to the joists.
Trimmer beams and headers should be of
greater width than the other beams, depend-
ing on the distance of the headers from the wall,
and the number of tail beams framed into it.
The New York Building Act requires all headers
should be hung in stirrup irons (fig. 23), and not framed in. It also re-
Fig. 23.
218
ARCHITECTURAL DRAWING.
quires all girders to be not less than 10 by 12 inches square, and that the
posts supporting them, shall be placed at intervals of not more than 10
feet.
Floors. In New York it is usual to lay single floors, of tongued and
grooved boards, but in the Eastern States, double floors are more common ;
the first floor consists of an inferior quality of boards, unmatched, laid
during the progress of the work as a sort of staging for the carpenter and
mason, and in finishing, a second course is laid on them of better material,
generally tongued and grooved, but ^sometimes only jointed. Ceilings
should always be furred ; that is, laths should never be nailed directly
to the joists; the usual furrings are of inch board, two inches wide,
and twelve inches from centre to centre, nailed across from joist to
joist.
Fig. 2-i. represents a section of a mill floor. The girders or beams,
generally in pairs with a space
of about an inch between them,
are placed at a distance of from
Fig. 24. seven to nine feet from centre
to centre, and are of from twelve to sixteen inches in depth. On these, a
rough plank floor of from three to four inches thick is laid ; the plank
are dowelled together, that is, put together with pins or dowels, like a
barrel head. Above the plank is laid the usual top floor, and beneath a
sheathing of thin boards.
For extended bearings and for heavy loads, it is often found necessary
to truss the girders or beams. Fig. 25 represents a bracing truss of
Fig. 25.
wrought iron between a double girder ; often a simple piece of arched iron
is let into the wood, half on each side, and the beams bolted strongly to-
Fig. 26.
gether. Fig. 26 represents a truss by suspension ; in this case, the strength
depends upon the cohesive force of the iron.
AKCHITECTTJKAL DRAWING. 219
Fire-proof floors. Fig. 27 represents a section of the fire-proof flooring
Fig. 27.
constructed by Cooper & Hewitt. The girders or beams are of wrought
iron, with arches of a single course of brick in cement between them,
resting on their lower flanches. The seven-inch deep beams are placed at
a distance of from three 'to five feet from centre to centre ; extreme width
of span, between side walls, fifteen feet. Strips of plank are fastened
lengthways at the side or on top of the beams, to receive the floor. Fur-
rings for the ceiling may be attached crossways to the bottom of the beams,
or the soffits of the arches may be plastered without any preparation.
Fig. 28 represents a section of one of the French systems of fire-proof
floors. It Consists Of J
girders, placed at a dis-
tance of one metre (39.38
inches), from centre to
centre, slightly cambered Fig. 28.
or curved upwards in the centre ; the depth of the girders to depend
upon the span. Stirrups of cast iron are slid upon the girders, into which
the ends of flat iron joists, set edgeways, pass and are secured by pins ;
the ends of the joists take a bearing also on the bottom flanges of the
girders. The joists are placed at a distance of one metre from centre to
centre. Upon the joists rest rods of square iron, which in this way form a
grillage for the support of a species of rough cast and the ceiling. By
this and other very similar systems, the French have succeeded in reducing
the cost of such floors to that of wooden ones.
Floors are sometimes constructed of brick in single or in groined
arches, the thrust being opposed by the weight of the abutments, but
owing to its expensiveness, and the amount of room occupied by the ma-
terial, this kind of construction is not at present very common in edifices.
Partitions are usually simply studs set at intervals of twelve or six-
teen inches, these spaces being adapted to the length of the lath (forty-
eight inches). The sizes of the studs are generally 2 x 4, 3 x 5, or 3 x 6
inches, according to the height of the partition ; for any high partitions,
greater depth may be required for the studs, but three inches will be suf-
ficient width. Partitions should be bridged like floors with herring-bone
bridging.
220
ARCHITECTURAL DRAWING.
'I g-LL
JJ LL
Fig. 29 represents the frame of the side of a wooden house, in which
A A are the posts, B
the plate, C C girts or
interties, ~D D braces, E
sill, F window posts or
studs, G G studs.
Usual dimensions of
timber for frame of com-
mon dwelling houses :
sills 6x8, posts 4x8,
studs 2 x 4 or 3 x 4,
girts 6 x the depth of
floor joists, plates 4 x
6, the floor joists (J fig.
30), are notched into the
girts ; more frequently
the girts are omitted.
Tig. 29.
Fig. ML
Fig. 31.
ID n
The studs are of the same length
as the posts, and the floor joists are
supported by a board, a, 3 or 4 x 1
let into the studs (fig. 31), and firmly
nailed ; the joists are also nailed
strongly to the studs. The posts and
studs are tenoned into the sills and
girts. Fig. 32 represents a tenon,
& c, in side and end elevation, and
mortice, a; the portions of the end
of the stud resting on the beam
are called the shoulders of the
tenon.
Hoofs. The roofs of city dwellings and stores are generally flat, that
is, with but very little inclination, from half an inch to two inches per foot,
merely sufficient to discharge the water. The beams are laid from wall to
wall, the same as floor timbers, but usually of less depth, or at greater dis-
tances between centres, and with one or two rows of bridging. The roof
is laid with tongued and grooved boards, and mostly covered with tin.
Figs. 1, 2,3, PI. XLV. represent side or portions of side elevations of
the usual form of framed roofs. The same letters refer to the same parts
in all the figures of the plate. T T are the tic, beams, E R the main rafters^
Fig. 32.
AECHITECTCEAL DRAWING. 221
rr the jack rafters, P P the plates, pp the purlines, KK the king posts,
k k king bolts, q q queen bolts, botli are called suspension bolts, C C the
collar or straining beams, B B braces or studs, b b ridge boards, c c cor-
bels.
The pitch of the roof is the inclination of the rafters, and is usually
designated in reference to the span as \, 1, f , &c., pitch, that is, the height
of the ridge above the plate is , |, |, &c, of the span of the roof at the
level of the plate. The higher the pitch of the roof, the less the thrust
against the side walls, the less likely the snow or water to lodge, and con-
sequently, the tighter the roof. For roofs covered with shingles or slate,
in this portion of the country, it is not advisable to use less than i pitch ;
above that, the pitch should be adapted to the style of architecture
adopted. The pitch in most common use is 1 the span.
Fig. 1 represents the simplest framed roof; it consists of rafters, resting
upon a plate framed into the ceiling beam ; this beam is supported by a
suspension-rod, k, from the ridge, but if supported from below, this rod
may be omitted. This form of construction is sufficient for any roof of
less than 25 feet span, and of the usual pitch, and may be used for a 40
feet span by increasing the depth of the rafters to 12 inches ; deep rafters
should always be bridged. By the introduction of a pmiine extending
beneath the centre of the rafter, supported by a brace to the foot of the
suspension rod, as shown in dotted line, the depth of the rafters may ob-
viously be reduced. It often happens that the king-bolt may interfere with
the occupancy of the attic ; in that case the beam is otherwise supported.
Again, it may be necessary that the tie beam, which is also a ceiling and
floor beam, should be below the plate some 2 to 4 feet ; in that case, the
thrust of the roof is resisted (fig. 4) by bolts, l> b, passing through the
plate and the beam, and by a collar plank, C, spiked on the sides of the
rafters, high enough above the beam to afford good head room. For roofs
| pitch and under 20 feet span, the bolts are unnecessary, the collar alone
being sufficient.
Fig. 2 represents a roof, a larger span than fig. 1 ; the frame may be
made very strong and safe for roofs of 60 feet span. King-bolts or sus-
pension-rods are now oftener used than posts, with a small triangular
block of hard wood or iron, at the foot of the bolts, for the support of the
braces. The objection to this form of roof is that the framing occupies
all the space in the attic ; on this account the form, fig. 3, is preferred for
roofs of the same span, and is also applicable to roofs of at least 75 feet
span, by the addition of a brace to the rafter from the foot of the queen-
bolt. The collar beam (fig. 6), is also trussed by the framing similar to
222 ARCHITECTURAL DRAWING.
fig. 2. In the older roofs, queen posts are used (fig. 33), with the foot
secured by straps or joint bolts to the tie beam.
In many church and barn roofs the tie beam is cut off,
fig. 5 ; the queen post being supported on a post, or itself
extending to the base, with a short tie rod framed into it
from the plate.
Figs. 7 and 8 are representations of the feet of rafters
on an enlarged scale. In fig. 7, the end of the rafter does not
project beyond the face of the plate ; the coving is formed
by a small triangular, or any desirable form of plank,
Fig. 33. framed into the plate. The form given to the foot of the
rafter is called a crowfoot. In fig. 8, the rafter itself projects beyond the
plate to form the coving. Fig. 9 represents a front and side elevation and
plan of the foot of a main rafter, showing the form of tenon, in this case
double ; a bolt passing through the rafter and beam retains the loot of the
former in its place. Fig. 10 represents the side elevation of the foot of a
main rafter with only a small portion of the beam, the remainder being
supplied by a rod. In fig. 7, of a similar construction to fig. 1, the tie rod
passes directly through the plate. In general, when neither ceiling nor
flooring is supported by the tie beam, a rod is preferable.
Roofs are now very neatly and strongly framed by the introduction of
cast-iron shoes and abutting plates for the ends of the braces and rafters.
Fig. 11 represents the elevation and plan of a cast-iron king head for a
roof similar to fig. 2. Fig. 12, that of the brace shoe ; fig. 13, that of the
rafter shoe for the same roof. Fig. 14, the front and side elevation of the
queen head of roof similar to fig. 3, and fig. 15,
elevation and plan of queen brace shoe.
Fig. 34 represents the section of a rafter shoe
for a tie rod ; the side flanches are shown in dotted
rig. 34. line.
On the size and the proportions of the different members of a roof:
Tie beams are usually intended for a double purpose, and are therefore af-
fected by two strains ; one in the direction of their length from the thrust
of the rafters, the other a cross strain, from the weight of the floor and
ceiling. In estimating the size necessary for the beam the thrust need not
be considered, because it is always abundantly strong to resist this strain,
and the dimensions are to be determined as for a floor beam merely,
each point of suspension being a support. When tie rods are used, the
strain is in the direction of their length only, and their dimensions can be
calculated, knowing the pitch, span, and weight of the roof per square
ARCHITECTURAL DRAWING. 223
foot and the distance apart of the ties, or the amount of surface retained
by each tie.
Rule. Multiply one half the weight of the roof by one half the span,
and divide the product by the pitch.
Example. What is the strain upon the tie rod of a roof 40 feet span
and 15 feet pitch ?
The weight of the wood-work of the roof may be estimated at 40 Ibs.
per cubic foot, or on an average at about 12 Ibs. per foot square, slate at 7
to 9 Ibs., shingles at 1 to 2 Ibs. The force of the wind may be assumed
at 15 Ibs. per square foot. The excess of strength in the timbers of the
roof as allowed in all calculations, will be sufficient for any accidental and
transient force beyond this. If, therefore, the roof be like fig. 1, Plate
XLV., without ceiling beneath, and retained by a tie rod, we may consider
as the weight per square foot for a slate roof: 12 + 7 + 15 = 34 Ibs. The
length of the rafter is ^20* + 15 r =: 25 feet; hence, if the tie rods are 10
feet apart, the amount of surface on each incline, or one half of the roof,
supported by the tie is 25 x 10 = 250 square feet, which multiplied by
the weight per square foot, or 250 x 34 = 8,500 Ibs. ; applying the rule,
8500 x 20 -5- 15 = 11,333 Ibs., the thrust on the tie rod. If we estimate the
strength of wrought-iron at 10,000 Ibs. per square inch of section, or 8,000
1 1 33S
Ibs. when a thread is cut upon the end, then, = 1.416 square inches,
8000
or a rod a little exceeding 1 T 5 F inches in diameter.
The rafters (fig. 1, PI. XLV.) may be considered as jack rafters of long
bearings, or as a beam supporting transversely the weight of the roof, and
the accidental pressures, and may be estimated by resolving the direction
of these pressures into a line perpendicular to the direction of the rafters.
The pressure on main rafters (figs. 2 and 3) is in the direction of their
length, when they are supported by braces at or very near the points where
the purlines rest ; but in addition to the weight of the roof, they support a
portion of the weight of the tie beam, and whatever may be dependent
upon it. If the frame is like fig. 2, that is, with a king^-bolt or post, and
the weight is uniformly distributed upon the beam, then one half the
weight is supported by the bolt or post, and consequently by the rafter,
and the other half by the side walls. Under the same circumstances, the
suspension rods (fig. 3), support each of the weight of the beam, &c., and
the side walls each . But in general, where the attic is made use of, the
load is not uniformly distributed, by far the greatest part is suspended
upon the rods.
224: AKCHITECTUEAL DK AWING.
To find the pressure on the main rafter. Multiply one half the weight
of the roof, and that portion of the weight of the beam and its load which
may depend upon it, by the length of the rafter, and divide the product
by the pitch.
Example. What is the pressure upon the main rafter of a slate roof
of 56 feet span, 21 feet pitch, frames 10 feet between centres, and form
like fig. 3, with an uniformly distributed load on the beam of 8,400 Ibs.,
and a load between the suspension rods of 10,000 Ibs. ?
The length of rafter is */28' + 21' = 35 feet.
Assuming the load per square foot upon the rafter at 40 Ibs. per
square foot,
then 40 x 31 x 10 = 12,400 Ibs., the weight of roof.
i of the uniform weight = 2,800 Ibs.
1 0000
\ the weight between rods * = 5,000 Ibs.
Total, 20,200 Ibs.
20200_x35 =33)6( , 61te _
If we now assume the resistance of wood at 740 Ibs. per square inch,* or
600 Ibs. for length exceeding 13 times their thickness,
= 56.111 square inches of section.
oUU
The proportion of the depth to the width is generally about 10 to 8 or 5
to 4.
Hence |/|? x 5 = 1.67 x 5 = 8.35 inches = depth.
1.67 x 4 = 6.68 inches width.
Gwilt, in his Architecture, recommends the following dimensions for
portions of a roof:
* Weisbach.
ARCHITECTURAL DRAWING.
225
SPAN.
FORM OF EOOF.
RAFTERS.
BRACES.
POSTS.
COLL'R BEAMS.
feet.
inches.
inches.
inches.
inches.
25
Fig. 2, Plate XLVI.
5x4
5x3
5x5
30
u
6x4
6x3
6x6
35
Fig. 3, "
5x4
4x2
4x4
7x4
45
"
6x5
5x3
6x6
7x6
50
2 Sets of Queen Posts
8xG
5x3
( 8x8)
9x6
(8x4)
GO
14
8x8
6x3
j 10 x 8 )
11 x 6
( 10 x 4 1
These dimensions, for rafters, are somewhat less than the usual practice
in this country ; no calculations seem to have been made for using the attic.
An average of common roofs here, would give the following dimensions
nearly: 30 feet span, 8x5 inches; 40 feet, 9 x 6 ; 50 feet, 10 x 7; 60
feet, 11x8; collar beams the same size as main rafters. Eoof frames
from 8 to 12 feet from centre to centre.
Dimensions for jack rafters 15 to 18 inches apart.
For a bearing of 6 feet,
<t if g U
" " 10
12 "
20 "
3 x 2| inches.
4x3 "
5x3
6x3 "
10 x 3
Purlines :
LENGTH OF BEARING.
DISTANCES APART IN FEET.
feet.
6
8
10
12
6i
6 x 3
61 x 4
7x5
8x5
8
7x5
8x5
9x5
9x6
10
9x5
10 x 5
10 x 6
11 x 6
12
10 x 6
11 x 6
12 x 7
13 x 8
The pressure on the plates is transverse from the thrust of the rafters,
but in all forms except fig. 1, owing to the notching of the rafters on the
purlines, this pressure is inconsiderable. The usual size of plates for figs.
1 and 2, is 6 x 6 inches. For forms, fig. 1, the size depends on the mag-
nitude of roof, and the distance between, the ties. The width in all such
cases to be greater than the depth ; 4 to 6 inches may be taken as the
depth, 8 to 12 for the width.
Joints. As timber cannot always be obtained of sufficient length for
the different portions of a frame, it is often necessary to unite two or more
15
226
ARCHITECTURAL DRAWING.
pieces together by the ends, called scarfing or lapping. Figs. 35 and 36
are the most common means of lapping or halving,
( J~r~l / which methods may be employed when there is
not much longitudinal compression or extension.
When such an effect is to be provided for, the
upper as well as the lower timber should be let
F ig. 86. into each other. Figs. 37, 38, 39, 40, 41, are differ-
Fig. 85.
Fig. 40. Fig. 41.
ent methods to obtain this result. In figs. 37, 38, the joint is brought to a
bearing by a key driven in tight. Fig. 39 represents a scarf suited for a
beam supported at this joint by a post, and where there is tensile strain,
the timber should be joint-bolted or anchored. The centre of the post
should be beneath the extreme edge of the lower joint. Figs. 40 and 41
are long scarfs, in which the parts are bolted through and strapped, suited
for tie beams. Joints are also often made by abutting the pieces together
and bolting splicing pieces on each side ; still further security is given by
cutting grooves in both timbers and pieces and driving in keys.
Varieties of Roofs. Roofs of country edifices especially are considera-
bly varied in outline ; without central supports the forms of frames are but
modified examples of those already given ; with central supports, many, like
the roofs of city buildings, may be considered as the framing of floors, or the
sidings of wooden houses. City roofs are generally composed of beams laid
from wall to wall, or girder to girder, with a slight incline of from %" to 1"
per foot, with the drip and gutter at the rear, with the front finished with
a cornice, and sometimes a false French roof, or a show of a steep pitch.
Square houses are often framed with hipped roofs, that is, with a pitch
on each side, as at A and B, figs. 42 and 43, which are called hips, whilst
C and D are gables. The fig. 43 represents the plan of a roof as usually
drawn in strong or close line or deep shade at the ridge, and lighter at the
ARCHITECTURAL DRAWING.
227
eaves. The Gambrel or Mansard roof (fig. 44) is a roof with two kinds of
pitch ; the theory of its construction is that of the polygon of rods (p. 118),
but in general they have central support from partitions, and their outlines
are much varied by curves in the lower rafters cut from plank.
Fig. 48.
\
The roofs of cotton and woollen factories are often framed with two
pitches, with a small upright part at the angle for windows, extending
the whole length of the roof. It may be framed as two roofs, the lower,
as in fig. 3, Plate XLY, and the upper, plain rafters projecting suffi-
ciently beyond the purline, and over the roof, to receive the window-
frame.
In the framing of roofs it has been the author's practice of late years to
disregard entirely purlines, jack -rafters, and plates, and make the covering
of plank 2" to 3", according to space between frames, tongued and grooved,
and spiked strongly to the main rafters. The. roofs of the gate-houses at
New Croton Reservoir are modified examples of this ; with plates trussed
laterally, as the angles of the roofs were cut by ventilating towers.
Circular Roofs. Circular roofs have often been constructed of strips
of boards, cut to the width of the rafters, bent into the form of the projected
arc, and nailed to the depth required for the span of the roof. In northern
climates they are objectionable on account of the flatness of the top and the
unequal distribution of a load of snow. Very large-span cylindrical roofs
have been made for station-houses of Howe's truss (fig.58), with circular
chords. Small cylindrical roofs, say not exceeding 30 feet span, are otten
made of curved and corrugated iron, with tie-bolts just above the eaves.
Eaves of roof are finished with cornices, of various mouldings appro-
priate to the style of the building. The gutters of eave-troughs are gene-
228
ARCHITECTURAL DRAWING.
rally formed in the cornice (fig. 45) ; sometimes on the top of the roof (fig.
46), and sometimes by raising a parapet (fig. 47), and forming a valley.
Fis. !<i. Fig. 45. Fig. 47.
Iron Roofs. PL XL VI. fig. 1, represents the half elevation of an iron
roof of a forge at Paris ; figs. 2, 3, 4, details on a larger scale. This is a
common form of iron roof, consisting of main rafters, E, of the J section,
fig. 4, trussed by a suspension rod, and tied by another rod. The pmiines
are also of I iron, secured to the rafters by pieces of angle iron on each
side ; and the roof is covered with either plate iron resting on jack rafters,
or corrugated iron extending from purlin e to puiiine. The rafter shoe, A,
and the strut, S, are of cast iron, all the other portions of the roof are of
wrought iron.
The surface covered by this particular roof, is 53 metres (164 feet) long,
and 30 metres (98 feet) wide. There are 11 frames, including the two at
the ends, which form the gables.
The following are the details of the dimensions and weights of the dif-
ferent parts :
Ibs.
2 rafters, 0.72 feet deep, length together, 99.1 feet, . . . 1,751
5 rods, 0.13 feet diameter, " " 131.4 "... 882
16 holts, " 79
8 bridle-straps, 0.24 x .05 ...... 123
2 pieces, .046 thick, connecting the rafters at the ridge, >
4 " " at the foot of the strut . . >'
4 " .036 thick, uniting the rafters at the junction in the strut together
with their bolts and nuts, . . . . . 1 76
2 cast iron struts, ....... 308
2 rafter shoes, . ..... 287
Total of one frame, . . . . . . . 3,695
16 purlines, 1 ridge iron, each 0.46 deep, 17.2 long, .... 2,985
Bolts for the same, ....... 64
16 jack rafters, I iron, 0.16 deep, ...... 2,489
Weight of iron covering, including laps, per square foot, . . 2.88
ARCHITECTURAL DRAWING. 229
The weight of iron in this roof could be reduced by substituting corru-
gated iron for the covering, even of less weight per square foot, and omit-
ting the jack rafters.
Roofs are sometimes made with deep corrugated main rafters with flat
iron between, or purlines and corrugated iron for the covering. The great
objection to iron roofs, lies in the condensation of the interior air by the
outer cold, or, as it is termed, sweating ; on this account they are seldom
used for other buildings than boiler houses or depots, except a ceiling be
made below to prevent the contact of the air inside with the iron.
Fig. 5 is an elevation of nearly two of the three panels of one of the
cast iron girders for connecting the columns, and carrying the transverse
main gutters, which supported the roof of the English Crystal Palace.
Figs. 6, 7, 8, 9, 10, 11, sections of various parts on an enlarged scale.
The depth of the girder was 3 feet, and its length was 23 ft. 3f inches.
The sectional area of the bottom rail and flange in the centre (fig. 7), was
6J- square inches ; the width of both bottom and top rail (fig. 6), was re-
duced to 3 inches at their extremities. It will be observed that the section
of the braces and ties are such as to give great stiifness, and the- section of
the braces at ~b ~b (fig. 8), is greater than at c c (fig. 9) ; the section of the
tie (fig. 10), is the same as the brace at c c ; they are all formed with a
draft, that is, with a taper from the centre to the outside of from ^ to T V
of an inch on a side, according to the depth of the feather.
The weight of these girders was about 1,000 Ibs., and they were proved
by a pressure of 9 tons, distributed on the centre panel.
A second series of girders were made of similar form to fig. 5, but of
increased dimensions in the section of their parts. Their weight averaged
about 1,350 Ibs., and they were proved, as above, to 15 tons.
A third series, of increased section of parts, weighed about 2,000 Ibs.,*
and were proved to 22 tons.
Fig. 12 represents an elevation of two of the nine panels of one of the
wrought iron trusses which carry the lead flat and arched roof across the
nave of the Crystal Palace. These trusses are 72 feet long and 6 feet deep.
The top rail G, shown in section fig. 13, consists of two angle irons 4^
inches deep, 3^ inches wide, and | of an inch thick, with a plate 9 inches
wide and f thick, riveted on top. A space of 2 inches is left between the
angle irons. The angle irons are in five lengths, and are connected by
eight rivets passing through them, and through a plate or plates intro-
duced between them. The top plate is in seven lengths, connected by i
inch rivets, through the angle irons, the plate, and a joint plate. The top
plate is riveted to the angle irons by 1 inch countersunk rivets, 5 inches
230 ARCHITECTURAL DRAWING.
apart. The bottom rail consists of two flat wrought iron bars, 6 inches
deep, with a 2 inch space between them. It is in four lengths, jointed by
six 1 inch rivets passing through joint plates 6 x 18 x jf inches on the
outside, and three plates 17 X f inches. The bars forming the central
lengths of the bottom rail are of an inch thick, and those forming the
side lengths are f of an inch thick.
The end standards are of cast iron, 3 inches wide, 4 inches deep, and
1 inch thick, of a T form of section, secured to the column by six l inch
bolts. The standard is 2 inches thick at top and bottom where it receives
the rails. Two sockets are formed in the middle, to receive the diagonals
I and J. I, being exposed to compression, is made of four angle irons,
2i x 2i x T V inch, riveted together in pairs with inch rivets. The
diagonal J is formed of two bars, 4 x \ inch, and is secured at each end
by a 1 f inch rivet. The ends are thickened by short plates riveted to them
to make up in a measure the loss of strength from the large rivet hole.
The diagonal K is formed of two bars 4| inches deep by 1 inch thick, and
is fixed at each end by a 2 inch bolt and nut. The other diagonals, being
exposed to much less strain, are formed of single bars 4| x \ inch, and
are secured at each end by a 1 inch rivet.
The standards B and C consist each of four angle irons, 2i x 2i x \
inch, riveted together in pairs, and the two pairs riveted together with six
small cast iron distance pieces between them. The next standard, that is,
the third from each end, but not shown in the drawing, is of cast iron. It
is of + section, being at the centre 6x6 inches, thickness of metal f to
of an inch. The base, which rests upon the base of the bottom rail, is 18
x 4 inches, and the top is 18 x 3 inches. Triangular projections enter
the top and bottom rail, where they are secured by 1 inch rivets. In the
Centre is a socket or slot through which pass the two light diagonals.
The main strength of the truss consists in the top and bottom rails, the
diagonals I, J, K, the first wrought iron standard, B, and the cast iron
standard, D.
On the General Principles of Bracing. Let fig. 42 be the elevation
of a common roof truss, and let
a weight, W, be placed at the
foot of one of the suspension
rods. Now, if the construction
consisted merely of the rafter
C' B, and the collar-beam C' C, F1 g- 43 -
resting against some fixed point, then the point B would support the whole
downward pressure of the weight ; but in consequence of the connection
ARCHITECTURAL DRAWING.
231
of the parts of the frame, the pressure must 'be resolved into components
in the direction C' A and C' B, C' 5 will represent the pressure in the
direction C' B, C' w the portion of the weight supported at B, C' a the
pressure in the direction C' A, and w W the portion of the weight sup-
ported on A. The same resolution obtains to determine the direction and
amount of force exerted on a bridge truss of any number of panels, by a
weight placed at any point of its length (fig. 43.) In either case, the
effect of the oblique form C' A, upon the angle C, is evidently to force
r. c' it upwards ; that is, a weight placed at one
side of the frame has, as in case of the arch,
a tendency to raise the other side. The effect
Fi s- ^ of this upward force is a tension on a por-
tion of the braces, according to the position of the weight ; but as braces,
from the manner in which they are usually connected with the frame, are
not capable of opposing any force of extension, it follows that the only
resistance is that which is due to the weight of a part of the structure.
Figs. 44 and 45 illustrate the results of overloading at single points
such forms of construction.
Fig. 45.
Fig. 46.
To remedy this effect, if counter braces be introduced, as shown in dot-
ted lines (fig. 46), the tendency of a weight moving across the structure is
to compress the counters and extend the braces. But since, as we have
said, braces are not usually framed, especially in wooden structures, to re-
sist a tensile strain, it is necessary to overcome this force in another way ;
that is, by introducing wedges between the end of the counter braces and
the joints against which they abut, or by means of the counters and the
suspension rods, in any way straining the structure so that there may be
an additional compression upon the brace more than the upward or tensile
force exerted by any passing weight. In this case, therefore, the passage
of a load would produce no additional strain upon any of the timbers, but
232 ARCHITECTURAL DRAWING.
would tend to relieve the counters. The counter braces do not, of course,
assist in sustaining the weight of the structure; on the contrary, the
greater the weight of the structure itself, the more will the counter braces
be relieved.
If, instead of the counter braces, the braces themselves are made to act
both as tie and as a strut, as has been done sometimes in iron bridges and
trusses, then the upward force will be counteracted by the tension of the
brace, but counter braces are preferable, as it is better that the force
exerted against any portion of the structure should always be in one direc-
tion.
It follows, from what has been shown of the eifect of a variable load,
that no bridge, either straight or arched, intended for the passage of heavy
vehicles or trains, should ever be without counter braces or diagonal ties.
On the Truss ~by Tension Rod (fig. 53). Since the limit of the elas-
a, ._*.. ,'.. I
Fig. 53.
ticity of iron is very small in comparison with wood, when iron is thus
used to truss timber, the rods must break before the beam reaches the de-
flection that the weight should produce. It is evident, therefore, that in
construction the beam should not be cambered by the tension of the
rod, but that the top of the beam should be arched, and be permitted to
settle with the weight before it strains the rod at all. The rods should be
depended on to resist the whole of the tension. To estimate the strain
upon the "suspension rod, multiply the weight supported at the point c or o f
by the length of the rod a d or d' 5, and divide the product by the length
of the strut c d. The length of the rod and of the strut may be measured
from any horizontal line which completes the triangle.
Suppose a system to be composed of a series of suspension trusses, as
in fig. 54, in which the load is uniformly distributed. If we represent the
load at each of the points, 4, 3, 2, 1, 2' &c., by 1, the load at 4 will be
supported 3 upon a and | upon 3 ; hence the strut 3 will have to support
a load of 1 + .5 = 1.5 ; of this, f will be supported by 2 and % by a ; f
of 1.5=1, 1 + 1=2, load on strut 2 ; of this load, or 1.5, will be supported
at 1, and since from the opposite side there is an equal force exerted at 1,
therefore the strut 1 supports 1 + 1.5 + 1.5 = 4; the tension on the rod
c 2 is 4; on 2 3, 4 + 1 = 5; on 3 4-, 5 + 1 = 6; on 4 a, 6 + 1 = 7; and
the rod should therefore be increased in strength in these proportions from
ARCHITECTURAL DRAWING.
233
the central point c, to the point of suspension, a. The tension on the rods
3 4, 2 3, 1 2, may be easily resolved from their direction and the load upon
the several struts.
If this construction be reversed, the parts which now act as ties must be
made as braces, and braces, ties ; then we have a roof truss, and the force
exerted on the several parts may be estimated in a similar way as for the
suspension truss.
To frame a construction so that it may be completely braced, that is,
under the action of any arrangement of forces, the angles must not admit
of alteration, and consequently the shape cannot. The form should be
resolvable into either of the following elements : Figs. 55, 56, 57.
Fig. 55.
In these figures, lines
compression ; lines -
Fig. 56.
Fig. 57.
represent parts required to resist
parts to resist tension only ; lines
parts to resist both tension and compression.
In a triangle (fig. 55), an angle cannot increase or diminish, without
the opposite angles also diminishing or increasing. In the form fig. 56, a
diagonal must diminish ; in fig. 55 a diagonal must extend, in order that
any change of form may take place. Consequently all these forms are
\
Fig. 58.
Fig. 59.
completely braced, as each does not permit of an effect taking place, which
would necessarily result from a change of figure. Hence also, any system
234: ARCHITECTURAL DRAWING.
composed of these forms, properly connected, breaking joint as it were
into each other, must be braced to resist the action of forces in any direc-
tion ; but as in general all bridge trusses are formed merely to resist a
downward pressure, the action on the top chord being always compression,
it is not necessary that these chords should act in both capacities. As
illustrations of bridge trussing, a few panels, fig. 58, of Howe's truss, and
of Pratt's, fig. 59, are given.
On the, Size and Proportion of Rooms in general. " Proportion and
ornament," according to Ferguson, " are the two most important resources
at the command of the architect, the former enabling him to construct or-
namentally, the latter to ornament his construction." A proportion to be
good, must be modified by every varying exigence of a design, it is of
course impossible to lay down any general rules which shall hold good in
all cases ; but a few of its principles are obvious enough. To take first
the simplest form of the proposition, let us suppose a room built, which
shall be an exact cube of say 20 feet each way such a proportion must
be bad and inartistic ; (and besides,) the height is too great for the other
dimensions. As a general rule, a square in plan is least pleasing. It is
always better that one side should be longer than the other, so as to give a
little variety to the design. Once and a half the width has been often
recommended, and with every increase of length an increase of height is
not only allowable, but indispensable. Some such rule as the following
meets most cases : " The height of the room ought to be equal to half its
width plus the square root of its length ; " but if the height exceed the
width the effect is to make the room look narrow; again, by increasing the
length we diminish, apparently, the other two dimensions. This, however,
is merely speaking of plain rooms with plain walls ; it is evident that it will
be impossible, in any house, to construct all the rooms and passages to
conform to any one rule of proportion, nor is it necessary, for in many
rooms it would not add to their convenience, which is often the most desir-
able end ; and if required, the unpleasing dimensions may be counteracted
by the art of the architect, for it is easy to increase the apparent height by
strongly marked vertical lines, or bring it down by horizontal ones. Thus
if the walls of two rooms of the same dimensions be covered with the same
strongly marked striped paper, in one case the stripes being vertical, and
in the other horizontal, the apparent dimensions will be altered very con-
siderably. So also a deep bold cornice diminishes the apparent height of a
room. If the room is too long for its other dimensions, this can be reme-
died by breaks in the walls, by the introduction of pilasters, &c. So also,
as to the external dimensions of a wall, if the length is too great it is to be
AECHITECTUEAL DRAWING. 235
remedied by projections, or by breaking up the lengths into divisions.
This will be understood by reference to elevations of " Country Houses,"
Plates LXV. to LX1X. In this view, as variety in form adds greatly to the
picturesque, it is far better in designing a country house, where one is not
restricted to room, to mark out the rooms to the size which we wish them to
be, cutting out slips of paper of the dimensions, according to some scale, to
arrange them then in as convenient an order as possible, and again modifying
the arrangement by the necessities of construction and economy. Thus
the more the enclosing surface, in proportion to the -included area, the
greater the number of chimneys, unnecessary extent of passages; all of
course conduce to an excess of expense. Again, the kitchen should be of
convenient access to the dining room, both should have large and commo-
dious pantries, and all rooms should have an access from a passage, with-
out being compelled to pass through another room ; this is particularly
applicable to the communication of the kitchen with the front door. Out-
side doors for common and indiscriminate access should open into passages
and never into important rooms.
As to the size of the different rooms, they must of course depend on
the purposes to which they are to be applied, the class of house, and the
number of occupants. To commence with the kitchen, for the poorer class
of houses it is also used as an eating room, and should therefore be of con-
siderable size to answer both purposes ; for the richer houses, size is neces-
sary for the convenience of the work ; in New York City houses the aver-
age will be found to be about 15 x 18 feet, for medium houses in the
country they are in general less, say 12 x 14. A back kitchen, scullery,
or laundry, should be attached to the kitchen, arid may serve as a passage
way to out of doors.
The dining or eating rooms. The width of dining tables vary from 3
to 5 feet 6 inches, the depth occupied by the chair and person sitting at the
table is about 18 inches on each side ; the table space, for comfort, should
be not less than 2 feet for each person at the sides of the table, and
considerable more at the head and foot ; hence we may calculate the space
that will be necessary for the family and its visitors, at the table. If we
now allow a farther space of 2 feet at each side for passages, and some 3
to 5 at the head for the extra tables or chairs, we can mark out the mini-
mum of space required : but, if possible, do not confine the dining room to
meagre limits, unless for very small families ; let not the parties be lost in
the extent of space, nor let them appear crowded.
The show room parlors, if there are any intended for such in the house,
may be made according to the rules given above, not square, but the length
236
ARCHITECTURAL DRAWING.
about once and a half the width ; if much longer than this, break up the
walls by transoms or projections. As to the particular dimensions no rules
can be given, it must depend on every person's taste and means. 20 x 16
may be considered a fair medium size for a regular living room parlor, not
a drawing room. The same size will answer very well for a sleeping room.
The usual width of single beds is 2 feet 8 inches, of three-quarter 3 feet
6 inches, of whole 4 feet 6 inches, the length 6 feet 6 inches, so that if
adequate means of ventilation are provided, it is easy to see into how small
quarters persons may be thrust. The bed should not stand too near the
fire, nor between two windows ; its most convenient position is head against
an interior wall, with a space on each side of at least 2 feet.
Pantries. Closets for crockery should not be less than 14 inches in
width in the clear ; for glass 8 inches, and for the hanging ^up of
clothes, not less than 18 inches. For medium houses, the closets
of large sleeping rooms should be at least 3 feet wide, with hanging room,
and drawers and shelves. There should also be blanket closets, for the
storing of blankets and linen ; these should be accessible from the entries,
and may be in the attic. Store closets should also be arranged for gro-
ceries and sweetmeats.
Passages. The front entries are usually 6 feet wide in the clear ; com-
mon passage ways are usually 3 feet wide ; these are what are required, but
ample passages give an important effect to the appearance of the houses.
The width of principal stairs should be not less than 3 feet, and all first
class houses, especially those not provided with water-closets and slop
sinks on the chamber floor, should have two pair of stairs, a front and
a back pair ; the back stairs may not necessarily be over 2 feet 6 inches
in width.
The Height of Stories. It is usual to make the height of all the rooms
on each floor equal, it can be avoided by furring down, or by the break-
ing up of the stories, by the introduction of a mezzonine or intermediate
story over the smaller rooms. Both remedies are objectionable ; the more
artistic way is to obviate the appearance of disproportionate height by
means stated above.
The average height of the stories for such city dwellings as we have
given plans of are : cellar 6 feet 6 inches, common basement 8 to 9 feet,
English basement 9 to 10, principal story 12 to 15, first chamber floor 10
to 12, other chamber floors 8 to 10 feet, all in the clear. For country
houses the smaller of the dimensions are more commonly used. Attic
stories are sometimes but a trifle over 6 feet in height, but are of course
objectionable.
AKCHITECTURALte DRAWING.
237
Fig. 60.
Details of parts. Stairs consist of the tread or step on which we set
our feet, and risers, upright pieces supporting the treads each tread and
riser forms a stair. If the treads are parallel they are called fliers, if less
at one end than the other, they are called winders; f and w, fig. 62. The
top step, or any intermediate wide step, for the purpose of resting, is called
a landing. The height from the top of the nearest step to the ceiling
fa above is called the headway. The rounded edge of
the step is called a nosing, a, tig. 60 ; if a small hol-
low, 5, be glued in the angle of the nosing and riser,
it is called a moulding nosing. The pieces which sup-
port the ends of the stairs are called strings, that
against the wall the wall string, the other the outer
string, ^Besides the strings, pieces of timber are framed and placed beneath
the fliers, called carriages. The opening on plan (which must occur be-
tween the outer strings, if they are not perpendicular over each other) is
called the well hole, W, fig. 62.
The breadth of stairs in general use is from 9 to 12 inches. In the best
staircases, the breadth should never be less than 12 inches, nor more than
15. The height of the riser shoul d
be the more, the less the width of
the tread ; for a 15 inch tread the
riser should be 5 inches high, for
12 inches, 6f, for 9 inches, 8. In
laying out the plan of stairs, hav-
ing determined the starting point
either at bottom or top as the case
may be, find exactly the height of
the story ; divide this by the height
you suppose the riser should be.
Thus (fig. 61), if the height of
the story and thickness of floor be
9 feet, and we suppose the riser
should be 7 inches high, then
108 inches, divided by 7 = 15f .
It is clear that there must be
an even number of steps, either 16 or 15 ; to be near to what we have sup-
posed to be the height of the riser, adopt 15, then
-y- 8 - = 7 T 3 j inches, height of riser.
For this particular case we assume the breadth of the step as 10 inches,
and the length at 3 feet, a very usual length, seldom exceeding 4 feet in
238
AECI1ITECTU RAL DK A WING.
the best staircases of private bouses. For tbe plan lay off tbe outside of
the stairs, two parallel lines 3 feet apart, and space off from the point of
beginning 14 treads of 10 inches each, and draw the cross parallel lines.
To construct the elevation, the line of the stair in plan mav be pro-
jected, and the height be divided into the number of risers, 15 of 7} inches
each, and cross parallels drawn through these points ; or the same points
may be determined by intersection of the projections of the plan with a
single inclined line drawn through the nosing of top and bottom steps.
It is to be observed that the number of treads is always one less than the
number of risers, the reason of which will appear by observing the elevation.
The drawing of the elevation of stairs is in general necessary, to deter-
mine the opening to be framed in the upper floor, to secure proper head-
way. Thus (fig. 61), the distance between the nearest stair and the ceiling
at a should not be less than 6 feet 6 inches ; a more J||||fL_
ample space improves the look of the stairway; ' \S i
but if we are confined in our limits, this will de-
termine the position of one trimmer, the other will
be of course at the -top of the stairs. "When one
flight is placed over another, the space required for
timber and plastering, under the steps, is about 6
inches for ordinary stairs.
When the stairs are circular, or consist in part
of winders and fliers, as in fig. 62, the width of the
tread of the winders should be measured on the
central line. The construction of the elevation is
similar to that of the straight run (fig. 61), by divid-
ing the space between the stories by a number of
parallel lines equal to the number of risers, and in-
tersecting the parallels by projections from the plan.
Fig. 63 represents a circular
flight of stairs without a well
hole, the narrow ends of the winders being mortised into
a central shaft or newel, N. The same term is also ap-
plied to the first laluster or post of the hand rail. The
objection to all circular stairs of this form, or with a
small well hole, is that there is too much difference be-
tween the width of the tread, but a small portion being of a suitable size.
The handsomest and easiest stairs are straight runs, divided into landings,
intermediate of the stories, and either continuing then in the same line, or
turning at right angles, or making a full return.
Fig. 62.
ARCHITECTURAL DRAWING.
239
The top of the hand rail should, in general, be about 2 feet 8 inches
above the nosing, and should follow the general line of the steps. The
angles of the head rail should always be eased off. A hand rail, affording
assistance in ascending or descending, should not be
wider than the grasp of the hand, fig. 64 ; but where,
for architectural effect, a more massive form may be
necessary, it is very convenient, and may
be very ornamental, to have a sort of
double form, that is, a smaller one
planted on top of the larger, fig. 65. Fig. 64. Kg.es.
J)oors. Fig. 60 represents the elevation, and fig. 67 the horizontal sec-
tion of a common inside door. A A are the stiles, B, C, II, D, the bottom,
lock, parting, and top rail, E the panels, and F the muntmj the combina-
tion of mouldings and offsets around the door, G, is called the architrave /
in the section, a a are the partition studs, 5 b the door jambs.
With regard to the proportions of internal doors, they should de-
pend in some degree on the size of the apartments; in a small room
a large door always gives it a diminutive appearance, but doors lead-
ing from the same entry, which are brought into the same view, should
be of uniform height. The smaller
doors which are found on sale are 2 ft.
4 in. x 6 feet; for water closets, or
very small pantries, they are some-
times made as narrow as 20 inches,
but any less height than 6 feet will
not afford requisite head room. 2 ft.
9 in. x 7 ft., 3 ft. x 7 ft. 6 in., or 3 ft.
6 in. x 8 ft., are well proportioned 6
panelled doors. But the apparent
proportions of a door may be varied
by the omission of the parting rail,
making the door 4 panelled, or nar-
rowed still more by the omission of
the lock rail, making a 2 panelled
door. Sometimes the muntin is omit-
ted, making but one panel ; but this of
course will not add to the appearance
of width, but the reverse. Wide panels
i are objectionable, as they are apt to
Fig. 67. " shrink from the mouldings and crack.
Fig. 66.
240
AKCHITECTDKAL DRAWING.
When the width of the door exceeds 5 feet, it is generally made in two
parts, each part being hung to its side of the frame, or one part hung to
the other, so as to fold back like a shutter ; or the parts may be made to
slide back into pockets or grooves in the partition, as shown in plan and
horizontal section, figs. 68 and 69. One of the doors in the drawing is
shown as a sash door, the other close panels, so as to give twp illustrations
in the same diagram ; the same may be said of the architrave. It may be
unnecessary to say that in construction both sides and doors should be uni-
form. The upper panels of the close door may be made of glass ; the finish
around this half of the door is with an architrave, as in fig. 66, but with
Fig. 69.
different mouldings. The finish over the other half of the door is an entab-
lature, supported by pilasters A, commonly called by carpenters antse, though
not correctly so, the antse being pilasters at the end of a projecting wall.
Figs. 70 and 71 are the elevation and horizontal section of an antse-
finished outside door, with the side lights C C, and a top, fan, or transom
light B. The bar A is called a transom, and this term is applied generally
to horizontal bars extending across openings, or even across rooms.
Fig. 72 is the elevation of an outside folding door. The plan (fig. 73)
shows a vestibule ~V, and an interior door. The outer doors, when open,
fold back into the pockets or recesses, p p, in the wall. This is the present
ARCHITECTURAL DRAWING,
241
usual form of doors for first-class houses in this city. The fan lights are
made semicircular, and also the head of the upper panels of the door ;
these panels in the interior or vestibule door are of glass.
Windows are apertures for the admission of light to the building, for
ventilation, and for looking out. When used for the admission of light
only, the sashes may be stationary, as they
sometimes are in churches, but for most po-
sitions they are intended for all these pur-
poses, and therefore the sashes are made to
open, either by sliding vertically, or laterally,
or like doors. The first is the common form
of window, the sashes are generally balanced
by weights; the second, except in a cheap form
in mechanics' shops, are seldom used ; the third
are called casements, or French windows.
Figs. 74
and 75 repre-
sent in eleva-
tion and plan
the parts of
the common
sash window
and its shut-
ters, in bro-
-ken lines, so
as to show
the details on
Fig. 78.
a large scale. S designates the sill of the
sash frame, W the stone sill, with a wash
to discharge the water, B is the bottom rail
of the sash, M the, meeting rails, and T the
top rail, H is the head of the sash frame,
A the architrave similar to that around
doors. In the sectional plan C C' are the
window stiles, F the pulley stile, w w the
sash weights, p the parting strip, and D D
double fold shutters. Sash windows for
dwellings are almost always made with Fi s- 75.
twelve lights, six in each sash. The height of the window must of course
depend on the height of the room. Unless the windows begin from, or
16
242
ARCHITECTURAL DRAWING.
nearly from, the floor, the point a (fig. 74) may be fixed at a height of
about 30 inches above the floor, and the top of the window sufficiently be-
low the ceiling to allow space for the architrave or other finish above the
window, and for the cornice of the room, if there be any ; a little space be-
tween these adds to the effect. For common windows, the width of the
sash is 4 inches more than that of the glass, and the height 6 inches more ;
thus the sash of a window 3 lights wide and 4 lights high, of 12 x 16 glass,
is 3 feet 4 inches wide, and 5 feet 10 inches high. In plate glass windows
more width is taken for the stiles and rails. The most usual sizes of glass
are 7 x 9, 8 x 10, 9 x 12, 10 x 12, 10 x 14, 11 x 15, 12 x 16, 12 x 18, 12 x 20,
14 x 20, but glass may be had of intermediate or of much larger sizes. Plate
glass, either polished or rough, may be had of a size as large as 14 x 7 feet.
Fig. 76 represents the elevation of half of a French window, also in broken
lines, the same letters designate similar parts as in fig. 68. A transom bar
is often framed between the meeting rails, and in this case the upper sash
may be movable ; in the fig. it is fixed. An upright, called
a mullwn, is often introduced in the centre, against which
the sash shuts. Fig. 77 is an elevation on a smaller scale.
For convenience of ^~ j
egress and ingress, the
lower sashes should
not be less than 5 feet
6 inches high, that is,
when the window opens
on a stoop or balcony.
It will be seen that in
both forms of sash the
bottom rail is the widest,
and that for the same
aperture the French
window admits the least
light. The chief objec-
tion to this window lies
in the difficulty of keep-
ing out the rain at the
bottom in a driving
storm. To obviate this, the small mould-
ing d, with a drip or undercut, is nailed
to the bottom rail ; but the more effec-
tual means is the patent weather strip, the same as used on outside doors.
Fig. 76.
Pig. 7T.
ARCHITECTURAL DRAWING.
243
The most simple exterior finish for windows, in brick or stone houses,
is a plain stone cap and sill, the height of the cap for common apertures
being from four to five courses of brick, and the sill three courses, the
latter always to project from one to two inches beyond the line of
brickwork. Usually in wooden structures, and often in stone and brick,
an architrave is formed around the window (figs. 78, and 79). For brick
houses the facings are made of stone. The architrave should not project
so much as to interfere with the shutting back of the blinds. Blinds are
commonly tlutee-eighths of an inch narrower, and one inch longer than
the sash.
Fig. ?a
U LI
Fig. 79.
Fig. SO.
Fig. 80 represents a section of the finish around the bottom of the wall
of a room. A, is the base, consisting of a plain strip or skirting, with a
moulding above it. B, is the surbase or chair rail ; between these, it is
not unusual to have a panelled or plain board, called the dado. The
rough plastering is usually continued to the floor, the skirting and surbase
are then nailed on, the hard finish is next put on, and lastly the base
moulding. The panels of the dado are imitated in oil or distemper ; the
surbase is seldom used but in dining rooms or offices.
For the finish of the angle of the wall and ceiling, the cornice adds
often to the architectural effect. It consists of a series of mouldings similar
to the style of finish of the houses, extending around the room. The
effect of the cornice is to diminish the apparent height of the room ;
for low rooms, if adopted, it should extend in width on the ceiling, and
244
AKCHITECTTJKAL DBAWING.
but little in depth on the wall, and the reverse where an opposite effect
is desired. Figs. 81, 82, 83.
Fig. 81.
Fig. 82.
Fig. 83.
Fire-Places. Fire-places for wood are made with flaring jambs of the
form shown in plan fig. 84 ; the depth from 1 foot to 15 inches, the width
of opening in front from 2 feet 6 inches to 4 feet, according to the size of
the room to be warmed ; height 2 feet 3 inches to 2 feet 9 inches, the
width of back about 8 inches less than in front ; but at present fire-places
for wood are seldom used, stoves and grates having superseded the fire-
place. The space requisite for the largest grate need not exceed 2 feet in
width by 8 inches in depth. The requisite depth is given by the projec-
C
Fig. 85.
Fig. 86.
tion of the grate, and the mantel-piece. Ranges require from 4 feet 4 inches
to 6 feet 4 inches wide x 12 inches to 20 inches deep ; jambs 8 inches to
12 inches. Fig. 86 represents the elevation of a mantel-piece of very usual
proportions. The length of the mantel is 5 feet 5 inches, the width at
base 4 feet 6 inches, the height of opening 2 feet 7 inches, and width 2 feet
9 inches. A portion of this opening is covered by the iron sides or archi-
trave of the grate, and the actual open space would not probably exceed
18 inches in width by 2 feet in height. The sizes of flues are 8 x 8, 4 x 12,
and 8 x 12 inches. In brick or stone houses the flues are formed in the
thickness of the wall, but when distinct they have an outside shell of a
half brick or 4 inches. The flues of different fire-places should be distinct,
those from the lower stories pass up through the jambs of the upper fire-
ARCHITECTURAL DRAWING. 24:5
places, and keeping side by side with but 4-inch, brick work between them,
are topped out above the roof, sometimes in a double and often in a single
line 16 inches wide by a breadth required by the number of flues. The
chimney is usually capped with stone, sometimes with tile or cement pots.
As an architectural feature, the chimney is often made to add considerably
to the effect of a design.
Privies, Water-Closets, and Out-Houses. The size of privies must
depend greatly on the uses of the building to which they are to be at-
tached, its position, and the character of its occupants. Allowing nothing
for evaporation or absorption, the entire space necessary for the excremen-
titious deposits of each individual, on an average, will be about seven
cubic feet for six months, of which three-quarters is fluid. In the country,
vaults are usually constructed of dry rubble-stone ; and the fluid matters
are expected to be filtered through the earth, the same as in cess-pool
waste ; but great care must be taken that they neither vitiate the water supply
nor the air of the house. A brick and cement vault, air and water tight,
with a ventilating pipe into a hot chimney-flue, is the best preventive,
and may even be built within the house. In all other cases there should
be free air space between the house and privy. In the city, where there
is adequate water supply and sewerage, the water-closet should be adopted,
except in houses occupied by many ignorant and irresponsible tenants,
who throw extraneous matters into the hoppers, and obstruct the sewer-
pipes. In these, tight privy-vaults, with trapped sewer connections, and
with all the house waste and roof water discharging into them, are the
easiest kept in order. Water-closet, or privy, with a single seat, should
occupy a space not less than 4 ft. x 2 ft. 6 in. The rise of seat should
be about 17 in. high ; and the hole egg-shaped, 11 x 8 in.
Cess-Pools. The house waste, when there ia no system of sewerage,
can only be got rid of by cess-pools, which permit its absorption in the
earth, or by cistern, and using the water for irrigation and manuring.
Their size will depend entirely on the quantity of this waste. They
should be placed as far as possible from the house, and the connec-
ctions with them should be trapped, that no effluvia may escape into the
house.
Traps are of various kinds. The general principle of their construc-
tion is by bend or basin in the sewer-pipe to make a water closure, through
which water may pass, but not air.
For Wood or Coal Sheds. In estimating the size of these accessories,
it may only be necessary to state that a cord of wood contains 128 cub. ft.,
and a ton of coal occupies a space of about 40 cub. ft.
24:6 ARCHITECTURAL DRAWING.
DRAWING.
The reader is probably now sufficiently acquainted with the use of
drawing instruments to construct, with but little explanation, most archi-
tectural drawings. The first thing to be done toward the execution of a
drawing is to determine the scale upon which the drawing is to be made.
The usual scale for plans and elevations in architectural drawings is
either 4 ft. or 8 ft. to the inch, and especially in working drawings it
is necessary that the scale should be on some marked division of the com-
mon two-foot rule. Working details are generally drawn on coarse paper,
and on as large a scale as possible, often full size.
To construct the Ist-story plan (fig. 87), end (fig. 88), and side (fig. 89)
elevation of a house. Select a scale of say 4 ft. to the inch, and commence
with the plan. Lay off a base line A B, on this measure of 20 ft.
for the length of the house, and erect perpendiculars at the extrem-
ities thus measured. Lay off 16 ft. for the width on each of the perpen-
diculars, and connect these points ; the line will be parallel to A B, and the
outline of the house will be defined. The thickness of the wall for a house
of this size, if of brick, will be 8 inches for wall and 2 inches for furring
and plastering, or 10 inches ; if of wood, the studs should be 2 x 4 ; \ \
inches for outside boarding and 1 for lath and plaster, or 6^ in total. Lay
off on the inside of the outline the thickness of the wall, and draw the in-
terior lines. Lay off the partitions marking the rooms, by a single line
or by two lines. The thickness of partition for such an edifice will be
from 4 to 5 inches. The dimensions, as generally marked, should be from
outside lines to centres of partitions, or from centres to centres of parti-
tions, as more determinate for the carpenter to work from than dimensions
in the clear, that is, actual spaces in the rooms. Lay off the position of the
windows, which are to be 3 ft. wide, preserving uniformity both in inside
and outside appearance. Thus, the front window should be, as near as
possible, in the centre of the room, and on the outside, uniform with the
door. The side of the door may be 4 ft. from the end of the house ; mark
its position and width ; but the side of the window, if in the centre of the
room, will be 5 ft. 4 in. from the opposite corner ; make it 5 ft., and lay
off the window in the rear opposite to the front window. The openings
for windows are distinguished from doors by straight lines drawn across
the aperture. The window in the end of the house, to present a uniform
appearance outside, should be in the centre, and this suits the purpose for
which the room is designed, a blank corner being necessary for the bed.
AECHITECTUEAL DKAWING.
247
Lay off the position of the fireplace in the centre of the end of the
room, the opening to be 3 ft, the width of the jambs 8 in. each, and
the width of the back 2 ft. 4 in. Draw lines for a partition flush with
the face of the chimney to one side of the room, with an opening for
a door to be for a closet or pantry ; all the inside doors to be 2 ft. 10 in.
wide, and to be represented by openings in the partitions merely. Lay
off the stairs, as shown in the lobby, 2 ft. 9 in. wide. Fill the space be-
tween the interior and exterior outlines of the walls and partition in black,
leaving the openings for the doors and windows, and the plan is complete.
The filling is not indispensable, but it adds to the distinctness of the plan.
248 AECHITECTUKAL DRAWING.
To construct the front elevation (fig. 88), project the various points or
position of the corners, window and door, as shown on the plan, and extend
the lines of projection as high as may be necessary above the line CD;
on these lines mark off the height of window and door, the height of the
eaves in projecting must be determined from the end elevation ; finish the
window and door with lines to represent caps and sills, and whatever other
lines may be necessary for the style of finish of the drawing. On the line
C D erect also the outlines of the end elevation, taking the horizontal di-
mensions from the plan, and projecting, as far as possible, the vertical ones
from the front elevation. Establish the ridge of the roof by setting off
the height of the pitch on the centre line above the eaves, and draw the
various lines to represent the boarding and cornice. Draw the chimney
in the centre above the ridge. It is to be observed that this chimney is at
the opposite end of the house, and may be represented in lighter lines than
. the rest of the drawing. The base of the chimney and the eaves are to
be projected on the front elevation. It is to be observed that many lines
may be projected, either from the front to the side elevation, or vice versa ;
the true way is to construct both elevations together.
The outline of the section may be taken from the end elevation, and
the other plans from Ist-story plan, or constructed directly from the dimen-
sions.
Plates XL VII. to LI. inclusive are plans and elevations of a house, and
contain as full representations as are usually given by architects for the
purposes of estimate, accompanied by specifications. The size of our page
has compelled the titles to be put within the body of the drawings ; place
them outside, and give good margin. On Plate LI. the section and end
elevation are given together. This is also for economy of space; but
should be copied by the draughtsman in two distinct drawings.
Page 249 represents plans of familiar forms of houses, all drawn to the
same scale, as illustrations to the student, and as examples to be copied on
a larger scale. The same letters of reference are used on all the plans, for
rooms intended for similar purposes. Thus, K K designate kitchens, cook-
ing rooms, or laundries, D D eating rooms, S S sleeping rooms, P P draw-
ing rooms, parlors, or libraries, p p pantries, china or store closets, or
clothes-presses, c c water-closets and bath rooms.
Fig. 1 is the 1st floor plan of a very small square house. Figs. 2 and 2'
are 1st and 2d floor plans of a still larger house. Fig. 3 is the floor plan of the
same house differently arranged, the kitchen being in the basement. Fig.
4, the same, with an L in the rear for the kitchen. These plans are all of
ARCHITECTURAL DRAWING.
249
Fig. 8.
33'
Fig. 9.
I
Fig. 1.
. 1
250 AKCHITECTURAL DEAWIXG.
square houses, and although not picturesque in their elevations, are yet
very convenient and economical structures; they are intended for the
country. The sheds and back offices should be beneath a different roof,
but attached or not to the main building as may be desired.
Figs. 5, 6, and T, are first floor plans of houses of a different outline,
but yet uniform, or nearly so ; in subsequent plates will be found illus-
trations of more varied forms of houses. In the cities, houses are
mostly confined to one form in their general outline, a rectangle. Figs.
8 and 9 may be taken as the usual type of New York City houses. Figs.
8, 8', 8", are the basement, first, and second floor plans of a Basement
house, three rooms deep. There is usually a cellar beneath the basement,
but in some cases there are front vaults, entered beneath the steps to the
front door ; the entrance to the basement itself is also beneath the steps.
The front room of the basement may be used as an eating room, for the
servants' sleeping room, billiards, or library. The usual dining room is on
the first floor ; a dumb waiter being placed in the butler's pantry j?,
for convenience in transporting dishes to and from the kitchen. The ob-
jection to three room deep houses is that the central room is too dark, being
lighted by sash folding doors between that and the front or rear rooms or
both. Fig. 8"' is a modification to avoid this objection, the dining room,
or tea room as it is generally called, being built as L, so that there is at
least one window in the central room opening directly out-doors. Figs. 9,
9', 9", 9 /// , are plans of the several floors of an English Basement house so
called, distinguished from the former in that the principal floor is up one
flight of stairs. The first story or basement, is but one or two steps above
the street, and contains the dining room, with its butler's pantry and dumb
waiter, a small sitting room, with, in some cases, a small bed room in the
room in the rear of it. The kitchen is situated beneath the dining room,
in the sub-basement. The grade of the yard is in general some few steps
above the floor of the kitchen. Vaults for coal and provisions are exca-
vated either beneath the pavement in front or beneath the yard. The
advantages of this form of house are the small sitting room on the first
floor, which in small families, and in the winter months, is the most fre-
quently occupied of any in the house ; the spaciousness of its dining room
and parlors in proportion to the width of the house, which is often but 16
feet 8 inches in width, or three houses to two lots, and not unfrequently of
even a less width. The objections to the house are the stairs, which it is
necessary to traverse in passing from the dining rooms or kitchen to the
sleeping rooms, but this objection would, of course, lie against any house
of narrow dimensions, where floor space is supplied by height.
ARCHITECTURAL DKAWLNGt
251
MOULDINGS.
It will have been observed that for the finish of most of the parts of an
edifice, mouldings are found necessary ; so much so, that they should
be classed among useful rather than ornamental members. These mould-
ings are drawn either directly or indirectly from the Grecian orders of
architecture.
The regular mouldings are eight in number : Fillet or Band, Torus,
Astragal or Bead, Ovolo, Cavetto, Cyma Recta or Ogee, Cyma Reversa or
Talon, Scotia.
To construct a Fillet. Thejillet, a (fig. 68), is the smallest rectangular
member employed in any composition of mouldings. When it stands on
a flat surface, its projection is usually made equal to its height. It is em-
ployed to separate members.
To describe a Torus, or an Astragal. The
torus and astragal are semicircles in form pro-
jecting from vertical diameters, as in fig. 69. Fig.es. Fig. 69.
Bisect the vertical diameter aft, on which the figure is projected; on the
centre <?, describe a semicircle with c a as radius. The astragal is described
like the torus, and is distinguished from it in the same order by being
made smaller. The torus is generally employed in the bases of columns ;
the astragal, in both the base and capital.
To describe an Ovolo. The ovolo is a member strong at the extremity,
and intended to support. The Roman ovolo consists of a quadrant or a
less portion of a circle ; the Greek ovolo is elliptic.
First, the Roman ovolo. When the projection is equal to the height.
Draw a b for the height, and b c at right angles
and equal to it, for the projection^ On the
centre b describe the quadrant ca.
When the projection is less than the height.
Draw a b and b c (fig. 71), as before, equal to Fi s- 70 - Fi ?- "
the height and the projection. On centres a and c, with radius ab, describe
arcs cutting at d / and on d with same radius describe
the arc a c to form the ovolo.
Second, the Greek ovolo. Draw df from the lower
end of the proposed curve, at the required inclination ;
draw the vertical g ef to define the projection, the
point 6 being the extreme point of the curve. Draw Fi s- T2 -
e h parallel to df, and draw the vertical d h &, such that d h is equal to
fc
~ - .
252
ARCHITECTURAL DRAWING.
Fig. 73.
Fig. 74.
h Jc. Divide e Ji and ef into the same number of equal parts ; from d
draw straight lines to the points of division in ef, and from k draw lines
to meet those others successively. The intersections so found are points in
the curve, which may be traced accordingly.
To describe a Cavctto. The cavetto is described like the Roman ovolo :
by circular arcs, as shown in
figs. 73 and 74. Sometimes
it is composed of two circular
arcs united (fig. 75) ; set off
be, two- thirds of the projec-
tion, draw the vertical b d
equal to b e, and on d describe the arc b i. Join e d and produce it to p /
draw in perpendicular to ed, set off no equal to ni, and draw the
horizontal line op meeting ep , on p describe the arc io to complete the
curve.
To describe a Cyma recta, or Ogee. The ogee (fig. 76), is compounded
of a concave and a convex surface. Join a
and b, the extremities of the curve, and bisect
a b at c j on a, c, as centres, with the radius
a c, describe arcs cutting at d j and on b, c,
Fig - 76 - Fig - 77 - describe arcs cutting at e. On d and e, as
centres, describe the arcs a c, c b, composing the moulding.
To describe a Cyma reversa, or Talon. The talon (fig. 77), like the
ogee, is a compound curve, and is distinguished from the other by having
ei.-~: ;--T~~l the convex part uppermost. It is described in the
same manner as the ogee.
Note. If the curve be required to be made quicker,
a shorter radius than ac must be employed. The projec-
tion of the moulding n b (fig. 76), is usually equal to the
height a n.
Second, the Greek talon. Join the extreme points a, b (fig. 78) ; bisect
a b at c, and on ac, c b, describe semicircles. Draw
perpendiculars do, &c., from a number of points in
ac, cb, meeting the circumferences ; and from the same
points set off horizontal lines equal to the respective
perpendiculars : o n equal to o d for example. The
curve line b n a, traced through the ends of the lines,
will be the contour of the moulding.
To describe a Scotia. Divide the perpendicular a b
Fig. 79. (fig. 79^ i n to three equal parts ; and with the first, a e,
ri s- T8 -
ARCHITECTURAL DRAWING.
as radius, and on centre e, describe the arc aflij on the perpendicular co,
set off cl equal to ae, join el and bisect it by the perpendicular ?,
meeting c o at o. On centre 0, with radius c, describe the arc c h to
complete the figure.
ORDERS OF ARCHITECTURE.
Order, in architecture, is a system or assemblage of parts subject to
certain uniform established proportions, regulated by the office each part
has to perform. An order may be said to be the genus, whereof the spe-
cies are Tuscan, Doric, Ionic, Corinthian, and Composite ; and consists of
two essential parts : a column and entablature.
These are subdivided, the first into three parts, namely : the base, the
shaft, and the capital. The second also into three parts, namely : the
architrave, or chief beam, C, Plate LIL, which stands immediately on
the column ; the frieze B, which lies on the architrave ; and the cornice A,
which is the crowning or uppermost member of an order. In the subdi-
visions certain horizontal members are used, which from the curved form
of their edges are called mouldings, the construction of which has already
been explained, and their application may be seen on the Plate ; thus a is
the ogee, 5 the corona, c the ovolo, d the cavetto, which with fillets com-
pose the cornice,yy the fasciae. The capital of the column consists of the
upper member or abacus g, the ovolo moulding c, the astragal ii, and
the neck h. The base consists of the torus &, and the plinth I. The
character of an order is displayed, not only in its column, but in its general
forms and detail, whereof the column is, as it were, the regulator ; the ex-
pression being of strength, grace, elegance, lightness, or richness. Though
a building be without columns, it is nevertheless said to be of an order,
if its details be regulated according to the method prescribed for such
order.
In all the orders a similar unit of reference is adopted for the construc-
tion of their various parts. Thus, the lower diameter of the column is
taken as the proportional measure for all other parts and members, for
which purpose it is subdivided into sixty parts, called minutes, or into two
modules of thirty minutes each. Being proportional measures, modules
and minutes are not fixed ones like feet and inches, but are variable as to
the actual dimensions which they express larger or smaller, according to
the actual size of the diameter of the column. For instance, if the diam-
eter be just five feet, a minute, being one-sixtieth, will be exactly one inch.
254: ARCHITECTURAL DRAWING.
Therefore before commencing to draw an elevation of any one of the
orders, determine the diameter of the column, and from that form a scale
of equal parts, by sixty divisions, and then lay off the widths and heights
of the different members according to the proportions of the required
order as marked in the body or on the sides of the plates.
Plate LIT., presents an illustration of the Tuscan order, considered by
architects as a spurious or plain sort of Doric, and hardly entitled to re-
mark as a distinct order, e, in the frieze corresponding to the triglyph,
illustrates still further the connection of the two orders ; but by many
architects this member is not introduced. Fig. 1 is an elevation of cap-
ital and entablature, fig. 2 of the base, and fig. 3 of a detached capital.
Our example is constructed according to the rules given by Yincent
Scamozzi.
Examples of two capitals are given, differing merely in the number of
mouldings in the abacus. In fact, this introduction of simple mouldings
is about the only variety allowable in the order. Ornament is not admit-
ted, nor are the pillars ever fluted.
A slightly convex curvature, or entasis, is given in execution to the
outline of the shaft of a column, by classic architects, just sufficient to
counteract and correct its appearance, or fancied appearance, of curvature
in a contrary direction (i. e., concavely), which might else take place, and
cause the middle of the shaft to appear thinner than it really is.
Fig. 4: represents the form of a half column from the Pantheon at
Home. In fig. 5, another example of entasis, the lower third of the shaft
is uniformly cylindrical ; the two upper thirds are divided into seven equal
parts. On the semicircle shown in the figure, is a chord cut off parallel
to the diameter, the length of which is fifty-two parts, only one-half being
shown. Divide that part, a J, of the circumference between the diameter
and chord into seven equal parts, and draw parallel lines from each division
to those of the upper part of the column, which will give the diameter of
the shaft at each division ; by increasing the number of the divisions,
more diameters for different parts of the shaft may be found.
PI. LIII. exhibits an example of the Doric order, from the temple of
Minerva in the Island of Egina. The dimensions are given in parts of the
diameter, as in the preceding plate, and the same capital letters denote
corresponding parts. Fig. 1 is an elevation of the capital and the entab-
lature. Fig. 2 of the base, and a part of the Podium. Fig. 3 shows the
forms of the flutes at the top of the shaft, and fig. 4 at the base. Fig. 5,
the outline of the capital on an enlarged scale.
The Doric order may be said to be the original of the Greek orders,
ARCHITECTURAL DRAWING. 255
of winch there are properly but three : the Doric, Ionic, and Corinthian,
which differ in the proportion of their parts, and in some of the ornaments
or mouldings. Of the Doric, the mutules a a, the triglyphs b b, the guttoe
or drops d d of the entablature, the echinus/", and the annulets g g of the
capital, may be considered characteristic. With regard to the arrangement
of the triglyphs, one is placed over every column, and one or more inter-
mediately over every intercolumn (or span between two columns), at such
a distance from each other that the metopes c, or spaces between the tri-
glyphs, are square.
In the best Greek examples of the order, there is only a single triglyph
over each intercolumn. One peculiarity of the Grecian Doric frieze is,
that the end triglyphs, instead of being, like the others, in the same axis
or central line as the columns beneath, are placed quite up to the edge or
outer angle of the frieze. The mutules are thin plates or shallow blocks
attached to the under side or soffit of the corona, over each triglyph and
each metope, with the former of which they correspond in breadth, and
their soffits or under-surfaces are Avrought into three rows of guttse or
drops, conical or otherwise shaped, each row consisting of six guttse, or the
same number as those beneath each triglyph. Though a few exceptions to
the contrary exist, the shaft of the Doric column was generally what is
technically called fluted. The number of channels is either sixteen or
twenty, afterwards increased in the other orders to twenty-four ; for they
are invariably of an even number, capable of being divided by four ; so
that there shall always be a centre flute on each side of the column.
PI. LIY. presents an example of the Ionic order, taken from the temple
of Minerva Polias at Athens. Fig. 1 is an elevation of capital and entab-
lature, fig. 2 of the base, fig. 3 is a half of the plan of the column at
the base and the top, fig. 4 an elevation of the side of the capital. In the
proportions of its shaft, which are more slender, and the addition of a base,
it differs from the Doric ; but the capital is the indicial mark of the order,
by which it is immediately recognized. It is far more complex and irreg-
ular than the other orders of capitals ; instead of showing four equal sides,
it exhibits two fronts, with spirals or volutes parallel to the architrave, and
narrower baluster sides (fig. 4), as they are termed, beneath the architrave.
When a colonnade was continued in front and along the flanks of the
building, this form of capital in the end column occasioned an offensive
irregularity ; for while all the other columns on the flanks showed the vo-
lutes, the end one showed the baluster side. It was necessary that the
end column should, therefore, have two adjoining volute faces, which was
effected by placing the volute at the angle diagonally, so as to obtain there
256 ARCHITECTURAL DRAWING.
two voluted surfaces placed immediately back to back. This same diag-
onal disposition of the volutes is employed for all the capitals alike, in
Koman and Italian examples of this order.
The capital admits of great diversity of character and decoration it
sometimes is without necking, sometimes with ; which may either be plain
or decorated, to suit the entire design. The capital may also be modified
in its proportions, first as regards its general proportion to the column ;
secondly, as regards the size of the volutes compared with the width of
the face. In the best Greek examples, the volutes are much bolder than
in the Roman. The spirals also of the volutes may be either single or
manifold, and the eye or centre of the spiral may be made larger or smaller,
flat or convex, or curved as a rosette.
Plate LY. represents an example of the Corinthian order, from the Arch
of Hadrian, at Athens. This order is distinguished from the Ionic, more
by its deep and foliaged capital than by its proportions, the columns of
both have bases differing but little from each other, and their shafts are
fluted in the same manner.
Although the order itself is the most delicate and lightest of the three,
the capital is the largest, being considerably more than a diameter in
height, varying in different examples from one to one and a half diame-
ter, upon the average about a diameter and a quarter.
The capital has two rows of leaves, eight in each row, so disposed that
of the taller ones, composing the upper row, one comes in the middle, be-
neath each face of the abacus, and the lower leaves alternate with the
upper ones, coming between the stems of the latter ; so that in the first
or lower tier of leaves there is in the middle of each face, a space between
two leaves occupied by the stem of the central leaf above them. Over
these two rows is a third series of eight leaves, turned so as to support the
small volutes which, in turn, support the angles of the abacus. Besides
these outer volutes, which are invariably turned diagonally, as in the
y / four-faced Ionic capital, there are two other
smaller ones, termed caulicoU, which meet each
other beneath a flower on the face of the abacus.
The abacus itself is not, properly speaking, a
square, although it may be said to be so in its
general form. But instead of being straight,
the sides of the abacus are concave in plan, be-
ing curved outwards so as to produce a sharp
Fig ^ point at each corner, which is usually cut off.
Fig. 80 represents one of the capitals of the Tower of the Winds, show-
ARCHITECTURAL DRAWING. 257
ing the earliest formation of the Corinthian capital. In this example the
abacus is square, and the upper row of leaves of the kind called water
leaves, from their resemblance to those of water plants, being broad and
flat, and merely carved upon the vase or body of the capital.
The proper Corinthian base differs from that of the usual Ionic or Attic,
in having two smaller scotise, separated by two astragals : however, both
kinds are employed indiscriminately. The shaft is fluted, in general, simi-
larly to that of the Ionic column, but sometimes the flutes are cabled as it
is called, that is, the channels are hollowed out for only about two-thirds
of the upper part of the shaft, and the remainder cut so that each channel
has the appearance of being partly filled up by a round staff or piece of
rope, whence the term cabling.
The cornice is very much larger than in the other orders, larger as to
height, and consequently as to projection also.
From this greatly increased depth of cornice, it consists of a greater
number of mouldings beneath the corona, for that and the cyinatium over
it invariably retain their places as the crowning members of the whole
series of mouldings. In our illustration, square blocks or dentels are
introduced, but often to the dentels is added a row of modillions^ imme-
diately beneath and supporting the corona. These modillions are orna-
mental blocks, curved in their under surface somewhat after the manner
of the letter S turned thus, QQ ; and between them and the dentels, and
also below the latter, are ether mouldings, sometimes cut, at others left
plain. Sometimes a plain uncut dentel land is substituted for dentels ;
sometimes, in simpler cornices, that is omitted altogether, and plainer
blocks are employed instead of modillions ; or else both dentels and mo-
dillions are omitted. The dentel is not peculiar to this order, but is con-
sidered as more properly belonging to the Ionic.
The Composite Order is hardly to be considered as a distinct order,
being but a union of the Ionic and Corinthian. Its capital consists of a
Roman Ionic one, super-imposed upon a Corinthian foliaged base, in which
the leaves are without stalks, placed directly upon the body of the vase.
In general, the entablature is Corinthian but in a few examples it is Ionic.
Although columns and entablatures do not of themselves, properly
speaking, constitute an order, except they enter into the organization of a
structure ; yet, as the Greek edifices as such, are almost entirely inappli-
cable to purposes of the present day, we have confined our illustrations
of the orders to the pillars and entablatures merely, remarking, that how-
ever the Greek temples differed from each other as to the treatment of the
order adopted, the number of columns and mere particulars of that kind,
17
258 ARCHITECTURAL, DRAWING.
they resemble each other.' Not only were their plans invariably parallel-
ograms, but alike also as to proportion, forming a double square, or being
about twice as much in length as in breadth. The number of the columns
in front was invariably an even one, so that their might be a central inter-
column ; but on the flanks of the edifice, where there was no entrance, the
number of intercolumns was an even, and that of the columns an uneven
one, so that a column came in the centre of these side elevations.
As to the mode in which the front influenced the sides, by determining
the number of columns for them, the established rule seems to have been
to give the flanks twice as many intercolumns as there were columns at
each end : thus the Parthenon, which is octastyle or eight columns in front,
has sixteen intercolumns, and consequently seventeen columns, on each
flank. In like manner, a hexastyle temple would have twelve intercol-
umns and thirteen columns on each side.
In the Doric order, the distances between the columns is governed
entirely by the triglyphs of the frieze, so that there can be no medium
between monotriglyphic and ditriglyphic intercolumniation, accordingly
as there is either one or two triglyphs over each intercolumn. But in the
other orders there is no such restriction ; in them the intercolumns may be
made wider or narrower, as circumstances require, from one diameter and a
quarter to a half in width. Close spacing carries with it the expression of
both richness and strength, whilst wide spacing produces an effect of open-
ness and lightness, but also partakes of meagreness and weakness, owing
to the want of sufficient apparent support for the entablature.
Another mode of columniation and intercolumniation which has some-
times been practised by Modern Architects, consists in coupling the
columns and making a wide intercolumn between every pair of columns,
so that as regards the average proportion between solids and voids, that
disposition does not differ from what it would be were the columns
placed singly. Supercolimmiation, or the system of piling up orders, or
different stages of columns one above another, was employed for such
structures merely as were upon too large a scale to admit of the applica-
tion of columns at all as their decoration, otherwise than by disposing them
in tiers. This method was afterwards adopted by the Architects of the Pal-
ladian School. Sometimes all the three orders are employed in as many
tiers of columns or pilasters. In other cases, the two extreme orders,
that is, the Doric and Corinthian, are brought together ; in other cases
but a single order.
In one or two instances the Greeks employed human figures to support
entablatures or beams ; the female figures or Caryatides, are almost uni-
AECniTECTUEAL DRAWING. 259
formly represented in an erect attitude, without any apparent effort to
sustain any burden or load ; whilst the male figures, Telamones or Atlantes,
manifestly display strength and muscular action. Besides entire figures,
either Hermes' pillars or Termini are occasionally used as substitutes for
columns of the usual form, when required to be only on a small, at least a
moderate scale. The first mentioned consist of a square shaft with a bust
or human head for its capital ; the latter of a half-length figure rising out
of, or terminating in, a square shaft tapering downwards. Hermes' pillars
seem to be in great favor with modern German architects, they having not
unfrequently employed them for the decoration of windows.
The Greek orders may be considered as the rudiments of modern
architecture, but the forms of their buildings are almost entirely inappli-
cable to modern purposes. The Eomans developed and matured the
Corinthian order, and also worked out a freer and more complex and com-
prehensive system of architecture. They introduced circular forms and
curves not only in elevation and section, but in plan ; and while, among
the Greeks, architecture was confined almost exclusively to external ap-
pearance and effect, in the hands of the Romans it was made to minister
to internal display also. The true Roman order consists, not in any of the
columnar ordinances, but in an arrangement of two pillars placed at a dis-
tance from one another nearly equal to their own height, and having a
very long entablature, which, in consequence, required to be supported in
the centre by an arch springing from piers. This, as will be seen from
fig. 1, Plate LVI. was, in fact, merely a screen of Grecian architecture
placed in front of an arcade. Though not without a certain richness of
effect, still as used by the Romans, these two systems remain too distinctly
dissimilar for the result to be pleasing, and their use necessitated certain
supplemental arrangements by no means agreeable. In the first place the
columns had to be mounted on pedestals, or otherwise an entablature pro-
portional to their size would have been too heavy and too important for a
thing so useless and so avowedly a- mere ornament. A projecting key-
stone was also introduced into the arch. This was unobjectionable in
itself, but when projecting so far as to do the duty of an intermediate cap-
ital, it overpowered the arch without being equal to the work required of
it. The Romans used these arcades with all the three orders, frequently
one over the other, and tried various expedients to harmonize the construc-
tion with the ornamentation, but without much effect. They seem always
to have felt the discordance as a blemish, and at last got rid of it, remov-
ing the pier altogether, and substituting in its place the pillar taken down
from its pedestal. This of course was not effected at once, but was the
260 ARCHITECTURAL DRAWING.
result of many trials and expedients. One of the earliest of these is ob-
served in the Ionic Temple of Concord, in which a concealed arch is
thrown from the head of each pillar, but above the entablature, so as to
take the whole weight of the superstructure from off the cornice between
the pillars. When once this was done it was perceived that so deep an
entablature was no longer required, and that it might be either wholly
omitted, as was sometimes done, in the centre intercolumniation, or at all
events very much attenuated. There is an old temple at Talavera, in
Spain, which is a good example of the former expedient ; and the Church
of the Holy Sepulchre, built by Constantino at Jerusalem, is a remarkable
example of the latter. There, the architrave is cut off so as merely to
form a block over each of the pillars, and the frieze and cornice only are
carried across from one of these blocks to the other, while a bold arch is
thrown from pillar to pillar over these, so as to take any weight from off a
member which has at last become a mere ornamental part of the style.
Figs. 2, 3 and 4, Plate LYI. from the Palace of Diocletian at Spalatro,
are illustrations of the different modes of treatment of the arch and entab-
lature.
Perhaps the most satisfactory works of the Romans are those which we
consider as belonging to civil engineering rather than to architecture;
their aqueducts and viaducts, all of which, admirably conceived and exe-
cuted, have furnished practical examples for modern constructions, of which
the High Bridge across Harlem River may be taken as an illustration.
The whole history of Roman architecture is that of a style in course
of transition, beginning with purely Pagan or Grecian, and passing into a
style almost wholly Christian. The first form which Christian art took in
emancipating itself from the Pagan was the Romanesque, which afterwards
branched off into the Byzantine and the Gothic.
The Romanesque and Byzantine, as far as regards the architectural
features, are almost synoymous; in the earlier centuries there is an orna-
mental distinction, the Romanesque being simply a debasement of Roman
art the Byzantine being the art combined with the symbolic elements
introduced by the new Christian religion. As commonly used, the dome
is also considered a characteristic of the Byzantine, but this will be found
among Roman examples. In its widest signification, the Romanesque is
applied to all the earlier round arch developments, in contradistinction to
the Gothic or later pointed arch varieties of the North. In this view the
Norman is included in the Romanesque, and this distinction will be suffi-
cient for our purpose.
The general characteristics of the Gothic are these : it is essentially
ARCHITECTURAL DRAWING. 261
pointed or vertical in its tendency, in its details geometrical, in its win-
dow tracery, in its openings, in its cluster of shafts and bases, in its suits of
mouldings, and by the universal absence of the dome, and the substitution
of the pointed for the round arch.
The Romanesque pillars are mostly round or square, and if square,
generally set evenly, whilst the Gothic square pillar is set diagonally.
Figs. 5, 6, 7, 8 and 9, Plate LYI., represent sections of Gothic pillars ;
fig. 10 is half of one of the great western piers of the Cathedral of Bourges,
measuring 8 feet on each side.
Figs. 11 and 12 are the elevations of capitals and bases and the sec-
tions of Gothic pillars, one from Salisbury, the other from Lincoln Cathe-
dral. Fig. 13 is a Byzantine capital from the church of St. Sophia at
Constantinople ; fig. 14 one from the palace at Gelnhausen ; fig. 15, a Nor-
man one, from Winchester Cathedral, and fig. 16 a Gothic capital and base
from Lincoln Cathedral.
Mouldings. " All classical architecture, and the Eomanesque which is
legitimately descended from it, is composed of bold independent shafts,
plain or fluted, with bold detached capitals forming arcades or colonnades
where they arc needed, and of walls whose apertures are surrounded by
courses of parallel lines called mouldings, and have neither shafts nor
capitals. The shaft system and moulding system are entirely separate, the
Gothic architects confounded the two ; they clustered the shafts till they
looked like a group of mouldings, they shod and capitalled the mouldings
till they looked like a group of shafts."
Gothic Mouldings appear in almost every conceivable position ; from
the bases of piers and piers themselves, to the ribs of the fretted vaults
which they sustain, scarce a member occurs which is incapable of receiv-
ing consistent decoration by this elegant method.
Jamb Mouldings. In the earliest examples of Nor-
man doorways, the jambs are mostly simply squared
back from the walls; recessed jambs succeeded, and are
common in both Norman and Gothic architecture ; and
when thus raised detached shafts were placed in each
angle (fig. 81). In the later styles, the shafts were almost
invariably attached to the structure. The angles them- F i g . si. *
selves were often cut or chamfered ofF, and the mouldings attached to the
chamfer plane. The arrangement of window jambs during the successive
periods was in close accordance with that of doorways.
In the richer examples small shafts were introduced, which, rising up to
the springing of the window, carried one or several of the arch mouldings.
262
ARCHITECTURAL DRAWIXG.
Yet mouldings are not nevertheless essential accessories ; many windows
of the richest tracery have their nmllions and jambs composed of simple
chamfers.
Arch Mouldings, even when not continuous, partook of the same
general arrangement as those in the jambs, with greater richness of detail.
"When shafts were employed, they carried groups of mouldings more
elaborate than those of the jambs, though still falling on the same planes.
Capitals were either moulded, or carved with foliage, animals, &c. ;
they always consisted of three distinct parts (fig. 82), the
head mould A, the bell B, and the neck mould C. In
Norman examples the head mould was almost invariably
square ; in the later styles it is circular, or corresponding
to the form of the pillar.
Bases consist of the plinth and the base mouldings.
The plinth was square in the Norman style, afterwards
octagonal, then assuming the form of the base mouldings,
it bent in and out with the outline of the pier. Base mouldings were also
extensively used round the buttresses, towers and walls of churches.
String Courses. The most usual and\ perhaps essential position of the
string course is under the windows. Iti the Norman styles they were
usually heavy in the outline, and displayeq no particular beauty of arrange-
ment. In the later styles they were remarkably light and elegant ; from
restraint or horizontally, they now rose close under the sill of the window,
and then suddenly dropping to accommodate themselves to the arch of a
low doorway, and again rising to run immediately under the adjoining win-
dow. In this way, the string courses frequently served the purpose of a
drip stone or hood moulding over doors ; occasionally the hood mould was
continued from one window to the other. But in the later styles they were
generally terminated in heads, flowers, or some quaint device, or simply
returned at the springing of the arch.
Cornices are not an essential feature in Gothic architecture. In the
Norman and early English styles, the cornice was a sort of enlarged string
course formed by the projection of the upper part of the wall, which was
supported on brackets or corbels, and hence termed the corbel table.
The earliest moulding in Norman work, is a circular bead strip worked
out of the edges of a recessed arch, called a circular bowtel
(fig. 83). From a circular form the bowtel soon became
pointed, and, by an easy transition, into the bowtel of one,
two, or three fillets, all of which, with their numerous varieties,
performed important parts in the Gothic moulding system.
ARCHITECTURAL DRAWING.
263
Fig. 84 is the scroll moulding, being, in fac^ a simple filleted bowtel,
with the fillet undeveloped on one side, as shown by the
dotted lines. If this moulding be cut in half, through
the centre of the fillet, we have on the developed side
the moulding now termed by carpenters the rule joint,
which, by rounding off the corners by reverse curves, becomes the wave
moulding.
The ogee is also used very generally in Gothic architecture, both single
and double, the latter formed by the junction of two ogees.
Figs. 85 and 86 are examples of
groupings of mouldings, fig. 85 being
of the earlier Gothic, the filleted bow-
tel with alternate hollows, fig. 86, of
the perpendicular style, the hollow in
the one case being made prominent,
and dividing individual mouldings ;
in the latter insignificant, and as a
separation of groups of mouldings.
Arches are generally divided into the triangular-headed arch, the
round-headed arch, and the pointed arch. Of round-headed arches there
are four kinds, the semicircular, segmental, the stilted, and the horse-shoe.
The stilted arch, fig. 87, is semicircular, but the sides are carried down-
wards in a straight line below the spring of the curve, till they rest upon
the imposts. In the horse-shoe arch, the sides are also carried down be-
low the centre, but follow the same curve (fig.
Fig. 86.
Fig- 87. Fig. 83. Fig. 89. Fig. 90.
The pointed arch may be divided into two classes, those described from
two centres, and those described from four. Of the first class there are
three kinds, the equilateral, the lancet, and the obtuse. The equilateral
(fig. 89), is formed of two segments of a circle, of which the radii are
equal to the breadth of the arch. The radii of the lancet segment are
longer than the width of the arch, and of the obtuse, shorter.
Of the complex arches, there is the Ogee (fig. 90), and the Tudor (fig.
92). The Tudor arch is described from four centres, two on a level with
the spring and two below it.
264
AECHITECTUEAL DRAWING.
Of foiled arches, there are the round-headed trefoil (fig. 91), the pointed
trefoil (fig. 93), and the square-headed trefoil arch (fig. 94).
Fig. 91.
Fig. 92.
Fig. 93.
Fig. 94
The semi-circular arch is the Roman Byzantine and Korman arch, the
ogee and horse-shoe is the profile of many Turkish and Moorish domes, the
pointed and foliated arches are Gothic. *
Domes and Vaults. Both domes and vaults are found in Eoman
works, but with the decline of Roman power the art of vaulting was lost,
and the churches of all Roman Christendom remained with nothing but
timber roofs. But among the Greek Christians, or Byzantines, it was re-
tained, or else re-invented ; but the Greek vaulting consisted wholly of
spherical surfaces, whilst the Roman consisted of cylindrical ones. Figs.
95 and 96 illustrate this distinction, fig. 95 being the elevation of a Roman
cylindrical cross vault, and fig. 96, the elevation of the roof of the church
Fig. 95.
Fig. 96.
of St. Sophia at Constantinople ; and the sprouting of domes out of domes
continues to characterize the Byzantine style, both in Greek churches and
Turkish mosques, down to the present day. This system of vaulting has
also been adopted in St. Paul's, London, and at St. Genevieve, Paris. As
a constructive expedient the -cross vault is to be preferred, as the whole
pressure and thrust are collected in four definite resultants, applied at the
angles only, so that it might be supported by four
flying buttresses, no matter how slender, provided
they were placed in the direction of these result-
ants, and were strong enough not to be crushed by
the pressure.
Fig. 97 represents a compartment of the sim-
plest Gothic vaulting, , a, groin ribs, J, 5, 5, side ribs.
The Romans introduced side ribs, appearing
Fig. 97.
ARCHITECTURAL DRAWING. 265
on the inside as flat bands, and harmonizing with the similar form of
pilasters in the walls, but they never nsed groin ribs ; the Gothic build-
ers introduced these, and deepened the Roman ribs. The impenetra-
tion of vaults, either round or pointed, produces elliptical groin lines,
or else lines of double curvature. Yet the early Gothic architects rarely
made their groin ribs elliptical, and never deviating from a vertical plane.
These ribs were usually simple pointed arches of circular curvature,
thrown diagonally across the space to be groined, and the four side arches
were equally simple, the only care being that all the arches should have
their vertices at the same level. The shell between, therefore, was no
regular geometric surface. The strength depended on the ribs, and the
shell was made quite light, often not more than six inches, while Roman
vaults of the same span would have been three or four feet. The differ-
ence- of principle being, that the Romans made their vault surfaces geo-
metrically regular, and left the groins to take their chance ; while the early
Gothic architects made their groins geometrically regular, and let the in-
termediate surfaces take their chance.
In the next step the groin ribs were elliptical, and when intermediate
ribs or tiercerons were inserted, these ribs had also elliptical curvatures,
but different from the groins, in order that the vault of cut stone built
upon them might have a regular cylindrical surface. In augmenting the
number of tiercerons, and making them ramify, combinations of circular
arcs were substituted for the elliptic curves ; the surfaces of these vaults
could not be cylindrical, but the ribs were placed very near each other, in
order that the portion of the vault between each pair might practically be
almost cylindrical. In the formation of the compound circular ribs three
conditions were to be observed : 1st, that the two arcs should have a
common tangent at the point of meeting. 2d. That the feet of all the
ribs should have the same radius, up to the level at which they completely
separate from each other. 3d. That from this point upwards, their curva-
tures should be so adjusted as to make them all meet their fellows on the
same horizontal plane, so that all the ridges of the vaults may be on one
level.
The geometrical difficulty of such works led to what is called fan
tracery vaulting. If similar arches spring from each side of the pillars
(fig. 97), it is easy to perceive that the portion of vault springing from
each pillar would have the form of an inverted concave-sided pyramid,
its horizontal section at every level being square. Now the later archi-
tects converted this section into a circle, the four-sided pyramid became a
conoid, and all the ribs forming the conoidal surface became alike in cur-
266 ARCHITECTURAL DRAWING.
vature, so that they all might be made simple circular arcs ; these ribs
are continued with unaltered curvature till they meet and form the
ridge ; but in this case the ridges are not level, but gradually descend
every way from the centre point (fig. 98).
In the figure this is not fully carried out,
for no rib is continued higher than those
over the longer sides of the compartment,
so that a small lozenge is still left, with a
boss at its centre. When the span of the
main arch 1) a, was large in proportion to
that of I GJ the arch c became a very acute
'rig. 98. lancet arch, and scarcely admitting win-
dows of an elegant or sufficient size. To obviate this, the compound curve
was again introduced, and the ribs were made less curved in their upper
parts than in the lower. Hence the four- centred or Tudor arches.
The four-centred arch is not necessarily flat or depressed, it can be
made of any proportion, high or low, and always with a decided angle at
the vertex. In general, the angular extent of the lower curve is not more
than 65, nor less than 45. The radius of the upper curve varies from
twice to more than six times the radius of the lower, but generally speak-
ing, the greater their disproportion, the less pleasing is the sudden change
of curvature. The projecting points of the trefoil arch are sometimes
called cusps, often introduced for ornament merely, but serving construc-
tively both in vaults and arches, as a load for the sides, to prevent them
rising from the pressure on the crown. This property of arches has been
explained, depending on the principle (p. 118), that if a polygon of rods
be reversed, the position in which it will stand is that which it will assume
for itself when loaded with the same weights and suspended ; and perhaps
the equilibrium of some of the boldest vaultings was insured by experi-
ments on systems of rods representing the ribs inverted ; and for any archi-
tect who may wish to introduce pendants or cusps in his vaultings, this
rule of trial will be found particularly useful.
As vaultings, in general, were contrived to collect the whole pressure
of each compartment into four single resultants, at the points of springing,
leaving the walls so completely unloaded that they are required only as
enclosures or screens, they might be entirely omitted or replaced by win-
dows. Indeed, the real supporting walls are broken into narrow slips,
placed at right angles to the outline of the building, and called luUresses.
As to the enclosing walls, being not for support, they may be placed as
AECHITECTTJKAL DRAWING. 267
the architect pleases, either at the outer or inner edge of the buttresses.
The one method, being that adopted by the French architects, gave to
their interiors those deep recesses, whilst the other, or English method,
served only to produce external play of light and shade.
The Gorman buttress resembles a flat pilaster, being a mass of masonry
with a broad face, slightly projecting from the wall. They are, generally,
of but one stage, rising no higher than the cornice, under which they
often, but not always, finish with a slope. Sometimes they are carried up
to, and terminate in, the corbel table.
Fig. 1, Plate LYII. represents a buttress in two stages, with simple
slopes as set-offs ; this example is somewhat narrower and projects more
than the Norman buttress.
Fig. 2 is a buttress of the Early English style, having a plain triangu-
lar or pedimental head. The angles were sometimes chamfered off, and
sometimes ornamented with slender shafts. In buttresses of different
stages, the triangular head or gable is used as a finish for the intermediate
stages.
In the Decorated style, the outer surfaces of the buttresses are orna-
mented with niches, as in fig. 3. In the Perpendicular style, the outer
surface is often partially or wholly covered with panel-work tracery
(fig. 4).
It has been said that the buttress was a constructive expedient to re-
sist the thrust of vaulting, but to resist the thrust of the principal vault,
or that over the nave or central part of the church, buttresses of the re-
quisite depth would have filled up the side aisles entirely. To obviate
this, the system of flying buttresses 'was adopted, that is, the connection
of the interior with the outer buttress, by an arch or system of arches, as
shown in fig. 5. To add weight, and consequently solidity, to the outer
piers, they were surmounted by pinnacles, rendering them thus a suffi-
ciently steady abutment to the flying arches, which, in their turn, abutted
the central vaults.
An easy transition leads us from pinnacles to spires, the latter being
but the perfect development of the former, and each requiring the assist-
ance of the other in producing a thoroughly harmonious effect. Yet the
spire never was a constructive expedient, or useful in any way. From the
tower, the spire arose first as a wooden roof, and as height was one of the
great objects to be attained, it was carried to an elevation beyond the
mere requirements of a protection against the weather.
The earlier towers of the Romanesque style were constructed without
spires. All are square in plan, and extremely similar in design. Fig. 6,
268 ARCHITECTURAL DRAWING.
Plate LYIL, is an elevation of the tower attached to the church of Sta.
Maria, in Cosmedin, and is one of the best and most complete examples
of this style. Its dimensions are small, being but 15 feet broad and 110
feet high ; a sufficiency of height, where buildings are not generally tall,
to give prominence, without overpowering other objects, which renders
these towers not only beautiful structures in themselves, but singularly
appropriate ornaments to the buildings to which they were attached.
Those towers are the types of the later Italian campaniles, or bell-towers,
most generally attached to some angle of churches, but sometimes de-
tached, yet so placed that they still form a part of the church design.
Sometimes they are but civic constructions, as belfries, or towers of de-
fence. In design, the Gothic towers differ from the Italian campaniles.
The campanile is square, carried up without break or offset, to two-thirds,
at least, of its intended height ; it is generally solid to a considerable height,
or with only such openings as serve to admit light to the staircases. Above
this solid part one round window is introduced in each face, in the next
story, two, in the one above this, three, then four, and lastly, five, the
lights being separated by slight piers, so that the upper story is, virtually,
an open loggia.
The Gothic towers have projecting buttresses, frequent offsets, lofty
spires, and a general pyramidal form. Fig. 7 is the front elevation of a
simple English Gothic tower ; here the plain pyramidal roof, rising at
an equal slope on each of the four sides, is intersected by an octagonal
spire of steep pitch. The first spires were simple quadrangular pyramids,
afterwards the angles were cut off, and they became octagonal, and this is
the general Gothic form of spire. Often instead of intersecting the square
roof as in the figure, the octagonal spire rests upon a square base, and the
angles of the tower are carried up by pinnacles, or the sides by battlements,
or by both, as in fig. 8, to soften the transition between the perpendicular
and sloping part.
In general the spires of English churches are more lofty than those on
the Continent. The angle at the apex in the former being about 10 and in
the latter, about 15. The apex angle of the spires of Chichester and Lich-
field, are from 12 to 13, or a mean between the two proportions, and ac-
cording to Ferguson, more pleasing than either; although having more lofty
spires, yet the English construction is much more massive in appearance,
than the Continental ; the apertures are less numerous, and the surfaces are
less cut up, and covered with ornaments. The spires of Friberg Church
and many others on the Continent are made open work, a precedent fol-
lowed sometimes in this country, but not in the same material wood
ARCHITECTURAL DRAWING. 269
rather than stone. In cast iron, the same effect would be obtained at a less
cost, and equally durable with stone.
Sometimes the central spires of the tower were omitted ; each of the
pinnacles at the angles being converted as it were into spires. Sometimes
the tower is abruptly ended by mere battlements around its sides. In the
poorer churches, a bell-cot was made to serve the purpose of a bell tower ;
this was formed by carrying up the gable wall, as in fig. 9, and making
apertures for the reception of the bell. When the wall was not of the re-
quisite thickness, the cot was either supported by buttresses from beneath,
or the corbels were projected from each side of the wall.
Fig. 10 represents the upper portion of the tower of Ivan Yeliki at
Moscow. The Russian towers are generally constructed independent of
their churches, and are intended for the reception of their massive bells.
Windows. Before the use of painted glass, very small apertures suf-
ficed for the introduction of the required quantity of light into a church ;
as a consequence the windows of the Romanesque churches were gener-
ally small, and devoid of tracery. Again, as the Byzantine architects
adorned their walls with paintings, they could not make use of stained
glass ; neither in their climate, did they require large apertures ; they fol-
lowed in general form the Romanesque window, apertures with circular
heads, either single or in groups (fig. 1, Plate LVIII. or fig. 6, Plate LYII).
The Korman windows were also small, each consisting of a single light,
semicircular in the head, and placed as high as possible above the ground ;
at first splayed on the inside only, afterwards the windows began to be
recessed with mouldings and jamb shafts in the angles, as in fig. 2.
The Lancet in general use in the early Gothic period was of the sim-
plest arrangement : in these windows the glass was brought within three
or four inches of the outside of the wall, and the openings were widely
splayed in the interior. The proportions of these windows vary consider-
ably ; in some the height being but five times the width, in others as much
as eleven ; eight or nine times may be taken as the average. Lancet win-
dows occur singly : in groups of two, three, five and seven, rarely of four
and six. The triplet, fig. 3, is the most beautiful arrangement of lancet
windows. It was customary to mark with greater importance the central
light, by giving it additional- height, and in most cases increased width
also. In some examples the windows of a lancet triplet are placed within
one dripstone forming a single arch, thus bearing a strong resemblance to
a single three-light window. The first approximation to tracery appears to
have been the piercing of the space over a double lancet window com-
prised within a single dripstone ; in place of the customary simple
270 ARCHITECTURAL DRAWING.
arch head, in some examples of lancet windows, the head of the light is
foiled.
From the combination and foiling, or cusping, of distinct lancets, a
single window divided by mullions and tracery derives its origin.
A traceried window may be justly regarded as a distinctive character-
istic of Gothic architecture. With the decided establishment of the prin-
ciple of window tracery, it became a recognized constructive arrangement
to recess the mullions from the face of the wall in which the window arch
was pierced, and the fine effect thus produced was, as the art advanced,
speedily enhanced by the introduction of distinct orders of mullions, and
by recessing certain portions of the tracery from the face of the primary
mullions and their corresponding tracery bars. The tracery bars are those
portions of the masonry of the window head which mark out the principal
figures of the design ; from these the minor and more strictly decorative
parts of the stone work may be distinguished under the title of Form
pieces.
Decorated window tracery has been generally divided into two chief
varieties, Geometrical and Flowing ; the former consisting of geometrical
figures, as circles, trefoils, quatrefoils, curvilinear triangles, lozenges, &c.
&c. ; while in flowing tracery, these figures, though still existing, are
gracefully blended together in one design. In its most perfect state, geo-
metrical tracery invariably exhibits some large figure of a distinct and de-
cided character, which occupies the entire upper part of the window-head.
Fig. 4 represents a quatrefoil window, fig. 5, a pointed trefoil in out-
line ; with the centres of the different circles indicated, and such lines as
may be necessary to explain the way in which they are described. These
forms and modifications of them, will be found of general application in
traceried windows. Fig. 6 represents two forms of circular windows, or
roses tournantes.
Fig. 7 represents an example of the earlier decorated tracery window-
head, consisting of two foiled lancets, with a pointed quatrefoil in the
spandrel between them. One half of the windows in this, as in some of
the following figures, is drawn in skeleton to explain their construction.
Fig. 8 is another example of Decorated tracery.
Fig. 9 is an example of the English leaf, tracery ; fig. 10 of the French
flamboyant. The difference between the two styles is, that while the
upper ends of the English loops or leaves are round, or simply pointed,
the upper ends of the latter terminate like their lower ones, in angles of
contact, giving a flame-like form to the tracery bars and form pieces.
In England the Perpendicular style succeeded the Decorated ; the mul-
ARCHITECTURAL DRAWING. 271
lions instead of diverging in flowing or curvilinear lines, are carried up
straight through the head of the windows ; smaller mullions spring from
the head of the principal lights, and thus the upper portion of the window
is filled with panel-like compartments. The principal as well as the sub-
ordinate lights are foliated in their heads, and large windows are often di-
vided horizontally by transoms. The forms of the window arches vary
from simple pointed, to the complex four-centred, more or less depressed.
Fig. 11 is an example of Perpendicular windows.
Fig. 12 is a square-headed window, such as were usual in the clere
stories of Perpendicular architecture.
Figs. 13 and 14 are quadrants of circular windows, used more especial-
ly in France, for the adornment of the west ends and transepts of the ca-
thedrals.
Besides the tracery characteristic of Gothic architecture, there is a
tracery peculiar to the Saracenic and Moorish style, of which fig. 15 may
be taken as an example it being a window of one of the earliest mosques.
The general form of the window and door-heads of this style is that of the
horse-shoe, either circular or pointed.
Doorways. Plate LIX. fig. 1, is the elevation of a circular-headed
doorway, which may be considered the type of many entrances both in
Romanesque, Gothic, and later styles. It consists of two or more recessed
arches, with shafts or mouldings in the jambs. In the earlier styles the
arches were circular, in the later Gothic, generally pointed, but some-
times circular ; in the earlier, the angles in which the shafts are placed are
rectangular ; in the later, the shaft is often moulded on a chamfer plane,
that is, a plane inclined to the face of the wall, generally at an angle of
45 ; often the chamfer and rectangular planes are used in connection.
Fig. 2 is a simple head of a depressed four-centred or Tudor-arched
doorway, with a hood moulding. Fig. 3 represents the incorporation of
a window and doorway. Sometimes the doorway pierces a buttress ; in
that case, the buttress expands on either side forming a sort of porch. The
Gothic architects placed doors where they were necessary, and made them
subservient to the beauty of the design.
Fig. 4 is an example of a gabled doorway with crockets and finials.
Fig. 5, of a Perpendicular doorway, with a label or hood moulding above,
and ornamented spandrels.
Fig. 6 is an example of a Byzantine, and fig. 7 of a Saracenic doorway.
The Renaissance style succeeded the Gothic, being, originally, but the
revival or a fair rendering of the classical orders of architecture, with or-
naments from the Byzantine and Saracenic styles.
272 ARCHITECT-HEAL DRAWING.
It was in Italy tliat this revival took place, and Garbett divides this
style into three Italian schools, the Florentine, Venetian, and Roman, ex-
hibiting a certain analogy to the three orders of ancient architecture.
The Florentine, corresponding to the Doric, admits of little apparent
ornament, but any degree of real richness, preserving in its principal
forms severe contrast; powerful masses self-poised without corbelling,
without arching ; breadth of every thing, of light, of shade, of ornament,
of plain wall ; depth of recess in the openings, of perspective in the whole
mass, of projection in the cornice. To these add a sort of utilitarianism,
or absence of features useless to convenience or stability, an absence of
sacrifice of material, admitting of great plainness, of very florid enrich-
ment. On the whole, the Florentine may be called the plain, common-
sense school.
Very different in principle was the Venetian school, which, like its
prototype, the Corinthian, superseded its sober rivals. Its aim was splen-
dor, variety, show, and ornament ; not so much real as effective ornament.
Thus, it rarely contains so much carving or minute enrichment as the Flo-
rentine admits ; but it has larger ornaments, constructed (or built) orna-
ments, great features useless except for ornament, as inaccessible porticoes,
detached columns, and architraves supporting no ceiling, towers built only
for breaking an outline. Its decoration is spread equally over the whole
work. Rectangular severity gives place to curved elegance, in arches,
domes, circular and oblique-angled plans, true grandeur to effect, intellec-
tual sense of fitness to eumorphic beauty.
The Roman school, holding the same place as the Ionic, is intermediate
in every respect between the two other schools. It is better adapted to
churches than to any other class of buildings. This fitness arises from the
grand, simple, and unitory effect of one tall order, generally commencing
at or near the ground, and including, or rather obliterating, the distinction
'of two or three stories, making a high building appear a single story.
To describe these schools technically, the Florentine is mostly astylar,
the style of finistration and rustic quoins ; the Roman, the style of pilas-
ters ; and the Venetian, that of columns. In calling the Florentine asty-
lar, a total absence of external orders is not implied, but their absence as
main features, or on a considerable scale. Their chief application is to
windows and doors, and the greatest orders never include so much as the
height of a single story. In the Roman school, the great scale of the
principal order renders it chiefly an order of pilasters. The outer pilasters
of the great order were often filled in with smaller and columnar ones, in
two tiers, while a still smaller set decorated the openings. As the Vene-
ARCHITECTURAL DRAWING. 273
tians did not use such large orders, they easily made them more columnar,
and introduced hanging entablatures. In this school there is (except in
churches) no principal story or order, if there be more than one, all are
nearly equal, or equally important.
General plan and outline, in the Florentine, is of the utmost simplicity,
rendering it fitter for town than country buildings ; in the Koman, slightly
more varied ; in the Venetian, whenever the site will admit, broken,
complex, and picturesque.
We have thus briefly treated of the distinguishing features, according
to Garbett, of the three modern schools ; there are many other distinctive
styles and names, but they may mostly be included under the one or the
other of these schools, their claim for a distinctive name resting rather on
the peculiar style of ornament or tracery used, than any great distinctive
architectural feature.
Ornament. Architectural ornament is of two kinds, constructive and
decorative. By the former is meant all those contrivances, such as capi-
tals, brackets, vaulting shafts, and the like, which serve to explain or give
expression to the construction ; by the latter, such as mouldings, frets,
foliage, &c., which give grace and life, either to this actual constructive
forms, or to the constructive decoration. It is to the latter class that we
wish to call attention ; mouldings of the different styles have been already
treated of; we therefore propose to give now what are even more merely
decorations of a style.
First, as to Grecian orders. By reference to Plate LIII. we see that
the Doric has the triglyph mutules and guttse. By reference to Plate
LIV., the Ionic, we find various mouldings of the cornice frieze, abacus,
and neck of the column enriched. The principal ornament of the neck
of the column is the anthemion, commonly known, in its most simple
form, as the honeysuckle or palmetto ; in the anthemion as represented in
the figure, the palmetto alternates with the lily or some analogous form.
The ornament of the abacus is the egg and dart, shown on a large scale,
fig. 9, Plate LX., where may be found also the
ornament of the frieze and cornice, fig. 7. Fig. 99,
the fret, and fig. 100, the guilloche, are also com-
mon Greek ornaments, used to adorn the soffits , Fi s- "
of beams, and ceilings. The acanthus is the dis-
tinctive ornament of the Corinthian, of which a
leaf is represented on a large scale in front and
side view, figs. 1, 2, 3, Plate LX. These figures
illustrate, also, the way in which ornaments of
18
274: ARCHITECTURAL DRAWING.
irregular figure are copied by the draughtsman. Thus, suppose it were
required to draw fig. 2, and in a reversed position ; circumscribe around
the given figure, a parallelogram ; divide this parallelogram into any
number of equal squares, and in the position required for the copy, as
fig. 3 for instance, construct a similar parallelogram. For convenience
of reference, we have marked the vertical divisions of the squares by let-
ters, a, 5, c, d, e,f, g, and the horizontal divisions by figures, 1, 2, 3, 4 ; this
is done with both parallelograms. But it must be remarked, that if the
copy is to be a reverse of the original, the figures marking the horizontal
divisions of the copy must be the reverse of the original, as may be seen
in figs. 2 and 3. Now mark on the different vertical and horizontal lines
of the corresponding squares, the relative positions of the parts of the
leaf, and through these points thus established, construct the leaf required.
A similar method may be used in constructing a copy to an enlarged
or to a reduced size of the original, by enlarging or reducing the compar-
ative sizes of the squares of the parallelogram of the copy on a scale pro-
portioned to the enlargement or reduction required. In general, the inter-
sections of the portions of the leaf, or other figure, with the vertical or
horizontal lines, are measured and transferred by the eye ; the larger the
number of squares, therefore, the greater probability of the copy coinciding
with the original. Figs. 4, 5, and 6, are the side elevation, front elevation,
and section of a Greek bracket, the principal ornaments of which are taken
from the anthemion and acanthus.
Fig. 1, Plate LXL, is an elevation of a portion of an enriched cornice
from the temple of Jupiter Stator at Rome, of the Corinthian order of ar-
chitecture. Fig. 2, is the under side of the modillion.
The chief characteristic of Roman ornament, is its uniform magnifi-
cence. As a style it is not original, but rather an enlargement or enrich-
ment of the Greek. There is, further, this distinction between the two
styles, that the most rarely used elements among the Greeks are the most
characteristic of the Roman decorations, the scroll and the acanthus. In-
deed, every form which will admit of it, is habitually enriched with acan-
thus clothing or foliations. The acanthus of the Greeks is the narrow
prickly acanthus ; that of the Roman, the soft acanthus. For capitals the
Roman acanthus is commonly composed of conventional clusters of olive
leaves. The Greek scroll is seldom elaborated, but the Roman is seldom
without acanthus foliations. Fig. 3, represents a Roman acanthus scroll.
The free introduction of monsters and animals is likewise a character-
istic of Greek and Roman ornament, as the sphinx, the triton, the griffin,
and others ; they occur however more abundantly in the Roman.
ARCHITECTURAL DRAWING. 275
As the Christian art succeeded the Pagan, symbols became the founda-
tion of decorations in the Byzantine and Romanesque. The early sym-
bols were the monogram of Christ, the lily, the cross, the serpent, the fish,
the aureole, or vesica piscis, and the circle or nimbus, the glory of
the head, as the vesica is of the whole body. These are very important
'elements in Christian decoration, especially the nimbus, which is the ele-
ment of the trefoil and quatrefoil ; the first having reference to the Trinity,
the Jfecond to the four Evangelists, as the testimony of Christ, and to the
Cross, at the extremities of which we often find four circles, besides the
circle in the centre, which signifies the Lord. Occasionally the symbolic
images of the Evangelists, the angel, the lion, the ox, and the eagle, are
represented within these circles.
The hand in the attitude of benediction, and the lily (the fleur-de-lis),
the emblem of the virgin and purity, are common in Christian decoration.
This last symbol was eventually elaborated into the most characteristic
foliage of Byzantine and Romanesque art. Conspicuous in their foliage
also is a peculiar formed leaf, somewhat resembling the leaf of the ordinary
thistle. The serpent figures largely in Byzantine art as the instrument of
the fall, and one type of the redemption.
As paganism disappeared, their ornaments, under certain symbolic
modifications, were admitted into Christian decorations. Thus the folia-
tions of the scroll were terminated by lilies, or by leaves of three, four and
five blades, the number of blades being significant ; and in a similar way,
the anthemion and every other ancient ornament. In the Byzantine the
symbolism is seldom or ever absent, however much it may be modified or
disguised. An important feature, always to be observed in the Byzantine,
is that all their imitations of natural forms were invariably conventional ;
it is the same even with animals and the human figure, every saint had his
prescribed colors, proportions and symbols.
The Saracenic was the period of gorgeous diapers, for their habit of
decorating the entire surfaces of their apartments was highly favorable to
the development of this class of design. The Alhambra displays almost
endless specimens, and all are in relief and enriched with gold and color,
chiefly blue and red. The religious cycles and symbolic figures of the By-
zantine are excluded. Mere curves and angles or interlacings were now
to bear the chief burden of a design, but distinguished by a variety of co-
lor. The curves however very naturally fell into standard forms and floral
shapes, and the lines and angles were soon developed into a very charac-
teristic species of tracery, or interlaid strap work, very agreeably diversified
by the ornamental introduction of the inscriptions, which last custom of
ARCHITECTURAL DRAWING.
elaborating inscriptions with their designs was peculiarly Saracenic. Al-
though flowers were not palpably admitted, yet the great mass of the
minor details of Saracenic designs are composed of flower forms disguised,
the very inscriptions are sometimes thus grouped as flowers ; still no ac-
tual flo'wer ever occurs, as the exclusion of all natural images is funda-
mental to the style in its purity.
Fig. 3, is a specimen of Alhambra diaper.
All the symbolic elements of the Byzantine are continued in the Gallic.
Ornamentally, the Gothic is the geometrical and pointed element elaborated
to the utmost ; its only peculiarities are its combinations of details ; at first
the conventional and geometrical prevailing, and afterwards these com-
bined with the elaboration of natural objects in its decoration. The By-
zantines never did this, their ornaments are purely conventional ; while
in the finest gothic specimens, not only the traditional conventional orna-
ments, but also elaborate imitations of natural plants and flowers are found.
The most striking feature of all Gothic work is the wonderful elaboration
of its geometric tracery ; vesicas, trefoils, quatrefoils, cinquefoils, and an
infinity of geometric varieties besides. The tracery is so paramount a
characteristic, that the three English varieties, the early English, the deco-
rated, and the perpendicular, and the French flamboyant, are distinguished
almost exclusively by this feature. See Plate LYIII.
Under the head of Gothic, the Norman is often included, but it is rather
a transition style between the Romanesque or Byzantine, and the Gothic.
The ornamental mouldings used in the decorative details of this style are
numerous, among which the more common is the chevron or zig-zag, (fig.
1, plate LXII.,) simple as the indented, or duplicated, triplicated or quad-
rupled ; the billet, the prismatic billet, the square billet, and the alternate
billet (fig. 2) ; the star fig. 3, the fir cone ; the cable, fig. 4 ; the embat-
tled, fig. 5 ; the nail head, fig. 6. In the early English style we find the
dog-tooth, fig T ; a kind of pyramid-shaped flower leaves ; the ball flower,
fig. 8, and the serpentine vine scroll, are the most characteristic ornamen-
tal mouldings of the decorated style. The mouldings of the perpendicu-
lar are not peculiar ; they are less enriched than the preceding styles, and
the same panelling which is found in the windows is spread over every
surface of the building.
In the early English we have the first development of geometrical tracery,
flying buttresses, crocketed pinnacles, columns clustered, and an extensive
application of foliage with the trefoil leaf, as the most characteristic orna-
ment ; sometimes formed as a clover leaf, at other times very irregularly
formed.
ARCHITECTURAL DRAWING. 277
The early English is characterized, besides its tracery, by the ogee and
the pinnacled canopied recesses of its buttresses and other parts producing
a prominence of diagonal lines. There is also more copying of nature in
its ornamental details.
In the Perpendicular, the new features are the horizontal line and the
panellings, and the substitution of perpendicular for flowing tracery.
The crocket, in its earliest form, was the simple arrow head of the
Episcopal, pastoral staff; subsequently finished with a trefoil, and after-
wards still further enriched. Figs. 9 and 10 are early English crockets ;
fig. 11 a decorated one. Fig. 12 is a finial of the same style ; both finials
and crockets in detail display a variety of forms ; some resembling the
botanical productions of one class, some of another.
The Parapets of the early English style are often a simple horizontal
course, supported by a corbel table, sometimes relieved by a series of sunk
blank trefoil-headed panels ; sometimes a low embattled parapet crowns
the wall. In the decorated style, the horizontal parapet is sometimes
pierced with trefoils, sometimes with wavy flowing tracery (fig. 13). Gro-
tesque spouts or gargoyles discharge the water from the gutters. The
parapets of the perpendicular style are frequently embattled (14), covered
with sunk or pierced panelling, and ornamented with quatrefoil, or small
trefoil-headed arches ; sometimes not embattled but covered with sunk or
pierced quatrefoils in circles, or with trefoils in triangular spaces as in
fig. 15.
Amongst the varieties of ornamental work, the mode of covering small
plain surfaces with diapering (fig. 16), was sometimes used ; the design
being in exact accordance with the architectural features and details of
the style. The rose, fig. 17, the badge of the houses of York and Lancas-
ter, is often met with in the perpendicular style ; and tendrils, leaves and
fruit of the vine, are carved in great profusion in the hollows of rich cor-
nice mouldings, especially on screen work in the interior of a church.
Fig. 18, in its original type, a Byzantine ornament, an alternate lily and
cross, is a common finish to the cornice of rich screen work in the latest
Gothic, and is known under the name of the Tudor flower.
Figs. 19, 20, 21, are examples of ornamental crosses used as finials,
either for spires or pinnacles.
The Ornaments of the Renaissance. The term Renaissance is used in a
double sense ; in a general sense implying the revival of art, and specially,
signifying a peculiar style of ornament. It is also sometimes, in a veiy
confined sense, applied in reference to ornament to the style of Benvenuto
Cellini ; or, as it is sometimes designated, the Henry II. (of France) style.
ARCHITECTURAL DRAWING.
The mixture of various elements is one of the essentials of this style.
These elements are the classical ornaments ; unnatural and natural flowers
and foliage, the former often of a pure Saracenic character; man and ani-
mals, natural and grotesque ; cartouches, or pierced and scrolled shields, in
great prominence ; tracery independent, and developed from the scrolls of
the cartouches ; and jewel forms. Fig. 1, and 3, Plate LXIII.
The Elizabethan is a partial elaboration of the same style, the only
difference being that what we now term the Elizabethan exhibits a very
striking preponderance of strap and shield work, but the earlier and pure
Elizabethan is much nearer allied to the continental styles of the time ;
classical ornaments but rude in detail, occasional scroll and arabesque
work, and strap work, holding a much more prominent place than the
pierced or scrolled shields. Fig. 2 is an example of the style from the old
guard chamber, Westminster.
Of the earliest and transition styles of Renaissance ornament, are the
Tricento and the Quatrecento ; the great features of the first are its intricate
tracery and delicate scroll work of conventional foliage, the style being
but a slight remove from the Byzantine and Saracenic. Of the second
are, in addition, elaborate natural imitations of fruit, flowers, birds or
animals (fig. 4), all disposed simply with a view to the ornamental ; also
occasional cartouches, or scrolled shield work.
In all these styles, the evidence of their Byzantine or Saracenic origin
is constantly preserved, in the tracery, in the scroll work and foliage, and
in the rendering of classical ornaments. The Eenaissance is, therefore,
something more approximative to a combination of previous styles than a
revival of .any in particular. Yet it is a style that was developed solely
on aesthetic principles, from a love of the forms and harmonies themselves,
as varieties of effect and arrangements of beauty, not because they had
any particular signification, or from any superstitious attachment to them
as heirlooms.
Fig. 5 is an example of ornament in the Cinquecento style. The ara-
besque scroll work is the most prominent feature of the Cinquecento, and
with this in its elements, it combines every other feature of classical art,
with the unlimited choice of natural and conventional imitations from
the entire animal and vegetable kingdom, both arbitrarily disposed and
combined. Absolute works of art, such as vases and implements, and in-
struments of all kinds, are prominent elements of the Cinquecento ara-
besque, but cartouches and strap work wholly disappear from the best
examples. Another chief feature of the Cinquecento is the admirable
play of color in its arabesques and scrolls, and it is worthy of note that
ARCHITECTURAL DRAWING. 2 79
the three secondary colors, orange, green, and purple, perform the chief
parts in all the colored decorations.
Fig. 6 is an example of the Louis Quatorze style of ornament. The
great medium of this style was gilt stucco work, and this absence of color
seems to have led to its most striking characteristic, infinite play of light,
of shade ; color, or mere beauty of form in detail, having no part in it
whatever. Flat surfaces are not admitted ; all are concave or convex : this
constant varying of the surface gives every point of view its high lights
and brilliant contrasts.
The Louis Quinze style differs from that of Louis Quatorze chiefly in
its absence of symmetry ; in many of its examples it is an almost random
dispersion of the scroll and shell, mixed only with that peculiar crimping
of shell work, the coquillage.
The ornaments of which we have thus given examples are, in general,
applied to interior decorations, to friezes, pilasters, panels, architraves, the
faces and soffits of arches, ceilings, &c., to furniture and to art manufac-
tures in general. For exteriors these ornaments are sparingly applied ;
shield and scroll work, of the later Elizabethan or Kenaissance style, is
sometimes used, but very seldom tracery.
Of common exterior ornament, the baluster is peculiarly modern, with
all the refinement of a classic model. Balustrades are sometimes of real
use in building, and at other times merely ornamental. Such as are in-
tended for use, as when they are employed on steps or stairs, before win-
doM's, or to enclose terraces or other elevated places of resort, must always
be nearly of the same height, from three to three and a half feet, so that
a person of ordinary height may, with ease, lean over them without the
danger of falling. But those that are principally designed for ornament,
as when they finish a building ; or even for use and ornament, as when
they form the railing over a large bridge, should be proportioned to the
architecture they accompany, and their height ought never to exceed four-
fifths of the entablature on which they are placed ; nor should it be less
than two-thirds, without counting the plinth, the height of which must be
sufficient to leave the whole balustrade exposed to view.
Figs. 101, 102, 103, 104, 105, and 106,
represent various figures of balusters and of va-
rious proportions, suited to the various orders
they may serve to finish. The double-bellied
balusters (figs. 101 and 102) are the lightest,
and, therefore the best adapted to windows Fig ' 101 - Fig ' m F1&m Fis ' 104 -
or other compositions of which the parts are small and the profiles
280 AKCHITECTUEAL DRAWING.
delicate. The base and rail may be of the same profile but not so large
as for single-bellied ones.
In balustrades, the distance between two balus-
ters should not exceed the half of the diameter of
the thickest part of the baluster, nor less than one-
third of it. The pedestals, if possible, should be at
intervals of about nine balusters, but as the pedes-
tals must be placed over the centre of the piers, the
intervals must frequently contain more balusters.
Fig. 105. Fig. 106. -pig. 105 shows the arrangement of a baluster
with inclined rails and bases.
"When used in interiors, either for decoration or use, the forms of the
baluster are much varied and enriched ; this is especially observable in
constructions in iron.
ELEVATIONS OP HOUSES.
Having thus given a brief abstract of the characteristics of various
prominent styles of architecture, we continue our article on houses by
giving elevations, either suited to plans already exhibited, or to other
plans which will be found on the same plate as the elevations. It must be
perceived that, in general, in modern constructions, pure ancient art is
seldom exhibited, nor would it, in domestic architecture, be found suitable,
the requirements and appliances being very different, and he may be
called an architect, who, conversant with ancient and modern practice,
can adapt them in unity and harmony to modern necessities.
PL LXIY. represents the front elevation of a basement house with the
general characteristics of the Florentine style, uniting richness and gran-
deur of effect, admirably suited to the locality and purpose for which it
is designed, a first-class house, or even what might be termed a palatial
residence. This building has been constructed in Fifth Avenue, New
York, after designs of T. Thomas & Son, architects ; the specifications of
which will be found in a subsequent chapter.
English basement houses are generally constructed with a rusticated
basement as in the preceding example, (PL LXIV.) with a balustrade
marking the distinction between it and the principal story. The entrance
is generally raised not to exceed three steps, and seldom with a projecting
porch ; the intention being to make the basement subordinate to the prin-
cipal story, the usual finish of the door-head is similar to that of the win-
dow. In general English basement houses are intended for narrow lots ;
AECHITECTUEAL DRAWING. 281
showing in front but two windows to the stories above the basement, and
one basement window. Circular heads are almost invariably used for the
basement openings, and for the windows above either square or slightly
arched lintels. Sometimes a species of Romanesque window of clustered
openings is adopted.
"When the architect is not controlled by the form or size of the lot,
much picturesqueness may be given by varieties of form and irregularities
of outline in the construction of edifices.
Plate LXY. is an elevation of a house from Holly's " Country Seats,"
after a style of architecture usually designated here as the French, from
the form of the Mansard roof and its dormer windows, rather than any dis-
tinctive features in the main elevation or its ornaments. This style of roof
is very effective, and has become very popular ; it is well adapted both for
city and country residences.
Plates LXYL, LXYIL, LXYIII., and LXIX. are elevations and plans of
country residences from " Downing's Country Houses," drawn in perspec-
tive, the principles of which will be given in a subsequent chapter. They
may be taken as beautiful illustrations of modern constructions.
Plate LXYI. are the plans and elevation of a Farm House in the Eng-
lish Rural Style.
Plate LXYIL is an elevation and plan of a plain timber cottage villa,
after designs of Gervase Wheeler, Architect, of Philadelphia. "The
construction, though simple, is somewhat peculiar. It is framed in such a
manner that on the exterior the construction shows. At the corners are
heavy posts, roughly dressed and chamfered, and into them are morticed
horizontal ties, immediately under the springing of the roof ; these, with
the posts and the studs and the framing of the roof showing externally.
Internally are nailed horizontal braces at equal distances apart, stopping
on the posts and studs of the frame, and across these the furring and lath-
ing cross diagonally in different directions. On these horizontal braces,
the sheathing composed of plank placed in a perpendicular position is sup-
ported and retained in its place by battens, two and a half inches thick,
and made with a broad shoulder. These battens are pinned to the hori-
zontal braces, confining the planks, but leaving spaces for shrinking and
swelling, thus preventing the necessity of a single nail being driven
through the planks. A representation is given (fig. 107) of the batten B,
and the mode of framing.
Fig. 108 represents the usual form of vertical boarding, which is less
expensive than the first illustration, and, in general, will be found suffi-
ciently secured for the class of buildings to which it is applied.
282 ARCHITECTURAL DRAWING.
Plate LX YIII. is a villa in what Mr. Downing designates as the Rural
b
Gothic style, designed by him-
self. Figs. 109, 110, 111, and
112, represent some of the de-
tails on a larger scale.
Fig. 109 is an elevation of
the bay window with a balco-
ny over it, to the scale of one-
quarter of an inch to the foot.
Fig. 110, the verge board of
the small gable over this bal-
cony. Fig. Ill, part of the
verge board of the gable over
the porch. Tig. 112 are chim-
ney tops, such as can be ob-
tained of Garnkirk clay.
Fig.
Fig. 111.
ABCHITECTUKAL DEAWING.
283
w
wW
?m
ra 3
C4$
i
Fig. 112.
a a B Q
Fig. 113.
284
ARCHITECTURAL DRAWING.
Plate LXIX. is the elevation and principal floor plan of a villa in the
Italian style, as constructed after plans of Mr. Upjohn. " It is one of the
most successful specimens of the Italian style in the
United States."
This villa is built of brick, painted externally
of a light freestone color, and the window dressings,
string-courses, cornices, brackets, &c., are all free-
stone. Figs. 113 and 114 are the front and side
elevation of the balcony window in the front of the
house, drawn to a scale of one-quarter of an inch
to a foot.
Stables. Under this general name are included
the barn, or the receptacle of hay and fodder, the
carriage-house and the stable proper, or lodging-
house for horses and cows. The first two may be
included under one roof, the carriages on the 1st
floor, hay in the loft, and oats in the cellar ; but
the lodging-place should be distant in a wing attached
to the barn, that the odors from the animal may not
impregnate the food, or the cloth-work of the car-
riages, or the ammonia tarnish their mountings.
Hay in bulk, in the mow, occupies about 7 c.
ft. per ton ; bales average 2'.-l" x 2 / .6 // x 4/, and
weigh, from 220 to 320 Ibs. The door space for a
load of hay in the bulk should be from 12 to 13 ft.
high and 12 ft. wide. The floor beneath the hay
should be tight, so that dust and seed may not drop
on the carriage. A door for carriage should be 10 ft. 6 in. high x 9 ft. wide.
The horse is to be treated with greater care than any other domestic
animal. His stable is to be carefully ventilated, that he may have fresh
air without being subject to cross-drafts. Preferably the floor should be on
the ground, that there may be no cold from beneath. He should stand as
near as possible level ; and for this purpose a grated removable floor, with
small interstices, should be laid over a concrete bottom, with a drip towards
the rear of the stall, and the urine should be collected in a drain, and dis-
charged into a trapped manure-tank outside the stable. The manure
should never be deposited beneath the stable, but should be wheeled out,
and deposited in a manure-yard or tank daily. It is as essential that all
excrements should be removed entirely from the stable as that the privy
should be placed outside the house. ,
Fig. 114.
ARCHITECTURAL DRAWING.
285
The breadth of stalls should be from 4 ftfci 6 in. to 5 ft. in the clear ;
the length, 7" ft. 6 in. to 8 ft : the rack and feed-box require two feet in
addition, to which access is given in the best stables by a passage in front.
Hack and feed-box are often made of iron, and the upper part of stalls
fitted with wrought-iron guards. Box-stalls, in which horses are shut up,
but not tied, in cases of sickness or foaling, are about 10 ft. square.
Grain and hay are delivered from the loft to the stable-floor by shoots,
or boxes with slides.
in PH.
tin
Fig. 115.
Fig. 115 is one-half the second-story plan, and Plate LXX. an eleva-
tion in perspective of a tenant-house, built after designs by John "W.
Hitch, architect.
In its construction it is almost entirely fire-proof; the staircases are of
iron, the hall floors are constructed with iron beams, brick arches, and blue
flagging ; the dividing floors and walls are deafened ; every alternate wall
is of brick ; every window has inside shutters, and every room is ventilated
by air-flues to the roof.
On the south is a spacious flagged court-yard, 12 by 188 feet, which is
used by the inmates for washing and drying their clothes the families
on each floor having the exclusive use of it for specified days of the week.
The yard connects with the main hall, by cross-halls, and is shut oft' from
the streets by high gates that are kept closed, except fuel is brought to the
premises. The cellar is divided into 94 compartments, that is, one for
each tenement, with a lock and key for each.
On the upper floor are two large adjoining rooms, 53' x 50' each,
which can be thrown into one or disconnected at pleasure. They
are designed for lectures, concerts, or moral and educational uses for
286
ARCHITECTURAL DRAWING.
the inmates during the wee*:, and for Sunday school and religious observ-
ances on the Sabbath.
The exterior is of brick with brown stone window-sills, and in its style
Is an excellent example of the architectural effect that may be produced
in our most common materials, and in an unpretending edifice, by break-
ing up the monotony of fagade by even slight projections, by the clustered
and circular heads of the windows, and by an appropriate and varied
cornice. This style is becoming very popiilar and is particularly appli-
cable to the construction of mills and workshops.
Store and Warehouses. Plate LXX1., is an elevation of a store front,
and figs. 117 and 118 plans of first story and basement.
These plans may be taken as a type of the general class of large whole-
sale or retail stores covering but one lot. In this city there is usually
beneath the sidewalks two stories, the basement and sub-cellar. These
are generally let with the first story, and the upper stories together by
themselves. The depth of the stores are mostly from 100 to 200 feet, on
an average about 150 feet. The centre is lighted by a skylight in the roof,
and by well-holes, B, beneath, in the several floors. In front of the en-
trance is a platform, A, which is either an iron grating, or, when the base-
ment extends through into the front vaults, covered with patent vault
lights. To protect the vaults from moisture the walls are laid hollow, and
the outside covered with asphalte. The hoistway t6\ basement and sub-
cellars, is by a trap in the grating front of the window, usually a plat-
form supported by chains at the four corners, and raised vertically, often
ARCHITECTURAL DRAWING. 287
by a car sliding on an incline, if there are outisde stairs leading to the
basement. In the rear an area of some ten to fifteen feet in width is dug
out, and the two lower stories show full. All the rear windows are pro-
tected by iron shutters.
The floor of the first story is often laid with a rising grade, of about
1 foot in 100 towards the rear, to prevent the appearance which a long
level sometimes has of descending, and to afford more light in the rear to
the basement. The offices are in the rear on this floor. The safe is some-
times built into the wall, or into a projection from it, or the safe is mova-
ble ; or, what is rare at present, a book vault is made in the front vault.
The front windows and doors are mostly protected by revolving shutters
rolling up like a curtain in the box lintels above. Separations are made
between tenants occupying different floors by iron framed skylights over
the well-holes.
C is the entry way to the second story, separated from the store by a
glass partition protected by a wrought iron screen or guard. Above this
entrance in the second floor, is the hoistway for goods, generally about five
feet square. The second floor does not differ in plan from the first, and
so with the stories above, except in some cases the well-holes are wider
in the upper stories. The floors are all level.
The water closets are mostly on the third floor, and in the front base-
ment vault. The heating is either by stoves, hot air furnaces, or steam.
The shelvings, counters and other furniture depend, of course, on the class
and kind of business.
Front Elevation. Various styles are adopted, but in one particular
there is almost an uniformity ; that is, the whole front is supported on posts
of cast iron in the first story, with iron lintels and cornice ; the great ob-
ject being to get as much light as possible in this story. These posts are
sometimes square or rectangular in plan, with a small sunk panel on the
face, and shield-like ornaments containing the number of the store, and
capitals at the top ; sometimes a sort of Corinthian column, and some-
times two posts, the inside one circular, and the outside square. As there
is but little chance for ornament, the building seldom assumes any distinc-
tive expression till it reaches the second story. The great ornament of
the first story is the plate glass. The elevation and plans represent the
usual form of the wholesale stores with but three openings in the first story
one window and two doors. In the retail stores occupying a full lot
there are generally four openings, the door to the first floor, central between
two windows, and the side door leading to the second story ; but where
all the stories are occupied by the same trade, the side door is usually
288 ARCHITECTURAL DRAWING.
omitted. The door of the retail store is generally recessed, with show
windows at the sides to admit of the greater display of goods. The glass
of the windows are sometimes of one plate, as large as 8 x 14 feet even,
but more usually in four squares ; seldom more in number.
Above the first story, the front begins to assume an architectural ex-
pression, though seldom perhaps very significant of any-intention or de-
sign for a specific purpose inside. The example selected may be con-
sidered a fair average of the class. It is to be remarked that where
various businesses are to be carried on in the same building, and where
large signs may be necessary to designate them, there will be but little
room, as there will but little necessity, for much ornamental detail.
Plate LXXII. is a chaste and beautiful fagade of two stores, erected
on Broadway, from designs by J. B. Snook, architect.
Plate LXXIII. is an elevation of a storefront executed in cast iron byD.
D. Badger & Co. of this city. The style is Venetian, and when the front is
more than fifty feet in width, the effect is imposing. It is rather more ap-
propriate for stores with offices above, or for stores designed for but one
purpose, as signs larger than could be placed in the panels would mar the
effect. Iron was first introduced for house fronts by Mr. Bogardus, and
it has much to recommend it. Ornaments can be applied profusely, and
at the same time cheaply, and in durability it exceeds our common free-
stones. The chief objection at present lies in this, that few wish to go to
the expense of new patterns : the result is that the forms become too ste-
reotyped, especially objectionable when much ornament is used. The color
which it should be painted has been a subject of much discussion ; the
prevailing tint at present is a sort of cream color, with brown trimmings
of the windows.
School Houses. PL LXXIY. contains a plan and elevation of a dis-
trict school house, with seats for forty-eight scholars. There are two en-
trances, one for each sex, with ample accommodations of entry or lobby
room for the hanging up of hats, bonnets and cloaks. A side door leads
from each entry into distinct yards, and an inside door opens into the
school-room. The desk, T, of the teacher, is central between the doors,
on a platform, P, raised some six or eight inches above the floor. In
the rear of the teacher's desk is a closet or small room, for the use of
the teacher. The seats are arranged two to each desk, with two alleys
of eighteen inches, and a central one of two feet ; the passages around
the room are three feet. The scale is eight feet to the inch. The eleva-
tion is in a very plain Romanesque style, to be constructed of brick with
hollow walls.
AKCIIITECTUEAL DRAWING. 289
On the Requirements of a School-House. Every scholar should have
room enough to sit at ease, his seat should be of easy access, so that he
may go to and fro, or be approached by the teacher without disturbing
any one else. The seat and desk should b.e properly proportioned to each
other and to the size of the scholar for whom it is intended. The seats as t
furnished by the different makers of school furniture, vary from nine to
sixteen inches in height ; and the benches from seventeen to twenty-eight
inches ; measuring on the side next the scholar. The average width of the
desk is about eighteen inches, and is formed with a slope of from one and
a half to two and a half inches, with a small horizontal piece of from two
to three inches at top. There is a shelf beneath for books, but it should
not come within about three inches of the front. The width of the seat
varies from ten to fourteen inches, with a sloping back, like that of a chair ;
it should, in fact, be a comfortable chair. It will be
observed that, in the plate, two scholars occupy one [ p I p j~~p f]
bench ; fig. 119 represents another arrangement, in
which each scholar has a distinct bench ; and, in h rj ! h
many respects it is preferable, but is not quite so ' ' "
economical in room. In primary schools, desks are [ L , I L , I L , f|
not necessary ; and in many of the intermediate | H I H I \ II
schools, the seat of one bench is formed against the j i i i i i rt
back of the next bench ; but distinct seats are pre- I U IP P
ferable. The teacher's seat is invariably on a raised rig. im
platform, and had better be against a dead wall than where there are win-
dows. The best light is undoubtedly a skylight, but as this is seldom con-
venient, the lights at the side should be high above the floor. Blackboards
and maps should be placed along the walls: Care should be taken in the
warming and ventilation ; the room should not be less than ten feet high ;
the best method of heating is by furnaces in the- cellar, warm air should
be introduced in proportion to the number of scholars, and ventiducts
should be formed to carry off the impure air.
In cities the school-houses are made of a number of stories the pri-
maries being in the lower stories, and, in some cases, play rooms also, and
the grammar-schools occupying floors above. In these cases the teachers
are numerous, and separate rooms are prepared for the hearing of recita-
tions.
Plate, page 290, is a view of one of the largest of the New York
City schools, of which fig. 1, p. 291, is the plan of grammar-department
floors, and fig. 2 plan of the same floors of another house of a different
outline.
290
ARCHITECTURAL DRAWING.
AKCHITECTUKAL DKAWISG.
291
Fig. 1.
Fig. 2.
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g
CUSS I I ROOM
a c
CUSS | | RQUN
SCHOOL ROO
D:D:D:Q:C:D:D:D:D:0:D:D:D:D:D:
0:0:0:0:rj:[:D:fl:G:Q:fl:C:0:C:0:
RDROB
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r J-gCE^J
.CUSS . 1 ROC
CT3 CZ3
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292 ARCHITECTURAL DE AWING.
Lecture Rooms, Churches, Theatres, Legislative Halls.- To the proper
construction of rooms or edifices adapted for these purposes some know-
ledge of the general principles of acoustics, and their practical application,
is necessary. In the case of lecture rooms and churches, the positions of
the speaker and the audience are fixed ; in theatres, one portion of the
enclosed space is devoted to numerous speakers, and the other to the
audience ; in legislative halls, the speakers are scattered over the greater
part of the space, and also form the audience.
The transmission of sound is by vibrations, illustrated by the waves
formed by a stone thrown into still water ; but direction may be given to
sound, so that the transmission is not equally strong in every direction ;
thus, Saunders found that a person reading at the centre of a circle of one
hundred feet in diameter, in an open meadow, was heard most distinctly
in front, not as well at the sides, but scarcely at
all behind. Fig. 120 shows the extreme dis-
tance every way at which, the voice could be
distinctly heard : ninety-two feet in front, seventy-
five feet on each side, and thirty-one feet in the
rear. The waves of sound are subject to the
same laws as those of light, the angles of reflec-
tion are equal to those of incidence ; therefore,
in every enclosed space, there are reflected sounds
more or less distinct, according to the position of -the hearer, and to the form
and condition of the surfaces against which the waves of sound impinge.
Thus, of all the sounds entering a parabolic sphere, the reflected sounds
are collected at the focus. Solid bodies reflect sound, but draperies absorb
it. As, in all rooms, the audience can never be concentrated at focal
points, nor is it possible in any construction to make calculation for
all positions, it is in general best to depend on nothing but the direct force
of the voice, and not to construct larger than can be heard directly without
aids from reflected sounds.
There is great difference in the strength of voice of different speakers ;
the limits as given in the figure are for ordinary reading in an open space.
In enclosed spaces, owing to the reflected sounds or some other cause, there
are certain pitches or keys peculiar to every room, and to speak with ease,
the speaker must adapt his tone to those keys. The larger the room, the
slower and more distinct should be the articulation.
It has been observed, that the direction of the sound influences the ex-
tent to which it may be heard. The direction of the currents of air through
which the sound passes effects the transmission of the sound, and this may
ARCHITECTURAL DRAWING. 293
be made useful when the rooms are heated by hot air, by introducing the
air near the speaker, and placing the ventilators or educts at the outside
of the rooms, and by placing their apertures rather nearer the bottom
of the room than at the top. It would seem much better and easier
to make a current of air a vehicle of sound rather than depend on re-
flection.
The best form for a lecture room is the semicircle, or three-fifths of a
circle, fig. 120, the speaker in the one case at the centre, in the other, at
the point A, on a platform raised some two or three steps above the floor,
the audience being ranged in concentric seats, rising from the centre
outwards. The room should be no higher than requisite for beauty
or for ventilation. The ceiling should be slightly curved, not flat nor half
globe.
On the space occupied by seats in general. A convenient arm-chair oc-
cupies about twenty inches square, the seat itself being about eighteen
inches in depth, and the slope of the back two inches. Eighteen inches
more affords ample space for passage in front of the sitter : this accommoda-
tion would be ample. In the arrangement of seats at the Academy of Mu-
sic the bottom turns up, and twenty-nine inches only is allowed for both seat
and passage, and eighteen inches for the width of seat, which may be taken
as the average allowance in width to each sitter in comfortable public
rooms. In lecture rooms stalls are often used, the space there occupied by
seat and passage being about two feet six inches. The alleys should
be at the sides of the room, with two intermediate, dividing the seats
into three equal benches, and not one in the centre, except in very,
large rooms, as the space thus left is the best for hearing and seeing the
speaker.
In the earlier churches, ceremonies and rites formed a very large part
of the worship, the sight was rather appealed to than the hearing, and for
this purpose, churches were constructed of immense size, and with all the
appliances of ornament and construction, with pillars, vaults, groins and
traceried windows. In the churches of this country, the great controlling
principle in the construction of a church, is its adaptation to the comfort-
able hearing and seeing the preacher. In this view alone, the church is
but a lecture room : but since even the character of the building may tend
to devotional feelings in the audience, and since certain styles and forms
of architecture have long been used for church edifices, and seem particu-
larly adapted for this purpose, it has been the custom to follow these time-
honored examples, adapting them to the modern requirements of church
worship.
294
ARCHITECTURAL DRAWING.
Fig. 121, is a plan of an ancient basilican or Romanesque church ;
fig. 122, a sectional elevation of the same. Fig. 123 is a plan of a Gothic
church in which C is the chancel, usually at the eastern extremity, TT the
transept, and N" the nave. In general elevation the Gothic and Roman-
esque agree ; a high central nave and low side aisles. In the later Roman-
esque 'the transept is also added.
The basilicas aggregated within themselves all the offices of the Rom-
ish church. The circular end or apex, and the raised platform, or dais in
front of it, was appropriated entirely to the clergy ; beneath was the crypt
or confessional where were placed the bodies of the saints and martyrs,
and pulpits were placed in the nave, from which the services were said or
sung by the inferior order of clergy.
The plan, fig. 123, is that of the original Latin cross, the eastern limb
or chancel being the shortest, and the nave the longest. Sometimes the
eastern limb was made equal to that of the transepts, sometimes even lon-
ger, but never to exceed that of the nave. In the Greek cross all the
limbs are equal. In most of the French Gothic churches the eastern end
is made semicircular, often enclosed by three or more apsidal chapels, that
is, semi-cylinders, surmounted by s^mi-domes.
The Byzantine church consisted internally of a large square or rectan-
gular chamber, surmounted in the centre by a dome, resting upon massive
piers ; an apse was formed at the eastern end. Circular churches were
built in the earlier ages for baptisteries, and for the lombs of saints and em-
perors.
Having thus briefly treated of the general form of ancient churches,
we proceed now to the consideration how far they may be applied to
the requirements of modem church services. The prime necessities are
those of the lecture room ; comfortable seats, convenient for hearing and
seeing the preacher ; and proper provision for ventilation. In addition, an
eligible position for the choir, a small withdrawing room for the clergy-
man, and a room suitable for Sunday Schools and for parish meetings.
AECHITECTUKAL DRAWING. 295
Seats are arranged by pews or stalls, the width of each pew being in gen-
eral about two feet ten inches. The length of pews is various, being gen-
erally of two sizes, adapted to either small or large families, say from seven
feet six inches, to eleven feet six, eighteen inches being allowed for each sit-
ter. In arrangement it is always considered desirable that there should be a
central aisle, and if but four rows of pews, two aisles against the wall ; if
six rows, one row on each side will be wall pews. Few churches are now
without an organ ; its dimensions should of course depend on the size of
the church. In form it may be adapted somewhat to the place which may
be appropriated to it. In general it is oblong in form, the longer side be-
ing with the keys. The dimensions suited to a medium sized church are
about nine feet by fifteen, and twelve feet in height. The withdrawing
room for the clergyman may be but of very small dimensions, and should
be accessible from without. The Sunday School, in general, requires in
plan about half the area of the church.
As city residences differ from those in the country, from the same ne-
cessities do the city churches differ from the rural ones. A very common
form of city church is, in plan, that of the Latin cross, with extremely
short transepts and chancels ; sometimes the roof is supported by pillars,
with imitated vaults in plaster, but often with a double pitch roof, and
open timber finish in the inside. The organ loft is sometimes in one of
the transepts, sometimes at the back of the congregation over the door of
entrance.
A sort of basilican church is also very common : rectangular in form
with a small semicircular niche behind the preacher, and small withdraw-
ing rooms or vestries at each side of it. The ceilings are finished after the
Greek style, with sunk panels, sometimes coved, with pilasters but seldom
pillars, except short ones, to support the galleries which are adopted in
this style of buildings, but not so commonly in the Gothic. The rooms for
Sunday Schools are almost invariably in the basement of the city churches.
The basilican form is evidently the most economical in its occupation
of land ; if the church be situated at the corner of two streets, it can cover
the whole lot, one side, or a portion of one side being left blank of win-
dows. If an elevation similar to fig. 122 be adopted, the light can be ta-
ken in at clere-story windows. But this form is objectionable as requiring
pillars in construction, which, unless made of iron, and of small size, very
much interfere with sight and hearing.
The position of our city churches is usually as we have said at the cor-
ner of streets, but if they can be placed so far in the centre of a lot as to
receive the light from the back areas, the position is preferable as removed
296 ARCHITECTURAL DRAWING.
from the noise of passing vehicles. In that case the church proper is ap-
proached by a long aisle, above which may be the room for the Sunday
School. This room should be fitted with water-closets, in fact they would
be often of great convenience connected with all churches.
In elevation, city churches are Greek with porticoes in front, Roman-
esque and Gothic, occasionally Byzantine. The Greek have no tower but
often a spire above the portico ; the Romanesque and Gothic generally one
tower, over the central door of entrance, or at one corner ; sometimes two,
one at each side of the principal door, almost invariably surmounted by
spires, high and tapering, usually of wood, but in some instances of stone.
Plate LXXV., is a design for a church in the English Decorated
Gothic style. It will be observed in the design that there is a side en-
trance with its appropriate gable ; in a similar way, small edifices may be
attached to the main one, for necessary offices, parsonages, or Sunday
schools, adding much to the picturesque effect, and particularly appropri-
ate to country churches.
Plate LXXVI. is the original design (by James Renwick, architect) of
the front elevation of the Roman Catholic cathedral now being built in
Fifth avenue. The style is the French Decorated Gothic, and the faQade
is more extensive than that in process of construction.
Plate LXXYII. is a design in the Romanesque or Byzantine style, by
Messrs. Renwick & Sands, architects. It is now being built in Fourth
avenue ; but the drawing is incomplete, being partially a working one,
and a campanile forms a part of the design.
It is the custom in Episcopal churches to place, if possible, the chancel
at the eastern end, and often a large window at the extremity of the chan-
cel. The light from this window should be very much subdued, as it comes
full in the eyes of the congregation ; for the comfort of preacher and peo-
ple a side or top-light in the chancel would be much better. A very beau-
tiful effect is produced by skylight in the apse of the Romanesque church,
which, being high above the congregation, does not interfere with them,
and affords the best light to lead the services. The light in churches
should not be garish, but subdued and well diffused, which will be best
effected by light from windows, placed as high up as possible. A single
north window, in many small churches, would be sufficient for all purposes,
would not injure the eyes of the congregation by cross lights, would add
very much to the effect when the walls are painted in fresco or distemper,
and, if suitable means are provided for summer ventilation, no other windows
would be necessary. In some recent city churches the light in the daytime
is taken entirely from skylights, and at night from gas lights placed in
ARCHITECTURAL -DKAWING. 297
the roof of the church, and reflected below through the same apertures in
the ceiling.
Theatre. The requirements of theatres and opera houses, differ essen-
tially from those of lecture rooms and churches, in that the audience
themselves form an important part of the exhibition. It is not only ne-
cessary that the audience should have a good position for hearing and see-
ing the performance upon the stage, but also to see each other. The most
approved form, now, for the body of the house, is a circular plan, the
opening for the stage occupying from one-fourth to one-fifth of the circum-
ference, the sides of the proscenium being short tangents. The circular
form is well adapted for both hearing and seeing, and also for lighting.
In the general position of the stage, proscenium, orchestra, orchestra
seats, parquette, and boxes, but one plan is followed. "We proceed to give
briefly the usual arrangements of seats, and some other requirements, and a
small table of the proportions of different houses. The line of the front of
the stage, at the foot lights, is generally slightly curved, with a sweep, say,
equal to the depth of the stage, and the orchestra and parquette seats are
arranged in circles concentric with it : of the space occupied by seats we have
already spoken. The entrance to the parquette may be through the boxes,
near the proscenium, and often centrally, but better at the sides, dividing the
boxes into three equal benches ; the seats in the boxes are usually concentric
with the walls, and more roomy than those of the parquette. The orches-
tra seats are of a height to bring the shoulders of the sitter level with the
floor of the stage, and the floor of the parquette rises to the outside, 1 in
15 to 18. The floor of the first row of boxes is some 2 to 3 feet above the
floor of the parquette at the front centre, and rises by steps at each row,
some 4 inches ; in the next tier of boxes the steps are considerably more
in height, and so on in the boxes above. In general, three rows of boxes
are all that is necessary ; in front, above the second, the view of the stage
is almost a bird's eye view. The floor of the stage descends to the foot-
lights at the rate of about 1 in 50. In large theatres it is of the utmost
importance that all the lobbies or entries should be spacious, and the
means of exit numerous and ample. The staircases broad, in short flights
and square landings, and not circular, as, in case of fright, the pressure of
persons behind may precipitate those in front the whole length of the
flight. Ladies' drawing rooms should be placed convenient to the lobbies,
of a size adapted to that of the theatre, arranged with water closets ;
there should also be provided rooms for the reception of gentlemen's canes
and umbrellas, with water closets attached. The box-office should be, of
course, near the entrance, but so arranged as to interfere as little as possi-
298
ARCHITECTURAL DRAWING.
ble with the approach to the doors of the house. At the entrance there
should be a very spacious lobby, or hall, so that the audience may wait
sheltered against the weather ; if possible, there should be a long portico
over the sidewalk, to cover the approach to the carriages. But single en-
trances are necessary to distinct parts of the house, but the greater the
number of, and the more ample places for exit, at the conclusion of the
piece, the better.
COMPAEATIVE TABLE OF THE DIMENSIONS OF A FEW THEATRES.
DISTANCE, IX FEET.
HEIGHT, IN FEET.
NAiffi AXD LOCATION.
Bet. boxes and
foot-lights.
l!
fl
Bet. curtain and
back of stage.
Create at breadth
of pit.
s
f .3
11
S
P3
Breadth of stage
but. eiUo walls.
From floor of pit
ti
=1
li
I s
Alexandra, St. Petersburg,
65
11
84
53
56
75
53
58
, Berlin,
62
16
76
51
41
92
43
47
La Scala, Milan,
77
IS
73
71
49
86
CO
64
San Carlo, Naples, ....
77
18
74
74
52
66
81
83
Grand Theatre, Bordeaux,
46
10
69
47
37
80
50
57
Salle Lepellctier, Paris,
67
9
82
66
43
73
52
66
Co vent Garden, London,
66*
55
51
32
86
54
Drury Lane, "
64*
80
56
32
48
60
Boston, Boston,
53
18
63
46
87
55}
53
Academy of Music, New York, .
74
13
71
62
48
83
74
Pike's Opera House, "
54
8}
63}
4S
44
76
52
67
Opera House, Philadelphia,
61
17
72
66
43
90
64}
74
* These dimensions include the distance between the foot-lights and curtain.
Although much has been written about the construction of legislative
halls, in relation to acoustic principles, there yet seems to be great disa-
greement in practical examples,*and in the deductions of scientific men.
The Chamber of French Deputies was constructed after a report of most
celebrated architects, in a semicircular form, surmounted by a flat dome,
but as the member invariably addresses the house from the tribune, at the
centre, in its requirements it is but a lecture room. Mr. Mills, Architect,
of Philadelphia, recommends for legislative or forensic debate, a room cir-
cular in its plan, with a very slightly concave ceiling. Dr. Reid, on the
contrary, in reference to the Houses of Parliament, gave preference to the
square form, with a low, arched ceiling. The Hall of Representatives nearly
completed at Washington, is 139 feet long, by 93 feet wide, and about 36
feet high, with a spacious retiring gallery on three sides, and a reporter's
ARCHITECTURAL DRAWING. 299
gallery behind the Speaker's chair. The members' desks are arranged in a
semicircular form. The ceiling is flat, with deep sunk panels, openings for
ventilation, and glazed apertures for the admission of light. The ventila-
tion is intended, in a measure, to assist the phonetic capacity of the Hall,
the air being forced in at the ceiling and drawn out at the bottom.
In .reviewing the general principles of acoustics, it will be found that
those rooms are the best for hearing in which the sound arrives directly to
the ear, without reflection ; that the sides of the room should not be re-
flectors, not sounding boards, and that surfaces absorbing sound are less
injurious than those that reflect. Slight projections, such as ornaments of
the cornices and shallow pilasters, tend to destroy sound, but deep alcoves
and recessed rooms produce echoes. Let the ceiling be as low as possible,
and slightly arched or domed; all large external openings should be
closed ; as M. Meynedier expresses it, in his description of an opera house,
" Let the hall devour the sound ; as it is born there, let it die there."
Plate LXXVIII. is the interior perspective of the New York Crystal
Palace, an illustration of the earlier form of a class of buildings which
originated with Paxton, in his design for the London Exhibition of 1851,
and of which some of the details of construction have been already
given. These* buildings are composed wholly of iron, p;lass and wood, but
no large pieces of either material are used ; in this consists their great pecu-
liarity. Stiifness and tenacity of material are applied rather than mass, to
counteract incidental strains ; and, on this account, they are not as suitable
as walls of brick and stone for permanent structures, nor are they as cheap ;
and in this respect, an improvement has been made in the French exhi-
bition building ; but for a structure easily moved and put together, as it was
intended, and for green or hot-houses, it seems especially adapted ; and as
a practical example of the application of iron, and an economical applica-
tion, it has been of great importance.
Before concluding the article on architectural drawing it may be ap-
propriate to speak briefly of materials as applied in the exteriors of edifi-
ces. Sufficient has already been said of their strength, we now refer
merely to their fitness to architectural ornament.
Brick in cities is by far the most common of all materials, nor do we
know, of any more suited to workshops and factories, for appropriateness,
economy, and durability (when hard burned), nor do we know of any style
of architecture more fitted to the material than the Komanesque, as in
Plate LXX. Stone in the rough or rubble walls, laid in cement or mortar,
are often used for these structures, but in that case the lintels should be
square, and if possible of a different shade of stone.
300 ARCHITECTURAL DRAWING.
For city residences, and stores, the exteriors are composed of all sorts
of building materials, with the exception of wood, from its insecurity in
case of fire ; brick, with marble, freestone, iron or terra cotta lintels and
sills for openings, red brick and straw-colored bricks, brick on rusticated
basements, and sometimes brick in alternate stripes with marbles ; free-
stone in a great variety of shades, mostly of a reddish brown, oftert fawn
and drab ; marbles white and veined ; native and foreign granite ; and iron,
the use of which in fronts is the invention of our age, and is destined to
modify our style of architecture.
All materials are suited for country residences except iron; stone
houses may be kept in their native color, but brick or wood should be
painted. We extract from Downing the following on the color of country
houses. " We think all buildings in the country should be of those soft
quiet shades called neutral tints, such as fawn, drab, gray, brown, etc., and
that all positive colors, such as white, yellow, red, blue and black should
always be avoided ; neutral tints harmonizing best with nature and, posi-
tive colors most discordant.
In the second place, we would adapt the shade of color as far as possible,
to the expression, style or character of the house itself. A Jarge mansion
may receive a somewhat sober, dignified hue ; a house of moderate size, a
lighter and more pleasant tone ; small cottages should always have a cheer-
ful, lively tint, not much removed from white. Country houses thickly
surrounded by trees, should always be of a lighter shade than those stand-
ing exposed. In proportion as a house is exposed to view, let its hue be
darker ; and wbere it is much concealed by foliage, a very light shade of
color is to be preferred.
" A species of monotony is produced by using the same neutral tint
for every part of the exterior of a country house. A certain sprightliness
is bestowed on a building in neutral tint by painting the bolder projecting
features of a different shade. The simplest practical rule that we can sug-
gest for effecting this in the most satisfactory manner, is the following : if
the tint selected for the body of the house be a light one, let the facings
of the windows, cornices, etc., be painted several shades darker of the
same color. The blinds may either be a still darker shade than the fa-
cings, or else the darkest green. If on the other hand, the tint chosen is
a dark one, then let the window dressings, etc., be painted of a much
lighter shade of the same color."
Thus far Mr. Downing. Most persons must be struck with the justness
of his remarks in general, but all are not prepared entirely to ignore white
as a color for country houses. We have always fancied in contemplating
ARCHITECTURAL DRAWING. 301
an extensive landscape that jottings of white enlivened the scene, and
prefer a whitewashed cottage, carrying an air of cleanliness, to the least
admixture of neutral tint : neither seems it high art to harmonize always
with nature, it often makes a very flat picture.
However we build, or whatever built of, let the building express the
purpose, and let the material be suited to it. Xet those which are intend-
ed for time be of lasting materials, but those that are temporary, be of
that most convenient ; let not one imitate the other.
Ventilation and Warming. To the proper construction of all edifices
some knowledge of the principles of ventilation and warming are neces-
sary, as the arrangements for this purpose are to be made in planning the
building. Air is deteriorated in apartments by the respiration and perspi-
ration of people, and by combustion in heating and lighting. At least 3
cubic feet per minute of fresh air should be supplied for each person occu-
pying the room, this quantity being deteriorated by respiration and per-
spiration. As to combustion, 1 pound of carbon or charcoal, in burning,
consumes 2.6 pounds of oxygen, which is that contained in between 13
and 14 pounds of atmospheric air ; and 1 pound of hydrogen, consumes 8
pounds of oxygen, which is that contained in about 40 pounds of atmos-
pheric air. Now tallow, wax and oil contain upon the average from YY to
80 per cent of carbon, and from 11 to 14: per cent, of hydrogen : the per
centage of carbon in anthracite and bituminous coal is more various, but the
same calculations may be used. 100 cubic feet of air weighs about Y
pounds, so from the above data the approximate consumption of oxygen
by any given quantity of the above, combustibles, is easily calculated.
The combustion of coal gas generally spoils thrice its bulk of oxygen, or,
fifteen times that of air.
The ventilation of a building does not necessarily imply the warming,
but the warming should include 1 means of ventilation, as ventilation is as
necessary in summer as winter, which, in warm weather, is mostly effected
by windows and -doors ; but warming necessarily implies the closing of
these apertures, and the provision of others through which the passing air
may be warmed before coming in contact with the occupants of the
building. Yentilation requires something besides large rooms and open
spaces ; means of circulation must be provided. A nuisance in a fenced
yard, open to the sky, may exist for a long time before the smell becomes
diluted by mixture with other air ; an attic without windows, in summer
would be oppressive, with all its flooring removed. Air may be regarded
as a liquid like water ; it is moved more readily, but the laws of its motion
are the same, and for artificial movement or ventilation the application of
302 AECHITECTTJKAL DRAWING.
heat is generally the readiest means. Air, as it is heated, rises, and colder
air rushes in to supply its place ; if the upper strata alone be heated, the
lower strata will still remain cool. Let the heat be so applied that circu-
lation is promoted, and that there be as few eddies, or corners, as possible,
in which the air may be stagnant. When heat is applied for ventilation
only, as in mines, a fire is built in a flue near the top, and the air necessary
for combustion is drawn from the mines ; the flue extends from the bottom
of the mine, capped by a chimney above, and ducts are led from the
bottom of the flue to the face of the workings, the cold air for ventilation
being drawn down through the working-shafts and drifts. Dry air is
almost a perfect non-absorbent of heat ; the absorbence is mostly due to
the moisture in it, which acts also as a governor or preserver of the heat,
preventing surfaces from being too readily heated or cooled. Air in
winter is very dry, but as its volume is enlarged by heat, it draws a
supply of moisture from every thing with which it conies in contact from
the skin and lungs, creating that parched and feverish condition expe-
rienced in many furnace-heated houses; from furniture and wood-work,
snapping joints and making unseemly cracks.
The following table gives the amount in grains Troy of moisture con-
tained in one cubic foot of air when saturated :
Degrees
Fahrenheit.
10 ...
Grains of
in cubic
8
20
1.3
30
2
40
2.9
50
. 4.
60
6
To
8.
80
10
90
15.
100...
. 19.
Thus, taking the air at 10, and heating it to 70, the ordinary temper-
ature of our rooms requires about 9 times the moisture contained in the
original external atmosphere, and, if heated to 100, as most of our hot-air
furnaces heat the air, it would require about 23 times.
Methods of Heating. The open fireplace heats by radiation, commu-
nicating heat to objects which by contact transfer it to the air. Persons
coming in contact with rays are themselves heated, while the air around
them is cool and invigorating for breathing; the bright glow has a
cheering and animating effect upon the system, somewhat like that of
AKCHITECTUKAL DRAWING. 303
sunliglit. As a ventilator, an open fire is one of the most important,
drawing in air not only for the support of combustion, but also, by the
heat of the fire and flue, making a very considerable current through the
throat of the chimney above the fire. From this cause, although there is
a constant change of air, yet there arises one great inconvenience of disa-
greeable drafts, especially along the floor, if the air supply be drawn
directly from the outer cold air ; but in connection with properly regulated
furnaces or stoves, the open fireplace becomes the most perfect means of
heating and ventilation ; and here it may be proper to remark, in regard
to ventilation, that a mere flue does not imply that a current will be cre-
ated in it ; if hot, there may be an upward current, if cold, perhaps the
reverse, or, like the neck to a bottle, with no movement in it at all. If
there be an open fire in the house, the air will rush to supply it through
the readiest openings ; the parlor fire may be fed by a current of cold air
down staircases and well holes, and the chamber flues, or up warm with
the smell from the kitchen.
Stoves. Open stoves heat by direct radiation, and by heating the air in
contact with it ; close stoves by the latter way only ; as economical means
of heating, they are the best, and, when properly arranged, give both a
comfortable and wholesome atmosphere. There should be some dish of
water upon them to supply a constant evaporation of moisture, sufficient to
compensate for increased capacity of the air due to its increased heat. In
the hall there will be no objection to a close stove, letting it draw its sup-
ply of air as it best can; but in close rooms the open stove is best, on the
plan of the old Franklin stove, or, if a close stove, somewhat on the plan
of a furnace, with an outer air supply for combustion and ventilation, with
an opening into the chimney-flue at the bottom for the escape of foul air,
and at the top another opening with a register valve for the escape of the
air when it becomes too hot.
Hot-air furnaces are close cast-iron stoves, enclosed in air chambers of
brick or metal, intd which external air is introduced, heated, and distrib-
uted by tin pipes to the different rooms of a house. Furnaces have been,
of late, very much decried, but under proper regulation they are very
cheap, economical, and even healthful means of ventilation and warming.
The heating surface should be very large, the pot thick, or even encased
with fire-brick, that it may not become too hot ; there should be a plentiful
supply of water in the chamber for evaporation, perhaps also beneath the
opening of each register ; the air supply should always be drawn from the
outer air, and unobjectionable sources, through ample and tight ducts,
without any chance of draft from the cellar; the pot, and all joints in
304: ARCHITECTURAL DRAWING.
the radiator, should be perfectly gas-tight, so that nothing may escape
from the combustion into the air-chamber. With these provisions on a
sufficient scale, and proper means for distribution of the heated air and
escape of foul air, almost any edifice may be very well heated and ventilated.
The air should be delivered through the floor or the base-board of the
room, and at the opposite side from the flue for the escape of foul air,
making as thorough a current as possible across the room, and putting the
whole air in motion. In dwelling-houses the fireplace will serve the best
means of exit ; in public rooms distinct flues will have to be made for this
purpose, and they should be of ample dimensions, with openings at the
floor, and means should be provided for heating this flue. An architect, in
laying out flues for heating and ventilation, should, both in plan and ele-
vation, fix the position of hot and foul air flues, and trace in the current
of air, always keeping in mind that the tendency of hot air is to rise ; he
will then see that if the exit-opening be directly above the entrance-flue,
the hot air will pass out, warming the room but little ; if the exit-opening
be across the room and near the ceiling, the current will be diagonal,
with a cold corner beneath, where there will be very little circulation or
warmth. To heat the exit-flue, a very simple way is, to make the furnace-
flue of iron, and let it pass up centrally through the exit-flue.
Size for furnace and ducts, as given by an old established maker:
Largest pot, 22" diam. ; air-chamber, 4' x 4' 4" x 5' 6" ; air-box or cold-air
duct, 3 sq. ft. of sectional area, with a slide damper ; hot-air duct, main
riser, say 22" x 12", or equal area, branches according to the size of room
and position ; water-trough in air-chamber, about 3 cub. ft. capacity. This
furnace would be adequate to heat one of our city houses of 25 ft. fron.t x
60 ft. deep, with registers as high as the third story. Furnaces vary in the
size of the pot to as small as 16" diam., and the accessories may be reduced
proportionately.
. Steam and hoi-water circulation are applied to the heating of buildings
by means of wrought or cast iron pipes connected with boilers. In the
simplest form, as common in workshops and factories, steam is made to give
warmth without ventilation by direct radiation from wrought-iron pipes,
with or without coils or radiators. The general arrangement is by rows of
" or V pipe hung against the walls of the room as close to the base as
possible, one foot of f " pipe being considered adequate to heat 50 cub. ft.
of space ; if there are many windows in the room, or the building is very
much exposed, more length should be allowed. Steam is used at either
low or high pressures, exhaust or live steam ; the rising main should be as
near to boiler or engine as possible, and the heating pipes should have a
ARCHITECTURAL DRAWING. 305
drip or descent of about " in 10 ft., away from rising main, and the con-
densed water should be returned directly or indirectly to the boiler.
"When ventilation is combined with steam or hot- water heating, coils or
radiators are placed in air-chambers, through which the outer air is intro-
duced and distributed by ducts throughout the building, like a hot-air
furnace, and a great improvement over this is that the metallic surfaces
brought in contact with the air in the first case usually range from 200 to
250, whilst the pot of the air-furnace is often near a white heat. In a
sanitary point of view, hot-water or low-steam coils in air-chambers are by
far the healthiest means of warming and ventilation ; the greatest objec-
tion is their expense, the care requisite in attending them, the danger of
freezing and bursting the pipes if worked intermittently in winter. In the
arrangement it is usual, in dwelling-houses, to place the coils at different
points in the cellar, as near as possible beneath the rooms to be heated.
In public buildings frequently a very large space in the cellar is occupied
by the coils, into which the air is forced by a fan, and then distributed by
flues or ducts throughout the building. Hot-water circulation is very
much used in the heating of forcing or hot-houses, as the mass of water
once heated acts as a regulator, and there are less liabilities to change of
temperature by inattention to the fires.
All inlet or outlet ventilating flues should be provided with dampers
or registers, to control the supply or discharge of air, cutting it off when
sufficient heat is secured, or retaining the warmth when ventilation is not
required.
Much has been written on the subject of ventilation and warming, and
many expedients, undoubtedly adequate in themselves, have failed, from
the carelessness of servants and* from want of attention. The grand requi-
site seems to be, something that will be sure, and will not get out of order.
It is now but a half a century since gas was introduced for lighting; it has
been applied for cooking and warming, but not to a large extent eco-
nomically ; whether it may be brought into general use for this purpose
is a problem yet to be solved ; but steam, as now applied in most New
England factories, from a central set of boilers, could easily be applied to
the warming and ventilation of many houses, and for many culinary pur-
poses, and with gas would supply all requirements.
Ventilators. Patented expedients are numerous to excite drafts in flues
or ducts that will not draw, some movable and some immovable, and are to
be purchased at almost every tinsmith's. The simplest form is that in which
the wind, blowing across the top of the chimney, is given such a direction
that it creates a vacuum in the flue beneath ; but when adequate heat can
20
306 AECHITECTUEAL DBA WING.
be applied, either by a jet of burning gas, by steam coil, or direct fire, a
circulation can be assured. In many large public institutions, charitable,
educational, and legislative, forced circulation by means of fans is very
common ; in most cases, by drawing the air from the outside of the build-
ing, and forcing it under pressure into the interior, the escape being out-
ward through all flues and cracks, or ventilation by & plenum, as it is called ;
in other cases the fan drawing its air directly from the interior, and
reducing the pressure, to overcome which the air rushes in from the out-
side through the hot-air ducts and cracks, or ventilation by vacuum.
Ventilation has been treated as applicable to dwellings or buildings for
the occupancy of mankind, but the architect must understand that fresh air
is equally necessary for animals and even plants. In stables there must be
provision for- the escape of foul air and the introduction of fresh air, with-
out injurious drafts or too great reduction of temperature. Care should
also be taken, in the discharge of the foul air, that it does not escape into
the hay-loft, to be condensed on the fodder, to make it unpalatable. Barn
and stable should be under separate roofs ; the barn may be ventilated by
a cupola, with blinds, on the roof; but if the stable can be ventilated some-
what similar to the room of a dwelling-house, by taking in air at one side
and letting it out at the other, so much the better ; and as the animals
cannot have a radiant fire to warm themselves, give them sunlight.
Drainage, in a sanitary point of view, is almost as necessary as venti-
lation ; as applied to agriculture by means of subsoil tile drains, it increases
the crops, and conduces to the healthfulness of the country ; to dwellings,
through sewers and drains, it removes one of the most prolific causes of
disease. If the land be wet and springy, some provision for drainage must be
made below the level of the cellar. A simple and effectual way is by enclos-
ing the foundation, as far as may be necessary, by drains, and leading the
water off; if the cellar walls be laid in concrete and bottom cemented, the
precaution will be sufficient. But the foul water or house waste is also to
be disposed of ; in cities this is done by connection with street sewers, which
are usually placed deep enough to drain the cellar. The sewer pipes or
drain connections with street sewers are, in this city, established at 6"
diam., usually made of vitrified stone-ware pipe, and the pitch or descent
should be as uniform as possible, and at least ^' in 10 ft. Many make
use of cement pipe instead of stone-ware, as somewhat cheaper. Iron
pipe is used, of from 5" to 6" diam., when the pipe cannot be covered. In
cities the sewer pipe is used for the discharge of the rain water from the
leaders, but in country residences this is usually collected in cisterns for
washing purposes ; the sewer pipes, being merely for the discharge of the
ARCHITECTURAL DRAWING.
307
foul water of the house, are usually made about 4? diam. The purpose of
all house apparatus of water-closets, sinks, waste and sewer pipes, is for
the immediate and speedy conveyance of every particle thrown into them
out of the house into the final collecting place, and that there should be no
smell escape from them into the interior of the house or by return. For
the first the pipes should be laid with as few bends as possibje,, and with a
good descent; for the latter every waste pipe should be trapped. The
most efficient traps are water closures, which admit the passage of water,
but not the return of air. Thus, in Fig. 125, which is a section of a form
Fig. 125.
Fig. 126.
of privy vault, recommended by the Board of Health for city tenement
houses, the trap is made by the bending down of the sewer pipe S into the
surface of the vault, or a sipfion trap; and in fig. 126 by the wall W
extending below the level of the bottom of the sewer pipe, corresponding
to the catch-basin trap ; the same purpose is served in a waste pipe by
making a bend or S trap in it. The pipes L L are waste pipes from house
or roof leaders to flush the vaults.
The final receptacle of waste for country houses is one of the most diffi-
cult problems to be solved by the architect. If the premises are extensive, .
it may be made at any place on a descending slope from the house, the
walls laid in stone dry, so that the earth may absorb the liquid portion.
In this case it should be distant from the house and the water supply, so
that there may be no offence from the smell, nor any contamination from
infiltration ; a sink of this kind, even in the most porous earth, becomes,
in a few years, clogged, and another has to be dug in fresh earth. The
size of the sink will, of course, depend on the character of the soil and the
308 ABCHTTECTUBAL DRAWING.
quantity of waste ; the top of some are domed in brick and cement, but
the cheapest covering, which will endure as long as the sink continues to
perform its functions without overflow, is locust sleepers and flag covers,
with 1 ft. to 18" of earth above. In many cases a tight cesspool becomes
necessary to act as a sort of settling reservoir, whilst the liquid portions are
permitted to escape through an overflow pipe ; but this only mitigates the
evil, there is still the difficulty where to lead the overflow. In some few
instances the waste is pumped out of the cesspool and distributed as manure
over the land, no liquid being permitted to escape ; where this is done
daily there is no cause of offence, and the return is in increased vegetation.
Lighting. Make such provisions and arrangements of rooms and pas-
sages that they may secure as much as possible of their light from the
natural source, and direct without the aid of reflectors. Sunlight in the
dwelling conduces, with fresh air and good drainage, to health and happi-
ness. Let there be no dark nooks and corners in which artificial light is
necessary in the daytime. Make windows as required, with movable sash
for ventilation, and blinds for regulating the light.
Water Supply. In all dwellings there must be some source of water
supply, by springs, wells, cisterns, or aqueducts. Most of our cities have
sufficient water works, and in that case connection therewith to the interior
of houses is almost compulsory. In country houses, on the other hand, the
cases are very few in which water can be supplied directly by gravitation
to the upper stories. Wells and pumps, force or lift, are the most usual
expedients, and when well water is hard, the soft water for washing
may be drawn from cisterns, which receive the water from the roof. The
quantity necessary for each household varies with the wants and habits of
the occupants; from 7 to 10 gallons can be reckoned as amply sufficient for
each person if used legitimately. In the construction of wells to supply
a house by gravitation, it is a very good plan to dig horizontally into a fit
place on. the side-hill, till the supply running doWn the trench is sufficient,
then make a stone culvert from the upper end down as far as necessary,
closing the end by a cement wall, for a reservoir, and covering the whole
with earth, leaving a man-hole for access. The size of pipe depends upon
the head above the outlets, and the quantity of water required ; in this
city the tap permitted in the street mains for a dwelling-house is f " ; for
country houses the pipe should generally not be less than 1" diam.
Lead pipes are almost the only ones used in plumbing, but care should
be taken in testing the effect of water to be used on lead, as, in many cases,
it acts on the metal and becomes a most subtle poison. In such cases it is
better to use pure tin pipes, tin-cased pipes, or galvanized iron.
ARCHITECTURAL DRAWING. 309
Principles of Design. It is not intended in this book, professedly
treating of architecture only in its most mechanical phase of drawing, to
give a history of it as an art, or a treatise on the distinctions of styles.
To one anxious to acquire knowledge in this department we refer, as the
very best compendium within our knowledge, to Ferguson's " Hand-book
of Architecture." The study of this work will give direction to a
person's observation, but, without referring to actual examples, mere read-
ing will be of little nse. Drawings give general ideas of the character
of buildings, but no idea of size or of the surroundings of a building. Many
a weak design, especially in cast-iron buildings, acquires a sort of strength
by the number of its repetitions, giving an idea of extent ; and many a
beautiful design on paper has failed in its execution, being dwarfed by its
surroundings. With regard to the style of a building, there are none of
the ancient styles in their purity adapted to present requirements ; our
churches and theatres are more for the gratification of the ear than the
eye, and the comforts of our domestic architecture, and the requirements
of our stores and warehouses, are almost the growth of the present cen-
tury. For a design, look first to the requirements of the structure, the
purposes to which it is to be applied ; sketch the plan first, arrange the divi-
sions of rooms, the openings for doors and windows, construct the sections,
and then the elevations first in plain outline ; modify each by the exigencies
of construction.
" Construction, including in the term the disposition of a building in
reference to its uses, is by some supposed to be the common part of the art
of architecture, but it is really the bone, muscle, and nerve of architecture,
and the arts of construction are those to which the true architect will look,
rather than to rules and examples, for the means of producing two at least
of the three essential conditions of building well, commodity r , firmness, and
delight, which conditions have been aptly said to be the end of architecture
as of all creative arts.
" The two great principles of the art are : First, that there should be no
features about a building which are not necessary for convenience, con-
struction, or propriety ; second, that all ornament should consist of en-
richment of the essential construction of the building.
" The neglect of these two rules is the cause of all the bad architecture
of the present time. Architectural features are continually tacked on build-
ings with which they have no connection, merely for the sake of what is
termed effect, and ornaments are continually constructed instead of forming
the decoration of construction to which in good taste they should always be
subservient. The taste of the artist ought to be held merely ancillary to
310 ARCHITECTURAL DRAWING.
truthful disposition for structure and service. The soundest construction
is the most apt in the production or the reproduction, it may be, of real
art. The Eddystone lighthouse is well adapted to its uses ; it is commo-
dious, firm and stable almost to a miracle, and its form is as beautiful in
outline to the delight of the eye, as it is well adapted to break and miti-
gate the force of the sea in defence of its own structure. The Great Exhi-
bition Building of 1851 was most commodious for the purposes of an ex-
hibition, firm enough for the temporary purpose required of it, and there
was delight in the simplicity and truth of its combinations ; and all this
may be said to have grown out of propriety of construction, as applied to
the material, cast iron. The use of unfitting material, or fitting material,
inappropriately, leads almost entirely to incommodiousness, infirmity, and
offence, or some of them.
" Out of truth in structure, and that structure of a very inartificial sort,
grow the beautiful forms of the admirable proportions found in the works
of the Greeks ; and out of truth in structure with the strictest regard to
the necessities of the composition and of the material employed, and that
structure as full of artifice, as the artifice employed is of truth and simpli-
city, grew the classical works vulgarly called Gothic, but now character-
istically designated as Pointed, from the arch which is the basis of the style.
Structural untruth is not to be justified by authority ; neither Sir Chris-
topher Wren, nor the Athenian exemplars of Doric or Ionic in the Propy-
laeum and in the Minerva Polias, with their irregular and inordinately wide
intercolumniation, can persuade even the untutored eye to accept weak-
ness for strength, or what is false for truth.
" The Greek examples offer the most beautiful forms for mouldings, and
the^ Grecian mode of enriching them is unsurpassed. It should be borne
in mind that the object in architectural enrichment, is not to show orna-
ment, but to enrich the surface by producing an effective and pleasing
variety of light and shade ; but still, although ornament should be a second-
ary consideration, it will develop itself, and therefore should be of elegant
form and composition."
"We have quoted thus at some length from the article Architecture,
Encyclopaedia Britannica, because with many authority is necessary, and
they distrust their own powers of observation and analysis ; all must feel
the truth of the above, but in practice it is very little appreciated or carried
out. The present taste in architecture, as in the theatre, is for the spec-
tacular ; breadth or dignity of effect is not popular ; edifices are not only
covered with, but built up in ornament ; and construction is but secondary.
The French, having a building stone that is very easily worked, cut merely
ARCHITECTURAL DRAWING. 311
the joints, leaving the rough outer surface to be worked after it is laid ;
chopping out mouldings and ornaments almost as readily as though it were
in plaster, and the surface when finished is covered with enrichments in
low-relief. The fashion thus set is imitated in this country at immense
cost, in the most unfitting materials marble and granite. Our architec-
tural buildings express fitly our condition a rich country, recent and easily-
acquired wealth, and a desire and rivalry ,to exhibit it, or a display as a
means of advertising, and in this. truth of expression will have an archaeo-
logical interest ; although it does not contribute much to present excellence
in construction, it still has this value, that the architect or constructor need
be governed by no rules or principles, he can make experiments on a
pretty extensive scale, and out of much bad construction even forms and
ornament may spring up which will stand the test of time, and form a
nucleus of a new style adapted to the present wants.
A new building material for building purposes cast iron has recently
come into use, but, with the exception of exhibition-buildings before
mentioned, has not yet been treated distinctively; buildings erected with
it have been copies of those in stone, and have been even imitated in
color. For the first story of stores, where space is necessary for light and
the exhibition of wares, cast-iron columns are almost invariably used,
but are objected to architecturally, that they look too weak for the sup-
port of the piles of brick and stone above them. The objection should not
be to the use, but that the truth of the adequate strength of the cast iron
is not conveyed by the form or color. No one objects that the ancles of
Atlas look too light to support the massive figure and globe, or wishes it
seated to give the idea of stability ; so if the columns and lintels were some
other form, than Greek or Komari with immense intercolumniations, and
colored fitly, the appearance of weakness would be entirly lost sight of.
In conclusion. Some years since the author made an experiment on
building according to the principles above laid down, construction and
adaptation and expression of purpose. As regards convenience and
strength, it was found, on occupation, all that could be wished ; in regard
to the third requirement, delight, a sketch is given on page 312, and is
open to the reader's criticism. Some allowance should be made for absence
of color in the sketch, which contributed much to architectural effect.
Posts, lintels, window-frames and sash, and ornamental letters, were of iron,
and painted a very deep green; the structure was of brick, with sills and
bands of rubbed Ulster Milestone, roof of "Welsh slate. The building
occupied one corner of Greene and Houston Streets, in this city, but was
burned, and cannot, therefore, be referred to practically.
312
ARCHITECTURAL DRAWING.
SHADING AND SHADOWS. 313
SHADING AND SHADOWS.
LIGHT is diffused through space in straight lines, and the lines of light
are called rays. When the source of light is situated at a very great dis-
tance from the illuminated objects, as in the case of the sun with relation
to the earth, the rays of light do not sensibly diverge, and may be regarded
as exactly parallel to each other. Such is the case in mechanical draw-
ings, where the objects to be represented are always regarded as illumi-
nated by the solar light.
Light is called direct when it is transmitted to an object without the
intervention of any opposing medium. But as all bodies subjected to the
action of light possess, in a greater or less degree, the property of giving
out a certain portion of it to the surrounding objects, this reflected light
becomes in its turn, though with greatly diminished intensity, a source of
illumination to those objects which are deprived of direct light.
Everything which tends to intercept or prevent the direct light from
falling in upon a body, produces upon the surface of that body a degree
of obscurity of greater or less intensity ; this is called a shade or shadow.
Such effects are usually classified as direct shadows and cast shadows.
The shade proper, or direct shadow, is that which occurs on that por-
tion of the surface of a body which is situated opposite to the enlightened
part, and is the natural result of the form of the body itself, and of its posi-
tion with regard to the rays of light. The cast shadow, on the other hand,
is that which is produced upon the surface of one body by the interposi-
tion of another between the former and the source of light ; thus inter-
cepting the rays which would otherwise illuminate that surface. An
illustration of this distinction is afforded in the pyramid represented at
fig. 1, Plate LXXX., where the shade proper is shown upon that half of
the figure which is denoted by the letters D' E' G' F' in the plan, while the
cast shadow occupies the space comprised between the lines E 7 e and F ; d
on the horizontal plane of projection. Cast shadows may also obviously
314: SHADING AND SHADOWS.
be produced upon the surface of a body by the form of the body itself ; as,
for example, if it contain projecting or concave parts.
The limit of the direct shadow in any body, whatever may be its form
or position, is a line of greater or less distinctness, termed the line of sep-
aration between light and shade', or, more shortly, the line of shade this
line is, of course, determined by the contact of the luminous rays with the
surface of the body ; and if these rays be prolonged till they meet a given
surface, by joining all the points of intersection with that surface, we ob-
tain the outline of the shadow cast upon it by the part of the body which
is deprived of light.
The rays of light being regarded as parallel to each other, it is obvious
that in the delineation of shadows, 'it is only necessary to know the direc-
tion of one of them ; and as that direction is arbitrary, we have adopted
the usual and confessedly the most convenient mode of regarding the ravs
as in all cases falling in the direction of the diagonal of a cube, of which
the sides are parallel to the planes of projection. The diagonal in projec-
tion upon the vertical and horizontal planes lies at an angle of 45 with
the ground-line ; and thus the light in both elevation and plan appears at
the angle of 45. In illustration, let E, E' (fig. 1, pi. LXXIX.) be the pro-
jections of a ray of light in elevation and plan ; and let A, A', those of a
point of which the shadows are required to be projected upon the vertical
plane X Y. Draw the straight lines A a, A' a', parallel to the lines E, E',
and from a' ', where the line A' a' meets the plane X T, draw the perpen-
dicular a' a to meet the oblique line A a; then the intersection a is the
position of the shadow of the point A.
In the following illustrations, the same letter accented, is employed in
the plan as in the elevation, to refer to the same point or object.
The projections of the diagonals of the imaginary cube which denote
the direction of the rays of light being equal in both planes, it follows
that in all cases, and whatever may be the form of the surface upon which
the shadow is cast, the oblique lines joining the projections of the point
which throws the shadow, and that which denotes it, are also equal. Thus
the line A a in the elevation is equal to the line A! a' in the plan. Hence
it will in some cases be found more convenient to use the compasses
instead of a geometrical construction; as, for example, in place of project-
ing the point a' by a perpendicular to the ground-line, in order to obtain
the position of the required shadow , that point may be found by pimply
setting off upon the line A a a distance equal to A! a'.
Plate LXXIX., fig. 1. Required to determine the shadow cast upon
the vertical wall X Y ~by the straight line A B.
SHADING AND SHADOWS. 315
It is obvious that in tins case the shadow itself will be a straight line ;
hence, to solve the problem, it is only necessary to find two points in that
line. We have seen that the position of the shadow thrown by the point
A is at a; by a similar process we can easily determine the point b, the
position of shadow thrown by the opposite extremity B of the given line ;
the straight line a b, which joins these two points, is the shadow required.
It is evident from the construction of this figure, that the line a l> is
equal and parallel to the given line A B ; this results from the circum-
stance that the latter is parallel to the vertical plane X Y. Hence, when
a line is parallel to a plane, its shadow upon that plane is a line which is
equal and parallel to it.
Suppose now that, instead of a mere line, a parallel slip of wood or
paper, A B C D, be taken, which, for the sake of greater simplicity, we
shall conceive as having no thickness. The shadow cast by this object
upon the same vertical plane X Y is a rectangle a 1 c d, equal to that
which represents the projection of the slip, because all the edges of the
latter are parallel to the plane upon which the shadow is thrown. Hence,
in general, when any surface, whatever may be its form, is parallel to a
plane, its shadow tliroivn upon that plane is a figure similar to it, and simi-
larly situated. This principle facilitates the delineation of shadows in
many cases. In the present example, an idea may be formed of its
utility ; for, after having determined the position of any one of the points
a, o, c, d, the figure may be completed by drawing lines e'qual and parallel
to the sides of the slip, without requiring to go through the operations in
detail.
Fig. 2. "When the object is not parallel to the given plane, the cast
shadow is no longer a figure equal and similarly placed ; the method of
determining it remains, however, unchanged ; thus, take the portion A E
of the slip A B, which throws its shadow on the plane X Y ; draw the
lines' A a, E e, C c, F/, and A' a', E' e', parallel to the rays of light ; make
A a and C c equal to A! a'', and E e and F/ equal to E' e' \ connect a ef"c,
and we have the outline of the shadow of the slip A E.
By an exactly similar construction we have the shadow of the portion
E B on the plane Y Z, which being inclined to the plane of projection in
a direction contrary to X Y, necessarily causes the shadow to be broken,
and the part e d to lie in a contrary direction to af.
Fig. 3 still further illustrates the determination of the shadow of the
slip upon a moulding placed on the plane X Y parallel to the slip.
Fig. 4. To find the shadow cast by a straight line A B upon a curved
surface, either convex or concave, whose horizontal projection is repre-
sented by the line X e' Y.
316 SHADING AND SHADOWS.
It has been already explained, that the shadow of a point upon any
surface whatever is found by drawing a straight line through that point,
parallel to the direction of the light, and marking its intersection with the
given surface. Therefore, through the projections A and A' of one of the
points in the given straight line, draw the lines A a, A' a', at an angle of
45; and through the point ', where the latter meets the projection of the
given surface, raise a perpendicular to the ground-line ; its intersection
with the line A a is the position of the shadow of the first point taken ;
and so for all the reversing points in the line.
If it be required to delineate the entire shadow cast by a slip A B C D,
by the construction above explained, trace two equal and parallel curves
aeb, cfd, which will represent the shadows of the sides A B and C D ;
while those of the remaining sides will be found denoted by the vertical
straight lines a c and o d, also equal and parallel to each other, and to the
corresponding sides of the figure, seeing that these are themselves vertical
and parallel to the given surfaces.
Fig. 5. When the slip is placed perpendicularly to a given plane X Y,
on which a projecting moulding, of any form whatever, is situated, the
shadow of the upper side A' B', which is projected vertically in A, will be
simply a line A a at an angle of 45, traversing the entire surface of the
moulding, and prolonged unbroken beyond it. This may easily be de-
monstrated by finding the position of the shadow of any number of points
such as D', taken at pleasure upon the straight line A' B'. The shadow
of the opposite side, projected in C, will follow the same rule, and be de^
noted by the line C <?, parallel to the former. Hence, as a useful general
rule : in all cases where a straight line is perpendicular to a plane of pro-
jection, it throws a shadow upon that plane in a straight line, forming an
angle of 45 with the ground-line.
Fig. 6 represents still another example of the shadow cast by the slip
in a new position ; here it is supposed to be set horizontally in reference
to its own surface, and perpendicularly to the given plane X Y. Here
the shadow commences from the side D B,. which is in contact with this
plane, and terminates in the horizontal line a c, which corresponds to the
opposite side A C of the slip.
Plate LXXIX., fig. 7. Required to find the shadow cast upon a verti-
cal plane X Y by a given circle parallel to it.
Let C, C', be the projections of the centre of the circle, and E, B', those
of the rays of light.
It has been already shown, that when a figure is parallel to a plane,
its shadow cast upon that plane is a figure in every respect equal to, and
SHADING AND SHADOWS. 317
symmetrical with it ; therefore the shadow cast by the circle now under
consideration will be expressed by another circle of equal radius ; conse-
quently, if the position of the centre of this new circle be determined, the
problem will be solved. Now the position of the shadow of the central
point C, according to the rules already fully developed, is easily fixed at
c j from which point, if a circle equal to the given circle be described, it
will represent the outline of the required shadow.
Fig. 8. "Wlien the circle is perpendicular to both planes of projection,
its projection upon each will obviously be represented by the equal diam-
eters A B and C' D', both perpendicular to the ground-line. In this case,
to determine the cast shadow, describe the given circle upon both planes,
as indicated by the figures, and divide the circumference of each into any
number of equal parts ; then, having projected the points of division, as
A 2 , C 2 , E 2 , &c., to their respective diameters A B and C'D', draw from
them lines parallel to the rays of light, which, by their intersection with
the given plane, will indicate so many points in the outline of the cast
shadow.
Fig. 11. If the given circle be horizontal, its shadow cast upon the ver-
tical plane X Y becomes an ellipse which must be constructed by means
of points, as indicated by the figures referred to above ; that is to say, that
in the circumference of the circle a certain number of points are to be
taken, -such as A' D' B', &c., which are to be projected successively to
A, D, B, on the line A B, and through each of these points lines are to be
drawn parallel to the direction of the rays of light, and their intersection
with the given plane determined. The junction of all these points will
give the ellipse a d &, which is the contour of the required shadow.
Fig. 9 represents a circle whose plane is situated perpendicularly to
the direction of the luminous rays. In this example the method of con-
structing the cast shadow does not differ from that pointed out in reference
to fig. 11, provided that both projections are made use of. But it is obvious,
that instead of laying down the entire horizontal projection of this circle,
all that is necessary is to set off the diameter D' E' equal to A B, because
the shadow of this diameter, transferred in the usual way, gives the major
axis of the ellipse which constitutes the outline of the shadow sought,
while its minor axis is at once determined by a 5, equal and parallel to
AB.
Fig. 10 exhibits the case of a circle parallel to the vertical plane of pro-
jection, throwing its shadow at once upon two plane surfaces inclined to
each other. To delineate this shadow, all that it is necessary specially
to point out is, that the points d and e are found by drawing from Y a
318 SHADING AND SHADOWS.
line Y D', parallel to the rays of light, and projecting the point D' to D
and E.
Fig. 12 represents constructions similar to the foregoing, for obtaining
the form of the shadow cast by a horizontal circle upon a vertical curved
surface.
"We may here remark, that in every drawing where the shadows are to
be inserted, it is of the utmost importance that the projections which re-
present the object whose shadow is required should be exactly defined, as
well as the surface upon which this shadow is cast ; it is therefore advis-
able, in order to prevent mistakes and to insure accuracy, to draw the
figures in China ink, and to erase all pencil marks before proceeding to
the operations necessary for finding the shadows.
Plate LXXX., fig. 1. To find the outline of the shadow cast upon loth
planes of projection by a regular hexagonal pyramid.
In these figures it is at once obvious, that the three sides A' B' F',
A' B' C', and A' C' D' alone receive the light ; consequently the edges
A' F' and A' D f are the lines of shade. To solve this problem, then, we
have only to determine the shadow cast by these two lines, which is
accomplished by drawing from the projections of the vertex of the pyra-
mid the lines A b' and A' a! parallel to the ray of light, then raising from
the point V a perpendicular to the ground line, which gives at a' the
shadow of the vertex on the horizontal plane, and finally by joining this
last, point a' with the points D' and F' ; the lines D' a' and F' a' are the
outlines of the required shadow on the horizontal plane. But as the pyra-
mid happens to be situated sufficiently near the vertical plane to throw a
portion of its shadow towards the vertex upon it, this portion may be
found by raising from the point c, where the line A' a' cuts the ground-line,
a perpendicular c a, intersecting the line A b' in aj the lines a d and a e
joining this point with those where the horizontal part of the shadow
meets the ground-line, will be its outline upon the vertical plane.
Fig. 2. Required to determine the limit of shade on a cylinder placed
vertically ', and likewise its shadow cast upon the two planes of projection.
The lines of shade on a cylinder situated as indicated, are at once found
by drawing two tangents to its base, parallel to the ray of light, and pro-
jecting through the points of contact lines parallel to the axis of the
cylinder.
Draw the tangents D' d' and C' c' parallel to the ray E' ; these are the
outlines of the shadow cast upon the horizontal plane. Through the point
of contact C' draw the vertical line C^C' ; this line denotes the line of shade
upon the surface of the cylinder. It is obviously unnecessary to draw the
SHADING AND SHADOWS. 319
perpendicular from the opposite point D', because it is altogether con-
cealed in the vertical elevation of the solid. In order to ascertain the
points C' and T)' with accuracy, draw through the centre (y a diameter
perpendicular to the ray of light R/.
Had this cylinder been placed at a somewhat greater distance from the
vertical plane of projection, its shadow would have been entirely cast upon
the horizontal plane, in which case it would have terminated in a semi-
circle drawn from the centre o' ', with a radius equal to that of the base.
But as a portion of the shadow of the upper part is thrown upon the ver-
tical plane, its outline will be denned by an ellipse drawn in the manner
indicated in fig. 11, plate LXXIX.
Fig. 3. To find the line of shade in a reversed cone, and its shadow
cast upon the two planes of projection.
From the centre A! of the base draw a line parallel to, the ray of light ;
from the point a', where it intersects the perpendicular, describe a circle
equal to the base, and from the point A' draw the lines A' V and A' c', touch-
ing this circle ; these are the outlines of tlie shadow cast upon the horizon-
tal plane. Then from the centre A' draw the radii A 7 B' and A' C' parallel
to a! V and a! c' / these radii are the horizontal projections of the lines of
shade, the former of which, transferred to B D, is alone visible in the ele-
vation. But in order to trace the outline of that portion of the shadow
which is thrown upon the vertical plane, it is necessary to project the
point C' to C, from which, by a construction which will be manifest from
inspection of the figures, we derive the point c and the line c d as part of
the cast shadow of the line C' A'. The rest of the outline of the vertical
portion of the cast shadow is derived from the circumference of the base,
as in fig. 2.
Fig. 4. "When the cylinder is placed horizontally, and at the same
time at an angle with the vertical plane, the construction is the same as
that explained (fig. 2) ; namely, lines are to be drawn parallel to the ray
of light, and touching the opposite points of either base of the cylinder,
and through the points of contact A and C the horizontal lines A E and
C D are to be drawn, denoting the limits of the shade on the figure. The
latter of these lines only is' visible in the elevation, while, on the other
hand, the former, A E alone, is seen in the plan, where it may be found
by drawing a perpendicular from A meeting the base F' G' in A'. The
line A 7 E' drawn parallel to the axis of the cylinder is the line of shade
required. Project the shadow of the line A E on the vertical plane as in
previous examples, and the construction ^ will define the outline of the
shadow of the cylinder.
320 SHADING AND SHADOWS.
The example here given presents the particular case in which the base
of the cylinder is parallel to the direction of the rays of light in the hori-
zontal projection. In this case, all that is required in order to determine
the line A' E' is to ascertain the angle which the ray of light makes with
the projection of the figure. Draw a tangent to the circle F' A 2 G' (which
represents the base of the cylinder laid down on the horizontal plane), in
such a manner as to make with F' G' an angle of 35 16', and through the
point of contact A 2 draw a line parallel to the axis of the cylinder ; this
line E' A' will be the line of shade as before.
Fig. 5 represents a cylinder upon which a shadow is thrown by a rec-
tangular prism, of which the sides are parallel to the planes of projection.
The shadow in this case is derived from the edges A' D' and A' E', the
first of which, being perpendicular to the plane of projection, gives, accord-
ing to principles already laid down, a straight line at an angle of 45 for
the outline of its shadow, whereas the side A' E' being parallel to that
plane, its shadow is determined by a portion of a circle a b <?, described
from the centre o.
Figs. 6, 7. If the prism be hexagonal, or a cylinder be substituted for
it, the mode of construction remains the same. But it should be observed,
that it is best in all such cases to commence by finding the points which
indicate the main direction of the outline. To ascertain the point a at
which the shadow commences, draw from a' the line a' A' at an angle of
45, which is then to be projected vertically to a A. Then the highest
point 5 (fig. 7) should be determined by the intersection of the radius
O' B' (drawn parallel to the ray), with the circumference of the base of
the cylinder on which the required shadow is cast ; and finally, the point
c, where the outline of the cast shadow intersects the line of shade, should
be determined by a similar process.
Fig. 1 represents a hexagonal prism upon which a shadow is thrown
by a rectangular prism. Determine the point a as in fig. 5, pi. LXXX. ;
draw from the angular points J', c' lines parallel to the direction of the
light, intersecting the edge of the rectangular prism at B', C' ; project these
points, and draw the lines B 5, C c/ their intersections with the edges of
the hexagonal prism will be the limiting points &, <?, of the required
shadow.
Fig. 2 represents a hexagonal prism upon which a shadow is cast by
another hexagonal prism. The construction is precisely similar to the
preceding. Lines parallel to the direction of the light are drawn from
the angular points of both interior and exterior prisms ; these points are
projected, and the limiting points , 1>, c, d, of the shadow are determined.
SHADING AND SHADOWS.
321
Fig. 3 represents a hexagonal prism upon which a shadow is cast by a
cylinder, a variety of the preceding; but as in this case the outline
of the shadow is curved, in addition to the lines from the angular points
Fig. 1.
Fig. 2.
Fig. 8.
of the prism, parallels are also drawn from as many intermediate points
b' d f , as may be necessary to determine the outline of the curved shadow.
Plate LXXXL, fig. 7. To define the shadows cast upon tJie interior of
a hollow cylinder in section by itself and by a circular piston fitted into it.
The example shows a steam cylinder, A, in section, by a plane passing
through its axis, with its piston and rod in full.
Conceive, in the first instance, the piston P to be removed ; the shadow
cast into the interior of the cylinder will then consist, obviously, of that
projected by the vertical edge B C, and by a portion of the horizontal
edge B A. To find the first, draw through B 7 a line B' V at an angle of
45 with B 7 A ; the point 5 7 , where this line meets the interior surface of
the cylinder, being projected upwards to fig. 7, gives the line bf as the
outline of the shadow sought. Then, parallel to the direction of the light,
draw a tangent at F 7 to the inner circle of the base ; its point of contact
being projected to F in the elevation, marks the commencement of the out-
line of the shadow cast by the upper edge of the cylinder. The point 5,
where it terminates, will obviously be the intersection of the straight line
/ b already determined, with a ray B b from the tipper extremity of the
edge B C ; and any intermediate point in the curve, as e, may be found by
taking a point E 7 , between B 7 and 5 7 , projecting it to E, and causing rays
E <?, E' e', to pass through these points. The outline of the shadow required
will then be the curve F e b and the straight line bf. 'Suppose now the
piston P and its rod T to be inserted into the cylinder as shown. The
lower surface of the piston will then cast a shadow upon the interior sur-
face of the cylinder, of which the outline D d h o, may be formed in the
21
322
SHADING AND SHADOWS.
same way, as will be obvious from inspection of the figures, and compari-
son of the letters of reference. The piston-rod T being cylindrical and
vertical, it casts also its shadow into the interior of the cylinder ; it will
obviously consist of a rectangle ij I lc drawn parallel to the axis, and of
which the sides ij and & I are determined by the tangents I' i' and K/ Icf.
Figure 4. This example consists of a hollow cylinder, surmounted
by a circular disc or cover, sectioned through the centre, where it is also
penetrated by a cylindrical aperture. The construction necessary for find-
ing the outlines of the cast shadow will obviously be the same as already
laid down. In this case, however, it is proper to know beforehand what
parts of the upper and lower edges of the central aperture cast their
shadows into the interior of the cylinder ; if, then, we take the trouble to
construct the shadows of each of these edges separately, we shall find that
that of the upper edge is a curve 1} cf, and that of the lower a similar
curve a c e, cutting the former in c. This point limits the parts of each
curve which are actually visible ; namely, the portion 5 c of the first, and
the portion e c of the second ; hence it follows, that in order to avoid un-
necessary work, we should first determine the position of the point of inter-
section, c, of the two curves, which is in fact the cast shadow of the lowest
point in the curve D C, previously laid down in the circular opening of
the cover, in the manner indicated in fig. 7, plate LXXXI.
Fig. 4.
Fig. r,.
Fig. 5.-Let a cylinder in section to be set at an inclination to the horizon-
tal plane. To find the outline of the shadow cast into its interior, describe
upon the prolongation of the axis of the cylinder a semicircle A' a B',
representing its interior surface, and then, in any convenient part of the
paper, draw the diagonal in o parallel to the line of light A' E r , and con-
SHADING AND SHADOWS.
323
struct a square in n op (fig. 6) ; from one of the extremities, o, draw the
line o r parallel to A' B', and through the opposite extremity, m, draw a
perpendicular r s to this line, and set off on the perpendicular the distance
r s equal to the side of the square, and join s o. Now,
draw through the point A', in the original figure, a line
A' ', parallel to s o, intersecting the circle A' a! B' in the
point ', which being projected by a line parallel to the
axis of the cylinder, and meeting the line A , drawn at
an angle of 45, gives the first point a in the curve C d a.
The other .points will be obtained in like manner, by draw-
ing at pleasure other lines, such as T)' d', parallel to A' a'. Flg . 6 .
To find the outline of the shadow cast into the interior of a hollow
hemisphere.
Let A B C D (fig. 7) represent the horizontal projection of a concave
hemisphere. Here it is sufficiently
obvious, that if we draw through
the centre of the sphere a line per-
pendicular to the ray of light A C,
the points B and D will at once
give the extremities of the curve
sought. Construct the square (fig.
6), making m o the diagonal paral-
lel witli A C' at m ; erect the per-
pendicular in m 3 , making m m 3
equal to one side of the square ;
connect m 3 o. Take now upon the
prolongation of the line B D any
point O', from which, as a centre, describe a semicircle with the radius
A O, and from the point A' draw the straight line A' a parallel to om 3 ;
the point a! of its intersection with the circle A' a! C', projected to , will
be another principal point in the outline of the shadow.
By imagining similar sections, such as E F parallel to the former A C,
and laying down in the same way semicircles E F corresponding to them,
and drawing through E 7 lines parallel to o in 3 , and projecting their intersec-
tion with their semicircles to the corresponding sections E F on the plan, the
remaining points in the curve sought may be obtained. But as this curve
is an ellipse, of -which the diameter D B is the major axis, and the line a
the half minor axis, it follows that this last line being determined, the
curve may be constructed by the ordinary methods for ellipses.
Fig. T.
324:
SHADING AND SHADOWS.
There will now be no difficulty in constructing the cast shadow in the
interior of a concave surface (fig. 8), formed by the combination of a hollow
semi-cylinder and a quadrant of a hollow sphere, called a niche, as we
know the mode of tracing the shadows upon each of these figures separ-
ately. Thus, the shadow of the circular outline upon the spherical por-
tion is part of an ellipse
i c D, whose semi- axis major
is O D ; the semi-axis minor
is obtained by describing the
semicircle B 2 i' E with the
> y'\ ,' .*' V
;x v.-V r ra dius O B, drawing from
the point B 2 the straight line
B 2 i' parallel to a line o m 3 ,
found by a construction as before (fig. 6),
and finally projecting the point of intersec-
tion i' to i on the straight line B O. The
point e, where this ellipse cuts the horizontal
diameter A F, limits the cast shadow upon
the spherical surface ; therefore all the points
beneath it must be determined upon the cy-
lindrical part, Through A' in the plan draw
the line A' a' parallel to the ray of light ;
project a' till it intersects the line of light
A a in the elevation at a. The line of shadow
below a is the shadow of the edge of the cy-
linder, and must therefore be a straight line.
The line of shadow between a and e is pro-
duced by the outline of the circular part fall-
ing on a cylindrical surface, and is established as in previous constructions,
by drawing lines parallel to the rays of light through different points, as
B in the curved outline, and similar lines through the corresponding
points B 7 in the plan, and projecting their intersections V with the semir
circle till they intersect the first line at & as points in the line of shadow."
Plate LXXXL, figs. 1, 2. To find the line of shade in a sphere., and
the outline of its shadow cast upon the horizontal plane.
The line of shade in a sphere is simply the circumference of a great
circle of which the plane is perpendicular to the direction of the luminous
ray, and consequently inclined to the two planes of projection. This line
will, therefore, be represented in elevation and plan by two equal ellipses,
Fig. 8.
SHADING AND SHADOWS. 325
the major axes of which are obviously the diameters C.D and C' D', drawn
at an angle of 45.
To find the minor axes of these curves, assume any point O 2 , upon the
prolongation of the diameter of the perpendicular C' D' (fig. 2), draw
through this point the straight line O 2 o', inclined at an angle of 35 16',
to A' B' or its parallel, and erect upon it the perpendicular E 2 F 2 . The
projection of the two extremities E 2 and F 2 upon the line A' B', will give,
in the plan, the line E 7 F 7 for the length of the required minor axis of the
ellipse, i. e. of the line of shade in the plan ; and this line being again
transferred to the elevation, determines the minor axis E F of the line of
shade in the elevation.
Supposing it were required to draw these ellipses, not by means of
their axes, but by points, any number of these may be obtained by making
horizontal sections of the sphere. Thus, for example, if we draw the chord
G H, parallel to A' B', to represent one of these sections, and from the
point a, where it cuts the diameter E 2 F 2 , if we draw a perpendicular to
A 7 B', the points a' a', where it intersects the circumference of a circle
representing the section G H in plan, will be two points in the line of
shade required. These points may be transferred to the elevation, by
supposing a section g h to be made in fig. 1, corresponding to G H in fig.
2, and projecting the points a! a! by perpendiculars to g h, the line repre-
senting the cutting plane.
The outline of the shadow cast by the sphere upon the horizontal plane
is also obviously an ellipse ; it may be constructed either by means of its
two axes, or by the help of points, in the manner indicated in the figure.
Figs. 3, 4, and 5. To draw the line of shade on the surface of a ring
of circular section, in vertical section, elevation, and plan.
We shall first point out the mode of obtaining those primary points in
the curve which are most easily found, and then proceed to the general
case of determining any point whatever.
If tangents be draAvn to the circles represented in figs. 3 and 4, parallel
to the ray of light, their points of contact, a, I, c, d, will be the starting
points of the required lines of shade.
Again, the intersections of 'the horizontal lines a e, d g, c /, drawn
through these points, with the axis of the ring, will give so many new
points <?, g,f, in the curve. These points are denoted in the plan (fig. 5), by
setting off the distances a e and c f upon the vertical line g' D, on both
sides of the centre C'.
Farther, the diameter F 7 G', drawn at an angle of 45, determines, by
its intersections with the exterior and interior circumferences of the ring,
326 SHADING AND SHADOWS.
four other points F', t' , a?', and G', in the curve in question ; these points
are all to be projected vertically upon the line A B.
And, lastly, to obtain the lowest points I /, construct the two squares
(fig. 6), making the diagonals a' mf and o rn parallel severally to the ray
of light in plan and elevation ; revolve o' m upon the point o f until it be-
comes parallel with the vertical plane of projection; project to m 2 , and
connect o m? ; draw tangents to the circles represented in figs. 3 and 4,
parallel to the line o m 2 , and transfer the distances between the points of
contact, s, s, and the axis of the ring, to the radius E' C' (fig. 5), where
they are denoted by I' I' / these latter points are then to be projected ver-
tically to Z, I, upon the horizontal lines drawn through the same points s, s
(figs. 3 and 4).
The curves sought might now, in most instances, be traced by the
points thus obtained ; but should the ring be on a large scale, and great
accuracy be required, it may be proper to determine a greater number
of points. For this purpose, draw through the centre C', a straight line
I' H', in any direction, draw through o', one of the angular points of the
cube at fig. 6, a straight line or parallel to I' IF, and from the opposite
point tn' draw a perpendicular m' r' to o' T'. Then, having revolved the
point r'- to r 1 by means of a circular arc, in order to admit of this last
point being projected to ?', we join o T.
Applying this construction to the figures before us, we now draw tan-
gents to the circles represented in figs. 3 and 4, parallel to the line o r,
and, taking as radii the distances from their respective points of contact,
h and I, to the axis of the ring, we describe corresponding circles about
the centre C', fig. 4. We thus obtain four other points in the curves re-
quired, namely, I', *', A, and H, which may also be projected upon the
horizontal lines drawn through the points h or I.
By drawing the straight line J' K' so as to form with F' G' the same
angle which the latter makes with the line H' I', we obtain, by the inter-
section of that line with the circles last named, ftfur other points of the
curves in question.
Figs. 8 and 9. Of the shadows cast upon the surfaces of grooved pulleys.
The construction of cast shadows upon surfaces of the kind now under
consideration is founded upon the principle already announced, that when
a circle is parallel to a plane, its shadow , cast upon that plane, is another
circle equal to the original circle.
Take, in the first place, the case of a circular-grooved pulley (figs. 8
and 9) ; the cast shadow on its surface is obviously derived from the cir-
cumference of the upper edge A B. To determine its outline, take any
SHADING AND SHADOWS. 327
horizontal line D E upon fig. 8, and describe from the centre C' (fig. 9) a
circle with a radius equal to the half of that line ; then draw, through the
same centre, a line parallel to the ray of light, which will intersect the
plane D E in c ; lastly, describe from the point c', as a centre, an arc of a
circle with a radiifs equal to A C ; the point of intersection, #', of this arc,
with the circumference of the plane D E, will give when projected to a
(fig. 8), one of the points in the curve required.
To avoid unnecessary labor in drawing more lines parallel to D E than
are required, it is important, in the first place, to ascertain the highest
point in the curve sought This point is the shadow of that marked H on
the upper edge of the pulley, and which is determined by the intersection
of the ray C' H' with the circumference of that edge in the plan ; and it
is obtained by drawing through the point A (fig. 8) a straight line at an
angle of 35 16' with the line A B, and through the point e, striking a
horizontal line e.f, which by its intersection with the line H A, drawn at
an angle of 45, will give the point sought.
In fig. 9, the pulley is supposed to be divided horizontally in the centre,
and the shadow represented is derived from the smaller circle I K, and is
easily constructed by methods above described.
Plate LXXXII. To trace the outlines of the shadows cast upon the
surfaces of screws and nuts, both triangular and square-threaded.
Figs. 1 and 2 represent the projections of a screw with a single square
thread, and placed in a horizontal position, A' a' being the direction of the
ray of light. In this example, the shadow to be determined is simply that
cast by the outer edge, A B, of the thread upon the surface of the inner
cylinder ; therefore its outline is to be delineated in the same manner as we
have already pointed out, in treating of a cylinder surmounting another
of smaller diameter (page 320).
Figs. 3 and 4. The case of a triangular-threaded screw does not admit
of so easy a solution as the above, because the outer edge A C D of the
thread, in place of throwing its shadow upon a cylinder, projects it upon a
helical surface inclining to the left, of which the .generatrix is known. De-
scribe from the centre O (fig. 3) a number of circles, representing the
bases of so many cylinders, on the surfaces of which we must suppose
helical lines to be traced, of the same pitch with those which form the ex-
terior edges of the screw (see fig. 4). We must now draw any line, such
as B' E', parallel to the ray of light, and cutting all the circles described
in fig. 3 in the points B', F', G', E 7 , which are then to be successively pro-
jected to their corresponding helical lines in fig. 4, where they are denoted
by B 2 , F, G, and E. Then, transferring the point B' (fig. 3) to its appro-
328 SHADING AND SHADOWS.
priate position B on the edge A C D (fig. 4), and drawing through the
latter a line B 5 at an angle of 45, its intersection with the curve B 2 G E
will give one point in the curve of the shadow required. In the same
manner, by constructing other curves, such as H 2 J K, the remaining
points, as A, in the curve may be found.
Figs. 5 and 6. The same processes are requisite in order to determine
the outlines of the shadows cast into the interior surfaces of the nut cor-
^ responding to the screw last described, as will Be evident from inspection
of figs. 5 and 6. These shadows are derived not only from the helical
edge A B D, but also from that of the generatrix A C.
Figs. 7 and 8. The shadow cast by the helix ABC upon the concave
surface of the square-threaded nut is a curve a ~b C, which is to be deter-
mined in the same way as that in the interior of a hollow cylinder. The
same observation applies to the edges A A 2 and A 2 E, as well as to those
of the helix F G H and the edge H I. With regard to the shadow of the
two edges J K and Iv L, they must obviously follow the rules laid down
in reference to figs. 4 and 6, seeing that it is thrown upon an inclined heli-
cal surface, of which A L is the generatrix.
The principles so fully laid down and illustrated in the preceding pages
will be found to admit of a ready and simple application to the delineation
of the shadows of all the ordinary forms and combinations of machinery
and architecture, however varied or complicated ; and the student should
exercise himself, at this stage of his progress, in tracing, according to the
methods above explained, the outlines of the cast shadows of,pulleys, spur-
wheels, and such simple and elementary pieces of machinery. It must be
observed, that the student should never copy the figures as here repre-
sented, but should adopt some convenient scale somewhat larger than our
figures, and construct his drawings according to the description, looking to
the figures as mere illustrations ; in this way the principles of the con-
struction will be more surely understood, and more firmly fixed in his
mind.
MANIPULATION OF SHADING AND SHADOWS. METHODS OF TINTING.
The intensity of .a shade or shadow is regulated by the various peculi-
arities in the forms of bodies, and by the position which objects may
occupy in reference to the light.
Surfaces in the light. Flat surfaces wholly exposed to the light, and
at all points equidistant from the eye, should receive a uniform tint.
In geometrical drawings, where the visual rays are imagined parallel
to the plane of projection, every surface parallel to this plane is supposed
SHADING AND SHADOWS. 329
to have all its parts at the same distance from the eye ; such is the vertical
side of the prism a led (fig. 4, plate LXXXIII).
When two surfaces thus situated are parallel, the one nearer the eye
should receive a lighter tint than the other. Every surface exposed to the
light, but not parallel to the plane of projection, and therefore having no
two points equally distant from the eye, should receive an unequal tint.
In conformity, then, with the preceding rule, the tint should gradually in-
crease in depth as the parts of such a surface recede from the eye. This
" effect is represented in the same figure on the surface, a dfe, which, by
reference -to the plan (fig. 1), is found to be in an inclined position.
If two surfaces are unequally exposed to the light, the one which is
more directly opposed to its rays should receive the fainter tint.
Tims the face e' a' (fig. 1), presenting itself more directly to the rays of
light than the face a' 5', receives a tint which, although graduated in con-
sequence of the inclination of this face to the plane of projection, becomes
at that part of the 'surface situated nearest to the eye fainter than the tint
on the surface a 5.
Surfaces in shade. When a surface entirely in the shade is parallel to
the plane of projection, it should receive a uniform dark tint.
"When two objects parallel to each other are in the shade, the one
nearer the eye should receive the darker tint.
"When a surface in the shade is inclined to the plane of projection, those
parts which are nearest to the eye should receive the deepest tint.
The face ~b g ti c (fig. 4), projected horizontally at V g' (fig. 1), is situated
in this manner. It will there be seen, that towards the line I c the tint is
much darker than it is where it approaches the line g h.
If two surfaces exposed to the light, but unequally inclined to its rays,
have a shadow cast upon them, that part of it which falls upon the surface
more directly influenced by the light should be darker than where it falls
upon the other surface.
Exemplifications of the foregoing rules may be seen on various figures
in the plates.
In order that these rules may be practised with proper effect, we shall
give some directions for using the brush or hair-pencil, and explain the
usual methods employed for tinting and shading.
The methods of shading most generally adopted are either by the
superposition of any number of flat tints, or by tints softened off at their
edges. The former method is the more simple of the two, and should bo
the first attempted.
Shading ~by flat tints. Let it be proposed to shade the prism (fig. 4,
330 SHADING AXD SHADOWS.
plate LXXXIII.), by means of flat tints. According to the position of the
prism, as shown by its plan (fig. 1), the face a 1) c d (fig. 4) is parallel to the
plane of projection, and therefore entirely in the light. This face should
receive a uniform tint either of India ink or sepia. When the surface to be
tinted happens to be very large, it is advisable to put on a very light tint
first, and then to go over the surface a second time with a tint sufficiently
dark to give the desired tone to the surface.
The face l>g he being inclined to the plane of projection, as is shown
by the line V g' in the plan (fig. 1), should receive a graduated tint from '
the line 1) c to the line g h. This graduality is obtained by laying on a
succession of flat tints in the- following manner : First, divide the line
V g' (fig. 1) into equal parts at the points 1', 2', and from these points pro-
ject lines upon, and parallel to, the sides of the face b g h c (fig. 4). These
lines should be drawn very lightly in pencil, as they merely serve to cir-
cumscribe the tints. A greyish tint is then spread over that portion of the
face l>ghc (fig. 2), between the lines l> c and 1, 1. When this is dry, a
similar tint is to be laid on, extending over the space comprised within the
lines 1) c and 2, 2 (fig. 3). Lastly, a third tint covering the whole surface
1} c h g (fig. 4) imparts the desired graduated shade to that side of the prism.
The number of tints designed to express such a graduated shade depends
upon the size of the surface to be shaded ; and the depth of tint must vary
according to this number.
As the number of these washes is increased, the whole shade gradually
presents a softer appearance, and the lines which border the different tints
become less harsh and perceptible. For this reason the foregoing method
of representing a shade or graduated tint by washes successively passing
over each other is preferable to that sometimes employed, of first covering
the whole surface lygJic with a faint tint, then putting on a second tint
1) 2 2 c, followed, lastly, by a narrow wash 5 1 1 cy because, in following
this process, the outline of each wash remains untouched, and presents, un-
avoidably, a prominence and harshness which, by the former method, are
in a great measure subdued.
The face adfe is also inclined to the plane 'of projection, as shown by
the line a' e' in the plan (fig. 1) ; but as it is entirely in the light, it should
be covered by a series of much fainter tints than the surface Iff he, which
is in the shade, darkening, however, towards the line ef. The gradation
of tint is effected in the same way as on the face 1) g he.
Let it be proposed to shade a cylinder (fig. 12), by means of flat tints :
In shading a cylinder, it will be necessary to consider the difference in
the tone proper to be maintained between the part in the light and that in
SHADING AND SHADOWS. 331
the shade. It should be remembered that the line of separation between
the light and shade a 5 is .determined by the radius O a' (fig. 5), drawn
perpendicular to the rays of light R O. That part, therefore, of the cylin-
der which is in the shade is comprised between the lines a l> and c d. This
portion, then, should be shaded conformably to the rule previously laid
down for treating surfaces in the shade inclined to the plane of projection.
All the remaining part of the cylinder which is visible presents itself to
the light ; but, in consequence of its circular figure, the rays of light form
angles varying at every part of its surface, and consequently this surface
should receive a graduated tint. In order to represent with effect the
rotundity, it will be necessary to determine with precision the part of the
surface which is most directly affected by the light. This part, then, is
situated about the line e i (fig. 12), in the vertical plane of the ray of light
K O (fig. 5). As the visual rays, however, are perpendicular to the ver-
tical plane, and therefore parallel to V O, it follows that the part which
appears clearest to the eye will be near this line Y O, and may be limited
by the line T O, which bisects the angle Y O E and the line K O. By
projecting the points e' and m', and. drawing the lines e i and m n (fig. 12),
the surface comprised between these lines will represent the lightest part
of the cylinder.
This part should have no tint upon it whatever if the cylinder happen
to be polished : a turned iron shaft or a marble column for instance ; but
if the surface of the cylinder be rough, as in the case of a cast-iron pipe,
then a very light tint considerably lighter than on any other part may
be given it.
Again, let us suppose the half-plan of the cylinder/ 7 m' a' c' (fig. 5), to
be divided into any number of equal parts. Indicate these divisions upon
the surface of the cylinder by faint pencil lines, and begin the shading by
laying a tint over all that part of the cylinder in the shade a c d l> (fig. 6).
This will at once render evident the light and dark parts of the cylinder.
When this is dry, put on a second tint covering the line a 5 of separation
of light and shade, and extending oA r er one division, as shown in fig. 7.
A third tint should be spread over this division, and one on each side of
it, as in fig. 8. Proceed in this way until the whole of that part of the
cylinder which is in the shade is covered. The successive stages of this
process may be seen in figs. 9, 10, and 11.
Treat in a similar manner the part feig, and complete the operation
by covering the whole surface of the. cylinder excepting only the division
e m n i (fig. 12) with a very light tint ; the cylinder will then assume the
appearance presented by fig. 12.
3o2 SHADING m> SHADOWS.
Shading by softened tints. The great advantage which this method
possesses over the one just described, consists in imparting to the shade a
much softer appearance ; the limitations of the different tints being imper-
ceptible. On the other hand, it is considerably more difficult, requiring
longer practice, and greater mastery over the movements of the brush to
accomplish it with tolerable precision.
Let it be proposed to shade by this method the segment of the hexa-
gonal pyramid (fig. 8, plate LXXXIV.)
The plan of this figure is similar to that of the prism (fig. 4, plate
LXXXIII.) Its position in reference to the light is also the same. Thus
the face abed should receive a uniform flat tint. If, however, it be de-
sired to adhere rigorously to the preceding rules, the tint may be slightly
deepened as it approaches the top of the pyramid, seeing that the surface
is not quite parallel to the vertical plane.
The face b g h c being inclined and in the shade, should receive a dark
tint. The darkest part of this tint is where it meets the line b c, and grad-
ually becomes lighter as it approaches the line g h. To produce this
eifect, apply a narrow strip of tint to the side b c (fig. 6), and then, quali-
fying the tint in the brush with a little water, join another strip to this,
and finally, by means of another brush moistened with water, soften off this
second strip towards the line 1, 1, which may be taken as the limit of the
first tint. This is shown in fig. 6.
When the first tint is dry, cover it with a second, which must be simi-
larly treated, and should extend beyond the first up to the line 2, 2 (fig.
7). Proceed in this manner with other tints, until the whole face l> y h c
is shaded, as presented in fig. 8.
In the same way the face e a df is to be covered, though with a con-
siderably lighter tint, for the rays of light happen to fall upon it almost
perpendicularly.
It may be observed, that consistently to carry out the rules we have
laid down, the tint on these two faces should be slightly graduated from
e a tofd, and from c h to b g. But this exactitude may be disregarded
until some proficiency in shading has been acquired.
It is now proposed to shade the cylinder (fig. 4) by means of softened
tints. The boundary of each tint being indicated in a manner precisely
similar to that shown by fig. 5, plate LXXXIII., the first strip of tint must
cover the line of extreme shade a b, and then be softened off on each side,
as shown in fig. 13. Other and successively wider strips of tint are to fol-
low, and receive the same treatment as the one first put on. The results
of this process are shown in figs. 2, 3, and 4.
SHADING AND SHADOWS. 333
As this method requires considerable practice before it can be per-
formed with much nicety, the learner need not be discouraged at the fail-
ure of his first attempts, but persevere in practising on simple figures of
different sizes.
If, after shading a figure by the foregoing method, any very apparent
inequalities present themselves in the shade, such defects may be remedied
in some measure by washing off redundancies of tint with a brush or a
damp sponge, and by supplying a little color to those parts which are too
light.
Dexterity in shading figures by softened tints will be facilitated in prac-
tising upon large surfaces ; this will be the surest way of overcoming that
timidity and hesitation which usually accompany all first attempts, but
which must be laid aside before much proficiency in shading can be ac-
quired.
ELABOKATION OF SHADING AND SHADOWS.
Thus far the simplest primary rules for shading isolated objects have
been laid down, and the easiest methods of carrying them into operation
explained. It is now proposed to exemplify these rules upon more com-
plex forms, to show where the shading may be modified or exaggerated,
to introduce additional rules more especially adapted for mechanical color-
ing, and to offer some observations and directions for effectively shading
the drawing of machines in their entirety.
Whatman's best rough-grained drawing-paper is better adapted for
receiving color than any other. Of this paper, the double elephant size is
preferable, as it possesses a peculiar consistency and grain. A larger
paper is seldom required, and when the drawing to be made happens to be
small, a portion of a double elephant sheet should be used.
The paper for a colored drawing ought always to be strained upon a
board with glue or strong gum. Before doing this, care must be taken to
damp the face of the paper with a sponge well charged with water, in
order to remove any impurities from its surface, and as a necessary prepa-
ration for the better reception of the color. The sponge should merely
touch the paper lightly, and not rub it. The whole of the surface is to be
damped, that the paper may be subjected to a uniform degree of expansion,
thereby insuring, as it dries, a uniform degree of contraction. Submitted
to this treatment, the sheet of paper will present, when thoroughly dry, a
clean smooth surface, not only agreeable to work upon, but also in the
best possible condition to take the color.
The size of the brushes to be used will, of course, depend upon the
334: SHADING AND SHADOWS.
scale to which the drawing is made. Long thin brushes, however, should
be avoided. Those possessing corpulent bodies and fine points are to be
preferred, as they retain a greater quantity of color, and are more manage-
able.
During the process of laying on a flat tint, if the surface be large
though this is seldom the case except in topographical drawings the draw-
ing may be slightly inclined, and the brush well charged with color, so that
the edge of the tint may be kept in a moist state until the whole surface is
covered. In tinting a small surface, the brush should never have much
color in it, for if it have, the surface will unavoidably present coarse
rugged edges, and a coarse uneven appearance throughout. A moderate
quantity of color in the brush, well though expeditiously rubbed into the
. paper, is the only method of giving an even close-grained aspect to the
surface. In fact, for mechanical drawings, there is rarely occasion for well
charging the brush with color. The tint in the brush may be very dark
or very light, but there should seldom be much of it.
As an invariable rule let it be remembered, that no tint, shade, or
shadow is to be passed over or touched until it is quite dry.
In the examples of shading which are given in this work, it may be
observed that all objects with curved outlines have a certain amount of
reflected light imparted to them. It is true that all bodies, whatever may
be their form, are affected by reflected -light ; but, with a few exceptions,
this light is only appreciable on curved surfaces. The judicious degree
and treatment of this light is of considerable importance for the acquire-
ment of an. effective style of shading.
All bodies in the light reflect on those objects which surround them
more or less light according to the situation. Wherever light extends, re-
flection follows. If an object be isolated, it is still reached, by reflected
light, from the ground on which it rests, or from the air which surrounds it
In proportion to the degree of polish or brightness in the color of a
body, is the amount of reflected light which it spreads over adjacent ob-
jects, and also its own susceptibility of illumination under the reflection
from other bodies. A polished steam-cylinder, or a white porcelain vase,
receives and imparts more reflected light than a rough casting or a stone
pitcher.
Shade, even the most inconsiderable, ought never to extend to the out-
line of any smooth circular body. On a polished sphere, for instance, the
shade should be delicately softened off just before it meets the circumfer-
ence, and when the shading is completed, the body color intended for the
sphere may be carried on to its outline. This will give a transparency to
SHADING AND SHADOWS. 335
that part of tlio sphere influenced by reflected light, which it could not
have possessed if the shade tint had been extended to its circumference.
Very little shade should be suffered to reach the outlines even of rough
circular bodies, lest the coloring look harsh, and present a coarse appear-
ance quite at variance with its natural aspect. Shadows also become
lighter as they recede from the bodies which cast them, owing to the in-
creasing amount of reflection which falls on them from surrounding ob-
jects.
Shadows appear to increase in depth as their distance from the spectator
diminishes. In nature this increase is only appreciable at considerable
distances. Even on extensive buildings, inequalities in the depth of the
shadows are hardly perceptible ; much less, then, can any natural grada-
tion present itself in the shadows on a machine, which, supposing it to be
of the largest construction, is confined to a comparatively small space. It
is most important, however, for the effective representation of machinery,
that the variation in the distance of each part of a machine from the spec-
tator should at once strike the eye ; and an exaggeration in expressing the
varying depths of the shadows is one means of effecting that object. The
shadows on the nearest and most prominent parts of a machine should be
made as dark as color can render them, the colorist being thus enabled to
exhibit a marked difference in the shadows on the other parts of the ma-
chine as they recede from the eye. The same direction is applicable in
reference to shades. The shade on a cylinder, for instance, situated near
the spectator, ought to be darker than on one more remote ; in fact, the
gradation of depth for the shades follows that which depicts the shadows.
As a general rule, the color on a machine, no matter what it may be in-
tended to represent, should become lighter as the parts on which it is
placed recede from the eye.
Plates LXXXY. and LXXXYI. present some very good examples of
finished shading.
Plate LXXXV, represents, both in elevation and plan, different solids
variously penetrated and intersected. The rules for the projection of these
solids have been given under the head of Geometrical Projection, and illus-
trated in plates VI., VII., VIIL, IX., X. They are selected with a view
of exhibiting those cases which are of most frequent occurrence, and at the
same time elucidating the general principles of shading.
Plate LXXXVI. presents examples of shading and shadow.
Fig. 1 presents a hexagonal prism surmounted by a fillet. The most
noticeable part of this figure is the shadow of the prism in the plan view.
It presents a good example of the graduated expression which should be
336 SHADING AND SHADOWS.
given to all shadows cast upon plain surfaces. Its two extremities are re-
markably different in their tone. As the shadow nears the prism, it in-
creases rapidly in depth ; on the contrary, as it approaches the other end,
it assumes a comparatively light appearance. This difference is doubt-
lessly a great exaggeration upon what it would 1 naturally display. Any
modification of it, however, in the representation would destroy the best
effect of the shadow.
The direction which the shades and shadows take in all the plans of
the figures in this plate, is from the left hand lower corner. This is rigor-
ously correct, supposing the objects to remain stationary, whilst the spec-
tator views them in both a vertical and horizontal position. Nevertheless,
to many, this upward direction given to the shadows has an awkward
appearance, and, perhaps, in the plan of an entire machine, the shadows
may look better if their direction coincide with that which is given to
them in the elevation. If, however, the shadows be correctly projected,
their direction is an arbitrary matter, and may be left to the taste of the
draughtsman.
Figs. 2, 3, and 6 exemplify the complex appearance of shade and
shadow presented on concave surfaces. It is worthy of particular notice,
that the shadow on a concave surface is darkest towards its outline, and
becomes lighter as it nears the edge of the object. Reflection from that
part of the surface on which the light falls most powerfully causes this
gradual diminution in the depth of the shadow, the greatest amount of re-
flection being opposite the greatest amount of light.
It may be as well to remark here, that no brilliant or extreme light
should be left on concave surfaces, as such lights would tend to render it
doubtful at first sight whether the objects represented were concave or
convex. After the body-color which shall be treated in a subsequent
section has been put on, a faint wash should be passed very lightly over
the whole concavity. This will not only modify and subdue the light, but
tend to .soften any asperities in the tinting, which are more unsightly on a
concave surface fhan on any other.
The lightest part of a sphere (fig. 4) is confined to a mere point, around
which the shade commences and gradually increases as it recedes. This
point is not indicated on the figure referred to, because the shade tint on
a sphere ought not to be spread over a greater portion of its surface than
is shown there. The very delicate and hardly, perceptible progression of
the shade in the immediate vicinity of the light point should be effected
by means of the body-color of the sphere. If, for instance, the material
of which the sphere is composed be brass, the body-color itself should be
SHADING AND SHADOWS. 337
lightened as it nears the light point. In like manner all polished or light-
colored curved surfaces should be treated ; the part bordering upon the ex-
treme light being covered with a tint of body-color somewhat fainter than
that used for the flat surfaces. Again, if the sphere be of cast-iron, then the
ordinary body-color should be deepened from the light point until it meets
the shade tint, over which it is to be spread uniformly. Any curved un-
polished surface is to be thus treated ; the body-color should be gradually
deepened as it recedes from that part of the surface most exposed to the
light. Considerable management is necessary in order to shade a sphere
effectively. The best way is to put on two or three softened-off tints in
the form of crescents converging towards the light point, the first one
being carried over the point of deepest shade.
A ring (fig. 5) is a difficult object to shade. To change with accurate
and effective gradation the shade from the inside to the outside of the ring,
to leave with regularity a line of light upon its surface, and to project its
shadow with precision, require a degree of attention and care in their exe-
cution greater, perhaps, than the shade and shadow of any other simple
figure. The learner, therefore, should practise the shading of this figure,
as he will seldom meet with one presenting greater difficulties.
Figs. 7 and 8 show the peculiarities of the shadows cast by a conical
form on a sphere or cylinder. The following fact should be well noted in
the memory : That the depth of a shadow on any object is in proportion
to the degree of light which it encounters on the surface of that object.
In these figures very apt illustrations of this fact may be remarked. It
will be seen by referring to the plan (fig. 7), that the shadow of the apex
of the cone happens to fait upon the lightest point of the sphere, and is,
therefore, the darkest part of the shadow. So also the deepest portion of
the shadow of the cone on the cylinder in the plan (fig. 8) is exactly where
it coincides with the line of extreme light. Flat surfaces are ^similarly
affected, the shadows thrown on them being less darkly expressed, accord-
ing as their inclination to the plane of projection increases. The body-
color on a flat surface should, on the contrary, increase in depth as the sur-
face becomes more inclined to this plane.
Another notable fact is exemplified by these figures : that reflected
light is incident to shadows as well as to shades. This is very observable
where the shadow of the cone falls upon the cylinder. It may likewise
be remarked, though to a less extent, on other parts of these figures. The
reflected light on the cone from the sphere or cylinder is also worthy of
observation. This light adds greatly to the effect of the shadows, and
22
338 SHADING AND SHADOWS.
indeed to the appearance of the objects themselves. Altogether, these
figures offer admirable scope for study and practice.
The concentration within a small space of nearly all the peculiarities
and effects of light, shade, and shadow, may be seen on plate LXXXVIL,
in the examples of screws there given.
The parts of a highly-finished colored drawing of a machine are always
affected by a certain degree of indefinableness in their outline.
Notwithstanding the most careful exertions of the colorist to keep
every feature of a machine clear and distinct, some amount of uncertainty,
resulting unavoidably from the proximity and natural blending of the dif-
ferent parts, will pervade the lines which separate its component members.
For practical working purposes, therefore, a completely colored drawing
of a machine is unsuitable. On the other hand, a mere outline, although,
perhaps, intelligible enough to those who are familiarly acquainted with
the machine delineated, has an undecided appearance. As complete
coloring renders it difficult for the eye to separate the various parts of a
machine, owing to an apparently too intimate relationship between them ;
a line drawing, on the contrary, perplexes the eye to discover any relation
between them at all, or to settle promptly their configuration. The eye
involuntarily asks the question, is that part round or square, or is it even
a distinct part of the machine at all ? As a means of avoiding the inde-
finiteness presented by the outline in the former case, and the want of
adequate coherence and doubtfulness in the form of the different parts
amenable to the latter, recourse is not unfrequently had to a kind of semi-
coloring, or rather mere shading of the parts of a machine. Exemplifi-
cations of this practical style of representing machines may be observed
in plates XL. to XLIII. inclusive, " Drawing of Machinery," pages
204 and 206. Every figure looks complete without elaboration, and is
clearly Delineated without degenerating into the bareness of a mere skele-
ton. Outlines and forms are at once apprehended, and every member of
the machine is adjusted without hesitation to its proper place.
In such drawings shading only is allowed, and therefore but slight
scope is permitted for imparting effects ; and it is advisable to follow a
direction previously given, and to modify the color on every part accord-
ing to its distance from the eye. It may be as well also, for the purpose
of maintaining harmony in the coloring, and of equalizing its appearance,
to color less darkly large shades than small ones, although they may be'
situated at an equal distance from the eye. No very dark shading is per-
missible on this species of drawing ; indeed the tint should be very con-
siderably lighter than on finished colored drawings. Besides presenting
SHADING AND SHADOWS. 339
too violent a contrast between the parts colored and those without any
color at all, dark shading would produce, in some measure, the indistinct-
ness which is objectionable in completely tinted drawings.
At page 383, Plate XCYIII. is a photograph from a finished drawing
of the engine and a boiler of the steamer Pacific. Every shadow is care-
fully projected, every detail elaborated, and the execution perfect; it may
serve as a model of its class, not only for its accuracy and distinctness of
detail, but also for its vigor as a picture. It is seldom that so much labor
is devoted to a mechanical drawing, but the result is very satisfactory to
the designer.
FINISHED COLORING.
The coloring of drawings representing machinery requires a special
study, the process of its development being, in many essentials, very differ-
ent from that pursued in the artistic expression of other objects.
The parts of a machine being usually constructed with mathematical
accuracy, and always presenting a well-defined rigid outline, the same
unmistakable definiteness should be maintained in any attempt to picture
such an object on paper. There should be no "blending" of different
colors, no doubtful finish to a tint, no softening off into the imaginative ;
every part should present at once to the eye its form and position ; should,
in fact, supply the place of a model of the machine. So important a fea-
ture is this in mechanical coloring, that when correct shadows would mate-
rially obscure any part of a machine, they should either be entirely sup-
pressed, or, when such an omission would be very striking, so modified as
to lessen as much as possible the obscurity thus produced.
The color of cast-iron fresh from the foundry is commonly a very dark
bluish black, having blended with it an almost imperceptible brownish-
green tint or cast. To represent the casting on paper to the best advan-
tage, the following colors should be employed : Indian ink and indigo,
with a very slight admixture of lake. This last ingredient is necessary ;
for Indian ink, being actually only a very dark brown, it would, in con-
junction merely with a blue, impart too green a cast to the tint sought to
be realized.
Great care should be taken in mixing these colors. First, the lake
crimson is preferable should be rubbed on the pallet ; about half-a-dozen
turns of the hand are sufficient, as too much of this color would impart a
rusty appearance to the desired tint. The indigo may then be added,
and lastly the Indian ink. The quantity of lake being very inconsiderable,
about two-thirds of the mixture should be composed of Indian ink, and
34K) SHADING AND SHADOWS.
the remaining third of indigo. This proportion, however, will be best
ascertained by occasionally trying the tint on a scrap of drawing-paper
during the process of mixing. "When the tint appears to have approxi-
mated as near as possible, according to the colorist's judgment, to the tint
described above, its ingredients should be well mixed together with the
brush. The more intimate this intermixture of the colors can be rendered,
the better ; for if any considerable number of particles of the same color
remain together, the tint, when essayed, will present a streaky, semi-party-
colored appearance. The tint being thus prepared, should be left for a short
time untouched, so as to allow the grosser particles of color to settle at the
bottom of the saucer. Eo fear need be entertained of getting the tint too
dark, or of mixing too much ; on the contrary, it is better to compound a
considerable quantity and very dark in one saucer, and then gently pour
a little into one or two others, in which, with varying quantities of water,
different gradations of tint may be produced. The tint left in the first
pallet should be preserved for shading or for shadows, and when it has be-
come dry, should by no means be discarded, as it will always be service-
able, and indeed preferable for imparting the lesser dark effects to various
parts of the drawing.
With one or two exceptions, which will be pointed out later, this tint,
variously modified, is the only one to be employed for the representation
of cast iron. It is adapted as well for expressing the shades and shadows
as for depicting the body-color. If the shades and shadows be indicated
by Indian ink alone, the small amount of "brilliancy" which cast iron
naturally enjoys will disappear wherever covered with Indian ink, and
even the effect of the body-color will be very sensibly diminished.
The first parts of the drawing of a machine which it is usually most
judicious to color are those of a circular form cylinders, the more im-
portant shafts, &c. The rods and smaller shafts, especially where they
cross other parts of the machine, may be left until the other work is
finished.
Taking for granted that the learner has practised the art of shading
according to the simple methods previously described, and, therefore, that
he is somewhat acquainted with the use of the brush, let him now proceed
to color a circular casting, it being with cast iron only that we have to do
at present.
Imagine this casting to be a large cylinder. First draw two faint
pencil lines, to indicate the extremes of light and shade on its surface.
Pass the brush, moderately full of the darkest tint, down the line of deep-
est shade, spreading the color more or less on either side, according to the
SHADING AND SHADOWS. 341
diameter of the cylinder ; then, if possible, before this layer of tint is dry,
towards the line of extreme light, beginning at the top, and encroaching
slightly over the edge of the first tint, lay on another not quite so dark,
but about double its width. It may be observed, that it is not very essen-
tial to put on the second tint before the first is dry, for the latter should
be so dark and thick, that its edges may be easily softened at any time.
Whilst this second tint is still wet, with a much lighter color in the brush,
proceed in the same manner with a third tint, and so on, until the line of
extreme light is nearly attained. Repeat this process on the other side of
the first tint, approaching the outline of the cylinder with a very faint
wash, so as to represent the reflected light which progressively modifies
the shade as it nears that line. Then let a darkish narrow strip of tint
meet, and pass along the outline of the cylinder on the other side of the
extreme line of light, after which gradually fainter tints should follow,
treated in a manner similar to that which has been already described, and
becoming almost imperceptible just before arriving at the line of light.
This is a very expeditious way of shading a cylinder ; but even to the
most experienced colorist, it is not possible, by the above-described means
alone, to impart a sufficient degree of well-regulated rotundity to the ap-
pearance of such an object. Superfluities and deficiencies of color will
appear here and there. It will be necessary, therefore, to equalize to some
extent, by a species of gross stippling, the disparities which present them-
selves. This is done by spreading a little color over the parts where it is
deficient, and then passing very lightly over nearly the whole width of the
shade, with the brush supplied with a very light wash. This process may
be repeated to suit the degree of finish whichr it is desired to give the
drawing. In the same manner the shading of all curved surfaces is to be
treated.
Recourse is often had to what is called " washing " or " sponging," in
order to impart softness and circularity to certain forms. Beyond a very
limited extent, this is a most injudicious system. It robs the shade of all
the lightest and most brilliant particles of color, the natural position of
which is on the surface ; it destroys that " crisp " freshness, so essential
towards the beautiful appearance of all coloring ; and, what is still worse,
spreads a dirty appearance not only over the whole surface of the coloring,
but more or less on all the paper which surrounds it. Sponging should
never be adopted, and if a slight washing with the brush be. sometimes
attempted, it should be done very lightly, and, except on rare occasions,
not allowed to pass beyond those parts of the drawing covered with color,
otherwise that sharp cleanly appearance, which so enhances the effect of a
3i2 SHADING AND SHADOWS.
colored drawing, will be lost. Let it, then, be remembered, that the less
the color, whether as a shade, shadow, or tint of any kind, is touched after
it has reached the paper, the better. The system of shading by numerous
tints laid one over the other a system which almost universally prevails
is no doubt a very easy, and, therefore, advantageous one for the initia-
tion of beginners into a dexterous use of the brush and the grosser mysteries
of coloring ; but no highly effective mechanical drawing can be produced
in this manner.
The principal shadows are the next parts of the coloring which will now
claim attention. The outline of any shadow being drawn in pencil, along
its inner line the line which forms a portion of the figure of the object
whose shadow is to be represented lay on a strip of the darkest tint, wide
or narrow, according to the width of the shadow, and then, before it is
dry, soften off its outer edge. This may be repeated as often as the taste
of the colorist may dictate, but the color should not spread itself over
much more than half the space occupied by the shadow. These prelimi-
nary touches will add to the intensity of the proposed shadow, and neutralize
a certain harshness of appearance inevitable to all shadows made equally
dark throughout. The effect they give to the drawing is very pleasing,
and is, moreover, quite natural, for, as previously explained, the greatest
depth of a shadow is invariably that part of it immediately contiguous to
the object shadowed forth.
The representation of the casting is now to be completed by laying on
the body-color. This might be done by a single wash of tint if the ap-
pearance of cast iron were as light as it is usually depicted ; but its natural
color being, on the contrary, very opaque and heavy, two and sometimes
three washes are necessary, the first tint being rather darker than those
which follow. Each tint should pass over the shades and shadows when
they occur, care being taken to manoeuvre the brush at such parts very
lightly.
The most conspicuous fault observable in the generality of colored
mechanical drawings is a deficiency in the depth of the tints employed.
There appears to exist an undefinable fear of transferring to paper the
naturally dark appearance of iron ; the result is the production of tame,
ineffective representations, which, instead of looking as they should, like
models of iron machines, present mere faint shadows of such objects, or,
at best, machinery constructed of some unknown, light, and rather dirty
materials.
The sectional surfaces -of cast-iron are to be indicated by one light tint
of indigo.
SHADING AND SHADOWS. 343
The next most extensive and important component used in the manu-
facture of machines is wrought iron. Precisely the same colors are to be
employed to represent this material as have been pointed out for cast-iron.
The difference in the appearance of these metals is produced by altering
the proportion of the two principal colors, Indian ink and indigo. These
ingredients should be mixed, carefully and well mixed, in about equal pro-
portions, a very small quantity of crimson lake being first rubbed in the
saucer.
The same methods of shading and of laying on the shadows prescribed
for cast-iron are to be adopted in the case of wrought-iron, keeping, how-
ever, all parts of the latter lighter, particularly the body-color. The direct
and reflected lights must also present themselves more distinctly, and to a
much greater extent. Polished and semi-polished surfaces invariably
afford greater contrasts of light and shade than other surfaces. The steps
or rather glidings from one extreme to the other are, moreover, softer and
more delicately graduated, and, therefore, greater care is requisite in repre-
senting them on paper. These remarks are very effectively illustrated by
the fragments of large screws shown on pi. LXXXYII. and also by the
photograph of the steamer Pacific, Plate XCVIII.
For the parts of wrought iron in section a light tint of Prussian blue is
most suitable. This is the only service for which Prussian blue can pro-
perly be made available in coloring drawings of machinery. In conjunc-
tion with Indian ink or indigo its inherent brightness entirely disappears ;
an ill-assorted union with the former producing a dirty color, in appear-
ance not unlike that presented by the surface of a stagnant pool ; and
with the latter creating a tint bearing a striking resemblance to soiled
glass. For mechanical drawings, then, this color must never be used in
combination.
Brass, except in small quantities, seldom makes its appearance in
machinery. This is fortunate for the colorist, as there is no metal more
difficult to represent than brass. The body-tint is composed either of
gamboge and burnt sienna, or gamboge and crimson-lake ; the shading
and shadows being best expressed by burnt umber.
The most delicate and careful treatment is needed in making use of
these colors ; for, when on the paper, they are all of them very soft, and
therefore highly sensitive to every touch of the brush. For this reason
the shadows are best put on after the body-color, otherwise their edges
will inevitably present a smeary, indefinite appearance.
For representing brass and copper, the method of coloring we have
described in this section is particularly suitable. To attempt the produc-
3M SHADING AND SHADOWS.
tion of a shade with burnt umber, by means of a succession of tints, would
merely realize a complicated smear. We find, therefore, that the shades
and shadows of brass are usually represented by Indian ink ; but as gam-
boge almost invariably enters as an ingredient into the body-color of brass,
the result is that the bright gamboge over the brown-black Indian ink ex-
hibits a species of green, to which we cannot find any thing comparable,
but which commonly has a very unpleasant effect to the eye.
In shading circular surfaces great management is requisite. " Wash-
ing " is here entirely out of the question, for even the necessary softening
off with the brush is attended with much difiiculty. The brush should not
pass heavily or often over the shade tint, lest unseemly deficiencies and
streaks of color present themselves here and there, which prove rather
difficult blemishes to repair. The utmost care and experience, neverthe-
less, cannot wholly insure the colorist against the perplexities of such
partial failures. The only way to manage these defects is by delicate
stippling ; suiting the depth of tint to the various degrees of shade
affected, and then passing a soft brush, filled moderately with dark body-
color, very lightly over the whole shade.
A light tint of gamboge is to be used for the sections of brass.
The directions which we have given for the most advantageous treat-
ment of the colors representing brass, are equally applicable to those which
exhibit the nearest approach to copper, the colors to be used for this metal
opposing nearly an equal amount of difiiculty in their management. A
mixture of orange chrome and lake, or red-lead and lake, best represent
this metal ; its shades and shadows being indicated by sepia, whilst its
sectioning is shown by a light tint of orange chrome.
Such are the colors, and such is the manner of treating them, em-
ployed for depicting on paper each of the principal metals used in ma-
chinery.
Having explained in detail the tinting of machinery in reference both
to its shading and body-color, we propose to complete our remarks on
mechanical coloring with a few suggestions for imparting some peculiar
effects to the representations of masses of machinery.
We have already noticed that the shades and shadows of a machine
are modified in intensity as their distance from the eye increases. Its
body-color should be treated in a similar manner, becoming lighter and
less bright as the parts of the machine which it covers recede from the
spectator.
When the large circular members of a machine have been shaded, the
SHADING AND SHADOWS. 345
shadows, and even the body-color on those parts furthest removed from
the eye, are to follow, and the proportion of Indian ink in . the tint used
should increase as the part to be colored becomes more remote. A little
washing, moreover, of the most distant parts is allowable, as it gives a
pleasing appearance of atmospheric remoteness, or depth, to the color thus
treated.
The amount of light and reflection on the members of a machine should
diminish in intensity as the distance of such objects from the spectator in-
creases. As it is necessary, for effect, to render, on those parts of a m'a-
chine nearest the eye, the contrast of light and shade as intense as possible,
so, for the same object, the light and shade on the remotest parts should
be subdued and blended according to the extent or size of the machine.
A means of adding considerably to the definiteness of a colored me-
chanical drawing, and of promoting, in a remarkable degree, its effective
appearance, is obtained by leaving a very narrow margin of light on the
edges of all surfaces, no matter what may be the angles which they may
form with the surfaces that join them. This should be done invariably ;
we do not even except those edges which happen to have shadows falling
on them ; in such cases, however, this margin, instead of being left quite
white, which would have a harsh appearance, may be slightly subdued.
The difficulty of achieving this effect, of imparting a clear, regular, un-
broken appearance to these lines of light, seems very formidable, and,
indeed, nearly insuperable. The hand of the colorist may be as steady
and confident as a hand can be, and yet fail to guide the brush, at an
almost inappreciable distance from a straight or circular line, with that
precision and sharpness so requisite for the accurate delineation of this
beautiful effect. We shall, however, explain a novel and effective method
of arriving at this most desirable result.
Suppose the object about to receive the color to be the elevation of a
long flat rod or lever, on the edge of which a line of light is to be left.
Fill the drawing pen as full as it will conveniently hold with tint destined
to cover the rod or lever, and draw a broad line just within, but not touch-
ing, the edge of the lever exposed to the light. As it is essential for the
successful accomplishment of the desired effect that this line of color should
not dry, even partially, until the tint on the whole side of the lever lias
been put on, it will be as well to draw the pen again very lightly over
the same part, so that the line may retain as much tint as possible. Im-
mediately this has been done, the brash, properly filled with the same
tint, is to pass along and join the inner edge of this narrow strip of color,
and the whole surface of the lever filled in. Thus a distinct and regular
346 SHADING AND SHADOWS.
line of light is obtained, and, in fact, the lever, or whatever else the ob-
ject may be, covered in a shorter time than usual. A still more expedi-
tious way of coloring such surfaces is to draw a second line of color along
and joining the opposite edge of the lever or other object, and then expe-
ditiously to fill in the intermediate space between the two wet lines, by
means of the brush. In this manner a clear uniform outline to the tint is
obtained, which could not be effected in any other way. As celerity- in
the movements of the colorist is very necessary to carry out properly this
method of leaving a light edge to the boundaries of flat surfaces, and as
confidence in possessing the requisite ability to perform it must precede
success, a little practice is desirable before essaying it on any drawing
of importance. The blades of the drawing pen must not be sharp, and the
pen should be used with great precaution and delicate lightness, otherwise
the blades will cut more or less the paper and leave their course visible
an unsightly betrayal of the mechanical means employed to obtain such
regularity in the coloring. Flat circular surfaces may be treated in the
same manner, by using the pen-compass in place of the drawing pen.
When such surfaces are rather extensive, it will be judicious to color them
in halves, or in quadrantal spaces, taking great care, when joining the
parts together, that they may overlap or fall short of each other as little
as possible. The appearance of these junctions may be obliterated by
slightly washing them, or by going over the whole surface with a very
light tint, and, in passing, gently rubbing the seams with the brush. By
similar means the line of light on a cylinder, shaft, or other circular body,
may be beautifully expressed. To indicate this light with perfect regu-
larity is highly important, for if a strict uniformity be not maintained
throughout its whole length, the object will look crooked or distorted.
After having marked in pencil, or guessed the position of the extreme
light, take the drawing pen, well filled with a just perceptible tint, and
draw a line of color on one side the line of light, and almost touching it ;
then with the brush, filled with similar light tint, join this line of color
whilst still wet, and fill up the space unoccupied by the shade tint, within
which the very light color in the brush will disappear. Let that part of
the object on the other side of the line of light be treated in the same way,
and the desired effect of a stream of light clear and mathematically regu-
lar will be obtained. The effectiveness and expedition of this method will
be most obvious in coloring long circular rods of small diameter, where
the want of accuracy is more immediately perceptible. The extreme
depth of shade, as well as the line of light in such rods may, with great
effect, be indicated by filling the pen with dark shade tint, and drawing it
SHADING AND SHADOWS. 347
exactly over the line representing the deepest part of the shade. On
either side and joining this strip of dark color, another, composed of
lighter tint, is to be drawn. Others successively lighter are to follow,
until, on one side, the line of the rod is joined, and on the other the
lightest part of the rod is nearly reached. The line of light is then to be
shown, and the faint tint used on this occasion spread with the brush
lightly over the whole of that part of the rod situated on either side of
this line, thus blending into smooth rotundity the graduated strips of tint
drawn by the pen.
In all tinted drawings the more important parts, whether the machinery
or the structure, should be more conspicuously expressed than those parts
which are mere adjuncts. Thus, if the drawing be to explain the construc-
tion of the machine, the tint of edifice and foundations may be kept lighter
and more subdued than those of tne machine ; and if the machine, on the
contrary, be unimportant, it may be represented quite light, or in mere
outline, whilst the edifice is brought out conspicuously.
As has been stated, there are two methods of shading, by flat tints and
by softened tints; but in the work, the "Engineer and Machinist's Draw-
ing Book," from which the preceding Treatise on Shading and Shadows
has been taken, the process of coloring by flattened tints, or superposition
of tints, is ignored, and the method confined to that of softened tints ; and
very strong objection is made to washing, although " a little washing of the
most distant parts is allowable " (page 345). By this process recommended
for coloring, a distinct and even an artistic drawing of architectural or
mechanical objects could undoubtedly be made by a skilful draughtsman ;
but by the inexpert, we think that the process of coloring by flat tints will
be found much the more simple and readier way of producing a respect-
able drawing ; and the method given pages 330 and 331 applies equally
well to drawings in color.
"With regard to washings, the soft sponge is an implement not to be
neglected by the draughtsman ; it is an excellent means of correcting
great errors in drawing, better than rubber or an eraser, but care of course
must be taken to wash and not to rub off the surface, and for errors in
coloring washing is almost the only corrector. In removing or softening
color on large surfaces, the sponge is to be used, and for small spots the
brush. Whilst coloring, keep a clean, moist brush by you : it will be ex-
tremely useful in removing or modifying a color.
The immediate eifect of washing is to soften a drawing, an effect often
very desirable in architectural and mechanical drawings, and the process is
simple and easily acquired ; keep the sponge or brush and water used
348 SHADING AND SHADOWS.
clean ; after the washing is complete take up the excess of moisture by the
sponge or brush, or by a piece of clean blotting paper. Where great
vigor is required, let the borders of the different tints be distinct ; if the
strips are narrow, the effect in comparison with that obtained by softened
tints is as a line engraving compared with a mezzotint.
With regard to the colors to be used to represent different materials,
there are no conventional tints, none that draughtsmen have agreed upon
to be uniformly used, and we think that some improvement can be made
on those before recommended. India ink, it has been observed, is not a
black, but a brown, making with a blue a greenish cast, and with gamboge
a smear. A colored drawing is better without the use of India ink at all ;
any depth of color may be as well obtained with blue as with black ; there
is also an objection to gamboge, that it is gummy, and does not wash well,
and the effect is better obtained with yellow ochre. For the reds, the mad-
der colors are the best, as they stand washing.
For the shade tint of almost every substance a neutral tint, Payne's
grey, or madder brown subdued with indigo ; for the local color, or what
has improperly been designated as the body color, for cast-iron use a wash
of indigo, and for wrought-iron, of cobalt blue ; for sections of these sub-
stances, pure indigo for cast-iron, and cobalt blue for wrought-iron, with
hatchings of deep tints of the same color. For shadows on brass or cop-
per use madder brown ; local color for brass, yellow ochre or Indian yel-
low ; and for copper, a light wash of Venetian or light red ; for sections,
pure tints or washes, with deep hatchings of the same.
For building material, as granite or brick, imitate the color, but float
on the tints, leaving it in patches, and soften by washes of clean water, or
by some local tints which suit the material. The outer walls of houses, in
section, are often colored in a simple tint of carmine or madder brown.
For wood of a light color use a tint of burnt sienna ; for dark woods, a
mixture of burnt umber and sepia ; and for the shadows madder brown.
Plates LXXXVHI. and LXXXIX. are illustrations in chromo-lithogra-
phy from colored drawings. It has not been possible to express the effect
given by hand, but they may serve in a measure as models, with the text
as a guide. Every one wishing to become a draughtsman should make a
scrap-book or collection of such drawings as he can from time to time pick
up, to serve him as guides, study the effects which are given in water-color
drawings ; in the architectural department especially we know of nothing
cheaper and better than the illustrations of some of the London papers ;
whether in ink or color, they afford capital studies of design and of execu-
tion.
TOPOGRAPHICAL DRAWING. 34:9
TOPOGKAPHICAL DKAWING.
TOPOGRAPHICAL DRAWING is the delineation of the surface of a locality,
with the natural and artificial objects, as houses, roads, rivers, hills, etc., upon
it in their relative dimensions and positions ; giving, as it were, a miniature
copy of the farm, field, district, etc., as it would be seen by the eye moving
over it. Many of the objects thus to be represented can be defined by
regular and mathematical lines, but many other objects, from their irregu-
larity of outline, it would be very difficult thus to distinguish ; nor are
the particular irregularities necessary for the expression. Certain con-
ventional signs have therefore been adopted in general use among drafts-
men, some of which resemble, in some degree, the objects for which they
stand, whilst others are purely conventional. These signs may be ex-
pressed by lines, or by tints, or by both. "We commence with those in
lines, and in the latter part of our treatise, finish with examples in color.
Plate XC. fig. I, represents meadow or grass line, the J^r^-^* rr*.
short lines being supposed to represent tufts of grass ; fS'^^st^^
the base line of these tufts should always be parallel tb < 5? i *w-S--? p
the base of the drawing, no matter what may be &e^!i>C'W^*^
shape of the enclosure. Fig. 1 expresses the same thing M^.^J*^-*^-**-
on a larger and coarser scale.
Fig. II. represents an orchard ; fig. III. a forest or clump of forest trees.
In both these examples, the trees are represented in elevation ; this is a
very common method of representation, but not consonant with the other
parts of the plan. It seems better that trees should be represented in
plan, as in fig. TV. Orchards may be represented thus (fig. 2), and forests,
on a larger scale, by a sort of distinctive foliage, according to the kinds of
trees ; thus fig. 3 may represent chestnut, fig. 4 oak, fig. 5 pine and fir.
When trees occur upon a hill-side, the shading lines of the hill-side should
be interrupted to receive the body of the tree, but not its shadow, which
may be drawn independently of them when the slope is slight, but when
350
TOPOGKAPHICAL DRAWING.
it is steep the shadows may be omitted, and the trees shaded nearly as
dark as that of the slope, but the foliage should be represented rather
sparse.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. V. represents a house and cultivated ground ; the walks and roads
are in white, the buildings are marked by diagonal lines. The cultivated
land by parallel rows of broken and dotted lines, supposed to be furrows.
Sometimes signs are used to represent the crops.
Fig. VI. represents marsh land, water and bog. Fig. VII., a river
with mud and sand banks. Sand is represented by fine dots made with
the point of the pen ; mud in a very similar way, but the dots should be
much closer together.. Gravel is represented by still coarser dots, and
stones by irregular angular forms, imitating their appearance, as seen from
above.
Fig. VEIL represents a bold shore bounded by cliffs. "Water is almost
invariably represented in the same way, except in connection with bogs
(fig. VI.), by drawing a line parallel to the shore or coast, following its
windings and indentations, and as close to it as possible; then another
parallel a little more distant, a third still more so, and so on. Small ponds
are sometimes represented by parallel horizontal lines, but usually by the
curved lines of shore. Brooks, and even rivers, when the scale is small,
are represented by one or two lines. The direction of the current is shown
by arrows.
Fig. 6 represents a turnpike. If the toll-bar and marks for a gate
be omitted, it is a common highway. Fig. 7 represents a road as sunk or
cut through a hill. Fig. 8, one raised upon an embankment. Fig. 9 is a
Fig. 6.
Fig. 7.
Fig. 8.
Fig. 9.
railroad, often represented without the cross-tie, by two heavy parallel
lines, sometimes by but one.
TOPOGRAPHICAL DRAWING. 351
Fig. 10 represents a bridge with a single pier. Fig. 11, a swing or
JL
Fig. 10.
Fig. 11.
Fig. 12.
Fig. 13.
Fig. 14
draw bridge. Fig. 12, a suspension bridge, and
fig. 13 a ford. Fig. 14, a lock of a canal. Ca-
nals are represented like roads, except that in
the latter the side from the light is the shaded
line, in the former, the side to the light.
Fig. 15 represents dwellings, or edifices of Fig. is. Fig. ie.
any sort ; they are often made distinctive of their purpose by some small
prefix, as a pair of scales for a court-house, an elevation of a sign-post for
a tavern, a letter for a post-office, a horseshoe for a smithy, a small water-
wheel for a water-mill, and a chimney for a steam-mill.
Fig. 16 represents a church or cathedral ; this is sufficiently expressed
by its plan ; but usually, churches are represented according to their own
plan, with the distinctive prefix of a cross or a steeple.
The localities of mines may be represented by the signs of the planets,
which were anciently associated with the various metals, and a black circle
for coal. Thus Mercury, ? Copper, T? Lead, D Silver, O Gold, $ Iron,
It Tin, Coal.
On the Representation of Hills. The two methods in general use for
representing with a pen or pencil the slopes of ground, are known as the
Fig. IT
Fig. 13.
352
TOPOGRAPHICAL DRAWING.
vertical and the horizontal. In the first (fig. 17), the strokes of the pen
follow the course that water would take in running down these slopes.
In the second (fig. 18), they represent horizontal lines traced round them,
such as would be shown on the ground by water rising progressively by
stages, 1, 2, 3, 4, 5, 6, up the hill. The last is the most correct represen-
tation of the general character and features of the ground, and when ver-
tical levels or contours have been traced by level at equal vertical dis-
tances over the surface of the ground, they should be so represented ; or
when, by any lines of levels, these contours can be traced on the plans
with accuracy, the horizontal system should be adopted ; but where, as in
most plans, the hills are but sketched in by the eye, the vertical system
should be adopted, it affords but proximate data to judge of the slope,
whereas, by the contour system, the slope may be measured exactly. It is
a good maxim in topographical drawing, not to represent as accurate any
thing which has not been rigorously established by surveys. On this
account, for general plans, when the surface of the ground has not been
levelled, nor is required to be determined Avith mathematical precision,
we prefer the vertical to the horizontal system of representing slopes.
On drawing hills on the vertical system, it is very common to draw
contour lines in pencil as guides for the vertical strokes. If the horizontal
lines be traced at fixed vertical intervals, and vertical strokes be drawn
between them in the line of quickest descent, they supply a sufficiently ac-
curate representation of the face of the country for ordinary purposes. It
is usual to make the vertical strokes heavier the steeper the inclination,
and systems have been proposed and used, by which the inclination is
defined by the comparative thickness of the line and the intervening
spaces.
In describing ground with the pen, the light is generally supposed to
descend in vertical rays, and the illumination received by each slope is di-
minished in proportion to its divergence from the plane of the horizon.
Thus in fig. 19, it will be seen
that a horizontal surface receives
an equal portion of light with the
inclined surface testing upon it,
and as the inclined surface is of
Fls 19 . greater extent, it will be darker
than the horizontal in proportion to the inclination and consequent in-
crease of the surface, and on this principle varied forms of ground arc
represented by proportioning the thickness of stroke to the steepness of
the slope.
TOPOGRAPHICAL DRAWING.
353
In the German system as proposed by Major Lehmann, of representing
the slopes of ground by a scale of shade, the slope at an angle of 45, as
reflecting its light horizontally, is supposed to be the greatest ever required
to be shown, and is represented by black, whilst the horizontal plane re-
flecting all rays upward is represented by white, 'and the intermediate
slopes by different proportions of black in the lines to white in the spaces
intervening. " We have not thought it necessary to give an illustration of
this scale of shade, as it does not discriminate between slopes of greater
inclination than 45, preferring the modification as proposed for the IT. S.
Coast Survey, adapted to the representation of all necessary slopes, and
consonant with the demonstration, fig 19. Fig. 20 represents this scale of
shade tabellated, the following are the proportions of black and white for
different inclinations, and the construction may be
easily understood from fig 20. Thus form eight paral-
lel rectangles according to the number of slopes to
be represented ; divide each of these rectangles into
eleven parts, then the proportion of white to black in a
slope of 2i will be to make one of these parts black ;
of 5 two parts, of 10 three, and so on. Now thicken
the lines according to this proportion, and copy the
strokes till the hand becomes habituated to their for-
mation, and the eye so practised, that the graduation for all practical pur-
poses may be performed without direct reference to the scale.
On Drawing Hills ly Contour s.^-Draw first the curves which have
been traced on the ground by levels, and these should be distinguished
from the other lines by color, as red, or by size of lines. It should be ob-
served that whatever point has been actually established by survey, it
should not be confounded with sketching by eye. If there are no such
lines, but it is merely intended to sketch the hills as in the usual vertical
style, lay off the curves at equal vertical intervals, say 10, 20, 50 or 100
feet, according to the scale, and then proceed to fill in. The ground between
23
354:
TOPOGRAPHICAL DRAWING.
these fixed curves or sections, is supposed to slope uniformly. Divide the
space therefore equally, and draw within each set of curves as many lines
as may be suited to the scale of the map, and the vertical intervals be-
tween the curves. Draw the lines with firmness, and let them have a
length varying from one to three fourths of an inch, according to the great-
er or less degree of the slope. When the hill is steep the frjies should be
short and heavy, growing longer and lighter as the inclination becomes
less. The lines should nearly touch each other, so as to appear almost
consecutive, but not overlap, nor with a determinate interval between
their ends. Fig. 21 represents the half of the hill, fig. 18, and at double
scale, completed by drawing the intermediate contour lines.
Fig. 21.
Drawing hills by contours is of comparatively late introduction, and
is generally practised abroad, but little used here ; it is more difficult for
the draughtsman, and no more expressive of the features of the ground
than the vertical system, and has little to recommend except where actual
lines have jeen traced, and it becomes a record of facts. Certain lines
in pencil are necessary for the ^i'oper drawing according to the vertical
system, but when the drawing is complete, an implied line is merely left.
Hills are much more effectively expressed by the brush than the pen, and
much more readily, of which illustrations will be given further on.
In our list of conventional signs we have given but few, and I
only the most prominent. It is useless to tax the memory with many, as
the purposes for which an edifice or locality is intended will supply some
characteristic by which they are easily distinguished ; as in case of mills
already given, or as in case of a graveyard by a tombstone, a quarry by
a stone-hammer, a battle field by crossed swords, &c. "When there is no
obvious characteristic, the positions may be lettered or numbeix.l and ex-
plained bv marginal notes, if there be not room on the plan in its appro-
priate locality.
TOPOGRAPHICAL DEAWING. 355
PLOTTING.
Plotting is the making of the plan on paper from the measurements
taken in the field.
The rough sketch is usually made in the field-book, that is, the book
kept in the field, in which all the steps or observations of the survey are
noted on the spot. The field-book is generally ruled with a middle col-
umn, from one half to one inch in width. This middle column is intended
to represent the station line itself, and all lines crossing the station line
are not drawn directly across the middle column, but arrive at one side
and leave it on the other, at points precisely opposite. The middle column
is reserved entirely for the angles and measures, made in direction of the
station line. All measurements of offsets or angles other than those on
the direct line are entered in the marginal spaces at each side of the middle
column, according to the side of the station line on which they are taken.
The stations are marked thus 0/and the notes commence at the bottom of
the page.
Scales. The choice of the scale for the plot depends in a great measure
on the purpose for which the plan is intended. It should be large
enough to express all the details which it is desirable, modified by the cir-
cumstances, whether the map is to be portable, or whether space can be
afforded for the exhibition of a l^rge plan. We must adapt our plan for
the purposes which it is intended to illustrate, and the place it is to oc-
cupy.
Plans of house lots are usually named as being so many feet to the
inch ; plots of farm surveys, as so many chains to the inch ; maps of sur-
veys of States, as so many miles to the inch, and maps of railway surveys,
as so many feet to the inch, or so many inches to the mile.
For farm surveys, if of small extent, two chains to the inch is a con-
nt scale ; for larger farms three chains to the inch. This last seal" is
that prescribed by the English Tithe Commissioners for the first-cL....;
maps. One acre laid out in the form of a square, to the scale of
1 chain to the inch occupies 3.16 inches square.
U " " " 2.10 "
2 1.58 "
3 " " " 1.05 " "
and so on.
356 TOPOGRAPHICAL DRAWING.
Knowing how much is the area of the ground to be plotted, if the form
is square we can easily determine the side of the square occupied, by mul-
tiplying the square root of the area in acres by 3.16, and dividing the
product by the number of chains to the inch in the scale assumed. Thus
if 50 acres were to be plotted in a square, to the scale of 3 chains to the
OO Q_l
inch = V50 = 7.07. 7.07 x 3.16 = 22.34 -~- 7.45 inches, side
o
of the square of the plot on a scale of 3 chains to the inch. This rule will
assist the draughtsman in selecting a scale for figures not very irregular in
form.
State surveys are of course plotted on a smaller scale than those of
farms. On the U. S. Coast survey all the scales are expressed fractionally
and decimally. The original surveys are generally on a scale of one to ten
or twenty thousand, but in some instances the scale is larger or smaller.
The public surveys embrace three general classes: 1. Small harbor
charts. 2. Charts of bays, sounds, &c. 3. General coast charts.
The scales of the first class vary from 1 : 5,000 to 1 : 60,000, according
to the nature of the harbor and the different objects to be represented.
The scale of the second class is usually fixed at 1 : 80,000. Prelimi-
nary charts are, however, issued of various scales, from 1 : 80,000 to
1 : 200,000.
Of the third class the scale is fixed at 1 : 400,000 for the general chart
of the coast from Gay Head to Cape Henlopen, although considerations
of the proximity and importance of points on the coast may change the
scales of charts of other portions of our extended coast.
On all plots of large surveys, it is very desirable that the scales adopted
should bear a definite numerical proportion to the linear measurement of
the ground to be mapped, and that this proportion should be expressed
fractionally on the plan, even if the scale be drawn or expressed some
other way, as chains to the inch. The decimal system has the most to
recommend it, and is generally adopted in government surveys.
For Eailroad Surveys, the New York general railroad law directs the*
scale of map which is to be filed in the State engineer's office, to be 500
feet to one-tenth of a foot, 1 : 5,000.
For the Canal Maps, a scale of 2 chains to the inch, 1 : 1584 is em-
ployed. In England, plans and sections for projected lines of inland com-
munication, or generally for public works requiring the sanction of the
Legislature, are required, by the " standing orders," to be drawn to scales
not less than 4 inches to the mile, 1 : 15,840, for the plan, and 100 feet to
the inch, 1 : 1,200, for the profiles.
TOPOGRAPHICAL DRAWING. 357
In the United States engineer service, the following scales are pre-
scribed :
General plans of buildings, 10 feet to the inch, 1 : 120
Maps of ground with horizontal curves 1 foot apart, 50 " " 1 : 600
Topographical maps 1 miles square, . . .1 mile to 2 feet, 1 : 2,610
" comprising 3 miles square, .1 " 1 foot, 1 : 5,280
" " " bet. 4 and 8 miles, 1 " 6 in., 1 : 10,560
" " 9 miles square, .1 " 4 " 1 : 15,840
Maps not exceeding 24 miles square, . . .1 " 2 " 1 : 31,680
" comprising 50 " " . . ' . 1 " 1 " 1 : 63,360
100 " " ... .1 " i " 1 : 126,720
Surveys of roads and canals, . . . .50 feet to 1 " 1 : 600
The description of various scales and the principles of their construc-
tion will be found at pp. 17 and 18, to which the reader is referred.
In plotting from the field book, the first lines to be laid down are the
outlines or main lines of the survey. If the survey has been made by
triangles, the principal triangles are first laid down in pencil by the inter-
section of their sides, the length being taken from the scale and described
with compasses ; if the lines are longer than the reach of the compasses,
or the extent of the scale, lay off the length on any convenient line, and
measure and describe with beam compasses. The principal triangles being
laid down, other points will be determined by intersections in the same
manner as measured on the ground. In general, when the surveys have
been conducted without instruments to measure the angles, as the compass
or theodolite, the position of the points on paper are determined by the
intersection and construction of the same lines as has been done in the
field.
Surveys are mostly conducted by measuring the inclination of lines to
a meridian or to each other by the compass, or by the theodolim In the sur-
veys of farms, where great accuracy is not required, the compass is most
used. The compass gives the direction of a line in reference to the magne-
tic meridian. The variation from the true meridian, or a direct north and
south line, varies considerably in different parts of the country. In 1840,
the line of variation in which the needle pointed directly north, passed
in a nearly straight direction from a little west of Cape Hatteras,
N. C., through the middle of Virginia, about midway between Cleve-
land, Ohio, and Erie, Pa., and through the middle of lakes Erie and
Huron. At all places east of this line the variation is westerly, that is, the
needle points west of the line north. West of this line the variation is
'easterly.
358
TOPOGRAPHICAL DRAWING.
Fig. 22 represents the field notes of a survey by compass. Fig. 23 a
plot of the same, with the position of the protractor in laying
off the angles. In this way of plotting, a meridian is laid off
at the intersection of each set of lines. Sometimes the angles
are plotted directly from the determination of the angle of de-
flection of two courses meeting at any point, without laying
down more than one meridian : Figure 24. When the first
3.23
-(5)-
3.54
cn
-(4)-
2.22
-(3)-
1.29
-(2)-
2.78
Fig. 23.
letters of the bearing are alike, that is, both K. or both S., and the last let-
ters also alike, both E. or both W., the angle of deflection C B B' will be
the difference of the bearings, or, in this instance, 20.
w-
N
Fig. 24.
' Fig. 25. When the first letters are alike and the last different, the
angle C B B' will be the sum of the two bearings.
TOPOGRAPHICAL DRAWING.
Eig. 26. When the first letters are different and the last alike, sub-
tract the sum of the bearings from 180 for the angle C B B' : when both
the first letters and last are different, subtract their difference from 180
for the angle.
Instead of drawing a meridian through each station, or laying off the
angle of deflection, by far th^easiest way is to lay off but a single meridian
near the middle of the sheet ; lay off all the bearings of the survey from
some one point of it as shown in fig. 27, and number to correspond
with the stations from which
the bearings are taken, and
then transfer them to the
places where they are wanted
by any of the instruments used
for drawing parallel lines. For
the protracting of the rough
plan, sheets of drawing paper
can be bought with protractors
printed on them. When the
plans are large, it is often
convenient to lay out two or
three meridians on different
parts of the sheet and lay off
the bearings of lines adjacent to each meridian upon them.
In plotting from a survey by a theodolite or transit, it is generally
usual to lay off the angles of deflection of the different lines as taken
in the field, plotting all the tie lines as corrections.
When the plot of a survey does not close, that is, come together, or
return to the point of commencement, as it seldom does exactly, it may
be corrected o.r forced ; but first be sure that the bearings and distances as
given in the field book are laid down accurately, and then proceed to cor-
rect as follows : thus, let A B D E,
fig 28, be the boundary lines plotted
according to the notes, and suppose
the last course conies to E instead
of ending at A as it should. Sup-
pose also that there is no reason to
suspect any error more in one line
than another, that the measures
and bearings of all are equally cer- FJ-. 2^-
360
TOPOGRAPHICAL DRAWING.
tain; then the inaccuracy must be distributed among all the lines in
proportion to their length. Each point, B, C, D, E, must be moved in a
direction parallel to E A, by a certain distance. Thus add together the
length of all the lines, and this sum is to the line A B, as the error A E is
to the correction B B' ; for the next point, the whole sum is to A B, B C, as
the error is to the correction, E C ; and so o^ ; obtaining the second term
of the proportion by adding consecutively the different lines. This calcu-
lation may be much simplified by the use of the sector, according to the
rule given for finding a fourth proportional (p. 23). Take the error A E
from the plan, and open the sector until this quantity becomes the trans-
verse distance of the first term or sum of the lines; then the distance be-
tween the points corresponding to the consecutive sums will be the corre-
sponding error.
The best way of correcting errors and of plotting a survey, whether
made by compass or by theodolite, is by balancing the survey, or correct-
ing the latitudes and departure of the courses so that they shall be equal.
For the method of doing this, we refer to any of the recent works on sur-
veying. From Gillespie on Land Surveying, we have taken most of the
preceding on the plotting of angular surveys, and the following para-
graph on balancing.
Ita.
Total Latitude
from Sta. 1.
Total Departure
from Sta. 1.
1
0.00
0.00
2
+ 2.21 N.
+ 1.55E.
f
+ 2.36 N.
+ 2.S3E.
4
+ 1.15N.
+ 4.69 E.
5
- 1.78 S.
+ 2.69 E.
1
0.00
0.00
This table represents the
survey as given, fig. 22, bal-
anced.
To plot from this table,
draw a meridian through the
point taken for station 1, as
in fig. 29. Set off upward
from this along the meridian
the latitude 2.21 chains north
to A, and from A to the right or E. set off the departure 1.55 chains. This
gives the point or position of Station 2, join 1 and 2. From A again set off
upward 2.36 chains, and from B to the right perpendicularly set off
2.83 chains, which gives position of Station 3, join 2 and 3 : and so pro-
ceed setting off North latitudes upwards, and South downwards, East
TOPOGRAPHICAL DRAWING.
361
departures perpendicularly to the right, "and West perpendicularly to the
left.
In balancing surveys made by a theodolite, a meridian is assumed,
generally one of the lines of the survey. The most convenient will be
some long line of which ^he survey lies all to one side.
Fig 30.
The advantages of this method of plotting are
its accuracy, rapidity, the impossibility of an er-
ror in one point affecting the others, and the cer-
tainty of coming together.
The above explains the method of plotting the
main lines of the survey ; the filling in is from
points established from these main lines, either
by the construction of triangles, by measure, or
by angles, or by perpendiculars. In case of un-
important lines, as the crooked brook for instance,
fig. 30, offsets are taken to the most prominent
angles, as, a, a, a, and the intermediate bends are
sketched by eye into the field book. In copy-
ing them on the plan a similar construction is
adopted.
The most rapid way of plotting the offsets, is
by the use of a plotting and offset scale, fig. 31
the one being fixed parallel to the line A B from
which the offsets are to be laid off, at such a dis-
tance from it, that the zero line on the movable
scale coincides with it, whilst the zero of its own
scale is on a line perpendicular to the position of
the station A from which the distances were meas-
ured. It is to be observed that in the field book
all the measures are referred to the point of be- Fig. si.
ginning on any one straight line. Having placed the plotting scale, move
the offset scale to the first distance by the scale at which an offset has
been taken, mark off now on the offset scale the length of the offset on
362 TOPOGRAPHICAL DRAWING.
its corresponding side of the liner. Proceed then to the next distance, es-
tablishing thus repeated points, join the points by lines as they are on the
ground.
The plotting and offset scale must of course be of the same scale as the
rest of the drawing, on which account it may not always be possible to
obtain such scales adapted to those of the plan ; but they may be easily
constructed of thick drawing paper or pasteboard.
When a great deal of plotting to one scale is necessary, as in govern-
ment surveys, the offset scale may be made to slide in a groove upon the
plotting scale.
In protracting the triangles of an extended trigonometrical survey in
which the sides have been calculated or measured, it is better to lay down
the triangles from the length of their sides rather than by measuring the
angles, because measures of length can be taken with more accuracy from
a scale, and transferred to the plan with more exactness than angles can
be pricked off from a protractor ; but for ordinary surveys, the triangula-
tion is most frequently and expeditiously plotted by the means of a pro-
tractor.
The outlines of the survey having been balanced and plotted in, and
the subsidiary points, as established by offsets and by triangles, the filling
in of the interior detail is done by copying from the field book the natural
features of the ground, in their appropriate position, and according to the
conventional signs already described.
In many surveys, as of roads, rivers, canals, and boundaries, the plot
to be made is but a single line, with a few of the nearest local objects on
either side. In some instances the angles at each intersection are taken
merely with reference to the two lines forming the angle, and are therefore
to be plotted as shown in fig. 23, by laying the protractor at each inter*
section ; but in other instances, by the method of surveying, the directions
of all lines are referred to the first as meridian, or if the survey is exten-
sive, to some number of lines, and the plotting is then expeditiously per-
formed as in fig. 27. In this system of surveying, instead of fixing the
vernier at zero, for every back angle the preceding forward angle is re-
tained except for those lines intended as meridians.
Surveys for railways, like those above, are of lines extensive in length
but of very little width. In the surveys of preliminary or trial lines, the
curves at the intersection of lines are seldom introduced ; an(t in plotting
1 it is but the usual method of plotting surveyed lines, by either of the
methods, according as the survey may have been conducted, with the the-
odolite, or with the compass.
TOPOGRAPHICAL DRAWING.
363
In plotting curves on a line of location, lay off from the intersection
f tangents, as C, fig. 32, the distance of the tangent points A and B, and
find the centre O of the curve, by the erection of perpendiculars to these
Degree.
Radii, ft.
Central Ordinate.
1
5T29.65
0.218
20
286493
0.436
30
1910.08
0.655
40
1432.69
0.873
50
1146.28
1.091
63
955.8T
1.309
TO
819.02
152S
80
T16.7S
1.74G
90
63T.27
1.965
100
573.69
2.183
two points, or if the ra-
dius of the curve is
known, by describing
arcs with this radius
from the same points.
Railway curves are de-
signated by degrees or
according to the angle of deflection made by two ' chords of 100 feet
each.
Two curves often succeed each other having a common tangent, at the
point of junction. If the curves lie on opposite sides of the common tan-
gent, they form a reversed curve, ABC, (fig. 33,) and their radii may be
the same or different.
If they lie on the same
side of the common
tangent and have dif-
ferent radii, they form
a compound curve, A
BD. When the radii of
curvature are known,
the determination of
the centres is ob-
tained easily, by de-
scribing arcs with the established radii from the tangent points.
If the radii of curvatures are not known, and it is required to plot
a compound or reversed curve which shall be tangent at the points A
and D or C to other straight lines, the change of curvature taking
364r
TOPOGRAPHICAL DRAWING.
plac*e at B, then at A erect a perpendicular to the given line, find
some point on this perpendicular which is equally distant from A and B,
and this point will be the centre of the curve A B ; through this point
and B draw an indefinite line, its intersection by the perpendicular to the '
tangent at D will be the centre for the other potion of the compound
curve ; and its intersection by the perpendicular to the tangent at C will
give the centre for the reversed. The centres of curves tangent to each
other must lie in a straight line, passing through their point of con-
nection.
When the curves are larger than can be described by the dividers or
beam compasses, they can be plotted as shown in geometrical problems,
or points of a curve may be obtained by calculation of their ordinates,
and the curves drawn from point to point by sweeps and variable curves.
Approximately, knowing the central ordinate of the curve between two
points, the central ordinate of one half that curve will be one quarter of
the first. Thus, fig 32, G D is about one quarter of E F, hence by subdi-
visions as many points as are necessary may be obtained ; but it should
be observed, that the greater the number of degrees in the arc, the less
near to the truth the rule.
w 3l.43FcetFattpertiae x/ . Ltvel ^
Fig. 35 represents a plot of a railway line ; in this plot the curve is
represented as a straight line, the radius of curvature being written in.
This method is sometimes adopted when it is desirable to confine the plot
within a limited space upon the sheet, and it is convenient when plotted
thus directly beneath the profile or longitudinal section (fig. 34).
In plotting the section, a horizontal or base line is drawn on which are
laid off the stations- or distances at which levels have been taken ; at these
points perpendiculars or ordinates are erected, and upon them are marked
TOPOGRAPHICAL DRAWING. 365
the heights of the ground above the base, and the marks are joined by
straight lines. To express rock in a cut, it is generally represented by di-
agonal lines; rivers are represented in section by cross lines or 'colored in
blue ; the depth of the sounding in a mud bottom by masses of dots.
Since it would be in general impossible to express the variations of the
surface of the ground in the same scale as that adopted for the plan, it is
usual therefore to make the vertical scale larger than that of the horizontal
lines one, in proportion of 10 or 20 to 1. Thus, if the horizontal scale of
the plan be 400 feet to the inch, the vertical scale would be 40 or 20 feet
to the inch.
For the purpose of facilitating the plotting of profiles, profile paper is
prepared, on which are printed horizontal and vertical lines ; the horizon-
tal lines being ruled at a distance of gV of an inch from each other, every
fifth line being coarser, and every twenty-fifth still heavier than the others.
Each of the spaces is usually considered one foot. The vertical lines are
one quarter of an inch distant from each other, every tenth line being
made more prominent than the others ; these spaces in general represent
a distance of 100 feet, the usual distance between stations on a railroad.
Much time is saved by the use of this paper, both in plotting, and in read-
ing the measurements after they are plotted.
In the plotting of sections afcross the line, which are extended but little
beyond the line of the cut or embankment, equal vertical and horizon-
tal scales are adopted ; these plots are mostly to determine the position of
the slope, or to assist in calculating the excavation. To facilitate these,
cross section paper is prepared, ruled with vertical and horizontal lines,
forming squares of T V of an inch each. Every fifth line in each direction
is made prominent. "When cross sections are extended to show the grade of
cross road, or changes of level at considerable distance from the line of
rail, the same scales vertical and horizontal are adopted as in the longitu-
dinal section or profile.
It will be observed in fig. 34, that the upper or heavy line represents
the line of the rail, the grades being written above ; this is the more usual
way, but sometimes, as in fig. 36, the profile and plan are combined ; that
is the heights and depths above and below the grade line of the road are
366 TOPOGRAPHICAL DRAWING.
transferred to the plan, and referred to the line in plan, which becomes
thus a representation both in plan and elevation.
Cross sections, for grades of cross roads, etc., are usually plotted be-
neath or above the profile ; they may, if necessary, be plotted across the
line when plan and profile are combined.
Besides the complete plans as above, giving the details of the location,
land plans, so called, are required, showing the position and direction of
all lines, of fences and boundaries of estates, with but very few of the
topographical features. The centre line of road is represented in bold
line, and at each side, often in red, are represented the boundaries re-
quired for the purposes of way. In general, a width of five rods is the
amount of land set off, lines parallel to the central line being at a distance
of two and one half rods on each side ; but when, owing to the depth of the
cut or embankment, the slopes run out beyond this limit, the extent is de-
termined by plotting a cross section and transferring the distances thus
found to the plan, and enclosing all such points somewhat within the
limits as set off for railway purposes. These plans are generally filed in
the register's office for the county through which the line passes.
For Railway plans prepared for the English Parliament certain regulations are defined.
The plan must be upon a scale of at least four inches to a mile, and must describe the
line or situation of the whole of the proposed work, and the lands through which the
same will be made, and also any communication to be made with the proposed work. If
the plan is on a scale less than } of an inch to every 100 feet, there must be an additional
plan upon that scale (1 : 4,800) of any building, yard, and of any ground cultivated as a
garden.
The plan to exhibit thereon the distances in miles and furlongs, from one of the ter-
mini, with memoranda of the radii of all curves less than one mile in length, noted on the
plan in chains where the curve occurs. When a tunnel is intern 1 :-u to be constructed, it
must be marked in by a dotted line on the plan.
Each distinct property, divided by any visible boundary from another property, should
have a separate number ; with this exception, that any collection of buildings and grounds
within the curtilage of R building, belonging to one person and in one occupation, may be
tlescril C'l under one m; , V r ; thus, farm house, yn ' &r . TVhcn it is necessary to inter-
pose a ruirnber, a duplic-^ number should be addei thi 1 f><>. ?>a. 'he r.umbei.-! ig should
recommence in every parish.
All lands included within the limit of deviation, shown by lines drawn on plan, and
all lands which those lines touch, must be numbered and described. Public roads, and
private roads if fenced out, should have a separate number. Navigable and mill streams
. aratel;
It is sometimes usual, at the c^mple+ion of a railway, to make plans of
the works as finished ; and, if a profile of the line, to represent the differ-
TOPOGRAPHICAL DRAWING.
367
ent strata or rocks in the cuts, with their dip or inclination. This is more
properly a geological profile ; the different rocks are usually distinguished
by different colors and explained by marginal notes and squares, some-
times by marks, dots, and cross hatchings, as fig. 37, often by color in ad-
dition. Figs I., II., III., IY., represent the primary, secondary, tertiary,
and recent plutonic rocks. Fig. 1 represents the primary fossiliferous
strata ; figs. 2, 3, 4, the secondary, tertiary, and recent strata. In the plots
of geological sections, it is especially requisite that the different strata
should be accurately represented.
In plotting hydrometrical or marine surveys, the depths of soundings
are not expressed by sections, but by figures written on the plan, express-
ing the sounding or depth below a datum line, generally that of high
water. The low water line is generally represented by a single continued
line. The soundings are generally expressed in fathoms, sometimes in feet.
It is usual, in plotting from a field book, to make first but a rough
draft, and then naak^ a fin' hed copy on another sheet. In tL. first, many
lines of construction, balances of survey, and trial lines are drawn which
are unnecessary in the copy ; outlines of natural features are sketched
roughly, but the plotting of surveys, and such lines and points as are to
be preserved in the copy, must be ])]< I ' 1 i accuracy.
Tho "i .m- v.-ay of transferri . . topy, is by snperpo-
fc'aon o. the plan above the sheet intended for the copy, and pricking
through every intersection of lines on the plan and all such points as may
be necessary to preserve. The clean paper should be laid and fastened
smoothly on the drawing board, the rough draft should be laid on smoothly
and retained in it position by weig ! 's. The needle must
be held perpenulcular to the surface of the plan and pressed through both
sheets ; begin at one side and \\ rk with ~ys*em, so as not to prick through
each point but once, nor omit any ; make the important points a trifle the
368 TOPOGRAPHICAL DE AWING.
largest. For the irregular curves, as of rivers, make frequent points, but
very small ones. On removing the plan select the important points, those
denning leading lines ; draw in these, and the other points will be easily
recognized from their relative position to these lines. "When any point
has not been pricked through, its place may be determined by taking any
two established points adjacent to the cjne required, and with radii equal to
their distance, on the plan, from the point required, describing arcs, on
the copy, on the same side of the two points, their intersection, will be the
point desired. In this way, as in a trigonometrical survey, having estab-
lished the two extremes of a base, a whole plan may be copied. For this
purpose the triangular -compasses (p. 27) will be found very convenient.
In extensive drawings it is very common to prick off but a few of the
salient points, and fill in by intersections, as above, or by copying detached
portions on tracing paper, and transferring them to the copy ; the position
of each sketch being determined by the points pricked off, the transfer is
made by pricking through as above, or by transfer paper placed between
the tracing and the copy.
Tracing paper is a thin, transparent paper, prepared expressly for the
purpose of making copies of drawings. Placed above the drawing, every
line shows through, and is traced directly with the pen, in India ink.
These tracings are used mostly to preserve duplicates of finished drawings.
As tracing paper is of too slight a texture to bear much handling, cotton
cloth is prepared and sold under the name of " vellum tracing paper."
When it is necessary to use tracing paper drawings to work by, it is usual
to attach them to sheets of white paper, which serves both to bring out
the lines and to strengthen the paper.
Duplicates of drawings are now very neatly executed, and of course
accurately copied, by the Photographic process, but it is more applied to
mechanical and architectural drawings than topographical.
An accurate and rapid way of tracing, on drawing paper, plans of small
extent, is by means of an instrument called a copying glass. It consists
of a large piece of plate glass set in a frame of wood, which can be in-
clined at any angle in the same manner as a reading or music desk. On
this glass is first laid the original plan, and above, the fair sheet, and the
frame being raised to a suitable angle, a strong light is thrown by reflect-
ors or otherwise on the under side of the glass, whereby every line in the
original plan is seen distinctly through the fair sheet, and the copy is made
at once, in ink, as on tracing paper, and finished while being traced. This
same process, on a small scale, is adopted by putting the plans upon a
pane of glass in a window.
TOPOGRAPHICAL DRAWING. 369
Plans of great extent cannot be conveniently copied by means of the
copying glass. Moreover, being often mounted on cloth, which renders
them opaque, they do not admit of being traced in this way. In such
cases the copy may be made by means of transfer paper. The plan is first
traced in ink on tracing paper or cloth, black leaded or transfer paper is
then placed on the fair sheet, and the tracing paper copy is placed above.
All is steadied by numerous weights along the edges, or by drawing pins
fixed into the drawing board. A fine and smooth point is then passed
over each boundary or mark on the tracing with a pressure of the hand
sufficient to cause a clear, pencilled mark to be left on the fair sheet by
the black leaded or transfer paper. The whole outline is thus obtained,
and afterwards drawn in ink. The copyist should be careful in his manipu-
lations, so as not to transfer any other lines than those required, nor leave
smutches on the fair sheet.
Plans may be copied on a reduced or enlarged scale by means 6"f the
pentagraph, but the more usual way is by means of squares. Draw on
the plan to be reduced, a series of parallel and equi-distant lines, with
others perpendicular to them at similar intervals, thus covering the whole
surface with equal squares. On the clean sheet draw a similar set of
squares, but with their sides to the desired reduced scale ; one-half, one-
third, &c., as the case may be. Then copy into each small square all the
points and lines in the corresponding square on the plan, in their true po-
sition relative to the sides and corners of the square, reducing each
distance, by proportional dividers, or by eye as may be necessary, in the
given ratio. In reducing by the camera lucida, squares on the plan are
brought apparently to coincide with squares in the copy, and the details
as seen through the prism of the instrument, are then filled in with the
pencil. This instrument is used in the U. S. Coast Survey office, but it
cannot reduce smaller than one-fourth without losing distinctness, and it is
very trying to the eyes.
Finishing the Plan or Map. In general, in topographical as in ar-
chitectural and mechanical drawings, the light is supposed to fall upon the
surface in a diagonal direction from the upper, left-hand corner. This rule
is not uniform : by some draughtsmen the light is introduced at the lower
left, and hills are mostly represented under a vertical light, although the
oblique adds more to the picturesque effect. The plan is usually so drawn
that the top may represent the north, and the upper left-hand corner is
then the north-west.
In inking in, commence first with the light lines, since a mistake in
these lines may be covered by the shade lines. Describe all curves which
24
3<0 TOPOGRAPHICAL DRAWING.
are to be drawn with compasses or sweeps before the straight lines, for it
is easier to join neatly a straight line to a curve than the opposite. Ink
in with system, commencing say at the top ; ink in all light lines running
easterly and westerly, then all light lines running northerly and southerly,
then commence in the same way and draw in the shade lines. It will of
course be understood that elevated objects have their southern and eastern
outline shaded, whilst depressions have the northern and western ; thus
in conventional signs roads are shaded the opposite to canals. Having
inked in all lines that are drawn with a ruler or described with compasses,
commence again at one corner to fill in the detail, keeping all the rest of
the plan except what you are actually at work upon covered with paper,
to protect it from being soiled. The curved lines of brooks, fences, &c.,
are sometimes drawn with a drawing pen, sometimes with a steel pen or
goose quill. The latter are generally used in drawing the vertical lines
of hills.
Boundary lines of private properties, of townships, of counties, of
states, <fcc*., are indicated by various combinations of short lines, dots and
crosses, thus :
All plans should have meridian lines drawn on them, also scales. Plate
XCI. shows some designs for meridians. In these diagrams it will be
observed that both true and magnetic meridians are drawn ; this is desira-
ble ,when the variation is known, but in many surveys merely the magnet-
ic meridian is taken ; in these cases this line is simply represented with
half of the barb of the arrow at the north point, and on the opposite side
of the line from the tme rneridan. Scales are drawn or represented in va-
rious forms, as may be seen in the following plates, or the proportion of
the plan to the ground is expressed decimally, as the number of feet,
chains, etc., to the inch.
Lettering. The style in which this is done very much affects the gen-
eral appearance of the plan. Great care must be taken in the selection
and character of the type, and in the execution. The usual letters are the
R o M A x .
ABCDEFGHIJKLMNO
PQRSTUVWXYZA
TOPOGRAPHICAL DRAWING. 371
SMALL ROMAN.
abcdefghijklmnopqrstuv
wxyz,;:.
1234567890
ITALIC.
ABCDEFGHIJKL
MNQPQRSTUYW
abcdefghijklmnopqrst
uvwxyz;:
GOTHIC, OR EGYPTIAN.
ABCDEFCHIJKLMN
OPQRSTUVWX
&
372 TOPOGRAPHICAL DRAWING.
ABCDEGHJKLMORSTUWY
abcdefghijklmnopqrstuvwxy
a fe @ t g 1 1 m p E> s tm w
ABGDE&HJKLMORSTTJWXY
ABCDEFGHIJKLiNOPQRSTDfim
ABCDEGHJKMORSTUWXY
abcdefghijklmnopqrstuvwxyz
TOPOGRAPHICAL DRAWING.
OLD ENGLISH.
373
OLD ENGLISH SCRIBE BLACK.
GERMAN TEXT.
374r TOPOGRAPHICAL DRAWING.
ENGLISH OUTLINE. ,
9 c D a
PI. XCII. represents a mechanical method of constructing letters and
figures. This plate should be copied by the draftsman, and on a much
larger scale, by drawing first the system of squares or parallelograms, and
then sketching in the letters ; in this way well formed and proportioned
letters can always be made, and from a selection of alphabets the lettering
may be selected and transferred to the plan.
PL XCIII. are examples of titles, intended merely as an illustration of
the form of letters and their arrangement, the scale being much smaller
than that used on plans, except such as are drawn to a very small scale.
It will be observed that the more important words are made in promi-
nent type. The lower part of the title should always contain, in small
character, the name of the party making the survey, and also the name
of the draftsman, with date of the execution of the plan : if the survey
was made some time previous, the date of the survey should be given.
If the plan is compiled from several surveys, the authorities should, if
possible, be given. The lettering of the title should be in lines parallel to
the bottom of the plan, and, in general, the great mass of lettering in the
body of the plan is formed in similar lines ; but curved lines are often not
only essential, but they materially contribute to the beauty of the plan.
Thus on crooked boundaries on outlines of maps, the lettering should fol-
low the general curve of the boundary; also on crooked rivers, lakes,
seas, &c. ; on irregular or straggling pieces of land, in order to show the
extent, connection, or proprietorship thereof, the lettering should follow
the central line of such a tract ; and if pieces of land be very oblong in
form but regular in outline, the lettering will be central in the direction of
the longest side. The lettering of roads, streets, &c., is always in the di-
rection of the line of road. Curved lines of lettering are often introduced
into extended titles to take off the monotonous appearance presented by a
great number of straight lines of writing.
The direction of all lettering should be so as to be read from left to
right. If shades or shadows are introduced to give relief or break up the
monotony, they should be uniform with the rest of the plan.
On the Spacing of Letters. It will be observed that letters vary very
considerably in their width, the I being the narrowest, and the TFthe
TOPOGRAPHICAL DRAWING.
375
widest : if therefore the letters composing a word be spaced off at equal
distances from centre to centre, the interval or space between the letters
will be more in some cases than in others. Tims, in the word
R A I L W A Y
To avoid this, write in first one letter, and then space off a proper interval,
and then write in the next letter, and then space off the interval as before,
and so on, thus,
RAILWAY
When, as frequently happens, the words are very much extended, in order
to embrace and explain a large extent of surface or boundary, and the
space occupied by the letter is small in comparison with the interval, the
disparity of intervals will not be noticed, and the letters may be then laid
off at equal spaces from centre to centre, thus :
A
B
W
When the lines of lettering are curved, the same rules for spacing are to
be observed as above. If the letters are upright, as Eoman or Gothic, the
sides of each letter are to be parallel to the radius drawn to the centre of
the letter, and the bottom and top lines at right angles to it. If the let-
ters be inclined as Italic letters, then the side lines of the letters must be
inclined to the central radial line, as on a horizontal line they are inclined
to the perpendicular.
In laying off letters by equal intervals, it is usual to count the number
of letters in the word, and fix the position on the plan of the central one,
and then space off on each side : this is particularly important in titles,
when it is necessary that many lines should have their extremities at uni-
376 TOPOGRAPHICAL DRAWING.
form distances from the centre line. In laying off the title, we determine
what is necessary to be included in the title, the space it must occupy, the
number of lines necessary, and the style and arrangement of characters to
be used. Thus, if the title were, plan of a proposed terminus of the Har-
lem Railroad at New York, 1857, knowing the space to be occupied, we
can write the title thus :
of
We now draw parallel lines at intervals suited to the character of the type
we intend to employ for the different words. Harlem Railroad is the line
to be made most prominent ; this, calling the interval between the words
one letter, includes 15 letters ; or, if we consider /, with its proper interval,
but half a letter, (which will be found a very good rule in spacing,) 141 ;
hence the centre of the line will be 7^ letters from the beginning, or \ of
the space occupied by the letter H and its interval. Draw a perpendicu-
lar line at the centre, and write in R in such a character as may suit the
position to be filled, and lay off by letters and spaces the other letters.
The line Harlem Railroad is intended to occupy the whole length of
space ; that is, it must be the longest line in the title, and the lines above
and below must gradually diminish, forming a sort of double pyramid.
Proposed Terminus includes 16i letters, the I and interval between the
words being rated as above, we find the centre to be nearly midway be-
ween the words. These words including more letters, and being confined
within less space, must be in smaller character than the preceding ; and as
a further distinction, a different style should be adopted. Having deter-
mined this, we proceed to write in the letters as before, and in the same
way with the other lines, the prepositions as unimportant are always writ-
ten in small type.
TOPOGRAPHICAL DRAWING. 377
jof the.
HARLEM RAILROAD
.at.
NEW YORK
1857_
I
In general it is better that letters should be first written on a piece of
paper, distinct from the plan, as repeated trials may be necessary before one
is arranged to suit the draughtsman. Having formed a model title, it may be
copied in the plan by measures or by tracing and transfer paper. There
are some words, such as Plan, Map, Section, Scale, Elevation, &c., which, as
they are of constant occurrence, may be cut in stencil ; sometimes whole
alphabets are thus cut and words compounded. It will be found very con-
venient for a draughtsman if he makes tracing or copies of such titles as
he meets with, and preserves them as models ; for there is no manipulation
on a plan that contributes more to the effect than good lettering and arrange-
ment of titles, and considerable practice should be expended in acquiring
a facility in lettering, and for the first start, perhaps nothing will be found
more valuable than tracing good examples.
We have treated of mechanical methods by which most persons can
learn to form letters and words ; but it must be borne in mind that the
distances between letters on the plan are only intended to suit the eye ;
if therefore a person accustom himself to spacing, so that his eye is cor-
rect, there will be no necessity of laying off by dividers; in this mode,
such letters as A and V, L and T are brought nearer each other than the
regular interval. In general it may be observed in reference to to the
lettering of Topographical Drawings, stiff letters like those of stencil
should not be introduced, but there should be such variety, incident on
construction by the pen, as may be consonant with the rest of the drawing.
378 TOPOGRAPHICAL DRAWING.
TINTED TOPOGRAPHICAL DRAWING.
We have hitherto treated of the representation of the features of the
country by the pen only, but it may be done full as effectively and much
more expeditiously by means of the brush and water colors, either by
India ink alone, or by various tints, or by the union of both.
The most important colors for conventional tints are, (besides India
ink), Indigo (blue), Carmine (or crimson lake), and Gamboge (yellow),
used separately or compounded. Besides these, Burnt Sienna, Yellow
Ochre, and Vermilion are sometimes used, although the three first are
susceptible of the best combinations, and the others are generally used
alone.
The following conventional colors are used by the French Military En-
gineers in their colored topography. "Woods, yellow using gamboge and
a very little indigo. Grass land, green ; made of gamboge and indigo.
Cultivated land, brown y lake, gamboge, and a little India ink; "Burnt
Sienna " will answer. Adjoining fields should be slightly varied in tint.
Sometimes furrows are indicated by strips of various colors. Gardens are
represented by small rectangular patches of brighter green and Thrown.
Uncultivated land, marbled green and light 'brown. Brush, brambles, &c.,
marbled green and yellow. Heath, furze, &c., marbled green and pink.
Vineyards, purple; lake and indigo. Sands, a light "brawn ; gamboge
and lake ; " Yellow Ochre " will do. Lakes and rivers, light Hue, with a
darker tint on their upper and left hand sides. Seas, dark blue, with a lit-
tle yellow added. Marshes, the Hue of water, with spots of grass green,
the touches all lying horizontally. Roads, brown between the tints for
sand and cultivated ground, with more India ink. Hills, greenish brown ;
gamboge, indigo, lake and India ink. Woods may be finished up by
drawing the trees and coloring them green, with touches of gamboge to-
wards the light, (the upper and left hand side,) and of indigo on the op-
posite side.
In addition to the conventional colo*rs, a sort of imitation of the con-
ventional signs already explained are introduced in color with the brush,
and shadows are almost invariably introduced. The light is supposed to
come from the upper left hand corner, and to fall nearly vertical, but suf-
ficiently oblique to allow of a decided light and shade to the slopes of
hills, trees, &c. The shadow of any object will therefore surround its
TOPOGRAPHICAL DRAWING. 379
lower right hand outlines. After the shadow has been painted, the out-
line of the object is strengthened by a heavy black line on the side oppo-
site the light. The flat tints are first laid on as above, and then the con-
ventional signs are drawn in with a pencil and colored in with appropriate
and more intense tints ; the shadows are generally represented in India ink.
Hills are shaded, not as they would appear in nature, but on the con-
ventional system of making the slopes darker in proportion to their steep-
ness : the summit of the highest ranges being left white. This arrange-
ment, though obviously incorrect in theory, has the advantage of being
generally understood by those not accustomed to plan drawing, and is
also easy of execution. Wash the surface first with the proper flat
tint, trace in with a pencil, outlines ; then lay on in India ink tints pro-
portioned in intensity to the height of the hills and steepness of the
slopes. To soften the tints two brushes are used, one as a color brush, the
other as a water brush : the tints are laid on with the first, and softened
by passing the water brush rapidly along the edges. The water brush
must not have too much water, as it would in that case, lighten the tint
to a greater extent than is intended, and leave a ragged harsh edge. Tints
may be applied in very light shades, one tint over another, with the
boundary of the upper tint not reaching the extreme limit of the tint
below it. When depth of shade is required, it is best produced by appli-
cation of several light tints in succession : no tint is to be laid over the
other until the first is dry, and a little indigo mixed with the India ink im-
proves its color and adds to the richness of effect.
When woods have to be represented, the shading used for the trees
instead of interfering with the shadows due to the slopes, may be made to
harmonize with them, and contribute to the general effect by presenting
greater or less depth, according to the position of the woods on the sides,
or summits of the hills.
An expeditious and effective way of representing hills with brush, a
species of imitation of hills drawn with a pen on the vertical system, is
effected by pressing out flat the brush to a sort of comb-like edge ; draw-
ing this over a nearly dry surface of India ink, and then brushing lightly
or more heavily between the contours, according to the steepness of the
slope, each of the comb-like teeth making its mark.
Kivers and masses of water may be shaded in with a color and water
brush as above, or by superposition of light tints, a shadow may be
thrown from the bank towards the light, and the outline of this bank
strengthened with a heavy black line. The tints are to be in indigo, the
shadows in India ink.
380
TOPOGRAPHICAL DEAWING.
Topographical drawings may be made in water color with but one tint, as
India ink, or ink mixed with a little sepia. The conventional signs are in
imitation of pen drawings, the hills in softened tint, or drawn with the
comb-edged brush, and the rivers shaded with superposed tints.
Most artistic and effective drawings are made of hills as they would
appear in nature, under an oblique light : the sides of the hills next the
light receiving it more or less brilliantly, according as they are inclined
more or less at right angles with its rays, and the shades on the sides re-
moved from the light increasing in intensity as the slopes increase in steep-
ness. This style may be rendered most expressive by a skilful draughts-
man, especially when the character and strike of the hills are favorable to
the direction of the light, but with this style of representation the hills are
generally made to partake more or less of the same character, appearing
almost uniformly steepest on the sides removed from the light. It partakes
therefore more of the artistic character, more difficult to execute, and
conveying information in a more vague manner than by the common to-
pographical conventionalities.
TOPOGEAPHICAL DRAWING. 381
In preparing the paper for a tinted drawing it must be damp-stretched
upon the drawing board in such a manner that the moisture of the color
will not cause undulations or blisters on the surface : this process is pre-
viously, described at page 37. Having prepared a sheet of paper accord-
ing to the directions there given, first draw in the lines in pencil, and af-
terwards repeat them with a very light ink line : a soft sponge well satu-
rated should then be passed quickly over the surface of the drawing, in
order to remove any portions of the ink which would be liable to mix with
the tint and mar its uniformity. When the paper is dry proceed to lay on
the conventional tints.
Great care is necessary in preparing and combining the diiferent colors,
and attention to certain mechanical conditions and rules must be observed
in order to insure neatness and despatch in execution. The cakes of color
are quite brittle, and it is well to moisten the end and allow it to soften
slightly before using, then rub upon a perfectly clean palette, with a few
drops of pure water, a sufficient quantity of color to tinge to the proper
intensity as much water as will be required for the whole drawing, This
should be thoroughly mixed with the brush, and as often as the brush is
filled, to insure uniformity in the tint.
Previous to applying the tint, it is well to moisten the surface to be col-
ored with clean water, which will prevent the tint from drying too rapidly
at the edges. In tinting never allow the edge to dry until the whole sur-
face is covered: leave a little superfluous color along the edge whilst
filling the brush. Great caution is necessary in approaching the outlines
of the drawing, and the point of the brush should be used so as not to
overrun the lines.
In applying a flat tint to large surfaces, let the drawing board be in-
clined upwards at an angle of 5 or 6 degrees, so as to allow the color to
flow downwards over the surface. With a moderately full brush com-
mence at the upper outline, and carry the color along uniformly from left
to right and from right to left in horizontal bands, taking care not to over-
run the outlines, in approaching which the point of the brush should be
used, and at the lower outline let there be only sufficient color in the
brush to complete the tinting.
~No color should be allowed to accumulate in inequalities of the paper,
but should be evenly distributed over the whole surface.
Too much care cannot be given to the first application of color ; as any
attempt to remedy a defect by washing or applying fresh tints will be
found extremely difficult, and to generally make bad worse.
Erasers should never be used on a tinted drawing to remove stains or
382 TOPOGRAPHICAL DK AWING.
patches, as the paper when scratched, receives the tint more readily, and re-
tains a larger portion of color than other parts, thereby causing a darker tint.
Marbling is done by using two separate tints, and blending them at
their edges. A separate brush is required for each tint ; before the edge
of the first is dry, pass the second tint along the edge, blending one tint
into the other, and continue with each tint alternately.
In reference to the general effect to be produced in tinted topographi-
cal drawings, as to intensity, every thing should be subordinate to clear-
ness, no tint should be prominent or obtrusive. Tints that are of small
extent must be a little more intense than large surfaces, or they will appear
lighter in shade. Keep a general tone throughout the whole drawing.
Beginners will find it best to keep rather low in tone, strengthening their
tints as they acquire boldness of touch.
In lettering tinted drawings, let the letters harmonize with the rest of
the plan ; let them be in tint more intense than the topography, prominent
but not obtrusive.
Flourishes around the titles may be used on handsome estate maps, and
on engraved maps of countries. They should be used in proportion to the
degree of finish bestowed on the rest of the map ; and while they give
grace and elegance to the title when used in moderation, care should be
taken to prevent their having too prominent an appearance.
We would recommend to every one who wishes to make himself a
perfect draughtsman, that he should collect good charts and drawings,
study them, and in his leisure moments copy them. In this way he will
acquire a readiness of manipulation, and ease and freedom of expression.
Plate XCIY. is a map of the Harbor and City of New Haven, re-
duced from the charts of the U. S. Coast Survey, without the depth of
soundings or the marks of shoals.
Plates XCY. and XCYI. are examples of topographical drawings, the
one in ink and the other in color.
Plate XCYII. is a geological map from " Blake's Geological Survey of
California." On geological maps sections are similarly represented, and
plans are colored in patches according to the formation. Shades of India
ink usually represent coal measures ; of blue, limestone ; of pink, the igne-
ous rocks, as trap, granite, &c. In all cases, there are small blocks of
color at the margin of the map, to designate the mineral represented by
each.
TOPOOEAPHICAL DKAWING.
383
! Private houses (occupied by persons not in receipt of wages).
2. Offices and shops.
3. Houses occupied by persons in receipt of wages.
4. Warehouses.
5. Stables and outhouses.
6. Public buildings
7. Contours, vertical distances between lines, two feet.
8. Sewers.
9. Gas-pipes.
10. Water-pipes.
The above map is a portion of the city of London, taken from a Sanitary
Report by a Commission of Parliament ; and embodies in a graphic way
the details in regard to drainages, natural and artificial, contour lines and
street sewers ; position of gas and water mains, and occupancy of buildings.
On the original are also given the number of the houses and names of
streets.
Reference has been made to the drawing of hills by contours, and it
has not been recommended except when the lines have been accurately
384
TOPOGRAPHICAL DRAWIXG.
determined by level. When this is the case, they should always be used ;
it is the simplest and most explanatory record of facts, and if the facts have
been worth determining they are worth recording. When contour lines
are brought more closely together (as shown in the cut below from the
same Sanitary Eeport and of a larger portion of London), it produces the
effect of physical relief, and shows at a glance the lines of natural drainage,
and from it profiles can be made in any direction, for the grading of streets
or sewers. Were town and county maps thus drawn with contour lines,
much time and money would be saved in the location of highways and
railways.
Having thus illustrated within the limits of our page the different
kinds of topographical drawing, we should recommend to every one, who
wishes to become a good draughtsman, to collect good charts and drawings,
study and copy them, to acquire readiness of manipulation and ease and
freedom of expression.
PERSPECTIVE DEAWING.
385
PEESPECTIYE DRAWING.
Fig. 1.
THE science of Perspective is the representation by geometrical rules, upon
a plane surface, of objects as they appear to the eye, from any point of
view.
All the points of the surface of a body
are visible by means of luminous rays
proceeding from these points to the eye.
Thus, let the line A B (fig. 1) be placed
before the eye, C, the lines drawn from
the different points 1, 2, 3, 4, &c., repre-
sent the visual rays emanating from each
of these points. It is easy to understand
that, if in the place of a line a plane or
curved surface is substituted, the result will be a cone of rays.
Let A B (fig. 2) be
a straight line, and let
the globe of the eye be
represented by a cir-
cle, and its pupil by
the point C. The ray
emanating from A, en-
tering through C, will
proceed to the retina
of the eye, and be de-
picted at a. And as
it follows that all the
points of A B will send
rays, entering the eye Fig. 2.
through 0, the whole image of AB will be depicted on the retina of the
25
386 PERSPECTIVE DRAWING.
eye in a curved line a 3 b. Conceive the line AB moved to a greater
distance from the eye, and placed at A' B', then the optic angle will be
reduced, and the image a! 3 b' will be less than before ; and as our visual
sensations are in proportion to the magnitude of the image painted on the
retina, it may be concluded that the more distant an object is from the eye,
the smaller the angle under which it is seen becomes, and consequently
the farther the same object is removed from the eye the less it appears.
Observation has rendered it evident, that the greatest angle under
which one or more objects can be distinctly seen, is one of 90. If be-
tween the object and the eye there be interposed a transparent plane (such
as one of glass m n\ the intersection of this plane with the visual rays are
termed perspectives of the points from which the rays emanate. Thus a
is the perspective of A, b of B, and so on of all the intermediate points ;
but, as two points determine the length of a straight line, it follows that
a b is the perspective of A B, and a' b' the perspective of A'' B'.
It is evident from the figure that objects appear more or less great ac-
cording to the angle under which they are viewed ; and further, that ob-
jects of unequal size may appear equal if seen under the same angle.
For faswfg, and its perspective will be found to be the same as that of
A'B'.
It follows also, that a line near the eye may be viewed under an angle
much greater than a line of greater dimensions but more distant, and
hence a little object may appear to be much greater than a similar object
of larger dimensions. Since, therefore, unequally sized objects may ap-
pear equal in size, and equally sized objects unequal, and since objects are
not seen as they are in effect, but as they appear under certain conditions,
perspective may be defined to be a science which affords the means of rep-
resenting, on any surface whatever, objects such as they appear when seen
from a given point of view. It is divided into two branches, the one
called linear perspective, occupying itself with the delineation of the con-
tours of bodies, the other called aerial perspective, with the gradations of
colors produced by distance. It is tlie former of these only, that is pro-
posed here to be discussed.
The perspective of objects, then, is obtained by the intersection of the
rays which emanate from them to the eye, by a plane or other surface
(which is called the picture), situated between the eye and the objects.
From the explanation and definition just given, it is easy to conceive
that linear perspective is in reality the problem of constructing the section,
by a surface of some kind, of a pyramid of rays of which the summit and
the base are given. The eye is the summit, the base may be regarded as
PEESPECTIVE DRAWING.
387
the whole visible extent of the object or objects to be represented, and the
intersecting surface is the picture.
A good idea of this will be obtained by supposing the picture to be a
transparent plane, through which the object. may be viewed, and on which
it may be depicted.
In addition to the vertical and horizontal planes with which we are fa-
miliar in the operations of projection, several auxiliary planes are em-
ployed in perspective, and particularly the four following :
Fig. 3.
1. The horizontal plane A B (fig. 3), on which the spectator and the
objects viewed are supposed to stand, for convenience supposed perfectly
level, is termed the ground plane.
2. The plane M N", which has been considered as a transparent plane
placed in front of the spectator, on which the objects are delineated, is
called the plane of projection or the plane of the picture. The intersec-
tion M M of the first and second planes is called the line of projection, the
ground, or base line of the picture.
3. The plane E F passing horizontally through the eye of the spectator,
and cutting the plane of the picture at right angles, is called the horizontal
plane, and its intersection at D D with the plane of the picture is called the
horizon line, the horizon of the picture, or simply the horizon.
4. The plane S T passing vertically through the eye of the spectator,
and cutting each of the other planes at a right angle, is called the central
plane.
Point of view, or point of sight, is the point where the eye is supposed
to be placed to view the object, as at C, and is the vertex of the
optic cone. Its projection on the ground plane S is termed the station
point.
388
PEESPECTIVE DRAWING.
The projection of any point on the ground plane is called the seat of
that point.
Centre of view (commonly, though erroneously, called the point of
sight), is the point V where the central vertical line intersects the horizon
line ; a line drawn from this point to the eye would be in every way per-
pendicular to the plane of the picture.
, Points of distance, are points on the horizontal line, as remote from
the centre of view as the eye.
Vanishing points, are points in a picture to which all lines converge
that in the original object are parallel to each other.
Parallel perspective. An object is said to be seen in parallel perspec-
tive when one of its sides is parallel to the plane of the picture.
Angular perspective. An object is said to be seen in angular perspec-
tive when none of its sides are parallel to the picture.
To find the perspective of points, as the points m, s, (fig. 4) in the ground
Fig- 4
plane, the same letters designating similar planes and points as in fig.
3. From the point r/i draw a line to the point of sight C, and also to the
station point S, at the intersection of the line m S with the base line M S',
erect a perpendicular cutting the line in C, the intersection m 1 will be the
perspective projection of the point m, on the plane of the picture M V.
The point s being in the central plane, its projection must be in the in-
tersection of that plane by the plane of the picture, as the point s' the
intersection of the central vertical line by the line s C. The point v
being both in the central and horizontal plane, its projection in the plane
of the picture must be in the intersection of all three planes, or at the
DRAWING IN PERSPECTIVE.
389
point of view Y. The point h being in the horizontal plane, its projec-
tion must be in the intersection of this plane with the plane of the picture,
or the intersection h' of the horizon line by the line h 0. The points ti
and m being in the same vertical line, the points h' and m' must also be in
the same vertical line in the plane of the picture, and the position of h'
might be determined by the intersection of h C by the perpendicular to
the base line at its intersection by m S.
Connect the points hvsm, and also their projected perspective points
h' Y s' m', and we find that when an original line is parallel or perpendic-
ular to the lase of the picture, the perspective of that line will also be par-
allel or perpendicular to it.
Fig 5. Draw the diagonals Ms and m S', project as in preceding fig-
ure the points m and s into the plane of the picture, draw M m, M S' and
S' m' ; now since m and M are the extremities of a line perpendicular to
Fig. 5.
the plane of the picture, the line m' M must be the projection of this line
on the plane of the picture, and if this line be extended it will pass through
Y, which may be demonstrated of all lines perpendicular to the plane of
the picture ; hence the perspective direction of lines perpendicular to the
picture is to the centre of view.
If the line ra' S' be extended, it will pass through the point D, and if
M s' be extended it will pass through a point in the line of the horizon at
a distance from Y equal to Y D ; by construction D Y has been made equal
to Y C, and as this demonstration is applicable to other similar lines, and
since MmsS' is a square; hence the perspective direction of all lines,
390 DRAWING IN PERSPECTIVE.
making an angle of 45 with the plane of the picture, is towards the point
of distance.
Having thus illustrated the rules of parallel perspective, we now pro-
ceed to apply them to the drawing of a square and cube, PI. XCIX. The
same letters are employed in similar position as in preceding figures.
It is necessary to premise that the student should draw these examples
at least three times the size of those in the plate.
Let A and B (fig 1,) represent the plan, or situation upon the ground,
of two squares, of which a perspective representation is required. First
draw the line M M, which represents the base line of the picture ; make S
the station point or place of the observer, and draw lines or rays from all
visible angles of the squares, to S ; then draw the lines S M, parallel to the
diagonal lines of the squares. Now draw M' M' parallel to M represent-
ing the base line of the picture in elevation ; then draw S' Y, the vertical
line immediately opposite the eye ; let the distance, S' Y, be the height of
the eye from the ground, and draw D D the horizontal line ; Y being the
centre of view ; let fall perpendicular lines from the angles a and J of the
plan of the square A, and also from the point c, where the ray from the
angle e intersects the base line, M M, and from a' and 5', where a a' and 5 V
intersect the base or ground line M' M', draw lines to the centre of view,
Y; and e' where the perpendicular line from c intersects the line 5' Y, will
give the apparent or perspective width of the side b e ; from e' draw a
line parallel to a' 5', and the perspective representation of the near-
est square A, is complete. In order to prove the accuracy of this per-
formance, it is necessary to try if the diagonal lines, a' e f , and V f, incline
respectively to the points of distance, D D, on the horizontal line : if so, it
is correct. The square B is drawn in precisely the same manner, and will
be easily understood by observing the example.
The plans of the two cubes C and D, are the same as the plans of the
squares A and B. As neither of these cubes appears to touch the plane
of the picture M M, it will be necessary to imagine the sides I </, and ~k h,
to be continued until they do so ; now draw down perpendicular lines from
where the continuations of these sides intersect the base line, and set off
on them from the line M' M', the height of the cube, as 1 2 which
is the same as the width, and complete the square shown by the dotted
lines : from all four angles of this square draw lines to the centre of view
this will give the representation of four lines at right angles with the
picture carried on as far as it would be possible to see them ; then it only
remains to cut off the required perspective widths of the cubes by the per-
pendicular lines from the intersection of the visual rays with the plane of
PERSPECTIVE DRAWING. 391
the picture : the completion of this problem will be very easy, if the
drawing of the squares is well understood.
In such simple objects as these it will not be necessary to draw a plan ;
when one side is parallel to the picture, and dimensions are known. In
fig. 2, the same objects as those in fig. 1 are drawn without a plan thus :
Draw the ground line M M, then the vertical line S' Y, and the horizon-
tal line D D, at the height of the eye ; making D D the same distance on
each side of V, that the eye is from the transparent plane; for drawing
the squares mark off from S' to &', on the ground line, the distance that
the square is on one side of the observer ; let V a' be the length of one
side of the square ; from V and a' draw lines to Y, which represent the
sides of the square carried on indefinitely ; to cut off the required per-
spective width of the side I' e' of the square, lay off the width, a' V, from
b' top, then draw from^> to D on the left, and the point e' where the line
Dp intersects V Y, will give the apparent width required ; then draw/' e'
parallel to a' &', and the square is complete : this may be proved in the
same way as in fig. 1 . The further square may be obtained in a similar
manner, setting off the distance between the squares from^ to , and the
width of the square beyond that, and drawing lines to D as before:
some of the lines in this plate are not continued to the ground line,
in order to avoid confusion. Proceed with the cubes by the same rule.
Let 1, 2, 3, 4, be the size of one side of the cube if continued until touch-
ing the picture ; from these points draw rays to Y : from 3 to t set off the
distance the cube is from the picture, and from t to r, the width of the
cube ; draw from these points to D on the right, and their intersection of
the line 3 Y in m, <?, will give the perspective width and position of that
side of the cube : draw lines perpendicular to the ground line from m and
e>, and lines parallel to 4 2 from the angles of the cube, l',g f ,m; then
draw the side n A', and the cube is complete. The operation of drawing
the other cube is similar, and easy to be understood.
From the drawing of a square in parallel perspective, we deduce rules
for the construction of a scale in perspective. Let D M M D, (fig 6,) be
the plane of the picture, the same letters of reference being used as in
preceding figures. From S' lay off the distance o S' equal to some unit
of measure, as may/be most convenient ; from o draw the diagonal to D
the point of distance ; now draw 1 V parallel to the ground line M M,
again draw from I/ the diagonal V D, and lay off the parallel 2 2', pro-
ceed in the same way with the diagonal 2' D and the parallel 3 3', and
extend the construction as far as may be necessary. It is evident o S' 1 1/,
V 1 2 2', 2' 2 3 3' are the perspective projectors of equal squares, and
392
PEESPECTIVE DRAWING.
therefore o S', 1 1', 2 2' 3 3', etc., and S' 1, 1 2, 2 3, etc., are equal to each
other, and that if o S' is set off to represent any unit of measure, as one
foot, one yard, or ten feet, &c., each of these lines represents the same dis-
JIT
Fig. 6.
tance, the one being measures parallel to the base line, the others perpen-
dicular to it. In making a perspective drawing a scale thus drawn will be
found very convenient ; but as in the centre of the picture it might inter-
fere with the construction lines of the object to be put in perspective, it is
better that the scale be transferred to the side of the picture a M o, the di-
agonals to be laid off to a point to the right of D equal to the point of
distance.
The scales thus projected are for lines in the base or ground plane ; for
lines perpendicular to this plane the following construction is to be adopt-
ed ; upon any point of the base line removed from S', as a for instance,
erect a perpendicular, ad / on this line, lay off as many of the units o S'
as may be necessary ; in this example three have been laid off, that is, a d
=3 o S'. From a and d draw lines to the centre of view, and extend the
parallels 1 1', 2 2', 3 3 ' ; at the intersection of these lines with a Y erect
perpendiculars. The portions comprehended between the lines a Y and d
Y will be the perspective representations of the line a d, in planes at dis-
tances of 1, 2, 3, o S' from the base line, and as 5, c, d are laid off at inter-
vals equal to o S', by drawing the lines c Y and 5 Y six equal squares are
constructed, of which the sides correspond to the unit of measure, S'.
To determine the Perspective Position of any point in the Ground Plane.
Thus (fig. 7), to determine the position of the point p, which in plane
would be six feet distant from the plane of the picture, M M, and ten feet
from the central plane, to the left.
Lay off from S', to the left, the distance a S', equal to six feet on the
PERSPECTIVE DRAWING.
393
scale adopted ; draw the diagonal to the point of distance D, on the right ;
at its intersection a! with the vertical line V S', draw a parallel to M M
the base line; lay off from S', S' ~b equal to ten feet, draw 5 V; the inter-
section of this line p, with the parallel previously drawn, will be the posi-
tion of the point required.
Fig. 7.
By a similar construction the position of any point in the ground plan
may be determined. It is not necessary that the distances should be ex-
pressed numerically ; they may be shown on the plan and thence be trans-
ferred to the base line, and thrown into perspective by the diagonals and
parallels. As the intersections of the various lines of the outlines of ob-
jects are points, by projecting perspectively these points, and afterwards
connecting by lines, the perspective of any plane surface, on the ground
plane, may be shown.
If the point p were not in the ground plane, but in a position directly
above that already assumed, that is, the distances from the plane of the pic-
ture, and the central plane being the same, but its distances above the
ground plane were, say, five feet ; then at 5 erect a perpendicular, and lay
off b V equal to five feet, connect V Y, at p erect another perpendicular,
and its intersection p' with the line V Y will be the position of the point
required.
Or the plane of the point p 1 might be assumed as the position of the
ground plane, M' M' becoming the base line, and laying off from S", S" a"
and S"b f equal respectively to six and ten feet; drawing the diagonal
a"D and V Y and the parallel as before, the point p' will be determined.
To draw an Octagon in Parallel Perspective, Let A (fig. 8) represent
the plan of an octagon. Draw M M, S' Y, and D D, as before ; from the
points M, a, J, 0, draw rays to Y. Set off on M M from c to the right the
distances ce, cd, cf, from which draw diagonals to D on the left, and
at their intersection with the ray c Y, draw parallels e' g', d' A', ~k' I', to
the base line ; these points will correspond to the angles on the plan. Now
394:
PECTIVE DRAWING.
connect the angles on the perspective view, in the proper succession, and
the perspective projection is complete.
It will be observed, that in this construction the plan has been placed
forward of the plane of the picture, contrary to the position it should oc-
cupy, which should be the same relative position back of this plane ; but
it will be found much simpler in construction than if it were placed as in
PI- XCIX. and the points were all projected to the base line ; it is, of course,
equally correct in its perspective projection.
D
Fig. 8.
To draw a Circle in Parallel Perspective. Let C, (fig. 8) represent the
plan of a circle, round which let the square a e c m be described, two of
its sides being parallel to the base line M M ; draw diagonals across the
square, and where these intersect the circumference of the circle draw the
lines bk and dg parallel to the base line, and the lines o n and^?^ at right
angles thereto. Draw also the lines f I and c A at right angles to each
other through the centre of the circle, project the points #, 0, l,p, m, to
the base and draw rays to Y ; set off from a' to the left the distances a' ,
7>, a'c, a'd, a'e, and draw diagonals to the point of distance D on the
right ; at their intersection with the line a' V draw horizontal lines, or
parallels to the base, and there will be projected in perspective the square
aecm, with all the lines of parallels and perpendiculars ; connect the
intersections corresponding to the points c, n, f, g, h, k, Z, r, and we
PERSPECTIVE DRAWING.
395
have the perspective projection of the required circle, which will be an
ellipse.
To erect upon the octagonal base A an octagonal pillar or tower. This
construction resolves itself into simply constructing another octagon on an
upper plane, and connecting the visible angles by perpendiculars, or per-
pendiculars may be erected at the points M, #, J, c, and the heights of the
tower laid off upon them, and from these extremities rays drawn to the
centre of view ; the intersection of these rays by perpendiculars from the
angles of the octagon beneath will determine the projection of the upper
surface of the pillar; represent in full lines all visible outlines, and the
projection is complete.
In the same manner a pillar may be erected on the circular base. If
the pillars be inclined, the first method of projecting the upper outline on
a plane assumed at the height of the pillar, must be adopted.
To draw a Pyramid in Parallel Perspective. Let A (fig. 9) be the
plan of a pyramid, the diagonal lines represent the angles, and their in-
tersection the vertex ; project the plan as in previous examples of squares.
Draw diagonal lines from M to 5, and a to c, their intersection gives the
perspective centre of the square ; upon this point raise a perpendicular
line which is the axis of the pyramid ; draw a perpendicular line e /, in
D
Fig. 9.
the centre of the line M a, upon which set up the height of the pyramid
ef; from/ draw a line to V, and its intersection of the axis of the pyramid
at d will give the perspective height ; complete the figure by drawing
lines from d, the apex, to M, a, 5, the three visible angles. The other
two pyramids are drawn in a similar manner, by setting their distances
from the plane of the picture off from a, on the ground line to the right,
and drawing diagonals to the point of distance on the left.
396 PERSPECTIVE
To draw a Cone in Parallel Perspective. Let B (fig. 9) represent the
plan of a cone, apply the same lines of construction as to C (fig. 8) ; and
draw the perspective view of a circle, upon the perspective centre of
which draw a perpendicular line, a, J> / on the centre of the line d <?, raise
a perpendicular, upon which set up the height of the cone, from the
ground line to c j from c draw a ray to Y, and the point where this line
intersects the axis of the cone a &, in &, will give the perspective height
of the axis ; from 5 draw lines toy and g, and the figure is complete.
To draw the prism, C, which consists of two triangular ends and three
rectangular sides, place the length of the side a M upon the ground line,
and draw lines to V ; mark off the width of one end from a to the left upon
the ground line, and j daw to the point of distance on the right, which
gives the perspective width, ad; find the perspective centre f of the
side in the same way, and from f and d draw horizontal lines until they
intersect the line from M Y ; upon f and g draw perpendiculars ; set up
the height a 5, of the end of the prism, and from b draw a line lo Y, and
the point where^ it intersects f c, in c, will give the perspective height of
the end of the figure ; from G draw c e, parallel to a M, from c draw c d,
and c a, and the visible end is complete ; the other end is dotted in to show
the process only.
E is the representation of a cylinder, with one end towards the spec-
tator; its projection will be easily understood by examination.
To draw a Square and Cube in Angular Perspective. Plate C. Let
A (fig. 1) be the plan of the square, and B the plan of the cube, M M the
base or ground line, and S the station point. Draw M' M', and D D' par-
allel to M M, the one being the ground line and the other the horizon of
the plane of the picture ; project the point d on M M to d, on M' M'. It
has been shown in parallel perspective that the vanishing points of diago-
nals of squares lie in the points of distance ; if through the station point
S, in any of the preceding figures, lines be drawn parallel to the diagonals,
they will intersect the base lines at distances from the 'central plane equal
to the points of distance. In like manner to find the vanishing points of
lines in the ground planes, or in planes parallel to the ground plane, in-
clined to the plane of the picture, through the station point S draw lines par-
allel to the inclined lines, and project their intersection with the base line
to the horizon of the picture ; thus, in the present example draw S M, S M
parallel to ad, e A, and to dc, kg ; project their intersections M, M, with
the base line to D, D', the horizon of the picture, and D, D', will be the
vanishing points of all lines parallel to a d and d c. Draw d' D and d' D',
the perspective projection of d a will lie in the former of these lines and
PERSPECTIVE DKAWING. 397
d c in the latter. To determine the perspective position of the points a
and c, or the length of these lines, draw the rays a S and <?S, project their
intersection with the base M H, upon the lines 'd 1 D and d' D', and their in-
tersections a', c' will be the perspective projection of the points a and'c.
To complete the projection of the square, draw the lines a! D' and c D,
their intersection will be the perspective projection of the point 5, and the
square is complete. To prove the construction, draw the ray I S and pro-
ject its intersection with the base M M, and if the construction be correct
it will fall upon the point V .
As- the cube is placed at some distance from the plane of the picture,
it will be necessary to continue either e h or g h, or both, till they intersect
the base line M M at n and m; drop perpendiculars or project these points
upon M M' at n' and m' ; on these perpendiculars set up the height of the
cube m! o and n' s, draw the lines in' D', o' ~D' and n' D, s D ; connect the
intersections Ji and h" ; draw the rays S e and S #-, and project their inter-
sections with M M, to g' e' ; draw the lines e" D' and g" D ; if the construction
be correct, the projection of the intersection of the ray S/with the base
will fall upon/', and of the ray S h will fall upon h" and ti.
To Solve the Same Problem ly a Different Construction. Let A and
B, (fig. 1,) be as before the plans of the square and of the cube ; to pro-
ject them perspectively on the plane of the picture MD D'M, (fig. 2).
From the point M and M, (fig. 1,) set off distances equal to M S, M S,
to^> and^/ ; project these points upon D D' fig. 2, the pointy, fig 2, will be
that from which any number of parts may be laid off on lines vanishing in
D' ; the pointy will be the corresponding point for lines vanishing in D.
These points may be called the points of division. In parallel perspec-
tive the points of distance were the points of division, the one for the
other. To illustrate their application in the present example, project the
point d, (fig. 1,) to d' (fig. 2,) draw d' D and d' D', from d' on either lay
off a distance d' i, d' Tt, equal to the side of the square a d. Now since p
is the division point of lines vanishing in D from *', draw the line ip, and
its intersection with d' D cuts off a line d' o! equal perspectively to the
line d' i or ad measured on the base line. Again since p' is the division
point of lines vanishing in D', the line fop' cuts off on d' D', a line d' c' equal
perspectively to the line d' &, or a d measured on the base : having a' d' c,
the square is completed by drawing the lines c' V towards D, and a' b'
towards D'.
To construct the cube, project the point m, (fig. 1,) to m', (fig. 2) ; lay
off on the perpendicular forming the projection, the height m' o' of the
cube ; draw the lines m' D' and o' D'. Lay off the disfcnce m! r equal to
398 PERSPECTIVE DRAWING.
m h, (fig. 1,) and draw the line rp f , its intersection with m' D' will cut off
m! Ji f ) equal to m h, (fig. 1,) and establish the angle h of the cube. From
r lay o&rs, equal to hg, (fig. 1,) draw sp 1 , and its intersection with m! D'
establishes the angle g 1 '. From h' draw a line vanishing in D. Through.
A' extend a line p h' to , from t lay off to the left t a, equal to the side of
the cube he; draw ap, and its intersection with the line h' D, establishes
a third point e of the cube. Upon these points h' (f e' erect perpendicu-
lars ; those upon h' and g' will, by their intersection with o' D, determine
h" g". Draw h" D', its intersection with the perpendicular at G determines
e". Draw g" D' and e" D to their intersection, and the cube is complete.
To Draw the Perspective Projection of an Octagonal Pillar in Angu-
lar Perspective. Plate GI. Let A, (fig. 1,) be the plan of the pillar. En-
close it by a square. Let M M be the base line, and S the station point ; de-
termine the position of the vanishing points for the sides of the square as in
Plate 0., and project the square upon the plane of the picture M D D' M'
by either of the methods already explained. These lines of construction are
omitted, as on the necessarily small diagrams they would confuse the stu-
dent ; but in drawing these examples to the scale recommended, they might
be retained. From the angles of the octagon visible to the spectator
draw rays to the station point S project their intersection with the base
line MM, to the perspective square, (fig. 2,) which will thus determine on
the sides of the square the positions of the points #', 5', <?', d', e', correspond-
ing to the visible angles of the octagon ; connect these points by lines. To
construct the pillar upon this base, upon the perpendicular let fall from
the corner/" of the square upon M M' at/" set off the height of the pillar ;
from this pointy draw lines to the vanishing points D, D', and construct
three sides of an upper square similar to the lower one. The lines of this
square will determine the length of the sides of the tower, which are the
perpendiculars 10t fall upon a 1 V c' d' e'.
To Construct a Circular Pillar in Angular Perspective. Plate CI.
Let B, (fig. 1,) be the plan of the base ; enclose it with a square whose sides
are parallel respectively to S MandS M; project this square upon the plane
of the picture, (fig. 2,) divide the plan into four equal squares by lines par-
allel to the sides ; draw rays through the points h and i, and project their
intersection with M M upon the perspective square. From the points h f and
i' thus formed, draw lines to vanishing points D' and D, and the perspec-
tive square is divided similarly to the original, and there are four points of
the circle established : through these draw the perspective of the circle.
By the division of the base into smaller squares more points of the curve
might be determined, but for the present purpose they are unnecessary.
PERSPECTIVE DRAWING. 399
To determine the outline of the pillar, draw from S rays tangent to the
sides of the plan at Jc and *, the perpendiculars let fall from their intersec-
tion with M M will be the outline of the cylinder. To cut them off to the
proper height, and to determine the top of the cylinder, upon the perpen-
dicular let fall upon *', set off the height of the cylinder I' I", and upon this
plane project the square as before, and draw in through the points thus
determined the outline of the curve. As a still further elucidation of the
principle of projection, an enlarged cap is represented on the pillar, of
which the circumscribing circle (fig. 1,) is the plan. In this by extending
the central lines of the square, both in plan and perspective, we are en-
abled to project readily eight points in the larger circle through which
the curve may be drawn.
To Draw an Octagonal Pyramid in Angular Perspective. Plate CI.
Let/", (fig. 1,) be the base of the pyramid; project upon the plane of the
picture, (fig. 3,) the visible angles of the base, as in the case of the pillar.
Through the centre of the plan draw a line parallel to one of the sides and
intersecting MM at m / from this point let fall a perpendicular to mf on
M M x , (fig. 3,) ; on this perpendicular set off the height of the pyramid m'
o from m! and draw lines to D'. From the centre of the plan draw a ray
to S, and project its intersection with M M, upon the line o D', its intersec-
tion o' with this line will be the apex of the pyramid : from this point
draw lines to the angles of the base already projected, and the pyramid is
complete.
To Draw a Cone in Angular Perspective. Plate CI. Let the inner
circle B, (fig. 1,) be the base of the cone project its visible outline to fig.
3, as in case of the cylinder. To determine its height extend one of the
diameters of the plan to the base line atjp; from this point let fall a per-
pendicular to p' on MM', and set off upon \tp' g, the height of the cone ;
fromy and g draw lines to the* vanishing point D'. From the centre of
the plan, (fig. 1,) draw rays to S, and project its intersection with MM,
upon r' on the line g D', and / will be the apex of the cone : connect the
apex with the extremities of the perspective of the base, and the projec-
tion of the cone is complete.
To Draw the Elevation of a Building in Angular Perspective. Plate
Oil. For. example, take the school-house, PI. LXXIY. of architecture.
Plot so much of the plan of the building at it as may be seen from the po-
sition of the spectator at S. Draw a base line, and through the station
point draw parallels to the sides of the building cutting the base as at M
M : draw MM' for a base, and DD' for the horizontal line of the picture.
Project M and M to D and D', for the vanishing points, the one of the
400 PERSPECTIVE DRAWING.
lines parallel to a c, the other to a I ; extend a c, a I ; project d, <?, to
d, e' t and on d f d set off the height of the eaves d' o, f and of the ridge d
n ; from d', o and n draw lines to D', and from e to D, draw rays from c
and l> to S', and project their intersection with the base to the vanishing
lines just drawn. To find the perspective of the ridge draw a ray from
the centre of a &, and project its intersection with the base to r on the line
n D', the point is the apex of the gable, the line r D will be the perspective
of the ridge ; to determine its length erect a perpendicular at the intersec-
tion of t D' and s D, draw the sloping lines of the roof, and the outline of
the building is complete. The filling in of the details will be readily under-
stood ; it will only be necessary to keep in mind, that all lines parallel to
a 1) must meet in D', those to a c in D : all measures laid off on any lines
of the plan must be connected with the point of sight S, and their inter-
sections with the base projected. All vertical heights must be laid off on
the line d' d, and referred to the proper position by lines to D or D', as the
case may be.
As an example of the other method of constructing this same problem,
let the scholar lay off to the double of the present scale the plane of the
picture M D D' IF, and the division points p' and p, and without drawing
plan or elevation take the dimensions from Plate LXXIY.
To Draw an Arched Bridge in Angular Perspective. PL CIII. Let
A and B, (fig. 1,) be the plans of the piers; on the line a A, one of the sides
of the bridge, lay down the curve of the arch as it would appear in eleva-
tion, in this example an ellipse. Divide the width of the arch as at b. c.
d. e.f. g. A., carry up lines perpendicular to 5 h until they intersect the
curve of the arch, and through these points, draw lines parallel to b h as Jc.
I. m. / let o r be the height of the parapet of the bridge above the spring of
the arch. Through the station point draw lines parallel to the side a h
and end a a of the bridge, till they intersect the assumed base line M M :
project these intersections to the horizon line of the picture for the vanish-
ing points D, D' of perspective lines parallel to a h and a a. Let fall a
perpendicular from a to a', and on this perpendicular set off from a! the
heights s k, s I, s m, and s t ; from a! and r' draw lines to D and D', and from
the points ra', Z', &' to D'. Draw rays from the points a. 1). c. d. e.f. g. h.
to the station point S, and project their intersection with the base lines to
the perspective line a' D' as in previous examples : the intersection of the
lines Jc' D 7 , 1' D', m' D 7 by the perpendiculars thus projected, will establish
the points of the curve of the arch on the side nearest the spectator. To
determine the position of the opposite side of the arch, from a", the per-
spective width of the bridge, draw a" D', and from h' draw lines to D ;
PERSPECTIVE DRAWING. 401
the line h''p' will be the perspective width of the pier ; draw' V D ; and from
F, k* D ; from / the intersection of the curve of the arch by the perpen-
dicular to /, draw / D, the intersection with k" D' will be one point in
the curve of the arch on the opposite side of the bridge : in the same way,
from any point in the nearer arc draw lines to D, and the intersection
with lines in the same planes on the opposite side of the bridge, will fur-
nish points for the further arch : all below the first only will be visible to
the spectator.
To Draw in Parallel Perspective the Interior of a Room. PI. GUI.
We propose to construct this by scale without laying down the plan.
Draw the horizon line D Y D', and the base H M', making D and D' the
point of distance. Let the room be 20 feet wide, 14 feet high, and 12 feet
deep ; on the base M M', lay off the rectangle of the section in our figure
on a scale of 8 feet to the inch, 20 feet x 14 feet. From the four corners
draw lines to the centre of view V ; from S' lay off to the right or left on
M M' 12 feet, and through this point draw lines to D' or D as the case may
be : through the point of intersection a' of this line with S' V draw a line
parallel to M M' ; at the intersections of this line with M Y and M' Y erect
a perpendicular, cutting the vanishing lines of the upper angle of the
room at d and e / connect d e and the perspective of the room is complete.
To draw the aperture for a door or window on the side, measure off from
S' the distance of the near side from the plane of the picture, and in addi-
tion thereto the width of the aperture ; from these two points draw lines to
the proper point of distance, and at their intersection with S' Y, draw par-
allels to M M', cutting the lower angles of the room, and erect perpendic-
ulars, the height of which will be determined by a line drawn from/, the
height of the window above the floor measured on M D. Should the win-
dow be recessed, the farther jamb will be visible ; extend the farther par-
allel to M M', and cut it by a line g Y. M g being the depth of the recess,
the rest of the construction may be easily understood by inspection of the
figure. At the extremity of the apartment a door is represented half open,
hence as the plane of the door is at right angles to the plane of the picture,
the top and bottom lines will meet in the point of view ; if the door were
open at an angle of 45, these lines would meet in the points of distance ;
if at any other angle, the vanishing points would have to be determined
by constructing a plan, drawing a line parallel to the side of the door
through the station point, and projecting it upon the horizon line. The
chair in the middle of the room is placed diagonally, and the table parallel
to the plane of the picture ; their projection is simple.
To Draw in Perspective a Flight of Stairs. PL CIV. Lay off the
26
402 PEESPECTIVE DKAWING.
base line, horizon, centre of view, and point of distance of the picture,
construct the solid abed, efg A, containing the stairs, and in the required
position in the plane of the picture, divide the rise a c into equal parts ac-
cording to the number of stairs, four for instance ; divide perspectively the
line a b into the same number of parts ; at the points of division of this
latter erect perpendiculars, and through the former draw lines to the cen-
tre of view ; one will form the rise and the other the tread of the steps.
From the top of the first step to the top of the upper continue a line a d,
till it meets the perpendicular S' Y prolonged in v / this line will be the
inclination or pitch of the stair ; if through the top of the step at the other
extremity a similar line be drawn, it will meet the central perpendicular
at the same point v, and will define the length of the lines of nosing of
the steps, and the other lines may be completed. As the pitch lines of both
sides of the stairs meet the central vertical -in the same point, in like man-
ner v will be the vanishing point of all lines having a similar inclination
to the plane of the picture. The projection of the other flight of stairs
will be easily understood from the lines of construction perpendicular to
the base line or parallel thereto, lying in planes.
To Find the Reflection of Objects in the Water. PI. CIY. Let B be
a cube suspended above the water ; we find the reflection of the point a,
but letting fall a perpendicular from it, and setting off the distance a f w
below the plane of the water equal to the line a w above this line ; the line
wf will also be equal to the line wf; find in the same way the points V
and e', through these points construct perspectively a cube in this lower
plane, and we have the reflection of the cube above.
To find the reflection of the square pillar D removed from the shore :
suppose the plane of the water extended beneath the pillar, and proceed
as in the previous example.
It will be observed that those lines of an object which meet in the cen-
tre of view Y, in the original ; their corresponding reflected lines will con-
verge to the same point. If the originals converge to the points of distance,
the reflected ones will do the same. To find the reflection of any inclined
line, find the reflection of the rectangle of which it is the diagonal, if the
plane of the rectangle is perpendicular to the plane of the picture ; if the
line is inclined in both directions enclose it in a parallelopided and project
the reflection of the solid.
To find the Perspective Projection of Shadows. Plate CY. Let the
construction points and lines of the picture be plotted. Let A be the per-
spective projection of a cube placed against another block, of which the
face is parallel to the plane of the picture : to find the shadow upon the
PERSPECTIVE DRAWING. 403
block and upon the ground plane, supposing the light to come into the
picture from the upper left-hand corner and at an angle of 45. Since the
angle of light is the diagonal of a cube, construct another cube similar to
A, and adjacent to the face d c g ; draw the diagonal Ik, it will be the
direction of the rays of light, and Jc will be the shadow of I; connect/7;
and c7c,fk must be the shadow of the line If, and ck of b c; the one
upon the horizontal plane and the other in a vertical one : the former will
have its direction, being a diagonal, toward the point of distance D', the
other being a diagonal in a plane, parallel to that of the picture, will be
always projected upon this plane in a parallel direction.
Let B be a cube similar to A ; to find its projection upon a horizontal
plane, the shadow of the point V may be determined as in the preceding
example, but the shadow of the point c', instead of falling upon a plane
parallel to the picture, falls upon a horizontal one ; its position must be de-
termined as we did before by b. Construct the cube and draw the diag-
onal c'l; in the same way determine the point in 1 the shadow of df ; con-
nect c k' I m n, and we have the shadow of the cube in perspective on a
horizontal plane.
On examination of these projected shadows, it will be found that as the
rays of light fall in a parallel direction to the diagonal of the cube, the
vanishing point of these rays will be in one point V on the line D' M'
prolonged, at a distance below D' equal V D' ; and since the shadows of
vertical lines upon a horizontal plane are always directed towards the point
of sight, the extent of the shadow of a vertical line may be determined
by the intersection of the shadow of the ground point of the line by the
line of light, from the other extremity. Thus, the point k, cube A, is the
intersection of/ D' by 5 V ; the points k', I, m are the intersections of c D',
o D', n' D' by V Y 7 , G' Y'by df Y.' Similarly on planes parallel to that of the
picture, k, cube A is intersection of the diagonal c k, by the ray of light 5 V.
Applying this rule to the frame 0, fromr, s,p, draw lines to D' from /,
s ' iP ' > draw rays to Y'; their intersections define the outline of the shadow of
the post. To draw the shadow of the projection, the shadow upon the post
from t will follow the direction of the diagonal G k. Project u and v upon
the ground plane at u' andv'y from t' u' v' andj? draw lines to D'; from t,
u, v, w and x draw rays to Y', and the intersection of these lines with their
corresponding lines from their bases will give the outline required ; as v and w
are on the same perpendicular, their rays will intersect the same line v' Y'.
With reference to the intensity of." shade and shadow " and the neces-
sary manipulation to produce the required eifect, the reader is referred to
the article on this subject.
404 PERSPECTIVE DRAWING.
In treating of Perspective it has been considered not in an artistic
point, as enabling a person to draw from nature, but rather as a useful art to
assist the architect or engineer to complete his designs, by exhibiting them
in a view such as they would have to the eye of a spectator when con-
structed. In our examples, owing to size of the page, we have been limit-
ed in the scale of the figures, and in the distance of the point of view, or
distance of the eye from the plane of the picture, and as it was unimportant
to the mathematical demonstration, few of the figures extend above the
line of the horizon. In these particular points it is unnecessary that the
examples should be copied. The most agreeable perspective representa-
tions are generally considered to be produced by fixing the angle of vision
M S M', at from 45 to 50, and the distance of the horizon above the
ground line at about one-third the height of the picture.
Linear perspective is more adapted to the representation of edifices,
bridges, interiors, &c., than to that of machinery ; it belongs, therefore,
rather to the architect than to the engineer or the mechanic ; for the pur-
poses of the latter we would recommend Isometrical Perspective, uniting
accuracy of measures with graphic perspective representation.
ISOMETRICAL DRAWING. 4Q5
ISOMETKICAL DKAWING.
PROFESSOR PARISH, of Cambridge, has given the term Isometrical Per-
spective to a particular projection which represents a cube, as in fig. 1.
The words imply that the measure of the representations of the lines
forming the sides of each face are equal.
The principle of isometric representation con-
sists in selecting for the plane of the projection,
one equally inclined to three principal axes, at
right angles to each other, so that all straight lines
coincident with or parallel to these axes, are drawn
in projection to the same scale. The axes are
called isometric axes, and all lines parallel to them
are called isometric lines. The planes containing
the isometric axes are isometric planes ; the point Fi &-
in the object projected, assumed as the origin of the axes, is called the reg-
ulating point.
To draw the isometrical projection of a cube, (fig. 2,) draw the hori-
zontal line A B indefinitely ; at the point D erect the perpendicular D C,
equal to one side of the cube required; through D draw the line D5 and.
T)f to the right and left, making f D B and 5 D A each equal an angle of
30. Consequently the angles F ~Df and F D 5 are each equal to 60.
Make D 1) and ~Df each equal to the side of the cube, and at b and/" erect
perpendiculars, making 5 a and f e each equal to the side of the cube ;
connect F a and F e and draw e g parallel to a F, and a g parallel to F <?,
and we obtain the projection of the cube.
If from the point F, with a radius F D, a circle be described, and com-
mencing at the point D radii be laid off around the circumference, forming
a regular inscribed hexagon, and the points D a e be connected with the
406
ISOMETRICAL DRAWING.
centre of the circle F, we have an isometrical representation of a cube.
The point D is called the regulating point.
If a cube be projected according to the principles of isometrical per-
spective, in a similar manner as we have constructed one according to the
rules of linear perspective, the length of the isometrical lines would be to
the original lines as .8164 to 1, but since the value of isometrical perspec-
tive as a practical art lies in the applicability of common and known
scales t6 the isometric lines, in our constructions we have not thought it
necessary to exemplify the principles of the projection, but have drawn our
figures without any reference to what would be the comparative size of the
original and of the projection, transferring measures directly from plans and
elevations in orthographic projections, to those in isometry. It will be ob-
served that the isometric scale adopted applies only to isometric lines, as
F D, F a and F e or lines parallel thereto ; the diagonals which are abso-
lutely equal to each other, and longer than the sides of the cube, are the
one less, the other greater ; the minor axis being unity, the isometrical lines
and the major axis are to each other as, 1. -j/2. y3.
Understanding the isometrical projection of a cube, any surface or
solid may be similarly constructed, since it is easy to suppose a cube suffi-
ciently large to contain within it the whole of the model intended to be
represented, and as hereafter will be farther illustrated, the position of any
point on or within the cube, the direction of any line or the inclination of
any plane to which it may be cut, can be easily ascertained and repre-
sented.
In figs. 1 and 2 one face of the
cube appears horizontal, and the
other two faces appear vertical. If
now the figures be inverted, that
which before appeared to be the top
of the object, will now appear to
be its under side.
The angle of the cube formed
by the three radii meeting in the
centre of the hexagon, may be
made to appear either an internal or
external angle ; in the one case the
'"" faces representing the interior, and
in the other the exterior of a cube.
Figs. 3, 4, 5, illustrate the application of isometrical drawing to simple
ISOMETRICAL DRAWING. 407
combinations of the cube and parallelopipedon. The mode of construction
Figs. 3, 4, 5.
of these figures will be easily understood by inspection, as they contain no
lines except isometrical ones.
To draw Angles to the Boundary Lines of an Isometrical Cube.
40 SO 20 10
Fig. 6, 7.
Draw a square C (fig. 6,) whose sides are equal to those of the isome-
trical cube A, and from any of its angles describe a quadrant, which di-
vide into 90, and draw radii through the divisions meeting the sides of
the square. These will then form a scale to be applied to the faces of the
cube ; thus on D E, or any other, by making the same divisions along their
respective edges.
As the figure has twelve isometrical sides, and the scale of tangents
408
ISOMETRICAL DE AWING.
may be applied two ways to each, it can be applied therefore twenty-four
ways in all. "We thus have a simple means of drawing, on the isometrical
faces of the cube, lines, forming any angles with their boundaries.
Figs. 1, 2, 3, 4, 5, 6, PL CVI. show the section of a cube by single
planes, at various inclinations to the faces of the cubes. Figs. 7, 8 are the
same cube, but turned round, with pieces cut out of it. Fig. 9 is a cube
cut by two planes forming the projection of a roof. Fig. 10 is a cube with
all of the angles cut off by planes, so as to leave each face an octagon.
Fig. 11, represents the angles cut off by planes perpendicular to the base
of the cube, forming thereby a regular octagonal cylinder. By drawing
lines from each of the angles of an octagonal base to the centre point of
the upper face of the cube, we have the isometrical representation of an
octagonal prism.
As the lines of construction have all been retained in these figures,
they will be easily understood and copied, and are sufficient illustrations
of the method of representing any solid by enclosing it in a cube.
We have now to consider the application of this species of projection
to curved lines.
x
Figs. 8, 9.
Let A B (fig. 8,) be the side of a cube with a circle inscribed ; and
suppose all the faces of the cube to have similarly inscribed circles. Draw
the diagonals A B, CD, and at their intersection with the circumference
lines parallel to A C, B D. Now draw the isometrical projection of the
cube, (fig. 9,) and lay out on the several faces the diagonals and the par-
allels; the projection of the circle will be an ellipse, of which the diago-
nals being the axes, their extremities are defined by their intersections/ 6,
ISOMETRICAL DRAWING.
409
e 5, a 2, 1 1, ^3, c 4, by the parallels; having thus the major and minor
axis, construct the ellipse by the trammel, or since the curve is tangent at
the centre of the sides, we have eight points in the curve ; it may be put
in by sweeps or by the hand.
Fig. 10.
To Divide the Circumference of a Circle. First method. 'On the
centre of the line A B erect a perpendicular C D, making it equal to C A
or C B ; then from D, with any radius, describe an arc and divide it in
the ratio required, and draw the divisions radii from D meeting A B ;
then from the isometric centre of the circle draw radii from the divisions
on A B, cutting the circumference in the points required.
Second method. On the major axis of the ellipse describe a semi-cir-
cle, and divide it in the manner required. Through the points of division
draw lines perpendicular to A E, which will divide the circumference of
the ellipse in the same ratio. On the right hand of the figure both
methods are shown in combination, and the intersection of the lines give
the points in the ellipse.
Plate CYII. The upper figure is an isometrical projection of a bevel
wheel, with a half-plan beneath, and projected lines explanatory of the
method to be adopted in drawing the teeth, and of which only half are
shown as cut. It will be seen by reference to the second method given
above for the division of the circumference of a circle, that the semicircle
is described directly on the major axis of the ellipse ; in practice it will be
found more convenient, when a full drawing is to be made, to draw the
semicircle on a line parallel to the major axis, and entirely without the
410
ISOMETKICAL DRAWING.
lines of the main drawing. And also, as in the example of the bevel gear,
complete on the semicircle, or half-plan, the drawings of all lines, the
intersection of which with circles will be necessary to be projected on the
isometrical drawing.
The lower figure is an isometrical projection of a complete pillow
block, with its hold-down bolts. By reference to Plate XXII. and
figs. 238 and 239, p. 146, it will be seen how much more graphically
these forms of gearing are given by isometry than by the usual projec-
tion. As an exercise for the learner, it will be very good practice to
project isometrically the spur-gear, Plate XVIII., and the standard and
hanger, Plates XV. and XVI., of which sufficient details are given in
the plates.
Plate CVIIL is an isometrical projection of a culvert, such as were
built beneath the Croton Aqueduct, and is a good example of construc-
tion, and better illustrated by the drawing than it would be by plan and
elevations.
Fig. 11.
Figs. 11, 12, and 13 are further examples to show the applicability, of
isometry to other forms of construction.
ISOMETRICAL DKAWIXG. 411
Fig. 11 is a sectional view of a very complicated boiler, the construc-
tion of which could hardly be explained to a mechanic by any number of
sectional drawings, and only by isometry or perspective to one not con-
versant with such matters.
Fig. 12.
Fig. 12 is an elevation of a portion of a truss of an iron bridge, given
not as an example of construction, but merely of the projection.
Fig. 13 is an elevation of the roof-truss, fig. 2, Plate XL VI. ; no side
view is shown on the plate, but the dimensions of timber and spaces are
drawn as usual in practice.
Plate CIX. is an elevation and section in isometry of the district school-
house given in Plate LXXIV. The scale has been necessarily reduced to
bring the drawing within the limits of the page, but is given on the plate
as it should always be either drawn or written, on all drawings to a scale,
not intended for mere pictures or illustrations. The section is drawn at
the height of 8 feet above the base course, and higher than is usual in such
sections, but it was necessary on account of the extra height of the window-
sill above the floor, desirable in all school-rooms. As a plan it is more
graphic than that given in Plate LXXIV., and when there are staircases
one above the other in the drawing, they are more intelligibly expressed,
but there is nothing in the present drawing that cannot be nearly as well
shown by the plan ; and to a mechanic, for the purposes of construction, the
plan is the simpler.
412
ISOMETRICAL DRAWING.
By comparing the elevation, Plate CIX., with the perspective, GIL,
the former appears distorted, and out of drawing, but it is much more
readily drawn, and has this great convenience, that it is drawn and can be
measured by a scale, but only on the isometric lines all others are dis-
torted, too long or too short, as may be seen in the major and minor axes
of the bevel gear, Plate CYII., or the rake lines of the roof, Plate CIX.
We have multiplied examples of isometrical drawing, to show its ap-
plicability to varied forms of construction, mechanical and architectural.
The principles of this projection are easy and intelligible, and their use
should be extended. Isometrical projection is especially valuable to the
mechanical draughtsman, explaining many constructions that could hardly
be done by any amount of plans, elevations, and sections, and still uniting
with pictorial representation the applicability of a scale. For drawings
for the Patent Office it is especially desirable, in a simple and practical
form combining the requisites of many projections ; but as a drawing of
WINDLASS.
413
414
ISOMETEICAL DRAWING.
what could be absolutely seen by the eye it is not truthful, and therefore
when pictorial illustration only is requisite, the drawing should be in linear
perspective.
In confirmation of the above, we give below, and on page 413, two
drawings in perspective, in which the point of sight is above the plane
of the picture, and approaching in general appearance to drawings in
isometry ; and yet, having all the truthfulness of sight, are much better
suited to the purposes for which they were intended illustrations of a
specimen book of Ship-work, as manufactured by James L. Jackson &
Brother, of this city.
Centre-Board Winch.
ENGINEEKING DKAWESTG. 415
ENGINEERING DRAWING.
TEEDGOLD defines " civil engineering as the art of directing the great
sources of power in nature for the use and convenience of man, being the
practical application of the most important principles of natural philoso-
phy. * The most important object of civil engineering is, to im-
prove the means of production and of traffic in States, both for external
and internal trade. It is applied to the construction and management of
roads, bridges, railroads, aqueducts, canals, river navigation, docks, and
storehouses, for the convenience of internal intercourse and exchange; and
in the construction of ports, harbors, moles and breakwaters, and light-
houses; and in the navigation by artificial power for the purposes of
commerce.
" Besides these great objects of individual and national interest, it is
applied to the protection of property where natural powers are the sources
of injury, as by embankments for the defence of tracts of country from
the encroachments of the sea or the overflowing of rivers ; it also directs
the means of applying streams and rivers to use either as powers to work
machines or as supplies for the use of cities and towns, or for irrigation, as
well as the means of removing noxious accumulations, as by the drainage
of towns and districts to prevent the formation of malaria, and secure the
public health. This is only a brief sketch of the objects of civil engineer-
ing ; the real extent to which it may be applied is limited only by the
progress of science ; its scope and utility will be increased with every dis-
covery in philosophy, and its resources with every invention in mechanical
or chemical art."
The object of the engineer should be to arrange his material with re-
gard to economical effects. The artistic is considered to be the depart-
ment of the architect, who was originally, as the term implies, the chief
builder. It is not indispensable that the engineer should know any thing
of architecture in its signification of a fine art though it would be better,
in many instances, if he did ; when properly carried out, all his works
416 ENGINEERING DRAWING.
should show that beauty which is inseparable from truthfulness of design
and fitness for purpose. The engineer, according to Ferguson, should be
" the architect who occupies himself more especially with construction and
the more utilitarian class of works, and the architect the artist who at-
tends to the ornamental distribution of buildings, and their decoration
when erected."
There is little in the present volume, except the artistic portion of
Architectural Drawing, that might not well come under the general head
of Engineering Drawing ; and it is not, therefore, proposed to repeat what
has already been treated of, but to illustrate in a similar way other depart-
ments, especially of hydraulic and railway engineering, with numerous
examples, drawn mostly from practice in this country, reference being had
to the rules and principles of construction, before given, with such addi-
tions as may seem necessary.
There is no branch of engineering more important than the securing
of a good foundation, and none so difficult for the application of rules.
The general requirements are stated in the beginning of Architectural
Drawing, and the expedients commonly adopted, but the importance of
the subject demands further illustration.
Piles are used either as posts or columns driven through soft earth
to a hard bottom, or depending on their exterior frictional surface to give
the necessary support, either in earth naturally compact or made so by
the driving of the piles. In the first case, care must be taken that the
piles be driven sufficiently deep into the lower strata to secure their ends
from slipping laterally, and soundings should be made carefully to ascer-
tain the dip and character of this strata. In many places, from the hard-
ness and the inclined position of the lower strata, this kind of foundation is
inapplicable and unsafe.
In the case in which the support from the piles depends on the exterior
frictional resistance, the rule most generally adopted by engineers is that
of Major Saunders, published in the Journal of the Franklin Institute for
1851:
Multiply the weight of the ram by the distance which the ram falls, in
inches, at last blow, divided by 8 times the depth driven or set at that
blow. Thus, suppose the ram to be 1600 Ibs. weight, the fall 20 ft., or 240*,
and the set \ inch, then the safe load would be -. 5- =96,000 Ibs.
2"X O
According to Claudel, the weight on a pile should not exceed about
800 Ibs. to the square inch. The above pile should therefore be A |^^ JL =
120 square inches, or say 12 inches diameter.
ENGINEERING DRAWING.
41T
The usual weight of the ram, or hammer, employed on our public works,
varies from 1,400 to 2,400 Ibs., and the height of leaders, or fall, from
20 to 35 ft. Many have been made much higher ; but Mr.W. J. McAlpine
states, as the result of his experiments, that " there is no increased force of
blow obtained by a fall of more than 40 feet, as the friction on the ways is
increased so rapidly that no increased velocity is obtained by falling from a
greater height." The leaders required to drive piles on N. Y. State canals
were to be 35 feet high, and weight of hammer not less than 1,600 Ibs.
But there is an advantage gained by increasing the weight of the hammer
and reducing the fall ; there is less damage done to the head of the pile
and to the iron rings or hoops for the head, and the more frequent the
blows the more readily the pile is driven. Piles were driven by the Nas-
myth machine in 7 minutes, while an hour was required for a similar pile
by another machine.
In specifications piles are usually required to be smooth and straight,
and at least 10 inches diam. at the lower end, and that there should be only
so much set, say an average of 1" per blow for the last 4 blows. This last
will be left somewhat to the discretion of the engineer after watching the
driving a few of the piles. Piles driven beneath the surface of the ground,
and always kept wet, may be of any timber that is straight enough, and
will stand the blows of the hammer but exposed .piles should be either
oak or chestnut.
As usually driven, and of average size, when the whole weight is to be
supported by the pile, 10 tons may be reckoned as a safe load ; but where
additional support is received from the compacted earth, or from concrete,
Fig. 1. Fig. 2.
it would be impossible to assign a limit based on the pile alone ; to those
under the bridge of Neuilly, there is a load of 57 tons for each pile.
Sheet-piling (figs. 1, 2) is usually of plank 2 to 3 inches thick, set or
27
418
ENGINEERING DRAWING.
driven. For driving, the bottom of the plank should be sharpened to a
chisel-edge, a little out of centre toward the timber side, and cornered
slightly at the outer edge, that it may hug the timber and the plank before
driven. I have driven successfully, by a steam pile-driver, timber sheet-
piling 6 inches thick, and from 25 to 30 feet long. The necessities of the
case required prompt action timber was taken of such variety and such
width as could be procured each pile, of course, being uniform in width,
but varying from 6" to 18" for different piles. Southern pine was found to
drive the best. At first each pile was grooved V x Z" (fig. 3), and 2" x 1"
slip-tongue inserted, and strongly spiked, but as time pressed, the groove
was formed by 2" strips planted on.
Hollow cast-iron piles have been driven by exhausting the air 'from
the inside ; then the weight of the pile, and sometimes an added load,
cause the pile to settle into the earth : this is called the vacuum process.
The process by plenum is by expelling the water out of the pile by forcing
in air in excess of the pressure of the
surrounding water, and the workmen
descending within the pile and excavat-
ing the material. This last process was
adopted by Mr. McAlpine for the piers
at the Harlem River bridge. A section
of a pile is given (fig. 4). It consists
of a series of hollow cylinders 6 ft. in
diameter, 9 ft. longxli-" thick, bolted
;J together by flanges in the inside. On
^ the top of the pile is fastened a wrought-
H iron cylinder (a), with a cast-iron head,
called the air-lock. In the top and
bottom head there are man-holes (b b)
which can be closed at pleasure by plates
opening by hinges on the lower sides,
and lined with rubber at the joints; in
the same head there are also two cocks,
2" diam. Leading from the outside of
the air-lock, near its bottom, are two
curved tubes (c c), 4" diameter, passing
through the lower head, or diaphragm,
of air-lock, and are closed.by cocks. To
one of these tubes small air-pumps are
Fig 4 connected by flexible hose. After shut-
. ENGINEERING DRAWING. 419
ting the man-hole in the diaphragm, and starting on the pumps, the pressure
in the pile commences to increase, and the water is forced out beneath the
bottom, till the pressure is equal to that due to the head of water outside,
and the water is all out. The workmen now enter the air-lock, close the
man-hole in upper head, open the 2" cock in the diaphragm, and equalize
the pressure in the air-lock and pile. They now open the man-hole in the
diaphragm, and descend to the bottom of the pile, and commence the ex-
cavation of the material, which is raised in canvas bags by windlass into
the air-lock, and removed at any time outside by closing the diaphragm
man-hole, equalizing the air-lock pressure with that outside, and then
opening the upper head man-hole. By reversing the operation, as in the
descent of the workmen, material can be taken into the pile.
When the pile has been entirely cleaned to the bottom, care is taken to
see that no obstructions, such as boulders or logs, remain under the edge
of the pile. The workmen then pass out to the outer air as did before the
material. Men are then stationed at the guy -ropes, the 4" cock connec-
tion with the outer air is suddenly opened, the air rushes quickly out of
the pile and falls with an effect like that of a blow ; the water at the same
time rushing in beneath the bottom, bringing in the earth, and leaving an
excavation for the settlement of the pile. The amount of settling at one
time has sometimes amounted to 12 ft.
By repetition of this operation, the pile may be sunk as far as may be
required. It is then to be filled with concrete or masonry. But Mr. McAl-
pine, by driving under the loVer edge of the pile (wooden sheet piles 5 ft.
long x 3" wide x 1 thick) at an angle of about 30, and excavating be-
neath and filling in with concrete, has managed to extend the base very
considerably. As soon as the concrete or masonry has so far progressed
that an effectual stopper is made at the bottom of the pile, the air-lock is
removed, and the work is carried on on the open pile.
When the hard bottom is not very deep below the level of the water, it
is often usual to curb out the site of the foundation by sheet-piling, or by
a double curb of sheet-piling set one within the other, the space between
the two being filled in with clay or some compact earth, or a coffer-dam.
The water is then kept pumped down, and the material excavated to the
required depth, and the structure commenced and carried up. It will be
necessary to brace the coffer-dam or sheet-piling curb in the inside, to re-
sist the pressure of water and earth on the outside as the work progresses.
The general plan adopted by Mr. G. A. Parker, in the erection of the
piers of the Susquehanna bridge, was first to dredge away as much
as possible of the material in the bed of the river at the pier site. A f- "-
4:20
ENGINEERING DRAWING.
thick boiler-iron curb was then sunk and secured in its place. The curb
was about 30 ft. wide and 50 to 60 ft. long, and of sufficient height to
reach above the bed of the river. The material was then pumped by sand-
pumps out of the curb, which gradually undermined, and settled down to
the required depth, or on to the bed-rock. When stumps, logs, or boulders
were met with, they were removed by divers working in a bell. After the
rock had been thoroughly cleaned off, it was brought to a uniform level by
a solid bed of concrete extending over a greater space than the size of the
bottom of the pier, using the diving-bell for this purpose.
Three guide-piles on each side, and one at each end, were fixed firmly
in position. A strong platform of solid timber, the size of the bottom of
the pier, was then placed in position over the curb, and at the surface of
the water. On this was placed a caisson of iron large enough to contain the
pier, and with sides and ends high enough to reach to the level of high
water after the caisson is landed on the bottom. The caisson was then made
water-tight. The bottom was then floored over with masonry and stone,
and laid in mortar up the sides of the caisson to the top, thus constituting
a stone caisson inside of an iron one. This was secured to the guide-piles,
and the masonry of the pier proper was laid up, the caisson sinking as the
weight of masonry inside increased, until it finally settled upon the bottom
which had been prepared for it, as already described. At some of the
piers (figs. 5, 6) screw-rods were used to suspend the pier and gearing
ill
Fig. 5.
Fig. 6.
attached, governed by one man, who at pleasure could raise or lower with-
out assistance the whole pier. The rock was reached, and the masonry
founded upon it at five of the piers. At the other four, after the dredging
had been finished, and the curb was sunk to the required depth, wooden
. ENGINEERING DRAWING. 421
piles were driven, and cut off under water by machinery just above the
ground, and the platform, with its incumbent pier, lowered upon them.
Plate CX. is a transverse section of the river-wall Thames embank-
ment, Middlesex side. It may be said to be a wall of concrete, &c., faced
with granite, with a sewer and subway within the same, -both enclosed by
brick-work. In the drawings the different material is represented by dif-
ferent shadings; and letters g granite, bb brickwork, cc concrete. To
make a drawing effective it is often expressed by tints of col-
ors suited to the material, and the concrete with dots, thus
This plate and the following are not only to show the im-
portance of the work, and how it has been treated, but to call
attention to the extensive use of concrete, and show what dependence is
here put on a material that is very little used in such situations in this coun-
try. Extracts are given from the specifications, to explain the construction :
" The embankment-wall is to be formed within iron caissons or coffer-dams, as the
engineer may direct. Should caissons be adopted, the backs and fronts thereof are iu
all cases to be left in the ground, up to a level of 8 ft. below datum. Should coffer-dams
be adopted, the whole of the piles are to be cut off to such a level as the engineer may
point out, that no danger may arise to the several works from any drawing of piles. As
soon as the excavations shall have been made to the requisite depths, and the works
cleared of water, the trenches shall be filled up with concrete to a level of 12| ft. below
datum, and a bed dressed to the proper slope and level for the footings of the brick-
wall. This wall shall be formed thereon (when the concrete has become thoroughly
bard and consolidated) at a true slope in sets-off, as shown on drawing. The brick-
work generally shall be laid in courses at right angles to the face of the wall. The
low level sewer is to be formed on concrete foundation carried down as shown. The
sewer shall be 7' 9" in the clear diameter for a length of 1,820 ft., and 8' 3" in diameter
for the remainder of its length, the whole to be formed in brickwork 1 ft. 1J" thick.
The subway shall be formed 7' 6* high by 9 ft. wide in the clear, generally ; the side-
walls to be 18", the arch 1' 1|" thick. The subway-sewer and river-wall shall be tied
into each other, at intervals of 6 ft., by cross or counterfort walls 18* thick, extending
from the brickwork of the wall to a vertical line 9" beyond the side of the sewer farthest
from the said wall, and from footings 9 ft. below datum, which are to be bedded on a
concrete foundation 12" thick, up to the under side of the subway. The upper arch of
the subway, and all other similar arches, shall be coated on their outside circumference
with a layer of Claridge's Patent Seyssel Asphalte, 1" thick, laid on hot, and returned
up all spandrel walls rising above the arch to a height of 9". The river-wall shall be
faced with granite, generally to a level of 8 ft. below datum, and shall be surmounted
with a moulded parapet of solid granite ; the stones to be laid in courses, as far as prac-
ticable to the dimensions, in alternate headers and stretchers, and as shown in the draw-
ings. There are to be * lamp-pedestals, also cast-iron lions' heads, one to be fixed to
each pedestal, with cast and hammered gun-metal mooring-ring, 18" inner diameter, and
secured to the counterfort behind the subway by a cast iron washer-plate and a flat
422 ENGINEERING DRAWING.
wrought-iron mooring-bar. The lions' heads shall be thickly coated with copper on the
face and bed by the galvanic process, or cast in bronze, as may be directed by the engineer.
u The sand for the various works is to be very clean, sharp, washed river-sand. The
ballast for the whole of the concrete is to be very clean, and the material is to be well
washed before it is delivered on the works. The whole of the cement for these works
to be Portland cement of the best quality, ground extremely fine, and weighing not less
than 112 Ibs. to the imperial (striked) bushel. The whole of the concrete to be used
throughout the works to be formed of 1 measure of cement to 6 equal measures of ballast ;
for all foundation-works, and for all other work, 1 of cement to 8 of ballast the whole
to be thoroughly mixed by machines. Should any te allowed to be mixed by hand, no
greater proportion than 6 of ballast to 1 of cement will be permitted to be used. The
lime that may be required to be used in any part of the works shall be of a quality
equal to the best Aberthaw, brought in lumps fresh from the kiln, and ground upon the
works in mills under edge-runners. - The mortar to be mixed in proportions of 1 of lime
to 2 of sand, well tempered, and ground in similar mills for 25 minutes at least, adding
the necessary quantity of water from time to time. The bricks throughout to be of the
best and hardest quality ; all bats brought on the works will be at once rejected. The
brickwork to be executed in the most workmanlike manner, each course flushed in,
grouted, and finished solid ; the courses laid truly parallel, evenly, and uniformly to the
curvature of the works ; and centres in neat, close, and regular joints, kept straight or
regularly curved, as the case may be ; and the joints struck neatly and flush with the
face of the work. The work to be formed in rings, Old English or other bond, or
herring-bone courses, when and as may be directed ; the brickwork throughout to be
set in Portland cement, mixed with not more than an equal measure of sand, and which
shall not be remixed or used after it has become set. , Where, however, the works are
affected by water, or where pointing is required, neat cement is to be used.
" All granite to be used in the various works, especially when in exposed faces, to be
of uniform color, and free from stains, flaws or other imperfections, and all to be sound
and fully equal to the samples deposited at the office of the Board. In the embankment-
wall the stones are to be fine dressed, and axed on the face to the true curved batter, so as
to present a fair and perfect surface ; the beds and joints to be full and square for the
whole depth, so that, when set, the work may be close and solid throughout ; and no
joint to exceed ^" in thickness. The whole of the stones above the given level (111 ft.
above datum) to be dowelled together in bed and joints with slate-dowels, not less than
5 for every foot run of wall; each 2J* square at least, let fully 2^" into each stone, very
accurately fitted, and run in with neat cement ; the stones to be bedded and jointed in
cement, and the joints struck with neat cement.
" The cast iron shall be from the best gray cold-blast pig-iron, free from cinder,
mixed with of best scrap-iron, and smelted in the cupola. Sample-bars shall be
cast for the purpose of testing ; each bar to be 2" deep by V wide, placed on supports,
with a clear bearing of 3 ft., and to bear without breaking a weight of 3,200 Ibs. when
placed upon the centre between the supports. The wrought iron to be of the best
quality of Staffordshire, and to sustain without breaking a load of 22 tons per square
inch. The castings to be clean and sound, free from porous places, sand ( and air-holes ;
and these and the wrought-iron work to be free from all other imperfections. The whole
to be delivered on the works perfectly free from paint or other coatings."
ENGINEERING DBA WING. 423
Plate CXI., is an isometrical view of the overflow and outlet of the Vic-
toria and Eegent Street sewers in the Thames embankment. S is the
main sewer, and "W the subway shown in plate CX. ; s s s the street-sewers,
discharging into the overflow basin O ; w w the weirs over which the water
is discharged into the weir-chamber c c ; p is the penstock-chamber, which
is but a continuation of the weir-chamber. It has been attempted in the
drawing,. by breaks, to explain, as far as possible, the whole construction;
its purpose and mode of action are perfectly simple. Whenever, from
storms, the discharge from the street-sewers (s s s) is greater than can
be carried off by the main sewer (S), the water rises in the overflow-
chamber (0), passes over the weirs (w w) down into the weir-chamber (c),
then into the penstock-chamber, and through the flap-gates (g) into the river.
Brief extracts from the specifications will be found useful for further
explanation :
" The foundation to be of concrete, not less than 2 ft. in thickness ; upon this brick-
work shall be built for the flooring of the chambers, and for the side-end and weir-walls.
TLie weir-chamber shall be divided in the direction of its length, by a brick-wall,
into two rectangular overflow-channels, covered with cast-iron plates, 6' 8J" long, 3'
wide by Y general thickness, with strong ribs and flanges on the under side, properly
bolted together and jointed with iron cement, and bolted down to stones which
are to be built into the under side of the brickwork of the basement-chamber. Arches
on either side, running parallel thereto, and communicating with this chamber and
with the weirs which are to be formed, upon which weir-walls, divided so as to
correspond with these arches, are to be built in brickwork, capped with granite
blocks, 4' long, 2' deep, and 2' 3" in the bed. The floor of the penstock-chamber to
be formed with York landings, 6" thick, having a fall of 3" to the river. The outlets
for the penstock-chamber through the river-wall shall be formed by an arch-recess in
granite, and fixed with two tidal flaps, well hung, and firmly secured to the masonry
by strong bolts and screws.
" The subway is to be continued over the low level sewer, and across the overflow-
chamber, by cast-iron plates, curved to the form of the arch, I" general thickness, with
strong ribs and flanges on the upper side, properly bolted together, and strongly bolted
down to the brickwork ; jointed with iron cement, and covered with brickwork, to
form the floor of the subway. From a point of 10' 3" on either side of the central lon-
gitudinal line of the chamber, where the sewer and subway are farthest from the river-
wall, these are again to be brought into their general position by two curves, each not
less than 80 ft. in length.
" The whole of the cast iron shall receive one coat priming of red lead and linseed
oil, and three coats best coal-tar, before fixing ; and the accessible surfaces one further
coat best coal-tar, when fixed."
Fig. 7 is a section of the crib-pier erected on the west bank for
the Quarantine establishment for the port of New York, after plans by
424
ENGINEERING DRAWING.
Mr. J. "W. Ritch. The structure consists of an outer wall of crib-work,
with an interior filling of sand. In outline it is like a bridge-pier, 228 ft.
Fig. 7.
wide by 488 ft. long, measuring between the extreme points. The inter-
ties occur at intervals of 6 ft. spaces, or 7 ft. centres.
Extracts from specifications :
" The exterior wall to be built in blocks up to low water, of about 80 ft. in length,
sunk to a line, and to be filled up to low water with stone-filling, before the remaining
portion is commenced. From the low water the construction of the exterior wall will be
continuous, breaking the joints of the logs throughout the entire length. The base of
the blocks will be formed with timbers, 14" square ; two rows on the outside, held to-
gether with interties of timber, 12" square, each end dovetailed into the outside, and
shiplapped to the other timbers, secured at each end and intersection with iron bolts, 1"
square, 14" long, well driven home. The standards are to be 10" square at the lower end,
and long enough to reach above low water all to be placed about 40 ft. apart, and to be
secured to the timbers with 1" square bolts, 14" long.
"The cribs of the entire exterior wall, from the foundation to the top, to be built with
sound white pine or spruce timber, 12" square, laid so that they touch each other; se-
cured at every crossing or intersection, and in the centre between each crossing, with
iron bolts, 1" square, 20" long. Eacb end of each floor-log to be secured to the interties
with iron bolts, 1" square, 12" long. Deck-timber to consist of one under tier of cross-
timber, and one upper tier of cross-timber, each 12" x!2", one tier of ranging timbers,
each 14" x 14". The cross-timbers to be all in one length ; the ranging timbers to be in
lengths of not less than 46 ft. ; joints broken over the logs below. The cross-timbers to
be dovetailed at the ends, and shiplapped at intersections. The under tier of timbers to be
ENGINEERING DRAWING. 425
secured to the logs below, the ranging timbers to the under tier, and the upper tier to the
ranging timbers, as follows, at each end and every crossing with an iron bolt, 1" square,
21" long, well driven home. Also 4 stay-plates to each row of cross-timbers, each plate
to be |"x3i" iron, 8 ft. long, secured to timber and logs with 9 iron spikes, f square,
10" long. The string-piece to be 12" x 12", secured to timbers below, every 7 ft. with
iron bolts, 11" square, 30" long, well driven. Tlie entire deck to be covered with white
pine plank, averaging 12" wide, 4" thick, secured by -" iron spikes, 11" long. The entire
exterior to be close fendered, extending from the deck-plank to low water, with sawn
white-oak plank, 5" thick, and not over 12" wide ; each plank to be secured with 7 iron
bolts, 3" square, 15" long. The 6 corners of this fendering to have each 3 iron bands, 5
ft. long on each limb, 3|" x 1" counter-sunk holes to receive 5 iron bolts, J" square, 15"
long, in each limb.
" Each crib to be tilled, from the floor-logs to within 6" of the deck-plank, with stone,
granite, gneiss, or trap-rock ; none of the stone to be more than 2 ft. in any direction.
The entire exterior to be protected with stone, in large pieces .done in riprap, extending
from the base horizontally at least 13 ft., and perpendicularly up to high water, each
stone to be not less than 1 ft. thick, 5 ft. long, 2 ft. wide. The entire space inside the
wall to be filled, up to the level of the deck-plank, with sand to be dredged from the
shoal."
Plate CXII. is a section of the dam across the Connecticut Eiver, at
Holyoke, Mass. This dam is 1,017 ft. long between abutments, and
averages 30 ft. high by a base of 80 ft. It is constructed of timber crib-
work, loaded in with stone for about | its height. The foot of each rafter
is bolted to the ledge, and all timbers at their intersections are treenailed
together with 2" white-oak treenails. The inclined plank-face is loaded
with gravel, and the joint at the ledge covered with concrete. The
lower or base-tier of ranging timbers were 15" x 15" ; the other tim-
bers, 12" x 12". The rafters are placed vertically over each other, in
bents of 6 ft. between centres. The planking was of hemlock, 6" thick,
with oak cross-planking at crest of dam, 4" thick at bottom and 8" at top.
The crest was plated with iron, \" thick, 5 ft. wide. During the construction
the dam was planked first about 30 ft. on the incline ; a space was then left
of about 16 ft. width by sufficient length, through which the water flowed ;
and the balance of the dam was then completed. A plank-flap was then
made for the opening, and when every thing was ready, it was shut down,
and the pond filled. The dam was built under the direction of the late
Mr. John Chase, and since its construction the greatest depth of water
passing over the crest during a freshet was 12' 6".
Fig. 8 is a section of part of the dam across the Merrimack Eiver,
at Lowell, built under the direction of Mr. Jas. B. Francis. It was laid
dry, with the exception of the upper face and coping, which was laid full
in cement.
426
ENGINEERING DRAWING.
The horizontal joints at the crest were run in with sulphur. The coping-
stones were dowelled to the face and together, and clamped to an inclined
Fig. 8. Scale : {" = 1 ft.
stone on the lower slope, the end-joint between these stones was broken
by making every alternate lower stone longer, and the upper shorter, than
shown in the drawings.
The Cohoes Dam (fig. 9) was built under my direction, directly below an
old dam of somewhat similar construction to that of Holyoke. The old
dam had become very leaky and worn, and the overfall had in many
places cut deep into the
rock, and in some places
within the line of the dam.
It was therefore proposed
to make the new dam, as a
roll to the old one, to dis-
charge the water as far from
the foot of the dam as pos-
sible, and to keep the old
Fig. 9.-Scaie: T y = ift. dam for the protection of
ENGINEERING DRAWING. 427
the new. The exterior of the dam was of rock-faced ashlar ; the caps were in
single length of 10 ft., and none less than 15" thick and 2 ft. wide ; they were
dowelled together with two galvanized wrought-iron dowels each. The
whole work was laid full in cement, the 20" wall next the old dam being
laid distinct without bond into the rest of the work. The whole was
brought up to the outline, to receive the cap-stones, which were bedded in
cement ; the top-joints were then run or grouted in neat cement, to within
about 6" of the top of the stone, which was afterward run in with sulphur.
Entire length of overfall, 1,443 ft. ; average depth below crest of dam, 12 ft.
In the examples of dams given, the foundations of all have been upon
ledge, and where the body of water which may at any time discharge
over the dam is large and the fall high, it is especially desirable to secure a
location where the overfall can be upon solid rock. If there be ledge at
the side of the river, and none can be found in the channel, it is often bet-
ter to make a solid dike across the river and above the level of freshets, and
cut the overfall out of the bank. "When from any circumstances the dam
can have only an earth foundation, an artificial apron, or platform of tim-
ber or rock, is to be made, on which the water may fall, and break up a
high fall by a succession of steps. In some cases, a roll or incline, like
that given in the new Cohoes dam, is extended to the bed of the stream,
and continued by an apron. The water thus rolls or slides down, and
takes a direction, as it leaves the apron, parallel with that of the bed of the
stream. But care must be taken to protect the outer extremity of the
apron by sheet-piling and heavy paving, as the current, by its velocity,
takes with it gravel and all small rocks, and undermines the apron.
Fig. 10 is a section of the dam across the Croton Eiver, constructed
under the direction of Mr. John B. Jervis, for the supply of the aqueduct
for the city of New York. This dam was built on an earth foundation,
with curved roll in cut stone, extended by a timber-apron some 50 ft., sup-
ported by strong crib-work. Originally there was a secondary dam still
farther down, to throw back-water on this apron. In the erection of this
dam, excavation was made of all loose material; the cribs C and D
were built up, and the tops, were planked ; on this planking were carried
up the cribs F and G. Between these piers the space E, as well as e be-
low and on the cribs, was filled in with concrete ; on this the body of the
dam was erected in stone-masonry, laid in cement. The face-work of
granite is cut to admit of a joint, not exceeding T \ of an inch. Above
the dam is an 'earth embankment; its upper part protected by a rubble-
paving. The radius of the granite-face is 55 ft., and the dam 38 ft. high
from level of apron to crest of dam.
428 ENGINEERING DRAWING.
Fig. 10.- Scale: s y = 1 ft.
Dams are constructed to pond water for the supply of cities and towns ;
for inland navigation, by deepening the water over shoals, and the feeding
of canals ; and for power in its application to mills and workshops. To
whatever purpose the water is to be applied, there are two questions to
be settled : Whether the level will be raised high enough by the construc-
tion ; and whether the flow of the stream be sufficient for the purpose re-
quired ; and further, it may often be important to know how large a pond
will be thus formed, how ample a reservoir for unequal flow, or intermit-
tent use. If the pond be small, so that the water cannot be retained, but
the supply is only the natural run of the stream at a higher level, then the
minimum flow of the stream is the measure of its capacity.
For the measuring or gauging of small streams, a rectangular notch, or
weir, in the vertical plane side of a dam or sheet-piling across the stream,
is by far the most convenient apparatus ; for the calculation of the dis-
charge of which formulae have been established by extensive experiments,
both here and abroad ; none of which have been of more practical value
than those conducted by Mr. Jas. B. Francis, and published by him, in
" Lowell Hydraulic Experiments." The formula given by him, which is
most generally applicable, is
Q = 3.33 VTL (L 0.2 H) H
in which Q is the discharge, per second, in cubic ft. ; L the length of
weir ; and H the height of water above the edge of weir, in feet.
Thus, if the weir were 8 ft. long, and the depth over the weir 0.81
ft., then vH = 0.9 L 0.2 H = 8 0.162 = 7.838.
Q = 3.33 x 9 x- 7.838 x 0.81 = 19.03 cubic ft.
ENGINEERING DRAWING. 429
This rule is based on experiments in which the contraction was com-
plete both at sides and at bottom, the up-stream side of the dam was a
smooth and vertical plane for the distance of at least twice the depth
of H below the edge of the weir, and three times li on each side ; and
also the edge of the weir was not so thick, either at bottom or sides,
as to form a short pipe for contact with the water, but the aperture was
effectively as if cut in a thin plate. The water should also have a clear
overfall, so that a constant film of air may be beneath the whole sheet
of water. The depth H is to be measured in still water, say thrice the
depth H, from the weir.
0.2 H, in the above formula, is a correction of the length L,
for the effect due to the side-contraction ; if there be a contraction on
one side, it becomes L 0.1 IT, or simply L when there is no side-con-
traction.
To measure large streams of water, there are various methods of ap-
proximation by determining the top or a mean velocity in some portion
of their channel, when the sections are nearly uniform ; by finding the fall
in a determinate distance, and the average cross-section. But a course of
measurement even of a season will supply only a means of guessing, which
may as accurately be determined by an estimate of the area of the rain
shed supplying the stream, and taking one-half of the rainfall as the dis-
charge of the stream. From the knowledge of the country, we can judge
how speedily the rainfall may be discharged, and how much of the water
.of freshets can be reserved for use during the dry season. Streams
which flow from hilly and mountainous regions are subject to brief but
heavy freshets, and often to seasons of very low water ; but streams flow-
ing through plains, and connected with ponds and marshes, are slower in
discharging the rainfall and more steady in their supply. To overcome
the inconvenience resulting from fluctuations in the flow of streams, storage-
reservoirs are often constructed on the main stream or its brandies, or the
natural ponds or lakes enlarged and controlled by gates. In view of the
improvement of a stream by the construction of storage-reservoirs, they may
be fairly considered an element in the calculation of the value of a stream.
Many a city or town in this country is dependent for its supply on the
ponding of a stream which in summer months would be entirely inade-
quate. Mr. Ellet proposed to construct reservoirs on the tributaries of the
Ohio, for its supply for purposes of navigation during the dry season ; and
there was completed in 1866, near St. Etienne, in France, a dam across
the river Furens, which ponds all the water of the greatest rainfall, even
of a water-spout, to prevent the destruction of property which has often
430 ENGINEERING DRAWING.
resulted from severe freshets, and to reserve the water for the useful pur-
poses of mill-power and water-supply. The Croton Aqueduct Department
are now constructing on one of the tributaries of the Croton a dam to
pond water for the increase of the city-supply.
Blodget, in his tk Climatology of the United States," says that " in this
sense of permanence as a physical fact, we may consider the quantity of
rain for a year as a surface-stratum, on the Atlantic slope and in the cen-
tral States of 3i ft., which may be diminished to half this quantity, or in-
creased to twice as great a depth in the extreme years. But with such an
average and such a known range, we may deal with the quantity as defi-
nitely as with a stream of which we know the mean volume and the ex-
tremes to which it is liable, and for many departments of engineering
these climatological measures are as indispensable as those of tide or river
hydrography."
The evaporation from a reservoir-surface at Baltimore, during the sum-
mer months, was assumed by Colonel Abert to be double the quantity of rain-
fall. Dr. Holyoke assigns the annual quantity evaporated at Salem, Mass.,
to be 56" ; but from experiments made by the Croton Aqueduct Depart-
ment, in 1864, of the evaporation from a box set in the earth-bank, and
two afloat in the upper reservoir, the quantity was found to be severally
37.12, 3T.53, and 39.97 inches.
Head-gates are constructions necessary to control the flow from the
river-pond or reservoir into the canal or conduit by which the water is to
be conveyed and distributed for the purposes to which it is to be applied.
The top of the works should therefore be entirely above the level of the
highest freshets, that no water may pass, except through the gates ; and it
is better that the opening of the gates should be entirely below the level
of the top of the dam, to prevent as much as possible the passage of drift
or ice, which are often excluded by booms and racks placed outside the
gates.
Plates CXIII. and CXIY. are drawings, in plan and detail, of the
head-gates, and the machinery for hoisting them, at the Cohoes Company's
dam.
It will be seen, by reference to the plan, that there are 10 gates. The
dimensions of 4 are 8' x 6' 6" ; and 6, 8' x 9', in the clear all of which can
be hoisted by machinery connected with a turbine-wheel at a, or separately
by hand. At b there is an overfall, at the same height as the dam, over
which any drift that is brought against the gate-house is carried. At c,
there is a similar overfall within the gates, and another at d, by which any
sudden rise of the level of the canal is prevented. At e, there is a gate
ENGINEERING DRAWING. 431
for drawing down the pond, and another at f, for drawing off by the
canal, both raised and lowered like the head-gates.
The head-gates are of solid timber bolted together, moving in cast-iron
guides set in grooves in the stone; in front of these grooves there is an-
other set of grooves (g g), which are intended for slip-planks or gates, to
be put in whenever it is necessary to shut off the water from the gates
themselves in case of repairs. Hoisting apparatus. To each gate there
are strongly bolted two cast-iron racks, geared into two pinions on a shaft
extending across the gate-space, and suppor!ed on cast-iron standards on
the piers. At one extremity of this shaft, there is a worm-wheel, driven
by a worm or screw on a shaft perpendicular to the pinion-shaft. The
worm-shaft can be driven either by a hand-wheel at one end, or by the
friction-bevel at the other. The friction-bevel can be driven in either direc-
tion by being brought in contact with one or other of the friction-bevels on
a shaft extending the whole length of the gate-house, and in gear directly
with the small turbine at a. The small turbine draws its supply through
a pipe, built in the walls, and opening into the space between the gates and
the slip-plank groove.
The sections of canals depend upon the purposes to which they are
to be applied, whether for navigation or for power: if for navigation,
reference must be had to the class of boats for which they. are intended;
if for power, to the quantity of water to be supplied, and sundry pre-
cautions of construction.
Fig. 11 is a section of the Erie Canal : width at water-line, 70 ft. ; at
bottom, 28 ft. ; depth of water, 7 ft. ; width of tow-path, 14 ft. It will be
observed that the slopes are gravelled and paved, and that the edge of the
tow-path is paved with cobble-paving, and the path gravelled. The smaller
Fig. 11.
canals of this State and of Pennsylvania are generally 40 ft. wide at water-
line, and 4 ft. deep ; the Delaware and Karitan, 75' x 7' ; the Chesapeake
and Delaware, 66' x 10' ; the ship-canals of Canada, 10 ft. deep and from
70 to 190 ft. wide.
The dimensions for canals for the supply of mills depend first, on the
quantity of water to be delivered. Their area of cross-section should be
such, that the average velocity of flow should not exceed 2 ft. per second,
and 'in northern climates less velocity than this would be still better ; it
432
ENGINEERING DRAWING.
should always be such, that during the winter the canals may be frozen
over, and remain so, to prevent the obstruction from drift and anchor-ice
in the water-wheels. The usual depths of the larger canals are from 10 to
15 ft. ; with such depths, the cover of ice which reduces the section by the
amount of its thickness, does not materially increase the velocity of flow,
nor diminish, consequently, very perceptibly the available head.
Fig. 12 is a section of the Northern Canal, at Lowell, Mass., which
may be considered a model for large works. The width at water-line is
103 ft,, and the depth 16' ; and is intended for an average flow of 2,700
cubic ft. per sec. ; the fall in the whole length of 4,300 ft. is between
Fig. 13.
2" and 3" ; when covered by ice, about 4". The sides are walled in dry
rubble, and coped by split granite. It will be observed that the portion
above, and about 3 ft. below, the water-line, or between the limits of ex-
treme fluctuations of level, is laid plumb, that the ice may have as free a
movement as possible vertically.
Fig. 13 is a section, on a scale of |" = 1 ft., of the river-wall of this
Fig. 13.
ENGINEERING DRAWING.
433
same canal, where the canal passes out into and occupies a portion of the
river-channel, and the depth of water in the canal is greater than in above
section. The main wall is in dry masonry, faced on river-side with rough-
faced ashlar, pointed beds and end-joints. The inside lining is of two
courses of cement-wall, the dry rubble backing being first laid, then
pointed with cement, against which is laid the first cement lining, which is
plastered on the inside, and the interior wall is then laid ; the granite
inside wall, above lining, is also laid in cement.
Fig. 15. Scale: r y = 1ft.
Locks of Canals. Figs. 14 and 15 are portions of plan and vertical
section of locks, taken from the general plans for timber locks on the
Chemung Canal. They represent the half of upper gates.
28
434
ENGINEERING DRAWING.
Fig. 16 is a section of one
side of the lock of the same.
Fig. 17 is the plan of a por-
tion of one of the enlarged
locks of the Erie Canal, show-
ing one of the upper gates and
the side-walls.
Fig. 18 is a cross-section of
one of the same locks, showing
the culvert in the centre between
the locks, used for the supply of Fig. ie.
the waste of the lower level, and to preserve the proper height oi water in
this level.
Fig. 18. Scale : iV = 1 ft.
ENGINEERING DRAWING.
435
Fig. 19 is a drawing, in outline, of the hollow quoin of the lock-gate
on a scale of ft fall size (Chemung Canal).
Fig. 80.-A full Bize.
Fig. 20 is a plan and elevation of pintal for heel-post of lock, with a
section of the bottom of the post. The pintal is imbedded in bottom
timber or stone, as the case may be.
Fig. 21 is a plan and elevation of the strap for the upper part of heel-
post.
Extracts from lock specifications (New York State Canals, 1854) :
" I^ocks to be composed of hydraulic stone masonry, placed on a foundation of tim-
ber and plank. The chamber to be 18' wide at the surface of the water in the lower
level, and 110' long between the upper and lower gate-quoins. The side-walls to extend
21' above the upper gate-quoins, and 14' below lower gate-quoins. If the bottom is of
earth, and not sufficient to support the foundation, then bearing piles of hard wood, not
less than 10" diameter at small end, shall be driven, to support the foundation. There
shall be 4 rows of piles under each main wall, and 1 row in centre of lock; the piles
shall be driven in rows, at 3' from centre to centre. The piles to support the wing and
breast-walls and wing buttresses, and also under the mitre-sills, to be driven in rows to
conform to the form and shape of the same. The heads of the piles to be cut off smooth
and level, to receive the foundation timbers. The foundation timbers to be 12*xl2",
and of such lengths as will exttnd from and cover the outside piles, and to be trcenailed
with a 2" white-oak or white-elm treenail. 24" long, to each pile.
436 ENGINEERING DRAWING.
" If the bottom is of earth sufficiently compact and firm to support the foundation
without bearing piles, then the foundation shall be composed of timber, 12* thick and not
less than 10" wide, counterhewed on upper side, timbers to average 12* wide, to be
placed at uniform distance, according to their width, so as to occupy or cover at least
^ of the area of the foundation, and under the lower mitre-sill to be placed side by side :
in all cases to be of sufficient length to extend across the lock to the back line of the
centre buttresses, and at the head and foot to the rear or back line of wing walls. The
timber under the lower mitre-sill to be of white oak, white elm, or red 'beech, the other
foundation and apron timber to be of hemlock. The foundation to be extended 3' above
the face of the main wall at the head of the lock, and at the foot from 25' to 30' below
the exterior wing that portion of the spaces between the timbers in all cases to be
filled with clean coarse gravel, well rammed in, or concrete. In cases where rock com-
poses the bottom of the lock, the foundation timbers, if required, shall be 10* thick un-
der the lower mitre-sill, and 8" thick at other places. Where the rock is of such a char-
acter that timber is not required for the foundation, the same shall be excavated smooth
and level, and the first course of stone well fitted to the rock.
" Sheet-piling. In all cases where rock does not occur, there shall be a course at
the head of the foundation, under each mitre-sill, and at the lower end of the wings,
and at the lower end of the apron, to be from 4' to 6' deep, as may be required in each
to extend across the whole foundation. The sheet-piling to be of 2" hemlock plank, lined
with 1" pine boards. Ditches are to be excavated to receive the sheet-piling, which are
to be placed edge to edge, and the top well secured to the foundation timber ; the spaces
to be filled up with fine hard gravel, well puddled in, or with concrete.
"Flooring. A course of 2i* pine or hemlock plank to be laid over the whole of the
foundation timbers, except a space, 8' wide, under the face line of each wall, to be 2"
white oak: the whole to be well jointed, and every plank to be treenailed with 2 white-
oak treenails at each end, and at every 3' in length, to enter the timber at least 5", or
with wrought-iron spikes, treenails to fill 1" bore. Platform for the upper mitre-sill
to be 5' 10" wide, and 6' high above foundation, and to extend across from side-wall to
side-wall, to be composed of masonry, coped with white-oak timbers, which are to extend
6" into each side-wall. The timbers to be 12" deep and 14" wide, covered with 2 courses
of 1" white-oak plank. Mitre-sills to be of best white-oak timber, 9" thick, to be well
jointed, and bolted to the foundation or platform timbers, as the case may be, with bolts
of iron, 20" long, 1" x 1", well ragged and headed, 8 bolts to each side.
"Masonry. The main walls, for 21' 6" in length, from wing-buttresses at the head,
and 32' at lower end, to be 9' 8^" thick, including recesses, and for the intermediate
space, 7" 8|" thick, with 3 buttresses projecting back 2|', and 9' long at equal distances
apart. The quoin-stones, in which the heel-post is to turn, shall not be less than 4' 6" in
length in line of the chamber, to be alternately header and stretcher. The recesses for
the gates to be 20" wide at top of wall, 12' long, with sub-recesses, 9" wide, 6' high, 10'
long, for the valve-gates. Breast-wall to commence 5' below upper end of foundation,
5' wide, 8' high, finished with a coping of cut stone. The interior wing-walls, and ex-
terior wing from main walls to the termination of first curve, to be V 6" thick, and the
running curve of exterior wing to be 6' thick on the foundation.
" Culvert "between Locks. In such cases as may be required, a culvert shall be con-
structed, to pass the water from the upper to the lower level, as follows : A foundation
ENGINEERING DRAWING. 437
of suitable timber and plank, as for lock-walls, and covering all the space between the lock-
' foundations, shall be put down. Three apertures for the sluice-way shall be made in the
head-wall with cut-stone jambs, grooves to be cut in the jambs for the sluice-gates, and
the coping to form a recess, corresponding with the grooves in the jambs; grooves to be
cut on the top and bottom coping, 1" deep, to secure the jambs. The bottom of the aper-
ture to be of cut stone, with lower corner bevelled off, over which the water will fall into
the well, the bottom of which shall be covered with a sheeting of cut stone, 6" thick.
The apertures to be 3' 6" deep, placed immediately below the coping-stone, and 4' long.
Suitable gates of plank, for regulating the water in passing the sluice, to be prepared; the
well to commence on the foundation, to be made of substantial hydraulic masonry.
" Second flooring of seasoned 2" first-quality white-pine plank, to be well jointed, and
laid on the foundation between the walls, from the breast-wall to lower end of main wall,
and also on the floor of the well, to be close and firmly jointed to mitre-sills and walls, so
as to make a water-tight flooring. The plank to butt, or the end-joints to come to the
centre of a foundation timber, and each plank to be treenailed with 2 treenails at end and
2 at every 3' intermediate : treenails 10" long, to fill !]" bore.
" Gates. The framing to be made of best-quality white-oak timber ; the cross-bar to
he framed into heel and toe posts with double tenons, each tenon to be V" long, and thick-
ness equal to the thickness of the bar, and secured with wrought-iron Ts, well bolted.
The heel and toe posts to be framed to the balance-beam by double tenons, and secured
by a wrought-iron strap and balance-rod, from the top of the beam to the under side of
the upper bar. The lower ends of the heel-posts to be banded with wrought-iron bars ;
the collar and other hangings to be of wrought iron, secured together with a double nut and
screw, and to the coping by bedding the depth of the iron in, and by screw-bolts fastened
with sulphur and sand-cement. The pivots and sockets which support the heel-posts to
be of best cast iron; a chilled cast-iron elliptical ball, 2" horizontal, and 1" vertical
diameter, to be placed on the pivot and in the socket of each heel-post, to facilitate the
movement of the gate. The gates to be planked with seasoned first-quality 2" white-pine
plank, jointed, grooved, and tongued tongues of white oak the plank to be secured by
6" pressed spike. On the chamber-side of the gates, fenders of white-oak plank, to be
put on with pressed spike."
Water, ponded by dams, and conveyed by canals for use as mill-p6*wer,
is carried within the workshops or manufactories, to be applied on water-
wheels, by some covered channels. These channels, although of various
forms, are usually designated as flumes. The common form of a flume for
the conveyance of water to breast, overshot, or undershot wheels, is of a
rectangular section, framed with sills, side-posts, and cap, and, if large
section is required, intermediate posts are set in. The sills are set, and
earth well rammed in the spaces between them ; the bottom plank is then
laid, posts and cap framed with tenon and. mortice, set and pinned, and
the plank is then firmly spiked on outside of posts and caps. The planks
are usually nearly green, jointed, and brought to close joints ; the size of
timbers will depend on the depth beneath the soil, or the insistent load.
438 ENGINEERING DRAWING.
Within the mill, and just above the wheel, the flume is framed with-
out a cover, and the posts and side-planks are brought above the level
of the water. This open flume is termed the penstock, especially ne-
cessary in the class of wheel above referred to, to secure the full head of
water.
Many flumes are made of a circular section, pipes of iron, or wood.
For the conveyance of water to turbine-wheels, wrought-iron pipes are
almost invariably used. (Plates XLII. and XLIII.) Cast iron is also
sometimes used, with flange, or hub and spigot-joints. Plank-pipes are
also used, made with continuous staves, and hooped with wrought iron :
these constructions are much cheaper, and serve a very good purpose.
The head-gates of flumes are placed at the head of the flumes, in a recess
back from the face of the canal, with racks in front to prevent the passage
of any drift that might obstruct or injure the wheel. The total area of
passages through the racks should liberally exceed the area of cross-section
of the flume, not only on account of the extra lateral friction of the rack-
bars, but also on account of their liability to become obstructed. Some-
times two sets of racks are placed in front of the flumes, especially for
turbines and reacting wheels : a coarse rack with wide passages, say 2"
spaces outside, and a finer one inside, say of " to %" spaces. The head-
gates to the flume, directly back of the racks, in their function are like the
head-gates at the dam, and are similar in construction strong plank gates,
moving in slides, vertically or horizontally, with a paddle-gate in them,
to fill the flume when empty, so that the gates themselves may be opened
without any pressure due to a difference. of head outside and inside of the
gates, and also to prevent any damage to the flume by the water-ram,
which might result from a too sudden filling of the flume by the opening
of a large gate suddenly.
Plate CXY. is the elevation and section of the head-gates manufactured
at Holyoke, Mass. G, G are plank gates, sliding laterally, moved by two
pinions, working into racks on top and bottom of gates, turned by a hand-
spike. P is the paddle-gate ; E, the rack ; F, the flume, or plank-pipe ;
A, air-pipe, for the escape of air from the flume while being filled.
Conduits for the conveyance of water for the supply of cities and towns
should always be covered, and of a capacity adapted to the quantity to be
delivered. Capacity is determined by area of section, descent or loss of
head, and length and directness of conduit.
Fig. 22 is a cross-section of the main conduit of the Nassau Water- works
for the supply of the city of Brooklyn, L. I. The width is 10' at the springing
of the arch ; the side-walls 3 ft. in height ; versed sine of invert, 8" ; height of
ENGINEERING DRAWING.
439
conduit in centre, 8' 8" ; fall or inclination of bottom, 1 in 10,000 ; full capa-
city, 47,000,000 K Y. galls, in 24 hours. In preparation of the foundations
the contract specifications required a bed
of concrete to be first laid, 15' wide ; but,
when the water was very troublesome,
it was found necessary to lay a plat-
form of plank for the concrete. The
side-walls are of stone, except an interior
lining of 4" brickwork. The arch is
brick, 12", and the invert 4" thick. The
outside of arch, as it was finished, and
each wall, were plastered over on the out-
side with a thick coat of cement-rnortar.
The concrete was formed from clean bro- Fig. 22.-scaie, j" = i ft.
ken stone, broken so as to pass through a 2" ring ; 2 to 2 measures of broken
stones were mixed with 1 measure cement-mortar. The centres of the arch-
ing were not allowed to be struck until the earth had been well packed in
behind the side-walls and half-way up the arch. In both cuttings and em-
bankments- the arch was covered with 4 ft. of earth, with a width of 8 ft.
at top, and slopes on each side of 1 to 1, covered with soil and seeded with
grass.
Fig. 23 represents a section of the Croton Aqueduct, in an open rock-
cut. The width at spring of arch, T ; versed sine of invert, 6" ; height
of conduit, 8' 6" ; fall or inclination of
bottom, about 1 in 5,000 ; flow, where
there is 5' 10" in depth of water at
centre, 60,000,000 K Y. galls, per 24
hours. The bottom is raised w T ith con-
crete to the proper height and form
for the inverted arch, of a single
course of brick ; the side-walls are of
stone, laid in cement, plastered, and
laced with a single course of brick ;
the arch is semicircular, of brick two
courses thick, with spandrel backing nearly to the level of the crown, and
earth filled on the top. In earth-cuts or embankments, side-walls were
constructed of stone, in cement; and in embankments the whole structure
rested on dry rubble-walls, built up from solid earth-foundations.
At the crossing of the Harlem Kiver, as the bridge was depressed be-
low the level of the aqueduct, the water was conveyed by two cast-iron
440
ENGINEERING DRAWING.
pipes, a a, 3' in diameter, fig. 24 ; but, as the demand for water increased in
the city, the obstruction caused by lack of capacity in these pipes has
made necessary the introduction of a larger pipe, which has been made of
wrought iron, %" thick and T' 6" in diameter ; this is supported by cast-
Fig. 24.
iron columns which admit of a rocking movement, and slip-joints are also
made in the pipe, to compensate for any expansion or contraction by
change of temperature. The pipes are enclosed in a long chamber or pas-
sage, extending the whole length of the bridge, covered by a brick arch,
laid in cement, and protected by asphalt, with a brick pavement over all.
A A are the arch-stones of the bridge, and C C coping stones, in which
are inserted the posts of a cast-iron railing.
In large works, where there is considerable length of conduit, receiving
reservoirs, within or near the limits of the city, are necessary as a precaution
to guard against accidents which might happen to conduit or dam, and cut off
the supply, and also as a sort of balance against unequal or intermittent
draught among the consumers. The size of these reservoirs must depend on
the necessities of the case, and on the facilities for construction. The capa-
city of the Kidgewood reservoir, at Brooklyn, is 161,000,000 N. Y. gal-
lons, when full ; of the new Croton reservoir, about 1,000,000,000 gallons.
Both these reservoirs are made double, that is, in two compartment;.
Fig. 25 is a section of the division-bank of the new Croton reservoir.
It is made of earth, with a puddled ditch in the centre, and slopes pro-
tected by rock-paving.
ENGINEERING DRAWING.
441
Fig. 25.
A few extracts from the specification will explain the general con-
struction of the reservoir :
" The reservoir will be formed by an exterior bank forming the outer sides of the
basin. There will be a division-bank, dividing the reservoir into two basins. All the
banks will have the inner and outer slopes of \\ base to 1 perpendicular. All the inner
or water-slopes will be covered with 8" of broken stone, on which will be placed the
stone pavement, H ft. thick. The outer slopes will be covered with soil, 1 ft. thick.
The banks, when finished, to be 15 ft. on top, exclusive of the soil on the outer slope.
The top of the outer bank to be 4 ft. above water-line ; the top of the division-bank to
be 3 ft. below water-line. In the centre of all the banks a puddle-bank will be built, ex-
tending from the rock to the paving in the division-bank, and to within 2 feet of the top
of the outer bank. It will be 6' 2" wide at top in division-bank, and 14' wide at top in
exterior bank, and 16' wide at a plane 38' below top of exterior bank. In the middle
of the division-bank there will be built a brick-wall,* laid in cement-mortar, 4' high, 20"
wide, the top of the wall to be connected with the bottom of the stone pavement; 8"
thickness of concrete is to be laid on the top of the bank, on each side of, and connected
with, this wall. On the pavement 18" thick will be laid in concrete. The slope-wall on
each side of the division-bank, 10' in width, to be laid in cement.
" Puddle-ditches are to be excavated to the rock under the centre of all embank-
ments where the rock is not over 46' below top of exterior bank. "Where the rock is more
than 46', two rows of sheet-piling are to be driven to the rock, 16' apart, and the material
between them excavated, so as to remove all soil, muck, or vegetable matter. Sheet-piling
will be formed of spruce or pine plank, 6" thick, tongued and grooved : the tongue and groove
to be 11" x 1". The earth within the working-lines of interior slopes will be excavated
to the depth of 40' below top of exterior bank, rock 36'. The puddle-ditch will be
formed of clay, gravel, sand, or earth, or such admixture of these materials, or any of
them, as the engineer may direct, to be laid in layers of not more than 6", well mixed
with water, and worked with spades by cutting through vertically, in two courses at right
angles with each other : the courses to be 1" apart, and each spading to extend 2" into
the lower course or bed. Whenever the work is suspended, the puddle must be covered
* This wall was formed of concrete.
442 ENGINEERING DEAWING.
\vith boards or earth to prevent cracking, and, whenever cracks do occur in the puddle,
those parts must be removed and reworked. The puddle will extend to all the masonry
and pipes, and along and around it and them as the engineer may direct.
" The embankments will be formed in layers of not more than 6", well packed by
carting and rolling, and, in such places as the rollers cannot be effectually used, by ram-
ming. The embankments will be. worked to their full width as they rise in height and
not more than 2' in advance of the puddle. The interior slopes of all the banks will be
covered with 8" thickness of stone, broken to pass through a 2" ring. On this will be
laid the paving, 18" in thickness, of a single course of stones set on edge at right angles
with the slope, laid dry, and well wedged with pinners."
Distribution. Fig. 26 are plans of tlie various cast-iron pipes used in the
city of Brooklyn. Figs. 27, 28, 29, 30, 31, and 32, are sections of the spigot
and faucet ends of some of the same pipes. Of these pipes there were two
classes, A and B. The A pipes were designed for positions subject to an ex-
treme head of 120', the B pipes for positions below this level, subject to a head
of from 120 to 170 ft. It will be observed (fig. 26) that the 4", 6", and 8"
pipes have belts cast on them. This was to give thickness of metal to
hold the tapped in branch. The thicknesses adopted were greater than
given by the English or French rules, but consistent with the practice in
the United States, and are not now, after trial, considered too heavy.
The formulae given by Mr. Seville are t = .0024 (n + 10) d + .33 for
pipes cast horizontally, t = .0016 (n + 10) d + .32 for pipes cast vertically.
ByM. Dupuis of the Paris Water- works, t = .0016 n d + .013 d + .32, in
which t is the thickness in inches, n the number of atmospheric pressures, ta-
ken at 33' each, to which the pipe is to be subjected, and d the diam. in inches.
For the discharge of water through pipes, the formulae given by dif-
ferent hydraulic engineers are varied. Experiments on a large scale were
made by the Brooklyn engineers on the flow through some of the Croton
and Jersey City pipes, and a formula was deduced, which in its simplest
form is expressed V = 40 (^j^j d ; and is well adapted for ordinary
purposes, where pipe has been laid some time, and has been affected some-
what by corrosion.
Blackwell's formula is very nearly Y = 48 (^y- \ d.
V = velocity in ft. per second, and all other dimensions being expressed
in ft. ; thus, if the diameter of the pipe be 10" or .83 ft., the head 100',
and the length 1,000, then ^ - 1 TW^ Or l/ ' 83 = >289 ' Y =
40 x .289 = 11.56 ft. per second, and the discharge would be the area of a
10" pipe, or .545 square ft. x 11.56 = 6.3 cubic ft. per second.
ENGINEERING DRAWING.
443
Fig. 26.
444
ENGINEERING DKAWING.
Fig. 33 is a half-plan and half-section of a 12" x 8" 4-way branch, and
fig. 34 of a 36" sleeve.
Fig. 33. Fig. 34.
In Plates XXXVI., SXXVIL, and XXXVIII., are* given the details
of one of the large 48" stop-cocks.
From the specifications of " Cast-iron Distribution-pipes and Pipe-
mains, with their Branches," etc., Brooklyn, L. I. :
" All pipes of 20" diameter and upward to be formed so as to give a lead joint of not
Jess than " in thickness all round, and not more than T V; those of 12" diameter and
under, not exceeding f", and not less than T y. The straight pipes of 12" diameter and
upward shall be cast in dry sand moulds, vertically. The smaller pipes may be cast at
an angle with the horizon of not less than 12. The pipes shall be free from scoria, sand-
holes, air-bubble;!, cold-short cracks, and other defects or imperfections; they shall be
truly cylindrical in the bore, straight in the axes of the straight pipes, and true to the re-
quired curvature or form in the axes of the other pipes; they shall be internally of the
full specified diameters, and have their inner and outer surfaces concentric. No plugging
or filling will be allowed. They shall be perfectly fettled and cleaned ; no lumps or rough
places shall be left in the barrels or sockets. No pipes will be received which are defective
in joint-room. The spigot-ends of all the branches to have lugs or horns cast on each.
Every pipe-branch and casting shall pass a careful hammer-inspection, and shall be sub-
ject thereafter to a proof by water-pressure of 300 Ibs. to the square inch for all pipes
30" in diameter and under, and 250 Ibs. per square inch for all pipe-mains exceeding 30"
diameter. Each pipe, while tinder the required pressure, shall be rapped with a linrul-
hammer from end to end, to discover whether any defects have been overlooked. The
pipes shall be carefully coated inside and outside with coal-pitch and oil, according to Dr.
R. A. Smith's process, as follows :
" Every pipe must be thoroughly dressed and made clean from sand and free from rust.
If the pipe cannot be dipped presently after being cleansed, the surface must be oiled
with linseed-oil, to preserve it until it is ready to be dipped ; no pipe to be dipped after
ENGINEERING DRAWING.
445
rust has set in. The coal-tar pitch is made from coal-tar, distilled until the naphtha is en-
tirely removed and the material deodorized. The mixture of 5 or 6$ of linseed-oil is rec-
ommended by Dr. Smith. Pitch which becomes hard and brittle when cold will not
answer. The pitch must be heated to 300 Fahr., and maintained at this temperature
during the time of dipping. Every pipe to attain this temperature before being removed
from the vessel of hot pitch. It may then 'be slowly removed and laid upon skids to
drip."
Weights of 9 and 12 ft. pipe.
9 ft. 12 ft.
1 ft. laid.
"Din
|
T r\ ' * * t
A
B | A
B
A
B
Ibs.
H>8.
Ibs.
Ibs.
Ibs.
Ibs.
Ibs.
4'
200
38.077
41.538
6'
330
360
.
49.615
57.692
8.5
8'
430
500
570
660
76.286
92.556
10.
12'
890
1,080
16.
20'
2,100
2,500
30.3
30'
3,960
4,890
53.5
36'
4,750
69.5
48'
8,300
For the removal of waste water and raini'all from houses, sewers are
very convenient in towns and cities, even before the construction of water-
works; but after the introduction of a liberal arid regular supply of water,
sewers are indispensable in removing this water after it has been used.
The ruling principle in the establishment of sewerage-works is, that each
day's sewage of each street and of each dwelling should be removed from
the limits of city and town, as far as practicable, on the day of its produc-
tion, that it should pass off before decomposition begins, that it should not
be allowed to settle and fester in the sewers, producing those noxious gases
so prejudicial to health. To attain this end, the refuse fluids must be suf-
ficient in quantity to float and carry off the heavier matters of sewage.
If the rate of inclination of a sewer be not less than 1 in 440, the ex-
perience of Brooklyn, and other cities equally well supplied with water,
shows that the fluid domestic sewage is sufficient to carry off all the
heavier matters, and keep the sewers free and clean, provided the form is
such as to concentrate as much as possible the sewage waters. Less in-
clination than 1 in 440 will require some means of flushing. The Brook-
lyn system of sewers, adopted on the report and plans of Mr. J. "W. Adams,
has been as successful as any that I know of, and is entirely different from
the former practice in this country : the principle of construction has been,
to make the sewers as small as the service required of them will admit, to
maintain as much velocity of flow as possible, so that nothing may be de-
posited, without any provision for a man entering and passing through the
416
ENGINEERING DRAWING.
sewer, which has been found by experience unnecessary. The conditions
which govern the size of the sewer are the quantity of sewage to be dis-
charged, the quantity of rainfall, and the inclination of the sewer ; but
household sewage is small in comparison with the extremes of rainfall.
The rule adopted by Mr. Adams has been that given by Mr. J. ~\V. Bazal-
gette, engineer of the London drain age- works, viz. :
3 log. A -flog, y + 6.8
10
= log. D ;
or, more mathematically expressed,
D = ( A 3 x NX 6309574) :
in which A is the area, in acres, to be drained ; !N", the distance, in feet, in
which the sewer-pipe falls 1 ft. ; and D the diameter in inches.
The value of sewers depends on the correctness of their lines, uniformity
of descent, and smoothness of interior surface. The pipes used at Brook-
lyn have generally been strong glazed earthenware pipes of 12", 15", and
18" diameter. Many cement pipes have also been used, and, in such situa-
ations as required great capacity, brick sewers were used, the leading
forms of which are egg-shaped, as in fig. 35, of which the dimensions
are as follows, the longest diameter D and
the longest radius R' being alike in each
size, and E" $ R, and D 3 times R:
Area.
R
!
D
60" diameter circular
24:808 1 74.425
48" " "
19.8465
59.54
36" " "
14.886
44. Goo
24" " "
9.923
29.77
Thickness of brickwork, 8"; boards
shown at bottom only used in cases of
soft earth for convenience of construction.
Rule to determine area of egg-shaped sewer of above section :
= 4.5942 R 2 , or nearly multiply R 2 x 4.6.
In some locations the depth did not admit of the egg-shaped sectio'n.
A circular form of 6 ft. diameter was adopted for the Union Avenue
sewer, and one of a section similar to the main conduit of the water-works
10 ft. in width and 9 ft. high, in the clear, for Kent Avenue.
Man-holes are built along the line of sewers, at a distance of from 100
to 150 ft. apart, to give access to the sewers for purposes of inspection and
removal of deposit.
Figs. 36 and 37 are section and plan of the man-hole at present used by
ENGINEERING- DEAWIXG.
the Croton Sewer Department. It consists of a funnel-shaped brick well,
oval at the bottom, 4' x 3', circular at top, 2' diameter, curbed with cast-
iron frame and covered by cast-iron plate. Side-walls, 8" thick, through
which the pipe-sewers pass at the bottom of the well. Across the open
space, the sewer is formed in brick, whose bottom section corresponds to that
Fig. 30.
Fig. 37. Scale: J" = 1 ft.
of pipe, side-walls carried up perpendicular to top of sewer ; the flat spaces
at the sides of sewer are nagged. In the figure the main sewer is 12"
pipe, with a 12" branch entering at an acute angle, as all branches and
connections with a sewer should. The short lines on the left vertical wall
represent sections of f| staples, built in to serve for a ladder. In the Brook-
lyn sewer the covers of these man-holes were at first perforated for the venti-
lation of the sewer, but this has proved to be unnecessary, and they are
now made close. If a sewer performs the function for which it is in-
tended, of removing all waste before it becomes offensive, ventilation is
not necessary.
ENGINEERING DRAWING.
Wherever necessary, from the slope and confirmation of the ground,
to remove the surface or rain water directly from the street-gutters into
the sewers, catch-basins are placed generally at the corners of streets.
Figs. 38 and 39 are section and plan of the Croton sewer catch-basins, on
a scale of \" 1 ft. The intention of the catch-basin is to receive the
street washings, retain the heaviest portion in the basin, and let the liquid
escape into the sewer. The basin in the figure is rectangular in plan, with
a semicircular end, 3' 8" in width by 5' V long ; bottom of flag, and side-
walls of brick 12" thick. It will be observed that a piece of flag is built
into the side-walls from the top, extending about half way to the bottom ;
this divides the upper part of the basin into two parts ; the sewer enters
the basin 3 ft. above bottom flag ; the dividing flag comes to within 2' 6" ;
before any water can flow out through the sewer-pipe this flag must be
submerged 6" ; a trap is thus formed, which cuts off any smell from
ENGINEERING DRAWING. 449
the sewer escaping into the street. The water is received into the
basin through the channel C, which is curbed with granite, and protected
4)y a grating. The coping (b) is of granite, and forms a portion of the
sidewalk; through this there is a man-hole cut, 16" diameter, for access
to the basin, and for removal of the deposit ; it is covered by a strong cast-
iron plate.
Gas Supply. !N"ext in importance to the necessities of a city or town
for water supply and sewerage, is the luxury of gas supply. The gas-works
should always be placed remote from the thickly-populated part of a city,
for under the best regulations some gas will escape in the manufacture,
offensive and deleterious. They should be placed at the lowest level, for,
gas being light, readily rises, and the portions of the city below the works
are supplied at less pressure than those above. Gas-mains, like those for
water, are of cast iron, and put together in the same way ; but, as they
have to resist no pressure beyond that of the earth in which they are
buried, they are never made of as great thickness as those of water-pipes.
Weight of Gas-pipes per running foot.
The quantity of gas that can be delivered through a main depends on
the size and length of the pipe, and on the specific gravity of the gas and
the pressure with which it is driven through the mains. This last is usu-
ally estimated in inches of water. The specific gravity ranges from .4 to
.5, atmospheric air being 1. The pressure is reduced by friction and leak-
age of pipes, and increases or diminishes about T ^-g- of an inch for every
foot rise or fall in the level of the pipe.
The usual house-burner consumes about 4 cubic ft. per hour. Large or
Argand burners, from 6 to 10 cubic ft.
In the formula, D = \/ r-5L for determining the size of the small
wrought iron for the distribution throughout dwellings and street-mains,
the value of C varies from .073 to .063. Without illustrating by an exam-
ple, it will be sufficient for the purpose to give a calculated discharge for
different-sized pipes of one length and at one pressure, merely stating that
the discharge will vary directly as the square root of the pressure, and
inversely as the square root of the length, i. e., under a pressure of two
inches the same pipe will deliver 1.41 (/2) times as much gas as under a
12 Ibs.
10"
50 Ibs.
16 "
12"
62 "
27 "
16"
103 "
...40 "
20"...
..150 "
450 ENGINEERING DRAWING.
pressure of one inch, and under the same pressure, but with 4 times the
length, the same size of pipe will deliver one-half I I as much gas.
Gas Service-Pipes. Under a pressure of -3^- of an inch, and with 9
length of pipe 100 ft.,
a pipe of diameter f " %" f " I" 1J" l^"
will deliver 8.6 17.5 48.3 99. 173.3 272.6 cubic ft. per hour.
Gas-Mains. Under a pressure of one inch, and with a length of pipe
of 10,000 ft.,
a pipe of diameter of 3" 4" 6" 9" 12"
will deliver per hour 401.8 826.2 2276. 6274. 12879 cubic ft.
Itoads. Under this general term are included all routes of land-travel ;
but the term " streets " is applied mostly to city, town, and suburban roads,
while "roads" and "highways" are applied to those of the country. By
an " avenue " is generally understood a wide street ; although in this city all
the streets designated by numbers or letters, running north and south, are
called avenues, and those at right angles, streets. The term boulevard is
applied to very wide avenues in which there are rows of trees. The
term street in this city is legally applied to that portion which cannot be
built on that is, the street-lines, as laid out, are the established bounds on
which buildings may be erected. The street, therefore, technically in-
cludes the street or travelled way
for carriages, and the sidewalks and
area. Our numbered streets are
usually 60 ft. wide that is, 60 ft. be-
rlKl tween brick-lines of this 30 ft. are
.- . . .,,..,..
devoted to carriages, and 15 ft. on
each side to footwalks and areas ; the
avenues and wide streets are each 100 ft. wide, with 60 ft. for carriages and
20 ft. on each side. The space occupied by areas, or from street-line to
line of fence, is usually 5 ft. ; this may be enclosed by iron fence, but can-
not be included within the building above the level of sidewalk. The
stoop-line extends into the sidewalk beyond the area-line some V to 18",
fixing the limit for the first step and newel to a high stoop or platform.
The new boulevard which is to occupy the present line of Broadway and
the Bloomingdale Koad is 150 ft. wide, of which 100 ft. are to be carriage-
way, and 25 ft. on each side for sidewalk and area, the latter not to exceed
7 ft, ; one row of trees to be set within the sidewalk, about 2 ft, from the
curb. In Paris, there is no area; the sidewalk comes up to the house
or street-line, and there is a space for trees between sidewalk and street-
ENGINEERING DRAWING.
451
curb. This space is available for pedestrians, a part being a gravel, as-
phalt, or flagged walk. The following are the dimensions according to the
law of June 5, 1856 :
Entire width Width
of boulevard and of
Width
of
Width for trees ! of
Distance of row from
avenues.
carriage-way. : sidewalk.
j trees.
Street-line.
St.-curb.
Metres.
Metres.
Metres.
Metres.
Metres.
Metres.
26 to 28
12
1
5.5 to 6.5
1.5
30 " 34
14
1 j 6.5 " 8.5
1 5
36 38
12 to 13
3.5
8. to 8.5 2-
5. " 5.5
1.5
40
14
3.5
9.5 2
6.5
1.5
1 metre = 3.281 ft.
eet-
The footwalks in this city and vicinity are generally formed of flags,
or what is here termed blue-stone, laid on a bed of sand or cement-mortar.
The flags are from 2" to 4" thick. In the more important streets the up-
per surface is axed, the quality of the stone selected, and the dimensions
are sometimes as much as 10' X 14'. Brick are often used in towns, or
places where good flag cannot be readily obtained, usually laid flatways on
a sand-bed. Granite is very often employed in business streets, in lengths
the full width of the sidewalk, and about 1' in thickness, the inner ends
resting on a cast-iron girder, and the outer on the vault wall, forming in
this way a roof for the vault, and the outer ends a curb for the street.
Curb. Sidewalk.
Carriage-wa
Fig. 42.
Curbs here are generally of flag, about 4" thick by 20" deep, extending
10* above the gutter-stone ; but where the street is nearly level, and the
452 ENGINEERING DRAWIXG.
gutter-stones have to be raised to give sufficient descent for the flow of the
water, the curbs, in extreme cases, are not more than 4" exposed. The
gutter-stones are from 12" to 15" in width, and from 3" to 5" in thickness,
laid close, and bedded in cement. The bridge or crossing stones are of
blue-stone or granite, from 2' to 15" wide, and not less than 5" thick, laid
in double rows.
Carriage-way. Most of our streets and avenues are paved formerly
entirely of cobble-stone, and, if selected so as to be of a uniform size and
shape, and properly bedded in sand, and well rammed, they form in many
places a cheap and very fair roadway ; but the cubical block stone pave-
ment of trap or granite, often called the Belgian, is -in every way to be
preferred. Mr. Kneass, the engineer of the city of Philadelphia, recom-
mends
' ; That the blocks should not exceed 3" in width, 6" in depth, nor 8" in length ;
that, as to depth, they should be uniform within ", and in length be not less than 6".
For foundations the material should be taken out' to a depth of 20" below the proposed
surface of paying, and to be made to accurately conform to shape of finished road.
After being compactly rolled with a heavy roller, it should have a covering of clean
anthracite coal-ashes, placed upon it to a depth of 10", laid on in two layers, each well
rolled ; the ashes to be scrupulously clean, i. e. free from any organic matter. Upon
this should be laid a bed of clean gravel, 4" in depth, and rolled ; upon which again
should be a layer of sand, clean and sharp, or fine-screened gravel, in which to set and
bed the stone-blocks. Each layer of ashes and gravel should in surface conform to the
outline intended for the surface of the stone. The stone should be carefully assorted,
so that, when laid across the street, the joint-lines may be straight ; and each stone
should be set on its led fair and square, so that no edge shall extend above the general
level of the surface, and the surface of each stone shall be an extension of that lying
next to it. The joints I would not make smaller than ", to be filled with sand and
grouted with liquid lime. Before grouting, the entire surface should be rammed until
no impression can ~be made on it."
Broadway, as far as* Fourteenth Street, was originally paved with
Buss pavement, consisting of blocks of granite, 9" to 12" wide X 10" to
15" long X 10" deep, in diagonal rows, resting on a concrete base, with
sand enough merely to bring the blocks to a uniform surface. These
blocks are now being replaced by granite, in rows at right angles to
street, from 4" to 5" wide, 8" to 15" long, and 9". deep, with a bed of 3"
sand on the old concrete base. " In London, the usual practice is, to set
their blocks 3" wide X 9" deep, and from 6" to 12" long, on a bed of
gravel, with a base of broken granite 12" deep.
Wooden pavements have been tried of various kinds, but the only one
in any extensive use is the JSTicolson patent, which consists of pieces of
ENGINEERING DRAWING. 453
3" plank, G" long, set on a board base supported by a sand-bed. The
plank is set on end in lines perpendicular to line of street, with, a strip of
board V wide between the rows, nailed to the blocks : the top of strip
being some 2" to 3" below top of plank. Boiled coal-tar is used Jreely
while setting the blocks, and is poured into the interstices ; the V joint is
filled with gravel, wet with tar, and well rammed.
In Paris, of late years, asphalt has been used to a very great extent,
both for foot and carriage ways. The carriage-ways are composed of a
layer of asphalt, from lijr" to 2" thick, on . a bed of concrete, or on a worn
McAdam road, over which is spread a thin coat of cement. The cement
having become dry, the asphaltic rock, reduced to a powder, is spread over
the surface to a depth of about .40 per ct. more than the finished thickness ;
it is then rammed with rammers warmed by portable furnaces, beginning
gradually, and increasing the force of the blows as the work approaches
completion. For a footway the same concrete bed is used, and the layer
of asphalt is about f". "Walks and roads have been constructed in this
country with an artificial asphalt, prepared from coal-tar mixed with
gravel. None have yet stood the test of time sufficiently.
Roads and Highways. McAdam first, in England, made the con-
struction of broken-stone roads a science, and has given his name, in this
country, to all this class of roads. lie says that the whole science of arti-
ficial road-making consists in making a dry solid path on the natural soil,
and then keeping it dry by a durable water-proof covering. The road-
bed, having been thoroughly drained, must be properly shaped, and sloped
each way from the centre, to discharge any water that may penetrate to it.
Upon this bed a coating of 3" of clean broken stone, free from earth, is to
be spread on a dry day. This is then to be rolled, or worked by a travel
till it becomes almost consolidated; a second 3"-layer is then added, wet
down so as to unite more readily with the first ; this is then rolled, or
worked, and a third and fourth layer, if necessary, added. Mc^\ dam's
standard for stone was 6 oz. for the maximum weight, corresponding to a
cube of 1", or such as would pass in any direction through a 2%" ring.
The Telford road is of broken stone, supported on a bottom course or layer
of stone set by hand in the form of a close, firm pavement.
At our Central Park, after trials of the McAdam and Telford roads,
the grayel-road (of which fig. 43 is a cross-section of one-half) was adopted,
as being, according to the statement of their engineer, Mr. Grant, " the
easiest and most agreeable kind of road for both carriages and horses ; that
it is the cheapest at first cost, and can be kept in repair at an equal if not
less cost than any other equally satisfactory road." This road consists of
454
ENGINEERING DRAWING.
a layer of rubble-stones, about 7" thick, on a well-rolled or packed bed,
with a covering of 5" of gravel. C are the catch-basins for the reception
of water and deposit of silt from the gutters ; S is the main sewer or
drain, mnd s a sewer-pipe leading to catch-basin on opposite side of the
Fig. 43.
road. In wider roads each side has its own main drain, and there is no
cross-pipe s. The road-bed was drained by drain-tiles of from 1^-" to 4"-
bore, at a depth of 3' to 3|-' below the surface. The maximum grade of
the Park roads is 1 in 20. The grades of the streets of Paris vary from 1
in 20 to 1 in -200. The best grade is from 1 in 50 to 1 in 100 ; this gives
ample descent for the flow of water in the gutters. Many of our street-
gutters have a pitch not exceeding 1 ft. in the width of a block, or 200 ft.
The grade of a road is described as 1 in so many so many feet to the
mile, or such an angle with the horizon.
Inclination.
1 in 10
Feet per mile.
528
Angle.
5 43'
I
iclination.
in 30
Feet per mile.
176
Angle.
1 55'
1 " 11
462
5
" 40
132
1 26'
1 " 14
369
4
" 50
106
1 9'
1 " 20
264
2 52'
" 57
92
1
1 " 29
184
2
" 100
53
35"
The best transverse profile for a road on nearly level ground is that
formed by two inclined plains, meeting in the centre, and having the angle
rounded. The degree of inclination depends somewhat on the surface of
the road. A medium for broken-stone roads is about \" in 1', or 1 in 24 ;
but Telford, on the Ilolyhead Road, adopted 1 in 30 ; and McAdam, 1 in
3G, and even 1 in 60. For paved streets in Paris, a crown of $ of the
width is adopted, and for McAdamized, -3-^. The inclination of sidewalks
should not exceed $" in 1 ft., and, when composed of granite, the surface
should be roughened.
The necessity of a well-drained road-bed is as important beneath a rail
as on a highway. The cuts should be excavated to a depth of at least 2
ENGINEERING DRAWING. 455
ft. below grade, with ditches at the sides still deeper, for the discharge of
water. The embankments should not be brought within 2 ft. of grade :
this depth to be left in cut and on embankment for the reception of ballast.
The best ballast is McAdam stone, on which the cross-ties are to be
bedded, and finer-broken stone packed between them. Good coarse gravel
makes very good ballast ; but sand, although affording filtration for the
water, is easily disturbed by the passage of the trains, a great annoyance
to travellers, and injury to the rolling stock by getting into boxes and
bearings. The average length of sleepers on the 4.8 gauge railways is
about 8 ft. ; bearing surface, 1" ; distance between centres, 2' 4". English
railways, 9 ft. long, 10" wide, 5" thick, 3' centres, and on lines of very
heavy traffic, 2' 6" centres. Average width of New- York railways, of same
gauge as above, for single lines, in cuts IS', banks 13' ; for double lines,
cuts 31', banks 26'. The width between two lines of track was formerly
6 ft. ; but, with the increase in overhang of cars, it is often now made 7 ft.
Resistance of WJieel- Carriages on Roads.
Soft, gravelly ground \ = .143
Gravel road ^ = .067
Broken stone on firm foundation . J- = .029
Stone pavement -fa = .015
On level railway track, and be-
low speed of 10 miles per hour -^^ = .003
Sledge on hardened snow, about J$.
The tractive force which a horse can exert steadily and continuously, at
a walk, is estimated at about 120 Ibs.
There are many circumstances in the condition of roads and of car-
riages which may reduce these resistances a little, but more probably in
practice very much increase them. Below a speed of 10 miles per hour,
the resistance of railway trains represents only the effect of friction ; above
it, the concussion and resistance of the air must come into the calculation.
Bridges. Fig. 44 is an elevation of a pile pier for a bridge. Tenons
are cut on the top of the piles, and a cap (a) mortised on. The two outer
piles are driven in an inclined position, and the heads bolted to the piles
adjacent. The piles are made into a strong frame laterally by the
planks b and c, and plank braces d d on each side of the piles, bolted
through. The string-pieces of the bridge of dimensions adapted to the
traffic, and the distance between the sets of piles rest 011 the cap. Longi-
tudinal braces are often used, their lower ends resting on the plank b
which should be, then, notched on to the piles and their upper ends
coming together, or with a straining-piece between, beneath the string-
pieces, acting not only as supports to the load, but also as braces to pre-
vent a movement forward of the frames ; as the tendency of a moving train
is, to push the structure on which it is supported forward, in railway bridges
456
ENGESEEBIXG DRAWING.
especially, great care is taken to brace the structure in every way vertically
and horizontally, laterally and longitudinally. If the plank c be a timber-
sill, and the piles beneath be replaced by a niasonry-pier, the structure
will represent a common form of trestle.
rr rr n no
Fig. 45 is a plan of one of the stone piers of the railway bridge across
the Susquehanna, at Havre de Grace. To lessen as much as possible the
obstruction to the flow of the stream, and increase of the velocity of current
in the spaces, which might wash out the foundations of the piers them-
selves, it is usual to make both extremities of the piers pointed. Some-
times the points are right angles ; sometimes, angles of 60 ; often, a semi-
circle, the width of the pier being the diameter ; occasionally, pointed
arches, of which the radii are the width of the pier, the centres being alter-
nately in one side, and^heir arcs tangent to the opposite side. It will be
observed (fig. 45) that none of the stones break joint at the angle this is
important in opposing resistance to drift-wood and ice. "When, from its
position, the pier is liable to the blows of strong field-ice, the upper point
of the pier should be formed with a long incline, so that the ice may be
raised up out of the water, and break by its own weight. It is not unusual,
in very exposed, places, to make distinct ice-breakers above each pier usu-
ally of strong crib-work, with a plank-slope like a dam, of 45, and wtih
ENGINEERING DRAWING.
457
a width somewhat more than that of the pier a cheap structure, that may
preserve a costly bridge.
Bridges may be divided into three general classes :
1st. Arch bridges whether of stone, brick, or metal wherein the
parts of the arch exert a direct thrust upon the abutments, resisted by the
inherent weight of the latter, or its absolute fixed mass, as in the case of
natural rock abutments.
2d. Girders or truss bridges, which exert no thrust on the abutments,
but rest thereon by their weight simply, the combined action of thrust and
tension by which the bridge is sustained in place being wholly within the
framing of the bridge.
3d. Suspension bridges, wherein the weight rests upon chains and
cables, the latter being secured to the natural abutments, or the tension
strain resisted by masses of masonry, to which the chains are anchored.
The calculations required for determining the stress and strains, in the
first case, are among the most delicate researches of applied mathematics.
The theory of Coulomb best fulfils the varying conditions which are found
to exist in practice, but they are much too oomplicated for insertion in the
present work ; and it is considered of more value to give dimensions of
existing bridges, with some empirical rules determined from them.
Location.
Material. ! Form of Arch.
Span.
Rise.
Depth
at
crown.
Depth
at
spring.
High Bridge, Harlem River, N. Y..
Orleans and Tours R R
Stone
U
((
Brick
u
Stone
Semicircular
u
Segmental
Semicircular
Segmental
Elliptical
Segmental
Semicircular
it '
Segmental
Elliptical
Segmental
80
27.7
82.5
24.4
13.2
60.
65.
79.
18.
63.
30.
87.
128.
124.
40.
13.5
12.2
6.6
13.6J
21.
13.6
9.
31.6
15.
16.
24.3
6.11
2.8
2.7i
4.6
1.4
1.7i
3.6
3.9
3.6
1.6
3.
1.6
4.H
5.
2.8
Uniform
4.6
7.4
4.6
Uniform
2.3
Uniform
7.H
3.7
Chemin du Fer du Nord, sur 1'Oise.
D'Enghien R. R. du Nord
Du Crochet R R
Stirling Bridge
Carlisle "
Hutcheson "
Manchester and Birmingham R. R. .
u a u a
London and Brighton R. R
" " Blackwall "
Great Western R. R
Experimental arch, designed and
built by M. Vaudray, Paris
The arch last in the list was a very bold specimen of engineering, built
as an experiment, preliminary to the construction of a bridge over the
Seine. It was made of cut stone, laid in Portland cement, with joints of
f ", and left to set four months ; the arch was 12' wide ; the centres rested
on posts in wrought-iron boxes filled with sand, and, as the centring was
eased by the running out of the sand, the crown came down -fa* ; the
458 ENGINEERING DRAWING.
joints of one of the skew-backs opening y^/' during the first day, it came
down -j-L". It was then loaded with a distributed weight of 300 tons ;
under this load the crown settled ^" more. Since then nothing has stirred,
although it was afterward tested by allowing 5 tons to fall vertically V 6"
on the roadway over the keystone. This bridge will not come within any
of the rules laid down for other constructions. It will be observed that
the rise is about -fa the span, although the usual practice for segmental and
elliptical arches is more than -jl, or within the limits of J and -J-.
To determine the depth of the keystone, Rankin. gives the following
empirical rule, which applies very well to most of the above examples :
Depth of keystone for an arch of a series, in feet, = 4/.17 x radius at crown.
For a single arch, = 4/.12 x radius at crown.
To find the radius at crown of a segmental arch, add together the
square of half the span and the square of the rise, and divide their sum
by twice the rise
Thus, the Blackwall Railway-bridge has a span of 87 ft., and a rise of 10
43^+16^ _ 1892.25 + 256 _ 6 - -,
2x16 ~~32~
To find the radius of an elliptical arch, we proceed on the hypothesis
that it is an arch of 5 centres (fig. 164, p. 72), then the half span is a mean
proportional between the rise and the radius. Thus, for example, the
Great Western Railway bridge (p. 457) is 128' span, and 24.25' rise
64* = 24.25 xR
To find the depth of keystone, by rule above, as in one of a series
d = 1/17x169 = t/2ST73 = 5.33
It will be observed that the depth of the voussoir or arch-stone is in-
creased in most bridges from the keystone to the springing course, but not
always, nor can any rule be deduced from the examples given. It is safer
so to increase the depth, and adds but little to the expense increasing the
arch-masonry, and decreasing the spandrel-backing.
Every portion of an arch in perfect equilibrium is equally pressed, the
lines of pressures passing through the centres of the joints of the several
voussoirs a condition seldom found in practice, but compensated by an
extra depth of joint, so that the line of pressures will not pass without the
limits of the joint. If an arch be loaded too heavily at the crown, the line
ENGINEERING DRAWING. 459
of pressures passes above the extrados of the crown, and the key is thrown
down, while the line of pressures Mis below the line of intrados at the
haunches, and these are thrown upward, the arch separating into four
pieces, and vice versa if the arches are overloaded at the haunches. To
prevent such effects, especially from moving loads, in construction the
arches are loaded with masonry and earth, that the constant load may
be in such excess that there may be no dangerous loss of equilibrium by
accidental changes of load.
To find the level to which the spandrel-backing should be built up,
take a mean proportional between the radius of the intrados at the crown
and the depth of the arch-stone (prop. LIV., p. 67), which is the depth of
top of spandrel-backing below extrados at crown. In practice, the upper
surface of the spandrel-backing is never level ; a certain amount of pitch is
necessary to discharge the moisture which may percolate- through the
earth-load ; but the rules above given secure plenty of load, if the outer
edge be dropped somewhat, and the inner raised a little toward the crown.
To determine the horizontal thrust of an arch, divide the span into
four equal parts, and to the weight of one of the central parts add -^ of its
difference from the weight of one of the extreme parts, multiply the sum
by half the span, and divide by the rise of the arch.
Thus the weight of one of the middle quarters of the experimental
arch (table, p. 457) for 1 ft. in width may be estimated at 12,000 Ibs., an
extreme quarter at 13,800 Ibs., the quarter of the distributed load of 300
tons, or 672,000 Ibs., divided by 12, or the width in feet of the arch, would
give 14,208 Ibs. for the ring of 1 ft. width ; the sum would then be
= 26300
6
half span = 62 ft. ; rise 6.92
= 235 - 634 ' r6 > horizontal thrust -
To determine the size of a tie-rod to resist this thrust, it would only be
necessary to divide it by 10,000 (p. 227) ; but this
thrust is to be resisted by the masonry of the
abutment and the earth-load behind it.
Thus, if fig. 46 be a section of an abutment
of an arch, the horizontal thrust exerted at T
is resisted by the mass of masonry of the abut-
ment, the tendency is to slide back the abutment
on its base A C, or turn it over on the point A.
The sliding motion is resisted by friction, being Fig. 46.
460 ENGINEERING DEAWESTG.
a percentage, say from to f , of the weight of the abutment and of half
the arch which is supported by this base ; but, in turning over the abut-
ment on the point A, the action may be considered that of a lever, the
force T acting with a lever T C to raise the weight of the abutment on
a lever A B (G being the centre of gravity, and G B the perpendicular let
fall on the base), and the weight of half of the arch on the lever A C. That
is, to be in equilibrium, the horizontal thrust T x T C must be less than sum
of the weights of the abutment multiplied by A B, and the weight of the
arch multiplied by A C. Seldom, in practice, are the centres struck, and
the arch allowed to exert its thrust, till the abutment is backed in with
earth, well rammed ; and, in loading the arch, great care is taken in com-
pacting the earth and keeping the load on the sides balanced, and not too
much at the sides without any load on the crown.
SJcew Iwidges are those in which the abutments are parallel, but not at
right angles to each other, and the arches oblique. To construct these in cut
stone involves considerable intelligence, both in the designer and stone-
cutter ; but when the work is laid full in cement, so that the joints are as
strong as the material kself, this refinement of stone-cutting is not neces-
sary. The arch may safely be constructed as a regular cylinder of a diam-
eter equal to the rectangular distance between the abutments, with its ex-
tremity cut off parallel to the upper line of road. For such arches hard-
burnt brick is the best material, the outer voussoirs being cut stone.
CLASS 2. Girder or Frame Bridges, resting on Piers.
"Whatever may be the form of truss or arrangement of the framing, pro-
viding that its weight only acts on the abutment, the tension of the lower
chord, or the compression of the upper chord, at centre, may be determined
by this common rule :
ffule. The sum of the total weight of the truss, and the maximum dis-
tributed load which it will be called on to bear, multiplied by the length of
the span, and divided by 8 times the depth of the truss in the middle, the
quotient will be the tension of lower chord and compression of upper at
the middle. In nearly all the forms of diagonal bracing, if the uniform
load be considered as acting from the centre toward each abutment, each
tie or brace sustains the whole weight between it and the centre, and the
strain is this weight multiplied by the length of tie or brace, divided by
its height. Any diagonals, equally distant from the centre, sustain all the
intermediate load, if rods, as in fig. 48, by tension ; if braces, fig. 47, by
compression.
It follows, therefore, that in all these trusses the upper and lower chords
ENGINEERING DRAWING.
461
should be stronger at the centre than at the ends, while diagonals should
be largest at the abutments. Unless the weight of the bridge is great com-
pared with the moving loads, counter-braces become necessary (p. 231).
Y
Fig. 49 is a side elevation of one-half the gallows-frame of a draw-
bridge ; fig. 50 an elevation and part cross-section of same ; and fig. 51
plan of bottom chord.
fir u
Fig. 49.
<S>O
Fig. 50.
462 ENGINEERING DRAWING.
The whole length of bridge was 13SJ' giving two clear spaces or
water-ways on each side of central or draw pier, a little exceeding 50 ft.
The gallows-frame rested on a turn-table on this central pier. The truss is
the usual form of Howe's patent. The upright posts of gallows-frame are
WxlO"; the tension rods are in pairs, 2" in diam., one on each side of
the truss, and extending from the top of frame to the bottom of the truss,
at a distance of 21 ft. from the ends. Fig. 51 explains the construction of
the chords of this bridge, how the joints are broken, and how the timbers
are clamped and bolted together.
The general rule adopted in the construction of the Howe truss is, to
make the height of the truss -J- of the length up to 60 ft, span ; above this
span the trusses are 21 ft. high, to admit of a system of lateral bracing,
with plenty of head clearance for a person standing on the top of a freight-
car. From 1T5 ft. to 250 ft. span, height of truss gradually increased up to
25 ft. Moving load for railroad bridge calculated at 1 ton per running
foot,
Extract from specifications for (road) bridge on site of Macomb's Dam,
New York city, Howe's truss :
" The spans at the ends of the draw shall each be 175' long in the clear, resting 6'
upon the piers and abutments, on suitable bolsters. The height of truss to be 20' from
outside to outside of chords. The bottom chords will each be formed of 4 timbers,
7" x 14" each. The top chords each of 4 pieces, 7J" x 12". The main braces to ave-
rage 9" x 10" scantling. Counter-braces to average 7" x 9". The suspension-bolts, or tie
rods, of which there shall be 3 to each panel of truss, to be 1 1" diam. at end of bridge,
and gradually decrease to 1 J" diam. at centre. The roadway to be 20' wide between
the chords. The floor-timbers to be 80' 4" x 10'', placed at 2' from centre to centre, to
alternately project 4' outside the chords, to support sidewalks. The space between
trusses to be planked with 3" white-oak plank; sidewalks with 2'' pine plank."
Fig. 52.
Fig. 52 is side elevation, plan, and section of cast-iron girder, adopted
by Mr. Joseph Cubitt, C. E., for railway purposes, a pair of girders for
each track, the rails being supported on wooden cross-beams.
ENGINEERING DRAWING.
463
Dimensions for different Spans.
Opening.
Bearing on
abutment.
Height of gird-
er at centre.
Top flanch.
Bottom flanch
at centre.
.
At end.
Thickness of
middle web.
12 ft.
I'.G"
I'A"
3" X IV
1'.4" x IV
1'.8" xlV
IV
30 ft.
2'.G"
3'.
5" X 2"
I'.G" x 2"
I'.10"x2"
2"
45 ft.
2'.9"
3'.9"
7" x 2V
2'. x 2V
2'. x 2V 2"
The upper flanch is made somewhat stronger than Hodkinson's rule, to withstand
the lateral strain or vibration.
Fig. 53 is a side elevation of the widest span wrought-iron truss over the
Connecticut Eiver on the N.-EL, H. & S. E.-E., designed and built by
Mr. Laurie, C. E. :
"This girder is 177' long; there are others on the same bridge much less, but the
general form adopted in all but the two shortest spans is that of a truss composed of
Fig. 53.
rolled plate angle and T iron. There are three distinct varieties of this general form
adopted for the different length of spans, by which the use of bars beyond a certain
size is avoided in the longer spans. The difference consists in the arrangement of the
tie-bars. In the span of 177', the ties cross 3 of the panels formed by the vertical posts ;
in the 140' and 88$-' spans they cross 2 panels, while in the 76' span they cross but one
panel. Where the ties cross 3 panels diagonally, the truss partakes somewhat the char-
acter of a lattice, and the principle is capable of being ex-
tended still further for longer spans, by making the tics
cross more panels.
" The top and bottom chords, of which a section is
shown, fig. 54, are composed of horizontal plates, 26" wide,
varying in thickness. At right angles to these are 2 ver-
tical plates, placed 15" apart, and connected with the hor-
izontal plates by 4 angle irons in each chord, to which
both are riveted. The horizontal and vertical plates, ex-
cept at the ends of the girders, are mostly in lengths of
15' 0", the joints coming between the posts of the truss.
At the joints in the plates there are covers to make
strength uniform. Rivets in chords 1" diam., 8-}" apart.
Batween the lower edges of the vertical plates of the
upper chords there are wrought-iron distance pieces, one
Fig. 54.
46-i ENGINEERING DRAWING.
to each panel, by which the two plates are held securely in place, and give the chords
additional stiffness. The top chord at the middle has 2 horizontal plates, 26" x f " ; 4 angle
irons, 4" x 4" x ", and two vertical plates, 15" x f ", making a sectional area of 76.2 sq. in. ;
at the end there is one horizontal plate, 26" x f " ; angle irons, 4" x 4" x V ; vertical plate,
15" x ". The bottom chord at centre, horizontal plates, 26" x f", and 26" x f " ; at ends
one plate, 26" x". Length of girder, 177'.3", in panels of 5'.3"; height 16'.9" between
horizontal plates. Width between centres of girders, 10'.6". End-posts are composed
of 6 T bars, 5" x 3$-" x \", in pairs, with 2 side plates, 25$" x " ; 1 end plate, 12V * f".
The next post has 2 T bars, 5" x 3V x \" 2 side plates, 10" x ^".
Posts 3 to 5 each 2 T bars, . . . . . . 6" x 4" x f "
6 to 8 " 2 " G" x 4" x V
" 9 to 10 " 2 ' 5"x3VxV
" 10 to centre, 2 " 5" x 3V x T y
" All the posts have diagonal bracing between the T bars ; they are divided into 5
spaces between the chords by cross-plates, 5" x T 7 ff ", with diagonals, 2V x J". Between
the vertical plates of the chords the T irons are connected by plates 13"xl2"xV.
The posts are placed between and riveted to the vertical plates of the chords. Xear
the ends of the truss there are 10 rivets on each side, top and bottom; 2 being through
the angle irons, which connect the vertical and horizontal plates, the others through
the vertical plates. The number is diminished toward the centre, according to the sec-
tional area of the post. Rivets through posts and chord, 1" diam., in diagonal bracing,
f " diam.
" The ties are in pairs : the 1st from end crosses 1 panel ; the 2d 2 ; the 3d 3. The
ties extend 2 panels beyond the middle, shown, by calculation, to be all the counter-
bracing necessary. The ties vary in width from 8" at the end of bridge to 2V at centre,
uniform thickness, f ". The ties are riveted to the outside of the vertical plates of the
chords, part of the rivets also passing through the posts inside the vertical plates. The
number of rivets in each bar is so arranged as to make the sectional area of the rivets
fully equal to that of the bar, and so placed that the effective area of the tie-bar is only
diminished by the amount of metal taken out by one rivet-hole.
" The horizontal bracing across the top and bottom of the two trusses is formed of
T bars, placed at right angles to the girders, at intervals of 10' 6", or 1 on every 2d
post, varying in size from 6" x 4" x V, at ends, to 4" x 4" x f" at middle. Between these
are horizontal diagonals of round iron, varying from iy to iy diam. The vertical
diagonal tie-rods are iy diam. at ends, the rest IV diam. Both horizontal and verti-
cal tie-rods are fitted with nuts and screws for tightening when necessary. The ends
of the girders rest on cast-iron plates ; one end is firmly fixed to pier, the other resting
on rollers.
" The superstructure of the bridge is .formed of wooden floor-beams, 17' 9" long,
7" x 12" laid across the top of the girders, 20" from centre to centre. Upon these rest
the. longitudinal stringers, 9" x 15", which support the track." Desertion of the Iron
Bridge, etc., T. G. ELLIS, C. E.
For strength of \vrought-iron box-girders, and the comparative strength
of different spans and dimensions, see p. 127.
ENGINEERING DRAWING.
465
CLASS 3. In suspension bridges the platform of the bridge is sus-
pended from a cable, or chains, the ends of which are securely anchored
within the natural or artificial abutments. From the nature of the struc-
ture, the bridge accommodates itself to each change in the load, assuming
the position of equilibrium for each particular load to which it is tem-
porarily subjected.
The curve of a suspended chain is that known as the Catenary, and, if
the whole weight of the structure were in the chain itself, this would be
the curve of the chains of a suspension bridge ; but, as a large part of the
weight and the whole of the loading lies in the platform, the curve assimi-
lates to that of a parabola, and, in all calculations, it is so regarded.
Let fig. 55 represent a suspension bridge, with the roadway or platform
F L, and A D B C being the curve formed by the chain.
Fig. 53.
To determine the form of the chain, calculating the position of the
point in reference to the horizontal line F L : the data for this are the semi-
span A E, the deflection E B, and the length of the shortest suspension
rod B II.
To find the length of any suspension rod, viz., D G, subtract the length
of the shortest suspension rod *B II from the deflection E B, multiply the
remainder by the square of the horizontal distance D K, and divide by the
square of the semi-span A E ; to the quotient add the length of the short-
est rod B II, and it will give the length of the suspension rod D G. In
the same way, any number of points in the curve may be determined,
through which the curve can be determined.
For the strain of tension on the chain at the points of support A
andC:
Rule. Add together four^imcs the square of the deflection (E B) 3 and
the square of half-span (A E) 2 , and take the square root of this sum ; mul-
tiply this result by the total weight of one chain and all that is suspended
from it, including the distributed load, and divide this product by four
times the deflection (E B) of the cable at the centre, and the result will be
30
466
ENGINEERING DRAWING.
the tension on one chain, at each point of support, A. and C. The angle
made by the chain at the point of support, viz., angle POL and the angle
of the backstays, or continuation of the chain (angle L C ^s") should be
equal to each other, in order that there be no tendency to overset the tower
C L and A F.
The horizontal pull in the direction between A and C, if the chain v:ere
fastened there, would be the tension found by the first rule, multiplied by
the cosines of the angle P C E, or the tension at the point of support, mul-
tiplied by E 0, and divided by C P. Therefore, if the main chains and
backstays make unequal angles with the tower, the difference of the
cosines of these angles will be the tendency of the towers to overset.
Having determined the strain on the chains, it is easy to estimate the
size necessary to resist it ; but, when the chain is composed of links, the
size of pins and eyes is to be in some proportion to the body of the links.
By experiment, Sir Charles Fox established a rule, that the diameter of the
pin should be -f the width of the body of the link, and the width of the
two sides of the eyes should be about 10^ greater than the body of the
link, link and eye being of uniform thickness throughout.
Main
spans.
Deflection
of chain or
cable.
No. of
chains and
cables.
Total effective Mean weight Fixed load
section of cable of cable per ft. of per ft. of
in eq. inches. span (\bs.). span (Ibs.).
Breadth
of platform
in feet.
Menai
!
570 43
16
260
880
28
Chelsea. . . .
348 29
4
230
767
47
Pesth
666
47.6
4
507
1690
9892
46
Bamberg. . .
211
14.1
4
40.2
137
1581
30.5
Freyburg . .
870
63
4
49
167
760
21.25
Niagara
Falls
821
54 and 64
4
241.6
820
2032
24
Cincinnati. .
1057
89 1 2
172.6 '
516
2580 1 36
Steam-Engines. Under the head of " Mechanics," pp. 133-136, rules
are given for determining the effective power of a steam-engine, the vol-
ume of steam required under different pressures or tensions to supply this
power, the quantity of water required for this volume of steam, the prob-
able amount of coal to produce the evaporation of this quantity of water,
and the general relative proportions of grate and heating surface for a
boiler to do this economically. It will be observed that the effective power
of an engine is measured by the area of the piston, in square inches, mul-
tiplied by the average pressure of steam, in pounds per square inch on it.
multiplied again by its travel, in feet per minute, and the product iV
the effective power in Ibs. ft per minute ; or, if the product be divided
ENGINEERING- DRAWING. 467
by 33,000, the result is the horse-power (HP) of the engine. Hence, with
the very same engine, if either the steam-pressure or travel of piston
be varied, the power of the engine is changed ; thus, an engine of 10"
diameter (A = 78.54 square inches), 2 ft. stroke, with an average pressure
of 30 Ibs. per square inch, making 50 revolutions per minute (travel,
50x2x2), would give nearly 14f HP
78.54 x 30 x 200^33,000 = 145 ;
but if the average pressure were 60 Ibs., or the travel 400 ft., the HP would
be double ; or, if pressure and travel be both doubled, the HP would be
quadrupled; and, within the limits of practice, this same engine would
give an actual 100 HP, provided there-were a boiler of sufficient capacity
to supply the steam. It is evident, therefore, that when mechanics speak of
an engine as so many horse-power, it is merely a conventional term, mean-
ing a certain size in the idiom of a particular shop, but has no general
acceptation which defines the dimensions. Among the mechanics them-
selves it is very common to give the size in inches diameter by inches stroke ;
thus, in engine above, 10" x 24", the diameter is given first; but to their cus-
tomers they speak of so many horse-power. The term was first introduced
by Watt, who made experiments on the strength of horses, and rated his en-
gines at so many nominal HP by fixed rules. Since his time, his standard
of pressure and travel has been very much increased, and the engine de-
livered for so many nominal HP, according to Watt, will give 6 to 12
times as many actual or effective HP.
In the same way it is very usual to speak of boilers as so many HP,
meaning, probably, that they have the capacity of evaporation to supply
steam for so many HP ; yet there is no standard of volume of steam or
water to supply a IIP. Under various conditions of cut-off and condensa-
tion, one engine will require -| the steam of another to develop the same
power ; and, under the varying conditions of fuel and draught, the same
boilers will evaporate twice as much water, with but little difference in
economy of combustion ; and different types of boilers, with equal and
well-proportioned grates and heating surfaces, will consume very different
amounts of coal, and evaporate very different quantities of water.
" Thus, the Cornish boilers at Jersey City "Water "Works, Belleville, N. J., are 7 ft.
diameter x 84 ft. long ; grate-surface, 20.22 sq. ft. ; single central flue, 4' 4" diameter,
through which the products pass to the end of the boilers, and then return by wheel-
draught at the sides, and back beneath boiler. Cumberland coal burned per square foot
of grate was 4,839 Ibs. per hour, and evaporation 10.50 Ibs. of water per pound of
coal.
" The drop-flue boiler (fig. 5G) at the Hartford Water Works is 7' 6" diameter, 22' 8"
468
ENGINEERING DRAWING.
long; four direct flues, 18" diameter, 14' 4" long; 1 return-flue, 12" diameter; 2 9"
and 13 8" all 12' 3" long ; grate, 23 sq. ft. The products of. combustion pass through
the 18"-flues to the back-connections, thence, dropping down, pass through the smaller
Fig. 56.
return-flues to the front-connection, thence back beneath the boiler to the chirnney-flue.
Cumberland coal burned per square foot of grate, 5.683 Ibs. ; evaporation, 10.96 Ibs. of
water per pound of coal.
" Two tubular boilers, at the Nashua Manufacturing Company's mills, Nashua, N. H.,
each 5' diameter, 20' long; 55 tubes, 3V' diameter; grate beneath boiler-shell, 4' loy x
5' 6". The products of combustion pass beneath the boiler, return through the tubes,
thence back through the flues of a heater placed above and between the boilers ; fire-
surface in heater, 160 sq. ft. Coal, anthracite, burned per square foot of grate, 5.309
Ibs. ; evaporation, 9.18 Ibs. of water per pound of coal.
" Upright cylindrical boiler, in Massachusetts Cotton Mills, Lowell, Mass., 31J"
diameter, 12' high, 3" water-space whole height of boiler ; pot or drum in centre 19"
diameter, 9' 3" high, with flue, liy diameter, through the same ; grate, 3.14 sq. ft.
Coal, anthracite, burned per square foot of grate, 13.4 Ibs. ; evaporation, 8.65 Ibs. water
per pound of coal.
" Locomotive boiler, at the Boston Cotton Mills, Lowell, Mass. The products of
combustion pass from the fire-box through 64 tubes, 2f" diameter, 13' 9" long, to a
smoke-box surrounded by water, thence under the boiler through a heater ; grate-sur-
face, 16.66 sq. ft. ; heating surface, including heater, 748 sq. ft. Anthracite coal con-
sumed per square foot of grate, 7.52 Ibs. ; evaporation, 8.48 Ibs. of water per pound
of coal.
" The temperature of feed-water reduced to the standard of 100. The first two
experiments conducted by Messrs. Copeland and Worthen, the latter by Mr. Jas. B.
Francis."
In the above experiments, although the boilers were but doing their
usual work, if we except the upright boiler, the consumption of coal per
square foot of grate is much less than in practice. At the Ridgewood
Pumping Engines, where there are drop-flues similar to those at Hartford,
the usual consumption of coal is about 13 Ibs. per hour per square foot of
grate, but with somewhat less pounds of evaporation than in the above ex-
ENGINEERING- DRAWING.
469
periment. In general too little attention is paid to the economy of large
boiler-capacity, and it may be safe to estimate, for the general boilers in use,
an evaporation of 8 Ibs. of water per pound of coal, and a combustion of 12
Ibs. of coal per square foot of grate ; and, for high-pressure engines, a con-
sumption of 5 Ibs. of coal per IIP per hour ; steam-space in boilers nearly
equal to that of water-space.
The boilers of steamships and locomotives, from want of space, have
necessarily the heating surface very much concentrated, and an artificial
draught to increase the combustion.
The following is a table of English experience, from Rankin, and gives
the comparative rate under different boilers :
Per sq. ft. of
grate per hour.
4 Ibs.
10 "
.12 to 16 "
.10 " 24 "
With Chimney-draught.
Slowest rate of combustion in Cornish boilers
Ordinary " " " " "
" " " " factory "
" " " marine "
With draught produced by blast-pipe or fan.
Locomotives 40 " 120 "
In deciding upon the size of engine and boilers to do a certain w^ork,
there is to be determined the amount of work in Ibs. ft., or HP; the Id ml
of work, whether the resistances are uniform, and, if so, what limits the pis-
ton travel, as water in a pumping-engine, or the proper speed of shaft for
a mill, or for a propeller ; or, if the resistance be not uniform, as in an
iron-rolling mill, what' velocity may be required in moving parts to over-
come the maximum resistance ; what is the space that can be occupied for
the purposes of power ; whether the engine is to be condensing or not ;
low or high pressure ; and whether economy in first cost is more desirable
than in maintenance.
The limits of ordinary travel of stationary engine-pistons is from 200'
to 500' per minute ; average pressure in low-pressure condensing engines,
from 10 to 15 Ibs. ; boiler-pressure, from 15 to 25 Ibs. Average pressure
of high-pressure stationary engines, from 25 to 50 Ibs. ; boilers, 45 to 75 Ibs.
" Fairbairn gives the following table of the safe working-pressure of boilers of dif-
ferent diameters, 44$ being allowed for loss of strength by rivet-holes :
Diam.
3
%"-plate.
118
157 25
Diam.
6.0
%"-plate.
.... 59
78.75
o 6
101
13475
6.6...
. 54.25
72.5
4
885
118
7.0
50.50
67.25
4 6
7875
104.75
7.6
47.
62.75
5
70.75
94.25
8
44.
59.
5.6...
. 64.75
85.75
8.6...
. . 41.5
55.5
470
ENGINEERING DRAWIXG.
"According to Holley ('Railway Practice'), the ordinary American plate is 4
stronger than that of the English, and for a 48"-shell it is not uncommon here to use
plates of from " to T y under a working-pressure of 120 Ibs. The thickness of fire-
plates does not seem to add to their durability. J" Lowrnoor are successfully used
here, while V Lowmoor fail after a few months' use in England. Joints of a double
thickness of metal, and rivet-heads in a fire-box, give way sooner than the single plate.
" Rivets are commonly from " to " diameter, and pitched at If" to 2" centres.
The maximum strength is obtained when the sectional area of the rivets is that of the
punched plates ; lap of plates, about 2".
u Mr. Fairbairn made experiments on the strength of flues or tubes, which he found
to be inversely as their diameters and their lengths, and directly as the 2.19-power of
fjv; 2 .10
the thickness of the plates ; collapsing pressure in Ibs. = 806,300 ; and gives the
Li J_)
following table of equal strength of cylindrical flues for a collapsing pressure of 450
Ibs. per square inch
Biain.
12"
Length of flues.
10' 20' 30'
Thickness of plate.
291" 399" 480"
Diam.
86"
Length of flues.
10' 20' 30'
Thickness of plate.
480" 659" 794'
18
350 480 578
42 .
516 707 Sol
01
399 548 659
48 .
.548 752 905
30 .-.
. .442 .607 .730
These calculations are based on the supposition that the flues are plain
cylinders ; but it is now the practice in England to make the joints as in
figs. 57 and 58. The circular joints
are made with T or fl iron, which,
in effect, is virtually shortening the
Fi s- 57 - m s-58. tubes. The T and U are in the
water-space ; a space is left between the sheets in the flue for calking ; the
horizontal joints are butt, with a single welt.
In boiler or other plate-work, where two
joints at right angles to each other are butt-
joints covered with a welt, the intersecting welts
are made by scarfing the one and chamfering the
other, as shown in section at b, fig. 59.
Plate CXYI. is longitudinal (fig. 1) and half-
\ transverse (fig. 2) sections of the fire-box of an
anthracite -burning locomotive from the !Kew-
Jersey R.-Tt., and illustrates very fully the differ-
ent kinds of stays in use in such construction.
The water-space is 4" wide in front, 3" on
In this example the in-
o o
Fig. 59.
sides, and 6" behind.
ENGINEERING DRAWING. -471
closing plates are parallel; but it is considered a very good practice to
make the inside plate overhang a little, giving a wider water-space at
top than bottom. The stay-bolts in water-space are |" diameter, and
4" centres ; they are screwed into inside, and riveted. It is common to
make these bolts of tubes, fastening them the same as the tubes in the
boiler, and closing up the end with an iron plug, except where it may
be convenient to introduce air through them into the fire-box. The bot-
tom of the water-space is made with wrought-iron ring, the inside plate be-
ing bent down a little, as shown, and the whole riveted strongly together.
The opening for the door is made by turning a fianche on the inside plate,
to which a plate-ring is riveted, which is also riveted to an angle-iron ring,
riveted to the outside plate. The crown-sheet of fire-box is supported by
cast-iron girders, extending across the boiler ; these girders are cast double,
with a space for the insertion of the bolts ; the ends of these girders rest
on the inner plates of the fire-box, but they are also supported by hangers
(h h) from the outer shell, and the inside of the steam-drum, being mutual
stays for the crown of the fire-box and roof of boiler. These hangers have
a fork at one end, through which a pin is passed to connect it with the foot
riveted to the boiler ; the other end passes into the space in the girder, and
a pin is passed through girder and hanger. It will be observed that there
are stays (s s) for the boiler-front, extending back to inclined part of the
shell, and a similar one (s) in the angle of the fire-box beneath the tubes.
For the staying of the fiat surface of boiler-fronts,
stays like the hangers are often used, at an angle con-
necting the end with upper shell or triangular plates
(fig. 60), called gussets, riveted in angle of shell. The
boiler has 130 2 // -tubes, and 26 S^'-tubes; space be-
tween tubes, -g" ; length, 10 ft.
Plate CXYII. are drawings of a tubular boiler,
with grate beneath. Fig. 1 is a side elevation of the
boiler, with walls and front in section. The boiler is Fi s- 60 -
represented broken, as the page would not, on this scale, admit the full
length.
Fig. 2 is a half-front elevation, and half section through grate and front
flue. B is the boiler ; D the steam-drum ; v is the safety-valve ; s the pipe
for steam-connection ; b for steam blow-ofi", or waste from safety-valve ; m
the man-hole, and h the hand-hole ; S the saddle supporting the rear end
of boiler ; the front is supported on frame, cast with the front f. There
are also 3 brackets riveted on each side of boiler, and resting on wall ; g
are the grates, d the fire-door, and d' the ash-pit door, w the bridge-wall ;
472 ENGINEERING DRAWING.
p is the feed-pipe for supplying water to the boiler. By an arrangement
of valve the same pipe may be also used as a blow-off, or to draw off the
water from the boiler. The products of combustion arising from the grate
pass over the bridge and around the lower hemisphere of the boiler to its
rear, thence through the tubes to the front flue, and thence by the side flue
f ' into the chimney. Access is had to the flue f,' and the tubes, through
the door c hung on the front. It will be observed that a part of the
boiler projects over the flue f,' forming what is called by mechanics a hog-
nose; but by many this is omitted, and only the shell of the boiler projects
the entire circumference, with a pipe shown by the dotted line, connecting
with a larger horizontal circular flue, passing over and receiving like pipes
from other boilers in the row. To protect the boiler from loss of heat by
radiation, it is the most common practice to cover the upper half of the
shell with ashes to the depth of from V to IS", and the steam-drum by
felting. The side-walls by the grate, the bridge-wall, the front, and all
parts exposed to the direct action of the fire, are lined with fire-brick.
Many boilers, like locomotives, are not set in brickwork ; these it is
usual to cover with felting. The protection of all parts of boilers and
steam-pipes exposed to the air by some cover of a non-conducting ma-
terial adds much to economy in the consumption of coal, and dryness of
steam.
Plate CXVIII. are sections of two chimneys. Fig. 1 is a section of a
chimney attached to an English gas-house, taken from "Engineering,"
and fig. 2 a section of the chimney at the Ridgewood Pumping-eugine
House.
Fig. 1 is given as an example of a very neat chimney, uniform flue and
shell, additional strength being given by the buttresses shown in section,
fig. 3. It differs from the chimneys usually constructed, in having no in-
dependent flue inside, as shown in the section of the Ridgewood chimney,
which can freely expand with the heat without affecting the outer shell.
Fig. 4 is an elevation of the Ridge wood chimney, at the point where the
square base is changed into an octagonal. Fig. 5 is a section of the shaft,
but the flue should have been represented circular.
For the area of chimney-flues one square inch for every pound of coal
burnt per hour on the grate has been found to answer well in practice.
Chimneys are constructed of various sections, sometimes uniform through-
out their length, sometimes tapering at the top, and sometimes bell-
mouthed ; all answer the purpose. The great point to be observed is, that
there be no abrupt changes of section or direction, and that they be carried
well above all disturbing causes.
ENGINEERING DRAWING.
473
Plate CXIX. contains a side-elevation and some details of a simple
form of stationary engine. Fig. 1 is the elevation in which S is the steam-
cylinder, p the piston-rod attached at its outer end to a cross-head h, slid-
ing on the guides g ; the connecting rod r connects the cross-head and
crank c on the fly-wheel shaft. The fly-wheel is only shown in part ; on
the same shaft is an eccentric giving motion to the valve-rod r, and a pulley
to drive the governor with not shown on drawing. The rod e hooks on
to a rocker, r', to which is attached the valve-rod by means of the handle
h' ; the rod e can be unhooked from the rocker, and the valves moved by
hand. The steam is introduced through the steam-pipe s beneath the
cylinder, and exhausted through the pipe E. The cast-iron engine-frame
F rests upon a stone base, B, on a brick or stone foundation, to which it is
strongly bolted, and which is shown separately (plate CXX.).
Fig. 2 is a section of the cylinder ; P the piston, C the steam-chest, v v
the valves, S the steam-connection, and E the exhaust, Access may be had
to the valves by taking off the bonnets b b ; b' is the stuffing-box. Bolts
are not shown on the drawing, but the different pieces will be understood
by the cross-hatching. The valve-chest is bolted on face of cylinder, of
which fig. 3 is a plan of a part showing p' and
p", the steam and exhaust parts, and the bolt-
holes tapped in face to receive valve-chest bolts.
Fig. 61 is section and plan of stuffing-box for
steam piston-rod. It consists of two parts the
box b, which is attached to the cylinder-head,
and is bored out somewhat bigger than the pis-
ton-rod, except at the bottom ; and the gland g,
which is turned to fit the box, and bored to fit
the piston. The space in the box is filled by a
gasket, or other packing, and the gland is then
forced in by the screws, compressing the pack-
ing and making a tight fit around the piston-
rod. The dimensions vary with the diameter
of piston-rod, depth of box being from 3" to
12", and space for packing " to 1". In small
engines, the gland is often screwed into the
box, a follower or ring being placed above the
packing.
Fig. 62 is a sectional plan, and fig. 63 a sectional elevation, of a part
of the exterior of a piston, showing the common form of ring-packing,
which consists of a single interior ring (r), and two exterior rings (r" r"),
Fig. ei.
EXGDfEEKLNXJ DEAWLN'G.
each cut in two, and so fastened that the joints are always broken. The
packing is set out by springs, one of which is shown at s. F is the fol-
lower, which can be taken oif for the admission of the rings, and then
Fig. 62.
Fig. 63.
replaced and bolted to piston, making a close joint with end of rings.
The dimensions vary with the diameter of cylinder, the thickness of the
piston being from 3" to 9" at exterior, with a swell in the centre for large
pistons.
Plate CXX. Figs. 1, 2, and 3 are side and end elevation, and plan,
of the foundation of the stationary steam-engine (pi. CXIX.). F is the cast-
iron frame or bed-plate of the engine ; B the granite bed of engine, or
coping of foundation ; P the stone or brick pier, laid full in cement. The
granite bed is levelled accurately, and well hammered, to receive the engine-
frame. Strong wrought-iron bolts pass through frame, bed, and pier, with
nuts at each end ; and the whole is strongly bolted together. Pockets are
left in the pier near bottom for access to nuts, and these pockets are
covered by granite caps or iron plates.
PROJECTIONS OF THE GLOBE. 475
PEOJECTIONS OF THE GLOBE.
UNDER the head of " Topographical Drawing " are given plans of por-
tions of the earth's surface, and the conventional signs by which its
features are designated. It now remains to explain the common projec-
tions by which meridians and parallels of latitude are represented on
charts and maps.
If the sphere be projected orthographically, or in perspective, the rep-
resentation will be correct, but the parts will not admit of measure, except
for a small space in the centre, or directly beneath the eye, as will be
readily understood by referring to the principles given under " Geometrical
Projection " and " Perspective Drawing." Neither of these projections
are suited, therefore, to the purposes of maps, in which it is important
that the relative distance between different points in every position upon
the map should be represented as accurately as possible. Without going
fully into the principles of the different projections usually employed, it
will be considered sufficient for the present purpose to explain how the
meridians and parallels are projected.
GLOBULAR PROJECTION OF THE SPHERE. According to this method, the
circles of the sphere should be represented by ellipses ; but in practice,
and as employed in most school maps, they are represented approximately
by circles. f The following is the construction :
To project a hemisphere (fig. 1). Draw two lines, at right angles to and
intersecting each other, from the point C of their intersection as a centre,
with a radius equal to that intended for the hemisphere, describe a circle,
and mark the points N", S, W, E. N" and S will be the poles, the line N S
the central meridian, and W E the equator. Divide N S and W E into as
many equal parts as there are degrees or numbers of degrees to be rep-
resented in the figure in divisions of 30 and meridian and equator
into 6 equal parts, as the hemisphere embraces 180. Commence at
C, and divide the half lines into three equal parts. Divide the arcs
N TV, IS" E, S "W, and S 'E, each into 3 equal parts. There will be
4T6
PROJECTIONS OF THE GLOBE.
Fig. 1.
now determined 3 points in 2 parallels of north and south latitude, 30
and 60, through which to describe the arcs representing the parallels.
The centre of these arcs will
be in the line K" S ; describe
the arc, and with the same
radius from a centre on the
line N S, below the S pole,
describe a similar arc pass-
ing through the S 30 point
on the meridian. There-
fore, keeping the steel point
of the dividers on the line
N S, by trial radii may be
found of arcs which shall
pass through the points on
the central meridian and
on the circle. "With the
radii describe arcs for the
parallels in north and south
latitude. All the meridians pass through the IS" and S poles, and through
the divisions of degrees on the equator. There are 3 points, therefore, de-
termined in the arc of each meridian which may be described frojn centres
found by trial on the line E "W.
STEBEOGEAPHIC PROJECTION. To project the hemisphere on the plane
of the meridian (fig. 2). Draw central meridian, equator, and circle, as
in the preceding problem. To pro-
ject the other meridians (say every
10), divide the quadrant N" E into
9 equal parts ; from S to these points
of division, 10, 20, 30, draw lines in-
tersecting C E in 10, 20, 30. These
latter points are in the meridians
through which N and S arcs are to
be described from centres on the line
EW.
To find in like manner the 3
points in the parallels of latitude,
divide the quadrants into 9 parts,
80, TO, 60, and through these points draw lines to "W ; the inter-
sections with the central meridian 80, TO, 60, will with the points of
PBOJECTION3 OF THE GLOBE.
477
the quadrant furnish 3 points through which to describe arcs of paral-
lels of latitude.
To project the hemisphere on the plane of the equator (fig. 3}. Draw
two lines at right angles to each other ;
describe the circle and divide the cir-
cumference as before. The centre C
will be the projection of 1ST or S pole,
the lines at right angles to each other
will be meridians, as well as any other A[
diameters, as D H, F K, drawn through
some division of the circumference.
To project the parallels of latitude.
The circle represents the projection of
the equator, and the other parallels
must be arcs on the same centre C, Fi s-3.
of which the radii are to be determined by the intersections of the line
C B by lines drawn from A to the divisions of the circle 10, 20, 30.
To project the hemisphere on the plane of the horizon for a given
latitude (fig. 4). Draw the two lines at right angles to each other, and
describe the circle as before (fig. 4) ; and, to prevent confusion with the
constructive lines, draw a similar figure (fig. 5). The circle N W S E will
represent the horizon on which the sphere is projected, and K, "W, S, E are
the cardinal points. Lay off W P' equal to the given latitude, and draw
E' P' ; the point P" in which it intersects W C' will be the projection of
the pole, which, laid off on fig. 4, is represented by P. Draw the lines
478
PROJECTIONS OF THE GLOBE.
P' C 7 and F A' perpendicular to it at C' ; draw A 7 E 7 : its intersection A"
with C' S 7 will be a point in the arc of the equator. Transfer this point to
A (fig. 4), and through the 3 points W, A, E, describe arc for the projection
of the equator. For other parallels lay off (say) 20" and 40" each side of
P 7 ; draw E' 20, E 7 40, intersecting C' N 7 and C' S 7 in a', c', d', b 7 ; a 7 V
and c 7 d 7 are the diameters of the projected parallels, corresponding to 50
and 70 of latitude. Transfer these points to fig. 4, and, on a b and c d as
diameters, describe circles for the above parallels.
To project the meridians. The lines N P P S are the projections of the
opposite meridians which pass through the N and S points of the horizon.
Draw the tangent S 7 D, and extend P 7 p, intersecting it at D. . In fig. 4
take C B equal to D S 7 (fig. 5), and from B as centre describe a circle pass-
ing through P. It will pass through AY and E, and will be the projection
of the meridian of the place for which the projection is made. Draw
G H through B and perpendicular to C S ; at B lay off the angles which
the meridians make with each other (say 15), B P 15, B P 30 ; the
intersections 15, 30, on the line G H will be centres on which to de-
scribe through the point P the projected meridians required, as m P n 7 ,
n P m 7 , etc.
CONSTRUCTION OF MAPS BY DEVELOPMENT. The methods of projection
already explained are usually confined to the delineation of a hemisphere ;
but for the delineation of a single country the method of development is
employed, which exhibits with greater precision the correct distances be-
tween places, while for the purposes of navigation, where the bearings of
places, one from another, must be correctly and simply shown, the J/ier-
cator's Chart is used.
It is obvious that the surface of
a sphere cannot be precisely repre-
sented by a plane-surface, but that,
for small extents, it may very nearly
coincide with that of a cone or a
cylinder.
Following preceding constructions,
|E let N W S E (fig. 6) be the section of
a sphere on the plane of the meridian,
N S the axis, and W E the diameter
of the equator. Take any arc, E F,
and bisect it at G ; through G draw
a tangent intersecting the axis pro-
duced at L. If the hemisphere re-
w
PROJECTIONS OF THE GLOBE.
4T9
volvo about; the axis K S, it will generate a sphere,
while the tangent L d will generate a conical surface j
and it may be readily seen that the surface of the sphere
embraced between the two parallels f F and e E, and
the meridians passing through any two points, as II
and G on the central parallel g G, will diner but little
from the conical surface embraced between the lines
L H and L G, which is developed in fig. 7 by construc-
tion explained, pp. 102-10-i.
Fig. 7.
Table showing the Number of Geographic Miles in a Degree of Longi-
tude, under each Parallel of Latitude.
Parallel
of Latitude.
Geog. Miles in a
Degree.
Parallel
of Latitude.
Geog. Miles in a
Degree.
Parallel
of Latitude.
Geog. Miles in a
Degree.
o
60.00
30
52.00
60
30.07
1
59.99
31
51.47
61
29.16
2
59.96
32
50.93
62
28.24
3
59.92
33
50.37
63
27.31
4
59.85
34
49.79
64
26.37
5
59.77
35
49.20
65
25.43
6
59.67
36
48.60
66
24.47
7
59.56
37
47.97
67
23.51
8
59.42
38
47.34
68
22.54
9
59.26
39
46.69
69
21.56
10
59.09
40
46.02
70
20.58
11
58.90
41
45.35
71 -
19.59
12
58.70
42
44.65
72
18.60
13
58.47
43
43.95
73
17.59
14
58.23
44
43.23
74
16.59
15
57.97
45
42.50
75
15.58
16
57.69
46
41.75
76
14.56
17
57.39
47
40.99
77
13.54
18
57.08
48 .
40.22
78
12.51
19
56.75
49
39.44
79
11.48
20
56.40
50
38.64
80
10.45
21
56.04
51
37.83
81
9.42
22
55.66
52
37.01
82
8.38
23
55.26
53
36.18
83
7.34
24
54.84
54
35.34
84
6.29
25
54.41
55
34.40
85
5.24
26
53.96
56
33.63
86
4.20
27
28
53.50
53.01
57
58
32.75
31.87
87
88
3.15
2.10
29
30
52.52
52.00
59
60
30.98
30.07
89
90
1.05
0.00
480
PROJECTIONS OF THE GLOBE.
Each degree of latitude is _ always GO geographical miles, while a
degree of longitude is 60 miles only at the equator, and becomes less
toward the poles.
A map may quite accurately present a portion of a sphere by repre-
senting it as the development of a cylinder of which the parallels are
straight lines, and distant from each other on a scale of GO miles for each
degree, and the central meridian perpendicular to the parallels, and the
others inclined, and distant from the central one according to their posi-
tions in latitude. Thus, in fig. 8, which embraces 40 of latitude, and ex-
tends from 10 to 40 K latitude in divisions of 10 each, the parallels
Fig. 8.
are 600 miles apart, the meridians on 10 IS". 590.9 miles, on 40 K 460.2
miles. The meridians are here drawn as straight lines ; but it would be
more accurate if the length of meridians were laid off on each parallel,
and curved lines drawn. In addition, the parallels may be drawn in con-
centric circles, taking radii from the development of the cone (fig. 7).
To construct a Mercator's Chart (fig. 9). Draw two straight lines, "W E
and N S, intersecting each other at right angles at C. WE is the equator,
1ST S the meridian passing through the middle of the chart. From C set
off equal parts on the equator both ways, to represent degrees of longi-
PROJECTIONS OF THE GLOBE.
481
tude, subdivided into minutes if the size of the chart will admit of it.
Assuming the equator as a scale of minutes set off from toward N and
S, the number of minutes in the enlarged meridian corresponding to each
degree of latitude, as shown by the table of meridional parts. Draw lines
parallel to N S through the divisions of the equator for meridians, and
parallels to "W E through the divisions of 1ST S for parallels of latitude.
To find the bearing of any one place from another it is only necessary to
draw a straight line between the two points, and observe the angle it makes
with the meridians.
Table of Meridional Parts.
Latitude.
Meridional Parts.
Latitude.
Meridional Parts. Latitude.
Meridional Parts.
o
0.00
35
2244.29
70
5965.92
5
300.38
40
2629.69
75
6970.34
10
G03.07
45
3029.94 -
80
8375.20
15
910.46
50
3474.47
85
10764.62
20
1225.14
55
3967.97
90
Infinite
25
1549.99
GO
4527.37
30
1888.38
65
5178.81
01
482 SPECIFICATIONS.
SPECIFICATIONS.
IN the construction of works, buildings, tools, or machinery, it is im-
possible to make every thing intelligible by plans and drawings only.
This necessity is supplied . by written descriptions, called " specifications,"
which form, with the plans, a part of the contract.
The specifications define the work to be done, the materials of which it
is to be composed, the kind of workmanship ; exhibit dimensions and parts
not shown on plan, and are, in general, explanatory, so that contractor
or workmen may execute and finish agreeably to the intention of the de-
signer, who states the different works in the order they will be probably
executed, as, if a building, with the excavation, then masonry, carpentry,
plumbing, painting, etc. ; if machinery, a general description usually pre-
faces the specification, which then follows in some natural order. In Eng-
lish specifications, it is customary to make out tables of quantities of
materials, on which the contractor bases his estimates a far better way
than that here adopted, where every contractor makes out his own bill of
items, involving an unpaid trouble to him, and great liability to mistakes.
A full bill of items is more explanatory of the work than any thing else can
be, and, in my own practice, I have found that with it there is by far
greater uniformity in the bids received from responsible parties.
In the heading of the specification, the kind of work (building, ma-
chine, etc.), where it is to be erected or delivered, and that it is to be done
agreeably to the specification and accompanying plans made by .... (ar-
chitect, engineer, etc.), should be stated. Another clause specifies the party
usually the same as above to whose approval the work is subject, and
to whom the disputes between contracting parties are to be referred. If
possible, it is well to state some standard of similar work, to which " ma-
terials and workmanship shall be in every respect equal." A common
phrase is, to state that " all the several materials used are to be of the very
best quality, and all the work to be done in the best and most workman-
like manner."
Some provision is to be made for the commencement, finishing, and
delivering of the work ; also for any damages to persons or property, that
may arise during the execution of the work, and for insurance.
APPENDIX.
Extracts from the Law relating to Buildings in the City of New York.
THE footing, or base course, under all foundation-walls, shall be of stone or concrete,
and at least 12" wider than the bottom width of the foundation-walls. And if the
walls be built of isolated piers, then there must be inverted arches, at least 12" thick,
turned under and between the piers, or 2 footing courses of large stone, at least 10"
thick in each course. All foundation-walls other than those of dwellings shall be at
least 4" thicker than the wall next above them, to a depth 16' below the curb-level, and
shall be increased 4" for every additional 5' in depth below the said 16'. Foundation-
walls in dwelling-houses shall be, below the basement-floor beams, 4" thicker than the
walls next above them. By foundation-walls is meant that portion of the wall below
the level of the street-curb, and below the basement-floor beams in dwellings, and depth
shall be computed-from the curb-level downward.
In all dwelling-houses not above 30' in height, and not more than 20' in width, the
party and outside walls shall not be less than 8" thick ; hi all dwelling-houses from 30'
to 55' in height, and not more than 30' in width, the outside and party-walls shall not
be less than 12" ; and if above 55', the walls shall not be less than 16", to the top of the
2d-story beams ; provided the same is 20' above the curb, if not, then to the under side
of the 3d-story beams ; and also provided that portion of the 12"-wall shall not exceed
40' in height from said 16"-wall.
In all buildings other than dwelling-houses, not above 30' in height, and not more
than 25' in width, the outside walls shall not be less than 8" thick, and the party-walls
not less than 12" ; if above 30' and under 50' in height, the outside walls shall not be
less than 12", and the party-walls not less than 16" ; if above 50' and under 65' in
height, the outside walls shall not be less than 16" to the height of the 3d-story beams,
and not less than 12" from thence to the top, and the party-walls not less than 20" to
the height of the 2d-story beams, and not less than 16" from thence to the top ; and if
above 65' and under 80' in height, the outside walls shall not be less than 16" to the
height of at least 40', and up to the under side of the next-story beams above, and not
less than 12" from thence to the top, and the party-walls not less than 20" to the height
of the 3d-story beams, and not less than 16" from thence to the top. In all buildings
over 27' in width, and not having either brick partition-walls or girders, supported by
columns running from front to rear, the walls shall be increased an additional 4" in
thickness, to the same relative thickness in height as required under this section, for
every additional 10' in width of said building ; and in all buildings intended or used
484: APPENDIX.
for the purposes of storing or keeping heavy merchandise or materials, the walls shall
be an additional 4" thicker than above required. The amount of materials specified
may be used either in piers or buttresses, provided the walls between the same shall in
no case be less than 8" thick, to the height of 40', and if over that height, then 12" ;
and if over 65', 16", to the height of 20' from the curb.
All stone walls less than 24" thick shall have at least 1 header, extending through
the walls, in every 6 sq. ft. ; and if over 24", shall have 1 header for every 6 sq. ft. on
both sides of the wall, and running into the wall at least 2'. In every brick wall every
5th course of bricks shall be a heading course, except where walls are faced with brick,
in which case every 5th course shall be bonded into the backing. In all walls which
are faced with thin ashlar, anchored to the backing, the backing of brick shall not be
less than 8", laid up in cement mortar, and shall not be built to greater height than
prescribed for 8"-walls.
Every isolated pier less than 6 sq. ft. at the base, and all piers supporting a wall
built of rubble-stone or brick, or under any iron beam or arch-girder, or arch on which
a wall rests, or lintel supporting a wall, shall, at intervals of not more than 30" in
height, have built into it a bond-stone not less than 4" thick, of a diameter each way
equal to the diameter of the pier, except that in front piers above the curb, the bond-
stone may be 4" less than the pier in diameter ; the walls and piers under all compound
girders, iron or other columns, shall have a bond-stone every 30" in height from the
bottom, whether said piers are in the walla or not, and a cap-stone at least 12" in thick-
ness by the whole size of the bearing.
All walls shall be securely anchored with iron anchors, to each tier of beams. The
front, rear, side, end, and party-walls shall, if not carried up together, be anchored to
each other, every 6' in their height, by wrought-iron tie-anchors, 1 J" x f ", built into the
side or party-walls not less than 16", and into the front and rear walls at least i the
thickness of the front and rear walls ; and all stone used for the facing of any build-
ing, except where built with alternate headers and stretchers, shall be strongly anchored
with iron anchors, let into the stone at least 1". The side, end, or party-walls, shall be
anchored at each tier of beams, at intervals of not more than 8' apart, with f " x 1"
wrought-iron anchors, well built into the side-walls and fastened to the side of the
beams ; and where the beams are supported by girders, the ends of the beams resting on
the girder shall be strapped by wrought-iron scraps of the same size, and at the same
distance apart, and in the same beam as the wall-anchors.
Compound'beams with cast-iron arches and wrought-iron ties, used to span open-
ings not more than 10' in width, shall have a bearing of at least 12" by the thickness
of the wall to be supported, and for every additional foot of span over and above the
said 10', the bearing shall be increased 1", provided the same are supported at the ends
on brick or stone, and on the front of any building where the supports are of iron or
solid cut stone, they shall be at least 12" on the face, and the width of the thickness
of the wall to be supported, and shall rest upon a cut-granite base block at least 12"
thick by the full size of the bearing, and all compound beams or girders used in any
building shall be, throughout, of a thickness not less than the thickness of the wall to
be supported. All compound beams shall have a cast-iron shoe on the upper side, to
answer for the skew-back of a brick or cut-stone arch, to be turned over the same, and
in no case less than 8" in height by the width of the wall to be supported. Cut-stone
APPENDIX. 485
or hard brick arches, where the arch has not abutments of sufficient size to resist its
thrust, may be turned over any opening less than 40', provided they have skew-backs
of cut stone or cast or wrought iron, into which tension-rods are properly secured.
All chimneys and all flues in stone or brick walls, in any building, shall be properly
pargetted, or the joints shall be struck smooth on the inside. And flues or pipes of a
single thickness of metal, to convey heated air in any building, shall be so constructed
as to have a thickness of not less than 1" of plaster of Paris between the said metal and
any of the wood-work adjoining the same. No smoke-pipe in any building shall enter
any flue, unless the said pipe shall be at least 18" from either combustible floors or ceil-
ings ; and where smoke-pipes pass through stud or wooden partitions of any kind, they
shall be guarded by either a double collar of metal, with at least 4" air-space, and holes
for ventilation, or by a soapstone ring, not less than 3" in thickness and extending
through the partition, or by a solid coating of plaster of Paris, 3" thick, or by an
earthenware ring, 3" from the pipe. In no building shall any wooden beams or timbers
be placed within 8" of any flue. All wooden timbers in the party-wall of every buliding
shall be separated from the beam or timber entering in the opposite side of the wall by
at least 4" of solid mason-work. No floor-beam shall be supported wholly upon any
wood partition, but every beam, except headers and tail-beams, shall rest, at each end,
not less than 4" in the wall, or upon a girder.
All exterior cornices and gutters of all buildings shall be of some fire-proof material.
The planking and sheathing of the roof of every building shall in no case be extended
across the front, rear, side, end, or party-wall thereof.
All gas, water, steam, or other pipes, in any building other than a dwelling-house,
shall not be let into the beams, unless the same be placed within 36" of the end of the
beams, and not more than 2" in depth.
In all buildings, every floor shall be of sufficient strength in all its parts to bear
safely, upon every sq. ft. of its surface, 75 Ibs. ; and if used as a place of public assem-
bly, 120 Ibs. ; and if used for any manufacturing or commercial purposes, from 150 to
500 Ibs. and upward, in addition to the weight of the materials of which the floor is
composed ; and every column shall be of sufficient strength to bear safely the weight
of the portion of each and every floor depending upon it, in addition to the weight
required as above to be supported safely upon said portions of said floors. In all cal-
culations for the strength of materials to be used in every building, the proportion
between the safe weight and the breaking weight shall be as 1 to 3 for all pieces sub-
jected to a cross strain; and shall be as 1 to 6 for all vertical supports, and for tie-rods,
and pieces subjected to a tensile strain.
In all fire-proof buildings, either brick walls with wrought-iron beams, or cast or
wrought-iron columns with wrought-iron beams, must be used in the interior. The
met a rcolumns shall be planed true and smooth at both ends, and shall rest on cast-iron
bed-plates and have cast-iron caps, also planed true. If brick arches are used between
the beams, the arches shall have a rise of at least 1J" to each foot of span between the
beams. All arches shall be at least 4" thick. Arches over 4' span shall be increased in
thickness toward the haunches by additions of 4" in thickness of brick ; the first addi-
tional thickness shall commence at 2}' from the centre of the span ; the second addition
6V from the centre of the span; and the thickness shall be increased thence 4" for every
additional 4' of span toward the haunches.
486
APPENDIX.
tween the bearings of continuous shafts,
subject to no transverse strain except from
their own weights.
Distance between bearings,
Diameter of
shaft
in feet.
in inches.
If of
wrought iron.
If of steel.
1
12.27
12.61
2
15.46
15.89
3
17.70
18.19
4
19.48
20.02
5
20.99
21.57
6
22.30
22.92
7
23.48
24.13
8
24.55
25.23
9
25.53
26.24
10
26.44
27.18
11
27.30
28.05
12
28.10
28.88
Extracts from " Formulas and Tables for the Shafting of Mills and
Factories" by Mr. J. B. Francis, Journal of the Franklin Institute.
"Shafts for transmitting power are subject to two forces, viz. : transverse strain and
torsion. In shafts of wrought iron or steel, in which the bearings are not very near to
each other, the weight of the shaft itself will pro-
Table of the greatest admissible distances be- v
duce an inadmissible amount of deflection when-
ever this distance exceeds a certain amount, which
varies with the material and diameter of the shaft.
"In practice, long shafts are scarcely ever en-
tirely free from transverse strains ; however, in the
parts of long lines which have no pulleys or gears,
with the couplings near the bearings, the interval
between the bearings may approach the distances
given in the table. Near the extremities of a
line, the distances between the bearings should be
less than are given in the table. The last space
should not exceed 60$ of the distance there given,
the deflection in that space being much greater
than in other parts of the line.
"In factories and workshops, power is usually
taken off from the lines of shafting, at many points,
by pulleys and belts. "When the machines to be
driven are below the shaft, there is a transverse strain on the shaft, due to the weight
of the pulley and tension of the belt, in addition to the transverse strain due to the
weight of the shaft itself. When the power is taken off horizontally on one side, the
tension of the belt produces a horizontal transverse strain; and the weight of the
pulley acts with the weight of the shaft, to produce a vertical transverse strain. "When
the machinery to be driven is placed above the floor, to which the shaft is hung in the
story below, the transverse strain produced by the tension of the belt is in the opposite
direction to that produced by the weight of the pulley and shaft. To transmit the same
power, the necessary tension of a belt diminishes in proportion to its velocity; conse-
quently, with pulleys of the same diameter, the transverse strain will diminish in the same
ratio as the velocity of the shaft increases. In cotton and woollen factories with wooden
floors, the bearings are usually hung on the beams, which are usually about 8' apart; and
a minimum size of shafting is adopted for the different classes of machinery which has
been determined by experience as the least that will withstand the transverse strain.
This minimum is adopted independently of the size required to withstand the torsional
strain due to the power transmitted ; if this requires a larger diameter than the mini-
mum, the larger diameter is. of course, adopted. In some of the large cotton-factories in
this neighborhood (Lowell, Mass.), in which the bearings are about 8' apart, a minimum
diameter of 1^" was formerly adopted for the lines of shafting driving looms. In some
mills this is still retained, in others 2|" and 2 T y have been substituted. In the same
mills, the minimum size of shafts driving spinning machinery is from 2J" to 2y|". In
very long lines of small shafting, fly-wheels are put on at intervals, to diminish the vibra-
tory action due to the irregularities in the torsional strain.
APPENDIX.
4:87
"The following ttible gives the power which can be safely carried by shafts making
100 revolutions per minute. The power which can be carried by the same shafts at any
other velocity may be found by the following simple rule :
" Multiply the power given in the table ~by the number of revolutions made by the
shaft per minute ; divide the product l>y one hundred; the quotient will ~be the power
which can ~be safely carried"
DIA3IETER
IN
INCHES.
;orse-power which can be safelv carried
by shafts for prime movers and gears,
well supported by bearings, and making
100 revolutions per minute ; if of
Horse-power which can be safely transmit-
ted by shafts making 100 revolutions per
minute, in which the transverse strain, if
any, need not be considered ; if of
Wrought iron.
Steel.
Cast iron.
Wrought iron.
Steel.
Cast iron.
1.00
1.00
1.60
0.60
2.00
3.20
1.20
1.25
1.95
3.12
1.17
3.90
6.24
2.34
1.50
3.37
5.39
2.03
6.74
10.78
4.06
1.75
5.36
8.58
3.22
10.72
17.16
6.44
2.00
8.00
12.80
4.80
16.00
25.60
9.60
2.25
11.39
18.22
6.83
22,78
36.44
13.66
2.50
15.62
24.99
9.37
31.24
49.98
18.74
2.75
20.80
33.28
12.48
41.60
66.56
24.96
3.00
27.00
43.20
16.20
54.00
86.40
32.40
3.25
34.33
54.93
20.60
68.66
109.86
41.20
3.50
42.87
68.59
25.72
85.74
137.18
51.44
3.75
52.73
84.37
31.64
105.46
168.74
63.28
4.00
64.00
102.40
38.40
128.00
204.80
76.80
4.25
76.77
122.83
46.06
153.54
245.66
92.12
4.50
91.12
145.79
54.67
182.24
291.58
109.34
4.75
107.17
171.47
64.30
214.34
342.94
128.60
5.00
125.00
200.00
75.00
250.00
400.00
150.00
5.25
144.70
231.52
86.82
289.40
463.04
173.64
5.50
166.37
268.19
99.82
332.74
532.38
199.64
5.75
190.11
304.18
114.06
380.22
608.36
228.12
6.00
216.00
345.60
129.60
432.00
691.20
259.20
6.25
244.14
390.62
146.49
488.28
781.24
292.98
6.50
274.62
439.39
164.78
549.24
878.78
329.56
6.75
307.55
492.08
184.53
615.10
984.16
369.06
7.00
343.00
548.80
205.80
686.00
1097.60
411.60
7.25
381.08
609.73
228.65
762.16
1219.46
457.30
7.50
421.87
674.99
253.13
843.74
1349.98
506.26
7.75
465.48
744.77
279.29
930.96
1489.54
558.58
8.00
512.00
819.20
307.20
1024.00
1638.40
614.40
8.25
8.50
561.52
614.12
898.43
982.59
336.91
368.47
1123.04
1228.24
1796.86
1965.18
673.82
736.94
8.75
669.92
1071.87
401.95
1339.84
2143.74
803.90
9.00
729.00
1166.40
437.40
1458.00
2332.80
874.80
9.25
9.50
9.75
791.45
857.37
926.86
1266.32
1371.79
1482.98
474.87
514.43
556.12
1582.90
1714.74
1853.72
2532.64
2743.58
2965.96
949.74
1028.86
1112.24
10.00
1000.00
1600.00
600.00
2000.00
3200.00
1200.00
488 APPENDIX.
From Mr. Francises notes we extract the following on the power re-
quired to drive various machinery :
" At Lowell we commonly reckon for No. 14 cotton goods 33 dead-throstle spindles
and all the other machinery to a horse-power. The Rhode Island people, I believe,
reckon GO spindles to a horse-power on print-cloths, ring-warp and mule-filling. In
1850, Mr. John Newell weighed the power used to drive the machinery in a mill of
12,544 dead-throstle spindles making No. 14 yarn, including all the machinery, but not
the shafting, which he found to be, .... 175,860 Ibs. ft. per second.
Shafting (estimated), 43,965
Total, 319,825
or 31.4 horse-power per spindle.
" The proportion for each room he made as follows :
Picking-room, 0.12583
Card " 0.18557
Spinning " 0.47801
Drawing " 0.03821
Weaving " 0.17338
Total, 1.00000
" From experiments by Mr. Newell on Rock Bottom Flannel Mill, Stow, Mass. :
1 picker, 1154.18 Ibs. ft. per second.
7 sets of cards, 7769.79 " "
15 jacks (2859. sp.), 3628.07
18 plain looms, ,
28 twilled looms,
1 dresser, 2 warpers, 2 spoolers, 1 small pump,
1 rotating fulling-mill,
2 washers,
1 napper,
13$ for geering,
19346.44 = 35.17 HP.'
The Uses of Profile and Cross-section Paper. At page 365 mention
is made of the construction of profile and cross-section paper, and their
use with reference to railway plots ; but they are now of much more ex-
tended application. There are many facts involving two factors or consid-
erations, which can be expressed graphically in the form of a profile, the
abscissas representing one, and the ordinates the other ; and consequently
APPENDIX.
489
more readily by the paper already prepared for the purpose. Thus, the
fluctuations of gold, of coal, and the iron trade, are represented by ab-
scissas of time, and ordinates of price, or of amount of sales ; the sanitarian
represent time as above, and the number of deaths by ordinates ; the rail-
way engineer, speed of train by one, and resistance by the other, etc. ;
while the architect and mechanic find a very valuable use of cross-section
paper in designing or copying, making the squares scales of parts.
Fig. 1 represents the path of float in a wooden flume or channel, taken
from the last edition of Francis Lowell's " Hydraulic Experiments." The
cut was copied directly on the wood, and is therefore reversed. The
1 (\
^
J-~v
-o
Oi>
X
->~v
^
.X
~
V.
-
'
/
J
,
0,
3
,
'5
!
!
^
Q
Q
JE
-"
s
^ ; .
,
1
V
>
c
Fig. 1.
width of the cut represents the width of the flume, each abscissa being 1
ft. ; the ordinates are the speeds of float in divisions of 0.1 ft. per second ;
the o o are the floats in their observed path, and speed ; and the curved
line the average velocity in the different threads of the stream.
Fig. 2 is from Clarke's " Railway Machinery." The abscissas repre-
Miles per hour.
'
20 30 40 50 GO- fl '
0,
1
-.
"i-
i
-*
(}
1
->
-,
1
^^^^
i
a
>
v.
3
^
00
ii
v
li
90
Fig. 2.
sent the speed in miles per hour ; the ordinates the Ibs. per ton resistance
of a 100-ton train.
490
APPENDIX.
Fig. 3 is made up from time-table of ]ST. Y. & "N. H. K., showing the
movement of trains, two from ]STew York and two from Kew Haven, the
abscissas being cut off on a scale of miles for each station, the ordinates
being a scale of hours :
s : s s
Jn HI uiif i 1 il li
^3 ,3 ^ 8 | g| | ft o 2 =;=
Fig. 3.
IISTDEX.
A CANTHUS, 273.
-A_ Acoustics, principles of, 292.
Alhambra, 276.
Angle, greatest, under which objects can be seen,
386.
Angles taken by compass, how laid off by pro-
tractor, 358.
Antae, 240.
Aqueduct. Brooklyn, 438, 431 ; Croton, 439 ; iron
pips across Harlem River, 439, 440.
Aqueducts, 2 GO.
Arcades, 259.
Arch, dimension of, bridges, 457 ; depth of key-
stone, 458 ; radius at crown, segmental, ellip-
tical, 488 ; effect of unequal loads on, 458,
459 ; height of spandrel backing, 459 ; hori-
zontal thrust, 459.
Arches, terms applied to parts of, 215 ; rules to
determine the depth at crown of, 216 ; trian-
gular-headed, round-headed, stilted, horseshoe,
pointed, complex, foiled, 263, 264 ; four-centred
or Tudor, 266.
Architrave*, 28%, 242.
Ashlar (see MASONRY).
Asphalt pavement, 453.
Atlantes, 259.
Avenue, 450.
Axles, size of, 129.
BATTER (see MASONRY).
Balcony, 284.
Ballast for railway, 455.
Balusters, 279, 280.
Base and surbase of rooms, 243.
Basilicas, 294.
Bay-windows, 281.
Beams and girders, 124-128 ; \vooden, 216-218;
collar, tie (see ROOF).
Bearings for shafts, 142, 145, 146.
Belting, strain on, 155 ; strength of, 156.
Bevels (see GEERS).
Bevel-wheels, projections of, 179, 181 ; skew bev-
els, 181 ; isometrical view of, 410.
Blinds, 242.
Boarding, vertical, 281.
Boilers, evaporation of, 136 ; construction of,
467-471.
Bolts, size and proportions of, 190 ; thread, nuts,
191 ; washers, 192.
Bond (see MASONRY).
Boulevards, 450.
Bracing, principles of, 230, 231.
Bridges, girder or frame, 460 ; piers of, 455, 456 ;
arch, 457-460 ; skew, 460 ; suspension, 465, 466.
I Brushes for tinting, 333.
Buttresses, 266.
Buildings, expression of purpose in, 301 ; prin-
ciples of design of, of construction of, essential
conditions of good, truth in expression, present
taste, material for, 309-311.
Building act, extracts from New York, 483-485.
Byzantine architecture, 260.
CABLING, 257.
Camera lucida, 35.
Campanile, 268.
Cam-punch and shear, frame of, 192.
Canals, sections of, 431 ; for transportation, 430;
for the supply of mills, 431, 432.
Carriage-way, 452.
Caryatides, 259.
Catch-basins, 307 ; drawing of, trap of, 448.
Cathedral, pi. LXXVI.
Centre of gravity, 113 ; of surfaces, of solids, 1 14.
Central plane, 387.
Cesspools, 245, 307.
Chimneys, 245, 246, 472 ; chimney-tops, 282.
Chancel, 294.
Churches, 292, 294; Church of the Holy Sep-
ulchre, 260.
-92
INDEX.
Ciiiquecento, 278.
Coal, consumption of, 468.
Coffer-dam, 419 ; for Susquehanna bridge piers,
419, 420.
Colors, preparation of, for tints, 381 ; applica-
tion of, 381, 382.
Coloring, finished, 339 ; color suited to different
materials, examples of, and method of laying
on, selection of colors, 340-348.
Columns, cast-iron, 122 ; spaces between, 258 ;
super columnation, 258.
Compass, use of, in surveying, 357 ; plot of a
survey by, 358.
Compassers or dividers, 11, 12; triangular, 27;
proportional, 28 ; beam, 29 ; portable, 30 ;
tubular, large screw, 31.
Composite order, examples of, 257.
Concrete, use of, 421.
Conduits for water-supply, 438 ; cross-section of
main for, Brooklyn Water-works, 438, 439 ;
Croton, 439.
Cone, development of surface of, 103 ; of rays
of light, 385.
Cones, 90 ; penetrations of, 93-99 ; pulleys, 154.
Conic sections, construction of, 90, 91.
Connecting rods, drawings of, and proportions
of, 197, 198.
Contours, drawing of hills by, 353 ; map of por-
tion of city of London, 383 ; town and country
maps drawn with, 384.
Conventional signs, topographical, 349, 350.
Correction of errors in surveys, or balancing ex-
amples of, 359, 360.
Copying glass, 368 ; of large plans, 369.
Cornice, 234 ; interior, 243, 244.
Cornices, Gothic, Norman, English, 262.
Cornish Pumping Engine, 202, 462.
Corinthian order, examples of, 256.
Corbels, 262.
Couplings, face, 148 ; sleeve, screw, 149 ; clamp,
box, horned, 150; clutch, 151; friction cone,
152.
Country-house, drawings of, pi. LXV.-LXIX. ;
designing of, 235.
Crank, drawing of, table of relative size of eyes
in, 196.
Crockets, 277.
Culvert, isometrical view of, 410.
Curbs, 452.
Curves', definition of, 3-5 ; construction of, and
problem on circles, ellipse, parabola, hyperbo-
la, cycloid, epicycloid, involute, spiral, 49, 79 ;
catenary, 117; helix, 99; eccentric, 185.
Cylinder, projections of, 84 ; penetrations of, 92,
97 ; development of surface of, 103.
DADO, 243.
Dam across Connecticut River, at Holyoke,
425; across Herrimack River, at Lowell, 425,
426 ; across Mohawk, at Cohocs, 426, 427 ;
across Croton River, 427 ; for pondage, 428;
across the Furens, in France, 429.
Day's work, 131.
Design of buildings, principles of, 30.
Development of surfaces, 103.
Dimensions fcr different spans of truss, 463.
Dome, 260.
Domes and vaults, 264-266.
Doors, terms applied to parts of, proportions of,
and drawings of, 239, 240; of stores, 287.
Doorways, Norman, 261 ; circular-headed, Tudor-
arched, Gothic, gabled, perpendicular, Byzan-
tine, Saracenic, renaissance, Florentine, Vene-
tian, and Roman, 271, 272.
Doric order, examples of, 254, 255.
| Dowels, 218.
j Drainage, 306 ; of road-beds, 454, 455 ; and sewer-
! rage of part of city of London, 383.
Drawing, first design, working, 246 ; paper for
tinting, 333 ; table, board, paper, 35.
Drawings, skeleton,' partial, outline, 201, 202 ;
working, 202.
Drums, 154.
EAVES, finish of, 227.
Eccentrics, projections of, 185-189.
Edifices, purposes of, shown by appropriate sign
in topography, 351.
Egg and dart, 273.
Elizabethan style, 278.
Embankment, section of Thames, 421.
Engineer, object of, 416.
Engineering, definition of civil, 415.
Entablature, 253, 257, 259.
Entasis, 254.
Equilibrium of arches, 266 ; of the polygon of
rods, 117; of framework, 118.
1 Evaporation of different fuels, 135.
Extrados (see ARCHES)
Exterior of buildings, material, style, color, 299-
| 300.
tracery, 265.
Feed-water, temperature of, 468.
Field-book, 355.
Finishing of plans, 369.
Fireplace, 244, 246.
Floor?, 218; fire-proof, 219; of stores, 287.
Florentine school of architecture, 272.
Flues, 244 ; for heating and ventilating, 304 ;
dampers to, 305.
INDEX.
493
Flumes, construction of, 4-38.
Footwalk, 452.
Forces, parallel, 110; inclined, 111; resultants,
components, parallelogram of, 112 ; application
of, 113.
Formula?, Neville's and Dupius's, for cast-iron
pipes, 442 ; for the discharge of pipes, 442.
Foundations, 209, 210; for dams, 427; of con-
duit, Brooklyn Water-works, 439.
Frame of wooden house, terms applied to parts
of, 220.
Frames, application of iron to, 192-194.
Framing, 119 ; wood, as applied to buildings, 216 ;
flooring, bridging, headers, trimmers, and tail-
beams, 217 ; girders and joists, 217.
Friction, coefficient of, 115; limiting angle of re-
sistance to, 116 ; Morin's experiments on, JIG;
of surfaces, 117.
Fuel, table of evaporative powers of different
kinds of, 135.
Furnaces, hot-air, 303, 304.
GALLOWS, frame, of drawbridge, 461.
Gas, supply, 449 ; service-pipe, mains,
450.
Gauging flo\v of streams, 428, 429.
Geers, spur, mitres, 157 ; internal, rack geer and
pinion, bevel, trundle-face, leader, follower, size
of bevel gecrs, 158; skewed bevels, 160; pitch
and form of teeth of, 160-166 ; to calculate
strength of teeth of, 167; epicycloid teeth,
169 ; preparation and use of templates, 172,
173 ; involute teeth, 175 ; projections of wheels,
175-184.
Geological features, representations of, 382.
Geological map of California, pi. XCVII.
Geometrical definitions, 1-6 ; problems on lines,
42-49 ; arcs and circles, 49-55 ; parallel, 55 ;
circles, and rectilinear figures, 55, 67 ; on the
ellipse, parabola, hyperbola, cycloid, and epi-
cycloid, 68-76 ; the involute, 78 ; spiral, 79 ;
projections, 80 ; constructions, 85, 100.
Gib and cotter, 197.
Girder, 126, 232 ; form adopted for cast iron,
by Mr. Cubett, 462 ; of Crystal Palace, 229.
Globular projections of the sphere, 475.
Golden Gate, elevation and section of engines
of, 204, 205.
Gothic church, 294.
Grade of roads, 454.
Greek temples, plans of, 258.
Grillage, 219.
Groins, 265 (see ARCHES).
Ground plane, 387.
Guilloche, 273.
TTAND-RAIL (see STAIRS).
-Ll Hangers, 147, 148.
Headgates of canals, at Cohoes, 430 ; of flumes,
438.
Heating, 301 ; of stores, 287 ; of schools, 289.
Helix, projections of, 99-101.
j Hermes pillars, 259.
High Bridge, Harlem River, 260.
Hills expressed by brush-drawing, 354 ; repre-
sented under an oblique light, 380 ; how
shaded in topographical drawings, 379; topo-
graphical, 351-352.
Hoistways, 286.
Hoisting apparatus at Cohoes headgates, 430.
House, elevations of, from Holly's country-houses,
pi. LXV. ; country-residence, Downing, LXVI.,
LXVIII., LXIX. ; cottage, Gervase Wheeler,
LXVII. ; villa, Upjohn, LXIX. ; elevation of,
English basement, 280, LXII.
Houses, plan and elevations of, pi. XLVII.-LI. ;
various plans, 249, 250 ; city, basement, 250.
Housing?, 193.
Hooks, forms of, 192.
Hot water, heating by, 304.
Hydrostatic press, 105, 192.
TNCLIXED plane, 105, 109.
-J- Ink, China, 318 ; China, to draw lines in, 40 ;
inkles, 41.
Internal wheel and pinion projections, 183, 184.
Intrados (see ARCHES).
Ionic order, examples of, 255, 256.
Iron fronts, 288 ; Works, Althause, 312.
Isometrical view of Victoria and Regent Street
sewers, Thames embankment, 423; drawing,
principles and examples of, 405-414 ; projec-
tions of a cube, 405-408 ; of a prism, 408 ; of
curved lines, 409 ; of a bevel-wheel, 409 ; of a
a pillow-block, 410 ; of a culvert, 410 ; of a
boiler, truss-bridge, roof-truss, school-house,
411.
"TACK-SCREW, drawing of, 192.
*J Joints in plates, 463.
Journals, size of, 133-141.
T ECTURE-ROOMS, 292.
-L^ Legislative halls, 292 ; Chamber of French
Deputies, 298 ; Mr. Mills and Reid on form
of, 298 ; House of Representatives, Washing-
ton, 298, 299.
Lettering, 370 ; various kinds of letters and type,
370-374 ; irregular, of roads, of curved lines,
spacing of letters, 374 ; examples of, 375 ; of
tinted drawing, 382.
494
Lever, 105-107, 108-113.
Light, how diffused, reflected, 31D ; direction of,
on plans and elevations, 314 ; in topographical
drawings, 352 ; how it is supposed to fall on
topographical drawings, 369 ; in churches,
296.
Lightning, 308.
Line drawing?, 138.
Lines of series, secants, tangents, semitangents,
rhumbs, longitude, 21 ; of chords, 23; of poly-
gons, 24 ; of sines, lines, tangents, 26 ; scale
of, of circles of planes, 28; dotted full and
broken, 42; boundary, how designated, 370;
of horizon, 387.
Locks of canals, 433 ; plans and details of, 434-
437.
Locomotive, 201, 202 ; sections of boiler of, 204 ;
boiler, 468.
Louis Quinze style, 279.
"jy/TACHIKES, location of, 198; examples of
JJ-L two weaving-rooms in plans and sections,
199, 200.
Man-holes of sewers, 446 ; covers of, 447.
Mansard, or gambrel roof, 227.
Mantels, 244.
Maps by development, 478.
Marbling, how done, 382.
Masonry, terms applied to, 212-215.
Materials, mechanical properties of, 120; resist-
ance to compressure, 121-123 ; to tension,
124; to transverse strains, 124-128; to de-
tention, to torsion, 128, 129 ; for exterior of
edifices, 299 ; style of, 299.
Mechanical powers, 105-107.
Mechanical work, 130; unit of, 130; terms of,
131 ; of men and animals, 131.
Mercator's chart, 481.
Meridian?, 370.
Metals, strength of, 121 ; topographical marks
for, 351.
Moisture, table of grains in one cubic foot of air,
302.
Motion, transmission of, 158.
Motors, 131.
Mortar, 215.
Mouldings, names of, and construction of, 251,
252; Gothic, 261; arch, 262; hood, 262;
scroll, 263.
-*~ Xew Haven, map of harbor and city of,
pi. XCIV.
Xew World, frame of engine, 193 ; working-beam,
194.
/~\RDERS of architecture, 253.
V-' Ornaments, constructive, decorative, Gre-
cian, Roman, Symbolic, Byzantine, Saracenic,
Gothic, Renaissance, Elizabethan, 272-279.
TDACIFIC, frame of engine, 193.
-L Palace, of Diocletian, 260; Crystal, origin
and description of, 299.
Pantheum, column, 254.
Paper, drawing, 35 ; tracing, 36 ; stretching, 37 ;
mounting, 38 ; cleaning, 40 ; preparing, for tints,
381 ; profile, and cross section, 365, 489 ; tra-
cing, 368.
Parapets, 277.
Partitions, 219 ; bridging, 219 ; drawing of, 246.
Pavements, 452, 453.
Pentograph, 33.
Perspective, angular, 388 ; parallel, 389-401 ;
drawing, 385 ; linear and aerial, 386.
Pier, crib, quarantine for port of New York, 423,
424; pile, 455; trestle, 456; form of ends
Susquehanna river bridge, 456.
Pile piers, 455.
Pile-driver, 417 ; Nasmyth, 417.
Piling sheet, how driven ; material for, 417, 418.
Piles, how used, 416; material of, 417; hollow,
cast-iron, how driven, example from Harlem
bridge, 418, 419.
Pillars, sections of Gothic, pi. LVI., 261, 262,
Pillow-block, 146 ; isometrical view of, 410.
Pinnacles, 267-269.
Pipes, cast iron, plans and section of, Brooklyn
water, 442 ; formula for strength of, 442 ; ex-
amples of branches and stems, 444 ; specifica-
tion for Brooklyn, 444; table of weights of,
445 ; gas, weight of, 449.
Piston and rods, 473, 474.
Pitch (see GEERS, SCREWS, ROOFS).
Plane of the picture, 387.
Planing-machine, frame of, 192.
Plot, how to close, 359 ; of railway line, 364.
Plotting, 355 ; offsets, 361 ; offsets scale, 362 ; of
roads, canal railway survey, 362 ; from a sur-
vey by compass, by theodolite, 359.
Point of view, station, 387 ; of distance, vauish-
! ing points, 388.
Power, absolute, effective, 107 ; water, steam, 132 ;
horse, 133 ; required to drive machinery, 488.
{ Prism, projections of, 88-90.
Privies, 245.
i Profile, of road, 454 ; of railway line, 365 ; paper
| scales for, 365 ; paper, 365, 489.
| ProtrUctor, 19 ; circular, 31.
j Pulleys, 153 ; line, 154 ; width of face, 155; fast
! and loose, 156; position of, 157.
495
Purlincs (see ROOF).
Pyramids, projections of, 85-88.
RACK and pinions, projections of, 182.
Racks, 438.
Rafters (see ROOFS).
Railway, curves, designation of and plotting of,
3G3, 364 ; width of, ballast for, length of cross-
ties, 455.
Rain-fall, 429 ; rain-shed, how much can be cal
culated, 430.
Ranges, 244.
Reflection of objects in water, in perspective,
402.
Reservoirs, size of, 440 ; Ridgewood, New Cro-
ton, 440 ; division bank in Croton, 441.
Resistance of wheel-carriage?, of sled, 455.
Road, McAdam, Telford, Central Park, 450-454.
Romanesque architecture, 260.
Romanesque church, 294.
Roman school of architecture, 272.
Roofs, drawing of various, 221-230 ; terms applied
to parts of, pitch of, bridging of rafters of, 221 ;
form of foot of rafter of, 222 ; application of
cast iron to sheer and abutting-plates of, size
and proportion of different numbers of, 222;
determination of strain on different parts of,
223, 224 ; iron, elevation of, 228.
Roof-truss, isometrical view of, 40.
Rooms, size and proportions of, 234; dining, 235 ;
parlors, 235 ; pantries, passages, 236 ; details
of parts of, 237 ; finish of, 243.
Rubble (see MASONRY).
Ruler, common, 7; parallel, 10.
SAFES, 287.
Scale, selection of, 17 ; paper, 17; diagonal,
18; plotting, verneir, plain, 19; plain, double-
sectoral, 22; to form, 23; Marquois's, 26; in
plotting, 355 ; house-lots, farm surveys, Eng-
lish Tithe Commission maps, 355 ; State
surveys, harbor charts, decimal system of rail-
road surveys, canal maps, English standing or-
ders, 356 ; U. S. engineer service, 357 ; to be
adopted, 328 ; of shades for slopes of ground,
353 ; to be drawn on maps, 370 ; in perspec-
tive, 392.
School-houses, 288 ; plans and elevation of, pi.
LXXIV. ; on the requirements of, location of
desks, maps, black-boards, 259; heating of,
plan of New York City schools, 289-291 ; re-
quirements of Sunday, 295 ; isometrical vie
of, 411.
Sector, 22-26.
Screw, 110; projections of, 189, 190.
Sewers, 306 ; principle in establishing, 445 ; incli-
nation to be given to, 445 ; rule to determine
size of, 446 ; value of, pipe, brick, to determine
area of egg-shaped, 446 ; Union Avenue, Brook-
lyn, 446 ; angle of branches, 447 ; Victoria and
Regent Street, 423.
Shade-lines, 83 ; French system, 85 ; topographi-
cal, 349; on a cylinder, 318; on a reversed
cone, 319 ; of a ring, 325 ; surfaces in, 329.
Shading, methods of, 328 ; of a cylinder by flat
tints, 330; of a prism, of a cylinder, by soft-
ened tints, 332 ; examples of finished, 335, 336.
Shades and shadows, 313.
Shadows cast by a straight line on a vertical wall,
314 ; upon a curved surface, 315 ; of a circle on
a vertical plane, 317 ; on surfaces inclined to
each other, 317; cast by a pyramid, 318 ; by
and on a cylinder, 319; by one prism on an-
other, 320 ; by a cylinder on a prism, 321 ; on
the interior of a hollow cylinder, 321 ; steam
cylinder, 321-223 ; in interior of sphere, 324 ;
by the sphere on a plane, 325 ; upon the sur-
face of grooved pulleys, 326, 327 ; upon the
surfaces of screws and nuts, 327; in topo-
graphical drawings, 378, 379 ; projections of,
in perspective, 402, 403.
Shafts, torsion of, 129.
Shafting, wooden, cast-iron, fastened, tubular, 1st,
2d, and 3d movers, 137 ; strain on, 138 ; size
of, 138, 485; load on, 139; torsional strain on,
140 ; size of wrought-iron, 141 ; upright, 144.
Sheds for wood or coal, 24C.
Sheet-piling, 210.
Shipper, 152.
Shutters, revolving, 287.
Skeleton drawing, Mandoley and Field's marine
engine, 201.
Sketch, rough topographical, 355.
Skew backs (see ARCHES).
Skylight in apse, 296.
Slopes of ground by scale of shades, German sys-
tem, 353.
Spandrels (see ARCHES).
Specifications, extracts from, for Thames embank-
ment, 421-423 ; extracts from, for quarantine
pier, 424, 425 ; extracts from, for lock, New
York State canals, 435-437 ; extract from, for
construction of new Croton reservoir, 441, 442 ;
form of, 482.
Sphere, development of surface of, 103, 104 ; pro-
jections and penetrations of, 94-99 ; projec-
tions of, 475-481.
Spires, 267-269.
Spurs (see GEERS); projections of, 175-179.
Stables, 284.
496
INDEX.
Stairs, terms applied to parts of, proportions of,
and laying out of, 237-239, 246.
Steam, effective pressure on piston, expansion,
133 ; pressure at different densities, 134 ; table
of pressures, temperature, and volumes, 135 ;
heating by, 304; cylinder, 321 ; engines, 466.
Steps, 142, 145.
Stereographic projection, 476.
Stirrup-irons, 217.
Stop-cocks, elevations, sections, and plans of 48''-
gate, Brooklyn Water-works, 203.
Store, elevation of, by J. B. Snook, pi. LXXII. ;
cast-iron front of, LXXIII.
Stores, height of, 236 ; and warehouses, 286, pi.
LXXI.
Stoves, 302.
Streets, 450 ; pave of, 452.
String-courses, 262.
Stuffing-box, 473.
Surveys, how conducted, 357.
Susquehanna, framing of engine, 194.
Suspension bridges, 465.
Sweeps, 1 ; variable curves, 10.
Symbols as ornament, 275.
rpEETH of wheel (see GEERS).
J- Temple, of Concord, 260 ; at Talavera, 260.
Tenons and mortices, dra\vings of, 220.
Templates (see GEERS).
Theatres, 292; requirements of, 297 ; dimensions
of several, 298.
Thread (see SCREWS AND BOLTS).
Timber, dimensions of, for frames of dwelling-
houses, 220.
Tinting, methods of, ?28-335.
Tinted topographical drawings, conventional tints
and signs, 378.
Titles, flourishes around, 382 ; of plans, 376,
377.
Toggle-joint, 105.
Topographical drawing, 349 ; in single tint by
brush, 380.
Towers, 267-269.
Tracing of drawings, 368.
Tractive force of a horse, 455.
Traps, siphon, catch-basin, S, 307.
Transept, 294.
Transfer of plots, 368.
Truss (see ROOF) ; by suspension-rod, 232.
Trussing girders or beams, 218.
Trestles, 456.
T square, 9.
Tubular boilers, 468.
Tudor flower, 277.
Turbine wheel, plan and section of turbine and
wheel post, Tremont Manufacturing Company,
Lowell, Mass., 205 ; description of, rules for
the proportioning of parts of, drawings of
guides in disk, and floats in wheel, 206.
Tuscan order, examples of, 254.
~T~TARIATION of magnetic needle, 357.
V Vaults and domes, 264-266.
Vaults, to protect, from moisture, 286.
Venetian school of architecture, 272.
Ventilation, 301-306 ; of sewers, 447.
Ventilators, 302, 305.
Verge-boards, 281.
Viaducts, 260.
Volutes, 256.
Voussoirs (see ARCHES).
WALLS, terms applied to parts of, 212, 213 ;
construction of, bond, batter, brick, stone,
returning, offsets, 212, 213 ; walls of buildings,
London Building Act, 213 ; of canal, 432, 433 ;
enclosing, 266.
Water-closets, 245, 287.
Water supply for dwellings, pipes, size and kinds
of, 308.
Water-wheel, wooden, 138.
Weaving-room, 198-200.
Wedge, 109.
Wheel and axle, 186.
Wheels, toothed (see GEERS); water, 138; tur-
bine, 205.
Windows, terms applied to parts of, proportions
of, and drawings of, 240-242, 246; Roman-
esque, Norman, lancet, trained, decorated,
quatrefoil, flamboyant, perpendicular, square-
headed, 269-271.
Window-jambs, 261.
Woods, strength of, 121.
Working-beam, dimensions and drawing of, 195,
196.
Worm wheel and screw, projections of, 183.
THE E2O).
'353
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