Okt. O '|C^AA.i-» '^c^/iaX^ Ua^J2/vOi-
THE
PRACTICAL DRAUGHTSMAN'S
BOOK OP INDUSTRIAL DESIGN,
AND
MACHINIST'S AND ENGINEER'S DRAWING COMPANION:
FORMING A COMPLETE COURSE OP
Jlttjatitcal, ^Kgiiweriitg, m)i ^xt^ikdmil graining.
TRANSLATED FROM THE FRENCH 0»
M. ARMENGAUD, THE ELDER,
PROFESSOR OF DESIGN IN THE CONSERVATOIRE OF ARTS AND INDUSTRY, PARIS,
AND
MM. ARMENGAUD, THE YOUNGER, AND AMOUROUX,
CIVIL ENGINEERS.
REWRITTEN AND ARRANGED, WITH ADDITIONAL MATTER AND PLATES, SELECTIONS FROM AND EXAMPLES 0?
THE MOST USEFUL AND GENERALLY EMPLOYED MECHANISM OF THE DAY.
BY
WILLIAM JOHNSON, Assoc. Inst., C.E ,
EDITOR OF "THE PRACTICAL MECHANIC'S JOURNAL."
PHIL AD E LP II 1 A:
H E N K Y CAREY B A I R D,
INDUSTRIAL PUBLISHER,
No. 4 6 WALNUT STREET.
1871.
I o ^ '=\ E 7
/
■o
V
PREFACE.
Industrial Design is destined to become a universal language ; for in our material age of rapid transition
from abstract, to applied, Science — in the midst of our extraordinary tendency towards the perfection of the
means of conversion, or manufacturing production — it must soon pass current in every land. It is, indeed,
the medium between thought and Execution ; by it alone can the genius of Conception convey its meaning to
the skill which executes — or suggestive ideas become living, practical realities. It is emphatically the
exponent of the projected works of the Practical Engineer, the Manufacturer, and the Builder ; and by its
aid only, is the Inventor enabled to express his views before he attempts to realise them.
Boyle has remarked, in his early times, that the excellence of manufactures, and the facility of labour,
would be much promoted, if the various expedients and contrivances which lie concealed in private hands,
were, by reciprocal communications, made generally known ; for there are few operations that are not
performed by one or other with some peculiar advantages, which, though singly of little importance, would,
by conjunction and concurrence, open new inlets to knowledge, and give new powers to diligence ; and
Herschel, in our own days, has told us that, next to the establishment of scientific institutions, nothing has
exercised so powerful an influence on the progress of modern science, as the publication of scientific periodicals,
in directing the course of general observation, and holding conspicuously forward models for emulative
imitation. Yet, without the aid of Drawing, how can this desired reciprocity of information be attained ; or
how would our scientific literature fulfil its purpose, if denied the benefit of the graphic labours of tlie
Draughtsman ? Our verbal interchanges would, in truth, be vague and barren details, and our printed
knowledge, misty and unconvincing.
Independently of its utility as a precise art. Drawing really interests the student, whilst it instructs him.
It instils sound and accurate ideas into his mind, and develops his intellectual powei's in compelling him to
observe — as if the objects he delineates v/ere really before his eyes. Besides, he always does that the best,
which he best understands ; and in this respect, the art of Drawing operates as a powerful stimulant to
progress, in continually yielding new and varied results.
A chance sketch — a rude combination of carelessly considered pcncillings — the jotted memoranda of a
contemplative brain, prying into the corners of contrivance — often form the nucleus of a splendid invention.
An idea thus preserved at the moment of its birth, may become of incalculable value, when rescued from the
desultory train of fancy, and treated as the sober olVspring of reason. In nice gradations, it receives the
refining touches of leisure — becoming, first, a finislied sketch, — then a drawing by the practised hand — so that
many minds may find easy access to it, for their joint counsellings to improvement — unlil it linally emerges
from the workshop, as a practical triunipli of mechanical invention— an illustrious I'Mmipio oi' a happy
PREFACE.
corabination opportunely noticed. Yet many ingenious men are barely able even to start this train of
production, purely from inability to adequately delineate their early conceptions, or furnish that transcript of
their minds which might make thdr thoughts immortal. If the present Treatise succeeds only in mitigating
this evil, it will not entirely fail in its object ; for it will at least add a few steps to the ladder of Intelligence,
and form a few more approaches to the goal of Perfection —
"Thou hast not lost an hour whereof there is a record;
A written thought at midnight will redeem the livelong day."
The study of Industrial Design is really as indispensably necessary as the ordinary rudiments of learning.
It ought to form an essential feature in the education of young persons for whatever profession or employment
they may intend to select, as the great business of their lives ; for without a knowledge of Drawing, no
scientific work, whether relating to Mechanics, Agriculture, or Manufactures, can be advantageously studied.
This is now beginning to receive acknowledgment, and the routines of study in all varieties of educational
establishments are being benefited by the introduction of the art.
The special mission of the Practical Draughtsman's Book of Industrial Design may almost be gathered from
its title-page. It is intended to furnish gradually developed lessons in Geometrical Drawing, applied directly
to the various branches of the Industrial Arts : comprehending Linear Design proper ; Isometrical Perspective,
or the study of Projections ; the Drawing of Toothed Wheels and Eccentrics ; with Shadowing and Colour-
ing ; Oblique Projections ; and the study of parallel and exact Perspective ; each division being accompanied
by special applications to the extensive ranges of Mechanics, Architecture, Foundry-Works, Carpentry,
Joinery, Metal Manufactures generally, Hydraulics, the construction of Steam Engines, and Mill-Work. In
its compilation, the feeble attraction generally offered to students in elementary form has been carefully
considered ; and after every geometrical problem, a practical example of its application has been added, to
facilitate its comprehension and increase its value.
The work is comprised within nine divisions, appropriated to the different branches of Industrial Design.
The first, which concerns Linear Drawing only, treats particularly of straight lines — of circles — and their
application to the delineation of Mouldings, Ceilings, Ploors, Balconies, Cuspids, Rosettes, and other forms,
to accustom the student to the proper use of the Square, Angle, and Compasses. In addition to this, it affords
examples of different methods of constructing plain curves, such as are of frequent occurrence in the arts, and
in mechanical combinations — as the ellipse, the oval, the parabola, and the volute ; and certain figures,
accurately shaded, to represent reliefs, exemplifying cases where these curves are employed.
The second division illustrates the geometrical representation of objects, or the study of projections.
This forms the basis of all descriptive geometry, practically considered. It shows that a single figure is
insufficient for the determination of all the outlines and dimensions of a given subject ; but that two projections,
and one or more sections, are always necessary for the due interpretation of internal forms.
The third division points out the conventional colours and tints for the expression of the sectional details
of objects, according to their nature ; furnishing, at the same time, simple and easy examples, which may at
once interest the pupil, and familiarise him with the use of the pencil.
In the fourth division are given drawings of various essentially valuable curves, as Helices, and different
kinds of Spirals and Serpentines, with the intersection of surfaces and their development, and workshop
applications to Pipes, Coppers, Boilers, and Cocks. This study is obviously of importance in many professions,
and clearly so to Ironplate-workers, as Shipbuilders and Boiler-makers, Tinmen and Coppersmiths.
PREFACE.
The fifth division is devoted to special classes of curves relating to the teeth of Spur Wheels, Screw?
and Racks, and the details of the construction of their patterns. The latter branch is of peculiar importance
here, inasmuch as it has not been fully treated of in any existing work, whilst it is of the highest value to the
pattern maker, who ought to be acquainted with the most workmanlike plan of cutting his wood, ami
eflecting the necessary junctions, as well as the general course to take in executing his pattern, for facilitating
the moulding process.
The sixth division is, in effect, a continuation of the fifth. It comprises the theory and practice of
drawing Bevil, Conical, or Angular Wheels, with details of the construction of the wood patterns, and
notices of peculiar forms of some gearing, as well as the eccentrics employed in mechanical construction.
The seventh division comprises the studies of the shading and shadows of the jn/incipal solids — Prisms,
Pyramids, Cylinders, and Spheres, together with their applications to mechanical and architectural details,
as screws, spur and bevil wheels, coppers and furnaces, columns and entablatures. These studies naturally
lead to that of colours — single, as those of China Ink or Sepia, or varied ; also of graduated shades produced
by successive flat tints, according to one method, or by the softening manipulation of the brush, according
to another.
The pupil may now undertake designs of greater complexity, leading him in the eighth division to various
figures representing combined or general elevations, as well as sections and details of various complete
machines, to which are added some geometrical drawings, explanatory of the action of the moving pai-ts
of machinery.
The ninth completes the study of Industrial Design, with oblique projections and parallels, and exact
perspective. In the study of exact perspective, special applications of its rules are made to architecture and
machinery by the aid of a perspective elevation of a corn mill supported on colunms, and fitted up with all
the necessary gearing. A series of Plates, marked a, b, &c., are also interspersed throughout the work, as
examples of finished drawings of machinery. The Letterpress relating to these Plates, together with an
illustrated chapter on Drawing Instruments, will form an appropriate Appendix to the Volume. The general
explanatory text embraces not only a description of the objects and their movements, but also tables and
practical rules, more particularly those relating to the dimensions of the principal details of machinery, as
facilitating actual construction.
Such is the scope, and such are the objects, of the Practical Draughtsman's Book of Industrial Design.
Such is the course now submitted to the consideration of all who are in the slightest degree connected
with the Constructive Arts. It aims at the dissemination of those fundamental teachings which arc so
essentially necessary at every stage in the application of the forces lent to us by Nature for the conversion
of her materials. For ■' man can only act upon Nature, and appropriate her forces to his use, by coniprohcnding
her laws, and knowing those forces in relative value and measure." All art is the true apitlication of
knowledge to a practical end. We have outlived the times of random construction, and ihc mvvc lu^apinL;
together of natural substances. We must now design carefully and delineate accurately before we prooood
to execute — and the quick pencil of the ready draughtsman is a proud possession for our purpose. Let the
youthful student think on this ; and whether in the workshop of the Engineer, the studio of the Architool.
or the factory of the Manufacturer, let him remember that, to spare the blighting of his fdmlosi hoiu's. auil
the marring of his fairest prospects — to achieve, indeed, his higher a8i)iratioiis, and vei-ity his loftier thoughts.
which point to eminence — he must give his days and nights, his business and his leisure, to the study of
5 n I) u 5 1 1 i a I SI c s i 9 ti .
ABBREVIATIONS AND CONVENTIONAL SIGNS.
In order to simplify the language or expression of arithmetical and geometrical operations, the following conventional
wfijns are used : —
The sign + signifies plus or mo/v, and is placed between two or mure terms to indicate additior.
Example : 4 + 3, is 4 plus 3, that is, 4 added to 3, or 7.
The sign — signifies minus or less, and indicates subtraction.
Ex. : 4 — 3, is 4 minus 3, that is, 3 taken from 4, or 1.
The sign X signifies multiplied by, and, placed between two terms, indicates multiplication.
Ex. : 5 X 3, is 5 multiplied by 3, or 15.
When quantities are expressed by letters, the sign may be suppressed. Thus \ve .write, indifierently —
a X b, or ab.
The sign : or (as it is more commonly used) -j-, signifies divided by, and, placed between two quantities, indicati»s division.
Ex.
12
12 : 4, or 12 4- 4, or — , is 12 divided by 4.
The sign = signifies equals or equal to, and is placed between two expressions to indicate their equality.
Ex. : 6 + 2 = 8, meaning, that 6 plus 2 is equal to 8.
The union of these signs, I ;: I indicates geometrical proportion.
Ex. : 2 : 3 :: 4 : 6, meaning, that 2 is to 3 as 4 is to 6.
The sign \ ~ indicates the extraction of a root ; as,
■v/ 9 = 3, meaning, that the square root of 9 is equal to 3.
The interposition of a numeral between the opening of this sign, y', indicates the degree of the root. Thus —
{/27= 3, expresses that the cube root of 27 is equal to 3.
The signs z. ^^^ 7 indicate respectively, smaller than and greater than.
Ex. : 3 / 4, = 3 smaller than 4 , and, reciprocally, 4 7 3, = 4 greater than 3.
Fig. signifies figure ; and pi., plate.
'
FRENCH AND ENGLISH LINEAR MEASURES COMPARED.
French.
English
1 Millimetre
=
•0394 Inches.
11) Millimetres
==
1 Centimetre
=
•3937 "
10 Centimfetres
=
1 Decimetre
=
3-9371 "
10 Decimetres
=
1 Metre
= •
3-2809 Feet.
1-0936 Yards.
10 Metres
=
1 Decametre
=
1-9884 Poles or Eods.
10 Decametres
-=
1 Hectometre
=
19-8844 "
10 Hectometres
=
1 Kilometre
-i
49-7109 Furlongs.
6-2139 MDes.
10 Kilometres
=
1 Myriametre
62-1386 "
Bullish.
12 Inches
3 Feet
53 Yards
40 Poles
8 Furlongs )
1760 Yards
1 Inch.
= 1 Foot
= 1 Yard
= 1 Pole or Rod
= 1 Furlong
= 1 Mile
French.
. ( 25-400 Millimetrea.
' \ 2-540 Centimetres.
: 3-048 Decimetres.
: 9-144
5-029 Metres.
12 Decametres.
: rblO Hcctometrea
C N T E N T S.
1
Preface, ....
Abbreviations and Conventional signs,
VJidB
iii
vi
CHAPTER I.
LINEAR DRAWING,
Definitions and Problems : Plate I.
Lines and surfaces, - . . - .
Applications.
Designs for inlaid pavements, ceilings, and balconies :
Plate II., . . . . .
Sweeps, sections, and mouldings : Plate III.,
Elementary Gothic forms and rosettes : Plate IV.,
Ovals, Ellipses, Parabolas, and Volutes : Plate V.,
Rules and Practical Data.
Lines and surfaces, - - . . .
CHAPTER IT.
THE STUDY OF PROJECTIONS, -
Elementary Principles : Plate VI.
Projections of a point, - - -
Projections of a straight line, ...
Projections of a plane surface, ...
Of Prisms and Other Solids : Plate VII., -
Projections of a cube : Fig. /^, - -
Projections of aright scpiare-based prism, or rectan-
gular parallclopipod : Fig. [g, -
Projections of a quadrangular pyramid; Fig. (g, -
Projections of a right prism, partially hollowed, as
Fig. ©,
Projections of a right cylinder : Fig. [1,
Projections of a right cone : Fig. [^, - .
Projections of a sphere : Fig. (g, -
Of shadow lines, . - . . .
Projections of grooved or fluted cylin(h'rs and
ratchet-wheels: rt.Ai'E VUL., . . .
The elements of architecture : Plate IX., -
Outline of the T'uscan order, ...
Rules and Practical Data.
The measurement of solids, - - . .
ib.
11
13
14
15
19
22
lb.
23
tb.
24
ib.
25
tb.
ib.
ib.
■ib.
26
ib.
27
28
29
30
CHAPTER in.
ON COLORING SECTIONS, WITH APPLICA-
TIONS.
Conventional colors, ....
Composition or mixture of colors: Plate X.,
Continuation of the Study of Projections.
Use of sections — details of machinery : Plate XL,
Simple applications — spindles, shafts, couplings,
■wooden patterns : Plate XII., . . .
Method of constructing a wooden model or pattern
of a coupling, .....
Elementary applications — rails and chairs for rail-
ways : Plate XIII., ....
Rules and Practical Data.
Strength of materials, ....
Resistance to compression or crushing force,
Tensional resistance, ....
Resistance to flexure, ....
Resistance to torsion, . ^ . .
Friction of surfaces in contact, ...
35
ib
36
39
40
41
42
ib.
43
44
46
49
CHAPTER IV.
THE INTERSECTION AND DEVELOPMENT OF
SURFACES, WITH APPLICATIONS, - . 49
The Intersections of Cylinders and Cones ; Plate
XIV.
Pipes and boilers, - - - . - 50
Intersection of a cone with a sphere, . . ib.
Develoi'ments, - - - - - - ib
Development of the cylinder, . - - 51
Development of the cone, - . . - ib
The Delineation and Devklovment of IIelioks,
Screws, and Seki-entines : Plate XV.
Helices, - - - - - - 62
Dovclopiiu'iit of the helix, . . - . 5U
Screws, ... . . ,7,_
Inlcrniil screws, ..... fi.j
Sci'pciililii'S. -.-... |7).
.\]>plicaliiin of the helix the construction of a
staircase: I'l.Mir \\'l., - - - - 5^
Iv
CONTENTS.
The intersection of surfaces — applications to stop-
cocks : Plate XVII., . - - .
EuLES AND Practical Data.
Steam, ......
Unity of heat, . . . . .
Heating surface, . - . . .
Calculation of the dimensions of boilers,
Dimensions of firegrate, - - . .
Chimneys, .-.--.
Safety-valves, . . . . .
CHAPTER V.
THE STUDY AND CONSTRUCTION OF TOOTHED
GEAR,
Involute, cycloid, and epicycloid : Plates XVIII.
and XIX.
Involute : Fig. 1, Plate XVIII., -
Cycloid: Fig. 2, Plate XVIII., -
External epicycloid, described by a circle rolling
about a fixed circle inside it : Fig. 3, Plate XIX.,
Internal epicycloid : Fig. 2, Plate XIX., -
Delineation of a rack and pinion in gear : Fig. 4,
Plate XVIII., - . - . .
Gearing of a worm with a worm-wheel : Figs. 5 and
6, Plate XVIII.,
Cylindrical ok Spdr Gearing : Plate XIX.
External delineation of two spur-wheels in gear :
Fio- 4 ..... .
Delineation of a couple of wheels gearing internally :
Fig. 5, - _
Practlcat delineation of a couple of spur-wheels :
Plate XX.,
The Delineation and Construction of Wooden Pat.
terns for Toothed Wheels : Plate XXI.,
Spur-wheel patterns, ....
Pattern of the pinion, .
Pattern of the wooden-toothed spur-wheel, -
Core moulds, .....
Bulks and Practical Data.
Toothed gearing, .....
Angular and circumferential velocity of wheels.
Dimensions of gearing, ....
Thickness of the teeth, ....
Pitch of the teeth, .....
Dimensions of the web, ....
Number and dimensions of the arms,
Wooden patterns, .....
CHAPTER VI.
CONTINUATION OF THE STUDY OF TOOTHED
GEAR.
Conical or bevil gearing, . , .
Design for a pair of bevil-wheels in gear : Plate
XXII.,
Construction of wooden patterns for a pair of bevil-
wheels : Plate XXIII., . . . .
Involute and Helical Teeth : Plate XXIV.
Delineation of a couple of spur-wheels, with involute
teeth : Figs. 1 and 2, - . - -
Helical gearing : Figs, 4 and 5, -
Contrivances FOE Obtaining DifferentialMovements.
56
58
59
60
ib.
61
ib.
62
63
ib.
64
65
tb.
ib.
67
ib.
68
69
70
ib.
71
ib.
72
74
75
ib.
76
77
ib.
- ib.
ib.
80
82
83
The delineation of eccentrics and cams: Plate
XXV., 85
Circular eccentric, - ... ib.
Heiirt-sliaped cam : Fig. 1, - - . . ib.
Cam for producing a uniform and intermittent
movement : Figs. 2 and 3, ... ih.
Triangular cam : Figs. 4 and 5, - - -86
Involute cam : Figs. 6 and 7, - . . ib
Cam to produce intermittent and dissimilar move-
ments : Figs. 8 and 9, - - - .87
Rules and PnACTicAi, Data.
Moclianical work of effect, . . - - 88
The simple machines, - - . - 91
Centre of gravity, - - . - - 93
On estimating the power of prime movers, - - ib.
Calculation for the brake, .... ib.
The fall of bodies, - - - - - 95
Momentum, - . . - - ib.
Central forces, - - - - - ib.
CHAPTER VII.
ELEMENTARY PRINCIPLES OF SHADOWS, -
Shadows of Prisms, Pvramids, and Cylinders : Plate
XXVI.
Prism, - - * -
Pyramid, - - - -
Truncated pyramid, . . . . .
Cylinder, --....
Shadow cast by one cylinder on another, -
Shadow cast by a cylinder on a prism,
Shadow cast by one prism on another.
Shadow cast by a prism on a cylinder,
Principles of Shading : Plate XXVII.,
Illumined surfaces, - - . - .
Surfaces in the shade, . . . .
Flat-tinted shading, - . . . .
Shading by softened washes, . . .
Continuation of the Study of Shadows : Plate
XXVIII.
Shadow cast upon the interior of a cylinder.
Shadow cast by one cylinder upon another,
Shadows of cones, .....
Shadow of an inverted cone.
Shadow cast upon the interior of a hollow cone.
Applications, . . . . .
Tuscan Order : Plate XXIX.
Shadow of the torus, - - - -
Shadow cast by a straight line upon a torus, or
quarter round, . . - - .
Shadows of surfaces of revolution, ...
Rules and Practical Data.
Pumps, ....
Hydrostatic principles, ....
Forcing pumps. .....
Lifting and forcing pumps, - . . .
The hydrostatic press, ....
Hydrostat-ical calculations and data — discharge of
water through different orifices, -
Gauging of a water-course of uniform section and fall,
Velocity of the bottom of water-courses,
Calculation of the discharge of water through rect-
angular orifices of narrow edges, -
96
ib.
97
98
ib.
ib.
ib.
99
ib.
100
ib.
ib.
101
102
103
ib.
^^.
104
105
ib.
ib.
i07
ib.
108
ib.
ib.
109
?"&.
?7).,
Ill)*
ib
. HI
CONTENTS.
Calculation of the discharge of water through over-
shot outlets, ..... 114
To determine the width of an overshot outlet, - tb.
To determine the depth of the outlet, - - ib.
Outlet with a spout or duct, - - - -116
TO TOOTHED
CHAPTER VIII.
APPLICATION OF SHADOWS
GEAR: Plate XXX.
Spur-wheels : Figs. 1 and 2, -
Bevil-wheels : Figs. 3 and 4, - - ■
Application of Shadows to Screws : Plate XXXI.
Cylindrical square-threaded screw : Figs. 1, 2, 2"
and 3, ----- •
Screw with several rectangular threads : Figs. 4 and .5
Triangular-threaded screw : Figs. 6, 6°, 7, and 8, ■
Shadows upon around-threaded screw : Figs. 9 and 10, 119
Application of Shadows to a Boiler and its Furnace :
Plate XXXII.
Shadow of the sphere : Fig. 1, - - -
Shadow cast upon a hollow sphere : Fig. 2,
Applications, . . . . -
Shading in Black — Shading in Colours : Plate
XXXIIT.,
ib.
117
118
ib.
ib.
tb.
ib.
120
ib.
122
CHAPTER IX.
THE CUTTING AND SHAPING OF MASONRY :
Plate XXXIV., . - . .
The Marseilles arch, or ari-ilre-voussure : Figs. 1
and 2, - - - -
Rules and Practical Data.
Hydraulic motors, - - . . -
Undershot water-wheels, with plane floats and a
circular channel, - - - - -
Width,
Diameter, ......
Velocity, . . . . ^ -
Number and capacity of the buckets.
Useful effect of the water-wheel, . . .
Overshot water-wheels, - r - -
Water-whecis, with radial floats, . . -
Water-wheels with curved buckets.
Turbines, ......
Realakks ON Machine Tools, . . , .
- 123
ib.
126
ib.
ib.
127
ib.
ib.
ib.
128
129
130
ib.
131
OHAPTEU X.
THE STUDY OF MACHIN ERY AND SKETCHING.
Various applications and combinations, - - 133
The Sketching of Macuinkry : Plates XXXV. and
XXXVI., ib.
Drilling Machine, - - - - - ib.
Motive Machine.s.
Water-wheels, . . . - . i;!.'j
Construction and setting up of water-wheels, - ib.
Delineation of watcr-wliecls, ... i;i()
Design for a water-wheel, - - - - 1117
Sketch of a water-wheel, . . - - ib.
(Wkrshot Water-Whei;i,s : Fig. 12, - - - ib.
Delineating, sketching, and designing overshot
watcr-whcels, - - - - - 13H
Water-Pumps : Plate XXXVII.
Geometrical delineation, - - - - 138
Action of the pump, . - - . - 139
Steam Motors.
High-pressure expansive steam-engine : Plates
XXXVIII., XXXIX., and XL., - - - 141
Action of the engine, .... 142
Parallel motion, - - - . ib.
Details of Construction.
Steam cylinder, - - - 143
Piston, .... • ib.
Connecting-rod and crank, - - - > ib.
Fly-wheel, - - - - - • ib.
Feed-pump, ----- ib.
Ball or rotating pendulum governor, - 144
Movements oftheDistribution and Expansion Valves, ib.
Lead and lap, - . . . . 145
Rules and Practical Data.
Steam-engines : low pressnre condensing engine
without expansion valve, - - - - 146
Diameter of piston, . - - . - 147
Velocities, .----- 148
Steam-pipes and passages, - - - - ib.
Air-pump and condenser, - - - ■ ib.
Cold-water and feed-pumps, - - - . 149
High pressure expansive engines, - - - ib.
Medium pressure condensing and expansive steam-
engine, ...... 151
Conical pendulum, or centrifugal governor, - 153
CHAPTER XI.
OBLIQUE PROJECTIONS.
Application of rules to the delineation of an oscilla-
ting cylinder : Plate XLL,
CHAPTER XII.
PARALLEL PERSPECTIVE.
Principles and applications : Plate XLIL,
- 154
.55
CHAPTER Xin.
TRUE PERSPECTIVE.
Elementary principles : Plate XLIIL, - - 11)8
First problem — the perspective of a hollow prism :
Figs. 1 and 2, - - - - - ib.
Second problem — the perspective of a cylinder:
Figs. 3 and 4, - - - - - ir>a
Third problem — the perspcclivc of a regular solid.
when the point of sight is situated in a piano
passing through its axis, and perpendicular to the
plane of the picture : Figs. 5 and 6, - - ItiO
Fourlh problem— the perspective of a bearing brass,
phiced with its axis vertical: Figs. 7 and 8, - il>.
Fifth jirobleni — the perspective of a stojnuH'li with
a splieric-al boss : Figs. 9 and 10, - - - ib.
Sixtli problem — the i)erspectivo of an object iilaced
in any i)osition with regard \i.t the ])lane of tlie
]>icture : Figs. 11 and 12. - - - - lOl
A])plications — Hourinill driven by belts: Plavks
XLIV. and Xl-V.
nescription of the mill, - - - - ib.
CONTENTS.
Representation of the mill in perspective, -
Notes of recent improvements in flour-mills,
Schicle's mill, ....
Mulliu's " ring millstone," ...
Barnett's millstone, . - - -
Hastie's arrangement for driving' mills,
Currie's improvements in millstones,
Rur,ES AND Practiciai- Data.
Work performed by various machines.
Flour-mills, - - - - -
Saw-mills, - - . - .
Veneer sawing machines, - . -
Circular saws, . . - .
PAGE
163
164
ih.
165
166
ib.
ih.
168
170
171
ib.
CHAPTER XIV.
EXAMPLES OF FINISHED DRAWINGS OF MA-
CHINERY.
Example Plate [^, balance water-meter, - - 172
Example Plate [g, engineer's shaping machine, - 174
Example Plate ©, [o), 1^, express locomotive engine, 178
Example Plate [^, wood planing machine, - - 180
Example Plate (g, washing machine for piece goods, 182
Example Plate [X], power-loom, ... ih.
Example Plate 0, duplex steam boiler, - - 183
Example Plate (J), direct-acting marine engines, - 184
CHAPTER XV.
DRAWING INSTRUMENTS, -
186
INDEX TO THE TABLES.
ench and English linear measures compared,
ultipliers for regular polygons of from 3 to 12 sides.
Approximate ratios between circles and squares, -
Comparison of Continental measures with French millime-
tres and English feet, . . . - -
Surfaces and volumes of regular polyhedra.
Proportional measurements of the various parts of the
(modern) Doric order, . . . - .
Proportional measurements of the various parts of the
Tuscan order, ...---
Weights which solids, such as columns, pilasters, supports,
will sustain without being crushed, . . .
Weights which prisms and cylinders will sustain when
submitted to a tensile strain, - - - -
Diameters of the journals of water-wheel and other shafts
for heavy work, ------
Diameters for shaft journals calculated with reference to
torsional strain, ------
Ratios of friction for plane surfaces, - - -
Ratios of friction for journals in bearings, -
Pressures, temperatures, weights, and volumes of steam, -
Amount of heat developed by one kilogramme of fuel.
Thickness of plates in cylindrical boilers, -
Dimensions of boilers and thickness of plates for a pressure
of five atmospheres, - - - - -
Diameters of safety-valves, . - - - -
Numbers of teeth, and diameters of spur gear.
Pitch and thickness of spur-teeth for different pressures, -
PAGE
6
19
20
21
30
33
34
42
44
46
49
ib.
58
59
ib.
60
•62
73
77
PA3B
Dimensions of spur-wheel arms, - - - - 78
Average amount of mechanical effect producible by men
and animals, - - - - - - 89
Heights corresponding to various velocities of falling bodies, 94
Comparison of French and English measures of capacity, - 110
Discharges of water through an orifice one mfetre in width, 111
Discharge of water by overshot outlets of one mfetre in width, 113
Discharge of water through pipes, - - - - 115
Dimensions and practical results of various kinds of turbines, 131
"Velocity and pressure of machine tools or cutters, - 132
Diameters, areas, and velocities of piston in low pressure
double-acting steam engines, with the quantities of steam
expended per horse power, - - - - 147
Force in kilogrammfetres given out with various degrees of
expansion, by a cubic mfetre of steam, at various pres-
sures, ------- 150
Proportions of double-acting steam-engines, condensing and
non-condensing, and with or without cut-off, the steam
being taken at a pressure of four atmospheres in the con-
densing, and at five atmospheres in the other engines, - 152
Proportions of medium pressure condensing and expansive
steam-engines, with two cylinders, on Woolf 's system —
pressure four atmospheres, - - - - ib.
Dimensions of the arms, and velocities of the balls of the
conical pendulum, or centrifugal governor, - - 153
Power, quantity of wheat ground, and number of pairs of
stones, with their accessory apparatus, required in flour-
mills. 16S
PRACTICAL DRAUGHTSMAN'S
BOOK OF INDUSTRIAL DESIGN.
CHAPTER I.
LINEAR DRAWING.
I N Drawing, as applied to Mechanics and Architecture, and to the
Industrial Arts in general, it is necessary to consider not only the
aiere representation of objects, but also the relative principles of
action of their several parts.
The principles and methods concerned in tliat division of the
art which is termed linear drawing, and which is the foundation of
all drawing, whether industrial or artistic, are, for the most part,
derived from elementary geometry. This branch of drawing has
for its object the accurate delineation of surfaces and the con-
struction of figures, obtainable by the studied combinations of
lines ; and, with a view to render it easier, and at the same time
more attractive and intelligible to the student, the present work
has been arranged to treat successively of definitions, principles,
and problems, and of the various applications of which these are
capable.
Many treatises on linear drawing already exist, but all these,
considered apart from their several objects, seem to fail in the due
development of the subject, and do not manifest that general ad-
vancement and increased precision in details which are called for
at the present day. It has therefore been deemed necessary to
begin with these rudimentary exercises, and such exemplifications
have been selected as, with their varieties, are most frcqiioutly
met with in practice.
Many of the methods of construction will be necessarily such as
are already known ; but they will be limited to those which are
absolutely indispensable to the development of the principles
and their applications.
DEFINITIONS.
OF LINES AND SURFACES.
PLATE I.
In Geometry, space is described in (he terms of its three dimen-
sions — length, oreadth or thickness, iiiul height or (lc|)th.
The combination of two of these dimensidiis rciircsi^iils siirfnce,
tnd one dimension takes the form of a lltw.
Lines. — There are several kinds of lines used in drawing —
straight or right lines, curved lines, and irregular or broken lines.
Right lines are vertical, horizontal, or inclined. Cmred lines are
circular, elliptic, parabolic, <^c.
Surfaces. — Surfaces, which are always bounded by lines, are
plane, concave, or convex. A surface is plane when a straight-edge
is in contact in every point, in whatever position it is applied to it.
If the surface is hollow so that the straight-edge only touches at
each extremity, it is called concave ; and if it swells out so that
the straight-edge only touches in one point, it is called convex.
Vertical lines. — By a vertical line is meant one in the position
which is assumed by a thread freely suspended from its upper ex-
tremity, and having a weight attached at the other ; such is the
line A B represented in fig. A. This line is always straight, and
the shortest that can be drawn between its extreme pomts.
Plumb-line. — The instrument indicated in fig. A is called a
plumb-line. It is much employed in building and the erection
of machinery, as a guide to the construction of vertical lines and
surfaces.
Horizontal line. — When a liquid is at rest in an open vessel, its
upper surface forms a horizontal plane, and all lines drawn upon
such surface are called horizontal hnos.
Levels. — It is on this' principle that what are called fiuid levels
are constructed. One description of fluid level consists of two
upright glass tubes, connected by a pipe comiiuinicating with the
bottom of each. Wlien the instrument is partly filled with water,
the wafer will stand at the same height in both tubes, and thereby
indicate the true level. Another form, and one more generally
used, denominated a spirit level — spirit being usually eniployeil —
consists of a glass tube (fig. ©) enclosed in a metal case, <»,
attached by two supports, b, to a plate, c. The tube is almost
filled \vith liquid, and the bubble of air, </, which remains, is
always exactly in the centre of the tube wlion any surface, c P, ou
which the iiistrnmeiit is placed, is perfectly level.
Masons, carpenters, joiners, and other mechanics, are in iJio
habit of using the instrument represented in fig. ©, consisting
sinqily of a ])lniiil) line attached to the point of junction of the two
inclined side pieces, ab, be, of equal lengtli, and connected iic:ir
their free ends hy the cross-piece, a n, which has a marl; :il [if
THE PRACTICAL DRAUGHTSMAN'S
cuiitiv. VVIiLMi the plumb lino coincidus with this mark, the object,
c u, on which the instrument is placed, is exiictly horizontal.
Perpendiculars. — If the vertical line, a b, fig. 1, be placed on
(he horizontal line, c d, the two lines will be perpendicular to, and
Torm right angles with, each other. If now we suppose these lines
to be turned round on the point of intersection as a centre, always
preserving the same relative position, they will in every position
'x! perpendicular to, and at right angles with, each other. Thus
the line, i o, fig. 5, is at right angles to the line, e f, although
neither of them is horizontal or vertical.
Broken lines. — It is usual to call those lines broken, which con-
sist of a series of right lines lying in different directions — such as
tiie lines b, a, e, h, f, n, fig. 14.
Circular lines — Circumference. — The continuous line, e f g H,
fig. 5, drawn with one of the points of a pair of compasses — of
which the other is fixQd — is called the circumference : it is evidently
equally distant at all points from the fixed centre, o.
Radius. — The extent of opening of the compasses, or the dis-
tance between the two points, o, f, is called a radius, and conse-
quently all lines, as o e, o f, o a, drawn from the centre to the
circumference are equal radii.
Diameter. — Any right line, l h, passing through the centre o,
and Ihnited each way by the circumference, is a diameter. The
di'imeter is therefore double the length of the radius.
Circle. — The space contained within the circumference is a
jdane surface, and is called a circle : any part of the circumference,
e r F, or F L G, is called an arc.
Chords. — Right lines, e f, f g, connecting the extremities of
arcs, are chords ; these lines extended beyond the circumference
oecome secants.
Tangent. — A right line, a b, fig. 4, which touches the circumfer-
ence in a single point, is a tangent. Tangents are always at right
angles to the radius which meets them at the point of contact, b.
Sector. — Any portion, as b o h c, fig. 4, of the surface of a
circle, comprised within two radii and the arc which connects their
outer extremities, is called a sector.
Segment. — A segment is any portion, as e f i, fig. 5, of the
surfiice of a circle, comprised within an arc and the chord which
subtends it.
Right, continuous, and broken lines, are drawn by the aid of the
square and angle ; circular lines are delineated with compasses.
Angles. — We have already seen that, when right lines are per-
pendicular to each other, they form right angles at their inter-
sections: when, however, they cross each other without being
perpendicular, they form acute or obtuse angles. An acute angle
is one which is less than a right angle, as f c d, fig. 2 ; and an
obtuse angle is greater than a right angle, as g c d. By angle is
generally understood the extent of opening of two intersecting
anes, the point of intersection being called the apejc. An angle is
rectilinear when formed by two right lines, mixtihnear when formed
by a right and a curved line, and curvilinear when formed by two
curved lines.
Measurement of angles. — If, with the apex of an angle as the
centre, we describe an arc, the angle may be measured by the por-
tion of the arc cut off by the lines forming the angle, with reference
to the whole circle ; and it is customary to divide an entire circle
into 360 or 400* equal parts, called degrees, and mstrumests called
protractors, and represented in figs. ©, g, are constructed, whereby
the number of degrees contained in any angle are ascertain-
able. The first, fig. [6), which is to be found in almost every set
of mathematical instruments, being that most in use. consists of a
semicircle divided into 180 or 200 parts. In making use of it,
its centre, b, must be placed on the apex of the angle in such a
manner that its diameter coincides with one side, a b, of the angle,
when the measure of the angle will be indicated by the division
intersected by the other side of the angle. Thus the angle, a be,
is one of 50 degrees (abbreviated 50°), and it will always have this
measure, whatever be the length of radius of the ai-c, and conse-
quently whatever be the length of the sides, for the measuring arc
must always be the same fraction of the entire circumference
The degree is divided into 60 minutes, and the minute (or 1') into
60 seconds (or 60") ; or when the circle is divided into 400 degrees,
each degree is subdivided into 100 minutes, and each minute into
100 seconds, and so on
The other protractor, fit. E, of modern invention, possesses the
advantage of not requiring access to the apex of the angle. It
consists of a complete circle, each half being divided on the inner
side into 180 degrees, but externally the instrument is square. It is
placed against a rule, R, made to coincide with one side, c e, of the
angle — ^the other side, d c, crosses two opposite divisions on the
circle indicating the number of degrees contained in the angle. It
will be seen that the angle, dee, is, one of 50°.
Oblique lines. — Right lines, which do not form right angles with
those they intersect, are said to be oblique, or inclined to each
other. The right lines, g c and f g, fig. 2, are oblique, as referred
to the vertical line, k c, or the horizontal line, c j.
Parallel lines. — Two right lines are said to be parallel with each
other when they are an equal distance apart throughout their
length ; the lines, i k, a b, and l m, fig. 1, are parallel.
Triangles. — The space enclosed by three intersecting lines is
called a triangle ; when the three sides, as d e, e f, and f d, fig. 12,
are equal, the triangle is equilateral; if two sides only, as g h.
and G I, fig. 9, are equal, it is isosceles ; and it is scalene, or irregular,
when the three sides are unequal, as in fig. 6. The triangle is
called rectangular when any two of its sides, as d l and l k, fig. 10,
form a right angle ; and in this case the side, as d k, opposite to, or
subtending the right angle, is called the hypothenuse. An instrument
constantly used in di-awing is the set-square," more commonly
called angle; it is in the shape of a rectangular triangle, and is
constructed of various proportions ; having an angle of 45°, as fig.
©, of 60° as fig. [nl, or as fig. 0, having one of the sides which form
the right angle at least double the length of the other.
Polygon. — A space enclosed by several lines lying at any angle
to each other is a polygon. It is plane when all ttie lines lie in one
and the same plane; and its outline is called its perimeter. A
polygon is triangular, quadrangular, pentagonal, hexagonal, tiepta-
gunal, octagonal, ^c, according as it has 3, 4, 5, 6, 7, or 8 sides
A square is a quadrilateral, the sides of which, as a b, b c, c d,
* As another step towards a decimal notation, it was proposed, in 1790, to divide
the circle into 400 parts. The suggestion was again revived in 1840, and actually
adopted by several distinguished individuals. The facility afforded to calculators
l»y the many sul)niultiples possessed hy the number 360, however, accounts (or the
still vary general use of the ancient system of division.
BOOK OF INDUSTRIAL DESIGN.
aai D A, fig. 10, are equal and perpendicular to one another, the
angles consequently also being equal, and all right angles.
A rectangle is a quadiilateral, ha\ing two sides equal, as a b
and F N, fig. 14, and perpendicular to two other equal and parallel
sides, as a j" and b n.
A parallelogram is a quadrilateral, of which the opposite sides
and angles are equal ; and a lozenge is a quadrilateral with all the
sides, but only the opposite angles equal.
A trapezium is a quadiilateral, of which only two sides, as h i
and ivi L, fig. 9, are parallel.
Polygons are regular when all theii- sides and angles are equal,
and are otherwise irregular. All regular polygons are capable of
being inscribed in a circle, hence the great facility with which they
may be accurately delineated.
OBSERVATIONS.
Wi; have deemed it necessary to give these definitions, in order
to make our descriptions more readily understood, and we propose
now to proceed to the solution of those elementary problems with
which, from then- frequent occurrence in practice, it is important
that the student should be well acquainted. The first step, how-
ever, to be takon, is to prepare the paper to be drawn upon, so
that it shall be well stretched on the board. To effect this, it
must be slightly but equally moistened on one side with a sponge ;
the moistened side is then applied to the board, and the edges of
the paper glued or pasted down, commencing with the middle of
the sides, and then securing the corners. When the sheet is dry,
it wilJ be uniformly stretched, and the drawing may be executed,
being first made in faint pencil lines, and afterwards redelineated
with ink by means of a drawing pen. To distinguish those lines
which may be termed working lines, as being but guides to the
foimation of the actual outlines of the drawing, we have in the
plates represented the former by dotted lines, and the latter by full
continuous lines.
PROBLEMS.
1. To erect a perpendicular on the centre of a given right line, as
c D, Jig. 1 . — From the extreme points, c, d, as centimes, and with a
radius greater than half the line, describe the arcs which cross each
other ill a and b, on either side of the line to be divided. A line,
A B, joining these points, will be a perpendicular bisecting the line,
c D, in G. Proceeding in the same manner with each half of the line,
c G and G D, we obtain the porpendicuiars, i K and l im, dividing
the line into lour equal parts, and we can thus divide any given
light lino into 2, 4, 8, I(i, &c., equal parts. This problem is of
constant application in drawing. For instance. In order to obtain
the principal lines, vx and yz, wiiich divide the sheet of paper
into four equal parts; with the points, rsiw, taken as near the
edge of the jiaper as possible, as centres, wo describe the airs
which intersect each other in p and q; and with these hist as
centres, describe also the arcs wiiich cut each other in y, z. Tiie
right lines, v X kiuI y /, drawn Ihrnugh Hk; poinis, r, (j, ;ind ij. z.
respectively, are perpendicular to each oilier, and servo as guides
in drawing on diflerent parts of the paper, and are merely pencilled
in, to bo afterwards effaced.
2. To erect ajierpnuUciildr on anu given jmini, as ii, in the line c i),
Jig. 1 — Mark oflT on the line, on each side of the point, two equal
distances, as c h and h g, and with the centres c and g describe
the arcs crossing at i or k, and the line drawn through them, and
through the point h, will be the Une required.
3. To let fall a perpendicular from, a point, as l, apart from
the right line, c d. — With the point l, as a centre, describe an
arc which cuts the line, c d, in g and d, and with these points as
centres, describe two other arcs cutting each other in M, and the
right line joining l and M will be the perpendicular required. In
practice, such perpendiculars are generally drawn by means of an
angle and a square, or T-square, such as fig. H".
4. To draw parallels to any given lines, as v x and T z. — For
regularity's sake, it is well to construct a rectangle, such as r s tu,
on the paper that is being drawn upon, which is thus done : — From
the points v and x, describe the arcs r, s, t, u, and apphing the
rule tangentially to the two first, draw the line r s, and then in the
same manner the line t u. The lines r t and s u are also obtained
in a similar manner. In general, however, such parallels are more
quickly di-awn by means of the T-square, wiiich may be slid along
the edge of the board. Short parallel lines may be drawn with
the angle and rule.
5. To divide a given right line, as a b. Jig. 3, into several equal
parts. — We have already shown how a line may be di\ided into 2
or 4 equal parts. We shall now give a sunple metliod for dividing
a line into any number of equal parts. From the point a, dra^
the line a c, making any convenient angle with a b ; mark off on a c
as many equal distances as it is wished to divide the line a b into ;
in the present instance seven. Join c b, and trom the several
points mai-ked off on a c, draw parallels to c b, using the rule and
angle for this purpose. The line a b will be divided into seven
equal parts by the intersections of the parallel lines just drawn.
Any line maldng any angle with a B, as A J, may be employed in-
stead of A c, with exactly the same results. This is a very usefui
problem, especially applicable to the formation of scales tor the
reduction of drawings.
6. A scale is a straight line divided and subdivided into feet,
inches, and parts of inches, according to English measures; or into
metres, decimetres, centimetres, and millimetres, according to
French measures ; these divisions bearing the same proportion to
each other, as in the system of measurement from w hich they are
derived. The object of tlie scale is to indicate the proportion the
drawing bears to the object represented.
7. To construct a scale. — The French scale being the one
adopted in this work, it will be iioccssary to state that the rnitre
( = 39-371 English inches) is the unit of nieasurouK'nt, and is.
divided into 10 decimetres, 100 coiitiuii'lros, and 1000 niillinu'triw*.
If it is intended to execute the drawing to a scale of i or J ; Uie
metro is divided by 4 or 5, one of the divi.sions being tlio length of a
mitre on the reduced scale. A lino of #this length is drawn on the
pMjier, and is divided into reduced dccimetros, &c., just :is the
ineti-e is itself. Fig. 7 is pai't of a scale for reducing a drawing to
one-liflli. Ill this .scale an e.xtra division is placed to the loi^ of
/eio, which is subdivided, to fncilitalo the oblainmont of any re-
quired measure. For oxamplo, if we want a length correspondini.-
to 32 ceiitiint>tres, we place one point of liio oonipassos on tin-
division marked 3 to the right of zero, and tlie other on fho second
10
THE PRACTICAL DRAUGHTSMAN'S
division to the left, and the length comprised between these points
^\'ill be 3 decimetres, 2 centimetres, — 32 centimetres.
The diagonal scale. — When very minute measurements are re-
quired, greater precision is obtained witli a diagonal scale, such as
fig. 8. It is thus constructed : — Ha\ing drawn a line and divided
it, as in fig. 7, draw, parallel and equal to it, ten other lines, as
c, d, e,f, &c., at equal distances apart, crossing these with perpen-
diculars at the decimetre divisions. From one of the smaller
divisions to the left of zero, di-aw the diagonal, b i, and draw
parallels to it from the remaining centimetre dinsions, 1', 2', 3', &e.
From the division corresponding to 1 decimetre, draw a diagonal
to the point on the extreme parallel, i 4, cut by the zero perpen-
diculai-, and di-aw also the parallel diagonals, 1 — 2, 2 — 3, and 3 — 4.
It will be evident, that as in the space of the ten horizontal lines,
the diagonal extends one di\asion to the left, it will intersect each
intermediate line, as the 1st, 2d, 3d, &c., at the distance of 1, 2, 3,
&c., tenths of such division, in the same direction, so that the
diagonal line, 2', will cut the 5th line at a point 2^ of a diWsion
distant from zero. Thus, one point of the compasses being placed
on the point Z, and the other on the intersection of the same
horizontal line with the perpendicular of the decimetre di\ision 3,
the measure comprised between them wUl be 3 decimetres, 2 cen-
timetres, and ^, or 5 millimetres = 325 millimetres.
8. To divide a given angle, as y c D,Jig. 2, inlo two equal angles.
— With tlie apex, c, as a centre, describe the arc, h i, and with the
two points of intersection, H, i, as centres, describe the arcs cutting
each other in j ; join j c, and the right line, j c, will divide the
angle, f c d, into two equal angles, h c j and j c i. These may
be subdivided in the same manner, as shovvn in the figm-e. An
angle may also be divided by means of either of the protractors,
9. To draw a tangent to a given circle, o b d h, fig. 4. — ^If it is
required to draw the tangent through a given point, as D, in the
circle, a radius, c d, must be di-awn meeting the point, and be pro-
duced beyond it, say to e. Then, by the method already given,
draw a line, f g, perpendicular to c e, cutting it in d, and it
will be the tangent required. K, however, it is required to draw
the tangent through a given point, as a, outside the circle, a
straight line must be drawn joining the point, a, and the centre, c,
of the circle. After bisecting this line in the point, o, with tliis
point as a centre, describe a circle passing through a and c, and
cutting the ^ven circle in b and h; right lines joining ab and
A H will both be tangents to the given circle, and the radii c b and
c H will be perpendiculars to a b and a h respectively.
10. To find the centre of a given circle, or that loiih which a given
arc, as E F G, fig. 5, is draion. — With any three points, e, f, g,
as centres, describe arcs of equal cu'cles, cutting each other, and
ttu-ough the points of intersection draw right lines, i o and l o ;
o, the point of intersection of these two lines, is the required
centre.
11. To describe a circle through any three points not in a right
line. — Since only one circle can pass through the same tlu-ee points,
and since any circle may be described when the centre is found
and a point in the circumference given — this problem is solved in
exactly the same manner as the preceding.
12. To iriscribe a circle in a given triangle, as ab c, fig. 6. —
A eh'cle is said to be inscribed in a figure, when all the sides of the
latter are tangents to it. Bisect any two of the angles by right
lines, as A o, B o, or c o ; and from the point of intersection, o, let
fall perpendiculars to the sides, as o e, o f, and o g. These per-
pendiculars will be equal, and radii of the required circle, o being
the centre.
13. To divide a triangle, as gk i,fig. 9, into two equal parts. — ^If
the parts are not required to be similar, bisect one side, as g i, in the
point, 0, with which, as a centre, describe the semicii-cle, g k i, of
which G I is the diameter. This semicucle wll be cut in the point,
K, by the perpendicular, k o ; mark off on g i a distance, g l, equal
to G K, and draw the line, l m, parallel to h i. The triangle, g l M,
and the trapezium, h i l m, will be equal to each other, and each
equal to half the triangle, g h i. If the given triangle were g n i,
it would also be divided into two equal parts by the line, l m.
14. To draw a square double the size of a given square, a b c d,
fig. 10. — After producing from diflerent corners any two sides
which are at right angles to each other, as d a and d c, to H and l,
with the centre, d, and radius, d b, describe the quadrant or quarter
of a circle, fee; and through the points of intersection, f and e, -nith
the lines, d a and d l, di-aw parallels to d l and d a respectively, or
tangents to liie quadrant, fee; the square, f g e d, will be double
the area of the given square, a b c d ; and in the same manner-a
square, held, may be drawn double the area of the square,
F G e D. It is evident that the diagonal of one square is equal to
one side of a square twice the size.
15. To describe a circle half the size of a given circle, as acbd,
fig. 11. — Draw two diameters, a b and c d, at right angles to
each other; join an extremity of each, as a, c, by the chord, a c
Bisect this chord by the perpendicular, e f. The radius of tha
required circle will be equal to e g. It follows that the annulai
space shaded in the figure is equal to the smaller circle within it.
16. To inscribe in given circles, as in fig. 12, an equilateral
triangle and a regular hexagon. — ^Draw any diameter, g f, and with
G, as a centre, describe the arc, doe, its radius being equal to that
of the given circle ; join d e, e f, and e d, and d e f will be the
triangle required. The side of a regular hexagon is equal to the
radius of the cu-cumscribing circle, and, therefore, in order to in-
scribe it in a circle, all that is necessary is to mark off on the cir-
cumference the length of the radius, and, joining the points of in-
tersection, as K I L H M J, the resulting figure will be the hexagon
required. To inscribe figures of 12 or 24 sides, it is merely ne-
cessary to divide or subdivide the arcs subtended by the sides
obtained as above, and to join the points of intersection. It is
frequently necessary to draw very minute hexagons, such as screw-
nuts and bolt-heads. This is done more quickly by means of the
angle of 60°, [H] , which is placed against a rule, (^, or the square,
in different positions, as indicated in fig. 12.
17. To inscribe a square in a given circle, as acbb, fig. 13. —
Draw two diameters, as a b, c d, perpendicular to one another, ana
join the points of intersection with the circle, and acbd vv-iU be
the square required.
18. To describe a regular octagon about a circle having a given
radius, as o e, fig. 13. — Having, as in the last case, drawn two
diameters, as e f, g h, draw other two, i j, k l, bisecting tne
angles formed by the former ; through the eight points of interseo-
BOOK OF INDUSTRIAL DESIGN.
II
tinn with the circle draw the tangents, e, k, g, j, f, l, i — these tan-
gents will cut each other and form the regular octagon required.
This figure may also be drawn by means of the square, and angle
of 45°, ©.
19. To construct a regular octagon of wMch one side is given, as
A. ^,Jig- 14. — Draw the perpendicular, o d, bisecting a b ; draw a f
parallel to o d, produce a b to c, and bisect the angle, c a f, by the
line E A, making e a equal to a b. Draw the line o a, perpendi-
cular to, and bisecting e a. o g will cut the vertical, o d, in o,
which wUl be the centre of the circle circumscribing the required
octagon. This may, therefore, at once be drawn by simply mark-
ing otf arcs, as E H, H f, &c., equal to a b, and joining the points,
E, H, F, &c. By dividing and subdividing the arcs thus obtained
we can draw regular figures of 16 or 32 sides. The octagon is
a figure of frequent application, as for dravvdng bosses, bearing
brasses, «&c.
20. To construct a regular pentagon in a given circle, as a b c d f,
also a decagon in a given circle, as e r T!d,Jig. 15. — The pentagon
is thus obtained ; draw the diameters, a i, e j, perpendicular to each
other ; bisecting o e in k, with k as a centre, and k a as radius,
describe the arc, a l ; the chord, a l, will be equal to a side of the
pentagon, which may accordingly be drawn by making the chords
.which form its sides, as a e, f d, d c, c b, and b a, equal to A l.
By bisecting these arcs, the sides of a decagon may be at once
obtained. A decagon may also be constructed thus : — Draw two
radii perpendicular to each other, as o m and o r ; next, the tan-
gents, N M and N R. Describe a circle having n m for its diameter ;
join R, and p the centre of this circle, the line, R p, cutting the
eircle in a ; R a is the length of a side of the decagon, and applying
tt to the circle, as r b, &-c., the required figure will be obtained.
The distance, r a or r c, is a mean proportional between an entire
radius, as r n, and the difference, c n, between it and the radius.
A mean proportional between two lines is one having such relation
to them that the square, of which it is the one side, is equal to the
rectangle, of which the other two are the dimensions.
21. To construct a rectangle of ichich the sides shall he mean pro-
portionals between a given line, as a c, fig. 16, and one a third or
two-lhirds of it. — a c, the given line, will be the diagonal of the
required rectangle; mth it as a diameter describe the circle ab c d.
Divide a c into three equal parts in the points, m, n, and from these
points draw the perpendiculars, m d and n b ; the lines wMch join
the points of intersection of these lines with the circle, as a b, a d,
c B, c D, will form the required rectangle, the side of which, c d, is
a mean proportional between c m and c a, or —
Cm: CD::CD :CA;
that is to say, the square of which c d is a side, is equal to a rec-
tangle of which c A is the length, and c m the height, because
CDxCD = CmxCA*
In like manner, a d is a mean proportional between c A and m a.
This problem often occurs in practice, in measuring timber. Thus
the rectangle inscribed in the circle, fig. 16, wiiich may bo con-
sidered as representing the section of a tree, is Uio loriii of tlu^
b(?am of the greatest strength which can be obtained IVoni (lui
tree.
* S«o the noted mid rulcn givim at the end of thin rhniitor
APPLICATIONS.
DESIGNS FOR INLAID PAVEMENTS, CEILINGS, AND BALCONIES.
PLATE II.
The problems just considered are capable of a great variety of
applications, and in Plate II. will be found a collection of some of
those more frequently met \vith in mechanical and architectural
constructions and erections. In order, however, that the student
may perfectly understand the ditferent operations, we would
recommend him to draw the various designs on a much larger
scale than that we have adopted, and to which we are necessarily
limited by space. The figures distinguished by numbers, and
showing the method of forming the outlines, are drawn to a larger
scale than the figures distinguished by letters, and representing the
complete designs.
22. To draw a pavement consisting of equal squares, figs. ^ and
1. — Taking the length, a b, equal to half the diagonal of the
required squares, mark it off a number of times on a horizontal
line, as from a to b, b to c, &c. At a erect the perpendicular
I H, and draw parallels to it, as d e, g f, &c., through the several
points of division. On the perpendicular, i h, mark off a number
of distances equal to a b, and draw parallels to a b, through the
points of division, as h g, i f, &,c. A series of small squares will
thus be formed, and the larger ones are obtained simply by di-aw-
ing the diagonals to these, as shown.
23. To draw a pavement composed of squares and interlaced
rectangles, figs. [|3 and 2. — Let the side, as c d, of the square be
given, and describe the circle, l m q b, the radius of which is equal
to half the given side. With the same centre, o, describe also
the larger eircle, k n p i, the radius of which is equal to half the
side of the square, plus the breadth of tlie rectangle, a b. Draw
the diameters, a c, e d, perpendicular to each other; draw tan-
gents through the points, a, d, c, e, forming the square, j H f g ;
draw the diagonals j f, g, h, cutting the two circles 'n the points,
I, B, K, L, M, N, p, Q, through which draw paraUels to thv.^ diagonals.
It will be perceived that the lines, a e, e c, c d, and d a, are
exactly in the centre of the rectangles, and consequently serve to
verify their correctness. The operation just described is repeated,
as far as it is wished to extend the pattern or design, many of tlie
hues being obtained by smiply prolonging those already dra^^^l.
In inking this in, the student must be very careful not to cross tlie
lines. This design, though analogous to the first, is somewhat
different in appearance, and is applicable to the construction of
trellis-work, and other devices.
24. To draw a Grecian border or frieze, figs. © and 3. — On two
straight lines, as a b, a c, perpendicular to each other, mark ofl',
as often as necessary, a distance, a i, representing the w idtli, c/,
of tlie ribbon forming the iiattorn. Through all the points ot'
di\ isioii, diaw paralk'ls to A B, a c — thus forming a series of small
sciuairs, guided by wiiich the pattern may bo at once inked in,
(•([ual ilistances being maintained betweeu tiio sets of linos, as in
fig. ©. This ornament is frequently mot with iu arcJutooturo,
biiiig used lor ceilings, cornices, railings, and b!iloonios ; jUso in
cabinet work and maciiinory for borders, and for wood and iron
gratings.
25. '/') draw a \mvcincnt comixyscd of squarts <iful rn;ular i^Utt-
12
THE PRACTICAL DRAUGHTSMAN'S
gnns,figs. [D) and 4. — With a radius, e o, equal to half tlie width,
E F, of tlie octagon, describe a circle, e g f h, and, as was shown
in reference to fig. 13, Plate I., draw the octagon circumscribing
it — the squai-e, a b c d, being first obtained, and its diagonals,
A c, B D, drawn cutting the cu-cle In the points, i, J, k, l, tangents
being then drawn through these points. The octagon may also
be formed by marking off' from each corner of the square, a, b, c, d,
a distance equal to a o, or half its diagonal — and thereby will be
obtained the points of junction of the sides of the octagon. The
pattern is extended simply by repeating the above operation, the
squares being formed by the sides of four contiguous octagons,
which are inclined at an angle of 45° to the horizontal lines.
This pattern is generally produced in black and white marble,
or in stones of different colours, whereby the effect is distinctly
brought out.
26. To draw a fakement composed of regular hexagons, figs. [1
and 5. — With a radius, a o, equal to a side, a b, of the hexagon,
describe a circle, in which inscribe the regular hexagon, a b c d e f.
The remaining hexagons wiU readily be obtained by producing, in
different directions, the sides and diagonals of this one. In fig. d,
the hexagons are plain and shaded alternately, to show their
aiTangement ; but in practice they are generally all of one colour.
27. To draw a pavement composed of trapeziums, combined in
squares, figs. ^ and 6. — Draw the square, a b c d ; also its diago-
nals, A c, b d; construct the smaller square, abed, concenti'ic with
the fii'st. On the diagonal, b d, mark the equal distances, or, cf, and
through e and/ draw parallels to the diagonal, a c; join the points
of intersection of these with the smaller squares by the lines, k I,
m n, which ■nill give all the lines required to form the pattern,
requiiing merely to be produced and repeated to the desired
extent. Very beautiful combinations may thus be formed in
different kinds of wood for furniture and panels.
28. To draw a panel design composed of lozenges, figs. © and 7. —
On a straight line, a b, mark off the length of a side of the lozenge
twice ; construct the equilateral triangle, a b c ; draw the line c d,
perpendicular to a b ; and draw a e and b f parallel to d c, and
E F parallel to a b. Construct the equilateral triangle, e d f, cut-
ting the triangle, a b c, in g and h, and join g h. In this manner
are obtained the lozenges, a g h d and e g h c, and by continuing
the lines and drawing parallels at regular distances apart, the
remainder of the pattern will be readily constructed — this being
repeated to any desired extent.
29. To draw a panel pattern composed of isosceles triangles, figs. [L,
and 12. — Kin the last-mentioned fig. ©, we draw the longitudinal
diagonal of each lozenge, we shall obtain the type of the pattern
[L,. We will, however, suppose that the base, ab, of the triangle is
given, instead of the side of the lozenge. Mark off this length
twice on the line, a b, and construct the equilateral triangle, a c d,
just as in the preceding case ; also the second similar triangle,
D E F, thus obtaining the points g and h. Join a h, g b, e h, and
G F, &c., and each point of intersection, as i, l, &c., vAW be the
apex of three of the isosceles triangles. The pattern, [L,, is pro-
duced by giving these triangles various tints.
Tlie patterns we have so far given are a few of the common
arrangements of various regular polygons. An endless variety of
patterns may be produced by combining these different figures,
and these are of great use in many arts, particularly for cabmet
inlaid mosaic work, as well as for pavements and other ornamental
constructions.
30. To draw an open-work casting, consisting of lozenges and
rosettes, figs. ^ and 8. — The lozenge, a &c<^, being given, the points,
a, b, c, d, being each the centre of a rosette, draw and indefinitely
produce the diagonals, ac,bd, which must always be perpendicular
to each other. Through the points, a, b, c, d, draw parallels to
these diagonals, also an indefinite number of such parallels at
equal distances apart. The intersections of these lines will be the
centres of rosettes and lozenges alternately, and the former may
accordingly be drawn, consisting merely of circles with given radii.
The centres of the rosettes are joined by straight lines, and to
right and left of these, at the given distances, fg, fh, parallels to
them are drawn, thereby producing the concentric lozenges com-
pleting the pattern.
31. To draw a pattern for a ceiling, composed of small squares or
lozenges, and irregular but symmetrical octagons, figs. Q and 9. —
The rectangle, a b c d, being given, its corners forming the centres
of four of the small lozenges, draw the Unes, e f, g h, dividing the
rectangle into four equal parts ; next mark off the semi-diagonals
of the lozenges, as a i, a o, and join i and o. The centre lines of
the pattern being thus obtained, the half-breadths, fg, fh, are
marked on each side of these, and the appropriate parallels to them
drawn. In extending the pattern by repetition, the points con-es-
ponding to i and o will be readily obtained by di-awing a series of
parallel lines, as 1 1 and o o. By varying the proportions between
the lozenges and the octagons, as also those between the different
dimensions of each, a number of patterns may be produced of very
varied appearance, although formed of these simple elements.
32. To draw a stone balustrade of an open-work pattern, com-
posed of circular and straight ribbons interlaced, figs. S and 10. —
Construct the rectangle, a b c d. its corners being the centres of
some of the required circles, which may accordingly be di-awn, with
given radii, as Ab, cd; after bisecting a b in e, and dra\ving the
vertical E g, make e f equal to e a, and with f as a centre, draw
the cu-cle having the radius, Fg, equal to a J, drawing also the
equal circles at c, b, e, &c. Draw verticals, such as g h, tangents
to each of the circles,- which wUl complete the lines required for
the part of the pattern, S, to the left. The rosettes to the right
are formed by concentric cu'cles of given radii, as e e, e/. The
duplex, fig. S, may be supposed to represent the pattern on the
opposite sides of a stone balustrade. Where straight lines are rim
into parts of circles, the student must be careful to make them
join well, as the beauty of the drawing depends gi-eatiy on this
point. It is better to ink in the circles first, as it is practically
easier to draw a straight line up to a circle than to draw a cu'cle to
suit a straight line.
33. 7'o draw a pattern for an embossed plate on- casting, composed
of regular figures combined in squares, figs. K and 11. — Two
squares being given, as a b c d and f g h i, concentric, but with the
diagonals of one parallel to the sides of the other, draw first the
square, abed, and next the inner and concentric one, efgh. The
sides of the latter being cut by the diagonals, a c and b d, in the
points, i,j, k, I, through these di'aw parallels to the sides of the
square, a b c d, and finally, with the centre, o, describe a smaV
BOOK OF INDUSTRIAL DESIGN.
13
cin:le, the diameter of which is equal to the width of the indented
crosses, the sides of these being di-awn tangent to this circle.
Thus are obtained all the lines necessary to delineate this pattern ;
the relievo and intaglio portions are contrasted by the latter being
shaded.
In the foregoing problems, we have shown a few of the many
varieties of patterns producible by the combination of simple
regular figures, lines, and cu-cles. There is no limit to the multi-
plication of these designs ; the processes of construction, however,
being analogous to those just treated of, the student will be able
to produce them with every facility.
SWEEPS, SECTIONS, AND MOULDINGS.
PLATE III.
34. To draw in a square a series of arcs, relieved by semicircular
mouldings, figs. A and 1. — Let a b be a side of the square ; draw
the diagonals cutting each other ia the point, c, through which
draw parallels, d e, c f, to the sides ; with the corners of the
square as centres, and with a given radius, a g, describe the four
quadrants, and with the points, d, f, e, describe the small semicncles
of the given radius, n a, which must be less than the distance, n b.
This completes the figure, the symmetry of which may be verified
by drawing circles of the radii, c g, c h, which should touch, the
former the larger quadrants, and the latter the smaller semi-
circles. If, instead of the smaller semicucles, larger ones had
been drawn ndth the radius, d b, the outline would have formed a
perfect sweep, being free from angles. This figure is often met
wth in machinery, for instance, as representing the section of a
beam, connecting-rod, or frame standard.
35. To draw an arc tangent to two straight lines. — First, let the
radius, a b, fig. 2, be given ; with the centre, a, being the point
of intersection of the two lines, a b, a c, and a radius equal
to a b, describe arcs cutting these lines, and through the points of
intersectton di-aw parallels to them, b o, c o, cutting each other in
0, which will be the centre of the required arc. Draw perpen-
diculars from it to the straight lines, a e, a c, meeting them in
u and E, which will bo the points of contact of the required arc.
Secondly, il a point of contact bo given, as n, fig. 3, the lines
being a b, a c, making any angle with each oilier, bisect the
angle by the straight line, a d ; draw b o perpendicular to a b, from
the point, B, and the point, o, of its intersection with a d, \\ill be
the centre of the required arc. If, as in figs. 2 and 3, we draw
arcs, of radii somewhat less than o b, we shall form conges, which
Btand out from, instead of being t^ingents to, the given straight
lines. This problem meets with an application in drawing fig. |3,
which represents a section of various descriptions of castings.
3G. To draw a circle tangent to three given straight lines, wlilch
make antj angles tvilh each other, fig. 4. — Bisect the angle of the
lines, a b and a c, by the straight lino, a e, and the angle formed
by a D and c a, by lluj line, c f. a e and c f will cut each other
in the point, o, which is at an equal distance from each side, and
is cons('(incntly tim centre fif the r<'(|nir('(l circle, wiiicli may !)('
drawn willi a i-adiiis, i'{|ii:il lo a lirn- IVuiii tlic |i(iiiil, o, |iri|H'M(li-
cular to any of the sides. This |)iiil)l(iii is necessary lor llic cuni-
pietion of tig. y.
37. To draw the section of a stair rail, fig. ©. — This gives riso
to the problems considered in figs. 5 and 6. First, let it be re-
quu-ed to draw an arc tangent to a given arc, as a b, and to the
given straight line, c d, fig. 6 — d being the point of contact with
the latter. Through d draw e f perpendicular to c d : make f d
equal to o b, the radius of the given arc, and join of, thiough
the centre of which draw the perpendicular, g e, and the point, e, of
its intersection with e f, will be the centre of the required arc,
and E D the radius. Further, join o e, and the point of intersec-
tion, B, with the arc, a b, will be the point of junction of the two
arcs. Secondly, let it be required to draw an arc tangential to a
given arc, as a b, and to two straight lines, as b c, c d, fig. S.
Bisect the angle, b c d, by the straight line, c e ; v.ith the centre, c,
and the radius, c h, equal to that of the given arc, o a, describe
the arc, a G ; parallel tone di-aw i h j, cutting e c in j. Join o j,
the line, o j, cutting the arc, h g, in g ; join c g, and draw o K pa-
rallel to c G ; the point, k, of its intersection with e j, will be
the centre of the required arc, and a line, k l or k m, perpendicular
to either of the given straight lines, will be the radius.
38. To draw the section of an acorn, fig. ©. — This figure calls
for the solution of the two problems considered in figs. 9 and 10.
Fiist, it is required to draw an arc, passing through a given point,
A, fig. 9, in a line, a b, in which also is to be the centre of the
arc, this arc at the same time being a tangent to the given arc, c.
Make a d equal to o c, the radius of the ^ven arc ; join o d, and
di-aw the perpendicular, f b, bisecting it. b, the point of inter-
section of the latter line, with a b, is the centre of the required aic,
A E c, a B being the radius. Secondly, it is required to di-aw an arc
passing through a given point, a, fig. 10, tangential to a given arc,
BCD, and having a radius equal to a. With the centre, o, of the
given arc, and with a radius, o e, equal to o c, plus the given radius,
a, draw the arc e ; and witli the given point, a, as a centre, and
with a radius equal to a, describe an arc cutting the former in E
— E wall be the centre of the requued arc, and its point of contact
with the given arc will be in c, on the line, o e. It will be seen
that in fig. ©, these problems arise in drawing either side of the
object. The two sides are precisely the same, but reversed, and
the outline of each is equidistant from the centre line, which should
always be pencilled in when drawing similar figures, it being ditli-
cult to make them symmetrical without such a guide. This is ;m
ornament frequently met with in macliinery, smd in articles of
various materials and uses.
39. To draw a wave curve, formed by arcs, septal and tangent lo
each other, and parsing through given points, a, b, their radius being
equal lo half the distance, A B, figs. E and 7. — Join a n, and draw
tlie pri|)enilicular, E F, bisecting it in c. ^V'itll the centres, A luid
c, and radius, A c, describe arcs cutting each other in G, and with
the centres, b and c, other two cutting each other in ii; g imd h
will be the centres of tlio required anis, forming the curve or
sweep, A c B. This curve is very oonuuon in architecture, and is
styled the ci/ma recta.
40. To draw a similar curve to the preceding, but formed by
arcs of a given radius, as a. i.figs. [? ami 11. — Divide tlio stniigh*
line into fo\ir eiiual parts by the pi'i-jKiudiculiirs, K F, g h. mui
c D ; then, with the centre, a, and given radius, a i, which uuist
idways be greater tiuui tlio quarter of a u, describe tho urv
14
THE PRACTICAL DRAUGHTSMAN'S
cutting CD in c ; also with the centre, b, a similar arc cutting
G H in H ; c and h will be the centres of the arcs forming the re-
quired curve. Wliatever be the given radius, provided it is not
too small, the centres of the arcs will always be in the lines, c d
and G H. It will be seen that the arcs, c i and h l, cut the straight
lines, c d and g h, in two points respectively. If we take the
second points, k l, as centres, we shall form a similar curve to the
last, but with the concavity and convexity transposed, and called
the ajma reversa. The two will be found in fig. [F, the first at a,
and the second at b. This figure represents the section of a
door, or window frame — it is one well known to carpenters and
masons.
The little instrument known as the " Cymameter," affords a
convenient means of obtaining rough measurements of contoiu^
of various classes, as mouldings and bas-reliefs. It is simply a
light adjustable frame, acting as a species of holding socket for a
mass of parallel slips of wood or metal — a bundle of straight
wires, for example. Previous to applying tiiis for taking an im-
pression of measurement, tlie whole aggregation of pieces is
dressed up on a flat surface, so that their ends form a perfect
plane, like the ends of the bristles in a square cut brush ; and
these component pieces are held in close parallel contact, vvith just
enough of stiff friction to keep them from slipping and falling
away. The ends of the pieces are then applied well up to the
moulding or surface whose cavities and projections are to be mea-
dui'ed, and the frame is then screwed up to retain the slips in the
position thus assumed. The surface thus moulds its sectional
contour upon the needle ends, as if the surface made up of these
ends was of a plastic material, and a perfect impression is there-
fore carried away on the instrument. The nicety of delineation is
)bviously bounded by the relative fineness of the measuring ends.
41. To draw a baluster of a duplex carUour, fgs. @ and 8. — ^It is
here necessary to draw an arc tangent to, or sweeping into two
known arcs, a i and c d, and having its centre in a given hori-
zontal, e i. Extend e i to h, making i h equal to g d, the radius
of the arc, c d. Join g h, bisecting g h by a perpendicular ; this
will cut e H in the point, e, which is the centre of the required arc —
e i being its radius. A line joining e g cuts c d in c, the point of
contact of the two arcs. The arc, d f, which is required to be a
tangent to c r», and to pass tlu-ough the point, f, is drawn with the
centre, o, obtained by bisecting the chord, d f, by a perpendicuJiir
which cuts the radius of the arc, c n. This curve has, in fig. @, to
be repeated both on each side of the vertical line, m n, and of the
horizontal line,/g'.
42. To draw the section of a baluster of simple outline, as fig. ^.
— ^We have here to draw an arc passing through two points, a, b,
fig. 12, its centre being in a straight line, b c; this arc, moreover,
requiring to join at d, and form a sweep with another, d e, having
jts centre in a line, f d, parallel to b c. Joining b a, a perpendicu-
lar bisecting b a, will cut b c in o, which will be the centre of the
first arc, and that of the second may now be obtained, as in
problem 37, fig. 6.
43. The base of the baluster, fig. frO, is in the form of a curve,
termed a scolia. It may be di-awn by various methods. The
following are two of the simplest — according to the fii'st, the
wrve may be formed by arcs sweeping into each other, and tan-
gents at A and c to two given parallels, a, p, c d, fig. 13. Through
A and c draw the perpendiculars, c o and a e, and divide the latter
into three equal parts. With one division, f a, as a radius,
describe the first arc, a g h ; make c i equal to f a, join i f, and
bisect I F by the perpendicular, o k, which cuts c o in o. o will
be the centre of the other arc required. The line, o h, passing
through the centres, o and f, will cut the arcs in the point of junc-
tion, H. It is in this manner that the curve in fig. [>[] is obtained.
The second method is to form the curve by two arcs sweeping
into each other and passing through the given points, a b, fig. 14,
their centres, however, being in the same horizontal line, c d,
parallel to two straight lines, e f and b c, passing through the
given points. Through a, draw the perpendicular, a i. i, its point
of intersection with c d, is the centre of one arc, a d. Next draw
the chord, b d, the perpendicular bisecting which, will cut c d in o,
the centre of the other arc, the radius being o d or o B. This curve
is more particularly met with in the construction of bases of the
Ionic, Corinthian, and Composite orders of arcliitecture.
With a view to accustom the student to proportion his designs
to the rules adopted in practice in the more obvious applications,
we have indicated on each of the figs. ^, [B, ©, <Sic., and on the
corresponding outlines, the measurements of the various parts, in
millimetres. It must, however, at the same time be understood,
that the various problems are equally capable of solution with
other data ; and, indeed, the number of applications of which the
forms considered are susceptible, wiU give rise to a considerable
variety of these.
ELEMENTARY GOTHIC FORMS AND ROSETTES.
PLATE rv.
44. Having solved the foregomg problems, the student may
now attempt the delineation of more complex objects. He need
not, however, as yet, anticipate much difficulty, merely giving his
chief attention to the accurate determination of the principal lines,
which serve as guides to the minor details of the drawing.
It is in Gothic architecture that we meet with the more numer-
ous applications of outlines formed by smoothly joined circles and
straight lines, and we give a few examples of this order in Plate IV.
Fig. 5 represents the upper portion of a window, composed of a
series of arcs, combined so as to form what are denominated cuspid
arches. The width or span, a b, being given, and the apex, c ,
joining a c, c b, draw the bisecting perpendiculars, cutting a b in
D and e. These latter are the centres of the sundiy concentric
arcs, which, severally cutting each other on the vertical, c f, form
the arch of the window. The small interior cuspids are drawn
in the same manner, as indicated Ln the figure ; the horizontal,
G H, being given, also the span and apexes. These interior arches
are sometimes surmounted by the ornament, M, termed an ceil-de-
iceuf consisting simply of concentric circles.
45. Fig. 1 represents a rosette, formed by concentric circles,
the outer interstices containing a series of smaller circles, forming
an interlaced fillet or ribbon. The radius, a o, of the cu'cle, con-
taining the centres of all the small circles, is supposed to be given.
Divide it into a given number of equal parts. With the points of
division, 1, 2, 3, &c., as centres, describe the circles tangential to
BOOK OF INDUSTRIAL DESIGN.
15
each other, forming the fillet, making the radii of the alternate
ones in any proportion to each other. Then, with the centre, o,
describe concentric circles, tangential to the larger of the fillet
circles of the radius, Ab. The central ornament is formed by
arcs of circles, tangential to the radii, drawn to the centres of the
fillet circles, their convexities being towards the centre, o ; and the
arcs, joining the extremities of the radii, are drawn with the actual
centres of the fillet circles.
46. Fig. 6 represents a quadi-ant of a Gothic rosette, distin-
guished as radiating. It is formed by a series of cuspid arches and
radiating muUions. In the figure are indicated the centre Unes of
the several arches and mullions, and in fig. 6^, the capital, con-
necting the mullion to the arch, is represented drawn to double
the scale. With the given radii, a b, a c, a d, a e, describe the
different quadrants, and divide each into eight equal parts, thus
obtaining the centres for the trefoil and quadrefoil ornaments in
and between the different arches. We have drawn these orna-
ments to a larger scale, in figs. 6*, &", and 6°, in which are indi-
cated the several operations required.
47. Fig. 4 also represents a rosette, composed of cuspid arches
and trefoil and quadi-efoil ornaments, but disposed in a different
manner. The operations are so similar to those just considered,
that it is unnecessary to enter into further details.
48. Fig. 7 represents a cast-iron grating, ornamented with
Gothic devices. Fig. 7» is a portion of the details on a larger
scale, from which it will be seen that the entire pattern is made
np simply of arcs, straight lines, and sweeps formed of these two,
the problems arising comprehending the division of lines and
angles, and the obtainment of the various centres.
49. Figs. 2 and 3 are sections of tail-pieces, such as are sus-
pended, as it were, from the centres of Gothic vaults. They also
represent sections of certain Gothic columns, met with in the
architecture of the twelfth and thirteenth centuries. In order to
draw them, it is merely necessary to determine the radii and centres
of the various arcs composing them.
Several of the figures in Plate FV. are partially shaded, to
indicate the degree of relief of the various portions. We have in
this plate endeavoured to collect a few of the minor difficulties,
our object being to familiarize the student to the use of iiis instru-
ments, especially the compasses. These exercises will, at the same
time, qualify him for the representation of a vast number of forms
met with in machinery and architecture.
OVALS, ELLIPSES, PARABOLAS, VOLUTES, &c.
PLATE V.
50. The ove is an ornament of the shape of an egg, and is
formed of arcs of circles. It is frequently employed in architecture,
and is thus drawn: — The axes, ab and c d, fig. 1, being given,
oorpondicular to each olhcr; with Iho point of intersection, o, as a
centre, first describe the circle, cade, half of whicli forms tlio
upper portion of the ove. Joining b c, make c f equal to b e,
the difference between the radii, o c, o b. Bisect F n by the per-
pendicular, G n, cutting c d in Ji. ii will be the centre, and n c
the radius of the arc, c j ; and i, the point of intersection of h c
with A n, will bo the centre, and i b the radius of tho anialler arc,
I B K, which, together with the arc, h k, described with the centre,
L, and radius, l d, equal to h c, form the lower portion of the
required figure ; the lines, g h, l k, which pass through the
respective centres, also cut the arcs in the points of junction,
J and K. This ove will be found in the fragment of a cornice,
fig. ^- A more accurate and beautiful ove may be diTiwn by
means of the instrument represented in elevation and plan in the
annexed engraving.
The pencil is at a, in an adjustable holder, capable of sliding
along the connecting-rod, b, one end of which is jointed at c, to a
slider on the horizontal bar, D, whilst the opposite end is similarly
jointed to the crank arm, e, revolving on the fixed centre, f, on
the bar. By altering the length of the crank, and the position of
the pencil on the connecting-rod, the shape and size of the ovo
may be varied as required.
51. The oval differs from the ove in having the upper portion
symmetrical with the lower ; and to di-aw it, it is only necessary to
repeat the operations gone thi'ough in obtaining the curve, l b y
fig. 1.
52. The ellipse is a figure which possesses the following ptv»
perty : — The sum of the distances from any pomt, a, fig. 2, in Oio
circumference, to two constant points, b, c, in the longer axis, is
always equal to that axis, d e. The two points, b, c, are termed
foci. The curve forming the ellipse is symmetric wit/ rsfereuc*
both to the horizontal line or iLxis, d e, and to the vertical line, i g,
bisecting the former in o, the centre of the ellipse. Linos, as
B A, c A, b F, c F, &c., joining any point in the chcumfcrenco Willi
the foci, B and c, are called rectors, and any pair proceeding from
one point are together equal to the longer axis, d e, wliich is lalled
the transverse, f g being tho conjugate axis. There are diflerent
methods of drawing this curve, which we will proceed to in
dicato.
53. First Method. — This is based on the definition given above,
and requires that tho two axes bo given, as d e and f o, fig. 2.
The foci, b and c, are first obtained by describing an :uv. \\ iiii tlie
extremity, g or f, of the conjugjite axis as a centre, and with a
radius, f c, equal to half the transverse axis; the an- will cut tho
latter in the points, n and c, llu' I'oii. It" now we divide n c
inuMiu.'iUy in H, and with the radii, n u, v. n, and the foci as ceiitn};!,
wo describe arcs .severally cutting each otlior in i, J, K, A ; the ii
four points will lie in the circumference. If, I'urtlier, we «g;uB
iinequ.Mlly divide n E, siiy in i,, we can similarly ohtjiin t\uir ot'iei
16
THE PRACTICAL DRAUGHTSMAN'S
poin Is in the circumference, and we can, in like manner, obtain any
number of points, when the ellipse may be traced through tliem
by hand. The large ellipses which are sometimes required in con-
structions, are generally drawn with a trammel instead of compasses,
the trammel being a rigid rule with adjustable points. — The
gardener's ellipse: To obtain this, place a rod in each of the foci
of the required ellipse ; round these place an endless cord, which,
when stretched by a tracer, will form the vectors ; and the ellipse
■will be drawn by carrjing the tracer round, keeping the cord
always stretched.
54. Second Method. — Take a strip of paper, having one edge
straight, as d b, and on this edge mark a length, a b, equal to half the
transverse axis, and another length, b c, equal to half tlie conju-
gate axis. Place the strip of paper so that the point, a, of the
longer measurement, lies on the conjugate, f g, and tlie other point,
c, on the transverse axis, d e. If the strip be now caused to
rotate, always keeping the two points, a and c, on the respective
axes — the other point, b, vsill, in every position, indicate a point
in the circumference which may be marked %\-ith a pencil, the
ellipse being afterwards traced through the points thus obtained.
55. Third Method, fig. 3. — It is demonstrated, in that branch of
geometry which treats of solids, as we shall see later on, that
if a cone, or cylinder, be cut by a plane inclined to its axis, the
resulting section will be an ellipse. It is on this property that
the present method is based. The transverse and conjugate axes
being given, as a b and c d, cutting each other in the centre, o,
draw any line, a e, equal in length to the conjugate axis, c d,
ajid on A E, as a diameter, describe the semicircle, e g a. Join
E B, and through any number of points, taken at random, on e a,
as 1, 2, 3, &c., draw parallels to e e. Then, at each point of
division, on b a, erect perpendiculars, 1 a, 2 5, 3 c, &c., cutting
the semicircle, and, at tlie corresponding di\Tsions obtained on
A B, erect perpendiculars, as 1' a', 2' b, 3' c, &c., and make them
equal to the corresponding perpendiculars on e a. A line traced
through the various points thus obtained, that is, the extremities,
a', b', c, &c., of the lines, wiU form the required ellipse.
56. Fourth Method. — On the transverse axis, a b, and with the
centre, o, describe the semicircle, a f b, the axis forming its dia-
meter ; and with the diameter, h i, equal to the conjugate axis,
describe the smaller semicircle, h d i. Draw radii, cutting the
two semicircles, the larger in the points, i^j, k, I, &c., and the
smaller in the points, i',j', k', T, die. It is not necessary tliat
the radii should be at equiangular distances apart, though they
are di-awn so in the plate for regularity's sake. Through the latter
points draw parallels to the ti-ansverse axis, a b, and thi'ough the
former, parallels to the conjugate axis, c d, the points of intersection
of these lines, as q, r, s, t, &.C., 'will be so many points in the required
ellipse, which may, accordingly, be traced through them. It
follows from this, that, in order to draw an ellips-e, it is sufHcient
to know either of the axes, and a point in the cu-cumference. Let
the axis, A b, oe ^ven, and a point, r, in the circumference, which
must always lie within perpendiculars passing through the extre-
mities of the given axis. Through r draw a line, rj', parallel to
A. B, and a line, rj, perpendicular to it; with tlie centre, o, and
•^us, o A, equal half the given axis, describe the arc, cutting
J ui J ; join J o, and the line, j (\, will cut rj' in j' : oj' will be
equal to half the conjugate axis, on. If the conjugate axis, c d,
be given, proceed as before ; the arc, however, in this case, ha\ing
the smaller radius, OD, and cutting rj' 'mj'; then join oj,' pro.
ducing the line till it cuts rj, which will be in _;", and oj will equal
half the transverse axis, a b. It has already been shown how to
describe an ellipse, when the two axes are given.
We may here give a method invented a short time back by Mr.
Crane of Bii-mingham, for constructing an ellipse with the com-
passes. Tliis method applies to all proportions, and produces as
near an approximation to a true ellipse, as it is possible to obtain
by means of four arcs of circles.
By appljing compasses to any true ellipse, it will be seen that
certain parts of the curve approach very near to arcs of circles,
and that these parts are about the vertices of its true axes ; and by
the natm-e of an ellipse, the cm've on each side of either axis is
equal and simOar; consequently, if arcs of circles be drawn
through all the vertices, meeting one another in four points, the
opposite arcs being equal and similar, the resulting figure will bo
indefinitely near an ellipse. Four circles, described from four
different points, but with only two difi'erent radii, are then required.
These four points may be aU within the figure ; the centres of the
two greater circles may either be within or without, but the
centres of the two circles at the extremities of the major axis
must always be within, and, consequently, the w^hole four points
can never be without the figure. Again, the proportions of the
major and minor axes may vary infinitely, but they can never be
equal ; therefore, any rule for describing ellipses must suit all pos-
sible proportions, or it does not possess the necessary requirements.
Moreover, if any rule apply to one certain proportion and not to
another, it is evident that the more the proportions differ from that
one — whether crescendo or diminuendo — the greater will be the
difference of the result from the true one. From this it follows,
that if a rule applies not to aU, it can only apply to one propor-
tion ; and also, that if it apply to a certain proportion and not to
another, it can only be correct in that one case.
Let A B be any major axis, and c d any minor axis ; produce them
both in either direction, say towards f and h, and make a f equal
to c G ; then join c a, and through f draw f h parallel to c a.
Set off B I, A J, and c k, equal to h c ; join j K, and bisect it in k,
and at k erect a perpendicular, cutting c d, or c d produced, at M ;
then make g e equal to g m ; j, e, i, m, wiU be the centres of the
four circles required. Through the points, j and i, draw m n, m o
E p, E Q, each equal to m c ; then M n and e p will be the radii of
the greater circles, and j k, x o, of the less : the points of contact
will therefore be at n, o, p, q, and the fig-ure drawn through a, h
c, 0, B, Q, D, p, will be the required ellipse.
BOOK OF INDUSTRIAL DESIGN.
n
T\
Several instruments have been invented for drawing ellipses,
many of them very ingeniously contrived. The best known of
tlii.'se contrivances, are those of Farey, WUson, and Hick — ^^the
l;ist of which we present in the annexed engraving. It is shown
as in working order,
with a pen for drawing
ellipses in ink. It con-
sists of a rectangular base
plate, A, having sharp
countersunk points on
its lower surface, to hold
the instrument steady,
and cut out to leave a
sufficient area of the
paper uncovered for the
traverse of the pen.
It is adjusted in position
by four index lines,
setting out the trans-
verse and conjugate
axes of the intended
ellipse — these lines
being cut on the inner
edges of the base. Near
sne end of the latter, a vertical pillar, b, is screwed down, for the
(jurpose of carrying the traversing slide-arm, c, adjustable at any
height, by a milled head, d, the spindle of which carries a pinion
in gear with a rack on the outside of the pillar. The outer end
of the arm, c, terminates in a ring, with a universal joint, e,
through which the pen or pencil-holder, f, is passed. The pillar,
B, also carries at its upper end a fixed arm, g, formed as an ellip-
tical guide-frame, being accurately cut out to an elliptical figure, as
the nucleus of all the varieties of ellipse to be drawn. The centre
of this ellipse is, of course, set directly over the centre of the
universal joint, e, and the pen-holder is passed through the guide
and through the joint, the flat-sided sliding-pieee, h, being kept in
contact with the guide, in traversing the pen over the paper. The
pen thus turns upon its joint, E, as a centre, and is always held in
its proper line of motion by the action of the slider, h. The dis-
tance between the guide ellipse and the universal joint determines
the size of the ellipse, which, in the instrument here delineated,
ranges from 2| inches by 1|, to -pV ^Y I '"'^'i- ^'^ general, how-
over, these instruments do not appear to be suflicienlly simple, or
convenient, to bo used with advantage in geometrical drawing.
57. Tangents to ellipses. — It is frequently necessary to deter-
mine the position and inclination of a straight lino which shall
be a tangent to an elliptic curve. Three cases of this nature
occur : when a point in the ellipse is given ; when some external
point is ^ven apart from the ellipse ; and when a straight lino
is given, to which it is necessary that the fangcnl, should \h'
parallel.
Fii'st, then, let the point, A, in the ellipse, fig. 2, bo given ;
draw the two vectors, c a, n a, and produce the latter to m ; liiscct
the angle, M a c, by the stniight line, n r ; this lino, N p, will Iki
Vie tangent re(|uiri^d; that is, it will Idmi-Ii the (•urv(^ in the point,
A, and in that point alone.
Secondly, let the point, l, be given, apart from the ellipse, fig
3. Join L with i, the nearest focus to it, and with l as a centre,
and a radius equal to l i, describe an arc, M i n. Next, with the
more distant focus, h, as a centre, and with a radius equal to the
transverse axis, a b, describe a second arc, cutting the first in m
and N. Join m h and n h, and the ellipse wiU be cut in the points
V and X ; a straight line drawn through either of these points from
the given point, l, ^vill be a tangent to the ellipse.
68. Thndly, let the straight line, q r, fig. 2, be given, paralle
to which it is required to di-aw a tangent to the ellipse. From the
nearest focus, b, let f;vll on q k the perpendicular, s b ; then with
the further focus, c, as a centre, and with a radius equal to the
transverse axis, D E, describe an arc cutting b s in s ; join c s, and
the straight line, c s, will cut the ellipse in the point, t, of contact
of the required tangent. All that is then necessary is, to draw
through that point a line parallel to the given line, q e, the
accuracy of which may be verified by observing whether it bisects
the line, s b, which it should.
59. — Tlie oval of five centres. Jig. 4. — As in previous cases, the
transverse and conjugate axes are given, and we commence by
obtaining a mean proportional between their halves; for tliis
purpose, with the centre, o, and the semi-conjugate axis, o c, as
radius, we describe the arc, c i k, and then the semi-ckcle, a l k, of
which A K is the diameter, and further prolong o c to l, o l being
the mean proportional required. Next construct the parallelo-
gram, A G c 0, the semi-axes constituting its dmiensions : joining
c A, let fall from the point, g, on the diagonal, c a, the per-
pendicular, G H D — which, being prolonged, cuts tiie conjugate axis
or its continuation in d. Havmg made c m equal to the mean
proportional, o l, with the centre, d, and radius, d m, describe
an arc, a m b ; and having also made a n equal to the mean pro-
portional, o L, with the centre, h, and radius, h n, describe tlie
arc, N a, cutting the former in a. The points, h, a, on one side,
and ii', b, obtained in a similar manner on the other, together
with the point, d, will be the five centies of the oval ; and straight
lines, R H a, s h' b, and p a d, q i d, passing through the respective
centres, wll meet the curve in the points of junction of the various
component arcs, as at R, p, Q, s.
This beautiful curve is adopted in the construction of many
kinds of arches, bridges, and ■s'aults ; an example of its use is given
in fig. ©.
60. The parabola, fig. 5, is an open curve, that is, one wiiich
does not return to any assumed starting point, to however great
a length it may bo extended ; and wiiich, consequently, can never
enclose a space. It is so constituted, that any point in it, r>, is at
an equal distance from a constant point, c, termed the focia, and
in a perpendicular direction, from a straight line, a b, called the
(llrrr/rix. The straight line, r c, iH'r[)t'ndicular to the directrix,
A 1!, and jiassing tlinnigh the focus, r. is the a.vis o\' the curve,
which it divides into two synunetricil portions. The point, .\.
midway between F and c, is the apex of the curve. There are
several methods of drawing this curve.
(il. First method: — This is based on the dotinition just given
,111(1 rciiiiires that the foeua and directrix be known, as c, and .v D
'I'aKc any points on the directrix, A n, as a, k, h, i, and throu-ji
liiem (haw jiarallols to iJio 'Lxis, K u, as also tlio straight line*
• c
18
THE PRACTICAL DRAUGHTSMAN'S
A c, E c, H c, I c, joining them with the focus. Draw pei-pen-
dicular^ bisecting these latter lines, and produce them until they
cut the corresponding parallels, and the points of intersection, /),
^, D, e, will bo in the required cur\e, wliich may be traced through
them.
62. The sti-aight lines which were just drawn, cutting the
parallels in different points of the curve, are tangents to the curve
at the several points. If, then, it is required to di-aw a tangent
througli a given point, c, it is obtained smiply by joining c c,
making h a equal to c c, and bisecting the angle, h c c, by the
straight line, c d, which will be the required tangent. K the point
gix'en bo apart from the curve, the procedure will be the same,
but the line corresponding to h c will not be parallel to the axis.
63. Second method: — We have here given the axis, a g, the
apex, a, and any point, Z, in the ciu-ve. From the point, I, let fall
on the axis the perpendicular, I g, and prolong this to r, making
c e equal to I a. Divide I g into any number of equal parts, as
m the points, i, J, k, through which draw parallels to the axis ;
divide also the axis, a g, into the same number of equal parts, as
in the points, f, gih; through these draw lines radiating from the
point, e, and they will intersect the parallels in the points, m, n, o,
which are so many points in the curve.
64. If it is required to draw a line tangent to a given parabola,
and parallel to a given line, J k, we let fall a perpendicular, c l,
on this last ; this perpendicular will cut the dii-ectrix in p, and p n
drawn parallel to the axis will cut the curve in the point of con-
tact, n.
We find frequent applications of this curve in constructions and
machinery, on account of the peculiar properties it possesses,
which the student will find discussed as he proceeds.
65. The objects represented in figs, [o), [o)', are an example of
the application of this curve. They are called Parabolic Mirrors,
and are employed in philosophieal researches. The angles of
incidence of the vectors, a h, a c, a d, are equal to the angles of
reflection of the parallels, b b', c c', d d'. It follows from this pro-
perty, that if in the focus, a, of one min-or, b /, the flame of a
lamp, or some incandescent body be placed, and in the focus, a', of
the opposite mirror, &'/', a piece of charcoal or tinder, the latter
will be ignited, though the two foci may be at a considerable dis-
tance apart; for all the rays of caloric falling on the muTor, b f,
are reflected from it in parallel lines, and are again collected by
the other mirror, b' f, and concentrated at its focus, a'.
66. To draw an Ionic volute, fig. 6. — The vertical, a o, being
given, and being the length from the summit to the centre of the
volute, divide it into nine equal parts, and with the centre, o, and
a radius equal to one of these parts, describe the circle, abed,
which forms what is termed the eye of the volute. In this circle
(represented on a larger scale in fig. 7) inscribe a square, its dia-
gonals being vei-tical and horizontal ; through the centre, o, draw
the lines, 1 — 3, and 4 — 2, parallel to the sides, and divide the half
of each into three equal parts. With the point, 1, as a centre,
and the radius, 1 a (fig. 6), di-aw the arc, a e, extending to the
norizontal line, 1 e, which passes through the point, 2. With this
iatter pomt as a centre, and a radius equal to 2 e, draw the nex-t
arc, extending to the vertical line, 2 /, which passes through the
pomt, 3, tne next centre. The points, 4, 5, 6, &c., form the sub-
sequent centres ; the arcs in all cases joining each other on a line
passing through then- respective centres. The internal curve is
drawn in the same way; the points, 1', 2', 3', &c., fig. 7"'% bemg
the centres of the component arcs. The first arc is drawn with a
radius, 1' a', a ninth less than 1 a, and the others are consequently
proportionately reduced, as manifest ui fig. 6. The application of
the volute will be found in fig. [U.
67. To draiu a curve tangenlially joining two slraigM lines, a b
and B c,fig. 8, the points a and c being the points of junction. — Join
a c, and bisecting a c in d, join d with b, the point of intersection
of the lines, a b, b c. Bisect b d in e', which will be a point in the
curve. Join e c, e a, and bisect the lines, e c, e a, by the per
pendiculars, a b, c d; make e/and e' f equal to a fourth part of
E D ; /and/' will be other two points in the curve. Proceed in
the same way to obtain the points, g h and g' h', or more if desir
able, and then trace the curve through these several points. Thi«
method is generally adopted by engineers and constructors, and A^il]
be met with in raOways, bridges, and embankments, and wherever
it is necessary to connect two straight lines by as regular and per-
fect a curve as possible. It is also particularly applicable where
the scale is large.
RULES AND PRACTICAL DATA.
LINES AND SURFACES.
68. The square metre is the unit of surface measurement, just
as the linear m^tre is that of length. The square mfetre is sub-
divided mto the square decimetre, the square centimetre, and the
square millimetre. WhOst the linear decimetre is a tenth part of
the metre, the square decimetre is the hundredth part of the square
metre. In fact, since the square is the product of a niunber mul-
tiplied into itself,
O'lm. X O'lm. = O'Ol square metres.
In the same manner the square centimetre is the ten-thousandth
part of the square metre ; for
0-01 m. X 0-01 m. = 0.0001 square metres.
And the square mOlimetre is the millionth part of the square
m^tre ; for,
0-001 m. X 0-001 m. = 0-000001 square metres.
It is in this way that a relation is at once determined between
the units of linear and sm-face measurement.
Similarly in English measm-es, a square foot is the ninth part of
a square yard ; for
1 foot X 1 foot =: \ yard X ^ yard = ^ square yai-d.
A square inch is the 144th part of a square foot, and the 1296th
part of a square yard ; for
1 inch X 1 inch = ^ foot X ]\ foot = -plj square foot,
and
1 inch X 1 inch =: ^ yard x 3*5 yard = y-g^ square yard.
This illustration places the simplicity and adaptability of the
decimal system of measui-es, in strong contrast with the complexity
of other methods.
69. Measurement of surfaces. — The surface or area of a square,
as well as of all rectangles and parallelogi-ams, is expressed by the
product of the base or length, and height or breadth measured
BOOK OF INDUSTRIAL DESIGN.
19
perpendicularly from the base. Thus the area of a rectangle, the
base of which measures 1-25 metres, and the height "75, is equal to
1-25 X -75 = -9375 square metres.
The area of a rectangle being known, and one of its dimensions,
llie other may be obtained by dividing the area by the given
dimension.
Example. — The area of a rectangle being -9375 sq. m., and the
base 1-25 m., the height is
•^^ = -75 m.
125
This operation is constantly needed in actual construction ; as, for
instance, when it is necessary to make a rectangular aperture of a
certain area, one of the dimensions being predetermined.
The area of a trapezium is equal to the product of half the sum
of the parallel sides into the perpendicular breadth.
Examjile. — The parallel sides of a trapezmm bemg respectively
1-3 m., and 1-5 m., and the breadth -8 m., the area will be
1-3 +1-5 Q , ,„
— X -8 = 1-12 sq. m.
The area of a triangle is obtained by multiplying the base by
tialf the perpendicular height.
Example. — The base of a triangle being 2-3 m., and the perpen-
dicular height 1-15 m., the area will be
2-3 X — = 1-3225 sq. m.
2 ^
The area of a triangle being known, and one of the dimensions
given — that is, the base or the perpendicular height — the other
dimension can be ascertained by dividing double the area by the
^ven dimeusion. Thus, in the above example, the division of
(1'3225 sq. m. x 2) by the height V\b m. gives for quotient the base
2-3 m., and its division by the base 2-3 m. gives the height ri5 m.
70. It is demonstrated in geometry, that the square of the
hypothenuse, or longest side of a right-angled triangle, is equal to
the sum of the squares of the two sides forming the right angle.
It follows from this property, that if any two of the sides of a
right-angled triangle be ^ven, the third may be at once ascertained.
First, If the sides forming the right angle be given, the hypo,
thenuse is determined by adding together their squares, and
extracting the square root.
Example. — The side, a b, of the triangle, a b c, fig. 16, PL I,
being 3 m., the side b c, 4 m., the hypothenuse, a c, will be
A c = V32-1-42 = 4/9 + 16 = ^2b = 5 m.
Secondly, If the hypothenuse, as a c. be known, and one of
the other sides, as A b, the third side, b c, will be equal to the
square root of the difference between the squares of a c and A b.
Thus assuming the above measures —
BC= ■^'25 — 9 = V16 = 4m.
The diagonal of a square is always equal to one of the sides mul
tiplied by ■>^2; therefore, as V2 = 1-414 nearly, the diagonal is
obtained by multiplying a side by 1-414.
Exajnple. — The side of a square being 6 meti-es, its diagonal
= 6 X 1-414 = 8-484 m.
The sum of the squares of the four sides of a parallelogram is
equal to the sum of the squares of its diagonals.
71. Regular polygons. — The area of a regular polygon is
obtained by multiplying its perimeter by half the apothegm or per
pendicular, let fall from the centre to one of the sides.
A regular polygon of 5 sides, one of which is 9-8 m., and tne
perpendicular distance from the centre to one of the sides 5-6 m.,
will have for area —
9-8 X 5 X —= 137-2 sq.m.
2 ^
The area of an irregular polygon will be obtained by dividing it
into triangles, rectangles, or trapeziums, and then adding togetlier
the areas of the various component figures.
TABLE OF MULTIPLIERS FOR REGULAR POLYGONS OF FROM 3 TO 12 SIDES.
Names.
.Multipliers.
B
D
Area
I side = 1.
Internal Anglo.
F
Apothegm
or
Perpendicular.
Triangle, . .
Square, . . . .
Pentagon, . ,
Ht^xagon, . .
Heptagon, . ,
Oclagon, . ,
Krmcagon, .
DccMgon, . ,
(Indccagon,
IJuodecagon,
3
4
5
6
7
8
9
10
11
12
2-000
1-414
1-238
1-156
1.111
1-080
1-062
1-050
1-0 10
1-037
1-730
1-412
1-174
radius.
-867
•765
-681
-616
•561
-516
-579
•705
•852
side.
M60
1-307
1-470
1-625
1-777
1-940
-433
1-000
1-720
2-598
3-634
4-8-28
6-182
7-694
9-365
11-196
60" 0'
90° 0'
108° 0'
120° 0'
128° 34'!;
135° 0'
140° 0'
144° 0'
147° 16',-',
150° 0'
-2SS6751
•6000000
•6881910
-8660254
1-0382607
1-2071069
1-37373S7
1-53SS4I8
1-7028436
1-8660254
By means of (his liibli^ we cnn easily .solve iiiaiiy inlrrcsliiig
problems connected with regular polygons, fi-oiii iJin triuiigln \\\> (o
the duodecagon. Sutth are the following : —
First, The width of a polygon being given, to find tlie radius of
(he circumscribing circle. — Wlion the number of sides is oven, the
width is understood a-s tho iierptindicular diatanco between two
opposite and i)arallel sides; when the nuinl)(<r is unin-en, it is
twice tiio perpendicular distiuice from tiio contru to one side.
Rule. — l\liilti|ily h;ilt' llic widlii of the iioly;;-.in l>y llie l-ii-ti.r in
(•(iliinin A, corrcspoiniiiig to the nuiiiher tit' .sides, !ind the prodiu-l
will he, the reciiiircd radius.
E.vamiilc. — I-et 18-5 in. lie tiie width of nil oi-lai^nni ; then,
^^J' X 1-08 =9-99ni.;
or say 10 iniXres, tlie radius of tiie ciiriiinM-i-ibing cirpJe.
so
THE PRACTICAI. DRAUGHTSMAN S
Second, The radius of a circle being given, to find the length of
the side of an inscribed polygon.
Rule. — iliiltiply the radius by the factor in colximn B, corre-
sponding to the number of sides of the required polygon.
Example. — The radius being 10 m., tlie side of an inscribed
octagon will be —
10 X -"Go = 7-65 m.
Third, The side of a polygon being given, to find the radius of
the circumscribing circle.
Rule. — Multiply tlie side by the factor in column C, corre-
sponding to the number of sides.
Example. — Let T'eS m. be the side of an octagon ; then
7-65 X 1-307 = 10 m., nearly.
Fourth, TTie side of a polygon being given, to find the area.
Rvle. — Multiply the given side by the factor in column D,
corresponding to the nimiber of sides.
Example. — The side of an octagon being 7'65 m., the area will
be—
7-65 X 4-828 ~ 36-93 sq. m.
THE CmCtlMFERENCE AND AJIEA OF A CIRCLE.
72. If the oircumferenee of any circle be divided by its diame-
ter, the quotient will be a number which is called, the ratio oj
the circumference to the diameter. The ratio is found to be (ap-
proximately) —
3-1416, or 22 : 7;
that is, the circumference equals 3-1416 times the length of the
diameter. It is expressed, in algebraic formulas, by the Greek
letter ji (jn). Thus, if C represents the circumference of a circle,
and D its diameter, the following formula,
C = jt D, or C = 3-1416 x D,
expresses the development of the circumference. Thus, if tlie
diameter of a circle, or D, = 2-7 m., or the radius R = 1-35 m.,
the circumference will be equal to —
3-1416 X 2-7, or 3-1416 x 1-35 x 2 = 8-482 m.
The circumference of a circle being known, its diameter, or radius,
is found by dividing this circumference by 3-1416 for the former,
or 6-2832 for the latter. Thus, the diameter, D, of a circle, the
circimiference of which is 8-482 m., is —
8-482
3-1416
:= 2-7 m.
and the radius, R, is-
8-482
1-35 m.
6-2832
The area of a circle is found by multiplying the circumference by
half the radius. — Tliis rule is expressed in the following formula : —
R
The area of a circle = 2 « R x .
R=
This term, Tt R-, is merely the simplification of tlie formula.
The number 2 being both multiplier and divisor, maj- be can-
celled, and the product of R Into R is expressed by R-, or the
square of the radius. It follows, then, that the area of a circle is
equal to the square of the radius multiplied by the circumference,
or 3-1416.
Example. — The radius of a circle being 1-05 m., the area ■will
be—
3-1416 X 1-05 X 1-05 = 3-4635 sq. m.
The area of a cu-cle being known, tlie radius is determined by
dindiag the area by 3-1416, and extracting the square root of thti
quotient.
Example. — The area of a circle being 3-4635 sq. m., the radius
y
3-4635
= 1-05 m.
3.1416
The area of a circle is derived from the diameter ; thtis—
Area =
,-t D X D
rt D2
then, since H or ^lHi^= -7854,
4 4
the formula resolves itself into
Area ;= -7854 x D-.
That is to say, if we multiply the fraction, -7854, by the square of
the diameter, the product will be the area.
Example. — The area of a circle, the diameter of which mea-
sures 2-1 m., is —
-7854 X 2-1 X 2-1 = 3-4635 sq. m.
It follows from this, that if the area of a square is kno^ii, that
of an inscribed circle is obtainable, by multiplying by -7854 ; that
is, the area of a square is to the area of the inscribed circle, as,
4 : 3-1416, or 1 : -7854.
TABLE OF APPKOXniATE KATIOS BETWEEN CIECLES AND SQUARES.
The diameter of a circle
The circumference of a circle, . .
The diameter,
The circumference,
The area of a circle,
The side of an. inscribed square.
8. The side of a square,
This table aifords a ready solution of the following amongst
other problems : —
First, The diameter of a circle being -125 m. or 125 ^1^ (mUli-
metres), the side of a square of equal area is
125 X -8862 = 110-775"/„.
X
X
X
X
X
X
X
X
X
-8S62 ) _
•2821 S ~
-7071 /
-2251 f
-6366
1-4142
4-4430
1-1280
3-5450
the side of a square of equal area.
the side of the inscribed square.
the area of the inscribed square.
the diameter of the circumscribing circle.
the circumference of the circumscribing circle.
the diameter of an equal circle.
the circumference of an equal circle.
Second, The circTimference of a circle being SGC^/n, the side of
the inscribed square is
860 X -2251 = 193-586°/„.
Third, The side of a square being 215-86'°/„, the diameter of
the circumscribing circle is
215-86 X 1-4142 =305-27"/„.
BOOK OP' INDUSTRIAL DESIGN.
21
'ITie radii and diameters of circles are to each other as the cir-
cumferences, and vice, versa. The areas, therefore, of circles are to
each other as the squares of their respective radii or diameters.
It follow? hence, that if the radius or diameter be doubled, the
eircumference will only be doubled, but the area will be quadrupled ;
Ihus, a drawing reduced to one-half the length, and half the
()readth, only occupies a quarter of the area of that from which it
Is reduced.
73. Sectors — Segments. — In order to obtain the area of a sector
n( segment, it is necessary to know the length of the arc subtend-
ing it. This is found by multiplying the whole circumference by
the number of degi-ees contained in the arc, and dividing by 360°.
Example. — The circumference of a circle being 3'5 m., an arc of
45° wdll be
3-5 X 45
360
•4375 m.
The length of an arc may be obtained approximately when the
chord is known, and the chord of half the arc, by subtracting the
chord of the whole arc from eight times the chord of the semi-arc,
and taking a third of the remainder.
Example. — The chord of an arc being -344 m., and that of half
the arc -198, the length of the arc is
•198 X 8— -344 .,„„
= '4133 m.
3
The area of a sector is equal to the length of the arc multiplied
into half the radius.
Example. — The radius being -169 m., and the arc -266;
•266 X -169
2
= ^0225 sq. m., the area of the sector.
The area of a segment is obtained by multipljing the wdth ; that
is, the perpendicular between the centre of the chord, and the
centre of the arc, by -626, then adding to the square of the pro-
duct the square of half the chord, and multiplpng twice the
square root of the sum by two-thirds of the width.
Example. — Let 48 m. be the length of the chord of the arc,
and 18 m. the width of the arc, then we have
18 X ^626 = 11-268, and (11-268)2 = 126-9678; whilst ■
(?)'
576; therefore, 2 x V' 126-9678 + 576 x -^ = 636-24
sq. m., the area of the segment.
The area of a segment may also be obtained very approximately
by dividing the cube of the width by twice the length of the chord,
and adding to the quotient the product of the width into two
thirds of the chord. Thus, with the foregoing data, we have
18 =
48 x 2
60-7
and,
48 X 2 X 18
= 576-0
Total, 636-7 sq. m.
A still simpler method, is to obtain the area of the sector of
wliich the segment is a part, and then subtract the area of the
COMPARISON OF CONTINENTAL MEASURES, WITH FRENCH MILLIMETRES AND ENGLISH FEET.
Country.
Austria, .
Spain, . . .
Pupal StatcB,
Frankfort,
llunilnirg, .
Hanovor, . .
llessc, ....
Designation of Measure.
(Vienna) Foot or Fuss = 12
iiiclies =: 144 lines,
(Bohemia) Foot,
(Venice) Foot,
Foot (Palme)
" Foot (Architect's Meas.)
(Carlsruhe) Foot (new) = 10
inches = 100 lines,
(Munich) Foot = 12 inches =
144 lilies,
(Augsburg) Foot,
(Brussels) Ell or Aune = 1 m6tre,
Foot,
(Bremen) Foot = 12 inches =
144 lines
(Brunswick) Foot =: 12 inches
= 144 lines
(Crucow) Foot,
(Copeiiliageii) Foot,
(iMadii(l) Foot (according to Loli-
miiM,
<Justiliaii Vara( " Ijiscar),
(iravaiia) Vara= ;5 Madrid toot,
(Uoiiio) Koot,
Arohiteot's Span = I foot, . . .
Aiioieut Foot,
Foot,
l<'oot = 8 spans = 12 inches =:
96 parts,
(Ilaiiovor) Foot ^ 12 inches
111 lines,
(Daniistiidt) l''()ol,= 10 inciios =
100 linos
Value in
Millimetres.
31G-103
2'JU-416
435-185
347-398
396-500
300-000
291-859
296-168
1,000-000
285-588
289-197
285-362
356-421
313-821
282-655
835-906
817-965
297-896
223-422
294-216
284-61(>
286-490
291-995
300-000
Value in
Feet.
1-037
-970
1-460
1-140
1-301
•984
•958
•972
3-281
-937
•949
•936
1-169
1-029
•927
2-742
2-782
•977
•783
•965
•938
•940
•958
•981
Country.
Holland,.
Lubeck,
Mecklenburg,
Modena,
Ottoman Empire
Parma
Designation of Measure.
Poland, . ..
Portugal, .
Prussia, . .
Russia, . . .
Sardinia, .
Saxe,
Sicilies, . .
S-wedon,
Switzerland,
1 usonny
Wiirtoniburg,.,.
(Amsterdam) Foot = 3 spans
= 11 inches,
(Rhine) Foot
(Lubeck) Foot,
Foot,
(Modena) Foot,
(Reggio),.
(Constantinople) Grand pie,. .
Ai-ms-]ongth= 12 inches = 1728
atonii
(Varsovic) Foot = 12 inches ^
144 lines,
(Lisbon) Ft. (Arohitoot'sMoasnre)
Vara = 40 inohos,. .
(Berlin) Foot =12 iuoiios, . . .
(St. Petersburg) Russian Footj
" Arcliino,
(Oiigliari) Span
(Woiniar) Foot,
Span ==12 iiulus (oimoes = 60
niinuti),
(Stockholm) Foot,
(Bale and Ziirioli) Fool,
(Borne and Jsoiit'oliiMol) Foot =
12 iiK'hos,
(Gonovii) Foot,
(l.iiusaniio) Foot = 10 inolios =
loil linos
(I.uooriioiind ulluT I'lmtons) I't.,
l''oot, ...
Foot = 10 inches = 100 linos,..
Value in
Millimetres.
283-056
313-854
291-002
291-002
523-048
530-898
669-079
544 670
297-769
838-600
1,008-368
809-726
538-151
711-480
•202-578
281-972
208-670
296-888
804-587
298-2.'>S
487-900
30(1-000
8L'iS54
5481 67
280 4 90
Value in
Foot.
■928
1-080
•954
•964
1-716
1-742
2^195
1787
-977
Mil
8-686
1-016
1-705
2884
•604
•926
•865
•974
•999
•902
1 •OOti
•984
1-080
1-798
•940
22
THE PRACTICAL DRAUGHTSMAN'S
trianele constituting the diiference between the sector and
segment.
To find the area of an annular space contained between two
concentric circles, multiply the sum of the diameters by their
difference, and by the fraction -7854.
Example. — Let 100 m. and 60 m. be the respective diameters ;
then, (100 + 60) X (100 — 60) x -7854 = 5026-56 sq. m. the
area of the annular space.
The area of a fragment of such annular space will be found
by multiplying its radial breadth by half tlie sum of the arcs, or,
more correctly, by the arc which is a mean proportional to them.
CraCUMFERENCE AND AREA OF AN ELLIPSE.
74. The circumference of an ellipse is equal to that of a circle,
of which the diameter is a mean proportional between the two axes ;
therefore, it will be obtamed by multipljing such mean propor
tional by 3-1416, the ratio between the diameter and circumferenco
of a circle.
Example. — Let 10 m. and 6-4 m. be the lengths of the respec-
tive axes ; then,
Via X 6-4 X 3-1416 = 25-13-28 m.
The area of the ellipse is obtained by multiplmg the product
of the two axes by -7854, the ratio between the diameter and the
area of the circle.
Example.— \0 X 6-4 X -7854 = —50-2656 sq. m.
These rules meet with numerous applications in the indus-
trial arts, and particularly in mechanics, as will be seen further
on. The examples given will enable the student to understand
the various operations, as well as to solve other analogous
problems.
CHAPTER n.
THE STUDY OF PROJECTIONS.
75. To indicate all the dimensions of an object by pictorial deli-
neation, it is necessary to represent it imder several different
aspects. These various views are comprehended imder the general
denomination of Projections, and usually consist of elevations, plans,
and sections. The object, then, of the study of projections, or
descriptive geometry, is the reproduction on paper of the appeai--
ances of all bodies of many dimensions as viewed from different
positions.
It is customary to determine the projections of a body on two
principal planes, one of which is distinguished as the horizontal
plane, and the other as the vertical plane, or elevation. These two
planes are also called geometric projections or plans. They are
annexed to each other, the horizontal plan being the lower ; the
line intersecting them is called the base line, and is always parallel
to one of the sides of the drawing.
It is of great importance to have a thorough knowledge of the
elementary principles of descriptive geometry, in order to be able
to represent, in precise and determinate forms, the contours of all
kinds of objects ; and we shall now enter upon such explanatory
details as are necessary, commencing primarOy with the projections
of a point and of a line.
ELEMENTARY PRINCIPLES.
THE PROJECTIONS OF A POINT.
PLATE VI.
76. Let A B c D, figs. 1 and 1°, be a horizontal plane — repre-
senting, for example, the board on which the drawing is being
made, or perhaps the surface of a pavement. Also, let a B E f be
a vertical plane, such as a wall at one side of the piece of pave-
ment; the straight lice, which is the intersection of these two
planes, is the base line. Finally, let o be any point in space, the
representation of which it is desired to effect. If, from this point,
0, we suppose a perpendicular, o o, to be let fall on the hori2ontal
plane, the point of contact, o, or the foot of this perpendicular,
will be what is understood as the horizontal projection of the
given point. Similarly, if from the point, o, we suppose a per-
pendicular, o 6, to be let fall on the vertical plane, a b e f, the
point of contact, 6, or foot of this perpendicular, will be the
vertical projection of the same point. These perpendiculars are
reproduced in the vertical and horizontal planes, by drawing lines,
6 n and n o, respectively parallel and equal to o o and o o.
77. It follows from this construction, that, when the two pro-
jections of any point are given, the position in space of the point
itself is determinable, it being necessarily the point of intersection
of perpendiculars erected on the respective projections of the
point.
As in di-awing, only one surface is employed, namely, the sheet
of paper, and we are consequently limited to one and the same
plane, it is customary to suppose the vertical plane, a b e f, fig. 1,
as forming a continuation of the horizontal plane, a b c d, being
turned on the base line, a b, as a hinge, so as to coincide vfith it —
just as a book, half open, is fully opened flat on a table. We
thus obtain the figure, d c e f, fig. 1°, representing on the paper
the two planes of projection, separated by the base fine, a b, and
the points, o, 6, fig. 1% represent the horizontal and vertical pro-
jections of the given point.
It will be remarked, that these points lie in one line, perpen-
dicular to the base line, a b ; this is because, in the turning down
of the previously vertical plane, the line, n 6, becomes a prolonga-
tion of the fine, n o. It is necessary to observe, that the fine, n 6,
measures the distance of the point from the horizontal plane,
whilst n o measures its distance from the vertical plane. In
other words, if on o we erect a perpendicular to the plane, and
BOOK OF INDUSTRIAL DESIGN.
23
measure the distance, n o', on this perpendicular, we shall obtain
the exact position of the point in space. It is thus obvious, that
the position of a point in space is fully determinable by means of
two projections, these being in planes at right angles to each
other.
' THE PROJECTIONS OF A STRAIGHT LINE.
78. In general, if, from several points in the given line, perpen-
diculars be let fall on to each of the planes of projection, and
their points of contact wdth these planes be joined, the resulting
lines will be the respective projections of the given line.
When the line is straight, it will be sufficient to find the pro-
jections of its extreme points, and then join these respectively by
straight lines.
79. Let M o, fig. 2, represent a given straight line in space,
which we shall suppose to be, in this instance, perpendicular to
the horizontal, and, consequently, parallel to the vertical plane of
projection. To obtain its projection on the latter, perpendiculars,
M m', o o', must be let fall from its extremities, m, o ; the straight
line, m' o', joining the extremities of these perpendiculars, will be
the required projection in the vertical plane, and in the present
case it will be equal to the given line.
The horizontal projection of the given line, M o, is a mere
point, m, because the line lies wholly in a perpendicular, m 711, to
the plane, and it is the point of contact of this line which consti-
tutes the projection. In drawing, when the two planes are
converted into one, as indicated in fig. 2°, the hoiizontal and ver-
tical projections of the given right lines, m 0, are respectively the
point, m, and the right line, m' 0'.
80. K we suppose that the given straight line, M 0, is horizon-
tal, and at the same time perpendicular to the vertical plane, as
in figs. 3 and 3", the projections will be similar to the last, but
transposed ; that is, the point, (/, will be the vertical, whilst the
straight line, m 0, will be the horizontal projection.
In both the preceding cases, the projections lie in the same
perpendicular line, m m, fig. 2% and 0' 0. fig. 5°.
8L When the given straight line, m 0, is parallel to both the
horizontal and the vertical plane, as in figs. 4 and 4", its two pro-
jections, m and m' 0', will be parallel to the base line, and they
will eiich be equal to the given line.
82. When the given straight line, M o, figs. 6 and 5", is parallel
to the vertical plane, a b e f, only, the vertical projection, m' o',
will be parallel to the given line, whilst the hoiizontiil projection,
m o, will be parallel to the base line. Inversely, if the given straight
line be parallel to the horizontal plane, its horizontal projection will
be i)arallel to it, whilst its vertical projection will bo parallel to
the base line.
83. Finally, if the given straight line, m o, figs. 6 and 6", is
inclined to both planes, the projections of it, m 0, m' o', will both
be inclined to the base line, a b. These projections are in all
cases obtained by letting fall, fiuiii each extremity of the line, per-
pendiculars to each plane.
The projections of a straight line being given, its position in
sjjace is determined by erecting porpcndiculars to the liorizonliil
plane, from the extremities, 7/1 o, of the jjrojcctcd lini^ and nialiing
llicrn (•(lual to the verti('als, ?i in' .miicI /; n'. The same result
riillows, ir fidiii IIk^ piiiiils, //(', o', ill till' vcitiiMl |il:ni(', we erect
perpendiculars, respectively equal to the horizontal distances
m n and p o. The free extremities of these perpendiculars meet
each other in the respective extremities of the line in space.
THE PROJECTIONS OF A PLANE SURFACE.
84. Since all plane surfaces are bounded by straight lines, as
soon as the student has learned how to obtain the projections of
these, he will be able to represent any plane surface in the two
planes of projection. It is, in fact, merely necessary to let fall
perpendiculars to each of the planes, from the extremities of the
various lines bounding the surface to be represented; in other
words, from each of the angles or points of junction of these lines,
by which means the corresponding points will be obtained in the
planes of projection, which, being joined, will complete the repre-
sentations. It is by such means that are obtained the projections
of the square, represented in different positions in figs. 7, 7*, 8, 8",
and 9, 9'. It will be remarked, that, in the two first instances, the
projection is in one or other of the planes an exact coimterpart of
the given square, because it is parallel to one or other of the
planes.
85. Thus, in fig. 7, we have supposed the given surface to be
parallel to the horizontal plane ; consequently, its projection in
that plane will be a figure, m p q, equal and parallel to itself,
whilst the vertical projection will be a straight line, p' 0', parallel to
the base line, a b.
86. Similarly, in fig. 8, the object being supposed to be paralle.
to the vertical plane, its projection in that plane wOl be the equa.
and parallel figure, m! d p' q', whilst that in tlie horizontal plane
will be the straight line, mo. When the two planes of projection
are converted into one, the respective projections will assume the
forms and positions represented in figs. 7°, 8°.
87. If the given surface is not parallel to either plane, but yet
perpendicular to one or the other, its projection in the plane to
which it is perpendicular will still be a straight line, as p' o', figs.
9 and 9", whiist its projection in the other plane will assume tlie
form, mop q, being a representation of the object somewhat fore-
shortened in the direction of the inclination.
The cases just treated of have been those of rectangular sur-
faces, but the same principles are equally applicable to any poly-
gonal figures, as may be seen in figs. 12 and 12', wiiich will b*
easily understood, the same letters in various ch;u-acters indicating
corresponding points and perpendiculars. Nor does the obtain-
ment of the projections of surfaces bounded by curved lines, aa
circles, require the consideration of other principles, as we shall
proceed to show, in reference to figs. 10 ;i8(l 1 1.
88. In the first of these, fig. 10, the circular tlisc, M r q, is su]v
posed to be parallel to the vertical plane, A B e f, and its projec-
tion on that plane will be a circle, m' o' p' q', equal luid fwiniilel to
itself, whilst its projection on the horizontal pUine, a b c d, will be
a straight lino, q m o, equal to its (Uainoter. If, on the other hand,
the disc is parallel to the horizontal plane, jus in fig. 11, its vortical
projection will be the straight lino, /)' (Z /«', whilst its horizontal
projection will be (lu^ circle, op m q.
It' the given circular disc bo inclined to oillu'r piano, its projoc
tiiin in lli.'il |)l:uu' will bo an ellipse ; and if it Is inoliuod to both
phinrs, Imlli prcijocliiins will l>o olli[isos. This will bo niadooxi-
24
THE PRACTICAL DRAUGHTSMAN'S
dent ty obtaining the projections of various points in the circum-
ference.
89. When constructing the projections of regular figures, it
facilitates the process considerably if projections of the centres and
centre lines be first found, as in figs. 10, 11, and 12.
In general, the projection of all plane surfaces may be found,
when it is known how to obtain the projections of points and lines.
And, moreover, since solids are but objects bounded by surfaces
and lines, the construction of then- projections follows the same
rules.
PRISMS AND OTHER SOLIDS.
PLATE VII.
90. Before entering upon the principles involved in the repre-
sentation of soKds, the student should make himself acquainted
with the descriptive denominations adopted in science and art,
with reference to such objects ; and we here subjoin such as will
be necessary.
Definitions. — A solid is an object having three dimensions;
that is, its extent comprises knglh, width, and height. A solid
also possesses magnitude, volume, or capacity.
There are several forms of solids. The polyhedron is a solid,
bounded by plane surfaces ; the cone, the cylinder, and the sphere,
are bounded by curved surfaces. Those are termed solids of re-
volution, which may be defined as generated by the revolution of a
plane about a fixed straight line, termed the axis. Thus, a ring,
or annular torus, is a solid, generated by the revolution of a circle
about a straight line, lying in the plane of the cucle, and at right
angles to the plane of revolution. A prism is a polyhedron, the
lateral faces of which are parallelograms, and the ends equal and
parallel polygons. A prism is termed right, when the lateral
^es, or facets, are perpendicular to the ends; and it is regular,
when the ends are regular polygons. A prism is also called a
parallelopiped, when the ends are rectangles, or parallelograms ;
and when it is formed of six equal and square facets, it is termed
a cube, or regular hexahedron. This solid is represented in fig. ^.
Other regular polyhedra, besides the cube, are distinguished by
appropriate names ; as, the tetrahedron, the octahedron, and the
icosahedron, wliich are bounded externally, respectively, by four,
eight, and twenty equilateral triangles; and the duodecahedrcm,
which is terminated by twelve regular pentagons. A pyramid
13 a polyhedron, of which all the lateral facets are triangles,
uniting in one point, the apex, and having, as bases, the sides of a
polygon, which is the base of the pyramid, as fig. ©. The prism
and ppamid are triangular, quadrangular, pentagonal, hexagonal,
&c., according as the polygons forming the bases are triangles,
squares, pentagons, hexagons, &c.
By the height of a pyramid is meant the length of a perpendi-
cular let fall from the apex on the base ; the pyramid is a right
pyramid when this perpendicular meets the centre of the base.
A truncated pyramid, or the frustum of a pyramid, is a solid
wliich may be described as a pyramid ha\dng the apex cut off by
a plane parallel, or inclined to the base.
A cylinder is a solid which may be described as generated by
a straight line, revolving about, and at any given distance from, a
rectilinear axis to which it is parallel. A cylinder wliich is genc^
rated by a rectangle, revolving about one of its sides as an axis,
is said to be a right cylinder ; such a one is represented in fig. [E.
A cone, fig. [p", is a solid generated by a triangle, revolving about
one of its sides as an axis. A truncated cone is one which is
terminated short of the apex by a plane parallel, or inclined to
tlie base. This solid is also called the frustum of a cone. A
cone is said to be right when its base is a circle, and when a per-
pendicular let fall from the apex passes through the centre of
the base.
A sphere is a solid generated by the revolution of a scniiciicle
about its diameter as an axis, as fig. @.
A spheric sector is a solid generated by the revolution of a plane
sector, as o' l e', about an axis, a h, which is any ladius of llie
sphere of which the sector forms a part. When the axis of
revolution is exterior to the generating sector, the spheric sector
obtained vvill be annular or zonic. The zone described by the
arc, L e', is the base of the spheric sector. The zone becomes a
spheric arc when the axis of revolution is one of the radii forming
the sector.
A spheric wedge, or ungula, is any portion, as i H g y, fig. T,
comprised between two scmicucular planes inclined to each other
and meeting in a diameter, as i g, of the sphere. That portion of
the surface of the sphere which forms the base of the ungula, is
termed a gore.
A spheric segment is any part of a sphere cut off by a plane,
and may be considered as a solid of revolution generated by the
revolution of a plane segment about its centre line. The plane
surface is the base of the segment. Wlien the plane passes
through the centre of the sphere, two equal segments are obtained,
termed hemispheres.
A segmental annulus is a solid generated by the revolution of a
plane segment, d' b' k, fig. 7, about any diameter, a b, of the sphere,
apart from the segment, d' k is the chord, and m n, its projection
on the axis, is the height of the segmental annulus.
A zonic segment of a sphere is the part, l n k d', of a sphere
comprised between two parallel planes.
A spheric pyramid, or pyramidal sector, is a pyramid of which
the base is part of the surface of a sphere, of which the apex is
the centre ; the base may be termed a spheric polygon.
THE PKOJECTIONS OF A CUBE, PIG. ^.
91. A cube, of which two sides are respectively parallel to
the planes of projection, is represented in these planes by equal
squares, a b c d, and a' b' e' f', figs. V and 1.
This is indeed but a combination of some of the simple cases
already given. We have seen that when a side, such as a b e f,
fig. A, is parallel to the vei-fical plane, its projection on the hori-
zontal plane is reduced to a straight line, a b, fig. P, its projection
on the vertical plane being a figure, a' b' e' f', fig. 1, equal to itself.
Similarly, the side, a b c d, which is parallel to the horizontal
plane, is projected on the vertical plane in the line, a' e', *ig. 1,
and by the figure, a b c d, fig. 1°, in the horizontal plane, llie
sides, A D H E and b c g e, fig. A, which ai-e perpendicular to both
the horizontal and the vertical plane, are represented in both oy
sti-aight lines, as a n and b c, fig. 1, and a' f' and b' e', fig. 1,
BOOK OF INDUSTRIAL DESIGN.
25
hp.ing respectively in the same straight lines perpendicular to the
base line, l t. It will also be perceived, that the base, f e g h,
fig. A> cannot be represented in the horizontal projection, nor the
side, D c G H, in the vertical, since they are respectively immedi-
ately behind and hidden by the sides, a b c d and a b e f, repre-
sented, in the projections by the squares, a b c d, fig. 1°, and
a' b' e' f', fig. 1. They are, however, indicated in the planes to
which they are perpendicular, by the straight lines, f' e' and d c.
92. It will be evident from these remarks, that in order to
design a cube so that a model may be constructed, it is sufficient
to know one of the sides, for all the sides are equal to each other.
When the plans are intended to be used in the actual construc-
tion of machinery or buildings, the objects should be represented
in the projections as having then- principal sides parallel or per-
pendicular to the horizontal and vertical plane respectively, in
order to avoid the foreshortening occasioned by an oblique or in-
clined position of the object with reference to these planes, because
the actual measurements of the different parts cannot be readily
ascertained where there is such foreshortening.
To obtain, then, the projections of the cube, fig. ^, a square
must be constructed, as a b c d, fig. 1°, having its sides equal to
the given side or edge, the sides a b and d c being disposed parallel
to the base line ; next, the square must be reproduced as at a' b' e' f',
fig. 1, on the prolongations of the sides, A d and b c, which are
perpendicular to the base line.
THE PROJECTIONS OF A RIGHT SQUARE-BASED PRISM, OR
RECTANGULAR PARALLELOPIPED, FIG. [B.
93. The representation of this solid is obtained in precisely the
same manner as that of the cube, the sides being supposed to be
parallel or perpendicular to the respective planes of projection.
The base of the prism being square, its horizontal projection is
necessarily also a square, a b c d, fig. 2°; but its vertical pro-
jection will be the rectangle, a' b' e' f', fig. 2, equal to one of the
sides of the prism. For the construction of these projections, the
same datum as in the preceding case is required ; namely, a side of
the base, and in addition, the height of the parallelepiped, or
prism.
the projections of a quadrangular pyramid, fig. ©.
94. This pyramid is supposed to be inverted, and having its
base, A b c d, parallel to the horizontal plane : it follows upon this
assumption, that its horizontal projection is represented by the
square, A b c d, fig. 3". The axis, or centre line, o s, wliich is
supposed to bo vertical, and consequently passes through the centre
of the base, is projected on the horizontal plane as a point, o, fig.
'<', and on the vertical plane as a line, o' s ; drav/mg througli tlio
point, o', of this line, the horizontal line, a' b', equal to a side of
llu! base, wiiicii is supposed to be parallel to the vortical plane, wo
shall obtain the vertical projeclion of the base ; and joining a's, ii's',
that of tho whole pynuiiid, tiio points a' and d' may bo found by
prolonging the parallels, A D, B c, fig. 3". This may bo conveniently
done with tho square, and the operation is usually termed squaring
over a measurement — that is, from one projection to another. Tho
lateral f'acels, s n c arid sab, are rcprosenttid in the vortical ])ro-
jection by tho straight linos, a's, n' s, lig. 1, since Ihoy are per-
pendicular to the vertical plane ; and the projection of the facet.
D s c, is identical with a' s b', that of the front facet, a s b, imme-
diately behind which it is. Since each of the inclined converging
facets is liidden by the base, they cannot be dra\vn in sharp lines
in the horizontal projection ; we have, however, indicated their
positions in faint lines, fig. S". Were these lines full, the projec-
tion would be that of a pyramid with the apex uppermost, or of
a hollow, baseless pyramid, in the same position as fig. ©.
THE PROJECTIONS OF A RIGHT PRISM, PARTIALLY HOLLOWED,
AS FIG. ©.
95. The vertical and horizontal projections of the exterior of
this solid, are precisely the same as those of fig. \B ; they are re-
presented respectively by the square, a b c d, fig. 4°, and the rec-
tangle, a' b' e( f', fig. 4. It will be perceived, that the internal
surfaces of this figure are such as may be supposed to form some
of the sides of a smaller prism ; the sides, g h i j and k l m n, are
parallel to the vertical plane, and g k n j and h i m l perpendicular
to it, and it follows that the projections of this lesser figure
wUl assume the forms, g' h' i' j', fig. 4, and g h l k, fig. 4*.
The lines, k g, l h, are faint dotted lines, instead of being sharp
and full, as being hid by the base, a b c d, of the external prism.
These lines wll be found to be different to the projection lines, or
working Unes. The latter are composed of irregular dots, whilst
those which indicate parts of the figure which actually exist, but
are hidden behind more prominent portions, are composed of regu-
lar dots. This distinction has been adhered to throughout the
entire series of Plates.
97. On examining the examples just treated of, it vdU be ob-
served, from the horizontal projections, that the contour, or out-
line, is in every case square, whilst, from the vertical projections,
it will be seen that each object is different. This demonstrates
that one projection is not sufficient for the determination of all tlie
dimensions of an object ; and that, even in the simplest cases, two
different projections are absolutely necessary. It \\ill, moreover,
be seen, as we advance, that in many cases, three, and at times
more, projections are required, as well as sections through two or
more pKanes.
the PROJECTIONS OF A RIGHT CYLINDER, FIG. d.
98. The axis, o M, of this cylinder is supposed to be verfio^xl,
and its bases, a b, e f, will consequently be horizontal. Its pro-
jections in figs. 5 and 5° are represented by tho rectangle, a' b' e' f',
on the one hand, and the circle, a c b d, on tho otlier. It is evi-
dent, that to draw (hose figures, it is quite sutliciont to know the
radius, o a, of the base, and tho height, o iM ; with the giver,
radius, wo describe the circle, a c b d, which is tho horizontal {>i<.
jection of the whole cyliiulor; then making the vertical, o' m, equju
to tiic given lu'iglit, .Mild squaring o\or by iiicans of tho panUlfls,
a a', 1! n', the diameter of tho circle, we draw, through o' and
M, the liori/outals, a' b', e' f', com[)letiiig tho ivinillelognuu.
a' li' r.' i', which is the vertical projection of the cyliiulor.
Tin; puoJKCTiONs of a right cone, Flu. [?.
!>!>. The projections of a right coiu' ditVor from tluise of tho cviiii-
dor solely as far as reganls tho vertical piano. Thus it will be .>ieou,
1>
26
THE PRACTICAL DRAUGHTSMAN'S
m figs. 6 and 6', that the horizontal projection of the cone, s a b, is
exactly the same as that of a cylinder having an equal base ; but
the vertical projection, s' a' b', in place of being a rectangle, is an
isosceles triangle, of which the base is equal to the diameter of the
circle, forming the horizontal projection, whilst tlie height is that
of the cone. Similarly to the preceding case, in order to construct
these projections, it is sufficient to know the radius of the cu'cular
biLse and the height.
THE PROJECTIONS OF A SPHERE, FIG. @.
100. A sphere, in whatever position it may be with reference
to the planes of projection, is invariably represented in each by a
circle, the diameter of which is equal to its own ; consequently, if
the two projections, o and o, figs. 7" and 7, of the centre be given,
we have mei'ely to describe circles with these centres, with a
radius equal to that, o a, of the ^ven sphere.
It would seem from this, that one projection should be sufficient
to indicate that the object represented is a sphere ; but on referring
to figs. 5°, and 6°, and T, it wiU be seen that a circle is one pro-
jection of three very different solids — namely, the cylinder, the
cone, and the sphere. This is a further illustration of the in-
adequacy of one projection to give a faitliful representation of
any solid form. It is true, that by shading the single projection,
we approach nearer to the desired representation ; but still,
such shaded projection would equally represent that of a cylinder
with a hemispherical termination. The same remark applies to
the shaded projections of cylinders and cones, and, indeed, to all
solid bodies.
OF SHADOW-LINES.
101. To distinguish and relieve those parts of a drawing which
are intended to indicate the more prominent portions of the object
represented, it is customary to employ fine sharp fines for that
part of the outline on which the fight strikes in fuU, and strong
and heavy lines for the parts which are at the same time in refief
and in the shade ; the latter description of fines are called shadow-
lines.
For the maintenance of uniformity, it is obviously expedient
to suppose the light to strike any object in some constant and
particular direction. The assumed direction should be inclined,
m order that some parts of the object may be thrown into shade,
whfist others are more strongly iUuminated. Hitherto, a uniform
method has not been generaUy adopted with regard to the assumed
direction of the rays of fight. Some authors have recommended
that it should be, as it were, paraUel to that diagonal, a g, of the
cube, fig. ^, of which the projections are a c and a' e', figs. 1 and
:"; others, however, cause the ray of light to take the direction
a' e' in the vertical, and d b in the horizontal plane of projection,
and some liive proposed that the ray should strike the object in
a direction perpendicular to either of the planes. We have
adopted the fu-st mentioned system, and we shall shortly indicate in
what points it is superior, and on what account it is preferable, to
any other.
The line which we take as the diagonal of the cube, is that
which extends from the comer, a, of the front facet of the cube,
fifif. ^ tn the extreme and opposite corner, g, of the posterior
facet. The projections of this straight fine in the representiitiuns
of the cube, figs. 1 and 1°, are respectively the lines, a c and a e',
lying ea«h at an angle of 45° with the base line. Thus, in gen-
eral, in our dravvdngs the objects are supposed to receive the light
in the du'ection of the arrows, r and r', in fig. 8, according as the
projection is in the vertical or horizontal plane.
102. We must observe, that the actual incUnation of the straight
fine thus adopted, is not that of 45° with respect to either plane
of projection ; the angle of incUnation is in fact somewhat less,
and may be determined by means of the diagram, fig. 9. For
this purpose, it is necessary to suppose the perpendicular plane iu
which the ray or line fies, as turned or folded down upon the ver-
tical or horizontal plane, the turning axis being perpendicular to
the base line. Let us, in the first place, suppose the two pro-
jections, E and r', of the ray, to meet in the point, o, in the base
line, L T ; taking any point in this ray, as projected in the horizontal
plane at a, and in the vertical at a', with the point, o, as a centre,
and radius, a o, describe the arc, a c a', cutting the base fine in the
point, c; through this point draw the perpendicular, b b', limited
each way by the lines, a b, a' b', di-awn paraUel to the base fine
through the points, a, a . Joining o b and o b', the lines thus
obtained indicate the position and inclination of the ray, when
folded down, as it were, on either plane of projection ; and on
applying a protractor, it will be found that the actual angle of in-
cfination is one of 35° 16' nearly. Having, then, fixed upon the
direction of the rays of fight, which ai-e, of course, supposed to
be paraUel amongst themselves, it wUl be easy to determine which
part of an object is illuminated, and which is in the shade. It
will be perceived, for example, in figs. 1 and 1°, that the illumi-
nated portions are those represented by the lines, a b and a d, on
the one hand, and a' b' and a' f' on the other ; and that those in
the shade are represented respectively by the fines, b c, c d, and
b' e', f' e'. It must be observed, that according to this system,
whatever part of the object is represented as Ulumtnated in one
projection, is equaUy so in the other; the shaded parts corre-
sponding in a simUar manner. What has just been said with
reference to the cube, is equaUy appficable to all prisms or sofids
boimded by sharp definite outlines, care being taken to employ
heavy shadow-lines only on the outlines of parts which are both
promident and in the shade — such shadow-fines separating the
facets which are Uluminated, from those which are not.
103. With regard to round bodies, the projections of the lateral
portions being bounded by fines which should not indicate
prominent and sharply defined edges, so fuU a shadow-line should
not be employed as that forming the outfine of a plane and pro-
minent surface. Thus, in figs. 5, 6, and 7, the lines, b' e', s' b',
and c' b' d', are not nearly as strong as the corresponding lines, b' e',
in figs. 1 to 4. Nevertlieless, these lines should not be as fine as
those on the Uluminated side of the object, but of a medium
strength or thickness, to indicate the portion of the object which
is in the shade. In other words, a sharp fine fine indicates the
fully Uluminated outfine, a fuUer line the portion in shade, and
a shadow-line still stronger that portion which is both in the
shade, and has 9 prominent sharply defined edge. The straight
lines, f' e' and a' b', figs. 5 and 6, wiU necessarily be fuU shadow
lines, as representing the edges of planes entirely in the shade.
BOOK OF INDUSTRIAL DESIGN.
27
In the liorizontal projection of the cylinder, fig. 5°, the illumi-
nated portion corresponds to the semi-circle, adb, whilst that in
the shade is the other semi-cu'cle, a ch; the points, a, h, of
separation of the two halves, are obtained by drawing through
Uie centre, o, a diameter, a b, perpendicular to the ray of light,
d o, or by drawing a couple of tangents to the circle parallel
to this ray. The straight line, a b, is inclined to the base line
at an angle of 45°. Great care is necessary in producing the
circular shadow-line, acb, and the nibs of the drawing-pen should
be gradually brought closer as the extremities, a and b, of the
shadow-line are approached, so that it may gi-adually die away
into the thickness of the illuminated line. By inclining the
drawing-pen, or by pressing it sideways against the paper, the
desired effect may be produced ; the exact method, however, being
obtained rather by practice than by following any particular in-
structions. A very good effect may also, in some instances, be
produced, by first dravdng the entire circle with the fine line, and
then retracing the part to be shadow-lined with a centre slightly
to one side of the first centre, and repeating this untQ the desired
strength of the shadow-line is obtained.
104. In the plan of a cone, fig. 6°, the part in the shade is always
less than the part illuminated; but it requires an especial con-
struction, which will be found in the chapter treating of Shadows,
for the determination of the lines of separation, s e ; and it is sel-
dom that such extreme nicety is observed in outline drawings, the
shadow-line of the plan of the cone being generally made the
same as that of a cylinder, or perliaps a little less, according to
tlie judgment of the artist. Yet, if the height of the cone be
less than the radius of the base, the whole conical surface will be
illuminated, and consequently its outUne should have no siiadow-
Jine.
105. In explanation of the motives which have guided us in
the adoption of the diagonal of a cube, as projected in the lines,
R, r', fig. 8, as the direction of the rays of light, in preference to
the other systems proposed, we shall proceed to point out some
of the inconveniences attending the latter.
In the first place it must be observed, that if we adopt, as the
direction of the rays of light, the diagonals projected in a' e' and
D B, figs. 1 and r, that part of the object which is represented in
the plan as illuminated, does .not correspond with the part repre-
sented as illuminated in the elevation : in such case, the shadow-
lines would be AB and b c in the horizontal projection, and f' e',
d' e' in the vertical, so that there is no distinction made betwec^n
the plan and the elevation; whereas, according to the system
adopted by us, it is at first sight apparent wliich is (he plan, and
whicli the elevation, from (he mere .shadow-lines, whidi are in Ihc
latter at (ho lower [)arts of (ho object; whilst, in (lie Coi-nier, (licy
arc, on tiie contrai-y, at (ho up|)cr j)mi'Is. K Is nol nntin-al, moic-
ovcr, ((• snpposc, lliat in Hk; I'l-prcscnlnlidii iifinn- (ilijccf, llic light
can li(! made to come as it were from behind llu^ ol)jc<'(, tor in (liat
case (he side nearest (he spec(a(()r would evidently be in (he
miaiiu; and y(^t it is only on such a siii)posilii]n that the prdjci'llnns
ot the ray of light can be such as u u and a' k'. Thus i\ dimble
inconvenience may bcs urged against (his system.
If, on the otIi'T hand, (h(i rays oC light are Hni)]M)scd (o be pir-
pondioilar to either plane, such confusion will resuK as (o rendei-
it impossible to ascertain, by any reference to the shadow-fines
what is, or what is not, illuminated, and thus the object of employ-
ing shadow-lines would be lost sight of. Thus let us suppose, for
example, that the light is perpendicular to the vertical plane,
whence it follows that the whole of the anterior facet, figs. 1 to 4,
is fully illuminated ; but, at the same tune, all the facets perpen-
dicular to the vertical plane are equally in the shade, and it would
consequently be necessary to use shadow-lines all roimd, or else not
at all ; and whichever plan was adopted, would be quite unintelligi-
ble. Besides this, it is unnatural to suppose that the spectator
should place himself between the light and the object. Indeed, it
is unquestionable that the most appropriate du-ection to be given to
the ray of light is as before stated, that of the diagonal of a cube,
of which the facets are respectively parallel to the two planes of
projection ; and the projections of this diagonal are, consequently,
inclined to the base fine at an angle of 45°, but proceeding from
above in the vertical projection, and from below in the horizontal
projection, as shown by the arrows, e and e', fig. 8.
PROJECTIONS OF GROOVED OR FLUTED CYLINDERS
AND RATCHET WHEELS.
PLATE VIII.
106. The various diagi'ams in this plate are designed principally
with the view of making the student practically conversant with
the construction of the projections of objects ; and, besides teach-
ing him how to deUneate their external contours, to enable him to
represent them in section, that their internal structure may also be
recorded on the drawing.
Figs. 1 and 1° are, respectively, the plan and elevation of a
right cylinder, which is grooved on its entire external surface. The
grooves on one-half of the circumference are supposed to be
pointed, being formed by isosceles triangles of regular dimen
sions, and may represent the rollers used in flax machinery, in
apparatus for preparing food for animals, and in many other
machines. The other half of the circumference is formed into
square or rectangular grooves, the lateral faces of which are cither
parallel to the centre lines which radiate from the centre, or are
(hemsehx's radiating.
107. To construct the horizontal projeelina nt' this cylinder,
that is, as seen from above, wo must first ascertiiin how many
grooves are contained in the whole circumference ; then di-awing a
circle with a radius, a o, wliich should idways bo gi-eater tlian thM
of the given cylinder, divide it into twice as many equal pjirts a.s
(here are grooves. If (he student will refer back to the section
treating of linear drawing, illusVated in Plate 1.. he will liml simple
methods of dividing circles into '2, 3, 4, (!, 8, and I'J equal parts,
and, further, of subdi\iding these. Thus, as the cylinder, lijj. 1,
contains 24 grooves, its cireumforcnco must bo divided into 48 equal
par(s. To obtain these, begin by drawing two dimnoters, a n, c i>.
perpendicular to each odier ; then, from e.'icli oxtromily, mark oil" tJio
liii'jth of (ho radius, a o, thus <)b(aining the ("our points numocreil
s on one side, and (lu> points nundiered I on the otiior — making,
wiih tlu' points of intersection iif the two dianiotors with the cir-
enmCerence, in all, I'J points. It remiiiiiii simply to bisivt ojid)
spac(>, as A — t, U — I, or 4 — 8, &c.. as \\\'\\ as the leaser sivaces
I
28
THE PRACTICAL DRAUGHTSMAN'S
thus found; tliis will give the 48 divisions required. Tlirough
the points of division draw a series of radii, whieli ^\ill divide
the inner circle described, with the radius, o f, into the same
number of equal parts. Tlie depth of the grooves is limited
by the circle described with the radius, o E, whilst the outside
of the intervening ridges is defined by the cu-cle of the radius, o f.
All the operations which we have so far indicated, are called for
in the construction of both the triangular and rectangular grooves.
In proceeding, we must, in the former case, join the points of in-
tersecti ~>n, a, b, c, d, which are in each circumference alternately ;
whilst ir the latter ease we require no fresh lines, but have simply
to ink in alternate portions of the two circles, as well as the radial
lines joining these.
108. To draw the vertical projection, fig. 1', it is necessary that
the depth should be given, say m' n' = 54. First set out the
two horizontals, m' p', n' q', limiting the depth of the figure ; then,
to obtain the projection of the grooves and ridges, square over
each of the points, e,f,g,-'h,&.c., and di-aw parallels through
the po'iits thus found in fig. 1°, as e',f',g',h'. This completes
the elevation, and represents the whole exterior of that part of the
cylinder below the horizontal, m p.
109. It has already been observed, that two projections are not
always sufficient to form a complete representation of an object ;
thus it wiW be evident, from a consideration of figs. 1 and 1", that
a third view is necessary to explain the interior of the cylinder.
The radius, o g = 42, of the central circular opening, is not ap-
parent in fig. 1", it is only to be found in the plan; whereas we
have already seen, that, to determine its exact position, it should
be represented in two projections. From figs. 1 and 1°, it is im-
possible to see if the opening exists throughout the depth of the
cylinder, or if its radius be the same dowTi to the bottom ; and the
same remark applies to the key-way. In consequence of this, it
is expedient to draw the object as sectioned — for example, through
the centre line, m p — ^by a plane parallel to the projection. Such
a section is represented in fig. 1° ; and from it, it is at once manifest
that the central eye or opening, as well as the key-way, extend
equally throughout the depth of the cylinder. The outline of
these parts is formed by the verticals dra\vn through the points,
g', m\ v', h', obtained by squaring over the corresponding points,
G, m, n, H, in the plan. This ^■iew also shows that the external
grooves are equal throughout their depth, as Lodlcated by the ver-
ticals drawn through m', l', e', p'. When the outlines of the in-
terior of an object are few and simple, they may be indicated
in an elevation, such as 1", by dotted lines. But if the outUnes are
numerous or complex, too great a confusion would result from
this method ; and it is far better, in such case, to give a sectional
\iew.
That portion of the solid mass of the cylinder, through which
the sectional plane passes, is indicated in fig. 1', by a flat^tinted
sihading, so as to distinguish it from the parts which the plane
does not meet: this is the plan generally adopted to show the
parts in section ; the strength of the shade, or sectioning, is varied
according to the nature of the material. Thus, for cast-iron, a
darker shade is used, whilst a lighter one indicates wood or stone ;
and as an example of tliis distinctive use of various degrees of
snadc, we have to point out that the sectioning in fig. 1', indicates
the object to be made of copper, wliUst thi^t in fig. 2' corresponds
to cast-iron, and in fig. 3' to wood or masonry.
110. It must be observed, that the section lines, of whatever
description they may be, are always inclined at an angle of 45"
with the base line ; this is to distinguish the sectioning from flat
tints frequently employed in elevations, to show that one surface is
less prominent than another: this latter flat-tinting is generally
produced by perpendicular or horizontal lines. The line, t j', which
indicates the base of the mternal cylinder, g mn, should not be
a shadow-line equal in strength to the bases of the sectional parts,
for the latter are more prominent. This point is seldom attended
to as it should be ; gi-eater beauty and effect, however, would
result if it were. This remark appUes equally to all projections
of objects, of which one portion is more prominent than another.
Thus, in figs, l", 2", S", the vertical lines passing through f are
considerably more pronounced than those passing t'li-ough p' q'
and lying in a posterior plane. It is the more important to observe
these distinctions in representations of complex objects, so as to
assist as much as possible a comprehension of the dravring. After
the preceding consideration of fig. 1 on this plate, figs. 2 and 3,
representing ratchet wheels and fluted cylinders, will be quite
intelligible to the student ; such operations as are additional, being
I rendered quite obvious by the views themselves.
THE ELEMENTS OF ARCHITECTURE.
PLATE IX.
111. Columns of the different orders of architecture are fre-
quently employed in buildings, and also in mechanical constructions,
as supports, where it is desired to combine elegance with strength.
The ancient orders of architecture number five ;* as,
1. The Tuscan.
2. The Doric.
3. The Ionic.
4. The Corinthian.
5. The Composite.
A sixth order is sometimes met svith, denominated the Poestum-
Doric.
112. Each order of architecture comprises three priacipal
parts: the pedestal, the column, and the entablature, in all ftie
ordei-s, the pedestal is a third of the length of the shaft in height,
and the depth of the entablatm-e is a fom-th of the shaft. The
proportion between the diameter and height of the column varies
in each order. The height of the Tuscan column is seven times
the diameter at the lowest part ; the Doric, eight times ; the Ionic,
nine times; the Corinthian and Composite, ten times. The
pedestal is frequently altogether dispensed with. All the differ-
ent parts, in the various orders, bear some proportion to a module,
which is half the diameter of tlie lower part of the column. This
module may be termed, the imit of proportion. It is di\-ided
* We have adhered to the classification wnich, from being of more ancient
date, is supported by superior authority ; but we do not profess, in this work, to
decide which carries more reason with it. Reason frequently runs counter to
authority Modern architects say there are only three orders — the first comprising
Ancient, Modern, and Tuscan Doric ; the second, Greek, Roman, and Modern
Ionic ; and the third, Corinthian and Composite.
BOOK OF INDUSTRIAL DESIGN.
29
into 12 parts, in the Tuscan and Doric orders; and into 18 parts,
in the Ionic, Corinthian, and Composite. The whole height of
the Tuscan order is 22 modules 2 pails, apportioned as follows : —
The column is 14 modules; the pedestal, 4 modules 8 parts ; and
the entablature, 3 modules 6 parts. The whole height of the
Doric order is 25 modules 4 pai-ts — the column being 16 modules;
tlie pedestal, 5 modules 4 parts ; and the entablature, 4 modules.
The whole height of the Ionic order is 28 modules 9 parts — the
pedestal, 6 modules; the colmnn, 18 modules ; and the entablature,
4 modvdes 9 parts. The whole height of the Coiinthian and
Composite orders is 31 modules 12 parts — of which 6 modules 12
parts form the pedestal, 20 modules the cohmin, and 5 modules
the entablatmx'.
As we do not propose to treat especially of arcliiteuture, we
Iiave not given drawings of all the various orders, but have con-
fined ourselves to the Tuscan, as being the simplest, as well as the
one more generally adopted in the construction of machinery.
At the end of this Chapter, will be found tables of the dimensions
of the various components of the Tuscan order, and we also there
give a similar tabla for the Doric order.
OUTLINE OF THE TUSCAN ORDER.
113. The whole height being given, as m n, the proportions of
the different parts may always be determined. Let this height
be, for example, 4 metres 272 millimetres, fig. 7. First, divide it
into 19 equal parts, then take 4 such parts for the height of the
pedestal, 12 for that of the column, and the remaining 3 for the
entablature. Then,' according to the order which it is intended to
follow, the height, m n, of the column, is divided into 7, 8, 9, or 10
equal p.arts, and the diameter of the lower part of the column \vill
be equal to one of these divisions : thus, in the Tuscan order, the
diameter, a b,is ^ of the height, ?« n ; the half of this diameter,
or the radius of the shaft, is the unit of proportional measurement,
called the module, and with which all the components of the order
are measured : it follows then, that in tlie Tuscan order this mo-
dule is jj- of the height of the column, in the Doric -^, in the
Ionic Jj, and 4^ in the Corinthian and Composite. -
114. The three members of an order are each subdi\ided into
three divisions. Thus the Pedestal is composed of the Socle,
or lower Plinth, a ; of the Dado, b ; and Cornice, c : the column
consists of the Base, or Plinth, d ; the Shaft, e ; and the Capital, f ;
and in the entablature are the Architrave, g ; the Frieze, ii ; and
the Cornice, l
115. Bofjre proceeding to delineate these different parts, and
the mouldings witli whicli they are ornamented, it is expedient to
set off a scale of modules, detennined in the manner just stated,
the module being, of course, subdivided into 12 equal parts.
To make the mouldings and various details more intelligible,
we have drawn tlic various portions of the order, sepai-ately, to a
larger scale. Thus the socio and pedestal of the column are ropre-
.lentc'! in elevation in fig. 2, and in plan in (ig. 3, to a scale 2.'- times
tiiat of the comj)leto view, fig. 1, and the modulo will, of coui'se, be
proportionately larger. All the numbers indicated on these figures,
give the exact measurements of each part and eiu'h moulding, so
that they may bo drawn in i)ert'ect ac^cordance with tJio scale given.
Ft conduces considerablj to the symmetry and cxactitudu of the
drawing, to set off all the measurements from the axis or centre
line, c d. The module being but an aibitrary measmement, it ia
necessaiy, in practically caiTying out any design, to ascertain the
different measures in metres and parts of metres ; and for this
reason we have given additional scales in metres, to correspond to
those in modules ; and we have also expressed in millimetres, on
each figure, the measxu-ements of the various details, placing the
metrical in juxtaposition with the modular ones. And, to give a
distinct idea as to the degrees of prominence or relief of the
various members, a part of the elevation is shown as sectioned
by a plane passing through the axis of the shaft, this part being
sufficiently distinguishable from the sectional flat-tinting. In
the horizontal projection, fig. 3, are also represented portions of
sections in two different planes, one being at the height of the
line, 5 — 6, and the other at that of 7 — 8. The first shows that the
shaft is round, as well as the fillet,/, and the torus, g, wliilst the
base, 7(, and cornice, i j, are square : the second section shows,
in a similar manner, that the dado, b, the socle, a, and its fillet, f.
are square. The flat-tintings sufficiently indicate the parts in
section.
Fig. 4 represents the entablature and tlie capital of the column
La elevation and in section. Fig. 5 is a horizontal section of the
column with its capital, as it were inverted, and is supposed to be
half tlii-ough the line, 1 — 2, and half through 3 — 4. The whole
is what is termed a false section, the parts in section being in
paraUcl, but not identical planes. The different measurements
are given in modules and metres, as in the other figures ; they
indicate the respective distances from the axis, c' d'.
116. The execution of this design offers little or no dilficidtv ;
but all the operations required, as well as the pai-ts to which the
measurements apply, are carefully indicated. It is, therefore,
unnecessary to enter into further detaOs, except as far as relates
to such parts as involve some peculiarity ; the shaft of the column,
for example, and one or two of the mouldings.
Referring, in the first place, to the column, it is to be observed
that it is customary to make the shaft cylindi'ical for one tliird of
the height, that is, of equal diameter tlu-oughout tliat extent:
above that point, however, it diminishes gradually in diiuneter up
to the capital. This taper is not regular tiiroughout, being
scarcely perceptible at the lower part, and becoming more and
more convergent towards the top. Its contour is consequently a
curve, instead of a straight line. This curvature constitutes what
is termed the eniasis, and is employed to correct the appaient
narrowness of a recti]inc;u- column at the middle. Such defooti\o
appearance only takes i)lace when the cyliniirical piece, or column,
is between a pedestal and an entablature having jilano surfaces
A cylinder, or sphere, always seems to occupy less sjwce tiian a
plane surface equiil to its greatest section. Thus llie outline of a
cylinder or sphere, appeai-s to grow less wiien it is shaded. Now,
where the column is in contact wilii the plane surface of the pe-
destal or entablature, it cannot appear less in proportion, the
proximity "f the latter correcting sucii appeanince, whilst th.u
influence is loss felt at the central [kwU wliich is furthest from lli»
I)edestal and entablature. A true cyliiuler, therefore, in sucn
])()silion, appi'ars to bo thinner at the middle, and this is corrected
by the entasis, or curved contour.
30
THE PRACTICAL DRAUGHTSMAN'S
But many autJaorities consider tliis a fastidious nicety, and it
IS frequently disregarded, particularly in designing short tliick
columns for machinery, and also where the other extreme is reached,
and thf' columns become mere rods.
\Vlaat may be termed the mechanical entasis, is, moreover, em-
ployed in beams, levers, and connecting-rods of all descriptions ;
tlie object of this convexity, and Lucreased width in tlie middle
part, in such cases, being to obtain strength and rigidity, whilst it
undoubtedly adds to the beauty of form.
To determine the amount of the entasis in the Tuscan column,
divide the line, c d, fig. 6, which represents two-thirds the height
of the shaft, iato any nimiber of equal parts, say six. With the
point, d, as centre, and a radius, d e, equal to one module, draw
an are of a circle ; next, having made c i; equal to 9^ parts, draw
through V a line, v x, parallel to the axis, c d\ this parallel will
cut the arc in the point, x ; divide the ai'c, e x, into six equal parts,
and then tlirough the points, 1, 2, 3, &c., thus obtained, draw par-
allels to the axis. These parallels \vill intersect the horizontal
lines drawn through the divisions, q, r, s, t, of the axis, respectively,
in the poiats, 1', 2', 3', &c., and through these will pass the re-
quked curve, forming the contour of the shaft. This curve, being
symmetrically reproduced on the opposite side of the axis, c d, will
complete the outline of the shaft.
In the entablatm-e and pedestal will be found two similar
mouldings, termed cymatia ; they are both examples of the cyma
reversa, discussed in reference to Plate 3. The slight peculiarities
in their construction, will be easOy understood from the enlarged
view, fig. 8. Tlie quarter rounds and accompanying minor mould-
jQgs belonging to the capital and entablature, are also represented
separately, and on a larger scale, in figs. 9 and 10.
RULES AND PRACTICAL DATA.
THE MEASUREMENT OF SOLIDS.
117. We have already seen that the volume or solidity of a body,
»8 the extent of space embraced by its three dimensions — length.
Avidth, and height ; tlie last of these being frequently termed depth,
or thickness. The volume of a solid is deteixnined when it ia
ascertained what relation it bears to, or how many times it contains,
any cube which is adopted as the unit of the measurement. Such
a unit is the cubic metre, just as a square metre is employed to
measure surface, and a linear metre length.
The subdivisions of the cubic metre are the cubic decimetre,
the cubic centimetre, and the cubic millimetre. The relations
these bear to the linear subdivisions will be obvious from the
following comparison.
Whilst 1 metre = 10 decimetres = 100 centimMres = 1000
millimetres.
1 Cubic metre = (10"^ x 10" x 10"- =) 1000 cubic deci-
metres — (100'=- X 100'- X lOO'- =) 1,000,000 cubic centimetres
= ( 1000 "/^ X 1000 ■"/„ X 1000"/„ = ) 1,000,000,000, cubic
millimetres ; consequently, 1 cubic decimetre — "001 or j^,'^,, cubic
meti-e, the cubic centimetre = -0000001 or -yovWot cubic m^tre.
SimOaiiy, we measure volume by cubic yards, feet, or inches, just
as we measure surface by square, and length by linear yards, feet,
and inches. A cubic foot is -^^ of a cubic yard, for —
1 cubic foot = i yard x J yard X J yard = ^V J^^ >
and an inch, or
Jj foot X 7^ foot X ^2 foot = -^jL^ foot.
119. Parallelopipeds. — The volume of a parallc'.opiped is equal
to the product of its base multiplied into its height.
Example.— Fig. ©. PI. 7. Let A F = 2 feet, F E = 1-4 feet,
and F H = 1-4 feet. Then the base = 1-4 X 1-4 = 1-96, and
1-96 X 2 = 3-92 cubic feet; or more simply, 1-4 X 1-4 X 2 =
3'92 c. ft. A cube itself, having all its dimensions equal — ^it?
volume is expressed by the third power of the measure of one of
its sides; that is, by the product of one side three times into
itself.
Thus the cube, fig. A, of which one side measures, say 1-4
feet, contains r4 x 1*4 X 1*4, or 1-4' = 2-744 cubic feet.
In general, the volume of a right prism, whatever be its base, is
equal to the product of the base into the height.
table of surfaces, and volumes of regular polthedea.
Number of Sides. Name. Surface. Voldme
4 Tetrahedron, . 1-7820508 •1178519
6 Hexahedron, or Cube, 6-0000000 1-0000000
8 Octahedron, 3-4641016 -4714045
12 Dodecahedron, 20-6457788 7-6631189
20 Icosahedron, 8-6602540 2-1816950
120. Pyramids. — The volume of a polygonal pyramid is equal
to its base multiplied into a third of its height.
Example. — Let S O, fig. © = 2 inches, A B and A D e:ich =
1-4 inches; the cubic contents of the pyramid are —
1-4 X 1-4 X 2 „ „„ , . . ,
X = 1-3066 cubic mches.
Thus the volume of a pyramid is one-third of that of a right
•irism. having an equal base, and being of the same height.
The volume of a truncated pyramid, with parallel bases, is
equal to the product of a third of the height, into the sum of the
two bases added to the square root of tlieu- product.
Thus, if V represent the volume of a truncated pyramid, of
which the height, H, = 3 feet, the lower base, B,= 6 square feel^
the upper, B', =: 4 square feet ; we have —
V = 1^ X (B + B' + VWB') =
— X (6 s. f. + 4 s. f. + -^6 X 4) = 14-898 sq. feet.
BOOK OF INDUSTRIAL DESIGN.
31
111 practice, when there is little difference between the areas of
the bases, a close approximation to the volume is obtained by
taking the half of the sum of the bases, multiplied into the
hdio-ht. Thus, with the preceding data, we have
/B -t- B'\
V = Hx(^^ — J = 15 sq.ft.
121. Cylinders. — The cubic contents of aHy cylinder, as fig. [^,
is equal to the product of the base mto the height. Thus, in the
case of a cylinder of a circular base, we have B = rt R^ (72) ;*
consequently, the volume, V, = n; R^ X H.
First Exam-pie. — What is the volume of a cast-ii-on cylinder,
of which the radius, R, = 20 inches, and the length, H, = 108
inches ?
¥= 3-1416 X 20- X 108 = 135,717 cubic inches.
The volume may also be derived from the diameter of the cylin-
der, in wliich case we have —
V= fl^xH; or,
4
V = -7854 X 402 X 108 = 135,717 cubic in.
The convex surface of a right cylinder, when developed, is equal
to the area of a rectangle, having for base the rectilinear develop-
ment of the circumference, and for height that of the cylinder.
It is therefore obtained by multiplying the circumference into
the height or length. With the data of the preceding case, the
convex surface is expressed by the formula —
S = 2rtRxH, orrtD XH= 3-1416 X 40 X 108 =
13,571-7 cubic inches.
The volume of a hollow cylinder is equal to the difference between
that of a solid cylinder of the same external radius, and that of
one whose radius is equal to the internal radius of the hollow
cylinder. Or, it is equal to the product of the sectional area into
the height, such area being equal to the difference between two
circles of the external and internal radius, respectively.
Example. — It is requu-ed to find the volume, V, and the internal
surface, S', of a steam-en^ne cyhnder, including its top and bottom
flanges in the volume. Let the following bo the dimensions : —
External diameter, D, = 56 inches; internal diameter, D', = 60
inches; length or height, H, = 120 inches; external projection
of the flanges, F, = 5 inches, and their thickness, E, = 4 inches.
Then, for the internal surface, wo have —
S' = 3-1416 X 60 X 120 = 18,850 sq. in.
For the volume of the body of the cylinder, we have —
rt562 ^502
V' = — ^ -^ X 120= (-7854 X 66=)— (-7854 x 50-) x
120 = 60,000 cubic inches.
And for the additional volume of (lie flanges —
r„ _H (56 + 10)- _rt. 'J6
V" = ^
X 4 X 2 = (-7854 X 66-) —
=: 7666 cubic inches.
4 4
(•7864 X .66=) X
Whence the whole volimio —
V + V" = 67,666 cubic inches,
• Whon wo wi»h to rol'or tho «tU(!on(. U> uny riilo or priiiriplo ulrondy given,
nre do »n by iiiimiiih of tho iiumbor of llui |mragru|ih RontiiiiiliiK kuoIi rulo or priiioi-
ple. Ill the ]>rvBciil inalnuco, whut is rofurrud to will bu found nt jiugii 30.
122. Cones. — Tlie cubic content of a cone is equal to the pro-
duct of its base into a third of its height ; or,
V = B X 5.
3
In the right cone, fig. P", of which the base is circular—
V;t = R^xH_lD3>,H;
3 "~ 4 3
and as rt, or 3-1416 -;- (4 x 3) := -2618, the formula resolves
itself into —
V = -2618 x D2 X H.
Example. — What is the volume of a right cone, of which the
height, H, = 24 inches, and the diameter of the base, or D, =
17 inches 1
We have —
V=-2618 X 172 X 24= 1816 cubic mches.
As we shall demonstrate, at a more advanced stage, the
development of the convex surface of a right cone is equal to ine
sector of a circle, of which the radius is the generatrix, and tlie
arc the circumference of the base of the cone — consequently, the
conical surface is equal to the product of the circumference of
the base into the half of the generatrix: whence is derived tlie
following formula : —
G
S = 2rt R X — =rtR X G.
With the data of the foregoing example, and allomng the
generatrix to be equal to 25| inches, we have —
S = 3-1416 X 8-5 X 25-5 = 681 cubic inches.
123. Frustum of a cone. — The volume of the frustum of auuno
may be obtained in the same manner as that of the truncated
pyramid (120). The convex smface of a truncated cone is equal
to the product of half the generati-ix of the frustum into the sum
of the circumferences of the bases, and is expressed in the follow-
ing formula :
S=h X 2rt (R + R') = L X rt (R + R').
Example. — Let the length, L, of the generatrix of tho couio
frustum, = 14 inches; tho radius, R, of the lower base, = 8-5
inches ; tho radius, R', of the upper base, = 3-8 inches ; then tlio
convex surface —
S = 14 X 3-1416, X (8-5 + 3-8) = 54 square-inches.
124. Sphere. — Tho volume of a sphere may be asoerfjiinod a*
soon as its radius is known. Its surface is equiU to four times
that of a circle of equal diameter. This is expressed by the
formula —
S = 4rt R3 = rt D=' = 3-1416 X D^
or tho square of tho dimnetor multiplied by 3-1416.
The volume is t'nual to the iiroduct of tiio surface into ono-third
of the radius, as in the I'ormula —
V = 4rt R=' X ^=4 ^ "* ^^''' *"■ ^' = •'■'^'^ ^ ^^ '•
or, if we enq>loy tlie di:iini'tral ratio —
V = rtD^
D
= -6236 X D'
32
THE PRACTICAL DRAUGHTSMAN'S
Example. — We would know what is the surface and the volume
of a sphere, of which the diameter measures 25 inches.
The surface —
S= 25- X 3-1416 = 1963-5 sq. inches
Tlie volume —
V = -5236 X 25=' = 8181-25 cubic inches.
To find the radius or diameter of a sphere, of which the volume
is Known, it is sufficient to invert the preceding operations, the
formulas becoming as follows —
3V V .
4-188 '
R=
4rt
3
Wlience,
Similarly,
whence.
R
V 4-188
D3
V
•5236'
^ = V -.5236 '
which, with the preceding data, gives R = 12-5 inches, and D =
25 inches.
Tlie radius is derived from the surface by means of the follow-
ing formula: —
S
R- =
whence,
'4x3-1416'
1-5664
D^ =
S
3-1416'
whence,
" = \/3
S
3-1416*
125. Spheric sectors, segments, and zones. — The surface of a
zone or spheric segment, is equal to the product of the circiun-
ference of a circle of the sphere, luto the height of the zone or
segment; or,
S = 2rt R X H.
Example, — The height, H, of a spheric segment being 1-5
inches, and the radius, R, of the sphere, 7-5 inches, the surface —
S = 2 X 3-1416 X 7-5 X 1-5 = 70-686 sq. inches.
The volume of a spheric sector is equal to the product of the sur-
face of its spherical base, into one-third the radius of the sphere
of wMch it is a portion.
The corresponding formula is therefore^
R 2
V=2rtRxHx - = -rtxR2H= 2-094 x R^ x H.
o o
Example. — The volume of the spheric sector, whose spheric
base is equal to the surface considered in the pre%'ious example, is —
V = 2-094 X 7-52 X 1-5 = 176-68 cubic mches.
The volume of a spheric segment is equal to the product of
tne arc of the circle of which the chord is radius, into one-sixth
of the height of the segment; or,
V = rt r= X - = -5296 x r^ x H.
6
Example. — Let r = 6-5 inches, and H 1-5 inches; the tuen
volume —
V = -5296 X 6-52 X 1-5 = 33-56 cubic inches.
The volume of a spheric ungula is equal to the product of thts
gore, which is its base, into a tliird of the radius.
The formula b —
V=f A X R2;
3
where A — the area of the gore.
The volume of a zonic segment is equal to half the sum of
its bases, multiplied by its height, plus the volume of a sphere
of which that height is the diameter ; whence the formula —
\={ =1 )xH+-
IP
6~"
126. Observations. — The volumes of spheres are proportionaj
to the cubes of their radii, or diameters. Let V = 14-137
cubic inches, and v = 4-188 cubic inches. It will be found
that the respective radii are —
R =
4-188
and
_ -^ /14-137 _ 5.
~V "4088-^^-
'•=VfT88=V^
^ 4-188
4-188
= 1;
and, consequently, D = 3 and d = 2.
The cubes of these numbers, that is, 27 and 8, have the same
ratio to each other as the volumes given ; that is to say —
27 : 8 :: 14-137 : 4-188.
When of equal height, cylinders are to each other, as well as
cones, as the squares of the radii of their bases.
When of equal diameter, these solids are to each other as
their heights.
First, then, we have —
V = rt R2 X H, and v = n r^ x H ;
whence,
V : D :: R2 : 7-2
And, secondly,
V = rt R2 X H, and D = rt R2 X ft;
whence,
Y:v:'.H:h.
The volume of a sphere is to that of the circimiscribed cylinder
as 2 to 3. A sphere is said to be inscribed in a cylinder, when
its diameter is equal to the height and diameter of the cylinder.
The volume of an annular torus, or ring, is equal to the product
of its section into the mean circumference. We have pointed
out (90) that an annular torus is a solid, generated by the revolu-
tion of a circle about an axis, situated in the plane of the circle,
and at right angles to the plane of revolution.
Let R be the radius of the generating circle, and r the distance
of its centre from the axis, we have —
V = rt R2 X 2rt r = 19-72 R^ x r.
BOOK OF INDUSTRIAL DESIGN.
3a
PROPORTIONAL MEASUREMENTS OF THE VARIOUS PARTS OF AN ENTIRE ORDER.
THE (modern) DORIC ORDER.
Designations of the Members and Mouldings
constituting the Order.
Pi
<
Eh
&3
Cornice,
'Reglet, . .
Cavetto, .
Fillet, . .
Cymatium,
Corona, .
Fillet, . .
Mutules, .
Guttae, . .
FUIet, . .
Cymatium,
Capitals of the Triglyphs,
A ( Taenia,
ARCHITRAVE, jp^j^^'
o
Reglet, .
Cymatium, ,
^ Abacus, .
Capital,., -j e.^i^^; _
Shaft, ,
Base, .
H
W
Cornice, .
Dado,
Bas£,.
Three Annulets,
^Necking, ....
f Beading or Astragal,
Cincture,
' ' I Sh£
Shaft Proper,
'Fillet,. .
Beading,
Torus, .
Plinth, .
•i
f Reglet,
Quarter-Round,
Fillet,
Corona, ....
Cymatium, . . .
fFillct,
Beading,
Cyina Reversa, . . .
Plinth
Sul)-l'linth or Socio,
Total height of the Order,
Measurements according to Vignoles,
in Modules of 12 Parts.
Amount of
Projection
from the Axis
of the Shaft.
M. P.
2 10
2 7
6i
6'
5
4i
2"
n
3"
1
Oi
lU
11"
10
111
10
3^
2^
2
lU
loi
10
1
11^
10'
1
1 U
1 2
1 5
1 5
1 11
1 10?
1 9|
1 9
1 6,V
1 6J
1 5
6
7
7
81
9
9.V
Heights.
M. P.
1
3
li
4
2
2
1 6
2
10
M. P.
1 6
- 4
1 6
1
13 lOJ
■ 14
1
Vie
1
X
IJ
4
I
l"
2
oi
"•J
4
4
10
)■ 5 4
Measures in Decimals
The Module = 1.
Amount of
Projbc.'on
from the
Axis of the
Shaft.
2-833
2.583
2-542
2-500
2-417
2-375
2-167
2-125
1-250
1-083
1-042
•959
•917
•833
•959
•833
r292
1-271
1-188
1-167
1-146
-959
-875
-833
1-000
-959
•833
rooo
M04
1-167
1-417
1-417
1-017
1-889
1-806
1-750
1-512
1-459
1-417
1-500
1-583
1-708
1 -750
1-792
Heights.
13
083
250
042
125
333
042
042
209
042
166
166.
500
167
833
y 1-500
1-500
1-000
4-000
042"
083
209
209
124
333
083"
042
875
1-000
14-000
0831
083 1
334 f
500 J
1-000
► 16 000
•042 "
■083
•042
•209
. -500
•124
•000
4-000
V 1(1(100
•042 1
•083
•1()6
•209
•333
8-333
25-338
34
THE PRACTICAL DRAUGHTSMAN'S
PROPORTIONAL MEASUREMENTS OF THE VARIOUS PARTS OF AN ENTIRE ORDER.
THE TUSCAN ORDER.
Designations of the Members and Mouldings
constituting the Order.
Amount of
Projection
from the Axis
of the Shaft.
Pi
<:
<
o
Quarter-Round, . .
Beading,
Fillet,
Larmier or Corona,
Fillet,
Cymatium,
Fkheze,
Architrave
O
H i
Q
CL,
•1
listel,
Facia,
H
W
P5
{Listel,
Abacus,
Echinus, or Quarter-Round, . .
Fillet,
Necking,
l^^"^" ISul',::;
Shaft Proper,
(Fillet,.
< Torus,
( Plinth,
o
O
( Listel,
1 Cymatii;
Dado, .
j Listel,
Socle or Plinth,
Measurements according to Vignoles,
in Modules of 12 Parts.
Measurements in Decimals.
The Module = 1.
M. P.
2 3i
2
1 lU
1 10|
1 7i
10"
9l
1 2i
1 n
1 r
lOi
9l
11
lOi
9i
1 O'
1 U
1 41
1 4i
1 8i
1 8"
1 6
4^
6i
Heights
M. P.
4
1
I
1
4
1 2
2
10
1 4
1 2
h »J
I 3 6
1 1
3
3
1
4
1
11 10
II
1
M4
8 I
3 8 13 8
6
► 4 8
Amount of
Projection
from the
Axis of the
Shaft.
2-292
2-000
1-959
1-875
1-625
1-125
-833
•792
•959
•792
1-709
1-667
1-417
1-375
1-542
1-709
Heightg.
•333 "~1
•083
•042
•500
•042
•333
1^167
•167
•833
1-333
1-167
1-000
3-600
1-209
•083]
1-125
•250
1-083
•250
-875
•083
•792
•334 J
•917
-083'
-042
•875
•792
1-000
\
11-875
1-125
•083
1-375
•417
1-375
•500
1-000
12-000
1-000
Y 14000
•500
•167 1
I -333 \
3-661 I 3-667
•083 ) ..„„
•417 f ^0^
4-667
Total height of the Order,
22 2
22-167
With the help of these tables we can easily determine the proper
measurement for any member or moulding, in feet, inches, or
fnetres, when the height of the whole order is given. For this
• When two measurements are given, the first applies to the tipper portion, the
■"wnd lo the lower.
purpose the given height must be divided by the decimal measure-
ment in the tables for the total given height ; the quotient is the
measurement of the module proportioned to such height. Then
that of any required member is found by multiplying this module
into the decimal in the table con-esponding to such member.
First Example. — It is required to know what is the diameter of
BOOK OF INDUSTRIAL DESIGN.
35
the lower part of the shaft according to the Tuscan order, the
height of the entire order being 15 feet.
The height of the entire order being 22-167 when the module
22-167
:1. we have-
15
= 1-4778 the module, and 1-4778 x 2 = 2-9556
feet, the diameter of the lower part of the shaft.
Second Example. — \Vhat is the height of the socle or lower
plinth according to the Tuscan order, supposing the module to be
1-4778 feet 1 We have 1-4778 x -417 = -616.
In like manner the dimensions of all the other details may be
easily determined according to the Tuscan or Doric order.
CHAPTER m.
ON COLOURING SECTIONS, WITH APPLICATIONS.
CONVENTIONAL COLOURS.
127. Hitherto we have indicated the sectional portions of
objects by means of linear flat-tinting. This is a very tedious
process, whilst it demands a large amount of artistic skill — only
obtainable by long practice — to enable the draughtsman to pro-
duce pleasing and regular effects ; and although, by vai;ying the
strength or closeness of the lines, as we have already pointed out,
it is possible to express approximately the nature of the material,
yet the extent of such variation is extremely limited, and the dis-
tinction it gives is not sufficiently intelligible for all purposes. If,
however, in place of such line sectioning, we substitute colours
laid on with a brush, we at once obtain a means of rapidly tinting
the sectional parts of an object, and also of distinctly pointing
out the nature of the materials of which it is composed, however
numerous and varied such materials may be. Such colours are
generally adopted in geometrical drawings ; they are conventional
— that is, certain colours are generally understood to indicate par-
ticular materials.
In Plate X. we give examples of the principal materials in use,
with their several distinctive colours ; such as stone and brick, steel
and cast-iron, copper and brass, wood and leather. We propose
now to enter into some details of the composition of the various
colours given in this plate.
THE COJMPOSITION OR MIXTURE OF COLOURS.
PLATE X.
128. Stone. — Fig. 1. This material is represented by a light dull
yellow, which is obtained from Roman ochre, with a trifling addi-
ton of China ink.
129. Brick. — Fig. 2. Alight red is employed for this material,
and may 1)0 obtained from vermilion, which may Hometimi's be
brighten(HJ by the addition of a little carmine. A pigiiirnt fiiuinl
in most colour-boxes, and tcniieil LIfrhi lied, may also be use.! w In ii
great |)urity and brigiitness of tint is not wanted. If it is desired
to distinguish firebrick from the ordinary kind, since the former is
lighter in colour and inclined fo yellow, somo gamboge mu.st bo
mixed vvifh the vermilion, the wlioh^ being laid on more faintly.
In (external vi(^ws it is customary to iiidieate the outlines of the
Individual bricks, but in the section of a mass of brickwork lliis
"efinoHK^iit may he dispensed with, except in ciisi's where it is
intended to show the disposition or method of building up. Thus,
in furnaces, as also in other structures, the strength depends greatly
on the method of laying the bricks. When vermilion is used in
combination with other colours, the colour should be constantly
mixed up by the brush — as, from its greater weight, the vermilion
has a tendency to sink and separate itself fi-om the others ; and if
this is overlooked, a varying tint of unpleasing effect will be
imparted to the object coloured.
130. Steel or Wrought Iron. — Fig. 3. The colour by which these
metals are expressed is obtained from pm-e Prussian blue laid on
light — being lighter and perhaps brighter for steel than for wrought-
iron. The Prussian blue generally met with in cakes has a con-
siderable inclination to a greenish hue, arising from tlic gum with
wliich it is made up. Tliis defect may be considerably ameuded
by the addition of a little carmine or crimson lake — the proper
proportion depending on the taste of the artist.
131. Cast-iron. — Indigo is the colour employed for this metal;
the addition of a little carmine improves it. The colours termed
Neutral Tint, or Payne's Gray, are frequently used in place of the
above, and need no further mixture. They are not, however, so
easy to work with, and do not produce so eipiable a tint.
132. Lead and Tin are represented by similar means, the
colour being rendered more dull and gray by the ;uklition ol' ("hiiii
ink and carmine or lake.
133. Cupper. — Fig. 5. For tliis metal, pure carmine or crimson
lake is ])roper. A more exact imitation of the reidity may be ob-
tained by the mixture, with either of these colours, of a little China
ink or burnt sieiuia — the carmine or lake, of course, considerably
predominating.
134. Brass or Bronze. — Fig. 6. These are expressed by an orange
colour, the former being the brighter of the two; burnt Ri>man
ochre is the simplest pignu'iit for producing this colour. Where,
however, II very bright tint is desire^l, a mixture should be made
of gamboge with a little vermilion — i-ire being taken to keep il
constantly agitated, as before recomiuended. Many draughtsmen
use simi)le gamboge or other yellow.
i:!5.— »\'i'(«/.— Fig. 7. It will be observable, from l>reee.iini;
e\;iliiplrs, (liMt the tints li:i\i' been elio-.ell W illl iel'er<nee lo tlu)
actual eoloms of the materials wliieli they are iutcmled to express —
carrying out tlii> same jiriueiple, we should ha\e a very wide niUjiri)
ill the e.ise ivf wood. 'I'lu' eolour i,'i'iu rally used, however, U
36
THE PRACTICAL DRAUGHTSxMAN'S
burnt umber or raw sienna ; b it the depth or strength with which
it is laid on, may be considerably varied. It is usual to apply
a, light shade first, subsequently showing the graining with a
darker lint, or perhaps with burnt sienna. These points are sus-
ceptible of great variation, and very much must be left to the
judgment of the aiiist.
136. Leaiher, Vulcanized India-Rubber, arid Gutla Percha. —
?ig. 8. These are all represented by very similar tints. Leather
Dy light, and gutta percha by dark sepia, whilst vulcanized india-
rubber requires the addition of a little indigo to that colour.
We may here remark, that if the student is unwilling to obtain
an extensive stock of colours, he may content himself with merely
a good blue, a yellow, and a red — say Prussian blue, gamboge or
yellow ochre, and crimson lake. With these three, after a little
experimental practice, he may produce all the various tints he
needs; but, of course, with less readiness and facility than if his
assortment were larger.
137. The Manipulation of the Colours. — We have seen by what
mixtures each tint may be obtained, and we shall proceed to give
a few hints relative to their application. It may be imagined that
it is an easy matter to colour a geometrical drawing — that is, sunply
to lay on the colours; but a little attention to the following
observations will not be misplaced, as the student may thereby at
once acquire that method which conduces so much to regularity
and beauty of effect, and which it might otherwise require some
practice to teach.
The cake of colour should never be dipped in the water, as this
causes the edges to crack and crumble off, wasting considerable
quantities. Instead of this, a few drops of water should be first
put in the saucer, or on the plate, and then the required quantity
of coloiu' rubbed down, the cake being wetted as little as is
absolutely necessary. The strength or depth of the colour is
obtained by proportioning the quantity of water, the whole being
well mixed, to make the tint and shade equable throughout.
When large surfaces have to be covered by one shade, which it
is desired to make a perfectly even flat tint, it is well to produce
the required strength by a repetition of very light washes. These
washes correct each other's defects, and altogether produce a soft
and pleasing effect. This method should generally be employed
by the beginner, as he will thereby more rapidly obtain the art of
producing equable flat tints. The washes should not be applied
before each preceding one is perfectly dry. When the drawing-
paper is old, partially glazed, or does not take the colour well,
its whole surface should receive a wash of water, in which a very
small quantity of gum-arabica or alum has been dissolved. In
proceeding to lay on the colour, care should be taken not to fill
the brush too full, whilst, at the same time, it must be replenished
before its contents are nearly expended, to avoid the difference in
tint which would otherwise result. It is also necessary first to
try the colour on a separate piece of paper, to be sure that it will
produce the desired effect. It is a very common habit with
water-coloiir artists to point the brush, and take off any super-
rtuous colour, by passing it between their lips. This is a very
bad and disagreealjle haoit, and should be altogether shunned.
Not only may tne colour which is thus taken into the mouth be
injurious to health, but it is impossible, if this is done, to produce
a fine even shade, for the least quantity of saliva which may be
taken up by the brush has the effect of clouding and altogether
spoiling the wash of colour on the paper. In place of this un-
cleanly method, the artist should have a piece of blotting-paper at
his side — the more absorbent the better. By passing the brush
over this, any superfluous colour may be taken off, and as fine a
point obtained as by any other means. The brush should not be
passed more than once, if possible, over the same part of the draw-
ing before it is dry ; and when the termination of a large shade is
nearly reached, the brush should be almost entirely freed from the
colour, otherwise the tint will be left darker at that part. Care
should be taken to keep exactly to the outline ; and any space
contained within definite outlines should be wholly covered at one
operation, for if a portion is done, and then allowed to dry, or become
aged, it will be almost impossible to complete the shade, without
leaving a distinct mark at the junction of the two portions. Fmally,
to produce a regular and even appearance, the brush should not be
overcharged, and the colour should be laid on as thin as possible ;
for the time employed in more frequently replenishing the brush,
because bf its becoming sooner exhausted, will be amply repaid by
the better result of the work under the artist's hands.
CONTINUATION OF THE STUDY OF PROJECTIONS.
THE USE OF SECTIONS — DETAILS OF MACHINERY.
PLATE Xr.
138. We have already shown, when treating of the illustrations
in Plate VIIL, that it is advisable to section, divide or cut
through, various objects, so as to render their internal organiza-
tion clearly intelligible ; and we may now proceed to demonstrate,
with the aid of sundry examples, brought together in Plate XI.,
that in particular cases sections are indispensable, and even more
necessary, than external elevations. It is with this object that,
in many of our geometrical drawings, w"e have given representa-
tions of objects, cut or sectioned through then- axes or centres, so
as to accustom the student to this description of projections, the
impoilance and utility of which cannot be overrated.
139. Footstep Bearing. — Figs. 1 and 1° are the representations,
in plan and elevation, of a footstep, formed to receive the lower
end of a vertical spindle or shaft. This footstep consists of seve-
ral pieces, one contained within the other ; and it is evidently
impossible to say, from the external views, what their actual
entire shape may be, although a part of each is seen in the hori-
zontal projection, fig. 1. If, however, we suppose the whole to
be divided by a vertical plane in^^the line, 1 — 2, fig. 1, we shall be
enabled to form another vertical projection, fig. 1', showing the
internal structure, and which is termed a vertical section, or sec-
tional elevation. This figure shows, first, the thickness of the
external cup-piece, or box, a, as also the dimensions of the open-
ing, a, which is made in its base, for the introduction of a pin, to
raise the footstep proper, b, when necessary ; secondly, the thick-
ness and internal depth of the footstep, b, as also the internal
vertical grooves, b. which serve for the introduction of the key,
c; thirdly, the form and manner of adjustment of the centre-bit,
c, which sustains the foot of the vertical spindle or shaft. This
BOOK OF INDUSTRIAL DESIGN.
37
centre-bit, which, of coiu-se, should not turn with the spindle-foot,
is prevented from doing so by means of the key, c, which fits
into a cross groove in its under side, the key itself being held
firmly by the grooves, b, into which its projecting ends are made
to fit. Of these details, the cup-piece, a, is of cast-iron, the
footstep, B, of gun-metal or brass, the centre-piece, c, of tempered
steel, and the small key, c, of wi-ought-iron. Therefore, bearing
in mind what has already been said, we may indicate these various
materials in the sections, either by line-shading, of different
strengths, as in the figure, or by means of colours, corresponding
to those employed in Plate X.; and we may here remark, that
where line-sectioning is used, brass, gun-metal, or bronze, is
fi-equently expressed by a series of lines, which are alternately
full and dotted. There are, besides, many ways of varying the
effect produced by line-shading. For example, the spaces between
the lines may be alternately of different widths, or the lines may
be alternately of different strengths.
Strictly speaking, figs. 1 and 1' are all that are necessary for
the representation of the object under discussion. The cup-
piece, A, however, which is externally cylindrical, has, at four
points, diametrically opposite to each other, certain projectmg
rectangular plane surfaces, d, which are provided to receive the
thrust of the screws which adjust the footstep accurately in the
centre. The width of these facets is shown in the plan, fig. 1
whilst their depth is obtainable from the elevation, fig. 1°. If,
instead of these facets, d, being, as they are, tangential to the
cylinder, a, they had projected, in the least, at their centres, their
depth would necessarily have been given in the section, fig. l*,
and in such case the elevation, fig. l", might have been altogether
dispensed with. Whilst referring to the representation of the
projecting facets, in connection with the cylinder, a, we may
remark, that when a cyb'nder is intersected by a plane, which is
parallel to its axis, the line of intersection is always a, straight
line, as ef, figs. 1 and 1°.
140. SluJfi.ng-box cover, or gland. — In pumps and steam-engine
cylinders, the cover is furnished, at the opening through which
the piston-rod passes, vdth a stufiing-box, to prevent leakage.
The hemp, or other mateiial used as packing, is contained in an
enlargement of the piston-rod passage, and is tightly pressed down
by a species of hollow bush with flanges, as rcoresented in plan
in fig. 2, and in elevation in fig. 2°. In this instance, the neces-
sity of a sectional view is still more obvious than in the case of
the footstep already treated of. In the vertical section, fig. 2\
it is shown, that the internal diameter is not unilbnn throughout,
and that there is a ring or ferule, b', lot in at the lower part of
the interior. The cylindrical opening, a, of the gland, coincides
exactly with llie dianuiter of the piston-rod ; the internal diameter
of a i)ortion, li, of lii<^ ring, b', is also the same. Tlu^ [)art, c,
aowever, comprised lictwccii llicso two, is gi-catcr in (liiuncter, so
us to lessen Iho (jxiciit of Murfiico in frictional cipiiImcI wilji the
piston-rod, and it also serves for (ho lodgiiiciil of luliiic'iting
tiialtcr. It is furllicr disceriiihic in tli(> section, that the (laiigcs
ov lugs, (/, which project on either side of tlu^ n|i|)er portion of (Ik^
gland, have e:ic-li a eylimh-icnl (ipening, i; IhiMiunleiut, their whuli'
ili'|pth. 'i'heso arc the hoU^s for the holts, wiiich forces down the
el.uul, and Hccuro it to the corresponding tlaiiges, or lugs, on tho
stulling-box. The aimular hollowing out, /, at the upper ard
internal part of the gland, acts as a reservoir, into which the
lubricating oil is first poured, and whence it gradually oozes ^to
the interior. The ring, b', is forcibly fitted into the bottom
the gland, and terminates below in a wedge, in the same manner
as the gland itself, the double wedge jamming the packing against
the piston-rod and the sides of the stuffing-box, and thus forming
a steam-tight joint. The ring, b', is generally made of brass,
both with a view to lessen the friction, and to its being replaced
with facility when worn, without the necessity of renewing the
whole gland. The latter is generally made of cast-iron, though
the whole is sometimes made of brass or gun-metal.
141. SpliericalJoirU. — In some cases, a locomotive receives water
from its tender by means of pipes which are fitted with spherical
joints, as a considerable play is necessary in consequence of the
engine and tenddr not being rigidly connected together, and also to
obviate any difficulty of attachment from the pipes in the locomo-
tive not being exactly opposite to those in the tender. This speeit-s
of joint, represented in plan and elevation in figs. 3 and 3", gives
a free passage to the water, in whatever position, within certain
limits, one part may be with respect to the other. For its con-
struction to be thoroughly understood, the vertical section fig. 3' is
needed. This view, indeed, at once explains tho various compo-
nent parts, consisting — first, of a hollow sphere, a, of the same
thickness as the pipe, b, of which it forms the prolongation ; and,
second, of two hemispherical sockets, c, d, which embrace the ball,
a, and which are firmly held together by bolts passing through
lugs, a, a. When this species of joint is used of a small size, as at
the junction of a gas chandelier wth the ceiling, the two half-
sockets are simply screwed togetlier — this method, indeed, being
adopted in many locomotives. It must be borne in mmd, tliat our
object in this work is simply to instruct Uie student to accurately
represent mechanical and other objects, and for this purpose Ave
employ both precept and example ; but such examples do not
necessarily comprise the latest and most improved or efficient
forms. The half-socket, c, forms part of the continuation, e, of
the feed-pipe, whilst the half-socket, d, is a detached piece, neces-
sarily moveable, to allow of the introduction of the spherical pjirt,
A. This half-socket, d, is partially cut away at tho lower part,
and does not fit closel}' to the neck of the ball. This allows the
pipe, B, to move to a slight extent from side to side in any direction ;
and tho upper end of tho ball, A, is cut away to a corresponding
extent, to prevent any diminution of tho opening into tho pipe, e,
when tho two portions are thus inclined to each other. The pi[H',
K, with its half-socket, c, is an example of the coud>ination of a
cylinder with a s|iliere, and gives us occasion to observe, that tho
intersection fornieil by the nu'eting or junction of these solids is
always a circle in one projection, and a straight line in the other.
The subject of sueli iiiteisietioi\s will be discussed nioni -n det.iil
in reference to Plate .\l\'.
The sockets, and i), are liirnied wiili four external nigs, or evi"-
pieees, /i. for connection by bolts, as l<*>fon< stated. The eiirved
onijines of these lugs, which glide taiigeiitially inio (hat of tho
JMidy of the socket, give rise to the solution of a probloiu wliicli
may he thus jiut : To draw with ii niirn rndiiis an arc Iniiiirnlial In
Iwii giivn tiirs. The solution is (Inis dhtainiHl: with tho eendes,
38
THE PRACTICAL DRAUGHTSMAN'S
o, o, and nidii equal to tlioso of the respective arcs given, plus
th.it of the required arc, describe arcs at about the position in
which, the centres of the required ares should be ; the intersections
of these arcs will give the exact centres, as f, &c., and the lines
joining f, with the centres, o, o. give the points of junction of the
«ic, G H, with tl;e other two. This spherical joint, which requires
gr<;at accuracy of adjustment of the different parts, is generally
•jast in brass, being finished by turning and grinding.
142. Safely-Valve. — To insure, as far as is practically possible,
the safe and economical working of steam boilers, they are usually
titted with pressure gauges, level indicators, alarm whistles, and
safety-valves. The object of the safety-valve is to give an outlet
to the steam as soon as it reaches a greater pressure than has been
determined on, and for which the valve is loaded. Figs. 4. 4°, and
4', respectively, represent a horizontal section, elevation, and ver-
tical section of a safety-valve in common use. This apparatus
consists of two distinct parts : first, the cast-iron seat, a, per-
manently fixed to the boiler-top by three or four bolts, the joint
being made perfectly steam-tight by means of layers of canvas and
cement ; second, the valve itself, b, which is sometimes cast-iron,
sometimes brass. The valve-piece, b, is cast with a central spindle,
c, hollowed out laterally into the form of a triangle with concave
sides, for the purpose of giving a passage to the steam, and, at the
same time, of lessening the extent of frictional contact of the spindle
with the sides of the passage— some contact, however, being neces-
sary for the guidance of the valve. The method of drawing the
horizontal section of this valve-spindle is similar to that given for
fig. A, Plate III. (34), with tliis difference, that it is drawn in an
equilateral triangle, mstead of in a square. The base of the valve-
piece, or the part by which it rests on the seat, a, consists of a
very narrow annular surface ; the upper edge of the seat is bevelled
off internally and externally, so that the surface on which the valve-
piece rests exactly coincides with that of the latter. The upper
external surface of the valve-piece is hollowed out centrally to
receive the point of the rod through which the weighted lever
acts upon the valve ; this lever is adjusted and weighted to corre-
spond with the pressure to which it is deemed safe to submit the
boiler, so that, when this pressure is exceeded, the valve rises,
and the steam blows off as long as relief is necessary.
143. Equilibrium or Double-heat Valve. — Steam engines of large
dimensions, such as those for pumping, met with in Cornwall, as well
as marine engines, are often furnished with a species of double-beat
or equilibrium valve in place of the ordinary D slide. An example
of this description of valve is given in figs. 5, 6", and 5'. It possesses
the property of giving a large extent of opening for the passage of
the steam, with a very little traverse, and very little power is
required to work the valve. The valve here represented consists of
a fixed seat, a, of cast-iron or brass, and forming part of the valve
chamber ; and a bell-shaped valve-piece, b, also in brass, fitted
with a rod, c, by means of which it is moved. The contact of the
valve with its seat is eflTected at two places, a and b, which are
foiTOed into accurate conical surfaces — one, a, being mternal, and
the other, b, external. When the valve is closed, these surfaces coin-
cide with similar ones on the seat, and when it is lifted, as in fig. 5',
two annular openmgs are simultaneously formed, thus giving a
double exit to the steam — which issues from the upper opening.
through the central part of the valve-piece, b. The rod or spindle
of this piece is fixed to a centre-piece cast in one with the valve^
piece, and connected to it by four branches, c. The seat is simi-
larly constructed. The external contour of this valve presents a
series of undulations, involving the following problems in tlieii
delineation : — To draw the curved Junction of the body of the valve
with the upper cylindrical part. This is similar to the one treated
of in reference to fig. 5, Plate III. (37), and may easily be drawn
with the assistance of the enlarged detail, fig. 5' (Plate XI). Next,
for the junction of the branch, c, with the more elevated boss, we
require : To draw an arc tangent to a given straight line, and passing
through a given, point. The solution of this is extremely simple :
we have merely to erect a perpendicular on the line, ef, fig. 5°, r.l
the point of contact, e, of the tangential arc ; to join e and the
given point, g, through which the arc is to pass ; on the centre of
the line, e g, to erect a perpendicular, h i, and the point, i, of inter-
section of this line with the perpendicular, i e, wOl be the centre of
the arc sought, eh g, and i e, the radius.
The central leaves or feathers of the seat, a, are drawn accord-
ing to a problem already discussed (38).
The student will now see the imperative necessity of internal
views or sections for the perfect intelligibility of the construction
and action of various pieces of mechanism. With reference also
to the examples collected together in this plate, a little considera^
tion will show that the internal formation could not generally be
sufficiently indicated by dotted lines ; for, besides the complication
and confusion that would result from such a method, many such
lines would confound themselves with full ones representing some
external outline.
We have not thought it necessary to enter more into detail
respecting the methods of constructing the various outlines, being
persuaded that the dotted indications we have given will be quite
sufficient for the student who has advanced thus far, the more so
since the requisite operations bear great resemblance to those
treated of in reference to Plate III.
SIMPLE APPLICATIONS.
SPINDLES, SHAFTS, COUPLINGS, WOODEN PATTERNS.
PLATE XII.
144. For the conveyance of mechanical action, under the form
of rotatory, or partial rotatory motion, details, technically kno%vn
as shafts or spindles, of ^vrought and cast-iron and wood, are used.
Shafts of the latter description, namely, cast-iron and wood, are
employed chiefly in hydraulic njotors, water and wind mills, and
in all machines where a great strain has to be transmitted, render-
ing considerable bulk necessary. Of these two kinds, wooden
shafts, being more economical, have been preferred in some cases,
particularly when the length is great, since they will better sustain
severe shocks. Wrought-iron shafts are employed for the trans-
mission of motion in factories and workshops, and for the main
paddle-shafts of steam-vessels. Wrought-iron has the advantage
of being less brittle than cast, and of possessing gi-eater tenacity
and elasticity.
145. Wooden Shaft. — Figs. 1, 4, 5, and 6, represent f^ifFerent
BOOK OF INDUSTRIAL DESiGN.
39
projections of a woodeu shaft, such as is used for a water-wheel.
Fig. 4»^hows, on one side, a lateral elevation of the shaft, furnished
witii its iron ferules or collars, and its spindle ; and at the same
tim(!, on the other side, a vertical section, passing through the
centre of the shaft, giving the ferules in section, but supposing
the central spindle, with its feathers, to be in external elevation,
(xenerally, in longitudinal sections of objects enclosing one or more
pieces, the innermost or central piece should not be sectioned,
unless it has some internal peculiarity — ^the object of a section be-
ing to show and explain such peculiarity where it exists, and being
quite unnecessary where the object is simply solid. In the same
manner, it is not worth while sectioning the various minutias of
machinery, such as bolts and nuts, simple cylindrical shafts and
rods and screws, unless these are constructed with some intrinsic
peculiarity.
Fig. 5 is a transverse section through the middle of the shaft,
and merely shows that it is solid, and that it has the external con-
tour of a regular octagon. Fig. 6 is an end view of the same
shaft, showing the fitting of the spindle, with its feathers, into the
socket and grooves, formed in the end of the shaft to receive
them, and the binding of the whole together by the ferules or
hoops. These views are what are required to determine all the
vaiious parts of the shaft. It manifestly consists, in fact, of a
long prismatic beam of oak, a, of an octagonal section, and of
which the extremities, b, are rounded, and .slightly conical. The
spindles, b, which are let into the ends, are each cast with four
feathers, c, and a long tail-piece, d, uniting and strengthening
them. Some en^neers construct the spindles without the addi-
tional tail-piece, d. Though this simplifies the thing considerably,
it is an arrangement which does not possess so much strength as
when the spindle is longer. The beam-ends are turned out and
grooved to receive these spindles, the grooves for the feathers
being made rather wider than the feathers themselves. When
the spindles are introduced into the sockets, b, thus formed for
them, the whole are bound together by means of the iron hoops,
f, which are forced on whilst hot. After this, hard wooden
wedges are jammed in on each side of the feathers, thus tightening
and solidifying the whole mass. In addition to this, iron spikes,
g, are sometimes hammered in, to jam up the fibres of the wood
still closer. Fig. 1, which is a shaded and finished elevation of
one end of the shaft, givfts an accurate idea of its ai)pc'arance when
complete and ready for adjustment.
146. Cast-iron Shaft. — There are several descriptions of cast-
iron shafts. Some are cast hollow, otiicrs (juito solid, and cylin-
drical or prismatical in cross section. Such as ai'c intended to
sustain very great strains, arc generally strengthened by the addi-
tion of feathers, which project more towards the middle. These
pve groat rigidity to the piece. A shaft of this description is
represented in elevation in fig. 7, half being sectioned tiirougli the
irregular lino, 1 — 2 — 3 — 4, and lialf in external elevation. Fig. 8
is an end-view of it ; and fig. 9 a transverse section through the
lino, !) — G, in fig. 7. In pr;icticc, it is not considered al)s(iliit('ly
necessary that a section should follow a straight line. Ficipiciitly
a much greater amount of c.xiil.in.ilion inny bo given in one view,
by supposing the obj(!ct sciclioncMJ liy portions of plnncs at diU'cr-
ent jyarts, and solid and easily (^ornpreluindcd portions are geneiiilly
shown in elevation, as the feathers of the shaft, in the present in-
stance, or the spokes of a spur-wheel or pulley. The shaft under
consideration is such a one as is employed for hydraulic motors.
The body, a, is cylindrical and hollow, and it is cast with four
feathers, b, disposed at right angles to each other, and of an ex-
ternal parabolic outline, so as to present an equal resistance to
torsion and flexure throughout. Near the extremities of these
feathers, four projections are cast, for the attachment of the bosses
of the water-wheel. These projections are formed with facets,
so as to form the corners of a circumscribing square, as shown in
fig. 8 ; and they are planed to receive the keys, i, by which they
are fixed and adjusted to the bosses or naves, which are grooved
at the proper places to receive them. The spindles, d, which
terminate the shaft at each end, are cast with it, and are afterwards
finished by turning. The shaft thus consists of only one piece, or
casting.
147. Although we have already shown the method of dra%\nng a
parabola, in Plate V., the outline of the shaft feathers affords a
practical exemplification, which it will be useful to illustrate. We
here also give the method generally adopted — ^because of its sim-
plicity — when the curve is a very slow or obtuse one, such as is
given to the feathers of shafts, beams, side-levers, connecting-rods,
and similar pieces. It is understood, in these cases, that two points
in the curve are given ; the one, a, fig. 7, being at tlie summit,
and at the same time in the middle of the piece, and the other, />,
situated at the extremity. In the present instance, we suppose
the heights, a c and b d, from the axial line, m n, of the shaft to be
given. This line, m n, may also be taken as the centre line of a
beam, or connecting-rod. After having di-awn through tlie point,
b, a line, e b, parallel to the axis, divide the perpendicular, a e, into
any number of equal parts, and transfer these divisions to the line,
b i, the prolongation of the line, db; then draw lines from the
points, 1, 2, 3, to the summit, a. Further, divide also the length,
c d, into the same number of equal parts as the perpentlicular,
and, through the divisions, 1', 2,' 3', di-aw other perpendiculars,
the respective points, /,«-, h, of intersection of these with the lines
already drawn, will be points in tlie required curve. As the
lower feather is an exact counterpart of tlio upper one, the perpen-
diculars may be prolonged downwards, and corresponding distances,
as 1'^', 2' — g', 3 — li, set olTon them. To draw, also, the half
of each feather to the left, it is merely necessary to erect perpen-
diculars of corresponding lengths, at corresponding points in the
axis. A different method of dramng this curve is soiiiotiiiios
adopted ; namely, the one which wo have already given in Plato
IX., for the entasis of the Tuscan column. As, iiowever, it does
not possess the advantages of the true parabolic fonu, and as the
curve becomes too sudden towards the extremities, wo think the
method given in Plato XII. is to be preferred. .
Fig. 3 represents a portion of the shiitY just discussed, shaded
and finished, the lines running in ditlerent directions, the bettor to
distinguisii the tl.il iVoni llu' round surfaces.
1 IS. Shall (^tuiiliiiii. — 111 extensive factories, and other works,
will re consi(liM!ible lengths of .sliafling lire necessary, they havo
lo lie conslniiii'd in sr\iT:il pii'ces, and coupli'tl tt>gether. Theso
couplings are genenilly ot' ciist-iron, and toniied of one or inoro
pieces, according to their size. One form consists of a spivios ^>f
4U
THE PRACTICAL DRAUGHTSMAN'S
cylindiiciil socket, aceuralely turned internally, which receives
the ends of the two shafts to be connected, these being scarfed or
halved into each other, so as to be bound well together, and re-
volve like one continuous piece. According to another form, two
sockets are employed, of increased diameter at the part where they
meet, and formed at this part into quadrant-shaped clutches,
gearing witli each other. The coupling represented in side eleva-
tion, in fig. 10, is of this Ivind ; and in front elevation, as separated,
in fig. 11.
This coupling was designed for a shaft, of which the diameter
at the collars was 28 centimetres. The socket, a, of the coupling,
is adjusted on the end of the first part of the shaft, c. The other
socket, a', is similarly adjusted on the end of the other part, c',
of the shaft. These two socket-pieces gear with each other,
when brought together, by means of the projections or clutches,
B and b', concentric portions, however, being adapted to fiit the
one into the other, to insure the coincidence of their centres.
Fig. 11, which is a front view of the socket- piece, a, shows the
exact shape and dimensions of these projecting clutches, each
occupying a quadi-ant of the circle on the face of the socket-piece.
Those of the second piece, a', are precisely the same, occupying,
however, the intervals of those on a, so that the two may fit
exactly into each other, as shown in fig. 10. The perfect and
accurate union of these coupling-pieces with the two portions of
shafting is obtained, in the first place, by means of two keys, a,
diametrically opposite to each other, and let half into the shaft,
and half into the socket-piece ; and secondly, by screws, h, one of
which is visible in fig. 10. The keys, a, are for the pui-pose of
fixing the coupling-pieces to the two portions of shafting, making
them solid therewith ; and the screws, b, prevent the longitudinal
separation of the two halves of the coupling.
THE METHOD OF CONSTRUCTING A WOODEN MODEL OF. PATTERN
OF A COUPLING.
149. After the design for any piece of mechanism which it is
proposed to cast has been decided on, it is generally necessary to
construct a model or pattern in wood, by which to form the
moulds for the casting. The proper formation of such a pattern is
no easy matter, and requires considerable skill on the part of the
pattern-maker, as also a knowledge of the kind of wood most ap-
propriate, and of the various precautions needed to insure success
when the mould comes to be prepared.
It is customary to construct the patterns of deal, because of its
cheapness. Sometimes, however, plane-tree or sycamore or oak
is used ; and for small patterns, and such as require great precision,
mahogany, box, or walnut. Whatever kind of wood is used, it
should be perfectly dry, and well seasoned. The pattern is made
solid, or holj£w and built up, according to the dimensions of the
object. For a drum, for instance, a column of any considerable
width, a steam-engine cylinder, or for a coupling of large size,
such as that represented in figs. 12, 13, and 14, the pattern is
generally hollow, for economizing the wood, and reducing the
■veight of the piece. Also, if built up, there is less risk of warp-
ing or alteration of form, from changes of temperature. In fig.
13, the pattern of a coupling-piece is represented, pai-tly in exter-
nal side elevation, and partly in longitudinal section, being cut by
a vertical plane, passing thi-ough the axis. Fig. 12 is a frout
elevation, showing the projecting clutches. It is easy to see from
these views, that the pattern is formed of two boards, d d', round
the circumferences of which are fitted a series of staves, e, secured
to the boards by screws. The wood for these staves is first cut
up into pieces of the requu-ed thickness, and the sides are then
bevilled off, to coincide with the radii, c d, c e, fig. 14. They are
then fitted to the boards, d d', and at this stage present the ap-
pearance of the left-hand portion of fig. 14. The di-um is after-
wards put into the lathe, and the cu'cumference is reduced to a
cylindrical surface, Uke the right-hand portion of the same figure.
On one of the ends, d, of this drum, is fixed the projecting
clutch-piece, b, which has been previously cut out of a board of
greater thickness, so as to present the outline of fig. 12. On the
opposite end, d', of the drum, are fixed several discs, or thicknesses,
of wood, F, which are turned down to a diameter proportionate to
the central socket which it is intended to form in the coupling-
piece. After having been turned where necessary, the pattern is
treated with sand-paper to make the surface as smooth as possible,
and to prevent the adherence of the loam of which the mould is
formed. Such patterns, particularly when of small size, are more-
over coated wdth black-lead, well rubbed in, to ^ve a polish
and hardness to the surface. The diameter of the core-piece, f,
is less than that of the shaft to which the coupling is to be fitted,
so as to leave some margin in the casting for turning and giinding
dovm the socket to the exact dimensions. The core itself, which
gives form to the socket, is a cylinder of loam placed in the centre
of the mould, fitted into the recess formed for that purpose by the
piece, F. As in the present example, the mould would be con-
structed on end, and the core is very short, it would not require
further support ; but where a core is very long, or placed Ln a
horizontal position, it requires to he supported at both ends, and,
further, to be strengthened by wires or rods passing through its
centre. For this reason, a core-piece, as f, is only attached to
one end of the drum. It wiU be observed that this is slightly
conical ; the drum is so also, but to a less extent ; the core itself,
however, is quite cylindrical.
150. Draw, or Taper, and Shrink, or Allowance for Con-
traction. — In order that the pattern may be lifted from the mould
without bringing away portions of the sides, it is necessary to form
its sides with a slight taper, or draw, a.i it is technically called.
For example, the diameter of the core-piece, f, as well as that of
the drum itself, must be less at the lower extremity, or at the part
first introduced into the mould, than at the opposite extremity.
A very slight difference of diameter is sufBcient for the purpose.
Cast-iron, as is the case with all the metals of the engineer, is of
less bulk when cold than when in a state of fusion, and, because of
this contraction, it is necessary to make the patterns of somewhat
larger dimensions than the casting is to be when finished. It
follows, then, that when the pieces to be cast have afterwards to
be planed, turned, gi-ound, or grooved, it is necessary to bear in
mind, in constructing the wooden pattern, not only tke after re-
duction due to the contraction, or, as it is termed, the shrink, of
the metal, but also that whicb is occasioned by the reducing pro-
cesses involved in finishing the article. In general, grey iron
requires an allowance for shrink of from 1 to 1 jV per cent. ; whiU
BOOK OF INDUSTRIAL DESIGN.
41
iron, however, requires a much larger allowance. The allowance
to oe made for the reduction caused by the finishing processes,
depends entirely on their nature.
When, with a view to avoid the expense of constructing a
pattern, the mould is formed from the actual object which is to bo
reproduced or multiplied, the mould-makers obtain the necessary
margin by shifting the model slightly during the formation of the
mould. This, of course, can only bo done with advantage when
the piece is not of intricate shape.
ELEMENTARY APPLICATIONS
RAILS AND CHAIES FOR RAILWAYS.
PLATE XIII.
151. In railways, the two iron rails on which the trains ran are
placed at the distance apart, or gauge, of li metres, and are
generally formed of lengths of 4| to 5 metres. In England, the
gauge is generally 4 feet 8i inches, and the rails are rolled in
lengths of from 12 to 15 feet. These rails are supported by
cast-iron chairs, placed at from 9 to 10 decimetres asunder, and
adjusted and bolted on oak sleepers, lying across the rails, imbed-
ded even with the surface. Those chairs which occur at the
junctions of the lengths of rails are made wider at the base, and
of greater length, so as to embrace the end of each length of rail,
and render their rectilinear adjustment and union as perfect as
possible.
In Plate XIII. we give details of a very common form of rail and
chair. There are many different forms in use ; but the method of
drawing or designing each will be similar, and may be thoroughly
understood from the exemplification here given. Figs. 1 and 2
represent the elevation and plan of a chair, with a portion of
the rail which it supports. Fig. 3 is a vertical section through
the line, 1 — 2. in the plan ; but supposing the chair to be turned
round, or to belong to the right-hand rail — showing, in connexion
with fig. 1, the relative positions of the two lines of rails, with
their respective chairs. Fig. 4 is a side view of the chair alone, and
fig. 5 is an end view of a length of rail. This chair, which is de-
signed wdth the view of combining solidity and strength with
economy of material, consists of a wide base, a, by which it is
seated on the sleeper, and of two lateral jaws, b, b', strengthened
by double feathers, c, c'. The base, b, is perforated at a — the
holes being cylindrical, and slightly rounded at their upper edges.
These holes are for the reception of the bolts which secure tho
chair to tho sleeper. Tho space between tho jaws of tho ciiair is
for tho reception of tho rail, d, and the wooden wedge, e, whii'ii
holds it in position.
In this oxiimplo, (ho vertical section of the rail, d, presents an
outline which is synimetiical with rclereiico both to tho vortical
centre lino, & c, and also to the liori/oiil.'il lino, (/c, (ig. 6. This
permits of tho niil being turncid when one of the running surliu'cs
■s worn. Tho section of the wedge, k, is also syninictrieal with
reference to Its diagonals, so that it is immaterial which way it
18 introduced, whilst it also fits equally well to tho rail wiien the
latter is reversed.
The outline of the rail is composed of straight lines and arcs,
which are geometrically and evenly joined, as shown in fig. 5.
The necessary operations are fully indicated on the drawing itself.
These operations are, for the most part, but the repetition and
combination of the problems treated of in the first division of the
subject. We have, moreover, given some of the problems de
tached, and on a larger scale, in figs. 6, 7, 8. Fig. 6 recalls the
problem (35), which has for its object the dra\ving of an arc, ij k.
tangent to the straight lines, fg and gh^ the radius, o/r, being
given, equal to 31-5"'/„. (fig. 2, Plate III.) This problem meets
with an application at fg h, fig. 5. In fig. 7 we have the problem
(37), which requires that an arc, Imn, be dra\vn tangent to a
sti-aight line, n p, and to a given arc, qr I, the point of contact, n,
being known (fig. 6, Plate III.) This meets with its application at
Imn, fig. 5.
The problem illustrated by fig. 8 is, to draw a tangent, g^/',
to two given circles of radii, s t and o' k^, respectively. In
this problem (9), we require to find a common point, u, on the
line, OS, which joins the centres of the two circles. To effect
tliis, we di-aw through the centres, o and s, any two diameters, v x
and ■»' x', parallel to each other. Join two opposite extremities of
each, as v and x', by the straight line, v x', which will cut the line,
o' s, in the point, u. The problem then reduces itself to the draw-
ing of a tangent to any single circumference (fig. 4, Plate I.), from
a given point, u. The tangents obtained, in the present instance,
will lie in one straight line, and be the line required — tangent to
both circles. The application of this problem is at xk^t in
fig. 4.
On fig. 9 we have also indicated the solution of a problem (41),
which is — to draw an arc, y z, of a given radius, a' b', tangent to
two other ai-cs, having the radii, c' d' and e'/" (fig. 8, Plate III.)
This problem is called for in drawing the outline of tiie jaw of tho
chair, where it runs into the base, a, near the edge of the bolt-
hole, a, fig. 3.
To complete the outline of the chiiir, it remains for us to show
how to determine the lines, g' h', wliich represent the intersections
of portions of cylindrical surfaces, as will be gathered from tigs. 1
to 4. To avoid a confusion of lines, wo have reproduced this
portion in figs. 10, 11, and 12, wiiich represent — tlie two former,
vertical sections of each cylindrical portion, and tlio latter, the lino
of intersection in plan.
We must first determine on fig. 12, which corresponds to lig. 2,
the horizontal projection, i', of any point, as i, taken on tho arc,
g' h', in fig. 10; letting fiill from this point, on tho b:iso line, L T, a
perpendicular, i i', and also drawing IVom it a iiorizontal line, i i^.
This latter line meets tho cylindrical outline, g' h', fig. 11, in i*.
Project t' in i' on the biiso line, triuisferring it to ii lino at riglit
angles to tlie hasu line, by means of a quadrant of a oia-lo, and
(Ir.iw through the point thus obtainoil a line pjirallel to the b;iso
line, and meeting the line 1 1' in i', which will he a point in tlio
curve required ; other points, as ;', ii', are I'onml in a simihir
manner.
It. must be observed, lli;it when iJie two i'\ lindriciil portions !UX>
of ciinal di:nneter.s, their intersection with each other, j;' A', ns will
lie (l.inonslr.iteil hereatler, is projwied hori/.outjUly as a stniight
lini ; the e,,;iler Ijie ilitVerenoo botwoon iJio two cylinders, the more
42
THE PRACTICAL DRAUGHTSMAN'S
cun'ed \vill tlie line of their intersection be, as is apparent in figs.
10, 11, 12.
The outlines of the feathers, c and c', glide into that of the
base. A, with a curve which, in the plan, is projected in the arc,
k' v. The operation necessary to determine these curves, is quite
analogous to that treated of in reference to the preceding figures,
and will be found sufficiently explained in figs. 13, 14, and 15. We
should here remark, that we have given explanatory diagrams of
all the sweeps or combinations of curves, both that the student
may be well exercised in many of the problems already discussed,
and also vrith a view of collecting, in one plate, several of the diflfi-
culties which more frequently meet the draughtsman in the course
of his practice. In such objects as that chosen for exemplification,
very little of the nicety here carried out is observ^ed, and the
curves are generally obtained by measurements with callipers from
the object itself, or are formed of arcs determined by the eye.
The rails are not adjusted in their chairs perpendicularly, but
are inclined sUghtly towards each other, in such a manner that
their centre lines, c 5, form a slight angle with the vertical, c IP: this
inclination is given to counteract any tendency that the carriages
may have to run off the rails, as is the case more particularly in
curves, from the effort made by the wheels to run in a straight
line. The expedient of laying the outer or convex rail at a level
slightly higher than the other, is also resorted to in quick curves,
for the like purpose of keeping the ti-ains on the line.
RULES AND PRACTICAL DATA.
STRENGTH OF MATERIALS.
152. The various materials employed in mechanical and other
constructions, differ considerably in their several natures, both vtdth
reference to the amoimt of force they ■nlll bear or resist uninjm-ed,
and the description of force or mode of applying it, to which they
offer the greatest resistance.
Such forces are termed, according to the mode in which they
are applied — tension, compression, flexure, and torsion.
A series of practical rules have been deduced from often re-
peated experiments, which serve as guides for readily calculating
the dimensions of any piece of mechanism, with reference to the
description and degree of force to which it will be subjected.
RESISTANCE TO COMPRESSION OR CRUSHING FORCE.
153. Compression is a force which strives to crush, or render
more dense, the fibres or molecules of any substance which is sub-
mitted to its action.
According to Rondelet's experiments, a prism of oak, of such
dimensions that its length or height is not greater than seven times
the least dimension of its transverse section, will be crushed by a
weight of from 385 to 462 kilogrammes, to the square centimetre
of transverse section, or a weight of from 5,470 to 6,547 per square
inch of transverse section.
In general, with oak or cast-iron, flexure begins to take place in
a piece submitted to a crushing force, as soon as the length or
height reaches ten times the least dimension of the transverse
section. Up to this pomt, the resistance to compression is pretty
regular.
Wrought-iron begins to be compressed under a weight of 4,90f)
kOog. per square centimetre, or of nearly 70,000 per square uich,
and bends previously to crushing, as soon as the length or height
of the piece exceeds three times the least dimension of the trans-
verse section.
We show, in the following table, to what extent per square inch
we may safely load bodies of various substances.
Table of the Weights which Solids — such as Columns, Pilasters,
Supports — will sustain without being crushed.
WOODS AND METALS.
Proportion of 1
ength to least dimension.
Description
of
Material.
Up to 12.
Above 12.
Above 24.
Above 48.
Above 60.
lb.
lb.
lb.
lb.
lb.
Sound oak, ....
426-'750
355-625
213-375
71-125
35-562
Inferior oak, . . .
210-215
119-490
7M25
"
"
Pitch pine, ....
533-437
440-975
266-007
106-687
"
Common pine, .
137-982
116-645
69-702
"
«
"Wrought-iron, .
1422O-000
11877-875
7 112-501.
2375-575
1194-900
Cast-iron
28450-000
23755750
14225-000
4741-666
2375-575
Rolled copper, .
11707-175
'■
*'
"
"
STONES, BEICKS, AND MOKTAKS.
Description of Material
Basaltic Marble, Swedish and Anvergnese,
Granite from Normandy,
" green, from Vosges,
" grey, from Bretagne,
" " from Vosges,
" ordinary,
Marble, hard,
" white and veined,
Freestone, hard,
" soft,
Stone from Chatillon, near Paris,
Very hard freestone, or lias, from Bagneux, near Paris,
A softer stone, from the same place,
Stone from Areueil, near Paris,
" ' from Saillancourt, near Pontoise, best quality, .
" from Conflans, much used at Paris
Hard calcareous stone,
Ordinary calcareous stone,
Calcareous stone from Givry, near Paris,
" " ordinary, from the same place,
An inferior stone, termed Lambourde,
Bricks, very hard,
" inferior,
" hard and well-baked,
" red,
A soft stone, Lambovrde vergetee,
Plaster, mixed with water,
" mixed with lime-water,
Mortar, best, eighteen months old,
" ordinary, eighteen months old,
" " of lime and sand,
Cement,
" Eoman, or Neapolitan
Lenfftli being
less than
12 times least
dimension.
lb.
2845
996
882
925
697
569
1422
427
1280
6
242
726
185
355
199
128
711
427
441
171
33
171
57
213
85
85
71
104
57
36
50
68
53
Rule. — To find, by means of this table, the greatest compress-
ing weight to which any piece may be submitted ^\ith safety : —
Multiply the transverse sectional area of the piece by the number ta
BOOK OF INDUSTRIAL DESIGN.
43
the table, corresponding to the material, and to the proportionate
'ength of the piece. And inversely, from the weight which a piece
is to support, its smallest transverse section may be determined.
By dividing this weight, expressed in pounds, by the number in tlie
table corresponding to the material, and to the proportionate length.
First Example. — What weight can be pat vrtth safety upon a
pillar constructed of ordinary bricks, the pUlar being of a rectan-
gular section, of 50 inches by 60, and the height being below 12
times the length of this cross section ?
We have 50 x 60 = 3000 square inches of transverse sectional
irea. Then, according to the table, we have —
3000 X 67 = 171,000 lbs.
Second Example. — What must be the transverse sectional area
.)f a square post of sound oak, 19 feet 8 inches in height, and
rt'hich will safely bear a load of 60,000 lbs. ?
According to the table, if we suppose the length to be not more
than 12 times the least cross section, the number or coefficient of
compression, in pounds per square inch, is 426-75.
Then,
and
60,000
426-75
^140 square inches;
■•^140 = 11-8 inches, the length of the supposed side.
Comparing tins 11-8 inches with the given height, we find that
19 ft 8 in.
11-8 in.
236
11-8
20.
Tliis shows — and we have constructed the example with this
riow — that in this mstance the proportionate length has not been
sorrectly estimated ; and therefore, instead of taking the number
426"75, as in the first column, we must take that in the second
column, for a proportionate length of between 12 and 24 times the
cross section. The calculation wOl, consequently, have to be
rectified thus —
60,000 „„ . .
155^625 = ^^^'^ '^"^'■^ "'*'''
and VlSS-l = 13 inches nearly, the proper dimension for the
cross section of the post.
Third Example. — Wliat is the greatest load that can be borne
with safety by a solid cast-iron column, 3 inches in diameter, and
12 feet in height?
It is, in the first pl.ace. evident that the ratio of the diameter to
the height is 12 feet, or 144 inches, -f- 3 inches = 48.
Consequently,
the section -785 x 3^ x 4741-666 = 33,.500 lbs.
In shops and warehouses, builders employ solid caaf-iron
columns, instead of brick pillars, so as to take up less .space. These
columns arc generally calculated to support loads of above
33,000 lbs. each. They are usually about 3 inches in diameter,
and 12 feet high. In which case, supposing a cubic foot of cast-
iron weighs 452 lbs., they will weigh (3 inches being equal to -25
foot)—
•785 X •26=» X 12 X 452 = 266 lbs.
If, instead of these columns being massive or solid, wo employ,
m place of two of them, a hollow one, to support tho proportional o
load of 66,000 lbs., and lu'lng 6 iiiclics in (iiainctcr, this incrc;is(i
in the diameter makes the ratio of the length to it 24, instead of
48 ; and the coefficient to be taken from the table will conse-
quently be 14,225, mstead of 4742.
Now, 66,000 -^ 14,225 = 4-64 square inches, would be the
cross section of a soUd pOlar, equivalent to that of which tho
thickness is sought. Since, however, the diameter of the latter is
6 inches, its section of solidity would be —
•785 X 62 = 28-26 square inches.
Then, deducting from this area 4-64 square inches, as above
determined, we have 28-26 — 4-64 = 23^62, for the cross sectional
area of the central hollow. From this we deduce the Latemal
diameter, thus —
^23-62
/23-62
V "^785
= 5-485 inches.
And, finally, the thickness of the column will be —
6 — 5-485
2
= -2575 inches.
The weight of such a column, if 12 feet in height, will be —
4-64
-^^ X 12 X 452 = 174-77 lbs.
144
This result shows very markedly how great an economy results
from the employment of hollow in place of solid cast-iron columns.
The thickness, determined as above, of -2575 inch, is theoretically
sufficient, but in practice we seldom find such castings under half
an inch thick.
In the above examples, too, the mouldmgs usually added to tlie
columns are not taken into the account. With these, the weight
will be a tenth or so more, according to the description of moulding.
TENSIONAL RESISTANCE.
155. A tensQe force is one which acts on a bod}' in the direc-
tion of its length, tending to increase the length, and when carried
to a sufficient extent, to cause rupture.
As vidth reference to compression, many experiments have been
made to determine the sectional area to be given to bodies of
various materials submitted to a tensile strahi, so that they may
safely resist a given force.
First Example. — Required the sectional area for four square
tension rods of wrought-iron, to connect the top and bottom of a
hydraulic press, in winch tho force which teuds to separate tliese
two ends, and consequently to rupture tlie rods, is equal to 500,000
lbs.
Each rod must be capable of resisting
500,000 ,, „ „
— = 125,000 lbs.
According to the fjible, the best wrought-iron may be sjifoly
subjected to a strain of 14,225 lbs. per squiu-o inch of cross section.
We have, consequently,
125,000 „„ . ,
— — = 8-79 square inches.
14,225
for the area of tim cross section ; and
V 8-79 = 2-964, or iicirly 3 iiiohos,
for a side of tho square rod. If tho rod were round, wo sliould havo— •
/8-79
3-346 inches, lor ilio diwuotor.
44
THE PRACTICAL DRAUGHTSMAN'S
In tlie same manner, the diameter proper for steam-engine
piston-rods may be calculated, when the pressure on the piston is
known.
Table of Weights which Prisms and Cylinders will sustain when
submitted to a Tensile Strain.
Description of Material.
Oak,
Deal,
Woods.
.,, ,, . ( sound, . .
with the eram, ■< -,■
^ ( ordmarj',
across the grain,
■with the grain, .
across the grain.
Ash, with the grain, . . .
Elm,
Beech, " . . .
Per square
inch of
cross section:
Metals.
Wrought or ( PP^rior and select samples,
bar iron 1 'if6rior,indiscriminatelyselected
' ( medium.
Sheet J in the direction in which it was rolled,
iron, ( in the direction perpendicular to thi;
Hoop iron, soft
' Be Laigle, -009 inch, or -23 ■"/
in diameter,
inferior, and of considerable
diameter
best, from -02 to -04 inch, or -o
to 1 "/„ in diameter, . .
medium quality, -04 to -12 inch,
_ or 1 to 3 '°/n, in diameter
Ii'on. wire rope,
T„ „„ui„, ( ordinary, oblong links, . . .
Iron cables, -^ , ,,•'' j > ° . '
( strengthened by stays.
Unannealed
iron wire.
Grey cast iron, | ™n yert,ically, strongest kind.
r
un horizontally, inferior,
cast or wrought, selected,
Steel i ^'^^^^}°\' hadly tempered, taken indis-
'I criminately,
1^ medium,
Gun metal, average,
r rolled lengthwise
Copper, \ f superior quality,
^^ ' I hammered,
Unannealed
copper wire,
l_ cast,
'superior, under -04 inch, or 1"/
in diameter,
medium, from -04 to -08 inch, or
1 to 2 ""/„ in diameter,
inferior,
Yellow copper, or fine brass
Unannealed ( superior, under -04 inch, or 1-/,
brass wire, j ^^^ diameter,
' ( medium,
Platinum ( hardened unannealed, -0045 inch,
wire 1 °'' ^"^ ^^ diameter,
' ( hardened, annealed
Cast tin
Cast zinc
Sheet zinc,
Cast lead,
Sheet lead,
Cordage.
Hawsers and cables of Strasburg hemp, "5 to -6
inch, or 13 to 14 ""/o, in diameter,
Do. of Lorraine hemp
Do. of Lorraine or Strasburg hemp, -9 inch, or
23 "/„ in diameter,
Do. of Strasburg hemp, 1-5 to 2 inch, or 40 to
54 ■"/„ in diameter,
Old rope, -9 inch, or 23 "/„ in diameter, . .
Black leather bands,
lbs.
1,138
853
228
1,138 to;
1,280 \
60
1,707
1,479
1,138
14,225
5,917
9,474
9,957
8,535
1,069
21,337
11,850
18,996
Per square
centimetreof
cross section.
kilog.
80
60
16
80 to 90
4-2
120
104
80
1,000
416
666
700
600
750
1,500
833
1,333
14,225
1,000
7,112
500
5,690
400
7,587
533
3,201
225
3,087
217
23,702
1,667
8,535
600
17,781
1,250
5,448
383
4,979
350
6,117
433
5,932
417
3,314
233
16,600
1,167
11,850
833
9,488
667
2,987
210
20,143
1,416
11,850
833
27,511
1,933
8,066
567
711
50
1,422
100
1,185
83-3
303
21-3
320
22-5
6,259
440
4,623
325
4,267
300
3,912
275
2,987
210
284
20
Second Example. — Required the amount of tensile or tractive
force which can safely be resisted by a carriage draw-shaft, made
of ash, and having a cross-sectional area of 15'5 square inches.
According to the table, we have —
1707 lbs. X 15-5 = 26-460 lbs.
155. Pulley Bands. — The following simple formula may gene
rally be employed in practice, to determine the dimensions proper
for pulley bands : —
20 X H P .
V
in which L is the width of the band in inches ; H P, the force in
horse power ; and v, the velocity in feet per minute.
By horse -power is meant a force equal to 33,000 lbs., raised
1 foot high in a minute. French engineers call it 75 kilogrammes,
raised a meti-e high per second, which is very nearly the same as
the English measure. To suit the French system, the above for
mula would be —
1500 X H P
Li — ,
V
V, in this case, signifying the velocity in centimetres per second,
and L, the width in centimetres. The thickness of the leather is
supposed to be that of strong ox-liide, say about -2 inch, or 5'°/n,.
The above formula gives rise to the following rule : — Multiply
the horse power by the constant multiplier, 20, and divide the pro-
duct by the velocity in feet per minute, and the quotient will be the
loidlh of the band in inches.
Example. — Let H P = 2 horse power, d = 10 feet per minute,
then,
20 X 2
L =
10
4 inches.
This formula satisfies the following conditions — that tne band
do not slide round the pulley which it embraces ; that it be not
liable to increase perceptibly in length ; and that it be capable of
resisting the strain transmitted by it.
It is found advisable never to make the respective diameters
of two pulleys, coupled together, in a greater ratio to each other
than 1 : 3.
RESISTANCE TO FLEXURE.
156. The resistance of a piece of any material to flexure, is the
effort which it opposes to all strain acting upon it in a direction
perpendicular to its length, as in the case of levers, beams, or
shafts.
Bodies may be submitted to the strain of flexure in several
ways. Thus, the piece may be tu-mly fixed in a wall by one end,
whilst the straining weight or force is applied at the other ; or it
may be securely fixed at both ends, and the weight apphed in the
centre ; or it may be supported at the centre, and have the weight
applied at both extremities.
We shall first consider the case of a piece fixed by one end,
and subjected to a strain at the other.
Let W be the weight in pounds, placed at a distance, L, in
inches from the wall, in which the piece under experiment is
fixed ; C, a coefficient varying vidth the material ; a, the horizontal
dimension in inches of cross section; b, the vertical dimension,
similarly expressed ; — then the greatest weight that the piece wiL
BOOK OF INDUSTRIAL DESIGN.
45
bear, without undergoing alteration, will be determiaabie by the
following formuia, Inv piece oeing of rectangular section, and fixed
at one end, and weighted at the other.
Now, C = 8,535, for wrought-iron ;
10,668, for cast-iron ;
854, for oak and deal.
Substituting these values of C, in the preceding formula, we
shall have, for pieces of rectangular section, according to the
material —
For Wrought- Iron,
u 8535 Xab^ . , 1422-5 Xab^
P = ^T^ , or more simply, = .
TX"
For Cast-Iron,
n 10,668 X a 6-, • ,
P — — '- — -~ -', or more simply,
6 L
1778 xab^
For Wood.
854 X a &2 142 X ffl b^
P = — -f •, or more simply, = ^ .
These formulae lead us to the follovwng rule for pieces of rect-
angular section: — Multiply the horizcmial dimension in inches of
cross section, by the square of the vertical dimension in inches, and
by a coefficient depending on the material : then divide the product
by the length in inches, and the quotient will be the wp.ight in pounds,
which the piece will sustain without alteration.
This rule is derived from the fact, that the transverse resistance
of pieces submitted to a deflective strain is inversely as their
length, directly as then* width, and as the square of their vertical
thickness.
According to this, pieces fixed at one end, and intended to bear
a strain at the other, should be placed on edge ; in other woras,
the greatest cross section should be parallel to the direction of the
strain.
First Example. — What weight can be suspended, without causing
deflection, to the free end of a wrought-iron bar, fixed horizontally
into a wall at one end, and projecting 5 feet {— 60 inches) from it;
the bar being of a rectangular cross section, having its horizontal
dimension, a = 1-2 inches, and its vertical dimension, & = 1-6
mclies 1
We have
1422-5 X 1-2 X 1-6=
P =
60
=: 72-8 lbs.
This result is obtained on the supposition that the bar is placed
on edge ; but what would bo the weight, other things being equal,
supposing the bar to bo placed on its side — that is, when 1-6 inches
is its horizontal dimension, a, and 1-2 its vertical diinciision, i?
We have, in this case,
1422-5 X 1-6 X 1-23
P =
60
64-6 lbs.
This inferior result shows the advantage of placing the bur on
edge.
When the pi(H-,o under oxperiiuont is of s(iiiui-o instead of oblong
section, a nocossariiy = b, and a //•* bccoinos //', and this is conse-
quently to bo substituted in the fonnual for Mio lornior.
If, however, the piece is cylindrical, the formula wOl be — D
representing the diameter,
854 X D3
For wrought-iron, P =
For cast-iron, P =
For wood, P =
L
1066 X D^
L
85 X D3
T ■
In each of the cases just referred to, the transverse sectional
dimensions of pieces fixed at one end, and submitted to a strain
at the other, are determined by the following formulas : —
Material.
Wrought-iron, . .
Cast-iron,
Wood,
FoEM OF Section.
Rectangular.
a J-
a b~
a Z)-
PL
1,422-5
PL
1,778
PL
142
Square.
PL
1,422-5
PL
1,778
PL
OiiLular.
142
PL
854
PI
1066
PL
85
The rule derivable from these formulae for the determination of
the transverse section, whether rectangular, square, or circular, of a
bar or beam fixed by one end and loaded at the other is thus stated : —
Multiply the weight in pounds by its distance in inches from the sup-
port ; divide the product by a coejjicient varying wilh the material
and form of section ; and extract the cube root, which mil give in
inches the vertical dimension, the side of the square, or the diameter
of tJie circle, according as the bar or beam is rectangular, square, or
circular in cross section.
First application : WHiat should be the transverse section of a
rectangular wrought-iron bar, intended to carry at its free end, and
at a distance of 5 feet from its support, a weight of 728 lbs., Ino
bar being supposed to be placed on edge ?
We have hero,
72-8 lbs. X 60 in.
1,422-5
then, if a bo taken — 1-2 inches.
=3-071;
/3-071
1-6 indu's, the vortical dimension.
Second application : ^Vhat should the side of the cross swliou
measure, of a square bar, under sunilar circumstances otJierwiso ?
b^ =
72-8 X 60
~Ti22-5
= 3-071, and
J = V3-071 = 1-454 in., the thickness of the bar.
157. Obsf.kvations. — W'Iumi the bar, or beam, under oxjH'n
nu-nt possesses in itsclt' any wiight cajwblo of inllui-iu-ing its
resistance ; or, besides the \\ I'i^^lit suspended or acting at one end,
lias a weight equally distributed throughout itslengili; the trans-
verse-sectional dinu iisimis are, in the lirst [ijace. deterniir.vHl \\iih.
46
THE PRACTICAL DRAUGHTSMAN'S
out taking the additional weight into consideration. This is,
then, calculated approximately, and the half of it added to the
suspended load, a fresh calculation being made with this sum as
a basis.
A bar, or beam, fixed by one end, and loaded at the other, has
always a tendency to break off at the shoulder, or point of junc-
tion, with its support, because it is on that point that the weight
or strain acts with the greatest leverage. When, therefore, the
transverse section of the piece has been determined, in accordance
witli the formulae given above, which are calculated for the
dimensions of the piece at the shoulder ; the section may be bene-
ficially diminished towards the free extremity, thereby economis-
ing the material, and lessening its own overhanging weight. The
curve proper to give the outline is the parabola, as described in
reference to Plates V. and XII. It may also be obtained in the
following manner, for the particular case imder consideration : —
Calculate the transverse section for different lesser lengths of the
piece, the other data remaining as before, and the required cm-ve
wUl be one which passes through the outline of each section, when
they are placed at distances from the load equal to the lengths for
which they are calculated. This curve is also given to bars,
beams, or shafts, fixed at both ends and loaded in their middle, or
sustaining a uniform weight throughout their length. The cast-
b'on shaft represented in Plate XII. may be taken as an example
of this. Steam-engine beams and side levers are also formed with
feathers of this shape, as it gives them a uniform resistance
throughout, so tliat they are not liable to break or give way ir
any one point rather than another.
A bar, or beam, supported in the centre, and loaded at either
end, win support double the weight capable of being canied by
one of similar dimensions, supported at one, and loaded at the
other end; it is, indeed, e\ident that each weight will only act
^\ith half the leverage, being only half the distance from the point
of support.
Similarly, a bar, or beam, freely supported at both extremities,
and loaded in the centre, will support a weight double that sus-
tained by a piece of the same dimensions ILxed at one, and loaded
at the other end. Therefore, in calculating the proportions for
these two last-mentioned cases, it is necessary simply to double
the coefficient, c, given for the first case.
A bar, or beam, firmly and solidly fixed by both ends, will sup-
port a load four times as great as one of the same dimensions
fLxed at one end, and loaded at the other extremity. It will, con-
sequently, be necessary to quadruple the above coefficient in this
case.
For calculating the diameters of the spindles, or journals, of
cast-ii-on shafts for hydraulic motors, which are intended to sus-
tain great weights, the following particular formula may be
employed : —
D= VWx -1938,
where D expresses the diameter in inches, and W the weight to
be sustained in pounds.
TABLE 01
THE DIAMETERS OF
THE JOUKNALS OF WATEE-AV^HEEL AITD OTHEE
SHAFTS FOR HEAVY
WORK.
Diameter of Journal in Inches.
Diameter of Jonmal in Inches.
Total load in
Total load in
Pounds.
Pounds.
Cast-Iron.
Wrought-Iron.
Cast-iron.
Wrought-Iron.
17*2
*
•4315
70343
8
6-9040
137-4
1
•8630
84373
H
7-3355
463-7
n
1-2945
100156
9
7-7670
1099-0
2
1-7260
117793
H
8-1985
2146-7
2i
2-1575
137388
10
8-6300
3709-5
3
2-5890
158604
m
9-0615
6890-5
H
3-0205
182864
11
9-4930
8805-6
4"
3-4520
208950
111
9-9245
12619-5
H
3-8835
237296
12
10-3560
17176-5
5
4-3150
268012
121
10-7875
22858-0
51
4-7465
311666
13'
11-2190
29676-0
6
5-1780
338026
m
11-6505
37730-0
6i
5-6095
376993
14
12-0820
43873-0
7
6-0410
418845
14i
12-5135
68915-7
7i
6-4725
463685
15
12-9450
According to this formula, the diameter of the cast-iron spindle,
or journal, is found by extracting the cube root of the weight, or
strain, in pounds, and multiplying it by the constant, -1938, the
product being the diameter in inches.
The diameter for wrought-iron spindles, or journals, may be
derived from that for cast-iron, by multiplying the latter by -863 ;
or, directly, by employing the multiplier, -1673, in the above for-
mula.
Example. — Of what diameter must the spindle of a water-
wheel shaft be, the total strain being equivalent to 70,000 lbs. 1
Here,
D= ^70,000 X -1938 =7-987, or 8 inches nearly.
A ^^^-ought-u-on spmdle of (7-987 X -863 =) 6-9 inches, will
answer the same purpose.
RESISTANCE TO TORSION.
158. When two forces act in opposite directions, and tangen-
tially to any solid, tending to turn its opposite ends in different
BOOK OF INDUSTRIAL DESIGN.
47
directions, or to twist it, it Ls said to be subjeoted to torsion, and
offers more or less resistance to tliis action according to its form
and composition. Taking, for example, the main shaft of a steam-
engiae, at one end of which the power acts through a crank, set
at right angles to it, and at the other the load, by means of wheel
gear — the resistance which this load presents, on the one hand,
and on the other, the power applied to the crank, represent two
forces which tend to twist the shaft, subjecting it to the action of
torsion.
In machinery, all shafts and spindles which communicate power
by a rotatory, or partially rotatory, movement on their axes, are
subject to a torsional strain. Those which sustain the greatest
torsional efforts are those shafts denominated first movers, the
first recipients of the power. Such are the fly-wheel shafts of
land engines, and the paddle-shafts of marine engines. In these
the action is fui'ther complicated and heightened by the irregularity
with which, in reciprocating engines, the power is communicated
to them. Such shafts as carry very heavy toothed gearing, but
receive and transmit the power in an equable manner, and without
a fly-wheel, are termed second movers ; and finally, such as carry
only pulleys, or comparatively small toothed wheels, are comprised
in the class of third movers. Such shafts, again, as meet v^th an
intermittent resistance, as is the case with all cam movements,
require increased strength to meet this irregularity of action.
In constructing formulae for the determination of the diameters
of shafts, regard must always be had to the class to which they
belong, and also to the description of work they have to perform.
As the journals are the parts of a shaft on which the greatest
strain is concentrated, it is obviously to the determination of their
dimensions that our investigation should be directed. The prac-
tical formula, for ascertaining the diameter proper for the journal
of a cast-iron first-mover shaft, is —
= ^-
HP
R
X 419.
Here, d = diameter of journal in inches.
H P = the horse power transmitted by the shaft.
R = the number of revolutions of the shaft per minute.
This formula is expressed in the following rule.
1.59. To determine the diameter at the journals of a cast-iron
first-mover shaft : — Divide the horse poiver of the engine by tlie
numoe.r of revolvXions of the shaft per minute, muUiply the quotient
by the constant, 419, and extract the cubic root, which will be llie
diameter required in incites.
For the journals of cast-iron shafts wliich are second movers,
the formula is —
VHP
V-R-
X206;
and for third movers —
s/Tj-p
ThcHO are, in fiu't, siiiiilar to the formula given for first tnovors,
willi tlio (^\•coption, that for tlicso tho (constant Mniltii>li('r is 11!),
whilst for tho latter it. is 'MHi and 10(i, rospcctivoiy.
160. For tlio journals of wrouj^'lit iroii sliafis tho saino foriniiho
are (employed, tho iiiu]li|)!i('rs only l)(Mng changod ; llicse are iJI9 for
first movers, 1.34 for secornl tnovcis, and ()7'li lor third movers.
If, v(ath a view of suppressing the radical sign in the above for-
mulae, we raise both sides of the equation to their third or cubic
power, and further express the multiplier by m, we have
J3 HP
from which formula it wtR be seen, that the cube of the diameter
of the jom-nal is proportional to the force transmitted. Similarly,
the resistance of a journal is proportional to the cube of its dia-
meter. In other words, one journal, of which the diameter is
double that of another, is capable of sustaining a strain eight times
greater, since the cube of 2 is 8.
161. As, in consequence of the necessity of extracting cubic
roots, the calciriation, according to these formulae, becomes very
tedious and complex, we have rendered it much simpler by
means of the table on the next page.
We may, however, first observe, that the formula.
d= =
HP
-^ Km,
may be
put
in the form-
d^
m
HP
~ R '
or again
, reversing the terms.
m
17^
R
~HP
If now we divide the coefficient, m, by the cubes of the series,
1, 2, 3, 4, &c., representing the diameters of the journals in inches,
we shall obtain a series of numbers corresponding to
R
HP"
Thus, if 419 be successively divided by the cubes, 1, 8, 27, 64,
&c., the numbers in the second column of the table will be
obtained; and by dealing with fho other multipliers in hke manner,
the numbers in the 3d, 4th, 5th, 6th, and 7th columns, will be
found.
Rule. — ^When the table is used, the rule for determining the
diameter of the journal of a shaft is thus stated : —
Divide tlie number of revolutions per minute of the shaft by the
liorse power, and find (lie number in the table which is nearest to the
quotient thus obtained, bearing in mind the class and 7naterial, ami
the corresponding number in the first column will be tlie diamcttr
required.
First Example. — \Vliat should be tho di.Tmctcr at (ho journals
of a ca.st-iron first^motion shaft, for an engine of 20 horse powei,
the shaft in question to make 31 rovolutions per minute?
We have —
R
HP
- i' - 1-55
- 20 ~ ^ ''''•
It will bo observed that this quotient is tho nearest to tiio
number 1-526 in the second colunm of Uio table, and that 1-526 is
opposite to 6^ ; tho diameter, d, of the shaft journal should ooii-
so(iU('nlly bo 6.'. inches in dianiotor.
If ;i sIkH'I lor tlio siuno purpose .is tho above bo niado of
w roui^ht-iiMn, we must look in the liClh cohinin for ll>e nunilnT to
wliiih 155 approaches nearest. It will l>i> observed that it lios
bolwi'on tho numbers 1-99'J and \W1, respeelively >>pp(>site to 5
and 5,'| inelios ; the iiiMin(>tor of tho journal should consoquciitly lio
lielween these — siiy about 5 J inches.
4S
THE PRACTICAL DRAUGHTSMAN'S
TABLE OF DIAMETERS FOR SHAFT JOURNALS, CALCULATED WITH REFERENCE TO TORSIONAL STRAIN.
Journals of Cast-iron Shafts.
Journals of Wrought-Iron Shafts.
Diameter
in
laches.
First Movers.
Second Movers.
Third Movers.
First Movers.
Second Movers.
Third Move: s.
i
3352-000
1568-000
848-000
1981-200
1072-000
540-800
1
419-000
206-000
106-000
249-000
134-000
67-600
H
124-133
61-037
31-408
73-778
39-704
20-030
2
52-375
25-750
13-250
31-125
16-750
8-450
2*
26-816
13-190
6-790
15-872
8-576
4-327
3
15-519
7-630
3-922
9-222
4-963
2-504
3i
9-773
4-805
2-475
5-808
3-123
1-577
4
6-547
3-219
1-656
3-891
2-094
1-563
44
4-598
2-266
1-163
2-732
1-475
•742
6
3-352
1-648
•848
1-992
1-072
•541
54
2-519
1-239
•637
1-497
•806
•406
6
1-940
•954
•491
1-153
•620
•313
6}
1-526
•750
•386
•906
•488
•246
7
1-222
•601
•309
-726
•391
•197
7i
1-002
•493
•253
•595
•325
•162
8
•838
•402
-207
•487
•261
•130
8i
•682
•335
•173
•405
•218
•110
9
•575
•282
•145
•341
•184
•093
9i
•489
•240
•124
•290
•156
•079
10
•419
•206
•106
•249
•134
•068
10^
•362
•178
•092
•215
•116
•058
11
•314
•155
•079
•187
•101
•051
111
•275
•135
•069
•163
•089
•044
12
•242
•119
•061
•144
•078
•039
12^
•214
•105
•054
•127
•068
•034
13
•191
•094
•049
•114
•061
•031
13i
•170
•084
-043
•101
•054
•027
14
•153
•075
•038
•091
•049
•024
144
•137
•067
•035
•082
•044
•022
15
•124
•061
•031
•074
•039
•020
1
2
3
4
6
6
7
Second Example. — ^We require to ascertain the diameter proper
for the journals of a shaft of the second class, intended to trans-
mit a force, equal to 15 horse power, at the rate of 40 revolutions
per minute.
Here,
A _ 40 _ 2-67
j^___267.
This quotient lies between the numbers 3-219 and 2-266 in the
third column, and between 3-123 and 2-094 m the sLxth. It follows,
then, that if the shaft is to be of casl^u-on, its journals must be
between 4 and 4^ mches in diameter ; or, if it is to be of wrought-
iron, between 3i and 4 inches, there being about half an inch of
difference between the two materials in this instance.
Third Example. — A shaft, intended for a third mover, is to
transmit a force equal to 6 horse power, at a velocity of 50 revo-
lutions per minute, what should be the diameter of its journals in
cast or wrought-iron 1
Here,
-5:- = ££ = 8-333.
HP 6
This number, in the third column, lies between 13-25 and 6-79;
therefore the diameter for cast-iron should be between 2 and 2J,
say, 2 1 inches. For wrought-iron the diameter should be 2
inches, as the number, 8-45, opposite to this in the seventh column,
almost coincides with the quotient above obtained.
The length of the journals and their bearings should always be
greater than their diameter. For large sizes, it should be l-2d
to l-Ad, and for smaller sizes, Vbd to 2d. Thus, the length of a
vn-ought-iron journal, 1-5 inches in diameter, should be from
(1-5 X 1-5 =) 2-25 to (1-6 x 2 =) 3 mches.
When shafts have to resist both a torsional and a lateral or
transverse strain, the diameter of their journals should be determined
with reference to that strain which is the greatest, or which of
itself would require the greatest dimensions.
When shafts are not of any great length — 3 to 6 feet, for
example — their diameter need not be above a tenth greater than
that of their journals. Solid cast-iron shafts of above six feet in
length should have a diameter one-fifth, or even one-fourth greatei
than that of their journals.
BOOK OF INDUSTRIAL DESIGN.
49
FRICTION OF SURFACES IN CONTACT.
162. Friction is the resistance which one surface offers to
another — movbg or sliding on it. Friction may be distinguished
^is sliding friction, and the friction of rotation. The former is that
which arises from the simple rubbing of one surface upon another ;
the latter, from the rotation of one surface upon another.
The friction caused by the rubbing of plane surfaces is inde-
pendent of the extent of surface or velocity of movement ; it
depends essentially on the weight of the body, or, more accurately,
the pressure binding the two surfaces together. It may therefore
be said, that the fiiction is in proportion to the pressure.
Similarly, the friction of a journal in its bearings is independent
of the length of these, but is proportional to the diameter and to
the pressure.
We give tables for each of those classes of friction, indicating
the ratio of the friction to the pressure, and consisting of a series
of coefficients, whereby the pressure must be multiplied in order
to ascertain the amount of resistance due to friction.
Table of the Ratios of Friction for Plane Surfaces.
Description of Materials
Disposition
of the Fibres.
Condition
of the Surfaces.
Ratio of Friction
to Pressure.
At
Starting.
In
Motion.
Oak on oak, ......
Parallel.
Do. 1
Across.
Do.
Endwise
(on one piece).
Parallel.
Do.
Do.
Do.
Do.
Flat or on edge.
i Flat.
Dry.
Lubricated with
dry soap.
Dry.
Wet with water.
{ Dry.
Do.
Do.
Do.
Do.
Wet with water,
j Do.
( Oiled or greased.
Dry.
Do.
Do.
1
1
62
44
44
71
43
38
53
80
62
65
62
12
47
28
16
14
•48
•16
•34
•25
•19
Ash, lieech, or deal on oak, . .
JEIempen cord on o;ik
Wrought-iron on oak,
•38
•52
•49
•22
Pump leather on cast-iron,. . .
Belt i "" oaken dru ms,
t on cast-iron pulleys,. . .
Cast-iron on cast-iron
•27
•15
•19
Example. — What power is necessary to raise an oaken flood-
gate weighing 30 lbs., and against which a pressure is exerted
equal to 700 lbs. 1
We have
(•71 X 700 =) 497 -f- 30 = 527 lbs. at starting,
and (-25 x 700 =) 175 -f 30 = 205 when in motion,
supposing the pressure continues the same.
Table of the Ratios of Friction for Journals in Bearings.
Description of Materials.
Cast or wrought-iron journals,
in cast or wrought-iron, brass, ■
or gun-metal bearings, '
Cast-iron in cast-iron
Cast-iron in brass or gun-metal.
Cast-iron in lignumvitse, -
Wrought-iron in brass or gun- i
metal 'i
Wrought-iron in lignumvitiB, . ■
Condition of the
Surfaces.
Lubricated with oil }
or lard, '^
Similarly lubricated )
and wet with water f
Lubricated with or- j
dinary oil, and wet >
with water, )
Greased,
Greased and wet, . . . .
Unlubrieated,
Lubricated with oil and
lard
Lubricated with a pre
paration of lard and
plumbago,
Greased,
Greased and wet, . .
Badly lubricated, . .
Lubricated with oO and
lard
Lubricated with ordi
nary oil
Ratio of Fric-
tion to Pres-
sure, with re-
gular Lubrica-
tion.
•07 to -08
•08
■14
•16
•19
•18
•10
•14
•09
•19
•25
•11
•19
Rule. — To determine the frictional pressure, f, acting on tno
bearings of a journal, always bearing in mind the weight of the
shaft and the gear carried by it, the power transmitted, as also the
resistmg load : Multiply the product of these, p, by the coejicient, c,
to obtain the amount of friction ; next multiply this by the constant
•08, and by the diameter d., in inches, (or by -OSd.,) m obtain the
amount per revolution; and, finally, multiply this by the number of
revolutions in a minute, which will give the amount of power con-
sumed by friction during this unit of lime.
Example. — What amount of power, a, is absorbed by the
friction of the journals of a cast-iron shaft revolving in bearings ;
also, of cast-iron, under the following conditions ?
The diameter at (ho journals = 5 inches.
The pressure of the shaft and gear = 20,000 lbs.
The velocity = 5 revolutions per minute.
According to the table, the coefficient, c, is -076.
Here we have —
F=-08d X a xP,
= -08 X 5 X ^075 X 20,000 = 3,000 lbs.
CHAPTER IV.
THE INTERSECTION AND DEVELOPMENT OF SURFACES, WITH APPLICATIONS.
Ifi3. Nowhere is descriptive geometry moro useful, in its appli-
cation to the industrial arts, than in the dotcrinination of the lines
of intersection, or junction, of the various solids, whether the in-
tersection bo that of two similar solids with each other, as a
fVlindof with a cylinder ; or of dissimilar ones, as a cylinder with
a sphere or a cone. With the sud, however, of this branch ot
geometry, wo can dotormino, in tho most o.\act nnuinor, the pro-
portions of all tho curves — of double as of single cun'ature — which
may bo produced by tho intcrsottions of surfaces of revolution,
the constructive or generative data of which are knowu.
50
THE PRACTICAL DRAUGHTSMAN'S
The apijlieations of forms in which such curves occur are exceed-
ingly numerous; they abound in the works of the brazier, the
tinsmith, the joiner, the carpenter, the builder, and the architect,
IS well as in all descriptions of machinery. This treatise would,
indeed, be incomplete, were we not to render the delineation of
these curves quite familiar to the student. Intimately connected
also with this branch of the subject, is the development of curved
surfaces ; that is, the determination of the dimensions of such
surfaces when extended in a plane, so that the workman may be
able to cut out such pieces with the certainty of their taking the
form, and fitting the place assigned for them.
The study of projections, moreover, comprises the methods of
delineation of such curves as helices, spirals, and serpentines,
which frequently occur in mechanical and architectural con-
struction.
THE INTERSECTIONS AND DEVELOPMENT OF
CYLINDERS AND CONES.
PLATE XIV.
PIPES AND BOTLEES.
164. The intersections of cylindrical or conical surfaces may
be curves of either single or double curvature. A curve is said
to possess a double curvature, when it is not situated wholly in one
plane. The problem to be considered in representing the lines of
intersection, reduces itself to the determination in succession of the
projections of several points in the curve, and the completion, by
tracing the projections of the entire curves through the points
thus found. The principle generally to be followed in these cases
is, to imagine the two cylinders, for instance, to be cut by one and
the same plane — their intersection in that plane being a point in
the line sought. Thus, to obtain a point in the line of intersection
of two right cylinders, a and b, represented in figs. 1 and 2, draw
any plane, c d, parallel to the axes of both cylinders. This plane
cuts the vertical cylinder, a, as projected horizontally in the points,
e'/, and as projected vertically through the lines, e' e^, and/'/^.
The horizontal cylinder, b, is cut in c rf in the horizontal projection,
and in c' d' in the vertical projection, which latter is obtained as
follows : — The semi-base, g i, of the cylinder, b, is drawn in plan
as in fig. l"; then prolonging c d until it cuts this plan in c^, itwUl
give the distance, c^ h, of the cutting plane from the axis of the
cylinder ; this distance is then transferred to i c', fig. 2, and through
c the line, c' d', is drawn as required. The points, e-, /-, of
Intersection of this line, c' d', with the lines, e' e-,/'/-, are points
in the intersecting line respectively on each side of the vertical
cylinder, a.
By repeating this operation vidth another plane, as m n, parallel
to the first, other two points will obviously be found, as P and o-.
Tlie extreme points, a', k'. are naturally determined by the inter-
section of the outlines of the cylinders. As for the point, b', it is
found by means of the imaginary plane, g p, tangential to the ver-
UcaI cylinder, a, and also to the horizontal one, b, when, as in this
wse. the two are of equal diameter.
As many points having been found as constitute a sufficient
guide, according to the scale or size of the dramng, and the pro-
portion between the intersecting solids, their reunion into on<"
continuous line completes the delineation of the curve of intersec-
tion. It will be observed that, in fig. 2, the vertical projection of
this curve is a straight line, as a' b', or b' k', these two being at
right angles to each other ; this results from the fact, that the
cylinders, a and b, chosen for this illustration are of equal diameter,
and have their axes situate in the same plane, and at right angles
to each other, in such a manner that the curves of intersection,
which are elliptical, lie in a plane at right angles to the plane of
the vertical projection. It follows, then, that this peculiarity being
known, all that is necessary in similar cases — that is, when two
equal right cylinders, having their axes in the same plane, cut each
other — is simply to draw lines from the extreme points, a' and k\
to the summit, b', the projection of the point of intersection of
their axes — the operations above described being, in such cases,
altogether dispensed with.
165. When the cylinders are of unequal diameters, the curve
of intersection becomes one of double curvature, notwithstanding
that the axes of the cylinders may lie in the same plane. Thus,
figs. 7 and 8, which represent two intersecting cylinders, a and b,
of very different magnitudes, show that the curve of intersection,
a' b' k', drawn according to the method before given, is one of
double curvature, becoming the more flattened at b', according
as the diameters of the cylinders differ more. To render it quite
plain that the operation is the same, we have indicated the
various points obtained, by the same letters which mark the cor-
responding points in figs. 1 and 2. We must further remark,
that in figs. 7 and 8 the curve is determined with the assistance
of two elevations, taken at right angles to each other ; whilst, in
"figs. 1 and 2, an elevation and a plan were employed, similarly, at
right angles to each other.
We show the application or exemplification of this cui-ve iv
figs. 4 and 5, which represent a steam-engine boiler, c, seen partly
in elevation and partly in section. The tubular piece, d, which
is a species of man-hole, is supposed to be cylindiical, and is
attached to the body of the boiler by means of a flange, which
gives rise to the external intersectional curves, a b, c d, and the
internal one, ef.
THE INTERSECTION OF A CONE WITH A SPHERE.
166. Whenever a cone is cut by a plane parallel to its base, the
section presents an outlme simOar to that of the base ; ihen, when
the cone under consideration is a right cone of a circular base, all
such parallel sections are circles. Thus, in figs. 3 and 3°, repre-
senting a right cone, a' b' s', the plane, a' b', parallel to its base,
a' b', cuts the cone so as to present a circle, of which the diameter
is exactly contained within the extreme generatrices, a' $', b' s'.
If, then, with the centre, s, and radius, a s = a' b'-~ 2, we describe
a circle, it will be the outline- of that part of the cone intercepted
by the cutting plane.
The section of a sphere, c, by any plane whatever, is also a
circle. When this cutting plane, a' b', for instance, is perpen-
dicular to the plane of projection, it is necessarily projected as a
straight line, as in fig. 3", and as a circle, as in fig. 3, in the plane
BOOK OF INDUSTRIAL DESIGN.
5»
of projection to which it is parallel. It follows, from the existence
of thebe respective properties, that we have at hand a very simple
method for determining the curve of intersection of a cone with a
sphere, whatever may be the relative position of their axes. This
method consists ia supposing a series of parallel planes to cut
both the cone and the sphere, so as to produce circular sections
of both — the intersections of the outlines of which will conse-
quently be points in the curve sought, as indicated in fig. 3.
The iutoi'section, a' b', is a circle, the diameter of which is limited
by the extreme generatrices, s' a', s' b, of the cone, where they
encounter the gi-eat circle of the sphere, c. The same method
holds good when the cone is cut by any plane, a' g, inclined to the
base, the outline of the section being ia this case an ellipse, which
is projected in the plan, fig. 3, by the line, a i" g' n', the resultant
of the various intersections m the planes adopted in the construc-
ts iii and obtainment of the curve.
The same occurs with the intersection of a cone, a' b' s', with a
cylinder, a'b'df; and when their axes lie in the same straight
line, the intersection, a b', is also a circle, the diameter of which
is equal to that of the cylinder.
167. If their axes are parallel, though not in the same straight
line, the intersection of these two surfaces becomes a curve of
double curvature, which may be determined either according to
the method adopted in reference to figs. 1 and 3, or by supposing
a series of planes to cut the cone parallel to its base, and conse-
quently at right angles to the generatrix lines of the cylinder; by
this means circular sections will be produced, those of the cylinder
being always the same, but those of the cone varying according to
the distance of the planes from its apex. The points of intersec-
tion of the various circles representing, respectively, sections of
the cone and cylinder, wOl be points in the curve of intersection
sought.
DEVELOPMENTS.
168. By this term is meant the unrolling, extending, or flatten-
ing out upon a plane, of any curved surface, in order to ascertain
its exact superficial measurement.
The more generally used surfaces or forms which are capable
of development in this manner, are — the cylinder, the cone, prisms,
pyramids, and tho frusta, or fragments of these solids.
Tin and copper-smiths and boiler-makers, who operate upon
metals which come into their hands in the foi-m of thin sheets,
have continually to transform these sheets into objects which are
analogous in form to ihese solids.
To do their work with skill and exactitude, and not by mere
guess, and also to avoid the cutting of the material to waste, tiny
should make plans of the whole or part of tho object as finished,
so that they may calculate the exact development of the surface,
both as to form and size, and cut it at once from tho sheet of
nielal with all possible precision.
THE DEVELOPiVIENT OF THE CYLINDER.
1()9. Here, taking fig. 2, which wo have on a former occasion
considered as a coui)le of solid cylinders, to roprescnit, in the
present case, two pipes or hollow cylinders formed of thin sheet
motiil, let us set about ascertaining what should he the sliapi' .•uui
size of the pieces of metiil as extended out Hat, of which Ihc m'
two cylinders are to be formed. It is to be observed, in the first
place, that the rectification or development into a straight fine of
a circle, is equal to its diameter multiplied by 3'1416, &c. ; whence
the development of the base, p q, of the right vertical cj linder, a.
fig. 2, of which the diameter measures "322 metres, is obviously
equal to -322 x 3-1416 = 1-012 m.
This length, then, 1-012 m., is set off on the line, m m, fig. 10,
and the circumference having been divided into a number of equal
parts, as was done to obtain the curve of intersection of the two
cylinders in fig. 1 ; the line, m m, is divided into the like number
of equal divisions. Through each of these points of division,
perpendiculars are erected upon the line, m m, representing so
many generatrix lines corresponding to those of the cylinder, a,
fig. 2 ; and for the sake of greater intelligibility, we have marked
the corresponding lines by the same letters. Ne.xt, on each of
these are set oft' distances, m b', e' e^, I' P, p d,p p, o' o-, Q k', <Sic.,
equal to the respective distances in fig. 2. By this means are
obtained the points, b', e', I', &c., in fig. 10, through which the curve
passes which forms the contour of intersection corresponding to tliat
portion of the semi-cylinder, b' a b", fig. 1, which is intersected by
tlie horizontal cyhnder, b ; and as the other half of the cylinder is
precisely the same, the cur\ e has simply to be repeated, as shown
in fig. 10.
It is unnecessary to detail the method of finding this develop-
ment of the horizontal cylinder, b, as it is identical hi principle
to that just discussed.
It may be gathered from the above exemplification, that tlie
principle generally to bo followed in obtaining the development of
cylindrical surfaces, is, first, to unfold it in a direction at right
angles to one of its generatrices, or in the direction the generatrix
takes in the construction of the solid, and theu to set oft" from the
straight line thus produced, at equal distances apai-t, any number
of distances previously obtained from the projections of the out-
line or line of intersection when the cylmder is joined to anotlier,
Of of its section when cut by any plane. The curve of this out-
line is finally obtained by tracing a line through the extremities
of the generatrices, di-awn perpendicular to the base.
THE DEVELOPMENT OF THE CONE.
170. As in the case of the cyUndcr, so likewise, in order to find
the development of the cone, do wo unfold it, as it were, in Uie
direction of motion of its generatrix. Now, as lUI the generatrices
of a right cone are equal, and converge to one point, the apex, ii
follows that, when tho conical surface is d^'veloped upon a iilane,
these generatrices will form radii ol' a poiiion of a circle ; coiise-
(juently, If with one of tho generatrices, as a railius, we describe
a circle, and cut otVas much of the circumference as sli:ill be equal
to that of the base of the developid cone, wo sliall obtain .1
sector of a circle eqiuil in area to the lateral s\irl'ace of the cone, as
developed upon a plane.
Fi"-. 9 represents Ilu> development of tlie Irustum, or truucateo
cone, a' b', a' r,', as projected in lig. 3*. and of which the :i|h>\
would be s', were the cono ontiu>. The operation is as fol-
lows: —
We shall suiiposc the coni> to be developol in the direction
lak.'M by Iho jMiuiadix, s' a', lig. 3*; therefor.', wiih a radius egual
52
THE PRACTICAL DRAUGHTSMAN'S
to s' a', and with the centre, s', describe the fragment of a cu-cle,
a' b' a-, fig. 9.
Having divided the circular base, a b, fig. 3, of the cone into
some arbitrary nmnber of equal parts, say 16, and having drawn
the respective generatrices, 1 s, 2 s, 3 s, &c., set oflf on the arc,
A' B A-, fig. 9, an equal number of arcs, each equal to the respec-
tive arcs obtained by the subdivision of the circle, a b, fig. 3.
From the points thus obtained, 1' 2' 3', &c., fig. 9, ivaw the radii,
1' s', 2' s', 3' s, dz-c, representing generatrices corresponding to
those projected in fig. 3.
By this operation we obtain the development of the entire cone,
and find that it produces the figure s' a' b' a-, fig. 9, the circular
perimeter of which is equal to that of the base of the cone itself
The cone, however, under consideration, is divided by a plane, a' h',
fig. 3, parallel to its base, which reduces it to a frustum of a cone ;
the development of the conical surface of which is equal to the
space contained between the arc, a' b' a^, corresponding to the
base of the cone, as just determined, and the arc, a c i, of lesser
radius, drawn with the same centre, s', and with a radius equal to
the generatrix, s' a', of the portion of the cone taken away.
The development, then, of the truncated cone, is the fragment of
an annular space, distinguished in fig. 9 by a flat tinted shade.
171. In the case of a truncated cone, of which the dividing plane
is in(;liiied to the base, as a' g, fig. 3°, instead of being parallel, or
if the cone is joined to a cylindrical or spherical body, and the line
of intersection is curved in any way, the development of this edge
of the conical surface will no longer take the form of the arc, a c b,
fig. 9. It* true representation will be obtainable by setting off", on
the several radii, fig. 9, lengths corresponding to the respective
generatrices as intercepted by the plane, or curve, of intersection.
In order to obtain the lengths of the respective generatrices, which
can be done from the vertical projection, fig. 3°, each intermediate
one, as 4' t, &c., must be squared across to an extreme one, as at
a' i', and indicated by the horizontal lines ; this will give the exact
length of each — being otherwise, as projected, considerably fore-
shortened. Thus, the division of the cone by the inclined plane,
a' g, fig. 3", produces an ellipse, the development of which, in fig. 9,
takes the form of the curve, a i g b.
In the ccnstructicn of the boiler, represented in figs. 4 and 5,
wliich is formed of several pieces of sheet metal, we shall find
extensive applications of the principles just discussed. It must be
borne in mind that, in calculating the development, or the size and
shape of the component pieces, an allowance must be made for the
lap of the pieces over each other, for the purpose of joining them
together, as indicated in the drawing.
172. In cylindrical steam boilers, the extremities are generally
constructed in the shape of hemispheres — this form offering the
greatest resistance to internal or external pressure.
As the sphere cannot be developed upon a plane, these hemi-
spherical ends cannot be made in a single piece, unless cast or
forged. In practice this difficulty is overcome, by forming this
portion in from 5 to 8 gores, according to the size of the boiler,
these being sunnounted by a central cap-piece. After being cut
out, these several pieces are hammered to give them the necessary
sphericity.
In fig. 6, we give a practical method of approximately determining
the development of one of such gores; this consists in drawing
with the centre, o', an arc, m n, corresponding to the radius of the
hemisphere. On this arc, from m to n, set off the circular length
of the gore ; set off", also, the length, p q, corresponding to one of
the six divisions, as seen in the end view, fig. 5. On the arc, m n,
fig. 6, mark an arbitrary number of points, at equal distances
asunder, as 1, 2, 3, 4, 5 ; draw through these horizontal lines, cut-
ting the vertical, m o', thereby giving the various radii, o' 1", (/ 2",
o' 3", &c., with which arcs are di-awn as indicated ; the rectifies^
tion, or development, of these arcs, contained between the radii,
p o', g o', are then obtained, and transferred to perpendiculars
di-awn through the points, 1', 2', 3', &c., on the line, o' n', which is
the rectification, or development, of the arc, m n, with its series of
divisions. Thus, from the arc, p q, is obtained the line, p' q', and
similarly the entire figure, p' n' q', which is an approximation to
the surface of the gore, as supposed to be flattened out. The
necessary allowance for lap is superadded, as shown by the flat
tinting in the drawing, fig. 6. The gore cut to this outline in sheet
metal is then hammei-ed to a proper form upon a mandril, or
anvU, with a spherical surface.
THE DELINEATION AND DEVELOPMENT OF HELICES,
SCREWS, AND SERPENTINES.
PLATE XV.
173. That curve is called a cylindrical helix, which may be
said to be generated by a point caused to travel round a cylinder,
having, at the same time, a motion in the direction of tlie length
of the cylinder — this longitudinal motion bearing some regular
prescribed proportion to the circular or angular motion. The dis-
tance between any two points which are nearest to each other,
and in the same straight line parallel to the axis of the cylinder,
is called the pitch — ^in other words, the longitudinal distance tra-
versed by the generating point during one revolution.
This definition at once suggests a method of drawing the
lateral projection of this curve, when the two projections of the
cylinder and the pitch are known. This method consists in
dividing the circumference of the base of the cylinder into any
number of equal parts, and drawing parallels to the axis thi'ough
the points of division projected on the vertical plane ; at the same
time a portion of the axis, equal to the pitch, must be divided
into the like number of equal parts, and as many lines must be
di-awn perpendicular to the axis. The intersections of the cor-
responding lines of each set will be points in the curve.
Let A and a', figs. 1 and 2, be the horizontal and vertical pro-
jections of a right cylinder, and a' — a^ the length of the pitch of
the helLx, generated by the traverse, as already defined, of the
point projected in a and a'.
The circle described with the radius, a o, and representing the
base of the cylinder, is divided into 12 equal parts, starting from
the point, a. Through each of the points thus obtained, a ver-
tical line is di-awn. The pitch, a' a-, is similarly divided into 12
equal parts, and a corresponding number of horizontal lines are
drawn to cut the vertical ones in the points, 1', 2', 3', &c. ; thea*
BOOK OF INDUSTRIAL DESIGN.
53
points are next connected by the contmuous line, a', 1', 3', 6', 9',
a^, which forms the vertical or lateral projection of the helix.
Half of this curve is indicated by a sharp full line, as being on
the front surface, a, 3, 6, of the cylinder, whilst the other half is
in dotted lines, representing the portion of the curve which is on
the other side, a, 9, 6.
The number of divisions of the circumference of the cylinder
is a matter of indifference as regards the accurate delineation of
the curve, and it is therefore natural to choose a number that
calls for the simplest operations — an even number, for example,
as 6, 8, or 12; and in the present instance, wherein the starting
point, a, lies in the horizontal diameter, a 6, of the base, it will
be observed that two points occur in the same vertical line, as
2 — 10, which gives the points, 2', 10', in the vertical projection.
The operations wOl be similar, if the given starting point be
diametrically opposite to a, as 6', the pitch, b^ b-, being equal to
174. The conical helix is different from the cylindrical one,
simply in that it is described on the surface of a cone instead
of on that of a cylinder, and the operation consequently differs
very slightly from the one before described; the horizontal and
vertical projections of the cone are given, and also the pitch.
Fig. 3 is the vertical projection of a truncated cone, c, the bases
of which, a' b', c' d', are represented in the plan, fig. 1, by the
concentric circles described, with the respective radii, a o and c o.
The outer circle having been divided as already shown, radii are
dravra to the centre, o, from all the points of division, 1, 2, 3,
&c., which cut the inner circle in the points, e,f,g, &c. These
points are then projected upon the upper base, c' d', in fig. 3, those
on the outer circle being similarly projected on the lower base,
a b' ; the respective points in each base are next joined, thus forming
a series of generatrices of the cone, as 1- — e^, 2- — /-, o' — o-, &c.,
which would all converge in tlie apex, if the cone were complete.
These lines are cut by horizontals drawn through a corresponding
number of divisions in the length of the given pitch, a' c', and the
points of intersection thus obtained lie in the curve which it is
required to draw. The horizontal projection of the curve is then
obtained by letting fall from the points of intersection last
obtained, a series of verticals which cut the respective radii in the
plan, fig. 1. This produces a species of spu-al, or volute, e^,/^, g^,
h^, 2^, &c. By following out the same principles, helices may bo
represented as lying upon spheres, or any other surfaces of revo-
lution.
THE DEVELOPMENT OF THE HELI.X.
175. It will be recollected that a cylinder, and also a cone, are
capable of being developed upon a plane surface, and that the
base of either, when rectified, or converted into a straight line, is
equal to the diameter multiplied by 3'1416. Lot, then, a 6, fig.
4, bo a portion of the development of the base of the cylinder,
A, figs. 1 and 2; to obtain the development of (he helix drawn
upon this cylinder, we must first divide it off into ienglhs, corrc-
sponding and equal to the arcs obtained by the division of llie
circle, (I (>. On each of the divisions thus obtained, as 1, 2, 3,
&<•., we then erect perpendiculars, making thoni equal rospecliveiy
to the distances from the starting point, a, of tho several divisions
of the pitch. The extremities of these perpendiculars, as 1', 2',
3', &c., will be found to lie in the same straight line, a 6', which
consequently represents the development of a portioD of the
helix. In general, the development of a helLx is a straight line,
forming the hypothenuse of a right-angled triangle, the base of
which is equal to the circumference of the cylinder, and the
height to the pitch of the helix.
Several helices di-awn upon the same cylinder, and having tho
same pitch, or a helix which makes several convolutions about a
cylinder, is represented by a series of parallel curves, the distance
between which, measured on any fine parallel to the axis, is
always equal to the pitch.
The development of the conical helix may be obtained by means
of an operation analogous to that employed for the development
of tne cone (Plate XIV.) ; and in this case the result will be a
curve, instead of a straight line.
We meet with numerous applications of the helical curve in the
arts, for all descriptions of screws ; and staircases, and serpentines.
176. Screws are employed in machinery, and in mechanical
combinations, either for securing various pieces to each other, so
as to produce contact pressure, or for communicating motion.
Screws are formed with triangular, square, or rounded threads or
fillets.
A screw is said to have a triangular thread, when it is generated
by a triangle, isosceles or not, the three angles of which describe
helices about the same given axis, situate ui the same plane as the
triangle. Figs. 5 and 5° represent the projections of a triangular-
threaded screw, such as would be generated by the helical move-
ment of the triangle, a' b' c', of which the apex, a', is situate on
a cylinder of a radius equal to a o, and of which the other angles.
b', c', are both situate on the internal cylinder, having the radius,
b o, which is called the core of the screw, and is concentric with
the first. The difference, a b, between the radii, a o and b o, indi-
cates the depth of the thread.
When, as in the case taken for illustration, the screw is single-
threaded, the pitch is equal to the distance between the two point.s,
b' and c' ; that is, to the base of the triangle. The screw is one of
2, 3, 4, or 5 threads, according as tho pitch is equal to 2, 3, 4, or
5 times the base of the generating triangle. From what iins
already been discussed, tho method of delineating the triangular-
threaded screw will be easily comprehended; for all tliat is nives-
sary is to draw the helices, generated by tho three angular points,
in the manner shown in reference to figs. 1 and 2. \\"e liavo,
notwithstanding, given tho entire operation for one semi-convolu-
tion of tho thread, in figs. 5 and 6*. When one of tlio curves, as
a' 3' 6', is obtained, it is repeated as many tinie< as there are cor.
volutions of tho thread on tlie length of the screw. To do thir,
with facility, and without reiieating the entire operation, it is.
customary to cut out a i)atlern of the curve in iianl cuii-buani,
or, by picl'erenoe, in veneer wood ; then sotting liiis pattern to tlio
points of ilivision, (/'(•'/', ])reviously set (tlV, tho curves arc easilv
(Irnwn parallel to one another. Tho s;nuo may bo done \\il!i lhv>
inner liolioiil curves.
It must be obsorvod, that, to comploli' tiio outline of the .vrow,
04
THE PRACTICAL DRAUGHTSMAN'S
tlioso vai-ious curves require to be joined by the portions, h> d', d' V,
vvhicii, thouijli in fig. 5° they are dj-awn as simple straight lines,
should, if it is wished to bo precise, be shown by lines slightly
curved and tangential to the curves passing througli the points, a
and h', as in fig. 5'. These curves are the result of a series of
helices, traced by the component points of the Imes forming the
sides, a' b', a' c', of the generating triangle. In practice, this nicety
is disregarded, and simple straight lines are employed.
177. A screw is termed square-threaded when it is generated
by a square or by a rectangle, the parallel sides of which lie in
right concentric cylinders, and the angles or corners of which
describe helices about the axes of these cylinders. Figs. 6 and 6",
represent projections of a square-threaded screw — the thread being
generated by the square, a' c' b' d'. The horizontal side, a' c',
determines the depth. The height, a' d', marks the width of the
thread, and d' d is the width of the interval, which is generally
equal to that of the thread.
When the screw is single-threaded, the pitch, d e', is equal to
the sum of the wdths of the thread and of the interval, or, in the
case before us, to twice the side of the squai-e. Of course, when
the pitch is equal to 2, 3, or 4 times a' e, the screw is 2, 3, or 4^
threaded, in all cases having as many intervals as threads. The
opei'ations called for in delineating the screw of a single square
thread, are fully indicated upon the figures. The delineation of a
screw of several threads does not possess any additional points of
difficulty.
INTERNAL SCREWS.
178. An internal screw, or nut, is a screw in intaglio, cored out
of a solid body — instead of being in relief, and having the material
cut away from it — in such a manner that its more indented portions
correspond to the more elevated portions of the common or
external screw, whilst the more indented portions of the latter
correspond to the more elevated portions of the former. In order
to represent the helical fillets or threads of the internal screw, it is
necessary to section it by a plane passing through its axis ; it is in
this manner that we have represented the nuts, m n p q, in figs.
6° and 6°, the first having a triangular thread fitting to and embrac-
ing the screw, d, which is represented as just introduced into it ;
the other has a square thread similarly adapted to the screw, e.
It follows, from these nuts being represented in section, that we
only see the half of each corresponding to the posterior portion of
tlieir respective screws, d and e ; and in consequence of this, the
helical curves are inclined in the opposite direction to those repre-
senting the anterior portions of the screws.
Those screws are distinguished as right-handed screws, of which
the thread in relief rises from the left to the right, as in the screws,
r, e ; and as left-handed when the thread takes the direction from
right to left — that is, for example, in the direction taken by the
curves representing the nuts, d, e.
SERPENTINES.
170. Serpentine is the name given in practice to a pipe or tube
bent to the helicoidal form ; but, in geometry, it is the term given
to the solid generated by a sphere, the centre of which traces a
helicoidal path.
This form is often employed, whether hollow, as for pipes, such
as the worm of a distilling apparatus, or solid, as for metal springs
The first thing to be done in delineating this solid, is to deter
mine the helix traced by the centre of the generating sphere, ita
pitch, and the radius of the cylinder on which it runs, being given
Tlie helix having been di-awn, a series of circles are described witj
the radius of the sphere, and with various points of the curves asi
centres ; curves drawn tangential to these cu'cles, will then form the
outline of the object. Figs. 7 and 7° represent the plan and ele-
vation of a serpentine formed in this manner. The circle di'awn
with the radius, a o, is the base of the cylinder, on which lies the
helix generated by the traverse of the point projected vertically in
a'. Next is given the radius, a' b', of the generating sphere, and
the pitch, a* a^, of the helix. Tliis helix is then projected accord-
ing to the operation indicated on the di-awing, and already
described, by the curve, a', 1', 2', 6', 9', and a^ ; it may be con-
tinued indefinitely, according to the number of convolutions desired.
With diiferent points of tliis curve as centres, and with the radius,
a' b', are then described a series of circles pretty near to each other,
and two curves are drawn tangential to these, as shown on fig. 7'.
In going over this figure with ink, it is of importance to limit
these curves to the portion of the outline, which is quite apparent
or distinct ; thus, for the anterior portion, a, 1,2, 3, 6, fig. 7, of the
serpentine, the lower curve, c efg, ends at the point of contact, c,
with the circle whose centre is a~, whOst the upper one, hid, ends
in the point, d, on the circle described with the centre, 6'. The
posterior portions of these curves are limited by the points, g and i,
where the bend of the serpentine goes behind, and is hid by the
anterior portion.
The horizontal projection of the serpentine is always comprised
within two concentric circles, the distance asunder of which is
equal to the diameter of the generating sphere, as in fig. 7.
Fig. 7- represents a tubular serpentine, which is supposed to be
divided by a plane, 1 — 2, in fig. 7, passing through its axis. It is,
consequently, the posterior portions that are visible, and they are
inclined from right to left ; the section at the same time shows tue
thickness of the tube, or pipe.
In the arts, we also see serpentines, both solid and hollow,
generated by conic, or other helices ; of this description are the
springs employed in the moderator lamps, and the forms of distil-
lery worms are sometimes varied in this way.
180. Observation. — The curves representing tlie outline of screws
and serpentines, the rigorously exact delineation of which we have
just explained, are considerably modified when these objects come
to be represented on a very small scale ; thus the triangular-
threaded screw may be represented, as shown in fig. 8, by a series
of parallel straight, instead of curved, lines — these being inclined
from side to side to the extent of half the pitch. These lines
should be limited by two parallels to tl e axis on each side, mark-
ing the amount of relief of the thread. When the scale of the
drawing is still smaller, and greater simplicity desirable, the
draughtsman is content with a series of parallel lines, as in fig. 9,
limited by a single line on each side parallel to the axis.
For the square-threaded screw, the helical curves may similarly
be replaced by straight lines, as in fig. 10. The same also applies
to the serpentine, as shown in fig. 11.
BOOK OF INDUSTRIAL DESIGN.
55
THE APPLICATION OF THE HELIX.
THE CONSTEUCTION OF A STAIRCASE.
PLATE XVI.
181. The staircases, which afford a means of communication
oetween the various floors of houses, are constructed after various
systems, the greater number of which comprise exemplifications
of the helix. The cage, or space set apart for the staircase, varies
m form with the locality. It may be rectangular, circular, or
elliptic.
Figs. 1 and 2 represent a staircase, the cage of which is rec-
tangular this space being provided for the construction of the
main frame of the stair, with its steps and balustrade, and with a
central space left sufficient for the admission of Ught from above.
In the case of a cylindrical cage, the curve with which the steps
rise is helical from bottom to top ; but in a staircase within a rec-
tangular cage, the steps rise for some distance in a straight line,
and only take the helical twist at the part forming the junction
between the rectilinear portions running up alternate sides of the
rectangle. Stairs are sometimes made without this curved part,
a simple platform, or " restbg-place," connecting the two side
portions.
For the division of the steps, we take the line, efg h i, passing
through the centre of their width, and taking exactly the direction
it is intended to give the stairs. The first or lowest step, a,
which lies on the gi'ound, is generally of stone, and is larger and
wider than the others.
For the stairs, as for the helLx, the pitch or height, say 3-38 m.,
from the basement to the floor above, is divided into as many
equal parts as it is wished to have steps. The centre line, efg h i,
is also divided into a like number of equal pai-ts. In general, the
number of steps should be such, that the height of each does not
exceed 19 or 20 centunetres. The larger the staircase is, the
more may this height be reduced — say, perhaps, as low as 15 or
16 centimetres. The width, 1 — 2, of the step should not be under
18 to 20 centimetres.
If, for example, in the given height of 3-38 m., we wish to make
21 steps, we must divide this height into 21 equal parts, and draw
a series of horizontal lines through the points of division, which
will represent the horizontal sui-fiiccs of the steps.
For those steps which lie parallel to each other, it is simply
requisite to erect verticals upon the points of division on the
centre line in the plan. The points of intersection of these with
the horizontals above, as 1, 2, 3, 4, fig. 2, indicate the edges of these
steps. For the turning steps, however, or winders, as those steps
are called which are not parallel to each other, a particular opera-
tion is necessary, termed the balancing of the steps, the object of
which is to make the steps as nearly equal in widtli as possitilo,
without, at the same time, making thcin very narrow on the inner
edges, or rendering the twist or curve too sliarp or sudden.
Where' fho stairs are narrow, as in the case we have illustrated,
the balancing should conuiience a step or two before rcacliiiig the
curved portions. This balancing may bo obtMijicil in the follow-
ing manner: — A i)art of the rectilinear ])<)rtioii, /) /, (([ual to three
«t«ps, is d(;veloii<(l, and then a part of the lurvcd pmlion, / ;;/ n,
equal to three more steps. On the vertical, I q, fig. 3, set ofi' the
heights of the first three steps; and through the point, q, draw
the horizontal, q 4, representing the development of the widths of
the steps in a straight line. Also, on the prolongation, i q'. of the
vertical, t q, set off" the heights of other three steps. Through the
point, q', draw a horizontal, and make q' 10 equal to the arc, I m ru,
in the plan, fig. 1, as rectified. The straight Une, t 10, will then
be tlie development corresponding to the curve of the framepiece.
At the latter point, n, erect a perpendicular on this line, and at the
point, 6, a perpendicular to the straight line, t 4. The point ot
intersection, o, of these two perpendiculars, gives the centre of
the arc, p k n, which is drawn tangentially to these lines. Then,
through each point of division on the vertical, q </, draw hoiizon-
tals, meeting this curve in the points, _;', k, I, m, through which
diaw parallels to q q'. Then transfer the respective distances.
J 6, k t, I 8, m 9, comprised between the arc and the two straight
lines, I 4 and t u, on the line of the framepiece, p k n,in the plan,
fig. 1, as at _;' k, k I, I m, m n. Next, draw straight lines through
the points,y, k, I, m, and through the respective points of division,
6, 7, 8, 9, on the centre line, which will give the proper incli-
nation for the steps as balanced. The second half of the curved
portion is obviously precisely the same as the first in plan, and
may easUy be copied from it.
Having thus determined the position of the steps in tlie hori-
zontal projection, they must next be projected on the vertical
plane, by means of a series of verticals, wliich cut the respective
horizontals drawn through the points of division, 1, 2, 3, 4. As
in fig. 2, the anterior wall of the cage is supposed to be cut away
by the line, a 6' 10', in the plan, the outer edges of the steps are
seen and are determined by erecting verticals on the correspond
ing points, 6', 7', 8', 9', &c.
The perpendicular portions, v v', of the steps, which are over-
hung by the horizontal portions, and consequently invisible in the
plan, fig. 1, are, nevertheless, indicated there in dotted lines,
parallel to the edges of the steps. To render them quite distinct,
however, and at the same time to show the manner in which tliey
are fitted into the framepiece, we have represented diem, in fig. 4,
without the actual steps, supposing them to be cut in successioiij
horizontally, through then- middles.
The framepiece is the principal piece in the staircase. It is
situated m the centre of the cage, and sustains each step, and,
consequently, must bo constructed very accurately, for upon it, in
a great measure, depends the strengtli and solidity of tJio staircase,
For a staircase of proportions, like those of the one represented ir.
the plate, the framepiece is generally made of oak, in three pieces ;
the middle piece, c, corresponding to the curved portion, whils*
the other two, B and d, jouied to that one, form the rectilinea'
portions. A special set of diagrams is necessary, to determine
tlie shape and proportions of the various jiart-s of this fr^iopioce.
The method lure to be followed is, in the first place, to draw the
joints, by w liith the vertical portions of llio steps are attached to
the frainepiece. These can easily be obtained by squaring tiiem
over from tig. 4 to tig. 5, in which hust are the iiori/.ontal dixision
lines, corresponding to tig. 2. It will be observed, tliat tiie joints
referred to are bevilled ofl", so as not to be npjifirent externally,
The faces on the IVuniepicco arc seen on tig. 6, at llie jwuts, b. c.
50
THE PRACTICAL DRAUGHTSMAN'S
unci tlio method of obtaining them is sufficiently indicated by the
dotted lines. The framepiece has a certain regular depth through-
out, and is cut on the upper side, to suit the form of the steps,
and below, to the curvilinear outline, d b' c' d' e' f, which is
nothing but the combination of a helix with a couple of straight
lines. These straight lines, a' h' and e' f, are naturally parallel to
the curve passing through the edges of the steps, 1, 3, 5, 13, 16,
19. The curved part, 6' cS which corresponds to the anterior
face, h- c", of tlie framepiece, is drawn in precisely the same manner
iis a common cylindrical helix. It is the same with the part, d' e',
which corresponds to the interior portion, d- e-. If, in order to
better indicate the space occupied by the framepiece, it is wished
10 construct an end view of it, this may be done, as in fig. 6, from
the data furnished by figs. 4 and 5.
In figs. 12, 13, and 14, we have given, on a larger scale, the
different views of the curved part, c, of the framepiece, so as to
show its construction more plainly, as well as the form of the joint
connecting the three parts of the framepiece together. Each of
these figures is inscribed in a rectangle, indicated by dotted lines,
and representing the rectangular parallelepiped, in which the piece
may be said to be contained. To strengthen the junction of this
piece with the portions, e and d, they are connected by iron straps
or binding-pieces let in, or by bolts passing through, the entu-e
thickness of the wood.
Figs. 7 and 8 represent, in plan and elevation, the details of
the landing-stage, which forms the top step of one flight of the
stairs, and is on a level with the upper floor. It is with this
piece that the upper portion, d, of the framepiece is connected, by a
joint similar to that uniting the other portions. Fig. 9 is a sec-
tion, through the centre of this step-piece, through the line, 1 — 2,
in the plan ; whilst fig. 10 is another section, through the line, 3 — 4 ;
and fig. 11a tliird, through the line, 5 — 6. The form, dimensions,
and joint of tliis piece, are all fully mdicated in this series of
figures.
The shaft of the staircase, or the open space left in the centre
of the cage, is partially occupied by a balustrade, formed by a
number of rods of iron or wood, attached at their lower extremities
to the framepiece, as shown in fig. 15, and united above by a flat
bar of iron, surmounted by a hand-rail of polished furniture-wood,
of the form given in Plate III., fig. ©. The position of the rods,
as given in the plan, fig. 1, is sufficient for the determination of
theur vertical projection or elevation.
THE INTERSECTION OF SURFACES. •
APPLICATION TO STOPCOCKS.
PLATE XVII.
182. We have already discussed several examples of intersec-
tions of surfaces, as in pipes and boilers, and we shall now pro-
ceed to give some others, which are pretty generally met with,
p;vrticularly in the construction of stopcocks ; and for that reason
we take one of these contrivances as an illustration.
A stopcock is a mechanical arrangement, the function of which
Is to establish or interrupt at pleasure the communication through
pipes, for the passage of gases or liquids. It consists of two dis-
tinct parts, one called the cock, and the other — adjustable and
moveable in the first — ^termed the key or plug.
Stopcocks are generally made of brass, compositioii-metal, or
cast-iron, and the cock is formed with or without flanges, for
attachment to vessels or piping. The key is generally conical, so
as to fit better in its seat. The degree of taper given to the key
varies with different constructors. We have shown in dotted Tmes.
in fig. 2', various degrees of taper to be adopted, according as
greater tightness or greater facility of movement is wanted. The
part of the cock which receives the key is, of course, turned out
to a corresponding conical surface. This portion of the cock is
connected to the tubular portions by shoulders, of a slightly ellip-
tical contour. A stopcock answering to this description is repre-
sented, in plan and elevation, in figs. 1 and V.
In these figures wOl be seen the conical part, a, which embraces
the key, the cylindrical portions, b, united to the former by the
shoulders, d, and terminated by the flanges, c.
The conical key, e, adjusted in the cock, is surmounted by a
handle, f, by means of which it is turned. The key is retained
in the cock by a nut, g, working on a screw, formed on the lower
projecting end of the key. Fig. 1' represents an end view of this
cock, and fig. 2' is a vertical projection of the key alone. Fig. 2*
is a view of the key, looking on the lower end. Fig. 2 is a hori-
zontal section through the line, 1 — 2, in fig. r ; and fig. 2° is a
vertical section through the line, 3 — 4, fig. 1.
It will be easy to see, from these various views, that, in order
to represent the stopcock with exactitude, it has been necessary
to find, on the one hand, the projection of the intersection of the
elliptic shoulder, d, with the external conical surface of the central
part. A, of the cock ; and on the other, that of the cylindrical sur-
faces of the handle, f, of the key, as well when this is placed in a
position parallel to the vertical plane, as in fig. 2°, or inclined to
this plane, as in fig. 1°. We have, moreover, to determine the
intersection with the external surface of the key of the rectangular
opening, h, provided for the passage of fluids ; and also the inter-
section of a prism with a sphere, which occurs in the shape of the
nut, G, which secures the key m its place. The various opera,
tions here called for are indicated on the figures, which we shall
proceed to explain.
Figs. 3 and 3" show the geometrical construction of the inter-
section of the horizontal cylinder, f', of the handle, with the verti-
cal cylinder, e', of the key. The curve is obviously obtained
according to the method already described in reference to figs. 1
and 2, Plate XIV. We have, however, repeated the operations,
the exemplifications being a variation from that previously given.
183. When the horizontal cylinder, f', figs. 3' and 3°, becomes
inclined to the vertical plane, its curve of intersection with the
vertical cylinder, e', assumes a different appearance as projected
in this plane. Its construction, however, is precisely the same, as
follows : — To obtain any point in the curve, we proceed just as in
the preceding example, drawing the vertical plane, d' e', parallel
to the axes of the cylinders, this plane cutting the vertical cylinder
through the lines, d f and e g. This same plane cuts the horizon-
tal cylinder, as projected in plan, at d' e', whence the vertical pro-
jection is obtained, after drawing the semi-cylindrical end view, as
m fig. 8. The distance, i V, being set off' on h h', the horizontal
BOOK OF INDUSTRIAL DESIGN.
57
iinc, d e, drawn through the point, W, represents the mtersection
of the plane with the horizontal cylinder, ft h', being, of course,
measured from its r xis. It will be seen that the line, d e, is cut
by the vertical lines, d f and e g., in the points, d, e, which lie in
die curve sought ; and the same construction will apply to every
other point in the curve, dbec-
184. Figs. 4 and 5 represent the intersection of an elliptical
cylinder wth a right cone of circular base, corresponding to the
external conical surface, a, of the cock, at its junction with the
shoulders, d. Fig. 5 is a plan, looking on the cock from below,
and which shows the horizontal projection of the intersectional
curves.
The solution of the problem requiring the determination of these
curves consists in appljing a method already given — namely, in
taking any horizontal plane which cuts the cone, so as to present a
circular section on the one hand, and the cylinder in two straight
lines on the other — the points of intersection of these straight lines
with the circle representing the section of the cone. Thus, by
drawing the plane, c d, fig. 4, we obtain a circle of the diameter, c d,
which is projected horizontally, as with the centre, o, fig. 5 ; we
have also two generatrices of the cylinder, both projected in the
vertical plane in the line, a b, and in the horizontal plane in the lines,
a h' and a- h-. Hanng dra\vn the semibase of the cyhnder, d e, as
at d'fe', and having taken the distance,/^, fig. 4°, and set it off, in
fig. 6, from the axis, as from g> to a', and to a-, we thereby obtain
the generatrices, a' V and a- b^, which cut the circle of the dia^
meter, c' d', in the four points, h' i', which are squared across to,
and projected in, the vertical plane in the points, h, i. lu the
same manner we obtain any other points, as m, n; k I being the
plane taken for this purpose. The extreme points of the curve
are obtained in a very simple and obvious manner, as /, g, r, s,
being the points of intersection of the extreme generatrices, or
the outlines of the two solids. With regard to the points, t, u,
which form the apices of the two curves, their position may be
obtained from the diagram, fig. 4°, by drawing from the point, s,
which would be the apex of the entire cone, a tangent, s i, to the
base, d' fe', of the cylinder, and projecting the point of contact, t,
in the line, x y, representing a plane cutting the cone, which must
be projected in the horizontal plane. Then, making g' x', fig. 5,
equal to t V, fig. 4% and drawing horizontals through x', their
intersections, i' u', with the circular section of the cone, will
be the points sought, which are accordingly squared over to
fig. 4. The operations just described are analogous, it will be
observed, to those employed in obtaining the intersection of two
cylinders.
If, in the case of the cone and cylinder, the latter h.id been one
of circular instead of elliptical base, as is frequently the case, still
the constructior, as a little consideration will show, must be pre-
. isely the .same, and the resulting curves would be analogous — tliat
is. when the diameter of the cylinder is less than that of the cone
at .he part where it meefs the lowest generatrix of the cylinder ;
tne curves, however, assume a different appearance when the dia-
meter of the cylinder exceeds this, as is .siiown in figs. 6 and 7.
In this case the intersections are represented by the cur\-es, s I r
:ind p 11 q; the method of obtaining these is fully indicated on the
diii'Tiims.
185. The opening or slot, h, cut through the key of the stop-
cock, is generally rectangular, rather than circular, or similar to
the tubular portions of the cock. The object of this shape is to
make the key as small as possible, and yet retain the required
extent of passage. This rectangular opening gives rise, in fig. 2',
to the intersectional curves, a b, c d, which are portions of the
hyperbola, resulting from the section made by a plane, cutting the
cone parallel to its axis. The operations whereby they are deter-
mined are indicated in figs. 10, 11, and 12.
To render the character of the curve more apparent, we have,
in these figures, supposed the generatrices of the cone to make a
greater angle with the axis than in fig. 2'. The line, a b, repre-
sents the vertical plane in which the curve of intersection lies.
It is evident that, if we delineate a series of horizontal planes, as
c d, ef, ghfik, fig. 10, we shall obtain a corresponding series of
circles in the horizontal projection, these circles cutting the plane,
a b,m the points, I', m', n', p', &c. These points are squared over
to the vertical projection, fig. 10, giving the points, l,m,n,p; and
the apex, o, of the curve is obtained, by drawing in the plan,
fig. 12, with the centre, s, a circle tangent to the plane, a b, and
then projecting this on the vertical plane, fig. 10, as shown.
From these diagrams, it is easy to see that the opening, h, will be
partly visible when the key is seen from below% as in fig. 2'.
186. Figs. 8 and 9 represent the vertical and horizontal pro-
jections of the nut, g, which secures and adjusts the key in its
socket. This nut is hexagonal, being terminated by a portion of
a sphere, the centre of which lies in the axis of the prism. Each
of the facets of the prism cuts the surface of the sphere, so as to
present at their intersection portions of equal circles, which should
be determined in lateral projection. The diameter of the sphere
is generally three or lour times that of the circle circumscribing
the nut, but, to render the curves more distinct, we have adopted
a smaller proportion in the case under exammation. The sphere,
y, is represented by tw'o circles of the radius, o a ; and the nut by
an hexagonal prism, the a.xis of which passes through the centre of
the sphere. The anterior facet, a! b', of this nut, cuts the sphere,
so as to show a circle of the diameter, c' d'. This circle, projected
vertically on fig. 8, cuts the straight lines, a e and h f, of the
prism, in the points, a and b ; and the portion of the circle com-
prised between these two points, consequently, represents the
intersection of this facet witii the sphere. The other two facets,
a g' and b' h', which are inclined to the vertical piano, also cut
the sphere, .so as to produce, at their intersection with the surface,
arcs of equal radii with tiiat of tlie facet, a b. From their incli-
nation, these arcs become slightly elliptical, being comprised, on
the one hand, between the points, a and g, and b, h, on tlie other.
The summits of these ellipses are obtained by drawing horizontal
lines tangential to the arc, a b, and cutting it in the points, V, /,
by peqK'ndiculars drawn through the middle of the lateral fa«'-ts.
In practice, it is quite sutlicicnt to descrila' cia'ular arcs. pas,sing
through the points, g, k, a, and b, I, h.
Wo have already seen, in reference to Plato XIV., that the
intersection of a right cylinder witJi a sphoro, througli tlio centre
of which its axis passes, gives a circle pr<>joctod laforally !lh a
straight line. Thus, the opening o', which pas-scs throiigli tlio nut,
being cylindrical, produces, by Ha intersection with the sphcrr, a
I
J
58
THE PRACTICAL DRAUGHTSMAN'S
.Mivlo of the diameter, m' n', in tlie plan, projected vertically in the
straight line, m n.
Fig. 8° indicates the analogous operations required to determine
the same intersections when the nut is seen with one of the
angles in the centre, and only two facets visible, as represented in
fig. V.
The elliptic curve, b^ P h-, corresponding to the one, b 1 h, must
obviously be comprised between the same two horizontal lines
passing through these points, and an arc is drawn through them
as before.
We may here observe, that the proticient draughtsman will,
doubtless, deem it unnecessary, except in extraordinary cases, to
enter into such minute details of construction for the various
intersectional curves as those wc have discussed, being guided
simply by his own judgment, and the appearance presented by
ditferent experimental proportions. All draughtsmen, however,
will find that some practice in obtaining the exact representation
of the various curves, according to the methods here given and
rules laid down, will be of immense advantage to them — enabling
them, from possessing a tliorough theoretical knowledge of the rela-
tions of the various forms of solids to each other, to approach
much nearer truth, when, at a more advanced stage, they relinquish
tlie aid of such constructive guides.
RULES AND PRACTICAL DATA.
STEAM.
187. All liquids become changed to vapour when their tem-
perature is sufficiently elevated. When water, contained in a close
vessel, is elevated to a temperature of 212° Fahrenheit, it produces
steam of a pressure or force equal to that of the atmosphere.
The pressure of the atmosphere is a force capable of sustaining,
in a vacuum, a column of water 33 feet high, or a column of mer-
cury 30 inches high. This force is equal to a weight of about
15 lbs. per square inch.
Thus, taking the square inch as the unit of superficial measure-
ment, the pressure or tension of the steam at 212° Fahrenheit is
also equal to 15 lbs.
When the containing vessel is hermetically closed, as in a boiler,
if the temperature be increased, the steam becomes endued with
more and more expansive force — this increase of force, however,
not being directly proportionate to the increase of temperature.
The tension or expansive force of steam, as also of gases
generally, is inversely as the volume ; thus, at the pressure of one
atmosphere, for example, the volume of the steam or gas being
one cubic foot, the same quantity of steam would occupy only
half the space at a pressure of two atmospheres, and reciprocally.
TABLE OF PRESSURES, TEMPERATURES, WEIGHTS, AND VOLUMES OF STEAM.
Pressure in
Pressure in Inches
Pressure per Square
Temperature
Temperature
Weight of a Cuhic Foot
Volume of a Pound
Atmospheres.
of Mercury.
Inch.
Fahrenheit.
Centigrade.
of Steam.
of Steam.
Degrees.
Degrees.
Lb.
Cubic Feet.
1-00
30-0
15-00
212
100-0
-3671
2-7236
1-25
37-6
18-75
224
106-6
•4508
2-2183
1-50
45-0
22-50
234
112-4
•6332
1-8852
1-75
62-5
26-25
243
117-1
•6144
1-6281
2-00
60-0
30-00
250
121-5
•6936
1-4433
2-25
67-5
33-75
258
125-5
•7729
1-2938
2-50
75-0
37-50
264
128-8
-8491
1-1777
2-75
82-5
41-25
270
132-1
•9284
1-0771
3-00
90-0
45-00
275
135-0
1-0058
•9942
3-25
97-5
48-75
280
137-7
1-0826
-9237
3-50
105-0
52-50
285
140-6
1-1582
■8634
4-00
120-0
60-00
294
145-4 -
1-3086
•7642
4-50
135-0
67-50
301
149-1
1-4572
•6862
5-00
150-0
75-00
308
153-3
1-6033
•6236
5-50
165-0
82-50
314
156-7
1^7494
•5716
6-00
180-0
90-00
320
160-0
1^8946
•5273
6-50
195-0
97-50
326
163-3
2^0359
•4912
7-00
210-0
105-00
331
166-4
2-1777
•4583
8-00
240-0
120-00
342
172-1
2-4562
•4071
1
With the assistance of this table, we can solve the following
problems : —
First Example. — What is the amount of steam pressure acting
on a piston of 10 inches diameter, corresponding to a temperature
of 275 degrees? It will be seen that the pressure corresponding
to 275° is equal to three atmospheres, or to 45 lbs. per square
inch.
The area of a piston of 10 inches in diameter is equal to
102 X .7854 = 78-54 sq. inches;
consequently,
78-54 X 45 = 3534-3 lbs.
Thus, to solve the problem, we look in the table for the pressure
corresponding to the given temperature, and multiply it by the
area of the piston expressed in square inches.
Second Example. — Wliat weight of steam is expended during
each stroke of the piston, the length of stroke being 3 feetl
We first obtain the volume expended,
78-54
12
X 3 = 19-635 cubic feet.
BOOK OP INDUSTRIAL DESIGN.
59
At a pressure of three atmospheres, a cubic foot of steam weighs
I'OOoS lb.; consequently,
19-635 X 1-0038 = 19-75 lbs.
To sc)he this problem, then, we ascertain the volume expended
in cubic feet, and multiply it by the weight corresponding to the
gi\en temperature, or pressure — the product is the weight in
pounds.
UNITY OF HEAT.
188. With a \iew to facilitate various comparisons connected
with the subject of steam, the French experimentahsts have
adopted the term calorie, or unity of heat. This is defined as the
amount of heat necessar}' to raise the temperature of a kUogramme
(= 2-203 lbs.) of water, one degree centigrade.
Thus, a kilogramme of water at 25° contains 23 unities of heat ;
■Mi'l, in the same manner, 60 kilog. of water at 30' contains
30 X 60 = 3000 unities.
The number of unities of heat is obtained by multiplying its
weight in kilogrammes by the temperature in degrees centigi-ade.
The amount of heat developed by different descriptions of fuel
varies according to their quality, and according to the construction
of the furnaces.
According to M. Pdelet, the mean quantity of heat developed
by a kilogramme of coal is equal to 7300 calories, or unities of
\ieat.
According to M. Berthier, that developed by a kilogramme of
wood charcoal varies from 5000 to 7000 unities.
In the following table will be found the results of experiments
with different descriptions of fuel : —
Table of ihe Amount of Heat developed hy one Kilogramme of Fuel.
Description of Fuel.
Number of
unities of heat
developed by
1 kilo^.
Quantity of
steiim practically
obtainable from
1 kilojf.
Wood Charcoal
Coke
Medium Coal *^ . . .
Dry Turf,
Common Turf, containing 20 per cent, j
of water, ]
Inferior Turf^
Dry Wood of all descriptions,
Common wood, containing 20 per i
cent, of water,
Turf Charcoal
6000 to 7000
6000
7500
4800
3000
1500
3600
2800
5800
Kilog.
-6 to 6
" 8
•75 " 7
•8 " 2
3-7
2-7
1-8 to 3
In the last column of this table, we have given the quantities
of steam produced by the combustion of one kilogramme of fuel,
being such as are practically obtainable in apparatus most com-
monly met with.
Example. — What is the quantity of coal necessary for the sup-
ply of a furnace intended to produce 250 kilog. of steam ?
The average produce of 1 kilog. of coal being 65 kilog., wo
have
250
6-5
— 84 kilog. of coal.
of the shape represented in figs. 4 and 5, Plate XIV.— that is,
cylindrical, and terminated by hemispheres. They are frequentlv
accompanied by two or three tubular pieces in connection with the
main portion of the boiler by pipes. BoOei-s answering to thLs
description are termed French boilers, being of French origin ; they
are found very effective, and are much used in the manufacturing
districts of England. These boilers are made of plates of wrought^
iron, the thickness of which varies, not only according to the size
of the boilers, but also according to the pressure at which it is
intended to produce steam.
The proper thickness for the plates of cylindrical boilers may
be determined by the following formula, which is the one adopted
by the French Government in their police regulations : —
_ 18 X d X p
10
+ 3;
where
189. The boilers in which the .steam is to be produced, may bo
T = thickness in millimetres;
d = diameter of boiler in metres ;
p = pressure in atmospheres, less one.
The rule derivable from the formula is — ■
To multiply the eff'ective pressure of the steam in atmospheres
by the diameter of the boiler, and by the constant 18, dividing the
product by 10, and augmenting the quotient by 3, which will give
the thickness in millira^tres.
To simplify these calculations, we give a table showing the
thickness proper for boiler plates, calculated up to a diameter
of 2 metres, and to a pressure of 8 atmospheres above the atmo-
sphere : —
Table of Thicknesses of Plates in Cylindrical Boilers.
Diameter
of Bailer.
Pressure of Steam in Atmospheres.
2.
3.
4.
6.
6.
7.
8.
Metres.
Millim.
Millim.
Millim.
Millim.
Millim.
Ml lim.
MiUim.
-50
3-9
4-8
5-7
6-6
7-5
8-4
9-3
•55
4-0
5-0
6-0
7-0
7-9
8-9
99
•60
4-1
51
6-2
73
8-4
95
105
•65
4-2
53
6-3
7-7
8-8
10-0
112
•70
4-3
55
68
8^0
9-3
10-5
11-8
•75
4-3
5^7
7-0
8-4
9-7
11-1
12-4
•80
4-4
3-9
7-3
8^8
10-2
11-6
13-1
•85
4-5
6-1
7-6
91
10-6
12-2
13-7
•90
4-6
6-2
7-9
9-5
111
12-7
14-3
•95
4-7
6-4
8-1
9-8
11-5
13-3
160
1-00
4-8
6-6
8^4
10-2
12-0
13-8
15-6
MO
5-0
7-0
8-9
10-9
12-9
14-9
"
1-20
5-2
7-3
9-5
11-6
13-8
160
u
1-30
6-3
7-7
100
124
14-7
il
"
1-40
5-5
8-0
10-6
131
16-6
U
u
1-50
5-7
8-4
HI
13-8
i(
11
u
1-60
5-9
8^8
11-6
145
11
it
u
1-70
6-1
91
12-2
15-2
*^
u
It
1-80
6-2
95
13-7
160
11
u
u
1-90
6-4
9-8
13-3
(i
"
u
u
2-00
6-6
10-3
13-8
U
"
11
u
To suit English measures, the fomiuln is-
18_x_d_><ji
10,000
1182.
60
THE PRACTICAL DRAUGHTSMAN'S
And liere,
T = thickness in inches ;
d = diameter in inches ;
f = pressure in atmospheres, less one.
HEATIN& SURFACE.
190. In priictice it is generally calculated that a square m^tre
of heating surface will produce from 18 to 25 kilog. of steam per
hour, whatever be the form of boiler, whether cylindrical, with or
without additional tubes, or waggon-shaped.
The amount of heating surface, per horse power, generally
adopted, is from 1 to \b sq. m.
In this surface is not only included that which is directly
exposed to the action of the foe, but also that which receives heat
from the smoke and gases which traverse the flues ; this last being,
of course, mucli less eftective in the production of steam.
According to circumstances, one-half or a third of this surface
may be exposed to the du-ect action of the foe, which will give,
for the whole heating surface, two-thirds of the entire smface of
the boiler.
In the following table we give the principal dimensions, corre-
sponding to given horses power of boilers of the French descrip-
tion — that is, cylindrical with two tubes or smaller cylinders
below.
Table of Dimensions of Boilers and Thickness of Plates for a
Pressure of five Atmospheres.
Horses
Power.
Length of
Boiler.
Len^h of
ihe
two Tubes.
Diameter
of Boiler.
Diameter
of the
Tubes.
Thickness
of Plates for
the Boiler.
Thickness
of I*lates for
the Tubes.
jr.
M.
M.
M.
"/m.
"/..
2
1-65
1-75
-66
•28
8
8
4
2-10
2-20
•70
•30
8
8
6
2-VO
2-85
•75
•35
9
10
8
3-40
3-60
•80
•35
9
10
10
4-10
4-30
-80
■38
10
10
12
4-80
5-00
•80
•38
10
10
15
5-60
5-80
•80
•45
10
10
20
6.60
6-80
•85
•50
10
10
25
8-00
8-20
•85
•50
10
10
30
8-30
8-50
1-00
•60
10-5
10
35
9-50
9-70
1-00
•60
11
10
40
10-00
10-30
1-00
•70
11
10
In cylindrical boilers, without additional tubes, the water should
occupy two-thirds of the whole space, and in boilers, vidth the
tubes, it should occupy about one-half the main cylindrical body
of the boiler, in addition to the tubes.
In order that the steam may not carry along with it small
quantities of water, which action is tei'med '^priming," the boiler
is surmounted by a cylindrical chamber, or dome, in which the
steam collects, and from the highest part of which it makes its exit,
quite out of reach of the water thrown up by the ebullition.
CALCULATION OF THE DIMENSIONS OF BOILERS.
First Example. — What is the proper length for a cylindrical
boiler, without additional tubes, capable of supplying an engine of
6 horses power, supposing the diameter to be '8 m., and the heat-
ing surface 1-3 sq. m., per horse power?
We have ] •S x 6 = 7^8 sq. m., total heating surface.
Now, as the heating surface should be two-thirds of the entire
surface of the boiler, it follows that
2 ^ 4rt R
7-8 sq. m. = L X 2 rt R X — = L X — - — ,
where L represents the length sought, and R the semi-diametei,
= •4 m.; so that, substituting for rt and R their respective
numerical values, we have —
4
7-8 sq. m. =:: L X 3^14 x -4 x ^ = ^ x r676 ;
whence,
L =
_T8_
r675
X 4-65 m.
As it may be well to know the capacity of the boiler for watc;
and steam, this may be ascertained according to the rules pre-
viously given for the contents of cylinders, spheres, &c. (121 — 124^.
The boiler being terminated by hemispheres, the length of the
cylindrical portion will be equal to
4-65 — 04 X 2)= 3^85.
We shall have, then, for the volume corresponding to the
cylindi-ical portion —
V= Y X 3^14 X
•i- X 3^85 — 1^29 cub. ra.
and for that of the hemispherical ends —
2 4
— X — X 3^14 X •4=' = •ng cub. m.
The whole volume of water is, consequently,
r29 -1- -179 = 1^469 cubic metres.
The remainder of the volume, which is occupied by the steam
is obviously
1-469
= -734 cubic metres.
and the contents of the entire boiler,
1^469 + -134: — 2-203 cubic metres.
This result might have been obtained by the following general
formula : —
4
L X rt R^
+ -.R3=
3^85 X 3^14 X -42 -t- — X 3^14 x
2^203 cubic metres.
191. We here quote the portion of the regulations enforced by
the French Government, relating to the steam-boilers, as showing
what conditions are deemed necessary in France for the insurance
of public safety, and also as forming a good basis for calcula-
tions : —
" (33.) The boUers are divided into four classes.
" The capacity of the boiler, including that of the tubes, if
there be any, must be expressed in cubic metres, and the maxi-
mum steam pressure must be expressed in atmospheres ; and these
two quantities multiplied into each other.
" If the product exceeds fifteen, the boiler is of the first clasSy
It is of the second class, if the product is more than seven, but
does not exceed fifteen. Of the third, when more than three, and
not exceeding seven. And of the fourth, when not exceeding
three.
" If two or more boilers are arranged to work in concert, and
have any communication with each other, direct or indii-ect, the
BOOK OF INDUSTRIAL DESIGN.
61
term taken to represent the capacity must be the sum of the
capacities of each.
" (34.) Steam-boilers of the fost class must be stationed outside
of all dwelling-houses or workshops.
" (35.) Nevertheless, in order that the heat, which would other-
wise be dissipated by radiation, may be better economised, the
officer may allow a boQer of the first class to be stationed inside a
workshop, provided this does not form part of a dwelling-house.
"(36.) Whenever there is less than 10 metres in distance
between a boUer of the first class and a dwelling-house or public
road, a wall of defence must be built, in good and solid masonry,
and 1 m6tre thick. The other dimensions are specified in article
41. This wall of defence must, in all cases, be distinct from the
masonry of the furnaces, and separated from it by a space of at
least 50 centimeti-es in width. It must, in a like manner, be
separated from the intermediate walls of the neighbouring houses.
" If the boiler be sunk into the ground, in such a manner that
no part of it is less than 1 metre below the level of the gi-ound,
the wall of defence shall not be required, unless the boiler is within
5 metres of a dwelling-house or of the public road.
" (38.) Steam-boilers of the second class may be stationed inside
a workshop which does not form part of a dwelling-house, or of
a factory or establishment consisting of several stories.
" (39.) F a boiler of the second class be within 5 mefres from a
dwelling-house or the public road, an intermediate wall of defence
shall be erected, as prescribed in article 36.
" (41.) The authority given by the inspecting-officer for boilers
of the first and second class, shall indicate the situation of the
boiler, its distance from dwelling-houses and the public roads, and
shall determine, if there be space enough, the direction to be given
to the axis of the boUer.
" This authority shall also specify the situation and dimensions,
as to length and height, of the wall of defence, where this is
niquired, in conformity with the above regulations.
" In determining these dimensions, regard must be had to the
capacity of the boiler, to the pressure of the steam, as well as to
all circumstances tending to make the boiler more or less danger-
ous or inconvenient.
" (42.) Steam-boilers of the thii-d class may be stationed in
workshops which do not form parts of dwelling-houses, and it is
not necessary to erect a wall of defence.
•* (43.) Steam-boilers of the fourth class may be stationed in
aoy workshop, even if this forms part of a dwelling-house.
" In this case, the boilers must bo furnished with an open
manometer.
" (44.) The furnaces of steam-boilers of the third and fourth
class shall be entirely separated, by a space of at least 50 centi-
metres, from any dwelling-house."
According to these regulations, a boiler of the dimen.sions taken
for illustration, and supposing the maximum pressure to bo 3
atmospheres, would be in the second class, for — 2*203 x 3 =
•i-609, which is below 7.
Second Example. — What should be the (liiiiensions of a cyliii-
Jrical boiler, with two additional tubes, intended to supplv an
engine of 16 horses power, the diameter of the main body bcmg
■9 m., and that of the tubes -45 m. ?
Assuming 1-2 sq. m., per horse power, for the heating surface,
we shall have 1-2 x 16 = 19-2 sq. m., for the entire heating sur-
face. Half of the surface of the main cylinder of the boiler, and
three-fourths of that of the tubes, is the best disposition of this
heatmg surface. These data give rise to the following formula : —
,.„ 2rtRxL ^ , 3
19-2 sq. m. = -fr27i;xLx2x — =
2 4
rt R L -f Srt r L — .
L here represents the length of the boiler and tubes, which is
the only unknown term.
Substituting for R and r their numerical values, -45 and -225,
we have —
19-2 sq. m. = 3-14 x -45 x L -|- 3 x 3-14 x -225 x L; or
19-2 sq. m. = L (3-14 x -45) -f 3 x 3-14 x •225 —
L (1-413 -f 2-12);
whence,
19-2 19-2
L =
= 5-43 m.
1-413 -f- 2-12 3-533
Thus, the total length of the boiler is 5-43 m., but the ends
being hemispherical, the length of the cylindrical portion is
equal to —
5-43 — -9 = 4-53 m.
The tubes usually project in front of the main body, to a dis-
tance of about 50 centimetres; but, for convenience in constructing
the return flues, they do not extend as far back, so that they aro
of about the same length as the main body.
192. In distillery boilers, a horse power is understood to mean
the capability of evaporating 25 kilogi-ammes of water in an hour.
Thus, a boiler of 10 horses power should be capable of evaporating
250 kilog. of water in that time. Now, assuming 1-12 sq. m. of
heating surface, per horse power, for a steam-engine, we should
only have an evaporation of from 18 to 20 kilog. per hour, per
horse power, and per square metre of heating surftice.
DIMENSIONS OF FIRE-GRATE.
193. In practice, 1 square metre of grate will burn from 40 to
45 kilog. of coal per hour. Thus, a boiler intended to produce
280 kilog. of steam per hour, will require, for this purpose —
assuming that 1 kilog. of coal produces 6-65 of stoam —
280
"TTTTT = 43 kilogrammes of coal ;
6-65 •=
and the fui-nace of this boiler should have a grate, measurin;,' 1
square metro.
The grate-bars aro generally of cast-iron, of from 30 to 35
millimetres in width, but having between lluni a siuico of only
7 or 8 millimetres, so that tho intervjJs only t>i'cnpy a fourth or
a filth of tho whole area.
It has been found that greater strength and dnrahility i.-»
obtained by making tho bars straight above, and strengthened by
])arab(ilic featliers below.
CHIMNEYS.
191. 'riu> luit^ht of i-liiuMieys is very varialile, and oannol Ih»
subjected to ;tny fixed rule. 'I'lie cross section at the sunuiiit
depenils upi»n the size of iJio grille luul i.s generally iduuil u sixth of
62
THE PRACTICAL DRAUGHTSMAN'S
this. In the followmg application will bo found calculations
respecting chimneys, and examples of the various rules we have
just given.
APPLICATION.
We propose calculating the dimensions of the furnace of a
boiler, with its chimney, for an engine of 8 horses power, for
example, to be worked on the high-pressure system, consuming, as
a maximum, 5 kilogrammes of coal, per horse power, per hour, the
amount of heating surface being taken at 1-52 sq. m., per horse
power.
For 8 horses power, the heating surface will be —
1-62 X 8 = 12-16 sq. m.
Each square metre of heating surface producing, at an average,
18 kilogrammes of steam, we have —
12-6 X 18 = 218-88 kilog. of steam.
As 5 kilog. of steam are produced by 1 kilog. of coal, then —
218-88 „,.,
— - — — 43-8 kilog.,
representing the quantity of coal consumed per hour.
The grate area, corresponding to this consumption, assuming
that one square decimetre is sufficient for 1-2 kOog. per hour, will
be—
43-8
1-2
= 36 square decimetres,
supposing a fourth of this area to be free to the passage of air.
It now only remains to calculate the cross sectional area of the
chimney. With reference to this we must remark, that 18 cubic
metres of air are required for the consumption of 1 kilog. of coal ;
therefore, 43-8 kUog. will require
43-8 X 18 = 788-4 cubic metres.
This air, in traversing the fire, relinquishes a portion of its
oxygen, which is partially replaced by carbonic acid gas and steam.
If the gases escape from the chimney at a mean temperature of
300° centigrade, the volume being, according to M. Peclet, at the
rate of 38-54 cubic m. per kilog. of coal, will be 43-8 x 38-54 =
1 688 cubic metres per hour. If we divide this by 3600, we shall
obtain the quantity which escapes per second ; namely,
1688
3600
= -4689 cubic m.
K we assume, as is usually the case with boilers of the propor-
tions here discussed, that the chimney is 22 metres high, the
external atmosphere being at a temperature of 15° centigi'ade, the
rate of exit of the gases may be obtained by the following formula : —
V = V2g H a (t'—t).
In the case under consideration, H =: 22 m., a is the constant
multiplier, -00365, t' = 300°, t = 15°, and 2g — 19-62. Substi-
tuting, then, for the letters their numerical values, we have
V = i^ 19-62 X 22 X -00365 x (300 — 15) = 21.
This signifies that the gas will escape from the chimney-top at
the rate of 21 metres per second, if it meets with no resistance
from the lateral surfaces of the flues and chimney; the actual
rate, however, is only 70 per cent, of this — or,
21 X -7 = 14-7 m.
If we divide the volume of gas which escapes per second by
^Jie rate at wliich it escapes in that time, as just determined, we
shall obtain the cross sectional area proper for the upper part 01
the chimney ; as thus —
•4689 „„ J. • .^
=3-2 square decmaetres.
Thus the chimney, which is supposed to be square, will only
require to measure, internally, something less than two deci-
metres each way at the point of exit; this, however, is a mini-
mum dimension, and it vidll be advisable to give it gi-eater dimen-
sions than these. Thus it might be made 25 centimetres square,
or even 30 or 35 centimetres, if there is any likelihood of the
power of the boiler being increased afterwards, such mcrease being
frequently called for in manufactories. A damper, however,
should always be provided at the base of the chimney, by means
of which the draught may be suited to the requu-ements.
SAFETY VALVES.
Table
oj Diameters of Safety
Valves.
Pressu
re in Atmosph
sres.
Extent of
Henting
Surface.
H
2
H
3
3*
4
5
6
7
8
Sq. M.
"/n.
"/m.
""L
"/m.
«/m.
"/m.
"/m
"/m.
-"■/n,.
"/m
1
25
21
18
16
15
14
12
11
10
9
2
35
29
25
23
20
19
17
15
15
13
3
48
86
31
29
26
24
21
19
17
15
4
50
41
36
32
29
27
24
22
20
19
5
56
46
40
36
33
30
27
24
22
21
6
61
60
44
39
36
34
30
27
25
23
7
66
54
48
43
39
36
32
29
27
2S.
8
70
58
51
46
42
39
34
81
29
27
9
75
62
54
48
44
41
36
33
30
28
10
79
65
67
51
47
43
38
35
32
80
11
83
68
60
54
49
45
40
36
33
31
12
87
71
62
56
51
47
42
38
85
33
13
90
74
65
58
53
49
44
40
86
34
14
98
77
67
60
55
51
45
41
37
35
15
96
80
70
62
57
53
47
42
38
36
16
100
82
72
65
59
55
48
44
40
38
17
103
85
74
67
61
56
50
45
42
39
18
106
87
76
68
63
58
51
47
43
4C
19
109
90
78
70
64
60
53
48
44
41
20
111
92
80
72
66
61
54
49
45
42
21
114
94
82
74
68
63
56
50
46
43
22
117
97
84
76
69
64
57
51
47
44
23
119
99
86
77
70
66
58
53
48
45
24
122
101
88
79
72
67
59
54
49
46
25
125
103
90
81
74
69
60
55
50
47
26
127
105
91
82
75
70
62
56
51
48
27
129
107
93
84
77
71
63
57
52
49
28
132
109
95
85
78
73
64
58
53
50
29
134
111
97
87
80
74
65
59
54
51
80
136
113
98
88
81
75
66
60
55
52
32
140
116
100
90
82
76
67
62
57
53
34
145
119
104
94
86
79
69
64
59
55
36
149
122
107
96
87
82
71
65
61
57
88
151
125
110
97
90
83
74
66
62
58
40
156
130
113
101
92
86
75
69
64
59
45
167
137
119
107
97
91
80
73
68
63
50
174
1«
125
113
104
96
84
76
70
67
55
184
151
132
119
107
101
88
80
75
70
60
193
158
137
121
118
106
94
84
78
73
195. Steam-engine boilers are always provided with various
accessories, as safety valves, manometers, floats, alarm whistles.
The manometer is an instrument which serves to indicate the
pressure of the steam inside the boUer in atmospheres, and frac^
tions of atmospheres. These instruments are constructed aftei
various systems.
BOOK OF INDUSTRIAL DESIGN.
63
The float serves to indicate the level of the water, and the
whistle to give the alarm when the water is much below the pro-
per level.
The safety valve pro\ides an exit for the steam when the pres-
sure is too high.
We have given a drawing of one at fig. 4, Plate XI. Their
diameters vary with the dimensions of the boilers and the pres-
sure of the steam.
The regulations of the French Government contain the following
rules and the above table for their determination. To find the proper
diameter for the safety valve, the heating surface of the boUer,
expressed in square metres, must be divided by the maximum
pressure of steam intended to be mamtained, expressed in atmo-
spheres, previously diminished by the constant -412: the square
root of the quotient being extracted, is to be multiplied by 2-6,
and the product will be the diameter sought, expressed in centi-
metres. This rule may be put as a formula, thus : —
d= 2-6
V 71 — -412 '
where d is the diameter of the \alve in centimetres, s the heating
surface of the boiler, including both fire and flue surface, e.xpressed
in square metres, and n the number expressing the pressure Id
atmospheres.
CHAPTER V.
THE STUDY AND CONSTRUCTION OF TOOTHED GEAR.
196. Toothed gear is a mechanical expedient, universally
employed for the transmission of motion. It is met with of all
proportions, from the minute movements of the watch, to the
^gantic fittings of manufacturing workshops. Toothed gear is
generally constructed with a view to the following principle of
action — that the lateral acting-surfaces develop the same arc during
the same duration of contact, whilst their angular velocities
vary inversely as their diameters. By the angular velocity of
any body, turning about a centre, is meant the angle passed
through by the body in a unit of time ; whilst the real or linear
velocity of any point is the space passed through by this pomt,
whether the direction of motion be rectilinear or circular. Thus,
various points on a crank, taken at different distances from the
centre of the shaft, have all the same angular velocity, whilst their
actual velocity differs considerably, because of their respective
distances from the centre. It is the same with a pendulum, which
vibrates through an angle, or has an angular motion about its
centre of suspension. The angular velocity of a body is greater,
JUS the angle passed through m the same time is greater. Two
points may have the same angular velocity, although the space
passed through by each may be very different. Thus, all the
points in the pendulum are affected with an equal angular motion,
wliilst their actual velocities, or the course traversed by each,
vary as the distance from the centre of motion.
This description of gear consists of a series of projections, or
teeth, regularly arranged on straight, cylindrical, or conical sur-
fswes, termed webs, and disposed so as to act on each other during
a limited time.
In order, however, that the gearing action may take jilaco in a
regular, even manner, it is indispensably necessary that tiic sur-
faces of the teeth should bear upon each other (aiigcntiaiiy,
throughout the entire duration of their contact ; and lor this pur-
pose, far from being arbitrarily designed, their I'niin should b((
determined with the utmost geometrical exactitude, lor on their
form entuely depends their accurate and easy working. It is.
therefore, obviously incumbent on the student to give particular
attention to the delineation of these teeth.
The curves generally adopted in practice for the outline of
teeth, are the involute, the cycloid, and the epicycloid.
It is useful to Investigate the nature and construction of tnese
curves, both on account of their application to the teeth of wheels,
and also because of their employment in several other mecnanical
contrivances.
INVOLUTE, CYCLOID, AND EPICYCLOID.
PLATES XVIII. AND XIX.
INVOLUTi:.
Figure 1.— Plate XVIII.
197. \Vlien a thread is unwound from the circumference of a
circle, and is kept uniformly extended, its extremity will describe
the curve known as the involute.
This definition serves as a basis for obtaining the geometrical
delineation of the involute. Let a b c be the given ciri-le of
the radius, a o, and a the extremity of a tlire.id wound upon it.
Starting from the point, a, mark off, at equal distances apart,
several points, as a, h c, so near to each other, that the interven-
ing arcs may be taken for straight lines without sensible error.
Through each of these points draw tangents to the cin'le, or jH>r-
pendiculars to the corresponding radii ; and ou these t.angents sot
ofi' distances, equal to the rectifications of tlio respective arcs, a a,
A b, A c, &c. ; by which means are obtained the points, a', b\ r', &o.,
and the curve passing through these points is a portion at' the
tnolulc. Hy continuing Uio development or unwinding of the
hrcad, the curve may bo extended to a series of convolutions,
lUTcasing more and more in radius, and becoming a spwies of
spiijil. .M'ler one complete evolution of the cin-uniference, the
shortest distance between two oonsivutivo convolutions is always
the sjune, and eipinl to Uie development or rectification of tJio
64
THE PRACTICAL DRAUGHTSMAN'S
i-ia-umference of the generating circle, which forms the nucleus of
tlie curve.
The points, a, i, c, being taken at equal distances apart on the
circumference, the tangents are respectively double, triple, >&c.,
that of tlie first, a a ; and if, as we directed, these points are suffi-
ciently near to each other, the curve may be dra\vn, with closely
approximate accuracy, by describing a succession of arcs, having
these tangents for radii. Thus, witli the point, a, as centre, and
radius, a a', the first arc, a a', is di-awn ; and ^vlth the centre, h,
and radius, b b', the second arc, a' b', in like manner; and similarly
with the rest.
We shall show the application of the involute to toothed gear,
worm wheels, and also for cams and eccentrics.
FiGUKK 2.— Plate XVIIL
198. When a circular disc is rolled upon a plane surface in a
rectilinear direction, any point in the circumference of this disc
generates the curve called the cycloid. Thus, any point taken on
the outside of a locomotive wheel in motion, describes as many
repetitions of the curve as the wheel makes revolutions.
In order that the curve may be perfect and true thi'oughout,
it is necessary that the motion should take place without any slid-
ing upon the plane ; in other words, the length of the straight line
forming the path of the disc should be equal to the portion of the
circumference which, during the motion, has been applied to, or in
contact TOth, the plane throughout that length.
We propose to delineate the cycloid generated by the point, a,
of a circle of the given radius, a o, and rolling upon a given
straight line, b c.
There are several methods of sohing this problem.
Isi Solution. — Set off on the circumference, starting from the
point, A, a number of distances equal to a a, so small that the arcs
so divided may be taken as straight lines. Set off the same dis-
tance a like number of times along the straight line, a c, and at
the points, a b c, erect perpendiculars, cutting the line, o o', gene-
rated by the centre of the rolling circle, and parallel to the given
straight line, b c. Li this way are obtained the points of inter-
section, o, o^, 0-, which are the centres of the circle when in the
positions corresponding to the points of contact, a, b, c, d. With
each of these points as centres, then describe portions of circles,
on each of which successively set off the lengths of the arcs,
a a', A b', A d, &.C., from a to a", 6 to a^, and from c to b^, and so
on throughout. The curve, a a" a- b^ c-, passing through the
points thus obtained, is the c)-cloid required.
2d Soluiion. — The points in this curve may also lie obtained by
drawing horizontal lines through the points of division, a' b' c', of
the original circle, and then intersecting these by the arcs drawTi
with the respective centres, o, o*, o-.
3d Soluiion. — In place of dravring arcs of circles -nith the various
centres, as indicated on the right-hand side of fig. 2, the curve
may be obtained by setting off successively from the vertical, a o,
on the horizontals, as before drawn, distances equal to those
respectively contained between the original circle and the perpendi-
culars thi-ough the several corresponding positions of the centre ;
thus, the distances, e e'.,ff,gg', h h', &c., are set off from 1 to a-
2 to b", 3 to C-, &c.
To avoid confusion, we have constructed the diagram appertain
ing to this last solution to the left-hand side of fig. 2, which showa
a portion of a second cycloid similar to the first.
When the generating circle has made half a revolution, the sum-
mit of the curve is obtained, as at d', the point corresponding to
the diameter, a d. The length, a c, of the given straight line, is
obviously equal to the rectification of the semi-circumference of the
generating circle, whose radius is a o.
By continuing the construction, a complete curve may be
obtained, having equal and s3"mmetrical portions on either side of
the vertical, c d, and havmg for its base a line double the lengtli
of A c, and cons3quently equal to the rectification of the entire
circumference of the generating circle.
The cycloid is the curs'e more generally given to the teeth of
wheel gear and endless screws.
EXTERNAL EPICYCLOID.
FiGUSE 1. — Plate XIX.
199. The epicycloid only differs from the cycloid, in that the
generating circle, instead of rolling along in a straight line, does
so ai-ound a second cu-cle, which is fixed. When the two circles
are in the same plane, the point taken generates a right or cylindri-
cal epicycloid ; when the two circles are situate in different planes,
but maintaining a uniform ang-le to each other, the generated curve
becomes a spheric epicycloid; in this case the generating circle is
supposed to revolve about a fixed centre, at the same time rolling
along the circumference of the stationary circle.
1st Soluiion. — For the delineation of the right epicycloid, the
methods of construction to be adopted are analogous to that ^ven
for the cycloid. Thus, let a o be the radius of the generating
circle, and a c the radius of the fixed circle ; divide the former into
a number of equal parts in the points, a', b', c', d', &c., and on the
latter divide off as many arcs equal to the arcs of the former, start-
ing from A, as at a, b, c, d, &c. Through these latter points of divi-
sion draw radii, ca,cb,cc, and prolong them so as to cut a circle,
the radius of which is c o ; this circle being generated by the centre
of the moving one during its rotation about the stationary one ; in
this way are obtained the points, o, o', o-, o^, which are the succes-
sive positions of the centre of the generating circle, as during its
rotation it is successively in contact at the points, a, b, c, d, of the
fixed circle. Then, with these points as centres, describe the
several arcs of equal radii \\'ith the generating ch-cle, making them
severally equal to the corresponding arcs, a a', a 6', a c', as from
a to a-, b to b-, cto c-, &c. The curve passing through the points,
a-, b~, C-, is the epicycloid requu-ed.
2d Solution. — The points of this curve may also be determined
by drawing, with the centre, c, arcs passing through the points of
division, a', b', c', d', and cutting the arcs described with the various
centres, o, oS o-, o^, in a-, b^, c", d-, which are so many points in
the epicycloid.
3d Soluiion. — The curve may also be delineated by transferring
the distance between the points, e, f, g, &c., of the generating
cucle in its original position, and the radii, c ^ c i, c k, passing
BOOK OF INDUSTRIAL DESIGN.
66
through the different points of contact on the stationary circle,
measured upon the arcs described with the centre, c, to the same
arcs, but so that the extremities of the whole may lie in the pro-
longation of the radius, c b. Thus the distances, e e',ff, gg\ &c.,
are set off, from 1 to a-, 2 to J -, 3 to c-, &c. The diagram refer-
ring to this construction forms the right-hand portion of fig. 1.
When the generating circle has made an entire revolution, the
em've obtained is an entire epicycloid, a d b, comprising two equal
md symmetrical portions on either side of the line, d e, which is
»qual to the diameter of the moving circle.
EXTERNAL EPICYCLOID DESCRIBED BY A CIRCLE ROLLING ABOUT
A FIXED CIRCLE INSIDE IT.
Figure 3.— Plate XIX.
200. For this diagram, which is analogous to the preceding one,
the radii of the circles are given, c a being that of the fixed circle,
and B A that of the moving one. Divide the first circle into any
number of equal parts, in the points, a, i, c, d, &c., and divide off,
on the larger circle of the radius, b a, a like number of arcs, equal
to those on the other circle, as from a to a', a' to b, &c. Then
with the point c as centre, and with the radius b c, describe a
circle, cutting the radii, c a, c a, c ft, c c, in the points, b, b', b', b^,
and with each of the last as centres, and with the radius, a b,
describe arcs, which will be tangents to the tixed circle, at the
different points of contact, a, b, c, in succession. Then, with the
centre, c, describe arcs, passing successively through the points,
«', b', c', d', on the moving circles, as in its first position. These
Vist vvill cut the arcs tangential to the given circle, in the points,
t^, 6', c^, d^, and the curve passing through these points is the
ipicycloid sought.
The other two methods given, of drawing the common epicy-
' )■ J, are also applicable to this last case.
INTERNAL EPICYCLOID.
Figure 2. — Plate XIX.
201. The epicycloid is termed internal, when the generating
circle rolls along the concave side of the cu-cumference of a fixed
circle.
Let c A be the radius of the fixed circle, and b a that of the
generating circle. As in preceding cases, so also here, wo
commence by dividing the moving circle into a certiiin number of
equal parts, and then dividing the fixed circle correspondingly, so
that the arcs thus obtained in each may be equal. We then pro-
ceed as in the case of the external epicycloid, according to which-
ever of the three solutions we propose adopting, all being alike
applicable. The operations are fully indicated on fig. 2, and the
same distinguishing letters are employed as in fig. 1.
When the generating circle is equal to half of the fixed circle,
the epicycloid generated by a point in the circunif'eronco is a
straight line, equal to the diameter of the fixed circle. Thus, in
fig. 3, Plato XVIII., the epicycloid generated by (ho point, a, of
the moving circle of the radius, A c, after a semi-revolution, coin-
i'i(l(^H exactly willi the diameter, a b.
II', wilh circles of the same i)roi)or1i(ins iis those in fig. 3,
Plate XVIII., we take a point, d, outside tlu^ generating circle,
Inii preserving a constant distjinco from it, the epicycloid generated
by it will be the ellipse, d f e g, having for its transverse axiS the
line, D E, equal to the diameter, a b, of the fixed cu-cle, augmented
by twice the distance, d a, of the point, d, from its extremity ; and
for conjugate axis, the line, g f, equal to twice the same distance,
D A, alone. If it is wished to determine this curve according to
its properties as an epicycloid, and without having recourse to the
methods given in reference to Plate V., and proper to the ellipse,
it may be done by adding the distance, a d, to that of the radius,
c A, in each successive position occupied by the generating circle
during its rotation. If the generating point be taken mside the
moving circle, the curve produced wUl also be an ellipse.
The epicycloid is the curve most employed for the form of
the teeth, whether of external or internal spur or be\'il wheels.
Toothed gearing may be divided generally into two categories ;
namely, right, cylindrical, or " spur " wheels, and conical, angular,
or " bevil " wheels. In the first are comprehended the action of a
rack and pinion, that of a worm or tangent-screw with a worm-
wheel, and finally, that of two wheels. We may remark, that in
all these modes the teeth are so formed and arranged, as to act
equally well whichever of each couple be the driver, and in which-
ever dii-ection the motion takes place.
THE DELINEATION OF A RACK AND PINION IN GEAR.
Figure 4.— Plate XVIII.
202. A rack is a species of straight and rigid rod or bar, formed
with teeth on one side, so as to take into or gear with the teetn of
a right wheel, generally of small diameter, and in such case tenned
a pinion. Such a rack is represented at a b in the figure.
In proceeding to construct this design, as well as for all kmds
of toothed gear, it is necessary to have determined beforehand
the thickness, a b, of the teeth, as this dimension varies accord-
ing to the power or strain to be transmitted; and rules ;md tabies,
for this purpose, will be found at the end of the chapter.
When the rack and pinion are made of tlie same met;xl, tiie
thickness of the teeth should be the same in both. The spaces or
intervals between the teeth ought also to be equal in such case.
Theoretically speaking, the intervals should be equal to the thick-
ness of the teeth ; but in practice, they arc made a little mder, to
admit of freer action.
203. The pitch of the teeth comprises the width of tin liuitli
and that of the interval. In a wheel this pitch is measured upon
a circle of a given radius, termed the pi'imilire or pitch circle, and
in the rack on a straight line tangent to the pitch I'ircle of the
pinion, and also called the priiniliie or pitch line.
204. Let o c be the radius of the pitch circle of a pinion L>v;ir-
ing with a rack, of whicii the jtitch line is a b. Wo i>roposo, ni
tho first place, to detei-inino the curve of the teeth of the pinion,
so as to gear with and drive the rack, and we shall subsequently
deterniino the curve of the teeth of the r;u'k, enabling it to gear
with and drive tlu^ pinion.
'J'lie (i|)eiati()ns consist in rolling the straight line, A c, uiugon-
tially to the i>itch circle, o c : during this movement, tiio ponu, c.
will "■enerato an involute, c n, whicii may be drawn in liio niannor'
inilicaled in lig. 1 — a constrnctioii wliifli is t'uitlier tvpeatcd ut
ii' (/'. (Ml one of the teeth of the pinion, lii,'. 4.
This cuiN e possesses this property, that if tiio toetli are t'ornu-d
t
66
THE PRACTICAL DRAUGHTSMAN'S
to it, and the pinion be turned on its axis, the point of contact, c,
uill always be in the straiglit line, a, b, traversing this line at pre-
cisely the same velocity as the pinion at that distance from the
ccnti-e, that is, at the pitch circle; consequently, we divide this
pitch circle into as many equal parts as there are to be teeth and
intervals in the pinion, and at each of the points of division repeat
the involute curve, c d, which will, of course, fulfil the same con-
ditions at the various positions ; then, each of these divisions recti-
fied is set off on the pitch line, a b, of the rack, as many times as
is necessary. For each tooth the curves are placed symmetrically
with reference to the radius which passes through their centres, as
indicated at o df, so that the pinion may act equally well when
turning in one direction as in tlie other.
205. Smce the teeth cannot have an indefinite length, they may
be limited as far as is compatible with the following considera-
tions: — The tooth of the wheel, which is the di-iver, should not
relinquish contact with the one upon which it acts, until the tooth
immediately succeeding it has taken up its original position, which,
in the working of two wheels, corresponds to the line joining the
centres, and in that of a pinion and rack, to the radius, o c, per-
pendicular to the pitch line, a b.
Thus, supposing the pinion to move in the direction indicated
by the arrow, the tooth, e, which is acting on the tooth, h, of the
rack, should continue to impel it until the following tooth, g, shall
have taken its place, when it will itself have taken the place of the
tooth, F, having made the tooth, h, of the rack traverse to i. It
will be observed that the curved part of the tooth is in contact at
the point, c, on the pitch line, a b ; the tooth might be cut away at
this point; but in practice, in order that the pinion teeth may act
thi-ough a somewhat greater interval, and to avoid the play results
ing from wear, they are truncated at a little beyond this point, c,
a circle being described with the centre, o, cutting the cm-ves of
all the teeth at equal distances from the centre.
To allow of the passage of the curved portion of the teeth of
the pinion, the rack must be grooved out, so as to present bearing
surfaces, which are determined simply by the perpendiculars, bf,
c d,g b, to the pitch line, a b, and passing through the points of
division already set out on this line.
These perpendiculars, at the same time, form the sides or flanks
of the rack teeth.
Rigorously speaking, the depth of the intervals on the rack
should be limited by the straight line, m n, tangential fo the exter-
nal circle of the pinion ; but, to prevent the friction of the teeth
against the bottom, it is preferable to augment the depth of the
tioUows by a small quantity, joining the sides of the teeth ^vith
the bottom by small quadi-ants, which, avoiding sharp angles,
£rives greater strength to the teeth.
206. As in practice, toothed gear is constructed so as to drive,
or be driven, indifferently, we require yet — to complete the design
under consideration — to give to the teeth of the rack such cxu-va-
ture as is necessary to enable them to drive the pinion with which
they are m gear in their turn, always fulfilling the conditions of a
-egular and uniform motion, both of the rack at its pitch Hue, and
if the pinion at its pitch circle.
With a view to the determination of this curve, we may remark,
w at if. with the radius, o c, as a diameter, we describe the circle,
o L c, and cause it to roll along the straight line, a b, the point of
contact, c, will generate a cycloid, c k, which may be constructed
according to the methods indicated in fig. 2.
If the same circle is made to roll along the mterior of the pitch
circle, g c j, of the pinion, the same point, c, will generate a right
epicycloid, coinciding with the radius, o c, as has been seen in
reference to fig. 3.
Then, if we give to the teeth of the rack the curve, c k, and to
the flanks of the pinion teeth the straight line, c o, the arrange-
ment will exactly fulfil the condition sought ; that is to say, that,
in impelling the pinion teeth from right to left, the curve, c k, of
the rack teeth will constantly apply itself to the straight Ime, o c,
being always tangential to it.
For example, suppose the curve, c k, to be traversed to the
position, c' l, the radius, o c, will then be in the position, o l ;
then, if from the point, l, the straight line, l c, be drawn, the
angle, o l c, will be a right angle ; that is to say, the line, o l, will
be perpendicular to l c, and, consequently, tangential to the curve,
L c', in the point, l. If, therefore, the motion of the rack is
regular and uniform, that of the pinion will be equally so. The
same curve, c k, is drawn at each of the points of division of the
pitch line of the rack, as was already done for the teeth of the
j pinion.
To find the proper length to give to the teeth, all that is neces-
sary is to place, in the generating circle, o l c, a chord, l c, equal
to twice c &-, and through the point, l, thereby obtained, to draw
a straight line, m n, parallel to a b. If, through all the points of
division in the pitch circle of the pinion, are drawn radii con-
verging in the centre, o, they will give the flanks of the teeth, as
i J, k I, &c., which are limited by a cu'cle described with the centre,
o, and tangential to the straight line, m n ; for the same reason as
that assigned in the case of the rack, however, the spaces between
the teeth are made a little deeper, and the sides of the teeth are
joined to the bottoms by quarter circles, the circle in which the
bottoms lie being described with a radius somewhat less than that
of the circle last drawn.
As it would be a tedious process to repeat the operations for
determining the curves in the case of each individual tooth, it is
a convenient plan to cut a piece of card or thin wood to the curve,
so as to form a pattern or template, by the application of which to
each of the points of division, the sides of the teeth may be drawn,
care being taken to make the two sides of each perfectly symme-
trica] with reference to the centre line of the tooth.
Even the labour of making a template or pattern is often dis-
pensed with, and, in place of the ciirve, a simple circular arc is
employed for the side of the tooth, the arc being of such a radius
as to approximate as near the true curve as possible. With this
view the arc should be timgential to the side of the tooth, and
passing through the external comer. Thus, supposing it is
wished to substitute an arc for the true curve of the rack teeth,
such as o r of the tooth, p, since this arc has to pass through the
point, r, corresponding to l, and obtained by making r' r equal to
L q, and to be a tangent at o, to the vertical, o p, draw the chord,
o r, and bisect it by the perpendicular, s t, and its point of inter-
section, s, with the pitch line, a b, wOl be the centre of the
required arc, and the sides of all the teeth may afterwards be drawn
BOOK OF INDUSTRIAL DESIGN.
67
VFith the same raoius, care being taken to Keep the centres in the
line, A B.
An analogous operation will give the proportions of the arc,
suhstituting the curve of the pinion teeth.
THE GEARING OF A WORM WITH A WORM-WHEEL.
Figures 5 and 6. — Plate XVIII.
207. This system of gear is constructed on the same principles
as that of a rack and pinion, which method requires that, in the
first place, the worm and worm-wheel be supposed to be sectioned
by a plane passing through the axis of the former, and at right
angles to that of the latter. The representation of this section
becomes analogous to the diagram, fig. 4 ; that is to say, the pitch
circle, g c j, of the worm-wheel being given, and also the straight
pilch line, a b, of the worm tangential to this circle, and parallel
to the axis of the worm, the involute curve, c d, is sought for the
teeth of the wheel, and the cycloid, c k, for those of the worm.
The lengths of these curves are limited, as in the preceding e.\am-
ple, and when the whole is complete, an outline will be produced
similar to the tinted portions of fig. 6. It is in this manner that
the gearing of the worm and worm-wheel is made to depend upon
the same principles as that of a rack and pinion, and the same
method may be employed in construction in determining the outline
of the teeth, as we h.ave shown.
To represent the worm and worm-wheel geometrically in exter-
nal elevation, instead of a section of the teeth alone, it is necessary
to know the diameter and pitch of the worm on the one baud, and
the thickness of the worm-wheel, fig. 5, on the other.
Let m' a' be the distance of the pitch line, a b, from the axis,
m' n, of the worm, and a b the width of the wheel. When the
worm is single-threaded (177), the pitch of the helix is the same
as that of the teeth, and, therefore, the thickness of a tooth, added
to the width of an interval. In this case, each revolution of the
worm turns the wheel to the extent of one tooth, and this is the
arrangement represented in the figures. If the worm, however,
is double or triple-threaded, its helical pitch will be correspond-
ingly two or three times the pitch of the teeth; and in such case,
each revolution will turn the wheel to the extent of two or three
teeth.
The worm-wheel being of a certain thickness, and requiring to
gear with the convolutions of the worm, must necessaiily have its
teeth inclined to correspond with the obliquity of the worm-thread.
It is further to be observed, that the sides of the wheel-teelh
being simply tangential to the worm-thread, contact cannot,
rigorously speaking, take place in more than one point of each
tooth and convolution. This point constantly changes wilii liic
motion, but always lies in the plane, o' m', of the section.
In delineating the convolutions of the worm-thread, helices
have to be drawn passing tiu'ough the external corners, d, e, and
internal corners, /, g-. Wo have repeated these points to the left-
hand side of fig. 6, where the required ojjeralions are fully indi-
eated, in connection with tho i)r()jeclion, fig, 0, and in accordMiicu
vvilh Iho principles already explained (173). The corresponding
points in tiio two figs. (5 and (i) are distinguishod by llu' s.-iiuc
k'tters and numbers.
208. For the representation, in external elevation, of the teeth
of the worm-wheel, it is required to develop a portion of the
cylindrical surface generated by the revolution of the pitch-line,
A B, about the axis of the worm, and containing the portion,
A ik I m, tor example, of the helix, described by the central point
of contact, A. To obtain this, make the line, e' a', fig. 7, equal to
the semi-circumference, a' m e^, rectified. At the point, e', erect
the perpendicular, c' e', and make it equal to c e, fig. 6, or half
the pitch, and join e' a', \vhereby will be obtained the actual
inclination of the worm-thread. On each side of the point, m, on
e' a', mark distances, m a' and m b', equal to m' a and m' b, fig. 5,
and through these points draw parallels to c' e', and tLe poi-tion,
■p q, of the enclosed line compiised within them, wiU serve to deter-
mine the width and inclination of the teeth of the worm-wheel.
Through the points, p, r, draw p I and r s parallel to e' a', and
mark oflT the distances, t s and s q, which are equal, on the pitch
circle of the wheel, fig. 6, from s to i and q, after having drawn
through the points, s, but only m faint pencil or dotted lines, the
contours of the teeth as sectioned at f and g'. It is then sufficient
to repeat these outlines through the points, i and q, limiting their
length by the same internal and external circles.
Finally, the edge view of the worm-wheel, fig. 5, being the
lateral projection of the teeth, is determined by squaiing across the
points, w, r, a;, to m', 1)', a;', which give the interioi-s of the teeth ;
and the points, u-, v^, x", being squared over to w^, v^, x^, give their
exterior edges.
Worm-wheels are sometimes constructed with the form of the
teeth concave, and concentric with the axis of the worm, with the
view of their being in contact with the convolutions of the worm-
thread throughout a certain extent, in place of only touching at
single points.
This arrangement, which requires a particular operation for its
construction, is generally adopted when great precision is required,
and when it is wished to avoid, as much as possible, any play
between the teeth and the woi'ui-lhread during the transmission of
motion.
CYUNDRICAL OR SPUR (iEARING.
PLATE XIX.
THE EXTERNAL DELINEATION OF TWO SFUR-WHEELS IN GEAli.
FlUUUE 4.
209. Spur-toothed wheels are sucii as have fheir teetli parallel,
and lying upon a cyliiuirical surface or web. When a couple (if
such wheels are of unc(|ual size, the smaller one is generally
called a pinion, and the larger one a spur-wheel. Two wiieols,
which are intended to gear together, cannot work satislactorily in
concert, unless their radii or pitcli circles are e.xai-tly proportional
to the number of teeth contained by each. Consequently, in
onlir to coMstiiict ilesigns for couples of tootliod wheels, it In
necessary to know — the niunher of teeth of each, and the radius
of one or other of tluiu ; or the radii or diameters of both, and tl»o
uuuiher of teeth iil'oni'; or the distanct> between their COntres, and
the radius or lUMulur ol' teeth i>i' olio; or linally. the nuinbor of
revolutions of catli in the .>iunu> lime, and liie distance beiwoon
68
THE PRACTICAL DRAUGHTSMAN'S
their centres, or the radius and number of teeth of one of them.
In the rules and data at the end of this chapter, mil be found the
solution of the several problems involved in these various cases.
If we assume the followng data, A B = 240, and B C = 400,
these being the respective radii of the pitch cu-cles of two right
wheels, and n = 24, the number of teeth of the pinion— we at
once ascertiiin the number of teeth, N, of the spur-wheel, by the
following proportional formula : —
A B : B C :: n : N, or 240 : 400 :: 24 : N = 40.
Then describe the pitch circles of the radii, a b and b c, and
divide tliem respectively into 24 and 40 equal parts, thereby
ibtainiug the pitch, or the central point of each tooth, wliich is
exactly the same on both pitch circles. Next subdivide the pitch
into four equal parts, to obtain the centres of the intervals, and, at
the same time, the points through which the flanks of the teeth
pass. K, with the line, a b, on the line of the centres, a c, as a
diameter, we describe a circle, the centre of which is at o, and
suppose this circle to roll round the pitch circle, d b e, of the
spur-wheel, the point b, at present in contact, \v\l\ generate an
epicycloid, b f, as shown previously in reference to tig. 1 ; and
this curve is the one proper to give to the side of the teeth of the
spur-wheel, and it is accordingly repeated symmetrically on each
side of the several teeth, as shown in the diagram. If, fm'ther,
we suppose the same circle of the radius, o b, to roll round the
interior of the pitch ciicle, g b h, of the pinion, we shall obtain
the internal epicycloid (sometimes called hypocycloid), b o, as
already explained in reference to fig. 3, Plate XVIU., and a por-
tion, B a, of this, forms the flank of the pinion tooth.
Supposing the curve, b c, to form a part of the wheel, turning
about the centre, c, in the direction of the arrow, i, it will fulfil
the condition of impelling the flank, b a, which forms part of the
pinion, so as to turn about the centre, a, in the like uniformity.
In other words, the space passed through by the point, b, on the
pitch circle, g b h, shall be exactly the same as that passed through
oy the same point, b, considered as belonging to the spur-wheel,
on the pitch circle, e b d.
210. In proceeding to the determination of the length to give to
the tooth, it is first to be observed that the epicycloidal curve
should be sufficiently long to bear upon the side of the tooth,
through an extent of circumferential movement equal to the leno-th
of the pitch from the line of centres ; that is to say, until the flank,
at present in the position, e a, shall have arrived to the position,
c. d. At this moment, it will be observed that the curve, b f,
has reached the position, hf, and is in contact with the flank of the
pinion tooth in the point,/, on the circumference of the generatino-
circle of the radius, a o. It will thus be ob\ious that the pomt,/,
may be obtained by simply cutting off, on the generating circle, an
arc, b/, equal to the length of the pitch. Through this point,/,
describe a circle having c for its centre, and it will cut all the teeth
at the proper length.
The depth of the intervals is theoretically determmed by
describing, with the centre, a, a circle tangential to the fij-st; but in
practice, as it is necessary to leave a slight space between the ends
of the teetli and the bottoms of the intervals into which these work,
the circle in question is described witi; a somewhat smaller radius,
a& A a.
211. Hence it is manifest, on the supposition that the spur-wheel
is intended always to be the driver, without being diiven at any time
by the pinion, the teeth of the spur-wheel would only require to be
of the form indicated at J, and those of the pinion, like the portion of
a tooth, K, slightly tinted for the sake of distinction ; but generally,
and for obvious reasons, all spur gear is so constructed as to act
reciprocally, and equally well, whichever be the driver, and we
must, therefore, shape the teeth of the pinion, so that it may, in
turn, perform tliat function.
With this view, describe a circle with the centre, o', of the radius.
B c, taken as a diameter ; and suppose tliis ckcle to roll round the
pitch circle, h b g, of the pinion, the point, b, at present in con-
tact, will generate the epicycloid, b l, which is the proper curve
to be given to the teeth of the pinion. The same point, b, con-
sidered as on the spur-wheel, will, as we have seen, generate a
straight line, b' o', when rolling in the same manner round the
I interior of the circle, e b d, and this line forms the flank of the
tooth of the spur-wheel. The operation proceeds in the same
manner as for the pinion, the length of the teeth of which is deter-
mined by making the arc, b/', equal to the length of the pitch, and
describing, with the centre, a, a cii'cle passing through the point,/'.
The depth of the intervals of the spur-wheel is, in like manner,
limited by a circle described with the centre, c, and radius, c g,
which is somewhat short of being a tangent to the external circle
of the pinion, so as to allow a little play to the teeth in their
passage, as already explained. In this manner are obtained tlie
complete forms of the teeth, which are regular, symmetrical, and
similar to each other, and satisfy the conditions of reciprocal
gearing.
in the graphic operations here discussed, we have supposed the
intervals between the teetli to be exactly equal in width to the
teetli themselves; but as, in practice, it is necessary to allow of
some play between the teeth, in order that they may work into
each other with facility, this object is attained by reducing the
thickness of the teeth a little ; and in the drawing, when the scale
is not very large, it will be sufficient to delineate the ink lines just
within the thickness of the pencil lines. Where it is wished to
be more precise, this allowance may be calculated at about Jjth
or Jjth of the pitch. To give strength to the teeth, the interior
angles of the intervals aie roimded, as shown at each tooth in
fig. 4.
WTien the pinion is but of small diameter, the web, m, which
carries the teeth, is c;ist soUd with the boss, the interval being
filled up with a disc ; but when the wheel is larger, as in the case
of the spur-wheel, the web, Ji', is attached to the boss, p', by arms,
Q, which are strengthened by feathers, rounded in at the angles,
as represented in fig. 4.
delineation of a couple of "WHEELS GEARING INTERNALLY.
Figure 5. — Plate XIX.
212. The principles observed in determining the relative num-
bers of the teeth, with reference to the example just discussed,
apply in like manner to the case before us ; that is, such numbers
must be in the exact ratios of the diameters of the pitch circles.
The curvature of the teeth is also determinable by means of the
BOOK OF INDUSTRIAL DESIGN.
69
same operations, modified to suit the diiferent positions of the
parts with respect to each other. Thus the curve, b l, of the
pinion tooth, is generated by the rolling round the pitch circle,
& B H, of the circle described with the centre, o, and radius, o b,
equal to the half of b c, the radius of the pitch circle, d b e, of the
larger wheel. This is an application of the operations explained
in reference to fig. 3. The flanks, b a, or the sides of the teeth,
are obtained by simply drawing radii, or lines converging in the
point, c.
In the same manner, the curve, b f, of the teeth of the large
wheel, is generated by rolling along the interior of its pitch circle,
B p E, a circle described from the centre, o', and radius, b o', equal
to half the radius, b a, of the pitch circle, g b h, of the pinion.
These curves being obtained, the outlines of the teeth are com-
pleted in the manner explained in reference to fig. 4. It may,
however, be observed that, in the diagram, fig. 5, though the teeth
might be cut otf by a circle passing through the point, /, and
described with the centre, a, they are prolonged beyond that, so
that the teeth remain longer in contact, and a greater number of
teeth are, consequently, engaged at one time, allowing the strain
to be distributed over a greater number of points. It is the fact
of the curvatures of the two lines of teeth being in the same direc-
tion, which admits of a greater number of teeth being engaged at
once, without that increase of friction, and other disadvantages,
which would result from such an arrangement with wheels like
fig. 4.
THE PRACTICAL DELINEATION OF A COUPLE OF SPUR-WHEELS.
Plate XX.
213. In the cases treated of in the preceding sections, which
comprehend the general principles involved in rack and wheel
gearing, we have assumed that the rack and pinion, or pinion and
spur-wheel, are constructed of the same material, and in this case
tlie thickness of the teeth is the same in any two working together.
It very often happens, however, in actual construction, that one
of the two has wooden, and the other cast-iron teeth, or of other
dissimilar material. When this is the case, the thickness of the
one description must necessarily be greater than that of the other,
to compensate for the difference in the strength of the materials.
The pitch, however, will still be the same for both wheels ; for,
since the intei-vals on one wheel correspond to the teeth on the
other, a tooth and an interval on one must obviously be equal to
an interval and a tooth on the other. A couple of wheels of this
description are represented in plan and elevation, in figs. 1 and 2.
We here assume the wheels to be in the ratio to one another of
3:4; whence, giving the pinion 36 teeth, the spur-wheel must
have 48. After dividing the pitch circle of the spur-wheel, drawn
with the radius, c b, into 96 equal parts, \\w points of division
representing the centres of the teeth and of (he intervals, and the
|)ilcli circle of the pinion drawn with the radius, A b, likewise, into
72 equal parts — with the centres, o and o', doscribo the circles
wliicn generate the epicycloidal curves, b f and b l. Take \\ of
the pitch, b c, for the thickness of the wooden tooth, d e, and >,",
for that of the cast-iron tooth, allowing the remaining ..', for (lie
play in working. Next draw a .series of radii, to indicate the
flanks of the teeth, both of the pinion and spur-wheel, and at the
point of their junction with the pitch circle, di-aw the curved por-
tion of each, with the aid of a small pattern or template, cut to
the curves, b l and b f ; and, finally, limit the lengths of the teeth
and the depths of the hollows in the manner already pointed out,
in reference to Plate XIX.
As draughtsmen are generally satisfied with representing the
epicycloidal curves by arcs of circles which almost coincide with
them, and nearly fulfil the same conditions, such arcs must be
tangential to the radial sides of the teeth at their points of inter-
section with the pitch circle. They are determined in the follow-
ing manner: — Let fig. 10 represent one of the pinion teeth, drawn
to a larger scale. Through the point of contact, b, draw a taa-
gent, B o, to the pitch circle ; then bisect the chord, b n, whieh
passes through the extremities of the curve, by a perpendicular,
which wUl cut the tangent, b o, in the point, o. This is the centre
of the arc, b m n, which very nearly coinc.des with the epicycloidal
curve. The same arc is repeated for each side of all the teeth of
the pinion, the radius, b o, being preserved throughout. .\n
analogous operation determines the radius of the arc to be substi-
tuted for the curve in the teeth of the spur-wheel.
It is generally advisable to make wooden teeth about three-
fourths as long as the pitch, and cast-iron teeth about two-thirds
as long. In no case, however, should the lengths of the teeth in
the two wheels geared together be less than those obtained by
calculation, and determined by the points, /, f, situated on the
circles described with the centres, o, o', by which the epicycloids
are generated. The ratio of the curved external portion, n vi, «»f
the tooth to the flank, n ^, is 4 : 5. In other words, tlie wluilo
height or length of the tooth being divided into 9 equal parts, 4 of
these are to be taken for the length of the curved portion, :md 5
for the rectilinear flanks. When the teeth are of cast-iron, the
thickness, p q, of the web should be equal to the tliickness, r s, of
the tooth. Sometimes it is made only Jths of this ; but in that
case it is strengthened by a feather on the interior.
For wooden-toothed wheels, since it is necessary that the tenon,
t, of the tooth be firmly secured, the web is made of a thickness,
p q, often double that of the tooth. The tenons of the teetli must
be adjusted very carefully and accurately in the web. They are
made with a slight taper, and are secured on the interior of the
web either by iron pegs, as at u, passing through them, or by a
series of wooden keys or wedges, v, driven in between them, and
forming strong dove-tiiil joints. Those two methods of fixing the
teeth are shown at diflerent parts on fig. 1, and moie in det;ul in
fig. 7. There is a third modification, which also possesses some
advantages. We have represented it at t, tig. 3, whence it will
be seen that it con.sists in forming the teeth wiih a couple of
•shoulders, z, which allow of the tenons, /, being niaiie nuicli
stronger, and also tjiko away llurcby some of the weight of nutai,
two ol)j(!cts of great imporlance.
'J'he width, .i' y, of (ho (ee(li is equal (o two or three limes (heir
pilch. In \\liriU enlii<'l\ of I'Mst-iroii, the web is of the sjunc
width as Ihe tci lb; bui ii is nuich broader wiien (lie teeth jl-o ot
wood, for it requires to be morti.sod, to rocoivo tlio tenons of liui
(I'tlli, and sluiuld have n width equ;d to that t.A' tiu> leetli, jilus ;iii
amount equal to once and n half or twice their '.iiickMes,s. Wu
70
THE PRACTICAL DRAUGHTSMAN'S
have already mentioiiecl, that in wheels of moderate size, the
web, M', is attached to the boss, p', by arms, q. The number of
these arms varies, 4, 6, or 8 being used according to the diameter.
In the present case the wheels have sLx arms ; this number, amongst
otlier reasons, being more particularly convenient, because the
number of teeth are divisible by 6. Wlience it follows, that the
feathers wiiich strengthen the arms on either side of the wheel,
can be made to lie between two of the teeth, at each of the six
points of attachment to the web.
The feathers are joined to the body of each arm by cavetto or
concave quarter-round mouldings, with or without fillets, as indi-
cated in figs. 5 and 6, which reoresent sections of the arms taken
through 1—2, 1—2, in fig. 1
At other times the feathers are united to the body of the arm
by plain chamfer portions, as shown in fig. 8; or, even more sin^ply
still, and without filling up the angle formed, as in fig. 9, ihe
feathers being united, as it were, to the body of the arm without
any additional moulding.
In all cases, however, these feathers are made wth a taper,
being thicker at their point of union with the bod}-, and gradually
decreasing in thickness outwardly.
Figs. 3 and 4 represent cross sections of the wheels, taken
through the irregular line, 3 — 4 — 5, on fig. 1. We may observe,
in reference to these sections, that at the upper part of each the
plane of section is supposed to be parallel to the arm, or the arm is,
as it were, turned so as to be parallel to the plane, c c', or a a',
fig. 1, in order that it may be projected in the sectional view with-
out foreshortening. At the low-er parts of these views, however,
tlie arms are projected, as in the oblique position represented in
fig. 1.
In this description of drawings, these oblique projections are
generally dispensed with, and are, indeed, avoided, as they do not
readily give the exact measurements of the parts represented.
The operations indicated on the figures complete the general
design of Plate XX., whether of the plan, elevation, or sections.
THE DELINEATION AND CONSTRUCTION OF WOODEN
PATTERNS FOR TOOTHED WHEELS.
PLATE XXI.
SPUR-WHEEL PATTERNS.
214. If, as we have already endeavoured to impress upon the
student, great care is required in the construction of wooden pat-
terns in general, above all is this care and extreme accuracy called
for in the execution of the patterns of toothed wheels, because of
the great exactitude absolutely needed in the proportions of the
various parts — as tliat, for example, between their diameters and
numbers of teeth.
The pattern-maker must make allowance, not only for the shrink-
ing of the cast-iron, but also for the quantity of metal to be taken
away by turning and finishing afterwards. Moreover, the pattern,
w^hich is necessarily in many pieces, must be joined together so
strongly and solidly, that it may not run the risk of changing its
shiipe during the construction of the mould.
For wooden-toothed wheels, the web must be pierced with a
number of openings or mortices to receive the tenons of the teeth.
But in place of producing these mortices on the wooden patterns —
which system, besides weakening it, would render the formation of
the mould much more difficult — small projections corresponding to
the teeth are fixed externally to the web. These projections form
sockets in the mould, in which the actual loam cores are fixed,
which foi-m the mortices w^hen the piece is cast.
Bearing in mind these various considerations, we may proceed
to the construction of the patterns for two spur-wheels, such as are
represented in Plate XX.
PATTERN OF THE PINION.
215. Figs. 1 and 2 show a half plan and a vertical section of
tlie wooden pattern of the pinion. It Ls composed of many prin-
cipal pieces — namely, the web, or crown, and its teeth ; the boss,
witli its core-pieces ; and the arms, or spokes, with their feathers.
We shall proceed to examine these various parts in succession.
WEB OR CROWT*.
The pattern-maker takes planks, of from 25 to 30 millimetres
in thickness, and cuts out of it a series of arcs, a, of a uniform
radius, corresponding to tliat to be given to the pinion, with the
addition of the allowance for shrinkage and loss from finishing.
These arcs are built up like brickwork, the joints of one layer, oi
series, being opposite to solid portions of the contiguous layers,
as shown in fig. 3. This arrangement prevents the liability to
warp or change the fonn, from variation in the humidity of the
atmosphere, as would be the case were the crowTi made of a single
piece.
This piece being finished and glued together, and the joints
quite dry, is put into a lathe, and there turned quite true, both
externally and internally. The two surfaces are here made per-
fectly parallel, and the whole is reduced to the exact dimensions
determined on, and shown upon a large working drawing of the
actual size, previously prepared, generally by the pattern-maker
himself
At this stage, the external surface of the crown, is divided ofl"
by lines, showing the positions of the teeth, which are then some-
times sunply screwed or nailed on. It is, however, much prefer-
able, and conduces very much to the soUdity of the wiieel, to cut
out grooves of a trifling depth on the periphery, into which the
teeth are fixed, being formed with a dovetail for that purpose, as
shown at b, in fig. 1.
BOSS.
The boss is made in two pieces, each one solid block of wood,
D, except when the w heel is of a large size, in which case the boss
requires to be built up of several pieces.
These blocks are each turned separately to the exact dimen-
sions given in the plans, and they secure between them the thick-
ness of the body part of tlie arm.
ARMS OR SPOKES.
The body of each arm, c, fig. 4, is also cut out of planks of a
uniform thickness, being formed not only to the external contour
of that part of the arm which is afterwards the only part visib.o
BOOK OF INDUSTRIAL DESIGN.
11
in the casting, but also comprising, above and beyond this, the
projections by which, in the pattern, it is attached to the boss on
tlie one hand, and to the crown on the other. The extremity, a,
of the boss end of the arm is in the form of a sector, correspond-
ing to a sixth part of the cu-cle of the boss, the pinion having six
arms ; the lateral facets, b, of this part are grooved out, to receive
small tongue-pieces, or keys, c, fig. 1, so as to form a strong joint
when glued together. The other extremity, d, of the arm is cut
circularly, to the form of the crown, or web, into which it is fitted,
penetrating to a slight extent, the crown being previously formed
with a socket to receive it.
Next, the feathers have to be attached to the body, c, of the
arm. These feathers, b, are each cut out in sepai-ate pieces, to
(he shape indicated in fig. 5: they have supplementary projec-
tions, e and /, at their opposite extremities, whereby they are fixed
into the crown and boss. When all these feathers are in their
place, and the arms glued into the crown, the two portions, d, d,
of the boss are fixed to them, the grooves for the reception of the
ends of the feathers being glued, as well as the other parts, to
give greater solidity. Finally, the boss is surmounted by the
conical projecting pieces, f, f, which serve to produce in the mould
the cavities, or sockets, which retain the loam core in position, the
core being provided to produce the eye of the wheel, into which
<Jie shaft is fitted.
To give compactness and strength to the whole, a bolt, g, is
passed through the centre; and this method of securing permits
of the core projections, f, f, being changed for larger or smaller
ones, if desired, without having to pull the entire wheel to pieces.
[f, to add to the elegance of the shape of the wheel, it is wished to
ornament the arms with mouldings, as at i, these are applied at
the angles of junction of the feathers with the body of the arm.
These are simply glued or naOed on. The sectional view, fig. 6,
she. .s the form and position of these mouldings.
It is to be observed that, in wheels of a moderate size, when
cast-iron teeth are to work on cast-iron, they are at once cast to
the exact shape, and the pattern is constructed accordingly ; but
it is almost always indispensable, where cast-iron and wooden
t«eth have to work together, to finish and reduce the former after
being cast ; and the projections, b, on the pattern answering to
them, must consequently be made of larger proportions every way,
to provide for the quantity of metal taken away in the finisliing
process.
PATTERN OF THE WOODEN-TOOTHED SPUR-WIIEEr..
216. Figs. 7, 8, and 9 i-epresent, in elevation, plan, and vertical
section, the wooden pattern of the spur-wheel, which gears with
the pinion just described. It consists, like that wheel, of the
crown or web, the boss, and the arms ; and these various parts,
which are designated by letters corresponding to those employed
in the preceding example, are constructed exactly in the same
manner.
There is, however, an essential difTerencc in the exterior of the
crown : in place of this carrying the projections, b, cut to the
shape of the teeth, and such as will actually be iJifxhiccd on Ihe
castii.g, it has other projections, b', of a simpler fortii, intended to
produce in the mould the sockets for receiving the corc-pioccs
which form the mortises in the casting, to receive the tenons of
the wooden teeth. These projections are let into the crown, or
simply applied thereto, and fixed by nails, as at Z, or by screws, aa
at m, the latter method being preferable, as it has the advantage
of pei-mitting the nmnber of teeth to be changed without injury to
themselves or to the crown. In the wooden pattern, the length
of the projections, b', is carried to the edge of the face of the
crown, on that side which descends into the lower half of the
mould-frame, to allow of the more accurate adjustment of the core-
pieces, and also to facilitate the recovery of the pattern from the
mould. These core-pieces, however, are so formed as to make the
moi-tises no wider than is necessary, and to leave a sufficient thick-
ness of metal for the strength of the crown, as already pointed out
in reference to Plate XX.
CORE-MOULDS.
217. The core-pieces for the mortises should not only be piaced
at equal distances apart throughout the circumference of the crown,
but they must all also be of precisely the same form and dimen-
sions throughout, so that the mortises may be perfectly equal.
With this view, a wooden core-box or mould is made ; and there
are several methods of doing this. Thus, fig. 10 represents a face
view, and fig. 11a horizontal section, through the line 3 — 4 in
fig. .10, of one form of core-mould, consisting of a single piece. iTie
portion, n, of the cavity corresponds to the projecting core-piece,
b', outside the crown, and the portion marked o, to the mortise, or
hollow socket, in the crown : this last has the same section as the
crown in the width of the cut-out part. The moulder fills the
cavity of the core-mould with loam, pre%"iously prepared, and after
pressing it well in, levels it off" with a straight-edged doctor or
scraper ; he finally inverts the mould, thus releasing the core com-
plete. The operation is repeated as many times as there are teeth ;
and when the cores are all dry, they are placed with great care in
the mould, their supplementary projections, b', being let into the
sockets formed to receive them — thereby insuring the accuracy of
their adjustment.
Figs. 12, 13, and 14, show another construction of wooden core-
mould, formed in two separate pieces, h and i. These have be-
tween them the cavity, n o, corresponding to that in the one just
described. In this last case, the surface of the core which re-
quires to be levelled off with a scraper, is only at one of the extre-
mities instead of on the lateral faces, as in the other, and the cores
are released by separating the two pieces, h, i, which are rendered
capable of accurate adjustment to each other by moans of marking-
pins, k.
To return to the wheel itself: when it is of very large dimen-
sions, the blocks, d, of the boss are secured together by two or
more bolts, g, in place of one.
The mould for the wheel is in two i>iecos, the lower frame, or
"drag," being let into the ground in the moulding shop; the uppor
frame or top part, is moveable, and it will bo obvious that very
great caro is required to lift this oflFthe |witteni, .so !us not to injure
the regnlarity and ,shar|inoss of llio impression ; and for this i)ur-
pose, sullicieut " draw " or taper must bo ifivon to tlio vjirious ixarta.
as the crown, the boss, and tin- fe.ntluTs on (ho amis, as ftln'.idv
pointed out.
72
THE PRACTICAL DRAUGHTSMAN'S
WTien the patterns are hea^y, two screw-staples, or "draw-
plates," L, fig. 1, 8, and 15, of iron or brass, are countersunk into
the crown, and into these draw-handles are screwed, by which the
pattern is lifted out of the mould.
In figs. 1, 2, 8, and 9, are combined, in smgle views, several
different projections, to avoid repetitions of the diagrams, and to
simplify the whole drawing, and bring it into a small space. This
system is very much used in dra^\ings, or plans, made for actual
( onstruction.
RULES AND PRACTICAL DATA.
TOOTHED GEARKG.
218. It has been already laid dowTi, as a fundamental rule, that
in order to work well, all toothed wheels coupled together must
have the same ratio between the numbers of their teeth as between
their diameters.
It follows from this principle, that when we know the radii of
the pitch cu-cles of t^o wheels, and the number of teeth of one of
them, we can determine that of the other, and reciprocally.
Thus, putting x to represent the number of teeth of a wheel of
the radius, e ; and n to represent the number of teeth of a wheel
of the radius, r, we have the direct proportionals, a : n : : r : r ;
whence we can, at any time, ascertain any one of the terms when
the other three are known.
First Example. — Let the radius of the pitch circle of a spur-
wheel be 12 inches, and the number of teeth on it 75, what should
be the number of teeth on a pinion gearing ^^ith it, the radius of
the pitch circle of which is 8 inches 1
We have
75 : n :: 12 : 8; whence
75 X 8
12
= 50 teeth.
Second- Example. — Let 75 and 50, respectively, be the number
of the teeth of a spur-wheel and pinion, and 12 inches the radius
of the pitch circle of the former, the radius of the pitch circle of
the latter may be found by means of the proportion —
75 : 50 :: 12 : r; whence,
50 X 12 „ . ,
r = — i^^ — = 8 mches.
7o
219. The velocities of rotation, or the numbers of revolutions
of the shafts of a spur-wheel and pinion in gear with each other,
are in the inverse ratio of the respective diameters, radii, or num-
bers of teeth of the two.
Consequently, putting V to represent the velocity of rotation of
the pinion shaft, the radius of the pitch circle of which equals r,
and the number of the teeth n, and putting v to represent the velo-
city of the spur-wheel shaft, of which the pitch circle radius equals
R, and number of teeth N, we have the inverted proportions —
V : r :: r : R,
and
V : t; :: 71 : N.
In either of these proportions, we can determine, as in the former
axample, any one term when the three others are known.
First Example. — A spur-wheel, the pitch circle radius of wliioh
is 10 inches, has a velocity of 25 revolutions per minute ; whit is
the pitch circle radius of a pinion to gear with it, and make 60
revolutions in the same time 1 By the inverse proportion,
25 : 60 :: r : 10;
whence,
25 X 10
60~^
4g- inches,
the pitch circle radius of the pmion.
A spur-wheel has 60 teeth, and is requhed to run at 25 revolu-
tions per minute, and at the same time to drive a pinion at the rate
of 75 revolutions per minute, what should ^e the number of teelii
of the latter ?
Here,
75 : 25 : : 60 : n ;
whence,
25 X 60
n= — —— =20,
the number of teeth the pinion must havt*
These principles apply equally to puUeys or drums put in com-
munication with one another by cords or belts, and kno\vn as belt-
gearing.
Sometimes, in systems of geared spur-wheels, aU that is known
is the distance apart of their centres, the number of teeth which
they are to carry, or the number of their revolutions in the same
time. In this case we have, on the one hand, an inverse proportion
between the distance of their centres, the sum of then- revolutions,
and between their respective radii and revolutions ; and, on the
other hand, a direct proportion between the distance of the centres,
the sum of the teeth on both wheels, and their respective radii, oi
the ntunber of teeth of each.
Let D be the distance apart of the centres of a spur-wheel and
pinion of the respective radii, R, r, and number of teeth, N, n, or
the reciprocal velocities, r and V; we have first the following
inverse proportion,
D: V+i::R: V;
and, secondly, the direct proportion,
D:N + 7i::N:R.
First Example. — Let 45 inches be the distance between the
aentres of a spur-wheel and pinion, the former of which is to make
22 revolutions per minute to the other's 15i; what should be their
respective radii?
We have, first,
45 : 22 + 15-5 :: R: 22;
whence,
„ 45 X 22
R = _____ = 26-4 mches,
and
whence,
22 + 15-5
45 : 22 + 15-5 :: r : 15'5;
45 X 15-5
22 + 15-5 ■
18'6 inches.
^Vhen the pitch cu-cle radius of one of the wheels is ascertameO,
it is evidently unnecessarj' to search for the other radius by means
BOOK OF INDUSTRIAL DESIGN.
73
iiT tlie second proportion, for it is sufficient to subtract the one
found from the sum of both; thus,
45 — 26-4 =: 18-6; or,
45—18-6 = 26-4.
Second Example. — The distance, d, between the two centres
oeing known =: 45 inches, and one wheel carrying 31 teeth and
She other 44, what are their respective radii 1
We have here, in the first place,
45 : 31 + 44 :: R : 44;
whence,
45 X 44
"-31 + 44-^""'
and
45 : 31 +44 :: r : 31;
whence,
45 X 31
'•=31 + 44 = ^«-«'
or, more
simply,
r = 45 — 26-4 = 18-6 inches
In like manner, the respective radii of a spur-wheel and pinion,
to gear together, may be determined geometrically, when the dis-
tance between their centres is known, as well as the numhera of
revolutions of each, by the following rule : —
Divide the distance into as many equal parts as there are of any
measure contained exactly in the sum of the velocities, such mea-
sure being also contained exactly any number of times in each of
the velocities alone. Then, for the pinion radius, take as many of
these measures as are contained in the lesser velocity, and for the
radius of the spur-wheel, the remainder of them.
Example. — Let 16 inches be the distance between the centres
of a spur-wheel and pinion which make 6 and 4 revolutions re-
spectively, or any equi-multiples or equi-submultiples of these, as
12 and 8, or 3 and 2. Divide the distance into 10 equal parts,
and take 4 of these for the pinion radius, and 6 for the spur-wheel
radius.
This rule is of very simple application when the ratios of the
numbers of revolutions are whole numbers, such as 1 : 4, or 2 : 5 ;
for all that is necessary is to add the two together, to divide the
distance between the centres to correspond, and to take the re
spective numbers of measures for each wheel.
The following table will be of great assistance in the solution
of various problems connected with systems of gearing, when tho
number of teeth, the pitch, or the radius are known.
TABLE FOR CALCULATING THE NUMBERS OF TEETH AND DL&.IHETERS OF SPUR GEAR, FROM THE PITCH, OR VICE VERSA.
Number.
Coefficient.
Nnmber.
Coefficient.
Number.
Coefficient.
^fumber.
Coefficient.
Number.
Coefficient
10
3-183
39
12-414
68
21-644
97
30-875
126
40-106
11
3-501
40
12-732
69
21-963
98
31-193
127
40-424
12
3-820
41
13-050
70
22-281
99
31-512
128
40-742
13
4-138
42
13-369
71
22-599
100
31-830
129
41-061
14
4-456
43
13-687
72
22-917
101
32-148
130
41-379
15
4-774
44
14-005
73
23-236
102
32-467
131
41-697
16
5-093
45
14-323
74
23-554
103
32-785
132
42016
17
5-411
46
14-642
75
23-872
104
33-103
133
42-334
18
5-729
47
14-960
76
24-191
105
33-421
134
42-652
19
6-048
48
15-278
77
24-509
106
33-740
135
42-970
20
6-366
49
15-597
78
24-827
107
34-058
136
43-289
21
6-684
50
15-915
79
25-146
108
34-376
137
43-607
22
7-002
51
16-233
80
25-464
109
34-695
138
43-925
23
7-321
52
16-552
81
25-782
110
35013
139
44-244
24
7-639
53
16-870
82
26-100
111
35-331
140
44-562
25
7-957
54
17-188
83
26-419
112
35-650
141
44-880
26
8-276
55
17-506
84
26-737
113
35-968
142
45-199
27
8-594
56
17-825
85
27-055
114
36-286
143
45-517 ■
28
8-912
57
18-143
86
27-374
115
36-604
144
45-835
29
9-231
58
18-461
87
27-692
116
36-923
145
46-153
30
9-549
59
18-780
88
28-010
117
37-241
146
46-472
31
9-867
60
19-098
89
28-329
118
37-559
147
46-790
32
10186
61
19-416
90
28-647
119
37-878
148
47-108
33
10-504
62
19-734
91
28-965
120
38-196
149
47-4-27
34
10-822
63
20-053
92
29-284
121
38-514
150
47-745
35
11-140
64
20-371
93
29-602
122
.38-833 1
151
48-063
36
11-4.59
65
20-689
94
29-920
123
39-151
152
4S-3S2
37
11-777
66
21 -DOS
95
30-238
124
39-l(:9
153
48-700
38
12-095
67
21-32()
96
30-557
125
39-788
154
49020
RULES CONNECTED WITH THE PRECEDING TABLE.
[. To find tho diameter of a spur-whoel, when the number and
pikh of the toeth are known.
Mulli/ilij Ihc ciMffirirnl in l)w lahh; corrfspomliitg to Oie tiutiiber if
teeth, by the given pilch in feet, imhes, mdtrcs, or other measures, and the
product will he llw di'imeler in feel, inches, or jnttres, to corresponiL
74
THE PRACTICAL DRAUGHTSMAN'S
First Example. — What is the diameter of a spur-wheel, of 63
teeth, having a pitch of H inches?
Opposite the number 63, in the table, we find the coefficient,
20053. Then—
20-053 X 1-5 = 30-08 inches,
the diameter of tlie spur-wheel.
Second Example. — \Vliat are the diameters of two wheels, of 41
and 150 teeth respectively, their pitch being | inch?
On the one hand, we have
13-05 X -75 = 9-7875 inches,
the diameter of the pinion of 41 teeth; and on the other,
47-745 X -75 = 35-8 inches,
the diameter of the spur-wheel of 150 teeth.
n. To find the pitch of a spur-wheel, when the diameter and
number of teeth are known.
Divide the given diameter by the coejicienl in the table correspond-
ing to the number of the teeth, and the quotient will be the pitch
sought.
First Example. — What is the pitch of a wheel of 30-08 inches
diameter, and of 63 teeth ?
Here —
30-08 : 20-053 = 1-5 inch,
the pitch required.
Second Example. — It is required to construct a spur-wheel, of
126 teeth, to work with the preceding, what must be its diameter?
Here—
1-5 X 40-106= 60-159 inches,
the diameter of a wheel of 126 teeth, and of the same pitch.
in. To find the number of teeth of a wheel, when the pitch
and diameter are known.
Divide the given diameter by the given pitch, the number in the
table corresponding to the quotient will be the number of teeth sought.
If the quotient is not in the table, take the number correspond-
ing to that nearest to it.
First Example. — The diameter of a spur-wheel is 30-08 inches,
and the pitch of the teeth is 1-5 inch, what number of teeth should
the wheel have ?
30-8 : 1-5 = 20-53;
which quotient corresponds to 63 teeth.
Second Example. — What should be the number of teeth of a
pinion, the diameter of which is 875 miUimetres, and which is
intended to gear with a rack, of which the pitch is 25 millimetres ?
875 : 25 = 35.
The number most nearly corresponding to this is 110, the
number of teeth to be given fo the pinion.
ANGULAR AND CIRCUMFEKENTIAL VELOCITY OF WHEELS.
619. When it is known what is the angular velocity of the shaft
of a fly-wheel, spur-wheel, or pulley, the circumferential velocity
may be found by the following rule : —
Multiply the circumference by the number of revolutions per
minute, and the product will give the space passed through in the
same time ; and this product being divided by 60, will give the velo-
ii.y of the circumference per second.
Fxample. — Let the diameter of a wheel be 4 feet, and the
number ot its revolutions per minute 20, what is the velocity a*
the circumference ?
The circumference of the wheel =4 x 3-1416 = 12-5664;
then
12-5664 X 20 = 251-328 feet,
the space passed through per minute by any pomt m the circum-
ference ; and
251-328
60
:4-2,
the velocity in feet per second.
When the velocity at the circumference is known, the angulai
velocity, or the number of turns in a given time, may be ascer-
tained by the following rule : —
Divide the circumferential velocity by the circumference, and ine
quotient will be the angular velocity, or number of revolutions in the
given time.
In the preceding case, 4-2 feet being the circumferential velocity
per second, and 4 feet the diameter, we have
4-2 _
4 X 3-1416 ~ ■^^'*'
the angular velocity per second ; and
•334 X 60 = 20,
the number of revolutions per minute.
In practice, it is easy to ascertain the velocity of a wheel, the
motion of which is uniform. With this view, a point is marked
with chalk on the rim of the wheel, and note is taken of how often
this point passes a fLxed point of observation in a given time ; then
this number of revolutions is multiphed by the circumference
described by the marked point, and the product divided by the
duration of the observation expressed in seconds. The result
will be the velocity of the circumference of the wheel. Every
other point on the wheel will have a diiferent velocity, propor-
tioned to its distance from the centre of motion.
Example. — A wheel, 2 feet in diameter, having, according to
observation, made 75 revolutions per minute, what is its circum-
ferential velocity (per second) ?
"5 X 3-14 X 2
60
= 7-83 feet.
circumferential velocity of the wheel.
Reciprocally, when the circumferential velocity (per second) is
known, the number of revolutions per minute is found by means
of the formula —
V X 60
^~3-14xD'
or, with the data of the preceding case.
N =
7-83 X 60
= 75 revolutions per minute.
3-14 X 2
When several spur-wheels or pulleys are placed on the same
shaft, the circumferential velocity of every one of them is found
in the same manner, by multiplying the number of revolutions by
the respective circumferences, and dividing the products by 60.
Example. — Three wheels or pulleys, a, b, c, are fixed on one
shaft; the radius of the pulley, a, is equal to 1-1 feet; that of the
pulley, b, 1-6 feet; and that of the pulley, c, 2-15 feet; and the
shaft makes 12 turns per minute, — what is the circimiferential
velocity of these three pulleys ?
BOOK OF INDUSTRIAL DESIGN,
76
For the pulley, a, we have —
6-28 X M X 12
V = ;::: = 1-38 feet per minute;
60
for the pulley,
,,, 6-28 X 1-6 X 12
V' = — = 2 feet;
60
and for the pulley, c —
6-28 X 2-15 X 12
V" = — = 2-7 feet
oO
DIMENSIONS OF GEARING.
220. In designing tooth-gearing of all descriptions, it is neces-
sary to determine — first, the strength and dimensions of the teeth ;
second, the dimensions of the web which carries the teeth ; and,
third, the dimensions of the arms.
THICKNESS OF THE TEETH.
221. The resistance opposed to the motion of the wheel or the
load, may be considered as a force applied to the crown, to pre-
vent its turning, and the power, during its greater strain, as applied
to the extremities of the teeth. The teeth then should be con-
sidered as solids fixed at one end, and loaded at the other ; and
the equation of equilibrium for then; is —
Fxh^^kxfxw;
m which formula, P signifies the pressure in kilogrammes at the
extremity of the tooth ; h, the amount of projection of the teeth
from the web in centimetres ; k, a numerical coefficient ; t, the thick-
ness of the teeth in centimetres ; w, their width in centun^tres.
In this formula, the numerical coefficient, k, which is calculated
with reference to the motion of toothed geaiing, varies with the
material of which the teeth are constructed.
From Tredgold's experiments with well-constructed cast-iron
wheels, this coefficient has been calculated to be 25 for that
metal ; and adopting it, the preceding formula will then become
P X h — 25 X {^ X w;
wTience,
_ 25 X ^w
^- h ■'
a lormula in which three dimensions are variable.
The following ratios usually exist between these quantities : —
w varies between 3 t and 8 t.
h = 1-2 t to 1-5 t.
Let, then, w = 5t, and h = 1-2 t, so that, substituting these values
in the equation, it becomes —
„ lb X 5 X t xC
P= 1-2 X. =104x.';
whence,
P
«» = -— and / = -098 -^/p]
104
It me above ratio between the thickness, t, and width, ir, !»■
adopted for all proportions ; for low pressures or small loads, we
shall have teeth much too thin and small ; and for hii,'li pres-
sures, on the other hand, the defects of too great lliickncss jind
pitch. To retain, then, the thicknesses within convenient lunits, it
is well to vary the ratio of t to w, according to the pressures ; and
in order that the pitch may not be too great, the width of the teeth
is determined at the outset, according to the pressure or load
which they have to sustain, in the follovraig manner : —
L For 100 to 200 lb., make 2^ = 3 i; when Z = -126 ^F
n. « 200 300 " M=3-5; " t — -\\l\'Y
III. " 300 400 " w= At " < = -110^P'
IV. " 400 500 " w = A-5t " t = -104 ^P"
V. " 500 1,000 « w= ot " t — -098 ^ P"
VL " 1,000 1,500 « w = 5-5t " t — -093.^ F
VII. " 1,500 2,000 " M! = 6 i! « t = -089 ^P"
Vm. " 2,000 3,000 " w = 6-5t " <=-084*'P"
IX. " 3,000 5,000 " w= It " < = -082 ^'¥'
X. " 5,000 and upwards, " w= 8i " t = -Oil V'p'
The height, or projection, h, should be comprised between 1-2 t
and 1-5 t, the latter applicable to low powers or loads, and the
former to high ones.
For teeth of wood, which are ordinarily made of beech or ehn,
the coefficient should be augmented by a third in each of the last
given formulae, which become —
L t = -168 W making w = 3-0 t.
IL t—-156VP' « w=3-5t.
in. i—-lnVP' " w = 'l-Oi.
IV. i^-lSdW " w = 4-5t.
V. t — -13\W " w=o-Ot.
VL t = -124:VF " w = 5-5t.
VIL t=z-n9VP' " w=6-0t.
vm. t = -U2VP' " w=6-5t.
IX. i — -109 W " w = 7-0 1.
X. t = -103\^ " w; = 8-0^
All these formulae are constructed on the supposition that,
although there are generally several teeth in contact at the same
time, yet each should be capable of sustaining the whole strain as if
there were only one in contact, and they should be strong enough
to compensate for wear, and sustain shocks and irregularities ui the
strain for a considerable length of time.
The pressure, P, on the teeth may be determined according to
the amount of power transmitted by the wheels per second at tho
pitch circumference.
This pressure is obtained by ditiding the strain to be transmitted,
expressed in kilogranwuirr, by the lelncily ]>tr sirond of the pilch
circumference. A kilognunmetro is a term corrt'spondiiig to llut
English expression, " one pound raised one foot liigJi per minute."
A kilogrammetro is equal to one kilognunme raised one mt^tru
high per second : it is written sliorlly thus — k. ni.
First Example. — A spur-wlicol is intended to tnmsinit n forto
equal to a power acting nt tho pitch oirxMunference of 600 kilo-
gramnietres, at the rate of 209 ni. per second, wluit pressure huvo
tho teeth to sustain .'
76
THE PRACTICAL DRAUGHTSMAN'S
Here,
500 k. m.
2-09
: 239 kiloff.,
the strain that each tooth must be capable of resisting without
risk of breakage, even after considerable use and wear.
Second Example. — A spur-wheel, 2 metres in diameter, transmits
a force equal to 20 horses power, and makes 25 revolutions per
minute, what is the pressure on the teeth ?
We have, in the first place,
20 H.p. = 75 X 20 = 1500 kilogrammetres,
and
V =
3-14 X 2 X 25
60
=: 2-62 m. per second ;
whence,
1500 ^^^ , .,
-^^g^ = 573 kilog..
the pressure on the tooth.
When the power that a wheel has to sustain at its circumference
is known, the thickness proper for the tooth may be calculated by
one of the preceding formulas, according to the material of which
it is constructed.
Thus, in the former of the last two examples, in which P = 239
kilog., the thickness of the tooth, if of cast-iron, should be making
w = S-5t:
t — -lin V239 = 1-8 cent. = 18 millimetres.
And, in the second example, where P = 573 kil., the thickness
will be, supposing the teeth to be of beech, and w = 5t,
i =: -131 4/573 = 3-23 c, or 32-3 millimetres,
w = 5 X 32-3 = 161-5 millimetres.
Third Example. — A water-wheel of 4-2 metres diameter makes
4^ revolutions per minute, and transmits a force equal to 25 horses
power by means of a spur-wheel, the radius of which is 1-65 m.,
it is required to determine — ^first, the pressure on the teeth of this
spur-wheel ; and, secondly, the thickness of their teeth.
In the first place,
25 X 75 = 1875 kilogrammetres,
ana
V =
1-65 X 2 X 3-14 X 4-5
60
= -777 m. ;
whence,
1875
•777
2413 kilogrammetres ;
consequently, making w = 6-5 /, the thickness of the tooth will be
t — -084 V2413 = 3-7 c = 37 millim.,
and
w = 37 X 6-5 = 240-5 millim.
Fourth Example. — The cast-iron pinion of a powerful machine
is 1-06 m. in diameter, it is fixed on a shaft which should transmit
an effective force of 200 horses power, at the rate of 45 revolutions
per mmute, what is the pressure on the teeth and their dimensions ?
The power transmitted is
200 X 75 = 15,000 kilogrammetres,
•ind
_ 1-06 X 3-14 X 45
» — ^7j = 2-37 metres per second.
The pressure on the teeth is —
consequently, making w = 8 l; we have, for the thickness of the
teeth, in cast-iron,
i = -077 4/6333 = 61-2 "■/„.
and w — S x 61-2 = 489-6 ■"/„.
For a pinion of the above proportions, actually constructed, the
thickness was made 75 millim., and the width 525 millim.
PITCH OF THE TEETH.
222. It will be recollected (203) that the pitch of cast-iron
spur-wheel teeth, measured on the pitch circumference, comprises
the thickness, t, of the tooth, and the width of the interval, which
last is, in ordinary cases, made equal to i, augmented by one-tenth;
this gives, ^ = 2-1 1.
Thus, with the data of the preceding examples —
In the 1st, ^=2-1x18 = 27-8 ■"/„.
3d, ^ = 2-1 X 37 = 77-7 "/^
4th, p = 2-l X 61-2 = 128-5 "/^
When the spur-wheel is intended to carry wooden teeth, as in
the second of the preceding examples, it will generally be coupled
with a pinion, having cast-iron teeth, which should be of about
three-fourths the thickness of the wooden ones ; in this case the
pitch will be equal to
t + -151 + -11= I + ■85t= 1-85 t.
Thus, in this example, we should have —
p=32-3 X 1-85 = 59-8°/^.
After this is done, that is, when the pitch is ascertained, which,
as has already been observed, should be precisely the same on the
pitch circles of any two wheels working together, the number of
teeth of one of the wheels may be obtained by the following
formula —
P
where N signifies the number of teeth of the spur-wheel ; R, the
radius of the pitch circle ; and p, the pitch, measured on this circle.
First Example.— What is the number of teeth on a spur-wheel
of two metres diameter, and the pitch of which is -0278 metres ?
Here —
,, 2 X 3-14 X 1
^^ -0278 =225 teeth.
It vAW be easily understood, that the fraction arising from the
operation must be neglected, since we cannot have a part of a
tooth. In cases, therefore, where there is a fraction, the pitch
must be slightly increased. Thus, in the example xmder consi-
deration, the pitch becomes —
2rt R 6-28
^==T = -225=-02^9'"-'
instead of 0-278 m.
Second Example. — It is required to determine the number of
wooden teeth to be carried by a spur-wheel of two metres diameter^
the pitch being -0598 m.
Here,
T.T 3-14 X 2
^ = ^059^ = ^«^-
BOOK OF INDUSTRIAL DESIGN.
77
Wlien a spur-wheel is to have wooden teeth, it is necessary
that the number of these be some multiple of the number of arms
of the wheel, in order that they may be conveniently attached to
the web ; thus, in the present example, if the wheel is to have
6 arms, the number of teeth must be 102 or 108, to be divisible
by that number ; and if the former be adopted instead of 105, the
pitoh will be slightly augmented in consequence.
To obviate the necessity of making long and tedious calcula-
tions, a table is subjoined, showing the thickness and pitch of
teeth of spur-wheels, in which is adopted the coefficient '105 of
M. Morin, which makes the formula,
t = -105 V^
for cast-iron teeth, and
^ = •145 VT
for wooden teeth : the width being constantly equal to nearly 4-5
the thickness.
TabU of the Pilch and Thickness of Spur Teeth far different
Pressures.
Of Cast-Ii-on.
Of Wood.
Pressure in
Kilogrammes.
Thickness of
Teeth in
Millimetres.
Pitch in
Millimetres.
Thickness of
Teeth in
Millimetres.
Pitch in
Millimetres.
5
10
15
20
30
40
50
60
70
80
90
100
125
150
175
200
225
250
275
300
350
400
500
600
700
800
900
1000
2-3
3-3
4-0
4-6
5-7
6-6
7-4
8-1
8-7
9-4
9-9
10-5
11-6
12-8
13-8
14-8
15-7
16-6
17-3
18-2
19-6
21-0
23-4
25-7
27-7
29-7
31-5
33-2
4-9
6-9
8-5
9-7
12-0
18-9
15-6
17-0
18-4
19-7
20-8
22-0
24-4
26-9
29-1
31-1
33-0
34-8
36-3
38-1
41-2
43-2
49-1
54-0
58-2
62-4
66-1
69-6
3-2
4-7
5-6
6-4
7-9
9-1
10-2
11-2
12-1
12-9
13-7
14-5
16-1
17-7
19-1
20-2
21-7
22-9
23-9
251
27-1
29-0
32-4
35-5
37-2
41-0
43-8
45-8
5-9
8-7
10-4
11-8
14-4
16-9
18-9
20-8
22-4
23-9
25-3
26-8
29-8
32-7
34-8
37-4
40-1
42-4
44-2
46-4
50-1
53-6
69-9
65-7
69-1
75-8
83-0
84-7
With the assistance of this table, and the preceding rules, wo
can always determine, not only the thickness and pit<'h of the teeth,
but also their height and width, since these are in proportion to
their thickness.
DIMENSIONS OF THE WEB.
223. The width of the web is ordinarily equal to that of the teoth
when the whole is of cast-iron. Nevertheless, in some cases — such
as. for example, where very groat irregularities in the pressure and
speed, and reiterated shocks have to bo borne in the heavy nia-
ininerv in engine shops — the web ia made wider than the teeth,
projecting also on either side of the teeth, so that these are whoUy
or partially imbedded, which increases their power of resistance
very considerably. These lateral webs are generally each made
of about half the thickness of the tooth.
The thickness of the web, or crowTi, is never made less than
three-fourths that of the tooth, and very frequently it is further
strengthened by an internal feather, as already mentioned.
213. When the teeth are of wood, the web is much thicker, to
give sufficient hold to the tenons of the teeth ; it is generally
made about 1-5 to 2- times the thickness of the tooth.
NUMBER AND DIMENSIONS OF THE AEMS.
224. The number of arms, or spokes, which a spur-wheel ought
to have, has not, up to the present time, been precisely ard
scientifically determined. According to general expenence, up lo
a diameter of 1 metre, or about 3 feet, four arms are sufficient;
from 1 metre to 2 metres, or 3 feet to 6 or 7 feet, sLx are
necessary and sufficient ; beyond 2-5 m., or 8 feet, eight arms aie
used: and for 5 m., or 16 feet, ten are given; it is seldom this
last number is exceeded, except for wheels of extraordinary di-
mensions.
The section of the arms of the wheel is always in the form ot
a cross, the stronger portion of which lies in the plane of the cir
cumferential strain, whether these arms are cast in one piece with
the boss and the crown, as is the case with wheels ol small
diameter — that is, of such as have not a greater radius than 2 m. or
64 feet ; or whether they are cast in separate pieces, and after-
wards fitted together. The thicker part, then, of the arm must be
strong enough to bear the circumferential strain. Experience
has shown, that when a spur-wheel is in motion, and acted upon
by a considerable force, this strain has a tendency to make the
arms assume a twisted shape, and produce on them a lateral
inflexion. It is to obviate and prevent this, that the arms arc
strengthened by feathers.
The power acts with greatest eflfect near the boss of the wheel,
so that it is necessary to make them wider at this part than near
the crown, so as to approximate to the form which presents au
equal resistance throughout. This will be observed in the
figures in Plate XX. Tlie boss must have such a thickness as
will allow of the wheels being solidly fixed on the shaft. A
thickness of 5 inches may be considered a maximum for the bosses
of moderately-sized wheels. The dimensions of the arms should
be in proportion fo the widtii of the web or crown, their thickness
being ordinarily about J that of the crown. Tiiis proportion is n
good one for wheels under 6i feet in diameter. For larger sizes,
i the width of the web is considered sufficient.
The lateral feathers sliould have, at tlie very most, only tlu'
thickness of the arm. GeiicniUy. the wiilili ol' ilic arm near llio
web is made about 5 of its width near the boss. The lollowing
table, calculated from Tredgold's experiments, shows the pro|)or-
tions to bo given to the arms or spokes of spur-wheels, noeonliiig
to the strain acting at their circumferences; supposing the dia-
meter of tiio wheels to bo 1 in., ami the nunibi>r of anus 6. their
tlii<'kiu'ss being taken equal to \ the widlii of the crown. The
liiiiicnsiiuis triveii are the averages, or those to he applied to tho
:inM. hall-wnv belwi'cn the boss and the crown.
78
THE PRACTICAL DRAUGHTSMAN'S
Table cf the Dimensions of Spur-wheel Arms.
Timgentia\ Strain on the
Width of the Arm in
Width over all, of the
Whtel iu kilog.
centimetres.
Feathers in centimetres.
10
4-20
1-21
10
6-00
2-00
80
8-00
3-00
158
8-50
3-90
>244
9-70
4-85
«36
10-67
6-30
430
11-64
6-80
680
12-12
8-25
730
13-10
8-73
870
13-80
9-70
1100
14-50
10-67
1210
15-50
11-64
1500
16-00
12-60
1750
16-50
13-68
2200
17-00
14-06
2300
17-50
16-50
2660
18-00
17-00
2840
18-50
17-95
3220
19-00
19-50
3500
19-50
19-40
To apply the numbers in this table to wheels of other diameters,
they must be multiplied by V R, R being the radius of the wheel
for which the dimensions are to be calculated.
WOODEN PATTERNS.
225. When a casting has not to be turned, oi otherwise re-
duced, about 1 per cent, must be allowed in the dimensions oi tne
pattern, and if the piece has to be turned, a little more than this.
It is, however, impossible to give any rule in this last case, as the
allowance to be made depends entirely upon the nature and desti-
nation of the piece. The larger the piece, the greater should be
the per centage given.
No piece can be cast with mathematical precision — ^whether it
is, that, on the one hand, the pattern loses its true shape, and
lines, which have been made perfectly straight or circular, become
twisted, notwithstanding that every precaution has been taken in
perfecting it; or, on the other hand, that, in lifting it from the
loam, the moulder is forced to move it laterally, to some slight
extent, so that the casting becomes larger at one part, or twisted
at another ; or, again, that the metal does not shrink equally at all
parts. With regard to the last-mentioned source of error, it has
often been found that the diameter of a wheel, measured through
the line of the arms, is sensibly less than as measured across the
centres of the spaces between the arms. This diflFerence is indeed
so great, that in wheels of 10 to 15 feet diameter, it reaches an
eighth or a sixth of an inch.
It is manifest, that all these considerations must be borne in
mind when constructing "ivooden patterns for castings ; otherwise,
errors of considerable magnitude will arise.
CHAPTER VI.
CONTINUATION OF THE STUDY OF TOOTHED GEAR.
CONICAL OR BEVIL GEARING.
226. Cylindrical or spur-wheels are only capable of transmitting
motion between shafts which are parallel to each other ; and when
the shafts are inclined, or form any angle with each other, the
wheels require to be made conical, and are then called bevil-
wheels.
In order that this description of gear may be capable of working
well and regularly, and of transmitting considerable power when
needed, as with spur gear, it is essential that the shafts or axes of
any pair working together be situated in the same plane ; in this
case, the axes will meet in a point which is the apex common to
the two wheels.
Formerly, when it was required to transmit power through
shafts intersecting each other at right angles, a species of lantern-
wheel was employed for one of the wheels, consisting of a couple
of discs with cylindrical bars for teeth, passing from one to the other
parallel to the axis ; and the wheel to gear with this one was formed
with similar teeth, also parallel with the axis, but projecting up
from a single disc or ring. This form of gearing is still to be found
m old mills ; but it is very defective, and very inconvenient when
any speed is required.
Sometimes, as for some descriptions of spinning machinery — the
cotton-spmner's fly or roving-frame, for example — bevil-wheels are
used, in which the axes are not situate in the same plane ; these are
tei-med " skew bevils," from the teeth having a hyperbcloidal twist
in order that they may act properly on each other. This kind of
wheel does not work well, and is seldom employed, except where
the size is very small, or where a small power only has to be trans-
mitted; the peculiar form of their teeth also renders them very
difficult to construct. Their use is so limited, that further details
respecting them are uncalled for. Indeed, they ought rather to be
avoided, since there are very few cases in which common bevil-
wheels cannot be substituted for them with advantage.
The teeth of bevil-wheels are made of wood or metal, similarly
to spur-wheel teeth, and their geometrical forms are determined on
the same principles.
DESIGN rOR A PAIR OF BEVIL-WHEELS IN GEAR.
PLATE XXII.
227. We propose, in the present example, to give the largei
wheel wooden teeth, and the smaller ones cast-iron ones, as was
done with the pair of spur-wheels last described.
Let A B and a c, figs. 1 and 2, be the axes of the two whe'^ls
assumed here to be at right angles to each other ; though we
BOOK OF INDUSTRIAL DESIGN.
must observe, that what follows will apply equally well to the
i.onstruction of a couple of wheels, the axes of wMch make any
angle with each other, acute or obtuse.
Let B D = -220 m., and e f = -440 m., the radii of the pitch
i-ircles of the two wheels. It is, in the first place, necessary to
determine the position these circles should occupy on their respec-
tive axes. With this view, on any point, b, taken on the axis, a b,
erect a perpendicular, b d, and make it equal to the radius of the
smaller wheel, and through the extremity, d, draw a line, d l,
parallel to this axis ; in the same way, at any point, e, taken on
the axis, a c, erect the perpendicular, e f, equal to the radius of
the larger wheel, and through the extremity, f, draw f h parallel
to A c. The point of intersection, g, of these two lines, f h and
p L, is the point of contact of the two pitch circles, the radii of
which are g i and g k. Make i h and k l, respectively, equal to
the radii, and join the points, h g l, to the common apex, a, thereby
determining what are termed the " pitch " cones, a h g and a g l,
of the two wheels, the straight lino or generatrix, a g, being the
line of contact of the two cones. These pitch cones possess the
same properties as the pitch circles, or, more correctly, pitch
cylinders, of spur-wheels ; that is to say, their rotative velocity is
in the inverse ratio of their diameters, and their diameters are pro-
portional to the respective numbers of their teeth.
The proportions of the pitch cones being thus obtained, with the
centres, o and o', figs. 2 and 3, taken on the prolongation of the
^ven axes, describe the pitch circles, a h' i' and g' k' l'. Divide
these circles into as many equal pai-ts as there should be tooth ;
(hat is to say, in the present case, 24 and 48, respectively, which
operation will give the pitch ; each part is then bisected to obtain
the centres of the teeth and of the intervals, and on each side of
the centre lines are set off the demi-widths of the teeth, regard
bemg had to the difference to be made between the wooden and
cast-iron teeth, as already explained (213).
The external contours of the teeth will be situated in cones, the
generatrices of which are perpendicular to those of the pitch cones ;
they are obtained by drawing through the point of contact, g, on
the line, a g, a perpendicular, b c, meeting the axis of the smaller
wheel in b, and that of the larger one in c ; the points, b and c, are
the apices of the two cones, b h g and c g l.
If these last-mentioned cones be developed upon a plane, it will
be easy to draw upon it the exact forms of the teeth. Now, we
have seen (170) that the development of a cone on a plane surface
takes the form of a sector of a circle, which has for radius the
generatrix of the cone, and for arc the development of the base of
the cone. As it is unnecessary to develop the entire cone in the
present case, it is sufficient to describe with any point, b', fig. 4,
with a radius equal to b g, an arc, a e h, on which, starting from
the point, c, are divided off distances — one, c d, equal to the thick-
ness of the tooth of the smaller wheel, fig. 3, and the other, c e, to
that of the tooth of the larger wheel, fig. 2. The same operation
IS performed for the larger wheel ; that is, with tlio jiaiiit, c',
situated on the prolongation of b' c, and with a radius equal to c g,
describe the arc, f c g, on which are meiusurod the distances,
respectively, equal to the former ones, e (/ and c e.
This done, the outlines of the tccrtli are obtained liy means of
precisely the «>ame operations as those oxpl.iincii in refcronco to
the spur-wheels. Thus, on the radius, b' c, considered as a
diameter, describe a circle, i cj, which, in rolling round the circle,
fcg, considered as the pitch circle of the larger wheel, determines
the epicycloid, e h, which gives the curvature of the teeth of the
larger wheel ; in the same manner, the circle, k e I, described on
the radius, e c', as a diameter, and rolling round the circle, a c b,
gives the epicycloid, c m, which is taken for the curve of the teeth
of the smaller wheel. After having repeated these curves sym-
metrically on each side of the teeth, these are limited by drawing
chords in the generating circles from the point, e, each equal to
the pitch of the teeth, as c n, c k, and then with c' and b' as centres,
describe circles passing, one just outside the point, n. and the other
just outside the point, k ; and to indicate the line of the web, describe
a second couple of circles, nearly tangents to the preceding. Then
project the points, o and p, which indicate the depth and extre-
mities of the teeth, over to the line, b c, in o and p' ; through these
last draw straight lines to the apex, a, which wBl represent the
extreme generatrices of the teeth, as in vertical section.
As all the teeth converge in one point, it is obvious that thtj
contour of the inner ends of the teeth cannot be the same as that
of the outer ends ; the difference is the greater, according as the
width, G r, on the generatrix line of contact is itself greater, ii;
proportion to the entne cone, and to the greater or less angle
formed by the extreme generatrices.
In other respects, this contour is determined in the same manner
as the first. Thus, through the point, r, is drawTi the straight
line, s /, perpendicular to a g, wliich cuts the two axes, and gives
the proportions of the two cones, on the surface of wliich lie the
contours of the inner ends of the teeth. Continuing the opera,
tion as above, portions of the cones are developed, arcs being
described with the points, s' and t', as centres, and radii equal to
r s and r i. The diagram, fig. 5, which is analogous to fig. 4, fully
explains what further is to be done.
What has been said so far, has referred only to one tooth of
each wheel. In proceeding with the execution of the design, after
cutting out templates to the form of the teeth as obtained by
means of them, the outline is repeated, as often as is necessiiry, on
the external cones, the generatrices of which are b g and g c, for
the outer ends of the teeth, and on the internal cones, the genera-
trices of which are r s and r t, for the outlines of the inner ends of
the teeth. At the same time, and in order that the oix-ration
may be performed with regularity, a series of lines should bo
drawn through the points, o, p, of the two wheels, lying on the
surface of tbe external cones, a H g, a g l, and uniting at tlie
apex, a, by means of a "false square," of a form analogous to that
represented at X, in fig. 3, Plato XXIII., for the smaller wheal,
and like that rei)resented at T, fig. 4, of the sjmio Plate, for the
larger wheel.
The forms of the teeth being thus obtained, the partial .section,
fig. 1, of the two wheels is drawn, tlio radii of the slial^s Ix-ing
given, as well as the thickness of the bosses and webs, tlie jiropor
tions employed in the present example being indicated on the
drawings. It will be observed that those teeth wliicii are of wooa
nr(^ adjusted in liie weh of the larger wheel, in the sjune niannoi
as in the spur or cylindrical wheels, the forms of the tenons U-iiiv
nuidilied, so thai their .sides nil incline to the common !1|h«x, a
80
THE PRACTICAL DRAUGHTSMAN'S
The sections, together with the developments, figs. 4 and 5, are
sufficient for the purposes of construction, as all the required mea-
surements can be obtained from them ; but when It is desired to
produce a complete external elevation of the two wheels, it will be
necessary to find the projections of the teeth and other parts.
With this view, the teeth are first actually drawn upon the planes
of projection parallel to the bases of the wheels, as showTi in figs.
2 and 3. It will be recollected, that divisions have already been
made on the pitch cii-cles, a h' i', and l' k' g', indicating the centre
lines, as well of the teeth as of the intervals, and marking the
positions of the flanks; and, consequently, all that remains is to
draw the external outlines and the curved portions. For the
bmaller wheel, the operation consists in projecting to y, in fig. 3,
tlie point, p, fig. 1, which limits the lower and outer edge of the
tooth, and in describing with the centre, o, and radius, o p', a circle
hmiting the whole of the teeth externally, and cori-espondlng to the
section of the cone in which the points, p p', fig. 1, lie. In this
circle, also, terminate the curves of the outer portions of the teeth,
and tlieir exact points of intereection are obtained by measuring on
each side of the centre lines, r o, distances, v u, v p, equal to the
corresponding distances, t' u', i' p', in fig. 4. Then, through the
points. It, p, draw a series of lines, converging to the centre, o ; and
through the points, e, e, found in a similar manner, draw similarly
converging lines, indicating the inner angles of the intervals.
Further, find the circular arc to represent the epicycloidal curve,
passing through the points, w, p, and tangential at the same time
to the lines, o e, at the points, e, e.
The method of doing this is shown in fig. 3 : it consists in draw-
ing through the point, e, a line, e z, at right angles to the radius,
o e, and in bisecting the chord, e w, by a perpendicular cutting,
e z in z, which will be the centre of the required arc. Arcs of the
same radius, e z, are employed for the curves of all the teeth on the
smaller wheel, and the outline of these is completed by determin-
ing, in a similar manner, the arcs for the corresponding curved
portions of the inner ends of the teeth, after ha\Tng projected and
drawn circles through the points corresponding to r and p'', of fig.
1. Finally, the lines of the web between the teeth — that is, the
bottom lines of the intervals — are drawn, the projections of the
points, y and y', being found, and circles described with the centre,
o, passing through them. It will be observed that, on a portion
of fig. 3, is represented a view of a quarter of the lower and inner
side of the wheel, whilst the other portion of the figure is an
external view, showing the teeth as in plan ; in the former case, the
outline resembles that of a spur-wheel, for, as it is the larger ends
of the teeth and web on which we are looking, the narrower and
mclined portions are hid behind.
The lateral projection of the teeth of the small wheel, fig. 1, is
obtained, first, by successively projecting or squaring over, from
the plan, fig. 3, the points, e, e, to the pitch line, g h; and
secondly, by similarly squaring over the points, p p, to the
gxteri'al line, p p'. Through the points, e, e, draw a series of lines
eonverging at the apex, b, and representing the flanks of the teeth,
and liinited by the line, y y; then draw curves tangential to the.'^e
flanks at the points, w, p, making them pass through the extreme
points, e, e. Where the scale of the drawing" is very large, and it
is wished to be particularly precise in delineating these curves,
points intermediate between m, p, and e, e, may be obtained by
describing intermediate circles in fig. 3, representing sections of
the cone, projected in straight lines in fig. 1, over to which are
projected the points of intersection of the curves, \vith the circles
in fig. 3. Through the points, u p, draw straight Unes, converg-
ing in the apex, a, and find the lateral projection of the inner end?
of the teeth, supposing planes to pass through the points, r, y'
and p' ; these points in the circular projection of the planes, fig. 3,
being squared over to the corresponding rectilinear projections
in fig. 1. The inner ends of the teeth are then completed by
drawing the flanks, e', y', all converging in the apex, s, and joined
by arcs passing through the points, e', p'.
The upper left-hand quadi-ant, ivi, of fig. 2, is a face view of the
teeth of the larger bevU-wheel, with wooden teeth, the whole
being drawn in the same manner as in fig. 3. The method of find-
ing the centre, z, of the arc, which is substituted for the cuE\-ed
portion of each tooth, is shown in fig. 2°. From this view (fig. 2)
are obtained the various points required to produce the lateral
projection of the teeth in fig. 1. The operations are precisely the
same as those just described in reference to fig. 3, and the smaller
wheel ; the same distinguishing letters are also used to point out
the simOarity.
The same figure (2) also comprehends at n a second quadrant
of the wheel, drawn as seen from the under side, so as to show a
face view of the tenons of the wooden teeth, the sides of which all
converge in the point, o'. A third quadrant, p, gives a view of
the outer side of the web or crown, the teeth being supposed to be
removed, so that the mortises are seen. The last quadrant, e.
gives a back view of the web, also without the teeth.
Fig. 6 is a section of one of the arms of the larger bevil-wheel,
made through the line, 1 — 2, in fig. 2. Fig. 7 is a section of the
web made through the line, 3 — 4, fig. 2, passing through the
centre of the mortise ; and fig. 8 comprehends a lateral projection
and two end views of one of the wooden teeth.
BevU, as well as spur-wheels, are fixeci on their shafts by means
of keys, and pressure screws, v, are often added to insure their
perfect adjustment centrally.
The measurements given in the diagrams will enable the student
to form an accurate idea of the actual proportions of the variou.--
parts.
the coksteuction of wooden patterns for a pair
of bevtl-wheels.
Plate XXTTT.
228. The observations we have already made with reference to
the patterns of spur-wheels, are evidently equally applicable to the
construction of patterns for bevD-wheels ; stUl, at the same time,
the difference in the form of the latter calls for further details,
more especially appertaining to them.
PATTERN OF THE SMALLER BEVIL-WHEEL.
229. F^gs. 1 and 2 represent the two projections of the pattern
of the smaller of the tw"o wheels in the preceding Plate. Fig. 3
is a vertical section through the line, 1 — 2, of fig. 2, showing on
one side the layers of wood put roughly together, and intended tn
BOOK OF INDUSTRIAL DESIGN.
8}
form the crown ; and on the other, a view of the same as finished,
with the arm and its feathers.
It will be seen from these figures that the crown is built up in
tlie same manner as that of the pinion in Plate XXI. ; the layers
of wood are, however, in steps, increasing in diameter dovmwards,
so as to give the required conical form when turned. When these
pieces are glued together, the whole is turned externally and
internally in such a manner as to conform exactly to the full-sized
drawing, previously made on a board planed smooth for the pur-
pose. " Squares," also, should be made from the dravidngs, to
serve as guides in producing the correct conical inclination.
After turning the top face, V b', perpendicular to the axis of the
cone, the pattern-maker proceeds to turn the external conical
surface, a' V, of the web or crown. As a guide in doing this, he
takes a "false square," t, fig. 4, of which one side, h b, corresponds
to the plane face, b' b', and the other, a b, to the inclination
of the conical generatrix, a' b' : it is very easy with this to take
off just as much of the wood as is necessary, without the liability
of going too far. It is also necessary to determine the inclination
of the generatrix, b a', of the outer cone, perpendicular, it will be
recollected, to the contact generatrix, g r, by means of the square,
X, fig. 3, the side, a b, of which is applied exactly to the conical
surface, a' b', and the side, a c, then gives the inclination of the
conical surface, a' c' ; and the same square being turned round wdll
give the inclination of the internal conical surface, b' d', the gene-
ratrix of this, the smaller cone, being s r, parallel to b g, that of
the larger one.
Finally, the thickness at a' c' and b' d', is measured on the
wooden web, so as to obtain the proportions of the internal conical
surface, c' d', to be turned out in a similar manner.
Mortises have now to be cut in the crown to receive the ends of
the arms, c, and their feathers, e. As the wheel under considera^
tion is of very small diameter, the number of arms is limited to
four ; these arms are so placed inside the crown that the feathers
are all on one side, and towards the wider end of the cone. Their
attachment to the web is by means of a circular groove or mortise,
seen at e'/', fig. 2, and at g' d', fig. 3, and they are united at the
centre to each other and to the boss, in the same manner as the
arms of the pinion, described in reference to Plate XXI. The
arras are not placed in the middle of the boss, as in the spur-wheel
and pinion, but are simply applied to the base of the boss, which
may, consequently, bo of a single piece ; and the feathers are let
into a groove extending their whole length, and are fixed into the
boss and (M'own at either extremity. The boss is slightly coned,
so as to give the " draw " necessary in the construction of the
mould. Its outer edges are indicated by the lines, mn,mn, whilst
the other lines, o p, which are, on the contrary, parallel to the axis,
«how the depth of the grooves cut to receive the feathers of the
arm. Those last, as shown in the section, fig. 10, ;uo tliicker near
the arms.
The core pieces, e, are added on either end of the boss, and lln^
whole is held firmly together by means of a central bolt.
The pattern being so far advanced, the external conical surface
if* divided into as many equal parts as there are to bo teoth and
uitervals, and, with the assistance of the " false sciuaro," T, lines
wlmii represent generatrices of the cone, are drawn through the
points of division, to indicate the positions of the teeth or of the
grooves to receive them.
Each tooth is cut out separately according to the full-size draw
ing made, as already mentioned, which, besides containing the ver-
tical section, fig. 3, should also show the exact form of each end of
the tooth, b', and of the dovetail joint attaching them to the web
Fig. 5 shows a portion of this drawing for the larger ends of the
teeth.
PATTERN OF THE LARGER BEVIL WHEEL.
230. Figs. 6 and 7 represent the elevation and plan of the
pattern of the larger bevU wheel, with wooden teeth, represented
in Plate XXJI. Fig. 8 is a vertical section through the axis of the
wheel, showing on one side the arrangement of the pieces of wood
built up upon one another, and forming the crown, a, and on the
other side, the same piece, turned and finished, attached by the
arm, c, to the boss, d.
Fig. 9 represents the false square, T, employed as a guide for
giving the proper incUnation to the external conical surface, a' b',
of the crown.
Fig. 11 is a transverse section of one of the arms, or spokes,
taken through the line 7 — 8 in fig. 7.
Whatever explanations are called for regarding the construction
of the crown, a, the arms, c, and the boss, d, as well as the uniting
of these parts with each other, have already been given in reference
to preceding examples. We have distinguished all corresponding
parts and working lines by the same letters.
The only difference between this last and the preceding example
consists in tlie disposition of the tooth pieces, b', placed on the out-
side of the crown, to form the sockets in the mould for receiving
the core pieces for the mortises, into which the wooden teeth are
to be fixed after the piece is cast.
It must be observed, in the fii'st place, that these projections
must be shaped so that the end, k I, is inclined to the surface, b' a',
instead of being perpendicular to it. This inclination must be
sufliicient to allow of the easy disengagement of the piece from
the mould. This disposition is necessary, because the lower
half of the mould takes the hnpression of the outside of the crown,
with the tooth pieces and the upper portions of the arms, wliilst
the top part of the mould takes the inside of the crown, the
feathers of the arms, and the boss, the position of the whole being
the reverse of that in which they are represented in the drawing.
The core pieces for the teeth are formed by the moulder in core
boxes, similar to those described in reference to figs. 10 mid 11,
Plate XXI., which wo have reproduced in Plato XXIII., figs. 12,
13, and 14, as modified to suit the dillerent form of tooth. Fig.
12 is a face view, and figs. 13 and 14 are sections made through
the lines 9—10 and 11—12 otMig. 12. It will be observed that,
at the larger (Mil (it llic tonlli. the part to project is foniieci with
an iiuTniatioli corrt'sponding to /,■ /, in tig. 7, already reforn-d lo Jis
rciiuircd in this case.
The operations (■•■illcd for in dclincMling those piiftorns are all
fully indicated, and are analogous to those in the pn>ceding plalos.
Tli<« observations, also (It!) and 214), already made, with referenco
to calculating the allowance to bo made for shrinking, and for tJio
turning and tiuishing proces.ses, are eiiually jipplicablc to Iho awo
before us.
h9,
THE PRACTICAL DRAUGHTSMAN'S
INVOLUTE AND HELICAL TEETH.
PLATE XXIV.
DELISEATIO.N OF A COUPLE OF SPUR-WHEELS WITH INVOLUTE
TEETH.
FlGUBES 1 AND 2.
231. In the various systems of gearing just discussed, wherein
epicyeloidal teeth have been employed, it will have been observed —
1st. That the outline of the teeth of one wheel depends on the
diameter of the other wheel with which it is in gear.
2d, That the distance between the centres of any couple of
wheels cannot be altered in the slightest degree without deteriorat-
ing the movement.
3d. That the distance from the respective centres of the point
of contact varies throughout the duration of the contact; from
which must obviously result irregularity in the action and inequality
in the amount of friction.
The practical defects arising from these causes have induced a
search after other forms, and amongst these a modification of the
involute has been tried. The form in question possesses the fol-
lowing advantages : —
Ist The form of the teeth of such a wheel is quite independent
of the diam.ter of the wheel with which it is to gear.
2d. The distance between the centres of the wheels may be
varied without disadvantage.
Some authors also attribute to this form the property of trans-
mitting the pressure uniformly throughout the duration of the eon-
tact. This, however, cannot be the case altogether, for the distance
of the point of contact from the centres of the wheels is constantly
varying — ^the variation not being accurately proportional in the two
wheels. This system of gearing is constructed on the following
principles : —
Let the centres, o and o', fig. 1, of the two wheels be given,
and the radii, o a and a o', of the respective pitch circles ; also,
let o B be the radius of any circle described with the centre, o ; to
the circumference of this last draw a tangent, a b, passing through
the point, a, and prolong it indefinitely in either direction. From
the centre, o', let fall on this line a perpendicular, o' c, on which
will accordingly be the radius of a second circle tangent to the
same line. These circles of the radii, o b and o' c, are those from
which are derived the involute curves, a b and c d, forming the
outline of the teeth. For the rest, the wheels are drawn just as
in Plate XVIH. (197.)
It must be observed that the curve, a b, which is the involute
of the circle of the radius, o b, is that for the tooth of the spur-
wheel, the centre of which is the same, and the radius, o a : and, in
like manner, the curve, c d, the involute of the smaller circle, is
that for the teeth of the pinion of the radius, o' a. It thus follows
that the form of the teeth of the spur-w^heel is quite independent
of the diameter of the pinion, whilst that of the pinion teeth is
independent of the diameter of the spur-wheel. From which it
follows, that wheels constructed in this manner may be set to gear
with any wheels whose teeth are formed on the same principle,
and whose pitch is the same, whatever may be their respective
ntameters. The epicyeloidal system does not admit of this,
although, when the wheels are large, and there is not much dif-
ference between their 4iameters, a slight de\iation from strict
mathematical proportions is not found practically inconvenient.
The mvolute curves, a b and c d, are repeated sjinmetrically
on either side of the division lines representing the centre lines of
the teeth. If we now suppose the two involutes, a b' and a c', to
be in contact at the point, a, on the line of centres, o o', and we
measure off" on the common tangent, a B, a distance, a e, equal to
the pitch, /g', as measured at the pitch circle, and then, with the
centre, o', describe a circle passing through the point, e, this circle
will be the external limit of the pinion teeth.
In like manner, if, on the other portion, a c, of the tangent, we
measure a distance, a e', also equal to the pitch, /^, and with the
centre, o, describe a circle passing thi-ough tlie point, e', it will
be the limi t of the spur-wheel teeth. It is further obnous, that
circles passing a little within the point, e', on the one hand, and e,
on the other, will determine the depth of the intervals, or the line
of the web of the pinion and spur-wiieel respectively.
Fig. 3 is a diagram to show — first, how that the point of con-
tact of the two involute curves is always in the line of the common
tangent, b c. Thus, referring again to fig. 1, and supposing the
pinion to turn in the direction of the arrows, the point of contact,
as A of the involute, a 6, is grad'ially removed away from the
centre, o', of the pinion, whDst it approaches nearer and nearer to
the centre, o, of the spur-wheel. Returning to fig. 3, it is showTi,
in the second place, that the distance between the two centres,
o, o', may be varied without its being necessary to alter the cunes ;
but, in such case, the inclination of the tangent \viU be different,
becoming, for example, as b c', when the two centres are brought
nearer together.
In practice, instead of determining the radius, o B, arbitrarily,
and then deri%ing the other radius, o c, from it, or vice versa, the
circles which sene for generating the involutes may be found, as
w'ell as the inclination of the tangent, by the following method : —
On one of the pitch circles, that of the pinion, for example, take
an arc, a ?', equal to the pitch of the teeth ; draw the radius, o' i,
and on it let fall a perpendicular, a m, from the point, a ; o m will
then be the radius of the generating circle for the involute curve f i
of the teeth of the pinion, and by prolonging ot a to n, wliich is, in M
fact, the common tangent, and drawing the radius, o tj, perpendi-
cular to it, or, what is the same thing, parallel to o' m, o n will bo
the radius of the generating circle for the involute of the teeth of
the spur-wheel.
If this rule is applied to wheels of large diameters, it will give
curves dififeiing very slightly from epicycloids.
By taking for the generatrng circles, as in the first case, radii,
o B and o' c, sensibly less than the radii of the pitch circles, the
inclination of the common tangent to the line joining the centres is
greater, and the resulting form of tooth possesses greater propoi-
tionate width and strength at the roots, which is desirable foi
gearing intended to transmit great or irregular strains.
It will be observed further, that, according to this system, tho
rectilinear portion of the flank of the tooth is almost reduced to
nothing, indeed the cur^-e may be continued down to the line of the
web with advantage, as the tooth will, in consequence, be much
stronger near the w-eb, which is not the case with the epicyeloidal
BOOK OF INDUSTRIAL DESIGN.
83
toeth, for in these the flanks all converge towards the centre of the
wheel, and the tooth is, in consequence, narrower at the neck, close
to the web, than at the pitch circle.
Fig. :3 is a fully shaded elevation, or vertical projection of the
spur-wheel separated from the pinion. The portions of these
wheels not particularly referred to, are constructed on the same
general principles as those previously discussed.
helical gearing.
Figures 4 and 5.
232. If to a worm-wheel we apply, instead of a worm, a pinion
with teeth helically inclined to correspond to the similarly inclined
teeth of the worm-wheel, we shall have a spur-wheel and pinion
constructed on the helical principle.
This system, invented in the seventeenth century by Hooke,
but reproduced since by White and others, claims to possess two
properties which have been often thought to be incompatible with
each other — ^namely, uniformity of angular velocity, and freedom
from other than rolling friction between the teeth. In other words,
the arcs described by driver and follower will be equal in equal
times, and the contact between the teeth will resemble that of
circles rolling on planes.
Added to these properties, and consequent to them, are the
advantages of a constant contact, and of an insusceptibility to the
play between the teeth, which invariably exists more or less
palpably in gearing constructed according to the systems before
described.
The form of the helical teeth, as taken in a sectional plane at
right angles to the axis of the wheel, may be derived either from a
couple of epicycloids, or a couple of involutes ; it is only the sides
which, in common spur-gearing, are parallel to the axis that here
follow the inclination of a succession of helices coming in contact
one after the other. The arrangement is such that the contact of
each tooth commences at one side of the wheel and crosses over
to the other, and does not cease until the following tooth shall
have commenced a fresh contact.
Thf helicoidal system may 1)0 applied either to wheels having
their axes parallel, as spur-wheels, or intersecting, as bevil-whcels,
or again inclined, but not intersecting, as skew bevils.
In figs. 4 and 5 are represented, in face and edge view, a spur-
wheel and pinion, constructed according to this system of Hooke's,
this being its simplest iind most common application : — Let a o an.l
a' o be the radii of the respective pitch circles of the two wheels,
these radii being, of course, in the same ratio as the numbers of
the teeth, as in common gearing. The radii are supposed to lie
in a vertical plane, b' c', and it is on this plane, as turned at rigiit
angles, that the operations represented in fig. 4 are sujjposed to bo
performed.
Tiiese operations have for their ohjoet the ohtainnient of the
outline of the teeth, and are i)recisely the same as for any oilier
epicycloidal system of gearing. Thus, the curves, a I> and a c,
are derived from the generating circles, o n a and a n' o', as also
fho flanks, a (/ and a /;; hut it is uiineccssai-y to repeat a, dclailecl
explanation of the proceeding.
Supposing, thi^n, the outline of the (cetli lo jx? .Irauii as nn lln^
plane, b' c', representing say the anterior face or base of the wheels,
next draw the line, e f, (fig. 5,) representing the opposite face, and
parallel to the first, limiting also the breadth of the wheels.
To proceed methodically, the teeth should also be drawn as seen
on this plane, e f being behind the outlines of the anterior ends of
the teeth, a distance equal to a a', or rather more than the pitch.
These last outlines need only be represented in faint dotted or
pencil lines in fig. 4, as the parts they represent are not actually
seen in that view when complete. Thus, starting from the point,
a', on the pitch circle of the spur-wheel, and from the point, a',
on the pitch circle of the pinion, we repeat the contours of the
teeth, as obtained at e a i and d a n, respectively.
As the result of this disposition, it will be observed, that if the
curve, A i, of the tooth, a, of the spur-wheel is in contact, at the
pitch circle, with the flank, g d, of the tooth, g, of the pinion at
the anterior face, b' c', and if the wheels be made to turn to a cer-
tain extent in the direction of the arrows, the curve, a' i', on the
opposite face, e f, wUl in time be found to be in contact with the
corresponding flank, g' d', of the pinion. In other words, if the
space between the curves, a i and a' i', be filled up by a helicoidal
surface, as also the space between the flanks, g d and g' d', all the
points of one such surface will be in contact successively with the
corresponding points on the other ; so that when, for example, the
curve A i', shall have reached the position, a^ P; that is, when it
shall have passed through a distance equal to a a', the posterior
curve, a' i, will have assumed the position held originally by a t ;
or rather, a position directly behind this in the plane passing
through the axis, and the point of contact between a' i' and a' d'
will then obviously be in the line of centres, o o'. It thus follows,
that any'tvv'o teeth which act on each other will be constantly in
contact on the line of centres throughout a space equal to a a'.
This space, a a', is, as before stated, somewhat greater than the
pitch of the teeth, so as to allow a following couple of teeth to act
on each other, and be in contact on the line of centres before the
couple in advance shall be quite free, and thus a constant contact
on the line of centres is preserved throughout the entire revolu-
tion.
In order to delineate the lateral projection, fig. 5, it w ill bo
necessary to find the curves which i'oiin the outline of the helicoidal
surfaces of the teeth. The princi])h', according to which this is
to be done, is precisely what has already been explained (208). In
the present case, however, as we have but IVaginonts of helices to
draw, in place of finding the pilch ut llic lirlix. and then (li\idiii^
it and the circnnd'erfnce proportionately, it will be sullieient to
divide tho width, b' e, of the wheels, into a certain nundier of
equal parts; and through the points of liivision, to draw lines
parallel to b' c'. Further, tho arcs, a a', v c', / /', must be divided
into a like number of equal parts.
To render the diagram dearer, these divisions are Iranst'orrcil
to 1, 2, 3, 4, &c., and 1', 2', 3', 4', &c. (fig. 4.) Each point, 1,2, 3, I,
beinf squared over, in succession, to tiio corresponding linos m
fi,r. 5 — namely, the lines of division first obtained, and lying
]>arallcl to the faci-s of thi< wheels, the operation will give tho
iMirvc, 1, 3, 6, 6, (fig. ft,) corresponding to tho ontlino of tho exter-
nal edge, extending from i to i'. The curve. 1', 3', !i\ fi', siiniliirly
gives the other edee. it is alse oli\ ious thai ihe lino of junction
84
THE PRACTICAL DRAUGHTSMAN'S
of the tooth with the web will be represented by the helical curve,
a a\ (fig. 5), having the same pitch as the last, but lying on a cylin-
der of a somewhat smaller diameter.
The lateral projections of all the teeth are determined in the
same manner, but they will, of course, assume various aspects,
from the different positions in which they lie with respect to the
vertical plane.
233. In construction, in order to determine the exact inclination
of the teeth, the following proportional formula is employed. The
four terms of the formula being, the radius of the wheel, its width,
the given circumferential distance, corresponding to a a', and the
pitch of the helix ; that is, a a' : a o : : b' e : x, x being the heli-
cal pitch for the spur-wheel, or the quantity sought. It may be
obtained geometrically, simply thus : — Make the straight line, M n,
(fig. 6,) equal to the arc, a a', as developed ; at the extremity, n, of
this line, erect a perpendicular, n l, equal to the width, e' e, of
the wheels ; join l m, which will give the mean inclination of the
tooth, corresponding to the pitch cu-cle. Then make n i equal
to the arc i i', rectified, and n j equal to the arc, e e', rectified,
which will give the inclinations, l i and l j, of the helices, passing
through the extremity, i, of the tooth, and the line, e, of junction
of the tooth to the web.
It will be understood that the helices of the pinion-teeth will
have the same inclination as those of the spur-wheel teeth, with
which it is in gear, and the helical pitch is, in consequence, differ-
ent ; for, the radius is smaller, and the corresponding proportional
formula becomes a a' or a g' : a o' : : b' e : x.
The motion of wheel-work, constructed according to the helical
system, is remarkably smooth, and free from \ibratory action,
but it has the defect of producing a longitudinal pressure upon the
axes, from the obliquity of the surfaces of contact to the plane of
rotation. This, however, may be obviated, and the longitudinal
action balanced, by making the wheels duplex ; that is, as if two
wheels, on each axis, were joined together — the inclination of the
helices being in contrary directions, or right and left handed.
Such wheels, though duplex, need not be wider or thicker, in
proportion, than simple ones ; for the arrangement would permit
of a much greater obliquity of the teeth, the only limit, indeed, to
the degree being the tendency to jam, which would arise were the
inclination very great.
When the wheels are placed on axes which are inclined to each
other, as in common bevil-wheels, the helices become such as are
described upon conical surfaces, and require to be drawn in the
manner already shown (174), the form of the tooth being previously
determined, for each end, by means of the developed planes of the
opposite faces of the wheels.
Besides the epicycloid and involute and their various combina-
tions, other and more complex curves have at different times been
proposed for the forms of wheel teeth. The most worthy of
notice amongst these is that derived from the "hour-glass " curve,
the properties of which have lately been investigated in a very
scientific manner by Professor Sang of the Imperial School at
Constantinople.
If a couple of discs, with their pitch circles touching, be made
to revolve at a rate proportionate to the required number of teeth
in each, a point may be imagined as travelling along a curve,
returning upon itself in such a manner that it will describe the
forms of the respective teeth on each disc. In the system of
teeth alluded to, this point is made to travel along the "hour-
glass " curve, a curve similar to that described by the piston-rod
attachment in Watt's parallel motion, and also exhibited by the
vibration of a straight wire, whose breadth is double its thickness.
The form of tooth obtained in this manner is demonstrated by its
inventor to be theoretically superior to all others yet known. The
chief advantage appears to be, that whilst according to the epicy-
cloidal and involute systems, the form of the entire tooth is made
up of two ciu-ves of different natures, whose junction cannot, in
consequence, be perfectly smooth or fluent, the point of inflexion
or passage from one curve to the other, occurring, moreover,
precisely where the best action would otherwise be. The " hour-
glass " curve produces one continuous analytic curve for the entire
outline of the wheel, thereby avoiding all sudden transitions, such
outline, at the same time, allowing of the interchange, in any way,
of wheels of the same pitch.
The great exactness and nicety obtainable by and called for in
the construction of teeth on this system, is, however, far beyond
the requirements of ordinary machinery. Indeed the practical
engineer and machinist will not be at the trouble of emplo}ing
even epicycloidal or involute curves, but contents himself with arcs
of circles approximating pretty nearly to these curves. The
method generally pursued in determining the best proportions for
the radii of these substitutive ares is as follows : A pair of templets
or thin boards are cut to the curvature of the pitch circle and
generating circle, respectively, of the whe^l, the shape of whose
teeth is sought. The generating templet carries a point which is
made to describe the outline of the tooth on an additional board,
by rolling its edge on that of the pitch templet. The operator then
finds by trial with a pair of compasses, a centre and radius which
will give an arc agreeing as nearly as possible with the curve
traced by the templet. Through the centre thus found he describes
a circle concentric with the pitch circle, and in which the centres
for the arcs of all the teeth will ob%iously lie, and retaining the
radius, he steps from tooth to tooth in both directions, until all the
teeth are marked out.
A very ingenious and useful scale was invented some years ago
by Professor Willis, which renders unnecessary this preliminary
operation for obtaining the radii and centres. This scale, termed
the " Odontograph," is now largely emplo}'ed, and is found to give
very excellent forms of teeth. Its application is very convenient.
A graduated side of the instrument has a certain inclination to
another, which is first made to coincide with a radius of the
wheel, whilst its point of intersection with the first is placed in the
pitch circle. The graduated side gives the direction in which the
centres lie, whOst the lengths of the radii are obtained from tables
calculated for the purpose, indicating the respective distances on
the graduated scale, and corresponding to the given pitch and
number of teeth.
Wheels with teeth formed according to this scale are capable
of being interchanged, which is not the case with those in which
the arcs are determined according to other rules.
After going through the explanations given, and rules laid down
in the last few sections, the student should be quite competent
BOOK OF INDUSTRIAL DESIGN.
85
to design practical arrangements and combinations of toothed gear
according to whichever of the systems may be preferred.
CONTRIVANCES FOR OBTAINING DIFFERENTIAL
MOVEMENTS.
THE DELINEATION OF ECCENTRICS AND CAMS.
PLATE XXV.
234. Eccentrics and cams are employed to convert motion,
whilst toothed-wheel work is for the simple transmission of it.
Endued themselves \vith a continuous circular movement, they
are so constructed as to give to what they act upon, an alternate
rectilinear movement, or an alternate circular movement, as the
case may be, the motion so produced being obtainable in any
desired direction.
CIRCULAR ECCENTRIC.
235. There are several descriptions of eccentrics. The simplest
and most generally employed, consists of a circular disc, completely
filled up, or open and with arms, according to its size, and made to
turn in a uniform manner, being fixed on a shaft which does not
pass through its centre. Such eccentrics are represented in
Plate XXXIX.
The stroke of such a piece of mechanism is always equal to
twice the distance of its centre from that of the shaft on which it
turns ; that is to say, to the diameter of the circle described by its
centre during a revolution of the shaft. The motion of the piece
acted upon is uninterrupted during either back or forward stroke,
but it is not uniform throughout the stroke, although that of the
actuating shaft is so ; the velocity, in fact, increasing during the
fii'st half of the stroke, and decreasing during the second half.
heart-shaped cam.
Figure 1.
236. When it is required to produce an alternate rectilinear
motion which shall be uniform throughout the stroke, the shape of
the eccentric or cam is no longer circular; it is differentially
curved, and its outline may always be determined geometrically
when the length of the stroke is known, together with the radius
of the cam, or the distance of its centre from the nearest point of
contact.
An example of this form of cam is represented in the figure.
Let a a' be the rectilinear distance to be traversed, and o, the
centre of the shaft on which the cam is fixed, it is required to
make the point, a, advance to the point, a', in a uniform manner
during a semi-revolution of the siiaft, and to return it to its original
position in the same manner during a second semi-revolution.
With the centre, n, and radii, o a, and o a', describe a couple of
circles, and divide them into a certain number of equal jiarts by
radii passing through the points, 1, 2, 3, 4, &c. Also divide the
length, a a', into half as many equal parts as the circles, as in the
points, 1', 2', 3', &.C. D(^scribe circles passing through these
points, and concentric with tiie first. Th(^so circles will succc^s-
sively intersect the radii, o 1, o 2, o 3, &.C., in the points, b, c, </, c,
&<!., and the continuous curve passing through those points vvili
be the theoretical outline of the cam, which will cause the poLntj
a, to traverse to a', in a uniform manner, for the equal distances,
a' V, V 2', 2' 3', &c., passed through by the point, a, correspond
in succession to the equal angular spaces, a' 1, 1 — 2, 2 — 3, &c.,
passed through by the cam during its rotation.
As it is not possible to employ a mathematical point in prac-
tice, it is usually replaced by a friction roller of the radius, a i,
which has its centre constantly where the point should be ; and it
will be seen, that in order that this centre may be made to travel
along the path already determined, it wDl be necessary to modify
the cam, and this is done in the following manner : — With each of
the points, b, c, d, &c., on the primitive curve as a centre, describe
a series of arcs of the radius, a i, of the roller, and draw a cun-e
tangent to these, and such eurv'e will be the actual outline to be
given to the cam, b.
It will be seen from the drawing, that the curv'e is symmetrica],
with reference to the line, a e, which passes through its centre ;
in other words, the fii-st half which pushes the roller, and conse-
quently the rod, a, to the end of which the roller is fitted, from
a to a', is precisely the same as the second half, with which the
roller keeps iij contact during the descent of the rod from a' to a.
Thus the regular and continuous rotation of the cam, b, produces
a uniform alternate movement of the roller, and its rod, a, which
is maintained in a vertical position by suitable guides.
In actual construction, such a cam is made open, and with one
or more arms, like a common wheel, or filled up, and consisting of
a simple disc, according to its dimensions ; and it has a boss, by
means of which it is fixed on the shaft. When it is made open,
it is cast with a crown, of equal thickness all round, and strength-
ened by an internal feather, curved into the boss at one side, and
into the arm or arms at the other.
Examples of the heart-shaped cam are found in an endless
variety of machines, and particularly in spinning machinery.
cam for producing a uniform and intermittent movemesl.
Figures 2 and 3.
In certain machines, as, for example, in looms for the " picking
motion," eases occur where it is necessary to produce a uniform
rectilinear and alternate motion, but with a pause at each extremity
of the stroke. The duration of tiie pause may bo equal to, or
greater, or less, than that of the action. Fig. 2 represents tho
plan of a cam designed to produce a movement of this description ;
and in this case the angular space passed through by the cam, in
making the point, a, traveree to the position, a', is supposed to bo
equal to half the angular space described by it, whilst the point, a,
is stationary, whether in its position nearest to tlie centre, or
its furthest, a', from it. For this reason, tlie cin-les of tho
ra(hi, o a, and o a' are caili (li\ idiil into six (•<|ual ]iarts in iho
points, a', 1, /, ^'S h, and J. Of these portions, tfii' two opposit.-.
1 /"and j h, corresi)ond to the eccentric curves, b f and / A, which
produce the niovenu'Ml, whilst the otlu'r portions correspond to tliu
l)auses.
After liavini,'' diawn tlu' dianu'lers, 1 /i, and /" /, the ecceniru'
curves, b f, and / /i, an- deterniined in precisely tlie sjuno manner
as tile contiinious enrve :ihi'ady discussed, and represented in fiy.
1. 'J'liat is t(i s:iy. Hie aii's I /', and j /i, are U- he ilivided into «
80
THE PRACTICAL DRAUGHTSMAN'S
certain number of equal parts by radial lines; and the line, a a',
being divided into a like number of equal parts in the points, 2',
3', 4', &c., concentric cu-cles are to be di-a\\-n through those points,
which will be intersected in the points, c, d, e, by the radial division
lines. Lines passing through these points of intersection will be
the curves sought, bf, and I h.
The arcs, b a I, and fg h, which unite the extremities of the
curves, are concentric with the shaft, and consequently, as long as
the point remains in contact with these arcs, it will continue with-
out motion, although the cam itself continue its rotation.
The observation made with reference to the preceding example
of a cam, applies equally to the one, c, under consideration — that
is, with regard to the actual shape to be given to it, which is derived
from the substitution of a friction roller of the radius, a i, for the
mathematical point, a. The operation is fully indicated on the
diagram.
This eccentric not being intended to overcome any great resist-
ance, is made very light, a considerable portion of the metal being
cut away, and merely a couple of arms left for stiffhess. The
crown, arms, and a great part of the boss, are, in fact, all of a
thickness, as will be more plainly seen in fig. 3, which is simply
a section made through the line, 1 — 2, in fig. 2. Fig. 3 also shows
the proportions of the roller, and its spindle.
When the moving point, or the roller, is constrained to move
through an arc, instead of a straight line, being, for example, at
the end of a vibratory lever, the curves of the cam are no longer
s3Tnmetrical, but the operations by which they are determined are
stOl the same, the difference arising from the divisions of the arc,
which takes the place of the straight line, a a'.
TEIANGIILAR CAM.
FiGUKES 4 AXD O.
238. A species of cam, in the form of a curvilinear equilateral
triangle, is sometimes employed in the steam-engine, to give mo-
tion to the slide valve. This valve is generally of cast-iron, of a
rectangular form, as at t, figs. A and ©. It is hollowed out in
its inner side, to form a passage, and it applies itself, with its inner
planed edges, to a surface, a b, on the cylinder, d, also planed
true, and called the valve face. Its function is to allow the steam
to pass alternately to the upper part of the cylinder, by the port,
c, or to the lower part, by the port, d, whilst the hollow part of
the valve forms a communication alternately between either of
these ports, c, d, with the escape pipe, e. To obtain the desired
effect, it is necessary that the slide valve be actuated with an alter-
nate rectilinear reciprocatory movement; for this purpose it is
attached to a vertical rod, i, passing through a stuffing-box in
the valve casing, and connected to the rod, w, represented in fig.
©, and forming one piece, with the rectangular frame, f, inside,
which works the triangular cam, g.
It is the last piece which has to effect the raising and lowerinor
of the valve a certain distance and intermittently, in such a man-
ner that the port, c, for example, may be open to the entering
steam for a certain time, whilst the other, d, is in communication
with the escape pipe, and reciprocally.
Let e, fig. 4, be the whole stroke of the valve, or the dist<mee
Uirough which it traverses; with the centre, o, and with this
distance, o e, for a radius, describe a circle, and divide it into six
equal parts, in the points e, 1, 2, 3, 4, and 5. With any two adja-
cent points, as 1 and 2, and with the same radius, o e, describe two
arcs, o 2, and o 1, so as to form the curvilinear triangle, o — 1 — 2,
which is exactly the outline of the eccentric, g, each side of which .
is equal to a sixth of the circumference.
Draw the parallels, 5 — 1 and 4 — 2, tangential to the two side."
of the triangle, g, and we shall thus obtain the upper and lower
internal surface of the frame, f.
The cam is made of steel, as well as the two sides of the frame,
F, which bear upon it. It is adjusted and secured by the screw-
bolt, h, to the disc, h, keyed on tlie shaft, j, as shown in the hori-
zontal section, fig. 5, taken through the line, 3 — 4, in fig. 4.
It wOl be easOy conceived, that if the shaft turns in the direc-
tion of the arrow, the curved side, o 1, of the cam, acting against
the upper side of the frame, will cause it to rise, carrying with it
the rod, u, in such a" manner, that when the point, 1, shall have
reached the position, e ; that is, when the cam shall have made a
sixth of a revolution, this side of the frame will occupy the posi-
tion, m n. thereby indicating that the slide-valve has been elevated
to a distance equal to half o e, and that, in consequence, the port,
d, is uncovered, so as to allow the steam to enter the lower part
of the cylinder (fig. (B) ; W"hilst, on the other hand, a communica-
tion is established between the upper port, c, and the escape orifice,
E, so that the steam can pass out from the upper end of the cylin-
der. If the movement of the cam be continued during a second
sixth of a revolution, the slide-valve will remain in the same posi-
tion, because the arc, 1 — 2, w-hich is concentric with the axis, does
not change the position of the frame, as long as it is in contact
with its side, m n. As soon, however, as the point, 2, of the cam
reaches the position, e, the side, o 1, will be in the position, o 5,
and it will, in consequence, be in contact with the low"er side of
the frame, w"hich is in the position of the horizontal centre-line,
3 — 4. The further revolution of the cam, therefore, makes the
frame descend from its pressure on the lower side, until the side,
o 1, of the cam, occupies the position, o 3, w^hen the lower side of
the frame will occupy the position, m' n', corresponding to the
position of the valve, represented in fig. [B.
It follows from this aiTangement, that the valve will remain
stationary when it arrives at each extremity of its stroke, and the
pause each time wiU be of a duration corresponding to one-sixth
of a revolution of the cam shaft. The upward and downw'ard
movements each take place during a third of a revolution, and the
velocity of the valve will not be uniform, although the rotation of
the cam-shaft is so. In actual construction, the angles of the
cam are slightly rounded off, to avoid a too sudden change of
motion, and to prevent the too rapid wear of the sides of the frame.
involute cam.
Figures 6 axd 7.
239. In certain industrial arts, an instrument is employed for
pounding, crushing, and reducing substances, such as plaster or
tanbark, for example, in which the direct-acting force is the w-eight
of the instrument itself brought into play by its descent through
a determined height. The mechanical forge-hammer is a well
known working application of this expedient.
BOOK OF INDUSTRIAL DESIGN.
87
In these cases, the stamp, or hammer, has to be raised or tilted
up p.eparatory to each succeeding stroke, and it is obvious that
this may be most economically done in a gi-adual manner. It is
fcnci-ally eflt'ected by a cam, the outline of which is the involute
c'.ir.e already described ; this form being preferable on account of
iiie uniformity of its action.
The office, then, which the cam under consideration has to
fulfil, is the raising of the stamp, or load, to a certain height, and
t'len the letting it fall, without impediment, upon the object sub-
n itted to its action.
The diameter of the cam-shaft being predetermined, as well as
Ir at of the generating circle, which last is usually the same as that
ol the boss of the cam, the design is proceeded with as follows : —
Letting a be the cam-shaft, and taking a o as the radius of the
ge lerating circle, whilst a a! is the height to which the projection,
K, r.g. ©, formed on the stamp, c, is to be raised, develop the
circuiiiference (197) of the circle of the radius, a o, by means
of a seiies of tangents which give the points, c, d, e, &c., the curve
passing through which forms the involute, hfi. The inner por-
tion, h o, is not a continuation of the involute, but simply joins the
boss with a circular turn, because the stamp projection, b, does
not approach the cam-shaft, a, nearer than the point, a, to which
6 'oriesponds. Through the point, a, draw the vertical, a a', and
make it equal to the height to which the stamp has to be raised;
then with the centre, a, and a radius equal to a a', describe the
arc, a' m i, which will cut the involute in the point, i, and this
pomt is consequently the outer limit of the cam. A little con-
sideration will show that if the cam-shaft, a, be turned in the
direction of the arrow, supposing that it is originally placed, so
that the point b, coincides with a, it must necessarily raise the
litting-piece, b, the lower side of which is indicated by the line,
ni a, and will carry it by equal increments up to the position, m' a'.
'J'lie point, i', will then have attained the position, a', and the rota-
tion continuing, the next moment it will pass it, when the cam will
be entirely clear of the lifting-piece, b, and this last being unsup-
ported, must fall by its own weight.
The involute might have been derived from a generating circle
of the radius, a a, and had this radius been adopted, the resulting
curve would have been shorter, notvv'ithstanding that it would give
the same extent of lift. The angular space passed over would
also be less, and this would admit of a higher velocity of the cam-
shaft, and the strokes might be given in more rapid succession,
whilst on the other hand, a greater power would be required to
raise the same weight.
The cam we have represented in fig. [o), is such as is employed
to actuate the chopping stamp of mills for reducing oak, or other
bark, for the preparation of tan. The bark is placed in a kind of
wooden trough, e, solidly fixed* into the floor. The stamps are
armed with a series of cutters, n, in the form of crosses. The side
of the trough next to the stamp is vertical, whilst the opposite side
is elliptical in shape, and the matter under operation ha.s, con-
sequently, always a tendency to fall under the stump. The stamps
are kept vertical bv slides in which they work. They arc generally
from 450 to 700 pounds Weight, and fall through a height of from
16 to 20 indicH.
Fig 7 is ,'i pl.-iii i.l' llic cMrii as seen Cri in hVl.jw, .■ii'il t'lilly
indicates the width of the rim, and of the boss, and the thickness
of the feather or disc uniting the two.
A series of such cams are frequently employed in different
planes on the same shaft, actuating a corresponding series of
stamps, and in such case they are arranged in steps so as to come
into action one after the other. Two or more are also sometimes
employed in the same plane, and working a single stamp. In this
latter case, the generating circle requires to be of much larger
diameter in proportion, but the principle of construction Ls how-
ever the same.
cam to produce intermittent and dissimilar movements.
Figures 8 and 9.
240. In certain examples of steam engines, the valve movement
is obtained from a species of duplex cam, which bemg formed of
two distinct thicknesses, affords a means of adjustment whereby
the valve may be made to move intermittently and at different
rates, the proportions of which are variable at pleasure. The
object of this is to form and shut off the communication between
the cylinder with the steam-pipe, at any required point of the
stroke. In other words, the arrangement permits of the working
of the machine on the expansive principle, and of varying the
" cut-off " point at pleasure within certain limits. We shall see,
at a more advanced period, what is to be understood by the fore-
going expressions. In designing cams of this class, we primarily
determine the radius o a, of the cam boss, and the entire length,
b c, of the stroke to be given to the valve-rod. This distance,
which in the present mstance we shall take as equal to three tunes
the height of the port, must not be traversed at one movement.
On the contrary, a third only of this is at first passed through,
with some rapidity, and the remaining two-thirds are traversed at
a later period, in a continuous manner : in other words, after a
third of the stroke has been traversed, a slight pause takes place
before the remainder is traversed, and a second pause also occurs
before the commencement of the return stroke.
After describing a couple of concentric circles with the respec-
tive radii, o a, and o c, and having determined tlie angular spaces,
a d, and f g, corresponding to the times during which the valve is
to remain stationary, and the spaces, g h, and of, corresponding to
the duration of the movements ; divide the whole stroke, b c, into
three equal parts in the points, i,j, through which describe circles
concentric with the preceding. Through the points,/, g, h, draw
radii, and produce them to/, g', and h'.
As the cam will act on two friction rollers, g, diamotrioally
opposite to each other, their radius is determined, -.is a e: one is
di-awn with its centre, e, on the radius, o a produced, iuul tangen-
tial to the circle described with that radius: the other, witli its
centre, e , on the radius, o c produced, is, in like mnnnor, tangentia'
to the circle described wth this railius.
Between the two points, d and A-, and oomprisod within tiiei
given angle, g o h. a curve, k I rf, is drawn and united by tangcnfiai
arcs at either oxtieniily with tho cirilos of liio radii. </ o and » \,
respectively, in such a iiiMniier as to avoid any sudden changi> of
direction. Next di\ ide tho aiv, i,-' /i, into a certdn nuinlH-r of equiU
parts in Ilie imiiils. 1, 2, \-c., and cnriy tiio radii jicross to 1'. "J ,
^•i'. ; Hull, on .Mcli (if llu'se ladii, as a centre line, dosi>ritH> an are
88
THE PRACTICAL DRAUGHTSMAN'S
corresponding to the radius of the roller, g, and tangential to the
curve, k I d. By this means will be obtained the points, t s I,
indicating the successive positions of the centre of the roller on
the line, e e', when unpolled by the curve, kid. If, then, starting
fi-om these several points, we measure on the con-esponding cross
'ines, 1 — r, 2 — 2', &c., distances equal to e e', which is obviously
constant, we shall obtain the positions, r', s', l!, of the centre of the
opposite roller, g', corresponding to those of the first. Then, with
these points, r, s, t, as centres, describe arcs of the radius of the
roller, and draw the curve, d' V f, tangential to them, and unite
them to the circles of the i-adil, c o and/ o, in a similar manner to
the opposite curve. The curve, d' I' f, will obviously, from its
construction, be in contact with the roller, g', whilst the fii-st, dlk,
is in contact with the other roller, g.
In order that the rollers, g, g', may maintain then- relative
position, and move in the same rectilinear du-ection, they are carried
m bearings, h, forming, with four tierods, i, a frame which em-
braces the cam and cam shaft, the middle of the rods being planed
to rest and slide upon the latter.
To one end of the frame is bolted the cast-iron connecting rod,
J, fig. @, jointed to the be!I-crank lever, K. This last vibrates
on the centre, u, and by its second arm actuates the link, v, con-
nected to the rod, x, of the valve t, fig. H, above. In the position
given to the cam and roller frame, in fig. 8, the valve is not cover-
ing the upper part, c', and this remains open whilst the cam
rotates through the angle, a o d, because the arc, a d, and its oppo-
site, c d, are both concentric with the axis of the cam shaft, o.
When, however, the point, d, shall have arrived at the position,
a, supposing the cam shaft to continue to turn in the direction of
the arrow, the cam will shortly pass through the angle, dog, and
the projecting curve, dlk, will push the roller, g, to the right, and
the opposite roller, g', being drawn in the same direction, will roll
along the corresponding curve, d' I' g'. This movement will cause
the valve to be raised to the extent of a third of its stroke, cor-
responding exactly to the width of the port, c'. This port will, in
fact, be completely closed when the radius, o k, of the cam shall
have reached the position, o e. At this point, the valve is required
to remain stationary for a short time, during which the cam, in
continuing to revolve, describes the angle, g of. As soon, how-
ever, as the radius, of, reaches the position, o e, the valve, and its
actuating gear', will again move, and continue to do so, until the
lower port, c', be completely open. This movement wOl take place
whilst the cam describes the angle, foe, and is caused by the
curve, a' b' c', which pushes the roller, g, and the frame still further
to the right. The curve, a' b' c', is united by a gradual turn to the
circles of the radii, o k and o c, in the same manner as the curves
previously di-awn. The opposite and corresponding curve, a mn,
is obtained in the same manner as d' I' g', opposite to, and derived
from, the first curve, d I k. The operations in both cases are fully
indicated on the diagram, and it has merely to be borne in mind
that the object is to keep the two rollers, g and g', in contact with
the cam in every position of the latter.
After the cam has passed through the angle, foe, the valve,
with its gear, remains stationary during another interval, in which
the angle, c o d", is traversed, and then the first curve will com-
mence to act upon the roller, g', and cause it, with the frame, to
return from right to left, and the movements and intervals will take
place in the same order as to time as in the up stroke of the valve
already described in detail ; but the direction will be reversed —
that is, the valve will perform its return stroke — until it reaches
its original position, as represented in fig. [1.
To proceed : it is easy to conceive the cam, as constructed in
two pieces, precisely alike in all respects, and laid upon one another,
as M and m', fig. 9, one M being fast to the shaft, whilst the other,
m', is capable of being adjusted to the first in any relative position.
Since the rollers, g, g', are long enough to be in contact with both,
it wOl follow they will, in any given position, be acted upon by
that half of the cam which projects most at that particular point;
so that, if the curved portion, d I k, of one is turned slightly in
advance, it will come into action sooner, and, by consequence, will
cause the valve to shut off the communication between the steam
pipe and the cylinder at an earlier period of the stroke. In this
manner is obtained a means of varying the rate of expansion at
which the engine is worked.
Fig. 9 is a horizontal section, showng the two halves, m m', of
the eccentric, and the arrangement of the details of the friction
rollers, g g', and frame, h i.
Fig. i?" is a front view of the valve face of a steam-engine cylin-
der, showing the disposition of the ports.
An innumerable variety of movements may be produced by the
agency of cams ; but the principles of their construction are mostly
the same as those just discussed, and the examples given will be
a sufficient guide in designing others.
RULES AND PRACTICAL DATA.
MECHANICAL WORK, OR EFFECT.
24 L To work, considered in the abstract, is to overcome, during
any certain period of time, a continuously replaced resistance, or
series of resistances. Thus, to file, to saw, to plane, to draw
bm-dens, is to work, or produce mechanical effect.
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 destroys, 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, for example, the pressure exerted be
equal to 4 lbs., and the velocity 1 foot per second, the amount of
work ■wUl be expressed by 4 x 1 = 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 lbs. — the work will be, 8 x 2 = 16, or
the quadruplicate of its original amount.
In the term " velocity," " time " is understood ; so that, in fact,
just as space or solidity is represented in teims of three dimensions,
BOOK OF INDUSTRIAL DESIGN.
89
length, breadth, and depth, so also is mechanieal effect defined by
the three terms representing pressure, distance, and time. This
analogy gives rise to the possibDity of treating many questions
and problems, relating to mechanical effects, by means of geometri-
cal diagrams and theorems.
The unit of mechanical effect (corresponding to the geometrical
cubical unit) adopted in England, is the horse power, which is
equal to 33,000 lbs. 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 kilogrammetre, which is equal to a
kilogi'amme, raised one metre high In a second. Thus, supposing
the pressure exerted be 20 kilog., and the distance traversed by
the point of application be 2 metres in a second, the mechanical
effect is represented by 40 k. m. ; that is, 40 kilog., raised 1 metre
high. This unit is much more convenient thau the English one,
from its lesser magnitude. Indeed, when small amounts of me-
chanical effect are spoken of in English terms, it is generally said
that they are equal to so many pounds raised so many feet high.
That is to say, this takes place in some given time, as a minute,
for example. The time must always be expressed or understood.
It is impossible to express or state intelligibly an amount of
mechanical effect, without indicating all the three terms — pressure,
distance, and time. It is to the losing sight of this indispensable
definition, that we may attribute the vagueness and unintelligibility
of many treatises on this subject. The French engineers make
the horse power equal to 75 kilogrammetres ; that is, to 75 kilog.,
raised one metre high per second.
The motors generally employed in manufactures and industrial
arts are of two kinds — living, as men and animals ; and inanimate,
as air, water, gas, and steam.
The latter class, being subject only to mechanical laws, can
continue their action without limit. This is not the case with
the first, which are susceptible of fatigue, after acting for a certain
length of time, or duration of exeilion, and require refreshment
and repose.
What 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 relation to the velocity and duration
which will yield the largest possible product, 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 mechanical
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.
TABLE OF THE AVERAGE AMOUNT OF MECHANICAL EFFECT PRODUCIBLE BY MEN AND ANIMALS.
Nature of the Work.
Mpin weight
elevated or
effort exerted.
Velocity or
distance per
second.
Mechauical
effect per
second.
Duration per
diem.
Mechanical effect
per diem.
A man ascending a slight incline, or a stair, without a burden, his work
consisting simply in the elevation of his own weight,
A labourer elevating a weight by means of a cord and pulley, the cord
being pulled downwards,
A labourer elevating a weight directly, with a cord, or by the hand,
A labourer lifting or carrying a weight on his back, up a slight incline, or
stair, and returning unladen,
A labourer carrying materials in a wheel-barrow, up an incline of 1 in 12,
and returning unladen,
A labourer raising earth with a spade to a mean height of five feet,. .
ACTION ON MACHINES.
A labourer acting on a spoke-wheel, or inside a large drum.
At the level of the axis,
Near the bottom of the wheel,
A labourer pushing or pulling horizontally,
A labourer working at a winch handle, ; • • •.
A labourer pushing and pulling alternately in a vertical direction,. . .
A horse drawing a carriage at an ordinary pace,
A horse turning a mill at an ordinary pace,
A horse turning a mill at a trot,
An ox doing the same at an ordinary iiaco,
Amnio do. do.
\n ass do. do.
Lbs.
143
40
44
143
132
60
Feet.
•50
•65
•66
•13
•06
•13
132
•60
26
2-30
26
1-97
17i
3^4
11
3-61
154
2-95
90
2-95
66
6-56
M3
1-97
66
2-96
31
2-62
Lbs. raised
1 foot high.
71-5
26-0
24-6
18-6
8-5
7^8
66-0
59-8
51-2
43-0
39-7
454-3
292-0
4330
281-7
194-7
81-2
Hours.
8
6
6
10
10
8
8
8
8
10
8
8
8
Lbs. raised
1 foot high
2,059,201'
561,600
531,360
401,760
306,000
280,000
1,900.800
1,722,240
1,474.560
1,238,400
1,143,360
16,354,800
8,409,600
7,014,600
8,ir.i,960
5,607,360
2,338,560
It may be gathered from this table that a laboui-cr turning a
winch handle can make its extremity pass through a distance of
2-46 feet per second, or 60 x 2-46 = 147-6 feet per minute.
Then, supposing the handle has 13,? inches, = 1-147 feet radius,
vhich corresponds to a cin-^imferenco of 6-28 x 1-147 = 7-2 feet
at (he |)oint of aitplication, the labourer is cjiiwblo of JUi !ivi>ra£f«»
velocity of
1476
7-2
= 20 turns (nearly) \h'v niinufo.
Also, (he sjinu' labiMtror exerting ii fon-o equal to 171 Hw- ^^l**
90
THE PRACTICAL DRAUGHTSMAN'S
the velocity of 2*46 feet per second, will produce a mechanical
effect equal to
17i X 2-46 = 43 lbs. raised 1 foot high per second, or of
43 X 60 = 2580 per minute, and
2580 X 60 = 154,800 lbs. raised 1 foot high per hour.
And as he can work at this 8 hours per diem, the total mechanical
effect during this time will be, as indicated in the table, equal
to 1,238,400 lbs. raised 1 foot high.
We may then calculate that, as a day's work, a labourer
tiM-ning a winch-handle can elevate in a continuous manner 17^ lbs.
2-46 feet high per second ; when, however, the labourer has only
to apply his strength at intervals to a crane, a windlass, or a
capstan, he can develop a much greater force for a few minutes.
According to experiments tried in England with a discharging
crane, a man can in 90' raise a load of 1048'6 lbs. to a height of
16| feet. Now, to compare this with the tabulated quantities,
we must multiply the weight raised, 1048-6 lbs., by the height,
16i feet, and divide the product by the duration of the action, or
90"; the quotient, 192, indicates the number of pounds raised
1 foot high in a second, to which the mechanical effect is equal.
It has been proved by experiments, that under the most
favourable circumstances, an Irish labourer of extra strength can,
by great exertion, raise to the same height, 16^ feet, a load of
1474 lbs. in 132', which is equal to a mechanical effect of
1474 X 16-5 „ .,,,.,
— = 184-25 lbs. raised 1 foot high per second.
A man can evidently only exert such a force during a very
limited period ; we cannot, therefore, compare this kind of labour
with that which continues through several consecutive hours.
Although the load and velocity as given in the table are those
most conveniently proportioned to each other, still, when the case
requires it, they might be altered to some extent ; thus, if it is
necessary to apply a force of 25 lbs. to the extremity of the
winch-handle instead of 17|, then the velocity would be reduced,
and would become
— := 1-72 feet per second, instead of 2-46.
It has been ascertained from actual observations, that a horse,
going at the respective rates of 1, 3, 5, and 10 miles per hour,
cannot exert a greater tractive force than the corresponding
weights, 194, 143, 100, and 24 lbs., and cannot draw an/thing
appreciable when going at the rate of 15 miles per hour.
Thus, when it is \vished to increase the force exerted, a decrease
takes place in the velocity ; and reciprocally, when it is wished
to gain time and speed, it can only be done at the expense of the
load. Thus, in the case of the winch-handle, the two factors
must always produce an effect equal to 43 lbs. raised 1 foot high
per second, whatever ratio they may have to each other.
In all cases of the direct action of forces, a certain velocily is
impressed, for without movement there could not be the action of
a force.
There are two kinds of motion — uniform and varied motion.
243. Unitorm Motion. — A body is said to have a uniform
motion when it passes through equal distances in equal times.
Thus, for example, if a body traverses 5 feet in the first second,
6 feet in the second, and so on throughout, its motion is uniform.
Putting D to represent the distance, V the velocity, and T the
time, the formula, D = V x T, indicates that the distance Ih
equal to the velocity multiplied by the time.
First Example. — The velocity of a body subject to a uniform
motion is 3 feet per second, through what distance will it havo
passed in 15 seconds ?
D = 3 X 15 = 45 feet.
D
From the preceding formula, D = V x T, is obtained V = Tj^;
that is to say, the velocity per second is equal to the distance
divided by the time.
Second Example. — The distance passed through in 15 seconds
is 45 feet, what is the velocity ?
15
3 feet.
The wheel-gear of machinery, as well as many other instru-
ments of transmission, is, for the most part, actuated in a uniform
manner.
244. Varied Motion. — When a body passes in equal times
through distances which augment or decrease by equal quantities,
the motion is called uniformly varied.
The distance in motion uniformly varied is equal to half the
sum of the extreme velocities multiplied by the time in second
First Example. — What is the distance passed through in 4
seconds by a body in motion, the velocity of which is 2 feet per
second at starting, and 6 feet per second at the termination ?
T^ 2 + 6
D = — - — X 4
16 feet.
Second Example. — What is the distance passed through in 4
seconds by a body in motion, which at starting has a velocity of
6 feet per second, but which is gradually reduced to 2 feet 1
6 + 2
D:
X 4 = 16 feet.
It will be seen from these two examples, that, with like condi
tions, the total distance is the same for motions uniformly acce
lerated or retarded.
The velocity at the end of a given time, in uniformly acceler-
ated motion, is equal to the velocity at starting, plus the product
of the increase per second into the time in seconds.
First Example. — What velocity will a body have at the end of
8 seconds, supposing the initial velocity = 1, and that it increases
at the rate of 3 feet per second 1
V = 1 + (8 X 3) = 25 feet.
The velocity which, at the end of a given time, a body uniformly
retarded should have, is equal to the initial velocity minus the
product of the diminution per second, multiplied into the time in
seconds.
Second Example. — A body in motion starts with a velocity of
22 feet per second, and its velocity decreases at the rate of 2 feet
per second, what will be the velocity at the end of 10 seconds?
V = 22 — - (2 X 10) = 2 feet.
245. The motions of which the various parts of machines are
capable are of two principal kinds — continuous, and alternate or
back and forward motion.
BOOK OF INDUSTRIAL DESIGN.
91
These two kinds of motion may take place eitlier in straight or
curved lines, the latter generally being circular.
In the actual construction of machinery, we find that, from
diese principal descriptions of motions, the following combinations
are lerived : —
( Continuous rectilinear.
Continuous rectilinear motion is converted into < Continuous circular.
( Alternate circular.
Alternate rectilinear motion is converted into
Continuous circular motion is converted into
Alternate circular motion is converted into
Continuf)Us rectilinear.
' Continuous circular.
( Alternate circular.
r Continuous rectilinear.
J Alternate rectilinear.
] Continuous circular.
L Alternate circular.
C Alternate rectilinear.
< Continuous circular.
' Alternate circular.
THE SIMPLE MACHINES.
iJ46. This term is applied to those mechanical agents which
enter as elements into the composition of all machinery, whether
their function be to elevate loads, or to overcome resistances.
The simple machines are generally considered to be six — the
lever, the wheel and axle, the pulley, the inclined plane, the
screw, and the wedge.
A much more scientific and comprehensive arrangement of the
elementary machines is that lately suggested by Mr. G. P. Ren-
shaw, C.E., of Nottingham. According to his system, the elemen-
tary machines, or mechanical powers, are five — namely, the lever,
the incline, the toggle or knee-joint, the pulley, and the ram.
The wheel and axle, of the first system, is evidently but a
modification of the lever, and the screw and wedge are modifi-
cations of the inclined plane ; whilst no mention is made of tlie
toggle-joint and ram — the last so well represented by the hydro-
Btatie press.
All these machines act on the fundamental principle, known as
that of virtual velocilies. According to this principle, the pressure
or resistance is inversely as the velocity or space passed through,
or that would be passed through, if the piece were put in motion.
The nwmenlum of the power and resistance is equal when the
machine is in equilibrio. By momentum is understood the pro-
duct of the power by the space passed through by the point of
application.
Time is occupied in the transmission of all mechanical force.
In any mechanical action we do not see the effect and the cause
at the same instant. Thus, in continuous motion, in which the
time expended is not apparent at first sight, each succeeding por-
tion of the motion is due to a portion of the impelling action
exerted a certain time previously. This will be more obvious on
observing the commencement and termination of any motion.
The motion does not commence at the instant that the power is
applied, nor does it cease at the exact moment of the power's
cessation. The fiction of the vis inerliic has been invented to
account for these latter observed facts, but it explains them very
awkwardly. Thus, bodies are said to possess a certain force
which is opposed to a change of stiito, whether from rest to motion
or motion to rest. If such a resistive force existed, it wduld
require an effort to overcome it, in addition to what is actually
accounted for by the motion. If it is said that this is again given
back at the termination of tlie motion, another fiction is required
to account for it in the meantime, that is, during the continuation
of the motion. Moreover, there is nothing analogous to it
throughout the entire range of physical science.
The facts are described in a much more simple and philosophi-
cal manner, when they are said to arise from the time taken in the
transmission of motive force. Why there should be this expen-
diture of time is a more abstruse question. It probably arises
from the elasticity of the component particles of bodies and
resisting media, and is regulated by the laws which govern the
relation to time of the vibrations of the pendulum.
In all machines, a portion of the actuating power is tost or
misapplied in overcoming the friction of the parts.
247. The Lever. — The lever, in its simplest form, is an
inflexible bar, capable of oscillation about a fixed centre, termed
the fulcrum. A lever transmits the action of a power and a
resistance, or load ; the distance of the power, or load, from the
centre of oscillation, is called an arm of the lever.
There are two kinds of power levers, distinguished by the posi-
tion of the fulcrum as regards the power and the resistance.
These become speed levers, by transposing the power and resist-
ance. By a power machine, is meant one which gives an increase
of power at the expense of speed, and by a speed machine, one
that gives an increase of speed at the expense of power, and all
the simple machines are one or the other, according to the relative
position of the power and resistance.
In all cases of the lever, the power and the resistance are in the
inverse ratio to each other of their distances from the centre of oscil-
lation. That is to say, that when, in equilibrio, the momentum
of the power, P x A, or the product of this power into the space
described by the lever arm. A, is equal to the product, R x B, of
the resistance, into the space described by the lever arm, b : whence
the following inverse proportion : —
P : R:: B: A;
Any three of which terms being known, the f(»urth can be found
at once.
248. The wheel and axle is a perpetual lever. As a jiowor. the
advantage gained is in the proportion of the radius of the circum-
ference of the wheel to that of the axle. That is to sjiy, the power,
r, is to the resistance, R, as the radius, b, of the axle, is to the
radius, a, of the wheel, or the length of the winch handle — in the
simpler form of this machine, consisting of an axle and a winch
handle. The same rules and formuke obviously apply to this, as
to the first described form of lever.
Thus, multiply the resistance l>y the radius of tJie a.\lc, and
divide by that of the handle, and the quotient will bo the power.
In windlasses and cranes, consisting of a .system of wlioel-work,
the power is applied to a handle fixed on the spiiulle of a pinioi.,
which transmits the power to a spur-wheol, fixed in tlio s|)indlo ot
the barrel, about which the cord, or rope, carrying the load to be
raised, is wound.
Where there arc scxcnil pairs ol" such wlu'cls, it is neocss;iry to
include in the caKulalions tiio ratios of the pinions to the spui .
92
THE PRACTICAL DRAUGHTSMAN'S
Tlie proportional formula will, in tliis case, be the same as for a
system of levers : —
F : R :: b X b' X b' : a X a X a" ;
Or, the power is to the resistance, as the produst of the radii of
the pinions and barrel is to the product of the radii of the spur-
wheels and handle.
From this we derive the following rules : —
I. MuUiply the load to be raised by the product of the radius of
the barrel into the radii of the pinions, and divide the sum obtained
by ike product of the radius of the handle into the radii of the spur-
wheels, and the quotient will be the power, which, when applied to the
handle, irill balance the load.
n. Multiply the power applied by the radius of the handle, and
by the radii of the spur-wheels, and divide the prodvct by the radius
of the barrel, and by the radii of the pinions, and the quotient will
be the resistance ichich will balance the poicer.
in. Multiply the radii of the pinions and barrel, and divide the
prodjici by the radii of the handle and spur-wheels, and the quotient
will be the ratio of the power to the resistance.
249. The Isclexe. — Wlien a body is raised up a vertical
plane, its whole weight is supported by the elevating power, and
this power is consequently equal to the weight elevated.
When a body is drawn along a horizontal plane, the tractive
power has none of the weight of the body to sustain, but merely
to overcome the friction of the surface.
If, however, a body is drawn up an inclined plane, the power
required to elevate it is proportionate to the inclination of the
plane, in such a manner, that
If the power ads parallel to the plane, the length of the plane will
be to the load as the height is to the power.
The advantage gained by the use of the inclined plane, as a
power, is the greater the more its length outmeasures its height;
it is then the ratio of the length to the height which determines
that of the power to the resistance, whence we obtain the follow-
ing rules : —
I. The resistance, multiplied by the height and divided by the length
cf the 'plane, is equal to the power required to balance a body on the
inclined plane.
II. The power, multiplied by the length of the plane and divided
by the height, is equal to the resistance.
ni. The resistance, multiplied by the height of the plane and
divided by its length, is equal to the load on the plane.
The wedge and the screw are noticeable modifications of the
incline. An incline wrapped round a cylinder generates a screw.
\Vlien used as a power-machine, it is generally combined with a
lever, as in presses. The advantage gained depends upon the
length of the lever and the pitch of the screw. Multiply the
actuating force by the cu-cumference described by the end of the
arm or lever, and divide the product by the length of the pitch
of the screw ; the quotient, minus the friction, which is very con-
siderable m these machines, will be the pressure exerted by the
screw, and the velocity will, of course, be in the mverse ratio of
the theoretical pressure to the actuating force.
250. The Toggle. — This is met with chiefly in punching
presses. Deflected springs and rods are also examples of it, and
also the twisted cords used by carpenters to stretch their saws in
frames. As in the other machines, the resistance is to the power,
as the space passed through by the latter is to the space passed
through by the former.
251. The Pullet. — There are two kinds of pulleys, the one
turning on fixed centres, the other on traversing centres.
The pulley, which turns on a fixed centre, serves simply to
change the direction of the motive force, without altering the
relations of power and velocity. It is, in fact, only the moveable
pulleys which can be classed amongst the elementary machines.
A single moveable pulley, acting as a power, doubles it at the
expense of the speed ; thus, if a weight of 10 lbs. be suspended to
one extremity of the cord, it will balance 20 lbs. hung to the axis
of the pulley. This arises from the fact that, from the arrange-
ment of the cord, the pulley only rises through half the height
passed through by the motive force ; thus, if the latter pass
through 6 feet, the pulley will only rise 3 feet, and the resulting
momentum of the power, 10 x 6, WHl be equal to that of the
resistance, or 20 x 3, so that the two will be in equilibrium.
Though the stationary pulley cannot be considered as a
mechanical power, yet, in changing the direction of the motion,
it affords great facilities in the application of force ; thus, it Ls
easier to pull downwards than upwards, as the labourer brings his
weight to bear in the former case.
WTien several pulleys or sheaves are placed on one axis Lq a
suitable frame, it is called a block. Where two or more blocks
are employed, it is only the moveable ones which increase the
power, and this increase is equal to double the number of sheaves,
or pulleys, in the block or blocks.
The mechanical advantage of the block, as a power, arises from
the fact, that the space traversed by the motive power is equal to
the sum of the doublings of the cord round the pulleys, whilst the
load is only elevated to a distance corresponding to this space
divided by the number of these doublings.
A clock line and weight, in which the line goes round a pulley
fLxed in the weight, is an example of a speed-pulley, that is, one
in which the power, or resistance, is transposed, for the weight,
or motive power, causes the moveable end of the cord to pass
through twice the space it passes through itself.
252. The Ram. — This is the most economical tugmentor of
power that we have. It is freer from friction and other disad- I
vantages than the other simple machines, and it ia, in its action,
very closely allied to the pulley. Each derives its advantage
from the division of the points of support, for tlie proportionate
area of the piston in the Bramah press represenfej the number of
points over which the pressure, or resistance, is dilused.
253. Remarks. — ^It is essential that, to avoid illusive mistakes,
the student should perfectly understand, that wl.en, in emplojincr
mechanical forces, the effect of the power appLed is augmented,
the distance passed through by the resistance, or load, is dimin-
ished, with reference to that passed through by the power, in
exactly the same ratio that this is increased. This is true in all
cases, and may be stated thus : What we gain in force, by means
of machinery, we lose in speed, and reciprocally.
It follows from this, that the true object of maehinery cannot
be to augment the work performed by the motive agent, but ta
convert any primary action in a manne.- appropriate to the
BOOK OP INDUSTRIAL DESIGN.
93
circumstances in which the power is to be used. Thus we can
make a very small force, as that of a man, elevate an enormous
weight, but with a speed proportionately slow.
Finally, The mechanical effect developed in a given time by a given
force, through the instrumentality of machinery, must alivays equal the
useful effect obtained, plus the amount lost in overcoming frictional
and other resistances ; and the useful effect of machinery will be the
greater, according as the causes of these resistances are diminished.
CENTRE OF GRAVITY.
254. All bodies are equally subjected to the action of weight.
Gravity, or weight, is the action of that universal attraction which
draws all bodies towards each other, and by which, in the case of
bodies on the surface of the earth, these are drawn towards its
centre. The power, of whatever nature it may be, which balances
this action, is equal to the weight of tne body.
The curvature of the surface of the earth being quite inap-
preciable for small distances, gravity is considered as acting in
parallel lines, and its direction is given by the plumbline.
The centre of gi-avity is that point in any body in which the
action of its entire weight may be said to be concentrated. If
the body be suspended by this point it will be in equilibria, in
whatever position it is put.
The position of the centre of gravity depends upon the nature
and form of any body ; it may generally be found in the follow-
ing manner : —
Suspend the body by a thread attached to any point whatever
in it; when the body is motionless, the line of the suspension
thread will pass directly through the centre of gravity. Suspend
the body by any other point, and the centre of gravity will also
be in the continuation of the line of the thread, so that the actual
centre must be at the point of intersection of the two lines thus
obtained. This simple expedient reminds us of the application of
the square to the finding of the centres of circles — the unknown
centre on the endface of a shaft, for example — where the inter-
section of any two lines, drawn along the blade of the square,
when the head is laid against the periphery of the shaft in two
different positions, gives the required point of centre.
The centre of gi-avity of regular bodies, as spheres, cylinders,
prisms, is in the centre of their configuration.
The centre of gi'avity of an isosceles tiiangle is one third up
the centre line which bisects the base.
The centre of gravity of a pyramid, with a triangular or poly-
gonal base, is one fourth up the line which joins the summit with
the centre of gravity of the base. It is the same with a cone.
The centre of gravity of a hemisphere is situated three-eighths
up the radius at right angles to the base.
The centre of gravity of an ellipse is in the point of intersection
of the axes.
When a body is placed in a vertical or inclined position on a
plane, it is necessary, in order that it may rest upon it in that po-
sition without falling, that the vertical lino passing through the
centre of gravity shall fall within the external outline of the side
in contact with the plane. This limit, however, allows of consi-
derable deviation from the vertical in the geiu^ral contour of bodies,
as is instanced in the case of leaning or inclined cdiliccs. The
stability of bodies increases as the extent of their bases is g^reater
in comparison with their height, and also, as the vertical line,
passing through the centre of gravity, meets the plane on which
the body rests nearer to the centre of the base. A body is said
to be more stable when it requires a greater force to overturn it.
A cone is more stable than a cylinder of the same height and base.
The stability of walls depends greatly on the kind of foundations
given to them, and on the proportionate extension of their bases.
ON ESTIMATING THE POWER OF PRIME MOVERS.
255. As we shall see further on, the power of prime movers
may be calculated from the dimensions of the various parts of the
engine. Still, the many different modes of construction tend to
modify considerably the actual useful effect, and engineers have
endeavoured to construct an apparatus, by means of which the
actual power, or useful effect of engines, may be measured with
exactitude.
Prony's brake, which is the instrument most generally used for
this purpose, acts on the principle of the lever, and consists of a
cast-iron pulley in two halves, united by screws. This is fixed on
the main shaft of the prime mover, the force of which it is wished
to measure. It is embraced by two jaws, which may be tightened
down upon the pulley by screws. To the lower jaw is attached
a long lever, from the end of which is suspended a scale for
weights. If it is known what power the engine was designed to
possess, it is simply necessary to put into the scale the weight
corresponding to this power, that is, the weight which, by the
action of the lever, will give a pressure equal to the supposed
power of the machine.
Having fixed the apparatus on the engine, and provided means
of efficiently lubricating the frictional surface of the pulley with
soap and water, and having balanced the apparatus in such a
manner that it will not be necessary to take into the calculation
anything but the weight placed in the scale, the steam may bo
gradually let on. The engine will perhaps shortly acquire a
greater velocity than that for which it was designed ; if this is the
case, the jaws are gradually screwed closer and closer upon the
pulley. As the friction thereby increases, the velocity will dimi-
nish, and full steam may be let on. After a short time, and when
the friction is so great that the lever is raised slightly above
the horizontal line, and the engine is going at its proper velocity,
and the pressure of the steam at its correct point, so that i\w
power of the engine balances the load on the lever, it may be con-
cluded that the engine develops the i)ower lor which it was in-
tended. If the lever rises considcnihly, it will he necessary to
increase the weight in the scale, so as to obtain the actual uiaxi-
mum power of the engine ; and, on the contrary, if the engine does
not appear to have the desired power, the weight must be reduced,
by which nu\nis its actual power will be ascertainable.
CALCULATU)N 1-OK THIv BKAKK.
25(). The weight which will balance the foive of a macliinc may
be cuk'ulaled when ihc len^lli of llie Icmt arm is known, or. more
conectly, the radius tVoin the centre of the shalt to the jioinl of
s\ispension of the weight, and the nominal lior^o-pon i<r, by U»o
I'dllowing rule : —
94
THE PRACTICAL DRAUGHTSMAN'S
Multiply the nominal hmse-power by 33,000, ami divide the pro-
duct by llie circumference described by the end of the leitr, and by
the number of revolutions per minute, and tlie quotient will be the
■weiglU sought.
Let us take, for example, the main shaft of a steam-engine of
16 horse-power, which runs at the rate of 30 revolutions per
ruinutc, the radius of the brake being nine feet —
16 X 33,000
Here we have w ^
■ = 311-4 lbs.
6-28 X 9 X 30
Such is the net weight to be suspended from the end of the lever,
the brake being previously balanced by being suspended on its
centre of gravity.
The actual power, or maximum effect of an engine, may like-
wise be calculated by means of the following rule : —
Multiply the circumference described by the lever, by the number
of revolutions of the shaft per minute and by the weight in the
scale, and divide the product by 33,000 and the quotierd will be
the actual force of the engine hi horses power.
For example, let us suppose that the main shaft of a steam-
engine makes 30 revolutions per minute, that the radius of the
lever is 9 feet, and that the net weight in the scale is 311-4 lbs.,
what is the maximum force of the engine ?
^ 6-28 X 9 x30 X 311-4
^= 33:000 -^^H-P-
TABLE OF HEIGHTS CORRESPONDING TO VARIOUS VELOCITIES OF FALLING BODIES.
Velocity.
Height.
Velocity.
Height.
Velocity.
Height.
Velocity.
Height.
Velocity.
Height.
Inches
Inches.
Inches.
Inches.
Inches.
Inches.
Inches.
Inches.
Inches.
Inches.
•1
•0001
5-7
•165
16-5
1^388
44-5
10-094
72-5
26-794
•2
•0002
5-8
•171
17-0
1-473
45-0
10-322
73-0
27-164
•3
•0005
5^9
•177
17-5
1-561
45-5
10-553
73-5
27-538
•4
•0009
6-0
•184
18-0
^651
46-0
10-786
74-0
27-914
•5
■0013
6^1
•190
18-5
1-745
46-5
11-022
74-5
28-292
•6
-0019
6^2
•196
19-0
1-840
47-0
11-260
75-0
28-673
■1
•0026
6^8
•202
19-5
1-938
47-5
11-501
75-5
29-057
•8
•0034
6^4
•209
20-0
2-039
48-0
11-744
76-0
29-443
•9
•0043
6-5
•215
20-5
2-142
48-5
11-990
76-5
29-832
1-0
•0051
6-6
•222
21^0
2-248
49-0
12-239
77-0
30-223
1-1
•0062
6-7
•229
21^5
2-356
49-5
12-490
77-5
30-617
1-2
•0074
6-8
•236
22-0
2-467
50-0
12-744
78-0
31-013
1-3
-0087
6-9
•243
22-5
2-580
50-5
lS-000
78-5
31-412
1-4
•0101
7-0
•250
23-0
2-696
51-0
13-258
79-0
31-813
1-5
•0115
7-1
-257
23-5
2-815
51-5
13-520
79-5
32-217
1-6
•0131
7-2
-264
24^0
2-936
52-0
13-784
80^0
32-624
1-7
•0148
7-3
-272
24^5
8-060
52-5
14-050
80^5
33-033
1-8
•0166
7-4
•279
25^0
3-186
53-0
14-319
SVO
33-445
1-9
•0185
7-5
•287
25^5
3-315
53-5
14-590
81^5
33-859
2-0
•0204
7-6
•295
26^0
3-446
54-0
14-864
82-0
34-275
2-1
•0225
7-7
■302
26-5
3-580
54-5
] 5-141
82^5
34-695
2-2
-0247
7-8
•310
27-0
3-716
55-0
15-420
83^0
35-116
2-3
•0270
7-9
•318
27^5
3-855
55-5
15-701
83^5
35-541
2-4
•0294
8-0
•326
28^0
8-996
56-0
15-986
84-0
35-968
2-5
•0319
8-1
■334
28^5
4-140
56-5
16-272
84-5
36-397
2-6
■0345
8-2
■343
29^0
4-287
57-0
16-562
85-0
36-829
2-7
-0372
8-3
■351
29^5
4-436
57-5
16-854
85-5
37-264
2-8
-0400
8-4
■360
SO^O
4-588
58-0
17-148
86^0
37-701
2-9
-0429
8-5
•368
30^5
4-742
58-5
17-445
86^5
38-141
3-0
■0459
8-6
■377
31^0
4-899
59^0
17^744
87^0
38-583
3-1
•0490
8-7
•386
31^5
5-058
59-5
18-046
87^5
39-028
3-2
•0522
8-8
•395
32^0
5-220
60-0
18-351
88^0
39^475
3-3
-0555
8-9
•404
32-5
5-384
60-5^
18G58
88-5
39-925
3-4
•0589
9-0
•413
33^0
5-551
61-0
18-968
89^0
40-377
3-5
-0624
9-1
■422
38-5
5-721
61-5
19-280
89-5
40-832
3-6
-0660
9-2
•431
34-0
5-893
62-0
19-595
90-0
41-290
3-7
-0697
9-3
•441
34^5
6-067
62-5
19-912
90-5
41-750
3-8
-0735
9-4
•450
35^0
6-244
63-0
20-232
91-0
42-212
3-9
-0775
9-5
•460
35^5
6-424
63-5
20-554
91-5
42-677
4-0
•0816
9-6
•470
36-0
6-606
64-0
20-879
92-0
43-145
4-1
■0856
9-7
•480
36^5
6-791
64-5
21-207
92-5
43-615
4-2
■0899
9-8
•490
37^0
6-978
65-0
21-537
93-0
44-088
4-3
■0942
9-9
•500
37^5
7-168
65-5
21-869
93-5
44-563
4-4
■0986
10-0
■510
38^0
7-361
66-0
22-205
94-0
45-041
4-5
•1032
10-5
■562
38-5
7-556
66-5
22-542
94-5
45-522
4-6
■1078
11-0
•617
39^0
7-753
67-0
22-888
95-0
46-005
4-7
■1125
11-5
•674
39^5
7-953
67-5
23-225
95-5
46-490
4-8
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12-0
•734
40^0
8-156
68-0
23-571
96-0
46-978
4-9
•1228
12-5
•797
40-5
8-361
68-5
23-919
96-5
47-469
5-0
•1274
13-0
•861
41^0
8-569
69-0
24^969
97-0
47-962
5-1
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13-5
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41^5
8-779
69-5
24-622
97-5
48-458
5-2
•1378
14-0
•999
42^0
8-992
70-0
24-978
98-0
48-956
5-3
•1431
14-5
vol 2
42^5
9-207
70-5
25-336
98-5
49-457
5-4
•1486
15-0
M47
43^0
9-425
71-0
25-696
99-0
49-960
; 5-5
•1541
15-5
1-225
43-5
9-646
71^5
26-060
99-5
50-466
5-6
•1598
160
1^305
44-0
9-869
72^0
26-425
100-0
50-975
BOOK OF INDUSTRIAL DESIGN.
93
THE FALL OF BODIES.
258. When bodies fall freely of their own weight, the velocities
wliich they acquire are proportionate to the time during which
they have fallen, whilst the spaces passed through are as the
squares of the times.
It has been ascertained by experiment that a body falling freely
from a state of rest, passes through a distance of 16 feet and a
small fraction, in the first second of time. At the end of this
time it has a velocity equal to twice this distance per second.
From this it follows that if the times of observation are — •
1"
32 ft...,
2'
64 ft...
3" 4"
96 ft.... 128 ft.
16
16'
--... 64 "
...48"
.. 144 '
80'
256 "
112"
3, 4
3, 4
9, 16
5, 7
The corresponding velocities will be
The spaces passed through fi-om the )
coinmencement, J
The spaces passed through during each (
second, )
That is to say, that the times are as the numbers, 1, 2,
The velocities also as, 1, 2,
The spaces passed through as the squares, 1, 4,
And the space for each interval as the odd numbers, 1, 3,
These principles apply equally to all bodies, whatever may be
their specific gravity, for gravity acts equally on all bodies ; the
effect, however, being modified by the resistance of the media
through which the bodies pass, which is greater in proportion, as
the specific gravity Is less.
259. The velocity which a body will acquire in a given time
when falling freely, will be found by multiplying the time ex-
piessed in seconds by 32 feet.
Example. — Let it be required to ascertain the velocity acquired
by a body falling during 12 seconds.
V = 12 X 32 = 384 feet per second.
When a body falls from a given height, H, the ultimate velo-
city, or that acquired by the time the base is reached, will be given
by the formula (g being the velocity gravity causes a body to
acquire in the first second)
V = VJgH, or V =3 4^64 x H,
which leads to the following rule : — Multiply the ^ven height in
feet by 64, and extract the square root, which will be the velocity
in feet per second by the time the body shall have fallen through
the height, H, not taking resistance into consideration.
Example. — What will be the ultimate velocity of a body falling
a distance of 215 feet ?
V = 1^64 X 215 = 117-3 feet per second.
From the above formula.
we obtain V
v = vTpr,
2g-H, then
ya ya
^^2^ "^64'
whence we have this rule : — Divide the square of the velocity in
feet per second by 64, and the quotient will express the height
through which a body must fall unimpeded, from a state of rest, in
order to obtain that velocity.
Example. — A body has acijuired a velocity of 117'3 feet per
second, through what height must it have fallon ?
117:5'
H:
64
= 215 feet, the height of the fall.
To obviate the necessity of calculating the corresponding heights
and velocities, we give a very extensive tabic, calculated for
tenths of inches The niiinher.H, however, being (•<|nully correct
as representing feet or yards, those of both columns bemg of the
same denomination.
MOMENTUM.
260. The force with which a body in motion strikes upon
one in a state of rest, is equal to the product of the mass of
the moving body multiplied into the velocity ; this product is
termed its momentvun. If a body with a mass, m, is animated
with a velocity, v, its momentum is equal to m v. The term, nt,
however, may be taken as signifying the mechanical effect of a
weight falling during a second of time, or through 32 feet, there-
w
fore, OT = —, that is, the weight in pounds divided by 32 feet,
w y. V
whence, mv ^^ •
g
What distinguishes the simple momentum or force of impact of
a body from the mechanical effect of a prime mover is, that
whilst the former is due to a single impulse, we have in the latter
to consider the continuous action of the impelling force.
261. When a motive force Imparts continuously a certain velo-
city to a body, the result of its action is what may be termed vis
viva, or continuous momentum ; it is numerically the product of
the (moving) mass multiplied into the square of the velocity im-
parted to it.
Putting M to represent the mass of a body, and V the velocity
impressed upon it,
WV»
M V» or
g
is- the expression of the vis viva of the body. This force is double
that developed by gravity. For, in fact, when a body of the
weight, W, falls from a height, H, it acquires from its fall as
ultimate velocity, V, which we have already shown to be equal to
V2gH =
^g
and the mechanical effect, W H, is consequently expressed by
WV»
^g '
MV
now, putting for P, its value, ^l g, the formula becomes — g— •
Thus, the mechanical effect developed by gravity is equal to
half the vis viva unparted to a body.
CENTRAL FORCES.
262. When a body revolves freely about an axis, it is said to
be subjected to two central forces ; vne one, termed " centripetal,"
tends to draw the body to the axis ; the other, termed " ccntri-
fufral," or tangential, and duo to the tendency of bodies in motion
to proceed in straight lines, strives to carry tlie body away from
the centre. Tiiese forces are equal, and act trjmsvcrsciy to eai-h
other.
The centrifugal etVort exerted by a body in rotative motion, and
which tends to separate the component p;irticlos, is expressed by
the following formula :—
WV
F =
gxR'
in which W r( I'rescnt.s the weight of tlu- body : V, the verocify in
0(j
THE PRACTICAL DRAUGHTSMAN'S
teet per second ; and R, the radius, or distance of tlie centre of
motion from the centre of the revolving body.
Example.— Let a ball of the weight W = 23 lbs., attached to a
radius, R, measuring 5 feet, rotate with a velocity, V = 40 feet
per second, what is the centrifugal eflFort, or the pull of the ball on
the radius 1
23 X 40 X 40
F =
32 X 5
■ = 230 lbs. raised 1 foot high per second.
CHAPTER Vn.
ELEMENTARY PRINCIPLES OF SHADOWS.
263. We have already, when treating of shadow lines, laid it
down as a rule to be observed generally, in mechanical or geome-
trical drawing, that the objects represented shall be supposed to
receive the light in parallel rays, in the direction of the cubic dia-
gonal, running from the upper left hand corner of the anterior face
of the cube, do\vn to the lower right hand corner of the posterior
face.
We have also shown that the horizontal and vertical projections
of this cubic diagonal make angles of 45° with the horizontal or
base line.
The advantages of this assumption of the direction of the rays
of light will, no doubt, have been appreciated. Amongst these,
it has the merit of at first sight plainly pointing out the relative
degrees of prominence of the various parts of an object, even with
the aid of a single projection or view.
264. This point, then, being determined, on considering an ob-
ject of any form whatever, as receiving in this way the parallel rays
of light, it may be conceived that these rays will form a cylindri-
cal or prismatical column, the base of which wUl be the illumined
outline of the object. The part met by these rays of light will
be fully illumined, whilst the portions opposite to this will be as
entirely void of light. The absence of light on this latter part
may be termed the shadow proper of the object — ^that is, its own
shadow upon itself.
265. If, further, we suppose the Imninous rays surrounding the
object to be prolonged imtil Intercepted by the surface upon, or
adjacent to which it lies, a portion of such surface wUl be unUlu-
mined, because of the interception of some of the rays by the
object ; the outline of this unillumined portion will be limited by,
and depend upon the contour of the object, and it is termed the
shadow cast, or ihroum, by an object on any surface.
The line which separates the UlumLned from the unillumined
portion is termed the line of separation of light and shade, or the
outline of the shadow. This is modified by the form of the reci-
pient surface, as well as by that of the object which gives rise to it.
It is always bounded by straight lines when the generating surfaces
are planes ; and by curves when either or both are cylindrical,
conical, spherical, or otherwise curved.
266. As a general rule, the determination of the outlines of
shadows proper, and cast, reduces itself to the problem of finding
the point of contact of a straight line representing a luminous ray,
with a piane or other surface. The application, however, of this
general principle, though apparently so simple, gives rise to many
riifticulties it practice, from the variety of cases presented by the
'•aVerent forms of objects and it is necessary to give several special
examples, to explain the most simple and expeditious expedients
which may be employed in such cases, always with a due regard
to geometrical accuracy.
We shall primarily choose for these applications objects of sim-
ple form, and bounded by plane surfaces ; next, such as are wholly
or partially cylindrical ; and we shall proceed, in succession, to
objects of more complex forms. The objects which we have taken
in preference, as examples, are such as are most frequently met
with in machinery and architecture ; they will, notwithstanding,
afford quite sufficient illustration in connection with what has to
be said respecting the study of shadows.
SHADOWS OF PEISMS, PYRAMIDS, AND CTLINDEES.
Plate XXVI.
PRISMS.
267. Let the figures 1 and 1 a be given, the horizontal and ver-
tical projections of a cube, it is required to determine the form
of the shadow cast by this cube on the horizontal plane.
In the position given to this cube it is easy to see that the sides
which are in the light are those represented by a d and a c, in the
horizontal projection, and projected vertically in a' e' and a' c'.
The opposite faces, b c and b d, fig. 1, and b' c', b' e', fig. 1 a, are
consequently in the shade ; as, however, these latter faces are
reduced to mere lines in the representations, the shadow proper
can only be shown by a thick shadow line, produced by China ink
in line drawings, and by a narrow stroke of the brush in water-
colour drawings.
These lines, which distinguish the UlumLned sides of the cube
from those wliich are not so, are termed, as we have said, the lines
of separation of light and shade. It now only remains to find the
shadow cast by the cube on the plane, l t.
268. When the object rests on the horizontal plane, as supposed
in this case, and is at a greater distance fi-om the vertical plane than
is equal to its height, the entire shadow cast by it will be in the
horizontal plane ; and to determine its outline here, it is merely
requisite to draw straight lines from each comer of the cube,
representing the rays of fight, as c c, B &, d d, parallel to R, and to
find the points, c, b, d, in which these lines meet the plane.
To effect this, through the points, c', and a', fig. 1 a, the pro-
jections of the two first, b d, draw the rays, c' c', and a' b', paral-
lel to r', and meeting the base line, l t, in c' and b'. If now,
through these points, we draw perpendiculars to the base line, as c' c
b' b, these will cut the first rays in c, b, and d. The contour of
the shadow cast is, in consequence, limited by the lines cc, cb,
b d, and d d.
BOOK OF INDUSTRIAL DESIGN.
97
The face, e b', being that on which the cube rests, has no pro-
minence, and cannot therefore cast any shadow. It follows, then,
that the shadow, as above determined, is all that is apparent. It
is generally represented by a fiat, uniform shade, laid on with the
brush, and produced by a gi-epsh wash of China ink.
269. It will be observed that the lines, d b and h c, are parallel
to the straight lines, d b and b c. This is because these are
themselves parallel to the horizontal plane ; for when a line is
parallel to a plane (82), its projection on this plane is a line paral-
lel to itself; and hence we have this first consequence, that —
When a straight line is parallel to the plane of prq/'eciion, it casts
a shadow on the plane, in the form of an equal and parallel straight
line.
270. It will also be observed, that the straight lines, v> d, b b, c c,
which are the shadows cast by the verticals, projected in d, b, and
c, are inclined at an angle of 45° to the base line ; whence we
derive the second consequence, that —
When a straight line is perpendicular to the plane of projection,
it casts a shadow on the plane in the form of a straight line, parallel
to the rays of light, and consequently inclined at an angle of 45" io
the base line.
271. These observations suggest a means of considerably simpli-
fying the operations. Thus, in place of searching separately for
each of the points, c, b, d, where the rays of light pierce the hori-
zontal plane, it is sufficient to determine one of these points, such
as b, for example, and through it to draw the straight lines, b d, be,
parallel and equal to the sides, d b and b c, of the cube and
intersecting lines, inclined at an angle of 45° drawn from the
points, D c.
In the actual case before us, we may even entirely dispense v«th
the vertical projection, fig. 1°, since it would bave been sufficient
to prolong the diagonal, a b, to b, making b b equal to b a, or to
make the inclined lines, b d, or b b, equal to the diagonal, a b ;
because the vertical projection, c' c', and horizontal projection, c c,
of the same ray of light, are always of the same length, which fol-
lows from our having taken the diagonal of the cube for the direc-
tion of this ray, the two projections, a b and a' b', of this diagonal
being obviously equal. Whence follows the third consequence,
that—
If, through any point of which the two projections are given, we
draw a straight line, representing the ray of light, and if loe ascer-
tain the point in which this ray meets either plane, the length of the
ray in the other plane of projection will be the same.
272. Finally, it is to be observed that the distance, b d, taken
on the prolongation of the vertical line, c b, is equal to the entire
lieight, c' b', namely, that of the cube ; and consequently, in place
of employing the diagonal to obtain the various points, d, b, c, we
may make the distance, B d, equal to the height of tho cube, and
draw, through d, a. straight lino, d b, parallel and equal to d n,
and through b a second, b c, parallel and c(|n;il to B c, and then
join d V, c c.
Thus the shadow cast on a piano by a point, is at a distance
from the projection of tho )>oint, equal to IIk^ distance of the point
itself from the plane.
273. Figs. 2 and 2" niprosent a prism of he.vagonal base, su])-
|)o;icd to be elevated above the base line, hut at the same time at
such a distance from the vertical plane, that all the shadow cast
will be in the horizontal plane.
It will be seen that the vertical faces, a b, b c, and a f, are
illumined, whilst the opposite ones, e d, d c, and e f, are in the
shade.
Of these latter faces, c d is the only one visible in the vertical
projection, fig. 2", and represented by the rectangle, c' d' h g,
which should be shaded to a deeper tint than the cast shadows,
to distinguish it.
274. The operation by which we determine the shadow cast
upon the horizontal plane, is evidently the same a-s in the preced-
ing ease ; still, since the lower base, j h, does not rest upon the
horizontal plane, it will not be sufficient merely to draw the rays
of light through the points, c, d, e, f, of the upper side ; it is, in
addition, necessary to draw corresponding rays through the points,
J, I, G, H, of the base.
It is to be observed, as in the preceding case, that as these two
faces are parallel to the horizontal plane, the shadow cast by each
upon this plane will be a figure equal and parallel to itself; so that,
in place of seeking all the points of the shadow, it would have
been quite sufficient to obtain one of these points, as d, for
example, of the upper side, and A:, of the lower side, and then,
starting from them, to draw a couple of hexagons, parallel and
equal to a b c d e f.
It will also be understood, that as it is only the outside lines,
those of the separation of the light and shade, which make up tho
contour of the shadow, it is not necessary to determine the points
which fall within this contour, and correspond to those points
in the object itself which do not lie in the lines of separation of
the illuminated and shaded parts.
275. Thus it is unnecessary to find the points, a, b, e,h; and
generally, in making drawings, we do not seek the shadows cast
by points fully illuminated, or within the borders of the shaded
portion ; and the contour of the shadow is derived simply from
points lying in the line of separation of *he light and shade on tho
object.
276. From what we have already explained, it will be gaihored,
that the projection in one plane of tho shadow, cast by a point, cau
be obtained by drawing the diagonal of the square, a side of which
is equal to the distance of the point from tho piano, as shown in
the other projection.
For example, the shadow, /,-, on the hoiizontal piano of tho
point, tho two projections of which are f and i, figs. 2 and 2*, may
1)0 got by forming the square, F Ik, a side of which, F /, is equal
to tho distance, i /', of tho point from tho horizontal plane.
In the same way, wo have tho points, g, i,f, correspcmding te
G, I, F, which are tUo same height as the first above tho piano.
For th<:- points, <.•,</, ^,/, which corrosi)ond to iho upper side,
a' 1)', of tlu^ prism, wi> draw the diagonal, n' </', of tho squ;iri\
having for a side llio lioiglit, n' ;;i, of tho jioi::!, 1)', iilovo tho
horizontal piano, and sot out this diagonal fnun C to c, D to a
E to (', &,o.
277. When several straight linos oonvorgo to a point, fno-
shadows thoy oast on oithor jilaiio of projivtion must mvcssarilj
THE PRACTICAL DRAUGHTSMAN'S
also, converge to a point. Thus, in the pyramid, figs. 3 and 3°,
the apex of which is projected in tlie points, s and s', the edges of
all the sides Lemi; directed to this point, cast shadows on the
norizontal plane, bounded by lines converging to the point, s, the
shadow east by the apex on the same plane. In order, then, to
find the shadow cast by a pyramid, on either of the planes of pro-
jection, it is sufficient to draw tlie ray of light through the apex,
and ascertain the pomt at which this ray meets the plane; then to
draw lines to this point from all the angles of the base of the
pyramid, if this rests upon the plane. If, however, tlie pyramid
is raised above the plane, it will be necessary to find the shadows
cast by the various angles of the base, and then draw straight
lines from these to the shadow of the apex.
TRUXCATED PYRAMID.
278. When we have only the frustum of a pyramid to deal
with, and the apex is not given, it is necessary to find the shadows
cast both by the angles of the base, and by those of the surface of
truncation. Thus, the points, e, f, g, h, of the upper side, cast
their shadows on the horizontal plane, in the points, e,/, g, h,
which are obtained by drawing through each point, in the vertical
projection, e', f', g', h', the rays, inclined at an angle of 45°, meet-
ing the base line in the points e',/,^, h', which are squared over
to the horizontal projection, so as to meet the corresponding rays,
drawn through the points, e, f, g, h. Then, if we draw lines
from the points, e,f, g, h, to the angles, a, b, c, d, situated in the
horizontal plane, we shall obtain the shadows cast by each of the
lateral edges of the pyramid.
For the same reason that these edges are diversely inclined to
the horizontal plane, the shadows cast by them on this plane have
also difTerent inclinations to the base line : but the edges of the
upper side or surface of truncation being parallel to this plane,
cast a shadow, which in figure is equal and parallel to this side ;
this would not have been the case had it been inclined to the
plane. It is e\ident that, in the position in which the pyramid is
represented with regard to the rays of light, the two faces, a e h d
and A E F B, are in the light, whilst their opposites, d h g c and
c G F B, are in the shade. This last, which is the only one visible
in fig. 3", is there distinguished by a moderate shade of colour.
CTLISDER.
279. A cylinder with a circular base being a regular solid, all
that is wanted, in determining the lines of separation of light and
shade, is, when the cylinder is vertical, to draw a couple of planes
tangential to it, and parallel to the rays of light, as in figs. 4 and
4°. These tangential planes are projected in the horizontal plane,
in the lines, a a, si, tangents to the circle, and inclined at the
angle of 45°. By their points of contact with the circle, these
tangents give the lines of separation of light and shade, which are
projected vertically in a' c and b' d. One of these lines is appa-
fcnt on this view, but the other is not. We have thus the portion,
A E b, of the cylinder, in the Ught, and its opposite, a r b, in the
shade. A very small portion of this last is seen in fig. 4°, and is
there siightiy shaded.
• 280. Wltn reference to the cast shadow, it is to be remarked,
tliat for the \ erv reason that the lines of separation of light and
shade are vertical, the shadows they will cast on the horizontal
plane will be in two lines, c a and d b, with an inclination of 45',
as already explained, these lines being identical v\ith the prolonga-
tion of the tangential rays. The two bases of the cylinder being
parallel to the horizontal plane, their shadows will be circles equai
to themselves ; and all that is required is to find the shadows, n, o,
cast by their respective centres, n and &, and with the points, n, o,
as centres, to describe circles, with a radius equtxl to o a. The
entire shadow cast by the cylinder is comprised between the two
semicircles and the two tangents, c a, d h.
SHADOW CAST BY ONE CYLINDER UPON ANOTHER.
281. Hitherto we have only considered the shadow cast by an
object upon one of the planes of projection. It frequently hap-
pens, however, that one body casts a shadow on another, or that
the configuration of the body itself is such, that one part of it casts
a shadow on another.
Let fig. 6 be the vertical projection of a short cylinder, a, with
a concentric cylindrical head, b. We have, in the first place, to
find the line of separation of light and shade upon these two
cylinders ; and for this purpose w"e require to draw a second ver-
tical projection, fig. 6", at right angles to the first, and in the line
of its axis. In this figure, the projection of the ray of light also
makes an angle of 45° with the base line. We must, consequently,
draw the two straight lines, c' c and d' d', tangential to the circles,
a' and B , and project, or square over, the points of contact, c' and
d', to fig. 6, drawing the lines, a b and d d, which separate the
light from the shaded part of the objects. Instead of drawing
these tangents, we can directly obtain both points of contact, by
drawing the radius, o c' d', at right angles to the ray of light.
282. The shadow cast by the projecting head, b, upon the
cyKnder, a, is limited to that due to the portion, d' c' h', of the
circmnference. Different points in the outline of this shadow are
determined, by first taking any points, c', e', f', g', upon the arc,
d' c' h', and drawing through each of them lines, representing the
parallel rays of light, and meeting the circumference of the cylin-
der, a', in the points, c, e', f, g'. Having projected the first-
mentioned points on the base, d h (fig. 6), draw through the points,
c, E, F, G, a series of lines parallel to the first, and likewise
representing the rays of light, and square over the points of con-
tact, c',e',f,g', which will give the points, c,e,f,g, of the curve,
which is the outline of the shadow upon the cylinder, a.
As seen in a former example, instead of squaring over the
points, c', e',f,g', w^e can obtain the same result by making the
corresponding rays, c c, e e, f/, Gg, equal to the lines, c' c', e' e',
shadow cast by a cylinder upon a prism.
283. Figs. 7 and 7" represent two vertical projections of a prism,
A, of an octagonal base, having a cylindrical projecting head, b.
As in the preceding case, draw the radius, o d', perpendicular
to the ray of light, thereby obtaining the point of contact, d', and,
in consequence, the line of separation, d d. of light and shade on
the cylindrical head, b.
The inclined facet, c i, of the prism, being in the direction of
the ray of light, and, consequently, inclined at an angle of 45° wltli
BOOK OF INDUSTRIAL DESIGN.
99
the vertical plane, is considered to be completely in the shade.
The edge line, a b, fig. 7, is therefore the line of separation of
light and shade on the prism-shaped portion of the object, and
the surface, a bid, in consequently tinted. The shadow cast upon
the prism by the overhanging head, b, reduces itself to that due to
the portion, c' f' h', merely, of the circumference of the latter, and
it falls upon the two faces, c'/' and/' h', of the latter.
The lines indicated on the diagram, with their corresponding
letters, when compared with those of the preceding example, will
show that the operations are precisely the same m both cases, and,
ic the latter, the curves, cef and fgh, are the resulting outlines
of the shadow. In general, it >s unnecessary to obtain more than
the extreme points of the o^rve, and another near the middle.
Through the three points thu'; obtained, arcs of circles can then be
drawn. The curves are bowever, in reality elliptical.
SHADOW CAST BY ONE PRISM UPON ANOTHER.
284. Figs. 8 and f" represent a couple of vertical projections, at
right angles to ea^ih other, of a prism of an octagonal base, sur-
mounted by a similar and concentric, but larger prism. Although
the operation." c.-xlled for in this case are precisely the same as in the
kwo preceding', still it is axi exemplification which cannot be omitted;
ind its chief use is to show, that
27(6 •ihadoio cast by a straight line upon a plane surface is inia-
liubhj c, straight line ; and, consequently, it is sufficient to determine
k<i extreme points, in order to obtain the entire shadow in any one
ylane.
Thus, the straight line, e' c', casts a shadow upon the plane
facet, /c, which is represented by the straight line, ec.
it is further obvious, that
The shadow cast upon a plane surface, by any line parallel to it,
must be parallel to that line.
Thus, the straight line, e' g', of the larger prism, b, being jiaral-
lel to the plane faccit, f g', of the prism, a, casts a shadow upon
the latter, which is represented by the straight line,/g-, parallel to
the line, f g, the vertical projection of the edge, f' g'. It is not,
however, the same with the portion, ef because the corresponding
portion, e' f', of tlie edge of the larger prism, is not parallel to the
facet,/' e!.
SHADOW CAST BY A PRISM UPON A CYLINDER.
285. Figs. 9 and 9" represent vertical projections, at right angles
to each other, of a portion of an iron rod, a, surmounted by a con-
centric head, b, of a hexagonal base. The main object of this
diagram is to show, that
When a right cylinder is parallel, or perpendicular, to a plane of
projection, any straight line, which is perpendicular to tlie axis of the
cylinder, and parallel to the plane of projection, casts a. shadoio upon
the cylindrical surface, which is represented by a curve, similar to
the cross section of such surface.
If, therefore, the cylinder is of circular base or cross section, as
we have supposed in the present case, the shadow cast upon i(
will be a portion of a. circle, of the same radius as the cylindci-.
Tl,\is. the straight line, »' f', sitiialcfd in a, piano, at right jiiiglcM In
til" axis of the cylinder, a, and Ix^ing, at llu; same lime, parallel lo
<lif' vortical plane, casts a shadow upon the cyliiulci-, which is re-
presented by the portion, c ef, of a cu-cle, the centre, o', of which is
obtained- by drawing through the point, o, a line, o i, representing
the ray of light, and extending to the prolongation of the edge, d'
f'. The line, o i, cuts the circumference of the cylinder in the
point, i', which is squared over to i, upon the other projection, h i.
fig. 9, of the ray, o i. The lower point, c, is obtained from the
upper one, i, being symmetrical with reference to the axis of the
cylinder. The ray, h i, being continued to the axis, cuts it in the
point, o', which is, consequently, the centre of the arc, cei, the
radius, i o' or c o', of wliich is equal to that, o i', of the cylinder.
286. The edge, f' h', although situated in a plane perpendicular
to the axis of the cylinder, is not parallel to the vertical plane, and
does not, therefore, cast a shadow of a circular outline upon the
cylinder, but one of an elliptical outline, as fg h, which is obtain-
ed by means of points, the operations being fully indicated on the
diagrams. If the head, b, which casts a shadow upon the cylin-
der, were square, instead of hexagonal, as is often the case, one
of the sides of the square, as i h', fig. 9°, being perpendicular to
the vertical plane, would cast a shadow on the cylinder, having
for outline the straight line, h i, making the angle of 45° with the
axis.
Thus, ivhenever a straight line is perpendicular to the plane of
pyrojection, not only is its shadow, as cast upon this plane, a straiglU
line, inclined at the angle of 45°, but it is also the same on an object
projected in this plane, no matter of what form.
Observation. — In the four examples last discussed, we have only
represented half views of the objects in the auxiliary vertical pro-
jections, figs. 6", 7," 8°, and 9", this being quite sufficient for deter-
mining the shadow, as it is only that produced by this half which is
seen. It is obvious, that the same operations will answer the pur-
pose, whether the axis of the object be horizontal or vertical.
SHADOW cast by A CYLINDER IN AN OBLIQUE POSITION.
287. In figs. 5 and 5', we have given the horizontal niul vertical
projections of a right cylinder, having its axis horizontal, but
inclined to the vertical plane. As in this oblique jjrojcction we
cannot obtain the points of contact of the luminous rays w itli the
base in a direct manner, it becomes necessary to make an especia.
diagram, in order to determine the lines of sejiaiation of light and
shade, which are always straight lines, paiallcl lo Ihc axis of ilui
cylinder.
To this ert'ect, we shall make use of a general constriiclion, sus- ■
eeptible of ap|)lication to a variety of such cases. This construction
consists in determining the projection of the luniinous ray, in any
given plane, perpendicular to either of the geoinetriciU planes, w lienco
may bo derived its form and aspect in either of the latter planes.
It follows, that if we have any curve in (he given plane, wo can
easily lind llio point of separation of tlu' light and shade situated
upon this curve, by drawing a couple of tangents to it, [virallel to
the ray of light projected in this plane, and transferred to the other
plane of projection.
Thus, let li o and i;' o' be the projections of the luminous ray .
it is projwsed to liiul llie projccticui ,.l'lhis ray upon llic plane, d />,
of the base of llie cylin.lrr. To obtain this, project the u.iliit. v.
lo r, by means of a pi-rpciulicular lo a b. and r i icprosout.s tlio
horizontal projcctitm i>l' the ray of li;^lit \\\w\\ (he plane, we , ftiuf
100
THE PRACTICAL DRAUGHTSMAN'S
the vertical projection, r' o', is obtained by squaring over the point,
o to 0', on the base line, and the point, r to r', on the horizontal,
k' e', and then joining o' r . Next, draw tangents to the ellipses,
which represent the vertical projections of the ends of the cylinder,
fig. 5', making these tangents parallel to the ray of light, / o'.
Their points of contact give, on the one hand, the first line, c' i\
of separation of light and shade, which is visible in the vertical pro-
jection, and, on the other hand, the second line, e f\ which is not
visible in that projection.
By squaring over these points of contact, respectively, to the two
ends, a b and g h, of the cyliuder, in the horizontal projection, we
obtain the same lines of separation of light and shade, cd and/e,
as in this projection ; the former of wliich lines is invisible, whilst
the latter is visible.
The same lines, c d and/e, fig. 5, can be obtained independently
of the vertical projection, fig. o", in the following manner : — Draw
an end view of the cylinder, as at a' b\ having its centre in the con-
tinuation of the cylinder's axis. Upon this end view, also, draw
the ray of light, as projected upon the base, after describing the
circle, a' m b, with the radius, o a ; make r r" equal to the height of
the point, r', above the bottom of the cylinder, thereby obtaining
the line, o r^, representing the ray of light upon the end view of the
cylinder. Next, draw a couple of tangents to the circle, a" m b\
parallel to o r^, and their points of contact, c\f^, will represent the
end view of the lines of separation of light and shade, which are
transferred to the horizontal projection, fig. 5, by perpendiculars
drawn from them to the straight line, a b.
288. When the shadow proper, of the cylinder, has been thus
determined, it \vill not be difficult to find the outline of its shadow
cast upon the horizontal plane. In the first place, the shadows of
the two bases are found, being in the form of ellipses; and next,
those, c' d'' and /' e", cast by the lines, c d and fe ; namely, those
of the separation of light and shade upon the object itself. These
lines vnll necessarily be tangents to the ellipses, representing the
shadows of the bases. It may be observed, that the transverse
axes of the ellipses are parallel to the line, r* o.
If the cylinder were inclined at an angle of 45° to the veitical
plane, stUl remaining parallel, however, to the horizontal plane, the
lines of separation of light and shade would, in the horizontal pro-
jection, be confounded with the extreme generatrices, or outlines,
of the cylinder, the visible semicylinder being wholly in the light,
and the opposite semicylinder wholly in the shade. In the vertical
projection, the line of separation would be in the line of the axis,
and would divide the figure horizontally iiito two equal parts.
PRINCIPLES OF SHADING.
Plate XXVH.
289. Before proceeding to the further study of shadows, we must
observe that shade ws^ proper and cast — which are simply represent-
ed by flat tints, so as not to render the diagrams confused — should
be modified in intensity according to the form of the objects, and
the position of their surfaces with reference to the light.
The study of shading carries us somewhat into the pr.)vince of
toe non-mechanical pamter, who is guided by his taste rather than |
b} mathematical rules : still, whilst we acknowledge the difficulty
of lapng down an exact theory on this subject, we would recom-
mend the following systematic methods, which will render the first
difficulties of the study more easily surmountable.
In painting, and in every description of drawing, the eifects of
light and shade depend upon the following principles : —
ILLU.MISED SURFACES.
290. When an illumined- surface Tias all ils poinis at an equal
distance from ilte eye, il must receive a clear shade of uniform inten-
sity throughout.
In geometrical drawing, where all the visual rays are supposed
to be parallel and perpendicular to the plane of projection, all sur-
faces parallel to this plane have all their points equally distant from
the eye : such is the plane and vertical surface, a b a d, of the
prism, fig. A.
291. Of two such surfaces, disposed parallel to each other, and
illumined in the same manner, thai which is nearer to the eye should
receive a shade of less intensity.
292. Any illumined surface, inclined to the plane of the picture,
having its points at varying distances from the eye, should receive a
shade of varying intensity.
Now, according to the foregoing principle, it is the most advanced
portion of an object which ought to be the lightest Lq colour ; this
effect is produced on the face, a dfe, which, as shown in the plan,
fig. 1, is inclined to the vertical plane of projection.
293. Of two illumined surfaces, that which is more directly pre-
sented to the rays of light should receive a shade of less intensity.
Thus, the face, e' a', fig. 1, being presented more directly to the
light than the face, a' b', is covered with a shade which, being gra-
duated because of the inclination to the plane of the picture, is
still, at the more advanced portion, of less intensity than that of tlie
latter face. It is near the edge, a d, that the difference is more
sensible.
SURFACES IN THE SHADE.
294. When a suiface in the shade is parallel to the plane of p^u^
jection, or of the picture, il must receive a deep tint of uniform inie>i-
sity throughout.
An exemplification of this will be seen on the fillet, b, fig. ©,
Plate XXVIII., which is parallel to the vertical plane : the differ-
ence of shade upon this fillet, in comparison ^\•ith that upon the
more projecting portion, a, which is parallel to it, but in the fight,
distinctly points out the ditference between an illumined surface and
one in the shade, in conformity with the two principles, 290 and
294.
295. Of two parallel surfaces in the shade, that nearer the eye
should receive the deeper tint.
Thus, the shadow cast upon the fillet, b, fig. ©, Plate XX\aiI.,
is sensibly deeper than that cast by it upon the vertical plane, which
is more distant.
296. When a surface in the shade is inclined to the plane of the
picture, the part nearest to tlie eye should receive the deepest tint.
The face, bghc, fig. A, Plate XXVIT., projected horizontally in
b' g', fig. 1, is thus situated. The shade is made considerably
deeper neai- the edge, b c, than near the more distant one, g h.
BOOK OF INDUSTRIAL DESIGN.
lOi
297, When two surface!' in the shade are uneqiuzlly inclined,
umh reference to the, direction of the rays of light, the shadow cast
by any object should he deeper upon that which receives it more
direcily.
Thus, the shadow, a dfe, cast upon the face, f, of the prism,
fig. [1, Plate XXVI., should be slightly stronger than that cast
upon the face, g, because the first is more directly presented to
the light than the second, as showTi by the lines, /' h' and /' c',
fig. T.
These first principles are exemplified in the finished figures on
Plate XXVI., XXVIL, and subsequent ones.
As, in order to produce the gradations of shades, it is important
to have some knowledge of actual colouring or shading by means
of the brush, we shall proceed to give a few short explanations of
this matter.
Two methods of producing the graduated shades are in use —
one consisting in laying on a succession of flat tints ; the other, in
softening off the shade by the manipulation of the brush.
We have already said two or three words about the laying on of
flat tints, when treating of representing sections by distinguishing
colours. (137.) These first precepts may serve as a basis for tJie
first method of shading, which is the less diflScult of the two for
beginners. In fact, according to it, the gi-aduated shade is produced
by the simple superposition of a number of flat tints.
FLAT-TINTED SHADING,
298. Let it be required to shade a prism, A, Plate XVII., with
flat tints : —
According to the position of this prism, with reference to the
plane of projection, as seen in fig. 1, it appears that the face, a' h'->
is parallel to the vertical plane, and is fully illumined ; it should,
consequently, receive a clear uniform tint, spread over it by the
brush, and made either from China ink or sepia, as has been done
upon the rectangle, a, b, c, d, fig. A. When the surface to bo
washed is of considerable extent, the paper should first be prepared
by a very light wash, the full intensity required being arrived at by
a second or third. (137.)
The face, b' g', being inclined to the vertical plane, and com-
pletely in the shade, should receive a tint (294) deepest at the
edge, b c, and gradually less intense towards g h ; this is obtained
by laying on several flat shades, each of different extent. For this
purpose, and to proceed in a regular manner, we recommend the
student to divide the face, b' g', fig. 1, into several equal parts, as
in the points, 1', 2', and through these points to draw lines parallel
to the sides, b c, g h, fig. A. These linos should be drawn very
lightly indeed, in pencil, as they are merely for guides. A first
greyish tint is then spread over the surface comprised botwocu tho
first line, 1 — 1, and tho side, b c, as in fig. 2; when this is quite
dry, a second like it is laid on, covering tho first, and extending
from the side, b c, to tho line, 2 — 2, as in fig. 3. Finally, these
are covered with a third wash, as in fig. Z^, extending to llie outer
edge, g li, and completing the gniduatcnl sliado of the rectangle,
bcgh.
The number of washes by which tho gradation is expressed,
evidently depends upon tho width of tho surface to bo shaded ;
and it will be seen that ilw c:''oator the number of washes used,
the lighter they should be, and the lines produced by the edges of
each will be less hard, and a more beautiful effect will resdt.
The student must remember to efface the pencilled guide-lines,
as soon as the washes are sufficiently diy.
299. This method of overlaying the washes, and covering a
greater extent of surface at each succeeding time, is preferable to
the one sometimes adopted, according to which the whole surface,
b g h c,is first covered by a uniform wash; a second being then
laid over b 2 — 2 c; and finally, a third over the narrow strip,
b ] — 1 c. When the shade is produced in this manner, the edges of
the washes are always harder than when the washes are laid on as
we recommend — the narrovi'est first — for the subsequent washes,
coming over the edge of each preceding one, soften it to a consi-
derable extent.
The fiice, e' a', fig. 1, being likewise inclined to the vertical plane,
but being wholly illumined, should receive a very light shade (292),
being, however, a little bolder towards the outer edge, ef, fig. A.
The shade is produced in the same way as that of the face, U g",
but with much fainter washes.
300. Let it be proposed to shade a cylinder, fig. [B, with a series
of flat tints : —
In a cylinder, it is necessary to give the gradations of shade, both
of the illumined and of the non-illumined portion. In reference to
this, it will be recollected that the line of separation, a b, of light
and shade, is determined by the radius inclined at an angle of 45°,
as o a, fig. 4, perpendicular to the ray of light ; consequently, all
the shadow proper, which is apparent in the vertical projection, fig.
d, is comprised between the lino, a b, and the extreme generatrix,
c d. Consequently, according to the principle already laid down
(296), the shade of this portion of the surface should be graduated
from a b to c d, as was the case with the inclined plane surface, b' g',
fig. 1, the greater intensity being towards a b.
On the other hand, all that part of the cylinder comprised be-
tween the line, a b, and the extreme generatri.x,/g-, is in the light ;
at the same time, from its rounded form, each generatrix is at a
different distance from the vertical plane of projection, and makes
different angles vvdth the ray of light. Consequently, this portiou
of the surface should receive graduated sh.idos. (292.) To express
the etfect in a projjcr manner, it is necessary to know what {Mrt of
the surface is tho clearest and most brilliant : and this is cviilcnlly
the part about tho generatrix, c i, \\g. ©, situated in the vertical
plane of tho ray of light, r o, fig. 4. In consequence, however,
of the visual rays being perpendicular to the vertical plane and
parallel to tho line, v o, tho portion which a|)pears to tho eye to
bo tho clearest will be nearer to this line, v o, and may be limited,
on tho one hand, by the line, x o, bisecting the angle made by the
linos, R o and v O, and on the other, by the line, k o; squaring
over, then, tho points, c' and m', tig. I, and tlrawiii'j; the lines, c i
and m n, fig. ©, we obtain the suilace, c / m n, wliirli is the most
illumined.
301. This surface is bright, and rcni;iiiis white, wlun the cy.indej
is iKilislu'd, as :i t iii iicd iron shaft, for e\,Mm|ile, or a marble coliuuii ■
it is covered willi n \\>j:Ui shade, being always clearer, however, timn
the rest of liie suillu'e, when the eyliiuler is iinpulished, as a cast-
iron l)i|ie.
302. .\tler these pieliniinary observations, we ni;iy prooeod t<
102
THE PRACTICAL DRAUGHTSMAN'S
shade the cylimlcr, /' m' a! e', fig. 4, dividing it into a certain num-
ber of equal parts, the more numerous according as the cylinder is
•rrcater. These divisions are squared over to the vertical projection,
and straight lines drawn lightly with the pencil, as limiting guides
for the colour. We then lay a light gray shade on the surface,
acdb, fig. 5, to distinguish at once the part in the shade ; when
this is dry, we lay on a second covering, the line, a b, of separation
of lio^ht and shade, and extending over a division on either side of
it, as shown in fig. 6 ; we afterwards lay on a third shade, covering
two divisions to the right and to the left, as in fig. 7 ; and proceed
in the same manner, covering more and more each time, always
keeping to the pencil lines. The diflferent stages are represented
in figs. 8, 9, and 10.
303. We next shade the part,/e ig, laying on successive shades,
but lighter than the preceding, as indicated in figs. 8, 9, and 10.
The operation is finally terminated by laying a light wash over
the whole, leaving untouched only a very small portion of the bright
surface, e m n i, fig. [B. Tliis last wash has a beautiful and soften-
mg effect.
SHADING BY SOFTENED WASHES.
304. This system of shading differs from the former in producing
the effects of light and shade by imperceptible gradations, obtained
by manipulation with the brush in the laying on of the colour : this
system possesses the advantage over the first, of not leaving any
lines, dividing the dift'erent degrees of shade, which sometimes ap-
pear harsh to the eye, and seem to represent facets or flutings,
which do not exist.
For machinery, however, the former system is very effective,
bringing out the objects so shaded in a remarkable manner. In-
deed, we recommend all machinery to be shaded in this manner,
whilst architectural subjects will look better treated according to
the second system.
In this, the laying on of the shade is much more difficult, requii--
ing considerable practice, which will be aided by proceeding in the
following systematic course.
305. Let it be proposed to shade a truncated hexagonal pyramid,
fig. ©, Plate XXVII.
The position of this solid, with reference to the vertical plane
of projection, is the same as that of the prism, fig. A. Thus the
face, abed, should receive a uniform flat shade of little intensity ;
rigorously keeping to rules, this should be slightly graduated from
top to bottom, as the face is not quite parallel to the vertical
plane.
The face, b g h c, being inclined, and also in the shade, should
receive a deep shade, graduated from b c to g h; to this effect ap-
ply a fiirst light shade to the side, b c, fig. 16, softening it off to the
right, taking the line, 1 — 1, as a limiting guide in that direction:
this softening is produced by clearing the brush, so that the colour
may be all expended before the lighter side is reached ; and when
the shade is wide, a little water should be taken up in the brush
once or twice, to attenuate the colour remaining in it. By these
means an effect will be produced like that indicated in fig. 15., care
oeing tasec not to extend the wash beyond the outline of the
object.
When this first wash is well dry, a second is laid over it, produc-
ed exactly in the same manner, and extending further to the right,
covering the space, /; c 2 — 2, as shown in fig. 16. Proceeding
in the same manner, according to the number of divisions of the
face, we at length cover the whole, producing the graduated shade,
bghc, fig. ©.
The operations are the same for the face, e a df, which is nearly
perpendicular to the rays of light, but is considerably inclined to the
plane of projection.
In rigorously following out the established principles, the shade
on this face should be graduated, not only from e/to a d, but also
from eatofd. Also, on the face, b g h c, in the shade, the tint
should be a trifle darker at the base, c h, being graduated off
towards b g. But for objects so simple in form as the one under
consideration, this nicety may be neglected — at any rate, by the
beginner — as only increasing his difficulties ; the proficient, on the
other hand, is well aware how attention to these refinements assists
in producing effective and truthful representations.
306. Let it be proposed to shade a cylinder with softened washes,
fig. ©, Plate XXVII.
By following the indications given in fig. 4, for the regular im-
position of the shades, as explained with reference to the flat-wash
shading, the desired effect may be similarly produced by substituting
the softened washes. It is scarcely necessary to divide the circum-
ference into so many parts as for the former method ; a first shade
must be laid on at the line, a b, of separation of light and shade,
and this must be softened off in both directions, as in fig. 11; a
second and a third wash must then be applied and similarly softened
off, and in this manner we attain the effects rendered in figs. 12,
13, and ©.
We have not deemed it necessary to give diagrams of all the
stages, as the method of procedure will be easily understood from
preceding examples. The student should practise these methods
upon different objects of simple form, and he will thereby rapidly
acquire the necessary facility.
307. Wlien spots or inequalities arise in laying on a wash, from
defects in the paper or other accidents, they should be corrected
with great care. If they err on the dark side, they should, if pos-
sible, be washed out ; the best means of doing this, in very bad cases,
is to let the drawing become perfectly dry, and then slightly moisten
the spots, and gently rub off the colour with a clean rag. Lights
may be taken out in this way, where, from their minuteness or
intricate shape, it would be difficult to leave them whilst laying on
a flat shade, in the midst of which they may happen to be. A de-
fect on the light side is more easily corrected, by applying more
colour to the spots in question — being careful to soften off the
edges, and to equalize the whole wash.
Figs. ^, [U, ©, ©, [1, Plate XXVI., represent several shaded
objects, the shadows of which have already been discussed, as indi-
cated in figs. 1 to 9. These may serve as guides, also, in shading
with washes of colour, although the shades in that plate are pro-
duced by lines, whilst the figures in Plate XXVII. represent t^--
actual appearance of the wash-shading method. '
Finally, we have to recommend the adoption of a much larger
scale for practice, as it is desirable to be able to produce largo
washes with regularity and smoothness of effect.
BOOK OF IXDUSTRL\L DESIGN.
103
CONTINUATION OF THE STUDY OF SHADOWS.
Plate XXYIIL
shadow cast tjpos the dtteeior of a ctirsdee.
308. When a hollow cylinder, as a steam-engine cylinder, a cast-
ir )n column, or a pipe, is cut by a plane passing through its axis,
we hare, on the one hand, a straight projecting edge, and, on the
other, a portion of one of the ends, which cast shadows upon the
internal surface of the cylinder.
We propose, then, to determine the form, as projected, of the
shadow cast upon its interior by a steam-engine cylinder, a, sec-
tioned by a plane passing through its axis, figs. 1 and 1°. In the
first place, we seek the position of the shadow cast by the rectilinear
projecting edge, b c, which is, in fact, produced by the intersecting
plane, b' a'. This straight line, b c, being vertical, is projected
horizontally in the point, b', and casts a shadow upon the cylinder,
as represented by the straight line, 6/, which is also vertical, and
Ls determined by the point, h , of intersection of the ray of light,
b' V, with the surface of the cylinder, e' V of. Thus, when a
straight line is parallel to a generatrix of the cylinder, the shadow
cast by it will he a straight line parallel to the axis. It b, therefore,
evidently quite sufficient to find a single point, whence the entire
shadow may be derived.
309. We next proceed to determine the shadow cast upon the
interior of the cylinder by the circular portion, b' e' r', of the upper
end. If we take any point, e', on this circle, and square it over to
e in the vertical projection, and draw through this point a ray of
light, e' e*, e e, it will be found to meet the cylindrical surface in
the point- e', which is squared over to e, the length of the ray being
equal in both projections, according to the well known rule. This
applies to any point in the arc, e' f'. The extreme point on one
side is obtained by a tangent to the circle in the point, f', ^ving
the point, r, in the vertical projection ; the opposite extreme point,
b, being already given as the top point of the straight edge, b c ; we
have, therefore, the curve, i eb, for the upper outline of the shadow
due to the circular portion, b' e' r'.
310. If , as in figs. 1 and 1', we suppose the piston, p, with its
rod, T, to be retained unsectioned in the cylinder, we shall have to
determine the form of the shadow cast by the projecting part of
the piston upon the interior of the cylinder, and represented by the
cune, dho. For this purpose we take any points, b', h', &, on the
circumference of the piston, and draw through them, in both pro-
jections, the rays of light which meet the surface of the cylinder,
b' y 0, in the points, b' h' o', which are projected vertically in dho:
the curve passing through these points is the outline of the shadow
sought The curved portions of these shadows are elliptical.
The piston-rod, t, being cylindrical and vertical, casts a shadow,
of a rectangular form, upon the interior of the cylinder, the vertical
sides, i). k Z, being determined by the luminar tangents, i' i", k' k',
parallel to the axis.
SHADOW cast by O.XE CYLI.VDER tTPOX A.NOTHEB.
311. Let figs. 2 and 2* be the projections of a convex scroi-
cylinder, a, tangential to a concave semicylinder, b. funning a pat-
tern often met with in mouldings.
This problem, which consists in determining the shadow proper
of a convex cylinder, together with that cast by it upon the surface
of a concave cylinder, in addition to that cast by the latter upon
itself, is a combination of the cases discussed in reference to figs. 4
and 4', Plate XX\T., and to figs. 1 and 1* in the present plate. The
operations called for here are fully indicated on the diagram ; and
we have merely to remark, that it is always well to start by deter-
mining the extreme points, as c', d,' which limit the shadow proper
c G, and cast shadow, v c g: these points may be obtained more
exactly, as already pomted out, by drawing the radii, o c' and d' e,
perpendicular to the luminous rays.
SHADOWS OF COSES.
312. In this branch of the study, we propose to determine, first,
the shadow proper, or the line of separation of light and shade upon
the surface of the cone ; second, the shadow cast by the cone upon
the vertical plane of projection ; and- third- the shadow cast upon
the cone, and upon the vertical plane of projection, by a prism of a
square base, placed horizontally over the cone.
313. First: We have laid it down as a general principle, that, in
order to determine the shadow proper of any surface, it is neci-ssary
to draw a series of parallel luminous rays tangential to this surface.
When, however, the body is a solid of revolution generated by a
straight line, as a cylinder or a cone, it is sufiicient to draw tangen-
tial planes parallel to the Inminotis rays, to obtain the lines of sepa>-
ration of light and shade.
In the ca.=e of the cone represented in figs. 3 and 3*, and of which
the axis, s t, is vertical, the operation consists in drawing from the
apex, s and s', two lines, making angles of 45°, as s s and s' s',
gi\Tng, in the point, s', the shadow cast by this apex uj>on the hori-
zontal plane. From this point we draw a straight line, a s', tan-
gential to the base, a' c' b', of the cone. This straight line repre-
sents the plane, tangential to the cone, as intersecting the horizontal
plane of the base ; and the contact generatrix is then obtained by
letting fall from the centre, s', a radius, s' a', perpendicular to the
line, a* s' ; and this line, s' d, is the horizontal projection of one of
the lines of separation of light and shade. The vertical projection
of this straight line is obtained by squaring over the p<^)int of con-
tact, a',, to a, and then drawing the straight line, s a. The other
line, s b, of the separation of light and shade, is similarly obtainea
by means of the tangent, s' b'. Its vertical projection is, however,
not apparent in fig. 3*.
314. Second : The shadow cast by the cone upon the vertical
plane is limited, on the one hand, by the line of separation of light
and shade, and, on the other, by the portion of the illummed Laso
comprised between the two separation lines. Now, the straight
line, s a, casts a shadow, represented by the straight ILae, s* a*, aa
indicated in the diagram ; and the base, a' e' c' b', casts a shadow,
represented by the elliptic cur%e, / e d a', which is detennintil by
points, as in the ca.«e considered in reference to fig. 5, Plate X.Wl
315. Third: The shadow cast by the lower aide, o u, of iho
rectangular prism, r, upon the convex surface of the cone, is found
in oiTordance with the principle already onunciat*-*! — that when n
stmight line Ls jwirillel to a plane of pr»jo»tion, it oAsts a sJuulow
104
THE PRACTICAL DRAUGHTSMAN'S
upon this plane, which is represented by a straight line, equal and
jMirallel to itself. It follows, then, that if we cut the cone by a plane,
M N, parallel to its base, the shadow cast by the straight line, g h,
upon this plane, will be found by drawing from the point, i, of the
base, situated upon the axis of the cone, and projected horizontally
in the point, s', a luminous ray, which meets this plane, m n, in
the point, i, projected horizontally in the point, i', upon the projec-
tion of the same ray. If, next, we make i' i^ equal to s' j',
and through i' draw the straight line, g" h^ this last will be the
shadow cast by the edge, g h, of the prism, upon the plane, M n.
This plane, however, cuts the cone in a cii'cle, the diameter, m n,
of which is comprised between the extreme generatrices, whilst the
circle is projected horizontally in ivi' l' n'. The intersection of
the straiglit line, g" h°, with the circle, gives the two points, i^ and
i', which being projected vertically in i, i", upon the straight line,
M N, constitute two points in the outline of the shadow cast upon
the cone.
Continuing the operations in this manner, and taking any other
intersecting plane parallel to m n, any number of points may be
obtained. It will be observed that these planes are taken at a
convenient height, when the projections of the straight line, g h,
cut the corresponding circles ; and with regard to this, much use-
less labour may be avoided, by at tirst determining the limiting
points of the curve. Thus, in the example before us, we get the
summit, g, of the curve, by making i j, fig. 3°, equal to j' s', fig. 3.
Through the point, j, we then di-aw a luminous ray, and the point,
(r", at which it meets the extreme generatrix, a s, of the cone, is
squared over to the generatrix, s t, by means of the horizontal,g-"^,
whence g is the summit of the curve. We next obtain the extreme
points, h, h', of the same shadow, by making s' g", fig. 3, equal to
s' g', and Squaring over g" to g', in fig. 3°. Through g^ draw the
straight line, g' h', parallel to the luminous ray, as situate in a
vertical plane, passing through it ; the ray, as we have already seen,
making, in this plane, an angle of 35° 16' with the base line. The
point, h', at which this ray meets the extreme generatrix, s a, de-
termines the plane, h' h, which is intersected by the luminous rays,
making angles of 45°, and drawn through the points, i and g, in the
points, h, h', the limits of the curve sought.
The shadow cast by the prism, p, upon the vertical planes, pre-
sents no peculiarity apart from the principles already fully explained.
SHADOW OF AN INVERTED CONE.
316. When the cone, instead of resting upon its base, has its
apex downwards, as is the case with the one represented in figs. 4
and 4", the rays of light illumine a less portion of its surface ; and
the lines of separation of light and shade are determined by draw-
ing from the apex, s s', lines at an angle of 45°, which are prolonged
towards the light, until they intersect the prolongation of the plane
of the base, a b.
It will be observed, that the points of intersection, s, s', lie to
the left, instead of to the right of the cone. Through the point,
s', the horizontal projection of the point, s, draw a couple of lines,
s" a', s' b', tangential to the circumference, a' b' d', of the base.
The radii, s' a' and s' b', drawn to the points of contact, represent
the horizontal projection of the two lines of separation of light and
phade, and show that the illumined portion of the cone, consisting
of the surface, b' g^ a' s', is smaller than the portion, b' d' a s', in
the shade. In the case of the cone \vith its apex uppermost, the
contrary would be observed, the portion in the shade being there
less than that in the light ; and the method given of determining
the proportion of shadow of the inverted cone is suggested by the
consideration, that this proportion must be exactly the reverse of
that for the cone with its apex uppermost.
The first-mentioned line, s' a, is the only one apparent in the
vertical projection, fig. 4°. It is found by squaring over the point,
a', to a, and joining this last to the apex, s. As the cone is trun-
cated by the plane, d e, the line of separation obviously terminates
at the point, c, of its intersection with tliis plane.
317. The cone, thus inverted, is surmounted, moreover, by a
square plinth, the sides of which, f g' and c' h', cast shadows upon
its convex surface. The side, f g', as projected vertically in g', fig.
4", is perpendicular to the vertical plane, and consequently its cast
shadow is a straight line, making an angle of 45°, as of. The
extreme limit,/, is determined by proceeding as in previous exam-
ples ; that is to say, by making i G^ in fig. 4", equal to s' g', in fig.
4, and then drawing through the point, g', the straight line, g^ h",
parallel to the ray of light, as in the diagonal plane, that is, at an
angle of 35° 16' to the horizon ; next draw the horizontal line, h" h,
and it will be intersected by the straight line, g f, in the poiat,/,
which is consequently the shadow cast by the corner, g.
The following method, although more complicated, is of more
universal application : — Draw the vertical projection of the outline
of the body which receives the shadow, as sectioned by the verti-
cal plane, in which the ray of light lies, which passes through the
point whose shadow is sought; draw the same ray of light as
projected in the vertical plane, and its intersection with the projec-
tion of the sectional outline will be the projection of the shadow
of the point.
Thus, in the present instance, as the plane of the ray, g' s',
passes through the apex of the cone, the latter will present a trian-
gular section, the vertical projection of which may be obtained by
squaring over the point, g^ fig. 4, to the base line, a b, fig. 4° ;
then, if a straight line is drawn from the vertical projection of this
point to the apex, s, it will represent the projection of the section
of the cone, and it will be intersected by the luminous ray, g/, in
the point, /, which is the point sought.
If the plane passing through the point does not likevidse pass
through the axis of the cone, the section will be a parabola, which
may be di-awn according to methods already discussed. If the
objeet is a sphere instead of a cone, the section wall be a circle,
whether the plane passes through the centre or not, and the verti-
cal projection will, in all cases, be an ellipse. As a good idea of
the whereabouts of the point sought may always be formed on
inspection, it will generally be sufficient to find one or two points
in the parabola or ellipse, near the supposed position, when a suffi-
cient length of the curve may be drawn to give the intersection of
the luminous ray, as g/.
As the plinth is square, the summit, g, of the curved outline,
corresponding to the shadow of the front edge, g h, is obtained
directly by the intersection in g" of the line, g/, with the extreme
generatrix, a s, the horizontal line, g" g, being drawn through this •
point. Any oth^v point in the curve, as i", is afterwards found
BOOK OF INDUSTRIAL DESIGN,
105
by means of the sectional plane, m n ; g' h, fig. 4, is the shadow of
Ihe edge, g h, in that plane, and it cuts the circle representing the
section of the cone in the same plane in the point, i", which is ob-
viously a point in the outline of the shadow.
SHADOW CAST UPON THE INTERIOR OF A HOLLOW CONE.
318. Fig. 5 represents a plan of a hollow truncated cone, and
tig. 5" is a vertical section through the axis of the object. It is
required to determine the horizontal projection of the shadow cast
upon the internal surface of the cone by the portion of the edge,
a' b c, and the vertical projection of the shadow cast by the sec-
tional edge, D s, and by the small circular portion, a' d', projected
vertically in a d.
It is to be observed, in the first place, that the straight line, d s,
which is a genejatrix of the cone, casts a shadow upon the latter,
in the form of a straight line, for the plane parallel to the ray of
light, and passing through this line, d s, must cut the cone in a
generatrix ; "we therefore draw through the point, d', the ray of
light, d' d', making an angle of 45° with the base line, and from
the centre, s', let fall the perpendicular, s' e', this straight line
representing the horizontal projection of the intersection of the cone
by the plane passing through the line, d' s', and at the same time
parallel to the ray oi ngnt. By squaring over the point, e', to e,
fig. 5°, and joining e s, we have the vertical projection of this line
of intersection, and consequently the shadow cast by the line, D s.
The diagonal ray of light, d d, drawn through the point, d, deter-
mines the limit, d, of the shadow. The horizontal projection of the
extreme points, a' and c, of the curved outline of the shadow, is
also obtained by means of the tangents, s' a' and s' c, drawn from
the point, s, in which the ray of light passing through the apex
intersects the plane of the base of the cone. The determination
of the central or symmetrical point, b', of the same curve, is derived
from the straight line, d b, drawn from the point, D, parallel to the
ray of light, s e", as in the diagonal plane, that is, as at s r" ; the
point, b, in which this straiglit line meets the generatrix directly
opposite to that passing through the point, D, is projected horizon-
tally in the point, b', upon the prolongation of the diiigonal ray of
light, s' s'.
319. The operation for finding any intermediate point in the
curve, is based on principles already explained ; namely, that when
a line or a surface is parallel to a plane, the shadow cast is also
a line or a surface equal and parallel to the first. If, then, we
draw a plane, M n, parallel to the base, d f, of the cone, the
shadow cast by this base upon the plane, m n, will be a circle ; it
will consequently be sufficient to draw through the centre, o, fig.
6°, a ray, o a, which will meet the plane, m n, in a, which must
bo squared over to a', on the horizontal projection of the same
ray. Next, with the point, a', for a centre, and with a radius
(Hjual to D o, describe a circle, h' i j; this will represent the entire
shadow that would bo cast by the base, D f, of the cone upon the
plane, m n ; this plane, however, cuts the cone in the circle, of
(vhich M n is the diameter and vertical projection, whilst h' m' j n'
« (he horizontal projection ; this circle is cut by the former in the
points, ii' and j, wliicli are consequently two points in the outlines
of the shadow in fig. 5, and the one of these which is nwn in llic
vertical |)rojection is squared over to H, upon the line, ivi n.
apflications.
320. In this plate, as well as in Plate XXVI., we have given
shaded and finished representations of several objects, which serve
as applications of the several principles we have just pointed out,
whether referring to shadows proper, or cast, or to graduated shad-
ing. Thus, fig. A represents the interior of a steam-engine c\'linder
with piston and rod. In this example, regard has been had to the
general principle, that shadows are the stronger the brighter the
surfaces on which they fall would be, if illumined — that is, when
such surfaces are perpendicular to the rays of light, any shadow
cast upon them will be most intense ; the shade is consequently
made deepest about the generatrix, corresponding to g ft, in fig. 1°,
and situate in the vertical plane of the rays of light passing through
the axis of the cylinder : to the right and left of this line, the shade
is softened oft'.
321. In the graduation of the shade, regard has also been had
to the eflfects of the reflected light, which prevents a surface in the
shade from being quite black. In a hollow cylinder, for the por-
tion in the shade, it is the generatrix, f f', fig. r, which should
receive the shade of least intensity, as it receives the reflected
rays of light more directly. It will be recollected that the point,
f', is obtained by means of the radius, t f', perpendicular to the
ray of light.
Fig. [B represents a portion of a common moulding, and shows
how the distinction made between the shadow proper, and the cast
shadow, tends to bring out and show the form of the object.
Fio-. © is an architectural fragment from the Doric order, given
as an application of shadows cast upon cones, as well as those cast
by cones upon a vertical plane.
This example also shows how necessary it is, in producing an
effective representation, to make a diflference in the intensity of
shadows cast upon planes parallel to the plane of projection, and at
different distances from the eye ; and also to give gradations to
such shadows when cast upon rounded surfaces.
Fig. [B) is a combination of a cylinder with a couple of cones.
with their apices in opposite directions, showing how differently the
eflfects of light and shade have to be rendered upon each.
There is less shadow upon the upper cone than uj)on the cylin-
der, whilst there is more upon the lower cone ; the reasons of those
differences have already been explained in reference to figs. 3'
and 4°.
Fig. H represents an invc^rtcd and truncated oono, showing the
manner of shading the same, and the form of the shadow cast by
t!lio square tablet above ; and tig. P is a view of a hollow cone, sec-
tioned across the axis, presenting a further variety of combinations
TUSCAN ORDER.
Platk XXIX.
SHADOW OF THE TOKUS.
322. In geometry, the torus is a solid, gonoratod by a circle, ro.
volving about an axis, continuing constantly in the plane ol iJiia
axis, ill such a iiiamur, that all sections made by phmes [ViUs-sinji
thidiinh the axis are equal ciivles, and all sections by pianos perjH>n-
iliiiilai- In the axis v.ill also be cin-les, but of variable diiunetciv
106
THE PRACTICAL DRAUGHTSMAN'S
Wc have seen, that in architecture, the torus is one of the essen-
tial parts of the base, and of tlie capital of the column, of each
order. It nill, therefore, be useful to give the methods of deter-
mining the shadows upon it, or cast b}' it, in the quickest and most
accurate manner.
Figs. 1 and 1° represent the two projections of a torus, a, sup-
posed to be generated by the semicircle, af c, revolving about the
vertical axis, o p ; namely, that of the column.
323. We propose to determine the shadow proper of this torus,
(ir the line of separation of light and shade upon its external sur-
face. It will be convenient, in the first place, to seek the principal
points, which, for the most part, present little difficulty. Thus, by
drawing, parallel to the ray of light, r o', a couple of tangents to
the semicircles, afc^ which limit the contour of the torus in the
vertical projection, we at once obtain the two extreme points, 6, rf,
of the cuned line of separation. These points are more exactly
defined by letting fall perpendiculars from the centres, o, o', of the
semicircles, upon tlie tangents. Then, by drawing through the
point, h, the horizontal, i e, the middle point, e, of the cm-ve will be
obtained upon the vertical line, o p.
To obtain the curve in the horizontal projection, square over the
points, b, e, d, of fig. T, to b', e', d', fig. 1, which wUl lie in a circle,
having b e ox o d for radius. An additional point, g', is obtained
by drawing the diagonal, o' g', perpendicular to the ray, r' o' ; this
radius cuts the outer circumference,/' ft/', of the torus. La the point,
g'. This circxmiference,/' h /', is projected vertically in the hori-
zontal line,//, passing through the centres, o, o , of the semicircles,
and the point, g', is squared over to g, in fig. 1°.
To find the point, ?', which seems to be the lowest in the curve,
and which is situated in the vertical plane passing through the
luminous ray, i o , of the horizontal projection, fig. 1 — 2, we pro-
ceed as follows : — Suppose the vertical plane, i o', to be turned
about the axis, o p, so as to coincide with the plane of projection,
when the section of the torus by the plane, i' o', being obviously a
semicircle, wiU coincide with the semicircle, afc, draw a tangent,
k i'', to the last, parallel to the ray of light, as in the vertical plane,
i o' — that is, at an angle of 35° 16', as has been already explained
— the point of contact, ■?, is the one sought. But it has to be
transferred to the original position of the vertical plane ; and for
this purpose it is squared over to ^^ in the horizontal projection.
Then o' i' is made equal to o' i^, and the point, i', again squared
over to i, in the horizontal, t" i, drawn through f".
It is generally sufficient to find five principal points, as b, i, e. g,
and d, in the curved line of separation of light and shade ; but if,
because of the large scale of the drawing, it is wished to obtain
intermediate points, this may be done by drawing planes passing
through the axis ; such, for instance, as o' b', which cuts the torus
in a circle of the same radius as the generating circle. We then
proceed to find the point of contact of the ray of light, according
to the method indicated in figs. 5 and 5°, Plate XXVT. ; that is
to say, we seek the projection of the luminous ray upon this plane,
o' b'. For this purpose, we let fall up"n this plane a perpendicu-
lar, r' i, from any point taken upon the luminous ray, r' o', and
we obtain a face view of this ray, as projected upon the plane, o' b',
by supposing tne latter to be turned about the axis, o', until it
coincides with / o', r' then coinciding with r\ Then, as the
height, r' r, of the point, r', above the horizontal plane, is equal to
that of the point, r, the line joining r o will be the face \iew of
the projection of the ray in the plane, o b'. In turning round tlie
plane, o' b', the section of the torus will become coincident with the
semicircle, afc, as in a previous operation. K, therefore, we draw
a straight Ime, m n, tangential to this semicircle, and parallel to the
ray, r o', the point of contact, n, will be the point of separation of
light and shade, as in the plane, o' b'. Finally, we square n over
to n', fig. 1 ; make o' n^ equal to o' n', by describing an arc with
the centre, o', and radius, o' n', and cutting the line, o' b', in n*.
This point, n\ we again square over to n^, upon the horizontal, n ri',
in fig. 1°, and rf is the point sought. Or we might have drawn the
vertical projection of the section of the torus by the plane o' b',
which would have been an ellipse, similar to that in fig. 5°, Plate
XXVI. ; and we might have proceeded, as shown in reference to
that figure, the result being the same in both cases.
If the circular arc be prolonged to beyond the radius, o' g', and
upon it, g I be made equal to g" n\ another point, I', will be ob-
tained, symmetrical with w', with reference to the radius, o' g ,
which is at an angle of 45° to the base line, and perpendicular to
the luminous ray. This point, T, is to be squared over to I, in the
vertical projection, and upon the horizontal, p I, drawn at a distance,
q p, above the centre line, / /, equal to the distance, q s, of the
horizontal passing through the point, n\ below it.
324. When the shadow proper of a torus is known, it is very
easy to determine the shadow which it will cast upon the horizontal
plane — the plinth or pedestal below it, for instance — by drawing,
through any points in the line of separation of light and shade, a
number of lines parallel to the luminous ray, and then finding the
points at which these lines intersect the horizontal plane. Thus,
in figs. 2 and 2°, a portion of the torus, a, casts a shadow upon the
horizontal plane, b c, the outline of which is a curve ; but the por-
tion, a' b' c', of this curve is all that is visible.
Any point, b, b', in this curve is determined by the meeting of the
ray of light drawn from the point, 1 1', with the plane, b c.
The half of the line of separation of light and shade upon the
posterior portion of the torus, fig. 1°, which is not seen in the front
elevation, is similar to the anterior half; it is indicated in dotted
lines, the portion, o b, being simDar to the front portion, e d, whilst
d\s similar to e b.
325. When the torus is surmounted by a cylindrical fillet, the
line of separation of light and shade upon the latter will cast a
shadow upon the surface of the torus. Thus, in figs. 2 and 2°, this
will be the case with the fillet, d, the line of separation of light and
shade of which is. fh. This Ime, being vertical, casts a shadow,
which is a straight line,/' ?', parallel to the luminous ray, and de-
termined bv drawing through the point,/, a luminous ray,/ -u meet-
ing the horizontal plane, a a, in i, which point, i, is squared over to
the horizontal projection. It remains to determine the shadow cast
by the circular portion,/'/', which is in the shade: this may be
done according to the general method explained in reference to
figs. 3, 4, and 5, of Plate XXVT., and which ^ve shall have occasion
to repeat on figs. 3 and 3" of the present plate. This method is
also applicable for the determmation of the shadow, nj, n'j', cast
by the cylinder or shaft, e, upon the annular gorge, which unite^s
this cvlinder with the fillet, d.
BOOK OF INDUSTRIAL DESIGN.
107
SHADOW CAST BY A STRAIGHT LINE UPON A TORUS OR
QUARTER-ROUND.
326. Fig. 3 represents the horizontal projection, as seen from
below, of a fragment of a Tuscan capital, of which fig. 3" is the
vertical projection, the object of these figures being to show the
form of the shadow cast by the larmier, f, which is a square prism,
ujxin the quarter-round, a, which is annular.
We yet again recall the general principle, that when a straight
line is parallel to a plane, its shadow upon this plane is a straight
line parallel to itself For the rest, it will be sufficient to compare
the operations indicated with those of figs. 3 and 3", Plate XXVIII.,
to see that they are precisely the same : thus, on the one hand, we
have the diagonal, g /, for the shadow cast upon the quarter-round,
where it is limited by the curve, b e I, the line of separation of light
and shade upon this ; and, on the other hand, we have the curve, .
i" g i', likewise limited by the same curve, for the shadow cast by
the edge, g h, of the larmier upon the quarter-round.
Figs. 3 and 3" complete what refers to the shadow of the capital
of a column ; they show the operations necessary to determine the
shadow cast by the line of separation of light and shade of the
quarter-round upon a cylinder, as well as that east on the same
jylinder by a portion of the larmier. The operation, in fact, sunply
consists in drawing the luminous rays through various points, i" e,
in a portion of the line of separation of light and shade upon the
quarter-round, finding their intersection with the cylindrical surface
of the shaft, e, by "means of tlie horizontal projection. There is
no peculiarity or difficulty in this procedure, and the whole being
fully indicated upon the diagrams, we need not pause to detail it
further.
To render the diagi-ams just discussed more generally applicable
»nd intelligible, we have not given to the different paits the precise
proportions prescribed by this or that architectural order ; such pro-
portions, however, will be found in fig. /^, which represents the
model fully shaded and finished, being the entablature and eolunm
©f the Tuscan order. A double object is intended to be gained by
this beautiful example of drawing ; namely, to show the application
of the principles laid down regarding shadows, and the distinctness
and niceties to be observed in the various intensities of the washes,
and in the general shading.
SHADOWS OF SURFACES OF REVOLUTION.
3'27. It will be recollected, that a solid or surface of revolution
is that which may be said to bo generated by a straight or curved
line, caused to turn about a given fixed axis, and maintaining a uni-
form distance therefrom ; thus, the cylinder, the cone, the sphere,
the torus, are all sui'faces of revolution ; so, also, is the surfkco
generated by the curve, a b c, revolving about the axis, a b, figs. 4
and 4". It follows, from the above definition, that every section
iiiiido ])crpendicularly to the axis will bo a circle, and all such sec-
tions will be parallel. Every section made l)y a piano passing
through the axis will give an oulliiic ((lual to the generating curve,
and which may be termed a mcridinii.
3'28. The shadow of a surface of rcvohilidii lUMy Ixi (lelrriiiincd
in two differont ways : by drawing sectional plnncs |)erpi luliciihn-
to the axis, and then considering the sections made by tluse
planes aa bases of so many right cones ; or by imagining a sciirs
of planes passing through the axis, and then projecting the ray of
light upon these planes, so as to draw lines tangential to the dif-
ferent parts of the outline, and parallel to the projections of the
ray of light, the points of contact of which will be points in the
line of separation of fight and shade sought. This latter method
having been applied in the preceding figs. 1 and l", Plate XXIX.,
and figs. 3 and 4, Plate XXVIII., we deem it more useful, in the
present instance, to explain the operations called for in the first
method.
Take, then, any horizontal plane, b d, figs. 4 and 4% cutting the
surface of revolution in a circle, the radius of which is b e, and the
horizontal projection, b' e' d', through the points, b and d, draw a
couple of tangents to the generating curve which fonns the outline
of the surface of revolution. These tangents will cut each other
in the point, s, upon the axis, tiiis point being the apex of an im-
aginary cone, s b d; through this apex draw a luminous ray, s /
and a' b', meeting the horizontal plane of the section, b df,'mf,f';
from this latter point, the horizontal projection, draw two straight
lines,/' g' and/' i', tangents to the circle, b' c' d' ; then the points
of contact, g' and i', will be the two points of the line of separa-
tion of light and shade intersected by the plane, b d, and they are
therefore squared over to the vertical projection, fig. 4°, i, only
being there visible.
It is in a similar manner that the points, h and _;', are deter-
mined, these points being situated in planes, c d and e f, parallel
to the first. It is to be observed, however, that, in these two last
cases, the imaginary cones will be inverted, and the luminous raj
must consequently be drawn to the left instead of to the right, as
has already been explained in reference to figs. 3" and 4°, Plate
XXVIII.
329. Wlien the tangents to the generating curve are vertical, as
is the case with the sectional planes, m n and a I, the points, m and
n, of the line of separation of light and shade, are determined by
lines, inclined at an angle of 45°, and tangential to the circular sec-
tions in the horizontal projection, because these circular sections are
the bases of imaginnry cylinders and not cones.
When a sufficient number of points have been obtained in this
manner, as in fig. 4°, a curved line is drawn through them all, which
will give the visible portion, ?/( / 7i hj e, of the line of sejiaration
of light and shade upon the surface of revtilution. This method is
o-eneral, and may be applied to surfaces of rcxohition of any outline
whatever.
As it is well to determine directly the lowest point, /;, of this and
similar curves, it may be done in the same numner as for the torus-
figs. 1 and r, namely, by drawing the ray of light, k b, at the ssmio
inclination to the base line, as it is in the diagonal aiul vertical
plane, and then drawing parallel to it a tangcTit to the outline of
the surface of revolution, llic projccliou I'oi- the niomoiil being
supposed to be in ii plane parallel to the ray of light, k' a'; the
(lislance of the jxiint of contact, A. from the axis, being thou mea-
sured upon the horizontal projection, r' a', of the huninou8 my
lijves the jioiiil, /.'. which is finally squared over to A', in the hori-
zontal line in the virlicMl projection pa.ssing through the simie poinl
ofconlJicl.
A |)orlioM of llii-i cnrvi', n.uinly, the lower part, i: A /, Ci».Hl!« «
.shadow upon llie cylindrical lillcl.co; lo .lelirniiue this shadow
108
THE PRACTICAL DRAUGHTSMAN'S
it will, in the first place, be necessary to delineate the horizcmtal
projection of the curve, e kj, and then to diaw luminous rays
through one or two points in the latter, to meet the circle, c' o',
the horizontal projection of the fillet. The points in which the
luminous rays intersect the circle, are then to be squared over to
the vertical projection of the same rays, whence is derived the
curve, c p q. The various operation lines are not indicated on the
figures, to avoid confusion, but the proceeding will be easily com-
prehended.
330. Fig. 4° represents the vertical projection of a baluster, such
as is often seen in balconies of stone or marble, and sometimes
also in macliinery, serving as an isolated standard, or as a portion
of the framing. Below the fillet, c o, is an annular gorge, upon the
surface of which the base of the fillet casts a shadow. It is easy
to see that this shadow is obtained in precisely the same manner
as those occurring in figs. 3 and 3°, Plate XXVIIL, as well as in
subsequent diagrams.
Figs. © and © represent the shaded models of two descriplions
of baluster, consisting of surfaces of revolution. We recommend
the student to draw them upon a large scale, and to determine the
outline of the shadows in rigorous accordance with the principles
which we have laid down. Such balusters are generally made of
stone, and are susceptible of various sizes and proportions. We
have, however, supposed them to be drawn to a scale of one-tenth
their actual size.
Many forms and combinations, of which we have said nothing,
will be met with in actual practice ; but our labours would be
interminable were we to give them all. Our exemplifications involve
all the principles that are needed, and each case will suggest the
modification of operations applicable to it.
RULES AND PRACTICAL DATA.
PUMPS.
331. There are three kinds of pumps.
I. Lifting pumps, in which the piston or bucket lifts the water,
first drawing it up by suction. We engrave one of this kind in
Plate XXXVn.
II. Forcing pumps, in which the piston presses or forces the
water to any distance. The feed pumps of steam-engines are of
this class, and one is represented in Plate XXXIX.
III. Lifting and forcing pumps, in which both the above actions
are combined.
HYDROSTATIC PRINCIPLES.
332. Whatever be the height at which a pump delivers its water
• — whatever be the calibre or inclination of the suction or delivery
pipe — the piston has always to support a weight equal to a column
of water, the base of wWch is equal to the area of the piston, and
the height is equal to the difference of level of the water below,
from which the pump draws its supply, and the point of delivery
above.
Thus, putting H to represent the ditference in the level, D for
62-5
X H X 1-08;
the diameter of the piston, and P for the weight or pressure on the
piston —
p _rtD^
~ 4
To express this pressure in pounds, it must be multiplied by
62-5, that being the weight in pounds of a cubic foot of water ; the
formula then becomes —
rtD'H
P = 62-5 lbs.,
4 '
the measurement being expressed in feet.
333. Independently of this load, which corresponds to the useful
effect of the machine, the power employed in elevating the piston
has other passive resistances to overcome, namely —
1st. The friction of the piston against the sides of the pump.
2d. The fi-iction of the water itself in the pipes.
3d. The retardation of the water in its passage to the pump by
the suction valve.
4th. The weight of this valve.
These resistances can only be determined approximately. Still
it follows, from the experiments of M. d'Aubisson, that the load to
be overcome in raising the piston is equal to
rtD^
4
or, more simply,
52-5 D' H.
It is sufficient to add to this the weight of the piston and rod.
The power exerted in depressing the piston, being assisted by
the weight of itself and the rod, is always less than that required
to raise it.
334. In ordinary pumps, the volume of water delivered for each
stroke of the piston, instead of being given by the formula,
—^ X I, or -785 D- 1,
where I is the length of stroke, is determined by an expression
which varies between
•6 D^ I and -7 D' I.
The velocity of the piston generally ranges between a minimum of
60 feet and a maximum of 80 feet per minute. The diameter of
the suction and discharge pipes is generally equal to f or f of that
of the body of the pump.
It may be remarked, that the height to which liquids rise in vacuo,
by the pressure of the atmosphere, is in the inverse ratio of their
specific gravities. Thus this pressure, which is equal to 15 lbs. to
the square inch, makes water rise to 33 feet, whilst mercmy only
rises to 30 inches, its specific gravity being 13-59 times that of
water. If the atmosphere presses on a liquid lighter than water, it
will cause it to rise higher in vacuo than 33 feet, in proportion to
the difference of the specific gravity. In practice, more than 29 or
30 feet cannot be calculated on for the lift of the pmnp, because
of the difficulty of obtaining a perfect vacuum.
FORCING PUMPS.
335. What has been here said of lifting pumps, applies as well
to forcing pumps. The resistance, however, to be overcome, is
somewhat greater in the latter case — for instance, at the moment
of opening the discharge valve ; and in general this occm-s v«"itli
BOOK OF mDUSTRIAL DESIGN.
109
all valves having a great body of water above them, and with their
upper surface greater than the area of the orifice above.
LIFTING AND FORCING PUMPS.
336. A pump of this description ordinarily consists of a cylinder
with a short suction pipe, a discharge pipe, a solid piston, termed a
plunger, and suction and discharge valves.
Two such pumps are frequently coupled together, in which case
n single suction and discharge pipe serves for both.
337. The power necessary to work one or more pumps is ex-
pressed by 52-3 D^ Hv; or, taking into account the force necessary
to work the piston by itself, 55-7 D" H d ; v signifying the velocity
in feet per minute.
This velocity is obviously obtained by multiplying the number
of strokes per minute by the length of stroke ; thus —
V = 2n I,
n being the number of back-and-forvvard movements per minute j
consequently, the power required is equal to
55-7 D' H X 2nl = 111-4 D= H n Z;
this product representing pounds raised one foot high per minute,
the measurements being in feet.
With these premises, we can solve such problems as the fol-
lowing : —
First : What force, F, is required to work a pump, having a
piston 6 inches in diameter, a stroke of 18 inches, and a velocity of
15 double-strokes per minute ; the whole height between the well
and the point of delivery being 70 feet ?
The velocity v=2nl — S0 x 1^ = 45 feet. Then F = 53-7
D^ X H X i; = 53-7 X -23 x 74 x 43 = 46,997 lbs. raised one
foot high per minute.
To express this in horses power, we must simply divide it by
33,000 ; therefore,
46,997
F = „„ „ = li horses power, nearly.
Second : What quantity will the same pump raise in ten hours 1
Assuming, according to the formula (333), the effective volume,
V = -6 DW, or V = -6 X -23 x 1-3 = -223 cubic feet per
stroke ;
and the volume per minute,
•225 X 15 = 3-375 cubic feet;
and per hour,
3-375 X 60 = 202-5 cubic feet.
The quantity of water raised in ten hours will consequently be
202-5 X 10 = 2,025 cubic feet.
Third: What diameter should be given to the piston of a pump
which raises 202-5 cubic feet of water per hour, the velocity being
45 feet per miimte, the length of stroke 18 inches, and the height
to which the water is raised 75 feet ?
The formula above, relative to the effective discharge per stroke,
V = -6 D'^ X /,
by transposition, becomes
^ -6 X /.
Now, the volume, 202-5 cubic feet, discharged per hour, is, per
minute, 202-5
This last again reduces itself to
2 X 1-5 X 3-375
43
= -225 cubic feet per stroke;
consequently,
Wlience,
D= =
•223
•6 X /.
6:
= 3-376 ciHc feet.
X. / '225
^=V -6x 1-5 = 6 inches.
THE HYDKOSTATIC PRESS.
338. This powerful machine is an application of the lifting and
forcing pump. It consists of a bulky piston, or plunger, termed a
ram, working in a cylinder to correspond, and communicating, by a
pipe of small bore, with a small but very strong forcing pump.
To the top of the large piston is fixed a table or platform, which
compresses or crushes what is submitted to the action of the
machine.
The pressure exerted upon the water by the smaller piston, is,
by means of the fluid contained in the pipe, transmitted to the base
of the ram ; and as, according to the well-known hydrostatic law,
the pressure is equal on all points, the total force acting on each
piston vidll be in proportion to their area ; so that if, for example,
the diameters of the pistons are to each other as 1 to 5, the prc-s-
sure on the larger one, the ram, will be 25 times as great as that
exerted by the pump-piston. Suppose a man can apply a force
equal to 60 lbs. to the end of a lever 3 feet long, and that the poin*
of connection with the piston-rod is only 1| inch from the fuli-rum,
the leverage of the power will be 24 times as great as that of the
resistance, and the pressure upon the ram will consequently be
24 X 25 X 60 = 36,000 lbs., an effort equal to that of 600 men
acting at once.
In the hydrostatic press, we have, consequently, to consider two
mechanical advantages — that of the simple machine, tlie lever, and
that of the ram : these advantiiges are, however, necessarily com-
pensated for by the diminution in the velocity of the nun.
On these principles, enormously powerful presses and lifting-
machines have been constructed. The one capable of lifting 18,000
tons, at the Menai Tubular Bridge, is an unparalleled example.
HYDROSTATICAL CALCULATIONS AND DATA — DISCHARGE OF
WATER THROUGH DIFFERENT ORIFICES.
339. The discharge of a volume of water, in a given fiine. vanc.-^
according to Ihe velocity of the water, and dcpeiuls upon the area
imd form of the discharge orifice.
Surface Velocity. — The velocity of \\-ator at (ho surface of a wator-
coui'so or river, of which it is wisiiod to asi-ortain the tliscliargc. is
obtained by means of a lloal, whii'li is llnowii into ilu' p:irl wlicro
the current is strongest. As the w iml, if there is .-iny. alVccts the
ri'sult very considerably, the lloni must project above the surface tus
little as possible. A distance of as great a length as coiuenieni is
measured on the part of the stream where the current is most regu-
lar, and the tinui occupied by the llont in passing that distance is
noted by a secoiuls watch. The space passeii tiirougli is then diviiU'd
by the time expressed in seconils, nud the quotient will be the sur
face velocity per second.
no
THE PRACTICAL DRAUGHTSMAN'S
It is usual to try several floats in different parts of the current.
ExampZe.— Suppose the space passed through by each float is
150 feet in 35 seconds, what is the surface velocity 1
150
V =
35
• = 4-28 feet per second.
If the velocity is not uniform throughout the len^fth of the canal,
the velocity at any point may be obtsuned by means of a small
paddle-wheel, the floats of which just dip into the water. The
number of revolutions per minute of this instrument boing multi-
plied by its mean circumference— that is, the circumference corres-
ponding to the centre of the immerged part of the float — the pro-
duct expresses the velocity per minute ; and, by dividing by 60, the
surface velocity per second is obtained.
Example.— SnpTpose that the wheel makes 120 revolutions pel
minute, and that the mean circumference is equal to H foot, what
i'5 the surface velocity of the current 1
120 X 1-5 „ ^ ^ ,
. = 3 feet per second.
60 ^
340. Mean Velocity. — The velocity above obtained is only that
at the surface ; now, the mean velocity, V, of the whole body of
water, which is what is necessary to know for the gauging of the
river or canal, is deduced fiom the first, by multiplying it by a
coefficient, which varies in the following proportions : —
For a surface velo- j
city equal to )
Tlie ratio of V to j
V is i
•5 ft.
15 ft.
•78
3 ft.
■81
5 ft.
•83
6-5 ft.
•85
8 ft.
10 ft.
•87
ll-5ft
13 ft.
Example. — ^What is the mean velocity of a current of which the
surface velocity is 5 feet per second ?
It is equal to -83 X 5 = 4^15 feet.
The mean velocity of water in an open water-course or river of
aniform cross-section is determined by the following formula : —
/a H
W_x -22
V p T.
V = 56-86 x \ / — X -236.
P L
This formula requires the obtainment of the exact level of the
surface of the water throughout a certain length, L, the greater the
oetter ; the cross-sectional area, A ; the form of the immerged
perimeter or profile of the bed ; and the height of the fall, H, cor-
responding to the length, L.
Example. — ^What is the mean velocity of the water in a water-
course of uniform rectangular cross-section, having a width of 35
feet, a depth of 12 feet, and with a fall of -8 feet in a distance of
1400 feet?
The cross-sectional area. A,
= 36 X 12 = 420 square feet.
The immerged profile. P,
= 35 + (2 X12) = 59 feet.
Then,
V = 56-86 X
V
420 sq. ft. -8
59
1400
•236 = 3-39
feet per second.
Thus, according to this formula, it is necessary to extract the
square root of the product of the quantities placed under the radi-
••al sign y ; next to multiply this root by the co-efBcient 56-86;
and, finally, to subtract from the product "236 feet. When the
measurements are in metres this last item is -072.
COMPAKISON OF FRENCH AND ENGLISH MEASURES OF CAPACITY.
The French litre is equal to a cubic metre, and therefore tc
10-76 cubic feet, or -220 gallon. The gallon is equal to 4-543
litres or cubic metres, and the cubic foot to -9929 litres or cubic
metres.
THE GAUGING OF A WATER-COURSE OF UNIFORM SECTION
AND FALL.
341. When we know the mean velocity of a water-course of
regular section and uniform fall, the discharge per second can be
obtained by the following formula : — D = A x V , in which D
signifies the discharge per second; A, the cross-sectional area
and V, the mean velocity.
Example. — What is the discharge of a water-course, the cross-
section of which is 4-2 square metres, and the mean velocity 1-065
metres 1
D -- 4-2 X r065 = 4-473 cubic metres, or 4-473 litres per second
VELOCITY AT THE BOTTOM OF WATER-COURSES.
342. The velocity of water at the bottom of water-courses ls>
still less than the mean velocity.
Putting V to represent the surface velocity, \ tne mean velo-
city, and V the ground velocity, the relation of the three will be
expressed by V = 2 V — V. That is to say, the velocity at the
bottom of a canal is equal to twice the mean velocity minus the
surface velocity.
Example. — The surface velocity of a water-way is found to be
2 metres, and the mean velocity calculated to be 1-55 metres, what
is the ground velocity 1
V = 2 X 1-55 — 2 = 1-10 metres.
Too gi-eat a velocity at the bottom of a water-course tends to
loosen and carry away the bed, undermining the sides and causing
a great deal of damage ; too small a velocity, on the other hand, by
allowing the matter suspended in the water to settle, is a cause of
obstruction.
The following table shows the limit of velocity accormng to the
nature of the bed, which cannot be exceeded without danger : —
Nature of the Bed.
Soft brown earth,
Soft claj'
Sand,
Gravel,
Flint stones,
Shingle,
Agglomerated stones, soft schist,
Rock fragments,
Solid rock,
Limit of the Velocity per second.
Metres.
-076
•152
-305
-609
-614
1-220
1-52S
1-830
3-050
Feet.
-25
■49
1-00
2-00
2^02
4-00
5-00
6-00
10-00
343. Prony's measure. — The produce of any source ma)-' als:
be measured by damming up the entire width of the stream with
thin planks pierced with holes of 20 mUUmetres in diameter, dis-
posed in a horizontal line. These holes are at first covered, and
are opened in succession, until the level of the water within them
BOOK OF INDUSTRIAL DESIGN.
in
is maintained above their centres ; so that when this is effected,
the discharge is calculated from the number of orifices which
require to be open.
The quantity of water discharged by each orifice of -02 m. in
diameter, in a board -017 m. thick, and under a column -03 m.
above the centre, is 20 cubic metres in 24 hours.
Another method of gauging a stream of water, consists in setting
up an under or overshot sluice-gate at a similar dam, the discharge
being calculated accordiug to the following rules in reference to
this subject : —
CALCITLATION OF THE DISCHARGE OF WATER THROUGH
RECTANGULAR ORIFICES OF NARROW EDGES.
344. As it is of importance, in a majority of circumstances, to bo
able to calculate the discharge of water by sluice-gates, or by Iho
vertical discharge-gates of hydraulic motors, so as to know the
volume, and, consequently, the value of a stream of water, we snail
commence by giving a table, which enables us to determine this
discharge in a very simple manner, and places these operations
within the capacity even of labourers and working mechanics.
TABLE OF THE DISCHARGES OF WATER THROUGH AN ORIFICE ONE METRE IN WIDTH.
Height
of the
Vol
ame disch
argeU in litres per second, corresponding
to the heights : —
orifices
in centi-
metres.
•2 m.
■3 m.
•4 m.
■5 m.
•6 m
•7 m. 1
•8 m.
1^0 m.
r2m.
r4m.
rem.
rsm.
2 m.
2-5 m.
3'0 m.
3^5 m.
4-0 m.
4
50
61
71
79
86
93
99
110
121
130
138
146
154
172
188
201
215
6
62
76
88
98
107
116
124
138
161
162
173
182
191
214
255
251
268
6
75
91
107
117
128
139
148
165
181
194
207
218
229
257
281
301
321
7
86
106
122
136
148
161
172
192
210
226
241
255
267
299
327
350
374
8
98
120
139
155
170
184
196
219
240
268
275
290
305
341
374
400
427
9
109
135
156
174
191
208
220
246
267
289
309
326
343
382
420
450
481
10
1-22
149
173
193
212
228
246
272
298
321
342
362
380
424
466
500
533
11
133
164
189
212
230
249
267
299
327
353
376
398
418
466
511
650
587
12
145
178
206
230
251
272
291
326
356
384
409
434
455
507
557
699
640
13
157
192
222
249
272
294
314
352
386
416
443
469
492
649
602
647
693
14
168
206
238
267
292
316
338
379
414
446
476
504
630
590
648
697
745
15
179
220
255
285
312
338
361
405
443
477
609
539
666
631
693
747
799
16
190
234
271
304
330
360
385
432
472
509
642
574
603
673
739
797
852
17
201
248
287
322
350
382
414
456
601
540
676
610
638
715
784
847
905
18
213
262
304
340
370
403
432
484
529
671
608
644
677
757
830
896
958
19
223
276
324
358
392
425
454
510
558
601
641
680
715
799
876
946
1011
20
235
291
337
377
414
447
■ 486
636
686
627
675
715
763
841
922
996
1065
21
247
305
354
396
431
470
512
563
615
664
708
751
790
884
968
1046
1118
22
259
320
370
417
451
492
638
590
645
696
742
787
828
926
1014
1096
1171
23
271
334
388
434
472
615
550
616
674
726
776
823
865
968
1060
1146
1224
24
282
348
404
452
492
537
674
643
703
758
809
869
903
1010
1106
1195
1278
25
294
363
420
471
516
659
598
670
733
790
843
895
941
1062
1152
1245
1331
26
306
377
437
490
538
681
626
697
762
822
877
930
978
1094
1198
1295
1384
27
318
392
454
509
569
604
646
724
791
853
911
966
1016
1136
1245
1346
1437
28
329
406
471
527
573
626
679
740
820
886
944
1001
1054
1172
1291
1395
1491
29
340
421
487
546
602
649
693
777
850
916
978
1037
1092
1220
1337
1444
1544
30
353
434
504
564
624
670
718
804
880
948
1010
1073
1129
'1262
1385
1494
1597
31
364
449
521
683
635
• 694
741
831
909
980
1046
1109
1167
1305
1 129
1544
16.50
32
376
463
538
602
656
716
765
857
939
1011
1079
1144
1206
1366
1475
1594
1703
33
388
477
555
622
676
737
789
884
969
1043
1113
1180
1212
1389
1521
1644
1756
34
400
491
572
640
696
759
813
911
998
1074
1147
1216
1279
1431
1668
1693
1810
35
415
507
688
659
717
782
837
938
1027
1103
1180
1252
1317
1473
1614
1743
1 863
36
424
520
605
677
737
804
861
965
1057
1138
1214
1288
13.56
1516
1660
1793
1916
37
436
534
622
696
758
826
886
981
1086
1169
1248
1324
1392
1.567
1706
1843
1969
38
450
549
638
715
778
849
909
1018
1115
1201
1283
1359
1430
1599
17.52
1893
2023
39
462
664
663
734
798
872
933
1045
1145
1232
1315
1395
1 168
Kill
1798
1943
2076
40
484
577
671
753
819
894
957
1070
1174
1266
1351
1431
1.506
1683
1814
1992
2129
41
591
688
772
840
915
981
1097
1203
1298
1384
1467
1543
1725
1890
2012
21S2
42
"
606
705
790
860
936
1005
1124
1233
1329
1419
1503
1581
17(i8
1936
2092
2236
43
5*
620
722
809
881
961
1028
1151
1262
1361
1 163
1538
1618
1809
1982
2142
2289
44
"
635
737
828
901
983
1053
1171
1291
1393
1486
1671
1656
1851
2029
2 1 92
2313
45
"
619
754
847
920
1005
107(>
120 1
1321
1 124
1520
1609
1694
1894
2075
22 1 1
2391
4f)
''
()63
771
866
911
1 028
1 10(1
1231
1350
1 156
1554
1()36
1731
1936
2121
2291
2449
47
15
677
787
885
961
1050
1121
12,57
1380
1 188
1588
1681
1769
1978
2167
234 1
2504
48
»1
691
804
903
982
1072
11 18
1281
1409
1519
1622
1716
1807
2020
22 1 3
2391
2559
49
11
706
820
922
1002
1095
1172
1311
1438
1551
1656
1753
1845
2062
2339
24 10
26 1 4
50
11
719
836
940
1023
1115
1194
1337
1468
1583
1690
1789
1882
2104
239r)
2490
2669
112
THE PRACTICAL DRAUGHTSMAN'S
This table has been calculated by means of the follo-vving for-
mula : —
T) = wh X ^ 2gH X 1000;
in \vhich
D represents the volume of water discharged in litres per second ;
w, the width of the orifice in metres;
/(, the height of the orifice ;
H, the column, or the height of the pressure, in metres, measured
from the centre of the orifice to the upper level of the re-
servoir ;
g, signifies the action of gravity, being equal to 9'81 metres;
r, = V 2gli, the velocity due to the height H (see 258) ; and,
finally,
m is a coefficient, which varies in practice according to the heights,
h and H, from -59 to -66, supposing the contraction of the
orifice to be complete ; that is to say, it occurs on all four
sides of the orifice.
In the first column of the table we give the heights of the orifices
in centimetres, and in the foUowng columns the results of the dis-
charge effected, in litres per second, for various heights of the
column of pressure, from -20 to 4 metres.
By means of this table we can now determine, by a very simple
operation, the volume of water discharged through a vertical flood-
gate, or through a rectangular orifice, of which the edges are nar-
row ; the level of the reservoir being above the top of the orifice,
and the contraction complete. We have, in fact, sunply to find the
number in the table corresponding to the given height of the orifice,
and to the column of water acting at its centre, and then multiply this
number by the given width.
Example. — ^What is the volume of water discharged by the oi-ifice
of a vertical water-gate, 1'5 metres wide, the height of the orifice
being -25 m., and the height of the column, from the centre of the
orifice to the upper level in the reservoii-, 2-5 m., and the contrac-
tion being complete ?
In the table, on a line \vith the height, 25 centimetres, and in
the column corresponding to 2'5 m., wOl be found the number
1052.
We have, therefore,
1"5 X 1052 = 1578 litres for the actual discharge per second.
It will be equally easy to estimate very approximately the dis-
charge of water, corresponding to data which do not happen to be
in the table.
First Example. — What is the volume of water discharged by a
vertical sluice-gate, -8 m. in width, the height of the orifice being
16 centim., and the column upon the centre 2*75 m. ?
This height of column, 2.75 m., is not m the table, but it lies
between that corresponding to 2'5 m. and 3 m. ; consequently, the
discharge for the height of orifice, 16 c, will be comprised between
the numbers 673 and 739, and it will be about 706 ; therefore the
discharge ^v^ll be 706 x -8 = 664'8 Utres per second.
Second Example. — Suppose the height of the orifice to be
6'6 c, instead of 16, the other data remaining the same. As
this height is comprised between 16 and 17 centimetres, the dis-
charge effected will evidently be between the numbers 673 and
715, corresponding to a column of 2'5 m., and between the num-
bers 839 and 784, for a column of 3 metres. It will therefore be
very nearly a mean between these four numbers;
673 + 715 + 739 +784
or-
• = 727-75 litres.
i be
Whence we obtain 727-75 x -8 = 582-2 litres for the effective
discharge.
345. Incomplete Contraction. — When one or more sides ot
the orifice are simply the prolongation of the sides of the reservoir
or stream, the contraction is sensibly diminished, and the corre-
sponding coefficient is consequently greater.
In this case, in order to calculate the effective discharge, the
numbers must be multiplied bv
1-125, if the contraction is only on one side.
1-072, " " " two sides.
1-035, " " « three sides.
Example. — Required the volume of water discharged by an ori-
fice of -25 m. in height, 1-3 m. in width, and with a column of -8 m.)
measured from the centre of the orifice, the bottom of the openuig
being in a Hne with the bottom of the reservoir ; that is to say, the
contraction taking place only on the three sides 1
It will be found, according to the table, that the effective discharge
is 598 litres for a width of one metre, and consequently 598 x 1-3
= 777 litres, is the discharge for 1.3 m., when the contraction is
complete. We have, therefore, 777 x 1-035 = 804 litres, the
actual discharge sought.
346. Inclined Sluicegate. — It very often happens that the
sluicegate is inclined. In this case, if there is no contraction on the
sides or bottom of the orifice, the coefficient needs to be considera-
bly augmented. Thus, to calculate the effective discharge, it is
necessary to multiply tlie numbers in the preceding table by 1-33,
if the sluice is inclined at an angle of 45°, or with 1 m^tre of base
to 1 in height, and by 1-23, if the inclination is 60°, or 1 metre of
base to 2 in height.
Example. — It is desired to know the volume of water discharged
tiu-ough an orifice inehned at an angle of 45°, having 17 m. in
height vertically, 1-25 m. in width, and at a distance of 1-2 m. below
the surface of the reservoh ; the two vertical sides and the bottom
being in a line with the sides of the reservoir.
From the table we shall find 398 x 1-25 = 622-5 litres for the
discharge with a vertical orifice and complete contraction ; conse-
quently, 622-5 X I'S^ = 828 litres will be the effective discharge
sought.
347. When vertical floodgates have their lower edges very near
the bottom of the reservoir, as is generally the case, to determine
the discharge,
Multiply the numbers given in the table by 1-04.
Example. — What is the volume of water discharged per second
by a sluice, the orifice of which is opened to a height of -38 m.,
having -8 m. in width, and 2-5 m. being the distance from the centre
to the upper level ?
The table gives 1,599 litres for the discharge effected through an
orifice of a metre in width. Whence 1,599 x -8 x 1-04 = 1,330-0
litres, the effective discharge sought.
When two sluices are at not more than three metres distance
from each other, and are open at the same time, the discharge will
be obtained by
BOOK OF INDUSTRIAL DESIGN.
113
Multiplying the numbers given in ike table by -9 15.
Example. — K the orifices of two sluices, situated at a couple of
metres distance from each other, have together a width equal to
1-5 m., and are both opened to a height of '45 m., the column of
water upon their centres being 1-8 m., what will be the effective
discharge of the two together per second ?
In the table, we find that 1609 litres corresponds to a column
of 1-8 m., and a width of 1 m. Therefore, 1609 x 1-5 a 915
= 2208-35 litres is the required discharge.
TABLE OF THE DISCHARGE OF WATER BY OVERSHOT OUTLETS OF ONE METRE IN "WIDTH.
Heights
Discharge
Heights
Discharge
Heights
Discharge
of the
in
of the
in
of the
in
reservoir
litres per second.
reservoir
litres per second.
reservoir
litres per second.
level above
the bottom
level above
the bottom
level above
the bottom
of the outlet.
1st Case.
2d Case.
of the outlet.
1st Case.
2d Case.
of the outlet.
1st Case.
2d Case.
6-0
20
21
28-6
259
283
52-0
639
698
6-5
23
24
29-0
266
290
62-6
648
708
6-0
26
27
29'5
273
298
63-0
658
718
6-5
29
31
30-0
280
306
53-5
667
728
7-0
32
34
30-5
287
313
64-0
676
738
7-5
36
38
31-0
293
321
54-5
685
748
8-0
40
42
31-5
301
329
55-0
694
758
8-5
43
46
32-0
309
337
55-5
704
769
9-0
47
60
32-5
315
344
66-0
713
779
9-5
51
64
33-0
323
353
66-5
724
790
10-0
56
69
33-5
330
361
57-0
733
800
10-5
60
63
34-0
338
369
57-6
743
811
11-0
64
68
34-5
346
377
58-0
753
822
11-5
68
73
35-0
353
385
58-5
762
832
12-0
72
77
35-5
360
393
59-0
771
842
12-5
77
82
36-0
368
402
69-5
781
853
13'0
82
87
36-5
376
410
60-0
791
864
13-5
86
92
37-0
382
419
60-5
801
875
14-0
92
98
37-5
392
428
61-0
811
886
14-5
97
103
38-0
399
436
61-6
821
896
15-0
101
108
38-5
408
445
62-0
831
907
16-5
107
114
39-0
416
453
62-5
841
918
16-0
111
119
39-5
423
462
63-0
851
929
16-5
117
125
40-0
431
471
63-5
861
940
17-0
121
130
40-5
439
479
64-0
871
951
17-5
127
136
41-0
447
488
64-5
882
963
18-0
132
142
41-5
465
497
65-0
892
974
18-5
138
148
42-0
463
506
65-5
902
985
19-0
143
154
42-6
472
615
66-0
912
996
19-5
149
160
43-0
481
625
66-5
922
1007
20-0
154
166
43-5
488
533
67-0
932
1018
20-5
160
173
44-0
497
543
67-5
943
1030
21-0
166
179
44-6
506
652
68-0
954
1042
21-5
171
185
45-0
614
661
68-5
965
1054
22-0
176
192
45-5
623
671
69-0
976
1066
22-5
182
199
46-0
631
581
69-5
987
1078
23-0
188
205
46-5
540
590
70-0
998
1090
23-5
194
212
47-0
549
599
70-6
1008
1101
24-0
202
219
47-5
558
609
71-0
1019
1113
24-5
207
226
48-0
667
619
71-6
1030
1125
25-0
212
233
48-5
576
629
72-0
1011
1137
25-5
220
240
49'0
584
638
72-5
1052
1149
26-0
226
247
49-5
593
648
73-0
1063
1161
26-5
233
254
60-0
603
658
735
1073
1172
27-0
2.39
261
60-6
612
668
74-0
1084
1184
27-5
245
268
51-0
621
678
71-6
1095
1196
28-0
253
276
51-5
630
688
76-0
1106
1208
114
THE PRACTICAL DRAUGHTSMAN'S
CALCULATION OF THE DISCHARGE OF WATER THROUGH OVER-
SHOT OUTLETS.
348. The practical formula employed by engineers to determine
the quantity of water which escapes in a second of time, through
an overshot or open-topped outlet, is the following : —
D = W X H X V29H X »i X 1000;
in which formula,
D represents, as before, the discharge in litres per second ;
W, the width of the outlet in metres ;
H, the depth of the outlet, as measured vertically from its bottom
edge, to the level of the water in the reservoir.
The following table is calculated by means of this formula, it
being supposed.
First, That the wadth of the outlet is 1 metre.
Second, That the heights of the outlet increase at the rate of
•005 m., from -05 m. up to -75 m. These heights are expressed in
centimetres in the first column of the table, the corresponding ve-
locities being given in the table at page 94.
Third, That the outlet is supposed to be narrower than the reser-
voir, or water-course, in which case, MM. Poncelet and Lesbros
give the following numerical values for the term m. : —
For the height, H, of . . .
The term, m, is
m.
m.
m.
m.
m.
m.
m.
•03
•04
•06
■08
•10
•15
•20
•412
•407
•401
•397
•395
•393
•390
•22
•385
The corresponding discharges in this case are given in the
second column of the table. They are expressed in litres per
second.
Or, fourth, that the outlet is virtually of the same wdth as the
reservoir, or water-course, ha\ing its lower edge only a little, if
anything, above the bottom. In this case, according to M. d'Au-
buisson (M. Costal's experiments), the coefficient, m, is equal to -42
on the average. The corresponding discharges will be found in the
third column of the table.
Rule. With the aid of this table, the calculation for determin-
ing the effective discharge of water by an overshot outlet, reduces
itself to the following: —
Multiply the width of the outlet, expressed in metres, by the number
given in the second column, and corresponding to the height of the
outlet in the first column, when the outlet is narrower than the water-
course, and when the water b discharged freely into the air ;
And by the number in the third column corresporuiing to the sam£
height, when the water-course is of the same width as the outlet,
its depth, likewise, not being sensibly greater than that of the lower
edge of the outlet.
First Example. — It is necessary to determine the volume of
water discharged per second by an overshot outlet, the width of
which is 2-5 m., and the height of the overflow -22 m., the case
being supposed of the first description.
It will be seen from the second column of the table, that the dis-
charge effected through an outlet of a metre in width, and of -22
m. in depth, is 176 litres per second ; whence we have
176 X -lb — 44» litres, the volume sought.
Second Example. — Required to determine the discharge with the
same data ; the case being supposed of the second description.
In the third column, the number corresponding to the depth of
•22 m. will be found to be 192 litres ; whence,
192 X 25 = 480 litres, the volume sought.
Remark. — If the given height happen to fall between some of
the numbers given in the table, it will be necessary to take a mean
proportional between the two corresponding results, in order to
obtain the actual discharge.
Example. — What is the quantity of water discharged by an
overshot outlet of 3 metres in width, and of a depth equal tt>
•183 m.?
In the first case, the discharge effected, for 1 metre in width, wil>
be between 132 and 138 litres, the mean between which is very
nearly 136.
Consequently, 136 x 3 = 408 litres, the effective discharge per
second.
And in the second case, the discharge effected for 1 metre in
width, being comprised between the numbers 142 and 148, will be
about 146.
Whence, 146 x 3 = 438 litres effective discharge.
TO DETERMINE THE WIDTH OF AN OVERSHOT OUTLET.
349. W^hen the volume of water to be discharged per second is
knowTi, and it is wished to calculate the width to be given to an
overshot outlet, or sluice-gate, so as to effect the desired discharge
wth a given height of water, this may be done in the following
manner : —
Take from the table the number corresponding to the given height
(this number expressing the discharge for a width of 1 metre), and
divide the given volume, expressed in litres, it will give the required
2tidth in metres.
First Example. — What width must be given to an outlet, re-
quired to discharge 600 litres per second, with a depth above the
bottom edge of '12 m. 1
In the second column of the table, and opposite •12 m., will be
found the number 72.
We have, then —
600 -^ 72 = 8^33 m., the width sought.
Second Example. — ^What width must be given to an open sluice,
required to discharge 448 litres of water per second, wth a depth
of -205 m. ?
From the table, we find that 1 60 litres is the effectual discnarge,
corresponding to a width of 1 m6tre.
Wlience —
448 -4- 160 = 2-8 m., the width sought.
TO DETERMINE THE DEPTH OF THE OUTLET.
350. Cases may occur where we are limited as to width. It is
then necessary to ascertain the least depth necessary to effect tho
required discharge, which may be done by means of the foUowing
rule : —
Divide the discharge expressed in litres per second, by the width in
metres, and take the nujnber in the second column tohich is nearest to
the quotient obtained, the number in the first column corresponding
tvill give the depth sought, or very nearly so.
BOOK OF INDUSTRIAL DESIGN.
115
Example. — With what depth of outlet will a discharge of 350
litres per second be effected, the width being limited to 2 metres 1
Wo have
350 -4- 2 = 175 litres.
In the second column of the table will be found the number 176,
corresponding to a height of -22 m. in the first column, which will
therefore be the required height, witiiin a millimetre.
351. Observation. — When it is not possible to measure the
depth, H, with exactness, the lesser depth, h, must be taken im-
mediately over the lower edge of the outlet, and multiplied by rn?*,
so as to obtain the actual value of H, corresponding to the num-
bers given in the table, according as the outlet is narrower than the
reservoir, or water-course, or equal to it in width.
First Example. — Determine the discharge effected through an
outlet, 4 metres wide, the depth, ft, immediately above the lower
edge being equal to •!! m., the width being about four-fiftlis of
that of the reservoir.
We have '11 m. x 1'178 = -IS m., for the assumed neight, 11,
of the reservoir level.
Corresponding to this height, we have, in the second column, the
quantity, 82 litres.
Then 82 x 4 = 328 litres, the effective discharge sought
TABLE OF THE DISCHARGE OF WATER THROUGH PIPES.
Diameters of the Pipes.
Mean
velocity
in
•10
n.
•15
m.
•20 m.
■25
m.
•30
ra.
metres
per
Discharge
Fall
Discharge
Fall
Discharge
Fall
Discharge
Fall
Discharge
Fall
second.
in
per me ire
in
per metre
in
per metre
in
per metre
in
per metre
litres
in length
litres
in length
litres
in length
litres
in length
litres
in length
per
in
per
in
per
in
per
in
per
in
second.
nentimetres.
second.
centimetres.
second.
centimetres.
second.
centimetres.
second.
centimetres.
0-10
0-8
0-02
1-8
0-01
3-1
0-01
4-9
0-01
7-07
0-01
0-15
1-2
0-04
2-6
0-03
4-7
0-02
7-4
0-02
10-60
0-01
0-20
1-6
0-07
3-5
0-05
6-3
0-03
9-8
0-03
14-14
0-02
0-25
2-0
0-10
4-4
0-07
7-8
0-05
12-3
0-04
1767
0.03
0-30
2-3
0-15
5-3
0-10
9^4
0-07
14-7
0-06
2r20
0-05
0--35
2-7
0-19
61
013
11-0
0-10
17-2
0-08
24^74
0-07
0-40
31
0-25
7-1
0-17
12-6
0-12
19-6
0-10
28^27
0-08
0-45
3-5
0-31
8-0
0-21
141
0-16
22-0
0-12
31-81
010
0-50
3-9
0-38
8-8
0-25
15-7
0-19
24-5
0-15
35^34
013
0-55
4-3
0-46
9-7
0-30
17-3
0-23
27-0
0-18
38^88
0-15
0-60
4-7
0-54
10-6
0-36
18-8
0-27
29-4
0-22
42-41
0-18
0-65
51
0-63
11-5
0-42
20-4
0-32
319
0-25
45-95
0-21
0-70
5-5
0-73
12-4
0-49
22-0
0-36
34-4
0-29
49-48
0^24
0-75
5-9
0-83
13-2
0-56
23-6
0-42
36-8
0-33
5301
0^28
0-80
6-3
0-95
14-1
0-63
251
0-47
39-3
0-38
5655
031
0-85
6-7
1-06
150
0-71
26-7
0-53
417
0-43
60^08
0-35
0-90
7-0
M9
15-9
0-79
28-3
0-59
44-2
0-48
63-62
©•40
0-95
7-5
1-32
16-8
0-88
29-8
0-66
46-6
0-53
67-15
044
1-00
7-8
1-46
17-7
0-97
31-4
0-73
491
0-58
70-7
049
MO
8-6
1-76
19-4
1-17
34-5
0-88
54-0
0-70
77-7
059
1-20
9-4
2-09
21-2
1-39
37-7
1-04
58-9
0-83
84-8
069
1-30
10-2
2-44
23-0
1-63
40-8
1-22
63-8
0-98
91-9
0^81
1-40
iro
2-82
24-7
1-88
44-0
1-41
68-7
113
98-9
094
1-50
11-8
3-24
26-5
2-16
47-1
1-62
73-6
129
106-0
108
1-60
12-6
3-68
28-3
2-45
50-3
1-84
78-5
1-47
113-1
1-22
1-70
13-3
4-14
30-6
2-76
53-4
2-07
83-4
1-66
120-2
1-38
1-80
141
4-64
31-8
3-09
56-5
2-32
88^3
1-85
127-2
165
1-90
14-9
516
33-6
3-44
59-7
2-58
93-3
2-06
134-3
172
2-00
1.5-7
5-71
35-3
3-80
62-8
2-85
98^2
2-28
141-4
1-90
2-10
16-4
6-29
371
419
66-0
3- 14
1031
2-51
1484
210
2-20
1T2
6-89
38-9
4-60
69-1
3-45
108-0
2-76
1555
230
2-30
18-0
7-53
40-6
5-02
72-2
3-76
112-9
301
1626
2-50
2-40
18-8
8-19
42-4
5-46
75-4
4-09
117-8
3-28
169-6
a-T*
2-50
19-6
8-88
44-2
5-91
78-5
4-44
122-7
3-65
176-7
a-9o
2-60
20-4
9fi0
45-9
6-40
81-7
4-80
127-6
3-83
183-8
3-20
2-70
21-2
10-34
47-7
6-89
84-8
517
132-5
4- 1 4
190-8
3-44
2-80
220
11-11
49-4
7-41
88-0
5-56
137-4
4-.i:. ,
197-9
3-70
2-90
22-8
11-92
51-2
7-94
fll-l
5-95
142-3
4-77
205-0
3-97
3-00
23-6
12-74
530
8-50
94-2
6-37
147-3
5- 10
3121
4-25
116
THE PRACTICAL DRAUGHTSMAN'S
Second Example. — ^With the like data, what would be the effec-
tive discharge, supposing the outlet to be of the same width and
depth as the reservoir ?
We have, as before, -11 x M78 = -13 m. for the depth, H,
to which 87 litres is the corresponding discharge, as in the third
column.
Whence —
87 X 4 = 348 litres, the actual discharge.
OUTLET WITH A SPOUT, OR DUCT.
352. It may happen that a spout or duct, slightly inclined, or
even horizontal, is fitted to the outlet, and that it is more contract-
ed, both at the bottom and at the sides, than the reservoir. In such
case, the discharge is sensibly different ; and to determine it, it is
necessary to multiply the numbers in the second column of the table
by •83, when the height is -2 m., or upwards ; by '8, when the
height is -IS m. ; and by '76, when the height is only '1 m.
PIPES FOR THE CONDUCTION OF WATER.
353. The formulas employed in calculating the proportions of a
conduit for water of uniform section, consisting of cylindrical tubes,
are the followins; : —
V = 53-58
/ dY
y/-^- 0-025;
and
D = S V =
Tid"
xV.
In which,
V is the mean velocity ;
D, the volume in litres;
d, the internal diameter of the conduit :
F, the fall per metre, or the length, L, of the conduit, divided by
the difference between the levels at either extremity ; and
S, the section of the conduit.
In order to abridge the calculations, we jrive a table, with the
aid of which, various questions relative to the laying down of
water-ducts, formed by cylindrical tubes, may be solved very
speedily.
First Example. — What fall must be given to a conduit, -1 m. in
diameter, in order that it may discharge 11 litres of water per
second 1
From the table it will be seen, that the fall, in this case, should
be -1 c, or 1 millimetre, per metre.
SecoTid Example. — What diameter must be given to a conduit,
500 metres in length, in order that it may discharge 168 cubic metres
of water per hour, the whole fall being -265 m. ?
We have 168 cubic metres, or 168,000 litres, -t- (60 x 60) =
46-65 litres, discharged per second ;
and -265 -^ 500 = -53 c, the fall per metre.
It will be seen from the table, that the diameter necessary foi
this discharge, and with this fall, is -25 m., or 25 centimetres.
CHAPTER Vm.
APPLICATION OF SHADOWS TO TOOTHED GEAR.
PLATE XXX.
SPUR "WHEELS.
Figures 1 and 2.
354. We have already pointed out, that before shading an object
in a finished manner, it is generally necessary to lay down the out-
lines of all the shadows, proper and cast, which may happen to be
occasioned by the form of each part.
Thus, before proceeding to apply the finishing shades to the spur-
wheel and pmion, fig. A, we must first detei-mine, separately, on
each wheel, both the shadow proper of the external surface of the
web, and the shadows of the teeth upon it, and also upon them-
selves. The operations called for with one of the wheels are indi-
cated in the figures.
The external surface, a c, of the web, a, of the spur wheel,
being cylindrical, the line of separation of light and shade wUl be
obviously determined by a tangent parallel to the luminous ray,
or better, by the radius, o d, at right angles to this. By squaring
over the point of contact, d, in the horizontal projection, we obtain
the line, d' e, in the vertical projection. Similarly, by squaring
over the point, e, we get the straight line, f' g, for the line of
separation of light and shade on the outer ends of the teeth.
which are likewise cylindrical. A portion of the lateral surface of
the teeth is also in the shade, as will easily be determined, by draw-
ing lines through the extreme angles, as a, 6, c, &c., parallel to the
luminous rays. Thus the surfaces, a d, b e, and cf, do not receive
any light, and are, therefore, shaded in the elevation, as within the
outlines, a' d' g h, b' e' ij, and c'/' k I.
Each of these teeth, also, easts a shadow upon the cylindrical
surface of the web; and as their edges, a' h, b' j, d I, are vertical,
their shadows on the web are also vertical. These last are deter-
mined by drawing the luminar lines through the points, a, b, c, and
a', b', c', and then squaring over the points of contact, 7n, n, o, to
m', n', d .
To complete the shadows of the teeth upon the web, it is further
necessary to obtain the outline corresponding to the edges, ad, b e,
c /, &c. We already have the extreme points, d', d, f, and w', n', d,
and in most cases these are sufficient. Where, however, greater
exactness is required, it is well to find a few intermediate points.
The lower edge of the tooth, also, casts a shadow upon the web,
which is obtained in the same manner, by drawing luminar lines
through the points, p, q, r, meeting the surface of the web in points
projected vertically in r', d'.
BOOK OF INDUSTRIAL DESIGN.
ll";
Some of the teeth, also, cast shadows upon each other ; but as
their surfaces are vertical, these shadows are simply determined by
the contact of the luminar lines with them. Thus, the edges pro-
jected in s, t, y, &.C., have for shadows the straight lines projected
vertically in u' u', x' x\ z' z'.
Finally, when we have drawn the horizontal projection of the
wheel, as in the present example, we have to determine the shadow
cast by the web upon the tenons of the teeth, and upon the arms,
or spokes. All these surfaces being horizontal and parallel, the
shadow cast upon each will be a circle equal to the one, h i l,
which is the projection of the inner edge of the web. All that is
necessary, then, is to draw through the centre, o, o', a line parallel
to the luminous ray ; and to find the points of intersection, o^ and
o', with the planes, m o" and n o"', in which lie the upper surfaces
of the tenons and of the arms, and to describe arcs with the points,
o" and o^ as centres, and with the common radius, o h (280). In
the same manner we obtain the shadows cast by the boss of the
wheel, and by the feathers upon the arms.
When we have thus gone through the requisite operations for
each wheel, we proceed vwth the shading, according to the prin-
ciples laid down (289, et seq.), covering first the portions which
require a more pronounced shade, and leaving the lighter parts to
the last.
The specimen, fig. A, which we recommend to be copied on a
larger scale, indicates the various gradations of shade required to
produce the proper effect, according to the different positions of the
planes, and to the contour of the suifaces. These wheels are also
supposed to be mounted upon their shafts, which are shaded as
polished cylinders.
bevil wheels.
Figures .3 and 4.
355. The procedure here called for will be the same as in the
preceding case — that is to say, we must first draw the outlines of
the shadows, proper and cast, for each wheel. The figures repre-
sent a horizontal and vertical projection of a bevil wheel with cast-
iron teeth, the shadows being indicated on the different surfaces.
The external surfaces of the teeth and of the web being conical,
the shadows proper are determined in the same manner as for the
cone, by drawing through the apex a plane parallel to the luminous
ray, and finding the generatrix at which this plane touches the
conical surface (313).
It is in this manner that, for the outer ends of the teeth, we ob-
tain the generatrix projected in o a, fig. 3, and for the outer surface
of the web, that projected in o b. These generatrices, which are the
lines of separation of light and shade, are projected vertically in the
straight lines, o' a' ai\d d' b', converging to the apex of the cone ;
since, however, these lines occur between two teeth in the present
example, they are not apparent in fig. 4.
Some of the teeth have their lateral faces in the shade, whilst all
the lower conical surface corresponding to the wider ends of the
teeth is in deep shade, as indicated in fig. 4 by a darker tint.
Wo have, besides, merely to determine the shadows cast hy
the outer edges, a d, b e, c f, and by the curved portions, d g, e h,
and f i. Now, the outer edges, a d, b c, c f, cast Hhadows upon
the conical surface of the web. which are n'pr(^s('nled hy straight
lines coinciding with generatrices on this surface ; and therefore,
to determine them, we must draw through the corresponding edges
a series of planes parallel to the luminous ray ; the whole of these
necessarily passing through the common apex, o, it is simply re-
quisite, therefore, to find the shadow cast by any one point in
these edges. Let us take, for example, the points, d, e, f, all
situate in the same circle, e d f ; the operation, then, is to find
the shadow of this circle upon the conical surface, and is the same
as that which we have already indicated and explained several
times ; it consists, in fact, in drawing any planes, g h and i j, per-
pendicular to the cone's axis, and, consequently, parallel to the plana
of the circle, e c? F.
356. We have seen that the shadow cast by the circle, n d f..
upon each of the planes, will be a circle equal to itself; and it ia,
therefore, simply necessary to find the shadow cast by the centre, .
o, o'. This shadow falls in o, o', on the plane, g h, and in o'^ o', un
the plane, i i ; if, then, with the points, o and f/, as cenbes, and
with the radii, o k' and o^ j', equal to the radius, o e, we drav- a
couple of arcs, these arcs will cut the circles, g' k' h' and i' l' j'. Hi ;
projections of the sectional planes, in the points, k' and j', wh'-j ,
being squared over to the vertical projection in the points, k apd ,
will give two points in the curve, j k M n, representing the shado v
cast by the circle, e d f, upon the conical surface of the v/oj.
Consequently, if we draw tb*^ luminar lines through the poinis,
d', e',f, &c., the respective points of their intersection with the
curve, as m, p, q, will represent their shadows cast upon the web
surface. These points are squared over to lu', p', Q', in the h iri-
zontal projection.
The points, g, h, i, situated upon the upper base of the cone,
obviously cast no shadows, the shadows of the teeth, however,
springing from them ; if it is wished to determine any points be-
tween these and those already found, it will be necessary to de-
scribe an imaginary circle, such as g', k', h', passing between the
points, d and g, the outer and inner angles of the teeth. Tlio
curve, R s T, as projected in the elevation, will be found to repre-
sent the shadow cast by this circle upon the conical surface of the
web.
As the edges, a d,b e,cf, cast shadows which coincide witl^
generatrices of the cone, they may be obtained simply by drawing
straight lines through the several points, w', q', and p', converging
in the apex of the cone in both planes of projection.
Finally, the shadows cast by some of the outer edges of tl.o
teeth, such as/ c, upon the teeth inunediately behind, are defined
by drawing the luminar line,//, through the point,/, meeting the
flank, I m, of the other tooth, wliich lies in a vortical plane. This
point of contact is projected vertically in /', on the vertical projec-
tion,/' /', of the luminar line. It now remains to draw a line, /' ;»,
through this point, /', and through the apex of the cone, and this
lino will represent the shadow cast by the cdge,/c.
357. In the case where \W himinar lino passing through the
extremity of the tooth— as that, for examiile, drawn through tlio
point, f — falls upon a c\u-ved portion of the tooth behind, it is
necessary, if great accuracy is reciuin
■il, to iiuagiuo a vortical
plane jiassiug through this jioint and lluoui^li the luminar line.
Miul then to find the intersection of this plane with the curved sui-
face of llie tooth. This would letiuiu' a separate diagnuu ; hu»
118
THE PRACTICAL DRAUGHTSMAN'S
the operatiiiii is very simple, and lias been explained in reference to
previous examples (287).
358. The example, fig. [B, represents the application of finished
shading to a bevil wheel with wooden teeth, in gear with a pinion
on each side, each with cast-iron teeth. It is to be remarked, that
altliough the shadows are not the same upon each of these wheels,
because of their difl:'erent positions with regard to the light, these
are, nevertheless, determined by means of the same operations as
tliose which we have just explained.
In shading this example, the principles and observations already
discussed must be borne in mind, and note taken of the various
lights and shades. It is also to be observed, that, from the posi-
tions of the two pinions, the inner end of the shaft of one is com-
pletely in the shade, whilst the inner end of tlie other is illumi-
nated.
APPLICATION OF SHADOWS TO SCREWS.
PLATE XXXI.
359. It has already been shown, that a screw may be generated
by a triangle, a rectangle, or by a circle, the plane of which passes
through the axis of the screw, the generating movement being along
a helical path. The screw is, consequently, called triangular,
square, or round-threaded. In each of these cases, the outer edges
cast shadows upon the core of the screw, or upon the twisted sur-
face of the consecutive convolutions of the thread itself. If the
screvif is surmounted by a head, there will be, in addition, the
.shadow east by this upon the outer surface of the thread, as well
as upon the other parts. We shall proceed to explain the methods
of determining the various shadows upon these different kinds of
screws.
cylindrical square-threaded screw.
Figures 1, 2, 2°, and 3.
360. The limit of the shadow proper upon the screw, is obtained
in the same manner as that upon a right cylinder, by drawing the
radius, o a, at right angles to the ray of light, r o, and then squar-
ing over the point, a, to a' and a', and drawing a line through these
parallel to the axis of the cylinder. In the same manner we obtain,
by projecting the point, b, the line of separation of light and shade,
b' B^ upon the surface of the core.
The shadows cast by the outer edge of the threads upon the
cylindrical surface of the core, are simply determined by means of
the straight lines c c,r) d, drawn parallel to the luminous ray, r o,
and meeting the circle, e (^ b, the projection of the core, in c and d;
then, by squaring over the points, c d, to c' d', and drawing through
the latter the straight lines, c c', d d', parallel to the ray of light,
r', we obtain the points, c', d', for the shadows sought. We can,
in the same manner, obtain as many points as ai-e necessary to
complete the curved outline of the shadow.
When the threads of the screw are inclined to the left, as in figs.
2* and 3, instead of being inclined to the right, as in figs. 1 and 2,
the operations necessary for determining the curve of the shadows
are still the same. This is rendered sufficiently plain by the em-
ployment of the same letiers to represent similar and sjTiimetrical
points ; it only requires to be observed, that the end view, fig. 3
is that of the right half of the screw, whilst that in fig. 1 is one of
the left half, or, one may be supposed to be the turning over of the
end of the screw to the right, whilst the other is to the left ; the
ray of light is similarly respectively represented to the right and
left ; this, however, does not make any difference, as it is the length
of the line merely, as d d, which is required. The luminous ray,
in both cases, makes an angle of 45° with the axis of the screw,
vi'hich is horizontal.
The polygonal head, f g h i, which separates the right-handed
from the left-handed portion of the screw, casts a shadow upon a
part of the latter, represented by curves, which will be easily de-
termined, in accordance wth previous examples (285 and 286), and
the principal points in which are/, g, in figs. 2° and 3.
screw with several rectangular threads.
Figures 4 and 5.
36 L The construction of the shadows of a rectangular-threaded
screw is the same, whether it be in a horizontal or vertical position,
or whether it be right-handed or left-handed. Thus, the screw
with several rectangular threads, represented in figs. 4 and 5, has
in the first place a shadow proper, limited by the vertical line, a' a'',
as squared over from the point, a, and next, the shadow, c' d'f'j
cast upon the core by the outer edge of the thread, c' d' e' ; there
is, moreover, a portion of the shadow cast by the circular shoulder,
G H I, upon the threads, and also upon the core. The outlines of
these shadows are found in precisely the same maimer as those in
figs. 1, 2, and 3 (361).
TRLANGULAR-THREADED SCREW.
Figures 6, C, 1, and 8.
362. When the screw is generated by an isosceles triangle, such
as c a d, fig. 6, of which the height, a b, is greater than the half of
the base, c d, there will be a shadow cast by the outer edge of the
thread upon the twisted surface of the succeeding convolution. In
proceeding to determine the outline of this shadow, in accordance
with the general method, which consists in finding the points of
contact of the luminous rays with the surface, we are led to seek,
in the first place, the curve of intersection of this surface, with a
plane passing through the luminous ray, and parallel to the axis of
the screw.
For this purpose, let e o, fig. 7, be the sectional plane ; its in-
tersection with the outer edge, c g' p, of the screw-thread will be
in the point, e, e', figs. 6 and 7, and similarly its intersection with
the inside, a I g, will be in the point, r, r'. To obtain interme-
diate points of the sectional curve, we must describe various circles
with the centre, o, and radii, o m, o n, representing the projections
of so many cylinders, on which lie the helices comprised between
the inner one, a I g, and outermost, d b" s, being of the same pitch
as these latter. We thereby obtain the points of intersection, h, i,
fig. 7, which are to be squared over to li', i', in fig. 6, and then, by
joining the several points, E^ h', i', r, we get the cur\'e of inter-
section of the plane with the helical surface of the thread ; so that,
if we draw a luminar line, e' e', through the point, e', in the same
i
BOOK OF INDUSTRIAL DESIGN,
119
plane, its intersection with the curve, e^ h' i' r, will give a point, e',
in tlie outline of the shadow sought.
In like manner, by drawing other planes, as f h and g i, parallel
to tiie first, E 0, we shall obtain the intersectional curves, f^/' h'
and ir'J' g I, and further upon these the points,/' and^', of the
outline of the shadow. By proceeding thus, we can obtain as many
points as may be deemed necessary for the construction of the
shadow cast by the outer edge, c g' p, of the thread, and the curve
obtained is, of course, repeated on the several convolutions of the
thread. We would remark, that there is no shadow ca.st when the
depth of the thread is such, only that a h, fig. 6, is less than the half
of the base, c d, of the generating triangle.
The diagrams, figs. 6" and 8, which represent a portion of a left-
handed screw, will show that the operations required Lq this modi-
fication, to determine the outlhies of the shadows, are precisely the
same as those last explained.
The core, n, which separates the two portions of the double
screw, as well as the end, n', receives a shadow cast by the outer
edge of the adjacent convolution of the thread.
shadows upon a round-threaded screw.
Figures 9 and 10.
363. These figures represent a species of screw generated by a
circle, ah c d, the plane of which passes through the screw's axis,
and of which each point describes a helix about the same axis.
The intervals or hollows between the convolutions of the thread
are also formed with a helical surface generated by a semicircle,
d e f, tangential to the first. We have, then, to determine the
limiting line of the shadow proper upon the screw, and the shadow
cast by this line upon the hollows.
The projecting thread being a species of spiral torus or serpen-
tine, the determination of its shadow will be similar to that of the
shadow of the ring (323).
Thus, if the screw be sectioned by a vertical plane, g o, passing
through its axis, itb intersection with the thread will evidently be a
circle, as projected in^' V, fig. 9. This circle, being inclined to the
vertical plane of projection, fig. 10, is projected therein in the form
of an ellipse, the principal points,/, It, I, of which are obtained by
squaring over the points, y, fe', V, respectively, upon the helices cor-
respondmg to the points, a, b, c. If, then, upon the plane, g o,
which we suppose to be reproduced at o g, fig. 10", we project the
luminous ray, r o, it will be sufficient to determine the point of
contact of this ray with the curve,/ k I; for this purpose, find the
projection of the ray upon the vertical plane in g' &', fig. 10"; then
draw a line, g^ o^, tangential to the ellipse,/ /,: I, and parallel to the
straight line, g" o', its point of contact, m, with the ellipse will be a
point in the line of separation of light and shade upon the outer
surface of the screw-thread. By proceeding in this manner, any
number of points in this line may be obtained.
By continuing the sectional plane, g o, across the hollow of tlie
screw, we .shall likewise obtain the elliptic curve, n o', the principal
points in which are equally situated ui)on the helices .which pass
througli the points, d, e,f; itMS sufficient to prolong the luininar
line, g' o', until it cuts the ellipse, n o'' p, so as to obtain the point,
o', which is the shadow cast by the corresponding j)uiiil, ?/(, of the
line of separation of light and shade upon the hollows oi intervals
between the convolutions of the thread.
It is to be remarked, that the prolongation of the line of separa-
tion of light and shade, s t, casts a shadow upon the outer surface
of the convolution immediately below ; and, in the same manner,
the shoulder above casts a shadow over the projection and hollow
of the adjacent thread.
APPLICATION OF SHADOWS TO A BOILER AND ITS
FURNACE.
PLATE XXXII.
shadow of the sfheee.
Figure 1.
364. It will be recollected, that a sphere is a regular solid, gene-
rated by the revolution of a semicircle about its diameter. From
this definition it follows, that its convex or concave surface, ac-
cording as it is considered solid or hollow, is a surface of revolution,
of which every point is equally distant from the centre of the gene-
rating circle. To determine, then, the shadow proper, upon the
surface of a sphere, we can proceed according to the general prin-
ciple (328) ; but, in this particular case, the follomng will be the
simpler method.
Let us suppose the sphere to be enveloped in a right cylinder.,
having its axis parallel to the luminous ray ; this cylinder will touch
the sphere at a great circle, which is, in fact, the line of separation
of light and shade, and the plane of which is perpendicular to thp
luminous ray, and, consequently, inclined to the planes of projec-
tion ; it follows, therefore, that the projection of this line upon those
planes will be an ellipse.
Thus, let fig. 1 represent the horizontal projection of a sphtre,
whose radius is o a, the projections of the extreme generatrices, b c
and D E, of the cylinder, parallel to the luminous ray, touch tho
external contour of the sphere in the points, c, e, which are dia-
metrically opposite to each 'Other, and are the extremities of tho
transverse axis of the ellipse.
As, in general, this curve can be drawn when its two axes aro
determined, it merely remains to find the length of its conjugate
axis. To this effect let us imagine a vertical plane to pass through
the luminous ray, R o, and let us take two lines tangent to tho
section of the sphere in this plane, and parallel to tlie luminous
ray ; if now we turn this plane about the line, r o, considered as
an a.xis, so as to fold it over upon, and nuiko it coincide with, tho
horizontal plane, the great circle, which is its section with thu
sphere, will obviously coincide with tho cin-le dniwn with tlm
radius, a o. The luminous ray will, as already seen (287), bt(
turned over to r' o, making an angle of 35° 16' witli the lino, h o.
It may also be obtained by making tho lino, r' r, jx^rj>ondicular to
R 0, and equal to a side of tho square, .ns g k, and then joining
r' o. Tho two luminous rays tangential to tho sphere will then
coincide with tho straight linos, ii l and m n, luirallol to k'o,
their points of contact with llio great circle will be tho oxtn-niitios
of the diiunotor. l n, iHT|H'iHliculnr lo k' o. If now wo iiruigine
the plane to l»e returned to its original j »lt' >n, tho iH>inl.s, l ;uui
1'20
THE PRACTICAL DRAUGHTSMAN'S
N, will be projected in l' and n', thereby giving the length, l n, of
the conjugate axis of the ellipse sought.
365. If, in place of constructing this ellipse by the ordinary
methods, it is preferred to determine the various points by means of
a series of analogous sections, with their subsequent operations,
it will be sufficient, for example, to draw the plane, a b, parallel to
R o, and then to turn over, as it were, the section of the sphere
formed by this plane, so that it shall coincide with the horizontal
plane, making its centre, at the same time, to coincide with the
centre, o, of the sphere, the section in question being a circle,
described with the radius, c o, equal to a c. Next draw the tan-
gent, e d, parallel to r' o, and then project the point, d, on the
diameter, l n, to d', upon the original line of section, and a point
in the elliptic curve. In this manner, as many points may be ob-
tained as are wished. The distance, c d\ being made equal to c d', d'
will be the symmetrical point in the opposite, and now apparent
part of the ellipse.
If the projection of the sphere were supposed to be upon the
vertical plane, the operations would be identical, only the transverse
axis of the ellipse, instead of being in the direction, c e, would, on
the contrary, be in the direction, a i, perpendicular to it, as seen on
fig. [B, representing the hemispherical end of a boUer, shaded and
finished.
shadow cast upon a hollow sphere.
Figure 2.
366. When a hollow sphere is cut by a plane passing through its
centre, and parallel to the plane of projection, the inner edge of the
section will cast a shadow upon the concave surface, the outline
of which will be an elliptic curve, which may be determined in ac-
cordance with the general principle of parallel sections, already
explained (287), or by means of the simpler system of sections and
auxiliary views adopted in the preceding example, and of which we
shall proceed to give another instance, in fig. 2.
This figure represents the projection of a hollow sphere upon
the vertical plane, being, in fact, the section through the line, 1 — 2,
of the boiler, represented in figs. 4 and 5. If this hemisphere be
sectioned by a diametrical plane, a b, parallel to the luminous ray,
the section represented in the auxiliary view, fig. 3, will be a semi-
circle, a' c' b'. The luminous ray, lying in this plane, and passing
through the point, a, a', will, in. fig. 3, be represented by the line,
a' c', parallel to the line, r' a, obtained, as indicated in fig. 2, by
the method already explained. This straight line, a' c', cuts the
circle, a' c' b', in the point, c', wliich must be squared over to
c, on the line, a b, fig. 2, when c will be the shadow cast by the
point, A.
In the same manner we obtain the points, h, d, by means of the
sectional planes, ah, c d, parallel to a b, and cutting the sphere in
semicircles, represented by a' b' e, in fig. 3. This semicircle is cut
in the point, b', by the line, a' b', parallel to a' c'. The extreme
pomts, D, E, are obviously situated at the extremities of the diame-
ter, D E, perpendicular to the luminous ray, r o, and representing
ihe transverse axis of the ellipse. These shadows are frequently
met with in architectural and mechanical subjects ; as, for example,
'0 mcJies, domes, and boilers.
APPLICATIONS.
367. Fig. 4 represents a longitudinal section, at the line, 3 — 4,
in fig. 5, of a cylindrical wrought-iron boiler, with hemispherical
ends, and surmounted by a couple of cylindrical chambers, one of
which serves for a man-hole, and has a cover fitted to it.
Fig. 5 is a plan of the same boiler, looking down upon it, and
showing the cylindrical chambers.
Fig. 6 is a transverse section, made at the line, 5 — 6, in figs. 4
and 5.
For this boiler, we have to determine —
First, In fig. 4, the shadows, b d c and e J c, cast upon the
spherical surfaces at either end of the boiler, as well as those, c gi
anij k I, upon the cylindrical surface, together with the shadows
cast on the interior of the cylindrical chambers.
Second, In fig. 5, the shadows proper of the external cylindrical
and spherical portions of the boiler, and the shadows cast upon _
these by the cylindrical chambers. I
In fig. 7, we apply the same letters as to the analogous diagram,
fig. 3, this view being drawn for the purpose of obtaining the elliptic
curve, D (^ c, of the shadow cast by the circular portion, a c d, upon
the internal spherical surface of the end of the boiler.
In the same manner is obtained the portion, e J c Z, by means of
the diagram, fig. 8, observing that the sections made parallel to the
ray of light, above the line, a b, give semicircles, whilst those made
below, such as f c, give the circular portion to the right of the
line, a o, but an elliptical portion to the left of this line, m conse-
quence of this portion of the plane cutting the cylindrical part of
the boiler obliquely. It must be remarked, that the cylindrical
chambers, situatea on tne top of the boiler, give rise to the inter-
sections, I J K F, which cast shadows upon the interior of the boiler,
instead of the rectiUnear portion, i f, of the extreme generatrix of
the cylinder, which would have cast a shadow, had the cylindrical
chambers not been there.
The shadow cast by the edge of these intersecting surfaces is
limited to the curves, J K f, which may be easily delineated with
the aid of the section, fig. 6, by squaring over the points, j, k, l, f,
to the arc, j' k' l' f', and then drawing a series of luminar lines
through these points; that is, lines parallel to the ray of light.
These vvtII meet the internal surface of the cylinder in the points,
J', k', r, which are squared over again to the longitudinal section,
fig. 4, by means of horizontals, intersecting the luminar lines, drawn
through the corresponding points in the edges of each chamber, in
the points, _;', k, I. The rectilinear portion, F i, of the uppermost
generatrix of the cylinder, has for its shadow, on the internal sur-
face thereof, the similar and equal straight line, I i, which coincides,
in the projection, with the axis, o o (308).
A part of the extreme left-hand generatrix, i n, of each cylindri-
cal chamber, likewise casts a shadow upon the internal surface of
the boiler, the outline of which is a curve, i mj, which is simply an
arc of a chcle, described with the centre, o, and with the radius, o i,
equal to that, c o, of the boiler. This shadow is chcular, because
the straight line, i n, which casts it, is perpendicular to the axis of
the cylinder; whilst the axis and itself lie in a plane, parallel to the
plane of projection.
We can, however, determine the points, i, m, j, of the curve.
BOOK OF INDUSTRIAL DESIGN.
12i
iiuleDendently, with the assistanne of the auxiliary projection, fig. 6,
at nght angles to fig. 4.
It is the same with the curve, n p q, which is likewise an arc of
(I circle, because the straight line, n p, the edge of the cover which
closes the top of the chamber, is at right angles to the axis of the
latter, and at the same time parallel to the vertical plane of projec-
tion. The edges, n r and r m, being vertical, have for shadows
upon the internal surface of the chamber, a couple of vertical
straight lines, parallel to themselves (309). The chamber to the
right having a circaiar opening in the cover, has a shadow upon its
internal surface, necessarily different from that in the other cham-
ber. It is, however, easily obtained, and in the same manner as in
figs. 1 and 1°, Plate XXVIII. It must be observed, however, that
a portion, .s t w, of this shadow is due to the under edge, s t u, of
the cover-piece ; whilst the other part, s v, takes its contour from
the upper edge, v x, of the same piece. A comparison of figs. 4
and 5 will render these points easy of comprehension.
There remains, finally, the curve, c e g h, and the rectilinear por-
tion, h i, together extending from the first, a d c,to the straight
line, i i, and which represents the shadow cast by the arc, a f, g h,
of the hemispherical end of the boiler, and the straight part, h i, of
the upper edge of the cylindrical portion.
The whole curve, d d,c g i, representing the shadow cast by
(be edge of the section of the boiler upon the internal surface of
the latter; is precisely the same as that distinguished ib architecture
by the name of the niche shadow. It is to be observed, however,
that the position in this case is different, as the axis of the niche is
vertical.
We have now to draw the shadows, proper and cast, upon the
outer surface of the boiler, as seen in horizontal projection, fig, 5.
As for the shadow proper, it consists partly in that limited by the
line of sepaiaiion of light and shade, d d, obtained by the tangential
line, making an angle of 45° with the horizon, and touching the
uircle in the point, c, and partly in that bounded by the elliptic
curves, c d and d c e, upon the spherical ends of the boiler, the
manner of determining which has already been thoroughly discussed
in reference to fig. 1.
368. As to the shadows cast by the cylindrical chambers, either
on their neck pieces, or upon the outside of the boiler itself, they
are simply represented by lines inclined at an angle of 45°, as a' d',
b' e', drawn tangential to the outsides of the cylinders, and which
are prolonged in straight lines, as far as the line of separation of
light and shade, upon the cylindrical portion of the boiler ; that is,
in case they stand out far enough from the boiler surface. If, on
the contrary, they do not rise very high, as exemplified in the
end view, fig. 9, it will be necessary to determine the outline of
the shadow cast by a portion of the upper edge, b' c', as lying
either upon the cylindrical part of the boiler, or upon one of tlio
spherical ends. To find the shadow in this last case, we have
supposed an imaginary vertical plane to pass through tho luminous
ray, r' o', fig. 5, producing an elliptical section of the cylinder,
and a circular one of the spheiical part. This plan being repro-
duced at V? o", fig. 9, and turned about, to coincide with the hori-
zontal |tlano, we have the curve, f" g' h'^ representing the section
in ()ucsti()n. Tho point of contact, b', being transferred to b', is
mIso turned down, as it were, upon the horizontal phuui, to tho
point, b'; so that if we draw a line, b' i', through this point, e',
parallel to the luminous ray, r' o', similarly brought into the hori
zontal plane, this line, b' i'', will cut the interscctionai curve in the
point, 1^ ; the horizontal projection, i', of this point, upon the line,
r' h^ being obtained by letting fall the perpendicular, i" I^ upon
the latter. The corresponding point, i', in fig. 5, is taken at a dis-
tance from b', equal to b" i^ in fig. 9. Proceed in the same manner
with another sectional plane, parallel to the first, and passing
through the point, c', in order to obtain a second point, c', of the
shadow. The operations necessary for determining the intersec-
tional curves are sufficiently indicated in figs. 5 and 9.
369. The cylindrical steam-boiler, represented in longitudinal
section in fig. A, in end elevation in fig. ©, and in transverse
section in fig. ©, conjoins the various applications of shadows, of
which we have been treating, in reference to .spheres and cylin-
ders ; whilst, at the same time, they serve as examples of shading,
by lines or by washes, indicating the effects to be aimed at, and to
be attained by the following out of the various principles already
laid down.
370. We must remind the student, that, in order to produce tnese
effects, he must not always confine himself to the representation
of the shadows proper and cast merely. He must, further, show
the gradations of the light or shadow upon each part, as has already
been explained with reference to solid and hollow » vlinders. As
upon a cylinder or a cone, there is always a line of pre-eminent
brilliancy, so likewise, upon the surface of a sphere, will there be a
point of greater brilliancy than the rest.
This point is actually situated upon the luminous ray, passing
through the centre of tho sphere, fig. 1. Since, however, the visual
rays are not coincident with the luminous rays, the apparent posi-
tion of this point is somewhat changed. Thus, if we bring the
vertical plane, b o, fig. 1, into the horizontal plane, the luminous
ray will coincide, as has been seen, with the line, r' o', and, conse-
quently, its point of intersection with the sphere will coincide with
the point, i. On tlie other hand, the visu.il rays which are perpen-
dicular to the horizontal plane will coincide with parallels to o o,
when brought into the horizontal plane. This latter line intersects
the sphere in the point, c ; and as the light is reflected from any
surface in the direction of the visual rays, so as to make the angle
of incidence equal to the angle of reflection, if we divide the angle,
i o c, into two equal angles, by tho line, n o, the point, 7i, will bo
that which will appear to the eye most brilliantly illuminated. Tho
positions, n' and i', in the vertical piano of the point.s, n and i", are
obtained by letting fall (icrpiMulii-iilars u[)on llie line, o a, repre-
senting this i)!an(!.
In shading up a drawing it is preferalile to place llie bright or
li'ditest part between the two points, n' and i", a more pleasing elVect
being obtained thereby. When the si)here is polishctl, as a steel,
brass, or ivory ball, a circular spot, of pure white, nuist bo left about
the i)oint in (piestion. When, however, the body is rough, as i>)
.supposed in fig. 03, (his part is always lighter than tho rest ; but, nl
the same lime, it is covered by a faint wa.sh.
In llie case of a hollow spiiere, figs. 2 and 3, wo have to liear in
mind, not only to indiiate Iho position of the bright spot, which
is |)roj('cted, in the same manner, upon the luminous niy, a ii. ••uul
lies between tho points, n', i', but also tho point in tho oast sliadow,
122
THE PRACTICAL ])RAUGHTSMAN'S
which should be the least prominent. This latter will be found
to be at m, fig. 2, as determined by the radius, o' m, fig. 3, di-awn
perpendicularly to the ray of light, a' c', as brought into the same
plane as fig. 3.
371. The boiler is represented as placed in its furnace, which
is built entirely of bricks, with a diaphragm passing down the
middle of its length, to oblige the flames and gases issuing from
the grate to pass along the flue to the left, then to return by that
to the right, and passing through a third flue, before it reaches
the chimney. In this third flue is placed an auxiliary boiler, full
of water, and in communication with the main boiler by a pipe
passing to the bottom of each. In this auxiliary boiler, the feed-
water becomes heated before entering the main boiler, so as not
to reduce the temperature of the latter to a serious extent, upon
its introduction into it.
The main boiler is represented as half full of water. It should
generally be two-thirds full, but is delineated as but half full ; so
that a greater portion of the shadow cast upon its interior may be
visible. The remainder of the space, as well as the cylindrical
chambers, is supposed to be filled with steam. The base of the
chimney is of stone, whilst the stiilk is of brick. The foundations
of the furnace are likewise of stone.
Besides this present example of a boiler, we give a further ex-
ercise for finished shading in Plate XXXIII., the objects in which
we recommend the student to copy, on a scale two or three times
as large, so as to acquire the proper skill and facility of treatment.
SHADING IN BLACK.— SHADING IN COLOURS.
PLATE XXXIII.
372. In a great number of drawings, and particularly in those
termed working drawings, and intended for use in actual construc-
tion, the draughtsman contents himself by shading the objects
with China ink — sometimes, perhaps, covering this with a faint
wash of colour, appropriate to the nature of the material. The
shading, on the one hand, brings out the parts in relief, and ren-
ders the forms of the object intelligible to the eye; whilst, on
the other hand, the colours indicate of what material they are
made. This duplex artistic representation makes th« drawing
much more life-like, and more easily comprehended. A drawing
may be coloured in several ways. The simplest plan is first to
shade up the various surfaces with China ink, having due regard
to the respective forces and gradations of tone, according to the
lights and shades, as has been done in the preceding plates. The
entire surface of each object is then covered with an especial
wash of colour, the line of which is quite conventional. It must
be laid on in flat washes, according to the instructions given in
reference to Plate X. This first method of operating may suffice
in many cases, but it leaves out much to be desired in the effective
appearance of the drawing, its aspect being generally idthout
vigour, cold and monotonous. A better result is obtainable by
not carrying the China ink shading to so great a depth, and by
covering the surfaces by two or three washes of colour, laid on in
irradations, as was done with the China ink itself, so as to produce
« sufficient strength of colour at the darker parts, whilst the light
parts are left very faint ; and where the objects are polished, a puro
white line or spot is left, which will add considerably to the bril-
liancy of the whole. A softer and more harmonious effect can be
produced by the use of a warm neutral tint, instead of the China
ink, for the preliminary shading. This colour, however, is very
difficult to mix, and to keep uniform.
When a little practice has given some skill and facility in the
preparation and combination of the colours, the di-aughtsman may
proceed, at the outset, in a more direct and vigorous manner, leaving
out altogether the preliminary shading with China ink, and laying
on at once the successive coloured washes, rendering, at the same
time, the effects of light and shade, and indicating the nature of the
material. This last method has the merit of giving to each part
of the dramng a richer translucence, more warmth, and a more
satisfactory fulfilment of all desirable conditions.
In general, all drawings intended to be shaded should be deli-
neated with faint gray instead of black outlines, as for a simple
outline drawing ; the faintness of such lines avoids the necessity
of making them very fine, and their greater breadth affords a much
better guide to the shading-brush. A black outline, however fine
it may be, always produces a too sharp and hard appearance, whilst
there is much gi-eater risk of overstepping it in laying on the
washes.
373. In Plate XXXIII. we give a few good examples of objects
shaded in colours, comprising the materials most in use in con-
struction.
Fig. 1 represents the capital of a Doric column in wood. Al-
though the woods are naturally very different in colour, still, in
mechanical drawings, a single tint is used indiscriminately : it is,
as we have said, entirely conventional.
In fixing upon these colours, the object in view has been to avoid
confusion, and to employ a distinct and intelligible colour for
the representation of each substance, without seeking to copy the
natural colour in all its varieties.
In colouring this wooden capital, after the preliminary opera-
tions which we have mentioned, for determining the outlines of
the shadows, proper and cast, it is first shaded throughout with
China ink, and when this shading has reached a convenient depth,
and is thoroughly dry, the whole surface is to be covered with a
light wash, which may be a mixture of gamboge, lake, and China
ink, or burnt umber alone. The colour, in fact, should be analo-
gous to that of fig. 4, Plate X. ; it should, however, always be
fainter than in that example, which represents the material in sec-
tion, and is, therefore, stronger.
This proceeding may be easily modified, and made to resemble
the effect of the second method, by leaving certain parts of the
object uncoloured, and by softening off the shade in those places
where the light is strongest, with a nearly dry brush. If, however,
the draughtsman has become somewhat familiarized with the use
of the brush and the mixture of the colours, he may, as we have
said, omit the preliminary shading in black, by modifying eacli
shade as laid on, mixing the China ink directly with the colours,
and then gradually bringing up the shades, either according to 1 he
system of flat washes, or the more difficult one of softened shades.
Care must be taken in laying on these shades to commence at the
deeper parts, and then to cover these over again by the subse
BOOK OF INDUSTRIAL DESIGN.
123
quent washes, which gradually approach the bright part of the
object ; for in this way a more brilliant and translucent effect will
bo obtained.
When the objects are of wood, it is customary to represent the
graining in faint irregular streaks, care being taken to make these
as varied as possible. A general idea of the effect to be produced
will be obtained from fig. 1.
Following out these principles, the draughtsman may proceed to
colour various other objects composed of different materials, merely
varying the mixtures of colour according to the instructions given
in reference to Plate X.
Fig. 2 represents the top of a chimney of brickwork, the form
being circular. In this external view, the outline of each brick is
indicated ; and to render them more distinct from each other, a
line of reflected light has been shown on the edges towards the
light, near the brighter part of the chimney. Indeed, it is generally
advisable to leave a narrow, pure white light at the edges of an
object which are fully illuminated, as it gives an effective sharp
appearance.
Fig. 3 represents the base of a Doric column in stone, showing
the flutings. This being an external view, the tint to represent the
stone is not made nearly so strong as for the sectional stone-work,
represented in Plate X. A yellowish grey may be used for it, made
by mixing gamboge, the predominant colour, with a little China ink,
adding a little lake to give warmth.
These three examples of wood, brick, and stone, represent bodies
with rough surfaces, and which, therefore, can never receive such
brilliant lights as objects in polished metal ; no part, indeed, should
be entirely free from some faint colour.
Fig. 4 represents a nut or bolt-head of wrought-iron ; and, as
we have supposed it to be turned and planed, and polished upon
its entire surface, it has been necessary to leave pure white lights
at the brighter parts, to distinguish the surfaces from those which
are rough and dull. It is the same in the example, fig. 5, repre-
senting the base of a polished cast-iron column, and in the lateral
projection, fig. 6, of polished brass upper and lower shaft-bearmgs
or brasses.
We would hope that the principles of shadows and shading, ex-
plained and exemplified in the last two chapters, may serve as
sufficient guides for the various applications which may present
themselves to the draughtsman — whether his skill be called forth
to render the simple effects of light and shadow, or to produce the
gradations of shade and colour due to roundness or obliquity of
surface — to the various positions of the objects in their polished or
unpolished state, and to the various materials of which they may
be composed.
Thus, it will be understood, that although two objects are pre-
cisely alike in material and form, if they are situated at unequal
distances from the spectator, the nearer one of the two must be
coloured more strongly and brilliantly than the more distant, more
force and depth being given to the darker shades.
CHAPTER IX.
THE CUTTING AND SHAPING OF MASONRY.
PLATE XXXIV.
374. The operation of stone-cuttmg has for its object, the pre-
paring and shaping stores in such manner that they may be built
up into any desired form in a compact and solid manner ; great
care and skill, as well as mathematical knowledge, is more parti-
cularly required in the preparation of stones for arches, vaults,
arcades, and such like structures.
The study of the shaping of stones is based entirely upon descrip-
tive geometry, being indeed but a particular application or branch
<)f it, and in it have to be considered the genei'ation of surfaces, as
well as their intersections and developments.
In proceeding to adapt the stones to the position they are to
occupy, the mason should prepare a preliminary drawing of the
actual size of each stone, as well as a general view of the entire
erection, indicating the joints of each stone ; these, according to
the various positions to be occupied by them, are called liey sloncs,
arch stones, &c.
It is not our intention to give a complete treatise on the shaping
of masonry; but, as this study seems to belong, in ])art, to geonic-
trical drawing, we have thought it (|iiit(r vvilhiii llu' design of llic
pres(nit work to give a few ap|)lic:ilions, siiflicicint to show (lie line
of jnocedure to be followed out in operations of this naliuc.
the marseilles arch, or arriere-vo^ssure.
Figures 1 and 2.
375. We propose to prepare the designs for llie b:iy and arch
of a door or window, to be built of stonework, the upper part
being cut away, so as to present a twisted surface, analogous to
that known as the arriere-voussure of Marseilles.
This surface Is such as would be generated by a straight lino,
c A, kept constantly upon the horizontal, c' k', projected vertically
in the point, c, and moved, on lln' one hand, upon tiie semi-baso,
BED, of a right cylind(>r, having c' k' for its axis : and, on tJio
other, upon the circular arc, f k a, situated in a piano pimillol to
that, of the base, bed.
The lateral faces, f B l n and a r Q d, of tiio bay, are vortionl,
and are projected horizontally in v' n' and a' d', fig. 2. Tiioso
faces intersect the twisted surface at the curves, f h and .\ p, which
w(! shall proceed to detorinino.
For this purpose, the lirst thing to ho done is to seeil llio ]no.
jections of tiic straight generator lino, o a. a.s oocupymg dilToroiu
positions, so as to obtain tlioir points of iiitorsocuon wiui ouo «i
the oblique planes.
1
124
THE PRACTICAL DRAUGHTSMAN'S
We may remark, that if the are, f k a, be prolonged to the right,
for example, of fig. 1, and a number of lines be diawn through the
point, c, as c j, c a. c b, and c c, they will represent so many ver-
tical projections of the generatrix, c a, in diftorent positions.
These straight lines meet the semicircle, b e d, in the points, i, c, d, e,
which are projected horizontally in the points, i', c', d\ e'. These
same lines also cut the circular are, f a g, in the points, j, a, b, g,
which are projected upon the line, f' g', the horizontal projection
of this line, in the points, j', a', b', a'. By drawing lines thi-ough
these last, and through those first obtained, i', c', d', e, we obtain
the straight lines, c" j', c' a', c* b', c^ g', which are the horizontal
projections of the generatrix, c a, and correspond to the vertical
projections in fig. 1.
These straight lines cut the plane, a' d', in the points, m'/' i',
which are then projected vertically to m,/, i, upon the straight lines,
c J, c a, c b ; the curve, a ni/i d, passing through each of these
points, is the line of intersection sought, and it is reproduced sym-
metrically at f b, to the left hand of fig. 1, so as to avoid the neces-
sity of repeating the diagram.
To obtain this line of intersection full size, it is necessary to
bring the plane, d' a', into the plane of the picture, by supposing
it to turn about the vertical, d q, projected horizontally in d', as an
axis ; during this movement, each of the points, a', m', /', mil
describe an arc of a circle about the centre, d', finally coinciding
•with the points, M^ a", f, and -P. Through the corresponding
points. A, M,/, i, in the vertical projection, we must draw a series
of horizontal lines, a a", m m',//", upon which, square over the
preeiiding points, m", A^_p, i^, by which means will be obtained the
cm-ve, a" m" f d, representing the exact form or parallel projection
of the line of intersection.
376. The preliminary design thus sketched out, gives nothing
but the outline of the surface of the erection, and it now remains
to divide it into a certain number of parts, to represent the indivi-
dual stones of which it is to be built up.
The number of divisions necessarily depends upon the nature of
the material and sizes of stone at the mason's disposal ; the number
snould, however, in all eases, be an odd one, so that a central
space may be reserved for the principal piece, known as tlie key-
stone.
The di\dsions are struck upon the semicircle, b e d, by a series
of radii converging in the centre, c ; it is these lines which repre-
sent the divisions of the stones. Below the arched part, the regular
pieces, as X, consist of a series of stones of equal dimensions, the
joints of which are horizontal.
The horizontal projections of each of the stones forming the arch
are straight lines, because the joints lie in planes perpendicular to
the vertical plane, whose intersection with the twisted surface is
always a straight line correspondiag to a generatrix ; thus, the
planes, o c and p c, of the joints on either side of the key-stone, e,
are perpendicular to the vertical plane, and pass through the axis,
c k' ; and the portions of them, n h and n I, comprised between the
directing circles, bed and f k a, are represented in the horizontal
projection by the straight lines, n' h' and o' I', which are the joints
of the stones as seen from below — that is, the lines of their iQter-
Bection with the tw isted surface.
It IS the same wim the planes, m c and j c, in which lie the
joints of the corner stones, t z ; the portions, k g and 1 1»., of tbt
joints falling upon the twisted surface, are likewise projected hori-
zontally at the straight lines. A' g' and m' i'.
377. The design of the erection being thus completed, the shaper
should delineate each individual stone as detached from the arch,
in such a manner as to represent all the faces of the joints, and he
then takes for each, a stone of the most convenient dimensions
from amongst those at his disposal, which are generally hewm out
roughly in the shape of rectangular parallelopipeds; on each of
these pieces he marks off the parts to be cut away, to reduce the
stone to the requu'ed form and dimensions.
Thus, supposing he commences with the key-stone, k, for ex-
ample, detailed in front view' and vertical section through the
middle, in figs. 3 and 4; he takes a parallelepiped, of which the
base, p q r s, is capable of circumscribing the two parallel faces of
the upper part of the key-stone, and of which the height is at lea.st
equal to the length, I u'. After having cut and finished the two
vertical faces, f d and u r, of the prism, as well as the horizontal
face, i! u', he measures off upon the anterior face, fig. 3, the parallel
and vertical sides, t o and u p, and then the oblique lines, o h and
p I, which, it will be remembered, converge to the same point, c,
the axis of the voussure. He next sets off upon a template the
arcs, n o and h I, fig. 1, and reproduces them thence upon the paral-
lelepiped, fig. 3, at n o and h I. After this preliminary marking
out, the stonecutter reduces and takes away all the material upon
the sides of the parallelopiped which lies outside the lines, o h and
p 1 ; these faces being finished, the shape-designer lays out upon
them the lines projected at nh and o I. In order that the form of
this joint may be more easily comprehended, we have brought the
face, p I, into the plane of the picture, representing it in fig. 4^', as
parallel to the plane of projection.
This view, it will be seen, is easily obtained ; for, on the one
hand, we have the line, p' y, equal to k K^ representing the thick-
ness of the wall or of the arch ; and, on the other hand, all the
other dimensions, as projected horizontally in fig. 2, so that the
inclination of the line, d I', can be determined with the most rigor-
ous exactness.
Tliis straight line, as well as the corresponding one on the op-
posite face, o 7i, serves as a guide to the stonecutter in reducing
the twisted portion of the surface of the key-stone, comprised be-
tw^een them ; and as affording a means of verification, it may be
remarked, that this surface should be cut in such a manner, that a
rule or straight edge may be applied to all parts of it, being guided
by the arcs, n o and h I ; the former of which springs from the
point, o', and the latter from the point, I', on the face, p' x,
fig. 4*.
To determine the faces of the joints of either of the two corner
pieces, z, represented in detail, and detached in figs. 5 and 6, but
on which the faces are not represented in then- full dimensions, it
is necessary to proceed in the same manner as before, bringing
each face into the plane of projection — that is, delmeating auxiliary
views of them, as if parallel to this plane.
Thus, to obtain the actual dimensions of the face of the joint
projected at o h, with the point, w, as a centr«, describe a series of
arcs, with the respective radii, o w,n ic, h w, so as to reproduce thj
points, o, ?i, K at 6\ 7^^ h\ upon the vertical, o^ w ; then, by settmg
BOOK OF INDUSTRIAL DESIGN.
12A
off, h^ h', equal to n' h', fig. 2, and joining tlie points, h' n', fig. 7,
we get the inclination of the generatrix line, n" h', which is project-
ed vertically at n h, fig. 1. The form of the joint face is completed
by drawing the horizontal lines, o" 20 , h' z', y' v', and the verticals,
u' v' and z' j/, which last are already given full size in fig. 6. It
will be observed that fig. 7 is on the plate removed a little to the
right of the vertical, o' w ; but this is a matter of no importance, and
is merely done for convenience sake.
The same system of auxiliary projections is applicable to the
determination of the dimensions'of the other face of this piece —
namely, that projected at m g, which is brought round to the hori-
zontal, m' g\ and drawn with full dimensions in fig. 8 ; only, for
this last face, it is necessary to bear in mind the portion of the line
of intersection of the sides with the arched part which it contains,
and which is obtained in its actual propoi-tions, as at a^ m^, by
means of a template formed to the curve, a" m", in fig. 1. The
stone, Y, beneath, of course, contains the remainder of the inter-
sectional curve.
The methods just explained, in regard to the shaping of the key-
stone and one of the corner-stones, may be extended, without difli-
culty, to the remaining portions of this Marseilles arch.
In this application it has been necessary to determine the propor-
tions of the twisted bay of the arch, as well as the faces of the
joints ; but in the more general case of straight bays, such as that
represented in fig. 1"', the operations are considerably simplified,
and the designer has merely to attend to the form of the joint faces,
making use, for this purpose, of the auxiliary projections, as above
described. The delineation of the various parts of this figure pre-
senting no new peculiarity, it need not further detain us.
378. Let it be proposed to delineate a circular vault with a full
centering, bounded by two plane surfaces oblique to its axis, figs.
9 and 10. This example is taken from the entrance to the tunnel
on the Strasbourg Railway, near the Paris terminus, and it is a form
frequently met with in the construction of railways.
In the representation of this vault, we have supposed one of the
oblique planes to be parallel to the vertical plane of projection, and
it consequently follows that the axis of the arch is inclined to this
plane.
Let A B be this axis, and c d the horizontal projection of a
plane at right angles to it; with the point, b, as a centre, describe
the semicircle, cad, representing the arch in its true proportions,
as brought into the plane of the picture. Let us suppose this
semicircle to be divided into some uneven number of equal parts,
as in the points, a, h, c, d, e,/; through each of these points draw
straight lines, passing also through the centre, b, and representing
the joints, a g, b h, c i, of the arch stones, being, of course, normal
to the circular curvature of the arch, and being limited in depth, as
wo shall suppose, by the second outer semicircle, g i I, concentric
with the first. Each of these joint faces intersects the centering
of the arch in a straight line parallel with its axis, and the horizon-
tal projections of these intersections, as seen from Ix'low, arc ob-
tained simply by drawing through the points, a, h, c, d, lines parallel
to the axis, b a ; those last extend as far as the vertical plane, a r,
which bounds a portion of the vault. The external faces of (ho
key and arch stones are limited by straight vertic:»' lines, such as
in h, i n, oj, and horizontals, as m i and n o.
We have now to obtain the projections on the vertical plane,
fig. 10, of the intersection of each of the arch stones by the plane,
A E.
We may remark, in the first place, that since this plane is
oblique to the axis of the cylindrical arch, it produces an elliptical
section, having for its semi-transverse axis the length, c' a, and for
its semi-conjugate axis, the length, a b , equal to the radius, a' b.
As much of this ellipse as is required is drawn according to one
or other of the many methods given (53, ei seq.) — say as at c' b^ b'
fig. 9, which curve is reproduced at c' 0' a", in the elevation,
fig. 10.
If we, in like manner, obtain the projection of the semicircle, f i I,
which limits the radial joints, we shall also obtain the portion of an
ellipse, f" g" i", and we have further merely to project the points,
a', b', c', upon the first ellipse, in a" b" c" ; as also on the second
one, the points, g", h", i", corresponding to g h and i. The straight
lines, f" c", g" a", h" b", i" c", represent the intersections of the
faces of the arch stone joints, with the plane, a e.
The vault being supposed to extend no furthc baok than the
plane, c d, it will be necessary to represent the iniersection of this
last with the arch stones which extend thus far upon this plane,
c D. We have, therefore, to project the elliptic curves, c'" b'" c"
and f'" ^" i'", corresponding to the quarter circles of the radii,
F B and c B. As the arch stones cannot extend the entire length
of the vault, they are limited by planes, m n, perpendicular to the
axis, and, consequently, parallel to c d. so that the projections of
these joints will be but repetitions of portions of the same elliptic
curve : care is taken so to dispose the blocks of stone, that no two
joints form a continuous line, the joints in one course being brought
between those in the adjacent ones, as is customary in all brick and
stone work.
379. We have now to determine the intersection of the oblique
plane, a g, with the remaining half of the same circular vault, a id
then to obtain the projection of this intersection upon the vertical
plane.
The plane, a g, also produces an elliptical section of the vau't;
this is represented at g' gt, as brought into the picture in the auxiliary
diagi-am, fig. 11, which gives its actual proportions; the semi-trans-
verse axis, g' 0, of this ellipse is equal to G a, and its semi-oonjug:ito
axis is equal to the radius, a' b, of the vault.
After having divided this curve into a certain uneven number of
equal parts, draw normals,* p u, q v, r x, s y, and t 2, through the
points of division representing the joints of the arch stones, the
remaining sides of the external faces of which arc limited by l.ori-
zontals and verticals, as before.
If the vault is supjiosed not to cxtciul beyond the piano, a g,
the arch stones will have to be shaped as facing stones, and their
joints will require to be set oil' upon the first ellipse, g' q t, and to
be limited by the second, 11' (/' /^ ohiained from the interscotional
plane, H i, drawn parallel (o ,\ c ; by drawing strmght lines from
the points of division ohtniiinl u|i"n the ellipse, g'(//, to the cenlre.
o, \\e obtain the points, ji', (/^ i'\ .•>'', of iniersection of these line^
uiuin the second ellipse, and thr slraii^lit linos, /. /-', q 7', r r\ s .v'. re-
presenting the intersections of the ;neli >ion<s with the insi.lo of the
• A Hm> Is Hiilit to 1)0 naniiiil to ii ourvi-. wlu-n il Is pi-rpctnllc»lnr lo n t iiwenl ••
tlic curve piisslnit llirmmli Us point of lutiTbccUoii wllh the oiirvo n3).
126
THE PRACTICAL DRAUGHTSMAN'S
vault ; these straight lines are projected horizontally in f j)", q' q",
r' r", &c., fig. 9, where they are visible, because the diagram is
supjiosed to be a projection of the vault, as seen from below ; the
two diagrams, figs. 9 and 11, will render the determination of the
vertical projection, fig. 10, very easy, the same lines there being
designated by the same letters.
To limit the arch facing stones, and unite them conveniently with
the regular courses of the vault, they must be cut by planes, such
as J K and l p, fig. 9, perpendicular to the a.xis, a b. The intersec-
tions of these planes with the vault produce portions of circles, which
are projected as ellipses in fig. 11, such as l' 5° v and u' r* x', for
the inside of the vault, and j' q r' and l' s' p', for their outer ex-
tremities, these various ellipses corresponding to the radii, c b and
F B. The joints of these stones are finally completed by planes,
such as k' p" s q, fig. 11, passing through the axis, a b, and through
horizontal lines, y' z' and o s, in the vault, the latter and inner one
of which only is visible in the elevation, fig. 10.
380. In constructing this vault, it is necessary to make detailed
drawings of each particular stone, showing the dimensions of all the
faces. In figs. 12 to 15, we have represented one of these arch
stones, ©, in plan and elevation, as detached from the erection, fig.
10, and showing more particularly such lines as are not apparent in
fig. 11. Thus, in these views may be distinguished: —
1st. The anterior face, v q r x, which is projected horizontally
upon the line of the plane, a g ; this face, it will be remembered,
intersects the vault at the elliptic curve projected at q r.
2d. The face of the joint, a; r r" w, of which the one edge, x r, is
projected upon the same plane, g a, at x' r', whilst the opposite
edge, w r^ is projected upon the line, v" r", parallel to g a ; and
the lower edge, r r', the line of intersection of this face with the
interior of the vault, is projected in the line, r' r\ whilst, finally, the
upper and fourth edge, x w, is projected at x' v)f.
3d. The second joint face, v q(f w, is opposite to the first, and
projected at v/ ({ 5* w.
4th. The face, q s z y, the horizontal projection of which is
q' s' z' y' ; this face is situated in a plane passing through the axis
of the vault, and is additionally represented in the diagram, fig. 14,
on the radius, p q".
5th. The species of dovetail joint, q q^ y z r^ r, of which the
edges, q q^ and r r\ are projected, as has been seen, at q q* and
/ r" ; whilst the sides, q r and z r', are similarly projected at </ r
and z' r", and finally the side, q" y a.t q* y'.
6th. Lastly, the posterior face, q, of which it will be easy to
render an account by means of the distinctive letters, which are
invariably the same for the same points, although additional special
marks are superadded to obviate confusion amongst the various
figures. To render this vertical projection more intelligible, we
nave added the subsidiary view, fig. 14, representing the projection
of the block m the plane, l p, as brought into the plane of the
picture ; we have thus the actual proportions of the faces projected
In the planes, s' r" and p' q". To obtain the various points seen in
lias view, it is sufficient to set oflT the vertical distances from the
Ime, G o, of the elevation, figs. 11 and 13, obtaining in this man-
ner, for instance, the points, r\ q\ v^, x\ &c., corresponding to
r', (f, V, and X.
'^'nd examples chosen for this plate (XXXIV.) combine the more
difficult problems and applications met with in the shaping and
arrangement of stonework, and will make the student acquainted
with the operations upon which designs for these purposes ai-e
based, as well as with the general methods to be adopted La obtain-
ing oblique projections by the employment of auxiliary projections,
taken, as it were, in planes parallel to the different surfaces, and
then brought into the plane of the picture ; this system, at the same
time, being of much use in ascertaining the exact proportions of
various surfaces, such as the joints of masonry.
RULES AND PRACTICAI. DATA.
HYDRAULIC MOTORS.
381. The fall of a stream of water varies with the locality, and
gives rise to the employment of different kinds of hydraulic motors,
which are denominated as follows, according to their several pe-
culiarities.
First, Undershot water-wheels, which receive the water below
their centres, and the buckets or floats of which pass through aa
enclosed circular channel, at the part where the water acts upon
them.
Second, Overshot water-wheels, which receive the water from
above.
Thud, Wheels with vertical axes, known as turbines, and which
are capable of working at various depths.
Fourth, Water-wheels, with plane floats or buckets, receiving
the water below their centres, and working in enclosed channels,
through a portion of their circumference.
Fifth, Similar wheels, with curved buckets.
Sixth, Hanging wheels, mounted on barges, and suspended in
the current.
UNDERSHOT WATER-WHEELS, WITH PLANE FLOATS AND A
CIRCULAR CHANNEL.
382. The most advantageous arrangement that can be adopted
in the construction of an undershot water-wheel, with plane floats,
and working in an enclosed circular channel, is that in which the
outlet is formed by an overshot sluice-gate, and when the bottom
of this outlet is -2 to '25 m., or about 8 inches, below the general
level of the reservoir.
Let it be required to determine the width of an undershot water-
wheel, with the following data : —
First, The discharge of water is 1,200 litres per second.
Second, The height of the fall is 2'475 metres.
Thii-d, The depth of the water at the sluice-gate is to be -23 m.
WIDTH OF THE WHEEL.
It will be seen, in the table at page 113, that, with an outlet of
■23 in depth, a discharge can be effected of 188 litres of water per
second, for a width of 1 metre ; consequently, the width to be
given to the sluice, to enable it to discharge 1,200 litres per second,
should be —
1200 -T- 188 = 6-38 metres.
BOOK OF INDUSTRIAL DESIGN.
la:
DIAMETER OF THE WHEEL.
383. The diameter to be given to a wiieel of this description has
not been accurately determined, because it has not a direct influence
u[>f.n the useful effect that may be obtained from it. Nevertheless,
it is manifest that it should not be too small ; for in that case the
water would be admitted too nearly in the horizontal line passing
through the centre, or even above it, which would cause great loss
of power. Neither should it be too great, for in that case the ex-
aggerated dimensions would but involve an increased bulk and
weight, and, consequently, a greater load and more friction, without
any compensating advantage.
In general, for a fall of from 2 to 3 metres, it is advisable to make
the extreme radius of the wheel at least equal to the mean height
of the fall, augmented by twice the depth of the water upon the
edge of the outlet.
Thus, in the case before us, the height of the fall being limited
to 2-476 m., the outer radius of the wheel should not be less than
2-475 m. plus twice the depth of the overflowing body of water
when at its fullest — say -6 m.; that is to say, in all, 3-075 m., which
corresponds to a diameter of 6-15 metres.
Water-wheels, on the same system, with a fall of water of from
2-6 to 2-7 metres, have often an extreme diameter no greater than
tlus.
VELOCITY OF THE WHEEL.
384. Theoretically speaking, the velocity which it is convenient
lo give to an undershot water-wheel should be equal to half tiiat
due to the height of the overflow of the water ; that is to say, equal
to from r to 1-1 m. in the present case. Nevertheless, practice
shows that this rule may be departed from without inconvenience,
and the wheel may be made to attain a velocity of from r5 to 1 -6 m.
per second at pleasure, which is a very great advantage in many
circumstances.
If the wheel makes three turns per minute, the mean velocity at
the outer circumference, and at the edges of the floats, will be —
6-15 X 3-1416 X 3
60
= 1-021 m. per second.
Thus, when the height of the overflow is -24 m., the correspond-
ing velocity of the water being 2-17 m. nearly, as shown in the
table at page 94, which gives the heights, 23-66 and 24-67 cent.,
iherefore, the ratio of the velocity of the wheel to that of the water
is -47 : 1.
If the height of the overflow were reduced to -15 m., which sup-
poses that the discharge would only be
101 litres x 6-32 m. — 638 litres per second,
the corresponding velocity of the water would not be more than
1-72; and in this case, the ratio of the velocity of the wheel, sup-
posing it to be still the same, to that of the water, would be —
•5^5 : 1.
NCMBEF AND CAPACITV OF THE BUCKETS.
385. Although the number of buckets cannot bo delerniiiicd in
accordance with any exact rule, it is, ncvcrtiielcss, of importance
that their pitch should not be much greater than the depth, or
thickness, of the overflowing body of water iicting upon them. It
is also necessary that the number of the buckets should be divisible
by that of the arms of the wheel, so that the whole may be put
together conveniently.
Now, since the outer circumference of the wheel is
6-15 X 3-1416= 19-32 metres,
we can very conveniently give it 8 arms and 64 buckets ; and the
pitch of these last will be -32 m. With this distance between the
buckets, there should not generally be a greater depth of overflow
than -25 or -26 m. ; because, at -27 m. the water will begin to choke,
as it will not be admitted easily into the buckets, and will rebound
against the interior of the channel, giving rise to a continual shak-
ing action.
Thus, then, in determining the number of buckets for an under-
shot water-wheel, receiving the water from an overshot outlet, it is
necessary to calculate the spaces between them, so as to be about
a third, or at least a fourth, gi-eater than the depth of the water at
the outlet, whilst their number must be divisible by the number of
arms of the wheel.
For water-wheels of from 3-5 to 4-75 metres in diameter, six
arms for each rim or shrouding ; for wheels of 5 to 7 metres in
diameter, there should always be eight arms for each ishrouding ;
and the number of arms should obviously increase for wheels of
greater diameters than 7 metres, of which, however, there are but
few examples.
With regard to the capacity of the buckets, and the channel,
taken together, it should be equal to at least double the volume of
water discharged. Therefore, on this basis, we can always easily
determine the depth to be given to the buckets, when the maximum
discharge is known.
Thus allowing, in the present instance, the maximum discharge
to be 1,340 litres per second, instead of 1,200, since the velocity at
the outer circumference is 1-021 m. per second, the number of
buckets contained in this space is equal to
1-021 ^-32= 3-19.
Then—
1-340 H- 3-19 = -43 cubic metres nearly,
the quantity which should be in each bucket during the revolutioa
of the wheel. If, then, the capacity is to be double this, it will be
equal to -86 cubic metres. The product, however, of the width,
6-38 m., of the wheel, multiplied by the space between two consecu-
tive buckets, -32 m., is equal to 2-022 m.
We have, then, -86 -r- 2-022 = -42 in., for the depth of the
buckets. The distance between the buckets, however, is not tho
same at the inside as at the extremities, and the capacity is also
further diminished by the thickness of the sides of tho buckets, and
by the inner portions, which make an angle of 15 ' with the outer
portions. For these reasons, tho depth should be somewhat
increased. When tho discharge of water is considorable, and w <>
arc limited as to the width of the wheel, it is prel'erablo to do awav
with the inner inclined iiortion of the buckets, as indicated in tho
drawiii"-, Plate XXXVl., prolonging tlu'in i-onsiderahly toward"
the centre of the wheel.
USEFUL EFFECT OF THE WATKlt-WHEEL.
386. The absolute force of a stream of water is the product
of the water discimrgcd per second, exiiressid in kilog;nunmes, t»*
128
THE PRACTICAL DRAUGHTSMAN'S
the height of the fall expressed in metres, or the weight in pounds
by the height in feet.
Thus, when the discharge is 1300 litres or kilog. per second, and
the total height of the fall 2-475 m., the product of 1300 kilog. by
2-475 m. expresses in kilogrammetres the absolute force ; this may
be converted into horses-power by dividing the result by 75 : we
have, therefore —
1300 X 2-475 = 3217 km., and 3217 -=- 75 = 43 horses-power.
Undershot water-wheels, with plane-bottomed buckets and circu-
lar channels, when well constructed, are capable of utilizing from
70 to 75 per cent, of the absolute force of a stream of water.
OVERSHOT WATER-WHEELS.
387. Let it be proposed to construct a water-wheel to receive the
water from above, under the following circumstance : —
1st. The effective vertical height of the fall, or the distance
between the upper and lower level, is 4-56 m., without sensible
variation.
2d. The quantity of water discharged per second is supposed to
be almost uniform, and is measured by a vertical sluice-gate, with
complete contraction at the outlet.
3d. The width of the sluice-gate is "5 m., the height of the open-
ing -14 m., and the charge or height of the reservoir level above
the centre of the outlet, -55 m.
Solution. — From the table at page 111, of the discharges of
water, we find that 280 litres per second is the quantity which
escapes at an orifice -14 m. in height, by 1 metre wide ; and with
a pressure upon the centre due to a height of -55 m.. We have, con-
sequently,
280 X -5 = 140 litres.
This discharge being known, if we are not limited to any parti-
cular width of wheel, it may be constructed thus, for it gives
as great a useful effect as can be expected in ordinary circum-
stances.
In such case, the velocity, v, should be regulated to about one
metre per second at the circumference, because the advantage that
might result from a less velocity would be counterbalanced by the
consequent increase in the width of the wheel.
If we adopt the velocity, v, of 1 metre, we find that V, of the
water, at its point of escape from the outlet, should be 2 metres per
second, to act with proper effect upon the buckets ; now it %vill be
seen in the table, at page 94, that this velocity corresponds to a
height of -205 m. above the centre of the orifice.
This height has to be deducted from the total fall.
For small discharges of water, it is advisable to make the height
of the orifice as small as possible, so that the depth of the water
may be trifling, which will permit of its entering the buckets much
more freely : it may be taken at -06 m., or H ^ -06 m.
The half of this height, or -03 m., must be added to the first
-205 m., to give the whole height of the water in the duct behind
the outlet ; that is, from the upper level to the lower edge of the
orifice.
Taking also -01 for the extent of the trifling fall of the small
spov?* i-eafhing from the front of the outlet to the top of the
wheel, and -01 m. for the play-space which may be supposed to
flidst between the end of the spout and the latter ; after deduct-
ing all these quantities from the total fall, namely, 4-56, we shall
have remaining —
4-56 — (-205 + -03 +-01 4- -01) = 4-305 m.
for the extreme diameter, d, of the wheel.
The channel which conducts the water to the wheel, and the
width of the outlet orifice, should be disposed as much as possible,
so that it may not meet with contraction from the lateral or bottom
edges of the sluice-gate. Referring again to the table on page
111, the discharge, 140 litres, must be divided by the number 76,
corresponding to the height -06 m., and to the charge -2 m. ;
it must also be divided by the coefficient, 1-125, when we shall
have —
140
_ ~ 1-125= 1-66 m.
75
for the width of the outlet orifice.
By adding -1 m. to this, we have 1-76 m. for the width of the
wheel.
The depth of the buckets is determined thus : —
-140 m.
d =
8 X
= 214 m.;
3 X 1-78 m. X 1 m.
consequently, the internal diameter, d, of the wheel becomes —
d' — 4-305 — (-214 X 2) = 3-877 m.
By augmenting this depth about a fifth, which will make it
•257m., we get the distance to be allowed between the buckets;
so that, as the internal circumference is equal to —
3-14 X 3-877 = 12-174 m.,
dividing this by -257, it becomes
12-174 , ,
.t,K„ = 47-3 ; say 47 buckets.
For a water-wheel, however, of 4-305 metres in extreme diameter,
there should be eight arms ; and if it is intended to make the
shroudings of cast-iron, and in segments, it is advisable that the
number of buckets be divisible by 8; it will, therefore, be convenient
to have 48 instead of 47, and in this case the space allowed be-
tween each will be reduced to —
12-174-^48 = -254m.
It now merely remains to draw the wheel ; for this purpose, the
concentric internal and external circles are described ■with the de-
termined radii : the first is tlien to be divided into 48 equal parts,
and radii are drawn through each point of division, as indicated in
Plate XXXVI. ; on each of these, outward from the internal
cu-cle, is marked off a distance equal to a little more than half the
depth of the buckets, say -12 m., to indicate the bottoms of the
buckets.
The water-wheel, when constructed in this manner, may give
off 79 or 80 per cent, of the absolute force of the fall of water.
Now this force, expressed in horses-power, is equal to —
140 x 4-56
75
= 8-51 horses-power.
Deducting 5 or 6 per cent, at the most, for the friction of water-
wheel shaft in its bearings, we may still calculate, with certainty,
that the power utilized and transmitted by this wheel will be equal
to 74 or 75 per cent., or
8-51 X -75 = 6-38 horses-power.
The number of revolutions which this wheel should make per
minute is —
BOOK OF INDUSTRIAL DESIGN.
129
60 -=- 4-305 X 3-14 = 4-44,
since its velocity, v, is 1 metre per second, or 60 metres per
minute.
In tracing out the preceding solution, it will have been seen that
the width to be given to the wheel is 1"76 m. ; a much less width
might have been obtained, by making the wheel revolve faster, and
by augmenting the velocity of the water also. Let us suppose, for
example, that the question lias to be solved on the hypothesis, that
the velocity of the water-wheel is to be 1'5, instead of 1 metre, per
second; it wUl then be necessary, in order that the water may
escape from the orifice at double this velocity, that it be equal to 3
metres per second.
For this velocity, the height of the upper level, above the centre
of the orifice, should be '46 m.
Allowing "06 m. for the height of the open part of the sluice-gate,
the whole height above the wheel will be
•46 -1- -03 + -02 = -51 metres ;
consequently, the outer diameter of the latter should be
d = 4-56 — -51 = 4-05 metres,
the width of the sluice-gate, or
w = = I'll metres,
•06 X 3 X -7
and consequently the width of the wheel
= 1-11 4- -10 = 1-21 metres.
This width, it will be seen, is considerably less than that first
calculated. This wheel, however, which is narrower, and revolves
at the rate of 1'5 metres per second, will not be capable of trans-
mitting so great a useful eflfcct, by four or five per cent. Never-
theless, it may be preferable in many circumstances to adopt this
lesser width, either to render the wheel lighter and less costly in
construction, or to avoid the necessity of much intermediate gear
between the wheel and the machinery to be actuated. Thus, it is
evident that this wheel should make
(60 X 1-5) -r- (4-305 X 3-14) = 6-66 revolutions per minute,
whilst the first wheel only made 4'44.
The other parts of the wheel are proportioned according to the
above rules ; they will, however, differ but slightly from those of
the first wheel.
The proportions of the water-wheel might still be otherwise
modified; thus, the depth of water at the outlet might be allowed
to be greater than that taken for a basis in the preceding cal-
culations. Thus, the outlet might be opened to the height of
•1 m. instead of only -06 m. : in this ease, the width of the outlet
and of the wheel would be much less. But this arrangement
would have many disadvantages, for it would be necessary to make
the buckets more open ; that is to say, the angle made by the
outer portion of the bottom of the bucket with the tangent to the
circumference passing through its extremity, instead of being 15°
or 16°, as is usual, would have to be 30^^ or 32°; the buckets
would have to be deeper and more capacious ; they would empty
thctnsiilvcs sooner: from all which causes would follow a decrease
in the useful eH'cct given out, which might roach oven to 15 per
cent.
It la true, on tlio other hand, that the width of the outlet would
bo reduced to 1 metre, supposing the wheel to revolve at the
rate of 1 metre per second, and that it would not be more than
•67 m. when the wheel revolves at the rate of 1-5 m. per second ;
in which case, the depth of the buckets would be about -34, and
the spaces between them -4 m. each.
It will be easily conceived that such an arrangement cannot bo
advantageously adopted, except where there is plenty of water to
spare, and when the constructor is limited as to the width of the
wheel.
WATER-WHEELS WITH EADIAL FLOATS. •
388. In old mills we sometimes meet with water-wheels w^hicti
have plane floats placed radially, working in straight inclined chan-
nels, with a vertical outlet more or less distant from the centre of
the wheel.
These wheels generally give out 25 to 35 per cent, of useful
effect of the absolute force of the stream. In them the floats are
three or four centimetres clear of the sides of the channel ; when
a greater space than this is allowed, the useful effect is sensibly
diminished. Generally, the width of such wheels is equal to that
of the outlet.
At the present day, water-wheels are never constructed with
plane floats arranged in this way. When a wheel is required to
have a great velocity, it is preferable to construct it to work in an
enclosed circular channel, and to receive the water from above, or
from an orifice with a sufficient column above it, to give the proper
tionate velocity to the water.
Such wheels are constructed in the same manner and with the
same care as undershot water-wheels; in fact, they do not differ
from the latter, except in that these receive the water from an open
topped or overshot duct. The useful effect given out by them
varies from 40 to 50 per cent., according as the sluice-gate is more
or less near to the upper level of the water. Thus, the nearer the
channel approaches to the upper level, the more like the wnoel
becomes to a common undershot one, and, in consequence, the use-
ful effect is greater.
In the construction of a water-wheel of this kind, the same rules
are followed as are already laid down for conimou undershot w heels
with open outlets.
Thus, let it be proposed to construct a wheel for a i;ill of 1-75
metres, and with a discharge of 440 litres of water per second : lot
the centre of the outlet orifice bo at '4 m. below tlie upper level,
and the height of the orifice itself, 'IS m.
By referring to the table on page 111, it will be scon that t\\r
discharge of water through an orilice, under those circunistancos, is
255 litres per second for a width of one metre, and it w ill thorel'oro
be evident that the wlieol should have
440
■rr-p = 1-72 metres in wultli.
255
The velocity of the water at the sluico-gale, corresi)ondin;j to
the column of '4 in., is 2802 per second ; consequontly, if wo uiake
the Ncldoily of ihi' whorl oiiual (o •,').') liinos tiiat of tho water, it
will bo
'J'S02 X '55 = rSl nu^lros por second.
The ili.-unolor of ilio wliool is of itsolf a mailor of inditVorenco ;
it .slmuM lio rocluroil ;is imioli as possililo, so ii.s to lessen tho cost
of consli lution ; iiolw iilistuudiiig, it should never bo loss liiiui iw ioo
130
THE PRACTICAL DRAUGHTSMAN'S
2-7 metres.
llie wholi) height of the fall ; thus, in the present example, it
should not be less than 4 metres.
It has often been asserted that the power is increased by in-
creasing the diameter; it seems incontrovertible, however, that the
power transmitted must be in proportion to the height of the fall,
and to the quantity of water discharged. If the diameter of the
wheel is increased, the angular or rotative velocity is diminished,
and, consequently, the momentum and actual force communicated
remain the same.
Taking the diameter at 4 metres, we have
■ • = 7'2, the number of revolutions per minute.
4 X 3-1416 ^
If a wheel of this diameter were adapted to an open-topped or
overshot duct, with a depth of water at the sluice-gate equal to
•2 m., the velocity of the water being then reduced to 1-981 metres,
The velocity at the circumference of the wheel would not be more
than
1-981 X -55 = 1-09 m.,
and, consequently, the number of turns only
1-09 X 60
4 X 3.1416 = 5-2 per minute.
But then, as the discharge in such case, at an overshot outlet of
•2 m. in depth and 1 metre in width, is 166 litres per second (see
table, page 113), the width of the wheel must be made equal to
440
166
Thus, it will be seen that the water-wheel which revolves more
rapidly is narrower than the one with the same discharge of water
by an open outlet, and it is, consequently, less costly in construc-
tion ; but then, it only gives out, as useful effect, about 50 per
cent, of the absolute force of the stream, whilst that given out by
the other description may reach, as we have seen, as much as 70
per cent.
With regard to the other dimensions of the wheel, we have
merely to refer to what has been said about the common under-
shot water-wheel.
WATER-WHEELS WITH CURVED BUCKETS.
389. These wheels are fitted with inclined ducts for the water,
the inclination being equal to a base of 1 metre for every 1 or 2
metres in height — that is to say, to 45 to 600 ; they are enclosed
for a short distance Ln a circular channel, and between two side
walls.
They are seldom constructed except for low falls of from -5 to
1-3 metres, and when a great velocity is requii-ed ; the useful effect
they give out varies from 45 to 65 per cent.
It is of importance that the water duct be brought as close up to
the circumference of the wheel as possible, and that, at its lower
part, it should have an enlargement of 10 or 15 centimetres, to
facilitate the disengagement of the water, and render its action
freer ; this enlargement should commence at a distance from the
vertical line passing through the centre of the wheel, equal to the
space between two consecutive buckets. The velocity of the
wheel should be from -5 to -55 times that of the water at its exit
"rom the duct.
The width of these wheels is to be calculated in the same manner
as that for the preceding ones ; as to the diameter, it may be re-
duced in proportion to the fall, but it should never be less than
threo times the height of the latter.
The depth of the curved buckets, or the width of the shrouding
in the direction of the radii of the wheel, should be equal to mie-
fourlli of the fall augmented by the height of orifice open.
For falls of less than 1-2 m., the height of the orifice, or the depth
of the outflowing water, should be from -2 to -22 m. ; it may be
reduced to -18 or -16 m. for falls of from 1-2 to 1-5 m.
The buckets or floats are in the form of a cylindrical curve,
being a single circular arc, tangential to the radius at the inner part,
and making an angle of about 24° or 25° with the stream of water
flowing towards the inside of the crown of the wheel. The space
between two consecutive buckets is measured at the outer circum-
ference by an angle of 25°, and their thickness is 24 to 28 hun-
dredths when made of wrought-iron plates, and 32 to 35 when of
wood. The bottom of the channel should have a fall or inclina-
tion of about -jLth or y'jth — that is to say, equal to that of the
hypothenuse of a triangle, the base of which is 12 or 15, and the
height 1 metre.
TURBINES.
390. Among the varieties of turbines which receive the action
of the water throughout their whole circumference, may be distin-
guished those which discharge the water at their outer circumfer-
ences, and those which allow it to escape behind. The useful effect
given out by these wheels varies from 55 to 65 per cent, of the
absolute force of the stream of water.
For these descriptions of wheel, the discharge of water is cal-
culated in accordance with the rules and tables already cited. For
the first kind, termed centrifugal turbines, the internal diameter is
determined by multiplying the fourth or fifth of the velocity due to
the total fall by 785-4 ; then dividing the quantity of water to be
discharged by the result obtained, and finally extracting the square
root of the quotient.
Example. — Let us suppose that the fall is 2-2 m., and the dis-
charge of water 800 litres per second. It will be gathered from
the table on page 111, that the velocity due to the height,
2-2 m. — 6-57 m.
1-314;
We have then,
6-57
4
- 1-642, and ^'f^ -
5
and further.
D=>
/ ^^°
V 785-4 X 1-642 "
or, ■
./
r>
800
787
for the internal diameter of the cylindrical tank above the turbine.
Add 4 or 5 centimetres for the Internal diameter of the latter,
which gives
•82 to -91 m.
The external diameter should be equal to the internal diameter
multiplied by 1-25 or 1-45, and is, therefore,
1-025 to 1-189 m.;
BOOK OF INDUSTRIAL DESIGN.
131
or,
1-137 to 1-319 m.
When the height of the fall and the discharge of water are
■\-ariable, the diameters should be calculated for the extreme cases,
so that the most advantageous proportion may be adopted — that is,
the one which will give the best result throughout the greater part
of the year.
If the variation is very considerable, there should be two or more
(urbines employed, some calculated for the lowest discharges, others
for the mean, and others again for the maximum discharges.
The height of the buckets — that is to say, the vertical distance
between the two discs which form their top and bottom — is gene-
rally about a fifth, or a fourth at most, of the radius of the interior
of the wheel.
Thus, in the case before us, the diameter being -787 or -874, the
radius is -3985 or -437, and, consequently, the height of the buckets
should be -1 to 11 m.
The buckets being cylindrical in form, their entrance is normal
to the conducting channels which direct the water against them,
and for these low discharges of water should make angles of 68° to
70° with the internal circumference of the wheel — that is to say,
the conducting channels should make angles of 20° to 22° with the
circumference. When the discharges are large, this angle may be
increased to 30° or 45° ; thus, for a discharge of 600 to 700 litres
per second, it is considered that the angle should be about 30°.
In order to obtain the maximum useful effect, the velocity of the
wheel should be equal to about -7 times that of the water ; in prat>
tice, one-tenth may be added to this ratio, or one-fifth to one-sixth,
without materially diminishing the useful effect.
The space between each bucket, taken at the internal circum-
ference, should be nearly equal to the distance between the top and
bottom discs of the turbine; it should, however, never exceed 18
to 20 centimetres. The internal and external distances between
the buckets are necessarily in the ratio of the internal and external
diameters of the wheel.
In the following table we give the principal dimensions, data, and
results of several descriptions of tui-bines, constructed, within the
last few years, by MM. Fourneyron, Fontaine, and Andre KcEchlin.
These results have been selected under circumstances where the
best useful effects were given out ; —
FABLE OF DIMENSIONS AND PRAUTICAL RESULTS OF VARIOUS
KINDS OF TURBINES.
Data and Rpsults.
Tiiial fiill,
Discliarf^e per second
Kxtttrnal diami.'ter,
I-)<'plh of shrouding,
IIiMKlit of outlet,
NiiiiiliLT of l>uckels,
NijinlM:r of director curves
i>luijiljiT of revolution.H per minute,
Usr-ful effect
Ratio of useful effect to absolute
force, ■
Names
of the Turbines and c
Constructors.
f their
Nfoussay
Mulbach
Bouchet
Viidenay
Turbine.
Turbine.
Turbine.
Turbine.
Fourneyron.
Fourn«yron.
Fontaine.
Fonlolni'.
7-Ofi m.
.■i-45 m.
100 m.
r40 m.
52T lit.
2.500 lit.
218 lit.
1400 lit.
■850 m.
1-9 m.
1 33 m.
1-940 m.
•110 m.
335 m.
•23 ni.
•071 m.
■270 m.
•04 ni.
32
58
B4
24
24
32
185
55
30
35 II. v.
90 11. i>.
2 II. P.
18 11. P
7U°.'o
70O/o
Tl-Zo
Tl"/,
Data and Results.
Total fall,
Discharge per second,
Exteroal diameter, ,
Width of buckets,
Number of buckets, ,
Area of outlets,
Area of escape outlet below the wheel
Number of revolutions per minute,. . . ,
Useful effect,
Ratio of useful effect to absolute force,
Jonval Turbines, constructed byM
Andre Kcechlin, Mulhonse
2-720
m.
684 lit. 1
•800
m.
•410
m.
16
•290
sq. m.
■450
sq. m.
90 to 158 1
13
H. P.
2-77 m.
470 lit.
•800 m.
.100 m.
18
•220 sq. m.
•45 sq. m.
90 to 168
15 H. p.
72°/„
r70 m
355 lit.
•810 m.
•120 m.
18
0706 sq. m.
■2977 sq. ni.
90
6 H. P.
72°/„
REMARKS OX MACHINE TOOLS.
VELOCITY OF THE TOOL, OR OPERATING PIECE, IN MACHINES
INTENDED TO WORK IN WOOD AND METAL.
391. The principal machine tools, employed in machine shops,
are —
1. The simple lathe, the self-acting lathe, and the wheel-cutting
lathe, with adjustable table.
2. Boring machines of various dimensions, and radial drUiing
machines.
3. Horizontal and vertical shaping machines.
4. Planing machines with a fixed tool, or with a moveable one,
so as to work both ways.
5. Mortising or slotting machines, having a vertical tool with a
revolving table below.
6. Machines for finishing nuts and screws.
7. Machines for cutting screws and bolts.
8. Dividing engines, for dividing and cutting toothed-wheela
of all dimensions.
9. Straight and curved shears, for shearing plates.
10. Punching and, riveting machines.
11. Steam and other hammers.
12. Straight and circular saws.
The velocity of the cutting tools, in tluse niacliincs, varie;.
according to the nature of llie material, and tin- quality of work
desired.
In general, for soft cast-iron, it is convenient to give a velocity
of seven to eight ceiitiinetres per second to ihn tool, in such ma-
chines as latlius, and planing and slotting machines. This velocity
should be reduced, at least, to four or five centimetres in shaping,
drilling, and screwing machines. W'Irii the cast-iron is liani, the
velocity is considerably diminished.
For wrought-iron, the velocity may be advantagoously increased
one half, because the tool is kci)t well lubricated with oil, or with
soap and water ; thus, in turiiiug or planing, the velocity may bo
raised to eleven or twelve cent iiiu^'l res : and iu shaping and screw-
ing, to about six ceiitiiiiL'tres per second.
For copper, bra.ss, and other analogous iiiotiils, with which llu'
tool (hies not become heated whilst working, the velocity may be
very much greater; and for wood, its only limit.s are tiioso dotor-
minod by the si/c tif the tool, and by the pnwors of Uio nmciiino.
Witii regard to the pressure and rate of ndvanco of tho tool jht
revolution, or per stroke, it necessHrily varies uccoi-ding to tho
dimensions of the macliine itaell', and ulsi. iiccordinjf to tlio dogn<o
y
132
THE PRACTICAL DRAUGHTSMAN'S
of finish vvhich is to be given to the surface ; we evidently cannot
give as much pressure to the tool upon a small lathe as to that
upon a large one — to a small drill, as to a powerful shaping
roachme. This variation extends, for the different metals, from a
tenth of a millimetre, in some cases, to as much as two millimetres
in others. Amongst other things, the following table shows tho
rotative velocity to be given to the tool — when it revolves, and ths
work is fixed; or to the latter when it revolves, and the cutting
tool does not ;^in lathes, and shaping and drilling machines.
TABLE OF VELOCITY AND PRESSURE OF MACHIIfE TOOLS OR CUTTERS.
Turning.
Drilling an
d Shaping.
Diameter
Number
Work performed per hour
Number
Work performed per hour
in
of revolutions
with
of revi.
lutions
wi
th
centimetres.
per minute.
i ""/m of pressure.
per minute.
i "Im of
pressure.
Cast
Wrought
Cast
Wroug-ht
Cast
Wrought
Cast
Wrought
Iron,
Iron.
Iron.
Iron.
Iron.
Iron.
Iron.
Iron.
cent.
cent.
cent.
cent.
1
152-9
229-4
458-5
687-8
76-4
114-6
229-2
343-9
2
76-4
114-6
229-2
343-9
38-2
57-3
114-6
171-9
3
50-9
76-4
152-8
229-2
25-5
38-2
76-4
114-6
4
38-2
57-3
114-6
171-9
191
28-7
57-3
85-9
6
30-6
45-8
91-7
137-5
15-3
22-9
45-8
68-7
6
25-5
38-2
76-4
114-6
12-7
19-1
38-2
57-3
8
19-1
28-7
57-3
85-9
9-5
14-3
28-6
42-9
10
15-3
22-9
45-8
68-7
7-6
11-5
22-9
34-3
12
12-7
19-1
38-2
67-3
6-4
9-5
19-1
28-6
15
10-2
15-3
30-5
45-8
5-1
7-6
15-2
22-9
With 1 "/m of pressure.
Withln'lm
of pressure.
20
7-6
11-5
45-8
68-7
3-8
5-7
22-9
34-3
25
61
9-2
36-6
55-0
3-0
4-6
18-3
27-4
30
5-1
7-6
30-5
45-8
2-5
3-8
15-2
22-9
35
4-4
6-5
26-1
30-0
2-2
3-3
13-0
19-6
40
3-8
5-7
22-9
34-3
1-9
2-9
11-4
17-1
45
3-4
5-1
20-3
30-5
1-7
2-5
10-1
15-2
60
31
4-6
18-3
27-4
1-5
2-3
91
13-7
55
2-7
4-2
16-2
24-9
1-4
2-1
8-2
12-6
60
2-5
3-8
15-2
22-9
1-3
1-9
7-6
11-4
65
2-3
3-5
14-1
21-1
1-2
1-8
7-0
10-5
70
2-2
3-3
13-0
19-6
11
1-6
6-5
9-7
75
2-0
3-0
12-1
18-3
1-0
1-5
6-0
9-0
80
1-9
2-9
11-4
17-1
•9
1-4
5-7
8-5
90
1-7
2-5
10-1
15-2
•8
1-3
5-0
7-6
100
1-5
2-3
9-1
13-7
•8
11
4-6
6-8
110
1-4
2-1
8-2
12-6
•7
1-0
4-1
6-2
120
1-3
1-9
7-6
11-4
•6
•9
3-7
5-7
130
1-2
1-8
7-0
10-5
•6
-9
3-4
5-2
140
1-1
1-6
6-5
9-7
•5
•8
3-2
4-8
150
1-0
1-5
6-0
9-0
•5
•8
3-0
4-5
175
•9
1-3
5-1
7-8
•4
•6
2-6
3-9
200
•8
11
4-5
6-8
•4
•6
2-2
3-4
225
•7
1-0
4-0
6-0
•3
•5
1-9
3-0
250
•6
•9
3-6
5-4
•3
•4
1-8
2-7
275
•5
•8
3-3
4-9
•3
•4
1-6
2-4
300
•5
•7
3-0
4-5
■2
•4
1-5
2-2
350
•4
•6
2-5
3-9
•2
•3
1-2
1-9
400
•3
•5
2-2
3-4
•2
•3
1-1
1-6
This table will serve as a guide in designing machine tools for
!ne various combinations of movements, the application of which
may be called for according to the nature and dimensions of the
work to be submitted to their action. Thus, a lathe which is only
intended to turn articles of from four to twenty or thirty centi-
metres in diameter, should have a considerable rotative velocity,
whilst one that is to be cliiefly applied to turning and shaping
bulky pieces, or such as measure from one to two metres in dia-
meter, should, on the contrary, be actuated by a combination of
very slow, but, at the same time, very powerful movements.
BOOK OF INDUSTRIAL DESIGN.
133
CHAPTER X.
THE STUDY OP MACHINERY AND SKETCHING.
VAEIOUS APPLICATIONS AND COMBINATIONS.
392. Hitherto we have had to occupy ourselves with industrial
drawing, as regards only the geometrical delineatfon of the princi-
pal elements of machinery and architecture. This preliminary study
being of great importance, we have thought it well to dwell more
particularly upon it, since also it is the very basis of all designing,
with a view to actual construction, comprehending not only the
mere outline of objects, but also the proportions between the vaii-
ous pai'ts, as dependent upon the functions which each is required
to perform.
Machines are, indeed, but well calculated and thoughtfully ar-
ranged combinations of these elements, and afford innumerable
applications of the rules and instructions laid down in reference to
them. The study, therefore, of machines in their complete state,
naturally suggests itself as the next step to be taken.
393. Machines may, in general, be classified under three catego-
ries — machine tools, productive or manufacturing machinery, and
prime movers.
By machine tools are meant those by the instrumentality of
which we work upon raw materials, as wood, metal, stone ; lathes,
wheel-cutting machines ; drilling, boring, and shaping machines ;
mortising, slotting, planing, and grooving machines ; riveting ma-
chines; shears, saws, hammers — are of this class. The movements
)f these machines should be so combined, that the tool or cutting
instrument — that is, that part which attacks the material — should
move wdth a velocity properly proportioned to the nature of the
work.
In the few notes accompanying our text will be found some
experimental deductions, which may serve as guides for adjusting
the movements in designing and constructing machinery of this
description.
Amongst productive or manufacturing machinery, are comprised
spinning, weaving, and printing machines; pumps, presses, corn,
and oil mills ; and, finally, prime movers consist of those worked by
animals ; windmills, \\'ater-wheels, turbines, and steam-engines.
For the study of complete machines, we have selected from each
of these categories those possessing most interest and generality —
as a drilling machine, an instrument so very useful and so much
employed in machine-shops and railway works; a pump for raising
water, serving for domestic purposes as well as for important manu-
facturing establishments ; two examples of water-wheels, showing
various arrangements and forms of floats or buckets ; a high pres-
sure expansive steam-engine, with geometrical diagrams, deterniin-
ing the relative positions of the principal pieces in various ciicuin-
stances ; and, finally, a set of belt-driven flour mills, coiislruclcd on
a system recently adopted.
Before proceeding to the description of these machines, it will
bo necessary to habituate the student to draw from the rc^iliiy,
for up to the present time ho will have done notiiing but cdiiy
the various graj)liic uxainiilcs to this or that .scale. Tiie opc^ialicm
in question consibts in drawing with the hand, the elevation, |)l!ui,
sections, and details of a machine, preserving, as much as possible,
the forms and proportions of each part; and then taking the actual
measurement of each part, and laying it down in figures in its par-
ticular position upon the drawing : this duplex operation of sketch-
ing and measuring constitutes the study of the rough draughting
of machinery.
THE SKETCHING OF MACHINERY.
PLATES XXXV, AND XXXVI.
394. Before commencing the sketch or rough draught of a
machine, it is absolutely necessary to look carefully into its organi-
zation, the action of the various working parts, the motion of the
intermediate mechanical connections, and finally, its object ai-d
results. The object of this preliminary examination is to give the
draughtsman a good general idea of the more important parts —
those which he will have to render most prominent and detailed
when he comes to make a complete drawing of the whole ; such
drawing comprising a series of combined views, together with sepa-
rate diagrams of such details as may not be apparent in the former,
or require to be drawn to a different scale to render them intelligi-
ble. In fact, this must be done in such a manner, that, with iho
aid of the sketch, a perfect representation of the machine may be
got up, which, if necessary, may serve La the construction of otlier
similar machines.
DRILLING MACHINE.
PLATE XXXV.
395. In order to give an exact idea of the manner of sketching
machinery, we take a simple machine as an example ; this we suiv
pose to be represented in perspective* in fig. ^A, this view being
instead of the machine itself
This machine is for drilling metals : it consists of a verticil c;isl-
iron column, a, which forms part of the building or workshop.
This column is hollow, and rests by an enlarged base upon a stono
plinth, B, imbedded in the ground, and at its upper end it supports
the beam, c.
Upon one side of this culunui is cast the vertical face, p. which
is planed to receive three cast-irou brackets, e, f, g, attached to
it by bolts. To the opposite side, d', of tiie sjinie column, is in
liki^ niaiuier attaciied the bracket, ii, whici), with the middle one,
K, on the other side, serves to carry tlio hori/.ontiil spindU>, i.
This s|)indle caiiies on one side the cone-pulley, j, over which
pnsxs the ll^i^ill^-lH■ll, k, ;iiuI oh the other extremity it Hjis
kr\r<\ iipoii il tlir Ih'\ ilpiiiioii, I., which gears with ii larger
liiNil-wlucl, M. This hist is allachcil to the verlical sliart, N,
• In n sulim-qiu'iit I'liaplor, wo sliuU ixiilalii tlic in'iu'nil iiiliu-'i>Ir» of i>:iriUlol «n4
131
THE PRACTICAI. DRAUGHTSMAN'S
\vlii'-li is, in fact, the drill-holder, and is moveable in the bracket-
bearings, F and G. This shaft receives a duplex movement, that of
continuous rotation, which is more or less rapid, according as the
belt, K, is on the less or greater diameter of the cone-pulley ; and
tlie other vertical and rectilinear, due to the action of the screw, o,
which works in an internal screw in the end of the bracket, e.
This screw carries at its upper end a spur-wheel, p, gearing with a
tsmall pinion, q, the shaft, k, of which is prolonged downwards, and
terminates in a small hand-wheel, s.
The object to be drilled is held between a pair of jaws, a, a', set
In grooves upon the table, t, and capable of adjustment back and
forward by means of the screw, J, the head of which has a sliding
jandle. The table, x, is made in two pieces, so as to form a collar
about the column, a, and it is fixed at any convenient height upon
this column by means of the pressure screw, c ; the exact distance
of the table from the drill, d, according to the thickness of the piece
of metal to be drilled, is settled by means of the vertical rack, u,
n-hieh is fitted to the front of the column, and into which gears a
[>inion on the shaft, e, carrying the handle, /, at its extremity. The
rotation of this handle and the pinion necessarily causes the ascent
or descent of the table, t.
The diillmg machine, then, fulfils the following conditions : on
the one hand, the drill, d, is woi'ked at a greater or less speed of
rotation, whilst it descends vertically with a very slow motion,
which latter varies, of com-se, with the nature of the material acted
upon ; and, on the other hand, the table which carries the object to
be drilled is capable of being set at the most convenient height,
according to the forms and dimensions of the objects, whilst it may
rIso be set eccentrically, when necessary, by turning it to the re-
quired extent round the column.
396. After having thus taken note of the construction and action
of each individual piece and element of the machine, the draughts-
man may proceed to make his sketch. He should commence by
drawing a rough general view, indicating, in mere outline, the rela-
tive positions of the various pieces.
For example, in fig. 1 will be seen the geometrical elevation
'^i the column, a, with the positions of the brackets and the table,
which are merely in outline. It is advisable, even in this rough
draught, as well as in the finished drawing, with or without the
assistance of a rule, to draw the centre lines for guides ; thus,
after drawing the first centre line, g h, of the column, a, draw
upon each side of it the portions forming its contour ; then draw
parallel to it the centre line, ij, of the drill-stock, n ; then the hori-
zontal, k 1, which represents the centre line of the bevil-wheel, l,
and the driving-pulleys, j, and likewse the straight lines, m n,
o p, q r, which are the centre lines of the brackets, e, f, and
G ; finally, draw the lines, s t and u v, of the table, t, and
of the bracket, h ; and liliewise the extreme lines, x y, w z, of
the bottom and top of the column. At this stage it is neces-
sary to lay down the measurements upon the sketch. The
column being fitted with the principal parts of the drilling appa-
ratus, so that no clear space can be found upon it for the
height to be measured close to it, a plumb-line is suspended
from the point, z, on the beam, c, which rests upon the column,
imd this line is measured either by a foot-rule, or by a mea^
Buring-tape. and the measurement in feet, or metres, and frac-
tions thereof, may bo written upon the centre line, g h, of the
column.
The draughtsman must next measure tlie diameters at the base
and summit of the column, as well as those of the various mould-
ings. These diameters may be measured with callipers, which
open to such an extent that they can be applied to the place in
question, the amount of opening being then measured upon the
rule, and written down upon the corresponding place in the sketch;
or the diameters may be obtained by applying a cord, or a very
flexible rule, to the circumference. This latter method is always
employed for cylinders of large diameter, when it is not possible to
obtain the measurement from either base. In this case, to obtain
the actual diameter, it is necessary, as has been seen (72). to divide
the circumference found, by 3-1416.
To obtain the distance of the line, i j, from g h, the centre line
of the column, place the extremity of the rule at i', against the
surface of the column, and let it lie across the centre, i, of the
spur-wheel, p, or screw, o ; the measurement read off the rule will
be that of i' i, to which must be added the radius, i' t°, of the
column.
If the centre, ?, were not approachable with the rule, we should
have to take the internal distance between the surface of the
column and that of the screw, and then add the respective radii
of the screw and column. When these distances are greater
than the length of the measuring-rule, a rod or tape must be '
employed. When, indeed, the draughtsman has attained a
reasonable amount of skill, he may take the measurement, i i^,
directly, by applying the rule upon the surface of the column
opposite to its centre, and also opposite to the axis of the screw,
in such a manner that the rule shall be tangential to the column,
when the space between the two points will be the measurement
sought.
It is further necessary to quote upon the sketch, the vertica
distances between the different horizontal lines, m n, o p, q r, s t.
The measurements indicated upon fig. 1 will show how all these
are obtained.
The preceding operations will allow of the finished drawing being
commenced, by laying off the relative position of the main parts
which go to compose the machine to be sketched. We have next
to sketch and measure al) the minor details of each separate piec(i
of the machine. To this effect, and to avoid confusion, it is neces-
sary to treat each of these pieces as detached, and to draw different
views of them, upon which the dimensions of every part may be
properly indicated.
Figs. 2 and 3 represent, in elevation and plan, the detail of tlio
principal bracket, f, which supports the shafts, i and n, with tlie
bt'vil-wheels, l and m. Even these views are not sufficient to
represent thoroughly all the dimensions of this bracket; thus it
is necessary to draw a section such as that made at the line,
1 — 2, and projected in fig. 4, so as to show the exact form of the
feathers of the bracket ; it is likewise necessary to make a side
view (fig. 5) of the bearing, b', which holds up the shaft, n, to the
bracket, and also a vertical section (fig. 6) made at the line,
3 — 4, to show the brasses which embrace the journal of the
shaft, I. These details should always, if possible, be drawn to
a larger scale, so as to indicate the adjustments clearly, and to
BOOK OF INDUSTRIAL DESIGN.
135
give room for the measurements ; and it may be observed, that, for
a draughtsman who has not much practical knowledge of machine-
ry details, it will be necessary to take down or separate various
parts, such as the cap and the upper brass. With regard to wheel-
work, it will be sufficient to give the section of the web and boss,
as indicated in figs. 2 and 7, and a section, as fig. 8, of one of the
arms when the wheel has any, and then the numbers of teeth and
Arms must be counted and set down.
When all the parts of any detail are thus sketched out in
elevation, plan, or section, the draughtsman must take the
measurements of each, and set them down in their appropriate
positions upon the sketches, as indicated in the figures ; being
mindful to see that the principal measurements coincide with those
laid ofl" in the complete general view already commenced. The
measurements of the diameter of the pitch-circle, and of the width
of the teeth, will be sufficient, in addition to what has already
been directed to be done in reference to wheel-work, the proper
ratios being maintained between those in gear with each other.
As many parts of machinery require to be in proportion to
each other, a knowledge of such relations will enable the
draughtsman to dispense with a great deal of tedious measuring
and sketching, as in the case just alluded to, of wheels working
together.
The remaining parts of the machine are to be detailed in the
same manner. Thus, figs. 9, 10, and 11, represent a vertical sec-
tion, a plan, and a side view of a portion of the table, t, with its
solding-jaws, and its elevating pinion and shaft. Fig. 12 is a ver-
tical section of the lower extremity of the drill-stock, or spindle, n,
with the drill, d, in elevation. Fig. 13 is a section of the cone-
pulley, J. Figs. 14 and 15 show, in vertical and horizontal section,
the manner of jointing the screw, o, into the upper end of the
spindle, n. Finally, figs. 16 and 17 give a complete detail of the
mechanism for elevating the table, t, as well as that for fixing or
adjusting it at any required height.
397. On all the preceding details, we have quoted the measure-
ments of the different parts exactly as they should be upon an
actual machine. These measurements are expressed in millimetres,
as in former examples, this measuring unit being adopted because
its minute scale renders fractions unnecessary. We have also
slightly shaded various parts, as is generally done where the com-
plication and variety of forms would otherwise lead to confusion
and error. Besides, in this manner, a few touches of the pencil
show at once whether this or that portion is round or square, and,
in many instances, the labour of drawing additional views will
thereby bo dispensed with.
In order to facilitate the proceedings of beginners in sketching,
we would recommend them to delineate the main centre lines with
the aid of a rule, and the circles with compasses, though the dimen-
sions of the latter need not bo exact. This will give the sketch a
much neater appearance, and render the various objccta or details
more regular. It is with this view that sketchers frequently employ
cross-ruled paper, with horizontal and veitical lines 0(|uaily spaced.
That poition of IMate XXXV., upon which are sketehetl figs. 9, 10,
and 11, is of this- description.
It will 1)0 understood, that, if th(^ lines ruled upon Iho paper arc
at equal distances apart, corresponding to one or more units of Iho
scale to which the sketches are being m-ide, these may be drawn in
correct proportions at once, in which case it will be unnecessary to
write on the various measurements.
The example which we have given as an introduction to the
study of sketching machines, will have s^omewhat familiarized the
student with his operations even now. The applications contained
in the subsequent examples will suffice to complete this study,
which is one of gi-eat importance to the draughtsman and construct-
ive engineer.
MOTIVE MACHINES.
WATER-WHEELS.
PLATE XXKVI.
398. The water-wheel, represented in fig. 1, has plane floats, and
works, through a portion of its circumference, in a concentric cir-
cular channel. It receives the water from over a sluice-gate, a
little below Its centre, and is of the undershot description.
The wheel is composed of several parallel shroudings, a, in
which are fitted the radial wooden bearers, b, carrying the floats, C.
When the shroudings are of cast-iron, as is supposed in the preseut
example, they are cast in one piece with the arms, d, and centr;d
boss, E, and are firmly secured by keys, a, upon the shaft, f, also
of cast-iron.
The head of the channel, g, which embraces the lower part of
the wheel, is constructed with a piece, h, in cast-iron, called the
neck-piece, which is fitted upon the cross timber, i, and let into the
two lateral walls. Agamst this neck-piece works the wooden
sluice, J, above which overflows a certain depth of water, falling,
in succession, upon each float of the wheel as it comes round,
causing it to turn in the direction of the arrow. The rotatory
movement of this water-wheel is taken off by the cast-iron spur-
wheel, K, mounted upon the end of the shaft, f, and gearing with
the cast-iron pinion, l, the shaft of which communicates with the
machinery to be set in motion.
In giving this example, our object has been to examine this motor,
not only with reference to its accurate delineation, but also with a
view to sketching similar wheels, as well as to constructing and
setting them up, with their channel and sluice gear.
THE CONSTRUCTION AND SETTING UP OF THE WATER-WHEEI..
399. The channel, G,is built up of hewn stones, the lateral joints
of which converge towards the centre, o, of the wheel, imd they are
imbedded upon a foundation of ordinary stone-work. All li.i-
masonry is put together with mortar, made with hydraulic limo,
tiie joints being finished with Roman cement. In some localities
the channel is of bricks or freestone, and sometimes oven of wootl.
The apparent concave surfaco of the channel should be perleclly
cylindrical, and concentric with the exteruiil circunil'criMice of llio
wlieil. Also, before placing the latter in iis proper posili'"), llii.t
surlace should be finished, and rendered quite suioolh and true,
which may be done with the assistance of a tomponiry xhal^, o.
with the actual shiil't of liie wheel, in the followinfj manner: —
The shaft, k (IKi)- ''^ adjusted to the exact height at wliich it i.-
to bo afterwards, and it is made capable of rotation m appiv
\?,r,
THE PRACTICAL DRAUGHTSMAN'S
piiiite bearings, adjusted upon iron plates, let into and firmly bolted
to the lateral walls.
Upon this shaft are fitted the shroudings, a, each connected to
its boss by eight arms. To the outside of these arms are then
temporarily attached two radial pieces of wood, having a cross piece
attached to them, the outer edge of which is made true and parallel
to the shaft, and coincident with the external edges of tlie complete
wliu(il. It will be evident that, if the shaft is now made to revolve,
this frame upon it will describe a cylindrical surface, which is
precisely that which the channel should possess; it will serve,
therefore, as an accurate guide in giving the channel its appropriate
contour.
The lower part of the channel is continued on in a straight line,
commencing at the vertical, o b, and in a direction, b c, slightly
inclined to a short distance away from the wheel, to facilitate the
escape of the water.
The cross timber, i, which smmounts the masonry of the chan-
nel, and which receives the neck-piece, h, is also rendered concave
internally, like the channel, so as to allow the sluice-gate to be
brought closer up to the wheel. The neck-piece, h, which forms
the crest of the channel, is more frequently constructed of cast-iron
than of either wood or stone, as that material does not require to
be so thick, for resisting the pressure of the water. The top of the
neck-piece is at a distance below the upper water-level, corre-
sponding to the greatest depth of water which it is proposed to ad-
mit to the wheel at any time. This depth of water varies very
considerably, according to the quantity of water to be discharged,
and the width which it is wished to give the wheel. Behind the
neck-piece, a cavity, m, is formed in the masonry, which is intended
to receive the sluice, J, when lowered, and is of a sufficient size to
allow of its being cleaned out, so that it may not become choked
op with sediment. The small raised portion of masonry behind
this, again, serves to arrest floating bodies, as trees, &c., independ-
ently of a grating placed further behind, and preventing their re-
acting, and injuring the wheel.
The sluice consists of two strong oaken planks, having grooved
and tongued joints, and being made thicker at the middle than at
the extremities, where the wheel is of a greater width than 1|
m&tre. The amount of inclination of this sluice, J, is determined
by drawing a perpendicular to the extremity of the radius, o f,
drawn near the middle, or, perhaps, two-thirds of the depth of the
overflowing body of water. The sluice is moveable in grooves,
in two wooden side-posts, n, entirely imbedded in the lateral walls.
At the upper parts of these are iron bearing-pieces, to receive two
straight cast-iron racks, o, which rise above the cross timber, p,
attached to the two side-posts, n. These racks rest, on one side,
upon the friction-pulleys, h, which also guide them, and, on the
other side, they gear \vith the pinions, g, keyed upon one horizon-
tal axis. This latter is at one end prolonged, to receive the worm-
wheel, Q, actuated by the worm, e, which may be worked at
pleasure from above — a winch-handle, or hand-wheel, being fixed
upon the upper extremity of its vertical spindle for this purpose.
This arrangement permits of the regulation of the position of the
■'luice, and, consequently, of the depth of the overflowing water,
with the greatest nicety, as well as of the total shutting off" of the
water from the wheel.
The shrouding, a, of the wheel, being of cast-iron, the weight
has been reduced, by making panels in it, as at h, i, shown in the
elevation, fig. 1, and section, fig. 2, made at the circular line, 1 — 2.
It is also cast with mortises, to receive the tenons, or ends, of the
carrier-pieces, b, to which the floats are bolted.
When the wheel has counter-floats, as represented at the lower
part of figure 1, which is only the case when the discharge, and
consequently the depth, of water at the sluice-gate is very small,
the carrier-pieces are very short, and do not project far upon the
inner side of the shrouding. But when the wheel is without
counter-floats, which is the case when the discharge, and conse-
quently the depth, of water at the sluice is considerable, the floats, _
c, and theu- carrier-pieces, b, are prolonged to a considerable dis- I
tance inside the shroudings, as has been supposed to be the case
in the upper part of fig. 1. In both cases, the tenons, or ends, of
the carrier-pieces, always lie in the direction of radii from the centre
of the wheel, and they are retained by iron-keys, y, upon the inside M
of the shroudings. Sometimes, in order to facilitate the adjust- ^
ment of the carrier-pieces upon the shroudings, in place of fitting
them into closed mortises, they are received into slightly dovetailed
recesses, formed upon the side, as shown in figs. 3 and 4, being
retained in position by wedges, j. When this last arrangement is
adopted, it is unnecessary to cut holes in the carrier-pieces for the
reception of the keys.
When the shroudings are of wood, they must necessarily be
composed of several pieces, which are fitted together with mortise
and tenon joints, as shown in figs. 5 and 6 ; and to consolidate the
joint, iron straps, k, are added, secured at one side by bolts, and
at the other by keys, or tightening screws, bj' means of which the
perfect union of the component pieces can at all times be obtained,
should they begin to get loose. In this system, the carrier-pieces
are adjusted with tenons, keyed on the inside of the shrouding, as
indicated in figs. 7 and 8, and the oaken arms are joined to the
shrouding with tenons, being further secured by iron straps, as
shown in figs. 9 and 10.
The floats, c, of the wheel, are formed of oaken boards, and are
attached to the carrier-pieces, b, by means of the bolts, 1.
The counter-floats, s, extend from the inner ends of the floats,
c, to the bottom pieces, s', and are nailed down upon the small
ti'iangular pieces, m. The open spaces left between the ends of
the floats and the bottom pieces serve for the escape of the air.
When the floats are lengthened, they ai-e, of course, formed of
several boards, joined edge to edge.
DELINEATION OF THE WATER-WHEEL.
400. The explanations just given will have enabled the student
to comprehend the details and peculiarities of construction of the
wheel, channel, and sluice apparatus. He should now proceed to
delineate these various objects in the following manner : — Place
the centre, o, of the wheel, at the intersection of two lines which
form a right angle ; and with this centre, describe a first circle,
with a radius equal to that of the wheel and channel. Divide
this circle into as many equal parts as there are to b& floats. The
number of the floats should always be divisible by that of the
arms of the shrouding, so as not to be restricted as to space m
fittino- in the carrier-pieces. Through each point of division draw
BOOK OF INDUSTRIAL DESIGN.
137
lines passing through the centre, and representing the sides of the
carrier-pieces upon which each float is placed. Two circles must
next be described, expressing the depth of the shrouding. Then
the complete outline of one of the carrier-pieces must be drawn,
with the dimensions quoted on the figure ; and the key and bolts
may also be indicated upon it. Afterwards, to complete the
drawing, it will be sufficient to describe a series of circles, passing
through the bolts, the ends of the floats and canier-pieces of the
key, and of the counter-float. With regard to the floats, and to
the arms of the shrouding, as well as to the spur-gear for trans-
mitting the motion, the student may refer back to the diagrams
and explanations already given concerning similar objects. The
same remark applies to the lifting apparatus of the sluice-gate,
which is also composed of gearing already treated of in the course
of the studies.
DESIGN FOR A WATER-WHEEL.
401. If it is in contemplation to make a design for the con-
struction of a water-wheel, analogous, we shall suppose, to the one
above described, it is simply necessary to ascertain the height of
fall, and the amount of discharge per second, of the water at our
disposal, and to refer to the calculations and practical rules which
accompany our text, to be able to determine, on the one hand, the
diameter and width of the wheel, and, on the other, the depth and
interstices of the floats, and their number. By referring back, also,
to the tables and notes relating to the resistance of materials
(Chapter III.), we shall be able to complete the remaining dimen-
sions for the shaft and its journals, the shrouding and its arms.
The study of water-wheels of this description will be much sim-
plified, if we consider that certain dimensions, such as the thickness
of the floats, the section of the carrier-pieces and shrouding, and
the diameter of the bolts, as well as the details of the sluice appa-
ratus, do not sensibly vary ; and for them the draughtsman may
refer entuely to those indicated upon the drawing, which are
themselves examples of actual construction.
SKETCH OF A WATER-WHEEL.
402. The sketch of a water-wheel, already constructed and set
up, is a very simple matter ; for the apparatus consists of a repeti-
tion of various pieces, and it is sufficient to obtain the measure-
ments of one only of each kind. Thus, after having measured the
diameter and extreme width of the wheel wilh the aid of a long
rule or tape, and counted the number of buckets or floats, of the
shroudings, and of the arms, we have merely to take Iho sketch of
a single float, with its carrier-piece and accompaniments, then to
make a section of one of the shroudings, another of one of the
arms, and, finally, a third of the boss and shaft.
Th(^ details given in figs. 2 to 10 show the various jiarts of
vvliicli llic sketches have to bo made, as detached, togcllicr with Iho
I'.orrcNponding measurements. Fig. 20 is a transverse section of
one of the arms, d, of cast-iron, taken near the boss.
The sketching of the sluice apparatus consists in making ii
Rcction of the side-posts, with their cap-piece, and of tiie sluice
itnelf ; then a detailed view of one of the racks, with its pinion
and fVictioii-pulley, and of the worm-wheel and worm. As to the
Hinoiitit of inclioiition of the sluice and side-posts, it has already
been seen that it is determined by a perpendicular to the radius,
entering near the middle of the depth of water at the outlet, at
the circumference of the wheel. It may, however, be found by
means of a plumb-line, let fall from one of the edges of the cap-
piece down to the level of the water, by measuring the horizontal
distance, r s, of the plumb-line, from one of the sides of the side-
post, and then the vertical height, r t. By applying a rule against
the side-post, and down to the neck-piece, h. we can always obtain
the actual distance of the top of the latter, either from the pro-
longation of the horizontal, r s, "r from the caji-piece, p, of the
sluice. To obtain the horizontal distance, r s, with exactitude, it
should generally be taken at a given distance above the level of
the water, and chalked upon one of the side-walls of ihe channel ;
it is also advisable to make use of a spirit-level. (Plate I.)
In order to take an accurate sketch of the neck-piece and the
channel, it is almost always necessary to stop the water behind
by means of a dam, so that the parts requiring to be examined
may be dry and open. The sluice must also be taken away, as
well as a few of the floats of the wheel. We may remark, that
this labour may be avoided, when it is known that the height and
thickness of the neck-piece are nearly always equal to those indi-
cated in fig. 11; and as to the arrangement of the masonry or
brickwork, of which the channel may be constructed, it will bo
recollected that all the lateral joints are pointed towards the centre
of the wheel.
OVERSHOT WATER-WHEEL.
Figure 12.
construction of the wheel, and its sluice apparatus.
403. Overshot water-wheels, with buckets, receive the water
from a duct placed immediately above them, and allow it to escape
from as low a part only as possible. They are constructed of
wood, or of cast-iron. In the first case, which is the simi>ler and
more economical, the shaft, the arms, and the shroudings are of
oak. The lower part of the wheel, represented in the drawing,
fig. 12, is of this description. The buckets and the inner-rim are
likewise of oak, or of iron plates. As this wheel is of small dia-
meter, its shaft, F, has only six sides ; and consequently, each
shrouding, a, of the wheel has only six arms, D, which are recessed
into, and bolted ui)an, a central cast-iron frame, e, which is itself
keyed upon the shaft. The transverse section, (ig. 13, shows fiio
manner in which the arms arc attached to this I'ranu'. The wouitoti
shroudings, a, are generally coniposeil of two rings, jjlaccd one on
the other in such a manner that the joints of each are opposite lo
solid ])ortions ol" the other, to "break bond," and obvi.-ite the teu-
dency to wari). A portion of the shrouiiing is represented as de-
tached in ligs. 14 and 15. The-so rings are held together by sitcws,
r, or bv nails or pegs ; and at their junction with the anus, a couple
of bolts are passed throiigh all, as indicated in the transverse si>c-
tion, fig. 16. The buckets, c, are either let into grooves of sniai.
depth, upon the inrur l':ue of the slirouding, as seen at c',in figs. 14
and 1.5, or they are retained by bracket-pieces, c: and, nddt^l ti> fliiii.
strong feiision-nnls, li, hold the w hole together, being .s»vurwl lo
fiS
THE PRACTICAL DRAUGHTSMAN'S
the shroudingrs, a, on either side of tlie buckets. These tension-
rods are fixed, when the inner rim, s, or bottom of the buckets, has
l)een nailed or screwed to the inner edges of the shroudings. The
shroudings are further strengthened externally by a circular iron
strap, G, similar to the felloe of an ordinary wheel, and covering up
the joints of the duplex shrouding.
Sometimes the buckets are partly of wood., and partly of iron
plate, to give them greater strength. The edges, indeed, should
always be defended with metal, as they arc most apt to wear soon.
The lower portion of the di-awing, fig. 12, shows three dilibrent
ways of constructing these buckets.
When the wheel is of cast-iron, if it is not of a very givat di.-i-
raeter, but of the size represented in the upper part of fig. 12, the
arms and the boss, e', may be cast in one piece \\ itii the shroua.
ings, a'. Where the diameter is considerable, these consist of
several pieces bolted together. The bottom piece, or inside rim,
s', and the buckets, c', are of iron plates, of about Jth inch in thick-
ness. To secure the Mter, a series of feathers, figs. 12, 17, and
19, are cast upon the inner faces of the shroudings, to which they
are fixed by screw-bolts, I In the width of the wheel, the buckets
and the bottom rim are riveted together, as at t, or are fixed together
by small screw-bolts, i', figs. 17 and 18.
The advantage of making the buckets of iron plates, consists in
the being able to give them a curved form, which enlarges their
capacity, and allows of a more favourable introduction of the water;
whilst the w"ooden buckets necessarily consist of two rectilinear
portions, one of which is directed towards the centre of the wheel,
whilst the other is inclined.
The water is conducted by the wooden channel, ."vi, to the top of
the W'heel, and its outflow is regulated by a sluice, j, moving in side
gi-ooves, and worked by means of a couple of vertical racks, o, and
pinions, d, the shaft of which last carries the winch-handle, q. The
tW'O vertical sides, n, of the channel, are prolonged beyond the
actual summit of the wheel, and their distance asunder should be a
little less than that of the tw'o shroudings of the wheel, with the
twofold object of better directing the water into the buckets, and
of avoiding the splashing and loss of water by allowing the air to
escape laterally. The depth of the outflow of water depends on
the distance of the lower edge of the sluice above the bottom of
the channel, and should always be less than the smallest distance
existing between tw^o consecutive buckets. The pressure of the
w-ater upon the buckets, produces the rotation of the wheel in the
direction of the arrow, and this motion is given off by an internally-
toothed wheel, attached to the outside of one of the shroudincrs.
''^his wheel, which in the drawing is simply indicated by its pitch
circle, k, goars with the pinion, l, mounted on the extremity of the
shaft, which communicates with the machinery in the interior of
the factory or workshop.
DELINEATING, SKETCHING, AKD DESIGNIXG 0\"ERSHOT -WATER-
■WrSEELS.
404. The delineation of the principal parts of an overshot
Ducket water-wheel, is effected in the same w^ay as that of the
undershot wheel with floats, the only difference being, in fact in
the receptacles for the water. It has been seen, that when these
buckets arc of wood, they are composed of two boards, one of
w hich lies in the direction of a radius of the wheel, the other being
inclined according to the direction of the water, and make an angle
of 15 or 30 degrees, as the case maybe, with the tangent, to the
outer circumference of the wheel, drawn through its extremity, as
will be seen by the angle, a 6 c, in fig. 17. When the bucket is
made of iron plates, the same angle is adopted near the outer edge,
although the whole contour is a continuous curve, which may be
made up of two or three arcs of circles, as show"n in figs. 12, 17,
and 18.
In sketching this wheel, the directions given in the preceding
case n?ay be likewise followed here, by counting the number of
buckets, and taking an accurate sketch of one of them, together
with the accompanying measurements. We must also measure
the internal and external diameters of the sliroudings ; then the
least space existing between two consecutive buckets ," also the
depth from b to d, fig. 17. Finally, if it is required to obtain the
exact form or curvature of the bucket, it will be necessary to take
one down, and to make a pattern of it, by applying a sheet of
paper against one edge, and pencilling out the shape, as is done
for the forms of wheel-teeth, or other curves, which are difficult to
measure. As to the sketch of the other parts of the wheel, such
as the boss, the arms, and also the sluice apparatus, no peculiarity
or difficulty can present itself which need detain us here. The
drawing, moreover, indicates all the figures and measurements
which are necessary.
In designing an overshot water-wheel, it is necessary to know
the height of fall, and the daily discharge of the water. With
regard to these particulars, we must simply refer to our accomna-
nying Rules and Practical Data.
WATER-PUMPS.
PLATE XXXYII.
GEOMETRICAL DELINEATION.
405. We have already indicated, in preceding notes and calcul*
tions, the various classes of pumps, with their proper dimensions,
in proportion to the quantities of water to be flimished by them.
We now propose to enter upon more detailed and complete expla-
nations, with regard to their construction, action, and performance.
For this purpose we have selected, by preference, a combined
lifting and forcing pump, the discharge of which is ahnost continu-
ous, although its construction is analogous to what is termed a
single-acting pump.
Figure 1, on Plate XXXVU., represents a vertical sectioii.
taken through the axis of this pump. It consists of a cast-iron
cylinder, a, turned out for the greater portion of its length, and
resting upon a feathered base, b, cast in one piece with the suction
or lift-pipe, c, below. This base is bolted down either to stout
timbers, d, or to a stonew'ork foundation. It encloses the valve-
seat, E, which consists of a rectangular frame, divided by a central
partition, a, and having the sides formed so as to present two in
clined edges, upon which the brass clack, f, rests when shut.
The pipe, c, terminates below" in a flange, by means of which the
suction-pipe is attached, extending down to the water to be ele-
vated. Towards the upper part of the pump cylinder, a, is cast
BOOK OF INDUSTRIAL DESIGN.
139
a curved outlet, g, likewise terminating in flanges, to which the
discharge-pipe is secured. The piston, or bucket, of this pump is
composed of a brass ring, or short cylinder, h, upon the outer cir-
cumference of which is formed a groove, b, (fig. 2,) to receive a
packing-ring, c, which fits, air-tight, to the inside of the pump
cylinder. The bucket, h, has also a central partition, d, to the top
of which are jointed the two clacks, i, which rest upon inclined seats,
formed by the elevated sides, e, of the bucket. This is further cast
with a bridle,/, perforated in the middle, to receive the screw-bolt,
g, which secures it to the stout hollow piston-rod, j. This rod,
which, in the generality of pumps, is made of but small diameter,
like the upper part, k, of the one represented in the plate, is, in the
present instance, of a sectional area, equal to half that of the pump
.•ylinder. It follows from this, as will be more particularly ex-
,)!;dned further on, that the water is discharged during both the up
m J down stroke of the piston.
The chicks, F, have projections, h, cast upon them, which pre-
vent their opening too far, and falling over against the sides of
the casing, b, so as not to shut again when required to do so.
The clacks, i, in the bucket, have similar projections, i, for a like
purpose, these projections striking against the top of the bridle,/,
when the clacks open. It will have been observed, that the seats
of these valves are inclined at an angle of 45°, with the view of
facilitating their opening movement, and diminishing the concus-
sive action of their own weight. The edges of the valve-seats are
generally defended with a strip of leather, to facilitate their tight
closing.
Figure 2 represents, detached and in elevation, the bucket, h,
with its clacks, i. Fig. 3 is a horizontal section of the bucket, taken
at the line, 1 — 2. Figs. 4 and 5 give the details of the valve-seat,
E, in elevation and plan, the clacks being removed.
To prevent the entrance of air to the pump cylinder, it is closed
at the top by a cast-iron cover, l, which is fitted with a stuffing-
box for the passage of the piston-rod ; the packing is compressed
by the gland, m, similar in general form to that represented in
Plate XI. (81.)
ACTION OF THE PUMP.
406. The upper extremity of the piston-rod, k, carries a cross-
head, I, (fig. 6,) and is there jointed to the lower extremity of a
connecting-rod, n, which is itself jointed to the pin of a crank, o ;
this latter is mounted on the end of a horizontal shaft, p, actuated
by a continuous rotatory movement. This movement is trans-
formed by means of the crank and connecting-rod into an alternate
rectilinear motion — that is, into the up-and-down strokes of Ihe
pump bucket — this last being forced to move in a straight line, the
cross-head, I, sliding in vertical guide-grooves, to maintain the
piston-rod, k, in the same line with it.
It follows, from this disposition of parts, that when the crank,
0, is in the position, r — o, fig. 6, the piston will be at the bottom
of its stroke, that is, at ii' ; consequently, during (he tinu^ tlu^
crank turns, the piston must ris(^ tending to leave a vacuum below
it, because the spac(! I)etween the clacks, f, and its under side
increases, as well as the vohimi^ of air that may be Ihercin en-
closed. Consequently, the pressure of this air upon the clacks
is diminished, whilst that upon the surface of the water remains
the same, and causes the water to rise up the suction-pipe, and,
raising the claclcs, f, to enter the pump cylinder, filling it up nearly
to the under side of the piston ; or if the apparatus is in a perfectly
air-tight condition, it will rise quite up to the piston.
When the crank has reached the position, p — 12, — that is,
when it shall have described a semi-revolution, — the piston itself
will likewise be at the highest point of its stroke, and, in this
position, all the space left behind it in the body of the pump will
be filled with water; if now the crank, continuing its rotation,
makes a second semi-revolution, the piston will descend, and,
pressing upon the water below it, will cause the clacks, f, to
shut. Now, as the water is incompressible, it must find an e.xit,
or else prevent the descent of the piston ; and it therefore raises
the bucket-clacks, i, thus opening up tor itself a passage through
the piston, h, above which it then lodges. But as the piston-rod,
J, is of a large diameter, and therefore occupies a considerable
space in the pump cylinder, a part of the water must necessarily
escape through the outlet, g, in such a manner that, when the
piston shall have reached the bottom of its stroke, there will not
remain in the pump cylinder more than half the quantity of water
which was contained in it when the piston was at the top of its
stroke.
Such is the effect produced by the first turn of the crank, which
corresponds to a double stroke of the piston — that is, an ascent and
a descent.
At the second turn, when the piston again rises, it sucks up, as
it were, anew, a volume of water about equal to the length of
cylinder through which it passes, because the suction-clacks, f.
which were shut, now open again, and the bucket-clacks, i, whiuli
were open during the descent, are now shut by the upward move-
ment of the piston.
During this stroke, all the water which previously remained above
the piston, finds itself forced to pass off through the pipe, g, so
that, with this arrangement of piston and rod, or plunger, of large
diameter, it follows that, at each up-stroke of the piston, the quan-
tity of water which rises into the pump is equal to the length of
cylinder through which the piston passes, the half of which quantity
rises in the discharge-pipe during the descent, and the o.hcr half
during the subsequent ascent of the piston, and the jet is conse-
quently rendered almost continuous and uniform.
When, or the contrary, the piston-rod is made very small in
diameter, as in ordinary pumps (fig. 6), the discharge of the water
only takes place dui'ing the ascent of the piston, and it is conse-
quently intermittent.
In a pump, as in all other machines in wli'eli an alternate ree-
tilinear is derived from a continuous riitatory motion, by means
of a crank and connecting-rod, the spaces passed through in a
straight line by the i)iston do not correspond to the angular
spaces described by the crank-pin ; in fact, it will be soeu I'roiu
the diagram, lig. (i, that if Ihe crank-pin is sujjposed to desoribi'
a series of etpial arcs, beginning from the point, 0, the corn>spond-
ing distances, 0' 1', 1' 2', 2' 3', jiassed iIuouliIi bv the piston will
not he unil'orm ; very small al llie roinmeiieemeiil o\' tlio slroko,
(hey will gradually increa.-e towards llu' middle, alter im-ssiu^;
whii-li they will similarly decrease whilst the piston approaches
111,, olher end ol its stroke. The slU'COSsivO posltior)s of iho
140
THE PRACTICAL DRAUGHTSMAN'S
piston may bo obtained by describing with eacli of tlio points,
1, 2, 3, 4, upon the circumference traced by the crank-pin as cen-
tres, and with a radius equal to the length of the connecting-rod, a
series of arcs or circles cutting the vertical, passing through the
centre, p, in the points, 0^ l^ 2', which indicate upon this line the
various po-itions of the point of attachment, Z, of the connecting-
rod to the piston-rod: these points are then repeated at 0', 1', 2',
on the same line, at distances from the points, 0^ 1", 2^ equal to
the length of the piston-rod, measured from the point, I, to the
bottom of the piston.
It will be easily understood, that, in consequence of this irre-
gularity In the motion of the piston, the force and volume of tlio
jet of water will vary throughout the whole stroke. We have
endeavoured to show the nature of this variation in the diagiam,
fig. 7, which represents the comparative volumes of the jet of water
at successive periods for a single-acting pmnp, such as the one in
fig. 6.
This diagram is constructed by laying off upon any line, x y. as
many equal parts as we have taken in divisions on the circle de-
scribed by the crank-pin; then through each of these points, as 1, -J,
3, 4, drawing perpendiculars to x y. As during the ascent of ilie
piston from to 12 (figs. 6 and 7) there is no discharge, as
the piston only sucks up the water, there is nothing to indicate
upon these first divisions ; as soon, however, as the crank-pin passes
the highest point, and the piston begins to descend, it will pro-
duce the jet of water ; it is considered then, that when it has
passed through the first rectilinear space, 12' to 11', the quantity
of water forced out by it may be represented by its base multiplied
by the height, 11' — 12'. It is this distance which is set off from
13 to a, upon the perpendicular drawn through the point, 13; in
the same manner, when the piston descends from 11' to 10', it is
also taken as represented by its base multiplied by the height,
11' — 10', which last is therefore set off from the point, 14 to b. It
^fill be seen from this, that, in proceeding with the diagram, it is
simply necessary to set off upon each of the perpendiculars drawn
thi-ough the points of division, 15, 16, 17, the successive distances
passed through by the piston during its descent, so as to represent
intelligibly the actual volumes of water discharged for each f)ori:ion
of the stroke, since these volumes are proportional to the distances
passed through by the piston, the section of the cylinder remaining
constant.
If, through the various points, a, b, c, d, fig. 7, obtained in this
manner, we trace a curve, we shall obtain the outline of a surface
which we have distinguished by a flat shade, and which will give
a good idea of the amounts of water discharged in correspond-'nce
with any position of the crank. On continuing the rotation of
the crank, the piston next ascends and sucks up the water,
consequently the jet of water is interrupted during this up-
stroke, but recommences on the down-stroke due to the sub-
sequent part of the revolution; the quantity of water then
discharged is indicated in fig. 7, by a curve equal to the first,
and on which the same points are distinguished by the same
letters.
To avoid this irregularity in the discharge, pumping apparatus
is sometimes constructed with two, or with three, distinct cylinders,
in which the disposition of the pistons is such, that the points of
attachment to the several crank-pins divide the circle described by
them into two or three equal parts.
Figure 8 represents a geometrical diagram of the performance
of a two-cylinder pump ; it is evident that the product of each of
the pistons is alternately the same, since one descends whilst the
other rises ; it is thus that one of the pistons, having produced a
jet corresponding to, and expressed by, the curve, a' b' d, the other
one immediately afterwards produces a jet, expressed by the curve,
abed; so that this diagram only differs from fig. 7, in that the
unoccupied intervals, from to 12, and 24 to 12, in the latter, are
in the former filled up by an equal figure, covered by an equal flat
shade.
This diagram of the performance of a two-cylinder pump may
also be considered as representing that of the pimap, fig. 1, which,
because of its trunk piston-rod, acts as a double-acting pump, as
already explained.
Fig. 9 represents the diagram of the performance of a three-
cylinder pump, of which the pistons, h, h', h^ represented, for
convenience' sake, as in the same cylinder, occupy the positions
corresponding to those of the three crank-pins, 0, 0', 0", as placed
at the angles of an equilateral triangle, inscribed in the circle
described by them with the centre, p. In consequence of this
disposition, there are at one time two pistons ascending and one
descending, and at another time, on the contrary, only one
ascending and two desi^ending. It is easy to represent the com-
bined performance of these punips in a diagram, by using diffeieut
colours, or different dep.lis of shade, for the performance of ci.vli,
as dependent upon the siiccf^sive pcisitions, 1, 2, i^, 4, taken up by
U'lcir successive crank-pins. By this means all contusion v\iii uo
avoided, and it will be necessary to find the positions, n^ n', k", of
the attachment of the connecting-rod to the piston-rod upon the
vertical line passing through the centre, p, only as for one cylinder,
as the distances wDl be the same for all, being merely placed at
different parts of the diagram.
In the diagram, fig. 10, we have laid down the performance of
each of the three pumps, supposing them all to be of the same
diameter, and taking care, when two pumps are discharging to-
gether, to add together their performance ; thus, for example, when
one of the pistons elevates a quantity of water, corresponding to
the perpendicular, 13 a, that which is also discharging at the same
time furnishes a quantity expressed by the distance, a a' ; conse-
quently, the total volume of the discharge at this instant is repre-
sented by the total height, 13a' ; when, on the other hand, only one
of the three pumps discharges, whilst the pistons of the other
two are ascending, as in fig. 9, the volume discharged is represent(;d
by a single length of perpendicular, such as 18/. Now, it will be
observed, that it is precisely at the moment when only one pump
is discharging that it gives out its maximum performance ; from
which it follows, that the jet of water is continuous, and almost
uniform throughout its duration, as will be very evident from a
consideration of the diagram, fig. 10, the outline of which is detei-
mined by perpendiculars, or ordinates, reaching nearly to the
straight line, m n, throughout.
To compare the combined effect of a three-cylinder pump with
that of two or of three double-acting pumps, we have, in figs. 8 and
11, repeated the corresponding diagranas for the -wo last arranirf
BOOK OF INDUSTRIAL DESIGN.
141
ments ; and it \vill be remarked, that although, with cylinders of
an equal sectional area, we necessarily obtain a much larger dis-
charge, yet the regularity of volume is not so great as in the
previous example.
STEAM MOTORS.
HIGH-PRESSURE EXPANSIVE STEAM-ENGINE.
Plates XXXVIIL, XXXIX., and XL.
407. When the steam generated in a boiler is led into a vase or
cylinder which is hermetically closed, it acts with its entire expansive
foi-ce upon the sides and ends of the cylinder, so that, if this en-
closes a diaphragm, or piston, capable of moving through the cylin-
dei- in an air-tight manner, the force of the steam, in seeking to
enlaige its volume, will make the piston ino\c. It is in this way
that a mechanical effect is derived from the expansive action of the
steam, and it is on the same principle that the generality of steam-
engines are constructed.
Thus, in most apparatus to which this name is given, the action
of the steam is caused to exert itself alternately on the upper and
under surface of the piston, enclosed in the cylinder, thereby caus-
ing it to make a rectilinear back and forward movement or stroke.
(187.)
Steam-engines are said to be low or high pressure engines,
according as the tension of the steam is only of about 1 atmosphere
on the one hand, or of 2, 3, and upwards, on the other. Low-
pressure engines are generally condensing engines, and high-pres-
sure ones non-condensing ; so that the terms, low pressure or con-
densing, high pressure or non-condensing, are used indiscriminately,
although, in modern engineering practice, what are called high-
pressure condensing engines are extensively employed.
When the steam is made to act alternately above and below the
piston, the engine is said to be double-acting ; and of this description
are most of those employed at the present day; but if the steam
acts only on one side of the piston, as is the case in many mine-
pumping engines, the engine is called a single-acting one.
Low-pressure engines are generally also condensing engines;
that is to say, that after the steam has exerted its expansive action
upon the piston, and is on its way out of the cylinder, it passes into
a chamber immersed in cold water, and termed a condenser, whore
it is condensed or reduced to the state of water. This condensa-
tion produces a partial vacuum in the cylinder, and consequently
considerably diminishes the resistsince to the movement of tlio
piston.
In high-pressure engines, the steam which has produced its effect
upon the piston escapes directly to the atmosphere, so that the pis-
ton has always to overcome a resistance equal to one atmosphere,
or about 15 lbs. per square inch, acting in a direction opposite to
its motion.
Steam-engines are further distinguished as expansive and non-
exjiaiisivo ; of Iho hitter descripfion arc those wherein the steam
enters the cylinder throughout the entire stroke of the piston ; so
that the pressure is uniform, since the volume of steam of a given
pressure which enters is always equal to the space passed through
by the piston.
In expansive engines, on the contraiy, the steam is only allowed
to enter the cylinder during a portion of the stroke ; so that the
expansive power of the steam is called into action during the
remainder of the movement.
The machine detailed in Plates XXXVIIL, XXXIX., and XL.,
is a high-pressure engine, with a variable expansion valve.
Fig. 1, Plate XXXVIIL, represents an external elevation or front
view of the machine, the frame of which consists of a hollow column,
with lateral openings.
Fig. 2 is a horizontal section, taken at the height of the line,
1—2.
Fig. 3 is an elevation of a fragment of the lower part of the
column.
Fig. 4 is another horizontal section, taken at the line, 3 — 4; and
fig. 6 is an elevation of the capital of the column.
Figs. 6 and 7 are diagrams, relating to the movemeiit of the
governor, with its balls.
Fig. 8, Plate XXXIX., represents a vertical section, taken
through the axes of the column and the steam cylinder, at the plane,
5 — 6, parallel to that of the fly-wheel.
Fig. 9 is another vertical section, at right angles to the preceding
figure.
And, finally, fig. 10 is a horizontal section, taken at the broken
line, 7 — 8—9—10.
This machine consists of a cast-u-on cylinder, a, truly ooied out,
and enclosing the piston, b. On one side of the cylinder are i ast
the passages, a, b, by which the steam enters alternately above :.nd
below the piston. These passages are successively covered over
by a cup or valve, d, the details of which are given in figs. 28 to
31, Plate XL.; and the valve is itself contained in the cast-iron
chamber, e, called the valve casing, and communicating witn a
second chamber, F, called the expansion-valie casing ; it is imo
this latter chamber that the steam is first conducted by the pipe,
G, from the boiler. The communication between the two vah u
ca!-ings is intercepted for short periods during the action of tlu
machine, by the expansion valve, h, detiiiled in figs. 38 to 41'
Plate XL.
The vertical rod, i, of the piston, b, is attached at its upper ex-
tremity to a short cross pin, e\ which connects it to the wrouglit-
iron connecting-rod, j, hung on the pin, /, of the crank, k ; this is
adjusted and keyed upon the extremity of the horizontal shaft, i.,
which carries on one side the fly-wheel, M, and on the other the
eccentrics, n, o, p. The first of these eccentrics is intended to ac-
tuate the distributing valve, d, the rod, 5-, of which is connected to
it by the intermediate adjustable rod, n'. The second works tlio
exi)ansion valve, h, by means of the rods, 0' and h ; and, tinally. Iiu«
third eccentric, i", gives an altern;ite movement to the i>i8ton or
plunger, q, of the feed-pump, u.
The steam cylinder is bolti'd in a tirm and .solid manner, by
its uj)per flanges, to the top of the hollow cast-iron plinlii or (h--
dostal, s, on which also rests, and is bolted, tJie column, r. 'l'lu>
pedestal is square ; and, at the corners of its base, lugs are cast,
by means of which it is lirmly bolted down to 11 solid stone
foniulaliori.
The column, T, is cast liollnu, ami wilh four large lateral open-
ings diametrically opposite to each other, llnlr obitvt W'inji t«
142
THE PRACTICAL DRAUGHTSMAN'S
diminish the weight of the column, and to afford the necessary
passages for the introduction of the various pieces when being put
together, or when talcen down. This column also serves as a
f'lame for the entire machine, and above the capital is placed a cast-
iron pillow-block, u, furnished with bearing brasses to receive the
principal journal of the first motion shaft, as well as the supporting
brackets, k, k', of the spindle, /, of the ball governor. To its inner
side are also bolted the two supports, i, of the parallel motion, and
guide, y, of the valve-rod, g.
ACTION OF THE MACHINE.
408. Before proceeding further, we shall give some idea of the
general action of the machine. As already mentioned, the steam
is generated in a boiler, such, for example, as that represented in
Plate XIV. (189), and is conducted by the steam-pipe, g, into
liie first chamber, f ; when the valve, h, in this chamber uncovers
tne orifice, or port, d, the steam finds its way into the valve-casing,
E, whence it passes either to the upper or to the lower end of
the cylinder, accordingly as the valve, d, uncovers one or other
of the two ports or passages, a, b. Now, when the piston is,
for example, at the top of its stroke, the passage, a, is almost fully
open, whilst the channel, b, is in communication with the exit
orifice, c, from which the two pipes, e*, conduct it to the atmo-
sphere. If, on the introduction of the steam to the cylinder, it has
a pressure of, say four atmospheres, it follows that it will act upon
the piston with all this force to cause it to descend ; since, how-
ever, the lower part of the cylinder is at this time in communica-
■,ion with the external atmosphere, there is a resistance equal to
jne atmosphere opposed to its movement, therefore the actual
effective pressure acting on the top of the piston will be equal to
three atmospheres.
It is the same when the piston reascends ; the valve uncovers the
port of the passage, b, to allow the steam to enter the lower end
of the cylinder, whilst the port, a, is put in communication with the
exit orifice, c, by the cup of the valve, to give an outlet for the
steam which has just acted on the upper side of the piston during
the down-stroke.
It is to be remarked, that if the introduction of the steam takes
place during the entire up-and-down stroke of the piston, which
might be the case if the steam-pipe, g, communicated directly with
the valve-casing, e, and if the valve, n, kept one of the ports unco-
vered throughout the entire stroke, the pressure of the steam would
remain constant ; in such case, it would be said that the machine
was a high-pressure non-expansive engine — that is to say, that it
vvorked with a full allowance of steam.
In the machine, however, which at present occupies our atten-
tion, the steam first introduces itself into the casing, f, the valve,
H, of which, at each stroke, closes the passage, d, communicating
vvith the second casing, e, before the piston reaches either end of
(ts stroke. It follows, that the steam contained in the cylinder
at the time of closing the passage, d, must augment in volume or
expand, whilst its pressure will consequently decrease during the
remaining advance of the piston: the engine is then said to be
working expansively ; and in this case a quantity of steam is
» upended for each stroke, equal only to a third, half, or two-
tiiirds of the capacity of the cylinder, according as the intro-
duction of the steam is intercepted at one-third, one-half, or two-
thirds of the stroke ; it is the ratio between the quantity of full
steam-pressure introduced, and the entire capacity of the cylin-
der, which expresses the degree of expansion at which the engine
works.
PARALLEL MOTION.
409. The rectilinear alternate movement of the piston is trans-
formed into a continuous circular motion on the first motion shaft,
L, by the intervention of the connecting-rod, j, and crank, ic; but
with this arrangement there is naturally a lateral strain upon the
top of the piston-rod, i, and in order that its movement may be
perfectly rectilinear and vertical, it is jointed to a system of articu-
lated levers, forming what is termed a parallel motion.
This mechanism is composed of two wrought iron rods, v (figs
1, 4, and 8), which oscillate on the fixed centres, i, and are articu-
lated at their opposite extremities to the levers, x, near their middle,
by means of the pin, n. The levers, x, are also of wrought-iron.
and are jointed at one end to the cross pin, e", fig. 9, of the piston-
rod end ; and at the other to the rod, y, attached to a cross spindle,
0, and oscillating in bearings, in a couple of cast-u-on brackets, z,
bolted to the lower part of the frame.
The head of this last-mentioned oscillating rod is detailed sepa-
rately, in figs. 21 and 22, Plate XL. It has brasses, to embrace
the journal of the spindle, p*, by which it is connected to the ends
of the levers, x.
The combination of this mechanism is such, that the point of
attachment, e, constantly moves in a straight line throughout the
entire stroke. It may be designed on geometrical principles, as
indicated in the diagrams, figs. 8 and 11. To this effect we have
supposed, that after having drawn the horizontal line, e" p, and the
vertical, e e', distances, e e* and e e^ are set off on the latter, equal
to the half stroke of the piston, or to the radius of the crank; theni
with the points, e e^ describe an arc, with a radius, e p, equal to the
length of the lever, x, which is taken at pleasure, but should never
be less than the stroke of the piston. If we next lay off this
distance from e" to p\ the space, p p', will express the amount of
oscillation of the rod, y, the centre of oscillation, o, of which we
place below, on the vertical line, drawn at an equal distance from
and between the two points, p, p^. We next fix the point, n, of
attachment of the rods, v, to the lever, x. This point, n, during
the movement of the parallel motion, necessarily describes a circu-
lar arc, of which it is requisite to find the centre. In investigating
this problem, it is to be observed that, whatever may be the posi-
tion of the lever, the point, n, is always at an equal distance from
the extremity, p, or the other one, e. If, then, we in succession
draw the lines, p e, p' e', /^^ e", p^ e^ indicating the different posi-
tions of the lever, corresponding to those, e, e\ e', e', of the piston,
rod end, we shall, on each of these lines, obtain the several posi.
tions, n, n', ?^^ ?i', by laying off on them either of the distances,
p n or e n. We can then very easily find the centre of the arc
passing through these points. (10.) »
Fig. 10 represents the diagram of an analogous parallel motion,
but one in which the rods, v, are so disposed, that their point of
attachment is exactly in the middle of the levers, x ; ar d in this
BOOK OF INDUSTRIAL DESIGN.
143
case, their axis of oscillation lies in a plane passing through the
vertical axis, e e^.
DETAILS OF CONSTRUCTION.
' STEAM CYLINDER.
410. The cylinder is cast in one piece with its bottom cover and
lateral steam passages. As it should be bored with great care, so
as !o be perfectly cylindrical in the interior, a central opening is
made in the bottom for the passage of the spindle of the boring
tool; this opening, however, is afterwards closed by the small
cover, a', cemented at its junction surfaces, and bolted down to the
bottom of the cylinder. The upper end of the cylinder is closed
by a cast-iron cover, a', which is formed into a stuffing-box in the
centre, to embrace the piston-rod, which works steam-tight through
it. The packing is compressed or forced down for this purpose
by a gland (140), bolted to the stuffing-box, and hollowed out at
the top to receive the lubricating oil. The valve-face, on the out-
side of the cylinder, and on which the valve works, is planed very
carefully, so as to be a true plane throughout. The same is done
with the valve-casing at the flanges, where it is fitted to the valve-
face.
PISTON.
The piston (figs. 8, 9, 19, and 20) is composed of two cast-iron
plates, which have an annular space between them for the recep-
tion of two concentric cast or wrought-iron or brass packing-rings,
c'. These rings are cut through at one side, and are placed one
within the other in such a manner, that the breaks in each are
diametrically opposite to each other; their thickness gradually
diminishes on each side towards the break, and they are hammered
on the inside in a cold state, which renders them elastic, giving
them a constant tendency to open. Since the diameter of the
outer ring is equal to that of the cylinder when the two edges are
brought together, the elasticity of the iimer ring, combining with
that of the outer one, tending constantly to enlarge them, it
follows that there must be a perfect coincidence between the
outside of the ring and the inside of the cylinder throughout the
whole extent of the latter. Thus the contact of the piston with
the sides of the cylinder only fakes place through the packing-
ring, and not by the plates, which are of a slightly less diam';ter.
To prevent the passage of the steam through the break in the
outer packing-ring, a rectangular opening is made in the two edges
of the ring, and in this is placed a small tongue-piece, ffl^ screwed
to the inner ring, this piece serving to close or break the joint
without preventing the play of the rings. The principal plate of
the piston is fixed to the piston-rod by means of a key (fig.' 9)-
The pjston-rod is consequently of incre;ised diameter at its lower
end. The upper end of the piston-rod is likewise fixed in a socket,
l', (figs. 9 and 13,) which terniinalcs in two vertical brnnchos to
receive the middle of the spindle, e'^ whicli is held down by
means of a key.
CONNECTING-ROD AND CKANK.
411. Tno connecting-rod, J, (figs. 8, 9, 14, and 1.5,) terminates
at its lower end in a fork, by means of which it is jointed to tlio
spindle, e^ brasses being fitted in either side, and secured by bridle-
pieces passing under them and keyed above. The fork is jointed
to the spindle, e', on each side of the piston-rod head, sufficient
space, however, being left between them for the levers, x. The
head of the connecting-rod (figs. 15 and 16) is likewise fitted with
brasses to embrace the pin, /, of the crank ; these brasses are
tightened up by means of the pressure screw,/'.
The crank, k, like the connecting-rod, j, is of wrought-iron,
being adjusted on the end of the first motion shaft, and secured to
it by a key. This crank is very often made of cast-iron in sta-
tionary engines, but in marine and locomotive engines it is gene-
rally forged, so as to be better suited for resisting severe strains
and shocks.
The first motion shaft, l, is likewise either of cast or ^vrought-
iron. In the notes, we have already given u.bles and rules for
determining the respective dimensions of this detail. It is not
only supported by the brasses of the pillow-block, u, but also by
those of a similar one, fixed, we shall suppose, upon the wall
which divides the engine-house from the workshop or factory. It
should always be larger in diameter where it receives the fly-
wheel, M.
FLY-WHEEL.
412. The fly-wheel is of cast-iron — of a single piece in the pre-
sent example, because its diameter is only 3-5 metres. When of
larger dimensions, the rim and the arms are cast in separate pieces,
and then bolted together. For wheels of from 5 to 8 metres in
diameter, the rim is made in several pieces, and the arms are also
cast separate from the boss, and all the parts are then bolted
together. The arms are sometimes made of wrought-iron of small
dimensions, with the view of reducing the weight near the centre,
without reducing the effect of the wheel.
FEED-PUMP.
413. This pump serves to force into the boiler a certain qu.inlity
of water, to replace that which is converted into steam and expend-
ed in actuating the engine. It is a simple force-pump, consisting
of a cylinder, R, in which works the solid piston or plungei, g.
The piston is not in contact with the sides of the pump cylinder,
and the latter consequently only requires to be turned out at its
upper part, where it is formed into a stuffing-box and guide for
the plunger, being necessarily air-tight.
On one side of the pump is cast a short pipe, lO which is atl..>.h-
cd the valve-box, R-, generally made of brass. To the lower part
of this is secured the suction-pipe, t', communicating witn ii cistern
of water, and having a stopcock, s', uiion it, like the one rcprc^
sented in detail in Plato XVH. To one siiU> of the valvo-bo\ is
likewise fitted the discharge-pipe, carrying a similar stopcock, s*;
this last pipe is generally [tassed through the pipe which carries
off the waste steam, so that the water may take up some of the
heat of this steam before oiiteriii!.;- Ilie boiler.
It will bo seen, from tigs. !» and 23, that tliis viilvo-box contaitia
two valves, a', s'; the lower one of which, s', is the suction viilvo.
and the iip|Hr one, n\ is the discharge-valve. The latter is much
larger in diiiniel.r Ili.in \\\v former, so that its sent may hv wide
enom'h for the low.r \:ilve to hr passed through it. The up|HM
144
THi; PRACTICAL DRAUGHTSMAN'S
i^nd of the valve-box is closed by a cover, which is fiiinly held
down by the screw, r', and iron bridle, q'.
Both valves are made conical at the seat, as in fig. 24, so as to
tit more easily. The under part of the valve is cylindrical, so as
to guide it ; but it is cut away at the sides, to allow of the passage
of the water when it rises. It is from the appearance this gives
that these valves are called lantern lalves.
The pump cylinder is further furnished with a safety-valve, s',
of which fig. 25 is a detailed view. The object of this is to permit
the air to escape, when it accumulates within to such an extent as
to destroy the action of the pump. This valve is horizontal, and
is kept in its place by the bell-crank lever, u, upon the horizontal
arm of which is suspended a weight, sufficient to counterbalance
the internal pressure. (195.)
Tlie top of the plunger is surmounted by a small rod, /, adjust-
able in the socket which terminates the long wrought-iron rod, p',
tigs. 9 and 18, the upper extremity of which is formed into a collai',
embracing the circular eccentric, p. (142.)
The action of this pump is analogous to that of the pumps of
which we have already given a description. Thus, when the
machine is working, and the stopcocks, s' and s^ are open, the
water rises from the cistern by the pipe, r', the valve, s', opening,
to give it passage into the body of the pump, into which it flows as
long as the piston ascends. When, however, the piston descends,
the water is driven back, and, closing the lower valve, s', necessarily
opens the upper one, s^ and proceeds along the discharge-pipe to
the boOer. The quantity of feed-water is regulated by means of
the stopcocks, and may be entirely shut off by closing them ; but
then, in such case, as the eccentric, p, with its rod, p', will continue
to move, it will be necessary to loosen the plunger, q, which is
done by unscre-\ving the thumb-screw, v, by which the piston-rod
is attached to the eccentric rod, in such a manner that the socket,
&', fig. 18, at the end of the eccentric rod, p', will sin:ply slide up
and down the rod, without moving it.
BALL OR ROTATING PENDULTTM GOVERNOR.
414. The object of this piece of mechanism is the regulation of
the velocity of the machine, in proportion to the resistances to be
overcome ; and, accordingly, to this effect it opens or shuts a valve,
c\ placed in the steam-pipe, g, and called a throttle-valve. Just
as a freer or narrower passage is left for the steam by the opening
or closing of this throttle-valve, which is contained in an especial
box, to facUitate adjustment, and is actuated by a rod, passing
through a stuffing-box at the side — so is the quantity of steam
which finds its way to the cylinder more or less ; and, similarly, the
consequent acceleration or retardation of the motion of the piston,
as well as of the first motion shaft in connection with it.
It is composed, as seen in fig. 1, of a vertical spindle, Z, stepped,
at its lower extremity, in the end of a small bracket support, k', and
is held higher up by a second bracket, k. To its upper end are
jointed two symmetrical side rods, m', each terminating in cast-
iron or brass spheres, o'. These side rods are also connected by
means of the intermediate links, Z', to the wrought-iron or copper
socket or ring, i', moveable upon the main spindle.
A rotatory motion being given to the vertical spindle, and the
Vails being carried round with it, will have a constant tendency
to fly off from the vertical hne, by reason of the centrifugal force
due to the rotation (262) ; as long as the rotative velocity remains
the same, the balls wnW tend to occupy the mean position indicated
upon the drawing, and corresponding to the normal velocity ; that
is to say, the velocity to which the apparatus is regulated. When
this velocity is exceeded — in consequence, as we may suppose, for
example, of some parts of the machinery being put out of gear —
the balls will fly asunder, and occupy the extreme position, o-,
indicated upon fig. 6. In this position of the balls, the socket, i',
will be lifted up. Now this socket is embraced, at its circular
groove, by the prongs of the forked lever, /, fig. 9, which is con-
nected to the vertical rod, h, and this, by a suite of levers and bell-
cranks, g', g', g■^ g*, and g^, communicates with the throttle-valve,
c', drawn with its box, a^ in figs. 26 and 27. It follows, from the
combination of these connections, that, as the socket rises, the
valve will be shut. If, on the contrary, the velocity should be
reduced below the proper point, owing to an increased resistance,
the balls will approach each other, and assume the position given
in fig. 7. The socket, i', will descend, and, consequently, the
throttle-valve will become more open, so as to allow a greater
quantity of steam to enter the valve-casing, and thence pass into
the cylinder. The extreme positions of the governor arms, beyond
which they cannot go, are determined by the guides, m^ fixed upon
the spindle, Z.
The motion of this spindle is derived from the first motion shaft
L, by means of the grooved pulley, ^^ fixed upon the intermediate
spindle, r^ placed close to the capital of the column, t, and by the
bevil-wheels, r^ receiving their motion from the pulley, r*, so that
a constant ratio is maintained between the rate of the machine and
that of the governor.
The geometrical diagram, figs. 6 and 7, will sufficiently explain
the respective positions of each of the pieces of the pendulum, and
will show how the rising of the socket upon the spindle is caused
by, and is in proportion to, the flying asunder of the balls, accord-
ing to the number of revolutions of the spindle, and the length of
the suspending arms.
MOVEMENTS OF THE DISTRIBUTION AND EXPAN-
SION VALVES.
DISTRIBUTION VALVE.
415. We have seen that the valve, D, represented in different
positions in figs. 28 and 31, and in horizontal section, fig. 32,
is attached, by its rod, g, to the vertical rod, n', which is joined to
the rod, n^ of the circular eccentric, n, figs. 33 and 34. When,
as w-as customary until lately, the centre of the eccentric lies in a
radius perpendicular to the direction of the crank, the movements
of the steam piston and valve are diflTerent to each other — that is
to say, when the crank passes from the left horizontal to the right
horizontal position, the piston makes a corresponding rectilinear
movement ; tlie eccentric, however, passes from the lower extremity
of the vertical line, drawn through the centre of the first motion
shaft, to the upper extremity, or rice rersd,and consequently gives
the valve a rectilinear movement quite different to that of the pis-
ton, in such a manner that, when the latter is at the middle of ita
stroke, the valve, on the other hand, is at the end, and the .steam-
BOOK OF INDUSTRIAL DESIGN.
145
ports are consequently fully open, to give the steam the freest
passage into the cylinder.
Whilst the piston is accomplishing its stroke in one direction,
'-he valve moves up or down, and returns again to its central po-
sit' on, the part which it covered being opened and again shut;
when, however, the crank makes two fourths of a revolution in
different directions, the piston rises and falls half a stroke each
vav, whilst the valve makes a single rectilinear movement in one
arection.
Finally, for each of these movements, whilst the velocities of the
piston are increasing from the commencement towards the middle
of its stroke, those of the valve are decreasing, and reciprocally. It
therefore follows, that the maximum space passed through by the
piston, for a given portion of a revolution of the crank, corresponds
to the minimum passed through by the valve.
LEAD AND LAP OF THE VALVE.
416. Of late years, engineers have recognised the advantage of
inclining the radius of the eccentric, with regard to the radius of the
crank, instead of placing them perpendicular to one another, in such
a manner that, at the dead points — that is, the extreme high and
low positions of the piston — the valve shall already have passed
the middle of its stroke to a slight extent ; it is this advance of the
valve which is termed the lead.
The effect of giving this lead to the valve, is to facilitate the
introduction of the steam into the cylinder at the commencement
of the piston's stroke, and at the same time to allow a freer exit
to the v/aste steam on the other side of the piston ; a greater uni-
formity of motion is in consequence obtained, whUst les.*^ force is
lost.
In order to avoid as much as possible the back pressure due to
the slow exit of the waste steam, it is likewise customary, in addi-
tion to the lead, to give the valve more or less lap ; that is to say.
to make the width of that part of the valve which covers the ports,
a, b, fig. 28, sensibly greater than that of the ports themselves.
In explanation of the effects due to the lead and lap of the valve,
we have, in fig. 35, given a geometrical diagram, indicating the
relative positions of the crank, the piston, the eccentric, and of the
valve.
Let o o represent the radius of the crank ; with this distance as
a radius, and with the centre, o, describe a semicircle, which divide
into a certain number of equal parts. From each of the points of
division, let fall perpendiculars upon the diameter, o c. The points
of contact, 1, 2, 3, 4, &c., represent upon this diameter, considered
as the stroke of the piston, the respective positions of the piston,
corresponding to those, 2", 3', 4', &c., of the crank pin. It is un-
necessary to take into account the length of the connecting-rod,
which connects the latter to the piston, because, in the present caao,
Uie connecting-rod is supposed to be of an indefinite length, and
to remain constantly parallel to itself, so that it cannot modify the
results.
With the centre, o, likewise describe a circle with a radius,
I) a', e(jual to that of the eccentric, n. Wo have assumed the
point, a', to be the position the centre of the eccentric siiould
have at the moment when the piston i« at the end of its stroke —
that is to say, at o ; the distance of this point, a', from the vertical
m n, expresses the lead of the valve, and consequently the angle
m o a', is called the angle of lead. The position of the point, a'
may likewise be obtained, after the following data are decided on—
namely, the height of the ports, a, b, fig. 28, the width, r s, of the
flange of the valve, which is equal to the height of opening, t r
which properly expresses the amount of lead given to the port
augmented by twice the lap, together with the amount of the intro-
duction of the steam to the cylinder, and the amount of opening,
s' i', expressing the lead given to the escaping steam, and which is
always greater than the former, so that the exit passages may be in
communication as long as possible.
The diameter of the eccentric, n, is equal to the height of the
port, augmciuted by the width, r s, of the flange of the valve, and
the difference which exists between the two amounts of lead, s' t'
and r / ; it is, then, with the half of this as radius that the circle,
a' b' c' d', must be described ; and we then obtain the point, a', by
setting off from the centre, o, to the right of the vertical, m n, a.
distance equal to the lead of introduction, r t, augmented by the lap.
Starting from this point, a', we then divide this circle into as many
equal parts as we previously divided the one mto, described by the
crank pin, and then through each of the points of division we draw
perpendiculars to the vertical, m n.
We further draw the straight line, a' g', parallel to m n, when
the distance of the several points of division from this line ^\'ill
indicate the successive positions of the valve in relation to those
of the piston. Thus, after having drawn the horizontals, r u,
through the extreme point, r, of the valve, at the moment when
the piston is at the extremity of its stroke, make I'' — 1' equal to
b' b^, and the point, 1', indicates how far the valve has descended
during the time the piston has traversed the space, o 1, whilst the
crank has described the first arc, o 1'. In like manner, set off
the distances, c' c', d' d\ &c., which correspond to the third and
fifth divisions, reckoning from the horizontal line, ru, from i' to 3',
and from h^ to 5', on the verticals corresponding to the third and
fifth positions of the piston, and consequently the positions, 3^
and 6^ of the crank. It will then be seen that the valve con-
tinues to descend until the moment the centre of the eccentric
reaches the point,/', upon the horizontal line, of, corresponding
to the sixth position, and the valve then wholly uncovers the
port, a, as shown in fig. 29. During the continued revolution of
the eccentric, on passing this point tiie distances of the points of
division from the line, a g', diminish, and the valve reiiscends, in
such a manner as that, when the centre attains the point, p — that
is to say, when the crank shall have performed a semi-revolution,
and the piston have arrived at 18, at the other end of its stroke —
the valve will occupy the position indicated in tig. 30. This
figure shows that it uncovers tlio lower port, b, for tiie introduc-
tion of the fresh steam, and the upper one, a, for the escape of
the used steam. If the respective positions, U', 7', 8', 9', &c., of
the valve, be determined tliroughoul I lie .ntire stroke, by sotting
oir upon the verticals, 6, 7, 8, S), \c., the distances of tlio points
of division of tiio eccentric from the straight line, u' i.-', lus already
explained, a curve will br t'oriued, ;is at u 3' G' 9' 18', which is a
species of ellipse. This diaeTani lias the advantage of bringing
into a single \ie\\ the relative positions el' the crank, piston, wcon-
1 '.s
THE PRACTICAL DRAUGHTSMAN'S
trie, and valve, and facilitatea the determination of the position of
the valve, corresponding to any position of the piston.
Thus, to obtain the position of the valve to correspond to that,
y, of the steam-piston, it is sufficient to draw the vertical, y x',
which will cut the curve in the point, v'. The distance, i;' x', of
this point, from the horizontal, t u', passing through the upper
edge of the introduction port, «, shows how much of this is un-
covered by the valve. It will be seen, also, that the curve is cut
by the horizontal, t u', in the point, y\ which indicates the moment
at which the valve closes the port. In this position the piston will
only as yet have reached the point, xf, of its stroke ; and it has,
consequently, to traverse the distance, y'' 18, before it can receive
any more steam from the boiler, which shows that, with a valve
which has lead and lap, we actually work the steam expansively to
a slight extent. In the case before us, the steam is cut off at four-
fifths of the stroke.
It will be understood that, if the machine continues its action,
the piston will retrace its stroke, the centre of the eccentric which
had reached j) will continue to ascend, and the valve will shortly
attain the position indicated in fig. 31, this taking place as soon as
the centre of the eccentric reaches the point, z. In this position,
the ports, a, h, are completely open — the first to the exit aperture,
the other to the introduction of the steam, whilst the valve is at
the highest point of its stroke, as will also be found by continuing
the curve, u 9' 18', of which the prolongation, 18' 24' 30', is exactly
symmetrical with regard to the inclined line, u 18'. On the same
diagram, tig. 35, we have delineated a second elliptic curve, 0" 9" 18",
equal and parallel to the first, and which indicates the respective
positions of the point, s', of the lower flange of the valve, in relation
to the port, h, so as to have, at first sight, the respective positions
of this second flange. This outline is evidently obtained by setting
oflT the constant distance, r s', of the valve, fig. 28, upon the verti-
cals, drawn through 1, 2, 3, 4, &c.
It may be remarked, that the distance between the ports, a and
h, is arbitrary. It is, however, advisable to reduce it as much as
possible, in order to diminish the surface of the valve, and, conse-
quently, the pressure of the steam acting on the back of it In
all cases, it is necessary that the height of the exit port, c, should
be greater than that of the introduction ports, by a quantity at
least equal to the difference which exists between the lap and the
lead, l! s' and t r.
EXPAMSION VALVE.
417. The action of the expansion eccentric, o, is analogous to
rhat of the ordinary valve eccentric, except that the position of its
centre is not regulated in the same manner.
We may observe, in the first place, that this eccentric is not
immovably fixed upon the main shaft, l, as is the case with the
preceding one. It is only attached to the adjustable collar-piece.
F^ figs. 36 and 37, by screws, v?. This arrangement allows of its
throw being increased or diminished ; that is, of its centre being
placed further from or nearer to that of the shaft, according to the
icngth of stroke which it is wished to give it. To this end, its
central opening is oblong in shape, and the holes for the securing
screws are oblong likewise.
If the centre of this eccentric happens to be in the same diree-
tion as the crank, the expansion valve, h — the rod, hfoi which is
guided by the socket-bracket, 7i*, attached to the pedestal, s, and
drawn more detailed in fig. 43 — is wholly open when the piston is
at the end of its stroke ; but we have supposed, as indicated in fio-.
35, that the centre of this eccentric is in the point, a*, upon the
circle described with the centre, o, and radius, l a*, of the eccentric,
and that the valve does not therefore wholly uncover the entrance
port, d, at this moment, so that the time of closing it may be later
than would otherwise be the case.
As in the preceding case, we divide this circle into equal parts,
starting from the point, a*; through the point, aS draw a vertical
line, and then set off on the various verticals, 1, 2, 3, 4, &c., the
distances of the points of division from this line, measuring these
from the horizontal passing through the upper edge, r", of the
valve, H ; we thus obtain a second elliptic curve, u m' n' p', the
inside of which is flat — tinted with a slightly stronger shade than 1 J
the ellipse corresponding to the distribution valve, so as to rendei • '
the diagram more distinct. This curve cuts the horizontal line
drawn through the upper edge of the port, d, in the point, n, whi<;h
indicates at what time the valve, H, closes the entrance port, fig. 39.
It wUl be seen that this point corresponds to the position, b', of the
steam-piston, thereby signifying that the cut-off takes place when
the piston has performed no more than a fourth of its stroke. Con-
tinuing the movement, it will be observed that the valve, H, rises
higher and higher, so that it begins to uncover the entrance port a
little before the piston reaches the end of its stroke ; but it is evi-
dent that the steam cannot find its way into the cylinder at this
point, for the distribution valve is in its turn closed, as soon as the
position, y y', is passed ; no inconvenience, therefore, will be caused
by the fact of the valve, h, being open before reaching the end of
its stroke, as indicated in figs. 38 and 40, and as shown also in the
diagram, fig. 35.
By varying the radius of the eccentric, o, and the position of its
centre relatively with the radius of the crank, it will be easily un-
derstood, that *Fithin certain limits we can alter the time when the
valve, H, opens and closes the entrance port, and are consequently
enabled to vary the degree of expansion.
Figures 41 and 42 show that the rod of the valve is attached to
it by a T joint, which leaves the valve sufficiently free for the steam
to press it constantly against the planed valve face ; and a similar
adjustment is adopted v\'ith the distribution valve.
The general explanations which we have ^ven in the preceding
pages, with reference to the construction and action of this engine,
evidently apply to other systems, which merely differ in some of
the arrangements and forms of the component pieces. Moreover,
in our notes, the student will find the rules and tables concerned
in the calculations and designs of these engines.
RULES AND PRACTICAL DATA.
STEAM-ENGINES.
LOW-PRESSURE CONDENSING ENGINE, WITHOUT EXPANSION VALVE.
418. In those engines which are called low-pressure engines,
the steam is produced at a temperature very little over that of
boiling water, or 100° centigrade (212° Fahrenheit) — it is, in fact,
BOOK OF INDUSTRIAL DESIGN.
147
generally 106° cent. — in which case the tension of the steam will
sustain a column of mercury of 90 centimetres in height ; that is to
say, 14 centimetres above that due to atmospheric pressure. It is,
consequently, equal to a pressure of ri7 atmospheres, or I'ii kilog.
per square centimetre. It is for this pressure that what are gene-
rally known as Watt's engines, without cut-off valves, are calculated;
and the one we have been examining is regulated upon this datum.
There is, however, a great difference between the pressure of
the steam in the boiler, and that to which the effective power of the
machine is due. It is evident that a part of the pressure will be
absorbed by the back pressure due to an imperfect vacuum, as well
as by the friction of the piston, and other moving parts, and the
leakage and condensation in the steam passages. So that, taking
into consideration these various causes of loss, the effective force
may be estimated at '5 kilog. only, per square centimetre, in the
majority of engines, whilst it may reach, perhaps, -65 kilog. in the
most efficient.
The rule for calculating the power of low-pressure steam-engines
consists in —
Multiplying the mean effective pressure of the steam upon the piston
by the area of the latter, expressed in square centimetres, and the pro-
duct by the velocity in metres per second.
The result of this calculation will be the useful effect of the
engine in kilogrammetres.
To obtain the horses power, this result must be divided by 75.
Thus, the diameter of the cylinder of a low-pressure non-expan-
sive steam-engine being -856, and its section 5755 square centi-
metres, if the effective pressure upon the piston is -63 kilog. per
square centimetre, and the velocity 1-1076 —
We have
•63 X 5755 X 1-1076 = 4015-67 k. m.
Whence —
4015-67 -^ 75 = 53-54 H. P.
But the effective pressure upon the piston is not always -63
kilog. per square centimetre ; it is more frequently below than above
this amount. It varies not only according to the power of the
machine, but also according to the state of repair. Thus, some-
times the effective pressure will not be more than -45 kUog. in
small engines, whilst in large, powerful ones, it may at times reach
•65 kilog.
Single-acting engines, such as are employed in mines, are of the
same dimensions as double-acting ones, but of only half the power.
Thus, the cylinder of a low-pressure steam-engine, of 50 horses
power, and only single-acting — that is to say, receiving the action
of the steam during the descent only of the piston — is exactly the
same as in a machine of 100 horses power, in which the steam acts
alternately on both sides of the piston.
In the following table, which applies to this kind of steam-engine,
we have given the diameters and velocities of the steam-piston
from 1 to 200 horses power.
TABLE OF DIAMETERS, AREAS, AND VELOCITIES OF PISTONS, IN LOW-PRESSURE DOUBLE-ACTING STEAM-ENGINES, WITH
THE QUANTITIES OF STEAM EXPENDED PER HORSE POWER.
Horses
Diameter
Area of Piston.
Length of
Number
of
revolutions.
Velocity of
piston
per second.
Velocity of
piston
per minute.
Effective
pressure on
the piston
per square
centimetre.
Weight of
steam ejpend-
power.
of
piston.
Total.
Per
stroke.
ed per horse
power per hour
hor
se power.
cent.
sq. m. s
q. cent.
m.
per 1'.
III.
ni.
kilo?.
kiloj.
1
•15
•018
181
•62
50
-85
51
-49
38-81
2
-21
•036
178
•61
42
•86
62
-49
38^77
4
-30
-068
171
•76
34
-90
54
•49
38^-7
6
•35
■098
163
•91
31
-94
57
•49
38-72
8
■40
•128
160
r07
27
-96
68
•49
38-72
10
•45
•169
159
r22
24
•98
69
•49
38-64
12
•49
•189
157
V22
24
•98
59
•49
38-64
16
•55
•240
150
1-31
22
1-01
60
•50
37-80
20
•61
•292
146
1-62
20
1-02
61
•51
37-38
24
-66
•346
144
1^69
18
1-02
61
•62
3688
30
-73
•414
137
r83
17
1-04
62
53
36-04
40
•83
•535
134
1-99
16
1-06
64
63
36-70
50
•91
•658
132
2-13
15
1-07
64
64
36-32
60
TOO
•779
130
2^28
14
1-07
64
64
34-94
70
ro7
•903
129
2^44
13
1-06
63
•65
3436
80
M4
1-032
129
2-44
13
1-06
63
•56
3431
90
1-21
1-138
126
2-59
12
1-04
62
•57
•58
3301
3297
100
1-27
1-264
126
2-59
12
1-04
()2
120
1-39
i-512
126
2-74
11
1-00
60
•69
3 1 -92
160
r60
2-005
125
3-00
10
1-00
60
•60
3 1 •t)7
200
r78
2-480
124
3-00
10
1-00
(iO
•(il
31-47
DIAMETER OF THE PISTON.
By means of the above table, wo can, in a very simple inamuM-,
Aetoriiiino the diaiiietor and velocity of the piston of a iow-picsMiiro
ilouhlc-acting steam-ongino, supposing tlio nUmu to ho of tlio
pn^ssuro of 1-17 iilmosiihcrfs in tlu« hoilor, i-orrespoiiding U» a
coluiuii of iiicivinv "t!Hl .•.■iiliiii.ln'N in li.'iH:ht.
148
THE PRACTICAL DRAUGHTSMAN'S
Rule. — It is sufficient to obtain from tlie table the area of piston
per horse power, and to multiply it by the number of horses power
of the engine to be constructed, which will then determine the cor-
responding area of piston.
Example. — What should be the diameter of the piston of a low-
pressm-e double-acting steam-engine of 25 horses power ?
In the fourth column of our table, it will be seen that the area
to be given to the piston should be 144 square centimetres per
horse power for 24 to 26 horses, with a velocity of 1-02 m. per
second.
We have, therefore, 144 x 25 = 3600 sq. cent, for the total
area of the piston.
Whence —
■V'3600 X -7854 = 67-7 cent.
Thus, the diameter of the piston must be -677 m.
VELOCITIES.
The velocities per second, and per minute, given in the seventh
and eighth columns of the table, are what are generally adopted as
the regular working rates in establishments and manufactories
where steam-engines are employed, whatever may be the number
of revolutions of the crank, or strokes of the piston, per minute, for
this number varies according to the length of stroke which it is
wished to give to tne piston. Thus in stationary engines, the stroke
of the piston is generally longer ; and, therefore, fewer strokes are
made per minute than in marine engines, since in these latter the
engineer seeks, as much as possible, to reduce the height of the
machinery ; and the stroke is, consequently, much shorter for the
same amount of power.
The length of stroke of the piston is regulated at pleasure by
the constructor, according to what he may find most advantageous
in the transmission of the power to the machinery ; and he calcu-
lates so that the crank may make a few revolutions per minute
more or less, without occasioning any very sensible difference in
the velocity of the piston, with regard to the velocities laid down
in the table.
If, notwithstanding, it is wished to construct an engine to work
with a velocity somewhat less, or somewhat greater, than that given
in the table, it will evidently be necessary to take this difference
into consideration, and to augment or diminish the area of the
piston in proportion, so as always to obtain the required power.
The proper amount of alteration may be determined by a very
smaple operation.
Example. — Let it be proposed to construct our example engine
of the effective power of 25 horses, wdth a velocity of piston of 1
metre per second, in place of 1'02 m.
It will be sufficient to calculate the following inverse propor-
tion : —
1 : 1-02 : : 144 sq. c. : x.
Whence —
X = 144 X 1-02 = 146-8 sq. c,
the area, per horse power, to be given to the piston.
Consequently,
146'8 X 25 = 3675 sq. centimetres for the total area;
and
V3675 "=" '7854 = 68-4 cent., for the diameter of the piston.
As complemental to this table, we have given the expenditure
of steam corresponding to the different powers, as well as the de
ductions from this of the expenditure per horse power per houi.
It will be observed from the last column, which gives the expendi-
ture of steam that it is considerably more for engines of small
force than for more powerful ones — the reason of which is self-evi-
dent. Thus, for an engine of 12 horses power, the expenditure of
steam is 38-64 kilog. per horse power per hour ; whilst for an
engine of 100 horses power, the e.xpenditure only reaches 32-97
for a like power in the same time.
The expenditures or weights of the steam have been calculat^a
from the following formula : —
W = AxSx2ox2Nx60.
A representing the area per horse power ;
S, the stroke of the piston ;
w, the weight of a cubic metre of steam at the pressure employed;
N, the number of revolutions.
We need not here take into consideration the loss of steam re-
sulting from leakage and condensation in the steam pipes and
pa-ssages, which is generally estimated at one-tenth of the whole
expenditure, as this item should e\-idently enter into the calcula-
tions respecting the boiler.
STEAM-PIPES AND PASSAGES.
The section of the pipe which conveys the steam to the cylindei,
as well as that of the introduction ports and passages, should be
equal to a twentieth of the area of the piston.
Whence it follows, that the diameter of the steam-pipe should
be one-fifth of that of the cylinder.
We must, however, remark, that the greater the velocity of the
engine, the gi-eater should be the sectional area of the steam-pipes
and passages. It is because of this that, in locomotive engines,
this section is sometimes made a tenth or a ninth of that of the
cylinder, and at the same time the pressure of the steam is much
greater, being generally equal to 5 or 6 atmospheres, and sometimes
more, in such engines.
AIR-PUMP AND CONDENSER.
The stroke of the air-pump piston is equal to half that of the
steam-piston ; and as it gives the same number of strokes, but does
not discharge in ascending, it can only raise a quantity of air and
water equal to its own cubic contents, at each double stroke.
Now, the sectional area of the pump is -2827 sq. m. ; and the
length of stroke, -923 m. Its capacity is, therefore, -261 cubic
metres; and as twice the cubic contents of the steam-cylinder is
2-125 cubic m., it follows that the pump discharges only a little
more than an eighth of the volume sent out by the steam cylinder.
This capacity is quite sufficient for the effective action of the
engine.
The sectional area of the condenser is the same as that of the
pump, and its length is about 1 metre ; so that its capacity is, at
least, as great.
As the quantity of water to be injected into the condenser varies
according to the temperature of the injection water, it will be well
lo know how to regulate it.
To this end, the following rule will answer : —
BOOK OF INDUSTRIAL DESIGN.
149
Rule. — Take the excess of the temperature of the steam over that
of the injected water, and, after adding 550 to it, multiply it by the
weight of steam to be condensed, and divide the product by the differ-
ence of temperature between the discharged and the injected water.
The quotient will be the weight of cold water to be injected.
Thus, let w represent the weight of the steam to be condensed;
t, its temperature ; W, the weight of the cold water to be injected
into the condenser ; i', its temperature ; and T, that of the water
discharged : —
We have
w(o50 + t-T)
T — t'
If we make w = 26-16, t' — 12° cent., T = 38°, and i = 105°,
we shall have
„, 26-16 (550 + 105° — 38°)
38° — 12°
Whence, W = 621 kilog. or litres, for the expenditure per
minute of cold water in the condenser.
That is to say, the quantity of water to be injected into the con-
denser should, in this case, be about 24 times the weight of the
steam expended.
If the discharged water were of the temperature of 55°, the cold
water remaining at 12° —
We should then have
_ 26-16(550 + 105° — 55°)
55° — 12°
Whence —
W = 365 kUog. or litres.
That is to say, that in the la-st case the water injected would not
be more than 14 times the steam expended.
But it is to be remarked, that in this case the force of the steam
in the condenser, at a temperature of 55°, is equal to a column of
mercury of 12-75 centimetres in height; whUst, in the first case, it
would only be equal to a column of 5-5 cent. There is, therefore,
an advantage in employing sufficient injection-water to produce the
lower of the two temperatures.
From the preceding results, we may deduce what follows : —
First, That the stroke of the air-pump piston, in low-pressure
double-acting steam-engines, is ordinai-Uy equal to half the stroke
of the steam-piston.
Second, That the diameter of the air-pump piston is equal to
about two thirds of the diameter of the steam piston ; and, conse-
quently, its area is about half that of the latter.
Third, That the effective displacement of the air-pump piston —
that is, the cubic contents of the cylinder generated by the disc of
the piston — is equal to an eighth, or at least a ninth, of the contents
of the cylinder generated by a double stroke of the steam-piston.
Fourth, That the capacity of the condenser is at least equal to
that of the air-pump.
Fifth, That the sectional area of the passjigc communicating
between the condenser and air-pump is equal to one-fourth the area
of its piston.
Sixth, That the quantity of cold water to be injected into the
condenser varies according to its temperature, and to the tcnijxni-
ture of the water discharged.
Seventh, Tha' this quantity is equal to 24 times the weight of
steam expended by the cylinder, where the mean temperature of
the cold water is 12°, and that of the water of condensation 38",
which are generally what exist in low-pressure double-acting
engines.
COLD-WATER AND FEED PUMi-s.
The capacity of the cold-water pump should be the 24th or 18th
of that of the .steam cylinder. The capacity of the feed or hot-
water pump should be the 230th or 240th, at least, of that of the
steam cylinder.
HIGH-PRESSURE EXPANSIVE ENGINES.
Let the following dimensions be given for an engine analogous
to that which we have just described: —
Diameter of the cylinder, = -275 m.
Stroke of the piston, = -680 m.
Area of the piston, = -0594 square m.
Number of double strokes per minute, = "40
Let us suppose, in the first place, that when the steam reaches
the cylinder, its pressure is equal to 5 atmospheres, and that it is
cut off during three-fourths of the stroke ; that is to say, that the
cylinder only receives the steam during the first quarter of the
stroke.
This pressure of 5 atmospheres is equal to 5 x r033 = 5-165
kilog. per square centimetre. Consequently, the total pressure
exerted upon the surface of the piston is —
5-165 X 594 sq. cent. = 3068 kilog.
And as with this pressure the piston passes through a space
equal to one-fourth of its stroke, or
•680 -=- 4 = -170 m.,
it is capable, theoretically speaking, of transmitting an amount of
force expressed by
3068 X -17 = 521-56 kilogrammStres.
Next, dividing the length, -51 m., or the remaining three-fourths
of the stroke, into an even number of equal parts — as four, for
example — each of these parts will be equal to
51 , -
—- = -12/0 m.
4
Now we know that, according to Mariotte's law, the successive
volumes of a given quantity of any gas are in the inverse ratio of
their tension or pressure, provided the gas is in the same condition
throughout. This principle may be reg-arded a.s quiie true in steam-
engines, because the expansion is never carried very far, and as tlio
steam passes through the cylinder with groat rapidity, and is con-
tinually being renewed, alter a certain time and wiieii the i-yliiuiiT
has become warm, its teniperaturo is very little below that of Ilio
steam itself, and the latter suffers no appreciable clinngo in y>ass\\\<;
through it. Putting P for the pressure, 3068 kilog., as found lor
the first (juarter of the stroke, we may state the relation.s of tlio
vdlunus iuul pressures in the following manner: tlmt is, at Hio
points, 1, 2, 3, l, 5, of the stroke, or lor the sum'ssivo ep*c«»,
-170 ni., -295 m., -i'lb ni., 5525 m., -080 in.
150
THE PRACTICAL DRAUGHTSMAN'S
The corresponding pressures will be —
P -1700 -170 - 1700 -170
3068 A:, -2975 ' -425 ' -5525 ' -680 '
or finally,
3068 k, 1764 k, 1227 k, 944 k, 767 k.
Next, according to Simpson's method, we have
The sunt of the extreme pressures, = 3068 + 767 = 3835
Twice the pressures at the odd intervals, = 2X 1227 = 2454
Four times the pressures of the even intervals, =4(1764+ 944)= 10632
Total, 17121
Taking the third of this quantity, and multiplying it by -1275,
we shall have the work given out during the cut-off. Thus —
17121 X -1275
g = 727-64 k. m.
Adding to this 521-56 k. m., the work given out before the cut-
off, we shall have the total of the work given out by the steam
during the entire stroke of the piston —
= 1249-2 k. m.
Deducting now from this the effect of the atmospheric pressure,
which resists the motion of the piston throughout the stroke, and
which is equal to
1-033 k. X 594 sq. c. x -68 m. = 417-25 k. m.,
there remains for the effective force of the piston —
1249-2 — 417-25 = 832 k. m.,
nearly, for each stroke ; and as the piston gives 40 double or 80
single strokes per minute, the effective force per minute becomes
832 X 80 = 56560 k. m. ; that is, 56660 kilogrammes, raised one
metre high.
The effective povver of this, as well as of most other expansive
steam-engines, will be obtained in a much more simple and less
tedious manner, by taking advantage of the following table : —
TABLE OF THE FORCE, EST KILOGRAMMETRES, GIVEN OUT WITH VARIOUS DEGREES OF EXPANSION BY A CUBIC
ilETRE OF STEAM AT VARIOUS PRESSURES.
■
Force
3;iven out, corresponding- with tlie pressure of
Volume
when
expanded.
1
^
2
2^
3
4
5
6
atmosph.
atmosph.
atmosph.
atmosph.
atmosph.
atmosph.
atmosph.
atmosph.
Mibic metres.
k. ra.
k. m.
k. m.
k. m.
k. m.
k. m.
k. m.
k. m.
1-00
10333
15500
20666
25833
31000
41333
51666
62000
1-25
12639
18968
26278
31597
37917
50566
63196
75834
1-50
14523
21784
29046
36257
43668
68092
72615
87138
1-75
16116
24174
32232
40290
48348
64464
80580
86696
2-00
17496
26244
34992
43740
62488
69984
87480
104976
2-25
18713
28069
37426
46782
66139
74852
93565
112278
2-50
19802
29703
39604
49605
59406
79208
99010
118812
2-75
20787
31180
41574
61967
62361
83148
103936
124722
3-00
21686
32629
43372
54216
65058
86744
108430
130116
3-25
22613 '
33769
45026
56282
67539
90052
112666
135078
3-50
23279
34918
46658
58197
69837
93116
116395
139674
3-75
23992
35988
47984
59980
71976
95968
119960
143952
4-00
24658
36987
49316
61645
73974
98632
123290
147948
4-25
25285
37927
60570
63212
76855
101140
126425
151710
4-50
26875
38812
51750
64687
77625
103500
129375
165250
4-75
26434
39651
62868
66085
79302
105736
132170
158604
6-00
26964
40446
63928
67410
80892
107866
134820
161784
5-25
27467
41200
64934
68667
82401
109868
137335
164802
6-60
27949
41923
55898
69872
83847
111796
139745
167694
6-76
28408
42612
66816
710-20
85224
113632
142040
170448
6-00
28848
43272
67696
72120
86544
116392
144240
173088
6-25
29270
43905
58540
73175
87810
117080
146350
175620
6-50
29675
44512
69350
74187
89025
118700
148375
178060
6-75
30065
46097
60130
76162
90195
120260
150326
180390
7-00
30441
45661
60882
76102
91323
121764
152205
182646
7-25 ■
30804
46206
61608
77010
92412
123216
164020
183224
7-50
31164
46731
62308
77886
93462
124616
165770
186924
7-75
31494
47239
62986
78732
94479
125972
167465
188958
8-00
31820
47730
63640
79550
96460
127280
159100
190920
8-25
32139
48208
64278
80347
96417
128556
160695
192835
8-50
32447
48670
64894
81117
97341
129788
162235
194682
8-75
32747
49120
66494
81867
98241
130988
163735
196482
9-00
33038
49557
66076
82595
99114
132162
165190
198228
9-25
33321
49981
66642
83302
99963
133284
166605
199926
9-60
33597
50395
67194
83992
100791
134388
167985
201582
9-75
33866
50797
67730
84662
101595
136460
169325
203190
10-00
34127
61190
68254
85317
102381
136508
170635
204762
BOOK OF INDUSTRIAL DESIGN.
151
According to this table, if we have to calculate the force acting
upon the piston in this engine, in the same circumstances, we must,
in the first place, ascertain the original volume of the steam intro-
duced into the cylinder during the first quarter of the stroke of the
piston. This volume is equal to
•0594 X -17 — -010098 cubic metres.
Now it will be seen from the table, that the force given out when
a cubic metre of steam, of a pressure of 5 atmospheres, expands to
four times its original volume, is equal to
123290 k. m.
Consequently, that corresponding to a volume of -010098 cubic
metres will be —
123290 X -010098 = 1245 k. m.,
And deducting from this the atmospheric pressure, which resists
the motion of the piston, we have
1245 — 417 = 828 k. m.,
a quantity which differs very little from that obtained by the more
tedious calculation. Thus, the calculation for determining the
effective power of a steam-engine, of which we know the diameter
and stroke of the piston, the pressure of the steam, and the amount
of cut-off, reduces itself to the following rule : —
Rule. — Multiply the area of tlie piston by the portion of the length
of the stroke, during which the steam acts with full pressure, and you
will determine the volume of steam expended. Multiply this volume
by the amount of kilogrammetres in the table, corresponding to the
pressure of the steam and to the final volume, and then deduct froin
the product the amount, in kilogrammetres, of the atmospheric pressure
opposed to the piston during the entire stroke, and the result will be
^he theoretic amount of force, in kilogrammetres, given out by the
steam during a single stroke of the piston.
A MEDIUM-PRESSURE CONDENSING AND EXPANSIVE STEAM-
ENGINE.
Let the following data be assumed : —
The diameter of the steam-cylinder = -330 m.
The stroke of the piston = -650 m.
The diameter of the air-pump = -180 m.
The stroke of its piston = -325 m.
The diameter of the feed-pump . . . . = -035 m.
The stroke of its plunger = -235 ra.
It follows, from these dimensions, that we shall have — •
The area of the steam-piston = 855-30 sq. cent.
The area of the air-pump piston . . . = 254-47 "
The area of the feed-pump = 9.62 "
And for the displacement, or volumes of the cylinders generated
by the pistons —
That of the steam cylinder . . . . = 65-594 cubic dccim.
That of the air-pump = 8-270 "
That of the feed-pump — -226 "
Wo shall suppose that, when tho engine is in rogiii.ir working
••iiiidition, tho pressure of tho sicam is 3J atmoHpiuM-es ; and we
iiiiihI ascertain what is tho actual loivo given out, sM|)poHing the
HtcuMi to be cut off during three-fourlhs of the stroke of the piston.
That is to say, that the steam is admitted into the cylinder only
during a quarter of the stroke, which corresponds to -1625 m.
Since the sectional area of the cylinder is -0885 m., the volume
of steam expended during a fourth of the stroke will be equal to
-0885 X -1625 = -0139 cubic metres; or,
13-9 cubic decimetres.
Now, according to the table of the amounts of force given out by
the steam at various pressures, it will be found that the force due
to a cubic metre of steam, of an initial pressure of 3^ atmospheres,
when allowed to expand to four times its volume, is equal to 86303
kilogrammetres. As the table does not give the actual amount for
3i atmospheres, it may be taken by adding together that for 21
and 1 atmospheres. Thus —
61645 + 24658 = 86303 k. m.
We have, therefore, in the present case —
-0139 X 86303 = 1199-6 k. m.,
as the force due to a single stroke of the piston.
From this quantity, however, we must deduct the back pressure
due to the imperfect vacuum in the condenser. This back pressure
is, in the generality of cases, equal to about -27 kilog. per square
centimetre, when the temperature of the water of condensation is
about 65" cent.
Allowing this to be the case in the present example, we shall
have to deduct from the preceding result the action of this back
pressure upon the whole surface of the piston, and during the entire
stroke. This is
-27 X -0885 X -65 x 150-1 k. m.,
We have, consequently,
1199-6 — 150-1 — 1049-5 k. m.,
for the actual force given out by the piston during a single stroke ;
and if this engine works at the rate of 42 revolutions per minute,
which supposes the velocity of the piston to be -9 m. per second,
we shall find that the mechanical effect per minute will be equal to
1049-5 X 84 = 981588 k. m ; or,
881598 -f- 4500 = 19-59 horses power.
It is well known, however, that this amount is far from being all
transmitted by the first-motion shaft, for a portion is absorbed in
overcoming the friction of the various moving parts of the engine,
and there are also other causes of loss.
If we reckon that the force which is really utilised is not more
than four-tenths of that theoretically due to the steam, in which
case we must suppose that six-tenths are completely lost, we shall
have for the effective force transmitted to the first-motion shall—
19-69 X -4 = 7-84 horses power;
or almost 8 horses power, of 75 kilogrammetres each.
If it is desired to know the quantity of fuel consiuuod per hour
in producing this mechanical efVoct, we may remark, that a cuhiv
metro of steam, at a pressure equal to 3i atmospheres, weighs 1-8518
kilog. ; and at a pressure of 4 atmospheres, it weighs 2-0'291 kilog.
Now, although wo have supposed tho pressure in the cyliudor to
be 3'f atmospheres, wo, novertlieless, allow tiiat it will bo consid»»-
rably more in tho boiler, to compeusnto for tho leakjigo iu tho valve-
casing, passages, and valves.
Taking 1 atnios|ihores as the iirossiiro hi tho boiler, it will Ik<
foiMul that the weight of stonm expended for each .singlo stroko of
the piston is —
152
THE PRACTICAL DRAUGHTSMAN'S
•0139 X 2-091 = -0291 kilog. ;
and per hour —
•0291 X 84 X 60 = 146-204 kilog.
From which it follows, upon the hypothesis that one kiiog. of
2oal generates 6 kilog. of steam, that the quantity of fuel consum-
ed will be
146-64 -4- 6 = 24-44 kilog. per hour.
And since the power obtained is 7^84 horses power —
We have
24^44 -f- 7^84 = 3^1 kilog.
for the quantity of coal consumed, per horse power, per hour-
To complete the rules here given, we add the two following
tables, relating to the principal dimensions given to steam-engines
of different kinds : —
TABLE OF PROPORTIONS OF DOUBLE-ACTING STEAM-ENGINES, CONDENSING AND NONCONDENSING, AND WITH OR
WITHOUT CUT OFF, THE STEAM BEING TAKEN AT A PRESSURE OF 4 ATMOSPHERES IN THE CONDENSING, AND AT
5 ATMOSPHERES IN THE OTHER ENGINES.
Condensing en^nes,
Noncondensing expansive en-
Noncon
Jensing
Stroke
of
Velocity
of piston
Number of
revolutions
cutting off at one-fourth of the stroke.
gines, cutting oiF at one-fourth.
nonexpansive engines.
Horses
power.
piston.
per second.
per minute.
Diameter
Area of piston
Weight of
Diameter
Area of piston
Diameter
Weight of
of the
per
steam per horse
of the
per
of
steam per horse
piston.
horse power.
power per hour.
piston.
horse power.
piston.
power per hour.
cent.
cent.
cent.
sq. cent.
kil..g.
cent.
sq. cent.
cent.
kilog.
1
40
70
52-5
16
189
24-90
14
148
10
50-76
2
50
75
45-0
20
160
22-62
19
135
14
49-56
4
60
80
40-0
27
148
22-38
25
124
18
46-98
6
70
85
36-4
32
138
22-08
31
123
21
45-30
8
80
90
33-7
36
127
21-54
33
106
23
42-00
10
90
95
31-7
39
119
21-36
36
100
25
41-.34
12
100
100
30-0
42
112
21-18
38
92
26
40-86
16
110
105
28-6
46
104
20-58
42
87
29
39-96
20
120
110
27-5
49
94
20-28
45
81
31
38-82
25
130
115
26-5
54
92
19-80
49
76
34
38-52
30
140
120
25-7
57
86
19-32
52
72
36
37-56
35
150
125
25-0
59
77
18-54
55
68
38
37-38
40
160
130
24-3
62
75
18-06
57
64
39
36-60
50
170
135
23-8
67
70
17-28
62
60
43
36-18
60
180
140
23-3
72
68
17-22
66
58
46
35-76
75
190
145
22-9
78
67
17-16
72
54
50
35-04
100
200
150
22-5
85
57
16-62
84
66
56
34-08
TABLE OF PROPORTIONS OF MEDIUM PRESSURE CONDENSING AND EXPANSIVE STEAM-ENGINFS, WITH TWO
CYLINDERS ON WOOLFS SYSTEM; PRESSURE, 4 ATMOSPHERES.
Diameter of cylinders in
centimetres.
Area of pistons in square centimetres.
Stroke of pist
ons in metres.
Horse
Revolutions
power.
d.
D
Per horse power.
i.
S.
per minute.
a.
A.
a.
A.
4
16
27
201
572
50
143
•67
•90
30-0
6
19
35
283
962
47
160
67
•90
30-0
8
21
38
346
1134
43
141
75
1-00
30-0
10
23
42
415
1385
41
138
75
1-00
30-0
12
25
46
491
1662
40
138
82
1-10
27-3
16
28
52
616
2124
38
133
90
1-20
27-5
20
30
54
707
2290
35
114
97
1-30
25^4
24
32
59
804
2734
33
113
97
1-30
25-4
30
34
63
908
3117
30
103
20
1-60
21-6
36
37
67
1075
3526
29
98
20
1-60
21-6
40
37
67
1075
3526
26
88
27
1-70
22-1
45
39
71
1194
3959
26
87
27
1-70
22-1
50
41
75
1320
4418
26
88
35
1-80
20-8
60
45
82
1590
5281
26
88
35
1-80
20-8
70
48
87
1809
5945
25
84
50
2-00
19-5
80
51
93
2043
6793
25
84
50
2-00
19-5
90
54
99
2290
7698
25
85
57
2-10
18-6
100
57
104
2552
8495
25
85
57
2-10
18-6
110
60
109
2827
9331
25
84
57
2-10
18-6
120
62
114
3019
10207
25
85
57
2-10
18-6
130
65
118
3318
10936
25
84
1-67
2-10
18-6
BOOK OF INDUSTRIAL DESIGN.
1.-53
CONICAL PENDULUM, OK CENTRIFUGAL GOVERNOR.
The centrifugal ball-governor is compared, in physics, to a
pimple pendulum, the length of which is equal to the distance of
the point of suspension from the horizontal plane passing through
the centres of the balls ; and the duration of an entire revolution of
the ball-governor is equal to that of a complete oscillation of the
pendulum.
The formula for determining the vertical height or the distance
of the point of suspension above the plane of the balls is, conse-
quently, the same as that employed to find the width of a pendu-
'um, of which we know the number of oscillations. It may be
reduced to the following rule : —
Rule. — Divide the constant number, 89,478, by (he square of the
number of revolutions per minute. The quotient will give the height
in centimetres.
Example. — What is the vertical height or distance of the point
of attachment, from the horizontal plane passing through the
centres of the balls of a governor, revolving at the rate of 40 turns
per minute ?
We have 40' = 1600,
and 89478 -f- 1600 = 56 centimetres,
for the height sought.
With this rule, it will be easy for us to calculate the heights of
conical pendulums, from the velocity of 25 revolutions per minute,
to that of 67 ; and within these will be found the rates of combi-
nations more generally met with in practice. We have given
them in the following table, adding a column, which gives the
difference in height for each revolution. And as the angle which
the arms of the governor make with the spindle is generally one
of 30°, when the balls are in a state of repose, or are going at
their minimum velocity, we have given, in the fifth column of the
table, the lengths of these arms, from their point of suspension to
the centres of the balls, assuming the angle of 30°, and making
them to correspond with the number of revolutions given in the
first column.
In calculating the lengths of the arms, we have employed the
following practical rule : —
Rule. — Divide the constant number, 103,320, hy the square of the
number of revolutions per minute, and the quotient will be the length
in centimetres.
Example. — Assuming the angle to be 30°, what should be the
length of the arms of a conical pendulum, making 37 revolutions
per minute?
Wo have 37' = 1369.
10-3320
Then—
1369
75-46 ceutim^tres,
for the length of the arms of the pendulum, or the diameter of the
circle described by the balls.
It is evident, that if, on the otiicr hand, the length of the arms,
with this angle of 30°, is known, the number of revolutions which
the balls make in a minute, will bo found bij dividing the number,
103,320, by the length of the arms expressed in centimetres, ami then
extracting the square root of the quotient.
The weight of the balls, according to the resistance they have
to encounter, is as important to detcnnino as the length of the
suspending-arms, in order that the governing action of the pendu-
lum may be suflSciently powerful and quick. It often happens, in
badly designed engines, that the governor produces no effect, be-
cause the length of the suspending-arms is not proportionate to
the velocity, or because the weight of the balls is not proportionate
to the resistance to be overcome.
We have considered that it would be a great convenience to
engineers and artisans to possess a table, sho\^ing at sight the ve-
locities and corresponding lengths, for the conical pendulums, or
ball-governors, generally employed in steam-engines, so as to
enable them to determine with certainty the exact proportions to
be given them, in relation to their spindles and driving-gear.
When these points are determined, the weights of the balls may be
easily adjusted.
TABLE RELATIVE TO THE DIMENSIONS OF THE ARMS AND TO THE
VELOCITIES OF THE BALLS OF THE CONICAL PENDULUM OR
CENTRIFUGAL GOVERNOR.
Number of
Square of the
Velocities.
Length of
Difference of
Length of Armt
Revolutions
per Minute.
Pendulum in
Centimetres.
Length for one
Revolution.
with an Augle
of SU".
Cent.
Mill.
Cent.
25
625
143-1
108
16
26
676
132-4
96
153
27
729
122-7
86
142
28
784
114-1
77
132
29
841
106-4
70
123
SO
900
99-4
63
116
31
961
93-1
67
107
32
1024
87-3
52
101
83
1089
82-1
48
95
34
1156
77-4
44
89
35
1225
73-0
40
84
36
1-296
69-0
37
80
37
1369
65-3
34
75
38
1444
61-9
31
71
39
1521
58-8
29
68
40
1600
55-9
27
64
41
1681
53-2
25
61
42
1764
50-7
23
68
43
1849
48-4
22
56
44
1936
46-2
20
63
45
2025
44-2
19
51
46
2116
42-3
18
49
47
2-209
40-5
17
47
48
2304
38-8
16
45
49
2401
37-3
16
43
50
2500
35-8
14
41
51
2601
34-4
13
40
52
2704
33-1
12
38
63
2809
81-8
12
37
54
2916
30-7
11
88
55
3025
29-6
10
84
56
3136
28-5
10
S5
57
3249
27-5
9
83
58
3364
26-6
9
8]
59
3481
25-7
8
80
60
8600
24-8
8
29
61
3721
•24-8
8
28
62
3844
23-3
7
27
63
3969
22-6
1
20
64
4096
21-9
7
26
65
4225
21-2
6
24
66
4356
•20-5
6
24
67
4489
1 9 • 9
6
28
68
4624
19-;i
23
Note.— With an nnglo of 80", tlio oontrifugivl force is tho 8*me for
nil loiitrlliH of pondiiliim.
Tills l;il)lf may also Ix' ooiisulloil in llu> oiiso of singlo-nrmod pondu-
linns, wlnchiii.MHc'ii-^ioiuilly .inployod, instciid of oontrifuicnl govor-
inns.
154
THE PRACTICAL DRAUGHTSMAN'S
CHAPTER XI.
OBLIQUE PEOJECTIONS.
APPLICATION OF RULES TO THE DELINEATION OF AN OSCILLATING STEAM CTLINDEK.
PLATE XLI.
419. In geometrical drawing, the planes of projection on which
the objects are represented, are chosen, when possible, so as to be
[larallel to the faces of such objects ; from which it follows, that
these are expressed in their exact shapes and dimensions. It is
often, however, that the position of certain parts of the machine or
apparatus to be drawn, are inclined in regard to the other parts, so
that all the surfaces cannot be parallel to the geometrical planes.
The projections of the inclined parts are oblique, and, consequently,
are seen as foreshortened.
The general method employed in projections is evidently appli-
cable to the delineation of oblique projections. It is, however,
necessary first to represent the objects as if parallel to the plane
of the drawing, so as to obtain the exact proportions and dimen-
sions, such views being auxiliary to the production of the oblique
representations.
420. Thus it is proposed to represent a hexagonally-based prism
or a six-sided nut, the edges of which are inclined to both the hori-
zontal and vertical plane.
We first of all represent this nut, in fig. 1, as placed with its base
parallel to an auxiliary horizontal plane, represented by the line,
L T, fig. 3. This gives the regular hexagon, abed ef.
If we were to make the vertical projection of this prism on a
vertical plane, parallel to one of the faces, or to a d, we should, in
this second auxiliary plane, have the projection of the edges,
abed.
The straight line, l' t', fig. 3, indicates the line of intersection
of these two auxiliary planes, when placed in their actual position
with regard to the nut ; and it is, therefore, the base line of the two
projections. This line forms, we shall suppose, the angle, l o l',
with the base line of the actual drawing in hand, which angle,
likewise, expresses the amount of inclination of the top and bottom
of the prism, with the actual horizontal plane ; whilst the angle,
y o o^, formed by the perpendiculars, drawn to each of the lines
through the point, o, expresses the amount of inclination of the
edges and axis of the prism with regard to the vertical plane. After
this, it is merely necessary, in order to obtain the points, a', b', c', d',
to set otf to the right and left of the point, o, on the line, l' t', the
distances, a o or d o, and b g or e g, derived from fig. 1. Drawing
peipendiculars to the line, l' t', through each of the points, a\ b\ c", d\
and limiting them by the lines, a' d' and a' d', fig. 2, parallel to the
former, we obtain the entire vertical projection of the prism, as
upon the auxiliary plane, parallel to one of the faces, as e b. When
one of the bases of the nut is rounded, or terminated by a spherical
portion, which is generally the case, as already seen (186), its con-
tour is limited by circular arcs, expressing the intersection of each
race "'ith the sphere.
We can then, by means of the two projections, figs. 1 and 2,
obtain the oblique projection, fig. 4, upon the vertical plane, l t ;
fig. 1 giving the widths, the distances of each of the points from
the axial line, a d, which passes through the centre, o, and fig. 2-,
defining the vertical heights or distances of the various points
above the horizontal plane.
To this end, through any of the points, as c, for example, ex-
pressing the horizontal projection of the edge, c" c', erect a vertical
line, and through the corresponding points, c° c', fig. 2, draw a
couple of horizontal lines, cutting the vertical in c" and c"'. The
same operation is performed with regard to the points, b, a, d, &c.,
which are projected in b'", a'", d", d'", fig. 4. The whole matter
consists, therefore, in drawing vertical lines through each of the
points in fig. 1, and horizontal lines through the corresponding
points in fig. 2. The intersections of these lines give the projec-
tions of the extremities of each of the edges in the oblique view,
fig. 4.
If it is wished to obtain the projections of the circular outlines
with minute exactness, it will be necessary to determine, at least,
three points in each arc ; and as we have the extremities already,
we only require now to find the middle of each. It is the same for
the circle representing the central opening of the nut. Its oblique
projection is necessarily an ellipse, the proportions of which are
obtained by the projection of the two diameters perpendicular to
one another, one of which, m n, is parallel to the vertical plane, and
does not alter in magnitude ; consequently, giving the transverse
axis of the ellipse, whilst the other is inclined and foreshortened,
and gives the conjugate axis.
421. In general, the oblique projection of any circle is always an
ellipse, the transverse axis of which is equal to the actual diameter
of the circle, whilst the conjugate axis is variable, according to the
inclination or angle which the plane of the circle makes with one
of the planes of projection. The application of this principle wall
be seen in figs. 5, 6, and 7. The two first of these figures repre-
sent the horizontal and vertical projections made upon the auxiliary
planes of a portion of the cylindrical rod. A, of the piston, b, work-
ing in the oscillating steam-cylinder, c ; and the last, fig. 7, is the
oblique projection of this part of the piston-rod upon the vertical
plane, corresponding to that of the drawing.
It will be remarked, that the upper part of the fragment of the
rod being limited by a plane, k I, perpendicular to its axis, is pro-
jected as an ellipse, the transverse axis, p q, of which is equal to
k I, whilst its conjugate axis, I' k', is equal to the projection of this
line, k I, on fig. 7. The cylindrical fillets, r s, t u, &c., of this rod,
are projected obliquely, as similar ellipses, of which portions only
are apparent. For the torus, or ring, which is comprised between
these two fillets, the oblique projection is a curve, which results
from the intersection of an elliptical cylinder, the generatrices of
which are horizontal, and tangent to the external surface of the
torus. If, therefore, we wish to determine this curve with great
precision, we must use the very same method adopted in determining
the shadow proper of the external surface of the torus (323). In
BOOK OF INDUSTRIAL DESIGN.
155
practice, however, when the drawing is on but a small scale, we
may content ourselves with determining the principal points in the
eurve, by projecting first the point, v, situated upon the middle of
tne diameter, y y', of the torus, and drawing through it the line,
ii' t)', equal to the diameter ; and, secondly, di'awlng the horizontal
lines touching the external contour of the torus in the points, z, z',
fig. 6, over to z% z^ upon the axial line, I' o', fig. 7 ; then di-aw an
ellipse with these two lines, d' d' and z^ z^ for the transverse and
conjugate axes respectively. The key, d, which passes through
the rod, a, being rectangular in section, is projected in fig. 7, by
a couple of rectangles, as indicated by the dotted projection lines.
422. Proceeding upon these principles, we can make oblique
projections, in a very simple manner, of various objects, more or
less complicated in form, when we have already the projections of
these objects upon auxiliary planes, making any known angle with
the actual plane of the drawing. Thus, figs. 10 and 13 are the
oblique projections of an oscillating steam-cylinder, the first repre-
senting the cylinder in external elevation, whilst the second is a
section made through the axis of the cylinder.
It is easy to see that these projections have been obtained in the
same manner as those already given in figs. 4 and 7 ; that is to say,
the external projection, fig. 10, is derived from the two right pro-
jections, figs. 8 and 9 — one made upon an auxiliary vertical plane,
parallel to the axis of the piston-rod, and perpendicular to the axial
lines of the trunnions, and the other upon a horizontal plane,
parallel to the cylinder ends, and, consequently, perpendicular to its
axis. All the diiferent parts of this cylinder are, in fig. 10, project-
ed by straight lines and ellipses, accordingly as they are rectilinear
cr circular in contour. It is the same with the section, fig. 13,
ind the horizontal projection, fig. 14, which are derived from the
two right projections, figs. 11 and 12, made upon auxiliary planes ;
one vertical, and passing through the axis of the cylinder, and
through the valve-casing, whilst the other is perpendicular to this
axis, and passes through the line, 1 — 2, fig. 11. The dotted work-
ing lines, indicated upon the various figures, show sufficiently
clearly the various constructions necessary to obtain these oblique
projections. We have, moreover, applied numbers to the diflerent
parts projected, and more particularly to the axes or centre lines,
which show at sight what parts correspond with each other upon
the diiferent projections.
423. These drawings represent the cylinder of a steam-engine.
different from that which we have already described. The pre-
sent one is called an oscillating steam-engine, because, instead of
the cylinder being vertical and immovable, it oscillates during the
motion of its piston, b, upon the two trunnions, e, carried in suita-
ble bearings in the engine-framing. This arrangement of oscillat-
ing cyhnder has the advantage of dispensmg ^\-ith the parallel
motion, and of attaching the rod, a, of the piston, directly to
the crank-pin, to which its motion is transmitted, without the
intervention of any connecting-rod. In the head, h, of the rod,
there is, consequently, formed a bearing, which embraces the
crank-pin.
The bottom of the cylinder is cast in the same piece with it, but
it has a small central opening, for the passage of the spindle of the
boring tool, by means of which the interior of the cylinder is turned
smooth and true. This opening is closed by a cast-iron cap, f,
bolted to the bottom of the cylinder. Against a planed face, upon
one side of the cylinder, is fitted the valve-casing, g, which receives
the steam direct from the boiler, and has within it the valve, h,
which has an alternate rectilinear movement, at the same time
oscillating along with the cylinder. During this movement, the
valve alternately uncovers the ports, a, b, fig. 11, which conduct
the steam to the top and bottom of the cylinder. A blade
spring, I, attached to the inside of the valve-casing, at the back of
the valve, constantly keeps the latter well up against the planed
valve face.
The steam coming from the boiler introduces itself into the cas-
ing through the passage, c, fig. 12, which communicates with ono
of the trunnions, e, and the escape of the steam, when it has acted
upon the piston, is effected through the e.xit channel, d, which com-
municates with the other trunnion.
The piston, b, is composed of a cast-iron body, on the outer sur-
face of which is cut out a groove, to receive the hempen packing,
i, partly covered by an elastic metal ring, h, coinciding exactly with
the inside of the cylinder.
Oscillating cylinder-engines have always been admired for their
simplicity and beautiful action ; but it is only of late years, and
now that such superior workmanship is attainable, that such en-
gines have been constructed of considerable size. The aptness of
this arrangement for engines of the largest size has lately been
demonstrated by Penn, in the case of the Great Britain, and other
large vessels.
CHAPTER XII.
PARALLEL PERSPECTIVE
PKINCIPLES AND AITLICATIONS.
PLATE XI,1T.
424. Wo give the name of parallel perspective to (ho represen-
tation of objects by oblique projections, which diirer from the
preceding, in so far that the visual rays, which we have hilluilo
supposed to bo always iiorpeiidicular to the geometrical i)lMncs,
form, oil the contrary, a ccirlain angle with tluwc pl;iims, r<'iii;iin-
ing, however, coiistnnlly parallel to owh other ; from which it
follows, that all the strMiirht lines, which •.no panillol in the object,
niainl.'iiii their p;ii;illclisin in tin- iMcIure, according to this syslciu
of |u'rsi)octivc. .Mthougii, in genonil, it is imin:itcri:il wiiat ln»
ani'lo of inclination is, it is lU'Vcrllieh'ss prol'crable, in ri'jfuiai
156
THE PRACTICAL DRAUGHTSMAN'S
drawing, to adopt some particular angle as a matter of convention,
which will have the advantage of giving the entire dimensions of
the object in a single projection or view.
l^et A B and a' b', figs. 1 and 2, be the two projections of a
straight line, to which we wish to make the visual rays all parallel ;
the vertical projection, a b, of this straight line, forms an angle,
CAT, with the ground line, l t, which angle is, we shall suppose,
equal to 30°, and its horizontal projection, a' b', is such, that the
distance of the point, a', from the point, a, where it touches the
horizontal plane, is equal to twice the length, a b, of its vertical
projection, the pouit, b, being that at which it touches the vertical
plane.
We shall proceed to show, by means of the various figures in
Plate XT.TT., that, in taking the above straight lines as directrices
for the visual rays, a single projection will be quite sufficient to
express all the dimensions of any object. Instead of making the
directrices of the different objects, represented in this plate, to coin-
cide with the actual projection of the straight lines which we have
just indicated, we have, by preference, chosen the mere setting of
these same lines round at an angle of 30°. Thus, the lines,
c' d", k' i', fig. 3, are the straight lines perpendicular to the planes
of projection, set round to the angle in question ; whUst, on the
contrary, the straight line, y z, represents the projection of these
lines properly parallel to the horizontal projection, a' b', in fig. 2.
425. The finished view, fig. z^, is the representation in parallel,
or, as it is sometimes called, ya/se perspective, of a prism, e, with a
square base, resting upon a plinth, f, also prism-shaped and square.
In the first place, we suppose this prism to be represented in the
horizontal projection, fig. 3, by two concentric squares, a' d' ef and
h' i' k I; and in the vertical projection, fig. 4, by the rectangles,
abed and g li ij. These projections are made upon the suppo-
sition, hitherto acted upon, that the visual rays are perpendicular to
the geometrical planes.
If now, on the contrary, the visual rays make with the planes of
projection an angle equal to that of the given straight line, figs. 1
and 2, each of the faces parallel to the vertical plane continues to
be parallel to this plane, and is represented by a figure equal to
itself, whilst all the faces perpendicular to the two planes are in
projecting rendered oblique, in such a manner, that the lines hori-
zontally projected become parallel to a' b', fig. 2 ; and those verti-
cally projected, to a b, fig. 1. Consequently, if through the
points of the projecting angles, a, h, i,j, fig. 4, are drawn straight
lines, parallel to a b, they will express the directions of all the
edges perpendicular to the vertical plane. Since then, as we
have already stated, the length of the projection, a b, fig. 1, is
equal to one-half the perpendicular, a a', fig. 2, if from the points,
a, /i, i, j, and on each side of them, we measure oflf, upon the
oblique lines just drawn, the distances, a a' and a/", li h^ and h P,
i i?, and i Ir, &c., respectively equal to half the lengths, a' m and
h' n, &c., we shall, in fig. 4, have the perspective representation of
the various straight lines perpendicular to the vertical plane ; and
as all the other edges are parallel to this same plane, such lines as
are actually vertical are represented as vertical, whilst all lines
parallel to the base line remain horizontal. Thus, the edges, a b,
h g, d c, ij, being vertical, are in the parallel perspective repre-
sented bv the verticals, a' b", h^ g^. i' ;", d^ a' ; and likewise the
edges, a d, b c,h. i, &.C., which are parallel to both planes of pro
jection, are rendered by the straight lines, a" d?, b' c', K' -P, &c.,
parallel to the base line.
It \\i\\ be easOy seen, that by adopting the angle we have indi
cated for the direction of the visual, or more correctly termed,
representative ray, that the one single view in parallel perspective
is suflicient to make known all the dimensions of the object : for,
on the one hand, we have the exact widths and heights of those
faces which are parallel to the plane of the projection, as if the
perspective view did not differ from an ordinary geometrical pro-
jection, in which the representative rays are supposed to be per-
pendicular to the plane ; and, on the other hand, the oblique lines
representing all the edges actually perpendicular to the vertical
plane, and which are exactly equal to half the actual lengths of
the latter.
We may here observe, that the base of the prism being square,
the sides, li I and i' k, are equal to the sides, h' i' or I k. Conse-
quently, in order to construct the perspective or oblique projection,
fig. 4, the plan, fig. 3, is not needed, since it would have answered
the purpose equally well to have made the lines, d'^ e' or i^ /f", equal
to the half of a d or h i.
426. The shaded view, fig. [B, represents a frustum of a regular
pyramid, g, resting upon an octagonal base, h, the horizontal pro-
jection of which is indicated in full, sharp lines, in fig. 5, and tin
vertical projection in dotted lines, in fig. 6.
According to the principle thus laid down, the perspective view
is obtained, in the first place, by dra\nng all the lines which are
perpendicular and parallel to the base line, fig. 5, and passing
through the opposite angles of each of the octagons, representing
the upper and lower bases of the pyramidal frustum, and of the
plinth. Of these lines, all, a' d'.f e', h' i', &c., which are parallel
to the base line, as well as the sider, p' q[, t' u, x! x', which are
likewise parallel to the former, remain horizontal in the perspective
^iew, fig. 6, whilst, on the other hand, all the straight lines, such as
y r', i' y', v' z', as well as the sides, a' f, h' I', i' k!, which are per-
pendicular to the base line, become inclined at an angle of 30° from
this line, as in fig. 6, or, in other words, parallel to the straight
line, A B, fig. 1.
If now, through the points, a, g, p, q, &c., and the points, li, o, i,
of the two bases of the pyramidal frustum, we draw parallels to the
straight line in question, and then mark off from each of those
points, aud on each side of them, the distances, a a', g g'jp p\ h ^
&c., respectively, equal to the semi-lengths of the corresponding
sti-aight lines, mf, g' n, &.C., of fig. 5, we shall have the parallel-
perspective representation of all these lines ; and consequently, by
joining the extreme points of each of them, we shall also have all
the lines representing the contours of the two bases ; and further,
by joining the angles of these bases, we define the lateral faces,
and complete the view, fig. 6.
427. The pai-allel-perspective representation of a cylindrical
object, of which the axis is perpendicular to the vertical plane, as
in the finished example, fig. ©, may be determined without the
assistance of the horizontal projection : that is, when the length of
the cylinder is known, as well as that of any other pai-t which may
be perpendicular to the vertical plane.
i Let ab c dgfe, fig. 7, be the vertical projection of this object,
BOOK OF INDUSTRIAL DESIGN.
157
the perspective of its base, abed, will be parallel to a b. The
circles which have their centres at o, being parallel to the vertical
plane, are represented in perspective by two circles equal to them-
selves ; and their position is obtained by drawing through the
point, 0, the straight line, o' o% parallel to a b, fig. I, and marking
off a distance, lying equally on both sides of the point, o, equal to
half the length of the cylinder, measured in the direction of the
axis, perpendicular to the vertical plane. Then with the points,
o\ 0°, as centres, describe the circles with the equal radii, o'/' and
o' i', straight lines,/'/'' and t' i", drawn tangential to the circles,
and parallel to the axis, o' o^ express in perspective the genera-
trices of the two cylinders forming the contour of the object.
The cylindrical pieces which join the cylinder to the base are
determined in the same manner by means of the line, n ri\ drawn
through the centre, n, of the circle, d g, parallel to o' o", and by
the distances, n w', n n', together equal to half the actual length of
these cylindrical surfaces. The base is drawn as in the preceding
example.
428. The example, fig. E), represents a cone resting upon a
cylindrical base, both cone and base having the same axis perpen-
dicular to the horizontal plane. This cone and cylinder are pro-
jected on the plan, fig. 8, in sharp lines, and in the elevation, fig. 9,
in dotted lines.
The circles, fig. 8, representing the bases of the cone and cylin-
der, are to be divided into a certain number of equal parts ; and
through the points of division, 1, 2, 3, &c., perpendiculars are drawn
to the ground line, and are prolonged as far as the horizontal line,
a' o', which is the vertical projection of the two bases. Through
the points, a', b', c', o', are drawn straight lines parallel to a b, fig.
1 ; and on each of these are set off the distances, a' 2', b' 3', c' 4',
o' 5', &c., respectively equal to half the lengths of the perpendicu-
lars, 2 ffl, 3 6, 4 c, 5 0. &c., which oi)eration gives the points, 2', 3',
4', 6', &e., through which an ellipse must be traced, to represent
the perspective of the base of the cylinder. In the same manner
we obtain the points through whicli passes the ellipse, representing
the perspective of the base of thS cone.
The heights of the cone and its base remain precisely what they
really are, in consequence of their common axis being parallel to
the vertical plane ; but this is not the case with their bases, which,
being horizontal, are projected obliquely, in the form of the ellipses
wo have just drawn. The apex of the cone, at the upper extre-
mity of the vertical axis, does not change, and for the generatrices,
or sides of the cone, it is simply necessary to draw through the
apex the straight lines, o^ m and ff' n, tangents to the ellipse repre-
senting the base of the cone, whilst the generatrices of the cylinder
are tangents to the two ellipses representing its upper and lower
bases.
429. The example, fig. (E, is the parallel-perspective reprcsenta^
tion of a metal sphere or knob, attached to a polygonal base by a
circular gorge, forming altogether an ornamental iicad for a screw
Figs. 10 and 11 are the horizontal and vertical prujictions of this
piece.
Wo must, in the first place, remark that tiio s])li('ro, tlio radius
of which is o a, may be determined in its (lerspoctivo represeiitji-
tion in several ways. First, by imagining the hori/ontal sections,
rt h, c d, ef, which give in plan the circles, with the ladii, «' o,
c' o', and e' o', and the perspectives of each of these circles may
be obtained by operations similar to the preceding, v/hich will give
a series of ellipses, to be circumscribed by another ellipse, tangen-
tial to them all. Second, by drawing the planes, g h, ij, parallel
to the vertical plane, and which are projected in perspective as
circles, with the radii, I" g, i m, the centres being upon the oblique
line, n n', parallel to the line, a b, fig. 1, and passing through the
centre, o, of the sphere. The external curve, drawn tangential to
all these circles, will be elliptical, as in the preceding method, and
represent the perspective of the sphere. Thirdly, by at first
drawing through the centre, o, an oblique line, n n' ; then a per-
pendicular, e e", passing through the same point. Then set out
from the centre, o, and on each side, the distances, o e, o e", equal
to the radius, o a, of the sphere, which gives the conjugate axis of
the ellipse. To obtain the transverse axis, it will be necessary to
draw tangents to the great circle of the sphere, parallel to the
oblique ray of projection, a b, fig. 1, as brought into the vertical
plane. This line is brought into the vertical plane, as at a' b. in
the following manner : — At the point, a, a perpendicular to a b is
erected, and the distance, a" a, is made equal to a a', when a" b is
joined.
These tangents touch the sphere in the points,/, /', which may
be obtained directly by drawing through the centre, o, the line,//,
perpendicular to a" b. These tangents, further, meet the line, n n',
in the points, n, n' ; and the distance between these points is, con-
sequently, the transverse axis of the ellipse, which represents the
perspective of the sphere, and which may be drawn according to
any of the known methods.
This last method is evidently the shortest and simplest of the
three for obtaining the perspective of the sphere, but it is confined
in its application ; for any other surface of revolution, as, for exam-
ple, the gorge, which unites the sphere to the hexagonal base,
cannot be defined in this way. In cases where the axis of the
surface of revolution is vertical, as in this example, it will be neces-
sary to adopt the first general method, which consists in taking
horizontal sections. When, on the other hand, the axis of the
surface of revolution is horizontal, we must employ the second
process, which consists in taking sections parallel to the vertical
plane, or perpendicular to the axis. The perspective of the thread
of the screw, of which the sphere is the ornamental head, is do-
terminable in a manner analogous to that of a circle. It is sufli-
cient, in fact, first to draw the two geometrical projections, figs. 8
and 12, of one or two convolutions of the thread, and to find the
perspectives of the very points wiiich iiavo served for tiic construc-
tion of the helices. Thu.s, for example, wo put llio circle,/;)!/,
fig. 8, into perspective, as at I* i>* q*, fig. 13, rotaining tor this jmr-
pose the same points, /, /;, </, &c., which wore oinployod in dolino-
ating the screw, fig. 12. 'J'hrough llicso points, l\p\q*, &c., draw
viutical lines; and upon (Ikiii .set oil" the distances, V P, /' /', yi' ;<',
&c. Then through the points, l\ f, &c., draw liio curve, whicli
will he the p.i siuctivc of Ihc outi'r holix of Iho scrow-tliroad.
liy giiiiiu' llniHinh Ihi' sniuc o|irra(ion for tlio iiinor circle, r s I, li.,'.
10, wo shall ohlain ihc sinilhiily |>c|-Niicclivo oullilio of llu> inilor
helix.
An I'Miiiiiiialion nt lig. 13 will furlhcr show iIimI iho hoiglil.>* on
(lie vertical linos are precisely Uio siuiio for Ik li lulicos, for th«<>
l68
THE PRACTICAL DRAUGHTSMAN'S
are taken upon radii common to the two circles, l" p" and rst, con-
sidered as bases.
We have deemed it unnecessary to enter further into the deve-
lopment of this species of perspective, of which we have already
given a general application in the boring-machine, represented at
fig. 1, Plate XXXV., an example in which are collected almost all
the various forms which present themselves for delineation amongst
mechanical elements and machinery.
In that example, as well as in the figures in Plate XLIL, which
we have just been studying, we have supposed the representative
or visual rays to be in all cases parallel to each other, and to be
inclined at an angle of 30° with the ground line in the vertical pro-
jection, in such a manner that a single view serves to express all
that two, or even three, geometrical projections can do, showing
not only the external contours of the objects, but also whatevei-
may be upon their surface.
It will be easy to comprehend the utility of this system of giv-
ing, at a single view, a general and precise idea of the actual relieve
appearance of any object. It is a manner of representation often
more intelligible to the generality of people than a series of geo-
metrical projections, and in many cases it will greatly simplify the
process of sketching buildings or machinery.
CHAPTER XIII.
TRUE PERSPECTIVE
PRINCIPLES AND APPLICATIONS. — ELEMENTARY PRINCIPLES.
PLATE XLIII.
430. Perspective, properly so called — ^but here defined as true or
exact perspective — differs from parallel or false perspective, in its
being founded upon the actual manner in which vision takes place ;
that is, that instead of being parallel to each other, the visual rays
converge to a point. An object is said to be drawn in perspective
when, on viewing the drawing from a particular point, it presents
the same appearance to the organs of vision as the object repre-
sented itself does when similarly viewed.
The visual rays, or impressions, travel from the object in
straight lines, converging to a point at the eye, and forming a cone
of rays. Let us suppose this cone to be intercepted by a trans-
parent plane, or diaphragm, of any form — then, noting the points
where the rays from the various parts of the object pierce this
diaphragm, let us paint upon it the outline, and complete the
picture with colours, which to the eye shall have the exact appear-
ance of those of the object itself, modified, as they may be, by
distance, position, form, and by their being in the light or shade.
After doing this, we may remove the object, but the picture upon
the diaphragm will make the same impression upon the eye that
the object would itself. It is to the representation of objects in
this exact and natural manner, that the art of drawing in perspec-
tive is devoted.
The fixed point, to which the rays converge, is called the point
of sight ; and in diagrams explanatory of perspective, it is shown,
together vsath the converging rays, as projected into the plane of
the picture. To determine the perspective delineation of an
object mathematically, we must have given us the horizontal and
vortical projections of the object ; also those of the point of sight,
and the position of the plane of the picture ; and the general
problem of perspective then reduces itself to conceiving the visual
rays as passing from the various points of the object to the point
of sight, ana to find the intersection of these rays with the plane
of the picture.
FIRST PROBLEM.
THE PERSPECTIVE OF A HOLLOW PRISM.
431. Let A and a', figs. 1 and 2, be the horizontal and vertical
projections of the prism which we wish to delineate in perspective,
the point of sight being projected in v and v', and the plane of the
picture, in t and t', being supposed to be perpendicular to the
planes of projection, and vertical, as is generally the case.
Through the point, v, in the horizontal projection, draw visual
rays from each of the points, a, b, c, d, appertaining to the external
contour of the prism. The intersection of these points with the
plane of the picture determines the points, b^, a", n\ cP, which give,
upon the horizontal projection, t o, of the latter, the perspective
of the points, a, b, c, d.
In like manner, through the point, v', in the vertical projection,
draw the visual rays, a! v', b' v',/v', e v', intersecting the plane of
the picture in the points, a", b",f", and e" ; these last being, conse-
quently, the vertical projections of the perspective of the points,
a', b', e,f.
As, because of the position given to the plane of the picture,
the perspective of the object is not visible, since all the points are
situated in the vertical line, t t', the plane being represented on
edge, we must imagine this plane as turned over upon the vertical
plane of the drawing; whilst we must suppose the line, t o, the
horizontal projection of the picture-plane, as turned about the
point, 0, as a centre, through a right angle, bringing it to coincide
with the ground line, l m. When this is done, the points, a", i^
c^ d'', will describe arcs of circles, and, finally, coincide with the
points, a\ b*, c\ d\ on the ground line. Next, upon these last
erect perpendiculars, to meet the horizontals, drawn through the
points, a", b",f", e" ; the points of intersection of these cross lines
will be the perspectives, a*, b*, c*, d*, of the corners, a, b, c, d, ot
BOOK OF INDUSTRIAL DESIGN.
159
(he top of the prism. We have, likewise, the points, e^/^ g, h, for
the perspectives of the corners of the bottom of the prism, which
is parallel to the top; consequently, by joining all these points
together in couples, as indicated in fig. A, we obtain the entire
perspective of the external outline of the prism. As the prism is
hollow, we shall see in the perspective view the outline, i' m! v! o',
corresponding to the edges of the part hollowed out.
The point of sight, of which v and v' are the geometrical projec-
tions, is projected upon the picture-plane in the points, d, d° ; and
when the picture-plane is turned over, the point will be found at d",
which is the position of the point of sight upon the perspective
drawing.
Tt must be observed, that in this example the lines, a* Z»S a* d^,
and 6' c*, which express the perspective of the corresponding lines,
ah,a d, and h c, are the intersections with the plane of the picture,
of planes passing through these lines, and through the point of sight.
Now, since the intersection of two planes is always a straight line,
the following conclusions may be drawn ; that,
432. First, The perspective of a straight line upon a plane is a
straight line.
It may also be remarked, that the verticals, such as b* e\c*k,d*g,
are the perspectives of the vertical edges, projected in the points,
b, c, d; whence we deduce that,
433. Secondly, The perspectives of vertical lines are verticals,
when the plane of the picture is itself vertical.
It will further be seen, that the horizontals, I* c*, d* a*, ^ h,f^ g,
of the perspective view, correspond to the straight lines projected
horizontally in 6 c and d c, which are parallel to the picture ;
whence it may be gathered, that,
434. Thirdly, The perspective of any straight horizontal line,
parallel to the picture, is itself horizontal.
Further, it follows from the two preceding principles, that all
lines parallel to the plane of the picture are represented, in per-
spective, by lines parallel to themselves.
Finally, the straight lines, a* b*, d* c*, ^ f, and h g, which all
converge to the same point, v", the projection of the point of sight
upon the plane of the picture, correspond to the edges, ab, dc, ef,
which are horizontal, but perpendicular to the plane of the picture;
whence it follows, that,
435. Fourthly, The perspectives of lines which are horizontal, but
perpendicular to the plane of the picture, are straight lines, which
converge to the point of sight, and are consequently foreshortened.
It will be seen, from figs. 1 and 2, that the whole width of the
perspective representation, fig. A, is comprised between the points,
i' and d^, which lie on the outermost visual rays, drawn in the
horizontal projection ; and that its height is limited by the two
points, a" and e", which correspond to the extreme visual rays in
the vertical projection. The angle formed by the extreme visual
rays is termed the optical angle. In the present example, this angle,
U' V d'', in the horizontal projection, dilfers from the angle, a" v' e",
in the vertical projection.
The positions of the object and point of sight being given, the
ditncnsions of the per.spoctivo rci)r(!scnt;iti()n vary acconling to
the position of the plane of the picture. It will thus bo seen, on
teforring to fig. 1, that if this plane bo removed from t x' to < t',
loarer to the object, the limits to the porspuctivo roprusentatlon
by the extreme visual rays will be enlarged ; whilst, on the con-
trary, if we remove the plane of the picture to the position, f i\
nearer to the point of sight, the limits will be sensibly narrowed.
Again, if, in place of moving the plane of the picture, the point
of sight is removed further off, or brought nearer to, the size of the
perspective outline will thereby be augmented or diminished. It
may therefore be concluded, that,
436. Fifthly, The dimensions in the perspective representation do
not wholly depend, either on the actual size of the object, or on the
distance from which it is observed, but also on the relative distances
of the point of sight and of the object from the plane of the picture.
Thus the sides, d a and c b, fig. 1, are actually equal, but the
former is furtiier from the plane of the picture than the latter; so
that whilst this is represented by the space, c* b*, fig. A, that is
limited to the much smaller space, d'^ a*, in the perspective view.
SECOND PROBLEM.
the perspective of a cxlinder.
Figures 3 and 4.
437. To obtain the perspective outline of a vertical cylindei, sucn
as the one projected horizontally in b, fig. 4, and vertically in li ,
fig. 3, we proceed, as in the preceding example, to draw througn
the point of sight, v' v, a series of visual rays, extending to the
various points, a, b, c, d, e, taken on the upper end of the cylinder,
by preference at equal distances apart. These lines intersect the
.plane, t t', of the picture, in the points, d'', c', a', g', &<;., in the
horizontal projection, and in the points, a", c", d". Sic, in the verti-
cal projection.
By bringing the plane, t t', of the picture, into that of the
diagram before us, or what is equivalent to it, by finding the points,
g\ a\ &c., by means of arcs, drawn with tlie centre, o, on wliich
the plane is supposed to turn, and drawing the horizontal lines
through the corresponding points in the vertical projection of the
picture-plane, we obtain the points, c*, d*, a*, g*, Slc., which are
points in an elliptic curve representing the top of the cylinder in
perspective, which is visible, in consequence of the point of sight
being above it.
The same points, a, b, c, d, of the horizontal projection, give, in
combination with their vertical projections, g^',/', c", &c., the per-
spectives, d°, e°,/', g•^ of the bottom of tlio cylinder, of which,
obviously, only a part is visible.
The two vertical generatrices, d* d'',g* g'', being drawn tangential
to the upper and lower ellipses, complete tho perspective outlino
of the cylinder, fig. S.
As this cylinder is hollow, nn operation similar to tho preceding
will 1)0 called for in delineating the upper visible edge of Uie hol-
lowod-out i)()rti()n.
It in\ist 1)0 observed that, in taking an oven nmnbor of divisions,
at eiiu:il ili.stain< - ai)iirt, upon tiie horizontal projection of tho
cylinder, and setting Ilieni off from tho dinniofor, c g, immllel to
tho plane of the picture, wo have always ii couplo of points situated
upon tho same perpendicular to the jilane, and of which the [ht-
speetives are, conseiiuently, situated on the siune stniight lini-
drawn through tho projection, i', of the point of sijjht Thui
160
THE PRACTICAL DRAUGHTSMAN'S
occurs with the points, b, d, the perspectives, b*, d*, of which, are
situated upon the line, v' d\ so that we have a means of verifying
the preceding construction.
THIRD PROBLEM.
the perspective of a regular solid, when the point of
sight is situated in a plane passing through its axis,
and perpendicular to the plane of the picture.
Figures 5 and 6.
438. Let V and v' be the projections of the point of sight, situ-
ated in the vertical plane, v o, passing through the axis, o o', of
the solid, c c', and perpendicular to the plane, t t', of the picture.
It will be seen at once, that in the perspective view, fig. ©, this
point must be projected on the vertical line, v" v\ representing the
axis of the object, and in relation to which all the lateral edges of
the object are synametrical. Such are the sides, a b and c d, which
are perpendicular to the plane of the picture, and which are repre-
sented in perspective by the Imes, a* b* and d^ c*, both directed to
the point of sight, v". It is the same with the edges, /^, h i, of
which the perspectives, /* g*, Ti* i*, likewise converge to the point
of sight, i;'.
As for the vertical edges, they retain their vertical position in
the perspective, and the horizontal lines, a d,l m, nk, c b, parallel
to the plane of the picture, are rendered in the perspective view by
parallels, such as a* d*, I* m*, n* k*, c* b*.
It may be gathered from the solution of this problem, that when-
ever the point of sight is in a plane, passing through the axis of a
regular solid, and perpendicular to the plane of the picture, the
perspective representation will be symmetrical with reference to the
centre line, and that it is, therefore, quite sufficient to go through
the constructive operations for one side only of the figure.
FOURTH PROBLEM.
the perspective of a bearing-brass, placed wath its axis
vertical.
Figures 7 and 8.
439. The pomt of sight being situated, as in the preceding
example, in a vertical plane, passing through the axis of the object,
and perpendicular to the picture-plane, the perspective vnil like-
wise be symmetrical in reference to the centre line, v" v" ; and it is
unnecessary to notice this peculiarity further. The inside of the
brass being cylindrical, and being terminated by horizontal semi-
circular bases, the perspective of these bases will be rendered by
a couple of regular semi-ellipses, of which it will be sufficient to
determine the axes. The transverse axis, a* c*, is equal to the
perspective of the straight line, a c, which is horizontal and parallel
to the picture-plane ; and the semi-conjugate axis, b* d', is equal to
Ine perspective of the line, b d or b' a', which is also horizontal,
but is perpendicular to the plane of the picture, and is conse-
quentlv foreshortened, whilst it coincides with a line passing
througn the point of sight, v". It will be remarked, that the
transverse axis of the ellipse, corresponding to the upper end of
the semi-cylinder, is equal to a* c*, but that the conjugate axis
is foreshortened to a greater extent than that, b* d*, of the lower
one, in consequence of being at a less distance below the point of
sight. The effect of the perspective, in this case, is well rendered
in fig. E).
FIFTH PROBLEM.
THE PERSPECTIVE OF A STOPCOCK, WITH A SPHERICAL BOSS.
FiGDKES 9 AND 10.
440. In can-ying out the general principle, it will be conceived
that, in obtaining the perspective representation of a sphere, we
should draw through the point of sight a series of rays, tangential
to the outer surface ; but in order to ascertain the points of con-
tact of these tangents, it will be necessary to imagine a series of
planes as passing through the sphere, producing circular sections,
and then to find the perspective of each of these circles. These
being found, a curve, drawn to circumscribe them, will be the
perspective of the sphere.
In the example, figs. 9 and 10, the point of sight being still
chosen as before — that is, as situated in a plane passing through
the centre of the sphere, and perpendicular to the picture-plane —
the perspective of the sphere wUl, on this account, simply be an
ellipse, having for conjugate axis the base, a? ¥, of the optical
angle, a' v b, in the horizontal projection ; and for transverse axis,
the base, c" d\ of the angle, c" v' d", in the vertical projection ;
because the right cone, formed by the series of visual rays tan-
gential to, and enveloping the sphere, is obliquely intersected by
the plane of the picture. If, however, the point of sight were
situated upon a horizontal line, passing through the centre, o, of
the sphere, the perspective of the latter would evidently bo a
circle.
The spherical part of the stopcock is traversed by a horizontal
opening, for the reception of the key, and the edge, e /, of this
opening, being situated in a plane parallel to the picture-plane,
■will, in the perspective, fig. [g, be rendered by a circle, of which the
diameter is e'f\
As to the cylindrical flanges on either side of the stopcock, for
forming its junction with a line of piping, the semicircle, a g b,is
represented by a portion of an ellipse, having the horizontal lioe,
a* b*, corresponding to a b; and the vertical, g' &, being the per-
spective of ^ o" . The circle, alib g, the horizontal projection of
the surface of the uppermost flange, is represented in the perspec-
tive view by a perfect ellipse, as is also the upper visible edge oi
the inner tubular portion. But this is the same as the case in fig.
3, and the ellipses may be formed in a similar manner.
The perspective representation of a circle, however situated,
otherwise than parallel with regard to the plane of the picture, is
always a perfect ellipse ; but the transverse axis of the eOipse does
not coincide with, or is not the representation of, any diameter of
the circle ; for it is evident, that the more distant half of the circle
must be more foreshortened in the perspective, and must occupy
a less space than the anterior half, whilst the ellipse is equally
divided by its transverse axis.
BOOK OF INDUSTRIAL DESIGN.
IP.l
When the point of sight is in a line perpendicular to the circle,
the rays from the latter will form a right cone ; and if the plane of
the picture is not parallel to the circle, the section determined by
it will be an ellipse, as is well known. Again, if the circle is not
perpendicular to the central visual ray, the cone of rays will be
nlliptical, and the sections of such cone will be ellipses, of various
proportions, that parallel to the circle, however, being a circle.
The transverse axis of the perspective ellipse is the perspective
of that chord in the original circle, which is subtended by the arc,
between the points of contact of the extreme visual rays, as pro-
jected iu the plane of the circle.
SIXTH PROBLEM.
the perspective of an object placed in ant position with
kegard to the plane of the picture.
Figures 11 axd 12.
441. In each of the preceding problems, we have supposed one
or other of the surfaces of the objects to be parallel or perpendicular
to the plane of the picture ; but it may happen that all the sides of
the object may form some angle with this plane. It is this case
which we propose examining in figs. 11 and 12.
Let a b c d he the horizontal projection of a square, of which
the sides are inclined to the plane, t t', of the picture, and of
which it is proposed to determine the perspective. The point of
sight being projected in v and v', if we employ the method
adopted in figs. 1 and 2, we shall find the points, a^ b^, c', d%
to be the horizontal projections, and a", b", c", d", the vertical
projections of the corners of the square ; and when we have
brought the plane of the picture, T t', into the plane of the
present diagram, as before, we shall find the actual positions of
these points to be at a*, b*, c*, d*. If we join these points, we
shall have a quadrilateral figure, of which the two opposite sides,
a* b*, c* d\ converge to the same point, /, whilst the other two
sides converge to the point, /'. These two points are termed
vanishing points. They are determined geometrically, by drawing
fhrough V, the horizontal projection of the point of sight, the
itraight lines, v t and v t', parallel to the sides, a b and b c, of the
given square, and prolonging these lines until they cut the line,
T t', representing the plane of the picture. Having drawn through
v' the horizontal, v' v', termed the horizontal lino or vanisliins
-plane, set off the distance, v t, from v" to /, and the distance,
1) t', from v" to /', and / and /' will be the required vanishing
points.
It follows from the preceding, that
When the straight lines which are inclined to the plane of the
Yiclure are parallel to each other, their perspectives will converge in
one point, situated on the horizontal line, and termed the vanishing
voint.
When several faces or sides, situated in difterent planes, are
parallel to each other, their perspectives all converge to the same
vanishing point, which allows of a great simplification of tho
operations.
Tims, fhe edges of the horizontal faces, h' i' and h" i", of tho
<|iia(lrangular prism, being respectively parallel to the .sides of tho
square, abed, are represented in perspective by the straight lines,
i'e\ i* e*, converging to the first vanishing point,/, and the straight
lines, i" g', i* g*, converging to the second point,/'.
The cone, f f', which is traversed laterally by the prism, has its
apex projected horizontally in the point, s, fig. 12, and vertically in
the point, s', which, with its axis, appertains to fig. 11. The per-
spective of the point, s s', on the plane of the picture, is found, in
the usual way, to be at s in the horizontal projection, and at s' in
the vertical projection ; and when the plane of the picture is brought
into the plane of the diagram, these points are represented by the
points, s" and s", upon the same vertical, s' o', which is, consequently,
the perspective of the axis, o s', of the cone, and the point, s", is
the perspective of fhe apex of the cone.
If we draw the perspectives of the two bases, k I and m n, of the
frustum of a cone, according to the methods already given, it will
only remain to draw through the point, s', the two lines, s" m' and
s'' n, tangential to the ellipses, representing the bases, which will
complete the perspective of the entire object, as in fig. [P".
APPLICATIONS.
FLOUR- MILL DRIVEN BY BELTS.
PLATES XLIV AND XLV.
442. The elementary principles of perspective which we have
laid down, will admit of application to the most complicated sub-
jects, and, among other things, to complete views of mechanical
and architectural constructions. In Plate XLV. we have griven
an example, which will enable the student to form a general idea
of this branch of drawing. This Plate is the perspective repre-
sentation of the machinery of a flour-mill driven by belts, and a^
fitted up by M. Darblay, at Corbeil. Before proceeding, however,
to discuss this as a study of perspective, we propose to desi'ribe
the various details of the mechanism composing the mill, and
which we have represented, in geometrical projection, in Plato
XLIV.
The construction of fiour-mills has latterly undergone very i".
portimt improvements, as well in reference to the principal drivinj^
machinery, as to the minor movements, and the cleaning Jind
dressing apparatus. As such machinery belongs to a most impor-
timt class, we have selected a mill, as an illustrative example of tho
subject before us, giving all the recently improved modifications
now at work, both in this country and on fhe Continent.
443. Before the introduction of what is known as the American
system, very large uncovered millstones, of upwards of six feet, or
two metres, in diameter, were employed ; and these gave what woro
then considered very good and economical results. Tlieso mills
were worked by water-wheels or wind-wheels ; but as iniprovc-
meuts gradually crept in, not only was the entire internal mechanism
changed, but also the motor, and tho description of stones. Tho
AnurL:')) flour-mills, commonly known on fhe Continent n.«
Enirlish mills, iliflered Irom the older mills in the employment of
smaller stones, Wit!. furrowed surfaces, and in their being driven
at a much greater speed, requiring, ill consequenoo. more wliet>l.
o-cjM- to bring up the speed. A mill on tho old sy.stem, with lartf*
162
THE PRACTICAL DRAUGHTSMAN'S
stones of six feet in diameter, ordinarily goes at the rate of 55 to
60 revolutions per minute, being moved, we shall suppose, by a
water-wheel, making 10 or 12 turns in the same time. Such a
nilH will only require a large toothed-wheel and a lantern-wheel ;
oi' better, a large bevil-wheel and a bevil pinion, in the ratio of 5 or
6 to 1.
But a modern mill, in which the stones are generally 4 feet or
rS m. in diameter, should make 115 or 120 revolutions per minute ;
whilst it may be impelled by an overshot water-wheel, making only
3 or 4 revolutions per minute ; so that it is necessary to employ
multiplying gearing betvi'een the power and the work. When this
multiplication is obtained by gearing, two or three pairs of wheels
are generally employed. The essential features of this gearing,
consisting of a large horizontal spur-wheel, driving a spur-pinion
on the mill-stone spindle, have recently been superseded, in many
instances, by belt and pulley gearing. This arrangement has the
advantage of rendering the motions more easy, and of allowing of
the stoppage of a single pair of stones, without stopping the prime
mover and the whole mill, which is a very essential point, more
particularly in a large and important mill, where many pairs of
stones are at work.
The drawing, Plate XLIV., represents a mill of this descrip-
tion, driven by belts, and erected by M. Darblay, at Corbeil. It
eomprises 10 pairs of stones, placed in two parallel rows. The
establishment contains several sets of stones, exactly like these.
Each set of mills is driven by a hydraulic turbine, on Fourneyron's
system.
Fig. 1 represents the plan of a portion of the principal gearing,
and one of the rows of stones. Fig. 2 is an elevation, prolonged
as far as the vertical shaft of the turbine. Fig. 3 is a transverse
vertical section, taken at right angles to the horizontal driving-
shaft.
At A, in fig. 2, is represented the upper end of the vertical
wrought-iron shaft, upon which the turbine is fixed lower down.
This shaft is supported by a step-bearing at its lower extremity, and
in a brass collar bearing, a, at its upper end ; this bearing being in
two pieces, adjusted in the top of the hollow casting, b, resting upon
the foundation-plate, c, and also bolted by a bracket to the cast-
iron pedestal, d, of the bearing, which receives the end journal, 6,
of the main driving-shaft, e.
This shaft, e, has fixed upon it, in the fi)'st place, the bevil-pinion,
F, with sti-ong thick cast-iron teeth, driven by the horizontal bevil-
wheel, G, fitted with wooden teeth, and keyed upon the end of the
turbine-shaft, a. This shaft is connected by a coupling-box, c, to a
wrought-iron shaft, a', which passes up to the higher floors of the
building, where it serves to drive the various accessory apparatus
of the mill, such as the sack-hoists, pressing, washing, and dress-
ing machines, and endless-chain elevators. At each floor, the shaft
i= supported by a collar-bearing, like that shown in section at cl, in
fig. 2.
The horizontal shaft, e, which drives the two rows of stones, is
in several pieces, joined together by cast-u-on couplings, as at e,
and it is supported at diflTerent points of its length by the pedestal-
bearings,/ each with its oil-receiver at the top, and bolted down
to the base of the arched cast-iron standards, H. These standards
»i-e foimeu mto receptacles at their tops, to receive the brass
footstep-bearings and steel pivot-piece, to support the lower case
hardened extremities, g, of the vertical shafts, i. An adjusting-
screw, i, is introduced from below, to raise the bearing when
necessary ; and the upper journal of the shaft revolves in the in-
verted cup-bearing, j, which is bolted to the cross beams of the
ceiling.
Each of the vertical shafts, i, has keyed upon it a bevil-pinion,
K, the teeth of which are cast upon it, as well as two horizontal
pulleys, L, of the same diameter. The pinions gear with the
wooden-toothed bevil-wheels, k', keyed upon the horizontal driving-
shaft, E ; and the pulleys are put in communication by means of the
leather belts, h. with other similar pulleys, l', of the same diameter.
These last are each keyed upon the cast-iron shaft of a pair of
stones, M. A tension pulley, n, upon a short vertical spindle, sup-
ported by the two arms of a second vertical spindle, o, serves to
stretch the belt of each pair of stones to the requisite degree of
tightness. For this purpose, a lever, k, is fitted to the vertical
spindle, o, and to its free extremity is attached a cord, passing over
a couple of guide-pulleys, I, and sustaining a small weight, m.
Thus, in the position given to each of the levers, h, in the drawing,
the weights are supposed to be acting ; and the belts are, conse-
quently, in a stretched state, and the motion of the pulleys, l, is
communicated to the pulleys, l'. But if the weight be lifted up,
so as not to act, the levers, k, will be set free, and also the tension-
pulleys, N ; and the belts will be slack, so that the motion will no
longer be communicated, and the pulleys, l', and consequently the
pairs of stones, will be stopped. The vertical spindles, o, are sup-
ported in bearings, carried by cast-iron brackets, f, bolted to the
under side of the cross beams. The tension-pulleys can thus
assume various positions, whilst their supporting-arms vibrate upon
the vertical spindles, o. In order that the belts may not fall when
they are slack, iron fingers, n, are placed at intervals, attached to
vertical rods, o, depending from the ceiling.
As the millstone shaft is generally made of cast-iron, its lower
end is fitted with a case-hardened step, which revolves upon a steel
pivot-piece, q, adjusted in the bottom of a brass footstep-bearing,
fig. 4, which is itself contained in a cylindrical cast-iron cup, r,
carried by the box, ra', formed in the casting, p', which surmounts
the solid masonry, o', upon which the entire framing of the mill is
supported. Screws are introduced through the sides of the foot^
step-bearing receptacle, by means of which the centre of the shaft
is adjusted ; whilst the shaft is adjusted vertically, and, consequently,
the distance between the stones, by means of the screwed spindle,
s, which has a small spur-wheel, (, keyed upon it, with which a
small pinion, u, is in gear, this last being actuated by the handle, r
upon its vertical spindle. By turning this handle to the right oi
the left, the small wheels are set in motion ; and as the spindle, s,
cannot otherwise turn, it is forced to rise or fall, and with it the
bearing and footstep of the millstone shaft. It is in this manner
that the pitch, or distance, between the two stones is adjusted with
all desirable precision, according to the kind of work required from
the stones.
The upper end of the millstone shaft is also case-hardened, and
is entered a certain distance into the boss of the centre-piece, w,
fig. 3, which is fired across the eye of the upper stone, or runner,
Q, and firmly imbedded into the stone at either side. On the
BOOK OF INDUSTRIAL DESIGN.
16S
top of the boss of the centre-piece, w, is a species of metal saucer,
.nto whicii dips the lower end of the pipe which conducts the grain
down from the funnels, R, generally made of copper. These fun-
nels, which receive the grain, communicate by the pipes, y, with a
single hopper above, and rest upon the wooden cross-pieces, s, tig.
2, held down on one side by a hinge, z, and on the other by a
vertical iron rod, z', by means of which they are raised or lowered
at pleasure, so as to set the bottom of their pipes at a greater or
less distance above the bottom of the saucer below. The object
of this arrangement is to allow more or less grain to enter between
the stones. The supports of the cross-pieces, s, are fixed upon a
wooden casing, t, which covers each pair of stones, a space being
left inside all round the stones, into which the produce of the
grinding falls, as it issues from between the stones. It is thence
conducted, by suitable channels, either to receiving-chests, or to the
elevators, by which it is carried to the upper part of the building,
to undergo the subsequent processes.
The lower immoveable stones, q', of the same diameter as the
runners above, are fitted with metal eyes, h', furnished with
brasses, which embrace the shafts of the runners, and assist in pre-
serving their perfectly vertical position. These stones are grooved,
as indicated in the plan of one of them, fig. 1 ; thai is to say, shal-
low channels are cut out of their working surfaces, so as to present
on one side a sharp edge, and act with the runner hke scissors,
cutting each grain as it comes upon them. The fine close-lined
dressing, which is given to the surface between these channels,
completes the fracture and crushing of the grain. These lower
atones rest upon the cast-iron plates, u, but with the intervention
of the three adjusting screws, a', which allow of the obtainment
of an exact level; whilst four lateral screws, a', fig. 1, entered
through the lateral cast-iron frame, serve to adjust with accuracy
the centre of the stone.
The base plate and side frames are not only bolted to the cross
beams of the building, but they are also supported at intervals by
cast-iron columns, v, placed between each pair of stones, and
resting upon the plates, o', and the solid masonry below them.
The ceiling is additionally supported by the solid wooden columns,
X, placed at the ends and between the two rows of stones. An
iron railing, y, is placed on each side of the driving-gear, to prevent
accidents which might arise from persons passing too near the
heavy wheels. Cavities are constructed in the masonry, for the
reception of the mechanism for adjusting the footstep-bearings of
the runner-shafts, already described. These openings are usually
covered by suitable doors.
THE REPRESENTATION OF THE MILL IN PER-
SPECTIVE.
PLATE XLV.
445. It was stated, in the preliminary instructions relating to
perspective drawing, that the perspective dimensions depend on tho
position, both of the; point of sight and of the object, from the piano
of till! picture, which is necessarily limited in si/e.
In tho perspective delineation of one or more objects, we shduld
consider, not only from what distance tho object shoultl lie virwiil,
but also at what height the eye, or the horizontal line, should be
placed. In the example selected, we have supposed the point of
sight to be placed at the height of a man's eye ; but it is e\ndeut
that this height of horizon is not invariable. It depends, more or
less, on what part of the object we wish to develop more particu-
larly in the perspective representation. Thus, for a machine of but
little height, the point of sight should be lower ; whilst it should,
in all cases, be at a sufficient height to enable the spectator to take
in the entire object, without changing his position.
In architectural subjects, the horizontal line should never be
taken at a less height than that of a man's eye ; whilst, in general,
a good effect may be anticipated, when the distance of the spectator
from the picture is equal to about one and a half times, or twice
the width of the paper, provided there is, at least, as great a distance
between the plane of the picture, and those parts of the object
which are nearest to it.* Taste and practice in drawing in perspec-
tive will be the best guides in the choice of the dispositions leadiag
to the happiest effects.
We have at 1 1\ figs. 1 and 2, Plate XLIV., indicated the position
assumed for the plane of the picture, which is supposed to be brought
into the plane of the diagram in fig. 5, Plate XLV.
The point of sight, agreeably to the recommendation we have
given, is supposed to be placed, with reference to the picture, at a
distance equal to about twice the width occupied by the machinery
of the mill in the vertical projection. It does not lie within the
limits of the paper in the geometrical projections, Plate XLIV. ;
but it is projected into the plane of the perspective picture in tho
point, i;', fig. 5.
In laying out the main design of this perspective picture, wo
must commence by finding the positions of the axial lines ol all
the columns, iron shafts, horizontal and vertical, and, in general,
of all symmetrical objects. Thus, through the points, 1, 2, 3, 4,
&.C., fig. 1, we must draw a series of visual rays, converging to the
point of sight, and cutting the projection, 1 1', of the plane of tho
picture, in the points, 1", 2", 3", 4", &c., which, in the picture
itself, fig. 5, Plate XLV., are represented by the points, 1', 2', a', 1',
&c.
Tho vertical lines, drawn through each of these points, will bo
the axial lines sought. We next obtain the porspeclive of tho
objects situated nearest to the picture plane, as the column, x, for
example. This column being very near the plane of tiie picture,
and the visual rays tangential to each side of tho cylindrical sui-
fiice, being both very much inclined to tho same side, its ditunotor
in its perspective plane seems proportionately greater than it is
in reality; but this is corrected by tho obliquity with which this
part of the perspective i)icturo should bo viewed. For it must bo
borno in mind, that all porspoctivo pictures must bo viewed from
the single; and precise point of sight in relation to wliicli thoy nro
drawn; othorwiso, tlu> licturcs will have an untrue and distorted
appearance.
We next determine tho porsi)eclivo of tho oolunuis, v, tho lUios
of all which are silualod in u piano iiorpondicular to tho piano of
• Wo do not SCO llic forci" of tills lonmrk, for wliy ulioulil not ii pcnpoctivc r«pn-.
sciitiition bo drawn full slue, or oven to n Inrtior »c«lo? In one c«»o, the oljrcl niu»t
lie siiiHWHwd na clone to llu> dIiuio of tho plrtni o, und, In llio otlior. «» lictwocn tt *i.il
tlK' point of 9lllllt.-TUAS.Sl.A10H AND Kl)IT>H.
164
THE PRACTICAL DRAUGHTSMAN'S
the picture : so that they will diminish gradually in height, being
limited by a couple of lines, converging in the point of sight, i'. It
follows hence, that when the perspective of the first column has
been obtiined, it will be sufilcient to draw through the principal
pomts of the mouldings, and other parts, a series of lines, converg-
mg in the point of sight, r', which will give the perspective posi-
tions of the corresponding points on the other columns, together
with their heights.
It is the same \\"ith the perspective of each of the runner shafts,
tlie pulleys, and other details, the centres of all which lie in the
vertical plane, passing through the axes of the columns, v.
As to the bevil-wheels, K, the axes of which are vertical, and are
projected horizontally in the pomts, 2, 10, 11, fig. 1, Plate XUV.,
and consequently represented in perspective by the vertical lines,
2', 10', ir, &.C., fig. 5, Plate XLV. — it is well to find the perspec-
tive of the apex, s', of the cone, of which the web carrying the teeth
is a frustum ; because the perspectives of all the edges of the teeth
converge to this point.
The edges of the teeth of the bevil- wheel, k', the axis of which is
horizontal, are also represented in perspective by straight lines con-
verging in this same point, s' : and as to the lines defining the sides
or flanks of the teeth, as they also lie in cones, the surfaces of
which, however, are perpendicular to the first, their perspectives
likewise converge to a single point — ^the perspective of the apex of
the cone, on the surface of which they lie.
The centre lines of the arms of the bevil-wheel, k', converging
to the centre of this wheel, are represented in perspective by lines
which are directed towards the point, 0% the perspective of that
centre.
Since the perspectives of the objects which are repeated, and of
which the axes lie in the same plane perpendicular to the picture
plane, are always alike, and difter only in being of diminished
dimensions according to their distance from the plane of the picture,
it will be easily understood that when the perspective of one of
them has been determined, the operations for determining the per-
spectives of the rest may be much simplified, especially by prolono--
ing the various lines to the point of sight. This observation applies
to the wheel gearing, the bearings and frames which support the
various shafts, the pulleys, and other often-repeated details. It is
thus that, in fig. 5, are drawn the p