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APPLETONS'
CYCLOPEDIA OF TECHNICAL
DRAWING.
EMBRACING
Jjrinnplts of
AS APPLIED TO
PRACTICAL DESIGN.
WITH NUMEROUS ILLUSTRATIONS OF
TOPOGRAPHICAL, MECHANICAL, ENGINEERING,
ARCHITECTURAL, PERSPECTIVE, AND
FREE-HAND DRAWING.
EDITED BY
W. E. WORTHEN, C. E.
NEW YORK:
D. APPLETON AND COMPANY,
1, 3, AND 5 BOND STREET.
1892.
COPYRIGHT, 1885,
BY D. APPLETON AND COMPANY.
PREFACE.
" AT the suggestion of the publishers, this work was undertaken to form
one of their series of dictionaries and cyclopedias. In this view, it has been
the intention to make it a complete course of instruction and book of refer-
ence to the mechanic, architect, and engineer. It has not, therefore, 'been
confined to the explanation and illustration of the methods of projection, and
the delineation of objects which might serve as copies to the draughtsman,
matters of essential importance for the correct and intelligible representation
of every form ; but it contains the means of determining the amount and
direction of strains to which different parts of a machine or structure may be
subjected, and the rules for disposing and proportioning of the material em-
ployed, to the safe and permanent resistance of those strains, with practical
applications of the same. Thus, while it supplies numerous illustrations in
every department for the mere copyist, it also affords suggestions and aids
to the mechanic in the execution of new designs. And, although the arrang-
ing and properly proportioning alone of material in a suitable direction, and
adequately to the resistance of the strains to which it might be exposed,
would produce a structure sufficient in point of strength for the purposes for
which it is intended, yet, as in many cases the disposition of the material
may be applied not only practically, but also artistically, and adapted to the
reception of ornament, under the head of Architectural Drawing, the general
characteristics of various styles have been treated of and illustrated, with
brief remarks on proportion and the application of color." . . . 1857.
Since its first publication, this work has been subjected from time to time
to revision. It has now been deemed necessary to almost entirely rearrange
and rewrite it ; to add largely to the subject-matter and to the illustrations,
introducing examples of later practice and experience ; to extend the scope
of the work, and make it more nearly a cyclopaedia of drawing and design.
There are no changes in the principles of projection as applied to drawing,
and no marked improvement in drawing-instruments ; but in the present
practice finished drawings in shade and color are exceptional. It is suffi-
cient, for almost every purpose, for the draughtsman to make accurate projec-
tions with pencil on paper, and trace them afterward on cloth. The pencil-
drawings can be readily altered or amended, and, where there are many repe-
iv PREFACE.
titions of the same parts, but a single one may be drawn. On the tracing
they are made complete, and these are preserved as originals in the office,
while sun-prints of them are used for details of construction in the shop, or
distributed as circulars among customers.
In the sale of former editions of this work, it has been found that its
success has been largely due to its value as a book of design. Great attention
has therefore been given to secure practical illustration of constructions and
machines of recent date ; the nature of materials in common use has been
more extensively treated, and the character and effect of stresses and strains,
their kind and direction, more fully explained, with as simple rules as possible
to determine them for practical application.
Of late years the science of " graphics " has become of great importance,
and is here fully illustrated in its varied applications, showing not only this
method of recording the facts of the statistician, and affording comparisons
of circumstances and times, the growth of population, the quantities and cost
of agricultural and mechanical production, and of their transport, movements
of trade, fluctuations of value, the atmospheric conditions, death-rates, etc.,
but also in its application to the plotting of formulae for their ready solution,
by the draughtsman and designer. For many of the rules in this work the
results of the formulae of various authors have been plotted graphically, and
the rule given one deemed of the greatest weight, not always by average,
but most consistent with our own experience.
In astronomical calculations every decimal may have its importance. It
is not so in those of the mechanical or architectural designer ; solutions by
graphics are sufficient for their purpose, and, simpler than mathematical cal-
culations, they are thus less liable to error ; it is very good practice to use one
as a check on the other. It is to be remarked that inaccuracy in facts, and
carelessness in observation, are not eliminated from an equation by closeness
of calculation, and when factors are not established within the limits of units
it is useless to extend the results to many places of decimals. It is of the
utmost importance to know at first well the object and purposes of the
design, the stresses to which its parts are to be subjected, and the strength
and endurance of the materials of which it is to be composed. In establish-
ing rules for ourselves, be sure of the facts, and that there are enough of
them for a general application. Rules are necessary, but their application
and usefulness depend largely on the experience of the user, and life must
be a record of applications and effects. It is comparatively easy to make
a work strong enough ; but to unite economy with proportion is difficult.
To make the work complete in itself, so as to form a sort of single book
for most of the purposes of the draughtsman and designer embracing the
profession of surveyor, engineer, and architect tables of logarithms, latitudes
and departures, squares and cubes, and square and cube roots, weights and
measures, and weights of material, have been added.
w.
CONTENTS.
PAGES
CONSTRUCTION OF GEOMETRICAL PROBLEMS 1-39
Drawing of lines straight, curved, and parallels, angles, perpendicular ; bisecting
angles; arcs and circles, 15. On polygons and circles; triangles, parallelograms,
squares ; circles, angles ; polygons ; inscribed and described circles ; pentagons, hexa-
gons, octagons ; table of polygonal angles, 23. On the ellipse, parabola, hyperbola,
cycloid, epicycloid, involute and spiral, 33. Use of triangle and square, 33. Areas
of figures, 37. To draw squares of given proportionate sizes, 39.
DRAWING INSTRUMENTS 40-77
Description and use ; rulers ; triangles; T-square; parallel ruler ; sweeps and vari-
able curves ; drawing pens ; dotting point ; pricking point ; compasses ; dividers ;
beam compasses ; porportional dividers ; scales ; scale guard ; diagonal scales ; ver-
nier scales ; sector ; protractors ; pantagraphs ; camera lucida ; drawing table and
board, 56. Drawing paper ; tracing paper ; tracing cloth ; mouth glue ; damp stretch-
ing paper ; mounting paper and drawings, 59. Management of the instruments ;
ink ; exercises with drawing pen ; various letters and numerals ; cross-section paper ;
diagrams showing use of cross-section paper, 77.
ORTHOGRAPHIC PROJECTION 78-109
Definitions; points; straight line; solid, 81. Simple bodies; pyramid; prism,
87. Construction of the conic sections, 90. Penetration or intersection of solids ;
cylinders, cone, and sphere ; cylinder and ring ; sphere and prism ; cone and prism ;
cone and cylinder, 102. Of the helix, 104. Development of surfaces ; cylinder ;
cone; sphere, 107. Shade-lines, 109.
SHADES AND SHADOWS 110-136
Of a point : straight line ; solid ; circle ; pyramid ; cylinder ; cone ; shadows cast
upon a cylinder by various-shaped caps ; shadows cast upon a prism ; shadows upon
the interior of a cylinder, hollow hemisphere, a niche, a sphere ; line of shade on the
surface of a ring, grooved pulley, square-threaded nut and screw, triangular-threaded
nut and screw, 126. Manipulation of shading and shadows methods of tinting;
surfaces in the light ; surfaces in shade ; shading by flat tints ; by softened tints,
129. Elaboration of shading and shadows; examples of finished shading; on con-
cave surfaces, spheres, ring, cone, flat surfaces ; colors for tints ; expeditious way of
shading a cylinder ; body color ; margin of light ; washing ; conventional tints for
materials, 136.
PLOTTING ; 137-148
Scales ; scales prescribed by different commissions, 138. Variation of compass,
1 39. Plotting compass surveys ; balancing error ; plotting by latitudes and depart-
ures ; area by latitudes and departures ; area by triangles ; plotting by offsets, 147.
System of division of United States land, 148.
VI
CONTENTS.
PAGES
TOPOGRAPHICAL DRAWING 149-180
Conventional signs ; representation of hills ; contour lines, 156. Railway surveys ;
profiles; sections, land plans, 159. Hydrometrical or marine surveys, conventionali-
ties, 160. Geological and statistical features, 162. Transferring ; pricking through;
by tracing; blue-print process; copying-glass; transfer-paper; pantagraph, 165.
Map projections ; perspective projection on planes ; developed perspective projec-
tions ; projections by developing elements ; projections conforming to some arbi-
trary condition ; polyconic adopted by United States Coast Survey ; De Lorgne's
projection; M creator's chart, 171. Colored topography ; conventional colors ; direc-
tions; finishing; lettering; titles, 180.
MATERIALS 181-199
Earth and rocks, 182. Building materials ; wood, 185. Stones; technical terms
masonry; granitic stones, argillaceous stones ; sandstones ; limestone, 188. Artificial
building material ; bricks ; tile ; terra-cotta ; mortars ; limes ; cement ; concrete ;
plastering, 191. Metals; conventional hatchings ; iron; steel; other metals; specific
gravity ; weight ; melting-point ; resistance to crushing and tension ; results of Prof.
Thurston's tests of metals, 196. Sulphur; glass; rubber; paints; coals, 199.
MECHANICS 200-219
Force ; center of gravity ; levers ; wheel and axle ; pulley ; inclined plane ; wedge ;
screw ; inclined forces ; parallelogram of forces ; hydraulic press ; velocity of falling
bodies; friction, 212. Mechanical work or effect; horse-power, etc.; water-power;
wind ; steam ; steam worked expansively ; cut-offs ; compound engines ; indicator
cards, 219.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS 220-361
Stress ; dead load ; strength of posts and columns ; Phoenix columns, etc. ; shear-
ing stresses ; torsional stress ; transverse stress ; strength of beams ; tables of dimen-
sions of channel beams and angle-iron ; composite beams ; bolts and nuts ; strength
of bolts ; washers ; shafts and axles ; journals ; keys ; car-axles ; shafting ; bear-
ings ; couplings; clutch; pulleys; belts; ropes, 275. Gearing; epicycloidal teeth;
projections of a spur-wheel and bevel-wheel, 294. Drawing of screws, 297. Fric-
tional gearing, 299. Ropes and chains ; hooks ; levers ; cranks ; connecting-rods ;
steam-engine ; working-beam ; parallel-motion links ; steam-cylinders and pistons,
331. Valves ; hydrants, 342. Riveted joints for boilers ; boilers, 351. Wrought-
iron pipe connections, 355. Frames ; governors ; fly-wheels ; air-chambers, 361.
ENGINEERING DRAWING 362-460
Foundations ; sheet-piling ; retaining-walls ; foundations for piei^s, etc., 375.
Dams ; locks of canals ; conduits ; reservoirs, 395. Water-pipes, 398. Sewers, 401.
Gas-supply, 402. Roads, 407. Roofs and bridges; piers, 432. Arch-bridges; sus-
pension-bridges, 438. Boiler-setting ; chimneys, 444. Location of machines ; ma-
chine foundations, 449. Tunnels, 453. Railway stock, 458. Wave-line principle of
ship construction, 460.
ARCHITECTURAL DRAWING
Details of construction ; concrete walls, 468. Frames and floors ; fire-resisting
floors, 474. Groined ceilings, 476. Doors ; windows ; moldings, 485. Stairs, 492.
Fireplaces ; flues ; roofs ; gutters ; plastering, 495. Proportions and distribution of
rooms and passages, 500. Plans and elevations of buildings ; stores and warehouses,
521. School-houses, 531. Churches, theatres, lecture-rooms, music and legislative
halls ; hospitals, 542. Stables ; cow-houses ; greenhouses, 547. Ventilation and
warming, 555. Plumbing, 564. Greek and Roman orders of architecture, 596. Or-
naments of the Renaissance ; principles of design, 601.
461-601
CONTENTS.
vn
PAGES
PERSPECTIVE DRAWING 602-624
Angular perspective, 610. Parallel perspective, 624.
ISOMETRICAL DRAWING 625-638
FREE-HAND DRAWING
Geometrical figures and design, 643. Proportions of the human frame, 647.
Figure drawing, 650. Forms of animals, 653. Illustrations from different artists,
664.
APPENDIX
Extracts from New York building laws, 670. Patent-Office drawings, 670. Men-
suration ; properties of triangles, 672. Lineal measure, 672. Table of inches in
decimals of a foot, 673. Table of measures of surface, 673. Table of measures of
capacity ; dry measure ; weights ; cubic measure, 674. Table of weight of rolled
iron, 675. Table of weight of wrought-iron and brass plates and wire, 676. Table
of weight of wrought-iron welded tubes ; boiler tubes ; driven-well tubes ; heavy
wrought galvanized iron spiral riveted pipes, 678. Table of copper and brass rods,
678. Table of number of Burden's rivets in 100 pounds, 679. Table of number of
wrought spikes to a keg, 679. Table of length of cut nails and spikes, and number
in a pound, 680. Table of weights of lead pipe per foot, 680. Table of the weight
of a cubic foot of water at different temperature?, 680. Table of properties of satu-
rated steam, 682. Table of mean pressures in steam cylinders at different rates of
expansion, 683. The flow of water, with table of discharge over weir, 685. Flow
of water through pipes and sewers, 689. Flow of air through pipes, 690. Table of
circumferences and areas of circles, 695. Table of squares, cubes, square and cube
roots of numbers, 703. Table of latitudes and departures, 709. Table of natural
sines and cosines, 718. Logarithms of numbers, 735.
INDEX.
DESCRIPTION OF PLATES.
PLATES I TO XIV.
SCRAPS.
639-664
665-735
DESCRIPTION OF PLATES.
PLATE
I. Shading of prism and cylinder by flat tints. Referred to on pages
126-7.
II. Shading of cylinder and segment of hexagonal pyramid. Referred to
on pages 128-9.
Ill, IV. Finished shading and shadows of different solids. Referred to on
page 131.
V. Shades and shadows on screws. Referred to on page 126.
VI. Example of topographical drawing, done entirely with the pen.
VII. The same, with the brush, in black.
VIII. The same, with the brush, in color. Referred to on page 174.
IX. Contoured map of Staten Island, shaded by superimposed washes,
the washes increasing in intensity or strength as required to pro-
duce the effect.
X. Geological map of part of New Jersey, colored to show the different
formations.
XI. Finished, shaded sectional view, colored to show the different metals,
of a balanced leather cup-valve. The body is of cast-iron ; the
piston, brass ; packing, leather ; piston-rod, wrought-iron this
last not distinctively colored.
XII. Finished perspective drawing, with shades and shadows, of a large
bevel-wheel and two pinions, with shifting clutches.
XIII. Front elevation of a building, in color.
XIV. Perspective view of Gothic church, finished in color.
XV. Plan, elevation, and section of bevel-wheel, pinion, and clutches,
shown in perspective Plate XII.
XVI to XX. Details of progressive perspective projections of Plate XV, as
shown completed in Plate Xll.
APPLETONS'
CYCLOPAEDIA OF DRAWING,
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
MOST persons, at some time, have made use of the simple drawing instru-
ments, pencils, straight-edges or rulers, and compasses or dividers with change-
able points, and many suppose that there can be no skill in their use ; but to
one critical in these matters there are great differences to be observed even in
common drawings, in the straightness and uniformity of the lines, and in the
care of the surface of the paper.
Select for the geometrical problems and
for usual drawings a No. 3 or H H H pen-
cil. It should be sharpened to a cone-point
(Fig. 1). Where a pencil is used for drawing
lines only, some draughtsmen sharpen the
pencil to a wide edge, like a chisel.
In drawing a straight line, hold the ruler
firmly with the left hand ; with the right
hand hold the pencil lightly but without
FlG - 1 - slackness, and a little inclined in the direc-
tion of the line to be drawn, keeping the pencil against the edge of the
ruler, and in the same relative position to the edge during the whole operation,
of drawing the line.
2 CONSTRUCTION OF GEOMETRICAL PROBLEMS.
To draw a clean line and preserve the point of the pencil, the part of the
cone a little above the point of the pencil should bear against the edge of the
ruler, and the pencil should be carried steadily while drawing. Any oscilla-
tion will throw the point farther from or nearer the ruler, and the line will
not be straight (Fig. 2).
FIG. 2.
In the use of the compasses do not make a hole through the paper with the
needle or sharp point, but only into the paper sufficient to maintain the
position.
Keep the paper clean, and use rubber as little as possible.
As drawing is based on geometrical principles, we commence with geo-
metrical definitions and problems to give the learner some knowledge of
the science of geometry, with a valuable exercise in the use of drawing
instruments.
Geometrically a point is defined as position merely : in drawing, the posi-
tions of points are marked on the paper by prick-marks of a needle or sharp
point, and by the dot of a pencil ; sometimes inclosed O, sometimes designated
by the intersection of two short lines X >.
Geometrically lines have but one dimension, .length, and the direction
of a line is the direction from point to point of the points of which the
line is composed : in drawing, lines are visible marks of pencil or pen upon
paper.
FIG. 3.
A straight line is such as can be drawn along the edge of the ruler, and is
one in which the direction is the same throughout. In drawing a straight line
through two given points, place the edge of the ruler very near to and at equal
distances from the points, as the point of the pencil or pen should not be in
contact with the edge of the ruler (Fig. 3).
Lines in geometry and drawing are generally of limited extent. A given
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
3
FIG. 4.
or known line is one established on paper or fixed by dimensions. Lines of
the same length are equal.
To draw Curved Lines. Insert the pencil-point in the compasses, and open
them to a suitable extent. With the needle or sharp point resting on the
paper describe a line with the pencil around this point ; the line thus
described is usually called a circle more strictly it is the circumference of a
circle the circle being the space inclosed. A portion
of a circumference is an arc. The point around
which the circumference is described is the center
of the circle (Fig. 4).
If a line be drawn from the center to the circum-
ference it is called a radius. As it is the length
embraced between the points of the compasses, it is
often called by mechanics the sweep.
If a line be drawn through the center, and limited
by the circumference, it is called the diameter, and is
equal to two radii.
A radius is a semi-diameter ; a diameter is the longest line that can be con-
tained within a circumference. Lines limited by the circumference, and which
are not diameters, are chords.
It will be observed that arcs are lines which are continually changing the
directions, and are called curved lines, but there are other curved lines than
those described by compasses, of which the construction will be explained
hereafter.
Besides straight and curved lines there are often lines, in drawing, which
can neither be drawn by rulers or compasses, as lines representing the direc-
tions of brooks and rivers, the margins of lakes and seas, points in which are
established by surveys, defined on paper, and connected by hand-drawing.
These may be called irregular or crooked lines.
Where it is necessary to distinguish lines by names, we place at their
extremities letters or figures, as A B, 1 2 ; the line A B, or 1 2.
But in lines other than straight, or of considerable extent, it is often necessary
to introduce intermediate letters and figures, as a a a.
In the following problems, unless otherwise implied or designated, where
lines are mentioned, straight lines are intended.
If we conceive a straight line to move sideways in a single direction, it will
sweep over a surface which is called a plane. All drawings are projections on
planes of paper or board.
Two lines drawn on paper, and having the same direction, can never come
any nearer each other, and must always be at the same distance apart, however
far extended. Such lines are called parallel lines.
4: CONSTRUCTION OF GEOMETRICAL PROBLEMS.
PEOB. L To draw a line parallel to a given line, and at a given distance
from it (Fig. 5).
Draw the line A B for the given line, and take in the compasses the dis-
tance A C the distance at which the other line is to be drawn. On A, as a
Jj
FIG. 5.
FIG. 6.
center, describe an arc, and on B, as a center, another arc ; draw the line C D
just touching these two arcs, which will be the parallel line required.
PKOB. II. To draw a line parallel to a given line through a given point
outside this line (Fig. 6).
Draw the given line A B, and mark the given point C. With C as a centei,
find an arc that shall only just touch A B ; and with B as a center, and the
same radius, describe an arc D. Draw through the point C a line just touching
this last arc, and the line C D will be the parallel line required.
Two lines in the same plane, not parallel to each other, will come together
if extended sufficiently far. The coming together, cutting, or intersection of
two lines, is called an angle (Fig. 7).
If but two lines come together, the angle may be designated by a single
letter at the vertex, as the angle E.
But, if three or more lines have a common vertex, the angles are designated
by the lines of which they are composed, as the angle D B C of the lines D B
D
FIG, 8
and B C ; the angle A B C of A B and B C ; the angle A B D of A B and B D.
The letter at the vertex is not repeated, and must always be the central letter.
Describe a circle (Fig. 8). Draw the diameter A B. From A and B
as centers, with any opening of the compasses greater than the radius,
describe two arcs cutting each other as at D. Through the intersection
of these arcs and the center C, draw the line D E. D E makes, with the
diameter A B, four angles, viz., A C D, D B, B C E, and E C A. Angles
A*
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
are equal whose lines have equal inclination tfc^ach other, and whose lines, if
placed one on the other, would coincide. By construction, the points C and D
have, respectively, equal distances from A and B ; the line D C can not, there-
fore, be inclined more to one side than to the other, and the angle A C D must
be equal to the angle BCD. Such angles are called right angles. It can be
readily proved that all the four angles, formed by the intersection of D E with
A B, are equal, and are right angles.
The angles A C D and D C B, on the same side of A B, are called adjacent
angles ; as also DOB and B C E, on the same side of D E.
When a line, standing on another line, makes the two adjacent angles equal,
the angles are right angles, and the first line is perpendicular to the other.
If the second or base line be parallel with the surface of still water,
it is called an horizontal line, and the perpendicular line is called a ver-
tical line.
Draw the line C F. It will be observed that the angle F C D is less than
a right angle, and it is called an acute angle ; the angle F C A is greater than
a right angle, and it is called an obtuse angle. It will be observed that, no
matter how many lines be drawn to the center, the sum of all the angles on
the one side of A B can only be equal to two right angles, and, on both sides
of A B, can only be equal to four right angles. It will be observed that the
angles at the center include greater or less arcs between their sides, according
to the greater or less inclination of their sides to each other ; that the right
angles intercept equal arcs, and that, no matter how large the circle, the pro-
portion of the circle intercepted by the sides
of an angle is always the same, and that the
arcs can therefore be taken as the measures
of angles. For this purpose the whole cir-
cumference is supposed to be divided into
three hundred and sixty degrees (360), each
degree subdivided into sixty minutes (60'),
and each minute into sixty seconds (60*).
Each right angle has for its measure one
quarter of the whole circumference (-^p-),
or 90, and is called a quadrant.
PROB. III. To construct an angle equal
to a given angle (Fig. 9).
Draw any angle, as C A B, for the given
angle, and the line a b as the base of the
required angle. From A, with any suitable
radius, describe the arc B C, and from a,
with the same radius, describe the arc b c.
Measure the length of the arc B C, or rather
the chord, that is, the distance in a straight line from B to C, and lay off the
same distance on the arc b c. Draw the line a c, and the angle cab will be
equal to C A B.
PROB. IV. To construct an angle of sixty degrees (Fig. 10).
Lay off any base-line, and from A, with any radius, describe an arc, and
Fio. 9.
6 CONSTRUCTION OF GEOMETRICAL PROBLEMS.
from B, with the same radius, describe another arc cutting the first, as at C.
Draw the line C A, and the angle CAB will be an angle of sixty degrees.
The reason of this construction will be readily understood if, on the cir-
\
FIG. 10.
;#
Fio. 11.
cumference of any circle, chords equal to the radius are stepped off succes-
sively. Six will exactly complete the circle, making 360, or each 60, and
the angle corresponding will be 60.
PKOB. V. To draw a perpendicular to a line from a point without the
line (Fig. 11).
Draw a line, and mark the given point outside it, A. From A as a center,
with a suitable radius, describe an arc cutting the line at G and F. From G
and F, as centers, describe arcs cutting each other. The line drawn through
the point A, and the point of intersection E, will be perpendicular to the
line G F.
The radial line A E divides the chord G F and the arc G E F into two
equal parts ; and, conversely, the line perpendicular to the middle point of a
chord of a circle is radial passes through the center of that circle.
PROS. VI. To draw a perpendicular to a line from a point within that
line (Fig. 12).
1st Method. Draw a line, and take the point A in the line. From A, as
a center, describe arcs cutting the line on each side at B and C. From B and
Nr
,'D
A
FIG. 12.
C l
Fia. 13.
C, as centers, describe intersecting arcs at D. Draw a line through D and A>
and it will be perpendicular to the line B C at A.
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
\C/
2d Method (Fig. 13). Draw the line, and mark the point A as before.
From any center F, without the line, and not directly over A, with a radius
equal to F A, describe more than a half-circle cutting the line, as at D. From
D, through the center F, draw a line cutting the arc at E. Draw A E, and it
will be the perpendicular to the line A D.
It will be observed that the line D E is the diameter of a circle, and that
the angle DAE, with its vertex at A in the circumference, would embrace
with its sides half a circle, had a full circle been described. It has been shown
that angles at the center of a circle have for their measure the arc embraced
by their sides. It is easily demonstrable that angles, with their vertices in the
circumference, have for their measure half the arc embraced by their sides,
and, consequently, angles embracing
half a circumference are right an-
gles, and their sides are perpendicu-
lar to each other.
PROB. VII. To draw a perpen-
dicular to the middle point of a line
(Fig. 14).
From the extremities A and B
of the line, as centers, describe in-
tersecting arcs above and below the
line. Through these intersections
draw the line D. It will be per-
pendicular to the line A B, and bi-
sect or divide it into two equal parts.
If the line A B be considered the chord of a circle, its center would lie in
the line C D.
This construction is sometimes used merely to divide a line into two equal
parts, or bisect it ; but if we have dividers or compasses, with both points
sharp, it can be more readily done with them (Fig. 15).
Place one point of the dividers on one end of the line, and open the
dividers to a space as near as may be half the line. Turn the dividers on the
central point ; if the other point then falls exactly on the opposite extremity
DIE
FIG. 14.
FIG. 15.
of the line, it is properly divided ; but, if the point falls either within or with-
out the extremity of the line, divide the deficit or excess by the eye, in
halves, and contract or extend the dividers by this measure. Then apply the
dividers as before, and divide deficit or excess till a revolution exactly covers the
length of the line. By accustoming one's self to this process, the eye is made
accurate, and one estimate is sufficient for a correct division of any deficit or
8
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
excess. By a similar process it is evident that a line can be divided into any
number of equal parts, by assuming an opening of the dividers as nearly as
possible to that required by the division, and, after spacing the line, dividing
the deficit or excess by the required number of parts, contracting or expanding
the dividers by one of these parts, and spacing the line again, and so on till it
is accurately divided.
PKOB. VIII. To bisect a given angle (Fig. 16).
Construct an angle, and from its vertex A, as a center, describe an arc
cutting the two sides of the angle at B and C. From B and C, as centers,
describe intersecting arcs. Draw a line through A and the point of intersec-
tion D, and this line will bisect the angle.
-B
6
I)
FIG. 16.
FIG. 17.
PROB. IX. To Used an angle when the vertex is not on the paper (Fig. 17).
Draw two lines, A B and E C, inclined to each other, but not intersecting.
Draw two lines intersecting each other, a b and a c, inside and parallel to A B
and E C. Bisect b a c by the line a d, and this line will also bisect the angle
whose vertex is not on the paper.
PROB. X. Through two given points to describe an arc of a circle with a
given radius (Fig. 18).
From B and C, the two given points, with an opening of the dividers equal
to the given radius, describe two arcs crossing at A. From A, as a center,
with the same radius, describe an arc, and it willbe the one required.
FIG. 18.
FIG. 19.
PROB. XL To find the center of a given circle, or of an arc of a circle.
Of a circle (Fig. 19). Draw the chord A B. Bisect it by the perpen-
dicular C D, whose extremities lie in the circumference, and bisect C D. Gr,
the point of bisection, will be the center of the circle.
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
Of an arc, or of a circumference (Fig. 20). Select the points A, B, and C
in the circumference, well apart. With the same radius from A and B as
centers, and then from B and C as centers, describe arcs cutting each other ;
draw the two lines D E and F G through their intersections. The point 0,
where these lines meet, is the center required.
PKOB. XII. To describe a circle passing through three given points
(Fig. 20).
Proceed, as in the last problem, to find the center 0. From 0, as a
center, with a radius A, describe a circle, and it will be the one required.
FIG. 20.
PKOB. XIII. To describe a circle passing through three given
where the center is not available.
1st Method (Fig. 21). From the extreme points A and B, as centers,
describe the arcs B G and A H. Through the third point, C, draw A E and
B F, cutting the arcs. Divide the arcs A F and B E into any number of equal
parts, and set off a series of equal parts of the same length on the upper por-
tions of the arcs beyond E and F. Draw straight lines, B L, B M, etc., to the
points of division in A F, and A I, A K, etc., to the points of division in E G ;
the successive intersections N, 0, etc., of these lines are points in the circle
required between the given points A and C, which may be filled in accord-
ingly. Similarly, the remaining part of the curve, B C, may be described.
Zd Method (Fig. 22). Let A, D, and B be the given points. Draw A B,
A D, and D B. Draw e f parallel to A B. Divide D A into a number of
equal parts at 1, 2, 3, etc.,
and from D describe arcs "'
through these points to meet
ef. Divide the arc A e into
the same number of equal
parts, and draw straight
lines from D to the points
of division. The intersec-
tions of these lines successively with the arcs are points in the circle, which
may be filled in as before.
Note. The second method is not perfectly true, but sufficiently so for arcs
less than one fourth of a circle.
10
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
To describe the arc mechanically. Let a, c, I be the three points of a curve ;
transfer these points to a piece of stout card-board, and draw the lines a c and
c I, and extend them beyond a and I. Cut out the card-board along these
FIG. 23.
lines. Insert upright pins on the points a and I of the drawing, and placing
the edges of the cut card-board against them, and maintaining the contact of
the edges of the card-board with the pins, slide the card each way. Dot the
positions of the vertex of the angle c, and the dots will be points in the curve.
PROB. XIV. To draw a tangent to a circle from a given point in the cir-
cumference.
1st Method (Fig. 24). Through the given point A draw the radial line
A C. The perpendicular F G- to that line will be the tangent required.
FIG. 24.
FIG. 25.
2d Method (Fig. 25). From the given point A set off equal arcs, A B and
A D. Join B and D. Through A draw A E parallel to B D, and it will be
the tangent required. This method is useful when the center is inaccessible.
PROB. XV. To draw tangents to a circle from a point without it (Fig. 26).
From the given point A draw A to the center of the circle. From D, the
FIG. 26.
FIG. 27.
intersection of A C with the circle, describe an arc, with a radius D C, cutting
the circle at E and F. Draw A E and A F, and they will be the tangents required.
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
11
To construct within the sides of an angle a circle tangent to these sides at a
given distance from the vertex (Fig. 27). Let a and b be the given points
equally distant from the vertex A. Draw a perpendicular to A C at a, and to
A B at I. The intersection of these perpendiculars will be the center of the
required circle.
In the same figure, to find the center, the radius being given, and the
points a and b not known. Draw lines parallel to A C and A B, at a distance
equal to the given radius, and their intersection will be the center required.
PROB. XVI. To describe a circle from a given point to touch a given
circle (Figs. 28 and 29).
D E being the given circle, and B the given point, draw a line from B to
the center C, and produce it, if necessary, to cut the circle at A. From B,
FIG. 28.
FIG. 29.
as a center, with a radius equal to B A, describe the circle F G, touching the
given circle, and it will be the circle required.
The operation is the same whether the point B is within or without the
circle.
It will be remarked that, in all cases of circles tangent to each other,
their centers and their points of contact must lie in the same straight line.
PROB. XVII. To draw tangents to two given circles.
1st Method (Fig. 30). Draw the straight line ABC through the centers
of the two given circles. From the centers A and B draw parallel radii, A D
FIG. 30.
and B E, in the same direction. Draw a line from D to E, and produce it to
meet the center line at C ; and from C draw tangents to one of the circles by
Problem XV. Those tangents will touch both circles as required.
2d Method (Fig. 31). Draw the line A B connecting the two centers.
Draw in the larger circle any radius, A H, on which set off H G, equal to the
12
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
radius of the smaller circle. On A describe a circle with the radius A G, and
draw tangents, B I and B K, to this circle from the other center, B. From A
FIG. 81.
and B draw perpendiculars to these tangents. Join C and D, also E and F.
The lines D and E F will be the required tangents.
Note. The second method is useful when the diameters of the circles are
nearly equal.
PKOB. XVIII. Between two inclined lines to draw a series of circles
touching these lines and touching each other (Fig. 32).
Bisect the inclination of the given lines A B and C D by the line N ; this
is the center line of the circles to be
inscribed. From a point, P, in this
line, draw P B perpendicular to the
line A B ; and from P describe the
circle B D, touching the given lines,
and cutting the center line at E.
From E draw E F perpendicular to
the center line, cutting A B at F ;
from F describe an arc, with a ra-
dius, F E, cutting A B at G ; draw
G H parallel to B P, giving H the
center of the second touching circle, described with the radius H E or II G.
By a similar process the third circle, I N, is described. And so on.
Inversely, the largest circle may be described first, and the smaller ones in
succession.
Note. This problem is of frequent use in scroll-work.
PROB. XIX. Between tivo inclined lines to draw a circular arc to fill up
the angle, and touching the lines (Fig. 33).
Let A B and D E be the inclined lines. Bisect the inclination by the line
F C, and draw the perpendicular A F D to define the limit within which the
circle is to be drawn. Bisect the angles A and D by lines cutting at C,
and from C, with radius C F, draw the arc H F G, which will be the arc
required.
PROB. XX. To fill up the angle of a straight line and a circle, with a cir-
cular arc of a given radius (Fig. 34).
On the center C, of the given circle A D, with a radius C E equal to that
of the given circle plus that of the required arc, describe the arc E F. Draw
FIG. 32.
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
13
G F parallel to the given line H I, at the distance G H, equal to the radius
of the required arc, and cutting the arc E F at F. Then F is the required
H I
FIG. 34.
center. Draw the perpendicular F I, and the line F C, cutting the circle
at A ; and, with the radius F A or F I, describe the arc A I, which will be the
arc required.
PKOB. XXI. To fill up the angle of a straight line and a circle, with a
circular arc to join the circle at a given point (Fig. 35).
In the given circle B A draw the radius to A, and extend it. At A
draw a tangent, meeting the given line at D. Bisect the angle A D E, so
formed, with a line cutting the radius, as extended at F ; and, on the center
F, with radius F A, describe the arc A E, which will be the arc required.
PKOB. XXII. To describe a circular arc joining two circles, and to touch
one of them at a given point (Fig. 36).
Let A B and F G be the given circles to be joined by an arc touching one
of them at F.
Draw the radius E F, and produce it both ways ; set off F H equal to the
radius, A 0, of the other circle ; join C to H, and bisect it with the perpen-
dicular L I, cutting E F at I. On the center I, with radius I F, describe the
arc F A, which will be the arc required.
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
PROB. XXIII. To find the arc which shall be tangent to a given point on a
straight line, and pass through a given point outside the line (Fig. 37).
Erect at A, the given point on the given line, a perpendicular to the line.
From C, the given point outside the line, draw C A, and bisect it with a per-
pendicular. The intersection of the two perpendiculars at a will be the center
of the required arc.
a.
FIG. 37.
FIG. 38.
PROB. XXIV. To connect two parallel lines by a reversed curve composed
of two arcs of equal radii, and tangent to the lines at given points (Fig. 38).
Join the two given points A and B, and divide the line A B into two equal
parts at C ; bisect C A and C B by perpendiculars ; at A and B erect perpen-
diculars to the given lines, and the intersections a and b will be the centers of
the arcs composing the required curve.
PROB. XXV. To join two given points
in two given parallel lines by a reversed
curve of two equal arcs, whose centers lie
in the parallels (Fig. 39).
Join the two given points A and B,
and divide the line A B in equal parts at
C. Bisect A C and B C by perpendicu-
lars ; the intersections a and b of the
parallel lines, by these perpendiculars,
will be the centers of the required arcs.
PROB. XXVI. On a given line, to construct a compound curve of three
arcs of circles, the radii of the two side ones being equal and of a given length,
FIG.
'/b
H /
D
FIG. 40.
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
15
and their centers in the given line ; the central arc to pass through a given
point on the perpendicular, bisecting the given line, and to be tangent to the
other two arcs (Fig. 40).
Let A B be the given line, and C the given point. Draw C D perpen-
dicular to A B ; lay off A a, B b, and C c, each equal to the given radius of the
side arcs ; draw a c, and bisect it by a perpendicular ; the intersection of this
line with the perpendicular C D will be the required center of the central arc
e C e r . Through a and b draw the lines D e and D e' ; from a and b, with the
given radius equal to a A or b B, describe the arcs A e and B e f . From D, as
a center, with a radius equal to C D, and, consequently, by construction, equal
to D e and D e', describe the arc e C e'. The entire curve A e C e' B is the
compound curve required.
t It will be observed in all the preceding problems that, when a line is tangent
to a curve, the center of that curve must be in the perpendicular to the line
at its tangent point ; and that, when two curves are tangent to each other, their
centers must be in the same radial line passing through the point of tangency.
PROBLEMS ON POLYGONS AND CIRCLES.
Three lines inclosing a spa^ce form a triangle (Fig. 41). If two of the sides
are of equal length, it is an isosceles triangle ; if all three are of equal length,
FIG. 41.
FIG. 42.
it is an equilateral triangle. If one of the angles is a right angle, it is a right-
angled triangle, and if no two of the sides are of equal length, and not one of
the angles a right angle, it is a scalene
triangle.
PROB. XXVII. To construct an isos-
celes triangle (Fig. 42).
FIG. 43. FIG. 44.
Draw any line as a base, and, from each extremity as a center, with equal
radius, describe intersecting arcs. Draw a line from each extremity of the
base to this point of intersection, and the figure is an isosceles triangle.
PROB. XXVIII. To construct an equilateral triangle (Fig. 43).
16
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
Draw a base line, and from each extremity as a center, with a radius equal
to the base line, describe intersecting arcs. Draw lines from the extremi-
ties of the base to this point of intersection, and the figure is an equilateral
triangle.
PKOB. XXIX. To construct a right-angled triangle (Fig. 44).
Construct a right angle by any one of the methods before described.
Draw a line from the extremity of the one side to the extremity of the other
side, and the figure is a right-angled triangle.
It will be evident, in looking at any right-angled triangle, that the side
opposite the right angle is longer than either of the other or adjacent sides ;
this side is called the hypothenuse.
PROB. XXX. To construct a triangle equal to a given triangle.
Let ABC (Fig. 45) be the given triangle.
1st Method (Fig. 46). Draw a base line, and lay off its length equal to
FIG. 45.
FIG. 46.
A B ; from one of its extremities, as a center, with a radius equal to A C,
describe an arc ; and, from its other extremity, with a radius equal to B C,
describe an arc intersecting the first. Draw lines from the extremities to the
point of intersection, and the triangle equal to A B C is complete.
2d Method (Fig. 47). Draw a base line, as before, equal to A B. At one
C
FIG. 47.
FIG. 48.
extremity construct an angle equal to C A B, and at the other an angle equal to
C B A. The sides of these angles will intersect, and form the required triangle.
3d Method (Fig. 48). Construct an
angle of the triangle equal to any angle
of A B C, say the angle A C B. On
one of its sides measure a line equal to
C A, and on the other side one equal to
C B ; connect the two extremiities by a
line, and the triangle equal to A B C is
~~ complete.
FIG. 49. From the above constructions it will
CONSTRUCTION OF GEOMETRICA
17
FIG. 50.
foe seen that, if the three sides of a triangle, or two sides and the included an-
gle, or one side and the two adjacent angles are known, the triangle can be
constructed.
Construct a triangle, ABC (Fig. 49). Extend the base to^E;"8m^Sraw
B D parallel to A C. As A C has the same inclination to C B that B D has,
the angle C B D is equal to the angle A C B. As A C has the same inclina-
tion to A E that B D has, the angle D B E is equal to C A B. That is,
the two angles formed outside the triangle are equal to the two inside at A and
C ; and the three angles at B are equal to the three angles of the triangle, and
their sum is equal to two right angles. There-
fore, the sum of the three angles of a trian-
gle is equal to two right angles.
On one side of a triangle (Fig. 50) con-
struct a triangle equal to the first, with op-
posite sides parallel.
The exterior lines of the two triangles
form a four-sided or quadrilateral figure, of which the opposite sides are equal
and parallel, and the opposite angles equal. This figure is called a parallelo-
gram, and the line C B, extending between opposite angles, is a diagonal.
On the hypothenuse of a right-angled triangle (Fig. 51) construct another
equal to it, and the exterior lines form a parallelogram, which, as all the angles
are right angles, is called a rectangle. If
the four sides are all equal, it is called a
square.
A parallelogram in which all the sides
are equal, but the angles not right angles,
is called a rhombus (Fig. 52) ; if only the
opposite sides are equal, it is called a rhom-
boid ; if only two sides are parallel, the
figure is a trapezium (Fig. 53).
Describe a circle (Fig. 54). Draw a diameter, and erect on its center C the
perpendicular C F. Draw at any angle with the diameter the line C A. Draw
D H and A B perpendicular to the diameter, the first from the intersection of
the line C A with the circumference, the other from the extremity B of the
FIG. 51.
FIG. 52.
FIG. 53.
diameter. Draw D G and E F perpendicular to the radius C F, one from the
point D, the other from the extremity of the radius C F. The angles DOG
and D C H are complements of each other ; that is, together they form a
right angle, as it completes with it a right angle. The line D H is the
2
18
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
sine of the angle D C H and the cosine of D C G. D G is the sine of the
angle D C G and the cosine of D C H. A B is the tangent of the angle
.DOB and the cotangent of D C G. E F is the tangent of the angle DOG
and the cotangent of D C H. A C is the secant of the angle D C H and
the cosecant of D C G. C E is the secant of the angle DOG and the cosecant
of D C H. H B is the versed sine of the angle D C H, and G F of D C G.
It will be observed that the angles of the triangle D H are equal to those
of A C B, and that, if we suppose C A to be the radius of a larger circle, the
arcs, and consequently the half -cords or sines D H and A B, will be propor-
tionate to the radii ; that is, D H will
A be to A B as C D is to C A.
Triangles which have equal angles
have their sides proportional, and are
called similar. This is demonstrable of
other triangles than the right-angled
ones in the figure.
Take any figure (Fig. 55) of more
than three sides bounded by straight
FIG. 54.
FIG. 55.
lines, and from any angle draw lines to the opposite angles. The figure will
be divided into as many triangles as there are sides, less two, and the sum of
the angles of the figure will be equal to as many times two right angles as
there are sides, less two.
If another figure were made with similar triangles, similarly placed, the
two figures would be similar.
Polygons, or many-sided figures, are similar when their angles are equal to
each other and similarly placed, and their homologous sides, or sides including
these angles, proportional.
FIG. 56.
FIG. 57.
FIG. 58.
FIG. 59.
On this principle of similarity of figures the science of drawing is based.
With a scale of equal parts, one inch on paper, for instance, representing a
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
19
foot, a yard, or a mile, in nature, the figure drawn on that scale will represent
the object accurately in reduced form ; and measurements may be made in de-
tail by the scale as well as from the natural object in the shop or on the estate.
Polygons, with their sides and angles equal, are called regular polygons
(Figs. 56, 57, 58, 59).
Regular polygons are easily constructed by means of circles, whose circum-
ferences are divided into the number of sides required, with chords drawn
representing the sides. As the circle is then
outside the polygon, the circle is said to be
described about it, while the polygon is in-
scribed within the circle. If the polygon is
described about the circle, its sides are tan-
gent to it.
PROB. XXXI. To describe a circle about a
triangle (Fig. 60).
Bisect two of the sides A B, A 0, of the tri-
angle at E, F ; from these points draw perpen-
diculars cutting at K. From the center K,
with K A as radius, describe the circle ABC,
as required.
PROB. XXXII. To inscribe a circle in a triangle (Fig. 61).
Bisect two of the angles A, 0, of the triangle A B C, by lines cutting at
D ; from D draw a perpendicular D E to any side, as A ; and with D E as
radius, from the center D, describe the circle required.
When the triangle is equilateral, the center of the circle is more easily found
by bisecting two of the sides, and drawing perpendiculars, as in the previous
problem. Or, draw a perpendicular from one of the angles to the opposite
side, and from the side set off one third of the length of the perpendicular.
FIG. 60.
FIG. 62.
PROB. XXXIII. To inscribe a square in a circle ; and to describe a circle
about a square (Fig. 62).
To inscribe the square. Draw two diameters, A B, D, at right angles,
and join the points A, B, 0, D, to form the square as required.
To describe the circle. Draw the diagonals A B, C D, of the given square,
cutting at E ; on E as a center, with E A as radius, describe the circle as
required.
In the same way, a circle may be described about a rectangle.
20
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
PROB. XXXIV. To inscribe a circle in a square ; and to describe a square
about a circle (Fig. 63).
To inscribe the circle. Draw the diagonals A B, C D, of the giver square,
cutting at E ; draw the perpendicular E F to one of the sides, and with the
radius E F, on the center E, describe the circle.
To describe the square. Draw two diameters A B, C D, at right angles,
and produce them ; bisect the angle D E B at the center by the diameter F G,
and through F and G draw perpendiculars A C, B D, and join the points A, D,
and B, C, where they cut the diagonals, to complete the square.
PROB. XXXV. To inscribe a pentagon in a circle (Fig. 64).
Draw two diameters, A C, B D, at right angles ; bisect A at E, and
from E, with radius E B, cut A C at F ; from B, with radius B F, cut the
F
FIG. 63.
B
FIG. 64.
FIG. 65.
circumference at G and H, and with the same radius step round the circle to I
and K ; join the points so found to form the pentagon.
PROB. XXXVI. To construct a regular hexagon upon a given straight
line (Fig. 65).
From A and B, with a radius equal to the given line, describe arcs cutting
at g; from g, with the radius g A, describe a circle ; with the same radius set
off from A the arcs A G, G F, and from B the arcs B D, D E. Join the
points so found to form the hexagon.
PROB. XXXVII. To inscribe a regular hexagon in a circle (Fig. 66).
J>
FIG. 66.
FIG. 67.
Draw a diameter, A B ; from A and B as centers, with the radius of the
circle A C, cut the circumference at D, E, F, G ; draw straight lines A D,
D E, etc., to form the hexagon.
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
21
The points of contact, D, E, etc., may also be found by setting off the
radius six times upon the circumference.
PROB. XXXVIII. To describe a regular hexagon about a circle (Fig. 67).
Draw a diameter, A B, of the given circle. With the radius A D from A
as a center, cut the circumference at C ; join A C, and bisect it with the
radius D E ; through E draw F G parallel to A C, cutting the diameter at F,
and with the radius D F describe the circle F H. Within this circle describe
a regular hexagon by the preceding problem ; the figure will touch the given
circle as required.
PROB. XXXIX. To construct a regular octagon upon a given straight line
(Fig. 68).
Produce the given line A B both ways, and draw perpendiculars A E, B F ;
bisect the external angles at A and B by the lines A H, B C, which make
A B
FIG. 68.
FIG. 69.
equal to A B ; draw C D and H G parallel to A E and equal to A B ; and
from the centers G, D, with the radius A B, cut the perpendiculars at E, F,
and draw E F to complete the octagon.
PROB. XL. To convert a square into a regular octagon (Fig. 69).
Draw the diagonals of the square intersecting at e; from the corners A, B,
C, D, with A e as radius, describe arcs cutting the sides at g h, etc. ; join the
points so found to complete the octagon.
PROB. XLI. To inscribe a regular octagon in a circle (Fig. 70).
FIG. 70.
FIG. 71.
Draw two diameters, A C, B D, at right angles ; bisect the arcs A B,
B C, etc., at e, /, etc.; and join A e, e B, etc., for the inscribed figure.
22
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
PKOB. XLII. To describe a regular octagon about a circle (Fig. 71).
Describe a square about the given circle A B ; draw perpendiculars h k,
etc., to the diagonals, touching the circle. The octagon so formed is the
figure required.
Or, to find the points h, k, etc., cut the sides from the corners of the
square, as in Prob. XL.
PKOB. XLIII. To inscribe a circle within a regular polygon.
When the polygon has an even number of sides, as in Fig. 72, bisect two
opposite sides at A and B, draw A B, and bisect it at C by a diagonal D E
drawn between opposite angles ; with the radius C A describe the circle as
required.
When the number of sides is odd, as in Fig. 73, bisect two of the sides
at A and B, and draw lines A E, B D, to the opposite angles, intersecting
at C ; from C, with C A as radius, describe the circle as required.
FIG. 72.
FIG. 73.
PROB. XLIV. To describe a circle without a regular polygon.
When the number of sides is even, draw two diagonals from opposite
angles, like D E (Fig. 72), to intersect at C ; and from C, with C D as radius,
describe the circle required.
When the number of sides is odd, find the center C (Fig. 73) as in last
problem, and, with C D as radius, describe the circle.
The foregoing selection of problems on regular figures is the most useful
in mechanical practice on that subject. Several other regular figures may be
constructed from them by bisection of the arcs of the circumscribing circles.
In this way a decagon, or ten-sided polygon, may be formed from the penta-
gon by the bisection of the arcs in Prob. XXXV, Fig. 64. Inversely, an
equilateral triangle may be inscribed by joining the alternate points of division
found for a hexagon.
The constructions for inscribing regular polygons in circles are suitable
also for dividing the circumference of a circle into a number of equal parts.
To supply a means of dividing the circumference into any number of parts,
including cases not provided for in the foregoing problems, the annexed table
of angles relating to polygons, expressed in degrees, will be found of general
utility. In this table, the angle at the center is found by dividing 360, the
number of degrees in a circle, by the number of sides in the polygon, and by
setting off round the center of the circle a succession of angles by means of
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
23
the protractor, equal to the angle in the table due to a given number of sides :
the radii so drawn will divide the circumference into the same number of
parts. The triangles thus formed are termed the elementary triangles of the
polygon.
TABLE OE POLYGONAL ANGLES.
Number of Sides of Kegu-
lar Polygon ; or number
Angle at
Number of Sides of
Angle at
of equal parts of the cir-
Center.
Kegular Polygon.
Center.
cumference.
No.
Degrees.
No.
Degrees.
3
120
12
30
4
90
13
27*
5
72
14
25f
6
60
15
24
7
51-f
16
22J
8
45
17
21*
9
40
18
20
10
36
19
18|f
11
32*
20
18
CONSTRUCTION OF THE ELLIPSE, PAEABOLA, HYPERBOLA, CYCLOID, EPICY-
CLOID, INVOLUTE, AND SPIRAL.
An ellipse is an oval-shaped curve (Fig. 74), in which, if from any point,
P, lines be drawn to two fixed points, F and F', foci, their sum will always be the
same. The line A B passing through
the foci is the transverse axis, and
the perpendicular C D at the cen-
ter of it is the conjugate axis.
PROB. XLV. To construct
an ellipse, the axes being known
(Fig. 75).
1st Method. Let the two axes
be the lines A B and C D. From
as a center, with a radius equal
to E B (half the transverse axis),
describe an arc cutting this axis
at two points, F and F', which are
the foci. Insert a pin in each of the foci, and loop a thread upon them, so
that, when stretched by a pencil inside the loop, the point of the pencil
will coincide with C. Move the pencil round, keeping the loop evenly
stretched, and it will describe an ellipse. This construction follows the
definition above given of an ellipse, that the sum of the distances of every
point of the curve from the foci is equal. It is seldom used by the draughts-
man, as it is difficult to keep a thread evenly stretched ; but for gardeners,
laying out beds or plots, it is very convenient and sufficiently accurate.
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
2d Method. Carpenters, almost invariably, lay out an ellipse by means
of a trammel (Fig. 76), which consists of a rectangular cross, E Gr F H,
with guiding grooves, in which
metal rods, attached to slides
on a bar, are fitted so as to
move easily and uniformly. In
describing the ellipse, the tram-
mel is placed with its grooves
on the lines of the axes. Ad-
just the metal points, Tc and Z,
which slide in the grooves, so
as to have between them a dis-
tance equal to half the conju-
gate axis, and make the dis-
tance from k to m (the position
on the bar of the pencil or marker) equal to half the transverse axis. Kevolve
the bar, keeping the points Tc and I always in the grooves, and the pencil will
describe an ellipse. Xeat
instruments of this sort are
made for the use of the
draughtsman, but, for of-
fices where curves of this
sort are required but little,
a substitute for the tram-
mel can be had in a strip of
paper (Fig. 77), by marking
the straight edge at a and b
and c, the distance c a being
made equal to half the trans-
verse axis, and the distance
c b to half the conjugate
FIG. 77.
CONSTRUCTION OF GEOMET
25
axis. Revolving the strip of paper, keeping b on the line of the transverse
axis, and c on the line of the conjugate axis, and dotting the positions of a at
short intervals, enough points of the curve will be determined through which
the ellipse may be drawn readily.
PEOB. XLVI. To describe an ellipse approximately, by means of cir-
cular arcs.
First, with arcs of two
radii (Fig. 78). Take the
difference of the transverse
and conjugate axes, and set
it off from the center to /
a and c, on A and C ;
draw a c, and set off half a c
to d; draw d i parallel to
a c, set off e equal to d,
join e i, and draw e m, d m,
parallels to d i, i e. On cen-
ter m, with radius m C, de-
scribe an arc through C, and
from center i describe an arc
through D ; on centers d y e,
also, describe arcs through A and B. The four arcs thus described form
approximately an ellipse. This method does not apply satisfactorily when the
conjugate axis is less than two thirds of the transverse axis.
Second, with arcs of three radii (Fig. 79). On the transverse axis A B,
draw the rectangle B G, equal in height to C, half the conjugate axis.
Extend C above and below the rectangle. Draw Gr D perpendicular to A 0,
intersecting C extended at D. Set off K equal to C, and on A K as a
diameter describe the semicircle A N K ; draw a radius parallel to 0,
26 CONSTRUCTION OF GEOMETRICAL PROBLEMS.
intersecting the semicircle at N and the line G E at P ; set off M equal to
P N, and on D as a center, with a radius D M, describe an arc ; from A and
B as centers, with a radius L, intersect this arc at a and b. The points
H, #, D, b, H', are the centers of the arcs required ; produce the lines a
H, D a. D b, b H', and the spaces inclosed determine the lengths of each arc.
This process works well for nearly all proportions of ellipses. It is em-
ployed in striking out vaults and stone bridges.
PROB. XLVII. To draw a tangent to an ellipse through a given point in
the curve (Fig. 80).
From the given point T draw straight lines to the foci F, F'; produce F T
beyond the curve to c, and bisect the exterior angle c T F' by the line T d.
This line T d is the tangent required.
PROB. XLVIII. To draw a tangent to an ellipse from a given point with-
out the curve (Fig. 81).
From the given point T as center, with a radius equal to its distance from
the nearest focus F, describe an arc ; from the other focus F', with the trans-
;** \K
verse axis as radius, cut the arc at K, L, and draw K F', L F', cutting the
curve at M, N ; then the lines T M, T N, are tangents to the curve.
The Parabola.
The parabola may be defined as an ellipse whose transverse axis is infinite 5
its characteristic is that every point in the curve is equally distant from the
directrix E N, and the focus F (Fig. 82).
PROB. XLIX. To construct a parabola when the focus and directrix are
given.
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
27
1st Method (Fig. 82). Through the focus F draw the axis A B perpendicu-
lar to the directrix E N, and bisect A F at e, then e is the vertex of the curve.
Through a series of points, C, D, E, on the directrix, draw parallels to A B ;
connect these points, C, D, E, with the focus F, and bisect by perpendiculars
the lines F C, F D, F E. The intersections of these perpendiculars with the par-
allels will give points, C', D', E', in the curve, through which trace the parabola.
2d Method (Fig. 83). Place a straight-edge to the directrix E N, and
apply to it a square LEG; fasten at G one end of a cord, equal in length
FIG. 82.
FIG. 83.
ri
to E G ; fix the other end to the focus F ; slide the square steadily along
the straight-edge, holding the cord taut against the edge of the square by a
pencil, D, and it will describe the curve.
PKOB. L. To construct a parabola when the vertex, the axis, and a point
of the curve are given (Fig. 84).
Let A be the vertex, A B be
the axis, and M the given point
of the curve.
Construct the rectangle A B-
M 0. Divide M into any num-
ber of equal parts, four, for in-
stance ; divide A C in like man-
ner ; draw A 1, A 2, A 3 ; through
1', 2', and 3', draw lines parallel
to the axis. The intersections I, II, and III, of these lines are points in the
required curve.
The Hyperbola.
An hyperbola is a curve from any point P, in which, if two straight lines
be drawn to two fixed points, F, F', the foci, their difference shall always be
the same.
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
PKOB. LI. To describe an hyperbola (Fig. 85).
From one of the foci F, with an assumed radius, describe an arc, and from
the other focus F', with another radius exceeding the former by the given
difference, describe two small arcs, cutting the first as at P and p. Let this
operation be repeated with two new radii, taking care that the second shall
exceed the first by the same difference as before, and two new points will be
determined ; and this determination of points in the curve may thus be con-
tinued till its track is obvious. By making use of the same radii, but trans-
posing, that is, describing with the greater about F, and the less about F', we
have another series of points equally belonging to the hyperbola, and answer-
ing the definition ; so that the hyperbola consists of two separate branches.
FIG. 85.
FIG. 86.
The curve may be described mechanically (Fig. 86). By fixing a ruler
to one focus F', so that it may be turned round on this point, connect the
other extremity of the ruler R to the other focus F by a cord shorter than the
whole length F 7 R of the ruler by the given difference ; then a pencil P keep-
ing this cord always stretched, and at
the same time pressing against the
edge of the ruler, will, as the ruler
revolves around F', describe an hy-
perbola, of which F F' are the foci,
and the differences of distances from
these points to every point in the
curve will be the same.
PEOB. LII. To draw a tangent to
any point of an hyperbola (Fig. 87).
Let P be the point. On F' P lay
off P p, equal to F P ; draw the line
F p ; from P let fall a perpendicular
Fio 87 on this line, P p, and it will be the
tangent required.
The three curves, the ellipse, the parabola, and the hyperbola, are called
conic sections, as they are formed by the intersections of a plane with the sur-
face of a cone. See CONSTRUCTION OF THE CONIC SECTIONS.
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
If the cone be cut through both its sides by a plane not parallel to the
base, the section is an ellipse ; if the intersecting plane be parallel to the side
of the cone, the section is a parabola ; if the plane have such a position that,
when produced, it meets the opposite cone, the section is an hyperbola. The
opposite cone is a reversed cone formed on the apex of the other by the con-
tinuation of its sides.
The Cycloid.
The cycloid is the curve described by a point in the circumference of a
circle rolling on a straight line.
PKOB. LIII. To describe a cycloid (Fig. 88).
Draw the straight line A B as the base ; describe the generating circle tan-
gent at the center of this line, and through the center draw the line E E
parallel to the base ; let fall a perpendicular from C upon the base ; divide
the semi-circumference into any number of equal parts, for instance, six ; lay
off on A B and E distances C I/ V 2'. . ., equal to the divisions of the
circumference ; draw the chords
D 1, D 2. . . ; from the points
1', 2', 3'. . .on the line C E,
with radii equal to the generat-
ing circle, describe arcs ; from
the points 1', 2', 3', 4', 5', on
the line B A, and with radii
equal successively to the chords
D 1, D 2, D 3, D 4, D 5, describe
arcs cutting the preceding, and
the intersections will be points
of the curve required.
2d Method (Fig. 89). Let
9' be the base-line, 4 9 the
half of the generating circle ;
divide the half circle into any
number of equal parts, say nine,
and draw the chord 1, 2, Fl0 ' 89 *
3, etc. ; lay off on the base 1', I' 2', 2' 3' , equal respectively to the
length of one of the divisions of the half circle 1 ; draw through the points
30
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
1', 2', 3' lines parallel to the chords 1,0 2, 3 ; the intersections
I, II, III of these lines are centers of the arcs a, al), I c , of which
the cycloid is composed.
The Epicycloid.
The epicycloid is formed by a point in the circumference of a circle revolv-
ing either externally or internally on the circumference of another circle as
PKOB. L1V. To describe an epicycloid.
Let us in the first place take the exterior curve. Divide the circumfer-
ence A B D (Fig. 90) into a series of equal parts 1, 2, 3 , beginning from
the point A ; set off in the same manner, upon the circle A M, A N, the divis-
ions 1', 2', 3' equal to the divisions of the circumference A B D. Then,
as the circle A B D rolls upon the circle A M A N, the points 1, 2, 3 will
coincide successively with the points 1', 2', 3'; and, drawing radii from the
point through the points 1', 2', 3', and also describing arcs of circles from
the center 0, through the points 1, 2, 3 , they will intersect each other
successively at the points c, d, e Take now the distance 1 to c, and set
it off on the same arc from the point of intersection i, of the radius A C ;
in like manner, set off the distance 2 to d, from b to A 2 , and the distance 3 to
e, to A 8 , and so on. Then the points A 1 , A 2 , A 3 , will be so many points in the
epicycloid ; and their frequency may be increased at pleasure by shortening
^^\.
CONSTRUCTION OF GEOMETRICAL PROBLEMS. 31
the divisions of the circular arcs. Thus the form of the curve may be deter-
mined to any amount of accuracy, and completed by tracing a line through
the points found.
As the distances 1 to c, which are near the commencement of the
curve, must be very short, it may, in some instances, be more convenient to
set off the whole distance i to 1 from c, and in the same way the distance b
to 2 from d to A 2 , and so on. In this manner the form of the curve is the
more likely to be accurately defined.
2d Method. To find the points in the curve, find the positions of the
center of the rolling circle corresponding to the points of contact 1', 2', 3',
etc., which may be readily done by producing the radii from the center 0,
through the points 1', 2', 3', to cut the circle B C. From these centers
describe arcs of a circle with the radius of C A, cutting the corresponding
arcs described from the center 0, and passing through the points 1,2, 3,
as before. The intersections of these arcs at A 1 , A 2 , A 3 , . . . .give points of the
curve.
When the moving circle A B D is made to roll on the interior of the cir-
cumference A M, A N, as shown (Fig. 91), the curve described by the point
x
\
FIG. 91.
A is called an interior epicycloid. It may be constructed in the same way as
in the preceding case, as may be easily understood, the same figures and letters
of reference being used in both figures.
The Involute.
The involute is a curve traced by the extremity of a flexible line unwind-
ing from the circumference of a circle.
32
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
PROB. LV. To describe an involute.
Divide the circumference of the given circle (Fig. 92) into any number of
equal parts, as 0, 1, 2, 3, 4, ; at each of these points draw tangents to the
FIG. 92.
given circle ; on the first of these lay off the distance 11', equal to the arc
1 ; on the second lay off 2 2', equal to twice the arc 1 or the arc 2 :
establish in a similar way the points 3', 4', 5', as far as may be requisite,
which are points in the curve required.
It may be remarked that, in all the problems in which curves have been
determined by the position of points, the more numerous the points thus
fixed, the more accurately can the
curve be drawn.
The involute curve may be
described mechanically in several
ways. Thus, let A (Fig. 93) be
the center of a wheel for which
the form of involute teeth is to
be found. Let m n a be a thread
lapped round its circumference,
having a loop-hole at its extrem-
ity, a; in this fix a pin, with
which describe the curve or in-
volute a b h, by unwinding
the thread gradually from the circumference, and this curve will be the proper
form for the teeth of a wheel of the given diameter.
The Spiral
The spiral is the involute of a circle produced beyond a single revolution.
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
33
PROB. LVL To describe a spiral (Fig. 94, and Fig. 95 of the primary on
a larger scale).
Divide the circumference of the primary into any number of equal parts,
say not less than eight. To these points of division o, e, f, i, etc., draw tangents,
and from these points draw a succession of circular arcs ; thus, from o e lay
FIG. 94.
FIG. 95.
off o g, equal to the arc a o reduced to a straight line, and connect a and g
by a curve ; from e, with the radius e g, describe the arc g h ; from / the next
arc, and so on. Continue the use of the centers successively and repeatedly
to the extent of the revolutions required. Thus the point a in the figure is
used as a center for three arcs, b I, c m, d n.
USE or TKIAKGLE A:NT> SQTJABE.
Right-angled triangles constructed of wood, hard rubber, or metal, are very
useful in connection with a straight-edge, or ruler, in drawing lines parallel
or perpendicular to other lines.
To draw lines parallel to each other, place any edge of the triangle in close
contact with the edge of the ruler. Hold the ruler (Fig. 96) firmly with the
3
34 CONSTRUCTION OF GEOMETRICAL PROBLEMS.
thumb and little finger of the left hand, and the triangle with the other three
fingers ; with a pencil or pen in the right hand, draw a line along one of the
free edges of the triangle ; withdraw the pressure of the three fingers upon the
FIG. 96.
triangle, and slide it along the edge of the ruler, keeping the edges in close
contact ; a line drawn along the same edge of the triangle, as before, will be
parallel to the first line. If the edge of the hypothenuse of the triangle be
placed in contact with the ruler, lines drawn along one edge of the triangle
will be at right angles to those drawn along the other.
FIG. 97.
PROB. LVII. Through a given point to draw a line parallel to a given
line (Fig. 97).
Place one of the shorter edges of the triangle along the given line A B, and
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
35
bring the ruler against the hypothenuse ; slide the triangle up along the edge
of the ruler until the upper edge of the ruler is sufficiently near to the given
2k
'V
4
-\
E D
r^
FIG. 98.
FIG. 99.
point C to allow a line to be drawn through it. Draw the line, and it will be
parallel to A B.
If the triangle be slid farther up along the edge of the ruler, and a line be
drawn through C along the other edge of the triangle (Fig. 98), this line will
be perpendicular to A B. If the triangle be slid still farther up along the
edge of the ruler, and a third line be drawn touching A B, the figure con-
structed will be a rectangle ; and if E D
be laid off on A B, equal to C E, the fig-
ure inclosed is a square (Fig. 99).
It will be seen that the triangle and
ruler afford a much readier way of draw-
ing parallel lines, and lines at right an-
gles, than the compasses and ruler, and
may be used in solving the following
problems :
The area of a figure is the quantity
of space inclosed by its lines.
Construct a right angle (Fig. 100). Divide the base and the perpendicular
by dividers into any number of equal spaces ; for instance, eight on the one
and five on the other. Construct a rectangle with this base and perpendicu-
lar, and through the points of division lay off lines parallel to the base and
perpendicular. The rectangle will be divided into forty equal squares, and
its measure in squares will be the divisions eight in the base, multiplied by
the five in the perpendicular. If the division were inches, then the area of
FIG. 100.
FIG. 101.
B
FIG. 102.
this rectangle would be forty square inches ; if feet, then forty square feet.
If there were but five divisions in the base and five in the perpendicular,
the surface would be twenty-five squares. Therefore, a rectangle has for its
measure the base multiplied by its adjacent side or height.
36 CONSTRUCTION OF GEOMETRICAL PROBLEMS.
Draw a diagonal, and the rectangle is divided into two equal triangles.
Each triangle must therefore have for its measure the base multiplied by half
the perpendicular, or, as is usually said, by half the altitude.
Take any triangle (Fig. 101), and from its apex draw a line perpendicular
to the base. The triangle is divided into two right-angled triangles, which
must have for their measure A D x C D, and D B x ^ C D, and the sum of
the two must be A B x C D.
If the perpendicular from the apex falls outside the triangle (Fig. 102),
then the triangles B D C and ADC will have for their measure B D x J C D
and A D x C D ; and as the origi-
nal triangle A B C is the difference
of these two triangles, its measure must
be A B x C D. Every triangle must
have for its measure the base multi-
plied by half the altitude, and it makes
no difference which side is taken as the
base.
Construct the right-angled triangle
A C B (Fig. 103), and let fall the FlG - 103 -
perpendicular C D. As will be seen by the equality of the angles compos-
ing the triangles, the perpendicular divides the original triangle into two right-
angled triangles, similar to each other and to the original triangle. Therefore
FIG. 104.
A D is to C D as C D is to B D, or, expressed by signs, A D : C D : : C D :
B D ; therefore, by the Rule of Three, A D x B D = C D 2 ; that is, C D is a
mean proportional between A D and B D. So that the perpendicular let fall
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
from the vertex of a right angle upon the hypothenuse of the triangle, is a
mean proportional between the two parts of the hypothenuse into which it is
divided by the perpendicular.
In comparing the two triangles with the original triangle, A C is a mean
proportional between A D and A B, and B C is a mean proportional between
B D and A B ; that is, A C 2 =A Dx A B
BC 2 =BDxAB
adding the two, A C a +B C 2 = (A D+B D)xA B
and as A D + B D = A B, we have A C a + B C 2 = A B 2 ; that is, the square
on the hypothenuse is equal to the sum of the squares on the other two
sides.
Construct squares on the three sides of a right-angled triangle (Fig. 104).
b
FIG. 105.
FlCr. 106.
PROB. LVIII. To construct a square equal to one half of a given square
(Fig. 105).
FIG. 107.
FIG. 108.
Construct the given square, and draw diagonals in it. The square, abed,
constructed on one half of one of these diagonals will be equal to one half the
given square.
38
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
PKOB. LIX. To construct a square equal to double a given square
(Fig. 106).
Construct a square on one of the diagonals in the given square, or en-
close the square with parallels to the diagonals of the square.
PEOB. LX. To construct a square equal to three times a given square
(Fig. 107).
Extend the base of the given square, and lay off on it the length of its
diagonal. Draw a line from the point at which this diagonal ends to the ex-
treme angle of the square, and upon this line erect a square, which will be the
square required.
For a square four times the size of a given square, make the base double
that of the given square.
PKOB. LXI. To construct a square equal to five times a given square
(Fig. 108).
Extend the base of the given square, making the extension to d equal to
the base of the given square. From d draw a line to a, and on this line con-
struct a square, abed, which will be the square required.
FIG. 109.
Assuming the side of the given square in Figs. 105, 106, 107, and 108 to
be the radius (or diameter) (Fig. 109) of a given circle, then the side of the
square to be constructed half, twice, three, four, or five times the size of the
given square will be the radii (or diameters) of the circles half, twice, three, four,
or five times the size of the given circle.
CONSTRUCTION OF GEOMETRICAL PROBLEMS.
39
PKOB. LXII. To determine how much is added to a given square by
extending its base and constructing a square thereon (Fig. 110).
p
c
FIG. 110.
H
K
J
Let a represent the length C D of the base of the given square. Its square
will be a X a or a? .
Extend the base C D by a certain length, D G, represented by I. Then
the new square (a + b) x (a + b) will be made up of the old square, or a 2
and two rectangles, D G E H and P E K L, or 2 (a x b) or 2 a b
and one square, E H K J, or b x b or b 2
PROB. LXIII. To determine how much is taken from the area of a given
square, by reducing its base and constructing a square (Fig. 110).
Let a represent the length C G of the base of the given square. Reduce
C G by a certain length, G D, to be represented by b.
Then the new square (a b)* will be the old square, or a 9
diminished by two rectangles, D G J K and P L J H, or 2 a b
excepting one square, E H J K, or b x b or 4- b*
The last two constructions, in default of a table of squares, may often be
found of use.
DRAWING INSTRUMENTS.
THE simple drawing instruments, already illustrated and applied in the
construction of the preceding problems, together with scales of equal parts,
a protractor and a drawing pen, are all the instruments essential for topo-
graphical or mechanical drawing. It is often convenient, for facility in work-
ing, to have compasses of varied sizes and modifications, and these, together
with an assortment of rulers, triangles, squares, scales, and protractors,
adapted to varied work, are included in boxes of drawing instruments as
furnished by dealers. The smaller rulers and triangles, as furnished, are
generally of hard rubber, and the larger of wood. As it is often incon-
venient to carry long rulers, and difficult to procure them ready-made, the
draughtsman may have to depend on a carpenter or joiner for them.
Eulers should be of close-grained, thoroughly - seasoned wood, such as
mahogany, maple, pear, etc. They should be about -J of an inch thick in
the square or slightly rounded edges, 1 to 2% inches wide, according to their
length. As the accuracy of a drawing depends greatly on the straightness of
the lines, the edge of the ruler should be perfectly straight. To test this,
place a sheet of paper on a perfectly smooth board ; insert two very fine
needles in an upright position through the paper into the board, distant from
each other nearly the length of the ruler to be tested ; bring the edge of the
ruler against these needles, and draw a line from one needle to the other ;
reverse the ruler, bringing the same edge on the opposite side and against
the needles, and again draw a line. If the two lines coincide, the edge is
straight ; but, if they disagree, the ruler is inaccurate, and must be re-jointed.
When one ruler has been tested, the other can be examined by placing their
edges against the correct one, and holding them between the eye and the
light.
Triangles may be made of the same kinds of wood as the ruler, and some-
what thinner, and of various sizes. They should be right-angled, with acute
angles of 45, or of 60 and 30. The most convenient size for general use
measures from 3 to 6 inches on the side. A larger size, from 8 to 10 inches
long on the side, is convenient for making drawings to a large scale. Circular
openings are made in the body of the triangle for the insertion of the end of
the finger to give facility in sliding the triangle on the paper. Triangles are
sometioies made as large as 15 to 18 inches on the side ; but in this case they
are framed in three pieces of about 1J wide, leaving the center of the triangle
open. The value of the triangle in drawing perpendicular lines depends on
the accuracy of the right angle. To test this (Fig. Ill), draw a line with an
DRAWING INSTR
41
accurate ruler on paper. Place the right angle of the triangle near the center
of this line, and make one of the adjacent sides to coincide with the line ; now
draw a line along the other adjacent side, which, if the angle is strictly a
right angle, will be perpendicular to the first line. Turn the triangle on this
perpendicular side, bringing it into the posi-
tion ABC'; if now the sides of the triangle
agree with the line B C' and A B, the angle
is a right angle, and the sides straight. If
they do not agree, they must be made to do
so with a plane, if right angles are to be
drawn by the triangle. The straightness
of the hypothenuse or longest side can be
tested like a common ruler.
The T square is a thin " straight edge " or ruler, a (Fig. 112), fitted at one
end with a stock, b, applied transversely at right angles. The stock being so
formed as to fit and slide against one edge of the drawing-board, the blade
reaches over the surface, and presents an edge of its own at right angles to
FIG. 111.
FIG. 112.
that of the board, by which parallel straight lines may be drawn upon the
paper. The stock should be long enough to give sufficient bearing on the
edge of the board, and heavy enough to act as a balance to the blade, and to
relieve the operation of handling the square. The blade should be sunk flush
into the upper half of the stock on the inside, and very exactly fitted. It
should be inserted full breadth, as shown in the figure ; notching and dove-
tailing is a mistake, as it weakens the blade, and adds nothing to the secu-
rity. The upper half of the stock should be about \ inch broader than the
lower half, to rest firmly on the board and secure the blade lying flatly on the
paper.
One half of the stock, c (Fig. 113), is in some cases made loose, to tarn
FIG. 113.
upon a brass swivel to any angle with the blade a, and to be clenched by a
screwed nut and washer. The loose stock is useful for drawing parallel lines
42 DRAWING INSTRUMENTS.
obliquely to the edges of the board, such as the threads of screws, oblique-
columns, and connecting-roads of steam-engines.
In many drawing-cases will be found the parallel ruler (Fig. 114), consist-
ing of two rulers connected by two bars moving on pivots, and so adjusted
that the rulers, as they open,
form the sides of a parallelo-
e^ G^ | gram. The edge of one of
the rulers being retained in
a position coinciding with, or
parallel to, a given line, the
\ \> I other ruler may be moved,
and lines drawn along its
edge must also be parallel to
the given line. This instrument is only useful in drawing small parallels, and
in accuracy and convenience does not compare with the triangle and ruler, or
T square.
An improvement on the above parallel ruler has been patented by Lieuten-
ant-Commander Sigsbee, U. S. N. (Fig. 115), in which the blades are* made
FIG. 115.
with hinges, by which, holding one blade on the paper, the other may be raised
over creases or torn edges of the paper, or over thumb-tacks. One blade can
be raised, if necessary, at right angles to the other, still preserving the parallel-
ism of lines that may be drawn along these edges. Small cushions of rubber
inserted in the blades, pressed by the fingers, prevent the slipping of the
blades.
FIG. 116.
SWEEPS AND VARIABLE CURVES.
For drawing arcs of a large radius, beyond the range of ordinary com-
passes, and lines not circular but varying in curvature, thin slips of wood,
DRAWING INSTRUMENTS.
FIG. 117.
termed sweeps (Figs. 116 and 117), are usually employed. These two forms
are of very general application, but others of
almost every form, and made of hard rubber,
can be purchased. Whatever be the nature of
the curve, some portion of the sweep will be
found to coincide with its commencement, and
it can be continued throughout its extent by
applying, successively, such parts of the sweep
as are suitable, care being taken that the parts are tangent to each other,
and that the continuity is not injured by unskillful junction.
No varnish of any description should be applied to any of the wooden
instruments used in drawing, as the best varnish will retain dust, and soil the
paper. Use the wood in its natural state, keeping it care-
fully wiped. Various other materials besides wood have
been used, as steel for the blades of the T square and the
ruler ; the objection is the liability to soil the paper. Glass
is frequently used for the ruler and the triangle, and retains
its correctness of edge and angle, but it is too heavy, and
liable, of course, to fracture.
Thin splines are also to be had, which, held in position
by leaden weights, serve admirably for a guide to the pen in
describing curves (Fig. 118). For the same purpose a thin,
hard rubber ruler, with soft rubber backing, answers well,
and, as it can be readily rolled up, is extremely portable.
The weights above shown are very convenient in holding
the drawing-paper on the board, but the drawing-pins (Fig.
119), steel points, or tacks, with large, flat heads, are in
general use.
Elliptic and parabolic curves are furnished in sets, but
the draughtsman can readily make a model out of thick
card-board, with which he can draw a very uniform curve.
For the drawing of ellipses, very neat trammels or com-
passes, with elliptic guides or patterns, may be purchased.
The drawing-pen (Fig. 120) is used for drawing straight lines. It consists
of two blades with steel points fixed to a handle ; and they are so bent that a
sufficient cavity is left between them for the ink, when the ends
of the steel points meet close together, or nearly so. The blades
are set with the points more or less open by means of a mill-
headed screw, so as to draw lines of any required fineness or
thickness. One of the blades is framed with a joint, so that by
taking out the screw the blades may be completely opened, and
the points effectively cleaned after use. The ink is to be put
between the blades by a common pen, and in using the pen it should be
slightly inclined in the direction of the line to be drawn, and care should be
taken that both points touch the paper ; and these observations equally apply
to the pen-points of the compasses before described. The drawing-pen should
be kept close to the ruler or straight edge, and in the same direction during
FIG. 118.
44
DRAWING INSTRUMENTS.
the whole operation of drawing the line. Care must be taken in holding the
straight edge firmly with the left hand, that it does not change its position.
For drawing close parallel lines in mechanical and
architectural drawings, or to represent canals or roads, a
double pen (Fig. 121) is frequently used, with an adjust-
ing screw to set the pens to any required small distance.
This is usually called the road-pen.
Border-pens, for drawing broad lines, are double pens
with an intermediate blade, and are applicable to the
drawing of map-borders. The same work may be done
by drawing the outer lines with the common drawing-pen,
and filling in with a goose-quill, cut as shown in Fig. 122.
In drawing with this pen, incline the drawing-board so
that the ink will follow the pen.
The curve-pen (Fig. 123) is especially designed for the
ready drawing of curved lines.
The dotting-point (Fig. 124) resembles a drawing-pen,
except that the points are not so sharp. On the back
blade, as seen in the engraving, is a pivot, on which may
be placed a dotting-wheel, , resembling the rowel of a
spur ; the screw ~b is for opening the blades to remove the
wheel for cleaning after use, or replacing it with one of
another character of dot. The cap c, at the upper end of
the instrument, is a box containing a variety of dotting-
wheels, each producing a different-shaped dot. These are
used as distinguishing marks for different classes of bound-
aries on maps ; for instance, one kind of dot distinguishes
county boundaries, another kind town boundaries, a third
kind distinguishes that which is both a county and a town boundary, etc., etc.
In using this instrument, the ink must be inserted between the blades above
FIG. 120. FIG. 121.
FIG. 122.
the dotting-wheel, so that, as the wheel revolves, the points shall pass through
the ink, each carrying with it a drop, and marking the paper as it passes.
It sometimes happens that the
wheel will revolve many times
before it begins to deposit its
ink on the drawing, thereby
leaving the first part of the line
altogether blank, and, in attempting to go over it again, the first-made dots
are liable to get blotted. This evil may be mostly remedied by placing a piece
of blank paper over the drawing to the very point the dotted line is to com-
FIG. 123.
DRAWING INSTRUMENTS.
mence at, then begin with drawing the wheel over the blank paper first, so
that, by the time it will have arrived at the proper point of commencement,
the ink may be expected to flow over the points of the wheel, and make the
dotted line perfect as required.
The best pricking-point (Fig. 125) is a fine needle held in a pair of for-
ceps, and is used to transfer drawings by pricking through at the points of a
drawing into the paper placed beneath. When drawings are transferred by
I
FIG. 124.
FIG. 125.
FIG. 126.
tracing a prepared black sheet being placed between the drawing and the
paper to receive the tracing the eye-end of the needle forms a good tracing-
point.
Compasses, in addition to pencil-points, as before shown, are fitted with
movable ink-points and lengthening bars, so that larger circles may be drawn.
Compasses should have joints in the legs, so that the points, pencil, and pen
may be set perpendicular to the planes in which the
circles are described (Fig. 126). Compasses of this
general form may be had in sizes of 3 to 7 inches.
For the measurement and laying off of small spaces,
and the describing of small circles, there are small bow-
compasses (Fig. 127). These are sometimes made with
jointed legs.
For the measurement or laying off of distances the
plain dividers are convenient, but for ready and close
adjustment the hair dividers (Fig. 128) are most suit-
able. The only difference is that, in the hair dividers,
FIG. 127.
DRAWING INSTRUMENTS.
one of the points is attached to the body by a spring, and by means of the
screw b it can be moved toward or from the fixed point a very small amount
more accurately than by closing or opening the dividers. In dividing a line
into equal parts especially, it enables one to divide the excess or
deficit readily.
Large screw dividers (Fig. 129) are used for the same purpose,
but they belong rather to the mechanic than to the draughtsman.
For convenience of carrying in the pocket, there are portable
or turn-in compasses (Fig. 130).
FIG. 128.
FIG. 129.
For setting off very long lines, or describing circles of large radius, learn
compasses are used (Fig. 131). These consist of a mere slip of wood, A
FIG. 130.
which is readily procured ; two brass boxes, B and 0, which can easily be
attached to the beam, and connected with the brass boxes are the two points
of the instrument, G and H. The object of this instrument is the nice adjust-
ment of the points G and H at any definite distance apart ; at F is a slow-
motion screw, by which the joint G may be moved any very minute quantity
after the distance from F to G has been adjusted as nicely as possible by the
hand alone. The important parts of this instrument can be carried in a very
small compass.
There are beam compasses in which the beam is graduated, and in which
the boxes corresponding to B and 0, in Fig. 131, are fitted with vernier or
reading plates, to afford the means of minutely subdividing the divisions on
the beam.
DRAWING INSTRUMENTS.
47
Proportional dividers (Fig. 132), for copying and reducing drawings, are
found in most cases of instruments.
Closing the dividers and loosening the screw 0, the slide may be moved up
in the groove until the mark on the slide or index corresponds with the
required number; then clamping the screw, the space inclosed
between the long points, A B, will be as many times that between
the short points, E D, as is shown by the number opposite the in-
dex. If the lines are to be reduced, the distances are measured
with the long points, and set off by the short ones ; if the lines
.are to be enlarged, then vice versa.
It often happens that the length of the points becomes re-
FKI. 131.
FIG. 132.
duced by use or accident, and the division on the instrument then becomes
useless, but the purpose may be served by trial on paper, moving the slide up
or down until a measured line is reduced or enlarged, as required*
SCALES.
Practically, a two-foot rule, with its division into inches, half inch, quarter
inch, eighth inch, and sixteenth inch, may be made use of as a scale of equal
parts, the inch or any of its parts being taken as the unit to represent a foot,
a yard, or a mile ; but among drawing instruments, scales especially adapted
to the purpose are found in great varieties of form, division, and material.
Fig. 133 represents the usual scale to be found in the common boxes of
drawing instruments. It contains, on its two sides, simply divided scales a
diagonal scale on the reverse side and a protractor along the edges. The
simply divided scales consist of a series of equal divisions of an inch, which
are numbered 1, 2, 3, etc., beginning from the second division on the left
hand ; the upper part of the left division in each is subdivided into 12 equal
parts, and the lower part into 10 equal parts. In Fig. 134 the scales are
marked 30, 35, 40, etc., and the subdivisions of tenths can be considered as
units, one mile, or one chain, or one foot, then each primary division will
48
DRAWING INSTRUMENTS.
represent ten units, ten miles, ten chains, or ten feet, and the scale is said to
be 30, 35, 40 (according to the scale selected) miles, chains, or feet to the
inch. Thus, suppose that it were required,
on a scale of 30 feet to the inch, to lay off
47 feet. On the scale marked 30, place one
point of the compasses or dividers at 4, and
bring the other point to the seventh lower
subdivisions, counting from the right, and
we have the distance required. Each of the
primary divisions may be regarded as unit,
one foot for instance ; then the upper sub-
divisions are twelfths of a foot or inches, and
the lower subdivisions tenths of an inch.
In Fig. 133 the scales are marked at the
left, 1 inch, f , , ; the primary divisions
are 1 inch, f, -J, and i of an inch. These
scales are more generally used for drawings of
machinery and of architecture, while those of
Fig. 134 are for topographical drawings. The
applications of these scales are similar to those
already described. When the primary divis-
ions are considered inches, then the drawings
will be each full, f, -J, or \ size, according to
the scale adopted.
On the selection of the scale. In all work-
ing architectural and mechanical drawings,
use as large a scale as possible ; neither de-
pend, even in that case, that the mechanics
employed in the construction will measure
correctly, but write in the dimensions as far
as practicable. For architectural plans, the
scale of J- an inch to the foot is one of very
general use, and convenient for the mechanic,
as the common two-foot rule carried by all
mechanics is subdivided into ^ths, ^ths, and
sometimes sixteenths of an inch, and the dis-
tances on a drawing to this scale can therefore
be easily measured by them. This fact should
not be lost sight of in working drawings.
When the dimensions are not written, make
use of such scales that the distances may be
measured by the subdivisions of the common
two-foot rule ; thus, in a scale of i or ^ full size, 6 inches or 3 inches rep-
resent one foot ; in a scale of an inch to the foot or twelfth full size, each
i an inch represents 6 inches, i of an inch, 3 inches ; but when or T V an
inch to the foot, or any similar scale, is adopted, it is evident that these
divisions can not be taken by the two-foot rule. The scale should be writ-
FIG. 133.
DRAWING INSTRUMENTS.
49
ten on every drawing, or the scale itself should be drawn on the margin.
In topographical and geodesic drawings the latter is essential, as the scale
adopted frequently has to be drawn for the specific purpose, and the paper
^
t
t
[
8
Jo
i : i,
I
j.
i
50
;ip
i
|
i
i
I
L
-? -, L
-1 -
4
;
*
it 5
-i^p-
i
2
i
s 1 ,1
I
,
it
frfl
"p
1,
[
IG
Is 1
l\0
[
35
-^p
5
2
j
L
1
L
30
* 1 1 1
^
FIG. 134.
itself contracts or expands with every atmospheric change, and the measure-
ments will therefore not agree at all times with a detached scale ; and, more-
over, a drawing laid down from such a detached scale, of wood or ivory, will
not be uniform throughout, for on a damp day the measurements will be too
short, and on a dry day too long. Mr. Holtzapffel has sought to remedy this
inconvenience by the introduction of paper scales ; but all kinds of paper do
not contract and expand equally, and the error is therefore only partially cor-
rected by his ingenious substitution of one material for another.
tn 1 '
1 2
1 3
4
\ 6
1 6
7
| 96 3 |
X
n
yi
Si 8
tj e
9j i
S
t
01 I
1 Sit 8;l
fr I S
m
III
1 \ 1
1 1 1
1 1 1
1 1 1
1
FIG. 135.
Plotting scales (Fig. 135) are scales of equal parts, with the divisions on a
fiducial edge, by which any length may be marked off on the paper without
using dividers. There are also small offset scales, for use of which see " Topo-
graphical Drawing."
Sometimes these scales are made with edges chamfered on both sides, and
graduated to four different scales. Sometimes the section of the scale is tri-
angular (Fig. 136), with six scales on the different edges. Both of these scales
are convenient as portable instruments. To avoid the objection that having
A
FIG. 136.
many scales on one ruler leads the draughtsman into error by the confusion of
the scales, the triangular has a small slip of metal, A, readily put on, which
covers partially the scales not in use.
50
DRAWING INSTRUMENTS.
To divide a given line into any number of equal parts (Fig. 137).
Let A B be the line, and the number of parts be ten. Draw a perpendicu-
lar at one extremity, A, of the line ; with a plotting scale place the zero at
the other extremity, B, of the line ; make the mark
10 on the scale coincide with the perpendicular ;
draw a line along the edge of the scale, and mark
the line at each division of the scale 1 to 9 ; draw
perpendiculars through these marks to the line A B,
and they will divide A B into ten equal parts.
The construction is based on the principle of
the proportions of parts between similar triangles,
and it is evident that if the perpendicular at 1 be
taken as a unit, that at 2 will be two units, and so on. This way of dividing
a line will often be found convenient in practice. The lines may be at any
angle to each other, and the lines connecting the divisions must be parallel to
the line completing the triangle. The above figure illustrates the construction
of diagonal scales. The simply divided scales give only two denominations,
primaries and tenths, or twelfths ; but more minute subdivision is attained by
the diagonal scale, which consists of a number of primary divisions, one of
which is divided into tenths, and subdivided into hundredths by diagonal
lines (Fig. 138). This scale is constructed in the following manner : Eleven
FIG. 137.
Fm. 138.
parallel lines are ruled, inclosing ten equal spaces ; the length is set off into
equal primary divisions, as D E, E 1, etc. ; the first D E is subdivided, and
diagonals are then drawn from the subdivisions between A and B, to those
FIG. 139.
between D and E, as shown in the diagram. Hence it is evident that at every
parallel we get an additional tenth of the subdivisions, or a hundredth of the
DRAWING INSTRUMENTS;
stlfli
51
primaries, and can therefore obtain a measurement with great exactness to
three places of figures. To take a measurement of (say) 168, we place one foot
of the dividers on the primary 1, and carry it down to the ninth parallel, and
then extend the other foot to the intersection of the diagonal, which falls
from the subdivision 6, with the parallel that measures the eight-hundredth
part (Fig. 139). The primaries may, of course, be considered as yards, feet, or
inches ; and the subdivisions as tenths and hundredths of these respective
denominations.
The diagonals may be applied to a scale where only one subdivision is
required. Thus, if seven lines be (Fig. 140) ruled, inclosing six equal spaces,
7/V
s / V
9/ -V-
7 \2
/ \1
1 \
0/2
FIG. 140.
and the length be divided into primaries, as A B, B 0, etc., the first primary,
A B, may be subdivided into twelfths by two diagonals running from 6, the
middle of A B, to 12 and 0. We have here a very convenient scale of feet and
inches. From C to 6 is 1 foot 6 inches ; and from C on the several parallels
to the various intersections of the diagonals we obtain 1 foot and any number
of inches from 1 to 12.
Vernier scales are preferred by some to the diagonal scale already de-
scribed. To construct a vernier scale (Fig. 141) by which a number to three
places may be taken, divide all the primary divisions into tenths, and number
10
2 4
6 8
f
f l f ||
I I I I
I I I I
I I I I I I I I I
I I I I I I i I I I I I I I I I I I I I
._ I
I I I I I I
100 fr a 6 J 4 2
FIG. 141.
these subdivisions 1, 2, 3, from left to right. Take off now with the com-
passes eleven of these subdivisions, set the extent off backward from the end
of the first primary division, and it will reach beyond the beginning of this
division, or zero point, a distance equal to one of the subdivisions. Now
divide the extent thus set off into ten equal parts, marking the divisions on
the opposite side of the divided line to the lines marking the primary divisions
and the subdivisions, and number them 1, 2, 3, etc., backward from right to
left. Then, since the extent of eleven subdivisions has been divided into ten
equal parts, so that these ten parts exceed by one subdivision the extent of ten
subdivisions, each one of these equal parts, or, as it may be called, one division
of the vernier scale, exceeds one of the subdivisions by a tenth part of a sub-
division, or a hundredth part of a primary division ; thus, if the subdivision
be considered 10, then from to the first division of the vernier will be 11 ; to
the second, 22 ; to the third, 33 ; to the fourth, 44 ; to the fifth, 55, and so
on, 66, 77, 88, 99.
52
DRAWING INSTRUMENTS.
To take off the number 253 from this scale, place one point of the dividers
at the third division of the vernier ; if the other point be brought to the pri-
mary division 2, the distance embraced by the dividers will be 233, and the
dividers must be extended to the second subdivision
of tenths to the right of 2. If the number were 213,
then the dividers would have to be closed to the sec-
ond subdivision of tenths to the left of 2. To take
off the number 59 from the scale, place one point of
the dividers at the ninth division of the vernier ; if
the other point be extended to the mark, the di-
viders will embrace 99, and must therefore be closed
to the fourth subdivision to the left of 0.
These numbers, thus taken, may be 253, 25 '3,
2-53 ; 213, 21 -3, 2'13 ; 59, 5 -9, .59, according as the
primary divisions are taken as hundreds, tens, or
units.
The construction of this scale is similar to that
of the verniers of theodolites and surveying instru-
ments ; but, in its application to drawing, is not as
simple as the diagonal scales (Figs. 138, 140).
The sector (Fig. 142), now seldom used, consists
of two flat rulers united by a central joint, and open-
ing like a pair of compasses. It carries several plain
scales on its faces, but its most important lines are
in the pairs or double scales, running accurately to
the central joint.
The principle on which the double scales are con-
structed is that similar triangles have their like sides
proportional (Fig. 143). Let the
^ ^C lines A B, AC, represent the legs
of the sector, and A D, A E, two
equal sections from the center ;
then, if the points B and D E
be connected, the lines B C and
D E will be parallel ; therefore,
the triangles A B C, A D E, will
be similar, and, consequently, the
sides A B, B C, A D, D E, propor-
tional that is, as A B : B C : :
A D : D E ; so that if A D be the
half, third, or fourth part of A B,
then D E will be a half, third, or
fourth part of B C ; and the same
holds of all the rest. Hence, if
D E be the chord, sine, or tangent
of any arc, or of any number of degrees to the radius A D, then B C will be
the same to the radius A B. Thus, at every opening of the sector, the trans-
DRAWING INSTRUMENTS.
53
verse distances D E and C B from one ruler to another are proportional to
the lateral distances, measured on the lines A B, A C. It is to be observed
that all measures are to be taken from the inner lines, since these only run
accurately to the center.
On the scale in common boxes of drawing instruments, the edge of one
side is divided as a protractor, for the laying out of angles, whose use
will be readily understood from the description of the instrument, when by
itself. It consists of a semicircle of thin metal or horn (Fig. 144), whose cir-
cumference is divided into 180 equal parts or degrees (180). In the larger
protractors each of these divisions is subdivided.
Application of the protractor (Fig. 144). To lay off a given angle from a
given point on a straight line, let the straight line a b of the protractor coin-
cide with the given line, and the point c with the given point ; now mark on
the paper against the division on the periphery coinciding with the angle
required ; remove the protractor, and draw a line through the given point and
the mark.
For plotting field-notes expeditiously, drawing paper can be obtained with
large, full circular protractors printed thereon, on which the courses can be
readily marked, and thus transferred to the part of the paper required by a
parallel ruler, or by triangle and ruler. These sheets are of especial use in
plotting at night the day's work, as, on account of the large size of protractor,
angles can be laid off with greater accuracy than by the usual protractor of
a drawing-instrument case, with less confusion of courses, and more expe-
ditiously.
For accurate plotting of angles, the circular protractor (Fig. 145) is one of
the best. It is a complete circle, A A, connected with its center by four radii,
a a a a. The center is left open, and surrounded by a concentric ring or collar,
&, which carries two radial bars, c c. To the extremity of one bar is a pinion,
d, working in a toothed rack quite round the outer circumference of the pro-
tractor. To the opposite extremity of the other bar, c, is fixed a vernier,
which subdivides the primary divisions on the protractor to single minutes,
54 DRAWING INSTRUMENTS.
and by estimation to 30 seconds. This vernier is carried round the pro-
tractor by turning the pinion d. Upon each radial bar, c c, is placed a branch,
ee, carrying at their extremities a fine steel pricker, whose points are kept
above the surface of the paper by springs placed under their supports, which
give way when the branches are pressed downward, and allow the points to
FIG. 145.
make the necessary punctures in the paper. The branches e e are attached to-
the bars c c with a joint which admits of their being folded backward over
the instrument when not in use, and for packing in its case. The center of
the instrument is represented by the intersection of two lines drawn at right
angles to each other on a piece of plate glass, which enables the person using
it to place it so that the center or intersection of the cross-lines may coincide
with any given point on the plan. If the instrument is in correct order, a line
connecting the fine pricking points with each other would pass through the
center of the instrument, as denoted by the before-mentioned intersection of
the cross-lines upon the glass. In using this instrument, the vernier should
first be set to zero (or the division marked 360) on the divided limb, and then
placed on the paper, so that the two fine steel points may be on the given line
(from whence other and angular lines are to be drawn), and the center of the
instrument coincides with the given angular point on such line. This done,
press the protractor gently down, which will fix it in position by means of very
fine points on the under side. It is now ready to lay off the given angle, or any
number of angles that may be required, which is done by turning the pinion d
till the opposite vernier reads the required angle. Then press downward the
branches e e, which will cause the points to make punctures in the paper at
opposite sides of the circle ; which being afterward connected, the line will
pass through the given angular point, if the instrument was first correctly set.
In this manner, at one setting of the instrument, a great number of angles may
be laid off from the same point.
The pantagraphs are used for the copying of drawings either on the same
scale, on a reduced scale, or on an enlarged scale, as may be required. The
DRAWING INSTRUMENTS.
55
form of pantagraph as shown in Fig. 146 consists of a set of jointed rulers,
A, B, and another, C, D, about one half the length of the former. The free
ends of the smaller set are jointed to the larger at about the center. Casters
are placed at a a, etc., to support the instrument and to allow an easy move-
ment over the paper. The rulers A and C are divided with a scale of propor-
tional parts, marked i, -J, etc. These arms are also provided with movable
indices, E, F, which can be fastened at any division by clamp screws. Each
index is provided with a socket adapted to carry either a pencil or a tracing
point.
Fig. 146 represents the instrument in the act of reducing the plan H to h,
one half the size. The tracing point is placed in the socket at E, the pencil at
F, and the fulcrum at G. The indices, E, F, are clamped each at on the
scales. If the instrument is correct, the points E, F, G, are in a straight line.
Pass the tracing point delicately over the plan H, and the pencil point F will
trace h, one half the original size.
If the object had been to enlarge the drawing to double its scale, then the
tracer must have been placed at F, and the pencil at E. And if a copy be
required, retaining the scale of the original, then the slides E and F must be
placed at the divisions marked 1. The fulcrum must take the middle sta-
tion, and the pencil and tracer those on the exterior rules A and B of the
instrument. Another form of this instrument is shown in Fig. 147.
FIG. 147.
The camera lucida is sometimes used for copying and reducing topograph-
ical drawings. A description of the use of this instrument will be found under
the head of topographical drawing.
The drawing table and drawing board. The usual size of the drawing
table should be from 5 to 6 feet long and 3 feet wide, of 1|- or 2-inch white
pine plank well seasoned, without any knots, closely joined, glued, doweled,
and clamped. It should be fixed on a strong, firm frame and legs, and of such
56
DRAWING INSTRUMENTS.
a height that the draughtsman, as he stands up, may not have to stoop to his
work. The table is usually provided with a shallow drawer to hold paper or
drawings. Drawing tables are made portable by having two horses for their
supports, and a movable drawing board for the top ; this board is made similar
to the top of the drawing table, but of inch boards, and barred at the ends.
Various woods are used for the purposes, but white pine is by far the cheapest
and best. Drawing boards should be made truly rectangular, and with per-
fectly straight sides for the use of the T square. Two sizes are sufficient for
common purposes, 41 X 30 inches to carry double elephant paper with a mar-
gin, and 31 X 24 inches for imperial and smaller sizes. Boards smaller than
this are too light and unsteady in handling.
Small boards are occasionally made, as loose panels fitting into a frame, flush
on the drawing surface, with buttons on the back to secure them in position.
The panel is mostly of white pine, with a hard-wood frame.
DKAWIKG PAPER.
Hand-made drawing paper is usually made to certain standard sizes about
as follows :
Demy ........... 20 inches by
inches.
Medium
Eoyal
22|
24
' 17*
' 191
Super Royal
Imperial
27i
30
j. t/ ^
; 19^
' 22
Elephant
28
' 23
Columbier 35 inches by 23^ inches.
Atlas 34 " 26 "
Double Elephant. 40 27
Antiquarian 53 '* 31
Emperor 68 " 48 "
But of late machine-made papers are the most used, and are furnished in
rolls of widths up to 58 inches, and wider can be obtained by order.
Whatman's white paper is the quality most usually employed for finished
drawings ; it will bear wetting and stretching without injury, and, when so
treated, receives color readily. For ordinary working drawings, where damp-
stretching is dispensed with, cartridge paper, in rolls of a coarser, harder, and
tougher quality, is preferable. It bears the use of India-rubber better, receives
ink on the original undamped surface more freely, shows a fully better line,
and, as it does not absorb very rapidly, tinting lies better and more evenly
upon it. For delicate small-scale line-drawing, the thick blue paper, such as
is used for ledgers, etc., imperial size, answers exceedingly well ; but it does
not bear damp-stretching without injury, and should be merely pinned or
waxed down to the board. With good management, there is no ground to fear
the shifting of the paper. Good letter paper receives light drawing very well ;
of course, it does not bear much fatigue.
Drawings destined for rough usage and frequent reference should be on
sheet or roll drawing paper, backed with cotton cloth, which can be purchased
at the stationer's.
Tracing paper is a preparation of tissue paper, transparent and qualified to
receive ink lines and tinting without spreading. When placed over a drawing
already executed, the drawing is distinctly visible through the paper, and may
be copied or traced directly by the ink instruments ; thus an accurate copy may
DRAWING INSTRUMENTS. 57
be made with great expedition. Tracings may be folded and stowed away very
conveniently ; but, for good service, they should be mounted on cloth, or on
paper and cloth, with paste.
Tracing paper may be prepared from thick tissue paper by sponging over
one surface with a mixture of one part raw linseed oil and five spirits of tur-
pentine ; five gills of turpentine and one of oil will go over from forty to fifty
sheets of paper.
Tracing cloth is a similar preparation of linen, and is preferable for its
toughness and durability. Tracing paper and cloth are usually to be had in
rolls, and tracings on cloth are now preserved as originals, and copies are made
from them by some sun process.
Mouth Glue, for the sticking of the edges of drawing paper to the board, is
made of glue and sugar or molasses ; it melts at the temperature of the mouth,
and is convenient for the draughtsman.
Drawing paper may be fixed down on the drawing board by the pins at the
corners, by weights, or by gluing the edges. The first is sufficient when 110
shading or coloring is to be applied, and if the sheet is not to be a very long
time on the board ; and it has the advantage of preserving the paper in its
natural state. For shaded or tinted drawings, the paper must be damped and
glued at the edges, as the partial wetting of paper, loose or fixed at the corners
merely, by the water-colors, distorts the surface.
Damp-stretching is done as follows : The edges of the paper should first be
cut straight, and, as near as possible, at right angles with each other ; also, the
sheet should be so much larger than the intended drawing and its margin as
to admit of being afterward cut from the board, leaving the border by which it
is attached thereto by glue or paste, as we shall next explain.
The paper must first be thoroughly and equally damped with a sponge and
clean water, on the opposite side from that on which the drawing is to be made.
When the paper absorbs the Water, which may be seen by the wetted side be-
coming dim, as its surface is viewed slantwise against the light, it is to be laid
on the drawing board with the wetted side downward, and placed so that its
edges may be nearly parallel with those of the board ; otherwise, in using a J
square, an inconvenience may be experienced. This done, lay a straight flat
ruler on the paper, with its edge parallel to, and about half an inch from, one
of its edges. The ruler must now be held firm, while the said projecting half-
inch of paper be turned up along its edge ; then a piece of solid or mouth glue,
having its edge partially dissolved by holding it in boiling or warm water for a
few seconds, must be passed once or twice along the turned-up edge of the
paper, after which, by sliding the ruler over the glued border, it will be again
laid flat, and, the ruler being pressed down upon it, that edge of the paper will
adhere to the board. If sufficient glue has been applied, the ruler may be re-
moved directly, and the edge finally rubbed down by an ivory book-knife, or by
the bows of a common key, by rubbing on a slip of paper placed on the draw-
ing paper, so that the surface of the latter may not be soiled, which will then
firmly cement the paper to the board. This done, another but adjoining edge
of the paper must be acted upon in like manner, and then the remaining edges
in succession ; we say the adjoining edges, because we have occasionally ob-
58 DRAWING INSTRUMENTS.
served that, when the opposite and parallel edges have been laid down first,
without continuing the process progressively round the board, a greater degree
of care is required to prevent undulations in the paper as it dries.
Sometimes strong paste is used instead of glue ; but, as this takes a longer
time to set, it is usual to wet the paper also on the upper surface to within an
inch of the paste mark, care being taken not to rub or injure the surface in the
process. The wetting of the paper in either case is done for the purpose of
expanding it ; and the edges, being fixed to the board in its enlarged state, act
as stretchers upon the paper, while it contracts in drying, which it should be
allowed to do gradually. All creases or undulations by this means disappear
from the surface, and it forms a smooth plane to receive the drawing.
To remove the paper after the drawing is finished, cut oif inside the pasted
edge, and remove the edge by warm water and the knife.
With paneled boards, the panel is taken out, and the frame inverted ; the
paper, being first damped on the back with a sponge slightly charged with
water, is applied equally over the opening to leave equal margins, and is pressed
and secured into its seat by the panel and bars.
MOUNTING PAPER AND DRAWINGS, VARNISHING, ETC.
When paper of the requisite quality or dimension can not be purchased
already backed, it may be mounted 011 cloth. The cloth should be well
stretched upon a smooth flat surface, being damped for that purpose, and its
edges glued down, as was recommended in stretching drawing paper. Then
with a brush spread strong paste upon the canvas, beating it in till the grain
of the canvas be all filled up ; for this, when dry, will prevent the canvas from
shrinking when subsequently removed ; then, having cut the edges of the paper
straight, paste one side of every sheet, and lay them upon the canvas sheet
by sheet, overlapping each other a small quantity. If the drawing paper is
strong, it is best to let every sheet lie five or six minutes after the paste is put
on it, for, as the paste soaks in, the paper will stretch, and may be better spread
smooth upon the canvas ; whereas, if it be laid on before the paste has moist-
ened the paper, it will stretch afterward and rise in blisters when laid upon
the canvas. The paper should not be cut off from its extended position till
thoroughly dry, which should not be hastened, but left in a dry room to do
so gradually, if time permit ; if not, it may be exposed to the sun, unless in
the winter season, when the help of a fire is necessary, provided it is not
placed too near a scorching heat.
In joining two sheets of paper together by overlapping, it is necessary, in
order to make a neat joint, to feather-edge each sheet ; this is done by care-
fully cutting with a knife half way through the paper near the edges, and on
the sides which are to overlap each other ; then strip off a feather-edged slip
from each, which, if done dexterously, will form a very neat and efficient joint
when put together.
For mounting and varnishing drawings or prints, stretch a piece of linen
on a frame, to which give a coat of isinglass or common size, paste the back of
drawing, which leave to soak, and then lay it on the linen. When dry, give it
at least four coats of well-made isinglass size, allowing it to dry between each
DRAWING INSTRUMENTS. 59
coat. Take Canada balsam diluted with the best oil of turpentine, and with a
clean brush give it a full flowing coat.
MANAGEMENT OF THE INSTRUMENTS.
In constructing preparatory pencil-drawings, it is advisable, as a rule of
general application, to make no more lines upon the paper than are necessary
to the completion of the drawing in ink ; and also to make these lines just so
dark as is consistent with the distinctness of the work. With respect to the
first idea, it is of frequent application : in the case, for example, of the teeth
of spur wheels, where, in many instances, all that is necessary to the drawing
of their end view in ink are three circles, one of them for the pitch line, and
the two others for the tops and bottoms of the teeth ; and again, to draw the
face view of the teeth that is, in the edge view of the wheel we have only
to mark off by dividers the positions of the lines which compose the teeth, and
draw four pencil lines for the two sides, and the top and bottom of the eleva-
tion. And here we may remark the inconvenience of that arbitrary rule, by
which it is by some insisted that the pupil should lay down in pencil every line
that is to be drawn before finishing it in ink. It is often beneficial to ink in
one part of a drawing before touching other parts at all ; it prevents confusion,
makes the first part of easy reference, and allows of its being better done, as the
surface of the paper inevitably contracts dust and becomes otherwise soiled in
the course of time, and therefore the sooner it is done with the better.
Circles and circular arcs should, in general, be inked in before straight lines,
as the latter may be more readily drawn to join the former than the former
the latter. When a number of circles are to be described from one center, the
smaller should be inked first, while the center is in better condition. When a
center is required to bear some fatigue, it should be protected with a thickness-
of stout card glued or pasted over it, to receive the compass-leg.
India-rubber is the ordinary medium for cleaning a drawing, and for cor-
recting errors in the pencil. For slight work it is quite suitable ; that sub-
stance, however, operates to destroy the surface of the paper ; and, by repeated
application, it so ruffles the surface, and imparts an unctuosity to it, as to spoil
it for fine drawing, especially if ink shading or coloring is to be applied. It is
much better to leave trivial errors alone, if corrections by the pencil may be
made alongside without confusion, as it is, in such a case, time enough to
clear away superfluous lines when the inking is finished.
For cleaning a drawing, a piece of bread two days old is preferable to India-
rubber, as it cleans the surface well and does not injure it. When ink lines to
any considerable extent have to be erased, a small piece of damped soft sponge
may be rubbed over them till they disappear. As, however, this process is apt
to discolor the paper, the .sponge must be passed through clean water, and ap-
plied again to take up the straggling ink. For ordinary small erasures of ink
lines, a sharp rounded pen-blade, applied lightly and rapidly, does well, and the
surface may be smoothed down by the thumb-nail. In ordinary working draw-
ings, a line may readily be taken out by damping it with a hair-pencil and
quickly applying the India-rubber ; and to smooth the surface so roughened, a
light application of the knife is expedient. In drawings intended to be highly
60 DRAWING INSTRUMENTS.
finished, particular pains should be taken to avoid the necessity for corrections,
as everything of this kind detracts from the appearance.
In using the square, the more convenient way is to draw the lines off the
left edge with the right hand, holding the stock steadily but not very tightly
against the edge of the board with the left hand. The convenience of the left
edge for drawing by is obvious, as we are able to use the arms more freely, and
we see exactly what we are doing.
To draw lines in ink with the least amount of trouble to himself, the me-
chanical draughtsman ought to take the greater amount of trouble with his
tools. If they be well made, and of good stuff originally, they ought to last
through three generations of draughtsmen ; their working parts should be care-
fully preserved from injury, they should be kept well set, and, above all, scru-
pulously clean. The setting of instruments is a matter of some nicety, for
which purpose a small oil-stone is convenient. To dress up the tips of the
blades of the pen or of the bows, as they are usually worn unequally by the
customary usage, they may be screwed up into contact in the first place, and
passed along the stone, turning, upon the point in a directly perpendicular
plane, till they acquire an identical profile. Being next unscrewed and exam-
ined to ascertain the parts of unequal thickness round the nib, the blades are
laid separately upon their backs on the stone, and rubbed down at the points,
till they be brought up to an edge of uniform fineness. It is well to screw
them together again, and to pass them over the stone once or twice more, to
bring up any fault ; to retouch them also on the outer and inner side of each
blade, to remove barbs or fraying ; and, finally, to draw them across the palm
of the hand.
The China ink which is commonly used for line-drawing ought to be
rubbed down in water to a certain degree, avoiding the sloppy aspect of light
lining in drawings, and making the ink just so thick as to run freely from the
pen. This medium degree may be judged of after a little practice by the ap-
pearance of the ink on the palette. The best quality of ink has a soft feel when
wetted and smoothed ; free from grit or sediment, and musky. The rubbing
of China ink in water tends to crack and break away the surface at the point ;
this may be prevented by shifting at intervals the position of the stick in the
hand while being rubbed, and thus rounding the surface. Nor is it advisable,
for the same reason, to bear very hard, as the mixture is otherwise more evenly
made, and the enamel of the palette is less rapidly worn off. When the ink, on
being rubbed down, is likely to be for some time required, a considerable quan-
tity of it should be prepared, as the water continually vaporizes ; it will thus
continue for a longer time in a condition fit for application. The pen should
be leveled in the ink, to take up a sufficient charge ; and, to induce the ink to
enter the pen freely, the blades should be lightly breathed upon before immer-
sion. After each application of ink, the outsides of the blades should be
cleaned, to prevent any deposit of ink upon the edge of the squares.
To keep the blades of his inkers clean is the first duty of a draughtsman
who is to make a good piece of work. Pieces of blotting or unsized paper and
cotton velvet, wash-leather, or even the sleeve of a coat, should always be at
hand while a drawing is being inked. When a small piece of blotting paper is
DRAWING INSTRUMENTS. 61
folded twice so as to present a corner, it may usefully be passed between the
blades of the pen now and then, as the ink is liable to deposit at the point and
obstruct the passage, particularly in fine lining ; and for this purpose the pen
must be unscrewed to admit the paper. But this process may be delayed by
drawing the point of the pen over a piece of velvet, or even over the surface of
thick blotting-paper ; either method clears the point for a time. As soon as
any obstruction takes place, the pen should be immediately cleaned, as the
trouble thus taken will always improve and expedite the work. If the pen
should be laid down for a short time with the ink in it, it should be unscrewed
to keep the points apart, and so prevent deposit ; and, when done with alto-
gether for the occasion, it ought to be thoroughly cleaned at the nibs. This
will preserve its edges and prevent rusting.
For the designing of machinery, it is very convenient to have some scale of
reference by which to proportion the parts ; for this purpose a vertical and
horizontal scale may be drawn on the walls of the room.
EXERCISES WITH THE
Before proceeding to the construction of finished drawings, skill should be
acquired in the use of the drawing-pen, supplemented often by the steel pen.
Beginning with lines, outlines of figures, alphabets, and the like, the draughts-
man should strive to acquire the habit of readily drawing clean, uniform lines,
without abruptness or breaks, where straight lines connect with curved ones.
Draw straight lines of different grades :
as, fine -
medium - - --
coarse ^ ^^
at first, lines of indefinite length, taking care that they are drawn perfectly
straight and of uniform width or grade ; then draw lines of definite length
between assumed points, taking care to terminate the lines exactly at these
points. Lines as above are full lines, the grades depending on the effect which
the draughtsman wishes to give.
Draw dotted lines, broken lines, and broken and dotted lines, of different
grades :
Draw fine lines at uniform distances from each other
DRAWING INSTRUMENTS.
To give uniform appearance, the lines must be of uniform grade and equally
spaced. Practice in lines of this sort is important, as they are much used in
drawing to represent sections, shades, and conditions, as soundings on charts,
density or characteristics of population, areas of rain, temperature, and the like.
Draw lines as in Fig. 148. These lines are diagonal with the border-lines, and
FIG. 148.
are used to represent sections of materials. In the figure, lines differently in-
clined represent different pieces of the same material.
Sections of different materials may be represented in different kinds of
lines, as in Figs. 149, 150, 151.
FIG. 149.
FIG. 150.
FIG. 151.
These particular ones are used to represent sections of wrought-iron, steel,
and cast-iron ; but they may be used to represent different colors, the location
of different mineral or agricultural products, etc.
To represent cylindrical surfaces (Fig. 152).
Draw a semi-circumference, and mark on it a number of points, at equal
distances apart, and through these points draw lines perpendicular to the
FIG. 152.
FIG. 153.
diameter across the surface to be represented. It is not absolutely necessary
that the central space should be equal to the others ; it will be more effective
to leave out two of the lines, and make it to this extent wider.
DRAWING INSTRUMENTS.
63
To construct a mass of equal squares (Fig. 153).
Lay off a right angle, and on its sides mark as many points, at equal dis-
tances apart, as may be necessary ; through these points draw lines parallel to
the sides.
Or, construct a rectangle ; mark on its sides as many
points, at equal distances apart, as may be necessary ;
through these points draw the lines.
To construct the squares diagonally to the base (Fig.
154).
Mark on the sides of the right angle as many points,
at distances apart equal to the diagonal of the required
squares, as may be necessary. Con-
nect these points by lines as shown,
and through the same points draw
lines at right angles to the others.
Or, as above, construct a rec-
tangle, and mark on its sides points
at distances apart equal to the di-
agonal of the required squares.
To cover a surface with equi- FlG 154
lateral triangles (Fig. 155).
Construct an angle of 60, and mark on its sides points at distances apart
equal to the side of the triangle. Connect these points ; and through these
points draw lines parallel to the sides of the angle.
Figures composed of two triangles, with the same base, are called lozenges.
Six triangles may be arranged as
a hexagon. The whole surface
may be arranged in lozenges or
hexagons.
To cover a surface with octa-
gons and squares (Fig. 156).
Lay off the surface in squares
having sides equal to the width
of the octagons. Corner the outer
squares to form octagons, as by
Prob. XL., page 21. Extend the
sides of these octagons across the
other squares, and similar corners
will be cut off, and the octagons
and squares required will be com- FlG 155
plete.
With the aid of paper thus covered with squares, triangles, and lozenges,
various geometrical designs may be readily constructed, pleasing in their effect,
and affording good practice to young draughtsmen.
In the examples given of designs constructed on squares or triangles, if it is
desired to increase or diminish the size of the original designs, it is only neces-
sary to make the sides of the squares or triangles larger or smaller, and taking
64 DRAWING INSTRUMENTS.
relatively the same squares for the construction of the figures. In transferring
designs and drawings from books or plates, on which squares can not be drawn,
it is very convenient to have a square of glass, with squares upon it, which may
be laid on the drawing, and thus serve the same purpose as if squares had been
C
F
drawn. The glass may be readily prepared by painting one of its surfaces with
a thin coat of gum, and drawing squares upon it with the drawing-pen ; if
every fifth or tenth line be made fuller or in a different color, it will be still
more convenient for reference.
Fig. 157 is the front view and side of an acanthus-leaf, of which the sur-
faces are covered with squares, somewhat larger than would be recommended
FIG. 157.
FIG. 158.
in practice, but sufficient to illustrate the principle, which may be done by
the learner on the same or other sized squares. If the same size, the intersec-
tions of the lines of the figure with those of the squares are easiest transferred
by a straight-edged slip of paper, placed along a line, and making all the inter-
sections at once, and then transferring the marks to the copy.
DRAWING INSTRU
65
Fig. 158 is the side-view of the acanthus-le^f, in a reversed position from
the original (Fig. 157) ; that is, right-handed, while the original is left-handed.
It will readily be understood how this may be done by observing the letters on
the side and the numerals at the top of the squares.
Fig. 159 represents the construction of Gothic letters and numerals on a
system of squares. These letters are formed mechanically by the drawing-pen
and dividers.
Fig. 160 are Italic letters, drawn on rhombs, in which the upright lines
are inclined to horizontal.
On pages 66, 67, 68, 69, are specimens of type taken from the printer's
font, which can be readily transferred to a drawing, by covering them with
a bit of glass or horn, laid off in squares, as described above. Printers' let-
ters are in general well proportioned, but it is customary often to distort
letters, to call attention to them, or to adapt them to the position in which
they are to be placed. Spaces between the letters are in printing uniform,
but in drawing, when such letters come together as F and A, L and T, one
wide at top and the other at bottom, the spacing between them may be
reduced a little. The acquisition of a ready hand in lettering enables a
draughtsman to give a finish to a good drawing or map which might other-
wise be spoiled by poor lettering.
FIG. 159.
66 DRAWING INSTRUMENTS.
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DRAWING INSTRUMENTS. 67
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DRAWING INSTRUMENTS.
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DRAWING INSTRUMENTS. 69
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Paper printed in squares is used by designers of figures for calicoes, silks,
and woolens. For the engineer, there is a class of papers called cross-section
papers, sold in sheets or rolls, and of various scales, originally intended, as the
name implies, for cross-sections of railway or canal cuts, but now extensively
employed by the architectural and mechanical designer for the rough sketches
of works either executed or to be executed ; by the sanitarian for the plotting of
death-rates ; for thermometric and hygrometric readings ; by the broker and
merchant for the graphic representation of the prices of gold, stocks, or articles
of merchandise, during a term of years ; by the railway superintendent for the
movement of trains ; and for multitudes of other uses. These may hardly be
considered in the light of drawings ; but, as they involve the drawing of lines,
shading of spaces, and lettering, and as there is no head of drawing under
70
DRAWING INSTRUMENTS.
which this use of cross-section paper can be classed, it seems proper to give
here a few illustrations, which will show its general application.
Fig. 161 shows a graphical method of determining the equivalent values of
the metric system of measurements in United States units, or vice versa. The
vertical scale represents the metric units, and the horizontal the common or
oo TJI to so
UNITED STATES UNITS.
FIG. 161.
United States units. The method of using the diagram can be best shown by
taking one or two examples.
What is the equivalent value of seven kilometres in miles ? Read upward
on the metric scale to 7, then read on that horizontal line to the point of in-
tersection with the line designated "MILES & KILOMETRES," that is, at
the point on the United States scale of units representing 4 '35 ; therefore,
seven kilometres are equal to 4*35 miles.
What is the value of five pounds in kilogrammes ? The process is the same
as the foregoing, except that, to change United States units into the metric
units, first read horizontally, then upward. The result will be in this case that
five pounds is found equal to 2*25 kilogrammes. The divisions may represent
single units, ten units, one hundred units, etc. ; that is, if we had wished to
find the equivalent of 500 pounds, it would have been 225 kilogrammes.
DRAWING INSTRUMENTS.
71
Fig. 162 is a diagram illustrating graphically the difference charged on a ton
of merchandise per mile } on the New York Central and Hudson River Railroad
and the Erie Canal, for every year between 185? and 1880 ; the values being
FIG. 162.
published in the Report of the United States Bureau of Statistics for 1880.
The higher values in every case represent the railroad rates and the lower the
canal rates. The black band shows the difference between these values. In
1865, for instance, the railroad rates were 3 '30 cents, and the canal 1'02 cents,
the difference being 2 -28 cents.
Fig. 163 is made up from the time-table of the New York, New Haven, and
Hartford Railroad, showing the movement of trains, two from New York and
two from New Haven, the abscissas (horizontal lines) being cut off on a scale
of miles for each station, the ordinates (vertical lines) being a scale of hours.
DRAWING INSTRUMENTS.
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taken from the last edition of Francis's "Lowell Hydraulic Experiments."
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DRAWING INSTRUMENTS. 73
width of the cut represents the width of the flume, each abscissa being one
foot ; the ordinates are the speeds of float in divisions of 0*1 foot per second ;
the o o on the cut are meant to represent the floats in their observed path and
speed ; and the curved line the average velocity in the different threads of the
stream.
Fig. 165 is from Clarke's " Railway Machinery." The abscissas represent
the speed in miles per hour ; the ordinates the pounds per ton resistance of a
100- ton train.
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Fig. 166 is a diagram illustrating the daily mortality during the month of
November, 1873, in New York City. The figure is a copy of a portion of the
chart published in the Report of the Metropolitan Board of Health for that
year. The lower irregular line shows the daily mortality. The upper single
irregular line shows the daily average temperature. The terminal cross-lines at
the ends of perpendicular bars show the daily range of temperature. The
double irregular line shows the daily humidity, saturation being 100 on the
scale of temperatures. The black bands in the upper portion of the diagram
give the daily rain- fall in inches. This method of representing the rain-fall
will do for this chart, but, for most meteorological purposes, is insufficient.
The time of the commencement and end of the rain-fall should be given where
any effect due to the rain is to be detected. These few diagrams illustrate the
method of graphical representation, so that any one should with little difficulty
be able now to make them for such cases as he may see fit.
On pages 75, 76, 77, are some designs, showing other uses to which squared
or quadrille paper can be put. The execution of such ornamental designs is
greatly facilitated by the use of this paper. The figure on page 77 illustrates
how color may be represented in a design, by different grades and directions,
of black lines and white spaces.
DRAWING INSTRUMENTS.
NOVEMBER, 1873.
FIG. 166.
DRAWING INSTRUMENTS.
75
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&* OR6
#V
ORTHOGRAPHIC PROJECTION.
ARCHITECTURAL and mechanical drawings are usually the delineation of
bodies by orthographic projection, the representation on a sheet of paper hav-
ing only two dimensions, length and breadth, of solids having three, length,
breadth, and thickness ; and on such scales that dimensions can be taken from
the parts, and structures and machines constructed therefrom.
Place any surface for instance, a sheet of paper or a drawing-board at
right angles to the sun's rays. This may be readily done by inserting a pin into
the surface, and making it vertical to the surface in every direction by a right-
angled triangle ; then place the surface in the direct rays of the sun, and in
such a position that there will be no shadow on the surface from the pin ; the
sun's rays are then perpendicular to the surface. Take a wafer or a circular
bit of paper, and hold it over the paper by means of a long pin or wire, and we
obtain shadows, as above, varying with the inclination of the wafer to the
plane of the paper. When parallel with the plane, the shadow is a complete
circle ; when at right angles, a line ; and varying between them as the wafer is
inclined. These shadows are the orthographic projections of the wafer ; no
line can be longer than it is naturally, but, if inclined or vertical, it is reduced
in length till it becomes a point only. The orthographic projection of the pin
which has determined the position of the surface is merely the shadow of the
head. If the pin be inclined at all, the body of the pin is projected as a shadow
by a line ; if the pin be laid on the surface, its shadow, or projection, is that of
the whole length of the pin. The sun's rays act as perpendiculars, which
will be hereafter spoken of as projecting the points of an object upon a surface
which will represent the object itself in drawing ; and, should any confusion
occur to the draughtsman of how an object is' to be projected or drawn, if he
can make the outline of the object on any convenient scale in wire and get its
shadows by the sun's vertical rays on a plane, he can readily see how the object
should be drawn.
Since the surfaces of all bodies may be considered as composed of points,
the first step is to represent the position in space of a point, by referring it to
planes whose position is established. The projection of a point upon a plane
is the foot of the perpendicular let fall from the point on the plane. If, there-
ORTHOGRAPHIC PROJECTION.
79
iore, on two planes not parallel to each other, whose positions are known, we have
the projections of a point, the position of this point is completely determined by
erecting perpendiculars from each plane at the pro-
jected points : their intersection will be the point.
If from every point of an indefinite straight line,
A B (Fig. 167), placed in any manner in space, per-
pendiculars be let fall on a plane, L M N 0, whose
position is given, then all the points in which these
perpendiculars meet the plane will form another
indefinite straight line, a b : this line is called the
projection of the line A B on this plane. Since two
points are sufficient to determine a straight line, it
is only necessary to project two points of the line,
and the straight line drawn through the two projected points will be the pro-
jection of the given line. The projection of a straight line, itself perpendicular
to the plane, is the point in which this perpendicular meets the plane.
If the projections a 1) and a' V of a straight line on the two planes L M N
.and L M P Q (Fig. 168) are known, this line A B is determined ; for if,
FIG. 167.
FIG. 168.
FIG. 169.
through one of its projections, a #, we suppose a plane drawn perpendicularly
to L M N 0, and if through a' V another plane be drawn perpendicular to
L M P Q, the intersection of the two planes will be the line A B.
To delineate a solid, as the form of a machine, for instance, it must be
referred to three series of dimensions, each of them at right angles to the plane
of the other.
Thus, let a b c (Fig. 169) be a parallelepiped in an upright position, of
80
ORTHOGRAPHIC PROJECTION.
which the plane a b is horizontal, and the planes a c and c ~b vertical. Let d e,
d f, and d g, be the planes of projection. The sides of the body being parallel
to these planes, each to each, let the figure of the parallelepiped be projected
on them. For this purpose draw parallel lines from the angles of the body
perpendicular to the planes, as indicated by the dotted lines ; then upon the
plane d e we shall have a' 1)', the projection of the surface a I : this is called
the plan of the object. Upon the plane dfwe have a' c', the projection of the
surface a c, the front elevation ; and upon the plane d g, the projection I' c'
of the surface b c, the side elevation. We have then three distinct views of
the regular solid a b c delineated on plane surfaces, which convey an accurate
and sufficient idea of its form. Indeed, any two of these representations are
sufficient as a description of the object. From the two figures a' c', 5' c', for
example, the third figure a' b' may be compounded, by merely drawing the
vertical lines c' h b' i, and a' k, c' I, to meet the plane d e, and by producing
them horizontally till they meet and form the figure a' b'. Similarly, the
figure b' c' may be deduced from the other two by the aid of the lines Ji, i,
from a' b 1 ', and the lines m, n, from a' c' .
It is in this way that a third view of any piece of machinery is to be found
from two given views ; and in many cases two elevations, or one elevation and
a plan, may afford a sufficiently corn-
plete idea of the construction of a
machine. In other cases, many parts
may be concealed by others in which
they are inclosed ; this suggests the
occasional necessity of views of the
interior, in which the machine is sup-
posed to be cut across by planes, ver-
tically or horizontally, so as properly
to reveal its structure. Such views
are termed sections, and, with refer-
ence to the planes of section, are de-
nominated vertical and horizontal sec-
tions. To all such drawings is given
the general title of geometrical draw-
ings, as distinguished from perspective
drawings.
In practice, the drawings are done
upon one common surface, the plane
of paper, and we may readily suppose
the plane d g (Fig. 169) revolved back
into the position d g r , and d e also
moved to d e', both of these positions
being in the plane of d f. This being done, we have the three views depicted on
one plane surface (Fig. 170). In this figure, the same letters of reference are
employed as in Fig. 169 ; d I and d m are the ground and vertical lines. It is
evident that the positions of the same points in a' c' and a' b' are in the same
perpendicular from the ground-line : that, in short, the position of a point in
FIG. 170.
ORTHOGRAPHIC PROJECTION. 81
the plane may be found by applying the edge of the square to the same point
as represented in the elevation. The same remark is applicable as between the
two elevations. Hence the method of drawing several views of one machine
upon the same surface of paper in strict agreement with each other.
PROJECTIONS OF SIMPLE BODIES.
In most of the following examples, the projections of the bodies are given
both with and without the construction lines.
Right projections of a regular hexagonal pyramid (Fig. 171). It is evident
that two distinct geometrical views are necessary to convey a complete idea of
the form of the object : an elevation to represent the sides of the body, and to
express its height ; and a plan to express the form horizontally.
Draw a horizontal straight line L T through the center of the sheet to rep-
resent the ground-line. Then draw a perpendicular S S' to the ground-line to
represent the axis of the pyramid. For the sake of preserving the symmetry of
the drawing, the centers of the horizontal projections of Figs. 171 and 172 are
in the same straight line A' S', drawn parallel to the ground-line.
In delineating the pyramid, it is necessary, in the first place, to construct the
plan. Take any point, S', on the line S S' as the center of the figure, and from
this point, with a radius equal to. the side of the hexagon which forms the base
of the pyramid, describe a circle, cutting A' S' at A' and D'. From these points
with the same radius, draw four arcs of circles, cutting the primary circle in
four points. These six points being joined by straight lines, will form the figure
A' B' 0' D' E' F', the base of the pyramid ; and the lines A' S', B' S', etc., will
represent the projections of its edges shortened as they would appear in the plan.
By the help of this plan the vertical projection of the pyramid may be easily
constructed. Since its base rests upon the horizontal plane, it must be pro-
jected vertically upon the ground-line ; therefore, from each of the angles at
A', B', C', and D', erect perpendiculars to that line. The points of intersec-
tion, A, B, C, and D, are the true positions of all the angles of the base ; and
it only remains to lay off the height of the pyramid, from the point G to S, and
to draw S A, S B, S C, and S D, which are the only edges of the pyramid visi-
ble in the elevation. Of these it is to be remarked that S A and S D alone,
being parallel to the vertical plane, are seen in their true length ; and, more-
over, that from the assumed position of the solid under examination, the points
F' and E' being situated in the lines B B' and C C', the lines S B and S C are
each the projections of two edges of the pyramid.
To construct the projections of the same pyramid, having its base set in an
inclined position, but with its edges S A and S D still parallel to the vertical
plane (Fig. 172).
It is evident that, with the exception of the inclination, the vertical projec-
tion of this solid is precisely the same as in the preceding example, and it is
only necessary to copy that elevation. To do this, fix the position of the point D
upon the ground-line, through which draw D A, making with L T the desired
inclination of the base of the pyramid. Make D A equal to the A D of the
preceding figure, and on this erect the vertical projection S A D of that figure.
Since the edges S A and S D are still parallel to the vertical plane, and
6
82
ORTHOGRAPHIC PROJECTION.
FIG. 171.
FIG. 172.
ORTHOGRAPHIC PROJECTION.
83
the point D remains unaltered, the projection A' of the point A will still be in
the line M N. The remaining points B', C', etc., in the projection of the base,
are found by the intersections of perpendiculars let fall from the corresponding
points in the elevation, with lines drawn parallel to M N, at a distance equal
to the width of the base. By joining all the contiguous points, we obtain
A' B' C' D' E' F', the horizontal projection of the base, two of its sides,, how-
ever, are dotted, being concealed by the body of the pyramid. The vertex S
having been similarly projected to S', and joined by straight lines to the several
angles of the base, the projection of the solid is completed.
To find the horizontal projection of a transverse section of the same pyramid,
made by a plane perpendicular to the vertical, but inclined at an angle to the
horizontal plane of projection; and let all the sides of the base be inclined to the
ground-line (Fig. 173).
FIG. 173.
Since none of the sides of the base are to be parallel with the ground-line,
draw a diameter A' D' making the required angle with that line, and from the
points A' and D' proceed to set out the angular points of the hexagon as in the
figure. Then, in order to obtain the projections of the edges of the pyramid,
join the angular points which are diametrically opposite ; and, following the
ORTHOGRAPHIC PROJECTION.
method pointed out in reference to Fig. 171, project the figure thus obtained
upon the vertical plane, as shown in the elevation.
Now, if the cutting plane be represented by the line a d in the elevation, it
is obvious that it will expose, as the section of the pyramid, a polygon whose
angular points, being the intersections of the various edges with the cutting
plane, will be projected in perpendiculars drawn from the points where it
meets these edges respectively. If, therefore, from the points a, f, b, etc., we
let fall the perpendiculars a a', //', b b f , etc., and join their contiguous points
of intersection with the lines A' D', F' C', B' E', etc., we shall form a six-sided
figure, which will represent the section required. The edges F S and E S, being
concealed in the elevation, but necessary for the construction of the plan, have
been expressed in dotted lines, as also the portion of the pyramid situated above
the cutting plane, which, though supposed to be removed, is necessary in order
to draw the lines representing the edges. We have here introduced the ordi-
nary method of expressing sections in purely line-drawings, by filling up the
spaces comprised within their outlines with a number of parallel straight lines
drawn at equal distances called section-lines.
PROJECTIONS OF A PEISM.
B
a H
C
D
I K
FIG. 174.
ORTHOGRAPHIC PROJECTION.
85
Required to represent in plan and elevation a regular six-sided prism in an
upright position (Fig. 174).
Lay down the ground-line G K and draw the axis of the prism S S'. De-
scribe the hexagonal plan A' B' C' D' E' F', as in the previous example. From
each of the angular points, A', B', etc., erect perpendiculars to the ground-line,
and on one of these perpendiculars set off A G, the height of the prism, and
draw a parallel A D to the ground-line, which completes the vertical projection.
The face, B H I, being parallel to the vertical plane, is seen in its true size.
B' C' being equal to one half of A' D', therefore H I is equal to one half of
G K. We have then G H and I K equal each to one half of H I. This enables
us to draw the elevation of such a prism situated as is this one without con-
structing the plan. This fact should be remembered in the drawing of nuts,
bolt-heads, etc., in machine-drawing, where it is of frequent application.
To form the projections of the same prism, supposing it to have been moved
round the point G, in a plane parallel to the vertical plane (Fig. 175).
Copy the elevation (Fig. 174) on the inclined base G K. Let fall perpen-
FIG. 175.
86
ORTHOGRAPHIC PROJECTION.
diculars from all the angles in the elevation, and, joining the contiguous points
of intersection with the horizontal lines appropriate to these points respectively,
the plan of course remaining the same width as before, we obtain the polygon
A' B' 0' D' E' F' as the projection of the upper surface, and G' H' I' K' L' M'
as that of the base of the prism. Finally, it will be observed that all the edges
are represented, in the horizontal projection, by equal straight lines, as D' K',
A' G', etc., and that the sides A' B', G' H', etc., remain still parallel to each
other, which will afford the means of verifying the accuracy of the drawings.
As the upper surface and the base are seen obliquely in this projection, of
course they do not appear as true hexagons in the plan.
Required the projections of the same prism set into a position inclined to
loth planes of projection (Fig. 176).
FIG. 176.
Assuming that the inclination of the prism upon the horizontal plane is
the same as in the preceding figures, for the sake of simplifying the operation,
the first process is to copy the plan of Fig. 175 on an axis A' K' inclined to the
vertical plane of projection.
ORTHOGRAPHIC PROJECTION.
87
Now, since the prism has been supposed to have preserved its former inclina-
tion to the horizontal plane, it is obvious that every point in it, such as A, has,
in assuming its new position, simply moved in a horizontal plane, and will
therefore be at the same distance above the ground-line that it was in the
elevation (Fig. 175), and it will also be in the perpendicular A' A ; the point
of intersection A is, therefore, its projection in the elevation. The remaining
angular points in this view are all determined in the same manner, and, having
joined the contiguous points, and the corresponding angles of the upper and
lower surface, we obtain the complete vertical projection of the prism in its
doubly-inclined position.
CONSTKUCTION OF THE CONIC SECTIONS.
The plan of the cone (Fig. 177) is simply a circle, described from the center
S', with a diameter equal to that of the base. Its elevation is an isosceles tri-
angle, obtained by drawing tangents A' A, B' B, perpendicular to and inter-
X
X
FIG. 177.
88 ORTHOGRAPHIC PROJECTION.
secting the ground-line ; then set off upon the center line the height C S, and
join S A, S B. These lines are called the exterior elements of the cone.
Given the projections of a cone, and the direction of a plane X X, cutting it
perpendicularly to the vertical, and obliquely to the horizontal plane ; required
to find, first, the horizontal projection of this section; and, secondly, the out-
line of the ellipse thus formed (Figs. 177, 178).
Through the vertex of the cone draw a line S E to any point within the
base A B ; let fall a perpendicular from E, cutting the circumference of the base
in E', and join E' S' ; then another perpendicular let fall from e will intersect
E' S' in a point e f , which will be the horizontal projection of a point in the
curve required ; and so on for any required number of points.
The exterior generatrices A S and B S being both projected upon the line
A' B', the extreme limits of the curve sought will be at the points a' and b f on
that line, which are the projections of the points of intersection a and b of the
cutting plane with the outlines of the cone. And since the line a' b' will
obviously divide the curve symmetrically into two equal parts, the points /',
g ' , h' , etc., will be readily obtained by setting off above that line, and on their
respective perpendiculars, the distances d' d*, e' e*, etc. A sufficient number
of points having thus been determined, the curve drawn through them (which
will be found to be an ellipse) will be the outline of the section required.
This curve may be obtained by another method, depending on the principle
that all sections of a cone by planes parallel to the base are circles. Thus, let
the line F G represent such a cutting plane ; the section which it makes with
the cone will be denoted on the horizontal projection by a circle drawn from
the center S', with a radius equal to half the line F G ; and by projecting the
point of intersection H of the horizontal and oblique planes by a perpendicular
H H', and noting where this line cuts the circle above referred to, we obtain
two points H' and I' in the curve required. By a similar construction, as
exemplified in the drawings, any number of additional points may be found.
As the projection obtained by the preceding methods exhibits the section as
fore-shortened, and not in its true dimensions, we shall now proceed to the
consideration of the second question proposed. Let the cutting plane X X be
conceived to turn upon the point b, so as to coincide with the vertical line b k,
and (to avoid confusion of lines) let b Tc be transferred to a' b', which will rep-
resent, as before, the extreme limits of the curve required. Now, taking any
point, such as d, it is obvious that, in this new position of the cutting plane, it
will be represented by d?, and, if the cutting plane were turned upon a' b' (Fig.
178) as an axis till it is parallel to the vertical plane, the point which had been
projected at d* would then have described round a' b' an arc of a circle, whose
radius is the distance d' d? (Fig. 177). This distance, therefore, being set off
at d' and f on each side of a' b' , gives two points in the curve sought. By
a similar mode of operation any number of points may be obtained, through
which, if a curve be drawn, it will be an ellipse of the true form and dimen-
sions of the section.
To find the horizontal projection and actual outline of the section of a cone,
made by a plane Y Y parallel to one side or element, and perpendicular to the
vertical plane (Figs. 179, 180).
ORTHOGRAPHIC PROJ
89
Determine by the second method laid down in the preceding problem any
number of points, as F', G', J', K', etc., in the curve representing the horizon-
tal projection of the section specified. The horizontal plane passing through
M gives only one point M', which is the vertex of the curve sought.
FIG. 180.
In order to determine the
actual outline of this curve,
suppose the plane Y Y to turn
as upon a pivot at M, until it
has assumed the position M B,
and transfer M B parallel to
itself to M 2 B 2 (Fig. 180). The
point F will thus have first
described the arc F E till it
reaches the point E, which is
then projected to E 2 ; suppose
the given plane, now represent-
ed by M 2 B 2 , to turn upon that
line as an axis, until it assumes
a position parallel to the ver-
tical plane, the point E 2 , which is distant from the axis M' B' by the distance
F' S' (Fig. 179), will now be projected to F 2 and G a , two points in the curve
required, which is & parabola.
To draw the vertical projection of the sections of two opposite cones made ly
a plane parallel to their axis (Fig. 181).
Let C E D and C B A be the two cones, and X X the position of the
cutting plane. Project in plan either of the cones, as I E' D' ; from its center,
with a radius equal to L H, describe a circle, and draw the tangent la; la
will be the horizontal projection of the cutting plane. Draw the line H' M'
parallel to the cutting plane ; H', M' corresponding in position to the inter-
90
ORTHOGRAPHIC PROJECTION.
sections H, M, of the plane with the cones. From H' and M' lay off distances
equal to L K, K I, and the length of the cone, and through these points draw
perpendiculars, as f e\ d' c r , V a', etc., which must be made equal to the
chords f e, d c, b a, made by the cutting plane a b, with circles whose radii are
G K, -I F, and the radius of the base of the cone. Through the points a', c',
e, H', /', d',V, draw the curve, and we have the projection required. A similar
construction will give the sectional projection of the opposite cone at M'. The
curve thus found is the hyperbola.
PENETRATIONS OR INTERSECTIONS OF SOLIDS.
On examining the minor details of most machines, we find numerous ex-
amples of cylindrical and other forms, fitted to, and even appearing to pass
through, each other in a great variety of ways. The examples given are selected
with a view of exhibiting those cases which are of most frequent occurrence,
and of elucidating general principles.
Represent the projections of two cylinders of unequal diameters (Fig. 182)
meeting each other at right angles ; one by the rectangle ABED in the ver-
tical, and by the circle A' H' B' in the horizontal projections ; the other, which
is supposed to be' horizontal, is indicated in the former by the circle L P I N,
and in the latter by the rectangle L' I' K' M'. From the position of these two
solids it is evident that the curves formed by their junction will be projected
horizontally in the curves 0' H' P', R' S' T', and vertically by L P I N.
But, if the position of these bodies be changed into that represented by Fig.
183, the lines of their intersection will assume in the vertical projection a
totally different aspect, and may be accurately determined as follows :
Through any point taken upon the plan of Fig. 183 draw a horizontal line
a' V, which is to be considered as indicating a plane cutting both cylinders
parallel to their axes ; this plane would cut the vertical cylinder in lines drawn
perpendicularly through the points c' and d'. To find the vertical projection
of its intersection with the other cylinder, conceive its base I' L', after being
ORTHOGRAPHIC PROJECTION.
91
FIG. 182.
FIG. 183.
92 ORTHOGRAPHIC PROJECTION.
transferred to I 2 L a , to be revolved about I 2 L 2 as an axis parallel to the hori-
zontal plane ; this is expressed in part by simply drawing a semicircle of the
diameter I 3 L 1 . Produce the line a' V to # a ; then set off the distance a? e' on
each side of the axis I K, and draw straight lines through these points parallel
to it. These lines a b, g h, denote the intersection of the plane a' V with the
horizontal cylinder, and therefore the points c, d, m, o, where they cut the
perpendiculars c c', d d', are points in the curve required. By passing other
horizontal planes similar to a' V through both cylinders, and operating as
before, any number of points may be obtained. The vertices i and k of the
curves are obviously projected directly from i' and Jc', the intersections of the
outlines of both cylinders. When the cylinders are of unequal diameters, as
in the present case, the curves of penetration are hyperbolas.
When the diameters of the cylinders are equal (Fig. 184), and when they
cut each other at right angles, the curves of penetration are projected vertically
in straight lines perpendicular to each other. In the figure, most of the points
are indicated in elevation and plan by the same letters of reference.
To delineate the intersections of two cylinders of equal diameters at right
angles, when one of the cylinders is inclined to the vertical plane (Fig. 185).
Supposing the two preceding figures to be drawn, the projection c of any
point such as c' may be ascertained by observing that it must be situated in
the perpendicular c' c, and that, since the distance of this point (projected at c
in Fig. 184) from the horizontal plane remains unaltered, it must also be in
the horizontal line c c. Upon these principles all the points indicated by literal
references in Fig. 185 are determined ; the curves of penetration resulting
therefrom intersecting each other at two points projected upon the axial line
L K, of which that marked q alone is seen. The ends of the horizontal cylin-
der are represented by ellipses, the construction of which will also be obvious
on referring to the figure.
To find the curves resulting from the intersection of two cylinders of un-
equal diameters, meeting at any angle (Fig. 186).
For the sake of simplicity, suppose the axes of both cylinders to be parallel
to the vertical plane, and let A B E D and N Q P be their projections upon
that plane. In constructing, in the first place, their horizontal projection,
observe that the upper end A B of the larger cylinder is represented by an
ellipse A' K' B' M', which may easily be drawn by the help of the major axis
K' M' equal to the diameter of the cylinder, and of the minor A' B', the projec-
tion of the diameter. The visible portion of the base of the cylinder being
similarly represented by the semi-ellipse L' D' II', its entire outline will be com-
pleted by drawing tangents L' M' and H' K'. The upper extremity P N of the
smaller cylinder will also be projected in the ellipse p' i' N'.
Conceive a plane, as a' g', to pass through both cylinders parallel to their
axes ; it will cut the surface of the larger cylinder in two straight lines, passing
through the points/' and g' on the upper end of the cylinder ; these lines will
be represented in the elevation, by projecting the points/' and g' to/, g ; and
drawing a f and c g parallel to the axis. The plane a' g' will in like manner
cut the smaller cylinder in two straight lines, which will be represented in the
vertical projection by d h and e i, and the intersections of these lines with af
ORTHOGRAPHIC PROJECTION.
93
FIG. 184.
FIG. 185.
ORTHOGRAPHIC PROJECTION".
and c g will give four points ?, &, m, and n, in the curves of penetration. Of
these points, one only, that marked I, is visible in the plan, where it is denoted
by/'.
To find the curves of penetration in the elevation without the aid of the plan
(Fig. 186).
Let the bases D E and Q of both cylinders be conceived to be revolved
parallel to the vertical plane after being transferred to any convenient distance,
as D 2 E 2 and Q 2 O 2 , from the
principal figure ; they will
then be vertically projected
in the circles D 2 H 2 E 2 and
Q 2 G' 0". Now draw 2 e*
parallel to D E, and at any
suitable distance from the
center I ; this line will rep-
resent the intersection of the
base of the cylinder with a
plane parallel to the axes of
both, as before. The inter-
section of this plane with the
base of the smaller cylinder
will be found by setting off
from R a distance R p, equal
to I o, and drawing through
(i^G . \0" the point p a straight line
G[_ \ JR, .--' '\ '| parallel to Q 0. It is obvious
that the intersection of the
supposed plane with the con-
vex surfaces of the cylinders
will be represented by the
lines a f., c g, and d h, e i ;
and that, consequently, the
intersections of these lines
indicate points in the curves
sought. These points may
be multiplied indefinitely by
conceiving other planes to
pass through the cylinders,
and operating as before.
To find the curves of penetration of a cone and sphere (Fig. 187).
Let D S be the axis of the cone, A' L' B' the circle of its base, and the tri-
angle'A B S its projection on the vertical plane ; and let C, C', be the projec-
tions of the center, and the equal circles E' K' F' and E G F those of the
circumferences of the sphere.
This problem, like most others similar to it, can be solved only by the aid of
imaginary intersecting planes. Let a b represent the projection of a horizontal
plane ; it will cut the sphere in a circle whose diameter is a ~b, and which is par-
FIG. 186.
K
ORTHOGRAPHIC PROJECTION.
95
tially drawn from the center C' in the plan, as a' f V. Its intersection with
the cone is also a circle described from the center S' with the diameter c d as
c'f d' ; the points e' and/', where these two circles cut each other, are the hori-
zontal projections of two points in the lower curve, which is evidently entirely
hidden by the sphere. The points referred to are projected vertically upon the
line a b at e and/. The upper curve, which is seen in both projections, is ob-
FIG. 187.
tained by a similar process ; but it is to be observed that the horizontal cutting
planes must be taken in such positions as to pass through both solids in circles
which shall intersect each other. For our guidance in this respect it will
be necessary, first, to determine the vertices m and n of the curves of pene-
tration.
For this purpose, conceive a vertical plane passing through the axis of the
cone and the center of the sphere ; its horizontal projection will be the straight
line C' L' joining the centers of the two bodies. Let us also make the suppo-
sition that this plane is turned upon the line C C' as on an axis, until it be-
96
ORTHOGRAPHIC PROJECTION.
comes parallel to the vertical plane ; the points S' and L' will now have assumed
the positions S 2 and L a , and consequently the axis of the cone will be projected
vertically in the line D' S 3 ,
and its side in S 3 L 3 , cut-
ting the sphere at the
points p and. r. Conceive
the solids to have resumed
their original relative po-
sitions, it is clear that the
vertices or adjacent lim-
iting points of the curves
of penetration must be in
the horizontal lines p o
and r q, drawn through
the points determined as
above ; their exact posi-
tions on these lines may
be ascertained by project-
ing vertically the points
m' and ri, where the arcs
described by the points p
and r, in restoring the
cone to its first position,
intersect the line S' L'.
It is of importance,
further, to ascertain the
points at which the curves
of penetration meet the
outlines A S and S B of the
cone. The plane which
passes through these lines,
being projected horizon-
tally in A' B', will cut the
sphere in a circle whose
diameter is i' f ; this cir-
cle, described in the ele-
vation from the center C,
will cut the sides A S and
S B in four points, at
which the curves of pene-
tration are tangent to the
outlines of the cone.
To find the lines of
penetration of a cylinder
and a cylindrical ring or
torus (Fig. 188).
Let the circles A' E' B', F' G' K', represent the horizontal, and the figure
ORTHOGKAPHIC PROJECTION. 97
A C B D the vertical projection of the torus, and let the circle H'/' L', and
the rectangle II I M L be the analogous projections of the cylinder, which
passes perpendicularly through it. Conceive, as before, a plane, a b, to pass
horizontally through both solids ; it will obviously cut the cylinder in a circle
which will be projected in the base H' /' L' itself, and the ring in two other
circles, of which one only, part of which is represented by the arc/' b* b', will
intersect the cylinder at the points/' and b 3 , which, being projected vertically,
will give two points /and V* in the upper curve of penetration.
Another horizontal plane, taken at the same distance below the center line
A B as that marked a b is above it, will evidently cut the ring in circles coin-
ciding with those already obtained ; consequently the points/' and b 9 indicate
points in the lower as well as in the upper curves of penetration, and are pro-
jected vertically at d and e. Thus, by laying down two planes at equal dis-
tances on each side of A B, by one operation four points in the curves required
are determined.
To determine the vertices m and n, following the method explained in the
preceding problem, draw a plane n', passing through the axis of the cylinder
and the center of the ring, and conceive this plane to be revolved about the
point until it has assumed the position B', parallel to the vertical plane ;
the point n', representing the extreme outline of the cylinder in plan, will now
be at r', and, being projected vertically, that outline will cut the ring in two
points jo and r, which would be the limits of the curves of penetration in the
supposed relative position of the two solids ; and by drawing the two horizontal
lines r n and p m, and projecting the point n r vertically, the intersections of
these lines, m and n, are the vertices of the curves in the actual position of the
penetrating bodies.
The points at which the curves are tangents to the outlines H I and L M of
the cylinder, may readily be found by describing arcs of circles from the center
through the points H' and L', which represent these lines in the plan, and
then proceeding, as above, to project the points thus obtained upon the eleva-
tion. Lastly, to determine the points, as j, z, etc., where the curves are tan-
gents to the horizontal outlines of the ring, draw a circle P' s' j' with a radius
equal to that of the center line of the ring, namely, P. D ; the points of inter-
section z' and j' are the horizontal projections of the points sought.
Required to represent the section which would be made in this ring by a
plane, N' T', parallel to the vertical plane.
Such a section will be represented in its actual form and dimensions in the
elevation. To determine its outlines, let two horizontal planes, g q and i k,
equidistant from the center line A B, be supposed to cut the ring ; their lines
of intersection with it will have their horizontal projections in the two circles
g' o' and h' q 1 , which cut the given plane N' T' in o' and q'. These points being
projected vertically to 0, q, k. etc., give four points in the curve required. The
line N' T' cutting the circle A' E' B' at N', the projection N of this point is
the extreme limit of the curve.
The circle P' s'j', the center line of the rim of the torus, is cut by the planes
N' T' at the point s' 9 which, being projected vertically upon the lines D P and
C I, determines s and I, the points of contact of the curve with the horizontal
98
ORTHOGRAPHIC PROJECTION.
outlines of the ring. Finally, the points t and u are obtained by drawing from
the center a circle, T' v', tangent to the given plane, and projecting the point
of intersection v 1 to the points v and x, which are then to be replaced upon C D
by drawing the horizontals v t and x u.
Required to delineate the lines of penetration of a sphere and a regular hex-
agonal prism whose axis passes through the center of the sphere (Fig. 189).
FIG. 189.
The centers of the circles forming the two projections of the sphere are, ac-
cording to the terms of the problem, upon the axis C 0' of the upright prism,
which is projected horizontally in the regular hexagon D' E' F' G' H' I'. Hence
it follows that, as all the lateral faces of the prism are equidistant from the cen-
ter of the sphere, their lines of intersection with it will necessarily be circles of
equal diameters. The perpendicular face, represented by the line E' F' in the
plan, will meet the surface of the sphere in two circular arcs, E F and L M,
described from the center C, with a radius equal to c' V or a' c'. And the in-
tersections of the two oblique faces D' E' and F' G' will obviously be each
projected in two arcs of an ellipse, whose major axis d g is equal to e 1 f, and
the minor axis is the vertical projection of e' /'. But, as it is necessary to
ORTHOGRAPHIC PROJECTION.
99
draw small portions only of these curves, the following method may be em-
ployed :
Draw D G through the points E, F ; divide the portions E F and F G re-
spectively into the same number of equal parts, and, drawing perpendiculars
through the points of division, set off from F G the distances from the corre-
sponding points in E F to the circular arc E C F, as points in the elliptical
arc required. The remaining elliptical arcs can be traced by the same method.
Required to draw the lines of penetration of a cylinder and a sphere, the
center of the sphere being without the axis of the cylinder (Fig. 190).
FIG. 190.
Let the circle D' E' L' be the projection of the base of the given cylinder,
and let A B be the diameter of the given sphere. If a plane, as c' a", be drawn
parallel to the vertical plane, it will evidently cut the cylinder in two straight
lines, G G', H H'. This plane will also cut the sphere in a circle described
from the center C with a radius of half the line c' d' ; its intersection with the
lines G G' and II H' will give so many points in the curves sought, viz.,
G, H, I, K.
The planes a' l> and e'f, which are tangents to the cylinder, furnish respect-
100
ORTHOGRAPHIC PROJECTION.
ively only two points in the curves ; of these points, E and F alone are visible,
the other two, L and M, being concealed by the solid ; therefore, the planes
drawn for the construction of the curves must be all taken between a' V and
e' f. The plane which passes through the axis of the cylinder cuts the sphere
in a circle whose projection upon the vertical plane will meet at the points D,
N, and g, Ji, the outlines of the cylinder, to which the curves of penetration
are tangents.
To find the lines of penetration of a frustum of a cone and a prism
(Fig. 191).
The frustum is represented in the plan by two circles described from the
center C' ; and the horizontal lines M N" and M' N' are the projections of the
FIG. 191.
axis of a prism of which the base is square, and the faces respectively parallel
and perpendicular to the planes of projection.
In laying down the plan of this solid, it is supposed to be inverted, in order
that the smaller end of the cone and the lines of intersection of the lower sur-
face, F G, of the prism may be exhibited. According to this arrangement, the
letters A' and B' ought, strictly speaking, to be marked at the points I' and H',
and conversely ; but, as it is quite obvious that the part above M' N' is exactly
ORTHOGRAPHIC PROJECTION.
101
symmetrical with that below it, the distribution of the letters of reference
adopted in our figures can lead to no confusion.
The intersection of the plane F G- with the cone is projected horizontally
in a circle described from the center 0', with the diameter F' G'. The arcs
.!' F' A' and H' G' B' are the only parts of this circle which require to be
drawn. In the vertical projection the extreme points K, L, A, B need only
be found, for the lines of intersection are here projected straight.
To describe the curves formed by the intersection of a cylinder with the
frustum of a cone, the axes of the two solids cutting each other at right angles
(Fig. 192).
M
FIG. 192.
The projections of the solids are laid down in the figure precisely as in the
preceding example. The intersections of the outlines in elevation furnish, ob-
viously, four points in the curves of penetration ; these points are all projected
horizontally upon the line A' B'. Now pass a plane, as a b, horizontally through
both solids ; its intersection with the cone will be a circle of the diameter c d,
while the cylinder will be cut in two parallel straight lines, represented in the
elevation by a ft, and whose horizontal projection may be determined in the fol-
lowing manner: Conceive a vertical plane, f g, cutting the cylinder at right
102
ORTHOGRAPHIC PROJECTION.
angles to its axis, and let the circle g e f thereby formed be described from the
intersection of the axes of the two solids ; the line j li will now represent, in
this position of the section, the distance of one of the lines sought from the
axis of the cylinder. Set oif this distance on both sides of the point A',
and through the points k and a' thus obtained, draw straight lines parallel to
A' B' ; the intersections of these lines with the circle drawn from the center C'
of the diameter c d will give four points m', p', n, and o, which, being projected
vertically upon a b, determine two points, m and p, in the curves required.
In order to obtain the vertices or adjacent limiting points of the curves,
draw from the vertex of the cone a straight line, t e, touching the circle g e f,
and let a horizontal plane be supposed to pass through the point of contact e.
Proceed according to the method given above to determine the intersections
of this plane with each of the solids in question, the four points i' 9 r', q, and s,
which, being projected vertically upon the line e r, determine the vertices i and
r required.
THE HELIX.
The Helix is the curve
described upon the surface
of a cylinder by a point re-
volving round it, and at the
same time moving parallel
to its axis by a certain in-
variable distance during
each revolution. This dis-
tance is called the pitch of
the helix or screw.
Required to construct
the helical curve described
- by the point A 1 upon a cyl-
inder projected horizontally
in the circle A' C' F', the
pitch being represented by
the line A 1 A 8 (Fig. 193).
Divide the pitcli A 1 A 3
into any number of equal
parts, say eight; and
through each point of di-
vision, 1, 2, 3, etc., draw
straight lines parallel to the
ground-line. Then divide
the circumference A' C' F'
into the same number of
equal parts ; the points of
division, B', C', E', F', etc.,
will be the horizontal pro-
jections of the different po-
sitions of the given point
ORTHOGRAPHIC PROJECTION.
103
during its motion round the cylinder. Thus, when the point is at B' in the
plan, its vertical projection will be the point of intersection B of the perpen-
dicular drawn through B' and the horizontal drawn through the first point of
division. Also, when the point arrives at C' in the plan, its vertical projection
is the point C, where the perpendicular drawn from C' cuts the horizontal
passing through the second point of division, and so on for all the remaining
points. The curve A 1 B C F A 3 , drawn through all the points thus obtained,
is the helix required.
To draw the vertical elevation of the solid contained between two helical sur-
faces and two concentric cylinders (Fig, 193).
A helical surface is generated by the revolution of a straight line round the
axis of a cylinder, its outer end
moving in a helix, and the line
itself forming with the axis a
constant and invariable angle.
Let A' C' F' and K' M' 0'
represent the concentric bases
of the cylinders, whose com-
mon axis S T is vertical ; the
curve of the exterior helix A 1
C F A 3 is the first to be drawn
according to the method above
shown. Then, having set off
from A 1 to A 2 the thickness
of the required solid, draw
through A 2 another helix
equal and similar to the for-
mer. Now construct, as above,
similar helices, K C and K 2
C 2 O 2 , of the same pitch as the
last, but on the interior cylin-
der. The lines A' K', B' L',
C' M', etc., represent the hori-
zontal projections of the va-
rious positions of the generat-
ing straight line, which, in
the present example, has been
supposed to be horizontal ;
and these lines are projected
vertically at A 1 K, B L, etc.
It will be observed that in
the position A 1 K the generat-
ing line is projected in its Flo> 194
actual length, and that at the
position C' M' its vertical projection is the point C. The same remark applies
to the generatrix of the second helix. The parts of both curves which are
visible in the elevation may be easily determined by inspection.
104 ORTHOGRAPHIC PROJECTION.
To determine the vertical projection of the solid formed by a sphere moving
in a helical curve (Fig. 194).
Let A' C' E' be the base of a cylinder, upon which the center point C' of a
sphere whose radius is a' C' describes a helix, which is projected on the vertical
plane in the curve A C E J, determined as before. From the various points
A, B, C, D , in this curve, as centers, describe circles with the radius
a' C' ; these denote the various positions of the sphere during its helical mo-
tion ; and, if lines be drawn touching them, the curves thereby formed will
constitute the figure required. One of these curves disappears at 0, but reap-
pears again at I. The exterior and interior circles of the plan represent the
horizontal projection of the solid in question.
The conical helix differs from the cylindrical one in that it is described on
the surface of a cone instead of on that of a cylinder ; but the construction
differs but slightly from the one described. By following out the same prin-
ciples, helices may be represented as lying upon spheres or upon any other
surfaces of revolution.
In the arts are to be found numerous practical applications of the helical
curve, as wood and machine screws, gears, and staircases, the construction of
which will be still further explained under their appropriate heads.
DEVELOPMENT OF SURFACES.
The development of the surface of a solid is the drawing or unrolling on a
plane the form of its covering ; and if that form be cut out of paper, it would
exactly fit and cover the surface of the solid. Frequently in practice, the form
of the surface of a solid is found by applying paper or thin sheet-brass directly
to the solid, and cutting it to fit. Tin and copper smiths, boiler-makers, etc.,
are continually required to form from sheet-metal forms analogous to solids ;
to execute which they should be able to construct geometrically the develop-
ment of the surface of which they are to make the form.
The development of the surface of a cylinder is evidently but a flat sheet,
of which the circumference is one dimension while its length is the other.
To develop the surface of a cylinder formed by the intersection of another
equal cylinder, as the knee of a stove-pipe (Fig. 195).
Let A B C D be the elevation of the pipe or cylinder. Above A B describe
the semicircle A' 4' B' of the same diameter as the pipe ; divide this semicir-
cle into any number of equal parts, eight for instance ; through these points,
1', 2', 3', etc., draw lines parallel to the side A of the pipe, and cutting the
line C D of the intersection of the two cylinders. Lay off A" B" equal to the
semicircle A' 4' B', and divided into the same number of equal parts ; through
these points of division erect perpendiculars to A" B", and on these perpen-
diculars lay off the distances A" C", 1" 1", 2" 2", 3" 3", and so on, corresponding
to A C, 1 1, 2 2, 3 3, etc., in preceding figure. Through the points C", 1", 2",
D", draw connecting lines, and we have the developed surface required.
It is to be remarked that this gives but one half of the surface of the pipe, the
other being exactly similar to it.
To develop the surface of a cylinder intersected 1 by another cylinder, as in
the formation of a "[-pipe (Fig. 196).
ORTHOGRAPHIC PROJECTION.
105
7 (! 5 4 3 2 1 A
7" 6" 5" 4" 3" 2" 1" A"
B* T 6" 5" A
r 3" " /" ^4
y^
Ss
6
X
5
\
FIG. 195.
The construction is similar to the
preceding, and, as the same letters
and figures are preserved relatively,
the demonstration will be easily un-
derstood from the foregoing.
To develop the surface of a right
cone (Fig. 197).
From C' as a center, with a ra-
dius, C' A', equal to the inclined
side A C of the cone, describe an arc
of a circle A' B' A" ; on this arc lay
off the distance A' B' A" equal to
the circumference of the base of the
cone ; connect A' 0' and C' A", and
A' B' A" C' is the developed surface required.
To develop the surface of the frustum of a cone, D A B E (Fig. 197).
D' E' D" is the development of the cut-off cone C D E as shown by the pre-
ceding construction, and we therefore have A' B' A" D" E' D as the developed
surface of the frustum.
To develop the surface of a frustum of a cone, when the cutting plane a h
is inclined to the base (Fig. 197).
On A B, the base, describe the semicircle A 3' B ; divide the semicircle into
any number of equal parts, six for instance ; from each point of division, 1', 2',
3', 4', 5', let fall perpendiculars to the base ; at 1, 2, 3, 4, 5, connect each of
these last points with the apex 0. Divide now the arc A' B' A*, equal to the
base A 3 B, into twelve equal parts ; each of these parts by the construction
is equal to the arc A 1', V 2' ; connect these points of division with the
point C' ; on C' A' take C' a' equal to C a, a being the point at which the
106
ORTHOGRAPHIC PROJECTION.
E
plane cuts the inclined side of the cone ; in the same way on C' B', lay off
C' b' equal to C b.
It is evident that all the lines connecting the apex C with the base, included
within the two inclined sides, are rep-
resented as less than their actual length,
and must be projected on the inclined
sides to determine their absolute di-
mensions ; project, therefore, the points
1", 2", 3", 4", 5", at which the cutting
plane intersects the lines C 1, C 2, C 3,
04, 05, by drawing parallels to the
base through these points to the in-
clined side C B. Now lay off C' 1"",
0' 2"", etc., equal to C 1'", C 2", etc.;
connect the points a', I"", 2"", - - b 1 ,
] A a", and we have the developed sur-
face a' A' B' A" a" b' required.
A
B'
FIG. 198.
FIG. 199.
ORTHOGRAPHIC PROJECTION.
107
To develop the surface of a sphere or ball (Figs. 198, 199).
It is evident that the surface can not be accurately represented on a plane.
It may be done approximately by a number of gores. Let CAB (Fig. 199)
be the eighth of a hemisphere ; on C D describe the quarter circle D A c ;
divide this arc into any number of equal parts, six for instance ; from the points
of division 1, 2, 3, ... let fall perpendiculars on C D, and from the intersec-
tions with this line describe arcs 1' 1", 2' 2", 3' 3", . . . cutting the line C B at
1", 2", 3", ; on the straight line C' D' (Fig. 198), lay off C' D' equal to
the arc D A c, with as many equal divisions ; then
from either side of this line lay off 1'" 1"", 2'" 2""
D' B' equal to the arcs 1' 1', 2' 2" D B
(Fig. 199). Connect the points C', 1"", 2, and
C' A' B' is approximately the developed surface.
It is to be remarked that, in the preceding demon-
strations, the forms are described to cover the surface
only ; in construction, allowance is to be made for
lap by the addition of margins on each side as neces-
sary. It is found difficult, in the formation of hemi-
spherical ends of boilers, to bring all the gores to-
gether at the apex ; it is usual, therefore, to make them, as shown (Fig. 200),
by cutting short the gores, and surmounting the center with a cap-piece.
SHADE-LINES.
In outline drawings, or drawings which consist simply of the lines employed
to indicate the form of the object represented, the roundness, the flatness, or
the obliquity of individual surfaces, is not indicated by the lines, although it
may generally be inferred from the relation of different views of the same part.
The direct significance of an out-
\
FIG. 200.
line drawing may, however, be con-
siderably increased, by strengthen-
ing those lines which indicate the
contours of surfaces resting in the
shadow ; and this distinction also
improves the general appearance of
the drawing. The strong lines, to
produce the best effect, ought to
be laid upon the sharp edges at
the summits of salient angles ; but
bounding lines for curve surfaces
should be drawn finely, and should
be but slightly, if at all, strength-
ened on the shade side. This dis-
tinction assists in contrasting flat
and curve surfaces. To understand
and apply the shade-lines, however, we must know the direction in which
the light is supposed to fall upon the object, and thence the locality of the
shadows.
FIG. 201.
108
ORTHOGRAPHIC PROJECTION.
It is necessary, for the explicitness of the drawing, that, firstly, the light be
supposed to fall upon the object in parallel lines, that all the parts may be
shade-lined according to one uniform rule ; secondly, that the light should be
supposed to fall upon the object obliquely, as in this way both the horizontal
and vertical lines may be relieved by shading.
Fig. 201 represents the drawing of a cube, with its projections on a vertical
FIG. 202.
FIG. 203.
and on a horizontal plane, or in elevation and plan, all in perspective. The
arrows show the directions in which the light is supposed to fall : in space
diagonally through the body of the cube and in projection diagonally through
the squares representing the plan and elevation of the cube. The projections
of the rays, therefore, form -angles of 45 with the ground-line, which line is
represented in the figure by B D. In the old method, still used in topograph-
ical and by many in mechanical drawings, the light is supposed to fall in space,
as if A D were the ground-line, but the shade-lines in the vertical plane are the
same in both methods.
Copies of a few of the preceding projections are here given, with the
proper shade-lines, according to the first or French method (Fig. 201). The
outlines to be shaded can be determined, ordinarily, by mere inspection and by
using a 45 triangle. Such a triangle gives the direction of the projected rays,
ORTHOGRAPHIC PROJECTION.
109
and determines the surfaces in shadow. Fig. 202 is a reproduction of Fig. 171 ;
Fig. 203 is a reproduction of Fig. 176 ; Fig. 204 is a reproduction of Fig. 184 ;
Fig. 205 is a reproduction of the plan of Fig. 188. The outlines on which
the light falls are represented by fine lines, the
others by coarse lines. In general, it is not cus-
tomary to use more than two grades of lines,
one for the outlines in light, and the other for
those in shade ; but, for lines parallel with the
FIG. 204.
FIG. 205.
rays of light, medium lines are sometimes used, and sometimes the shade-lines
are proportioned to the depth of the surfaces to which they belong, below the
original surfaces from which the shadows arise.
SHADES AND SHADOWS.
LIGHT is diffused through space in straight lines, and the lines of light are
called ray*. When the source of light is situated at a very great distance from
the illuminated objects, as in the case of the sun with relation to the earth, the
rays of light do not sensibly diverge, and may be regarded as exactly parallel to
each other. Such is the case in mechanical drawings, where the objects to be
represented are always regarded as illuminated by the solar light.
Light is called direct when it is transmitted to an object without the inter-
vention of any opposing medium. But, as all bodies subjected to the action of
light possess, in a greater or less degree, the property of giving out a certain
portion of it to the surrounding objects, this reflected light becomes in its turn,
though with greatly diminished intensity, a source of illumination to those
objects which are deprived of direct light.
Everything which tends to intercept or prevent the direct light from falling
upon a body, produces upon the surface of that body a degree of obscurity
of greater or less intensity ; this is called a shade or shadow. Such effects are
usually classified as direct shadows and cast shadows.
The shade proper, or direct shadow, is that which occurs on that portion of
the surface of a body which is situated opposite to the enlightened part, and is
the natural result of the form of the body itself, and of its position with regard
to the rays of light. The cast shadow, on the other hand, is that which is
produced upon the surface of one body by the interposition of another between
the former and the source of light ; thus intercepting the rays which would
otherwise illuminate that surface. Cast shadows may also obviously be pro-
duced upon the surface of a body by the form of the body itself ; as, for exam-
ple, if it contain projecting or concave parts.
The limit of the direct shadow on any body, whatever may be its form or
position, is a line of greater or less distinctness, termed the line of shade ; this
line is, of course, determined by the contact of the luminous rays with the
surface of the body ; .and, if these rays be prolonged till they meet a given sur-
face, by joining all the points of intersection with that surface, we obtain the
outline of the shadoiv cast upon it by the part of the body which is deprived of
light.
The rays of light being regarded as parallel to each other, it is obvious that,
in the delineation of shadows, it is only necessary to know the direction of one
of them ; and, as that direction is arbitrary, we have adopted the usual and
confessedly the most convenient mode of regarding the rays as in all cases
falling in the direction of the diagonal of a cube, of which the sides are parallel
SHADES AND SHADOWS.
Ill
to the planes of projection. This is graphically shown in Fig. 201 of the pre-
ceding chapter. The projections of the ray form each an angle of 45 with
the ground-line. This is not true of the ray itself in space, for that forms
an angle of 54 44' with the ground-line, and an angle of 35 16' with each of
the planes of projection.
To find the shadow of a point, as A, A' (Fig. 206), on either plane of pro-
jection, the vertical, for instance, we draw a line through the horizontal pro-
jection of the point A' at an angle of 45 with the ground-line, and at the
point of intersection of those lines, a', erect a perpendicular to intersect the
vertical projection of the ray through A, which will be at the point a, the
shadow in question.
This, as may readily be seen, is simply finding the point of intersection of
the ray passing through the point and the vertical plane of projection. The
converse of this method will as easily determine the shadow of the point on
the horizontal plane.
The line A a in the elevation being equal in every case to the line A' a' in
the plan, it will in some cases be found more convenient to use the compasses
instead of a geometrical construction ; for example, in place of projecting the
point a' by a perpendicular to the ground-line, in order to obtain the position
of the required shadow a, that point may be found by simply setting off upon
the line A a a distance equal to A' a'.
In the following illustrations the same letter accented is employed in the
plan as in the elevation to refer to the same object or point.
Required to determine the shadow cast upon the vertical wall X Y by the
straight line A B (Fig. 206).
It is obvious that in this case the shadow itself will be a straight line ;
hence, to solve the problem, it is only necessary to find two points in that line.
We have seen that the position of the shadow thrown by the point A is at a;
lillllillililiilliiiililiillllilllllllll liliillllillllllllllllll
FIG. 206.
by a similar process we can easily determine the point #, the position of the
shadow thrown by the opposite extremity B of the given line ; the straight
line a b, which joins these two points, is the shadow required.
It is evident, from the construction of this figure, that the line a b is equal
112
SHADES AND SHADOWS.
and parallel to the given line A B ; this results from the circumstance that the
latter is parallel to the vertical plane X Y. Hence, when a line is parallel to a
plane, its shadow upon that plane is a line ivhich is equal and parallel to it.
Suppose now that, instead of a mere line, a parallel slip of wood or paper,
A B C D, be taken, which, for the sake of greater simplicity, we shall conceive
as having no thickness. The shadow cast by this object upon the same verti-
cal plane X Y is a rectangle a b c d, equal to that which represents the projec-
tion of the slip, because all the edges of the latter are parallel to the plane upon
which the shadow is thrown. Hence, in general, when any surface, whatever
may be its form, is parallel to a plane, its shadow thrown upon that plane is a
figure similar to it, and similarly situated. This principle facilitates the de-
lineation of shadows in many cases. In the present example, an idea may
be formed of its utility ; for, after having determined the position of any one
of the points a, b, c, d, the figure may be completed by drawing lines equal and
parallel to the sides of the slip, without requiring to go through the operations
in detail.
When the object is not parallel to the given plane, the shadow cast is no
longer a figure equal and similarly placed ; the method of determining it re-
mains, however, unchanged ; thus (Fig. 207), take the portion A E of the slip
A B, which throws its shadow on the plane X Y ; draw the projections of the
rays of light A a, E e, C c, F/, and A' a'. E' e', and project a' vertically to a, c,
and e' to e, f ; connect a, e, f, c, and we have the outline of the shadow of the
slip A E.
By an exactly similar construction we have the shadow of the portion E B
on the plane Y Z, which, being inclined to the plane of projection in a direction
contrary to X Y, necessarily causes the shadow to be broken, and the part e d
to lie in a contrary direction to af.
The determination of the shadow of the slip upon a molding placed on
the plane X Y parallel to the slip (Fig. 208) can be readily determined by an
inspection of the figure.
E'
FIG. 210.
FIG. 211.
When the slip is placed perpendicularly to a given plane, X Y (Fig. 209), on
which a projecting molding, of any form whatever, is situated, the shadow of
the upper side A' B', which is projected vertically in A, will be simply a line,
SHADES AND SHADOWS.
113
A a, at an angle of 45, traversing the entire surface of the molding, and pro-
longed unbroken beyond it. This may easily be demonstrated by finding the
position of the shadow of any number of points such as D', taken at pleasure
upon the straight line A' B'. The shadow of the opposite side, projected in C,
will follow the same rule, and be denoted by the line C c, parallel to the
former. Hence, as a useful general rule : in all cases where a straight line is
perpendicular to a plane of projection, it throws a shadow upon that plane
in a straight line, forming an angle of 45 ivith the ground-line.
When the slip is set horizontally in reference to its own surface, and per-
pendicularly to the given plane X Y (Fig. 210), the shadow commences from
the side D B, which is in contact with this plane, and terminates in the hori-
zontal line a c, which corresponds to the opposite side A C of the slip.
To find the shadow cast by a slip, A B C D, upon a curved surface, either
convex or concave, whose horizontal projection is represented by the line X e' Y
(Fig. 211).
This construction is similar to the foregoing illustrations, and requires no
explanation more than the figure.
Required to find the shadow cast upon a vertical plane, X Y, by a given
circle parallel to it (Fig. 212).
Let C, C', be the projections of the center of the circle, and R, R/, those of
the rays of light.
The position of the shadow of the center C, according to the rules already
fully developed, is easily fixed at c ; from which point, if a circle equal to the
given circle be described, it will represent the outline of the required shadow,
according to the principle previously enunciated on page 112.
When the circle is perpendicular to both planes of projection (Fig. 213), its
Li'
FIG. 213.
D'
FIG. 214.
projection upon each will obviously be represented by the equal diameters A B
and 0' D', perpendicular each to the ground-line. To determine the cast
shadow, describe the given circle upon both planes, as indicated in the figures,
and divide the circumference of each into any number of equal parts ; then,
having projected the points of division, as A a , E 2 , C 2 , etc., to their respective
8
114
SHADES AND SHADOWS.
diameters A B and C' D', draw from them lines parallel to the rays of light,
which, by their intersection with the given plane, will indicate so many points
in the outline of the cast shadow.
If the given circle be horizontal (Fig. 214), its shadows cast upon the
straight and curved portions of the vertical plane X Y become ellipses, which
must be constructed by means of points, as indicated in the figure.
If the plane of the circle is situated perpendicularly to the vertical projec-
tion of the luminous rays (Fig. 215), the method of constructing the cast
shadow does not differ from that pointed out in reference to Fig. 214. It is
obvious that, instead of laying down the entire horizontal projection of this
circle, all that is necessary is to set off the diameter D' E' equal to A B, be-
cause the shadow of this diameter, transferred in the usual way, gives the
major axis of the ellipse which constitutes the outline of the shadow sought,
while its minor axis is at once determined by a b, equal and parallel to A B.
To delineate the shadow
of a cirde paralhl to the
vertical plane of projection,
throwing its shadow at once
upon two plane surfaces in-
clined to each other (Fig.
216), all that it is neces-
sary specially to point out
is, that the points d and e
are found by drawing from
Y a line Y D', parallel to
the rays of light, and pro-
jecting the point D' to D
andE.
We may here remark
that, in every drawing
where the shadows are to be inserted, it is of the utmost importance that the
projections which represent the object whose shadow is required should be
exactly defined, as well as the surface upon which this shadow is cast ; it is
therefore advisable, in order to prevent mistakes and to insure accuracy, to
draw the figures in India ink, and to erase all pencil-marks before proceeding
to the operations necessary for finding the shadows.
To find the outline of the shadow cast upon both planes of projection by a
regular hexagonal pyramid (Fig. 217).
It is obvious that the three sides A' B' F, A' B' C', and A' C' D', alone
receive the light ; consequently the edges A' F' and A' D' are the lines of shade.
To solve this problem, then, we have only to determine the shadow cast by
these two lines, which is accomplished by drawing from the projections of the
vertex of the pyramid the lines A b' and A' a' parallel to the ray of light, then
raising from the point b' a perpendicular to the ground-line, which gives at a'
the shadow of the vertex on the horizontal plane (on the other side of the
ground-line), and finally by joining this last point a' with the points D' and F' ;
the lines D' a' and F' a' are the outlines of the required shadow on the hori-
FIG. 215.
ir
FIG. 216.
SHADES AND SHADOWS.
115
2ontal plane. But, as the pyramid happens to be situated sufficiently near the
vertical plane to throw a portion of its shadow toward the vertex upon it, this
portion may be found by raising from the point c, where the line A' a' cuts the
ground-line, a perpendicular c a, intersecting the line A V in a ; the lines a d
and a e joining this point with those where the horizontal part of the shadow
meets the ground-line, will be its outline upon the vertical plane.
FIG. 217.
FIG. 218.
To determine the limit of shade on a cylinder placed vertically, and likewise
its shadow cast upon the two planes of projection (Fig. 218).
The lines of shade on a cylinder situated as indicated are at once found by
drawing two tangents to its base, parallel to the ray of light, and vertically
projecting through the points of contact lines parallel to the axis of the cyl-
inder.
Draw the tangents D' d' and C' c' parallel to the rays of light ; these are the
outlines of the shadow cast upon the horizontal plane. Through the point of
contact C' draw the vertical line C C' ; this line denotes the line of shade upon
the surface of the cylinder. It is obviously unnecessary to draw the perpen-
dicular from the opposite point D', because it is altogether concealed in the
vertical elevation of the solid. In order to ascertain the points C' and D' with
accuracy, draw through the center 0' a diameter perpendicular to the rays of
light.
Had this cylinder been placed at a somewhat greater distance from the
vertical plane of projection, its shadow would have been entirely cast upon the
horizontal plane, in which case it would have terminated in a semicircle drawn
from the center o', with a radius equal to that of the base. But, as a portion
of the shadow of the upper part is thrown upon the vertical plane, its outline
will be denned by an ellipse drawn in the manner indicated in Fig. 214.
To find the line of shade in a reversed cone, and its shadow cast upon the
two planes of projection (Fig. 219).
116
SHADES AND SHADOWS.
From the center A' of the base draw a line parallel to the ray of light ;
from the point a' where it intersects the perpendicular, describe a circle equal
to the base, and from the point A' draw the lines A' b' and A' c' 9 tangent to
this circle ; these are the outlines of the shadow cast upon the horizontal plane.
Then from the center A' draw the radii A' B' and A' C' parallel to a' V and
a' c' ; these radii are the horizontal projections of the lines of shade, the former
of which, transferred to B D, is alone visible in the elevation. But, in order
to complete the outline of the shadow, it is necessary to project the point C' to
C, from which, by a construction which will be manifest by inspecting the
figures, we derive the point c and the line c d as part of the cast shadow of the
line C' A'. The rest of the outline of the vertical portion of the cast shadow
is derived from the circumference of the base, as in Fig. 218.
To find the line of shade and the shadow of a horizontal cylinder inclined to
the vertical plane (Fig. 220).
FIG. 219.
FIG. 220.
The construction in this case is the same as that explained by Fig. 218. Of
the horizontal lines of shade A B and C D, the latter alone is visible in the
elevation, while, on the other hand, the former, A B alone, is seen in the plan,
where it may be found by drawing a perpendicular from A meeting the base
F' G' in A'. The line A' E', drawn parallel to the axis of the cylinder, is the
line of shade required. Project the shadow of the line A B on the vertical
plane as in previous examples, and the construction will define the outline of
the shadow of the cylinder.
The example here given presents the particular case in which the bases of
the cylinder are parallel to the direction of the rays of light. In this case, to
determine the line A' E', lay off the angle A' L A 2 equal to 35 16', which the
ray of light makes with the horizontal plane, so that the side A a L shall be
tangent to the circle F' A 2 G' (which represents the base of the cylinder laid down
on the horizontal plane) ; through the point of tangency A 2 , draw a line, A' E',
parallel to the axis of the cylinder, which will be the line of shade, as before.
SHADES AND SHADOWS.
117
To determine the shadows cast upon a cylinder by various shaped caps.
Fig. 221 represents a cylinder upon which a shadow is thrown by a rec-
tangular prism, of which the sides are parallel to the planes of projection.
The shadow in this case is derived from the edges A' D' and A' E', the first
of which, being perpendicular to the plane of projection, gives, according to
principles already laid down, a straight line at an angle of 45 for the outline
of its shadow, whereas the side A' E' being parallel to that plane, its shadow
is determined by a portion of a circle, a b c, described from the center o.
If the prism be hexagonal (Fig. 222), or a cylinder be substituted for it
J)'~-
7?r/. for
Jj' 6\
. K
' --J \
\----r
\
C'\
V
(A
FIG. 221.
FIG. 222.
FIG. 223.
(Fig. 223), the mode of construction remains the same. But it should be
observed that it is best in all such cases to commence by finding the points
which indicate the main direction of the outline. To ascertain the point a
at which the shadow commences, draw from a' the line a' A' at an angle of
45, which is then to be projected vertically to a A. Then the highest point Z
(Fig. 223) should be determined by the intersection of the radius 0' B' (drawn
parallel to the ray) with the circumference of the base of the cylinder on
which the required shadow is cast ; and, finally, the point c, where the outline
of the cast shadow intersects the line of shade, should be determined by a simi-
lar process.
To determine the shadows cast upon a hexagonal prism ~by the same caps.
Fig. 224 represents a hexagonal prism upon which a shadow is thrown
by a rectangular prism.
Fig. 225 represents a hexagonal prism upon which a shadow is cast by
another hexagonal prism.
Fig. 226 represents a hexagonal prism upon which a shadow is cast by a
cylinder. These three cases do not materially 'differ from the preceding three,
and can easily be understood from an examination of the figures.
To define the shadows cast upon the interior of a hollow cylinder, in section,
by itself, and by a circular piston fitted into it (Fig. 227).
The figure shows the section of a steam-cylinder, by a plane passing through
its axis, with its piston and rod in full.
Conceive, in the first instance, the piston P to be removed ; the shadow cast
118
SHADES AND SHADOWS.
into the interior of the cylinder will then consist, obviously, of that projected
by the vertical edge B C, and by a portion of the horizontal edge B A. To find
the first, draw through B' a line, B' V, at an angle of 45 with B' A' ; the point
D
\i
J e
FIG. 224.
FIG. 225.
FIG. 226.
hA
b', where this line meets the interior surface of the cylinder, being projected
vertically, gives the line b f as the outline of the shadow sought. Then, paral-
lel to the direction of the light, draw a tangent at F' to the inner circle of the
base ; its point of contact, being projected to F in the ele-
vation, marks the commencement of the outline of the
shadow cast by the upper edge of the cylinder. The point
b, where it terminates, will obviously be the intersection
of the straight line / b already determined, with a ray, B b,
from the upper extremity of the edge B C ; and any inter-
mediate point in the curve, as e, may be found in precisely
the same way. The outline of the shadow required will
then be the curve F e b and the straight line b f. Sup-
pose, now, the piston P and its rod T to be inserted into
the cylinder, as shown. The lower surface of the piston
will then cast a shadow upon the interior surface of the
cylinder, of which the outline D d h o may be formed in
the same way, as will be obvious from inspection of the
figures and comparison of the letters of reference. The
FIG. 227. piston-rod T being cylindrical and vertical, it casts also
its shadow into the interior of the cylinder ; it will obviously consist of the
rectangle ij I k drawn parallel to the axis.
To find the shadow cast in the interior of a hollow cylinder, surmounted by
a circular disk or cover, sectioned through the center, where it is also penetrated
by a cylindrical aperture (Fig. 22S).
The construction necessary for finding the outlines of the cast shadow will
obviously be the same as already laid down. To know beforehand what parts
of the upper and lower edges of the central aperture cast their shadows into
the interior of the cylinder, in order to avoid unnecessary work, we should first
determine the position of the point of intersection, c, of the two curves b cf
and ace, shadows of these edges, which is the cast shadow of the lowest point,.
SHADES AND SHADOWS.
119
0, in the curve D C, previously laid down in the circular opening of the cover,
in the manner indicated in the previous example.
To find the shadow cast in the interior of a cylinder, in section, inclined to
the horizontal plane (Fig. 229).
In any convenient part of the paper, draw the diagonal m o parallel to the
line of light A' E, and construct a square m n o p (Fig. 230) ; from one of the
0'
extremities, o, draw the line o r parallel to A' B', and through the opposite
extremity, m, draw a perpendicular, r s, to this line, and set oif on the perpen-
dicular the distance r s equal to the side of the square, and join s o. Now,
draw through the point A', in the original figure, a line, A' #', parallel to s o,
intersecting the circle A' a' B' in the point a', which, being projected by a line
parallel to the axis of the cylinder,
and meeting the line A a, drawn at
an angle of 45, gives the first point
a in the curve C d a. The other
points will be obtained in like man-
ner, by drawing at pleasure other
lines, such as D' d', parallel to A' a 1 .
To find the outline of the shadow
cast into the interior of a hollow
hemisphere (Fig. 231).
Let A B D represent the hori-
zontal projection of a concave hem-
isphere. Here it is sufficiently ob-
vious that, if we draw through the
center of the sphere a line perpen-
dicular to the ray of light A C, the
points B and D will at once give
the extremities of the curves sought. On any point of B D produced as 0',
construct the semicircle A' a' C' with a radius. A' 0', equal to A 0. At A'
draw the line A' a', making an angle of 35 16' with A' C'. This angle, as has
been said before, is equal to that made by the ray of light in space with the
FIG. 231.
120
SHADES AND SHADOWS.
/ft-
0'
r \
planes of projection. The point of intersection of the line with the semicircle
at a' projected to a, gives a point of the outline of the shadow. Similar sec-
tions, as E F parallel to A C, will give other points. But, as this outline cover
of the shadow is an ellipse whose axes are B D and twice a, it may be
constructed, when the point a is determined, by the ordinary methods for
ellipses.
To construct the outlines of the shadow in the interior of a concave sur-
face, formed l>y the combination of a hollow semi-cylinder and a quadrant of
a hollow sphere, called a niche
(Fig. 232).
We already know the mode
of tracing the shadows upon
each of these figures separately.
\ \X \ Thus, the shadow of the circu-
.}'-. >- - \ lar outline upon the spherical
,%>-. >-B portion is part of an ellipse,
;^->. _ i c D, found precisely as in the
previous example. The point
e, where this ellipse cuts the horizontal diam-
eter A F, limits the cast shadow upon the spher-
ical surface ; therefore, all the points beneath it
must be determined upon the cylindrical part.
Through A' in the plan draw the line A' a' par-
allel to the ray of light ; project a' till it inter-
sects the line of light A a in the elevation at a.
The line of shadow below a is the shadow of
the edge of the cylinder, and must therefore be
a straight line. The line of shadow between a
and e is produced by the outline of the circular
part falling on a cylindrical surface, and is es-
tablished as in previous constructions.
To find the line of shade in a sphere, and
the outline of its shadow cast upon the hori-
zontal plane (Fig. 233).
The line of shade in a sphere is simply the
circumference of a great circle, the plane of
which is perpendicular to the direction of the
luminous rays, and consequently inclined to the two planes of projection. This
line will therefore be represented in elevation and plan by two equal ellipses,
the major axes of which are obviously the diameters C D and C' D', drawn at
an angle of 45.
To find the minor axes of these curves, assume any point, O 2 , upon the pro-
longation of the diameter of the perpendicular C' D', draw through this point
the straight line O 2 o', inclined at an angle of 35 16', to A' B' or its parallel, and
erect upon it the perpendicular E 3 F 2 . The projection of the two extremities
E 9 and F 2 upon the line A' B' will give in the plan the line E' F' for the
length of the required minor axis of the ellipse, i. e., of the line of shade in the
FIG. 232.
SHADES AND SHADOWS.
121
plan ; and this line, being again transferred to the elevation, determines the
minor axis E F of the line of shade in the elevation.
Supposing it were required to draw these ellipses, not by means of their
axes, but by points, any number of these may be obtained by making horizontal
sections of the sphere. Thus, for example, if we draw the chord G- H parallel
to A' B', to represent one of these sections, and from the point a, where it cuts
the diameter E a F 2 , if we draw a perpendicular to A' B', the points a' a', where
FIG. 233.
it intersects the circumference of the circle G' a' H', representing the section
G H in plan, will be two points in the line of shade required. These points
may be transferred, by supposing a section, g h, to be made in the elevation cor-
responding to G- H, and projecting the points a! a' by perpendiculars to g h,
the line representing the cutting plane.
The outline of the shadow cast by the sphere upon the horizontal plane is
also obviously an ellipse ; it may be constructed either by means of its two axes
or by the help of points, in the manner indicated in the figure.
To draw the line of shade on the surface of a ring of circular section, in
vertical section, elevation, and plan (Fig. 234).
122
SHADES AND SHADOWS.
AVe shall first point out the mode of obtaining those primary points in the
curve which are most easily found, and then proceed to the general case of de-
termining any point whatever.
If tangents be drawn to the
circles represented in both ele-
vation and section, parallel to
the rays of light, their points
of contact, #, b, c, d, will be
the starting-points of the re-
quired lines of shade.
Again, the intersections of
the horizontal lines ae,dg, cf,
drawn through these points,
with the axis of the ring, will
give so many new points, e, g,
f, in the curve. These points
are denoted in the plan by
setting off the distances a e
and c f upon the vertical line
g' D, on both sides of the cen-
ter C'.
Further, the diameter F'
G', drawn at an angle of 45,
determines, by its intersec-
tions with the exterior and
interior circumferences of the
ring, four other points, F', t',
x', and G', in the curve in
question ; these points are all
to be projected vertically upon
the line A B.
And, lastly, to obtain the
lowest points, I l y draw tangents
to the circles in elevation and
section at an angle of 35 16' with the ground-line, and transfer the distances be-
tween the points of contact, s, s, and the axis of the ring, to the diameter E' J',
where they are denoted by I' I' ; these points are then projected to
1) I, upon the horizontal lines drawn through the same points s, s.
To determine any other points, draw through the center C' a
diameter, I' H', in any direction. Draw through o', one of the
angular points of the horizontal projection of the cube, made at
any convenient size (Fig. 235), a straight line, o' r', parallel to
I' H', and from the opposite point m' draw a perpendicular, m' r',
to o' r'. Then, having revolved the point r' to r a , and projected
r 2 to r, join o and r.
Applying this construction to the figures before us, we now draw tangents
to the circles represented in elevation and section parallel to the line o r, and.
234.
SHADES AND SHADOWS.
123
taking as radii the distances from their respective points of contact, h and I, to
the axis of the ring, we describe corresponding circles about the center C' of
the plan. We thus obtain four other points in the curves required, namely,
I', i', li' ', and H', which may also be projected upon the horizontal lines drawn
through the points h or I.
By drawing the straight line J' K' so as to form with F' G' the same angle
which the latter makes with the line H' I', we obtain, by the intersection of
that line with the circles last named, four other points of the curves in
question.
To determine the shadows cast upon the surfaces of grooved pulleys
(Fig. 236).
The construction of cast shadows upon surfaces of the kind now under con-
sideration is founded upon the principle, already announced, that when a circle
is parallel to a plane, its shadow, cast upon that plane, is another circle equal
to the original circle.
Take the case of a circular-grooved pulley ; the cast shadow on its surface
is obviously derived from the circumference of the upper edge A B. To deter-
mine its outline, take any horizontal line
D E in upper fig. and describe from the cen-
ter C' a circle with a radius equal to the half
of that line ; then draw through the same
center a line parallel to the ray of light,
which will intersect the plane D E in c ;
lastly, describe from the point c', as a cen-
ter, an arc of a circle with a radius equal to
A C ; the point of intersection, a', of this
arc, with the circumference of the section
D E, will give, when projected' to a, one of
the points in the curve required.
To avoid unnecessary labor in drawing
more lines parallel to D E than are required,
it is important, in the first place, to ascer-
tain the highest point in the curve sought.
This point is the shadow of that marked H
on the upper edge of the pulley, and which
"is determined by the intersection of the ray
C' H' with the circumference of that edge in the plan ; and it is obtained by
drawing through the point A a straight line at an angle of 35 16' with the line
A B, and through the point e, striking a horizontal line ef, which by its inter-
section with the line H h, drawn at an angle of 45, will give the point sought.
In the plan, the pulley is supposed to be divided horizontally in the center,
and the shadow represented is derived from the smaller circle, and is easily
constructed by methods already described.
To trace the outlines of the shadows cast upon the surfaces of a square-
threaded nut and screiv (Figs. 237, 238).
Fig. 238 represents the projections of a screw with a single square thread,
and placed in a horizontal position, A' a' being the direction of the ray of light.
FIG. 236.
124
SHADES AND SHADOWS.
In this example, the shadow to be determined is simply that cast by the outer
edge, A B, of the thread upon the surface of the inner cylinder ; therefore, its
outline is to be delineated in the same manner as we have already pointed out,
in treating of a cylinder surmounting another of smaller diameter (Fig. 223).
A E'K'
FIG. 237.
FIG. 238.
The shadow cast by the helix ABC upon the concave surface of the square-
threaded nut is a curve, a ~b C (Fig. 237), which is to be determined in the same
way as that in the interior of a hollow cylinder. The same observation applies
to the edges A A 2 and A 8 E, as well as to those of the helix F G H and the
edge H I. With regard to the shadow of the two edges J K and K L, they
will follow the rules laid down in reference to the following figures, seeing that
they are thrown upon an inclined helical surface, of which A L is the gene-
ratrix.
To determine the outlines of the shadows cast upon the surfaces of a triangu-
lar-threaded nut and screw (Figs. 239, 240).
Fig. 240 represents the case of a triangular-threaded screw, and does not
admit of so easy a solution as the square-threaded, because the outer edge,
A D, of the thread, in place of throwing its shadow upon a cylinder, pro-
jects it upon a helical surface inclining to the left, of which the generatrix
is known. Describe from the center a number of circles, representing the
bases of so many cylinders, on the surfaces of which we must suppose helical
lines to be traced, of the same pitch as those which form the exterior edges
SHADES AND SHADOWS.
125
of the screw. We must now draw any line, such as B' E', parallel to the ray of
light, and cutting all the circles described in the plan in the points B', F', GT,
E', which are then to be successively projected to their corresponding helical
lines in the elevation, where they are denoted by B 2 , F, Gr, and E. Then,
transferring the point B' to its appropriate position, B, on the edge A C D, and
FIG. 239.
FIG. 240.
drawing through the latter a line, B #, at an angle of 45, its intersection with
the curve B 2 G E will give one point in the curve of the shadow required. In
the same manner, by constructing other curves, such as H 2 J K, the remaining
points, as Ji, in the curve may be found.
The same processes are requisite in order to determine the outlines of the
shadows cast into the interior surfaces of the corresponding nut, as will be
evident from an inspection of Fig. 239. These shadows are derived not only
from the helical edge A B D, but also from that of the generatrix A C.
The principles so fully laid down and illustrated in the preceding pages
will be found to admit of a ready and simple application to the delineation of
the shadows of all the ordinary forms and combinations of machinery and
architecture, however varied or complicated ; and the student should exercise
himself, at this stage of his progress, in tracing, according to the methods
above explained, the outlines of the cast shadows of pulleys, spur-wheels, and
such simple and elementary pieces of machinery. It must be observed that
the student should never copy the figures as here represented, but should adopt
some convenient scale somewhat larger than our figures, and construct his
126 SHADES AND SHADOWS.
drawings according to the description, looking to the figures as mere illustra-
tions ; in this way, the principles of the construction will be more surely under-
stood, and more firmly fixed in his mind.
MANIPULATION OF SHADING AND SHADOWS. METHODS OF TINTING.
The intensity of a shade or shadow is regulated by the various peculiarities
in the forms of bodies, and by the position which objects may occupy in refer-
ence to the light.
Surfaces in the Light. Flat surfaces wholly exposed to the light, and at
all points equidistant from the eye, should receive a uniform tint.
In geometrical drawings, every surface parallel to the plane of projection
is supposed to have all its parts at the same distance from the eye ; such is
the vertical side of the prism abed (Fig. 4, PL I). When two surfaces thus
situated are parallel, the one nearer the eye should receive a lighter tint than
the other. Every surface exposed to the light, but not parallel to the plane
of projection, and therefore having no two points equally distant from the eye,
should receive an unequal tint. The tint should gradually increase in depth
as the parts of such a surface recede from the eye. This effect is represented
in the same figure on the inclined surface, a dfe.
If two surfaces are unequally exposed to the light, the one which is more
nearly perpendicular to the rays should receive the fainter tint.
Thus, the face e' a' (Fig. 1, PI. 1), presenting itself more directly to the rays
of light than the face a' b', receives a tint which, although graduated in conse-
quence of the inclination of this face to the plane of projection, becomes at
that part of the surface situated nearest to the eye fainter than the tint on the
surface a' b' .
/Surfaces in Shade. When a surface entirely in the shade is parallel to the
plane of projection, it should receive a uniform dark tint.
When two objects parallel to each other are in the shade, the one nearer
the eye should receive the darker tint.
When a surface in the shade is inclined to the plane of projection, those
parts which are nearest to the eye should receive the deepest tint. This can be
seen on the face bg h c (Fig. 4), where the tint is much darker toward the line
b c, than where it approaches the line g h.
If two surfaces exposed to the light, but unequally inclined to its rays, have
a shadow cast upon them, that part of it which falls upon the surface more
directly influenced by the light should be darker than where it falls upon the
other surface.
Exemplifications of the foregoing rules may be seen on the various figures
of Plates I to V.
In order that these rules may be practiced with proper effect, we shall give
some directions for using the brush or hair-pencil, and explain the usual meth-
ods employed for tinting and shading.
The methods of shading most generally adopted are either by the superpo-
sition of any number of flat tints, or by tints softened off at their edges. The
former method is the more simple of the two, and should be the first attempted.
)
SHADES A^D SHADOWS. 127
To shade a prism by means of flat tints (PL I).
According to the position of the prism, as shown by its plan, the face abed
(Fig. 4) being parallel to the plane of projection, should receive a uniform tint
either of India ink or sepia. When the surface to be tinted happens to be
very large, it is advisable to put on a very light tint first, and then to go over
the surface a second time with a tint sufficiently dark to give the desired tone
to the surface.
The face b g h c being inclined to the plane of projection, should receive a
graduated tint from the line b e to the line g li. This gradation is obtained by
laying on a succession of flat tints in the following manner : First, divide the
plan b' g' into equal parts at the points 1', 2', and from these points project
lines upon, and parallel to, the sides of the face b g h c. These lines should be
drawn very lightly in pencil, as they merely serve to circumscribe the tints.
A grayish tint is then spread over that portion of the face b g h c (Fig. 2),
between the lines b c and 1, 1. When this is dry, a similar tint is to be laid
on, extending over the space comprised within the lines b c and 2, 2 (Fig. 3).
Lastly, a third tint, covering the whole surface bclig (Fig. 4) imparts the
desired graduated shade to that side of the prism. The number of tints
designed to express such a graduated shade depends upon the size of the sur-
face to be shaded ; and the depth of tint must vary according to this number.
As the number of these washes is increased, the whole shade gradually pre-
sents a softer appearance, and the lines which border the different tints become
less harsh and perceptible. For this reason the foregoing method of represent-
ing a shade or graduated tint by washes successively passing over each other
is preferable to that sometimes employed, of first covering the whole surface
bg he with a faint tint, then putting on a second tint b 2 2 c, followed, lastly,
by a narrow wash b 1 1 c ; because, in following this process, the outline of each
wash remains untouched, and presents, unavoidably, a prominence and harsh-
ness which, by the former method, are in a great measure subdued.
The face a dfe being also inclined to the plane of projection, should, as it
is entirely in the light, be covered by a series of much fainter tints than the
surface bghc, which is in the shade, darkening, however, toward the line ef.
The gradation of tint is effected in the same way as on the face b g h c.
To shade a cylinder by means of flat tints (PL I).
In shading a cylinder, it will be necessary to consider the difference in the
tone proper to be maintained between the part in the light and that in the shade.
It should be remembered that the line of separation between the light and
shade a b (Fig. 6) is determined by the radius a' (Fig. 5), drawn perpendicu-
lar to the rays of light E 0. That part, therefore, of the elevation of the cylin-
der which is in the shade is comprised between the lines a b and c d. This
portion, then, should be shaded conformably to the rule previously laid down
for treating- surfaces in the shade inclined to the plane of projection. All the
remaining part of the cylinder which is visible presents itself to the light ; but,
in consequence of its circular figure, the rays of light form angles varying at
every part of its surface, and consequently this surface should receive a gradu-
ated tint. In order to represent with effect the rotundity, it will be neces-
sary to determine with precision the part of the surface which is most directly
128 SHADES AND SHADOWS.
affected by the light. This part, then, is situated about the line e i (Fig. 12),
As the visual rays, however, are perpendicular to the vertical plane, and there-
fore parallel to V 0, it follows that the part which appears clearest to the eye
will be near this line V 0, and may be limited by the line T 0, which bisects
the angle V K. By projecting the points e' and m', and drawing the lines e i
and m n (Fig. 12), the surface comprised between these lines will represent the
lightest part of the cylinder.
This part should have no tint upon it whatever if the cylinder happen to
be polished a turned iron shaft or a marble column, for instance ; but if the
surface of the cylinder be rough, as in the case of a cast-iron pipe, then a very
light tint considerably lighter than on any other part may be given it.
Again, let us suppose the half-plan of the cylinder /' m' a' c' to be divided
into any number of equal parts. Indicate these divisions upon the surface of
the cylinder by faint pencil-lines, and begin the shading by laying a tint over
all that part of the cylinder in the shade a c d b (Fig. 6). This will at once
render evident the light and dark parts of the cylinder. When this is dry,
put on a second tint covering the line a 1) of separation of light and shade, and
extending over one division, as shown in Fig. 7. Proceed in this way until
the whole of that part of the cylinder which is in the shade is covered. The
successive stages of this process may be seen in Figs. 6 to 12.
Treat in a similar manner the part/e ig (Fig. 12), and complete the opera-
tion by covering the whole surface of the cylinder excepting only the division
e m n i with a very light tint ; the cylinder will then assume the appearance
presented by Fig. 12.
To shade the segment of an hexagonal pyramid by means of softened tints-
(PI. n).
The plan of this figure is similar to that of the prism (PI. I). Its position
in reference to the light is also the same. Thus, the face abed should receive
a uniform flat tint. If, however, it be desired to adhere rigorously to the pre-
ceding rules, the tint may be slightly deepened as it approaches the top of the
pyramid, seeing that the surface is not quite parallel to the vertical plane.
The face b g li c being inclined and in the shade, should receive a dark tint.
The darkest part of this tint is where it meets the line b c, and gradually be-
comes lighter as it approaches the line g h. To produce this effect, apply a
narrow strip of tint to the side b c (Fig. 6), and then, qualifying the tint in
the brush with a little water, join another strip to this, and finally, by means
of another brush moistened with water, soften off this second strip toward the
line 1, 1, which may be taken as the limit of the first tint.
When the first tint is dry, cover it with a second, which must be similarly
treated, and should extend beyond the first up to the line 2, 2 (Fig. 7). Pro-
ceed in this manner with other tints, until the whole face bg he is shaded, as
presented in Fig. 8.
In the same way the face e a d f is to be covered, though with a consid-
erably lighter tint, for the rays of light happen to fall upon it almost perpen-
dicularly.
It may be observed that, consistently to carry out the rules we have laid
down, the tint on these two faces should be slightly graduated from eatofd,
SHADES AND SHADOWS. 129
and from c li to b g. But this exactitude may be disregarded until some pro-
ficiency in shading has been acquired.
To shade a cylinder ~by means of softened tints (PL II).
The boundary of each tint being indicated in a manner precisely similar to
that shown in PL I, the first strip of tint must cover the line of extreme shade
a I, and then be softened off on each side. Other and successively wider strips
of tint are to follow, and receive the same treatment as the one first put on.
The results of this process are shown in the figures.
As this method requires considerable practice before it can be performed
with much nicety, the learner need not be discouraged at the failure of his first
attempts, but persevere in practicing on simple figures of different sizes.
If, after shading a figure by the foregoing method, any very apparent ine-
qualities present themselves in the shades, such defects may be remedied, in
some measure, by washing off excesses of tint with a brush or a damp sponge,
and by supplying a little color to those parts which are too light.
Dexterity in shading figures by softened tints will be facilitated in prac-
ticing upon large surfaces ; this will be the surest way of overcoming that
timidity and hesitation which usually accompany all first attempts, but which
must be laid aside before much proficiency in shading can be acquired.
ELABORATION OF SHADING AND SHADOWS.
Thus far the simplest primary rules for shading isolated objects have been
laid down, and the easiest methods of carrying them into operation explained.
It is now proposed to exemplify these rules upon more complex forms, to show
where the shading may be modified or exaggerated, to introduce additional
rules more especially adapted for mechanical coloring, and to offer some obser-
vations and directions for effectively shading the drawing of machines in their
entity.
Whatman's best rough-grained drawing-paper is better adapted for receiving-
color than any other. Of this paper, the double-elephant size is preferable, as
it possesses a peculiar consistency and grain. The face of the paper to be used
is the one on which the water-mark is read correctly.
The paper for a colored drawing ought always to be strained upon a board
with glue or strong gum. Before doing this, care must be taken to dampen the
face of the paper with a sponge well charged with water, in order to remove
any impurities from its surface, and as a necessary preparation for the better
reception of the color. The sponge should merely touch the paper lightly, and
not rub it. The whole of the surface is to be dampened, that the paper may
be subjected to a uniform degree of expansion, thereby insuring, as it dries,
a uniform degree of contraction. Submitted to this treatment, the sheet of
paper will present, when thoroughly dry, a clean, smooth surface, agreeable
to work upon .
The size of the brushes to be used will, of course, depend upon the scale to
which the drawing is made. Long, thin brushes, however, should be avoided.
Those possessing corpulent bodies and fine points are to be preferred, as they
retain a greater quantity of color, and are more manageable.
9
130 SHADES AND SHADOWS.
During the process of laying on a flat tint, if the surface be large though
this is seldom the case except in topographical drawings the drawing may be
slightly inclined, and the brush well charged with color, so that the edge of
the tint may be kept in a moist state until the whole surface is covered. In
tinting a small surface, the brush should never have much color in it, for, if it
have, the surface will unavoidably present coarse, rugged edges, and a coarse,
uneven appearance throughout.
In the examples of shading which are given in this work, it may be observed
that all objects with curved outlines have a certain amount of reflected light
imparted to them. It is true that all bodies, whatever may be their form, are
aifected by reflected light ; but, with a few exceptions, this light is only appre-
ciable on curved surfaces.
All bodies in the light reflect on those objects which surround them more
or less light according to the situation. Wherever light extends, reflection
follows. If an object be isolated, it is still reached, by reflected light, from
the ground on which it rests, or from the air which surrounds it.
In proportion to the degree of polish or brightness in the color of a body, is
the amount of reflected light which it spreads over adjacent objects, and also
its own susceptibility of illumination under the reflection from other bodies.
A polished steam-cylinder or a white porcelain vase receives and imparts more
reflected light than a rough casting or a stone pitcher.
Shade, even the most inconsiderable, ought never to extend to the outline
of any smooth circular body. On a polished sphere, for instance, the shade
should be delicately softened off just before it meets the circumference, and,
when the shading is completed, the body-color intended for the sphere may be
carried on to its outline. This will give a transparency to that part of the
sphere influenced by reflected light, which it could not have possessed if the
shade-tint had been extended to its circumference. Very little shade should
be suffered to reach the outlines even of rough circular bodies, lest the coloring
look harsh, and present a coarse appearance quite at variance with its natural
aspect. Shadows also become lighter as they recede from the bodies which cast
them, owing to the increasing amount of reflection which falls on them from
surrounding objects.
Shadotvs appear to increase in depth as their distance from the spectator
diminishes. In nature this increase is only appreciable at considerable dis-
tances. Even on extensive buildings, inequalities in the depth of the shadows
are hardly perceptible ; much less, then, can any natural gradation present
itself in the shadows on a machine, which, supposing it to be of the largest
construction, is confined to a comparatively small space. It is most important,
however, for the effective representation of machinery, that the variation in
the distance of each part of a machine from the spectator should at once strike
the eye ; and an exaggeration in expressing the varying depths of the shadows
is one means of effecting that object. The shadows on the nearest and most
prominent parts of a machine should be made as dark as color can render them,
the colorist being thus enabled to exhibit a marked difference in the shadows
on the other parts of the machine as they recede from the eye. The same
direction is applicable in reference to shades. The shade on a cylinder, for
SHADES AND SHADOWS. 131
instance, situated near the spectator, ought to be darker than on one more
remote ; in fact, the gradation of depth for the shades follows that which de-
picts the shadows. As a general rule, the color on a machine, no matter what
it may be intended to represent, should become lighter as the parts on which it
is placed recede from the eye.
Plates III and IV present some very good examples of finished shading.
Plate III represents, both in elevation and plan, different solids variously
penetrated and intersected. The rules for the projection of these solids have
been given under the head of Orthographic Projection. They are selected with
a view of exhibiting those cases which are of most frequent occurrence, and at
the same time elucidating the general principles of shading.
Plate IV presents examples of shading and shadow.
Fig. 1 presents a hexagonal prism surmounted by a fillet. The most notice-
able part of this figure is the shadow of the prism in the plan view. It presents
a good example of the graduated expression which should be given to all shad-
ows cast upon plain surfaces. Its two extremities are remarkably different in
their tone. As the shadow nears the prism, it increases rapidly in depth ; on
the contrary, as it approaches the other end, it assumes a comparatively light
appearance. This difference is doubtlessly a great exaggeration upon what it
would naturally display. Any modification of it, however, in the representa-
tion would destroy the best effect of the shadow.
The direction which the shades and shadows take, in all the plans of the
figures in this plate, is from the left-hand lower corner. This is rigorously
correct, supposing the objects to remain stationary, while the spectator views
them in both a vertical and a horizontal position. Nevertheless, to many, this
upward direction given to the shadows has an awkward appearance, and, per-
haps, in the plan of an entire machine, the shadows may look better if their
direction coincide with that which is given to them in the elevation. If, how-
ever, the shadows be correctly projected, their direction is an arbitrary matter,
and may be left to the taste of the draughtsman.
Figs. 2, 3, and 6 exemplify the complex appearance of shade and shadow
presented on concave surfaces. It is worthy of particular notice that the
shadow on a concave surface is darkest toward its outline, and becomes lighter
as it nears the edge of the object. Reflection from that part of the surface on
which the light falls most powerfully causes this gradual diminution in the
depth of the shadow, the greatest amount of reflection being opposite the great-
est amount of light.
It may be as well to remark here that no 'brilliant or extreme light should
be left on concave surfaces, as such lights would tend to render it doubtful at
first sight whether the objects represented were concave or convex. After the
body-color which shall be treated in a subsequent section has been put on, a
faint wash should be passed very lightly over the whole concavity. This will
not only modify and subdue the light, but tend to soften any asperities in the
tinting, which are more unsightly on a concave surface than on any other.
The lightest part of a sphere (Fig. 4) is confined to a mere point, around
which the shade commences and gradually increases as it recedes. This point
is not indicated on the figure referred to, because the shade-tint on a sphere
132 SHADES AND SHADOWS.
ought not to be spread over a greater portion of its surface than is shown there.
The very delicate and hardly perceptible progression of the shade in the imme-
diate vicinity of the light-point should be effected by means of the body-color
of the sphere. If, for instance, the material of which the sphere is composed
be brass, the body-color itself should be lightened as it nears the light point.
In like manner all polished or light-colored curved surfaces should be treated ;
the part bordering upon the extreme light being covered with a tint of body-
color somewhat fainter than that used for the flat surfaces. Again, if the
sphere be of cast-iron, then the ordinary body-color should be deepened from
the light point until it meets the shade-tint, over which it is to be spread uni-
formly. Any curved unpolished surface is to be thus treated ; the body-color
should be gradually deepened as it recedes from that part of the surface most
exposed to the light. Considerable management is necessary in order to shade
a sphere effectively, ^he best way is to put on two or three softened-off tints
in the form of crescencs converging toward the light-point, the first one being^
carried over the point of deepest shade.
A ring (Fig. 5) is a difficult object to shade. To change with accurate and
effective gradation the shade from the inside to the outside of the ring, to leave
with regularity a line of light upon its surface, and to project its shadow with
precision, require a degree of attention and care in their execution greater,
perhaps, than the shade and shadow of any other simple figure. The learner,
therefore, should practice the shading of this figure, as he will seldom meet
with one presenting greater difficulties.
Figs. 7 and 8 show the peculiarities of the shadows cast by a conical form
on a sphere or cylinder. The following fact should be well noted in the mem-
ory : That the depth of a shadow on any object is in proportion to the degree
of light which it encounters on the surface of that object. In these figures
very apt illustrations of this fact may be remarked. It will be seen, by refer-
ring to the plan (Fig. 7), that the shadow of the apex of the cone happens to
fall upon the lightest point of the sphere, and is, therefore, the darkest part of
the shadow. So also the deepest portion of the shadow of the cone on the
cylinder in the plan (Fig. 8) is exactly where it coincides with the line of ex-
treme light. Flat surfaces are similarly affected, the shadows thrown on them
being less darkly expressed, according as their inclination to the plane of pro-
jection increases. The body-color on a flat surface should, on the contrary,
increase in depth as the surface becomes more inclined to this plane.
Another notable fact is exemplified by these figures that reflected light is
incident to shadows as well as to shades. This is very observable where the
shadow of the cone falls upon the cylinder. It may likewise be remarked,
though to a less extent, on other parts of these figures. The reflected light
on the cone from the sphere or cylinder is also worthy of observation. This
light adds greatly to the effect of the shadows, and, indeed, to the appearance
of the objects themselves. Altogether, these figures offer admirable scope for
study and practice.
The concentration within a small space of nearly all the peculiarities and
effects of light, shade, and shadow, may be seen on Plate V in the examples,
of screws there given.
SHADES AND SHADOWS. 133
Under the head of Topographical, Mechanical,, and Architectural Drawing,
will be given examples of drawings in shade and shadow, and in varied colors
expressing conditions of surfaces or materials of composition. In the topo-
graphical and architectural examples, often a certain amount of artistic effect
can be introduced, but, in the mechanical, distinctness of outline and accuracy
of expression are essential ; but, to maintain harmony in the coloring, and to
equalize the appearance of the drawing, large shades should be colored less
darkly than small, as they may be situated at the same distance from the eye,
and no very dark shading is permissible.
In preparing colors for tints, great care should be used in grinding. The
end of the cake should be slightly wetted and rubbed on a porcelain palette,
and then transferred by a wet brush to another saucer, and water added to
bring to the required tint. Mixed colors should be intimately blended by the
brush. Grind in excess enough of all the tint required, and let it stand in the
saucer till the grosser particles have settled and the liquid is of clear and uni-
form tint. It is very common to make little boxes or bag-like receptacles of
waste drawing-paper to hold the colors instead of saucers ; the gross matters,
settling on the bottom, are not then so readily disturbed.
Instead of hard cakes of color, moist colors are used, either in cakes or
collapsible tubes, which preclude the necessity of grinding. For flat tints or
washes, aniline colors, dissolved in water and kept bottled, afford the readiest
means of coloring, but are not applicable to finished work.
Sometimes the surface of the paper is, as it were, greasy, and resists colors ;
in that case, dissolve a piece of ox-gall, the size of a pea, in a tumbler of
water, and use this solution with the colors instead of plain water.
When the brush is too full, as it comes toward the limit of the tint, take up
the surplus moisture on a wet sponge or piece of cloth or blotting-paper.
An expeditious way of shading a cylinder or expressing the shores of a
stream or lake, is by drawing with a brush full of the darkest tint along the
sides of cylinder or shores of water, and then, with a wet brush, modifying
this tint toward the light from the sides, so as to give a shaded appearance.
For this purpose, two brushes will be necessary, one with color, the other with
water ; also, a tumbler of water, and a piece of blotting-paper, to take up the
excess of moisture from paper or brush. Often a single line of dark color
blended this way will express all that is necessary, but the effect may be im-
proved by a sort of stippling with the color-brush and extending the line of shade.
The same effect is obtained better by drawing two faint pencil-lines on the
elevation of the cylinder, for instance, to indicate the extremes of light and
shade on its surface. Pass the brush, moderately full of the darkest tint, down
the line of deepest shade, spreading the color more or less on either side, accord-
ing to the diameter of the cylinder ; then, if possible, before this layer of tint
is dry, toward the line of extreme light, beginning at the top, and encroaching
slightly over the edge of the first tint, lay on another not quite so dark, but
about double its width. It may be observed that it is not very essential to put
on the second tint befor the first is dry, for the latter should be so dark and
thick that its edges may be easily softened at any time. While this second tint
is still wet, with a much lighter color in the brush, proceed in the same man-
134: SHADES AND SHADOWS.
ner with a third tint, and so on until the line of extreme light is nearly
attained. Repeat this process on the other side of the first tint, approaching
the outline of the cylinder with a very faint wash, so as to represent the re-
flected light which progressively modifies the shade as it nears that line. Then
let a darkish narrow strip of tint meet and pass along the outline of the cylin-
der on the other side of the extreme line of light, after which gradually fainter
tints should follow, treated in a manner similar to that which has been already
described, and becoming almost imperceptible just before arriving at the line
of light.
This is a very expeditious way of shading a cylinder ; but even to the most
experienced colorist it is not possible, by the above-described means alone, to
impart a sufficient degree of well-regulated rotundity to the appearance of such
an object. Superfluities and deficiencies of color will appear here and there.
It will be necessary, therefore, to equalize to some extent, by a species of gross
stippling, the disparities which present themselves. This is done by spreading
a little color over the parts where it is deficient, and then passing very lightly
over nearly the whole width of the shade, with the brush supplied with a very
light wash. This process may be repeated to suit the degree of finish which it
is desired to give the drawing. In the same manner the shading of all curved
surfaces is to be treated.
The shades being put in, that of the shadows follows. The outline of any
shadow being drawn in pencil, along its inner line the line which forms a
portion of the figure of the object whose shadow is to be represented lay on a
strip of the darkest tint, wide or narrow, according to the width of the shadow,
and then, before it is dry, soften off its outer edge. This may be repeated as
often as the taste of the colorist may dictate, but the color should not spread
itself over much more than half the space occupied by the shadow. These
preliminary touches will add to the intensity of the proposed shadow, and
neutralize a certain harshness of appearance inevitable to all shadows made
equally dark throughout.
The finish is made by a light wash or two of the body-color, and in passing
over the shades and shadows care must be taken to maneuver the brush at
such parts quickly and lightly.
The shades and shadows of a machine are modified in intensity as their
distance from the eye increases. Its body-color should be treated in a similar
manner, becoming lighter and less bright as the parts of the machine which it
covers recede from the spectator.
When the large circular members of a machine have been shaded, the shad-
ows, and even the body-color on those parts farthest removed from the eye, are
to follow, and the proportion of India ink in the tint used should increase as
the part to be colored becomes more remote. A little washing, moreover, of
the most distant parts is allowable, as it gives a pleasing appearance of atmos-
pheric remoteness, or depth, to the color thus treated.
The amount of light and reflection on the members of a machine should
diminish in intensity as the distance of such objects from the spectator in-
creases. As it is necessary, for effect, to render, on those parts of a machine
nearest the eye, the contrast of light and shade as intense as possible, so, for
SHADES AND SHADOWS. 135
the same object, the light and shade on the remotest parts should be subdued
and blended according to the extent or size of the machine.
A means of adding considerably to the definiteness of a colored mechanical
drawing, and of promoting, in a remarkable degree, its effective appearance,
is obtained by leaving a very narrow margin of light on the edges of all sur-
faces, no matter what may be the angles which they may form with the sur-
faces that join them. This should be done invariably ; but the margin of
those edges which happen to have shadows falling on them, instead of being
left quite white, may be slightly subdued.
To effect this, suppose the object about to receive the color to be the eleva-
tion of a long, flat rod or lever, on the edge of which a line of light is to be
left. Fill the drawing-pen as full as it will conveniently hold with tint des-
tined to cover the rod or lever, and draw a broad line just within, but not
touching, the edge of the lever exposed to the light. As it is essential for the
successful accomplishment of the desired effect that this line of color should
not dry, even partially, until the tint on the whole side of the lever has been
put on, it will be as well to draw the pen again very lightly over the same
part, so that the line may retain as much tint as possible. Immediately this
has been done, the brush, properly filled with the same tint, is to pass along
and join the inner edge of this narrow strip of color, and the whole surface of
the lever filled in. Thus a distinct and regular line of light is obtained, and,
in fact, the lever, or whatever else the object may be, covered in a shorter time
than usual. A still more expeditious way of coloring such surfaces is to draw
a second line of color along and joining the opposite edge of the lever or other
object, and then expeditiously to fill in the intermediate space between the two
wet lines by means of the brush. By similar means the line of light on a
cylinder, shaft, or other circular body, may be beautifully expressed. To indi-
cate this light with perfect regularity is highly important, for, if a strict uni-
formity be not maintained throughout its whole length, the object will look
crooked or distorted. After having marked in pencil, or guessed the position
of the extreme light, take the drawing-pen, well filled with a just perceptible
tint, and draw a line of color on one side the line of light, and almost touch-
ing it ; then with the brush, filled with similar light tint, join this line of color
while still wet, and fill up the space unoccupied by the shade-tint, within
which the very light color in the brush will disappear. Let that part of the
object on the other side of the line of light be treated in the same way, and
the desired effect of a stream of light clear and mathematically regular will be
obtained. The extreme depth of shade, as well as the line of light in such rods,
may, with great effect, be indicated by filling the pen with dark shade-tint, and
drawing it exactly over the line representing the deepest part of the shade.
On either side and joining this strip of dark color, another, composed of
lighter tint, is to be drawn. Others successively lighter are to follow, until,
on one side, the line of the rod is joined, and on the other the lightest part of
the rod is nearly reached. The line of light is then to be shown, and the faint
tint used on this occasion spread with the brush lightly over the whole of that
part of the rod situated on either side of this line, thus blending into smooth
rotundity the graduated strips of tint drawn by the pen.
136 SHADES AND SHADOWS.
In all tinted drawings the more important parts, whether the machinery or
the structure, should be more conspicuously expressed than those parts which
are mere adjuncts. Thus, if the drawing be to explain the construction of
the machine, the tint of edifice and foundations may be kept lighter and more
subdued than those of the machine ; and if the machine, on the contrary, be
unimportant, it may be represented quite light, or in mere outline, while the
edifice is brought out conspicuously.
With regard to washings, the soft sponge is an implement not to be neg-
lected by the draughtsman ; it is an excellent means of correcting great errors
in drawing, better than rubber or an eraser, but care of course must be taken
to wash and not to rub off the surface, and for errors in coloring washing is
almost the only corrector. In removing or softening color on large surfaces,
the sponge is to be used, and for small spots the brush. While coloring, keep
a clean, moist brush by you : it will be extremely useful in removing or modi-
fying a color.
The immediate effect of washing is to soften a drawing, an effect often very
desirable in architectural and mechanical drawings, and the process is simple
and easily acquired ; keep the sponge or brush and water used clean ; after the
washing is complete, take up the excess of moisture by the sponge or brush, or
by a piece of clean blotting-paper. Where great vigor is required, let the
borders of the different tints be distinct.
There are no conventional tints that draughtsmen have agreed upon to be
uniformly used, to represent different materials. India ink is not a black, but
a brown, making with a blue a greenish cast, and with gamboge a smear. A
colored drawing is better without the use of India ink at all ; any depth of
color may be as well obtained with blue as with black ; there is also an objec-
tion to gamboge, that it is gummy, and does not wash well, and the effect is
better obtained with yellow ochre. For the reds, the madder colors are the
best, as they stand washing ; for the shade-tint of almost every substance a
neutral tint, Payne's gray, or madder brown subdued with indigo.
PLOTTING.
PLOTTING is the laying out on paper in plan or in horizontal projection
the boundaries of lots, estates, farms, etc., portions of the earth's surface of
greater or less extent, from the notes of surveys or other records. When the
extents are large, beyond the usual limits of personal property, and embracing
degrees of latitude and longitude, the plots are designated as maps ; but if of
small extent as lots, estates, and farms they are usually designated as plans
or plots. After completing the outlines, it is usual to fill up the plot, with the
characteristic features, geographical, geological, agricultural, industrial, and
domestic, which are expressed more or less conventionally, as will be shown
under the head of " Topographical Drawing."
Scales. The choice of the scale for the plot depends in a great measure on
the purpose for which the plan is intended. It should be large enough to
express all the details desirable, modified by the circumstances whether the
map is to be portable or whether space can be afforded for the exhibition of
a large plan. We must adapt our plan for the purposes which it is intended
to illustrate, and the place it is to occupy.
Plans of house-lots are usually named as being so many feet to the inch ;
plots of farm-surveys, as so many chains to the inch ; maps of surveys of
States, as so many miles to the inch ; and maps of railway-surveys, as so many
feet to the inch, or so many inches to the mile.
Formerly the lines of farms were measured by the four-rod chain ; latterly
the 100-foot chain is more usually adopted. Two to three chains to the inch
was then a very common scale.
State surveys are of course plotted on a smaller scale than those of farms.
On the United States Coast Survey all the scales are expressed fractionally and
decimally. The original surveys are generally on a scale of one to ten or twenty
thousand, but in some instances the scale is larger or smaller. The public sur-
veys embrace three general classes : 1. Small harbor-charts. 2. Charts of bays,
sounds, etc. 3. General coast-charts.
The scales of the first class vary from 1 : 5,000 to 1 : 60,000, according to the
nature of the harbor and the different objects to be represented.
The scale of the second class is usually fixed at 1 : 80,000. Preliminary
charts, are, however, issued of various scales, from 1 : 80,000 to 1 : 200,000.
Of the third class the scale is fixed at 1 : 400.000 for the general chart of the
coast from Gay Head to Cape Henlopen, although considerations of the prox-
imity and importance of points on the coast may change the scales of charts of
other portions of our extended coast.
138 PLOTTING.
On all plots of large surveys, it is very desirable that the scales adopted
should bear a definite numerical proportion to the linear measurement of the
ground to be mapped, and that this proportion should be expressed fractionally
on the plan, even if the scale be drawn or expressed some other way, as chains
to the inch. The decimal system has the most to recommend it, and is gener-
ally adopted in government surveys.
For railroad-surveys, the New York general railroad law directs the scale of
map which is to be filed in the State Engineer's office, to be 500 feet to one
tenth of a foot, 1 : 5,000.
For the canal-maps, a scale of two chains to. the inch, 1 : 1,584 is employed.
In England, plans and sections for projected lines of inland communication, or
generally for public works requiring the sanction of the Legislature, are re-
quired, by the "standing orders," to be drawn to scales not less than four
inches to the mile, 1 : 15,840, for the plan, and 100 feet to the inch, 1 : 1,200,
for the profiles.
In the United States engineer service the following scales are prescribed :
General plans of buildings 10 feet to the inch, : 120
Maps of ground with horizontal curves 1 foot apart . . 50 " " 1 : 600
Topographical maps li mile square 1 mile to 2 feet, : 2,640
Topographical maps comprising 3 miles square . . . 1 " 1 foot, : 5,280
Topographical maps comprising between 4 and 8 miles . 1 " 6 in., : 10,560
Topographical maps comprising 9 miles square . . . 1 " 4 " : 15,840
Maps not exceeding 24 miles square 1 " 2 " 1 : 31,680
Maps comprising 50 miles square 1 " I inch, 1 : 63,360
Maps comprising 100 miles square 1 " \ " 1 : 126,720
Surveys of roads and canals 50 feet to 1 " 1 : 600
In cities and towns, lots and squares are generally rectangular, and they can
be readily plotted on any convenient scale.
Fig. 241 is a plan of the usual New York city lot, 25 x 100, on a scale of
20 feet to the inch, or -%fa full size.
Fig. 242 is a city square containing thirty-two of these lots, on a scale of
100 feet to the inch, or T -gVo- ^ ne mos t accurate way is to plot the large rec-
tangle 400 x 200 feet, and then subdivide it.
Fig. 243 is a plan of the same city squares, with the inclosing streets, on a
scale 200 feet to the inch, or s ^.
But there are many lots, and most estates, which are not rectangular, the
angles of which are recorded, which must be plotted by the aid of a pro-
tractor.
If the survey has been made by triangles, the principal triangles are first
laid down in pencil by the intersection of their sides, the length being taken
from the scale and described with compasses. In general, when the surveys
have been conducted without instruments to measure the angles, as the com-
pass or theodolite, the position of the points on paper are determined by
the intersection and construction of the same lines as has been done in
the field.
Surveys are mostly conducted by measuring the inclination of lines to a
meridian or to each other by the compass or by the theodolite. In the sur-
PLOTTING.
139
veys of farms, where great accuracy is not required, the compass is most used.
The compass gives the direction of a line in reference to the magnetic meridian.
The variation from the true meridian, or a direct
north-and-south line, varies considerably in different
parts of the country. In 1875 the line of variation
in which the needle pointed directly north, passed in
a nearly straight direction from Wilmington, North
FIG. 241.
FIG. 242.
Carolina, to Cleveland, Ohio. At all places east of this line the variation is
westerly, that is, the needle points west of the line north. West of this line
the variation is easterly.
Fig. 24A represents the plot of a compass survey, with the positions of the
protractor in laying off the angles. To the left of the figure are given the
field-notes. In this way of plotting, a meridian is laid off at the intersection
of each set of lines. Sometimes the angles are plotted directly from the deter-
mination of the angle of deflection of two courses meeting at any point, with-
out laying down more than one meridian (Fig. 245). When the first letters
of the bearing are alike, that is, both N. or both S., and the last letters also
alike, both E. or both W., the angle of deflection C B B' will be the difference
of the bearings, or, in this instance^ 20.
140
PLOTTING.
When the first letters are alike and the last different (Fig. 246), the angle
C B B' will be the sum of the two bearings.
j i
L
1 [
r
FIG. 243.
When the first letters are different and the last alike (Fig. 247), subtract
the sum of the bearings from 180 for the angle C B B' ; when both the first
letters and last are different, subtract their difference from 180 for the angle.
-(5)-
3.55
GO
-(4>-
222
od
-(3)-
1.29
H
-(2)-
2.70
-dh-
FIG. 244.
Instead of drawing a meridian through each station, or laying off the
angle of deflection, by far the easiest way is to lay off but a single meridian
PLOTTING.
141
near the middle of the sheet ; lay off all the bearings of the survey from some
one point of it, as shown in Fig. 248, and number to correspond with the sta-
tions from which the bearings are taken, and then transfer them to the places
FIG. 246.
FIG. 247.
where they are wanted by any of the instruments used for drawing parallel
lines. For the protracting of the rough plan, sheets of drawing-paper can be
bought with protractors printed on them. When the plans are large, it is
FIG. 249.
often convenient to lay out two or three meridians on different parts of the
sheet, and lay off the bearings of lines adjacent to each meridian upon them.
In plotting from a survey by a theodolite or transit, it is generally usual to
lay off the angles of deflection of the different lines as taken in the field, plot-
ting all the tie-lines as corrections.
When the plot of a survey does not close that is, come together, or return
to the point of commencement, as it seldom does exactly it may be corrected
or forced ; but first be sure that the bearings and distances as recorded are laid
down accurately, and then proceed to correct as follows :
If the plot of the last line does not close up the outline of the figure exactly,
by its extremity falling upon the point of beginning of the plot, as upon the point
a (Fig. 249), instead of upon 1, either the survey or the plotting is incorrect.
142
PLOTTING.
If the latter be correct, the error of the survey must be balanced, or distributed
through the lines and angles of the plot. Connect 1 with a, and draw lines
parallel to 1 a through 2, 3, 4, 5, of the plot. Draw an indefinite line, 1 1} (Fig.
250), and on this, with any convenient scale, lay off consecutively the lines of the
survey, 1-2, 2-3, 3-4, 4-5, 5-a. Erect perpendiculars at the extremities of the
lines, 2, 3, 4, 5, and b. On the perpendicular a b, lay off 1 a from the plot and con-
nect 1 1. The intersections of the perpendiculars by this line will determine how
CJ
much each of the points of the plot are to be moved on the parallels to 1 a to dis-
tribute the error. The dotted lines on the figure show the corrected outline.
By the aid of the Traverse Table a survey may
be balanced and accurately plotted. The Traverse
Table (see appendix) is a table of differences of
latitudes and departures, the difference of latitude
between two stations being the difference north
and south between them ; the difference of depart-
ure, the difference east and west.
s Thus, ]\ r S (Fig. 251) being the meridian, A
is the difference of latitude between A and B, and
A D the departure.
The differences vary according to the length
of A B, and the angle it makes with the meri-
dian.
Taking the field-notes of the previous survev, we make a table as follows :
W-
STATION.
Bearing.
Distance.
Latitude.
N. S.
Departure.
E. W.
1
N. 35 E.
N. 83| E.
S. 57 E.
S. 34iW.
N. 56| W.
2-70
1-29
2'22
3-55
3-23
2 -21
15
1-78
1-21
2-93
1-55
1-28
1-86
2-00
2-69
2
3
4
5. . . .
4-14
4-14
4-69
4-69
In the Traverse Table, on the line with 35, and in
column 2, latitude = 1 '638 departure = 1 '147
" 7, " = -573 " = '401
2-211 1-548
Again, on the line with 83-j- , in
column 1, latitude = -113 departure = '994
= -0226 " = -1987
" = -01019 " = -08942
14579 1-28212
And in the same manner the table is completed.
PLOTTING.
143
The table following is constructed by adding up the northings and sub-
tracting the southings for the latitude, and by adding up the eastings and
subtracting the westings for the departures.
STATION.
Total latitude from
station.
Total departure from
station 1 -.
1
o-oo
o-oo
2
+ 2-21 N.
+ 1'55 E
3
+ 2'36 N.
+ 2 '83 E.
4
+ ri5 N.
+ 4'69 E.
6
1'78 S
+ 2 '69 E
1
O'OO
o-oo
From this table the survey can be readily plotted (Fig. 252). Draw the
meridian through the point taken for station 1 ; measure to the north 2*21
chains to A ; draw an easterly line, or one perpendicular to the meridian at
A, and lay off on it 1 -55 chains, and we have station 2 ; measure again from 1
northerly 2*36 chains to B, and lay off from B due easterly 2 -83 chains for
station 3 ; measure again from 1 northerly 1 -15 chains to C, and lay off from
C due easterly 4*69 chains for station 4 ; measure again from 1 southerly
1'78 chains to D, and layoff from D easterly 2*69 chains for station 5.
Connect 1, 2, 3, 4, 5, and 1 for the complete plot.
In this survey the latitudes balance, 4'14 to 4 '14, and the departures bal-
ance, 4*69 to 4 -69, but this seldom happens. Generally there is a difference
which must be balanced before plotting. For instance, in this survey, had the
northings been J and had the southings been | 2 -95 the difference would
1*70
14-05
144
PLOTTING.
have been *14 to be divided, in proportion to their lengths, between the north-
ings and the southings, adding to the former and deducting from the latter.
The total northings and southings is 4'05 -1- 4'19 = 8*24 chains, in which an
error of 14 links is to be balanced, or about -017 chain to each chain. In
the 2-20 N. the correction will be 2 '20 x -017 = '0374, or about -04, and with-
out much calculation we can see that
f2'24 p. 22
the corrected northings will be J and the corrected southings will be -j %'9Q
-j '
14*12
I 4-12
The same calculation is applied to the departures when there is a difference
in the total eastings and westings.
The errors are to be balanced before the survey is plotted.
When a field has been plotted, it can be divided into triangles, and its area can be calculated ;
but, having the latitudes and departures balanced and tabulated, the area can be calculated as fol-
lows:
STATION.
Latitude.
Departure.
Double
longitude.
Double area.
N +
S. -
E. +
W. -
N. + | S. -
1
2-21
15
1-78
1-11
2-93
1-55
1-28
1-86
2-00
2-69
+ 1-55
+ 4'38
+ 7-52
+ 7-38
+ 2-69
3-4255
0-6570
4-7882
9-0992
21-6234
2
3
4
5
4-14
4-14
4-69
4-69
8-8707
30-7226
8-8707
Content = 1A., OR., 15P.
2)21-8519
10-9259 =
1 09259 acres.
The first five columns are from the preceding tables. To construct the column of double longi-
tudes : the double longitude of the first course is equal to its departure.
The double longitude of the second course is equal to the double longitude of the first course,,
added to the departure of that course, added to the departure of the second course.
The double longitude of the third course is equal to the double longitude of the second course^
added to the departure of that course, added to the departure of the third course.
The double longitude of any course is equal to the double longitude of the preceding course added
to the departure of that course, added to the departure of the course itself ; the double longitude of
the last course is equal to its departure.
Thus, the double longitude of first course is its departure = 1 '55
add the departure of first course 1'55
add the departure of second course 1 '28
and we have the double longitude of second course = 4-38
add the departure of second course 1'28
add the departure of third course 1'86
and we have the double longitude of third course = 7.52
add the departure of third course 1*86
subtract the departure of fourth course 2.00
and we have the double longitude of fourth course = 7'38
subtract the departure of fourth course 2'00
subtract the departure of fifth course 2'69 4'69
and we have the double longitude of fifth course = 2-69
PLOTTING.
145
Multiply the double longitude of each course by the latitude of that course, placing the north
products in one column and the south products in another ; subtract the lesser total of the one
column from the greater total of the other,
and divide the difference by two. The prod-
uct will be in square chains, which, divided
by ten, will give the result in acres and
decimals.
The area of an irregular figure can be
calculated most conveniently, and with suf-
ficient accuracy, by dividing it into triangles,
measuring the height and base of each, cal-
culating the. area of each, and adding the
r
FIG.
253.
areas together.
Or, the polygon may be resolved readily
into a single triangle, and its area calculated. For instance, take the five-sided polygon, 1, 2, 3, 4, 5
(Fig. 253). Call the side 5 1 the base, and extend it. Join 1 and 3. Draw 2 1' parallel to 1 3.
Join 1' and 4. Draw 2' 3 parallel to 1' 4. Join 2' and 4. The triangle 2' 4 5 will be a triangle
equal to the polygon.
The same construction will apply to a figure of a greater number of sides.
The area of a triangle can be calculated graphically (Fig. 254). Let the scale be two chains to
the inch. Prepare a strip of drawing-paper one inch wide, and divide it by perpendicular lines in
FIG. 255.
20ths of an inch. Apply it to the triangle A B C so that one edge will fall upon A, and the other
at B. Keeping the same points on the extended line A' B, slide the scale up till its upper edge
arrives at the point C. The line A' C in divisions of
the scale is the area of the triangle in square chains.
If the scale had been three chains to the inch, the
strip should have been f of an inch in width ; if four
chains to the inch, then f of an inch in width, and so on.
When the lines of a plot are irregular, as in Fig.
255, draw across it a number of equidistant parallel
lines, and with a strip of paper measure these lines, one after another, till the sum of their lengths
is marked on the edge of the strip. Cut the strip at the last mark, and fold it in two. This measure
(half the length of the strip), multiplied by the uniform width between the parallel lines, will give
very nearly the area.
Having completed the plot that is, the main lines of the survey the
filling of other points may in general be done on paper, the same way that
they have been established in the field. Intersections of the main lines by
10
146
PLOTTING.
roads, streams, fences, and the like, are measured off ; other points not inter-
secting, are usually fixed by triangles or by offsets from the main lines, or lines
run on purpose by angles from the main lines.
fixed Sea
FIG. 256.
In case of unimportant lines, as the crooked brook, for instance (Fig. 256),
offsets are taken to the most prominent angles, as, a, a, a, and the intermediate
bends are sketched by eye into the field-book. In copying them on the plan a
similar construction is adopted.
The most rapid way of plotting the offsets is by
the use of a plotting and offset scale (Fig. 257), the
one being fixed parallel to the line A B from which
the offsets are to be laid off, at such a distance from
it, that the zero-line on the movable scale coincides
with it, while the zero of its own scale is on a line
perpendicular to the position of the station A from
which the distances were measured. It is to be ob-
served that in the field-book all the measures are re-
ferred to the point of beginning on any one straight
line. Having placed the plotting-scale, move the
offset-scale to the first distance by the scale at which
an offset has been taken, mark off now on the offset-
scale the length of the offset on its corresponding
side of the line. Proceed then to the next distance,
establishing thus repeated points, join the points by
lines as they are on the ground.
The plotting and offset scale must of course be of
the same scale as the rest of the drawing, on which
account it may not always be possible to obtain such
scales adapted to those of the plan ; but they may be
easily constructed of thick drawing-paper or paste-
board.
When a great deal of plotting to one scale is
necessary, as in government surveys, the offset-scale
may be made to slide in a groove upon the plotting-
scale.
In protracting the triangles of an extended trigo-
nometrical survey in which the sides have been cal-
culated or measured, it is better to lay down the
triangles from the length of their sides rather than
by measuring the angles, because measures of length
can be taken with more accuracy from a scale, and transferred to the plan
with more exactness than angles can be pricked off from a protractor ; but,
PLOTTING.
147
for ordinary surveys, the triangulation is most frequently and expeditiously
plotted by the means of a protractor.
The outlines of the survey having been balanced and plotted in, and the
subsidiary points, as established by offsets and by triangles, the filling in of the
interior detail, with the natural features of the ground, from the skeleton or
suggestions in the field-book or other records, is done according to imitative
and conventional signs, to be shown under "Topographical Drawing."
The public lands of the United States are surveyed, mapped, and divided
into nearly square tracts, according to the following system :
Ranges. Standard lines must first be determined, from which to measure.
Accordingly, in each land-district some meridian-line is run due north and
south ; this is called the Principal Meridian. From some point of the Principal
Meridian is also run a line due east and west, called the Base-Line.
Other lines are then run in the same direction as the Principal Meridian, at
distances of six miles (measured on the Base-Line) on each side of it. The
strip between the Principal Meridian and the first line thus run east of it is
known as Range 1 East ; the second strip is Range 2 East, etc. And so on
the west ; the successive strips running north and south, six miles wide, are
called Range 1 West, Range 2 West, etc. This division is shown in Fig. 258.
i
r
Tp.2
North
be
.
few
i?
t?
i 3
i
I
Ki
k*
1
Tp.l
North
BASE
LINE
Tp.l
a
f
.
1
South
e
2
9
Tp.2
South
P
fe
(
e^
-j
!
FIG. 258.
FIG. 259.
Townships. In -like manner, lines are run north and south of the Base-
Line at intervals of six miles. These lines cut at right angles those which
separate the ranges, and with them form squares six miles on each side, called
townships. Each township contains thirty-six square miles.
The township nearest the Base-Line on the north is known as Township 1
North, of whatever range it may be in ; the next farther north is Township 2
North, of that range and so on. In like manner, going south from the
Base-Line, we have in succession Township 1 South, Township 2 South, etc.
(Fig. 259).
148
PLOTTING.
Sections. Each township is divided into thirty-six squares, called Sections,
each one mile long and one mile wide, and therefore having an area of one
square mile. The sections of a township are numbered 1, 2, 3, etc., up to 36,
beginning at the northeast, and running alternately from right to left and from
left to right, as shown in Fig. 260.
6
5
4
3
2
1
7
8
9
10
11
12
18
17
16
15
14
13
19
20
21
22
23
24
30
29
28
27
26
25
31
32
33
34
35
36
1 mile.
E
FIG. 260.
FIG. 261.
A section may be subdivided into half-sections, quarter-sections, eighths,
and sixteenths, designated as in the example that follows :
Let F G (Fig. 261) be Section 3 of Township 2 North, in Range 1 West ;
then
A is N. (north) -J of Section 3, Township 2 North, Range 1 West.
B is S. W. (southwest) of Section 3, Township 2 North, Range 1 West,
C is W. (west) | of S. E. (southeast) i of Section 3, Township 2 North,
Range 1 West.
D is N. E. i of S. E. i of Section 3, Township 2 North, Range 1 West.
E is S. E. i of S. E. i of Section 3, Township 2 North, Range 1 West.
Correction- Lines. If the north-and-south (meridian) lines were parallel to
each other, the townships and sections would be exact squares. But as these
lines gradually converge toward the north, meeting at the pole, the townships
deviate somewhat from squares, being narrower on the north than on the south ;
and the northern sections of a township are a little smaller than the southern
ones.
In order that the townships of a range may not thus keep getting smaller
and smaller as we go toward the north, a new base-line, called a Correction-
Line, is taken at intervals (differing in length in different land-districts), and
new north-and-south lines are run at distances of six miles measured on the
Correction-Lines.
The system of survey described above is not used in Texas, the public lands
there being State property.
TOPOGRAPHICAL DRAWING.
TOPOGBAPHICAL DRAWING is the delineation of the surface of a locality,
with the natural and artificial objects, as houses, roads, rivers, hills, etc., upon
it in their relative dimensions and positions, giving, as it were, a miniature
copy of the farm, field, district, etc., as it would be seen by the eye moving
over it. Many of the objects thus to be represented can be defined by regular
and mathematical lines, but many other objects, from their irregularity of out-
line, it would be very difficult thus to distinguish ; nor are the particular
irregularities necessary for the expression. Certain conventional signs have
FIG. 262.
FIG. 263.
FIG. 264.
FIG. 265.
therefore been adopted in general use among draughtsmen, some of which
resemble, in some degree, the objects for which they stand, while others are
purely conventional. These signs may be expressed by lines, or by tints, or
by both.
Figs. 262 and 263 represent meadow or grass land, the short lines being
supposed to represent tufts of grass ; the bases of the tufts should always
ft
I I
li!
I!
Ill rj
LU.il
FIG. 266.
FIG. 267.
FIG.
be parallel to the base of the drawing, whatever may be the shape of the in-
closure.
Figs. 264, 265, 266, 267, give various methods of representing trees. Figs.
264 and 265 represent in plan a forest and an orchard, while Figs. 266 and 267
show the same in elevation. The latter method of representing trees is not
150
TOPOGRAPHICAL DRAWING.
consonant with the projection of the plan, but to many is more expressive and
intelligible.
Fig. 268 represents cultivated land. The lines are supposed to represent
plow-furrows, and adjacent fields should be distinguished from each other by
different inclinations of lines.
Figs. 269 and 270 represent marsh or bog land. Fig. 269 is the more ordi-
nary mode of representing fresh-water bog, and Fig. 270 of salt-marsh.
FIG. 269.
FIG. 270.
FIG. 271.
Fig. 271 represents a river, with mud and sand banks. Sand is designated
by fine dots, made with the point of the pen ; mud in a similar way, but the
dots should be much closer together. Gravel is represented by coarser dots,
and stones by irregular angular forms.
Water is almost invariably represented in the same way, except in connec-
tion with bogs, by drawing a line parallel to the shore, following its wind-
ings and indentations closely ; then another parallel a little more distant ; a
FIG. 272.
FIG. 273.
third still more so ; and so on. Brooks, and even rivers, when the scale is
small, are represented by one or two lines. Fig. 272 gives a plan and sec-
tional view of water, in which the white curves represent the character and
direction of the flow of streams, retarded at bottom and sides, and more rapid
TOPOGRAPHICAL DRAWING.
151
near the surface and at center, therefore convex down stream. The direction
of the current may also be shown by arrows, as in Fig. 271.
Fig. 273 represents a bold shore bounded by cliffs.
Fig. 274 represents a turnpike. If the toll-bar and marks for a gate be
omitted, it is a common highway. Fig. 275 represents a road as sunk or cut
3M Bar
FIG. 274.
FIG. 275.
FIG. 276.
FIG. 277.
through a hill. Fig. 276, one raised upon an embankment. Fig. 277 is a
railroad, often represented without the cross-ties by two heavy parallel lines,
sometimes by but one.
FIG. 278.
FIG. 279.
FIG. 280.
FIG. 281.
A
Saw-mill,
-
Wind-mill, (
&
Steam-mill,
m
Furnace,
ml
Woolen-factory,
&
Cotton-factory,
t
Dwellings, f
X
Churches, m
^%
O
Grave-yards,
Fig. 278 represents a bridge with a single pier. Fig. 279, a swing or draw
bridge. Fig. 280, a suspension bridge, and Fig. 281 a ford. Fig. 282, a lock
of a canal. Canals are represented like roads, except
that in the latter the side from the light is the shaded
line ; in the former, the side to the light. FlG - 282 -
The more important objects that are likely to need representation on a map
have conventional signs, as follows :
Signal of Survey,
Telegraph,
Court-house,
Post-office,
Tavern,
Blacksmith's shop,
Guide-board,
Quarry,
Grist-mill,
The localities of mines may be represented by the signs of the planets,
which were anciently associated with the various metals, and a black circle for
152
TOPOGRAPHICAL DRAWING.
coal. Thus, $ Mercury, ? Copper, ^ Lead, D Silver, O Gold, 6 Iron,
K Tin, Coal.
The Representation of Hills. The two methods in general use for rep-
resenting with a pen or pencil the slopes of ground, are known as the vertical
and horizontal. In the first (Fig. 283), the strokes of the pen follow the course
that water would take in running down these slopes. In the second (Fig. 284),
FIG. 283.
FIG. 284.
they represent horizontal lines traced round them, such as would be shown on
the ground by water rising progressively by stages, 1, 2, 3, 4, 5, 6, up the hill.
The last is the more correct representation of the general character and features
of the ground, and, when vertical levels or contours have been traced by level
at equal vertical distances over the surface of the ground, they should be so
represented ; or when, by any lines of levels, these contours can be traced on
the plans with accuracy, the horizontal system should be adopted : but where,
as in most plans, the hills are but sketched in by the eye, the vertical system
should be adopted ; it affords but proximate data to judge of the slope, whereas,
by the contour system, the slope may be measured exactly. It is a good maxim
in topographical drawing not to represent as accurate anything which has not
been rigorously established by surveys. On this account, for general plans,
when the surface of the ground has not been leveled, nor is required to be
determined with mathematical precision, we prefer the vertical to the hori-
zontal system of representing slopes.
On drawing hills on the vertical system, it is very common to draw contour-
lines in pencil as guides for the vertical strokes. If the horizontal lines be
traced at fixed vertical intervals, and vertical strokes be drawn between them
in the line of quickest descent, they supply a sufficiently accurate representa-
tion of the face of the country for ordinary purposes. It is usual to make the
vertical strokes heavier the steeper the inclination, and systems have been pro-
posed and used, by which the inclination is defined by the comparative thick-
ness of the line and the intervening spaces.
TOPOGRAPHICAL DRAWING.
153
In describing ground with the pen, the light is generally supposed to de-
scend in vertical rays, and the illumination received by each slope is dimin-
ished in proportion to its divergence
from the plane of the horizon. Thus,
in Fig. 285, it will be seen that a hori-
zontal surface receives an equal por-
tion of light with the inclined surface
resting upon it, and, as the inclined
FIG. 285.
surface is of greater extent, it will be
darker than the horizontal in propor-
tion to the inclination and consequent increase of the surface, and on this
principle varied forms of ground are represented by proportioning the thick-
ness of stroke to the steepness of the slope.
FIG. 286.
In the German system, as. proposed by Major Lehmann, of representing the
slopes of ground by a scale of shade, the slope at an angle of 45, as reflecting
its light horizontally, is supposed to be the greatest
ever required to be shown, and is represented by black,
while the horizontal plane reflecting all rays upward is
represented by white. Fig. 286 gives the intervening
proportions of black and white.
A modification of Lehmann's method, proposed by
the United States Coast Survey, has the advantage of
discriminating between slopes of greater inclination
than 45. The table gives the proportions of black
and white for different inclinations, and the construc-
tion may easily be understood from Fig. 287.
Contour- Lines. Conceive a hill to be completely covered with water.
Then suppose the water to be drawn down, say five feet at a time. Each line
of contact of the hill and the water will be a contour-line, or a line every point
of which is at the same height or level above a fixed horizontal plane, called
the datum-plane. For a small hill, stake out the ground in squares of say
fifty feet to the side, and take levels at each point of these squares, and as many
intermediates as the change of slope makes necessary. To draw the map, lay off
these squares to a scale, and mark the elevation of each point and the interme-
diates in pencil. Then by the eye draw in the contours at such vertical dis-
Slope.
Proportion of
Black. White.
24 or 2
1
10
5 or 6
2
9
10 or 11
3
8
15 or 16
4
1
25 or 26
5
6
35
6
5
45
7
4
60
8
3
75
9
2
154
TOPOGRAPHICAL DRAWING.
tances apart as the requirements of the map call forth. For a large survey,, say
of a mountain, such a method is impracticable. In this case, the surveyor
FIG. 287.
fixes a number of points at the same level, the points being absolutely estab-
lished by the transit or compass so that they can be plotted accurately. Con-
nect all points at the same level, and fill in the distances between by the eye,
on the supposition that the slope is uniform between these lines. The lines
absolutely established and those merely sketched in must not be confounded,
and should be distinguished apart either by color, by size of lines, or by dot-
ting. The contour-lines denoting every even five, ten, etc., feet above the
datum or plane of reference may be numbered with such height. This is an
effective way of representing hills, but is only to be recommended when lines
FIG. 288.
have been traced and it becomes a record of facts. Fig. 288 represents, on
double the scale, the half of the hill, Fig. 284, with one half completed by
drawing the intermediate contour lines.
The objection to the drawing of hills by any system is that the depths of
shade representing different slopes conflict with the lights and shades of the
drawing, and are therefore confusing. The plan adopted by Von Eggloffstein
in his maps was to form a model and then put in the hills as they appeared,
with the rays of light inclined 45 to the plan of the drawing. He adopted a
ready way of forming his model. The contours were cut out of sheet-wax
under the needle of a sewing-machine, then properly superimposed on one
another. A mold was then taken from them in plaster. A model from the
mold, also in plaster, was then taken. This was watered while fresh by a verti-
cal rain from a water-pot, which broke down the vertical edge of the contours,
and gave natural lines of water shed. This model would then be photographed
TOPOGKAPHICAL DRAWING.
155
FIG. 289.
156
TOPOGRAPHICAL DRAWING.
Degree.
Radii, ft.
Central
Ordinate.
1
5729-65
0-218
2
2864-93
0-436
3
1910-08
0-655
4
1432-69
0-873
5
1146-28
1-091
6
955-37
1-309
7
819-02
1-528
8
716-78
1-746
9
637-27
1-965
10
573-69
2-183
under an inclined light, and gave an admirable projection. When a model
was not made, the hills are represented in the same way under an inclined
light of 45.
Fig. 289 is a map of the harbor and city of New Haven, reduced from the
charts of the United States Coast Survey.
Plate VI is a map of a farming country. These two maps illustrate the
practical applications of topographical conventionalities.
Railway surveys are usually plotted by tangents. The curves are then put
in, and the topographical features for the width necessary. The curves are
designated by degrees, as a curve of 1, 2, 3, etc.,
according as the angle subtended at the center by a
100-feet chord is 1, 2, 3, etc.
Knowing the tangent points, it is easy to plot in
the curve, as the center of the curve must be the
intersection of the perpendiculars to the tangents at
these points. .Or, if we know one point of tangency
and the radius, erect a perpendicular at this point,
and lay off the radius on it to get the center of the
curve.
When the curves are larger than can be described
by the dividers or beam compasses, they can be plotted as shown in geometrical
problems, or points of a curve may be obtained by calculation of their ordinates,
and the curves drawn from point to point by sweeps and variable curves. Ap-
proximately, knowing the central ordinate of the curve between two points, the
-X
31.43FcdFallpcrMilt
Level
FIG. 290.
central ordinate of one half that curve will be one quarter of the first ; but it
should be observed that, the greater the number of degrees in the arc, the less
near to the truth is the rule.
Fig. 291 represents a plot of a railway line ; in this plot the curve is repre-
sented as a straight line, the radius of curvature being written in. This method
is sometimes adopted when it is desirable to confine the plot within a limited
!AL DRA
TOPOGRAPHICAL DRAWING. 157
'
space upon the sheet, and it is convenient when plotted thus directly beneath
the profile or longitudinal section (Fig. 290).
In plotting the section, a horizontal or base line is drawn on which are laid
off the stations or distances at which levels have been taken ; at these points per-
pendiculars or ordinates are erected, and upon them are marked the heights of
the ground above the base, and the marks are joined by straight lines. To
express rock in a cut, it is generally represented by diagonal lines ; rivers are
represented in section by cross-lines or colored in blue ; a mud-bottom by
masses of dots.
Since it would be in general impossible to express the variations of the sur-
face of the ground in the same scale as that adopted for the plan, it is usual
therefore to make the vertical scale larger than that of the horizontal, usually
in proportion of 10 or 20 to 1. Thus, if the horizontal scale of the plan be 400
feet to the inch, the vertical scale would be 40 or 20 feet to the inch.
For the purpose of facilitating the plotting of profiles, profile-paper can be
obtained from stationers, on which are printed horizontal and vertical lines ;
the horizontal lines being ruled at a distance of -$ of an inch from each other,
every fifth line being coarser, and every twenty-fifth still heavier than the
others. Each of the spaces is usually considered one foot. The vertical lines
are one quarter of an inch distant from each other, every tenth line being
made more prominent than the others ; these spaces in general represent a
distance of 100 feet, the usual distance between stations 011 a railroad. Much
time is saved by the use of this paper, both in plotting, and in reading the
measurements after they are plotted.
In the plotting of sections across the line, which are extended but little
beyond the line of the cut or embankment, equal vertical and horizontal scales
are adopted ; these plots are mostly to determine the position of the slope, or
to assist in calculating the excavation. To facilitate these, cross-section paper
is sold, ruled with vertical and horizontal lines, forming squares of T V of an
inch each. Every fifth line in each direction is made prominent. When
cross-sections are extended to show the grade of cross-road, or changes of level
at considerable distance from the line of rail, the same scales, vertical and hori-
zontal, are adopted as in the longitudinal section or profile.
It will be observed, in Fig. 290, that the upper or heavy line represents the
line of the rail, the grades being written above ; this is the more usual way,
FIG. 292.
but sometimes, as in Fig. 292, the profile and plan are combined ; that is, the
heights and depths above and below the grade-line of the road are transferred
to the plan, and referred to the line in plan, which becomes thus a representa-
tion both in plan and elevation.
158
TOPOGRAPHICAL DRAWING.
Cross-sections, for grades of cross-roads, etc., are usually plotted beneath or
above the profile ; they may, if necessary, be plotted across the line when plan
and profile are combined.
Besides the complete plans as above, giving the details of the location, land
plans, so called, are required, showing the position and direction of all lines
of fences and boundaries of estates, with but very few of the topographical feat-
TIG. 293.
ures. The center line of the road is represented in bold line, and at each side,
often in red, are represented the boundaries required for the purposes of way.
In general, a width of 100 feet is the amount of land set off, lines parallel to
the central line being at a distance of 50 feet on each side ; but when, owing
to the depth of the cut or embankment, the slopes run out beyond this limit,
the extent is determined by plotting a cross-section and transferring the dis-
tances thus found to the plan, and inclosing all such points somewhat within
TOPOGRAPHICAL DRAWING.
159
the limits as set off for railway purposes. These plans are generally filed in
the register's office for the county through which the line passes.
Hydrometrical or Marine Surveys. In plotting hydrometrical or marine
surveys, the depths of soundings are seldom expressed by sections, but by
figures written on the plan, expressing the sounding or depth below a datum-
line, generally that of high water. The low-water line is usually represented
by a single continued line. The soundings are generally expressed in fathoms,
sometimes in feet.
Fig. 293 is a map of Cape Cod Bay plotted by this method. The depths
are expressed in fathoms (six feet), and the dotted lines inclose depths between
certain fixed limits so as to plainly indicate a channel or bar, as the case may be.
Another and an exceedingly effective way of making a marine chart is to
express the different depths by lines varying in direction, distance apart, width,
Depth under 5 Fathoms. 5 to 10 Fathoms.
10 to 20 Fathoms
Over 20 Fatlioma.
5 Miles.
FIG. 294.
etc. Fig. 294 is a chart of the Isle of Wight and the surrounding water,
with the depths expressed as shown at the bottom of the cut. Sections are often
used for rivers, especially for those like our Western ones, that have a very
changeable bottom. By plotting sections, taken at different times, over one
another, distinguishing them apart by a difference in color and variety of line,
160
TOPOGRAPHICAL DRAWING.
the changes that take place in the bottom of the river, and the erosion of the
banks, are more boldly shown than by the use of any other method. The
ordinary marine conventionalities are as follows :
DIRECTION OF THE CURRENT
Anchorage for ships,
Anchorage for coasters, j^
Rocks always covered, j^
Buoys, 1 1
Wrecks, ,
Harbors,
Light-house,
Signal-house,
Channel-marks,
Representation of Geological and Statistical Features. The geological feat-
ures of a country may be readily expressed on a map by the use of lines as in
W.oPGr
1. Alluvia. 2. Upper Tertiary. 3. London Clay, &c. 4. Chalk. 5 & 6. Greensand and Gait,
10 & 11. Triassic, &c.
16. Silurian.
12. Permian.
13. Carboniferous.
FIG. 295.
marine charts. Fig. 295 is a geological map of Southeastern England, and
will be easily understood by inspection.
TOPOGRAPHICAL DRAWING.
161
A geological profile may be represented in the same way. The different
rocks or formations are usually distinguished by color and explained by mar-
ginal notes and squares, but more often by marks, dots, or cross-hatchings, as
FIG. 296.
in Fig. 296, which exhibits the geological features of the United States east of
the Rocky Mountains and Canada to the south of the St. Lawrence.
Fig. 297 is a section from Pennsylvania to Canada, showing the relations of
the subdivisions to each other.
11
162
TOPOGRAPHICAL DRAWING.
Fig. 298 represents an ideal diagram of the principal groups in American
geology, in the order of their superposition.
Ideal Section north and south from Canada to Pennsylvania : A, Archaean ; L S and U S, Silurian ; D, De
vonian ; C 1 , Carboniferous.
ERAS.
3. P&ychozoic.
4. Cenozoic
3. Mesozoic ..<
3. Palaeozoic . .<
1. Archaean,..
Carboniferous.
Huronian.
Lanrentian.
FIG. 298.
Ideal General Section of the Whole Series of Strata,
stowing the Principal Divisions and Subdivisions.
Still another form of a topographical
and statistical map is shown in Fig. 299,
which is a portion of the city of Lon-
don, taken from a sanitary report by
a commission of Parliament ; and em-
bodies, in a graphic way, the details in
regard to drainages, natural and arti-
ficial, contour-lines and street-sewers ;
position of gas and water mains, and
occupancy of buildings. On the origi-
nal are also given the number of the
houses and names of streets.
Eeference has been made to the
drawing of hills by contours, and it has
not been recommended except when the
lines have been accurately determined
by level. When this is the case, they
should always be used ; it is the sim-
plest and most explanatory record of
facts, and if the facts have been worth
determining they are worth recording.
When contour-lines are brought more
closely together (as shown in Fig. 300,
which is from the same sanitary report,
and of a larger portion of London), it
produces the effect of physical relief,
and shows at a glance the lines of natu-
ral drainage, and from it profiles can
be made, in any direction, for the grad-
ing of streets or sewers. Were town
and county maps thus drawn with con-
tour-lines, much time and money would
be saved in the location of highways and
railways.
Transferring. It is usual, in plot-
ting from a field-book, to make first but
a rough draft, and then make a finished
copy on another sheet. In the first,
many lines of construction, balances of
TOPOGRAPHICAL DRAWING. 163
survey, and trial lines are drawn, which are unnecessary in the copy ; outlines
of natural features are sketched roughly, but the plotting of surveys, and such
lines and points as are to be preserved in the copy, must be done with accuracy.
FIG. 299.
Private houses (occupied by persons not in receipt of wages).
Offices and shops.
Houses occupied by persons in receipt of wages.
Warehouses.
Stables and outhouses.
Public buildings.
Contours ; vertical distances between lines, two feet.
= Sewers.
Gas-pipes.
_,__i_ 5 _i- Water-pipes.
The most common way of transferring, for a fair copy, is by superposition
of the plan above the sheet intended for the copy, and pricking through every
intersection of lines on the plan, and all such points as may be necessary to
preserve. The clean paper should be laid and fastened smoothly on the draw-
ing-board ; the rough draft should be laid on smoothly, and retained in its
164
TOPOGRAPHICAL DRAWING.
position by weights, glue, or tacks. The needle must be held perpendicular
to the surface of the plan, and pressed through both sheets ; begin at one side
and work with system, so as not to prick through each point but once, nor
omit any ; make the important points a trifle the larger. For the irregular
FIG. 300.
curves, as of rivers, make frequent points, but very small ones. On removing
the plan, select the important points, those defining leading lines ; draw in
these, and the other points will be easily recognized from their relative position
to these lines. When any point has not been pricked through, its place may
be determined by taking any two established points adjacent to the one re-
quired, and with radii equal to their distance, on the plan, from the point
required, describing arcs, on the copy, on the same side of the two points ; their
intersection will be the point desired. In this way, as in a trigonometrical
survey, having established the two extremes of a base, a whole plan may be
copied. In extensive drawings it is very common to prick off but a few of the
salient points, and fill in by intersections, as above, or by copying detached
portions on tracing-paper, and transferring them to the copy ; the position of
each sketch being determined by the points pricked off, the transfer is made
by pricking through as above, or by transfer-paper placed between the tracing
and the copy.
If tracing paper or cloth (pages 56, 57) be placed above the drawing, every
line will show through, and can be traced directly with the pen, in India ink.
These tracings are used mostly to preserve duplicates of finished drawings.
Duplicates of drawings, contracts, estimates, etc., on paper allowing the
light to pass through are readily made by the use of .ferro-prussiate paper, or
the blue-print process. Paper can be prepared by washing it with a mixture
TOPOGRAPHICAL DRAWING. 165
of 1 ounces of citrate of iron and ammonia with 8 ounces of water, and 1
ounces of red prussiate of potash and 8 ounces of water, dissolved separately
and mixed. The mixture and prepared paper should be kept from the light.
The prepared paper in close rolls can be readily purchased. For the manipu-
lation there is needed plate-glass, and a blanket a little larger than the draw-
ing, a shallow tin dish, that the drawing can be placed in flat for washing.
Lay down the blanket on a drawing-board, above that the ferro-prussiate
paper, next the drawing, and then the glass. Expose to the sunlight for about
ten minutes if the drawing is on tracing-paper or cloth, and longer for thicker
paper ; when done, the background should be a metallic gray. Now lay the
ferro-prussiate paper in the tin dish, cover with water, and leave it for five to
ten minutes ; wash thoroughly and dry. The lines will be white on a blue
ground. The negative of ferro-prussiate paper gives blue lines on a white
ground, and other processes black lines on a buff ground.
An accurate and rapid way of tracing, on drawing-paper, plans of small
extent, is by means of an instrument called a copying-glass. It consists of a
large piece of plate-glass set in a frame of wood, which can be inclined at any
angle. On this glass is first laid the original plan, and above, the fair sheet,
and the frame being raised to a suitable angle, a strong light is thrown by re-
flectors or otherwise on the under side of the glass, whereby every line in the
original plan is seen distinctly through the fair sheet, and the copy is made
at once, as on tracing-paper. This same process, on a small scale, is adopted
by putting the plans upon a pane of glass in a window.
Plans mounted on cloth, or on opaque paper, do not admit of being traced
in this way. In such cases the copy may be made by means of transfer-paper.
The plan is first traced on tracing-paper or cloth, black-leaded or transfer paper
is then placed on the fair sheet, and the tracing-paper copy is placed above.
All is steadied by numerous weights along the edges, or by drawing-pins fixed
into the drawing-board. A fine and smooth point is then passed over each
boundary or mark on the tracing with a pressure of the hand sufficient to
cause a clear, penciled mark to be left on the fair sheet by the black-leaded
or transfer paper. The whole outline is thus obtained, and afterward drawn
in ink. The copyist should be careful in his manipulations, so as not to
transfer any other lines than those required, nor leave smutches on the fair
sheet.
Plans may be copied, on a reduced or enlarged scale, by means of the pan-
tagraph (Figs. 146, 147), or by the method of squares (pages 63, 64).
Map Projections. For a farm or other small survey, the surface of the earth
can be conceived to be flat, and the map a horizontal projection of the plane
surface on a reduced scale ; the error being practically insignificant, while
the labor is greatly reduced by making this assumption. But, for large maps
of countries, States, rivers, etc., where the meridians and parallels of latitude
are represented, such a system would be so erroneous as to be impracticable.
The surface of the earth being a sphere, it is incapable of development on a
plane, so that it becomes necessary to make the best approximation possible in
form, relation, and proportional area of the portions to be represented on a
map or chart. There are many different kinds of projection, all more or less
166
TOPOGRAPHICAL DRAWING.
imperfect, but most of which possess advantages for some descriptions of maps
or charts. They may be divided into four classes, as follows :
Class I. Perspective projection on planes.
" II. Developed perspective projections.
" III. Projections by developing elements.
" IV. Projections conformed to some arbitrary condition.
In Class I, the more important kinds are the globular or equidistant, and
the stereographic.
Globular or Equidistant Projection of the Sphere. According to this
method the eye is placed at a distance from the center of the earth, equal to
1.707 x radius. The plane of projection passes through the center perpen-
dicular to the central ray. This method is quite common in school maps.
The following is the construction :
Draw two lines (Fig. 301), at right angles to and intersecting each other ;
from the point of their intersection as a center, with a radius equal to that
FIG. 301.
intended for the hemisphere, describe a circle, and mark the points N", S, W, E.
N and S will be the poles, the line N S the central meridian, and W E the
equator. Divide N S and W E into as many equal parts as there are degrees
or numbers of degrees to be represented in the figure in divisions of 30 and
meridian and equator into six equal parts, as the hemisphere embraces 180.
Commence at C, and divide the half-lines into three equal parts. Divide the
arcs N W, N E, S W, and S E, each into three equal parts. There will be now
determined three points in two parallels of north and south latitude, 30 and
60, through which to describe the arcs representing the parallels. The center
of these arcs will be in the line N S ; describe the arc, and with the same radius
from a center on the line N S below the S pole, describe a similar arc passing
through the S 30 point on the meridian. Therefore, keeping the steel point
of the dividers on the line N S, by trial radii may be found of arcs which shall
TOPOGRAPHICAL DRAWING.
167
pass through the points on the central meridian and on the circle. With the
radii describe arcs for the parallels in north and south latitude. All the me-
ridians pass through the N and S poles, and through the divisions of degrees on
the equator. There are three points, therefore, determined in the arc of each
meridian which may be described from centers found by trial on the line E W.
Stereographs Projection. In this method the eye is taken at the center of
the earth, at the pole of the great circle used as a plane of projection. Circles
are stereographically projected into circles. An increasing exaggeration out-
ward from the center is its principal defect. To project stereographically the
hemisphere on the plane of the meridian, draw the central meridian, equator,
FIG. 303.
and circle (Fig. 302), as in the preceding problem. To project the other me-
ridians (say every 10), divide the quadrant N E into nine equal parts ; from
S to these points of division, 10, 20, 30, draw lines intersecting C E in 10, 20,
30. These latter points are in the meridians through which N and S arcs are
to be described from centers on the line E W.
To find in like manner the three points in the parallels of latitude, divide
the quadrants into nine parts, 80, 70,, 60, and through these points draw lines
to W ; the intersection? with the central meridian 80, 70, 60, will with the
points of the quadrant furnish three points through which to describe arcs of
parallels of latitude.
To project the hemisphere on the plane of the equator (Fig. 303). Draw two
lines at right angles to each other ; describe the circle and divide the circum-
ference as before. The center will be the projection of N or S pole, the lines
at right angles to each other will be meridians, as well as any other diameters,
as D H, F K, drawn through some division of the circumference.
To project the parallels of latitude. The circle represents the projection of
the equator, and the other parallels must be arcs on the same center C, of
which the radii are to be determined by the intersections of the line C B by
lines drawn from A to the divisions of the circle 10, 20, 30.
In Class II, instead of projecting directly on planes, an intermediate cone
or cylinder is employed to receive the projection, which is then developed on a
168
TOPOGRAPHICAL DRAWING.
tangent plane. The cylinder or cone must always be employed, because they
are the only surfaces that can be developed on a plane. The eye is always con-
ceived to be at the center of the earth in all the projections of this class.
In Class III the portions of the earth's surface are mapped by being divided
into small or differential elements which are successively developed. This
method admits of greater accuracy than any of the four classes. The two most
important subdivisions are Bonne's and the Polyconic.
In Bonne's projection, assume a central meridian, and a central parallel
with a cone tangent along the latter. The central meridian is then developed
on that element of this cone to which it is tangent, and the cone is then de-
veloped on a tangent plane. The parallel, by this process, becomes an arc
with its center at the vertex of the cone, and the meridian becomes a graduated
line. Conceive concentric circles to be traced through points on this meridian
at elementary distances apart. The zones of the sphere situated between the
parallels through these points are then conceived to be developed each between
its corresponding arcs. In this way all the i:ones of the sphere are developed
on a plane surface in their true relation to each other and the central, each
having the same length, width, and relation to its neighboring zone that it did
on the spherical surface. The areas are not changed by the development, and
distances along the parallels are correct, while those along the meridians are
slightly increased, except those along the central meridian, which are strictly
correct. The scale is nearly uniform over the whole map, and, for moderate
areas, the intersections are nearly rectangular. Bonne's method is almost
universally applied to the detailed topographical maps based on the trigonomet-
rical surveys of the different states of Europe.
The Polyconic has been adopted by the United States Coast Survey, and
all their maps are projected by this method. Each parallel is supposed to be
represented on a plane by the development of a cone having the parallel for its
base, and its vertex at the point of intersection of a tangent to the parallel and
the earth's axis. The map thus becomes the development of the surface of
successive cones, and the degrees of the parallel preserve their true length.
The following tables are given for use in projecting large maps. Their use
will be explained in an example. For making small maps, with a great de-
gree of accuracy, tables are published by the United States Coast Survey.
Co-ordinates of Curvature in Miles for Maps of Large Extent.
Latitude 20.
Latitude 24".
Latitude 28.
Latitude 32.
LONGITUDE.
D. M.
D. P.
D. M.
D. P.
D. M. D. P.
D. M.
D. P.
2.
130-0
0-8
126-4
0-9
122-2
1-0
117-4
M
4.
260-0
3-1
252-8
3-6
244-4
4-0
234-8
4-3
6.
390-0
6-9
379-2
8-1
366-5
9-0
352-0
9-8
8.
620-0
12-4
505-5
14-4
488-6
16-0
46S-3
17-3
10.
649-8
19-4
631-7
22-4
610.4
25-0
586-3
27-1
12.
779-7
27-8
757-9
32-2
732-4
36-0
703-5
39-1
14
909-2
38-0
883-6
43-9
853*7
49-0
819-6
53-1
16.
1039-2
49-6
1009-9
57-4
975-7
64-1
936-8
69-5
18.
1168-1
62-8
1134-8
726
1096-0
80-9
1051-9
87.8
20.
1298-0
77-6
1261-2
89-7
1218-8
100-1
1169-2
108-6
R.
10892
8905
7458
6348
TOPOGRAPHICAL DRAWING. 169
Co-ordinates of Curvature in Miles for Maps of Large Extent. (Continued.)
Latitude 36.
Latitude 40.
Latitude 44.
Latitude 48.
LONGITUDE.
I). M.
D. P.
D. M.
D. P.
D. M.
D. P.
D. M.
D. P.
2 .
112-0
1-2
106-1
1-2
99-7
1-2
92-7
1-2
4. .
224-0
4-6 212-2
4-8
198-9
4-8
185-4
4-8
6. .
335-9
10-3
318-1
10-7
298-7
10-9
277-9
10-8
8..
447-7
18-4
423-9
18-9
398-0
19-3
370-3
19-2
10..
659-2
28-7
529-4
29-7
497-1
30-2
462-3
30-0
12..
670-5
41-3
634-7
42-8
595-9
48-4
654-1
43-2
14. .
781-6
56-2
739-7
58-2
694-3
59-1
645-6
68-8
16..
892-3
73-4
844-3
76-0
792-3
77'1
736-5
76-7
18..
1002-6
92-8
948-5
96-1
889-9
97-5
827-0
97-0
20..
1112-5
114-5
1052-3
1185
986-9
120-2
916-9
119.6
R.
5461
4729
4110
3575
Length of a Degree of Longitude at Different Latitudes, and at Sea-Level.
Deg.
of
Lat.
Miles.
?
Lat.
Miles.
Deg.
of
Lat.
Miles.
"of'
Lat.
Miles.
Deg.
of
Lat.
Miles.
Deg.
of
Lat.
Miles.
69-16
14
67-12
28
61-11
; 42
51-47
56
38-76
70 1 23-72
2
69-12
16
66-50
30
59-94
44
49-83
58
36-74
72
21-43
4
68-99
18
65-80
32
58-70
46
48-12
60
34-67
74
19-12
6
68-78
20
65-02
34
57-39
48
46-36
62
32-55
76
16-78
8
68-49
22
64-15
36
56-01
50
44-54
64
30-40
78
14-42
10
68-12
24
63-21
38
54-56
52
42-67
66
28-21
80
12-05
12
67-66
26
62-20
40
53-05
54
40-74
68
25-98
82
9-66
Lengths for intermediate degrees can be found accurately by proportion.
At the equator, 1 of latitude = 68 '70 miles ; at latitude 20 = 68 '78 ; at 40
= 69-00 ; at 60 = 69-23 ; at 80 = 69-39 ; at 90 = 69-41.
To draw a map according to the tables, we lay off on the straight line (Fig.
304) N" S, representing the middle meridian, the lengths representing the ten
degrees of latitude between 20 and 30, 30 and 40, etc. Through these
points draw circular arcs with the radii designated by R in the preceding tables.
On these arcs lay off the lengths of ten degrees of longitude for each correspond-
ing latitude on each side of the center meridian. Through the points thus
formed draw the meridians, which will be found slightly concave toward the
middle one. If the scale is so large that it is impossible to draw the circular
arcs with beam-compasses, erect perpendiculars at the points 20, 30, 40, and
50, and on them lay off the values d m from the tables. At each of the points
so found erect perpendiculars, and set off on them the corresponding values of
d p. Through the points thus found draw the parallels and meridians. The
principal advantages of this projection are a minimum amount of distortion
at any portion of the map ; a scale of degrees and minutes of the parallels
and meridians, by means of which, positions, determined by their latitudes and
longitudes, may be readily inserted on the maps ; the use of a linear scale in
any portion or direction of the map ; and the intersection of parallels and
meridians at nearly right angles.
In Class /Fsome arbitrary mathematical condition is imposed, for some
practical purpose, usually giving rise to distorted maps.
iro
TOPOGRAPHICAL DRAWING.
For polar projections, De Lorgne's has much merit. Calculate first a cir-
cle with an area equivalent to that of the hemisphere to be projected. Draw
N
such a circle and connect the graduations of the circumference with the center.
These represent the meridians. The radius can be divided into ninety equal
N
W
80
eo
<.o
20
160
/40
120
100
80
60
W
20
20
p
20
VO
60
80
Too
120
/to
160
.0
60
80
s
FIG. 305.
parts ; but, where it is possible, the chords of the polar distances of the par-
allels should be used for determining the parallels.
TOPOGRAPHICAL DRAWING.
m
Mercator's chart is especially valuable to the navigator. By it he can lay
off his course accurately on the chart in a straight line. It has little value for
the other purposes of a chart. Meridians are represented by equidistant par-
allel straight lines, and the parallels by a perpendicular set of parallel straight
lines, whose distances from each other increase from the equator outward in
the same ratio as the corresponding longitudinal degrees diminish. By this
means the relation between the latitude and longitude measurements on the
chart is preserved uniformly as on the earth's surface.
To construct a Mercator's Chart (Fig. 305). Draw two straight lines, W E
and N S, intersecting each other at right angles at C. W E is the equator, N S
the meridian passing through the middle of the chart. From set off equal
parts on the equator both ways, to represent degrees of longitude, subdivided
into minutes if the size of the chart will admit of it. Assuming the equator
as a scale of minutes, set off from C, toward N and S, the number of minutes in
the enlarged meridian corresponding to each degree of latitude, as shown by
the table of meridional parts. Draw lines parallel to N S through the divisions
of the equator for meridians, and parallels to W E through the divisions of
N S for parallels of latitude.
To find the bearing of any one place from another, it is only necessary to
draw a straight line between the two points, and observe the angle it makes
with the meridians.
Table of Meridional Parts.
Latitude.
Meridional parts.
Latitude.
Meridional parts.
Latitude.
Meridional parts.
o-oo
35
2244-29
70
5965-92
5
300-38
40
2629-69
75
6970-34
10
603-07
45
3029-94
80
8375-20
15
910-46
50
3474-47
85
10764-62
20
1225-14
55
3967-97
90
Infinite.
25
1549-99
60
4527-37
30
1888-38
65
5178-81
COLORED TOPOGRAPHY.
Topographical features may be represented effectively and expeditiously by
means of the brush and water-colors, either by India ink alone, or by various
tints, or by the union of both.
The most important colors for conventional tints are (besides India ink),
indigo (blue), carmine (or crimson lake), and gamboge (yellow), used separately
or compounded. Besides these, burnt sienna, yellow ochre, and vermilion are
sometimes used, although the first three are susceptible of the best combina-
tions, and the others are generally used alone.
The following conventional colors are used by the French military engineers
in their colored topography : Woods, yellow ; using gamboge and a very little
indigo. Grass-land, green ; made of gamboge and indigo. Cultivated land,
brown ; lake, gamboge, and a little India ink ; "burnt sienna" will answer.
Adjoining fields should be slightly varied in tint. Sometimes furrows are in-
dicated by strips of various colors. Gardens are represented by small rectangu-
172 TOPOGRAPHICAL DRAWING.
lar patches of brighter green and brown. Uncultivated land, marbled green
and light brown. Brush, brambles, etc., marbled green and yellow. Heath,
furze, etc., marbled green and pink. Vineyards, purple ; lake and indigo.
Sands, a light brown; gamboge and lake; "yellow ochre "will do. Lakes
and rivers, light blue, with a darker tint on their upper and left-hand sides.
Seas, dark blue, with a little yellow added. Marshes, the blue of water, with
spots of grass green, the touches all lying horizontally. Eoads, brown ; be-
tween the tints for sand and cultivated ground, with more India ink. Hills,
greenish brown ; gamboge, indigo, lake, and India ink. Woods may be finished
Up by drawing the trees and coloring them green, with touches of gamboge
toward the light (the upper and left-hand side), and of indigo on the opposite*
side.
In addition to the conventional colors, a sort of imitation of the conven-
tional signs is introduced in color with the brush, and shadows are almost
invariably introduced. The light is supposed to come from the upper left-hand
corner, and to fall nearly vertical, but sufficiently oblique to allow of a decided
light and shade to the slopes of hills, trees, etc. After the shadow has been
painted, the outline of the object is strengthened by a heavy black line on
the side opposite the light. The flat tints are first laid on as above, and
then the conventional signs are drawn in with a pencil and colored in with
appropriate and more intense tints ; the shadows are generally represented in
India ink.
Hills are usually shaded, not as they would appear in nature, but on the
conventional system of making the slopes darker in proportion to their steep-
ness ; the summits of the highest ranges being left white an arrangement
incorrect in theory, but generally understood by those not accustomed to plan-
drawing, and is easy of execution. Wash the surface first with the proper flat
tint, trace in with a pencil outlines, then lay on in India ink tints propor-
tioned in intensity to the height of the hills and steepness of the slopes. To
soften the tints, two brushes are used, one as a color-brush, the other as a water-
brush : the tints are laid on with the first, and softened by passing the water-
brush rapidly along the edges. The water-brush must not have too much
water, as it would in that case lighten the tint to a greater extent than is in-
tended, and leave a ragged, harsh edge. Tints may be applied in very light
shades, one tint over another, with the boundary of the upper tint not reaching
the extreme limit of the tint below it. When depth of shade is required, it is
best produced by application of several light tints in succession ; no tint is to
be laid over the other until the first is dry.
When woods have to be represented, the shading used for the trees, instead
of interfering with the shadows due to the slopes, may be made to harmonize
with them, and contribute to the general effect by presenting greater or less
depth, according to the position of the woods on the sides, or summits of the
hills.
An expeditious and effective way of representing hills with a brush, a spe-
cies of imitation of hills drawn with a pen on the vertical system, is effected by
pressing out flat the brush to a sort of comb-like edge ; drawing this over a
nearly dry surface of India ink, and then brushing lightly or more heavily be-
TOPOGRAPHICAL DRAWING. 173
tween the contours, according to the steepness of the slope, each of the comb-
like teeth making its mark.
Kivers and masses of water may be shaded in with a color and water brush
as above, or, by superposition of light tints, a shadow may be thrown from the
bank toward the light, and the outline of this bank strengthened with a heavy
black line. The tints are to be in indigo, the shadows in India ink.
Topographical drawings may be made in water-color with but one tint, as
India ink, or ink mixed with a little sepia. The conventional signs are in
imitation of pen-drawings, the hills in softened tint, or drawn with the comb-
edged brush, and the rivers shaded with superposed tints.
Most artistic and effective drawings are made of hills as they would appear
in nature, under an oblique light ; the sides of the hills next the light receiving
it more or less brilliantly, according as they are inclined more or less at right
angles with its rays, and the shades on the sides removed from the light, increas-
ing in intensity as the slopes increase in steepness.
Having damp-stretched the paper upon the drawing-board, first draw in
the lines in pencil, and afterward repeat them with a very light ink-line ; a
soft sponge, well saturated, should then be passed quickly over the surface of
the drawing, in order to remove any portions of the ink which would be liable
to mix with the tint and mar its uniformity.
The moistened surface will prevent the tint from drying too rapidly at the
edges. In tinting, "never allow the edge to dry until the whole surface is cov-
ered ; leave a little superfluous color along the edge while filling the brush. In
applying a flat tint to large surfaces, let the drawing-board be inclined upward
at an angle of five or six degrees, so as to allow the color to flow downward over
the surface. With a moderately full brush, commence at the upper outline,
and carry the color along uniformly from left to right and from right to left in
horizontal bands, taking care not to overrun the outlines, in approaching which
the point of the brush should be used, and at the lower outline let there be
only sufficient color in the brush to complete the tinting.
No color should be allowed to accumulate in inequalities of the paper, but
should be evenly distributed over the whole surface.
Too much care can not be given to the first application of color ; as any
attempt to remedy a defect by washing or applying fresh tints will be found
extremely difficult, and to generally make bad worse.
Erasers should never be used on a tinted drawing, as the paper, when
scratched, receives the tint more readily, and retains a larger portion of color
than other parts, thereby causing a darker tint.
Marbling is done by using two separate tints, and blending them at their
edges. A separate brush is required for each tint ; before the edge of the first
is dry, pass the second tint along the edge, blending one tint into the other,
and continue with each tint alternately.
In reference to the general effect to be produced in tinted topographical
drawings, as to intensity, everything should be subordinate to clearness ; no
tint should be prominent or obtrusive. Tints that are of small extent must be
a little more intense than large surfaces, or they will appear lighter in shade.
Keep a general tone throughout the whole drawing. Beginners will find it best
174 TOPOGRAPHICAL DRAWING.
to keep rather low in tone, strengthening their tints as they acquire boldness
of touch.
Plate VIII gives an example of colored topography.
In lettering tinted drawings, let the letters harmonize with the rest of the
plan ; let them be in tint more intense than the topography, prominent but
not obtrusive.
Finishing the Plan or Map. In general, in topographical drawings, the
light is supposed to fall upon the surface in a diagonal direction from the
upper left-hand corner. This rule is not uniform ; by some draughtsmen the
light is introduced at the lower left, and hills are mostly represented under a
vertical light, although the oblique adds more to the picturesque effect. The
plan is usually so drawn that the top may represent the north, and the upper
left-hand corner is then the northwest.
In inking in, commence first with the light lines, since a mistake in these
lines may be covered by the shade-lines. Describe all curves which are to be
drawn with compasses or sweeps before the straight lines, for.it is easier to join
neatly a straight line to a curve than the opposite. Ink in with system, com-
mencing say at the top ; ink in all light lines running easterly and westerly,
then all light lines running northerly and southerly, then commence in the
same way and draw in the shade-lines. It will of course be understood that
elevated objects have their southern and eastern outline shaded, while depres-
sions have the northern and western ; thus, in conventional signs, roads are
shaded the opposite to canals. Having inked in all lines that are drawn with
a ruler or described with compasses, commence again at one corner to fill in
the detail, keeping all the rest of the plan except what you are actually at work
upon covered with paper, to protect it from being soiled. The curved lines of
brooks, fences, etc., are sometimes drawn with a drawing-pen, sometimes with
a steel pen or goose-quill. The latter are generally used in drawing the verti-
cal lines of hills.
Boundary-lines of private properties, of townships, of counties, of States,
etc., are indicated by various combinations of short lines and dots, thus :
All plans should have meridian lines drawn on them ; also scales, and the
dates on which the plans were finished. Page 175 gives several designs for
meridians and borders. In these diagrams it will be observed that both true
and magnetic meridians are drawn ; this is desirable when the variation is
known, but in many surveys merely the magnetic meridian is taken ; in these
cases this line is simply represented with half of the barb of the arrow at
the north point, and on the opposite side of the line from the true meridian.
Scales are drawn or represented in various forms, or the proportion of the plan
to the ground is expressed decimally, as the number of feet, chains, etc., to
the inch.
Lettering. The style in which this is done very much affects the general
appearance of the plan. Great care must be taken in the selection and char-
acter of the type, and in the execution.
TOPOGRAPHICAL DRAWING.
175
176 TOPOGRAPHICAL DRAWING.
MAP OF
EXPLORATIONS AND SURVEYS
IN
NEW MEXICO AND UTAH
made under the direction of die
SECRETARY OF WAR
by
CAPT. J.N.MACOMB TOP^ENG". 8
assisted by
C,H.DIMMOCK, C.ENG*
I860
Scale of 12 Miles to one Inch or 1:760320
fc"-" -4 " ' -*-i 53 20 x "' >o
In the chapter on Drawing Instruments examples of the method of con-
structing letters, as well as some alphabets, are given.
Titles* On this pageare given some examples of titles, intended merely as an
illustration of the form of letters and their arrangement, the scale being much
smaller than that used on plans, except such as are drawn to a small scale. It
TOPOGRAPHICAL DRAWING. 177
will be observed that the more important words are made in prominent type.
The lower part of the title should always contain, in small character, the name
of the party making the survey, and also the name of the draughtsman, with
date of the execution of the plan : if the survey was made some time previous,
the date of the survey should be given. If the plan is compiled from several
surveys, the authorities should, if possible, be given. The lettering of the title
in lines parallel to the bottom of the plan is preferable, and, in general, the
great mass of lettering in the body of the plan should be formed in similar
lines ; but curved lines are often not only essential, but they materially con-
tribute to the beauty of the plan., Thus, on crooked boundaries, on outlines
of maps, the lettering should follow the general curve of the boundary ; also
on crooked rivers, lakes, seas, etc. ; on irregular or straggling pieces of land,
in order to show the extent, connection, or proprietorship thereof, the lettering
should follow the central line of such a tract ; and, if pieces of land be very
oblong in form but regular in outline, the lettering will be central in the
direction of the longest side. The lettering of roads, streets, etc., is always
in the direction of the line of road. Curved lines of lettering are often intro-
duced into extended titles to take oif the monotonous appearance presented
by a great number of straight lines of writing.
The direction of all lettering should be so as to be read from left to right.
If shades or shadows are introduced, they should be uniform with the rest of
the plan.
It will be observed that letters vary very considerably in their width, the /
being the narrowest, and the W the widest ; if, therefore, the letters composing
a word be spaced off at equal distances from center to center, the interval or
space between the letters will be more in some cases than in others. Thus, in
the word
R A I L W A Y
To avoid this, write in first one letter, and then space off a proper interval,
and then write in the next letter, and then space off the interval as before,
and so on, thus :
RAILWAY
When, as frequently happens, the words are very much extended, in order to
embrace and explain a large extent of surface or boundary, and the space occu-
pied by the letter is small in comparison with the interval, the disparity of
intervals will not be noticed, and the letters may be then laid off at equal
spaces from center to center, thus :
R A I L W A Y
When the lines of lettering are curved, the same rules for spacing are to be
observed as above. If the letters are upright, as Roman or Gothic, the sides
of each letter are to be parallel to the radius drawn to the center of the letter,
and the bottom and top lines at right angles to it. If the letters be inclined,
12
178
TOPOGRAPHICAL DRAWING.
as Italic letters, then the side-lines of the letters must be inclined to the central
radial line, as on a horizontal line they are inclined to the perpendicular.
In laying off letters by equal intervals, it is usual to count the number of
letters in the word, and fix the position on the plan of the central one, and
then space off on each side ; this is particularly important in titles, when it is
necessary that many lines should have their extremities at uniform distances
from the center line. In laying off the title, we determine what is necessary
to be included in the title, the space it must occupy, the number of lines neces-
sary, and the style and arrangement of characters to be used. Thus, if the
title were, Plan of the Proposed Terminus of the Harlem Railroad at New
York, 1857, knowing the space to be occupied, we can write the title thus :
an
We now draw parallel lines at intervals suited to the character of the type we
intend to employ for the different words. Harlem Railroad is the line to be
made most prominent ; this, calling the interval between the words one letter,
includes 15 letters ; or, if we consider /, with its proper interval, but half a
letter (which will be found a very good rule in spacing), 14 ; hence the center
of the line will be 7i letters from the beginning, or \ of the space occupied by
TOPOGRAPHICAL DRAWING. 179
the letter R and its interval. Draw a perpendicular line at the center, and
write in R in such a character as may suit the position to be filled, and lay off
by letters and spaces the other letters. The line Harlem Railroad is intended
to occupy the whole length of space ; that is, it must be the longest line in
the title, and the lines above and below must gradually diminish, forming a
sort of double pyramid. Proposed Terminus includes 16 letters ; the / and
interval between the words being rated as above, we find the center to be nearly
midway between the words. These words, including more letters, and being
confined within less space, must be in smaller character than the preceding ;
and, as a further distinction, a different style should be adopted. Having de-
termined this, we proceed to write in the letters as before, and in the same
way with the other lines ; the prepositions, as unimportant, are always written
in small type.
i
.of the.
FIOIPOSEB TE1MIIUS
of the
HARLEM RAILROAD
.at.
NEW YORK
1857
In general, it is better that letters should be first written on a piece of paper,
distinct from the plan, as repeated trials may be necessary before one is arranged
to suit the draughtsman. Having formed a model title, it may be copied in
the plan by measures or by tracing and transfer paper. There are some words,
such as plan, map, section, scale, elevation, etc., which, as they are of constant
occurrence, may be cut in stencil ; sometimes whole alphabets are thus cut
and words compounded. It will be found very convenient for a draughtsman
if he makes tracing or copies of such titles as he meets with, and preserves
them as models ; for there is no manipulation on a plan that contributes more
to the effect than good lettering and arrangement of titles, and considerable
practice should be expended in acquiring a facility in lettering, and, for the
first start, perhaps nothing will be found more valuable than tracing good ex-
amples.
We have treated of mechanical methods by which most persons can learn
to form letters and words ; but it must be borne in mind that the distances
between letters on the plan are only intended to suit the eye ; if, therefore, a
180
TOPOGRAPHICAL DRAWING.
person accustom himself to spacing, so that his eye is correct, there will be no-
necessity of laying off by dividers ; in this mode, such letters as A and V, L
and T, are brought nearer each other than the regular interval. In general, it
may be observed, in reference to the lettering of topographical drawings, stiff
letters like those of stencil should not be introduced, but there should be such
variety, incident on construction by the pen, as may be consonant with the
rest of the drawing. Of late, rubber type have been introduced, of fair forms,
much used on common drawings, by which lettering is very rapidly executed,
and is an improvement on that of most draughtsmen.
MATEEIALS.
VAEIED materials enter into the composition of structures and machines,
or form their supports, which are not only to be represented by the draughts-
man, but he should also understand the composition and properties of these
materials, that he may use them appropriately in his designs, and devise proper
forms to resist adequately and economically the strains to which they are to be
subjected. The earths and rocks, in their natural position, serve as the sup-
ports of structures and machines ; they may be represented as shown under the
head of "Topographical Draw-
ing," or by a closer imitation of
nature, with or without color.
Fig. 306 represents a plan and
section of an earth-bank of a canal,
with a paved rock-slope. A break-
water, of which the base A is a
mass of loose stone, is represented
by Fig. 307. A base of rock may
be represented by a stratification
(Fig. 308). For the foundation
of a structure, nothing is better
than solid rock, but the ' rock
should either have a horizontal
bed or be cut in horizontal steps,
,so that the walls resting on it
may not slide. The base of the
wall need not be widened. Sand
and gravel are also very good foundations, but the base resting on the earth
should in general be about double the width or thickness of the wall rest-
ing on it. For extensive buildings it is important that the areas of the bases
Plan
FIG. 306.
J ........
FIG. 307.
of its different parts should be proportioned to the weights upon them, and
it is also important that soundings should be made to determine whether
182 MATERIALS.
there are any compressible or sliding strata below. A stratum of 3 to 5>
feet of gravel upon a stony stratum is sufficient foundation to support 1 to 1^
tons per square foot ; but, if the sand rests upon rock, even at a very great
depth, it is not unusual to load it with 2 to 5 tons per square
foot. On sand and gravel, the building may settle somewhat,
but with proper bases uniformly ; on wet clay, it is more un-
certain ; the building may settle by displacement, as on a fluid ;
and, if the stratum is inclined, it is extremely apt to slide under
its load. There are others still more fluid, as quicksand and
marsnv deposits, where support must be obtained by extending
the bases. On water itself, it is obtained by means of a scow
or tight box, the displacement being equal to the weight of box and structure.
Earth, when first dug, occupies more space than when in its natural con-
dition, but, after a time, it shrinks and becomes more compact. The earth
dug out of a hole, when settled, will not fill the hole. Sand, gravel, loam, and
clay, will occupy from 8 to 12 per cent less space than when in the natural cut.
Clay can be puddled to occupy 25 per cent less.
Loose, dry sand weighs from 90 to 100 pounds per cubic foot ; compacted,
110 ; gravel, about the same ; clay, 120 pounds. Fresh water, at 60 Fahr.,.
weighs about 62 -4 pounds, and salt water about 64'1 pounds per cubic foot.
Sands and gravels are excellent material for embankments and fills, but clays
are much affected by the weather. The slopes of the former in cuts and fills
are usually 1-J horizontal to 1 perpendicular ; no fixed slope can be predicated
of clays. Sands and gravels are readily drained, and, when dry, are but little
affected by frost. The clays are hard to drain, heave with the frost when wet,
and, under the influence of a thaw or excess of water, become fluid. Very fine
sand, with gravel, and perhaps some admixture of clay, forming the glacier till
of geologists, is known as hard-pan\>y engineers, very difficult to be moved with
the pick, and often requiring blasting. The same material without the gravel
in low bottom forms a quicksand a jelly-like material from which, if a spade-
ful be taken out, the hole closes up at once, and excavation shows but little
visible sign of a depression, the space being made good from the entire mass.
This same material, dry, is a species of hard-pan. There is another material,
called quicksand, which is rather a running sand even when not wet, it rests
with a very flat slope ; the particles are very fine, and flow like the sands in an
hour-glass.
Sands and gravels are large components of mortars, betons, and concrete ;,
clay, of brick, tile, and pottery.
BUILDING MATERIALS.
The natural building materials of civilized communities are wood and stone,
which are to be worked or fashioned to the purposes to which they are to be
applied.
Figs. 309, 310, 311, are drawings of wood, longitudinal and sectional, in
which the grain of the wood is imitated, but wood is more often represented in
plain outline, and the cross-section of a timber thus (Fig. 312), or by mere
MATERI
183
FIG. 811.
FIG. 312.
hatching. When distinguished by color, burnt sienna is used commonly for
wood, but sometimes the color of the wood is imitated.
The draughtsman, for his designs,
will probably have to confine himself
to the timber within his reach. But
he should know what is best for his
purpose, reference being had to econ-
omy in cost and maintenance. For
most purposes, wood should be sea-
soned, so that joints may not open
under this operation after the ma-
terial is in the structure. But, for
work under water, wood should be
but slightly seasoned, as a swelling
of the wood may be disastrous. Sea-
soning of timber may be done by exposure for a time to outer air-currents ; if in a kiln,
it can be done speedily with heated air, or by steam. For beams, girders, and the like,
there should be few knots, especially on the outer edges for posts, small ones are not
objectionable; while for sidings and under-floors, firm, large knots do not impair the
work ; but no smooth work can be made with knotty lumber.
The trunk of the tree is composed of sap-wood and heart-wood : the one soft, readily
rotting; the other more dense and durable. In most specifications, lumber is "to be
square-edged, without sap, and large or loose knots."
In selecting lumber for a permanent structure, the life and endurance of the material are
to be considered. Most of the woods, sheltered from the wet and exposed to air-
currents, will last for a very long time ; but many will check and warp and become dis-
torted. All lumber in earth beneath the level of water will last indefinitely. In salt
water, above the earth, all are subject to the attacks of the worm the Teredo and Lim-
noria and, where the water is pure, the destruction is very rapid. Sewer-water and fresh
water are both destructive to the worm.
The life of lumber, in situations exposed to wet and dry, can be prolonged by im-
pregnating it with creosote or with various metallic salts, as the chlorides of zinc,
mercury, pyrolignite of iron, and others.
OHAEACTEEISTICS AND USE.
White Pine. A wood of the most general application in the market; is light, stiff,
easily worked, nails are easily driven into it, and takes paint well, warps and checks but
little in seasoning, endures well in exposed situations ; clear stuff, of best quality, useful
for patterns and models, for interior finish of houses, doors, window-sashes, furniture. It
forms base or inner core of the best veneered work, holds glue well, and the composite
structure is better than single solid wood. The cheaper kinds of pine are used for frames
of buildings, posts, girders, and beams. Even with large knots is well adapted for board-
ings, and is extensively used for goods-boxes.
Southern Pine. A heavy, strong, resinous, lasting wood, clear and mostly without
knots, hard to be worked by hand-tools, and when seasoned difficult to nail. The surfaces,
from their resinous character, do not hold paint well. It is used very largely for girders,
beams, and posts of mills and warehouses, and for floors of the same, when exposed to
heavy work or travel. For the first, it can be obtained of almost any dimension to suit;
for floors, it is sold in long strips, from two to six inches wide, of varied lengths, tongued
and grooved, and when laid is blind-nailed, toeing the nail through the tongue, so that the
nail-head does not show.
184: MATERIALS.
Canadian Red, Norway, and Silver Pines. Are resinous woods, like the Southern
pine, and are used for similar purposes, but are not as valuable woods less straight in
the grain, and with more knots.
Spruce. A light, straight-grained wood, with but few knots, which are small and often
decayed. It does not last well exposed to the weather, and checks and warps badly in
seasoning. It is the most common wood here for floor-beams and common floors, but it
must be well braced and nailed, and is not fitted for joiner-work.
Hemlock is similar to the spruce, and, when selected, is less liable to check and twist
in seasoning. It is often of a very poor quality, brash and shaky. Exposed, it is but little
better, if any, than the spruce. For stables, it is well adapted for grain-boxes, as the fiber
prevents the gnawing of rats.
Ash. Some of the ashes are of exceeding toughness. A straight, close-grained wood.
It is used for carriage and machine frames, and for interiors, doors, wainscot, floors,
when no paint is used.
Chestnut. Somewhat like the ash in appearance, but coarser-grained, and very endur-
ing in exposed positions. It is most largely used for cross-ties of railways. As a roof-
frame exposed in the inside, and in general interior finish without paint, the effect is
very good. The closer-grained woods are very often thus used.
Black Walnut. Is, in the trunk, a straight-grained, gummy wood, clogging the plane
a little in its working ; the knots are useful for veneer. Were the wood cheap enough, it
would undoubtedly make a good frame. It is used here for desks and counters, for fur-
niture and interior finish, as an ornamental wood.
Butternut. Similar to the black walnut, less commonly used, but fully equal as an
ornamental wood.
Hickory. A strong, tough wood ; is used for cogs of mortise-wheels, handspikes, axe-
helves, and wheelwrights' work.
Beech. A close-grained wood, but of little application in this market. Sometimes used
for cogs of wheels, for small tool-handles, and in marquetry.
Oak, Live. A very strong, tough, enduring wood, used industrially almost entirely for
ship-building. Ornamentally, in marquetry and panels.
Oak, White. A very valuable, strong, tough wood, with great endurance. It is heavy,
and hard to work, and was formerly used largely for the frames of houses, but has been
superseded by the white pine. It is u?ed in ship-yards and in water-works for the frames
of flumes, penstocks, and dams, and for the planking of the latter, for dock-buffers and
piles, and for railway and warehouse platforms. The red and black oaks may in general
be considered a cheaper and poorer quality of the white oak. All have a handsome grain,
that adapts them to ornamental work.
Bans, Poplar, White-wood, are light woods, mostly used in the manufacture of fur-
niture, for drawer-bottoms, cabinet-backs, panels; they are very clear stock, easily worked,
and can be readily obtained in thin, wide boards.
Cedar. A straight-grained, light wood, of great endurance, valuable for posts, sills,
shinies ; used for pails and domestic utensils. The red variety, from its odor, is admirable
for drawers and chests, preserving their contents from moths.
Locust is in the market only in small sticks; is of extreme endurance. It is used
almost invariably here for the sills of the lowest floors of buildings, where there can be no
ventilation, and for treenails of ship-planks.
J57 W ._ Although a tree of wide diffusion, is but little used as lumber. It is kept
for an ornamental tree, beyond its usefulness for any other purpose but fuel. Well
selected, it is said to be an enduring timber, useful for piles and places exposed
to wet.
Maples are tough, close-grained woods, rather to be considered among the orna-
mental woods, for fnrniture and interior finish. The same may be said of the cherry,
MATERIALS. 185
plum, and apple tree, of which the denser woods are admirably adapted for the handles of
small tools, for bushings of spools and bobbins.
The list of imported woods is extremely large, mostly for ornamental purposes; but
the mahogany is one of the very best of woods for patterns and small models, as it changes
but little in seasoning ; and the lignum-vitas, a very hard and heavy wood, is used for pul-
ley-sheaves, packing-rings of pumps, water-wheel steps, and shaft-bushings.
The weight and strength of the several woods are usually given in tables, but speci-
mens of the same wood differ essentially in both particulars. For all practical purposes,
the weight per cubic foot of white-pine, spruce, hemlock, poplar, bass, and cedar, may be
taken at from 23 to 30 pounds. Ash, cherry, chestnut, black-gum, black-walnut, and
butternut, from 32 to 45 pounds. Birch, beech, and elm, from 40 to 50 pounds. The
oaks, except live, from 40 to 55 pounds. Locust and hickory, from 50 to 60 pounds. Live-
oak and pitch-pine, from 60 to 70 pounds. Lignum- vitse, from 75 to 80 pounds. Their
resistance to crushing varies from 4,000 to 11,000 pounds per square inch, und to tension,
from 1,100 to 4,000 pounds, but their practical use will be given in future illustrations.
STONES.
In selecting the form of construction, and the stones of which it is to be
composed, the draughtsman must be governed by the fitness for the purpose
and the cost. He must select from what he can readily get, and arrange the
form to suit the material. He must know what is to be the exposure, and
what the effect will be on the stones. Almost any stone will stand in a pro-
tected wall, but many of the sandstones and slates disintegrate and exfoliate
under the influence of the weather, heat, cold, frost, and moisture. Even the
granites are liable to serious decomposition when the feldspars are alkaline ;
.and the limestones (dolomites), of which the English Houses of Parliament are
composed, have failed in the sulphurous air of London smoke, while at South-
well Minster they have stood for over 800 years. Chemical tests of stone to
determine endurance are deceptive. The safe way is to see how the material
has stood in like situations to the one in which it is to be employed, and, if this
is not possible, go to the quarry, and see how the stones have weathered there.
The strength of stones to resist crushing, as determined by experimental
cubes, is even in the weaker stones much in excess of what would be required
in structures, but most stones are weak under cross-strains, and failures in con-
struction are more likely to occur by faulty workmanship or design, by which
the stones are subjected to unequal strains, and for which they are not adapted.
The weight should not be brought on the outer edges or arrises, as the faces will
chip readily ; nor should most stones be used for wide-span lintels, unless they
form a part of the masonry above the opening, so that the whole is a beam.
TECHNICAL TERMS OF MASONRY.
We follow the nomenclature recommended in " Transactions of the Ameri-
can Society of Civil Engineers," November, 187? :
Rubble masonry includes all stones which are used as they come from the
quarry, prepared at the work by roughly knocking off their corners. It is
called uncoursed rubble (Fig. 313) when it is laid without any attempt at regu-
lar courses ; coursed rubble, when leveled off at specified heights to a horizon-
tal surface (Fig. 314).
186
MATERIALS.
FIG. 313.
FIG. 314.
Square-stoned Masonry. Square stones cover all stones that are roughly
squared and roughly dressed on bed and joints, and when the joints, when laid r
are one half inch or more.
Quarry-faced stones are those which are left untouched as they come from
the quarry.
Pitch-faced stones are those on which the arris is clearly defined beyond
which the rock is cutting away by pitching-tool.
Drafted stones are those in which the face is surrounded by a chisel-draft.
FIG. 315.
FIG. 316.
If laid in regular courses of about the same rise throughout, it is range-
work (Fig. 315). If laid in courses that are not continuous, it is broken range
(Fig. 316).
Cut stones or ashlar covers all squared stones with smoothly-dressed bed
and joints. Generally, all the edges of cut stone are drafted, if the face is not
entirely fine cut, but they may be quarry-faced or pitch-faced ; as a rule, the
FIG. 317.
FIG. 318.
courses are continuous (Figs. 317, 318), but, if broken by the introduction of
smaller stones of the same kind, it is called broken ashlar (Fig. 316). If the
courses are less than one foot in height, it is small ashlar (Fig. 317).
Square-stoned masonry is usually backed up with rubble masonry. Any of
this masonry may be laid dry, or with mortar or cement, which is to be spe-
cified.
The joints in one course should not come directly over those of another :
MATERIALS. 1ST
there should be a lap or bond, and, in connecting the front or face with the
backing, headers must be introduced for bond. Headers are stones extending
into the wall, stretchers running with the face.
In addition to the classes of stone- work, there is an old form lately come
into use called hock and ham by old English builders ; it is a species of rub-
ble, in which there are no courses. The stones are very carefully selected in
size and shape, so as to make an ornamental work ; the joints are close, but
have no uniformity of direction.
For rubble-work, all varieties of sound stone are used, and of almost any
size. In dry work, for foundations and for heavy revetment-walls, the stones
are laid with derricks, but they must have fair beds and builds. If bowlders,
they must be split, and cobbles in the filling are worse than useless.
For rubble laid in mortar, the usual size is such as can be laid by hand.
GRANITIC STONES.
Granite and syenite are by builders classed as granites. The granite in general rifts in
any direction, and works well under the hammer and points. From these circumstances-
it is more desirable than the syenites, which are much harder to be worked. Both are
admirable stones for heavy dock-walls, bridge-abutments, river-walls, either as rubble-
squared stones or cut work, and are very enduring. They are also used for the faces of
important buildings, either as fine-cut, quarry, or pitched-face. Ornamental work of the
simpler kind is readily produced ; more elaborate is expensive, but it is about the only
stone in this climate in which foliage and sharp undercut work will stand the weather
without exfoliating. These stones, especially the syenites, admit of a high polish, and are
used considerably for columns and panels in buildings, and in monumental work. Gneiss
is of the granitic order, but a cheaper, poorer stone. It splits with difficulty, except parallel
with line of bed. It has a foliated structure, and is not adapted for ashlar, but is very good
for squared-stone masonry and rubble-work, and often used for sidewalk-covers of vaults.
AKGILLACEOUS STONES.
The slates or stones thus designated by builders were formerly in very common use as
roofing material, and were almost entirely from Wales, but latterly they are taken from
Vermont and Pennsylvania, and other parts of the United States. They are also used,
in thicknesses of one inch and above, for floors, platforms, facing of walls, mantels, and
for wash-tubs by plumbers. Soap-stone may be classed under the clay stones ; also, used
for tubs, for stoves, and for the lining of grates and furnaces.
The Ulster, or North River blue stone of this market, is a coarser slate, a very strong
and enduring stone ; it can be quarried of varying thickness up to twelve inches, and of
any dimension that can be transported. It can be readily cut, hammer-dressed, axed,
planed, and rubbed. Is generally used for sidewalks under these various forms. It ia
used as bond-stones in brick piers, for caps, sills, and string-courses.
THE SANDSTONES.
Sandstones, called also freestones, from the ease with which they are worked ; and from
their colors, are very popular for the fronts of edifices. In general, they are not very
enduring stones, and when laid must be set parallel to their natural beds, as otherwise
they flake off under the influence of the weather. The sandstones are not all of the
same quality; those in which the cementing material is nearly pure silex, are strong,
enduring stones, but not those in which the cementing material is alumina, or lime. By
examining a fresh fracture, the character of the stone can generally be detected. A clear,
shining surface with sharp grains indicate a good stone; while rounded grains, a dull.
188 MATERIALS.
mealy surface, indicate a soft, perishable stone. None of the sandstones in this locality
are used for heavy pier or abutment work and the like, but there are sandstones in other
localities adapted to it.
LIMESTONE.
The coarser calcareous stones are of great variety ; some are well adapted for building
stones, being hard and compact, while others are soft and friable. They are more easily
worked than granite, but are not considered as enduring. They are well adapted to the
same class of heavy work, and the locks of the Erie and Northern Canals and the dam
across the Mohawk, at Cohoes, are built from limestone on the line of the canals.
The finer kinds of limestones are classed under the head of marbles. They are easily
worked, sawed, turned, rubbed, and polished. Marble is not popular as a building material,
although more enduring than most sandstones, but is susceptible to the action of sulphurous
gases in the smoky air of cities ; and it is said that the Capitol at Washington, D. C.,
built of marble, is suffering from disintegration. But, for interior finish, as tiles, wain-
scots, architraves, mantels, linings of walls, it is admirably adapted, and from its richness,
deanliness, and variety of color, it is very ornamental and effective.
ARTIFICIAL BUILDING MATEEIAL.
The most common and useful are bricks. They are generally made of clay, with an
admixture of sand, well incorporated together, and mixed with water to the consist-
ency of a smooth, strong, viscous mud, pressed into molds, dried, and burned, *the best
quality being those in the interior of the kiln. The exteriors are light, friable bricks
adapted to walls supporting but little weight and not exposed to wet. The brick forming
the arches are very hard-burned, dark in color, often swelled and cracked ; but by proper
selection, they can be used for foot-walks. A good brick is well burned throughout ;
when struck, it gives a ringing sound, and is of uniform shape.
Bricks vary somewhat in size and weight in different localities from 8 to 8^ inches
long x 3 to 4 inches broad x 2 to 2 inches thick ; in general, the thickness of a wall
with the joints is called some multiple of 4", as 8", 12", 16". Here an 8-inch wall by 1
foot face contains 14 bricks ; 12-inch, 21 ; 16-inch, 28 bricks 9 courses high are equal to 2
feet. In the Eastern States the brick is somewhat thinner 5 courses to the foot. The
best front brick are pressed, and are a little larger than the common brick. Philadelphia
and Baltimore pressed brick are distinguished by their clear, cherry-red color ; Milwaukee
are of a pale-straw color.
Bricks are laid in mortar, of lime, lime and cement, or cement only all with an ad-
mixture of sand ; in common walls, in lime ; in walls of heavy buildings, above-ground, in
lime and cement ; beneath, and in wet, exposed positions, in cement only. The common
bond of the different courses of brick is by header-courses every fifth or seventh course.
When bricks are laid in arches they are set on edge, and turned in 4-inch rings, sometimes
without any bond between the different rings; sometimes with a bond of brick length-
ways, when two courses come on the same line.
Bricks set on edge, as in arches or in a level course, are here termed rollocks.
Arch brick, between iron beams, to reduce the weight, are often made hollow, and laid
in flat arches ; that is, the joints are radial, but the upper and lower surfaces are level.
Hollow brick are also used for walls and partitions.
Fire-brick can be made of any size and pattern, but are usually 9 x 4 x 2f. They are
used for the lining of furnaces, flues, and chimneys, exposed to the action of flame or great
heat. Fire-clay, with an admixture of sawdust, which is burned out in the firing, leaves a
light, porous, spongy mass, which can be sawed in sheets or strips, and is well adapted
for covering the exposed parts of iron beams and girders, and, as it admits of nailing, is
convenient for partitions.
Enameled Brick. The English size is that of fire-brick the American is that of com-
MATERIALS. 189
mon brick. The brick, on the faces to be exposed, are covered with glaze of varied colors
and designs, and fired. They make a handsome ornamental face for walls, do not absorb
moisture, and can be washed.
Tile are a species of brick, with or without enamel. The latter were originally used
for roof-covering, but now are used in flooring walks and the like. The enameled or
encaustic tile are generally in squares, 4" x 4", 6" x 6", 8" x 8", but there are smaller
ones for tessellation, and rectangular strips for borders. They can be obtained of any
color or design, forming beautifully ornamented floors and wall-panels.
Terra-cotta, a kind of brick, is now largely used for exterior decoration. It is molded
in every variety of capitals, cornices, caps, friezes, and panels. It is a good, strong brick,
with all the good qualities of such a material.
Mortars. Brick are never laid dry, except in the under part of drains, to admit of the
removal of ground-water. Stone-work, except in rough, heavy, rubble-work, is also gen-
erally laid in mortar. Where cut-work is backed with rubble, the joints in the latter
should be as close as possible, and full of mortar, that the settling of the wall in itself
may not be more in the backing than in the face. Some lay the rubble dry, and fill in
with cement grout, or cement mortar made liquid to flow into the interstices, but the sand
is apt to separate and get to the bottom of the course.
By mortar, is usually understood a mixture of quicklime and sand, but mortar may
have an addition of cement to the lime, or it may be cement only with sand.
Lime, or properly quicklime, is made by the calcination of limestone, shells, and sub-
stances composed largely of carbonate of lime, carbonic-acid gas, water of crystallization,
and organic coloring-matter. Quicklime, brought in contact with water, rapidly absorbs it,
with a great elevation of temperature, and bursting of the lime into pieces, reducing it to
a fine powder, of from two to three and a half times the volume of the original lime. This
is slaked lime. It may be slaked slowly by exposure to the air, from which it will take
the moisture. This is air-slaked lime. Barrels of lime exposed to rain often take fire
from the heat caused by slaking. The paste of slaked lime may be kept uninjured for a
considerable time, if protected from the air, and this may readily be done by a covering of
sand, and it is customary, in some places, to hold it over one season, as an improvement
to the uniformity of quality in the paste. But, in general, the lime is used soon after
slaking, and is thoroughly mixed with sand, in various proportions, generally about two
of sand to one of lime. The theory of the mixture is, that the lime should fill the void
spaces in the sand, and the space occupied by the mortar is a little in excess of that occu-
pied by the sand alone.
The sand should be sharp, clean, silicious grains, from one twelfth to one sixtieth of an
inch in diameter. Close brick-joints do not admit of as coarse sand as those of cut stone
work, and, in rubble-work, sand coarser than the above can be used, and there will be
considerable saving of lime in using a mixture of coarse and fine sand.
The hydraulic limes contain a small proportion of silica, alumina, and magnesia ; slake
with but little heat, and small increase of volume ; are more or less valuable, according"
to the property which they have for hardening under water ; but, in this particular, are
not equal to the hydraulic cements, which contain a larger proportion of silica, alumina,
and magnesia. They are made by calcining natural rocks, or Jby the combination of clay
and soft carbonate of lime, or chalk, calcining and grinding. The cements make a paste
with water, with little or no heat on slaking, and set, in open air or under water, with
more or less rapidity ; but this is not a sure criterion of the value of the cement, when time
comes in as an element before the work is subject to stress.
Cement is mixed with sand in varied proportions from 1 to 1 to 1 to 3 it is stronger
without any admixture of sand, but is seldom used neat, except in pointing, and for very
close joints. By experiments of Mr. F. 0. Norton (Trans. Am. Soc. of C. E's.) it was
found that Portland cement, with two volumes of sand, was equal to that of Rosendale (or
190 MATERIALS.
native cement) with one part of sand. In purchasing cement, it is usual that it should be
required to be up to a certain standard ; that is, made np into a ball with water at 65
temperature; it should set in water to withstand a pressure of say one pound on a one-
quarter inch wire within so many minutes. By many, cements are required to be of a
oertain degree of fineness that only a very small per cent should be left on the screen, say
of one sixtieth-inch mesh, and that it should weigh about 80 pounds to the bushel, and
that it should have a certain tensile strength after so long a set.
Cement is used in all masonry in exposed and wet situations. With a small admixture
of lime, it works better under the trowel, and for brick-work it does not sensibly impair
its value. Cement adds to the strength of lime-mortar, and gives it an amount of hydrau-
licity. To increase the quick-setting of cement, it may sometimes be necessary to add a
little plaster-of-Paris, but it is preferable to get a quick-setting cement.
Concrete or beton are terms now used for the same material. It consists of cement,
sand, and gravel, or broken stone, which may be intimately mixed, in varied proportions,
according to the quality of the cement and the character of the inert materials. For the
blocks of the New York city docks, the proportions were :
Portland cement 3 volumes.
Sand, damp 5
Broken stone 10 "
This is a strong mixture. It is not uncommon to make a cement of Rosendale cement 1,
sand 2 to 3, and broken stone or clean gravel as much as can be well covered by the mix-
ture, but it should have time to set.
Concrete is used for the base course or foundations of walls, and is formed in situ,
that is, depositing and ramming it in the trench where it is to be left; or by forming in
molds, in immense blocks, for docks or break-water, or in the smallest forms of brick and
moldings.
The bituminous cements are formed of natural bitumens, or artificial from coal-tar
mixed with various proportions of gravel and inert material. The mixture is usually
heated, put down in layers, and rolled or rammed. It is used for roads and sidewalks,
and for water-proof covering of vaults. For the covering of roofs, coarse paper, sat-
urated with bitumen, is put on in layers, one over the other, breaking joints, cemented
with the bitumen, the last coat being of bitumen, in which gravel is imbedded. For
an anti-damp course in a wall, or for the joints in the bricks of a wet cellar-floor,
or on top of a roof, bitumen is used as a cementing material the bricks must be dry,
bitumen hot.
Plastering. Coarse-stuff is nothing more than common brick-mortar, with an admix-
ture of bullock's hair. When time can not be given for the setting it is gauged, that is,
mixed with some plaster-of-P;iris. Fine-stuff is made of pure lump-lime with an admix-
ture of fine sand, and perhaps plaster-of-Paris. Hard-finish is composed of fine-stuff and
plaster-of-Paris. One-coat work is of coarse-stuff, which may be rendered, that is, put on
masonry, or laid on laths. Two-coat work is a coat of coarse-stuff, or scratch,-coat ; that
is, after the coat, is partially dry it is scored or scratched for a back for the next or fine
coat. In three coats, the first coat is a scratch-coat, the second the brown-coat, and the
third is hard-finish, or stucco. Keene^s cement, for the last finish, gives a very hard
surface, which admits of washing.
A single brick weighs between 4 and 5 pounds ; but a cubic foot, well laid in cement,
with full joints, will weigh about 112 pounds. They have resisted, in an experimental
test, as high as 13,000 pounds to the square inch, but 12 tons should be the limit to the
load per square foot ; and the brick should be uniform, well burned, and closely laid in
cement, and without cross-strain. In lime mortar, the load should not exceed 3 tons per
square foot.
MATERIALS.
191
The granites weigh from 160 to 180 pounds per cubic foot ; the limestones from 150
to 175 ; the sandstones from 130 to 170 ; the slates from 160 to 180 ; mortar, set, about
100 pounds ; masonry, laid full in mortar, according to the quality of the stone and the
percentage of mortar, from 150 to 170 pounds. Some of the granites have withstood a
crushing strain of 15,000 pounds per square inch, and, when structures are important, and
subject to great strains, specimens of the stones to be employed should be tested ; but, for
practical purposes, common mortar-rubble is not considered equal in strength to a brick
wall, as it is seldom laid with equal care, and the joints are not as likely to be well filled,
and the load as evenly distributed ; but cut stones will sustain more, and ashlar, up to 50
tons per square foot for sound, strong stones.
METALS.
Metals are often to be shown distinctively by the draughtsman. If lie can
use color, he will in a measure imitate that of the material. For cast-iron,
India-ink, with indigo, and a slight admixture of lake ; for wrought-iron, the
same colors, with stronger predominance of the blue ; steel, in Prussian-blue ;
brass, in a mixture of gamboge and burnt sienna ; copper, gamboge and crim-
son lake. But it is often requisite to express distinctive metals in drawings
where no color is admissible. When the drawings may be required for photo-
graphing, or reproduced in printing, some conventional hatchings are used to
represent sections of metals, but none have been so established as to have a
universal application. The following are submitted to represent the most com-
mon industrial metals :
^\VS\*N$
Cast Iron.
FIG. 319.
Wrought Iron.
FIG. 320.
Steel.
FIG. 321.
Brass.
FIG. 322.
Lead.
FIG. 323.
Under the term iron may be included cast-iron, wrought-iron, and steel, differing from
"each other in the percentage of carbon contained, and in the uses to which they are applied.
Oast-iron contains more carbon than the others, say from two to five per cent. It can
be cast in varied forms in molds, but can not be welded or tempered. The usual molds are
192
MATERIALS.
in sand or loam, in which the pattern is imbedded, and when drawn out the space is filled
with molten metal. The drawing of patterns for molding involves a knowledge of the art
of founding. The shrinkage of the metal, usually about one per cent, for which provision
must be made in increased size of pattern, is provided for by the pattern-maker, the
draughtsman giving finished sizes, but the draughtsman must know whether the pattern
can be drawn from the sand, and by what system of cores voids can be left ; or it may
often happen that castings, designed as a whole, will have to be made in a number of
pieces, involving flanges and bolts. In cooling, the shrinkage takes place the soonest in
the thinnest parts, and, if great care be not taken by the molder in exposing the thicker
parts to the air first, the parts will shrink unequally, and there will be a strain induced
which will materially weaken the casting, and it may even break in the mold. The
draughtsman, in his design, should make the parts of as uniform thickness as possible.
Castings cool from the outside inward, in annular crystals perpendicular to the face,
as in Figs. 324 and 325. Now, if the casting consist of a right angle (Fig. 326), there will
evidently be a weak place along the line A B, but, if the angle be eased by a curve, the
FIG. 324.
FIG. 325.
FIG. 326.
FIG. 327.
crystallization takes place as in Fig. 327, and the line of weakness is avoided. This is
effected by a very small easement of the angle, and a cove is almost invariably introduced.
In castings, in almost all metals, the same effects result from cooling, and therefore the
changes of direction should not be abrupt.
"When castings are ordered for important structures, iron of certain tensile strength is
called for, and specimens of the metal, in small rectangular bars, are required, cast at the
same time and under as nearly the same conditions as the casting which may be subjected
to test.
If the casting be made in dry sand, it cools slowly, and the surface is comparatively
soft; if in greensand sand somewhat moist the surface becomes harder; but if cast
on an iron plate, or chill, some irons become as hard as the hardest steel, useful in
surfaces exposed to heavy wear, as the treads of rail way- wheels. Cast-iron, in general,
is brittle under the blows of a hammer, but some mixtures, under a process of annealing,
become malledble iron, used largely for steam -fittings, parts of agricultural machines, forms
requiring the toughness of wrought-iron, but difficult to forge.
\\ rought-iron is produced from cast-iron by removing the carbon and impurities by
puddling, squeezing, heating, and rolling. As a material, it is sold in all sizes of wire,
rods, shafts, bars, plates, shapes girders and beams, chains and anchors. Its applica-
tion industrially is well known. When hot, it can be welded, forged, drawn, and swaged
into almost any required shape. Under the steam-hammer, the largest shafts, anchors, -and
cranks can be built, or by hand or by machinery it can be wrought into tacks, nuts, bolts,,
nails, or drawn into the finest wire.
For shafts of mills it is generally turned in a lathe and polished, but of late it can he-
bought, up to four inches diameter, cold-rolled, which adds very considerably to the strength,
and is ready for use.
Bessemer and Siemens-Martin metals are made by burning out the carbon from a
melted iron, and then reintroducing a known quantity, say from 0'03 to 0'6 per cent of car-
bon. There are other patents covering somewhat different irons, but the above are the best
known. All are commonly classed as steel, but by many are called homogeneous metal :
MATERIALS. 193
first-class iron, of very uniform texture and great strength, but not equal to that of the best
steel.
Steel is produced from pure wrought-iron by what is called cementation heating the
bars in contact with charcoal, by which a certain amount of carbon is taken up. The bars,
when taken out, are covered with blisters, apparently from the expansion of minute bub-
bles within ; hence called blistered steel. From this shear-steel can be produced by piling,
heating, and hammering, or cast-steel from melting in a crucible.
Steel, when broken, does not show the fibrous character of wrought-iron. The frac-
ture of shear-steel is fine, with a crystalline appearance. The fracture of cast-steel is very
fine, requiring very close inspection to show the crystals or granulations ; its appearance
is that of a fine, light, slaty -gray tint, almost without luster. Steel is stronger than any of
the other iron products, and especially applicable for the piston-rods of steam-engines, and
positions requiring great strength and stiffness, with the minimum of space. But it is the
way in which steel can be hardened and tempered which adapts it to its peculiar appli-
cations.
When the malleable metals are hammered or rolled, they generally increase in hard-
ness, elasticity, and denseness, and some kinds of steel springs are made by the process
of hammer-hardening ; but the usual process of hardening and tempering is by heating
the steel to a degree required by the use to which it is to be applied, and cooling it
more or less suddenly by immersing in water or oil. The greater the difference between
the heated steel and the cooling medium, the greater the hardness, but too much heat
may burn the steel, and too sudden cooling make it too brittle. Steel, in tempering, is
heated from 430 Fahr. to 630. The temperature is shown by the color from a pale
yellow to deeper yellow, light purple to a dark purple, dark blue to a light blue, with a
greenish tinge.
Steel is used for the edges of all cutting-tools, faces of hammers and anvils, and is gen-
erally welded to bodies of wrought-iron, but often composing the entire tool ; for saws,
springs, railway tires, pins, and can be bought in the form of wire, rods, bars, sheets, and
plates, in varied forgings and castings.
All irons are very liable to rust, and must be protected where exposed to moisture.
Polished surfaces are kept wiped and oiled, others painted, others galvanized or plated
with some less oxidizable metal, generally tin, zinc, or nickel. Of late, a process has
been introduced of coating them with black oxide, but is yet of no general application.
Antimony, bismuth, copper, lead, tin, and zinc, are used more or less industrially, and
alloys of them are extremely useful. They may be hardened somewhat by the process
of rolling and hammering, but can not be welded. Joinings are made by soldering or
brazing or burning that is, melting together.
Antimony expands by cooling. With tin, in equal proportions, it makes speculum-
metal, and is used, with lead, to make type. Type metal makes a very good bearing for
shafts and axles.
Bismuth is chiefly used as a constituent of fusible metal : 3 bismuth, 5 lead, and 3 tin,
is an alloy which melts at 212. Other mixtures are made, increasing the melting-point
to adapt the metal for fusible plugs in boilers, or lowering the melting-point, so that, in
case of fire in a building, a heat of say 140 melts the joint made by the metal, and lets
water through sprinklers, to automatically put out the fire.
Copper is very malleable and ductile. In sheets, it is used for the cover of roofs, gut-
ters, leaders, lining of bath-tubs, kettles, stills, and kitchen utensils. It is worked more
easily than iron, and is stronger than lead or zinc, but it is much more costly than either
of these metals, and its oxide is so poisonous that, without great care and cleaning, it can
not be used to transmit or contain anything that may be used as food, without a cover of
tin. It oxidizes slowly, and is used extensively for ships' fastenings and for bottom-sheath-
ing. It is the most important element in all the brass and bronze alloys.
13
194 MATERIALS.
Brass, in common use, covers most of the copper alloys, no matter what the other
components are, whether zinc, tin, or lead, or all three.
Copper and zinc will mix in almost any proportions. The ordinary range of good
yellow brass is from 4| to 9 ounces of zinc to the pound of copper. With more zinc it
becomes more crystalline in its structure, but, as zinc is very much cheaper than copper,
the founder is apt to increase the percentage of zinc, with the addition of a small per-
centage of lead. Muntz metal, in its best proportion, contains lOf ounces of zinc to the
pound of copper.
Copper and tin mix in almost any proportion. The composition of ancient bronzes is
from 1 to 3 ounces of tin to the pound of copper. Ten parts of tin to 90 of copper is the
usual mixture for field-pieces, and this is used in steam-engine work, often under the
name of composition. Bell-metal is from 4 to 5 ounces of tin to the pound of copper ; Bab-
bit-metal, for journal-boxes, 90 of tin to 10 of copper.
Copper and lead mix in any proportion up to nearly one half lead, when they separate
in cooling.
An addition of from one quarter to one half ounce of tin to the pound of yellow brass
renders it sensibly harder. A quarter to one half ounce of lead makes it more malleable.
German-silver is 50 copper, 25 zinc, and 25 nickel.
Holzapfel gives the following alloys :
1-J- ounce tin, ounce zinc, to 16 ounces copper, for works requiring great tenacity.
1-J- to If ounces tin, 2 ounces brass, to 16 ounces copper, for cut wheels.
2 ounces tin, 1-| ounce brass, to 16 ounces copper, for turning-work.
2J ounces tin, 1 ounce brass, to 16 ounces copper, for coarse-threaded nuts and bearings.
2 ounces tin, 2 ounces zinc, to 16 ounces copper, Sir F. Chantry's mixture, from which
a razor was made, nearly as hard as tempered steel.
Professor R. H. Thurston, of Stevens Technological Institute, has tested various alloys
of copper, tin, and zinc, and, by a graphic method, determines the best alloy for toughness
as well as strength to be copper 55, tin 2*5, zinc 44'5.
There are various other alloys, as phosphate bronze, aluminium bronze, Sterro-metal, of
which the strength will be given hereafter in a table.
Lead is a very soft metal, that can be readily rolled into sheets and drawn into pipes,
and is so flexible that it can be readily fitted in almost any position. It is, therefore,
especially adapted to the use of plumbers, for the lining of cisterns and tanks, and for pipes
for the conveyance X)f water and waste. For pipes for conveying pure water for drinking
purposes, or for cisterns containing it, it is objectionable, as it oxidizes, and the oxide is a
dangerous and a cumulative poison, but, in common waters which are more or less hard,
the insides of the pipes become covered with a deposit which protects them. It is well,
before drinking from a lead pipe in which the water has stood for a time, to draw off all
the water, and, in lead-lined cisterns exposed more or less to the air, to protect them by a
coating of asphalt varnish. Lead expands readily, and has so little tenacity that, in many
positions, if heated, it has not strength in cooling to bring it back to its original position.
It remains in wrinkles on roofs, and, for pipes conveying hot water, unless continuously
supported, it will hang down in loops, continuously increasing under variations of tem-
perature, to rupture. But it makes a very good plating for sheet-iron for roofs, and its
oxides are the most valuable of all pigments.
Tin, in a pure state, is used for domestic utensils, as block-tin, and has also been used
for pipes in the conveyance of water by parties who feared the poisonous qualities of lead
pipe. But its chief use is for the covering of sheet-iron, which is sold under the name of
tin or tin-plate, and is of universal application for architectural, industrial, and domestic
purposes. Its oxide is not injurious, and it is so little affected by air and moisture that
roofs, in many places, covered with it, need no painting, and oxidization takes place in the
iron beneath only from deficiency in plating, or from the abrasion or breaks in it.
MATERIALS.
195
Zinc, in the pure form of spelter, is crystalline and brittle, but, at a temperature be-
tween 210 and 300, it is so ductile and malleable that it can be readily rolled into sheets,
and of late has been used as a cheap substitute for sheet-copper ; but, under considerable
variations of temperature, as for lining of bath-tubs, it takes permanent wrinkles, and, for
coverings of roofs, suitable provision must be made for its expansion. But as a plating of
iron, under the name of galvanizing, it affords un admirable protection, cheaply, and ex-
tends the use of iron in sheets, bolts, and castings, where it would not otherwise be appli-
cable. Zinc, as a pigment, does not discolor, like lead, under the action of sulphureted
hydrogen, but is objected to by painters for its want of body or cover.
METALS.
METALS AND ALLOYS.
Specific
gravity.
Weight
per c. ft.
Melting-
point.
Resistance in pounds per square inch.
To crushing.
To tension.'
Aluminum-bronze
Fahr.
73,000-96,000
1,060
3,250
18,000
22,000
13,'000-25',000
1,800
22,000-50,000
55,000
40,000
85,000-145,000
4,600
2,500
40,000- 60,000
70,000-120,000
50,000-100,000
120,000-200,000
50,000- 85,000
Antimony cast
4-500
9-900
8-500
8-726
19-238
7'20
11-479
280
617
530
537
1,200
450
716
932
476
1,873
4,587
5,237
18,000
594
Bismuth
Brass
50,000-160,000
117,000
82,000-14'5,000
7,000
'Copper
<rold
Iron
Lead
Phosphor-bronze .
Platinum, cast. .
21-500
10-480
7-800
7-250
7-215
7-77
7-85
1,340
654
486
450
450
485
490
3,080
3,677
442
700
2,822
2,462
Silver "
Steel "
125,000-295,000
15,500
Tin "...
Zinc "
Iron forced
40,000- 65,000
100,000-180,000
Steel "
Iron wire (unannealed) .
Steel wire " i
Sterro-metal
7
Fig. 328 is an admirable illustration of the graphic representation of facts
adopted by Professor R. H. Thurston of exhibiting the results of his tests on
the strength of alloys which not only exhibits the results, but enables others
to judge the probable strength of other mixtures. The apices of the triangle
marked copper, tin, and zinc, represent the points of pure metal, 100 per cent.
The lines opposite the apex of any metal represent the of such metal thus the
base opposite copper represents an alloy of tin and zinc only, without any cop-
per, and every line drawn above this base, and parallel to it, will contain a per-
centage of copper increasing by regular scale, from the base to the apex, and so
with lines opposite tin and zinc ; the first contains only copper and zinc, the
latter tin and copper, and the percentages of tin and zinc increase with the
distance from their opposite lines to their vertices. It will be seen that the
intersections of these percentage parallels define the percentages of each metal,
their sum always making 100 per cent. If, then, the strength of such alloy,
as obtained by test, be supposed to represent an ordinate or elevation, on any
convenient scale, and be represented by this height at its opposite intersection
of percentage, a contour map, as in the figure, may be formed which the pro-
fessor has not only done, but made a model from it. The summit, 65,000 on
the figure, represents the position of the strongest alloy found : if through the
196
MATERIALS.
scales marked copper on each side, we find the parallel to the base, which passes
through this summit, it will be found to be about 55, that is, 55 per cent cop-
per. In like manner, the parallel to the o zinc base, intersecting this summit,
Avill be about 43 per cent zinc ; and, in the same way, tin is 2 per cent. If we
<VZ
-%
FIG. 328.
wish to find the probable strength of any mixture, it is only necessary to find
the contour intersected by the triple parallels representing the percentages
which we are investigating. It is said probable strength, because the care and
manipulation of the founder are such important factors in the result.
Sulphur, when used in sufficiently large masses as to show on a drawing, may be repre-
sented by a reddish-yellow tint, or some distinctive hatching. It melts at 248 Fahr., and,
from its fluidity, answers admirably for the filling of joints between stones, beneath the
balls of iron columns, between wood and stone, and around anchor-bolts in stone, forming,
when cold, a strong, uniform bearing, and adapting itself to the roughness of the material,
and is detached with difficulty. It is used largely for the bases of engines, and for the
joints of the cap-stones of dams. On the dam across the Mohawk, at Cohoes, many tons
were used in these joints, the depth of sulphur being about 6 inches, and now, after
seventeen years' use, but few of the joints are little worn, and there has been no injurious
effect from the sulphur on the limestone, of which the apron or capping is composed. It
is better for most of the above purposes than lead, being cheaper, more fluid when molten,.
MATERIALS.
197
shrinks less in cooling, is less affected by temperature, and its crushing strength is adequate
to any of the positions of use above, but it is brittle under blows. It sometimes rusts the
bolts or iron with which it is brought in contact, but this is prevented by an addition of
about 20 per cent of coal tar. This mixture is used as a cement to fasten lights in illumi-
nated tile and vault covers.
When heated to about 300, sulphur begins to grow viscid, and at 428 it has the con-
sistency of thick molasses. Above this, it begins to grow thin again. Heated to 518,
and thrown into cold water, it becomes for a time plastic, and is used for taking molds or
casts.
Sulphur, in powder, mixed in proportions of one sal-ammoniac, two sulphur, and fifty
of iron-filings, makes a mastic which is used for calking the joints of iron pipes, especially
gas-pipes. The joint is called a rust-joint.
Glass, in drawing, is represented by a bluish tint or by different shades
or hatchings, expressive of the effect of light upon it, whether the light is
reflected or transmitted.
Fig. 329 represents a portion of a mirror
when the light is reflected. The exterior of
windows is often represented in the same way,
but with deeper shades, and often with a piece
of curtain behind in white with dim outline.
A window viewed from inside is represented in
shades less than in the figure, or as transpar-
ent, which is conveyed by the dimness of out-
line of figures or skies seen beyond.
Fig. 330 represents a glass flask.
Fig. 331 represents a glass box with glass
sides.
Fig. 332 represents a glass jar containing
fluids of different densities.
Figs. 333 and 334 represent spars, which may be taken for any transparent
substances, as glass, ice, and the like.
FIG. 329.
FIG. 330.
FIG. 331.
Common window-glass is blown in the form of cylin-
ders (hence called cylinder-glass), flatted out, and cut
in lights of varying dimensions, from 6 x 8 up to 30 x 30
inches, and put up in boxes containing about fifty square
feet. It is classed as single-thick (about T V inch) and
double-thick (-J- inch). When the squares are large, or used for sky-lights, they should be
the latter. Plate-glasspolished plate is used for windows of stores and first-class build-
FIG.
198 MATERIALS.
ings. It can be got of almost any dimensions, and of a thickness from T \ to f of an inch.
Rough plate is largely used for floor-lights and sky-lights. It is cut to required sizes, and
of a thickness from f to one inch.
Single thick cylinder-glass cuts off from about 8 to 15 per cent of the light.
Double-cylinder, from 12 to 20 per cent of the light.
Polished plate, three sixteenth inch thick, from 5 to 7 per cent of the light.
Rough plate, one half inch thick, from 20 to 30 per cent of the light.
Rough plate, one inch thick, from 30 to 40 per cent of the light.
This is when the glass is clean ; but there is always a film of moisture on its surface r
which attracts dust, and impairs very much the transmitted light. Rough plate more
readily retains the dirt, and, when it is used as floor-lights, becomes scratched. It is
therefore usual, in the better class of buildings, to use a cast white glass, set in iron frames.
In outer, or platform lights, these lights are in the form of lenses, set in cast-iron frames,
with an asphalt putty, or resting on iron frames and imbedded in Portland cement.
Fi&. 333. FIG. 334.
Rubber, mixed and ground with sulphur, subjected to heat, becomes vulcanized, and is
not affected by moderate variations in temperature. Soft rubber, most extensively used for
industrial purposes, is subjected to a heat of from 265 to 300, and for a time can withstand
a temperature a little below this without losing its elasticity; after a time it will harden.
Soft rubber is classed as pure rubber, and fibrous rubber, or rubber with cloth. Pure
rubber contains about fifty per cent of rubber and fifty per cent of compound, white lead
and sulphur. It is used for the buffers and springs of railway-carriages, and for the faces
of valves and seats of water-pumps, but it is not well suited for the pumping of hot water,
especially above 212, as it is liable to lose its elasticity ; and, although some valves may
stand a considerable time, it is almost impossible to secure uniformity in the rubber.
Fibrous rubber rubber ground with cotton or other fiber, or spread on cloth, on more or
less thicknesses is used for the packing of faced joints of pipes and gaskets for water or
steam. It makes a stanch joint, and, even when hardened under heat, it still preserves
it. Rubber cloth is also used for belting and hose-pipes. When used for the convey-
ance of steam, the inner coat is the first affected, and it may be some time before the
whole pipe suffers. In buying rubber, explain the purpose to which it is to be applied r
and depend on the guarantee of the vender. Rubber is often to be designated by the
draughtsman, which it may be by a bluish-black tint, or by lines across it parallel to its
length.
Paints are used for a twofold purpose for covering and preserving the material to
which they are applied, and for ornamentation. The best and the most general is white-lead
ground with linseed-oil, either used by itself or mixed with various other pigments, a&
ochre, chrome, lamp-black, etc. It is often adulterated with barytes. For the covering
of iron, or for the packing of close joints in it, nothing is better than pure red-lead, but
many of the oxides of iron, red or yellow, form good covers of iron, and, as cheap and
good paints, are used on tin roofs. All the leads and pigments are ground in oil : if the
oil is raw, it dries slowly ; driers, as litharge, are added to hurry the process, but, with
MATERIALS.
199
boiled oil, no drier is necessary. Almost any inert substance, as cement, chalk, or sand,
if fine enough, can be ground with oil for a paint, and make a good cover, and for these
fish-oil will answer. The general specification for painting is " paint with good coats of
white-lead, of such color as may be directed." The priming-coat of new wood- work
requires more oil than paint. For the next coats, one-half pound of paint to the square yard
would be considered a good coat. If the paint is on old work, or that which has been
already painted, there will be a little less lead required. Wood should be fairly dry before
the application of paint, so that it may properly adhere and not inclose moisture that may
rot the wood. The knots should be Trilled, that is, covered with shellac varnish or similar
preparation, to prevent the exuding of the resin. The heads of nails should be sunk, and
the holes and cracks filled with putty, and the surface of the wood smoothed.
Coals and other minerals are represented like rocks or stones, in varied shades of tones
and colors. Fig. 334a represents the fire-box of a locomotive, with coal in the state of
ignition in its usual type. In color, flame is represented in streaks of red-yellow, with
dark tints for smoke. Water occupies the lower half of the boiler ; but, as steam under
FIG.
FIG. 3346.
pressure is invisible like gas, the space occupied by it is shown as empty. If the direction
of its movement is desired, it is indicated by arrows. Steam issuing into atmosphere, or
boiling in an open kettle, has the appearance of a very light smoke or cloud (Fig. 3345).
There are many substances used in such masses in construction, or to be shown in the
processes of manufacture, that must be graphically represented by the draughtsman by a
general imitation of their natural appearance, or conventionally with explanatory marginal
blocks and legends.
MECHANICS.
THE draughtsman, in designing a structure, should be conversant not only with the
nature of the material, but also with the forces to which it is to be subjected their mag-
nitude, direction, and points of application, and their effects; that is, he should know the
iirst principles of mechanics, the science of rest, motion, and force to wit, Statics, Dynam-
ics, and Kinematics. Statics treats of balanced forces, or rest ; dynamics, of unbalanced
forces, where motion ensues ; and kinematics, of the comparison of motions with each
other. Considering statical forces simply in the abstract, the bodies to which they are
applied are assumed as perfectly rigid, without breaking, binding, twisting, or in any wise
changing by the application of such forces.
Force is a cause tending to change the condition of a body as to rest or motion. Force
is measured by weight. In England and the United States the unit of force is the pound,
on the Continent the gramme. All bodies fall, or tend to fall, to the earth. This force is
called the attraction of gravitation. Its direction is shown by that of a string from
which a weight is suspended (Fig. 335). It is called a vertical line, and its direction is
toward the center of the earth. Practically, these lines are considered
parallels. Let a mass, P (Fig. 336), be suspended by a cord. Each particle
is acted on by gravity, and the resultant of all these parallel forces is the
force resisted by the cord, or the entire weight of the body. If a mass
(Fig. 337) be suspended from two different points, P and Q, the directions
of the string will meet at a point C, which is called the center of gravity,
where all the weight may be considered to be concentrated. When a body
of uniform density has a center of symmetry (a point which bisects all
straight lines drawn through it), this point coincides with the center of
FIG. 335.
FIG. 336.
FIG. 337.
FIG. 338.
gravity, as the middle of a straight line, the center of a circle, the intersection of the
diagonals of a parallelogram, the intersection of lines drawn from any two angles of a
triangle to the centers of the opposite sides ; in solids, the center of a sphere, the middle
point of the axis of a cylinder, and the intersection of the diagonals of a parallelepiped.
The center of gravity of the triangular pyramid, Fig. 338, is in the straight line A E,
connecting the apex A with the center of gravity of the base triangle BCD, and distant
i of the length of the line A E from E.
MECHANICS.
201
The center of gravity of solids, which may be divided into symmetrical figures and
pyramids, as for all practical purposes most may be, can be found by determining the
center of gravity of each of the solids of which it is compounded, and then compound-
ing them, observing that each center of gravity represents the solid contents of its own
mass or masses of which it may be composed. The center of gravity of bodies enclosed
by more or less regular contours, as a ship for instance, is determined by dividing it into
parallel and equidistant sections, finding the center of gravity of each, and compounding
them into a single one.
The center of gravity of a body may be determined practically, as shown above, by its
suspension from different points. It can be done generally more readily by balancing the
body in horizontal positions on different lines of support ; the center of gravity will lie
in the intersection of planes perpendicular to these lines. A body placed in a horizontal
position will fall over, unless the vertical line from the center of gravity falls within the
FIG. 339.
FIG. 340.
FIG. 341.
FIG. 342.
base of support ; as Fig. 339 will stand, while Fig. 340 will fall over. A person car-
rying a weight insensibly throws a portion of the body forward, backward, or laterally, to
balance the load. Thus, in Fig. 341, the body is thrown back, so that the vertical from
the center of gravity ^, compounded of the center of gravity G of the woman and of the
load H, falls within the base of the feet.
When a figure rests in such a position that its center of gravity is in its lowest position,
it is said to be in stable equilibrium. It may, like a ball, rest in any position, as the center
of gravity is neither depressed nor raised by movement ; but, in the ellipsoidal form (Fig.
FIG. 343.
FIG. 344.
FIG. 345.
342) or in the toy (Fig. 343), any movement tends to raise the
center of gravity, and, on the cessation of the force, the body
returns to its original position. The ellipsoidal form (Fig. 344),
placed on its pointed end, is balanced, but the slightest move-
ment lowers the center of gravity, and, without the applica-
tion of an outside force, it can not be raised, and therefore falls. This is called unstable
equilibrium. In the toy (Fig. 345), the body of the figure is light, and the weight of the
balls brings the center below the point of support. This will admit of great oscillation,
and return to its original position. A cork with two forks inserted in it, like the wires
of the balls, and resting on the top of a glass, will illustrate this readily.
202
MECHANICS.
FIG. 346.
When two parallel forces, F F', are applied at the extremities of a straight line (Fig. 346),
they have a resultant, K, equal to their sum, and acting at a point, 0, which divides the line
inversely proportional to the forces. If the forces are equal,
the point will be at the center of the line ; if the force F is
double that of F', C A will be equal to one half C B. This is
called the principle of the lever.
Levers, in practice, are called of the first (Fig. 347), second
(Fig. 348), and third class (Fig. 349), according to the position,
weight, W, power applied, P, and fulcrum, support or turning-
point, C, of the lever. They are all forces, and only vary in
name. The two extreme forces must always act in the same
direction ; the middle one must act in the opposite direction, and be equal to the sum
of the other two; and the magnitude of the extreme forces be Diversely proportional to
their distances from the middle one. Let the middle force C be measured by a spring-
balance (Fig. 350) ; it will mark the sum of the
weights a and 5. Call the distance from a to c, #,
and from 5 to c, y, then the weight a will be to
the weight 5 as y is to 35, or a x = 5 y. Suppose the
weight a to be 6 pounds and at 5 3 pounds, at c it
FIG. 347.
W F
FIG. 348.
a
FIG. 350.
.Q
FIG. 349.
FIG. 351.
will be 9 pounds, and a c or x will be to & c or y as 6 to 3, or, if the lever is 48 inches,
& c will be 16 inches and ac 32 inches.
To find graphically the fulcrum, or point, at which a lever should be sup-
ported to sustain in equilibrium weights, or equivalent forces, acting at the
extremities of the lever. Let A B (Fig. 351) be the lever. At A and B let
fall and erect perpendiculars to the lever. Lay off from A, on any con-
venient scale, A B', corresponding to the weight applied at B ; and at B, on
the same scale, B A', the weight applied at A ; draw the line A' B' ; its inter-
MECHANICS.
203
F'
FIG. 352.
section, F, with the lever will be the position of the f ulci^^/^ T^his is on the
hypothesis that there is no weight to the lever, or that, after determining the
position of the fulcrum, the lever itself is balanced on the point by the addi-
tion of weight on the short arm F A, or the reduction of weight on the long
one F B. If the lever is of uniform weight,
on perpendiculars to C, the center of the
lever (Fig. 352), and to F, the fulcrum, as
before determined, lay off F C', the weight
of the lever, and C F', the sum of the
weights applied at A and B ; draw C' F'.
Its intersection, F", will be the actual ful-
crum, taking into consideration the weight
of the lever in addition to the weights sus-
pended at the extremities.
The Wheel and Axle. If a weight, P, be sus-
pended from the periphery of a wheel (Fig. 353),
while another weight, W, is suspended on the op-
posite side of a barrel or axle attached to the
wheel, the principle of action is the same as that
of the lever. P multiplied by its length of lever
or radius ca of the wheel is equal to W multiplied by its length of lever or radius
of the axle cb ; the axis c is the fulcrum. If a movement downward be communicated
to P, as shown by the dotted line, a rotary motion is given to the wheel and axle; the
cord of P is unwound while that of W is wound up, but P is
still suspended from a and W from & ; the leverage, or dis-
tance from the fulcrum, of each is the same as at first. The
wheel and axle is a lever of continuous and uniform action.
Since the wheel has a larger circumference than the axle, by
their revolution more cord will he unwound from the former
than is wound up on the latter, P will descend faster than TV
is raised, in the proportion of the circumference of the wheel
to that of the axle, or of their radii ca to c &. When P has
reached the position P', W will have reached W. If c a be
four times c 5, then P will have moved four times the dis-
tance that W has. The movement is directly as the length
of the levers, or the radii of the points of suspension. It
will be perceived, therefore, to move a large weight by the
means of a smaller one, that the smaller must move through
the most space, and that the spaces described are as the op-
posite ends of the lever, or inversely as the weights.
It is the fundamental principle of the action of all me-
chanical powers, that whatever is " gained in power," as it
is said, is lost in space traveled; that, if a weight is to be
raised a certain number of feet, the force exerted to do
this must always be equal to the product of the weight
by the height to which it is to be raised ; thus, if 200
pounds are to be raised 50 feet, the force exerted to do this
must be equal to a weight, which, if multiplied by its fall,
will be equal to the product 200 x 50, or 10,000 ; and it is immaterial whether the
force be a weight of 10,000 pounds falling 1 foot, or 1 pound 10,000 feet.
FIG. 353.
204
MECHANICS.
It is now common to refer all forces exerted to a unit of pounds-feet, that is, 1 pound
falling 1 foot ; and the effect to the same unit of pounds-feet, 1 pound raised 1 foot.
Thus, in the example above, the force exerted or power is 10,000 pounds-feet falling ; the
effect 10,000 pounds-feet raised. In practice, the pounds-feet of force exerted must always
be more than the pounds-feet of effect produced ; that is, there must be some excess of
the former to produce movement, and to overcome resistance and friction of parts.
The measure of any force, as represented by falling weight, is termed the absolute power
of that force ; the resulting force, or useful effect for the purposes for which it is applied, is
called the effective power.
The Pulley. The single fixed pulley (Fig. 354) consists of a single grooved wheel
movable on a pin or axis, called fixed, because the strap through which the pin passes is
attached to some fixed object. A rope passes over the wheel in the groove; on one side
the force is exerted, and on the other the weight is attached and raised. It may be con-
sidered a wheel and axle of equal diameters, or as a lever in which the two sides are equal,
the pin being the fulcrum. P, the force exerted, must therefore be equal to the weight
"W, raised ; and, if movement takes place, W will rise as much as P descends.
The fixed pulley is used for its convenience in the application of the force ; it may be
easier to pull down than up, for instance ; but the pounds of force must be equal to the
pounds of effect. The tension on the rope is equal to either the force or weight.
Fig. 355 is a combination of a fixed pulley, A, and a movable pulley, B. The simplest
way to arrive at the principle of this combination is to consider its action. Let P be pulled
down, say two feet; the length of rope drawn to this side of the pulley must be furnished
from the opposite side. On that side there is a loop, in which the movable
pulley, with the weight W attached, is suspended. Each side of this loop, 2
and 3, must go to make up the two feet for the side or end 1.
will therefore furnish each one foot. As these cords are
shortened one foot, the weight W is raised one foot, and, as
Cords 2 and 3
FIG. 354.
FIG. 355.
FIG. 356.
FIG. 357.
the movement of W is but one foot for the two feet of P, W must be twice that of P,
because the two pounds-feet of P must equal the pounds-feet of W.
In the combination of pulleys (Fig. 356), let P be pulled, say three feet; then this
length of rope, drawn from the opposite side of the pulley, is distributed over the three
cords, 2, 3, 4, and the weight W is raised one foot ; consequently, the weight W is three
times that of P. The cord 1 supports P, the cords 2, 3, 4, the weight W, or three times
P; consequently, the tension on every cord is alike. The same rope passing freely around
pulleys must have the same tension throughout ; so that, to determine the relation of W
to P, count the number of cords which sustain the weight. Thus, in Fig. 357, the weight
is sustained by four cords ; consequently, it is four times the tension of the cord, or four
times the force P. In order not to confuse the cords, the pulleys are represented as in the
figures ; but, in construction, the pulleys, or sheaves, are usually of the same diameter,
and those in connection, as A and B, and C and D, run on the same pin.
MECHANICS.
205
The Inclined Plane. To support a weight by means of a single fixed pulley, the force
must be equal to the weight. Suppose the weight, instead of hanging freely, to rest upon
an inclined plane b d (Fig. 358) ; if motion ensue, to raise the weight W the height a 5, the
rope transferred from the weight side of the pulley will be equal to 5 d, and P will have,
consequently, fallen this amount ; thus, if b d be six feet, and a ft one foot, while W is raised
one foot, P has descended six feet, and, as pounds-feet of power must equal pounds-feet of
effect, P will be one sixth of W ; and, by reference to the figure, P is to W as a 5 is to 5 d,
or as the height of the incline is to its length. If the end of the plane d be raised, till it
becomes horizontal, the whole weight would rest on the plane, and no force would be
necessary at P to keep it in position; if the plane be revolved on 5, till it becomes per-
FIG. 358.
FIG. 359.
pendicular, then the weight is not supported by the plane at all, but it is wholly depen-
dent on the force P, and is equal to it. Between the limits, therefore, of a level and a
perpendicular plane, to support a given weight W, the force P varies from nothing to an
equality with the weight.
The construction (Fig. 359) illustrates the principle of the wedge, which is but a mova-
ble inclined plane ; if the wedge be drawn forward by the weight P, and the weight
"W be kept from sliding laterally, the fall of P a distance equal to a d will raise the
weight W a height cl. P will therefore be to W as c 5 is to a d. For example, if the
length of the wedge a d be ten feet, and the back c ~b two feet, then P will be to W as
two to ten, or one fifth of it.
Let the inclined plane a & d (Fig. 359) be bent round, and attached to the drum A
(Fig. 360), to which motion of revolution on its axis is given, by the unwinding of the
turns of a cord from around its periphery, through the action of a weight P suspended
from a cord passing over a pulley. If the weight W
be retained in its vertical position, by the revolution
of the drum, it will be forced up the incline, and
when the cord has unwound one half turn from the
drum, and consequently the weight P descended a
distance, c e, equal to
one half the circum-
ference of the drum,
the weight W has been
raised to the height a &
by the half revolution
of the plane ; P must
therefore be to W as
a 5 is to one half the
circumference. Extend
the inclined plane so as
to encircle the drum (Fig. 361). The figure illustrates the mechanism of the screw, which
may be considered as formed by wrapping a fillet-band or thread around a cylinder at a
uniform inclination to the axis. In practice, the screw or nut, as the case may be, is moved
by means of a force applied at the extremity of a lever, a complete revolution raises the
FIG. 360.
FIG. 361.
206
MECHANICS.
weight the distance from the top of one thread to the top of the one above, or the pitch.
If the force be always exerted at right angles to the lever (Fig. 362), the lever may be con-
sidered the radius of a wheel, at the circumference of which the force is applied. Thus,
if the lever be three feet long, the diameter of the circle would be six feet, and the cir-
FIG. 362.
cumference 6 x 3-1416, or 18 T 8 /o- feet ; if the pitch be one inch, or one twelfth of a foot,
then the force would be to the weight as one twelfth is to 18-85 ; and if the force be one
pound, the weight would be 226*20 pounds.
The resultant of two forces of exertion, as has been seen, is their sum, and counter-
balances the forc^ of resistance, which must be applied at a point intermediate between,
and distant from each of them, inversely as the forces exerted.
The resultant of any number of parallel forces acting in one direction is equal to
their sum acting in the same direction at some intermediate point ; that is, the effect of
all these forces is just the same as if there were but one force, equal to their sum,
acting at this point, and is balanced by an equal force acting in the opposite direction.
This central point may be determined by finding the resultant, i. e., the sum, and the
point of application for any two of the forces, as shown graphically in Figs. 351, 352, and
then of other two, the resultants thus determined being again added together like simple
forces.
Inclined Forces are those whose directions are inclined to each other. When two
men of equal strength pull directly opposite to each other, the resultant is nothing. Let
a third take hold of the center of the rope (Fig. 363), and pull at right angles to the
rope; he will make an angle in the rope, and the
other two now pull in directions inclined to each
other. The less the force exerted at the center, the
less the flexure in the rope ; but when it becomes
equal to the sum of the forces at the ends, the two,
to balance it, must pull directly against it, bringing
the ends of the rope together, and acting as parallel
forces.^ Between the smallest force and the largest that can be exerted at the center and
maintain a balance or equilibrium, the ends of the rope assume all varieties of angles,
which angles bear definite relations to the forces.
Represent these forces by weights (Fig. 364). Let P and P' be the extreme forces act-
ing over the pulleys M and N, and tending to draw the rope straight, which the weight P"
prevents. Lay off the weight of P (90 pounds) along A B, and the weight of P' (60 pounds)
along A 0. Draw En parallel to A C, and Cn parallel to A B. Connect n with A. If
this is measured with the same scale that A B and A were laid off with, it will be found
that it equals 120 pounds, which will be found to be the same as the weight P". An, there-
FIG. 363.
MECHANICS.
207
fore, gives the amount and direction of the resultant of the two forces P and P', which
resultant is balanced by P". In the same way the resultant of any number of inclined
forces (Fig. 365) may be found by compounding the resultant of any two forces with a
third, and so on.
As two forces may be compounded into a single resultant, so conversely one force may
be resolved into two components ; thus, let the weight P (Fig. 366) be supported by two
inclined rafters, C A and C B.
Each resists a part of the force
exerted by the weight P. To
find the force exerted against
the abutments A and B, in the
direction of C A and C B, draw
c A' (Fig. 367) parallel to C A,
c B' to C B, and c d, a paral-
lel to the line C P, the direc-
tion in which the weight P
acts ; lay off. c d from a scale
of equal parts, a length which
FIG. 367.
will represent the number of pounds, or whatever unit of weight there may be in the
weight P ; draw d a parallel to c B', and d 5 parallel to c A' ; c a, measured on the scale
208 MECHANICS.
of equal parts adopted, will represent the pounds or units of weight exerted against A
in the direction of A, and c b the pounds or units of weight exerted against B in the
direction of C B.
This method of finding the resultant of two forces, or the components of one force, is
called the parallelogram of forces. If two sides of a parallelogram represent two forces in
magnitude and direction, the resultant of these two forces will be represented in magni-
tude and direction by the diagonal of the parallelogram and conversely.
The sum of ac and c & is greater than c d ; that is, the weight P exerts a greater force
in the direction of the lines C A and C B, against A and B, than its own weight ; but the
down pressure upon A and B is only equal to the weight of P and of the rafters which
support it, which last, in the present consideration, is neglected. Ptesolve c 5, the force
acting on B in the direction of eB', into g ~b or ce the downward pressure, and eg or eb
the horizontal thrust on the abutment B, and ca into cf&ndfa. To decompose a force,
form a triangle, with the direction of the other
forces, upon the line representing the magnitude
and direction of the given force ; c e represents
the weight on B, c f the weight on A ; c d, or
c e + d e, the whole weight P ; therefore, the
weight upon the two abutments A and B is
equal to the whole weight of P.
The steelyard (Fig. 368) is a lever, from the
short arm of which a dependent hook or scale
supports the article to be weighed ; while, on
the long arm, a fixed weight, P, is slid in or
out from the fulcrum till it balances the article ; the distance as marked on a scale on the
long arm determines the weight. In platform-scales, when very heavy weights are bal-
anced by small weights on a graduated arm, combinations of levers are used, the principle
of which can be understood from Fig. 369. Thus, suppose PF to be 7", a F 2", a F' 9" r
&F'2", &F"11", F"W 3".
P is to force a as a F to P F, or as 2 to 7
Force a is to 6 as b F' to a F, or as 2 to 9
& is to W as F" W to b F", or as 3 to 11
P is to W as 12 to 693
The differential axle, or Chinese capstan, consists of an axle with two different diameters
(Fig. 370), the weight W being suspended in the loop of a cord wound around these axles
in opposite directions by a single turn of the axle. The weight is only raised or low-
FIG. 369.
ered by the difference between these two circumferences ; one takes up while the other
lets out, and the P, to balance W, must be as these differences of circumference of axles
is to the circumference of the wheel from which P is suspended.
The differential screw (Fig. 371) consists of an exterior screw, A, and an interior screw,
B. By the revolution of the arm, the screw A is moved through the plate D in propor-
tion to its pitch, but the interior screw B moves inward its pitch, and the movement of
W is only the pitch of A less th#t of B, and the power applied is to the weight moved as
the difference of these pitches is to the circumference described by the power.
MECHANICS.
FIG. 370.
FIG. 371.
As the lever (Fig. 372) moves under the action of power or weight, the lever be-
comes inclined to the direction of the forces, but the forces are still parallel. The rela-
tions of the forces to each other are not changed, but the absolute action of each is only
FIG. 373.
that due to the length a 5 and 5 c, to which the directions of the forces are perpendicu-
lar. In the bent levers (Figs. 373 and 374) the action of the forces is estimated- on
lengths of arms, determined by the
perpendiculars a b and b c let fall
from the fulcrum on the directions
of the forces.
The toggle-joint (Fig. 375) is
much used for presses. The force
is exerted in the direction of the
arrow at 0, and the effective force
o
FIG. 874.
14
FIG. 375.
210
MECHANICS.
is to separate the plates A and B. The action is as shown in Fig. 376. Equal movements,
as 0-1, 1-2, 2-3, correspond to unequal movements at A and B. as A a', a' a?, a? a 3 . The
nearer the force C is to the line A B, the less the movement a 2 a 3 ; and, consequently, the
force C exerts greater effects in intensity, but the latter is less in movement.
C
lib.
FIG. 376.
Fig. 377 exhibits the principle of the hydraulic press. The small plunger or piston may
be considered the application of the force, and the large one the weight to be raised to
balance each other ; the pressure per square inch of surface must be the same, and the
force must be to weight as the surface of its piston is to that of the weight-piston. If
16 j. motion takes place, the force will move through space cor-
responding to the area of weight-piston, while the weight
will move that of the area of the force-piston. And this is
the great principle of all mechanism in the transmission of
force : there can be no total gain. What is gained in force is
lost in movement, and in many complicated machines the
theoretical comparison of force applied and resultant force
may be ascertained by the measures of their movements.
The resultant effects of forces, as heretofore treated, have
been without motion, or static. But when motion is produced,
the forces are called dynamic. A weight suspended or sup-
ported exerts a force, which is balanced by the resistance of
the suspending or supporting medium ; but a falling weight
acquires an increasing velocity with every unit of time or
space passed. All bodies would fall with the same velocities were it not for the different
resistances from the air due to their different bulk in proportion to their weight. Dense
articles, as stones and metals, acquire a velocity in this latitude of about 32'2 feet in each
second, called the intensity of gravity, or g. The value of g at the equator is 32'088 ; at
the poles, 32-253. A body
FIG. 377.
Starting with a velocity
Falls during the 1st second
Acquiring a velocity of
Falls during the 2d second
Acquiring a velocity of twice 32, or
Falls during the 3d second.
Acquiring a velocity of 3 X 32=
Falls during the 4th second
Acquiring a velocity of 4 X 32 =
Falls during the 5th second
Acquiring a velocity of 5 X 32 =
\32 feet per second.
32+ 16\=
\
\64 feet per second.
Ft. Tot. Fall.
16 16
48 64
32+32+16\= 80
96 feet per second.
32+
32+J32+ 32+ 16\= 112 256
j \128 feet per second.
32+32 +
16\ = . 144 400
' J 160 feet per second.
MECHANICS.
211
FIG. 378.
Calling s the space passed over, the terminal velocity in feet, t the time in seconds of
falling, * = igt*, v=gt or = V64.4a. In determining the velocity of issuing water under a
head A, corresponding to s in the equation, it is generally near enough to reckon as eight
times the square root of the head (VA).
The motion of falling bodies is a uniformly accelerated one, but there are also uni-
formly retarded motions in which the velocity is decreased by equal losses in equal times.
There are also uniform motions when bodies are impelled
by a constant force and opposed by constant resistances.
In Fig. 378, o s represents the trace of a body impelled
horizontally by a uniform, but falling through the action of
gravity with an accelerated, force. This curve, a parab-
ola, represents approximately the curve of the thread of
stream issuing from an orifice, or flowing.
It will be seen that to produce twice the velocity the
body must fall through four times the space; that there is
four times the force stored in the body. But to main-
tain this velocity uniformly, only twice the force is neces-
sary. The momentum of a body is its mass multiplied by
its velocity, but its inertia is as the square of the velocity.
It is an established principle of mechanics that the results
must be proportional to the causes: if a body has to be
raised four feet to obtain a double velocity in falling, the
destructive result of that fall must also be four times.
Under statics, it has been shown that forces may be
resolved and compounded. The same may be done dynamically- that which has been
treated as weight must now be considered as momentum.
In treating of dynamic forces the resultants have been considered as equal to the exer-
tion, without any losses by resistances. This never happens in practice ; the resist-
ances are a very large element. Resistances from the medium in which the bodies are
moved are from the surfaces oh which the bodies are supported ; resistances due to
the displacement of the fluid in which the bodies move, and fric-
tional resistances, or what is termed skin-resistances, of bodies
moving through air or water; and the surface-resistance of bod-
ies sliding or rolling on each other. Suppose a weight to rest
on a horizontal surface it will take a certain force to move the
insistent weight depending on the amount of this weight and
the kind of surfaces in contact, and the force that will just
cause motion overcomes the friction, or frictional force, and is
equal to it. The frictional force is only a percentage of the in-
sistent force of the body, and this percentage is called the co-efficient of friction. If the
horizontal surface of support be raised at one end, so as to make the surface inclined, it
will after a time become so steep that the insistent body will
slide down the surface. Thus, in Fig. 379, if the body Q is
ready to slip on the surface A B, the angle BAG, which rep-
resents the angle of the surface with the horizontal, is called
the angle of repose, or limiting angle of frictional resistance;
or thus (Fig. 380), if the force acting in the direction P" M
is just sufficient to produce motion of the mass M along the
plane F Q, the angle P M P" is the limiting angle of resist-
ance.
General Morin has made an elaborate course of experiments on friction, the results of
which are given in the table on page 212. It was formerly held that friction was directly
M
FIG. 379.
212
MECHANICS.
as the weight, without regard to the amount of surface or velocity of movements. And
M. Morin's experiments, as rather applicable to the friction of quiescence and slow move-
ments, come within this rule. But in practice it has been found that the co-efficient of
friction with unguents is reduced by increase of velocity and temperature ; that extent
of surface maybe prejudicial; and that careful selection of unguents, according to the
work to be done, must be made to economize power by the reduction of friction.
Mr. 0. I. H. Woodbery, in his experiments on the driving of cotton-spindles, found the
co-efficient of friction to be from 7 to 20 per cent, the lond being from one to five pounds
per square inch ; while Professor Thurston, with heavy loads of 1,000 pounds per square
inch, as on the crank-pins of the North River steamboat-engines, found the co-efficient of
friction was one half of one per cent, the unguent being sperm-oil. Practically it may be
said that the co-efficient of friction for light-running spindles should not exceed 10 per
cent, and for the usual work in shops, of say 100 to 200 pounds, should not exceed from
2 to 3 per cent.
EXPERIMENTS ON FRICTION, BY M. MORIN.
SURFACES OF CONTACT.
WITHOUT UNGUENTS.
UNCTUOUS SURFACES.
FRICTION OF
MOTION.
FKICTION OF
QUIESCENCE.
FRICTION OF
MOTION.
FRICTION OF
QUIESCENCE.
Co-efficient
of friction.
Limiting
angle of
resistance.
Co-efficient
of friction.
Limiting
angle of
resistance.
Co-efficient
of friction.
Limiting
angle of
resistance.
Co-efficient
of friction.
Limiting
angle of
resistance.
Oak upon oak, fibers parallel to the
motion
0-478
0-324
0-246
25 33'
17-58
13-50
0-625
0-540
0-376
32 1'
28-28
20-87
0-108
0-143
0-136
0-140
6 10'
8 9'
7-45
7-59
0890
0-314
21 19'
17 26'
Oak upon oak, fibers of the moving
body, perpendicular to the motion. . .
Oak upon elm, fibers parallel
Wrought-iron upon oak |
0-619
0-133
0-194
0-172
0-195
0-152
0-147
0-217
0-161
0201
0-296
31 47'
7-52
10 59'
9-46
11-3
8-39
0-619
0-137
0-194
0-i62
31 47'
7-49
10-59
9-is
" wrought-iron
41 cast-iron
44 " brass
0-177
o'-ieo
0-125
0-144
0-143
0-132
0-107
103
9-6'
7-8
8-12
8-9
7-32
6-7
o'-iis
6-44
Cast-iron on elm ....
'" " cast-iron
44 " wrought-iron
44 " brass
8-22
12-15
9-9
11-22
1630
....
Brass upon cast-iron !
44 " brass ... . -
....
....
0-184
7-88
o-iei
9-19
14-57
Leather ox-hide, well tanned, on oak..
u on cast-iron, wetted. . j
belts on oaken drums '
41 cast-iron pulleys
Common building - stones upon the
same
0-229
12-54
2-67
0-27
0-28
( 0-38 to
|0-65
0-47
20-49-
882
0-65
0-75
38-2-
36-53
MECHANICAL WORK OK 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 direc-
tion to the point of application of the resistance ; it is, therefore, the product of two indis-
pensable qualities or terms :
First. The effort, or pressure exerted.
Second. The space passed through in a given time, or the velocity.
The unit of force in England and here is represented by the pound, and the unit of
space by the foot.
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 pounds, and the velocity one foot per second, the amount
of work will be expressed by 4x1 = 4. If the velocity be double, the work becomes
4x2 = 8, or double also ; and if, with the velocity double, or 2 feet per second, the press-
ure be doubled as well that is, raised to 8 pounds the work will be 8x2 = 16 pounds
MECHANICS. 213
feet. It is more usual to write foot-pounds, but we invariably use the former, following
the Continental idiom of kilogrammetre, in which the unit of force, kilogramme, precedes
that of space, the metre.
In comparison of motors with each other, it is usual to speak of them as so many horse-
power equivalent to 550 pounds feet per second, or 33,000 pounds feet per minute. The
Continental horse-power is equal to 75- kilograinmetres or 54:2*48 pounds feet per second.
It is very common to use other units of force and space, as tons-miles ; and train-miles,
in railway practice.
The time must also be expressed or understood. It is impossible to express or state
intelligibly an amount of mechanical effect, without indicating all the three terms force,
space, and time.
The motors generally employed in manufactures and industrial arts are of two kinds
living, as men and animals ; and inanimate, as water and steam.
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 ex-
erting a certain effort at a certain velocity, than he will by exerting a greater effort at a
less velocity, or a less effort at a greater velocity, and the proportion of effort and velocity
which will yield the maximum effect is different in different individuals.
In the manner and means in which the strength of men and animals is applied, there
.are three circumstances which demand attention :
1. The power, when the strength of the animal is exerted against a resistance that is
at rest.
2. The power, when the stationary resistance is overcome, and the animal is in motion.
And,
3. The power, when the animal has attained the highest amount of its speed.
In the first case, the animal exerts not only its muscular force or strength, but at the
same time a very considerable portion of its weight or gravity. The power, therefore,
from these causes must be the greatest possible. In the second case, some portion of the
power of the animal is withdrawn to maintain its own progressive motion ; consequently,
the amount of useful labor varies with the variations of speed. In the third case, the
power of the animal is wholly expended in maintaining its locomotion; it therefore can
carry no weight.
Weisbach calls the mean effort of an animal one fifth its weight, which may serve as a
general rnle, but, in practice, will be considerably modified, when applied to the indi-
vidual, depending upon the exertions, and the conditions and circumstances under which it
is made. A man-power is usually estimated at one sixth of a horse-power (H. P.) ; yet, if
the muscular force of a man be required for an effort of short duration, it will exceed one
liorse-power. Thus, n horse-power is equal to 33,000 pounds feet per minute, or 550
pounds feet per second; and, if a man weighing 150 pounds move up-stairs at the rate of
four feet per second, he exerts a force of 600 pounds feet, which he can readily double for
a few seconds.
The force of a man is utilized mechanically through levers, as in pumping or rowing,
or at a vertical capstan, or at a crank, carrying or dragging loads, shoveling, etc. In
continuous work at the lever he will exert from 25 to 30 pounds ; at the crank, from 15
to 20 pounds.
The muscular force of horses is utilized in the draft of carriages, in hoisting, and in
horse-powers, either moving in a circle round a central shaft or on a revolving platform,
or on an endless belt. The draught of a horse varies with the speed of movement and its
214:
MECHANICS.
duration. Trautwine gives the draught of a horse at two and a half miles per hour for 10
hours per day, 100 pounds; 8 hours, 125 pounds; 6 hours, 166f pounds; 5 hours, 200
pounds. The omnibus-horses here average nearly six miles per hour, and make 16 to 24
miles per day; the average will not exceed 16 miles. At the Manhattan Gas Works, a
span of horses hoist from the lighter 200 tons gross in 10 hours to the height of say 25
feet, with charges of 6 to the ton, in a bucket weighing 150 pounds, the rope passing over
a single block and through a snatch-block. On a horse-power, the force exerted by a single
horse is from 125 to 175 pounds, at an average speed of about three miles per hour, and for
eight hours per day. Beyond a speed of four miles per hour, the pounds foot of work of a
horse will decrease in an increasing ratio up to the limits of his speed, when the whole
work done will be used up in locomotion. In proportioning levers, cranks, traces, chains,
through which animal force is transmitted to machines, or for mechanical purposes, it ia
not safe to estimate the stress as the average force ; there are impulses and stresses in
action which will exceed the weight of the animal.
Water-Power. Water, used for the purposes of power, moves machinery either by its-
weight, by pressure, by impact, or by reaction, and is applied through various forms of
wheels. However used, the mechanical effect inherent in water is the product of its
weight into the height from which it falls ; but there are many losses incurred in its appli-
cation, so that only a portion of the mechanical effect becomes available ; and the com-
parative efficiency of any water-wheel or motor is represented by this percentage of the
absolute effect of the water applicable to power.
The quantity of water supplied to the mills at Lowell, permanently, for the working
hours per day is about 4,000 cubic feet per second, and the entire fall 33 feet. In the dis-
tribution of the water by the canals about two feet of fall is lost, and the mill-powers, as
leased to the mills, would be about 4,000 cubic feet per second, on a 31 -foot fall. In the
passage of the water through the trunks or pent-stocks to the wheels, and from the wheels
to the river or other canals, there is still another loss of head, which may be considered
about one foot, so that the net fall is only 30 feet.
4,000 cu. ft. x 62-33 weight of water per cu. ft. x 30 ft. fall
550 Ibs. ft. per H. P. per sec.
= 13,600 horse-power.
But only a percentage of this power is available for mechanical power. The efficiency
of the turbines, the wheels now generally in use here, may be taken at 80 per cent of the
gross horse-power. The net horse-power will then be 13,600 x '80 = 10*880 horse-power.
Wind is applied for the purposes of power ; but, as there is no constancy in its action,
its use is mostly confined to the purpose of raising water
by means of pumps into cisterns or reservoirs.
Steam is the elastic fluid into which water is converted
by a continuous application of heat. It is used to pro-
duce mechanical action almost invariably by means of a
piston movable in a cylinder. Thus, in Fig. 381, the steam
entering through the lower channel-way, or port, presses
against the under side of the piston in the direction of the
arrow, the piston is forced upward, the steam above the
piston escaping through the exhaust-channel o. When
the piston reaches the top of the cylinder, the valve is
changed by mechanism, the steam enters above the pis-
ton, and the steam below it escapes through the exhaust.
In this way a reciprocating movement is established. To
determine the horse-power of a steam-engine, multiply the area of the piston in square
inches by the effective pressure in pounds on each square inch of piston, and the product
by the travel in feet through which the piston moves per minute, and divide this last
FIG. 381.
MECHANICS.
215
product by 33,000. The travel is the length of stroke multiplied by the number of strokes,
or double the number of revolutions per minute.
Example. Let the diameter of the piston be 18 inches, the effective pressure 45 pounds
per square inch, the stroke 30", the revolutions 60, or 300 feet travel per minute, what will
be the horse-power of the engine ?
Area of piston, 254-46 square inches.
254-46 x 45 x 300
-337000 = 104 ' h <^-Pwer.
As steam in its passage through channels and in the cylinder is subject to various
losses of pressure, and as the steam is worked under more or less expansion, and as the
exhaust steam is discharged under more or less pressure, whether into the air or into a
condenser, it is impossible to determine the effective pressure except by the means of an
indicator.
The principle of working steam expansively is as follows : If a cubic foot of air of the
atmospheric density be compressed into the compass of half a cubic foot, its elasticity will
be increased from 15 pounds on the square inch to 30 pounds ; if the volume be enlarged
100
10
to two cubic feet, the pressure will be one half, or Y| pounds. The same law holds in all
other proportions for gases and vapors, provided their temperature is unchanged.
Fig. 382 illustrates this graphically. Suppose the piston in the cylinder to have made
one tenth of its stroke, and to be at .1, and the pressure at 100 pounds above the absolute
216
MECHANICS.
(or vacuum) to which expansion is referred, and not to the atmospheric line representing
nearly 15 pounds pressure: if the steam-valve be now closed, and the piston he moved to
the position .2, the space occupied by the steam will be double what it was at first, and
the pressure one half, J-f 2 -, or 50 pounds. If the piston be moved to .3, the pressure will
be -J-, or 33$- pounds ; to .4, J, or 25 pounds ; and so on to .5, .6, .7, .8, .9, .1.0, the pressure
will be , -J, -f, , |-, T V ; and, at the end, the expansion will be said to be ten times, and
the cut-off (or where the steam was shut off from the cylinder), at ^ of the stroke.
When the steam is cut off, if there be no leak through the valves or by the piston, this
quantity may be considered constant, although there are losses by condensation from the
surfaces of cylinder and piston, and the conversion of heat into work. But it will generally
be found that the weight of steam, as represented by the volumes, will be greater at the
end of the stroke than at the cut-off, owing to re-
evaporation of condensed or conveyed water by the
cylinder surfaces.
To illustrate the theoretical advantages of a cut-
off, draw lines across the card (Fig. 382) at 40 and 20
pounds. The portions of the card below these lines
will represent the card of an engine, working at an
initial pressure of 40 pounds, and cutting off at .25,
or stroke. The portions below the 20-pound line,
the card of an engine, with this initial pressure cut-
ting off at .5, or stroke. The original card and these
other cards use equal quantities of steam, but the work
is very different; in the first, all the work is below the 40-pound line, and in the other
all below 20 pounds.
Of late compound steam-engines have become very popular. They consist of two cyl-
inders, a high-pressure (h. p. c.)and a low-pressure (i. p. c.) one. Fig. 383 shows the gen-
eral arrangement, but without the valves. The h. p. c. (A, B, 0, D) draws its steam from
the boiler and exhausts into the 1. p. c. (A', B', C', D') ; the top of the h. p. c. into the bottom
of the 1. p. c., and vice versa, so that the pressure on the pistons of the two cylinders is in
the same direction.
For the comparison of the theoretical effect of the single cylinder and compound en-
gines, construct the card of a single cylinder, Fig. 384, shown in dotted line, in which 0'8 is
the length of stroke, 50 pounds the initial pressure, and .2 the point of cut-off. If 50, C, .2, .0
represent the h. p. c. of a compound, and the cylinder be filled at the pressure of 50 pounds,
MECHANICS.
217
the quantity of steam used at each stroke will be the same as in the single cylinder, and,
to expand equally with this, the stroke of the 1. p. c. is represented by .2, .1, .0. When the
piston of the h. p. c. is about to commence its stroke downward, for instance, the cylinder
beneath it is full of steam at 50 pounds. As the steam rushes in above the piston from
the boiler, the steam below the piston begins to exhaust into the upper part of the 1. p. c.,
and consequently falls off, while the pressure above the h. p. c. piston in connection with
the boiler maintains its 50 pounds.
When expansion commences in the 1. p. c., the pressure is the same as in the h. p. c.,
50 pounds ; but the expansion takes place differently from that in a single cylinder. At the
end of the first eighth of the stroke, the space in the 1. p. c. is equal to ^ that of the h. p. c.,
but its space has been reduced by the movement of the piston ; therefore, the space now
occupied by the steam is + , or ^ of what it was before expansion, and the 50 pounds
becomes - = 36-4 nearly. At stroke, the space in the 1. p. c. is equal to that of the
h. p. c., and in the h. p. c. it is reduced to f; the total space is now 1 + = , and the
expansion is , or 28-6 pounds nearly. At stroke the space is 1| x , and the
pressure, consequently, 23'5 nearly. At the end of the stroke the space of the h. p. c. is
entirely shut off, and that of the 1. p. c. filled with expanded steam, at 12 pounds, J of
the initial pressure. The full line, C T', represents the expansion as it has taken place in
the 1. p. c. ; but, as said above, the pressure below the piston in the h. p. c. falls off as
expansion goes on in the 1. p. c. The pressure in the 1. p. c. at the top is the same as the
h. p. c. at the bottom, and, if these pressures be transferred to the h. p. c. card, there will be
a curve, 50 T", which will represent the back pressure beneath the h. p. c. piston. The back
pressure is shown in the shaded portion, above which is the net pressure on the piston ; if
these net pressures be divided by 4, and plotted, as shown, above the 1. p. c. expansion,
curve C T', then the curve C H will represent the curves of pressures of the united h. p. c.
and 1. p. c. on the same scale as that of the single cylinder.
Figs. 385, 386 represent these cards, both on the same scale, and it will be observed
that, theoretically, there is no difference in effect between steam used in a single cylin-
FIG. 385.
FIG.
der or in a compound. But, practically, the compound is, for many purposes, found the
most economical, due in part to the smaller condensation, since the surfaces in the h. p. c.
are never cooled below the limit of expansion, in example 12 pounds (204), while the
1. p. c. and the single cylinder are cooled to the limit of condensation, or probably about
126.
In addition, comparing the two cards (Figs. 385, 386), it will be observed that the forces
in the compound cylinders are less irregular than in the single cylinder, and the necessi-
ties of a fly-wheel, to equalize forces and resistances, are less.
The cards of the compound engines above drawn do not take into consideration the
loss of pressure in the channels between the h. p. c. and 1. p. c., and there is a class
of compound engines in which the h. p. c. exhausts into an intermediate chamber, be-
218
MECHANICS.
tween it and the 1. p. c., to which the construction of cards given is not applicable. They
can best be determined from practical examples.
The above illustrations represent purely the theoretical card. The vacuum is perfect,
and the steam in the cylinders at full pressure, both in introduction and at relief, without
any wire-drawing, reduction, or rounding, incident on actual practice.
Fig. 387 represents a real card taken from a steam-cylinder of a condensing engine.
To determine the mean effective pressure, divide the atmospheric line, embraced in the
card, into 20 equal parts, and draw ordinates through the .1, .3, .5, .7, .9, .11, .13, .15, .17,
.19th divisions. The lines embraced be-
tween the card outlines represent the pres-
sure at different parts of the stroke .05,
.15, and so on, on the scale of the indicator-
spring ; these, added together and divided
by 10, give the mean effective pressure (in.
e. p.) on this card, 43'4 pounds.
The mean effective pressure multiplied
by the area of piston, in square inches, by
the length of stroke, and number of strokes
per minute, gives the pounds-feet of work
per minute, which, divided by 33,000, will
give the indicated horse-power (i. h. p.) of
the engine.
To determine whether a steam-engine is
working properly, it is necessary to compare
the absolute card with the theoretical one.
Fig. 388 represents an indicator-card, as
taken from a steam-cylinder in which there
is no condensation ; the exhaust is directly into the air. On this is shown the construction
of the isothermal curve. It will be observed that there is a line, A B, to the top of the card.
The space between this and the card represents the spaces between the cylinder-head and
piston, and between the steam-valves and the cylinder, called the clearance, which are
estimated in percentages of the capacity of the cylinder, and is thus plotted on the indica-
tor-card. On the indicator-card, as taken by the instrument, the absolute can not be
taken, but only that of the atmosphere, the will be at a distance below this, correspond-
ing to the barometric pressure, usually 14'8 pounds. Draw the line parallel to the at-
mospheric line, the clearance line perpendicular to it, a line parallel to the line, at the
height of the initial pressure, and a line parallel to the clearance line at the point of cut-
FIG. 387.
FIG. 388.
FIG. 389.
off on the initial pressure line. Any point on the expansion line, as l a , 2 2 , 3 2 , may be de-
termined by drawing lines B 1, B 2, B 3, and then horizontal lines 1 3 li, 2 a 2i, 3 2 3i from,
their intersections li, 2i, 81. With the cuj-off line, parallel to the line, and perpen-
diculars from 1, 2, 3, the intersections of these two lines, 1 2 . 2 a , 3 2 , will be the points in the
MECHANICS.
curve. The curves in the outline of the cards, at the times of admission, cut-off, and
exhaust, show the action of the valves and time occupied in change of condition. The
stroke commences at A, cuts off at C, commences to exhaust at E ; about D the exhaust-
valve closes, and the steam between the piston and the ends of the cylinder begins to be
compressed, and the curve developed is called the curve of compression.
In expanding, steam does not maintain the same temperature ; there is a fall of tem-
perature, and consequently less space occupied than shown by the isothermal curve; the
curve thus developed is called the adiabatic curve. In Fig. 389 the construction of this
curve, the line Ce, through the point of cut-off, is inclined to the line A B, 1 43', to which
the lines 1 l a , 2 2 a are drawn parallel, but otherwise the same as in the preceding figure ;
practically, the isothermal curve corresponds more nearly with that formed by the cards,
as, especially near the end of the stroke, there is considerable transmission of heat from
the cylinder surfaces to the steam, more than that lost by mere expansion.
The indicator-cards show very fairly the amount of power exerted on the piston, but
they do not show the economy of the whole machine including the boilers. The boilers
may be faulty, in that they do not evaporate sufficient water for the coal consumed, or that
the ebullition is too local and violent, without sufficient steam-space, so that water is
taken off with the steam ; or the steam-cylinder and its working may be faulty, in that the
steam is condensed therein without doing any work. The economic value of the boiler
may be determined by the measure of the quantity of water pumped into the boilers, and
the quality of the steam.
MACHINE DESIGN AND MECHANICAL
CONSTRUCTIONS.
IN the designing of new machines and mechanical constructions, the draughtsman must
draw from his knowledge of well-known forms and parts, and combine them; but, to pro-
portion them properly, and adapt them to the purposes required, he must understand the
stresses to which they are to be subjected, and the action and endurance of the material to
be used, to withstand these stresses.
In the present technical application of the term, stress is confined to a force exerted
between two bodies or parts of a body, such as a pull, push, or twist. Strain is the altera-
tion produced by a stress. Stress is the cause, strain the effect ; the first is measured
by the load, the latter by the deformation of the body produced by the first. A stress, not
greater than the elastic limit of the material acted upon, produces a strain which disappears
as soon as the load is removed : up to this limit the strain is proportional to the stress ;
beyond, the strain increases faster than the stress, up to the point of rupture. The elastic
limit is a percentage of the breaking strain, varying with the kind of material and applica-
tion of stress. Stress is usually designated as load, meaning thereby the sum of all the
external forces acting on the member or structure, together with its weight.
Dead load, or weight, is a steady, unchangeable load. Live loads are variable, alternately
imposed and removed, or varying in intensity or direction. It is usual, in designing con-
structions, to proportion the parts to resist a much greater load than will be brought on
them in the structure ; the load is multiplied by a factor termed factor of safety, as a secu-
rity against imperfections in material and workmanship, contingencies of settlement, and
other incidental stresses. But it must be observed that these imperfections are such as
can not be seen and met ; there can be no factor of safety to provide for poor and unknown
material and defective workmanship.
The factor of safety adopted for dead loads varies but little with the same kind of ma-
terial ; but, for live loads, the factor varies not only with the material, but with the char-
acter of the stresses, whether they are applied and relieved gradually or suddenly ; whether
they only vary in intensity, or also in direction, alternately compressive or tensile. In
this latter case the load should never be considered less than the sum of the stresses, with
a large factor of safety. Vibrations, shocks, and changes in the direction of stresses, con-
centrate the strains at the weakest point of the construction, and rupture takes place at
these points, which would be adequate to the strain if the form throughout were uniform
with that at these points. Thus, boiler-plates show wear just at the edge of the lap of
the sheets, and car-axles (Fig. 390), with sharp angles at the journals, are known to break
after a time, while under the same stresses an axle of uniform size with the journal would
not break; nor if a slight curve or rim inch radius (Fig. 391) be made in the angle to
distribute stress.
Besides provisions for strength, the draughtsman should understand the necessities of
the construction, and the character of the material to be used. He should know what
parts of the design are to be forged, cast, framed, and how it is to be done. He should
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 221
know what wear is to be met, and what waste, as rust or rot, to be provided for. This
knowledge can only be arrived at by reference to examples of practice and by observation
of results under similar conditions of use and time.
The stresses to which constructions and parts of constructions are subjected are the
tensile or stretching stress, tending to lengthen a body in the direction of the stress ; the
compressive or crushing stress, tending to shorten a body in the direction of the stress;
the shearing or cutting stress, tending to elongate, compress, and deflect; the torsional or
FIG. 390. FIG. 391.
twisting stress, the effect being an angular deflection of the parts of the body ; and the
transverse or lateral stress, tending to bend the body or break it across.
At page 195 is given a table of the strength of various metals to resist compression and
tensional stresses, and examples will hereafter be given of varied constructions, with their
usual or required factors of safety ; but, for a practical rule for the common necessities of
the above stresses, under dead loads, 10,000 pounds per square inch for wrought-iron may
be considered perfectly safe.
Posts in structures are subjected to compressive stresses ; but, as the action is modified
somewhat by a tendency to bend, depending on the proportion of the length to the diame-
ter, and the material of which they are composed, the usual tables of crushing strength
are not generally applicable, and the formulas to be depended on are those deduced from
practical tests. The best tests of wooden posts are those made by Professor Lanza, for the
Boston Manufacturers' Mutual Fire-insurance Company, and the following are the results :
" That the strength of a column of hard pine or oak, with flat ends, the load being uni-
formly distributed over the ends, is practically independent of the length, such columns
giving way by direct crushing, the deflection, if any, being very small. Tests were on
columns 6" to' 10" diameter x 12 feet. The average crushing strength of very highly-
seasoned, hard pine was 7,386 pounds per square inch. Some very slow-growth and
highly-seasoned, 9,339 pounds; very wet and green, 3,015 pounds; seasoned about three
months, 3,400 pounds; not very well seasoned and not very green, 4,400 to 4,700 pounds.
The average of two specimens of thoroughly-seasoned white-oak, 7,150 pounds; for green
and knotty, average, 3,200 pounds. Spruce, nearly 5,000 pounds. Whitewood, 3,000
pounds.
"That it is a mistake to turn columns, taper, or even turn them at all, square columns
being much stronger, cheaper, and better, and that accuracy of fitting is of great conse-
quence, that the stress may be directly vertical." The professor recommends that longitu-
dinal holes be bored through the center of columns to allow of the circulation of air (in
the experiments the holes were I'l" diameter), and that iron caps be used instead of wooden
bolsters, as the wooden bolster will fail at a pressure far below that which the column is
capable of resisting, and the unevenness of pressure brought about by the bolster is some-
times so great as to crack the column. He also recommends horizontal holes in the iron
caps to connect the longitudinal ones in the column with the outer air.
From the whole of the experiments, we estimate the safe load, for fair-grained, well-
seasoned oak or yellow-pine columns to be from 1,000 to 1,500 pounds per square inch ; for
the more imperfect and green specimens, from 300 to 500 pounds ; for good specimens of
whitewood, about 300 pounds ; and of spruce, about 500 pounds.
Cast-Iron. For the columns of buildings where the load is dead, cast-iron is very gen-
erally used. They are, in interiors, mostly of circular section, but for outer columns forms
are used suited to the necessities of their position or style of architecture. They admit of
222
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
considerable ornamentation and finish direct from the mold ; but, as they are liable to de-
fects not readily detected in the process of casting, the factor of safety is usually taken as
high as 5. To protect them against the effects of fire and water in conflagrations, they
are often covered with an outer shell of cast-iron or plaster, or of both.
The experiments of Hodgkinson are the usual basis of all formulae on the strength of
circular cast-iron columns, and the ends of all columns are now required to be faced by
architects and by the rules of building departments, since Mr. Hodgkinson states this rule,
that " in all long columns, of the same dimensions, the resistance to fracture by flexion is
three times greater when they are flat and firmly bedded than when they are rounded and
capable of moving."
Table of the safe load of solid cylindrical columns, with flat ends calculated with a fac-
tor of safety of 5.
TABLE OF SAFE LOADS FOR SOLID CAST-IRON COLUMNS, WITH FLAT ENDS.
Diam.
8'
1,000
Ibs.
9'
1,000
Jbs.
10'
1,000
Ibs.
11'
1,000
Ibs.
12'
1,000
Ibs.
13'
1,000
Ibs.
14'
1,000
Ibs.
16'
1,000
Ibs.
16'
1,000
Ibs.
17'
1,000
Ibs.
18'
1,000
Ibb.
19'
1,000
Ibs.
20'
1,000
Ibs.
21'
1,000
Ibs.
22'
1,000
lb.
23'
1,000
Ibs.
24'
1,000
Ibs.
8"
29-
23- j 20-
IT-
14-
13-
ii-
10-
9-
8-
7-
T-
6-
6-
5-
5-
4-
3*"
40-
81-
26- 22-
19-
17-
is-
13-
12-
If
10-
9-
8-
T-
7-
8-
6-
8?"
SO-
41-
84- 29- 25-
22-
iy
17- 15-
14-
12-
ii-
10-
10-
9- S-
8-
3J"
63-
54"
43- 37" 32-
23-
24-
2-2- 19' 18-
16-
is-
13-
12-
11- ! 11
10-
4"
77-
66- 54- 46 ; 40'
35-
31-
27- 24- 22-
20-
18-
17- 15-
14- 18-
12-
44"
li-
80- i 70-
57- ; 49-
4:3-
38-
34- 80- 27"
25
23-
21
19-
18-
16-
15-
4i"
no-
96- j 84-
74- 61'
53-
47" 41-
87'
S3"
30-
28-
25-
23-
22'
20-
19-
4*'
130-
113- 99-
88- ! 73-
64-
56-
50- 45-
41-
87-
34
31-
28-
ze-
24-
23-
5
152-
183-
117-
103- ] 92'
77"
68- 60'
54'
49-
44-
40-
37-
34-
st
29-
27'
54"
1715-
154- 136- 1 121-
10S'
97"
81- 72-
64-
8-
53-
48-
44-
40-
37-
85-
32-
4'
201-
177- 157- 140-
125-
113
95'
85-
76-
68'
62"
57'
52-
48'
44-
41-
38-
5J"
230-
203- 180- 161-
144-
13J-
US' 99- I 89-
80-
73-
66'
01-
se-
52-
48-
45-
6 '
260-
2W- 205- 183-
165-
149-
135-
115- 1(13-
93'
84'
77'
71-
es-
60-
56'
52-
tij"
2J2-
260-
232- i 203-
187'
169-
154' 140' lilt-
108-
98-
89-
82"
75-
69-
64'
60-
*'
827-
292" 2H1- 234-
212-
192-
174- ! 159- 146-
124'
112-
102-
94*
86-
80-
74-
09-
3"
304-
326'
292- i 263- 238'
216-
197-
180- 165-
141-
128'
117-
107-
99-
91-
85-
7'J-
7'
404-
362-
325- 293-1 266'
242-
221-
202- ! 186-
ni-
146'
133-
122-
112-
104-
96-
90-
74'
445-
400-
861-
326-
296-
269-
246- 226-
208-
192-
177-
151-
138- 127'
IIS- 109'
101-
If'
489-
441-
3)8-
861-
328-
299-
274- 251-
231- 214-
198-
170- 15(5- 148-
183- 123- 114'
"*'
536-
4S4-
438-
398-
862-
331-
303-
278'
257' '2:;7-
220'
294' 1 175-
Iftl-
149- 138- 128-
8''
584-
529-
480-
436-
398-
364-
334-
80S'
284- 263-
244"
227-
i '.); iso-
167- 155-
144'
^i '
689-
626-
571-
521-
477'
437-
402'
871-
343-
818-
296-
275-
257- 241-
2(17' 192- 178-
9 '
802-
733-
670-
614-
564-
519-
479
442'
410-
881-
3^4- 331-
309- 290-
272- 235- ' 218-
*''
926-
849-
780-
717-
660-
609'
563-
522-
484- 451-
420- 393-
867- 34.V
824- 305- 265-
10 '
1058-
975-
898-
829-
765-
708'
650-
61)9-
566- 528-
493' |4G1-
432-
406-
382- :36U- 840-
10*"
1195-
1108-
1026'
957'
892-
848-
779- 740-
693- 658-
610- 580-
546- 511-
485- 459' 1433-
11"
1359-
1264-
1159-
1083-
1017-
950'
889' 846'
793- 7. r )l-
703-
665'
627-
589-
561- 542- 513-
11*"
1517-
1413-
1319-
1226-
1147-
1080-
1018- 956-
904 '
852-
810- 758-
727-
691-
655- (518- 587-
12''
1674-
1583- ! 1470-
1880-
1289- 1221-
1142- 1074' 1018-
973-
916-
871- 746" 701-
667- 645-
Gil'
1 i i
Solid columns are very seldom used in constructions; they are almost invariably made
hollow, the shell being i" to 2" thick. To determine the safe load of a hollow column, it
will be sufficiently accurate to take from the table the safe load of a column equal to that
of the exterior diameter, and subtract from this the safe load of a column of a diameter
equal to the core.
Example. To find the safe load of a column 12 feet long, 8" exterior diameter, shell ".
Safe load of 8" column 398,000 Ibs.
" " u 6V " 212,000 "
" " required column 186,000 "
For square box-columns, it will be safe to estimate that a square column will support as
much as a round one, the side of the one being equal to the diameter of the other, and the
thickness of shell the same.
For a star-column (Fig. 392), the load should be about less than on a cylindrical col-
umn of same diameter and same area of section.
Wrought-Iron Columns. With the decrease in the cost of the manufacture of shapes in
wrought-iron, columns of this material have largely superseded those of cast-iron in con-
MACHINE DESIGN AND MECHANIC
structions liable to varying loads and shocks. Fig.
column, Fig. 394 of the Piper, Fig. 395 of the Keystone
The Phoenix columns vary in the number of segments,
223
ws the section of a Phoenix
FIG. 393. FIG. 394.
TABLE OF PHOENIX COLUMNS.
FIG. 395.
MAEK OF COLUMN.
Thickness in
inches.
Area in square
inches.
Weight in pounds
per foot.
Internal diam-
eter.
A
,
2'8
9'3
4 segments
A
5'8
19'4
*|
B
A
5'0
16-7
4 segments.
A b
14'8
51
*tf
A
5-8
19'4
4 segments
f
17'
58'6
6tt
C
A
8'8
30-3
4 segments. . . .
IS
40*
138
*A
D
1
14-0
48'2
5 segments. .
4
26-
89'7
H
B..? :"::'.:
i
16'
55'2
6 segments. . . .
il
60'
207'
11
F
4
24'5
84'5
7 segments
A
36'4
125'6
13
G
A.
24-
82'8
8 segments
IX
80-
276-
Uf
TABLE OF PIPER AND KEYSTONE COLUMNS.
4-iNcu COLUMN.
6-iNCH COLUMN.
8-iNCH COLUMN.
10-INCH
COLUMN.
Piper.
Keystone.
Piper.
Keystone.
Piper.
Keystone.
Piper.
Keystone.
Area, Weight
Area,
Wight Area,
Weight Area,
Wight
Area,
Weight Area,
Wight
Area,
Weight
Area,
Weight
sq. in. per ft.
sq. in.
per ft.
sq. in.
per ft. sq. in.
per ft.
14. to.
per ft.
sq. in.
per ft.
sq. in.
per ft.
sq. in.
per ft.
A
5-2
17-4
5-6
18-7
i
6' 20-
7-3
24-3
7-1
23-8
11-
36-6
9'8
32-6
T^g-
6-8 22-7
8-4
28-1
8-7
28-9
12-5
41-7
11-8
39-3
16"
53-3
14-2
47-4
|^
7-6 25-3
7-1
23-7
9-0
31-8 10-2
34-
14-
46-8
13-8
46'
17-9
59-7
16-6
55-3
A
8-4 ; 28'
8-2
27-3
10-7
35-6 11-7
39-1
15-6
51'8
15-8
52-8
19-8
66-
18-9
63-1
i
9'3
30-9
11-8
39-4 13-3
44-2
17-1
56-9
17-9
59-5
21-7
72-3
23-7
78-9
A
14-8
49-3
18-6
62-
19-9
66-2 j 23-6
78-7
26-
86-7
f
16-3
54-4
20-1
67-1
21-9
72-9 25-5
85-
28-4
94-6
H
23-9
79-6
27-4
91-3
30-7
102-4
25-9
86-4
29-3
97-7
33-1
110-3
if
35-5
118-2
Figs. 396-399 are sections of box-columns; the covers of 398 and 399 must be made in
short pieces, to admit of the inside riveting, and with close butt- joints to preserve the
strength. The thickness of the webs should exceed ^ of the width, to prevent buckling
under stress.
224 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
FIG. 396.
FIG. 397.
FIG. 398.
FIG. 399.
. /-\ r^ r\.
IF
FIG. 401.
FIG. 402.
FIG. 403.
FIG. 404.
FIG. 405.
FIG. 406.
FIG. 407.
FIG. 408.
FIG. 400.
FIG. 409.
FIG. 410.
FIG. 411.
Fig. 400 shows the elevation and section of an open or lattice column, common in bridge
and railway work. In estimating strength by area of section, in lattice- columns, the areas
of continuous support, as of the channel-irons, a 5 and c d in the figure, are only considered.
Figs. 401-403 are sections of other open columns.
Figs. 404-411 are sections of various forms of made-up columns.
The caps and bases are usually of cast-iron and molded to the requirements of po-
sition.
On the Strength of WrougJit-Iron Columns. The upper curve, Fig. 412, represents
graphically the average breaking load, taken from experiments on the Phoenix, Keystone,
Piper, and open columns, with flat ends. Horizontal distances give the proportions of
lengths of columns to diameters, or - , the vertical distances the loads in pounds.
diameter'
The lower curves represent the safe loads, under factors of safety of 3, 4, and 5. In look-
ing at these curves, it will be observed that, within the common limits of practice, of 15
to 35 -^- - , these lines may be considered straight ; that with iron of a breaking
strength of 52,000 pounds per square inch, and within the above limits, and a factor of
safety of 3, the safe load may be taken at 11,000 per square inch ; with a factor of safety
of 4, at 8,000 pounds ; with a factor of safety of 5, at 6,500 pounds ; and that for common
and usual purposes 10,000 pounds per square inch is a safe load.
It has generally been considered that columns with pin or cylindrical ends had about
f of the resisting strength of flat ends, but if the pin-ends are closely fitted, so that the
strains are uniformly in the direction of the length of the column, the difference is but
little between the two kinds of ends.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
225
The sectional areas of I, channel, and angle irons, of which the above posts are com-
posed, will be given hereafter.
50'
Shearing Stresses. Parts of machines and of constructions subjected to these stresses
have often the resistances modified by friction, combined with other stresses. The sizes
of parts necessary to resist such stresses practically, as in the cases of bolts, rivets, and the
like, will be hereafter illustrated by examples and determined by particular rules. In
general, the strength to resist shearing stress is, in wrought-iron and steel, from 70 to 80
per cent of its tensile strength ; in cast-iron, about 40 per cent of its crushing strength.
The softer woods, as spruce, white pine, hemlock, resting on walls or girders, will safely
sustain a load of 200 to 300 pounds per square inch of bearing surface, and the harder
woods, as oak and Southern pine, 300 to 500 pounds. By experiment, oak treenails, 1" to
If" diameter, were found to have an ultimate shearing strength of about two tons per
square inch of section ; but, according to Rankine, the planks thus connected together
should have a thickness of at least three times the diameter of the treenails. In 3" planks,
If" treenails bore only 1'43 tons per square inch of section; in 6" plank, l'T3 tons.
Torsional Stress. Every shaft through which power is transmitted, whether through
gears, cranks, or pulleys, is subjected to a torsional stress, of which the power acting tan-
gentially to the shaft in one direction is resisted by the load in an opposite direction.
When this stress exceeds a certain limit depending on the material, the fibers are twisted
asunder, but ranch below this limit the elasticity of the shaft may be too great to transmit
power uniformly.
The length of the axle subjected to torsion does not affect the actual amount of press-
ure required to produce rupture, but only the angle of torsion which precedes rupture,
and therefore the space through which the pressure must be made to act.
A torsional deflection of 1 in a length equal to twenty diameters of the shaft, is a good
working limit of deflection that is, -yfa part of a full turn. D. V. Clark gives the follow-
ing rule: "To find the diameter of a shaft capable of transmitting a given torsional stress
within good working limits. Divide the torsional stress in foot-pounds by 18'5 for cast-
15
226
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
iron; 27'7 for wro light-iron ; and 57'2 for steel. The cube root of the quotient is the
diameter of the shaft in inches.
Example. On the teeth of a 4^-foot gear, the force exerted is 2,800 pounds. What
should be the diameter of a wrought-iron shaft to transmit this force safely?
The torsional stress will be 2,800 pounds multiplied by the radius of leverage, 2J feet,
6,300
or 6,300 foot-pounds = -? == 228, ^228 = 6-1.
27*7
Transverse Stress. If a beam supported at its extremities be loaded with a weight, W,
Fig. 413, the beam is subjected to a bending movement, or transverse stress, composed of a
tensile stress on the lower part of the beam and compressive stress on the upper part, as
will be readily seen by the figure. In addition, the weight of the beam and its load, sup-
ported on the abutments, act at these points as shearing stresses.
FIG. 413.
FIG. 414.
The strength of a square or rectangular beam to resist transverse stress is as the
breadth and the square of the depth ; and inversely as the length, or the distance from or
between the points of support. Thus a beam twice the breadth of another, other propor-
tions being alike, has twice the strength ; or twice the depth, four times the strength ; but
twice the length, only half the strength.
It is evident, therefore, that, with the same area of section, the deeper a beam the
stronger it will be, if the breadth is sufficient to prevent lateral buckling.
To cut the best beam from a log, Fig. 414, the section of which is a circle : draw a diam-
eter, divide it into three equal parts, erect perpendiculars at the points of division 1, 2,
and they will intersect the circumference at the corners of the beam, of which the ex-
tremities of the diameter are the other two.
Q * j 2
For the transverse strength of rectangular beams the general formula is W = - , in
which W is the breaking weight ; S, a number determined by experiment on different
materials; 5, the breadth, and d, the depth in inches; and Z, the length in feet.
Figs. 415 to 422 represent the usual methods of loading beams, and the loads as drawn
represent the comparative strength of beams under these different conditions. Thus, in
FIG. 415.
V////A
FIG. 416.
Fig. 415, the beam supports but one unit of load, while Fig. 416 supports twice as much.
The formulae given represent the safe dead loads with a factor of safety of 6, deduced from
experiments of Mr. C. J. H. Woodbury on Southern pine. For spruce the co-efficient
would be about \ less, and for live loads the factor of safety should be 12.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
227
Beams fixed at one end and loaded at the other (Fig. 415).
Id*
Safe load = 30
I
Beams fixed at one end and load distributed uniformly, not as represented in the
figure, as the two units of weight would be spread over the whole length of the beam
(Fig. 416).
o cL
Safe load = 60 .
Beams supported at the extremities and loaded at the middle (Fig. 417).
Id*
Safe load = 120 .
FIG. 417.
FIG. 418.
Beams supported at the extremities and the load uniformly distributed (Fig. 418).
Id*
Safe load = 240 - .
Beams, one end firmly fixed, the other supported, and loaded at the middle (Fig. 419).
t -j g
Safe load = 160-.
I
^5^
^^
5 V r /'/ / /V
''/fty
^^
'/"'
yy>'yfl
s^\^\
;|
1
/x x^
^~
FIG. 419.
FIG. 420.
Beams with one end fixed, the other supported, and load uniformly distributed (Fig. 420).
Id*
Safe load = 240 .
I
This formula, although given by good authorities, is evidently too small ; it should be
Id*
probably about 300 .
FIG. 421,
FIG. 422.
228
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
Beams with both ends fixed, and loaded at center (Fig. 421).
Safe load 240 = .
L
Beams with both ends fixed, and load uniformly distributed (Fig. 422).
Safe load = 360 -^.
t
If the load on the beam be neither at its center nor distributed as in Fig.
423, lay off on any convenient scale an inclined line, A C, between the abut-
ments, equal to the weight of the
load. Let fall a perpendicular from
the bearing-point of the load to this
line ; it will divide it inversely pro-
portional to the load on the abut-
ments. In the figure, the load is
1,200 pounds ; the perpendicular in-
tersects the scale-line beneath at 900 ;
900 pounds is therefore the load on
the abutment at B, and the balance
of the weight, or 300 pounds, on the
abutment A. To determine the size
of beam of uniform section to resist
the bending movements of the loads,
multiply the loads on the abutments together, and divide by one quarter of the
sum of the two loads. Thus, in the figure,
Fm. 423.
W'"'
= = 900 > the
load at the center of the beam, and the size can be readily determined by the
formula or tables given.
If the load is not distributed symmetrically, Fig. 424, the bending move-
ment and shearing stresses may be readily determined graphically. Let loads
equal to 100, 365, 850, and 125 pounds be supported as shown by the beam A B
(say, 12 feet). At one side, on a line a b, perpendicular to the beam, lay off on
any convenient scale, 100, 365, 850, 125, to represent the loads on the beam ; from
1, 2, 3, 4, 5 draw lines meeting at some point, C. The point C can be chosen
anywhere, but, for reasons that will be hereafter self-evident, it will be better to
take C at a horizontal distance C D of either 10, 100, 1,000, etc., measured on the
same scale as the loads on the line a 1). From the points of support of the
loads on the beam A B, let fall perpendiculars ; from any point C, on line A C /
draw the line C, I, parallel to C 1, 1, 2, parallel to C 2, 2, 3, parallel to C 3,
3, 4 / parallel to C 4, and 4, F, parallel to 5. Connect C, and F, and draw
the line C F parallel to this. The distance 1 F, measured on the scale of loads,
will give the reaction in pounds on the abutment equal to 530, and 5 F = 910
pounds will be the reaction on the other abutment B. These are shearing
stresses, and their sum in every case should equal the sum of the loads in this
case, 1,440 pounds. The point of greatest stress in the beam will be imme-
diately above the longest ordinate in the polygon C / 1, F, C,. In this case
it will be at the point of support of the 850 pounds, 3, 3,, being the longest or-
dinate in the polygon. This ordinate, 2*7, measured on the scale of the beam
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
229
FIG. 424.
multiplied by the horizontal ordinate C D (taken here at 1,000), will give 2,700.
This number, divided by 3, one quarter of the span A B of the beam, will give
the center load, equal to 900, for which the size can be determined by the
formula or tables as before.
TABLE OF THE SAFE CENTRAL LOAD OF YELLOW-PINE BEAMS, CALCULATED
FEOM THE FORMULA 120
bd*
I '
Span in
feet.
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
DEPTH IN INCHES OF YELLOW-PINE BEAMS, ONE INCH WIDE.
Sins.
tins.
5 ins.
GillS.
7 ins. 8 ins.
9 ins.
10 ins.
11 ins.
12 ins.
13 ins.
14 ins.
15 ins.
16 ins.
7680-
270-
480-
750- 1080-
1470' 1920-
2430"
3000'
3630'
4320' 5070-
5880-
6750-
216-
180-
154-
135-
120-
384-
600-
500-
864-
720-
616-
1176'
980-
840'
735-
1636-
1280-
1097-
960'
853'
1944'
1620-
1389"
1215'
1080-
2400-
2000-
1714-
1500-
1333-
2904-
2420-
2074'
1815-
1613.
3456-
2880'
2469'
2160-
1920'
4056"
3380'
2897'
2535-
2253-
4704-
3920'
3360'
2940-
2613'
5400'
4500-
3857-
3375-
3000-
6144-
5120-
4388-
3840-
3413-
320-
274-
240-
213-
430-
375-
333'
540-
480-
653'
108-
192'
175-
BOO-
STS-
432-
392-
588'
535-
768-
700-
972-
1200-
1092-
1452.
1320-
1728-
1571-
2028'
1844.
2352-
2140-
2700-
2457-
3072-
2793-
882-
160-
250-
360-
490'
640-
810-
1000-
1210-
1440'
1690"
I960-
2250-
2560-
230-
21S-
332-
308-
452"
420"
592-
648"
747-
693"
923-
860-
1117-
1328- 1560-
1234- 1448'
1808-
1680-
2070-
1928-
2363-
2192-
1037'
288"
392-
512-
648-
800-
968-
1155' [1352-
1568-
1800-
2048'
270-
254-
368-
346'
480-
452-
607'
566"
748-
704-
907'
854-
1080' 1267'
1016" 1193-
1470'
1688-
1588-
1920'
1808-
1384-
327-
427*
540-
668'
806-
960- 1126'
1307- 1500-
1707'
404'
384-
512'
486"
632-
600'
764-
726-
909'
864-
1067"
1014-
1238-
1176-
1422-
1350-
1616-
1536-
463-
572-
691-
823-
966-
1120-
1287'
1463-
442-
546'
660'
785' 922-
1070-
1228-
1395-
522-
631-
752- 882-
1023'
1178-
1329-
600-
605'
720' 845-
980'
1125-
1280-
581-
691- 811-
940-
1080-
1230'
558'
665- 780'
904-
1035' 1182-
230
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
This table is deduced from Mr. Woodbury's experiments on yellow pine,
of good quality and practical sizes. For spruce he takes loads of about one
fifth less.
The table is intended to be used as a unit by which the strength of timber of
usual depths and spans can be estimated, by multiplying by such widths as are
found in practice ; widths of less than two inches are not used. The strength
given in the table is in excess of the stiffness, and in permanent constructions
it is necessary to proportion the beam to bear its load with a certain limited
deflection. Mr. Woodbury established this limit in wooden beams at three
quarters of an inch for 25-feet span, and his formula is E = ] y in which
/ 1* CL
E, the modulus of elasticity per square inch is for Southern pine 2,000,000,
and for spruce 1,200,000 : W central load in pounds, I the span in feet, I the
breadth, h the depth, and d the deflection of beam, all in inches. Using this
formula, we have drawn marks in each column of depth, above which the
loads will be supported stiffly, and below less so than recommended.
It is to be observed that the formula is applicable to seasoned wood.
Wooden and wrought-iron beams are of uniform section for their entire
span, but cast-iron can be readily adapted in form to the load to be sus-
tained.
The forms of beams which afford equal strength throughout are parabolic
(Figs. 425, 426, 427), of which the axis A B and the vertex A are given, and
A
FIG. 425.
FIG. 426
the points M determined by calculations. Figs. 426, 427 are oftener used
when the force is applied on alternate sides of A B.
A beam subjected to a transverse stress, as shown in Fig. 413, one side is
compressed, while the other side is extended ; and therefore, where extension
terminates and compression begins, there is a lamina or surface, g h, which
is neither extended nor compressed, called the neutral surface. As the
strains are proportional to the distance from this surface, the material of
which the beam is composed should be concentrated as much as possible at
the outer surfaces, as can readily be done in beams of cast and wrought iron.
Acting on these principles, Mr. Hodgkinson has determined the most econom-
ical form for cast-iron beams or girders, of which the section is given (Fig.
428); it has been found that the strength of cast-iron to resist compression is
about six times that to resist extension ; the top web is therefore made only
one sixth the area of the lower one. The depth of the beam is generally about
one sixteenth of its length, the deeper of course the stronger ; the thickness of
the stem or the upright part should be from -J an inch to 1 inch, according
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
231
to the size of the beam. The rule for finding the ultimate strength of beams
of the above section is : Multiply the sectional area of the bottom flange in
square inches by the depth of the beam in inches, and divide the product by
the distance between the supports in feet, and 2 '4:2 times the quotient will be
the breaking weight in tons (2,000 pounds). As has already been shown
FIG. 428.
above, the section thus determined need only be that of the greatest strain,
and can be reduced toward the points of support, either by reducing the
width of the flanges to a parabolic form (Fig. 428), or by reducing the thick-
ness of the bottom flange ; the reduction of the girder in depth is not in
general as economical or convenient.
For railway structures subject to an impulsive force, Mr. Joseph Cubitt,
C. E., recommends that the section of the upper flange should be one third
that of the lower.
Fig. 429 is side elevation, plan, and section of cast-iron girder, adopted by
FIG. 429.
him for railway purposes, a pair of girders for each track, the rails being
supported on wooden cross-beams.
DIMENSIONS FOR DIFFERENT SPANS.
Opening.
Bearing on '
abutment
Height of girder
at center.
Top flange.
Bottom flange
at center.
At end.
Thickness of
middle web.
12ft.
30ft.
l'-6"
2'-6"
l'-3"
3'-
3" x H"
5" x 2"
l'-4" x If
l'-6" x 2"
l'-8" X 1|"
l'-10" x 2"
H"
2"
45ft.
2'-9"
3'-9"
7" x 2f
2'' x 2|"
2'' x 2f
2"
232 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
Some years since the bow-string girder was in very common use in this city
for span openings of from fifteen to twenty-five feet in the fronts and rears of
stores and warehouses. The bow was made of cast-iron, in a x-fora 1 ? and
the strings, or tension-rods, were of wrought-iron. In this composite struct-
ure it was impossible to calculate the strength of the girder, to decide how
much was borne by the bow and how much by the string. The strings were
forged with heads, and it was intended that the fit should be an easy one,
so that some compression should be put on the bow before tension should be
put on the rods. But, with the diminished cost of wrought-
iron, cast-iron girders have given way to rolled beams and box-
girders of wrought iron.
Rolled or I beams, Fig. 430, may be taken as the type.
They are made at many rolling-mills. The depths of the
beams and the widths, B ? of bottom and top flanges do not
vary much with the different makers for the same class of
beams ; the thickness of the stems varies somewhat more pro-
portionally. For each depth there are usually two weights
the light and heavy and are thus classed in the trade, as light
twelves and heavy twelves, and lighter or heavier weights may
be made to order.
There is considerable difference in the strengths of these beams as given in
the tables of the different makers : in the table on page 233 we have tried to
modify these discrepancies as far as possible, adopting that of no single maker ;
and to give dimensions such as will suffice for the purpose of the draughtsman
in illustration, with tables of strength which can be relied on as practical.
We have discarded the usual practice of stating strength in tons, and have
taken 100 pounds instead, so that 00 need only be added to the tabulated
figures to give the safe distributed load in pounds.
It is assumed in these tables that proper provision is made for preventing
the beam from deflecting sideways. They should be held in position at dis-
tances not exceeding twenty times the width of the flange, but this is usually
effected by the necessities of the construction, the brick arches between the
beams, or the wooden joists resting on them. The beams will support the
loads as given in the tables, but the deflection may be too much for the
purposes to be served. A line is drawn in each column in the tables, at
which the deflection is -j-J-g-, or one inch for every thirty feet of span, beyond
which, if the beams carry plastered ceilings, the deflection is apt to crack the
plastering.
A common formula for determining the strength of a wrought-iron beam
SV(a + ^)S
is W = - , in which W is the load in pounds, equally distributed
L
on the beam, D the effective depth between the centers of gravity of the flanges,
and L the clear span, both in the same unit, feet or inches ; a the area of the
top or bottom flange in square inches ; a' the area of the stem.
To find the sectional area of a beam-plate or rod from its weight, divide
the weight per yard by 10 ; and, conversely, to determine the weight per
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 233
^gSo-Ss:
) C3 Ci t O iO to I
CNCN (MC
SSg ^^SS S^^
*
I *
OOO iO'-^ OO i (CO SOO T^ Oi^O
-io cct- oo r^cis <N04 S ooo
> CM O CO CO Tfl
SCO C03S) (?)S
<H 31 ,_ ^H
I r t-O QO * IOCO O OS CO 00 CO OJ O <?J O5 O -rtM
iO t-iO cocsi i O CiOO cot- t-co O O O O O O
oo oo OI-H oo ot*r-<o oco CMOO OCM o
j^t- ooo i o osoo cot- t-co oo oo o
^ I 3T
V> OSCO CCrfi i-ih- OG-1 OC)0
t- ?OO OO O Tf -^ rj< ^ CO CO
h-<N OS O COonicOO O CO rH O t- O
0500 0?0 O^ ! T}<-* coco CO*) OJCN
'OOS COOS COCO O30 OO
OT -*CO COCO COCN CN.-N
w
Sg
g^ CS S SS SS r^
O0 t-00 00 rH<3 CO-* 00 b-00 OS!
rH <N SO ^C OOt-OOO>O r-i (M CO * O O t- CO OS
J CNC3(M CN<M<N<MCNCO CO CO CO OS CO CO CO CO CO
I
KI
30iIVJL8ICI
234
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
linear foot from the sectional area, multiply the area by 10 and divide the
product by 3.
Thus, if a bar a yard long weigh 40 pounds, its sectional area will be 4
Q \/ 1 A
square inches : and a bar of 9 square inches section will weigh - - = 30
pounds per foot.
For naval constructions, deck-beams, Fig. 431, are rolled at different mills,
from 3" to 12" deep, and varied widths of flanges and thicknesses of stem ; in
general, not quite up to the grades of heavy and light I-beams in weight, but
they can be rolled to order to any desirable dimensions within the limits of
depth given. Properly proportioned, they should be equal in strength to the
I-beams.
Coupled I-Beams. When the load is beyond the strength of a single I-
beam, two or more may be united, as shown in Fig. 432. A cast-iron block, or
FIG. 432.
FIG. 433.
FIG. 434.
FIG. 431.
FIG. 435.
FIG. 436.
FIG. 437.
FIG. 438.,
FIG. 439.
FIG. 440.
FIG. 441.
FIG. 442.
FIG. 443.
FIG. 444.
separator, is inserted between the beams, and two bolts, passing through them
and the block, add lateral strength. The bolt-holes, if placed at some distance
from the center of the span, do not reduce the transverse strength.
It is not unusual to strengthen an I-beam by the riveting of a plate on top
(Fig. 433). It adds to the areas of the flanges by the area of the plate, less
that of the rivet-holes in both plate and flange.
Box-girders are sometimes made up in the same way by two Fs and plates
across top and bottom (Fig. 434) ; but, as the access to the inside for holding
the rivets is usually impossible, channel-beams (Fig. 435) are preferred for
these forms, within the limits to which these beams are rolled.
Channel-beams can be furnished of depths the same as I-beams, from three
to fifteen inches, of varied grades of light and heavy, and within any desirable
limits of weight.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
235
TABLE OF DIMENSIONS OF CHANNEL-BEAMS IN INCHES.
DEPTH.
Web.
Thickness.
FLANGE.
Width.
Thickness.
3
2 to '3
1-51 to 1-61
i to fer
4
24 u "39
1-74
1-89
"i
5
25 " '55
1-93
2'23
6
23
53
1-98
2-28
A " f
7
30
55
2-30
2-55
A " f
8
30
75
2-30
2-75
f "it
9
31
71
2-43
2-83
1 " *
10
31
76
2-56
' 3-01
f "If
12
46
96
2-71
' 3-21
ft " l
15
53
93
3-53
' 3-93
i "i
It may be desirable, on account of position, to finish a box-girder as in Fig.
436 ; in this case the dimensions must be such as to admit of a helper inside
to hold the rivets. Fig. 437 shows a closed box-beam made of channel-bars
and plates. The lower channel is first riveted, and the upper one afterward.
This form gives a clean surface below, but the lower channel-bar can be re-
versed and riveted the same as the upper.
Where the purpose can be served by I-beams, either single, or coupled,
as in Fig. 432, or in numbers, they afford the best and cheapest construction.
But, where the spans are large and loads heavy, it is often economical to obtain
greater depth by means of plate-girders, as in Figs. 438, 439, 440, 441, or per-
haps from requirements of position, as in Fig. 442, subject as above to the
necessities of large inside dimensions. These girders are made up of plates
of uniform thickness, and angle-irons riveted together.
Angle-irons are made of varied dimensions, and are classed as equal-legged
(Fig. 443), unequal-legged (Fig. 444), and square-root angles when the thick-
ness of the iron is uniform throughout, and consequently the interior angle a
complete right angle without rounding. The following table gives the dimen-
sions and weights of the angles to be found at different mills, but weights can
be increased to order :
ANGLE-IRON. WEIGHT IN POUNDS PEE FOOT.
SIZE, INCHES.
AVERAGE THICKNESS.
t"
A"
"
&"
t"
A"
*"
A"
f"
B"
i"
H"
1"
EQUAL LEGS.
6 x 6
19'2
21-7
24-2
26-7
29-2
31-7
34-95
4x4
9-5
11'2
12'9
14-5
16-2
17-9
19'5
3 x 3
8-3
9-7
11-2
12'7
14'1
15'6
17-0
34 x 34% .
7-7
9-0
10-4
11-7
13'1
14-4
15'8
3x3
5-9
7-2
8-4
9'7
10'9
12-2
2 x 2
5-4
6-5
7'7
8'8
24 x 2
4'9
5-9
7-0
8-0
24* x 24-
3-5
4'5
5'4
6'4
7'3
2x2
3-1
4-0
4-8
5-6
1& x 1&. .
2-1
2-8
3-5
4-3
5'0
14 x 14
1-8
2-4
3-0
3-6
14 x 14..
1-0
1-5
2-0
14 x 14
0-9
1-4
1'8
1x1
0'8
1-2
1-6
& x .. ,
0'6
0-9
236
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
ANGLE-IKON. WEIGHT IN POUNDS PEE FOOT.
SIZE, INCHES.
iVERAC
E THI
CKNE88.
1"
A"
*"
&"
1"
A"
i"
A"
1"
tt"
i"
W
r
UNEQUAL LEGS.
6 x 4
14-6
16-6
18'6
20-6
22-7
24-7
26-7
28'7
6x4
13-9
16'0
18'1
20-2
22-3
24-4
26 4
28'4
6 x 3
11-3
13'2
15'0
16-6
18-4
20-2
22-1
5 x 4
10-8
12'7
14-5
16'4
18-3
20-2
22'0
5x3^..
10-2
11'9
13-7
15'5
17-2
19'0
20-8
5x3..
9'5
11-2
19.-9
14-5
16'2
17'9
19'5
4 x 3-J
8-9
10-5
i?,-o
13'6
15-2
16'7
18'3
4x3..
8'3
9'7
11-2
12'7
14-1
15-6
17'0
3^ x 3
7-7
9'0
10'4
11-7
13'1
14-4
15-8
3i x 2
4-2
5-3
6-4
7'4
8 - 5
3x2^
4'4
5-5
6-7
7-8
9-0
3 x 2
4-0
5-0
6-0
7*1
8'1
2 x 2 ...
3-5
4-5
5-4
6'4
7'3
21 x 14k .
2-5
3-0
3'8
4-5
2 x If
2-0
2-6
3-3
4-0
T-irons (Fig. 445) may be used for top and bottom flanges
in the manufacture of plate-girders, by riveting a web on one
side of the T, or on both sides, with a separator between of the
thickness of the stem E ; but, as the areas of section of T-irons
to be had are small,, the flanges will be too slight in propor-
tion to the webs at depths above that of rolled beams. Angle-
irons are then to be preferred for flanges. The T-irons are well
adapted in many positions as struts or braces, and can be bought of varied
dimensions and weights, from widths, B, of from 2 to 5 inches, and equal or
less depths, A, and thicknesses from -f$" to f ".
Rivets for plate-girders are usually from " to " diameter, and pitched or
spaced not more than 6" nor less than 3" between centers. The number of rivets
through flange and stem are the same, but alternating. Usually angle irons and
plates can be had of the full length of girder, but, where joints are necessary,
they should be butt, with a splicing-piece to make the strength as nearly as pos-
sible uniform. Stiffeners are often necessary for the webs, which may be of
band, angle, or T iron, and one should always be placed at each end, where the
shearing stress is the greatest.
To construct a diagram from the formula, W =
8 D (a + ) S
in which
the relation of the factors may be shown. Let S be 10,000, on account of loss
of strength by rivet-holes, then W = X (a + -) 80,000. On a sheet of cross-
L 6
section paper, from a corner, 0, lay off on the line of ordinates, 5, 10, 15, 20,
25, representing the factor a + -. From the same 0, on the line of abscissas,
D D
i iV> -h, ih>> ih, A> A, iV> representing . Suppose -- = &, thenW =
/ , _L Lt
(a+ -) 2,000. If a + - be = 10, then W = 20,000. From the intersection of
6 6
ordinate on line of 3^, and abscissa line of 10, draw a line to the point 0. This
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
237
line will represent the safe distributed load W, and its intersections of the or-
dinates and abscissas will represent the relative proportions of the two factors
- and a -\ under this load. On the abscissa line 15, and ordinate fa, AY
L 6
= 30,000, on line 20, 40,000, and so on, and lines drawn from these intersec-
tions to will represent W.
Fig. 446 is thus constructed, but lines below 5 and above 30 on line of ordi-
nates are erased, as within these limits may be found most of the proportions
required in practice.
25
20
v
/s
y
/IS
20 /25
FIG. 446.
30
"We should recommend to every draughtsman who needed this sort of table
to construct one for himself on cross-section paper. ,
Application of the Diagram. What will be the area of section a+ -- of a
girder, 40-foot span, depth 32", distributed load 90,000 pounds?
D in the formula represents the distance between the centers of gravity of
the flanges, which will be somewhat less than the depth of beam. Approxi-
mately we assume it at 30", = - = -jV, and tne intersection of the line
L 30" ,
of load, 90,000, with the ordinate -fa, will be 18, on the line of a + -. A fair
6
proportion of a to a' is 5 to 6, therefore + - = 18 or a' = 18. = 0. 6" =
66 30
thickness of web, and a = f of 18 = 15, or weight per foot of one flange
-- = 50 pounds, which is slightly in excess of the weight of two angle-
3
irons 6 X 4 X f, compensated by thickness of web outside centers of gravity.
238
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
This calculation is sufficiently near for all practical purposes, but D can be
found more accurately by plotting the angle-irons as above, on thick card-
board, cutting out and then balancing for the center of gravity.
Composite Beams. Often, in constructions where the beams or girders are
of wood, and on account of extent of spans and loads, the stress is beyond the
strength and stiffness of beams of this material, of readily available dimen-
sions, it is usual to supplement by some application of iron. A simple form, in
which the iron is not exposed to view, is by bolting a plate or flitch of wrought-
iron between two beams, of the full length and depth of the beams, and of
such thickness as may be necessary. In bolting them together, let the bolt-
holes be so bored that the weight of the beam may primarily be on the wood ;
the stress will then be better adjusted between the two materials when in ser-
vice. It is usual to make the holes zigzag, in two lines about one quarter the
depth of beam from each edge, the holes closer together nearer the ends. The
safe-distributed load for the iron may be estimated from the formula : W. =
, b breadth, h depth, I length all in inches.
Fig. 447 represents a bracing truss of wrought-iron between two beams,
which should be let into the wood. As it is held firmly laterally, the factor of
FIG. 447.
safety may be considered about one third of the crushing resistance of the ma-
terial. The load on each inclined bar will be one half the load on the center,
multiplied by the length of the bar and divided by the rise. Instead of wrought-
iron, cast-iron or wood is used.
In Fig. 448 the beams are strengthened by a tension-rod, of which the
strength may be determined by that of the material ; allowing the usual factor
FIG. 448.
of safety, the load is obtained as in the example above. The deeper the block
beneath the center of the beam, the less the stress on the rods for the same
load. In construction, the beam should not be cambered by the screwing up
of the rod ; but, if the beams are crowning, the convex side should be placed up-
ward, the nut turned by hand just to a bearing, and the tension put on by the
settlement of the beams under the load.
Fig. 449 represents the trussing of a beam by two struts and a tension-rod.
The stress on the tension-rod is the load on c, multiplied by the length a d,
divided by c d.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
239
FIG. 449.
The theory of trusses will be treated and illustrated under "Bridges" and
"Roofs," and the proportions of rivets and forms of plate-iron joints under
"Boiler Construction."
Bolts and nuts are of such universal application that their manufacture
forms the center of large industries. Much thought has been given to their
FIG. 450.
proportions and the forms of thread, but without producing complete uni-
formity in the practice of different countries and makers. The old form of
thread was the A or sharp pitch (Fig. 451), still used by some, especially when
the threads are cut in a lathe. In this country the standard U. S. thread is
FIG. 451.
that recommended by the Franklin Institute in 1864 (Fig. 452). The angle is
60, with straight sides and flat surface at top and bottom, equal to one eighth
the pitch, or distance from center to center of threads.
In England, the standard thread for bolts and nuts is the Whitworth (Fig.
453) ; the angle is 55, with top and bottom rounded.
Z.oo
FIG. 455.
The square and rounded threads (Figs. 454 and 455) are only made to order
and used in presses and the like as parts of machines.
240
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
Figs. 456, 457, and 458 represent the proportions of the various parts of
English nuts to the diameters of bolts, as 1, or unity. Fig. 457 is a flange-
nut, in which a washer-like flange is forged with the nut.
R
1
FIG. 456.
FIG. 457.
FIG. 458.
Fig. 459 is a cap-nut, in which the thread does not go through the nut, to
prevent leaking along the thread, and a soft copper washer is introduced to pre-
vent leakage below the nut.
Figs. 460 and 461 are circular nuts, in one of which holes are drilled to
insert a rod for turning, and in the other grooves for a spanner.
FIG. 459.
FIG. 460.
FIG. 461.
FIG. 462.
Lock-nuts (Fig. 462) are intended to prevent the gradual unscrewing of
nuts subjected to vibration, which is to a great extent prevented by the use of
double nuts, the lock-nut being one half the thickness of the common nut.
The usual practice is as shown, the lock-nut being outside ; the better way is
inside.
The following figures are from trade circulars ; the limits of sizes given are
such as can usually be found in stock.
Figs. 463, 464, and 465 are machine-bolts, from i" to f" diameter, and 1"
FIG. 463.
to 4" long, but not flanged, as in Fig. 463, unless expressly ordered -, the dot-
ted line shows the radius of curvature of a finished head. The diagonal lines
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
241
beneath the head (Figs. 464 and 465) represent square bolts tapering into round
bolts, as shown by the curved lines.
Fio. 464.
FIG. 465.
Figs. 466 and 467 are tap-bolts and set screws, from i" to f " diameter, and
from 1" to 3" long.
FIG. 466.
FIG. 467.
Fig. 468 is a carriage-bolt, from %' to f " diameter, and from 1" to 16" long.
Fig. 469 is a plow-bolt, from f" to " diameter, and from 1" to 4" long.
FIG. 468.
Fig. 470 is a stove-bolt, from " diameter and from f" to 3" long.
Figs. 471 and 472 are machine-screws without nuts ; the holes in the metals
are tapped to receive them ; Fig. 471 is button-headed ; Fig. 472 a counter-
sunk head both slotted to admit of driving by a screw-driver. They are
M
Fia. 470.
FIG. 471.
FIG. 472.
T
FIG. 473.
FIG. 474.
FIG. 475.
made of various sized wire and lengths, and sold by the gross like the common
wood-screw (Fig. 473). The wood-screw is for connecting pieces of wood to-
gether, or metal to wood. They are of very great variety, usually with a
gimlet-point, so that they can be driven into the wood, without any holes being
previously made. When made of rods, with a square or hexagonal head (Figs.
16
242
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
474 and 475) to admit of the use of a wrench, they are called lag-screws. It
will be seen that wood-screws differ in their thread from bolts and machine-
screws. The thread is a very sharp V, flatter on the upper surface, and the
flat space between the threads wide as the thread, making it of easier introduc-
tion into the wood, and retaining as much strength in the iron as in the wood.
Fig. 476 is a stud-bolt, which is screwed firmly into one of the pieces of
connected metal ; the other is bored so as to slip over the bolt, and the nut
then brought down upon it. It is in common use for holding on the bonnets
of steam-chests and water-chambers, the bolt remaining permanent.
FIG. 476.
FIG. 478.
Fig. 477 is a hook-bolt ; it relieves the necessity of a bolt through the bot-
tom-piece, and may be turned like a button, to loose or hold the bottom-plate.
Fig. 478 is another kind of button -bolt ; the lower end can revolve on a
stud or pin if the nut be raised enough to clear the cap or upper plate. By
this arrangement there is no necessity of taking off the nut entirely ; the bolt
lies in a slot in the cap, and the nut bears on three sides.
FIG. 479.
FIG. 480.
FIG. 481.
Figs. 479, 480, and 481 show expedients to prevent the bolt from turning
when the nut is screwed on or off.
FIG. 483.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
243
Fig. 482 is an anchor-bolt, flattened and jagged, introduced into a hole in
masonry, and then leaded or sulphured in ; but the more common way is to
split the lower end of the bolt, insert a wedge into the cleft, place the bolt in
the hole, and drive the wedge in against the bottom of the hole, thus keying
the bolt in the hole.
Fig. 483 is a bolt with a fang-nnt or corner turned down and driven into
the wood to prevent turning ; the screwing to be done at the head.
It is often convenient to use bolts with two nuts, as in Fig. 484, or collar-
bolts, which are readily made to order, and of any dimensions.
Fig. 485 is a hanger-bolt ; the lag-
screw part is screwed into the wooden
beam, the hanger then put over the bolt,
and the nut put on.
Figs. 486 and 487 represent forms of
turn-buckles, and the swivel and pipe, sometimes designated as swivels. Turn-
buckles are very useful in straining tierods, where neither end of the bolts can
be got at. By turning the buckle, the rod can readily be made longer or
shorter. In the pipe-swivel, rigid and left threads are cut on the bolts, so
that each turn of the pipe shortens or lengthens the tie by double the pitch of
the screw. The turn-buckle is also made in the same way, with two screws
instead of a head at one end.
FIG.
FIG. 485.
FIG. 487.
Screws, unless otherwise ordered, are made right-handed ; that is, turning
the nut to screw up, the hand moves from left to right, the apparent motion of
the sun.
On the Strength of Bolts. The strength of a bolt depends on its smallest
section that is, between the bottom of the threads. It is very common,
therefore, especially in long bolts, to upset the screw-end, so that the screw may
be cut entirely from this extra boss, or re-enforce. Bolt-ends (Fig. 488) are
sold either with or without re-enforce, to be welded to bolts. It will be
observed that the ends of the pipe-swivel bolts (Fig. 487) are thus upset.
FIG. 488
In the following table, the sizes and dimensions "of bolts and nuts are from
the United States standard, and the strength, or safe-load of the bolts, is
computed from the report of the committee on the test of wrought-iron and
chain-cables to the United States Government in 1879. Nuts and heads as
furnished are either hexagonal or square. Columns 4, 5, and 6 apply equally
to either.
244 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
Diameter of
screw in
inches.
Diameter at
root of
thread.
Thread per
inch of
length.
Short diam-
eter of nut
and head.
Thickness of
nut.
Thickness of
head.
Safe-load of
upset bolts.
Safe-load of
plain bolts.
i
TV
i
A
f
185
20
3
TIT
if
t
240
294
18 if
16 ii
t
H
1,700
P
344
400
13 r
TV
i
i!
TV
1,900
2,200
TV
494
12 f
TV
If
2,500
f
507
11
1A
f
2,800
f*
620
10
H
ii
ii
f
If
Is
6,000
3,200
3,600
I 1
731
9
Hi
1A
H
Ii
7,000
8,000
4,300
5,100
1
837
8
if
i
ii
10,000
7,000
li
940
7
lit
il
14
12,000
9,000
li
1-065
7
2
il
i
15,000
11,000
H
1-160
6
2A
if
iA
18,000
13,500
li
1-284
6
2f
IA
21,000
16,000
If
If
1-389
1-490
51
5
It
if
if
|r
24,000
28,000
19,000
22,300
11
1-615
5
ii
32,000
25,500
2
1-712
4|
8f
2
IA
36,000
29,300
21
8 A
21
114
40,000 33,000
2i
1-962
4*
81
If
45,000
37,000
2f
3ii
2f
iff
50,000
41,500
2i
2-175
4
31
21
55,000
46.000
2f
4rV
2f
2^L
2f
2-425
4
41
2f
2 i
21
4A
21
2^ 2
3
2-629
81
4|
3
^A
81
2-879
81
5
31
2 i
81
3-100
31
5f
3i
2^ \
3
3-317
3 of
3f
21
4
3-567
3
61
4
8A
41
3-798
21
6i
41
81
41
4-028
2f
61
41
8A
4f
4-255
2f
71
4f
8f
5
4-480
2i
*jf
5
51
4-730
2i
8
51
4 lb
51
5-058 2|
8f
51
4 T\
5f
5-203
2f
8f
5f
H
6
5-423 21
6
Washers (Fig. 489) in common use to provide seatings for nuts which
would otherwise rest on rough metallic surfaces, and also to adapt bolts to
shorter spaces than their lengths are sold for bolts up to 2" diameter. Cir-
FIG. 489.
FIG. 490.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
245
cular in form, their diameter is slightly in excess of that of the largest diam-
eter of the nut, and the hole that of the bolt, and thickness from $" to -J-",
according to the diameter of the bolt. The square washer is used under both
head and nut on surfaces of wood, and of dimensions suited to the stress.
That they may neither sink into the wood, nor bend or break, cast-iron is fre-
quently used, and often, as shown in Fig. 490, for roof-frames.
Shafts and Axles. Short shafts, revolving in bearings or boxes, or fastened
with pulleys, drum, or wheels revolving on them, are called axles ; but shafts
of large dimension or extent, and revolving, are usually termed shafts, as water-
wheel shafts and fly-wheel shafts. These may be independent ; that is, a sin-
gle shaft, revolving in its bearings, or they may be coupled together, forming
what is termed a line of shafting. The small shafts, as in clock-work and
spinning-machinery, are termed pins and spindles.
Shafts and axles are made
of wood and metal, and of va-
ried sections and form.
Wooden shafts are polygo-
nal, circular, or square section FIG. 491.
(Fig. 491).
Wrought metal, iron, or steel shafts, are almost invariably circular in sec-
tion, but sometimes square.
Cast-iron is used in great variety of section and form for shafts (Fig. 492) ;
without uniformity longi-
tudinally, but adapted to
their position and load.
Formerly, either wood
or cast-iron was invariably
used for water-wheel shafts ; FIG. 492.
but a change of motors,
from the breast, over-shot and under-shot wheels to reactors or turbines, has
involved an entire change of construction, and now only wrought-iron is used.
Still, wooden shafts are often used in machines subject to wet or shock, and
often from greater convenience in procuring the material ; and, from the same
cause, the bearings or bushings on which the shafts revolve are of the same
material, and serve a good purpose where the movements are not continuous
or rapid. But it is usual to make metal boxes, in which the rounded ends of
shafts revolve ; these ends are called journals or gudgeons. The diameters
and lengths of journals depend on the weight to be supported, the material of
shafts and bearings, and the velocity at which the shafts are run.
TABLE OF DIAMETER OF JOURNALS FOE HEAVY WORK.
Total load in
DIAMETER IN INCHES.
Total load in
DIAMETEK IN INCHES.
Total load in
DIAMETER IN INCHES,
pounds.
Cast-
Wrought-
pounds.
Cast-
Wrought-
pounds.
Cast-
Wrought-
iron.
iron.
iron.
iron.
iron.
iron.
1,100
2
1-7
30,000
6
5-1
137,000
10
8'6
3,700
3
2-5
44,000
7
6-0
183,000
11
9-4
8,800
4
3-4
70,000
8
6-9
237,000
12
10'3
17,000
5
4-3
100,000
9
7-7
312,000
13
11-2
246
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
PS.
FIG. 493.
FIG. 494.
The usual length of journals is from once to
twice their diameters, but this is to be modified
by the speed at which the shafts are run ; if
slow-moving, one diameter in length is ample,
but at very high speed, and small shafts, like
those of circular saws, from 4 to 6 diameters is
not uncommon. If the boxes are of cast-iron,
they will sustain the load given in the above
table for large journals ; but when the boxes
are lined with brass and composition, or Babbit-
metal, the first should not be loaded beyond 500
pounds per square inch, on half-circumferential
section, or 750 pounds on the axial section. Bab-
bit-metal should have a somewhat less load, say
500 pounds on the axial section.
Wooden shafts are sometimes fitted with wood-
en journals and boxes, but the usual practice is
to insert cast-iron journals.
Figs. 493 and 494 represent different views of
a wooden shaft. Fig. 493 shows at one end the
side elevation of the shaft, furnished with its iron
ferules or collars and its gudgeon ; at the other
end, the shaft is shown in sections, giving the
ferules in section, but showing the central spin-
dle with its feathers in an external elevation.
Generally, in longitudinal sections of objects in-
closing one or more pieces, the innermost or cen-
tral piece is not sectioned unless it has some in-
ternal peculiarity, the object of a section being
to show and explain peculiarities, and being there-
fore unnecessary when the object is solid ; on
this account, bolts, nuts, and solid cylindrical
shafts are seldom drawn in section. Fig. 494 is
an end view of the shaft, showing the fitting of
the spindle B and its feathers into the end of
the shaft, and the binding of the whole by ferules
or hoops, a a. The spindles B, which are let
into the ends, are cast with four feathers or
wings, c. The tail-piece ~b is by most millwrights
omitted. The ends of the beam are bored for
the spindle, and grooved to receive the feathers ;
the casting is then driven into its place, hooped
with hot ferules, and after this hard-wood wedges
are driven in on each side of the feathers, and
iron spikes are sometimes driven into the end of
the wood.
Figs. 495, 496, and 497 represent different.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
247
FIG. 496.
views of a cast-iron shaft of a water-wheel. Fig.
495 is an elevation of the shaft, with one half in
section to show the form of the core ; Fig. 496, an
end elevation ; Fig. 497, a section on the line c c
across the center. The
body is cylindrical and hol-
low, and is cast with four
feathers, c c, disposed at
right angles to each other,
and of an external para-
bolic outline. Near the
extremities of these feath-
ers four projections are
cast, for the attachment of
the bosses of the water-
wheel. These projections
are made with facets, so as to form the corners of a
circumscribing square, as shown in Fig. 496, and
they are planed to receive
the keys by which they are
fixed to the naves which
are grooved to receive them.
The shaft is cast in one en-
tire piece, and the journals
are turned.
It will be observed that
although no weight is sup-
ported at the center, yet
there is an increase in the
diameter of the feathers at
this line ; the weight of the shaft itself is a consid-
erable factor.
Fig. 498 represents the section of a portion of a
breast water-wheel, with a cast-iron shaft, formerly
much in use in this country, in which stiffness was
given to the wheel and shaft by wooden trusses.
These shafts are cast circular, in two lengths, con-
nected at the center, with circular bosses on which
the naves of the wheel are keyed.
Journals of independent shafts are always of less
diameters than those of the rest of the shafts, and if
the load on each is nearly equal the diameters of the
two journals are equal ; but, if the load is not cen-
tral, the diameters are proportioned to their several
loads, as shown on pages 228, 229.
Shafts of wrought-iron, of less diameter than six
inches, are fitted by turning down the journals
FIG. 497.
FIG. 495.
248
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
only. Large shafts are generally
forged, mostly in steps, as in Fig.
499, with the largest boss beneath
the gear or pulley-hub, and suffi-
ciently above the next boss, on each
side, to admit of the planing or cut-
ting of the key-seats.
To determine the size of a shaft,
considered as a beam merely, but
with a shafting load as by the rev-
olution of the shaft each longitu-
dinal line of surface has to undergo
successively tension and compres-
FlG 498 sion. The safe load of wrought-
iron is estimated at 6,000 pounds
per square inch, and the formula on which the graphic diagram (Fig. 500) is
constructed is d = '06 |/ w I, d being diameter, I = length between bearings,
FIG. 499.
l)oth in inches, w the load in pounds ; the load is not only the weight of shaft
and pulleys or gears, but also the stress in transmitting the power.
Use of Diagram. Suppose w = 50,000 pounds, and I = 6 feet = 72", then
14'
13
12
111
I
10
56789
Product of Load and Span in Millions.
FIG. 500.
10
12,000,OuO
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
249
w I = 3,600,000, the ordinate of 3 '6 cuts the curve on the abscissa 9 '2, which is
the required diameter of the shaft in inches.
FIG. 501.
FIG. 502.
FIG. 503.
FIG. 504.
Keys are pieces of metal, usually steel, employed to secure the hubs of pul-
leys, gears, and couplings to shafts. They may be sunk keys (Fig. 501), flat
keys (Fig. 502), and hollow keys (Fig. 503). The shaded
circle represents the shaft. The breadth of the key (Fig.
504) is uniform, but the thickness is tapered about one
eighth of an inch per foot. The shoulder h is for the pur-
pose of drawing out the key. Sunk keys are not necessa-
rily taper. Some prefer them of uniform section, and to
force the hub on over the key.
PROPOKTIONS OF SUNK KEYS.
DIAMETER OF SHAFT, IN INCHES.
1
2
3
4
5
6
Breadth of key. .
o
4
1
1*
j5
Thickness of key
25
'34
'43
52
61
'71
Depth sunk in shaft
10
125
'15
175
20
225
Depth sunk in wheel
15
215
28
'345
41
485
DIAMETEE OF SHAFT, IN INCHES. 7
8
9
10
11
12
Breadth of key 1
01
O8
O5
9i
qi
Thickness of key . .... i '80
^$
89
*f
98
*f
*
1'lfi
*t
1 -9fi
Depth sunk in shaft -25
275
30
'325
'35
Q'JK
Depth sunk in wheel -55
615
68
745
'81
'875
Car- Axles. Fig. 505 is the form and dimensions of axle
adopted as standard by the American Master Car-Builders'
Association for wrought-iron and steel.
Shafting. Thus far, independent shafts or axles have
been treated of, and the dimensions have been established
mostly by the load acting transversely; but, in transfer-
ring power to machines, lines of shafting are necessary,
almost invariably of wrought-iron or steel bars, which are
subject not only to transverse but also torsional stress.
When there are no pulleys or gears on the shafts between
the bearings, and the couplings are close to the bearings,
there is still an amount of deflection due to the weight of
the shaft. James B. Francis, C. E., puts the maximum
distances between bearings for shafts of wrought-iron or
steel, under these conditions, as follows :
SE
1
7
f
V
r 2
^
^->
.*.
:
r-
i
i
^
.-ji *
3
250
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
Diameter Distance between Diameter Distance between Diameter Distance between
of shaft.
1"
2
3
4
bearings.
12ft.
15
18
20
of shaft.
5*
6
7
8
bearings.
21 ft.
22
24
25
of shaft.
9"
10
11
12
bearings.
26ft.
27
28
28
The diagram (Fig. 506) is one established by J. T. Henthorn, M. E. of the
Corliss Steam-Engine Company, to determine the size of wrought-iron shaft-
ing, to transmit a fixed amount of horse-power.
400
Horse-Power.
FIG. 506.
Use of Table. To find the size of a shaft making 150 revolutions, and trans-
mitting 350 horse-power.
The intersection of the ordinate of 350 with the abscissa of 150 is between
the diagonals 5 and 5, and the diameter of the shaft may be taken safely at 5J".
Mr. Francis has constructed a table from his own experiments, of which
the following is a synopsis :
'' The following table gives the power which can be safely carried by
shafts making 100 revolutions per minute. The power which can be carried
by the same shafts at any other velocity may be found by the following simple
rule :
"Multiply the power given in the table by the number of revolutions made
by the shaft per minute ; divide the product by 100 ; the quotient will be the
power which can be safely carried."
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
251
Horse-power which can be safely transmitted by shafts making 100 revolutions per minute, in which the trans-
verse strain, if any, need not be considered ; if of
Diameter
in inches.
Wrought-
iron.
Steel.
Diameter
in inches.
Wrought-
iron.
Steel.
Diameter
in inches.
Wrought-
iron.
Steel.
1-
2-0
32
4'5
182-
291-
7-5
843'
1350-
1-5
6-7
10-7
5-
250-
400'
8'
1024'
less-
2-
16-0
25-6
5'5
332-
532'
8'5
1228-
ees-
2'5
31-2
50"
6'
432'
691"
9-
1458'
2332'
3-
54-0
86-4
6'5
549'
878-
9-5
1714-
2743'
3-5
85'7
137-
7-
686'
1097'
10
2000"
3200'
4'
128-
204'
The diagram and table given are applicable to shafts which are called sec-
end movers, subject to no sudden shock. For first movers, Mr. Francis takes
but one half the horse-power given in the table for any diameter of shafts.
Of late, cold-rolled shafts can be procured in the market, which are much
stiffer than turned shafts, but not equal to that given for steel in the table.
It is usual to make the shafts of second and third movers throughout manu-
factories and shops of uniform diameter, without reduction at the journals, the
end-slip being prevented by collars keyed or fastened by set-screws. The usual
length between bearings is from 7 to 10 feet ; but that they may run smooth,
and not spring intermediately, it is desirable that they should never be less
than 2 inches diameter, and that the pulleys or gears through which the power
is transmitted to the next mover or to the machine should be as near as pos-
sible to the bearing.
Fig. 507 represents a line of shafting. A is an upright shaft ; a a, bevel-
gears ; b b, bearings for the shafts ; c, coupling or connection of the several
pieces of shafting. These shafts are intended to be of wrought-iron. No re-
duction is made for the journal, no bosses for pulleys or gears. As the power
is distributed from this line of shafting, the torsional strain diminishes with
FIG. 507.
the distance from the bevel-gears or first movers, and the diameter of each
piece of shafting may be reduced consecutively, if necessary ; but uniformity
will generally be found to be of more importance than a small saving of iron.
The drawing given is of a scale large enough to order shafting by, but the
dimensions should be written in.
In laying out lines of shafting, the position of the bearings is usually fixed,
and the lengths of shafts must be determined thereby, with as few couplings
as possible. When there is no necking or reduction of the shafts, which is
usually the case, the orders given for shafting will be so many lengths and of
such diameters, and so many couplings and hangers. When 'there is to be a
252
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
necking, the sketch for the order may be very simple, showing length and
diameter of shaft, and position, length, and diameter of bearing.
The joints or couplings are generally made near the bearings, and it is also
usual to bring the pulleys as near the bearings as possible. It frequently hap-
pens, therefore, that the coupling and pulley are needed at the same point ;
to remedy this, as the position of the pulley depends on the machine which it
is required to drive, it frequently can not be moved without considerable in-
convenience or loss of room ; the shaft will have, therefore, to be lengthened
or shortened, to change position of coupling ; or, if the couplings are plate
couplings, the coupling and pulley may be made together.
When a horizontal shaft is supported from beneath, its bearing is usually
called & pillow- or plumber --block, or standard ; if suspended, the supports are
called hangers.
FIG. 509.
Figs. 508 and 509 are the elevation and plan of a pillow-block. It consists
of a base plate, A, the body of the block B, and the box C. The plate, as in
the step, is bolted securely to its base, the surface on which the block B rests
being horizontal. A and B are connected by bolts passing through oblong
holes, so as to adjust the position in either direction laterally. The box or
bush C is of brass, in two parts or halves, extending through the block, and
forming a collar by which it is retained in its place. The cap of the block is
retained by the screws o o o ; in the figure there are two screws on one side
and one on the other ; often four are used, two on each side, but most fre-
quently but one on each side.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 253
F
FIG. 511.
The standard is simply a modification of the pillow-block, being employed
for the support of horizontal shafts at a considerable distance above the founda-
tion-plate. Fig. 510 is a front elevation ; Fig. 511, a plan ; and Fig. 512, an
end elevation of a standard. Like the pillow-block, the plate A is fastened to
254:
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
the foundation itself, and the upper surface is placed perfectly level in both
directions. On these bearing surfaces, a a a, the body of the standard rests, and
can be adjusted in position horizontally, and then clamped by screws to the
foundation-plate, or keyed at the ends.
Elevations and plan are usually drawn in such positions to each other that
lines of construction can be continued from one to the other, which not only
simplifies the drawings, but makes them more readily intelligible. Letters and
dotted lines in these figures illustrate this sufficiently.
It will be observed that the sides of the elevations are represented as broken;
this is often done in drawing, when the sides are uniform, and ecpnomy of
space on the paper is required.
Suspended bearings or hangers for horizontal shafts are divided into two
general classes, side-hangers (Figs. 513, 514) and sprawl-hangers ; the figures
will sufficiently explain the distinction. The side-hanger is the more conven-
ient when it is required to remove the shaft, and when the strain is in one
FIG. 513.
FIG. 514.
direction, against the upright part ; they are generally used for the smaller
shafts, but sprawl-hangers, affording a more firm support in both directions,
are used as supports for all the heavier shafts. Hangers are bolted to the floor-
timbers, or to strips placed to sustain them, the centers of the boxes being
placed accurately in line, both horizontally and laterally.
Fig. 515 represents the elevation of a sprawl-hanger ; Fig. 516, the plan
looking from above, with cover of box off ; Fig. 517, a section on the line A B,
Fig. 515.
Fig. 518 represents the elevation of a bracket, or the support of a shaft
bolted to an upright ; the box is movable, and is adjusted laterally by the set-
screws. Fig. 519 is a front elevation of the back plate cast on the post ; it will
be seen that the holes are oblong, to admit of the vertical adjustment of the
bracket.
Figs. 520-523 represent different views of what may be called a yoke-hanger.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 255
FIG. 517
/
i
i
1
/
5
A
r
o
i
3
c
J
1
7
~v
FIG. 510.
Fig. 520 is a front and Fig. 521 a side elevation; Fig. 522 a plan of the hanger,
looking up ; and Fig. 523 a plan of the yoke, looking down upon it. A is
the plate which is fastened to the beam, E is the yoke, and B the stem of the
256 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
FIG. 520.
FIG. 521.
FIG. 522.
FIG. 523.
yoke, cut with a thread so as to admit of a vertical adjustment ; the box D of
the shaft C is supported by two pointed set-screws passing through the jaws of
the yoke ; this affords a very flexible bearing, and a chance for lateral adjust-
ment.
For Upright Shafts. Footstep, or Step, for an Upright Shaft. Fig. 524
represents an elevation, Fig. 525 a plan of the step. It consists of a founda-
tion or bed-plate, A. a box, B, and a cap or socket, C. The plate A is firmly
fastened to the base on which it rests ; in the case of heavy shafts, often to a
base of granite. The box B is placed on A, the bearing surface being accu-
rately leveled, and fitted either by planing or chipping and filing ; h, b, #, are
what are commonly called chipping-pieces, which are the bearing surfaces of
the bottom of B. A and B are held together by two screws ; the holes for
these are cut oblong in the one plate at right angles to those of the other ; this
admits of the movement of the box in two directions to adjust nicely the lat-
eral position of the shaft, after which, by means of the screws, the two plates
are clamped firmly to each other. 0, the cup or bushing, which should be
made of brass, slips into a socket in B. Frequently circular plates of steel are
dropped into the bottom of this cup for the step of the shaft. The cup 0, in
MACHINE DESIGN AND MECHANICAX CONSTKUCTIONS. 257
FIG. 525.
case of its sticking to the shaft, will revolve with the shaft in the box B ; if
plates are used, these also admit of movement in the cup.
Fig. 526 represents the elevation of a bearing for an upright shaft, in which
FIG. 526.
the shaft is held laterally by a box and bracket above the step. The step B is
made larger than the shaft, so as to reduce the amount of wear incident to a
heavy shaft. The end of the shaft and the cup containing oil are shown in
258
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
dotted line. The bed-plate A rests on pillars, between which is placed a pil-
low-block or bearing for horizontal shaft.
Figs. 527 and 528 represent the elevation and vertical section of the suspen-
sion bearing used by Mr. Boy den for the support of the shaft of his turbine-
wheels. It having been found difficult to supply oil to the step of such wheels,
it was thought preferable by him to suspend the entire weight of wheel and
shaft, where it could be easily attended to. The shaft (see section) is cut into
FIG. 527.
FIG. 528.
necks, which rest on corresponding projections cast in the box ~b ; the spaces in
the box are made somewhat larger than the necks of the shaft, to admit of Bab-
bitting, as it is termed, the box ; that is, the shaft being placed in its position
in the box, Babbitt, or some other soft metal melted, is poured in round the
shaft, and in this way accurate bearing surfaces are obtained ; projections or
holes are made in the box to hold the metal in its position. The box is sus-
pended by lugs b, on gimbals c, similar to those used for mariners' compasses,
i
FIG. 529.
FIG. 530.
which give a flexible bearing, so that the necks may not be strained by a slight
sway of the shaft. The screws e e support the gimbals, consequently the shaft
and wheel ; by these screws the wheel can be raised or lowered, so as to adjust
its position accurately ; beneath the box will be seen a movable collar, to adjust
the lateral position of shafts.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
259
Figs. 529 and 530 are the plan and elevation for the step, or rather guide
{as it bears no weight), of the foot of the shaft of these same turbines. The
plate A is firmly bolted to the floor of the wheel-pit ; the cushions C, holding
the shaft, are either wooden or cast-iron, and admit of lateral adjustment by
the three rows of set-screws.
In construction, the hanger and
guide of Mr. Boyden were found
FIG. 531.
to be too expensive, and wooden
steps (Fig. 531) are now almost FIG. 532.
universally used for turbines.
They are made either conical or a portion of a sphere, of various woods, usually
lignum-vitae, but oak and poplar are preferred by some. The load is from
fifty to seventy-five pounds per square inch. The fibers of the wood are
placed vertically, and afford a very excellent bearing surface. Water is some-
times introduced into the center of the wood, or into a box around it, from the
upper level of water. When cast-iron or steel is used for
the step, it is usual to incase the box and supply oil by lead-
ing a pipe, sufficiently high above the surface of the water,
to force the oil down.
FIG. 534.
For long, upright shafts, it is very usual to suspend the upper portion by a
suspension-box, and to run the lower on a step, connecting the two portions
by a loose sleeve or expansion coupling, to prevent the unequal meshing of the
260
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
bevel- wheels, incident to an alteration of the length of shaft by variations of
temperature. The suspension is frequently made by a single collar at the top
of the shaft.
Figs. 532 and 533 are perspectives of the hangers made by William Sellers
& Co., of Philadelphia ; and Fig. 534 a section showing the adjustment of the
boxes in Fig. 532.
The boxes are of cast-iron, long in proportion to the diameters of shafts ;
the center bearings are spherical, and are adjusted in position vertically by the
screws d and e ; b I are cups, to contain grease, which will melt if the bearings
become heated, but the lubrication depends on an oil-cup dripping oil into the
center of the bearing ; / is a cast-iron drip-pan to catch the waste oil from the
journal.
Fig. 533 is a view of side hanger adapted to a counter shaft, and the square
slot a is for the shipping-bar. This form of hanger is more common than
that shown in Figs. 513 and 514 ; the cap in this last is held down by a wedge,
in Fig. 533, by a lateral screw ; but with most makers the screw is vertical,
clamping the cap to the lower part of the box.
Couplings are the connections of shafts, and are varied in their construction
and proportions often according to the mere whim of the mechanic making
them.
The Face Coupling (Fig. 535) is the one in most general use for the con-
necting of wrought-iron shafts ; it consists of two plates or disks with long,
strong hubs, through the center of
which holes are accurately drilled
to fit the shaft ; one half is now
drawn on to the shaft, and tightly
keyed ; the plates are faced square
with the shaft, and the two faces
are brought together by bolts. The
number and size of the bolts depend
upon the size of the shaft ; never
FIG. 535. less than 4 for shafts less than 3
inches diameter, and more as the
diameter increases ; the size of the bolts varies from f to 1 inch in diameter.
The figure shows a usual proportion of parts for shafts of from 2 to 5 inches
diameter ; for larger than these, the proportion of the diameter of the disk to
that of the shaft is too large.
Fig. 536 is a rigid sleeve coupling for a cast-iron shaft ; it consists of a solid
hub or ring of cast-iron hooped with wrought-iron ; the shafts are made with
bosses, the coupling is slipped on to one of the shafts, the ends of the two are
then brought together ; the coupling is now slipped back over the joint, and
firmly keyed. This is an extremely rigid connection. Some makers use keys
without taper, and force the couplings on the shafts.
Fig. 537 is a screw coupling for the connecting of the lighter kinds of shafts.
It will be observed that this coupling admits of rotation but in one direction,
the one tending to bring the ends of the shafts toward each other ; the reverse
motion tends to unscrew and throw them apart, arid uncouple them.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
261
FIG. 536.
FIG. 537.
Fig. 538 is a clamp coupling for a square shaft.
William Sellers & Co., Philadelphia, make a double-cone vice coupling,
which is largely used (Fig. 539). It is shown complete on shaft at A. B is
the outer shell or sleeve, C the two
<3ones, and D the bolts. The sleeve is
cylindrical outside, but bored with a
double taper inside, smallest at center.
The cones are bored to fit the shaft, and
turned outside to fit the interior cones
of the sleeve. There are three bolt-
grooves in the cones and sleeve, and one is cut through to give elasticity to the
cones. The sleeve and cones are adjusted over the joint of the shafts, leaving
at an easy fit some f " between the ends of the cones ; if now the bolts be intro-
FIG. 538.
FIG. 539.
duced and screwed up, the cones are brought nearer to each other, and the
shafts are securely clamped together. Fig. 540 shows the coupling in section.
In many cases it occurs that rigid couplings, such as we have given, are
objectionable ; they necessarily imply that, to run with the least strain possible,
the bearings should be in accurate line ; any displacement involves the spring-
ing of the shaft, and, if considerably moved, fracture of shaft or coupling.
FIG. 540.
FIG. 541.
Wherever, then, from any cause the alignment can not be very nearly accu-
rate, some coupling that admits of lateral movement should be adopted. The
simplest of these is the box or sleeve coupling (Fig. 541), sliding over the end
of two square shafts, keyed to neither, but often held in place by a pin passing
through the coupling into one of the shafts. For round shafts, the loose sleeve
262
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
coupling is a pipe or hub, generally 4 to 6 times the diameter of the shaft in
length, sliding on keys fixed on either shaft.
Fig. 542 represents a horned coupling. The two parts of the coupling are
counterparts of each other, each firmly keyed to its respective shaft, but not
fastened to each other ; the horns of the one slip into the spaces of the other ;
if the faces of the horns are
accurately fitted, it affords
an excellent coupling, and
is not perfectly rigid.
It often happens that
some portion of a shaft or
machine is required to be
stopped while the rest of
the machinery continues in
motion. It is evident that,
if one half of a horned coupling be not keyed to the shaft, but permitted to
slide lengthways on the key the key being fixed in the shaft, forming in this
case what is more usually called a feather by sliding back the half till the
horns are entirely out of the spaces of the other half, communication of motion
will cease from one shaft to the other.
FIG. 542.
FIG. 543.
Fig. 543 represents a coupling of this sort for a large shaft, from the Corliss
Steam-Engine Company. The horns are 8 in number on each part, and are
thrown readily in or out of action by the handle h turning the loose part of the
clutch on the screw cut on the shafts.
Fig. 544 is another form of disengaging a large pulley from a main shaft,
from the Corliss Steam-Engine Company. The pulley is fastened to a cast-
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 263
iron pipe or sleeve p through which the main shaft s passes. The two are
attached by means of the coupling c, one half of which is attached to the shaft
and the other to the sleeve. When bolted together, the pulley and main shaft
264
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
move together ; but if the bolts be removed, then the pulley becomes stationary
even if the shaft is running. Shaft and sleeve have independent bearings.
A (Fig. 544) is a section of the coupling on a larger scale, and shows the strong
taper of the bolts without head.
Couplings are made on this principle, called slide or clutch couplings,
when the motion is required but in one direction. The general form of this
coupling is given in Fig. 545. A represents the half of the coupling that is
keyed to the shaft, B the
6 ^.-^ sliding half, c the handle or
lever which communicates
the sliding movement ; the
upper end of the lever ter-
minates, in a fork, inclosing
the hub of the coupling, and
fastened by two bolts or pins
to a collar c' round the neck
of the hub ; 1) is a box or
bearing for the shaft A ; to
support B the end of its shaft extends a slight dis-
tance into the coupling A. Shafts can not be en-
gaged with this form of coupling while the driving
shaft is in motion, without great shock and injury to the
machinery. To obviate this, other forms of coupling
are requisite ; one of these is represented (Fig. 546).
On the shaft B is fixed a drum or pulley, which is
embraced by a friction band as tightly as may be
found necessary ; this band consists of two straps of
iron, clamped together by bolts, leaving ends project-
ing on either side ; the portion of the coupling on the shaft A is the common
form of bayonet clutch ; the part c c is fixed to the shaft, and affords a guide
to the prongs or bayonets b b, as they slide in and out. Slipping these prongs
forward, they are thrown into gear with
the ears of the friction band ; the shaft
A being in motion, the band slips round
on its pulley till the friction becomes
equal to the resistance, and the pulley
gradually attains the motion of the
clutch.
But of all slide couplings, to engage
and disengage with the least shock and
at any speed, the friction cone coup-
ling (Fig. 547) is by far the best. It
consists of an exterior and interior
cone, a, b ; a is fastened to the shaft
A, while b slides in the usual way on the feather / of the shaft B ; pressing b
forward, its exterior surface is brought in contact with the interior conical sur-
face of a ; this should be done gradually ; the surfaces of the two cones slip on
FIG. 545.
FIG. 546.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
265
each other till the friction overcomes the resistance, and motion is transmitted
comparatively gradually, and without danger to the machinery. The longer
the taper of the cones, the more
difficult the disengagement ; but
the more blunt the cones, the more
difficult to keep the surfaces in
contact. An angle of 8 with the
line of shaft is a very good one
for surfaces of cones of cast-iron
on cast-iron. When thrown into
ifaas^^sai
FIG. 547.
FIG. 548.
gear, the handle of the lever or shipper is slipped into a notch, that it may not
be thrown out by accident.
The objection to this coupling is that it will work out of gear unless the
shipper-handle is held firmly in its
position, and producing considerable
friction against the collar. To obvi-
ate this the shipper is made to act on
a toggle-joint fastened to the shaft,
and, once thrown, the pressure is
self-continued, and preserved with-
out any action of the shipper, and
without friction.
Fig. 548 represents a double-fric-
tion clutch, of the Weston-Oapen
patent. The clutch G is slid over
the toggle, and the friction cone is
forced into the pulley and engaged
therewith. In the figure, D' is thus
engaged with A', while D and A are
not in contact.
Fig. 549 is a perspective of the
Mason clutch, in which two toggles FIG. 549.
are attached to the sliding hub F.
By the action of the shipper moving the hub inward the two toggles force the
two segments E E against the inner periphery of the pulley, which is turned
266
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
parallel with the axis. The toggles are so adjusted that when forced in they are
a trifle within the straight line, so that there is no tendency for them to fly out.
Pulleys are used for the transmission of motion from one shaft to another
by the means of belts ; by them every change of velocity may be effected. The
speed of two shafts will be to each other in the inverse ratio of the diameter of
their pulleys. Thus, if the driving shaft make 100 revolutions per minute,
and the driving pulley be 18 inches in diameter, while the driven pulley is 12
inches, then, ^ . lg .. 100 . 15Q .
that is, the driven shaft will make 150 revolutions per minute. Where there
is a succession of shafts and pulleys, to Jind the velocity of the last driven
shaft : Multiply together all the diameters of the driving, pulleys by the speed
of the first shaft, and divide the product by the product of the diameters of all
the driven pulleys.
FTG. 550.
FIG. 551.
FIG. 552.
Pulleys are made of cast-iron and of every diameter, from 2 inches up to
20 feet. The number of arms vary according to the diameter ; for less than
8 inches diameter the plate pulley is preferable (Fig. 550) ; that is, the rim is
attached to the hub by a plate ; for pulleys of larger diameters, those with
arms are used, never less than 4 in number. The arms are made usually
straight (Fig. 551), sometimes curved (Fig. 552).
FIG.
553.
FIG. 554.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
267
Fig. 553 represents a portion of the elevation of a pulley sufficient to show
the proportion of the several parts, and Fig. 554 a section of the same. The
parts may be compared proportionately with the diameter of shaft ; thus the
thickness of the hub is about -J- the diameter of the shaft ; this proportion is also
used for the hubs of couplings ; the width of the arms from to full diame-
ter ; the thickness half the width ; the thickness of the rim from 1 to -J- the
diameter ; the length of hub the same as the width of face.
Fig. 555 is a large pulley of the Southwark Foundry pattern. The hub
is cast with four divisions, to admit of contraction in cooling, and the rim is in
268
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
halves, to admit of the pulley being put on the shaft without removing it from
its bearings. This is now very common practice with large pulleys. Wrought-
iron rim-pulleys have lately been introduced in which the spider that is, the hub
and arms are of cast-iron, and a wrought-iron plate-
rim is bolted to flanges on the extremities of the arms.
Fig. 556 represents a faced coupling pulley, an
3 expedient sometimes adopted when a joint occurs
where a pulley is also required ; the two are then
combined ; the pulley is cast in halves two plate
pulleys, with plates at the side instead of central,
faced and bolted together.
Wooden pulleys are commonly called drums ; these are now but seldom
used except for pulleys of very wide face. Fig. 557 represents one form of
construction in elevation and longitudinal section. It consists of two cast-iron
pulleys A A, with narrow rims ; they are keyed on to the shaft at the required
FIG. 556.
w
I -
M^-i
FIG. 557.
distance from each other, and plank or lagging is bolted on the rims to form
the face of the drum ; the heads of the bolts are sunk beneath the surface of
the lagging, and the face is turned.
Fig. 558 represents a wooden pulley which may be termed a wooden plate
pulley. The plate consists of sectors of
inch boards firmly glued and nailed to-
gether, the joints of the boards being
always broken. The face is then formed
in a similar way, by nailing and gluing
arcs of board one to another to the re-
quired width of face ; these last should
be of clear stuff. The whole is retained
on the shaft by an iron hub, cast with a
plate on one side, and another separate
plate sliding on to the hub ; the hub is
placed in the. center of the pulley, the
two plates are brought in contact with the sides of the pulley, and bolted
through ; the face of the pulley is now turned in the lathe. A similar arrange-
ment of hub is used for the hanging of grindstones.
FIG. 558.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 269
V^>
Cone pulleys are used to change the speed of the driven shaft. Fig. 559
represents a cone pulley c on a shaft with a fast pulley d, or one attached to the
shaft, while the other, e, is loose and revolves on it. The cone pulleys are for
changes of speed on the machine, which has upon it another set of cone pul-
leys, but in reverse position, the small one being opposite the large one on the
FIG. 559.
counter shaft. A counter shaft is one disconnected from a main or leading
shaft, for the purpose of driving a machine. This counter is connected with
the main shaft by a belt from a pulley on this shaft passing over the fixed or
loose pulley. When on the fixed pulley, the counter shaft is moved ; when on
the loose pulley it revolves on the shaft, and the shaft is still. To move the
belt, there is a fork between which the sides of the belt approaching the coun-
ter passes, and a movement of this fork by a shipper throws the counter in or
out of movement. The faces of the fast or loose pulleys are made flat, and
provision is to be made for oiling the inside of the hub of the loose pulley,
which is done by oil-holes and grooves.
FIG. 560.
It is often necessary to reverse the motion of a machine. This is readily
done by a system of fast and loose pulleys, as shown in the plan and elevation,
Fig. 560, in which A is a drum or wide-faced pulley on the driving-shaft, B a
fast pulley on the driven shaft, and C and D loose pulleys on the same. The
action will be understood from the direction of the arrows. The driving-shaft
270
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
revolves always in the same direction, but on the driven shaft the loose pulley
of the straight belt is drawn from the bottom, and partakes of the same motion
as the driving-pulley ; while by the cross-belt the draft is at the top of its pul-
ley, and the motion reversed. If the straight or open belt be shipped on to the
fast pulley B, the motion given to the shaft is like that of the driving-shaft ; if
the cross-belt be shipped on to the fast pulley, the motion of the shaft is re-
versed. It will be observed in the elevation, the lower side of the open belt is
straight, while there is a sag in the upper ; the first is called the tight or lead-
ing belt, it being the belt through which the power is transmitted, while the
upper side is the loose or slack belt. The stress on the tight belt is equal to
that of the power transmitted, and the stress with which the belt is stretched
over the pulleys, so that it will not slip in conveying this power.
When the belt is shifted, while in motion, to a new position on a drum or
pulley, or from fast to loose pulley, or vice versa, the lateral pressure must be
applied on the advancing side of the belt, on the side on which the belt is ap-
proaching the pulley, and not on the side on which it is running off. It is only
necessary that a belt, to maintain its position, should have its advancing side
in the plane of rotation of that section of the pulley on which it is required to
remain, without regard to the retiring side. On this principle, motion may be
conveyed by belts to shafts at any angle to each other. Let A and B (Fig.
561) be two shafts at right angles to each other, A vertical, B horizontal, so
that the line run perpendicular to the direction of one axis is also perpendicu-
lar to the other, and let it be required to connect them by pulleys and a belt,
FIG. 561.
FIG. 562.
that their direction of motion may be as shown by the arrows and their veloci-
ties as 3 of A to 2 of B. On A describe the circumference of the pulley pro-
posed on that shaft ; to this circumference draw a tangent a b parallel to m n ;
this line will be the projection of the edge of the belt as it leaves A, and the
center of the belt as it approaches B ; consequently, lay off the pulley b on each
side of this line, and of a diameter proportional to the velocity required. To
fix the position of the pulley on A, let Fig. 562 be another view taken at right
angles to Fig. 561, and let the axis B have the direction of motion indicated by
the arrow, then, the circle of the pulley being described, and a tangent a' V
drawn to it perpendicular to the axis B as before determined, the position of
the pulley on the shaft A is likewise fixed.
The positions of the two pulleys are thus fixed in such a way that the belt
is always delivered by the pulley it is receding from into the plane of rotation
of the pulley toward which it is approaching. If the motion be reversed, the
belt will run off.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
271
Figs. 563 and 564 are the plan and elevation, on a large scale, of a similar
arrangement of pulleys and belts.
It is not an essential condition that the shafts should be at right angles
to each other to have motion transferred by a belt. They may be placed at
Fm. 564.
any angle to each other, provided the shafts lie in parallel planes, so that
the perpendicular drawn to one axis is perpendicular to the other. If other-
wise, recourse must be had to guide-pulleys, which should be considerably
convex on their face.
Fig. 565 is an arrangement adopted in port-
FIG. 566.
FIG. 565.
FIG. 567.
272
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
able grist-mills for driving the vertical shafts , I, of mill-stones, from pulleys
on a horizontal shaft. Here it is thought necessary to use guide-pulleys.
Figs. 566 and 567 are the elevation and plan of another arrangement of
pulleys and guide-pulleys ; a ~b is the intersection of the middle plane of the
principal pulleys. Select any two points a and b on this line, and draw tan-
gents a c, b d, to the principal pulleys. Then c a c and a b d are suitable direc-
tions for the belt. The guide-pulleys must be
placed with their middle planes coinciding with the
planes c a c and a b d. The belt will run in either
direction.
Fig. 568 is a perspective of a hanger of William
Sellers & Co., in which the guide-pulleys can be ad-
justed to revolve in the required plane.
It has been said that it is necessary to stretch the
belt over the pulleys, so that it will not slip while con-
veying the power. If the pulleys are horizontal, the
weight of the belt itself may provide for this friction,
and this friction diminishes with the inclination of
the belt till it becomes vertical, when the friction of
the stretch is the only factor of
the adhesion of the belt to the
lower pulley ; and, as the belt
lengthens by use, the value of
this friction becomes nothing.
This position of pulleys should
not obtain if it can be avoided ;
but if not, the friction-stress
should be by means of a binder on the loose belt. The
binder (Fig. 569) hangs in a loose frame or links, and
rests on the belt, so that the weight of the binder and
frame tends to take up the slack of the belt. Sometimes
the binder is forced against the belt by a screw acting on
its frame. By the relief of the binder the belt becomes
slack, and the friction of the belt on the pulleys may be-
come nothing, and motion stopped. On many machines
and lines of shafting this arrangement for engaging and
disengaging is made use of. Binders are a necessity where
the two pulleys are near to each other, either to increase
the bearing surface of the belt on the pulleys or to make
up for the slight weight of a short belt. Belts run the
best when their length and position are such as to give the
frictional stress without much stretching on the pulleys,
and without binders. It is also necessary that the surface of belt in contact
with the pulleys should be large, as the frictional stress varies with the surface.
The widths of belt hereafter given are based on the usual surface of about 180,
or half the circumference of the pulleys. On account of the friction and wear
it is usual to put the hair side of the belt next the pulley.
FIG. 568.
FIG. 569.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
273
In determining the necessary length for any position, the simplest way is to
measure it, if the construction is complete ; if not, to make a drawing of the
pulleys in position to a scale, and measure on the drawing.
The width of the belts should always be a little less than the face of the
pulley ; both are to be determined by the power to be transmitted and the
velocity of movement. For the lighter stress of belt a single thickness is only
necessary, but for belts from prime movers, transmitting great power, double
belts are used.
For single belts, embracing 180 of the circumference, with a velocity of 10
feet per second, one horse-power can be transmitted for each inch in width of
belt, with a maximum stress on the belt of 50 pounds, and pressure on journals
of about 85 pounds per inch of width of belt.
D X TT X R
John T. Henthorn's formula for double belts
is
450
= H. P.
per inch in width, in which D is the diameter of pulley in feet, E the revolu-
tions per minute. This is expressed graphically in Fig. 570.
Diameter of Pulleys, shown by Diagonals.
Horse-Power per Inch of Width.
FIG. 570.
Use of Diagram. To find the horse-power that can be transmitted by a
24" belt on a 20-foot pulley making 100 revolutions per minute : The abscissa
line 100 intersects the diagonal 20 on the ordinate line 14 ; 14 X 24 = 336 =
horse-power transmissible.
To find the belt necessary to transmit 100 horse-power through a 10-foot
pulley and 120 revolutions per minute of shaft : The abscissa 120 cuts the
diagonal 10 on the ordinate line 8; - - = 12" width of belt. If the pulley
18 8
274 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
were 12-foot instead of 10, it will be seen by the diagram, the intersection of
100
diagonal would be at 10, and the width of belt = 10".
The above rules are applicable to leather belts, but belts of India-rubber
and canvas are largely used, and can be procured of any desirable dimensions,
and the strength and adhesion are generally considered greater than those of
leather belts. They are especially valuable in situations exposed to wet, where
leather is not admissible.
It is the present practice to run belts at high speed ; 5,000 to 6,000 feet is
admissible with suitable pulleys and position.
The use of ropes instead of belts has not obtained largely in this country,
but in a late report of Mr. Edward Atkinson on English practice, he says
that in first-class mills ropes instead of leather belts have taken the place of
the upright shafts and gears which were formerly used. He instances in
one mill,
" The main wheel on the 2d-motion shaft from the engine or driving-pulley
is grooved ; it is 12 feet in diameter, 104 revolutions per minute, and has 20
ropes. "
" The rules for rope-driving have been given me as follows :
"1. Never use pulleys of less diameter than six feet for main work.
<( 2. The greater the velocity of the rope per minute the greater the effi-
ciency, up to 5,000 feet per minute.
"3. For great power, ropes Scinches diameter, 2 inches when stretched,
are best ; cable-laid with 3 strands, and each strand of 3 finer strands. Where
small power is required it is not necessary to have the rope cable-laid, or so
great in diameter. For ropes of small diameter smaller diameters of pulleys
may be used than 6 feet, and cotton ropes are preferable to hemp. For large
ropes or outside work, hemp is better than cotton. Cotton ropes made from
yarn, counts about 20 to 30, are better than those made from rovings.
" 4. With a rope 2% inches diameter, and pulleys above 6 feet, each rope
will drive 10 indicated horse-power every 1,000 feet of rope-speed per minute.
" 5. Whenever circumstances will allow, the slack side of the ropes ought
always to be on the top, so as to keep the rope tight in the groove where it
stretches.
1 'The tarred cotton rope and tarred spindle banding, thoroughly impreg-
nated with pine tar, are reasonably supple, perfectly free from stickiness, and
are said to be very non-elastic and substantially free from
the effects of humidity."
Fig. 571 represents a cross-section of single-grooved rim
for a cotton or hemp rope as used in this country, the
groove being simply turned and polished.
Fig. 572 is a cross-section of the rim of wheel for wire
FIG. 571. FIG. 572. rope, showing the rubber lining contained in a dovetailed
recess at the bottom of the groove.
From the circular of Messrs. Roebling & Sons we make the following
table of transmission of power by wire, the number of revolutions per minute
being 100 :
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
275
Diameter of
wheel in
feet.
Diameter of rope.
Horse-power.
Diameter of
wheel in
feet.
Diameter of rope.
Horse-power.
4
f
3-3
10
ttt
68-7
/TO.
4
f
4-1
5
A
8-6
11
f
81-1
94-4
6
7
*
I a 6
13-4
21-1
12
Ht
116-7
124-1
8
i
27-5
13
HI
140-
153-2
9
1% 1
50-
51-9
14
f *
185'
176-
10
IH
55-
58-4
15
* *
259'
259-
Gearing. The term gearing, in general sense, is applied to all arrange-
ments for the transmission of power ; it is also used in a particular sense, as
toothed gearing.
Toothed gearing may be divided into two great classes spur and level
wheels. In the former, the axes of the driving and driven wheels are parallel
to each other ; in the latter they may be situated at any angle ; if of equal size
and at right angles, they are called miter-gears.
Spur-wheels, strictly so called, consist of wheels of which the teeth are dis-
posed at the outer periphery of the wheel (Fig. 583), in direction of radii from
their centers.
Internal gearing, in which the teeth are disposed in the interior periphery
of the wheel, in direction of radii from their centers (Fig. 596).
Rack-gear and pinion are employed to
convert a rotatory into a rectilinear mo- T '
tion, or vice versa. In this arrangement
the pinion is a spur-wheel, acting on teeth
placed along a straight bar (Fig. 595).
FIG. 573.
Fm. 574.
Bevel-gearing, strictly so called, consists of toothed wheels formed to work
together in different planes, their teeth being disposed at an angle to the plane
of their faces (Fig. 591).
276
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
Trundle-pins or wheels (Fig. 578) are constructed with cylindrical pieces,,
called staves or pins, instead of teeth. Fig. 573 is an illustration of trundle-
gears with wooden pins ; the pinion with double plates is called a lantern.
This construction is very useful when iron gears can not be easily got or re-
paired. The trundle may be used either with a spur-wheel to transmit motion
to parallel shafts, or with /ace or crown wheels.
The primary object of toothed gears is the uniform transmission of power
supposed to be constant and equal ; the one wheel conducts the other, and
they are designated severally as driver and driven, or leader and. follower. There
must be a central line of contact of the teeth, when the surfaces move with the
same velocity. In spur-wheels this line of contact is represented by circles, as
A and B (Fig. 574). These circles are called pitch-circles they must have
the same angular velocity, and the number of revolutions of each wheel in a
given time must be inversely as their diameters.
To find the relative radii of two wheels whose number of revolutions are
known : Divide the distances between their centers into parts inversely pro-
portional to the number of revolutions which the wheels are to make in the
same unit of time. Thus, let A and B (Fig. 574) be the given centers, the
ratio of their velocities being respectively two and three ; if the line joining the
centers A and B be divided into 2 -f 3 = 5 equal parts, that is, into as many
equal parts as there are units in the terms of the given ratio, the radius of the
wheel upon A will contain three of these parts, and the radius of the pinion on
B will contain the remaining two parts.
The sizes of a pair of bevels are, however, limited to no particular diame-
ters as when the axes are parallel ; the wheels may be made of any convenient
sizes, and the teeth consequently of any breadth, according to the stress they
are intended to bear. The question is the mode of determining the relative
sizes of the pair ; and this resolves itself into a division of the angle included
between the two axes inversely as
the ratio of their angular velojci-
ties. Let B and C (Fig. 575) be
the position of the two given axes,
and let them be prolonged till
they meet in a point A. Further,
let it be required that C make
seven revolutions while B makes
four. From any points D and E
in the lines A B, AC, and per-
pendicular to them, draw D cl and
E e of lengths (from a scale of
equal parts) inversely as the num-
ber of revolutions which the axes
are severally required to make in
the same unit of time. Thus, the angular velocity of axis B being 4 (Fig.
575), and that of the axis C being 7, the line D d must be drawn = 7, and the
line E e = 4. Then through d and e parallel with the axes A B and A C draw
d c and e c till they meet in c. A straight line drawn from A through c will
Fm. 575.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
277
then make the required division of the angle BAG, and define the line of
contact of the two cones, by means of which the two rolling frusta may be pro-
jected at any convenient distance from A.
Otherwise, having determined the relative perimeters, diameters, or radii, of
the pair, then the lines D d and E e are to each other directly as these quanti-
ties. B F and C F are radii of the pitch-circle.
The case in which the axes are neither parallel nor intersecting admits of
solution by means of a pair of bevels upon an intermediate axis, so situated as
to meet the others in any convenient points.
When the contiguity of the shafts is such as to permit of their being con-
nected by a single pair, skewed bevels are sometimes employed.
When the axes are at right angles to each other, and do not intersect, the
wheel and screw may be employed to connect them. The velocity of motion is
in this arrangement immediately deduced from that of the screw, its number
of threads, and the number of teeth in its gearing- wheel. Thus, if it be required
to transmit the motion of one shaft to another, contiguous and at right angles
100 1,000 2,000 3,000 4,000 5,000 6,000
Stress in Pounds.
FIG. 576.
7,000 8,000 9,000
to it the angular motions being as 20 to 1 then, if the screw be a single-
threaded one, the wheel must have 20 teeth ; but if double-threaded, the num-
ber of teeth will be increased to 40, for 2 teeth will be passed at every revolu-
tion. If the screw have few threads compared with the number of teeth of the
wheel, it must always assume the position of driver on account of the obliquity
of the thread to the axis ; and in this respect its action is analogous to that of
a traveling rack, moving endwise one tooth, while the screw makes one revo-
lution on its axis.
If the pitch-circle be divided into as many equal parts as there are teeth to
be given to the wheel, the length of one of these parts is termed the pitch of
278
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
the teeth. One of these arcs comprehends a complete tooth and space, meaning
by space the hollow opening between two contiguous teeth.
The pitch depends on the power to be transmitted or the stress on each
tooth. The diagram (Fig. 576) is by John T. Henthorn, M. E., in which
pitch and face, represented by multiples of the pitch, are proportioned to the
stress in pounds.
If the pitch be known, the number of teeth in a wheel can be determined
approximately by dividing the circumference of the wheel by the pitch, but
there must be no remainder in the quotient there can be no fraction of a
pitch either the pitch or diameter of wheel must be changed if necessary to
produce this result ; generally the latter, as gears are usually made of determi-
nate inches and fractions, as given in the table, by which also calculation for
diameters and number of teeth is much simplified.
Example 1. Given a wheel of 88
teeth, 2i-inch pitch, to find the di-
ameter of the pitch-circle. Here the
tabular number in the second column
answering to the given pitch is '7958,
which multiplied by 88 gives 70 -OS
for the diameter required.
2. Given a wheel 33 inches diam-
eter, If -inch pitch, to find the num-
ber of teeth. The corresponding fac-
tor is 1*7952, which, multiplied by
33, gives 59-242 for the number of
teeth that is, 59 teeth nearly. Now
59 would here be the nearest whole
number, but as a wheel of 60 teeth
may be preferred for convenience of
calculation of speeds, we may adopt
that number, and find the diameter
corresponding. The factor in the
second column answering to If pitch
is -557, and this multiplied by Go
gives 33'4 inches as the diameter
which the wheel ought to have.
Another mode of sizing wheels in
relation to their pitches, diameters,
and number of teeth, is adopted in
some machine shops, by dividing the
diameter of the pitch-circle into as
many equal parts as there are teeth
to be given to the wheel. To illus-
trate this by an arithmetical example,
let it be assumed that a wheel of 20
inches diameter is required to have 40
teeth ; then the diametral pitch,
p
7T
~ IT
N = p x D.
PITCH IN
HULK. To find the
KTTLE. To find the
INCHES
AND
diameter in inches,
number of teeth,
PARTS OF
multiply the number i multiply the given
AN INCH.
of teeth by the tabu- diameter in inches
lar number answer- by the tabular mim-
ing to the given ber answering to the
pitch.
given pitch.
Values of
P.
Values of -jp
Values of-p-
6
1-9095
5236
5
1-5915
6283
4|
1-4270
6981
4
1-2732
7854
3
1-1141
8976
3
9547
1-0472
2f
8754
1-1333
a*
7958
1-2566
a*
7135
1-3963
2
6366
1-5708
1*
5937
1-6755
If
5570
1-7952
If
5141
1-9264
H
4774
2-0944
1$
4377
2-2848
li
3979
2-5132
H
3568
2-7926
1
&18S
3-1416
i
2785
3 ' 5904
t
2387
4-1888
I
1989
5-0266
I
1592
6-2832
1
1194
8-3776
i
0796
12-5664
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
279
20 l
- =
40 m
that is, the diameter being divided into equal parts corresponding in number
ttf the number of teeth in the circumference of the wheel, the length of each
of these parts is | an inch, consequently m = 2 ; and according to the phrase-
ology of the workshop, the wheel is said to be one of two pitch.
In this mode of sizing wheels, a few determined values are given to m, as
20, 16, 14, 12, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, which includes a variety of pitches
from -J-inch up to 3 inches, according to the following table, which shows the
value of the circular pitches corresponding to the assigned values of m.
VALUES OF m.
1.
2.
8.
4.
5.
fi.
7.
8.
9.
10.
12.
14.
16.
20.
Corresponding eir- }
cular pitch in dec- >
3-142
1-571
1-047
785
628
524
449
393
349
314
262
224
196
157
imals of an inch. )
Fundamental principle. In order that two circles A and B (Fig. 577) may
be made to revolve by the contact of the surfaces of the curves m m and n n of
their teeth precisely as they would by the friction of their circumferences, it is
necessary and sufficient that a line drawn from the point of contact t of the
teeth to the point of contact c of the circumferences (pitch-circles) should, in
every position of the point t, be perpendicular to the surfaces of contact at that
point ; that is, in the language of mathematicians, that the straight line be a
QT&
normal to both the curves m m and n n. The principle here announced ex-
hibits a special application of one particular property of that curve known to
mathematicians as the epicycloid (see page 30).
Of epicycloidal teeth. The simplest illustration of the action of epicycloidal
teeth is when they are employed to drive a trundle, as represented in Fig. 578.
280
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
Let it be assumed that the staves of the trundle have no sensible thickness ;
that the distance of their centers apart, that is their pitch, and also their dis-
tance from the center of the trundle, that is their pitch-circle, are known.
The pitch-circles of the trundle and wheel being then drawn from their respec-
tive centers B and A, set off the pitches upon these circumferences, correspond-
ing to the number of teeth in the wheel and number of staves in the trundle ;
let five pins, ale, etc. , be fixed into the pitch-circle of the trundle to represent
the staves, and let a series of epicycloidal arcs be traced with a describing cir-
cle, equal in diameter to the radius of the pitch-circle of the trundle, and meeting
in the points Iclm n, etc., alternately from right and left. If, now, motion be
given to the wheel in the direction of the arrow, then the curved face m r will
press against the pin #, and move it in the same direction ; but as the motion
continues, the pin will slide upward till it reaches m, when the tooth and pin
will quit contact. Before this happens, the next pin a will have come into
contact with the face a I of the next tooth, which repeating the same action,
will bring the succeeding pair into contact ; and so on continually.
To allow of the required thickness of staves, it is sufficient to diminish the
size of the teeth of the wheel by a quantity equal to the radius of the staves
(sometimes increased by a certain fraction of the pitch for clearance), by draw-
ing within the primary epicycloids, at the required distance, another series of
curves parallel to these. In practice, a portion must be cut from the points of
the teeth, and also a space must be cut out within the pitch-circle of the dri-
ver, to allow the staves to pass ; but no particular form is requisite, the con-
dition to be attended to is simply to allow of sufficient space for the staves to
pass without contact.
It is a common practice of shops to take as the diameter of the rolling circle
the radius of the smallest pinion which will ever be used for gears of this pitch,
and constructing the epicycloids for different diameters of this pitch, and allow-
ing arcs of circles corresponding very closely to these epicycloids. On this
principle, Robert Adcock, 0. E., constructed a table of radii for these arcs, for
rolling circles of pinions of 8, 10, and 12 teeth. We give the last only as an-
swering the conditions of practice :
3
SMALLEST PINION,
3
e
SMALLEST PINION,
1 5
JJ
SMALLEST PINION,
1
*i
TWELVE TEETH.
3
ll
TWELVR TEETH.
3
*!
TWELVE TEETH.
^
ii
Eadiiofthe
Eadii of the
*
s -a
Radii of the
Radii of the
1 o
o
:3 -a
Radii of the
Radii of the
X
5 "
facee
of
flanks of
3
!*
faces of
flanks of
-SB
faces of
flanks of
a
w' 5 *
teeth.
teeth.
M*
teeth.
teeth.
1
' a
teeth.
teeth.
12
1-93
1-880-75
27
4-31
23
84
4-68
41
42
6'69
6-601 -89
6-91
20
13
2-09
2-04
0-76
7-45
7-14
28
46
39
85
37
38
43
85
76 '89
7-06
20
14
2-25
19
77
4-86
4-27
29
62
55
85
92
36
44
7-01
92
89
22
19
15
2-40
35
78
3-92
3-04
30
78
70
86
5-07
34
45
17
7-07
89
38
18
16
2-56
50
78
62
3-53
31
94
86
86
21
32
46
33
23
90
53
18
17
2-72
66
79
58
2-22
32
5-10
6-02
86
37
30
47
49
39
90
09
17
18
2-88
82
80
59
2-02
33
26
18
86
52
29
48
64
55
90
84
16
19
3-04
97
81 -63
1-87
34
42
34
87
67
28
49
80
71
90
8-00
16
20
3-20
3-13
81 -73
76J
35
58
49 -87
82
26
50
96
86
90
96
16
21
3-35
29 -82
83
68
36
74
65! '87
97
25
51
8-12
8-02
91
31
16
22
3-51
44
82
95
61
37
90
81 '88
6-13
24
52
28
18
91
47
15
23
3-67
60
83
4-07
56
38
6-05
97
88
29
23
53
44
34
91
63
15
24
3-83
76
83
21
51
39
21
6-13
88
44
23
54
60
50
91
79
14
25
3-99
91
84
34
47
40
37
28
88
60
22
55
76
66
91
95
14
26
4-15
4-07
84
48
44
41
53
44 -89
75
211
56
92
81
91 9-10
14
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
281
A
"
SMALLEST PINION, .cj a
SMALLEST PINION,
1 a
SMALLEST PINION,
+3
1
O -g
TWELVE TEETH.
3
-S
-1
TWELVE TEETU.
1
<^1 TWELVE TEETH.
s
x
41
Eadii of the Kadii of the
faces of flanks of
o
11
Radii of the
faces of
Eadii of the
flanks of
*
-
-a Eadii of the
32 faces of
Eadii of the
flanks of
&
IB.
teeth.
teeth.
1
' a
teeth.
teeth.
to
(g ^ teeth.
teeth.
67
9-08
8-97
91
9-26 '13
120 19-10
18-99
19-10
183 i>9 13 29-00 '97
29-27
68
23
9-13
91
42 1 '13
121
26 19-15;
42 '06
184 -28 -16
43
1-03
69
37
29
92
67 '13
122
42
30 -95
67
185 ! -44 -32
59
60
55
45
92
73 -13 123
58
46
73
06
J186 I '60 -48 -97
74
1-03
61
71
61
92
89
12
124
74
62
89
187 i -76 j -64
90
62
87 -77
92
10-05
12
125
90
78
95
20-05
06
188 -921 -80
30-06
63
10-03 -92
92
20
12
126
20-05
94
21
189 30-08 -96 '98
22
1-03
64
19
10-08
92
36
12
127
21
20-10
37
05
190 -24 30-12
38
65
35
24
92
52
12 128
37 -26
96
53
|191
40! '28 -98 '54
1-02
66
51
40
92
68 -11 129
53 -42
69 -06
192 -55i -43 1 -70
67
67
se
92
84
11
130
68
58
84
193 -71 -59
86
6S
83
72
92
99
11
131
84
74
96
21-00
05
194 -87 '75 -98
31-02
1-02
69
98
88
93
11-15
11
|132
21-00
89
16
195 31-03 '91;
18
70
11-14
11-04
93
31
11
133
17
21-05
321 -06
i!96 -1931-07
33
1-02
71
30
11-20
93
47
1-10 134
33
21
96
48
197 '35 -23 -98
49
72
46
35
93
63 '10!!l35
49
37
64 -05
198 -51
39
65
1-02
73
62
51
93
79 -10 136
65
53
96
80
199 -67
55
81
74
78
67
93
95 '10:137
81
69
96 -05
200 -83 -71 -98
97
75
94
83
93
12-10 -10; 138
96
85
22-11
201 -99 -87
32-13
76 J12-10
11-99
93
26 -0911139
22-12
22-01
96
27 -05
202 32-15 32-02|
29
77
26 12-15
42
09 iUO
28
17
43 -05
j 203 -30! -18
45
78
42
30
93
58
09 141
44
33
59
204 -46 -34
61
79
58 -47
74
09 j 142
60
48 '96
75 '05 205 -62! '50
77
80
73 -63
93
90
09,1143
76
64
91
i 206
78 -66}
92
81
89
79
13-06
09! 144
92
80
23-07 '05 207
94 -82 33-08
S2
13-05
94
93
22
09 145 123-08
96 '96
23
208
33-10 -98
24
83
21
13-10
38
09 i 146
24
23-12
38| 1-04 209
2633-14
40
84
37
26
94
53
08 ! 147
40
28J
54
1210
42
30
56
S5
53
42
94
69 -08 148
56
44!
70 -04 211
58
46
72
86
69
58
85 -08
1149
72
60 '96
86
212
74
61
88
87
85
74
94
14-01
08
J150
87
76
24-02 '04 213
90
77
34-04
88
14-01
90
94
17
08
151
24-03
92
18
214
34-06
93
20
9
17
14-06
33
08
152
19
24-07 -96
34]
215
21 34-09
36
90
33
22
94
49
08 153
35
23
501 -04 216
37 -25
51
91
49
38
94
65
08 154
51
39
65
217 -53
41
67
92
64
53
81
08 155
67
55 '96
81
04 218 -69
57
83
93
80
69
94
97
08 156
83
71
98|
219
85
73
99
94
96
85
94
15-12
07 157
99
87
25-13
04 220
35-01
89
35-15
96
15-14
15-01
30
07' 158
25-15
25-03 '97
29
221
17 35-05
31
96
28
17
94
44
07
159
31
19
45 -04 222
331 -20
'47
J7
44
33
60
M)7 160
47
35
61
223
49 -36
63
98
60
49
94
76
07 161
62
51 '97
77
! 224
65 -52
79
99
76
65
92 -07 1 162
78
66
93
04 225
80
68
95
100
92
81
95
16-08 -07 163
94
82
26-09
226
96
84
36-10
101
16-08
97
24 -07 164
26-10
98 -97
25
04 227
36 12 36-00
26
102
24
16-13
40
165
26
26-14
42
228 -28 '16
42
103
39
28
95
56
07 166
42
30
56
04 229 '44 '32
58
104
55
44
72
* 167
58
46
72
Ii230
59 -48
74
105
71
60
87
07
168
74
62 -97 '88
03H231
75 -64
90
106
87
76 '9517-03
169
90
78 27-04
1232
91 -79
37-06
107
17-03
92 -19 '06
170
27-06
94 -22
03II233 37-08
95
22
108
19
17'08 -35
171
2227-10 -97 '36
'234 -2437'H
58
109
35
24 -95 -51 -06
172
38! -25
52 -03 235 -40
27
54
110
51
40
67
173
53 -41
68
236 I -56
43
69
111
67 '56
83 -06 174
69
57
97 '84
i 237
72
59
85
112
83 -71
96
99
175
85
73
1-00
03 238
87
75
38-01
113
99
87
18-15
06
176
28-01
89
28-16
239
38-03
91
17
114
18-15
18-03
95
30
177
17
28-05
97
31
03 240
19
38-07
33
115
30 -19
46
06
178
33
21
47
241
35
23
69
116
46
35
26
179
48
37
63
! 242
51
38
85
117
62
51
96 '62
06
180
64
53
97
79 '03 243
67! -54
39-01
118
78
67
78
181
80
69
95
244
83 -70
17
119 -94
83
95
94
1-06
182 -97 '84l
29-11 1-03 '245
99 '86
33
282
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
No. of teeth.
Radius of
pitch-circle.
SMALLEST PINION,
TWELVE TEETH.
No. of teeth.
Radius of
pitch-circle.
SMALLEST PINION,
TWELVE TEETH.
No. of teeth.
Radius of
pitch-circle.
SMALLEST PINION,
TWELVE TEETH.
Radii of the
faces of
teeth.
Radii of the
flanks of
teeth.
Radii of the
faces of
teeth.
Radii of the
flanks of
teeth.
Radii of the Radii of the
faces of flanks of
teeth. teeth.
246
39-15
3902
39-28
265
42-1742-04
42-31
284
45-19
45-06 45-33
247
31
18
44
266
33 -20
46
285
35
.22
49
248
47
34
60
267
49 '36
62
286
51
38
64
249
64
50
76
268
64 '52
78
287
67
54
80
250
78
86
92
269
80 '68
94
288
83
70
96
251
94
82
40-08
270
97 '84
43-10
289
99
86
46-12
252
40-10
97
24
271
43-13
1-00
26
290
46-15
46-02
28
253
26
40-13
40
272
28
43-15
42
291
30
17
44
254
42
30
56
273
44
31
58
292
46
33
60
255
59
45
72
274
60
47
74
'293
62
49
76
256
74
61
87
275
76
63
90
294
78
65
82
257
90
77
41-03
276
92
79
44-05
295
94
81
98
258
41-06
93
20j
277
44*08
96
21
296
47-10
97
i 47'13
259
22
41-09
36
278
24
44-11
37
297
25
47-13
29
260
38
25
51
279
40
27
53
298
42
29
45
261
53
41
67
280
55
43
69
299
58
45
61
262
69
56
83
281
71
59
85
300
74
61
77
263
85
72
99
282
87
74
45-01
Rak
129
1-000-129
1-00
264
42-01
89
42-15
283
45-03
90
17
Rule. Seek in the first column of the table for the number of teeth it is
proposed that the wheel shall contain. In a line with such number of teeth
take from columns 2, 3, 4, 5, and 6 the numbers that are in them ; and in
every case multiply such numbers by the pitch. The products will be the
number of inches and parts of inches to which the compasses must be opened
to describe the circles and parts of circles that are required.
Example. Suppose that a wheel is to be made to contain thirty teeth, and
that the pitch of the teeth is to be 2| inches, proceed as follows : Seek in col-
umn 1 for 30, the number of proposed teeth, and take from column 2 the
numbers 4*783, which multiply by 2 inches, the product will be 11"*957. Open
the compasses, therefore, to this radius and describe a circle, which will be the
" pitch-circle." On an arc of this circle lay off 2*5 X '48 = 1*2" for the'thick-
ness of a tooth, and 2*5x 5'2 1*3" for the space. Having determined the
number of teeth and pitch, next, in column 3, and in the same line with 30
teeth, will be found the numbers 4*704, which multiply by 2% inches the
product will be 11*75. With the compasses opened to this distance, and from
the same center as the last, describe another circle, which will be the paths of
centers for the curves of the faces of the teeth. From column 4 similarly take
the numbers 0*865 and multiply by &J inches. The product is 2*15, to which
distance the compasses must be opened to describe the faces of the teeth.
Again, in column 5, multiply 5*07 X 2*5 = 12"- 675, and from the center,
with this radius, describe another circle for the paths of centers of flanks of the
teeth, from column 6, 1*34 X 2*5 = 3*35, the radius of the flanks of the teeth.
For the height of a tooth a common proportion is T 3 of pitch outside of
pitch-circle, and -fa of pitch within, which leaves -^ pitch for clearance at the
bottom, where usually small arcs are described to connect the teeth with the
wheel.
Having described a few teeth of any gear to its full size, the rest may belaid
off from a templet, or cutters made by which the teeth may be accurately
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
283
formed. In the illustration (Fig. 579) the teeth and spaces are proportioned
to a common form, but there is considerable variation in proportion, as
Thickness of teeth, from '45 to -48 pitch.
Space between teeth, from -55 to '52 pitch.
Height of teeth outside of pitch-circle, from '2 to *3 pitch.
Depth of teeth inside of pitch-circle, from *3 to *4 pitch.
rH-ir-
~~ _
( i """"-*.
- I
sift
'i
I ii
FIG. 579.
It is not uncommon to make one of the set of gears with wooden teeth,
mortices being cast in the periphery of the wheel for the insertion of these
teeth hence called mortise wheels the elasticity of the wood diminishes the
effect of shocks, and they
run with less noise.
The usual proportions
and construction of mor-
tise wheels are shown in
Fig. 580, a section across
and with the rim of the
wheel. The figures rep-
resent the proportions to
pitch as unity ; b is from
2 to 3 p. The teeth are
held in position by wood-
en dovetailed keys.
Fig. 581 is a section
across the rim of mortised
bevel-gear ; the figures are
as before in ratios to p.
In this illustration the
teeth are held in by pins,
not unusual also in spur-
mortise-gears.
It is unusual in drawings to complete gears with teeth according to the ex-
amples given ; it is sufficient for the purposes of pattern-making that the pitch-
circle, pitch and form of one tooth be given. For lines of shafting, spur-gears
FIG. 580.
FIG. 581.
284 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
may be represented, like plain pulleys, of the diameters of the pitch-circle,
with the pitch and number of teeth written in : bevel-gears, as in Fig. 582.
But, as in finished drawings all the detail is necessary, we proceed to give the
simplest forms of describing spur- and bevel-gears
with sufficient accuracy for all practical purposes.
Projections of a Spur- Wheel To draw side ele-
vation (Fig. 583), an edge view (Fig. 584), and a
* vertical section (Fig. 585) of a spur-wheel with 34
teeth and a pitch of two inches :
Determine the radius of the pitch-circle from the
U table, page 278 ; draw the central line A C B and
> the perpendicular D E ; on C as a center, with a
U radius 17*19, describe the pitch-circle, and divide it
FIG. 582. into 54 equal parts. To effect this division, with-
out fraying by repeated trials that part of the paper
on which the teeth are to be represented, describe from the same center c,
with any convenient radius, a circle abed ; with the same radius divide its
circumference into six equal parts, and subdivide each sixth into nine equal
parts, and draw radii to the center c ; these radii will cut the pitch-circle at
the required number of points. Divide the pitch (2 inches) into 10 equal
parts ; mark off beyond the pitch-circle a distance equal to 3 of these parts,
and within it a distance equal to 4 parts, and from the center describe cir-
cles passing through these points ; these circles are projections of the cylinders
bounding the points of the teeth and the roots of the spaces respectively.
In forming the outlines of the teeth, the radii, which, by their intersections
with the pitch-circle, divide it into the required number of parts, may be taken
as the center lines of each tooth. The thickness of the tooth, measured on the
pitch-circle, is '46 pitch, and the width of .the space is equal to '54 p. These
distances being set off, take in the compasses the length of the pitch, and from
the center g describe a circular arc h i ; and from the center /, with the same
radius, describe another arc lik touching the former ; these arcs, being termi-
nated at the circles bounding the points of the teeth and the bottoms of the
spaces respectively, form the curve of one side of a tooth. The other side is
formed in a similar manner, by drawing from the center I the arc m n, and
from the center^ the arc mo, and so on for all the rest of the teeth.
The teeth having been thus completed, we proceed to the delineation of the
rim, arms, and eye of the wheel. The thickness of the rim is usually made
equal to that of the teeth, say ^ of the pitch, which distance is accordingly set
off on a radius within the circle of the bottoms of the spaces, and a circle is
described from the center C through the point q thus obtained. Within the
rim, a strengthening feather q r, in depth about of the thickness of the rim,
is generally formed, as shown in the plate. The eye, or central aperture for
the reception of the shaft, is then drawn to the specified diameter, as also the
circle representing the thickness of metal round the eye, which is usually made
equal to the pitch of the wheel.
To draw the arms, from the center C, with the radius C u equal to the
pitch, describe a circle ; draw all the radii, as C L, which are to form the cen-
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 285
\
Fio. 534.
286 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
ter lines of the arms, and set off the distance L v, equal to pitch, on each side
of these radii at the inner circumference of the rim ; and through all the points
thus obtained draw tangents to the circle passing through u. The contiguous
arms are rounded oft' into each other by arcs of circles, whose centers are ob-
tained by the following construction : Taking, for example, the arc M P Q, it
is obvious that its center is situated in the straight line E which divides
equally the interval between two contiguous arms. Having fixed the point P
(which should be at the same distance from t as the breadth of the feather at
the back of the rim) draw through it a perpendicular K P to the line C E ; the
question now becomes simply a geometrical problem, to draw a circle touching
the three straight lines M N, P R, and S Q. Divide the angle P R M into two
equal parts by the straight line R 0, which cuts C E in the point 0, the center
of the circle required ; its radius is the line M perpendicular to M N. If
now a circle be drawn from the center C with the radius C 0, its intersection
with the radii bisecting all the intervals between the arms will give the remain-
ing centers, such as 0', of the arcs required ; and the circle passing similarly
through M marks all the points of contact M Q M', etc. To draw the small
arcs terminating the extremities of the arms, set off upon the line C E, within
the point r, the required radius of the arcs, and from the center C with a
radius C w describe a circle ; the distance r w being then transferred to the
extremities of the arms at the points where they are cut by the circle, as at Sx,
will give the centers of the arcs required. Draw the central web of the arm by
lines parallel to their radii, making the thickness about f inch for wheel of
about this size.
Having thus completed the elevation, the construction of the edge view and
vertical section becomes comparatively simple. Draw the perpendiculars F G
and H I (Figs. 584 and 585) as central lines in the representations ; set off on
each side of these lines half the breadth of the teeth, and draw parallels ; pro-
ject the teeth of Fig. 583 upon Fig. 584, by drawing through all the visible
angular points straight lines parallel to A B, and terminated at either extremity
by the verticals representing the outlines of the breadth of the wheel ; project
in like manner the circles of the hub ; lay off half length on each side of F G,
and draw parallels to it. The section (Fig. 585) is supposed to be made on the
line D E of the elevation ; project, as in Fig. 584, those portions which will be
visible in this section, and shade those parts which are in section. The arms
are made tapering in width, and somewhat less than the face of the wheel.
Since the two projections (Figs. 583 and 585) are not sufficient to exhibit
fully the true form, a cross-section of one of them is given at Fig. 586 ; this
section is supposed to be made by a plane passing through X X' and Y Y'.
The points y, z, in Fig. 583, and corresponding lines in Fig. 585, represent the
edges of key-seat.
Oblique Projection of a Spur-Wheel. In drawing a spur-wheel or other ob-
ject in an oblique position with respect to the vertical plane of projection, it is
necessary, in the first place, to lay down the elevation and plan as if it were
parallel to that plane, as represented in Figs. 587 and 589. Then transfer the
plan to Fig. 590, giving it the same inclination with the ground line which the
wheel ought to have in relation to the vertical plane ; and assuming that the
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 287
fe
288 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
horizontal line A B represents the axis of the wheel, both in the parallel and
oblique positions, the center of its front face in the latter position will be de-
termined by the intersection of a perpendicular raised from the point C' (Fig.
590) with that axis. Now, it is obvious that if we take any point, as a in Fig.
587, the projection of that point on Fig. 589 must be in the line a a, parallel to
A B ; and further, this point being projected at a' (Fig. 590), it must be in
the perpendicular a' a ; therefore the intersection of these two lines is the point
required. Thus all the remaining points b, c, d, etc., may be obtained by the
intersections of the perpendiculars raised from the points b ', c', d', etc. (Fig.
590) respectively, with the horizontals drawn through the corresponding points
in Fig. 587. It will also be observed that since the points e and/, in the fur-
ther face of the wheel, have their projections in a and b (Fig. 587), their oblique
projections will be situated in the lines a a and b b, but they are also at e and
/; consequently, the lines ea and fb are the oblique projections of the edges
a' e' and b'f. We have now to remark that all the circles which, in the rec-
tangular elevation (Fig. 587), have been employed in the construction of this
wheel are projected in the oblique view into ellipses, the length and position
of whose axes may be determined without any difficulty ; for since the plane
F' G', in which these circles are situated, is vertical, the major axes of all the
ellipses in question will obviously be perpendicular to the line A B, and equal
to the diameters of the circles of which they are respectively the projections ;
and the minor axes, representing the horizontal diameters, will all coincide
with the line A B. Thus, to obtain the ellipse into which the pitch-circle
is projected, it is only necessary to set off upon the vertical D E (Fig. 589),
above and below the point C, the radius of the pitch-circle, whose horizontal
diameter ij being at i'f (Fig. 590) is projected to ij (Fig. 589) ; and thus
having obtained the major and minor axes, the ellipse in question may easily
be constructed. The intersection of the horizontal lines g g, hh, etc., with
this circle gives the thickness of the teeth at the pitch-line ; and, by projecting
in the same manner the circles bounding the extremities and roots of the teeth,
these points in each individual tooth may be determined by a similar process.
If strict accuracy is required, a greater number of points is necessary for the
construction of the curvature of the teeth, and two additional circles m n and
op maybe drawn on Fig. 587, and projected to Fig. 589, and the points of their
intersection with the curves of the teeth projected to Fig. 589, where the cor-
responding points are indicated by the same letters.
Projections, of a Bevel -Wheel. Fig. 591 is a face view, Fig. 592 an edge
view, and Fig. 593 a vertical transverse section. For the determination of the
division of the angle of inclination of the axes of a pair of bevel- wheels, see
Fig. 575) ; for their size and proportion, the rules given for spur-wheels ; thus,
consider the base of the cone A B (Figs. 592 and 593) as the diameter of the
pitch-circle of a spur-wheel, and proportion the pitch, form, and breadth of
teeth, according to the stress to which they are to be subjected.
Having determined and laid down, according to the required conditions,
the axis S of the primitive cone, the diameter A B of its base, the angle
A S which the side of the cone makes with the axis, and the straight lines
A o, D 0', perpendicular to A S, and representing the sides of two cones, be-
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 289
290 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
tween which the breadth of the wheel (or length of the teeth) is comprised,
the first operation is to divide the primitive circle, described with the radius
A C, into a number of equal parts corresponding to the number of teeth or
pitch of the wheel. Then upon the section (Fig. 593) draw with the radius
o A or o B, supposed to move parallel to itself, outside the figure, a small por-
tion of a circle, upon which construct the outlines of a tooth M, and of the rim
of the wheel, with the same proportions and after the same manner as we have
explained in reference to spur-wheels ; set off from A and B the points a, d,
and/, denoting respectively the distances from the pitch-line to the points and
roots of the teeth, and to the inside of the rim, and join these points to the
vertex S of the primitive cone, terminating the lines of junction at the lines
D o', E o' ; the figure abed will represent the lateral form of a tooth, and the
figure cdfe a section of the rim of the wheel, by the aid of which the face
view (Fig. 591) may easily be constructed.
The points a, b, c, d, and e, having been projected upon the vertical diam-
eter A' B', describe from the center C' a series of circles passing through the
points thus obtained, and draw any radius, as C' L, passing through the center
of a tooth. On either side of the point L set off the distances L &, L I, making
up the thickness of the tooth M at the point, and indicate, in like manner,
upon the circles passing through the points B' and d', its thickness at the pitch-
line and root; then draw radii through the points i, I, Tc, g, m, etc., termi-
nating them respectively at the circles forming the projections of the corre-
sponding parts at the inner extremity of the teeth ; these radial lines will repre-
sent the rectilinear edges of all the teeth. The curvilinear outlines may be
delineated by arcs of circles, tangents to the radii g C' and i C', and passing
through the points obtained by the intersections of the radii and the various
concentric circles. The radii of these circular arcs may in general, as in the
case of spur-wheels, be taken equal to the pitch, and their centers upon the
interior and exterior pitch-circles ; thus the points g and i, n and o, for exam-
ple, are the centers for the arcs passing through the corresponding points in
the next adjacent teeth, and vice versa.
The drawing of the teeth in the edge view (Fig. 592), and of such portions
of them as are visible in the section (Fig. 593), is sufficiently explained by in-
spection of the lines of projection introduced into the plate for this purpose.
In the construction of these views, observe that every point in the principal
figure from which they are derived is situated upon the projection of the circle
drawn from the center 0', and passing through that point. Thus the points
g and i, for example, situated upon the exterior pitch-circle, will be determined
in Fig. 592 by the intersection of their lines of projection with the base A B of
the primitive cone ; and the points Tc and I will be upon the straight line
passing through a a (Fig. 593), and so on. Farther, as the lateral edges of
all the teeth in Fig. 591 are radii of circles drawn from the center C', so in
Fig. 592 they are represented by lines drawn through the various points found
as above for the outer extremities of the teeth, and converging toward the
common apex S ; while the center lines of the exterior and interior extremities
themselves all tend to the points o and o' respectively.
Skew-Bevels. When the axes of wheels are inclined to each other, and yet
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
291
do not meet in direction, and it is proposed to connect them by a single pair of
bevels, the teeth must be inclined to the base of the frusta to allow them to
come into contact. Set off a e (Fig. 594) equal to the shortest distance between
the axes (called the eccentricity], and divide it in c, so that a c is to e c as the
mean radius of the frustum to the mean radius of that with which it is to work ;
draw c m d perpendicular to a e. The line c m d gives the direction of the teeth ;
and, if from the center , with radius a c, a circle be described, the direction of
any tooth of the wheel will be a tangent to it, as at c. Draw the line d e per-
pendicular to c m d, and with a radius d e equal to c e describe a circle ; the
direction of the teeth of
the second wheel will ~"~~
be tangents to this last,
as at d.
System composed of a
Pinion driving a Rack
(Fig. 595). The pitch-
line M N of the rack
and the pitch-circle A
B D of the pinion being
laid down touching one
another, divide the lat-
ter into twice the num-
ber of equal parts that
it is to have of teeth, and set off the
common distance of these parts upon
the line M N, as many times as may
be required ; this marks the thick-
ness of the teeth and width of the
spaces in the rack. Perpendiculars
drawn through all these points to
the solid part of the rack will rep-
resent the flanks of the teeth upon
which those of the pinion are to be
developed in succession. The curva-
ture of these latter should be an in-
volute A c of the circle A B D. The
teeth might be cut off at the point of
contact d upon the line M N, for at
this position the tooth A begins its action upon that of the rack E ; but it is
better to allow a little more length ; in other words, to describe the circle
bounding the points of the teeth with a radius somewhat greater than C d.
With regard to the form of the spaces in the rack, all that is required is to
set off from M 1ST, as at the point e, a distance slightly greater than the differ-
ence A a of the radius of the pitch-circle, and that of the circle limiting the
points of the teeth, and through this point to draw a straight line F G parallel
to M N. From this line the flanks of all the teeth of the rack spring, and their
points are terminated by a portion of a cycloid A b, which, however, may in
FIG. 594.
292 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
E
VJ
P
c --
FIG. 595.
most instances be replaced by an arc of a circle. The depth of the spaces in
the pinion obviously depends upon the height of this curved portion of the
FIG. 596.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
293
teeth ; their outline is formed by a circle drawn from the center C, with a
radius a little less than the distance from this point to the straight line bound-
ing the upper surface of the teeth of the rack.
System composed of a Rack driving a Pinion. In this case the construc-
tion is in all respects identical with that of the preceding example, with this
exception, that the form proper to be given to the teeth of the rack is a cycloid
generated by a point A in the circumference of the circle AEG rolling on the
line M N. The curvature of the teeth is an involute as before.
System composed of an Internal Spur- Wheel driving a Pinion (Fig. 596).
The form of the teeth of the driving-wheel is in this instance determined by
the epicycloid described by a point in the circle A E 0, rolling on the concave
circumference of the primitive circle M A N. The points of the teeth are to
be cut oif by a circle drawn from the center of the internal wheel, and passing
FIG. 597.
through the point E, which is indicated, as before, by the contact of the curve
with the flank of the driven tooth.
The wheel being supposed to be invariably the driver, the curved por-
tion of the teeth of the pinion may be very small. This curvature is a
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
part of an epicycloid generated by a point in the circle M A N rolling upon
BAD.
System composed of an Internal Wheel driven by a Pinion (Fig. 597).
This problem involves a different mode of treatment from that employed in
the preceding cases. The epicycloidal curve A a, generated by a point in the
circle having the diameter A 0, the radius of the circle M A N, and which
rolls upon the circle BAD, can not be developed upon the flank A b, the line
described by the same point in the same circle in rolling upon the concave cir-
cumference MAN; and for this obvious reason, that that curve is situated
without the circle BAD, while the flank, on the contrary, is within it. It
becomes necessary, therefore, in order that the pinion may drive the wheel
uniformly according to the required conditions, to form the teeth so that they
shall act always upon one single point in those of the wheel. This may be
most advantageously effected by taking for the curvature of the teeth of the
pinion the epicycloid A d, described by the point A in the circle MAN rolling-
over the circle BAD. It will be observed that, as in the preceding examples,
the tooth E of the pinion begins its action upon the tooth F of the wheel at
the point of contact of their respective primitive circles, and that it is un-
necessary that it should be continued beyond the point c, because the succeed
ing tooth H will then have been brought into action upon G ; consequent!}
the teeth of the wheel might be bounded by a circle passing through the point c.
It is, however, one of the practical advantages which this species of gearing has
over wheels working externally that the surfaces of contact of the wheel and
pinion admit of being more easily increased ; and, by making the teeth some-
what longer than simple necessity demands, the strain may be distributed over
two or more teeth at the same time. The flanks of the teeth of the wheel are
formed by radii drawn to the centre 0, and their points are rounded off to en-
able them to enter freely into the spaces of the pinion.
DRAWING OF SCKEWS.
Projections of a Triangular-threaded Screw and Nut (Fig. 598). Having
drawn the ground line A B, and the center lines C C' of the figures, from as
a center, with a radius equal to that of the exterior cylinders, describe the
semicircle a 3 6 ; describe in like manner the semicircle bee with the radius of
the interior cylinder. Now draw the perpendiculars a a" and 6 6", b b" and e e",
which will represent the vertical projections of the exterior and interior cylin-
ders. Then divide the semicircle a 3 6 first described into any number of equal
parts, say 6, and through each part draw radii, which will divide the interior
semicircle similarly. On the line a' a" set off the length of the pitch as many
times as may be required ; and through the points of division draw straight
lines parallel to the ground line A B. Then divide each distance or pitch into
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
295
twice the number of equal parts that the semicircles have been divided into,
and, following instructions already laid down (page 102), construct the helix
a' 3' 6 both in the screw and nut.
Having obtained the point b' ', by the intersection of the horizontal line pass-
ing through the middle division of a' a with the perpendicular b b", describe the
helix b' c' e', which will represent the bottom of the groove. The apparent out-
G
FIG. 599.
lines of the screw and its nut will then be completed by drawing the lines b' a f ,
a' b' 9 etc. , to the curves of the helices ; these are not, strictly speaking, straight
lines, but their deviation from the straight line is, in most instances, so small
as to be imperceptible, and it is therefore unnecessary to complicate the drawing.
When a long series of threads have to be delineated, they should be drawn
mechanically, by means of a mold or templet constructed in the following
296
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
manner : Take a small slip of thin wood or pasteboard, and draw upon it the
helix a' 3' 6 to the same scale as the drawing, and pare the slip carefully
and accurately to this line. By applying this templet upon Fig. 598, so
that the points a' and 6 on the plate shall coincide with a' and 6 on the draw-
ing, the curve a' 3' 6 can be drawn mechanically, and so on for the remain-
ing curves of the outer helix. The same templet may be employed to draw
the corresponding curves in the screw-nut by simply inverting it ; but for the
interior helix a separate one must be cut, its outlines being laid off in the same
manner.
Projections of a Square-threaded Screw and Nut (Fig. 599). The depth
of the thread is equal to its thickness, and this latter to the depth of the groove.
The construction is similar to the preceding, and will be readily understood
from the drawing, the same letters and figures marking relative parts. The
parts of the curve concealed from view are shown in dotted lines.
TJ 1 -------
FIG. 600.
System composed of a Wheel and Tangent, or Endless Screw. In laying out
the work, the pitch of the teeth is to be determined by the stress, as for spur-
wheels, and the number of the teeth in the wheel by the number of turns of
the screw for each revolution of the wheel. Suppose these determined, and
(Fig. 600) to be the center of the wheel, E F the axis of the screw, C A the
radius of the pitch-circle of the wheel, and G A that of the pitch-cylinder of
the screw ; the line M N drawn through A, parallel to E F, will be the gen-
eratrix of that cylinder, which will serve the purpose of determining the form
of the teeth. The section is made through the axis, and is obviously the case
of a rack driving a pinion ; consequently the curve of the teeth, or rather
thread, of the screw should be simply a cycloid generated by a point in the cir-
cle AEG, described upon A C as a diameter, and rolling upon the straight line
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
297
M N. The outlines of the teeth are helical surfaces described about the cylin-
der forming the screw, with the pitch A b equal to the distance, measured upon
the primitive scale, between the corresponding points of two contiguous teeth.
These curves are expressed by dotted lines. The teeth of the wheel are set at
angle to the plane of its face, and with surfaces corresponding to the inclination
and helical form of the thread of the screw. Usually the points of the teeth
and bottoms of the spaces are formed of a concave outline, adapted to the con-
vexity of the screw, in order to present as much bearing surface as possible to
its action. In this kind of gearing it is invariably the screw that imparts the
FIG. 601.
FIG. 602.
motion ; but in the proportions adopted by the Yale & Towne Manufacturing
Co. for worm gearing, the wheel under the weight will revolve the screw slowly.
This angle of the teeth is found to be the best adapted for economy of power.
In the wheel the teeth in section are those of a spur-wheel, cut with a chasing
cutter, and in the screw turned in a lathe.
Figs. 601 and 602 are two views, worm and wheel, with such lines of con-
struction dotted as will explain the manner of drawing.
functional Gearing. When motion is not continuous for along time, either
having frequently to be stopped and started or reversed, frictional gearing is
298
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
very often used. The starting is with as little shock as with belting, and un-
der the proper conditions of pressure it is fully as positive, and by the usual
appliances this pressure may be applied gradually. The simplest form of fric-
tional gearing is that in which the surfaces in
contact correspond to that of the pitch-circles
(Fig. 603).
Fig. 604 is a bevel frictional gear, such as is
used in Dow's grain stores, Brooklyn, N. Y. One
half is shown in section. The surface of the
upper or larger gear is of cast-iron, that of the
lower of paper, in washers compressed by a hy-
draulic press and firmly held together by bolts.
The bevel in section is in contact with the large
wheel-surface, the other is disengaged. A slight motion to the right will throw
out that in contact, and not throw in the other, and motion ceases in the
large driven wheel ; a still further motion throws in the left pinion, and the
motion of the driven wheel is reversed.
FIG. 603.
The mode in which this is done is shown in Fig. 605. The shipper consists
of a bell-crank, controlled by a screw. The screw works in a stand, on the top
of which is a hand wheel ; the hand wheel can be moved in either direction,
and any desirable pressure can be brought upon the frictional surfaces by means
of the screw. It is not unusual, instead of two pinions to have one pinion,
with a little clearance on each side, revolving between two wheels, a slight
lateral motion, in either direction, bringing it in contact with one or the other
of the wheels. Some provision, by a loose coupling or otherwise, must be made
to admit of this lateral movement in the pinion shaft. Straight pulleys, or
what would correspond to spur-gears without teeth, are constructed, as in the ex-
ample given, and are thrown in or out of gear by a lateral motion of the pinion.
In proportioning the face of the pulleys it has been found safe to consider
it the same as belts, given in the table (page 273). The pressure can be
applied according to the requirements of driving, and there is no falling off in
the friction. The frictional surfaces are not always paper ; wood, leather, and
prepared rubber are frequently used.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
299
Wedge Gearing, or Robertson Grooved- Surf ace
Frictional Gearing. Fig. 606 is the cross-section of
the rims of two wheels of this gearing. The angle
recommended by Robertson is 50 (usually not over
30 in our practice), and the pitch to vary somewhat
with the velocity and power to be transmitted. The
adhesion, under a pressure equal to that of the ten-
sion of a belt, is proved to be greater, and it would
be safe to make the
horizontal face equal
to that of a belt
under the same cir-
cumstances of trans-
fer of power.
FIG. 605.
The use of ropes as belts has been treated of (page 274), but they are often
used, as in Fig. 607, for a reciprocating power. The ropes are not endless,
but consist of two ropes, the ends of which are
attached to two drums parallel with each other,
each having several turns on the barrels or
drums, but in opposite directions, so that, by
the motion of the drums, one rope will un-
wind from one drum and wind up on the oth-
er, and vice versa, the length of the recipro-
catory movement being measured by the turns
on one of the drums. This arrangement is
FIG. 606. sometimes applied to run the barrels of a hoist ;
the barrels being attached to one drum and
the power applied, at the other, and in this form the application may be at
considerable distance apart.
FIG. 607.
A similar arrangement with chains, instead of ropes, was much used for the
reciprocating motion of the bed in the older type of planers.
300 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
The following table is from "Appletons' Cyclopaedia of Applied Mechanics"
TABLE SHOWING EOPES ANT) CHAINS OF EQUAL STRENGTH.
SIZES, IN INCHES, FOE EQUAL 8TBENGTH.
AVERAGE WEIGHT PEE FOOT.
Working
Strain.
Crucible
Steel Eope.
Charcoal
Iron Eope.
Hemp Rope.
Iron Chain.
Steel Hope.
Iron Eope.
Hemp Eope.
Jron Chain.
Cir.
Cir.
Cir.
Diam.
Lbs.
Lbs.
Lbs.
Lbs.
Tons.
....
I'OO
2f
h
14
0'34
0-50
0-3
....
1-18
3
i
....
0-21
0-46
0-65
0-4
00
1-39
H
3 E 2
0-17
0-28
0-67
0-81
0-5
26
1-57
*J
A
0-25
0-33
0-75
0-96
0-6
45
1-77
4i
1
0-30
45
0-83
1-38
0-8
57
1-97
5
A
0-35
0-57
1-16
1-76
1-0
77
2-19
i
if
0-45
0'70
1-20
2-20
1 3
1-96
2-36
6f
i
0-59
0-83
1-60
2-63
1-5
2-36
2-75
6f
f
0-85
1-08
2-00
4*21
2-3
2'75
3-14
n
tt
1-10
1-43
2'65
4-83
3-1
2-95
3 53
8f
T28
1-80
3'35
5-75
3'8
3-14
3-93
9f
*
1-45
2'30
4-00
7'50
4-8
3'53
4-32
10|
if
1-83
2-94
4 92
9-33
5-9
3-93
4-71
111
ih
2'33
3-56
5-83
10-6
7-0
4'32 5-10
12
H
2 98
4-00
6-20
11-9
8-2
4-71 5-50
14f
H
3-58
4-80
8-70
14-5
9-5
4-81
5-89
15*
if
3'65
5 60
9-00
17-6
11-0
5-10
6'28
15|
H
4 04
6-30
10-1
20*0
12-5
5-89
7-07
17*
if
5-65
7'95
18 7
22-3
15-9
6-35
7-85
m
if
6 50
9'81
16-4
24-3
19-6
Endless chains are often used for the transmission of power, where the
stress is great and the movement slow. When the chain used is of the com-
mon form, the wheels must be fitted with depressions or caps to receive the
flat links, with a slot for the vertical links, as in Fig. 617. A chain com-
1
:
~i P
J
.
1 1
1 1
I i 1
1
1
I
i r
1
n r
: ! J
1
J U
1
r "T
J
FIG.
posed of punched links, as in Fig. 608, admits of a tooth between the links,
and the wheels on which these run have therefore teeth adapted to the chain,
which is composed of links of uniform length.
But the chief application of ropes and chains is for the purpose of hoisting
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
301
or lowering heavy weights or loads, by the means of pulleys and blocks, or bar-
rels and capstans.
Rope for running-rigging is usually made of hemp or manilla, and wire-
rope for this purpose is mostly made with hemp centers.
A simple rule for the working-strength of
these ropes is to multiply the square of the
girth or circumference of the rope by 100 for
hemp or manilla, 600 for iron- wire rope, and
FIG. 609.
FIG. 610.
1,000 for steel-wire rope, and the result will be the working-strength in pounds.
Fig. 609 is the front and side view of a common wooden block, iron-
strapped. The pulley or sheave is shown in Fig. 610 ; the section shows a bush-
ing at the center for the pin ; the sheaves are of lignum vitae.
FIG. 611.
FIG. 612.
FIG. 613.
FIG. 614.
Figs. 611, 612, 613, 614 are wrought-iron tackle-blocks of the Yale & Towne
Manufacturing Company's pattern. The lower block of every set is always
sent with a becket attached, as shown in Fig. 612.
Diameter of sheave
In.
21
In.
31
In.
4
In.
4|
In.
5
i,.
6&
In.
Y
In.
8
In.
q
In.
10
In.
11
Will take rope diameter.
1
1
1
1
U
u
H
9,
91
9,1
Will take chain diameter
3
1
5
A
JL
A
4
4
Grin-blocks (Fig. 615). These blocks are made with wrought- and malle-
able-iron frames and wrought swivel-hook.
302
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
In. 1 In.
In.
In.
In.
In.
In.
Diameter of wheel
10 12
14
16
18
90
22
Will take rope, diameter
1 1
11
H
14-
H
li
Ton. Ton.
Ton.
Ton.
Ton.
Ton.
Ton.
Will carry about
1 U
11
0,
2i
2i
21
*
FIG. 615.
Winding-drums or barrels must have their diameters pro-
portioned to the diameters of the rope or chain to be used (see
table of sheaves above), and their length to the length of rope
or chain to be taken in, and when the coils or turns of the rope
are numerous provision must often be made for keeping the
rope or chain so that one coil may not ride on another. This is done by spiral
grooves in the barrel, or shifting the barrel or the rope-guide automatically.
Fig. 1504 shows the way in which a chain cable is taken in with but few coils
on the barrel. The coils are sufficient for the friction of taking up the cable ;
the tight cable is wound on the larger part of the barrel, and as the coils are
unwound on the slack side the tight coil slips down to a smaller diameter ;
the weight of the chain on the slack side, as it drops into the locker, is suffi-
cient to preserve the friction ; but with a rope, a man takes in the rope and
exerts at the same time a little strain. The application of a barrel of this form
for hoisting is very common ; by exerting a slight stress the man can hoist a
weight on a revolving barrel, and by slacking he can lower without changing
the direction of motion or speed of the barrel.
Chain-wheels with pockets, which have been spoken of in their application
to the transmission of power, are also especially applicable to the purpose of
hoisting, requiring a width only slightly greater than that of the chain, and
a diameter sufficient to give the proper engagement with it.
FIG. 617.
FIG. 616.
Flat punched links are of uniform length, and can be purchased of any de-
sirable sizes, and put together in multiples ; common chain has not that uni-
formity in length to adapt it nicely to the pockets of the wheel. The Yale
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
303
& Towne Manufacturing Company have made a spiral chain, of common form
but of uniform length, especially adapted to hoists, and Figs. 616, 617, and
618 illustrate the construction of their chain-wheel. A is a pocketed chain-
wheel, made of soft cast-iron, mounted on a frame B. is the chain-guide
enveloping the lower half of the chain-wheel. The inner curved surface of the
FIG. 619.
FIG. 620.
FIG. 621.
FIG. 622.
FIG. 623.
FIG. 624.
chain-guide is grooved, and is of such a shape as to leave a space between it and
the periphery of the chain-wheel merely sufficient to admit the chain ; it must
then enter properly and continue engaged with the chain- wheel. E is a chain-
guide roller, that delivers the slack chain into the box or locker. D is the
chain-stripper, bolted also to the plate B, with a tongue or rib projecting into
the center groove of the wheel which disengages the chain.
The usual forms of chain-cables are represented by the open circular link
(Fig. 619), the open oval (Fig. 620), oval with pointed stud (Fig. 621), oval
with broad-headed stud (Fig. 622), an obtuse angled stud-link (Fig. 623), and
the parallel-sided stud- link (Fig. 624). The usual proportions of chain-links are
6 diameters of the iron in length
by 3 in width. The end links,
which terminate each 15 fathoms
of chain, are 6*5 in length to 4*1
in breadth, and the iron about
1-2 the diameter of the rest of
the chain.
Hooks.Fig8. 625 and 626
(from Redtenbacher) represent
two wrought-iron hooks, in which
the material is distributed accord-
ing to the strain to which the
parts may be subjected. The
following are the proportions on
which Fig. 625 is constructed :
Assuming the neck of the hook
as the modulus or 1, the diam-
eter of journals of the traverse
are 1*1 ; width of traverse at center, 2 ; distance from the center of the hook
to the center of the traverse, 7*5; interior circle of the hook, 3*4; greatest
FIG. 625.
FIG. 626.
304:
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
FIG. 627.
thickness of the hook, 2 '8. Assuming (Fig.
626) the diameter of the wire of the chain
as 1 : interior circle of hook is 3 '2, and
greatest thickness of hook 3 *5.
Fig. 627 represents a hook as made by the
Yale & Towne Manufacturing Company.
This hook is fitted in a cross-head ; the diam-
eter at A is that of iron from which the hook
is forged, and the section shown hatched at
the center of the hook is equal to that of the
round iron.
It has been shown that hooks, of the pro-
portions but with a much greater load than
given in the following table, yield by the
gradual opening of the jaw, giving ample
notice before rupture.
Capacity of hook
Ton.
1
Ton.
i
Ton.
1
Ton.
1
Ton.
Ton.
2
Ton.
3
Ton.
4
Ton.
5
Ton.
6
Ton.
8
Ton.
10
Dimensions of A
In.
4
In.
if
In.
4
In.
1-iV
In.
11
In.
15-
In.
14
In.
2
In.
9,1
In.
<>4
In.
25-
In.
31
Dimensions of D
11
1*
H
14
9,
9,1
9,4
31
3
41
51
61
All parts of the hook are expressed in parts of A, and can readily be de-
termined from the scale above.
Figs. 628 and 629 are side and front elevations of an ordinary straight lever
on a shaft ; both are shown broken, either because the
length is indefinite, or because it is inconvenient to put on
the paper. The handle should be from 5 to 6 inches long,
and 1^ diameter. The bar beneath the handle to be square,
and of uniform width on one side of the lever and a taper
on the other, as shown, of about y in 4 feet on each side.
The sides of the square at the handle to be i |/ length in
inches, or say for 30" lever, $" for 4 feet, and 1" for 5
feet. The neck of the shaft to be, as proportioned in the
drawing, about T ^- of the greatest width of the lever, and
the diameter of hub 1^-. The stress exerted by a man may
be from 75 to 100 pounds, and the size of the shaft will
depend on the torsion al stress between the hub of the lever
and the point of resistance.
Fig. 630 is a hand-lever forming one arm of a bell-crank
a bolt passing through a slot in the frame and the arm of
the lever, and the two are clamped together by a thumb-
FIG. 628. FIG. 629. nut, n, by which the lever can be held in any position.
The same purpose is often effected by notches in the
frame, into which the arm of the lever is caught, or by spring latches, as in
Fig. 631,
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
305
Figs. 632 and 633 are side view and plan of a foot-lever. The foot-plate is
8" X 5" X f ", and as the lever is subject to double the stress of the hand-lever
above, the dimension should be somewhat increased. The side of square next
FIG. 630.
FIG. 631.
FIG. 632. FIG. 633.
the foot-plate should be, say for a lever of 30", V ; of 4 feet, 1^ ; of 5 feet, 1;
the form and taper as in the hand-lever.
Figs. 634 and 635 are views of a hand-crank. The diameter of the handles,
for convenience in grasping, should not be less than 1-J" ; if for the force of two
men, l-j-", and from the diameter of the handle the rest may be proportioned as
in the figure. The length
of handle for a single man
should be from 10" to 12" ;
for two men, from 20 to
24 : the crank from 15" to
FIG. 634.
FIG. 635.
18", and the height of shaft
above the foot support for
the men from 2' 10" to 3' 2".
Engine - Cranks. Fig.
636 is a graphic represen-
tation made from a table
from Bourne's " Handbook
of the Steam-Engine," for
determining "the diame-
ters of wrought crank-shaft journals" i. e., of the large eye of the crank.
The ordinates are diameters in inches of the steam cylinder, the inclined lines
the stroke in feet, and the abscissas the diameters of the eye in inches.
Use of the Table. To find the diameter of large eye of crank of a steam-
engine 40" cylinder and 4-foot stroke. Find on what line of abscissa is the in-
tersection of the ordinate 40" with the diagonal 4' of stroke, which will be
about S-J", the diameter of crank-eye.
20
306
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
The table is calculated on a steam pressure in the cylinder of 25 pounds ;
not the average pressure, but the maximum. This pressure is much less than
present practice, but the table can be readily adapted to any pressure. For
Diameter of Eye.
Ao' ~"
jj .
~ ~ = * * *----;;=-" - "\^\\\" " ~tj^
i. r~ ~ ~ --- - .,
'"""" -\~T " '" _-r-r " j_--'
1 "" ~ " " ~^~ie:"-l
_--="' ----""i--- I" " ~
," r -,ijf "_--*'" ;SE3"^*E
_-- ;; = ;; " - - -
u EE;;:^= ;:!;=!:::::=:::
if \]^"^^^^^^?^\\^\\\ \\\\\\
::!=!;;"=;:
l =::::
20"
jo* w* .w
;0' (9/r tt 5^
Diameter of Steam-Cylinder in Inches.
FIG. 636.
most stationary engines the pressure is from 75 to 100 pounds ; for 75, the area
on diagram must be three times what it is for 25 pounds. Thus, for a steam-
cylinder of 30" diameter and under 75 pounds pressure, multiply the area of
75
30"D. X 25 = 706-9 X 3=2120'7 = area of 52" diameter, which use for determin-
ing the diameter of
eye instead of 30".
It will agree very
nearly with com-
mon practice for
stationary engines
to multiply the di-
ameter of cylinder
in diagram by 2, for the diameter
to be used, and for locomotives, by
For the small eye of the crank,
under the same conditions of pres-
sure, Bourne gives the rule : Multi-
ply diameter of cylinder by -142.
This is too small for the present prac-
tice, which is from -17 to '25 or -J
to \ the diameter of the cylinder.
The crank-pins are made of steel or FIG. 63Y. FIG. 638.
iron case-hardened. Eyes are bored
by hydraulic or screw press to a very tight fit, and forced on to the shaft or
pin, or heated and shrunk on.
Figs. 637 and 638 are two views of a wrought-iron crank, and Figs. 639
and 640 of a cast-iron crank,* both proportioned in their parts to the diameter
* " Elements of Machine Design," Unwin.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
307
.5 G.U
of the large eye
as unity, but, as
shown by the di-
agram and rule
following, these
figures can only
apply to a single
throw of crank,
as the diameters of the two eyes
vary as their distances apart.
Taking the diameter of the large
eye of the crank, Eedtenbacher
gives in the table the relative sizes
of central and end eyes of cranks,
depending on the proportion be-
tween the length of crank and the
diameter of central eye. The first
column exhibits the number of
times the diameter of eye is con-
tained in the length of crank ; the
second and third columns give the suitable diameters of crank-pins.
Figs. 641 and 642 represent a side and front elevation of a crank, such as
FIG. 639.
FIG. 640.
is used on engines of American river boats.
The main body of the crank is of
cast-iron, with two horns a a
projecting from the central hub,
and the whole is bound with a
strap of wrought-iron.
DIAMETEE OF EYE, BEING UNITS.
-tr
FIG. 641.
For wrought-
iron shafts.
Cast-iron
shafts
2
85
0-62
3
0-69
0'51
4
0-60
0-44
5
0-54
0-39
6
0-49
0-36
7
0-45
0-33
8
0-42
0-31
9
0-40
0-29
10
0-38
0-28
11
0-36
26
12
0-34
0-25
10
0-33
0-24
FIG. 642.
The diameters of crank-pins
as above given are on the basis of a length of from 1 to 1-J of the diameter ;
if the length be increased beyond this the diameter should be increased in the
ratio of 1 to the square root of the diameter.
308
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
Disk-cranks are circular disks of cast-iron, with crank-pins of iron or steel,
and as much strength of metal around the pin as in the crank. They are bet-
ter than the crank, in that there is no unbalanced crank and pin, and part of
the weight of the connection can be balanced by a proper disposition of metal
within the area of the disk.
Fig. 643 is a plan of a double crank-axle, although by the projection the
FIG. 644.
FIG. 645.
FIG. 646.
lower axle A appears as a straight shaft. The dimensions given are from an
axle in use. In construction the cranks are rectangular in section, of which
the width is $ the depth, and the depth 1*5 the diameter of crank-journal
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
309
Cranks are usually forged solid, and the slot for the crank cut out ; that shown
in the figure was cast in steel for a double compound engine, 7 X 15 X 15, and
although there is often great condensation in the cylinders, it has worked
satisfactorily for many years.
Eccentrics. An eccentric is a modified crank ; the crank-pin is enlarged so
as to include the crank-shaft ; motion is conveyed through the crank to the
pin, and not through the pin to the shaft.
Fig. 644 represents a front view, Fig. 645 the side view, and Fig. 646 a sec-
tion, of a form of eccentric usually adopted in steam-engines for giving motion
to the valves regulating the action of the steam upon the piston. A ring or
hoop, eccentric strap, is accurately fitted within projecting ledges on the outer
circumference of the eccentric, so that the latter may revolve freely within it ;
this ring is connected by an inflexible rod with a system of levers, by which the
valve is moved. It is evident, that as the shaft to which the eccentric is fixed
revolves, an alternating rectilinear motion will be impressed upon the rod, its
amount being determined by the eccentricity, or distance between the center of
the shaft and that of the exterior circle. The throw of the eccentric is twice
the eccentricity C E ; or it may be expressed as the diameter of the circle de-
scribed by the point E. The nature of the alternating motion generated by
the circular eccentric is identical with that of the crank.
FIG. 647.
Fig. 647 is a common form of eccentric strap and rod adapted to the draw-
ing of the eccentric given ; it is usually fitted with a composition bush, and a
pan must be provided beneath to catch any oil that may drip from the eccen-
tric. This last may be avoided by the use of an eccentric strap, Figs. 648, 649,
650, in which it will be seen that the strap forms a cup-section (Fig. 650)
which secures a projecting ring on the eccentric, and retains the oil. These
figures represent the eccentric strap of a locomotive, and are made entirely of
cast-iron ; the bolts are very long, and the strap exceedingly rigid.
In practice, the term eccentric is generally confined to the circular eccen-
tric ; all others, with exception of that last described, being called cams or
wipers.
Projections of Eccentrics. The term eccentric is often applied in general
to all such curves as are composed of points situated at unequal distances from
a central point or axis.
310
MACHINE DESIGN AND MECHANICAL CONSTEUCTIONS.
FIG. 650.
FIG. 648.
FIG. 649.
Fig. 651. To draw the eccentrical symmetrical curve called the heart, which
is such as, when revolving with a uniform motion on its axis, to communicate
to a movable point A, a uniform rectilinear motion of ascent and descent.
Let C be the axis or center of
rotation upon which the eccentric is
fixed, and which is supposed to re-
volve uniformly ; and let A A' be
the distance which the point A is
s required to traverse during a half
\ revolution of the eccentric. From
\ the center C, with radii respectively
ft
equal to C A and C A', describe two
circles ; divide the greatest into any
number of equal parts (say 16), and
draw through these points of di-
vision the radii 01, C 2, 03, etc.
Then divide the line A A' into the
same number of equal parts as are
contained in the semicircle (that is,
into 8 in the example now before
us), and through all the points 1',
2', 3', etc., draw circles concentric with the former ; the points of their inter-
section B, D, E, etc., with the respective radii C 1, C 2, C 3, etc., are points
in the curve required, its vertex being at the point 8.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
311
It will now. be obvious that when the axis, in its angular motion, shall have
passed through one division ; in other words, when the radius 1 coincides
with C A', the point A, being urged upward by the curvature of the revolving
body on which it rests, will have taken the position indicated by 1' ; and fur-
ther, when the succeeding radius C 2 shall have assumed the same position, the
point A will have been raised to 2', and so on till it arrives at A', after a half
revolution of the eccentric. The remaining half, A G F 8, of the eccentric,
being exactly symmetrical with the other, will enable the point A to descend
in precisely the same manner as it is elevated. It is thus manifest that this
curve is fitted to impress a uniform motion upon the point A itself, but in
practice a small friction roller is usually interposed between the surface of the
eccentric and the piece which is to be actuated by it. Accordingly, the point
A is to be taken as the center of this roller, and the curve whose construction
we have just explained is replaced by another, similar to and equidistant from
it, which is drawn tangentially to arcs of circles described from the various
points in the primary curve with the radius of the roller. This second curve
is manifestly endowed with the same properties as the other ; for, supposing
the point e, for example, to coincide with A, if we cause the axis to revolve
through a distance equal to one of the divisions the point/, which is the inter-
section of the curve with the circle whose radius is C 1', will then obviously
have assumed the position V ; at the next portion of the revolution, the point g
(which is such that the angle/ C g is equal to e C/) will have arrived at 2', and
so on. Thus it is plain that the point a will be elevated and depressed uni-
formly by means of the second curve, in the same manner as that denoted by
A is actuated by the first.
It is obvious that the movable point a must, in actual working, be held
in contact with the surface of the ^
eccentric ; this is generally accom-
plished by the action of a weight
or of a spring ; but in forms simi-
lar to Fig. 651, in which all the
diameters, as A A 8, B F, D G, etc.,
are equal, two frictions connected
and placed diametrically opposite
each other may be used, which will
be thus alternately and similarly
impelled ; in many cases an eccen-
tric groove is cut, and the friction
roll or point a is made to slide in
this groove.
Fig. 652. To draw a double and
symmetrical eccentric curve, such as
to cause the point A to move in a
straight line, and with an unequal
motion ; the velocity of ascent being accelerated in a given ratio from the start-
ing-point to the vertex of the curve, and the velocity of descent being retarded
in the same ratio.
FIG. 652.
312
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
Upon A A' as a diameter describe a semicircle, and divide it into any num-
ber of equal parts ; draw from each point of division 1, 2, 3, etc. , perpendicu-
lars upon C A' ; and through the points of intersection 1', 2', 3', etc., draw
circles having for their common center the point C, which is to be joined, as
before, to all the points of division on the circle (A' 48). The points of inter-
section of the concentric circles with the radii 01, 02, 03, etc., are points in
the curve required.
Fig. 653. To construct a double and symmetrical eccentric, which shall
produce a uniform rectilinear motion, with periods of rest at the points nearest
to, and farthest from, the axis of rotation.
The lines in the figure above referred to indicate sufficiently plainly, with-
out the aid of further description, the construction of the curve in question,
which is simply a modification of the eccentric represented at Fig. 651. In
the present example, the eccentric is adapted to allow the movable point A to
remain in a state of rest during the first quarter of a revolution B D ; then,
FIG. 653.
FIG. 654. FIG. 655.
during the second quarter, to cause it to traverse, with a uniform motion, a
given straight line A A', by means of the curve D G- ; again, during the next
quarter E F G, to render it stationary at the elevation of the point A' ; and
finally, to allow it to subside along the curve B E, with the same uniform mo-'
tion as it was elevated, to its original position, after having performed the entire
revolution.
Fig. 654 represents an edge view of this eccentric, and Fig. 655 a vertical
section of it.
If but one side of this were constructed, and the motion only equal to that
of the arc and reciprocating, it would raise and lower every point resting on it,
and would be called a wiper. The wiped surface is generally flat, an arm
extending out from the rod to be raised, and a curve D Gr may be formed
adapted to any height of lift, and action during the lift.
Connections. Figs. 656 and 657 are sections of cottered joints of wrought-
iron bars, the first made with a socket and the end of one of the bars ; the
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
313
latter by a sleeve connecting the two bars. The bars in the socket and sleeve
are upset, to give more section than the bars themselves, so that the slots cut
for the cotters c c will not reduce the strength below that of the bars. The
cotters must have sufficient shearing strength and bearing surface, and at the
same time diminish as little as possible the section of the parts connected.
The proportions given in the figures are drawn to a scale of the diameter of
the enlarged part as the unit, and the proportions given in figures are such as
obtain in practice for wrought-iron. If the cotter be of steel, its breadth may
be f of that given, preserving the other dimensions the same ; the thickness is
*25 of the unit.
The knuckle-joint (Fig. 658) is given in dimensions of the bar as a unit,
and adapted to usual work. If there is much motion at the joint, the wearing-
surface should be larger, by increasing
the width of the eyes and the length
of the pin. The pin in the drawing is
through the collar ; usually the pin is
extended, and the pin passes through
the bolt outside the collar.
Connecting-rods, in their applica-
tion to steam-engines, are the rods
connecting the piston through the
cross-head to the crank. When two
cranks are connected it is called a
coupling-rod.
Figs. 659, 660, and 661 are side
plan and end views of a connecting- FIG. 658.
rod, as made by the South wark Foun-
dry, of Philadelphia, and used on their fast-running Porter- Allen engines.
The cross-head end is a strap-end, while that of the crank is a box-end, and
the latter is made of larger diameter than the former on account of the
application of the stress to the crank-pin, and the wear, this pin is made larger
than the pin of the cross-head. The length of the page does not admit of the
representation of the full length of the connecting-rod on the scale ; it is
314: MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
315
therefore shown broken, with the dimensions figured in. The sections of the
two ends are drawn in on the rods ; the circular section A is the same as that
FIG. 662.
of the piston-rod, and both are represented in the conventional hatching of
cast-iron. This is of wrought-iron. The gib g and key or cotter v at the strap-
FIG. 664.
end are of steel, and the key is fastened when in position by a set-screw through
the head. At the box-end, a wedge and screw forces the box into position.
316
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
It will be observed on the plan that this rod is drawn as though it were
flat on top ; but as the tops are curved, it is more accurately represented in
Fig. 662.
FIG. 665.
Fig. 663 is a strap-end of a connecting-rod, from the Corliss Steam-Engine
Company. The peculiarity is the adjusting-screws connected with the boxes.
Fig. 664 is the strap-end of a locomotive connecting-rod in which the wear
of the boxes is taken up by a cotter at the end ot the strap.
FIG. 667.
In Fig. 665 the key is between the bolts ; the weakness from bolt-holes or
cotter-slot is compensated by the width of the strap.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 317
Fig. 666 is a cast-iron eccentric strap ; the bolts are very long and the con-
nection very rigid. The box is fitted with metalline, which is put in small
disks ; oiling is thereby avoided.
The bolts for the large end are bored up for the greater part of their length,
to reduce their sectional area to that of the screwed portion and thus secure
equal elasticity ; with these long bolts no check-nuts are necessary.
In many marine engines the boxes of both crank and cross-head pins are
made similar to this, with the bolts strong and heavy, and connecting the two
boxes without any other rod.
FIG. 669.
FIG. 670.
Fig. 669 is the box-end of a locomotive ; the section (Fig. 670) is expressed
in shade line merely, without hatching.
Fig. 671 is the stub end of a coupling-rod. The bushes are solid, of brass,
and kept from turning round by taper-pins, which are secured by set-screws
pressing on the larger end ; taper, ^ in 3 inches.
Fig. 672 represents the forked end of a cast-iron connecting-rod of an Eng-
lish type, the end of the working-beam coming within the forks. Wrought-
iron connecting-rods of this kind are most generally used. One side of the fork
is shown in section, with its bosses, a #, and the cotters, c c.
318
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
Fig. 673 is a section of the lower end of the same beam.
The lower box n is held in position by a spherical boss, fill-
ing a recess in the rod, the upper brass by the cotter ; there
FIG. 671.
is a cover c over the box and crank-pin. The small
channel in the upper box is for the introduction of oil.
Cast-iron connecting-rods are now very seldom used.
In some cases of vertical-beam pumping engines, it is
necessary that the water-load of the pump should be
counterbalanced by some dead weight of material, and
it is then sometimes convenient to make use of a heavy
pump-connection. The wrought-iron crank connec-
FIG. 672.
FIG. 673.
tions of American river-boat engines are peculiar in their
construction. They are made as light as possible, with
very great stiffness. Fig. 674 represents the side ele-
FIG. 674.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
319
vation of such a connecting-rod. The means adopted to give the required
stiffness consist of a double-truss brace, a a, of round iron, which is fastened
by bolts to the rod near each end ; struts b #, cut with a screw, and furnished
with nuts, pass through the center of the brace, by which means the braces
are tightened. The connecting-rod at its smallest part near the extremities is
Fia. 675.
FIG. 676.
FIG. 679.
of the same diameter as the piston-
rod ; the boss in the center is from
one to two inches more.
Fig. 675 is the front view of the
forked end of the rod, which is fitted
with the usual straps, gibs, and cotters. Fig. 676 is the side view of the brace-rod.
The cross-head is the link between the piston-rod of the steam-engine and
the connecting-rod to the crank.
Figs. 677, 678 and 679 represent the plan, end view, and section of the
cross-head adopted by the Southwark Foundry for their high-speed engines.
It is of cast-iron, with large, flat faces, the pin p for the connecting-rod
being in the middle of the length. This pin is of wrought-iron, large and
flattened on top and bottom, so that the boxes of the rod can never bind on
the pin at the extreme of the vibrations of the rod ; usually these pins are
round. The pin is formed with large squares at the ends, by which it is fitted
into the jaws of the cross-head, where it is secured by a steel pin passing
through the cross-head. The bearing surfaces of the head and those of the
guide-bars are finished by scraping to true planes ; there are no means of ad-
justment, as there is no wear if kept clean.
It is to be understood that the piston-rod moves in a straight line, and that
the stress on the connecting-rod pin is mostly oblique. Guides are to be pro-
vided, between which the cross-head slides, to take the oblique stress off the
piston-rod.
Figs. 680 and 681 are elevation and plan of guide-bars which are in common
use for both vertical and horizontal engines. Lugs or ears are cast on the steam-
cylinders, and on the frames to which the bars are bolted, and between which the
cross-head slides. The grooves or notches across the guide-bars, at the ends of
the stroke, are to throw off any grease or dirt that may be carried along by the
head and prevent their accumulation. The stress on the guide-bars is due to
the pressure of the steam on the piston acting obliquely on the crank through
320
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
the connecting-rod, and is the greatest when the crank is at right angles to the
piston. It can be determined by multiplying the pressure on the piston by
the length of the crank, and dividing the product by the length of the con-
necting-rod, which will be the stress tending to separate the guides. If the
FIG. 680.
31
FIG. 681.
connecting-rod be 3 times the stroke, or 6 times that of the crank, which is the
usual proportion, then the stress is -J- the pressure on the piston. Sometimes
the proportion of connecting-rod to stroke is 2% to 1. When a portion of the
force of the steam is opposed directly to the resistance, as in direct-acting
pumps, and only the irregularities in the steam-pressure are transmitted through
the connecting-rods, the proportion of rod to stroke may be still smaller. In
this case the force transmitted to the fly-wheel is retransmitted to the cross-
FIG. 682.
head, whenever the resistance in the pumps exceeds the pressure of the steam,
thus utilizing the expansive properties of the steam by a cut-off.
When the top of the engine-frame is horizontal it may form the lower
guide of the cross-head. In many engines the guides are formed in the frame
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
321
itself (Fig. 682), in which the bearing surfaces of the guides are arcs of circles
within a pipe, open on the face, which forms a part of the frame and is bored
at the same time with the cylinder, and conse-
quently in true line. On locomotives it is not
unusual to have the guide on one side, as in Fig.
683, where the slide-bars are of wrought-iron and
the slide-block is fastened between the two plates
of the cross-head by bolts. It is the most com-
mon practice in this country to use guides with
vertical engines, even when the connection is with ** 688
working-beams, but abroad the parallel motion is
more popular. The working-beam is seldom applied to stationary engines, but
only to marine and pumping engines.
FIG. 684.
Fig. 684, elevation of engine of the " New World," may be taken as the
type of a North Kiver steamboat engine. The frame-work is composed of four
21
322
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
pieces of heavy pine timber, d d, which are formed into two triangles, and in-
clined slightly laterally to each other ; their lower ends rest on the keelsons,
and upon their upper extremities are placed the pillow-block c of the work-
ing-beam. They are solidly fastened together and to the boat by numer-
ous horizontal and diagonal timbers, which are secured by wooden knees and
keys, and are heavily bolted. The two front legs are bolted to flanges cast on
the sides of the condenser, and the other end of the framing is attached to a
large mass of timbers, which support the shaft pillow-block b ; the framing is
further steadied by two additional timbers, and rods running from the beam
pillow-blocks outside the shaft to the keelsons of the boat. The guides a are
bolted at the bottom to the cylinder-flange, and retained in their vertical po-
sition by wrought-iron braces connected with the framing. The height of the
frame is 46 feet, width at bottom 31 feet.
Figs. 685 and 686 are views of the working-beam on a larger scale. It is
composed of a skeleton frame of cast-iron, round which a wrought-iron strap
FIG. 686.
is fitted and fastened. This strap is forged in one piece, and its extreme ends
are formed into large eyes, which are bored to receive the end -pins or journals.
The skeleton frame is a single casting, and contains the eyes for the main cen-
ter and air-pump journals ; the center hub is strengthened by wrought-iron
hoops shrunk upon it. At the points of contact of the strap and skeleton, key-
beds are prepared. Small straps connect the frame and main-strap at these
points, keyed to the frame keys riveted over. The frame is further braced by
wrought-iron straps, C C, which tie the middle of the long arms to the ex-
tremities of the shorter ones. The following are the general dimensions : From
center to center of end-journals, 26 feet ; this is somewhat less than the usual
proportion to length of stroke, being slightly less than double the stroke ;
length of center hub, 26", a a ; diameter of main center eye c, 15f " ; of air-pump
journal-eye d, 6f " ; of end -journals e e, 8-J.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
323
Fig. 687 is the side elevation, Fig. 688 a plan, and Fig. 689 a section
through the hub of a cast-iron working-beam. The proportions are as in prac-
tice, but the end as shown is not usual. Fig. 690 shows the way in which the
FIG. 687.
Fm. 688.
connection-rod is attached, the dotted lines showing the head, which passes
over the end pivot. The common form of the end is like that of the working-
beam (Fig. 685).
FIG. 690.
From the following table of practical examples from "Architecture of Ma-
chinery," it is safe to assume as a rule for the working-beams of land engines,
that the depth at center should be the diameter of the cylinder, and the length
of beam three times the length of stroke. The outline is parabolic, having for
the vertex the extremity of the beam and the point B in the curve at the center.
The sectional area may be estimated from rules already given, knowing the
load at the extremity, that is, the pressure on the piston, the weight of the
same and its connections, and also the force required to drive the air-pump,
estimated at the extremity of the lever. As an engine is subject to shocks, the
load should be estimated at six times the absolute load. Five per cent of the
324
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
nominal power of the engine may be considered the maximum of power required
to drive the air-pump.
Diameter of
cylinder.
Length of
stroke.
Description of
work.
Length of beam
from center.
Depth at
center.
Sectional
area.
Inches.
Ft. In.
Ft. In.
Inches.
Square inches.
tff
8
Rolling,
12 4
48
240
40f
7
Pumping,
10 4
36
162
39|
6 9
Blowing,
9 6
38i
96
36f
6 3
Rolling,
9 3
30
60
24f
5
Mill-work,
8
25
50
18*
4
6 10
22^
50
42
4
Marine,
6 3
23
138
42
4 2
u
6 6
27
216
32
3
II
6
22
132
Double plates or flitches of wrought-iron are often used in the construction
of working-beams and side-levers. Fig. 691 is the section between the two
plates of a beam of this kind, attached to the compound pumping engines at
Milwaukee, Wis. The plates are each 30 feet long, by 6' 4" deep at center, by
If" thick. The connections between the two, shown in section in the figure,
are cast-iron pipes with wide flanges at each end riveted or bolted to the plates.
The main center and other small journal-pins are rods of wrought-iron, passing
through the pipes, and extending outside the plates to form the journals ; c is
FIG. 691.
the section of the pin for crank connection, p for that of pump, li for that of
high-pressure cylinder, I for that of low-pressure cylinder, m for main center-
pin, and g for the parallel-motion links. This last is usually the position of
the air-pump center, but in this engine the air-pump is below the high-pressure
cylinder, and its piston-rod is extended to the air-pump piston. The dimen-
sions are H. P., 36" X 62" ; L. P., 58" X 8 feet. The action of the parallel
motion, in keeping the cross-head of the low-pressure cylinder in a vertical line,
will be understood by the arcs described from the main center m and from the
fixed point a, or the journal of the radius bar #. The point e, the angle of a
parallelogram formed of rods and links, must partake of the motions of these
two arcs, and for a portion of movement it is in a straight line parallel to
that of the motion of the piston-rod cross-head. It is usual to make the
radii of these arcs equal.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
325
Fig. 692 is a general mode of finding the length of the radius rod g c. F is
the main center of the beam, a c is a strap or link attached to the beam at a,
the piston-rod to be attached to some point nearly central on the link, which
must move in a straight line. Moving the beam up and down, keep the
point b on the vertical line, and mark the positions of the lower end of the
FIG. 692.
FIG. 693.
link c c c ; find an arc which will pass through these points, and the center of
this arc will be the fixed center g of the radius bar, and the radius that of
this bar.
Steam- Cylinders. Fig. 693 is a sectional plan of a common form of small
steam-cylinder. A is the cylinder, B the piston, b the piston-rod, D the slide-
valves, d the valve-rod, C the valve-chest, c the chest-cover, s s the steam-ports,
e the exhaust-port, S the stuffing-box of the piston-rod, s' that of the valve-
rod. H is the front head and H' the back head of the cylinder. The bolts
attaching the heads to the body of the cylinder are not shown.
Length of Cylinder. It is the present practice, in the construction of
stationary engines for driving machinery, to make the stroke not over twice
the diameter of the cylinder, and for diameters above 24" about 1^ times the
diameter of the cylinder, and invariably to place the cylinders horizontally
with a direct connection with the crank, without the intervention of a work-
ing-beam.
Fig. 694 is the longitudinal section of a Corliss steam-cylinder which has
two steam- valves, s s, and two exhaust-valves, e e. The steam -pipe S is at-
tached to the top of the steam-chest, and the exhaust E to the bottom of the
exhaust-channel ; the bolts on cylinder-heads or stuffing-box are not shown.
The thickness of shell, Mr. Hawthorne finds by many examples in Corliss's
large practice, to conform to the formula t = *268 Vd ; t and d being in
inches. Thus the thickness of the shell of 16" cylinder will be ^16 X *268 =
4 X '268 = 1-072, a little more than 1". The thickness of flanges should ex-
ceed that of the shell by -J to its thickness. The bolts should not be less
than y and seldom more than 1". It is better to increase the number of bolts
than their diameter, the breadth of flange about 3 times the diameter of the
bolts, and the pitch of the bolts, or the distance between centers, about 6 times
the diameter of the bolts.
326
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
Fig. 695 is a sectional elevation of a Cornish pumping engine's steam-cylin-
der. The valves are in pipes outside the cylinder, as in most of our North
Eiver boats, and there is what is called a steam-jacket that is, a shell, jj,
outside the shell of the main cylinder, inclosing a narrow space filled with
steam by a pipe connection directly from the boiler, and with a pipe at the
bottom, through which the condensed water is returned either directly to the
boiler or discharged into the hot well. All steam-cylinders, whether with or
without jackets, should be clothed that is, covered with some preparation to
prevent the escape of heat from contact with air. The usual clothing is hair-
felt, with a lagging, ~b I that is, an exterior shell of some wood, usually black-
walnut.
Figs. 696 and 697 represent sections of two types of water-cylinders. In
Fig. 696 the pump-barrel is long and the piston short ; in Fig. 697 tho pump-
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
327
barrel is only about equal to the diameter of the piston in length, but the length
of the piston is equal to that of the stroke of the pump and that of the pump-
barrel. The figures are taken
from the Worthington pump, and
represent his arrangements of
valves and passage-ways. 1 1 are
FIG. 696.
FIG. 695.
the inlet chambers, i i the
lower valves, and o o the upper
ones. A is the air-chamber.
Pistons are of great variety
and of different proportions,
according to the work to be
done, the medium in which
they move, and the friction
FIG. 697.
328
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
due to their weight on the sides of the
cylinders.
Fig. 698 is the cast-iron piston of
a locomotive. The spring or snap-
rings forming the packing are of cast-
iron, 1|" wide by |" thick, of uni-
form section. The split is made with
a half lap, and the splits of the two
rings are on opposite sides of the pis-
ton. The outsides of the rings are
turned to a diameter slightly in excess
of that of the cylinder, and are sprung into recesses of the piston fitted to
receive them.
Fig. 699 is a sectional plan and Fig. 700 is a sectional elevation of the ex-
terior of a piston-ring, showing another common form of ring packing, which
consists of a single exterior ring r and two exterior rings r" r" , and each cut in
r"
FIG. 698.
FIG. 699.
FIG. 700.
two and so fastened that the joints are always broken. The packing is set out
by springs s s, one of which is shown. F is the follower, which can be taken
off for the admission of the rings and springs, and then replaced and bolted
to the piston, making a close joint with the end of the rings. The depth of the
piston at the exterior is from 3" to 9", varying with the diameter of the piston.
Figs. 701, 702, and 703 are sections of the exterior rings of pistons adapted
more particularly to water-pumps. Fig. 701 depends on the closeness of fit of
FIG. 701.
FIG. 702.
FIG. 703.
the exterior of the piston with the inner surface of the cylinder, and when accu-
rately turned and fitted the leak is very inconsiderable. By the use of grooves
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
329
in the piston (Fig. 702) this leak is still further reduced, as the thread of the
water in passing through the joint is broken by the grooves.
In Fig. 703 the joint between the piston and the cylinder is made tight by
a gasket, usually of hemp, compressed by a joint ring or follower, a, in the
pocket between piston and cylinder.
When the water-pressure is very great, as in hydraulic presses, peculiar
packing-rings of leather are used.
FIG. 704.
FIG. 705.
FIG. 706.
Fig. 704 is a cup leather packing, and Fig. 705 is a U-packing. The ap-
plication of the first will be understood from Fig. 706, in which the piston is
packed with two cup leathers, in this case to
withstand pressure in both directions. Were
the piston single-acting, but one cup would
be necessary and if from beneath the piston,
this would be the lower cup. The flexible
flange is pressed against the inside of the
cylinder, and the joint is perfectly stanch.
Fig. 707 shows the application of the U-
packing ; it is put into a recess in the cylin-
der by bending the packing into a saddle-bag
form, and then allowing it to spring back
into the recess.
Hemp packings are made to serve the
same purpose, as shown in Fig. 703. They
are more easily made than the U-packing, but
they require a follower or cap to retain them
in position.
Packings can be obtained from hydraulic-
pump and press manufacturers, and are kept
in stock of all the usual sizes. Their depths
are from I" to 1J" for diameters varying from 4" to 14", and the space occu-
pied by the thickness in the U from f to f ". A filling of flat braided hemp
is placed inside the IT to keep it tight when not under pressure. The pack-
ings are made by steeping the leather in warm water, forcing them into a
mold, and leaving them to dry and harden. The molds are made of either
metal or wood; frequently the rings are of metal, and the piston over which
the cup is formed, of wood.
Clearances in cylinders include, in general signification, not only the spaces
between the piston and cylinder-heads at the ends of the stroke, but also the
FIG. 707.
330
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
spaces between the cylinder and the valves ; and as those spaces are voided in
a steam-cylinder at each stroke for which adequate work from the steam is not
obtained, they are usually made as small as possible. If the steam is fairly dry,
from y to 1" will be sufficient for end-clearances that is, minimum distance
between piston and cylinder-head.
Piston-rods are proportioned to the stress on them, usually one square inch
of section to each 5,000 pounds of stress. In Fig. 698 the tapered end fits a-
taper hole in the piston, and is riveted over. It is more usually held by a nut,
and some use a shoulder on the inner end of the piston-rod instead of a taper,
and the nut brings the piston strongly up against this shoulder.
Piston-rods are made either of steel or hammered iron, some makers of
engines preferring one and some the other material.
Stuffing-boxes are the mechanisms to prevent the leakage of steam, air, or
water, in the passage of the piston or other rod out of the cylinder or chest.
They consist of an annular chamber around the rod, filled most generally with
gaskets of hemp, which is forced down by a ring or gland into close contact
with the rod and the sides of the box. In Fig. 693 there are two stuffing-
boxes shown, one for the main piston-rod, the other for the valve-rod. In the
latter the cap of the gland is fitted with a screw to connect it with the side of
the stuffing-box, by which the gasket may be more or less compressed. This
is the general form of stuffing-box for small stems or pistons used on steam-
valves, but sometimes with a ring or follower on the top of the gasket, which
is forced down by the gland without turning the ring or gasket. In the figure
the stuffing-box is made of brass, and screwed
into the end of the steam or valve-chest.
The stuffing-box to the piston is cast
with the head of the cylinder, and is bored
FIG. 708.
FIG. 709.
out, and a brass bushing fitted and driven into the end of the box. The
hole through the bushing in most boxes fits the piston-rod accurately. The
gland is of cast-iron, turned to fit the stuffing-box, and bored to fit the
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 331
piston-rod ; after packing the box the gland is forced in and retained by
screws.
Fig. 708 is the plan and section of a common stuffing-box, in which the
thickness of packing is from \" to 1J", and the depth from 1 to 2 times the
diameter of the piston-rod. The number of bolts vary with the diameter of
the piston seldom more than four, and, for the size of engines mostly in
use, but two.
Fig. 709 is the section of a stuffing-box of the proportions adopted by the
Southwark Foundry. Taking the diameter of piston-rod A as the unit, B is
2, C 3, D 2, all scant up to a 3" rod, or 22" cylinder. For a 28" X 42", A =
4, with an allowance ^of >" for clearance, B 6f, C 9, D 6^".
It has been said that hemp gaskets were in most common use for the pack-
ing of stuffing-boxes, and they can be procured readily ; but there are a very
great variety of packings, patented or otherwise, which are very good, adapted
to common stuffing-boxes ; and there are also metallic packings which have
given great satisfaction, and can be easily procured.
Valves Steam- Cylinder Valves. The simplest and most common is the
slide D, shown in Fig. 693. The function of the valve is to admit the steam
alternately into the ends of the steam-cylinder, and, while steam is being ad-
mitted through one port to one end of the cylinder, the other end is being
exhausted or the steam discharged through the other port.
It is absolutely necessary (Figs. 710, 711, 712, 713) that
one port should be closed before the other is opened, that
the steam may not be admitted to both ends of the cylin-
der at the same time, nor that it may flow through from
either end into the exhaust. The simplest form of valve
is shown in diiferent positions in the sections. The face
of the valve-seat is shown in Fig. 713 ; s and s' are the
steam-ports, and e the exhaust-port. The valve only just
covers the ports, so that there is no leak, and in Fig. 712
it is in the position in which the steam can neither enter
nor escape from the cylinder. Suppose there be a move-
ment of the valve to the left, the steam will be admitted through the steam-
port s', and the steam can escape through the other port s into the exhaust ;
at the end of the movement of the valve it will be as shown in Fig. 711, with
full opening of steam into s', and full exhaust through s. If the motion be now
alternated the ports will be gradually closed till the valve returns to its first
position (Fig. 712), and then, as the valve continues its movement, the port s
begins to take steam, and the port s' to connect with the exhaust, till at the
end of the motion in this direction the valve will be in the position shown in
Fig. 710. With this valve there can be no economical use of steam ; it follows
to the end of the stroke without cut-off, without benefit of expansion, except
that which may come from throttling, that is, impeding the flow resulting from
the gradual contraction of the steam openings.
Of the Size of Ports or Openings. Under "Steam-pipes" will be given
the formula for the flow of steam, but the general rule of proportioning the
ports of a cylinder is to consider the velocity of steam 100 feet per second, and
332
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
of the exhaust 80 feet per second. It will be seen from the movement of the
slide-valve that the opening is made gradually, and closed in the same way,
thereby throttling the flow of the steam. To avoid this, Mr. Corliss, in his
engine (Fig. 694), has made his ports long and narrow ; the steam-valves open
quickly and close at once by a drop. It will be seen that the valves have
cylindrical faces and seats, and are moved by a central
rocking-bar.
From the great size of the common slide-valve in pro-
portion to its port, the bearing-surface extending all round,
there ensues a great pressure on the surface, tending to wear it, and also mak-
ing the movement of the valve more difficult. Various expedients have been
adopted to relieve this pressure, which is especially desirable in quick-running
engines.
Fig. 714 is a horizontal section of cylinder, through steam and exhaust-
valves, of a Porter- Allen engine, and Fig. 715 a vertical cross-section through
cylinder and valves. The valves are four in number, one for admission and one
for exhaust, at each end of the cylinder, and on opposite sides. They stand
vertically so as to drain the cylinder. The valves work between opposite par-
allel seats ; the exhaust-valves nearly and the admission-valves wholly in equi-
librium. The action of the back plate, and how the wear is taken up, will be
understood from the section (Fig. 715), which passes through the middle of one
pressure-plate. It is made hollow, and most of the steam supplied to two of
the openings passes through it. It is arched to resist the pressure of the steam
without deflection. It rests on two inclined supports, one above and the other
below the valve. These inclines are so steep that the plate will move down
under steam pressure ; and also that it may be closed up to the valve with only
a small vertical movement, the pressure-plate is held in its correct position by
S
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
333
projections in the chest on one side and tongues projecting from the cover in
the other,, which bear against it at the near end, as shown. Between these
FIG. 715.
guides it is capable of motion up and down and back and forth from -fa" to -J".
The pressure of the steam on this plate tends to force it down the inclines to
rm
JTTL
_(TTL
rrn
rest on the valve. By the means of the screw the plate is forced up and away
from the valve, and can be so nicely adjusted that the valve works freely and
334:
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
perfectly steam-tight. When the pressure is greater in the cylinder than in the
chest, the pressure-plate is forced back, to the instant relief of the cylinder.
Cylindrical Valves. Fig. 716 represents the section of the steam-cylinder
of an Armington & Sims' steam-engine with a cylindrical valve. The steam-
chest S is central and incloses the valve ; the exhaust chambers E E are at the
ends of the valve, and are connected through the hollow stem or body of the
valve. The valve depends on its accuracy of fit for its tightness. The valve-
FIG. 717.
FIG. 718.
chamber is bored out and ground, the valve is turned, ground, and carefully
worked by hand, to so close a fit that there is no loss of steam in action, and
the valve is completely balanced.
There is a form of balanced valves, called the double-beat, much used both
for steam and water valves. Fig. 717 is a sectional elevation of a steam valve
of this kind, and Fig. 718 a plan of the lower seat , with the valve-guides g g
in section. There are two seats, a and ~b, and two faces on the valve corre-
sponding to them. The balance depends upon the relative diameters of the
bearing-lines of the two faces. In the figure, if the exterior of the bearing at
I and the interior at a are both tight, the valve is balanced under any pressure,
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
335
except as to its own weight ; s is the valve-stem, and the hole r is for a bolt to
fasten the valve-seat to the casting of the steam-chest. The scale is -J full size.
Fig. 719 is another form of balance, consisting of two equal poppet-valves
connected together the steam passage to the cylinder being central, and the
steam-chest at each end, and connected.
Automatic valves, that are moved by the action of the fluid in which they are
placed. Figs. 720 and 721 are the plan and section of a disk valve for the air-
pump of a condensing steam-engine. The valve
consists of a disk of rubber lying on a flat grating
or perforated plate of brass, held in position be-
tween the grating and a spherical guard by a
-central bolt. The shape of the guard gives a
FIG. 720.
FIG. 721.
uniform flexure to the rubber in lifting, and an easy flow to the current of air
and water. The rubber is not closely clamped between the guard and plate, as
will be seen in the figure. The lower nut, after being screwed home, is riveted,
and the upper nut usually pinned to prevent turning. The size of the apertures
in the grating are adapted to the thickness of the rubber. With an external
diameter of opening of 6", and rubber -J-" thick, the exterior ring of openings may
be f " by f", the lands or spaces between openings J" wide, and exterior lap of
the rubber \ inch. With larger diameters and larger openings thicker rubber
must be used. This valve is often made of a long strip or flap of rubber, on a
suitable grating, with a curved guard attached on one side. For the common
air-pump pressure, " rubber is sufficient for apertures 1" x 4". With the use
of backing and face plates on the rubber flaps, the gratings may be dispensed
with. Thimbles are inserted in the rubber, and the rivets connecting the two
plates pass through these thimbles. The valves to the Boston sewage pumping-
engines are of this description. Clear opening in seats 13" x 4", rubber "
336
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
SPACE OCCUPIED BY THE
VALVES.
thick, toe of guards curved to a 2-" radius for the hinge of the rubber ; the
guards have leather pads for the valves to cushion on in their lift.
Fig. 722 is the section of a metallic flap-valve or check-valve of the Ludlow
Valve Manufacturing Company pattern. Body and valve are of cast-iron, with
valve faces and seats of bronze. The bottom of the case B is flattened and
raised toward the seat, so that gravel and stones may not lodge against it.
In Fig. 723, section of a like valve, there is a small secondary valve on the
exterior of the main valve, which, being lighter than the latter, opens earlier
and closes later, and prevents shock to the main and to the valve.
Check-valves are placed outside of large pumps to prevent the return of
water in cases of accident to the pumps, and for facility of their examination.
Valves of this kind open from the pressure of
water beneath, and, from a state of rest, with
some suddenness and shock. To prevent this in
large valves, there is a valve and small by-pass
pipe, from one side to the other of the valves, by
opening which the pressure an the two sides of
the valve may be equalized, and the excess due
to the starting of the pump distributed. At
many pumping works the by-pass is kept open
except when necessary to get at the pumps. In
case of accident to the pumps the flow through
the by-pass would be comparatively small, and
readily shut oif.
Fig. 724 is a section of a poppet-valve ; the-
body is of cast-iron, but the valve and seat are
of brass. The valve is guided in rising and falling by three feathers on the
valve. The lift of the valve is controlled by the projection on the cover ; a
screw is often substituted for this, as it admits of adjustment to varied lifts.
Poppets are often guided by stems.
Fig. 725, ball-valve, guided in its movement by an open cage, c, shown in
Measure-
Measure-
SIZE.
ment from
face to face
ment from
end to end
of flange.
of hub.
Inches.
4
Hi
13|
5
14*
16
6
w*
16
8
17|
19
10
21|
24
12
24!
26-J
16
29
31
18
33
35
20
35i
38
24
39f
39
FIG. 724.
FIG. 725.
section and attached to the cover. Ball-valves are usually small metallic balls-
on metallic or wooden seats, or rubber balls on metallic seats ; and cylindrical
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
337
FIG. 726.
yalves have been made of the same section as in the figure ; the body of the
valves of brass pipes with rubber jackets.
Fig. 726 is a section of a rubber disk-valve in very common use in direct-
acting pumps and small pumping-engines ; sometimes with a thimble in the
rubber as a guide ; usually with a metallic plate on top of the rubber for the
bearing of the spring ; valve-seat generally of composition,
with spindle riveted or screwed into it. Sometimes the rub-
ber is held in a metallic plate or cup.
Large valves, either poppets or disks, are objectionable
from the great lift required for an outlet, proportionate to the
area of opening in the seat, making shocks both in the lifting
and seating. Consequently, these kinds of valves are made
small, the requisite area of outlet being made up by the number of the valves.
The balance-valve (Fig. 717) is commonly used in Cornish and large pump-
ing-engines. From its two beats, the lift is about one half that of a plain valve.
There must be difference enough in the faces to admit of the lift of the valve
by the pressure of water acting on this diiference. The seats of the valves are
often made of wood, set endways. , Automatic valves should have springs at
their backs to cushion the blow on the lift, and to start the valve downward
promptly on the check of the water-flow at the end of the stroke. The
great desideratum of water-valves is that there should be little lift but ample
water-way.
Valves controlled by Hand. Fig. 727 represents a side view of a water bib-
cock, called a /^ose-bib, because the outlet end is fitted with a screw to adapt
it to a hose. Without this screw it is a plain Mb.
If both ends of the cock are in the same line, it is
FIG. 727.
FIG. 728.
FIG. 729.
called a stop-cock. The ends may not be fitted with screws, as in the figure ;
the screws are sometimes female screws, and often with taper ends, to solder
lead pipe to, or to drive into a cask. These cocks come under the common
designation of plug-cocks, from their interior construction, which will be
readily understood from the section given in Fig. 728. They are used in both
steam and water pipes, but not in the former when the use is frequent and
daily, and then usually not over 2" in diameter of passage.
Fig. 729 is the side view of a compression water-bib, used when the press-
ure of the water is great. The section is somewhat similar to that of Fig.
732, in which a rubber disk is forced against a metallic seat to shut off the flow.
22
338 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
Figs. 730 and 731 are side views of common air-cocks for boilers and steam
work ; they are plugs in their construction, as are the cocks used in gas-fitting ;
size of air-cocks to f " diameter.
Fig. 732 is the section in part of a globe steam or water valve with a rubber
disk ; soft rubber for cold water, hard rubber for hot water or steam. The
FIG. 730. FIG. 731.
fluid enters below the diaphragm ana passes up through the aperture in it,
which is controlled by the valve ; a screw in the stem, below the stuffing-box,
bringing it in close contact with the face, or raising it to any height required.
FIG. 732. FIG. 733.
They are called globe-valves from the shape inclosing the valve (Fig. 733).
They are not necessarily rubber disks ; the smaller sizes are metallic poppet-
valves.
The dimensions of straightway globe-valves in common use are as follows,
from " Warming Buildings by Steam " (Briggs) :
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
339
Diameter of open-
ing in seat.
Body gun-metal
or cast-iron.
Nozzles tipped
or flanged.
Length over all.
Diameter of
flanges.
Number of bolts
each flange.
Inches.
Inches.
1
Gun-metal.
M
Tipped.
u
T45
1-78
i
u
u
2-20
f
U
u
2-65
1
n
u
3-30
1*
a
a
3-85
H
u
u
4-35
Cast-iron.
(c
6-10
2
Gun-metal.
u
5-30
Cast-iron.
U
5-90
u
Flanged.
5-75
6
4
2*
Gun-metal.
Tipped.
6'75
Cast-iron.
u
7-30
u
Flanged.
7-25
7
4
3
Gun-metal.
Tipped.
7'75
Cast-iron.
u
9*25
u
Flanged.
9-25
n
4
8*
a
Tipped.
10-25
"
Flanged.
10-25
8
5
4
1
u
11-25
9
5
5
(
u
13-25
10
6
6
I
U
15-25
11
6
8
1
19-
13^
8
10
1
II
23-
16
10
12
1
u
27-
19
10
Figs. 734 and 735 are eleva-
tions of valves of the same type
as the last, but from their form
are called angle and cross valves.
Figs. 736 and 737 are the
plan and section of a steam valve
of the Southwark Foundry pat-
tern. Its construction and action
will be readily understood from
the drawing. The valve is with
inclined faces, and seat ground
to a fit, and is guided in its
movement by three wings, w, w.
This is a common type of throt- FlG - ?34. Fio. 735.
tie-valve for steam use.
It will be observed that in the section (Fig. 737), and especially in that of
the globe-valve (Fig. 732), the flow of the fluid passing through them is very
disturbed and impeded ; to avoid this, straightway gates are almost invariably
used on water mains, in which the gate is raised entirely out of the line of
pipe, so as to leave the flow unobstructed.
Fig. 738 is a section of one of the oldest types of this kind of valve, the
Coffin valve, with double disks, d, d, self-adjusting on their seats. The screw
works within a long pipe or nut, and when raised the disk-valves are above the
line of pipe within the large circular chest.
In " Scraps " is a perspective view of a similar valve of another maker.
340 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
341
Fig. 740, the Safety- Valve. The illustration is of the common type ; a
poppet-valve, with a stem bearing on the top, and this weighted by a scale-
beam, by which any desirable pressure can be put on
the valve. To every boiler it is absolutely indispen-
sable that there should be such a valve attached di-
rectly, without any means of shutting it off, as in
Fig. 740, where B is
the boiler, S the steam-
pipe, and 1) the blow-
off from safety-valve.
The United States rules
require for the safety-
valves of this pattern,
FIG. Y?8.
B
FIG. 740
for ocean and river service, that they "shall have an area of not less than one
square inch for every two square feet of grate-surface."
" But when safety-valves are used, the lift of which will give an effective
area of one half of that due the
diameter of the valve, the area re-
quired shall not be less than one
half of one square inch to two feet
of grate-surface. "
Fig. 741 is what is termed a pop
safety-valve ; the steam issuing as
the valve rises, impinges on a cup
surface to force the valve further
open. The valve is held down by
a spring, but the valve can be raised
by the lever I. Valves of this kind
are often inclosed in a locked box,
that they may not be tampered
with.
Hydrants. For water - service
in connection with high-pressure
mains.
Fig. 742 is a section of the
Matthews post-hydrant, one of the
best known of the type. The valve
v consists of a series of leather disks
bolted together and turned coni- FIG. 741.
342
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
cal, which is brought in contact with a corresponding seat by the valve-rod
and its screw at the top of the hydrant. The valve is opened by being
forced down into the cavity of a branch of the pipe-main ; n is the nozzle
for the coupling of the hose ; outside the main pipe of the
hydrant there is a case, extending from near the line of
valve to the ground line, called the hydrant or frost case,
which prevents the hydrant from being lifted by the frost.
Were the water left in the hydrant, it would freeze in most
exposures during winter ; the hydrant, when not in use,
is therefore kept empty. This is effected by a small hole
at v, which, when the valve is closed, is opened, and the
water in the hydrant, if any, is discharged. This vent is
closed by a slide attached to the valve-rod, when this last
is moved down to open the main valve. Instead of leather
for the valve-face, many valves are fitted with rubber ; and
there is also a great variety of valves for hydrant purposes
slides, poppets, disks but in arrangement of hydrants the
illustration is almost universally followed; often, though,
without the hydrant case.
Riveted Joints, as used in the construction of Boilers.
Tigs. 743-749 are forms of rivets with their proportions re-
FIG. 744.
FIG. 749.
FIG. 742.
ferred to the diameters next the heads. The thickness of the plate connected
by rivets will be given in a table hereafter. Figs. 744 and 745 are the usual
finish of rivets in hand-riveting ; Figs. 746 and 747, when done by machines.
Fig. 748 is a counter-sunk rivet, the head being flush with the outside of the
plate. Fig. 749 is the head of a rivet, in which a narrow strip at the edge is
burred down by a chisel, or calked, to make the joint between rivet and plate
tight.
Fig. 750 is a plan and section of a single riveted lap-joint. Joints of this
kind fail from the tear of the plate on the line of rivets if the rivets are too
MACHINE DESIGN AND MECHANICAL CONSTKUCTIONS.
343
close ; by the shear of the rivets if too few ; or by the bursting of the plate
from the rivet to the outside if the space is too small. The great difference in
the quality of boiler-plates and rivets, and the uncertainty as to the effect of
O -Q--S0-
6 We-
FIG. 751.
punching plates, prevent any accurate determination of the exact proportion
of riveted joints. We insert the tables from a practical " Treatise on High-
Pressure Steam-Boilers " by William M. Barr. Dimensions in inches :
TABLE SHOWING DIAMETER AND SPACING OF RIVETS IN SINGLE-EIVETED
LAP-JOINTS.
Thick-
ness of
plate.
Diameter
of rivet.
Length of
rivet.
Center of
rivet to
edge of
plate.
Center to
center of
rivets or
pitch.
Thick-
ness of
plate.
Diameter
of rivet.
Length of
rivet
Center of
rivet to
edge of
plate.
Center to
center of
rivets or
pitch.
A
B
C
D
A
B
C
D
A
i
1
HI
H
i
|
2i
If
H
i
f
li
i
14
A
|
2|
If
2 i
A
f
H
i
1*
f
1
24
IT\
2f
1
f
14
1A
2
T
1
3
*A
21
A
4
2
1A
2|
4
H
3i
14
3
Single-riveted joints have the strength of about 56 per cent of the solid
plate ; double-riveted joints about 70 per cent. Fig. 751 is the plan and sec-
tion of a double riveted joint, and the proportions given in the table are those
recommended by Barr :
TABLE SHOWING DIAMETER AND SPACING OF EIVETING IN DOUBLE-RIVETED
LAP-JOINTS.
Thickness of
plate.
Diameter.
Length.
Center to edge.
Pitch.
Center to center.
Center to center
D
E
F
i
1
li
1
2
1*
1A
A
f
H
1
H
2
i
t
f
if
1*
B*
H
iff
TV
f
2
1*
H
2i
H
i
1
2i
If
3
*&
iff
T 9 ir
1
H
H
3i
*&
2
1
1
2|
IA
H
2|
2i
H
1
3
i*
3|
21
2A
1
u
3i
if
4
3
2i
344
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
For the most part in this country, rivet-holes are punched ; some drill them.
By punching first a small central hole, and then using a pin or teat-drill, an
annular washer is taken out, leaving a clean hole, and a ready means of test-
ing the quality of the material by bursting the washer by a drift. It is the
practice here to make no boilers less than " thick, and beyond this to use a
factor of safety of six, as shown in
the table.
The stress on the circumferential
seams of a boiler is the circular or
end area in square inches multiplied
by the pressure per square inch, and
this is to be met by the circumferen-
tial section of the shell. The longi-
tudinal stress can be estimated by
multiplying the diameter of the boiler
in inches by the pressure per square inch, and this stress is to be resisted by one
inch in length on each side of the boiler, or by a section of plate 2" wide by its
thickness, and with a proper factor for riveted joints.
Fig. 752 is the plan and section of a single-riveted butt-joint, and Fig. 753
the same of a double-riveted one. The two plates are brought close to each
other, and the joint is made by a cover,
proportioned in the pitch of the rivets
Strength of solid
SAFE WORKING LOAD.
plate pounds per
square inch.
Single-riveted.
Double-riveted.
50,000
4,700
5,800
60,000
5,600
7,000
70,000
6,500
8,200
o
o
o o
j)
FIG. 752.
FIG. 753.
and distances of centers from edges of plates, as in rules above given; and
although this form of joint in some cases is convenient, it has not been found
practically stronger than the lap-joint.
But butt-joints with double covers, one on each side of the plates, increase
the shearing resistance of the rivets, so that rupture always takes place in the
FIG. 754.
FIG. 755.
plates ; and as these can not bend, and there is considerable frictional resistance
between the plates, the strength of the joint has been found to be more than
that due to the net section of the plates between the rivets.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
345
Fig. 754 is a plan and section of a combined lap and butt joint. The pitch
of the exterior rows is double that of the central one ; for a f " plate, 4" for the
former and 2" for the latter.
Fig. 755 is the plan and section of a butt-joint when the cover is of T-iron
a not uncommon form of strengthening flues to resist collapse.
cjj
o
(J-vS.NI
\s_ss\
-
*(//>A
Lvvxl
FIG. 756.
FIG. 757.
FIG. 758.
Junction of more than two plates, shown in plans and sections (Figs. 756,
757, and 758). These become necessary when cross-joints intersect longitudi-
nal ones. At these joints one or more of the plates are thinned or drawn out
by forging.
Fig. 759 is the plan and section of an angular connection of plates by the
means of angle-iron ; this should be a little thicker than the plates, and its
width four times the diameter of the rivets.
FIG. 759.
FIG. 760.
FIG. 761.
FIG. 762.
Figs. 760, 761, and 762 are sections of angular connections by flanging the
plates. The iron should be good and the curvature easy ; inside radius at least
four times the thickness of the plates.
FIG. 763.
FIG. 764.
FIG. 765.
Figs. 763 and 764 are sections of joints of cylinders of unequal diameters,
or surfaces not in line with each other.
Figs. 765, 766, and 767 are sections of fire-box legs.
In all connections provisions are to be made for the means of holding the
head of the rivet, and for riveting and for calking the joints.
346
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
Fig. 768 is the perspective view of a boiler of the type most commonly used
when the fuel is anthracite, and often also when bituminous, called the horizon-
tal tubular. The proportions of the boiler vary with the requirements of their
position, and with the views of the mechanical engineer or maker constructing
them. Those in most extensive use are with shells of 4 to 5 feet inside diame-
ter and 3" to 3" tubes, 14 to 16 feet long. The line of the top of the upper
tubes is usually about -^ of the diameter of the boiler above its center ; tubes
arranged in vertical rows, with distance between tubes of their diameter. In
my own practice I have kept the average distance the same, but making them
farther apart at the top row, say J diameter, and the lowest J- diameter, so that
the line of tubes is radial instead of vertical.
FIG.
The following table is from Barr, showing the greatest number of tubes
which should be put in a given head, no tube to come nearer to the shell than
2" for boilers of small diameter, 2J" for medium, and 3" for the larger series :
Diameters of
bodies inside,
in inches.
NUMBER OF TUBES (outside diameter).
Sin.
8Jtn.
3^ in.
3| in.
4 in.
|
4J in. 5 in.
36
26
23
20
19
16
12 10
40
34
34
25
23
20
14 14
44
48
36
32
25
25
20 16
48
50
38
36
30
26
21 18
52
57
50
48
38
32
26 21
56
72
57
55
48
41
32 23
60
80
68
62
55
46
36 30
i
A (Fig. 768) is the man-hole, to enable the mechanic to get into the boiler
to examine it. It consists of a cast-iron frame, bolted to the shell of the boiler s
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
347
with an elliptical opening usually 9" X 15" in the clear ; the valve laps about
1" on each side. In closing the opening the valve is passed down into the
boiler, and is brought up against the valve-seat, where it is held by its stem
passing up through a movable yoke, and brought up tight by a nut and screw.
The joint is made with a gasket or with sheet-rubber. The man-hole is often
placed in one of the boiler heads. B is the hand-hole, of the same general con-
struction as the man-hole, but smaller, to enable the fireman to clean the boiler.
Formerly this hand-hole was quite small, but of late the practice is to make
them 6" X 8", or even as large as 8" X 12" for large boilers in fact, a man-
hole. There should be a hand-hole in the other end of the boiler, so that by
taking off both hand-holes one can look directly through the boiler. As this
hand-hole is exposed to the flame and products of combustion, it is well to
make it smaller than the front one, say 3" X 5"; III are lugs by which the
boiler is supported on brick- work. It will be observed that in the head above the
tubes there are rivet-heads, and also in the sides back of the first seams at each
end. These are for the attachment of diagonal stays. The tubes themselves
serve as stays in the lower part of the boiler, but above the flat surface needs
something to prevent the head from moving out under pressure. The stays
are made of round or flat iron, bolted directly to the shell, the round part
being flattened, and connected by a yoke and pin to a crow-foot or piece of
angle-iron attached to the head. The stays are from f " to 1^" diameter or
equivalent sections.
BARK'S PROPORTIONS FOR STAY-BOLTS FOR FLAT SURFACES.
CENTER TO CENTER OF STAY-BOLTS IN SQUARE INCHES.
Pressure per
square inch.
*" plate.
iV plate.
|" plate.
A" Plate.
*" plate.
60
H
*f
n
H
9
80
4f
6*
6*
n
w
100
4*
4f
Bi
6i
7
120
8|
4i
5
Bf
H
140
3f
4*
f
5i
6
Eigs. 769 and 770 are a longitudinal and a half transverse section of an
anthracite-burning locomotive from the New Jersey Railroad, which illustrates
the stays used in such forms of boilers. Water-spaces are 4" wide in front, 3"
at sides, and 6" at rear ; stay-bolts in water-spaces J" diameter, 4" centers.
The crown-sheet of fire-box is supported by double cast-iron girders, extending
across the boiler, ends resting on the inner plates of fire-box, and also sup-
ported by hangers h h from the outer shell, and the inside of the steam-drum.
These hangers have a fork at one end, through which a pin is passed to con-
nect it with the foot riveted to the boiler ; the other end passes into the space
between the double girder, and they are pinioned together. The crown-sheet
is held by bolts passing down through the double girder. The bottom of the
water-space is made with a wrought-iron ring. The opening for the door is
made by turning a flange on the inside plate, to which a plate ring is riveted,
348
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
349
ind the joint with the outside plate is made by a ring of angle-iron. The
ooiler has 130 2" tubes, and 26 8J" tubes.
Fig. 771 is a gusset stay, used in angles, con-
sisting of a triangular plate with the edges flanged
and riveted to the shell.
FIG. 771.
FIG. 772.
FIG. 773.
Flue Boilers. Where bituminous coal is used, small tubes become clogged
with soot ; it was therefore customary
to construct boilers with larger tubes
or flues of boiler-iron riveted together,
which sometimes failed from collapse,
their resistance being uncertain, due
largely to the defect of an accurate cir-
cular section. Mr. Fairbairn made ex-
periments on the resistance of tubes to
collapse, but it has been demonstrated
that the rule does not apply within the
limits of length adapted for boiler-flues,
and it may be considered ample to make
the tubes subject to outside stress
fifty per cent thicker than for
bursting, especially for the large
drawn tubes now made. From
Mr. Fairbairn's experiments it was
considered necessary to make the
joints of tubes subject to collapse
as in Figs. 772 and 773, which
may be useful against deteriora-
tion of force in riveted boiler-flues,
and might in long mains be of
importance, especially if of the
form of Fig. 773, which, besides
strengthening the tube, provides
for expansion.
Fig. 774 is a section of the
Shapley boiler, as made by the
Knowles Steam - Pump
Works a good form
of upright boiler, with
the crown -sheet simply
FIG. 774. stayed and well covered
350
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
by water. It is an admirable illustration for the draughtsman of how a boiler
in action may be represented.
The usual form of upright boiler consists of a fire-box, extending a little
above the door, and tubes extending from the crown-sheet to the top-head,
over which there is a bonnet to secure the smoke, which is led off by a smoke-
pipe. These are very convenient forms of boilers for furnishing small power,
as they occupy comparatively little space. They are not as economical in their
combustion, and they are very apt to prime that is, take up water with the
steam.
The common vertical boilers are from 2 feet 6" to 4 feet 6" outside diameter
of shell, with water space in legs of 2%" to 3" ; extreme height of boiler from 2
to 2J times the outside diameter of fire-box ; tubes from 2" to 2-J-" diameter,
and spaced from V to H" apart. Water-line from 10" to 15" above crown-sheet.
On account of small ground-space, vertical boilers are popular with some
makers, and are made with varied appliances to secure good evaporative re-
sults and to protect the upper joints of the tubes from being overheated.
There is supposed to be a proportion between the tube sectional area and
the grate-surface, say from \ in the horizontal to ^ in the vertical ; but this
rule is entirely empirical, as the length of the tube is a large factor in the dis-
charge of products of combustion (see Sturtevant tables in appendix). There
is also a proportion of grate to heating surface ; but
only the same class of boilers can be compared with
each other, as fire-box surface and that exposed di-
rectly to the flame is much more effective than that
of the tubes, and the products of combustion escape at
much different temperatures in different boilers.
Pipe Connections. Fig. 775 is the section of a
flanged connection of a cast-iron pipe of the most usual
form, but some thicken or reinforce the pipe a little for
1" to 2" in length next the flange ; but if there is a
good fillet in the angle of the flange it is unnecessary.
The proportions of flanges to the thickness of the pipe at the joint are
given below :
DIMENSIONS OF CAST-IRON FLANGED PIPE TO WITHSTAND SAFELY A PRESS-
URE OF ONE HUNDRED POUNDS PER SQUARE INCH.
FIG. 775.
Diameter of pipe
4
6
8
SO
12
16
20
Thickness of pipe
JL
1
9
|
|
Number of bolts
5
6
8
10
10
14
18
Diameter of bolts
A
4
|
1
|
|
1-
The flanges are almost invariably faced, and joints made with red and white
, or a sheet-rubber washer. Large cast-iron flanged pipe is but little used
for street mains ; water-service socket-pipes are invariably used, and for steam
connections wrought-iron pipe is to be preferred, and it can now be purchased
of any necessary diameter ; and when steam-drums are requisite, or very large
connections, they are made of riveted plate.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
351
Fig. 776 is a section of the joint used by Sir William Armstrong for the
pipes of his accumulator. For a working pressure of 800 pounds per square
inch, pipes of 5" diameter are made 1" thick and tested to 3,000 pounds per
square inch. The flange is elliptical, and there are but two bolts ; one pipe
slightly enters the other, forming a dovetailed recess in which is placed a gutta-
percha ring i" diameter.
FIG. 776.
FIG. 777.
FIG. 778.
Figs. 777 and 778 are sections of two other forms of cast-iron flanged pipes,
both with projections fitting into grooves. The packing in Fig. 778 is a ring
of lead. In Siemens's air reservoirs, where the pressure sustained by steel rings
is 1,000 pounds per square inch, the joint is made by turning a V groove in
the face of the rings, and placing in it a ring of annealed copper -fo' diameter.
This form is adopted by many mechanics for forming flanged joints even for
steam purposes.
Wrought-Iron Pipe Connections. With the present cost of wrought-iron
pipes they arc almost invariably used for the conveyance of steam, but are more
liable to rust for water purposes than
cast-iron. Wrought-iron pipes are
either butt-welded or lap-welded. It
is a mere question of manufacture.
It is difficult to make a lap-welded
tube less than 1-J" diameter, and,
therefore, below this size they are
usually butt-welded ; but this size
and above, lap-welded.
Wrought-iron pipes in continuous length are connected by socket or sleeve
couplings, shown partly in section (Fig. 779), which are almost invariably of
wrought-iron. A thread is cut on each end of the pipes, and
internal threads in the coupling. The coupling is screwed on to
the end of one pipe and the other pipe screwed into the coupling.
The screw in the coupling is tapped parallel usually, but the ends
of the tubes are cut with a taper thread, uniform with all makers,
of 1 in 32 to the axis. The length of the screwed portion varies
with the diameter.
Fig. 780 is the longitudinal section of tapering tube-end with the screw-
thread as actually formed, and considered standard by the late Robert Briggs,
C. E., in his " Treatise on Warming Buildings by Steam." It is shown in the
figure double full size for a nominal 2-J-" tube.
FIG. 779.
352
MACHINE DESIGN AND MECHANICAL CONSTEUCTIONS.
FIG. 780.
DIMENSIONS OF WROUGHT TUBES AND COUPLINGS.
DIAMETER OF TTTBE.
CIRCUMFERENCE.
Nomi-
nal in-
side.
Actual in-
side.
Actual out-
side.
Inside.
Outsid
In.
In.
In.
In.
In.
i
0-27
0-41
0-85
1-2^
i
0-36
0-54
1-14
1-7C
| 0-49
067
1-55
2-15
i 0-62
0-84
1-96
2-en
f 0-82
1-05
2-59
3'(
1 1-05
1-31
3-29-
4'U
li
1-38
1-66
4 33
5-21
H 1 61
1-90
5-06
5-9'
2 2-07
2-37
6-49
7'4(
2i
2-47
2-87
7-75
9-0;
3-07
3-50
9-64
11-0(
i
3-55
4-00 11-15
12-6*
4
4-03
4-50 12-65
14-1^
4*
4-51
5-00 14-15
15-71
5
5-04
5-56
15-85
17-4'
6
6-06
6-62
19-05
20-8
7
7-02
7-62
22-06
23-9,
8 | 7-98
8-62
25-08
27'K
9
9-
9-69
28-28
30-4
10
10-02
10-75
31-47
33-7
Weight
per foot
in length.
COUPLINGS.
of
Outside
diameter.
Length.
Lbs.
In.
In.
0'2t
0-55
* '
0-42
0-70
1
0-56
0-83
1
0-84
1-01
1ft
1-13
1-24
If
1-67
1-53
If
2-26
1-89
If
2-69
2-17
2
3-67
2 68 ! 2
5-77
3-19
2f
7-55
3-87
3
9-06
4 40
H
10-73
4 99
3J
12-49
5-49
3f
14-56
6-19
3 i
18-77
7-24
8*
23-41
8-36
4
28 35
9-49
4
34-08
10-54
4
40-64
11-72
5
When pipes are thus put together in lengths, with couplings, it is frequently
impossible to take out a length of pipe for repairs or alterations without break-
ing a coupling or fitting ;
provision is made for discon-
nections by the insertion of
a union or unions in the line.
Fig. 781 is an exterior
view, and Fig. 782 a section,
of the common malleable-
iron union ; p and p' are the
781. FIG. 782. halves into which the tube
is screwed, and the joint is
made by a male and female coupling. The male, #, turning on a flange on
the tube p, is screwed to the other half of the coupling, and the joint is made
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
353
tight by a rubber washer, shown in black. These unions are used only in the
smaller sizes of pipes. The flange coupling (Fig. 783) is preferred by most
fitters, and they are made of diameters up to 14" ; the thickness is about one
half that of the length of a coupling of the same diameter. The bolts are from
f" to f ", and spaced somewhat larger than that given for cast-iron flanges.
The width of flange is such as to admit of the head and nut of the bolt without
projection beyond the edge of the flange.
FIG. 784.
FIG. 785.
FIG. 783.
FIG. 786.
Fig. 784 is a common cast-iron flange, and with about the same proportions
as in Fig. 783. When the lines are long, and provision can not be made by
bends for the expansion and contraction of pipes under changes of temperature,
a fitting like a stuffing-box is often used, the end of one of the tubes being
attached to the box, and the other sliding in and out like a piston-rod ; some-
times expansion is permitted by two flexible flanges, admitting of a sort of bel-
lows-like movement ; sometimes by a connection between pipes of a ring, as
in Fig. 773, or a succession of corrugations.
FIG. 787.
Fig. 785 is a soldering union ; the ring b is like that of the male coupling
(Fig. 782), which is screwed directly to the wrought-iron pipe, while a is a
23
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
brass tube, with a shoulder on the bottom on which the coupling a turns, and
a lead pipe is soldered to the tube. If it is not necessary to break the joints, a
soldering nipple (Fig. 786) only is necessary, one end of which is screwed into
the wrought-iron pipe, and the other soldered to the lead pipe.
Figs. 787, 788, 789 are taken from Briggs's treatise, and give the dimen-
sions of the parts of elbow, tees, crosses, and branches. Fig. 788 shows the
parts of an elbow designated by letters in Fig. 787, and Fig. 789 shows the
applicability of the same to tees and crosses. The scale is one quarter full
size ; if much used, it would be better for the draughtsman to construct one
of full size. The dimensions are obtained by measuring from the base or zero
to the inclined lines, on ordinates corresponding to the inside diameter of pipe
required.
Fig. 790 is a close nipple ; Fig. 791 is a shoulder nipple.
If the uncut part of the tube is longer than in the figure, it is called a long
nipple ; they serve the purpose of short pipes.
FIG. 792.
FIG. 790.
FIG. 791.
Fig. 792 is a bushing. There is a thread cut inside. It is screwed into a
coupling, and the pipe that is screwed into the bushing must be smaller in
diameter than that connected with the coupling. The service of the bushing
is to connect pipes of different diameters, but the reduction of one side or arm
of a coupling, tee, or cross is better.
Fig. 793 is a plug to close up the end of a pipe by screwing it into the
coupling ; caps are used for the same purpose ; half -couplings with one end
closed, or blank flanges that is, flanges without any hole through them
bolted to a flange on the end of a pipe.
It will be seen by Fig. 788 that the cast-iron elbow
makes a very short turn, with considerable obstruction
to the flow of the fluid through it.
Fig. 794 is an elbow in which the obstruction is very
much reduced. The angle is a piece of wrought-iron
pipe curved to an easy radius ; and, as a general rule, it
may be said that for the connection of pipes not in a line
with each other, it is better to bend the pipe, if possible,
than make angles by cast-iron elbows.
Figs. 795 and 796 are a tee and a cross as used in
connections of hydraulic presses, made of composition.
The tubes are of wrought-iron, extra thick. The usual dimensions for such
are as follows :
FIG. 794.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 355
Outside diameter.
Inside diameter . .
r
8>f
I"
] II
The joints are made by leather washers,
square ends on square seats.
FIG. 795.
FIG. 796.
FRAMES.
Fig. 797 is the section of a common jack-screw, in which the pressure is
vertical ; the base is made extended to give it stability.
FIG. 797.
FIG. 798.
FIG. 799.
FIG. 801.
356
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
Figs. 798 and 799 are side and end views of a cast-iron housing for rolls.
The screw exerts the pressure on the box of the roll-journal, and the reaction
is a tensile pressure on the sides of the frame ; but there is in addition much
percussion and intermittent stress that is to be provided against.
Fig. 800 is the elevation of a hydraulic press, and Fig. 801 the plan of top
and bottom plates. The stress on the rods is tensile, and they must be of sec-
tional area to resist securely the power exerted on the press. The plates are
beams held at the four corners, and the
stress central. The platen p attached
to the ram is braced by triangular flanges
from the hub. The cylinders of large
hydraulic presses were formerly made
of cast-iron, sometimes hooped with
FIG. 802.
FIG. 803.
wrought-iron, but now it is the practice to make them of cast-steel. The cyl-
inders of hydraulic jacks and the smaller presses are made of drawn steel or
wrought-iron tubes.
Fig. 802 represents the (side elevation) cam-punch and shear ; in this case,
the force exerted while the machine is in the operation of punching or shearing
tends to open the jaws a a ; and the tendency increases with the depth of the
jaw, the stress obviously being the greatest at the inmost part of the jaw. The
Irame consists of a plate of cast-iron, with two webs around its edges ; the front
web, being subjected to a tensile strain, should be in the area of its section about
six times that of the rear web, which is subjected to a compressive force.
Fig. 803 is the side-frame of a planing-machine. The force here exerted is
horizontal against the cutter, which can be raised or lowered at pleasure, ac-
cording to the magnitude of the work to be planed ; the upright has, therefore,
to be braced, which is done in a curved form for beauty of outline.
Fig. 684, p, is a wooden frame supporting the working-beam and shafts of
a river-boat engine.
Fig. 682, p, is a side elevation of a horizontal engine, of the type of engine-
frame introduced by Mr. Corliss ; (Fig. 804) is a plan of the same. The old
MACHINE DESIGN AND MECHANICAL CONS
[ONS.
351
type of steam-engine frame was a rectangular cast-ipn frame ; the steam-
cylinder resting on the top side flanges, the pillow-block being bolted on the top
of one side flange, and the crank and connecting-rod forking centrally be-
tween the sides.
FIG. 804.
Fig. 805 is a side view of the inclined wrought-iron box-frame of the
war steamer Susquehanna. The steam-cylinder rests between the frames, and
is bolted to them. The two frames are securely stayed to each other, and
bolted to the keelson and the bottom of the ship. For small inclined engines,
^ as used on ferry-boats, the frames are of wood,
( ^\. as also in many of the horizontal engines of
boats on Western waters.
Governors. In the running of all
machinery there are variations of
speed, due to varying powers
and resistances, caused by
increase or decrease in
the pressure pro-
ducing the
FIG. 805.
power, as of steam or water, or in the resistances of the machinery, from
more or less being brought into action, or through inequalities of work done.
To maintain the speeds at as much uniformity as possible, governors are
used, which, applied to steam-engines or water-wheels, open or close valves
or gates, and increase or reduce the supply of steam or water to the cylin-
ders or wheels, according to the varying necessities. The ordinary gover-
nor (Fig. 806) consists of two heavy balls, suspended by links from a spindle,
and caused to revolve by some connection with the shaft of the motor. In the
figure the governor is driven by a belt-connection to the pulley, p, bevel-geared
to the governor. When at rest, the balls hang close to the spindle, but when
in motion the balls rise by the centrifugal force. When the motor is running
at its established speed, the links assume a position nearly at 45 with the
358
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
spindle. If the speed falls off, the balls fall, and, acting on the lever, as
shown in side view, open the valve or gate controlling the passage of steam to
the cylinder or water to the wheel ; if the speed rises, the balls rise and close
the valve or gate. The lever does not always connect directly with the gate,
nor is there always a lever, but the rise or fall of the balls acts on some
mechanism which performs the function of reducing or increasing the supply
of steam or water.
The size of the balls depends somewhat on the work to be done, the resist-
ance to be overcome in the movement of the gate and connections, and may be
much reduced if this work is thrown on some other mechanism, which is usu-
ally the case in the regulation of water-wheels ; while for steam-engines the
FIG. 806.
FIG. 807.
work to be done by the governor is reduced by balancing the steam- valve, or
to the merely setting a trip, that will permit the movement of the valve at any
point of cut-off.
In the Porter governor (Fig. 807) the balls of the governor are compara-
tively light, but they are connected to a heavy central weight by levers, the
same as those connecting the balls with the spindle.
Fly- Wheels. In most machinery there would be great inequality of move-
ments, from the great difference in power exerted or resistances to be overcome,
and in the application of the force, as through cranks. To obviate this, fly-
wheels are used, which absorb energy in one part of their revolution and give
it out at another, or by their mass in movement overcome resistances, as in the
punching, shearing, and rolling of metal, which comes only periodically, and
is much in excess of that usually required. In addition, fly-wheels give gov-
ernors time to act, and consequently the motion is more uniform and constant.
For the speed and weight of fly-wheels the conditions vary so much at differ-
ent times, even with the same engines, that it is impossible to get data for an
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
359
estimate by any mathematical formula embracing the conditions. From the
experience of the best mechanical engineers, and from published examples of
constructions, are deduced the following rules, applicable to common practice
for the fly-wheels of steam-engines : The diameter of fly-wheel to be 4 times
FIG. 808.
that of the stroke of the engine, and the entire weight of the wheel 40 times
the square root of the diameter, its exterior velocity being about 5,000 feet per
minute ; if less or more, increase or reduce the weight inversely as the veloci-
ty. The rim is generally a little less than f of the whole weight. For rolling-
360
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS.
mill engines, Mr. C. B. Kichards takes the weight of the fly-wheel at 60 times
the square of the diameter of the cylinder, and the diameter of the wheel 5
times that of the stroke, and rim
velocity not to exceed 125 feet per
second.
In most stationary engines the
fly-wheel is a pulley or band wheel
or gear driving the machinery, but
FIG. 810.
uJ LU UJ LU ' Lf
iTi iTi rfi ffi ffi
FIG. 811.
often the fly-wheel is independ-
ent. Fig. 808 is the elevation
and section of such a wheel, as
built by the Southwark Foun-
dry. The construction will be
understood from the drawings,
but the wrought-iron links con-
necting the segments, shown on
a larger scale (Fig. 809), do not project, but are counter-sunk in the sides of
the rim.
Air-Chambers. The action of the air-chamber is very similar to that of a
JJj
JLLJ
_____
n
3D
FIG. 812.
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 361
fly-wheel ; it tends to make the outflowing or inflowing pressure of the fluid
uniform, and cushions or prevents the reaction that takes place from the fluid
in reciprocating pumps, especially crank-pumps ; but pumps in which the pis-
tons or plungers start very slowly and stop equally so, require but little air-
chamber. Cornish engines are usually provided with a stand-pump instead of
an air-chamber that is, a vertical pipe of considerably larger diameter than
that of the pump, and high enough to contain the water-column.
Fig. 810 is the section of a copper air-chamber for the smaller size of steam-
pumps or hand-pumps. It is screwed into the top of the pump-chamber.
Fig. 811 is the elevation of an air-chamber for power pumps of larger size.
It may be entirely of cast-iron, or a cast-iron base with a copper chamber. A
flange is cast on the top of the pump-chest, and the chamber is bolted to it.
Fig. 812 is the elevation of an air-chamber of one of the Brooklyn pumping-
engines.
It will be observed that the lower end of the small air-chamber is necked,
or of smaller diameter than the main part of the chamber. This prevents a
too sensitive reaction of the air and prevents its escape. In chambers like
that of the Brooklyn engine it is good practice, for the same purpose, to put
a diaphragm across the inside of the chamber, perforated with holes. When
the inlet column is long, whether suction or under pressure, it is well to put
an air-chamber on it.
Air-chambers should be from 10 to 15 times the capacity of the pump-cylin-
der, with glass gauges to show the quantity of air in them for large pumps,
and some provision to supply and maintain the air at such levels as will be
found by experiment suited to the easiest working of the pump.
ENGINEERING DRAWING.
THERE is no part of engineering more important than that of securing a
good foundation for the structure. Where likely to be disturbed by frost, the
structure should start below it, unless, as in the extreme northern regions where
frost is permanent at certain depths, the support should be in it. In preparing
the foundation for any structure, there are two sources of failure which must be
carefully guarded against : viz., inequality of settlement, and lateral escape of
the supporting material ; and, if these radical defects can be guarded against,
there is scarcely any situation in which a good foundation may not be obtained.
It is therefore important that, previous to the commencement of the work,
soundings should be taken to ascertain the nature of the soil and the lay of the
strata, to determine the kind of foundation ; and, the more important and
weighty the superstructure, the more careful and deeper the examination. But
it must be understood that in general it is not an unyielding but a uniformly
yielding stratum that is required, and that a moderate settlement is not objec-
tionable, but an inequality of settlement.
In good sand or gravel, the common load per square foot is from three to
five tons. Many soils are very compressible, not supporting one ton per square
foot ; if the structure is important, the bearing resistance of the strata should
be tested by experiment. The base of the wall is extended to secure the requi-
FIG. 813.
FIG. 814.
FIG. 815.
site area of bearing-surface, either by a base-stone (Fig. 813), by a bed of con-
crete (Fig. 814), or by extending the wall by steps (Fig. 815), with or without
concrete base, or the weight may be distributed by inverted arches between
walls and piers. The walls themselves should sustain from three to ten tons
per square foot.
When the foundation is beneath water, the base may be made of plank, or a
grillage of plank and timber (Fig. 816). But the character of the soil must be
well understood. There are positions, as in the foundation of the Custom-
ENGINEERING DRAWING.
363
House and other public buildings at New Orleans, in which it would appear
that there could be no practicable area of surface that would secure a perma-
nent foundation for an extensive building. A pile
foundation in such earth is more satisfactory, but
all timber should be covered with water to prevent
rot.
Figs. 817 and 818 represent plan and elevation
of a pile foundation ; the piles are usually from
10" to 14" diameter, and driven at about 3 feet be-
tween centers. The tops are cut off square, and
capped with timber ; the caps treenailed or rag-
bolted to the piles, and plank spiked to the timber.
In the figure a sheet-piling, s s, is shown, inclosing
the piles ; the spaces between piles and timbers are often filled with concrete,
small stone, or closely packed earth.
Piles are used either as posts or columns driven through soft earth to a hard
bottom, or depending on their exterior frictional surface to give the necessary
support, either in earth naturally compact or made so by the driving of the piles.
In the first case, care must be taken
that the piles be driven sufficiently
deep into the lower strata to secure
their ends from slipping laterally,
and soundings should be made care-
FIG. 816.
O O
O O
o _o
o o
o o
o o
fully to ascertain the dip and char-
FIG. 817.
FIG. 818.
acter of these strata. In many places, from the hardness and the inclined
position of the lower strata, this kind of foundation is inapplicable and unsafe.
Where a firm foundation is required to be formed in a situation where no
firm bottom can be found within an available depth, piles are driven, to con-
solidate the mass, a few feet apart over the whole area of the foundation, which
is surrounded by a row of sheet-piling to prevent the escape of the soil ; the
space between the pile-heads is then filled to the depth of several feet with
stones or concrete, and the whole is covered with a timber platform on which
to commence the solid work.
In the case in which the support from the piles depends on the exterior
frictional resistance, the rule most generally adopted by engineers is that of
Major Saunders, published in the "Journal of the Franklin Institute" for 1851:
Multiply the weight of the ram by the distance which the ram falls, in
inches, at last blow, divided by 8 times the depth driven or set at that blow.
364
ENGINEEKING DRAWING.
Thus, suppose the ram to be 1,600 pounds weight, the fall 20 feet, or 240", and
the set 1 inch, then the safe load would be - = 48,000 pounds.
1x8
The usual weight of the ram or hammer employed on our public works va-
ries from 1,400 to 2,400 pounds, and the height of leaders or fall from 20 to 35
feet ; but there is a great advantage in reducing the fall, increasing the weight
of the hammer, and the frequency of the blows. As generally driven, and of
average size, when the whole weight is to be supported by the pile, ten tons
may be considered a usual load, but when additional support is received from
compacted earth, broken stone, or concrete between piles and caps, this bear-
ing-surface should also be taken into consideration. In some loose, sandy soils
piles are set, not by driving, but by the water- jet ; a 1" or 1-J* pipe is lashed to
the whole length of the pipe, and a force of water through this pipe clears out
a hole for the settlement of the pile. When in position, the pile is held, the
pipe is withdrawn, and the sand settles around the pile.
Iron pipes with a cast-iron foot are sunk also by a water-jet, the water being
forced into the pile and out beneath the foot.
Hollow cast-iron piles have been driven by exhausting the air from the in-
side ; then the weight of the pile, and sometimes an added load, cause the pile
to settle into the earth ; this is called the vacuum process. The process by
plenum is by expelling the water out of the pile by forcing in air in excess of
the pressure of the surrounding water, and the workmen descending within the
pile and excavating the material.
Sheet-piling (Figs. 819 and 820) is used to keep water out from a foundation,
or to prevent the passage of water
through the earth, as in an em-
bankment or levee. It is usually
of plank two to three inches thick,
FIG. 821.
set or driven. For driving, the
bottom of the plank should be
sharpened to a chisel-edge, a lit-
FIG. 819. FIG. 820. tie out of center toward the tim-
ber side, and cornered slightly at
the outer edge, that it may hug the timber and the plank while being driven.
Fig. 821 is the section of a timber sheet-piling, in which a tongue and
groove forms the guide, the grooves being either made in the timber, as shown
at a a, or planted on, b b. The pile should be of uniform thickness, but the
widths may be random ; six inches thick is a good practical thickness, driving
well under short and frequent blows ; the tongue should be of hard, straight-
grained wood, 2 inches by 2 inches, and well spiked to the pile.
ENGINEERING DRAWING.
365
Frequently, to secure the foundation from water, a wall is constructed of
two rows of sheet-piling, driven one within the other, and the space between
the two filled with clay or some compact earth. This is called a coffer-dam; the
two pilings are stayed to each
other by bolts, and if the wall
is wide enough no other stays
or braces will be necessary.
Retaining '-walls are such as
sustain a lateral pressure from
an embankment or head of wa-
ter (Figs. 822 and 823). The
width of a re tain ing- wall de-
pends upon the height of the
embankment which it may have
to sustain, the kind of earth of
which it is composed (the steep-
er the natural slope at which
the earth would stand, the less the thrust against the wall), and the compara-
tive weight of the earth and of the masonry. The formula given by Morin
for ordinary earths and masonry is I = -285 Ji -f- h' ; that is, to find the breadth
of a wall laid in mortar, multiply the whole height of the embankment above
F IG 323
the footing by
1000
for dry walls make the thickness one fourth more.
Most retaining- walls have an inclination or batter to the face, sometimes
also the same in the back, but offsets (Fig. 822) are more common. The usual
batter is from one to three inches horizontal for each foot vertical. To deter-
mine the thickness of a wall having a batter, "determine the width by the
rule above, and make this width at one ninth of the height above the base."
Fig. 824 is one section of the bulk-head wall, as constructed by the Depart-
ment of Docks of the City of New York, on the North Eiver side, and in
positions where the mud is deep.
The site of the wall is first dredged to hard mud compacted with sand. The vertical
piles are then driven, and small cobble-stones mixed with coarse gravel put around and
among the piles to the height of the under side of the binding frames, and rip-rap stone
placed outside the piles, in front and rear.
The binding frames are then slid down to their places. These binding frames were
made of two pieces of spruce plank 5x10 inches, placed edgewise one over the other, and
running from front to rear of the piles between the rows. An oak beam 8x8 inches is
let through these planks in front of the front row and in rear of the rear row of piles, and
an oak wedge block fitted and placed by the divers between the oak beam and each pile
nearest it. The duty of these frames is to hold the front rows of piles firmly, in case there
should be any tendency in them to tilt outward.
More cobble-stone is then put in to the height of the bottom of the base blocks of the
\vall, weighting the binding frames and preventing any tendency to floating.
The bracing piles are then driven on a slope of six inches horizontal to twelve inches
vertical, between the rows of vertical piles, and spaced three feet from center to center
longitudinally and transversely. All the piles are staylathed and adjusted in position as
soon as they are driven.
366
ENGINEERING DRAWING.
ENGINEERING DRAWING.
367
The bracing piles are cut off at right angles to their axis, about one foot below mean
low water, and capped with twelve inch square timber, running longitudinally. The sides
of the caps are kept horizontal and vertical, and a sloping recess or notch made to receive
the head of each bracing pile, and give it a good bearing.
The six rear rows of vertical piles are cut off at two inches above mean low water, and
notched front and rear to give an eight inch wide bearing across their tops for the trans-
verse caps.
The three front rows of vertical piles are cut off by a circular saw, suspended in the
ways of a pile-driver, at 15*3 feet below mean low water mark, to receive the concrete
base-blocks of the wall. It being impossible to cut off piles at this distance below the sur-
face of the water to exactly the same height, and as the bottom of the concrete base-blocks
would rest only upon the highest piles of those under them, a mattress of burlap, con-
taining freshly mixed soft mortar, in a layer about two inches thick, placed on a network
FIG. 825.
FIG. 826.
of marline stuff, supported by a plank frame about its edges, is lowered upon the tops of
these piles immediately before setting the base-blocks upon them. The diver then cuts the
netting between the edge of the mattress and the plank frame, and the frame floats to
the surface of the water.
368
ENGINEERING DRAWING.
The base-block is then immediately placed in position upon the mattress of mortar
resting on the piles, and the excess of mortar is pressed out from between the head of the
pile and the bottom of the base-block, until each pile has a well and evenly distributed
portion of the load to carry.
The concrete base-blocks for this section are T feet wide at the bottom and 5 feet wide
at the top ; on the front the vertical height is 13 feet, and on the rear 14 feet. The top
has a step on the rear of 1 foot height and 1 foot wide, extending the entire length of the
block, for the purpose of giving the mass concrete backing of the granite superstructure a
good hold upon the block. For handling, grooves for chains are molded in the end, and
a longitudinal hole, 2 feet in the clear above the bottom, connects them, with the corners
rounded, to enable the chain to render easily. The face is curved inward, to save material
while giving a broad base ; their length is 12 feet.
After the blocks are set, the vertical chain-grooves in each block, coming opposite
to each other, are filled in with concrete in bags, well rammed into place. This closes
the joints between the blocks, and also acts as a tongue set into the grooves in the
blocks.
As soon as the base-blocks are set, and the groove filled in, the cross-caps resting on
the tops of the vertical piles, and on the longitudical caps of the bracing piles, reaching
about half way across the base-blocks, are placed and fastened. Oak treenails are used in
all fastenings. The small cobbles are then filled in around and among the piles to the top
of the caps, and the rip-rap placed in the rear of them.
Figs. 825 and 826 are the elevation and plan of a crib with dock or pier.
Below the level of the water, as here shown, the logs are round and locked
to the cross timbers ; above the water the timber is squared, the exterior
walls presenting a tight, smooth surface into which the cross timbers are dove-
tailed.
FIG. 827.
Fig. 827 is a section of the outer wall of the crib-pier erected on the West
Bank for the Quarantine Department of the Port of New York by Mr. J. W.
ENGINEERING DRAWING. 369
Ritch. The structure consists of an outer wall of crib-work, with an interior
filling of sand, 228 feet wide by 488 feet long. The interties occur at inter-
vals of 6 feet spaces, or 7 feet centers.
Extracts from specifications :
" The exterior wall to be built in blocks up to low water, of about 80 feet in length,
sunk to a line, and to be filled up to low water with stone-filling. From the low water
the construction of the exterior wall will be continuous, breaking the joints of the logs
throughout the entire length. The base of the blocks will be formed with timbers 14 inches
square ; two rows on the outside, held together with interties of timber 12 inches square,
each end dovetailed into the outside, and shiplapped to the other timbers, secured at each
end and intersection with iron bolts, 1 inch square, 14 inches long, well driven home.
"The cribs of the entire exterior wall to be built with sound timber 12 inches square,
laid so that they touch each other, secured at every crossing or intersection, and in the
center between each crossing, with iron bolts, 1 inch square, 20 inches long. The cross
timbers to be all in one length ; the ranging timbers to be in lengths of not less than 46
feet ; joints broken over the logs below. The cross-timbers to be dovetailed at the ends,
and shiplapped at intersections. The under tier of timbers to be secured to the logs below,
the ranging timbers to the under tier, and the upper tier to the ranging timbers, as fol-
lows : at each end and every crossing with an iron bolt, 1 inch square, 21 inches long,
well driven home. The entire exterior to be close fendered. extending from the deck-
plank to low water, with sawn white-oak plank, 5 inches thick, and not over 12 inches
wide ; each plank to be secured with 7 iron bolts, 3 inches square, 15 inches long. The 6
corners of this fendering to have each 3 iron bands, 5 feet long on each limb, 3f- inches by
1 inch counter-sunk holes to receive 5 iron bolts, f inch square, 15 inches long, in each
limb.
"Each crib to be filled, from the floor-logs to within 6 inches of the deck-plank, with
stone, granite, gneiss, or trap-rock ; none of the stone to be more than 2 feet in any direc-
tion. The entire exterior to be protected with stone, in large pieces, done in riprap."
Fig. 828 is a transverse section of the river-wall Thames embankment, Mid-
dlesex side. It may be said to be a wall of concrete, etc. , faced with granite,
with a sewer and subway within the same, both inclosed by brick-work. In
the drawings the different material is represented by different shadings and
letters : g, granite ; hi, brickwork ; cc, concrete.
Extracts from specifications :
"The embankment- wall is to be formed within iron caissons or coffer-dams, as the en-
gineer may direct. As soon as the excavations shall have been made to the requisite
depths, and the works cleared of water, the trenches shall be filled up with concrete to a
level of 12| feet below datum, and a bed dressed to the proper slope and level for the foot-
ings of the brick wall. This wall shall be formed thereon (when the concrete has become
thoroughly hard and consolidated) at a true slope in sets- off, as shown on drawing. The
brick-work generally shall be laid in courses at right angles to the face of the wall. The
low level sewer is to be formed on concrete foundation carried down as shown. The sewer
shall be 7 feet 9 inches in the clear diameter for a length of 1,820 feet, and 8 feet 3 inches
in diameter for the remainder of its length, the whole to be formed in brick-work 1 foot 11-
inch thick. The subway shall be formed 7 feet 6 inches high by 9 feet wide in the clear,
generally ; the side-walls to be 18 inches, the arch 1 foot H inch thick. The subway sewer
and river-wall shall be tied into each other, at intervals of 6 feet, by cross or counterfort
walls 18 inches thick, extending from the brickwork of the wall to a vertical line 9 inches,
beyond the side of the sewer farthest from the said wall, and from footings 9 feet belovr
24
370
ENGINEERING DRAWING.
datum, which are to be bedded on a concrete foundation 12 inches thick, up to the under
side of the subway. The upper arch of the subway, and all other similar arches, shall be
coated on their outside circumference with a
layer of Claridge's patent Seyssel asphalt, 1
inch thick, laid on hot, and returned up all
spandrel walls rising above the arch to a
height of 9 inches. The river-wall shall be
faced with granite, generally to a
level of 8 feet below datum, and
shall be surmounted with a mold-
ed parapet of solid granite ; the
stones to be laid in courses, in al-
ternate headers and stretchers.
" The beds and joints to be full
and square for the whole depth, so
that, when set, the work may be
close and solid throughout ; and
no joint to exceed inch in thick-
FIG. 828.
ENGINEERING DRAWING. 371
ness. The whole of the stones above the given level (ll^- feet above datum) to be dow-
eled together in bed and joints with slate-dowels, not less than 5 for every foot run of
wall ; each 2 inches square at least, let fully 2i inches into each stone, very accurately
fitted, and run in with neat cement ; the stones to be bedded and jointed in cement, and
the joints struck with neat cement.
"The whole iron-work to be delivered on the works perfectly free from paint or other
coatings."
Fig. 829 is an isometrical view of the overflow and outlet of the Victoria
and Regent Street sewers in the Thames embankment. S is the main sewer,
and W the subway shown in Fig. 828 ; s s s the street-sewers, discharging into
the overflow basin ; w w the weirs over which the water is discharged into
the weir-chamber c c ; p is the penstock-chamber, which is but a continuation
of the weir-chamber. It has been attempted in the drawing, by breaks, to
explain, as far as possible, the whole construction. Whenever, from storms,
the discharge from the street-sewers (s s s) is greater than can be carried off by
the main sewer (S), the water rises in the overflow-chamber (0), passes over
the weirs (w w) down into the weir-chamber (c), then into the penstock-cham-
ber, and through the flap-gates (g) into the river.
Extracts from the specifications :
" The foundation to be of concrete, not less than 2 feet in thickness; upon this brick-
work shall be built for the flooring of the chambers, and for the side-end and weir-walls.
The weir-chamber shall be divided in the direction of its length, by a brick wall, into two
rectangular overflow-channels, covered with cast-iron plates, 6 feet 8^ inches long, 3 feet
wide by & inch general thickness, with strong ribs and flanges on the under side, properly
bolted together and jointed with iron cement, and bolted down to stones which are to be
built into the under side of the brick- work of the basement-chamber. Arches on either
side, running parallel thereto, and communicating with this chamber and with the weirs
which are to be formed, upon which weir-walls, divided so as to correspond with these
.arches, are to be built in brick-work, capped with granite blocks, 4 feet long, 2 feet deep,
and 2 feet 3 inches in the bed. The floor of the penstock-chamber to be formed with York
landings, 6 inches thick, having a fall of 3 inches to the river. The outlets for the penstock-
chamber through the river-wall shall be formed by an arch-recess in granite, and fixed with
two tidal flaps, well hung, and firmly secured to the masonry by strong bolts and screws.
" The subway is to be continued over the low-level sewer, and across the overflow cham-
ber, by cast-iron plates, curved to the form of the arch, $ inch general thickness, with
strong ribs and flanges on the upper side, properly bolted together, and strongly bolted
down to the brick-work ; jointed with iron cement, and covered with brick-work, to form
the floor of the subway. From a point of 10 feet 8 inches on either side of the central
longitudinal line of the chamber, where the sewer and subway are farthest from the river-
wall, these are again to be brought into their general position by two curves, each not less
than 80 feet in length.
" The whole of the cast-iron shall receive one coat priming of red lead and linseed oil,
and three coats best coal-tar, before fixing ; and the accessible surfaces one further coat
best coal-tar, when fixed."
Foundations for piers and abutments of bridges beneath the surface of water
are formed by piles, by throwing down masses of stone or beton until the mass
reaches the surface of the water, by open caisson or by inclosing the space
within a coffer-dam, and proceeding as in common foundations, or by an in-
verted caisson and air-lock.
372
ENGINEEKING DEAWING.
ENGINEERING DRAWING.
373
An open caisson is a chest of timber which is floated over the site of the
work, and, being kept in its place, is loaded with stone until it rests firmly on
the ground. In some cases the stone is merely thrown in, the regular masonry
commencing with the top of the caisson, which is sunk a little below the level
of low water, so that the whole wood-work may be always covered, and the
caisson remains as part of the structure. In others the masonry is built on the
bottom of the caisson, and when the work reaches the level of the water the
sides of the caisson are removed.
The general plan adopted by G-. A. Parker, C. E., in the erection of the
piers of the Susquehanna bridge, was :
First to dredge away as much as possible of the material in the bed of the river at the
pier site. A f-inch thick boiler-iron curb was then sunk and secured in its place. The
curb was about 30 feet wide and 50 to 60 feet long, and of sufficient height to reach above
the bed of the river. The material was then pumped by sand-pumps out of the curb,
which gradually undermined, and settled down to the required depth, or on to the bed-
rock. When stumps, logs, or bowlders were met with, they were removed by divers work-
ing in a bell. After the rock had been thoroughly cleaned off, it was brought to a uniform
level by a solid bed of concrete extending over a greater space than the size of the bottom
of the pier, using the diving-bell for this purpose.
Three guide-piles on each side, and one at each end, were fixed firmly in position. A
strong platform of solid timber, the size of the bottom of the pier, was then placed in
position over the curb, and at the surface of the water. On this was placed a caisson of
iron large enough to contain the pier, and with sides and ends high enough to reach to the
level of high water after the caisson is landed on the bottom. The caisson was then made
water-tight. The bottom was then floored over with masonry and stone, and laid in mor-
tar up the sides of the caisson to the top, thus constituting a stone caisson inside of an iron
one. This was secured to the guide piles, and the masonry of the pier proper was laid up,
the caisson sinking as the weight of masonry inside increased, until it finally settled upon
the bottom which had been prepared for it, as already described. At some of the piers
r-
i
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FIG. 830. FIG. 831.
(Figs. 830 and 831) screw-rods were used to suspend the pier and gearing attached, gov-
erned by one man, who at pleasure could raise or lower without assistance the whole
pier. Wooden piles were driven and cut off by machinery just above the ground, and the
374: ENGINEERING DRAWING.
platform, with its incumbent pier, lowered upon them ; at other piers the foundation was
on rock.
Piers are sometimes made by sinking a wrought-iron curb, extending from the bottom
to above the level of the water, driving Within it the usual proportion of piles, and then
filling the spaces entirely with concrete.
Dams are constructed to pond water for the supply of cities and towns ; for
inland navigation, by deepening the water over shoals, and the feeding of ca-
nals ; for power in its application to mills and workshops ; and for irrigation.
To whatever purpose the water is to be applied, there are two questions to be
settled : Whether the level will be raised high enough by the construction, and
whether the flow of the stream be sufficient for the purpose required ; and fur-
ther, it may often be important to know how large a pond will be thus formed,
how ample a reservoir for unequal flow, or intermittent use. If the pond be small,
so that the water can not be retained, and the supply is only the natural run of
the stream at a high level, then the minimum flow of the stream is the measure
of its capacity.
The rule that obtains on the Merrimack River, at Lowell, and Lawrence,
where the pondage is more than the average, is that 1 cubic foot per second
per day of 12 hours per square mile of water-shed can be depended on for per-
manent mill-power. On very small streams it may often happen that pondage
may be secured, and the supply be equal to one half the rain-fall.
Blodgett, in his " Climatology of the United States," says that " in this
sense of permanence as a physical fact, we may consider the quantity of rain
for a year as a surface-stratum, on the Atlantic slope and in the central States
of 3| feet, which may be diminished to half this quantity, or increased to twice
as great a depth in the extreme years. But, with such an average and such a
known range, we may deal with the quantity as definitely as with a stream of
which we know the mean volume and the extremes to which it is liable, and
for many departments of engineering these climatological measures are as indis-
pensable as those of tide or river hydrography."
The evaporation from a reservoir-surface at Baltimore, during the summer
months, was assumed by Colonel Abert to be double the quantity of rain-fall.
Dr. Holyoke assigns the annual quantity evaporated at Salem, Mass., to be
56" ; but from experiments made by the Croton Aqueduct Department, in
1864, of the evaporation from a box set in the earth-bank, and two afloat in
the upper reservoir, the quantity was found to be severally 37*12, 37 '53, and
39*97 inches.
Fig. 832 is the section of a crib-dam in northeastern Colorado for the pond-
age of water for the purposes of irrigation. The crib-work is of round logs,
10" at least in diameter, joined at the ends as in ordinary log huts, with dove-
tail or tongue. Each crib is 18 feet long on the face, and the fastenings are
2" X 18" treenails. The cribs are set radially, forming a curve up-stream of
200 to 238 feet radius. The crib gives the stability, but the water-tightness
depends on a shutter, p, or vertical panel of timber, on the up-stream side of
which there is a filling of earth.
Crib or wooden dams, when the timber is not kept covered with water, fail
from the decay or rot of the timber.
ENGINEERING DRAWING.
375
Fig. 833 is a section of the dam across the Croton River, constructed under
the direction of Mr. John B. Jervis, for the supply of the aqueduct for the
city of New York. This dam was built on an earth foundation, with curved
roll in cut stone, extended by a timber-apron some 50 feet, supported by strong
P
FIG.
crib-work. Originally there was a secondary dam still farther down, to throw
back-water on this apron. In the erection of this dam, excavation was made
of all loose material ; the cribs C and D were built up, and the tops were
planked ; on this planking were carried up the cribs F and G. Between these
piers the space E, as well as e below and on the cribs, was filled in with con-
crete ; on this the body of the dam was erected in stone-masonry, laid in cement.
The face-work of granite is cut to admit of a joint, not exceeding ^ of an inch.
FIG. 833.
Above the dam is an earth embankment, its upper part protected by a rubble-
paving. The radius of the granite face is 55 feet, and the dam 38 feet high
from level of apron to crest of dam.
Fig. 834 is a section of the dam across the Connecticut River, at Holyoke,
Mass. This dam is 1,017 feet long between abutments, and averages 30 feet
376
ENGINEERING DRAWING.
high by a base of 80 feet. It is constructed of tim-
ber crib- work, loaded in with stone for about ^ its
height. The foot of each rafter is bolted to the
ledge, and all timbers at their intersections are
treenailed together with 2" white-oak treenails.
The inclined plank-face is loaded with gravel,
and the joint at the ledge covered with concrete.
The lower or base-tier of ranging timbers were
15" X 15" ; the other timbers, 12" X 12". The
rafters are placed vertically over each other, in
bents of 6 feet between centers. The plank-
ing was of hemlock, 6" thick, with oak cross-
planking at crest of dam, 4" thick at bottom
.and 8" at top. The crest was plated with
iron, y thick, 5 feefc wide. During the con-
.struction the dam w r as planked first about
30 feet on the incline ; a space was then
left of about 16 feet width by sufficient
length, through which the water flowed ;
and the balance of the dam was then
completed. A plank-flap was then made
for the opening, and when every thing
was ready, it was shut down, and the
pond filled. The dam was built under
the direction of the late Mr. John
Chase, and since its construction the
greatest depth of water passing over
the crest during a freshet was 12' 6".
Some years after the construc-
tion of this dam it was found that
the overfall of water from its
crest was wearing away the
ledge and jeopardized the foun-
dation of the dam. An apron
(Fig. 835) was therefore
constructed of crib-work,
sheathed with plank, add-
ing stability to the struct-
ure, and discharging
the water more nearly
in the line of the river
current.
Fig. 836 is a
section of part of
the dam across the
Merrimack Eiver, at
Lowell, built under
ENGINEERING DRAWING.
377
FIG. 836.
378
ENGINEERING DRAWING.
SCALE :
A inch = 1 foot.
. 837.
the direction of Mr. James B. Francis. It was laid dry, with the exception
of the upper face and coping, which was laid full in cement.
The horizontal joints at the crest were run in with sulphur. The coping-
stones were doweled to the face and together, and clamped to an inclined stone
on the lower slope ; the end-joint between these stones was broken by making
every alternate lower stone longer, and the upper shorter, than shown in the
drawings.
The Cohoes dam (Fig. 837) was built under my direction, directly below an
old dam of somewhat similar construction to that of Holyoke. The old dam
had become very leaky and worn, and the overfall had in many places cut deep
into the rock, and in some
places within the line of the
dam. It was therefore pro-
posed to make the new dam,
as a roll to the old one, to
discharge the water as far
from the foot of the dam as
possible, and to keep the old
dam for the protection of the
new. The exterior of the
dam was of rock-faced ash-
lar ; the caps were in single
lengths of 10 feet, and none
less than 15" thick and 2 feet wide ; they were doweled together with two
galvanized wrought-iron dowels each. The whole work was laid full in cement,
the 20" wall next the old dam being laid distinct without bond into the rest
of the work. The whole was brought up to the outline, to receive the cap-
stones, which were bedded in cement ; the top-joints were then run or grouted
in neat cement, to within about 6" of the top of the stone, which was after-
ward run in with sulphur. Entire length of overfall, 1,443 feet ; average
depth below crest of dam, 12 feet.
Where the body of water which may at any time discharge over the dam is
large and the fall high, it is especially desirable to secure a location where the
overfall can be upon solid rock. If there be ledge at the side of the river, and
none can be found in the channel, it is often better to make a solid dike across
the river and above the level of freshets, and cut the overfall out of the bank.
When from any circumstances the dam can have only an earth foundation, an
artificial apron, or platform of timber or rock, is to be made, on which the
water may fall, or the high fall may be broken up by a succession of steps. In
some cases, a roll or incline, like that given in Croton dam, is extended to the
bed of the stream, and continued by an apron. The water thus rolls or slides
down, and takes a direction, as it leaves the apron, parallel with that of the
bed of the streanio But care must be taken to protect the outer extremity of
the apron by sheet-piling and heavy paving, as the current, by its velocity,
takes with it gravel and all small rocks, and undermines the apron.
Dams or dikes are often made entirely of compacted earth ; sometimes with
a puddle-wall of clay in the center, as in the reservoir embankment (Fig. 860),
ENGINEERING DRAWING.
379
or a sheet-piling. Dikes across salt
marshes are made of material taken
from the marsh at some distance
from the site of the dike, well packed
in thin layers on a base prepared on
the soil without excavation. Sand
and gravel, being heavier than the
moist material, break through it and
settle to the bottom, involving often
the construction of a large embank-
ment, while, by the use of a homo-
geneous material, the foundation is
not displaced but compressed.
Fig. 838 is a section of the dike
or embankment for the Ashti Tank
or Reservoir, constructed for retain-
ing water for irrigation purposes in
India. The following is an abstract
of the description of the work given
in the "Minutes of the Proceedings
of the Institute of Civil Engineers,"
vol. Ixxvi :
u The net supply available for irriga-
tion may be calculated thus :
Available capacity
of tank 1,348,192,450 cub. ft.
Deduct loss by evap-
oration, etc 233,220,240 "
Net supply available
for irrigation. .. 1,114,972,210 "
" Area of catchment basin nearly 92
square miles."
The total length of the dam is
12,709 feet ; the breadth at the top,
which is uniform throughout, six
feet ; breadth at full supply-level,
42 feet ; height of the top of the
dam above full supply-level, 12 feet ;
greatest height of dam, 58 feet.
The seat of the dam throughout was
cleared of vegetable mold, stones,
and loose material, all trees and
shrubs with their roots being com-
pletely grubbed or dug out. The
puddle-trench laid in the natural
JUL
380 ENGINEERING DRAWING.
ground is rectangular in cross-section, 10 feet in width, excavated through
various materials to a compact water-tight bed, and then filled in with puddle
material, consisting of two parts of muram or sand, and three parts of black
soil, carefully mixed and worked by treading with the feet, and then kneaded
into balls and thrown or dashed into the trench in layers up to 12 inches in
thickness. The puddle was brought to a level of one foot above the ground.
Across the river the trench was cut down to the rock and filled with concrete.
The general distribution of the material of the dam is shown in the figure.
The central core is formed of the best black soil attainable ; on each side, ex-
tending to the surface of the mixed material, brown, reddish, or white earth is
used. The outer part of the dam is formed of a mixture of equal parts of black
soil and muram, but where muram was difficult to obtain, and sand plentiful,
the latter was substituted for muram in the mixture. The black soil may be
described as a clayey earth, tenacious and adhesive when wet a product of
the decomposition of volcanic rock. The brown and reddish soils are of a
clayey nature, but contain admixtures of fine sand, kunkun nodules, and thin
layers of fine grains of lime. The white soil consists of finely powdered parti-
cles of a grayish color, similar to wood-ashes, which when dry possesses little
adhesion, but when wet is adhesive.
The various soils were laid in the work in layers eight inches in thickness,
every layer being thoroughly watered and rolled with iron rollers. The outer
slope was protected by a mixture of
equal parts of soil and reddish mu-
ram, and with sods of grass, laid
about three feet apart, which in time
extended over the whole slope.
The inner slope is protected from
the action of the waves by being
pitched or faced with dry stone, set
by hand, and laid on a layer of
coarse muram. The stones of the
FlQ - 839 - pitching were bedded on the slope,
and were laid with their broadest
end downward (Fig. 839), each stone being roughly squared with the hammer,
and touching for at least three or four inches. The interstices were then
packed with small stone-chippings, and finished off with muranio
Head-gates are constructions necessary to control the flow from the river-
pond or reservoir into the canal or conduit by which the water is to be con-
veyed and distributed for the purposes to which it is to be applied. The top
of the works should therefore be entirely above the level of the highest freshets,
that no water may pass, except through the gates ; and it is better that the
opening of the gates should be entirely below the level of the top of the dam,
to prevent as much as possible the passage of drift and ice, which are often ex-
cluded by booms and racks placed outside the gates.
Figs. 840 and 841 are drawings, in plan and detail, of the head-gates, and
the machinery for hoisting them, at the Cohoes Company's dam.
It will be seen, by reference to the plan, that there are ten gates. The
ENGINEERING DRA 1
382
ENGINEERING DRAWING.
ENGINEERING DRAWING.
383
dimensions of four are 8' x 6' 6" ; and six, 8' x 9', in the clear all of which
can be hoisted by machinery connected with a turbine-wheel at a, or separately
by hand. At b there is an overfall, at the same height as the dam, over which
any drift that is brought against the gate-house is carried. At c there is a
similar overfall within the gates, and another at d, by which any sudden rise
of the level of the canal is prevented. At e there is a gate for drawing down
the pond, and another at /, for drawing off by the canal, both raised and low-
ered like the head-gates.
The head-gates are of solid timber bolted together, moving in cast-iron
guides set in grooves in the stone ; in front of these grooves there is another
set of grooves (gg}> which are intended for slip-planks or gates, to be put in
whenever it is necessary to shut off the water from the gates themselves in case
of repairs.
Hoisting Apparatus. To each gate there are strongly bolted two cast-iron
racks, geared into two pinions on a shaft extending across the gate-space, and
FIG. 842.
FIG. 843.
supported on cast-iron standards on the piers. At one extremity of this shaft,
there is a worm-wheel, driven by a worm or screw on a shaft perpendicular to
384 ENGINEERING DRAWING.
the pinion-shaft. The worm-shaft can be driven either by a hand-wheel at one
end, or by the friction -bevel at the other. The friction-bevel can be driven in
either direction by being brought in contact with one or other of the friction-
bevels on a shaft extending the whole length of the gate-house, and in gear
directly with the small turbine at a. The small turbine draws its supply
through a pipe, built in the walls, and opening into the space between the
gates and the slip-plank groove.
Figs. 842 and 843 are the front elevation and section of the gates of Farm
Pond, Sudbury River Conduit, Boston Water- Works. The main web or plate
of the gate is !* thick, the ribs 6" deep, the gate-stems %\" diameter. The
nuts by which the gates are raised are geared together, and actuated by a
double crank. For smaller gates it is usual to have but a single stem, and the
nut in a hand-wheel on top of the standard. The gates and guides are faced
with brass, about T y thick.
Gates of this form are very common, consisting of plates of cast-iron
strengthened by ribs ; the guides are also of cast-iron, bolted to the masonry.
The faces of the gates and guides are usually covered by brass plates, as iron
faces become rusty. When the gates are small, there is usually but one stem.
Often, instead of nuts and screws, racks and pinions are used ; and with heavy
wooden gates, requiring but little use, the gates are raised by chains over a bar-
rel, by hand-spikes, and ratchets to hold the gates in position as they are raised.
Canals. The sections of canals depend upon the purposes to which they
are to be applied, whether for navigation or for power ; if for navigation,
reference must be had to the class of boats for which they are intended ; if for
power, to the quantity of water to be supplied, and sundry precautions of con-
struction.
Fig. 844 is a section of the Erie Canal : width at water-line, 70 feet ; at
bottom, 28 feet ; depth of water, 7 feet ; width of tow-path, 14 feet. It will
be observed that the slopes are graveled and paved, and that the edge of the
FIG. 844.
tow-path is paved with cobble-paving, and the path graveled. The smaller
canals of this State and of Pennsylvania are generally 40 feet wide at water-
line, and 4 feet deep ; the Delaware and Raritan, 75 ; x 7'; the Chesapeake and
Delaware, 66' x 10'; the ship-canals of Canada, 10 feet deep and from 70 to
190 feet wide.
The dimensions for canals for the supply of mills depend first, on the
quantity of water to be delivered. Their area of cross-section should be such
that the average velocity of flow should not exceed two feet per second, and in
northern climates less velocity than this would be still better ; it should always
be such that during the winter the canals may be frozen over, and remain so,
to prevent the obstruction from drift and anchor-ice in the water-wheels. The
ENGINEERING DRAWING.
385
usual depths of the larger canals are from 10 to 15 feet ; with such depths the
cover of ice which reduces the section by the amount of its thickness does not
materially increase the velocity of flow, nor diminish, consequently, very per-
ceptibly the available head.
Fig. 845 is a section of the Northern Canal, at Lowell, Mass., which may
be considered a model for large works. The width at water-line is 103 feet,
FIG. 845.
and the depth 16', and is intended for an average flow of 2,700 cubic feet per
second. The fall in the whole length of 4,300 feet is between 2" and 3"; when
covered by ice, about 4". The sides are walled in dry rubble, and coped by
split granite. It will be observed that the portion above, and about three feet
below, the water-line, or between the limits of extreme fluctuations of level,
is laid plumb, that the ice may have as free a movement as possible vertically.
Fig. 846 is a sec-
tion, on a scale of -J"
= 1 foot, of the river-
wall of this same ca-
nal, where the canal
passes out into and
occupies a portion of
the river-channel, and
the depth of water in
the canal is greater
than in the above sec-
tion. The main wall
is in dry masonry,
faced on river -side
with rough-faced ash-
lar, pointed beds and
end-joints. The in-
side lining is of two
courses of cement-
wall, the dry rubble
backing being first
laid, then pointed FIG. 846.
with cement, against
which is laid the first cement lining, which is plastered on the inside, and the
interior wall is then laid ; the granite inside wall, above lining, is also laid in
cement.
25
386
ENGINEERING DRAWING.
FIG. 847.
SCALE : A" = 1 foot.
FIG. 848.
Locks of Canals. Figs. 847 and 848 are portions of plan and vertical sec-
tion of locks, taken from the general plans for timber locks on the Chemung
Canal. They represent the half of
upper gates.
Fig. 849 is a section , of one side
of the lock of the same.
Fig. 850 is the plan of a portion
of one of the enlarged locks of the
Erie Canal, showing one of the upper
gates and the side-walls.
Fig. 851 is a cross-section of one
of the same locks, showing the cul-
vert in the center between the locks,
FIG. 849. used for the supply of the waste of
ENGINEERING DRAWING.
the lower level, to preserve the proper height of
by gates in the upper level.
'
\\\
E3 v \
IA v
\
tf this level controlled
FIG. 850.
I L
J L
FIG. 851.
SCALE : -fs" = 1 foot.
FIG. 852.
full size.
FIG. 853.
Fig. 852 is a drawing, in outline, of the hollow quoin of the lock-gate, on a
scale of -^ T full size (Chemung Canal).
Fig. 853 is a plan and elevation of pintal for heel-post of lock, with a sec-
388 ENGINEERING DRAWING.
tion of the bottom of the post. The pintal is imbedded in bottom timber or
stone, as the case may be.
Fig. 854 is a plan and elevation of the strap for the upper part of heel-post.
Extracts from lock specifications (" New York State Canals," 1854) :
" Locks to be composed of hydraulic
stone masonry, placed on a foundation
of timber and plank. The chamber to
be 18' wide at the surface of the water
in the lower level, and 110' long be-
tween the upper and lower gate-quoins.
The side- walls to extend 21' above the
upper gate-quoins, and 14' below lower
gate-quoins. If the bottom is of earth,
and not sufficient to support the foun-
dation, then bearing-piles of hard wood,,
not less than 10" diameter at small end,
shall be driven, to support the founda-
- 854. tion. There shall be four rows of piles-
under each main wall, and one row in
center of lock ; the piles shall be driven in rows, at 3' from center to center. The piles to
support the wing and breast-walls and wing buttresses, and also under the miter-sills, to be
driven in rows to conform to the form and shape of the same. The heads of the piles to
be cut off smooth and level, to receive the foundation timbers. The foundation timbers
to be 12'' x 12", and of such lengths as will extend from and cover the outside piles, and
to be treenailed with a 2" white-oak or white-elm treenail, 24" long, to each pile.
" If the bottom is of earth sufficiently compact and firm to support the foundation
without bearing-piles, then the foundation shall be composed of timber, 12" thick and not
less than 10" wide, counterhewed on upper side, timbers to average 12" wide, to be placed
at uniform distance, according to their width, so as to occupy or cover at least of the area
of the foundation, and under the lower miter-sill to be placed side by side : in all cases to
be of sufficient length to extend across the lock to the back line of the center buttresses,
and at the head and foot to the rear or back line of wing-walls. The timber under the
lower miter-sill to be of white oak, white elm, or red beach, the other foundation and
apron timber to be of hemlock. The foundation to be extended 3' above the face of the
main wall at the head of the lock, and at the foot from 25' to 30' below the exterior wing
that portion of the spaces between the timbers in all cases to be filled with clean coarse
gravel, well rammed in, or concrete. In cases where rock composes the bottom of the
lock, the foundation timbers, if required, shall be 10" thick under the lower miter-sill, and
8" thick at other places. Where the rock is of such a character that timber is not required
for the foundation, the same shall be excavated smooth and level, and the first course of
stone well fitted to the rock.
" Sheet- Piling. In all cases where rock does not occur, there shall be a course at the
head of the foundation, under each miter-sill, and at the lower end of the wings, and at the
lower end of the apron, to be from 4' to 6' deep as may be required in each to extend
across the whole foundation. The sheet-piling to be of 2" hemlock plank, lined with 1"
pine boards. Ditches are to be excavated to receive the sheet-piling, which are to be
placed edge to edge, and the top well secured to the foundation timber ; the spaces to be
filled up with fine hard gravel, well puddled in, or with concrete.
" Flooring. A course of 2" pine or hemlock plank to be laid over the whole of the
foundation timbers, except a space, 3' wide, under the lace-line of each wall to be 2"
white oak : the whole to be well jointed, and every plank to be treenailed with two white-
ENGINEERING DRAWING. 389
oak treenails at each end, and at every 3' in length, to enter the timber at least 5", or with
wrought-iron spikes, treenails to fill 1J-" bore. Platform for the upper miter-sill to be 5'
10" wide, and 6' high above foundation, and to extend across from side-wall to side-wall,
to be composed of masonry, coped with white- oak timbers, which are to extend 6" into
each side-wall. The timbers to be 12" deep and 14" wide, covered with two courses of
\\" white-oak plank. Miter- sills to be of best white-oak timber, 9" thick, to be well jointed,
and bolted to the foundation or platform timbers, as the case may be, with bolts of iron,
20" long, 1" x 1", well ragged and headed, eight bolts to each side.
" Masonry. The main walls, for 21' 6" in length, from wing-buttresses at the head,
and 32' at lower end, to be 9' 8" thick, including recesses, and for the intermediate space,
V 8" thick, with three buttresses projecting back 2V, and 9' long at equal distances apart.
The quoin-stones, in which the heel-post is to tarn, shall not be less than 4' 6" in length in
line of the chamber, to be alternately header and stretcher. The recesses for the gates to
be 20" wide at top of wall, 12' long, with sub-recesses, 9" wide, 6' high, 10' long, for the
valve-gates. Breast-wall to commence 5' below upper end of foundation, 5' wide, 8' high,
finished with a coping of cut stone. The interior wing-walls, and exterior wing from main
walls to the termination of first curve, to be 7' 6" thick, and the running curve of exterior
wing to be 6' thick on the foundation.
" Culvert between Locks. In such cases as may be required, a culvert shall be con-
structed, to pass the water from the upper to the lower level, as follows: A foundation
of suitable timber and plank, as for lock-walls, and covering all the space between the
lock-foundations, shall be put down. Three apertures for the sluice-way shall be made
in the head-wall with cut-stone jambs, grooves to be cut in the jambs for the sluice-
gates, and the coping to form a recess, corresponding with the grooves in the jambs ;
grooves to be cut on the top and bottom coping, 1" deep, to secure the jambs. The
bottom of the aperture to be of cut stone, with lower corner beveled off, over which the
water will fall into the well, the bottom of which shall be covered with a sheeting of
cut stone, 6" thick. The apertures to be 3' 6" deep, placed immediately below the coping-
stone, and 4' long. Suitable gates of plank, for regulating the water in passing the sluice,
to be prepared ; the well to commence on the foundation, to be made of substantial hy-
draulic masonry.
" Second flooring of seasoned 2" first-quality white-pine plank, to be well jointed, and
laid on the foundation between the walls, from the breast-wall to lower end of main wall,
and also on the floor of the well, to be close and firmly jointed to miter-sills and walls, so
as to make a water-tight flooring. The plank to butt, or the end-joints to come to the
center of a foundation timber, and each plank to be treenailed with two treenails at end
and two at every 3' intermediate : treenails 10" long, to fill 1J" bore.
" Gates. The framing to be made of best quality white-oak timber; the cross-bar to
be framed into heel and toe posts with double tenons, each tenon to be 7" long, and thick-
ness equal to the thickness of the bar, and secured with wrought-iron Ts, well bolted.
The heel and the posts to be framed to the balance-beam by double tenons, and secured by a
wrought-iron strap and balance-rod, from the top of the beam to the under side of the upper
bar. The lower ends of the heel-posts to be banded with wrought-iron bars; the collar
and other hangings to be of wrought-iron, secured together with a double nut and screw,
and to the coping by bedding the depth of the iron in, and by screw-bolts fastened with
sulphur and sand-cement. The pivots and sockets which support the heel-posts to be of
best cast-iron; a chilled cast-iron elliptical ball, 2V horizontal, and 1" vertical diameter,
to be placed on the pivot and in the socket of each heel-post, to facilitate the movement
of the gate. The gates to be planked with seasoned first-quality 2" white-pine plank,
jointed, grooved, and tongued tongues of white oak the plank to be secured by 6"
pressed spike. On the chamber-side of the gates, fenders of white-oak plank, to be put on
with pressed spike."
390 ENGINEERING DRAWING.
Water, ponded by dams, and conveyed by canals for use as mill-power, is
carried within the workshops or manufactories, to be applied on water-wheels,
by some covered channels. These channels, although of various forms, are
usually designated as flumes. The common form of a flume for the convey-
ance of water to breast, overshot, or undershot wheels, is of a rectangular sec-
tion, framed with sills, side-posts, and cap, and, if large section is required,
intermediate posts are set in. The sills are set, and earth well rammed in the
spaces between them ; the bottom plank is then laid, posts and cap framed with
tenon and mortice, set and pinned, and the plank is then firmly spiked on the
outside of posts and caps. The planks are usually nearly green, jointed, and
brought to close joints ; the size of timbers will depend on the depth beneath the
soil, or the insistent load. Within the mill, and just above the wheel, the
flume is framed without a cover, and the posts and side-planks are brought above
the level of the water. This open flume is termed the penstock, especially neces-
sary, in the class of wheel above referred to, to secure the full head of water.
Many flumes are made of a circular section, pipes of iron, or wood. For
the conveyance of water to turbine-wheels, wrought-iron pipes are almost inva-
riably used. Cast-iron is also sometimes used, with flange, or hub and spigot-
joints. Plank-pipes are also used, made with continuous staves, and hooped
with wrought-iron ; these constructions are much cheaper, and serve a very
good purpose. The head-gates of flumes are placed at the head of the flumes,
in a recess back from the face of the canal, with racks in front to prevent the
passage of any drift that might obstruct or injure the wheel. The total area
of passages through the racks should liberally exceed the area of cross-section
of the flume, not only on account of the extra lateral friction of the rack-bars,
but also on account of their liability to become obstructed. Sometimes two
sets of racks are placed in front of the flumes, especially for turbines and react-
ing wheels : a coarse rack with wide passages, say 2" spaces outside, and a finer
one inside, say of f" to f " spaces. The head -gates to the flume, directly back
of the racks, in their function are like the head-gates at the dam, and are simi-
lar in construction strong plank gates, moving in slides, vertically or horizon-
tally, with a paddle-gate in them, to fill the flume when empty, so that the
gates themselves may be opened without any pressure due to a difference of
head outside and inside of the gates, and also to prevent any damage to the
flume by the water-ram, which might result from a too sudden filling of the
flume by the opening of a large gate suddenly.
Fig. 855 is the elevation and section of the head-gates manufactured at
Holyoke, Massachusetts. G G are plank gates, sliding laterally, moved by
two pinions, working into racks on top and bottom of gates, turned by a hand-
spike. P is the paddle-gate ; R, the rack ; F, the flume, or plank-pipe ; A,
air-pipe, for the escape of air from the flume while being filled.
Conduits for the supply of water to cities and towns are of masonry, or cast
or wrought iron pipes. Their capacity to deliver the required quantity depends
upon the area and form of cross section, and the velocity of flow due to the
loss of head or of pressure permissible ; this velocity being due primarily to
gravity, but largely modified by conditions of structure, as the kind and
amount of wetted surface, and length and directness of line.
ENGINEERING DRAWING.
391
392
ENGINEERING DRAWING.
Fig. 856 is a cross-section of the main conduit of the Nassau Water- Works
for the supply of the city of Brooklyn, Long Island. The width is 10" at the
springing of the arch ; the side-walls 3 feet in height ; versed sine of invert,
8* ; height of conduit in center, 8' 8" ; fall or inclination of bottom, 1 in
10,000.
In preparation of the foundations the contract specifications required a bed of concrete
to be first laid, 15' wide; but, when the water was very troublesome, it was found neces-
sary to lay a platform of plank for the concrete. The side-walls are of stone, except aD
interior lining of 4" brickwork. The arch is brick, 12", and the invert 4" thick. The
outside of arch, as it was finished, and each wall, were plastered over on the outside with
a thick coat of cement-mortar. The concrete was formed from clean broken stone, broken
so as to pass through a 2" ring; 2 to 2 measures of broken stones were mixed with 1
measure cement-mortar. The centers of the arching were not allowed to be struck until
the earth had been well packed in behind the side-walls and half-way up the arch. In
both cuttings and embankments the arch was covered with 4 feet of earth, with a width
of 8 feet at top, and slopes on each side of 1-J- to 1, covered with soil and seeded with grass.
Fio. 856.
FIG. 857.
Fig. 857 represents a section of the Oroton Aqueduct, in an open rock-cut.
The width at spring of arch, 7'; versed sine of invert, 6*; height of conduit,
8' 6"; fall or inclination of bottom, about 1 in 5,000.
The bottom is raised with concrete to the proper height and form for the inverted
arch, of a single course of brick ; the side-walls are of stone, laid in cement, plastered,
and faced with a single course of brick; the arch is semicircular, of brick two courses
thick, with spandrel backing nearly to the level of the crown, and earth filled on the top.
In earth-cuts or embankments, side-walls were constructed of stone, in cement; and in
embankments the whole structure rested on dry rubble-walls, built up from solid earth-
foundations.
At the crossing of the Harlem Eiver, as the bridge was depressed below
the level of the aqueduct, the water was conveyed by two cast-iron pipes,
a a, 3' in diameter, Fig. 858 ; but, as the demand for water increased in the
city, the obstruction caused by lack of capacity in these pipes has made nec-
essary the introduction of a larger pipe, which has been made of wrought-
iron, -J" thick and 7' 6" in diameter ; this is supported by cast-iron columns
ENGINEERING DRAWING.
393
which admit of a
rocking movement,
and slip- joints are also
made in the pipe to
compensate for any
expansion or contrac-
tion by changes of
temperature. The
pipes are inclosed in
a long chamber or
passage, extending
the whole length of
the bridge, covered
by a brick arch, laid
in cement with a
cover of asphalt, and
a brick pavement
over all, affording a
wide promenade pro-
tected on each side
by cast-iron railings,
fastened to the coping-stones, CO. A A are the arch-stones of the bridge.
Fig. 859 is a section of the conduit of the Boston Water- Works. The
inside section is equal to a circle 8 feet diameter, and is uniform throughout
FIG. 858.
FIG. 859.
except in tunnels. The exterior lines vary according to the material on
which it is built and the cover or load on the top. The section given may be
394
ENGINEERING DRAWING.
considered the general one, resting on a bed of concrete, with masonry sides ;
brick lining at sides and invert at bottom, with an 8" arch at top for a 4'
cover, and 12" for exceptional depths or under railway-tracks. The lower
corners were of brick, of the special form shown.
The inclination of the conduit is 1 foot per mile, and the flow 80,000,000
gallons per 24 hours when full or 5 feet above center of invert. The maximum
flow takes place when the depth of water is 7' 2", the delivery then being 109,-
000,000 gallons.
In large works, where there is considerable length of conduit, receiving
reservoirs, within or near the limits of the city, are necessary as a precaution
to guard against accidents which might happen to conduit or dam, and cut off
the supply, and also as a sort of balance against unequal or intermittent draught
among the consumers. The size of these reservoirs must depend on the neces-
sities of the case, and on the facilities for construction. The capacity of the
Kidgewood reservoir, at Brooklyn, is 161,000,000 gallons when full ; of the
new Croton reservoir, about 1,000,000,000 gallons. Both these reservoirs are
made double that is, in two compartments.
FIG. 860.
Fig. 860 is a section of the division-bank of the new Croton reservoir. It
is made of earth, with a puddled ditch in the center, and slopes protected by
rock-paving.
A few extracts from the specification will explain the general construction
of the reservoir :
" The reservoir will be formed by an exterior bank forming the outer sides of the
basin. There will be a division-bank, dividing the reservoir into two basins. All the
banks will have the inner and outer slopes of 1 base to 1 perpendicular. All the inner
or water-slopes will be covered with 8" of broken stone, on which will be placed the
stone pavement, H feet thick. The outer slopes will be covered with soil 1 foot thick.
The banks, when finished, to be 15 feet on top, exclusive of the soil on the outer slope.
The top of the outer bank to be 4 feet above water-line ; the top of the division-bank to
be 3 feet below water-line. In the center of all the banks a puddle-bank will be built, ex-
tending from the rock to the paving in the division-bank, and to within 2 feet of the top
of the outer bank. It will be 6' 2" wide at top in division-bank, and 14' wide at top in
ENGINEERING DRAWING.
395
exterior bank, and 16' wide at a plane 38' below top of exterior bank. In the middle of
the division-bank there will be built a brick wall,* laid in cement-mortar, 4' high, 20" wide,
the top of the wall to be connected with the bottom of the stone pavement ; 8" thickness
of concrete is to be laid on the top of the bank, on each side of, and connected with, this
wall. On the pavement 18" thick will be laid in concrete. The slope-wall on each side
of the division-bank, 10' in width, to be laid in cement.
" Puddle-ditches are to be excavated to the rock under the center of all embankments
where the rock is not over 46' below top of exterior bank. Where the rock is more than
46', two rows of sheet-piling are to be driven to the rock, 16' apart, and the material be-
tween them excavated, so as to remove all soil, muck, or vegetable matter. Sheet-piling
will be formed of spruce or pine plank, 6" thick, tongued and grooved; the tongue and
groove to be 1$" x 1". The earth within the working-lines of interior slopes will be ex-
cavated to the depth of 40' below top of exterior bank, rock 36'. The puddle-ditch will
be formed of clay, gravel, sand, or earth, or such admixture of these materials, or any of
them, as the engineer may direct, to be laid in layers of not more than 6", well mixed with
water, and worked with spades by 'cutting through vertically, in two courses at right
angles with each other; the courses to be 1" apart, and each spading to extend 2" into the
lower course or bed. Whenever the work is suspended, the puddle must be covered with
boards or earth to prevent cracking, and, whenever cracks do occur in the puddle, those
parts must be removed and reworked. The puddle will extend to all the masonry and
pipes, and along and around it and them as the engineer may direct.
"The embankments will be formed in layers of not more than 6", well packed by
carting and rolling, and, in such places as the rollers can not be effectually used, by ram-
ming. The embankments will be worked to their full width as they rise in height, and not
more than 2' in advance of the puddle. The interior slopes of all the banks will be
covered with 8" thickness of stone, broken to pass through a 2" ring. On this will be
laid the paving, 18" in thickness, of a single course of stones set on edge at right angles
with the slope, laid dry, and well wedged with pinners."
FIG. 861.
FIG. 862.
FIG. 863.
FIG. 864.
Distribution. Figs. 861 to 865 are sections of the spigot and faucet ends
of some of the pipes of the city of Brooklyn. Of these pipes there were two
classes, A and B. The A pipes were designed for positions subject to an ex-
* This wall was formed of concrete.
396
ENGINEERING DRAWING.
treme head of 120', the B pipes for positions below this level, subject to a
head of from 120 to 170 feet.
The thicknesses of these pipes is greater than those which now obtain in
practice. The following table, made from the average of formulas and of the
dimensions in use in different cities, may be considered safe for a static press-
ure of 100 Ibs., or 231 feet. But pipes should be tested at the manufactories
to three times this pressure. The weights given are the pipes as delivered in
lengths of 12' or 12' 5"; as laid, the laps are 5", and for running feet about
4 per cent should be added to the table-weights :
LEAD JOINT.
TV
Th.ick.n6ss
Weight
Depth.
Weight.
In.
Lbs.
4
42
18
1*
4 i
6
47
30
1|
6 i
8
52
44
1*
8*
10
58
60
If
10i
12
63
78
2
13
16
73
120
2 i
24^-
20
83
170
g
31
24
94
228
2f
38
30
1-10
330
2*
57
36
1-24
450
2^
30
48
1-44
700
H
111
The smallest water-pipe laid in large cities now is the 6"; the other sizes
given in the table are in common use and are found in stock, except the 10",
which can be obtained by order.
In laying, a hemp gasket is forced
down to the lower end of the bell
to prevent the molten lead escap-
ing into the pipe. The end of the
pipe is then stopped by the clay
roll, or a rope covered with clay,
or clay alone, and the melted lead
poured in through an aperture or
gate at the top. After cooling, the
lead is calked or compacted in the
joint.
Specials. All parts of a main
except the straight pipes are called
special castings.
Fig. 866 is a 12" X 8" 4-way
branch, shown full and in section.
J?IG. oob.
diagonally. The horns on the 4"
branch are for the straps which hold in the plug, or cap, or a connected short
or curved pipe. The 4-way branches are often called crosses, and the 3- way
ENGINEERING DRAWING.
397
T's, or single branches. The branches may be of any appropriate size. In
ordering, designate diameter of main pipe first, and then that of the branches.
It is very common in these pipes to make all the ends bell ends it saves
sleeves when pipes are cut, as they usually are at street
intersections.
Fig. 867 is a section of a sleeve for uniting cut
pipes or uncut spigot-ends ; a kind of double hub is
often used for the former. Some-
times sleeves are made in halves,
and bolted together.
Fig. 868 is the section of a re-
ducer for the connection of pipes
of unequal diameters.
Fig. 869 'is the section of a
bend; the horns on the outer cir-
cle are for straps between the pipes,
as the pressure is unbalanced.
Fig. 870 is a section of the con-
nection of two wrought-iron pipes
by a bell riveted to the end of one, and a fillet or ring
to the end of the other. FIG. 868.
House-services are usually through lead pipes ; the
taps allowed on the mains for house-connections being usually from -J" to f ".
FIG.
FIG. 869.
From the specifications of " Cast-iron Distribution-Pipes and Pipe-Mains,
with their Branches," etc., Brooklyn, L. I. :
" All pipes of 20" diameter and upward to be formed so
as to give a lead joint of not less than f" in thickness all
round, and not more than T y ; those of 12" diameter and
under, not exceeding |", and not less than T y. The straight
pipes of 12" diameter and upward shall be cast in dry sand
molds, vertically. The smaller pipes may be cast at an angle
with the horizon of not less than 12. The pipes shall be
free from scoria, sand-holes, air-bubbles, cold-short cracks,
and other defects or imperfections ; they shall be truly cylindrical in the bore, straight
FIG. 870.
398 ENGINEERING DRAWING.
in the axes of the straight pipes, and true to the required curvature or form in the axes
of the other pipes; they shall be internally of the full specified diameters, and have their
inner and outer surfaces concentric. No plugging or filling will be allowed. They shall
be perfectly fettled and cleaned ; no lumps or rough places shall be left in the barrels or
sockets. No pipes will be received which are defective in joint-room. The spigot ends
of all the branches to have lugs or horns cast on each. Every pipe-branch and casting
shall pass a careful hammer-inspection, and shall be subject thereafter to a proof by water-
pressure of 300 Ibs. to the square inch for all pipes 30" in diameter and under, and
250 Ibs. per square inch for all pipe-mains exceeding 30" diameter. Each pipe, while
under the required pressure, shall be rapped with a hand-hammer from end to end, to
discover whether any defects have been overlooked. The pipes shall be carefully coated
inside and outside with coal-pitch and oil, according to Dr. R. A. Smith's process, as
follows:
"Every pipe must be thoroughly dressed and made clean from sand and free from
rust. If the pipe can not be dipped presently after feeing cleansed, the surface must be
oiled with linseed-oil to preserve it until it is ready to be dipped ; no pipe to be dipped
after rust has set in. The coal-tar pitch is made from coal-tar, distilled until the naphtha
is entirely removed and the material deodorized. The mixture of five or six per cent of
linseed-oil is recommended by Dr. Smith. Pitch, which becomes hard and brittle when
cold, will not answer. The pitch must be heated to 300 Fahr., and maintained at this
temperature during the time of dipping. Every pipe to attain this temperature before
being removed from the vessel of hot pitch. It may then be slowly removed and laid
upon skids to drip."
Sewers. For the removal of waste water from houses and rainfall, sewers
are very convenient in towns and cities, even before the construction of water-
works ; but, after the introduction of a liberal and regular supply of water,
sewers are indispensable. The ruling principle in the establishment of sewer-
age-works is, that each day's sewage of each street and of each dwelling should
be removed from the limits of city and town, as far as practicable, on the day
of its production, that it should pass off before decomposition begins, and that
it should not be allowed to settle and fester in the sewers, producing those nox-
ious gases so prejudicial to health. To attain this end, the refuse fluids must
be sufficient in quantity to float and carry off the heavier matters of sewage.
There has been considerable discussion of late whether sewage and rainfall
should be carried off by a single system of pipes. This must depend largely on
the location, economy of construction, and the financial ability to carry out
the design. If the rainfall can be provided for by street gutters, the pipes for
the conveyance of house-waste may be very small.
If the rate of inclination of a sewer be not less than 1 in 440, the experience
of Brooklyn, and other cities equally well supplied with water, shows that the
fluid of domestic sewage is sufficient to carry off all the heavier matters, and
keep the sewers free and clean, provided the form is such as to concentrate as
much as possible the sewage waters. Less inclination than 1 in 440 will require
some means of flushing. In the Brooklyn system of sewers, adopted on the
report and plans of Colonel J. W. Adams, the principle of construction has
been, to make the sewers as small as the service required of them will admit,
to maintain as much velocity of flow as possible, so that nothing may be de-
posited ; and without any provision for a man entering and passing through
the sewer, which has been found by experience unnecessary.
ENGINEERING DRAWING.
399
The value of sewers depends on the cor-
rectness of their lines, uniformity of de-
scent, and smoothness of interior surface.
The pipes used in Brooklyn have generally
been strong glazed earthenware pipes, of
12", 15", and 18" diameter. Many cement-
pipes have also been used, and, in such
situations as required great capacity, brick
sewers were used, the leading forms of which
are egg-shaped, as in Fig. 871, of which the
dimensions are as follow : R (as in table)
the longest diameter D, and the longest
radius R', each 3 times R, and R" R.
FIG. 871.
AEEA.
K.
D.
fiO"
diameter circular . . . .
24-8
74-4
-18"
u u
19-8
59'5
Sfi"
u u
14-9
44'7
94"
u a
9-9
29'8
Thickness of brickwork, 8" ; boards shown at bottom only used in cases of
soft earth for convenience of construction. For area of egg-shaped sewer of
above section, multiply R 2 by 4 -6.
In some locations the depth did not admit of the egg-shaped section. A
circular form of 6 feet diameter was adopted for the Union Avenue sewer, and
one of a section similar to the main conduit of the water-works, 10 feet in
width and 9 feet high, in the clear, for Kent Avenue.
Fig. 872 is a section of the largest Washington sewer. The bottom course
of the large sewers, or where exposed to a strong current, are of stone ; the
ring-courses, of brick, are 3 for the 13-foot sewers and 2 for the 7-foot.
Man-holes are built along the line of sewers, at a distance of from 100 to
150 feet apart, to give access to the sewers for purposes of inspection and re-
moval of deposit.
Figs. 873 and 874 are section and plan of the man-hole at present used by
the Croton Sewer Department. It consists of a funnel-shaped brick well, oval
at the bottom, 4' X 3', circular at top, 2' diameter, curbed with cast-iron frame
and covered by cast-iron plate. Side-walls, 8" thick, through which the pipe-
sewers pass at the bottom of the well. Across the open space the sewer is formed
in brick, whose bottom section corresponds to that of pipe, side-walls carried
up perpendicular to top of sewer ; the flat spaces at the sides of sewer are
flagged. The top of the sewer is a heavy cast - iron frame, fitted with a
strong cover, which may or may not be perforated, for ventilation. In the
figure the main sewer is 12" pipe, with a 12" branch entering at an acute
angle, as all branches and connections with a sewer should. The short lines
on the left vertical wall represent sections of f\ staples, built in to serve for a
ladder.
400
ENGINEERING DRAWING.
Wherever necessary, from the slope and conformation
of the ground, to remove the surface or rain water direct-
ly from the street-gutters into the
3g sewers, catch-basins are placed gen-
H erally at the corners of streets.
Figs. 875 and 876 are section and
plan of the Croton sewer catch-basins.
ENGINEERING DRAWING.
401
on a scale of y = 1 foot. The intention of the catch-basin is to receive the
street washings, retain the heaviest portion in the basin, and let the liquid
escape into the sewer. The basin in the figure is rectangular in plan, with a
semicircular end, 3' 8" in width
by 5' I" long ; bottom of flag and
side-walls of brick, 12" thick. It
will be observed that a piece of
flag is built into the side-walls
from the top, extending about
lialf-way to the bottom ; this
divides the upper part of the
basin into two parts ; the sewer
enters the basin three feet above
bottom flag ; the dividing flag
comes to within 2' 6" ; before
any water can flow out through
the sewer-pipe this flag must be
submerged 6" ; a trap is thus
formed, which cuts off any smell
from the sewer escaping into the
street. This trap is sometimes
made of a cast-iron elbow, turned
down and bolted to the sewer- Fl - 8 ^ 5 -
pipe in the wall. The water is
received into the basin through the channel 0, which is curbed with granite,
and protected by a grating. The coping (b) is of granite, and forms a portion
of the sidewalk ; through this there is a man-hole cut, 16" diameter, for access
to the basin, for removal of the
deposit ; it is covered by a strong
cast-iron plate.
Gas- Supply. Next in impor-
tance to the necessities of a city or
town for water-supply and sewer-
age, is the luxury of gas-supply.
The gas-works should always be
placed remote from the thickly-
populated part of a city, for under
the best regulations some gas will
escape in the manufacture, offen-
sive and deleterious. They should
be placed at the lowest level, for, gas being light, readily rises, and the portions
of the city below the works are supplied at less pressure than those above.
Gas-mains, like those for water, are of cast-iron, and put together in the same
way ; but, as they have to resist no pressure beyond that of the earth in which
they are buried, they are never made of as great thickness as those of water-
pipes, but drips must be provided, and the pipe laid with such inclination to
them that the condensed tar may be received in them and pumped out.
FIG. 876.
402
ENGINEERING DRAWING.
WEIGHT OF GAS-PIPES PER RUNNING FOOT.
8" 12 Ibs.
4" 16 "
6".. . 27 "
40
10" 50 Ibs.
12" 62 "
16".. . 103 "
150
FIG. 877.
Roads. Under this general term are included all routes of land-travel ; but
the term "streets" is applied mostly to city, town, and suburban roads, while
"roads" and "highways" are applied to those of the country. By an "ave-
nue" is generally understood a wide street. In New York all the streets run-
ning north and south are called avenues, and those at right angles, streets, and
the term boulevard to very wide avenues in which there are rows of trees. The
terms street and avenue, as laid out,
are the established bounds within
which no buildings may be erected.
The street, therefore, technically in-
cludes the street or traveled way for
carriages, and the sidewalks and front
areas. New York streets above Four-
teenth Street are 60 and 100 feet wide,
avenues 100 feet, of which the carriage-way occupies one half, and the sidewalks
and area one quarter on each side. The space occupied by areas, is from 5 to
8 feet, which may be inclosed by iron fence, but can not be included within
the building above the level of sidewalk. The stoop-line extends into the
sidewalk beyond the area-line some 1' to 18", fixing the limit for the first step
and newel to a high stoop or platform. The boulevard in the old line of
upper Broadway and the Bloomingdale Eoad is 150 feet wide, of which 100
feet are to be carriage-way, and 25 feet on each side for sidewalk and area, the
latter not to exceed 7 feet ; one row of trees to be set within the sidewalk,
about 2 feet from the curb.
In Paris, there is no area ; the sidewalk comes up to the house or street-line,
and there is a space for trees between sidewalk and street-curb. This space is
available for pedestrians, a part being a gravel, asphalt, or flagged walk. The
following are the dimensions according to the law of June 5, 1856 :
Entire width of
boulevard and
avenues.
Width of
carriage-way.
Width of
sidewalk.
Width for
trees.
Rows of
trees.
DISTANCE OF EOW FROM
Street-line.
Street-curb.
Metres.
Metres.
Metres.
Metres.
Metres.
Metres.
26 to 28
12
1
5'5 to 6-5
1-5
30 " 34
14
1
6-5 " 8-5
1-5
36 " 38
12 to 13
3-5
8' to 8-5
2
5' " 5-5
1-5
40
14
3-6
9-5
2
6'5
1-5
1 metre = 3'281 feet.
The foot-walks in this city and vicinity are generally formed of flags, or
what is here termed blue-stone, laid on a bed of sand or cement-mortar. The
flags are from 2" to 4" thick. In the more important streets the upper surface
ENGINEERING DRAWING.
403
is axed, the quality of the stone selected, and the sidewalk often covered by a
single width of stone. Brick are often used in towns, or places where good
flag can not be readily obtained, usually laid flatways on a sand-bed. Granite
Carriage-way.
FIG. 8V8.
is very often employed in business streets, in lengths the full width of the side-
walk, and about 1' in thickness, the inner ends resting on an iron girder,
and the outer on the vault- wall, forming in this way a roof for the vault and the
outer ends a curb for the street.
Curls here are generally of flag, about 4" thick by 20" deep, extending
10* above the gutter-stone ; but, where the street is nearly level, and the
gutter-stones have to be
raised to give sufficient Curb - Sidewalk,
descent for the flow of the
water, the curbs, in ex-
treme cases, are not more
than 4" exposed. When
sidewalks are stone of
large dimensions, they ex-
tend over the curbs. The FIG. 879.
gutter-stones are from 12"
to 15" in width, and from 3" to 5" in thickness, laid close, and bedded in
cement. The bridge or crossing-stones are of blue-stone or granite, from 2'
to 15" wide, and not less than 5" thick, laid in double rows.
Carriage-way. Streets and avenues were formerly paved entirely of cobble-
stone, and, if selected so as to be of a uniform size and shape, and properly
bedded in sand and well rammed, they formed a cheap and very fair roadway ;
but the cubical block-stone pavement of trap or granite, often called the Bel-
gian, is in every way to be preferred. Mr. Kneass, the engineer of the city of
Philadelphia, recommends :
" That the blocks should not exceed 3" in width, 6" in depth, nor 8" in length ; that,
as to depth, they should be uniform within J", and in length he not less than 6". For
foundations the material should he taken out to a depth of 20" below the proposed surface
of paving, and to be made to accurately conform to shape of finished road. After being
compactly rolled with a heavy roller, it should have a covering of clean anthracite coal-
ashes placed upon it to a depth of 10", laid on in two layers, each well rolled; the ashes
to be scrupulously clean i. e., free from any organic matter. Upon this should be laid a
bed of clean gravel, 4" in depth, and rolled; upon which again should be a layer of sand,
clean and sharp, or fine-screened gravel, in which to set and bed the stone blocks. Each
layer of ashes and gravel should in surface conform to the outline intended for the surface
404 ENGINEERING DRAWING.
of the stone. The stone should be carefully assorted, so that, when laid across the street,
the joint-lines may be straight ; and each stone should he set on its ~bed fair and square, so
that no edge shall extend above the general level of the surface, and the surface of each
stone shall be an extension of that lying next to it. The joints I would not make smaller
than ", to be filled with sand and grouted with liquid lime. Before grouting, the entire
surface should be rammed until no impression can he made on it"
New York pavements are usually laid of granite or trap-blocks, 4" wide,
6" deep, and 8" to 12" long, set in sand simply, or on a concrete base. In
London the usual practice is to set their blocks 3" wide by 9" deep, and from
6" to 12" long, on a bed of gravel, with a base of broken granite 12" deep.
Wooden pavements of various kinds have been tried. The "Nicholson"
consists of pieces of 3" plank, 6" long, set on a board base supported by a
sand-bed. The plank is set on end in lines perpendicular to line of street,
with a strip of board I" wide between the rows, nailed to the blocks ; the top
of strip being some 2" to 3" below top of plank. Boiled coal-tar is used freely
while setting the bloqks, and is poured into the interstices ; the I" joint is filled
with gravel, wet with tar, and well rammed. Instead of plank, blocks of wood,
sawed from trunks or limbs, from 4" to 9" diameter, with the bark removed,
are set on a board or plank base, with the interstices filled with gravel, into
which coal-tar or melted asphalt is poured, and the top covered with gravel
and thoroughly rolled into the wood, so that the wear is on the gravel. In
all cases, although more expensive, it is better to make a concrete base.
Of late years, asphalt has been used abroad to a very great extent, both
for foot and carriage ways. The carriage-ways are composed of a layer of
asphalt, from 1" to 2" thick, on a bed of concrete, or on a worn Macadam road,
over which is spread a thin coat of cement. The cement having become dry,
the asphaltic rock, reduced to a powder, is spread over the surface to a depth
of about *40 per cent more than the finished thickness ; it is then rammed with
rammers warmed by portable furnaces, beginning gradually, and increasing the
force of the blows as the work approaches completion. For a footway the same
concrete bed is used, and the layer of asphalt is about |". Walks and roads
have been constructed in this country with an artificial asphalt, prepared from
coal-tar mixed with gravel.
Pavements of mineral asphalt have also been laid in many of our cities. In
Washington, the asphalt pavement consists of 6" of hydraulic cement concrete
and a wearing surface of bituminous mastic laid in two coats respectively -J" and
2" thick when compressed. The mastic is composed of the following parts by
weight :
Asphaltic cement (refined Trinidad asphalt) 100 parts, petroleum oil 20 parts. 15 to 18
Limestone powder 15 to 17
Sand . . TO to 65
100 to 100
Roads and Highways. Macadam was the first to reduce the construction of
broken-stone roads to a science, and has given his name, in this country, to all
this class of roads. He says that " the whole science of artificial road-making
sonsists in making a dry, solid path on the natural soil, and then keeping it
dry by a durable water-proof covering." The road-bed, having been thoroughly
ENGINEERING DRAWING.
405
drained, must be properly shaped, and sloped each way from the center, to dis-
charge any water that may penetrate to it. Upon this bed a coating of 3" of
clean broken stone, free from earth, is to be spread on a dry day. This is then
to be rolled, or worked by travel till it becomes almost consolidated ; a second
3"-layer is then added, wet down so as to unite more readily with the first ; this
is then rolled, or worked, and a third and fourth layer, if necessary, added.
Macadam's standard for stone was 6 ounces for the maximum weight, corre-
sponding to a cube of 1J", or such as would pass in any direction through a %y
ring. The Telford road is of broken stone, supported on a bottom course or
layer of stone set by hand in the form of a close, firm pavement.
At the New York Central Park, after trials of the Macadam and Telford roads,
the gravel-road (of which Fig. 880 is a cross-section of one half) was adopted, as
being, according to the statement of their engineer, Mr. Grant, "the easiest and
FIG.
most agreeable kind of road for both carriages and horses ; that it is the cheapest
at first cost, and can be kept in repair at an equal if not less cost than any other
equally satisfactory road." This road consists of a layer of rubble-stones, about
7" thick, on a well-rolled or packed bed, with a covering of 5" of clean gravel.
C is the catch-basin for the reception of water and deposit of silt from the
gutters ; S is the main sewer or drain, and s a sewer-pipe leading to catch-basin
on opposite side of the road. In wider roads each side has its own main drain,
and there is no cross-pipe s. The road-bed was drained by drain-tiles of from
iy to 4"-bore, at a depth of 3' to 3J below the surface. The maximum grade
of the Park roads is 1 in 20. The grades of the streets of Paris vary from 1 in
20 to 1 in 200. The best grade is from 1 in 50 to 1 in 100 ; this gives ample
descent for the flow of water in the gutters. Many of our street-gutters have
a pitch not exceeding 1 foot in the width of a block, or 200 feet.
The grade of a road is described as 1 in so many ; so many feet to the mile,
or such an angle with the horizon.
Inclination.
Feet per mile.
Angle.
Inclination.
Feet per mile.
Angle.
1 in 10
528
5 43'
in 30
176
1 55'
1 " 11
462
5
" 40
132
1 26'
1 " 14
369
4
" 50
106
1 9'
1 " 20
264
2 52'
" 57
92
1
1 " 29
184
2
" 100
53
35"
406
ENGINEERING DRAWING.
The best transverse profile for a road on nearly level ground is that formed
by two inclined planes, meeting in the center, and having the angle rounded.
The degree of inclination depends somewhat on the surface of the road. A
medium for broken-stone roads is about -J" in 1', or 1 in 24 ; but Telford, on the
Holyhead Road, adopted 1 in 30 ; and Macadam, 1 in 36, and even 1 in 60.
For paved streets in Paris, a crown of -fa of the width is adopted, and for
Macadamized, -3-^. The inclination of sidewalks should not exceed f" in 1
foot, and, when composed of granite, the surface should be roughened.
The necessity of a well-drained road-bed is as important beneath rails as on
a highway. The cuts should be excavated to a depth of at least 2 feet below
grade, with ditches at the sides still deeper, for the discharge of water. The
embankments should not be brought within 2 feet of grade ; this depth to be
left in cut and on embankment for the reception of ballast. The best ballast
is Macadam stone, on which the cross-ties are to be bedded, and finer broken
stone packed between them. Good coarse gravel makes very good ballast ; but
sand, although affording filtration for the water, is easily disturbed by the pas-
sage of the trains, raising a dust, an annoyance to travelers, and an injury to
the rolling-stock by getting into boxes and bearings. The average length of
sleepers on the 4.8J- gauge railways is about 8 feet ; bearing surface, 7"; dis-
tance between centers, 2 feet to 2' 6", except at joints, where they are as close
to each other as the necessity of tamping beneath them will admit. Average
width of New York railways of same gauge as above, for single lines, in cuts
18', banks 13' ; for double lines cuts 31', banks 26f.
U ix* .j
....AW. ,.'U'.':A'.J
FIG. 881.
CROSS SECTION GRAVEL BALLAST.
FIG. 882.
Figs. 881 and 882 are two standard sections of the permanent way of the
Pennsylvania Railroad, in which the width of cuts and top of embankments
are the same, 31' 4", and other dimensions equally ample.
Sections of rail are of infinite variety and weights, adapted to the class of
railroads on which they are to be used, and the loads and speed of trains to
which they are to be subjected. For roads of the common gauge, the weight
of rails is from 56 to 70 Ibs. per yard. The joints are made with a fish-plate.
Figs. 883, 884, and 885 are the elevation, section, and plan of the standard
rail-joint of the New York, West Shore and Buffalo Railroad.
ENGINEERING DRAWING.
407
FIG. 883.
FIG. 885.
ROOFS AND BRIDGES.
At pages 238 and 239 will be found illustrations of the trussing of wooden
beams. These are simple forms, which may be used in roofs or bridges, and
rules are given for the proportion of parts. Soiled I-beams or plate-girders
will serve also for floor-beams and moderate spans, but with modern necessities
much more complicated structures are required.
On the General Principles of Bracing. Let Fig. 886 be the elevation of a
common roof-truss, and let a weight, W, be placed at the foot of one of the sus-
pension-rods. Now, if the construction consisted merely of the rafter C'B,
and the collar-beam C' C, resting against some fixed point, then the point B
would support the whole downward pressure of the weight ; but in consequence
of the connection of the parts of the frame, the pressure must be resolved into
components in the direction 0' A and C' B ; C' b will represent the pressure in
the direction C' B, C' w the
portion of the weight sup-
FIG. 886.
FIG. 887.
ported at B, C' a the pressure in the direction C' A, and w W the portion of
the weight supported on A. The same resolution obtains to determine the
direction and amount of force exerted on a bridge-truss of any number of
408
ENGINEERING DRAWING.
panels, by a weight placed at any pointy of its length (Fig. 887). In either
case, the effect of the oblique form 0' A upon the angle C is evidently to force
it upward ; that is, a weight placed at one side of the frame has, as in case of
the arch, a tendency to raise the other side. The effect of this upward force is
a tension on a portion of the braces, according to the position of the weight ;
but as braces, from the manner in which they are usually connected with the
frame, are not capable of opposing any force of extension, it follows that the
only resistance is that which is due to the weight of a part of the structure.
FIG. 890.
FIG. 889.
Figs. 888 and 889 illustrate the effects of overloading at single points such
forms of construction. Such an unequal loading on trusses requires that a
portion of the load W be tranferred to each point of support inversely propor-
tionate to the distances of the weight from each
support. The above trusses are not prepared to
transfer this weight to but one support. To rem-
edy the difficulty, it will be necessary to add braces
running in the opposite direction, as shown by
dotted lines (Fig. 890), at every point subject to the above distortion. These
are called counter-braces.
To prevent the braces from becoming loose when the counter-braces are in
action, it is always customary to give the braces and counter-braces an initial
compression, by putting a moderate tension on the suspension-rods. In this case,
therefore, the passage of a load would produce no additional strain upon any of
the timbers, but would tend to relieve the counters. The counter-braces do
not, of course, assist in sustaining the weight of the structure ; on the contrary,
the greater the weight of the structure itself, the more will the counter-braces
be relieved.
If, instead of _ 4 ^ 2 / 2.' _._ 4-'
the counter-
braces, the braces
themselves are
made to act both
as ties and struts,
as has been
done sometimes
in iron bridges
and trusses, then
the upward force will be counteracted by the tension of the brace.
Suppose a system to be composed of a series of suspension-trusses, as in Fig.
891, in which the load is uniformly distributed. If we represent the load at
FIG. 891.
ENGINEERING DRAWING. 409
each of the points, 4, 3, 2, 1, 2', etc., by 1, the load at 4 will be supported ^ upon
a and % upon 3 ; hence the strut 3 will have to support a load of 1 -f- "5 = 1 *5 ;
of this, f will be supported by 2 and -J- by a ; f of 1*5 = 1, 1 + 1 = 2, load on
strut 2 ; f of this load, or 1 *5, will be supported at 1, and since from the op-
posite side there is an equal force exerted at 1, therefore the strut 1 supports
1+1-5 + 1-5 = 4.
The tension on the rod c-2 = 2
cl
2-3 = 2% "
" " 3-4 = 3 "
If this construction be reversed, the parts which now act as ties must be made
as braces, and braces, ties ; then we have a roof-truss, and the force exerted on
the several parts may be estimated in a similar way as for the suspension-truss.
It is evident that neither of these constructions would serve for a bridge-
truss, subject to the passage of heavy loads, but is only fit to support uniform
and equally distributed loads.
To frame a construction so that it maybe completely braced that is, under
the action of any arrangement of forces the angles must not admit of altera-
tion, and consequently the shape can not. The form should be resolvable into
either of the following elements (Figs. 892, 893, and 894) :
FIG. 892. FIG. 893. FIG. 894.
In these figures, lines - - represent parts required to resist com-
pression ; lines parts to resist tension only ; lines parts
to resist both tension and compression.
It is evident that, in a triangle (Fig. 892), an angle can not increase or
diminish, without the opposite angles also increasing or diminishing. In the
form Fig. 893, a diagonal must diminish ; in Fig. 894 a diagonal must extend,
in order that any change of form may take place. Consequently, all these
forms are completely braced, as each does not permit of an effect taking place,
which would necessarily result from a change of figure. Hence, also, any sys-
tem composed of these forms, properly connected, breaking joint as it were
into each other, must be braced to resist -the action of forces in any direction ;
but as in general all bridge-trusses are formed merely to resist a downward
pressure, the action on the top chord being always compression, it is not neces-
sary that these chords should act in both capacities.
Roofs. The roofs of city dwellings and stores are generally flat, that is,
with but very little inclination, from half an inch to two inches per foot,
merely sufficient to discharge the water. The beams are laid from wall to wall,
the same as floor-timbers, but usually of less depth, or at greater distances be-
tween centers, and with one or two rows of bridging.
Figs. 895, 896, and 897 represent side or portions of side elevations of the
usual form of framed roofs. The same letters refer to the same parts in all
10
ENGINEERING DRAWING.
the figures. T T are the tie-beams, R R the main rafters, rr the jack-rafters,
PP the plates, pp the purlines, K K the Icing-posts, kk Icing-bolts, q q queen-
bolts both are called suspension- bolts C C the collar or straining beams, B B
braces or struts, b b ridge-boards, e corbels.
FIG. 895.
The pitch of the roof is the inclination of the rafters, and is usually desig-
nated in reference to the span, as , ^-, f, etc., pitch ; that is, the height of the
ridge above the plate is ^, i, f , etc., of the span of the roof at the level of the
plate. The steeper the pitch of the roof, the less the thrust against the side-
walls, the less likely the snow or water to lodge, and consequently the tighter
the roof. For roofs covered with shingles or slate, in this portion of the coun-
Fia. 896.
try, it is not advisable to use less than J pitch ; above that, the pitch should be
adapted to the style of architecture adopted. The pitch in most common use
is the span.
Fig. 895 represents the simplest framed roof : it consists of rafters, resting
upon a plate framed into the ceiling-beam ; this beam is supported by a sus-
pension-rod, k, from the ridge, but, if supported from below, this rod may be
omitted. As shown, the rafters are to be spaced from 1 to 2 feet centers, and
the tie-beams at intervals of from 6 to 8 feet : the roof cover to be of boards
ENGINEERING DRAWING.
411
nailed directly to the rafters. This form of construction is sufficient for any
roof of less than 25 feet span, and of the usual pitch, and may be used for a 40-
foot span by increasing the depth of the rafters ; deep rafters should always be
bridged. By the introduction of a purline extending beneath the center of the
FIG. 897
rafter^ supported by a brace to the foot of the suspension-rod, as shown in dot-
ted line, the depth of the rafters may obviously be reduced. It often happens
that the king-bolt may interfere with the occupancy of the attic ; in that case
the beam is otherwise supported. Again, it may be necessary that the tie-
beam, which is also a ceiling and floor beam, should be below the plate some 2
to 4 feet ; in that case, the thrust of the roof is resisted (Fig. 898) by bolts,
b b 9 passing through the plate and the beam, and by a
collar-plank, C, spiked on the sides of the rafters, high
enough above the beam to afford good head-room. For
roofs f pitch and under 20 feet span, the bolts are un-
necessary, the collar alone being sufficient.
Fig. 896 represents a roof, a larger span than Fig.
895 ; the frame may be made very strong and safe for
roofs of 60 feet span. King-bolts or suspension-rods
are now oftener used than posts, with a small triangular block of hard wood
or iron, at the foot of the bolts, for the support of the braces. The objection
to this form of roof is that the framing occupies all the space in the attic ;
on this account the form, Fig. 897, is preferred for roofs of the same span,
and is also applicable to roofs of at least 75 feet span, by the addition of a
brace to the rafter from the foot of the queen-bolt. The collar-beam (Fig.
900) is also trussed by the framing similar to Fig. 896.
In many church and barn roofs the tie-beam is cut off (Fig. 899) ; the
queen-post being supported on a post, or itself extending to the base, with a
short tie-rod framed into it from the plate.
412
ENGINEERING DRAWING.
Figs. 901 and 902 are representations of the feet of rafters on an enlarged
scale. In Fig. 901, the end of the rafter does not project beyond the face of
the plate ; the cove is formed by a Fl - 90
small triangular, or any desirable form
of plank, framed into the plate. The form given to the foot of the rafter is
called a crow-foot. In Fig. 902, the rafter itself projects beyond the plate to
form the coving. Fig. 903 represents a front and side elevation and plan of
the foot of a main rafter, showing the form
of tenon, in this case double ; a bolt, passing
nrn
FIG. 901.
FIG. 902.
FIG. 903.
through the rafter and beam, retains the foot of the former in its place. Fig.
904 represents the foot of a main rafter, with a wooden shoe too short at #,
outside of the rafter ; it should be framed as in Fig.
903. In Fig. 901, of a similar construction to Fig. 895,
the tie-rod passes directly through the plate. In general,
when neither ceiling nor flooring is
supported by the tie-beam, a rod is
preferable.
FIG. 904.
n
L
FIG. 905.
FIG. 906.
FIG. 907.
Roofs are now very neatly and strongly framed by the introduction of cast-
iron shoes and abutting plates for the ends of the braces and rafters. Fig. 905
ENGINEERING DRAWING.
413
represents the elevation and plan of a cast-iron king-head for a roof similar to
Fig. 896 ; Fig. 906, that of the brace-shoe ; Fig. 907, that of the rafter-shoe
FIG. 908.
FIG. 909.
FIG. 910.
for the same roof ; Fig. 908, the front and -side elevation of the queen-head
of roof similar to Fig. 897 ; and Fig. 909, elevation and plan of queen brace-
shoe.
Fig. 910 represents the section of a rafter-shoe for a tie-rod ; the side
flanches are shown in dotted line.
On the size and the proportions of the different members of a roof : Tie-
beams are usually intended for a double purpose, and are therefore affected by
two strains : one in the direction of their length, from
the thrust of the rafters ; the other a cross-strain, from
the weight of the floor and ceiling. In estimating the
size necessary for the beam the thrust need not be
considered, because it is always abundantly strong to
resist this strain, and the dimensions are to be deter-
mined as for a floor-beam merely, each point of sus-
pension being a support. When tie-rods are used, the
strain is in the direction of their length only, and their dimensions can be
calculated, knowing the pitch, span, and weight of the roof per square foot,
and the distance apart of the ties, or the amount of surface retained by
each tie.
The weight of the wood-work of the roof may be estimated at 40 pounds
per cubic foot ; slate at 7 to 9 pounds, shingles at 1J to 2 pounds per square
foot. The force of the wind may be assumed at 15 pounds per square foot.
The excess of strength in the timbers of the roof, as allowed in all calculations,
will be sufficient for any accidental and transient force beyond this. Knowing
the weights, pressures, and their directions on parts of a roof, their stresses may
be determined by the parallelograms of forces and dimensions proportioned to
the strength of the materials of which the roof is composed. It will generally
be sufficient for the draughtsman to have practical examples of construction to
draw from. Dimensions are therefore given of the parts of wooden roofs already
illustrated. With further examples of actual constructions, the beams are usu-
ally proportioned to the weight that they are to sustain in floors and load, but
where tie-rods are used, the stress upon them may be determined by the follow-
ing rule :
Rule. Multiply one half the weight of the roof and load by one half the
span, and divide the product by the rise or height of ridge above eaves.
Gwilt, in his "Architecture," recommends the following dimensions for
portions of a roof :
414:
ENGINEERING DRAWING,
Span.
Form of Koof.
Kafters.
Braces.
Posts.
Collar-beams.
Feet.
Inches.
Inches.
Inches.
Inches.
25
Fig. 896,
5x4
5x3
5x5
30
u
6x4
6x3
6x6
35
Fig. 897,
5x4
4x2
4x4
7x4
45
u
6x5
5x3
6x6
7x6
50
2 sets of queen-posts,
8x6
5x3
~> O AC
9x6
{ 8 x 4 f
60
u a
8x8
6x3
j 10 x 8 j
"j 10 x 4 f
11 x 6
These dimensions, for rafters, are somewhat less than the usual practice in
this country ; no calculations seem to have been made for using the attic. An
average of common roofs here would give the following dimensions nearly : 30
feet span, 8X5 inches ; 40 feet, 9 X 6 ; 50 feet, 10 X 7 ; 60 feet, 11 X 8 ;
collar-beams the same size as main rafters. Roof -frames from 8 to 12 feet from
center to center.
Dimensions for jack-rafters, 15 to 18 inches apart :
For a bearing of 12 feet. ... 6x3 inches.
" " 10 " . . 9 x 3 "
For a bearing of 8 feet. ... 4x3 inches.
" " 20 " . ..10 x 3 "
Purlines :
Length of Bearing.
Distances apart in Feet.
Feet.
6
8
10
12
8
7x5
8x5
9x5
9x6
10
9x5
10 x 5
10 x 6
11 x 6
12
10 X 6
11 x 6
12 x 7
13 x 8
The pressure on the plates is transverse from the thrust of the rafters, but
in all forms except Fig. 895, owing to the notching of the rafters on the pur-
lines, this pressure is inconsiderable. The usual size of plates for Figs. 895 and
896 is 6 x 6 inches.
In the framing of roofs, it is now customary, for roofs of mills, to omit
purlines, jack-rafters, and plates, and make the roof-boards of plank stiff
enough to supply their places, from 2" to 3" thick (according to the space
between the frames), tongued and grooved, and strongly spiked to the main
rafters.
The dimensions of rafters depend on the distances between their supports
and between centers. The depth in all such cases to be greater than the width ;
2 to 6 inches may be taken as the width, 8 to 12 for the depth.
When there are no purlines, and the roof is covered with plank, there is no
need of plates ; the plank forms a deep beam, and, if the ends of the frame are
secured, there may be no need of intermediate ties.
Iron Roofs. Roofs of less than 30 feet span are often made of corrugated
iron alone, curved into a suitable arc, and tied by bolts passing through the
iron about 2 to 4 feet above the eaves.
ENGINEERING DRAWING.
415
Fig. 911 represents the half elevation of an iron roof of a forge at Paris ;
Figs. 912, 913, and 914, details on a larger scale. This is a common type of
iron roof, consisting of main rafters, E, of the I-section (Fig. 914),
trussed by a suspension-rod, and tied by another rod. The purlines
are also of I-iron, secured to the rafters by pieces of angle-iron on each
side ; and the roof is cov-
ered with either sheet-iron resting ^4
on jack-rafters, or corrugated iron
extending from purlin e to purline.
The rafter-shoe, A, and the strut,
S, are of cast-iron ; all the other portions
of the roof are of wrought-iron. In Amer-
ican practice it is usual
to make the strut of FIG. 913.
wrought-iron, with a
single pin connection at its foot, instead of as in the figure.
The surface covered by this particular roof is 53 metres
(164 feet) long and 30 metres (98J- feet) wide. There are
eleven frames, including the two at the ends, which form
the gables.
The following are the details of the dimensions and
FIG. 914. weights of the different parts :
416
ENGINEERING DRAWING.
Pounds.
2 rafters, 0'72 feet deep ; length together, 99*1 feet 1,751
5 rods, 0-13 feet diameter ; length together, 131'4 feet 882
16 bolts, 0-13 feet diameter 79
8 bridle-straps, 0-24 x -05 123
2 pieces, 0'46 thick, connecting the rafters at the ridge, > 8g
4 pieces, 0'46 thick, at the foot of the strut j
4 pieces, 0*36 thick, uniting the rafters at the junction in the strut together with
their bolts and nuts 176
2 cast-iron struts 308
2 rafter-shoes 287
Total of one frame 3,694
16 purlines, 1 ridge-iron, each 0'46 deep, 17'2 long 2,985
Bolts for the same 64
16 jack-rafters, I-iron, 0*16 deep 2,489
Weight of iron covering, including laps, per square foot 2-88
Koofs are sometimes made with deep corrugated main rafters with flat iron
between, or purlines and corrugated iron for the covering. The great objec-
tion to iron roofs lies in the condensation of the interior air by the outer cold,
or, as it is termed, sweating ; on this account, they are seldom used for other
_ buildings than boil-
**^ er-houses or depots,
except a ceiling be
made below to pre-
vent the contact of
the air inside with
-<^ the iron.
=_, Fig. 915 is an
FIG. 915. elevation of one of
the three panels of
one of the cast-iron girders for connecting the columns, and carrying the trans-
verse main gutters, which supported the roof of the English Crystal Palace.
Figs. 916 to 921 are sections of va-
1* rious parts on an enlarged scale.
The depth of the girder was 3
feet, and its length was 23 feet
3f inches. The sectional area of
the bottom rail and flange in the
center (Fig. 917) was 6^ square
FIG. 919.
FIG. 920.
FIG. 921.
inches ; the width of both bottom and top rail (Fig. 916) was reduced to 3
inches at their extremities.
ENGINEERING DRAWING.
417
27
4:18
ENGINEERING DRAWING.
The weight of these girders was about 1,000 pounds, and they were proved
by a pressure of 9 tons, distributed on the center panel.
A second series of girders were made of similar form, but of increased
dimensions in the section of their parts. Their weight averaged about 1,350
pounds, and were proved to 15 tons. A third series weighed about 2,000
pounds, and were proved to 22% tons.
Figs. 922 to 927 are the elevation and details for an iron roof-truss, for
FIG. 929.
ENGINEERING DRAWING. 419
wood, slate, or corrugated iron covers, built by the Missouri Valley Bridge and
Iron Works, A. S. Tulloch, engineer.
Figs. 928 and 929 are sections and details of the trusses for sustaining
the roof and floor of the new English High and Latin School Gymnasium,
Boston, Massachusetts. The object of sustaining the gymnasium-floor by rods
was to secure a drill-hall for the military exercises of the school, and trusses
were designed to have sufficient strength to resist the vibration of the floor.
As the trusses were to be in sight, a central column of cast-iron was introduced
to sustain the center of the top chord, instead of some wrought-iron construc-
tion less pleasing to the eye, with lattice between the main diagonals to enable
them to act as counters, instead of a more complicated construction introducing
counters, and a 3-J-inch gas-pipe for horizontal bracing-struts. The floor-sus-
taining rods all have upset ends, and at their tops pass through ornamental
foliated castings, but their connection with the trusses is wholly of wrought-iron.
The top chords consist of two nine-inch channel-irons weighing 50 pounds
per yard, and one plate 12 X f inches. The end-posts have the same section.
The bottom chord consists of four bars 2-J- X 1 inch at the shallow end of the
truss, and four bars 2^ X f of an inch at the deep end of the truss. The diago-
nals are two bars 3X1 inch at deep end of truss, and two bars 3 X i inch at
shallow end of truss. The pins are all 2-J inches diameter.
These trusses were designed and constructed by D. H. Andrews, C. E., of
the Boston Bridge Works.
In order to secure free space in the room beneath the roof, it is my practice
to construct a roof or bridge truss above, and suspend from it the roof framed
as a floor, with such pitch as is requisite to shed rainfall. In this form of
construction the span of the unobstructed space required is readily met by the
truss construction.
Fig. 930 is a half cross-section of a two-story freight-shed for the New York,
Lake Erie and Western Eailroad. It is a simple and cheap construction of
wood, readily framed and put together. The shed rests upon a pile-dock.
The platform for the reception of freight is 4 feet above the dock-planking,
and about 26 feet wide, with occasional inclined runs for the transfer of freight
to or from vessels.
Bridge-Trusses. Whatever maybe the form of truss or arrangement of
the framing, provided its weight is supported only at the ends, the tension
of the lower chord, or the compression of the upper chord at center, may be
determined by this common rule :
Rule. The sum of the total weight of the truss, and the maximum dis-
tributed load which it will be called on to bear, multiplied by the length of the
span, and divided by 8 times the depth of the truss in the middle, the quotient
will be the tension of lower chord and compression of upper at the middle. In
nearly all the forms of diagonal bracing, if the uniform load be considered as
acting from the center toward each abutment, each tie or brace sustains the
whole weight between it and the center, and the strain is this weight multiplied
by the length of tie or brace, divided by its height. Any diagonals, equally
distant from the center, sustain all the intermediate load : if rods, as in Fig.
932, by tension ; if braces, as in Fig. 931, by compression.
ENGINEERING DRAWING.
CROSS-SECTION OF ONE HALF OF A FREIGHT-SHED, NEW YORK, LAKE ERIE
AND WESTERN RAILROAD.
II U U
FIG. 930.
ENGINEERING DRAWING.
421
It follows, therefore, that in all these trusses the upper and lower chords
should be stronger at the center than at the ends, while diagonals should be
largest at the abutments. Unless the weight of the bridge is great compared
with the moving loads, counter-braces become necessary.
The general rule adopted in the construction of the Howe truss is, to make
the height of the truss -J of the length up to 60 feet span ; above this span the
trusses are 21 feet high, to admit of a system of lateral bracing, with plenty of
head clearance for a person standing on the top of a freight-car. From 175
feet to 250 feet span, height of truss gradually increased up to 25 feet. Moving
load for railroad -bridge calculated at 1 ton per running foot. Center to center
of panels not exceeding 11 feet.
Wooden Truss-Bridges. Fig. 931 is the elevation of a few panels of a Howe
truss, and Fig. 932 of a Pratt truss. The Howe truss is by far the most popu-
FIG. 931.
FIG. 932.
lar of all wooden trusses, being readily framed and put together, uniting great
strength with simplicity of construction.
Fig. 933 is the side elevation of three of the five panels of a Howe truss
highway-bridge of the New York, Lake Erie and Western Railroad. Fig. 934
is a cross-section. It will be observed that there is a section of 3" plank laid
close, and another beneath, laid with spaces ; these planks are laid diagonally
across the floor-beams, and at right angles to each other, and are made to act
as lateral bracing. Fig. 935 are the details of the abutment end of bridge ;
the foot of the brace rests on a cast-iron shoe. The length over all that is,
including the portions on the abutment is 81' 2 ff , or 75 feet between abut-
ments, usually designated as the span.
Figs. 936, 937, and 938 are the side elevation, floor cross-section, and plan
of floor and bottom chord of three of the twelve panels of a single-track rail-
way Howe truss. Their length is each 10' 10^". The center braces are two,
7" X 10" ; the center rods three, 1-J-* diameter. The counters, each one 6" X
8" ; lateral brace top and
bottom, 6" X 6" ; rods 1
inch ; top chord, four pieces,
7" X 12"; bottom chord,
four pieces, 1" X 15" ; floor-
beams, 1" X 16". The shoes,
splices, and blocks between
chord-timber are of cast-iron.
In the earlier practice the angle-blocks were of oak, and the splices made as
in Fig. 939. Both of these were satisfactory.
FIG. 939.
422
ENGINEERING DRAWING.
ENGINEERING DRAWING.
423
TXg
fl
=i
F
A
JZ
uU'll i i !,,i
.J9f ** <!
FIG. 938.
424
ENGINEERING DRAWING.
Combination Truss. Figs. 940 and 941 are the elevation and plan, and
Figs. 942 and 943 the details of the combination or composite truss, which
owes its name to the use of the two materials, wood and wrought-iron, in
somewhat near the same proportion in its
construction, the tension members being of
iron and the compression of wood. The cen-
tral braces, which are subjected alternately
to tensile and compression stresses, may be of
wood with iron rods, or wrought-iron only.
This class is entirely American in practice,
and embodies, as will be seen in the details,
an essentially American feature, of pin con-
nections. The bridge illustrated is in 30-foot
panels, six to the full length. The shoes and
splices are of cast-iron.
Iron Bridges. When the span is of mod-
erate extent, the load can be safely carried by
beams put together at the works and trans-
ferred to the road in complete form. Web or
lattice girders are used, put together with
rivets.
FIG 942 Figs. 944, 945, and 946 are the outside
elevation, plan of top bracing, and plan of
bottom bracing of one half a deck plate-girder railway-bridge, 42' 6" over all,
or 40 feet span or effective length. Figs. 947 and 948 are the end-elevation
ENGINEERING DRAWING.
425
FIG. 943.
FIG. 945.
ENGINEERING DRAWING.
FIG. 946.
FIG. 947.
FIG. 948.
and a section near the center. This and the following illustrations are taken
from " The American Engineer/' and the bill of material given is as follows :
BILL OF MATERIAL FOR DECK GIRDER, 42' 6" LONG OVER ALL.
No.
IHMKXCTOVS.
Weight.
For what used.
Pounds.
4
Bars, angle, 4" x 5" 14'2 Ibs. x 14' 0"
Top flanges.
4
" " " " x 28' 0"
2,386
(' U
4
" " 4" x 6" 24-lbs. x 14' 0"
Bottom flanges.
4
" ' " " x 23' 0"
4,116
u u
4
" ' 3-J-" x 5" 20-8 Ibs. x 2' 8"
222
Angle-covers.
4
" ' 3"x4" 12 Ibs. x 2' 8"
128
a a
32
' 3" X 4" 8-3 Ibs. x3 10J"
1,029
Ends and stiffeners.
16
" ' ' 3" x3" 7'2 Ibs. x7' 5"
Lateral.
2
" ' " " x5' 4"
931
Center-bracing.
2
" < 2-fc" x 2V 5 Ibs. x 5' 9"
" "
4
End-bracing.
4
303
" ''
4
Plates, 48" x f" x 21' 0"
5,040
Webs.
2
14" x |" x 29' 0"
1,691
Top flanges.
4
12" x 1" x 3' 4"
200
Joint-covers.
4
10" x , 5 6 ' x 6' 5"
267
End-bracing.
7
9" x 4" x 1' 9"
Lugs.
1
" x 2' 0"
c *
1
" x 1' 0"
172
i.
2
8" x |" x 1' 0"
20
"
4
14" x i" x 2' 0"
187
Bearing-plates.
32
Flat bars, 3" x f" x 3' 4"
667
Fillers.
24
" 3" x |" x 3' 4"
400
Inside stiffeners.
4
" 6" x I" x 2' 5"
97
Joints.
17,856
Rivets 6 per cent
1,070
18,926
Cast bearing-blocks @ 200
800
Total weight ...
19,726
ENGINEERING DRAWING.
OUTSIDE ELEVATION.
4:27
FIG. 949.
PLAN OF TOP BRACING.
FIG. 950.
PLAN OF BOTTOM BRACING.
CROSS-SECTION NEAR CENTER
FIG. 952.
Figs. 949, 950, and 951 are the out-
side elevation, plan of top and bottom
bracing of one half a deck lattice-girder
railway-bridge, the same span as Fig.
944 above, and intended to carry the
-same load rolling 4,000 pounds, and
FIG. 953.
JOINT % RIVETS.
o o o
o o o o
o o o o o o o"
00 o o o o
o o
FIG. 954.
4:28
ENGINEERING DRAWING.
dead load 900 pounds per lineal foot. Figs. 952 and 953 are the end elevation
and cross-section near center, and Fig. 954 one of the joints.
BILL OF MATERIAL FOR DECK LATTICE-BRIDGE, 42' 6" LONG OVER ALL.
No.
DIMENSIONS.
Weight.
For what used.
Pounds.
4
Plates, 12" x
i"
X
13' 6"
Chords webs.
4
M
X
28' 6"
3,360
a a
4
10" x
1"
X
22' 0"
Chords flanges.
2
u
X
12' 0"
" "
1
a
X
6'0"
1,475
Bearing-plates = 4.
4
10" x
iV'
x
6'1"
254
End-bracing.
1
18" x
X
9' 6"
285
Lugs on diagonals 8 12" x -J"
1
9" x
X
18' 8"
280
a a a g_ g n x y,
1
6" X
i"
X
16'0"
160
a a a g_ Q ,, x y
1
9" x
i"
X
16' 4"
184
Web-covers 8 2' 6"
1
1
' 7" x
6" x
1"
X
X
2 8"
9' 7"
53
96
" gussets 4 0' 8"
" fillers = 12.
1
' 7" x
iY'
x
1'4"
10
End-gussets = 2.
1
8" x
r
X
2' 9"
28
Sway brace-lugs, etc.
1
' 7" x
r
X
24' 0"
210
Lateral lugs.
1
8" x
X
9'0"
120
End-posts, 4 2' 3"
2
5" x
i"
X
10' 0"
83
End-post splices, 16 1' 3"
1
Flat bar, 3" x
r
X
12'0"
60
Fillers.
8
Bars, angle, 3
" X
3"
9-7 Ibs.
x
13'
6'
Chords.
8
u
II
a
X
28'
6'
3,249
a
8
a
a
10-8 Ibs.
X
6'
3'
540
Diagonals.
8
a
a
8-2 Ibs.
X
6'
0'
"
8
"
"
a
X
6'
3'
804
"
4
a
u
6-8 Ibs.
X
6'
3'
170
a
8
a
u
7'4 Ibs.
X
6'
3'
370
a
8
a
a
6 Ibs.
x
6'
0'
a
4
"
a
u
X
6'
3'
"
4
u
u
a
X
2'
2'
Top chords outside.
4
u
a
a
X
2'
6'
i a
4
"
a
a
X
2'
9'
i a
2
u
a
X
8'
()'
< a
4
u
a
a
X
4'
9'
' inside.
4
u
u
a
X
4'
6'
( a
4
a
X
2' 3'
i a
4
u'
U
a
X
2'
4'
i a
4
u
1
a
X
0'
r
End-bracing.
. 2
"
'
"
X
6'
o'
Cross-bracing.
2
(1
I
a
X
5'
5'
a a
8
u
1
a
X
4'
3'
End posts.
16
"
'
"
X
7'
6'
2,251
Lateral bracing.
8
Angle-covers, 2
X
2^_8 Ibs
X
no'
117
Chords.
14,159
Rivets 6 uer
P.PT1
h
"fU1
15,000
4
Cast bearing-blocks r). 1 00 Ihs
400
Total iron in poi
inds
15,400
Figs. 955 to 959 are details of portions of a wrought-iron truss-bridge, a
very good example of usual American practice. Fig. 955 is a side elevation of
one of the posts ; Fig. 956 a cross-section as far as the first rail of the road ;
Fig. 957 the lattice under side of the top chord the top is a plate. Fig. 958
is a top view of the top chord, showing the lateral bracing, consisting of a lat-
tice box-girder and diagonal rods. As the bridge is a skew, this box-girder is
ENGINEERING
429
430
ENGINEERING DRAWING.
Vl
\ /i AX^
YV
Z -
J
1
'5 .*"
ENGINEERING DRAWING.
431
n
not perpendicular to line of bridge, but parallel with abutment. Fig. 959 is
the side elevation of angle connection of end-brace and top chord.
Figs. 960 to 964 are illustrations of the landing-bridge common at New
York city ferries. Fig. 960 is a longitudinal section, showing a section of the
float, /, with its lever and stone counterpoise to balance the weight of the bridge,
the end of which is thrown to one side of the float. Fig. 961 is the front ele-
vation, and Fig. 962 the plan, one half being planked, and one half showing-
framing. It will be seen that there are two chain-barrels, on each side of the
bridge, worked by hand -wheels ; on the outer ones are the chains by which the
boat is drawn up to the bridge ; on the inner ones the chains by which the
bridge is adjusted to the load on the boat, and by which a part of the weight
of the bridge is held, the upper ends of the chains being attached to the frame
of the overhead. The details (Figs. 963 and 964) in section and plan explain
the construction of the land-hinge ; a cushion of rubber is introduced into
the joint to modify
the shock caused by n -
the boat striking the
bridge, and a flap of
wrought-iron to cover
the joint, for protec-
tion to travel, and se-
curity from dirt.
Piers. Fig. 965 is
an elevation of a pile-
pier for a bridge. Ten-
ons are cut on the top
of the piles, and a cap
(a) mortised on. The
two outer piles are
driven in an inclined
position, and the heads
bolted to the piles adja-
cent. The piles are made into a strong frame laterally by the planks b and c 9
and plank-braces d d on each side of the piles, bolted through. The string-
pieces of the bridge rest on the cap. Longitudinal braces are often used, their
lower ends resting on the plank b which should be, then, notched on to the
piles and their upper ends coming together, or with a straining-piece between,
beneath the string-pieces, acting not only as supports to the load, but also as
braces to prevent a movement forward of the frames. As the tendency of a
moving train is to push the structure on which it is supported forward, in rail-
way-bridges especially, great care is taken to brace the structure in every
way vertically and horizontally, laterally and longitudinally. If the plank c
be a timber-sill, and the piles beneath be replaced by a masonry-pier, the
structure will represent a common form of trestle.
Fig. 966 is a plan of one of the stone piers of the railway-bridge across the
Susquehanna, at Havre de Grace. To lessen as much as possible the obstruc-
tion to the flow of the stream, it is usual to make both extremities of the piers
FIG. 965.
432
ENGINEERING DRAWING.
pointed or rounded. Sometimes the points are right angles ; sometimes, angles
of 60 ; often, a semicircle, the width of the pier being the diameter ; occa-
sionally, pointed arches, of which the radii are the width of the pier, the cen-
ters being alternately in one side,
and their arcs tangent to the
opposite side. It will be ob-
served (Fig. 966) that none of
the stones break joint at the
angle this is important in op-
FlG - 966 - posing resistance to drift-wood
and ice. It is not unusual, in
very exposed places, to make distinct ice-breakers above each pier, usually of
strong crib- work, with a plank-slope like a dam, of 45, and with a width
somewhat greater than that of the pier a cheap structure as a protection to an
expensive bridge.
Fig. 967 is the plan and Fig. 968 the side elevation of a pier of a bridge
across the Missouri, on the Northern Pacific Kailroad at Bismarck, designed and
constructed by George S. Morison, C. E. In this design both ends of the
pier are rounded, but the upper extremity is extended beyond the main body
of the pier, and the upper edge is inclined and plated with iron between low
and high water mark. This is intended
not only to turn aside drift, but as an
ice-breaker ; the ice, moving up the in-
cline, is broken by its own weight.
It is now very common in railroad
practice to construct wrought-iron piers,
as in Fig. 969, of very great height ;
skeleton-piers, of four or more posts,
adequate to sustain the load, with lat-
tice girts and lateral rod-bracing.
Fig. 970 is a section of the founda-
tion of the Bismarck bridge, showing
the construction of the inverted caisson,
similar to that used for the Brooklyn
bridge pier and others. The caissons
are 74' long, 26' wide, and 17' high out-
side ; the working-chamber 7 feet high.
The caissons are built of pine, sheathed
with two thicknesses of 3" oak-plank.
Above this is crib-work filled in with
Portland cement concrete ; a a are the
air-locks. The sand was removed from
the caissons by water-ejectors.
Arch bridges are of stone, brick, or FlG - 969 -
metal ; the parts of the arch exert a
direct thrust upon the abutments, resisted by the inherent weight of the latter,
or its absolute fixed mass, as in the case of natural rock abutments.
ENGINEERING DRAWING.
433
FIG. 968.
434
ENGINEERING DRAWING.
FIG. 970.
Arch bridges, in masonry, are arcs of circle, semicircular (Fig. 972), segment-
al (Fig. 971), elliptic, or described from three or five centers (page 25). The
stones forming the arch are called voussoirs, or arch-stones ; those forming the
exterior face are called ring-stones, the inner line of arch the intrados, exterior
line the extrados. The stones at the top, which are those set last and complete
the arch, are key-stones. The courses from which the arches spring are called
skew-backs, and the first course the springing-course. The masonry on the
shoulders of the arch is called the spandrel-courses, or spandrel-backing. The
weight at the crown of a semicircular arch tends to raise the haunches. This
is counteracted by the spandrel-backing, and by the earth -load, which should
be carefully distributed on each side of the arch.
To determine the depth of the key-stone, Rankin gives the following em-
pirical rule, which applies very well to most of the above examples :
Depth at key, for an arch of a series, in feet, = V'll X radius at crown.
For a single arch, = V'12 X radius at crown.
To find the radius at crown of a segmental arch, add together the square of
half the span and the square of the rise, and divide their sum by twice the rise
ENGINEERING DRAWING. 435
Thus, the Blackwall Railway-bridge has a span of 87 feet, and a rise of 16
43^5' + 1 6 * ^ 1892-25 + 256 _
2 X 16 ~~32"
To find the radius of an elliptical arch, on the hypothesis that it is an arch
of five centers (Fig. 79, page 25), the half-span is a mean proportional be-
tween the rise and the radius. Thus, for example, the Great Western Rail-
way-bridge is 128' span, and 24-25' rise
1>4 2 = 24-25 x R
E =ISh 169feet -
To find the depth of key-stone, by rule above, as in one of a series
d = |/17 x 169 = 1/287^=5-33.
The depth of the voussoir is increased in most bridges from the key-stone to
the springing-course, but not always ; it is safer to increase the depth.
If an arch be loaded too heavily at the crown, the lines of pressure
pass above the extrados of the crown, and below the line of intrados at the
haunches, depressing the crown and raising the haunches, separating the
arch into four pieces, and vice versa if the arches are overloaded at the
haunches. To prevent such effects, especially from moving loads, in con-
struction the arches are loaded with masonry and earth, that the constant
load may be in such excess that there will be no dangerous loss of equi-
librium by accidental changes of load.
The horizontal thrust may be determined, according to Rankin, by the fol-
lowing approximate rule, which seldom errs more than 5 per cent :
The horizontal thrust is nearly equal to the weight supported between the
crown and that part of the soffit ivhose inclination is 45>
This thrust is to be resisted by the masonry of the abutment and the earth-
load behind it.
Thus, if Fig. 973 be a section of an abutment of an arch, the horizontal
thrust exerted at T is resisted by the mass of masonry of the abutment ; the
tendency is to slide back the abutment on its base
A C, or turn it over on the point A. The sliding
motion is resisted by friction, being a percentage,
say from 4 to f of the weight of the abutment and
of half the arch which is supported by this base ;
but, in turning over the abutment on the point A,
the action may be considered that of a lever, the
force T acting with a lever T C to raise the weight
of the abutment on a lever A B (G being the center
of gravity, and G B the perpendicular let fall on
the base), and the weight of half of the arch on the FIG. 973.
lever A C. That is, to be in equilibrium, the hori-
zontal thrust T x T C must be less than the sum of the weights of the abut-
ment multiplied by A B, and the weight of the arch multiplied by A C.
Skew bridges are those in which the abutments are parallel, but not at right
angles to the center line, and the arches oblique. To construct these in cut
436
ENGINEERING DRAWING.
stone requires intelligence and education both in the designer and stone-cutter ;
but, when the work is laid full in cement, so that the joints are as strong as
the material itself, this refinement of stone-cutting is not necessary. The arch
may safely be constructed as a regular cylinder of a diameter equal to the rect-
angular distance between the abutments, with its extremity cut off parallel to
the upper line of road. For such an arch hard-burned brick is the best mate-
rial, the outer voussoirs being cut stone.
In the rules above given no consideration is paid to the strength of the
cement in which the stones are bedded. When the cement is thoroughly set,,
the structure is in a measure monolithic, and the thrust is inconsiderable.
FIG. 974.
Fig. 974 is the elevation of one of the stone arches of the Minneapolis
Union Railway Viaduct, with the timber centers on which the arch was turned.
The arch is nearly semicircular, 97*82 ft. span, 50 ft. rise; width, 28 ft. ;
depth of arch at spring, 40" ; at key, 36". The piers are 10 ft. thick at spring-
ing line ; their up-stream ends are at angles to the main body of the piers, and
parallel to the thread of the stream. The whole length is 2,100 ft., composed
of 3 arches of 40 ft. span, 16 of 80 ft, and 4 of 100 ft. Height above water,
65 ft. ; total height, 82 ft.
The centers were very light frames. 5 to each arch ; the chords, timber
arches, and ties were each 12" X 12", the central braces 10" X 10", and the
shorter side-braces 8" X 8" ; the bolts, single, H" diameter.
The bridge was constructed after the designs and under the direction of
Charles C. Smith, C. E., Chief Engineer of the St. Paul, Minneapolis and
Manitoba Railway, and is an example of a very economical and stable con-
struction. The piers are of Minnesota granite, but above springing line the
masonry is of magnesian limestone. It was commenced in February, 1882,
and completed in November, 1883.
ENGINEERING DRAWING.
437
LOCATION.
Material.
Form of arch.
Span.
Rise.
Depth
at
crown.
Depth
at
spring
Manchester and Birmingham Railroad
u u a
London and Brighton Railroad .
Brick.
u
U
Semicircular.
U
it
18
63
30
9
31-6
15
1-6
3
1'6
Unif'rm
u
2-3
u " Blackwall "
SeTnental
87
16
4'U
Unif'rm
Great Western Railroad
u
Elliptical.
128
24-3
5
7'H
Chestnut Street (Philadelphia) Railroad. . .
High Bridge, Harlem River, New York . . .
St. Paul, Minneapolis & Manitoba Railroad
(largest arch), at Minneapolis
u
Stone.
u
Segmental.
Semicircular.
Segmental.
60
80
97'8
18
40
50
2-6
2-8
3
3-4
Cabin John Washington Aqueduct
u
Elliptical
220
57-3
4'2
Lickinf Aqueduct and Ohio Canal . . .
u
u
90
15
2*10
Monocacy "
u
u
54
9
2*6
Hutcheson u
a
Segmental
79
13'6
3'6
4'6
Chcmin du Fer du Nord sur 1'Oise
U
u
82'5
13*5
4'6
D'En^hien Railroad du Nord
u
Semicircular.
24-4
12'2
1'4
Du Crochet Railroad
M
it
13-2
6*6
1-7A
Experimental arch, designed and built by
M Vaudray Paris . ...
Se "mental
124
6*11
A '2
2*8
3'7
The arch last in the list was a very bold specimen of engineering, built as
an experiment, preliminary to the construction of a bridge over the Seine. It
was made of cut stone, laid in Portland cement, with joints of f", and left to
set four months ; the arch was 12' wide ; the centers rested on posts in wrought-
iron boxes filled with sand, and, as the centering was eased by the running out
of the sand, the crown came down T 6 " ; the joints of one of the skew-backs
opening 10 ' T 00 /!r during the first day, it came down y^h/'. It was then loaded
with a, distributed weight of 300 tons ; under this load the crown settled -f^"
more. Since then nothing has stirred, although it was afterward tested by
allowing five tons to fall vertically 1' 6" on the roadway over the key-stone.
This bridge will not come within any of the rules laid down for other construc-
tions. It will be observed that the rise is about -fa the span, although the
usual practice for segmental and elliptical arches is more than , or within the
limits of and
FIG. 976.
In suspension-bridges the platform of the bridge is suspended from cables,
or chains, the ends of which are securely anchored within the natural or arti-
ficial abutments.
438
ENGINEERING DRAWING.
The curve of a suspended chain is that known as the catenary, and, if the
whole weight of the structure were in the chain itself, this would be the curve
of the chains of a suspension-bridge ; but, as a large part of the weight and
the whole of the loading lies in the platform, the curve assimilates to that of a
parabola, and in all calculations it is so regarded.
Let Fig. 976 represent a suspension-bridge, in which A, B, C, are points in
a parabolic curve.
Rule. Add together four times the square of the deflection (E B) 2 and the
square of half-span (AE) 2 , and take the square root of this sum ; multiply this
result by the total weight of one chain and all that is suspended from it, in-
cluding the distributed load, and divide this product by four times the deflec-
tion (E B) of the cable at the center, and the result will be the tension on one
chain, at each point of support, A and 0. The angle made by the chain at
the point of support, viz., angle POL and the angle of the back-stays, or con-
tinuation of the chain (angle L C N) should be equal to each other, in order
that there be no tendency to overset the tower C L and A F.
BRIDGES.
Main
spans.
Deflection
of chain or
cable.
No. of
chains and
cables.
Total effective
section of cable in
square inches.
Mean weight
of cable per foot of
span (pounds).
Fixed load
per foot of
span (Ibs.).
Breadth of
platform
in feet.
Menai
570
43
16
260
880
28
Chelsea. . . .
348
29
4
230
767
47
Pesth
666
47'6
4
507
1 690
9 892
46
Bamberg
211
14-1
4
40-2
137
1,581
30-5
Freyburg
870
63
4
49
167
760
21-25
Niagara Falls.
821
54 and 64
4
241-6
820
2,032
24
Cincinnati . . .
1,057
89
2
172-6
516
2,580
36
Brooklyn ....
1,595
BOILER-SETTING.
Fig. 977 is a longitudinal section, Fig. 978 a plan with section of wall, and
Fig. 979 an elevation half-front and half-sectional of a boiler and setting as
recommended by the Hartford branch of the Hartford Steam-Boiler and In-
spection Company, showing the interior bracing, steam and water connections,
and brick-work. There are ten braces in each head, secured to pieces of T-iron,
placed radially, as shown in dotted lines (Fig. 979). The feed-pipe is through
the front-head, just above the line of tubes, extending to the back of the boiler,
with a perforated branch across it, that the water may be warmed in its passage
and distributed. The front is a projecting cut-away front, the boiler-head
being nearly on a line with the front below, diiferent from that given in Fig.
768, where the lower part of the shell projects beyond the head of the boiler,
and the cast-iron front covers the end. The doors giving access to the tubes
are usually semicircular, and hung on the top diameter, but it will be found
more convenient to form them in two quadrants, and hung so as to move hori-
zontally. The boilers are to be protected against radiation by a covering of
ashes, or a brick arch, resting on the side-walls. My own practice is to ninke
the boiler without lugs to support it on the side-walls, but to hang the boiler
ENGINEERING DRAWING.
439
13 2'
from wrought-iron cross-bars, resting
on the top of the side-walls, and put-
ting small bars across just above the
top of the boiler, to roof over with
sheet-iron and fill above with ashes,
leaving the spandrels as hot-air spaces.
It will be observed (Fig. 978) that
the manhole-frame is riveted to the in-
side of the boiler ; frequently it is on
the outside. For most positions I pre-
fer that the manhole should be placed
in the back-head, as easier of access,
and in my form of cover there is no
disturbance of ashes for access to the
manhole. It is often well to make the
blow-off pipe a circulating pipe by
FIG. 979
440
ENGINEERING DRAWING.
00
000
000
000
ooo
ooc
ooo
ooo
ooo
ooo
oooo
oooo
oooc
oooo
oooo
oooo
oooo
oooo
oooo
oooo
ooo oooo
ooo oooo
ooo oooo
ooo
ooo oooo
000 OOOO
000 OOOO
000 OOOO
ooo oooo
00 OOOO
ENGINEERING DRAWING.
44:1
Fm. 982.
FIG. 985.
442
ENGINEERING DRAWING.
FIG. 987.
FIG. 988.
connecting an inch pipe
inside the valve with the
upper water-space of the
boiler.
Fig. 981 is a longitu-
dinal section, and Fig.
980 half front elevation
and half cross-section of
a class of boilers usual-
ly designated as marine
boilers, but largely used
at the Philadelphia Wa-
ter-Works. The fire-
boxes and ash-pits are
contained within the
body of the boiler ; it is
set on a cast-iron or brick
base, and the shell is
covered with some prep-
aration of plaster or hair-
felt clothing. The front
smoke-box is of wrought-
iron, and similar to that
shown in Fig. 977.
Locomotive-boilers
are used as stationaries,
and are set like the pre-
ceding, but with some
non-conducting cover-
ing. The protection of
all parts of boilers and
steam-pipes exposed to
the air by some cover of
a non-conducting mate-
rial adds much to econ-
omy in the consumption
of coal and dryness of
steam.
Fig. 982 is a vertical
section of a chimney at
the Eidgewood Pumping-
engine House, Brooklyn,
K Y., and Fig. 983 an
elevation at the point
where the square base is
changed into an octago-
nal.
ENGINEERING DRAWING.
443
FIG. 991.
4:4:4: ENGINEERING DRAWING.
Fig. 984 is a section of the shaft at a b, but the flue should have been repre-
sented circular.
Fig. 985 is a vertical section of a chimney attached to an English gas-house,
taken from " Engineering," with a uniform flue and shell, additional strength
being given by the buttresses shown in section at c d (Fig. 986). No inde-
pendent flue inside is shown, but it is desirable, as it can freely expand with
the heat, without affecting the outer shell.
Fig. 988 is the cross-section of a buttressed chimney at 100 feet above base,
built for the Calumet & Hecla Mining Company, and designed by E. D.
Leavitt, Jr., M. E. The whole height of the chimney is 150 feet. The but-
tress walls are 16" and 12" thick, that of the body 12" and 8", and of the cen-
tral flue 8" and 4", offsetting into each other by 1" oifsets ; the taper is 4 inches
to 10 feet on each side. Fig. 987 is a half elevation and half section of the
cap and the cover of the interior flue by which its expansion is permitted.
Fig. 989 is a sectional elevation of a chimney 160 feet high, from John T.
Henthorne, M. E., with a cross-section (Fig. 990) midway of the height.
Figs. 991 and 992 are sectional elevation and cross-section of a chimney of
my own design and construction. The buttresses supporting the central flue
are inside the chimney. The diameter of the flue is 4 feet, and the height
about 100 feet.
It has not been my practice to build high chimneys 100 feet is usually suf-
ficient but they should extend above surrounding houses, woods, and hills,
which are near enough to influence the draught. For chimneys of this height
an area of chimney-flue of one square inch for every pound of anthracite coal
burned per hour on the grate has been found to answer well in practice. For
chimneys less than this height, it is well to increase the section, and perhaps
reduce for higher chimneys.
Chimneys are constructed of various sections, sometimes uniform through-
out their length, sometimes tapering at the top, and sometimes bell-mouthed ;
all answer the purpose. The great point to be observed is, that there be no
abrupt changes of section or direction, either in the main flue or its connec-
tions, and that they be carried well above all disturbing causes.
ON THE LOCATION OF MACHINES.
The construction of buildings for mills and manufactories (if any aesthetic
effect is intended) is usually left to the architect, but the necessities of the
construction, the weights to be supported, and the space to be occupied, must
be supplied by the mechanical engineer or millwright.
In the arrangement of a manufactory or workshop, it is of the utmost im-
portance to know how to place the machinery, both as to economy of space and
also of working. Where a new building is to be constructed for a specific pur-
pose of manufacture, it will be found best to arrange the necessary machines
as they should be, and then build the edifice to suit them. For defining the
position of a machine, the space it occupies in plan and elevation, the position
of the driven pulley or gear, of the operative, and spaces for the working and
access to parts, are required. To illustrate this subject, take a two-story weav-
ing-room, of which Fig. 993 is an elevation, and Fig. 994 a plan.
ENGINEERING DRAWING.
445
Lay down the outlines of an interior angle of the building, and dot in, or
draw in red or blue, the position and width of beams. This last is of impor-
tance, as it will be observed (Fig. 993) that no driving-pulley can come beneath
V/////X///M /IVM/M/M !44^yy>y///t^J^^^^
i rt\ J 1 1 j j ] i I
FIG. 993.
the beam, and also that this is the position for the hanger. Lay off now the
width of the alleys and of the machines. The first alley, or nearest the side-
wall, is a back alley ; that is, where the operative does, not stand, and so on
alternate alleys. Draw the lines of shafting central to the alleys, as in this
446
ENGINEERING DRAWING.
FIG. 994.
ENGINEERING DRAWING. 447
position the belts are least in the way. One operative usually tends four looms ;
they are therefore generally arranged in sets of four, two on each side of the
main alley, where the operative stands ; the twos are placed as close to
each other as possible, say one inch between the lays, a small cross-alley being
left between them and the next set. Lay off now the alley necessary at the end
of the room, and space off the length of two rows of looms with alleys at the
end of alternate looms, and mark the position of the pulleys. It will be ob-
served that looms are generally rights and lefts, so that the pulleys of both
looms come in the space where there is no alley. Should the pulley come be-
neath a beam, the loom must be either moved to avoid it, or the pulley may be
shifted to the opposite end of the loom. Parallel with the pulleys on the looms
draw the driving-pulleys on the shafts, that is, k parallel with &, b with b, f
with/, and so on. Proceed now to draw the third and fourth row of looms,
since the second and third rows are driven from the same shaft ; if they are
placed on the same line, it will be impossible to drive both from the same end,
and, as this is important, we move the third row the width of the pulley b, and,
for the sake of uniformity, the fourth row also. Lay off now the length of
looms and position of pulleys as before, and parallel with the pulleys the driving-
pulleys on the shaft, that is, c against c, ^against g, and so on. Having in
this way plotted in all the looms, every alternate set being on a line with the
third and fourth row, proceed now to lay down the position of the looms in
the floor above ; and since for economy of shafting it is usual to drive from the
lines in the lower rooms, to avoid errors, interference of belts and pulleys, it is
usual to plot the upper room on the same paper or board as the lower room,
using either two different colored inks, or drawing the machines in one room
in deep and in the other in light line, as shown in Fig. 994. If the width of
the rooms is the same, the lateral lines of looms and alleys are the same, and
it is only necessary, therefore, to fix the end lines. Now, as the first loom in
the outer row of looms, in the lower room, occupies for its belt the position k
on the shaft, the loom in the upper room must be moved either one way or the
other to avoid this ; thus the position i of the pulley on the loom must be made
parallel to the pulley i on the shaft, so in the other looms a to a, e to e, d to
d, and h to h.
Besides the plan, it is often necessary, and always convenient, to draw a
sectional elevation (as in Fig. 993) of the rooms, with the relative positions of
the driving-pulleys and those on the machines, to determine suitably the length
of the belts, and also to see that their position is in every way the most con-
venient possible. In the figure, one of the lower belts should have been a
cross-belt, and one of the upper ones straight : now, had the belts to the second
row of looms in the upper story been drawn as they should have been, straight,
the belt would have interfered a little with the alley, and it would have been
better to have moved the driving-shaft a trifle toward the wall.
From this illustration of the location of machines, knowing all the require-
ments, in a similar way any machinery may be arranged with economy of space,
materials, power, and attendance. These last two items are of the more im-
portance as they involve a daily expense, where the others are almost entirely
in the first outlay.
448
ENGINEERING DRAWING.
ENGINEERING DRAWING.
449
Machine Foundations. Figs. 995, 996, and 997 are side and end elevation,
and plan, of the foundation of the stationary steam-engine. F is the cast-iron
frame or bed-plate of the engine ; B the granite bed of engine, or coping of
foundation; P the stone or brick pier, laid full in cement. The sides and sur-
faces of granite exposed are usually fine-hammered, the upper bed or build to
receive the engine-frame, hammer-dressed and set level. Strong wrought-iron
bolts pass through frame, bed, and pier, with nuts at each end, and the whole
is strongly bolted together. Pockets are left in the pier near the bottom for
access to nuts, and these pockets are covered by granite caps or iron plates.
Few stationary steam-engines are now built with bed-plates extending the
whole length of the engine, but the illustration is applicable to the partial
plates supporting the cylinder and pillow-block, and to engines and machines
for which heavy foundations are necessary. It is not an uncommon practice
now, instead of granite caps, to use timber, as cushioning the shocks and blows
incident to most machinery.
Tunnels. Figs. 998 to 1007 are illustrations, with description, taken from
" Tunneling," a standard work on this subject by H. S. Drinker.
Figs. 998 to 1003 illustrate the principles of timbering applied to driving a
gallery through running material. Figs. 998 and 999 are parts of the construc-
ipqiuip ooaros aajoimnj
FIG. 998.
FIG. 999.
tion on a large scale, with the technical names of the parts. Each frame is
called a timber-set. Suppose a leading set (Figs. 1000 and 1001) is in place,
close to the face, and that the leading ends of the poling-boards resting above
this leading set are held up from the collar by wedges sufficiently high to allow
the insertion of the new poling-boards. In Fig. 1001 the sets e e, standing mid-
way between the front and the hind ends of the poling-boards, serve as middle
sets between the main sets d d. By turning to the plan (Fig. 1003) of a gal-
lery thus timbered it will be seen that, owing to the fact that the site-poling
has also to be wedged out at its leading end, just as the roof-poling is wedged
up, therefore the space to be filled across the top by the roof -poling is wider
over a front main-set than over a back one. Owing to this fact, the two outer
2'J
4:50
ENGINEERING DRAWING.
top poling-boards, as shown in Fig. 998, are made wider at their leading ends
than at their back ends. Now, to begin inserting the roof-poling, the miners,
at either corner of the face, remove the extreme end- wedges between the collars
and the poling, and into this space the new poling-boards (i. e., the ones shown
in Fig. 998) that are wider at their leading ends are driven. But, though the
FIG. 1000.
FIG. 1001.
wedges between the collar and the poling-boards serve well enough to keep back
the material, it would be dangerous thus to take any of them out were there no
other guard for the poling, as the board just above the wedge removed would
be pressed down ; a run might also be started, and all the other wedges forced
out, when the poling-boards would snap
down on the leading collar, and per-
FIG. 1002.
FIG. 1003.
haps break off ; in any event, it would be a matter of great trouble to get them
wedged up again. In order to guard against this trouble, a cross-board or
plank a (Fig. 999) is placed just under the poling-boards, and over the wedges.
Then, when one wedge is removed, this cross-connection holds in place the
poling-board that is immediately above the wedge removed, until the new board
ENGINEERING
FIG. 1004.
(Section of Fig. 1006, through A B, looking west.)
HOOSAC TUNNEL.
Timbering and arching through soft ground at 'the West End. Scale, 11' = 1*
FIG. 1005.
452
ENGINEERING DRAWING.
West.
FIG. 1006.
HOOSAC TUNNEL.
West End. Scale, 11' = 1.
FIG. 1007.
ENGINEERING DRAWING. 453
can be put in ; it also stays the tendency to any general movement. The new
poling-board being inserted, it is now driven ahead six or twelve inches, and
then temporarily stayed by wedges, b (Fig. 1001). The corner roof-polings
being thus in place, the middle ones (Fig. 998) are similarly inserted. Then
the top retaining-board in the face is cut out, and the material allowed to flow
into the heading through the space. As room is thus given ahead, the poling-
boards are gradually driven forward, say 24 or 30 inches, or about half the
length of a board, supposing they are 5 feet long. Whenever they are thus
tapped, the wedges I (Fig. 999) must be loosened, and then tightened again
after the driving. The side-poling is similarly thus advanced ; and we must
bear in mind that, as space is gained ahead, it must be protected by new face-
boarding, stayed by stretchers. Thus the work can be gradually carried down
to the floor of the heading, by successively taking out the face-boards. Often
the floor of the gallery also has to be planked, and, in very extreme cases, to be
poled similarly to the roof and sides.
We now have reached the point, shown in Fig. 1002, where the new poling-
board has been inserted for its half-length. During this operation the boards
have been held in place by the double support oifered by a and b (Fig. 1002).
The face retaining- boards are kept back by a vertical plank laid across them,
and stayed by stretchers. On this newly-excavated chamber the outside pressure
will be great, especially acting, as it does, on the front half length of the poling-
board c a, and, if the remaining work is not rapidly executed, the front ends of
the boards may be snapped beyond a ; then, if it were attempted to drive the
remaining portion of the board on, as soon as its back end left b it would snap
between a and b. A middle set is therefore required at once. The middle set
being in position, the work of excavating the face can be proceeded with as
before. The face-boards are removed, one by one, from top to bottom, and
the polings are driven in to their full length ; then in the new length ahead the
next main set is erected.
Such are the general principles of head ing-driving thro ugh running ground,
or sheet-piling in tunneling.
Figs. 1004 to 1007 show the English system of bar-timbering, as used at
the Hoosac Tunnel for the soft ground at the west end. The material was of
the worst character, and was exceedingly difficult to drive through. Figs.
1004 and 1005 are cross-sections, the one looking west from A B, the other
east. Fig. 1006 is a longitudinal section. Fig. 1007 is a cross-section of the
tunnel as completed with an invert, and the bars not drawn but bricked in.
Railway Stock. Figs. 1008 and 1009 are the elevation and plan of a stand-
ard box-car of the New York Central and Hudson River Railroad.
Figs. 1010 to 1013 are the plan and elevations of the truck for the same car.
Figs. 1014, 1015, and 1016 are end-elevations and cross-sections, Figs.
1017 and 1019 longitudinal sections, and Fig. 1018 plan of a standard passen-
ger-car of the Pennsylvania Railroad.
Figs. 1020 to 1023 are elevations, in full and parts, and Fig. 1024 a plan of
the trucks of the above car.
In the figures, both of standard box and passenger cars, the elevations and
plans are usually broken, to show the construction. When the two sides or
454:
ENGINEERING DRAWING.
ENGINEERING DRAWING.
455
B
or side Door-
!< -Spring Plank 2'-IO'/z"
{< Transom 3 -3 " -x Swing Bolster. 2 -10 fc" I *,
Oentercf Sffing 2-4" - ->J
Between Centers of Journal Bearing 6-3
FIG. 1013. 'rtr Axle e-'5 5 /
Ervd Elevation.
456
ENGINEERING DRAWING.
two ends of a car or truck are similar, it has not been considered necessary to
show both, but complete the figure, with a section of the other part, through a
different plane.
STANDARD PASSENGER CAR OF THE PENNSYLVANIA RAILROAD
ENGINEERING DRAWING.
457
TRUCK OF PENNSYLVANIA RAILROAD STANDARD PASSENGER CAR.
^Erg3^' ^"H- ^ -^-i 1 . .
Fio. 1024.
458
ENGINEERING DRAWING.
and technical names of similar parts
The following letters of reference
apply equally to all the figures :
a, Sill.
a', End-sill.
6, Intermediate floor-timbers.
6', Center floor-timbers.
c, Sill knee-iron or strap.
d, Body bolster.
e, Body bolster truss-rod.
/, Truck side-bearing.
<7, Center plate, body or truck.
A, Check-chain on the truck, hooking into
A', Check-chain eye on the car.
i, Body truss-rod.
i', Body truss-rod queen-post.
y, Cross-frame tie-timber.
The Wave-line Principle of Ship- Construction, from Russell's "Naval Archi-
tecture." The general doctrines arrived at by J. Scott Russell, F. R. S., from
numerous and long-continued experiments and practical tests, is " that the
form of least resistance for the water-line of the bow is horizontally the curve
of versed sines, and that the form of least resistance for the stern of the vessel
is the cycloid ; and you can either adopt the said cycloid vertically or horizon-
tally, or you can adopt it partly vertically and partly horizontally, according
to the use of the vessel or the depth of water. "
"That the length of entrance, or fore body, should be f, and that of the
run, or after body, f . "
"When it is required to construct the water-lines of the bow of a ship of
which the breadth and the length of the bow are given, so as to give the vessel
Draw-bar.
Journal-box.
Pedestal.
Pedestal tie-bar.
Pedestal stay-rod.
Pedestal arch-bar.
Pedestal inverted arch-bar.
Transom.
Truck bolster.
Spring-plank.
Swing-hanger.
Safety-beam.
Equalizing-bar.
FIG. 1025.
the form of least resistance to passage through the water, or to obtain the high-
est velocity with a given power : Take the greatest breadth, M M (Fig. 1025),
on the main section of construction at midship-breadth, and halve this breadth,
M ; at right angles to M M at draw the center line of the length of the
bow, X ; on each half -breadth describe a half -circle, dividing its circumfer-
ENGINEERING DRAWING.
ence into, say, eight equal parts. Divide the length X
into the same number of equal parts. The divisions of the
circle, reckoned successively from the extreme breadth, indi-
cate the breadths of the water-line at the successive corre-
sponding points of the line of length. Through the divis-
ions of the circles draw lines parallel to X, and through
the divisions of X lines parallel to M M. These, inter-
secting one another, show the successive points in the re-
quired water-line. The line traced through all these points
is the wave water-line of least resistance for a given length
of bow and breadth of body. "
To construct the water-lines of the after body or run of
a ship (Fig. 1027), the mid-section (Fig. 1026) being given :
The bow is constructed as in Fig. 1025, but the divisions are
12 on the center line ; for the run lay off 8 divisions, each
459
FIG. 1026.
equal to those of the bow ; divide the half circle into 8 equal
parts, and draw chords to these divisions from to 1, 2, 3, 4.
From the point 1 on the center line lay off an inclined line
equal and parallel to the chord 1 ; the point 1' will be in
the water-line. In the same way from the point 2 draw an
inclined line parallel and equal to the chord 2, for 2', and J
determine in the same way the points 3', 4', 5', 6', 7'. The
other circles drawn in the figure are described on semi-
diameters of the mid-section at different levels, and the
points of their wave-lines are determined on the same in-
clined lines 1 1', 2 2', but the lengths are those of the
chords of the different circles. In Fig. 1026, the elevations
of the mid body, the curved lines of sections are projected
from the plan.
Fig. 1028 is a body plan of a vessel adapted to speed ;
Fig. 1029 of one adapted to freight.
" To determine the after body it is expedient to construct
a vertical wave-line on the run as well as a horizontal one,
and in designing shallow vessels to give more weight to the
vertical wave-line."
" The wave system destroys all idea of any proportion of
breadth to length being required for speed. An absolute
length is required for entrance and run, but, these being
formed in accordance with the wave principle for any given
460
ENGINEERING DRAWING.
speed, the breadth may have any proportion to that which the uses of the ship
and the intentions of the constructor require. "
" The wave system allows us to give the vessel as much length as we please.
It is by this means that we can give to a vessel of the wave form the capacity we
may require, but which the ends may not admit. Thus, the Great Eastern,
which is a pure example of the wave form, has an entrance or fore body of 330',
a run or after body of 220', and a middle body of 120', which was made of this
length merely to obtain the capacity required. The lengths of the fore and
after body are indicated by the required speed, and if the beam is fixed, it is
only by means of a due length of middle body that the required capacity,
stability, and such other qualities are to be given as will make a ship, as a whole,
suit its use."
FIG. 1028.
FIG. 1029.
Length of entrance of a vessel for a 10-mile speed should be 42 feet, of run
30 feet ; for a 20-mile speed, 168 and 120 feet ; that is, the lengths increase as
the squares of the speed.
Under Isometrical Drawing are given illustrations of vessels constructed on
wave-lines.
AKCHITECTUKAL CHAWING.
IT is the duty of an architect to design a building to be suitable and con-
venient for the purposes for which it is intended ; to select and dispose of the
materials of which it is composed to withstand securely and permanently the
stresses and wear to which they may be subjected ; to arrange the parts to pro-
duce the artistic effects consistent with the use of the building and its location,
and to apply such appropriate ornament as may express the purpose and har-
monize with the construction.
In domestic architecture, by far the most extensive branch of the profession,
most persons can give some idea of the kind of building which they wish to
have constructed, and perhaps express by line the general arrangement of
rooms ; but it is left to the architect to settle the style of building appropriate
to the position, to adapt the dimensions and positions of rooms and passages
to the requirements, to determine the thickness of walls and partitions, and
arrange for drainage, heating, and ventilating. The graphical representation
is left to the draughtsman, and his assistance is the more valuable if he is not
only conversant with practical details, but understands the best proportions of
parts; the necessities of construction, and the requirements of building laws.
The draughtsman usually commences his education with the copying of
drawings. Such are furnished him.
For this purpose, in Figs. 1030 to 1034, inclusive, are given plans and eleva-
tions of a simple house, which contain representations sufficient for the informa-
tion of the owner, and for the purposes of estimate of cost, if accompanied with
full specifications. The size of our page has compelled the titles to be put within
the body of the drawings ; after copying, place them outside, and give good
margin. On Fig. 1034 the section and end-elevation are given together. This
is also for economy of space, but should be copied by the draughtsman in two
distinct drawings, each of the full width of the building.
Instead of hatching, it is usual to give the walls a shade of color or black,
or in full black often, as . the black representing the solid wall,
and the inner line that of the plastering.
Details of Construction. The necessities of a suitable foundation for every
structure have been treated of (page 362), and that a good foundation may be
secured in an uniformly yielding earth, as on a rigid rock. For the extent or
width of base, the draughtsman, if there are practical examples in the vicinity
of the proposed structure, will conform to the teachings of practice, and to the
building laws, if there are any in force. In general, for small buildings, cellar-
462
ARCHITECTURAL DRAWING.
J
ARCHITECTURAL DRAWING.
463
464
ARCHITECTURAL DRAWING,
LJ
LJ
ARCHITECTURAL
30
466
ARCHITECTURAL DRAWING.
END ELEVATION
SECTION.
SCALE : 4' = 1 inch.
FIG. 1034.
ARCHITECTURAL DRAWING.
467
walls, if of stone laid in mortar, should not be less than 18" thick ; if of brick,
16", and the base 6" to 12" wider. For walls above the cellar, it will be found
difficult to lay stone walls in mortar, with fair bond and face, less than 16*
thick. Brick walls may be as thin as 8" for exteriors, and for partitions 4".
Brick walls are usually bonded by heading-courses every fifth
to seventh course. Where the outside course is pressed or face
brick, these are laid on stretchers, and the bond with the back-
ing may be thin strap-iron, laid in the joints, or, by cutting
off the interior corners of the face-course, say every fifth
course, and laying common brick diagonally of the wall rest-
ing in this clipped corner (Fig. 1035). The face of buildings
is often built of thin ashlar, which is secured with iron an-
chors to the brick backing.
In most large cities there are building acts in force, defin-
ing thickness of walls and foundations, to which all construc-
tions within their limits must conform. Extracts from the
New York law may be found in the Appendix.
Openings in masonry-walls are covered by lintels or arches,
or both. It is usual to place a stone or cast-iron lintel in the exterior face
over openings for doors and windows, with a wooden lintel inside (Fig. 1036),
and a relieving arch above. For larger openings, brick arches are turned in
cast-iron skew-backs, of which the thrust is resisted by a tie-bolt (Fig. 1037),
or cast-iron lintels, box, or j 4 , or roller I-beams. But it is to be observed that,
when the cement is set, there is little or no thrust from the arch. The whole
dead work, or masonry without
an opening, forms a monolithic
FIG. 1035.
FIG. 1036.
FIG. 1037.
beam, and, if there is depth enough of this, the arch is of no account. It is
the custom in the north of Italy to construct flat lintels of brick, of consider-
able span, depending entirely on the mortar for strength.
To distribute the weight over the foundation or walls, it is very common to
turn inverted arches beneath openings.
In old houses, it was not unusual to make the exterior arches of an opening
flat or rectangular in outline, with the joints radial. This is now relegated to
ornamental construction.
Concrete Walls. It is common in many places where brick and stone are
expensive and gravel is abundant to make walls of concrete, in proportions of one
of cement to five to seven of gravel. The space requisite for the wall is inclosed
with plank, and is filled in with concrete, well rammed. Figs. 1038 and 1039
are plans of concrete walls with inclosing plank, and Fig. 1040 an elevation.
468
ARCHITECTURAL DRAWING.
The planks are held by bolts passing through wall and plank,
all of which are removed after the wall is set, and the bolt-
holes are then filled with cement. The thickness of walls
should be a little in excess of those of brick.
Wooden walls are framed. Fig. 1041 represents the frame
FIG. 1038.
FIG. 1039.
FIG. 1040.
of the side of a wooden house, in which A A are the posts, B the plate, C C
girts or interties, D D braces, E sill, F window-posts or studs, G G studs.
U
1
i i
II
1
ff i
1
II
II
-L
r
~~l
n
1 1
1
1
' 1
1 1
1 1
1 ;
D
o
y
M
U
14
5
U
^
^
Y\
A
n
o
9
,y
rC
o
J
V
Li
J [
M
J - L
J [
M
"\^
//
j
(
'
<
CEILING LINE
FIG. 1041.
FIG. 1042.
The studs at all door-openings should be set at least 2* wider, and 3" higher
than the size of the finished opening. It is not unusual to have double studs
(2" X 4 ff ) to inclose these openings (Fig. 1042). This leaves the doorway more
or less independent of the partition.
Usual dimensions of timber for frame of common dwelling-houses : sills
6" X 8", posts 4" X 8", studs 2" X 4" or 3" X 4", girts 6" X the depth of floor-
joists, plates 4" X 6" ; the floor-joists (J, Fig. 1043) are notched into the girts.
The posts and studs are tenoned into the sills and girts. Fig. 1045 represents
ARCHITECTURAL DRAWING.
469
a tenon, I c, in side and end elevation, and mortice, a ; the portions of the end
of the stud resting on the beam are called the shoulders of the tenon. In the
balloon-frame the girts are omitted ; the studs are of the same length as the
posts, and the floor-joists are supported by a board, a, 3" or 4" X I", let into
the studs (Fig. 1044), and firmly
nailed ; the joists are also nailed
strongly to the studs.
The frame is covered with boards
usually 1" thick, laid either horizon-
FIG. 1043.
FIG. 1044.
I
QD
FIG. 1045.
tally or diagonally, and nailed strong-
ly to the posts or studs. Fig. 1046
is the elevation of the end frame of
a house, showing by breaks the diag-
onal cover of boards and the inner
lathing. The lower story is sheathed
or ceiled with narrow boards, the up-
per shingled. With balloon frames,
the bracing depends largely on the
diagonal boarding.
Partitions are usually simply
studs set at intervals of 12 or 16
inches, these spaces being adapted to
the length of the lath (48 inches).
The sizes of the studs are generally
2X4, 3X5, or 3x6 inches, ac-
cording to the height of the parti-
tion ; for very high partitions, greater
depth may be required for the studs,
but three inches will be sufficient
width.
Partitions are usually cut in be-
tween sills placed on the floor-beams
(Fig. 1047), and similar caps above,
beneath the beams. Where parti-
tions of the second story are directly
above those on the first story it is
better to foot the studs on the caps
of the latter, and not on the beams
(Fig. 1048). Where there are double
floors, the sills are placed on the bot-
470
ARCHITECTURAL DRAWING.
torn floor, or on the floor without a sill. It may be important that the parti-
tions should be self-sustaining. This is effected by simple bridging, well
FIG. 1049
IG. 1048.
FIG. 1050.
nailed to the studs, as shown in Fig. 1049, or by herring-bone bridge, as
shown in plan of floor (Fig. 1051), or by a system of trussing, as in Fig. 1050.
This method of truss-
ing must vary with
the position of open-
ing. The foot of the
braces should rest
on a positive sup-
port.
The bridging
should be accurate-
ly cut and firnily
nailed. Bridging
distributes the
weight of the partition, but trussing concentrates it at the ends of the braces.
Flooring. The timbers which support the flooring-boards and ceiling of a
room are called the naked flooring.
The simplest form of flooring, and the one usually adopted in the construc-
tion of city houses and stores, is represented in plan and section (Fig. 1051).
It consists of a single series of beams or deep joists, reaching from wall to wall.
As a lateral brace between each set of beams a system of bridging is adopted,
of which the best is the herring-bone bridging, formed of short pieces of joists
about 2x3, crossing each other, and nailed securely to the tops and bottoms of
the several beams, represented by a and b ; and wherever a flue occurs, or a
stairway or well-hole prevents one or more joists from resting on the wall, a
header, II, is framed across the space into the outer beams or trimmer-beams
T T, and the beams cut off or tail-beams are framed into the trimmer.
Whenever the distances between the walls exceed the length that can safely
be given to joists in one piece, an intermediate beam or girder, running longi-
tudinally, is introduced, on which the joist may be set (Fig. 1052), notched on
(Fig. 1053), or boxed in (Fig. 1054), or both boxed and notched. They may
also be framed in with tenon and mortice ; the best form is the tusk-tenon
AKCHITECTURAL DRAWING.
471
FIG. 1051.
(Fig. 1055). Flooring is still further varied, by framing with girders longi-
tudinally ; beams crosswise, and framed into or resting on the girders ; and
joists framed into the beams, running the same direction as the girders. It is
\
FIG. 1052.
Fm. 1053.
Fia. 1054.
FIG. 1055.
evident, when the joists are not flush or level with the bottom of the beams or
girders, either that in the finish the beams will show, or that ceiling- joists or
furrings will have to be introduced.
On the Size of Joists. The following dimensions may be considered as safe
sizes for ordinary constructions, the distances from center to center being one
foot.
Joists in floors, clear bearing
Exceeding 7 feet, and not exceeding 10 feet, to be not less than 6X2 inches.
10
12
14
16
18
20
12 "
14 "
16 "
18 "
20 "
22 "
24 "
6X3
7X3
9X3
9X3
10 X 3
11X3
12 X 3
It is to be observed that lumber is seldom sawed to dimensions of fractions
of an inch.
472
ARCHITECTURAL DRAWING.
Trimmer-beams and headers should be of greater width than the other
beams, depending on the distance of the headers from the wall, and the num-
ber of tail-beams framed into it. The New York Building Act requires that
all headers should be hung in stirrup-irons (Fig.
1056), and not framed in.
Floors. In New York it is usual to lay single
floors of tongued and grooved boards directly on the
beams, but in the Eastern States double floors are
more common. The first floor consists of an inferior
quality of boards, unmatched, laid during the prog-
ress of the work as a sort of staging for the carpenter
and mason, and, in finishing, a second course is laid on them of better material,
generally tongued and grooved, but sometimes only jointed. Ceilings should
always be furred, and the laths be nailed to the strips. Furring-strips usually
are of inch board, 2" wide, and
r - , ___.. /' ^_^_^_^_ 12" from center to center, nailed
across from joist to joist.
Fig. 1057 represents a section
FIG. 1056.
of a mill-floor. The girders or
beams, generally in pairs, with a
space of about an inch between them, are placed at a distance of from seven
to nine feet from center to center, and are of from twelve to sixteen inches in
depth. On these, a tongued and grooved plank floor of from 3" to 4" thick
is laid.
Fig. 1058 is the section of a beam and mill-floor now adopted as a fire-retard-
ing construction, and considered superior to iron beams and brick arches. It
consists of the usual
beam and plank floor ;
both plastered on the
under side and on the
lateral surfaces. The
lathing consists of wire
cloth stapled through
furring strips f * to y,
and then the usual
three-coat plaster. In
addition, it is common
to lay roofing-felt on
the upper surface of FIG. 1058.
the plank, with 1" to
1%" of cement mortar ; with the usual floor on the top of this, the floor being
nailed to strips attached to the plank, and serving as guides to surface the
cement mortar. Various methods are given (page 238) of trussing beams
when the spans or loads are in excess of the strength of lumber of the usual
dimensions.
Joinings. As timber can not always be obtained of sufficient lengths for
the different portions of a frame, or to tie the walls of a building, it is often
AECHITECTURAL DRAWING.
473
necessary to unite two or more pieces together by the ends, called scarfing or
lapping. Fig. 1059 is a most common means of lapping or halving employed
when there is not much longitudinal stress, and when a post is to be placed
beneath the lower joint.
V
FIG. 1059.
FIG. 1060.
Fig. 1060 is a long scarf, in which the parts are bolted through and strapped,
suitable for tie-beams. Joints (Figs. 1061, 1062, and 1063) are also often made
by abutting the pieces together, and bolting splicing -pieces on each side ; still
further security is given by cutting grooves in both timbers and pieces, and
driving in keys, Tc k.
','
',' >
! It >.
jl
i 1 J
_j
I I
FIG. 1061.
O
o o
o o
o
O o
FIG. 1062.
EH
s
FIG. 1063.
Floor-beams in a building acting as ties are usually strapped, or anchored
together by iron bars, spiked to the top or bottom of the beams, often sunk
into the beam.
FIG. 1064.
FIG. 1065.
FIG. 1066.
Figs. 1064, 1065, and 1066 are common forms of anchors. The first two
for connecting beams, the last for beams and walls. In warehouses, it is usual
ARCHITECTURAL DRAWING.
to carry the anchors entirely through the wall, with a washer and nut outside.
The beams are often joint- bolted together like stair-rails.
Fire-resisting Floors. Flames spread through buildings by means of the
spaces left between floors and ceilings, and between walls and f urrings and hol-
lows in partitions, which act as flues. But when the wooden beams and plank
floors are protected beneath by wiie netting and plaster there are no air-spaces
for circulation, and sufficient stay is made in the progress of the flames to ad-
mit of the application of means for extinguishment. And if the beams are
placed close together, and the joints filled with cement, there is still greater
security. Experiments were made in Paris on asphalt floors laid on plank, and
they resisted for a very long time the spread of flames, both when fires were
kindled beneath the floors, and directly on top of the asphalt. In the latter case,
a thin layer carbonized, and afforded a good fire-proof material.
Iron beams and brick arches, as in Fig. 1067, are the usual form of fire-
proof floors, but when efficient protection against fire is desired the bottom
flange must be
covered entirely
with some fire-
proof material, to
prevent contact FIG.
with flame and
excess of local heat, tending to warp and twist the beams. Iron often be-
comes necessary for spans greater than can be met by wooden beams, and they
should be protected by some fire-proof covering.
Fig. 1068 represents a section of one of the French systems of fire-proof
floors. It consists of I-girders, placed at a distance of one metre (39 '38 inches)
from center to center, slight-
ly cambered or curved up-
ward in the center, the depth
of the girders to depend
upon the span. Stirrups of
cast-iron are slid upon the
FIG. 1068. girders, into which the ends
of flat iron joists, set edge-
ways, pass and are secured by pins ; the ends of the joists take a bearing also
on the bottom flanges of the girders. The joists are placed at a distance of
one metre from center to center. Upon the joists rest rods of square iron,
which in this way form a grillage for the support of a species of rough-cast
and the ceiling. By this and other very similar systems, the French have suc-
ceeded in reducing the cost of such floors to that of wooden ones.
The dimensions of beams and girders for the above constructions can readily
be determined from rules given (page 233). The brick arches (Fig. 1067) are
usually in single ring or rolock courses, and beams spaced from 3' to 6' cen-
ters. Strips of plank are fastened on the top or at the side of the beam to
receive the floor, and the spandrels leveled up with concrete.
Floors constructed of concrete, in plain cylindrical or groined arches (Fig.
1069), are cheap and efficient constructions. One of the warehouses of the
ARCHITECTURAL DRAWING.
475
FIG. 1069.
publishers is covered by arches of this last form. Posts of brick, 2 feet square,
13 feet centers, arches arcs of circles, depth of concrete at spring 21", at key
9" to a level floor, supporting presses.
In Italy ceilings are made in single courses of
brick, and groined, laid without centers, the arcs
being described on the side-walls, and the bricks
laid to a line in plaster. The spandrels may be lev-
eled up with concrete, when rooms above are to be
occupied, but often there is only the brick arch
forming the ceiling of the principal rooms, with a
light wooden roof above.
Figs. 1070 to 1073 are illustrations of Koman
constructions in masonry, from " Dictionnaire Raisonne de 1' Architecture,"
par M. Viollet Le Due.
Fig. 1070 is a perspective view of a cylindrical arch in process of construc-
tion. The cen-
ters A and lag-
ging B are quite
light, as the full
load of the arch
is never borne by
them. On the
lagging, B, a cov-
er of flat tile, C,
is laid in cement,
and above ribs,
D D, and girts,
E E, in brick ma-
sonry, shown on
a larger scale in
Fig. 1071, with
the plank P used
for the support of
the girt bricks E,
which is removed
after the mortar
is set. The pan-
els are now filled
with concrete.
Fig. 1072 rep-
resents rib and
portionsofgirtsof
a groin shown in
plan, Fig. 1073, ef
g h being that of
the rib; K, a tim-
ber of the center.
FIG. 1070.
\
476
ARCHITECTURAL DRAWING.
A similar construction also obtained
for domes, the girts being of the same
width as the ribs, and sunk panels formed
by furring up on the wooden lagging of
the centers.
Fig. 1074 is a perspective of a dome,
in which the brick skeleton, ribs, and
girts are curved, with panels, B B, of con-
crete.
Doors. In stud-partitions, the open-
ings for doors are framed as in Fig. 1043,
the door-frame being independent of the
studs.
Fig. 1075 represents the elevation and
Fig. 1076 the horizontal section of a
common inside-door. A A are the stiles,
B, C, H, D, the bottom, lock, parting, and
top rail, E the panels, and F the muntin ;
the combination of moldings and offsets
around the door, G, is called the archi-
trave ; in the sec-
tion, a a are the
partition-studs, b b
the door-jambs.
Fig. 1077 rep-
resents the forms
of the parts of a
door, and the way
in which they are
put together.
When the tenons
are to be slipped
into the mortises,
they are covered
with glue, and,
after being closed
up, keys are driv-
en in.
With regard to
the proportions of
internal doors,
they should de-
pend in some de-
gree on the size of
the apartments ;
in a small room a
large door always
FIG. 1073.
FIG. 1072.
FIG. 1074.
ARCHITECTURAL DRAWING.
477
gives it a diminutive appearance, but doors leading from the same room or
passage, which are brought into the same view, should be of uniform height.
The smaller doors which are found on sale are 2 feet 4 inches X 6 feet ; for
water-closets, or very small pantries, they are sometimes made as narrow as 15
inches, but any less height than 6 feet will not afford requisite head-room ;.
2 feet 9 inches X 7 feet, 3 feet X 7 feet 6 inches, or 3 feet 6 inches X 8 feet,
are well-proportioned, six-pan-
eled doors. But the apparent
proportions of a door may be
FIG. 1075.
12
2
FIG. lore.
FIG. 1077.
varied by the omission of the parting-rail, making the door four-paneled,
or narrowed still more by the omission of the lock-rail, making a two-pan-
eled door. Sometimes the muntin is omitted, making but one panel ; but
this, of course, will not add to the appearance of width, but the reverse.
Wide panels are objectionable, as they are apt to shrink from the moldings and
crack. The moldings are generally planted on, and nailed to the stiles and
rails, but sometimes formed on them.
When the width of the door exceeds five feet, it is generally made in two
parts, each part being hung to its side of the frame, or one part hung to the
other, so as to fold back like a shutter ; or the parts may be made to slide back
into pockets or grooves in the partition. The doors may be supported on
wheels, and run on tracks at the floor-level ; or the tracks may be above the
doors, and the doors suspended ; or they may be supported by levers, and be
moved parallel without rollers.
478
ARCHITECTURAL DRAWING.
Figs. 1078, 1079, and 1080 are the elevation, vertical and horizontal sections
of a pair of sliding-doors. There are no knobs, but countersunk pulls to the
FIG. 1078.
FIG. 1079.
FIG. 1080.
doors, that they may be slid entirely within the pockets, with a special handle
in the locks at the edges of the doors for withdrawing them.
ARCHITECTURAL DRAWING.
479
FEET
Figs. 1081 and 1082 are vertical and horizontal sections of the
same doors hung on butts or hinges.
Figs. 1083 and 1084 are the elevation and horizontal section
of an antae-finished outside-door, with the side-lights C 0, and a
top, fan, or transom light B. The
bar A is called a transom, and this term is applied generally to
horizontal bars extending across openings, or even across rooms.
Fig. 1085 is the elevation of an outside folding-door. The plan
(Fig. 1086) shows a vestibule, V, and an interior door. The outer
FIO. losi. doors open, as shown by the arcs, and fold back into the pockets
or recesses, p p, in the wall. This is a very common form of doors
for first class houses in this city. The fan-lights are made semicircular, and
also the head of the upper panels of the door ; these panels in the interior or
vestibule door are of glass.
Windows are usually understood to be glazed apertures. The sashes may be
stationary, but for most positions they are made to open either by sliding verti-
cally, or laterally, or like doors. The first is the common form of window, and
the sashes are generally balanced by weights ; the second, except in a cheap form
in mechanics' shops, are seldom used ; the third, often used for access to bal-
480
ARCHITECTURAL DRAWING.
conies or between
rooms, are called
casements, or
French windows.
Figs. 1087 and
1088 are the outside
elevation and hori-
zontal section of one
half of a common
box-frame, and Fig.
1089 a vertical sec-
tion of the same in
a wooden frame
house. S is the sill
of the sash-frame,
W the frame-sill,
with a wash to dis-
charge the water, B
the bottom rail of
the sash, M the
meeting rails, T the
top rail, H the head
of the sash-frame,
and A the archi-
trave similar to that
around doors. In-
stead of two sills,
S and W, one is
often used, and in-
clined to form the
wash. D is the
common outside
blind. In the sec-
tional plan (Fig.
1094), 0' are the
window-stiles, F the
pulley-stile, w w the
sash -weights, p the
parting strip, and D
D double-fold shut-
ters.
Figs. 1090 and
1091 are the inte-
rior elevation and
vertical section of a
box -frame window
in a masonry wall ;
Fra. 1087.
FEET
FIG. 1088.
FIG. 1089.
ARCHITECTURAL DRAWING.
481
Fig. 1092 is an exterior view of the same window,
and Fig. 1093 a horizontal section.
Unless the windows begin from, or nearly from,
the floor, the point a (Fig. 1089) may be fixed at a
31
FIG. 1090.
FIG. 1091.
482
ARCHITECTURAL DRAWING.
height of about
30 inches above
the floor, and the
top of the win-
dow sufficiently
below the ceiling
to allow space for
the architrave or
other finish above
the window, and
for the cornice of
the room, if there
be any; a little
space between
these adds to the
effect. For com-
mon windows,
the width of the
sash is 4 inches
more than that of
the glass, and the
height 6 inches
more ; thus the
sash of a window
3 lights wide and
4 lights high, of
12"X16" glass,
is 3 feet 4 inches
wide and 5 feet
10 inches high.
In plate -glass
windows more
width is taken
for the stiles and
rails. The usual
sizes of cylinder
glass are 7" X 9"
up to 24" X 36",
but single thick
glass may be had
up to 40" X 60";
double thick,
48"X62". Plate
glass, polished or
rough, may be had
of a size as large
as 14 X 8 feet.
ARCHITECTURAL DRAWING.
483
In Fig. 1087 the blind D is hinged to the hanging stile, and folds within
the opening in the masonry. The slats are movable on pin tenons, and those
of each half, upper and lower, are connected by a central bar, so that they are
moved together, and adjusted at any angle to the light. In Fig. 1093 the
blinds are inside, 4-fold, and folding back into pockets. It is more usual to
make the pockets for the blinds inclined to the window, as in Fig. 1094, giv-
ing to the interior more light, or ampler space for curtains.
Fig. 1095 is the
outside elevation of a
French window or case-
ment.
Fig. 1096 represents
the sectional elevation
777$}
FIG. 1095.
FIG. 1096.
FIG. 1094.
of the same window,
in broken lines, and
on a larger scale ; the
same letters designate
similar parts as in Fig.
1089. A transom-bar
is often framed between the meeting-rails, and in this case the upper sash may
be movable ; in Fig. 1096 it is fixed. An upright, called a mullion, is often
introduced in the center, against which the sash shuts.
For use as doors, the lower sashes should not be less than 5 feet 6 inches
high. It will be seen that in these forms of sash the rails and stiles are wide,
and that for the same aperture the French window admits the least light. The
chief objection to this window lies in the difficulty of keeping out the rain at
the bottom in a driving storm. To obviate this, the small molding d, with
a drip or undercut, is nailed to the bottom rail ; but the more effectual means
is the patent weather-strip, the same as used on outside doors.
Dormer or attic windows are framed and set as in an upright stud-partition.
In all architectural finish moldings are a necessity, the simpler forms of
which are taken from Greek or Roman examples.
Greek and Roman Moldings. The regular Greek moldings are eight in
484
ARCHITECTURAL DRAWING.
number : the Fillet or Band, Torus, Astragal or Bead, Ovolo, Cavetto, Cyma
Recta or Ogee, Cyma Reversa or Talon, and Scotia.
The fillet (, Fig. 1097) is a small rectangular member, on a flat surface,
whose projection is usually made equal to its height.
FIG. 1097.
FIG. 1098.
\J
FIG. 1099.
The torus and astragal are semicircles in form, projecting from vertical
diameters, as in Fig. 1098. The astragal is distinguished from the torus in
the same order by being made smaller. The torus is generally employed in the
bases of columns ; the astragal, in both the base and capital.
The ovolo is a member strong at the extremity, and intended to support.
The Roman ovolo consists of a quadrant or a less portion of a circle (Fig. 1099).
The Greek ovolo is elliptic.
To describe the Greek ovolo (Fig. 1100) : Draw df from the lower end of
the proposed curve, at the required inclination ; draw the vertical g ef to define
the projection, the point e being the extreme point of the curve. Draw e h
parallel to d /, and draw the vertical d h k, such that d h is equal to h k.
Divide e li and ef into the same number of equal parts ; from d draw straight
lines to the points of division in ef, and from k draw lines through- the divis-
ions in e h to meet those others successively. The intersections so found are
points in the curve, which may be traced accordingly.
The cavetto is described like the Roman ovolo by circular arcs, as shown
in Figs. 1101 and 1102. Sometimes it is composed of two circular arcs united
(Fig. 1103) ; set off b e, two thirds of the projection, draw the vertical b d equal
to b e, and on d describe the arc b i. Join e d and produce it to p ; draw i n
perpendicular to e d, set off n o equal to ni, and draw the horizontal line op
meeting ep ; on p describe the arc io to complete the curve.
FIG. 1100.
I
FIG. 1101.
FIG. 1102.
FIG. 1103.
The ogee, or cyma recta (Fig. 1104), is compounded of a concave and a con-
vex surface. Join a and b, the extremities of the curve, and bisect a b at c ; on
a, c, as centers, with the radius a c, describe arcs cutting at d; and on b, c,
describe arcs cutting at e. On d and e, as centers, describe the arcs a c, cb,
composing the molding.
The cyma reversa, or talon (Fig. 1105), is a compound curve, distinguished
from the ogee by having the convex part uppermost.
ARCHITECTURAL DRAWING.
485
If the curve be required to be made quicker, a shorter radius than a c must
"be employed. The projection of the molding n I (Fig. 1104) is usually equal
to the height a n.
To describe the Greek talon: Join the extreme points a, ~b (Fig. 1106) ; bisect
a b at c, and on a c, c b, describe the semicircles b d c and c a. Draw perpendicu-
lars d o, etc., from a number of points in a c, c b, meeting the circumferences ;
FIG. 1104.
FIG. 1106.
I <L
FIG. 1107.
and from the same points set off horizontal lines equal to the respective perpen-
diculars : o n equal to o d, for example. The curve line b n a, traced through
the ends of the lines, will be the contour of the molding.
To describe a scotia : Divide the perpendicular a b (Fig. 1107) into three
equal parts, and with the first, a e, for radius, on e as a center, describe the arc
afh, in the perpendicular c o set off c I equal a e y join e Z, and bisect it by the
perpendicular o d, meeting c o at o, on the center o, with o c for radius, complete
the figure by the arc c h.
These moldings, and combinations of them, are stuck in wood, and are to
be purchased in every variety. Fig. 1108 represents some of the common
forms always to be had, and of suitable sizes.
Stairs consist of the tread or step on which we set our feet, and risers,
upright pieces supporting the treads each tread and riser forms a stair. If
the treads are parallel they are called fliers ; if less at one end than the other,
they are called winders, f and w (Fig.
1115). The top step, or any interme-
diate wide step, for the purpose of rest-
ing, is called a landing. The height
to &/.
FIG. 1109.
FIG. 1110.
from the top of the nearest step to the ceiling above is called the headway.
The rounded edge of the step is called a nosing (a, Fig. 1109) ; if a small hol-
low (b) be glued in the angle of the nosing and riser, it is called a molded
486
ARCHITECTURAL DRAWING.
FIG. 1108.
ARCHITECTURAL DRAWING.
487
nosing. The pieces which support the ends of the stairs are called strings
(Fig. 1110) ; that against the wall the wall-string, the other the outer string.
Besides these strings, pieces of tim-
ber are framed and placed beneath
\ 1 1
j
\ \
j
d^
1
j
I
V
^
_.-/ .'
s
FIG. 1114.
488
ARCHITECTURAL DRAWING.
the fliers, when the stairs are wide (Fig. 1111), called carriages. Sometimes
the strings, instead of being notched out to receive the steps, have the upper
and lower edges parallel, with grooves cut in their inner faces to receive the
ends of the steps and risers (Fig. 1112). These are called housed strings.
Steps and risers are secured in the grooves by wedges covered with glue, and
driven in.
For the rough, strong strings of warehouses the carriages are made of plank,
with grooves to receive plank-treads, and without risers.
Figs. 1113 and 1114 are elevation and plan of a straight run of stairs, both
partly in section. N is the newel-post, n a baluster, li the hand-rail, w the
well. In the section of the floors, cleats are shown nailed to the beams ; on
these short boards are nailed to form a box for the reception of mortar for
deafening.
The opening represented in the plan (which must occur between the outer
strings, if they are not perpendicular over each other) is called the well (W,
Fig. 1115).
The breadth of stairs in general use is from 9 to 12 inches. In the best
staircases, the breadth should never be less than 11 inches, nor more than 15.
The height of the riser should be the more, the less the width of the tread ;
for a 15-inch tread the riser should be 5 inches high ; for 12 inches, 6^ ; for 9
FIG. 1115.
FIG. 1116.
inches, 8. In laying out the plan of stairs, having determined the starting-
point, either at bottom or top, as the case may be, find exactly the height of
the story ; divide this by the height you suppose the riser should be. Thus
(Fig. 1116), if the height of the story and thickness of floor be 9 feet, and we
suppose the riser should be 7 inches high, then 108 inches, divided by 7 = 15f.
ARCHITECTURAL DRAWING.
489
It is clear that there must be an even number of steps, either 16 or 15 ; to
be near the supposed height of the riser, adopt 15, then
yy~ = 7f^ inches, height of riser.
For this particular case, assume the breadth of the step as 10 inches, and
the length at 3 feet, a very usual length, seldom exceeding 4 feet in the best
staircases of private houses. For the plan lay oif the outside of the stairs, two
parallel lines 3 feet apart, and space off from the point of beginning 14 treads
of 10 inches each, and draw the cross-parallel lines.
To construct the elevation, project the lines of the steps in plan, and divide
the height, either on a perpendicular or by an inclined line, into the number
of risers (15), and draw cross-parallels through these points ; or the same
points may be determined by intersection of the projections of the plan with
a single inclined line drawn along the nosing of top and bottom steps. It is
to be observed that the number of treads is always one less than the number
of risers, the reason of which will appear by observing the elevation.
For the framing plan the drawing of the elevation of stairs is in general
necessary, to determine the opening to be framed in the upper floor, to secure
proper headway. Thus (Fig. 1116), the distance between the nearest stair and
the ceiling at a should not be less than 6 feet 6 inches ; a more ample space
improves the look of the stairway ; but if we are confined in our limits, this
will determine the position of one trimmer, the other will be of course at the
top of the stairs. When one flight is placed over another, the space required
for timber and plastering, under the steps, is about 6 inches for ordinary
stairs.
When the stairs are circular, or consist in part of winders and fliers, as in
Fig. 1115, the width of the tread of the winders should be measured on the
FIG. 1117.
490
ARCHITECTURAL DRAWING.
central line. The construction of the elevation is similar to that of the straight
run (Fig. 1116), by dividing the space between the stories by a number of par-
allel lines equal to the number of risers, and intersecting the parallels by pro-
jections from the plan.
The objection to all circular ELEVATION.
stairs of this form, or with a
small well-hole or central shaft,
is that there is too much differ-
ence between the width of the
tread, but a small portion being
of a suitable size. The hand-
somest and easiest stairs are
straight runs, divided into land-
ings, intermediate of the sto-
ries, and either continuing then
in the same line, or turning at
right angles, or making a full
return.
Fig. 1117 is the side eleva-
tion of a stairs with wrought-
iron string and rail. The string
is made of wrought-iron knees,
welded together continuously,
with a flat bottom-bar riveted
across the lower angle of the
knees. The construction is not
very stiff, and is usually sup-
ported by an intermediate round
bar-post.
Where posts can not be put
in, it is better that the bottom
bar should be a carriage or beam
of I or channel-iron, with knees
or cast-iron angle-blocks riveted
on the top of the beam.
It is not unusual to make
housed strings of plate-iron, with
angle-irons riveted on to receive
the treads and risers. If the
plate-iron be wide enough to serve
instead of balusters, it makes a
very strong and stiff carriage.
Figs. 1118 and 1119 are the plan and elevation of a cast-iron stairs, with a
central post or newel (this term is applied also to the first post of any stairs).
The newel-ring, tread, and riser of each step are cast in one piece, and they are
put together by placing one newel-ring upon that below and bolting the outer
extremity of the riser to the tread below.
FIG. 1118.
PLAN.
FIG. 1119.
ARCHITECTURAL DRAWING.
491
Fig. 1120 is a form of cast-iron stairs with a well instead of a newel ; the
step and riser are bolted together by the flanges. It will be seen that one
tread is wider than the others ; this is a landing.
FIG. 1121.
FIG. 1120.
It is at times fashionable to make the newel a prominent feature in the hall,,
often occupying valuable space. It is sufficient that it be large and stiff enough
for a support to the hand-rail.
The top of the hand-rail should, in general, be about 2' 8" to 3' above the
nosing, and should follow the general line of the steps. The angles of the hand-
rail should always be eased off. A hand-rail, affording
assistance in ascending or descending, should not be
wider than the grasp of the hand (Fig. 1121) ; but where,
for architectural effect, a more massive form may be
necessary, it is very convenient, and may be very orna-
mental, to have a sort of double form, that is, a smaller
one planted on top of the larger (Fig. 1122).
To a draughtsman conversant with the principles of
projection already given, it will not be difficult to draw
in the hand-rail of stairs, or to lay off the mold for its
construction. It will follow the line of stair-nosing, and
where there are changes of pitch they are made to con-
nect by curves tangent to these pitches, except where the
landings are square, and newels set at the head of the
landings, the rail is made to bolt into the newel. At the
bottom the rail is curved to the horizontal, when it
comes into or upon top of the newel.
Balusters are of great variety usually turned forms attached to the treads
by dovetails, covered with the returned nosing, or with pin-ends and holes in
FIG. 1122.
492
ARCHITECTURAL DRAWING.
FIG. 1123.
treads and under side of caps. Sometimes (especially in iron-
work) the baluster is set in a bracket from the face of the
string, as in Fig. 1123. These brackets are often very orna-
mental, and the balusters may be cast on the same piece with
the bracket.
Fireplaces. Fireplaces for wood are made with flaring
jambs of the form shown in plan (Fig. 1124) ; the depth from
1 foot to 15 inches, the width of opening in front from 2 feet
6 inches to 4 feet, according to the size of the room to be warmed ; height 2
feet 3 inches to 2 feet 9 inches, the width of back about 8 inches less than in
front ; but at present fireplaces for wood
are seldom used, stoves and grates hav-
ing superseded the fireplace. The space
requisite for the largest grate need not
FIG. 1124.
exceed 2 feet in width by 8 inches in _J
depth. The requisite depth is given by FIG. 1125.
the projection of the grate, and the man-
tel-piece. Ranges require from 4 feet 4 inches to 6 feet 4 inches wide X 12
inches to 20 inches deep ; jambs 8 inches to 12 inches.
Fig. 1125 represents the elevation of a mantel-piece of very usual propor-
tions. The length of the mantel is 5 feet 5 inches, the width at base 4 feet 6
inches, the height of opening 2 feet 7 inches, and width 2 feet 9 inches. A
portion of this opening is covered
by the iron sides or architrave of
the grate, and the actual open space
would not probably exceed 18 inch-
es in width by 2 feet in height. In
brick or stone houses the flues are
FIG. 1126.
formed in the thickness of the wall,
but when distinct they have an out-
side shell of a half-brick or 4 inches,
and sometimes 8" (Fig. 1126) ; the
withs or division-walls always 4".
FIG. 1127.
ARCHITECTURAL DRAWING.
493
FIG. 1128.
The size of house flues is usually 8" X 8", but some are 4" X 8", 4" X 12", ar.d
8" X 12". The flues of different fireplaces should be distinct. Those from,
the lower stories pass up through the jambs of the upper fireplaces, and,
keeping side by side with but 4-inch brick-work between them, are topped out
above the roof, sometimes in a double and often in a single line 16 inches wide
by a breadth required by the number of flues, as in Fig. 1126, or in Fig. 1127.
The latter is an illustration of how far
flues may be diverted from a vertical
line, but it is to be observed that the
construction must be stable, as any set-
tling or cracks not only injures the
draught of the chimney, but impairs the
security of the building against fire.
Changes of direction of flues should
never be abrupt. The back of the fire-
place may be perpendicular through its
whole height, but it is usual to incline
the upper half inwardly toward the
room, making the throat to the flue long and narrow. It is very common to
form the upper 3" to 4" of the inclined back by an iron plate, which can be
turned back or forward to increase or diminish the draught. Fig. 1128 repre-
sents the arrangement of frame and brick arch for the support of the hearth.
The chimney is generally capped with stone, sometimes with tile or cement
pots. As an architectural feature, the chimney is often
^^\^ made to add considerably to the effect of a design.
^\ Roofs. Framed roofs have been illustrated (page 410).
/ \ City roofs are usually flat, and timbered similarly to floors,
r but not so strongly, with a slight pitch to discharge rain-
I fall. Eoofs of country dwellings are usually framed like
FIG. 1129. stud-partitions, with inclined studs somewhat deeper than
if they were vertical, depending on the inclination from
the vertical ; if flat, depth like that of a floor. The theory of the construc-
tion of the gambrel or Mansard roof (Fig.
1129) is a roof with two kinds of pitch ; it is
that of the polygon of rods, and self-sup-
porting ; but, in general, they have central
support from partitions, and their outlines
are much varied by curves in the lower raft-
ers cut from plank.
Fig. 1130 is the plan of a roof as usually
drawn, shaded strongly at the ridges. The
transept roof is hipped at A and B.
Gutters are generally formed in the cor-
nice (Fig. 1131) ; sometimes on the roof
(Fig. 1132), and sometimes by raising a parapet (Fig. 1133) and forming a
valley. The intersection of two roofs forms a valley.
Fig. 1131 represents a form of gutter very common to city buildings, the
FIG. 1130.
494:
ARCHITECTURAL DRAWING.
FlG. 1131.
FIG. 1132.
FIG. 1133.
Toof boarding extending over the gutter ; but it is preferable to make the roof
pitch from both rear and front to the center of the building, and to carry the
leader down in the interior, where it may serve as a soil-pipe for the water-clos-
ets, basins, and baths, affording ventilation in fair weather and a scour in rains.
Fig. 1134 is a gutter of a cottage roof.
Fig. 1135 is the section of a Mansard roof, so called,
showing the side elevation of a dormer-window, with the
gutter below its sill.
FIG. 1134.
FIG. 1135.
It is to be observed that the sheet-metal forming the gutter must extend
well up or back beneath the shingles or felt, or be soldered to the tin of the
roof, to prevent water finding its way into the interior ; and at the sides
flashings of tin must extend on the walls above the roof and into the joints of
the brick.
Plastering. To prevent damp striking through the plastering of outer
walls, and cracks in ceilings, it is usual to fur walls and beams ; that is, to nail
vertical strips of wood to the walls, and across from beam to beam. Furring-
ARCHITECTUKAL DRAWIN<
'
495
strips are from 1-J* to 2" wide, and about J" thick, nailed at distances of 12" or
16" centers (usually the former), adapted to the length of the laths, which are
4 feet long, and about iy X i" = spaces between laths i" to f". The first coat
of mortar is the scratch-coat, which is forced through the interstices between
1
K
*^^
<=
\...
<^
^
5
<
FIG. 1136.
Fro. 1137.
the laths, to make a lock to retain it. This coat is about " tliick. The
next or brown coat is about -J-" thick, and if the last coat is a sand-finish,
it will be less than -J" thick ; while, if the last coat is a hard
finish, its thickness will be almost imperceptible. Figs. 1136
and 1137 are sections of furring and plastering.
The brown coat is usually carried down to the floor. Over this
is nailed the base-board, A (Fig. 1138), for the finish around the
bottom of the walls of the room. Above the base is a molding
forming a part of the base ; above this, there may be a molded
FIG. 1139.
FIG. 1140.
FIG. 1141.
iiliiiil
FIG. 1138.
rail, B, called the chair-rail, or surbase, and between a panel,
termed a dado. The walls of stores are generally ceiled up as
high as the surbase. For the finish of the angle of the wall
and ceiling, it is usual in the better rooms to form a cornice
in plaster. The cornices are moldings of varied forms, with
or without enrichments that is, plaster ornaments. Figs. 1139, 1140, and
1141 are sections of cornices. If the rooms are low, the cornice should ex-
tend but little on the wall, but well out on the ceiling.
Proportions and Distribution of Rooms and Passages. Rooms of dwell-
ing-houses are to be proportioned and arranged according to the necessities of
position and use, the space that can be occupied, the financial means available,
and often to suit the peculiar wishes of owners or occupants. In cities, the
limits of the lot restrict the arrangements to a small ground-space, and require
an increase in the number of stories. Use has established certain forms often
peculiar to different cities, beyond which there is little change ; but in the
country, where there is plenty of ground-space, and where many stories are
496 ARCHITECTURAL DRAWING.
usually injurious to the aesthetic effect, and where there are few canons in
architecture to be observed, there is little limit to the variety of forms and
arrangements of country-houses.
In designing a country-house, where one is not restricted to room, it is often
convenient to mark out the rooms of the desired size on slips of paper, accord-
ing to some scale, then cut them out and arrange them in as convenient an
order as possible, and modify the arrangement by the necessities of construction
and economy. Thus, the more the inclosing surface in proportion to the in-
cluded area, and the greater the number of chimneys and space used for pas-
sages, the greater the cost. The kitchen should be of convenient access to the
dining-room, both should have large and commodious pantries, and all rooms
should have an access from an entry, without being compelled to pass through
other rooms ; this is particularly applicable to the communication of the kitchen
with the front door. Outside doors for common and indiscriminate access-
should not open into important rooms.
As to the size of the different rooms, they must of course depend on the pur-
poses to which they are to be applied, the class of house, and the number of
occupants. The kitchen for the poorer class of houses is also used as an eat-
ing-room, and should therefore be of considerable size to answer both purposes ;
for the richer houses, size is necessary for the convenience of the work. In
New York city houses the average will be found to be about 16 X 20 feet ; for
medium houses in the country they are in general less, say 12 X 16. A back
kitchen, scullery, or laundry, should be attached to the kitchen, and may serve
as a passage-way out.
The Dining or Eating Rooms. The width of dining- tables varies from 3 to
5 feet 6 inches ; the space occupied by the chair and person sitting at the table
is about 18 inches ; the table-space, for comfort, should be not less than 2 feet
for each person at the sides of the table, and considerable more at the head and
foot ; hence the space that will be necessary for the family and its visitors at
the table may be calculated. Allow a further space of 2 feet at each side for
passages, and some 3 to 5 at the head for the extra tables or chairs, for the
minimum of space required ; but, if possible, do not confine the dining-room
to meager limits, unless for very small families ; let not the parties be lost in
the extent of space, nor let them appear crowded.
The show-room parlors, if there are any intended for such in the house,
should be made according to the rules given below, not square, but the length
about once and a half the width ; if much longer than this, break up the walls
by transoms or projections. As to the particular dimensions, no rules can be
given ; they must depend on every person's taste and means ; 20 X 16 may be
considered a fair medium size for a regular living-room parlor, not a drawing-
room. The same size will answer very well for a sleeping-room. The usual
width of single beds is 2 feet 8 inches ; of three-quarter, 3 feet 6 inches ; of
whole, 4 feet 6 inches ; the length, 6 feet 6 inches ; and as the other furniture
may be made to consist of but very few pieces, if adequate means of ventilation
are provided, it is easy to see into how small quarters persons may be thrust.
The bed should not stand too near the fire, nor between two windows ; its most
convenient position is head against an interior wall, with a space on each side
ARCHITECTURAL DRAWING. 497
of at least 2 feet. To the important bedrooms of first-class houses, dressing-
rooms should be attached, and, if there is water and sewer service, fitted with
set bowls and baths and water-closets. If possible, there should be windows
opening to the outer air, but always with flue-ventilation.
Pantries. Closets for crockery should not be less than 14 inches in depth
in the clear ; for the hanging up of clothes, not less than 18 inches, and should
be attached to every bedroom. For medium houses, the closets of large sleep-
ing-rooms should be' at least 3 feet wide, with hanging-room, and drawers and
shelves. There should also be blanket-closets, for the storing of blankets and
linen ; these should be accessible from the entries, and may be in the attic.
Store-closets should also be arranged for groceries and sweetmeats.
Passages. Front entries are usually 6 feet wide in the clear ; common pas-
sage-ways, 3 feet ; these are what are required, but ample passages give an
important effect to the appearance of the house. The width of principal
stairs should be not less than 3 feet, and all first-class houses, especially those
not provided with water-closets and slop-sinks on the chamber-floor, should
have two pairs of stairs, a front and a back pair ; the back stairs need not
necessarily be over 2 feet 6 inches in width.
The Height, of Stories. It is usual to make the height of all the rooms on
each floor equal ; it can be avoided by furring down, or by the breaking up of
the stories, by the introduction of a mezzonine or intermediate story over the
smaller rooms. Both remedies are objectionable.
The average height of the stories for common city dwellings is : Cellar, 6
feet 6 inches ; common basement, 8 to 9 feet ; English basement, 9 to 10 feet ;
principal story, 12 to 15 feet ; first chamber floor, 10 to 12 feet ; other chamber-
floors, 8 to 10 feet all in the clear. For country-houses, the smaller of the
dimensions are more commonly used. Attic stories are sometimes but a trifle
over 6 feet in height, but are, of course, objectionable.
Privies, Water- Closets, and Out- Houses. The size of privies must depend
greatly on the uses of the building to which they are to be attached, its position,
and the character of its occupants. Allowing nothing for evaporation and ab-
sorption, the entire space necessary for the excrementitious deposits of each
individual, on an average, will be about seven cubic feet for six months, of
which three quarters is fluid. In the country, vaults are usually constructed of
dry rubble-stone, and the fluid matters are expected to be filtered through the
earth, the same as in cesspool-waste ; but great care must be taken that they
neither vitiate the water-supply nor the air of the house. A brick and cement
vault, air and water tight, with a ventilating-pipe into a hot chimney-flue, is
the best preventive, and may even be built within the house. In all other cases
there should be free air-space between the house and privy. In the city, where
there is adequate water-supply and sewerage, the water-closet should be adopted,
except in houses occupied by many ignorant and irresponsible tenants, who
throw extraneous matters into the hoppers, and obstruct the sewer-pipes. In
these, tight privy-vaults, with trapped sewer connections, and with all the
house-waste and roof-water discharging in to them, are the easiest kept in order.
The water-closet, or privy, with a single seat, should occupy a space not less
than 4'x 2' 6". The rise of seat should be about 17" high ; and the hole egg-
32
4:98 ARCHITECTURAL DRAWING.
shaped, 11" X 8". The earth-closet, when properly taken care of, is an ex-
tremely useful appendage to a country-house, and the space requisite for it is
the same as that of a water-closet. It is the most common practice to place
the water-closet in the bath-room. A common bath-tub will occupy a floor-
space of 6' X 2', and 18" deep ; the French tub, so called, is much shorter,
often not over 4' 6", but deeper. The water-closet seat will occupy about 2
feet in width X 20 inches in depth.
The forms of modern water appliances, and the means to get rid of house-
waste, will be illustrated hereafter, under the heads of Ventilation and
Plumbing.
For Wood or Coal Sheds or Bins. In estimating the size of these accesso-
ries, it may only be necessary to state that a cord of wood contains 128 cubic
feet, and a ton of coal occupies a space of about 40 cubic feet.
On the Size and Proportion of Rooms in general. "Proportion and or-
nament," according to Ferguson, "are the two most important resources at
the command of the architect, the former enabling him to construct ornament-
ally, the latter to ornament his construction." A proportion to be good must
be modified by every varying exigence of a design ; it is of course impossible to
lay down any general rules which shall hold good in all cases ; but a few of its
principles are obvious enough. To take first the simplest form of the propo-
sition, let us suppose a room built, which shall be an exact cube of say 20 feet
each way such a proportion must be bad and inartistic ; and, besides, the
height is too great for the other dimensions. As a general rule, a square in
plan is least pleasing. It is always better that one side should be longer than
the other, so as to give a little variety to the design. Once and a half the
width has been often recommended, and with every increase of length an in-
crease of height is not only allowable, but indispensable. Some such rule as
the following meets most cases : " The height of the room ought to be equal to
half its width plus the square root of its length " ; but if the height exceed the
width the effect is to make the room look narrow. Again, by increasing the
length we diminish, apparently, the other two dimensions. This, however, is
merely speaking of plain rooms with plain walls ; it is evident that it will be
impossible, in any house, to construct all the rooms and passages to conform to
any one rule of proportion, nor is it necessary, for in many rooms it would not
add to their convenience, which is often the most desirable end ; and, if re-
quired, the unpleasing dimensions may be counteracted by the art of the archi-
tect, for it is easy to increase the apparent height by strongly marked vertical
lines, or bring it down by horizontal ones. Thus, if the walls of two rooms of
the same dimensions be covered with the same strongly marked striped paper,
in one case the stripes being vertical and in the other horizontal, the apparent
dimensions will be altered very considerably. So also a deep, bold cornice
diminishes the apparent height of a room. If the room is too long for its other
dimensions, this can be remedied by breaks in the walls, by the introduction
of pilasters, etc. So also, as to the external dimensions of a wall, if the length
is too great it is to be remedied by projections, or by breaking up the lengths
into divisions.
Understanding the general necessities of a dwelling, the proportions of
ARCHITECTURAL DRAWING.
500
ARCHITECTURAL DRAWING.
rooms, forms of construction, and space to be occupied, the draughtsman is
prepared to undertake designing, and for this purpose cross-section paper will
be found of very great use. Taking the side of a small square as a unit
one foot, for instance he can readily pencil in rooms and passages, and alter
and modify at pleasure.
Figs. 1142 to 1149 are illustrations of this form of designing, making rou.sh
sketches. It is to be observed that partitions are to be as much as possible
one over the other, and the posts or walls arranged in the cellar, for the sup-
port of these lines of partitions. For the sketch, it is sufficient to make door
and window openings 3 feet, unless for some particular purpose bow or mul-
lioned windows are required. In arranging the stairs, the clear space is roughly
about 12 feet, and from the foot of the stairs to the top H times the height
of the story from the top of the floor to the top of the floor, counting the
square landings as 1 foot each. In the sketch, the stair-head room to be pro-
vided for is that for the cellar-stairs, that lead from a small entry between
the kitchen and main hall. Chimney-breasts may be sketched as 4' X 2'.
When the sketch is transferred to drawing-paper, the spaces are then to be
more exactly arranged and plotted to a scale.
Figs. 1150 to 1165 represent plans of familiar forms of houses, all drawn to
the scale of 32 feet to the inch, as illustrations to the student, and as examples
to be copied on a larger scale. The same letters of reference are used on all
the plans, for rooms intended for similar purposes. Thus, K K designate
kitchens, cooking-rooms, or laundries ; D D eating-rooms ; S S sleeping-rooms ;
P P drawing-rooms, parlors, or libraries ; p p pantries, china or store closets,
or clothes-presses ; c c water-closets and bath-rooms.
FIG. 1150.
FT"" 1 ' -" '
S
1
S
'rff\
1
="3>:
1 "' M '
FIG. 1151.
FIG. 1152.
Figs. 1150, 1151, and 1153 are first-story plans of
square houses, or of square outline. Fig. 1152 is the sec-
ond story of Fig. 1151. This form of house has the great-
est interior accommodations for the outside cover, and,
although not picturesque in its elevation, is a very con-
venient and economical structure. The kitchen (Fig.
1153) is in the basement, and the connection with the
dining-room is by a dumb-waiter in the pantry (p). In
Fig. 1154 the plan is the same as in Fig. 1153, but the
kitchen (k) is in an L attached to the house ; there is a small opening be-
tween the pantry (p') and kitchen, through which dishes are passed to and
from the dining-room.
FIG. 1153.
ARCHITECTURAL DRAWING.
501
Fig. 1155 is the plan of a very small but convenient floor, of prettier outline
than the square ; v is a portico or veranda. No chimney is shown in the sleep-
ing-room S ; there should be one either against the stairs or the back wall.
Figs. 1156 and 1157 are first-story plans of houses still more extensive.
All of the above are adapted to the country, dependent on lights on all sides,
and ample spaces.
FIG. 1155.
P K
K
FIG. 1154.
FIG. 1156.
FIG. 1157.
In the cities, houses are mostly confined to one form in their general out-
line a rectangle. Figs. 1158 and 1162 may be taken as the usual type of New
York city houses. Figs. 1158, 1159, and 1160 are the basement, first and
second floor plans of a three-rooms-deep, high-stoop house, as the first floor is
L
JJ
D
FIG. 1158.
FIG. 1159.
FIG. 1160.
FTG. 1161.
reached by an outside flight of steps about 6 feet high. There is usually a
cellar beneath the basement, but in some cases there are front vaults, entered
beneath the steps to the front door ; the entrance to the basement itself is also
beneath the steps. The front room of the basement may be used as an eating-
502
ARCHITECTURAL DRAWING.
room, for the servants' sleeping-room, billiards, or library. The usual dining-
room is on the first floor ; a dumb-waiter being placed in the butler's pantry, p,
for convenience in transporting dishes to and from the kitchen. The objection
to three-rooms-deep houses is that the central room is too dark, being lighted
by sash folding-doors between that and the front or rear rooms, or both. Fig.
1161 is a modification to avoid this objection, the dining-room, or tea-room, as
it is generally called, being built as an L, so that there is at least one window
in the central room opening directly out-doors. This was an old fashion here,
and has lately been revived.
Figs. 1162 to 1165 are plans of the several floors of an English basement-
house, so called, distinguished from the former in that the principal floor is up
one flight of stairs. The first story or basement is but one or two steps above
the street, and contains the dining-room, with its butler's pantry and dumb-
FIG. 1162.
FIG. 1163.
FIG. 1164.
S
FIG. 1165.
waiter, a small sitting-room, with, in some cases, a small bedroom in the space
in the rear of it. The kitchen is situated beneath the dining-room, in the sub-
basement. The grade of the yard is in general some few steps above the floor
of the kitchen. Vaults for coal and provisions are excavated either beneath
the pavement in front or beneath the yard. The advantages of this form of
house are the small reception-room on the first floor, which in small families
and in the winter months is the most frequently occupied as a sitting-room of
any in the house ; the spaciousness of its dining-room and parlors in propor-
tion to the width of the house, which is often but 16 feet 8 inches in width, or
three houses to two lots, and not unfrequently of even a less width. The ob-
jections to the house are the stairs, which it is necessary to traverse in passing
from the dining-rooms or kitchen to the sleeping-rooms, but this objection
would, of course, lie against any house of narrow dimensions, where floor-space
is supplied by height.
In New York, outside access to the kitchen is from the front, as there is no
back street or alley. In Philadelphia, where the lots are deeper, and there is
a street in the rear, the kitchen is usually in a rear L, on the level of the first
floor, with the dining-room above it on a mezzonine or half-story between the
first and second floors.
ARCHITECTURAL DRAWING.
503
Figs. 1166 to 1171 are plans and elevations of a country-house in the Flem-
ish or Queen Anne style.
PLAN OF FIRST FLOOE.
B
504
ARCHITECTURAL DRAWING.
PLAN OF SECOND FLOOK.
FIG. 1167.
ARCHITECTURAL DRAWING.
505
FRAMING-PLAN OF FIEST FLOOR.
506
ARCHITECTURAL DRAWING.
ARCHITECTURAL DRAWING.
ELEVATION OF CHIMNEY OF DINING-EOOM. SECTION.
507
I i i
J FEET
508
ARCHITECTURAL DRAWING.
AROHITECTUEAL DRAWING.
509
Figs. 1172 to 1177 are plans and elevations of country residences, from
Downing's "Cottage Houses."
ELEVATION OF A TIMBER COTTAGE, BY GERVASE WHEELER.
FIG. 1172,
510
ARCHITECTURAL DRAWING.
The construction of Fig. 1172, though simple, is somewhat peculiar. It is
framed in such a manner that the construction is manifest on the exterior.
At the corners are heavy posts, roughly dressed and chamfered, and into them
are mortised horizontal ties, immediately under the springing of the roof ;
FIG. 1173.
FIG. 1174.
ENGLISH EURAL STYLE.
FIG. 1175.
ARCHITECTURAL DRAWING
511
these, with the posts and the studs, and the framing of the roof, show exter-
nally. Internally are nailed horizontal braces at equal distances apart, stop-
ping on the posts and studs of the frame, and across these the furring and
lathing cross diagonally in different directions. On these horizontal braces,
the sheathing, composed of plank placed in a perpendicular position, is sup-
ported and retained in its place by battens two and a half inches thick, and
RURAL GOTHIC STYLE.'
FIG. 1176.
512
ARCHITECTURAL DRAWING.
ITALIAN VILLA, BY UPJOHN.
FIG. 1177.
ARCHITECTURAL D
513
made with a broad shoulder. These battens are pinned to the horizonl
braces, confining the planks, but leaving spaces for shrinking and swellfflfg,
thus preventing the necessity of a single nail being driven through the planks.
Fig. 1173 represents the batten, B, and the mode of framing.
Fig. 1174 represents the usual form of vertical boarding, which is less ex-
pensive than the first illustration, and, in general, will be found sufficiently
secured for the class of buildings to which it is applied.
Fig. 1178 represents the front elevation of a high-stoop house of T. Thomas
design, New York city.
To accommodate the poor and people of small means in all cities, it was,
and to some extent still is, the custom to divide houses which were intended
for single occupation into small apartments for many families, or to let rooms
singly for this purpose. This was found to be objectionable to both occupants
and owners, and houses have been constructed especially for the poorer classes.
Virtually, they are now nearly all apartment-houses, each family having dis-
tinct rooms or suites to itself. But the term tenement-houses is applied to
the cheaper kind of apartments, occupied by the poorer class, and situated
in the least expensive localities. The common form of tenement-house con-
sists of two buildings, one in the front and one in the rear of the lot, with an
outer or air space between. A hall leads through the first story to the central
area ; on each side of this hall there may be small stores and apartments.
Stairs from the hall lead to the apartments above. The 25 feet is divided in
two, making two living-rooms on each front ; these are the only rooms opening
directly into the outer air. Bedrooms are attached to each of these rooms,
but take their light and air from the staircases, or small light-wells. In the
rear houses there are two tenements to each story ; they take their light and
air from the central and back areas. Water-closets or privies are in the central
area. These tenements are mostly occupied by work-people, largely of foreign
birth, dependent directly on small wages. But there is a large class, of limited
means, to whom these accommodations are insufficient ; parties who can not
well afford an entire house, but still wish for the privacy of one. Within the
limits of a lot 25' X 100' it has been found difficult to secure all the necessaries
of light and ventilation, with the number of suites of apartments adapted to
the means of the occupants, and satisfactory as an investment to the owners.
Fig. 1179 is a plan of one of the best of these designs. It provides for
83
514
ARCHITECTURAL DRAWING.
ARCHITECTURAL DRAWING.
515
four families on each story, although it will be observed by the plan of the
stairs that the front and rear tenements are not on the same flat ; they are
separated by the half flight of stairs. By means of the cross-shaped court be-
PLAN.
FIG. 1180.
tween the adjacent houses, every room, including the bath-room, has a window
to the open air. This is the most commendable feature of the plan. It is
516 ARCHITECTURAL DRAWING.
remarkable, also, however, for providing more conveniences than have been
customary in dwellings of this class, as, for instance, a small bath-tub as well
as a water-closet for each family, and two wash-tubs as well as a sink ; also,
a dumb-waiter (common to two families) for bringing up fuel, provisions, etc.
The large rooms have recesses for beds, which provide for an extra bedroom,
while detracting but little from their value as parlors, as the recess may be cur-
tained off in the daytime, or the bed turned up. The dimensions of the rooms,
as marked on the plans, are the average length and breadth. These suites are
much too restricted for a very large class, but apartment-houses somewhat on
this model are constructed in desirable localities, where the accommodations
and conveniences are equal to those of any private house, and not bounded by
the limits of a single lot nor single story, many unsurpassed in luxury of finish
and appointments.
The larger apartment-houses are often designated as French flats, or flats.
The building should be of fire-resisting construction. The suites are invariably
supplied with water, gas, and steam heat ; some few have been lighted by elec-
tric light.
Fig. 1180 is an illustration of a "flat" situated on the corner of a street,
and one suite takes its light exteriorly from the streets while the other depends
in a measure on the court. Resistance to fire, protection from vermin, and
privacy, have been secured by the absence of interior light-wells connecting
stories, solid timbering without furring or framing spaces. Kitchens, in the
figure, are attached to the suites ; the laundries are in the upper story. Many
flats are without kitchens or laundries, and meals are furnished either from
without or from restaurants in the building. It then corresponds very nearly
to a hotel without transient custom, with ample and separate suites. It
would seem that boarding-houses might be built on such plans less extensive
in their arrangements and adapted to small families of moderate means ; but
boarding-houses are almost invariably private houses, but little modified for the
more public use.
Stores and Warehouses. Fig. 1181 is the front elevation of a common
type of New York city store, occupying a single lot of 25 ^!eet in width. It
will be observed that there are two stories beneath the level of the sidewalk,
the basement and sub-cellar, and this construction still obtains largely ; but
deep basements are considered preferable by some, with extra stories at the
top rather than in the cellar. Fig. 1182 is a section of the front wall, showing
heights of stories, which of late years have been increased over former practice,
say to 16' for the first story, 13' for the second, and 12' and 11' for others, the
light for the interior being taken almost universally from the front and rear,
and skylights done away with.
Fig. 1183 is a plan of the first-story floor, with basement in front dotted
in ; five feet of this space, or that usually allotted for areas, is covered with
illuminating tile (Fig. 1184), that is, small glass lenses, set in iron frames, the
whole water-tight. In the extreme rear there is a small area, A, open to the
air, of about 5 feet, for light and air to the basement and cellar. The offices of
the first story are situated at B, over which there is usually a curved lean-to of
illuminating tile. The main wall above this story is on the line a I plain
ARCHITECTURAL DRAWING.
517
SIDEWALK.
FIG. 1181.
FIG. 1182.
518
ARCHITECTURAL DRAWING.
brick with iron shutters. When shutters are used to close the first-story
front they are mostly rolling shutters of sheet-steel. The hoist-way to the up-
FIG. 1183.
per stories is at c, a position somewhat objectionable as interfering with the use
of the stairs, when a common hoist-wheel is used ; but if it is a power-hoist,
then it is put close to the wall, guarded by a rail, with a passage round to the
FIG. 1184.
stairs. In 50 feet front stores the hoist is put on the opposite corner from the
stairs, as at D, but this cuts off considerable light from the first-story front. In
some the arrangement is as in Fig. 1185, in which the hoists c c are in the rear of
. L_
FIG. 1185.
ARCHITECTURAL DRAWING.
519
the stairs. The arrangement for offices in the rear of the first story is in a T,
with spaces at the sides for the ventilation and light of the lower stories. It
FIG. 1186.
will he observed that there is no central door, as in the elevation (Fig. 1181),
which last most usually obtains for wholesale stores. For retail stores, there
520
ARCHITECTURAL DRAWING.
are usually four openings in the 25 feet, as shown in the double stores (Fig.
1186), a design of J. B. Snook.
When lots are only 100 feet in depth, 85 feet can be utilized by the building
with sufficient light from the ends, but very often the stores run through from
street to street, or 200 feet. Formerly the central portion was lighted by sky-
ARCHITECTURAL DRAWING. 521
lights, but this was found very objectionable, and it is now usual to leave an
open-air shaft on one side, inclosed by brick walls, and the windows protected
by iron shutters. The space should be 30 to 40 feet long and 6 feet wide,
which may be covered in the first story with glass. If this recess is on the
side occupied by the staircases, it does not detract from the inside finish of the
stores.
Hoists now in large stores are power-hoists that is, worked by either
steam or water. The platform of a freight-hoist is usually 5 feet square ; for
passenger-hoists, in wholesale stores, somewhat less 4' X 5'. For the raising
of goods from the basement or sub-cellar to the sidewalk there is a hatch in
the front light platform, opposite some window, and the space is like that
of freight-hoists, 5' x 5' ; these may be power or hand hoists. For the de-
livery of goods into these stores there is often a slide or incline, iron-plated,
ending at the bottom with an easy curvo to the horizontal, down which boxes
and bales are slid.
Fig. 1187 is the elevation of an iron-front store 100 feet in width, among
the earliest built in ^N'ew York city, and in its effect is as satisfactory as any
since constructed.
Fig. 1188 is a perspective view of a machine and blacksmith shop, built by
the author many years since . It was built for a purpose, and to express the
purpose constructionally and economically. As regards convenience and
strength, it was found to be, on occupation, all that could be wished. Some
allowance should be made for absence of color in the sketch, which con-
tributed much to architectural effect. Posts, lintels, window-frames, sashes,
and ornamental letters, were of iron, and painted a very deep green ; the
structure was of brick, with sills and bands of rubbed Ulster bluestone, roof
of Welsh slate. The building occupied one corner of Greene and Houston
Streets, in this city, but was burned, and can not, therefore, be referred
to practically. The chimneys shown in front, although not dummies, were
never used. Power and heat were supplied by steam-boilers in the front vault,
with a long, slightly inclined flue leading to a chimney at the center of the
side blank wall. On each side of this chimney, and separated by a thin
with, there were flues. Forges occupied all the exterior walls of the base-
ment, front and side areas, and the draught was upward and then down into
the nearly horizontal flues connected with the central flues, and the draught
was invariably good. Care was taken that all angles, horizontal and vertical,
should be rounded.
School- Houses. Figs. 1189 and 1190 are an elevation and plan of a country
district school-house, with seats for forty-eight scholars. There are two en-
trances, one for each sex, with ample accommodations of entry or lobby-room
for the hanging up of hats, bonnets, and cloaks. A side door leads from each
entry into distinct yards, and an inside door opens into the school-room. The
desk, T, of the teacher, is central between the doors, on a platform, P, raised
some 6" or 8" above the floor. In the rear of the teacher's desk is a closet or
small room, for the use of the teacher. The seats are arranged two to each
desk, with two alleys of 18" and a central one of 2'. The passages around the
room are 3'.
522
ARCHITECTURAL DRAWING.
ARCHITECTURAL DRAWING.
523
H
l
FIG. 1190.
524
ARCHITECTURAL DRAWING.
FIG. 1192.
Figs. 1191 and 1192 are the elevation in perspective and plan of an English
country school-house, introduced as suggestive whether a one-story plan might
not be better suited, and of more beautiful effect in our own country towns,
CPLPCP
where there is plenty of ground space, than
many stories.
On the Requirements of a School- House. Every scholar snoma have room
enough to sit at ease, his seat should be of easy access, so that he may go to
and fro, or be approached by the teacher without dis-
turbing any one else. The seat and desk should be
properly proportioned to each other and to the size of
the scholar for whom it is intended. The seats, as "1 I I I I |j
furnished by the different makers of school furniture,
vary from 9" to 14" in height ; and the benches from
17" to 28" ; measuring on the side next the scholar.
The average width of the desk is about 18", and it is
formed with a slope of from 1" to 2-J", with a small
horizontal piece of from 2" to 3" at top. There is a
shelf beneath for books, but it should not come within
about 3" of the front. The width of the seat varies
from 10" to 14", with a sloping back, like that of a chair ;
it should, in fact, be a comfortable chair. It will be observed that, in the figure,
two scholars occupy one bench. Fig. 1193 represents another arrangement, in
FIG. 1193.
SCHOOL ROOM
12. x BO.
FIG. 1195.
526
ARCHITECTURAL DRAWING.
it
ARCHITECTURAL DRAWING.
527
which each scholar has a distinct bench ; this is more desirable, but not quite so
economical in room. In primary schools, desks are not necessary ; and in many
of the intermediate schools the seat of one bench is formed against the back of
the next bench ; but seats distinct are preferable. The teacher's seat is inva-
riably on a raised platform, and had better be against a dead wall than where
there are windows. Blackboards and maps should be placed along the walls.
Care should be taken in the warming and ventilation ; warm air should be in-
troduced in proportion to the number of scholars, and ventiducts should be
formed to carry off the impure air.
In cities and large towns it is almost indispensable to build school-houses
many stories in height, dividing the rooms in each story according to the neces-
sities of their occupancy. The management of schools differs in different
localities. This will be seen in the illustrations given below, showing the ar-
rangements of school-houses in the city of New York and of Cleveland, Ohio.
Fig. 1194 is an elevation in perspective of one of the largest of the New York
city schools, showing the yards around it. Fig. 1195 is the plan of the gram-
FIQ. 1196.
mar-department floors of this house ; and Fig. 1196 the plan of the same floors
of another house of a different outline.
Figs. 1197 to 1200 are plans of school -houses, built at Cleveland, Ohio, a
type inaugurated under the supervision of the then superintendent, Mr. A. J.
Rickoff. Figs. 1197, 1198, and 1199 are plans of the High-School house. Fig.
528
ARCHITECTURAL DRAWING.
ARCHITECTURAL DRAWING.
529
n n
n D DiJDJjn
D D nib; b n a
a D DifDa a
D D Djiajja D a
a a pijaiiq a a
a a a a a a a
a a D a a a a
a a a n a
a a n a a a a
a a a a a a a
n D D D n
n-i i i i i i FEET
34
530 ARCHITECTURAL DRAWING.
1197 is the plan of the third story ; Figs. 1198 and 1199 of those portions of
the second and first stories which differ from that of the third. There is a
rear vestibule in the first story to correspond with the one in front, shown in
the figure. In the whole building there are 14 session-rooms, each 37' X 30' x
16' ; each having its connecting cloak-room ; one general assembly-room, 94' X
56' X 38' high, with a seating capacity for at least 1,000 persons ; one lecture-
room, with seats for 100, with an apparatus-room ; one room for drawing, 30' X
55', with a room for models, drawing-boards, etc. ; two rooms for the principal
and reception-room ; five rooms for library and recitation-rooms.
Fig. 1200, a plan of one half of one story of the Walton Avenue School, on
a larger scale, explains more fully the arrangement of seats and the ventilation.
Four ventilating educts, of 8 square feet of section each, may be heated to any
required temperature for the purposes of circulation by four upright 2" steam-
pipes ; six ducts of 1 square foot section lead from different points in the floor
of each session-room (as shown in dotted lines in the figure) into the ventilating
educts. There are besides other registers opening directly into the educts. The
building is heated by steam coils or radiators placed under the windows of the
rooms, with provision for the admission of fresh air under the stone sills behind
the radiators. It will be observed that the main light of every room is admitted
at the left hand of the pupil, so that in writing the shadow of the hand does
not fall on the space to be written on. There are none of the cross-lights
that so seriously impair the vision. The wall facing the pupil and behind the
teacher is unbroken by windows, aifording large and convenient spaces for black-
boards.
Churches, Theatres, Lecture- Rooms, Music and Legislative Halls. To the
proper construction of rooms or edifices adapted for these purposes some knowl-
edge of the general principles of acoustics, and their practical application, is
necessary. In the case of lecture-rooms and churches, the positions of the
speaker and the audience are fixed ; in theatres, one portion of the inclosed
space is devoted to numerous speakers and the other to the audience ; in legis-
lative halls, the speakers are scattered over the greater part of the space, and
also form the audience.
The transmission of sound is by vibrations, illustrated by the waves formed
by a stone thrown into still water ; but direction may be given to sound, so that
the transmission is not equally strong in every direc-
., - - ^ tion; thus, Saunders found that a person reading at the
center of a circle of 100 feet in diameter, in an open
meadow, was heard most distinctly in front, not as well
at the sides, but scarcely at all behind. Fig. 1201
shows the extreme distance every way at which the voice
could be distinctly heard : 92 feet in front, 75 feet on
each side, and 31 feet in the rear. The waves of sound
are subject to the same laws as those of light, the angles
FIG. 12Q1. O f reflection are equal to those of incidence ; therefore,
in every inclosed space there are reflected sounds, more
or less distinct, according to the position of the hearer, and to the form and
condition of the surfaces against which the waves of sound impinge. Thus,
ARCHITECTURAL DRAWING.
531
of all the sounds entering a parabolic sphere, the reflected sounds are collected
at the focus. Solid bodies reflect sound, but draperies absorb it. As, in all
rooms, the audience can never be concentrated at focal points, nor is it pos-
sible in any construction to make calculation for all positions, it is in general
best to depend on nothing but the direct force of the voice, and not to con-
struct larger than can be heard directly without aids from reflected sounds.
There is great difference in the strength of voice of different speakers ; the
limits as given in the figure are for ordinary reading in an open space. In in-
closed spaces, owing to the reflected sounds or some other cause, there are cer-
tain pitches or keys peculiar to every room, and to speak with ease the speaker
must adapt his tone to those keys. The larger the room, the slower and more
distinct should be the articulation.
It has been observed that the direction of the sound influences the extent to
which it may be heard. The direction of the currents of air through which
the sound passes affects the transmission of the sound, and this may be made
useful when the rooms are heated by hot air, by introducing the air near the
speaker, and placing the ventilators or educts at the outside of the rooms, and
by placing their apertures rather nearer the bottom of the room than at the
top. It would seem much better and easier to make a current of air a vehicle
of sound rather than depend on reflection.
On the Space occupied by Seats in general. A convenient arm-chair occu-
pies about 20" X 20*, the seat itself being about 18" in depth, and the slope of the
back 2" ; 18" more affords ample
space for passage in front of the
sitter. In churches the seats are
arranged by pews or stalls ; the
width of each pew in general being
about 2' 10". In the arrangement
of seats at the Academy of Music
the bottom turns up (Figs. 1202
and 1203), and 29" only is allowed
for both seat and passage-way, and
18" for the width of seat, which
may be taken as the average allow-
ance in width to each sitter in
comfortable public rooms. In lec-
ture-rooms, benches and settees are often used, the space there occupied by
seat and passage being about 2' 6".
In the earlier churches, ceremonies and rites formed a very large part of the
worship, the sight was rather appealed to than the hearing, and for this pur-
pose churches were constructed of immense size, and with all the appliances of
ornament and construction, with pillars, vaults, groins, and traceried windows.
In the churches of this country, the great controlling principle in the construc-
tion of a church is its adaptation to the comfortable hearing and seeing the
preacher. In this view alone, the church is but a lecture-room ; but since even
the character of the building may tend to devotional feelings in the audience,
.and since certain styles and forms of architecture have long been used for church
FIG. 1202.
FIG. 1203.
532
ARCHITECTURAL DRAWING.
edifices, and seem particularly adapted for this purpose, it has been the custom
to follow these time-honored examples, adapting them to the modern require-
ments of church worship.
Fig. 1205 is a plan of an ancient basilicon or Romanesque church. Fig.
1204 is a sectional elevation of the same. Fig. 1206 is a plan of a Gothic
church, in which C is the chancel, usually at the eastern extremity, T T the
transept, and N the nave. In general elevation the Gothic and Romanesque
agree : a high central nave and low side aisles. In the later Romanesque the
transept is also added.
FIG. 1204.
FIG. 1205.
The basilicas aggregated within themselves all the offices of the Romish
church. The circular end or apse, and the raised platform, or dais, in front of
it, was appropriated entirely to the clergy ; beneath was the crypt or confes-
sional, where were placed the bodies of the saints and martyrs, and pulpits were
placed in the nave, from which the services were said or sung by the inferior
order of clergy.
The plan (Fig. 1206) is that of the original Latin cross, the eastern limb
or chancel being the shortest, and the nave the longest. Sometimes the eastern
limb was made equal to that of the transept, sometimes even longer, but never
to exceed that of the nave. In the Greek cross all the limbs are equal. In
most of the French Gothic churches the eastern end is made semicircular, often
inclosed by three or more apsidal chapels, that is, semi-cylinders, surmounted
by semi-domes.
The Byzantine church consisted internally of a large square or rectan-
gular chamber, surmounted in the center by a dome, which rested upon
massive piers ; an apse was formed at the eastern end. Circular churches
were built in the earlier ages for baptisteries, and for the tombs of saints and
emperors.
The Greek, Roman, and English churches conform in their cathedrals and
larger edifices nearly to the Romanesque or Gothic models. But as the general
requirements for church services now are those of a lecture-room comfortable
seats, convenient for hearing and seeing the preacher, with adequate means of
heating and ventilation, for which the older forms are not suited modern
churches are constructed adapted to these purposes, and, in cities, to the size
and form of the lots, with some ecclesiastical accessories of towers and steeples :
windows and doors and interior finish.
AECHITEOTURAL DRAWING.
533
534
ARCHITECTURAL DRAWING.
Figs. 1207 and 1208 are the elevation and plan of a London Wesleyan
chapel characteristic of the above.
FIG. 1209.
Figs. 1209 and 1210 are the elevations and plan of the English church at
the Hague, where aesthetic effect has been more studied than in the above ex-
ample, with less economy in the occupancy of the lot.
ARCHITECTURAL DEAWI
The length of pews is various, being generally of
small or large families, say from 7' 6" to 11' 6", IS"
ter. In arrangement it is always considered desirable
#sjzes, adapted to either
allowed for each sit-
there should be a
FIG. 1210.
central aisle, and if but four rows of pews, two aisles against the wall ; if six
rows, one row on each side will be wall-pews. Formerly it was the universal
practice to construct pews with doors, but of late it is more customary to omit
the doors, making the pews open stalls.
Few churches are now without an organ ; its dimensions should of course
depend on the size of the church. In form it may be adapted somewhat to
the place which may be appropriated to it either in a gallery over the main
entrance, or at the side of the chancel, as in Fig. 1210. In general, it is ob-
long in form, the longer side being with the keys. The dimensions suited to
a medium-sized church are about 9' X 15', and 12' in height.
The vestry-room, if used for the purposes of its meetings, should be adapted
in size to the purpose ; but if only for a withdrawing or robing room for the
clergyman, it may be of very small dimensions, and should be accessible from
without. The Sunday-school room, in general, requires in plan about half
the area of the church. From motives of economy it is usually placed in the
basement of the church ; bufc, in the country especially, it is better that it
should be a separate building, and form one of the group of church, parson-
age, and Sunday-school house.
In elevation, city churches are Greek with porticoes in front, Romanesque,
and Gothic, occasionally Byzantine. The Greek have no tower, but often a
spire above the portico ; the Romanesque and Gothic generally one tower, over
the central door of entrance, or at one corner ; sometimes two, one at each side
of the principal door, almost invariably surmounted by spires, high and taper-
ing, usually of wood, but in some instances of stone.
Fig. 1211 is the front elevation of the Roman Catholic cathedral in Fifth
avenue, New York city, from designs by James Renwick, architect. The style
is the French Decorated Gothic.
Fig. 1212 is a perspective view of the Episcopal church of St. Bartholomew,
corner of Forty-fourth Street and Madison Avenue, New York ; Renwick and
Sands, architects. The style is Romanesque ; the vestry and parsonage are con-
nected with the church.
536
ARCHITECTURAL DRAWING.
FIG. 1211.
ARCHITECTURAL DRAWING.
537
FIG. 1212.
538
ARCHITECTURAL DRAWING.
Fig. 1213 is the cross-section of a common form of small country church,
with nave n, aisles a a, and clear-story c. The effect, both inside and out, is
FIG. 1213.
good, but there are objections to the masonry-columns, which cut off the view
of the desk and the altar from many sitters, and to the windows of the clear-
story, that in the winter they act
as coolers to the air which de-
scends in draughts upon the heads
of the congregation beneath them.
Neither columns nor clear-story
are constructively necessary ; the
span can readily be met by a sin-
gle roof, and sufficient light can
be obtained from the sides.
Figs. 1214, 1215, and 1216
are examples of open-timbered
Gothic roofs of churches.
The technical names (Fig.
1214. 1214) are : 1, Principals ; 2, Pur-
AKCHITECTURAL DRAWING.
530
1215.
lines ; 3, Collars ; 4, Braces ; 5, Wall-pieces ; 6, Wall-plates ; 7, Struts ; 8,
Rafters. 4 and 5 are shown in section.
Theatres. In theatres and
opera-houses it is not only ne-
cessary that the audience should
have a good position for hear-
ing and seeing the performance
upon the stage, but also to see
each other. The most approved
form, now, for the body of a
dramatic theatre is a circular
plan, the opening -for the stage
occupying from one fourth to
one fifth of the circumference,
the sides of the proscenium be-
ing short tangents ; but for a
lyric theatre, where music only
is performed, and where, conse-
quently, hearing is easier, the
curve is elongated into an ellipse,
with its major axis toward the
stage.
In the general position of
the stage, proscenium, orches-
tra, orchestra-seats, parquette,
and boxes, but one plan is fol-
lowed. The line of the front
of the stage, at the foot-lights,
is generally slightly curved, with
a sweep, say, equal to the depth
of the stage, and the orchestra
and parquette seats are arranged
in circles concentric with it : of
the space occupied by seats we
have already spoken. The entrance to the parquette may be through the boxes,
near the proscenium, and centrally, but better at the sides, dividing the boxes
into three equal benches ; the seats in the boxes are usually concentric with the
walls, and more roomy than those of the parquette. The orchestra seats are of
a height to bring the shoulders of the sitter level with the floor of the stage,
and the floor of the parquette rises to the outside, 1 in 15 to 18. The floor of
the first row of boxes is some 2 to 3 feet above the floor of the parquette at the
front center, and rises, by steps at each row, some 4 inches ; in the next tier of
boxes the steps are considerably more in height, and so on in the boxes above.
In general, three rows of boxes are all that is necessary ; in front, above the
second, the view of the stage is almost a bird's-eye view. The floor of the
Btage descends to the foot-lights at the rate of about 1 in 50. In large theatres
it is of the utmost importance that all the lobbies or entries should be spacious,
JTia. 1216.
540
ARCHITECTURAL DRAWING.
FIG. 1217.
and the means of exit numerous and ample the staircases broad, in short
flights and square landings, and not circular, as, in case of fright, the pressure
of persons behind may precipitate those
in front the whole length of the flight.
Ladies' drawing-rooms should be placed
convenient to the lobbies, of a size
adapted to that of the theatre, arranged
with water-closets ; there should also
be provided rooms for the reception of
gentlemen's canes and umbrellas, with
water-closets attached. The box-office
should be, of course, near the entrance,
but so arranged as to interfere as little
as possible with the approach to the
doors of the house. At the entrance
there should be a very spacious lobby,
or hall, so that the audience may wait
sheltered from the weather ; if possi-
ble, there should be a long portico over
the sidewalk, to cover the approach to
the carriages. Only single entrances are necessary to distinct parts of the
house, but the greater the
number of, and the more PLAN.
ample places for exit at
the conclusion of the piece,
or for the contingency of
fire, the better.
Fig. 1217 is a plan
suggested by Ferguson of
keeping the center of the
boxes perpendicular over
one another, and then, by
throwing back each tier
of side-boxes till the last
is a semicircle, the whole
audience would sit more
directly facing the stage,
would look at it at a bet-
ter angle, and the volume
of sound be considerably
increased by its freer ex-
pansion immediately on
leaving the stage.
Fig. 1218 and 1219 are
a plan and section of Wag-
ner's theatre.
In cities, the auditoria
FIG. 1219.
AKCHITECTUKAL DRAWING.
541
of dramatic theatres conforming to the shape of the lots are rectangular in
their outline, and seldom exceed a seating capacity of 1,000. Lyric theatres
are much larger, seating often as many as 2,000, and conforming in their
interior outline to the art requirements. Lecture-rooms are usually arranged
with the audience-floor flat, room rectangular, with reading-desk or platform
raised, and with or without galleries. The same form usually obtains for
music-halls, only they are much greater in extent ; the first being capable of
containing from 500 to 800 persons ; whereas some music-halls will contain
2,000, and Ferguson thinks that a music-hall might be arranged so that even
10,000 might hear as well as in those of present construction. The lecture
and music halls are seldom devoted to a single purpose, but are used for
political meetings, for fairs, and dances, and the construction must be such as
to serve these other purposes.
COMPARATIVE TABLE OF THE DIMENSIONS OF A FEW THEATRES.
DISTANCE
IN FEET
HEIGHT,
IN FEET.
NAME AND LOCATION.
Between boxes
and footlights.
Between footlights
and curtain.
Between curtain
and back of stage.
Greatest breadth
of pit.
Breadth of cur-
tain.
Breadth of stage
between side-walls.
it.
||
If
o|
11
Alexandra, St. Petersburg
65
11
84
58
56
75
53
58
, Berlin
62
16
76
51
41
92
43
47
La Scala, Milan ....
77
18
78
71
49
86
60
64
San Carlo, Naples
77
18
74
74
52
66
81
83
Grand Theatre, Bordeaux
46
10
69
47
37
80
50
57
Salle Lepelletier, Paris
67
9
82
66
43
78
52
66
Covent Garden, London
66*
55
51
32
86
54
Drury Lane, London.
64*
80
56
32
48
60
Boston, Boston
53
18
68
46
87
554
58
Academy of Music, New York
74
13
71
62
48
83
74
Grand Opera- House, New York
54
84
63i
48
44
?6
52
67
Opera-House, Philadelphia
61
17
72
66
48
90
644
~
74
* These dimensions include the distance between the footlights and curtain.
Legislative Halls. Although much has been written about their construc-
tion in relation to acoustic principles, there yet seems to be great disagreement
in practical examples, and in the deductions of scientific men. The Chamber
of French Deputies was constructed after a report of most celebrated architects,
in a semicircular form, surmounted by a flat dome, but as the member inva-
riably addresses the house from the tribune, at the center, in its requirements
it is but a lecture-room. Mr. Mills, architect, of Philadelphia, recommends
for legislative or forensic debate, a room circular in its plan, with a very slightly
concave ceiling. Dr. Eeid, on the contrary, in reference to the Houses of Par-
liament, gave preference to the square form, with a low, arched ceiling. The
Hall of Representatives, at Washington, is 139 feet long by 93 feet wide, and
about 36 feet high, with a spacious retiring gallery on three sides, and a re-
porters' gallery behind the Speaker's chair. The members' desks are arranged
542 ARCHITECTURAL DRAWIXG.
in a semicircular form. The ceiling is flat, with deep-sunk panels, openings
for ventilation, and glazed apertures for the admission of light. The ventila-
tion is intended, in a measure, to assist the phonetic capacity of the hall, the
air being forced in at the ceiling and drawn out at the bottom.
In reviewing the general principles of acoustics, it will be found that those
rooms are the best for hearing in which the sound arrives directly to the ear,
without reflection ; that the sides of the room should neither be reflectors nor
sounding-boards, and that surfaces absorbing sound are less injurious than those
that reflect. Slight projections, such as ornaments of the cornices and shallow
pilasters, tend to destroy sound, but deep alcoves and recessed rooms produce
echoes. Let the ceiling be as low as possible, and slightly arched or domed ;
all large external openings should be closed ; as M. Meynedier expresses it, in
his description of an opera-house, "Let the hall devour the sound; as it is
born there, let it die there."
Hospitals. In large cities, hospitals, by necessity, are confined to narrow
spaces, but they should be placed, if possible, on river fronts or on open parks,
to secure as much open-air ventilation as possible. They are usually many sto-
ries in height, with large wards one above the other. Sir J. T. Simpson alleges
a very high rate of mortality in hospitals after surgical operations as compared
with the mortality after the same operations wheu performed at the homes of
the patients, and asserts that the mortality after operations performed in hos-
pitals containing more than 300 beds is in excess of that in hospitals containing
less ; that great hospitals are great evils in exact proportion to their magnitude,
and suggests the construction of smaller hospitals.
Figs. 1220 and 1221 are an elevation and plan of an English country hos-
pital.
Stables. Under this general name are included the barn, or the receptacle
of hay and fodder, the carriage-house, and the stable proper, or lodging-house
for horses and cows. The first two may be included under one roof, the car-
riages on the first floor, and hay in the loft ; but the lodging-place should be
distinct, in a wing attached to the barn, that the odors from the animals may
not impregnate their food, or the cloth-work' of the carriages, or the ammonia
tarnish their mountings.
Hay in bulk, in the mow, occupies about 340 cubic feet per ton ; bales aver-
age 2' 4" x 2' 6" x 4', and weigh from 220 to 320 pounds. The door-space for
a load of hay in the bulk should be from 12 to 13 feet high and 12 feet wide.
The floor beneath the hay should be tight, so that dust and seed may not drop
on the carriage. A door for carriages should be 10 feet 6 inches high by 9 feet
wide.
The horse is to be treated with greater care than any other domestic ani-
mal. His stable is to be carefully ventilated, that he may have fresh air without
being subject to cross-draughts. Preferably, the floor should be on the ground,
that there may be no cold from beneath . He should stand as near as possible
level ; and for this purpose a grated removable floor, with small interstices,
should be laid over a concrete bottom, with a drip toward the rear of the stall,
and the urine should be collected in a drain, and discharged into a trapped
manure-tank outside the stable. In Fig. 1222 the pitch of bottom of stalls is
ARCHITECTURAL DRAWING.
543
FIG. 1220.
GROUND PLAN.
FIG. 1221.
544
ARCHITECTURAL DRAWING.
to the center and outward. The manure should never be deposited beneath
the stable, but should be wheeled out and deposited in a manure-yard or tank
daily. It is as essential that all excrements should be removed entirely from
the stable as that the privy should be placed outside the house.
The breadth of stalls should be from 4 feet 6 inches to 5 feet in the clear ;
the length, 7 feet 6 inches to 8 feet ; the rack and feed-box require two feet in
addition, to which access is given in the best stables by a passage in front.
Rack and feed-boxes are often made of iron, and the upper part of stalls fitted
with wrought-iron guards. Box-stalls, in which horses are shut up but not
tied in cases of sickness or foaling, are about 10 feet square,
FIG. 1222.
In large stables in cities the first floors are often occupied by the carriages,
while the horse-stalls are in the basement or upper stories, with inclined ways
of access. In the basement provision must be made for light and ventilation.
Tool
House-
Open Shade
o
Box St&Us.
Carriage House
FIG. 1223.
ARCHITECTURAL DRAWING.
545
In the upper stories these may be secured more readily, but the floors must be
made tight and deafened, that the urine may not leak through, nor the cold
come through from below to make too cool a bed for the horse.
Fig. 1222 is an elevation in perspective of two first-class stalls, a box shown
with the door open, and a single stall. The lower part of the inclosures is of
plank, with wrought-iron guards and ramp above. The posts are of oak,
and the hay-boxes or mangers of cast-iron ; the hay-rack in the box-stall
is of wrought-iron. These are of common manufacture, and are of
varied patterns ; but in the country they are usually made of wood,
and connected with the stall.
Fig. 1223 is the plan of a small country stable, show-
ing the desirable passages around the stalls and ex-
terior windows in front of each stall, that
the horses may not only have light
and air, but can see out.
Coiv - houses, for cows
giving milk, should
be constructed
with care
FIG. 1224.
FEET
for ventilation, light, and cleanliness. Other cattle are usually left out, with
sheds under which they can go for shelter. For those housed, the spaces occu-
pied should be about the same per Head as the single horse-stall. The manger
35
546
ARCHITECTURAL DRAWING.
should be on the floor, 12" to 18" high, and about 18" wide. It is not
usual to have partitions, but there ought to be between every pair,
reaching from the manger half-way to the gutter behind. The
floor should be level, grated, with a drip beneath, and cleansed
by washing out. In England the partition and man-
gers are often of cast-iron, and are on sale, but
here they are of wood.
Greenhouses. Fig. 1224: is
section of a greenhouse, with
shelves for plants. The
the
{FEET.
FIG. 1225.
ARCHITECTURAL DRA\V|N. 547
floor is of concrete and the walls are of masonry ; E^oiorthern exposure is a
blank wall.
Fig. 1225 are the details of windows. The sides are box-sash, hung with
weights (w, w, Fig. 1226). The lower roof sash is firmly fixed, but the
upper one can be slid down ; it is usually retained in place by a cord attached
to the lower part of the sash, passing over a pulley on the upper bar of the
frame, with the loose end within reach of the gardener, who can fasten it to
a cleat.
Ventilation and Warming. The purposes of ventilation are not changes of
air merely, but the removal of foul and vitiated air, and the substitution there-
for of pure air ; and this air may be warm or cool according to the necessities
of the season and personal requirements. Open space is not necessarily well
ventilated ; there must be circulation, outward and inward, the latter from
purer sources than the former, or the change is useless. With an equal dis-
charge and supply of pure air, the smaller the room, the more frequent the
change of air, the better its distribution, and the better the ventilation. But
if the means of removal, supply, and distribution of air be proportioned to the
size of the room, then the larger the room the better. Apertures do not neces-
sarily mean circulation ; a flue may draw or it may not draw, it may be inert,
or the air may come down ; a window may be open, with little or no inward or
outward movement of air. In a house exposed to a fresh breeze, on the wind-
ward side there is an air-pressure ; on the leeward side there is an eddy or
vacuum. Air is forced in on the first through every crack of door and win-
dow often down chimney-flues and drawn out on the other side. This often
happens even with fires in the chimneys, and with heat in ventilating educts.
If one will make an experiment in cold weather, when the windows are closed,
and there are fires in some rooms, he will find that there is cold air coming
down the unused flues, and will feel the cold current flowing down the stairs,
and along the floors to the fires. Architects have placed kitchens in the base-
ment, and in the attic, and the smell of cooking will rise through the house,
usually from the one, but descend from the other when the air is light and
muggy.
Every room should have its separate flue ; for if the current is not upward
it will probably be downward, affording a fresh supply if there is an exit else-
where. A chimney-flue may be too large for the purposes of a fire ; for most
fires a flue 8" X 8" is amply sufficient, and, for the purposes of ventilation in
the common occupation of a house, this flue will answer all the purposes in
cold weather. It is usual to depend largely on windows for ventilation, but
the space on which they open may be too circumscribed to afford the requisite
change of air, or the outer air itself may be too hot, or too cold, or too mala-
rial or offensive, to make the change of air sanitary or pleasant. In tenement
or apartment houses care should especially be taken that the inner windows
on different flats open into as large air-shafts as possible, and that these
shafts should l^ave free opening to the outer air without sky-lights ; and that
the floors should be tight, so that the smells may not pass from one flat to
another. Nothing more surely shows faults in ventilation than the diffusion
of kitchen-smells or tobacco-smoke. For the separation of apartments, let
548 ARCHITECTURAL DRAWING.
every room have its own flue, and this flue extending independently well above
the roof, and not into an attic with a ventilating louver. In this case the air
may ascend one flue and descend another, and not out of the louver.
The quantity of air taken into and expired from the lungs by a single indi-
vidual is quite small, probably about 13 cubic feet on an average per hour.
The usual gas-burner delivers from 4 to 6 cubic feet per hour, under a pressure
of 1" and 2" of water. It will be seen, therefore, how small apertures are neces-
sary to supply the lungs of a person, if it could be provided directly to him
and taken away without vitiating other air. But, in addition, air is vitiated by
personal emanations, and consumed by lights. These last can readily be made,
not only to remove all their products of combustion, but also increase the cir-
culation in flues for the ventilation of the room.
All systems of ventilation are based on the idea that so many individuals
within a room and so many lights burning vitiate so much air, and that conse-
quently a very large quantity of outer air must be introduced to reduce the per-
centage of vitiation, and generally with very little consideration as to the distri-
bution of this air, although it is in every one's experience that the air in some
portions may be fresh, in others stifling ; that in hospital wards there are often
dead ends where the air does not circulate, and where patients do not as a rule
recover. The system is to provide somewhere in a room air enough,' and trust
to chance for its distribution.
Some architects make the educts at the ceiling, some at the floor, some at
both, with registers to control the openings. For sleeping-apartments, if there
is a fireplace, this is all that will be necessary ; if the air goes up or comes down,
it does not make draughts about the heads of the occupants.
To make flues draw, various forms of
T 1 r chimney-tops or cowls are adopted. The
I /"""[* k 68 ^ an d simplest are the Emerson (Fig.
MBE^ 1227), and a modification of the same (Fig.
mlP X JK \ 12 28) ; there are also various forms of self-
lll I IIP 1 ncting na P s > turn-cowls, etc., the principle
being to take advantage of the wind to make
a draught. With the wind blowing across
FIG. 1227. FIG. 1228. the top of a chimney, a bit of square-ended
iron pipe extending above the chimney will
answer as an expirator, but without a wind the draught must depend on cir-
cumstances within the dwelling and artificial draught. When sufficient cir-
culation can not be obtained from natural differences of temperature in the
atmosphere, or from winds, it is usual to have recourse to fans, to force air
into or draw it from a building, or by heat applied to the air in flues, ducts, or
chambers in the hot-air furnaces. Both the air and the heat are necessary.
When heat is applied for ventilation only, as in mines, a fire is built in a flue
near the top, and the air necessary for combustion is drawn from the mines ;
the flue extends from the bottom of the mine, with a chimney above the sur-
face of the ground, and ducts are led from the bottom of the flue to the face of
the workings, the cold air for ventilation being drawn down through the work-
ing-shafts and drifts. In buildings, steam-pipes and gas-burners are put in flues.
ARCHITECTURAL DRAWING.
549
Methods of Heating. The open fireplace grate heats by radiation, commu-
nicating heat to objects, which by contact transfer it to the air. Persons com-
ing in contact with rays are themselves heated, while the air around them is
cool and invigorating for breathing ; the bright glow has a cheering and ani-
mating effect upon the system, somewhat like that of sunlight. As a ventilator,
an open fire is one of the most important, drawing in air not only for the sup-
port of combustion, but also, by the heat of the fire and flue, making a very
considerable current through the throat of the chimney above the fire. From
this cause, although there is a constant change of air, yet there arises one great
inconvenience of disagreeable draughts, especially along the floor, if the air-
supply be drawn directly from the outer cold air ; but in connection with prop-
erly regulated furnaces or stoves, the open fireplace becomes the most perfect
means of heating and ventilation. As a heater merely, the open grate, in very
cold weather, is not satisfactory ; its influence is only felt in its immediate
vicinity, and but from. 10 to 15 per cent of the heat of the fuel is rendered
available.
Fig. 1229 represents an old form of open fire used in a tavern bar-room and
office, which answered admirably for heating and ventilation, and admitted
of access to many persons. It
consisted of a circular grate at the
level of the floor in the center of
the room. In the cellar beneath
was an ash-pit, a, in brick-work,
with an opening, o, to supply air for
the combustion of the fuel. Above
the grate was a counter-weighted
sheet-iron hood, h, connected by a
pipe with the chimney, which could
be raised or lowered, to suit the re-
quired draught. Around the grate
was a ring-guard to rest the feet on,
and the customers ranged them-
selves in a circle round the fire.
Stoves. Open stoves heat by FIG. 1229.
direct radiation, and by heating
the air in contact with them, and close stoves by the latter way only; as
economical means of heating, the latter are the best, and, when properly
arranged, give both a comfortable and wholesome atmosphere. There should
be some dish of water upon them to supply a constant evaporation, sufficient
to compensate for increased capacity of the air for moisture due to its in-
creased heat. In the hall there will be no objection to a close stove, letting
it draw its supply of air as it best can ; but in close rooms the open stove is
best, on the plan of the old Franklin stove, or, if a close stove, somewhat on
the plan of a furnace, with an outer air-supply for combustion and ventilation.
Hot-air furnaces are close cast-iron stoves, inclosed in air-chambers of brick
or metal, into which external air is introduced, heated, and distributed by
metal pipes to the different rooms of a house. Furnaces have been, of late, very
550
ARCHITECTUKAL DRAWING.
much decried, but under proper regulation they are very cheap, economical,
and even healthful means of ventilation and warming. The heating-surface
should be very large, the pot thick, or even incased with fire-brick, that it may
not become too hot ; there should be a plentiful supply of water in the cham-
ber for evaporation, perhaps also beneath the opening of each register ; the air-
supply should always be drawn from the outer air and unobjectionable sources,
through ample and tight ducts, without any chance of draught from the cel-
lar ; the pot, and all joints in the radiator, should be perfectly gas-tight, so
that nothing may escape from the combustion into the air-chamber. With
these provisions on a sufficient scale, and proper means for distribution of
the heated air and escape of foul air, almost any edifice may be very well
heated and ventilated. The air should be delivered through the floor or the
base-board of the room, and at the opposite side from the flue for the escape
of foul air, making as thorough a current as possible across the room, and
putting the whole air in motion. In dwelling-houses the fireplace will serve
the best means of exit ; in public rooms distinct flues will have to be made
for this purpose, and they should be of ample dimensions and well distrib-
uted, with openings at the floor and ceiling with registers, and means should
be provided for heating the flues. An architect, in laying out flues for heat-
ing and ventilation, should, both in plan and elevation, fix the position of
hot and foul air flues, and trace in the current of air, always keeping in mind
that the tendency of hot air is to rise ; he will then see that, if the exit-
opening be directly above the entrance-
flue, the hot air will pass out, warming
the room but little ; if the exit-opening
be across the room and near the ceiling,
the current will be diagonal, with a cold
corner beneath, where there will be very
little circulation or warmth. To heat the
exit-flue, a very simple way is to make the
furnace-flue of iron, and let it pass up cen-
trally through the exit-flue.
Fig. 1230 may be taken as a type of a
portable (so named on account of its small
size and metallic case) hot-air furnace.
The air is introduced at the bottom of the
case, passes up and around the stove, and
out through the ducts D, D, D to different
parts of the building. The water-pan p is
indispensable to the hot-air furnace, and
should be of capacity enough for a day's
supply, or have automatic means of keep-
ing up the supply.
Air in winter is very dry, but as its
volume is enlarged by heat, it draws a
supply of moisture from everything with which it comes in contact from the
skin and lungs, creating that parched and feverish condition experienced in
FIG. 1230.
ARCHITECTURAL DRAWING. 551
many furnace-heated houses ; from furniture and wood-work, snapping joints
and making unseemly cracks.
Thus, taking the air at 10, and heating it to 70, the ordinary temperature
of our rooms requires about nine times the moisture contained in the original
external atmosphere, and, if heated to 100, as most of our hot-air furnaces
heat the air, it would require about 23 times.
The portable furnace is not so economical as the furnace set in brick -work,
as more heat escapes through the metallic case. The former are usually made
from 12" to 24" diameter of pot, from 2' to 4' outside diameter, and 5' to 6'
height of case.
The brick-set furnaces are from 20" to 28" pot, outside brick-work from 5'
to 6' square, walls 4" thick, height 6' to 7'. The size of air-ducts is propor-
tioned to size of furnace. The inlet should be, say, equal to that of the grate,
and the sum of the outlets but little in excess of this area. It is difficult to
give any rule for the heating capacity. A 22" pot should be adequate for the
heating of a common 25' x 60' city house, and the higher the air-duct the less
its diameter.
Steam and hot-water circulation are applied to the heating of buildings by
means of wrought or cast iron pipes connected with boilers. In the simplest
form, as common in workshops and factories, steam is made to give warmth
without ventilation by direct radiation from wrought-iron pipes. The gen-
eral arrangement is by rows of 1" pipe hung against the walls of the room, or
suspended from the ceilings, 3' of 1" pipe being considered adequate to heat 200
cubic feet of space ; if there are many windows in the room, or the building is
very much exposed, more length should be allowed.
Steam, as a means of heating, is the most convenient and surest in its
application to extensive buildings and works. From boilers, located at some
central point, steam can be conveyed to points so remote that in many cities it
is matter of sale, both for heating and power purposes. The limits of the
extension of steam-pipes economically have not yet been determined, but within
the range of the buildings occupied by any single textile manufacturing in-
dustry steam-heating has proved satisfactory, and is of almost universal adop-
tion. For stores, warehouses, large buildings of all sorts, where there are
extensive or numerous rooms to be heated, steam has been long used, and the
appliances for its use can be as readily obtained in all our cities and large towns
as stoves or grates. Steam is used for heating at either high or low pressures ;
under 5 or 6 pounds would be considered low pressure. A low-pressure ap-
paratus may draw direct from a boiler, or be supplied from the exhaust of a
steam-engine ; if from the latter, a certain amount of back pressure must be
put on the engine to establish a circulation in the steam-heating pipes.
In the operation of heating by steam, the steam, in giving off its latent
heat through the pipes to the air of the room, returns to water ; the apparatus
would then be nothing but pipes to convey the steam to radiators to condense
it, and pipes to return the water to the boiler, were it not for air invariably in
water and steam. This necessitates a more complicated circulation ; there should
be a regular flow outward of steam from the boiler, and inward of water and
steam to it, both as far as possible together, and in the same direction. When
552
ARCHITECTURAL DRAWING.
hot water is used for heating, there must be circulation throughout the sys-
tem ; the water flows out from the top of the boiler, gives out its heat, and
returns, practically of the same bulk, cold to the bottom of the boiler, and
any radiator out of the line of this current is of no use. A single valve shuts
off the circulation in the hot-water apparatus, while two are necessary with a
steam apparatus, for the steam cut off on the direct pipes may back up through
the return-pipe.
Steam is used for heating rooms either directly or indirectly. Direct steam-
heating is like that of common stoves, without any considerations for ventila-
tion. Indirect steam-heating is like that of hot-air furnaces. Steam radiators
are inclosed in a box or chamber, into which air is drawn or forced, and then
distributed by ducts to the rooms to be warmed and ventilated. Thus, when
ventilation is combined with steam or hot-water heating, the metallic surfaces
brought in contact with the air usually range from 212 to 250, while the pot
4,0.
FIG. 1231.
of the air-furnace may be near a white heat. In a sanitary point of view, hot-
water or low-steam coils in air-chambers are a more surely healthy means
of warming and ventilation ; the greatest objection is their expense, the care
requisite in attending them, and the danger of freezing and bursting the pipes
ARCHITECTURAL DRAWING.
553
if worked intermittently in winter. In the arrangement it is usual, in dwell-
ing-houses, to place the coils at different points in the cellar, as near as possible
beneath the rooms to be heated. In public buildings frequently a very large
space in the cellar is occupied by the coils, into which the air is forced by a
fan, and then distributed by flues or ducts throughout the building.
All inlet or outlet ventilating flues should be provided with dampers or
registers, to control the supply or discharge of air, cutting it off when sufficient
heat is secured, or retaining the warmth when ventilation is not required.
Fig. 1231 (an illustration from "The Sanitary Engineer") is the plan of a
portion of a large building heated by steam. B B are two boilers, either of
which would be sufficient for the purpose ; the steam mains are shown by black
lines following those of the building, with the sizes marked upon them ; the
risers by inclined lines, with the square foot of radiating surface on each story,
marked. This is a very convenient form of drawing, explanatory of the sys-
tem. It is usual to draw the steam mains and risers in red, and the returns in
black, with the diameters on each.
Fig. 1232 is the elevation of
a small steam-heating apparatus,
illustrating the general action. B
is the boiler, and R and R' radia-
tors on different stories ; s is the
steam-pipe, and r r' return or drip
pipes. The steam is drawn from
the top of the boiler, and the re-
turns must be below the surface,
W. L., of the water in the boiler.
The circulation is simple and in-
telligible, and applicable to a hot-
water apparatus ; as a steam ap-
paratus, if it is required to shut
off the lower radiator, R, both
the inlet and outlet valves on the
radiator must be shut. If only
the top valve be shut, the steam
in the radiator will be condensed,
and the pressure from the boiler
will fill it with water. If the lower
valve only be shut, the radiator
will still act as a condenser till it
is filled with water. In the upper
radiator, R', there is no outlet-
valve, as the radiator is supposed
to be set at a level above the
height to which the water would be raised by the pressure of the steam in the
boiler. This arrangement of separate returns for each radiator is sometimes
used, but the usual practice is to have single returns, into which there are
branches from each radiator, controlled by valves. In low-steam apparatus,
FIG. 1232.
554
ARCHITECTURAL DRAWING.
the steam is introduced and the water removed by the same pipe, and con-
trolled by a single valve.
Fig. 1233 is an elevation, showing the usual arrangement of mains, s s, and
returns, rr, when the hori-
zontal distance from the boiler
is small and the risers few.
The inclination of the mains
is toward the boiler, and their
condensed water returns by
them to the boiler.
Fig. 1234 is the better
practice, and necessary if the
steam is high pressure, the
mains extended, and the branches numerous. The inclination of the mains, s s,
is from the boiler, and the condensed water flows down to the lowest angle,
where it is connected with the return, r, and is by this brought back to the boiler.
The size of the boiler for a steam-heating apparatus is based on the amount
of radiating surface, which must include that of the steam-mains, if not
clothed, and of the returns. But, as boilers vary so much in their proper-
FIG. 1233.
FIG. 1234.
tions, it is impossible to give a rule applicable to all of them. Some estimate by
boiler-grate surface, 500 square feet of radiating surface to each square foot of
grate ; some, 1 H. P. of boiler to each 200 square feet of radiating surface.
The amount of radiating surface depends on the cubic feet of air to be heated.
It is usual to estimate that from 150 to 200 cubic feet of room-space can be
heated from to 70 by 1 square foot of radiating surface ; or, say, 4 run-
ning feet of f " pipe or 3 feet of I" pipe. But this is to be modified very much
by the exposure of the room, the amount of glass surface, the thickness of
wall, and the temperature of surroundings. The effect of glass as a cooling
surface can be readily understood by the difference one experiences in the heat
of cars in motion or stopped, and the advantages of double windows in the
same conveyances.
Where the heating is indirect, as there are more cubic feet of air to be heat-
ed, the radiating surface is to be increased, usually to about three times that of
the direct heating.
C. B. Eichards says that, for direct radiators, 1 square foot of surface gives
off 3 heat-units for each degree (1) difference of temperature between the air
of the room and that of the steam in the radiator.
As the boiler must be proportioned to the requirements of heating, as de-
termined by the square feet of radiators, the sizes of mains and returns are
also measured by the same standard. A common rule is I" diameter of main
AKCHITECTUEAL DRAWING. 555
for each 100 square feet of radiating surface, varying with the squares of the
diameters 2", 400 square feet; 3", 900 ; 4", 1,600. In fact, the larger sizes
will be sufficient for a much larger radiating surface than given by this rule.
For the returns, one size less than that of the steam mains is the rule ;
thus, a f " return for a 1" pipe, but no pipe of less diameter than f " is used ;
for a 2" steam a 2" return, and a larger than 2" is seldom used. It may not
be always practicable to return the condensed water, as shown in the figures
above, by gravitation, but there are various forms of receivers or traps in which
the water is collected and returned, automatically as in the Albany trap, or by
pumping to the boiler.
Figs. 1235 to 1241 are common forms of radiators. Fig. 1235 is a bench
coil, often called a mitre coil, from the vertical or horizontal angle made at the
end of the pipes to admit of their unequal expansion. In the circulating coil
(Fig. 1236), often called the trombone, the circulation is alternately forward
and back ; when placed in rows, as in Fig. 1237, it is a box coil ; the ends of
the pipes at both top and bottom are connected in heads. Fig. 1238 is a hori-
zontal radiator, similar in its action. Figs. 1239 and 1240 are vertical radia-
tors, the first composed of wrought-iron pipes inserted in a hollow cast-iron
base, circulation being obtained by a sheet-iron division in the pipe as in the
Nason radiator, by an inside pipe, or by the connection of two pipes at top by
a return bend. Fig. 1240 is a Bundy radiator, in which there are twin cast-
iron pipes connected at the top and bottom. Fig. 1241 are cast-iron pin radia-
tors, so called from the projections, effective for indirect radiators. In meas-
uring the surface of circulating coils, include the lengths of angles and all
fittings ; in the vertical radiators, include the base.
On the radiator (Fig. 1240) a small pipe (p) will be seen, which is an air-
vent, often automatic, but indispensable for a prompt start of the circulation.
Plumbing. The conveniences for comfort in modern buildings require the
introduction of water and its removal. Most cities have water-supplies and a
system of sewers, and the plumber makes the connections with both. In the
country, for the better class of houses there are private expedients to supply
their places, largely by wells and pumping, and connections to cesspools. The
quantity used in each household varies with the wants and habits of the occu-
pants. An average bath will take 25 gallons ; each use of a water-closet from
2 to 3 gallons. A wash-tub will hold from 10 to 20 gallons. If the water is to
be pumped by hand, from 7 to 10 gallons will be reckoned as the use by each
person ; if from aqueduct, 30 to 50 gallons is ample.
The regulation size of taps for city mains is from " to -f-", and the pipes
leading into the house from f " to 1" diameter. The pipes are usually of lead,
as most waters are not affected sensibly by lead, if the pipes are always kept
full, and there is fair circulation. In some cases block-tin pipes are used ; or
iron, galvanized, or coated with some preparation of asphalt, or glass-lined.
The soil or house-sewer pipe connections with the main sewer or cesspool
are usually vitrified stone-ware pipe, from 4" to 6" diameter, as they are not
only for the discharge of the sewage, but also for the rainfall from the roof.
Within the house the pipe is either of stone-ware or cast-iron ; invariably of
the latter if the pipe is exposed. The rising pipe to the roof is here, also,
556
ARCHITECTURAL DRAWING.
FIG. 1241.
AKCHITECTURAL DRAWING.
557
FIG. 1242.
usually of cast-iron, and 4" diameter may be considered
ample for a common house ; the smaller branches may
also be of iron, but when as small as 2" are usually of lead.
Fig. 1242 is the perspective of a kitchen-range boiler
and sink : c is the cold-water pipe leading to the sink and
to the boiler ; it enters the top of the boiler, and is led
down nearly to the bottom. The hot water is drawn from
the top, through the pipe h, is led down to the sink and
up for distribution through the house. The water is heat-
ed in the boiler by the connection with the water-back of
the range, r ; the water flows through the pipe, I, is con-
nected with the lower part of the water-back, and returns
by the pipe, u, from the top of the water-back to a higher
point in the boiler ; b is the blow-off pipe.
It will be observed that at the draw-cocks over the
sink there are pipes, a a, turned up ; these are air-cham-
bers, to cushion the blow of the water-hammer when the
cocks are shut quickly. Beneath the sink there is a
trapped connection with the sewer-pipe.
Fig. 1243 is the elevation of a gal vani zed-iron boiler,
but those in general use here are of copper.
Fig. 1244 is the perspective drawing of a cast-iron
sink, of the usual form and material. They are to be
obtained of all suitable dimensions, rectangular, from 16" FIG
658
ARCHITECTURAL DRAWING.
X 12" X 5" deep, to 96" X 24"
X 10" deep ; also, half -circle
and corner sinks, and deep and
slop sinks.
In the kitchen, or a laun-
dry-room adjacent, tubs are
set for washing, with hot and
cold water service. The water-
pipe connections are usually
}", the waste connections 2". The tubs themselves are mostly of wood, but
there are many of cast-iron (Fig. 1245), galvanized or enameled, of slate, and
of earthenware.
FIG. 1244.
FIG. 1245.
In the butler's pantry there is usually a sink set of planished tinned-copper,
with hot and cold water connections.
In the chambers and dressing-rooms, bowls of earthenware are set, with like
connections. The sizes of
basins vary from 12" to
18" outside diameters.
Fig. 1246 shows the
usual form of setting of a
wash-basin in a counter-
sunk marble slab, with a
back of the same mate-
rial ; these are the com-
FIG. 124. mon ground key swinging
faucets for the supply of
hot and cold water, and the waste is closed by a metal or rubber plug, at-
tached to a chain, with the other end fastened to a pin in the marble slab.
ARCHITECTURAL DRAWING. 559
The sides are inclosed with wood, forming a closet beneath the basin, with
usually small drawers for towels at each side of the closet.
Fig. 1247 is a cast-iron bath-tub, of a simple pattern, with an overflow, o,
and apertures c and h near the bottom for hot and cold water connections ; the
waste is closed, as in the basin above, by a plug. When there is no overflow to
the tub, this plug is a hollow pipe, down which there is an overflow when its
lower extremity or annular plug closes the waste-pipe. Bath-tubs are more
generally made of planished tinned-copper in a wooden box or support, and
inclosed by wooden panels. The more expensive bath-tubs are made of porce-
lain, and may or may not be inclosed. In most bath-rooms there is a set wash-
hand basin and a water-closet often a foot-bath and bidet-pan. Formerly it
was the common practice to have but one trap beneath the water-closet, into
which all the waste-pipes discharged, but of late the water-closet connection
with the soil-pipe is independent of the others.
FIG. 1247.
It is preferable to make the water-closet in a separate room, distinct, with
Its own water and sewer-service and means of ventilation.
The construction of one form of water-closet, with all the modern appli-
ances for the removal of soil and for ventilation, will be understood from the
section (Fig. 1248). The seat is not shown, but is just above the basin, B,
which contains some water to receive the defecations, to prevent the soil attach-
ing to the side of the basin, and in a measure to check its offensive smell. T
is the trap or water-seal which prevents the smell from the soil-pipe S passing
up through the basin. The water-discharge from the pipe W is through a rim-
flush around the edge of the basin. ^The sudden discharge washes out the basin
B into the trap T, which is also cleaned by the rush of water. The soil-pipe S
extends up through the roof, and may or may not also serve as a rain-leader.
A sudden flow of water down the soil-pipe often acts as an ejector to draw the
water out of the trap T, and break the water-seal ; to prevent this, there is
an air connection, A, leading also to the top of the house. But as the offense
of a water-closet is largely due to its recent use, and as smell once getting into
560
ARCHITECTUKAL DRAWING.
the room is with difficulty and slowly removed, there is a ventilating-pipe, V,
connecting the basin B with a ventilating-flue. It will be observed that this is
the most important part of the apparatus ; connected with a chamber commode,
it would remove all smell, and if there were no trap to the soil-pipe, or were
w
FIG. 1248.
the water-seal broken, it would still prevent any offensive smell from pene-
trating the house. If the soil-pipe be made also a ventilating-pipe, as is fre-
quently done by its connection with the hot-air flue, then the trap and pipes
A and V are unnecessary.
Fig. 1249 is an elevation of the simplest
form of closet the hopper-closet and in many
respects the best. It is shown in section (Fig.
1250), with its water, soil-pipe, and water con-
nection. By pulling up the handle, A, the disk-
valve in the cistern is raised, and water sup-
plied to the closet-basin.
Fig. 1251 is the section of a pan-closet, for
many years the most popular closet. The cop-
per pan, when shut, cuts off the view of the
trap below and any odor from it ; with a small
flow of water the basin is readily kept clean, but soil is apt to lodge in the iron
receiver, and the odor to arise from it when the pan is down. There is an
FIG. 1249.
ARCHITECTURAL DRAWING.
561
annular ventilating-tube beneath the seat, with an air-shaft attached, but of
altogether inadequate dimension for the purpose, as may be said of all such
vents attached to water-closets. There is also the air- vent to prevent the water
being drawn from the trap. No water connections are shown in the figure.
SEAT
FLOOR
LEAD TRA
FIG. 1251.
Fig. 1252 is the section of a flap-
closet, in which a flap-valve supplies
the place of a pan.
FIG. 1250.
FIG. 1252.
JET CLOSET.
Fig. 1253 is the section of a siphon-jet closet. In addition to the fan
flush, /, into the basin, it has a jet-pipe j at its bottom, inducing a current in
the direction of the inclined leg of the
trap, and by flush and jet the water is
siphoned from the basin.
The use of traps has already been ex-
plained, but they are varied in their form,
all answering the same purpose, to cut off
the air-connection of the soil-pipe with
the room in which the appliance is
placed. The smaller traps are invariably
in lead.
36
FIG. 1253.
ARCHITECTURAL DRAWING.
Figs. 1254 to 1261 represent the usual forms of lead traps. It will be ob-
served that there are screw-plugs at the bottom of the traps, which can be taken
out to remove any obstruction. As the water may be drawn out of any trap
by the passage of water down the pipe with which it is connected, an air-vent,
as already described, in the water-closet trap, is put on these small traps. In-
stead of this, by inserting the rising pipe at #, so that water from the waste
above should drip a little into the lower trap, draft from it is prevented.
9*8.
SHORT BEND. LONG BEND.
li_P
FIG. 1254. FIG. 1255. FIG. 1256. FIG. 1257. FIG. 1258. FIG. 1259. FIG. 1260. FIG. 1261.
Figs. 1262 and 1263 are cast-iron traps, with a cap that may be removed to
clean the trap, or the aperture may be used for air-vent connection.
S-TRAP. TRAP WITH SIDE OUTLET.
FIG. 1262.
FIG. 1263.
FIG. 1264.
FIG. 1265.
Fig. 1264 is the section of a foZ?-trap, used on sinks, with a strainer, S, above it.
Fig. 1265 is a plate with plug, for the bottom of sinks and bath-tubs.
Figs. 1266 to 1271 are common cast-iron bends or angles.
QUARTER
BEND.
DOUBLE HUB,
QUARTER BEND.
EIGHTH
BEND.
SIXTH
BEND.
SIXTEENTH
BEND.
KETURN
BEND.
FIG. 1268.
FIG. 1269.
FIG. 1270. FIG. 1271.
HALF
Y-BRANCH. Y-BRANCH.
DOUBLE
Y-BRANCH.
DOUBLE HALF
Y-BRANCH.
FIG. 1272.
FIG. 1273.
FIG. 1274. FIG. 1275.
FIG. 127*.
FIG. 1277.
Figs. 1272 to 1277 are cast-iron branches. The T branch and cross-head
are objectionable, as the flows from the branches and mains are at right angles.
ARCHITECTURAL DRAWI
563
and mutually obstructive ; whereas in the Y, especially^ 6 full Y, the flows
are at acute angles with each other, and the currents con?eggr> Similar fit-
tings are used for water, but they are much
heavier.
Most water-closet basins are inclosed by
a lidded seat and riser, but the less wood-
work about a basin the better. The seat is
generally hung with hinges, so that it can
be raised, and the basin used as a urinal for
men ; the upper edge of the basin being ex-
tended or covered with an earthenware tray,
sloping toward the basin.
Urinals, of which one form is shown
(Fig. 1278), are often used in public build-
ings, and in airy situations ; although they
have water connection, w, and a rim flush,
it is almost impossible to keep them sweet ;
a cake of carbolic soap is often put in the
basin, but the most effectual means adopted
on many railway-cars is a piece of ice. As
the raising of the seat of the water-closet
makes this convenience a good urinal, the distinctive one is but little used in
private houses.
The Water- Service to Water- Closets. In the cheaper hoppers the supply is
often directly from the house-
service. In these the trap is
well down, and the flush, if
not certain, there is nothing
B
FIG. 1278.
FIG. 1279.
objectionable to sight. The
water may be let on by hand,
or by an automatic valve con-
nected with sitting down on
the seat, or by opening or shut- FIG. 1280.
ting the closet-door.
As the supply from the service is uncertain if there is a draught in an-
other quarter, it is now more common to have a cistern-supply, shown in
561 ARCHITECTURAL DRAWING.
section, Fig. 1279. B is a ball-cock, operating a valve in the water-pipe, by
which the water is admitted to the cistern whenever the water is below a cer-
tain level ; / is a lever, by which the discharge- valve is raised or lowered ; the
valve-opening is large and the water flows into a service-box, S, filling it, and
at the same time discharging through the p into the closet-basin. When the
valve is closed, the water still continues to flow from the service-box, vent being
given through the air-pipe, which in this case serves also as an overflow.
Fig. 1280 is the section of another cistern, in which the ball, B, or float,
operates a common plug-valve, A. The service-box, D, acts as a sort of a
measure of the quantity of water used. When it is filled by means of the
valve G, the valve H is closed, and then, when the valve H is raised for the
flush of the closet, G is closed. There is an air- vent around the chain or rod
of the valve H, and the overflow E is independent. The supply-pipe is ex-
tended nearly to the bottom of the cistern by a short, loose pipe, as shown at
L, to avoid noise from falling water.
Lighting. It may be needless to say that the light in a building should be
as much as possible from natural sources, as it conduces to health and cleanli-
ness, and economy in conducting any industrial pursuits. But, for artificial
lighting, the present permanent fixtures are usually for the use of gas. In the
distribution of gas through the building, wrought-iron pipes are invariably
used. The old English rule for the sizes of these pipes :
i" 6 feet long 1 outlet. 1" 70 feet long 35 outlets.
f" 20 " 3 outlets. 1" 100 " 60 "
i" 30 " 6 " 1-J" 150 " 100 "
f" 90 " 20 " 2" 200 " 200 "
The couplings to elbows are similar to those used in steam-fitting, but
lighter ; 'the cocks are the common plug-cocks.
Gas fittings are in all forms of brackets and pendants, with any number of
branches, with fixed, swing, and slide joints, and burners in great variety.
Bat-wing and fish-tail tips and Argand burners are the most used, with or
without globes and shades. The Argand must have a chimney. Consump-
tion of gas is commonly from 3 to 6 ft. per hour per burner.
GREEK AND ROMAN ORDERS OF ARCHITECTURE.
In themselves, and for the purposes of construction, the " orders of archi-
tecture " are now of little utility ; but, as examples of proportions of graceful
curves and outlines, they are useful as studies and manual practice for the
draughtsman.
The Tuscan, Doric, Ionic, Corinthian, and Composite orders, are systems
or assemblages of parts subject to certain uniform established proportions,
regulated by the office each part has to perform, consisting of two essential
parts, a column and entablature, subdivided into three parts each : the first
into the base, the shaft, and the capital ; the second into the architrave, or
chief beam, 0, Fig. 1281, which stands immediately on the column ; the
frieze, B, which lies on the architrave ; and the cornice, A, which is the
crowning or uppermost member of an order. In the subdivisions certain
*
ARCHITECTURAL DRAWING.
f -l i
r
c
;*-- -2z% -
r
FIG.
1281.
FIG. i 1282.
fi
FiG.h283.
565
*$
2tfl/
. 1284.J
FIG. 1285.
566 ARCHITECTURAL DRAWING.
horizontal members or moldings are used : thus, the ogee (a), the corona (b),
the ovolo (c), the cavetto (d), with the fillets, compose the cornice ; the fasciae
(ff)> the architrave ; the abacus (g), the ovolo (c), the astragal (ii), and the
neck (h), are the capital of the column ; the torus (k) and the plinth (/) (Fig.
1283) are the base. The character of an order is displayed, not only in its
column, but in its general forms and details, whereof the column is, as it were,
the regulator ; the expression being of strength, grace, elegance, lightness, or
richness. Though a building be without columns, it is nevertheless said to
be of an order, if its details be regulated according to the method prescribed
for such order.
In all the orders a similar unit of reference is adopted for the construction
of their various parts. Thus, the lower diameter of the column is taken as
the proportional measure for all other parts and members, for which purpose
it is subdivided into sixty parts, called minutes, or into two modules of thirty
minutes each. Being proportional measures, modules and minutes are not
fixed ones like feet and inches, but are variable as to the actual dimensions
which they express larger or smaller, according to the actual size of the
diameter of the column. For instance, if the diameter be just five feet, a
minute, being one sixtieth, will be exactly one inch.
To draw an elevation of any one of the orders, determine the diameter of
the column, and from that form a scale of equal parts by sixty divisions, and
then lay off the widths and heights of the different members according to the
proportions of the required order, as marked in the body or on the sides of the
figures.
Figs. 1281 to 1285 are illustrations of the Tuscan order : e, in the frieze
corresponding to the Doric triglyph, may or may not be introduced. Fig.
1281 is an elevation of the capital and entablature ; Fig. 1283 of the base ;
and Fig. 1282 of another capital.
A slightly convex curvature, or entasis, is given in execution to the outline
of the shaft of a column, by classic architects, to counteract a fancied appear-
ance of concave curvature, which might cause the middle of the shaft to ap-
pear thinner than it really is.
Fig. 1284 represents the form of a half-column from the Pantheon at
Rome. In Fig. 1285, another example, the lower third of the shaft is uni-
formly cylindrical. The entasis of the' two thirds is constructed by dividing
the arc, a ft, into equal parts, and the columns into the same number, and pro-
jecting the divisions of the arc on to those of the column. The upper diame-
ter of column or chord at b is 52 minutes.
Figs. 1286 to 1290 exhibit an example of the Doric order, from the Temple
of Minerva, in the Island of Egina. Fig. 1286 is an elevation of the capital
and the entablature ; Fig. 1287 of the base, and a part of the podium ; Fig.
1288 shows the forms of the flutes at the top of the shaft, and Fig. 1289 at
the base ; Fig. 1290 the outline of the capital on an enlarged scale.
The mutules, a a, the triglyphs, b b, the guttae or drops, d d, of the entabla-
ture, the echinus,/, and the annulets, g g, of the capital, may be considered
characteristic of the Doric. The triglyph is placed over every column, and
one or more intermediately over every intercolumn (or span between two
ARCHITECTURAL DRAWING.
567
568
ARCHITECTURAL DRAWING.
ARCHITECTURAL DRAWING. 569
columns), at such a distance from eacli other that the metopes, c, or spaces
between the triglyphs, are square.
In the best Greek examples of the order, there is only a single triglyph over
each intercolumn. The end triglyphs are placed quite up to the edge or outer
angle of the frieze. The mutules are thin plates attached to the under side or
soffit of the corona, over each triglyph and each metope, with the former of
which they correspond in breadth, and their soffits or under surfaces are
wrought into three rows of guttae or drops, conical or otherwise shaped, each
row consisting of six guttae, or the same number as those beneath each triglyph.
The shaft of the Doric column was generally fluted ; the number of channels
is either sixteen or twenty, afterward increased in the other orders to twenty-
four, a center flute on each side of the column.
Figs. 1291 to 1294 exhibit an example of the Ionic order, taken from the
Temple of Minerva Polias, at Athens. Fig. 1291 is an elevation of the capital
and entablature ; Fig. 1292, of the base; Fig. 1293 is a sectional half of the
plan of the column at the base and the top ; Fig. 1294 an elevation of the bal-
uster side of the capital. It differs from the Doric in the more slender pro-
portions of its shaft, and the addition of a base ; but the capital is the indi-
cial mark of the order.
When a colonnade was continued in front and along the flanks of the build-
ing, this form of capital in the end column occasioned an offensive irregularity ;
for while all the other columns on the flanks showed the volutes, the end one
showed the baluster side. It was necessary that the end column should, there-
fore, have two adjoining volute faces, which was effected by placing the volute
at the angle diagonally.
Figs. 1295 and 1296 represent an example of the Corinthian order, from
the Arch of Hadrian, at Athens. This order is distinguished from the Ionic
more by its deep and foliaged capital than by its proportions. The capital is
considerably more than a diameter in height, varying in different examples
from one to one and a half diameter, upon the average about a diameter and a
quarter, and has two rows of leaves, eight in each row, so disposed that of the
taller ones, composing the upper row, one comes in the middle, beneath each
face of the abacus, and the lower leaves alternate with the upper ones, coming
between the stems of the latter ; so that in the first or lower tier of leaves there
is in the middle of each face a space between two leaves occupied by the stem
of the central leaf above them. Over these two rows is a third series of eight
leaves, turned so as to support the small volutes which, in turn, support the
angles of the abacus. Besides these outer volutes, invariably turned diagonally,
there are two other smaller ones, termed caulicoli, which meet each other be-
neath a flower on the face of the abacus. The sides of the abacus are concave
in plan, being curved outward so as to produce a sharp point at each corner,
which is usually cut off.
Fig. 1297 represents one of the capitals of the Tower of the Winds, showing
the earliest formation of the Corinthian capital. In this example the abacus
is square, and the upper row of leaves, of the kind called water-leaves, are
broad and flat, and merely carved upon the vase or body of the capital.
The shaft is, in general, fluted, similarly to that of the Ionic column, but
570
ARCHITECTURAL DRAWING.
ARCHITECTURAL DRAWING.
571
FIG. 1297.
sometimes the flutes are cabled; that is, the channels are hollowed out for only
about two thirds of the upper part of the shaft, and the remainder cut so that
each channel has the appearance of being partly
filled up by a round staff or piece of rope, v^ t *
The cornice is very much larger than in the
other orders, in height and in projection, consist-
ing of a greater number of moldings beneath the
corona, for that and the cymatium over it are in-
variably the crowning members. In Fig. 1295
square blocks or dentels are introduced, but often
to the dentels is added a row of modillions (Fig.
1418), immediately beneath, and supporting the
corona ; and between them and the dentels, and
also below the latter, are other moldings, some-
times cut, at others left plain.
The Composite Order is a union of the Ionic and Corinthian orders. Its
capital consists of a Eoman Ionic one, superimposed upon a Corinthian
foliaged base, in which the leaves are without stalks, placed directly upon the
body of the vase.
The spacing between the columns, or intercolumn, is from one to one and
one half diameters, but modern architects have coupled the columns, making
a wide intercolumn between every pair of columns, so that as regards the
average proportion between solids and voids, that disposition does not differ
from what it would be were the columns placed singly. Supercolumniation,
or the system of piling up orders, or different stages of columns one above an-
other, was employed for such structures merely as were upon too large a scale
to admit of the application of columns at all as their decoration, otherwise than
by disposing them in tiers.
The Greeks seldom employed human figures to support entablatures or
beams ; the female figures, or Caryatides, are almost uniformly represented
in an erect attitude, without any apparent
effort to sustain any load ; while the male fig-
ures, Telamones or Atlantes, display strength
and muscular action. Besides entire figures,
either Hermes' pillars or Termini are occa-
sionally used as substitutes for columns of
the usual form, on a moderate scale. The
first mentioned consist of a square shaft with
a bust or human head for its capital ; the lat-
ter of a half-length figure rising out of, or ter-
minating in, a square shaft tapering down-
ward. Hermes' pillars are frequently em-
ployed by modern architects for the decora-
tion of window architraves.
The Romans introduced circular forms and
curves, not only in elevation and section, but in plan. The true Roman order
consists, not in any of the columnar ordinances, but in an arrangement of
572
ARCHITECTURAL DRAWING.
timOOBAIflBromiAW^
FIG. 1300.
two pillars (Fig. 1298) placed at a distance from one another nearly equal to
their own height, and having a very long entablature, which, in consequence,
required to be supported in the center by an arch springing from piers.
Figs. 1299, 1300, and
1301, from the Palace of
Diocletian at Spalatro, are
illustrations 7 of the differ-
ent modes of treatment of
the arch and entablature.
Perhaps the most satis-
factory works of the Ro-
mans are those which we
consider as belonging to civil
engineering rather than to
architecture their aque-
ducts and viaducts, all of
which, admirably conceived and
executed, have furnished practi-
cal examples for modern con-
structions, of which the High
Bridge across Harlem Eiver may
be taken as an illustration.
The history of Roman archi-
tecture is that of a style in course of
transition, beginning with purely pagan
or Grecian, and passing into a style
almost wholly Christian. The first
form of Christian art was ths Roman-
esque, which afterward branched off
into the Byzantine and the Gothic.
The Romanesque and Byzantine, as
far as regards the architectural features,
are almost synonymous ; in the earlier
centuries there is an ornamental dis-
tinction. In its widest signification,
the Romanesque is applied to all the
earlier round-arch developments, in contradistinction to the Gothic or later
pointed arch varieties of the North. In this view the Norman is included in
the Romanesque.
The general characteristics of the Gothic are its essentially pointed or ver-
tical tendency, its geometrical details, its window- tracery, its openings, its
cluster of shafts and bases, its suits of moldings, the universal absence of the
dome, and the substitution of the pointed for the round arch.
The Romanesque pillars are mostly round or square, and, if square, gener-
ally set evenly, while the Gothic square pillar is set diagonally.
Figs. 1302 to 1306 represent sections of Gothic pillars. Fig. 1307 is half
of one of the great western piers of the Cathedral of Bourges, measuring 8 feet
ARCHITECTURAL DRAWING.
573
FIG. 1302.
FIG. 1303.
FIG. 1304.
FIG. 1305.
FIG. 1306.
1
\
1
r
J j
i
a
i
-v-
=)
1
1
B
1
FIG. 1307.
FIG. 1308.
FIG. 1309.
FIG. 1311.
FIG. 1313.
FIG.
1314.
574:
ARCHITECTURAL DRAWING.
on each side. Figs. 1308 and 1309 are elevations of capitals and bases, and sec-
tions of Gothic pillars ; one from Salisbury, the other from Lincoln Cathedral.
Figs. 1310, 1311, and 1312 are examples of Byzantine capitals ; Fig. 1313
a Norman one, from Winchester Cathedral ; and Fig. 1314 a Gothic capital
and base, from Lincoln Cathedral.
FIG. 1315.
FIG. 1316.
FIG. 1317.
Arches are generally divided into the triangular-headed arch, the round-
headed arch, and the pointed arch. Of the round-headed arch, there are
semicircular, segmental, stilted (Fig. 1315), and horseshoe (Fig. 1316). Of
the two-centered pointed, the equilateral (Fig. 1317), the lancet, and the ob-
tuse. Of the first, the radii of the seg-
ments forming the arch are equal to the
breadth of the arch, of those of the lan-
cet longer, and of the obtuse shorter.
Of the complex arches, there are the
ogee (Fig. 1318) and the Tudor (Fig.
1319). The Tudor arch is described from
four centers, two on a level with the
spring and two below it.
Of foiled arches, there are the round-headed trefoil (Fig. 1320), the pointed
trefoil (Fig. 1321), and the square-headed trefoil arch (Fig. 1322). The points
c are termed cusps.
The semicircular arch is the Koman Byzantine and Norman arch ; the ogee
FIG. 1318.
FIG. 1319.
FIG. 1320.
FIG. 1321.
FIG. 1322.
and horseshoe are the profiles of many Turkish and Moorish domes; the pointed
and foliated arches are Gothic.
Domes and Vaults. The Greek vaulting consisted wholly of spherical sur-
faces, the Koman of cylindrical ones. Figs. 1323 and 1324 illustrate this dis-
tinction, Fig. 1323 being the elevation of a Roman cylindrical cross-vault, and
Fig. 1324 the elevation of the roof of the church of St. Sophia at Constantino-
ple ; and the sprouting of domes out of domes continues to characterize the
Byzantine style. As a constructive expedient the cross-vault is to be preferred,
ARCHITECTURA1
as the whole pressure and thrust are colL
plied at the angles only, so that it might be
placed in
strong enough
AWING.
:
no
in four definite resultants, a;
rted by four flying buttresses,
3tioii of these resultants, n
rushed by the p:
FIG. 1323.
FIG. 1324.
FIG. 1325.
Fig. 1325 represents a compartment of the simplest Gothic vaulting a, a,
groin ribs ; #, I, b, side ribs.
The Romans introduced side ribs, appearing on
the inside as flat bands, and harmonizing with the
similar form of pilasters in the walls, but they never
used groin ribs ; the Gothic builders introduced
these, and deepened the Roman ribs. The impene-
tration of vaults, either round or pointed, produces
elliptical groin lines, or else lines of double curva-
ture ; yet the early Gothic architects made their
groin ribs usually simple pointed arches of circular
curvature, thrown diagonally across the space to be
groined, and the four side arches were equally simple, the only care being that
all the arches should have their vertices at the same level. The strength de-
pended on the ribs, and the shell was made quite light, often not more than
six inches, while Roman vaults of the same span would have been three or four
feet. The Romans made their vault surfaces geometrically regular, and left
the groins to take their chance ; while the early Gothic architects made their
groins geometrically regular, and let the intermediate surfaces take their chance.
In the next step the groin ribs were elliptical, and when intermediate ribs
or tiercerons were inserted, these ribs had also elliptical or cylindrical curva-
tures, diiferent from the groins, and the ribs were placed near each other, in
order that the portion of the vault between each pair might practically be
almost cylindrical. In the formation of the compound circular ribs three con-
ditions were to be observed : 1. That the two arcs should have a common tan-
gent at the point of meeting. 2. That the feet of all the ribs should have the
same radius, up to the level at which they completely separate from each other.
3. That from this point upward their curvature should be so adjusted as to
make them all meet their fellows on the same horizontal plane, so that all the
ridges of the vaults may be on one level.
The geometrical difficulty of such works led to what is called fan-tracery
vaulting. If similar arches spring from each side of the pillars (Fig. 1325),
the portion of vault springing from each pillar would have the form of an in-
verted concave-sided pyramid, its horizontal section at every level being square.
The later architects, by converting this section into a circle, the four-sided
576
ARCHITECTURAL DRAWING.
FIG. 1326.
pyramid became a conoid, and all the ribs forming the conoidal surface became
alike in curvature, so that they all might be made simple circular arcs ; these
ribs are continued with unaltered curvature till they meet and form the ridge ;
but in this case the ridges are not level, but
gradually descend every way from the center
point (Fig. 1326).
In the figure this is not fully carried out,
for no rib is continued higher than those
over the longer sides of the compartment, so
that a small lozenge is still left, with a boss
at its center. When the span of the main
arch 1) a was large in proportion to that of
b c, the arch b c became a very acute lancet
arch, scarcely admitting windows of an ele-
gant or sufficient size. To obviate this, the
compound curve was again introduced.
The four-centered arch is not necessarily flat or depressed ; it can be made
of any proportion, high or low, and always with a decided angle at the vertex.
In general, the angular extent of the lower curve is not more than 65, nor less
than 45. The radius of the upper curve varies from twice to more than six
times the radius of the lower. The projecting points of the trefoil arch, or
cusps, are often introduced for ornament merely, but serve constructively, both
in vaults and arches, as a load for the sides, to prevent them rising from the
pressure on the crown.
As vaultings, in general, were contrived to collect
the whole pressure of each compartment into four sin-
gle resultants, at the points of springing, leaving the
walls so completely unloaded that they are required
only as inclosures or screens, they might be entirely
omitted or replaced by windows. Indeed, the real sup-
porting walls are broken into narrow strips, placed at
right angles to the outline of the building, and called
buttresses, and the inclosing walls may be placed either
at the outer or inner edge of the buttresses. The first,
that adopted by the French architects, gave deep re-
cesses to the interiors, while the other, or English
method, served to produce external play of light and
shade.
The Norman buttress (Fig. 1327) resembles a flat
pilaster, being a mass of masonry with a broad face,
slightly projecting from the wall. They are, generally,
of but one stage, rising no higher than the cornice,
under which they often, but not always, finish with a
slope. Sometimes they are carried up to, and terminate
in, the corbel table.
Fig. 1328 represents a buttress in two stages, with slopes as set-offs.
Fig. 1329 is a buttress of the Early English style, having a plain triangular
FIG. 1327.
ARCHITECTURAL DRAWING.
577
FIG. 1333.
or pedimental head. The angles were sometimes chamfered
off, and sometimes ornamented with slender shafts. In but-
tresses of different stages, the triangular head or gable is used
as a finish for the intermediate stages.
In the Decorated style, the outer surfaces of the buttresses
are ornamented with niches, as in Fig. 1330. In the Perpen-
dicular style the outer surface is often partially or wholly cov-
ered with panel-work tracery (Fig. 1331).
The buttress was a constructive expedient to resist the
thrust of vaulting ; but to resist the thrust of the principal
vault, or that over the nave or central part of the church,
buttresses of the requisite depth would have filled up the side
FIG. 1329.
FIG. 1330.
FIG. 1331.
FIG. 1332.
aisles entirely. To obviate this, the system of flying but-
tresses was adopted ; that is, the connection of the interior
with the outer buttress, by an arch or system of arches, as
shown in Fig. 1332. The outer piers were surmounted by
pinnacles, to render them a sufficiently steady abutment to
the flying arches.
The earlier towers of the Romanesque style were construct-
ed without spires. All are square in plan, and extremely
similar in design. Fig. 1333 is an elevation of the tower
attached to the church of Sta. Maria, in Cosmedin, and is one
of the best and most complete examples of this style. It is
15 feet broad and 110 feet high. These towers are the types
of the later Italian campaniles, generally attached to some
578
ARCHITECTURAL DRAWING.
angle of churches ; if detached, so placed that
they still form a part of the church design.
Sometimes they are but civic constructions, as
belfries, or towers of defense. The campanile is
square, carried up without break or offset to two
thirds, at least, of its intended height ; it is gen-
erally solid to a considerable height, or with only
such openings as serve to admit light to the stair-
cases. Above this solid part one round window
is introduced in each face ; in the next story,
two ; in the one above this, three ; then four,
and lastly five ; the lights being separated by
slight piers, so that the upper story is virtually
an open loggia.
The Gothic towers have projecting buttress-
es, frequent offsets, lofty spires, and a general
FIG. 1334.
FIG. 1335.
pyramidal form. Fig. 1334 is the front eleva-
tion of a simple English Gothic tower ; here the
plain pyramidal roof, rising at an equal slope on
eacji of the four sides, is intersected by an octag-
onal spire of steep pitch. The first spires were
simple quadrangular pyramids ; afterward the an-
gles were cut off, and they became octagonal, and
this is the general Gothic form of spire. Often,
instead of intersecting the square roof, as in the
figure, the octagonal spire rests upon a square
base, and the angles of the tower are carried up
by pinnacles, or the sides by battlements, or by
both, as in Fig. 1335, to soften the transition be-
tween the perpendicular and sloping part.
In general the spires of English churches are
more lofty than those on the Continent ; the
angle at the apex in the former being about 10,
and in the latter about 15. The apex angle of
ARCHITECTURAL DRAWING.
Fm. 1338. FIG. 1339. FIG. 1337-
579
FTG. 1341,
FIG. 1342. FIG. 1343.
FIG. 1340.
FIG. 1344.
580
ARCHITECTURAL DRAWING.
the spires of Chichester and Lichfield are from 12 to 13, or a mean between
the two proportions, and, according to Ferguson, more pleasing than either.
Although having more lofty spires, yet the English construction is much more
massive in appearance than the Continental ; the apertures are less numerous,
and the surfaces are less cut
up and covered with orna-
ments. The spires of Fri-
berg Church, and many oth-
ers on the Continent, are
open work.
Figs. 1336 and 1337 are
bell-cots. Figs. 1338 to 1344
are spires. Fig. 1345 is an
FIG. 1
FIG. 1347.
is applied
service of
FIG. 1349.
apse, or circular end of a
church, from German Gothic
examples.
Figs. 1346 and 1347 are
examples of spire finials, with
weather-cocks.
Figs. 1348 and 1349 are
examples of towers not con-
nected with church edifices.
Fig. 1350 is a tower of
very recent construction, and
to the utilitarian purpose of sustaining a water-tank for the highest
the Croton in New York city.
FIG. 1348.
ARCHITECTURAL DRAWING.
581
FIG. 1350.
Fig. 1351 represents the upper portion of the tower of
Ivan Veliki, at Moscow. The Russian towers are generally
constructed independent of their churches, and are intend-
ed for the reception of their massive bells.
FIG. 1352.
Pllnllil
FIG. 1351.
FIG. 1353.
Windows. Before the use of painted glass, as very
small apertures sufficed for the introduction of the required
quantity of light into a church, the windows of the Roman-
esque churches were generally small, and devoid of tracery ;
and as the Byzantine architects, adorning the walls with
paintings, could not use stained glass, they followed in gen-
eral form the Romanesque window, apertures with circular heads, either single
or in groups (Fig. 1353 or Fig. 1352). The Norman windows were also small,
each consisting* of a single light, semicircular in the head, and placed as high
as possible above the ground ; at first splayed on the inside only, afterward
the windows began to be recessed with moldings and jamb-shafts in the angles,
as in Fig. 1353.
The Lancet, in general use in the early Gothic period, was of the simplest
arrangement : in these windows the glass was brought within three or four
inches of the outside of the wall, and the openings were widely splayed in the
interior. The proportions of these windows vary considerably ; in some the
height being but five times the width, in others as much as eleven ; eight or
nine times may be taken as the average. Lancet windows occur singly (Fig.
1354), or in groups of two, three, five, and seven, rarely of four and six. The
triplet (Fig. 1355) is the most beautiful arrangement of lancet windows. It
582
ARCHITECTUEAL DRAWING.
was customary to mark with greater importance the central light, by giving it
additional height, and in most cases increased width also. In some examples
the windows of a lancet triplet are placed within one drip-stone forming a sin-
FIG. 1355.
FIG. 1354.
FIG. 1356.
gle arch, thus bearing a strong resemblance to a single three-light window.
The first approximation to tracery appears to have been the piercing of the space
over a double lancet window comprised within a single drip-stone (Fig. 1356).
A traceried window is a distinctive characteristic of Gothic architecture ;
with the establishment of the principle of window tracery the mullions were
recessed from the face of the wall in which the window arch was pierced, and
the fine effect thus produced was speedily enhanced by the introduction of dis-
tinct orders of mullions, and by recessing certain portions of the tracery from
the face of the primary mullions and their corresponding tracery bars.
Decorated window tracery is divided into two chief varie-
FIG. 1357.
FIG. 1358.
FIG. 1359.
ties, Geometrical and Flowing ; the former consisting of geometrical figures,
as circles, trefoils, quatrefoils, curvilinear triangles, lozenges, etc. ; while in
flowing tracery these figures, though still existing, are gracefully blended to-
gether in one design.
ARCHITECTURAL DRAWING.
583
Fig. 1357 represents a quatrefoil window, Fig. 1358 a pointed trefoil in out-
line with the centers of the different circles and such constructive lines indi-
cated as may be necessary. Fig. 1359 represents two forms of circular win-
dows, or roses tournantes.
Fig. 1360 represents an example of the earlier decorated tracery window-
head, consisting of two foiled lancets, with a pointed quatrefoil in the span-
drel between them. One half of
the windows in this, as in some of
FIG. 1360.
FIG. 1361.
the following figures, is drawn in skeleton, to explain their construction.
Fig. 1361 is another example of Decorated tracery.
Fig. 1362 is an example of the English leaf tracery ; Fig. 1363, of the
French flamboyant. The difference between the two styles is, that while the
upper ends of the English loops or leaves are
round, or simply pointed ; the upper ends of
FIG. 1362.
FIG. 1363.
FIG. 1364.
the latter terminate, like their lower ones, in angles of contact, giving a flame-
like form to the tracery bars and form pieces.
In England the Perpendicular style succeeded the Decorated ; the mullions,
instead of diverging in flowing or curvilinear lines, are carried up straight
through the head of the windows ; smaller mullions spring from the head of
the principal lights, and thus the upper portion of the window is filled with
panel-like compartments. The principal as well as the subordinate lights are
foliated in their heads, and large windows are often divided horizontally by
transoms. The forms of the window arches vary from simple pointed to the
complex four-centered, more or less depressed.
Fig. 1364 is an example of a Perpendicular window.
584
ARCHITECTURAL DRAWING.
FIG. 1365.
FIG. 1366.
FIG. 1367.
FIG. 1368.
Fig. 1365 is a square-headed window, such
as were usual in the clear-stories of Perpen-
dicular architecture.
Figs. 1366 and 1367 are quadrants of cir-
cular windows, used more especially in France,
for the adornment of the west ends and tran-
septs of the cathedrals.
Besides the tracery characteristic of Gothic
architecture, there is a tracery peculiar to the
Saracenic and Moorish style, of which Fig.
1368 may be taken as an example it being a
window of one of the earliest mosques. The
general form of the window and door-heads of
this style is that of the horse-shoe, either circular or pointed.
Doorways. Fig. 1369 is the eleva-
tion of a circular-headed doorway, which
FIG. 1370.
FIG. 1369.
FIG. 1371.
ARCHITECTURAL DRAWING.
585
may be considered the type of many entrances both in Romanesque, Gothic,
and later styles. It consists of two or more recessed arches, with shafts or
moldings in the jambs. In the earlier styles the arches were circular, in the
later Gothic, generally pointed, but sometimes circular ; in the earlier, the
angles in which the shafts are placed are rectangular ; in the later, the shaft
is often molded on a chamfer plane, that is, a plane inclined to the face of the
wall, generally at an angle of 45 ; often the chamfer and rectangular planes
are used in connection.
Fig. 1370 is a simple head of a depressed four-centered or Tudor-arched
doorway, with a hood molding.
Fig. 1371 represents the incorporation of a window and doorway. Some-
times the doorway pierces a buttress ; in that case, the buttress expands on
either side, forming a sort of porch. The Gothic architects placed doors
where they were necessary, and made them subservient to the beauty of the
design.
Fig. 1372 is an example of a gabled doorway with crockets and finial.
FIG. 1372.
FIG. 1373.
Fig. 1373 is an example of a perpendicular doorway, with a label or hood
molding above, and ornamented spandrels.
Fig. 1374 is an example of a Byzantine, and Fig. 1375 of a Saracenic
doorway.
The Renaissance style was, originally, but the revival or a fair rendering of
the classical orders of architecture, with ornaments from the Byzantine and
Saracenic styles.
Garbett divides this style into three Italian schools, the Florentine, Vene-
tian, and Roman. The Florentine admits of little apparent ornament, but
any degree of real richness, preserving in its principal forms severe contrast ;
powerful masses self-poised without corbeling, without arching ; breadth of
everything, of light, of shade, of ornament, of plain wall ; depth of recess in
the openings, of perspective in the whole mass, of projection in the cornice.
Absence of features useless to convenience or stability, admitting of great
plainness, or of very florid enrichment.
586
ARCHITECTURAL DRAWING.
The aim of the Venetian school was splendor, variety, show, and ornament ;
not so much real as effective ornament. Thus, it rarely contains as much
carving or minute enrichment as the Florentine admits ; but it has larger
ornaments, constructed (or built) ornaments, great features useless except for
ornament, such as inaccessible porticoes, detached columns, and architraves
supporting no ceiling, towers built only for breaking an outline.
FIG. 1374.
FIG. 1375.
The Roman school is intermediate in every respect between the two other
schools. It is better adapted to churches than to any other class of buildings.
This fitness arises from the grand, simple, and unitary effect of one tall order,
generally commencing at or near the ground, obliterating the distinction of
two or three stories, making a high building appear a single story.
Moldings. "All classical and Romanesque architecture is composed of
bold independent shafts, plain or fluted, with bold detached capitals forming
arcades or colonnades where they are needed, and of walls whose apertures are
surrounded by courses of parallel lines called moldings,
and have neither shafts nor capitals. The shaft system
and molding system are entirely separate ; the Gothic
architects confounded the two ; they clustered the shafts
till they looked like a group of moldings, they shod and
capitaled the moldings till they looked like a group of
shafts." The moldings appear in almost every conceivable
position ; from the bases of piers and piers themselves, to
the ribs of the fretted vaults which they sustain.
In the earliest examples of Norman doorways the jambs
are mostly simply squared back from the walls ; recessed
jambs succeeded, and are common in both Norman and Gothic architecture ;
and when thus recessed, detached shafts were placed in each angle (Fig. 1376).
FIG. 1376.
ARCHITECTURAL DRAWING.
587
In the later styles the shafts were almost invariably attached to the structure.
The angles themselves were often cut or chamfered off, and the moldings
attached to the chamfer-plane. The arrangement of window jambs, during
the successive periods, was in close accordance with that of doorways.
In the richer examples small shafts were introduced, which, rising up to
the springing of the window, carried one or several of the arch moldings. Yet
FIG. 1385.
FIG. 1384.
moldings are nevertheless not essential accessories ;
many windows of the richest tracery have their mull-
ions and jambs composed of simple chamfers.
Figs. 1377 to 1385 are examples of arch and
architrave moldings, which, even when not continuous, partook of the same
general arrangement as those in the jambs, with greater richness of detail.
When shafts were employed, they carried groups of moldings more elaborate.
than those of the jambs, though still falling on the same planes.
88
ARCHITECTURAL DRAWING.
Capitals were either molded or carved with foliage, animals, etc. ; they
always consisted of three distinct parts (Fig. 1386) the head mold (A), the
bell (B), and the neck mold (C). In Norman examples the
head mold was almost invariably square ; in the later styles
it is circular, or corresponding to the form of the pillar.
Bases consist of the plinth and the base moldings. The
plinth was square in the Norman style, afterward octagonal ;
then, assuming the form of the base moldings, it bent in
and out with the outline of the pier. Base moldings were
also extensively used round the buttresses, towers, and walls
of churches.
String Courses, of which Figs. 1387 to 1392 are exam-
ples, were horizontal courses in the face of a wall ; the most
usual position being under the windows. In the Norman styles they were usu-
ally heavy in the outline ; in the later styles they were remarkably light and
elegant ; free from restraint or horizontally they now rose close under the sill
of the window, and then suddenly dropping to accommodate themselves to the
FIG. 1386.
FIG. 1387.
FIG. 1388.
FIG. 1389.
FIG. 1390.
FIG. 1391.
FIG. 1392.
arch of a low doorway, and again rising to run immediately under the adjoin-
ing window. In this way the string courses frequently served the purpose of
a drip-stone or hood molding over doors ; occasionally the hood mold was con-
tinued from one window to the other.
Cornices are not an essential feature in Gothic architecture. In the Nor-
man and early English styles, the cornice was a sort of enlarged, projecting
string course, forming a drip-stone beneath the roof, which, if
supported on brackets or corbels, was termed the corbel table.
The earliest molding in Norman work is a circular bead
strip, worked out of the edges of a recessed arch, called a cir-
cular bowtel (Fig. 1393). From a circular form the bowtel soon
became pointed, and, by an easy transition, into the bowtel of
one, two, or three fillets.
Figs. 1394 to 1399 are sections of Eomanesque drip- or cap-
stones, adapted to different pitches of roof.
FIG. 1393.
ARCHITECTURAL DRAWING.
FIG. 1397.
FIG. 1398.
Fig. 1400 is the scroll molding ; a simple filleted bowtel, with the fillet
undeveloped on one side, as shown by the dotted lines. If this molding be
cut in half, through the center of the fillet, we have on the developed side the
molding now termed by carpenters the rule joint, which, by rounding off the
corners by reverse curves, becomes the
wave molding.
FIG. 1400.
Fig. 1401 is a Gothic example of
the filleted bowtel with prominent
alternate hollows. FIG. 1401. FIG. 1402.
Fig. 1402 is an example of the
perpendicular style, an insignificant hollow separating groups of moldings.
590
ARCHITECTURAL DRAWING.
Figs. 1403 to 1408 are examples of molded timbers, used largely in open-
timbered roofs and for exposed beams. It is still the custom, when the fram-
ing is not covered in with plastering or ceiling, to corner the edges of the joists
and beams, at an angle of 45, for about I" on each face, but not extending it
olose to the joint or wall ; this is called stop-chamfering.
FIG. 1403.
FIG. 1405.
FIG. 1407.
FIG. 1408.
Ornament. Architectural ornament is of two kinds, constructive and deco-
rative. By the former is meant all those contrivances, such as capitals, brack-
ets, vaulting-shafts, and the like, which serve to explain or give expression to
the construction ; by the latter, such as moldings, frets, foliage, etc., which
give grace and life, either to the actual constructive form, or to the construct-
ive decoration. Moldings of the different styles have been already treated of ;
it is proposed to give now what are even more purely decorations of a style.
In the Grecian orders the Doric (Fig. 1286) has the triglyph mutules and
guttae ; the Ionic (Fig. 1291) has various moldings of the cornice, frieze, abacus,
and neck of the column enriched. The principal ornament of the neck of the
column is the anthemion, commonly known, in its most simple form, as the
FIG. 1409.
FIG. 1410.
honeysuckle or palmetto ; in the anthemion, as represented in the figure, the
palmetto alternates with the lily or some analogous form. The ornament
of the abacus is the egg and dart (Fig. 1409) ; the ornament of the frieze and
ARCHITECTURAL DRAWING.
591
'cornice (Fig. 1410). The fret (Fig. 1411) and the guilloche (Fig. 1412) are
also common Greek ornaments, used to adorn the soffits of beams and ceilings.
FIG. 1413.
FIG. 1414.
The acanthus is the distinctive ornament of the Corinthian, of which a leaf is
represented in front and side view (Figs. 1413 and 1414).
1415.
FIG. 1416.
FIG
592
ARCHITECTURAL DRAWING.
Figs. 1415, 1416, and 1417 are the side elevation, front elevation, and sec-
tion of a Greek bracket, the principal ornaments of which are taken from the-
anthemion and acanthus.
Fig. 1418 is an elevation of a por-
tion of an enriched cornice from the
FIG. 1418.
FIG. 1419.
temple of Jupiter Stator, at Rome, of the Corinthian order of architecture.
Fig. 1419 is the under side of the modillion, on a larger scale.
The chief characteristic of Roman ornament is its uniform magnificence, an
enrichment of the Greek. The most used elements of the Roman decorations,
are the scroll and the acanthus. The acanthus of the Greeks is the narrow
prickly acanthus ; that of the Roman, the soft acanthus. For capitals the
Roman acanthus is commonly composed of conventional clusters of olive-leaves.
Fig. 1420 represents a Roman acanthus scroll.
FIG. 1420.
The free introduction of monsters and animals is likewise a characteristic
of Greek and Roman ornament, as the sphinx, the triton, the griffin, and oth-
ers ; they occur, however, more abundantly in the Roman.
Symbols are the foundation of decorations in the Byzantine and Romanesque.
The early symbols were the monogram of Christ, the lily, the cross, the ser-
pent, the fish, the aureole, or vesica piscis, and the circle or nimbus, the trefoil
and quatrefoil, the first having reference to the Trinity, the second to the four
Evangelists. Occasionally the symbolic images of the Evangelists, the
ARCHITECTURAL DRAWING.
593
the lion, the ox, and the eagle, are represented within these circles. The hand
in the attitude of benediction, and the lily (the fleur-de-lis), the emblem of the
virgin and purity, are common ; also a peculiarly formed leaf, somewhat resem-
bling the leaf of the ordinary thistle. The serpent figures largely in Byzantine
art as the instrument of the fall, and one type of the redemption.
Pagan ornaments, under certain symbolic modifications, were admitted into
Christian decorations. Thus the foliations of the scroll were terminated by
lilies, or by leaves of three, four, and five blades, the number of blades being
significant ; and in a similar way the anthemion and every other ancient orna-
ment. In the Byzantine, all their imitations of natural forms were invariably
conventional ; it is the same even with animals and the human figure ; every
saint had his, prescribed colors, proportions, and symbols.
FIG. 1421.
FIG. 1422.
The Saracenic was the period of gorgeous diapers (Figs. 1421 and 1422), for
their habit of decorating the entire surfaces of their apartments was highly favor-
able to the development of this class of design. The Alhambra displays almost
endless specimens, and all are in relief and enriched with gold and color, chiefly
blue and red. The religious cycles and symbolic figures of the Byzantine are
excluded. Mere curves and angles or interlacings were now to bear the chief
burden of a design, but distinguished by a variety of color. The curves, how-
ever, very naturally fell into standard forms and floral shapes, and the lines
and angles were soon developed into a very characteristic species of tracery, or
interlaid strap-work, very agreeably diversified by the ornamental introduction
88
594:
ARCHITECTURAL DRAWING.
of the inscriptions, which last custom of elaborating inscriptions with their
designs was peculiarly Saracenic. Although flowers were not palpably admit-
ted, yet the great mass of the minor details of Saracenic designs are composed
of flower forms disguised the very inscriptions are sometimes thus grouped as
flowers ; still, no actual flower ever occurs, as the exclusion of all natural im-
ages is fundamental to the style in its purity.
All the symbolic elements of the Byzantine are continued in the Gothic.
Ornamentally, the Gothic is the geometrical and pointed element elaborated to
the utmost ; its only peculiarities are its combinations of details ; at first the
conventional and geometrical prevailing, and afterward these combined with
the elaboration of natural objects in its decoration. The most striking feature
of all Gothic work is the wonderful elaboration of its geometric tracery ; vesi-
cas, trefoils, quatrefoils, cinquefoils, and an infinity of geometric varieties be-
sides. The tracery is so paramount a characteristic that the three English
varieties, the early English, the decorated, and the perpendicular, and the
French flamboyant, are distinguished almost exclusively by this feature. (See
Figs. 1360 to 1364.)
The ornamental moldings used in the decorative details are numerous,
among which the more common is the chevron or zigzag (Fig. 1423), simple
FIG. 1423.
FIG. 1424.
FIG. 1425.
FIG. 1426.
FIG. 1428.
FIG. 1429.
FIG. 1430.
as the indented, or du-
plicated, triplicated, or
quadrupled ; the billet,
the prismatic billet, the
square billet, and the
alternate billet (Fig. 1424) ; the star (Fig. 1425), the fir-cone ; the cable (Fig.
1426) ; the embattled (Fig. 1427) ; the nail-head (Fig. 1428), the dog-tooth
(Fig. 1429) ; the ball-flower (Fig. 1430), and the serpentine vine-scroll.
The crocket, in its earliest form, was the simple arrow-head of the episco-
pal pastoral staif ; subsequently finished with a trefoil, and afterward still fur-
ther enriched. Figs. 1431 and 1432 are early English crockets ; Fig. 1433 a
decorated one. Fig. 1434 is a finial of the same style. Both finials and crock-
ets in detail display a variety of forms.
The parapets of the early English style are often a simple horizontal course,
supported by a corbel table, sometimes relieved by a series of sunk blank trefoil-
headed panels ; sometimes a low embattled parapet crowns the wall. In the
decorated style the horizontal parapet is sometimes pierced with trefoils, some-
ARCHITECTURAL DRAWING.
595
times with wavy, flowing tracery (Fig. 1435). Grotesque spouts or gargoyles
discharge the water from the gutters. The parapets of the perpendicular style
FIG. 1431.
FIG. 1433.
FIG. 1432.
FIG. 1434.
are frequently embattled (Fig. 1436), covered with sunk or pierced paneling,
and ornamented with quatrefoil, or small trefoil-headed arches ; sometimes
FIG. 1436.
FIG. 1435.
not embattled but covered with sunk or pierced
quatrefoils in circles, or with trefoils in triangular
spaces, as in Fig. 1437.
Among the varieties of ornamental work, the
mode of covering small plain surfaces with diaper-
ing (Fig. 1438) was sometimes used ; the design
being in exact accordance with the architectural
FIG. 1438.
FIG. 1437.
K FIG. 1439
features and details of the style. The rose (Fig. 1439), the badge of the
houses of York and Lancaster, is often met with in the perpendicular style ;
and tendrils, leaves, and fruit of the vine are carved in great profusion in the
596
ARCHITECTURAL DRAWING.
hollows of rich cornice moldings, especially on screen-work in the interior of a
church. Fig. 1440, in its original type a Byzantine ornament, an alternate
lily and cross, is a common finish to the cor-
nice of rich screen-work in the latest Gothic,
and is known under the name of the Tudor
flower.
Sculptured foliage (Figs. 1441 to 1446)
is much used in capitals, brackets, corbels,
bosses, and crockets. Among the forms of
FIG. 1440 foliage the trefoil is most predominant.
FIG. 1441.
FIG. 1442.
FIG. 1443.
FIG. 1444.
FIG. 1445.
FIG. 1446.
FIG
1447.
The Ornaments of the Renais-
sance. The term Renaissance is
used in a double sense ; in a general
sense implying the revival of art,
and specially signifying a peculiar
style of ornament. It is also some-
times, in a very confined sense, ap-
plied in reference to ornament of
the style of Benvenuto Cellini ; or,
as it is sometimes designated, the
Henry II (of France) style.
The mixture of various elements
is one of the essentials of this style.
These elements are the classical or-
naments ; unnatural and natural
flowers and foliage ; men and ani-
mals, natural and grotesque ; car-
ARCHITECTURAL DRAWING.
597
touches, or pierced and scrolled shields, in great prominence ; tracery inde-
pendent, and developed from the scrolls of the cartouches ; and jewel forms
(Fig. 1447 and 1448).
The Elizabethan is a partial elaboration of the same style ; the present Eliza-
bethan exhibits a very striking preponderance of strap and shield work ; but
FIG. IMS.
the earlier is much nearer allied to the Continental styles of the time, classical
ornaments but rude in detail, occasional scroll and arabesque work, and strap-
work, holding a much more prominent place than the pierced or scrolled
shields. Fig. 1449 is an
example of the style from
the old guard chamber,
'Westminster. *~7^rjymx^\. ^^ ~w iv (<r~$ t\m/s*\> .
FIG. 1449.
FIG. 1450.
FIG. 1451.
Of the earliest and transition styles of Eenaissance ornament are the Tri-
cento and the Quatrecento ; the great features of the first are its intricate tra-
cery and delicate scroll-work of conventional foliage, the style being but a slight
598
ARCHITECTURAL DRAWING.
remove from the Byzantine and Saracenic ; of the second, elaborate natural
imitations of fruit, flowers, birds, or animals (Fig. 1450), all disposed simply
with a view to the ornamental ; also occasional cartouches, or scrolled shield-
work.
The Renaissance is something more approximative to a combination of pre-
vious styles than a revival of any in particular, developed solely on aesthetic
principles, from a love of the forms and harmonies themselves, as varieties of
effect and arrangements of beauty, not because they had any particular signi-
fication, or from any superstitious attachment to them as heirlooms.
Fig. 1451 is an example of ornament in the Cinquecento style. The ara-
besque scroll-work is the most prominent feature of the Cinquecento, and with
this in its elements, it combines every other feature of classical art, with the
unlimited choice of natural and conventional imitations from the entire animal
and vegetable kingdom, both arbitrarily disposed and combined. Absolute
works of art, such as vases and implements, and instruments of all kinds, are
prominent elements of the Cinquecento arabesque, but cartouches and strap-
work wholly disappear from the best examples. Another chief feature of the
Cinquecento is the admirable play of color in
its arabesques and scrolls ; and it is worthy of
note that the three secondary colors, orange,
green, and purple, perform the chief parts in
all the colored decorations.
Fig. 1452 is an example of the Louis Qua-
torze style of ornament. The great medium
of this style was gilt stucco-work, and this
absence of color seems to have led to its most
striking characteristic, infinite play of light,
of shade ; color, or mere beauty of form in
detail, having no part in it whatever. Flat
surfaces are not admitted ; all are concave or
convex : this constant varying of the surface
gives every point of view its high lights and
brilliant contrasts.
The Louis Quinze style differs from that
of Louis Quatorze chiefly in its absence of
symmetry ; in many of its examples it is an
almost random dispersion of the scroll and shell, mixed only with that peculiar
crimping of shell-work, the coquillage.
The ornaments of which we have thus given examples are, in general, ap-
plied to interior decorations, to friezes, pilasters, panels, architraves, the faces
and soffits of arches, ceilings, etc., to furniture, and to art-manufactures in
general. For exteriors these ornaments are sparingly applied ; shield and scroll
work, of the later Elizabethan or Renaissance style, is sometimes used, but
very seldom tracery.
Principles of Design. Professedly treating of architecture only in its most
mechanical phase of drawing, the history of it as an art, and the distinctions
of styles, have been but briefly treated. To one anxious to acquire knowledge-
FIG. 1452.
ARCHITECTURAL DRAWING. 599
in this department we refer, as the very best compendium within our knowl-
edge, to Ferguson's " Hand-Book of Architecture." The study of this work
will give direction to a person's observation, but, without referring to actual
examples, mere reading will be of little use. Drawings give general ideas of
the character of buildings, but no idea of size or of the surroundings of a
building. Many a weak design, especially in cast-iron buildings, acquires a
sort of strength by the number of its repetitions, giving an idea of extent ; and
many a beautiful design on paper has failed in its execution, being dwarfed by
its surroundings. With regard to the style of a building, there are none of the
ancient styles in their purity adapted to present requirements ; our churches
and theatres are more for the gratification of the ear than the eye, and the
comforts of our domestic architecture, and the requirements of our stores and
warehouses, are almost the growth of the present century. For a design, look
first to the requirements of the structure, the purposes to which it is to be
applied ; sketch the plan first, arrange the divisions of rooms, the openings for
doors and windows, construct the sections, and then the elevations, first in
plain outline ; modify each by the exigencies of construction.
" Construction, including in the term the disposition of a building in ref-
erence to its uses, is by some supposed to be the common part of the art of
architecture, but it is really the bone, muscle, and nerve of architecture, and the
arts of construction are those to which the true architect will look, rather than
to rules and examples, for the means of producing two at least of the three
essential conditions of building well, commodity, firmness, and delight, which
conditions have been aptly said to be the end of architecture as of all creative
arts.
" The two great principles of the art are : First, that there should be no
features about a building which are not necessary for convenience, construc-
tion, or propriety ; second, that all ornament should consist of enrichment of
the essential construction of the building.
" The neglect of these two rules is the cause of all the bad architecture of the
o
present time. Architectural features are continually tacked on buildings with
which they have no connection, merely for the sake of what is termed effect,
and ornaments are continually constructed instead of forming the decoration
of construction to which in good taste they should always be subservient. The
taste of the artist ought to be held merely ancillary to truthful disposition for
structure and service. The soundest construction is the most apt in the pro-
duction or the reproduction, it may be, of real art. The Eddystone Lighthouse
is well adapted to its uses ; it is commodious, firm and stable almost to a mira-
cle, and its form is as beautiful in outline to the delight of the eye, as it is well
adapted to break and mitigate the force of the sea in defense of its own struct-
ure. The Great Exhibition Building of 1851 was most commodious for the
purposes of an exhibition, firm enough for the temporary purpose required of
it, and there was delight in the simplicity and truth of its combinations ; and
all this may be said to have grown out of propriety of construction, as applied
to the material, cast-iron. The use of unfitting material, or fitting material
inappropriately, leads almost entirely to incommodiousness, infirmity, and
offense, or some of them.
600 ARCHITECTURAL DRAWING.
" Out of truth in structure, and that structure of a very inartificial sort,
grow the beautiful forms of the admirable proportions found in the works of
the Greeks ; and out of truth in structure, with the strictest regard to the
necessities of the composition and of the material employed, and that structure
as full of artifice as the artifice employed is of truth and simplicity, grew the
classical works vulgarly called Gothic, but now characteristically designated as
Pointed, from the arch which is the basis of the style. Structural untruth is
not to be justified by authority ; neither Sir Christopher Wren, nor the Athe-
nian exemplars of Doric or Ionic in the Propylaeum and in the Minerva Polias,
with their irregular and inordinately wide intercolumniation, can persuade
even the untutored eye to accept weakness for strength, or what is false for
truth.
" The Greek examples offer the most beautiful forms for moldings, and the
Grecian mode of enriching them is unsurpassed. It should be borne in mind
that the object in architectural enrichment is not to show ornament, but to
enrich the surface by producing an effective and pleasing variety of light and
shade ; but still, although ornament should be a secondary consideration, it
will develop itself, and therefore should be of elegant form and composition."
We have quoted thus at some length from the article " Architecture,"
" Encyclopaedia Britannica," because with many authority is necessary, and
they distrust their own powers of observation and analysis ; all must feel the
truth of the above, but in practice it is very little appreciated or carried out.
The present taste in architecture, as in the theatre, is for the spectacular ;
breadth or dignity of effect is not popular ; edifices are not only covered with,
but built up in ornament ; and construction is but secondary. The French,
having a building-stone that is very easily worked, cut merely the joints, leav
ing the rough outer surface to be worked after it is laid ; chopping out mold-
ings and ornaments almost as readily as though it were in plaster, and the sur-
face when finished is covered with enrichments in low relief. The fashion thus
set is imitated in this country at immense cost, in the most unfitting materials,
marble and granite. Our architectural buildings express fitly our condition
a rich country, recent and easily acquired wealth, and a desire and rivalry to
exhibit it, or a display as a means of advertising, and in this truth of expression
will have an archaeological interest ; although it does not contribute much to
present excellence in construction, it still has this value, that the architect or
constructor need be governed by no rules or principles he can make experi-
ments on a pretty extensive scale, and out of much bad construction even forms
and ornament may spring up which will stand the test of time, and form a
nucleus of a new style adapted to the present wants.
Cast-iron as a building material, with the exception of exhibition-buildings,
has seldom been treated distinctively ; buildings erected with it have been
copies of those in stone, and have been even imitated in color. For the first
story of stores, where space is necessary for light and the exhibition of wares,
cast-iron columns are almost invariably used, but are objected to architecturally,
that they look too weak for the support of the piles of brick and stone above
them. The objection should not be to the use, but that the truth of the ade-
quate strength of the cast-iron is not conveyed by the form or color. "No one
ARCHITECTURAL DRAWING. 601
objects that the ankles of Atlas look too light to support the massive figure and
globe, or wishes him seated to give the idea of stability ; so if the columns and
lintels were some other form than Greek or Koman with immense inter-
columniations, and colored fitly, the appearance of weakness would be entirely
lost sight of.
In conclusion, the draughtsman should be conversant with classic and later
styles, still, as he must design to suit the necessities of the times, and the
requirements of present tastes and fashions of buildings, he should keep him-
self posted on what is being done, and he will find it very convenient to have a
scrap-book of cuts from which to draw parts of a design, and afford him ready
means of combinations. He will find much in illustrated magazines and news-
papers, many cuts unpromising as a whole, yet fruitful in suggestions of parts ;
many an agreeable outline illy filled up ; many that are only valuable as showing
dimensions requisite for certain uses. But the larger the collection the better
for the draughtsman ; it will save time to know, as far as possible, what has
been done, that he may judge what forms and proportions it will be best for
him to use, and what to avoid.
It has been our practice to select, from papers and magazines, cuts which
we considered of value, and arrange them in scrap-books with appropriate
headings. In the Appendix a few pages of " scraps v are given as illustrations.
PERSPECTIVE DRAWING.
o
FIG. 1453.
THE science of Perspective is the representation by geometrical rules, upon
a plane surface, of objects as they appear to the eye, from any point of view.
All the points of the surface of a body
are visible by means of luminous rays pro-
ceeding from these points to the eye. Thus,
let the line A B (Fig. 1453) be placed before
the eye, C, the lines drawn from the differ-
ent points 1, 2, 3, 4, etc., represent the
visual rays emanating from each of these
points. It is easy to understand that, if in
the place of a line a surface is substituted,
the result will be a pyramid of rays.
Let A B (Fig. 1454) be a straight line,
and let the globe of the eye be represented
by a circle, and its pupil by the point C. The ray emanating from A, enter-
ing through C, will proceed to the retina of the eye, and be depicted at a.
And as it follows that
all the points of A B
will send rays, enter-
ing the eye through C,
the whole image of A
B will be depicted on
the retina of the eye
in a curved line a 3 b.
Conceive the line A B
moved to a greater dis-
tance from the eye, and
placed at A' B', then
the optic angle will be
reduced, and the image
a' 3 b' will be less than
before ; and as our vis-
Fio. 1454. ual sensations arc in
proportion to the mag-
nitude of the image painted on the retina, it may be concluded that the more
distant an object is from the eye the smaller the angle under which it is seen
becomes, and, consequently, the less it appears.
PERSPECTIVE DRAWING. 603
Observation has rendered it evident that the greatest angle under which
one or more objects can be distinctly seen is one of 90. If between the ob-
ject and the eye there be interposed a transparent plane (such as one of glass,
m n), the intersections of this plane with the visual rays are termed perspectives
of the points from which the rays emanate. Thus a is the perspective of A,
b of B, and so on of all the intermediate points ; but, as two points determine
the length of a straight line, it follows that a b is the perspective of A B, and
a" I" the perspective of A' B'.
It is evident from the figure that objects appear larger or smaller according
to the angle under which they are viewed ; and further, that objects of une-
qual size may appear equal if seen under the same angle. For, draw fg, and
its perspective will be found to be the same as that of A' B'.
It follows also that a line near the eye may be viewed under an angle much
greater than a line of greater dimensions but more distant, and hence a little
object may appear to be much greater than a similar object of larger dimen-
sions. Since, therefore, unequally sized objects may appear equal in size, and
equally sized objects unequal, and since objects are not seen as they are in
reality, but as they appear under certain conditions, perspective may be defined
to be a science which affords the means of representing, on any surface what-
ever, objects such as they appear when seen from a given point of view. It is
divided into two branches, the one called linear perspective, occupying itself
with the delineation of the contours of bodies, the other called aerial perspec-
tive, with the gradations of colors produced by distance. It is the former of
these only that is proposed here to be discussed.
The perspective of objects, then, is obtained by the intersection of the rays
which emanate from them to the eye, by a plane or other surface (which is
called the picture), situated between the eye and the objects.
From the explanation and definition just given, it is easy to conceive that
linear perspective is in reality the problem of constructing the section, by a sur-
face of some kind, of a pyramid of rays of which the summit and the base are
given. The eye is the summit, the base may be regarded as the whole visible
extent of the object or objects to be represented, and the intersecting surface is
the picture.
A good idea of this will be obtained by supposing the picture to be a trans-
parent plane, through which the object may be viewed, and on which it may
be depicted.
In addition to the vertical and horizontal planes with which we are familiar
in the operations of projection, several auxiliary planes are employed in perspec-
tive, and particularly the four following :
1. The horizontal plane A B (Fig. 1455), on which the spectator and the
object viewed are supposed to stand, for convenience supposed perfectly level,
is termed the ground plane.
2. The plane M N, which has been considered as a transparent plane placed
in front of the spectator, on which the objects are delineated, is called the plane
of projection or the plane of the picture. The intersection M M of the first
and second planes is called the line of projection, the ground, or base line of
the picture.
604
PERSPECTIVE DRAWING.
3. The plane E F passing horizontally through the eye of the spectator, and
cutting the plane of the picture at right angles, is called the horizontal plane,
and its intersection at D D with the plane of the picture is called the horizon
line, the horizon of the picture, or simply the horizon.
4. The plane S T passing vertically through the eye of the spectator, and
cutting each of the other planes at a right angle, is called the central plane.
Point of view, or point of sight, is the point where the eye is supposed to
be placed to view the object, as at 0, and is the vertex of the optical pyramid.
Its projection on the ground plane S is termed the station point.
The projection of any point on the ground plane is called the seat of that
point.
Center of view (commonly, though erroneously, called the point of sight),
is the point V where the central vertical line intersects the horizon line ; a line
drawn from this point to the eye would be in every way perpendicular to the
plane of the picture.
Points of distance are points on the horizontal line as remote from the
-centre of view as the eye.
M
M
Fia. 1455.
Vanishing points are points in a picture to which all lines converge that
in the original object are parallel to each other.
Parallel Perspective. An object is said to be seen in parallel perspective
when one of its sides is parallel to the plane of the picture.
Angular Perspective. An object is said to be seen in angular perspective
when none of its sides are parallel to the picture.
To find the perspective of points, as the points m, s (Fig. 1456), in the ground
plane, the same letters designating similar planes and points as in Fig. 1455.
From the point m draw a line to the point of sight C, and also to the station
point S ; at the intersection of the line m S with the base line MS', erect a per-
pendicular cutting the line m C, the intersection m' will be the perspective
projection of the point m, on the plane of the picture M V. The point s being
in the central plane, its projection must be in the intersection of that plane by
the plane of the picture, at the point s' the intersection of the central vertical
line by the line s C.
PERSPECTIVE DKAW
In the same way find the perspective h' m' of
when an original line is parallel or perpendicular
perspective of that line will also be parallel or perpen
605- :
e h m, and we find that
base of the picture, the
to it.
FIG. 1456.
Fig. 1457. Draw the diagonals M s' and m S', project as in the preceding
figure the points m and s into the plane of the picture, draw M m' M S', and
S' m' ; now, since m and M are the extremities of a line perpendicular to the
plane of the picture, the line m' M must be the projection of this line on the
plane of the picture, and if this line be extended it will pass through V, which
may be demonstrated of all lines perpendicular to the plane of the picture ;
hence the perspective direction of lines perpendicular to the picture is to the
center of view.
v/
FIG. 1457.
If the line m' S' be extended it will pass through the point D, and if M s'
be extended it will pass through a point in the line of the horizon at a distance
from V equal to V D ; by construction D V has been made equal to V C, and
606
PERSPECTIVE DRAWING.
as this demonstration is applicable to other similar lines, and since M m s S' is
a square ; hence the perspective direction of all lines, making an angle 0/45
with the plane of the picture, is toward the point of distance.
Having thus illustrated the rules of parallel perspective, we now proceed to
apply them to the drawing of a square and cube (Fig. 1458). The same letters
are employed in similar positions as in preceding figures.
It is necessary to premise that the student should draw these examples at
least three times the size of those in Fig. 1458.
Let A and B (Fig. 1458) represent the plan, or situation upon the ground,
PERSPECTIVE DRAWING.
607
of two squares, of which a perspective representation is required. First draw
the line M M, which represents the base line of the picture ; make S the station
point or place of the observer, and draw lines or rays from all visible angles of
the squares, to S ; then draw the lines S M, parallel to the diagonal lines of the
squares. Now draw M' M' parallel to M M representing the base line of the
picture in elevation ; then draw S' V, the vertical line immediately opposite the
eye ; let the distance, S' Y, be the height of the eye from the ground, and draw
D D the horizontal line ; V being the center of view ; let fall perpendicular
lines from the angles a and b of the plan of the square A, and also from the
point c, where the ray from the angle e intersects the base line, M M ; from a 1
and V draw lines to the center of view, V ; and e' where the perpendicular line
from c intersects the line V V, will give the apparent or perspective width b r e'
of the side b e ; from e' draw a line parallel to a' b', and the perspective repre-
sentation of the nearest square A is complete. In order to prove the accuracy
of this performance, it is necessary to try if the diagonal lines, a' e', and b'f,
incline respectively to the points of distance, D D, on the horizontal line : if
so, it is correct. The square B is drawn in precisely the same manner, and will
be easily understood by observing the example.
The plans of the two cubes C and D are the same as the plans of the
squares A and B. As neither of these cubes appears to touch the plane of the
picture M M, it will be necessary to imagine the sides I g, and Jc h, to be con-
tinued until they do so ; now draw down perpendicular lines from where the
continuations of these sides intersect the base line, and set off on them from
the line M' M', the height of the cube, as 1 2 which is the same as the width,
and complete the square shown by the dotted lines ; from all four angles of this
square draw lines to the center of view this will give the representation of
four lines at right angles with the picture carried on as far as it would be pos-
sible to see them ; then it only remains to cut off the required perspective
widths of the cubes, by the perpendicular lines, from the intersection of the
visual rays with the plane of the picture : the completion of this problem will
be very easy, if the drawing of the squares is well understood.
In such simple objects as these it will not be necessary to draw a plan
608
PERSPECTIVE DRAWING.
when one side is parallel to the picture, and dimensions are known. In Fig.
1459, the same objects as those in Fig. 1458 are drawn without a plan thus :
Draw the ground line M M, then the vertical line S' V, and the horizontal
line D D, at the height of the eye ; making D D the same distance on each
side of V that the eye is from the transparent plane ; for drawing the squares,
mark off from S' to b', on the ground line, the distance that the square is on
one side of the observer ; let b' a' be the length of one side of the square ; from
b' and a' draw lines to V, which represent the sides of the square carried on in-
definitely ; to cut off the required perspective width of the side b' e r of the
square, lay off the width, a' b', from b f to p, then draw from p to D on the left
and the point e' where the line Dp intersects b' V will give the apparent width
required ; then draw/' e' parallel to a' b', and the square is complete : this may
be proved in the same way as in Fig. 1458. The further square may be obtained
in a similar manner, setting off the distance between the squares from p to q,
and the width of the square beyond that, and drawing lines to D as before :
some of the lines in this plate are not continued to the ground line, in order to
avoid confusion. Proceed with the cubes by the same rule. Let 1, 2, 3, 4, be
the size of one side of the cube if continued until touching the picture ; from
these points draw rays to V ; from 3 to t set off the distance the cube is from
the picture, and from t to r, the width of the cube ; draw from these points to
D on the right, and their intersections of the line 3 V in m, o, will give the
perspective width and position of that side of the cube ; then finish the cube
as in the figure. The operation of drawing the other cube is similar, and easy
to be understood.
From the drawing of a square in parallel perspective, we deduce rules for
the construction of a scale in perspective. Let D M M D (Fig. 1460) be the
plane of the picture, the same letters of reference being used as in the preceding
M
figures. From S' lay off the distance o S' equal to some unit of measure, as
may be most convenient ; from o draw the diagonal to D the point of distance ;
now draw 1 1' parallel to the ground line M M, again draw from 1' the diagonal
I'D, and lay off the parallel 2 2', proceed in the same way with the diagonal
2' D and the parallel 3 3', and extend the construction as far as may be neces-
PERSPECTIVE DRAWING.
609
sary. It is evident o S' 1 1', 1' 1 2 2', 2' 2 3 3' are the perspective projections of
equal squares, and therefore o S', 1 1', 2 2' 3 3', etc., and S' 1, 1 2, 2 3, etc., are
equal to each other, and that if o S' is set off to represent any unit of measure,
as one foot, one yard, or ten feet, etc. , each of these lines represents the same
distance, the one being measured parallel to the base line, the others perpen-
dicular to it. In making a perspective drawing a scale thus drawn will be
found very convenient ; but as in the center of the picture it might interfere
with the construction lines of the object to be put in perspective, it is better that
the scale be transferred to the side of the picture a M o, the diagonals to be laid
off to a point to the right of D equal to the point of distance.
The scales thus projected are for lines in the base or ground plane ; for lines
perpendicular to this plane the following construction is to be adopted : Upon
any point of the base line removed from S', as a for instance, erect a perpen-
dicular, a d ; on this line, lay off as many of the units o S' as may be necessary ;
in this example three have been laid off, that is, a d = 3 o S'. From a and d
draw lines to the center of view, and extend the parallels 1 1', 2 2', 33'; at the
intersection of these lines with a V erect perpendiculars. The portions com-
prehended between the lines a V and d V will be the perspective representa-
tions of the line a d, in planes at distances of 1, 2, 3, o S' from the base line,
and as b, c, d are laid off at intervals equal to o S', by drawing the lines c V
and b V nine equal squares are constructed, of which the sides correspond to
the unit of measure o S'
To determine the Perspective Position of any point in the Ground Plane.
Thus (Fig. 1461), to determine the position of the point p, which in plane would
be six feet distant from the plane of the picture, M D, and ten feet from the
central plane, to the left.
Lay off from S', to the left, the distance a S', equal to six feet on the scale
adopted ; draw the diagonal to the point of distance D on the right : at its
intersection / with the vertical line V S' draw a parallel to M M ; lay off from
S', S' b equal to ten feet, draw b V ; the intersection of this line p, with the
parallel previously drawn, will be the position of the point required.
D
By a similar construction the position of any point in the ground plan may
be determined. It is not necessary that the distances should be expressed nu-
merically ; they may be shown on the plan and thence be transferred to the
base line, and thrown into perspective by the diagonals and parallels. As the
intersections of the various lines of the outlines of objects are points, by pro-
39
610
PERSPECTIVE DRAWING.
jecting perspectively these points, and afterward connecting by lines, the per-
spective of any plane surface on the ground plane may be shown.
If the pointy were not in the ground plane, but in a position directly above
the ground plane, say five feet, then at b erect a perpendicular, and lay off
Z V equal to five feet, connect V V, at p erect another perpendicular, and its
intersection p' with the line V V will be the position of the point required.
To draw an Octagon in Parallel Perspective. Let A (Fig. 1462) represent
the plan of an octagon. Draw M M, S' V, and D D, as before ; from the
points M, a, b, c, draw rays to V. Set off on MM from c to the right the dis-
tances ce, cd, cf, from which draw diagonals to D on the left, and at their
intersection with the ray c V, draw parallels e' g', d' h ', k' I', to the base line ;
these points will correspond to the angles on the plan. Now connect the an-
gles on the perspective view, in the proper succession, and the perspective pro-
jection is complete.
It will be observed, that in this construction the plan has been placed for-
ward of the plane of the picture, contrary to the position it should occupy,
which should be the same relative position back of this plane ; but it will be
found much simpler in construction than if it were placed as in Fig. 1458, and
the points were all projected to the base line ; it is, of course, equally correct
in its perspective projection.
To draw a Circle in Parallel Perspective. Let C (Fig. 1462) represent the
plan of a circle, round which let the square a e c m be described, two of its sides
being parallel to the base line M M ; draw diagonals across the square, and
where these intersect the circumference of the circle draw the lines bJc and dg
parallel to the base line, and the lines on and pg at right angles thereto.
Draw also the lines/? and ch at right angles to each other through the center
PERSPECTIVE DRAWING.
611
of the circle, project the points a, o, I, p, m, to the base and draw rays to V ;
set off from a' to the left the distances a' a, a' b, a' c, a' d, a' e, and draw diago-
nals to the point of distance D on the right ; at their intersection with the line
a' V draw horizontal lines, or parallels to the base, and there will be projected in
perspective the square ae cm, with all the lines of parallels and perpendicu-
lars ; connect the intersections corresponding to the points c, n, f, g, h, k, I, r,
.and we have the perspective projection of the required circle, which will be an
ellipse.
To erect upon the octagonal base A an octagonal pillar or tower. This con-
struction resolves itself into simply constructing another octagon on an upper
plane, and connecting the visible angles by perpendiculars ; or perpendiculars
may be erected at the points M, a, Z, c, and the heights of the tower laid off
upon them, and from these extremities rays drawn to the center of view ; the
intersection of these rays by perpendiculars from the angles of the octagon be-
neath will determine the projection of the upper surface of the pillar ; repre-
,sent in full lines all visible outlines, and the projection is complete.
In the same manner a pillar may be erected on the circular base. If the
pillars be inclined, the first method of projecting the upper outline on a plane
assumed at the height of the pillar must be adopted.
VLL
FIG. 1463.
To draw a Pyramid in Parallel Perspective. Let A (Fig. 1463) be the plan
of a pyramid, the diagonal lines represent the angles, and their intersection the
vertex ; project the plan as in previous examples of squares. Draw diagonal
lines from M to #, and a to c, their intersection gives the perspective center of
the square ; upon this point raise a perpendicular line which is the axis of the
pyramid ; draw a perpendicular line ef, in the center of the line M a, upon
which set up the height of the pyramid ef\ from /draw a line to V, and its
intersection of the axis of the pyramid at d will give the perspective height ;
complete the figure by drawing lines from rf, the apex, to M, a, b, the three
visible angles. The other two pyramids are drawn in a similar manner, by
setting their distances from the plane of the picture off from a, on the ground
line to the right, and drawing diagonals to the point of distance on the left.
To draw a Cone in Parallel Perspective. Let B (Fig. 1463) represent the
612 PERSPECTIVE DRAWING.
plan of a cone, apply the same lines of construction as to C (Fig. 1462) ; and
draw the perspective view of the circle, lay off the height and finish precisely
as in the preceding case.
To draw a Square and Cube in Angular Perspective. Let A (Fig. 1464)
be the plan of the square, and B the plan of the cube, M M the base or ground
line, and S the station point. Draw M' M', and D D' parallel to M M, the
one being the ground line and the other the horizon of the plane of the pict-
ure ; project the point d on MM, to d' on M' M'. It has been shown in
parallel perspective that the vanishing points of diagonals of squares lie in
the points of distance ; if through the station point S, in any of the preceding
figures, lines be drawn parallel to the diagonals, they will intersect the base
lines at distances from the central plane equal to the points of distance. In
like manner to find the vanishing points of lines in the ground planes, or in
planes parallel to the ground plane, inclined to the plane of the picture,
through the station point S draw lines parallel to the inclined lines, and pro-
ject their intersection with the base line to the horizon of the picture ; thus, in
the present example, draw S M, S M parallel to a d, e h, and to dc, Jig ; pro-
ject their intersections M, M, with the base line to D, D', the horizon of the
picture, and D, D', will be the vanishing points of all lines parallel to a d and
d c. Draw d' D and d' D', the perspective projection ofda will lie in the
former of these lines and d c in the latter. To determine the perspective po-
sition of the points a and c, or the length of these lines, draw the rays a S and
c S, project their intersection with the base M M, upon the lines d' D and d' D',
and their intersections a', c' will be the perspective projection of the points a
and c. To complete the projection of the square, draw the lines a' D' and c' D,
their intersection will be the perspective projection of the point b, and the
square is complete. To prove the construction, draw the ray b S and project
its intersection with the base M M, and if the construction be correct it will
fall upon the point b'.
As the cube is placed at some distance from the plane of the picture, it will
be necessary to continue either eh or g h, or both, till they intersect the base
line M M at n and m ; drop perpendiculars or project these points upon M' M'
at n' and m' ; on these perpendiculars set up the height of the cube m' o and
n' s, draw the lines m' D', o D', and n' D, s D ; connect the intersections h' and
h" ; draw the rays Qe and S</, and project their intersections with MM, to
g'e' ; draw the lines e"D f and g" D ; if the construction be correct, the projec-
tion of the intersection of the ray S/ with the base will fall upon /", and of
the ray S li will fall upon h" and h'.
To solve the Same Problem by a Different Construction. Let AB (Fig.
1464) be as before the plans of the square and of the cube ; to project them
perspectively on the plane of the picture M D D' M (Fig. 1465).
From the point M and M (Fig. 1464), set off distances equal to M S, M S, to
p and p' ; project these points upon D D' (Fig. 1465), the point p' (Fig. 1465)
will be that from which any number of parts may be laid off on lines vanishing
in D' ; the point p will be the corresponding point for lines vanishing in D.
These points may be called the points of division. In parallel perspective the
points of distance were the points of division, the one for the other. To illus-
PERSPECTIVE DRAWING.
613
FIG. 1465.
614: PERSPECTIVE DRAWING.
trate their application in the present example, project the point d (Fig. 1464)
to d 1 (Fig. 1465), draw d'D and d' D', from d' on either side lay off a distance
d' i, d f k equal to the side of the square a d. Now, since p is the division point
of lines vanishing in D,from i, draw the line ip, and its intersection with d'T>
cuts off a line d' a 1 equal perspectively to the line d' i or ad measured on the
base line. Again, since p' is the division point of lines vanishing in D', the
line k p' cuts off on d' D', a line d' c' equal perspectively to the line d' k, or a d
measured on the base : having a' d' c, the square is completed by drawing the
lines c' b' toward D, and a' b' toward D'.
To construct the cube, project the point m (Fig. 1464) to m' (Fig. 1465) ;
lay off on the perpendicular forming the projection, the height m' o of the cube ;
draw the lines m' D' and 0D'. Lay off the distance m' r equal to mh (Fig.
1464), and draw the line rp', its intersection with m' D' will cut off m' h', equal
to m h (Fig. 1464), and establish the angle h' of the cube. From r lay off r s,
equal to h g (Fig. 1464), draw sp', and its intersection with m D' establishes the
angle g'. From h' draw a line vanishing in D. Through h' extend a line p h'
to t, from t lay off to the left t a, equal to the side of the cube li e ; draw ap f
and its intersection with the line h'T) establishes a third point Y of the cube.
Upon these points h' g' e' erect perpendiculars ; those upon h' and g' will, by
their intersection with o D', determine h" g". Draw h" D, its intersection
with the perpendicular at e' determine e". Draw g" D and e" D' to their inter-
section, and the cube is complete.
To draw the Perspective Projection of an Octagonal Pillar in Angular Per-
spective. Let A (Fig. 1466) be the plan of the pillar. Inclose it by a square.
Let M M be the base line, and S the station point ; determine the position of
the vanishing points for the sides of the square as in Fig. 1464, and project the
square upon the plane of the picture M D D' M' by either of the methods already
explained. These lines of construction are omitted, as on the necessarily small
diagrams they would confuse the student ; but in drawing these examples to
the scale recommended, they might be retained. From the angles of the octa-
gon visible to the spectator draw rays to the station point S, project their inter-
section with the base line M M, to the perspective square (Fig. 1467), which
will thus determine on the sides of the square the positions of the points a', b',
c', d', e', corresponding to the visible angles of the octagon ; connect these
points by lines. To construct the pillar upon this base, let fall a perpendicular
from the corner/ of the square upon M M', at /set off the height of the pillar ;
from this point/' draw lines to the vanishing points D, D', and construct three
sides of an upper square similar to the lower one. The lines of this square will
determine the length of the sid,es of the tower, which are the perpendiculars
let fall upon a' b r c' d' e'.
To construct a Circular Pillar in Angular Perspective. Let B (Fig. 1466)
be the plan of the base ; enclose it with a square whose sides are parallel re-
spectively to S M and S M ; project this square upon the plane of the picture
(Fig. 1467) ; divide the plan into four equal squares by lines parallel to the
sides ; draw rays through the points h and i, and project their intersection
with M M upon the perspective square. From the points h' and i' thus formed,
draw lines to vanishing points D' and D, and the perspective square is divided
PERSPECTIVE DRAWING.
615
FIG. 1466.
\ \f
FIG. 1467.
'
--....r'
M m"
FIG. 1468.
r
616 PERSPECTIVE DRAWING.
similarly to the original, and there are four points of the circle established :
through these draw the perspective of the circle. By the division of the base
into smaller squares more points of the curve might be determined, but for the
present purpose they are unnecessary. To determine the outline of the pillar,
draw from S rays tangent to the sides of the plan at k and i, the perpendicu-
lars let fall from their intersection with M M will be the outline of the cylin-
der. To cut them off to the proper height, and to determine the top of the
cylinder, upon the perpendicular let fall upon i, set off the height of the cylin-
der /' r, and upon this plane project the square as before, and draw in through
the points thus determined the outline of the curve. As a still further eluci-
dation of the principle of projection, an enlarged cap is represented on the
pillar, of which the circumscribing circle (Fig. 1466) is the plan. In this, by
extending the central lines of the square, both in plan and perspective, we are
enabled to project readily eight points in the larger circle through which the
curve may be drawn. .
To draw an Octagonal Pyramid in Angular Perspective. Let A (Fig.
1466) be the base of the pyramid ; project upon the plane of the picture (Fig.
1468) the visible angles of the base, as in the case of the pillar. Through the
center of the plan draw a line parallel to one of the sides and intersecting M M
at m ; from this point let fall a perpendicular to m' on M M' (Fig. 1468) ; on
this perpendicular set off the height of the pyramid m- o from m' and draw
lines to D'. From the center of the plan draw a ray to S, and project its
intersection with M M, upon the line o D', its intersection o' with this line will
be the apex of the pyramid : from this point draw lines to the angles of the
base already projected, and the pyramid is complete.
To draw a Cone in Angular Perspective. Let the inner circle B (Fig.
1466) be the base of the cone, project its visible outline to Fig. 1468, as in case
of the cylinder. To determine its height extend one of the diameters of the
plan to the base line at p ; from this point let fall a perpendicular to p' on
M M', and set off upon it p' q the height of the cone ; from p' and q draw
lines to the vanishing point D'. From the center of the plan (Fig. 1466) draw
a ray to S, and project its intersection with M M upon r' on the line q D', and
r' will be the apex of the cone : connect the apex with the extremities of the
perspective of the base, and the projection of the cone is complete.
To draw the Elevation of a Building in Angular Perspective. For ex-
ample, take the school-house (Fig. 1469). Plot so much of the plan of the
building as may be seen from the position of the spectator at S. Draw a
base line, and through the station point draw parallels to the sides of the
building, cutting the base as at M M ; draw M M' for a base, and D D' for the
horizontal line of the picture. Project M and M to D and D', for the vanish-
ing points, the one of the lines parallel to a c, the other to a 1) ; extend a c, ab ;
project d, e, to d', e', and on d' d set off the height of the eaves d' o, and of the
ridge d r n ; from d', o and n draw lines to D', and from e' to D, draw rays from
c and b to S, and project their intersection with the base to the vanishing lines
just drawn. To find the perspective of the ridge draw a ray from the center
of a b, and project its intersection with the base to r on the line nD', the point
is the apex of the gable, the line r D will be the perspective of the ridge ; to
PERSPECTIVE DRAWING.
618 PERSPECTIVE DRAWING.
determine its length erect a perpendicular at the intersection of tD' and s D,
draw the sloping lines of the roof, and the outline of the building is complete.
The filling in of the details will be readily understood ; it will only be neces-
sary to keep in mind that all lines parallel to a b must meet in D', those to a c
in D : all measures laid oif on any lines of the plan must be connected with
the point of sight S, and their intersections with the base projected. All ver-
tical heights must be laid oif on the line d' d, and referred to the proper posi-
tion by lines to D or D', as the case may be.
As an example of the other method of constructing this same problem, let
the scholar lay off to the double of the present scale the plane of the picture
M D D'M', and the division points p' and p, and without drawing plan or ele-
vation take the dimensions from Fig. 1190.
To draw an Arched Bridge in Angular Perspective. Let A and B (Fig.
1470) be the plans of the piers ; on the line a p, one of the sides of the
bridge, lay down the curve of the arch as it would appear in elevation, in
this example an ellipse. Divide the width of the arch as at b c d e f g h,
carry up lines perpendicular to b h until they intersect the curve of the arch,
and through these points draw lines parallel tobh as k I m ; let o r be the
height of the parapet of the bridge above the spring of the arch. Through
the station point draw lines parallel to the side a h and end a a of the bridge,
till they intersect the assumed base line M M : project these intersections to
the horizon line of the picture for the vanishing points D, D' of perspective
lines parallel to a li and a a. Let fall a perpendicular from a to a' , and on this
perpendicular set off from a' the heights s k, si, s m, and s r ; from a' and r'
draw lines to D and D', and from the points m', I', k 1 to D'. Draw rays from
the points abed efg h to the station point S, and project their intersection
with the base lines to the perspective line a' D' as in previous examples : the
intersection of the lines k' D', I' D', m' D' by the perpendiculars thus pro-
jected will establish the points of the curve of the arch on the side nearest the
spectator. To determine the position of the opposite side of the arch, from a",
the perspective width of the bridge, draw a" D', and from h' draw lines to D ;
the line h' p' will be the perspective width of the pier ; draw k' D ; and from
k", k" D' ; from g" the intersection of the curve of the arch by the perpen-
dicular to g' 9 draw^D, the intersection with k"D' will be one point in the
curve of the arch on the opposite side of the bridge ; in the same way, from
any point in the nearer arc draw lines to D, and the intersection with lines in
the same planes on the opposite side of the bridge will furnish points for the
further arch ; all below the first only will be visible to the spectator.
To draw in Parallel Perspective the Interior of a Room (Fig. 1471). We
propose to construct this by scale without laying down the plan. Draw the
horizon line D V D', and the base M M', making D and D' the point of dis-
tance. Let the room be 20 feet wide, 14 feet high, and 12 feet deep ; on the
base M M' lay off the rectangle of the section in our figure on a scale of 8 feet
to the inch, 20 feet X 14 feet. From the four corners draw lines to the center
of view V ; from S' lay off to the right or left on M M' 12 feet, and through this
point draw lines to D' or D as the case may be ; through the point of intersec-
tion, a' of this line with S' V, draw a line parallel to M M' ; at the intersections
PERSPECTIVE DRAWING.
FIG. 1471.
620
PERSPECTIVE DRAWING.
of this line with M V and M' V erect a perpendicular, cutting the vanishing
lines of the upper angle of the room at d and e ; connect de and the perspect-
ive of the room is complete. To draw the aperture for a door or window on
the side, measure oil from S' the distance of the near side from the plane of
the picture, and in addition thereto the width of the aperture ; from these two
points draw lines to the proper point or distance, and at their intersection with
S' V, draw parallels to MM', cutting the lower angles of the room, and erect
perpendiculars, the height of which will be determined by a line drawn from/,
the height of the window above the floor measured 011 M D. Should the win-
dow be recessed, the farther jamb will be visible ; extend the farther parallel
to M M', and cut it by a line gV. M.g being the depth of the recess, the rest
of the construction may be easily understood by inspection of the figure. At
the extremity of the apartment a door is represented half open, hence as the
plane of the door is at right angles to the plane of the picture, the top and bot-
tom lines will meet in the point of view ; if the door were open at an angle of
45 these lines would meet in the points of distance ; if at any other angle,
the vanishing points would have to be determined by constructing a plane,
drawing a line parallel to the side of the door through the station point, and
projecting it upon the horizon line. The chair in the middle of the room is
placed diagonally, and the table parallel to the plane of the picture ; their pro-
jection is simple.
To draw in Perspective a Flight of Stairs (Fig. 1472). Lay off the base
line, horizon, center of view, and point of distance of the picture ; construct
FIG. 1472.
the solid abed, efg h, containing the stairs, and in the required position in the
plane of the picture ; divide the rise a c into equal parts according to the num-
ber of stairs, nine for instance ; divide perspectively the line a b into one less (8)
PERSPECTIVE DRAWING.
621
number of parts ; at the points of division of this latter erect perpendiculars,
and through the former draw lines to the center of view ; one will form the
rise and the other the tread of the steps. From the top of the first step to the
top of the upper continue a line a d, till it meets the perpendicular S' V pro-
longed in v ; this line will be the inclination or pitch of the stairs ; if through
the top of the step at the other extremity a similar line be drawn, it will meet
the central perpendicular at the same point v, and will define the length of the
lines of nosing of the steps, and the other lines may be completed. As the
pitch lines of both sides of the stairs meet the central vertical in the same
point, in like manner v will be the vanishing point of all lines having a similar
inclination to the plane of the picture. The projection of the other flight of
stairs will be easily understood from the lines of construction perpendicular to
the base line or parallel thereto, lying in planes.
To find the Reflection of Objects in the Water. Lei B (Fig. 1473) be a cube
suspended above the water ; we find the reflection of the point a, by letting
fall a perpendicular from it, and setting off the distance, a' w below the plane
of the water equal to the line aw above this line, the line wf will also be
equal to the line wf ; find in the same way the points V and e', through these
points construct perspectively a cube in this lower plane, and we have the re-
flection of the cube above.
To find the reflection of the square pillar D removed from the shore : sup-
pose the plane of the water extended beneath the pillar, and proceed as in the
previous example.
It will be observed that those lines of an object which meet in the center of
view V, in the original, have their corresponding reflected lines converging to
D
B
FIG. 1473.
the same point. If the originals converge to the points of distance, the reflected
ones will do the same. To find the reflection of any inclined line, find the re-
flection of the rectangle of which it is the diagonal, if the plane of the rectangle
is perpendicular to the plane of the picture ; if the line is inclined in both
directions inclose it in a parallelepiped and project the reflection of the solid.
PERSPECTIVE DRAWING.
To find the Perspective Projection of Shadows (Fig. 1474). Let the con-
struction points and lines of the picture be plotted. Let A be the perspective
projection of a cube placed against another block, of which the face is parallel
to the plane of the picture ; to find the shadow upon the block and upon the
ground plane, supposing the light to come into the picture from the upper
left-hand corner and at an angle of 45. Since the angle of light is the diagonal
of a cube, construct another cube similar to A, and adjacent to the face dcg ;
draw the diagonal b k, it will be the direction of the rays of light, and k will
be the shadow of b ; connect fk and c k, fk must be the shadow of the line
bf, and c k of b c ; the one upon the horizontal plane and the other in a verti-
cal one : the former will have its direction, being a diagonal, toward the point
of distance D', the other being a diagonal in a plane parallel to that of the
picture, will be always projected upon this plane in a parallel direction.
Let B be a cube similar to A ; to find its projection upon a horizontal plane,
the shadow of the point l> may be determined as in the preceding example, but
the shadow of the point c', instead of falling upon a plane parallel to the pic-
ture, falls upon a horizontal one ; its position must be determined as we did
before by b. Construct the cube and draw the diagonal c' I ; in the same way
determine the point m the shadow of d' ; connect ck' Im n, and we have the
shadow of the cube in perspective on a horizontal plane.
On examination of these projected shadows, it will be found that as the
rays of light fall in a parallel direction to the diagonal of the cube, the vanish-
ing point of these rays will be in one point V on the line D' M' prolonged, at
a distance below D' equal V D' ; and since the shadows of vertical lines upon a
horizontal plane are always directed toward the point of sight, the extent of
the shadow of a vertical line may be determined by the intersection of the
shadow of the ground point of the line by the line of light, from the other ex-
tremity. Thus, the point k, cube A, is the intersection of /D' by bV ; the
points k', I, ware the intersections of eD', oD', nW by V V, c'V d'V.
Similarly on planes parallel to that of the picture, k, cube A is intersection of
the diagonal c k, by the ray of light b V.
Applying this rule to the frame C, from r, s, p, draw lines to D' ; from r r ,
$', p' f draw rays to V ; their intersections define the outline of the shadow of
the post. To draw the shadow of the projection, the shadow upon the post
from t will follow the direction of the diagonal ck. Project u and v upon the
ground plane at u' and v' ; from t u' v' and p draw lines to D' ; from t' 9 u, v,
w and x draw rays to V, and the intersection of these lines with their cor-
responding lines from their bases will give the outline required ; as v and w
are on the same perpendicular, their rays will intersect the same line v' V.
With reference to the intensity of " shade and shadow," and the necessary
manipulation to produce the required effect, the reader is referred to the article
on this subject,
In treating of Perspective it has been considered not in an artistic point, as
enabling a person to draw from nature, but rather as a useful art to assist the
architect or engineer to complete his designs, by exhibiting them in a view
such as they would have to the eye of a spectator when constructed. In our
examples, owing to size of the page, we have been limited in the scale of the
PERSPECTIVE DRAWING.
623
624 PERSPECTIVE DRAWING.
figures, and in the distance of the point of view, or distance of the eye from,
the plane of the picture, and as it was unimportant to the mathematical demon-
stration, few of the figures extend above the line of the horizon. In these par-
ticular points it is unnecessary that the examples should be copied. The most
agreeable perspective representations are generally considered to be produced
by fixing the angle of vision M S M', at from 45 to 50, and the distance of
the horizon above the ground-line at about one third the height of the picture.
Linear perspective is more adapted to the representation of edifices, bridges,
interiors, etc., than to that of machinery ; it belongs, therefore, rather to the
architect than to the engineer or the mechanic ; for the purposes of the latter
we would recommend Isometrical Perspective, uniting accuracy of measures
with graphic perspective representation.
ISOMETRICAL DRAWING.
PKOFESSOR FARISH, of Cambridge, has given the term Isometrical Per-
spective to a particular projection which represents a cube, as in Fig. 1474,
The words imply that the measure of the representations of the lines forming
the sides of each face are equal.
The principle of isometric representation consists in selecting, for the plane
of the projection, one equally inclined to three principal axes, at right angles
to each other, so that all straight lines
coincident with or parallel to these 9_
axes are drawn in projection to the
a
FIG. 1474.
same scale. The axes are called iso-
metric axes, and all lines parallel to FIG. 1475.
them are called isometric lines. The
planes containing the isometric axes are isometric planes ; the point in the
object projected, assumed as the origin of the axes, is called the regulating-
point.
To draw the isometrical projection of a cube (Fig. 1475), draw the horizontal
line A B indefinitely ; at the point D erect the perpendicular D F, equal to one
side of the cube required ; through D draw the lines D b and D / to the right,
and left, making /D B and b D A each equal an angle of 30. Consequently,
the angles F D / and F D b are each equal to 60. Make D b and D / each
equal to the side of the cube, and at b and /erect perpendiculars, making b a
and/e each equal to the side of the cube ; connect F a and F e, and draw e g
parallel to a F, arid a g parallel to F e, and we obtain the projection of the
cube.
40
626
ISOMETRICAL DRAWING.
If from the point F, with a radius F D, a circle be described, and commenc-
ing at the point D radii be laid oif around the circumference, forming a regular
inscribed hexagon, and the points D a e be connected with the center of the
circle F, we have an isometrical representation of a cube. The point D is called
the regulating-point.
If a cube be projected according to the principles of isometrical perspective,
in a similar manner as we have constructed one according to the rules of linear
perspective, the length of the isometrical lines would be to the original lines as
8164 to 1, but, since the value of isometrical perspective as a practical art lies
in the applicability of common and known scales to the isometric lines, in our
constructions we have not thought it necessary to exemplify the principles of
the projection, but have drawn our figures without any reference to what would
be the comparative size of the original and of the projection, transferring meas-
ures directly from plans and elevations in orthographic projections to those in
isometry. It will be observed that the isometric scale adopted applies only to
isometric lines, as F D, F a, and F e, or lines parallel thereto ; the diagonals
which are absolutely equal to each other, and longer than the sides of the cube,
are the one less, the other greater ; the minor axis being unity, the isometrical
lines and the major axis are to each other as, 1. /y/2. <\/3.
Understanding the isometrical projection of a cube, any surface or solid may
be similarly constructed, since it is easy to suppose a cube sufficiently large to
contain within it the whole of the model intended to be represented, and, as
hereafter will be further illustrated, the position of any point on or wi fchin the
cube, the direction of any line, or the inclination of any plane to which it may
be cut, can be easily ascertained and represented.
FIG. 1476.
FIG. 1478.
In Figs. 1474 and 1475 one face of the cube appears horizontal, and the
other two faces appear vertical. If now the figures bo inverted, that which
ISOMETRIC AL DRAWING.
627
before appeared to be the top of the object will now appear to be its under
side.
The angle of the cube formed by the three radii meeting in the center of
the hexagon may be made to appear either an internal or external angle ; in
the one case the faces representing the interior, and in the other the exterior of
a cube.
Figs. 1476, 1477, 1478, illustrate the application of isometrical drawing to
simple combinations of the cube and parallelopipedon. The mode of construc-
tion of these figures will be easily understood by inspection, as they contain no
lines except isometrical ones.
To draiv Angles to the Boundary Lines of an Isometrical Cube. Draw a
square (Fig. 1479) whose sides are equal to those of the isometrical cube A
(Fig. 1480), and from any of its angles describe a quadrant, which divide
4U JO 20
FIG. 1479.
FIG. 1480.
into 90, and draw radii through the divisions meeting the sides of the
square. These will then form a scale to be applied to the faces of the cube ;
thus, on D E, or any other, by making the same divisions along their respec-
tive edges.
As the figure is bounded by twelve isometrical lines, and the scale of tan-
gents may be applied two ways to each, it can be applied therefore twenty-four
ways in all, affording a simple means of drawing, on the isometrical faces of
the cube, lines at any angles with their boundaries.
Figs. 1481 to 1486 show the section of the cube by single planes, at various
inclinations to the faces of the cubes. Figs. 1487 and 1488 are the same cube,
but turned round, with pieces cut out of it. Fig. 1489 is a cube cut by two
planes forming the projection of a roof. Fig. 1490 is a cube with all of the
angles cut off by planes, so as to leave each face an octagon. Fig. 1491 repre-
sents the angles cut off by planes perpendicular to the base of the cube, form-
ing thereby a regular octagonal prism. By drawing lines from each of the
angles of an octagonal base to the center point of the upper face of the cube,
we have the isometrical representation of an octagonal pyramid.
As the lines of construction have all been retained in these figures, they will
628
ISOHETRICAL DKAWING.
FIG. 1481.
FIG. 1482.
FIG. 1483.
FIG. 1484.
FIG. 1485.
FIG. 1486.
FIG. 1487.
FIG. 1488.
FIG. 1489.
FIG. 1490
ISOMETRICAL DRAWING.
629
be easily understood and copied, and are sufficient illustrations of the method
of representing any solid by inclosing it in a cube.
In the application of this species of projection to curved lines, let A B (Fig.
1492) be the side of a cube with a circle inscribed ; and that all the faces of a
cube are to have similarly inscribed circles. Draw the diagonals A B, C D, and
FIG. 1492.
FIG. 1493.
at their intersection with the circumference, lines parallel to A C, B D. Now
draw the isometrical projection of the cube (Fig. 1493), and lay out on the
several faces the diagonals and the parallels ; the projection of the circle will
be an ellipse, of which the diagonals being the axes, their extremities are de-
fined by their intersections/ 6, e5, a 2, bl, d3, c4, with the parallels ; having
thus the major and minor axis, construct the ellipse by the trammel, or, since
the curve is tangent at the center of the sides, we have eight points in the
curve ; it may be put in by sweeps or by the hand.
630
ISOMETRICAL DRAWING.
To divide the Circumference of a Circle. First method : On the center of
the line A B (Fig. 1494) erect a perpendicular, C D, making it equal to C A or
C B ; then from D, with any radius, describe an arc and divide it in the ratio
required, and draw through the divisions radii from D meeting A B ; then
from the isometric center of the circle draw radii from the divisions on A B,
cutting the circumference in the points required.
Second method : On the major axis of the ellipse describe a semicircle, and
divide it in the manner required. Through the points of division draw lines
perpendicular to A E, which will divide the circumference of the ellipse in the
same ratio. On the right hand of the figure both methods are shown in com-
bination, and the intersections of the lines give the .points in the ellipse.
Fig. 1495 is an isometrical projection of a bevel-wheel, with a half-plan
(Fig. 1496) beneath, and projected lines explanatory of the method to be
FIG. 1496.
adopted in drawing the teeth, and of which only half are shown as cut. It
will be seen, by reference to the second method given above for the division of
the circumference of a circle, that the semicircle is described directly on the
major axis of the ellipse. In practice it will be found more convenient, when
a full drawing is to be made, to draw the semicircle on a line parallel to the
major axis, and entirely without the lines of the main drawing ; and also, as in
the example of the bevel-gear, complete on the semicircle, or half-plan, the
ISOMETKICAL DRAWING. 631
drawings of all lines, the intersections of which with circles it will be necessary
to project on the isometrical drawing.
Fig. 1497 is an isometrical projection of a complete pillow-block, with its
hold-down bolts. By reference to Fig. 592, and Figs. 508 and 509, it will
be seen how much more graphically these forms of gearing are given by isom-
etry than by the usual projection. As an exercise for the learner, it will be
very good practice to project isometrically the spur-gear (Fig. 583), and the
standard and hanger (Figs. 510 and 515), of which sufficient details are given.
FIG. 1497
Fig. 1498 is an isometrical projection of a culvert, such as were built be-
neath the Croton Aqueduct, and is a good example of construction, and better
illustrated by the drawing than it would be by plan and elevations.
Fig. 829 is an isometrical view of the overflow and outlet of the Victoria
and Regent Street sewers in the Thames embankment.
Fig. 1499 is an isometric elevation of the roof-truss (Fig. 896). No side-
view is shown on the plate, but the dimensions of timber and spaces are drawn
as usual in practice.
Figs. 1500 and 1501 are the elevation and section in isometry of the district
school-house given in Figs. 1189 and 1190. To bring the drawing within the
limits of the page, the scale has been necessarily reduced, but it is given in
632
ISOMETRTOAL DEAWING.
ISOMETKICAL DRAWING.
633
the figure as it should always be, either drawn or written, on all drawings
to a scale, not intended for mere pictures or illustrations. The section is
drawn at the height of 8 feet above the base course, and higher than is
FIG. 1499.
nsual in such sections, but it was necessary on account of the extra height
of the window-sill above the floor, desirable in all school-rooms. Fig. 1501
is more graphic than the plan (Fig. 1190), and, when there are staircases
one above the other in the drawing, they are more intelligibly expressed ; but
there is nothing in the present drawing that can not be nearly as well shown
by the plan, and to a mechanic, for the purposes of construction, the plan is
the simpler.
By comparing the elevation (Fig. 1500) with the perspective (Fig. 1469),
the former appears distorted, and out of drawing, but it is much more readily
drawn, and has this great convenience, that it is drawn to and can be measured
by a scale, but only on the isometric lines : all others are distorted, too long or
too short, as may be seen in the major and minor axes of the bevel-gear (Fig.
1496), or the rake-lines of the roof (Fig. 1499).
Fig. 1502 is the isometrical projection, on the wave-line principle, of ship
construction, from Russell's "Naval Architecture" as explained and illus-
634
ISOMETKICAL DRAWING.
FIG. 1501.
ISOMETRICAL DRAWING. 635
trated on pages 458 and 459 and Fig. 1503, another isometrical drawing from
the same work.
We have multiplied examples of isometrical drawing, to show its applica-
\\\\\\\\\\
\\x\\\ w\
bility to varied forms of construction, mechanical, architectural, and naval.
The principles of this projection are easy and intelligible, and their use should
636
ISOMETRIOAL DRAWING.
1SOMETRICAL DRAWING.
638
ISOMETRICAL DRAWING.
be extended. Isometrical projection is especially valuable to the mechanical
draughtsman, explaining many constructions that could hardly be done by any
amount of plans, elevations, and sections, and still uniting with pictorial rep-
resentation the applicability of a scale. For drawings for the Patent Office it
is especially desirable, in a simple and practical form combining the requisites
of many projections ; but as a drawing of what could be absolutely seen by the
eye it is not truthful, and therefore, when pictorial illustration only is requisite,
the drawing should be in linear perspective.
FIG. 1505.
In confirmation of the above, in Fig. 1504 is given a drawing in perspec-
tive, in which the point of sight is above the plane of the picture, and ap-
proaching in general appearance to drawings in isometry ; and yet, having all
the truthfulness of sight, is much better suited to the purpose for which it
was intended. Fig. 1505 is another illustration of the same kind, in common
use for business circulars and catalogues.
FREE-HAND DRAWING.
A DRAUGHTSMAN", who has made himself conversant with the rules of pro-
jection as laid down in this book, and has applied these rules to practice, will
be capable of representing correctly such objects as have been illustrated, or
make up similar combinations of his own invention and design. But natural
objects, as animals, trees, rocks, clouds, etc., can not be imitated on paper
with the aid of drawing instruments ; outlines so varied can not be copied in
this mechanical way ; it can only be done by what is called free-hand drawing,
an educated eye that can recognize proportion and position, and an educated
hand that can execute and portray naturally things recognized by the eye, with
the aid of pencil, pen, crayon, or brush. A free hand adds largely to the effect
of drawings, where close measures are not requisite, giving grace and beauty to
mechanical designs, and is especially applicable to architectural ornaments and
accessories. It will be found impossible to draw many of these in any other
way, and there are few drawings that do not require some patching by hand
short curves, which can be thus done much more readily, and connections of
lines, which can not be done by drawing instruments. It has been said before
that the lettering of a plan or map contributes very much to its appearance,
and as the Italian and Koman characters are now almost universally used it is
only by free hand that they can be made ornamental or graceful.
The pencil or pen should be held by the thumb and first finger, and sup-
ported and guided by the second. The two fingers touching the pencil should
be placed firmly on it, and be perfectly straight, the end of the middle finger
at least one inch above the point of the pencil. In drawing, it is well to com-
mence, as in writing, with straight lines. Lines vertical, horizontal, and in-
clined, parallel to each other and at angles, light and strong short and long
lines, straight and curved, with pen, pencil, or crayon on paper, or chalk on a
board. Dot points, and draw lines between them, at a single movement, with-
out going over them a second time, and without patching. Besides direction,
lines have a definite length, and the draughtsman must practice himself in
drawing lines of equal lengths, or in certain proportions to each other.
Lines equal to each other :
Lines twice another line :
Divide a line into any number of equal parts :
I I I I I
640
FREE-HAND DRAWING.
The accuracy of these divisions may be tested by a strip of paper applied
along the line, marking off the divisions upon it, and then slipping it along
one division, and noting if the divisions on the paper and line still agree. By
practice, the eye will be able to make these divisions almost accurately. Having
acquired this skill, copy the triangles in the Geometrical Problems, in their
proper proportions, and afterwards squares and rectangles.
FIG. 1506.
FIG. 1507.
FIG. 1508.
Draw two lines (Fig. 1506) at right angles to each other, and mark equal
distances on each one. Through these points draw a circle and a square.
Draw a circle and divide each quadrant into two equal arcs, and connect
the chords to form an octagon (Fig. 1507). Or, draw a square, and cut off
the corners (Fig. 1508).
Divide a circle into six equal arcs, and connect the chords for a hexagon.
r
FIG. 1509.
FIG. 1510.
FIG. 1511.
FIG. 1512.
Draw lines at right angles to each other, with only the opposite arms equal,
and construct the ellipses (Figs. 1509 and 1510).
Draw an arc tangent to a straight line (Fig. 1511).
FREE-HAND DRAWING.
641
Draw two parallel lines (Fig. 1512), and connect them by two equal and
reversed arcs, tangent to each other, and to the parallel lines.
Draw a similar curve, with arcs perpendicular to the parallels (Fig. 1513).
Although it will be observed that in all these problems guide or construction
lines are used, it is not the intention that
any use should be made of drawing instru-
ments, but the construction should be
dependent entirely on eye and hand ; still
it will be found, whether the draughts-
man draws from copy or nature, that it is
almost impossible to get along well with-
out defining positions by some points in
the pictures, and sketching in some defined FIG. 1513.
lines which may serve as guides. All the
above examples are from "Geometrical Problems," and it will be found good
practice to copy others.
Following this practice of guide lines, it will be well to copy the outlines of
architectural moldings, of which most of the ornaments are conventional rep-
resentations of natural objects.
In design, " a true artistic end has been accomplished when well-observed
features of natural objects have been chronicled within the conventionalized
limits of a few geometric rules that include proportion, symmetry, and a proper
subordination of one part to another."
The following example is from the "Art Journal" (trefoil design):
" In the equilateral triangle (Fig. 1514), each side is divided by
a dot, and from the center of the triangle lines are drawn to each
angle, and from the dot in the middle of each side to the opposite sides of the
figure. The geometrical plan of the design is thus laid out, and the figure is
easily filled in by drawing simple curves from the center of the form to the
dot on each side of it, and,
lastly, filling in the form of
the trefoil a little below the
point of each corner of the
triangle.
"The square (Fig. 1515),
which is the next form, is
developed in much the same
manner. The sides are bi-
sected, and from a point in
the center lines are carried
to each angle, and to all the dots on the sides. As in the preceding figure,
slight curves are made on either of the side-lines, and the trefoil is added to
each angle, with the base of the middle leaf touching the transverse working-
lines between the sides. It will be seen that the pentagon (Fig. 1516) and the
hexagon (Fig. 1517) also are formed in the same general manner, but the pro-
portion of the top of the trefoil varies from its sides.
"In drawing the circular rosette (Fig. 1518), the circumference should be
41
FIG. 1514.
FIG. 1515.
642 FREE-HAND DRAWING.
constructed on a vertical and a horizontal diameter, with two other diameters
bisecting it at equal angles, which divide it into eight sections, the half diame-
ters, upon all of which curved lines and the top of the trefoil are made. A
FIG. 1516. FIG. 1517. FIG. 1518.
series of arcs may be added at the pleasure of the designer. In the two pieces
of molding (Figs. 1519 and 1520), the trefoil is inserted vertically to the sides
in one and horizontally in the other. In the latter, a half of the trefoil is
added upon the sides to enrich the elementary figure ; and the double line and
FIG. 1519.
the transverse lines which form the squares are repeated for the sake of sym-
metry, and as affording an impression of agreeable repose.
" It is from such a basis as this that all these various patterns are derived,
FIG. 1520.
and they produce a result which an inexperienced eye, unaccustomed to analyze
designs, could scarcely resolve into its elements. "
Figs. 1521-1524 are other illustrations of the same principle, of varieties
of rosettes constructed on a similar plan.
FREE-HAND DRAWING.
643
All of these designs can be constructed mechanically, but more grace is
given to the design by the filling in with free hand, and it is an excellent prac-
tice in the execution of the more elaborate Saracenic and Moorish diaper ; but
FIG. 1521.
FIG. 1522.
FIG. 1524.
in all of these where there are repetitions of the same figures it is usual to
draw but one, and then transfer this, but the finish must be in crayon or pencil.
"Proportions of the Human Frame." By Joseph Bonomi.
The following, with the illustrations, are taken from the above work :
"The human frame is (Figs. 1525 and 1526) divided into four equal meas-
ures, by very distinctly marked divisions on its structure and outward form :
" 1. From the crown of the head to a line drawn across the nipples.
" 2. From the nipples to the pubes.
" 3. From the pubes to the bottom of the patella (knee-pan).
"4. From the bottom of the patella to the sole of the foot.
"Again, four measures, equal in themselves, and equal to those just de-
scribed, and as well marked in the structure of the human body, are seen when
the arms are extended horizontally. They are the following :
" From the tip of the middle or longest finger to the bend of the arm is
one fourth of the height of the person.
" From the bend of the arm to the pit of the neck is another fourth.
" These two measures, taken together, make the half of the man's height,
and with those of the opposite side equal the entire height.
" In the figures, the differences in width between the male and female figures
are given from the tables of the Count de Clarac of the Apollino and the Venus
de Medici. The male figure is in thicker line than the female, and the measure-
ments referring to it are on your right hand, and those referring to the female
on your left.
" The measurements of length, according to Vitruvius and Leonardo da
Vinci, are the same in both sexes, and expressed in long horizontal lines run-
ning through both the front and profile figures.
"Almost innumerable are the varieties of character to be obtained by the
alterations of widths, without making any change in the measurements of
length ; nevertheless, some ancient statues differ slightly in these measure-
ments of length.
" No measurement is given in the figure of the width of the foot ; its normal
proportion should be one sixteenth of the height. The views of the foot (Fig.
1527) are those of the female.
644
FRiE-HAND DRAWING.
" The scale, V, used is 8 heads to the height ; parts, i of a head ; and min-
utes, T V of a part.
" The whole height is usually taken at 8 heads, but there are slight differ-
ences in the classic statues ; the height of the Venus de Medici is equal to 7
heads, 3 parts, 10 minutes, that of the Apollino of Florence, 7 heads, 3 parts,
6 minutes.
" When the student is acquainted with the forms of the body and limbs in
two aspects viz., the front and side views and the normal proportions they
bear to each other, then will follow the study of the characteristic features of
FREE-HAND DRAWING.
645
childhood, youth, and mature age, and those niceties of character that the
ancients invariably observed in the statues of their divinities, so that in most
cases a mere fragment of a statue could be identified as belonging to this or
that divinity as, for instance, the almost feminine roundness of the limbs of
the youthful Bacchus, the less round and distinctly marked muscles of the
Mercury, and of the statues of the Athletae. "
Figure Drawing. In the album of Villard de Hennecourt, which dates
from the middle of the thirteenth century, certain mechanical processes are
given to facilitate the composition and design of figures. According to these
sketches, geometry is the generator of movements of the human body, and that
of animals, and serves to establish certain relative proportions of the figures.
From the time of Villard sculptors have had these practical methods, which,
if they could not inspire the artisan with genius, yet prevented him from fall-
ing into gross faults. The pen sketch (Fig. 1528) is an example of this prac-
C V
FIG. 1528.
FIG. 1529.
tical process. In comparing this mode of drawing with figures in the vignettes
of manuscripts, with designs on glass, and even with statues and bas-reliefs, we
must recognize the general employment in the thirteenth and fourteenth centu-
ries of these geometrical means, suited to give figures not only their propor-
tions but also the justness of their movement and bearing. Rectifying the
canon of Villard in its proportions by comparison with the best statues, nota-
bly those in the interior of the western facade of the Cathedral of Reims, we
646
FEEE-HAND DRAWING.
obtain the Fig. 1529. The line A B, the height of the human figure, is divided
into seven equal parts. The upper division is from the top of the head to the
shoulders. Let C D be the axis of the figure, the line at the breadth of the
shoulders is f of the whole height A B. The point E is the center of the line
D ; draw through this point two lines, af and b e, and from the point g
two other lines, g e and g f. The line 1) h is the length of the humerus, and
the line of the knee-pan is on i k. The length of the foot is f of a division, A 1.
Having established these proportions, it will be seen by the following cuts how
the artisan gave movements to these figures when the movements were not in
absolute profile.
Suppose the weight of the figure to be borne upon one leg (Fig. 1530), the
FIG. 1530.
FIG. 1531.
line ge becomes perpendicular, and the axis op of the figure is inclined. The
movement of the shoulders and trunk follow this inflection ; the axis of the
head and the right heel are in the same vertical line.
In stepping up (Fig. 1531) the axis of the figure is vertical, and the right
heel raised is on the inclined line s t, while the line of the neck is on the line
I m, and the trunk is vertical.
In Fig. 1532 it will be seen how a figure can be submitted to a violent move-
ment and vet preserve the same geometrical trace. The figure is fallen, sup-
ported on one knee and one arm, while the other wards off a blow ; the head
is vertical.
FREE-HAND DRAWING.
647
In Fig. 1533, the left thigh being in the line af, to determine the position
of the heel c on the ground, supposed to be level, an arc is to be described from
the knee-pan ; the line ef is horizontal.
It is clear that, in adopting these practical methods, all the limbs can be
developed geometrically without shortening.
The above is from the " Dictionnaire raisonne de 1' Architecture " of Viollet
Le Due, and will supply to many a ready means of sketching the human figure
FIG. 1532.
in various attitudes, naked, or in the close-fitting dresses of the present fashion ;
but in the arrangement of drapery upon a figure, care must be taken that the
drapery should fall in graceful folds. " It is necessary to give the body certain
inflections which would be ridiculous in a person walking naked. The walk
should be from the hips, with wide-spread legs, and, by the movements of the
trunk, make the drapery cling on certain parts and float on others."
FIG. 1533.
In figures in repose, their centers of gravity must fall within the points of
support, but the body can be sustained by muscular exertion, and this should
648
FREE-HAND DRAWING.
FIG. 1535.
FIG. 1536.
FIG. 1540.
FIG. 1539.
FIG. 1541.
FREE-HAND DRAWING.
649
FIG. 1542.
FIG. 1546.
FIG. 1549.
FIG. 1544.
FIG. 1545c
FIG. 1551.
FIG. 1552.
650
FREE-HAND DRAWING.
be expressed in such cases by the tension of the muscles on which the position
depends. In the act of running, the body inclines forward, its weight assists
the movement, and the motions prevent its falling.
Figs. 1534-1538 are illustrations of portions of the human head and face,
with some guide-lines to assist the copyist.
Figs. 1539-1541 are drawings of female hands and arms.
Figs. 1542-1545 are drawings of male hands, Figs. 1546-1552 of legs and
feet, with guide-lines, and Figs. 1553-1556 are those of children.
FIG. 1553.
FIG. 1554.
FIG. 1555.
FIG. 1556.
The Forms of Animals. The bodies of most quadrupeds standing can be in-
cluded in rectangles as guide-lines ; that of the ox and horse in that of a square
(Figs. 1557 and 1558). The action of the limbs of quadrupeds is chiefly di-
rectly forward or directly backward, the power of lateral motion being limited.
The hinder limbs always commence progressive motion, as in the first position
FREE-HAND DRAWING.
651
of the walk (Fig. 1559), the fore foot of the same side advances next, then the
hind foot of the opposite side, and lastly the fore foot on that side, and so on.
In the trot, the hinder leg of one side and the fore leg of the other are raised
together (Fig. 1560). In the canter or gallop, both fore legs and one hind
FIG. 1557.
leg are raised together (Fig. 1561) ; when rapidly moving, the two fore legs,
and two hind legs appear to advance together (Fig. 1562). In fact, all the
movements are rather resultants, as they appear to us, but when instantaneous-
ly photographed the legs are wonderfully mixed.
FIG. 1558.
The forms of feet range under two great divisions hoofs (Fig. 1564) and
paws (Fig. 1565). All hoofs, whether whole or cloven, approximate to a right-
angled triangle, and all paws to a rhomboid.
652
FREE-HAND DRAWING.
12& *>r ~| ^
FIG. 1563.
FREE-HAND DRAWING.
653
the horse ; Fig. 1567,
ivori ; Fig. 1569,.
A
FIG. 1564.
ZZ7
TJie Noses of Animals. Fig. 1566 represents
fchat of the ox and deer tribe ; Fig. 1568, those of
those of the camel, sheep, and
goat tribes ; and Fig. 1570, those
of the hog tribes. The muzzles
of nearly all quadrupeds will be
found to range under one or other
of these classes, with minute varia-
tions to characterize the diiferent
species and individuals.
In looking over the varied
sketches and engravings of Land-
seer which have been published, it
will be noticed in how varied a
manner they are executed. Some-
times in mere outline with lead-
pencil, sometimes with a camel's-
hair pencil charged with Indian
ink or sepia for the outlines, giv-
ing effect to the subject by slight
tints or washes of the same color ;
in others, pen and ink have been
alone employed. Some are in oils,
others in water-colors ; frequent-
ly chalks, both black and colored,
were the vehicles used. " As
we look at some of these, we are
tempted to believe that, of all the
instruments that can be used by
the artist, there is none quite so
wonderful as the pen. A simple
sketch with a pen or lead-pencil is
naked, unadorned truth, bearing
witness to the skill or its opposite
of the hand which produced it."
The above quotation is given
to show the value of accurate
drawing the skeleton, as it were,
may be more suggestive, and con-
vey more skillfully effective truth
than the finished drawing, and
the first necessity is truth in draw-
ing. Nothing has yet been said
of drawing from nature. The
copies given are intended as rudiments, and the following illustrations from
the " Art Journal" of objects in art, and sketches and pictures of different
painters, will serve to show their varied treatment of subjects.
FIG. 1570.
654
FKEE-HAND DRAWING.
The illustrations given are for the education of the eye of the draughtsman,
in showing him the varied appearance of different subjects by different artists,
and their modes of expression ; and he can acquire facility of hand in copying
them. If he wishes to draw from nature, let him look at objects as if they
were a picture, If he looks through a window, the frame may be considered
the border of his picture ; if he can portray what he sees through a square of
glass truthfully, in position and proportion, with pencil, chalk, or brush, he
has made a picture. He must keep his eyes in one position, or at such a dis-
tance from the plane of his picture or the glass that he can not see more of an
object than is comprehended by one look. To enable one to judge of the pro-
portion of an object, and its position, it is very common to make use of the
pencil as a scale, holding it with an extended arm always at the same distance
from the eye ; to slide the thumb down on the pencil till the length of the
object or line is embraced between the end of the pencil and the thumb, and
transferring this length to the paper in its proper position. Practically, in
this way, one arrives at the knowledge of perspective, of which the principles have
been given in " Perspective Drawing." Aerial perspective, or the tones of lights
and shadows according to their distances from the observer and the sources of
the light, he will acquire by studies of pictures and observations of nature.
The rule in drawing from nature is to draw only what you see, and express it
in the most truthful form.
FREE-HAND DRAWING.
655
656
FREE-HAND DRAWING.
FREE-HAND DRAWING.
657
Bacchus and the Water- Thieves. JOHN PENNIEL.
ess
FREE-HAND DRAWING.
After a Pen-and-ink Design, by FORTUNY.
FREE-HAND DRAWING.
659
660
FREE-HANI) DRAWING.
I
Study of Oak-Trees. K. LAXDSEER.
FREE-HAND /DRAWING.
,^KSi!^ : , : ,.^?
* %^
Apple- -Blossom*. A. T. BRICHER.
662
FREE-HAND DRAWING.
Cattle going Home. JAMES M. HART.
FREE-HAND DRAWING.
663
Morning. H. W. BOBBINS.
664
FREE-HAND DRAWING.
m
I
, I
I
^
1
APPENDIX.
Extracts from the Acts relating to Buildings in the City of New Yor%.
3. All foundation walls shall be laid not less than 4' below the surface of the earth,
on a good solid bottom, and, in case the nature of the earth should require it, a bottom of
driven piles, or laid timbers, of sufficient size and thickness, shall be laid to prevent the
walls from settling, the top of such pile or timber bottom to be driven or laid below the
water line ; and all piers, columns, posts, or pillars resting on the earth, shall be set upon
a bottom in the same manner as the foundation walls. Whenever in any case the founda-
tion wall or walls of- any building that may hereafter be erected shall be placed on a rock
bottom, the said rock shall be graded off level to receive the same. . . .
4. The footing, or base course, under all foundation walls, and under all piers, col-
umns, posts, or pillars resting on the earth, shall be of stone or concrete ; and if under a
foundation wall shall be at least 12" wider than the bottom width of the said wall ; and if
Binder piers, columns, posts, or pillars, shall be at least 12" wider on all sides than the
bottom width of the said piers, columns, posts, or pillars, and not less than 18" in thick-
ness ; and if built of stone, the stones thereof shall not be less than 2' x 3', and at least 8"
in thickness ; and all base stones shall be well bedded and laid edge to edge ; and if the
walls be built of isolated piers, then there must be inverted arches, at least 12" thick,
turned under and between the piers, or two footing courses of large stone at least 10"
thick in each course. All foundation walls shall be built of stone or brick, and shall be
laid in cement mortar, and, if constructed of stone, shall be at least 8" thicker than the
wall next above them, to a depth of 16' below the curb level, and shall be increased 4" in
thickness for every additional 5' in depth below the said 16'; and if built of brick, shall
be at least 4" thicker than the wall next above them to a depth of 16' below the curb level,
-and shall be increased 4" in thickness for every additional 5' in depth below the said 16'.
5. In all dwelling-houses that may hereafter be erected not more than 55' in height,
the walls shall not be less than 12" thick, and if above 55' in height, and not more than
SO' in height, the outside walls shall not be less than 16" thick to the top of second story
floor-beams ; provided the same is 20' above the curb level, and if not, then to under side
of the third story beams, and also provided that portion of the wall, that is 12" thick shall not
exceed 40' above the said 16" wall; and in every dwelling-house hereafter erected more
than 80' in height, 4" shall be added to the thickness of the wall for every 15' or part thereof
that is added to the height of the building. All party-walls in dwellings over 55' in height
shall not be less than 16" in thickness.
6. In all buildings other than dwellings hereafter erected, the bearing walls shall not
be less than 12" thick to the height of 40' above the curb level ; if above 40' in height
and not more than 55' feet in height, the bearing walls shall not be less than 16" thick ; if
above 55' and not more than 70' in height, the bearing walls shall not be less than 20"
660 APPENDIX.
thick, to the height of 20' above the curb level or to the next tier of floor-beams above,
and not less than 16" from thence to the height of 55' above the curb level or to the next
tier of floor-beams, and not less than 12" thick from thence to the top ; and if above TO'
and not more than 85' in height, the bearing walls shall not be less than 24" thick to the
height of 12' above the curb level or the second story floor-beams, and from thence to the
height of 60' above the curb level, the said walls shall not be less than 20" thick, and from
thence to the top not less than 16'' thick ; and if above the height of 85', the bearing walls
shall be increased 4" in thickness for every 15', or part thereof, that shall be added to the
height of said wall above the 85'. In all buildings over 25' in width, and not having either
brick partition walls or girders supported by columns running from front to rear, the wall
shall be increased an additional 4" in thickness, to the same relative thickness in height as
required under this section for every additional 10' in width of said building, or any por-
tion thereof. It is understood that the amount of materials specified may be used either
in piers or buttresses, provided the outside walls between the same shall in no case be less
than 12" in thickness to the height of 40', and if over that height then 16" thick ; but in
no case shall a party wall between the piers or buttresses of a building be less than 16" in
thickness. In all buildings hereafter erected, situated on the street corner, the bearing
wall thereof (that is, the wall on the street upon which the beams rest) shall be 4" thicker
in all cases than is otherwise provided for by this act. All walls other than bearing walls
may be 4" less in thickness than required in the clauses and provisions of this section above
set forth, provided no wall is less than 12" in thickness.
7. Every building hereafter erected more than 30' in width, except churches, thea-
tres, school-houses, car-stables, and other public buildings, shall have one or more stone or
brick partition walls running from front to rear, or iron or wooden girders supported on
iron or wooden columns ; these walls shall be so located that the space between any two
of the bearing walls shall not be over 25'. In case iron or wooden girders, supported on
iron or wooden columns, are substituted in place of the partition walls, the building may
be 75' in width, but not more ; and if there should be substituted iron or wooden girders,
supported on iron or wooden columns, in place of partition walls, they shall be made of
sufficient strength to bear safely the weight of 250 Ibs. for every square foot of the floor
or floors that rest upon them, exclusive of the weight of material employed in their con-
struction, and shall have a footing course and foundation wall not less than 16" in thick-
ness, with inverted arches under and between the columns, or two footing courses of large,
well-shaped stone, laid crosswise, edge to edge, and at least 10" thick in each course, the
lower footing course to be not less than 2' greater in area than the size of the column ;
and under every column, as above set forth, a cap of cut granite, at least 12" thick, and of
a diameter 12" greater each way than that of the column, and must be laid solid and level
to receive the column. Any building that may hereafter be erected in an isolated position,
and more than 100' in depth, and which shall not be provided with cross walls, shall be
securely braced, both inside and out, during the whole time of its erection, if it can be
done ; but in case the same can not be so braced from the outside, then it shall be properly
braced from the inside, and the braces shall be continued from the foundation upward to
at least one third the height of the building from the curb level.
8. ... Every temporary support placed under any structure, wall, girder, or beam
during the erection, finishing, alteration, or repairing of any building, or part thereof, shall
be equal in strength to the permanent support required for such structure, wall, girder, or
beam. And the walls of every building shall be strongly braced from the beams of each
story until the building is topped out, and the roof tier of beams shall be strongly braced
to the beams of the story below until all the floors in the said building are laid.
9. All stone walls less than 24" thick shall have at least one header, extending
through the walls, in every 3' in height from the bottom of the wall, and in every 4' in
length ; and, if over 24" in thickness, shall have one header for every six superficial feet
APPENDIX. 66T
on both sides of the wall, and running into the wall at least 2'; all headers shall be at
least 18" in width and 8" in thickness, and shall consist of a good flat stone, dressed on all
sides. In every brick wall every sixth course of brick shall be a heading course, except
where walls are faced with brick, in which case every fifth course shall be bonded into
the backing by cutting the course of the faced brick, and putting in diagonal headers
behind the same, or by splitting face-brick in half, and backing the same by a continuous
row of headers. In all walls which are faced with thin ashlar, anchored TO the backing,
or in which the ashlar has not either alternate headers and stretchers in each course, or
alternate heading and stretching courses, the backing of brick shall not be less than 12"
thick, and all 12" backing shall be laid up in cement mortar, and shall not be built to a
greater height than prescribed for 12" walls. All heading courses shall be good, hard,
perfect brick. The backing in all walls, of whatever material it may be composed, shall,
be of such thickness as to make the walls, independent of the facing, conform as to thick-
ness with the requirements of sections five and six of this act.
10. Every isolated pier less than ten superficial feet at the base, and all piers sup-
porting a wall built of rubble-stone or brick, or under any iron beam or arch girder, or
arch on which a wall rests, or lintel supporting a wall, shall, at intervals of not less than
30" in height, have built into it a bond stone not less than 4" thick, of a diameter each
way equal to the diameter of the pier, except that in piers on the street front, above the
curb, the bond stone may be 4" less than the pier in diameter ; and all piers shall be built
of good, hard, well-burned bricks and laid in cement mortar, and all bricks used in piers
shall be of the hardest quality, and be well wet when laid ; and the walls and piers under
all compound, cast-iron, or wooden girders, iron or other columns, shall have a bond stone
at least 4" in thickness, and if in a wall at least 2' in length, running through the wall,
and if in a pier, the full size of the thickness thereof, every 30" in height from the bot-
tom, whether said pier is in the wall or not, and shall have a cap stone of cut granite, at
least 12" in thickness, by the whole size of the pier, if in a pier, and if in a wall it shall be
at least 2' in length, by the thickness of the wall, and at least 12" in thickness. In any
case where any iron or other column rests on any wall or pier built entirely of stone or
brick, the said column shall be set on a base stone of cut granite, not less than 8" in thick-
ness by the full size of the bearing of the pier, if on a pier, and if on a wall the full thick-
ness of the wall. In all buildings where the walls are built hollow, the same amount of
stone or brick shall be used in their construction as if they were solid, as above set forth ;
and no hollow walls shall be built unless the t\vo walls forming the same shall be con-
nected by continuous vertical ties of the same materials as the walls, and not over 24""
apart. The height of all walls shall be computed from the curb level. No swelled or-
refuse brick shall be allowed in any wall or pier ; and all brick used in the construction,
alteration, or repair of any building, or part thereof, shall be good, hard, well-burned
brick; and if used during the months from April to November, inclusive, shall be well
wet at the time they are laid.
12. In no case shall the side, end, or party wall of any building be carried up more
than two stories in advance of the front and rear walls. The front, rear, side, end, and
party walls of any building hereafter to be erected shall be anchored to each other every
6' in their height by tie-anchors, made of one and a quarter inch by three eighths of an inch
of wrought-iron. The said anchors shall be built into the side or party walls not less than
16", and into the front and rear walls at least one half the thickness of the front and rear
walls, so as to secure the front and rear walls to the side, end, or party walls ; and all
stone used for the facing of any building, except where built with alternate headers and
stretchers, as hereinbefore set forth, shall be strongly anchored with iron anchors in each
stone, and all such anchors shall be let into the stone at least 1". The side, end, or party
walls shall be anchored at each tier of beams, at intervals of not more than eight feet apart,,
with good, strong, wrought-iron anchors, one half inch by one inch, well built into the
68 APPENDIX.
-side walls, and well fastened to the side of the beams by two nails, made of wrought-iron,
at least one fourth of an inch in diameter ; and where the beams are supported by girders,
the ends of the beams resting on the girder shall be butted together end to end, and
strapped by vvrought-iron straps of the same size, and at the same distance apart, and in
the same beam as the wall-anchors, and shall be well fastened.
13. All walls of any buildings over fifteen feet high shall be built up and extended at
least 24" above the roof, and shall be coped with stone or iron. . . .
14. All iron beams or girders used to span openings over 6' in width, and not more
than 12' in width, upon which a wall rests, shall have a bearing of at least 12" at each
end by the thickness of the wall to be supported ; and for every additional foot of span
over and above the said 12', if the supports are iron or solid cut stone, the bearing shall
be increased half an inch at each end ; but if supported on the ends by walls or piers built
of brick or stone, if the opening is over 12' and not more than 18', the bearing shall
be increased 4" at each end by the thickness of the wall to be supported ; and if the space
is over 18' and not more than 25' then the bearing shall be at least 20" at each end by the
thickness of the wall to be supported ; and for every additional 5' or part thereof that the
space shall be increased, the bearing shall be increased an additional 4" at each end by the
thickness of the wall to be supported. And on the front of any building where the sup-
ports are of iron or solid cut stone, they shall be at least 16" on the face and the width of the
thickness of the wall to be supported, and shall, when supported at the ends by brick walls
or piers, rest upon a cut granite base block, at least 12" thick by the full size of the bear-
ing; and in case the opening is less than 12', the granite block may be 6" in thickness by
-the whole size of the bearing ; and all iron beams or girders used in any buildings shall
be, throughout, of a thickness not less than the thickness of the wall to be supported. All
iron beams or girders used to span openings more than 8' in width, and upon which a wall
rests, shall have wrought iron tie-rods of sufficient strength, well fastened at each end of
the beam or girder, and shall have cast-iron shoes on the upper side, to answer for the
skew-back of a brick or cut-stone arch, which said arch shall always be turned over the
same, and the arch shall in no case be less than 12" in height by the width of the wall to
be supported, and the shoes shall be made strong enough to resist the pressure of the arch
in all cases. Cut-stone or hard-brick arches, with two wrought-iron tie-rods of sufficient
strength, may be turned over any opening less than 30', provided they have skew-backs of
cut stone or cast or wrought-iron, with which the bars or tension-rods shall be properly
secured by heavy wrought iron washers, necks, and heads of wrought-iron, properly
secured to the skew-backs. The above clause is intended to meet cases where the arch
has not abutments of sufficient size to resist its thrust. All lintels hereafter placed over
openings in the front, rear, or side of a building, or returned over a corner opening, when
supported by brick piers or iron or stone columns, shall be of iron, and of the full breadth
of the wall to be supported, and shall have a brick arch of sufficient thickness, with skew-
backs and tie-rods of sufficient strength to support the superincumbent lateral weight,
independent of the cast-iron lintel. . . .
15. All openings for doors and windows in all buildings, except as otherwise pro-
vided, shall have a good and sufficient arch of stone or brick, well built and keyed, and
with good and sufficient abutments, or a lintel of stone or iron, as follows : . . . For an
opening exceeding 6' in width, and not more than 8' in width, the lintel shall be of iron or
stone, and of the full thickness of the wall to be supported : and every such opening 6' or
less in width in all walls shall be at least one third the thickness of the walls on which it
rests, and shall have a bearing at each end not less than 4" on the walls ; and on the inside
of all openings, in which the lintel shall be less than the thickness of the wall to be sup-
ported, there shall be a good timber lintel on the inside of the other lintels, which shall
rest at each end not more than 4" on any wall, and shall be chamfered at each end, and
-shall have a double rolock arch turned over said timber lintel; arches built of stone or
APPENDIX.
brick may be turned over openings on a center, which may be struck after the arch is
turned, provided the arch has a good and sufficient rise, and that the piers or abutments
are of sufficient strength to bear the thrust of the arch. . . .
17. All chimneys, and all flues in stone or brick walls, in any building hereafter
erected, altered, or repaired, without reference to the purpose for which they may be
used, shall have the joints struck smooth on the inside, and no parging mortar shall be
used on the inside ; and the fire-backs of all chimneys hereafter erected shall not be less-
than 8" in thickness ; ... no wooden furring or lath shall be placed against any flue, metal
pipe, or pipes used to convey heated air or steam in any building ; and when any wall shall
hereafter be furred or lathed with wood, the space between the lathing and wall shall be
filled with plaster between the top and underside of the floor-beams of each story, so as to-
prevent fire from extending from one floor to another. And no air-flue shall be used at
any time as a smoke-flue. No steam-pipe shall be placed within 2" of any timber or
wood-work as aforesaid ; when the said space of 2" around the steam-pipe is objectionable,
it shall be protected by a soap-stone or an earthen ring or tube. No base, or flooring, or
roofing, or any other wood- work shall be placed against any brick or other flue until the
same shall be well plastered with plaster-of-Paris behind such wood- work. . . .
18. No smoke-pipe, in any building with wooden or combustible floors and ceilings,,
shall hereafter enter any flue unless the said pipe shall be at least 18'' from either the
floors or ceilings ; and in all cases where smoke-pipes pass through stud or wooden
partitions of any kind, whether the same be plastered or not, they shall be guarded by
either a double collar of metal, with at least 4" air space and holes for ventilation, or by a.
soap-stone ring, not less than 3" in thickness and extending through the partition, or by a
solid coating of plaster-of-Paris, 3" thick, or by an earthenware ring 3" from the pipe. . . .
19. In no building, whether the same be a frame building or otherwise, shall any
wooden girders, beams, or timbers be placed within 12" of the inside of any flue, whether
the same be a smoke, air, or any other flue. All wooden beams and other timbers in the
party wall of every building hereafter to be erected or built, of stone, brick, or iron, shall
be separated from the beam or timber entering in the opposite side of the wall by at least
8" of solid mason- work. No floor-beam shall be supported wholly upon any wood par-
tition, but every beam, except headers and tail-beams, shall rest, at one end, not less than
4" in the wall, or upon a girder, as authorized by this act. And every trimmer or header
more than 4' long, used in any building except a dwelling, shall be hung in stirrup-irona,
of suitable thickness for the size of the timbers. . . .
20. In all buildings, every floor shall be of sufficient strength in all its parts to bear
safely upon every superficial foot of its surface 75 Ibs. ; and if used as a place of public
assembly, 120 Ibs. ; and if used as a store, factory, warehouse, or for any other manufact-
uring or commercial purposes, from 150 to 500 Ibs. and upward ; and every floor shall be
of sufficient strength to bear safely the weights aforesaid, in addition to the weight of the
materials of which the floor is composed ; and every column, post, or other vertical sup-
port shall be of sufficient strength to bear safely the weight of the portion of each and
every floor depending upon it for support, in addition to the weight required as above to-
be supported safely upon said portions of said floors. In all calculations fo^ the strength
of materials to be used in any building, the proportion between the safe weight and the
breaking weight shall be as one to three for all beams,' girders, and other pieces subjected
to a cross-strain, and shall be as one to six for all posts, columns, and other vertical sup-
ports, and for all tie-rods, tie-beams, and other pieces subjected to a tensile strain. And
the requisite dimensions of each piece of material is to be ascertained by computation by
the rules given by Tredgold, Hodgkinson, Barlow, or the treatises of other authors now or
hereafter used at the United States Military Academy of West Point on the strength of
materials, using for constants in the rules only such numbers as have been deduced from
experiments on materials of like kind with that proposed to be used. ...
670 APPENDIX.
21. In all fire-proof buildings hereafter to be constructed, where brick walls, with
wrought- iron beams or cast or wrought iron columns with wrought-iron beams, are used
in the interior, the following rules must be observed :
1. All metal columns shall be planed true and smooth at both ends, and shall rest on
cast-iron bed-plates, and have cast-iron caps, which shall also be planed true. If brick
arches are used between the beams, the arches shall have a rise of at least an inch and a
quarter to each foot of space between the beams.
2. Under the ends of all the iron beams, where they rest on the walls, a stone template
must be built into the walls ; said templates to be 8'' wide in 12" walls, and in all walls of
greater thickness to be in width not less than 4" less than the width of said walls, and not
to be, in any case, less than 4" in thickness and 18" long. . . .
22. All exterior cornices and gutters of all buildings, hereafter to be erected or built,
shall be of some fire-proof material. . . .
23. The planking and sheathing of the roof of every building, erected or built as afore-
said, shall in no case be extended across the front, rear, side, end, or party wall thereof,
and every such building, and the tops and sides of every dormer-window thereon, shall be
covered and roofed with slate, tin, zinc, copper, or iron, or such other equally fire- proof
roofing. . . .
PATENT-OFFICE DRAWINGS
must be made upon pure white paper, of a thickness corresponding to three-sheet Bristol
board, with surface calendered and smooth. Indian ink alone must be used.
The size of the sheet must be exactly 10 by 15 inches. I" from its edges single mar-
ginal lines are to be drawn, leaving the " sight " precisely 8" by 13". Within this margin
all work must be included. Measuring downward from the marginal line of one of the
shorter sides, a space of not less than 1 J inch is to be left blank for the heading of title,
name, number, and date.
All drawings must be made with the pen only. All lines and letters must be abso-
lutely black, clean, sharp, and solid, and not too fine or crowded. Surface shading should
be open, and used only on convex and concave surfaces sparingly. Sectional shading
should be made by oblique parallel lines, which may be about ^" apart.
Drawings should be made with the fewest lines possible conistent with clearness.
The plane upon sectional views should be indicated on the general view by broken or
dotted lines. Heavy lines on the shade sides of objects should be used, except where they
tend to thicken the work and obscure letters of reference; light to come from the upper
left-hand corner, at an anp;le of 45.
The scale of the drawing to be large enough to show the mechanism without crowd-
ing ; but the number of sheets must never be increased unless it is absolutely necessary.
Letters and figures of reference must be carefully formed, and, if possible, measure at
least -J-" in height, and so placed as not to interfere with a thorough comprehension of the
drawing, and therefore should rarely cross the lines. Upon shaded surfaces a blank space
must be left in the shading for the letter. The same part of an invention must always
"be represented by the same character, and the same character must never be used to
designate different parts.
The signature of the inventor, by himself or by his attorney, is to be placed at the
lower right-hand corner of the sheet, and the signature of two witnesses at the lower left-
hand corner, all within the marginal line. The title is to be written with pencil on the
back of the sheet. The permanent names and title will be supplied subsequently by the
office in uniform style.
Drawings should be rolled for transmission to the office, not folded.
APPENDi:
671
MENSURATION.
Properties of Triangles. It has been already shown in " Geometrical Problems " that
to construct a triangle three dimensions must be known the three sides, or two sides and
the included angle, or one side and the two adjacent angles. If only the three angles are
known, triangles of varied sizes may
be constructed, but all similar to each
other. To determine the length of the
side of a right-angled triangle by calcu- A
lation, the other two sides being known,
use these formulae :
CL
A 2 = B 2 + C 2 , or A = VB 2 + C 2
B = 4/A a ~-Tc 2 7or VA +~c~x~A"^~6
C = VA a B 2 7 or VA^B x A B. FIG. 1.
The side of any triangle (Figs. 1, 2, or 3) can be found by the following formulae :
B sin. a
, and consequently
sin. b
B sin. a
sm. =
A sin. 5
sm . a= ___.
The area of a triangle is equal to half the product of the
base by the height. Taking any side as the base, say B, the
height is readily obtained by multiply-
ing the length of the adjacent side A
by the natural sine of c. All figures
bounded by straight lines can be divided
into triangles, and their dimensions
readily calculated.
Properties of circles. The circum-
ference of a circle is equal to the diam-
eter multiplied by 3-1416, or TT (pi), or
approximately 3|.
The area of a circle is equal to the square of the radius multiplied by 3'1416 (TT), or the
square of the diameter multiplied by '7854.
The chord A (Fig. 4) forms, with the chords of
half the arc and the three radii, right-angled triangles
whose dimensions may be calculated as given above.
But the solution by table of natural sines is extremely
simple ; thus the chord is twice the sine of half the
angle A E at the center made by the radii to the
extremities of the chord. D E is the cosine of the
angle D E C or D E A, and the versed sine B D is
equal to radius less the cosine.
The versed sine F G of the half chord is equal to
about one quarter of the versed sine D B of the whole
chord.
The area of a sector A B E is to that of the
whole circle as the angle at the center A E is to 360*
the length of the arc ABC, will give the area.
FIG. 4.
or the radius, multiplied bj half
672
APPENDIX.
The length of an arc of one degree = radius x "017453.
u u " " " " " minute = u x '000291.
second =
x -000005.
The area of a segment A B D is equal to that of the sector less the area of the tri-
angle AEG formed by the chord and the two radii.
To find the circumference of an ellipse, divide the conjugate or short diameter by the
transverse or long diameter, and find the quotient in the first column in the accompanying
table ; take the corresponding number from the table, and multiply it by the long span.
2 = 2-10 -5 = 2-43 -8 = 2-84
3 = 2*20 -6 = 2 58 -9 = 2'99
4 = 2-30 -7 = 2-69 TO = 3'14
To find the area of an ellipse, multiply the conjugate by the transverse diameter, and
the result by '7854.
The area of a parabola is the product of the base by two thirds the height.
Mensuration of Solids. The solidity of parallelopipeds, cylinders, and prisms is found
by multiplying the base by the altitude.
The solidity of cones or pyramids is found by multiplying the base by one third the
vertical height ; of frustums of pyramids, the sum of the areas of the two ends added to
the square root of their product multiplied by one third the height.
The solidity of the sphere is the cube of the diameter multiplied by -5236.
The area of the surface is the square of the diameter multiplied by 3*1416 (w), or four
times the area of the great circle passing through the center.
The curved surface of a spherical segment is the product of the diameter of the sphere
by the height of the segment by 3-1416.
The solidity is three times the diameter of the sphere, less twice the height of the seg-
ment, multiplied by the square of the height, multiplied by -5236.
The solidity of the wedge is the length of the edge added to twice the length of the
back, multiplied by the height and by one sixth of the breadth of the back.
LINEAL MEASUEE.
Inches.
Feet.
Yards.
Fath-
oms.
Links.
Rods.
Chains.
Furlongs
Statute
miles.
Nautical
miles.
Metres.
1
08333
02778
0139
126
005
00126
000126
000016
0254
12 =
1
333
1667
1-515
0606
0151
00151
00019
....
0-3048
36 =
3
1
5
4-545
182
0454
00454
00057
0-9144
72 =
6
2
1
9-1
364
091
0091
00114
....
1-8289
7-92 =
0-66
22
11
1
04
01
001
000125
....
2012
198 =
16|
5
2f
25
1
25
025
003125
....
5-0294
792 =
66
22
11
100
4
1
10
0125
....
20-118
7920 =
660
220
110
1000
40
10
1
125
....
201-18
63360 =
5280
1760
880
8000
320
80
8
1
0-86755
1609-41
6086-07
2028-69
1-1527
1
1855-11
39-3685
3-2807
1-0936
5468
0-000621
1
Latin prefixes, as milli-, centi-, deci-, to the French units of length (metre), surface (are), weight
(gramme), or volume (litre), signify YIOOO, YIOO, or l / i0 of the unit; as, millimetre, YIOOO of a metre,
decigramme, l / i0 of a gramme. Greek prefixes, as kilo, hekto, deka, multiples of the unit by 1,000.
100, or 10, as kilometre = 1000 metres.
APPENDIX.
6Y3
TABLE OF INCHES AND SIXTEENTHS IN DECIMALS OF A FOOT.
Inches.
A
A
A
A
A
A
A
A
A
H
H
w
H
ooo
005
010
016
021
026
031
036
042
047
052
057
062
068
073
078
1
083
089
094
099
104
109
115
120
125
130
135
141
146
151
156
161
2
167
172
177
182
187
193
198
203
208
214
219
224
229
234
240
245
3
250
255
260
266
271
276
281
286
292
297
302
307
312
318
323
328
4
333
339
344
349
354
359
365
370
375
380
385
391
396
401
406
411
5
417
422
427
432
437
443
448
453
458
464
469
474
479
484
490
495
6
500
505
510
516
521
526
531
536
542
547
552
557
562
568
573
578
7....
583
589
594
599
604
609
615
620
625
630
635
641
646
651
656
661
8
667
672
677
682
687
693
698
703
708
714
719
724
729
734
740
745
9....
750
755
760
766
771
776
781
786
792
797
802
807
812
818
823
828
10
833
839
844
849
854
859
865
870
875
880
885
891
896
901
906
911
11....
917
922
927
932
937
943
948
953
958
964
969
974
979
984
990
995
MEASUKES OF SURFACE.
Sq. inches.
Sq. feet.
Sq. yards.
Sq. rods.
Eoods.
Acres.
Sq. miles.
Sq. metres.
Ares.
1
'00694
144 =
1
Ill
0037
....
....
....
0929
0009
1296 =
9
1
033
....
....
....
8361
0084
....
272|r
30J
1
025
00625
....
25-293
0-253
10890
1210
40
1
25
....
43560
4840
160
4
1
00156
4046-86
40-47
....
27878400
3097600
....
....
640
1
....
25899
1549-8 =
10-763
1-196
0395
0009
000247
....
1
01
1076-31
119-60
02471
100
1
MEASURES OF CAPACITY.
LIQUID MEASURE.
Gills.
Pints.
Quarts.
Gallons.
Imp. gallons.
Litres.
Cubic feet.
Cubic in.
Lbs. water
at 62.
1 =
0-25
0-125
03125
026
1183
0042
7-219
26
4 =
1
0-5
0-125
1041
4731
01671
28-875
1-0412
8 =
2
1
0-25
2083
0-9463
03342
57-75
2-0825
32 =
8
4
1
0-8331
3-7852
0-1337
231
8-33
38-4096 =
9-6024
4-8012
1-2003
1
4-5435
0-1605
277-27
10-00
8-4534 =
2-1133
1-0567
0-26417
0-2201
1
0-0353
61-0279
2-2007
239-36 =
59-84
29-92
7'48
6-232
28-320
1
1728
62-321
'138528 =
034632
017316
004329
0036
0-01639
0-000579
1
03606
01604
27'727
1
674
APPENDIX.
DRY MEASURE.
Pints.
Quarts.
Gallons.
Pecks.
Bushels.
1 =
0-50
0-125
0625
0-01562
2 =
1
0'25
0-125
0-0312
8 =
4
1
0-50
0-125
16 =
8
2
1
0-^5
64 =
32
8
4
i
The standard bushel contains 2150-42 cubic inches.
WEIGHTS.
APOTHECARIES'.
TROY.
Grains.
Scruples.
Drachms.
Ounces.
Pounds.
1 =
05
0167
0021
00018
20 =
1
333
042
0035
60 =
3
1
125
0104
480 =
24
8
1
083
5760 =
288
96
12
1
Grains.
Pennyweights.
Ounces.
Pounds.
1 =
042
0021
00018
24 =
1
05
0042
480 =
20
1
083
5760 =
240
12
1
AVOIRDUPOIS.
Drachms.
Ounces.
Pounds.
Hundred-weights .
Tons.
French grammes.
1 =
0625
0039
000035
00000174
1-771836
16 =
1
0625
000558
000028
28-34938
256 =
16
1
00893
000446
453-59
28672 =
1792
112
1
05
50802-
673440 =
35840
2240
20
1
1016041-6
It is common usage here to omit hundred-weights (cwt.) and rate tons at 2,000 pounds as net, and
2240 Ibs. as gross.
COMPAKISON OF WEIGHT.
DYNAMIC TABLE.
Pounds
apothecaries'.
Pounds
Troy.
Pounds
avoirdupois.
Kilo-
gramme.
1 =
1
0-8229
0-37324
1 =
1
0-8229
0-37324
1-2153 =
1-2153
1
0-4536
2-6792 =
2-6792
2-2046
1
Pounds,
fppt
Kilogramme-
metre.
Horse-
power.
French
horse-power.
i =
0-13825
00003
000031
7-2331 =
1
000219
000222
Per min.
33-000 =
4562-3
1
1-01386
32548-9 =
4500
0-98633
1
CUBIC OR SOLID MEASURE.
Cubic inches.
Cubic feet.
Cubic yards.
Cubic metres.
United States gallon.
1 =
00058
000021
000016
004329
1728 =
1
0-037
0-0283
7-48
46656 =
27
1
0-7646
201-97
61016 =
35-31
1-3078
1
264-141
231 =
0-1337
00495
00379
1
ss
34 C
53 o S
T-^ O
23 S!
sss
O5 oo ?2 Si o S
832
1ST
sss
O M CN 00 00 O O **
r-ccoo ascot- oo o
2 {2^2 SS5
IS g'SS
O O t- i
O Ob-GO ^O^H W^J {S^S rHCCub 05COI?-
283
eo-^ oo-o t- oo as oo-r-i cscoo toco is coio co
o o co i- o
CM CO CC CO -*
O^-OO OOO5O i ^H CO Tf i h-
*Sco 882 ^SI2 Ss3
GO ob 01 o i (Ncoo t- i <>4 oi?-o
ioco CO-*TX ooo
(MCOCO CO-*-
:-o Sp SS^ 2co?i SSp gSS ^"2"
rH rH O4 OJ CN <J1 * COCOCO 4j< * O O O ^- OO OS O
Si ^;
I <M CO CO rf I
3 ili
tetpm ui
-ratp
i TO cf c?^ :
676 APPENDIX.
WEIGHTS OF WROUGHT-IRON AND BRASS PLATES AND WIRE, SOFT ROLLED--
BIRMINGHAM GAUGE.
No. of
gauge.
AMERICAN GAUGE.
Plate iron.
Thickness of
each number.
Thickness
of each
number.
PLATKS PER SQUARE FOOT.
WIRE PER LINEAL TOOT.
Wrought
iron.
Brass.
Wrought
iron.
Brass.
Lbs.
Inch.
Inch.
Lbs.
Lbs.
Lbs.
Lbs.
17-025
454
0000
46
17-25
19-68
5607
6051
15-9375
425
000
4096
15-361
17-53
4447
4799
14-25
38
00
3648
13-68
15-61
3527
380&
12-75
34
3248
12-182
13-90
2797
8018
11-25
3
1
2893
10-848
12-38
2218
2393-
10-65
284
2
2576
9-661
11-02
1759
1898-
9-7125
259
3
2294
8-603
9-81
1395
1505
8-925
238
4
2043
7-661
8-74
1106
1193
8-25
22
5
1819
6-822
7-78
0877
0946
7-6125
203
6
1620
6-075
6-93
0695
0750
6-75
18
7
1442
5-410
6-17
0551
0595-
6-1875
165
8
1284
4-818
5-49
0437
0472.
5-55
148
9
1144
4-291 .
4-89
0347
0374
6-025
134
10
1018
3-820
4-36
0275
0296
4-5
12
11
0907
3-402
3-88
0218
0235
4-0875
109
12
0808
3-030
3-45
0173
0186-
3-5625
095
13
0719
2-698
3-07
0137
014&
3-1125
083
14
0640
2-403
2-74
0109
0117
2-7
072
15
0570
2-140
2-44
00863
00931
2-4375
065
16
0508
1-905
2-17
00684
00758
2-175
058
17
0452
1-697
1-93
00542
00585
1-8375
049
18
0403
1-511
1-72
00430
00464
1-575
042
19
0358
1-345
1-53
00341
00368
1-3125
035
20
(319
1-198
1-36
00271
00292.
1-2
032
21
0284
1-067
1-21
00215
00231
1-05
028
22
0253
9505
1-08
00170
0018&
9375
025
23
0225
8464
9660
00135
00145
825
022
24
0201
7537
8602
00107
00115
75
02
25
0179
6712
7661
00085
000916
675
018
26
0159
5977
6822
000673
000726-
6
016
27
0141
5323
6075
000534
000576
525
014
28
0126
4740
5410
000423
0004 5 T
4875
013
29
0112
4221
4818
000336
000362.
45
012
30
0100
3759
4290
000266
000287"
375
01
31
0089
3348
3821
000211
000228
3375
009
32
0079
2981
3402
000167
000180-
3
008
33
00708
2655
3030
000132
000143
2625
007
34
00630
2364
2698
000105
000113
1875
005
35
00561
2105
2402
0000836
00009015
15
004
36
005
1875
214
0000662
0000715
37
00445
1669
1905
0000525
00005671
38
00396
1486
1697
0000416
00004 4 96-
39
00353
1324
1511
0000330
0000356ft
40
00314
1179
1345
0000262
00002827"
Copper is about 5 per cent heavier than brass. Lead is about 47 per cent heavier than wrought
iron. Zinc is about 7 per cent lighter than wrought iron. Sheet copper is rated by weight at i
many ounces per square foot, and sheet lead at so many pounds per square 1
APPENDIX.
677
TABLE OF DIMENSIONS AND WEIGHT OF WEOUGHT-1RON WELDED TUBES.
Length of ; Length of
VT *
Nominal
diameter.
External
diameter.
Thick-
ness.
Internal
diameter.
Internal
circum-
ference.
External
circum-
ference.
pipe per
square
loot of
internal
pipe per
square
foot of
external
Internal
area.
Weight
per foot.
no. 01
threads
per
inch of
surface.
surface.
screw.
Inches.
Inches.
Inches.
Inches.
Inches.
Inches.
Feet.
Feet.
Inches.
Lbs.
V.
40
068
27
85
1-27
14-15
944
057
24
27
V*
54
088
36
I'M
1-7
10-5
7-075
104
42
18
3 /8
67
091
49
1-55
2-12
7-67
5'657
192
56
18
v>
84
.109
62
1-96
2'65
6-13
4-502
305
84
14
3 A
1-05
.113
82
2'59
3'3
4-64
3-637
533
1-13
14
i
1-31
134
1-05
3-29
4-13
3-66
2-903
863
1-67
iiVt
1V4
1-66
14
1-38
4-33
5-21
2-77
2-301
1-496
2-26
nV.
W
1-9
145
1-61
5-06
5-97
2-37
2-01
2-038
2-69
ll 1 /*.
2
2-37
154
2-07
6-49
7-46
1-85
1-611
3-355
3-67
iiVt
2/ 2
2-87
204
2-47
7-75
9-03
1-55
1-328
4-783
5-77
8
3
3-5
217
3-07
9-64
11-
1-24
1-091
7-388
7-55
8
*/
4-
226
3-55
11-15
12-57
1-08
0-955
9-887
9-05
8
4
4-5
237
4-07
12-69
14-14
95
0-849
12-73
10-73
8
4V*
5-
247
4-51
14-15
15-71
85
0-765
15-939
12-49
8
5
5-56
259
5-04
15-85
17-47
78
0-629
19-99
14-56
8
6
6-62
28
6-06
19-05
20-81
63
0-577
28-889
18-77
8
7
7-62
301
7-02
22-06
23-95
54
0-505
38-737
23-41
8
8
8-62
322
7'98
25-08
27-1
48
0-444
50-039
28-35
8
9
9'69
344
9-
28-28
30-43
42
0-394
63-633
34-08
8
10
10-75
366
10-02
31-47
33-77
38
0-355
78-838
40-64
8
Nominal
diameter.
Thickness,
extra strong.
Thickness, double
extra strong.
Actual inside diameter.
Extra strong.
Actual inside diameter.
Double extra strong.
Inches.
Inches.
O'lOO
Inches.
Inches.
0-205
Inches.
l/.
0-123
0-294
/ 4
0-127
0-421
v!
0-149
0-298
0-542
0-244
3 /4
0-157
0-314
0-736
0-422
1
0-182
0-364
0-951
0-587
I 1 /
0-194
0-388
1-272
0-884
I 1 /
0-203
0406
1-494
1-088
2
0-221
0442
1-933
1-491
V.
0-280
0-560
2-315
1-755
3
0-304
0-608
2-892
2-284
8 1 /.
0-321
0-642
3-358
2-716
4
0-341
0-682
3-818
3-136
BOILER TUBES.
External
diameter.
Thickness,
wire gauge.
Average
weight.
External
diameter.
Thickness,
wire gauge.
Average
Weight.
Inches.
No.
Lbs. per foot.
Inches.
No.
Lbs. per foot.
W*
16
1-
3
11
3-5
i 1 /.
15
Me
3 J /4
11
4'
! 3 /4
14
1-63
4
8
6'4
2
13
2-
5
7
9-1
2'/4
12
2-16
6
6
12-3
*v.
12
2-56
7
6
15'2
*"/I6
11
2-2
8
6
16-
678
APPENDIX.
HEAVY PIPE FOR DRIVEN WELLS.
Tested at 1200 pounds hydraulic pressure. Furnished in five-foot lengths.
Size (inches)
H
H
2
' 2J
O
3*
4
Weight per foot, Ibs..
3-62
2-75
3-75
6-00
7-75
9-25
11-00:
HEAVY WROUGHT GALVANIZED IRON SPIRAL RIVETED PIPES,
WITH FLANGED CONNECTIONS.
Tested at 150 pounds hydraulic pressure. Regalvanized after riveting.
Inside diameter (inches) .
3
4
5
6
7
8
9
10
11
12
Wire gauge, Nos
20
20
20
18
18
18
18
16
16
16
Nominal weight per foot, Ibs. . .
2i
4
5
6
7
8
9
12
13
14
Manufactured lengths, 20 feet or less. Elbows and other fittings, cast iron.
LIGHT PIPE, SUITABLE FOB HOUSE LEADERS, VENTILATING, AIR, AND BLOWER PIPES, ETC.
Inside dia,meter (inches) . ...
2
2A
3
3i
4
4-i-
5
5i
6
Nominal wei' r ht per foot Ibs .
f
1
1
14-
If
H
14
1
TABLE OF COPPER AND BRASS RODS ONE FOOT IN LENGTH.
To find the weight of copper or brass pipe, take the weight of the exterior diameter from the
table, and subtract from it the weight of a rod equal to that of the interior diameter, or bore.
Diamet'r
in
inches.
Copper.
Brass.
Diamefr
in
inches.
Copper.
Brass.
DiametT
in
inches.
Copper.
Brass.
'/
047
045
W*
7-993
7-593
4'A
55-62
52-27
8 A
106
101
1"/16
8-630
8-198
4 3 / 8
58-94
5539
'A
189
, -179
I'A
9-270
8-806
4V*
62-36
58-60
6 A
296
281
! 13 Ae
9-950
9-452
4 5 /e
65-87
61-90
%
426
405
iVs
10-642
10-110
4 3 / 4
69-48
6
V
579
550
I"/,.
11-370
10-801
4 7 / 8
73-19
68-77
V.
757
719
2
12-108
11-503
5
77-43
72-76
9 A
958
910
*'/
13-668
12-985
5'/ 8
80-89
76-00'
5 /8
1-182
1-123
2'A
15-325
14-559
5V4
84-88
79-76
"Ae
1-431
1-360
2 3 /s
17-075
16-221
5 3 / 8
88-97
83-60
8 A
1-703
1-618
2'A
18-916
17-970
5'/i
93-15
87-63
13 A
1-998
1-898
2 5 /e
20-856
19-808
5 5 /s
97-44
91-56
V-
2-318
2-202
2 3 / 4
22-891
21-746
5 3 / 4
101-81
95-68
15 Ae
2-660
2-527
r/.
25-019
23-768
5Y
106-29
99-88
1
3-027
2-876
3
27-243
25-881
6
110-85
104-15
i'A.
3-417
3-246
3'/s
29-559
28-081
6 ! A
12030
113-04
W
3-831
3-639
ffif*
31-972
30-373
6V,
130-10
122-26
i 3 A
4-269
4-056
3 3 /e
34-481
32-757
6 3 / 4
140-32
131-85
i'A
4-723
4-487
3 1 /,
37-081
35-227
7
150-86
141-76
l'/i.
5-214
4-953
3 5 /e
39-777
37-788
v ! A
161-87
152-10
1 3 A
5-723
5-437
*/
42-568
40-440
/
173-22
162-77
!'/.
6-255
5-943
r/,
45-455
43-182
7 3 / 4
184-97
173-81
1%
6-811
6-470
4
48-433
46-000
8
197-03
185-14
! 9 Ae
7-390
7-020
4V
52-40
49-24
APPENDIX.
6?$
NUMBER OF BURDEN'S RIVETS IN ONE HUNDRED POUNDS.
Lengths.
DIAMETER.
Lengths.
JS. B.
i
1
H
}
1
1,092
665
....
....
5
90
|
1,027
597
....
....
6|
85
1
940
538
450
....
6
80
H
840
512
415
....
6|
75
U
797
487
389
356
7
70
If
760
460
370
329
n
67
H
730
440
357
280
8
65
if
711
420
340
271
H
61
if
693
390
325
262
9
57
H
648
375
312
257
9 i
54
2
608
360
297
243
10
51
H
573
354
10!
47
2 i
555
347
280
232
H
525
335
260
220
2f
500
312
242
208
3
460
290
224
197
H
433
267
212
180
3|
413
248
201
169
H
395
241
192
160
4
....
230
184
158
4*
....
220
177
150
4 I
....
210
171
146
4
....
200
166
138
5
....
190
161
135
BJ
....
180
156
130
B*
....
172
151
124
5f
....
164
145
120
6
....
157
140
115
6 i
....
150
138
111
6
....
146
134
107
6f
....
143
129
104
7
140
125
100
WROUGHT SPIKES NUMBER TO A KEG OF ONE HUNDRED AND FIFTY POUNDS.
LENGTH.
i"
A"
1"
A"
i"
Inches,
3
2,250
3i!r
,890
1,208
4
,650
1,135
4-i .
,464
1,064
5
,380
930
742
6
,292
868
570
7
,161
662
482
445
306
8
635
455
384
256
9
573
424
300
240
10 ...
391
270
222
11
249
203
12..
236
180
680 APPENDIX.
LENGTHS OF CUT NAILS AND SPIKES, AND NUMBER IN A POUND.
Size.
3d.
Length.
No.
Size.
Length.
No.
Size.
Length.
No.
Inches.
1
420
Sd.
Inches.
2*
100
30d
Inches.
4
24
4
1*
270
10
3
65
40
4
20
5
If
220
12
8*
52
60
6
2
175
20
3|
28
WEIGHTS OF LEAD PIPE PEE FOOT IN LENGTH.
Caliber.
MASK.
AAA
AA
A
B
C
D
E
Lbs. oz.
2
10
9$
1 8
i
i
I
A
i
f
t
i
li
1*
if
2
H
3
8*
4
4*
5
6
Lbs. oz.
Lbs. oz.
Lbs. oz.
Lbs oz.
Lbs. oz.
Lbs. oz.
Lbs. oz.
2
1 12
1 5
1 2
1
14
7
3 ..
2 8
3 8
4 14
6 ..
6 12
8
2
1 10
1 3
1
10
12
1 4
1 3
2 4
2 8
3 8
3
3 10
4
3
12
1
2
2
2
2 12
3 3
4 8
5 12
7
2 8
3
4
4 11
6 4
2
2 3
3 4
3 11
5
1 7
1 12
2 8
3
4 4
10 11
8 8
8 14
6 7
7
5
6
4
5
THICKNESS.
WASTE.
1
A
i
4
A
16 11
19 9
22 8
25 6
*31 3
13 10
16
18 7
20 14
10 10
12 9
14 8
16 7
18 6
20 5
7 3
9 4
10 12
12 2
13 9
15
6
5
4
3 8
8
6
10 8
10 8
12
7 6
TABLE OF THE WEIGHT OF A CUBIC FOOT OF WATER AT DIFFERENT TEM-
PERATURES.
Fahren-
heit.
Centi-
grade.
Weight in
pounds.
Fahren-
heit.
Centi-
grade.
Weight in
pounds.
Fahren-
heit.
Centi-
grade.
Weight in
pounds.
Degrees.
32
Degrees.
62-42
Degrees.
95
Degrees.
35
62-06
Degrees,
167
Degrees.
75
60-87
39
4
62-42
104
40
61-95
176
80
60-68
41
5
62-42
113
45
61-83
185
85
60-48
50
10
62-41
122
50
61-69
194
90
60-27
59
15
62-37
131
55
61-55
203
95
60-04
68
20
62-32
140
60
61-39
212
100
59-83
77
25
62-25
149
65
61-23
86
30
62-16
158
70
61-06
APPENDIX.
631
PROPERTIES OF SATURATED STEAM,
FROM "RICHABDS'S STEAM-ENGINE INDICATOR," BY CHAS. T. PORTER.
ELASTIC
HEAT, IN DEGREES
$i
1
ELASTIC
HEAT, IN DEGREES
I- 1
FORCE.
FAHRENHEIT.
I
^
FORCE.
FAHRENHEIT.
- I
*S
. ~
V *
?
'"=> *
B
sr
1 I -
1*
M
* *
k
si
?
c 8
%
!
l\
c S
s 3
"o 5
!?
*\
a *"
!)
II
i
Jj
I 1
V
fi
\\
"c ~
J 5
If
s
V
1*
I 3
r
i
2-04
102-
1043-0
1145-0
0029
037
64
130-40
296-9
907-6
1204-5
1416 j 1-754
2
4-08
126-3
1026-1
1152-4
0057
071 65
132-44
298-0
906-8
1204-8
1436 -779
3
6-11
141-6
1015-4
1157-1
0084
104 66
134-48
299-0
906-1
1205-1
1456 1-804
4
815
153-1
1007-5
1160-6
0110
136
67
136-51
300-0
905-4
1205-4
14V6 1-829
5
10-19
162-3
1001-0
1163-4
0135
167
68
138-55
300-9
904-8
1205-7
1496
854
6
12-22
1701
995-6
1165-8
0160
198
69
140-59
301-9
904-1
1206-0
1516
879
7
14-26
176-9
990-9
1167-9
0185
228 ! 70
142-63
302-9
903-4
1206-3
1536 -904
8
16-30
182-9
. 986-7
1169-7
0209
258 71
144-66 303-9
902-7
1206-6
1556 1-929
9
18-34
188-3
983-0
1171-3
0233
238 72
146-70 304 8
902-1
1206-9
1576 1-954
10
20-38
193-2
979-6
1172-8
0257
318 , 73
148-74
305-7
901-5
1207-2'
1596 1-979
11
22-41
197-8
976-4
1174-2
0281
348 74 !
150-78
306-6
900-9
120Y-5
1616
2-004
12
24-45
201-0
973-5
1175-5
0304
377 75
152-81 307-5
900-3
1207-8
1636
2-029
13
26-48
205-9
970-8
1176-7
0327
406
76
154-85 308-4
899-6
1208-0
1656
2-054
14
28-53 209-6
968-2
1177-8
0350
435
77
156-89 309-3
899-0
1208-3
1676
2-079
14-7
atmos.
78
158-93
310-2
898-4
1208-6
1696
2-103
15
30-56
213-0
965-8
1178-9
0373
463
79
160-96
311-1
897-8
1208-9
1716 2-127
16
32-60
216-3
963-6
1179-9
0396
492
80
163-00
312-0
897-1
1209-1
1736
2-151
17
34-64
219-4
961-5
1180-9
0419
520 SI
165-04
312-8
896-6
1209-4
1756
2-175
18
36-68
222-4
959-4
1181-8
0442
548
J 82
167-08
313-6
896-1
1209-7
1776
2-199
19
38-71
225-2
957-5
1182-7
0465
576
i 83 169-11
314-5
895-4
1209-9
1795
2-223
20
40-75
228-0
955-5
1183-5
0487
604
84
171-15
315-3
894-8
1210-1
1814
2-247
21
42-79
230-6
953-7
1184-3
0510
632
85 173-19
316-1
894-3
1210-4
1833
2-271
22
44-83
233-1
951-9
1185-0
0532
660
86 175-23
316-9
893-8
1210-7
1852
2-295
23
46-86
235-5
950-2
1185-7
0554
688
87 177-26
317-8
893-1
1210-9
1871
2-319
24
43-90
237-9
948-6
1186-5
0576
715 88 179-30
318-6
892-5
1211-1
1891
2-343
25
50-94
240-2
947-0
1187-2
0598
742 1 89 1181-34
319-4
892-0
1211-4
1910
2-367
26
52-98
242-3
945-6
1187-9
0620
769 ! 90 183-38
320-2
891-4
1211-6
1930
2-391
27
55-01
244-4
944-1
1188-5
0642
796 91 185-41
321-0
890-8
1211-8
1950
2-415
28
57-05
246-4
942-7
1189-1
0664
823 1 92 187-45
321-7
890-3
1212-0
1970
2-439
29
59-09
248-4
941-3
1189-7
0686
850 93 189-49
322-5
889-8
1212-3
1990
2-463
30
61-13
250-4
939-9
1190-3
0707
877
94
191-53
323-3
889-2
1212-5
2010
2-487
31
63-16
252-3
938-5
1190-8
0729
904
1 95 !
193-56
324-1
888-7
1212-8
2030
2-511
32
65-20
254-1
937-3
1191-4
0751
931 | 96 i
195-60
324-8
888-2
1213-0
2050
2-535
33
67-24
255-9
936-1
1192-0
0772
958 97
197-64
325-6
887-7
1213-3
2070
2-559
34
69-28
257-6
934-9
1192-5
0794
985
1 98 J199-68
326-3
887-2
1213-5
2089
2-583
35
71-31
259-3
933-7
1193-0
0815
1-012
99 201-71
327-1
886-6
1213-7
2108
2-607
36
73-35
260-9
932-6
1193-5
0837
1-033 100 203-75
327-8
886-1
1213-9
2127
2-631
37
75-39
262-6
931-4
1194-0
0853
1-064 101 1205-79
328-5
885-7
1214-2
2147
2-655
38
77-43
264-2
930-3
1194-5
0879
1-090 102 1207-88
329-2
885-2
1214-4
2167
2-679
39
79-46
265-8
929-2
1195-0
0900
1-116 103
209-86
329-9
884-7
1214-6
2186
2-703
40
81-50
267-3
923-1
1195-4
0921
1-142 1 104
211-90
330-6
884-2
1214-8
2205
2-727
41
83-54
268-7
927-2
1195-9
0942
1-168 105 1213-94
331-3
883-7
1215-0
2224
2-751
42
85-58
270-2
926-1
1196-3
0963
1-194 106
215-98
331-9
883-3
1215-2
2243
2-775
43
87-61
271-6
925-2
1196-8
0983
1-220 i 107
218-01
332-6
882-8
1215-4
2262
2-799
44
89-65
273-0
924-2
1197-2
1004
1-246 108
220-05
333-3
882-3
1215-6
2281
2-823
45
91-69
274-4
923-2
1197-6
1025
272 109
222-09
334-0
881-8
1215-8
2300
2-847
46
93-73
275-8
922-2
1198-0
1046
298 110
224-13
334-6
881-4
1216-0
2319
2-871
47
95-76
277-1
921-3
1198-4
1067 ! -324 111
226-16
335-3
880-9
1216-2
2337
2-895
48
97-80
278-4
920-4
1198-8
1087 '350 112
228-20
336-0
880-4
1216-4
2355
2-919
49
99-84
279-7
919-5
1199-2
1108
376 113
230-24
336-7
879-9
1216-6
2374
2-943
50
101-88 281-0
918-6
1199-6
1129 | -402 114
232-28
337-4
879-4
1216-8
2392
2-967
51
103-91
282-3
917-7
1200-0
1150
1-428 115 i
234-31 .
338-0
879-0
1217-0
2410
2-990
52
105-95
283-5
916-9
1200-4
1171
1-454 116 236-35
338-6
878-6
1217-2
2428
3-013
53
107-99 284-7
916-1
1200-8
1192
1-479 117 238-39
339-3
878-1
1217-4
2446
3-036
54
110-03 285-9
915-2
1201-1
1212
1-504 118 ,240-43
339-9
877-7
1217-6
2465
3-059
55
112-06 287-1
914-4
1201-5
1232
1-529 119 !-242-46
340-5
877-3
1217-8
2484
3-082
56
114-10 288-2
913-6
1201-8
1252 1-554 120 244-50
341-1
876-9
1218-0
2503
3-105
57
116-14 289-3
912-9
1202-2
1272 i 1-579 121
246-54
341-8
876-4
1218-2
2522
3-130
58
113-18 290-4
9121
12025
1293
1-604 122 248-58
342-4
876-0
1218-4
2541
3-155
59
120-21 291-6
911-3
1202-9
1314
1-629 123
250-61
343-0
875-6
1218-6
2560
3179
60
122-25 292-7
910-5
1203-2
1335
1-654 124
252-65
343-6
875-1
1218-7
2579
3-203
61
124-29 293-8
909-8
1203-6
1356
1-679 125
254-69
344-2
874-7
1218-9
2598
3-227
62 126-33 294-8
909-1 1203-9
1376
1-704 126
256-73 344-8
874-3
12191
2617
3-251
63 1128-36 295-9
908-3 1204-2
1396
1-729 127
258-76 345-4
873-9
1219-3
2636
3-275
682
APPENDIX.
PEOPEETIES OF SATUEATED STEAM (Continued.)
ELASTIC
HEAT, IN DEGREES
l|
|
ELASTIC
HEAT, IN DEGREES
J
1
FORCE.
FAHRENHEIT.
rS
<U Q
FORCE.
FAHRENHEIT.
|l
IJ
S"
Jo
o e
i.
o .
D O
-3
"S
ft
Id
e a
k
C *
l|
si
.1
ft
* 2
li
P
II
1
I 1
I 1
II
ft
CO
jl
| J
"* 3
jl
I 1
-
J
I-
I 1
128 260-80
346-0
873-4
1219-4
2655
3-299
140
285-25
352-9
868-6
1221-5
2883
3-582
129 262-84
346-6
873-0
1219-6
2674
3-323
141
287-29
353-4
868-3
1221-7
2902
3-605
130 264-88
347-2
872-6
1219-8
2693
3-347
142
289-33
354-0
867-9
1221-9
2921
3-628
131 266-91
347-8
872-2
1220-0
2712
3-371
143
291-36
354-5
867-5
1222-0
2940
3-651
132
268-95
348-3
871-9
1220-2
2731
3-395
144
293-40
355-0
867-2
1222*2
2959
3-674
133
270-99
348-9
871-5
1220-4
2750
3-419
145
295-44
355-6
866-8
1222-4
2978
3-697
134 273-03
349-5
871-1
1220-6
2769
3-443
146
297-48
356-1
866-4
1222-5
2997
3-720
135 275-06
350-0
870-7
1220-7
2788
3-467
147
299-51
356-7
866-0
1222-7
3016
3-74S
136 277-10
350-6
870-3
1220-9
2807
3-490
148
301 -55
357-2
865-7
1222-9
3035
3-765
137 279-14
351-2
869-8
1221-0
2826
3-513
149
303-59
357-8
865-2
1223-0
3054
3-787
138 281-18
351-8
869-4
1221-2
2845
3-536
150
305-63
358-3
864-9
1223-2
3073
3-809
139
283-21
352-3
869-1
1221-4
2864
3-559
1
TABLE OF MEAN PEESSUEES IN STEAM CYLINDEES AT DIFFEEENT BATES OF
EXPANSION.
Portion of
stroke during
which steam
Mean press-
ure during
whole of
Portion of
stroke during
which steam
Mean press-
ure during
whole of
Portion of
stroke during
which steam
Mean press-
ure during
whole of
Portion of
stroke during
which steam
Mean press-
ure during
whole of
is admitted.
stroke.
is admitted.
stroke.
is admitted.
stroke.
is admitted.
stroke.
80
98
56
88
40
77
24
58.
77
97
54
87
38
75
22
55-
74
96
52
86
36
73
20
52
70
95
50
85
34
71
18
49-
68
94
48
83
32
68
16
45
66
93
46
82
30
66
14
42
62
92
44
80
28
64
12
37
60
90
42
78
26
61
10
33
58
89
Examples of Application of above Table. To find the mean pressure in a condensing
engine with an initial pressure, as shown by the gauge, of 75 pounds, and a cut-off at -20,
or stroke.
The actual initial pressure above is 75 + 15, or 90 pounds. Mean pressure at '20 cut-
off in table '52 for each pound of initial pressure, 90 x '52 = 46-8 mean pressure above
in cylinder ; but as the vacuum in the cylinder can never be perfect, an allowance of two
to three pounds is to be made; 46'8 2'8 = 44, which may be taken as the probable
actual mean pressure to be used in estimating the H. P. or Ibs. ft. of work of the engine
made up thus : Mean pressure x area of steam piston in square inches, less \ that of the
piston-rod x length of stroke in feet x number of strokes per minute = Ibs. ft. of work
per minute, and divided by 33,000 = II. P.
If the engine is non-condensing, then the deduction from the mean pressure would be
the whole atmospheric pressure, 14/7, and probably about 1*3 back pressure, or say, 16
pounds, and the mean effective pressure in the cylinder would be for the cut-off and initial
power as above, 46*8 16, or 30'8 pounds.
In estimating for the per cent of cut-off or steam follow, the clearances are to be esti-
mated with the stroke and cut-off.
It may often be convenient to estimate the amount of water and coal necessary for an
engine, which can be done approximately by taking the tension or pressure of the steam
at any part of the stroke after the cut-off, finding in table the weight of one cubic foot
APPENDIX. 68a
of steam corresponding to this pressure, and multiplying it by the number of cubic feet
in the cylinder at the point taken, which will be the weight of steam used per stroke.
Multiplying this product by the number of strokes per working day, will give the total
weight of water used as steam ; and if 8 pounds of steam be allowed for each pound of coal,
it will give a fair average of tho coal consumption during working hours. There will be
additional coal used for getting up steam or for banking, and more water will be used than
shown by the steam in the cylinder, as there will be water entrained with the steam, and
condensed in passages and cylinder, equal to 25 per cent more, say, in total, 10 pounds of
water for each pound of coal fed on the grates.
THE FLOW OF WATER.
The velocity of water in a stream or channel is often taken approximately by floats
along different threads of the current. If the channel be an artificial one of rectangular
section, the average velocity may be determined very nearly by a number of such experi-
ments, with a tube float, extending nearly to the bottom of the channel ; but in the rivers
and streams, if surface floats be used, allowance is to be made for the friction of water on
the bed of the stream, and want of uniformity in the flow. There are a variety of tachom-
eters to determine the velocities beneath the surface, and to afford data for averages.
In the flow of water through apertures the theoretic velocity in feet per second is
8'04 ^ h, h being the head or height of surface of water in feet above the center of the
aperture. But in all apertures the discharge is less than the product of the area of their
section by the theoretic velocity. There are contractions which reduce the effective sec-
tion. If the discharge be through a thin plate into air, in which the contractions are
around the entire periphery, the discharge is T 6 7 of that due to the section and theoretic
velocity. If the edges are rounded, or the discharge be through a short pipe or ajutage,
or beneath the surface of the water, the loss is less, and by suitable ajutages it may be
almost entirely eliminated.
For the common purpose of gauging or determining the discharge of large pumps or
small streams, the most accurate measure is by weirs, on which many experiments have
been made, but those of Mr. James B. Francis, G. E., which are embodied in " Lowell
Hydraulic Experiments," embrace a more practical range than any other, and are consid-
ered standard.
His general formula, on which the following table is calculated, is Q = 3'33 (I -2A) Af,
in which Q is the discharge in cubic feet per second, I the length of the weir, and h the
height of water above the crest of the weir, both in feet; h is taken either at the side of
the weir or a slight distance up stream ; usually, a pipe with small perforations is laid
parallel with the weir, on the bottom, and connected with a tight vertical box, in which
the oscillations of the water surface are reduced to a mean.
In the table, the discharge is given for one foot in length ; but as in weirs there are
usually two end contractions, virtually reducing the length, and met in the formula above
by -2&, a column of correction has been added, which is to be subtracted from the
product of discharge, as given in the other columns of the table, by the length in feet.
Example. Let the weir, with end contractions, be 5-3 feet long, and depth of water v
or h = 0-612.
By table the discharge for one foot in length is 1 -594
5-3
8-4482
Correction -196
Discharge in cubic feet per second 8-252,
684
APPENDIX.
DISCHARGE, IN CUBIC FEET PER SECOND, OF A WETR ONE FOOT LONG, WITH-
OUT CONTRACTION AT THE ENDS; FOR DEPTHS FROM 0-500 TO 0-999 FEET.
Correction
for con-
tractions.
Depth.
1
2
3
4
5
6
7
8
9
012
0-20
0-298
0-300
0-302
0-305
0-307
0-309
0-311
0-314
0-316
0-318
013
21
0-320
0-323
0-325
0-327
0-330
0-332
0-334
0-337
0-339
0-341
015
22
0-344
0-346
0-348
0-351
0-353
0-355
0-358
0-360
0-362
0-365
017
23
0-367
0-370
0-372
0-374
0-377
0-379
0-382
0-384
0-387
0-389
019
24
0-391
0-394
0-396
0-399
0-401
0-404
0-406
0-409
0-411
0-414
021
25
0-416
0-419
0-421
0-424
0-426
0-429
0-431
0-434
0-436
0-439
023
26
0-441
0-444
0-447
0-449
0-452
0-454
0-457
0-459
0-462
0-465
025
27
0-467
0-470
0-472
0-475
0-478
0-480
0-483
0-485
0-488
0-491
'028
28
0-493
0-496
0-499
0-501
0-504
0-507
0-509
0-512
0-515
0-517
030
29
0-520
0-523
0-525
0-528
0-531
0-534
0-536
0-539
0-542
0-544
033
0-30
0-547
0-550
0-553
0-555
0-558
0-561
0-564
0-566
0-569
0-572
036
31
0-575
0-577
0-580
0-583
0-586
0-589
0-591
0-594
0-597
0-600
039
32
0-603
0-606
0-608
0-611
0-614
0-617
0-620
0-623
0-625
0-628
042
33
0-631
0-634
0-637
0-640
0-643
0-646
0-649
0-651
0-654
0-657
045
34
0-660
0-663
0-666
0-669
0-672
0-675
0-678
0-681
0-684
0-687
-048
35
0-689
0-692
0-695
0-698
0-701
0-704
0-707
0-710
0-713
0-716
-052
36
0-719
0-722
0-725
0-728
0-731
0-734
0-737
0-740
0-743
0-746
-056
37
0-749
0-752
0-755
0-759
0-762
0-765
0-768
0-771
0-774
0-777
-059
38
0-780
0-783
0-786
0-789
0-792
0-795
0-799
0-802
0-805
0-808
063
39
0-811
0-814
0-817
0-820
0-823
0-827
0-830
0-833
0-836
0-839
-067
0-40
0-842
0-846
0-849
0-852
0-855
0-858
0-861
0-865
0-868
0-871
072
41
0-874
0-877
0-881
0-884
0-887
0-890
0-893
0-897
0-900
0-903
076
42
0-906
0-910
0-913
0-916
0-919
0-923
0-926
0-929
0-932
0-936
081
43
0-939
0-942
0-945
0-949
0-952
0-955
0-959
0-962
0-965
0-969
085
44
0-972
0-975
0-978
0-982
0-985
0-988
0-992
0-995
0-998
1-002
090
45
1-005
1-009
1-012
1-015
1-019
1-022
1-025
1-029
1-032
1-035
-095
46
1-039
1-042
1-046
1-049
1-052
1-056
1-059
1-063
1-066
1-070
100
47
1-073
1-076
1-080
1-083
1-087
1-090
1-094
1-097
1-100
1-104
-106
48
1-107
1-111
1-114
1-118
1-121
1-125
1-128
1-132
1-135
1-139
-111
49
1-142
1-146
1-149
1-153
1-156
1-160
1-163
1-167
1-170
1-174
-118
0-50
1-177
1-181
1-184
1-188
1-191
1-195
1-199
1-202
1-206
1-209
124
51
1-213
1-216
1-220
1-223
1-227
1-231
1-234
1-238
1-241
1-245
-130
52
1-249
1-252
1-256
1-259
1-263
1-267
1-270
1-274
1-278
1-281
136
53
1-285
1-288
1-292
1-296
1-299
1-303
1-307
1-310
1-314
1-318
143
54
1-321
1-325
1-329
332
1-336
1-340
1-343
1-347
1-351
1-355
-150
55
1-358
1-362
1-366
369
1-373
1-377
1-381
1-384
1-388
1-392
157
56
1-395
1-399
1-403
407
1-410
1-414
1-418
1-422
1-425
1-429
164
57
1-433
1-437
1-441
444
1-448
452
1-456
1-459
1-463
1-467
171
58
1-471
1-475
1-478
482
1-486
490
1-494
1-498
1-501
1-505
-178
59
1-509
1-513
1-517
521
1-524
528
1-532
1-536
1-540
1-544
-186
0-60
1-548
1-551
1-555
559
1-563
567
1-571
1-575
1-579
1-583
194
61
1-586
1-590
1-594
598
1-602
606
1-610
1-614
1-618
1-622
202
62
1-626
1-630
1-633
637
1-641
645
1-649
1-653
1-657
1-661
210
63
1-665
1-669
1-673
677
1-681
685
1-689
1-693
1-697
1-701
218
64
1-705
1-709
1-713
717
1-721
725
1-729
1-733
737
1-741
-227
65
1-745
1-749
1-753
757
1-761
765
1-769
1-773
777
1-781
'236
66
1-785
1-790
1-794
798
1-802
806
1-810
1-814
818
1-822
245
67
1-826
1-830
1-834
838
1-843
847
1-851
1-855
859
1-863
254
68
1-867
1-871
1-875
1-880
1-884
888
1-892
1-896
900
1-904
263
69
1-909
1-913
1-917
1-921
1-925
1-929
1-934
1-938
942
1-946
"273
0-70
1-950
1-954
1-959
1-963
1-967
1-971
1-975
1-980
1-984
1-988
283
71
1-992
1-996
2-001
2-005
2-009
2-013
2-017
2-022
2-026
2-030
-293
72
2-034
2-039
2-043
2-047
2-051
2-056
2-060
2-064
2-068
2-073
-303
73
2-077
2-081
2-085
2-090
2-094
2-098
2-103
2-107
2-111
2-115
-314
74
2-120
2-124
2-128
2-133
2-137
2-141
2-146
2-150
2-154
2-159
'324
75
2-163
2-167
2-172
2-176
2-180
2-185
2-189
2-193
2-198
2-202
APPENDIX.
685-
DISCHARGE, IN CUBIC FEET PER SECOND, OF A WEIR ONE FOOT LONG, WITH-
OUT CONTRACTION AT THE ENDS; FOR DEPTHS FROM 0-500 TO 0-999 FEET.
( Continued. )
Correction
for con-
tractions.
Depth.
1
2
3
4
5
6
7
8
9
335
76
2-206
2-211
2-215
2-219
2-224
2-228
2-232
2-237
2-241
2-246
346
77
2-250
2-254
2-259
2-263
3-267
2-272
2-276
2-281
2-285
2-290
358
78
2-294
2-298
2-303
2-307
2-312
2-316
2-320
2-325
2-329
2-334
369
79
2-238
2-343
2-347
2-351
2-356
2-360
2-365
2-369
2-374
2-378
381
0-80
2-383
2-387
2-392
2-396
2-401
2-405
2-410
2-414
2-419
2-423
393
81
2-428
2-432
2-437
2-441
2-446
2-450
2-455
2-459
2-464
2-468
406
82
2-473
2-477
2-482
2-486
2-491
2-495
2-500
2-504
2-509
2-513
413
83
2-518
2-523
4-527
2-532
2-536
2-541
2-545
2-550
2-554
2-559
431
84
2-564
2-568
2-573
2-577
2-582
2-587
2-591
2-596
2-600
2-605
444
85
2-610
2-614
2-619
2-623
2-628
2-633
2-637
2-642
2-646
2-651
457
86
2-656
2-660
2-665
2-670
2-674
2-679
2-684
2-688
2-693
2-698
470
87
7-702
2-707
2-712
2-716
2-721
2-726
2-730
2-735
2-740
2-744
484
88
2-749
2-754
2-758
2-763
2-768
2-772
2-777
2-782
2-786
2-791
498
89
2-796
2-801
2-805
2-810
2-815
2-819
2-824
2-829
2-834
2-838
512
0-90
2-843
2-848
2-853
2-857
2-862
2-867
2-872
2-876
2-881
2-886
526
91
2-891
2-895
2-900
2-905
2-910
2-915
2-919
2-924
2-929
2-934
541
92
2-938
2-943
2-948
2-953
2-958
2-963
2-967
2-972
2-977
2-982
555
93
2-986
2-991
2-996
3-001
3-006
3-011
3-015
3-020
3-025
3-030
570
94
3-035
3-040
3-044
3-049
3-054
3-059
3-064
3-069
3-074
3-078
586
95
3-083
3-088
3-093
3-098
3-103
3-108
3-113
3-117
3-122
3-127
601
96
3-132
3-137
3-142
3-147
3-152
3-157
3-162
3-166
3-171
3-176
677
97
3-181
3-186
3-191
3-196
3-201
3-206
3-211
3-216
3-221
3-226
632
98
3-231
3-235
3-240
3-245
3-250
3-255
3-260
3-265
3-270
3-275
648
99
3-280
3-285
3-290
3-295
3-300
3-305
3-310
3-315
3-320
3-325
Flow of Water through Pipes. Figs. 5 and 6 are diagrams showing, by inspection,
the million gallons delivered in 24 hours under varying resistance-heads or sines of
slopes ( j of c.ean cast-iron pipes of diameters from 6" to 36". They are calculated
from the table of velocities in J. F. Fanning's " Practical Treatise on Hydraulics and
Water-Supply Engineering.' 11
Illustration of the Application of Diagram. To determine the million gallons dis-
charged per 24 hours through a 12" pipe with '02 sine of slope. The intersection of the
horizontal of -02 by the curve of 12" is on the ordinate 4 millions, which will be dis-
charge to be determined.
Again, to determine the loss of head per foot in length of a 30" pipe in delivering 25
million gallons per 24 hours. The intersection of the ordinate of 25 millions with the 30"
curve is in the horizontal '0067, the loss of head to be determined.
It will be seen that a 36" pipe would deliver the same quantity with a loss of but '0025
feet per foot in length.
These diagrams are applicable to long mains with a uniform current.
Flow through Sewers. Fig. 7 is a diagram similar to the preceding, by which may
be readily determined the cubic feet per second that would flow through circular sewers
from 12" to 72" diameter, with various falls of from TT5 * Jff to y^ of a foot per foot of
length.
It is calculated by the formula given by A. Fteley, M. A. S. 0. E., in the description of
the " Additional Supply from Sudbnry River," for the Boston Water-Works, and deduced
from experiments made by him on those works.
686
APPENDIX.
2QE
i
9)
i
FIG. 5.
APPENDIX.
1
GALLONS //v 24- novas
FIG. 6.
688
APPENDIX.
^^1
-s--
\|
Fio. 7.
APPENDIX.
689
The formula is
in which V = velocity in feet per second,
C = coefficient varying with R, as given in the following table,
area
R = hydraulic mean radius = : ,
wetted perimeter
which in circular sewers is = J of the diameter.
total fall
I = sine of inclination = - .
total length
R
C
R
C
R
C
o-i
96-3
0-6
119-4
1-1
128-5
0-2
104-7
0-7
121-7
1-2
129-8
0-3
109-9
0-8
123-6
1-3
131-1
0-4
113-8
0-9
125-4
1-4
132-2
0-6
116-9
1-0
127-0
1-5
133-3
Example. To determine the cubic feet per second that would be discharged by a
sewer 4' or 48" diameter with a fall per foot of -006.
The intersection of the horizontal '006 with the 48 in. curve is on the ordinate 124, which
is the quantity per second which would be discharged under the conditions of the example.
On the other hand, to determine the fall per foot necessary to give a 60" sewer to dis-
charge 200 cubic feet per second. Following up the ordinate 200 to its intersection of
the 60" curve, its intersection will be found on the -0049 horizontal, which will be the fall
required.
For the same cubic feet of discharge per second, it will be seen by the diagram that a
72" sewer would require but -0018 fall per foot, and a 54" sewer, for the same discharge,
a fall of -0086 feet.
Flow of Gas through Cast-iron Mains. The usual formula found in hand-books is
Q = 1350
HD
GL*
in which Q = cubic feet per hour, D diameter, and H head of water-pressure, both in
inches, L length of pipe in yards, and Gr specific gravity of the gas ; if the last be taken at
42, L at 1 mile or 1,760 yards, and H one inch, then Q = 1200, D * and
D = Vl ? 440,000 Q 2 = 17-25 Qf .
It will be observed that, in the flow through the pipes, equivalent sections do not imply
equal discharges ; that, by the formula above, the flow through 4 pipes under the same
head is not equal to that of one pipe of double the diameter, but that the flow is as the
square root of the 5th power of the diameter (D*).
Flow of Air through Pipes. B. F. Sturtevant & Co., in the appplication of their
fans and connections, found it very convenient to have tables of the value of pipes of dif-
ferent diameters in conveying air under different pressures, and the practical economy in
this application in the matter of power for the transmission of air. On the following
page are the tables published by them of the results of their calculations.
44
690
APPENDIX.
IN
TABLE FOR EQUALIZING THE DIAMETER OF PIPES.
1
i
P
arties putting up blast pipes are
four 6-inch pipes is the same as
very liable to think, because the combined area of
one 12-inch pipe, that the four pipes will convey the
2
5.7| 2
3
16
2.7
3
4
32
5.7
2.
*
8
ame quantit
as it acti
have
rn
y of air with the same ease and freedom that the 12-inch will, where-
lally does take 5-7 almost six 6-inch pipes. Again, 16 3-inch pipes
the combined area of one 12-inch pipe, but in actual practice it takes
ust 32 3-inch pipes to do the work of one 12-inch.
5
56
9.8| 3.6
-8| 5
6
88
1C
5.7
.8| 1.6
6
7
129
23 J 8.3| .1| 2.3
l.S
8
180
32
12 J .7 3.2
2.1
|1.4
IM
8^ This is d
s| 1 the s
ae
m
Ph
tot
all i
e la
an
1
het
ipes
ge
lete
Tl
xce
ovc
Eigu
rsir
16 fl
111
ss of friction fo
r that in the lar
-es at the top ol
inches of the b
gures at the int
ic with the ve
pipes, of the
of the co
r every cubic foot of air in
ge-
each column give the di-
ranch pipes,
ersection of the horizontal
rtical give the number of
diameter given at the top
umn, that will be equal in
city for conveying air to
>ne given opposite in the
first column.
9
244
42
16
.6| 4.3) 2.8
10
317
56
20 | .9| 5.7
3.6! 2.4|
.-7| 1.3
lo] r
11
402
71
26
2
7.0| 4.5J 3.1|
.2| 1.7
12
501
88 | 32
1C
9.0[ B.Tj 3.8J
.8 2.0J l.C| 1.2| 12]
13
613
107
39 | 19
11
6.9
4.7
.4 2.5| 1.9f 1.5| l.?| 13
14
737
129
47
23
13
8.3|5.7|
.1 3.0| 2.3| 1.8| 1.5|
.5
15
876
152
56
27
16
9.9| 6.7|
.8| 3.6| 2.8| 2.2| 1.8|
.4| 1.2
15
16
1026
180 | 65
32
18 | 11
7.9|
.7 4.2
3.2J 2 ' G i M
.'
i 1"
1.2
16
17
1197
208
76
37
21
13
9.2
.6 4.9| 3.8| 2.9( 2.4|
| 1.6
1.4
1.2
17 1 capa
18
1375
239 | 88
43
24
1C
10 |
.7 5.7
4.8| 3.4[ 2.8|
.3| 1.9| l.G| 1.3
| l.2| 18] <
19
1580
275
.100
49
28 | 18
12
6.5| 5. | 3.9| 3.2|
.G
|2.2| 1.8)1.5
1.3| 1.2I19)
20
1797
313
114
56
32
20 i 14
.9 7.4
5.7| 4.5;.3.6|
.9|2.5|2.1|1.7
1.5| 1.3[ 1.1|20
22
2284
398
145
71
41
26 | 18 | 13 9.3
7.2J 5.7J 4.5|
.7| 3.l| 2.G| 2.2| P.9| 1.7| 1.4| 1.3
22 |
2|24|
24
2834
493 | 180
88
(0
32 | 22 | 16. 12
8.9| 7.6| 5.7|
.6| 3.8|3.2
26
3474
605
219 |108
G2
39
27 | 19 | 14
11 | 8.6[ 6/9|
.7
|4.7
4.0) 3.4
2.9| 2.5| 2.2) 1.9
5| 1.2|26|
28
4165
725
2G5
129
74
48
32 23 | 17 | 13 | 10 | 8.3|
:8| 5.7
4.8 4.1
3.5| 3.0| 2.C| 2.3
.S| i.s| i.2|28|
30
4963
864
315 |154
88
50
38 28 20 | 16 | 12 | 9.3|
,OJ 6.7
5.7
4.7
4.1| 3.G| 3.0J 2.6
.2| 1.7| 1.4|1.2|3O|
36
7818
361
497
243
139
83
60 43 32
25 | 19 | 16 |
3
11
8.9
7.6i C.5| 5.7J 5.0| 4.3
.4| 2.7| 2.2|1.!>|1.G|36|
42 J11488 J2000
730 [358 |205
129
8* 63 47
36 | 29 | 23 |
9
16
13
11 | 9.G| 8.5| 7.3| 6.4
.0] 4.1| 3.3|2.8;2.3|l.5|42]
48 J15989 |2792 J1081 J492
232
180
123 88 66
50 | 39 | 32 | 26 | 22 18
16 | 13 1 12 | 10 | 8.9
.0| 5.7,' 4.7;3.8,3.2|2.l|l.4|48|
64 J21560 |3753 |l3C8
671 |384 |244
166 119 88
68 | 53 | 43 | 35 | 29
24
21
18 | 16 | 15 | 12
.4| 7.6| G.2j5. 2[4. 3'2.8|l.9|l. 3)54
6O |27913 |4879 |l781
872 (499
314
215 J154 115
88 | 69 | 56 | 46
38
Si
27
23 | 20 | 18 | 16
12 | 9.9| 8.li6.7;5.7 ; 3.8(2.4|l.8|l.S
DIAMETER OF PIPES IN INCHES.
LOSSES OF PRESSURE PER 100 FEET MUST BE PROVIDED FOR BY EXTRA SPEED AND POWER ON THIS
BLOWER.
Hi
?j&
LOSS OF PRESSURE IN OUNCES PER SQUARE INCH.
1 inch.
2 inch.
3 inch.
4 inch.
6 inch.
8 inch.
10 inch.
12 inch.
14 inch
16 inch
18 inch
20 inch
22 inch
100
on
006
004
003
002
001
001
001
001
001
001
.-001
001
200
044
022
015
on
007
006
004
004
003
003
002
002
002
400
178
088
059
044
030
022
018
015
013
on
010
009
008
600
400
200
133
100
067
050
040
033
029
025
022
020
018
800
711
356
237
178
119
089
071
059
051
044
040
036
032
1000
1-111
556
370
278
185
139
111
092
079
069
062
056
051
1200
1-600
800
533
400
267
200
160
133
114
100
089
080
073
1400
2-178
1-089
726
544
363
282
218
181
156
136
121
109
099
1600
2-844
1-422
948
711
474
356
284
237
203
178
158
142
129
1800
3-600
1-800
1-200
900
600 | '450
360
300
257
225
200
180
164
2000
4-444
2-222
1-481
1-111 -741
556
444
370
317
278
247
222
202
2200
5-378
2-689 i 1-793
1-344
896
672
538
448
384
336
299
269
244
2400
6-400
3-200 2-133
1-600
1-067
800
640
533
457
400
356
320
291
2600
7-511
3-756
2-504 1-877
1-252
939
751
626
537
468
417
376
341
2800
8-711
4-356
2-904 2-178
1-452
1-089
871
726
622
544
484
436
396
3000
10-000
5-000
3-333 2-500
1-667
1-250
1-000
833
714
625
556
500
455
3200
11-378
5-689
3-792 2-844
1-896
1-422
138
948
813
711
632
569
517
3400
12-844
6-422
4-281 3-211
2-141
1-606
284
1-070
917
827
714
642
584
3600 14-400
7-200
4-800 3-600
2-400
1-800
440
1-200
1-029
900
800
720
655
3800 ! 16-044
8-022
5-349
4-011
2-674
2-006
604
1-337
1-146
1-003 | -891
802
729
4000
17-778
8-889
5-926
4-444
2-963
2'222
778
1-481
1-270
1-111
988
889
808
4400
10-705
7-175
5-353 3-569
2-676
2-141
1-784
1-537
1-344
1-189 1-071
973
4800
12-800
8-533
6-400 4-267
3-200
2-560
2-133
1-829
1-600 1-422
1-280
1-164
5200
15-022
10-015
7-511 5-007
3-756
3-004
2-504
2-146
1-871 1-670
1-502
1-366
5600
17-422
11-615
8-711 5-807
4-356
3-484 i 2-904
2-489
2-178
1-936
1-742
1-584
6000
20-000
13-333
10-000 6-667
5-000
4-000 i 3-333
2-857
2-500
2-222 2-000
1-818
APPENDIX.
691
TABLES OF THE CIRCUMFERENCES OF CIRCLES TO THE NEAREST FRACTION OF
PRACTICAL MEASUREMENT ; ALSO, THE AREAS OF CIRCLES, IN INCHES AN1>
DECIMAL PARTS, LIKEWISE OF FEET AND DECIMAL PARTS.
Circumfer-
Diameter
Area
Area
Circumfer-
Diameter
Area
Area
ence in feet
in
in square
in square
ence in feet
in
in square
in square
and inches.
inches.
inches.
feet.
and inches.
inches.
inches.
feet.
1 61-
6
28-27
196
20
iV
003
i H
6*
29-46
204
39
1
012
1 71
6*
30-68
212
59
"1%
028
1 8
6f
31-92
220
78
98
i
049
077
1 8f
6*
6|
33-18
34-47
228
237
1-18
f
110
1 9i
6|
35-78
246
1-37
T 2 ?
150
1 9|
6^
37-12
256
1-57
*
196
1 10
7
38-48
267
1-77
A
248
1 10|
7-|
39-87
277
1-96
f
307
1 10|
*7*
41-28
287
2-16
371
1 11*
71
42-72
297
2-36
f
442
1 11*
7*
44-18
307
2-55
518
1 11*
7f
45-66
318
2-75
f
601
2 Of
71
47-17
328
2-94
690
2 0|
7*
48-71
338
3*
785
0054
2 1*
8
50-26
349
3*
i.
994
0069
2 1*
8*
51-85
360
3|
^
1-23
0085
2 I*
8*
53-46
371
4*
f
1-48
0103
2 2*
8f
55-09
383
-4f
1
1'77
0123
2 2|
8*
56-74
394
5*
if
2-07
0144
2 3
8|
58-43
406
5*
If
2-40
0167
2 3|
8|
60-13
428
l
It
2-76
0192
2 3l
81
61-86
430
6*
2
3-14
0218
2 4i
9
63-62
442
6f
2*
3-55
0246
2 4f
gl
65-40
455
7
2*
3-98
0276
2 5
9*
67-20
467
7f
2f
4-43
0307
2 5f
Q3.
69-03
480
'71
2*
4-91
0341
2 5f
9*
70-88
493
8*
21
5-41
0376
2 6*
9f
72-76
506
8|
2f
5-94
0412
2 6f
9f
74-66
519
9
2*
6-49
0450
2 7
93
76-59
532
1
3
7-07
0490
2 7f
10
78-54
545
10!
1
7'67
829
0532
0576
2 7f
2 8*
10*
10*
80-51
82-52
559
573
10f
3f-
8-95
0621
2 8*
iof
84-54
587
11
3*
9-62
0668
2 9
10*
86-59
601
111
3f
10-32
0716
2 9f
iof
88-66
615
111
3f
11-04
0766
2 9f
iof
90-76
630
12*
81
11-79
0818
2 10*
10*
92-88
645
1 0*
4
12-57
087
2 10*
11
95-03
660
1 1
4*
13-36
093
2 101
11*
97-21
675
If
4*
14-19
099
2 11*
11*
99-40
690
if
4f
15-03
105
2 ll|
llf
101-62
705
2*
4*
15-90
111
3 0*
11*
103-87
720
2*
4|
16-80
118
3 0*
llf
106-14
736
2*
4f
17-72
124
3 0|
lit
108-43
752
3*
41
18-66
130
3 1*
111
110-75
768
Sti-
ff
5
19-63
136
3 If
12
113-10
785
4*
5*
20-63
*14b
3 2
12*
115-47
802
1 4*
5*
21-65
150
3 2*
12*
117-86
819
43-
Si
22-69
157
3 21
12f
120-28
836
5 i
6*
23-76
165
o 04.
12*
122-72
853
5f
5f
24-85
173
3 3|
12f
125-19
870
6
5f
25-97
181
3 4
12f
127-68
887
6f
5i
27-11
189 1
3 4|
123-
130-19
904
692
APPENDIX.
TABLES OF THE CIRCUMFERENCES OF CIECLES, ETC. (Continued.)
Circumfer-
Diameter
Area
Area
Circumfer-
Diameter
Area
Area
ence in feet
in
in square
in square
ence in feet
in feet and
in square
in square
and inches.
inches.
inches.
feet.
and inches.
inches.
inches.
feet.
3 4f
13
132-73
922
5 21
20
314-16
2-182
3 5i
131
135-30
939
5 34
201
318-10
2-209
3 5f
134
137-89
956
5 3f
322-06
2-237
3 6
13f
140-50
974
5 4
20|
326-05
2-265
3 6f
143-14
992
5 4f
20
330-06
2-293
3 6f
13f
145-80
1-011
5 41
20f
334-10
2-321
3 71
13f
148-49
1-030
6 5J
20J
338-16
2-349
3 7f
ul
151-20
1-050
5 o
201
342-25
2-377
3 8
14
153-94
1-069
5 6
21
346-36
2-405
3 8f
14*
156-70
1-088
5 6|
21*
350-50
2-434
3 8f
144
159-49
1-107
5 6|
214
354-66
2-463
3 91
14f
162-30
1-126
5 7^
21f
358-84
2-492
3Q-L
2
14^
165-13
1-146
5 7i
2l|
363-05
2-521
3 91
14f
167-99
1-166
5 71
21|
367-28
2-550
3 104
3 lOf
141
170-87
173-78
1-186
1-206
5 84
5 8f
211
371-54
375-83
2-580
2-610
3 111
15
176-71
1-227
5 9J
22
380-13
2-640
3 Hi
151
179-67
1-247
5 9J
221
384-46
2-670
3 Hi
154
182-65
1-267
5 91
224
388-82
2-700
4 04
15if
185-66
1-288
5 10*
22f
393-20
2-730
4 Of
15^
188-69
1-309
5 lOf
224
397-61
2-761
4 1
15f
191-75
1-330
5 11
22f
402-04
2-792.
4 1*
15f
194-83
1-352
5 Hi
22*
406-49
2-823
4 11
151
197-93
1-374
5 111
221
410-97
2-854
4 24
16
201-06
1-396
6 04
23
415-48
2-885-
4 2f
161
204-22
1-418
6 Of
231
420-00
2-917
4 3
164
207-39
1-440
6 1
23^-
424-56
2-949
4 3f
16f
210-60
1-462
6 If
23|
429-13
2-981
4 3|
16^
213-82
484
6 If
23|
433-74
3-013
4 44
16f
217-08
507
6 24
23f
438-36
3-045
4 4|
16f
220-35
530
6 2f
28f
443-01
3-077
4 5
16!
223-65
553
6 3
231
447-69
3-10&
4 5f
17
226-98
576
6 3|
2
452-39
3-142.
4 5f
171
230-33
599
6 41
2 04
461-86
3-207
4 61
233-70
622
6 41
2
471-44
3-273
4 6i
171
237-10
645
6 5<
2 Of
481-11
3-341
4 6l
240-53
669
6 6;
2 1
490-87
3-408
4 71
171
243-98
693
6 7;
2 14
500-74
3-477
4 71
17!
247-45
718
6 8-
2 H
510-71
3-547
4 8*
171
250-95
743
6 8 F
2 If
520-77
3-617
4 8^
18
254-47
767
6 9|
2 2
530-93
3-687
4 81
181
258-02
792
6 10k
2 24
541-19
3-758
4 94
184
261-59
817
6 114
2 2J
551-55
3-830
4 9f
4 101
18*
18*
265-18
268-80
842
868
7
7 01
2 2f
2 3
562-00
572-56
3-904
3-976
4 10i
18f
272-45
893
7 If
2 34
583-21
4-050
4 101
4 114
1?!
276-12
279-81
918
943
7 2f
2 3*
2 3f
59396
604-81
4-124
4-200
4 Hf
19
283-53
1-969
7 31
2 4
615-75
4-276
5
19i
287-27
1-995
7 4f
2 44
626-80
4-352
6 Of
194
291-04
2-021
7 5*
2 4*
637-94
4-430
5 01
294-83
2-047
7 64
2 41
649-18
4-508
5 14
19|
298-65
2-074
7 7
2 5
660-52
4-586
5 If
19s
302-49
2-101
7 7i
2 54
671-96
4-666
5 2
19f
306-36
2-128
7 83-
2 5|
683-49
4-747
6 2|
310-25
2-155
7 9
2 5f
695-13
4-827
APPENDIX.
693
TABLES OF THE CIECUMFEEENCES OF CIRCLES, ETC. (Continued.)
Circumfer- ' Diameter
Area
Area
Circumfer-
Diameter
Area
Area
nce in feet
and inches.
in feet and
inches.
in square
inches.
in square
feet.
ence in teet
and inches.
in feet and
inches.
in square
inches.
in square
feet.
7 10i
2 6
706-86
4-908
11 6J
3 8
1520-5
10-56
7 11
2 6J-
718-69
4-990
11 7
3 8
1537-9
10-68
7 llf
2 64
730-62
5-073
11 7f
3 8*
1555-3
10-80
8 0|
2 6f
742-64
6-157
11 8|
3 8f
1572-8
10-92
8 1|
2 7
754-77
5-241
11 9|
3 9
1590-4
11-04
8 2^r
2 7
766-99
5-326
11 10
3 9i
1608-1
11-17
8 2
2 7|
779-31
5-411
11 103 3 9|
1626-0
11-29
8 3f
2 7f
791-73
5-498
11 llf 3 9f
1643-9
11-41
8 4|
2 8
804-25
5-585
12 0|
3 10
1661-9
11-54
8 5|
2 8i
816-86
5-673
12 lj
3 101
1680-0
11-67
8 6j
2 8|
829-58
5-761
12 2
3 10!
1698-2
11-79
8 6
2 8f
842-39
5-849
12 2|
3 lOf
1716-5
11-92
8 71
2 9
855-30
5-939
12 8f
3 11
1734-9
12-05
8 8fr
2 9^
868-31
6-029
12 4|
3 iii
1753-4
12-18
8 9i
2 9|
881-41
6-120
12 5*
3 11J
1772-0
12-30
8 10
2 9f
894-62
6-212
12 6
3 llf
1790-8
12-43
8 lOf
2 10
907-92
6-305
12 6f
4
1809-6
12-57
8 11|
2 10 J
921-32
6-398
12 7|
4 0|
1828-5
12-70
9 Of-
2 10J
934-82
6-491
12 8|
4 0|
1847-4
12-83
9 1|
2 101
948-42
6-586
12 9|
4 Of
1866-5
12-96
9 13
2 11
962-11
6-681
12 93
4 1
1885-7
13-09
9 2f
2 11J
975-91
6-777
12 lOf
4 11
1905-0
13-23
9 3
2 H|
989-80
6-874
12 11|
4 l!
1924-4
13-36
9 4-J-
2 llf
1003-8
6-970
13 0|
4 If
1943-9
13-50
9 5
3
1017-9
7-069
13 1
4 2
1963-5
13-63
9 5J
3 Oi
1032-1
7-167
13 13
4 2i
1983-2
13-77
9 6|
3 0|
1046-3
7-266
13 2f
4 2|
2003-0
13-91
9 7i
3 Of
1060-7
7-366
13 3f
4 2f
2022-8
14-05
9 8i
3 1
1075-2
7-466
13 4J
4 3
2042-8
14-19
9 9
3 1J
1089-8
7-567
13 5
4 3
2062-9
14-32
9 9|
3 1J
1104-5
7-669
13 5f
4 3!
2083-1
14-46
9 lOf
3 If
1119-2
7-772
13 6|
4 3f
210S-3
14-61
9 llf
3 2
1134-1
7-876
13 7f
4 4
2123-7
14-75
10 0|
3 2|
1149-1
7-979
13 8i
4 4J
2144-2
14-89
10 OI-
3 2|
1164-2
8-085
13 8|
4 4|
2164-7
15-03
IO If
3 2|
1179-3
8-189
13 9f
4 4f
2185-4
15-18
10 2|
3 3
1194-6
8-295
13 10|
4 5
2206-2
15-32
10 3J
3O 1
"i
1209-9
8-403
13 ll|
4 5
2227-0
15-46
10 4
3 3
1225-4
8-509
14
4 5|
2248-0
15-61'
10 43
3 3f
1241-0
8-617
14 03
4 5|
2269-1
15-76
10 5|
3 4
1256-6
8-727
14 If
4 6
2290-2
15-90
10 6|
3 4\
1272-4
8-836
14 2f
4 6
2311-5
16-05
10 7i
3 4i
1288-2
8-946
14 3i
4 6|
2332-8
16-20
10 8
3 4f
1304-2
9-056
14 4
4 6f
2354-3
16-35
10 81
3 5
1320-2
9-169
14 4f
4 7
2375-8
16-50
10 Qk
3 5J-
1336-4
9-211
14 5
4 7i
2397-5
16-65
10 10J
3 5
1352-6
9-394
14 6|
4 7i
2419-2
16-80
10 11|
3 5f
1369-0
9-506
14 7|
4 7f
2441-1
16-95
10 113
3 6
1385-4
9-62
. 14 7
4 8
2463-0
17-10
11 Of
3 6i
1402-0
9-73
14 83
4 8|
2485-0
17-26
11 i|
3 6|
1418-6
9-84
14 9!
4 8|
2507-2
17-41
11 2
3 6f
1435-4
9-96
14 10i
4 8f
2529-4
17-56
11 3
3 7
1452-2
10-08
14 11
4 9
2551-8
17-72
11 33
3 7i
1469-1
10-20
14 H|
4 9|
2574-2
17-88
11 4|
3 7|
1486-2
10-32
15 Of
4 9|
2596-7
18-03
11 5|
3 7f
1503-3
10-44
15 If
4 9f
2619-3
18-19
694
APPENDIX.
TABLES OF THE CIRCUMFERENCES OF CIRCLES, ETC. (Continued.}
Circumfer-
Diameter
Area
Area
Circumfer-
Diameter
Area
Area
ence in feet
and inches.
in feet and
inches.
in square
inches.
in square
feet.
| ence in feet
i and inches.
in feet and
inches.
in square
inches.
in square-
feet.
15 2t
4 10
2642-1
18-35
18 10i
6
4071-5
28-27
15 3
4 10t
2664-9
18-51
18 10*
6 01
4099-8
28-47
15 3f
4 10
2687-8
18-66
18 llf
6 Oi
4128-2
28-67
15 4i
4 10$
2710-8
18-82
19
6 Of
4156-8
28-87
15 5{
4 11
2734-0
18-98
19 l|
6 1
4185-4
29-07-
15 61
4 iii
2757-2
19-15
19 2j
6 li
4214-1
29-27
15 6|
4 iii
2780-5
19-31
19 23
6 li
4242-9
29-47
15 7f
4 llf
2803-9
19-47
19 3f
6 1|
4271-8
29-67'
15 8i
5
2827-4
19-63
19 4i
6 2
4300-8
29-87-
15 91
5 Oi
2851-0
19-80
19 51
6 2i
4329-9
30-07
15 10
5 Oi
2874-8
19-96
19 6
6 2i
4359-2
30-27
15 lOf
5 Of
2898-6 j 20-13
19 6f
6 21
4388-5
30-47
15 llf
5 1
2922-5
20-29
19 7i
6 3
4417-9
30-68-
16 Of
5 It
2946-5
20-46
19 8f
6 31
4447-4
30-88-.
16 U
5 1
2970-6
20-63
19 9i
6 3$
4477-0
31-0&
16 2
5 If
2994-8
20-80
19 9|
6 3f
4506-7
31-3O-
16 2f
5 2
3019-1
20-96
19 lOf
6 4
4536-5
31-50-
16 3i
6 2i
3043-5
21-13
19 Hi
6 4i
4566-4
31-71
16 4t
5 2i
3068-0
21-30
20 01
6 4i
4596-3
31-92
16 5
5 2|
3092-6
21-48
20 li
6 4f
4626-4
32-13-
16 Si
5 3
3117-2
21-65
20 li
6 5
4656-6
32-34
16 6f
5 3i
3142-0
21-82
20 2f
6 5i
4686-9
32-55
16 7*
5 3|
3166-9
21-99
20 3i
6 ef
4717-3
3276.
16 8}
5 3f
3191-9
22-17
20 41
6 6f
4747-8
32-97-
16 9
5 4
3217-0
22-34
20 6
6 6
4778-3
33-18
16 9f
5 4t
3242-2
22-51
20 5|
6 61
4809-0
33-40"
16 10|
5 4
3267-5
22-69
20 6
6 6J
4839-8
33-61
16 ll|
5 4f
3292-8
22-87
20 7f
6 6|
4S70-7
33-82^:
17 Oi
5 5
3318-3
23-04
20 8
6 7
4901-6
34-04
17 1
5 Si
3343-9
23-22
20 8|
6 71
4932-7
34-25
17 If
5 5
3369-6
23-40
20 9f
6 7^
4963-9
34-47
17 2J
5 5f
3395-3
23-58
20 10}
6 7f
4995-1
34-69-
17 3|
5 6
3421-2
23-76
20 111
6 8
5026-5
34-91
17 4|
6 6
3447-2
23-94
21
6 81
6058-0
35-12'
17 4|
5 6
3473-2
24-12
21 0|
C oiy
5089-5
35-34
17 5
5 6f
3499-4
24-30
21 If
6 8f
5121-2
35-56
17 6i
5 7
3525-1
24-48
21 2f
6 9
5153-0
35-78.
17 7i
5 71
3552-0
24-67
21 31
6 91
6184-8
36-01
17 8
5 7i
3578-5
24-85
21 4
6 9i
5216-8
36-23
17 8f
6 7|
3605-0
25-03
21 4|
6 9j
5248-8
36-45
17 9f
5 8
3631-7
25-22
21 5|
6 10
5281-0
36-67'
17 lOf
5 8i
3658-4
25-40
21 6f
6 101
5313-2
36-89-
17 lit
5 8i
3685-3
25-59
21 7j
6 10|
5345-6
37-12-
17 11|
5 8|
3712-2
25-78
21 7|
6 lOf
6378-0
37-35
18 Of
5 9
3739-3
25-96
21 8|
6 11
5410-6
37-57
18 li
5 9i
3766-4
26-15
21 9
6 lit
5443-2
87-80*
18 2i
6 9i
3793-7
26-34
21 101
6 11
5476-0
38-03-
18 3i
5 9f
3821-0
26-53
21 111
6 llf
5508-8
38-2&
18 3
5 10
3848-5
26-72
21 llf
7
5541-7
38-48
18 4f
5 10i
3876-0
26-92
22 Of
7 01
5574-8
38-71
18 Si
5 10J-
3903-6
27-11
22 If
7 Oi
5607-9
38-94
18 6i
6 lOf
3931-4
27-30
22 21
7 9|
6641-1
39-17
18 7
6 11
3959-2
27-49
22 3
7 1
5674-5
39-41
18 7|
5 Hi
3987-1
27-69
22 3*
7 It
5707-9
39-64
18 8|
5 Hi
4015-2
27-88
22 4i
7 1?
6741-4
39-87
18 9|
5 llf
4043-3
28-08
22 Si-
7 If
5775-0
40'10>
APPENDIX.
TABLES OF THE CIRCUMFERENCES OF CIRCLES, ETC. (Continued.)
695
Circumfer-
Diameter
Area
Area
Circumfer-
Diameter
Area
Area
ence in feet
and inches.
in feet and
inches.
in square
inches.
in square
feet.
ence in feet
and inches.
in feet and
inches.
in square
inches.
in square
feet.
22 6J
7 2
5808-8
40-34
26 21
8 4
7853-9
54-54
22 61
7 2}
5842-6
40-57
26 6J
8 5
8011-9
55-64
22 71-
I A
5876-5
40-80
26 8|
8 6
8171-3
56-75
22 8
7 2|
5910-5
41-04
26 ll
8 7
8332-3
57-86
22 9
7 3
5944-6
41-28
27 2f
8 8
8494-9
58-99
22 10
7 3
5978-9
41-52
27 5t
8 9
8659-0
60-13
22 101
6013-2
41-76
27 9
8 10
8824-7
61-28
22 11|
7 3f
6047-6
42-00
28 01
8 11
8892-0
62-44
23 Of
7 4
6082-1
42-24
28 3^
9
9160-9
63-62
23 U
7 4i
6116-7
42-48
28 6|
9 1
9331-3
64-80
23 2
7 44
6151-4
42-72
28 9}
9 2
9503-3
66-00
23 2|
7 4|
6186-2
42-96
29 Of
9 3
9676-9
67-20
23 3f
7 5
6221-1
43-20
29 3f
9 4
9852-1
68-42
23 4f
7 6t
6256-1
43-44
29 7
9 5
10028-8
69-64
23 5
7 6*
6291-2
43-68
29 10J
9 6
10207-1
70-88
23 6
7 51
6326-4
43-93
30 1
9 7
10386-9
72-13
23 6|
7 6
6361-7
44-18
30 4f
9 8
10568-3
73-39
23 7
7 6i
6397-1
44-43
30 74
9 9
10751-3
74-66
23 8
7 6J
6432-6
44-67
30 lOf
9 10
10935-9
75-94
23 9|
7 6|
6468-2
44-92
31 If
9 11
11122-0
77-24
23 91
7 7
6503-8
45-17
23 10
7 7i
6539-6
45-41
31 5
10
11309-8
78-54
23 llf
7 7
6575-5
45-66
31 81
10 1
11499-0
79-85
24 0^
7 7f
6611-5
45-91
31 ll|
10 2
11689-9
81-18
32 2f
10 3
11882-3
82-52
24 1
7 8
6647-6
46-16
32 5|
10 4
12076-3
83-86
24 H
7 8J
6683-8
46-42
32 8f
10 5
12271-9
85-22
24 2*
7 8v
6720-0
46-67
32 llf
10 6
12469-0
86-59
24 3j
7 81
6756-4
46-92
33 21
10 7
12667-7
87-97
24 41
7 9
6792-9
47-17
33 6|
10 8
12868-0
89-36
24 41
7 9i
6829-4
47-43
33 9
10 9
13069-8
90-76
24 51
7 9J
6866-1
47-68
34 Of
10 10
13273-3
92-17
24 6
7 9*
6902-9
47-94
34 8}
10 11
13478-2 93-60
24 7i
7 10
6939-7
48-19
34 6$
11
13684-8
95-03
24 8
7 101
6976-7
48-45
34 9f
11 1
13892-9
96-48
24 8|
7 lOfc
7013-8
48-71
35 0^
11 2
14142-6 97-93
24 9|
7 10|
7050-9
48-96
35 41
11 3
14313-9 99-40
24 10|
7 11
7088-2
49-22
35 7J
11 4
14526-8
100-88
24 llf
7 11J
7125-5
49-48
35 lOf
11 5
14741-2
102-37
25
7 ll|
7163-0
49-74
36 l|
11 6
14957-2
103-87
25 Of
7 llf
7200-5
50-00
36 4f
11 7
15174-7
105-38
36 7f
11 8
15393-8
106-90
25 1
8
7238-2
50-26
36 101
11 9
15614-5
108-43
25 2f
8 OJ
7275-9
50-53
37 2
11 10
15836-8
109-98
25 3J
8 o|
7313-8
50-79
37 5J
11 11
16060-6
111-53
25 31
8 Of
7351-7
51-05
25 4|
8 1
7389-8
51-32
37 8f
12
16286-0
113-10
25 5
8 1
7427-9
51-58
37 ll}
12 1
16513-0
114-67
25 6i
8 H
7466-2
51-85
38 2f
12 2
16741-6
116-26
25 7
8 if
7504-5
52-11
38 5|
12 3
16971-7
117-86
38 81
12 4
17203-4
119-47
25 71
8 2
7542-9
52-38
39
12 5
17436-7
121-09
25 8$
8 21
7581-5
52-65
39 3J-
12 6
17671-5
122-72
25 9|
8 2
7620-1
52-92
39 6f
12 7
17907-9
124-36
25 lOf
8 2|
7658-8
53-19
39 9
12 8
18145-9
126-01
25 11
8 3
7697-7
53-46
40 Of
12 9
18385-4
127-68
25 llf
8 3|
7736-6
63-73
40 3f
12 10
18626-6
129-35
26 OJ
8 85
7775-6
54-00
40 61
12 11
18869-2
131-04
26 !$
8 3f
7814-7
54-27
696
APPENDIX.
TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS OF NUMBERS.
Squares.
Cubes.
No.
Square
roots.
Cube
roots.
Squares.
Cubes.
No.
Square
roots.
Cube
roots.
1 1
1
I'OOO
1-000
4096
262144
64
8-000
4-000
4 8
2
1-414
1-259
4225
274625
65
8-062
4-020
9 27
3
1-732
1-442
4356
287496
66
8-124
4-041
16 64
4
2-000
1-587
4489
300763
67
8-185
4-061
25
125
5
2-236
1-709
4624
314432
68
8-246
4-081
36
216
6
2-449
1-817
4761
328509
69
8-306
4-101
49
343
7
2-645
1-912
4900
343000
70
8-366
4-121
64 612
8
2-828
2-000
5041
357911
71
8-426
4-140
81 729
9
3-000
2-080
5184
373248
72
8-485
4-160
100 1000
10
3-162
2-154
5329
389017
73
8-544
4-179
121
1331
11
3-316
2-223
5476
405224
74
8-602
4-198
144
1728
12
3-464
2-289
5625
421875
75
8-660
4-217
169
2197
13
3-605
2-351
5776
438976
76
8-717
4-235
196
2744
14
3-741
2-410
5929
456533
77
8-774
4-254
225 3375
15
3-872
2-466
6084
474552
78
8-831
4-272
256 4096
16
4-000
2-519
6241
493039
79
8-888
4-290
289 : 4913
17
4-123
2-571
6400
512000
80
8-944
4-308
324
6832
18
4-242
2-620
6561
531441
81
9-000
4-326
361
6859
19
4-358
2-668
6724
551368
82
9-055
4-344
400
8000
20
4-472
2-714
6889
571787
83
9-110
4-362
441
9261
21
4-582
2-758
7056
592704
84
9-165
4-379
484
10648
22
4-690
2-802
7225
614125
85
9-219
4-396
629
12167
23
4-795
2-843
7396
636056
86
9-273
4-414
676
13824
24
4-898
2-884
7569
658503
87
9-327
4-431
625
15625
25
5-000
2-924
7744
681472
88
9-380
4-447
676
17576
26
5-099
2-962
7921
704969
89
9-433
4-464
729
19683
27
5-196
3-000
8100
729000
90
9-486
4-481
784
21952
28
5-291
3-036
8281
753571
91
9-539
4497
841
24389
29
5-385
3-072
8464
778688
92
9-591
4-514
900
27000
30
5-477
3-107
8649
804357
93
9-643
4-530
961
29791
31
5-567
3-141
8836
830584
94
9-695
4-546
1024
32768
32
5-656
3-174
9025
857374
95
9-746
4-562
1089
35937
33
5-744
3-207
9216
884736
96
9-797
4-578
1156
39304
34
5-830
3-239
9409
912673
97
9-848
4-594
1225
42875
35
5-916
3-271
9604
941192
98
9-899
4-610
1296
46656
36
6-000
3-301
9801
970299
99
9-949
4-626
1369
60653
37
6-082
3-332
10000
1000000
100
10-000
4-641
1444
54872
38
6-164
3-361
10201
1030301
101
10-049
4-657
1521
59319
39
6-244
3-391
10404
1061208
102
10-099
4-672
1600
64000
40
6-324
3-419
10609
1092727
103
10-148
4-687
1681
68921
41
6-403
3-448
10816
1124864
104
10-198
4-702
1764
74088
42
6-480
3-476
11025
1157625
105
10-246
4-717
1849
79507
43
6-557
3-503
11236
1191016
106
10-295
4-732
1936
85184
44
6-633
3-530
11449
1225043
107
10-344
4-747
2025
91125
45
6-708
3-556
11664
1259712
108
10-392
4-762
2116 97336
46
6-782
3-583
11881
1295029
109
10-440
4-776
2209
103823
47
6-855
. 3-608
12100
1331000
110
10-488
4-791
2304
110592
48
6-928
3-634
12321
1367631
111
10-535
4-805
2401 117649 49
7*000
3-659
12544
1404928
112
10-583
4-820
2500
125000 50
7-071
3-684
12769
1442897
113
10-630
4-834
2601
132651 51
7-141
3-708
12996
1481544
114
10-677
4-848
2704
140608
52
7-211
3-732
13225
1520875
115
10-723
4-862
2809
148877
53
7-280
3-756
13456
1560896
116
10-770
4-876
2916
157464
54
7-348
3-779
13689
1601613
117
10-816
4-890
3025
166375
55
7-416
3-802
13924
1643032
118
10-862
4-904
3136
175616
56
7-4S3
3-825
14161
1685159
119
10-908
4-918
3249
185193
57
7-549
3-848
14400
1728000
120
10-954
4-932
3364
195112
58
7-615
3-870
14641
1771561
121
11-000
4-946
3481
205379
59
7-681
3-892
14834
1815848
122
11-045
4-959
3600
216000
60
7-745
3-914
15129
1860867
123
11-090
4-973
3721
226981
61
7-810
3-930
15376
1906624
124
11-135
4-986
3844
238328
62
7-874
3-957
15625
1953125
125
11-180
6-000
3969
250047
63
7-937
3-979
15876
2000376
126
11-224
5-013
APPENDIX.
697
TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS OF NUMBERS ( Continued).
Squares.
Cubes.
No.
Square
roots.
Cube
roots.
Squares.
Cubes.
No.
Square
roots.
Cube
roots.
16129
2048383
127
11-269
5-026
36100
6859000 190
13-784
5-748
16384
2097152
128
11-313
5-039
36481
6967871 191
13-820
5-758
16641
2146689
129
11-357
5-052
36864
7077888 192
13-856
5-768
16900
2197000
130
11-401
5-065
37249
7189517 193
13-892
5-778
17161
2248091
131
11-445
5-078
37636
7301384
194
13-928
5-788
17424
2299968
132
11-489
5-091
38025
7414875
195
13-964
6-798
17689
2352637
133
11-532
5-104
38416
7529536
196
14-000
5-808
17956
2406104
134
11-575
5-117
38809
7645373
197
14-035
5-818
18225
2460375
135
11-618
5-129
39204
7762392
198
14-071
B'828
18496
2515456
136
11-661
5-142
39601
7880599
199
14-106
5-838
18769
2571353
137
11-704
5-155
40000
8000000
200
14-142
5-848
19044
2628072
138
11-747
5-167
40401
8120601
201
14-177
6-857
19321
2685619
139
11-789
5-180
40804
8242408
202
14-212
5-867
19600
2744000
140
11-832
5-192
41209
8365427
203
14-247
5-877
19881
2803221
141
11-874
5-204
41616
8489664
204
14-282
6-886
20164
2863288
142
11-916
5-217
42025
8615125
205
14-317
5-896
20449
2924207
143
11-958
5-229
42436
8741816
206
14-352
5-905
20736
2985984
144
12-000
5-241
42849
8869743
207
14-387
5-915
21025
3048625
145
12-041
5-253
43264
8998912
208
14-422
5-924
21316
3112136
146
12-083
5-265
43681
9129329
209
14-456
5-934
21609
3176523
147
12-124
5-277
44100
9261000
210
14-491
5-943
21904
3241792
148
12-165
5-289
44521
9393931
211
14-525
5-953
22201
3307949
149
12-206
5-301
44944
9528128
212
14-560
5-962
22500
3375000
150
12-247
5-313
45369
9663597
213
14-594
5-972
22801
3442951
151
12-288
5-325
45796
9800344
214
14-628
5-981
23104
3511008
152
12-328
5-336
46225
9938375
215
14-662
5-990
23409
3581577
153
12-369
5-348
46656
10077696
216
14-696
6-000
23716
3652264
154
12-409
5-360
47089
10218312
217
14-730
6-009
24025
3723875
155
12-449
5-371
47524
10360232
218
14-764
6-018
24336
3796416
156
12-489
5-383
47961
10503459
219
14-798
6-027
24649
3869893
157
12-529
5-394
48400
10648000
220
14-832
6-036
24964
3944312
158
12-569
5-406
48841
10793861
221
14-866
6-045
25281
4019679
159
12-609
5-417
49284
10941048
222
14-899
6-055
25600
4096000
160 12-649
5-428 '
49729
11089567
223
14-933
6-064
25921
4173281
161
12-688
5-440
50176
11239424
224
14-966
6-073
26244
4251528
162
12-727
5-451
50625
11390625
225
15-000
6-082
26569
4330747
163
12-767 5-462
51076
11543176
226
15-033
6-099
26896
4410944
164
12-806
5-473
51529
11697083
227
15-066
6-100
27225
4492125
165
12-845
5-484
51984
11852352
228
15-099
6-109
27556
4574296
166
12-884 5-495
52441
12008989
229
15-132
6-118
27889
4657463
167
12-922
5'506
52900
12167000
230
15-165
6-126
28224
4741632
168
12-961
5-517
53361
12326391
231
15-198
6-135
28561
4826809
169
13-000
5-528
53824
12487168
232
15-231
6-144
28900
4913000
170
13-938
5-539
54289
12649337
233
15-264
6-153
29241
5000211
171
13-076
5-550
54756
12812904
234
15-297
6-162
29584
5088448
172
13-114
5-561
55225
12977875
235
15-329
6-171
29929
6177717
173
13-152
5-572
55696
13144256
236
15-362
6-179
30276
5268024
174
13-190
5-582
56169
13312053
237
15-394
6-188
30625
5359375
175
13-228
5-593
56644
13481272
238
15-427
6-197
30976
5451776
176
13-266
5-604
57121
13651919
239
15-459
6-205
31329
5545233
177
13-304
5-614
57600
13824000
240
15-491
6-214
31684
5639752
178
13341
5-625
58081 13997521
241
15-524
6-223
32041
5735339
179
13-379
5-635
58564
14172488
242
15-556
6-231
32400
58S2000
180
13-416
5-646
59049
14348907
243
15-588
6-240
32761
5929741
181
13-453
5-656
59536
14526784
244
15-620
6-248
33124
6028568
182
13-490
5-667
60025
14706125
245
15-652
6-257
33489
6128487
183
13-527
5-677
60516
14886936
246
15-684
6-265
33856
6229504
184
13-664
5-687
61009
15069223
247
15-716
6-274
34225
6331625
185
13-601
5-698
61504
15252992
248
15-748
6-282
34596
6434856
186
13-638
5-708
62001
15438249
249
15-779
6-291
34969
6539203
187
13-674
5-718
62500
15625000
250
15-811
6-299
35344
6644672
188
13-711
5-728
63001
15813251
251
15-842
6-307
35721
6751269
189
13-747 5-738
63504
16003008
252
15-874
6-316
698
APPENDIX.
TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS OF NUMBERS (Continued},
Squares. Cubes.
No.
Square
roots.
Cube
roots.
Squares.
Cubes.
No.
Square
roots.
Cube
roots.
64009 16194277
253
15-905
6-324
99856
31554496 316
17-776
6-811
64516
16387064
254 15-937
6-333
100489
31855013 317
17-804
6-818
65025
16581375
255 15-968
6-341
101124
32157432 318
17-832
6-825
65536
16777216
256 ! 16-000
6-349
101761
32461759 319
17-860
6-832
66049
16974593
257 16-031
6-357
102400
32768000 320
17-888
6-839
66564
17173512
258 16-062
6-366
103041
33076161 321
17-916
6-847
67081
17373979
259
16093
6-374
103684
33386248 322
17-944
6-854
67600
17576000
260
16-124
6-382
104329
33698267 323
17-972
6-861
68121
17779581
261
16-155
6-390
104976
34012224
324
18-000
6-868
68644
17984728
262
16-186
6-398
105625
34328125
325
18-027
6-875
69169
18191447
263
16-217
6-406
106276
34645976 326
18-055
6-882
69696
18399744
264
16-248
6-415
106929
34965783 327
18-083
6-889
70225
18609625
265
16-278
6-423
107584
35287552
328
18-110
6-896
70756
18821096
266
16-309
6-431
108241
35611289
329
18-138
6-903
71289
19034163
267
16-340
6-439
108900
35937000
330
18-165
6-910
71824
19248832
268
16-370
6-447
109561
36264691
331
18' 193
6-917
72361
19465109
269
16-401
6-455
110224
36594368
332
18-220
6-924
72900
19683000
270
16-431
6-463
110889
36926037
333
18-248
6-931
73441
19902511
271
16-462
6-471
111556
37259704
334
18-275
6-938
73984
20123643
272
16-492
6-479
112225
37595375
335
18-303
6-945
74529
20346417
273
16-522
6-487
112896
37933056
336
18-330
6-952
75076
20570824
274
16-552
6-495
113569
38272753
337
18-357
6-958
75625
20796875
275
16-583
6-502
114244
38614472
388
18-384
6-965
76176
21024576
276
16-613
6-510
114921
38958219
339
18-411
6-972
76729
21253933
277
16-643
6-518
115600
39304000
340
18-439
6-979
77284
21484952
278
16-678
6-526
116281
89651821
341
18-466
6-986
77841
21717639
279
16-703
6-534
116964
40001688
342
18-493
6-993
78400
21952000
280
16-733
6-542
117649
40353607
343
18-520
7-000-
78961
22188041
281
16-763
6-549
118336
40707584
344
18-547
7-006
79524
22425768
282
16-792
6-557
119025
41063625
345
18-574
7-013
80089
22665187
283
16-822
6-565
119716
41421736
346
18-601
7-020
80656
22906304
284
16-852
6-573
120409
41781923
347
18-627
7-027
81225
23149125
285
16-881
6-580
121104
42144192
348
18-654
7-033
81796
23393656
286
16-911
6-588
121801
42508549
349
18-681
7-040
82369
23639903
287
16-941
6-596
122500
42875000
350
18-708
7-047
82944
23887872
288
16-970
6-603
123201
43243551
351
18-734
7-054
83521
24137569
289
17-000
6-611
123904
43614208
352
18-761
7-060
84100
24389000
290
17-029
6-619
124609
43986977
353
18-788
7-06Y
84681
24642171
291
17-058
6-626
125316
44361864
354
18-814
7-074
85264
24897088
292
17-088
6-634
126025
44738875
355
18-841
7-080
85849
25153757
293
17-117
6-641
126736
45118016
356
18-867
7-087
86436
25412184
294
17-146
6-649
127449
45499293
357
18-894
7-093
87025
25672375
295
17-175
6-656
128164
45882712
358
18-920
7-100
87616
25934836
296
17-204
6-664
128881
46268279
359
18-947
7-107
88209
26198073
297
17-233
6-671
129600
46656000
360
18-973
7-113
88804
26463592
298
17-262
6-679
130321
47045831
361
19-000
7-120-
89401
26730899
299
17-291
6-686
131044
47437928
362
19-026
7-126
90000
27000000
300
17-320
6-694
131769
47832147
363
19-052
7-133
90601
27270901
301
17-349
6-701
132496
48228544
364
19-078
7-140
91204
27543608
302
17-378
6-709
133225
48627125
365
19-104
7-146-
91809
27818127
303
17-406
6-716
133956
49027896
366
19-131
7-153
92416
28094464
304
17-435
6-723
134689
49430863
367
19-157
7-159-
93025
28372625
305
17-464
6-731
135424
49836032
368
19-183
7-166
93636
28652616
306
17-492
6-738
136161
50243409
369
19-209
7'172
94249
28934443
307
17-521
6-745
136900
50653000
370
19-235
7-17
94864
29218112
308
17-549
6-753
137641
51064811
371
19-261
7-185
95481
29503609
309
17-578
6-760
138384
51478848
372
19-287
7-191
96100
29791000
310
17-606
6-767
139129
51895117
373
19-313
7-198
98721
30080231
311
17-635
6-775
139876
52313624
374
19-339
7-204
97344
30371328
312
17-663
6-782
140625
52734375
375
19-364
7-211
97969
30664297
313
17-691
6-789
141376
53157376
376
19-390
7-217
98596
30959144
314
17-720
6-796 1 142129
53582633
377
19-416
7-224
99225
31255875
316
17-748
6-804 1 142884
54010152
378
19-442
7-230
APPENDIX.
699
TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS OF NUMBERS ( Continued}.
Squares.
Cubes.
No.
Square
roots.
Cube
roots.
Squares.
Cubes.
No.
Square
roots.
Cube
roots.
143641
54439939
379
19-467
7-236
195364
86350888
442
21-023
7-617
144400
54872000
380
19-493
7-243
196249
86938307
443
21-047
7-623
145161
55306341
381
19-519
7-249
197136
87528384
444
21-071
7-628
145924
55742968
382
19-544
7-255
198025
88121125
445
21-095
7-634
146689
56181887
383 19-570
7-262
198916
88716536
446
21-118
7-640
147456
56623104
384
19-595
7-268
199809
89314623
447
21-142
7-646
148225
57066625
385
19621
7-274
200704
89915392
448
21-166
7-651
148996
57512456
386
19-646
7-281
201601
90518849
449
21-189
7-657
149769
57960603
387
19-672
7-287
202500
91125000
450
21-213
7-663
150544
58411072
388
19-697
7-293
203401
91733851
451
21-236
7-668
151321
58863869
389
19-723
7-299
204304
92345408
452
21-260
7-674
152100
59319000
390
19-748
7-306
205209
92959677
453
21-283
7-680
152881
59776471
391
19-773
7-312
206116
93576664
454
21-307
7-685
153664
602S6288
392
19-798
7-318
207025
94196375
455
21-330
7-691
154449
60698457
393
19-824
7-324
207936
94818816
456
21-354
7-697
155236
61162984
394
19-849
7-331
208849
95443993
457
21-377
7-702
156025
61629875
395
19-874
7-337
209764
96071912
458
21-400
7-708
156816
62099136
396
19-899
7-343
210681
96702579
459
21-424
7-713
157609
62570773
397
19-924
7-349
211600
97336000
460
21-447
7-719
158404
63044792
398
19-949
7-355
212521
97972181
461
21-470
7-725
159201
63521199
399
19-974
7-361
213444
98611128
462
21-494
7-730-
160000
64000000
400
20-000
7-368
214369
99252847
463
21-517
7-736
160801
64481201
401
20-024
7-374
215296
99897344
464
21-540
7-741
161604
64964808
402
20-049
7-380
216225
100544625
465
21-563
7-74T
162409
65450827
403
20-074
7-386
217156
101194696
466
21-587
7-752
163216
65939264
404
20-099
7-392
218089
101847563
467
21-610
7-758
164025
66430125
405
20-124
7-398
219024
102503232
468
21-633
7-763
164836
66923416
406.
20-149
7-404
219961
103161709
469
21-656
7-769
165649
67419143
407
20-174
7-410
220900
103823000
470
21-679
7-774
166464
67917312
408
20-199
7-416
221841
104487111
471
21-702
7-780'
167281
68417929
409
20-223
7-422
222784
105154048
472
21-725
7-785
168100
68921000
410
20-248
7-428
223729
105823817
473
21-748
7-791
168921
69426531
411
20-273
7-434
224676
106496424
474
21-771
7-796
169744
69934523
412
20-297
7-441
225625
107171875
475
21-794
7-802
170569
70444997
413
20-322
7-447
226576
107850176
476
21-817
7-807
171396
70957944
414
20-346
7-453
227529
108531333
477
21-840
7-813
172225
71473375
415
20-371
7-459
228484
109215352
478
21-863
7-818
173056
71991296
416
20-396
7-465
229441
109902239
479
21-886
7-824
173889
72511713
417
20-420
7-470
230400
110592000
480
21-908
7-829
174724
73034632
418
20-445
7-476
231361
111284641
481
21-931
7-835
175561
73560059
419
20-469
7-482
232324
111980168
482
21-954
7-840'
176400
74088000
420
20-493
7-488
233289
112678587
483
21-977
7-846
177241
74618461
421
20-518
7-494
234256
113379904
484
22-000
7-851
178084
75151448
422
20-542
7-500
235225
114084125
485
22-022
7-856
178929
75686967
423
20-566
7-506
236196
114791256
486
22-045
7-862
179776
76225024
424
20-591
7-512
237169
115501303
487
22-068
7-867
180625
76765625
425
20-615
7-518
238144
116214272
488
22-090
7-872
181476
77308776
426
20-639
7-524
239121
116930169
489
22-113
7-878
182329
77854483
427
20-663
7-530
240100
117649000
490
22-135
7-883
183184
78402752
428
20-688
7-536
241081
118370771
491
22-158
7-889
184041
78953589
429
20-712
7-541
242064
119095488
492
22-181
7-894
184900
79507000
430
20-736
7-547
243049
119S23157
493
22-203
7-899
185761
80062991
431
20-760
7-553
244036
120553784
494
22-226
7-905
186624
80621568
432
20-784
7-559
245025
121287375
495
22*248
7-910
187489
81182737
433
20-808
7-565
246016
122023936
496
22-271
7-915
188356
81746504
434
20-832
7-571
247009
122763473
497
22-293
7-921
189225
82312875
435
20-856
7-576
248004
123505992
498
22-315
7-926
190096
82881856
436
20-880
7-582
249001
124251499
499
22-338
7-931
190969
83453453
437
20-904
7-588
250000
125000000
500
22-360
7-937
191844
84027672
438
20-928
7-594
251001
125751501
501
22-383
7-942
192721
84604519
439
20-952
7-600
252004
126506008
502
22-405
7-947
193600
85184000
440
20-976
7-605
253009
127263527
503
22-427
7-952
194481
85766121
441
21-000
7-611
254016
128024064
504
22-449
7'95&
TOO
APPENDIX.
TABLE OF SQUAKES, CUBES, SQUAKE AND CUBE ROOTS OF NUMBERS (Continued).
Squares.
Cubes.
No.
Square
roots.
Cube
roots.
Squares.
Cubes.
No.
Square
roots.
Cube
roots.
255025
128787625
505
22-472
7-963
322624
183250432
568
23-832
8-281
256036
129554216
506
22-494
7-968
323761
184220009
569
23-853
8-286
257049
130323843
507
22-516
7-973
324900
185193000
670
23-874
8-291
258064
131096512
508
22-538
7-979
326041
186169411
671
23-895
8-296
259081
131872229
509
22-561
7-984
327184
187149248
572
23-916
8-301
260100
132651000
510
22-683
7-989
328329
188132517
573
23-937
8-305
261121
133432831
611
22-605
7-994
329476
189119224
574
23-958
8-310
262144
134217728
512
22-627
8-000
330625
190109375
575
23-979
8-315
263169
135005697
513
22-649
8-005
331776
191102976
576
24-000
8-320
264196
135796744
514
22-671
8-010
332929
192100033
577
24-020
8-325
265225 136590875
515
22-693
8-015
334084
193100552
578
24-041
8-329
266256
137388096
516
22-715
8-020
335241
194104539
579
24-062
8-334
267289
138188413
517
22-737
8-025
336400
195112000
580
24-083
8-339
268324
138991832
618
22-759
8-031
337561
196122941
681
24-103
8-344
269361
139798359
519
22-781
8-036
338724
197137368
582
24-124
8-349
270400
140608000
520
22-803
8-041
339889
198155287
583
24-145
8-353
271441
141420761
521
22-825
8-046
341056
199176704
584
24-166
8-358
272484
142236648
522
22-847
8-051
342225
200201625
585
24-186
8-363
273529
143055667
623
22-869
8-056
343396
201230056
586
24-207
8-368
274576
143877824
524
22-891
8-062
344569
202262003
587
24-228
8-372
275625
144703125
525
22-912
8-067
345744
203297472
588
24-248
8-377
276676
145531576
526
22-934
8-072
346921
204336469
589
24-269
8-382
277729
146363183
527
22-956
8-077
348100
205379000
590
24-289
8-387
278784
147197952
528
22-978
8-082
349281
206425071
591
24-310
8-391
279841
148035889
529
23-000
8-037
350464
207474688
592
24-331
8-396
280900
148877000
530
23-021
8-092
351649
208527857
693
24-351
8-401
281961
149721291
531
23-043
8-097
352836
209584584
694
24-372
8-406
283024
150568768
532
23-065
8-102
354025
210644875
595
24-392
8-410
284089
151419437
633
23-086
8-107
355216
211708736
696
24-413
8-415
285156
152273304
534
23-108
8-112
356409
212776173
597
24-433
8-420
286225
153130375
535
23-130
8-118
357604
213847192
698
24'454
8-424
287296
153990656
636
23-151
8-123
358801
214921799
599
24-474
8-429
288369
154854153
637
23-173
8-128
360000
216000000
600
24-494
8-434
289444
155720872
538
23-194
8-133
361201
217081801
601
24-515
8439
290521
156590819
639
23-216
8-138
362404
218167208
602
24-535
8-443
291600
157464000
640
23-237
8-143
363609
219256227
603
24-556
8-448
292681
158340421
541
23-259
8-148
364816
220348864
604
24-576
8-453
293764
159220088
542
23-280
8-153
366025
221445125
605
24-596
8-457
294849
160103007
543
23-302
8-158
367236
222545016
606
24-617
8-462
295936
160989184
544
23-323
8-163
368449
223648543
607
24-637
8-467
297025
161878625
545
23-345
8-168
369664
224755712
608
24-657
8-471
298116
162771336
546
23-366
8-173
370881
225866529
609
24-677
8-476
299209
163667323
547
23-388
8-178
372100
226981000
610
24-698
8-480
300304
164566592
548
23-409
8-183
373321
228099131
611
24-718
8-485
301401
165469149
549
23-430
8-188
374544
229220928
612
24-738
8-490
302500
166375000
560
23-452
8-193
375769
230346397
613
24-758
8-494
303601
167284151
551
23-473
8-198
376996
231475544
614
24-779
8-499
304704
168196608
652
23-494
8-203
378225
232608375
615
24-799
8-504
305809
169112377
553
23-515
8-208
379456
233744896
616
24-819
8-508
306916
170031464
554
23-537
8-213
380689
234885113
617
24-839
8-513
308025
170953875
555
23-558
8-217
381924
236029032
618
24-859
8-517
309136
171879616
556
23-579
8-222
383161
237176659
619
24-879
8-522
310249
172808693
557
23-600
8-227
384400
238328000
620
24-899
8-527
311364
173741112
558
23-622
8-232
385641
239483061
621
24-919
8-531
312481
174676879
559
23-643
8-237
386884
240641848
622
24-939
8-536
313600
175616000
560
23-664
8-242
388129
241804367
623
24-959
8-540
314721
176558481
561
23-685
8-247
389376
242970624
624
24-979
8-545
315844
177504328
562
23-706
8-252
390625
244140625
625
25-000
8-549
316969
178453547
563
23-727
8-257
391876
245314376
626
25-019
8-554
318096
179406144
564
23-748
8-262
393129
246491883
627
25-039
8-558
319225
180362125
565
23-769
8-267
394384
247673152
628
25-059
8-563
320356
181321496
566
23-790
8-271
395641
248858189
629
25-079
8-568
321489
182284263
667
23-811
8-276
396900
250047000
630
25-099
8-572
APPENDIX.
701
TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS OF NUMBERS (Continued}.
Squares.
Cubes.
No.
Square
roots.
Cube
roots.
Squares.
Cubes.
No.
Square
roots.
Cube
roots.
398161
251239591
631
25-119
8-577
481636
334255384
694
26-343
8-853
399424
252435968
632
25-139
8-581
483025
335702375
695
26-362
8-857
400689
253636137
633
25-159
8-586
484416
337153536 696
26-381
8-862
401956
254840104
634
25-179
8-590
485809
338608873 697
26-400
8-866
403225
256047875
635
25-199
8-595
487204
340068392 698
26-419
8-870
404496
257259456
636
25-219
8-599
488601
341532099 699
26-438
8-874
405769
258474853
637
25-238
8-604
490000
343000000
700
26-457
8-879
407044
259694072
638
25-258
8-608
491401
344472101
701
26-476
8-883
408321
260917119
639
25-278
8-613
492804
345948408
702
26-495
6-887
409600
262144000
640
25-298
8-617
494209
347428927 703
26-514
8-891
410881
263374721
641
25-317
8-622
495616
348913664
704
26-532
8-895
412164
264609288
642
25-337
8-626
497025
350402625
705
26-551
8-900
413449
265847707
643
25-357
8-631
498436
351895816
706
26-570
8-904
414736
267089984
644
25-377
3-635
499849
353393243
707
26-589
8-908
416025
268336125
645
25-396
8-640
501264
354894912
708
26-608
8-912
417316
269586136
646
25-416
8-644
502681
356400829
709
26-627
8-916
418609
270840023
647
25-436
8-649
504100
357911000
710
26-645
8-921
419904
272097792
648
25-455
8-653
505521
359425431
711
26-664
8-925
421201
273359449
49
25-475
8-657
1 506944
360944128
712
26-683
8-929
422500
274625000
650
25-495
8-662
508369
362467097
713
26-702
8-933
423801
275894451
651
25-514
8-666
509796
363994344
714
26-720
8-937
425104
277167808
652
25-534
8-671
511225
365525875
715
26-739
8-942.
426409
278445077
653
25-553
8-675
512656
367061696
716
26-758
8-946
427716
279726264
654
25-573
8-680
514089
368601813
717
26-776
8-950
429025
281011375
655
25-592
8'684
515524
370146232
718
26-795
8-954
430336
282300416
656
25-612
8-688
516961
371694959
719
26-814
8-958
431649
283593393
657
25-632
8-693
518400
373248000
720
26-832
8-962
432964
284890312
658
25-651
8-697
519841
374805361
721
26-851
8-966
484281
286191179
659
25-670
8*702
521284
676367048
722
26-870
8-971
435600
287496000
660
25-690
8-706
522729
377933067
723
26-888
8-975
436921
288804781
661
25-709
8-710
524176
379503424
724
26-907
8-979
438244
290117528
662
25-729
8-715
525625
381078125
725
26-925
8-983
439569
291434247
663
25*748
8*719
527076
382657176
726
26-944
8-987
440896
292754944
664
25-768
8-724
528529
384240583
727
26-962
8991
442225
294079625
665
25-787
8-728
529984
385828352
728
26-981
8-995
443556
295408296
666
25-806
8'732
531441
387420489
729
27-000
9-000
444889
296740963
667
25-826
8*737
532900
389017000
730
27*018
9-004
446224
298077632
668
25-845
8-741
534361
390617891
731
27-037
9-008
447561
299418309
669
25-865
8-745
535824
392223168
732
27-055
9-012
448900
300763000
670
25-884
8-750
5372b9
393832837
733
27-073
9-016
450241
302111711
671
25-903
8-754
538756
395446904
734
27-092
9-020
451584
303464448
672
25-922
8-759
540225
397065375
735
27-110
9/024
452929
304821217
673
25-942
8-763
541696
398688256
736
27-129
9-028
454276
306182024
674
25-961
8'767
543169
400315553
787
27*147
9-032
455625
307546875
675
25-980
8-772
544644
401947272
738
27-166
9-036
456976
308915776
676
26-000
8-776
546121
403583419
739
27-184
9-040
458329
310288733
677
26-019
8-780
547600
405224000
740
27-202
9-045
459684
311665752
678
26-038
8-785
549081
406869021
741
27-221
9-049
461041
313046839
679
26-057
8-789
550564
408518488
742
27-239
9-053
462400
314432000
680
26-076
8-793
552049
410172407
743
27-258
9-057
463761
315821241
681
26-095
8-797
553536
411830784
744
27-276
9-061
465124
317214568
682
26-115
8-802
555025
413493625
745
27-294
9-065
466489
318611987
683
26-134
8-806
556516
415160936
746
27-313
9-069
467856
320013504
684
26-153
8-810
558009
416832723
747
27-331
9-073
469225
321419125
685
26-172
8-815
559504
418508992
748
27-349
9-077
470596
322828856
686
26-191
8-819
561001
420189749
749
27-367
9-081
471969
324242703
687
26-210
8-823
562500
421875000
750
27-386
9-085
473344
325660672
688
26-229
8-828
564001
423564751
751
27-404
9-089
474721
327082769
689
26-248
8-832
565504
425259008
752
27-422
9-093
476100
328o09000
690
26-267
8-836
567009
426957777
753
27-440
9-097
477481
329939371
691
26*286
8-840
568516
428661064
754
27-459
9-101
478864
331373888
692
26-305
8-845
570025
430368875
755
27-477
9-105
480249
332812557
693
26-324
8-849
571536
432081216
756
27-495
9-109
702
APPENDIX.
TABLE OF SQUARES, CUBES, SQUAEE AND CUBE EOOTS OF NUMBERS (Continued).
Squares.
Cubes.
No.
Square
roots.
Cube
roots.
Squares.
Cubes.
No.
Square
roots.
Cube
roots.
673049
433798093
757
27-513
9-113
672400
551368000
820
28-635
9-359
574564
435519512
758
27-531
9-117
674041
553387661
821
28-653
9-363
576081
437245479
759
27-549
9-121
675684
555412248
822
28-670
9-367
577600
438976000
760
27-568
9-125
677329
557441767
823
28-687
9-371
579121
440711081
761
27-586
9-129
678976
559476224
824
28-705
9-375
580644
442450728
762
27-604
9-133
680625
561515625
825
28-722
9-378
582169
444194947
763
27-622
9-137
682276
563559976
826
28-740
9-382
583696
445943744
764
27-640
9-141
683929
565609283
827
28-757
9-386
585225
447697125
765
27-658
9-145
685584
567663552
828
28-774
9-390
586756
449455096
766
27-676
9-149
687241
569722789
829
28-792
9-394
588289
451217663
767
27-694
9-153
688900
571787000
830
28-809
9-397
589824 .
452984832
768
27-712
9-157
690561
573856191
831
28-827
9-401
591361
454756609
769
27-730
9-161
692224
575930368
832
28-844
9-405
692900
456533000
770
27-748
9-165
693889
578009537
833
28-861
9-409
594441
458314011
771
27-766
9-169
695556
580093704
834
28-879
9-412
595984
460099648
772
27-784
9-173
697225
582182875
835
28-896
9-416
697529
461889917
773
27-802
9-177
698896
584277056
836
28-913
9-420
699076
463684824
774
27-820
9-181
700569
586376253
837
28-930
9-424
600625
465484375
775
27-833
9-185
702244
588480472
838
28-948
9-427
602176
467288576
776
27-856
9-189
703921
590589719
839
28-965
9-431
603729
469097433
777
27-874
9-193
705600
592704000
840
28-982
9-435
605284
470910952
778
27-892
9-197
707281
594823321
841
29-000
9-439
606841
472729139
779
27-910
9-201
708964
596947688
842
29-017
9-442
608400
474552000
780
27-928
9-205
710649
599077107
843
29-034
9-446
609961
476379541
781
27-946
9-209
712336
601211584
844
29-051
9-450
611524
478211768
782
27-964
9-213
714025
603351125
845
29-068
9-454
613089
480048687
783
27-982
9-216
715716
605495736
846
29-086
9-457
614656
481890304
784
28-000
9-220
717409
607645423
847
29-103
9-461
616225
483736625
785
28-017
9-224
719104
609800192
848
29-120
9-465
617796
485587656
786
28-035
9-228-
720801
611960049
849
29-137
9-468
619369
487443403
787
28-053
9-232
722500
614125000
850
29-154
9-472
620944
489303872
788
28-071
9-236
724201
616295051
851
29-171
9-476
622521
491169069
789
28-089
9-240
725904
618470208
852
29-189
9-480
624100
493039000
790
28-106
9-244
727609
620650477
853
29-206
9-483
625681
494913671
791
28-124
9-248
729316
622835864
854
29-223
9-487
627264
496793088
792
28-142
9-252
731025
625026375
855
29-240
9-491
628849
498677257
793
28-160
9-256
732736
627222016
856
29-257
9494
630436
500566184
794
28-178
9-259
734449
629422793
857
29-274
9-498
632025
502459875
795
28-195
9-263
736164
631628712
858
29-291
9-502
633616
504358336
796
28-213
9"267
737881
633839779
859
. 29-308
9-505
635209
506261573
797
28231
9-271
739600
636056000
860
29-325
9-509
636804
508169592
798
28-248
9-275
741321
638277381
861
29-342
9-513
638401
510082399
799
28-266
9-279
743044
640503928
862
29-359
9-517
640000
512000000
800
28-284
9-283
744769
642735647
863
29-376
9-520
641601
513922401
801
28-301
9-287
746496
644972544
864
29-393
9-524
643204
515849608
802
28-319
9-290
748225
647214625
865
29-410
9-528
644809
517781627
803
28-337
9-294
749956
649461896
866
29-427
9-531
646416
519718464
804
28-354
9-298
751689
651714363
867
29-444
9-535
648025
521660125
805
28-372
9-302
753424
653972032
868
29-461
9-539
649636
523606616
806
28-390
9-306
755161
656234909
869
29-478
9 542
651249
525557943
807
28-407
9-310
756900
658503000
870
29-495
9-546
652864
527514112
808
28-425
9-314
758641
660776311
871
29-512
9-650
654481
529475129
809
28-442
9-317
760384
663054848
872
29-529
9-553
656100
531441000
810
28-460
9-321
762129
665338617
873
29516
9-557
657721
533411731
811
28-478
9-325
763876
667627624
874
29-r><>::!
9-561
659344
535387328
812
28-495
9-329
765625
669921875
875
29-580
9-564
660969
537367797
813
28-513
9-333
767376
672221376
876
29-597
9'568
662596
539353144
814
28-530
9-337
769129
674526133
877
29-614
9-571
664225
541343375
815
28-548
9-340
770884
676836152
878
29-631
9-575
665856
543338496
816
28-565
9-344
772641
679151439
879
29-647
9-579
667489
545338513
817
28-583
9-348
774400
681472000
880
29-664
9-582
669124
547343432
818
28-600
9-352
776161
683797841
881
29-681
9-586
70761
549353259
819
28-618
9-356 !
777924
686128968
882
29-698
9-590
APPENDIX.
703
TABLE OF SQUARES, CUBES, SQUARE AND CUBE BOOTS OF NUMBERS ( Continued).
Squares.
Cubes.
No.
Square
roots.
Cube
roots.
Squares.
Cubes.
Square
Cube
roots.
roots.
779689
688465387
883
29-715
9-593
894916
846590586
946
30-757
9-816
781456
690807104
884
29-732
9-597
896808
849278m
947
30-773
9-820
783225
693154125
885
29-748
9-600
898704
851971392
948
30-789
9-823
784996
695506456
886
29-765
9-604
900601
854670349
949
30-805
9-827
786769
697864103
887
29-782
9-608
902500
857375000
950
30-822
9-830
788544
700227072
888
29-799
9-611
904401
860085351
951
30-838
9-833
790321
702595369
889
29-816
9-615
906304
862801408
952
30-854
9-837
792100
704969000
890
29-832
9-619
908209
865523177
953
30-870
9-840
793881
707347971
891
29-849
9-622
910116
868250664
954
30-886
9-844
795664
709732288
892
29-866
9-626
912025
870983875
955
30-903
9-847
797449
712121957
893
29-883
9-629
913936
873722816
956
30-919
9-851
799236
714516984
894
29-899
9-633
915849
876467493
957
30-935
9-854
801025
716917375
895
29-916
9-636
917764
879217912
958
30-951
9-857
802816
719323136
896
29-933
9-640
919681
881974079
959
30-967
9-861
804609
721734273
897
29-949
9-644
921600
884736000
960
30-983
9-864
806404
724150792
898
29-966
9-647
923521
887503681
961
31-000
9-868
808201
726572699
899
29-983
9-651
925444
890277128
962
31-016
9-871
810000
729000000
900
30-000
9-654
927369
893056347
963
31-032
9-875
811801
731432701
901
30-016
9-658
929296
895841344
964
31-048
9-878
813604
733870808
902
30-033
9-662
931225
898632125
965
31-064
9-881
815409
736314327
903
30-049
9-665
933156
901428696
966
31-080
9-885
817216
738763264
904
30-066
9-669
935089
904231063
967
31-096
9-888
819025
741217625
905
30-083
9-672
937024
907039232
968
31-112
9-892
820836
743677416
906
30-099
9-676
938961
909853209
969
31-128
9-895
822649
746142643
907
30-116
9-679
940900
912673000
970
31-144
9-898
824464
748613312
908
30-133
9-683
942841
915498611
971
31-160
9-902
826281
751089429
909
30-149
9-686
944784
918330048
972
31-176
9-905
828100
753571000
910
30-166
9-690
946729
921167317
973
31-192
9-909
829921
756058031
911
30-182
9-694
948676
924010424
974
31-208
9-912
831744
758550528
912
30-199
9-697
950625
926859375
975
31-224
9-915
833569
761048497
913
30-215
9-701
952576
929714176
976
31-240
9-919
835396
763551944
914
30-232
9-704
954529
932574833
977
31-256
9-922
837225
766060875
915
30-248
9-708
956484
935441352
978
31-272
9-926
839056
768575296
916
30-265
9-711
958441
938313739
979
31-288
9-929
840889
771095213
917
30-282
9-715
960400
941192000
980
31-304
9-932
842724
773620632
918
30-298
9-718
962361
944076141
981
31-320
9-936
844561
776151559
919
30-315
9-722
964324
946966168
982
31-336
9-939
846400
778688000
920
30-331
9-725
966289
949862087
983
31-352
9-943
848241
781229961
921
30-347
9-729
968256
952763904
984
3T368
9-946
850084
783777448
922
30-364
9-732
970225
955671625
985
31-384
9-949
851929
786330467
923
30-380
9-736
972196
958585256
986
31-400
9-953
853776
788889024
924
30-397
9-739
974169
961504803
987
31-416
9-956
855625
791453125
925
30-413
9-743
976144
964430272
988
31-432
9-959
857476
794022776
926
30-430
9-746
978121
967361669
989
31-448
9-963
859329
796597983
927
30-446
9-750
980100
970299000 990
31-464
9-966
861184
799178752
928
30-463
9-753
982081
973242271
991
31-480
9-969
863041
801765089
929
30-479
9-757
984064
976191488
992
3T496
9-973
864900
804357000
930
30-495
9-761
986049
979146657
993
31-511
9-976
866761
806954491
931
30-512
9-764
988036
982107784
994
31-527
9-979
868624
809557568
932
30-528
9-767
990025
985074875
995
31-543
9-983
870489
812166237
933
30-545
9-771
992016
988047936
996
31-559
9-986
872356
814780504
934
30-561
9-774
994009
991026973
997
31-575
9-989
874225
817400375
935
30-577
9-778
996004
994011992
998
31-591
9-993
876096
820025856
936
30-594
9-782
998001
997002999
999
31-606
9-996
877969
822656953
937
30-610
9-785
1000000
1000000000
1000
31-622
10-000
879844
825293672
938
30-626
9-788
1000201
1003003001
1001
31-638
10-003
881721
827936019
939
30-643
9-792
1004004
1006012008
1002
31-654
10-006
883600
830584000
940
30-659
9-795
1006009
1009027027
1003
31-670
10-009
885481
833237621
941
30-675
9-799
1008016
1012048064
1004
31-685
10-013
887364
835896888
942
30-692
9-802
1010025
10150751251 1005
31-701
10-016
889249
838561807
943
30-708
9-806
1012036
1018108216 1006
31-717
10-019
891136
841232384
944
30-724
9-809
1014049
1021147343! 1007
31-733
10-023
893025
843908625 945
30-740
9-813
1016064
1024192512 1008
31-749
10-026
704
APPENDIX.
LATITUDES AND DEPARTURES.
f"
]
I
j
1
a
i
4
I
ft
f*
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
1-000
o-ooo
2-000
o-ooo
3-000
o-ooo
4-000
o-ooo
5-000
90
01
1-000
0-004
2-000
0-009
3-000
0-013
4-000
0-017
5-000
89f-
o*
1-000
0-009
2-000
0-017
3-000
0-026
4-000
0-035
5-000
89-^
0|
1-000
0-013
2-000
0-026
3-000
0-039 ;
4-000
0-052
5-000
894:
r
1-000
0-017
2-000
0-035
3-000
0-052
3-999
0-070
4-999
89
ij
1-000
0-022
2-000
0-044
2-999
0-065
3-999
0-087
4-999
88*
i*
1-000
0-026
1-999
0-052
2-999
0-079
3-999
0-105
4-998
88-^-
if
1-000
0-031
1-999
0-061
2-999
0-092
3-998
0-122
4-998
88|-
2
0-999
0-035
999
0-070
2-998
0-105
3-998
0-140
4-997
88
9*
0-999
0-039
998
0-079
2-998
0-118
3-997
0-157
4-996
87*
2*
! 0-999
0-044
998
0-087
2-997
0-131
3-996
0-174
4-995
87*
2f
0-999
0-048
998
0-096
2-997
0-144
3-995
0-192
4-994
87i
3
0-999
0-052
997
0-105
2-996
0-157
3-995
0-209
4-993
87
8*
0-998
0-057
1-997
0-113
2-995
0-170
3-994
0-227
4-992
86f
3*
0-998
0-061
1-996
0-122
2-994
0-183
3-993
0-244
4-991
86-J-
3f
0-998
0-065
1-996
0-131
2-994
0-196
3-991
0-262
4-989
86^
4 f
0-998
0-070
1 995
0-140
2-993
0-209
3-990
0-279
4-988
86
4^
0-997
0-074
1-995
0-148
2-992
0-222
3-989
0-296
4-986
85*
41
0-997
0-078
1-994
0-157
2-991
0-235
3-988
0-314
4-985
85*
4f
0-997
0-083
1-993
0-166
2-990
0-248
3-986
0-331
4-983
85^
5
0-996
0-087
1-992
0-174
2-989
0-261
3-985
0-349
4-981
85
5^
0-996
0-092
1-992
0-183
2-987
0-275
3-983
0-366
4-979
84*
5*
0-995
0-096
1-991
0-192
2-986
0-288
3-982
0-383
4-977
84^
5f
0-995
o-ioo
1-990
0-200
2-985
0-301
3-980
0-401
4-975
84^
6
0-995
0-105
1-989
0-209
2-984
0-314
3-978
0-418
4-973
84
*>i
0-994
0-109
1-988
0-218
2-982
0-327
3-976
0-435
4-970
83*
6*
0-994
0-113
1-987
0-226
2-981
0-340
3-974
0-453
4-968
83^
6|
0-993
0-118
1-986
0-235
2-979
0-353
3-972
0-470
4-965
83*
7
0-993
0-122
1-985
0-244
2-978
0-366
3-970
0-487
4-963
83
7i
0-992
0-126
1-984
0-252
2-976
0-379
3-968
0-505
4-960
82*
?*
0-991
0-131
1-983
0-261
2-974
0-392
3-966
0-522
4-957 ;
82^
0-991
0-135
1-982
0-270
2-973
0-405
3-963
0-539
4-954
82J
8
0-990
0-139
1-981
0-278
2-971
0-418
3-961
0-557
4-951
82
8
0-990
0-143
1-979
0-287
2-969
0-430
3-959
0-574
4-948
81*
8*
0-989
0-148
1-978
0-296
2-967
0-443
3-956
0-591
4-945
81*
8f
0-988
0-152
1-977
0-304
2-965
0-456
3-953
0-608
4-942
81*
9
0-988
0-156
1-975
0-313
2-963
0-469
3-951
0-626
4-938
81
9
0-987
0-161
1-974
0-321
2-961
0-482
3-948
0-643
4-935
80*
9*
0-986
0-165
1-973
0-330
2-959
0-495
3-945
0-660
4-931
80-J-
9|
0-986
0-169
1-971
0-339
2-957
0-508
3-942
0-677
4-928
8 i
10
0-985
0-174
1-970
0-347
2-954
0-521
8-939
0-695
4-924
80
10
0-984
0-178
1-968
0-356
2-952
0-534
3-936
0-712
4-920
If
10
0-983
0-182
1-967
0-364
2-950
0-547
3-933
0-729
4-916
10|
0-982
0-187
1-965
0-373
2-947
0-560
3930
0-746
4-912
79i
11
0-982
0-191
963
0-382
2-945
0-572
3-927
0-763
4-908
79
Hi
0-981
0-195
962
0-390
2-942
0-585
8-923
0-780
4-904
7H*
11*
0-980
0-199
960
0-399
2-940
0-598 |
3-920
0-797
4-900
78
llf
0-979
0-204
958
0-407
2-937
0-611
3 916
0-815
4-895
78^-
ir
0-978
0-208
956
0-416
2-934
0-624
3-913
0-832
4-891
78
12J-
0-977
0-212
1-954
0-424
2-932
0-637
3-909
0-849
4-886
77f
12i
0-976
0-216
1-953
0-433
2-929
0-649
3-905
0-866
4-881
77*
12f
0-975
0-221
1-951
0-441
2-926
0-662
3-901
0-883
4-877
77|
13
0-974
0-225
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APPENDIX.
707
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2-192
7598
2-505
8-547
2-818
71*
IS*
1-587
5-690
1-904
6-638
2-221
7-587
2-538
8-535
2-856
71*
18*
1-607
5-682
1-929
6-629
2-250
7-575
2-572
8-522
2-893 71
19
T628
5-673
1-953
6-619
2-279
7-564
2-605
8-510
2-930 || 71
191
1-648
5-665
1-978
0-609
2-308
7-553
2-638
8-497
2-967
70f
19*
1-669
5-656
2-003
6-598
2-337
7-541
2-670
8-484
3-004
70*
19f
1-690
5-647
2-028
6-588
2-365
, 7-529
2-703
8-471
3-041
?<*
20
1-710
5-638
2-052
6-578
2-394
7-518
2-736 8-457
3-078
70
201
1-731
5-629
2-077
6-567
2-423
7-506
2-769 8-444
3-115
69*
20*
1-751
5-620
2-101
6-557
2-451
7-493
2-802
8-430
3-152
69*
.20*
1-771
5-611
2-126
6-546
2-480
7-481
2-834
8-416
3-189
691
21
1-792
5-601
2-150
6-535
2-509
7-469
2-867
8-402
3-225
69
211
1-812
5-592
2-175
6-524
2-537
7-456
2-900
8-388
3-262 68*
21*
1-833
5-582
2-199
6-513
2-566
7-443
2-932
8-374
3-299
68*
21f
1-853
5-573
2-223
6-502
2-594
7-430
2-964
8-359
3-335
681
22
1-873
5-563
2-248
6-490
2-622
7-417
2-997
8-345
3-371
68
221
1-893
5-553
2-272
6-479
2-651
7-404
3-029
8-330
3-408
67*
22*
1-913
5-543
2-296
6-467
2-679
7-391
3-C61
8-315
3-444
67*
22|
1-934
5-533
2-320
6-455
2-707
7-378
3-094
8-300
3-480
671
23
1-954
5-523
2-344
6-444
2-735
7-364
3-126
8-285
3-517
67 1
231
1-974
5-513
2-368
6-432
2-763
7-350
3-158
8-269
8-6C8
66*
234-
1-994
5-502
2-392
6-419
2-791
7-336
3-190
8-254
3-589
66*
23f
2-014
5-492
2-416
6-407
2-819
7-322
3-222
8-238
3-625
661
24 3
2-034
5-481
2-440
6-395
2-847
! 7-308
3-254
8-222
3-661
66 f
241
2-054
5-471
2-464
6-382
2-875
i 7-294
3-286
8-206
3-696
65f
24*
2-073
5-460
2-488
6-370
2-903
7-280
3-318
8-190
3-732
65*
24f
2-093
5-449
2-512
6-357
2-931
7-265
3-349
8-178
3-768
651
25
2-113
5-438
2-536
6-344
2-958
7-250
3-381
8-157
3-804
65
251
2-133
5-427
2-559
6-331
2-986
7-236
3-413 8-140
3-839
64*
25*
2-153
5-416
2-583
6-318
3-014
7-221
3-444
8-123
3-875
H4*
25f
2-172
5-404
2-607
6-305
3-041
7-206
3-476 j
8-106
3-910
i 641
26
2-192
5-393
2-630
6-292
3-069
7-190
3-507 i
8-089
3-945
64
261
2-211
5-381
2-654
6-278
3-096
7-175
3-538
8-072
3-981 63*
26*
2-231
5-370
2-677
6-265
3-123
7-160
3-570 !
8-054
4-016 63*
26|
2-250
5-358
2-701
6-251
3-151
7-144
3-601
8-037
4-051 1 63^
21
2-270
5-346
2-724
6-237
3-178
7-128
3-632
8-019
4-086
63
271
2-289
5-334
2-747
6-223
3-205
7-112
3-663
8-001
4-121
62*
27*
2-309
5-322
2-770
6-209
3-232
7-096
3-694
7-983
4-156
62*
27*
2-328
5-310
2-794
6-195
3-259
7-080
3-725
7*965
4-190
621
28
2-347
5-298
2-817
6-181
3-286
7-064
3-756
7-947
4225
68*
281
2-367
5-285
2-840
6-166
3-313
7-047
3-7S7
7-928
4-260
61*
28*
2-386
5-273
2-863
6-152
3-340
7-031
3-817
7-909
4294
61*
2 a f
2-405
5-260
2-886
6-137
8-367
7-014
3-848 j
7-891
4-329
611
29
2-424
5-248
2-909
6-122
3-394
6-997
3-*78
7-872
4-363
61
291
2-443
5-235
2-932
6-107
3-420
6-980
3-909.
7-852
4-398
60*
29*
2-462
5-222
2-955
6-093
3-447
6-963
3-939
7-833
4-432
60*
29*
2-481
5-209
2-977
6-077
3-474
6-946
3-970
7-814
4-466
601
30
2-500
5-196
3-000
6-062
8-500
6-928
4-000
7-794
4-500
60
&
c
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
ti
j
5
7
8
j
708
APPENDIX.
LATITUDES AND DEPARTURES.
1
I 2
3
4
5
tUG
i
Lat.
Dep.
Lat. Dep.
Lat.
Dep.
Lat. Dep.
Lat.
30
0-866
0-500
i 1-732
1-000
2-598
1-500
3-464
2-000
4-330
60
30
0-864
0-504
! 1-728
1-008
2-592
1-511
3-455
2-015
4-319
59*
30|
0-862
0-508
1-723
1-015
2-585
1-523
3-447
2-030
4-308
59*
30f
0-859
0-511
1-719
1-023
2-578
1-534
3-438
2-045
4-297
59i
31
0-857
0-515
1-714
1-030
2-572
1-545
3-429
2-060
4-286
59
31*
0-855
0-519
1-710
1-038
2-565
1-556
3-420
2-075
4-275
58*
31i
0-853
0-522
1-705
1-045
2-558
1-567
3-411
2-090
4-263
58*
31* !
0-850
0-526
1-701
1-052
2-551
1-579
3-401
2-105
4-252
58i
32
0-848
0-530
1-696
1-060
2-544
1-590
3-392
2-120
4-240
58
32i
0-846
0-534
1-691
1-067
2-537
1-601
3-383
2-134
4-229
57*
32*
0-843
0-537
1-687
1-075
2-530
1-612
3-374
2-149
4-217
57*
32f
0-841
0-541
1-682
1-082
2-523
1-623
3-364
2-164
4-205
57*
33
0-839
0-545
1-677
1-089
2-516
1-634
3-355
2-179
4-193
57"
33
0-836
0-548
1-673
1-097
2-509
1-645
3-345
2-193
4-181
56*
33
0-834
0-552
1-668
1-104
2-502
1-656
3-336
2-208
4-169
56*
33|
0-831
0-556
1-663
1-111
2-494
1-667
3-326
2-222
4-157
56*
34
0-829
0-559
1-658
1-118
2-487
1-678
3-316
2-237
4-145
56
34*
0-827
0-563
1-653
1-126
2-480
1-688
3-306
2-251
4-133
55*
34|
0-824
0-566
1-648
1-133
2-472
1-699
3-297
2-266
4-121
56*
34|
0-822
0-570
1-643
1-140
2-465
1-710
3-287
2-280
4-108
55*
85
0-819
0-574
1-638
1-147
2*457
1721
3-277
2-294
4-096
55*
35*
0-817
0-577
1-633
1-154
2-450
1-731
3-267
2-309
4-083
64*
354
0-814
0-581
1-628
1-161
2-442
1-742
3-257
2-323
4-071
64*
35f
0-812
0-584
1-623
1-168
2-435
1-753
3-246
2-337
4-058
64*
36 1
0-809
0-588
1-618
1-176
2-427
1-763
3-236
2-351
4-045
54
36
0-806
0-591
1-613
1-183
2-419
1-774
3-226
2-365
4-032
68*
36
0-804
0-595
1-608
1-190
2-412
1-784
3-215
2-379
4-019
53*
36f
0-801
0-598
1-603
1-197
2-404
1-795
3-205
2-393
4-006
53*
3T
0-799
0-602
1-597
1-204
2-396
1-805
3-195
2-407
3-993
53
37i
0-796
0-605
1-592
1-211
2-388
1-816
3-184
2-421
3-980
52*
37|
0-793
0-609
1-587
1-218
2-380
1-826
3-173
2-435
3-967
52*
37|
0-791
0-612
1-531
1-224
2-372
1-837
3-163
2-449
3-953
52i
38
0-788
0-616
1-576
1-231
2-364
1-847
3-152
2-463
3-940
52
38
0-785
0-619
1-571
1-238
2-356
1-857
3-141
2-476
3-927
61*
38i
0-783
0-623
1-565
1-245
2-348
1-868
3-130
2-490
3-913
51*
38|
0-780
0-626
1-560
1-252
2-340
1-878
3-120
2-604
3-899
61*
39 f
0-777
(V-629
1-554
1-259
2-331
1-888
3-109
2-517
3-886
51
39|r
0-774
0-633
1-549
1-255
2-323
1-898
3-098
2-i31
3-872
60*
39J-
0-772
0-636
1-543
1-272
2-315
1-908
3-086
2-544
3-858
50*
39|
0-769
0-639
1-538
1-279
2-307
1-918
3-075
2-558
3-844
50*
40
0-766
0-643
1-532
1-286
2-298
1-928
3-064
2-571
3-830
50
40i
0-763
0-646
1-526
1-292
2-290
1-938
3-053
2-584
3-816
49*
40
0-760
0-649
1-521
1-299
2-281
1-948
3-042
2-598
3-802
49*
40f
0-758
0-653
1-515
1-306
2-273
1-958
3-030
2-611
3-788
49*
41
0755
0-656
1-509
1-312
2-264
1-968
3019
2-624
3-774
49
41i
0-752
0-659
1-504
1-319
2-256
1978
3-007
2-637
3-759
48*
41*
0-749
0-663
1-498
1-325
2-247
1-988
2-996
2-650
3-745
48*
41|
0-746
0-666
1-492
1-332
2-238
1-998
2-984
2-664
3-730
48*
42*
0-743
0-669
1-486
1-338
2-229
2-007
2-973
2-677
3-716
48
42i
0-740
0-672
1-480
1-345
2-221
2-017
2-961
2-689
3-701
47*
42*
0-737
0-676
1-475
1-351
2-212
2-027
2-949
2-702
3-686
47*
42*
0-734
0-679
1-469
1-358
2-203
2-036
2-937
2-715
3-672
47*
43
0-731
0-682
1-463
1-364
2-194
2-046
2-925
2-728
3-657
4T*
43
0-728
0-685
1-457
1-370
2-185
2-056
2-913
2-741
3-642
46*
43*
0-725
0-688
1-451
1-377
2-176
2-065
2-901
2-753
3-627
46*
43|
0-722
0-692
1-445
1-383
2-167
2-075
2-889
2-766
3-612
46*
44
0-719
0-695
1-439
1-389
2-158
2-084
2-877
2-779
3-597
46*
44
0-716
0-698
1-433
1-396
2-149
2-093
2-S65
2-791
3-582
46*
44*
0-713
0-701
1-427
1-402
2-140
2-103 i
2-853
2-804
3-566
45i
44f
0-710
0-704
1-420
1-408
2-131
2-112
2-841
2-816 !
3-551
45i
45 f
0-707
1-707
1-414
1-414
2-121
2-121
2828
2-828
3-536
45
be
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
.5
]
1
9
3
4
ft
M
APPENDIX.
709
LATITUDES AND DEPARTURES.
sp
5
6
7
9
9
I
Dep.
Lat. Dep.
Lat.
Dep.
Lat.
Dep.
Lat.
Dep.
I
30
2-500
5-196
3-000
6-062
3-500
i 6-928
4-000
7-794
4-500
60
301
2-519
5-183
3-023
6-047
3-526
: 6-911
4-030
7-775
4534
59|
8o|
2-538
5'170
3-045
I 6-031
3-553
! 6-893
4-060
7-755
4-568
59*
30
2-556
5-156
3-068
6016
3-579
6-875
4-090
7735
4-602
591
31
2-575
5-143
3-090
6-000
3-605
6-857
4-120
7-715
4-635
59
sil
2-594
5-129
3 113
5-984
3-631
6-839
4-150
7-694
4-669
58f
811
2-612
5-116
3-135
5-968
3-657
6-821
4-180
7-674
4-702
58|
31f
2-631
5-102
3-157
5-952
3-683
6-803
4-210
7-653
4-736
581
32
2-650
5-088
3-180
5-936
3-709
6-784
4-239
7-632
4-769
58
321
2-668
5-074
3-202
5-920
3-735
6-766
4-269
7-612
4-802
57f
2J
2-686
5-060
3-224
5-904
3-761
6-747
4-298
7-591
4-836
57|
32f
2-705
5-046
3-246
5-887
3-787
6728
4-328
7-569
4-869
571
33
2-723
5-032
3"268
5-871
3-812
6-709
4-357
7-548
4-902
5T
331
2-741
5-018
3-290
5-854
8-838
6-690
4-386
7-527
4-935
56
33!
2-760
5-003
3-312
5-837
3-864
6-671
4-416
7-505
4-967
56!
33|
2-778
4-989
3333
5-820
3-889
6-652
4-445
7-483
5-000
56i
34
2-796
4-974
3355
6-803
3-914
6-632 ' 4-474
7-461
5-033
56
**i
2-814
4-960
3-377
5-786
3-940
i 6-613 4-502
7-439 5-065
55
34!
2-832
4-945
3-398
5-769
3-965
; 6-593 4-531
7'417 509*8
56!
34
2-850
4-930
3-420
5-752
8-990
i 6-573 4-560
7-396 5-130
551
35
2-868
4-915
3-441
5-734
4-015
6-553
4-589
7-372 5-162
55
351
2-886
4-900
3-463
5-716
4-040
6-533
4-617
7-350 5-194
54*
35!
2-904
4-885
3-484
5-699
4-065
6-513
4-646
7-327
5-226 54!
35J
2-921
4-869
3-505
5-681
4-090
6-493
4-674
7-304
5-268
41
36
2-939
4-854
3-527
5-663
4-115
6-472
4-702
7-281
5-290
54 f
361
2-957
4-839
3-548
5-645
4-139
6-452
4-730
7'2f8
5-322
53f
36^
2974
4-823
3-569
5-627
4-164
6-431
4-759
7-235
6-353
53!
36f
2-992
4-808
3-590
5-6<>9
4-188
6-410
4-787
7-211
5-385
3!
3T
3-009
4-792
3-611
5-590
4-213
6-389
4-815
7-188
5-416
53
371
3-026
4-776
3-632
5-572
4-237
6-368
4-842
7-164
5-448
52f
37!
3-044
4-760
3-653
5-554
4-261
6-347
4-870
7-140
5-479
2!
87|
3-061
4-744
3-673
5-535
4-286
6-326
4-898
7-116
5-510
621
38
3-078
4-728
3-694
5-516
4-310
6-304
4-925
7-092
5-541
52
381
3-095
4-712
3-715
5-497
4-334
6-283
4-953
7-068
5-572
61*
38'
3-113
4-696
3-735
5-478
4-358
6-261
4-980
7-043
5-603
51!
38f
3-130
4-679
3-756
5-459
4-381
6-239
5-007
7-019
5-633
511
39 3
3-147
4-663
3-776
5-440
4-405
6-217
6-035
6994
5-664
51
391
3-164
4-646
3-796
5-421
4-429
6-195
6-062
6-970
6-694
50f
39!
3-180
4-630
3-816
5-401
4-453
6-173
6-089
6-945
5-725
50!
39|
3-197
4-613
3-837
5-382
4-476
6-151
5-116
6-920
5-755
501
40 3-214
4-596
3-857
6-362
4-500
6-128
5-142
6-894
5-785
50
401
3-231
4-579
3-877
5-343
4-523
6-106
5-169
6-869
5-816 i 49
40!
3-247
4-562
3-897
6-323
4-546
6-083
5-196
6-844
5-845 W
40|
3-264
4-545
3-917
5-303
4-569
6-061
6-222
6-818
5-875
491
41
3-280
4-528
3-936
6-283
4-592
6-038
5-248
6-792
5-905
49 f
411
3-297
4-511
3-956
6-263
4-615
6-015
5-276
6-767
5-934
48f
41!
3313
4-494
3-976
6-243
4-638
6-992
6-301
6-741
5-964
48!
41*
3-329
4-476
3-995
5-222
4-661
5-968
6-327
6-715
5-993
481
42
3-346
4-459
4-015
5-202
4-684
5-945
5-363
6-688
6-022
48 f
421
3-362
4-441
4-034
6-182
4-707
5-922
5-379
6-662
6-051
47
42!
3-378
4-424
4-054
6-161
4-729
5-898
5-405
6-635
6'080
47!
42f
3-394
4-406
4-073
6-140
4-752
5-875
6-430
6-609
6-109 !
471
43
3-410
4-388
4-092
5-119
4-774
5-851
6-456 i
6-582
6-138 i
47
431
3-426
4-370
4-111
5-099
4-796
5-827
5-481 !
6-555
6167
46f
43!
3-442
4-352
4-130
6-078
4-818
5-803
5-507
6-628
6-195
46!
43|
3-458
4-334
4-149
5-057
4-841
5-779
6-532
6-501
6-224
461
44
3-473
4-316
4-168
5-035
4-863
5-755
5-557
6-474
6-252
46
441
3-489
4-298
4-187
5-014
4-885
5-730
5-582
6-447
6-280
45f
44!
3-505
4-280
4-206
4-993
4-906
5-706
5-607
6-419
6-308
45!
44|
3-520
4-261
4-224
4-971
4-928
5-681
6-632
6-392
6-336 1
451
45
3-536
4-243
4-243
4.950
4-950
6-657
5-657
6-364
6-364
45
*
Lat.
Dep.
Lat.
Dep. Lat.
Dep.
Lat.
Dep.
Lat.
1
5
6
7
8
9
i
710
APPENDIX.
NATURAL, SINES AND COSINES.
1
rj^Q
3
40
/
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine..
Cosine.
Sine.
Cosine.
r
00000
Unit.
01745
99985
03490
99939
05234
99863
06976
99756
60
1
00029
Unit.
01774
99984
03519
99938
05263
99861
07005
99754
59
2
00058
Unit.
01803
99984
03548
99937 i
05292
99860
07034
99752
58
3
00087
Unit.
01832
99983
03577
99936
05321
99858
07063
99750
57
4
00116
Unit.
01862
99983
03606
99935
05350
99857
07092
99748
56
5
00145
Unit.
01891
99982
03635
99934
05379
99855
07121
99746
55
6
00175
Unit.
01920
99982
03664
99933
05408
99854 >
07150
99744
54
7
00204
Unit.
01949
99981
03693
99932
05437
99852
07179
99742
53
8
00233
Unit.
01978
99980 !
03723
99931
05466
99851 |
07208
99740
52
9
00262
Unit.
02007
99980
03752
99930
05495
99849
07237
99738
51
10
00291
Unit.
02036
99979
03781
99929
05524
99847
07266
99736
50
11
00320
99999
02065
99979
03810
99927
05553
99846
07295
99734
49
12
00349
99999
02094
99978
03839
99926
05582
99844
07324
99731
48
13
00378
99999
02123
99977
03868
99925
05611
99842
07353
99729
47
14
00407
99999
02152
99977
03897
99924
05640
99841
07382
99727
46
15
00436
99999
02181
99976
03926
99923
05669
99839
07411
99725
45
16
00465
99999
02211
99976
03955
99922
05698
99838
07440
99723
44
17
00495
99999
02240
99975
03984
99921
05727
99836
07469
99721
43
18
00524
99999
02269
99974
04013
99919
05756
99834
07498
99719
42,
19
00553
99998
02298
99974
04042
99918
05785
99833
07527
99716
41
20 00582
99998
02327
99973
04071
99917
05814
99831
07556
99714
40
21 00611
99998
02356
99972
04100
99916
05844
99829
07585
99712
39
22
00640
99998
02385
99972
04129
99915
05873
99827
07614
99710
38
23
00669
99998
02414
99971 i
04159
99913
05902
99826
07643
99708
37
24
00698
99998
02443
99970 i
04188
99912
05931
99824
07672
99705
3d
25
00727
99997
02472
99969
04217
99911
05960
99822
07701
99703
35
26
00756
99997
02501
99969
04246
99910
05989
99821
07730
99701
34
27
00785
99997
02530
99968
04275
99909
06018
99819
07759
99699
33
28
00814
99997
02560
99967
04304
99907
06047
99817
07788
99696
32
29
00844
99996
02589
99966
04333
99906
06076
99815
07817
99694
31
30
00873
99996
02618
99966
04362
99905
06105
99813
07846
99692
30-
31
00902
99996
02647
99065
04391
99904
06134
99812
07875
99689
29
32
00931
99996
02676
99964
04420
99902
06163
99810
07904
99687
2&
33
00960
99995
02705
99963
044-19
99901
06192
99808
07933
99685
27
34
00989
99995
02734
99963
04478
99900
06221
99806
07962
99683
26
35
01018
99995
02763
99962
04507
99898
06250
99804
07991
99680
25
36
01047
99995
02792
99961
04536
99897
06279
99803
08020
99678
24
37
01076
99994
02821
99960 ;
04565
99896
06308
99801
08049
99676
23
38
01105
99994
02850
99959
04594
99894
06337
99799
08078
99673
22.
39
01134
99994
02879
99959
04623
99893
06366
99797
08107
99671
21
40
01164
99993
02908
99958
04653
99892
06395
99795
08136
99668
20
41
01193
99993
02938
99957
04682
99890
06424
99793
08165
99666
19
42
01222
99993
02967
99956
04711
99889
06453
99792
08194
99664
18
43
01251
99992
02996
99955 :
. 04740
99888
06482
99790
08223
99661
17
44
01280
99992
03025
99954
04769
99886
06511
99788 i
08252
99659
16
45
01309
99991
03054
99953
04798
99885
06540
99786
08281
99657
15
46
01338
99991
03083
99952
04827
99883
06569
99784
08310
99654
14
47
01367
99991
03112
99952
04856
99882
06598
99782
08339
99652
13
48
01396
99990
03141
99951
04885
99881
06627
99780
08368
99649
12:
49
01425
99990
03170
99950
04914
99879
06656
99778
08397
99647
11
50
01454
99989
03199
99949
04943
99878
06685
99776
08426
99644
10
51
01483
99989
03228
99948
04972
99876
06714
99774
08455
99642
9
52
01513
99989
03257
99947
05001
99875
06743
99772
08484
99639
8
53
01542
99988 03286
99946
05030
99873
06773
99770
08513
99637
7
54
01571
99988 03316
99945
05059
99872
06802
99768
08542
99635
6
55
01600
99987 i 03345
99944 I 05088
99870
06831
99766
08571
99632
5
56
01629
99987
03374
99943
05117
99869
06860
99764
08600
99630
4
57
01658
99986
03403
99942
05146
99867
06889
99762
08629
99627
3
58
01687
99986
03432
99941
05175
99866
06918
99760
08658
99625
2
59
01716
99985
03461
99940
05205
99864
06947
99758
08687
99622
1
60
01745
99985
03490
99939
05234
99863
06976
99756
08716
99619
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
/
80
88
87
86
85
/
APPENDIX.
711
NATURAL. SIXES AND COSINES.
>
C
J
9
ro
*
)
/
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
/
08716
99619
10453
99452
12187
99255
13917
99027
15643
98769
60
1
08745
99617
10482
99449
12216
99251
13946
99023
15672
98764
59
2
08774
99614
10511
99446
12245
99248
13975
99019
15701
98760
58
3
08803
99612
10540
99443
12274
99244
14004
99015
15730
98755
57
4
08831
99609
10569
99440
12302
99240
14033
99011
15758
98751
56
5
08860
99607
10597
99437
12331
99237
14061
99006
15787
98746
55
6
08889
99604
10626
99434
12360
99233
14090
99002
15816
98741
54*
7
08918
99602
10655
99431
12389
99230
14119
98998
15845
98737
53
8
08947
99599
10684
99428
12418
99226
14148
98994
15873
98732
52
9
08976
99596
10713
99424
12447
99222
14177
98990
15902
98728
51
10
09005
99594
10742
99421
12476
99219
14205
98986
15931
98723
50
11
09034
99591
10771
99418
12504
99215
14234
98982
15959
98718
49
12
09063
99588
10800
99415
12533
99211
14263
98978
15988
98714
48
13
09092
99586
10829
99412
12562
99208
14292
98973
16017
98709
47
14
09121
99583
10858
99409
12591
99204
14320
98969
16046
98704
46
15
09150
99580
10887
99406
12620
99200
14349
98965
16074
98700
45
16
09179
99578
10916
99402
12649
99197
14378
98961
16103
98695
44
17
09208
99575
10945
99399
12678
99193
14407
98957
16132
98690
43
18
09237
99572
10973
99396
12706
99189
14436
98953
16160
98686
42
19
09266
99570
11002
99393
12735
99186
14464
98948
16189
98681
41
20
09295
99567
11031
99390
12764
99182
14493
98944
16218
98676
40
21
09324
99564
11060
99386
12793
99178 !
14522
98940
16246
98671
39
22
09353
99562
11089
99383
12822
99175 i
14551
98936
16275
98667
38
23
09382
99559
11118
99380
12851
99171
14580
98931
16304
98662
37
24
09411
99556
11147
99377
12880
99167
14608
98927
16333
98657
36
25
09440
99553
11176
99374
12908
99163
14637
98923
16361
98652
35
26
09469
99551
11205
99370
12937
99160
14666
98919
16390
98648
34
27
09498
99548
11234
99367
12966
99156
14695
98914
16419
98643
33
28
09527
99545
11263
99364
12995
99152
14723
98910
16447
98638
32
29
09556
99542
11291
99360
13024
99148 i
14752
98906
16476
98633
31
30
09585
99540
11320
99357
13053
99144
14781
98902
16505
98629
30
31
09614
99537
11349
99354
13081
99141
14810
98897
16533
98624
29
32
09642
99534
11378
99351
13110
99137
14838
98893
16562
98619
28
33
09671
99531
11407
99347
13139
99133
14867
98889
16591
98614
27
34
09700
99528
11436
99344
13168
99129
14896
98884
16620
98609
26
35
09729
99526
11465
99341 !
13197
99125
14925
98880
16648
98604
25
36
09758
99523
11494
99337 t
13226
99122
14954
98876
16677
98600
24
37
09787
99520
11523
99334
13254
99118
14982
98871
16706
98595
23
38
09816
99517
11552
99331
13283
99114
15011
98867
16734
98590
22
39
09845
99514
11580
99327
13312
99110
15040
98863
16763
98585
21
40
09874
99511
11609
99324
13341
99106
15069
98858
16792
98580
20
41
09903
99508
11638
99320
13370
99102 |
15097
98854
16820
98575
19
42
09932
99506
11667
99317
13399
99098
15126
98849
16849
98570
18
43
09961
99503
11696
99314
13427
99094
15155
98845
16878
98565
17
44
09990
99500
11725
99310
13456
99091
15184
98841
16906
98561
16
45
10019
99497
11754
99307
13485
99087
15212
98836
16935
98556
15
46
10048
99494
11783
99303
13514
99083
15241
98832
16964
98551
14
47
10077
99491
11812
99300 i
13543
99079
15270
98827
16992
98546
13
48
10106
99488
11840
99297
13572
99075
15299
98823
17021
98541
12
49
10135
99485
11869
99293
13600
99071 !
15327
98818
17050
98536
11
50
10164
99482
11898
99290
13629
99067
15356
98814
17078
98531
10
51
10192
99479
11927
99286
13658
99063
15385
98809
17107
98526
9
52
10221
99476
11956
99283
13687
99059
15414
98805
17136
98521
8
53
10250
99473
11985
99279
13716
99055
15442
98800
17164
98516
7
54
10279
99470
12014
99276
13744
99051
15471
98796
17193
98511
6
55
10308
99467
12043
99272
13773
99047
15500
98791
17222
98506
5
56
10337
99464
12071
99269
13802
99043
15529
98787
17250
98501
4
57
10366
99461
12100
99265
13831
99039 |
15557
98782
17279
98496
3
58
10395
99458
12129
99262
13860
99035 !
15586
98778
17308
98491
2
59
10424
99455
12158
99258
13889
99031
15615
98773
17336
98486
1
60
10453
99452
12187
99255
13917
99027
15643
98769
17365
98481
t
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
8^
1
&
i
&
3
81
L
8<
>
/
712
APPENDIX.
NATURAL. SINES AND COSINES.
10
11
13
13
14
/
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine. Sine.
Cosine.
r
17365
98481
19081
98163
20791
97815
22495
97437
24192
97030
60
I
17393
98476
19109
98157
20820
97809
22523
97430
24220
97023
59
2
17422
98471 !
19138
98152
20848
97803
22552
97424 ,
24249
97015
58
3
17451
98466 i
19167
98146
20877
97797
22580
97417
24277
97008
57
4
17479
98461
19195
98140
20905
97791
22608
97411
24305
97001
56
5
17508
98455
19224
98135
20933
97784
22637
97404
24333
96994
55
6
17537
98450
19252
98129
20962
97778
22665
97398
24362
96987
54
7
17565
98445
19281
98124
20990
97772
22693
97391
24390
96980
53
8
17594
98440
19309
98118
21019
97766
22722
97384
24418
96973
52
9
17623
98435
19338
98112
21047
97760
22750
97378
24446
96966
51
10
17651
98430
19366
98107
21076
97754
22778
97371
24474
96959
50
11
17680
98425
19395
98101
21104
97748
22807
97365
24503
96952
49
12
17708
98420
19423
98096
21132
97742
22835
97358
24531
96945
48
13
17737
98414
19452
98090
21161
97735
22863
97351
24559
96937
47
14
17766
98409
19481
98084
21189
97729
22892
97345
24587
96930
46
15
17794
98404
19509
98079
21218
97723
22920
97338
24615
96923
45
16
17823
98399
19538
98073
21246
97717
22948
97331
24644
96916
44
17
17852
98394
19566
98067
21275
97711
22977
97325
24672
96909
43
18
17880
98389
19595
98061
21303
97705
23005
97318
24700
96902
42
19
17909
98383
19623
98056
21331
97698
23033
97311
24728
96894
41
20
17937
98378
19652
98050
21360
97692
' 23062
97304 24756
96887
40
21
17966
98373
19680
98044
21388
97686
: 23090
97298 24784
96880
39
22
17995
98368
19709
98039
21417
97680
23118
97291 24813
96873
38
23
18023
98362
19737
98033
21445
97673
23146
97284 24841
96866
37
24
18052
98357
19766
98027
21474
97667
23175
97278 24869
96858
36
25
18081
98352
19794
98021
21502
97661
23203
97271 : 24897
96851
35
26
18109
98347
19823
98016
21530
97655
23231
97264
24925
96844
34
27
18138
98341
19851
98010
21559
97648
23260
97257
24954
96837
33
28
18166
98336
19880
98004
21587
97642
23288
P7251
24982
96829
32
29
18195
98331
19908
97998
21616
97636
23316
97244
25010
96822
31
30
18224
98325
19937
97992
21644
97630
23345
97237
25038
96815
30
31
18252
98320
19965
97987
21672
97623
23373
97230
25066
96807
29
32
18281
98315
19994
97981
21701
97617
23401
97223
25094
96800
28
33
18309
98310
20022
97975
21729
97611
23429
97217
25122
96793
27
34
18338
98304
20051
97969
21758
97604
23458
97210
25151
96786
26
35
18367
98299
20079
97963
21786
97598
23486
97203
25179
96778
25
36
18395
98294
20108
97958
21814
97592
23514
97196
25207
96771
24
37
18424
98288
20136
97952
21843
97585
23542
97189
25235
96764
23
38
18452
98283
20165
97946
21871
97579
23571
97182
25263
96756
22
39
18481
98277
20193
97940
21899
97573
23599
97176
25291
96749
21
40
18509
98272
20222
97934
21928
97566
23627
97169
25320
96742
20
41
18538
98267
20250
97928
21956
97560
23656
97162
25348
96734
19
42
18567
98261
20279
97922
21985
97553
23684
97155
25376
96727
18
43
18595
98256
20307
97916
22013
97547
23712
97148
25404
96719
17
44
18624
98250
20336
97910
22041
97541
23740
97141
25432
96712
16
45
18652
98245
20364
97905
22070
97534
23769
97134
25460
96705
15
46
18681
98240
20393
97899
22098
97528
23797
97127
25488
96697
14
47
18710
98234
20421
97893
22126
97521
23825
97120
25516
96690
13
48
18738
98229
20450
97887
22155
97515 i 23853
97113
25545
96682
12
49
18767
98223
20478
97881
22183
97508
23882
97106
25573
96675
11
50
18795
98218
20507
97875
22212
97502
23910
97100
25601
96667
10
51
18824
98212
20535
97869 1 22240
97496
23938
97093
25629
96660
9
52
18852
98207
20563
97863 i 22268
97489
23966
97086
25657
96653
8
53
18881
98201
20592
97857 22297
97483
23995
97079
25685
96645
7
54
18910
98196
20620
97851
22325
97476
24023
97072
25713
96638
6
55
18938
98190
20649
97845
_>:55:j
97470
24051
97065
25741
96630
5
56
18967
98185 20677
97839
22382
97463
24079
97058
25769
96623
4
57
18995
98179
20706
97833
22410
97457
24108
97051
25798
96615
3
58
19024
98174
20734
97827
22438
97450
24136
97044
25826
96608
2
59
19052
98168
20763
97821
22467
97444
24164
97037
25854
96600
1
60
19081
98163
20791
97815 22495
97437
24192
97030
25882
96593
Cosine.
Sin,..
Cosine.
Sine. Cosine.
Sine. Cosine.
Sine.
Cosine.
Sine.
/
7O
T8 o 77-0 Ij 76
75
/
APPENDIX.
713
NATURAL, SINKS AND COSINES.
15
ie
17
18
1O
r
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
/
25882
96593
27564
96126
29237
95630
30902
95106
32557
94552
60
1
25910
96585
27592
96118 || 29265
95622
30929
95097
32584
94542
59
2
25938
96578
27620
96110 29293
95613
30957
95088
32612
94533
58
3
25966
96570
27648
96102
29321
95605
30985
95079
32639
94523
57
4
25994
96562
27676
96094
29348
95596
, 31012
95070
32667
94514
56
5
26022
96555
27704
96086
29376
95588
31040
95061 | 32694
94504
55
6
26050
96547
27731
96078
29404
95579
31068
95052
32722
94495
54
7
26079
96540
27759
96070
29432
95571
31095
95043
32749
94485
53
8
26107
96532
27787
96062
29460
95562
31123
95033
32777
94476
52
9
26135
96524
27815
96054
29487
95554
31151
95024
32804
94466
51
10
26163
96517
27843
96046
29515
95545
31178
95015
32832
94457
50
11
26191
96509
27871
96037
29543
95536
31206
95006
32859
94447
49
12
26219
96502
27899
96029
29571
95528
1 31233
94997
32887
94438
48
13
26247
96494
27927
96021
29599
95519 1 31261
94988 1 32914
94428
47
14
26275
96486
27955
96013
29626
95511 31289
94979 |j 32942
94418
46
15
26303
96479
27983
96005
29654
95502
31316
94970 32969
94409
45
16
26331
96471
28011
95997
29682
95493
31344
94961 32997
94399
44
17
26359
96463
28039
95989
29710
95485
31372
94952 |! 33024
94390
43
18
26387
96456
28067
95981
29737
95476 ; 31399
94943
33051
94380
42
19
26415
96448
28095
95972
29765
95467 : 31427
94933
33079
94370
41
20
26443
96440
28123
95964
29793
95459 j; 31454
94924
33106
94361
40
21
26471
96433
28150
95956
29821
95450
I 31482
94915
33134
94351
39
22
26500
9tf425 I 28178
95948
29849
95441
31510
94906
33161
94342 38
23
26528
96417 28206
95940 !
29876
95433
31537
94897
33189
94332 37
24
26556
96410 | 28234
95931
29904
95424 i 31565
94888
33216
94322 36
25
26584
96402 l \ 28262
95923
29932
95415 ' 31593
94878
33244
94313 35
26
26612
96394 !i 28290
95915
29960
95407 31620
94869
33271
94303
34
27
26640
96386 ! 28318
95907
29987
95398 31648
94860
33298
94293
33
28
26668
96379 j 28346
95898
30015
95389 31675
94851
33326
94284
32
29
26696
96371 | 28374
95890
30043
95380
31703
94842
33353
94274
31
30
26724
96363 j 28402
95882
30071
95372 1 31730
94832
33381
94264
30
31
26752
96355 28429
95874
30098
"95363 ! 31758
94823
33408
94254
29
32
26780
96347 28457
95865
30126
95354 31786
94814
33436
94245
28
33
26808
96340 28485
95857
30154
95345 :i 31813
94805
33463
94235
27
34
26836
96332 28513
95849
30182
95337 i 31841
94795
33490
94225
26
35
26864
96324 ; 28541
95841
30209
95328 31868
94786
33518
94215
25
36
26892
96316 28569
95832
30237
95319 31896
94777
33545
94206
24
37
26920
96308 28597
95824 l| 30265
95310 31923
94768
33573
94196
23
38
26948
96301 28625
95816
30292
95301 31951
94758
33600
94186
22
39
26976
96293
! 28652
95807
30320
95293 31979
94749
33627
94176
21
40
27004
96285
28680
95799
30348
95284
32006
94740
33655
94167
20
41
27032
96277
28708
95791
30376
95275
32034
94730
33682
94157
19
42
27060
96269
28736
95782 i 30403
95266
32061
94721
33710
94147
18
43
27088
96261
28764
95774
30431
95257
32089
94712
33737
94137
17
44
27116
96253
28792
95766
30459
95248
32116
94702
33764
94127
16
45
27144
96246
28820
95757
30486
95240
32144
94693
33792
94118
15
46
27172
96238
28847
95749
30514
95231
32171
94684
33819
94108
14
47
27200
96230
! 28875
95740
30542
95222
' 32199
94674
33846
94098
13
48
27228
96222 ! 28903
95732
30570
95213
32227
94665
33874
94088
12
49
27256
96214 > 28931
95724
30597
95204 || 32254
94656
33901
94078
11
50
27284
96206
28959
95715
30625
95195 : 32282
946-46
33929
94068
10
51
27312
96198
28987
95707
30653
95186 jj 32309
94637
33956
94058
9
52
27340
96190 29015
95698
30680
95177 32337
94627
33983
94049
8
53
27368
96182 29042
95690
30708
95168 32364
94618
34011
94039
7
54
27396
96174 j 29070
95681 | 30736
95159 32392
94609 i 34038
94029
6
55
27424
96166 | 29098
95673 30763
95150 32419
94599 34065
94019
5
56
27452
96158 29126
95664 30791
95142 32447
94590 i 34093
94009 4
57
27480
96150 29154
95656 I! 30819
95133 32474
94580 34120
93999
3
58
27508
96142 29182
95647 : ' 30846
95124 32502
94571
34147
93989
2
59
27536
96134
29209
95639 30874
95115
i 32529
94561
34175
93979
1
60
27564
96126
29237
95630 30902
95106
32557
94552
34202
93969
Cosine.
Sine.
Cosine.
Sine. Cosine.
Sine. Cosine.
Sine.
Cosine.
Sine.
/
7-4,0
7-30 | 730 || T-XO
7O
/
714:
APPENDIX.
NATURAL, SINES AND COSINES.
30
31
33
33
34
/
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
/
34202
93969
35837
93358
37461
92718
39073
92050
40674
91355
60
1
34229
93959
35864
93348
37488
92707
39100
92039
40700
91343
59
2
34257
93949
35891
93337
37515
92697
39127
92028
40727
91331
58
3
342S4
93939
35918
93327
37542
92686
39153
92016
40753
91319
57
4
34311
93929
35945
93316
37569
92675
39180
92005
40780
91307
56
5
34339
93919
35973
93306
37595
92664
39207
91994
40806
91295
55
6
34366
93909
36000
93295
37t>22
92653
39234
91982
40833
91283
54
7
34393
93899
36027
93285
37649
92642
39260
91971
40860
91272
53
8 34421
93889
36054
93274
37676
92631
39287
91959
40886
91260
52
9
34448
93879
36081
93264
37703
92620
39314
91948
40913
91248
51
10
34475
93869
36108
93253
37730
92609
39341
91936
40939
91236
50
11
34503
93859
36135
93243
37757
92598
39367
91925
40966
91224
49
12
34530
93849
36162
93232
37784
92587
39394
91914
40992
91212
48
13
34557
93839
36190
93222
37811
92576
39421
91902
41019
91200
47
14
34584
93829
36217
93211
37838
92565
39448
91891
41045
91188
46
15
34612
93819
36244
93201
37865
92554
39474
91879
41072
91176
45
16
34639
93809
36271
93190
37892
92543
39501
91868
41098
91164
44
17
34666
93799
36298
93180
37919
92532
39528
91856
41125
91152
43
18
34694
93789
36325
93169
i 37946
92521
39555
91845
41151
91140
42
19
34721
93779
36352
93159
37973
92510
39581
91833
41178
91128
41
20
34748
93769
36379
93148
37999
92499
39608
91822
41204
91116
40
21
34775
93759
36406
93137
38026
92488
39635
91810
41231
91104
39
22
34803
93748
36434
93127
38053
92477
39661
91799
41257
91092
38
23
34830
93738
36461
93116
38080
92466
39688
91787
41284
91080
37
24
34857
93728
36488
93106
38107
92455
39715
91775
41310
91068
36
25
34884
93718
36515
93095
38134
92444
39741
91764
41337
91056
35
26
34912
93708
36542
93084
38161
92432
39768
91752
41363
91044
34
27
34939
93698
36569
93074
38188
92421
39795
91741
41390
91032
33
28
34966
93688
36596
93063
38215
92410
39822
91729
41416
91020
32
29
34993
93677
36623
93052
38241
92399
39848
91718
41443
91008
31
30
35021
93667
36650
93042
38268
92388
39875
91706
41469
90996
30
31
35048
93657
36677
93031
38295
92377
39902
91694
41496
90984
29
32
35075
93647
36704
93020
38322
92366
39928
91683
41522
90972
28
33
35102
93637
36731
93010
38349
92355
39955
91671
41549
90960
27
34
35130
93626
36758
92999
38376
92343
39982
91660
41575
90948
26
35
35157
93616
36785
92988
38403
92332
40008
91648
! 41602
90936
25
36
35184
93606
36812
92978
38430
92321
40035
91636
] 41628
90924
24
37
35211
93596
36839
92967
38456
92310
40062
91625
41 655
90911
23
38
35239
93585
36867
92956
38483
92299
40088
91613
41681
90899
22
39
35266
93575
36894
92945
38510
92287
40115
91601
41707
90887
21
40
35293
93565
36921
92935
38537
92276
40141
91590
41734
90875
20
41
35320
93555
36948
92924
38564
92265
40168
91578
41760
90863
19
42
35347
93544
36975
92933
38591
92254
40195
91566
41787
90851
18
43
35375
93534
37002
92902
38617
92243
40221
91555
41813
90839
17
44
35402
93524
37029
92892
38644
92231 j 40248
91543
41840
90826
16
45
35429
93514
37056
92881
38671
92220
40275
91531
41866
90814
15
46
35456
93503
37083
92870
38698
92209
40301
91519
41892
90802
14
47
35484
93493
37110
92859 38725
92198
40328
91508
41919
90790 13
48
35511
93483
37137
92849 38752
92186 40355
91496
41945
90778 1 -1
49
35538
93472
37164
92838 1 38778
92175
40381
91484
41972
90766
11
60
35565
93462
37191
92827 1 38805
92164
40408
91472
41998
90753
10
51
35592
93452
37218
92816
38832
92152
40434
91461
42024
90741
9
52
35619
93441
37245
92805
38859
92141
40461
91449
42051
90729
8
53
35647
93431
37272
92794
38886
92130
40488
91437
42077
90717
7
54
35674
93420
37299
92784
38912
92119
40514
91425
42104
90704
6
55
35701
93410
37326
92773 i 38939
92107
40541
91414
42130
90692
5
56
35728
93400
37353
92762 1 38966
92096
40567
91402
42156
90680
4
57
35755
93389
37380
92751
38993
92085
40594
91390
42183
90668
3
58
35782
93379
37407
92740
39020
92073
40621
91378
42209
90655
2
59
35810
93368
37434
92729
39046
92062
40647
91366
42235
90643
1
60
35837
93358
37461
92718
39073
92050
40674
91355
42262
90631
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Stoe.
Cosine.
Sine.
Cosine.
Sine.
/
eo
68
G7
GG
015
/
APPENDIX.
715
NATURAL. SINES AND COSINES.
35
30
37 38 39
/
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
/
42262
90631 i 43837
89879
45399
89101
46947
88295
48481
87462
60
I
42288
90618 ' 43863
89867
45425
89087 !
46973
88281
48506
87448
59
2
42315
90606 : 43889
89854
45451
89074
46999
88267
48532
87434 ! 58
3
42341
90594 43916
89841
45477
89061
47024
88254 j 48557
87420
57
4
42367
90582 43942
89828
45503
89048
47050
88240
48583
87406
56
5 42394
90569 j 43968
89816
45529
89035
47076
88226
48608
87391 55
6 ! 42420
90557 ! 43994
89803
45554
89021
47101
88213
48634
87377 54
7
42446
90545
44020
89790
45580
89008
47127
88199
48659
87363
53
8
42473
90532
44046
89777
45606
88995
47153
88185
48684
87349
52
9
42499
90520
44072
89764
45632
88981
47178
88172
48710
87335 51
10
42525
90507
44098
89752
45658
88968
47204
88158
48735
87321 50
11
42552
90495
44124
89739
45684
88955
47229
88144
48761
87306 49
12
42578
90483
44151
89726
45710
88942 ;
47255
88130
48786
87292 48
13
42604
90470
44177
89713
45736
88928
47281
88117
48811
87278 47
14
42631
90458
44203
89700
45762
88915
47306
88103
48837
87264
46
15
42657
90446
44229
89687
45787
88902
47332
88089
48862
87250
45
16
42683
90433
44255
89674
45813
88888
47358
88075
48888
87235
44
17
42709
90421
44281
89662
45839
88875 :
47383
88062 ! 48913
87221
43
18
42736
90408
44307
89649
45865
88862 47409
88048 ! 48938
87207
42
19
42762-
90396
44333
89636 ! 45891
88848 47434
88034 :j 48964
87193
41
20
42788
90383
44359
89623
45917
88835 i 47460
88020 i! 48989
87178
40
21
42815
90371
44385
89610
45942
88822 i 47486
88006 ! 49014
87164
39
22
42841
90358
44411
89597
45968
88808 47511
87993 ! 49040
87150
38
23
42867
90346
44437
89584
45994
88795 47537
87979 I 49065
87136
37
24
42894
90334
44464
89571
46020
88782
47562
87965 -i 49090
87121
36
25
42920
90321 ;
44490
89558 !
46046
88768
47588
87951 ;
49116
87107
35
26
42946
90309
44516
89545 1 46072
88755 :
47614
87937
49141
87093
34
27
42972
90296
44542
89532 :! 46097
88741 | 47639
87923
49166
87079 33
28
42999
90284
44568
89519 46123
88728
47665
87909
49192
87064 32
29
43025
90271
44594
89506 : 46149
88715 :
47690
87896
49217
87050 31
30
43051
90259
44620
89493 j 46175
88701
47716
87882 jj 49242
87036 | 30
31
43077
90246
44646
89480 II 46201
88688
47741
87868 ! 49268
87021
29
32
43104
90233
44672
89467 :
46226
88674
47767
87854 i 49293
87007 28
33
43130
90221
44698
89454
46252
88661
47793
87840 | 49318
86993 27
34
43156
90208
44724
89441
46278
88647
47818
87826 jl 49344
86978 26
35
43182
90196
44750
89428
46304
88634
47844
87812 ! 49369
86964 ! 25
36
43209
90183
44776
89415
46330
88620
47869
87798 i 49394
86949 24
37
43235
90171 !
44802
89402
46355
88607
47895
87784 1 49419
86935
23
38
43261
90158 i
44828
89389 ;
46381
88698 47920
87770 !
49445
86921
22
39
43287
90146
44854
89376
46407
88580 47946
87756 i
49470
86906
21
40
43313
90133
44880
89363 !
46433
88566 ! 47971
87743
49495
86892
20
41
43340
90120
44906
89350 46458
88553 47997
87729
49521
86878
19
42
43366
90108
44932
89337 46484
88539 48022
87715 !
49546
86863
18
43
43392
90095
44958
89324
46510
88526
48048
87701
49571
86849
17
44
43418
90082
44984
89311
46536
88512
48073
87687
49596
86834
16
45
43445
90070
45010
89298
46561
88499
48099
87673
49622
86820
15
46
43471
90057 I 45036
89285
46587
88485
48124
87659 i
49647
86805
14
47
43497
90045
45062
89272 46613
88472 '
48150
87645
49672
86791
13
48
43523
90032 ;
45088
89259
46639
88458
48175
87631
49697
86777
12
49
43549
90019
45114
89245
46664
88445
48201
87617
49723
86762
11
50
43575
90007
45140
89232
46690
88431 i
48226
87603
49748
86748
10
51
43602
89994
45166
89219
46716
88417
48252
87589
49773
86733
9
52
43628
89981
45192
89206
46742
88404
48277
87575
49798
86719
8
53
43654
89968
45218
89193
46767
88390
48303
87561
49824
86704
7
54
43680
89956
45243
89180
46793
88377 i
48328
87546
49849
86690
6
55
43706
89943 !
45269
89167
46819
88363
48354
87532
49874
86675
5
56
43733
89930
45295
89153 !
46844
88349 j
48379
87518
49899
86661
4
57
43759
89918
45321
89140
46870
88336
48405
87504
49924
86646
3
58
43785
89905
45347
89127
46896
88322
48430
87490
49950
86632
2
59
43811
89892
45373
89114
46921
88308
48456
87476
49975
86617
1
60
43837
89879
45399
89101
46947
88295
48481
87462
50000
86603
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine. I
Cosine.
Sine.
/
G4,o
G3 63
01
G0
f
716
APPENDIX.
NATURAL, SINES AND COSINES.
30
31
33
33
34
'
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
7
50000
86603
51504
85717
52992
84805
54464
83867
55919
82904
60
1
50025
86588
51529
85702
53017
84789
54488
83851
55943
82887
59
2
50050
86573
51554
85687
53041
84774
54513
83835
55968
82871
58
3
50076
86559
51579
85672
53066
84759
54537
83819
55992
82855
57
4
50101
86544
51604
85657
53091
84743
54561
83804 i 56016
82839
56
5
50126
86530
51628
85642
53115
84728
54586
83788 : 56040
82822
55
6
50151
86515
51653
85627
53140
84712
54610
83772 j 56064
82806
54
7
50176
86501
51678
85612
53164
84697
54635
83756 i 56088
82790
53
8
50201
86486
51703
85597
53189
84681
54659
83740 1 56112
82773
52
9
50227
86471
51728
85582
53214
84666
54683
83724 ; 56136
82757
51
10
50252
86457
51753
85567
53238
84650
54708
83708 ' 56160
82741
50
11
50277
86442
51778
85551
53263
84635
54732
83692
56184
82724
49
12
50302
86427
51803
85536
53288
84619
54756
83676
56208
82708
48
13
50327
86413
51828
85521
53312
84604
54781
83660 56232
82692
47
14
50352
86398
51852
85506
53337
84588
54805
83645
56256
82675
46
15
50377
86384
51877
85491
53361
84573
54829
83629
56280
82659
45
16
50403
86369
51902
85476
53386
84557
54854
83613
56305
82643
44
17
50428
86354
51927
85461
53411
84542
54878
83597 1 56329
82626
43
18
50453
86340
51952
85446 i
53435
84526
54902
83581 i 56353
82610
42
19
50478
86325
51977
85431
53460
84511
54927
83565 56377
82593
41
20
50503
86310
52002
85416
53484
84495
54951
83549
56401
82577
40
21
50528
86295
52026
85401
53509
84480 i 54975
83533
56425
82561
39
22
50553
86281
52051
85385
53534
84464 ; 54999
83517
56449
82544
38
23
50578
86266
52076
85370
53558
84448 55024
83501
56473
82528
37
24
50603
86251
52101
85355
53583
84433 h 55048
83485
56497
82S11
36
25
50628
86237
52126
85340
53607
84417
55072
83469
56521
82495
35
26
50654
86222
52151
85325
53632
84402
55097
83453
56545
82478
34
27
50679
86207
52175
85310
53656
84386
55121
83437
56569
82462
33
28
50704
86192
52200
85294 j
53681
84370
55145
83421
56593
82446
32
29
50729
86178
52225
85279
53705
84355
55169
83405
56617
82429
31
30
50754
86163
52250
85264
53730
84339
55194
83389
56641
82413
30
31
50779
86148
52275
85249
53754
84324
55218
83373
56665
82396
29
32
50804
86133
52299
85234
53779
84308
55242
83356
\ 56689
82380
28
33
50829
86119
52324
85218
53804
84292
55266
83340
56713
82363
27
34
50854
86104
52349
85203
53828
84277
55291
83324
56736
82347
26
35
50879
86089,! 52374
85188
53853
84261
55315
83308
56760
82330
25
36
50904
86074 I 52399
85173 !
53877
84245
55339
83292
56784
82314
24
37
50929
86059 52423
85157
53902
84230
55363
83276
56808
82297
23
38
50954
86045 52448
85142 !
53926
84214
55388
83260
: 56832
82281
22
39
50979
86030 1 52473
85127
53951
84198
55412
83244
56856
82264
21
40
51004
86015
52498
85112
53975
84182
55436
83228
56880
82248
20
41
51029
86000
52522
85096
54000
84167
55460
83212
56904
82231
19
42
51054
85985
52547
85081
54024
84151
55484
83195
56928
82214
18
43
51079
85970
52572
85066
54049
84135
55509
83179
56952
82198
17
44
51104
85956
52597
85051
54073
84120
55533
83163
56976
82181
16
45
51129
85941
52621
85035
54097
84104
55557
83147
57000
82165
15
46
51154
85926
52646
85020
54122
84088 1 55581
83131
57024
82148
14
47
51179
85911
52671
85005
54146
84072 55605
83115
57047
82132
13
48
51204
85896
52696
84989
54171
84057 55630
83098
57071
82115
12
49
51229
85881
52720
84974
54195
84041
55654
83082
57095
82098
11
50
51254
85866
52745
84959
54220
84025
55678
83066
57119
82082
10
51
51279
85851
52770
84943
54244
84009
55702
83050
57143
82065
9
52
51304
85836
52794
84928
54269
83994
55726
83034
57167
82048
8
53
51329
85821
52819
84913
54293
83978
55750
83017
57191
82032
7
54
51354
85806
52844
84897
54317
83962
55775
83001
57215
82015
6
55
51379
85792
52869
84882
54342
83946
55799
82985
57238
81999
5
56
51404
85777
52893
84866
54366
83930 55823
82969
57262
81982
4
57
51429
85762
52918
84851
54391
83915 55847
82953
57286
81965
3
58
51454
85747
52943
84836
54415
83899
55871
82936
57310
81949
2
59
51479
85732
52967
84820
54440
83883
55895
82920
57334
81932
1
60
51504
85717
52992
84805
54464
83867
55919
82904
57358
81915
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
f
50
58
57-
56 ij 55
APPENDIX.
TIT
NATURAL. SINKS AND COSINES.
3.5
3G
37
38
3O
/
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
/
57358
81915
58779
80902
60182
79864
61566
78801
62932
77715
60
I
57381
81899
58802
80885
60205
79846
61589
78783
62955
77696
59
2
57405
81882
58826
80867 60228
79829
61612
78765
62977
77678
58
3
57429
81865
58849
80850
60251
79811
61635
78747
63000
77660 1 57
4
57453
81848
58873
80833
60274
79793
61658
78729
63022
77641
56
5
57477
81832
58896
80816
60298
79776
61681
78711
63045
77623
55
6
57501
81815
58920
80799
60321
79758
61704
78694
63068
77605
54
7
57524
81798
58943
80782 60344
79741
61726
78676
63090
77586
5S
8
57548
81782
58967
80765 60367
79723
61749
78658
63113
77568
52
9
57572
81765
58990
80748 60390
79706
61772
78640
63135
77550
51
10
57596
81748
59014
80730 l 60414
79688
61795
78622
63158
77531
50
11
57619
81731
59037
80713 : 60437
79671
61818
78604
63180
77513
49
12
57643
81714
59061
80696 60460
79653
61841
78586
63203
77494
48
13
57667
81698
59084
80679 60483
79635
61864
78568
63225
77476
47
14
57691
81681
59108
80662 60506
79618
61887
78550
63248
77458
46-
15
57715
81664
59131
80644 i 60529
79600
61909
78532
63271
77439
45
16
57738
81647
59154
80627 60553
79583
61932
78514
63293
77421
44
17
57762
81631
59178
80610 60576
79565
61955
78496
63316
77402
4S
18
57786
81614
59201
80593 60599
79547
61978
78478
63338
77384
42
19
57810
81597
59225
80576 ! 60622
79530
62001
78460
63361
77366
41
20
57833
81580
59248
80558 60645
79512
62024
78442
63383
77347
40
21
57857
81563
59272
80541 60668
79494
62046
78424
63406
77329
39
22
57881
81546
59295
80524 ; 60691
79477
62069
78405
63428
77310
38
23
57904
81530
59318
80507 j 60714
79459
62092
78387
63451
77292
37
24
57928
81513
59342
80489
60738
79441
62115
78369
63473
77273
36-
25
57952
81496
59365
80472
60761
79424
62138
78351
63496
77255
35
26
57976
81479
59389
80455
60784
79406
62160
78333
63518
77236 34
27
57999
81462
59412
80438 !
60807
79388
62183
78315
63540
77218
33
28
58023
81445
59436
80420
60830
79371
62206
78297
63563
77199
32
29
58047
81428
59459
80403 60853
79353
62229
78279
63585
77181
31
30
58070
81412
59482
80386 60876
i
79335
62251
78261
63608
77162
30
31
58094
81395
59506
80368 60899
79318
62274
78243
63630
77144
29
32
58118
81378
59529
80351 60922
79300
62297
78225
63653
77125
28
33
58141
81361
59552
80334 60945
79282
62320
78206
63675
77107
27
34
58165
81344
59576
80316 ;
60968
79264
62342
78188
63698
77088
26
35
58189
81327
59599
80299 !
60991
79247
62365
78170
63720
77070
25
36
58212
81310
59622
80282 j 61015
79229
62388
78152
63742
77051
24
37
58236
81293
59646
80264
61038
79211
62411
78134
63765
77033
2a
38
58260
81276
59669
80247
61061
79193
2433
78116
63787
77014
22
39
58283
81259
59693
80230 61084
79176
62456
78098
63810
76996
21
40
58307
81242
59716
80212 61107
79158
62479
78079
63832
76977
20
41
58330
81225
59739
80195 61130
79140
62502
78061 i
63854
76959
19
42
58354
81208
59763
80178 61153
79122
62524
78043
63877
76940
18
43
58378
81191
59786
80160 61176
79105
62547
78025 |
63899
76921
17
44
58401
81174
59809
80143 61199
79087
62570
78007
63922
76903
16
45
58425
81157
59832
80125
61222
79069
62592
77988
63944
76884
15
46
58449
81140
59856
80108
61245
79051
62615
77970
63966
76866 14
47
58472
81123
59879
80091
61268
79033 ;
62638
77952
63989
76847 la
48
58496
81106
59902
80073 ! 61291
79016 i
62660
77934
64011
76828
12
49
58519
81089
59926
80056 ! 61314
78998 '
62683
77916
64033
76810
11
50
58543
81072
59949
80038 ' 61337
78980
62706
77897
64056
76791 10
51
58567
81055
59972
80021 ' 61360
78962 | 62728
77879
64078
76772
9
52
58590
81038
59995
80003
61383
78944
62751
77861
64100
76754
8
53
58614
81021
60019
79986
61406
78926
62774
77843
64123
76735
7
54
58637
81004
60042
79968
61429
78908
62796
77824
64145
76717
6
55
58661
80987
60065
79951
61451
78891
62819
77806
64167
76698
5
56
58684
80970
60089
79934
61474
78873
62842
77788
64190
76679
4
57.
58708
80953
60112
79916
61497
78855
62864
77769
64212
76661
3
58
58731
80936
60135
79899
61520
78837
62887
77751
64234
76642
2
59
58755
80919
60158
79881
61543
78819
62909
77733
64256
76623
1
60
58779
80902
60182
79864
61566
78801
62932
77715
64279
76604
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
/
51
53o
53
51
50
/
718
APPENDIX.
NATURAL, SINES AND COSINES.
4,0
41
43
43 4,4=
/
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
/
64279
76604
65606
75471
66913
74314
68200
73135 1! 69466
71934
60
1
64301
76586
65628
75452
66935
74295 68221
73116 69487
71914
59
2
64323
76567
65650
75433
66956
74276 j 68242
73096
69508
71894
58
3
64346
76548
65672
75414
66978
74256 68264
73076
69529
71873
57
4
64368
76530
; 65694
75395
66999
74237
68285
73056
69549
71853
56
5
64390
76511
65716
75375
67021
74217
68306
73036
69570
71833
55
6
64412
76492
65738
75356
67043
74198 | 68327
73016
69591
71813
54
7
64435
76473
65759
75337
67064
74178
68349
72996
69612
71792
53
8
64457
76455
65781
75318
67086
74159
68370
72976
69633
71772
52
9
64479
76436
65803
75299
67107
74139
68391
72957
69654
71752
51
10
64501
76417
65825
75280
67129
74120
68412
72937
69675
71732
50
11
64524
76398
65847
75261
67151
74100
68434
72917
69696
71711
49
12
64546
76380
65869
75241
67172
74080
68455
72897
69717
71691
48
13
64568
76361
65891
75222
67194
74061
68476
72877
69737
71671
47
14
64590
76342
65913
75203
67215
74041
68497
72857
69758
71650
46
15
64612
76323
65935
75184
67237
74022
68518
72837
69779
71630
45
16
64635
76304
65956
75165
67258
74002
68539
72817
69800
71610
44
17
64657
76286
65978
75146
67280
73983
68561
72797
69821
71590
43
18
64679
76267
66000
75126
67301
73963
68582
72777 ! 69842
71569
42
IP
64701
76248
66022
75107
67323
73944
68603
72757 ' 69862
71549
41
20
64723
76229
66044
75088
67344
73924
68624
72737 :: 69883
71529
40
21
64746
76210
66066
75069
67366
73904
68645
72717 ] 69904
71508
39
22
64768
76192
66088
75050
67387
73885
68666
72697 69925
71488
38
23
64790
76173
66109
75030
67409
73865
68688
72677 69946
71468
37
24
64812
76154
66131
75011
67430
73846
68709
72657
69966
71447
36
25
64834
76135
66153
74992
67452
73826
68730
72637
69987
71427
35
26
64856
76116
66175
74973
67473
73806
68751
72617
70008
71407
34
27
64878
76097
66197
74953
67495
73787
68772
72597
70029
71386
33
28
64901
76078
66218
74934
67516
73767
68793
72577
70049
71366
32
29
64923
76059
66240
74915
67538
73747
68814
72557
70070
71345
31
30
64945
76041
66262
74896
67559
73728
68835
72537
70091
71325
30
31
64967
76022
66284
74876
67580
73708
68857
72517
70112
71305
29
32
64989
76003
66306
74857 ji 67602
73688
68878
72497 || 70132
71284
28
33
65011
75984
66327
74838
67623
73669
68899
72477 I 70153
71264
27
34
65033
75965
66349
74818
67645
73649 ;
68920
72457
70174
71243
26
35
65055
75946
66371
74799
67666
73629
68941
72437
70195
71223
25
36
65077
75927
66393
74780
67688
73610
68962
72417
70215
71203
24
37
65100
75908
66414
74760
67709
73590
68983
72397
70236
71182
23
38
65122
75889
66436
74741*
67730
73570
69004
72377
-70257
71162
22
39
65144
75870
66458
74722
67752
73551
69025
72357
70277
71141
21
40
65166
75851
66480
74703
67773
73531
69046
72337
70298
71121
20
41
65188
75832
66501
74683
67795
73511
69067
72317
! 70319
71100
19
42
65210
75813
66523
74664
67816
73491
69088
72297
i 70339
71080
18
43
65232
75794
66545
74644
67837
73472
69109
72277
! 70360
71059
17
44
65254
75775
66566
74625
67859
73452
69130
72257
! 70381
71039
16
45
65276
75756
66588
74606
67880
73432
69151
72236
70401
71019
15
46
65298
75738
66610
74586
67901
73413
69172
72216
70422
70998
14
47
65320
75719
66632
74567
67923
73393
69193
72196
70443
70978
13
48
65342
75700
66653
74548
67944
73373
69214
72176
70463
70957
12
49
65364
75680
66675
74528
67965
73353
69235
72156
70484
70937
11
50
65386
75661
66697
74509
67987
73333
69256
72136 : ! 70505
70916
10
51
65408
75642
66718
74489
68008
73314
69277
72116
70525
70896
9
52
65430
75623
66740
74470
68029
73294
69298
72095
70546
70875
8
53
65452
75604
66762
74451
68051
73274
69319
72075
70567
70855
7
54
65474
75585
66783
74431
68072
73254
69340
72055
70587
70834
6
55
65496
75566
66805
74412
68093
73234
69361
72035
70608
70813
5
56
65518
75547
66827
74392
68115
73215
69382
72015
70628
70793
4
57
65540
75528
66848
74373
68136
73195
69403
71995
70649
70772
3
58
65562
75509
66870
74353
68157
73175
69424
71974
70670
70752
2
59
65584
75490
66891
74334
68179
73155
69445
71954
70690
70731
1
60
65606
75471
66913
74314
68200
73135 i
69466
71934
70711
70711
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
Cosine.
Sine.
/
4O
48
47
40
45
/
APPENI)IX.// S
.
LOGARITHMS OP NUMBERS.
719
IV.
o
1
3
3
-A
5
7
S
9
I>.
100
00 0000
0434
0868
1301
1734
2166
2598
3029
3461
3891
432
101
4321
4751
5181
5609
6038
6466
6894
7321
7748
8174
428
102
* 8600
9026
9451
9876
+300
0724
1147
1570
1993
2415
424
103
01 2837
3259
3680
4100
4521
4940
5360
5779
6197
6616
419
104
* 7033
7451
7868
8284
8700
9116
9532
9947
+361
0775
416
105
02 1189
1603
2016
2428
2841
3252
3664
4075
4486
4896
412
106
5306
5715
6125
6533
6942
7350
7757
8164
8571
8978
408
107
* 9384
9789
4195
0600
1004
1408
1812
2216
2619
3021
404
108
03 3424
3826
4227
4628
5029
5430
5830
6230
6629
7028
400
109
* 7426
7825
8223
8620
9017
9414
9811
+207
0602
0998
396
110
04 1393
1787
2182
2576
2969
3362
3755
4148
4540
4932
393
111
5323
5714
6105
6495
6885
7275
7664
8053
8442
8830
389
112
* 9218
9606
9993
*380
0766
1153
1538
1924
2309
2694
386
113
05 3078
3463
3846
4230
4613
4996
5378
5760
6142
6524
382
114
* 6905
7286
7666
8046
8426
8805
9185
9563
9942
+320
379
115
06 0698
1075
1452
1829
2206
2582
2958
3333
3709
4083
376
116
4458
4832
5206
5580
5953
6326
6699
7071
7443
7815
372
117
* 8186
8557
8928
9298
9668
+038
0407
0776
1145
1514
369
118
07 1882
2250
2617
2985
3352
3718
4085
4451
4816
5182
366
119
5547
5912
6276
6640
7004
7368
7731
8094
8457
8819
363
120
* 9181
9543
9904
+266
0626
0987
1347
1707
2067
2426
360
121
08 2785
3144
3503
3861
4219
4576
4934
5291
5647
6004
357
122
6360
6716
7071
7426
7781
8136
8490
8845
9198
9552
355
123
* 9905
+258
0611
0963
1315
1667
2018
2370
2721
3071
351
124
09 3422
3772
4122
4471
4820 ,
5169
5518
5866
6215
6562
349
125
* 6910
7257
7604
7951
8298
8644
8990
9335
9681
+026
346
126
10 0371
0715
1059
1403
1747
2091
2434
2777
3119
3462
343
127
3804
4146
4487
4828
5169
5510
5851
6191
6531
6871
340
128
* 7210
7549
7888
8227
8565
8903
9241
9579
9916
+253
338
129
11 0590
0926
1263
1599
1934
2270
2605
2940
3275
3609
335
130
3943
4277
4611
4944
5278
5611
5943
6276
6608
6940
333
131
* 7271
7603
7934
8265
8595
8926
9256
9586
9915
+245
330
132
12 0574
0903
1231
1560
1888
2216
2544
2871
3198
3525
328
133
3852
4178
4504
4830
5156
5481
5806
6131
6456
6781
325
134
* 7105
7429
7753
8076
8399
8722
9045
9368
9690
+012
323
135
13 0334
0655
0977
1298
1619
1939
2260
2580
2900
3219
321
136
3539
3858
4177
4496
4814
5133
5451
5769
6086
6403
318
137
6721
7037
7354
7671
7987
8303
8618
8934
9249
9564
315
138
*9879
4194
0508
0822
1136
1450
1763
2076
2389
2702
314
139
143015 '
3327
3639
3951
4263
4574
4885
5196
5507
5818
311
140
6128
6438
6748
7058
7367
7676
7985
8294
8603
8911
309
141
*9219
9527
9835
+142
0449
0756
1063
1370
1676
1982
307
142
15 2288
2594
2900
3205
3510
3815
4120
4424
4728
5032
305
143
5336
5640
5943
6246
6549
6852
7154
7457
7759
8061
303
144
* 8362
8664
8965
9266
9567
9868
+168
0469
0769
1068
301
145
16 1368
1667
1967
2266
2564
2863
3161
3460
3758
4055
299
146
4353
4650
4947
5244
5541
5838
6134
6430
6726
7022
297
147
7317
7613
7908
8203
8497
8792
9086
9380
9674
9968
295
148
17 0262
0555
0848
1141
1434
1726
2019
2311
2603
2895
293
149
3186
3478
3769
4060
4351
4641
4932
5222
5512
5802
291
150
6091
6381
6670
6959
7248
7536
-7825
8113
8401
8689
289
151
* 8977
9264
9552
9839
+126
0413
0699
0985
1272
1558
287
152
18 1844
2129
2415
2700
2985
3270
3555
3839
4123
4407
285
153
4691
4975
5259
5542
5825
6108
6391
6674
6956
7239
283
154
*7521
7803
8084
8366
8647
8928
9209
9490
9771
+051
281
155
19 0332
0612
0892
1171
1451
1730
2010
2289
2567
2846
279
156
3125
3403
3681
3959
4237
4514
4792
5069
5346
5623
278
157
5900
6176
6453
6729
7005
7281
7556
7832
8107
8382
276
158
* 8657
8932
9206
9481
9755
+029
0303
0577
0850
1124
274
159
20 1397
1670
1943
2216
2488
2761
3033
3305
8577
3848
272
IV.
O
1
2
3
4,
5
G
7
8
9
j>.
720
APPENDIX.
LOGARITHMS OP NUMBERS.
2V.
1
3
3
4,
5
7
8
O
r>.
160
20 4120
4391
4663
4934
5204
5475
5746
6016
6286
6556
271
161
6826
7096
7365
7634
7904
8173
8441
8710
8979
9247
269
162
* 9515
9783
4051
0319
0586
0853
1121
1388
1654
1921
267
163
21 2188
2454
2720
2986
3252
3518
3783
4049
4314
4579
266
164
4844
5109
5373
5638
5902
6166
6430
6694
6957
7221
264
165
7484
7747
8010
8273
8536
8798
9060
9323
9585
9846
262
166
22 0108
0370
0631
0892
1153
1414
1675
1936
2196
2456
261
167
2716
2976
3236
3496
3755
4015
4274
4533
4792
5051
259
168
5309
5568
5826
6084
6342
6600
6858
7115
7372
7630
258
169
*7887
8144
8400
8657
8913
9170
9426
9682
9938
4193
256
170
23 0449
0704
0960
1215
1470
1724
1979
2234
2488
2742
254
171
2996
3250
3504
3757
4011
4264
4517
4770
5023
5276
253
172
5528
5781
6033
6285
6537
6789
7041
7292
7544
7795
252
173
*8046
8297
8548
8799
9049
9299
9550
9800
4050
0300
250
174
24 0549
0799
1048
1297
1546
1795
2044
2293
2541
2790
249
175
3038
3286
3534
3782
4030
4277
4525
4772
5019
5266
248
176
5513
5759
6006
6252
6499
6745
6991
7237
7482
7728
246
177
*7973
8219
8464
8709
8954
9198
9443
9687
9932
+176
245
178
25 0420
0664
0908
1151
1395
1638
1881
2125
2368
2610
243
179
2853
3096
3338
3580
3822
4064
4306
4548
4790
5031
242
180
5273
5514
5755
5996
6237
6477
6718
6958
7198
7439
241
181
7679
7918
8158
8398
8637
8877
9116
9355
9594
9833
239
182
26 0071
0310
0548
0787
1025
1263
1501
1739
1976
2214
238
183
2451
2688
2925
3162
3399
3636
3873
4109
4346
4582
237
184
4818
5054
5290
5525
5761
5996
6232
6467
6702
6937
235
185
7172
7406
7641
7875
8110
8344
8578
8812
9046
9279
234
186
* 9513
9746
9980
+213
0446
0679
0912
1144
1377
1609
233
187
27 1842
2074
2306
2538
2770
3001
3233
3464
3696
3927
232
188
4158
4389
4620
4850
5081
5311
5542
5772
6002
6232
230
189
6462
6692
6921
7151
7380
7609
7838
8067
8296
8525
22$
190
* 8754
8982
9211
9439
9667
9895
*123
0351
0578
0806
228
191
28 1033
1261
1488
1715
1942
2169
2396
2622
2849
3075
227
192
3301
3527
3753
3979
4205
4431
4656
4882
5107
5332
22$
193
5557
5782
6007
6232
6456
6681
6905
7130
7354
7578
225
194
7802
8026
8249
8473
8696
8920
9143
9366
9589
9812
223
195
29 0035
0257
0480
0702
0925
1147
1369
1591
1813
2034
222
196
2256
2478
2699
2920
3141
3363
3584
3804
4025
4246
221
197
4466
4687
4907
5127
5347
5567
5787
6007
6226
6446
220
198
6665
6884
7104
7323
7542
7761
7979
8198
8416
8635
219
199
* 8853
9071
9289
9507
9725
9943
4161
0378
0595
0813
218
200
30 1030
1247
1464
1681
1898
2114
2331
2547
2764
2980
217
201
3196
3412
3628
3844
4059
4275
4491
4706
4921
5136
216-
202
5351
5566
5781
5996
6211
6425
6639
6854
7068
7282
215
203
7496
7710
7924
8137
8351
8564
8778
8991
9204
9417
2ia
204
* 9630
9843
4056
0268
0481
0693
0906
1118
1330
1542
212
205
31 1754
1966
2177
2389
2600
2812
3023
3234
3445
3656
211
206
3867
4078
4289
4499
4710
4920
5130
5340
5551
5760
210
207
5970
6180
6390
6599
6809
7018
7227
7436
7646
7854
209
208
8063
8272
8481
8689
8898
9106
9314
9522
9730
9938
208
209
32 0146
0354
0562
0769
0977
1184
1391
1598
1805
2012
207
210
2219
2426
2633
2839
3046
3252
3458
3665
3871
4077
206
211
4282
4488
4694
4899
5105
5310
5516
5721
5926
6131
205
212
6336
6541
6745
6950
7155
7359
7563
7767
7972
8176
204
213
*8380
8583
8787
8991
9194
9398
9601
9805
4008
0211
203
214
33 0414
0617
0819
1022
1225
1427
1630
1832
2034
2236
202
215
2438
2640
2842
3044
3246
3447
3649
3850
4051
4253
202
216
4454
4655
4856
5057
5257
5458
5658
5859
6059
6260
201
217
6460
6660
6860
7060
7260
7459
7659
7858
8058
8257
200
218
* 8456
8656
8855
9054
9253
9451
9650
9849
4047
0246
199
219
34 0444
0642
0841
1039
1237
1435
1632
1830
2028
2225
198
3V.
1
3
3
4 5
078
r>*
APPENDIX.
721
LOGARITHMS OP NUMBERS.
N.
013
3
4,
5
O 7
8
9
r>.
220
34 2423 2620
2817
3014
3212
3409
3606
3802
3999
4196
197
221
4392 i 4589
4785
4981
5178
5374
5570
5766
5962
6157
196
222
6353
6549
6744
6939
7135
7330
7525
7720
7915
8110
195
223
* 8305
8500 8694
8889
9083
9278
9472
9666
9860
+054
194
224
35 0248
0442
0636
0829
1023
1216
1410
1603
1796
1989
193
225
2183
2375
2568
2761
2954
3147
3339
3532
3724
3916
193
226
4108
4301
4493
4685
4876
5068
5260
5452
5643
5834
192
227
6026
6217
6408
6599
6790
6981
7172
7363
7554
7744
191
228
7935
8125
8316
8506
8696
8886
9076
9266
9456
9646
190
229
* 9835
+025
0215
0404
0593
0783
0972
1161
1350
1539
189
230
36 1728
1917
2105
2294
2482
2671
2859
3048
3236
3424
188
231
3612
3800
3988
4176
4363
4551
4739
4926
5113
5301
188
232
5488
5675
5862
6049
6236
6423
6610
6796
6983
7169
187
233
7356
7542
7729
7915
8101
8287
8473
8659
8845
9030
186
234
* 9216
9401
9587
9772
9958
+143
0328
0513
0698
0883
185
235
37 1068
1253
1437
1622
1806
1991
2175
2360
2544
2728
184
236
2912
3096
3280
3464
3647
3831
4015
4198
4382
4565
184
237
4748 ! 4932
5115
5298
5481
5664
5846
6029
6212
6394
183
238
6577 i 6759
6942
7124
7306
7488
7670
7852
8034
8216
182
239
* 8398
8580
8761
8943
9124
9306
9487
9668
9849
+030
181
240
38 0211
0392
0573
0754
0934
1115
1296
1476
1656
1837
181
241
2017
2197
2377
2557
2737
2917
3097
3277
3456
3636
180
242
3815
3995
4174
4353
4533
4712
4891
5070
5249
5428
179
243
5606
5785
5964 ! 6142
6321
6499
6677
6856
7034
7212
178
244
7390
7568
7746
7923
8101
8279
8456
8634
8811
8989
178
245
* 9166
9343
9520
9698
9875
+051
0228
0405
0582
0759
177
246
39 0935
1112
1288
1464
1641
1817
1993
2169
2345
2521
176
247
2697
2873
3048
3224
3400
3575
3751
3926
4101
4277
176
248
4452
4627
4802
4977
5152
5326
5501
5676
5850
6025
175
249
6199
6374
6548
6722
6896
7071
7245
7419
7592
7766
174
250
7940
8114
8287
8461
8634
8808
8981
9154
9328
9501
173
251
* 9674
9847
+020
0192
0365
0538
0711
0883
1056
1228
173
252
40 1401
1573
1745
1917
2089
2261
2433
2605
2777
2949
172
253
3121
3292
3464
3635
3807
3978
4149
4320
4492
4663
171
254
4834
5005
5176
5346
5517
5688
5858
6029
6199
6370
171
255
6540
6710
6881
7051
7221
7391
7561
7731
7901
8070
170
256
8240
8410
8579
8749
8918
9087
9257
9426
9595
9764
169
257
* 9933
+102
0271
0440
0609
0777
0946
1114
1283
1451
169
258
41 1620
1788
1956
2124
2293
2461
2629
2796
2964
3132
168
259
3300
3467
3635
3803
3970
4137
4305
4472
4639
4806
167
260
4973
5140
5307
5474
5641
5808
5974
6141
6308
6474
167
261
6641
6807
6973
7139
7306
7472
7638
7804
7970
8135
166
262
8301
8467
8633
8798
8964
9129
9295
9460
9625
9791
165
263
* 9956
+121
0286
0451
0616
0781
0945
1110
1275
1439
165
264
42 1604
1768
1933
2097
2261
2426
2590
2754
2918
3082
164
265
3246
3410
3574
3737
3901
4065
4228
4392
4555
4718
164
266
4882
5045
5208
5371
5534
5697
5860
6023
6186
6349
iea
267
6511
6674
6836
6999
7161
7324
7486
7648
7811
7973
162,
268
8135
8297
8459
8621
8783
8944
9106
9268
9429
9591
162
269
*9752
9914
+075
0236
0398
0559
0720
0881
1042
1203
161
270
43 1364
1525
1685
1846
2007
2167
2328
2488
2649
2809
161
271
2969
3130
3290
3450
3610
3770
3930
4090
4249
4409
160'
272
4569
4729
4888
5048
5207
5367
5526
5685
5844
6004
159'
273
6163
6322
6481
6640
6799
6957
7116
7275
7433
7592
159'
274
7751
7909
8067
8226
8384
8542
8701
8859
9017
9175
158
275
* 9333
9491
9648
9806
9964
+122
0279
0437
0594
0752
158
276
44 0909
1066
1224
1381
1538
1695
1852
2009
2166
2323
157
277
2480
2637
2793
2950
3106
3263
3419
3576
3732
3889
157
278
4045
4201
4357
4513
4669
4825
4981
5137
5293
5449
156
279
5604
5760
5915
6071
6226
6382
6537
6692
6848
7003
155
IV.
1
3
3
4,
5
O
. 7
H
r>*
722
APPENDIX.
LOGARITHMS OP NUMBERS.
IV.
1
3
3
4,
5
G
7
8
9
I>.
280
44 7158
7313
7468
7623
7778 7933
8088
8242
8397
8552
155
281
* 8706
8861
9015
9170
9324
9478
9633
9787
9941
+095
154
282
45 0249
0403
0557
0711
0865
1018
1172
1326
1479
1633
154
283
1786
1940
2093
2247
2400
2553
2706
2859
3012
3165
153
284
3318
3471
3624
3777
3930
4082
4235
4387
4540
4692
153
285
4845
4997
5150
5302
5454
5606
5758
5910
6062
6214
152
286
6366
6518
6670
6821
6973
7125
7276
7428
7579
7731
152
287
7882
8033
8184
8336
8487
8638
8789
8940
9091
9242
151
288
*.9392
9543
9694
9845
9995
+146
0296
0447
0597
0748
151
289
46 0898
1048
1198
1348
1499
1649
1799
1948
2098
2248
150
290
2398
2548
2697
2847
2997
3146
3296
3445
3594
3744
150
291
3893
4042
4191
4340
4490
4639
4788
4936
5085
5234
149
292
5383
5532
5680
5829
5977
6126
6274
6423
6571
6719
149
293
6868
7016
7164
7312
7460
7608
7756
7904
8052
8200
148
294
8347
8495
8643
8790
8938
9085
9233
9380
9527
9675
148
295
* 9822
9969
*116
0263
0410
0557
0704
0851
0998
1145
147
296
47 1292
1438
1585
1732
1878
2025
2171
2318
2464
2610
146
297
2756
2903
3049
3195
3341
3487
3633
3779
3925
4071
146
298
4216
4362
4508
4653
4799
4944
5090
5235
5381
5526
146
299
5671
5816
5962
6107
6252
6397
6542
6687
6832
6976
145
300
7121
7266
7411
7555
7700
7844
7989
8133
8278
8422
145
301
8566
8711
8855
'8999
9143
9287
9431
9575
9719
9863
144
302
48 0007
0151
0294
0438
0582
0725
0869
1012
1156
1299
144
303
1443
1586
1729
1872
2016
2159
2302
2445
2588
2731
143
304
2874
3016
3159
3302
3445
3587
3730
3872
4015
4157
143
305
4300
4442
4585
4727
4869
5011
5153
5295
5437
5579
142
306
5721
5863
6005
6147
6289
6430
6572
6714
6855
6997
142
307
7138
7280
7421
7563
7704
7845
7986
8127
8269
8410
141
308
8551
8692
8833
8974
9114
9255
9396
9537
9677
9818
141
309
* 9958
+099
0239
0380
0520
0661
0801
0941
1081
1222
140
310
49 1362
1502
1642
1782
1922
2062
2201
2341
2481
2621
140
311
2760
2900
3040
3179
3319
3458
3597
3737
3876
4015
139
312
4155
4294
4433
4572
4711
4850
4989
5128
5267
5406
139
313
5544
5683
5822
5960
6099
6238
6376
6515
6653
6791
139
314
6930
7068
7206
7344
7483
7621
7759
7897
8035
8173
138
315
8311
8448
8586
8724
8862
8999
9137
9275
9412
9550
138
316
* 9687
9824
9962
+099
0236
0374
0511
0648
0785
0922
137
317
50 1059
1196
1333
1470
1607
1744
1880
2017
2154
2291
137
318
2427
2564
2700
2837
2973
3109
3246
3382
3518
3655
136
319
3791
3927
4063
4199
4335
4471
4607
4743
4878
5014
136
320
5150
5286
5421
5557
5693
5828
5964
6099
6234
6370
136
321
6505
6640
6776
6911
7046
7181
7316
7451
7586
7721
135
322
7856
7991
8126
8260
8395
8530
8664
8799
8934
9068
135
323
* 9203
9337
9471
9606
9740
9874
+009
0143
0277
0411
134
324
51 0545
0679
0813
0947
1081
1215
1349
1482
1616
1750
134
325
1883
2017
2151
2284
2418
2551
2684
2818
2951
3084
133
326
3218
3351
3484
3617
3750
3883
4016
4149
4282
4414
133
327
4548
4681
4813
4946
5079
5211
5344
5476
5609
5741
133
328
5874
6006
6139
6271
6403
6535
6668
6800
6932
7064
132
329
7196
7328
7460
7592
7724
7855
7987
8119
8251
8382
132
330
8514
8646
8777
8909
9040
9171
9303
9434
9566
9697
131
331
*9828
9959
+090
0221
0353
0484
0615
0745
0876
1007
131
332
52 1138
1269
1400
1530
1661
1792
1922
2053
2183
2314
131
333
2444
2575
2705
2835
2966
3096
3226
3356
3486
3616
130
334
3746
3876
4006
4136
4266
4396
4526
4656
4785
4915
130
335
5045
5174
5304
5434
5563
5693
5822
5951
6081
6210
129
336
6339
6469
6598
6727
6856
6985
7114
7243
7372
7501
129
337
7630
7759
7888
8016
8145
8274
8402
8531
8660
8788
129
338
* 8917
9045
9174
9302
9430
9559
9687
9815
9943
+072
128
339
53 0200
0328
0456
0584
0712
0840
0968
1096
1223
1351
128
IV.
1
3
3
4=
5
O
7
8
9
r>.
APPENDIX.
723
LOGARITHMS OF NUMBERS.
IV.
o
1
3
3
4,
5
6 7
8
9
I>.
340
53 1479
1607
1734
1862
1990
2117
2245
2372
2500
2627
128
541
2754
2882
3009
3136
3264
3391
3518
3645
3772
3899
127
342
4026
4153
4280
4407
4534
4661
4787
4914
5041
5167
127
343
5294
5421
5547
5674
5800
5927
6053
6180
6306
6432
126
344
6558
6685
6811
6937
7063
7189
7315
7441
7567
7693
126
345
7819
7945
8071
8197
8322
8448
8574
8699
8825
8951
126
346
* 9076
9202
9327
9452
9578
9703
9829
9954
+079
0204
125
347
54 0329
0455
0580
0705
0830
0955
1080
1205
1330
1454
125
348
1579
1704
1829
1953
2078
2203
2327
2452
2576
2701
125
349
2825
2950
3074
3199
3323
3447
3571
3696
3820
3944
124
350
4068
4192
4316
4440
4564
4688
4812
4936
5060
5183
124
351
5307
5431
5555
5678
5802
5925
6049
6172
6296
6419
124
352
6543
6666
6789
6913
7036
7159
7282
7405
7529
7652
123
353
7775
7898
8021
8144
8267
8389
8512
8635
8758
8881
123
354
*9003
9126
9249
9371
9494
9616
9739
9861
9984
+106
123
355
55 0228
0351
0473
0595
0717
0840
0962
1084
1206
1328
122
356
1450
1572
1694
1816
1938
2060
2181
2303
2425
2547
122
357
2668
2790
2911
3033
3155
3276
3398
3519
3640
3762
121
358
3883
4004
4126
4247
4368
4489
4610
4731
4852
4973
121
359
5094
5215
5336
5457
5578
5699
5820
5940
6061
6182
121
360
6303
6423
6544
6664
6785
6905
7026
7146
7267
7387
120
361
7507
7627
7748 | 7868
7988
8108
8228
8349
8469
8589
120
362
8709
8829
8948 9068
9188
9308
9428
9548
9667
9787
120
363
* 9907
+026
0146 0265
0385
0504
0624
0743
0863
0982
119
364
56 1101
1221
1340 1459
1578
1698
1817
1936
2055
2174
119
365
2293
2412
2531
2650
2769
2887
3006
3125
3244
3362
119
366
3481
3600
3718
3837
3955
4074
4192
4311
4429
4548
119
367
4666
4784
4903
5021
5139
5257
5376
5494
5612
5730
118
368
5848
5966
6084
6202
6320
6437
6555
6673
6791
6909
118
369
7026
7144
7262
7379
7497
7614
7732
7849
7967
8084
118
370
8202
8319
8436
8554
8671
8788
8905
9023
9140
9257
117
371
* 9374
9491
9608
9725
9842
9959
+076
0193
0309
0426
117
372
57 0543
0660
0776
0893
1010
1126
1243
1359
1476
1592
117
373
1709
1825 1942
2058
2174
2291
2407
2523
2639
2755
116
374
2872
2988
3104
3220
3336
3452
3568
3684
3800
3915
116
375
4031
4147
4263
4379
4494
4610
4726
4841
4957
5072
116
376
5188
5303
5419
5534
5650
5765
5880
5996
6111
6226
115
377
6341
6457
6572
6687
6802
6917
7032
7147
7262
7377
115
378
7492
7607
7722
7836
7951
8066
8181
8295
8410
8525
115
379
8639
8754
8868
8983
9097
9212
9326
9441
9555
9669
114
380
*9784
9898
+012
0126
0241
0355
0469
0583
0697
0811
114
381
58 0925
1039
1153
1267
1381
1495
1608
1722
1836
1950
114
382
2063
2177
2291
2404
2518
2631
2745
2858
2972
3085
114
383
3199
3312
3426
3539
3652
3765
3879
3992
4105
4218
113
384
4331
4444
4557
4670
4783
4896
5009
5122
5235
5348
113
385
5461
5574
5686
5799
5912
6024
6137
6250
6362
6475
113
386
6587
6700
6812
6925
7037
7149
7262
7374
7486
7599
112
387
7711
7823
7935
8047
8160
8272
8384
8496
8608
8720
112
388
8832
8944
9056
9167
9279
9391
9503
9615
9726
9838
112
389
* 9950
+061
0173
0284
0396
0507
0619
0730
0842
0953
112
390
59 1065
1176
1287
1399
1510
1621
1732
1843
1955
2066
111
391
2177
2288
2399
2510
2621
2732
2843
2954
3064
3175
111
392
3286
3397
3508
3618
3729
3840
3950
4061
4171
4282
111
393
4393
4503
4614
4724
4834
4945
5055
5165
5276
5386
110
394
5496
5606
5717
5827
5937
6047
6157
6267
6377
6487
110
395
6597
6707
6817
6927
7037
7146
7256
7366
7476
7586
110
396
7695
7805
7914
8024
8134
8243
8353
8462
8572
8681
110
397
8791
8900
9009
9119
9228
9337
9446
9556
9665
9774
109
398
* 9883
9992
+101
0210
0319
0428
0537
0646
0755
0864
109
399
60 0973
1082
1191
1299
1408
1517
1625
1734
1843
1951
109
N.
O
1
2
3
4
5
G 7
S
r>.
724
APPENDIX.
LOGARITHMS OF NUMBERS.
IV.
1
2
3
4,
5
7
8
9
r>.
400
60 2060
2169
2277 2386
2494
2603
2711
2819
2928
3036
108-
401
3144
3253
3361
3469
3577
3686
3794
3902
4010
4118
108
402 4226
4334
4442
4550
4658
4766
4874
4982
5089
5197
108
403
5305
5413
5521
5628
5736
5844
5951
6059
6166
6274
108
404
6381
6489
6596
6704
6811
6919
7026
7133
7241
7348
107
405
7455
7562
7669
7777
7884
7991
8098
8205
8312
8419
107
406
8526
8633
8740
8847
8954
9061
9167
9274
9381
9488
107
407
* 9594
9701
9808
9914
+021
0128
0234
0341
0447
0554
107
408
61 0660
0767
0873
0979
1086
1192
1298
1405
1511
1617
lOfr
409
1723
1829
1936
2042
2148
2254
2360
2466
2572
2678
106.
410
2784
2890
2996
3102
3207
3313
3419
3525
3630
3736
106
411
3842
3947
4053
4159
4264
4370
4475
4581
4686
4792
106-
412
4897
5003
5108
5213
5319
5424
5529
5634
5740
5845
105
413
5950
6055
6160
6265
6370
6476
6581
6686
6790
6895
105
414
7000
7105
7210
7315
7420
7525
7629
7734
7839
7943
105-
415
8048
8153
8257
8362
8466
8571
8676
8780
8884
8989
105
416
* 9093
9198
9302
9406
9511
9615
9719
9824
9928
+032
104
417
62 0136
0240
0344
0448
0552
0656
0760
0864
0968
1072
104
418
1176
1280
1384
1488
1592
1695
1799
1903
2007
2110
104
419
2214
2318
2421
2525
2628
2732
2835
2939
3042
3146
104
420
3249
3353
3456
3559
3663
3766
3869
3973
4076
4179
103.
421
4282
4385
4488
4591
4695
4798
4901
5004
5107
5210
103
422
5312
5415
5518
5621
5724
5827
5929
6032
6135
6238
103
423
6340
6443
6546
6648
6751
6853
6956
7058
7161
7263
103-
424
7366
7468
7571
7673
7775
7878
7980
8082
8185
8287
102
425
8389
8491
8593
8695
8797
8900
9002
9104
9206
9308
102
426
* 9410
9512
9613
9715
9817
9919
+021
0123
0224
0326
102
427
63 0428
0530
0631
0733
0835
0936
1038
1139
1241
1342
102
428
1444
1545
1647
1748
1849
1951
2052
2153
2255
2356
101
429
2457
2559
2660
2761
2862
2963
3064
3165
3266
3367
101
430
3468
3569
3670
3771
3872
3973
4074
4175
4276
4376
100-
431
4477
4578
4679
4779
4880
4981
5081
5182
5283
5383
100'
432
5484
5584
5685
5785
5886
5986
6087
6187
6287
6388
100
433
6488
6588
6688
6789
6889
6989
7089
7189
7290
7390
100-
434
7490
7590
7690
7790
7890
7990
8090
8190
8290
8389
99-
435
8489
8589
8689
8789
8888
8988
9088
9188
' 9287
9387
99
436
* 9486
9586
9686
9785
9885
9984
+084
0183
0283
0382
99
437
64 0481
0581
0680
0779
0879
0978
1077
1177
1276
1375
99
438
1474
1573
1672
1771
1871
1970
2069
2168
2267
2366
99
439
2465
2563
2662
2761
2860
2959
3058
3156
3255
3354
99
440
3453
3551
3650
3749
3847
3946
4044
4143
4242
4340
98
441
4439
4537
4636
4734
4832
4931
5029
5127
5226
5324
98.
442
5422
5521
5619
5717
5815
5913
6011
6110
6208
6306
98
443
6404
6502
6600
6698
6796
6894
6992
7089
7187
7285
98
444
7383
7481
7579
7676
7774
7872
7969
8067
8165
8262
98
445
8360
8458
8555
8653
8750
8848
8945
9043
9140
9237
97
446
* 9335
9432
9530
9627
9724
9821
9919
+016
0113
0210
97
447
65 0308
0405
0502
0599
0696
0793
0890
0987
1084
1181
97
448
1278
1375
1472
1569
1666
1762
1859
1956
2053
2150
97
449
2246
2343
2440
2536
2633
2730
2826
2923
3019
3116
97
450
3213
3309
3405
3502
3598
3695
3791
3888
3984
4080
96
451
4177
4273
4369
4465
4562
4658
4754
4850
4946
5042
9fr
452
5138
5235
5331
5427
5523
5619
5715
5810
5906
6002
96
453
6098
6194
6290
6386
6482
6577
6673
6769
6864
6960
96
454
7056
7152
7247
7343
7438
7534
7629
7725
7820
7916
96
455
8011
8107
8202
8298
8393
8488
8584
8679
8774
8870
95
456
8965
9060
9155
9250
9346
9441
9536
9631
9726
9821
95
457
* 9916
+011
0106
0201
0296
0391
0486
0581
0676
0771
95
458
66 0865
0960
1055
1150
1245
1339
1434
1529
1623
1718
95
459
1813
1907
2002
2096
2191
2286
2380
2475
2569
2663
95
IV.
O
1
2
3
4.
5
7
8
I>.
APPENDIX.
v
725
LOGARITHMS OP NUMBERS.
IV.
1 3
3
4r
5
7
8
r
r>.
460
66 2758
2852
2947
3041
3135
3230
3324 3418
3512
3607
94
461
3701
3795
3889
3983
4078
4172
4266 4360
4454
4548
94
462
4642
4736
4830
4924
5018
5112
5206 5299
5393
5487
94
463
5581
5675
5769
5862
5956
6050
6143
6237
6331
6424
94
464
6518
6612
6705
6799
6892
6986
7079
7173
7266
7360
94
465
7453
7546
7640
7733
7826
7920
8013
8106
8199
8293
93
466
8386
8479
8572
8665
8759
8852
8945
9038
9131
9224
93
467
* 9317
9410
9503
9596
9689
9782
9875
9967
+060
0153
93
468
67 0246
0339
0431
0524
0617
0710
0802
895
0988
1080
93
469
1173
1265
1358
1451
1543
1636
1728
1821
1913
2005
93
470
2098
2190
2283
2375
2467
2560
2652
2744
2836
2929
92
471
3021
3113
3205 3297
3390
3482
3574
3666
3758
3850
92
472
3942
4034
4126
4218
4310
4402
4494
4586
4677
4769
92
473
4861
4953
5045
5137
5228
5320
5412
5503
5595
5687
92
474
5778
5870
5962
6053
6145
6236
6328
6419
6511
6602
92
475
6694
6785
6876
6968
7059
7151
7242
7333
7424
7516
91
476
7607
7698
7789
7881
7972
8063
8154
8245
8336
8427
91
477
8518
8609
8700
8791
8882
8973
9064
9155
9246
9337
91
478
*9428
9519
9610
9700
9791
9882
9973
+063
0154
0245
91
479
68 0336
0426
0517
0607
0698
0789
0879
0970
1060
1151
91
480
1241
1332
1422
1513
1603
1693
1784
1874
1964
2055
90
481
2145
2235
2326
2416
2506
2596
2686
2777
2867
2957
90
482
3047
3137
3227
3317
3407
3497
3587
3677
3767
3857
90
483
3947
4037
4127
4217
4307
4396
4486
4576
4666
4756
90
484
4845
4935
5025
5114
5204
5294
5383
5473
5563
5652
90
485
5742
5831
5921
6010
6100
6189
6279
6368
6458
6547
89
486
6636
5726
6815
6904
6994
7083
7172
7261
7351
7440
89
487
7529
7618
7707
7796
7886
7975
8064
8153
8242
8331
89
488
8420 ; 8509
8598
8687
8776
8865
8953
9042
9131
9220
89
489
* 9309
9398
9486
9575
9664
9753
9841
9930
+019
0107
89
490
69 0196
0285
0373
0462
0550
0639
0728
0816
0905
0993
89
491
1081 1170 ! 1258
1347
1435
1524
1612
1700
1789
1877
88
492
1965
2053
2142
2230
2318
2406
2494
2583
2671
2759
88
493
2847
2935
3023
3111
3199
3287
3375
3463
3551
3639
88
494
3727
3815
3903
3991
4078
4166
4254
4342
4430
4517
88
495
4605
4693
4781
4868
4956
5044
5131
5219
5307
5394
88
496
5482
5569
5657
5744
5832
5919
6007
6094
6182
6269
87
497
6356
6444
6531
6618
6706
6793
6880
6968
7055
7142
87
498
7229
7317
7404
7491
7578
7665
7752
7839
7926
8014
87
499
8101
8188
8275
8362
8449
8535
8622
8709
8796
8883
87
500
8970
9057
9144
9231
9317
9404
9491
9578
9664
9751
87
501
* 9838
9924
+011
0098
0184
0271
0358
0444
. 0531
0617
87
502
70 0704
0790
0877
0963
1050
1136
1222
1309
1395
1482
86
503
1568
1654
1741
1827
1913
1999
2086
2172
2258
2344
86
504
2431
2517
2603
2689
2775
2861
2947
3033
3119
3205
86
-505
3291
3377
3463
3549
3635
3721
3807
3895
3979
4065
86
506
4151
4236
4322
4408
4494
4579
4665
4751
4837
4922
86
507
5008
5094
5179
5265
5350
5436
5522
5607
5693
5778
86
508
5864
5949
6035
6120
6206
6291
6376
6462
6547
6632
85
509
6718
6803
6888
6974
7059
7144
7229
7315
7400
7485
85
510
7570
7655
7740
7826
7911
7996
8081
8166
8251
8336
85
511
8421
8506
8591
8676
8761
8846
8931
9015
9100
9185
85
512
*9270
9355
9440
9524
9609
9694
9779
9863
9948
+033
85
513
71 0117
0202
0287
0371
0456
0540
0625
0710
0794
0879
85
514
0963
1048
1132
1217
1301
1385
1470
1554
1639
1723
84
515
1807
1892
1976
2060
2144
2229
2313
2397
2481
2566
84
516
2650
2734
2818
2902
2986
3070 i 3154
3238
3323
3407
84
517
3491
3575
3650
3742
3826
3910 i 3994
4078
4162
4246
84
518
4330
4414
4497
4581
4665
4749 4833
4916
5000
5084
84
519
5167
5251
5335
5418
5502
5586 5669
5753
5836
5920
84
N.
1
2
3
4. f
5 6
7
S
r>.
726
APPENDIX.
LOGARITHMS OP NUMBERS.
w.
1
3
3
4=
5 O
7
g
9
r>
520
71 6003
6087
6170
6254
6337
6421
6504
6588
6671
6754
83
521
6838
6921
7004
7088
7171
7254
7338
7421
7504
7587
83
522
7671
7754
7837
7920
8003
8086
8169
8253
8336
8419
83
523
8502 8585
8668
8751
8834
8917
9000
9083
9165
9248 83
524
* 9331
9414
9497
9580
9663
9745
9828
9911
9994
+077
83
525
72 0159
0242
0325
0407
0490
0573
0655
0738
0821
0903
83
526
0986
1068
1151
1233
1316
1398
1481
1563
1646
1728
82
527
1811
1893
1975
2058
2140
2222
2305
2387
2469
2552
82
528
2634
2716
2798
2881
2963
3045
3127
3209
3291
3374
82
529
3456
3538
3620
3702
3784
3866
3948
4030
4112
4194
82
530
4276
4358
4440
4522
4604
4685
4767
4849
4931
5013
82
531
5095
5176
5258
5340
5422
5503
5585
5667
5748
5830
82
532
5912
5993
6075
6156
6238
6320
6401
6483
6564
6646
82,
533
6727
6809
6890
6972
7053
7134
7216
7297
7379
7460
81
534
7541
7623
7704
7785
7866
7948
8029
8110
8191
8273
81
535
8354
8435
8516
8597
8678
8759
8841
8922
9003
9084
81
536
9165
9246
9327
9408
9489
9570
9651
9732
9813
9893
81
537
* 9974
+055
0136
0217
0298
0378
0459
0540
0621
0702
81
538
73 0782
0863
0944
1024
1105
1186
1266
1347
1428
1508
81
539
1589
1669
1750
. 1830
1911
1991
2072
2152
2233
2313
81
540
2394
2474
2555
2635
2715
2796
2876
2956
3037
3117
80
541
3197
3278
3358
3438
3518
3598
3679
3759
3839
3919
80
542
3999
4079
4160
4240
4320
4400
4480
4560
4640
4720
80
543
4800
4880
4960
5040
5120
5200
5279
5359
5439
5519
80
544
5599
5679
5759
5838
5918
5998
6078
6157
6237
6317
80
545
6397
6476
6556
6635
6715
6795
6874
6954
7034
7113
80
546
7193
7272
7352
7431
7511
7590
7670
7749
7829
7908
79
547
7987
8067
8146
8225
8305
8384
8463
8543
8622
8701
79
548
8781
8860
8939
9018
9097
9177
9256
9335
9414
9493
79
549
* 9572
9651
9731
9810
9889
9968
+047
0126
0205
0284
79
550
74 0363
0442
0521
0600
0678
0757
0836
0915
0994
1073
79
551
1152
1230
1309
1388
1467
1546
1624
1703
1782
1860
79
552
1939
2018
2096
2175
2254
2332
2411
2489
2568
2646
79
553
2725
2804
2882
2961
3039
3118
3196
3275
3353
3431
78
554
3510
3588
3667
3745
3823
3902
3980
4058
4136
4215
78
555
4293
4371
4449
4528
4606
4684
4762
4840
4919
4997
78
556
5075
5153
5231
5309
5387
5465
5543
5621
5699
5777
78
557
5855
5933
6011
6089
6167
6245
6323
6401
6479
6556
78
558
6634
6712
6790
6868
6945
7023
7101
7179
7256
7334
78
559
7412
7489
7567
7645
7722
7800
7878
7955
8033
8110
78
560
8188
8266
8343
8421
8498
8576
8653
8731
8808
8885
77
561
8963
9040
9118
9195
9272
9350
9427
9504
9582
9659
77
562
* 9736
9814
9891
9968
+045
0123
0200
0277
0354
0431
77
563
75 0508 i 0586
0663
0740
0817
0894
0971
1048
1125
1202
77'
564
1279
1356
1433
1510
1587
1664
1741
1818
1895
1972
77
565
2048
2125
2202
2279
2356
2433
2509
2586
2663
2740
77
566
2816
2893
2970
3047
3123
3200
3277
3353
3430
3506
77
567
3583
3660
3736
3813
3889
3966
4042
4119
4195
4272
77
568
4348
4425
4501
4578
4654
4730
4807
4883
4960
5036
76
569
5112
5189
5265
5341
5417
5494
5570
5646
5722
5799
76
570
5875
5951
6027
6103
6180
6256
6332
6408
6484
6560
76
571
6636
6712
6788
6864
6940
7016
7092
7168
7244
7320
76
572
7396
7472
7548
7624
7700
7775
7851
7927
8003
8079
76
573
8155
8230
8306
8382
8458
8533
8609
8685
8761
8836
76
574
8912
8988
9063
9139
9214
9290
9366
9441
9517
9592
76
575
* 9668
9743
9819
9894
9970
+045
0121
0196
0272
0347
75
576
76 0422
0498
0573
0649
0724
0799
0875
0950
1025
1101
75
577
1176
1251
1326
1402
1477
1552
1627
1702
1778
1853
75
578
1928
2003
2078
2153
2228
2303
2378
2453
2529
2604
75
579
2679
2754
2829
2904
2978
3053
3128
3203
3278
3353
75
3V.
1
3
3
4
5
e
7
H
r>-
APPENDIX.
727
LOGARITHMS OP NUMBERS.
IV.
o
133456
7
8
9
rK
580
76 3428
3503 3578
3653
3727
3802
3877
3952
4027
4101
75
581
4176
4251 4326
4400
4475
4550
4624
4699
4774
4848
75-
582
4923
4-998 5072
5147
5221
5296
5370
5445
5520
5594
75-
583
5669
5743 5818
5892 5966
6041
6115
6190
6264
6338
74
584
6413
6487
6562
6636 6710
6785
6859
6933
7007
7082
74
585
7156
7230
7304
7379
7453
7527
7601
7675
7749
7823
74
586
7898
7972
8046
8120
8194
8268
8342
8416
8490
8564
74
587
8638
8712
8786
8860
8934
9008
9082
9156
9230
9303
74
588
* 9377
9451
9525
9599
9673
9746
9820
9894
9968
+042
74
589
77 0115
0189
0263
0336
0410
0484
0557
0631
0705
0778
74
590
0852
0926
0999
1073
1146
1220
1293
1367
1440
1514
74
591
1587
1661
1734
1808
1881
1955
2028
2102
2175
2248
73
592
2322
2395
2468
2542
2615
2688
2762
2835
2908
2981
73
593
3055
3128
3201
3274
3348
3421
3494
3567
3640
3713
73
594
3786
3860
3933
4006
4079
4152
4225
4298
4371
4444
73
595
4517
4590
4663
4736
4809
4882
4955
5028
5100
5173
73
596
5246
5319
5392
5465
5538
5610
5683
5756
5829
5902
73
597
5974
6047
6120
6193
6265
6338
6411
6483
6556
6629
73
598
6701
6774
6846
6919
6992
7064
7137
7209
7282
7354
73
599
7427
7499
7572
7644
7717
7789
7862
7934
8006
8079
72
600
8151
8224
8296
8368
8441
8513
8585
8658
8730
8802
72
601
8874
8947
9019
9091
9163
9236
9308
9380
9452
9524
72
602
* 9596
9669 9741
9813
9885
9957
*029
0101
0173
0245
72
603
78 0317
0389
0461
0533
0605
0677
0749
0821
0893
0965
72
604
1037
1109
1181
1253
1324
1396
1468
1540
1612
1684
72
605
1755
1827
1899
1971
2042
2114
2186
2258
2329
2401
72
606
2473
2544
2616
2688
2759
2831
2902
2974
3046
3117
72
607
3189 3260
3332
3403
3475
3546
3618
3689
3761
3832 71
608
3904
3975
4046
4118
4189
4261
4332
4403
4475
4546 71
609
4617
4689
4760
4831
4902
4974
5045
5116
5187
5259
71
610
5330
5401
5472
5543 5615
5686
5757
5828
5899
5970
71
611
6041
6112
6183
6254 6325
6396
6467
6538
6609
6680
71
612
6751
6822 : 6893
6964
7035
7106
7177
7248
7319
7390
71
613
7460
7531 7602
7673
7744
7815
7885
7956
8027
8098
71
614
8168
8239
8310
8381
8451
8522
8593
8663
8734
8804
71
615
8875
8946
9016
9087
9157
9228
9299
9369
9440
9510
71
616
* 9581
9651
9722
9792
9863
9933
4004
0074
0144
0215
70
617
79 0285 0356
0426
0496
0567
0637
0707
0778
0848
0918
70
618
0988 1059
1129
1199
1269
1340
1410
1480
1550
1620
70
619
1691
1761
1831
1901
1971
2041
2111
2181
2252
2322
70
620
2392
2462
2532
2602
2672
2742
2812
2882
2952
3022
70
621
3092 1 3162
3231
3301
3371
3441
3511
3581
3651
3721
70
622
3790
3860
3930
4000
4070
4139
4209
4279
4349
4418
70
623
4488
4558
4627
4697
4767
4836
4906
4976
5045
5115
70
624
5185
5254
5324
5393
5463
5532
5602
5672
5741
5811
70
625
5880
5949
6019
6088
6158
6227
6297
6366
6436
6505
69
626
6574 6644
6713
6782
6852
6921
6990
7060
7129
7198
69
627
7268 7337
7406"
7475
7545
7614
7683
7752
7821
7890
69
628
7960 8029
8098
8167
8236
8305
8374
8443
8513
8582
69
629
8651
8720
8789
8858
8927
8996
9065
9134
9203
9272
69
630
9341
9409
9478
9547
9616
9685
9754
9823
9892
9961
69
631
80 0029
0098
0167
0236
0305
0373
0442
0511
0580
0648
69
632
.0717
0786
0854
0923
0992
1061
1129
1198
1266
1335
69
633
1404
1472
1541
1609
1678
1747
1815
1884
1952
2021
69
634
2089
2158
2226
2295
2363
2432
2500
2568
2637
2705
69
635
2774
2842
2910
2979
3047
3116
3184
3252
3321
3389
68
636
3457
3525
3594
3662
3730
3798
3867
3935
4003
4071
68
637
4139
4208
4276
4344
4412
4480
4548
4616
4685
4753
68
638
4821
4889
4957
5025
5093
5161 5229
5297
5365
5433
68
639
5501
5569
5637
5705
5773
5841 5908
5976
6044
6112
68
1
IV.
O
1 3
3
4
5 e
7
J>.
728
APPENDIX.
LOGARITHMS OP NUMBERS.
N.
1
3
3
4,
5
7
8
r>.
640
80 6180
6248
6316
6384
6451
6519
6587
6655
6723
6790
68
641
6858
6926
6994
7061
7129
7197
7264
7332
7400
7467
68
642
7535
7603
7670
7738
7806
7873
7941
8008
8076
8143
68
643
8211
8279
8346
8414
8481
8549
8616
8684
8751
8818
67
644
8886
8953
9021
9088
9156
9223
9290
9358
9425
9492
67
645
* 9560
9627
9694
9762
9829
9896
9964
+031
0098
0165
67
646
81 0233
0300
0367
0434
0501
0569
0636
0703
0770
0837
67
647
0904
0971
1039
1106
1173
1240
1307
1374
1441
1508
67
648
1575
1642
1709
1776
1843
1910
1977
2044
2111
2178
67
649
2245
2312
2379
2445
2512
2579
2646
2713
2780
2847
67
650
2913
2980
3047
3114
3181
3247
3314
3381
3448
3514
67
651
3581
3648
3714
3781
3848
3914
3981
4048
4114
4181
67
652
4248 4314
4381
4447
4514
4581
4647
4714
4780
4847
67
653
4913 4980
5046
5113
5179
5246
5312
5378
5445
5511
66
654
5578
5644
5711
5777
5843
5910
5976
6042
6109
6175
66
655
6241
6308
6374
6440
6506
6573
6639
6705
6771
6838
66
656
6904
6970
7036
7102
7169
723.",
7301
7367
7433
7499
66
657
7565
7631
7698
7764
7830
7896
7962
8028
8094
8160
66
658
8226
8292
8358
8424
8490
8556
8622
8688
8754
8820
66
659 8885
8951
9017
9083
9149
9215
9281
9346
9412
9478
66
660
* 9544
9610
9676
9741
9807
9873
9939
+004
0070
0136
66
661
82 0201
0267
0333
0399
0464
0530
0595
0661
0727
0792
66
662
0858
0924
0989
1055
1120
1186
1251
1317
1382
1448
66
663
1514
1579
1645
1710
1775
1841
1906
1972
2037
2103
65
664
2168
2233
2299
2364
2430
2495
2560
2626
2691
2756
65
665
2822
2887
2952
3018
3083
3148
3213
3279
3344
3409
65
666
3474
3539
3605
3670
3735
3800
3865
3930
3996
4061
65
667
4126
4191
4256
4321
4386
4451
4516
4581
4646
4711
65
668
4776
4841
4906
4971
5036
5101
5166
5231
5296
5361
65
669
5426
5491
5556
5621
5686
5751
5815
5880
5945
6010
65
670
6075
6140
6204
6269
6334
6399
6464
6528
6593
6658
65
671
6723
6787
6852
6917
6981
7046
7111
7175
7240
7305
65
672
7369
7434
7499
7563
7628
7692
7757
7821
7886
7951
65
673
8015
8080
8144
8209
8273
8338
8402
8467
8531
8595
64
674
8660
8724
8789
8853
8918
8982
9046
9111
9175
9239
64
675
9304
9368
9432
9497
9561
9625
9690
9754
9818
9882
64
676
* 9947
+011
0075
0139
0204
0268
0332
0396
0460
0525
64
677
83 0589
0653
0717
0781
0845
0909
0973
1037
1102
1166
64
678
1230
1294
1358
1422
1486
1550
1614
1678
1742
1806
64
679
1870
1934
1998
2062
2126
2189
2253
2317
2381
2445
64
680
2509
2573
2637
2700
2764
2828
2892
2956
3020
3083
64
681
3147
3211
3275
3338
3402
3466
3530
3593
3657
3721
64
682
3784
3848
3912
3975
4039
4103
4166
4230
4294
4357
64
683
4421
4484
4548
4611
4675
4739
4802
4866
4929
4993
64
684
5056
5120
5183
5247
5310
5373
5437
5500
5564
5627
63
685
5691
5754
5817
5881
5944
6007
6071
6134
6197
6261
63
686
6324
6387
6451
6514
6577
6641
6704
6767
6830
6894
63
687
6957
7020
7083
7146
7210
7273
7336
7399
7462
7525
63
688
7588
7652
7715
7778
7841
7904
7967
8030
8093
8156
63
689
8219
8282
8345
8408
8471
8534
8597
8660
8723
8786
63
690
8849
8912
8975
9038
9101
9164
9227
9289
9352
9415
63
691
* 9478
9541
9604
9667
9729
9792
9855
9918
9981
+043
63
692
84 0106
0169
0232
0294
0357
0420
0482
0545
0608
0671
63
693
0733
0796
0859
0921
0984
1046
1109
1172
1234
1297
63
694
1359
1422
1485
1547
1610
1672
1735
1797
1860
1922
63
695
1985
2047
2110
2172
2235
2297
2360
2422
2484
2547
62
696
2609
2672
2734
2796
2859
2921
2983
3046
3108
3170
62
697
3233
3295
3357
3420
3482
3544
3606
3669
3731
3793
62
698
3855
3918
3980
4042
4104
4166
4229
4291
4353
4415
62
699
4477
4539
4601 i 4664
4726
4788 4850
4912
4974
5036
62
! i 1
IV.
1
3 3
4 r,
7
S
r>.
APPENDIX.
729
LOGARITHMS OP NUMBERS.]
3V.
o
1
3
3
4,
5
6
7
s
9
i>.
700
84 5098
5160
5222
5284
5346
5408
5470
5532
5594
5656
62
701
5718
5780
5842
5904
5966
6028
6090
6151
6213
6275
62
702
6337
6399
6461
6523
6585
6646
6708
6770
6832
6894
62
703
6955
7017
7079
7141
7202
7264
7326
7388
7449
7511
62
704
7573
7634
7696
7758
7819
7881
7943
8004
8066
8128
62
705
8189
8251
8312
8374
8435
8497
8559
8620
8682
8743
62
706
8805
8866
8928
8989
9051
9112
9174
9235
9297
9358
61
707
9419
9481
9542
9604
9665
9726
9788
9849
9911
9972
61
708
85 0033
0095
0156
0217
0279
0340
0401
0462
0524
0585
61
709
0646
0707
0769
0830
0891
0952
1014
1075
1136
1197
61
710
1258
1320
1381
1442
1503
1564
1625
1686
1747
1809
61
711
1870
1931
1992
2053
2114
2175
2236
2297
2358
2419
61
712
2480
2541
2602
2663
2724
2785
2846
2907
2968
3029
61
713
3090
3150
3211
3272
3333
3394
3455
3516
3577
3637
61
714
3698
3759
3820
3881
3941
4002
4063
4124
4185
4245
61
715
4306
4367
4428
4488
4549
4610
4670
4731
4792
4852
61
716
4913
4974
5034
5095
5156
5216
5277
5337
5398
5459
61
717
5519
5580
5640
5701
5761
5822
5882
5943
6003
6064
61
718
6124
6185
6245
6306
6366
6427
6487
6548
6608
6668
60
719
6729
6789
6850
6910
6970
7031
7091
7152
7212
7272
60
720
7332
7393
7453
7513
7574
7634
7694
7755
7815
7875
60
721
7935
7995
8056
8116
8176
8236
8297
8357
8417
8477
60
722
8537
8597
8657
8718
8778
8838
8898
8958
9018
9078 60
723
9138
9198
9258
9318
9379
9439
9499
9559
9619
9679 i 60
724
* 9739
9799
9859
9918
9978
+038
0098
0158
0218
0278
60
725
86 0338
0398
0458
0518
0578
0637
0697
0757
0817
0877
60
726
0937
0996
1056
1116
1176
1236
1295
1355
1415
1475
60
727
1534
1594
1654
1714
1773
1833
1893
1952
2012
2072
60
728
2131
2191
2251
2310
2370
2430
2489
2549
2608
2668
60
729
2728
2787
2847
2906
2966
3025
3085
3144
3204
3263
60
730
3323
3382
3442
3501
3561
3620
3680
3739
3799
3858
59
731
3917
3977
4036
4096
4155
4214
4274
4333
4392
4452
59
732
4511
4570
4630
4689
4748
4808
4867
4926
4985
5045
59
733
5104
5163
5222
5282
5341
5400
5459
5519
5578
5637
59
734
5696
5755
5814
5874
5933
5992
6051
6110
6169
6228
59
735
6287
6346
6405
6465
6524
6583
6642
6701
6760
6819
59
736
6878
6937
6996
7055
7114
7173
7232
7291
7350
7409
59
737
7467
7526
7585
7644
7703
7762
7821
7880
7939
7998
59
738
8056
8115
8174
8233
8292
8350
8409
8468
8527
8586
59
739
8644
8703
8762
8821
8879
8938
8997
9056
9114
9173
59
740
9232
9290
9349
9408
9466
9525
9584
9642
9701
9760
59
741
* 9818
9877
9935
9994
+053
0111
0170
0228
0287
0345
59
742
87 0404
0462
0521
0579
0638
0696
0755
0813
0872
0930
58
743
0989
1047
1106
1164
1223
1281
1339
1398
1456
1515
58
744
1573
1631
1690
1748
1806
1865
1923
1981
2040
2098
58
745
2156
2215
2273
2331
2389
2448
2506
2564
2622
2681
58
746
2739
2797
2855
2913
2972
3030
3088
3146
3204
3262
58
747
3321
3379
3437
3495
3553
3611
3669
3727
3785
3844
58
748
3902
3960
4018
4076
4134
4192
4250
4308
4366
4424
58
749
4482
4540
4598
4656
4714
4772
4830
4888
4945
5003
58
750
5061
5119
5177
5235
5293
5351
5409
5466
5524
5582
58
751
5640
5698
5756
5813
5871
5929
5987
6045
6102
6160
58
752
6218
6276
6333
6391
6449
6507
6564
6622
6680
6737
58
753
6795
6853
6910
6968
7026
7083
7141
7199
7256
7314
58
754
7371
7429
7487
7544
7602
7659
7717
7774
7832
7889
58
755
7947
8004
8062
8119
8177
8234
8292
8349
8407
8464
57
756
8522
8579
8637
8694
8752
8809
8866
8924
8981
9039
57
757
9096
9153
9211
9268
9325
9383
9440
9497
9555
9612
57
758
* 9669
9726
9784
9841
9898
9956
+013
0070
0127
0185
57
759
88 0242
0299
0356
0413
0471
0528
0585
0642
0699
0756
57
IV.
1
3
3
4=
5
6
7
S
e
r>.
730
APPENDIX.
LOGARITHMS OF NUMBERS.
]V.
O
1
2 3
4. 5
G
7
8
O
r>
760
88 0814
0871
0928
0985
1042
1099
1156
1213
1271
1328
57
761
1385
1442
1499
1556
1613
1670
1727
1784
1841
1898
57
762
1955
2012
2069
2126
2183
2240
2297
2354
2411
2468
57
763
2525
2581
2638
2695
2752
2809
2866
2923
2980
3037
57
764
3093
3150
3207
3264
3321
3377
3434
3491
3548
3605
57
765
3661
3718
3775
3832
3888
3945
4002
4059
4115
4172
57
766
4229
4285
4342
4399
4455
4512
4569
4625
4682
4739
57
767
4795
4852
4909
4965
5022
5078
5135
5192
5248
5305
57
768
5361
5418
5474
5531
5587
5644
5700
5757
5813
5870
57
769
5926
5983
6039
6096
6152
6209
6265
6321
6378
6434
56
770
6491
6547
6604
6660
6716
6773
6829
6885
6942
6998
56
771
7054
7111
7167
7223
7280
7336
7392
7449
7505
7561
56
772
7617
7674
7730
7786
7842
7898
7955
8011
8067
8123
56
773
8179
8236
8292
8348
8404
8460
8516
8573
8629
8685
56
774
8741
8797
8853
8909
8965
9021
9077
9134
9190
9246
56
775
9302
9358
9414
9470
9526
9582
9638
9694
9750
9806
56
776
*9862
9918
9974
+030
0086
0141
0197
0253
0309
0365
56
777
89 0421
0477
0533
0589
0645
0700
0756
0812
0868
0924
56
778
0980
1035
1091
1147
1203
1259
1314
1370
1426
1482
56
779
1537
1593
1649
1705
1760
1816
1872
1928
1983
2039
56
780
2095
2150
2206
2262
2317
2373
2429
2484
2540
2595
56
781
2651
2707
2762
2818
2873
2929
2985
3040
3096
3151
56
782
3207
3262
3318
3373
3429
3484
3540
3595
3651
3706
56
783
3762
3817
3873
3928
3984
4039
4094
4150
4205
4261
55
784
4316
4371
4427
4482
4538
4593
4648
4704
4759
4814
55
785
4870
4925
4980
5036
5091
5146
5201
5257
5312
5367
55
786
5423
5478
5533
5588
5644
5699
5754
5809
5864
5920
55
787
5975
6030
6085
6140
6195
6251
6306
6361
6416
6471
55
788
6526
6581
6636
6692
6747
6802
6857
6912
6967
7022
55
789
7077
7132
7187
7242
7297
7352
7407
7462
7517
7572
55
790
7627
7682
7737
7792
7847
7902
7957
8012
8067
8122
55
791
8176
8231
8286
8341
8396
8451
8506
8561
8615
8670
55
792
8725
8780
8835
8890
8944
8999
9054
9109
9164
9218
55
793
9273
9328
9383
9437
9492
9547
9602
9656
9711
9766
55
794
* 9821
9875
9930
9985
+039
0094
0149
0203
0258
0312
55
795
90 0367
0422
0476
0531
0586
0640
0695
0749
0804
0859
55
796
0913
0968
1022
1077
1131
1186
1240
1295
1349
1404
55
797
1458
1513
1567
1622
1676
1731
1785
1840
1894
1948
54
798
2003
2057
2112
2166
2221
2275
2329
2384
2438
2492
54
799
2547
2601
2655
2710
2764
2818
2873
2927
2981
3036
54
800
3090
3144
3199
3253
3307
3361
3416
3470
3524
3578
54
801
3633
3687
3741
3795
3849
3904
3958
4012
4066
4120
54
802
4174
4229
4283
4337
4391
4445
4499
4553
4607
4661
54
803
4716
4770
4824
4878
4932
4986
5040
5094
5148
5202
54
804
5256
5310
5364
5418
5472
5526
5580
5634
5688
5742
54
805
5796
5850
5904
5958
6012
6066
6119
6173
6227
6281
54
806
6335
6389
6443
6497
6551
6604
6658
6712
6766
6820
54
807
6874
6927
6981
7035
7089
7143
7196
7250
7304
7358
54
808
7411
7465
7519
7573
7626
7680
7734
7787
7841
7895
54
809
7949
8002
8056
8110
8163
8217
8270
8324
8378
8431
54
810
8485
8539
8592
8646
8699
8753
8807
8860
8914
8967
54
811
9021
9074
9128
9181
9235
9289
9342
9396
9449
9503
54
812
* 9556
9610
9663
9716
9770
9823
9877
9930
9984
+037
53
813
91 0091
0144
0197
0251
0304
0358
0411
0464
0518
0571
53
814
0624
0678
0731
0784
0838
0891
0944
0998
1051
1104
53
815
1158
1211
1264
1317
1371
1424
1477
1530
1584
1637
53
816
1690
1743
1797
1850
1903
1956
2009
2063
2116
2169
53
817
2222
2275
2328
2381
2435
2488
2541
2594
2647
2700
53
818
2753
2806
2859
2913
2966
3019
3072
3125
3178
3231
53
819
3284
3337
3390
3443
3496
3549
3602
3655
3708
3761
53
3V.
O
1
3
3
4=
5
y
8
9
r>.
APPENDIX.
731
LOGARITHMS OP NUMBERS.
NT.
O
1
2
3
4,
5
6 7
8
O
r>.
820
91 3814
3867
3920
3973
4026
4079
4132
4184
4237
4290
53
821
4343
4396
4449
4502
4555
4608
4660
4713
4766
4819
53
822
4872
4925
4977
5030
5083
5136
5189
5241
5294
5347 53
823
5400
5453
5505
5558
5611
5664
5716
5769
5822
5875
53
824
5927
5980
6033
6085
6138
6191
6243
6296
6349
6401
53
825
6454
6507
6559
6612
6664
6717
6770
6822
6875
6927
53
826
6980
7033
7085
7138
7190
7243
7295
7348
7400
7453
53
827
7506
7558
7611
7663
7716
7768
7820
7873
7925
7978
52
828
8030
8083
8135
8188
8240
8293
8345
8397
8450
8502
52
829
8555
8607
8659
8712
8764
8816
8869
8921
8973
9026
52
830
9078
9130
9183
9235
9287
9340
9392
9444
9496
9549
52
831
* 9601
9653
9706
9758
9810
9862
9914
9967
+019
0071
52
832
92 0123
0176
0228
0280
0332
0384
0436
0489
0541
0593
52
833
0645
0697
0749
0801
0853
0906
0958
1010
1062
1114
52
834
1166
1218
1270
1322
1374
1426
1478
1530
1582
1634
52
835
1686
1738
1790
1842
1894
1946
1998
2050
2102
2154
52
836
2206
2258
2310
2362
2414
2466
2518
2570
2622
2674
52
837
2725
2777
2829
2881
2933
2985
3037
3089
3140
3192
52
838
3244
3296
3348
3399
3451
3503
3555
3607
3658
3710
52
839
3762
3814
3865
3917
3969
4021
4072
4124
4176
4228
52
840
4279
4331
4383
4434
4486
4538
4589
4641
4693
4744
52
841
4796
4848
4899
4951
5003
5054
5106
5157
5209
5261
52
842
5312
5364 5415
5467
5518
5570
5621
5673
5725
5776
52
843
5828
5879
5931
5982
6034
6085
6137
6188
6240
6291
51
844
6342
6394
6445
6497
6548
6600
6651
6702
6754
6805
51
845
6857
6908
6959
7011
7062
7114
7165
7216
7268
7319
51
846
7370
7422
7473
7524
7576
7627
7678
7730
7781
7832
51
847
7883
7935
7986
8037
8088
8140
8191
8242
8293
8345 51
848
8396
8447
8498
8549
8601
8652
8703
8754
8805
8857 51
849
8908
8959
9010
9061
9112
9163
9215
9266
9317
9368
51
850
9419
9470
9521
9572
9623
9674
9725
9776
9827
9879
51
851
* 9930
9981
+032
0083
0134
0185
0236
0287
0338
0389
51
852
93 0440
0491
0542
0592
0643
0694
0745
0796
0847
0898
51
853
0949
1000
1051
1102
1153
1204
1254
1305
1356
1407
51
854
1458
1509
1560
1610
1661
1712
1763
1814
1865
1915
51
855
1966
2017
2068
2118
2169
2220
2271
2322
2372
2423
51
856
2474
2524
2575
2626
2677
2727
2778
2829
2879
2930
51
857
2981
3031
3082
3133
3183
3234
3285
3335
3386
3437
51
858
3487
3538
3589
3639
3690
3740
3791
3841
3892
3943
51
859
3993
4044
4094
4145
4195
4246
4296
4347
4397
4448
51
860
4498
4549
4599
4650
4700
4751
4801
4852
4902
4953
50
861
5003
5054
5104
5154
5205
5255
5306
5356
5406
5457
50
862
5507
5558
5608
5658
5709
5759
5809
5860
5910
5960
50
863
6011
6061
6111
6162
6212
6262
6313
6363
6413
6463
50
864
6514
6564
6614
6665
6715
6765
6815
6865
6916
6966
50
865
7016
7066
7117
7167
7217
7267
7317
7367
7418
7468
50
866
7518
7568
7618
7668
7718
7769
7819
7869
7919
7969
50
867
8019
8069
8119
8169
8219
8269
8320
8370
8420
8470
50
868
8520
8570
8620
8670
8720
8770
8820
8870
8920
8970 ! 50
869
9020
9070
9120
9170
9220
9270
9320
9369
9419
9469
50
870
9519
9569
9619
9669
9719
9769
9819
9869
9918
9965
50
871
94 0018
0068
0118
0168
0218
0267
0317
0367
0417
0467
50
872
0516
0566
0616
0666
0716
0765
0815
0865
0915
0964
50
873
1014
1064
1114
1163
1213
1263
1313
1362
1412
1462
50
874
1511
1561
1611
1660
1710
1760
1809
1859
1909
1958
50
875
2008
2058
2107
2157
2207
2256
2306
2355
2405
2455
50
876
2504
2554
2603
2653
2702
2752
2801
2851
2901
2950
50
877
3000
3049
3099
3148
3198
3247
3297
3346
3396
3445
49
878
3495
3544
3593
3643
3692
3742
3791
3841
3890
3939
49
879
3989
4038
4088
4137
4186
4236
4285
4335
4384
4433
49
W.
O
1
9
3
4=
5
6
7
8
r>.
732
APPENDIX.
LOGARITHMS OP NUMBERS.
IV.
O
1
2
3
4=
5
7
8 9
I>.
880
94 4483
4532
4581
4631
4680
4729
4779
4828
4877 I 4927
49
881
4976
5025
5074
5124
5173
5222
5272
5321
5370
5419
49
882
5469
5518
5567
5616
5665
5715
5764
5813
5862
5912
49
883
5961
6010
6059
6108
6157
6207
6256
6305
6354
6403
49
884
6452
6501
6551
6600
6649
6698
6747
6796
6845
6894
49
885
6943
6992
7041
7090
7140
7189
7238
7287
7336
7385
49
886
7434
7483
7532
7581
7630
7679
7728
7777
7826
7875
49
887
7924
7973
8022
8070
8119
8168
8217
8266
8315
8364
49
888
8413
8462
8511
8560
8609
8657
8706
8755
8804
8853
49
889
8902
8951
8999
9048
9097
9146
9195
9244
9292
9341
49
890
9390
9439
9488
9536
9585
9634
9683
9731
9780
9829
49
891
* 9878
9926
9975
+024
0073
0121
0170
0219
0267
0316
49
892
95 0365
0414
0462
0511
0560
0608
0657
0706
0754
0803
49
893
0851
0900
0949
0997
1046
1095
1143
1192
1240
1289
49
894
1338
1386
1435
1483
1532
1580
1629
1677
1726
1775
49
895
1823
1872
1920
1969
2017
2066
2114
2163
2211
2260
48
896
2308
2356
2405
2453
2502
2550
2599
2647
2696
2744
48
897
2792
2841
2889
2938
2986
3034
3083
3131
3180
3228
48
898
3276
3325
3373
3421
3470
3518
3566
3615
3663
3711
48
899
3760
3808
3856
3905
3953
4001
4049
4098
4146
4194
48
ii
900
4243
4291
4339
4387
4435
4484
4532
4580
4628
4677
48
901
4725
4773
4821
4869
4918
4966
5014
5062
5110
5158
48
902
5207
5255
5303
5351
5399
5447
5495
5543
5592
5640
48
903
5688
5736
5784
5832
5880
5928
5976
6024
6072
6120
48
904
6168
6216
6265
6313
6361
6409
6457
6505
6553
6601
48
905
6649
6697
6745
6793
6840
6888
6936
6984
7032
7080
48
906
7128
7176
7224
7272
7320
7368
7416
7464
7512
7559
48
907
7607
7655
7703
7751
7799
7847
7894
7942
7990
8038
48
908
8086
8134
8181
8229
8277
8325
8373
8421
8468
8516
48
909
8564
8612
8659
8707
8755
8803
8850
8898
8946
8994
48
910
9041
9089
9137
9185
9232
9280
9328
9375
9423
9471
48
911
9518
9566
9614
9661
9709
9757
9804
9852
9900
9947
48
912
* 9995
+042
0090
0138
0185
0233
0280
0328
0376
0423
48
913
96 0471
0518
0566
0613
0661
0709
0756
0804
0851
0899
48
914
0946
0994
1041
1089
1136
1184
1231
1279
1326
1374
47
915
1421
1469
1516
1563
1611
1658
1706
1753
1801
1848
47
916
1895
1943
1990
2038
2085
2132
2180
2227
2275
2322
47
917
2369
2417
2464
2511
2559
2606
2653
2701
2748
2795
47
918
2843
2890
2937
2985
3032
3079
3126
3174
3221
3268
47
919
3316
3363
3410
3457
3504
3552
3599
3646
3693
3741
47
920
3788
3835
3882
3929
3977
4024
4071
4118
4165
4212
47
921
4260
4307
4354
4401
4448
4495
4542
4590
4637
4684
47
922
4731
4778
4825
4872
4919
4966
5013
5061
5108
5155
47
923
5202
5249
5296
5343
5390
5437
5484
5531
5578
5625
47
924
5672
5719
5766
5813
5860
5907
5954
6001
6048
6095
47
925
6142
6189
6236
6283
6329
6376
6423
6470
6517
6564
47
926
6611
6658
6705
6752
6799
6845
6892
6939
6986
7033
47
927
7080
7127
7173
7220
7267
7314
7361
7408
7454
7501
47
928
7548
7595
7642
7688
7735
7782
7829
7875
7922
7969
47
929
8016
8062
8109
8156
8203
8249
8296
8343
8390
8436
47
930
8483
8530
8576
8623
8670
8716
8763
8810
8856
8903
47
931
8950
8996
9043
9090
9136
9183
9229
9276
9323
9369
47
932
9416
9463
9509
9556
9602
9649
9695
9742
9789
9835
47
933
* 9882
9928
9975
+021
0068
0114
0161
0207
0254
0300
47
934
97 0347
0393
0440
0486
0533
0579
0626
0672
0719
0765
46
935
0812
0858
0904
0951
0997
1044
1090
1137
1183
1229
46
936
1276
1322
1369
1415
1461
1508
1554
1601
1647
1693
46
937
1740
1786
1832
1879
1925
1971
2018
2064
2110
2157
46
938
2203
2249
2295
2342
2388
2434
2481
2527
2573
2619
46
939
2666
2712
2758
2804
2851
2897
2943
2989
3035
3082
46
N.
O
1
2
3
4
.5
O
7
S
r>.
APPENDIX.
733
LOGARITHMS OF NUMBERS.
N.
1
2
3
4:
5
O
7
8
o ! r>.
940
97 3128
3174
3220
3266
3313
3359
3405
3451
3497
3543
46
941
3590
3636
3682
3728
3774
3820
3866
3913
3959
4005
46
942
4051
4097
4143
4189
4235
4281
4327
4374
4420
4466
46
943
4512
4558
4604
4650
4696
4742
4788
4834
4880
4926
46
944
4972
5018
5064
5110
5156
5202
5248
5294
5340
5386
46
945
5432
5478
5524
5570
5616
5662
5707
5753
5799
5845
46
946
5891
5937
5983
6029
6075
6121
6167
6212
6258
6304
46
947
6350
6396
6442
6488
6533
6579
6625
6671
6717
6763
46
948
6808 i 6854
6900
6946
6992
7037
7083
7129
7175
7220
46
949
7266
7312
7358
7403
7449
7495
7541
7586
7632
7678
46
950
7724
7769
7815
7861
7906
7952
7998
8043
8089
8135
46
951
8181
8226
8272
8317
8363
8409
8454
8500
8546
8591
46
952
8637
8683
8728
8774
8819
8865
8911
8956
9002
9047
46
953
9093
9138
9184
9230
9275
9321
9366
9412
9457
9503
46
954
9548
9594
9639
9685
9730
9776
9821
9867
9912
9958
46
955
98 0003
0049
0094
0140
0185
0231
0276
0322
0367
0412
45
956
0458
0503
0549
0594
0640
0685
0730
0776
0821
0867
45
957
0912
0957
1003
1048
1093
1139
1184
1229
1275
1320
45
958
1366
1411
1456
1501
1547
1592
1637
1683
1728
1773
45
959
1819
1864
1909
1954
2000
2045
2090
2135
2181
2226
45
960
2271
2316
2362
2407
2452
2497
2543
2588
2633
2678
45
961
2723
2769
2814
2859
2904
2949
2994
3040
3085
3130
45
962
3175
3220
3265
3310
3356
3401
3446
3491
3536
3581
45
963
3626
3671
3716
3762
3807
3852
3897
3942
3987
4032
45
964
4077
4122
4167
4212
4257
4302
4347
4392
4437
4482
45
965
4527
4572
4617
4662
4707
4752
4797
4842
4887
4932
45
966
4977
5022
5067
5112
5157
5202
5247
5292
5337
5382
45
967
5426
5471
5516
5561
5606
5651
5696
5741
5786
5830
45
968
5875
5920
5965
6010
6055
6100
6144
6189
6234
6279
45
969
6324
6369
6413
6458
6503
6548
6593
6637
6682
6727
45
970
6772
6817
6861
6906
6951
6996
7040
7085
7130
7175
45
971
7219
7264
7309
7353
7398
7443
7488
7532
7577
7622
45
972
7666
7711
7756
7800
7845
7890
7934
7979
8024
8068
45
973
8113
8157
8202
8247
8291
8336
8381
8425
8470
8514
45
974
8559
8604
8648
8693
8737
8782
8826
8871
8916
8960
45
975
9005
9049
9094
9138
9183
9227
9272
9316
9361
9405
45
976
9450
9494
9539
9583
9628
9672
9717
9761
9806
9850
44
977
* 9895
9939
9983
+028
0072
0117
0161
0206
0250
0294
44
978
99 0339
0383
0428
0472
0516
0561
0605
0650
0694
0738
44
979
0783
0827
0871
0916
0960
1004
1049
1093
1137
1182
44
980
1226
1270
1315
1359
1403
1448
1492
1536
1580
1625
44
981
1669
1713
1758
1802
1846
1890
1935
1979
2023
2067
44
982
2111
2156
2200
2244
2288
2333
2377
2421
2465
2509
44
983
2554
2598
2642
2686
2730
2774
2819
2863
2907
2951
44
984
2995
3039
3083
3127
3172
3216
3260
3304
3348
3392
44
985
3436
3480
3524
3568
3613
3657
3701
3745
3789
3833
44
986
3877
3921
3965
4009
4053
4097
4141
4185
4229
4273
44
987
4317
4361
4405
4449
4493
4537
4581
4625
4669
4713
44
988
4757
4801
4845
4889
4933
4977
5021
5065
5108
5152
44
989
5196
5240
5284
5328
5372
5416
5460
5504
5547
5591
44
990
5635
5679
5723
5767
5811
5854
5898
5942
5986
6030
44
991
6074
6117
6161
6205
6249
6293
6337
6380
6424
6468
44
992
6512
6555
6599
6643
6687
6731
6774
6818
6862
6906
44
993
6949
6993
7037
7080
7124
7168
7212
7255
7299
7343
44
994
7386
7430
7474
7517
7561
7605
7648
7692
7736
7779
44
995
7823
7867
7910
7954
7998
8041
8085
8129
8172
8216
44
996
8259
8303
8347
8390
8434
8477
8521
8564
8608
8652
44
997
8695
8739
8782
8826
8869
8913
8956
9000
9043
9087
44
998
9131
9174
9218
9261
9305
9348
9392
9435
9479
9522
44
999
9565
9609
9652
9696
9739
9783
9826
9870
9913
9957
43
N.
O
1
2
3
4
5
6
7
S
r>.
734 APPENDIX.
The Application of Logarithms. The logarithm of a number is set down as a decimal,
and addition of ciphers to numbers does not change the logarithm ; it is the same for 11,
110, 1100, but the value of the number is established by figures to the left of the decimal
point ; thus, if the number is among the units, the characteristic is ; if in the tens, 1 ;
in the hundreds, 2 ; thousands, 3 ; tens of thousands, 4, and so on ; if the number is a
decimal fraction and the first figure a tenth, the characteristic is 1, if hundredths 2, thou-
sandths 3~
Multiplication of two numbers is performed by the addition of their logarithms and
characteristics, and finding the number corresponding to their sum ; thus, to multiply 119
by 2760.
Characteristic of 119 2, logarithm. 2-075547
" 2760 3, " 3-440909
5-516456
3284 403
401 D = 132)53(401
328440-1 528
200
132
68
As the characteristic is 5, the result is 6 figures of whole numbers.
Division is performed by subtracting the logarithm of the divisor from that of the divi-
dend, and finding the logarithm of the remainder for the quotient. But if the divisor is
the larger, then the characteristic of the remainder is .
Thus, to divide 500 by 63008.
Logarithm of 500 2-698970
Logarithm of 63000 = 4-799341
Logarithm of 63008 4.799396
Corresponding number -007985 = 3-899574
Numbers are raised to any power by multiplying their logarithm by the exponents, and
roots are extracted by dividing the logarithm. Thus, to get the square of any number, its
logarithm is multiplied by 2, for the cube by 3, for the 4th power by 4 ; in like manner, to
obtain the square root of the number, divide the logarithm by 2 ; by 3 for */ ; by 4
for*/-
The roots of numbers are better expressed by fractional exponents, thus: V# by a 1/a >
tya by a 1 8 .
The raising of numbers to different powers is extremely simple, by logarithms, when
the numbers are whole numbers, but becomes somewhat more complicated when the num-
bers are decimals.
Thus, to find the 4th power of -07.
Logarithm -07 2-845098
_ 4
8 3-380392
Number -00002401 5-380392
To extract the 4th root of -07
Logarithm -07 2-845098
Add 2 to the characteristic to make it
divisible by 4, and a positive 2 to the 2-2-845098
logarithm to balance it. 4)4'2 -845098
Number -5143 1- 711274
APPENDIX. 735
The exponent of a root is often a decimal ; thus the */'07 may be expressed by *07' 8B .
Logarithm -07 2-845098
-25
4225490
1690196
"5-21127450
5-5
Number -5143 1-71127450
NOTE. In this example, *5 is added to the resultant characteristic to bring it to an integer, and
an equal positive amount to the logarithm to balance it.
The same logarithm as by dividing by 4- and corresponding to the number -5143. The
rule is to consider the logarithm as a plus quantity, and multiply by the exponent and the
characteristic as minus, and. after similar multiplication, subtract it from the first product.
When a characteristic has a minus sign (3), and it is to be subtracted, the sign is changed
-and added.
Thus, to divide 10- by T V
Logarithm 10- 1-00000
rV I _
Logarithm of 100- 2-000
To divide T V Logarithm 1-00000
b 2
_
Logarithm of 10- 1-0000
To divide y^ Logarithm 3-00000
by 100 2
Logarithm of -00001 5
INDEX.
Acoustics applied to rooms, etc., 530, 542
Adcock's table of teeth, 280.
Air, flow of, through pipes, 689.
Alphabets, 65.
Anchors for floor-beams, 473.
Animals, forms of, 650.
Apartment-houses, plans for, 516.
Apothecaries' weight, 674.
Arch bridges, 432.
table of, 437.
Arch, Roman cylindrical masonry, 475.
Arches, 574.
ARCHITECTURAL DRAWING, 461-601.
Architecture, orders of, Byzantine, 572.
Composite, 571.
Corinthian, 569.
Domestic, 461.
Doric, 566.
Gothic, 572.
Greek and Roman, 564.
Ionic, 569.
of Houses, 461.
Roman, 571.
Tuscan, 566.
Architectural ornament, 590.
Areas of circles, 691.
Ashti reservoir, 379.
Asphalt pavement, 404.
Atkinson, Edward, on use of ropes in place of
belts, 274.
Automatic valves, 335.
Averaging speed of floats by diagram, 72.
Avoirdupois weight, 674.
Axle, differential, 208.
Axles, 245.
Backing paper and drawings, 58.
Ballast for roads, 406.
Balloon frame, 469.
Barns, 542.
Bases, 588.
Basilicas, plans of, 532.
Bath-tubs, 498.
'47
Beams, strength of composite, 238.
of iron, 230.
of wood, 226.
Beam, working, of engine, 322.
Bearings for shafts, 245.
Bearing, suspension, for upright shaft, 258.
Bed-rooms, 496.
Belgian pavement, 403.
Belts, 270.
horse-power of, 273.
ropes instead of, 274, 299.
Bends or angles in pipes, 562.
Beton, 190.
Bevel-wheel, isometrical projection of, 630.
projections of, 288.
Bismarck Bridge foundation, 432.
pier, 432.
Bituminous cement, 190.
Blast-pipes, table of losses of pressure per 100
feet, 690.
table for equalizing diameter of, 690.
Blinds, framing, 479.
Blocks, gin, 301.
tackle, 301.
Blue-print process, 164.
Board, drawing, 55.
Boiler tubes, weight, etc., 677.
Boilers, flue, 349.
Hartford Steain-Boiler and Inspection Co., 438.
horizontal tubular, 346.
locomotive, 442.
marine, 442.
setting, 438.
Shapley, 349.
vertical, 350.
Bolts and nuts, 239.
Bonne's projection, 168.
Bonomi, Joseph, proportions of the human frame
by, 643.
Boston Water- Works conduit, 393.
Box-car, New York Central and Hudson River
Railroad, 453.
Bracing, general principles of, 407.
738
INDEX.
Branches in pipes, 562.
Brass, Thurston's graphic representation of
strength of, 195.
plates, weight of, 676.
rods, weight of, 678.
tubes, weight of, 678.
wire, weight of, 676.
Bridges, and roofs, 407.
arch, 432.
Bismarck, 432.
ferry-landing, 431.
Howe truss, 421.
iron deck lattice-girder, 427.
iron plate-girder, 424.
skew arch, 435.
suspension, 437.
table of arch, 437.
table of suspension, 438.
truss combination, 424.
trusses, rules for, 419.
wooden truss, 421.
wrought-iron truss, 428.
Bricks, 188.
weight of, 190.
Brooklyn, N. Y., conduit of water-works, 392.
sewers, 398.
pipe-joints, 395.
Building materials, 182.
artificial, 188.
Buildings in the city of New York, extracts from
acts relating to, 665.
Burden's rivets, weight of, 679.
Buttresses, 576.
Byzantine church plan, 532.
Campaniles, 577.
Canals. 384.
Erie, 384.
locks, 386.
locks, specifications for New York State canals,
388.
Northern, at Lowell, Mass., 385.
Capacity, measures of, 673.
Capitals, 588.
Car, box, New York Central and Hudson River
Railroad, 453.
Pennsylvania passenger, 453.
Carriage-house, 542.
Castings, 192.
Cast-iron columns, strength of, 221.
Ceilings, brick, Italian, 475.
Cement, 189.
bituminous, 190.
Central Park gravel roads, 405.
Center of gravity, 200.
Chain cables, 303.
Chain wheel, 302.
Chains and ropes of equal strength, 300.
Chimney-tops, 548.
Chimneys, 442.
for houses, 493.
Church, Gothic, plan, 532-538.
Romanesque, plan, 532.
Churches, English, at Hague, 534.
Greek, Roman, English, Byzantine, Basilica,
532.
London Wesleyan, 534.
Roman Catholic Cathedral, New York, 535.
St. Bartholomew, New York, 535.
Circles, properties of, 671.
table of circumference and areas, 691.
Classification of masonry, 185.
Closets, 497.
Clutch couplings, 264.
Coals, 199.
Coffer-dam, 365.
Cohoes dam, 378.
head gates, 380.
Columns, strength of cast-iron, 221.
strength of wrought-iron, 224.
Combination truss-bridge, 424.
Compound steam-engines, 216.
compasses, 45.
Composite order of architecture, 571.
Concrete floors arched and groined, 474.
Conduit of the Croton Aqueduct, New York, 392.
of the Boston Water-Works, 393.
of Nassau Water-Works, Brooklyn, 392.
Conduits for water, 390
Cone pulleys, 269.
Cones, solidity of, 31, 672.
Connecting-rods, 313.
Connections for rods, 312.
Contents, to calculate, 144.
Contour lines, 152, 153, 162.
Conventional colors for topography, 171.
Conventional signs, 149.
for metals, 191.
geological, 160.
marine, 160.
statistical, 162.
Copper plates, weight of, 676.
rods, weight of, 678.
tubes, weight of, 678.
wire, weight of, 676.
Copying by blue-print process, 164.
by ferro-prussiate process, 164.
by transfer-paper, 165.
Copying-glass, 165.
Corinthian order of architecture, 569.
Cornices, 588.
plaster, 495.
INDEX.
739
Counter-shaft, 269.
Couplings, slide or clutch, 264.
for shafts, 260.
Cow-houses, 545.
Cranks, engine, 305.
hand, 305.
Crib, dock, 366.
Cross-section paper, 157.
uses of, 69.
Cross-sections, railroad, 157.
Croton Aqueduct, conduit, 392.
dam, 375.
new receiving reservoir, 394.
Cube, isometrical projection of, 625.
Cube roots, table of, 696.
Cubes, table of, 696.
Cubic measure, 674.
Culvert, isometrical projection of, 631.
Curbs, 403.
Curved lenses, isometrical projection of, 629.
Curves, 42.
Cylinders, solidity of, 672.
steam, 325.
water, 326.
Dam, Ashti Reservoir, 379.
Dams, 374.
Cohoes, 378.
Croton, 375.
head-gates for, 380.
Holyoke, 375.
Lowell, Merrimack River, 376.
De Lorgne's projection, 170.
Design, principles of architectural, 598.
Development of surfaces, 104.
Differential screw, 208.
axle, 208.
Dining-rooms, 496.
Distribution water-works, 395.
Dividers, 45.
Dock, crib, 366.
Dome of brick and concrete, 476.
Domes and vaults, 574.
Domestic architecture, 461.
Doors, sliding, 478.
folding, 479.
framing, 476.
Doorways, 584.
Doric order of architecture, 566.
DRAWING INSTRUMENTS, 40-77.
Drawing-pen, exercises with, 61.
Drinker, H. S., method of timbering tunnels, 449.
Driven wells, tubes for, weight, etc., of, 678.
Dry measure, 674.
Dynamic force, 210.
table, 674.
Eccentrics, 309.
projections of, 309.
Elevators, 521.
Ellipse, to. find the area or circumference, 672.
Embankment, Ashti Reservoir, 379.
ENGINEERING DRAWING, 362-460.
English churches, 532.
Equalizing diameter of blast-pipes, table for, 690.
Erie Canal, 384.
rates compared with New York Central and
Hudson River Railroad by diagram, 71.
Evaporation from reservoirs, 374.
Expansion, table of mean pressures in steam-
cylinders at different rates of expansion, 682.
Falling bodies, velocity of, etc., 210.
Fanning, J. F., table of flow of water through
pipes, 685.
Farm Pond head-gates, Boston Water- Works, 384.
Ferro-prussiate paper for copying, 164.
Ferry-landing bridge, 431.
Figure-drawing, human, Bonomi, 643.
Villard de Hennecourt's, 645.
Finishing topographical map, 174.
Fire-places, 492.
Fire-proof French floors, 474.
concrete floors, 474.
Fire-resisting floors, 472, 474.
Flats. See Apartment-houses.
Flooring, 470.
Floors, brick arch and iron beams, 474.
concrete, arched and groined, 474.
fire-resisting, 474.
mill fire-retarding, 472.
single and double, 472.
Flow of water, 683.
Flue boilers, 3491
Flues, 547.
for houses, 492.
Flumes, 390.
Forces, parallelogram of, 208.
Foundations, 181.
Bismarck Bridge, 432.
coffer-dam, 365.
concrete, 362.
iron piles, 364.
machine, 449.
pile, 363.
sheet-piling, 364.
steam-engine, 449.
stone, 362.
Susquehanna Bridge, 373.
timber, 362.
under water, 371.
Frame, balloon, 469:
houses, 468.
740
INDEX.
Frames, 355.
Framing, 468.
doors, 476.
roofs, 493.
scarfing, lapping, 473.
windows, sash, and blinds, 479.
Francis, J. B., formula for flow over weirs, dia-
gram of, 683.
FREE-HAND DRAWING, 639-664.
drawing, elementary exercises in, 639.
Freight shed, wood, 419.
French flats. See Apartment-houses.
Friction, 211.
Morin's experiments on, 212.
Frictional gearing, 297.
Fteley, A., formula for flow through sewers, 685.
Furnaces, hot air, 549.
Galvanized iron, spiral riveted pipes, weight, etc.,
678.
Gas, for lighting, 564.
flow of, through cast-iron mains, 689.
supply, 401.
Gearing. 275.
frictional, 297.
mortise-wheels, 283.
projections of bevel-wheels, 288.
projections of spur-wheel, 284.
teeth of, 277.
wedge, 299.
worm, 296.
Geometrical definitions, 2.
GEOMETRICAL PROBLEMS, CONSTRUCTION OF, 1-39.
Gin-blocks, 301.
Glass, 197.
sizes of cylinder and plate, 482.
Globular or equidistant projection of the sphere,
166.
Glue, mouth, 57.
Gothic architecture, 572.
Gothic church-plan, 532-538.
Grade of roads (table), 405.
Graphic diagrams belts, the power of, 273.
charges for transport of merchandise on rail-
road and canal, 71.
crank eyes, 306.
movements of a float in a canal, 72.
rainfall, temperature, and mortality, 74.
speed and resistance of railway -trains, 73.
steam-expansion in single and compound cylin-
ders, 215.
strength of wrought-iron columns, 225.
strength of wrought-iron girders, 237.
strength of wrought-iron shafts, 248.
-teeth of wheels, 277.
Thurston's strength of alloys, 19(5.
Graphic diagrams time-table of railroad, 72.
water-flow through pipes, 686.
water-flow through sewers, 688.
weights and measures, 76.
Gravel-roads in Central Park, 405.
Gravity, center of, 200.
Greek architecture, orders of, 564.
Greek churches, 532.
Greenhouses, 546.
Groin, Roman, 475.
Gutters, forms of, 493.
Halls, music, 541 ; legislative, 541.
Hand-valves, 337.
Hangers, 254.
Seller's, 260.
Head-gates, 380.
Cohoes dam, 380.
Farm Pond, Boston Water- Works, 384.
made at Holyoke, Mass., 390.
Heating, methods of, 549.
open fires, 549.
steam and hot water, 551.
stoves, 549 ; hot-air furnaces, 549.
Helix, 102.
Hills, Von Eggloffstein's system of representing,
154.
representation of, 152.
Hoists, power, 521.
Holyoke dam, 375.
Hoofs of animals, 651.
Hooks, form of, 303.
Hoosac Tunnel, method of timbering, 453.
Horses, movements of, 651.
Horse-power, etc., 213.
of belts, 273.
Hospitals, 542.
Hot-water heating apparatus, 551.
Houses, architecture of, 461.
elevation of high-stoop houses, 513.
frame, 468.
plans for apartment, 516.
plans and elevations of country residences, 509.
plans and elevations of, in Queen Anne style,
503.
plans for rooms in, 498.
plans of tenement, 513.
Howe truss-bridges, 421.
Human frame, proportions of, by Joseph Bonomi,
643.
Hydrants, 341.
Hydraulic press, 210.
Hydrometrical surveys, 159.
Illuminating-tile, 516.
Inches in decimals of a foot, 673.
INDEX.
741
Inclined forces, 206.
plane, 205.
Indicator cards, 216.
Ink, China, 60.
Instruments, management of, 59.
Ionic order of architecture, 569.
Iron, weight of rolled, 676.
weight of wrought plates, 676.
wire, weight of, 676.
ISOMETRICAL DRAWING, 625-638.
Isometrical projection, 633.
Joinings of timber, 472.
Joints of Brooklyn pipes, 395.
riveted, 342.
Joists, size of, 471.
Journals, 245-251.
Keys, 249.
Land-plans, railroad, 158.
Lands, division of U. S., 147.
Lanza on strength of wooden posts, 221.
Lapping, timber, 473.
Latitudes and departures, table of, 704.
Lead pipe, weight of, 680.
plates, weight of, 676.
Lecture-rooms, 541.
Legislative halls, 541.
Lehmann'a system of representing slopes, 153.
Lettering for maps, 174.
Letters, samples of, 65.
Levers, 202.
form of hand, 304.
form of foot, 304.
Lineal measure, 672.
Liquid measure, 673.
Locks of canals, 386.
Locomotive boilers, 442.
Logarithms, application and use of, 734.
of numbers, table of, 71$.
Lowell dam, Merrimack River, 376.
water-power, 214.
Macadam roads, 404.
Machine and blacksmith shop, perspective view,
etc., 521.
MACHINE DESIGN AND MECHANICAL CONSTRUC-
TIONS, 220-361.
Machine-foundations, 449.
Machines, location of, 444.
Man-holes for sewers in New York, 399.
Mantel-piece, 492.
Map, finishing topographical, 174.
projections, 165.
Maps, lettering, 174.
Maps, railway, 156.
titles, 176.
transferring, 162.
United States Coast Survey, table for project-
ing maps, 168.
Marine boilers, 442.
Marine surveys, 159.
Masonry, classification of, 185.
technical terms of, 185.
MATERIALS, 181-199.
Materials, building, 182.
bricks, 189.
coal, flame, steam, 199.
glass, rubber, 198.
metals, 197.
mortars and cements, 191.
stone, 188.
wood, 185.
Mechanical work or effect, 212.
MECHANICS, 200-219.
Melting-point of metals, 195.
Mensuration, 67 i.
Mercator's projection, 171 .
Meridional parts, table of, 171.
Metals, 191.
conventional signs, 191.
crushing strength, 195.
melting-point, 195.
specific gravity, 195.
tensile strength, 195.
weight, 175.
Metric system, diagram of equivalent values, 70.
Moldings, 586, 589, 594.
Greek and Roman names and forms, 483.
Morin's experiments on friction, 212.
Mortality shown by diagram, 73.
Mortars, 189.
Mortise-wheels, 283.
Mounting paper and drawings, 58.
Music-halls, 541.
Nails, weight and length, 680.
Nature, drawings from, directions for, 653.
New York building laws, extracts from, 665.
New York Central and Hudson River Railroad
rates compared with Erie Canal by diagram,
71.
New York Central and Hudson River Railroad
box-car, 453.
New York city sewer catch-basins, 400.
sewer man-holes, 399.
streets, widths, etc., 402.
New York docks, bulkhead- walls, 365.
New York, New Haven and Hartford Railroad.
diagram of time-table, 71.
Northern Canal, Lowell, Mass., 385.
742
INDEX.
Noses of animals, 653.
Organs, 535.
Ornament, architectural, 590.
Ornaments of the Renaissance, 596.
ORTHOGRAPHIC PROJECTION, 78-1U9.
Paints, 198.
Pantagraph, 54.
Pantries, 497.
Paper, backing, 58.
cross-section, profile, 157.
fixing down, 57.
mounting, 58.
stretching, 57.
uses of cross-section, 69.
varieties and sizes of, 56.
Parabola, area of, 612.
Parallel motion, 324.
ruler, 42.
Parallelogram of forces, 208.
Parallelepipeds, solidity of, 672.
Parapets, 594.
Paris streets, 402.
Parlors, 496.
Partitions, 469.
Passage-ways in houses, 497.
Patent-Office drawings, directions for, 670.
Pavement, asphalt, 404.
Belgian, 403.
wooden, 404.
Paws of animals, 651.
Pen, drawing, 43.
exercises with drawing, 61.
Penetrations or Intersections of solids, 90.
Pennsylvania passenger-car, 453.
PERSPECTIVE DRAWING, 602-624.
Perspective, angular, 612.
parallel, 604.
scale for drawing, 608.
Pew, size of, etc., 531, 535. '
Piers of stone, 431.
stone, Bismarck Bridge, 432.
wooden pile, 431.
wrought iron, 432.
Pile foundation?, 363.
piers, 431.
Piles, iron, 364.
Piling, shoot, 364.
Pillow-block, 252.
isometrical projection of, 631.
Pinion and rack, 291.
Pipe-connections. 350.
Pipe, lead, weight of, 680.
joints of Brooklyn, 395.
Pipes, diagram of flow of water through, 685.
Pipes, galvanized iron, spiral, riveted, weight,
etc., 678.
table for equalizing the diameter of blast-
pipe, 690.
table of losses of, pressure per 100 feet in
blast-pipes, 690.
Pistons, 327.
Plastering, 190, 494.
PLOTTING, 137-148.
Plumber block, 252.
Plumbing, 555.
water-supply, 555-563.
Polyconnic projection, 168.
Posts, strength of wooden, 221.
Power, horse, 213.
steam, 214.
water, 214.
Press, hydraulic, 210.
Pressure, table of loss of, per 100 feet in blast-
pipes, 690.
Prisms, solidity of, 672.
Privies, 497.
Profiles, railroad, 157.
Profile paper, 157.
PROJECTION, ORTHOGRAPHIC, 78-109.
Bonne's, 168.
De Lorgne's, 170.
globular or equidistant, 166.
Mercator's, 171.
polyconic, 168.
stereographic, 167.
Projections of simple bodies, 81.
for maps, 165.
Protractor, 53.
Pulley, 204.
Pulleys, 266.
cone, 269.
speed of, 266.
Pyramids, solidity of, 672.
Queen Anne style, plans and elevations for house
of, 503.
Rack and pinion, 293.
Railroad cross-sections, 157.
land plans, 158.
profiles, 157.
Railroads, ballast for, 406.
sections of rail, 406.
Rails, sections for railroads, 406.
Railway maps, 156.
stock, 453.
Reservoir, new Croton Receiving, 394.
Reservoirs, Ashti, 379.
receiving, 394.
Retaining walls, 365.
INDEX.
743
Retaining walls, crib docks, 366.
New York docks, 365.
Thames embarkment, 369.
Riveted joints, 342.
Rivets, Burden's, weight of, 679.
Roads, 402.
ballast for, 406.
Central Park gravel, 405.
Macadam, 404.
table of grades, 405.
Robertson's grooved-surface frictional gearing,
299.
Rolled iron, table of weight of, 675.
Romanesque church-plan, 532.
Roman orders of architecture, 564, 571.
churches, 532.
cylindrical masonry arch, 475.
groin, 475.
Roofs, Gothic church, 538.
Roof-truss, isometrical projection of, 631.
Roofs and bridges, 407.
bracing for wooden, 409.
framing, forms of, 493.
of iron, 414.
wooden freight-shed, 419.
Rooms and passages, sizes, arrangement, and
proportions of, 495.
Ropes and chains of equal strength, 300.
strength of, 301.
used as belts, 274, 299.
Rubber, 198.
Rulers, 40.
Russell, J. Scott, wave-line principle of ship-con-
struction, 458.
Safety-valves, 341.
Sash, framing, 479.
.Saunders's experiments on sound, 530.
Scale of perspective drawing, 608.
guard, 49.
Scales, 47.
Scarfing, timber, 473.
School-house, isometrical projection of, 631.
School-houses, ventilation and light, 530.
plans and elevation of New York city,
527.
plans and elevation of Cleveland city, 527.
plans, elevations, etc., 521.
Screws, 205, 241, 294.
Screw, wheel and endless, 296.
differential, 208.
Seats in general, space occupied by, 531.
Sewers, 398.
catch-basins and man-holes of New York, 399,
400.
diagrams and formula of flow through, 685.
Sewers, isometrical projection of, in Thames Em-
bankment, 631.
large street, Washington, D. C., 399.
of Brooklyn, N. Y., 398.
overflow and outlet of the Victoria and Re-
gent Streets sewers, Thames Embankment,
371.
Sewer pipe-connections, 555.
SHADES AND SHADOWS, 110-136.
Shade-lines, 107.
Shading and shadows, manipulation of, 126.
elaboration of, 129.
Shaft, counter, 269.
Shafting, 249.
Shafts, 245.
couplings for, 260.
upright, 256.
Shapley boiler, 349.
Shearing stress, 225.
Sheet-piling, 364.
Ship-construction, wave-line principle by J. Scott
Russell, 458.
Sidewalks, 402.
Sines and cosines, table of natural, 710.
Sinks, 557.
Skew-arch bridges, 435.
Slide couplings, 264.
Slopes, United States Coast Survey system of rep-
resenting, 153.
Lehmann's system of representing, 153.
Solid measure, 674.
Soil-pipe connections, 555.
Sound, Saunders's experiments on, 530.
Specials, water-pipe, 396.
Specific gravity of metals, 195.
Sphere, globular or equidistant projection of, 166.
area of, 672.
solidity of, 672.
Spikes, wrought, weight of, 679.
Spires, 578.
Spur-wheel (internal) driving a pinion, 293.
driven by a pinion, 294.
projections of, 284.
Square roots, table of, 696.
Squares, T, 41.
table of, 696.
Stables, 542.
Stairs, iron, 490.
framing, etc., 485.
Stalls for horses, 543.
Standard, 253.
Static force, 200.
Steam-cylinders, 325.
table of mean pressures in, at different rates of
expansion, 682.
Steam-engine, 214.
INDEX.
Steam-engine, compound, 216.
foundation, 449.
indicator cards, 216.
Steam-heating apparatus, 551.
Steam-power, 214.
Steam, table of propei'ties of saturated, 681.
Step for upright shaft, 256.
Stereographic projection, 167.
Stones, 185.
varieties of, 187.
weight of, 191.
Streams, flow of, 374.
Strength of brass alloy, 195.
of cast-iron columns, 221.
of composite beams, 238.
of iron beams, 230.
of metals, 195.
of wooden beams, 226.
of wooden posts, 221.
of wrought-iron columns, 224.
Stores, plans and elevations of, 516.
Stories, height of, in houses, 497.
height of, in stores, 516.
String courses, 588.
Stoves, 549.
Streets, asphalt pavement, 404.
Belgian pavement, 403.
carnage- way, 403.
curbs, 403.
Macadam, 404.
of New York, widths, etc., 402.
of Paris, 402.
sidewalks, 402.
wooden pavement, 404.
Stress, 220.
shearing, 225.
torsional, 225.
transverse, 226.
Stuffing-boxes, 330.
Sulphur, 196.
Sunday-school room, 535.
Surface, measures of, 673.
Surveys, hydrometrical, 159.
marine, 159.
Suspension-bridges, 437.
table of, 438.
Susquehanna Bridge foundations, 373.
curves, 42.
Table for projecting maps, 168.
of meridional parts, 171.
traverse. See Appendix.
Tables of areas of circles, 691.
blast-pipes, equivalent areas of, 690.
blast-pipes, losses of pressure in, 690.
boilers, number of tubes, 346.
Tables of boilers, stay-bolts, 347.
boilers, weight of tubes, 677.
bolts and nuts, 244.
brass plates, tubes, rods, wire, weight of, 676.
bridges, arch, 432.
bridges, suspension, 438.
chains and ropes, equivalent strength, 300.
circles, circumferences and areas, 691.
copper plates, tubes, rod, wire, weight of, 676.
cubes and cube roots, 696.
expansion, mean pressures at different rates of,
cut off, 682.
gas-pipes, weight of, 402.
gears, teeth of, 280.
hooks, proportions of, 304.
iron angle and channel, 235.
iron plates, tubes, rods, wire, weight of, 675.
iron, safe load of cast-iron columns, 222.
iron, safe load of wrought-iron columns, 223.
iron, safe load of I beams, 233.
iron tubes and couplings, sizes of, 352.
journals, dimensions of, 245.
latitudes and departures, 704.
lead in joints of pipes, 396.
lead pipes, sizes and weights, 680.
logarithms, 719.
maps for meridional parts, 171.
maps for projections of, 168.
metals and alloys, weight and strength, 195.
nails, weight of, 680.
rivets, pitch of, 343.
rivets, weight of, 679.
sheaves, sizes of, 301.
sines and cosines, natural, 710.
spikes, weight of, 679.
squares and square roots, 696.
steam, properties of, 681.
theatres, dimensions of, 541.
valves, dimensions of, 336, 339.
water-discharge over weirs, 684.
water-pipes, dimensions of, 396.
water, weight of, 680.
weights and measures, 672.
wooden beams, safe loads, 229.
working-beams of engines, 324.
Tackle-blocks, 301.
Teeth of gearing, 277.
Adcock's table of, 280.
Tenement-houses, plans of, 513.
Thames Embankment, river wall, 369.
Theatres, plans, 539.
Ferguson's plan, 540.
Wagner's, 540.
table of dimensions, 541.
Thurston's graphic representation of strength of
brass alloys, 195.
INDEX.
745
Tile, illuminating, 516.
Tinting, methods of, 126.
Titles for maps, 176.
Topographical map-finishing, 174.
TOPOGRAPHICAL DRAWING, 149-180.
Topography, colored, 171.
conventional colors, 171.
Torsional stress, 225.
Towers, 577, 580.
Transfer-paper for copying, 165.
Transferring maps, 162.
Transverse stress, 226.
Traps in pipes, 562.
Traverse-table, 704, 710.
use of, in plotting, etc., 142.
Triangle and square, use of, 33.
Triangles, 40.
properties of, 671.
Troy weight, 674.
Truss, isometrical projection of roof, 631.
Trusses, bridge, rules for, 419.
Tubes for boiler, weight of, 677.
driven wells, weight of, 678.
weight, etc., of wrought-iron welded, 677.
Tubular boilers, 346.
Tunnels, method of working, 449.
Tunnel, Hoosac, method of timbering, 453.
Tuscan order of architecture, 566.
Type, samples of, 65.
Upright shafts, 256.
Urinals, 563.
United States Coast Survey tables for projecting
maps, 168.
system of representing slopes, 153.
United States lands, division of, etc., 147.
Valves, automatic, 335.
hand, 337.
safety, 341.
steam cylinder, 331.
Varnishing drawings, etc., 58.
Vaulting, Greek and Roman, 574.
Gothic, 575.
Vaults and domes, 574.
Velocity of falling bodies, 210.
Ventilation and warming in general, 547.
Vestry-rooms, 535.
Villard de Hennecourt's design of human figures,
645.
Von Eggloff stem's system of representing hills,
154.
Walls, retaining, 365.
brick, stone, concrete, 467.
bulkhead, of New York docks, 365.
Walls, construction of, 461.
of Northern Canal, at Lowell, Mass., 385.
wooden, 468.
Washers, 244.
Washington, D. C., largest sewer, 399.
Wash-tubs, 558.
Water-closets, 497, 559-563.
Water cylinders, 326.
Water, flow of, 683.
flow through pipes, 685.
power, 214.
power at Lowell, 214.
supply, 555.
table of discharge of weir one foot long,
684.
weight of, 680.
works distribution, 395.
works distribution, house services, 397.
works distribution, specials, 396.
works distribution, specifications for Brooklyn
pipe, 397.
Wave-line principle of ship-construction, by J.
Scott Russell, 458.
Weaving-room, location of machinery, 444.
Wedge, 205.
gearing, 299.
solidity of, 672.
Weights and measures, 672.
Weight, comparison of, 674.
of brick, 190.
of materials, 182, 191.
of metals, 195.
of stones, 191.
of woods, 185.
Weir, table of discharge of, one foot long, 684.
Wheel and axle, 203.
and endless screw, 296.
isometrical projection of bevel, 630.
Windows, various forms of, 581.
dormer, 479.
sash and blinds, 479.
Wooden posts, Lanza on the strength of, 221.
pavement, 404.
Woods, weight of, 185.
characteristics and use of, 183.
representation of, 182.
Working-beam of engine, 322.
Worm-gearing, 296.
Wrought-iroii columns, 222.
plates, weight of, 676.
strength of, 224.
welded tubes, weight, etc., of, 677.
Yoke-hanger, 254.
Zinc plates, weight of, 676.
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GEOLOGICAL MAP OF
NEW JERSEY
GEORGE. H.COOK,STATE GEOLOGIST
&P'
FORMATIONS
CONGLOMERATES
SHALES
TRAP
RED SANDSTONE
ANDTRAP ROCKS
GRAVELLY EART
GLASS SAND AND
SHALES. ROOFING SLATES
AND
SLATY SANDSTONES
SANDY CLAYS
UPPER MARL BED
CAUDA GALLI CRIT
ORISKANY SANDSTONE
MIDW-E MARL BED
RED/SAND RED
LOWER MARLBED
LAMINATED SAND
FOSSILIFEROUS LIMESTE
MAGNESIAN LIMESTONE
LR HELDERBERGLIME
STONE AND WAT- LIME
RED SLATESANDSAND
STONE
SANDSTONE AND
GONG. OF KITTATINNYMT
E.5LATYBRI
AND GREEN POND MT
CONGLOMERATE
AND CLAY MARLS
POTTERS, AND FIRE
CLAYS AND SANDS
GRISTALLINE LIMESTONE
PLJvL
OUTLET
PLOT.
'L.XDI.
PL XV.
PL. XVI.
PL XVII.
PL XVIII.
' PL XIX.
PL. XX.
SCRAPS.
IT has been my practice for many years to collect, from the circulars of
mechanics and their agents, and from illustrated newspapers and magazines,
varied illustrations of tools and machines, engineering structures, buildings,
etc., and arrange them under their appropriate heads in scrap-books. *They
have been found very useful in assisting me in designs, not only enabling me
the more readily to make drawings, but to convey to the draughtsman the
character and proportions of the design which I wish to have made. And those
parts which are of common use and purchasable in the market can be readily
arranged in position and executed more economically than from a new design.
There is a saving in the matter of drawing, and a saving in the cost of con-
struction.
By a proper combination and arrangement of parts which have practically
served a purpose, a more satisfactory design can be made than from attempts
at originality. Knowledge of what has been done is economy in all labor.
When the thing itself can not be seen bodily, its picture can supply its place,
and its details can be studied at leisure ; and, as the education of the eye is
of essential importance to the draughtsman, let him see as much as he can
practically, but yet acquire a good collection of scraps from which to design.
There are few constructions from which something of education can not be
drawn, parts if not a whole.
In this view a small collection of scraps has been made pertinent to the
book. Its page does not admit of the sizes which will be found in the illustrated
papers and magazines the quarto will be found much more generally useful .
and a library of such scrap-books will furnish material for a draughtsman which
can not be found in any encyclopaedia.
48
SCRAPS.
SCRAPS.
SCRAPS.
Hydraulic, Stop-Valve.
Hydraulic Release- Valve.
Lever- Gate.
Elliptic Spring.
Half Elliptic Spring.
Vose Graduated Spring.
Volute Spring.
Oval Bar Spring
SCRAPS.
Compound Steam Cylinders. H. M. S. Spartan.
Wrought-Iron Plates and Covers.
Compressed-Air Locomotive, St. Oothard Tunnel.
SCRAPS.
Three- Throw Crank.
Forged weight, 24 tons 11 cwt.
Finished " 15 " 8 "
Weight, 25 tons 10 cwt.
SCRAPS.
Screw Propeller.
Vessel, 1400 gross tons. Engines, 130 nominal English horse-power.
SCRAPS.
TURBINES.
CENTRAL DISCHARGE.
Turbine with Horizontal Shaft.
Longest draft tube, at Manchester, N. H. 26 feet from center of shaft to tail water. Fall, 40 feet.
SCRAPS.
SCRAPS.
AMERICAN LOCOMOTIVES.
The Consolidation.
The Mogul.
Twelve- Wheeler, Central Pacific Railroad.
SCRAPS.
ss
I
SCRAPS.
Third Avenue Elevated Railroad.
Curve at Eighth Avenue.
SCRAPS.
BUILDERS' HARDWARE.
Mortise-Lock, cover off.
Front
Boxed Strike, Front. Sli&ing-door
Loch.
Thumb-Piece.
Knob and Rose.
Escutcheons.
SCRAPS.
Sash-Lifts.
Hook and Eye.
Shutter-
Knob.
SCKAPS.
Examples of Ancient Hinges and Doors.
Balusters.
Cast-Iron Tread.
SCRAPS.
;!
J
S
i
ID I
n
X
1
fl
. \
I
f
i
ED
1 ~|
i
JEESf
FEET
Plan, Section, and Elevation of a Wooden Mantel and Fire-Place.
SCRAPS.
Examples of Inlaid Floors or Marquetry.
49
SCRAPS.
SCRAPS.
'
\\
E
i In
,
1
,
I
|
L,
r^^
x
g L^
2- J
SCRAPS.
Enameled Tile.
Terra Coiia.
SCRAPS.
SOKAPS.
SCKAPS.
SCRAPS,
SCRAPS.
SCRAPS.
SCRAPS.
SCRAPS.
SCRAPS.
SCRAPS.
SCRAPS.
SCRAPS.
SCRAPS,
SCRAPS.
SCRAPS.
Central Park, New York City.
SCRAPS.
Coney Island.
SCRAPS.
Corny Island.
<L
*Hh
14 DAY USE
RETURN TO DESK FROM WHICH BORROWED
LOAN DEPT.
This book is due on the last date stamped below, or
on the date to which renewed.
Renewed books are subject to immediate recall.
tef
LD 21A-60m-4,'64
(E4555slO)476B
General Library
University of California
Berkeley
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