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ENGINEERING
DESCRIPTIVE GEOMETRY

^Engineering
Descriptive Geometry

A Treatise on Descriptive Geometry as the Basis of Mechanical

Drawing, Explaining Geometrically the Operations

Customary in the Draughting Room

P. W. BARTLETT

commandIer, u. s. navy

HEAD OF DEPARTMENT OF MARINE ENGINEERING AND NAVAL CONSTRUCTION
AT THE UNITED STATES NAVAL ACADEMY

AND

THEODORE W. JOHNSON

A. B., M.E.

PROFESSOR OF MECHANICAL DRAWING, UNITED STATES NAVAL ACADEMY
MEMBER OF AMERICAN SOCIETY OF MECHANICAL ENGINEERS

or THf
UNIVERSITY

NEW YORK
JOHN WILEY & SONS

London : CHAPMAN & HALL, Limited
1910

1^3

fi%/Mt

BALTIMORE, MD., V. S. A.

'^ OF THE

UNIVERSITY

PEEFACE.

The aim of this work is to make Descriptive Geometry an
integral part of a course in Mechanical or Engineering Drawing.

The older books on Descriptive Geometry are geometrical rather
than' descriptive. Their authors were interested in the subject as a
branch of mathematics, not as a branch of drawing.

Technical schools should aim to produce engineers rather than
mathematicians, and the subject is here presented with the idea
that it may fit naturally in a general course in Mechanical Drawing.
It should follow that portion of Mechanical Drawing called Line
Drawing, whose aim is to teach the handling of the drawing instru-
ments, and sliould precede courses specializing in the various
branches of drawing, such as Mechanical, Structural, Architectural,
and Topographical Drawing, or the " Laying Off " of ship lines.

The various branches of drawing used in the different industries
may be regarded as dialects of a common language. A drawing is
but a written page conveying by the use of lines a mass of informa-
tion about the geometrical shapes of objects impossible to describe
in words without tedium and ambiguity. In a broad sense all these
branches come under the general term Descriptive Geometry. It
is more usual, however, tp speak of them as branches of Engineer-
ing Drawing, and that term may well be used as the broad label..

The term descriptive Geometry will be restricted, therefore, to-
the common geometrical basis or ground work on which the various
industrial branches rest. This ground work of mathematical laws,
is unchanging and permanent.

The branches of Engineering Drawing have each their own
abbreviations and special methods adapting them to their own
particular fields, and these- conventional methods change from time
to time, keeping pace with changing industrial methods.

Descriptive Geometry, though unchanged in its principles, has
recently undergone a complete change in point of view. In
changing its purpose from a matliematical one to a descriptive one,
or, from being a training for the geometrical powers of a mathema-
tician to being a foundation on which to build up a knowleclge of

203782

Ti Preface

some branch of Engineering Drawing, the number and position of
the planes of projection commonly used are altered. The object is
now placed behind the planes of projection instead of in front of
them, a change often spoken of as a change from the " 1st quad-
rant " to the " 3d quadrant," or from the French to the American
method. We make this change, regarding the 3d quadrant method
as the only natural method for American engineers. All tlie prin-
ciples of Descriptive Geometry are as true for one method as for the
other, and the industrial branches, as Mechanical Drawing, Struc-
tural Drawing, etc., as practiced in this country, all demand this
method.

In addition, the older geometries made practically no use of a
third plane of projection, and we take in this book the further step
of regarding the use of three planes of projection as the rule, not
the exception. To meet the common practice in industrial branches,
we use as our most prominent metliod of treatment, or tool, the
auxiliary plane of projection, a device which may be called the
draftsman's favorite method, but which in books is very little
noticed.

As the work is intended for students who are but just taking up
geometry of three dimensions, in order to inculcate by degrees a
power of visualizing in space, we begin the subject, not with the
mathematical point in space but with a solid tangible object shown
by a perspective drawing. No exact construction is based on the
perspective drawings which are freely used to make a realistic ap-
pearance. As soon as the student has grasped the idea of what
orthographic projection is, knowledge of how to make the projection
is taught by the constructive process, beginning with the point and
passing through the line to the plane. To make the subject as
tangible as possible, the finite straight line and the finite portion of
a plane take precedence over the infinite line and plane. These
latter require higher powers of space imagination, and are therefore
postponed until the student has had time to acquire such powers
from the more naturally understood branches of the subject.

F. W. B.
T. W. J.

Mabch, 1910.

CONTENTS.

CHAPTER PAGE

I. Nature of Orthographic Projection 1

II. Orthographic Projection of the Finite Straight Line. .. 18

III. The True Length of a Line in Space 27

IV. Plane Surfaces and Their Intersections and Develop-

ments 38

V. Curved Lines 49

VI. Curved Surfaces and Their Elements 62

VII. Intersections of Curved Surfaces 70

VIII. Intersections of Curved Surfaces; Continued 81

IX. Development of Curved Surfaces 91

X. Straight Lines of Unlimited Length and Their Traces. . 98

XI. Planes of Unlimited Extent: Their Traces 108

XII. Various Applications 121

XIII. The Elements of Isometric Sketching 133

XIV. Isometric Drawing as an Exact System 142

Set of Descriptive Drawings 149

OF THE

UNIVERSITY

=£d^lFORH^

CHAPTER I.
NATURE OF ORTHOGRAPHIC PROJECTION.

1. Orthographic Projection. — The object of Mechanical Draw-
ing is to represent solids with such mathematical accuracy and
precision that from the drawing alone the object can be built or
constructed without deviating in the slightest from the intended
shape. As a consequence the " working drawing " is the ideal
sought for, and any attempt at artistic or striking effects as in
" show drawings '' must be regarded purely as a side issue of minor
importance. . Indeed mechanical drawing does not even aim to
give a picture of the object as it appears in nature, but the views
are drawn for the mind, not the eye.

The shapes used in machinery are bounded by surfaces of mathe-
matical regularity, such as planes, cylinders, cones; and surfaces
of revolution. They are not random surfaces like the surface of a
lump of putty or other surfaces called " shapeless." These definite
shapes must be represented on the flat surface of the paper in an
unmistakable manner.

The method chosen is that known as *'*' orthographic projection/*
If a plane is imagined to be situated in front of an object, and
from any salient point, an edge or corner, a perpendicular line,
called a "projector," is drawn to the plane, this line is said to
project the given point upon the plane, and the foot of this perpen-
dicular line is called the projection of the given point. If all
salient points are projected by this method, the orthographic draw-
ing of the object is formed.

2. Perspective Drawing. — The views we are accustomed to in
artistic and photographic representations are " Perspective Views."
They seek to represent objects exactly as they appear in nature.
In their case a plane is supposed to be erected between the human
eye and the object, and the image is formed on the plane by sup-
posing straight lines drawn from the eye to all salient points of

Engineerixg Descriptive Geometry

the object. Where these lines from the eye, or " Visual Rays/' as
they are called, pierce the plane, the image is formed.

Fig. 1 represents the two contrasted methods applied to a simple
object, and the customary nomenclature.

An orthographic view is sometimes called an " Infinite Perspec-
tive View,'' as it is the view which could only be seen by an eye at
an infinite distance from the object. " The Projectors " may then
be considered as parallel visual rays which meet at infinity, where
the eye of the observer is imagined to be.

Projector

Perspectjve View.

Orthographic View.

Fig. 1.

3. The Regular Orthographic Views. — Since solids have three
" dimensions," length, breadth and thickness, and the plane of the
paper on which the drawing is made has but two, a single ortho-
graphic view can express two only of the thrjee dimensions of the
object, but must always leave one indefinite. Points and lines at
different distances from the eye are drawn as if h^ing in the same
plane. From one view only the mind can imagine them at dif-
ferent distances by a kind of guess-work. If two views are made
from different positions, each view may supplement the other in
the features in which it is lacking, and so render the representa-
tion entirely exact. Theoretically two views are always required
to represent a solid accurately.

To make a drawing all the more clear, other views are generally
advisable, and three views may be taken as the average requirement
for single pieces of machinery. Six regular views are possible,
however, and an endless number of auxiliary views and " sections "
in addition. For the present, we shall consider only the " regular
views," which are six in number.

Nature of Orthographic Projection

4. Planes of Projection. — A solid object to be represented is
supposed to be surrounded by planes at short distances from it, the
planes being perpendicular to each other. From each point of
every salient edge of the object, lines are supposed to be drawn
perpendicular to each of the surrounding planes, and the succes-
sion of points where these imaginary projecting lines cut the planes
are supposed to form the lines of the drawings on these planes.
One of the planes is chosen for the plane of the paper of the actual
drawing. To bring the others into coincidence with it, so as to
have all of them on one flat sheet, they are imagined to be unfolded
from about the object by revolving them about their lines of inter-
section with each other. These lines of intersection, called " axes
of projection," separate the flat drawing into different views or
elevations.

Fia. 2.

Engineering Descriptive Geometry

Fig. 2a.

Fig. 2 is a true perspective drawing of a solid object and the
planes as they are supposed to surround it. This figure is not a
mechanical drawing, but represents the mental process by which
the mechanical drawing is supposed to be formed by the projection
of the views on the planes. In this case the planes are supposed
to be in the form of a perfect cube. The top face of the cube shows
the drawing on that face projected from the solid by fine dotted
lines. Eemember that these fine dotted lines are supposed to be
perpendicular to the top plane. This drawing on the top plane is
called the " plan." On the front of the cube the " front view " or
" front elevation '^ is drawn, and on the right side of the cube is

Nature of Orthographic Projection 5

the " right side elevation/^ Three other views are supposed to be
drawn on the other faces of the cube, but they are shown on Fig.
2a, which is the perspective view of the cube from the opposite
point of view, that is, from the back and from below instead of
from in front and from above.

This method of putting the object to be drawn in the center of
a cube of transparent planes of projection is a device for the im-
agination only. It explains the nature of the "projections," or
" views," which are used in engineering drawing.

5. Development or Flattening Out of the Planes of Projection. —
Xow imagine the six sides of the cube to be flattened out into one
plane forming a grouping of six squares as in Fig. 3. What we

ir— -

Left Side
Elevat/on

Plan

^ pROfNT

Elevation

BOTTOM

View

(Right) Side
Elevation

^ Back
Elevation

Fig. 3.

have now is a descriptive or mechanical drawing of the object
showing six " views.'' The object itself is now dispensed with and
its projections are used to represent it. These six views are what
we call the "regular views." With one slight change they cor-
respond to the regular set of drawings of a house which architects
make.

Engineering Descriptive Geometry

The set of six " regular " projections would not be altered by
passing the transparent planes at nnequal distances from each
other, so long as they surround the object and are mutually per-
pendicular. They may form a rectangular parallelopiped instead
of a cube without altering the nature of the views.

It will be noticed also that views on opposite faces of the cube
differ but little. Corresponding lines in the interior may in one
case be full lines and in the other " broken lines.^' Broken lines
(formed by dashes about -J" long, with spaces of tV") represent
parts concealed by nearer portions of the object itself. All edges
project upon the plane faces of the cube, forming lines on the draw-

Plan

^ Front
Elevat/on

(Ri&HT) Side-
Elevation

Plan

(Right) Side
Elevation -

-RON!
EVAT

foN

Fig. 4.

Fig. 5.

ings, the edges concealed by nearer portions of the object forming
broken lines.

6. The Reference Planes and Principal Views. — In drawings of
parts of machinery six regular views are usually unnecessary. The
three views shown in Fig. 2 are the " Principal Views," and others
are needed only occasionally. The planes of those views are the
" Eeference Planes."

These views, when flattened from their supposed position about
the object into one plane, give the grouping in Fig. 4.

Another arrangement of the same views, obtained by unfolding
the planes of the cube in a different order, is shown in Fig. 5.
These two arrangements are standard in mechanical drawing, and
are those most used.

Nature of Orthographic Projection 7

7. The Nomenclature. — The nomenclature adopted is as follows :
The " Eeferenee Planes," or three principal planes of projection,
are called from their position, the Horizontal Plane, or fl, the
Vertical Plane, or V, and the (right) Side Plane, or g. The plane
S is by some called the " Profile Plane/' The point 0 (Fig. 2),
in which they meet, is the " Origin " of coordinates. The line
OX, in which H and V intersect, is called the " Axis of X" or
" Ground Line.'' The line OY, in which ff and S meet, is called
the "x\xis of F/' and the line OZ, in which V and S meet, is
called the " Axis of Z." The three axes together are called the
" Axes of Projection."

Since drawings are considered as held vertically before the face,
it is considered that the plane V coincides at all times with the
" Plane of the Paper." In unfolding the planes from their .posi-
tions in Fig. 2 to that in Fig. 4, it is considered that the plane H
has been revolved about the axis of X (line OX), through an angle
of 90°, until it stands vertically above V- ^^ the same way S is
considered to be revolved about the line OZ, or axis of Z, until it
takes its place to the right of V-

The arrangement in Fig. 5 corresponds to a different manner of
revolving the plane §. It is revolved about the axis of Y (OF)
until it coincides with the plane H, and is then revolved with H?
about the axis of X, until both together come into the plane of the
paper, or V-

The three other faces of the original cube of planes of projection
are appropriately called H', V'? and S'. On account of the simi-
larity of the views on them, to those on H, V and g, they are but
little used, g' alone is fairly common since a grouping of planes
ff, V and S' is at times more convenient than the standard group
H, V and S.

8. Meaning of " Descriptive Geometry." — The aim of Engineer-
ing or Mechanical Drawing is to represent the shapes of solid
objects which form parts of structures or machines. It shows
rather the shapes of the surfaces of the objects, surfaces which are
usually composed of plane, cylindrical, conical, and other surfaces.
In the drawing room, by the application of mathematical laws and
principles, views are constructed. These are usually Plan, Front

8 - Engineering Descriptive Geometry

■c

Elevation, and Side Elevation, and are exactly such views as would
be obtained if the object itself were put within a cage of trans-
parent planes, and the true projections formed.

It is these mathematical laws or rules which form the subject
known as Descriptive Geometry. A drawing made in such a way
as to bring out clearly these fundamental laws of projection, by the
use of axes of projection, etc., may be conveniently called a " De-
scriptive Drawing.'^

In the practical application of drawing to industrial needs,
short-cuts, abbreviations, and special devices are much used (their
nature depending on the special branch of industry for which the
drawing is made). In addition, the axes of projection are usually
omitted or left to the imagination, no particular effort being made
to show the exact mathematical basis provided the drawing itself
is correct. Such a drawing is a typical " Mechanical Drawing."
By the addition of axes of projection, and similar devices, it may
be converted into a strict " Descriptive Drawing."

9. The Descriptive Drawing of a Point in Space. — The imagi-
nary process of making a descriptive drawing consists in putting
the object within a cube of transparent planes, and projecting
points and lines to these planes. In practice the projections are
formed all on a single sheet of paper, which is kept in a perfectly
flat shape, by the application of rules of a geometrical kind de-
rived from the imaginary process. The key to the practical pro-
cess is in these rules. The first subject of exact investigation
should be the manner of representing a point in space by its pro-
jections and the fixing of its position as regards the " reference
planes " by the use of coordinate distances.

Figs. 6 and 7 show the imaginary and the practical processes of
representing P by its projections.

Fig. 6 is a perspective drawing showing the cube of planes, or
rather the three sides of the cube regularly used for reference
planes. The cube must be of such size that the point P falls well
within it. The perpendicular projectors of P are PPit, PPv and
PPs. The origin and the axes of projection are all marked as on
Fig. 2.

Nature of Orthographic Projection

In Fig. 7 the " field " of the drawing, that part of the paper
devoted to it, is prepared by drawing two straight lines at right
angles to represent the axes of projection, lettering the horizontal
line XOYs and the vertical one ZOYn. This field corresponds to
that of Fig. 4, the outer edges of the squares being eliminated
since there is no need to confine each plane to the size of any par-
ticular cube. If more field is needed, the lines are simply ex-
tended. It must be remembered that these axes are quite different
from the coordinate axes used in plane analytical geometry, or
graphic algebra. These divide the field of the drawing into four

i H

X~1e ^ o'

Pv ^ 9
z

— J

Fig. 6.

Fig. 7.

quadrants, of which three represent three different planes, mutu-
ally perpendicular, the fourth being useful only for the purposes
of construction.

Usually the upper left quadrant, the " North-West," represents
IHI ; the lower left quadrant, or " South-West," represents V> and
the lower right quadrant, or " South-East," represents g.

On occasion the axes may be lettered XOZs horizontally and
ZvOY vertically, to correspond to Fig. 5, the upper right quadrant
now representing §.

10. Coordinates of a Point in Space. — A point in space is
located by its perpendicular distances from the three planes of
projection, that is to say, by the length of its projectors. These

10 Engineering Descriptive Geometry

distances are called the coordinates of the point, and are designated
by X, y and z. In the example given, these values are 2, 3 and 1.
In Fig. 6 PPs, the S projector of P, is two units long, or x — %.
The perpendicular distance to the plane V^ "the V projector, PPv,
is three units long. y=^. In the same way PPn, the H projector,
is one unit long. z = l.

In describing the point P, it is sufficient to state that it is the
point for which a; =2, y='^y and 2 = 1. This is abbreviated con-
veniently by calling it the point P (2, 3, 1), the coordinates, given
in the bracket, being taken always in the order x, y, z.

The projectors, the true coordinate distances, are shown in Fig.
6 by lines of dots, not dashes.

If in each plane H, V and §, perpendicular lines are drawn
(dashes, not dots) from the projections of P to the axes, we shall
have the lines PhS and Pa/, P^e and Px-g, Psg and Psf. These lines
meet in pairs at e, g, and f, forming a complete rectangular paral-
lelopiped of which P and 0 are the extremities of a diagonal. The
other corners of the parallelopiped are Ph, Pv, Ps, e, f and g.

Each coordinate, x, y and z, appears in four places along four
edges of the parallelopiped, as is marked in Fig. 6.

The distances x, y and z are all considered positive in the case
shown.

In Fig. 7, the descriptive drawing of the point P, P itself does
not appear, being represented by its projections, Pn, Pv and P,.
The true projectors (shown in Fig. 6 by lines of dots) do not
appear, but in place of each coordinate tliree distances equal to it
do appear, so that in Fig. 1 x, y and z each appear in three places
as is there marked. Thus x appears as Phfn, eO, and P^g. As all
these are measured to the left from the vertical axis, ZOYn, it
follows that Phepv is a straight line, or Pn is vertically above Pv
It is often said that Pv " projects vertically " to Ph. In the same
way Pv ^' projects horizontally " to Pg. The distance y appears as
ePji, Ofhf Ofs, and gPg. The point / appears double due to the
axis of Y itself doubling. To represent the original coincidence of
fh and fs, a quadrant of a circle with center at 0 is often used to
connect them.

Nature of Orthographic Projection 11

11. Three Laws of Projection for H, V and S-— The three rela-
tions shown by Fig. 7 amount to three laws governing the pro-
jections of a point in the three views, and must always be rigidly
observed. They may seem easy and obvious when applied to one
point, but when dealing with a multitude of points it is not easy
to observe them unfailingly.

They may be thus tabulated :

(1) Ph must be vertically above Pi,.

(2) Pg must be on the same horizontal line as Pv.

(3) Ps must be as far to the right of OZ as Pn is above OX.
From these laws it follows that if two projections of a point are

given, the third is easily found. In Fig. 7, if two of the corners
of the figure PhfhfsPsPv are given, the figure can be graphically
completed. Much of the work of actual mechanical drawing con-
sists in correctly locating two of the projections of a point by plot-
ting or measuring, and of finding the other projection by the appli-
cation of these laws or of this construction. Constant checking
of the points between the various views of a drawing is a vital prin-
ciple in drawing.

On the drawing board the horizontal projection of Pv to P« is
naturally done by the T-square alone, and the vertical projection
of Ph to Pv by T-square and triangle. There are two methods of
carrying out the third law in addition to the graphical construc-
tion of Fig. 7. Fig. 8 shows a graphical method which makes use.
of a 45° line, OL, in the construction space, instead of the quad--
rant of a circle. It is easier to execute, but the meaning is not so
clearly shown. The third method is by the use of the dividers,
directly to transfer the x coordinate from whichever place it is.
fi^st plotted, to the other view in which it appears.

12. Paper Box Diagrams. — When studying a descriptive draw-
ing, such as Fig. 8, imagine as you look at Pv that the real point P
lies haclc of the paper, at a distance equal to ePn.

Whenever figures in the text following seem hard to grasp, carry
out the following scheme. Trace the figure on thin paper, or on
tracing cloth. Using Fig. 8 as an example, and supposing it to
have been traced ^on semitransparent paper, hold the paper before
you and fold the top half back 90° on the line XOYs. Then, view-

Engineering Descriptive Geometry

ing Ph from above, imagine the true point P to lie below the paper
at a distance equal to ePv, in the same way as you imagine P to
lie back of Pv at a distance equal to ePh,

After flattening the paper, fold the right half back 90° on the
line ZOYh, and, viewing Ps from the right, imagine P to lie back
of Ps a distance Pvff. Finally, crease the paper on the line OL,
OL itself forming a groove, not a ridge, and bend the paper on all

1 IH
i

1
j

X jC

1 ' 4

1 .. ...

~'9

_.. .. „ .. J
P3

•

FiQ. 8.

the creases at once, so that H and S fold back into positions at
right angles to V and to each other at the same time.

The " construction space '^ YhOYs is thus folded away inside
and OYh and OYs come in contact with each other. Fig. 9 shows
the final folding partly completed.

'No diagram, however complicated, can remain obscure if studied
from all sides in this manner.

To have a convenient name, these space diagrams may be called
" Paper Box Diagrams."

Nature of Orthographic Projection

Figs. 4 and 5 make good paper box diagrams, while Fig. 3 may-
be traced and folded into a perfect cube which, if held in proper
position, will give the exact views shown in Figs. 2 and 2a, omit-
ting the solid object supposed to be seen in the center of those
figures.

13. Zero Coordinates. — Points having zero coordinates are some-
times perplexing. If one coordinate is zero, the point in question
is on one of the reference planes, and indeed coincides with one of
its own projections. Since x is the length of the orthographic
projector of the point P upon the plane S> if a:=0, this projector
disappears and the point P and its S projection Ps coincide. If

Fig. 9.

the point Q (0, 3, 1) is to be plotted it will be found to coincide
with Ps in Fig. 6. The descriptive drawing will correspond with
Fig. 7 with all lines to the left of ZOYn omitted, and with the let-
tering changed as follows: For Ps put Qs (and Q), for fu put Qn,
for g put Qv JThe student should make this diagram on cross-
section paper and should study out for himself the similar cases for
the points Q' (2, 0, 1) [P^ in Fig. 6] and Q" (2, 3, 0) [Pn in Fig.
6] and should proceed from them to more general cases, assuming
ordinates at will, using cross-section paper for rapid sketch work
of this kind.

If two coordinates are zero, the point lies on one of the axes,
on that axis, in fact, which corresponds to the ordinate which is not
zero. Thus the point R (2, 0, 0) is the point e of Fig. 6, En and
Bv are at e, and Rg is at 0.

Fig. 10.

Ftg. 10a.

These wire-mesh cages are not essential for a clear understanding
of the course. Cross-section paper should be used in the solution
of the problems and folded to make " space " or " paper box " dia-
grams, to illustrate knotty points. These folded diagrams are
practically miniature cages. The full-size cages are very con-
venient for class-room demonstrations.

16 Engineering Descriptive Geometry

Wire-mesh Cage.

If possible, it is very desirable to have cages similar to Fig. 10,"
formed of wire-mesh screens, representing the planes ff, V? S and
S'. On these screens chalk marks may be made and the planes,
being hinged together, may afterward be brought into coincidence
with Vj as represented in Fig. 10a.

In order to plot points in space within the cage, pieces of wire
about 20 inches long, with heads formed in the shape of small
loops or eyes, are used as point markers. They may be set in holes
drilled in the base of the cage at even spaces of 1" in each direc-
tion, so that a marker may be set to represent any point whose x
or y coordinates are even inches. To adjust the marker to a re-
quired z coordinate, it may be pulled down so that the wire projects
through the base, lowering the head the required amount, z may
vary fractionally.

In Fig. 10 a point marker is set to the point P (11, 4, 6), and
the lines on the screens have been put on with chalk, to represent
all the lines analogous to those of Fig. 6.

Fig. 10a represents the descriptive drawing produced by the
development of the screens in Fig. 10. It is analogous to Fig. 7.

Several points may be thus marked in space and soft lead wire
threaded through the loops, so that any plane figure may be shown
in space, and its corresponding orthographic projections may be
drawn on the planes in chalk.

Problems I.

(For solution with wire-mesh cage.)

1. Plot by the use of the wire markers the three points, A, B
and C, whose coordinates are (5,12,11), (3,3,3), and (12,4,8),
and draw the projections on the screens in chalk. By joining point
to point a triangle and its projections are formed. Use lead wire
for joining the points, and chalk lines for joining the projections.

2. Form the triangle as above with the following coordinates:

(11, 3, 2), (12, 6, 12) and (14, 12, 7).

3. Form the triangle as above with the following coordinates:

(7,0,11), (9,9,0) and (2,2,3).

Nature of Orthographic Projection 17

4. Form the triangle as above with the following coordinates:

(0,11,13), (14,3,3) and (14,13,0).
(The following examples may be solved on coordinate paper, or
plotted in inches on the blackboard.)

5. Make the descriptive drawing of a triangle in three views by
plotting the vertices and joining them by straight lines. The
vertices are the points A (1, 10, 8), B (5, 6, 8), C (9, 2, 4).

6. Make the descriptive drawing as above using the points

A (12, 2, 5), ^ (0, 8, 6), (7 (4, 6, 0).

7. Make the descriptive drawing as above using the points

A (3,4,2), B (13,8,10), C (5,10,14).

8. The four points A (3, 3, 3), B (3, 3, 15), C (15, 3, 15), and
D (15,3,3) form a square. Make the descriptive drawing. Why
are two projections straight lines only? What are the coordinates
of the center of the square ?

9. The four points A (12,2,12), B (2,2,12), C (7,14,12),
and D (7, 6, 2) are the comers of a solid tetrahedron. Make the
descriptive drawing, being careful to mark concealed edges by
broken lines.

10. Make the descriptive drawing of the tetrahedron A (2, 3, 2),
B (9, 8, 3), (7 (4, 8, 9), i) (12,3,6), marking concealed edges by
broken lines.

11. Make the descriptive drawing of the tetrahedron A (3, 2, 4),
B (6,8,2), C (8,1,8),I> (2,7,8).

12. Plot the points A (12,7,7), B (8,13,5), C (2,9,2), and
D (6,3,4). Why is the V projection a straight line?

13. Make the descriptive drawing of the tetrahedron A (13, 5, 3),
B (1, 5, 3), C (7, 2, 6), D (7, 8, 6). To which axis is the line AB
parallel ? To which axis is CD parallel ?

14. Plot and join the points 4 (11, 3, 3), 5 (3, 3, 3), (7 (7, 9, 7),
and Z> (15, 9, 7). Bo AC and BD meet at a point or do they pass
without meeting?

CHAPTER II.

ORTHOGRAPHIC PROJECTION OF THE FINITE STRAIGHT

LINE.

14. The Finite Straight Line in Space : One not Parallel to any
Reference Plane, or an " Oblique Line." — A line of any kind con-
sists merely of a succession of points. Its orthographic projection
is the line formed by the projections of these points.

In the case of a straight line, the orthographic • projection is
itself a straight line, though in some cases this straight line may
degenerate to a single point, as mathematicians express it.

Fig. 11.

FiQ. 12.

To find the fi-jl, V and § projections of a finite straight line in
space, the natural course is to project the extremities of the line
on each reference plane and to connect the projections of the ex-
tremities by straight lines. We shall not consider this as requir-
ing proof here. It is common knowledge that a straight line cannot
be held in any position that will make it appear curved, and ortho-
graphic projection is, as shown by Fig. 1, only a special case of
perspective projection. The strict mathematical proof is not ex-
actly a part of this subject.

Orthographic Projectiox of Finite Straight Line 19

The projectors from the different points of a straight line form
a plane perpendicular to the plane of projection. This " projector-
plane," of course, contains the given line. If the straight line is
a limited or finite line the projector-plane, is in the form of a
quadrilateral having two right angles. Thus in Fig. 11 the M
projectors of the straight line AB form the figure AAnBhB, having
right angles at Ah and Bn. These projector-planes AAhBnB,
AAsBsB, and AAvBvB are shown clearly in this perspective draw-
ing, in which they are shaded.

Fig. 12 is the descriptive drawing of the same line AB which has
been selected as a "line in space," that is, as one which does not
obey any special condition. In such general cases the projections
are all shorter than the line itself. As drawn, the extremities are
A (1,1,5) and B (5,4,2).

-I — I — I — h

FiG. 13.

Fig. 14.

1 1 r t

As B.

15. Line Parallel to One Reference Plane, or Inclined Line. —

A line which is parallel to one reference plane, but is not parallel
to an axis, appears projected at its true length on that reference
plane only.

Figs. 13 and 14 show a line five units long, connecting the points
A (1,2,2) and B (5,5,2). AnBn is also five units in length but
AvBr: is but four and AgBg is three. The projector-plane AAhBhB
is a rectangle.

The student should construct on coordinate paper the two simi-
lar cases. For example: the line C (4, 2, 1), Z) (1, 2, 5) is parallel
to V; ^ (2,1,2), F (2,5,5) is parallel tog.

Engineering Descriptive Geometry

16. Line Parallel to One of the Axes and thus Parallel to
Two Reference Planes. — If a finite straight line is parallel to one
of the axes of projection, its projection on the two reference planes
which intersect at that axis, will be equal in length to the line
itself. Its projection on the other reference plane will be a single
point.

Fig. 15 is the perspective drawing and Fig. 16 the descriptive
drawing, of a line parallel to the axis of X, four units in length.
Its extremities are the points A (1,2,2) and 5 (5, 2, 2). In H

% .

H — I — I — 1-— I — h

FiG. 15.

Fig. 16.

and V its projections are four units long. The projector-planes
AAjiBhB and AAvBvB are rectangles. The § projector-plane de-
generates to a single line BAAg. It will be seen that the coordi-
nates of the extreme points of the line differ only in the value of
the X coordinate. In fact, any point on the line will have the
y and z coordinates unchanged. It is the line (x variable, 2, 2).

The student should construct for himself descriptive drawings
of lines parallel to the axis of Y and the axis of Z, using prefer-
ably " coordinate paper " for ease of execution. Good examples
are the lines C (1, 1, 1), P (1,5,1) and JS' (3,1,1), F (3,1,4).
Points on the line CD differ only as regards the y coordinate. It
is a line parallel to the axis of Y. EF is parallel to the axis of Z
and z alone varies for different points along- the line.

Orthographic Projection of Finite Straight Line 21

17. Foreshortening. — The projection of a line oblique to the
plane of projection is shorter than the original line. This is
called foreshortening. The H> V and S projections of Fig. 12,
and the V and S projections of Fig. 14, are foreshortened. It is
a loose method of expression, but a common one, to say that a line
is foreshortened when it is meant that a certain projection of a
line is shorter than the line itself. When subscripts are omitted
and AhBh is called AB, it is natural to speak of the line ^4^ as
appearing "foreshortened" in tlie plan view or projection on ff.
This inexact method of expression is so customary that it can
hardly be avoided, but with this explanation no misconception
should be possible.

18. Inclined and Oblique Lines. — The words Inclined and Obli-
que are taken generally to mean the same thing, but in this subject
it becomes necessary to draw a distinction, in order to be able to
specify without chance of misunderstanding the exact nature of a
given line or plane.

A line will be described as :
Parallel to an axis, when parallel to any axis. As a special case a

line parallel to the axis of Z may be called simply vertical.
Inclined, when parallel to a reference plane, but not parallel to an

axis. The line AB, Fig. 13, is an illustration.
Oblique, when not parallel to any reference plane or axis. The

typical " line in space" is oblique. AB, of Fig. 11, illustrates

this case.

19. Inclined and Oblique Planes. — A plane will be called:
Horizontal, when parallel to ff. The V projector-plane in Fig. 15

is of this kind.
Vertical, when parallel to V o^ S* The H projector-plane in Fig.

15 is of this kind.
Inclined, when perpendicular to one reference plane only. The H

projector-plane of Fig. 13 is of this kind.
Oblique, when not perpendicular to any reference plane. Planes

of this kind will appear later on.
The surface of the solid of Fig. 2 is composed of vertical, hori-
zontal, and inclined planes (but no oblique plane). Its edges are

Engineerixg Descriptive Geometry

lines, parallel to the axes of X, Y and Z; and inclined lines (be-
cause parallel to §) ; but no oblique lines.

20. The Point on a Given Line. — It is self-evident that if a
given point is on a given line, all the projections of the point must
lie on the projections of the line.

If the middle point of a line AB is projected, as C in Fig. 17, its
projections Cj,, Cv, and Ca bisect the projections of the line. The
reason for this appears when we consider the true shape of the

Fig. 17.

projector-planes, all three of which appear distorted in the per-
spective drawing. Fig. 17, and which do not appear at all on the
descriptive drawing. Fig. 18. In Fig. 17 AAhBhB is a quadri-
lateral, having right angles at ^^ and Bh, it is therefore a trape-
zoid. CCh is parallel to AAn and BBh, and since it bisects AB at C,
it must also bisect AhBh at Ch. The result of this is that in Fig.
18, where the projections which do appear are of their true size,
Ch bisects AjiBj,, Co bisects AvB^, and Cs bisects AgBg.

This principle applies to other points than the bisector. Since
all W projectors are parallel to each other, if any point divides AB
into unequal parts, the projections of the point will divide the
projectors of AB in parts having the same ratio. A point one-

Orthographic Projection of Finite Straight Line 23

tenth of the distance from A to 5 will, by its projections, mark
off one-tenth of the distance on AhBn, AvBv, etc.

The points illustrated in Figs. 17 and 18 are A (2,3,1),
^ (5, 5, 5) and C (3J, 4, 3). It will be noticed that the x coordi-
nate of C is the mean of those of A and B, and the y and z coordi-
nates of C also are the mean of the y and z coordinates of A and B.

Unless all three of the projections of a point fall on the pro-
jections of a line, the point is not in the given line. If one of the
projections of the point be on the corresponding projection of the
line, one other projection of both point and line should be ex-
amined. If in this second projection it is found that the point
does not lie on the line, it shows that the point in space lies in one
of the projector-planes.

Thus the point D in Fig. 18 has its V projection on AvBv, but
its fi and S projections are not on AhBh and AgBs. D is not a
point in the line AB but is on the V projector-plane of AB, as is
clearly shown on Fig. 17.

In the case illustrated, Dv bisects BvCv. The plotting of the V
projection of a point is governed only by its x and z coordinates.
Dv bisects BvC'v because its x and z coordinates are the means of
the X and z coordinates of B and C. The y coordinate of D, how-
ever, has no connection with the y coordinates of B and C.

21. The Isometric Diagram. — A device to obtain some of the
realistic appearance of a true perspective drawing without the
excessive labor of its construction is known as " isometric '' draw-
ing.

A full explanation of this kind of drawing will follow later,
but for present purposes we may regard it as a simplified per-
spective of a cube in about the position of that in Figs. 2, 6, 11,
etc., but turned a little more to the left. Vertical lines are un-
changed. Lines which are parallel to the axis of X, and which in
the perspective drawing incline up to the left at various angles, are
all made parallel and incline at 30° to the horizontal. In the-
same way lines parallel to the axis of Y are drawn at 30° to the
horizontal, inclining up to the right.

Engineering Descriptive Geometry

Fig. 19 shows the shape of a large cube divided into small unit
cubes. In plotting points the same scale is used in all three direc-
tions, that is, for distances parallel to all three axes. Fig. 19a
shows the point P (2, 3, 1) plotted in this manner, so that the
figure is equivalent to the true perspective drawing. Fig. 6.

It is not intended that the student should make any true per-
spective drawing while studying or reciting from this book. If any
of the space diagrams here shown by true perspective drawings

Fig. 19.

Fig. 19a.

must be reproduced, the corresponding isometric drawing should
be substituted.

For rapid sketch work, especially ruled paper, called " isometric
paper,'^ is very convenient. It has lines parallel to each of the
three axes. With such paper it is easy to pick out lines correspond-
ing to those of Fig. 19.

An excellent exercise of this kind is to sketch on isometric
paper the shaded solid shown in Fig. 2, taking the unit square of
the paper for 1" and considering the solid to be cut from a 10" cube,
the thickness of the walls left being 2", and the height of the tri-
angular portion being 6". The' solid may be sketched in several
positions.

Orthographic Projection of Finite Straight Line 25

Problems II.

(For solution with wire-mesh cage, or cross-section paper, or on

blackboard.)

15. A line connects the points A (5|^, 6) and B (5, 12, G).
What are the coordinates of the point C, the center of- the line?
What are the coordinates of D, a point on the line, one- tenth of
the way from A to Bf

16. Same with points A (6, 6, 2) and 5 (6, 6, 12).

17. Draw the line AB whose extremities are A (2, 7, 4) and
B (14, 2, 4). On what view does its true length appear? What is
this length? What are the coordinates of a point C on the line
one-third of its length from A ?

18. With the same line A (2,7,4), B (14,2,4), state what is
the true shape of the H projector-plane. Give length of each edge
and state what angles are right angles. Same for V projector-
plane.

19. Same as Problem 18, with line A (4, 2, 2), B (4, 11, 8).

20. With the line of Problem 19, state what is the true shape of
the H and S projector-planes, giving length of each edge, and
state which angles are right angles.

21. Same as Problem 17, with line A (0, 4, 8), 5 (9, 4, 1).

22. The H projection of C (8,2, 6) lies on the fil projection of
the line A (10, 1, 9), B (2, 5, 2). Is the point on the line? .

23. Same as Problem 22, with line A (2, 1, 8), B (8, 10, 5), and
point C (4,4,7).

24. A triangle is formed by joining the points A (6, 3, 1),
B (10,3,10) and C (2,3,10). In what view or views does the
true length oi AB appear? In what view or views does the true
length of BC appear? Mark the center of the triangle (one-third
the distance from the center of the base BC to the vertex A) and
give its coordinates.

25. Same with points A (5,9,6), B (5,3,1) and C (5,3,11).

26. Same with points A (10, 1, 4), 5 (7, 10, 4) and C (1, 4, 4).

27. The V projections of the points A (8, 1, 2), B (10, 3, 8),
C (4, 3, 10.) and D (2, 1, 4) form a square. Draw the projections
and connect them point to point. What are the coordinates of the
center where AC and BD intersect?

26 Engineering Descriptive Geometry

28. Plot the parallelogram A (11, 3, 3), B (3, 3, 3), C (7, 9, 7),
D (15, 9, 7). The diagonals intersect at E. Give the coordinates
of E. Describe the ff projector-planes of AB, AC, and CD, giving
the length of each end projector. Is the plane of the figure in-
clined or oblique? Is AC an inclined or an oblique line?

29. Plot the quadrilateral

A (11, 10,3), 5 (3,10, 11), 0 (7,2, 7), D (11,4, 3).

Is the plane of the figure horizontal, vertical, inclined or oblique?

Is the line AB horizontal, vertical, inclined or oblique ?

Is the line BC horizontal, vertical, inclined or oblique?

Is the line CD horizontal, vertical, inclined or oblique?

Is the line DA horizontal, vertical, inclined or oblique?

CHAPTER III.
THE TETTE LENGTH OF A LINE IN SPACE.

22. The Use of an Auxiliary Plane of Projection. — To find the
true length of a " line in space/' or oblique straight line, an auxil-
iary plane of projection is of great value, and is constantly used
in all branches of Engineering Drawing.

A typical solution is shown by Figs. 20 and 21. The essential
feature is the selection of a new plane of projection, called an

Fig. 20.

Fig. 21.

auxiliary plane, and denoted by \J, which must be parallel to the
given line and easily revolved into coincidence with one of the
regular planes of reference.

This auxiliary plane is passed parallel to one of the projector-
planes. In Fig. 20 the plane S' of the cube of planes has been
replaced by a plane \], parallel to the ff projector-plane, AAhBnB.
Like that plane, \J is also perpendicular to fi, and XM, the line
of intersection of U and H, is parallel to AhBn. The distance of
the plane U from the projector-plane may be taken at will and
in the practical work of drawing it is a matter of convenience,
choice being governed by the desire to make the resulting figures
clear and separated from each other. In Fig. 20 the auxiliary

28 Engineerixg Descriptive Geometry

plane U has been established by selecting a point X in H for it
to pass through. U is an " inclined plane," not an " oblique plane/'

23. Traces of the Auxiliary Plane \]. — ^The auxiliary piano U
cuts the plane V iii ^ ^i^e XN, parallel to the axis of Z. The
lines of intersection of \] with the reference planes, are called the
"traces'^ of \]. Since there are three reference planes, there may
be as many as three traces of UJ. In the case illustrated in Fig. 20,
there are, however, but two traces. Qnly one of these three' possible
traces of UJ can be an inclined line. In Fig. 20 the trace XM alone
is an " inclined " line.

We shall see later that the auxiliary plane may be taken per-
pendicular to V or to § as alternative methods. In every case
there is but one inclined trace, that on the plane to which U is
perpendicular. It is this trace which lias the greatest importance in
the process. For the sake of uniformitj^, M and N will be assigned
as the symbols for marking the traces of an auxiliary plane of
projection.

24. The U Projectors. — A new system of projectors, AAu, BBu,
etc., project the line AB upon the plane U- These projectors,
being perpendicular distances between a line and a plane parallel to
it, are all equal, and the projector-plane AAuBuB of Fig. 20 is in
reality a rectangle. AuBu is therefore equal in length to AB, or
AB is projected upon U without foreshortening.

25. Development of the Auxiliary Plane JJ- — The descriptive
drawing. Fig. 21, is the drawing of practical importance, which is
based on the perspective diagram. Fig. 20, which shows the mental
conception of the process employed. In practical work, of course.
Fig. 21 alone is drawn, and it is constructed by geometrical reason-
ing deduced from the mental process exhibited by Fig. 20.

In the process of flattening out or " developing " the planes of
projection, JJ is generally considered as attached or hinged to the
" inclined trace," XM in this example. In Fig. 21 U has been
revolved about XM, bringing it into the plane of H, the trace XN
having opened out to two lines. N separates into two points and is
marked Nu as a point in U and Nv as a point in V, analogous to
Yh and Ys in the development of the reference planes. The space
NuXNv, like YnOYs, may be considered as construction space.

The True Length of a Line in Space 29'

26. Fourth law of Projection — that for Auxiliary Plane, U. —

It will be seen from Fig. 20 that AAneAv is a rectangle and that
eAv is equal io AhA. On the descriptive drawing, Fig. 21, these
two lines, eAv and eAn, form one line perpendicular to OX. This
is in accordance with the first law of projection of Art. 11.

As the plane \J is perpendicular to H we have the same rela-
tion there, and AAnkAu, Fig. 20, is a rectangle. IcAu is therefore
equal to AjiA, and in the development. Fig. 21, AJc and hA^ form
one line AJcAn perpendicular to XM.

If from Au and Avy Fig. 20, perpendiculars are let fall upon the
intersection of U and V (the trace XN) they will meet at the
common point I, both IcAJX and XlAvC being rectangles. In the
descriptive drawing. Fig. 21, AJ is perpendicular to XNu, IJv is
the arc of a circle, center at X,- and l^Av is perpendicular to XNv.
The following law of projection governs the position of Au in
the plane U-

(4) From the regular projections of A draw perpendiculars
to the traces of \]. These lines continued into the field
of U intersect at Au. One of these lines is carried across
the construction space by the arc of a circle whose center
is the meeting point of the traces of U-

27. The Graphical Application of this Law to a Point. — The
procedure for locating the projection Au on the descriptive drawing,
Fig. 21, after the location of the plane U has been determined, is
as follows : From the adjacent projections of the point draw lines
perpendicular to the traces of the plane U- Continue one of these
lines across the trace. Swing the foot of the other perpendicular
to the duplicated trace, and continue it by a line perpendicular to
this trace to meet the line first mentioned. Their intersection is
the projection of the point on U- In Fig. 21, this requires AiJcAu
to be draw^n perpendicular to XM, and the line AvUuAu to be
traced as show*n.

28. The True Length of a Line. — The procedure for finding the
true length of a line consists in first drawing, Fig. 21, a line paral-
lel to one of the projections of the line to act as the trace of the
auxiliary plane. Where this trace intersects an axis of projection
perpendicular lines are erected, one perpendicular to the axis, one

Engineering Descriptive Geometry

perpendicular to the trace. These lines are the two developed
positions of the other trace of the plane \]. Then locate the ex-
tremities of the given line on the auxiliary plane HJ. The line
joining the extremities is the required projection of the line on \],
and is equal in length to the given line.

29. Alternative Method of Developing the Auxiliary Plane, HJ. —
A modification of this construction is shown in the descriptive
drawing. Fig. 22, in which the plane JJ has been revolved about
the vertical trace XN until it coincides with the plane V- ^^
separates into two lines, XMh and Zil/«. h^ of Fig. 20, becomes
Jch and ku) and the space MhXMu is construction space. A is on

Fig. 22.

"a horizontal line drawn through Av. Ankh is perpendicular to XMn-
Ichku is the arc of a circle having Z as a center, and fc„A„ is per-
pendicular to XMw ^M is thus located.

This method of development of the planes is much less common
in practical drawing than the other, because, as a rule, it is less
convenient than the first method. In such cases as occur it offers
no particular difficulty. Both Figs. 21 and 22 are solutions of
the problem of finding the true length of the oblique line AB by
projection on an auxiliary inclined plane, U-

30. Alternative Positions of the Plane U- — We saw that the
exact position of the plane U? so long as it remained perpendicular
to H and parallel to AnBn, was left to choice governed by practical

The True Length of a Line in Space 31

considerations. JJ itself, however, may be taken perpendicular to
V and parallel to AvBvj or it may be taken perpendicular to S and
parallel to AgBg. To get an entire grasp of the subject the student
is advised to trace Fig. 21 on thin paper, or plot it on coordinate
paper, points A, B, and X being (6,6,2), (10,10,8) and
(11, 0, 0), and fold the figure into a paper box diagram, the con-
struction spaces NuXNv and YhOYg being creased in the middle
and folded out of the way. Fig. 22 will serve equally well. The
final result will be a paper box exactly similar to Fig. 20.

The variation in which \J is perpendicular to V may be plotted,
passing the new inclined trace of U (lettered YM) through the
point (0, 0, 3) parallel to AvBv. Fold this figure into a paper box,
the paper being cut along a line YN perpendicular to YM.

The other variation may be plotted with the inclined trace of U
on the plane S, parallel to AgBa and passing through the point
(0,0,6) (/, of Fig. 21). Letter this trace YsM and draw YsN
perpendicular to it, inclining up to the right. The paper must be
cut on this line to enable it to be properly folded.

31. The Method Applied to a Plane Figure.; — The special value
of this use of the auxiliary plane is seen when one operation serves
to give the true length of a number of lines at once, and thus
shows a whole plane figure in its true shape.

In Fig. 23 the polygon ABODE is shown by its projection, the
point A alone being lettered. It is noticeable that in V the edges
all form one straight line. The V projector-planes of the various
edges are therefore all parts of the same plane, and the polygon
itself is a plane figure placed perpendicular to \. It may be said
the polygon is " seen on edge " in V-

An auxiliary plane JJ has been taken parallel to the plane of the
polygon, and therefore perpendicular to V- The trace XM being
parallel to the V projections of the edges, this auxiliary plane serves
to show the true length of all the edges at once. The projection on
HJ is the true shape of the polygon ABODE. In the case illustrated,
the U projection discloses the fact that the polygon is a regular
pentagon, a fact not realized from the regular projections, owing
to the foreshortening to which they are subject.

Engineering Descriptive Geometry

This figure is well adapted to tracing and folding into a paper
box diagram.

Fig. 23.

32. The True Length of a Line by Revolving About a Projector.

^-A second method of finding the true length of a line seems in a
way simpler, but proves to be of much less value in practical work.
The method consists in supposing an oblique line AB to be revolved
about a projector of some point in the line until it becomes parallel
to one of the planes of reference. In this new position it is pro-
jected to the reference plane as of its true length.

In Fig. 24 the V projector-plane of the line AB has been shaded
for emphasis (A is the point (1, 1, 5), and B is the point (5, 4, 2) ).
The projector AAv has been selected at will, and the V projector-
plane (of which the line AB is one edge) has been rotated about
AAv as an axis until it has come into the position AvB'vB'A. In
its new position, AB' projects to ff as AnB'],. This is the true
length of the line. During its rotation the point B has moved to
B', but in so doing it has not revolved about A as its center, but
about the point h on AvA extended. hAv is equal in length to BBv.

The True Length of a Line in Space

Bi, moves to B'v, revolving about J.^ as a center. In Fig. 25, the
corresponding descriptive drawing, the original projections are
shown as full lines and the projections of the line after the rota-
tion has occurred are shown by long dashes.

In Yy AvBv swings about /!« as a pivot until in its new position
AvB'v it is parallel to OX. In H? Bn moves in a line parallel to
OX (since in Fig. 24 the motion of B takes place entirely in the
plane of IBB', which is parallel to V)j and as B\ must be verti-
cally above B'v the motion terminates where a line drawn vertically

Fig. 24

up from B'v meets the horizontal line BhB\. Joining Ah and B'hy
the new H projection is the true length of the given line. The S
projection is of no interest in this case. The ff and V projections
of Fig. 25 show the graphical process corresponding to the theory
of* this rotation. In V^ ^v moves to B'v, whence a .vertical pro-
jector meeting a horizontal line of motion from Bh determines B'j,,
the new position of Bh. AnB'n is the true length of the line. The
arrow-heads on the broken lines make these steps clear.

33. Variations in the Method. — The method is subject to wide
variations. The same projector-plane AAvBvB, Fig. 24, revolving
about the same projector AAv, might start in the opposite direction
and swing to a position parallel to S. The graphical process of
Fig. 25 would then confine itself to V a^d S instead of V and H.

Engineeeixg Descriptive Geometry

In addition, the rotation might have been about BBv as an axis
or about the V projector of any point in AB or AB extended.
Finally, the H projector-plane or the S projector-plane might
have been selected and made to revolve into position. There are
six distinct varieties of the process, each one subject to great modifi-
cations.

This method can be applied to a plane figure which appears " on
edge'' in one of the regular views. ]n Fig. 26 a polygon lies in a

U:.^ N^/

■"\\

4V+4

Fig. 26.

plane perpendicular to V- There are two varieties of the process
applicable in this case. Choosing the Y projector of the point A
for the axis of rotation, the whole polygon may be rotated up par-
allel to H, thence its true shape projected upon f\, or it may be
revolved down until parallel, to §, thence its true shape 'projected
upon S- Both methods are shown, though of course in practice one
at a time should be enough.

34. A Projector-Plane Used as an Auxiliary Plane. — The two
processes for finding the true length of a line differ in this respect.

The True Length of a Line in Space 35

In one the line is projected on a plane which is revolved into
coincidence with one of the reference planes, by revolving about a
line in that reference plane. In the second process, a projection
plane is itself revolved about a projector, that is, about a line
perpendicular to one reference plane, to a position parallel to a
second reference plane. The line in its new position is projected
on the latter plane.

A method which is a modification of the first process is in many-
cases very simple. A projector-plane is itself used as an auxiliary
plane, and is revolved into coincidence with the plane to wliich it is
perpendicular by rotation about its trace in that plane. In Fig. 23,
for example, instead of passing XM parallel to AvCv, AvCv would be
extended to the axis of X, and used itself for the inclined trace of
the auxiliary plane. XN would be moved to the right and other
slight modifications made.

As in the second method, a projector-plane is here rotated; but
it is not rotated about a projector, but about a projection (its
trace), and the real similarity of the process is with the first
method, that of the auxiliary plane of projection. It is but a
special case of this kind.

In practical drawing, it rarely happens that one of the projector-
planes can be thus used itself with advantage as an auxiliary plane
of projection. It leads usually to an overlapping of views and it
will not be found so useful as the more general method.

For the continuation of this study, all these methods should be
at the students' finger ends.

35. The True Leng^th of a Line by Constructing a Right Tri-
angle.— These methods of finding the true length of a line are
generally used for the true lengths of many lines in one operation,
or for the true shape of a plane figure. When a single line is
wanted, the construction of a right triangle from lines whose true
lengths appear on the drawing is sometimes resorted to. In Fig. 24
the triangle ABh is a right triangle, AB being the hypotenuse and
AhB the right angle. In the descriptive drawing, Fig. 25, AvBv
is equal in length to hB of Fig. 24, and AjJ) is easily found, equal
to Ah of Fig. 24. These lines may be laid off at any convenient

36 Engineering Descriptive Geometry

place as the sides of a right triangle, and the hypotenuse measured
to give the true length of AB. Mathematically the hypotenuse is
the square root of the sum of the squares of the sides. In the case
illustrated AiBv is 5 (itself the square root of A^c +BvC , ur
^32 + 42 ) and Anl) is 3. The length AB is therefore VFTB^
= V34 = 5.83.

Problems III.

(For use with wire-mesh cage, cross-section paper, or blackboard.)

30. A square in a position similar to the pentagon of Fig.
26 has the corners A (10,12,2), B (2,12,8), C (2,2.8) and
D (10, 2, 2). Find its true shape by the use of an auxiliary plane.

31. A square is in a position similar to the pentagon of Fig. 23.
The comers are A (9,3,3), B (9,13,3), 0 (3, 13, 11), and
D (3,3,11). Find its true shape by revolving into a plane par-
alel to fi.

32. Plot the triangle A (11, 3, 2), B (12, 6, 12), C (14, 12, 7).
Find its true shape by the use of an auxiliary plane perpendicular
to M.

33. Plot the triangle A (13,15,8), B (10,11,0), C (7,7,8).
Find the true shape of the triangle by revolving it about AA%
until in a plane parallel to §. Find the true shape by projection
on a plane \], perpendicular to ff, whose inclined trace passes
through the point (16,0,0). (With the wire-mesh cage turn the
plane S' to serve for this auxiliary plane U-)

34. Same with triangle A (9, 7, 8), B (12, 11, 13), C (15, 15, 2).

35. Plot the right triangle A (14, 4, 3), B (14, 10, 3), C (6, 4, 9).
Revolve it about BBv into a plane parallel to ff and project its
true shape on J-J. (With the wire-mesh cage put markers at points
A, B, C and C, the new position of C.)

36. Plot the right triangle A (9, 3, 6), B (9, 3,0), C (15, 11, 6).
Eevolve it about AB until in a plane parallel to V and plot C, the
new position of the vertex. Revolve it about the same axis into a
plane parallel to S? and plot C", the new position of the vertex.
(With the wire-mesh cage put point markers at A, C, C and C".)

The True Length of a Lixe ix Space 37

37. Plot the square A (14,8,2), B (ll,2,7i), C (11,14,74),
D (8, 8, l'2f ). The diagonal is 12 units long. Eevolve the square
about AAv into a plane parallel to fl, and project its true shape on
J-J. (With the wire-mesh cage put point markers at A, B, C, D,
B\ r, and D'.)

38. Plot the triangle A (12,2,14), B (2,2,14), C (7,7,2).
Eevolve it about AB into a plane parallel to V? and project the true
shape on V- (With the wire-mesh cage put markers at points
A, B, C and C On coordinate paper or blackboard show the true
shape by projection on an auxiliary plane \] perpendicular to S?
through the point (0,8,0).)

(For use on coordinate paper or blackboard, not wire-mesh cage.)

39. The triangle A (3,7,11), B (13,2,13), C (5,2,1) is a
triangle in an oblique plane. Find its true shape as follows : BC
appears at its true length in V- Draw AvDv perpendicular to BvCv.
AD is an oblique line, but it is perpendicular to BC since its V
projector-plane AAvDvD is perpendicular to BC. Find the true
length of AD by any method. On V extend AvD^ to Ev, making
DvEv equal to the true length of AD. EiByCv is the true shape of
the triancrle ABC.

CHAPTER IV.

PLANE STIRFACES AND THEIR INTERSECTIONS AND
DEVELOPMENTS.

36. The Omission of the Subscripts, h, v, and s — In a descriptive
drawing a point does not itself appear but is represented by its
projections on the reference planes. This fact has been emphasized
in the previous chapters. In the more complicated drawings which
now follow it will save time and will prevent overloading the figures
with lettering, to omit the subscripts h, v, and s, and to refer to a
point and its projections by the same letter. Thus " ^u " or " the
point A in V ^^ 8,re expressions which call attention to the projec-
tion of A on V> but a diagram will show only the letter A at that
place. If at any time it is necessary to be more precise the sub-
scripts may be restored. They should be used if any confusion is
caused by their omission.

If the projections of two points coincide, it is sometimes advis-
able to indicate which point is behind the other in that view by
forming the letter of fine dots. Referring back to Fig. 16, the
projections of A arid B on g coincide. On this system subscripts
are omitted and the letter ^ (on § only) is formed of dots,
as in Fig. 27.

37. Intersecting Plane Faces. — Many pieces of machines and
structures which form the subjects of mechanical drawings, are
pieces all of whose surfaces are portions of planes, each portion or
face having a polygonal outline.

In making such drawings there arise problems as' to the exact
points and lines of intersection, which can be solved by applying
the laws of projection treated of in the preceding chapters. How
these intersections are determined from the usual data will now
be shown,

38. A Pyramid Cut by a Plane. — As a simple example let us
suppose that it is required to find where a plane perpendicular to
V, and inclined at an angle of 30° with fi, intersects an hex-

Plane Surfaces and Their Intersections

agonal pyramid with axis perpendicular to Hi. Fig. 27 is the
drawing of the pyramid, having the base ABCDEF and vertex P.
The cutting plane is an inclined plane such as we have used for an
auxiliary plane, and its traces are therefore similar to those of an
auxiliary plane. KL is the inclined trace on V and KK' and LL'
are the traces parallel to the axis of Y. The problem is to find the
shape of the polygonal intersection ahcdef in H and S, and its
true shape.

The method of solution of all such problems is to take into con-
sideration each edge of the pyramid in turn, and to trace the points
where they pierce the plane. Thus, the edge PA pierces the given
plane at a, whose projection on V is first located; for the given
plane is seen on edge in V> and PA cannot pierce the plane at any
other point consistent with that condition, a, once located in V>
can be projected horizontally to the line PA in S and vertically to
PA in H.

The true shape of the polygon ahcdef may be shown on an aux-
iliary plane, U^ whose traces are ZM and ZN. In Fig. 27 the
projection of the pyramid on HJ is incomplete. As it is only to
show the polygon dbcdef the rest of the figure is omitted.

Engineerixg Descriptive Geometry

39. Intersecting Prisms. — As an example of somewhat greater
difficulty let it be required to find the intersection of two prisms,
one, the larger, having a pentagonal base, parallel to fi ; and the
other a triangular base, parallel to §. The axes intersect at right
angles, and the smaller prism pierces the larger.

H
G'

— V-

■^ f'

p Jp^

The known elements or data of the problem are shown recorded
as a descriptive drawing in Fig. 28. It shows the projection of
the pentagon on lil, of the triangle on §, and of the axes on V-
The problem is to complete the drawing to the condition of Fig.
29, shown on a larger scale. The corners of the pentagonal prism
are ABODE f and A'B'C'D'E'f and its axis is PP'. The corners
of the triangular prism are FGH and F'G'B' and its axis is QQ'.

40. Points of Intersection. — The general course in solving the
problem of the intersection of the prisms is to consider each edge
of each prism in turn, and to trace out where each edge pierces the
various plane faces of the other prism. When all such points of

Plane Suefaces and Their Intersections 41

intersection have been determined, they are joined by lines to give
the complete line of intersection of the prisms.

To determine where a given edge of one prism cuts a given plane
face of the other prism, that view in which the given plane face is
seen as a line only, or is "seen on edge/' as is said, must be re-
ferred to. Taking the hexagonal prism first, the edges AA', CC,
and DD' entirely clear the triangular prism, as is disclosed by the
plan view on H where they appear " on end " or as single points
only. They, therefore, have no points of intersection with the
triangular prism and in V and § these lines may be drawn as
nninterrupted lines, being made full or broken according to the
rule at the end of Art. 5. BB\ as may be seen in H, meets the
small prism. This line when drawn in S, where the plane faces
FF'G'G and FF'ITII are seen on edge, meets those faces at h and
&'. From § these points are projected to V- The edge BB' con-
sists really of two parts, Bh and h'B'. EE' meets the same two
faces at points c and e' determined in the same way.

FF', when drawn in W, is seen to pierce the plane face AA'B'B
at / and AA'E'E at f. These points, located in W, are projected
vertically down to V- OG' in H pierces BB'C'C at g, and EE'D'D
at g'. h and h' on the line HTF are similarly determined first in
H and are projected down to V-

41. Lines of Intersection. — Having found the points of inter-
section of the edges, we determine the lines of intersection of the
plane surfaces by considering the intersections of plane with plane,
instead of line with plane. BB' is one line of the plane AA'B'B,
and pierces the plane FF'G'G (seen on edge in S) at &. h is there-
fore a point of both planes. FF' is a line of the plane FF'G'G, and
it pierces the plane AA'B'B (seen on edge in IHI) at /, / is also a
point common to both planes. Since these two points are in both
planes, they are points on the line of intersection of the two planes.
We therefore connect h and / by a straight line in \, but do not
extend it beyond either point because the planes are themselves
limited.

By the same kind of reasoning h and </ are found to be points
common to BB'C'C and FF'G'G, and are therefore joined by a
straight line, hg in V- 9^^ also is the line of intersection of two

Engineeeing Descriptive Geometry

planes, and the student should follow for himself the full process
of reasoning which proves it. e, f, and g' are points similar to l, f,
and g. Since the original statement required the triangular prism
to pierce the pentagonal one, gg', ff, and liTi' are joined by broken
lines representing the concealed portions of the edges GG' , FF', and
Eir of the small prism. Had it been stated that the object was
one solid piece instead of two pieces, these lines would not exist on
the descriptive drawing.

42. Use of an Auxiliary Plane of Projection. — To find the inter-
section of solids composed of plane faces, it is essential to have

Fig. 30.

views in which the various plane faces are seen on edge. To obtain
such views, an auxiliary plane of projection is often needed.

Fig. 30 shows the data of a problem which requires the auxiliary
view on \] in order to show the side planes of the triangular prism
"on edge.'' (These planes are oblique, not inclined, and therefore
do not appear "on edge" on any reference plane.) Fig. 31 shows
the complete solution, the object drawn being one solid piece and
not one prism piercing another prism, h and d are located by the
use of the view on U- Tn this case and in many similar cases in
practical drawing, the complete view on IjJ need not be constructed.

Plane Surfaces and Their Intersections

43'

The use of U is only to give the position of b and d, which are then
projected to V- The construction on HJ of the square ends of the
square prism are quite superfluous, and would be omitted in prac-
tice. In fact, the view U would be only partially constructed in
pencil, and would not appear on the finished drawing in ink.

After the method is well understood, there will be no uncertainty
as to how much to omit.

43. A Cross-Section. — In practical drawing it often occurs that
useful information about a piece can be given by imagining it cut
by a plane surface, and the shape of this plane intersection drawn.
In machine drawing, such a section showing only the material
actually cut by the plane and nothing beyond, is called a " cross^

«»

'— : — 1

1 — '

■ 1

n. ' 1

1 1

0
Z

32.

1 "V

i 11 i

1 1^

^ %^

Fig.

section." In other branches of drawing other names for the same
kind of a section are used. The " contour lines " of a map are of
this nature, as well as the " water lines " of a hull drawing in
Naval Architecture.

44. Sectional Views. — ^The cross-section is used freely in ma-
chinery drawing, but a " sectional view," which is a view of a cross-
section, with all those parts of the piece which lie beyond the plane
of the section as well, is much more common.

These sectional views are sometimes made additional to the
regular views, but often replace them to some extent. Fig. 32 is a

44 Engineering Descriptive Geometry

good example. It represents a cast-iron structural piece shown by
plan, and two sectional views. The laws of projection are not
altered, but the views bear no relation to each other in one respect.
One view is of the whole piece, one is of half the piece, and one is
of three-quarters of it. The amount of the object imagined to be
cut away and discarded in each view is a matter of independent
choice.

In the example the projection on V is a view of half of the
piece, imagining it to have been cut en a plane shown in lil by the
line mn. The half between mn and OX has been discarded, and the
drawing shows the far half. The actual section, the cross-section
on the line mn, is an imaginary surface, not a true surface of the
object, and it is made distinctive by "hatching." This hatching
is a conventional grouping of lines which show also the material
of which the piece is formed. For this subject, consult tables of
standards as given in works on Mechanical Drawing. This pro-
jection on V is not called a " Front Elevation," but a " Front Ele-
vation in Section," or a " Section on the Front Elevation."

The view projected on S is called a " Side Elevation, Half in
Section," or a " Half-Section on the Side Elevation." Since a
eection generally means a sectional view of the object with half
removed, a half-section means a view of the object wath one-quarter
removed. If, in H, the object is cut by a plane whose trace is np
and another whose trace is pq, and the N". E. comer of the object
is removed, it will correspond to the condition of the object as seen

inS-

Sections are usually made on the center lines, or rather on central
planes of the object. When strengthening ribs or " webs " are seen
in machine parts, it is usual to take the plane of the section just
in front of the rib rather than to cut a rib or web which lies on the
central plane itself. This position of the imaginary saw-cut is
selected rather than the adjacent center line.

AYhen the plane of a section is not on a center line, or adjacent
to one, its exact location should be marked in one of the views in
which it appears " on edge," and reference letters put at the ex-
tremities. The section is then called the " section on the line mn/*

The passing of these section planes causes problems in intersec-
tion to arise, which are similar to those treated in Articles 37-42.

Plane Surfaces and Their Intersections

45. Development of a Prism. — It is often desired to show the
true shape of all the plane faces of a solid object in one view,
keeping the adjacent faces in contact as much as possible. This
is called developing the surface on a plane, and is particularly
useful for all objects made of sheet-metal, as the development forms
a pattern for cutting the metal, which then requires only to be bent
into shape and the edges to be joined or soldered.

Development is a process already applied to the planes of pro-
jection themselves when these planes were revolved about axes until
all coincided in one plane. The same operation applied to the
surfaces of the solid itself produces the development.

The two prisms of Fig. 29 afford good subjects for development.
Fig. 33 shows the developed surface of the triangular prism, the
lines g-g and g'-g' showing the lines of intersection with the other
prism. In this figure it is considered that the surface of the
triangular prism is cut along the lines GG\ OF, G'F', GH, and

46 Engineering Descriptive Geometry

O'H' ', and the four outer planes unfolded, using the lines bound-
ing FF'H'H as axes, until the entire surface is flattened out on the
plane of FF'H'H.

Fig. 34 shows the development of the large prism of Fig. 29,
with the holes where the triangular prism pierces it when the two
are assembled. The surface of the prism is cut on the line AA',
and on other lines as needed, and the surfaces are flattened out by
unfolding on the edges not cut.

The construction of these developments is simple, since the sur-
faces are all triangles or pentagons whose true shapes are given;
or are rectangles, the true length of whose edges are already known.
. In Fig. 33 the distances Og, G'/, Ff, F'f are taken directly
from V in Fig. 29. The points h and e are plotted as follows:
The perpendicular distance hi to the line GF is taken from V>
Fig. 29, and Gl is taken from Gl in S, Fig. 29. The other points
are plotted in the same manner.

46. Development of a Pyramid. — Fig. 35 shows the development
of the point of the pyramid, Fig. 27, cut off by the intersecting

plane whose trace is KL. The base is taken from the projection
on IUj where its true shape is given. Each slant side must have its
true shape determined, either as a whole plane figure (Art. 31),
or by having all three edges separately determined (Art. 28 or
Art. 32). In this case Pa and Pd are shown in true length in V?
Fig. 27, and it is only necessary to determine the true lengths of
Pb and Pc (or their equivalents Pf and Pe) to have at hand all the
data fox laying out the development. The face Pef may be con-
veniently shown in its true shape on an auxiliary plane "W? Fig. 27,
perpendicular to § and cutting § in a trace YgN as shown.

Plane Surfaces and Their Intersections 47

Problems IV.

(For use with wire-mesh cage, or on cross-section paper or
blackboard. )

40. Plot the projections of the points A (9, 3, 16), B (6, 3, 16),
C (6, 8, 16), D (9, 8, 16), and E (0, 3, 4), i^ (0, 3, 8), G (0, 8, 8),
H (0,8,4). Join the projections, A to E, B to F, C to G, etc.
(With the wire-mesh cage use stiff wire to represent the lines AE,
BF, etc.) Show how to find the true shape of every plane surface
of the solid (a prism) thus formed. On cross-section paper or on
blackboard show how to draw the development of the surface of
the solid.

41. Same as Problem 40, with points A (10, 8, 0), B (8, 10, 0),
C (12, 14, 0), D (14, 12, 0) on fi and E (10, 8, 16), F (6, 12, 16),
O (8, 14, 16), H (12, 10, 16) on H'. (In developing the surface,
find the true shape of the quadrilateral BFGC by dividing it into
two triangles by a diagonal BG whose true length will appear on g.
Divide CGHD by the diagonal CH.)

42. Draw the tetrahedron whose four comers are A (16,2,13),
B (6, 2, 13), C (11, 14, 13) and D (11, 7. 1). It is intersected by
a plane perpendicular to V cutting V in a trace passing through
the origin, making an angle of 30° with OX. Draw the trace of
the plane on V* Where are its traces on fj and S ? Show the H
and S projections of the intersection of the plane and tetrahedron.

43. A solid is in the form of a pyramid whose base is a square of
10 units, and whose height is 8 units. The corners are A ( 16, 2, 10) ,
B (10,2,2), C (2,2,8) and D (8,2,16) and the vertex
E (9,10,9). It is intersected by a plane perpendicular to H^
whose trace on Hil passes through the origin making an angle of
30° with OX, Draw the V and S projections of the intersection
of the pyramid and plane. Where is the trace of the cutting plane
on V?

44. A plane H' is parallel to H at a distance of 16 units, A
square prism has its base in H- The corners are ^ (8,2,0),
B (3, 7, 0), C (8, 12, 0), D (13, 7, 0). Its other base is in H', the
comers A', B', etc., having the same x and y coordinates as above,
and the z coordinates 16.

48. Engineering Descriptive Geometry

A plane S' is parallel to S at a distance of IG. A triangular prism
has its base in S, points E (0, 5, 8), F (0, 13, 2), G (0, 13, 1-1) ;
and its other base in S', points E', F', G' having x coordinates 16,
and y and z coordinates unchanged. Make the drawing of the in-
tersecting prisms considering the triangular prism to be solid and
parts of the square prism cut away to permit the triangular one to
pass through.

(For use on cross-section paper or blackboard, not wire-mesh
cage.)

45. A sheet-iron coal chute connects a square port, A (2,4,2),
B (2,12,2), C (2,12,10), D (2,4,10), with a square hatch,
E (14, 6, 16), F (14, 10, 16), G (10, 10, 16), E (10, 6, 16). The
corners form lines AE, BF, CG, DH and the side plates are bent
on the lines AH and BG. Draw the development of the surface.

46. Draw the development of the tetrahedron of Problem 42
with the line of intersection marked on it.

47. Draw the development of the pyraniid of Problem 43 with
the line of intersection marked on it.

48. Draw the development of the square prism of Problem 44
with the line of intersection marked on it.

49. Draw the development of the triangular prism of Problem
44 with the line of intersection marked on it.

CHAPTEE Y.

CURVED LINES.

47. The Simplest Plane Curve, the Circle. — The geometrical
natures of the common curves are supposed to be understood. De-
scriptive Geometry treats of the nature of their orthographic pro-
jections. The curves now considered are plane curves, that is,
every point of the curve lies in the same plane. It is natural,
therefore, that the relation of the plane of the curve to the plane of
projection governs the nature of the projection.

Plane
of the
^ Circle

^=^^

Fig. 36.

Fig. 37.

The simplest plane curve is a circle. Figs. 36 and 37 show the
three forms in which it projects upon a plane. In Fig. 36, a per-
spective drawing, we have a circle projected upon a parallel plane
of projection (that in the position customary for V). The pro-
jectors are of equal length and the projection is itself a circle ex-
actly equal to the given circle.

On a second plane of projection (that in the position of §) per-
pendicular to the plane of the circle the projection is a straight
line equal in length to the diameter of the circle, AC. The pro-
jectors for this second plane of projection form a projector-plane.

Engineering Descriptive Geometry

In Fig. 37 the circle is in a plane inclined at an angle to the
plane of projection. The projectors are of varying lengths. There
must be one diameter of the circle, however, that marked AC,
which is parallel to the plane of projection. The projectors from
these points are of equal length, and the diameter AC appears of
its true length on the projection as AvCv.

The diameter BD at right angles to AC, has at its extremity B
the shortest projector, and at the extremity D the longest projector.
On the projection, BD appears greatly foreshortened as BvDcy
though still at right angles to the projection oi AC and bisected
by it.

— C7

_ . •*

■-,

i ! ^

i !

CA J>

Fig

38.

The true shape of the projection is an ellipse, of which A^Cv is
the major axis and BJ)v is the minor axis. No matter at what
angle the plane of projection lies, the projection of a circle is an
ellipse whose major axis is equal to the diameter of the circle.

For convenience the two planes of projection in Fig. 36 have
been considered as V and S, and the projections lettered accord-
ingly. The plane of projection in Fig. 37 has been treated as if
it were V? and the ellipse so lettered. It must be remembered that
the three forms in which the circle projects upon a plane, as a
circle, as a line, and as an ellipse, cover all possible cases, and the
relations between the plane of the circle and the plane of projec-
tion shown in the two figures are intended to be perfectly general
and not confined to V and § alone.

Curved Lines

48. The Circle in a Horizontal or Vertical Plane. — Passing now
to the descriptive drawing of a circle, the simplest case is that of
a circle which lies in a plane parallel to fl, V or §. The projec-
tions are then of the kind shown in Fig. 36, two projections being
lines and one the true shape of the circle. Fig. 38 shows the case
for a circle lying in a horizontal plane. The true shape appears in
Hi. The V projection shows the diameter AC, the S projection
shows the diameter BD.

Fig. 39.

49. The Circle in an Inclined Plane. — Fig. 39 shows the circle
lying in an inclined plane, perpendicular to Vj and making an
angle of 60° with ffil. The V projectors, lying in the plane of the
circle itself, form a projector-plane and the V projection is a
straight line equal to a diameter of the circle. As the plane of the
circle is oblique to H and §, these projections on ff and S are
ellipses Avhose major axes are equal to the diameter of the circle.
Of course, for any point of the curve, as P, the laws of projection
hold, as is indicated. The true shape of the curve can be shown by

Engineerixg Descriptive Geometry

projection on any plane parallel to the plane of the circle. It is
here shown on the auxiliary plane HJ, taken as required. If the
drawing were presented with projections H, V and g, as shown,
one might at first suspect that it represented an ellipse and not a
circle ; but, if a number of points were plotted on \], the existence
of a center 0' could be proved by actual test with the dividers.

50. The Circle in an Oblique Plane. — ^AVhen a circle is in an
oblique plane, all three projections are ellipses, as in Fig. 40. The
noticeable feature is that the three major axes are all equal in
length.

When an ellipse is in an oblique plane, its three projections are
also ellipses, but the major axes will be of unequal lengths. The
proof of this fact must be left until later. The fact that the three
projections have their major axes equal must be taken at present as
sufficient evidence that the curve itself is a circle.

51. The Ellipse: Approximate Representation. — The ellipse is
little used as a shape for machine parts. It appears in drawings
chiefly as the projection of a circle. Some properties of ellipses
are very useful and should be studied for the sake of reducing the
labor of executing drawings in which ellipses appear..

An approximation to a true ellipse by circular arcs, known as the
" draftsman's ellipse/' may be constructed when the major axis 2a
and the minor axis 2h, Fig. 41, of an ellipse are known.

Curved Lines

The steps in the process are shown in Fig. 41., The center of the
ellipse is at 0. The major axis is AC, equal to 2a. The minor
axis is DB, equal to 2&. From C, one end of the major axis, lay
off CE, equal to I. The point £' is at a distance equal to a—h from
0 and at a distance equal to 2a — h from A. This last distance is
the radius of a circular arc which is used to approximate to the
flat sides of the ellipse. It may be called the " side arc." Setting
the compass to the distance AE and using D and B as centers,
points H and 0 are marked on the minor axis, extended, for use
as centers for the " side arcs.'* These arcs are now drawn (passing
through the points D and 5), as shown in the 2nd stage of the
process.

\ 5^ 5fa^e

THE

3rd Stage

DRAFTSMAN'S ELLIPSE."
Fig, 41.

By use of the bow spacer, the distance OE is bisected and the
half added to itself, giving the point F (distant f (^— &) from 0).
F is the center of a circular arc which approximates to the end of
the true ellipse. With F as center, and FC as radius, describe this
arc. If this work is accurate, this " end arc " will prove to be
tangent to the side arcs already drawn, as shov/n in the 3rd stage
of the process. If desired, the exact point of tangency of the two
arcs, K, may be found by joining the centers H and F and extend-
ing the line to K. F is s^vung about 0 as center by compass or
dividers to F', for the center of the other ^^end arc." In inking
such an ellipse, the arcs must be terminated exactly at the points
of tangency, K and the three similar points.

This method is remarkably accurate for ellipses whose minor
5

Enginei;eing Descriptive Geometry

axes are at least two-thirds the length of their major axes. It
should always be used for such wide ellipses, and if the character
of the drawing does not require great accuracy, it may be used
even when the minor axis is but half the length of the major axis.
For all narrow ellipses, exact methods of plotting should be used.
52. The Ellipse: Exact Representation. — The true and accurate
methods of plotting an ellipse are shown in Figs. 42, 43, and 44.
Fig. 42 is a convenient method when the major axis AC and minor
axis BD are given, bisecting each other at 0. Describe circles with
centers at 0, and with diameters equal to AC and BD. From 0
draw ani/ radial line. From the point where this radial line meets
the larger circle draw a vertical line, and from the point where it
cuts the smaller circle draw a horizontal line. Where these lines

Fig. 42.

Fig. 43

meet at P is located a point on the ellipse. By passing a large
number of such radial lines sufficient points may be found between
D and C to fully determine the quadrant of the ellipse. Having
determined one quadrant, it is generally possible to transfer the
curve by the pearwood curves with less labor than to plot each
quadrant.

With the same data a second metho(L Fig. 43, is more convenient
for work on a large scale when the T-square, beam compass, etc.,
are not available.

Construct a rectangle using the given major and minor axes as
center lines. Divide DE into any number of equal parts (as here
sliown, 4 parts), and join these points of division with C. Divide
DO into the same number of equal parts (here, 4). From A
draw lines through these last points of division, extending them to
the first system of lines intersecting the first of the one system with

Curved Lines 55

the first of the other, the second with the second, etc. These inter-
sections, 1, 2, 3, are points on the ellipse.

The third method, an extension or generalization of the second, is
very useful when an ellipse is to be inscribed in a parallelogram, the
major and minor axes being unknown in direction and magnitude.
Lettering the parallelogram A'B'C'D' in a manner similar to the
lettering in Fig. 43, the method is exactly the same as before, D'E'
and D'O being divided into an equal number of parts and the lines
drawn from C and A\ The actual major and minor axes, indicated
in the figure, are not determined in any manner by this process.

53. The Helix. — The curve in space (not a plane curve) which
is most commonly used in machinery, is the helix. This curve is
described by a point revolving uniformly about an axis and at the
same time moving uniformly in the direction of that axis. It is
popularly called a " cork-screw ^' curve, or " screw thread," or even,
quite incorrectly, a " spiral."

The helix lies entirely on the surface of a cylinder, the radius of
the cylinder being the distance of the point from the axis of rota-
tion, and the axis of th? cylinder the given direction.

Fig. 45 represents a cylinder on the surface of which a moving
point has described a helix. Starting at the top of the cylinder, at
the point marked 0, the point has moved uniformly completely
around the cylinder at the same time that it has moved the length
of the cylinder at a uniform rate. The -circumference of the top
circle of the cylinder has been divided into twelve equal parts by
radii at angles of 30°, the apparent inequality of the angles being due
to the perspective of the drawing. The points of division are marked
from 0 to 11, point 12 not being numbered, as it coincides with
point 0. The length of the cylinder is divided into twelve equal
parts on the vertical line showing the numbers from 0 to 12, and
at each point of division a circle, parallel to the top base, is de-
scribed about the cylinder. The helix is the curve shown by a
heavy line. From point 0, which is the zero point of both move-
ments, the first twelfth part of the motion carries the point from 0
to 1 around the circumference, and from 0 to 1 axially downward,
at the same time. The true movement is diagonally across the
curved rectangle to the point marked 1 on the helix. This move-

ExGiNEERiNG Descriptive Geometry

ment is continued step by step to the points 2, 3, etc. In the posi-
tion chosen in Fig. 45, points 0, 1, 2, 3, 4, 12 are in full view,
points 5 and 11 are on the extreme edges, and the intermediate

Fig. 45.

points (from 6 to 10) are on the far side of the cylinder. The
construction lines for these latter points have been omitted, in order
to keep the figure clear.

Curved Lines 57

54. Projections of the Helix. — The projection of this curve on a
plane parallel to the axis of the C3'linder is shown to the left. The
circles described about the cylinder become equidistant parallel
straight lines. The axial lines remain straight but are no longer
equally spaced, and the curve is a kind of continuous diagonal to
the small rectangles formed by these lines on the plane of projec-
tion.

The projection of the helix on any plane perpendicular to the
axis of the cylinder is a circle coinciding with the projection of the
cylinder itself. The top base is such a plane and on it the projec-
tion of the helix coincides with the circumference of the base.

55. Descriptive Drawing of the Helix. — The typical descriptive
drawing of a helix is shown in Fig. 46. The^axis of the cylinder is
perpendicular to ff, and the top base is parallel to ff. The helix
in IHI appears as a circle. In V-it appears as on the plane of pro-
jection in Fig. 45, but this view is no longer seen obliquely as is
there represented.

This V projection of the helix is a plane curve of such import-
ance as to receive a separate name. It is called the "sinusoid."
Since the motion of the describing point is not limited to one com-
plete revolution, it may continue indefinitely. The part drawn is
one complete portion and any addition is but the repetition of the
same moved along the axial length of the curve. The proportions
of the curve may vary between wide limits depending on the rela-
tive size of the radius of the cylinder to the axial movement for one
revolution. This axial distance is known as the "pitch" of the
helix.

In Fig. 46 the pitch is about three times the radius of the helix.
In Fig. 47, a short-pitch helix is represented, the pitch being about
J the radius, and a number of complete rotations being shown.

The proportions of the helix depends therefore on the radius and
on the pitch. To execute a drawing, such as Fig. 46, describe first
the view of the helix which is a circle. Divide the circumference
into any number of equal parts (12 or 24 usually). From these
points of division project lines to the other view or views. Divide
the pitch into the same number of equal parts, and draw lines per-
pendicular to those already drawn. Pass a smooth curve through

Engineering Descriptive Geometry

the points of intersection of these lines, forming the continuous
diagonal. In Figs. 45 and 46 the helix is a " right-hand helix/'
The upper part of Fig. 47 shows a left-hand helix, the motion of
rotation being reversed, or from 12 to 11 to 10, etc. The ordinary

J-*

— ~

zTZ

Fig. 46.

Fig. 47.

screw thread used in machinery is a very short-pitched right-hand
helix. It is so short indeed that it is customary to represent the
curve by a straight line passing through those points which would
be given if the construction were reduced to dividing the circum-

Curved Lines

ference and the pitch into 2 equal parts. This is shown in the
lower part of Fig. 47, where only the points 0, 6 and 12 have been
used.

The concealed portion of the helix is then omitted entirely, no
broken line for the hidden part being allowed by good practice.

56. The Curved Line in Space. — A curve in space may some-
times be required, one which follows no known mathematical law,
but which passes through certain points given by their coordinates.
For example, in Fig. 48, four points, A (12,1,9), B (5,4,6),

Fig. 48.

C (2,4,4) and D (2,5,1), were taken as given and a "smooth
curve/^ the most natural and easy curve possible, has been passed
through them. It is fairly easy to pass smooth curves through the
projections of the 4 points on each reference plane, but it is essen-
tial that not only should the original points obey the laws of pro-
jection of Art. 11, but every intermediate point as well. The
views must check therefore point by point and the process of trac-
ing the curve must be carried out about as follows: The projec-
tions of the 4 points on V a^d § are seen to be more evenly ex-
tended than those on ff, and smooth curves are made to pass
through them by careful fitting with the draftsman's curves. The

60 Engineering Descriptive Geometry

view on ff cannot now be put in at random, but must be constructed
to correspond to the other views. To fill in the wide gap between
Ah and Bn an intermediate point is taken, as Ev on AvBv. By a
horizontal line Eg is defined. From Ev and Eg the ff projection
(Eh) is plotted by the regular method of checking the projections
of a point. As many such intermediate points may be taken as may
seem necessary in each case.

To define the sharp turn on the curve betw'een Cn and Dh, one
or more extra points, as Fh, should be plotted from the V and §
projections. Thus every poorly defined part is made definite and
the views of the line mutually check. The work of " la3'ing out "
the lines of a ship on the " mold-loft floor " of a shipbuilding plant
is of this kind, with the exception that the curves are chiefly plane
curves, not curves in space.

Problems V.
(For blackboard or cross-section paper.)

50. Make the descriptive drawing of a circle lying in a plane
parallel to S, center at 0 (3, 6, 7) and radius 5.

51. Make the descriptive drawing of a circle lying in a plane
perpendicular to V and making an angle of 45° with H (the trace
in V passing through the points (18, 0, 0), and (0, 0, 18)). The
center of the circle is at C (9, G, 9), and the radius is 5. (Make
the V projection first, then a, projection on an auxiliary plane \].
From these views construct the ff and S projections, using 8 or 9
points.

52. Make the descriptive drawing of a circle in a plane perpen-
dicular to H? the trace in ff passing through the points (12, 0, 0)
and (0,16,0). The center is at (6,8,10) and the radius is 8.
(Draw the plan and an auxiliary view showing true shape first, and
from those views construct the projections on V and §.)

53. An ellipse lies in a plane passing through the axis of Y and
making angles of 45" with ff and g. The ff projection is a circle,
center at (10, 10, 0) and radius 8. Prove that the S projection is
also a circle and find the true shape of the ellipse by revolving the
plane of the ellipse into the plane ff.

Curved Lines 61

54. An ellipse lies in a plane passing through the axis of Y and
making an angle of 60° with H and 30° with S- The H projec-
tion is a circle, center at (8, 8, 0), radius 6. Find the true shape
of the ellipse. Construct the view on § by projecting points for the
center and for the extremities of the axes of the ellipse. Pass a
draftsman's ellipse through those points. Show that no appreci-
able error can be observed.

55. Construct a draftsman's ellipse, on accurate cross-section or
coordinate paper, with major axis 24 units, and minor axis 12 units.
Perform the accurate plotting of the true ellipse on the same axes,
one quadrant by the method of Fig. 42 and one by the method of
Fig. 43, using 6 divisions for DE and OD. Note the degree of
accuracy of the approximate process.

56. On coordinate paper, plot an ellipse by the method of Fig.
43, the major axis being 16 units long and the minor axis 8 units.
Plot another ellipse whose major axis is 16 and whose minor axis
is 12. (To divide the semi-minor axis of 6 units into 4 equal parts,
use points of division on the vertical line CE instead of OD. CE
being twice as far from A as OD, 12 units must be used for the
whole length, and these divided into 4 parts.)

57. On isometric paper pick out a rhombus like the top of Fig.
19, but having 8 units on each side. Inscribe an ellipse by plotting
by the method of Fig. 44.

58. Make the descriptive drawing of a helix whose axis is per-
pendicular to S through the point (0,7,7). The pitch of the
helix is 12, and the initial point is (2, 7, 2). Draw the ff and V
projections of a right-hand helix, numbering the points in logical
order.

59. Connect the 4 points A (10, 8, 10), B (8, 10, 6), C (6, 9, 4)
and D (2, 2, 4) by a smooth curve, filling out poorly defined por-
tions in S from the ff and V projections.

CHAPTER VI.
CTTRVED SURFACES AND THEIR ELEMENTS.

57. Lines Representing Curved Surfaces. — To represent solids
having curved surfaces, it is not enough to represent the actual
comers or edges only. Hitherto only edges have appeared on de-
scriptive drawings, and it has been a feature of the drawings that
every point represented on one projection must be represented on
the other projections, the relation between projections being strictly
according to rule. We now come to a class of lines which do not
appear on all three views, lines due to the curvature of the surfaces.

The general principle, called the " Principle of Tangent Projec-
tors," governing this new class of lines is as follows: In projecting
a curved surface to a given plane of projection (by perpendicular
projectors, of course) all points, and only those points, whose pro-
jectors are tangent to the curved surface should be projected. A
good illustration of this principle is shown in Fig, 45, where the
cylinder is projected upon the plane of projection. The top and
bottom bases are edges, and project under the ordinary rules, but
along the straight line 0, 1, 2, . . . ., 12 the curved surface of the
cylinder is itself perpendicular to the plane of projection. If from
any point on this line a projector is drawn to the plane of projection
(as is shown in the figure for the points 1, 2, 3, etc.), this projector
is tangent to the cylinder. The whole line therefore projects to
the plane of projection. The projection of the cylinder on a plane
parallel to its axis is therefore a rectangle, two of its sides repre-
senting the circular bases and the two other edges representing the
curved sides of the cylinder.

58. The Right Circular Cylinder. — The complete descriptive
drawing of a cylinder is therefore as shown in Fig. 49. This cylin-
der is a right circular cylinder. Mathematicians consider that the
cylinder is " generated " by revolving the line A A' about PP'y the
axis of the cylinder. The generating line in any particular posi-

Curved Surfaces and Their Elements

tion is called an " element '' of the surface. Thus AA', BB', CC,
etc., are elements.

When the cylinder is projected upon V, AA' and CC are the
elements which appear in V because the V projectors of all points
along those lines are tangent to the cylinder, as can be seen from
the view on H- The elements which are represented by lines on
S are BB' and DD'.

The right circular cylinder may also be considered as generated
by moving a circle along an axis perpendicular to its own plane
through its center.

P ■

■ X

P 0

r ^

f'j

JD'

c'l^

B^^B

Fig. 49.

Fig. 50.

In Fig. 45 consider the top base of the cylinder to be moved
down the cylinder. Each successive position of the circle is a " cir-
cular element" of the cylinder. The circles through the points
1, 2, 3, etc., are simply circular elements of the cylinder taken at
equal distances apart.

59. The Inclined Circular Cylinder. — Fig. 50 shows an inclined
circular cylinder. It has circular and straight line elements as
before, though it cannot be generated by revolving a line about
another at a fixed distance, but can be generated by moving the
circle ABCD obliquely to A'B'C'D', the center moving on the axis
PP\ The straight elements are all parallel to the axis. The cross-
section of a cylinder is a section taken perpendicular to the axis.

Engineering Descriptive Geometry

In this case the cross-section is an ellipse, and for this reason the
Inclined Circular Cylinder is sometimes called the Elliptical Cyl-
inder.

60. Straight and Inclined Circular Cones. — If a generating line
AP, Fig. 51, meets an axis PP' at a point P, and is revolved about
it, it will generate a Straight Circular Cone. The cone has both
straight and circular elements, the circular elements increasing in
size as they recede from the vertex P. The base A BCD is one of
the elements.

The Inclined Circular Cone (Fig. 52) has straight and circular
elements, but it is not generated by revolving a line about the axis.
The circular elements move obliquely along the axis PP' and in-
crease uniformly as they recede from the vertex P.

61. The Sphere. — ^The Sphere can be generated by revolving a
semiciicle about a diameter. Each point generates a circle, the
radii of the circles for successive points having values varying
between 0 and the radius of the sphere. Since the sphere can be
generated by using any diameter as an axis, the number of ways in
which the surface can be divided into circular elements is infinite.

62. Surfaces of Revolution. — In general, any line, straight or
curved, may be revolved about an axis, thus creating a surface of
revolution. Every point on the " generating line " creates a " cir-

CuKVED Surfaces and Their Elements

cular element " of the surface, and the plane of each circular ele-
ment is perpendicular to the axis of the surface.

The straight circular cylinder is a simple case of the general
class of surfaces of revolution. To generate it a straight line is
revolved about a parallel straight line. The different points of the
generating line create the circular elements of the cylinder, and

'M^'

;

Fig. 53.

Fig. 54.

the different positions of the generating line mark the straight ele-
ments. The cone and the sphere are also surfaces of revolution, as
they are generated by revolving a line about an axis.

If a circle be revolved about an axis in its own plane, but en-
tirely exterior to the circle, a solid, called an " anchor ring," is
generated. A small portion of this surface, part of its inner surface,
is often spoken of as a " bell-shaped surface,'* from its similarity
to the flaring edge of a bell.

Any curved line may create a surface of revolution, but in de-

66 Engineering Descriptive Geometry

signs of machinery lines made np of parts of circles and straight
lines are most frequently used. Figs. 53 and 54 show two exam-
ples which illustrate well the application of the Principle of Tan-
gent Projectors. The generating line is emphasized and the cen-
ters of the various arcs are marked.

Any angular point on the generating line, as a (Fig. 53), creates
a circular edge on the surface. This edge appears as a circle on the
plan (as aa' on li), and as a straight line, equal to the diameter,
on the elevation (as aa' on V)- See also the point h (Fig. 54).
In addition, any portion of the generating line which is perpen-
dicular to the axis, as h (Fig. 53), even if for an infinitely short
distance only, creates a line on the side view, as hh' on V? but no
corresponding circle on H. A V projector from any point on the
circular element created by the point h is tangent to the surface,
and therefore creates a point on the drawing, but an f\ projector
is not tangent to the surface, e is a similar point, and so also is
j of Fig. 54.

Any point, as c, Fig. 53, where the generating line is parallel to
ihe axis for a finite, or for an infinitely small distance, generates
a circular element, from every point of which the J-f projectors are
tangent to the surface, but the V projectors are not. A circle c(f
appears, therefore, on the plan for this element of the surface of
revolution, but no straight line on the side view. J is a similar
point, as are also / and g, on Fig. 54.

63. The Helicoidal Surface. — If a line, straight or curved, is
made to revolve uniformly about an axis and move uniformly along
the axis at the same time, every point in the line will generate a
helix of the same pitch. The surface swept up is called a Heli-
coidal Surface.

The generating line chosen is usually a straight line intersecting
the axis. The surfaces used for screw threads are nearly all of
this kind. Fig. 55 gives an example of a sharp V-threaded screw,
the two surfaces of the thread having been generated by lines in-
clined at an angle of 60° to the axis. Fig. 56 shows a square
thread, the generating lines of the two helicoidal surfaces being
perpendicular to the axis. Any particular position of the straight
line is a " straight element " of the helicoidal surface.

Curved Surfaces and Their Elements

64. Elementary Intersections. — In executing drawings of ma-
cliinery it is often necessary to determine the line of intersection of
two surfaces, plane or curved. The simplest lines of intersection
are such as coincide with elements of a curved surface. They may

Fig. 55.

Fig. 56.

be called " Elementary Intersections." An elementary intersection
may arise when a curved surface is intersected by a plane, so placed
as to bear some simple relation to the surface itself.

In Fig. 49, any plane perpendicular to the axis of the cylinder
intersects it in a circular element of the cylinder, and any plane
parallel to the axis of the cylinder (or containing it) intersects

68 Engineering Desceipttve Geometry

it (if it intersects it at all) in two straight line elements of the
cylinder.

In Fig. 50 any plane parallel to the base of the cylinder inter-
sects it in a circular element, and any plane parallel to the axis,
or containing it, intersects it in straight elements of the cylinder.

In Fig. 51 or 52 any plane parallel to the base of the cone inter-
sects it in a circular element, and any plane containing the vertex
of the cone (if it intersects at all) intersects the cone in straight
elements.

In Fig. 53 or 54 any plane perpendicular to the axis of the sur-
face of revolution intersects it in a circular element.

In Fig. 55 or 56 any plane containing the axis of the screw inter-
sects the helicoidal surfaces in straight elements. The plane per-
pendicular to ff, cutting H in a trace PQ, and cutting V in a
trace QR, cuts the helicoidal surfaces at each convolution in straight
elements. Only ah and a'h are marked on the figure.

Problems VI.

(For blackboard or cross-section paper or wire-mesh cage.)

60. Draw the projections of a cylinder whose axis is P (6, 2, 6),
P' (6, 16, 6), and radius 5. Draw the intersection of this cylinder
with a plane parallel to fi, at 4 units from H, and with a plane
parallel to V? 10 units from V-

61. An inclined circular cylinder has its bases parallel to S- Its
axis is P (2, 7, 7), P' (14, 7, 13). Its radius is 5. Draw the V
and S projections and the intersection, with a plane parallel to §,
6 units from S, and with a plane parallel to V? 3 units from V»

62. Draw a cone with vertex at P (4,8,8), center of base at
P' (16, 8, 8), and radius 6, the base lying in a plane parallel to S-
Draw the intersection with a plane parallel to §, 12 units from S,
and with a plane perpendicular to §, whose trace in S passes
through the points (0, 8, 8) and (0, 14, 0).

63. An oblique cone has its vertex at P (16, 8, 4) and its base in
a plane parallel to ff, center at P' (8, 8, 16), and radius 5. Draw
the intersection with a plane parallel to H? 13 units from H, and
with a plane containing the axis and the point (16, 0, 16).

Curved Surfaces and Their Elements 69

G4. A cone has an axis P (8, 2, 2), F (8, 14, 10). Its base is in
a plane parallel to V? l-i units from \ , and its radius is 6 units.
Draw the intersection with a plane containing the vertex and the
points (0,14,12) and (16,14,12).

65. A surface of revolution is formed by revolving a circle, whose
center is at (12, 8, 8) and radius 3 units, lying in a plane parallel
to Vj about an axis perpendicular to W at the point (8, 8, 0). It
is cut by a plane parallel to H at a distance of 6 units from f\.
Draw the intersections.

QQ. A sphere has its center at (8, 8, 8) and a radius of 5 units.
Draw the intersection with a cylinder whose axis is P (8,8,0),
P' (8,8,16), and whose radius is 4 units, its bases being planes
perpendicular to its axis.

67. A sphere has its center at (8, 8, 8) and a radius of 5 units.
Find its intersection with a cone whose vertex is P (0, 8, 8), center
of base (16, 8, 8), and radius of base 6 units, the base being in a
plane §' parallel to S-

68. In Fig. 53 let the generating line Pahcde be revolved about
ee' as an axis. Assume any dimensions for the line and draw the

V and § projections of the surface of revolution thus formed.
Draw tbe intersection with a plane parallel to S just to the right
of d.

69. In Fig. 54 let the generating line Pfgh be revolved about
liW as an axis. Assume any dimensions for the line and draw the

V and § projections of the surface of revolution formed.

CHAPTEE VII.
INTERSECTIONS OF CURVED SURFACES.

65. The Method of the Intersection of the Intersections. — The

determination of the line of intersection of two curved surfaces (or
of a curved surface and a plane), when not an " Elementary Inter-
section," is of much greater difficulty and requires a clear under-
standing of the nature of the curved surfaces themselves, and some
little ingenuity in applying general principles.

The method generally relied upon for the solution is the use of
auxiliary intersecting planes so chosen as to cut elementary inter-
sections with each of the given surfaces. These elementary inter-
sections are drawn and the points of intersection of the intersec-
tions are identified and recorded as points on the required line of
intersection. This method is spoken of as " finding the intersec-
tion of the intersections.^' When a number of auxiliary planes
have been used in this way, a smooth curve is passed through the
points on the required intersection of the surfaces, as described in
Art. 55. It should not be necessary, however, to interpolate points
to fill out gaps as was done in Fig. 48 for E and F. This can be
done better by the use of more auxiliary intersecting planes. Ex-
amples of tliis method will make it clear.

66. An Inclined Circular Cylinder Cut by an Inclined Plane. —
In Fig, 57 an inclined cylinder, axis PP', is cut by a plane perpen-
dicular to Y> and inclined to ff. The traces of this plane are IJ
in H, JK in V, and KL in S-

It is an Inclined Plane (see Art. 19), not an Oblique Plane.
Having the descriptive drawing of the cylinder and the traces of
the plane given, the problem is to draw the line of intersection of
the surfaces. It is well-known that in this case the line of inter-
section is an ellipse, but the method of determining it permits the
ellipse to be plotted whether it is recognized as such or not. No
use is to be made of previous knowledge of the nature of the curve

Intersections of Curved Surfaces

of intersection of any of the cases treated in this and the next
chapter.

Two variations of the method are applicable in this case. In the
first method, auxiliary intersecting planes may be taken parallel to
the axis of the cylinder. The simplest method of doing this is to
take auxiliary planes parallel to V? since the axis itself is parallel

Fig. 57.

to V. Let R'R be the trace on H, and RR" the trace on S of a
plane parallel to V- We may call this plane simply "' R.''

Let e and / be the points where R'R cuts the top base of the cyl-
inder. Project these points from Hi to V and in V draw ee' and
ff parallel to PP\ These straight elements of the cylinder are the
lines of intersection of the auxiliary plane with the cylinder. As
a check on the work, e' and f , where R'R in H cuts the bottom
base of the cylinder, should project vertically to e' and f in V-

7^ Engineering Descriptive Geometry

The auxiliary plane cuts the given plane JK in a line of inter-
section whose projection on V coincides with JK itself.

The points ;' and h, where ee' and ff intersect JK, are the "" inter-
sections of the intersections/' and are therefore points on the line
cf intersection of the cylinder and the plane K. Project j and h
to R'R on H and to RR" on S. These are points on the required
curves in ff and g. By extending in H the projecting lines of
j and Ic as far above the axis PP' as ; and h are below it, ;' and k',
points on the upper half of the cylinder, symmetrical with ; and k
en the lower half, are found. The construction is equivalent to
passing a second auxiliary plane parallel to PP^ at the same dis-
tance from PP' as R, but on the other side.

By passing a number of planes similar to R, a sufficient number
of points are located to define accurately the ellipse ahcd in H

andS.

The true shape of this ellipse is shown in JJ? a plane parallel to
JK, at any convenient distance. In the example chosen, the plane
JK has been taken perpendicular to PP\ so that the ellipse abed is
the true cross-section of the cylinder. Nothing in the method de-
pends on this fact and it is perfectly general and applicable to any
inclined plane.

A variation may be made by passing the auxiliary planes per-
pendicular to V and parallel to PP\ ee' in V J^^ay be taken as the
trace of such a plane. The intersections of this auxiliary with both
surfaces should be traced and the intersection of the intersections
identified and recorded as a point of the curve required. / and /
are the points thus found. This method indeed requires the same
construction lines as before, but gives a different explanation to
them.

67. A Second Method Using Circular Elements of the Cylinder. —
A plane parallel to the base of the cylinder and therefore, in this
case, parallel to H, will cut the cylinder in a line of intersection
which is one of the circular elements of the cylinder. Let T'T and
TT", in Fig. 58, be the traces of a plane " T" parallel to H- The
axis of the cylinder PP' pierces the plane T at p. p is therefore
tl:e center of the circle of intersection of the auxiliary plane T with
the cylinder. Project p to ff, and using ,p as a center and with a
radius equal to pt, describe the circle as shown.

Intersections of Curved Surfaces

The planes T and JK are both, perpendicular to V or " seen on
edge " in V- Their line of intersection is therefore perpendicular
to V> or is " seen on end " in V? as the point j. Project /to H?
where it appears as the line ;;'. This line is the intersection of the
two planes.

The points j and ;', where this line of intersection ;/ meets the
circular intersection whose center is at p, are the " intersections of
the intersections/' and are points on the required curve.

N I

Fig. 58.

Planes like T, at various heights on the cylinder, determine pairs
of points on the curve of intersection on H- From H and V the
points may be plotted on S by the usual rules of projection, thus
completing the solution.

68. Singular or Critical Points. — It is nearly always found that
one or two points on the line of intersection may be projected di-
rectly from some one view to the others without new construction
lines. In this case a and c in \, Fig. 57, may be projected at once

Engineering Descriptive Geometry

to IM and S. They correspond theoretically to points determined
by a central plane, cutting fi in a trace PP'. h and d may also be
projected directly, as they correspond to planes whose traces in
IHI are BB' and DD\ These critical points sliould always be the
first points identified and recorded, though usually no explanation
will be given, as they should be obvious to any one who has grasped
the general method.

69. A Cone Intersected by an Inclined Plane. — Fig. 59 shows

the descriptive drawing of a right circular cone intersected by an
inclined plane whose traces are JK and KL, Two methods of solu-
tion are shown.

A plane R, containing the axis PP', and therefore perpendicular
to fi, is shown by its traces R'R and RR". It intersects the cone
in the elements Pj and Ph. From JHI project these points / and Ic
to Vj and draw the elements in V- The V projection of the inter-
section of R with the plane JK is the line JK, and the points e and
/ are the intersections of the intersections, e and / are now pro-

Intersections of Curved Surfaces

jected to the plan, where they necessarily lie on the line BE". Sym-
metrical points e' and f are also plotted and all four points trans-
ferred to the side elevation.

A plane T perpendicular to the axis PP' whose traces are T'T
and TT" may be used instead of R. Its intersection with the cone
is a circle, seen on edge in the front elevation as the line hh'. Its
center is g, and radius is gh. Draw this circle in the plan. The

intersection of T with the plane JK is a line, seen on end, as the
point / of the front elevation. Draw /'/ in IHI as this line. The
points / and /' are the intersections of the intersections.

70. Intersection of Two Cylinders. — Fig. 60 shows the inter-
section of two cylinders. Since they are right cylinders, and their
axes are at right angles, planes parallel to any one of the three
reference planes will cut only straight or circular elements of the
cylinders. By the solution. Fig. 60, auxiliary planes parallel to V

Engineerixg Descriptive Geometry

liave been chosen, the traces of one being R'R and BE". This plane
intersects the vertical cylinder in the lines IcTc' and IV, and, it inter-
sects the horizontal cylinder in the lines mm! and nn'. The inter-
sections of these intersections are the points marked r.

If the axes of the cylinders do not meet but pass at right angles,
no new complication is introduced. If the axes of the cylinders
meet at an angle, and one or both cylinders are inclined, the choice

of methods may be greatly reduced, but one method is always pos-
sible. To discover it, try planes parallel to the axes of both cylin-
ders, or parallel to one axis and to one plane of reference; or in
some manner bearing a definite relation to the nature of the sur-
faces.

71. Intersection of a Cylinder and a Sphere. — In Fig. 61 a
sphere is intersected by a cylinder, whose axis PP' does not pass
through the center of the sphere at §. In the solution. Fig. 61,

Intersections of Curved Surfaces

auxiliary planes parallel to V have been chosen, the traces of one of
them being E'E and EE". The plane E cuts the sphere in a circle
whose diameter is eg, as given by the plan. This circle is described
in V- The intersections of this circle with the elements of the
cylinder W and IV are the points marked r, points on the required
curve of intersection.

In this case the points are first determined on the front elevation
and then projected to the side elevation. Solutions by planes par-
allel to ffi or to S may be made, requiring however different con-
struction lines.

Pig. 62.

72. Intersection of a Cone and a Cylinder: Axes Intersecting. —

In Fig. 62 a cone and a cylinder intersect at right angles. The
solution chosen is by horizontal planes, as T.

An alternate solution is by planes perpendicular to S, and con-
taining the point P. The planes must cut both surfaces, and their
traces, where seen on edge, as PE, Fig. 62, must cut the projections
of both surfaces. These two solutions hold good even if the axes
do not meet but pass each other at right angles.

.78

Engineering Descriptive Geometry

If the axes are not at right angles, modifications must be made,
and the search for a system of planes making elementary intersec-
tions with hotli surfaces requires some ingenuity and thought.

73. Intersection of a Cone and Cylinder : Axes Parallel. — A
simple case is shown in Fig. 63. Two methods of solution are avail-
able. In one, horizontal planes are used. Each plane, such as T,

Fig. 63.

makes circular intersections, with both cone and cylinder, the inter-
sections intersecting at points t and t. A second method is by
planes perpendicular to H, containing the axis PP'. One plane
'' R " is shown by its traces WP in H and PP" in S. this plane
being taken so as to give the same point t on the curve and another
point t' . In the execution of drawings of this class it is natural to
take the auxiliary planes at regular intervals if the planes are
parallel to each other, or at equal angles if the planes radiate from
a central axis.

Inteksections of Curved Surfaces 79

Problems VII.

' 70. An inclined cylinder has one base in H and one in a plane
parallel to H- Its axis is P (11, 8, 0), F (5, 8, 16) . Its radius is
4 units. It is intersected by a plane perpendicular to Yy whose
trace passes through the points (5, 0, 0) and (11, 0, 16). Draw the
three projections and show one intersecting auxiliary plane by con-
struction lines.

71. A cone has its vertex in H at (6,6,0) and its base in a
plane parallel to IM, center at (6, 6, 12), and radius 5. It is inter-
sected by a plane containing the axis of Y and making angles of
45° with H and S. Draw the projections.

72. A cone has its vertex at (2, 14, 16) and its base is a circle
in flf center at (8, 8, 0), and radius 6. Find its intersection with
a vertical plane 4 units from g.

73. A right circular cylinder has its base in S, center at (0, 8, 8),
and radius 4. Its axis is 16 units long. Another right cylinder
has its base in H, center at (8, 8,0), radius 5, and axis 16 units
long. Draw their lines of intersection, the smaller cylinder being
supposed to pierce the larger.

74. A right circular cylinder has its base in S, center at (0, 7, 8),
and radius 4. Its axis is 16 units long. Another right circular
cylinder has its base in fl, center at (8, 9, 0), radius 5, and axis 16
units long. Draw their line of intersection, the smaller cylinder
being supposed to pierce the larger.

75. Two inclined circular cylinders of 3 units radius have their
bases in H and in fj' (16 units from ff). The axis of one is
P (4,8,0), P' (12,8,16), and of the other is Q (12,8,0),
Q' (4,8,16). Prove that their intersection consists of two parts,
one a circle in a plane parallel to H, and one an ellipse in a plane
parallel to §.

76. A sphere has its center at (8, 9,8), and radius 6^ units. A
vertical right circular cylinder has its top base in ff, center at
(8,6,0), radius 4, and length 16 units. Find the intersection of
the surfaces.

77. A right circular cylinder, axis P (0,8,9), P' (16,8,9),
radius 5, is pierced by a right circular cone. The base of the cone

80 Engineering Descriptive Geometry

is in a plane 16 iinits from ff, center at Q' (8, 8, 16), and radius
6. The vertex of the cone is at § (8, 8, 0). Find the lines of inter-
section.

78. An inclined cylinder has an oblique line P (0,11,5),
P' (16,5,11) for its axis. The radius of the circular base is 4
units and the planes of the bases are S, and S' parallel to S at 16
units' distance. The cylinder is cut by a plane parallel to V at 7
units' distance from V- Draw the three projections of the cylinder
and the line of intersection.

79. An inclined cylinder has an oblique line P,(0, 11, 5),
P' (16, 5, 11) for its axis. The radius of the circular base is 4
units, and the planes of the bases are §, and §' parallel to S at 16
units' distance. The cylinder is Qut by a plane perpendicular to
V, its trace passing through the points (2,0,0) and (14,0,16).
Draw the three projections.

CHAPTEE VIII.

INTERSECTIONS OF CURVED SURFACES; CONTINUED.

74. Intersection of a Surface of Revolution and an Inclined
Plane. — In Figs. 64 and 65 a surface of revolution is shown. It is

Fig. 64.

cut by an inclined plane perpendicular to H in the first case, and
by one perpendicular to V in the second case. The planes are
given by their traces, and the problem is to find the curves of inter-
section. Both solutions make use of cutting planes perpendicular
to PP', the axis of revolution of the curved surface.

Engineering Descriptive Geometry

In Fig. 64 a plane T, taken at will perpendicular to PP', cuts
the surface of revolution in a circular element seen as the straight
line at' in V- ^ is projected to H and the circle aW drawn. The
inclined plane whose traces are JK and KL is intersected by the
plane T in a line whose horizontal projection is the line JK itself.
t and f (on lil) are therefore the intersections of the intersections
and are projected to the front elevation, giving points on the re-
quired line of intersection. A system of planes such as T defines
points enough to fully determine the curve, mit'n.

In Fig. 65 the given plane has the traces IX and XZ. The plane
T intersects the surface of revolution on the circle atct', and it

intersects the plane in the line W, seen on end in V as the point t.
t and t' in ffi are points on the required curve of intersection, mit'n.

The point of this surface of revolution APC has been given a
special name. It is an " ogival point." The generating line AP
is an arc of 60°, center at C, and conversely the generating line PQ
has its center at A. The shell used in ordnance is usually a long
cylinder with an ogival point. A double ogival surface is produced
by revolving an arc of 120° about its chord.

75. Intersection of Two Surfaces of Eevolution: Axes Par-
allel.— This problem is illustrated in Fig. QQ, where two surfaces of

Intersections of Curved Surfaces

revolution are shown. A horizontal plane T cuts both surfaces in
circular elements. These elements are drawn in JHI as circles abed
and efgli. t and t' are the intersections of the intersections. From
H t and if are projected to V and S- The problem in Art. 73 is
but a special case of this general problem. In addition to the solu-
tion by horizontal planes another solution is there possible, due to
special properties of the cone and cylinder.

76. Intersection of Two Surfaces of Revolution: Axes In-
tersecting.— An example of two surfaces of revolution whose axes
intersect is given by Fig. 67. A surface is formed by the revolution
of the curve ww' about the vertical axis PP\ and another surface
by revolving the curve uQ about the horizontal axis QQ\ The in-

84 - Engineering Descriptive Geometry

terscction of the axes PP' and QQ' is the point p. The peculiarity
of this case is that no plane can cut both surfaces in circular ele-
ments. However, a sphere described with the point of intersection
of the axes as a center, if of proper size, will intersect both surfaces
in circular elements. V is parallel to both axes and on this pro-
jection a circle is described with p as center representing a sphere.
The radius is chosen at will. To keep the drawing clear, this
sphere has not been described on plan or side elevation, as it would
be quite superfluous in those views.

The sphere has the peculiarity that it is a surface of revolution,
using any diameter as an axis. The curve wiu' and the semicircle
mabn are in the same plane with the axis PP'. When both axes
are revolved about PP'y a and b, their points of intersection, gene-
rate circular elements, which are common to the sphere and to the
vertical surface of revolution. Therefore, these circles are the in-
tersections of the sphere and the vertical surface. The ff and S
projections of these circles are next drawn.

The curve uQ and the semicircle qcdr are in the same plane with
the axis QQ\ When both axes are revolved about QQ\ their inter-
sections, c and d, generate circles which are common to both sur-
faces, or are their lines of intersection. The circle generated by c
is drawn in ff and S, but that generated by d is not needed.

The three circles aa', bb', and cc' appear as straight lines on V*
but from them the points t and s, the intersections of the intersec-
tions, are determined. These are points on the required curve in

The circle aa' appears as a circle ata't' in H, and as a line tt'
in S- The circle cc' appears as a circle ctc't' in S, and as a line ee'
in H. These circles intersect in H at ^ and f, and in S at # and t'
and s and 5'. These are points on the required curves in H and §.

For the complete solution, a number of auxiliary spheres, differ-
ing slightly in radius, must be used.

77. Intersection of a Cone and a Non-Circular Cylinder. — A
non-circular cylinder is a surface created by a line which moves
always parallel to itself, being guided by a curve lying in a plane
perpendicular to the generating line. This curve, called the direc-
trix, is usually a closed curve. The cross-section of such a cylinder
is everywhere similar to the directrix.

UNIVERSITY

Intersections of Curved Surfaces

This fact may be utilized to advantage in some cases. In Fig.
68, an oblique cone and a non-circular cylinder intersect. The
directrix of the cylinder is a pointed oval curve, abed in H. Hori-
zontal planes, as T'T, intersect the cylinder in a curve identical in
shape with its directrix, so that its projection on H coincides with
the projection of the directrix on ff. The intersection with the
cone is a circle, mt'tn, and the intersections of the intersections are
the points t.

78. Alteration of a Curve of Intersection by a Fillet. — In Fig.

69 a hollow cone and a non-circular cylinder, abed in W, intersect.
On the left half the unmodified curve of intersection is traced by
the method of the preceding article, no construction lines being
shown however, as the case is very simple. On the right half the
curve is modified by a fillet or small arc of a circle which fills in
the angular groove. The fillet whose center is at g modifies that
point of the line of intersection marked e. The top of the circular
arc marks the point where an J-J or g projector is tangent to the
surface.
7

Engineering Descriptive Geometry

The corresponding crest to the fillet at other positions on the
curve of intersection is traced as follows : If a line drawn through
h and parallel to PG, the generating line of the cone, is used as
a new generator it will by its rotation about PP' create a new
cone, on the surface of which the required line of the crests of the

Fig. 69.

fillets must lie. If a line mn, parallel to cc', the generating line of
the cylinder, is moved parallel to cc', and at a constant distance
from the surface of the non-circular cylinder, it will generate a
new non-circular cylinder on the surface of which the required
path of the point Ic must lie. The directrix of this new cylinder is
drawn in fl, the line rms, as shown. The intersection of these two

Intersections of Curved Surfaces

Fio. 70.

88 Engineering Descriptive Geometry

new surfaces, found by the method used above (or by planes per-
pendicular to J-f through the axis PP'), is the required path of Ic
or the line which appears on V and §. The line rms, representing
the same path on H, is not properly a line of the drawing and is
not inked except as a construction line.

79. Intersection of a Helicoidal Surface and a Plane. — In Fig.
70 there is shown a long-pitched screw having a triple tliread, such
as is often employe.d for a " worm.^' To the left is shown a partial
longitudinal section giving the generating lines. In V the con-
cealed parts of the helical edges are omitted, except in the cases of
one of the smaller and one of the larger edges. The plane whose
trace on V is KL is perpendicular to the axis, and terminates the
screw threads. The intersection of this plane with the screw
threads is the curve of intersection to be drawn on H- It is deter-
mined by passing planes containing the axis of the worm. One of
these is shown by its traces PR and BR'.

From points a and h in the plan corresponding points are plotted
on the front elevation, a falling on the helix of small diameter
(extended in this case), and & on the helix of large diameter. This
element ab of the helix is seen to pierce the plane KL at Ic. This
point Ic is projected to the plan and is one of the points on the
required curve mTcn.

Problems VIII.

(For units, use inches on blackboard or wire-mesh cage, or small
squares on cross-section paper.)

80. An anchor-ring is formed by revolving a circle of G units
diameter about a vertical axis, so that its center moves in a circle
of 10 units diameter, center at Q (8,8,8). The anchor-ring is
intersected by a plane parallel to V passed through the point
A (8, 6, 8) and by another plane parallel to V through the point
B (8,4,8). Draw the projections of the ring, the traces of the
planes and the lines of intersection.

81. The same anchor-ring is intersected by a plane perpendicu-
lar to \, having a trace passing through the points C (0, 0, 2) and
B (8, 0, 8). Make the descriptive drawing and show the true shape
of the lines of intersection.

Intersections of Curved Surfaces 89

82. The same anchor-ring is intersected by a right circular
cylinder, axis P (12,8,0), P' (12,8,16), and diameter 4 units.
Make the descriptive drawing of the anchor-ring, imagining it to
be pierced by the cylinder.

83. An anchor-ring has an axis P (0,8,8), P' (16,8,8). Its
center moves in a plane 7 units from S, describing a circle of 8
units diameter. The radius of the describing circle is 3 units. It
is intersected by an ogival point whose axis is a vertical line
Q (7, 8, 3|), §' (7, 8, 16). The generating line of the ogival point
is an arc of 60°, with center at (0,8,16), and radius 14 units.
The point Q is the vertex and the point Q' is the center of the circu-
lar base of 7 units radius. The axes intersect at ;? (7, 8, 8). Draw
the projections and the line of intersection, front and side eleva-
tions only.

84. The line P (4, 13, 8), P' (16, 8, 8) is the chord of an arc of
90°, whose radius is 9.2 units. The arc is the generating line of a
surface of revolution of which PP' is the axis. Draw the projection
on IHI- Draw the end view on an auxiliary plane U perpendicular
to PP', the trace of U on H intersecting OX at (16, 0, 0). The
surface is intersected by a plane perpendicular to f\ and contain-
ing the line PP'. Draw the line of intersection on V-

85. The same surface is intersected by a plane perpendicular to
H whose trace in f\ passes through the points (4, 16, 0) and
(16,5,0). Draw the line of intersection on V-

86. The line P (3, 8, 8), P' (13, 8, 8) is the chord of an arc of
90°, radius 7.07 units. It is the axis of revolution of a surface of
which the arc is the generating line. It is intersected by a right
circular cone having its vertex at 9 (8, 8, 2), and center of base at
Q' (8, 8, 12), radius of base 5 units. Draw the line of intersection.

87. A non-circular cylinder has its straight elements, length 16
units, perpendicular to H- The directrix is a smooth curve
through the points A (14, 6, 0), B (12, 4, 0), C (10, 4, 0),
D (8, 5, 0), E (5, 8, 0), F (2, 13, 0). It is pierced by a cylinder
whose base is in \, whose axis is perpendicular to V ^.t the point
(8,0,8), whose radius is 5 units, and whose length is 14 units.
Find the line of intersection in §.

90 Engineering Descriptive Geometry

88. The line P (8, 8, 2), P' (8, 8, 14) is the axis of a right cir-
cular cjdinder of 6 units diameter. Projecting from the cylinder is
an helicoidal surface, of 12 units pitch, of which G (5,8,2),
G' (1,8,2) is the generating line. The helicoid is intersected
by a plane perpendicular to H whose trace in H passes through the
points (5,0,0) and (16,11,0). Draw the plan and front eleva-
tion of the cylinder and helicoid and plot the line of intersection
with the plane.

89. The helicoidal surface of Problem 87 is intersected by a right
circular cylinder whose axis Q (12, 8, 2), Q' (12, 8, 14) is parallel
to PP\ The radius of the cylinder is 3 units. Draw the line of
intersection.

CHAPTEE IX.
DEVELOPMENT OF CURVED SURFACES.

80. Meaning of Development as Applied to Curved Surfaces.—

Many curved surfaces may be developed on a plane in a manner
similar to the development of prisms and pyramids explained in
Articles 45 and 46. By development, is meant flattening out,
without stretching or otherwise distorting the surface. If a curved
surface is developed on a plane and this portion of the plane, called
" the development of the surface/' is cut out, this development may

Fig. 71.

be bent into the shape of the surface itself. The importance of
the process comes from the fact that many articles of sheet metal
are so made. If a sheet of paper is bent in the hands to any fan-
tastic shape, it will always be found that through every point of
the paper a straight line may be drawn on the surface in some one
direction, the greatest curvature of the surface at this point being
in a direction at right angles to this straight line element through
the point. The surfaces which can be formed by twisting a plane
surface without distortion are called surfaces of single curvature.
The curved surfaces, therefore, which are capable of development
are only those which are surfaces of single curvature and have
straight line elements, but not by any means all of tl^se. All forms

Engineerixg Descriptive Geometry

of cylinders and cones, right circular, oblique circular, or non-
circular, may be developed. The helicoidal surfaces, illustrated by
Figs. 55 and 56, though having straight elements, cannot be de-
veloped, nor can the hyperboloid of revolution, a surface generated
by revolving a straight line about a line not parallel nor intersect-
ing. Figs. 71 and 72 are perspective drawings showing the process
of rolling out or developing a right circular cylinder and a right
circular cone.

81. Rectification of the Arc of a Circle. — In developing curved
surfaces it frequently happens that the whole or part of the cir-
cumference of a circle is rolled out into a straight line. Since the
surface must not be stretched or compressed, the straight line must
be equal in length to the arc of the circle. This process of finding
a straight line equal to a given arc is called rectifying the arc. Xo

This angle not
to exceed 60

Fig. 73.

Fig. 74.

absolutely exact method is possible, but methods are known which
are so nearly exact as to lead to no appreciable error. These have
the same practical value as if geometrically perfect.

In Fig. 73, AB is the arc of a circle, center at C. For accurate
work the arc should not exceed 60°. It is required to find a
straight line equal to the given arc. Draw AH, the tangent at one
extremity, and draw AB, the chord. Bisect AB at D. Produce the
chord and set oK AE equal to AD. With £' as a center, and with
EB as a radius, describe the arc BF, meeting AH at F. Then
^F=arc AB, within one-tenth of one per cent.

In this figure, and in the two following ones, the arc and the
straight line equal to it are made extra heavy for emphasis.

Development of Curved Surfaces

82. Rectifying a Semicircle. — A second method, applicable par-
ticularly to a semicircle, was recently devised by Mr. George Pierce,
In Fig. 74 the semicircle AFB is to be rectified. A tangent BC,
equal in length to the radius, is drawn at one extremity. Join AC,
cutting the circumference at D. Lay off DE = DC, and join BE,
producing BE to the circumference at F. Join AF. Then the
triangle AEF, shown lightly shaded, has its periphery equal to the
semicircle AFB, within one twenty-thousandth part. The peri-
phery may be conveniently spread into one line by using A and E
as centers, and with AF and EF as radii, swinging F to the left to
0 and to the right to H on the line AF extended. GH is the recti-
fied length of the semicircle.

83. To Lay Off an Arc Equal to a Given Straight Line. — This
inverse problem, namely to lay off on a given circle an arc equal to

\^(jo exceed 60"

A B

Fig. 75.

Fig. 76.

a given straight line, frequently arises. In Fig. 75 a line AB is
given. It is required to find an arc of a given radius AC equal to
the given line AB. At A erect a perpendicular, making AC equal
to the given radius, and with C as a center describe the arc AF.
On AB, take the point B at one-fourth of the total distance from
A. With T> as center and DB as a radius, draw the arc BF, meet-
ing AF at F. AF is the required arc, equal to AB.

This process is also accurate to one-tenth of one per cent if the
arc AF is not greater than 60°. If in the application of this process
to a particular case the arc AF is found to be greater than 60°, the
line AB should be divided into halves, thirds or quarters, and the
operation applied to the part instead of to the whole line.

94 Engineering Descriptive Geometry

84. Development of a Straight Circular Cylinder. — In Fig. 60
let the intersecting cylinders represent a large sheet-iron ventilat-
ing pipe, with two smaller pipes entering it from either side. Such
a piece is called by pipe fitters a " cross." The problem is to find
the shape of a flat sheet of metal which, when rolled np into a
cylinder, will form the surface of the vertical pipe, with the open-
ings already cut for the entrance of the smaller pipes. Before
developing the large cylinder, it must be considered as cut on the
straight element BB\ After the pipe is formed from the develop-
ment used as a pattern, the element BB' will be the location of a
longitudinal seam.

A rectangle. Fig. 76, is first drawn, the height BB' being equal
to the height of the cylinder and the horizontal length being equal
to the circumference of the base BCD A. (This length may be best
found by Mr. Pierce's method, which gives the half-length, BD.)
On the drawing. Fig. 60, the base BCDA must be divided into
equal parts, 24 parts being usually taken, as they correspond to
■arcs of IS'', which are easily and accurately constructed with the
draftsman's triangles. Only 6 of these 24 parts are required to be
actually marked on Fig. 60, as the figure is doubly symmetrical
and each quadrant is similar to the others. On Fig. 76 the line
BCDAB is divided into 24 parts also, the numbering of the lines
of division running from 0 to 6 and back to 0 for each half-length
of the development. In V of Fig. 60, draw the elements corre-
sponding to the points of division. The elemnt IV already drawn
corresponds to No. 4, and BB' and CC correspond to Nos. 0 and 6.
The others are not drawn in Fig. 60, to avoid complicating the
figure, but would have to be drawn in practice before constructing
the development. On the four elements which are numbered 4 on
the development. Fig. 76, lay off the distances Ir equal to Ir in
Fig. 60. On the two elements. Fig. 76, numbered 6, lay off Cc or
Aa equal to Cc of Fig. 60, and imagine the proper distances to be
laid off on elements numbered 3 and 5. Smooth curves through
the points thus plotted are the ovals which must be cut out of the
sheet of metal to give the proper-shaped openings for the small
pipes.

When it is known in advance that the surface of such a cylinder

Development of Curved Surfaces

as that in Fig. 60 must be developed, it is often possible to so
choose the system of auxiliary intersecting planes used to define
the curve of intersection as to give the required equally spaced
straight elements for the development.

The smaller cylinder may be developed in the same way. A new
system of equally spaced straight elements would probably have to
be chosen for this cylinder.

85. Development of a Right Circular Cone. — The cone of Fig.
63 has been selected for this illustration. Imagine it to be cut on
the element PB and flattened into a plane. The surface takes the

Fig. 77.

form of a sector of a circle, the radius of the sector being the slant
height of the cone (or length of the straight element), and the arc
of the sector being equal in length to the circumference of the base
of the cone. Several means of finding the length of the arc of the
sector are available.

The most natural method is to rectify the circumference of the
base and then, with the slant height as radius, to draw an arc and
to lay out on the arc a length equal to this rectified circumference.
In Fig. 63 suppose that the semi-circumference ABC (in fil) has
been rectified by Pierce's method. In Fig. 77 let an arc be drawn
with radius PB equal to PB in S? Fig. 63, and from B draw a
tangent BE equal to one-half the rectified length of the semi-cir-
cumference. Find the arc BC equal to BE by the method of Art.

96 Engixeerixg Descriptive Geometry

83, Fig. 75. BC is one-fourth of the required arc, and corresponds
to the quadrant BC in W, Fig. 63. Divide the arc BC and the
quadrant BC into the same number of equal parts, numbering
them from 0 to 6, if 6 parts are chosen. Eepeat the divisions in
the arc CD (equal to BC), numbering the points of division from
6 down to 0, this duplication of numbers being due to the symmetry
of the H projection of Fig. 63, about the line APC. In Fig. 63,
as in Fig. 77, the points 0 to 6 are all supposed to be Joined to P,
the only straight elements actually shown there being PO, P4, and
P6.

On the elements P4 of the development lay off the true length
of the line Pt (and the true length of the line Pf also). Ft is an
oblique line, but if its H projector-plane {Pt in H? Fig. 63) be
revolved up to the position Pm, the point Mn V moves to m, and
Pm is the true length of Pt. The distance Pg (V? in Fig. 63) is
laid off on P& of the development.

When the proper distances have been laid off on the elements
P2, P3 and P5, a smooth curve may be drawn through the points.
The sector, with this opening cut in it, is the pattern for forming
the cone out of sheet iron or any thin material.

If the ratio of PA to P'A in \, Fig. 63, can be exactly deter-
mined, the most accurate method of getting the angle of the sector is
by calculation, for the degrees of arc in the development are to the
degrees in the base of the cone (360°) as the radius of the base of
the cone is to the slant height. In this case P'A is f PA. The
sector in Fig. 75 subtends f X360°, or 216°. In the use of this
method a good protractor is required to lay out the arc.

Problems IX.

90. Draw an arc of 60° with 10 units radius. At one end draw
a tangent and on the tangent lay off a length equal to the given
arc. On the tangent lay off a length of 8 units, and find the length
of arc equal to this distance.

91. An arc of 12 units radius, one of 9 units radius, and a
straight line are all tangent at the same point. Find on the tan-
gent the straight line equal in length to 45° of the large arc. Find
the length on the other arc equal to this length on the tangent and
show that it is an arc of 60°.

Development of Curved Surfaces 97

92. Rectify a semicircle of 10 units radius and compare this
length with the calculated length, 31.4 units.

93. A rectangle 31.4 units by 12 units is the developed area of a
cylinder of 10 units diameter. A diagonal line is drawn on the
development, which is then rolled into cylindrical form. Plot the
form taken by the diagonal and show that it is a helix of 12 units
pitch.

94. A right circular cone has a base of 10 units diameter, and
a vertical height of 12 units. Its slant height is 13 units. Calcu-
late the angle of the sector which is the developed surface of the
cone. Find this angle by rectifying the circumference of the base
of the cone, and by finding the arc equal to the rectified length.
(This last operation must be performed on one-third or one-quarter
of the rectified length, to keep the accuracy within one-tenth of
one per cent.)

95. A semicircle, radius 10 units, is rolled up into a cone. What
is the radius of the base? What is the slant height? What is the
relation between the area of the curved surface of the cone and the
area of the base ?

96. A right circular cylinder, such as Fig. 49, is of 7.59 units
diameter, and 12 units height. It is intersected by a plane per-
pendicular to V through the points C and A'. Draw plan, front
elevation and the development of the surface.

97. A right circular cone, like that of Fig. 51, has its front ele-
vation an equilateral triangle, each side being 10 units in length.
From Av a perpendicular is drawn to PvCv cutting it at E. If this
line represents a plane perpendicular to V? draw the development
of the cone with the line of intersection of the cone and plane traced
on the development.

98. A right circular cylinder, standing in a vertical position, as
in Fig. 49, diameter 7 units, and length 10 units, is pierced from
side to side by a square hole 3 J units on each edge, the axis of the
hole and the axis of the cylinder bisecting each other at right
angles. Draw the development of the surface.

99. A sheet of metal 22 units square with a hole 11 units square
cut out of its middle, the sides of the hole being parallel to the
edges of the sheet, is rolled up into a cylinder. Draw the plan,
front and side elevations of the cylinder.

CHAPTER X.

STRAIGHT LINES OF UNLIMITED LENGTH AND THEIR

TRACES.

86. Negative Coordinates. — ^We have dealt only with points hav-
ing positive or zero coordinates, and the lines and planes have been

limited in their extent, or, if infinite, have extended indefinitely
only in the positive directions. As it becomes necessary at times
to trace lines and planes in their course, no matter if they cross
the reference planes into new regions of space, the use and meaning
of negative coordinates must be explained. The value of the x
coordinate of a point is the length of the § projector or perpen-
dicular distance from the point to the side reference plane g. (See
Figs. 6 and 7, Art. 9.) If this value decreases gradually to zero,

Lines of Unlimited Length: Their Traces

the point moves towards S until it lies in S itself. If this value
becomes negative, it is clear that the point crosses the side reference
plane into a space to the right of it.

For example, a point P, having a variable x coordinate, but hav-
ing its y coordinate always equal to 4 and its z coordinate equal to
2, is a point moving on a line parallel to the axis of X. If x de-
creases to zero, it is on S at the point marked Ps in Fig. 78. If
the X coordinate decreases further, reaching a value of —3, it
moves to the point P in that figure. Fig. 78 is the perspective
( — 3,4,2). The y and z projectors cannot
H and V in their customary positions, but

drawing of a point P
project the point P to

• IH],ext.t\

j \

X ' 0

V 9

and ^v
^extended.

e, Js

:ti

9 p

Y. ^

i5 V

Fig. 79.

Fig. 80.

Fig. 81.

project it upon parts of those planes extended beyond the axes of
Y and Z, as shown. In Fig. 79, the corresponding descriptive
drawing, it must be understood that the plane IM, extended, has
been revolved with H? about the axis of X, into the plane of the
paper, V? and § has been revolved as usual about the axis of Z,
coming into coincidence with V? extended. This " development "
of the planes of reference is exactly as described in Art. 7. It is
noticeable that the x coordinate of P is laid off to the right of the
origin instead of to the left. Ph lies, therefore, in the quadrant
which usually represents no plane of projection, and Pv lies in the
quadrant which usually represents S- Ps lies in its customary
place, since both y and z, the coordinates which alone appear in S>
are positive.

100 Engineering Descriptive Geometry

It is evident that the laws of projection for ff, V ^^d §, Art.
11, have not been altered, but simply extended. Ph and Pv are in
the same vertical line; Pv and Ps are in the same horizontal line;
and the construction which connects Pn and Ps still holds good.

In Fig. 79 the space marked S represents not only S but V
extended as well.

In Fig. 80 is represented a point P (3, — 2, 3 ) , having a negative
y coordinate. The point is in front of V^ ^t 2 units' distance, not
behind V- The projection on fi, instead of being above the axis
of X a distancee of 2 units, is below it by the same amount. So also
the projection on S is to the left of the axis of Z, a distance of 2
units, instead of the the right of it. After developing the reference
planes in the manner of Art. 7, plane H? extended, has come into
coincidence with V? a^d plane §, extended, has also come into co-
incidence with V- Thus the field representing V represents also
the other two reference planes, extended.

In Fig. 81 a point P (2,2,-3) having a negative z coordinate
is represented. The point is above H 3 units, instead of below ff,
at the same perpendicular distance. P projects upon V on V
extended above the axis of X. After developing the reference
planes, plane ff comes into coincidence witti V extended. Pa is
on § extended above the axis of Y, and therefore after develop-
ment it occupies the so-called " construction space."

Points having two or^three negative coordinates may be dealt
with in the same manner, but are little likely to arise in practice.

It is evident that subscripts must be used invariably, to prevent
confusion whenever negative values are encountered.

87. Graphical Connection Between Ph and Ps- — In Figs. 79, 80
and 81, Ph and Ps are connected by a construction line PhfhfsPs in
a manner which is an extension of that shown by Fig. 7, Art. 9.
Xote that the quadrant of a circle connecting Ph and Ps must be
described alw^ays on the construction space or on the field devoted
to V, never on the fields devoted to IM or S-

88. Traces of a Line of Unlimited Length, Parallel to an Axis. —
A straight line w^hich has no limit to its length, but extends in-
definitely in either direction, must necessarily have some points
whose coordinates are negative. In passing from positive to nega-

Lines of Unlimited Length: Their Traces

101

tive regions the line must pass through some plane of reference
(having one of its coordinates zero at that point), and the point
where it pierces a plane of reference is called the trace of the lino
on that plane of reference, the word trace being used to indicate a
" track '' or print showing the passage of the line.

Lines parallel to the axes have been used freely already. An fil
projector is simply a vertical line or line parallel to the axis of Z.
Any perspective figure showing a point P and its horizontal pro-
jection Ph will serve as an illustration of this line, as PPh in Fig.
G, Art. 9.

Fig. 82.

Imagine PPh to be extended in both directions as an unlimited
straight line. Then Pn is the trace of the line on H. In Fig. 7,
the point Ph itself is the W projection of the line. Pt-<?. extended
in both directions, is the vertical projection and Psfs is the side
projection. Thus it is seen that a vertical line has but one trace,
that on the plane to which it is perpendicular. PPv may be taken
as an illustration of a line parallel to the axis of Y, and PPg of one
parallel to the axis of X. A better example of this latter case is
shown iH^igs. 15 and 16, Art. 16. The line BAAs, perpendicular
to S, has itfe trace on § at Aa,

102

Engineering Descriptive Geometry

89. Traces of an Inclined Straight Line. — An inclined line snch
as AB in Figs. 82 and 83 pierces two reference planes as at A and
B, but as it is parallel to the third reference plane, S, it has no
trace on S. The peculiarity of the descriptive drawing of this line.
Fig. 83, is the apparent coincidence of the H and V projections
as one vertical line. The S projection is required to determine the
traces A and B.

90. Traces of an Oblique Straight Line : The H and V Traces. —
An oblique line, if unlimited in length, must pierce each of the
reference planes, since it is oblique to all three. Any line is com-

FiG. 84.

Fig. 85.

Fig. 86.

Fig. 87.

pletely defined when two points on the line are given. If two
traces of a straight line' are given, the third trace cannot be assumed,
but must be constructed from the given conditions by geometrical
process. It will always be found that of the three traces of an
oblique line one trace at least has some negative coordinate.

As the complete relation between the three traces is somewhat
complicated, the relation between two traces, as, for instance, H
and V traces, must be considered first. Two cases are shown, the
first by Figs. 84 and 85, and the second by Figs. 86 and 87. The
line AB is the line whose traces are A (5,0,4) and B (2,4,0).
The line CD is the line whose traces are C (7, 0, 5) and D (2, 4, 0).

Lines of Unlimited Length: Their Traces

103

From the descriptive drawing of AB, Fig. 85, it is seen that the
H projection of the line cuts the axis of X vertically above the
trace on V^ ^^^ that the V projection cuts the axis of X vertically
under the trace on H. It may be noted that the two right triangles
AhBBv and BvAAn have the line AhB^ on the axis of X as their
common base. From the descriptive drawing of the line CD, Fig.
87, it is seen that the effect of the vertical trace C having a nega-
tive z coordinate simply puts C (on V) above Cn, instead of below it.
The two right triangles CnDDv and DvCCh have the line ChDv on
the axis of X, as their common base, but the latter triangle is above
the axis instead of in its normal position.

Fig. 88.

91. Traces of an Oblique Straight Line: The V and S Traces. —

Figs. 88 and 89 show two lines piercing V aii4 S-

The line AB pierces V at A and S at B. The two right triangles
AsABv and BvBAg have their common base AgBv on the axis of Z.
The line CD pierces V at (7 and S extended at D, the point D
having a negative y coordinate. The right triangles CgCDv and
DvDCs have their base DvCg in common on the axis of Z, but in the
descriptive drawing DvDCs lies to the left of the axis of Z instead
of to the right, owing to the point D having a negative y coordinate.

104

Engineerixg Descriptive Geometry

92. Traces of an Oblique Straight Line : The H and S Traces.—

Figs. 90 and 91 show two lines piercing ff and §.

The line AB pierces n at A and § at B. The triangles AsABh
and BhBAs have their common base AsBn on the axis of Y, Fig. 90,
but in the descriptive drawing the duplication of the axis of Y
causes this base AsBh to separate into two separate bases, one on
OYh and one on OYs. Otherwise, there has been no change.

The line CD pierces H at C and S extended at D, the point D
having a negative z coordinate. In Fig. 90 CsCDh and DhDCs have

Fig. 90

their common base CgDh on the axis of Y, but in the descriptive
drawing CgDh appears in two places. The triangle DnDCs lies above
S in the "construction space," or on S extended, since D has a
negative z coordinate.

93. Three Traces of an Oblique Straight Line. — Figs. 92 and 93
show an oblique straight line ABC piercing V at A, H at B, and
S extended at C. Since the line is straight, the three projections
of the line AB^Cv, AsBsC and AjtBCh are all straight lines. In the
perspective drawing. Fig. 92, part of the V projection is on V ex-
tended and part of the S projection on S extended.

Lines of Unlimited Length: Their Traces

105

In the descriptive drawing, Fig. 93, the relation between A and
B is the same as that in Fig. 85, as shown by the two triangles
AhABv and BvBAh, or the quadrilateral AhABvB. The relation
between A and C, as shown by the quadrilateral AgACvC, is the
same as that between A and B, Fig. 89, as shown by the quadri-
lateral AsABvB. The relation between B and C, Fig. 93, as shown
by the two triangles BsBCn and ChCBg, is the same as that between
C and D of Fig. 91, as shown by the triangles CgCDh and DhDCs.
Xo new feature has been introduced.

Small part
o/Y extended

Fig. 92.

94. Paper Box Diagram. — To assist in understanding Figs. 92
and 93, a model in space should be made and studied from all
sides. The complete relation of the traces is then quickly grasped.
Construct the descriptive drawing, Fig. 93, on coordinate paper,
using, as coordinates for A, B and C, (15,0,12), (5,12,0), and
(0, 18, —6). Fold into a paper box after the manner of Fig. 9,
Art. 12, having first cut the paper on some such line as mn, so that
the part of the paper on which C is plotted may remain upright,
serving as an extension to g. It will be found that a straight wire
or long needle or a thread may be run through the points A, B and
C, thus producing a model of the line and all its projections.

106 Engineering Descriptive Geometry

95. Intersecting Lines. — If two lines intersect, their point of
intersection, when projected upon any plane of reference, must
necessarily be the point of intersection of the projections on that
plane. For example, a line AB intersects a line CD at E. Project
E upon a plane of reference, as lil. Then Eh must be the point of
intersection of AhBn and ChDh. In the same way Ev must be the
point of intersection of AvBv and CvDv, and Eg of AgBg and CsDg.

To determine whether two lines given by their projections meet
in space or pass without meeting, the projections on at least two
reference planes must be extended (if necessary) till they meet.
Then for the lines themselves to intersect, the points of intersec-
tion of the two pairs of projections must obey the rules of pro-
jection of a point in space (Art. 11). Thus if AhBh and ChDn are
given and meet at a point vertically above the point of intersection
of AvBv and CvDv, the two lines really meet at a point whose pro-
jections are the intersections of the given projections. If this con-
dition is not filled the lines pass without meeting, the intersecting
of the projections being deceptive.

96. Parallel Lines. — If two lines are parallel, the projections of
the lines on a reference plane are also parallel (or coincident).
For, the two lines make the same angle with the plane of pro-
jection; their projector-planes are parallel; and the projections
themselves are parallel.

Thus if a line AB is parallel to another line CD, then AhBh must
be parallel to ChDh, AvBv to CvDv, and AgBg to CsDg. If the two
lines lie in a plane perpendicular to a plane of projection — for
example, perpendicular to H — then the ff projector-planes coin-
cide and the ff projections also coincide. The V and S projections
are parallel but not coincident.

If two lines do not fill the conditions of intersecting or of parallel
lines, they must necessarily be lines which pass at an angle without
meeting.

Lines of Unlimited Length: Their Traces 107

Problems X.

100. Plot the points 4 (8, 6, -4), 5 (7, -3, 5), (7 (-7, 0, 12).

101. Plot the points A (6, -10, 3), 5 (0,0,-8), C (-6,5,4).

102. Make a descriptive drawing of a line 26 units long from
the point P (—8,4,9), perpendicular to g. What traces does it
have ? What are the coordinates of its middle point ?

103. A line is drawn from P (12,5,16) perpendicular to H.
Make the descriptive drawing of the line, and of a line perpen-
dicular to it, drawn from Q (0, 0, 8). What is the length of this
perpendicular line, and where are its traces?

104. A straight line extends from A (8, 12, 0) through D
(8, 6, 8) for a distance of 20 units. Make the descriptive drawing
of the line. Where are its traces and its middle point ?

105. A straight line pierces H at A (8, 6, 0) and Y at B
(8, 0, 12). Draw its projections. Where is its trace on §? What
are the coordinates of D, its middle point ?

106. A straight line extends from E (15,6,16) through A
(3, 6, 0) to meet g. Make the descriptive drawing and mark the
traces on H and §.

107. Draw the lines A (16, 11, 8), B (4, 8, 2) ; C (12, 5, 10),
D (0,2,4); and E (11,3,0), F (5,15,8). Which pair meet,
which are parallel, and which pass at an angle? What are the
coordinates of the point of intersection of the pair which meet?

108. The points A (8, 0, 12), B (0, 8, 6) and (7 (-8, 16, 0) are
the traces of a straight line. Make the descriptive drawing of the
line.

109. The points A (8, -4,0), D (4,4,6) and E (2,8,9) are
on a straight line. Find the trace B where it pierces V and the
trace C where it pierces S.

CHAPTER XI.
PLANES OF UNLIMITED EXTENT: THEIR TRACES.

97. Traces of Horizontal and Vertical Planes. — The lines of
intersection of a plane with the reference planes are called its
traces. Planes of unlimited extent may be of three kinds, parallel
to a reference plane, inclined, or oblique. Unlimited planes of the
first two classes have been dealt with already, but for the sake of
precision may be treated here again to advantage.

A horizontal plane is one parallel to fi, and the trace of such a
plane on V is a line parallel to the axis of X, and the trace on §
is a line parallel to the axis of Y. These traces meet the axis of Z
at the same point and appear on the descriptive drawing as one
continuous line. There is. of course no trace on H- In Pig- 58,
Art. 67, the plane T, represented by its traces TT on V and TT"
on §, is a horizontal plane. These traces are not only the intersec-
tions of T with H and S, but T is " seen on edge " in those views.-
Every point of the plane T , when projected upon V^ lies somewhere
on the line T'T, extended indefinitely in either direction.

A vertical plane parallel to V ^las for its traces a line on ff
parallel to the axis of X, and on S a line parallel to the axis of
Z, with no trace on V- These traces meet the axis of Y at the
same point, and appear on the descriptive drawing as two lines at
right angles to this axis, the point on Y separating into two points
as usual. In Fig. 57, Art. ^Qi, a vertical plane U, parallel to V? is
represented by its traces WR on H and RU" on g.

A vertical plane parallel to § has for its trace on H a line paral-
lel to the axis of Y , and for its trace on V a line parallel to the
axis of Z, with no trace on g. These traces meet the axis of X at
the same point and appear on the descriptive drawing as one con-
tinuous line.

98. Traces of Inclined Planes. — Inclined planes are those per-
pendicular to one reference plane, but not to two reference planes.
The auxiliary planes of projection have been of this kind. In

Planes of Unlimited Extent: Their Traces

109

Fig. 20, Art. 22, the plane \], perpendicular to Hi, has the line
MX for its trace on ff, and XN for its trace on V- I^- the de-
scriptive drawing. Fig. 21, MX and XNv are these traces.

If in Fig. 20 both U and S are imagined to be extended towards
the eye, they will intersect in a line parallel to OZ. This § trace
will be on S extended, and every point of it will have the same
negative y coordinate. Of the three traces of U> two are vertical
lines, and one only, MX, is an inclined line. The plane in Fig. 64,
Art. 7-1, may be taken as a second example of an inclined plane
perpendicular to ff. The trace on § is not a negative line in this
case, but is a vertical line on § to the right of the axis of Z at a
distance equal to OJ.

In Fig. 57, Art. 66, IJ, JK and KL are the three traces of an
inclined plane perpendicular to V- ^^ every case of an inclined
plane the inclined trace is on that reference plane to which it is
perpendicular, and shows the angles of the inclined plane with one
or both of the other reference planes.

p\ ^ b]

Fig. 95.

99. Traces of an Oblique Plane: All Traces "Positive." — The

general case of an oblique plane is shown in Fig. 94. The plane
P is represented as cutting the cube of reference planes in the lines
marked PH, PV and PS. These lines are the traces of the plane
P, and may be understood to extend indefinitely, the plane itself
extending in all directions without limit. They are shown limited
in Fig. 94 in order to make a more realistic appearance. PH, PV,
and PS are used to define the three traces.

110

Engineering Descriptive Geometry

Where PH and PV meet we have a point common to three planes,
P, HI and V- Since it is common to fil and V it is on the line of
intersection of H and V> or in other words it is on the axis of X.
This point is marked a. In the same way PH and P8 meet at h
on the axis of Y, and PV and PS meet at c on the axis of Z.

The descriptive drawing, Fig. 95, is obvious from the explana-
tion of the perspective drawing. From Fig. 95 it is evident that
if two traces of a plane are given the third trace can be determined

Fig. 96.

Fig. 97.

by geometrical construction. Thus, if PH and PV are given, P8
may be defined by extending PH to & on the axis of Y and extend-
ing PV to c on the axis of Z. The line joining he is the required
trace of the plane on §. If any two points on one trace are given,
and any one point on a second trace, the whole figure may be com-
pleted. Thus any two points on PH define that line and enable a
and h to be found. A third point on PV, taken in conjunction
with a, defines PV, and enables c to be located. Ic, as before,
defines the trace PS. This is an application of the general prin-
ciple that three points determine a plane.

Planes of Unlimited Extent: Their Traces

111

100. Traces of an Oblique Plane: One Trace "Negative." — In

Figs. 94 and 95 the plane P lias been so selected that all traces have
positive positions. These are the portions usually drawn. Of
course each trace may be extended in either direction, points on
the trace then having one or more negative coordinates. Any
trace having points all of whose coordinates are positive, or zero,
may be called a positive trace.

In Fig. 96 a plane P is shown, intersecting H and V in the
"positive" traces PH and PV. The third trace, P8, in this case,
has no point all of whose coordinates are positive. In the descrip-
tive drawing, Fig. 97, the tw^o positive traces, meeting at a on the

^yfo. Y3

Fig. 98.

Fig. 99.

axis of Xj are usually considered as fully representing the plane P.
From these lines PH and PV, alone, the imagination is relied upon
to " see the plane P in space," as shown by Fig. 96.

In Fig. 98, the plane Q is represented. Ordinarily the positive
traces QV and ^^S", meeting at c on the axis of Z, are the only
traces shown in the descriptive drawdng, Fig. 99, and are considered
to indicate perfectly the path of the plane Q.

101. Position of the Negative Trace. — The negative trace PS,
in Fig. 96, is shown as one of the edges of the rectangular plate
representing the unlimited plane P. This line PS has been de-
termined by extending PH to meet the axis of Y (extended) at

112 EXGINEERIXG DESCRIPTIVE GEOMETRY

h, and by extending PV to meet the axis of Z (extended) at c.
The line joining h and c is the trace PS. It will be noted that in
finding the location of Pas' in Fig. 97, PV has been extended to cut
the axis of Z (extended np from ZO) at c and PH has been ex-
tended to cut the axis of Y (extended down from YO) at h. 1)
has been rotated 90° about the origin, and the points h and c thus
plotted (on S extended) have been found to give the line PS.
Every step of the process and the lettering of the figure have been
similar to those used in finding PS from PH and PF in Art. 98.

In Fig. 98, the negative trace is QH, the top line of the rect-
angular plate representing the unlimited plane Q. QH has been
determined as follows: QV extended meets the axis of X extended
at a, and QS extended meets the axis of Y extended at h. The line
ab is therefore the trace on H, or QH. In the descriptive drawing
the same process of extending QV to a and QS to h determines the
line QH, a line every point of which has some negative coordinate.
Of course QH must be considered as drawn on parts of the plane
lil extended over V^ S? and the so-called construction space. In
finding the negative traces, it is imperative to letter the diagrams
uniformly, keeping a for the intersection of the plane with the axis
of X, h for that with the axis of Y, and c for that with the axis of
Z. With this rule h will always be the point which is doubled by
the separation of the axis of Y into two lines, and the arc hh will
always be described in the construction space or in the quadrant
devoted to V? never in those devoted to H and g.

102. Parallel Planes. — If two planes are parallel to each other,
their traces on H, V and S are parallel each to each. This prop-
osition may be proved as follows: If we consider two planes P
and Q parallel to each other and each intersecting the plane H, the
lines of intersection with ff (PH and QH) cannot meet, for, if
they did meet, the planes themselves would meet and could not then
be parallel planes. PH and QH must therefore be parallel lines
described on H. Thus, if a plane P and a plane Q are parallel,
then PH and QH are parallel, PV and QV are parallel, and PS
and QS are parallel.

The method of finding the true length of a line by its projection
upon a plane parallel to itself, treated in Chapter III, is really the

Planes of Unlimited Extent: Their Traces

113

process of passing a plane parallel to a projector-plane of the given
line. Thus in Fig. 21, Art. 22, the auxiliary plane U has its hori-
zontal trace XM parallel to AhBn, and the vertical trace of the ff
projector-plane, if drawn, would be parallel to XNv.

103. The Plane Containing a Given Line. — If a line lies on a
plane, the trace of the line on any plane of reference (the point
where it pierces the plane of reference) must lie on the trace of the
plane on that plane of reference. Thus, if the line EF, Fig. 100,
lies on the plane P, then A, the trace of EF on ff, lies on PH, the
trace of P on H ; and B, the trace of EF on V, lies on PV, the
trace of P on V.

Fig. 100.

Fig. 101.

From this fact it follows that to pass a plane which will contain
a given line it is necessary to find two traces of the line and to pass
a trace of the plane through each trace of the line. As an infinite
number of planes may be passed through a given line, it is neces-
sary to have some second condition to define a single plane. For
example, the plane may be made also to pass through a given point
or to be perpendicular to a reference plane.

In Fig. 100, if only the line EF is given and it is required to pass
a plane P, containing that line, and containing also some point,
as a, on the axis of X, the process is as follows: Extend the line
EF to A and B, its traces on HI and V- Joi^i Ba and oA. These

114 Engineering Descriptive Geometry

are the traces of the required plane P. In the descriptive drawing,
Fig, 101, the corresponding operation is performed. A and B
must be determined as in Art. 90, and joined to a. These lines
represent the traces of a plane containing the line EF and the
chosen point a.

To pass a plane Q containing the line EF and also perpendicular
to H (Figs. 100 and 101), the trace of § on H must coincide with
the projection of EF on H, for the required plane perpendicular to
fi is the ff projector-plane of the line. Its traces are therefore
ABh and BnB,

The traces of a plane containing EF and perpendicular to V are
BAv and AvA.

104. The Line or Point on a Given Plane. — ^To determine whether
a line lies on a given plane is a problem the reverse of that just
treated. It amounts sim.ply to determining whether the- traces of
the line lie on the traces on the plane. Thus, in Fig. 101, if PV
and PH are given, and the line EF is given by its projections, the
traces of EF must be found, and if they lie on PH and PV the line
is then known to lie on the given plane P.

To determine whether a given point lies on a given plane is
almost as simple. Join one projection of the point with any point
on the corresponding trace of the plane. Find the other trace of
the line so formed, and see whether it lies on the other trace of the
given plane. Thus in Fig. 101, if the traces PH and PV and the
projections of any one point, as E, are given, select some point on
PH, as A, and join EjA and EvAv. Find the trace B. If it lies
on PV, the point E itself lies on P.

To draw on a given plane a line subject to some other condition,
such as parallel to some plane of reference, is always a problem in
constructing a line whose traces are on the traces of the given
plane, and which yet obeys the second condition, whatever it may be.

105. The Plane Containing Two Given Lines. — From the last
article, if a plane contains tivo given lines, the traces of the plane
must contain the traces of the lines themselves. The given lines
must be intersecting or parallel lines, or the solution is impossible.

In Fig. 102 two lines, AB and AC, are given by their projections.
They intersect at A, since Ah, the intersection of the fl projections,

Planes of Unlimited Extent: Their Traces

115

is vertically above Av, the intersection of the V projections. Ex-
tend the lines to E, F, G and H, their traces on ff and V- Join
the fi traces, E and G, and produce the line also to a on the axis
of Z. Join the V traces, H and F, and extend the line HF also
to a. Ea and aH are the traces of a plane P containing both lines,
AB and AC. The meeting of the two traces at a is a test of the
accuracy of the drawing.

This process may be applied to a pair of parallel lines, but not of
course to two lines which pass at an angle without meeting.

-L 1

\>xG

-^ V

Fig. 102.

106. The Line of Intersection of Two Planes. — If two planes
P and Q are given by their traces, their line of intersection must
pass through the point where the Qi traces meet and the point
where the V traces meet. Thus, in Fig. 103, FB. and QB. meet
at A and FY and QY meet at B. A and B are points on the
required line of intersection of F and Q, and since A is on H and
B is on Vj they are the H and V traces of the line of intersection.
AB}i and BAv are therefore the projections, and should be marked
FQn and FQ^,

.116

Engineering Descriptive Geometry

A.™ "

V >^ .

Fig. 103.

Planes of Unlimited Extent: Their Traces 117

107. Special Case of the Intersection of Two Planes : Two Traces
Parallel. — ^The construction must be varied a little in the special
case when two of the traces of the planes are parallel. In Fig. 104
the traces PV and QV are parallel. In carrj'ing out the construc-
tion as in Fig. 100, it is necessary to join Av with B. But the
point B is the intersection of PV and QV, which are parallel, and
is therefore a point at an infinite distance in the direction of those
lines, as indicated by the bracket on Fig. 104. To join Av with B
at infinity means to draw a line through Av parallel to PV and QV.

PH ^

r\ \

•\N \

^ POv

Ja^ %

'^%

A S

Qvy

Fig. 105.

From B, at infinity, a perpendicular must be supposed to be drawn
to the axis of X, intersecting it at Bh. Bn is therefore at an infinite
distance to the right on the axis of X (extended). To join the
point A with the point Bh means, therefore, to draw a line through
4 parallel to the axis of X. These lines are the required projec-
tions of PQ.

108. Special Case of the Intersection of Two Planes: Four

Traces Parallel. — Another special case arises when the four traces

(on two planes of projection) are parallel. It is then necessary to

refer to a third plane of projection. In Fig. 105 the planes P and

118

Engineering Descbiptive Geometry

Q have their four traces on H and V all parallel. The planes are
inclined planes perpendicular to §, and if their traces are drawn
on S, their intersection is the line PQ. In S both P and Q are
" seen on edge/' so their line of intersection is " seen on end."
From PQsf PQv and PQn are drawn by projection.

109. The Point of Intersection of a Line and a Plane. — The
simple cases of this problem have been previously explained and
used. If the plane is horizontal, vertical or inclined, there is

Fig. 106.

always one view at least in which it is seen on edge. In that view
the given line is seen to pierce the given plane at a definite point
from which, by the rules of projection, the other views of the point
of intersection are easily determined. Thus in Fig. 27, Art. 38,
the point a, where PA pierces the plane KL, is determined first in
V and then projected to H and §.

The general case of this problem may be solved as in Fig. 106.
A plane P is given by its traces PH and PV. A line AB is given
by its projections. It is required to find where AB pierces P. The

Planes of Unlimited Extent: Their Traces 119

solution is as follows: Let a plane perpendicular to V be passed
through the projection AvBv. x\ccording to Art. 103 the traces of
this plane are BvFv and FvF. Draw the line of intersection of this
plane with the plane P (Art. 106) as follows: BvFv and PV in-
tersect at E. F and E are the traces of the line of intersection of
the two planes. Complete the drawing of the line of intersection
in H, as FEh. ■

Referring to- the horizontal projection, AnBn is seen to intersect
FEh, the lil projection of the line of intersection, at Wh. Since
both FE and AB are lines which lie in the vertical projector-plane
through AB, this point of intersection, Wn, is the projection of the
true point of intersection, W, of those two lines. From Wh project
to Wv for the other projection of W. This point W which lies on
P and is on the line AB is the required point.

Problems XI.

(For blackboard or cross-section paper or wire-mesh cage.)

110. Plot the point A (4,7,9). Pass a horizontal plane P
through the point A, and draw the traces of P. Also a vertical
plane Q, parallel to V^ and draw its traces. Also an inclined plane
R, perpendicular to H, making an angle of 45° with OX.

111. Plot the line A (8,2,4), B (2,6,16). Pass an inclined
plane P perpendicular to H through this line and draw the traces
of P. At C, the middle point of AB, pass a plane Q perpendicular
to P and to H, and draw QH and QV.

112. The plane P cuts the axes at the points a (10,0,0),
I (0, 5, 0) and c (0, 0, 15). Pass a plane Q parallel to P, through
the point a' (6, 0, 0).

113. A plane P has its trace on H through the points
A (12,12,0) and h (0,6,0). Its trace on V passes through the
point c (0, 0, 12). Draw the three traces. Draw three traces of a
plane Q, parallel to P through the point c' (3, 0, 0).

114. An indefinite line contains the points A (11,2,6) and
B (5,6,0). Pass a plane P perpendicular to H containing this
line and draw the traces PH, PV and PS. Pass a plane Q con-
taining this line and the point a' (2,0,0). Draw the traces QII
and QV. Draw the negative trace QS on § extended over H.

120 Engineering Descriptive Geometry

115. A plane P cuts the axis of Z at a (4, 0, 0), the axis of Y
at & (0, 6, 0), and the axis of Z at c (0, 0,-12). Draw its traces.
Draw the V and S traces of a plane Q parallel to P and containing
the line /i (1,4, 11), 5 (4,1,14).

116. An inclined plane, perpendicular to W, has for its V and
§ traces lines parallel to OZ at positive distances of 15 and 5 units.
An inclined plane Q perpendicular to H has its V and § traces
parallel to OZ at distances of 12 units and 8 units. Draw all three
traces and the projections of PQ, their line of intersection.

117. Draw the traces of a plane P, containing the points
A (8, 1, 3), B (4, 5, 1) and C (2, 4, 3). Does the point D (4, 1, 5)
lie on this plane?

118. The traces of a plane P are lines through the points
a (10,0,0), h (0,15,0) and E (14,0,6). A plane Q has its
traces through the points a' (2,0,0), E, and F (7,5,0). Draw
the projections of their line of intersection, PQ.

119. The plane P cuts the axes at a (12, 0, 0), h (0, 12, 0) and
c (0, 0, 12). Where does the line K (1, 5, 12), L (5, 3, 6) pierce
the plane?

CHAPTER XII.

VARIOUS APPLICATIONS*.

110. Traces of an Inclined Plane Perpendicular to an Oblique
Plane. — One of the most general devices used in the drafting room
is the auxiliary plane of projection, and it is often advantageous
to pass this plane perpendicular to some plane of the drawing in.

UlS

> f

. '^

Fig. 107.

Fig. 108. Fig. 109.

order to get the advantage of showing that plane '' on edge." Thus
in Fig. 31, Art. 42, the plane U has been taken perpendicular to
the long rectangular faces of the triangular prism, in order to
show clearly where BB' and DD' pierce those planes. The manner
of passing the plane U was fairly clear in that case from the
simplicity of the figure. However, as it is not always clear how to
pass a plane perpendicular ,to an oblique plane, the general method
may well be explained here. In Fig. 107 the plane P, previously
shown in Fig. 94, is represented, and an auxiliary plane l!J> Per-
pendicular to it and to H? is shown. The traces of P are PH, PV
and PS as before, and the traces of U are UH and US, It must

122 Engineering Descriptive Geometry

be understood that the ff traces of these planes, PH and UH, are
perpendicular to each other, as this condition is essential if P and
HJ are to be planes perpendicular to each other.

Fig. 108 is the descriptive drawing corresponding to the per-
spective drawing. Fig. 107. At some point h on PH a line Mdh
has been drawn perpendicular to PH. This line is the inclined
trace of a plane U perpendicular to H. The other traces of U are
parallel to the axis of Z (Art. 98). One of these, the trace on§,
is shown by the line dgNg, parallel to OZ, dn and dg being two
representations of the same point d in Fig. 107, just as In and &,
represent the point h, duplicated. Mdn may be called UH and dsN^
may be called IJ8. UH and US are the traces of an inclined plane
\], perpendicular to the oblique plane P.

The proof that P and- U are perpendicular to each other is as
follows: If, in Fig. 107, a line hh' is drawn perpendicular to H
at the point h, it will lie in the plane JJ- The angle ahh' will
then be an angle of 90°, and by construction the angle ahd is also
90°. Thus the line ah is perpendicular to two intersecting lines de-
scribed in the plane \] and is therefore perpendicular to U itself.
The plane P contains the line PH and is thus perpendicular to \},

111. An Auxiliary Plane of Projection Perpendicular to an
Oblique Plane. — To utilize the inclined plane U as an auxiliary
plane of projection, its developed position must be shown by drawing
dhNu perpendicular to UH, This line is the duplicate position of
dsNs or US. In developing the planes, U is first revolved on UH
as an axis into the plane of H as shown in Fig. 109, and then with
H into the plane of the paper, V- The trace of P on \}, or PU, is
the line of intersection of the planes, and is shown clearly in Fig.
107. This line passes through h where PH and UH meet, and
through 5 where PS and US meet. In Fig. 108, dhS is laid off on
dfj^u, equal to dgS, and the line hs is the required. trace of P on U>
or PU. The actual line PU , in Fig. 108, is only that part of PU,
in Fig. 107, which is between h and s, shown as a broken line.

The important part in this process is that \] is taken perpen-
dicular to P, so that P is " seen on edge " on U- By this process
the plane P, which is ohlique when ff, V and S are considered.

Various Applications

123

becomes an inclined plane when only ff and \] are considered.
As it is easier to deal with inclined than with oblique planes, we

/i^

Fig. 110.

may now treat P as inclined to ff and perpendicular to \J in
further operations.

Fig. 108 is well adapted to making a paper box diagram which,

124 Engineerixg Descriptive Geometry

when folded, will give most of the lines of Fig. 107. To reconstruct
Fig. 108, plot the points a (18,0,0), I (0,18,0), c (0,12,0),
d (0,6,0), h (6,12,0) and s (0,6,8). The line dr,Nu is at an
angle of 45° with ZOYn and the construction space YsOdjiNu can
be folded away inside by creasing or cutting it on several lines.

112. Intersection of an Oblique Plane and a Cylinder. — An ex-
ample of the use of an auxiliary view on which an oblique plane iti
seen on edge is shown in Fig. 110. An inclined cylinder is inter-
sected by an oblique plane P given by its traces PH, PV and PS.
It is required to describe on the cylinder the curve of intersection
of the plane and the cylinder. The solution is as follows: An
auxiliary plane \J, perpendicular to P and to Qi, is chosen, and
PU is drawn upon \] as in Fig. 108. PU is the view of P " seen on
edge " in U- Auxiliary cutting planes parallel to Ji are used for
the determination of the required line of intersection. The traces
of one of the planes are drawn, as TT in V, Tr in S, and T'T'"
in U- This latter trace is parallel to duM (or UH), because T is
parallel to H, and the distance dhT" is equal to d^T' in S- T"r"
cuts the axis of the cylinder at p. p is projected to H, and the
circular element described in H, with p as a center, is the inter-
section of the auxiliary plane T and the cylinder. In HJ the planes
P and T are both " seen on edge," intersecting in a line " seen on
end." This point projected to ff gives this line of intersection of
P and T as tf.

The intersections of the intersections are therefore the points t
and f , where the circle and the straight line meet.

113. The Angle between Two Oblique Lines. — This problem of
finding the angle between two oblique lines is shown in Fig. 111.
Let two lines AB and AC, meeting at A, be given by their If and
V projections. It is required to find the true angle between them.

By the process of Art. 105, Fig. 102, the traces of the plane con-
taining AB and AC are found and the lines are all lettered accord-
ing to Fig. 102.

An auxiliary plane of projection, U, is passed perpendicular to
PV, and therefore perpendicular to both P and V? and is revolved
into the plane V- The projections of AB and AC on this plane

Various Applications

125

fall in the single line AuCuBu, since P, the plane of the lines, is
" seen on edge " on \]. A portion of the plane P is now revolved
about the JJ projector of the point A into a position parallel to
XM. In Uj Cu moves to C'u and j5„ to B'u, revolving about A as
their center. In Y> Bv moves to B'v and Cv to C'v, both parallel to
XM. This is the process of finding the true length of a line by
revolving about a projector, as in Art. 32. AvB'v is the true length

Fig. 111.

of AB; AvC'v is the true length of AC ; and B'vAvC'v is the true
angle between the lines.

This process makes it possible to find the true shape of any
figure described on an oblique plane.

114. A Plane Perpendicular to an Inclined Line. — It is often
advantageous to pass a plane perpendicular to a line in order to
use the plane as a plane of projection, on which the given line will
be seen on end as a point. The method of passing a plane perpen-
dicular to an inclined line is shown in Fig. 112. Let AB be an
inclined line, lying in a plane parallel to V; so that AnBh is parallel

126

Engineering Descriptive Geometry

to the axis of X, It is required to find the traces of a plane P
perpendicular to AB. The essential point is that the traces of the
plane must be perpendicular to the corresponding projections of
the line. Thus, choose some point p on the inclined projection of
the line, in this case on AvB^, and through p draw a perpendicular
to AvBv, to serve as the trace of F. At a, where this trace FY

S
^

Fig. 112.

meets the axis of X, erect a perpendicular to FTL. These lines FY
and FK are the traces of an inclined plane perpendicular to AB
and to V- It is noticeable that the inclined trace of the plane is
on that reference plane which shows the inclined projection of the
line.*

* A proof that F is perpendicular to AB is as follows : AB
is the line of intersection of its own H projector-plane, and its
own V projector-plane. F is perpendicular to both these projector-

Various Applications

127

115. Application of a Plane Perpendicular to a Line. — In Fig.
113 an application of an inclined plane perpendicular to an in-
clined line is made for -the purpose of finding the line of intersection
between an inclined cone and an inclined cylinder whose axes do
not meet.

Fig. 113.

If from P, the vertex of the cone, a line Pp is drawn parallel to
QQ', as shown, any plane which contains this line and cuts both

planes. For, P is perpendicular to V and therefore to the W pro-
jector-plane, which is parallel to V; the V projector-plane is per-
pendicular to V.^ so that it is seen on edge on V just as is P itself ;
apAv is therefore the true angle between these two planes, and by
construction is a right angle. P is therefore perpendicular to both
projector-planes and therefore to the line AB, which is their line
of intersection.

128 Engineering Descriptive Geometry

surfaces will cut only simple elements of the surfaces. Tor such
a plane contains the vertex of the cone, and therefore, if it cuts
the cone, will cut it in straight elements; and such a plane is
parallel to QQ', and therefore, if it cuts the cylinder, cuts only
straight elements. No other planes can be found which cut simple
elements and can be used to determine the line of intersection.

If a plane U is passed perpendicular to Pp at any point p, and
is used as an auxiliary plane of projection, Pp will be seen on end
as the point P, and any plane R through P, seen on edge in \}, as
RR\ will cut only straight elements on the two curved surfaces.
The complete projections of the cone and cylinder have been shown
on \]y and the plane R cuts the bases at a, h, c and d. These points
projected to V enable the elements to be drawn there, and the
intersections of the intersections are the four points marked r.
From V these points are projected to W and g. Two of these
points r have been projected to the other views to show the neces-
sary construction lines.

116. A Plane Perpendicular to an Oblique Line. — To pass a
plane perpendicular to an oblique line, it is only necessary to draw
the traces of the plane perpendicular to the corresponding pro-
jections of the line. In Fig. 114, let AB be an oblique line. At
any point on AhBh draw a perpendicular line PH. From a, where
PH meets the axis of X, draw PV perpendicular to AB*

A paper box diagram traced from Fig. 114, or constructed on
cWdinate paper, using the coordinates A (10, 4, 4) and B (6, 8, 2),
C (2, 12, 0) and D (14, 0, 6), and a (8, 0, 0), will assist materially
in understanding the problem.

The oblique plane P is not serviceable as an auxiliary plane of
projection.

117. The Application of Axes of Projection to Mechanical
Drawings. — Descriptive Geometry is a geometrical science, the
science dealing primarily with orthographic projectioi*-, while Me-
chanical Drawing is the art of applying these principles to the

* The proof of this construction is more difficult than in the
corresponding case of an inclined line, but it depends as before
on the line AB being the intersection of its H and V projector-
planes, and these planes themselves being perpendicular to P,

Various Applications

129

needs of engineers and mechanics in the pursuit of industries.
Mechanical Drawing includes therefore many abbreviations and
conventional representations, which seek to curtail unnecessary
work and often to convey information as to methods of manu-
facture and other such commercial considerations foreign to the
strict scientific study.

Fig. 114.

In Mechanical Drawing many lines necessary to the strict execu-
tion of a descriptive drawing are omitted as unnecessary to the
application of the principles, when once the principles have been
fully grasped. A noteworthy omission is the axes of projection,
which, though absent, still govern the rules for making the draw-
ing. Instead of measuring distances from the axes for every point

130 Engineering Descriptive Geometry

on the' drawing, the "center lines" of the different views (which
really represent central planes) are laid off and distances from
these center lines are thereafter used. This is the regular pro-
cedure in drawing-room practice. That this difference is purely-
one of omission is clear from the fact that axes of projection may
always be inserted in a mechanical drawing. If two views only of
a piece are presented, any line between them (perpendicular to the
lines of projection from one view to another) may be selected as
the axis of X, and any convenient point on that line as the origin
of coordinates.

If three views are given, as, for example, Fig. 32, Art. 44, sup-
posing the axes to be there omitted, a ground line XOYg may be
selected at will, dividing the fields of H and V- The other line
must be determined as follows : By the dividers take the vertical
distance from OX to the center line mn, and lay off this distance
horizontally to the left from the center line of the side elevation.
The line ZOYn may then be drawn. All y coordinates of points will
now check correctly, measured parallel to the two axes of Y, if the
original drawing itself is accurate.

It is thus evident that in applying Descriptive Geometry to prac-
tical mechanical drawing we may fall back upon the use of axes of
projection whenever the lack of them is felt.

118. Practical Application of Descriptive Geometry. — Many
draftsmen have picked up a knowledge of Descriptive Geometry
without direct study of the science. This is largely due to the fact
that, till very recently, all books on Descriptive Geometry were
based on a system of planes of projection which are analogous to
the methods of practical drawing in use on the continent of Europe,
but which are little used in England, and hardly at all in the United
States of America. It will be found, however, that in American
drafting rooms all the usual devices of draftsmen are applications,
sometimes almost unconscious applications, of the principles covered
in the preceding chapters. The favorite device is the application
of an inclined auxiliary plane of projection, suitably chosen; next
in importance is the rotation of the object to show some true shape ;
while other applications are used less frequently. The methods of
determining lines of intersection of planes and curved surfaces are
exactly those described in Chapters IV, VII and VIII.

Various Applications 131

Problems XII.

(For use on blackboard, with cross-section paper or wire-mesh.

cage.)

120. The plane P has its traces through the points a (14, 0, 0),
& (0,14,0) and c (0,0,7). Pass a plane Q, perpendicular to F
and to H, through the point J. (5, 7, 0). If § is to be used as an
auxiliary plane of projection, draw the trace of P on § when Q has
been revolved into coincidence with H.

121. Draw the traces of a plane P cutting the axes at the points
a (12,0,0), I (0,8,0) and c (0,0,12). Draw the traces of an
auxiliary plane, U? perpendicular to FlI at the point A (3, 6, 0).
Is the point B (6, 1, 4^) on the plane P?

122. The fi trace of a plane P passes through the points
^ (12, 5, 0) and P (6, 2, 0). Its V trace passes through (7 (9, 0, 6).
Pass an inclined plane perpendicular to H and perpendicular to
T, through the point P (5, 9, 7).

123. Of a plane F, PH, the horizontal trace, passes through the
points A (5,3,0) and B (13,9,0), and FV passes through
C (12, 0, 11). Complete the traces of P and draw the traces of a
plane perpendicular to FV at the point B (9, 0, 8). Prove that the
line E (9, 6, 1), F (7, 3, 2) lies on the plane P.

124. A sphere of radius 7 units has its center at C (8, 8, 8). A
plane P cuts the axes of projection at a (26, 0, 0), h (0, 13, 0) and
c (0,0,13). Pass an auxiliary plane of projection \}, perpen-
dicular to H and to F, cutting the axis oi X oi d (16, 0, 0). Draw
the trace of P on \]. The circle of intersection of the sphere and
the plane P is seen on edge in U- Show the elliptical projection
of this circle, on H, by passing auxiliary cutting planes parallel
to U- (If this problem is solved by use of wire-mesh cage, the
point a is inaccessible, but FH passes through E (16,5,0), and
FV through F (16,0,5). The plane S' can be turned to serve

asU)

125. Find the true shape of the triangle A (3, 2, 6), B (9, 6, 2),
C (8, 0, 4). Find the traces of two of the sides of the triangle and
pass the plane \] perpendicular to the plane of the triangle and
perpendicular to f\, and through the point D (0,7,0).

132 Engineering Desceiptive Geometry

126. Find the true shape of the triangle A (8, 6,1), B (4, 2, 9),
C (10,2,3). Find the traces of two of the sides of the triangle
and pass the plane U perpendicular to the plane of the triangle
and perpendicular to H^ and through the point D (0, 1, 0).

127. Draw the traces of a plane P perpendicular to V and to
the line A (2,6,9), B (8,6,5) at C (11,6,3). If this plane is
used as an auxiliary plane of projection, what is the projection of
A^onit?

128. Draw the traces of a plane P perpendicular to lil and ta
the line A (3, 9, 6), B (13, 4, 6), at C (17, 2, 6), a point on AB.
(If wire-mesh cage is used for the solution, turn §' to serve as \J
and draw on it the view of AuBu.)

129. Draw the three traces of a plane P perpendicular to the
oblique line A (8, 12, 5), 5 (14, 3, 7). Show that all three traces
are perpendicular to the corresponding projections of AB.

CHAPTEE XIII.
THE ELEMENTS OF ISOMETRIC SKETCHING.

119. Isometric Projection. — There is one special brancli of
Orthographic Projection which is of peculiar value for represent-
ing forms which consist wholly or mainly of plane faces at right
angles to each other. Ordinary orthographic views are projec-
tions upon planes parallel to the principal plane faces of the object,
as shown in Fig. 2, Art. 4. If, however, instead of the regular
planes of projection, the object is projected upon a new plane of
projection, making the same angle ivith each of the regular planes,
an entirely different result is obtained, called an '^isometric pro-
jection." This view has the useful property that it has all the air
of a perspective and ma}^, witli certain restrictions, be used alone
without other views as a full representation of the object.

In Art. 21 the method of converting the perspective drawings
of this treatise into isometric sketches was explained in a rough
and unscientific way. In this chapter there is explained the method
of making isometric sketches from models, as a step to making
orthographic drawings or isometric drawings.

120. Isometric Sketches of Rectangular Objects. — Figs. 19 and
19a are the isometric drawings of a cube. Since the line of sight
from the eye to the point 0 makes equal angles with fil, V and S,
the three planes must subtend the same angle at 0. XOY, YOZ
and XOZ are each 120°. though representing angles of 90° on the
cube. Since opposite edges of H are parallel, it follows that each
face of the cube is a rhombus and that the cube appears as a regular
hexagon, all edges appearing of exactly the same length. This
fact is the basis of the name "isometric," meaning "equal-
measured."

Figs. 115, IIG and 117 are sketches of other objects, all of whose
corners are right angles. The angles at these comers appear there-
10

134

Engineering Descriptive Geometry

fore like those of the cube, either as 60° or 120° on the isometric
sketch.

In making the isometric sketch from a model having rectangular
faces, the tirst step is to put the object approximately in the iso-

Brick
Fig. 115.

Half Joinf

FiG. 116.

Mortise 5* Tenon Joint
Fig. 117.

Position •for
Orthographic
Projection .

Fig. 118.

Turned 45
about a verti-
cal dxis.

Fig. 119.

Tilt-ed 35"-44-'
about an hori-
zontal o?<.is.

Fig. 120.

metric position. At any projecting corner imagine a line to project
from the corner so as to make equal angles with the three edges
which meet at the given corner. View the object by sighting along
this imaginary line and begin the sketch from that view.

The Elements of Isometric Sketching 135

If there is any difficulty in finding this line of vision directly,
the object may be turned horizontally through an angle of 45 ° and
tilted down through an angle of 35° 44'. This operation is the
basis of the method of finding the " isometric projection."

Figs. 118, 119 and 120 show the steps in passing from the ortho-
graphic position to the isometric position, the model used being a
rectangular block with, a lengthwise groove cut in one face.

121. Isometric Axes. — It will be noticed in the previous iso-
metric figures that all lines are drawn in one of three general direc-
tions. One of these directions is usually taken as vertical and the
other two directions make angles of 120° with the vertical. These
three directions are known as the isometric axes. In this sense
the word axis means a direction, not a line.

In plotting points from a selected origin, the x coordinates are
plotted up and to the left, the y coordinates up and to the right,
and the z coordinates vertically downward, as in Fig. 19a.

122. Isometric Paper. — Paper ruled in the direction of the iso-
metric axes is called isometric paper, and is of great assistance in
making isometric sketches. The lines divide the paper into small
equilateral triangles.

In sketching, the sides of these equilateral triangles are taken to
represent unit distances, exactly or at least approximately. Thus,
if the model shown in Fig. 120 is a block 3" x 3" X 8", with a 2" x 1"
groove lengthwise along one face, some point a on the paper is
selected, and from it distances are taken along the isometric axes,
so that each unit space represents one inch.

From a three units are counted vertically downward, eight up,
and to the right, and one unit, follov^ed by a gap in the, line of one
unit, and then a second unit, up to the left. Thus all lines of the
sketch follow the ruled lines as long as the dimensions of the model
are in even inches.

An isometric sketch made in this manner, particularly if spaces
have been exactly counted off according to the dimensions of the
piece, is practically an isometric drawing. If fully dimensioned, a
sketch on plain paper proportioned by the eye is nearly as good as
one in which spaces are counted exactly. Such sketches serve all

136

Engineering Descriptive Geometry

purposes, though of course more difficult to make than those on
isometric paper.

Fig. 121.

Fig. 122.

123. Non-Isometric Lines in Isometric Sketching. — Objects
which have a few faces and edges oblique to the principal plane
faces may still be shown by isometric sketching. In such cases it is
always well to circumscribe a set of rectangular planes about the

Fig. 123.

oblique parts of the object to aid the imagination. Dimension
extension lines should be used for this purpose. In using isometric
paper this squaring up is done by the lines of the paper.

The Elements of Isometric Sketching

137

Figs. 122, 123 and 124 are good examples of oblique lines and
faces. Figs. 123 and 124 show also the circumscribed isometric
lines which " square up '' the oblique parts.

124. Angles in Isometric Sketching. — In isometric sketching
angles do not, as a rule, appear of their true magnitude. Thus the
90° angles on the faces of the brick appear in Fig. 115 as 60° or
120°, but not as 90°. In general, the lengths of oblique or inclined
lines depend on position, and are not subject to measurement by-
scale.

The lines which square up oblique parts are useful in giving tSe
tangent of the angle of an oblique surface. Thus in Fig. 124, the
angle a differs in reality from the angle as it appears in either place

marked, but the tangent of a is -!i . In Fig. 123, 0=tsin-^ VI . In

2^ n

practice angles are often given by their tangents. Thus the slope of

Fig. 126.

Fig. 127.

a roof is given as " one in two " or the gradient of a railroad as
" three per cent.''

125. Cylindrical Surfaces in Isometric Sketching. — In ortho-
graphic drawings circles appear commonly on planes parallel to the
three planes of projection. To illustrate the position and appear-
ance of circles in isometric drawing in the three typical cases. Fig.
125 represents the isometric sketch of a cube, having a circle in-
scribed in each square face.

Each of the faces of the cube is perpendicular to the isometric
axis given by the intersection of the other two faces. Thus the
square ABCD is perpendicular to the edge BF. The circle ahcd.

138 Engineering Descriptive Geometry

inscribed in the square ABCD, appears as an ellipse, whose minor
axis, ef, lies on the diagonal BD of the square, BD appearing as a
continuation of the edge FB, In all three cases, then, the minor
axis of the ellipse lies in the same direction, on the sketch, as that
isometric axis to which the plane of the circle is in reality perpen-
dicular.

The major axis is necessarily perpendicular to the minor axis,
and lies on the other diagonal of the square.

Since the cylinder is the curved surface most used in engineering,
the rule may be applied to cylinders as follows : The ellipse which
represents the circular base of any cylinder must be so sketched
that its minor axis is in line with the axis or center line of the
cylinder. Fig. 126 is an isometric sketch of a piece composed of
cylinders. All the ellipses are seen to follow this rule.

In sketching cylindrical parts of objects, it is necessary to im-
agine them squared up by the use of isometric lines and planes.
Thus the first steps in sketching the piece of Fig. 126 are shown
in Fig. 127. The circumscribing of a square about a circle in the
object corresponds to circumscribing a rhombus about the ellipse
in the isometric sketch. It now remains to inscribe an ellipse in
the rhombus. This ellipse must be tangent to the rhombus at the
middle of each side. To sketch the ellipse, as for example the small
end in Fig. 127, draw the diagonals of the rhombus to get the
directions of the major and minor axes, and find the middle points
of the sides (by center lines, through the intersection of the diagon-
als). It is now easy to sketch the ellipse, having four points given,
the direction of passing through those points, and the directions of
the major and minor axes.

126. Isometric Sketches from Orthographic Sketches. — A good
exercise consists in making isometric sketches from orthographic
sketches or drawings. The three coordinate directions, x, y and z,
must be kept in mind at all times. Fig. 128, as an example, is most
instructive. From the orthographic sketches, Fig. 128, the iso-
metric sketch. Fig. 129, is to be made. A point a is selected to rep-
resent a point a on the orthographic views. The line ah is an x
dimension and is plotted up to the left ; ac is a ?/ dimension, and is
plotted up to the right; while a.^ is a 2 dimension, and is plotted

The Elements of Isometric Sketching

139

vertically downward. The semicircle is inscribed in a half -rhombus,
tangent at b, e and /.

— h 1 1 1 1 ^^^><^<^^l^s>^^H^<'^<'

J >\ ^<^ S<r ^^S<r ^ cT *^ccr^^ ^>N ^^ ^<^

' ^sC^ ^^'^^^^'^^>vT>c >c^'^^'^'

-^ /f\ P% x"vl^ S ^ ^ ^'NaTS'^ "^^^k^^-^^ ^''

^^^2^^sx"\^^ ^<^pk^ x\D^^!^ V ^'

L C^ ^^ ^<C / C !^^^ \ ^'^j^]^^^lS^ S^^'

£ lb ■^'^L^s^ ^^ v^''^<^\<'^iN^s

(L ^\^\ ^ ^ ^^>i^ >^^iiyf*^'\-^^-^ ^ /x

^ a_ /'^ 2^! ^ \ /'>;>^^ / "S^ Q^^^'\/>\ ^ C "

r~ c ~^^^^^^ ^ '^ ">^$ ^^< ^ <i^ S<'

^/\ ^\ ^ Vj^ \^ >< s ^1^ !^^4^D^ ^^^\ / \ !^ *

V / ^ ^^ / \ ^^^"v !i^^!^^ !!^4\ ^ V !^^^ \ ^

V, V ^ 3^ 3^ "^^ ^ sic ^\ s^kr ^ ^ "^^^ ^ ^

^^^^ <p /T^S^^^ *^ S 5^$

— -.- ^^^^'^^-^^c^'^^^c '^'^ ^ '^^'^'^'^ >

:i.^\^^^^\^\!^x!^\Ss^V^\!^\!^\

x^<^Sc ^<r ^c ^c ^<r^^ ^ ^ ^cT >cr ^<'

><J^<i><^><^>x>x>xPK5\>x>x

Fig. 128.

Fig. 129.

The cross-section lines of Fig. 128 and the isometric lines of Fig.
129 are represented as overlapping between the figures. Some iso-
metric paper is ruled in this manner, so that it may be used for
both purposes.

140

Engineering Descriptive Geometry

Problems XIII.

(For blackboard or isometric paper.)

130. Make an isometric sketch of the angle piece, Fig. 130, using
the spaces for 1" distances.

Fig. 130.

Fig. 131.

131. Measure the tool-chest, Fig. 131, scale, 5"=1 foot, and
make a bill of material, tabulating the boards used, and recording
their sizes, giving dimensions in the order : width, thickness, length,
thus :

Mark. Name. Size. Number.

A. Top of Chest. 14" x 1" x 24". 2.

132. A parallelopiped, 9" x 6" x 3", has a 3" square hole from
center to center of the largest faces, and a 2" bore-hole centrally
from end to end. Make an isometric sketch.

133. Let Fig. 3, Art. 5, represent a model cut from a 12" cube
by removing the center, leaving the thickness of the walls 3". Let
the angular point form a triangle whose base is 12" and altitude 8".
Make an isometric sketch.

The Elements of Isometric Sketching 141

134. A cube of 10" has a 6" square hole piercing it centrally from
cne side to the other, and a 4" bore-hole piercing it centrally from
side to side at right angles to the larger hole. Make an isometric
sketch.

135. A grating is made by nailing slats f"xi"xl2", spaced J"
apart, on three square pieces, IJ" square, 22" long, spaced 4^" apart.
Make an isometric sketch.

136. Make orthographic sketches of the bracket. Fig. 122. Views
required are plan and front elevation. (On cross-section paper use
the unit distance for the unit of the isometric paper. On black-
board let each unit of the isometric paper be represented by a dis-
tance of 2".)

137. Make isometric sketches of Fig. 11, Art. 14, and Fig. 24,
Art. 32.

138. Make isometric sketches of Fig. 13, Art. 15, and of Fig. 82,
Art. 89. In Fig. 82 let A be the point (9, 8, 0) and B the point
(9,0,12).

139. Make an isometric sketch of Fig. 71, Art. 80, the diameter
of the cylinder being 7 units and the length 14 units.

140. Make an isometric sketch of Fig. 92, Art. 93, using the
coordinates given in Art. 94.

CHAPTER XIV.
ISOMETRIC DRAWING AS AN EXACT SYSTEM.

127. The Isometric Projection on an Oblique Auxiliary Plane. —

The sketches previously considered have generally had no exact scale.
Those drawn on isometric paper have a certain scale according to
the distance which one unit space of the paper actually represents.

If the isometric projection is derived from an orthographic draw-
ing of the usual kind by the laws of projection, the isometric projec-
tion so formed has of course the same scale as the original drawing.

In Fig. 132 an isometric projection of a cube is derived from the
orthographic drawing by the use of an inclined plane of projection.

Isometric Drawing as an Exact System 143

HJ, and an oblique auxiliary plane of projection "W- The aim is
to produce the projection on a plane making the same angle with all
three edges of the cube meeting at any one corner. This plane must
be perpendicular to a diagonal of the cube. In Fig. 132 this di-
agonal is the line EC, a. true diagonal, passing through the center of
the cube, not a diagonal of one face of the cube.

The first, or inclined, auxiliary plane U is taken parallel to the
V projection of EC, and therefore perpendicular to V and making
an angle of 45° with H and S- The projection of EC on U shows
its true length.

The second, or oblique, auxiliary plane 'W is taken perpendicular
to EC. It is oblique as regards H and V? b^t, as EC is a line par-
allel to U, and W is perpendicular to EC, W is perpendicular to
U. As regards V and U? W is an inclined plane, having its in-
clined trace MN on U^ the trace on V being a line MLv, perpendic-
ular to ZM, the trace of U on V- The construction of this second
projection is therefore according to the usual methods. Any point,
as F, is projected by a perpendicular line across the trace MN and
the distance nFw is laid off equal to mFv.

The projection on 'W is the isometric projection of the cube and
is full-size if the plan and front elevation are full-size projections.
The edges are all foreshortened, however, and measure only j-i^\ of
their true length.

128. The Angfles of the Auxiliary Planes. — The plane U makes
an angle of 45° with the plane ff. The plane W makes an angle
of 35° 44' with g, or (90° -35° 44') with V- If the side of the
cube is taken as 1, the length of the diagonal of the face of the cube
is V2, and the length of the diagonal of the cube is \^. The

first angle is that angle whose tangent is - ^ , or whose sine is ~7^*
The second angle is that angle whose tangent is -— ^ and whose sine

129. The Isometric Projection by Rotating the Object. — In Fig.
134 is shown a method of deriving the isometric projection by turn-
ing the object. The plan, front, and side elevations are drawn with

144

Engineering Descriptive Geometry

the object turned through an angle of 45° from the natural posi-
tion (that in which the faces of the cube are all parallel to the
reference planes) . The side elevation shows the true length of one
diagonal of the cube, AG. Some point on AG extended, as K, is
taken as a pivot, and the whole object is tilted down through an
angle of 35° 44', bringing AG into a horizontal position, A'G'. The

new projection of the object in V is the isometric projection. This
process of turning the object corresponds to the turning of the
object in isometric sketching, as shown in Figs. 118, 119 and 120.

The isometric projection of the cube has all eight edges of the
same length, but. foreshortened from the true length in the ratio of
V3 to V2.

Any object of a rectangular nature may be treated by either
process to obtain the isometric projection.

Isometric Drawing as an Exact System 145

130. The Isometric Drawing. — To make a practical system of
drawing capable of representing rectangular objects in an unmis-
takable manner in one view, the fact that all edges are foreshort-
ened alike is seized upon, but the disagreeable ratio of foreshorten-
ing is obviated by ignoring foreshortening altogether.

An isometric drawing is one constructed as follows: On three
lines of direction, called isometric axes, making angles of 120° with
each other, the true lengths of the edges of the object are laid off.
These lengths, however, are only those which are mutually at right
angles on the object. All otlier lines are altered in shape or length.
An isometric drawing is distinct from an isometric projection, as
it is larger in the proportion of 100 to 83 (V3: V2). The iso-
metric drawing of a 1" cube is a hexagon measuring 1" on each
edge.

131.. Requirement of Perpendicular Faces. — An isometric draw-
ing, being a single view, cannot really give " depth," or tell exactly
the relative distances of different points of the object from the eye.
It absolutely requires that the object drawn shall have its most
prominent faces, at least, mutually perpendicular. The mind must
be able to assume that the object represented is of this kind, or the
drawing will not be " read " correctly. Even on this assumption,
in some cases isometric drawing of rectangular objects may be
misunderstood if some projecting angle is taken as a reentrant one.
Thus in Fig. 133 we have a drawing which might be taken as the
pattern of inlaid paving or other flat object. If it is taken as an
isometric drawing and the various faces are assumed to be perpen-
dicular to each other, it becomes the drawing of a set of cubes.
Curiously enough, it can be taken to represent either 6 or 7 cubes,
according as the point A is taken as a raised point or as a depressed
one. In other words, it even requires one to know just how the
faces are perpendicular to each other to be able to take the drawing
in the way intended.

This requirement of perpendicular faces limits the system of
drawing to one class of objects, but for that class it is a very easy,
direct, and readily understood method. Untrained mechanics can
follow isometric drawings more easily than orthographic drawings.

146

Engineering Descriptive Geometry

IsoMETKTC Drawing as an Exact System 147

132. The Representation of the Circle. — In executing isometric
drawings, the circle, projected as an ellipse, is the one drawback to
the system. To minimize the labor, an approximate ellipse must
be substituted for an exact one, even at the expense of displeasing
a critical eye. The system, if used, is used for practical purposes
where beauty must be sacrificed to speed. In Fig. 125 the rhombus
ABCD is the typical rhombus in which the ellipse must be inscribed.
The exact method is shown in Fig. 43, but requires too much time
for constant use. The following draftsman's ellipse, devised to be
exactly tangent to the rhombus at the middle point of each side, is
reasonably accurate. From B, one extremity of the short diagonal
of the rhombus, drop perpendiculars Bd and Be upon opposite
sides, cutting the long diagonal at k and I. With 5 as a center and
Bd as a radius, describe the arc dc. Similarly, with D as a center,
describe the arc ha. With Tc and I as centers, and hd as a radius,
describe the arcs ad and ch. The resulting oval has the correct
major axis within one-eighth of 1 per cent, and has the correct
minor axis within 3^ per cent.

This draftsman's ellipse is exact where required, namely, on the
two diameters ac and dh, which are isometric axes, and it is prac-
tically exact at the extremity of the major axis.

133. Set of Isometric Sketches. — Fig. 135 is a set of isometric
sketches of the details of the strap end of a small connecting-rod,
from which to make orthographic drawings. The isometric sketch
is much clearer than the corresponding orthographic sketch, and
the set shows clearly how the pieces are assembled.

The orthographic drawing of the assembled rod end is much
easier to make than the assembled isometric drawing. It is in fact
clearer for the mechanic than the assembled isometric drawing
would be, for the number of lines would in that case be quite con-
fusing. It illustrates well the fact that isometric sketches and
drawings should be limited to fairly simple objects.

Another noteworthy fact is that center lines, which should always
mark s}Tnmetrical parts in orthographic drawings, should be used
in isometric drawing only when measurements are recorded from
them.

The sketch as given is taken directly from an examination paper
used at the U S. Naval Academy for a two-hour examination. On

148 Engineering Descriptive Geometry

account of the shortness of the period, only one orthographic view,
the front elevation, is required, but if time were not limited, a plan
also should be drawn.

The following explanation of the sheet is printed on the original :
''Explanation of Mechanism. — ^The isometric sketches represent
the parts of the strap end of a connecting-rod for a small engine.
In assembling. A, B, C, and D are pushed together, with the thin
metal liners, G, filling the space between B and C. The tapered
key, E, is driven in the J" holes of A and D, which will be found to
be in line, except for a displacement of ^" which prevents the key
from being driven down flush with the top of the strap D. The two
bolts, F, are inserted in their holes, nuts H screwed on, and split
pins (which are not drawn) inserted in the -J" holes, locking the
nuts in place. In time the bore of the brasses B and C wears to
oval form. To restore to circular form, one or two liners would be
removed and the strap replaced. The key driven in would then
draw the parts closer by the thickness of the liners removed.

"Drawing (to he Orthographic, not Isometric). — On a sheet
14"xll" make in ink a working drawing of the front elevation of
the rod end assembled, viewed in the direction of the arrow. Put
paper with long dimension horizontal. Put center of bore of
brasses 4" from left edge of paper and 5" from top edge. No
sketch, no legend, no dimensions.''

Problems XIV.

140. An ordinary brick measures S"x4^"x2^". Make an ortho-
graphic drawing and an isometric projection after the manner of
Fig. 132, Art. 127. Contrast it with the isometric drawing made
according to Art. 130.

. 141. Make the isometric projection of the brick, 8"x4"x2^",
turning it through the angles of 45° and 35° 44', as in Fig. 134,
Art. 129.

142. From Fig. 135 make a plan and front elevation of the
strap D.

143. From Fig. 135' make a plan and front elevation of the stub
end A.

144. From Fig. 135 make a plan and front elevation of the
brass C.

SET OF DESCRIPTIVE DRAWINGS.

The following four drawing sheets are designed to be executed in
the drawing room to illustrate those principles of Descriptive
Geometry which have the most freouent application in Mechanical
or Engineering Drawing.

The paper used should be about 28"x22", the drawing-board of
the same size, and the blade of the T-square 30".

To lay out the sheets find the center, approximately, draw center
lines, and draw three concentric rectangles, measuring 24" x 18",
22"xl6", and 21"xl5". The outer rectangle is the cutting line
to which the sheets are to be trimmed. The second one is to be inked
for the border line. The inner one is described in pencil only as a
"working line," or line outside of which no part of the actual
figures should extend. The center lines and other fine lines, in-
cluding dimensions, may extend beyond the working line. In the
lower right corner reserve a rectangle 6" x 3", touching the working
lines, for the legend of the drawing.

In making the drawings three widths of line are used.

The actual lines of the figures must be " standard lines " or lines
not quite one-hundredth of an inch thick. The thin metal erasing
shield may be used as a gauge for setting the right-line pen, by so
adjusting the pen that the shield will slowly slip from between the
nibs, when inserted and allowed to hang vertically. Visible edges
are full lines. Hidden edges are broken lines; the dashes J" long
and spaces ^y" long.

The extra-fine lines ave described with the pen adjusted to as fine
a line as it will carry continuously. The axes of projection are
fine full lines. The dimension lines are long dashes, J" to 1" long,
with -J" spaces. The center lines are long dashes with fine dots
between the dashes, or are dash-dot lines. The construction lines
are long dashes wdth two dots between, or are dash-dot-dot lines.
When auxiliar}^ cutting planes are used, one only, together with its
corresponding projection lines, should be inked in this manner.
11

150 Engineering Descriptive Geometry

The extra-heavy lines are about two-hundredths of an inch thick,
and are for two purposes : for shade lines, if used ; and for paths of
sections, or lines showing where sections have been taken, as pq.
Fig. 32. These paths of sections should be formed of dashes
about J" long.

SHEET I: PRISMS AND PYRAMIDS.

Lay out the sheet and from the center of the sheet plot three ori-
gins : The first origin 5^" to the left and 4^" above the center of the
sheet; the second 8" to the right and 2-i" above the center; and the
third A:" to the left and 4V' below the center. Pass vertical and
horizontal lines through these points to act as axes of projection.

First Origin: Pentagonal Prism and Inclined Plane.

Describe a pentagonal prism, the axis extending from P (2%
IJ", \") to P' (2", IJ", 2f' )• The top base is a regular pentagon
inscribed in a circle of IJ" radius, one corner of the pentagon
being at A (2", J", \"). Draw three views of the prism. Draw the
traces of a plane P, perpendicular to V> its trace on V passing
through the point c (0", 0", 24") and making an angle of 60° with,
the axis of Z. Draw on the side elevation the line of intersection
of the prism and the plane P. Show the true shape of the polygonal
line of intersection on an auxiliary plane KJ? perpendicular to V>
its traces on V passing through the point (0", 0", 4^"). On JJ
show only the section cut by the plane. Draw the development
of the surface of the prism, with the line of intersection described
on it. Draw the left edge of the development [representing'
A (2", i", I"), A' (2'Vi", 2Y)^ as a vertical line V to the right of
the axis of Y, and use the top working edge of the sheet as the top
line of the development. Omit the pentagonal bases.

Second Origin: Octagonal Prism and Triangular Prism.

Describe an octagonal prism, the axis extending from P (2^,
If" i") to P' (21", If", 4i"). The octagonal base is circumscribed
about a circle of 2J'' diameter, one flat side being parallel to the

Set of Descriptive Drawings 151

axis of X. Describe a triangular prism, its axis extending from
Q (0r52, If", li") to Q' (3'.'98, If", 3^"), intersecting TF' at its
middle point and making an angle of 60° with it. The base is in
a plane perpendicular to QQ\ and is an equilateral triangle cir-
cumscribed about a circle of \" diameter. One corner is at
J (1", If", 0".38). Draw the H, V, and S projections of the
prisms and a complete projection on a plane JJ? taken perpendicular
to QQ' , and whose trace on V passes through the point \^" , 0", 0").
'Draw the triangular prism as if piercing the octagonal prism.

Third Origin: Hexagonal Pyramid and Square Prism.

Describe an hexagonal pyramid, vertex at P (IJ"? 2", J"), center
of base at F' (IJ", 2", 3") . The hexagonal base is in a plane parallel
to H and is circumscribed about a circle 2J" in diameter, one
corner being at A (If", 0'.'5G, 3"). Projecting from the sides of the
pyramid are two portions of a square prism, whose axis is Q (J",
2'', 3i"), Q' (3i", 2'V2i"). The square base is in a plane parallel
to S and measures V on each edge, and its edges are parallel to the
axes of Y and Z. Letter the edges GG\ HH', etc., the point G
being (i", IJ", If"), H (i", ^", 1%"), etc. Draw the object as if
cut from one solid piece of material, the prism not piercing the
pyramid.

The views required are plan, front elevation, and side elevation,
and also an auxiliary projection on a plane \], perpendicular to W.
The IMI trace of U makes an angle of 120° with the axis of X at
the point Z (2f", 0", 0").

Draw also the developments of the surfaces. Place the vertex of
the developed pyramid at a point J" to the right and 3J" above the
origin, and the point A -J" to the right and 0."36 above the origin.
Mark the line of intersection with the prism on this development.

Between the side elevation and the legend space, draw the de-
velopment of the square prism, placing the long edges, OG', RE',
etc., in a vertical position. Describe the line of intersection on the
development. Let the edge which has been opened out be GG', and
let the middle portion of the prism, which does not in reality exist,
be drawn with construction lines.

152 Engineering Descriptive Geometry

General Directions for Completing the Sheet.

In inking the sheet show one line of projection for the determi-
nation of one point on each line of intersection. Shade the figure,
except the developments.

In the legend space make the following legend :

SHEET I. (Block letters 15/32" high.)

DESCRIPTIVE geometry. (All caps 3/16" high.)

PRISMS AND PYRAMIDS. (All caps 9/32" high.)

Name (signature). ClasS. (Caps 1/8" high, lower case 1/12" high.)

Date. (Caps 1/8" high, lower case 1/12" high.)

SHEET II: CYLINDERS, ETC.

Lay out cutting, border, and working lines, and legend space as
before.

Plot four points of origin as follows : First origin, 6" to the left
and 4" above the center of the sheet ; second origin, 4 J" to the right
and 4J" above the center; third origin, 6 J" to the left and 3 J"
below the center; fourth origin, 6^ to the right and 4 J" below the
center.

First Origin: Intersecting Right Cylinders.

Draw the three views of two intersecting right cylinders. The
axis of one is P (2J", 2", Y),P' m", ^", H"), and^its diameter is
3". The axis of the other is Q {I", 1%", 2"), Q' (4^", If", 2"), and
its diameter 2f". Determine the line of intersection in V by planes
parallel to V at distances of %", 1% IJ", etc.

Second Origin: Inclined Cylinder and Inclined Plane.

Draw three views of an inclined circular cylinder, cut by a plane.
The axis of the cylinder is P (3.73", If", J"), P' (2", If", 3i").
The base is a circle, diameter 2^", in a plane parallel to H- The
plane cutting the cylinder is perpendicular to V^ and its trace in
V passes through the middle point of PP\ and inclines up to the

Set of Descriptive Drawings 153

left at an angle of 30° with OX. Plot the intersection in H, V,
and S and find the true shape of the ellipse by an auxiliary plane
of projection perpendicular to V through the point (3'', 0", 4").

Third Origin: Right Circular Cone and Inclined Plane.

Draw a right circular cone, vertex at F (2", IJ", ^'), center of
base at F' (2", If", 4"), diameter of base 3". The cone is inter-
sected by a plane perpendicular to §, having its trace in § parallel
to the extreme right element of the cone and through the point
(0", 2^", A"), Draw the line of intersection in plan and front
elevation, and show the true shape of the curve by projection on an
auxiliary plane U perpendicular to §, its trace passing through
the point (0", 2^", 0").

Fourth Origin: Ogival Point, Vertical Plane and Inclined Plane.

Let S li^ to the right of W and make no use of V- The problem
is to draw two views of a 3^" ogival shell, intersected by two planes.
The ogival point is generated by revolving 60° of arc of 3 J" radius
about an axis perpendicular to H at the point (2", IJ", 0"). The
initial position of the generating arc is as follows: The center is
at D (0", 3 J", 3J"), one extremity is at B (0", 0", 34"), and one is at
F (2", IJ", 0.46"). The cylindrical body of the shell extends from
the ogival point to the right in the side elevation, a distance of f ".
Two planes, T and R, intersect the shell. T is parallel to and 1-J"
from §. F is perpendicular to S, and its S trace passes through
the origin, and makes angles of 45° with the axis of Y and the
axis of Z. Draw : The traces of T and F; the side elevation ; the
line of intersection of T with the shell; and, on the plan, the line
of intersection of F with the shell.

General Directions for Completing the Sheet.

■ In inking the sheet show one cutting plane for the determination
of each line of intersection, and show clearly how one point is de-
termined in each view of each figure. Shade the figure except the
developments.

154 EXGINEEEIXG DESCRIPTIVE GEOMETRY

In the legend space make the following legend :

SHEET II. (Block letters 15/32" high.)

DESCRIPTIVE GEOMETRY. (All caps 3/16" high.)

INTERSECTIONS OF CYLINDERS, ETC. (AH caps 9/32" high.)

Name (signature). ClaSS. (Caps 1/8" high, lower case 1/12" high.)

Date. (Caps 1/8" high, lower case 1/12" high.)

SHEET III: SURFACES OF REVOLUTION.

Lay out center lines, cutting, border and working lines, and
legend space as before.

Plot five points of origin as follow^s: First origin, 6}" to the
left and 3f" above the center of the sheet; second origin, IJ" to
the right and 6" above the center ; third origin, Sj" to the right and
5 J" above the center; fourth origin, GV to the left and 4J" below
the center ; fifth origin, 7}" to the right of the center of the sheet
on the horizontal center line.

First Origin: Sphere and Cylinder.

Draw a sphere pierced by a right circular cylinder. The center
of the sphere is at (2", 2", 2"), its diameter sV'. The axis of the
cylinder is P (2",14",i"), P' (2", 1^,31"). Its diameter is H".
Praw the sphere and cylinder in fi, V and S, and determine the
line of intersection by passing planes parallel to V at distances of
i", r, n" and If".

Second Origin: Forked End of Connecting-Rod.

The forked end of a connecting rod has the shape of a surface of
revolution, faced off at the sides to a width of 1^'', as shown in Fig.
136. The centers a, I, and c are points (2'', 1", 0"), (2", 0", f"),
and (2", 0", 1"). The arc which has cZ as a center is tangent at its
ends to the adjacent arc and to the side of the 1" cylinder.

Determine the continuation of the line of intersection of the
plane and surface at w, by passing planes parallel to fi at distances
from H of 2i", 2f ", ^", 2f " and 2%". Draw no side view.

Set of Descriptive Drawings

155

Third Origin: Stub End of Connecting-Rod.

The stub end of a connecting-rod is a surface of revolution faced
off at the sides to a width of IJ", and pierced by bore-holes parallel
to its axis as shown in Fig. 137. Centers are at a (IJ", 1", 0"),

h {r, r, 0"), c (r, r, o"), d (sr, o^ if), and e (r, 0", ir).

Determine the continuation of the line of intersection at lu by
passing planes parallel to H at distances from ff of lyV"^ ^¥'>

Fig. 136.

Fig. 137.

1^", and If". Draw also the side view and determine the ap-
pearance of the edge marked u, where the large part of the bore-hole
intersects the surface of revolution, by means of the same system of
planes.

Fourth Origin: Right Circular Cylinder and Cone.

A right circular cone is pierced by a right circular cylinder, the
axes intersecting at right angles, as in Fig. 62, Art. 72. The axis

156 Engineering Descriptive Geometry

of the cone is P (2f', 2^", \"), P' (24", 2^", 2J"). The base, in a
plane parallel to iHI, is a circle of 3}" diameter. The axis of the
cylinder is Q {l\ 2^", If"), Q' (4", 2^", If"), and its diameter is

Draw three views of the figures, determining the line of inter-
section by planes parallel to H. It is best not to pass these planes at
equal intervals, but through points at equal angles on the base of
the cylinder. Divide the base of the cylinder in S ii^to arcs of 30°,
and in numbering the points let that corresponding to F, in Fig. 62,
be numbered 0 and let H be numbered 6. Insert intermediate
points from 1 to 5 on both sides, so that the horizontal planes used
for the determination of the curve of intersection are seven in
number, the lowest passing through the point 0, the second through
the two points 1, the third through the two points 2, etc. Determine
the curve of intersection by these planes.

Draw the development of the surface of the cylinder, cutting the
surface on the element 00' (or FF' in Fig. 62). Place this line of
the development vertically on the sheet, the point 0 being 1" to the
left and 7^" below the center of the sheet, and 0' being 1" to the
left and 3}" below the center of the sheet.

Draw the development of the surface of the cone Note that the
radius of the base, the altitude, and the slant height are in the
ratio of 3:4:5. To get equally spaced elements on the surface of
the cone, divide the arc corresponding to BC in H, Fig. 62, into
five equal spaces. Number the point B 0 and C 5, and the inter-
mediate points in series. Since the cone is symmetrical about two
axes at right angles, one quadrant may represent all four quadrants.
Put the vertex of the developed surface 3" to the left of the center
of the sheet and 1" below it, and consider it cut on the line PO or
PB.* Locate the point 0 8|" to the left of the center of the sheet and
1" below it. Divide the development into four quadrants and then
divide each quadrant into five parts, numbering the 21 points
0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0.

Fifth Origin: Cone and Double Ogival Point.

In this figure a right circular cone pierces a double ogival point.
The cone has a vertical axis, PP', the vertex P being at (3", IJ",

Set of Descriptive Drawings 157

I"), and F, the center of the base, at (3", IJ'', 3^"). The base is a
circle of 2^" diameter lying in a horizontal plane.

The ogival point has an axis of revolution, Q (|", 1^", 2"),
Q' (5f^ li", 2"), 5J" long. The generating line is an arc of 4"
radius of which QQ' is the chord, and in its initial position the arc
has its center at (3", IJ", 5'.'02). Draw three views of the cone
piercing the double ogival surface, and determine the line of inter-
section by means of three auxiliary cutting spheres, centered at p,
the intersection of PP' and QQ'. Use diameters of 2J", 2^", and
2iV"- This curve appears on the U. S. Navy standard 3" valve.

General Directions for Completing the Sheet.

In inking the sheet show one cutting plane or sphere for the
determination of each line of intersection, and show clearly how
one point is determined in each view of each figure. Shade the
figures, except the developments.

In the legend space record the following legend :

SHEET III. (Block letters 15/32" high.)

descriptive geometry. (All caps 3/16" high.) ^

INTERSECTIONS OF SURFACES OF ^ ''

REVOLUTION. (All caps 9/32" high.)

Name (signature). ClaSS. (Caps 1/8" high, lower case 1/12" high.)

Date. (Caps 1/8" high, lower case 1/12" high.)

SHEET IV: CONES, ANCHOR RING AND HEnCOIDAL
SURFACES.

Lay out center lines, cutting, border, working lines, and legend
space as before.

From the center of the sheet plot origins as follows: First
origin, 3^' to the left of the center and Syl" above the center;
second origin, 5 J" to the right of the center and 3" above the center ;
third origin, lOJ" to the right of the center and 4 J" above the center ;
fifth origin, 3" to the right of the center and 6" below the center.

158 Engineering Descriptive Geometry

First Origin: Intersecting Inclined Cones.

Draw two intersecting inclined cones. The first cone has its
vertex at P (1", If, i"). and the center of its base at P' {2",\\
11", 4|"). The base is a circle of 3f" diameter, lying in a plane
parallel to fi- The second cone has its vertex at Q (5", 1^", 2''46),
and the center of its base at Q' (J", IJ", 3J"). The base is a circle
of 3" diameter lying in a plane parallel to g. Draw plan, front
elevation, side elevation, and an auxiliary projection on a plane JJ^
perpendicular to the line PQ, the trace of U on V^ passing through
the point M (TJ", 0", 0"). Determine the line of intersection of the
cones by auxiliary cutting planes containing the line PQ, and treat
the problem on the supposition that the cone PP' pierces the cone

Second Origin: Helicoidal Surface for Screw Propeller.

A right vertical cylinder, IJ" in diameter, has for its axis
P (2^', n", \"), P' (2i", 21", 3f '). Projecting from the cylinder
is a line A (3i", ^'\ -i"), B (^", 2i", ^"). This line, moving
uniformly along the cylinder, and about it clockwise, describes one
complete turn of a helicoidal surface of 3" pitch. Draw plan and
front elevation of the figure. This helicoid is intersected by an
elliptical cylinder of which the generating line is perpendicular to
H and the directrix is an ellipse lying in H, having its major axis
C (2", 2V', 0"), D {Y, 24", 0"), and minor axis E {1\", 3", 0"),
F {1\", 2", 0"). Find the intersection of the two surfaces. Ink in
full lines only the circular cylinder and the intersection. This
portion of a helicoidal surface is similar to that which is used for
the acting surface of the ordinary marine screw propeller, of 3 or
4 blades.

Third Origin: Worm Thread Surface.

A worm shaft is a right cylinder, IJ" in diameter, its axis being
■P (If", IJ", i"), P' (IJ", If", 8f"). A triple right-hand worm
thread, of the same profile as in Fig. 70, projects from the cylinder
along the middle 6" of, its length. The pitch of the thread is 4J",

Set of Descriptive Drawings 159

so that each thread has more than a complete turn. The outside
diameter of the worm is 3". Make a complete drawing of the plan
and front elevation, as in Fig. 70, letting the worm thread begin at
any point on the circumference.

Fourth Origin: Anchor Ring and Planes.

An anchor ring, R, is formed by revolving a circle of 1 J" diameter,
lying in a plane parallel to V and with its center at A (1^", 2f",
J"), about an axis perpendicular to W and piercing Cil at the point
B (^f":> ^f'? 0")- I^raw plan, front elevation, right side elevation
(to the right of IHI), and left side elevation (to the left of H) on a
plane S', 4J" from §. A plane P, parallel to S at f " from S, cuts
the ring. Draw the trace of P on f\, and^the intersection of P and
the ring on S- A second plane P', parallel to S at IJ" distance,
cuts the ring. Draw the trace P'H and the intersection P'R on S-
A third plane Q is parallel to V at If" distance from V- Draw the
trace QTI and the intersection QR on V- A fourth plane, Q', is
parallel to V at 2" distance. Draw the trace Q'H and the inter-
section Q'R on V- An inclined plane T is perpendicular to S and
S', its trace on §' passing through the point C (4f'', 2f", J"), and
inclining down to the right at such an angle as to be tangent to the
projection on §' of the generating circle when its center is at
D (2f", li'\ i"). Draw the trace of T on §', and the intersection
TR on W. Find the true shape of TR by means of an auxiliary
plane of projection \] perpendicular to S', cutting §' in a trace
parallel to T8' through the point on S' whose coordinates are
E (4r, 0", IJ")-

• General Directions for Completing the Sheet.

Ink the sheet uniform with the preceding sheets, and in the
legend space record the following legend :

SHEET ly. (Block letters 15/32" high.)

descriptive geometry. (All caps 3/16" high.)

CONES, ANCHOR RING AND HELICOIDS. (All caps 9/32" high.)

Name (signature). Class. (Caps 1/8" high, lower case 1/12" high.)

Date. (Caps 1/8" high, lower case 1/12" high.)

Short-title Catalogue

OF THE

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CIVIL ENGINEERING.

BRIDGES AND ROOFS. HYDRAULICS. MATERIALS OF ENGINEER-
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$2 50

Soper's Air and Ventilation of Subways 12mo,

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Wait's Engineering and Architectural Jurisprudence 8vo,

Sheep,

Law of Contracts Svo,

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Architecture 8vo.

Sheep,
Warren's Stereotomy — Problems in Stone-cutting 8vo,

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* Enlarged Edition, Including Tables mor.

Webb's Problems in the Use and Adjustment of Engineering Instruments.

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Fowler's Ordinary Foundations Svo,

Greene's Arches in Wood, Iron, and Stone Svo,

Bridge Trusses Svo,

Roof Trusses Svo,

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Symmetrical Masonry Arches Svo,

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Morison's Memphis Bridge Oblong 4to, 10 00

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Waddell's De Pontibus, Pocket-book for Bridge Engineers 16mo, mor. 2 00

* Specifications for Steel Bridges 12mo, 50

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HYDRAULICS.

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Bovey's Treatise on Hydraulics Svo, 5 00

Church's Diagrams of Mean Velocity of Water in Open Channels.

Oblong 4to, paper, 1 50

Hydraulic Motors Svo, 2 00

5 00

Coffin's Graphical Solution of Hydraulic Problems 16mo, mor. $2 50

Flather's Dynamometers, and the Measurement of Power 12mo, 3 00

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Frizell's Water-power 8vo, 5 00

Fuertes's Water and Public Health 12mo, 1 50

Water-filtration Works 12mo, 2 50

Gangmllet and Kutter's General Formula for the Uniform Flow of Water in

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Hazen's Clean Water and How to Get It Large 12mo, 1 50

Filtration of Public Water-supplies 8vo, 3 00

Hazelhurst's Towers and Tanks for Water-works 8vo, 2 50

-Herschel's 115 Experiments on the Carrying Capacity of Large, Riveted, Metal

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Hoyt and Grover's River Discharge 8vo, 2 00

Hubbard and Kiersted's Water-works Management and Maintenance.

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* Lyndon's Development and Electrical Distribution of Water Power.

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Mason's Water-supply. (Considered Principally from a Sanitary Stand-
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Merriman's Treatise on Hydraulics 8vo, 5 00

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Morrison and Brodie's High Masonry Dam Design. (In Press.)

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Schuyler's Reservoirs for Irrigation, Water-power, and Domestic Water-
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* Thomas and Watt's Improvement of Rivers 4to, 6 00

Tiimeaure and Russell's Public Water-supplies 8vo, 5 00

Wegmann's Design and Construction of Dams. 5th Ed., enlarged 4to, 6 GO

Water-Supply of the City of New York from 1658 to 1895 4to, 10 00

Whipple's Value of Pure Water Large 12rao, 1 00

Williams and Hazen's Hydraulic Tables 8vo, 1 50

Wilson's Irrigation Engineering 8vo, 4 00

Wood 's Turbines Svo, 2 59

MATERIALS OF ENGINEERING.

Baker's Roads and Pavements Svo, 5 00

Treatise on Masonry Construction Svo, 5 00

Black's United States Public Works Oblong 4to, 5 00

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Bleininger's Manufacture of Hydraulic Cement. (In Preparation.)

* Bovey's Strength of Materials and Theory of Structures Svo, 7 50

Burr's Elasticity and Resistance of the Materials of Engineering Svo, 7 50

Byrne's Highway Construction Svo, 5 00

Inspection of the Materials and Workmanship Employed in Construction.

16mo, 3 00

Church's Mechanics of Engineering Svo, 6 00

Du Bois's Mechanics of Engineering.

Vol. I. Kinematics, Statics, Kinetics Small 4to, 7 50

Vol. II. The Stresses in Framed Structures, Strength of Materials and
Theory of Flexures Small 4to,

* Eckel's Cements, Limes, and Plasters Svo,

Stone and Clay Products used in Engineering. (In Preparation.)
Fowler's Ordinary Foundations Svo,

* Greene's Structural Mechanics Svo,

* HoUey's Lead and Zinc Pigments Large 12mo,

HoUey and Ladd's Analysis of Mixed Paints, Color Pigments and Varnishes.

Large 12mo,

* Hubbard's Dust Preventives and Road Binders Svo,

Johnson's (C. M.) Rapid Methods for the Chemical Analysis of Special Steels,

Steel-making Alloys and Graphite Large 12mo,

Johnson's (J. B.) Materials of Construction Large Svo,

Keep's Cast Iron Svo,

Lanza's Applied Mechanics Svo,

Lowe's Paints for Steel Structures 12mo,

$2 00

4 00

5 00

1 00

2 00

2 50

5 00

2 00

3 00

5 00

1 50

4 00

Maire's Modem Pigments and their Vehicles 12mo.

Maurer's Technical Mechanics 8vo,

Merrill's Stones for Building and Decoration 8vo,

Merriman's Mechanics of Materials 8vo,

* Strength of Materials 12mo,

Metcalf's Steel. A Manual for Steel-users 12mo,

Morrison's Highway Engineering Svo.

Patton's Practical Treatise on Foundations Svo,

Rice's Concrete Block Manufacture Svo,

Richardson's Modem Asphalt Pavement Svo,

Richey's Building Foreman's Pocket Book and Ready Reference. 16mo,mor.

* Cement Workers' and Plasterers' Edition (Building Mechanics' Ready

Reference Series) 16mo, mor.

Handbook for Superintendents of Construction 16mo, mor.

* Stone and Brick Masons' Edition (Building Mechanics' Ready

Reference Series) 16mo, mor. 1 50

* Ries's Clays : Their Occurrence, Properties, and Uses Svo, 5 00

* Ries and Leighton's History of the Clay-working Industry of the United

States Svo. 2 50

Sabin's Industrial and Artistic Technology of Paint and Varnish Svo, 3 00

* Smith's Strength of Material 12mo 1 25

Snow's Principal Species of Wood Svo, 3 50

Spalding's Hydraulic Cement 12mo, 2 00

Text-book on Roads and Pavements 12mo, 2 00

Taylor and Thompson's Treatise on Concrete, Plain and Reinforced Svo, 5 00

"Thurston's Materials of Engineering. In Three Parts Svo, 8 00

Part I. Non-metallic Materials of Engineering and Metallurgy. . . .Svo, 2 00

Part II. Iron and Steel Svo, 3 50

Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their

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"Tillson's Street Pavements and Paving Materials Svo, 4 00

* Trautwine's Concrete, Plain and Reinforced 16mo, 2 00

"Tumeaure and Maurer's Principles of Reinforced Concrete Construction.

Second Edition, Revised and Enlarged Svo, 3 50

Waterbury's Cement Laboratory Manual 12mo, 1 00

Wood's (De V.) Treatise on the Resistance of Materials, and an Appendix on

the Preservation of Timber Svo, 2 09

Wood's (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and

Steel Svo, 4 00

RAILWAY ENGINEERING.

Andrews's Handbook for Street Railway Engineers 3X5 inches, mor. 1 25

Berg's Buildings and Structures of American Railroads 4to, 5 00

Biooks's Handbook of Street Railroad Location 16mo, mor. 1 50

Butts's Civil Engineer's Field-book 16mc, mor. 2 50

Crandall's Railway and Other Earthwork Tables Svo, 1 50

Transition Curve 16mo, mor. 1 50

* Crockett's Methods for Earthwork Computations Svo, 1 50

Dredge's History of the Pennsylvania Railroad. (1879) Paper, 5 00

Pisher's Table of Cubic Yards Cardboard, 25

■Godwin's Railroad Engineers' Field-book and Explorers' Guide. . 16mo, mor. 2 50
Hudson's Tables for Calculating the Cubic Contents of Excavations and Em-
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Ives and Hilts's Problems in Sturveying, Railroad Surveying and Geodesy

16mo, mor. 1 50

Molitor and Beard's Manual for Resident Engineers 16mo, 1 00

Nagle's Field Manual for Railroad Engineers 16mo, mor. 3 00

* Orrock's Railroad Structures and Estimates Svo, 3 00

Philbrick's Field Manual for Engineers 16mo, mor. 3 00

Raymond's Railroad Engineering. 3 volumes.

Vol. I. Railroad Field Geometry. (In Preparation.)

Vol. II. Elements of Railroad Engineering Svo, 3 50

VoL in. Railroad Engineer's Field -Book. (In Preparation.)

Roberts' Track Formulae and Tables. (In Press.)

Searles's Field Engineering 16mo, mor. $3 00

Railroad Spiral 16mo, mor. 1 50

Taylor's Prismoidal Formulae and Earthwork 8vo, 1 50

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12mo, mor. 2 50

* Method of Calculating the Cubic Contents of Excavations and Em-
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Webb's Economics of Railroad Construction Large 12mo, 2 50

Railroad Construction 16mo, mor. 5 00

Wellington's Economic Theory of the Location of Railways Large 12mo, 5 00

Wilson's Elements of Railroad-Track and Construction 12mo, 2 00

DRAWING.

Barr's Kinematics of Machinery 8vo, 2 50

* Bartlett's Mechanical Drawing Svo, 3 00

* " " " Abridged Ed Svo, 150

Coolidge's Manual of Drawing Svo, paper, 1 00

Coolidge and Freeman's Elements of General Drafting for Mechanical Engi-
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Durley's Kinematics of Machines Svo, 4 00

Emch's Introduction to Projective Geometry and its Application Svo, 2 60

French and Ives' Stereotomy Svo, 2 50

Hill's Text-book on Shades and Shadows, and Perspective Svo, 2 00

Jamison's Advanced Mechanical Drawing Svo, 2 00

Elements of Mechanical Drawing Svo, 2 50

Jones's Machine Design :

Part I. Kinematics of Machinery Svo, 1 50

Part II. Form, Strength, and Proportions of Parts Svo, 3 00

* Kimball and Barr's Machine Design Svo, 3 00

MacCord's Elements of Descriptive Geometry Svo, 3 00

Kinematics", or. Practical Mechanism Svo, 5 00

Mechanical Drawing 4to, 4 00

Velocity Diagrams Svo, 1 50

McLeod's Descriptive Geometry Large 12mo, 1 50

"i" Mahan's Descriptive Geometry and Stone-cutting Svo, 1 50

Industrial Drawing. (Thompson.) Svo, 3 50

Moyer's Descriptive Geometry Svo, 2 00

Reed's Topographical Drawing and Sketching 4to, 5 00

Reid's Course in Mechanical Drawing Svo, 2 00

Text-book of Mechanical Drawing and Elementary Machine Design.. Svo, 3 00

Robinson's Principles of Mechanism Svo, 3 00

Schwamb and Merrill's Elements of Mechanism Svo, 3 00

Smith (A. W.) and Marx's Machine Design Svo, 3 00

Smith's (R. S.) Manual of Topographical Drawing. (McMillan.) Svo, 2 50

* Titsworth's Elements of Mechanical Drawing Oblong Svo, 1 25

Warren's Drafting Instruments and Operations 12mo, 1 25

Elements of Descriptive Geometry, Shadows, and Perspective Svo, 3 50

Elements of Machine Construction and Drawing Svo, 7 50

Elements of Plane and Solid Free-hand Geometrical Drawing. . . . 12mo, 1 00

General Problems of Shades and Shadows Svo, 3 00

Manual of Elementary Problems in the Linear Perspective of Forms and

Shadow 12mo. 1 00

Manual of Elementary Projection Drawing 12mo, 1 50

Plane Problems in Elementary Geometry 12mo, 1 25

Weisbach's Kinematics and Power of Transmission. (Hermann and

Klein. ) Svo, 5 00

Wilson's (H. M.) Topographic Surveying Svo, 3 50

* Wilson's (V. T.) Descriptive Geometry Svo, 1 50

Free-hand Lettering Svo, 1 00

Free-hand Perspective Svo, 2 50

Woolf's Elementary Course in Descriptive Geometry. . , . '. Large Svo, 3 00

ELECTRICITY AND PHYSICS.

* Abegg's Theory of Electrolytic Dissociation, (von Ende.) 12mo, $1 25

Andrews's Hand-book for Street Railway Engineering 3X5 inches, mor. 1 25

Anthony and Brackett's Text-book of Physics. (Magie.) ... .Large 12mo, 3 00
Anthony and Ball's Lecture-notes on the Theory of Electrical Measure-
men ts 12nio, 1 00

Benjamin's History of Electricity 8vo, 3 00

Voltaic Cell 8vo, 3 00

Betts's Lead Refining and Electrolysis 8vo, 4 00

Glassen's Quantitative Chemical Analysis by Electrolysis. (Boltwood.).8vo, 3 00

* CoUins's Manual of Wireless Telegraphy and Telephony 12mo, 1 50

Crehore and Squier's Polarizing Photo-chronograph 8vo, 3 00

* Danneel's Electrochemistry. (Merriam.) 12mo, 1 25

Dawson's "Engineering" and Electric Traction Pocket-book. . . .16mo, mor. 5 00
Dolezalek's Theory of the Lead Accumulator (Storage Battery), (von Ende.)

12mo, 2 50

Duhem's Thermodynamics and Chemistry. (Burgess.) 8vo, 4 00

Flather's Dynamometers, and the Measurement of Power 12mo, 3 00

* Getman's Introduction to Physical Science.. 12mo, 1 50

Gilbert's De Magnete. (Mottelay ) 8vo, 2 50

* Hanchett's Alternating Currents 12mo, 1 00

Hering's Ready Reference Tables (Conversion Factors) 16mo, mor. 2 50

* Hobart and ElUs's High-speed Dynamo Electric Machinery 8vo, 6 00

Holman's Precision of Measurements 8vo, 2 00

Telescopic Mirror-scale Method, Adjustments, and Tests.. . .Large 8vo, 75

* Karapetoff 's Experimental Electrical Engineering 8vo, 6 00

Kinzbrunner's Testing of Continuous-current Machines 8vo, 2 00

Landauer's Spectrum Analysis. (Tingle.) 8vo, 3 00

Le Chatelier's High-temperature Measurements. (Boudouard — Burgess.) 12mo, 3 00

Lob's Electrochemistry of Organic Compounds. (Lorenz.) 8vo, 3 00

* Lyndon's Development and Electrical Distribution of Water Power. .8vo, 3 00

* Lyons's Treatise on Electromagnetic Phenomena. Vols, I .and II. 8vo, each, 6 00

* Michie's Elements of Wave Motion Relating to Sound and Light 8vo, 4 OU

Morgan's Outline of the Theory of Solution and its Results 12mo, 1 00

* Physical Chemistry for Electrical Engineers 12mo, 1 50

* Norris's Introduction to the Study of Electrical Engineering 8vo, 2 50

Norris and Dennison's Course of Problems on the Electrical Characteristics of

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* Parshall and Hobart's Electric Machine Design 4to, half mor, 12 50

Reagan's Locomotives: Simple, Compound, and Electric. New Edition.

Large 12mo, 3 50

* Rosenberg's Electrical Engineering. (Haldane Gee — Kinzbrunner.). .8vo, 2 00

Ryan, Norris, and Hoxie's Electrical Machinery. Vol. 1 8vo, 2 50

Schapper's Laboratory Guide for Students in Physical Chemistry 12mo, 1 00

* Tillman's Elementary Lessons in Heat 8vo, 1 50

Tory and Pitcher's Manual of Laboratory Physics Large 12mo, 2 00

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LAW.

* Brennan's Hand-book of Useful Legal Information for Business Men.

16mo, mor. 5 00

* Davis's Elements of Law 8vo, 2 50

* Treatise on the Military Law of United States 8vo, 7 00

* Dudley's Military Law and the Procedure of Courts-martial. . Large 12mo, 2 50

Manual for Courts-martial 16mo, mor. 1 50

Wait's Engineering and Architectural Jurisprudence Svo, 6 00

Sheep, 6 50

Law of Contracts Svo, 3 00

Law of Operations Preliminary to Construction in Engineering and

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11

MATHEMATICS.

Baker's Elliptic Functions 8vo, $1 50

Briggs's Elements of Plane Analytic Geometry. (Bocher.) 12mo, 1 00

* Buchanan's Plane and Spherical Trigonometry 8vo, 1 00

Byerley's Harmonic Functions 8vo, 1 00

Chandler's Elements of the Infinitesimal Calculus 12mo, 2 00

* Coffin's Vector Analysis 12mo, 2 50

■Compton's Manual of Logarithmic Computations 12mo, 1 50

* Dickson's College Algebra Large 12mo, 1 50

* Introduction to the Theory of Algebraic Equations Large 12mo, 1 25

Emch's Introduction to Projective Geometry and its Application Svo, 2 50

Piske's Functions of a Complex Variable Svo, 1 00

Halsted's Elementary Synthetic Geometry Svo, 1 50

Elements of Geometry Svo, 1 75

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Synthetic Projective Geometry Svo, 1 00

Hancock's Lectures on the Theory of Elliptic Functions. (In Press.)

Hyde's Grassmann's Space Analysis Svo, 1 00

* Johnson's (J. B.) Three-place Logarithmic Tables: Vest-pocket size, paper, 15

* 100 copies, 5 00

* Mounted on heavy cardboard, 8 X 10 inches, 25

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Johnson's (W. W.) Abridged Editions of Differential and Integral Calculus.

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Curve Tracing in Cartesian Co-ordinates 12mo, 1 00

Differential Equations Svo, 1 00

Elementary Treatise on Differential Calculus Large 12mo, 1 60

Elementary Treatise on the Integral Calculus Large 12mo, 1 50

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Theory of Errors and the Metliod of Least Squares 12mo, 1 50

Treatise on Differential Calculus Large 12mo, 3 00

Treatise on the Integral Calculus Large 12mo, 3 00

Treatise on Ordinary and Partial Differential Equations. . .Large 12mo, 3 50

Xarapetoff's Engineering Applications of Higher Mathematics. (In Preparation.)

Laplace's Philosophical Essay on Probabilities. (Truscott and Emory.) . 12mo, 2 00

* Ludlow and Bass's Elements of Trigonometry and Logarithmic and Other

Tables Svo, 3 00

* Trigonometry and Tables published separately Each, 2 00

* Ludlow's Logarithmic and Trigonometric Tables Svo, 1 00

Macfarlane's Vector Analysis and Quaternions Svo, 1 00

McMahon's Hyperbolic Functions Svo, 1 00

Manning's Irrational Numbers and their Representation by Sequences and

Series 12mo, 1 25

-Mathematical Monographs. Edited by Mansfield Merriman and Robert

S. Woodward Octavo, each 1 00

No. 1. History of Modern Mathematics, by David Eugene Smith.
No. 2. Synthetic Projective Geometry, by George Bruce Halsted.
No. 3. Determinants, by Laenas Gifford Weld. No. 4. Hyper-
bolic Functions, by James McMahon. No. 5. Harmonic Func-
tions, by V/illiam E. Byerly. No. 6. Grassmann's Space Analysis,
by Edward W. Hyde. No. 7. Probability and Theory of Errors,
by Robert S. Woodward. No. 8. Vector Analysis and Quaternions,
by Alexander Macfarlane. No. 9. Differential Equations, by
William Woolsey Johnson. No. 10. The Solution of Equations,
by Mansfield Merriman. No. 11. Functions of a Complex Variable,
by Thomas S. Fiske.

Maurer's Technical Mechanics Svo, 4 00

Merriman's Method of Least Squares Svo, 2 00

Solution of Equations Svo, 1 00

Rice and Johnson's Differential and Integral Calculus. 2 vols, in one.

Large 12mo, 1 50

Elementary Treatise on the Differential Calculus Large 12mo, 3 00

Smith's History of Modern Mathematics Svo, 1 00

* Veblen and Lennes's Introduction to the Real Infinitesimal Analysis of One

Variable Svo. 2 00

* Waterbury's Vest Pocket Hand-book of Mathematics for Engineers.

2JX6I inches, mor. $1 00

* Enlarged Edition, Including Tables mor. 1 60

Weld's Determinants 8vo, 1 00

Wood's Elements of Co-ordinate Geometry 8vo, 2 00

Woodward's Probability and Theory of Errors 8vo, 1 00

MECHANICAL ENGINEERING.

MATERIALS OP ENGINEERING, STEAM-ENGINES AND BOILERS.

Bacon's Forge Practice 12mo, 1 50

Baldwin's Steam Heating for Buildings 12mo, 2 50

Barr's Kinematics of Machinery Svo, 2 50

* Bartlett's Mechanical Drawing Svo, 3 00

* " " " Abridged Ed Svo, 150

* Burr's Ancient and Modern Engineering and the Isthmian Canal Svo, 3 50

Carpenter's Experimental Engineering Svo, 6 00

Heating and Ventilating Buildings Svo, 4 00

* Clerk's The Gas, Petrol and Oil Engine Svo, 4 00

Compton's First Lessons in Metal Working 12mo, 1 50

Compton and De Groodt's Speed Lathe 12mo, 1 50

Coolidge's Manual of Drawing Svo, paper, 1 00

Coolidge and Freeman's Elements of General Drafting for Mechanical En-
gineers Oblong 4to, 2 50

Cromwell's Treatise on Belts and Pulleys 12mo, 1 50

Treatise on Toothed Gearing 12mo, 1 50

Dingey's Machinery Pattern Making 12mo, 2 00

Durley's Kinematics of Machines Svo, 4 00

Flanders's Gear-cutting Machinery ■ Large 12mo, 3 00

Flather's Dynamometers and the Measurement of Power 12mo, 3 00

Rope Driving 12mo, 2 00

Gill's Gas and Fuel Analysis for Engineers 12mo, 1 25

Goss's Locomotive Sparks Svo, 2 00

Greene's Pumping Machinery. (In Preparation.)

Hering's Ready Reference Tables (Conversion Factors) 16mo, mor. 2 50

* Hobart and Ellis's High Speed Dynamo Electric Machinery Svo, 6 00

Hutton's Gas Engine Svo, 6 00

Jamison's Advanced Mechanical Drawing Svo, 2 00

Elements of Mechanical Drawing Svo, 2 50

Jones's Gas Engine Svo, 4 00

Machine Design:

Part I. Kinematics of Machinery Svo, 1 50

Part II. Form, Strength, and Proportions of Parts Svo, 3 00

Kent's Mechanical Engineer's Pocket-Book 16mo, mor. 5 00

Kerr's Power and Power Transmission Svo, 2 00

* Kimball and Barr's Machine Design Svo, 3 00

* Levin's Gas Engine . Svo, 4 00

Leonard's Machine Shop Tools and Methods Svo, 4 00

* Lorenz's Modem Refrigerating Machinery. (Pope, Haven, and Dean). .Svo, 4 00
MacCord's Kinematics; or. Practical Mechanism Svo, 5 00

Mechanical Drawing 4to, 4 00

Velocity Diagrams Svo, 1 50

MacFarland's Standard Reduction Factors for Gases Svo, 1 50

Mahan's Industrial Drawing. (Thompson.) Svo, 3 50

Mehrtens's Gas Engine Theory and Design Large 12mo, 2 50

Oberg's Handbook of Small Tools Large 12mo. 3 00

* Parshall and Hobart's Electric Machine Design. Small 4to, half leather, 12 50

Peele's Compressed Air Plant for Mines Svo, 3 00

Poole's Calorific Power of Fuels Svo, 3 00

* Porter's Engineering Reminiscences, 1S55 to 1882 Svo, 3 00

Reid's Course in Mechanical Drawing Svo, 2 00

Text-book of Mechanical Drawing and Elementary Machine Design.Svo, 3 00

Richards's Compressed Air 12mo,

Robinson's Principles of Mechanism 8vo,

Schwamb and Merrill's Elements of Mechanism 8vo,

Smith (A. W.) and Marx's Machine Design 8vo,

Smith's (O.) Press-working of Metals 8vo,

Sorel's Carbureting and Combustion in Alcohol Engines. (Woodward and

Preston.) Large 12mo,

Stone's Practical Testing of Gas and Gas Meters 8vo,

Thurston's Animal as a Machine and Prime Motor, and the Laws of Energetics.

12mo,
Treatise on Friction and Lost Work in Machinery and Mill Work. . .8vo,

* Tillson's Complete Automobile Instructor 16mo,

* Titsworth's Elements of Mechanical Drawing Oblong 8vo,

Warren's Elements of Machine Construction and Drawing 8vo,

* Waterbury's Vest Pocket Hand-book of Mathematics for Engineers.

2|X5| inches, mor.

* Enlarged Edition, Including Tables mor.

Weisbach's Kinematics and the Power of Transmission. (Herrmann —

Klein.) 8vo,

Machinery of Transmission and Governors. (Hermann — Klein.) . .8vo,
Wood's Turbines 8vo,

MATERIALS OF ENGINEERING.

* Bovey 's Strength of Materials and Theory of Structures 8vo, 7 50

Burr's Elasticity and Resistance of the Materials of Engineering 8vo, 7 50

Church's Mechanics of Engineering 8vo, 6 00

* Greene's Structural Mechanics 8vo, 2 50

* Holley's Lead and Zinc Pigments Large I2mo 3 00

Holley and Ladd's Analysis of Mixed Paints, Color Pigments, and Varnishes.

Large 12mo, 2 50
Johnson's (C. M.) Rapid Methods for the Chemical Analysis of Special

Steels, Steel-Making Alloys and Graphite Large 12mo, 3 00

Johnson's (J. B.) Materials of Construction 8vo, 6 00

Keep's Cast Iron 8vo, 2 50

Lanza's Applied Mechanics 8vo, 7 50

Maire's Modem Pigments and their Vehicles 12mo, 2 00

Maurer's Technical Mechanics 8vo, 4 00

Merriman's Mechanics of Materials 8vo, 5 00

* Strength of Materials 12mo, 1 00

Metcalf 's Steel. A Manual for Steel-users 12mo, 2 00

Sabin's Industrial and Artistic Technology of Paint and Varnish 8vo, 3 00

Smith's ((A. W.) Materials of Machines 12mo, 1 00

* Smith's (H. E.) Strength of Material 12mo, 1 25

Thurston's Materials of Engineering 3 vols., 8vo, 8 00

Part I. Non-metallic Materials of Engineering 8vo, 2 00

Part II. Iron and Steel 8vo, 3 50

Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their

Constituents 8vo, 2 50

Wood's (De V.) Elements of Analytical Mechanics 8vo, 3 00

Treatise on the Resistance of Materials and an Appendix on the

Preservation of Timber 8vo, 2 00

Wood's (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and

Steel 8vo, 4 00

STEAM-ENGINES AND BOILERS.

Berry's Temperature-entropy Diagram 12mo, 2 00

Carnot's Reflections on the Motive Power of Heat. (Thurston.) 12mo, 1 50

Chase's Art of Pattern Making 12mo, 2 50

Creighton's Steam-engine and other Heat Motors 8vo, $5 00

Dawson's "Engineering" and Electric Traction Pocket-book. .. . 16mo, mor. 5 00

* Gebhardt's Steam Power Plant Engineering 8vo, 6 00

Goss's Locomotive Performance 8vo, 5 00

Hemenway's Indicator Practice and Steam-engine Economy 12mo, 2 00

Button's Heat and Heat-engines Svo, 5 00

Mechanical Engineering of Power Plants Svo, 5 00

Kent's Steam boiler Economy Svo, 4 00

Kneass's Practice and Theory of the Injector Svo, 1 50

MacCord's Slide-valves Svo, 2 00

Meyer's Modern Locomotive Construction 4to, 10 00

Moyer's Steam Turbine Svo, 4 00

Peabody's Manual of the Steam-engine Indicator 12mo, 1 50

Tables of the Properties of Steam and Other Vapors and Temperature-
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Thermodynamics of the Steam-engine and Other Heat-engines. . . . Svo, 5 00

Valve-gears for Steam-engines Svo, 2 50

Peabody and Miller's Steam-boilers Svo, 4 00

Pupin's Thermodynamics of Reversible Cycles in Gases and Saturated Vapors.

(Osterberg.) 12mo, 1 25

Reagan's Locomotives: Simple, Compound, and Electric. New Edition.

Large 12mo, 3 50

Sinclair's Locomotive Engine Running and Management 12mo,

Smart's Handbook of Engineering Laboratory Practice 12mo,

Snow's Steam-boiler Practice Svo,

Spangler's Notes on Thermodynamics 12mo,

Valve-gears Svo,

Spangler, Greene, and Marshall's Elements of Steam-engineering Svo,

Thomas's Steam-turbines Svo,

Thurston's Handbook of Engine and Boiler Trials, and the Use of the Indi-
cator and the Prony Brake Svo,

Handy Tables Svo,

Manual of Steam-boilers, their Designs, Construction, and Operation Svo,

Manual of the Steam-engine 2 vols., Svo,

Part I. History, Structure, and Theory Svo,

Part II. Design, Construction, and Operation Svo,

Wehrenfennig's Analysis and Softening of Boiler Feed-water. (Patterson.)
. Svo,

Weisbach's Heat, Steam, and Steam-engines. (Du Bois.) Svo,

Whitham's Steam-engine Design Svo,

Wood's Thermodynamics, Heat Motors, and Refrigerating Machines. . .Svo,

MECHANICS PURE AND APPLIED.

Church's Mechanics of Engineering Svo, 6 00

Notes and Examples in Mechanics Svo, 2 00

Dana's Text-book of Elementary Mechanics for Colleges and Schools .12mo, 1 50
Du Bois's Elementary Principles of Mechanics:

Vol. I. Kinematics Svo 3 50

Vol. II. Statics Svo, 4 00

Mechanics of Engineering. Vol. I Small 4to, 7 50

Vol. II Small 4to, 10 00

* Greene's Structural Mechanics Svo, 2 50

Hartmann's Elementary Mechanics for Engineering Students. (In Press.)
James's Kinematics of a Point and the Rational Mechanics of a Particle.

Large 12mo, 2 00

* Johnson's (W. W.) Theoretical Mechanics 12mo, 3 00

Lanza's Applied Mechanics Svo, 7 50

* Martin's Text Book on Mechanics, Vol. I, Statics 12mo, 1 25

* Vol. II, Kinematics and Kinetics. 12mo, 1 50

Maurer's Technical Mechanics Svo, 4 00

* Merriman's Elements of Mechanics 12mo, 1 00

Mechanics of Materials Svo, 5 00

* Michie's Elements of Analytical Mechanics Svo, 4 00

Robinson's Principles of Mechanism 8vo, $3 00'

Schwamb and Merrill's Elements of Mechanism 8vo, 3 00

Wood's Elements of Analytical Mechanics 8vo, 3 00

Principles of Elementary Mechanics 12mo, 1 25

MEDICAL.

* Abderhalden's Physiological Chemistry in Thirty Lectures. (Hall and

Defren.) Svo,

von Behring's Suppression of Tuberculosis. (Bolduan.) 12mo,

Bolduan's Immune Sera 12mo,

Bordet's Studies in Immunity. (Gay.) Svo,

Chapin's The Sources and Modes of Infection. (In Press.)
Davenport's Statistical Methods with Special Reference to Biological Varia-
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Ehrlich's Collected Studies on Immunity. (Bolduan.) Svo,

* Fischer's Physiology of Alimentation Large 12mo,

de Fursac's Manual of Psychiatry. (Rosanoff and Collins.).. . .Large 12mo,

Hammarsten's Text-book on Physiological Chemistry. (Mandel.) Svo,

Jackson's Directions for Laboratory Work in Physiological Chemistry. .Svo,

Lassar-Cohn's Practical Urinary Analysis. (Lorenz.) 12mo,

Handel's Hand-book for the Bio-Chemical Laboratory 12mo.

* Nelson's Analysis of Drugs and Medicines 12mo.

* Pauli's Physical Chemistry in the Service of Medicine. (Fischer.) ..12mo,

* Pozzi-Escot's Toxins and Venoms and their Antibodies. (Cohn.). . 12nio,

Rostoski's Serum Diagnosis. (Bolduan.) 12nio,

Ruddiman's Incompatibilities in Prescriptions Svo,

Whys in Pharmacy I2mo,

Salkowski's Physiological and Pathological Chemistry. (Omdorflf.) .. ..Svo,

* Satterlee's Outlines of Human Embryology 12mo,

Smith's Lecture Notes on Chemistry for Dental Students Svo,

* Whipple's Tyhpoid Fever Large 12mo,

* Woodhull's Military Hygiene for Officers of the Line Large 12mo,

* Personal Hygiene 12mo,

Worcester and Atkinson's Small Hospitals Establishment and Maintenance,
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METALLURGY.

Betts's Lead Refining by Electrolysis Svo, 4 00'

Bolland's Encyclopedia of Founding and Dictionary of Foundry Terms used

in the Practice of Moulding 12mo, 3 OO'

Iron Founder 12mo, 2 50

Supplement 12mo, 2 50

Douglas's Untechnical Addresses on Technical Subjects 12mo, 1 00

Goesel's Minerals and Metals: A Reference Book 16mo, mor. 3 00>

* Iles's Lead-smelting 12mo, 2 50

Johnson's Rapid Methods for the Chemical Analysis of Special Steels,

Steel-making Alloys and Graphite Large 12mo, 3 00*

Keep's Cast Iron Svo, 2 50-

Le Chatelier's High-temperature Measurements. (Boudouard — Burgess.)

12mo, 3 00

Metcalf's Steel. A Manual for Steel-users 12mo. 2 00

Minet's Production of Aluminum and its Industrial Use. (Waldo.). . 12mo, 2 50-

* Ruer's Elements of Metallography. (Mathewson.) Svo, 3 00-

Smith's Materials of Machines 12mo, 1 00'

Tate and Stone's Foundry Practice 12mo, 2 00'

Thurston's Materials of Engineering. In Three Parts Svo, S 00

Part I. Non-metallic Materials of Engineering, see Civil Engineering,

page 9.

Part II. Iron and Steel Svo, 3 50

Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their

Constituents Svo, 2 50>

00-

5fy

50*

5©

5 00

Ulke's Modern Electrolytic Copper Refining 8vo, $3 00'

West's American Foundry Practice 12mo, 2 5Q

Moulders' Text Book 12mo. 2 50

MINERALOGY.

Baskerville's Chemical Elements. (In Preparation.)

* Browning's Introduction to the Rarer Elements 8vo,

Brush's Manual of Determinative Mineralogy. (Penfield.) 8vo,

Butler's Pocket Hand-book of Minerals 16mo, mor.

Chester's Catalogue of Minerals Svo, paper,

Cloth,

P Crane's Gold and Silver . Svo,

Dana's First Appendix to Dana's New "System of Mineralogy". .Large Svo,
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Large Svo,

Manual of Mineralogy and Petrography 12mo,

Minerals and How to Study Them 12mo,

System of Mineralogy Large Svo, half leather,

Text-book of Mineralogy Svo,

Douglas's Untechnical Addresses on Technical Subjects 12mo,

Eakle's Mineral Tables Svo,

Eckel's Stone and Clay Products Used in Engineering. (In Preparation.)

Goesel's Minerals and Metals: A Reference Book 16mo, mor.

Groth's The Optical Properties of Crystals. (Jackson.) (In Press.)
Groth's Introduction to Chemical Crystallography (Marshall) 12mo,

* Hayes's Handbook for Field Geologists 16mo, mor.

Iddings's Igneous Rocks Svo,

Rock Minerals Svo,

Johannsen's Determination of Rock-forming Minerals in Thin Sections. Svo,

With Thumb Index 5 00

* Martin's Laboratory Guide to Qualitative Analysis with the Blow-

pipe 12mo, 60

Merrill's Non-metallic Minerals: Their Occurrence and Uses Svo,

Stones for Building and Decoration Svo,

* Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests.

Svo, paper.

Tables of Minerals, Including the Use of Minerals and Statistics of

Domestic Production Svo,

* Pirsson's Rocks and Rock Minerals 12mo,

* Richards's Synopsis of Mineral Characters 12mo, mor.

* Ries's Clays: Their Occurrence. Properties and Uses Svo,

* Ries and Leighton's History of the Clay-working Industry of the United

States Svo.

* Tillman's Text-book of Important Minerals and Rocks Svo,

Washington's Manual of the Chemical Analysis of Rocks Svo,

MINING.

* Beard's Mine Gases and Explosions Large 12mo,

* Crane's Gold and Silver Svo,

* Index of Mining Engineering Literature Svo.

* Svo, mor.

Ore Mining Methods. (In Press.)

Douglas's Untechnical Addresses on Technical Subjects 12mo,

Eissler's Modem High Explosives Svo,

Goesel's Minerals and Metals: A Reference Book 16mo, mor.

Ihlseng's Manual of Mining Svo,

* Iles's Lead Smelting 12mo,

Peele's Compressed Air Plant for Mines Svo,

Riemer's Shaft Sinking Under Difficttlt Conditions. (Coming and Peele.)Svo,

* Weaver's Military Explosive*. Svo,

Wilson's Hydraulic and Placer Mining. 2d edition, rewritten 12mo,

Treatise on Practical and Theoretical Mine Ventilation 12mo,

2 00

2 50

1 25.

SANITARY SCIENCE.

Association of State and National Food and Dairy Departments, Hartford

Meeting, 1906 8vo, $3 00

Jamestown Meeting, 1907 8vo, 3 00

* Bashore's Outlines of Practical Sanitation 12mo, 1 25

Sanitation of a Country House 12mo, 1 00

Sanitation of Recreation Camps and Parks 12mo, 1 00

Chapin's The Sources and Modes of Infection. (In Press.)

Folwell's Sewerage. (Designing, Construction, and Maintenance.) 8vo, 3 00

Water-supply Engineering. . 8vo, 4 00

Fowler's Sewage Works Analyses. 12mo, 2 00

Fuertes's Water-filtration Works 12mo, 2 50

Water and Public Health 12mo, 1 50

Gerhard's Guide to Sanitary Inspections 12mo, 1 50

* Modern Baths and Bath Houses 8vo, 3 00

Sanitation of Public Buildings 12mo, 1 50

* The Water Supply, Sewerage, and Plumbing of Modern City Buildings.

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Hazen's Clean Water and How to Get It Large 12mo, 1 50

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Kinnicut, Winslow and Pratt's Purification of Sewage. (In Preparation.)
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Mason's Examination of Water. (Chemical and Bacteriological) 12mo, 1 25

Water-supply. (Considered principally from a Sanitary Standpoint).

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* Merriman's Elements of Sanitary Engineering 8vo, 2 00

Ogden's Sewer Construction 8vo, 3 00

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Parsons's Disposal of Municipal Refuse 8vo, 2 00

Prescott and Winslow's Elements of Water Bacteriology, with Special Refer-
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* Price's Handbook on Sanitation 12mo, 1 50

Richards's Cost of Cleanness 12mo, 1 00

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Rideal's Disinfection and the Preservation of Food 8vo, 4 00

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Turneaure and Russell's Public Water-supplies 8vo, 5 00

Venable's Garbage Crematories in America 8vo, 2 00

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Ward and Whipple's Freshwater Biology. (In Press.)

Whipple's Microscopy of Drinking-water Svo, 3 50

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Jacobs's Betterment Briefs. A Collection of Published Papers on Or-
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Standage's Decoration of Wood, Glass, Metal, etc 12mo, 2 00

Thome's Structural and Physiological Botany. (Bennett) 16mo, -2 25

Westermaier's Compendium of General Botany. (Schneider) 8vo, ? 00

Winslow's Elements of Applied Microscopy 12mo, 1 50

HEBREW AND CHALDEE TEXT-BOOOKS.

Gesenius's Hebrew and Chaldee Lexicon to the Old Testament Scriptures.

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Full text of "Engineering descriptive geometry; a treatise on descriptive geometry as the basis of mechanical drawing, explaining geometrically the operations customary in the draughting room"

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