Edward Shaw.

# Shaw's Civil architecture; being a complete theoretical and practical system of building, containing the fundamental principles of the art online

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Font size jection, its projection upon the other will be upon the
grovmd line. Thus, if a line be situated upon a hori-
zontal plane, its vertical projection will be upon the
ground line ; and if this line were given upon the
vertical plane, its horizontal projection would be upon
the ground line.

Reciprocally, if one of the projections of a line be
upon the ground line, this line will be upon the plane
of the other projection. Thus, for example, if it be
the vertical projection of the line in question, which
is upon the ground, this line will be upon the hori-
zontal plane ; if, on the contrary, it were upon the
horizontal projection of this line which was upon the
ground line, this line would be upon the vertical
plane.

If a line be at any time upon the two planes of
projcctioM, the tA.vo projections of this line would be
upon the ground line, and the line in question would
coincide with this ground line. Reciprocally, if the
t^vo projections of a line were upon the ground line,
the line itself would be upon the ground line.

If two lines in space are parallel, their projections
upon each plane of projection are also parallel. Re-
ciprocally, if the projections of two lines are parallel
on each plane of projection, the two lines will be
parallel to one another in space.

If any two lines whatever in space cut each other,
the projections of their point of intersection will be

62

PRACTICAL GEOMETRY

upon the same perpendicular line to the ground line,
and upon the points of intersection of the projections
of these lines. Reciprocally, if the projections of
any two lines whatever cut each other in the two
planes of projection in such a manner that their
points of intersection are upon the same perpendic-
ular to tlie ground line, these two lines in question
will cut each other in space.

The position of a plane is determined in space
when we know the intersections of this plane with
the planes of projection.

The intersections A B, A C, of the plane in ques-
tion, witli the planes of projections, arc called the
traces of this plane.

The trace situated in the horizontal plane is called
the horizontal trace, and the trace situated in the ver-
tical piano is called the vertical trace.

A very important remark is, that the two traces of
a plane intersect each other upon the ground line.

If a plane be parallel to one of the planes of pro-
jection, this plane wiU have only one trace, which
will be paraDel to the ground line, and situated in
the other plane of projection. Reciprocally, if a
plane has a trace parallel to the ground line, this
plane will be parallel to the plane of projection,
which docs not contain this trace. Thus : â€”

1. If a plane be parallel to the horizontal plane,
this plane will not have a horizontal trace, and its
vertical trace will be parallel to the ground line.
Likewise, if a plane be parallel to the vertical plane,
this piano will not have a vertical trace, and its hori-
zontal trace will be parallel to the ground line.

2. If a plane has only one trace, and this ti-ace
parallel to the ground line, let it be in the vertical
plane ; then the plane will be parallel to the horizon-
tal plane. So if the trace of the plane be in the
horizontal plane, and parallel to the ground line, the
plane will be parallel to the vertical plane. .

If one of the traces of a plane be perpendicular to
the ground line, and the other trace in any position
whatever, this plane will be perpendicular to the
plane of projection in which the second trace is.
Thus, if it bo a horizontal trace which is perpendic-
ular to the ground line, the plane will be perpendic-
idar to the vertical plane of projection ; and if, on
the contrary, the vortical trace be that which is per-
pendicular to the ground line, tiien the plane will be
perpendicular to the horizontal plane.

Reciprocally, if a plane be perpendicular to one of

the planes of projection, without being parallel to the
other, its trace upon the plane of projection, to which
it is perpendicular, will be perpendicular to any posi-
tion whatever, and the other trace will be perpendic-
ular to the ground line. Thus, for example, if the
plane be perpendicular to the vertical plane, the ver-
tical trace will be perpendicular to the ground line.
The reverse will also be true, if the plane be perpen-
dicular to the horizontal plane.

If a plane be perpendicular to the two planes of
projection, its two traces will be perpendicular to the
ground line. Reciprocally, if the two traces of a
plane are in the same straight line perpendicular to
the ground line, this plane will be perpendicular to
both the planes of projection.

If the two traces of a plane are parallel to the
ground line, this plane will be also parallel to the
ground line. Reciprocally, if a plane be parallel to
the ground line, its two traces will be parallel to the
ground line.

When a plane is not parallel to either of the planes
of projection, and one of its traces is parallel to the
ground line, the other trace is also necessarily par-
allel to the ground line.

If two planes arc parallel, their traces in each of
the planes of projection will also be parallel. Recip-
rocally, if on each plane of projection the traces of
the two planes are parallel, the planes will also be
parallel.

If a line be perpendicular to a plane, the projec-
tions of this line will be in each plane of projection
perpendicular to the respective traces in this plane.
Reciprocally, if the projections of a line are respec-
tively perpendicular to the traces of a plane, the line
will be perpendicular to the plane.

If a line be situated in a given plane by its traces,
this line can only intersect the planes of projection
upon the traces of the plane which contains it.
Moreover, the line in question can only meet the
plane of projection in its own projection. Whence
it follows, that the points of meeting of the right
line, and the planes of projection, arc respectively
upon the intersections of this right line, and the
traces of the plane which contains it.

K a right line, situated in a given plane by its
ti-aces, is parallel to the horizontal plane, its horizon-
tal projection will be parallel to the horizontal trace
of the given plane, and its vertical projection will be
parallel to the ground line. Likewise, if the right

PRACTICAL GEOMETRY.

63

line situated in a given plane by its traces is parallel
to the vertical plane, its vertical projection will be
parallel to the vertical line of the plane which con-
tains it, and its horizontal projection will be parallel
to the ground line.

Reciprocally, if a line be situated in a given plane
by its traces, and, for example, its horizontal pro-
jection be parallel to the horizontal trace of the
given plane, this line will be parallel to the horizon-
tal plane, and its vertical projection wiU be parallel
to the ground line. Liliewise, if the vertical projec-
tion of the line in question be parallel to the vertical
trace of the given plane, this line will be parallel to
the vertical plane, and its horizontal projection wiU
be parallel to the ground line.

SECTION

On the Developments of the Surfaces of Solids.

PROBLEMS.
Plate 15.

PROBLEM I.

To find tlie development of the surface of a
right semi-cylinder.

Fig. 1. Let A C D E be the plane passing through
the axis. On A C, as a diameter, describe the semi-
circular arc ABC. Produce C A to B, and make
A F equal to the development of the arc ABC.
Draw F G parallel to A E, and E G parallel to
A F ; then A F G E is the development required.

PROBLEM II.

To find the development of that part of a semi-
cylinder contained between two perpendicular
surfaces.

Figs. 2, 3, 4. Let A B C D E be a portion of a
plane passing through the axis of the cylinder, C D
and A E being sections of the surface, and let D E
and F G be the insisting lines of the perpendicular
sTirface ; also, let A C be perpendicular to A E and
C D. On A C, as a diameter, describe the semicir-
cular arc ABC. Produce C A to H, and make
A H equal to the development of the arc ABC.
Divide the arc ABC and its development each

into the same number of equal parts, at the points
1, 2, 3.

Though the points 1, 2, 3, &c., in the semicircular
arc, and in its development, draw straight lines par-
allel to A E, and let the parallel lines through 1, 2, 3,
in the arc ABC, meet F G in p, rj, r, Sec, and A C
in k, I, m, &c. Transfer the distances Icp, Iq, vir, &c.,
to the development upon the lines 1 a,2 b,3 c, &c.
Through the points F, a, b, c, &c., draw the curve
line F c 1. In the same manner draw the curve line
E K ; then F E I K will be the development re-
quired.

PROBLEM III.

To find the development of the half surface of
a right cone, terminated by a plane passing
through the axis.

Fig. 5. Let A C E be the section of the cone pass-
ing along the axis A E, and C E the straight lines
which terminate the conic surface, or the two lines
which are common to the section C A E and the
conic surface ; and let A C be the line of common
section of the axal plane and the base of the cone.

On A C, as a diameter, describe a semicircle,
ABC. From E, with the radius E A, describe the
arc A F, and make the arc A F equal to the semi-
circular arc ABC, and join E F; then the sector
A E F is the development of the portion of the
conic surface required.

PROBLEM IV.

To find the development of that portion of a
conic surface contained by a plane passing
along the axes, and two surfaces perpendic-
ular to that plane.

Fig. 6. Let A C E be the section of the cone
along the axis, and let A C and G I be the insisting
lines of the perpendicular surfaces. Find the devel-
opment A E F, as in the preceding problem. Divide
the semicircular arc ABC, and the sectorial arc
A F, each into the same number of equal parts at
the points 1, 2, 3, &c. From the points 1, 2, 3, &c.,
in the semicircular arc, draw straight Lines, 1 k, 2 /,
3 m, &c., perpendicular to A C. From the points
k, I, m, &c., draw straight lines, A; E, Z E, m E, Sec,
intersecting the curve A C in j?, q, r, &c. Draw the
straight lines pt, qu, rv, &cc, parallel to one side, E C
meeting A C in the points t, u, v, Sec Also from the

64

PROJECTION OF PRISMS.

points 1, 2, 3, in the sectorial arc A F, draw the
straight lines 1 E, 2 E, 3 E, &c. Transfer the dis-
tances pt, qii, rv, &c., to 1 a, 2 b, 3 c, &cc. ; then,
through the points A, a, b, c, &c., draw the curve
A c F, and A c F is one of tlie edges of the develop-
ment, and by drawing the other edge, the entire de-
velopment, A G H, will be found.

Note. â€” This treatise on the subject of Geometry we have
thought best to insert in the form in which it was fiist written. It

is a system in a degree peculiar to its author, and it is, without
doubt, the production of a laborious research ; and although the
same conclusions would, in some cases, be arrived at by the more
direct process of demonstration used at the present day, yet, from its
extent and completeness, we have concluded, as a whole, that no
part could be materially changed for the better without seriously
intruding upon the theories of our venerable author, and that, too,
at the expense of interfering with liis style of writing, wliich will
at once be recognized as the original. We will state here, that we
have endeavored, as much as possible, to preserve every idea that
he has advanced, and that, toO; in his own language. â€” Editors.

PROJECTION or PRISMS.

In the annexed definitions and problems, the student will find
enough to give him a correct and sufficiently perfect idea of the
nature and importance of this useful branch of science ; and he
will also find occasion to apply the geometrical principles, a knowl-
edge of which he is presumed to have acquired from the preceding
pages of the present work.

DEFINITIONS.

1. When straight lines are drawn according to a
certain law from the several parts of any figure or
object cut by a plane, and by that cutting or inter-
section describe a figure on that plane, the figure so
described is called the projection of the other figure
or object.

2. The lines taken altogether, which produce the
projection of the figure, are called a syatem of rays.

3. When the system of rays are all parallel to
each other, and are cut by a plane perpendicular to
them, the projection on the plane is called the orlhog-
raphy of the figure proposed.

4. When the system of parallel rays is perpendic-
ular to the horizon, and projected on a plane parallel
to the horizon, the orthographical projection is then
called the ichnography, or plan of the figure proposed.

5. When the rays of the system are parallel to
each other and to the horizon, and if the projection
be made on a plane perpendicular to those rays and
to th(! horizon, it is called the elevation of the figure
proposed.

[In this kind of projections, the projection of any
particular point or line is sometimes called the seat
of that point or line, on the plane of projection.]

G. If a solid be cut by a plane passing quite

through it, the figure of that part of the solid which
is cut by the plane is called a section.

7. When any solid is projected orthographieally
upon a plane, the outline or boundary of the projec-
tion is called the contour or profile of the projection.

Note. â€” Although the term orthography signifies, in general, the
projection of any plane which is perpendicular to the projecting
rays, without regarding the position of the plane on which the ob-
ject is projected, yet writers on projection substitute it for elevation,
as already defined, by which means it will be impossible to know
when we mean that particular position of orthographical projection
which is made on a plane perpendicular to the hori/on.

Axiom. â€” K any point, line, or plane of any origi-
nal figure or object touch the plane on which it is
to be projected, the place where it touches the pro-
jecting plane is the projection of that point, line, or
plane of the original figure or object.

Proposition. â€” The orthographical projection of a
line which is parallel to the plane of projection is a
line equal and parallel to its original.

PROBLEMS.

Plate 16.

PROBLEM I.

To project the elevation of a prism standing on

a plane perpendicular to the projecting plane ;

giA'en, the base of the prism and its position

to the projecting plane.

Fig. 1. Let A B C D, No. 1, be the base of the
prism ; let H F be the intersection of the projecting
plane, with the plane on which the prism stands.

Draw lines from every angle of the base, cutting

T K

fw /.]â– â– â– /

PEOJECTION OF PRISMS.

65

H F at H, and F will be the projection of the points
A and C ; the angle D, touching H F at D, is its
projection.

From each of the points H, D, F, in No. 2, draw
the lines H I, D E, and F G, each perpendiciilar to
H F ; make D E eqnal to the height of the prism ;
through E di'aw I G, cutting H I and F G at I and
G, which will give the projection sought.

PROBLEM II.

To project the ichnograpliy and elevation of a
square prism, to rest upon one of its angles
upon a given point A, in the plane, on which
tlie ichnography is to be described ; given the
ichnograpliy A L, of an angle, wliich the two
under planes make with each other; the
angle ]\I a I, which the angle of the solid
makes with its ichnography A L ; the inter-
section A a of one of its ends with the plane
of the ichnography ; the angle D A Â«, which
one side of the end makes at A, with the in-
tersection A a of that end ; also given one of
the sides of the ends, and the length of the
prism.

Fig. 2. At the given point A, with the intersection
A a, make an angle a AT>, equal to the angle which
one of the sides of the end makes with A a; make
A D equal to one of the sides of the end; then on
A D consti-uct the square A B C D ; through the
angles of the square B, C, D, draw lines B H, C I,
and D M, parallel to A L ; then at the point a, in
the right line D M, make an angle M a 1, with a M,
equal to the angle of the solid, whose projection is
A L, with A L ; make a I equal to the length of the

9

solid ; through the points a and /, No. 1, draw the lines
a e and I i perpendicular to a I ; through the points
B and C, No. 2, draw B R and C S parallel to A a,
cutting D M, produced at E, and S ; on a, as a cen-
tre, with the distances a D, a R, and a S, describe
ares V g, R/, and S e, cutting a e, No. 1, at g-, /, e ;
through the points g, f, e, draw the lines g- k, f h,
and e i, parallel to a I, and No. 1 will be completed,
which will be the projection of tlie prism on a plane
parallel to A L. Through the points g, f, e, draw
the lines e E, / F, and g G, perpendicular to D M,
or A L, cutting D E, C I, and B H, respectively, at
G, E, F; also through the points I, k, h, i, cbaw the
lines I L, k K, h H, and i I, Hkewise parallel to A L,
cutting A L, G M, B H, and C I, respectively, at the
points L, K, H, and I ; join E F, E G, and H I,
I K, K L, L H, then will the planes E F H I,
E I K L, and H I K L represent the ichnogi-aphy
of the upper sides of the solid ; and if F A and A G
be joined, then will F A G E, F A L H, and G A
L K represent the sides of the solid next to the
plane of projection. Then to project the elevation
on a plane whose intersection is T U, from F, E, G,
A, H, I, K, and L, that is, from all the points in the
ichnography representing the solid angles, di-aw the
lines F /, E e, G^, A o, H h, I i, K k, and L I, per-
pendicular to the intersection T U, cutting T U at
P> 5') (^> Â§"> o, m, and k ; make p f^ q e, g g,n h, o i,
m I, and k k, at No. 3, respectively, equal to P/, Q e,
G g-, N A, O i, M I, and K k, at No. 1 ; then join
f a, a g, g e, e f; e i, i k, k g, k I, and la; and
fage, geik, gkla will be the elevations of the
outside planes of the solid; and by joining/ A, and
h i,fh i e,fh I a, and i h I k will be the elevations
of the planes of the solid next to the plane on which
the elevation is projected.

66

This' is one of the most interesting branches of architectural sci-
ence I or perhaps it may, with more propriety, be termed a branch
of geometry, for it is almost entirely dcpenilent on, and governed
by, geometrical principles.

From a knowledge of it the architect is enabled to draught his plans,
and to give them their true effect, or representation of light and
shade ; to construct his windows in order to receive light to the best
advantage, &c., &c. The art of keeping a proper gradation of light
and shade on objects, according to their several distances, colors,
and other circumstances, is of the utmost consequence to the artist.

THE EFFECT OF DISTANCE ON THE
' COLOR OF OBJECTS.

The art of giving a due diminution or degradation
to the strength of the light and shade and colors of
objects, according to the different distances, the quan-
tity of light which falls on their surfaces, and the
medium through which they are seen, is called
keeping;.

1. When objects are removed to a gi-cat distance
from the eye, the rays- of light Avhicli they reflect
will be less vivid, and the color will become more
diluted, and tinged wiih a faint bluish cast, by rea-
son of the great body of air through which they are
seen.

2. In general, the shadows of objects, according as
they are more remote from Hie eye, will be lighter,
and the light parts will become darker ; and at a
certain distance the light and shadow are not distin-
guishable from each other, for both will seem to ter-
minate in a bluish lint of the color of the atmosphere,
and will appear entirely lost in that color.

3. If the rays of light fall upon any colored sub-
stance, the reflected rays will be tinged with the color
of that substance.

4. If the colored rays be reflected upon any object,
the color of that object will then be compounded
of the color of the reflected rays and the color
of the object ; so that the color of the object which
receives the reflection will be changed into another
color.

5. From the closeness or openness of tlie place
where the object is situated, the light, being much

more variously directed, as in objects which are sur-
romidcd by buildings, will be more deprived of re-
flection, and, consequently, will be darker than those
which have no other objects in their vicinity, except
the surrounding objects are so disposed as to reflect
the rays of light upon them.

6. In a room, the light being more variously di-
rected and reflected than abroad in the open air, (for
every apertm-e gives an inlet to a different stream,)
which direction is various, according to the place and
position of the apertm-e, whereby every diflcrent side
of the room, and even the same side in such a situa-
tion, will be variously afTected with respect to their
light, shade, and colors, from what they would in an
open place when exposed to rays coming in the same
direction.

Some original colors naturally reflect light in a
greater proportion than others, though equally ex-
posed to the same degrees of it, whereby their degra-
dation at different distances will be different from
that of other colors which reflect less light.

on objects, according to their several distances, colors,
and other circumstances, is of the utmost conse-
quence to the artist.

In orthographical projections, where equal and
similar objects stand in the same position to the
plane of projection, they will be represented similar,
and of an equal magnitude at every distance from
that plane ; and, consequently, planes wliich are par-
allel to each other would not appear to have any dis-
tance, so that the representation of any number of
objects, at different distances from each other, would
be entirely confused, and no particular object could
be distinguished from the others ; but, by a proper
attention to the art of kccpitig;, every object will be
distinct and separate, and their respective distances
and colors from each other will be preserved. But
to be preserved, according to the respective distances
of objects from each other, artists in general take too
gi"cat liberties with nature : we frequently see in the
drawings of architects the art of keeping carried to
so great an extreme as to render their performances
ridiculous.

67

DEFINITIONS.

1. A body which is continually emitting a stream
of matter from itself, and thereby rendering objects
visible to our sense of seeing, is called a luminary;
such as the sun, or any other body producing the
same effect.

2. The stream of matter which is emitted from the
luminary is called light.

3. A substance or body which light cannot pene-
trate is called an opaque hodij.

4. If a space be deprived of light by an opaque
body, it is called a shade.

5. The whole or part of any sm-face on which a

6. A body which will admit of light to pass through
it is called a transparent substance.

7. A line of light emitted from the luminary is
called a ray.

Proposition 1. â€” The rays of light, after issuing
from the luminary, proceed in straight lines.

Proposition 2. â€” If the rays of light fall upon a re-
flectiiig plane, the angle made by any incident ray,
and a perpendicular to the reflecting plane, is called
the angle of incidence, and â– wHl be equal to the angle
that its reflected ray will make with the same per-
pendicular, called the angle of reflection ; these two
propositions are known lirom experiment.

Proposition 3. â€” If the rays of light fall upon any
curved surface, whether concave or convex, or mLxed
of the tvs^o, the angle of reflection will stiU be equal
to the angle of incidence.

Proposition 4. â€” Any uneven reflecting surface,
whose parts lie in various directions, will reflect the
rays of the sun in as many different directions.

Demonstration. â€” If any ray fall upon a part of
the surface which is perpendicular to that ray, it will
be reflected m the same line as the incident ray ; but
the more or less any part of the surface is inclmed to
a ray, falling upon that part of the surface, the greater
or less angle will the reflected ray make with the in-
cident ray. For imagine a perpendicular to be erected
to that part of the surface where any incident ray
impinges on the surface, it is evident that the meas-
ure of the angle of incidence is equal to the obtuse