PLATE 9.
Vol. 12.
PLATE 10.
VoL 12.
PLATE 11.
Vol. 12.
PLATE 12.
Vol. 12.
COM PrO UND BUGGY. — £ in. scale
Designed expressly for the New York Coach-maker's Magazine.
Explained on page 41.
ROAD PHAETON. — I in. scale.
Designed expressly for the New York Coach-maker's Magazine.
Explained on page 41.
DEVOTED TO THE LITERARY, SOCIAL, AND MECHANICAL INTERESTS OF THE CRAFT
Vol. XII.
1STEW YOEK, AUGUST, 1870.
No. 3
^iterator*.
TREATISE ON THE WOOD-WORK OF
CARRIAGES.
{Continued from page 21.)
Having the projections of the triangle on a plane S,
perpendicular to the axis of deployment, the question is
therefore reduced that we have solved (art. 74). When
the triangle is turned down upon the horizontal plane, the
arcs described by each of the angles (a a,), (c c¦), b b;),
pierce that place in points, whose vertical projections on
the plane S are a///,-cl//,d///,where the arcs described with
point el as center, and e/ an e, c(, e, bf, as radii, meet M N.
The horizontal projections of the same points are, for the
angle (a a/), the point A, the intersection of the lines a A
and an/ A/? the first being perpendicular to the axis d e,
and the second perpendicular to the intersection M N.
The same refers to the other angles, the arcs of which
pierce the horizontal plane in B and C. On joining the
points A, B, and C by the lines A B, A C, and B C, the
triangle ABC formed by them is the triangle sought for.
LXXIX. Instead of turning over the triangle (a b c,
a1 b1 c') on to the horizontal plane, it can be brought into
a parallel position to the first vertical plane Q, by two
rotary movements, and the new projections that will re-
sult on that plane can be constructed. The first rotary
movement, for instance, could be executed around a ver-
tical axis of the point d, until the line d e be brought par-
allel to X Y. In this movement, all the horizontal pro-
jections would preserve their relative positions. The
second movement would take place around d e, until the
triangle be brought into a vertical position. Then, in
order to construct the length of the radii of the arcs from
each point taken, it would require a plane perpendicular
to the axis, or deduct the lengths of the two projections of
each radius in like manner as that adopted hereafter in
reference to a line. This second method, by which to
construct the triangle in its full size, would be longer than
the first, because there would be an additional rotary mo-
tion. Therefore there would be two projections on the
horizontal plane, the projection a b c, and another after
having brought the line d e to bear parallel to X Y; there
would likewise be an auxiliary plane S, and two projec-
Vol. XII.—5
tions on the vertical plane Q; firstly, a1 b'c1, which al-
ready exists, and secondly, that which must be constructed
when the triangle is brought to bear parallel to that plane :
in all, five projections ; whereas, by deploying the trian-
gle on to the horizontal plane, there would be but four.
The constructions requiring the fewer projections must
in all cases be preferred, because, however great may be
the precision and attention exercised, in practice there are
at all times causes for error, either arising from the imper-
fection of the instruments or the physical means that are
employed. It therefore naturally follows that the causes
of error are multiplied in proportion to the number of
projections.
LXXX. Observations.—The table of all the graph-
ical constructions that will be presented by the operations
being for the purpose of determining in their size, the lines,
surfaces, and dihedral angles, will only offer a repetition
of the constructions that we have just executed in reference
to a triangle (art. 72 to 79).
According to the position of the triangle, or of any
other.plane surface under consideration, in respect to the
planes of projection, the operation being for the purpose of
bringing that surface either parallel to one of the two
planes of projection, or on one of those planes, it can be
carried out: firstly, by turning over or deploying (art.
74, 75, and 76); secondly, by a rotary movement (art.
77); thirdly, by a change of the plane of projection, and
turning down (art. 78); fourthly and lastly, by two rotary
movements and a change of the plane of projection (art.
79).
LXXXI. Deploying indicates the operation+hat con
sists in moving the surface considered in space, in order
to bring it to bear upon one of the two planes of projec-
tion. Then the axis of deployment is the common inter-
section of the plane of projection and the surface in space,
extended if necessary.
According to the position of the surface in space, the
axis of rotation can occupy four different positions on one
of the two planes of projection, in reference to their com-
mon intersection : it can be confused with that line, paral-
lel, perpendicular, or oblique to it.
LXXX1I. Under the name of axis of rotation, it is
more particularly designated a line perpendicular to one of
the two planes of projection, and passing through one of
the extreme points of the surface in question. Hence it
follows, that if the axis is perpendicular to the horizontal
34
THE NEW YORK COACH-MAKER'S MAGAZINE.
August,
plane, for instance, each point of the surface, in turning
around that axis, describes a horizontal arc, which is pro-
jected with its radius in full size on the horizontal plane,
and on a line parallel to the ground line, on the vertical
plane, and reciprocally. Then it is not at all necessary
to construct the two projections of the axis of rotation.
It suffices to indicate on the plane of projection perpen-
dicular to the axis the projection of the point through
which it passes. Therefore, in the article 79, it would
have sufficed to state that the triangle is turned around a
vertical axis of point c.
The axis of rotation furthermore infers a line parallel
to the intersection of the two planes of projection, and
taken on one of those planes, around which a surface in
space is moved, in order to bring in a position parallel
to the other plane of projection.
LXXXIJI. In reference to the operation having for
its object the bringing over of a surface on to a plane of
projection, or in a position parallel to one of those planes,
which is effected by deploying, by a rotary motion, or by
both, it is necessary to remark:
Firstly. That every point of the surface in question,
in turning around a fixed axis, describes a circumference
or an arc, the plane of which is perpendicular to the line
taken for the axis of rotation.
Secondly. That the circumference or the arc described
by each point is projected with its radius in their full
size, on a plane perpendicular to the axis, and by a line
perpendicular to the axis, on any plane passing through
that axis.
Thirdly. That the center of the circumference, or of
the arc described by each point, is included in the axis, or
in its extension, at the intersection of that axis and the
plane of the circumference or the arc.
length of a line suffices to secure the size of the plane sur-
face to which that line belongs.
To. find the Length of a Line, the two Projec-
tions of which are given.—If the proposed line is paral-
lel to one of the two planes of projection, its length is
determined by its projection on that plane (art. 52). Let
A B (fig. 54) be the proposed line, and parallel to the
vertical plane Q. Its projection a'b1 on that plane will
be equal and parallel to it. It is acknowledged that a line
LXXXIV. Operations on Straight Lines.—Opera-
tions on straight lines in space are for the purpose of fix-
ing their length. Frequently the determination of the
A B is parallel to one of the two planes of projection Q
when its projection a b, in the other plane P, is parallel td
the ground line X Y, so as to have a a0 equal to b b0. The
figure 54 is in perspective, but in that respect we give fig-
ure 55, that presents the same given points, with the
points marked with the same letters, on geometrical
planes. The line A B is represented on those two planes
by its two projections (a b, a' bJ).
If the line is inclined in any manner whatever, in re-
spect to the planes of projection, its length is greater than
that of each of its projections, which in that case are
oblique to the ground line. The length of the line is re-
duced by the aid of its projections, by one of the con-
structions effected later, the solution of which we give
below. &
Solution.—The proposed line, with its two projections
on either of the two planes of projection, and its projec-
tion on that plane, form a rectangular trapeze determined
by the two planes of projection. One of those planes
contains the projection that forms the basis of the trapeze,
and the other the length of the two adjacent sides that,
together with the basis, form the two right angles of the
trapeze ; the fourth side expresses the length of the pro-
posed line. It therefore remains but to construct that
trapeze on either of the planes of projection.
Suppose A B (fig. 56) to be the proposed line pro-
jected in perspective on two planes, one of which P is
supposed horizontal, and the other Q vertical. Let us
first consider the vertical trapeze formed by the line A B
by its two projectants A a, B b, and by its projection a b
1870.
THE NEW YORK COACH-MAKER'S MAGAZINE.
35
on the horizontal plane. The trapeze is now figured in
space, while it is only represented on the planes of projec-
tion by three of its sides: the horizontal projection a b,
which forms its basis, and the two adjacent sides to that
basis, the lengths of which are expressed by the lines
a'a0, bfba, lowered from the extremities arand ft/of the
other projection perpendicular to X Y, which (art. 46)
are respectively equal to the projectants A a and B b, each
one measuring the distance from the extremities of the
line A B to the horizontal plane.
In order to construct the trapeze on that plane, indefi-
nite perpendiculars a- A; and b B'are drawn to the pro-
jection a b, by its extremities a and b, on which the dis-
tances a'awb'ba of the plane Q are measured off from a
to k' and from b to B>. The line A'B'drawn by th%
points Ayand B/expresses the length desired. In reality,
the trapeze A'a bB', constructed on the horizontal plane,
is equal to the trapeze A a b B in space, having a common
basis a b, the two sides A'a, B'b respectively equal to
the two sides A a, B b, and perpendicular to the basis, the
fourth side of which A'B'is equal to A B.
The figure 57, on geometrical planes of projection,
expresses the construction indicated on the figure 56 in
perspective, with the same points marked with the same
letters. Here the line in space is represented by its two
projections (a b, a! b1), and the sides of the right an-
gles of the vertical trapeze that passes that line, and by
its horizontal projection a b, are represented by the ver-
tical lines a' a0, bJb0, which are respectively carried from
a to A' and from b to B;.
LXXXV. If the horizontal plane P (fig. 56) be ele-
vated until it touches the line A B at its lower end A, the
new horizontal projection of the line will be found trans-
ferred to A c parallel to a b, and the ground line in a'c'
parallel to X Y. Then the line A B is equal to the hy-
pothenuse of a rectangular triangle A c B, the sides of the
right angle being the horizontal projection A c and the
elevation Be or b' c' of the other extremity of the line
above the horizontal plane.
To construct this triangle on the horizontal plane, it
will be observed that the projectien a b being equal to
A c, one of the sides of the right angle, it suffices to draw
a pcrpenaicul.ir b O'io inat une. by one of ihe extremities
b, and to carry over on to that perpendicular the other
side of the right angle B c or b1 c' from b to D. The line
a D'thnat joins the points a and D;is the hypothenuse of
the triangle a b D;, equal to the triangle Ac B. The
same construction is made on the geometrical planes of
figure 57. •
The two planes of projection being rectangular, the
operations that have been effected on one (figs. 56and 57)
could have been carried out upon the other, and would
have yielded the same result. In general, whatever may
be the plane of projection considered, the length of a line
in space (a b, a b') (fig. 58), when oblique to two planes
.of projection, is equal to the hypothenuse A'b of a rectan-
36
THE NEW YORK COACH-MAKER'S MAGAZINE.
August,
gular triangle A a b, of which the sides of the right angle
are : 1°. The projection a b from the line on to one of
the two planes of projection; 2°. The difference a' c'
of the distances of its extremities a' a0, b' b0, to that
plane.
If the vertical projection a/6/of the line is taken for
one of the sides of the right angle, the other side will be
the difference d b of the distances of its extremities 6 b0,
a