1. science
2. mathematics
3. geometry

HOW
TO
DRAW
LECTURE
A
ON
STRAIGHT
LINKA.GES.
LINE;
'^
NATURE
HOW
SERIES.
TO DRAW
A STMI6HT
LECTURE
LINKAGES.
ON
B.
A.
LINE;
KEMPE,
B.A.,
r"
OF
UBUBER
AND
THE
THE
COUNCIL
LATE
SCHOLAR
or
WITH
NUMEROUS
INHEK
OF
OF
THE
TBHPLB,
LOKDON
TBINITIT
ESQ. ;
MATHEMATICAL
OOLLEOE,
SOCIETY
;
CAMBRIDOB.
ILLUSTRATIONS.
AND
MACMILLAN
CO.
1877.
is Reserved.]
[The Sightof Translal-ton and lieprcdwdicn
LONDON
lY,
SONS,
BREAD
QUEEN
AND
:
TAYLOR,
STREET
VICTORIA
HILL,
STREET.
PRINTEE8,
K4-
NOTICE.
This Lecture
teachers last
was
of the seriesdelivered to science
one
connection
in
summer
Collection of Scientific
Apparatus. I
opportunityafforded by
enlargeit and
I
me
to the
undertaken
its
in
and
drawingthem
by
able and
whose
Loan
the
Office
Collection
deliveryof
the
slightly
tions
For the illustra-
I
models furnished
could
hardly have
the Lecture,and
Row, Temple,
M510957
taken
in
co-operation
indefatigable
the
constructing
January 16tA,1877.
have
Loan
brother,Mr. H. E. Kjimpe,
publication.
1, Crown
the
to
publication
several notes.
indebted to my
am
without
to add
its
with
still less
HOW
TO
DEAW
A
STRAIGHT
LECTXJEE
ON
LINKAGES.
LINE;
DRAW
TO
HOW
LECTURE
A
The
various
the
us
These
processes.
termed,
be
is
great
any
straight
lines
example,
effected
be
have
no
these
circles
as
the
trisection
the
by employing
geometrical
by straight
It
construction
circles.
and
becomes
lines
preliminary
and
and
then
these
physical
to
many
of
angle
circles
to
requirements,
circumstances
which
effected
be
which
such
"
as,
said
are
be
cannot
by
readily
can
means,
they
the
to
solved
only.
interesting
straight
master-
refuse
problems
simple
since
solution,
would
"
are
And
this
to
can
Hence
other
certain
should
we
demonstration
a
than
an
effect
of
circles.
paid
who
many
Elements
that
and
is
to
requirements
lines
that
his
to
demand
straight
are
able
the
to
"geometrical"
other
requires
demonfitrating
in
be
as
said
there
that
of
designation
for
be
veneration
the
geometrician,
should
we
describe
to
before
contained
Postulates,
roughly
may
able
so
that
LINE:
LINKAGES.
Euclid,
propositions
requires
Geometry,
ON
geometrician
great
STRAIGHT
A
inquire
how
lines,
of the
how
we
can
with
as
we
can
describe
much
problems
will
effect
these
accuracy
admit
B
of"
HOW
2
As
DRAW
TO
A
circle we
regardsthe
Euclid's
and
definition,
that
surface
our
plane,(1)'we
a
pointpreserve
a
and
constant
be effected
pieceof
the
hole in the
a
distance
givenradius
shall
with
even
we
this rude
get
result
a
given
first is
the
; and
or
pivots,
when
truth which
the
pairof
a
the
Euclid
defines it
points." This
is
a
with
straightedge;
Now
We
These
are
be
pencil,
and
you
a
we
mechanical
back to
we
our
to
clearly
refer to
figures
Notes
shall
atus
apparThe
nothing
course
it is usual
and
but
to
a
say
compasses.
going to describe
its extreme
postulatea
going
we
must
to
that ]
text-books say
Our
ruler,the ruler
are
with
of its movement.
much.
Postulates
how
siderable
con-
employ
to
come
lyingevenlybetween
help us
able*
circle with
beggingthe question.If
come
I wish
^
we
the
postulates
second
But
surelythat
straightline
"
as
does not
that the firstand
?
straight
compasses,
line,how
straight
a
the
equal to
lathe affords,
we
and accuracy
that the third Postulate
But
the
unequalledperhaps among
of
througha
largeones, turned
even
apparatus I have justdescribed is of
simpleform
the
as
pencilthrough another
or
from
ease
for the smoothness
readily
can
centre
apparatus,to describe
very small holes and
all that marvellous
This
form, such
then, by moving
and
accuracy
of the circle
givencentre
at the
tracer
hole in it whose
;
the
tracing-
our
here, and passinga pivotwhich
given surface
piece,and
must,
we
make
requiredradius.
I have
cardboard
course
only to
flat pieceof any
by taking a
is fixed to the
have
distance from
equalto
of
wish to describe the circle is
we
we
Taking
difficulty.
no
assuming,as
that
see
LINE:
encounter
which
on
STRAIGHT
are
ruler
(2).
to draw
itself have
make
the
a
edge
starting-point.
understand
the
difference
at the end of the lecture.
A
between
the method
circle,and
If I
the
take
I
ON
circular
circle by passingthe
my
have the
for I should
first have
then
between
I
that
aware
do
we
such ; but
used
to describe
want
as
size),
remain
for
can
at the
be
beggingthe
no
I
circle and
a
the distance
of
am
course
simplecompass,
our
the
of
are
the
it fits are
they are
so, but
not
forced to
curves
different
a
spot. If
same
even
when
get results
be
in the machine
made
which
we
by
the
makes
the tracer
of
pivot
in
error
will be
can
those
to
holes
pivotsand
of very
employment
the
be made
employ largepivotsand
small
ones
them.
have
we
a circle,
describing
means
corresponding
small
very
perhaps varyingbeyond
then, that althoughwe
of
moving piece
dimensions
accurate, because
as
stitute
best sub-
the
truly circular,
not
circle of moderate
a
the
employ a
we
widlth of the thinnest line which
describe ; and
adopt the
point in
one
not
infinitesimal,
practically
method
in
because
not
making
of
description
It appears
itself circular..
moving piecewhich
are
the
holes may
straight-edge,
I have
invariable.
I should
the
hole, though they be
shall
the
simplyrequirethat
with
case
and
we
with
that
(theyare
finite dimensions, we
we
edge,and
trace
but because,
straight-edge,
of constructing
pivotsor holes of
impossibility
is the
through the
no
the
and
penny,
the lamina
employ circles
they are
a
employed involves
the hole in the
pivotand
I had
make
pointsshall
two
as
pencilround
it to trace one, but
use
to
firstassume
do not
I
question.
describinga straightline.
the description
of a circle,
lamina, such
to
the other method
But
of
that
difficulty
same
3
employed for describinga
ruler method
a
LINKAGES.
justnow
ruler method
appliedthe
I should
we
LECTURE
describinga
an
easy and
rate
accu-
have at firstsightno
straightline;and
B
2
TO
HOW
4
would
there
DRAW
to be
seem
questionhow
to
get
LINE:
substantial
a
call the
mathematicians
what
STBAmHT
A
simplestcurve,
the
that
so
becomes
difficulty
that
over
in producing
difficulty
of
one
a
decided theoretical interest.
of direct
importanceto
largenumber
of machines
one
a
that
requisite
in
a
the
the
with
is adopted,and
principle
In
scientific apparatus it is
and
littlefriction
as
questionis
mechanician.
practical
pointor points should
some
line
straight
ruler
only,for
is the interest theoretical
Nor
the
accurately
move
the
possible.If
as
pointis kept in
its
path
of making
have, besides the initial difficulty
by guides,we
the
the
guidestrulystraight,
the
friction of the
and
wear
tear
slidingsurfaces,and
produced by
the deformation
produced by changes of temperature and varying strains.
It
therefore
becomes
method
some
possible,
of
which
features, but
of
shall not
possess
the
obtain, if
to
consequence
characterise
which
movement
real
involve
these
accuracy
and
jectionable
obease
circle-producing
our
apparatus.
Turning
to
requisite
to
that
draw
radius is to have
tracer
apparatus, we
notice that
with
a
the distance
first
between
the second
radius
the firstcircle.
third
at the
at
any
if I
second
pivota
given
any
pivot and
the
second
point having
the
"
"
piece
tracer
a
is
as
the distance
piecehas-,by properlydetermining
circle whose
a
a
circle of
between
properlydetermined,and
to the fixed surface at
the
accuracy
all that
pieceto
and
I
pivot,
can
describe
second
a
that of
proportionI pleaseto
Now, removing the tracers,let me
bears any
these two
pointswhere
pointon
tracer
radial
the tracers
this third
or
as
pieces,
were,
I may
and let
me
pivot
call them,
fix
a
traversvngpiece. You
tracer
will
A
at
once
that
see
would
tracer
LECTURE
in the
LINKAGES,
5
if the I'adial
pieceswere
big enough the
describe circles or
though theyare
as
ON
in
motion,with the
of the
case
will not
but
curve.
complicated
This
and
however
describe
and if I wish to
I
which
in
reproduce
the
pivotsand
in both cases, and the
curves
I could of
on
course
go
get pointson
to accuracy
curve
certain
These
a
tracers
will also be
all the
the circles
second
ease
scribed,
de-
were
apparatusthe
and
the
accurately
the
accurately
the structure
but with
the
any
the correct determination
on
dependingsolely
same.
ticular
par-
of
a
of distances.
systems, built up
that the various
same
results
same
smoothness, the reproduction
of
number
definite
the
in
produced,describing
of
piecespointed or
and turningabout pivotsattached
together,
so
fixedsurface,
and
adding fresh piecesad libitumy
generalvery complicated
curves,
as
the
; the
produce with this,I have only to get
distances between
I should
circle on
a
with
of movement
which
theniy
and accuracy
salaie ease
will, however, be described with
curve
accuracy
CTU^es
circles on
simplecircle-drawing
apparatus
tracer
a
portionsof
pointson
the
to
a
pivoted
fixed base,
piecesall describe
definite
HOW
6
links."
STRAIGRT
A
propertiesof
them
apart from the base
word
"
"
is
linkage
these
which
to
employed to
denote
pivotedin
any
to
way
fixed
a
such
base,the
confined
being necessarily
is called
structure
of every
the motion
before
to these
which
point is
stated,termed
a
two
expressions
describes
being called
"
a
"
occupiedmuch
is
do not,
however, propose
shall have
quiteenough
That
be
not
they would
have
had
that modem
of
production
true
structures,cannot
have
not
at
to do if
only add
a
link-motion
curve
late years
much
among
of
side
deal
confine
we
ages"
link-
"
complexityand
mathematical
to
these
the
to-day,aid we
our
attention to
only mathematicians
could
have
results
cannot
well be that
may
to them
; and
a
valuable
are
value
their
I think
great beauty
be that
fifty
years
which, through the great
mechanicians
and
be ascribed
present,I think, been
tained.
ob-
importancewhich they
an
it may
planes,rulers
now
path being,
obtained,
these
reallypossess
in which
have
to attribute
some
''
mathematicians
doubted, though it
has led
do
purely
which
I believe
which
link-
paths,the
graph,"the
of
subjectof
a
questionI
and
pointson
I shall
"
a
attention
the
results
practical
of
of
properties
difficulty.With
the
is
point of
is called
of the various
mathematicians,and
combination
definite
some
the
:
of
(3).
gi*am
The consideration
has
more
combination
motion
link-motion."
"
curve
any
in
pivoted,the
linkwork
"
a
:
a
fixed
to
*'
linkwork
"
a
any
sideration
con-
regard
to
structures
they are
pieces pivoted together. "When
as
facilitates the
As, however, it sometimes
of the
it not
LINE:
"link-motions,"the pieces
beingtermed
I shall term
curves,
**
DRAW
TO
have
ago
provements
im-
effected in the
other exact mechanical
to them.
But
linkages
sufficiently
put
befoie
LECTUBE
A
the mechanician
be
i-eally
set
but few in
number,
plunginginto
we
we
require;
and
"
what
say
by
the
value
should
of
linkages
use
connected
closely
are
to consider them
of my
that
here is
get
results
our
'^
I
straight-
Before,however,
linkagesit
of the great
one
fact been
if I make
lecture.
of these
in
the
with
problem,and
construct
practically
can
can
we
will be useful
models
such
as
advantagesof
visiblybefore
us
so
our
very
for fixed
easily.Pins
for the other
cards for links,string
or cotton
pivots,
and a dining-room
pivots,
table,or a drawing-
if the former
board
and
midst
the
how
subject
obtained
the backbone
line motion"
7
motion/' having
straight-line
'^
led
naturally
know
to
us
of
duringthe investigation
discovered
shall be
LINKAGES,
them.
upon
problem of
to
enable
to
results
practical
The
are
ON
all
be
for
thought objectionable,
fixed
a
artistic be
require. If something more
the planadopted in the models exhibited by me
preferred,
Collection can be employed. The models were
in the Loan
constructed by my brother,Mr. H. R Kempe, in the following
base,
are
hard
very
ale
work
about
aveiy
are
thin
steel chisel
neatlyshaped
making them,
ten
minutes,
being formed
on
plate;
the
pivots;
at any
you
can
to
cardboard
knife
tunjs the
littlerivets made
of
face of
a
;
(itis
sharpenyour
the cardboard
edge
catgut,
heated
gut after it is passedthrough
gives a
durable
very
fiirm and
links may
in this
bootmakers
get the proper
largetool shop.
painted black
of thick
by pressingthe
must
pivot-holes
eyeletsused by
boards
have
the ends of the
working joint. More
as
out
you
as
the holes in the links ; this
the
deal
; the pivotsare
rapidly)
heads
the
bases
The
way.
the links
we
case
be
be
made
a
of tin-
punched, and
for laced boots
tools at
smoothly-
employed
trifling
expense
HOW
8
Kow,
TO
varioTis
have
I
as
But
complex.
they
them
described
link-motions
in
are
at
But
generalvery
we
perly
pro-
and get a pointon one of
simplicity
in a straight
line 1 That is what
accurately
can
fullest
the
to
go
of
extent
By
disposalwe
our
can
the
by
so.
necessarily
not
are
simple.
very
LINE:
curves
distances
the
choosing
make
said, the
these
pointson
STRAIGHT
A
DRAW
moving
them
going
are
we
investigate.
to
the
solve
To
: all the pointson
impossible
therefore go to the next
In this
case
distance
will
you
between
singlelink
is
it describe circles.
We
with
problem
simplecase
that
see
the fixed
the
pivots,
those
at
that
so
distances between
the
distance between
the
pivots
from
distances of the tracer
we
choose
move
in
Can
shall
tracing-point
our
must
the
disposal
our
in all six differentdistances.
distances
clearly
three-link motion.
our
"
have
we
pivotson the radial links,the
the traversing
on
link,and the
tl^ose
pivots;
our
a
line ?
straight
The first person who
James
Watt.
"
Watt's
1784, is well known
nearly every
"
radial
"
length
pivots; the
and
for
Parallel
to every
beam-engine.
simplestform, is
The
this
investigated
in
shown
bars
are
of
brevity,to
links of
course
the distance between
is such
that
when
the
that
was
"
Motion
(4),invented
and
engineer,
The
is
the
to its
Fig.2.
equallength,
"
denote
may
the
I
employ the
the distance
be of any
pivotsor
radial bars
traversingpiece. The
is,if the apparatus does
in
employed in
apparatus,reduced
not
curve
the
"
traversinglink
parallelthe line
the
are
described
deviate
between
word
lengthor shape,
to the
joiningthose pivotsis perpendicular
The tracing-point
is situate half-waybetween
on
great man
much
radial bars.
the
by
from
pivots
the tracer
its
mean
HOW
10
TO
to the truth
still not
I
than that
A
W
STRAIGHT
given by
Watt
LINE:
be
can
obtained,but
actual truth.
have here
examples
some
The firstof
Jvoberts
DBA
these,shown
of these
in
closer
Fig.3, is
due
tions.
approximato Richard
of Manchester.
^""
3.
Fig.
bars
radial
The
the fixed
between
of
the
a
pivotsand
tracer
the
The
that of the
twice
distance from the
is situate
pivotson
tracer
in
straightline joiningthe
half-waybetween them.
coincide at
between
equal length,the
the tracer
radial bars.
\vith the
of
pivotsis
and
piece,
traversing
at
piece,
are
any
the
when
other
fixed
pivots.
it passes
line.
straight
points,but
the
The
it
pivotson
the
on
distance
equalto
traversing
the
consequence
fixed
deviates
path
lengths
coincides
pivots at
It does
the
those
not, however,
very
described
slightly
by
deviates
pivotsaltogether
the
from
A
The
other
cheff of St.
bars
Petersburg. It
then
pivots or
long.
half-waybetween
I
"
had
we
draw
through the
tracer
will
the
well
distance
traversingbar
the
yet, well, if
the
in
be found
at the
a
in its
as
pivots
it
th6
between
The
draw
we
now
cannot
is
tracer
do
a
that
called
shown
in
as
position,
forming the fixed pivots,it
will coincide with that line at
through the
but,as
position,
long as
we
fixed
mean
that
motion, it coincides
parallel
is very small
the
distance
If
radial
The
straightline,popularlyso
verticals
mean
last.
Tchebi-
little model
in my
inches.
forgottenthat
that the tracer
pointswhere
the
Professor
Fig.4.
between
two
these
to
figure,pai'allel
as
by
is shown
four inches, and
be
straightline
"
The
11
LINKAGES.
invented
apparatus was
five inches
taken
ON
equal in length,being each
are
must
LECTURE
nowhere
remams
fixed
in the
pivotscut
case
it
as
of Roberts's
else,
thoughits deviation
betweeijithe verticals.
HOW
12
We
TO
that
A
have failed then with
the next
to
DRAW
we
case,
three
odd
an
number
apparatus describingdefinite
problem with
first accurate
five 1
his due
firstsolved
was
by
M.
in the
French
at
by
strict
His
substantial reward
from
Watt's
at
last been
"
Prix
M.
as
prizeof
the
give
must
find the
not
the problem
discovery,
officer of
discoverywas
student
the
not
into
Engineers
M.
mated
at first esti-
oblivion,and
named
the Russian
and
recognized,
great mechanical
not
was
5.
supposed originality.However,
has
an
solve the
we
did
on
observe
want
we
this
go
order.
chronological
an
Peaucellier,
Russian
a
; but
can
value,fell almost
its true
rediscovered
a
army.
Can
(althoughhe
Fig.
will
of links if we
and
discovered,
simplestway) and proceedin
In 1864, eightyyears after
must
we
for you
"
curves.
we
motion
parallel
first inventor
the
Well,
LINE:
links,and
five-link motion
a
have
must
STBAIGHT
lipkin,who
Government
Institute
of
got
for his
Peaucellier's
he has been
was
merit
awarded
the
France, the
Montyon."
Peaucellier's
you
see,
seven
apparatus is shown
piecesor links.
in
There
Fig.5.
are
It has,
first of all
LECTURE
A
long links
two
the
fixed
same
of
equal length.
The
rhombus
a
These
13
both
are
their other extremities
point;
oppositeanglesof
links.
LINKAGES.
ON
portion of
composed of
the
apparatus
pivotedat
pivotedto
are
four
I
equalshorter
have
far
thus
described,considered apart from the fixed base,is a linkage
termed
and
a
"Peaucellier
pivotit to
fixed
the
fixed
a
link
is
rhombus
at its
then
the
then
pointwhose
to which
point,that
length of
cell.** We
take
distance from
the cell is
is
pivoted,
link ; the other end
extra
pivoted to
; the other free
of
one
angleof
free
the
link,
extra
an
the first
the
same
as
of the extra
angles of
the rhombus
has
describe
pivot. That pencilwill accurately
a
a
the
pencil
straight
line.
I must
indulgein
now
that
absolutely
necessary
understand
may
In
Q, the other pivoton
to
that you
apparatus.
pivotedto
the fixed
the circle OCR,
C, describing
P
and
in order
so
M'
are
supposedto
be
point
The
perpendicular
MEQOM'.
angle 0 0 E, being the anglein a semicircle,is
0 C B, 0 M P are
rightangle. Therefore the triangles
Now
a
it
do
our
link
It is
simple geometry.
I should
the extra
M
little
of
principle
the
Fig. 6,QCis
lines P
straight
a
similar.
the
Therefore,
00
:
OB
::
OM
:
OP.
Therefore,
Oa'OP==OM'OE,
wherever
0 R
are
C
may
be
on
the circle.
both constant, if while G
That
moves
is,since 0
in
a
ci^leP
fifand
moves
A
DRAW
TO
HOW
14
STRAIGHT
always in
80
that
Oy C, P
are
so
that
00* OP
is
always constant;
P
M
the
line
straight
if
we
if OG'OF
straightline F'M',
take the
construction
This will be
cell,we
of the
line,and
if the
to OF
it must
"
proved yet
"
On is
equal
see
seen
straightline
to
our
an
is
a
that from
cell,0, C, P,
stillbe
that
P
then
-4
line 0
pointF
the other
the
be
portant.
be im-
drawing
skeleton
symmetry of
imaginaryone,
same
of
the
straigh
perpendicular
drawn
apparatus does draw
TtP,
on
Q.
to
presently
all lie in the
w
will describe
P' will describe the
is constant
Now, turningto Fig. 7, which
the Peaucellier
straightline,and
same
to the
perpendicular
It is also clear that
side of 0, and
the
LINE:
as
a
we
have
not
straightline
LECTURE
A
ON
LINKAGES.
15
Now,
0A^
On^
=
-f
AF^=^Fn^^
A7i^
An^
therefore,
OA^--AF^
=
On^^Pn^
+
[On-Fn]'[Chi
Fn]
=
OC'OF,
=
Thus
since 0
A
and
A
F
always constant,however
If then
the
pivot0
the
pivotC be made*
beingpivotedto the
all
satisfy
pivots will
of
quantityOA^
hope
the
the
"
you
if
end
a
modifications
C and F may
near
point0
of the extra
in
which
of
on
I
extra
now
of the cell. The
it
line from
the
the
figureby
will
in
move
the two
link
and
a
of the
pleasure.
elements
the
posing
com-
cell,and
wish to describe to you
extra
a
the fixed
magnitude
be varied at
may
clearlyunderstand
apparatus, the
and
Fig. 6,
fixed at F it will draw
pencilbe
depend
make
is
be to 0.
link,the pivotF
to
necessary
distance
part each plays,as
OCOF
constant
describe the circle in the
to
course
OF^
or
be fixed to the
The
straightline.
I
far
the conditions
straightline,and
both
are
link will remain
some
the
Goog
HOW
16
same
as
DRAW
TO
it is
before,and
STRAIGHT
A
LINE:
will
cell which
only the
undergo
Iteration.
If I take the
as
"
the
other
so
kite
"
two
and
that the
"
long links
of the
shall
together,we
If then
keep
we
the
between
"spear-head,"we
by that
multiplied
US
the
see
8.
same
that
height of
"
the
"
spear-headis
to
is
and
the other will
of
one
not
describe,
P' M\
short
if the
in the circle of
move
line
straight
fixed,and
kite
"
"
and
"kite'*
the
constant.
amalgamating the long links
of
linkageof Fig. 9;
in the
links,the
of the
instead
now,
links meet
made
short
7.
long links, or
the
ngles between
amalgamate the
linkages,
the two
the
those
get the originalcell of Figs.5 and
Fig.
Let
coincide with
one
the
on
other, and then amalgamate the coincident long links
of the
that
known
linkagesin Fig. 8, which are
and placeone
the
spear-head,"
the
In this
the
We
ones.
Fig. 6 by
free
the
straightline
form, which
is
a
F
get
the short
pivot where
other
then
of
pivotsbe
extra
M,
very
link,
but
the
compact
TO
HOW
18
the
kept the
links of
10 I fasten
to end.
the
Then,
the links
pointwhere
which
straightline.
a
des Arts
et Metiers
; the
M.
extra
an
by
cell
a
link,to
of
form
the Conservatoire
ship
workmanexquisite
is of
line.
alongthe straight
discoverywas introduced into England
to swim
pencilseems
Peaucellier's
Collection
of Paris, and
of
employing this
model
cell is exhibited in the Loan
pivotat
together, I get
employment
A
Fig.
that in
seen
if I fix the
cases,
fixed
are
be used, by the
may
describe
in the former
as
before,
was
end
figuretogether,
links of each
shorter
it
the
superimposing the
of
other,it will be
the
figureon
one
productof
the
same,
instead
Now
LINE:
be half what
will
figures
the two
still constant.
but
STRAIGHT
A
anglesbeing however
heightsof
the
DRAW
.
Professor
by
Sylvesterin
Royal Institution
of the
In
read
Academy
(6),in
which
The
If to the
of the
same
rather
an
shown
an
this
excited very
country.
British
of Woolwich
Hait
Mr.
year
at the
paper
Association
cell could be
only four
apparatus containing
links instead
Peaucellier
linkageis arrived
new
ordinaryPeaucellier
lengthas
in
the
Fig. 11.
cell I add
long
ones
it may
Now
at thus.
I
fresh links
get the double,
be used
Mr.
two
Hart
form
what
then
took
are
unequal,and
is called
four
between
a
found
the
crossed the
that
if he
the
which
links
so
as
contra-parallelogram.
Fig. 12,
points on
the
pivotsin
four
the
links
same
or
in four different
linkwork, in
ordinaryparallelogrammatic
adjacentsides
meeting
s
that M.
quadruple cell,for
ways,
took
a
same
he showed
replacedby
of six.
of the
the
at
of the consideration
the commencement
was
subjectof linkagesin
August
delivered
January, 1874 (5),which
in
great interest and
lecture he
a
dividingthe
to
and
tances
dis-
those
proportion,
A
LINKAGES.
ON
LECTURE
19
four
pointshad exactlythe same
propertiesas the four
pointsof the double cell. That the four pointsalways lie
in
straightline
a
is
thus
seen
:
consideringthe triangle
Fig. 11.
ahdy since aO
hd, and
to the
the
:
Ob
:
:
aP
:
Pd
distance
perpendicular
heightof
the
is
therefore OP
ahd
triangle
between
as
the
Oh is to ah
to
parallel
is
parallels
the
;
to the straight
line C0\ and since
reasoningapplies
c
d: O'd and the
heightsof
the
are
That
12.
therefore the distances oi 0
the same,
the
and
0 C P
0' lie in the
product OG'OP
oh: Oh::
ahd, chd,are clearly
triangles
Fig,
the same,
same
P
same
is constant
and
O'C from
h d
straightline.
appears
c
at
once
2
Googk
HOW
20
when
it is
half
TO
is half
it
;" similarly
constant, as also OC'CO'
Hart's
cell as
straightline
Loan
I
Professor
lead
model
you
another
three-6ar
wish
I think
I shall
by
three-bar
motion, and
the
to
me
The
was
proposition
obvious ; the
sooner
curves
were
0 and
let P be any
the
interesting
more
enabled
its
the
I do not
know
B^
and
pointon
and
traversingbar
versing
tra-
a
or
find whence
why,
one's
the
curves
in any
given
described
by
the
in which
the
bar
traversing
been made
to
than
radial bars
let D
it
ordinary
how
Ahe
the solution became
and
DA,
In
Fig. 13
turningabout
the
a
describing
lengthsof AB, BG, GD,
points on
change places.
similar.
precisely
the two
bring
to
age
importantlink-
very
stated
no
and B AhQ
fixed centres
on
those
of the radial bars had
one
have
we
you
radial bars and
two
back
motion, but
and
bars,
Sylvester.
"
the
pointson
similar three-bar
apparatus
those
consider the relation between
"
described
and
it will be
therebybe
it is often very difficult to go
let G D
Hart's
pointson
elegantand
very
originate to
of Mr.
apparatus by detailingto you
bar,it occurred
a
Mr.
bring before
to
of
consisting
motion
in the
simultaneously
by
the problem presentedby
considering
When
ideas
In
bars and
of Professor
^the discovery
"
five-link
extension
an
discovered
with
to this
as
history,especially
the
then
of this is exhibited
attention to
was
of bars.
up
Employing
is
Breguet.
apparatusI
instead
pieces
before
OP'FO\
Sylvesterand myself.
in the
if I
M.
only concerned
were
but
by
apparatus,which
Hart's
we
A
motion.
OaP
that CF'O'C
be shown
may
and
wish to call your
now
"
spear-head and
"
a
:
we
employedPeaucellier*s,
get a
we
Collection
LINE
STRAIGHT
A
that ObC
seen
"kite
a
DRAW
the
traversingbar,
curve
Now
depending
add to the
ON
LECTURE
A
tliree-barmotion
the bars CE
DA, and EA equalto CD,
if
and
an
in
on
and
G DA
GP, viz.,GD
the
GP' bears
:
Thus
GE.
same
the
that described
as
the
CE
proportion
CD
and
DA,
the
precisely
we
same
:
Thus
CD.
shall
is the
curves,
and the
On
he at
same
once
a
tbat P' is
fixed
fixed
point
to
proportion
a
by P' is precisely
by P, only it is lai^erin
described
13.
if
we
take away
only of
different
and this
described,
as
link
traversing
my
beingequalto
the bars
get a three-bar linkwork,describing
firstthree-bar motion
linkwork
CE
parallelogram,
drawn, cuttingEA produced
always a
curve
21
tben
Eis
Tig.
our
EAF,
imaginaryline CPP" be
P',it will at once be seen
and
produced,
EA
LINKAGES.
DA
the old with
magnitude,am
new
three-bar
the radial link
CD
(7).
interchanged
communicatingthis result to
that the propertywas
saw
Professor
one
not
Sylvester,
confined to
HOW
22
the
DRAW
TO
case
particular
"
piece
on
"
the
which
bar
A
E
P'y making
A
E
P
G
E,
the
C E
the
A
are
and the
P,
or
E
C
equal
to
Now,
A
so
a
before,add
as
Z", and
Z", and
PDA;
triangle
EA
this
::
"
"
angleP
G P
"grams"
AD
you
that
difficulty
the
lie.
to
:
easilyfrom
without
D
on
the
the
piece
triangle
that the
angles
equal,and
PE
It follows
jointsA
being equal
similar to the
AEPyADP
and
in
as
tracingpoint or "graph does not lie
A D, but is anywhereelse on
the joints
joining
the line
P
motion,
three-bar
Fig.l4tCDAB\BB,
In
Fig.13, but
"
pointslying on the traversingbar,
by three-piece
motion, but was possessed
of
in fact to three-5ar
motion.
LINE:
STRAIGHT
A
:
work
can
the
DP.
it out
ratio P'G
ia constant
described
; thus
by
the
:
P
for yourselves
G
is
stant
con-
paths of
"graphs,"P
the
LECTURE
A
and
P',are
is turned
Now
through an anglewith respectto
quiteindependentof
the
curve
particular
^, in both
cases
shall
we
Skew
enlargeor
them
P
be
reduce
made
to
G P' is made
forms
of the
made
to
round
same
as
deserves to be
a
figurethe
sion
exten-
it will do
it will
more,
I
see
have
any
0
in
the
angle
value.
required
180",we
or
get
value
which
If the
the
two
if it be
use;
common
to
is
a
sub-
can,
from
know,
put
pictureof
a
this that the instrument, which
of the
of
little model
by
constructed,
practically
been
never
into the hands
furnished
instiniment
also
please,and
we
of (7 P
ratio
by passingthe pointP each time
the point P reproduceit
pattern make
centre C after the fashion of a kaleidoscope.
you will
has, as far
here
first
beautiful
a
P', the
and
successively
any
the fixed
I think
P
pantagraphnow
assume
the
get rid
we
its inventor,the
Sylvester,
equal to
of 360",we
multiple
over
only affects
If
the
second
but
figures,
what
be made
can
angle P
in
P.
given
thi'oughany requiredangle,for by properly
can
C P*
have
it
Pantagraph, Like the Pantagraph,
choosingthe positionof
C P'
get
one
the other.
B, which
and
in the
and
sizes,
proofsI
P
by
it,called by Professor
Plagiographor
turn
the bar A
described
ordinarypantagraph,and
will
the two
will observe that
are
of
23
of different
similar,onlythey are
you
of ^
LINKAGES.
ON
to
me
designer. I give
form
possible
a
the
Loan
for the
Collection
by
request of Professor Sylvester(8).
After
him
and
it occurred to
of Professor Sylvester,
discovery
me
simultaneously our letters announcing our
this
to
"
discovery to
each
of
principle
the
Hart's
other
crossing in
plagiographmight
contra-parallelogram
; and
the
be
this
post that the
"
extended
to
discoveryI
Mr.
shall
HOW
24
DRAW
TO
A
now
proceedto explainto
more
easilyable
manner
to that
If we' take
bend
the
to do
so
in which
the
links at
straightline, or
STRAIGHT
I
you.
LINE:
however, be
sliall,
it
by approaching
I did when
in
a
difEerent
I discovered it.
of Mr. Hart, and
contra-parallelogram
the four points
which lie on the same
fod
as
they
are
sometimes
termed,
HOW
26
TO
line
straight
will be
and
pivotsO
afterwards
planelinks,such
the
various
Its
a
very
soon
I
taken.
in
in my
Oa,
half
aP
"
a
the kind
ing
express-
interest which
he took
regret that his departurefor
the
post
of Professor
in the
I must
besides
leavingthe
whose
much
Peaucellier cell and
that
seen
a
simplestlinkwork
our
circle of any
if
radius,and
of ten miles' radius the proper
ten-mile
link, but
as
with
a
much
that would
smaller
cell is mounted
line,as
straight
pivotsmust
we
modifications,
would
be to have
to
we
purpose
effect
can
of
same
as
the
lengthof
the
our
cellier
the Peau-
describinga
I told you, the distance between
be the
least,
say the
apparatus. When
for the
to
us
to describe
be,
that
enables
wished
course
to know
cumbrous, it is satisfactory
purpose
its
point out another important propertythey possess
that of furnishing
with exact rectilinear motion.
us
have
describe
a
new
investigations.
Before
one
is
name
Hopkins Universityhas deprived me of one
and encouragement helped me
valuable suggestions
We
form
kite,"you will
Johns
in my
I
of it.
my
to undertake
when
Fig.12, Oh, hC
apparatus,in which my
Professor Sylvester,
without
researches,and
America
in
Fig.16, as
for
deep gratitude
my
and
investigate,
this
associated with that of
which
"
get to the bottom
leave
of
This
difficult to
not
are
explainsthe name
Sylvester,the
Quadruplane."
and
"spear-head,"
cannot
I
simplicity
employed in Fig.1,on
those
as
to you that
pointout
for
fixed
strictly
speaking formed
course
Professor
are
properties
half
apparatus,which
pointsare
by
joiningthe
links which
fonjaed of four straight
as
four
it
LINE:
the line
to
parallel
bent, is of
given to
STRAIGHT
A
This
Q,
described
have
DRAW
"
the fixed
extra
"
link.
LECTURE
A
LINKAGES,
ON
If this distance be not
the
same
27
shall not
we
I
get straight
'
lines described
be
circles described will be of
slightthe
(forif
circles
the
portion which
the
not
be amiss to
In
Fig.17
If then
will be
to
parallel
S
he
P
Q',and
the
Q
C
O P'O
since
that
0
C describes
a
circle about
Taking then 0, C
cell in
It is
That
(7is constant.
Q does
and P
Fig. 7, we
how
see
hardlynecessary
P
0, C and
me
to
state
circles of
Of
"
cell "
the purpose.
The
and M.
Breguetare
purpose
of
thus
models
varying the
circles of
getting
may
exhibited
furnished
Fig.
constant,
Z",
to O
is constant
and
about
one
Q'.
the
a
circle.
importanceof
arts
for
drawing
the various modifications
course
described
I have
Q\
0
P of the Peaucellier
in the mechanical
of the
wHl
Z)
with
is
to describe
comes
for
P
CO
the Peaucellier compass
largeradius.
circles.
to
ratio
will describe
P
the
as
G
constant
a
is if 0
Q,
0
in connection
gave
bears
P
O
it may
proposition.
Q\ and
S
to
0 P
same
therefore
a
circles be at
two
7, it will be clear that the product 0 D'O
and
only
straightline through 0, D Q
any
to
proportion
the proofwe
considering
Now
pencil
that the
the radii of the
to
proportional
C P
O D
bear the
Q, Q of the
let the centres
0
knows
proofof
short
a
vexity
con-
0, if less
this will be clear,
but
circle,
a
givehere
distances from
the
be
course
mathematician,who
a
by
be towards
described)will
is
circle is
a
will of
large it
are
concavity.To
inverse of
the circle described
portionof
If the
Q C, the
greater than
Q 0, Fig.6, be made
of the
tude,
magni-
enormous
the difference increases.
size as
in
decreasing
distance
circles. If the difference
but
pencil,
the
by
with
distance
any radius.
all be
by
employed for
Conservatoire
the
sliding"
pivotsfor
between
0
and
the
Q, and
BOW
28
My
attention
TO
DRAW
was
lecture of Professor
A
8TBAIGHT
LINE:
first called to these linkworks
to
Sylvester,
which
I have
by
the
referred.
stated that there
in which it was
passage in that lecture
motions
were
probablyother forms of seven-link parallel
A
A
besides M.
then
Peaucellier's,
LINKAGES.
the
the subject,and
investigate
to
some
'
ON
LECTURE
to that
I shall
If I take two
that the
and
a
short
a
long one
one
twice
kites,one
long links
make
of each
of the
a
big
as
of the
short
then
and
large,
P'
can
importantproperty of
by moving
approach to
line
or
joiningthem
through
the
the links
recede
is
L Mhe
fixed
made
to move
one
this
in
the
lengthof
of
each
so
as
or
the short
kite lie
small
small
the
the
on
on
coincident
pointsP
link L
P' be
always to
fixed line.
by fixingF.
and
other,the imaginary
that
M.
remain
Fig. 19
a
drawn
It follows
fixed,and
line,the other pointwill describe
to the
perpendicular
made
motion
parallel
other, such
linkageis that,although
bottom
piVotsP
to
us
Fig.18.
about, make
from
the
as
always perpendicularto
pivotson
that if either of the
a
will lead
amalgamate the
links,I shall get the linkageshown
The
of these
importance.
link
laige,and
obtaining
bring two
twice the
are
long
one
of the
of
linkworks
other
some
in
me
different character
entirely
of them
notice,as the investigation
consider
we
known, led
only one
I succeeded
an
(9).
of M. Peaucellier
to your
ones,
of
motions
parallel
new
29
the
IitiV
parallelto
straightline
shows
you
the
It is unnecessary
for
HOW
30
to
me
how
point out
to the line
joiningthe
from
the linkwork
describe
fixed
pivots;
of
number
preserved
figure. The
the
pointP
is
we
cular
perpendiever,
how-
can,
links,make
point
a
inclined to the line
line
straight
a
is
the
by
two
the
increasing
without
on
is described
which
L M
oi
parallelism
the
link S Z, it is obvious
by adding the
line
straight
LINE:
STRAIGHT
A
DRAW
TO
\
"S P
at
angle,or
any
link
straight
C
piecemoving
that
In
a.
distance
from
(7.
vertical
the
same
Now
for the
same
as
have
we
a
the
be any
line ff P* is fixed in
all the other
the
P
fixed
move
C
is
C
pointson
; but
one
C
from
and
C
and
P
the
The
it all at
that
in
F
shown.
are
piercedin
Fig.19
P'
the
C
"
in
moves
Fig. 20 being
a
well-known
of the holes
piercedin
constant,thus the straight
and
position,
holes
P
distance P
if H
circle,
F
of
link
lengthsP
from
seen
Fig.19
was
piece,the angleH
through
link
in
it
as
the
has holes
straightline,the
property of
the
the
"
number
a
the
direction.
in every
circular and
same
for
by substituting
can,
only
Fig.20, for simplicity,
pieceis
new
we
get
planepiece,
piece substituted
new
a
F
rather
H
along it ;
larly
simi-
along in straightlines
ing
pass-
pivot P\
and
moves
we
line
get straight
LECTURE
A
motion
is
distributed in all directions.
called
worth
same
LINKAGES,
ON
noticingthat
it would
as
inside the
the motion
circle ;
reproduced
by
only requiremotion
in
except a portionforming
F, and
the
The
other
some
have,
all
double
not
a
pointson
kite
useful
of
moves
Fig. 18
linkworks.
singlepoint,but
it
move
in
a
bent
in
the
may
be
may
we
whole
cut
away
all
containing0,
arm
requireddirection.
employed
It is often
a
if
course,
20.
the disc
point which
Of
direction,we
one
it rolled
have, in fact,White's
we
linkwork.
Fig.
is
of the circular disc is the
been if the dotted circle on
have
It
"*
large dotted
motion
parallel
of motion
species
TMs
Sylvester tram-motion."
Professor
by
31
necessary
piecemoving
straightlines.
I
to
may
so
form
to
that
instance
HOW
32
the
slide
TO
rests
DRAW
in
A
STRAIGHT
LINE:
lathes, traversing tables, punches,
etc.
drills,drawbridges,
double
The
kite
enables
ns
Fig. 21,
to
producelinkworks
work
of
Fig.21, the
if
appreciated
link
you
moves
having this property.
construction
understand
to
and
of which
the
fro
as
double
if
In
the link-
will be at
once
zontal
kite, the hori-
slidingin
a
fixed
"FN
LECTURE
A
which
Tchebicheff,
camp-stool,the
motion
is not
will be
seen
bears
seat
ON
a
strong likeness
of which
rectilinear ;
strictly
by observingthat
is of invariable
it is
a
"
of Professor
added
to
variation
length,and
TchebichefE
keep
of
the
the
to
a
complioated
a
the
"
as
figure
25.
link
might
of the
therefore be
put
motions
parallel
given in Fig. 4, with
links
some
The
with the base.
parallel
horizontal
plane from a strictly
seat
upper
The
apparatusbeing
the thin line in the
of two
combination
36
has horizontal motion.
Fig.
where
LINKAGES.
D
2
HOW
36
TO
DRAW
A
is therefore
movement
8TBAIGET
LINE:
double
that
of the
similar
apparatus of
in
tracer
the
motion.
simpleparallel
Fig. 26 shows
how
a
much
simpler
construction,employingthe Tchebicheff approximateparallel
be made.
motion, can
distance
the
between
the fixed
between
link is one-half
27.
figurein
angle,so
angle.
made
which
that
The
by
of the
are
which
same
the four
forming
given before (Fig4).
the
the
moving
The
seat is half that
length of
the
remaining
is shown
description
bent
are
through
C 0 F
0 G'O P ia constant, and
pivotedto
of the extra
the radius
in
Fig.
/oa of the quadriplaneshown
the links
focus 0 is
meaps
the links
that of the radial links.
0, G, (y,F
in the
pivotson
and
pivots,
exact motion
An
been
have
motion
parallel
the
lengthsof
The
a
fixed
link Q C to
a
right
"
right
point,and
move
in
a
C
is
circle of
Q C ia equal to the pivot distance 0
A
J^
LECTUEE
in
consequentlymoves
five
ON
far
described
motion.
Sylvester-Kempeparallel
moving seat
of
and
which, Fit
the
and
E
remaininglink
0\
a
it will do
also.
so
advantagewhere
is
This
very
R
^, the
added
the
0',the pivotdistances
0
Q.
The seat in
and
as
F
sequence
con-
moves
straightline,every point on
apparatus might be
used
with
table
traversing
smoothly-working
required.
I
now
would
come
show
the fixed
a
a
0
the
constituting
tljisare
to Q 0,
parallel
horizontal
accuratelyin
To
equalto
are
always remains
37
to
straightline parallel
a
moving pieces thus
LINKAGES,
of the
to the second
you.
If I take
base,and make
the
a
motions
parallel
kite and
sharp end
line,passingthrough the
straight
links wiU
rotate
about
the fixed
pivotthe
move
fixed
I said I
blunt end to
up and
down
pivot,the
in
short
pivotwith equalvelocities
HOW
38
in
TO
DRAW
A
LINE:
STRAIGHT
if
direotions ; and, conversely,
opposite
the links rotate
the path of the
equal yelocitjin oppositedirections,
sharp end will be a straightline,and the same will hold
with
(m
good
if instead of
same
point they are
To
with
find
a
the
linkwork
equalvelocities
short
links
being pivoted to
pivotedto different
which
in
should
make
the
ones.
two
oppositedirections
was
links rotate
one
of the
A
LECTURE
first problemsI set
in
making
two
ON
myselfto
LINKAGES.
solve. There
links rotate with
39
was
no
equalvelocities in
difficulty
the
same
linkwork
direction, the ordinaryparallelogrammatic
em-
"
ployedin
locomotive
the
composed of the engine,
engines,
cranks,and the connectingrod, furnished
was
none
in
making
two
links rotate in
that ; and
two
there
directions
opposite
HOW
40
with
TO
A
DRAW
linkwork
required
trouble
I
that ;
After
to be discovered.
obtainingit by
in
succeeded
had
a
some
combination
of
largeand small contra-parallelogram
put togetherjustas
the two
kites
in the
were
is made
parallelogram
long links
of each
the thin line drawn
link,be
six-link
same
as
n
seen
to
twice
as
largeas
through the
be formed
The
usefullyemployed
31
contra-
other, and
(10).
each
pivotsin
by fixingdifferentlinks
two
They
for the
shown
as
mere
can
as
a
of the
contra-parallelograms
rotate with
thus,
the
will,by considering
fixed
pointedlinks
motions.
give parallel
the
the short
longas
and
Pigs.30
and
oppositedirections,
once
as
linkagecomposedof
juststated.
One
linkageof Fig.18.
twice
are
in
linkworks
The
be
LINE:
varyingvelocity
; the contra-parallelogram
gave
but the
a
STRAIGHT
equalvelocity
in
Fig. 28, at
of course,
purpose
of
however,
reversing
angular velocity(11).
An
extension of the
linkage employedin
these two
last
42
ones
HOW
to any
ones
we
TO
DRAW
A
LINE
STRAIGHT
angle,
you will see thafc the
exactlytruect the angle. Thus
:
desired
two
will
the
have hiad to call into
firstPostulate
"
operationin
^linkagesenables
"
us
mediate
interpower
order to efEectEuclid's
to
solve
a
problem
j
A
which
go
on
divide
has
LECTURE
LINKAGES.
''
no
''geometricalsolution.
extending my
an
ON
angleinto any
I could
linkageand get others
number
of
43
of
which
equalparts.It
course
would
is obvious
HOW
44
that
TO
these
for
works
DBA
W
STBAIGHT
A
linkagescan
same
also be
:
employed
link-
as
doubling,
trebling,
etc.,angularvelocity
(12.)
Another
form
of
"
Isoklinostat
"
is
is
appai'atus
the
so
discovered
"
construction
for
"
by Professor Sylvester^was
termed
LINE
apparent from Fig. 33.
It
by him.
The
has the great
of
advantage
being composed of links having only two
pivotdistances bearing any proportionto each other, but
it has a largernumber
of links than the other,and as the
of the links is
openingout
limited^it cannot
be
employed
for
multiplyingangularmotion.
of
Subsequentlyto the publication
have
the
of these linkworks
account
an
been
speaking,I pointedout
Royal Society(13)that
well
as
were,
depended
composedof
similar
two
in
of mine
a
paper
Peaucellier and
the
on
which
paper
tained
con-
of which
I
before
read
motions giventhere
parallel
of linl^works of
cases
particular
all of which
the
those of M.
as
the
a
Mr.
Hart, all
generalcharacter,
very
employment
iigures. I have
of
linkage
a
sufficienttime,
not
.
and
I think
here,
so
I
will leave
questionhas
this
At
we
been
mechanician
you
the
as
have
theoretical
shown
it
on
in the
original
paper.
productionof straight-
as
must
with,if theyare
what
I have
you that
we
can
to
want
go much
part of the question. The
been obtained
to deal
far
interest
an
excited to consider the
to-day is concerned, come
I have
in whom
hardly,for practical
purposes,
results that
have,
sufficiently
inviting
stands, and I think you will be of opinion
now
farther into the
I
those
be
character,to dwell
point the problem of
line motion
that
not
of its mathematical
account
on
subjectwould
the
be left to
now
of any
undertaken
to
describe
a
value,
practical
to
the end
the
bringbefore
of my
tether.
line,and
straight
A
Jiow
to
we
can,
your
of
all. I
makes
can
hope that
have
I have been able to show
are
hardlyfail to
no
you
still vast, and
make
I have
shown
this
led
doubt
the
that you whose
duty it is
will not let the
subjectdrop with
to
extend
I
(and
new
field
tance.
impor-
the
more
plored
unex-
investigator
earnest
I
But
much
but
to-day(14),
discoveries.
new
done
ns
you
and
great interest
possessing
Mathematicians
fields
problemhas
that
hope a belief)
that
is one
investigation
than
45
important piecesof apparatus.
this is not
attention
LINKAGES.
the consideration of the
and
some
investigate
hope that
ON
LECTUBE
hope therefore
the domain
the close of my
of science
lecture.
NOTES.
hole
(1) The
a
circle
**
"
But
to
where
points
Draw
ruler
lies
the
the
the
to
only
QPR
=
'*
F
point
of
angle
the
PP'
RV,
"m
PP'^
QRP
not
and
angle
an
=
the
angle
let
ruler.
Let
PP'
that
RQS,
PP'.
=
fit the
if
For
TSP
the
=
Sy
^ be
angle
the
half
is
the
through
RST.
angle
be
we
passes
angle
the
if
th
known,
PP'
2JtS
Then
edge
joined,
and
ST,
to
the
be
the
3),
that
By
is well
as
can,
angle,
the
trisects
RQ
half
Note
surface
one."
graduated
a
edge of the
that
of
part
fi^yen plane
a
perpendicular
so
latter
describe
to
plane.
he
RST
cut
RSTUV
and
the
Let
EV
and
or
be
to
graduations,
two
thus"
graduations
figure
ouRU
middle
a
assumed
the
made
be
can
passes
(see
circle
(it is carefully added)
parallel
RU
pencil
surface
any
be
course
with
ruler
a
the
describe
readily trisected,
be
P
of
must
of
use
wish
we
surface
(2)
of
independently
when
but
which
through
angle
RSQ.
solution
This
have
is of
indicated,
employed.
But
construction
?
Is
?
not
4, graduated
to
on
It
because
he
side
AB
A
and
B,
at
difficult
seems
which
^that
produced
by
point
line
second
to
any
confine
use
and
to
ABC
and
to
take
are
Postulates
this
Book
in
told
not
we
three
I
sense
fitting process
his
ruler
triangle
are
the
and
graduated
a
of the
length
a
it, and
Postulate.
in
**
a
that
of
fitting process
Proposition
I.
it up
by
prove
That
a
straight line,"
in the
First
rigid adherence
straight
it to
point
be
and
fit
can
or
"
Hue
it
as
without
is in
the
by
the
a
late
Postumay
be
he
did
why
rather,
Euclid's
which
then,
second
straight
Book
to
line
so,
the
employed
terminated
a
propositions
terminated
to
Euclid
why
see
requires
difficult
means
outside
with
to
the
it among
put
no
ruler
himself
the
in
one
DE%
"
not
not
the
"
geometrical
a
graduated
a
Euclid
does
Does
"
not
course
It
problem.
methods
the
same
employment
first
to
is
find
a
straight
of
Postulate,
the
be
NOTES.
48
joinedto
the
extremityof
the
and the process can be
Postulate circles can be drawn
line which
is thus
given straight
since by the
continued indefinitely,
with any
centre
and
(3) It is importantto notice that the fixed base
duced,
prothird
radius.
to which
the
pivots
link in the system. It would on that accoimt
consideration of the subject,
in a general
tp
scientific,
attached is really
one
are
be
perhapsmore
by calling
any combination of pieces(whether those pieces
be cranks,beams, connecting-rods,
or anythingelse)
jointedor pivpted
of
the
links
is confined to
a
linkage." "When the motion
together,
the
of
number
to
a
one
planes,
parallel
system is called
plane or
that
this lecture is confined
a
plane linkage." (It will be seen
solid
about
to planelinkages
linkageswill be found
; a few remarks
motion
selves
themof the links among
at the end of the note.) The
commence
**
**
in
a
linkagemay
be deteraainate
or
the motion
When
not.
the
is
and
of links must
be even,
^the number
the
motion
"When
is
determinate the
not
"complete."
linkageis said to have 1,2,8,etc. degreesof freedom,accordingto the
These
of liberty
the links possess in their relative motion.
amount
"secondary,"etc. linkages.Thus
linkages
may be termed "primary,**
if we
take the linkagecomposed of four links with
two pivotson
others
is determinate,
and
each,the motion of each link as regardsthe
the linkage
is a "complete linkage.**If one link be jointedin the
middle the linkagehas one
age."
degree of libertyand is a "primary linkSo by making fresh joints secondary or
ages
etc. linktertiary,"
be
formed.
These
be
formed
etc.
in
primary,
liukages
may
may
various other ways, but the example givenwill illustrate the reason
for the nomenclature.
When
link of a linkageis a fixed base the
one
linkwork.**
structure is called a
Linkwoiks, like linkages,
maybe
i.e. one
in
"primary," "secondary," etc. A "complete lirikwork,"
which the motion of every pointon the moving part of the structure is
link-motion.** The various
described by
is called a
definite,
grams
these link-motions are very difiicultto deal with.
I have shown, in a
the
in
the
London
Matherruitical
Proceedingsof
Society,1876,
paper
that a link-motion can be found to describe any given algebraic
curve,
determinate
said to
linkageis
be
'*
**
*'
**
"
but the
is
one
the
*'
converse
towards
**
problem,
most
tricirculartrinodal
eminent
Taking them
Given
the link-motion,what
the solution of which
motion,
three-piece
"
**
are
but littleway
is the
curve
?"
and
beenjpiade;
"
of the simple
grams
the consideration of somp
of our
which
sextics,*'
still under
has
are
the
"
mathematicians.
in their greatestgenerality,
the
theoretically
simplest
NOTES.
50
(9) See the Messengerof Mathematics, " On
1875, vol. iv.,p. 121.
Some
New
Linkages/'
(10)A reference to the paper referred to in the last note will show
that it is not necessary that the small kites and contra-parallelograms
should be half the size of the large
ones, or that the long links should
be double
the short ; the
in lecturing.
description
(11) By
an
make
can
two
particular
lengthsare
arrangement of Hookers
rotate with
axes
therefore
producean
**
"
exact
solid
joints,
pure
equal velocities
(SeeWillis's Principles
of Mechanism^
chosen
2nd
Ed.
for
ease
of
we
linkages,
in contrary directions
516, p. 456),
sec.
and
motion.
parallel
"
to the
are
contra-parallelogram
subject
inconvenience
(mathematicaUy very important)of having "dead
points." These can be, however, readilygot rid of by "employin
(See
pins and gabsin the manner
pointedout by Professor Beuleaux.
Professor
translated
B"nleaux's Kinematics
Kennedy,
ofMachinery,
by
(12)The
kite
and the
"
Macmillan, pp. 290-294.)
No. 163, 1875, "On
a
B"yaZ Society,
Linkwork."
Exact
Motion
Rectilinear
by
Obtaining
(13)Proceedings
of
Method
of
the
of pointing
out that the results there arrived
opportunity
extended from the following
greatly
simpleconsideration.
this
be
General
I take
at may
Fig. 34?.
link
any angleD with the straight
links we employ the pieces
0 A, and ifinstead of employingthe straight
and
if
the
'A'OA
A'OA, B'OB,
angle
equalsthe angleFOB, then the
If the
link
straight
0 B makes
NOTES.
angleB'OA^ equalsD,
enable
Mr.
us
of
recognition
this very obvious fact will
motion
from that of
Sylv"ster-Xempe
parallel
Hart.
(14) In addition
may
The
to derive the
51
be referred to
part of the
ArticuUeSf**
par M.
1876, pp. 620"660.
to the authorities
by
those who
subjectof
V.
alreadymentioned, the following
desire to know
**
more
about the mathematical
Linkages."
SysUnvesde Tiges
in
the
Nqttvelles
Annales, December,
Liguine,
**
Stir lea
"
On Three-bar Motion,'*
by Professor Cayley and Mr.
papers
Roberts, in the Proceedingsof the London, Mathematical
Society^
Two
S.
1876, vol. vii.,pp.
14 and
diort papers in the Proceedings
vols, v., vi, vii.,and the
Society
,
136.
Other
of the London Mathematical
Messengerof MatherruUicSfVols.
iv. and
v.
R.
BRRAD
CLAY,
STREET
SONS,
HILL,
TAA
AND
QUEEN
LOB,
VICTORIA
PRIMTKRS,
STREET.