Heron Alexandrinus
,
Mechanica
,
1999
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[3]
After
this
has
been
made
clear
in
this
introduction
,
let
us
rotate
two
equal
circles
,
namely
<hekd>
and
<zgqe>,
around
their
centers <
a
>, <
b
>,
while
they
touch
at
point
<
e
>.
If
they
now
move
from
point
<
e
>
for
the
same
time
for
half
their
extant
,
point
<
e
>
in
this
time
runs
through
the
arc
<ehd>
and
reaches
the
point
<
d
>
by
moving
like
the
point
<
g
>
on
the
arc
<gqe>.
Then
it
can
occur
that
points
move
in
the
same
direction
and
that
they
move
in
opposite
directions
.
The
ones
positioned
on
the
same
side
move
in
opposite
directions
,
the
ones
opposed
to
each
other
move
in
the
same
direction
.
It
may
occur
however
,
that
points
that
are
described
as
being
in
opposite
motion
go
in
the
same
direction
(
both
upward
or
both
downward
).
For
,
when
points
move
and
their
motion
starts
from
one
point
,
namely
the
point
<
e
>,
and
we
imagine
two
lines
<zaq>
and
<hbk>
perpendicular
to
the
line
<gd>,
then
the
motion
on
the
arc
<ez>
is
the
opposite
of
the
motion
on
the
arc
<
eh
>,
since
the
one
goes
to
the
right
,
the
other
to
the
left
side
.
The
motion
can
also
occur
in
the
same
direction
,
if
we
imagine
the
distance
of
the
points
staying
the
same
from
<zh> (
text
<zk>).
Likewise
when
the
motion
on
the
arc
<zg>
and
<hd>
towards
<
g
>
and
<
d
>
is
balanced
.
We
also
have
to
assume
the
same
for
the
arcs
<gq>, <dk>
and
for
the
arcs
<qe>
and
<ke>.
We
further
say
that
they
can
move
in
the
same
direction
.
For
we
say
that
the
points
<de>
move
in
the
same
direction
(
this
time
to
the
left
),
when
point
<
e
>
moves
on
the
arc
<ezg>
and
point
<
d
>
on
the
arc
<dke>,
and
their
distance
from
points
<
z
>, <
k
>
as
well
as
their
approach
to
them
remains
the
same
,
so
the
motion
is
still
called
opposite
(
because
<
e
>
moves
up
,
then
down
, <
d
>
down
,
then
up
).
Therefore
the
same
and
the
opposite
are
just
complementary
and
in
any
motion
one
has
to
distinguish
between
the
same
and
the
opposite
.
This
our
explanation
has
to
be
observed
with
equal
circles
.
As
for
different
circles
,
we
shall
demonstrate
it
in
the
following
.
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