Heron Alexandrinus
,
Mechanica
,
1999
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[5]
If
we
imagine
a
third
constructed
circle
which
touches
the
circle
with
the
center
<
b
>,
so
we
shall
prove
for
the
third
circle
what
we
mentioned
about
the
first
one
.
For
if
the
first
circle
is
moving
in
the
direction
opposite
from
the
second
one
,
the
second
one
however
moves
opposite
to
the
third
,
then
the
motion
of
the
first
circle
is
the
same
as
that
of
the
third
.
For
if
something
is
moving
in
the
same
manner
as
something
else
,
this
however
moves
in
the
opposite
direction
of
a
third
thing
,
so
is
the
first
thing
moving
in
the
direction
opposite
to
the
third
.
If
further
a
fourth
circle
is
present
,
we
proceed
after
the
same
method
.
In
general
,
what
ensues
from
the
three
circles
will
occur
with
all
circles
whose
number
is
odd
and
what
ensues
from
the
two
circles
takes
place
with
all
circles
whose
number
is
even
.
But
one
not
only
sees
with
two
and
more
circles
that
the
motion
is
now
equal
,
now
opposite
,
but
in
one
circle
one
sees
the
same
point
move
now
in
one
direction
,
now
in
its
opposite
.
For
when
the
moving
point
starts
moving
at
any
point
,
it
does
not
stop
moving
in
the
same
direction
until
it
has
run
through
a
semicircle
;
when
it
now
runs
through
the
second
semicircle
it
moves
in
the
direction
opposite
to
it
.
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