Heron Alexandrinus
,
Mechanica
,
1999
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[7]
Sometimes
however
the
motion
of
the
smaller
and
the
larger
circles
can
be
equally
fast
,
even
when
the
circles
are
attached
to
the
same
center
and
rotate
around
it
.
Let
us
assume
two
circles
attached
to
the
same
center
<
a
>
and
let
a
tangent
to
the
larger
circle
be
given
,
namely
the
line
<bb'>.
If
we
further
connect
the
points
<
a
>, <
b
>,
then
line
<
ab
>
is
perpendicular
to
line
<bb'>
and
line
<bb'>
is
parallel
to
line
<gg'>;
then
line
<gg'>
is
a
tangent
of
the
smaller
circle
.
If
we
further
draw
through
point
<
a
>
a
line
that
is
parallel
to
these
lines
,
namely
line
<aa'>,
then
if
we
imagine
the
larger
circle
rolling
on
line
<bb'>,
the
smaller
circle
will
roll
by
running
through
the
line
<gg'>.
When
now
the
larger
circle
has
made
one
rotation
,
we
see
that
the
smaller
one
also
has
made
one
rotation
,
so
that
the
position
of
the
circles
is
the
position
of
those
circles
whose
center
is
at
<
a
'>
and
the
position
of
line
<
ab
>
is
that
which
is
taken
by
line
<
a
'
b
'>.
Therefore
line
<bb'>
equals
line
<gg'>.
Line
<bb'>
however
is
the
line
on
which
the
larger
circle
rolls
when
it
makes
one
rotation
and
line
<gg'>
is
the
line
on
which
the
smaller
circle
rolls
when
it
makes
one
rotation
;
thus
the
motion
of
the
smaller
circle
is
equally
fast
as
that
of
the
larger
one
,
because
line
<bb'>
equals
line
<gg'>.
Things
that
run
through
the
same
distance
in
the
same
time
,
however
,
have
equal
speed
and
equal
motion
.
One
might
think
this
sentence
is
absurd
,
since
it
is
impossible
that
the
circumference
of
the
larger
circle
should
equal
the
circumference
of
the
smaller
one
.
We
now
say
that
not
only
the
circumference
of
the
smaller
circle
has
rolled
on
line
<gg'>,
but
that
the
smaller
circle
also
runs
through
the
path
of
the
larger
one
,
thus
we
see
that
the
smaller
circle
through
two
motions
reaches
the
same
speed
as
the
larger
one
;
then
,
if
we
imagine
the
larger
circle
rolling
,
the
smaller
one
,
however
,
not
rolling
,
but
only
attached
to
the
point
<
g
>,
then
it
will
in
the
same
time
cover
line
<gg'>;
then
the
center
<
a
>
covers
in
this
time
line
<aa'>.
This
however
equals
the
lines
<bb'>
and
<gg'>;
thus
the
continuous
rolling
of
the
smaller
circle
does
not
make
any
difference
in
the
motion
and
as
a
consequence
the
length
of
the
distance
of
the
larger
circle
is
the
same
as
that
covered
by
the
small
circle
;
for
we
see
that
the
center
,
without
rolling
,
covers
the
same
distance
,
due
to
the
motion
the
large
circle
is
in
.
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