The Sleepwalkers (158 page)

At
this
point
Kepler,
for
no
particular
reason,
became
interested
in
the
angle
at
M
–
the
angle
formed
between
the
sun
and
the
centre
of
the
orbit,
as
seen
from
Mars.
This
angle
was
called
the
"optical
equation".
It
varies,
of
course,
as
Mars
moves
along
its
orbit;
its
maximum
value
is
5°
18
′
.
This
is
what
happened
next,
in
Kepler's
own
words:
³¹

"...
I
was
wondering
why
and
how
a
sickle
of
just
this
thickness
(0.00429)
came
into
being.
While
this
thought
was
driving
me
around,
while
I
was
considering
again
and
again
that
...
my
apparent
triumph
over
Mars
had
been
in
vain,
I
stumbled
entirely
by
chance
on
the
secant
^*
of
the
angle
5°
18
′
,
which
is
the
measure
of
the
greatest
optical
equation.
When
I
realized
that
this
secant
equals
1.00429,
I
felt
as
if
I
had
been
awakened
from
a
sleep..."

It
had
been
a
true
sleepwalker's
performance.
At
the
first
moment,
the
reappearance
of
the
number
0.00429
in
this
unexpected
context
must
have
appeared
as
a
miracle
to
Kepler.
But
he
realized
in
a
flash
that
the
apparent
miracle
must
be
due
to
a
fixed
relation
between
the
angle
at
M
and
the
distance
to
S,
a
relation
which
must
hold
true
for
any
point
of
the
orbit;
only
the
manner
in
which
he
had
stumbled
on
that
relation
was
due
to
chance.
"The
roads
that
lead
man
to
knowledge
are
as
wondrous
as
that
knowledge
itself."

Now
at
last,
at
long
last,
after
six
years
of
incredible
labour,
he
held
the
secret
of
the
Martian
orbit.
He
was
able
to
express
the
manner
in
which
the
planet's
distance
from
the
sun
varied
with
its
position,
in
a
simple
formula,
a
mathematical
Law
of
Nature.
But
be
still
did
not
realize
that
this
formula
specifically
defined
the
orbit
as
an
ellipse
.
^†
Nowadays,
a
student
with
a
little
knowledge
of
analytical
geometry
would
realize
this
at
a
glance;
but
analytical
geometry
came
after
Kepler.
He
had
discovered
his
magic
equation
empirically,
but
he
could
no
more
identify
it
as
the
shorthand
sign
for
an
ellipse
than
the
average
reader
of
this
book
can;
it
was
nearly
as
meaningless
to
him.
He
had
reached
his
goal,
but
he
did
not
realize
that
he
had
reached
it.

____________________

The
result
was
that
he
went
off
on
one
more,
last,
wild
goose
chase.
He
tried
to
construct
the
orbit
which
would
correspond
to
his
newly
discovered
equation;
but
he
did
not
know
how,
made
a
mistake
in
geometry,
and
arrived
at
a
curve
which
was
too
bulgy;
the
orbit
was
a
via
buccosa
,
chubby-faced,
as
he
noted
with
disgust.

What
next?
We
have
reached
the
climax
of
the
comedy.
In
his
despair,
Kepler
threw
out
his
formula
(which
denoted
an
elliptic
orbit)
because
he
wanted
to
try
out
an
entirely
new
hypothesis:
to
wit,
an
elliptic
orbit.
It
was
as
if
the
tourist
had
told
the
waiter,
after
studying
the
menu:
"I
don't
want
côtelette
d'agneau
,
whatever
that
is;
I
want
a
lamb
chop."

By
now
he
was
convinced
that
the
orbit
must
be
an
ellipse,
because
countless
observed
positions
of
Mars,
which
he
knew
almost
by
heart,
irresistibly
pointed
to
that
curve;
but
he
still
did
not
realize
that
his
equation,
which
he
had
found
by
chance-plus-intuition,
was
an
ellipse.
So
he
discarded
that
equation,
and
constructed
an
ellipse
by
a
different
geometrical
method.
And
then,
at
long
last,
he
realized
that
the
two
methods
produced
the
same
result.