As
you
can
imagine
,
400
years
ago
,
navigating
the
open
ocean
was
difficult
.
The
winds
and
currents
pushed
and
pulled
ships
off
course
,
and
so
sailors
based
their
directions
on
the
port
they
left
,
attempting
to
maintain
an
accurate
record
of
the
ship
?
s
direction
and
the
____________________
sailed
.
This
process
was
known
as
dead
reckoning
,
because
being
just
half
a
degree
off
could
result
in
sailing
right
past
the
island
that
lay
several
miles
just
over
the
horizon
.
This
was
an
easy
mistake
to
make
.
Thankfully
,
three
____________________
made
modern
navigation
possible
:
sextants
,
clocks
and
the
mathematics
necessary
to
perform
the
required
calculations
quickly
and
easily
.
All
are
important
.
Without
the
right
tools
,
many
sailors
would
be
____________________
to
sail
too
far
from
the
sight
of
land
.
John
Bird
,
an
instrument
maker
in
London
,
made
the
first
device
that
could
measure
the
angle
between
the
sun
and
the
horizon
during
the
day
,
called
a
____________________
.
Knowing
this
angle
was
____________________
,
because
it
could
be
compared
to
the
angle
back
in
England
at
the
exact
same
time
.
Comparing
these
two
angles
was
necessary
to
determine
the
longitude
of
the
ship
.
Clocks
came
next
.
In
1761
,
John
Harrison
,
an
English
____________________
and
carpenter
,
built
a
clock
that
could
keep
accurate
time
at
sea
.
The
timepiece
that
could
maintain
accurate
time
while
on
a
pitching
,
yawing
deck
in
harsh
conditions
was
necessary
in
order
to
know
the
time
back
in
____________________
.
There
was
one
catch
though
:
since
such
a
timepiece
was
handmade
,
it
was
very
expensive
.
So
an
alternate
method
using
lunar
measurements
and
intense
calculations
was
often
used
to
cut
costs
.
The
____________________
to
determine
a
ship
?
s
location
for
each
measurement
could
take
hours
.
But
sextants
and
clocks
weren
?
t
useful
unless
sailors
could
use
these
tools
to
determine
their
position
.
Fortunately
,
in
the
1600s
,
an
amateur
mathematician
had
invented
the
____________________
piece
.
John
Napier
toiled
for
more
than
20
years
in
his
castle
in
Scotland
to
develop
____________________
,
a
calculation
device
.
Napier
?
s
ideas
on
logarithms
involved
the
form
of
one
over
E
and
the
constant
10
to
the
seventh
power
.
Algebra
in
the
early
1600s
was
not
fully
____________________
,
and
Napier
?
s
logarithm
of
one
did
not
equal
zero
.
This
made
the
calculations
much
less
convenient
than
logarithms
with
a
base
of
10
.
Henry
Briggs
,
a
famous
mathematician
at
Gresham
College
in
London
,
read
Napier
?
s
work
in
1614
,
and
the
following
year
made
the
long
journey
to
Edinburgh
to
meet
Napier
.
Briggs
showed
up
unannounced
at
Napier
?
s
____________________
door
and
suggested
that
John
switch
the
base
and
form
of
his
logarithms
into
something
much
simpler
.
They
both
agreed
that
a
base
of
10
with
the
log
of
one
equal
to
zero
would
greatly
simplify
everyday
calculations
.
Today
we
remember
these
as
Briggs
Common
Logarithms
.
Until
the
development
of
electric
____________________
machines
in
the
20th
century
,
any
calculations
involving
multiplication
,
division
,
powers
,
and
extraction
of
roots
with
large
and
small
numbers
were
done
using
logarithms
.
The
history
of
logarithms
isn
?
t
just
a
lesson
in
math
.
There
were
many
players
responsible
for
successful
navigation
.
Instrument
makers
,
astronomers
,
mathematicians
,
and
of
course
sailors
.
____________________
isn
?
t
only
about
going
deep
into
one
?
s
field
of
work
,
it
?
s
about
cross
-
pollination
between
disciplines
too
.
