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Nauticall
Questions
Decimall
fractions
a
Decimall
fraction
is
that
which
hath
for
his
Denominator
the
figure
with
one
or
moe
circles
after
it
,
a
separating
Line
the
number
of
circles
in
the
Denominator
of
any
Decimall
fraction
must
bee
equall
to
the
number
of
places
of
figures
following
the
separating
line
.
it
will
be
convenient
to
note
the
Vnite
place
with
a
circle
before
the
separating
Line
the
first
scruple
,
by
shewing
the
true
reason
of
Decimall
fractions
.
description
and
use
of
the
severall
circles
are
declared
:
the
working
of
proportions
foure
Considerations
Rules
for
finding
out
the
true
value
of
the
fourth
or
emergent
number
sought
for
:
Cubing
of
numbers
new
additament
for
Navigation
tenth
Circle
fourth
Circle
fourth
Circle
Instrumentall
difference
reall
or
true
difference
line
of
the
Radius
Instrumentall
difference
true
and
reall
line
of
the
Radius
fourth
Circle
Instrumentall
difference
reall
or
true
difference
line
of
the
Radius
fourth
Circle
Instrumentall
difference
to
make
the
reall
.
true
difference
true
difference
line
of
the
Radius
the
reall
or
true
difference
line
of
the
Radius
line
of
the
Radius
line
of
the
Radius
line
of
the
Radius
true
difference
true
difference
reall
difference
line
of
the
Radius
difference
of
degrees
(
viz
true
difference
reall
difference
line
of
the
Radius
by
the
fifth
rule
The
termes
of
a
ratio
being
proposed
,
to
find
the
reall
or
true
difference
betweene
their
places
in
the
fourth
circle
of
the
Instrument
.
If
either
the
numbers
given
be
of
the
same
degree
:
or
else
differ
but
one
degree
,
the
line
of
falling
betweene
them
:
they
differ
lesse
then
a
circuite
.
If
the
numbers
bee
not
of
the
same
degree
:
they
differ
so
many
whole
ciruits
as
they
doe
degrees
.
But
yet
if
the
line
of
fall
betweene
them
:
they
differ
one
circuit
lesse
.
The
antecedent
terme
of
a
ratio
being
given
,
together
with
the
reall
or
true
difference
of
the
termes
in
the
Instrument
:
to
find
out
the
consequent
terme
.
If
thereall
difference
be
lesse
then
one
circuit
,
and
the
line
of
fall
not
betweene
the
places
of
the
two
termes
:
the
numbers
are
both
of
the
same
degree
.
But
if
the
line
of
fall
betweene
the
places
:
they
differ
one
degree
.
If
the
reall
difference
containe
one
or
moe
circuits
:
the
numbers
differ
so
many
degrees
as
there
are
whole
circuits
.
But
if
the
line
of
fall
betweene
the
places
:
they
differ
one
degree
more
.
first
,
second
first
part
For
the
use
of
Navigation
are
added
two
circles
,
the
sixth
and
the
seventh
:
and
a
small
alteration
in
the
fifth
.
fifth
circle
two
circuits
first
circuit
second
circuit
figures
are
doubly
noted
neerer
side
first
circuit
further
side
second
circuit
Decimall
parts
seventh
circles
sixth
circle
seventh
circle
Elevations
of
the
Pole
.
Two
Latitudes
being
given
in
the
same
Hemisphere
,
that
is
both
Northerne
,
or
both
Southerne
,
to
find
the
summe
of
all
the
Secants
betweene
them
.
lesser
Latitude
line
of
the
Radius
fifth
circle
number
of
Secants
betweene
the
two
Latitudes
proposed
.
number
of
Secants
line
of
the
Radius
fifth
circle
number
of
Secants
betweene
the
two
Latitudes
proposed
.
The
summe
of
all
the
Secants
between
two
Latitudes
in
the
same
Hemisphaere
being
given
,
together
with
one
of
the
Latitudes
,
to
find
the
other
Latitude
.
line
of
the
Radius
summe
of
Secants
given
(
in
the
fifth
circle
)
:
line
of
the
Radius
summe
of
the
Secants
greater
Latitude
summe
of
Secants
line
of
the
Radius
fifth
circle
line
of
the
Radius
the
degrees
of
the
other
Latitude
sought
for
.
summe
of
Secants
greater
Latitude
line
of
the
Radius
fifth
circle
greater
Latitude
lesser
Latitude
if
the
two
Latitudes
be
in
the
severall
Hemisphaeres
,
that
is
one
Northerne
and
the
other
Southerne
,
the
manner
of
working
differeth
in
effect
but
little
from
the
former
.
summe
of
the
Secants
North
side
of
the
Aequinoctiall
South
side
the
line
of
the
Radius
either
of
the
Latitudes
given
the
line
of
the
Radius
the
line
of
the
Radius
other
Latitude
former
Latitude
line
of
the
Radius
fifth
circle
summe
of
Secants
sought
for
the
summe
of
all
the
Secants
betweene
two
Latitudes
,
of
which
one
is
on
the
North
side
of
the
Aequinoctiall
,
and
the
other
on
the
South
side
,
being
given
;
together
with
one
of
the
Latitudes
,
to
find
the
other
Latitude
:
summe
of
the
Secants
Latitude
degr
line
of
the
radius
fift
circle
line
of
the
radius
the
latitude
Latitude
sought
for
you
may
more
easily
find
out
the
summe
of
the
Secants
betweene
any
two
Latitudes
given
,
fifth
circle
:
keepe
it
fifth
circle
:
keepe
that
two
numbers
kept
in
mind
the
summes
of
the
Secants
for
the
two
Latitudes
given
:
subducted
one
out
of
the
other
,
Latitudes
are
both
in
the
same
Hemisphere
:
added
together
,
if
the
Latitudes
are
in
diverse
Hemisphaeres
.
The
summe
of
the
Secants
and
one
of
the
Latitudes
being
given
,
you
may
find
out
the
other
Latitude
Latitude
given
fifth
circle
adde
the
summe
of
the
Secants
,
lesser
of
the
two
Latitudes
out
of
it
subduct
the
summe
of
the
Secants
,
greater
of
the
two
Latitudes
two
Latitudes
are
in
the
contrary
Hemisphaeres
,
fifth
circle
summe
of
the
Secants
other
Latitude
in
what
place
the
ship
is
at
any
time
.
place
of
the
ship
at
sea
is
estimated
and
understood
by
comparing
it
with
any
knowne
place
:
that
is
how
much
the
same
is
situated
from
the
place
,
where
the
Ship
is
,
either
toward
the
North
or
South
,
which
is
called
the
difference
of
Latitude
:
or
else
toward
the
East
or
West
,
which
is
called
the
difference
of
Longitude
.
how
farre
wide
of
the
Aequinoctiall
how
farre
the
Meridian
is
distant
from
the
Meridian
true
situation
is
taken
by
observation
of
the
Meridionall
altitude
,
to
teach
the
use
of
my
Instrument
only
,
in
tracing
the
Ships
course
.
Master-piece
of
Nauticall
science
variation
of
the
Magneticall
needle
Meridian
is
difficultly
to
be
had
at
Sea
,
where
the
chiefest
use
of
Longitude
is
:
motion
of
the
Moone
true
place
of
the
Moone
,
to
be
observed
moment
of
the
Moones
comming
into
the
Meridian
,
true
motion
of
the
Moone
is
so
exactly
knowne
,
nor
observation
can
at
Sea
bee
so
precisely
made
,
subtile
a
businesse
water
is
more
probable
chymicall
spirit
three
glasses
difference
of
Longitudes
true
time
time
by
observation
difference
of
those
times
difference
of
Longitude
keeping
the
reckoning
of
Longitude
frequent
observations
course
of
the
Ship
Latitude
observed
foure
things
therfore
whereof
a
Sea-man
should
be
most
carefull
&
circumspect
,
the
angle
of
inclination
with
the
Meridian
,
on
which
the
Ship
maketh
her
course
:
which
angle
is
directed
by
the
Compasse
:
Rumbe
or
poynt
of
the
Compasse
.
A
point
containeth
degr
a
quarter
of
a
point
therfore
is
degr
:
table
of
Rumbes
measure
of
the
Ships
way
on
the
Rumbe
or
point
,
a
mile
a
minute
of
a
degree
miles
on
a
great
circle
give
the
difference
of
one
whole
degree
.
way
of
the
Ship
in
hundreth
parts
of
a
degree
,
measure
or
quantity
of
the
Ships
way
is
found
by
the
Logg-line
and
minute-glasse
.
observation
of
Latitude
as
oft
as
it
may
be
keeping
of
time
Rumbe
and
way
of
the
Ship
,
continuall
companions
and
faithfull
guides
of
the
Sea-man
,
exactly
made
reverend
Divine
and
learned
Mathematician
Master
William
Barlow
in
his
Navigators
supply
course
be
steered
aright
just
point
or
Rumbe
negligence
of
a
loose
and
idle
Steeresman
:
measuring
of
the
quantity
of
the
ships
way
,
It
must
first
be
knowne
how
many
English
feet
of
inches
to
the
foot
,
answer
to
one
degree
of
〈◊〉
great
circle
upon
the
earth
.
English
mile
by
statute
is
the
length
of
furlong
is
a
perch
is
feet
a
mile
containeth
feet
in
length
mistaken
,
that
of
such
miles
make
a
degree
English
mile
drawne
from
Barley
cornes
of
our
miles
answer
to
a
degree
,
Diverse
wayes
for
finding
out
the
true
compasse
of
the
earth
:
wrought
on
of
observation
too
short
dilligence
of
the
practiser
way
which
is
by
the
height
of
an
hill
,
and
a
tangent
line
from
thence
to
the
superficies
of
the
sea
,
refraction
of
the
vapours
be
an
excellent
worke
,
if
the
height
of
the
Pole
at
two
townes
of
this
Land
,
distant
North-ward
one
from
the
other
some
scores
,
or
rather
hundreds
of
miles
,
being
with
Instruments
of
sufficient
magnitude
by
some
learned
Artists
exactly
observed
:
there
were
also
imployed
certaine
skilfull
Surveyors
(
such
as
are
indeed
lovers
of
art
and
truth
)
to
take
the
true
distances
and
positions
from
place
to
place
betweene
the
said
townes
.
with
good
plaine
tables
with
much
facility
three
principles
that
if
with
a
levelling
Instrument
set
up
in
any
place
parallel
to
the
Horizon
a
man
take
a
true
levell
unto
another
place
:
the
visuall
line
by
which
he
levelled
,
shall
be
a
tangent
to
such
an
arch
of
a
great
circle
on
the
earth
,
as
is
contained
betweene
the
station
and
the
marke
:
Because
that
the
visuall
line
,
together
with
the
two
lines
imagined
,
out
of
the
center
of
the
earth
,
doe
include
a
right
angled
Triangle
▪
having
the
right
angle
at
the
levell
.
that
if
the
same
Instrument
he
set
just
even
with
the
former
mark
,
and
you
levell
backward
to
the
former
station
,
this
last
visuall
line
shall
overshoot
the
former
place
of
the
Instrument
:
and
shall
inclose
a
new
and
greater
right-angled
triangle
,
hauing
the
right
angle
at
the
second
station
.
the
tangent
of
the
arch
the
number
of
feet
answering
to
a
degree
upon
the
earth
.
concerning
the
Instrument
,
place
,
and
time
.
levelling
Instrument
a
channell
for
water
doth
continually
exhale
vapours
to
avoyd
any
drynesse
collect
it selfe
stand
in
a
heape
diffuseth
it selfe
in
pursuit
of
any
moysture
.
sight-hole
be
very
small
,
it
hindreth
our
seeing
:
large
,
it
admitteth
too
many
visiverayes
the
true
and
individuall
point
of
the
object
.
a
ledge
a
blacke
stroke
to
be
in
a
square
white
board
,
marke
to
levell
at
two
edges
of
the
ledge
place
for
the
tryall
of
this
experiment
,
a
plaine
field
feet
,
which
is
the
tenth
part
of
a
mile
.
ground
to
be
handsomely
plained
and
beaten
,
time
for
making
your
observation
Midsummer
,
in
a
seasonable
,
constant
,
drye
,
and
calme
weather
:
Easterne
station
eleven
a
clocke
in
the
forenoone
.
Westerne
station
one
a
clocke
in
the
afternoone
accurate
cautions
the
formes
of
the
levelling
Instrument
and
of
the
marke
.
feete
to
answer
to
a
degree
upon
the
earth
.
measure
of
the
Ships
motion
or
way
is
observed
by
the
watch-glasse
and
Log-line
:
the
number
of
seconds
which
the
Watch-glasse
runneth
number
of
feet
vered
in
the
Logg-line
while
the
glasse
is
running
,
a
rule
to
know
how
many
hundreth
parts
of
a
degree
the
Ship
sayleth
in
one
houre
so
many
feet
gone
in
an
houre
centesimes
of
a
degree
gone
in
an
houre
.
generall
rule
for
Centesimes
how
many
minutes
of
a
degree
the
Ship
sayleth
in
one
houre
:
sexagesimes
of
a
degree
gon
in
an
houre
.
generall
rule
for
sexagesimes
true
finding
out
of
the
way
,
which
the
Ship
maketh
in
an
houre
,
estimated
in
the
parts
of
a
degree
,
is
the
maine
ground
and
principle
,
by
which
the
place
of
her
being
both
for
longitude
and
latitude
is
argued
and
computed
:
in
two
Examples
centesimes
of
a
degree
(
viz
proportion
of
the
Ships
sayling
for
one
houre
way
the
Ship
hath
made
certaine
reductions
To
convert
degrees
or
houres
into
Minutes
,
to
convert
them
into
seconds
reduce
minutes
into
degrees
or
houres
,
to
reduce
seconds
into
degrees
or
houres
,
convert
minutes
of
degrees
or
houres
into
centesimes
reduce
centesimes
of
degrees
or
houres
into
minutes
:
motion
of
the
Ship
upon
a
Rumbe
is
either
circular
,
or
winding
with
a
kind
of
spirall
line
.
foure
cardinall
points
▪
North
or
South
East
or
West
East
or
West
All
great
circles
are
equall
one
to
another
,
and
have
equall
degrees
:
but
the
parallels
are
greater
or
lesser
one
then
another
;
and
consequently
have
greater
or
lesser
degrees
,
as
every
one
is
neerer
or
farther
distant
from
the
Aequinoctiall
.
degrees
both
of
the
greater
and
lesser
circles
,
Iust
Degrees
proper
degrees
of
such
a
parallel
.
Theoremes
for
the
computation
of
the
motion
of
the
ship
:
Iust
Degrees
Iust
Degrees
Iust
Degrees
a
Ship
saile
under
a
Meridian
,
South
Rumbe
the
Ship
sayle
under
the
Aequinoctiall
upon
the
very
line
it selfe
Eastward
or
Westward
:
if
the
Ship
sayle
directly
East
or
West
under
any
parallel
circle
,
Iust
Degrees
Iust
Degrees
sine
of
the
complement
of
the
parallel
,
way
of
the
ship
upon
that
parallel
in
just
degrees
,
difference
of
longitude
Iust
degrees
proper
degrees
difference
of
longitude
sought
is
degr
:
circular
sayling
foure
cardinall
points
great
circle
a
parallel
true
measure
of
the
ships
way
:
oblique
sayling
when
the
Ship
runneth
upon
some
Rumbe
between
any
of
the
foure
cardinall
points
,
making
an
oblique
angle
with
the
Meridian
:
a
Rumb
These
Helices
or
spirall
lines
ought
to
consist
of
most
minute
and
insensible
,
yea
indivisible
parts
:
the
spirall
line
of
the
ships
course
be
recalled
to
a
precise
exactnesse
.
ordinary
canon
of
the
Meridian
divided
according
to
Mercator
:
sixth
and
seventh
Circles
of
this
Instrument
,
ground
of
oblique
sayling
severall
questions
in
Navigation
way
of
the
Ship
in
degrees
difference
of
latitude
s
co
way
of
the
ship
in
degrees
summe
of
the
bases
of
all
the
triangles
intercepted
betweene
the
parallels
of
those
two
places
.
difference
of
latitude
summe
of
the
bases
of
all
the
triangles
intercepted
,
&c.
summe
of
the
secants
of
all
the
parallels
base
of
one
of
those
triangles
,
difference
of
longitude
sexagesime
or
centesime
,
&c.
summe
of
the
secants
of
all
the
parallels
summe
of
the
secants
of
all
the
parallels
difference
of
longitude
degrees
of
the
ships
way
differences
both
of
latitude
and
longitude
,
summe
of
the
secants
of
parallels
,
fourth
circle
degrees
and
Decimall
parts
of
degrees
.
in
their
owne
circles
to
worke
these
questions
with
your
pen
:
the
tables
for
the
division
of
the
Meridian
line
according
to
Mercator
:
Wrights
Errours
of
Navigation
Willibrordus
Snellius
his
Tiphys
Batavus
Gunters
Booke
latitudes
in
degrees
and
minutes
;
with
Decimalls
of
Minutes
,
Decimalls
of
Degrees
the
difference
of
latitude
An
Example
measure
of
the
Ships
way
difference
of
longitude
betweene
the
Meridians
of
the
two
places
.
fifth
circle
difference
of
longitude
An
Example
angle
of
the
inclination
of
the
Rumb
leading
between
those
two
places
,
with
the
meridian
:
An
Example
summe
of
the
secants
of
the
parallels
between
the
latitude
degr
:
given
,
and
the
latitude
of
the
place
wherein
the
Ship
is
.
summe
of
the
parallels
for
the
latitude
sought
:
two
operations
The
first
The
second
An
Example
difference
of
latitude
betweene
the
beginning
and
place
where
the
Ship
is
.
latitude
of
the
place
whither
he
Ship
is
come
.
(
viz
Which
arch
of
degr
is
the
difference
of
longitude
sought
for
.
two
operations
The
first
The
second
An
Example
(
viz
the
angle
of
the
Rumbe
arch
of
degr
is
the
difference
of
longitude
sought
for
.
two
operations
The
first
The
second
An
Example
the
angle
of
inclination
of
the
Rumbe
,
with
the
meridian
:
measure
of
the
Ships
way
in
degrees
.
two
operations
The
first
The
second
An
Example
is
the
summe
of
the
secants
of
the
parallels
betweene
the
latitude
of
degr
:
and
the
other
latitude
sought
is
the
quantity
of
the
Ships
way
in
degrees
.
rule
of
false
position
the
use
of
the
Instrument
in
the
solution
of
all
nauticall
Questions
:
certain
problemes
how
by
reasonable
conjecture
the
course
of
the
Ship
may
bee
most
probably
rectifyed
,
when
the
reckoning
thereof
by
the
Compasse
and
way
estimated
,
shall
bee
found
to
disagree
from
the
coelestiall
observations
:
hidden
motions
of
the
Seas
,
how
they
are
to
be
observed
,
and
how
to
be
considered
of
in
computing
the
motion
of
the
Ship
:
bias
)
of
the
Ship
Masters
of
Ships
Elias
Allen
Calculation
of
Triangles
resolution
of
Arithmetical
Quaestions
two
Rulers
S
,
T
,
N
,
E
,
line
of
the
Radius
Vnite
line
line
of
Sines
two
lines
of
Tangents
line
of
Numbers
line
of
Aequall
parts
a
socket
a
pinnicide
the
line
of
the
Radius
single
line
of
Sines
,
Tangents
,
line
of
Tangents
,
and
numbers
are
continued
double
divisions
a
line
of
Aequall
parts
line
of
Latitudes
Elevations
of
the
Pole
two
Rulers
Circles
of
Proportion
hold
the
Transversary
in
your
left
hand
,
with
the
end
at
which
the
line
of
the
Radius
or
Vnite
line
is
,
from
you
ward
:
turning
that
side
of
the
Ruler
upward
,
on
which
the
line
of
the
kind
of
the
first
terme
is
,
whether
it
be
Number
,
Sine
,
or
Tangent
:
and
therein
seeke
both
the
first
terme
,
and
the
other
which
is
homogene
to
it
.
Then
take
the
Staffe
in
your
right
hand
with
that
side
upward
,
in
which
the
line
of
the
kind
of
the
fourth
terme
sought
for
is
:
and
seeke
in
it
the
terme
homogene
to
the
fourth
.
Apply
this
to
the
first
terme
in
the
Transversarie
:
and
the
other
homogene
terme
shall
in
the
Staffe
shew
the
fourth
terme
.
As
the
difference
of
the
Tangents
of
the
arches
cut
in
either
Station
,
is
to
the
distance
betweene
the
stations
:
so
is
the
Tangent
of
the
lesser
arch
,
to
the
neerer
distance
from
the
Tower
.
As
the
Radius
is
to
the
Tangent
of
the
greater
arch
;
so
is
the
neerer
distance
found
,
to
the
height
.
(
that
is
by
applying
the
Vnite
line
of
the
Staffe
,
to
the
Tangent
in
the
Transversarie
,
if
the
arch
be
lesse
then
degr
:
but
if
the
arch
exceed
degr
:
by
applying
the
said
Vnite
line
,
unto
the
arch
it selfe
,
or
the
complement
thereof
,
which
in
the
Transversarie
is
all
one
)
As
the
difference
of
the
tangents
of
the
complements
of
the
arches
cut
in
either
station
,
is
to
the
Radius
So
is
the
distance
betweene
the
stations
,
to
the
altitude
.
As
the
Radius
is
to
the
Sine
of
So
is
the
distance
(
of
the
Sunne
from
the
next
Aequinoctial
point
)
in
minutes
,
or
Decimals
,
to
the
declination
in
minutes
,
or
Decimals
.
some
cases
wherein
the
worke
differeth
from
the
ordinary
manner
.
by
Quadrats
triplicated
proportions
The
difference
of
a
Quadrat
from
a
Quadrat
is
double
the
difference
between
their
sides
.
the
difference
of
a
Cube
from
a
Cube
is
triple
the
difference
betweene
their
sides
.
Example
in
pag.
of
Navigation
double
divisions
on
the
fourth
side
of
the
Staffe
,
line
of
Latitudes
or
eleuations
of
the
Pole
on
the
fourth
side
of
the
Staffe
:
line
of
Aequall
parts
line
of
Aequall
parts
the
degrees
serving
for
the
Crosse-staffe
are
placed
both
on
the
Staffe
and
Transversarie
,
on
the
same
side
on
which
the
line
of
Sines
is
.
the
Transversarie
is
to
be
set
in
the
Socket
so
that
it
may
stand
on
the
right
hand
of
the
Staffe
.
Gentle
Reader
,
by
reason
of
my
absence
,
whilest
this
Booke
was
in
the
Presse
,
it
is
no
mervaile
though
some
faults
have
escaped
,
which
you
will
be
pleased
to
amend
thus
.
third
circle
p.
lin.
ult.
and
And
also
in
the
Aequations
which
haue
a
magnitude
aequall
to
a
fraction
:
the
same
magnitude
together
with
the
note
of
aequality
,
ought
to
be
set
right
against
the
line
that
is
betweene
the
Numerator
and
Denominator
of
the
fraction
,
as
in
these
,
In
the
diagramme
of
right-lined
Triangles
the
leter
is
wanting
at
the
perpendiculer
.
And
in
the
the
angle
ought
to
have
been
marked
with
a
little
line
.
pag.
lin.
ult
sines
pag.
&
lin.
lin.
ult.
for
the
line
of
the
Radius
the
line
of
the
radius
Latitude
sought
for