THE
FIRST
BOOKE
OF
THE
CROSSE-STAFFE
.
CHAP.
I.
Of
the
description
of
the
Staffe
.
THe
Crosse-Staffe
is
an
instrument
wel
knowne
to
our
Sea-men
,
and
much
vsed
by
the
ancient
Astronomers
and
others
,
seruing
Astronomically
for
obseruation
of
altitude
and
angles
of
distance
in
the
heauens
,
Geometrically
for
perpendicular
heights
and
distances
on
land
and
sea
.
The
description
and
seuerall
vses
of
it
are
extant
in
print
,
by
Gemma
Frisius
in
Latin
,
in
English
by
Dr.
Hood
.
I
differ
something
from
them
both
,
in
the
proiection
of
this
Staffe
,
but
so
,
as
their
rules
may
be
applied
vnto
it
,
and
all
their
propositions
be
wrought
by
it
:
and
therefore
referring
the
Reader
to
their
bookes
,
I
shal
be
briefe
in
the
explanation
of
that
which
may
be
applied
from
theirs
vnto
mine
,
and
so
come
to
the
vse
of
those
lines
which
are
of
my
addition
,
not
extant
heretofore
.
The
necessary
parts
of
this
Instrument
are
fiue
:
the
Staffe
,
the
Crosse
,
and
the
three
sights
.
The
Staffe
which
I
made
for
my
owne
vse
,
is
a
full
yard
in
length
,
that
so
it
may
serue
for
measure
.
The
Crosse
belonging
to
it
is
26
inches
⅕
betweene
the
two
outward
sights
.
If
any
would
haue
it
in
a
greater
forme
,
the
proportion
betweene
the
Staffe
and
the
Crosse
,
may
be
such
as
360
vnto
262.
The
lines
inscribed
on
the
Staffe
are
of
foure
sorts
.
One
of
them
serues
for
measure
and
protraction
:
one
for
obseruation
of
angles
:
one
for
the
Sea-chart
;
and
the
foure
other
for
working
of
proportions
in
seuerall
kinds
.
The
line
of
measure
is
an
inch
line
,
and
may
be
knowne
by
his
equall
parts
.
The
whole
yard
being
diuided
equally
into
36
inches
,
and
each
inch
subdiuided
,
first
into
ten
parts
,
and
then
each
tenth
part
into
halfes
.
The
line
for
obseruation
of
angles
may
be
knowne
by
the
double
numbers
set
on
both
sides
of
the
line
,
beginning
at
the
one
side
at
20
,
and
ending
at
90
:
on
the
other
side
at
40
,
and
ending
at
180
:
and
this
being
diuided
according
to
the
degrees
of
a
quadrant
,
I
call
it
the
tangent
line
on
the
Staffe
.
The
next
line
is
the
meridian
of
a
Sea-chart
,
according
to
Mercators
proiection
from
the
Equinoctiall
to
58
gr.
of
latitude
,
and
may
be
knowne
by
the
letter
M
,
and
the
numbers
1.
2.
3.
4.
vnto
58.
The
lines
for
working
of
proportions
,
may
be
knowne
by
their
vnequall
diuisions
,
and
the
numbers
at
the
end
of
each
line
.
1
The
line
of
numbers
noted
with
the
letter
N
,
diuided
vnequally
into
1000
parts
,
and
numbred
with
1.
2.
3.
4.
vnto
10.
2
The
line
of
artificiall
tangents
is
noted
with
the
letter
T
,
diuided
vnequally
into
45
degrees
,
and
numbred
both
ways
,
for
the
Tangent
and
the
complement
.
3
The
line
of
artificiall
sines
,
noted
with
the
letter
S
,
diuided
vnequally
into
90
degrees
,
and
numbred
with
1.
2.
3.
4.
vnto
90.
4
The
line
of
versed
sines
for
more
easie
finding
the
houre
and
azimuth
,
noted
with
V
,
diuided
vnequally
into
about
164
gr.
50
m.
numbred
backward
with
10.
20.
30.
vnto
164.
Thus
there
are
seuen
lines
inscribed
on
the
Staffe
:
there
are
fiue
lines
more
inscribed
on
the
Crosse
.
1
A
Tangent
line
of
36
gr.
3
m.
numbred
by
5.
10.
15.
●ledge●nto
35
:
the
midst
whereof
is
at
20
gr
;
and
therefore
I
call
it
●ledge●he
tangent
of
20
;
and
this
hath
respect
vnto
20
gr.
in
the
Tangent
on
the
Staffe
.
2
A
Tangent
line
of
49
gr.
6
m.
numbred
by
5.
10.
15.
vn●ledge●o
45
;
the
midst
whereof
is
at
30
gr.
and
hath
respect
vnto
●ledge●0
gr.
in
the
Tangent
on
the
Staffe
,
whereupon
I
call
it
the
●ledge●angent
of
30.
3
A
line
of
inches
numbred
with
1.
2.
3.
vnto
26
;
each
inch
●ledge●qually
subdiuided
into
ten
parts
,
answerable
to
the
inch
line
●ledge●pon
the
Staffe
.
4
A
line
of
seuerall
chords
,
one
answerable
to
a
circle
of
●ledge●welue
inches
semidiameter
,
numbred
with
10.
20.
30.
vnto
●ledge●0
:
another
to
a
semidiameter
of
a
circle
of
six
inches
;
and
●ledge●he
third
to
a
semidiameter
of
a
circle
of
three
inches
;
both
●ledge●umbred
with
10.
20.
30.
vnto
90.
5
A
continuation
of
the
meridian
line
from
57
gr.
of
la●ledge●tude
vnto
76
gr
;
and
from
76
gr.
to
84
gr.
For
the
inscription
of
these
lines
.
The
first
for
measure
is
●ledge●qually
diuided
into
inches
and
tenth
parts
of
inches
.
The
tangent
on
the
Staffe
for
obseruation
of
angles
,
with
●ledge●e
tangent
of
20
and
the
tangent
of
30
on
the
Crosse
,
may
●ledge●l
three
be
inscribed
out
of
the
ordinary
table
of
tangents
.
The
●ledge●affe
being
36
inches
in
length
;
the
Radius
for
the
tangent
●ledge●
the
Staffe
will
be
13
inches
and
103
parts
of
1000
:
so
the
●ledge●hole
line
will
be
a
tangent
of
70
gr.
and
must
be
numbred
●ledge●
their
complements
,
&
the
double
of
their
complements
,
●ledge●e
tangent
of
10
gr.
being
numbred
with
80
and
160.
The
Radius
for
the
tangent
of
20
on
the
Crosse
,
will
be
6
inches
,
and
the
whole
line
betweene
the
sights
a
tangent
36
gr.
3
m.
according
as
it
is
numbred
.
The
Radius
for
the
●ledge●ngent
of
30
gr.
on
the
Crosse
,
will
be
22
inches
and
695
●ledge●rts
of
1000
:
so
the
whole
line
betweene
the
sights
wil
con●ledge●ine
a
tangent
of
49
gr.
6
m.
in
such
sort
as
they
are
num●ledge●ed
.
The
meridian
line
may
be
inscribed
out
of
the
Table
●ledge●hich
I
set
downe
for
this
purpose
in
the
vse
of
the
Sector
.
The
line
of
numbers
may
be
inscribed
out
of
the
first
Chiliad
of
Mr.
Briggs
Logarithmes
:
&
the
rest
of
the
lines
of
proportion
out
of
my
Canon
of
artificiall
sines
and
tangents
;
and
in
recompence
thereof
this
booke
will
serue
as
a
comment
to
explane
the
vse
of
my
Canon
.
CHAP.
II.
The
vse
of
the
lines
of
inches
for
perpendicular
heights
and
distances
.
IN
taking
of
heights
and
distances
,
the
Staffe
may
be
held
in
such
sort
,
that
it
may
be
euen
with
the
distance
,
and
the
Crosse
parallell
with
the
height
:
and
then
if
the
eye
at
the
beginning
of
the
Staffe
shall
see
his
marks
by
the
inward
sides
of
the
two
first
sights
,
there
will
be
such
proportion
betweene
the
distance
and
the
height
,
as
is
betweene
the
part
intercepted
on
the
Staffe
and
the
Crosse
.
Which
may
be
farther
explaned
in
these
propositions
.
geometric illustration
1
To
find
an
height
at
one
station
,
by
knowing
the
distance
.
Set
the
middle
sight
vnto
the
distance
vpon
the
Staffe
the
height
will
be
found
vpon
the
Crosse
.
For
As
the
segment
of
the
Saffe
vnto
the
segment
on
the
Crosse
:
So
is
the
distance
giuen
,
vnto
the
height
.
As
if
the
distance
AB
being
knowne
to
be
256
feete
,
it
were
required
to
find
the
height
BC
:
first
I
place
the
mid●le
sight
at
25
inches
and
6
parts
of
10
;
then
holding
the
Staffe
leuell
with
the
distance
,
I
raise
the
Crosse
,
parallell
vnto
the
height
,
in
such
sort
,
as
that
my
eye
may
see
from
A
the
beginning
of
the
inches
on
the
Staffe
by
the
sight
E
,
at
the
beginning
of
the
inches
on
the
Crosse
vnto
the
mark
C
:
which
being
done
,
if
I
find
19
inches
and
2
parts
of
10
intercepted
on
the
Crosse
betweene
the
sights
at
E
and
D
,
I
would
say
the
height
BC
were
192
feete
.
Or
if
the
obseruation
were
to
be
made
before
the
distance
were
measured
,
I
would
set
the
middle
sight
either
vnto
10
inches
,
or
12
,
or
16
,
or
20
,
or
24
,
or
some
such
other
number
as
might
best
be
diuided
into
seuerall
parts
,
and
then
worke
by
proportion
.
As
if
in
the
former
example
the
middle
sight
were
at
24
on
the
Staffe
,
and
18
on
the
Crosse
,
it
should
seem
that
the
height
is
¾
of
the
distance
;
and
therefore
the
distance
being
256
,
the
height
should
be
192.
2
To
finde
an
height
,
by
knowing
some
part
of
the
same
height
.
As
if
the
height
from
G
to
C
were
knowne
to
be
48
,
and
it
were
required
to
find
the
whole
height
BC
:
either
put
the
third
sight
or
some
other
running
sight
vpon
the
Crosse
betweene
the
eye
and
the
marke
G.
For
then
As
the
difference
betweene
the
sights
,
vnto
the
whole
segment
of
the
Crosse
:
So
is
the
part
of
the
height
giuen
,
vnto
the
whole
height
.
If
then
the
difference
betweene
the
sights
E
and
F
,
shall
be
45
,
and
the
segment
of
the
Crosse
ED
180
,
the
whole
height
BC
will
be
found
to
be
192.
3
To
find
an
height
at
two
stations
,
by
knowing
the
difference
of
the
same
stations
.
As
the
difference
of
segments
on
the
Staffe
,
vnto
the
difference
of
stations
:
So
is
the
segment
of
the
Crosse
,
vnto
the
height
.
Suppose
the
first
station
being
at
H
,
the
segment
of
the
Crosse
ED
were
180
,
and
the
segment
of
the
Staffe
HD
300
:
then
coming
64
feete
nearer
vnto
B
,
in
a
direct
line
,
vnto
a
second
station
at
A
,
and
making
another
obseruation
;
suppose
the
segment
of
the
Crosse
ED
were
180
as
before
,
and
the
segment
of
the
Staffe
AD
240
;
take
240
out
of
300
,
the
difference
of
segments
will
be
60
parts
.
And
As
60
parts
vnto
64
the
difference
of
stations
:
So
DE
180
vnto
BC
192
the
height
required
.
In
these
three
Prop.
there
is
a
regard
to
be
had
of
the
height
of
the
eye
.
For
the
height
measured
,
is
no
more
then
from
the
leuell
of
the
eye
vpward
.
4
To
find
a
distance
,
by
knowing
the
height
.
As
the
segment
of
the
Crosse
,
vnto
the
segment
of
the
Staffe
:
So
is
the
height
giuen
,
vnto
the
distance
.
So
the
segment
ED
being
18
,
and
DA
24
,
the
height
CB
192
,
will
shew
the
distance
AB
to
be
256.
5
To
find
a
distance
,
by
knowing
part
of
the
height
.
As
the
difference
betweene
the
sights
,
vnto
the
segment
of
the
Staffe
:
So
is
the
part
of
the
height
giuen
,
vnto
the
distance
.
And
thus
the
difference
betweene
E
and
F
being
45
,
and
the
segment
DA
240
;
the
part
of
the
height
GC
48
,
will
giue
the
distance
AB
to
be
256.
6
To
finde
a
distance
at
two
stations
,
by
knowing
the
difference
of
the
same
stations
.
As
the
difference
of
segments
on
the
Staffe
,
vnto
the
difference
of
stations
:
So
is
the
whole
segment
,
vnto
the
distance
.
And
thus
the
segment
of
the
Crosse
being
180
,
the
segment
of
the
Staffe
at
the
first
station
240
,
at
the
second
300
,
the
difference
of
the
segments
60
,
&
the
difference
of
stations
64
,
the
distance
AB
at
the
first
station
will
be
found
to
be
256
,
and
the
distance
HB
at
the
second
station
320.
7
To
find
a
bredth
by
knowing
the
distance
perpendicular
to
the
bredth
.
This
is
all
one
with
the
first
Prop.
For
this
bredth
is
but
an
height
turned
sidewayes
:
and
therefore
As
the
segment
of
the
Staffe
,
vnto
the
segment
of
the
Crosse
:
So
is
the
distance
vnto
the
bredth
.
And
thus
the
segment
of
the
Staffe
being
24
,
and
the
segment
of
the
Crosse
18
,
the
distance
AB
256
,
will
giue
the
bredth
BC
to
be
192.
8
To
find
a
bredth
at
two
stations
in
a
line
perpendicular
to
the
bredth
,
by
knowing
the
di●ference
of
the
same
stations
.
This
is
also
the
same
with
the
third
Prop:
and
therefore
As
the
difference
of
segments
on
the
Staffe
,
vnto
the
difference
of
stations
:
So
the
segment
on
the
Crosse
betweene
the
two
sights
,
vnto
the
bredth
required
.
And
thus
the
difference
betweene
the
stations
at
A
and
H
being
64
,
the
difference
of
segments
on
the
Staffe
60
,
the
segment
of
the
Crosse
180
,
the
bredth
BC
will
be
found
to
be
192.
In
like
maner
may
we
finde
the
bredth
GC
for
hauing
found
the
bredth
BC
the
proportion
will
hold
.
As
DE
is
vnto
FE
,
so
B
C
vnto
GC
.
Or
otherwise
,
As
H
a
vnto
HA
,
so
FE
vnto
GC
.
Neither
is
it
materiall
whether
the
two
stations
be
chosen
at
the
one
end
of
the
bredth
proposed
,
or
without
it
,
or
within
it
,
if
the
line
betweene
the
stations
be
perpendicular
vnto
the
bredth
:
as
may
appeare
if
in
stead
of
the
stations
at
A
and
H
,
we
make
choise
of
the
like
stations
at
I
and
K.
There
might
be
other
wayes
proposed
to
work
these
Prop.
by
holding
the
Crosse
euen
with
the
distance
,
and
the
Staffe
parallell
with
the
height
:
but
these
would
proue
more
troublesome
,
and
those
which
are
deliuered
are
sufficient
,
and
the
same
with
those
which
others
haue
set
downe
vnder
the
name
of
the
Iacobs
staffe
.
CHAP.
III.
The
vse
of
the
Tangent
lines
in
taking
of
Angles
.
geometric illustration
1
To
find
an
angle
by
the
Tangent
on
the
Staffe
.
LEt
the
middle
sight
be
alwayes
set
to
the
middle
of
the
Crosse
,
noted
with
20
and
30
,
and
then
the
Crosse
drawne
nearer
the
eye
,
vntill
the
marks
may
be
seene
close
within
the
sights
.
For
so
if
the
eye
at
A
(
that
end
of
the
Staffe
which
is
noted
with
90
and
180
)
beholding
the
marks
K
and
N
,
betweene
the
two
first
sights
,
C
and
B
,
or
the
marks
K
and
P
betweene
the
two
outward
sights
,
the
Crosse
being
drawne
downe
vnto
H
,
shall
stand
at
30
and
60
,
in
the
Tangent
on
the
Staffe
:
it
sheweth
that
the
angle
KAN
is
30
gr.
the
angle
KAP
60
gr.
the
on●
double
to
the
other
;
which
is
●he
r●ason
of
the
double
numbers
on
this
line
of
the
Staffe
:
and
this
way
wil
serue
for
any
angle
from
20
gr.
toward
90
gr.
or
from
40
gr.
toward
180
gr.
But
if
the
angle
be
lesse
then
20
gr.
we
must
then
make
vse
of
the
Tangent
vpon
the
Crosse
.
2
To
find
an
angle
by
the
Tangent
of
20
vpon
the
Crosse
.
Set
20
vnto
20
,
that
is
,
the
middle
sight
to
the
middest
of
the
Crosse
at
the
end
of
the
Staffe
,
noted
with
20
:
so
the
eye
at
A
,
beholding
the
marks
L
and
N
,
close
betweene
the
two
first
sights
,
C
and
B
,
shall
se●
them
in
an
angle
of
20
gr.
If
the
marks
sh●ll
be
nearer
together
,
as
are
M
and
N
,
then
draw
in
the
Crosse
from
C
vnto
E
:
if
they
be
farther
asunder
,
as
are
K
and
N
,
then
draw
out
the
Crosse
from
C
vnto
F
;
so
the
quantitie
of
the
angle
shal
be
still
found
in
the
Crosse
in
the
Tangent
of
20
gr.
at
the
end
of
the
Staffe
;
and
this
will
serue
for
any
angle
from
0
gr.
toward
35
gr.
3
To
find
an
angle
by
the
Tangent
of
30
vpon
the
Crosse
.
This
Tangent
of
30
is
here
put
the
rather
,
that
the
end
of
the
Staffe
resting
at
the
eye
,
the
hand
may
more
easily
remoue
th
Crosse
:
for
it
supposeth
the
Radius
to
be
no
longer
then
AH
,
which
is
from
the
eye
at
the
end
of
the
Staffe
vnto
30
gr.
about
22
inches
and
7
parts
.
Wherefore
here
set
the
middle
sight
vnto
30
gr.
on
the
Staffe
,
and
then
either
draw
the
Crosse
in
or
out
,
vntill
the
marks
be
seene
between
the
two
first
sights
;
so
the
quantitie
of
the
angle
will
be
found
in
the
Tangent
of
30
,
which
is
here
represented
by
the
line
GH
;
and
this
will
serue
for
any
angle
from
0
gr.
toward
48
gr.
4
To
obserue
the
altitude
of
the
Sunne
backward
.
Here
it
is
fit
to
haue
an
horizontall
sight
set
to
the
beginning
of
the
Staffe
,
and
then
may
you
turne
your
backe
toward
the
Sun
,
and
your
Crosse
toward
your
eye
.
If
the
altitude
be
vnder
45
gr.
set
the
middle
sight
to
30
on
the
Staffe
,
and
looke
by
the
middle
sight
through
the
horizontall
vnto
the
horizon
,
mouing
the
Crosse
vpward
or
downward
,
vntill
the
vpper
sight
doe
shadow
the
vpper
halfe
of
the
horizontall
sight
:
so
the
altitude
will
be
found
in
the
Tangent
of
30.
If
the
altitude
shal
be
more
then
45
gr.
set
the
middle
sight
vnto
the
middest
of
the
Crosse
,
and
look
by
the
inward
edge
of
the
lower
sight
through
the
horizontall
to
the
horizon
,
mouing
the
middle
sight
in
or
out
,
vntill
the
vpper
sight
do
shadow
the
vpper
halfe
of
the
horizontall
sight
:
so
the
altitude
will
be
found
in
the
degrees
on
the
Staffe
betweene
40
and
180.
5
To
set
the
Staffe
to
any
angle
giuen
.
This
is
the
conuerse
of
the
former
Prop.
For
if
the
middle
sight
be
set
to
his
place
and
degree
,
the
eye
looking
close
by
the
sights
as
before
,
cannot
but
see
his
obiect
in
the
angle
giuen
.
6
To
obserue
the
altitude
of
the
Sunne
another
way
.
Set
the
middle
sight
to
the
middle
of
the
Crosse
,
and
hold
the
horizontall
sight
downward
,
so
as
the
Crosse
may
be
parallell
to
the
horizon
,
then
is
the
Staffe
verticall
;
and
if
the
outward
sight
of
the
Crosse
do
shadow
the
horizontall
sight
,
the
complement
of
the
altitude
wil
be
found
in
the
tangent
on
the
Staffe
.
7
To
obserue
an
altitude
by
thread
and
plummet
.
Let
the
middle
sight
be
set
to
the
middest
of
the
Crosse
,
and
to
that
end
of
the
Staffe
which
is
noted
with
90
and
180
;
then
hauing
a
thread
and
a
plummet
at
the
beginning
of
the
Crosse
,
and
turning
the
Crosse
vpward
,
and
the
Staffe
toward
the
Sunne
,
the
thread
will
fall
on
the
complement
of
the
altitude
aboue
the
horizon
.
And
this
may
be
applied
to
other
purposes
.
8
To
apply
the
lines
of
inches
to
the
taking
of
angles
.
If
the
angles
be
obserued
betweene
the
two
first
sights
,
there
wil
be
such
proportion
between
the
parts
of
the
Staffe
and
the
parts
of
the
Crosse
,
as
betweene
the
Radius
and
the
Tangent
of
the
angle
.
As
if
the
parts
intercepted
on
the
Staffe
were
20
inches
,
the
parts
on
the
Crosse
9
inches
.
Then
by
proportion
as
20
vnto
9
,
so
100000
vnto
45000
the
tangent
of
24
gr.
14
m.
But
if
the
angle
shall
be
obserued
betweene
the
two
outward
sights
,
the
parts
being
20
and
9
as
before
,
the
angle
will
be
48
gr.
28
m.
double
vnto
the
former
.
In
all
these
there
is
a
regard
to
be
had
to
the
parallax
of
the
eye
,
and
his
height
aboue
the
Horizon
in
obseruations
at
Sea
;
to
the
Semidiameter
of
the
Sun
,
his
parallax
and
refraction
,
as
in
the
vse
of
other
staues
.
And
so
this
will
be
as
much
,
or
more
then
that
which
hath
been
heretofore
performed
by
the
Crosse-staffe
.
CHAP.
IIII.
The
vse
of
the
lines
of
equall
parts
ioyned
with
the
lines
of
Chords
.
THe
lines
of
equall
parts
do
serue
also
for
protraction
,
as
may
appeare
by
the
former
Diagrams
;
but
being
ioyned
with
the
lines
of
Chords
,
which
I
place
vpon
one
side
of
the
Crosse
,
they
will
farther
serue
for
the
protraction
and
resolution
of
right
line
triangles
;
whereof
I
will
giue
one
example
in
finding
of
a
distance
at
two
stations
otherwise
then
in
the
second
Cap.
geometric illustration
Let
the
distance
required
be
AB
.
At
A
the
first
station
I
make
choise
of
a
station
line
toward
C
,
and
obserue
the
angle
BAC
by
the
tangent
lines
,
which
may
be
43
gr.
20
m
;
then
hauing
gon
an
hundred
paces
toward
C
,
I
make
my
second
station
at
D
,
where
suppose
I
find
the
angle
BDC
to
be
58
gr.
or
the
angle
BDA
to
be
122
gr
;
this
being
done
,
I
may
finde
the
distance
AB
in
this
maner
.
1
I
draw
a
right
line
AC
,
representing
the
station
line
.
2
I
take
100
out
of
the
lines
of
equall
parts
,
and
pricke
them
downe
from
A
the
first
station
vnto
D
the
second
.
3
I
open
my
compasses
to
one
of
the
chords
of
60
gr.
and
setting
one
foote
in
the
point
A
,
with
the
other
I
describe
an
occult
arke
of
a
circle
intersecting
the
station
line
in
E.
4
I
take
out
of
the
same
line
of
chords
a
chord
of
43
gr.
20
m.
(
because
such
was
the
angle
at
the
first
station
)
and
this
I
inscribe
into
that
occult
arke
from
E
vnto
F
,
which
makes
the
angle
FAD
equall
to
the
angle
obserued
at
the
first
station
.
5
I
describe
another
like
arke
vpon
the
center
D
,
and
inscribe
into
it
a
chord
of
58
gr.
from
C
vnto
G
,
and
draw
the
right
line
DG
,
which
doth
meet
with
the
other
line
AF
in
the
point
B
,
and
makes
the
angle
BDC
equall
to
the
angle
obserued
at
the
second
station
.
So
the
angles
in
the
Diagram
being
equall
to
the
angles
in
the
field
,
their
sides
wil
be
also
proportionall
:
and
therefore
,
6
I
take
out
the
line
AB
with
my
compasses
,
and
measuring
it
in
the
same
line
of
equall
parts
,
from
which
I
tooke
AD
,
I
find
it
to
be
335
,
and
such
is
the
distance
required
.
CHAP.
V.
The
vse
of
the
Meridian
line
.
1
THe
Meridian
line
,
noted
with
the
letter
M
,
may
serue
for
the
more
easie
diuision
of
the
plane
sea-chart
,
according
to
Mercators
proiection
.
For
if
you
shall
draw
parallell
meridians
,
each
degree
being
halfe
an
inch
distant
from
other
,
the
degrees
of
this
meridian
line
on
the
Staffe
,
shall
giue
the
like
degrees
for
the
meridians
on
the
chart
,
from
the
Equinoctiall
toward
the
Pole
:
and
then
if
through
these
degrees
you
draw
streight
lines
perpendicular
to
the
meridians
,
they
shall
be
parallels
of
latitude
.
If
any
desire
to
haue
the
degrees
of
his
chart
larger
then
those
which
I
haue
put
on
the
Staffe
,
he
may
take
these
and
increase
them
in
a
double
,
or
treble
,
or
a
decuple
proportion
at
his
pleasure
.
2
This
meridian
line
being
ioyned
with
the
line
of
chords
,
may
serue
for
the
protraction
&
resolution
of
such
right
line
triangles
as
concerne
latitude
,
longitude
,
rumb
and
distance
in
the
practise
of
nauigation
.
As
may
appeare
by
this
example
.
Suppose
two
places
giuen
,
A
in
the
latitude
of
50
gr.
D
in
the
latitude
of
52
gr.
½
,
the
difference
of
longitude
betweene
them
being
6
gr.
and
let
it
be
required
to
know
,
first
what
Rumb
leadeth
from
the
one
place
to
the
other
;
secondly
how
many
degrees
distant
they
are
asunder
.
1
I
draw
a
right
line
AE
,
representing
the
parallell
of
the
place
from
whence
I
depart
.
2
I
take
6
gr.
for
the
difference
of
longitude
●●th●r
out
of
the
line
of
inches
,
allowing
halfe
an
inch
for
e●●●y
de●ree
,
o●
out
of
the
beginning
of
the
meridian
line
;
(
for
there
the
meridian
degrees
d
ffer
very
little
from
the
equinoct●all
degrees
)
and
these
6
gr.
I
pricke
downe
in
the
parallell
from
A
to
E.
3
In
A
and
E
,
I
erect
two
perpendiculars
,
AM
and
ED
,
representing
the
meridians
of
both
places
.
geometric illustration
4
I
take
the
difference
of
latitude
from
50
gr.
to
52
gr.
30
m.
out
of
the
meridian
line
,
and
pricke
it
downe
in
the
meridians
from
A
vnto
M
,
and
from
E
to
D
,
and
draw
the
right
line
MD
for
the
parallell
of
the
second
place
,
and
the
right
line
AD
for
the
line
of
distance
betweene
both
places
:
so
the
angle
MAD
shall
giue
the
Rumb
that
leadeth
from
the
one
place
to
the
other
.
5
To
finde
the
quantitie
of
this
angle
MAD
,
I
may
either
make
vse
of
the
Protractor
,
or
else
of
a
line
of
chords
,
and
so
I
open
my
compasses
vnto
one
of
the
chords
of
60
gr.
and
setting
one
foote
in
the
point
A
,
with
the
other
I
describe
an
occult
arke
of
a
circle
,
intersecting
the
meridian
in
F
,
and
the
line
of
distance
in
G
;
then
I
take
the
chord
FG
with
my
compasses
,
and
measuring
it
in
the
same
line
of
chords
as
before
,
I
find
it
56
gr.
¼
:
and
such
is
the
inclination
of
the
Rumb
to
the
meridian
,
which
is
the
first
thing
that
was
required
.
6
To
find
the
quantitie
of
the
line
of
distance
AD
,
I
take
it
out
with
my
compasses
,
and
measuring
it
in
the
meridian
line
,
setting
one
foote
beneath
the
lesser
latitude
,
and
the
other
foote
as
much
aboue
the
greater
latitude
,
I
find
about
4
gr.
½
intercepted
between
both
feet
:
and
such
is
the
distance
vpon
the
Rumb
,
which
is
the
second
thing
that
was
required
.
But
if
this
example
were
protracted
according
to
the
common
Sea-chart
,
where
the
degrees
of
the
equinoctiall
and
meridian
are
both
alike
;
the
Rumb
MAD
would
be
found
to
be
aboue
67
gr.
and
AD
the
distance
vpon
the
Rumb
about
6
gr.
½
.
Suppose
farther
,
that
hauing
set
forth
from
A
toward
D
,
vpon
the
former
Rumb
of
56
gr.
15
m.
NEbE
,
after
the
ship
had
runne
36
leagues
,
the
wind
changing
,
it
ran
50
leagues
more
vpon
the
seuenth
Rumb
of
EbN
,
whose
inclination
to
the
meridian
is
78
gr.
45
m.
And
let
it
be
required
to
know
what
longitude
and
latitude
the
ship
is
in
,
by
pricking
down
the
way
thereof
vpon
the
Chart.
Hauing
drawne
a
blanke
chart
as
before
,
with
meridians
&
parallels
,
according
to
the
latitude
of
the
places
proposed
.
1
I
would
make
an
angle
MAD
of
56
gr.
15
m.
for
the
Rumb
of
NEbE
,
which
is
done
after
this
maner
:
I
open
my
compasses
to
one
of
the
chords
of
60
gr.
and
setting
one
foote
in
the
point
A
,
with
the
other
I
describe
an
occult
ark
of
a
circle
,
intersecting
the
meridian
in
F
;
then
I
take
56
gr.
15
m.
out
of
the
same
line
of
chords
,
and
pricke
them
downe
from
F
vnto
G
:
so
the
right
line
AG
shall
be
the
Rumb
of
NEbE
.
2
I
would
take
36
leagues
out
of
the
meridian
line
,
extending
my
compasses
from
50
gr.
to
51
gr.
48
m.
or
rather
from
as
much
below
50
as
aboue
51
,
and
prick
them
downe
vpon
the
Rumb
from
A
vnto
I
;
so
the
point
I
,
shal
represent
the
place
wherein
the
ship
was
when
the
wind
changed
.
And
this
is
in
the
latitude
of
51
gr.
0
m.
and
in
the
longitude
of
2
gr.
21
m.
Eastward
from
the
meridian
AM.
3
By
the
same
reason
,
I
may
draw
the
right
line
IK
for
the
Rumb
of
EbN
,
and
pricke
downe
the
distance
of
50
leagues
from
I
vnto
K
:
so
the
point
K
shal
represent
the
place
whither
the
ship
came
,
after
the
running
of
these
50
leagues
and
this
is
in
the
latitude
of
51
gr.
30
m.
and
in
longitude
6
gr.
16
m.
Eastward
from
the
first
meridian
AM
,
and
therefore
16
m.
Eastward
from
the
second
meridian
ED.
But
if
these
two
courses
were
to
be
pricked
downe
by
the
common
sea-chart
,
the
point
I
would
fall
in
the
latitude
of
51
gr.
0
m.
and
the
point
K
in
the
latitude
of
51
gr.
30
m.
But
the
longitude
of
I
would
be
onely
1
gr.
30
m.
and
the
longitude
of
K
onely
3
gr.
57
m.
which
is
33
m
Westward
from
the
meridian
of
the
place
to
which
the
ship
was
bound
.
Such
is
the
difference
betweene
both
these
charts
.
CHAP.
VI.
The
vse
of
the
line
of
Numbers
.
1
Hauing
two
numbers
giuen
to
find
a
third
in
continuall
proportion
,
a
fourth
,
a
fift
,
and
so
forward
.
EXtend
the
compasses
from
the
first
number
vnto
the
second
;
then
may
you
turne
them
,
from
the
second
to
the
third
,
and
from
the
third
to
the
fourth
,
and
so
forward
.
Let
the
two
numbers
giuen
be
2
and
4.
Extend
the
compasses
from
2
to
4
,
then
may
you
turne
them
from
4
to
8
,
and
from
8
to
16
,
and
from
16
to
32
,
and
from
32
to
64
,
and
from
64
to
128.
Or
if
the
one
foote
of
the
compasses
being
set
to
64
,
the
other
fall
out
of
the
line
,
you
may
set
it
to
another
64
nearer
the
beginning
of
the
line
,
and
there
the
other
foot
will
reach
to
128
,
and
from
128
you
may
turne
them
to
256
,
and
so
forward
.
Or
if
the
two
first
numbers
giuen
were
10
and
9
:
extend
the
compasses
from
10
at
the
end
of
the
line
,
backe
vnto
9
,
then
may
you
turne
them
from
9
vnto
8.1
,
and
from
8.1
vnto
7.29
.
And
so
if
the
two
first
numbers
giuen
were
1
and
9
,
the
third
would
be
found
to
be
81
,
the
fourth
729
,
with
the
same
extent
of
the
compasses
.
In
the
same
maner
,
if
the
two
first
numbers
were
10
and
12
,
you
may
finde
the
third
proportionall
to
be
14.4
,
the
fourth
17.28
.
And
with
the
same
extent
of
the
compasses
,
if
the
two
first
numbers
were
1
and
12
,
the
third
would
be
found
to
be
144
,
and
the
fourth
to
be
1728.
2
Hauing
two
extreme
numbers
giuen
,
to
find
a
meane
proportionall
between
them
.
Diuide
the
space
betweene
the
extreme
numbers
into
two
equall
parts
,
and
the
foote
of
the
compasses
will
stay
at
the
meane
proportionall
.
So
the
extreme
numbers
giuen
being
8
and
32
,
the
meane
betweene
them
will
be
found
to
be
16
,
which
may
be
proued
by
the
former
Prop.
where
it
was
shewed
,
that
as
8
to
16
,
so
are
16
to
32.
3
To
find
the
square
roote
of
any
number
giuen
.
The
square
roote
is
alwayes
the
meane
proportionall
betweene
1
and
the
number
giuen
,
and
therefore
to
be
found
by
diuiding
the
space
betweene
them
into
two
equall
parts
.
So
the
roote
of
9
is
3
,
and
the
roote
of
81
is
9
,
and
the
roote
of
144
is
12.
4
Hauing
two
extreme
numbers
giuen
,
to
find
two
meane
proportionals
between
them
.
Diuide
the
space
betweene
the
two
extreme
numbers
giuen
,
into
three
equall
parts
.
As
if
the
extreme
numbers
giuen
were
8
and
27
,
diuide
the
space
betweene
them
into
three
equall
parts
,
the
feet
of
the
compasses
will
stand
in
12
and
18.
5
To
find
the
cubique
roote
of
a
number
giuen
.
The
cubique
roote
is
alwayes
the
first
of
two
meane
proportionals
betweene
1
and
the
number
giuen
,
and
therefore
to
be
found
by
diuiding
the
space
betweene
them
into
three
equall
parts
.
So
the
roote
of
1728
will
be
found
to
be
12.
The
roote
of
17280
is
almost
26
:
and
the
roote
of
172800
is
almost
56.
6
To
multiply
one
number
by
another
.
Extend
the
compasses
from
1
to
the
multiplicator
;
the
same
extent
applied
the
same
way
,
shall
reach
from
the
multiplicand
to
the
product
.
As
if
the
numbers
to
be
multiplied
were
25
and
30
:
either
extend
the
compasses
from
1
to
25
,
and
the
same
extent
will
giue
the
distance
from
30
to
750
;
or
extend
them
from
1
to
30
,
and
the
same
extent
shall
reach
from
25
to
750.
7
To
diuide
one
number
by
another
.
Extend
the
compasses
from
the
diuisor
to
1
,
the
same
extent
shall
reach
from
the
diuidend
to
the
quotient
.
So
if
750
were
to
be
diuided
by
25
,
the
quotient
would
be
found
to
be
30.
8
Three
numbers
being
giuen
to
find
a
fourth
proportionall
.
This
golden
rule
,
the
most
vsefull
of
all
others
,
is
performed
with
like
ease
.
For
extend
the
compasses
from
the
first
number
to
the
second
,
the
same
extent
shall
giue
the
distance
from
the
third
to
the
fourth
.
As
for
example
,
the
proportion
between
the
diameter
and
the
circumference
,
is
said
to
be
such
as
7
to
22
:
if
the
diameter
be
14
,
how
much
is
the
circumference
?
Extend
the
compasses
from
7
to
22
,
the
same
extent
shall
giue
the
distance
from
14
to
44
:
or
extend
them
from
7
to
14
,
and
the
same
extent
shall
reach
from
22
to
44.
Either
of
these
wayes
may
be
tried
on
seuerall
places
of
this
line
;
but
that
place
is
best
,
where
the
seete
of
the
compasses
may
stand
nearest
together
.
9
Three
numbers
being
giuen
to
finde
a
fourth
in
a
duplicated
proportion
.
This
proposition
concernes
questions
of
proportion
betweene
lines
and
superficies
;
where
if
the
denomination
be
of
lines
,
extend
the
compasses
from
the
first
to
the
second
number
of
the
same
denomination
:
so
the
same
extent
being
doubled
,
shall
giue
the
distance
from
the
third
number
vnto
the
fourth
.
The
diameter
being
14
,
the
content
of
the
circle
is
154
:
the
diameter
being
28
,
what
may
the
content
be
?
Extend
the
compasses
from
14
to
28
,
the
same
extent
doubled
will
reach
from
154
to
616.
For
first
it
reacheth
from
154
vnto
308
;
and
turning
the
compasses
once
more
,
it
reacheth
from
308
vnto
616
:
and
this
is
the
content
required
.
But
if
the
first
denomination
be
of
the
superficiall
content
,
extend
the
compasses
vnto
the
halfe
of
the
distance
,
betweene
the
first
number
and
the
second
of
the
same
denomination
:
so
the
same
extent
shall
giue
the
distance
from
the
third
to
the
fourth
.
The
content
of
a
circle
being
154
,
the
diameter
is
14
:
the
content
being
616
,
what
may
the
diameter
be
?
Diuide
the
distance
betweene
154
and
616
into
two
equall
parts
,
then
set
one
foote
in
14
,
the
other
will
reach
to
28
the
diameter
required
.
10
Three
numbers
being
giuen
to
find
a
fourth
in
a
triplicated
proportion
.
This
proposition
concerneth
questions
of
proportion
betweene
lines
and
solids
;
where
if
the
first
denomination
be
of
lines
,
extend
the
compasses
from
the
first
number
to
the
second
of
the
same
denomination
:
so
the
extent
being
tripled
,
shall
giue
the
distance
from
the
third
number
vnto
the
fourth
,
Suppose
the
diameter
of
an
iron
bullet
being
4
inches
,
the
weight
of
it
was
9
l
:
the
diameter
being
8
inches
,
what
may
the
weight
be
?
Extend
the
compasses
from
4
to
8
,
the
same
extent
being
tripled
,
will
reach
from
9
vnto
72.
For
first
it
reacheth
from
9
vnto
18
;
then
from
18
to
36
;
thirdly
from
36
to
72.
And
this
is
the
weight
required
.
But
if
the
first
denomination
shall
be
of
the
Solid
content
,
or
of
the
weight
,
extend
the
compasses
to
a
third
part
of
the
distance
betweene
the
first
number
and
the
second
of
the
same
denomination
:
so
the
same
extent
shal
giue
the
distance
from
the
third
number
vnto
the
fourth
.
The
weight
of
a
cube
being
72
l
,
the
side
of
it
was
8
inches
:
the
weight
being
9
l
,
what
may
the
side
be
?
Diuide
the
distance
betweene
72
and
9
,
into
three
equall
parts
;
then
set
one
foote
to
8
,
the
other
will
reach
to
4
,
the
side
required
.
CHAP.
VII
.
The
vse
of
the
lines
of
artificiall
Sines
.
THis
line
of
sines
hath
such
vse
in
finding
a
fourth
proportionall
,
as
the
ordinary
Canon
of
Sines
:
and
the
maner
of
finding
it
,
is
alwayes
such
as
in
this
example
.
As
the
sine
of
30
gr.
vnto
the
sine
of
52
gr.
So
the
sine
of
38
gr.
to
a
fourth
sine
.
Extend
the
compasses
in
the
line
of
sines
from
30
gr.
vnto
52
gr
;
the
same
extent
shall
giue
the
distance
from
38
gr.
vnto
76
gr.
Or
extend
them
from
30
gr.
vnto
38
gr.
the
same
extent
will
reach
from
52
gr.
vnto
76
gr.
which
is
the
fourth
proportionall
sine
required
.
And
thus
may
the
rest
of
all
sinical
proportions
be
wrought
two
wayes
.
The
minutes
which
are
wanting
in
the
first
degree
,
may
be
supplied
by
the
line
of
Numbers
.
CHAP.
VIII
.
The
vse
of
the
line
of
artificiall
Tangents
.
THis
line
of
Tangents
hath
like
vse
,
but
commonly
ioyned
with
the
line
of
sines
:
the
maner
of
working
by
it
,
may
appeare
by
this
example
.
As
the
Tangent
of
38
gr.
30
m.
is
to
the
Tangent
of
23
gr.
30
m.
So
the
Sine
of
90
gr.
to
a
fourth
Sine
.
This
Prop.
and
such
others
vpon
two
lines
,
may
be
wrought
two
wayes
.
For
extend
the
compasses
from
the
Tangent
of
38
gr.
30
m.
to
the
Tangent
of
23
gr.
30
m
;
the
same
extent
shall
giue
the
distance
from
the
sine
of
90
gr.
to
the
sine
of
33
gr.
8
m.
Or
else
extend
them
from
38
gr.
30
m.
in
the
Tangents
vnto
90
gr.
in
the
line
of
sines
;
the
same
extent
from
the
Tangent
of
23
gr.
30
m.
shall
reach
to
the
sine
of
33
gr.
8
m.
which
is
the
fourth
proportionall
sine
required
.
And
this
crossework
in
many
cases
is
the
better
,
in
regard
the
tangents
which
should
passe
on
from
40
gr.
to
50
gr.
and
so
forward
,
do
turne
backe
at
45
gr.
These
two
lines
of
Sines
and
Tangents
,
may
serue
for
the
resolution
of
all
sphericall
triangles
,
according
to
those
Canons
which
I
haue
set
downe
in
the
vse
of
the
Sector
.
Or
if
at
any
time
one
meete
with
a
secant
,
let
him
account
the
sine
of
80
gr.
for
a
secant
of
10
gr.
and
the
sine
of
70
gr.
for
a
secant
of
20
gr.
and
so
take
the
sine
of
the
complement
in
stead
of
the
secant
.
As
if
the
proposition
were
,
As
the
Radius
to
the
secant
of
51
gr.
30
m.
So
the
sine
of
23
gr.
30
m.
to
a
fourth
sine
.
Extend
the
compasses
from
the
Radius
that
is
the
sine
of
90
gr.
to
the
sine
of
38
gr.
30
m.
the
same
extent
will
giue
the
distance
from
the
sine
of
23
gr.
30
m.
both
to
the
sine
of
14
gr.
22
m.
and
to
the
sine
of
39
gr.
50
m.
But
in
this
case
,
the
sine
of
39
gr.
50
m.
is
the
fourth
required
.
For
the
first
number
being
lesse
then
the
second
,
that
is
,
the
Radius
lesse
then
the
secant
,
the
sine
of
23
gr.
30
m.
which
is
the
third
,
must
also
be
lesse
then
the
fourth
.
CHAP.
IX
.
The
vse
of
the
line
of
Sines
and
Tangents
ioyned
with
the
line
of
numbers
.
THe
lines
of
sines
and
tangents
haue
another
like
vse
ioyned
with
the
line
of
numbers
,
especially
in
the
resolution
of
right
line
triangles
,
where
the
angles
are
measured
by
degrees
and
minutes
,
and
the
sides
measured
by
absolute
numbers
,
whereof
I
will
set
downe
these
propositions
.
1
Hauing
three
angles
and
one
side
,
to
find
the
two
other
sides
.
As
the
side
of
the
angle
opposite
to
the
side
giuen
,
is
to
the
number
belonging
to
that
side
giuen
:
So
the
sine
of
the
angle
opposite
to
the
side
required
,
to
the
number
belonging
to
the
side
required
.
As
in
the
example
of
the
fourth
Cap.
of
this
booke
,
where
knowing
the
distance
betweene
two
stations
at
A
and
D
to
be
100
paces
,
the
angle
BAC
to
be
43
gr.
20
m.
and
the
angle
BDC
to
be
58
gr.
it
was
required
to
find
the
distance
AB
.
First
hauing
these
two
angles
,
I
may
find
the
third
angle
ABD
to
be
14
gr.
40
m.
either
by
substraction
or
by
complement
vnto
180.
Then
in
the
triangle
BAD
,
I
haue
three
angles
,
and
one
side
,
whereby
I
may
find
both
AB
and
DB.
I
know
the
angle
ABD
opposite
to
the
measured
side
AD
to
be
14
gr.
40
m.
and
the
angle
ADB
opposite
to
the
side
required
,
to
be
122
gr:
wherefore
I
extend
the
compasses
in
the
line
of
sines
from
14
gr.
40
m.
vnto
122
gr.
or
(
which
is
all
one
)
to
58
gr.
(
for
after
90
gr.
the
sine
of
80
gr.
is
also
the
sine
of
100
gr.
and
the
sine
of
70
gr.
the
sine
of
100
gr.
and
so
in
the
rest
)
so
shall
I
find
the
same
extent
to
reach
in
the
line
of
numbers
,
from
100
vnto
335.
And
such
is
the
distance
required
betweene
A
and
B.
geometric illustration
In
like
maner
if
I
extend
my
compasses
from
the
sine
of
34
gr.
40
m.
to
the
sine
of
43
gr.
20
m.
the
same
extent
will
reach
in
the
line
of
numbers
from
100
to
271.
And
such
is
the
distance
betweene
D
and
B.
Or
in
crosse
worke
,
I
may
extend
the
compasses
from
14
gr.
40
m.
in
the
sines
,
vnto
100
parts
in
the
line
of
numbers
:
so
the
same
extent
will
giue
the
distance
from
58
gr.
to
335
parts
,
and
from
43
gr.
20
m.
to
271
parts
.
2
Hauing
two
sides
giuen
,
and
one
angle
opposite
to
either
of
these
sides
,
to
find
the
other
two
angles
and
the
third
side
.
As
the
side
opposite
to
the
angle
giuen
,
is
to
the
sine
of
the
angle
giuen
:
So
the
other
side
giuen
,
to
the
sine
of
that
angle
to
which
it
is
opposite
.
So
in
the
former
triangle
,
hauing
the
two
sides
AB
335
paces
and
AD
100
paces
,
and
knowing
the
angle
ADB
,
which
is
opposite
to
the
side
AB
,
to
be
122
gr.
I
may
find
the
angl●
ABD
,
which
is
opposite
to
the
other
side
AD.
For
if
I
extend
the
compasses
from
335
to
100
in
the
line
of
numb●rs
,
I
shall
finde
the
same
extent
to
reach
in
the
line
of
sines
from
122
gr.
to
14
gr.
40
m
;
and
therefore
such
is
the
angle
ABD
.
Then
knowing
these
two
angles
ABD
and
ADB
,
I
may
find
the
third
angle
BAD
either
by
subtraction
or
by
complement
to
180
,
to
be
43
gr.
20
m
;
and
hauing
three
angles
and
two
sides
,
I
may
well
find
the
third
side
DB
,
by
the
former
Prop.
This
may
be
done
more
readily
by
crosse
worke
.
For
if
I
extend
the
compasses
from
335
parts
,
in
the
line
of
numbers
,
to
the
sine
of
122
gr.
the
same
extent
wil
reach
from
100
parts
to
the
sine
of
14
gr.
40
m.
and
backe
from
43
gr.
20
m.
to
271
parts
;
and
such
is
the
third
side
DB.
3
Hauing
two
sides
and
the
angle
between
them
,
to
find
the
two
other
angles
and
the
third
side
.
If
the
angle
contained
betweene
the
two
sides
be
a
right
angle
,
the
other
two
angles
will
be
found
readily
by
this
canon
.
As
the
greater
side
giuen
,
is
to
the
lesser
side
:
So
the
tangent
of
45
gr.
to
the
tangent
of
the
lesser
angle
.
So
in
the
rectangle
triangle
AIB
,
knowing
the
side
AI
to
be
244
,
and
the
side
IB
to
be
230
:
if
I
extend
the
compasses
from
244
to
230
in
the
line
of
numbers
,
the
same
extent
will
reach
from
45
gr.
to
about
43
gr.
20
m.
in
the
line
of
tangents
;
and
such
is
the
lesser
angle
BAI
,
and
the
complement
46
gr.
40
m.
shewes
the
greater
angle
ABI
.
The
angles
being
knowne
,
the
third
side
AB
may
be
found
by
the
first
Prop.
So
likewise
in
the
example
of
the
third
Cap.
of
this
booke
,
concerning
taking
of
angles
by
the
line
of
inches
,
where
the
parts
intercepted
on
the
Staffe
being
20
inches
,
and
the
parts
on
the
Crosse
9
inches
,
it
was
required
to
find
the
angle
of
altitude
.
For
I
may
extend
the
compasses
in
the
line
of
numbers
,
from
20
vnto
9
,
the
same
extent
will
reach
in
the
line
of
tangents
,
from
45
gr.
to
24
gr.
14
m.
Or
in
the
crosse
worke
,
I
may
extend
the
compasses
from
20
parts
in
the
line
of
numbers
to
the
tangent
of
45
gr
;
the
same
extent
shall
giue
the
distance
from
9
parts
vnto
the
tangent
of
24
gr.
14
m.
And
such
is
the
angle
of
altitude
required
.
But
if
it
be
an
oblique
angle
that
is
contained
betweene
the
two
sides
giuen
,
the
triangle
may
be
reduced
into
two
rectangle
triangles
,
and
then
resolued
as
before
.
As
in
the
triangle
ADB
,
where
the
side
AB
is
335
,
and
the
side
AD
100
,
and
the
angle
BAD
43
gr.
20
m
:
if
I
let
downe
the
perpendicular
DH
vpon
the
side
AB
,
I
shal
haue
two
rectangle
triangles
,
AHD
,
DHB
;
and
in
the
rectangle
AHD
,
the
angle
at
A
being
43
gr.
20
m.
the
other
angle
ADH
will
be
46
gr.
40
m
;
and
with
these
angles
and
the
side
AD
,
I
may
find
both
AH
and
D
H
,
by
the
first
Prop.
Then
taking
AH
out
of
AB
,
there
remaines
HB
for
the
side
of
the
rectangle
DHB
;
and
therefore
with
this
side
HB
and
the
other
side
DH
,
I
may
find
both
the
angle
at
B
,
and
the
third
side
DB
,
as
in
the
former
part
of
this
Prop.
Or
I
may
find
the
angles
required
,
without
letting
downe
any
perpendicular
.
For
As
the
summe
of
the
sides
,
is
to
the
difference
of
the
sides
:
So
the
tangent
of
the
halfe
summe
of
the
opposite
angles
,
to
the
tangent
of
half
the
difference
between
those
angles
.
As
in
the
former
triangle
ADB
,
the
summe
of
the
sides
AB
,
AD
,
is
435
,
and
the
difference
betweene
them
235
;
the
angle
contained
43
gr.
20
m
;
and
therefore
the
summe
of
the
two
opposite
angles
136
gr.
40
m.
and
the
halfe
summe
68
gr
20
m.
Hereupon
I
extend
the
compasses
in
the
line
of
numbers
from
455
to
235
,
and
I
find
them
to
reach
in
the
line
of
tangents
from
68
gr.
20
m.
vnto
53
gr.
40
m
;
and
such
is
the
halfe
difference
betweene
the
opposite
angles
at
B
and
D.
This
halfe
difference
being
added
to
the
halfe
summe
,
doth
giue
122
gr.
for
the
greater
angle
ADB
:
and
being
subtracted
,
it
leaueth
14
gr.
40
m.
for
the
lesser
angle
ABD
.
Then
the
three
angles
being
knowne
,
the
third
side
BD
may
be
found
by
the
first
Prop.
4
Hauing
the
three
sides
of
a
right
line
triangle
,
to
find
the
perpendicular
and
the
three
angles
.
Let
one
of
the
three
sides
giuen
be
the
base
,
but
rather
the
greater
side
,
that
the
perpendicular
may
fall
within
the
triangle
;
then
gather
the
summe
,
and
the
difference
of
the
two
other
sides
,
and
the
proportion
will
hold
.
As
the
base
of
the
triangle
,
is
to
the
summe
of
the
sides
:
So
the
difference
of
the
sides
to
a
fourth
,
which
being
taken
forth
of
the
base
,
the
perpendicular
shal
fall
on
the
middle
of
the
remainder
.
As
in
the
former
triangle
ADB
,
where
the
base
AB
is
335
,
the
summe
of
the
sides
AD
and
DB
371
,
and
the
difference
of
them
171.
If
I
extend
the
compasses
in
the
line
of
numbers
from
335
vnto
371
,
I
shall
find
the
same
extent
to
reach
from
171
vnto
189.4
.
This
fourth
number
I
take
out
of
the
base
335.0
,
and
the
remainder
is
145.6
,
the
halfe
whereof
is
72.8
,
and
doth
shew
the
place
H
,
where
the
perpendicular
shall
fall
,
from
the
angle
D
,
vpon
the
base
AB
,
diuiding
the
former
triangle
ADB
into
two
right
angle
triangles
,
DHA
and
DHB
,
in
which
the
angles
may
be
found
by
the
former
part
of
the
third
Prop.
And
this
may
suffice
for
right
line
triangles
.
But
for
the
more
easie
protraction
of
these
triangles
,
I
will
set
downe
one
proposition
more
concerning
chords
.
5
Hauing
the
semidiameter
of
a
circle
,
to
find
the
chords
of
euery
arke
.
As
the
sine
of
30
gr.
to
the
sine
of
halfe
the
arke
proposed
:
So
is
the
semidiameter
of
the
circle
giuen
,
to
the
chord
of
the
same
arke
.
As
if
in
protracting
the
former
triangle
ADB
,
it
were
required
to
find
the
length
of
a
chord
of
43
gr.
20
m.
agreeing
to
the
semidiameter
AE
,
which
is
knowne
to
be
3
inches
.
The
halfe
of
43
gr.
20
m.
is
21
gr.
40
m
;
wherefore
I
extend
the
compasses
from
the
sine
of
30
gr.
to
the
sine
of
21
gr.
40
m.
and
I
finde
the
same
extent
to
reach
in
the
line
of
numbers
from
3.000
parts
to
2.215
;
which
shewes
,
that
the
semidiameter
being
3
inches
,
the
chord
of
43
gr.
20
m.
will
be
2
inches
and
215
parts
of
1000.
In
like
maner
the
chord
of
58
gr.
agreeing
to
the
same
semidiameter
,
would
be
found
to
be
2
inches
and
909
parts
.
For
the
halfe
of
58
being
29
;
if
I
extend
the
compasses
in
the
line
of
sines
from
30
gr.
to
29
gr.
the
same
extent
will
reach
in
the
line
of
numbers
from
3.000
.
vnto
2.909
.
Or
in
crosse
worke
,
if
I
extend
the
compasses
from
the
sine
of
30
gr.
to
3.000
in
the
line
of
numbers
,
I
shall
find
the
same
extent
to
reach
from
21
gr.
40
m.
to
2.215
parts
,
and
from
29
gr.
to
2.909
parts
,
and
from
7
gr.
20
m.
to
765
parts
;
for
the
chord
of
14
gr.
40
m.
for
the
third
angle
ABD
.
CHAP.
X.
The
vse
of
the
line
of
versed
sines
.
THis
line
of
versed
sines
is
no
necessary
line
.
For
all
triangles
,
both
right
lined
and
sphericall
,
may
be
resolued
by
the
three
former
lines
of
numbers
,
sines
and
tangents
;
yet
I
thought
good
to
put
it
on
the
Staffe
for
the
more
easie
finding
of
an
angle
hauing
three
sides
,
or
a
side
hauing
three
angles
of
a
sphericall
triangle
giuen
.
Suppose
the
three
sides
to
be
,
one
of
them
110
gr.
the
other
78
gr.
and
the
third
38
gr.
30
m.
and
let
it
be
required
to
find
the
angle
,
whose
base
is
110
gr.
I
first
adde
them
together
,
and
from
halfe
the
summe
subtract
the
base
,
noting
the
difference
after
this
maner
.
The
base
|
110
gr.
|
0
m.
|
The
one
side
|
78
|
0
|
The
other
side
|
38
|
30
|
The
summe
of
all
three
|
226
|
30
|
The
halfe
summe
|
113
|
15
|
The
difference
|
3
|
15
|
This
done
,
I
come
to
the
Staffe
,
and
extend
the
compasses
from
the
sine
of
90
gr.
to
the
sine
of
78
gr.
which
is
one
of
the
sides
;
and
applying
this
extent
from
the
sine
of
the
other
side
38
gr.
30
m.
I
find
it
to
reach
to
a
fourth
sine
,
about
37
gr.
30
m.
From
this
fourth
sine
of
37
gr.
30
m.
I
extend
the
compasses
again
,
to
the
sine
of
the
halfe
summe
113
gr.
15
m.
(
which
is
all
one
with
the
sine
of
66
gr.
45
m.
)
and
this
second
extent
wil
reach
from
the
sine
of
the
difference
3.
gr.
15
m.
to
the
sine
of
4
gr.
54
m.
Ouer
against
this
sine
you
shal
find
146
gr.
in
the
line
of
versed
sines
;
and
such
is
the
angle
required
.
THE
THIRD
BOOKE
.
Of
the
vse
of
the
lines
of
Numbers
,
Sines
and
Tangents
for
the
drawing
of
Houre-lines
on
all
sorts
of
Planes
.
THere
are
ten
seuerall
sorts
of
Planes
,
which
take
their
denomination
from
those
great
circles
to
which
they
are
parallels
,
and
may
sufficiently
for
our
vse
be
represented
in
this
one
fundamentall
Diagram
described
before
in
the
vse
of
the
Sector
,
and
be
knowne
by
their
horizontall
and
perpendicular
lines
,
of
such
as
know
the
latitude
of
the
place
,
and
the
circles
of
the
sphere
.
1
An
horizontall
plane
parallell
to
the
horizon
,
here
represented
by
the
outward
circle
ESWN
.
2
A
verticall
plane
parallell
to
the
prime
verticall
circle
which
passeth
through
the
zenith
and
the
points
of
East
and
West
in
the
horizon
,
and
is
right
to
the
horizon
and
the
meridian
;
that
is
,
maketh
right
angles
with
them
both
.
This
is
represented
by
EZW
.
3
A
polar
plane
parallell
to
the
circle
of
the
houre
of
6
,
which
passeth
through
the
pole
and
the
points
of
East
and
West
,
being
right
to
the
Equinoctiall
and
the
Meridian
,
but
inclining
to
the
horizon
,
with
an
angle
equall
to
the
latitude
.
This
is
here
represented
by
EPW
.
4
An
equinoctiall
plane
parallell
to
the
Equinoctiall
,
which
passeth
through
the
points
of
East
and
West
,
being
right
to
the
Meridian
,
but
inclining
to
the
Horizon
,
with
an
angle
equall
to
the
complement
of
the
latitude
.
This
is
here
represented
by
EAW
.
geometric illustration
6
A
meridian
plane
parallell
to
the
meridian
,
the
circle
of
the
houre
of
12
,
which
passeth
through
the
zenith
,
the
pole
,
and
the
points
of
South
and
North
,
being
right
to
the
horizon
,
and
the
prime
verticall
.
This
is
here
represented
by
SZN
.
7
A
meridian
plane
inclining
to
the
horizon
,
parallell
to
any
great
circle
,
which
passeth
through
the
points
of
South
and
North
,
being
right
to
the
prime
verticall
,
but
inclining
to
the
horizon
.
This
is
here
represented
by
SGN
.
8
A
verticall
declining
plane
,
parallell
to
any
great
circle
,
which
passeth
through
the
zenith
,
being
right
to
the
horizon
,
but
inclining
to
the
meridian
.
This
is
represented
by
BZD.
9
A
polar
declining
plane
,
parallell
to
any
great
circle
,
which
passeth
through
the
pole
,
being
right
to
the
equinoctiall
,
but
inclining
to
the
meridian
.
This
is
here
represented
by
HPQ
.
10
A
declining
inclining
plane
,
parallell
to
any
great
circle
,
which
is
right
to
none
of
the
former
circles
,
but
declining
from
the
prime
verticall
,
and
inclining
both
to
the
horizon
and
the
meridian
,
and
all
the
houre
circles
.
This
may
be
here
represented
either
by
BMD
,
or
BFD
,
or
BKD
,
or
any
such
great
circle
,
which
passeth
neither
through
the
South
and
North
,
nor
East
and
West
points
,
nor
through
the
zenith
nor
the
pole
.
Each
of
these
planes
(
except
the
horizontall
)
hath
two
faces
whereon
houre-lines
may
be
drawne
;
and
so
there
are
19
planes
in
all
.
The
meridian
plane
hath
one
face
to
the
East
,
and
another
to
the
West
:
the
other
verticall
planes
haue
one
to
the
South
,
and
another
to
the
North
,
and
the
rest
one
to
the
zenith
,
and
another
to
the
nadir
:
but
what
is
said
of
the
one
,
may
be
vnderstood
of
the
other
.
To
find
the
inclination
of
any
Plane
.
For
the
distinguishing
of
these
Planes
we
may
finde
wheth●r
they
be
horizontall
,
or
verticall
,
or
inclining
to
the
horizon
,
and
how
much
they
incline
,
either
by
the
vsuall
inclinatorie
quadrant
,
or
by
fitting
a
thread
and
plummet
vnto
the
Sector
.
For
let
the
Sector
be
opened
to
a
right
angle
,
the
lines
of
Sines
to
an
angle
of
92
gr.
the
inward
edges
of
the
Sector
to
90
gr.
and
let
a
thread
and
plummet
be
hanged
vpon
a
line
parallell
to
the
edges
of
one
of
the
legs
,
so
that
leg
shall
be
verticall
,
and
the
other
leg
parallell
to
the
horizon
.
building with angles marked
If
the
plane
seeme
to
be
leuell
with
the
horizon
,
you
may
trie
it
by
setting
the
horizontall
leg
of
the
Sector
to
the
plane
,
and
holding
the
other
leg
vpright
:
for
then
if
the
thread
shall
fall
on
his
plummet
line
,
which
way
soeuer
you
turne
the
Sector
,
it
is
an
horizontall
plane
.
If
the
one
end
of
the
plane
be
higher
then
the
other
,
and
yet
not
verticall
,
it
is
an
inclining
plane
,
and
you
may
find
the
inclination
in
this
maner
.
First
hold
the
verticall
leg
of
the
Sector
vpright
,
and
turne
the
horizontall
le●
about
,
vntill
it
lie
close
with
the
plane
,
and
the
thread
fall
on
his
plummet
line
:
so
the
line
drawne
by
the
edge
of
that
horizontall
leg
,
shall
be
an
horizontall
line
.
Suppose
the
plane
to
be
BGED
,
and
that
BD
were
thus
found
to
be
the
horizontall
line
vpon
the
plane
;
then
may
you
crosse
the
horizontall
line
at
right
angles
with
a
perpendicular
CF
:
that
done
,
if
you
set
one
of
the
legs
of
the
Sector
vpon
the
perpendicular
line
CF
,
and
make
the
other
leg
with
a
thread
and
plummet
to
become
verticall
,
you
shal
haue
the
angle
betweene
the
verticall
line
and
the
perpendicular
on
the
Plane
,
as
before
in
the
vse
of
the
Sector
,
pag.
50.
and
the
complement
of
this
angle
is
the
inclination
of
the
plane
to
the
horizon
.
To
find
the
declination
of
a
Plane
.
The
declination
of
a
Plane
is
alwayes
reckoned
in
the
horizon
betweene
the
line
of
East
and
West
,
and
the
horizontall
line
vpon
the
Plane
.
As
in
the
fundamentall
Diagram
,
the
prime
verticall
line
(
which
is
the
line
of
East
and
West
)
is
ECW
;
if
the
horizontall
line
of
the
plane
proposed
shal
be
BCD
,
the
angle
of
declination
is
ECB
.
But
because
a
Plane
may
decline
diuers
wayes
,
that
we
may
the
better
distinguish
them
,
we
consider
three
lines
belonging
to
euery
Plane
:
the
first
is
the
horizontall
line
;
the
second
the
perpendicular
line
,
crossing
the
horizontall
at
right
angles
;
the
third
the
axis
of
the
plane
,
crossing
both
the
horizontall
line
,
and
his
perpendicular
,
and
the
plane
it selfe
at
right
angles
.
The
perpendicular
line
doth
help
to
find
the
inclination
of
the
plane
as
before
,
the
horizontall
to
finde
the
declination
,
the
axis
to
giue
denomination
vnto
the
plane
.
For
example
,
in
a
verticall
plane
here
represented
by
EZW
,
the
horizontall
line
is
ECW
,
the
same
with
the
line
of
East
and
West
,
and
therefore
no
declination
;
the
perpendicular
crossing
it
,
CZ
the
same
with
the
verticall
line
,
drawne
from
the
center
to
the
zenith
,
right
vnto
the
horizon
,
and
therefore
no
inclination
.
The
axis
of
the
plane
is
SCN
,
the
same
with
the
meridian
line
,
drawne
from
the
South
to
the
North
,
and
accordingly
giues
the
denomination
to
the
plane
.
For
the
plane
hauing
two
faces
,
and
the
axis
two
poles
,
S
and
N
;
the
pole
S
falling
directly
into
the
South
,
doth
cause
that
face
to
which
it
is
next
to
be
called
the
South
face
;
and
the
other
pole
at
N
,
pointing
into
the
North
,
doth
giue
the
denomination
to
the
other
face
,
and
make
it
to
be
called
the
North
face
of
this
plane
.
In
like
maner
in
the
declining
inclining
plane
here
represented
by
BFD
,
the
horizontall
line
is
BCD
,
which
crosseth
the
prime
verticall
line
ECW
,
and
therefore
it
is
called
a
declining
plane
,
according
to
the
angle
of
declination
ECB
or
WCD
.
The
perpendicular
to
this
horizontall
line
is
CF
,
where
the
point
F
falleth
in
the
plane
QZH
perpendicular
to
the
plane
proposed
,
betweene
the
zenith
and
the
North
part
of
the
horizon
,
and
therefore
it
is
called
a
plane
inclining
to
the
Northward
,
according
to
the
arke
FQ
,
or
the
angle
FCQ
.
The
axis
of
the
plane
is
here
represented
by
the
line
CK
,
where
the
pole
K
is
90
gr.
distant
from
the
plane
,
and
so
is
as
much
aboue
the
horizon
at
H
,
and
the
other
pole
as
much
below
the
horizon
at
Q
,
as
the
plane
at
F
is
distant
from
the
zenith
:
and
this
pole
K
here
falling
betweene
the
meridian
and
the
prime
verticall
circle
into
the
Southwest
part
of
the
world
,
this
vpper
face
of
the
plane
is
therefore
called
the
Southwest
face
,
and
the
lower
the
Northeast
face
of
the
plane
.
The
declination
from
the
prime
verticall
may
be
found
by
the
needle
in
the
vsuall
inclinatorie
Quadrant
,
or
rather
by
comparing
the
horizontall
line
drawne
vpon
the
plane
with
the
azimuth
of
the
Sunne
and
the
meridian
line
,
in
such
sort
as
before
we
found
the
variation
of
the
magneticall
needle
.
For
take
any
boord
that
hath
one
side
strait
,
and
draw
the
line
HO
parallell
to
that
side
,
and
the
line
ZM
perpendicular
vnto
it
,
and
on
the
center
Z
make
a
semicircle
HMO
:
this
done
,
hold
the
boord
to
the
plane
,
so
as
HO
may
be
parallel
to
BD
the
horizontall
line
on
the
plane
and
the
boord
parallell
to
the
horizon
;
then
the
Sunne
shining
vpon
it
,
hold
out
a
thread
and
plummet
,
so
as
the
thread
being
verticall
,
the
shadow
of
the
Sunne
may
fall
on
the
center
Z
,
and
draw
the
line
of
shadow
AZ
representing
the
common
section
,
which
the
azimuth
of
the
Sunne
makes
with
the
plane
of
the
horizon
,
and
let
another
take
the
altitude
of
the
Sunne
at
the
same
instant
:
so
by
resoluing
a
triangle
,
as
I
shewed
before
pag.
65.
you
may
find
what
azimuth
the
Sun
was
in
when
he
gaue
shadow
vpon
AZ
.
Suppose
the
azimuth
to
be
(
as
before
pag.
64.
)
72
gr.
52
m.
from
the
North
to
the
Westward
,
and
therefore
17
gr.
8
m.
from
the
West
,
we
may
allow
these
17
gr.
8
m.
from
A
vnto
V
,
and
draw
the
line
ZV
,
and
so
we
haue
the
true
West
point
of
the
prime
verticall
line
:
then
allowing
90
gr.
from
V
vnto
S
,
we
haue
the
South
point
of
the
meridian
line
ZS
,
and
the
angle
HZV
shall
giue
the
declination
of
the
plane
from
the
verticall
,
and
the
angle
OZS
the
declination
of
the
plane
from
the
meridian
.
Or
we
may
take
out
onely
the
angle
AZH
,
which
the
line
of
shadow
makes
with
the
horizontall
line
of
the
plane
,
and
compare
it
with
the
angle
AZV
,
which
the
line
of
shadow
makes
with
the
prime
verticall
.
And
so
here
if
AZV
the
Sunnes
azimuth
shall
be
17
gr.
8
m.
past
the
West
,
and
yet
the
line
of
shadow
AZ
7
gr.
12
m.
short
of
the
plane
,
the
declination
of
the
plane
shall
be
24
gr.
20
m.
as
may
appeare
by
the
site
of
the
plane
and
the
circles
.
If
the
altitude
of
the
Sunne
be
taken
at
such
time
as
the
shadow
of
the
thread
falleth
on
BD
or
HO
,
and
then
a
triangle
resolued
,
the
declinat●on
of
the
plane
will
be
such
as
the
azimuth
of
the
Sunne
from
the
prime
verticall
.
If
at
such
a
time
as
the
shadow
falleth
on
MZ
,
the
declination
will
be
such
as
the
azimuth
of
the
Sunne
from
the
meridian
.
If
it
be
a
faire
Summers
day
you
may
first
finde
what
altitude
the
Sunne
will
haue
when
he
cometh
to
be
due
East
or
West
,
and
then
expect
vntill
he
come
to
that
altitude
;
so
the
declination
of
the
plane
shall
be
such
as
the
angle
contained
betweene
the
line
HO
and
the
line
of
the
shadow
.
Hauing
distinguished
the
Planes
,
the
next
care
will
be
for
the
placing
of
the
style
and
the
drawing
of
the
houre-lines
.
The
style
will
be
as
the
axis
of
the
world
,
sometimes
parallell
to
the
plane
,
sometimes
perpendicular
,
sometimes
cut
the
plane
with
oblique
angles
.
The
houre-lines
will
be
either
parallell
one
to
the
other
,
or
meete
in
a
center
with
equall
angles
,
or
meete
with
vnequall
angles
.
If
the
style
be
perpendicular
to
the
plane
,
the
angles
at
the
center
will
be
equall
;
and
this
falls
out
onely
in
the
South
and
North
face
of
an
equinoctiall
plane
:
if
the
style
be
parallell
to
the
plane
,
the
houre-lines
wil
be
also
parallell
one
to
another
;
and
this
falls
out
in
all
polar
planes
;
as
in
the
East
and
West
meridian
planes
parallel
to
the
circle
of
the
houre
of
12
,
in
the
vpper
and
lower
direct
polars
parallell
to
the
circles
of
the
houre
of
6
,
and
in
the
vpper
and
lower
declining
polars
which
are
parallell
to
any
of
the
other
houre
circles
.
But
in
the
horizontall
and
all
other
planes
,
the
style
will
cut
the
plane
with
an
acute
angle
,
and
the
houre
lines
will
meet
at
the
root
of
the
style
,
and
there
make
vnequall
angles
.
CHAP.
I.
To
draw
the
houre-lines
in
an
equinoctiall
Plane
.
AN
equinoctiall
plane
is
that
which
is
parallell
to
the
equinoctiall
circle
here
represented
by
EAW
,
wherein
the
spaces
betweene
the
houre
circles
being
equall
,
there
is
no
need
of
further
precept
,
but
onely
to
draw
a
circle
and
to
diuide
it
into
24
equall
parts
for
the
24
houres
,
and
subdiuide
each
houre
into
halues
and
quarters
,
and
then
to
set
vp
geometric illustration
the
style
perpendicular
to
the
plane
in
the
center
of
the
circle
.
The
help
which
these
lines
of
proportion
doe
here
affoord
vs
,
is
onely
in
the
diuision
of
the
circle
,
which
may
be
done
readily
by
that
which
I
shewed
before
,
Pag.
29.
For
example
,
suppose
the
semidiameter
of
the
equinoctiall
circle
to
be
six
inches
,
and
that
it
were
required
to
know
the
distance
of
the
houre-points
each
from
other
:
here
each
houre
being
15
gr.
distant
from
other
,
I
extend
the
compasses
from
the
sine
of
30
gr.
vnto
the
sine
of
7
gr.
30
m.
the
halfe
of
15
gr.
and
I
find
the
same
extent
to
reach
in
the
line
of
numbers
from
6.00
vnto
1.56
.
Or
in
crosse
worke
I
extend
them
from
the
sine
of
30
gr.
vnto
6.00
in
the
line
of
numbers
,
the
same
extent
will
reach
from
the
sine
of
7
gr.
30
m.
vnto
1.56
in
the
line
of
numbers
;
which
shewes
that
in
a
circle
of
six
inches
semidiameter
,
the
geometric illustration
distance
of
the
houre-points
each
from
other
will
be
about
1
inch
and
56
centesmes
or
parts
of
100.
The
like
reason
holds
for
the
inscribing
of
all
other
chords
in
the
Prop.
following
.
CHAP.
II.
To
draw
the
houre-lines
in
a
direct
polar
plane
.
A
Direct
polar
plane
is
that
which
is
parallell
to
the
houre
of
6
,
here
represented
by
EPW
,
wherein
the
style
will
be
parallell
to
the
plane
,
and
the
houre-lines
parallell
one
to
the
other
,
and
therefore
may
be
best
drawne
by
that
which
I
haue
shewed
in
the
vse
of
the
Sector
.
They
may
be
also
drawne
by
the
helpe
of
these
lines
of
proportion
,
in
this
maner
.
First
draw
a
right
line
WE
for
the
horizon
and
the
equator
,
and
crosse
it
at
the
point
C
,
about
the
middle
of
the
line
with
CB
another
right
line
,
which
may
serue
for
the
meridian
and
the
houre
of
12
,
and
must
also
be
the
substylar
line
wherein
the
style
shall
stand
.
Then
,
to
proportion
the
style
vnto
the
plane
,
consider
the
length
of
the
horizontall
line
,
and
what
houre-lines
you
would
haue
to
fall
on
your
plane
.
For
the
distance
of
any
one
houre-line
from
the
meridian
being
knowne
,
we
may
finde
both
the
length
of
the
style
and
the
distance
of
the
rest
:
because
As
the
tangent
of
the
houre
giuen
,
is
to
the
distance
from
the
meridian
:
So
the
tangent
of
45
gr.
to
the
height
of
the
style
.
Suppose
the
length
of
the
horizontall
line
to
be
12
inches
,
and
that
it
were
required
to
put
on
all
the
houre-lines
from
7
in
the
morning
vnto
5
in
the
euening
.
Here
we
haue
5
houres
and
6
inches
on
either
side
the
meridian
.
Wherefore
I
allow
15
gr.
for
an
houre
,
and
extending
the
compasses
the
compasses
from
the
tangent
of
75
gr.
the
measure
of
5
houres
vnto
the
tangent
of
45
gr.
I
find
the
same
extent
to
reach
in
the
line
of
numbers
from
6.00
to
about
1.61
.
This
shewes
both
the
height
of
the
style
,
and
the
distance
of
the
houre-points
of
9
&
3
from
the
meridian
to
be
1
inch
,
61
parts
.
To
find
the
length
of
the
Tangent
betweene
the
substylar
and
the
houre-points
.
As
the
tangent
of
45
gr.
to
the
tangent
of
the
houre
:
So
the
height
of
the
style
to
the
length
of
the
tangent
line
betweene
the
substylar
and
the
houre-points
.
Thus
hauing
found
the
length
of
the
style
in
our
example
to
be
1.61
,
if
I
exend
the
compasses
from
the
tangent
of
45
gr.
vnto
the
tangent
of
15
gr.
the
measure
of
the
first
houre
from
the
substylar
,
I
shall
find
the
same
extent
to
reach
in
the
line
of
numbers
from
1.61
vnto
0.43
for
the
length
of
the
tangent
betweene
the
substylar
and
the
houre-points
of
11
and
1.
If
I
extend
them
from
the
tangent
of
45
gr.
vnto
the
tangent
of
75
gr.
the
measure
of
the
fift
houre
,
I
shall
finde
them
to
reach
in
the
line
of
numbers
from
1.61
vnto
6.00
for
the
length
of
the
tangent
from
the
substylar
to
the
houre-points
of
7
and
5.
For
howsoeuer
it
be
the
same
distance
in
the
line
of
tangents
from
45
vnto
75
,
as
from
45
vnto
15
;
yet
because
75
are
more
,
and
15
lesse
then
45
,
the
tangent
lines
that
answer
to
them
wil
be
accordingly
more
or
lesse
then
the
length
of
the
style
.
Ho.
|
An.
Po
|
Tang.
|
Gr.
M.
|
In.
Par
|
12
|
0
0
|
0
0
|
11.1
|
15
0
|
0
43
|
10.2
|
30
0
|
0
93
|
9.3
|
45
0
|
1
61
|
8.4
|
60
0
|
2
79
|
7.5
|
75
0
|
6
0
|
6.6
|
90
0
|
Infin
|
Againe
,
if
I
extend
them
from
45
gr.
in
the
tangents
vnto
30
gr.
the
measure
of
the
second
houre
,
I
shall
finde
them
to
reach
in
the
line
of
numbers
from
1.61
vnto
0.93
for
the
houre
of
10
and
2
:
if
I
extend
them
from
the
tangent
of
45
gr.
vnto
the
tangent
of
60
gr.
for
the
fourth
houre
,
I
shal
find
them
to
reach
in
the
line
of
numbers
from
1.61
vnto
2
79
,
and
such
is
the
length
of
the
tangent
line
from
the
substylar
vnto
the
houre
of
8
and
4.
And
the
like
reason
holdeth
for
the
inscribing
of
all
other
tangent
lines
in
the
propositions
following
.
But
for
such
tangents
as
fall
vnder
45
gr.
I
may
better
vse
crosse
worke
,
and
extend
the
compasses
from
the
tangent
of
45
gr.
vnto
1.61
in
the
line
of
numbers
,
so
shall
I
finde
the
same
extent
to
reach
from
30
gr.
in
the
tangents
,
to
93
parts
in
the
line
of
numbers
,
for
the
distance
of
the
second
houre
,
and
from
15
gr.
in
the
tangents
to
43
parts
for
the
distance
of
the
first
houre
from
the
meridian
.
Or
if
this
extent
from
45
gr.
backward
to
1.61
be
too
large
for
the
compasses
,
I
may
extend
forward
from
the
tangent
of
5
gr.
43
m.
to
1.61
parts
in
the
line
of
numbers
,
and
the
same
extent
shall
reach
from
15
gr.
in
the
tangents
,
to
43
parts
in
the
line
of
numbers
,
for
the
distance
of
the
first
houre
;
and
from
30
gr.
to
93
parts
,
for
the
distance
of
the
second
houre
,
as
before
.
Hauing
found
the
length
of
the
tangent
lines
in
inches
and
parts
of
inches
,
and
pricked
them
in
the
equator
on
both
sides
of
the
meridian
,
from
the
center
C
;
if
wee
draw
right
lines
through
each
of
those
points
,
crossing
the
equator
at
right
angles
,
they
shall
be
the
houre-lines
required
;
and
if
we
set
a
style
ouer
the
meridian
,
so
as
the
edge
of
it
be
parallell
to
the
plane
,
and
the
height
of
it
be
as
much
aboue
the
meridian
as
the
distance
between
the
meridian
and
the
houre-points
of
3
or
9
,
it
shall
represent
the
axis
of
the
world
,
and
be
truly
placed
for
the
casting
of
the
shadow
vpon
the
houre-lines
in
a
polar
plane
.
Horae
|
Ang.
Po
|
Tang
|
Gr.
M.
|
In.
Pa
|
6
|
0
|
0
|
|
3
45
|
165
|
|
7
30
|
1
32
|
|
11
15
|
1
99
|
7
|
15
0
|
2
68
|
|
18
45
|
3
39
|
|
22
30
|
4
14
|
|
26
15
|
4
93
|
8
|
30
0
|
5
77
|
|
33
45
|
6
68
|
|
37
30
|
7
67
|
|
41
15
|
8
77
|
9
|
45
0
|
10
00
|
|
48
45
|
11
40
|
|
52
30
|
13
03
|
|
56
15
|
14
97
|
10
|
60
0
|
17
32
|
|
63
45
|
20
28
|
|
67
30
|
24
14
|
|
71
15
|
29
46
|
11
|
75
0
|
37
32
|
|
78
45
|
50
27
|
|
82
30
|
75
96
|
|
86
15
|
152
57
|
12
|
90
0
|
Infin
.
|
CHAP.
III.
To
draw
the
houre-lines
in
a
meridian
plane
.
A
Meridian
plane
is
that
which
is
parallell
to
the
meridian
circle
here
represented
by
SZN
;
it
hath
two
faces
,
one
to
the
East
,
and
the
other
to
the
West
;
in
each
of
them
the
style
will
be
parallell
to
the
plane
,
and
the
houre-lines
parallell
one
to
the
other
,
as
in
a
polar
plane
,
the
difference
being
onely
in
the
placing
of
the
equator
&
in
numbring
of
the
houres
.
geometric illustration
For
supposing
the
length
of
the
style
CB
to
be
ten
inches
,
the
length
of
the
tangent
line
belonging
to
the
first
houre
wil
be
2
in
.
68
p.
the
length
of
the
second
5
in
.
77
p.
as
in
the
Table
.
Then
the
tangent
of
15
gr.
being
prickt
downe
in
the
equator
on
both
sides
from
6
,
shal
serue
for
the
houres
of
5
and
7
,
and
the
tangent
of
30
gr.
for
the
houres
of
4
and
8
,
and
so
in
the
rest
.
This
done
,
if
we
draw
right
lines
through
each
of
these
points
,
crossing
the
equator
at
right
angles
,
they
shall
be
the
houre-lines
required
:
and
if
we
set
a
style
ouer
the
houre
of
6
,
so
as
the
edge
of
it
may
be
parallell
to
the
plane
,
and
the
height
of
it
may
be
equall
to
the
distance
betweene
the
houres
of
6
and
9
in
the
equator
,
it
shall
represent
the
axis
of
the
world
,
and
be
truly
placed
for
the
casting
of
the
shadow
vpon
the
houre-lines
in
a
meridian
plane
.
CHAP.
IIII.
To
draw
the
houre-lines
in
an
horizontall
plane
.
AN
horizontall
plane
is
that
which
is
parallell
to
the
horizon
,
here
represented
by
the
outward
circle
ESWN
,
in
which
the
diameter
SN
drawne
from
the
South
to
the
North
,
may
go
both
for
the
meridian
line
and
the
meridian
circle
,
Z
for
the
zenith
,
P
for
the
pole
of
the
world
,
and
the
circles
drawne
through
P
for
the
houre-circles
of
1.2.3.4
.
&c.
as
they
are
numbred
from
the
meridian
.
These
houre-circles
considered
with
the
meridian
&
the
horizon
,
do
make
diuers
triangles
,
PN
1
,
PN
2
,
PN
3
,
in
which
we
haue
knowne
first
the
right
angle
at
N
,
the
North
intersection
of
the
meridian
and
the
horizon
;
secondly
the
side
PN
,
the
arke
of
the
meridian
between
the
pole
and
the
horizon
,
which
is
alwayes
equall
to
the
latitude
of
the
place
;
thirdly
the
angles
at
the
pole
,
made
by
the
meridian
and
the
houre-circles
,
the
angle
NP
1
being
15
gr.
NP
2
30
gr.
each
houre
15
gr.
more
then
other
,
each
halfe
houre
7
gr.
30
m.
each
quarter
3.
gr.
45.
m.
And
these
three
being
known
,
we
may
finde
the
arks
of
the
horizon
between
the
meridian
and
the
houre-circles
N
1
,
N
2
,
N
3
,
&c.
For
As
the
sine
of
90
gr.
is
to
the
sine
of
the
latitude
:
So
the
tangent
of
the
houre
to
the
tangent
of
the
houre-line
from
the
meridian
.
Ho.
|
Ang.
Po
|
Arc.
Pla
|
Gr.
M.
|
Gr.
M.
|
12
|
0
0
|
0
0
|
|
3
45
|
2
56
|
|
7
30
|
5
52
|
|
11
15
|
8
51
|
1
|
15
0
|
11
50
|
|
18
45
|
14
52
|
|
22
30
|
17
57
|
|
26
15
|
21
6
|
2
|
30
0
|
24
20
|
|
33
45
|
27
36
|
|
37
30
|
31
0
|
|
41
15
|
34
28
|
3
|
45
0
|
38
3
|
|
48
45
|
41
45
|
|
52
30
|
45
34
|
|
56
15
|
49
30
|
4
|
60
0
|
53
35
|
|
63
45
|
57
47
|
|
67
30
|
62
6
|
|
71
15
|
66
33
|
5
|
75
0
|
71
6
|
|
78
45
|
75
45
|
|
82
30
|
80
25
|
|
86
15
|
85
13
|
6
|
90
0
|
90
0
|
geometric illustration
Extend
the
compasses
from
the
sine
of
90
gr.
to
the
sine
of
the
latitude
,
so
the
same
extent
shall
reach
from
the
tangent
of
the
houre
,
to
the
tangent
of
the
houre-line
from
the
meridian
.
Thus
the
latitude
being
51
gr.
30
m.
I
extend
the
compasses
from
the
sine
of
90
gr.
to
the
sine
of
51
gr.
30
m.
&
find
the
same
extent
to
reach
from
the
tangent
of
3
gr.
45
m.
vnto
the
tangent
of
2
gr.
56
m.
for
the
distance
of
the
first
quarter
from
the
meridian
;
and
from
the
tangent
of
7
gr.
30
m.
vnto
the
tangent
of
5
gr.
52
m.
for
the
halfe
houre
;
and
from
the
tangent
of
11
gr.
15
m.
to
the
tangent
of
8
gr.
51
m.
for
the
third
quarter
;
and
from
the
tangent
of
15
gr.
0
m.
vnto
11
gr.
50
m.
for
the
first
houre
:
and
so
in
the
rest
.
geometric illustration
This
done
,
I
come
to
the
Plane
,
and
there
according
as
the
lines
do
fall
in
the
fundamentall
diagram
,
1
I
draw
a
right
line
SN
seruing
for
the
meridian
,
the
houre
of
12
and
the
substylar
.
2
In
this
meridian
I
make
choice
of
a
center
at
C
,
and
there
describe
an
occult
circle
representing
the
horizon
.
3
I
find
a
chord
of
11
gr.
50
m.
and
inscribe
it
into
this
circle
on
either
side
of
the
meridian
for
the
houres
of
11
&
1
;
in
like
maner
a
chord
of
24
gr.
20
m.
for
the
houres
of
10
and
2
;
and
a
chord
of
38
gr.
3
m.
for
the
houres
of
9
and
3
;
and
so
for
the
rest
of
the
houres
,
their
halues
and
quarters
.
4
I
draw
right
lines
through
the
center
and
the
termes
of
these
chords
,
and
these
lines
so
drawne
are
the
houre-lines
required
.
Lastly
I
set
vp
the
style
ouer
the
meridian
,
so
as
it
may
cut
the
plane
in
the
center
,
and
there
make
an
angle
with
the
meridian
equall
to
the
latitude
of
the
place
,
so
it
shall
represent
the
axis
of
the
world
,
and
be
truly
placed
for
casting
of
the
shadow
vpon
the
houre-lines
in
an
horizontall
plane
.
CHAP.
V.
To
draw
the
houre-lines
in
a
verticall
plane
.
A
Verticall
plane
is
that
which
is
parallel
to
the
prime
verticall
circle
here
represented
by
EZW
.
It
hath
two
faces
,
one
to
the
North
,
the
other
to
the
South
;
in
each
of
them
the
substylar
will
be
the
same
with
the
meridian
line
,
and
the
angle
of
the
style
aboue
the
plane
will
be
equall
to
ZP
the
complement
of
the
latitude
.
The
triangles
here
considered
are
made
by
the
verticall
,
the
meridian
,
and
the
houre-circles
,
in
which
we
know
the
side
ZP
,
the
angles
at
the
pole
,
and
the
right
angle
at
the
zenith
,
and
therefore
may
find
the
arks
of
the
verticall
,
between
the
meridian
and
the
houre-circles
after
this
maner
:
As
the
sine
of
90
gr.
is
to
the
cosine
of
the
latitude
:
So
the
tangent
of
the
houre
to
the
tangent
of
the
houre-line
from
the
meridian
.
Extend
the
compasses
from
the
sine
of
90
gr.
to
the
sine
of
the
complement
of
the
latitude
,
so
the
same
extent
shall
reach
from
the
tangent
of
the
houre
,
to
the
tangent
of
the
houre-line
from
the
meridian
.
Thus
in
the
latitude
of
51
gr.
30
m.
I
extend
the
compasses
from
the
sine
of
90
gr.
to
the
sine
of
38
gr.
30
m.
and
find
the
same
extent
to
reach
from
the
tangent
of
15
gr.
to
the
tangent
of
9
gr.
28
m
for
the
distance
of
the
first
houre
from
the
meridian
:
and
from
the
tangent
of
75
gr.
vnto
the
tangent
of
66
gr.
42
m.
for
the
fift
houre
;
and
so
in
the
rest
as
in
the
Table
following
.
geometric illustration
These
arks
being
knowne
,
I
may
come
to
the
plane
,
and
there
by
help
of
a
thread
and
plummet
draw
a
verticall
line
seruing
both
for
the
meridian
and
the
houre
of
12
,
and
the
substylar
;
then
may
I
draw
an
occult
verticall
circle
,
and
therein
inscribe
the
chords
of
those
former
arks
,
and
draw
the
houre-lines
,
and
set
vp
the
style
,
as
before
in
the
horizontall
plane
.
Ho.
|
Ang.
Po
|
Arc.
Pla
|
Gr.
M.
|
Gr.
M.
|
12
|
0
0
|
0
0
|
|
3
45
|
2
20
|
|
7
30
|
4
41
|
|
11
15
|
7
3
|
1
|
15
0
|
9
28
|
|
18
45
|
11
56
|
|
22
30
|
14
27
|
|
26
15
|
17
4
|
2
|
30
0
|
19
46
|
|
33
45
|
22
35
|
|
37
30
|
25
32
|
|
41
15
|
28
38
|
3
|
45
0
|
31
54
|
|
48
45
|
35
22
|
|
52
30
|
39
3
|
|
56
15
|
42
58
|
4
|
60
0
|
47
9
|
|
63
45
|
51
36
|
|
67
30
|
56
20
|
|
71
15
|
61
23
|
5
|
75
0
|
66
42
|
|
78
45
|
72
17
|
|
82
30
|
78
3
|
|
86
15
|
84
0
|
6
|
90
0
|
90
0
|
If
it
be
the
South
face
of
the
plane
,
the
center
will
be
vpward
,
and
the
style
must
point
downward
;
if
the
North
face
,
the
center
must
be
in
the
lower
part
of
the
meridian
line
,
and
the
style-point
vpward
in
all
such
places
as
are
to
the
Northward
of
the
equinoctiall
line
,
as
it
may
appeare
by
considering
how
the
lines
do
fall
in
the
fundamentall
Diagram
.
CHAP.
VI.
To
draw
the
houre-lines
in
a
verticall
inclining
plane
.
ALl
those
Planes
that
haue
their
horizontall
line
lying
East
and
West
,
are
in
that
respect
said
to
be
verticall
;
if
they
be
also
vpright
and
passe
through
the
zeni●h
,
they
are
direct
verticals
;
if
they
incline
to
the
pole-they
are
direct
polars
:
if
to
the
equinoctiall
,
they
are
properly
called
equinoctiall
planes
,
and
are
described
before
:
if
to
none
of
these
three
points
,
they
are
then
called
by
the
generall
name
of
inclining
verticals
.
These
may
incline
either
to
the
North
part
of
the
horizon
,
or
to
the
South
;
and
each
of
them
hath
two
faces
,
one
to
the
zenith
,
the
other
to
the
nadir
,
in
which
we
are
first
to
consider
the
height
of
the
pole
aboue
the
plane
,
by
comparing
the
inclination
of
the
plane
to
the
horizon
,
with
the
latitude
of
the
place
.
As
in
our
latitude
of
51
gr.
30
m.
if
the
inclination
of
the
plane
EIW
shall
be
13
gr.
Northward
,
that
is
,
if
IN
the
ark
of
the
meridian
between
the
plane
and
the
North
part
of
the
horizon
shall
be
13
gr.
we
may
take
these
13
gr.
out
of
PN
51
gr.
30
m.
the
eleuation
of
the
pole
aboue
the
horizon
,
and
there
wil
remain
PI
38
gr.
30
m.
for
the
eleuation
of
the
North
pole
aboue
the
vpper
face
of
the
plane
,
and
therefore
38
gr.
30
m.
for
the
height
of
the
South
pole
aboue
the
lower
face
of
the
plane
.
Or
if
the
inclination
of
the
plane
shall
be
found
to
be
62
gr.
to
the
Southward
,
we
may
number
them
in
the
meridian
from
S
the
Southpart
of
the
horizon
vnto
L
,
and
there
draw
the
arke
ELW
representing
this
plane
;
so
the
arke
of
the
meridian
PL
shall
giue
the
height
of
the
North
pole
aboue
the
vpper
face
of
this
plane
to
be
66
gr.
30
m.
and
therefore
the
height
of
the
South
pole
aboue
the
lower
face
of
the
plane
is
also
66
gr.
30
m.
In
like
maner
if
the
inclination
of
the
plane
EYW
shall
be
15
gr.
Southward
,
that
is
,
if
SY
the
arke
of
the
meridian
between
the
South
part
of
the
horizon
and
the
plane
,
shall
be
15
gr.
The
height
of
the
North
pole
aboue
the
vpper
face
of
the
plane
,
and
the
height
of
the
South
pole
aboue
the
lower
face
of
the
plane
,
will
be
also
found
to
be
66
gr.
30
m.
But
if
the
plane
shall
fall
betweene
the
zenith
and
the
North
pole
,
then
will
the
North
pole
be
eleuated
aboue
the
lower
face
,
and
the
South
pole
aboue
the
vpward
face
of
the
plane
,
as
may
appeare
by
the
proiection
of
the
sphere
in
the
fundamentall
Diagram
.
Then
in
the
triangles
made
by
the
plane
,
the
meridian
,
and
the
houre-circles
,
we
haue
the
side
which
is
the
height
of
the
pole
aboue
the
plane
,
together
with
the
angles
at
the
pole
,
and
the
right
angle
at
the
intersection
of
the
meridian
with
the
plane
,
by
which
we
may
find
the
arks
of
the
plane
betweene
the
meridian
and
the
houre-circles
,
after
this
maner
:
As
the
sine
of
90
gr.
is
to
the
sine
of
the
pole
aboue
the
plane
:
So
the
tangent
of
the
houre
to
the
tangent
of
the
houre-line
from
the
meridian
.
Thus
in
the
former
example
,
where
PI
the
height
of
the
pole
aboue
the
plane
was
found
to
be
38
gr.
30
m.
if
you
shall
extend
the
compasses
from
the
sine
of
90
gr.
to
the
sine
of
38
gr.
30
m.
the
same
extent
wil
reach
from
the
tangent
of
15
gr.
vnto
the
tangent
of
9
gr.
28
m.
for
the
distance
of
the
first
houre
from
the
meridian
,
and
from
30
gr.
vnto
19
gr.
46
m.
for
the
second
houre
,
and
so
forward
as
in
the
direct
vertical
.
And
for
the
two
last
examples
,
you
may
extend
the
compasses
from
the
sine
of
90
gr.
vnto
the
sine
of
66
gr.
30
m
:
so
the
same
extent
shal
reach
in
the
line
of
tangents
from
15
gr.
vnto
13
gr.
48
m.
for
the
first
houre
,
from
75
gr.
vnto
73
gr.
43
m.
for
the
fift
houre
,
from
30
gr.
vnto
27
gr.
54
m.
for
the
second
houre
,
from
60
gr.
vnto
57
gr.
48
m.
for
the
fourth
houre
,
and
from
45
gr.
vnto
42
gr.
31
m.
for
the
third
houre
from
the
meridian
.
These
arks
being
knowne
,
you
may
first
draw
the
horizontall
line
,
and
crosse
it
in
the
middle
with
a
perpendicular
that
may
serue
both
for
the
meridian
and
the
houre
of
12
,
and
the
substylar
;
then
knowing
which
pole
is
eleuated
aboue
the
plane
,
you
may
accordingly
make
choice
of
a
fit
point
in
the
meridian
for
the
center
of
your
houre-lines
,
and
thence
describe
an
occult
arke
of
a
circle
,
inscribe
the
chords
of
those
former
arks
,
and
draw
the
houre
lines
,
and
set
vp
the
style
,
as
I
shewed
before
in
the
horizontall
plane
.
CHAP.
VII
.
To
draw
the
houre-lines
in
an
verticall
declining
Plane
.
ALl
vpright
planes
whereon
a
man
may
draw
a
verticall
line
,
are
in
this
respect
said
to
be
verticall
;
if
they
shall
also
stand
directly
East
and
West
,
they
are
direct
verticals
;
if
directly
North
and
South
,
they
are
properly
called
meridian
planes
,
and
are
described
before
:
if
they
behold
none
of
these
foure
principall
parts
of
the
world
,
but
shall
stand
between
the
prime
verticall
and
the
meridian
,
they
are
then
called
by
the
generall
name
of
declining
verticals
.
These
haue
two
faces
,
one
to
the
South
,
the
other
to
the
Northward
,
which
may
be
distinguished
in
these
Northerne
parts
of
the
world
after
this
maner
.
If
the
Sunne
coming
to
the
meridian
shall
shine
vpon
the
plane
,
it
is
the
South
face
;
if
not
,
it
is
the
North
face
of
that
plane
.
Againe
,
if
the
Sunne
shall
shine
vpon
the
plane
at
high
noone
,
and
yet
longer
in
the
forenoone
then
in
the
afternoon
,
it
is
the
Southeast
face
;
if
longer
in
the
afternoone
then
in
the
forenoone
,
it
is
the
Southwest
face
of
the
plane
.
But
how
much
the
declination
cometh
to
,
is
best
found
as
before
.
When
the
declination
is
found
,
there
be
foure
things
more
to
be
considered
before
we
can
come
to
the
drawing
of
the
houre-lines
,
and
all
foure
represented
in
the
fundamentall
Diagram
.
1
The
meridian
of
the
plane
and
his
inclination
to
the
meridian
of
the
place
.
Let
the
arke
EZW
represent
the
prime
verticall
,
and
BZD
a
declining
verticall
,
according
to
the
angle
of
declination
EZB
,
the
meridian
of
the
place
is
represented
by
PZS
,
crossing
the
verticall
EZW
at
right
angles
at
the
zenith
in
the
point
Z
:
but
the
proper
meridian
of
the
plane
wil
be
PR
,
which
is
a
perpendicular
let
downe
from
the
pole
vnto
the
declining
verticall
,
and
crossing
it
with
right
angles
in
the
point
R
,
so
the
angle
RPZ
shall
shew
the
inclination
of
the
two
meridians
,
and
may
thus
be
found
.
In
the
triangle
PRZ
we
know
the
angle
at
R
to
be
a
right
angle
,
and
the
angle
at
Z
,
for
it
is
the
complement
of
the
declination
,
and
the
base
PZ
,
for
it
is
the
complement
of
the
latitude
.
And
therefore
geometric illustration
As
the
sine
of
the
latitude
is
to
the
sine
of
90
gr.
So
the
tangent
of
the
declination
to
the
tangent
of
the
inclination
required
.
2
The
height
of
the
style
aboue
the
plane
.
This
is
here
represented
by
the
perpendicular
arke
PR
,
and
may
be
found
by
that
which
we
haue
knowne
in
the
former
triangle
PRZ
.
For
As
the
sine
of
90
gr.
to
the
cosine
of
the
latitude
:
So
the
cosine
of
the
declination
to
the
sine
of
the
height
of
the
style
.
Or
if
you
please
to
make
vse
,
of
the
angle
of
the
inclination
of
the
two
meridians
,
the
proportion
will
hold
.
As
the
sine
of
90
gr.
to
the
cosine
of
the
inclination
of
meridians
:
So
the
cotangent
of
the
latitude
to
the
tangent
of
the
height
of
the
style
.
3
The
distance
of
the
substylar
from
the
meridian
.
This
is
here
represented
by
the
arke
ZR
,
and
may
be
found
by
that
which
we
haue
knowne
in
the
former
triangle
PRZ
.
As
the
sine
of
90
gr.
to
the
sine
of
the
declination
:
So
the
cotangent
of
the
latitude
to
the
tangent
of
the
substylar
from
the
meridian
.
4
The
distance
of
each
houre-line
from
the
substylar
.
The
distances
of
the
houre-lines
from
the
substylar
,
are
here
represented
by
those
arks
of
the
declining
verticall
belonging
to
the
plane
,
which
are
intercepted
betweene
the
proper
meridian
of
the
plane
and
the
houre-circles
.
To
this
purpose
we
haue
diuers
triangles
made
by
the
declining
plane
,
together
with
his
proper
meridian
and
the
houre-circles
.
In
these
we
haue
knowne
,
first
the
right
angle
at
the
intersection
of
the
proper
meridian
with
the
plane
;
then
the
side
which
is
the
height
of
the
pole
aboue
the
plane
;
and
thirdly
the
angles
at
the
pole
.
For
knowing
the
angle
of
inclination
betweene
the
meridian
of
the
plane
and
the
meridian
of
the
place
,
which
is
alwayes
the
houre
of
12
,
we
may
finde
the
angle
betweene
the
meridian
of
the
plane
and
the
houre
of
1
,
by
allowing
in
15
gr.
and
the
angle
betweene
the
meridian
of
the
plane
and
the
houre
of
2
,
by
allowing
in
30
gr.
and
so
for
the
rest
,
which
being
knowne
,
we
may
find
the
arks
of
the
plane
from
the
substylar
to
the
houre-circles
,
in
this
maner
.
As
the
sine
of
90
gr.
to
the
sine
of
the
height
of
the
pole
aboue
the
plane
:
So
the
tangent
of
the
houre
from
the
proper
meridian
,
to
the
tangent
of
the
houre-line
from
the
substylar
.
Thus
in
our
latitude
of
51
gr.
30
m.
if
the
declination
of
an
vpright
plane
shall
be
found
to
be
24
gr.
20
m.
from
the
prime
verticall
,
the
one
face
open
to
the
Southwest
,
the
other
to
the
Northwest
,
I
may
number
these
24
gr.
20
m.
in
the
horizon
of
the
fundamentall
Diagram
,
from
E
vnto
B
,
according
to
the
situation
of
the
plane
,
and
there
draw
the
verticall
BZD
,
which
shall
represent
the
plane
proposed
.
Then
taking
the
compasses
into
my
hand
,
1
I
may
extend
them
from
the
sine
of
the
latitude
51
gr.
30
m.
vnto
the
sine
of
90
gr.
the
same
extent
will
reach
in
the
line
of
tangents
from
24
gr.
20
m.
the
declination
giuen
,
to
about
30
gr.
and
such
is
ZPR
the
angle
of
inclination
between
the
meridian
of
the
place
&
the
meridian
of
the
plane
;
and
therefore
the
meridian
of
the
plane
will
here
fall
vpon
the
circle
of
the
second
houre
from
the
meridian
of
the
place
,
(
as
it
may
also
appeare
by
opening
the
compasses
to
the
nearest
extent
,
between
the
pole
and
the
plane
)
and
there
I
place
the
letter
R
to
make
this
rectangle
triangle
PRZ
.
2
I
extend
the
compasses
from
the
sine
of
90
gr.
vnto
the
sine
of
38
gr.
30
m.
the
complement
of
the
latitude
,
and
the
same
extent
will
reach
from
the
sine
of
65
gr.
40
m.
the
complement
of
the
declination
,
vnto
the
sine
of
34
gr.
33
m.
Or
I
may
extend
them
from
the
sine
of
90
gr.
vnto
the
sine
of
60
gr.
the
complement
of
the
inclination
of
the
meridians
,
and
the
same
extent
wil
reach
from
the
tangent
of
38
gr.
30
m.
the
complement
of
the
latitude
,
vnto
the
tangent
of
34
gr.
33
m.
and
such
is
the
arke
PR
,
the
height
of
the
pole
aboue
the
plane
.
3
I
may
extend
the
compasses
from
the
sine
of
90
gr.
vnto
the
sine
of
24
gr.
20
m.
the
declination
giuen
,
and
the
same
extent
will
reach
from
the
tangent
of
38
gr.
30
m.
the
complement
of
the
latitude
,
vnto
the
tangent
of
38
gr.
8
m.
and
such
is
the
arke
ZR
,
the
distance
of
the
substylar
from
the
meridian
.
4
That
I
may
finde
the
distance
of
each
houre-line
from
the
substylar
,
I
consider
the
angle
of
inclination
of
the
meridians
RPZ
,
and
there
see
how
that
PZ
the
meridian
of
the
place
,
which
is
the
houre
of
12
,
being
30
gr.
distance
from
PR
the
meridian
of
the
plane
,
and
that
one
face
of
the
plane
being
open
to
the
Southwest
,
and
the
other
to
the
Northeast
,
this
meridian
of
the
plane
falleth
to
be
the
same
with
the
houre
of
2
,
(
otherwise
with
the
houre
of
10
:
)
therefore
allowing
15
gr.
for
an
houre
,
the
houre
of
1
,
RPO
will
be
15
gr.
and
RFX
the
houre
of
11
,
45
gr.
distant
from
PR
the
proper
meridian
of
the
plane
:
and
so
I
gather
the
inclination
of
the
rest
of
the
houre-circles
toward
this
meridian
,
according
to
their
angles
at
the
pole
,
as
in
this
Table
.
Then
taking
my
compasses
in
my
hand
,
I
extend
them
from
the
sine
of
90
gr.
vnto
the
sine
of
34
gr.
33
m.
the
height
of
the
pole
aboue
the
plane
,
and
finde
them
to
reach
in
the
line
of
tangents
from
15
gr.
the
inclination
of
the
houre
of
1
,
to
8
gr.
38
m.
for
the
arke
of
1
,
from
the
substylar
,
and
from
30
gr.
vnto
18
gr.
8
m.
for
the
houre
of
12
,
agreeable
to
the
third
Prop.
and
from
45
gr.
vnto
29
gr.
33
m.
for
the
houre
of
11
,
and
so
the
rest
,
which
I
also
set
downe
in
a
Table
.
Hor.
|
Ang.
Po
|
Ar.
Pla.
|
Gr.
M.
|
Gr.
M.
|
8
|
90
0
|
90
0
|
9
|
75
0
|
64
42
|
10
|
60
0
|
44
30
|
11
|
45
0
|
29
33
|
12
|
30
0
|
18
8
|
1
|
15
0
|
8
38
|
2
|
Merid
|
substyl
|
3
|
15
0
|
8
38
|
4
|
30
0
|
18
8
|
5
|
45
0
|
29
33
|
6
|
60
0
|
44
30
|
7
|
75
0
|
64
42
|
8
|
90
0
|
90
0
|
These
arks
being
thus
found
,
wil
serue
for
the
drawing
of
the
houre-lines
,
both
on
the
Southwest
face
,
and
the
Northwest
face
of
the
plane
.
For
coming
to
the
plane
,
1
By
the
help
of
a
thread
and
plummet
I
draw
a
verticall
line
,
seruing
both
for
the
meridian
of
the
place
and
the
houre
of
12.
2
In
this
meridian
line
I
make
choice
of
a
center
at
C
,
in
the
vpper
part
of
the
line
,
if
it
be
the
South
face
,
as
here
we
suppose
it
,
that
the
style
may
haue
roome
to
point
downward
;
but
in
the
lower
part
of
the
line
,
if
it
be
the
North
face
of
the
plane
;
for
there
the
style
must
point
vpward
:
and
vpon
this
center
I
describe
an
occult
circle
,
representing
the
declining
verticall
belonging
to
the
plane
.
geometric illustration
4
According
to
the
Table
of
the
arkes
of
the
plane
from
the
substylar
,
I
find
a
chord
of
8
gr.
38
m.
and
inscribe
it
into
this
circle
,
from
the
substylar
toward
the
meridian
,
for
the
houre
of
1.
In
like
maner
a
chord
of
29
gr.
23
m.
for
the
houre
of
11
,
and
a
chord
of
44
gr.
30
m.
for
the
houre
of
10
,
and
so
for
the
rest
of
the
houres
,
their
halues
and
quarters
.
5
I
draw
right
lines
through
the
center
and
the
termes
of
these
chords
,
and
these
lines
so
drawne
are
the
houre-lines
required
.
Lastly
,
I
set
vp
the
style
ouer
the
substylar
,
so
as
it
may
cut
the
plane
in
the
center
,
and
there
make
an
angle
with
the
substylar
of
34
gr.
33
m.
according
to
the
height
of
the
pole
aboue
the
plane
;
so
it
shall
represent
the
axis
of
the
world
,
and
be
truly
placed
for
casting
of
the
shadow
vpon
the
houre-lines
in
this
declining
plane
.
A
second
example
.
After
the
like
maner
if
in
our
latitude
an
vpright
plane
shall
decline
85
gr.
from
the
prime
verticall
,
the
one
face
of
it
being
open
to
the
Northwest
,
and
the
other
to
the
Southeast
,
we
may
in
some
sort
represent
it
by
the
verticall
QZH
,
and
then
working
as
before
.
1
The
angle
ZPT
,
the
inclination
of
the
two
meridians
will
be
found
to
be
86
gr.
5
m.
so
that
PT
the
meridian
of
this
plane
,
will
here
fall
between
the
houre-circles
of
6
and
7
from
the
meridian
.
2
The
arke
PT
the
height
of
the
pole
aboue
the
plane
will
be
onely
3
gr.
6
m.
3
The
arke
ZT
the
distance
of
the
substylar
from
the
meridian
38
gr.
23
m.
4
The
Table
of
the
angles
at
the
pole
will
be
also
gathered
,
by
comparing
the
meridian
of
the
plane
with
the
rest
of
the
houre-circles
.
For
the
angle
TPZ
betweene
PT
the
meridian
of
the
plane
,
PZ
the
meridian
of
the
place
,
and
the
houre
of
12
,
being
86
gr.
5
m.
allowing
15
gr.
for
an
houre
,
the
houre
of
11
½
wil
be
78
gr.
35
m.
and
the
houre
of
11
71
gr.
5
m.
distant
from
the
meridian
of
the
plane
;
&
so
the
rest
of
the
houres
,
as
in
the
second
columne
of
this
Table
.
Hor.
|
Ang.
Po
|
Ar.
Pla.
|
C
F
|
C
G
|
Gr.
M.
|
Gr.
M.
|
In.
Par.
|
In.
Par.
|
12
|
86
5
|
38
23
|
91
08
|
79
21
|
|
78
35
|
15
3
|
30
92
|
26
89
|
11
|
71
5
|
9
6
|
18
42
|
16
02
|
|
63
35
|
6
13
|
12
52
|
10
89
|
10
|
56
5
|
4
36
|
9
25
|
8
05
|
9
|
41
5
|
2
42
|
5
43
|
4
72
|
8
|
26
5
|
1
31
|
3
05
|
2
65
|
7
|
11
5
|
0
36
|
1
20
|
1
05
|
|
Merid.
|
Substyl
|
0
0
|
0
0
|
6
|
3
55
|
0
13
|
0
44
|
0
38
|
5
|
18
55
|
1
4
|
2
15
|
1
86
|
4
|
33
55
|
2
5
|
4
18
|
3
64
|
3
|
48
55
|
3
33
|
7
13
|
6
20
|
2
|
63
55
|
6
20
|
12
77
|
11
10
|
|
71
25
|
9
10
|
18
56
|
16
14
|
1
|
78
55
|
15
28
|
31
82
|
27
67
|
|
86
25
|
40
55
|
99
67
|
86
68
|
Then
hauing
the
height
of
the
pole
aboue
the
plane
,
and
these
angles
at
the
pole
;
the
arkes
of
the
plane
,
betweene
the
substylar
and
the
houre-circles
,
will
be
found
as
in
the
third
columne
.
These
arks
being
found
,
will
serue
for
the
drawing
of
the
houre-lines
on
either
face
of
this
plane
.
1
By
the
help
of
a
thread
and
plummet
I
draw
ZC
a
verticall
plane
,
seruing
both
for
the
meridian
of
the
place
and
the
houre
of
12.
2
In
this
meridian
line
I
make
choice
of
a
center
in
the
vpper
part
of
the
line
,
if
it
had
been
the
Southerne
face
of
the
plane
,
but
here
in
C
the
lower
part
of
the
line
,
because
we
supposed
it
to
be
the
Northwest
face
of
the
plane
,
and
the
style
must
point
vpward
;
and
vpon
this
center
I
describe
an
occult
circle
representing
the
declining
verticall
belonging
to
this
plane
.
geometric illustration
4
The
substylar
being
drawne
,
I
may
inscribe
the
chords
of
the
arks
of
the
plane
from
the
substylar
,
and
draw
the
houre-lines
,
and
set
vp
the
style
as
in
the
former
plane
.
Or
the
arks
of
the
plane
from
the
substylar
being
found
as
before
,
we
may
draw
the
houre-lines
vpon
the
plane
otherwise
then
by
chords
.
For
hauing
drawne
the
houre-lines
as
in
the
last
figure
,
vpon
paper
or
paist-boord
,
we
shall
find
the
most
part
of
them
,
in
this
and
such
like
planes
that
haue
greater
declination
,
to
fall
so
close
together
,
that
they
can
hardly
be
discerned
:
wherefore
to
draw
them
at
large
to
the
best
aduantage
of
the
plane
,
I
leaue
out
the
center
,
and
draw
them
by
tangents
,
as
in
the
polar
plane
.
1
I
consider
the
length
and
bredth
of
the
plane
whereon
I
am
to
draw
the
houre-lines
,
which
I
suppose
to
be
a
square
,
whose
side
is
36
inches
,
and
find
that
the
little
square
ABDE
wil
contain
both
the
substylar
and
all
those
houre-lines
which
are
required
in
the
great
square
AZCQ
.
2
I
draw
two
parallell
lines
FN
,
GM
,
crossing
the
substylar
at
right
angles
in
the
points
F
&
G
,
so
as
they
may
best
crosse
all
the
houre-lines
,
and
yet
the
one
be
distant
from
the
other
as
far
as
the
plane
will
giue
me
leaue
;
and
I
find
by
the
sight
of
the
figure
that
if
AB
the
side
of
the
lesser
square
shall
be
36
inches
,
the
line
CF
will
be
about
115
inches
,
and
the
line
CG
about
100
inches
,
and
therefore
FG
15
inches
.
Againe
,
that
the
point
F
will
fall
about
6
inches
below
the
vpper
horizontall
side
AB
,
and
about
12
inches
from
the
next
verticall
side
BD
;
for
I
need
not
here
stand
vpon
parts
.
3
Because
these
two
parallell
lines
are
tangent
lines
in
respect
of
circles
drawne
vpon
the
semidiameters
CF
,
CG
,
and
such
tangents
as
belong
to
the
arkes
of
the
plane
,
betweene
the
substylar
and
the
houre-lines
,
the
proportion
will
hold
,
As
the
tangent
of
45
gr.
to
the
tangent
of
the
arke
of
the
plane
:
So
the
length
of
the
semidiameter
to
the
length
of
the
tangent
line
.
As
for
example
,
the
arke
of
the
plane
betweene
the
substylar
and
the
houre
of
1
,
is
15
gr.
28
m.
in
the
former
Table
,
the
semidiameter
CF
115
inches
,
and
the
semidiameter
CG
100
inches
:
wherefore
I
extend
the
compasses
from
the
tangent
of
45
gr.
vnto
the
tangent
of
15
gr.
28
m.
the
same
extent
will
reach
from
115
in
the
line
of
numbers
vnto
31
,
82
,
which
shewes
the
length
of
the
tangent
line
betweene
F
in
the
substylar
and
the
houre-line
of
1
,
to
be
31
inches
,
82
cent
.
or
parts
of
100.
Againe
,
the
same
extent
will
reach
from
100
vnto
27
,
67
;
and
such
is
the
length
of
the
lesser
tangent
from
G
to
the
houre
of
1.
The
like
reason
holds
for
the
length
of
the
other
tangents
from
the
substylar
to
the
rest
of
the
houres
,
as
in
the
Table
;
as
also
for
the
height
of
the
style
aboue
these
tangent
lines
;
and
so
the
angle
of
the
style
aboue
the
plane
being
3
gr.
6
m.
the
height
FK
will
be
found
to
be
6
inches
23
cent
.
and
the
height
GL
5
inches
42
cent
.
Where
the
Reader
may
obserue
,
that
if
the
extent
from
the
tangent
of
45
gr.
to
the
tangent
of
3
gr.
6
m.
or
to
115
in
the
line
of
numbers
,
be
too
large
for
his
compasses
,
he
may
vse
the
tangent
of
5
gr.
43
m.
in
stead
of
the
tangent
of
45
gr.
as
I
noted
before
Pag.
100.
4
Hauing
found
these
lengths
and
heights
,
and
set
them
downe
in
a
Table
,
I
come
to
the
plane
here
resembled
by
the
lesser
square
ABDE
,
where
I
begin
with
an
occult
verticall
FH
,
about
12
inches
from
the
side
BD
,
and
vpon
the
center
F
,
about
6
inches
below
the
side
AB
describe
an
occult
arke
of
a
circle
.
5
Into
this
arke
I
first
inscribe
a
chord
of
38
gr.
23
m.
the
distance
of
the
substylar
from
the
meridian
,
to
make
the
angle
HFG
equall
to
the
ZCT
;
so
the
line
FG
shall
be
the
substylar
:
and
then
another
chord
of
51
gr.
37
m.
the
complement
of
this
distance
,
to
make
vp
the
right
angle
GFN
;
so
the
line
FN
shall
be
the
greater
of
the
two
tangent
lines
before
mentioned
.
6
I
set
off
15
inches
from
F
vnto
G
,
toward
the
center
,
and
through
G
draw
the
lesser
tangent
line
GM
parallell
to
the
former
.
7
These
two
occult
tangent
lines
being
thus
drawne
,
I
looke
vnto
the
former
Table
for
the
houre
of
1
,
and
there
finde
the
arke
of
the
plane
betweene
the
substylar
and
the
houre
of
1
,
to
be
15
gr.
28
m.
and
the
length
belonging
to
it
in
the
greater
tangent
line
to
be
31
inches
82
cent
.
in
the
lesser
tangent
line
27
inches
67
cent
:
wherefore
I
take
out
31
inches
82
parts
,
and
pricke
them
downe
in
the
greater
tangent
from
F
to
N
,
and
then
27
inches
67
parts
,
and
prick
them
downe
in
the
lesser
tangent
from
G
to
M
,
and
draw
the
MN
for
the
houre
of
1
,
which
if
it
were
produced
would
crosse
the
substylar
FG
in
the
center
C
,
and
there
make
the
angle
FCN
15
gr.
28
m.
The
like
reason
holdeth
for
the
drawing
of
all
the
rest
of
the
houre-lines
.
Lastly
I
set
vp
the
style
right
ouer
the
substylar
,
so
as
the
height
FK
may
be
6
inches
23
cent
.
and
the
height
GL
5
inches
42
cent
.
then
shall
KL
represent
the
axis
of
the
world
,
and
if
it
were
produced
would
crosse
the
substylar
FG
in
the
center
C
,
and
there
make
the
angle
FCK
to
be
3
gr.
6
m.
and
so
be
truly
placed
for
casting
of
the
shadow
vpon
the
houre-lines
in
this
declining
plane
.
CHAP.
VIII
.
To
draw
the
houre-lines
in
a
meridian
inclining
plane
.
ALl
those
planes
wherein
the
horizontall
line
is
the
same
with
the
meridian
line
,
are
therefore
called
meridian
planes
:
if
they
be
right
to
the
horizon
,
they
are
called
by
the
generall
name
of
meridian
planes
without
farther
addition
,
and
are
described
before
:
if
they
leane
to
the
horizon
,
they
are
then
called
meridian
incliners
.
These
may
incline
either
to
the
East
part
of
the
horizon
,
or
to
the
West
,
and
each
of
them
hath
two
faces
,
the
vpper
toward
the
zenith
,
the
lower
toward
the
Nadir
,
wherein
knowing
the
latitude
of
the
place
,
and
the
inclination
of
the
plane
to
the
horizon
,
we
are
to
consider
.
-
1
The
inclination
of
the
meridian
of
the
plane
to
the
meridian
of
the
place
.
-
2
The
height
of
the
pole
aboue
the
plane
.
-
3
The
distance
of
the
substylar
from
the
meridian
.
-
4
The
distance
of
each
houre-line
from
the
substylar
.
And
all
these
foure
are
represented
in
the
fundamentall
Diagram
,
as
in
this
example
.
In
our
latitude
of
51
gr.
30
m.
a
meridian
plane
inclineth
Eastward
50
gr
;
these
50
gr.
I
number
in
the
verticall
circle
from
E
vnto
G
,
according
to
the
inclination
of
the
plane
,
and
there
draw
the
arke
SGN
representing
the
plane
proposed
.
Then
I
let
downe
a
perpendicular
arke
PV
from
the
pole
to
the
plane
,
seruing
for
the
meridian
of
the
plane
,
so
haue
I
a
rectangle
triangle
PVN
,
wherein
the
base
PN
is
the
height
of
the
pole
aboue
the
North
part
of
the
horizon
,
and
the
angle
PNV
the
complement
of
the
inclination
to
the
horizon
;
and
these
being
knowne
,
1
I
may
finde
the
angle
NPV
of
inclination
of
the
two
meridians
.
For
As
the
cosine
of
the
latitude
is
to
the
sine
of
90
gr.
So
the
tangent
of
inclination
to
the
horizon
,
to
the
tangent
of
inclination
of
meridians
.
Extend
the
compasses
from
the
sine
of
38
gr.
30
m.
the
complement
of
the
latitude
,
vnto
the
sine
of
90
gr.
the
same
extent
will
reach
from
the
tangent
of
50
gr.
0
m.
the
inclination
of
the
plane
to
the
horizon
,
vnto
the
tangent
of
62
gr.
25
m.
and
such
is
the
inclination
of
the
meridian
of
the
plane
to
the
meridian
of
the
place
;
which
being
resolued
into
time
,
doth
giue
about
4
houres
and
10
m.
from
the
meridian
,
for
the
place
of
the
substylar
among
the
houre-lines
.
2
The
height
of
the
pole
aboue
the
plane
is
here
represented
by
the
quantitie
of
the
arke
of
the
proper
meridian
PV
,
betweene
the
pole
and
the
plane
,
and
may
be
knowne
by
that
which
we
haue
giuen
in
the
former
triangle
PVN
.
For
As
the
sine
of
90
gr.
to
the
sine
of
the
latitude
:
So
the
cosine
of
the
inclination
to
the
horizon
,
to
the
sine
of
the
height
of
the
pole
aboue
the
plane
.
Extend
the
compasses
from
the
sine
of
90
gr.
vnto
51
gr.
30
m.
the
sine
of
the
latitude
,
the
same
extent
wil
reach
from
the
sine
of
40
gr.
the
complement
of
the
inclination
of
the
plane
to
the
horizon
,
vnto
the
sine
of
30
gr.
12
m.
Or
as
the
sine
of
90
gr.
to
the
cosine
of
inclination
of
meridians
:
So
the
tangent
of
the
latitude
to
the
tangent
of
the
height
of
the
pole
aboue
the
plane
.
Extend
the
compasses
from
the
sine
of
90
gr.
vnto
the
tangent
of
51
gr.
30
m.
the
latitude
of
the
place
,
the
same
extent
will
reach
from
the
sine
of
27
gr.
35
m.
the
complement
of
the
inclination
of
the
two
meridians
,
vnto
the
tangent
of
30
gr.
12
m.
And
such
is
PV
the
height
of
the
pole
aboue
the
plane
,
and
such
must
be
the
height
of
the
style
aboue
the
substylar
.
3
The
distance
of
the
substylar
from
the
meridian
is
here
represented
by
NV
the
arke
of
the
plane
betweene
the
two
meridians
,
and
may
be
found
by
that
which
we
haue
giuen
at
the
first
in
the
former
triangle
PVN
.
For
As
the
sine
of
90
gr.
to
the
sine
of
the
inclination
to
the
horizon
:
So
the
tangent
of
the
latitude
to
the
tangent
of
the
substylar
from
the
meridian
.
Extend
the
compasses
from
the
sine
of
90
gr.
vnto
the
tangent
of
51
gr.
30
m.
the
latitude
of
the
place
,
the
same
extent
will
reach
from
the
sine
of
50
gr.
the
inclination
of
the
plane
to
the
horizon
,
vnto
43
gr.
55
m.
And
such
is
the
arke
NV
the
distance
of
the
substylar
from
the
meridian
.
4
The
distances
of
the
houre-lines
from
the
substylar
,
are
here
also
represented
by
those
arkes
of
the
plane
,
which
are
here
intercepted
betweene
the
proper
meridian
and
the
houre-circles
,
and
may
be
found
by
that
which
we
haue
giuen
in
the
triangles
made
by
the
plane
,
with
his
proper
meridian
and
the
houre-circles
.
For
the
angle
at
V
,
betweene
the
plane
and
the
proper
meridian
,
is
well
knowne
to
be
a
right
angle
,
and
the
side
PV
is
the
height
of
the
pole
aboue
the
plane
,
and
the
angles
at
the
pole
betweene
the
proper
meridian
and
the
houre-circles
are
easily
gathered
into
a
Table
.
The
angle
VPN
betweene
VP
the
proper
meridian
of
the
plane
,
and
PN
the
generall
meridian
of
the
place
being
62
gr.
25
m.
the
angle
between
the
proper
meridian
and
the
circle
of
the
houre
of
11
,
will
be
77
gr.
25
m.
and
the
angle
belonging
to
the
houre
of
1
,
47
gr.
25
m
,
and
so
the
rest
of
the
angles
at
the
pole
.
Then
As
the
sine
of
90
gr.
to
the
sine
of
the
pole
aboue
the
plane
:
So
the
tangent
of
the
angle
at
the
pole
,
to
the
tangent
of
the
houre-line
from
the
substylar
.
Hor.
|
Ang.
Po.
|
Arc.
Pla.
|
Gr.
M.
|
Gr.
M.
|
11
|
77
25
|
66
4
|
12
|
62
25
|
43
55
|
1
|
47
25
|
28
41
|
2
|
32
25
|
17
43
|
3
|
17
25
|
8
58
|
4
|
2
25
|
1
13
|
|
Merid
|
Substyl
|
5
|
12
35
|
6
26
|
6
|
27
35
|
14
44
|
7
|
42
35
|
24
48
|
8
|
57
35
|
38
23
|
9
|
72
35
|
58
3
|
10
|
87
35
|
85
12
|
Wherefore
I
extend
the
compasses
from
the
sine
of
90
gr.
vnto
the
sine
of
30
gr.
12
m.
the
height
of
the
pole
aboue
the
plane
,
and
I
finde
the
same
extent
to
reach
in
the
line
of
tangents
from
77
gr
25
m.
vnto
66
gr.
4
m.
for
the
distance
belonging
to
the
houre
of
11
;
and
from
the
tangent
of
62
gr.
25
m.
to
43
gr.
55
m.
for
the
houre
of
12.
as
when
I
found
the
distance
of
the
substylar
from
the
meridian
.
And
so
for
the
rest
of
the
arks
of
plane
betweene
the
substylar
and
the
houre-circles
,
as
in
the
Table
.
These
arks
being
thus
found
,
wil
serue
to
draw
the
houre-lines
on
either
side
of
this
plane
:
but
supposing
it
to
be
the
vpper
side
,
1
I
draw
the
horizontall
line
CN
,
seruing
for
the
meridian
and
houre
of
12.
2
In
this
line
I
make
choice
of
a
center
at
C
,
and
thence
describe
an
occult
arke
of
a
circle
representing
the
plane
proposed
.
3
I
find
a
chord
of
43
gr.
55
m.
the
distance
of
the
substylar
from
the
meridian
,
and
inscribe
it
into
this
circle
from
N
vnto
A
,
according
as
I
finde
the
proper
meridian
PV
to
fall
in
the
fundamental
diagram
,
and
there
I
draw
the
line
CA
seruing
for
the
substylar
.
geometric illustration
4
The
substylar
being
drawne
,
I
may
inscribe
the
chords
of
the
arkes
of
the
plane
from
the
substylar
,
and
draw
the
houre-lines
,
and
set
vp
the
style
,
as
in
the
former
planes
.
CHAP.
IX
.
To
draw
the
houre-lines
in
a
polar
declining
Plane
.
THose
planes
wherein
a
line
may
be
drawne
parallell
to
the
axis
of
the
world
,
are
called
polar
planes
,
because
that
line
pointeth
vnto
the
poles
,
and
these
planes
are
always
parallell
to
some
one
of
the
houre-circles
.
If
they
be
parallell
to
the
houre
of
6
,
they
are
called
direct
polar
planes
;
if
to
the
houre
of
12
,
they
are
called
meridian
planes
;
and
both
these
are
described
before
:
if
to
any
other
of
the
houre-circles
,
they
are
then
called
by
the
name
of
polar
declining
planes
,
because
of
their
inclining
to
the
pole
,
and
declining
from
the
verticall
.
These
kind
of
planes
may
be
knowne
in
this
sort
:
First
consider
the
inclination
of
the
plane
to
the
horizon
,
which
in
these
parts
of
the
world
must
alwayes
be
Northward
,
and
more
then
the
latitude
of
the
place
.
Then
find
the
declination
from
the
verticall
.
These
two
being
knowne
,
if
the
proportion
hold
,
As
the
sine
of
90
gr.
to
the
cosine
of
the
declination
:
So
the
tangent
of
the
inclination
to
the
tangent
of
the
latitude
;
it
is
then
a
polar
declining
plane
,
otherwise
not
.
For
example
,
in
our
latitude
of
51
gr.
30
m.
a
plane
is
proposed
declining
from
the
verticall
65
gr.
40
m.
and
inclining
Northward
71
gr.
51
m.
the
vpper
face
being
open
to
the
Southeast
,
and
the
lower
to
the
Northwest
.
If
I
number
those
65
gr.
40
m.
in
the
horizon
of
the
fundamental
diagram
from
E
vnto
Q
,
and
draw
the
line
HCQ
,
it
shall
represent
the
horizontall
line
of
the
plane
;
then
crossing
it
at
right
angles
with
the
plane
BZD
drawne
through
the
zenith
,
I
number
71
gr.
51
m.
for
the
inclination
from
D
vnto
R
,
and
there
draw
the
circle
HRQ
,
this
circle
so
drawne
shall
represent
the
plane
proposed
;
and
because
it
also
passeth
through
the
pole
,
it
is
therefore
a
polar
plane
.
But
for
farther
triall
I
extend
the
compasses
from
the
sine
of
90
gr.
to
the
sine
of
24
gr.
20
m.
the
complement
of
the
declination
,
and
I
find
the
same
extent
to
reach
from
the
tangent
of
71
gr.
51
m.
the
inclination
proposed
,
vnto
the
tangent
of
51
gr.
30
m.
which
is
the
true
latitude
of
the
place
,
and
therefore
it
is
a
polar
plane
.
Here
then
the
style
will
be
parallell
to
the
plane
,
and
the
houre-lines
parallell
one
to
the
other
,
as
in
the
meridian
and
direct
polar
planes
.
But
that
we
may
know
how
to
draw
the
houre-lines
,
and
where
to
place
the
style
,
we
are
to
consider
1
The
arke
of
the
plane
betweene
the
horizon
and
the
houre-lines
.
In
a
meridian
plane
the
arke
betweene
the
horizon
and
the
hourelines
,
is
alwayes
equall
to
the
latitude
of
the
place
;
in
a
direct
polar
it
is
an
arke
of
90
gr
;
in
these
declining
polars
it
is
greater
then
the
latitude
,
and
yet
lesse
then
90
gr.
and
may
be
knowne
in
this
maner
.
As
the
sine
of
90
gr.
to
the
cosine
of
the
latitude
:
So
the
sine
of
the
declination
to
the
cosine
of
the
arke
betweene
the
horizon
and
the
houre-lines
.
Extend
the
compasses
from
the
sine
of
90
gr.
vnto
the
sine
of
38
gr.
30
m.
the
complement
of
the
latitude
,
the
same
extent
will
reach
from
the
sine
of
65
gr.
40
m.
the
declination
proposed
,
vnto
the
sine
of
34
gr.
34
m.
whose
complement
55
gr.
26
m.
the
arke
of
the
plane
betweene
the
horizon
and
the
substylar
,
to
which
all
the
houre-lines
must
be
parallel
.
2
The
inclination
of
the
meridian
of
the
plane
to
the
meridian
of
the
place
.
The
substylar
in
a
direct
polar
plane
is
alwayes
the
same
with
the
houre
of
12
:
in
a
meridian
plane
it
is
the
same
with
the
houre-line
of
6
:
in
these
declining
polars
it
must
be
placed
betweene
12
and
6
,
according
to
the
inclination
of
the
meridian
of
the
plane
to
the
meridian
of
the
place
,
which
is
thus
knowne
.
As
the
sine
of
90
gr.
to
the
sine
of
the
latitude
:
So
the
tangent
of
the
declination
of
the
plane
,
to
the
tangent
of
the
inclination
of
meridians
.
Extend
the
compasses
from
the
sine
of
90
gr.
to
the
sine
of
51
gr.
30
m.
the
latitude
of
the
place
,
the
same
extent
will
reach
from
the
tangent
of
65
gr.
40
m.
the
declination
proposed
,
vnto
the
tangent
of
60
gr.
and
such
is
the
angle
of
inclination
betweene
the
meridian
of
the
place
and
the
proper
meridian
of
the
plane
,
which
resolued
into
time
doth
make
foure
houres
;
and
so
the
substylar
must
here
be
placed
vpon
the
houre
of
8
in
the
morning
.
This
angle
being
knowne
,
the
rest
of
the
angles
at
the
pole
are
easily
gathered
.
For
if
the
houre
of
12
be
60
gr.
distant
from
the
meridian
of
the
plane
,
the
houre
of
1
will
be
75
gr.
and
the
houre
of
11
,
will
be
45
gr.
distant
,
and
the
rest
of
the
houres
,
as
in
the
Table
following
.
Then
comming
to
the
plane
,
1
I
draw
an
occult
horizontall
line
HQ
,
wherein
I
make
choice
of
a
center
H
,
and
describe
an
occult
circle
for
the
horizon
of
the
plane
.
2
I
find
a
chord
of
55
gr.
26
m.
and
inscribe
it
into
this
circle
,
from
Q
vnto
B
,
according
to
the
situation
of
the
plane
;
so
the
line
HB
shall
be
the
meridian
of
the
plane
:
and
therefore
the
substylar
and
the
line
AC
crossing
it
at
right
angles
,
shall
be
the
equator
.
3
I
consider
the
length
of
the
plane
,
and
how
many
houres
I
am
to
draw
vpon
it
,
that
so
I
may
proportion
the
height
of
the
style
;
and
I
finde
by
the
fundamentall
diagram
and
the
former
table
,
that
it
will
containe
all
the
houres
from
Sun
rising
vntill
1
afternoone
:
and
therefore
the
meridian
of
the
plane
falling
on
the
houre
of
8
in
the
morning
,
there
will
be
foure
houres
on
the
one
side
,
and
fiue
on
the
other
side
of
the
substylar
.
But
in
all
polar
planes
the
height
of
the
style
aboue
the
substylar
must
be
equall
to
the
distance
of
the
third
houre
from
the
substylar
,
or
about
4/7
of
the
fourth
houre
,
or
little
more
then
¼
of
the
fift
houre
,
and
thereupon
I
allow
the
height
of
this
style
to
be
equall
to
CB
,
which
you
may
suppose
to
be
ten
inches
.
geometric illustration
4
Because
the
equator
AC
is
a
tangent
line
in
respect
of
the
Radius
BC
,
and
the
parts
thereof
are
such
as
belong
to
the
angles
betweene
the
meridian
of
the
plane
and
the
hourelines
,
which
angles
are
set
downe
in
the
table
following
,
I
may
finde
the
length
of
each
seuerall
tangent
in
this
maner
.
As
the
tangent
of
45
gr.
is
to
the
tangent
of
the
houre
:
So
the
parts
of
the
Radius
,
to
the
parts
of
the
tangent
line
.
The
angle
ABC
betweene
the
meridian
of
the
plane
and
the
houre
of
12
,
the
meridian
of
the
place
is
60
gr.
in
the
former
table
,
and
the
Radius
BC
is
supposed
to
be
ten
inches
;
whereupon
I
extend
the
compasses
from
the
tangent
of
45
gr.
vnto
the
tangent
of
60
gr.
the
same
extent
will
reach
from
10
in
the
line
of
numbers
,
vnto
17.
32
,
which
shewes
the
length
of
the
tangent
AC
betweene
the
substylar
and
the
houre
of
12
,
to
be
17.
32
cent
.
The
like
reason
holds
for
the
rest
of
the
houres
.
Hor.
|
An.
Po.
|
Tangent
|
Gr.
M.
|
In.
Par.
|
4
|
60
0
|
17
32
|
5
|
45
0
|
10
00
|
6
|
30
0
|
5
77
|
7
|
15
0
|
2
68
|
8
|
Merid
|
Substyl
|
|
3
45
|
65
|
|
7
30
|
1
32
|
|
11
15
|
1
99
|
9
|
15
0
|
2
68
|
|
18
45
|
3
39
|
|
22
30
|
4
14
|
|
26
15
|
4
93
|
10
|
30
0
|
5
77
|
|
33
45
|
6
68
|
|
37
30
|
7
67
|
|
41
15
|
8
77
|
11
|
45
0
|
10
00
|
|
48
45
|
11
40
|
|
52
30
|
13
03
|
|
56
15
|
14
97
|
12
|
60
0
|
17
32
|
|
63
45
|
20
28
|
|
67
30
|
24
14
|
|
71
15
|
29
46
|
1
|
75
0
|
37
32
|
|
78
45
|
50
27
|
|
82
30
|
75
96
|
|
86
15
|
152
57
|
2
|
90
0
|
Infinit
|
5
These
lengths
being
thus
found
and
set
downe
in
the
table
,
I
take
out
17
inches
32
cent
.
and
prick
them
in
the
equator
from
C
vnto
A
for
the
houre
of
12
,
and
37
inches
32
cent
.
and
prick
them
downe
for
the
houre
of
1.
And
so
the
rest
of
the
houre-points
.
6
This
done
,
if
I
draw
right
lines
through
each
of
these
points
,
crossing
the
equator
at
right
angles
,
they
shall
be
the
houre-lines
required
:
and
if
I
set
the
style
ouer
the
substylar
,
so
as
the
edge
of
it
may
be
parallell
to
the
plane
,
and
the
height
of
it
be
ten
inches
equall
to
the
former
Radius
BC
,
it
shall
represent
the
axis
of
the
world
,
and
be
truly
placed
for
casting
of
the
shadow
vpon
the
houre-lines
in
this
declining
polar
plane
.
CHAP.
X.
To
draw
the
houre-lines
in
a
declining
inclining
plane
.
IF
a
plane
shall
decline
from
the
prime
verticall
,
and
incline
to
the
horizon
,
and
yet
not
lie
euen
with
the
poles
of
the
world
,
it
is
then
called
a
declining
inclining
plane
.
Of
these
there
are
seuerall
sorts
;
for
the
inclination
being
Northward
,
the
plane
may
fall
between
the
horizon
and
the
pole
,
as
the
circle
BMD
in
the
fundamentall
Diagram
;
or
betweene
the
zenith
and
the
pole
,
as
BFD
:
or
the
inclination
may
be
Southward
,
and
so
be
represented
by
BKD
,
it
may
also
fall
either
below
the
intersection
of
the
meridian
and
the
equator
,
or
aboue
it
;
and
each
of
these
haue
two
faces
,
the
vpper
toward
the
zenith
,
and
the
lower
toward
the
nadir
;
wherein
hauing
the
latitude
of
the
place
with
the
declination
and
inclination
of
the
plane
,
we
are
farther
to
consider
,
-
1
The
arke
of
the
meridian
betweene
the
pole
and
the
plane
.
-
2
The
inclination
of
the
plane
to
the
meridian
.
-
3
The
arke
of
the
plane
between
the
horizon
and
the
meridian
.
-
4
The
angle
of
inclination
betweene
both
meridians
.
-
5
The
height
of
the
pole
aboue
the
plane
.
-
6
The
distance
of
the
substylar
from
the
meridian
.
-
7
The
distances
of
each
houre-line
from
the
substylar
.
And
all
these
seuen
may
be
represented
in
the
fundamentall
diagram
,
as
in
this
example
.
In
our
latitude
of
51
gr.
30
m.
a
plane
is
proposed
,
declining
from
the
verticall
24
gr.
20
m.
and
inclini●g
Northward
36
gr.
the
vpper
face
lying
open
to
the
Southwest
,
the
lower
to
the
Northeast
.
If
I
number
these
24
gr.
20
m.
in
the
horizon
from
E
to
B
,
and
there
draw
the
line
BCD
,
it
shall
represent
the
horizontall
line
of
the
plane
:
then
crossing
it
at
right
angles
with
the
plane
HZQ
drawne
through
the
zenith
,
I
number
36
gr.
for
the
inclination
from
Q
vnto
M
,
and
there
draw
the
circle
BMD
,
crossing
the
meridian
in
the
point
a
;
this
circle
so
drawne
shall
represent
the
plane
proposed
;
and
because
it
doth
not
passe
through
the
pole
,
is
therfore
no
polar
,
but
an
ordinary
declining
inclining
plane
.
1
The
arke
of
the
meridian
of
the
place
between
the
pole
and
the
plane
,
is
here
represented
by
Pa
,
and
may
be
found
by
resoluing
the
triangle
DN
a
,
wherein
the
angle
at
N
is
knowne
to
be
a
right
angle
,
the
angle
at
D
is
the
angle
of
inclination
,
the
side
D
N
the
complement
of
the
declination
,
which
being
knowne
,
As
the
sine
of
90
gr.
to
the
cosine
of
declination
:
So
the
tangent
of
inclination
to
the
horizon
,
to
the
tangent
of
the
meridian
betweene
the
horizon
and
the
plane
.
Extend
the
compasses
from
the
sine
of
90
gr.
vnto
the
sine
of
65
gr.
40
m.
the
complement
of
the
declination
,
the
same
extent
will
reach
from
the
tangent
of
36
gr.
the
inclination
proposed
,
vnto
the
tangent
of
33
gr.
30
m.
and
such
is
the
ark
of
the
meridian
Na
between
the
horizon
and
the
plane
.
This
arke
Na
being
compared
with
the
arke
NP
,
which
is
the
eleuation
of
the
pole
aboue
the
horizon
,
and
is
here
supposed
to
be
51
gr.
30
m.
the
difference
N
a
cometh
to
18
gr.
and
such
is
the
arke
of
the
meridian
required
betweene
the
pole
and
the
plane
.
2
The
inclination
of
the
plane
to
the
meridian
is
here
represented
by
the
angle
N
aD
,
and
may
be
found
by
that
which
we
haue
giuen
in
the
former
triangle
DN
a.
For
As
the
sine
of
90
gr.
to
the
sine
of
the
declination
from
the
verticall
:
So
the
sine
of
inclination
to
the
horizon
,
to
the
cosine
of
the
inclination
to
the
meridian
.
Extend
the
compasses
from
the
sine
of
90
gr.
vnto
the
sine
of
24
gr.
20
m.
the
declination
of
the
plane
,
the
same
extent
will
reach
from
the
sine
of
36
gr.
the
inclination
giuen
,
vnto
the
cosine
of
76
gr.
And
such
is
N
aD
the
angle
of
inclination
betweene
the
plane
D
a
,
and
N
a
the
meridian
of
the
place
.
Or
As
the
sine
of
the
arke
of
the
meridian
betweene
the
horizon
and
the
plane
,
is
to
the
sine
of
90
gr.
So
the
cotangent
of
the
declination
to
the
tangent
of
the
inclination
to
the
meridian
.
Extend
the
compasses
from
the
sine
of
33
gr.
30
m.
the
ark
of
the
meridian
betweene
the
horizon
and
the
plane
,
vnto
the
sine
of
90
gr.
the
same
extent
will
reach
from
the
tangent
of
65
gr.
40
m.
the
complement
of
the
declination
vnto
the
tangent
of
76
gr.
And
such
is
the
inclination
of
the
plane
to
the
meridian
,
the
same
as
before
.
3
The
arke
of
the
plane
betweene
the
horizon
and
the
meridian
,
is
here
represented
by
D
a
,
and
may
also
be
found
by
that
which
we
haue
giuen
in
the
former
triangle
DN
a.
As
the
sine
of
90
gr.
to
the
cosine
of
inclination
to
the
horizon
:
So
the
cotangent
of
the
declination
to
the
tangent
of
the
plane
required
.
Extend
the
compasses
from
the
sine
of
90
gr.
vnto
the
sine
of
54
gr.
the
complement
of
the
inclination
of
the
plane
to
the
horizon
,
the
same
extent
will
reach
from
the
tangent
of
65
gr.
40
m.
the
complement
of
the
declination
,
vnto
the
tangent
of
69
gr.
54
m.
And
such
is
D
a
the
arke
of
the
plane
,
betweene
the
horizon
and
the
meridian
of
the
place
.
4
The
inclination
of
meridians
is
here
represented
by
the
angle
aPb
.
For
if
I
let
downe
a
perpendicular
Pb
from
the
pole
vnto
the
plane
,
this
perpendicular
shall
be
the
meridian
of
the
plane
;
and
we
shall
haue
another
triangle
abP
,
wherein
the
angle
at
b
is
a
right
angle
,
because
of
the
perpendicular
,
the
angle
at
a
is
the
inclination
of
the
plane
to
the
meridian
of
the
place
,
and
the
side
P
a
is
the
arke
of
the
meridian
betweene
the
pole
and
the
plane
,
which
being
knowne
,
As
the
cosine
of
the
arke
of
the
meridian
betweene
the
pole
and
the
plane
is
to
the
sine
of
90
gr.
So
the
cotangent
of
the
inclination
to
the
meridian
,
to
the
tangent
of
inclination
of
the
meridian
of
the
plane
to
the
meridian
of
the
place
.
Extend
the
compasses
from
the
sine
of
72
gr.
the
complement
of
the
arke
Pa
,
betweene
the
pole
and
the
plane
,
vnto
the
sine
of
90
gr.
the
same
extent
wil
reach
from
the
tangent
of
14
gr.
the
complement
of
the
inclination
of
the
plane
to
the
meridian
,
vnto
the
tangent
of
14
gr.
41
m.
And
such
is
the
angle
aPb
of
inclination
betweene
the
meridian
of
the
place
and
the
proper
meridian
of
the
plane
,
which
resolued
into
time
,
doth
make
about
59
minutes
,
and
so
the
substylar
must
here
be
placed
neare
the
houre
of
1
after
noone
.
5
The
height
of
the
pole
aboue
the
plane
is
here
represented
by
Pb
,
the
arke
of
the
proper
meridian
between
the
pole
and
the
plane
,
and
may
be
found
by
that
which
wee
haue
giuen
in
the
triangle
abP
.
For
As
the
sine
of
90
gr.
to
the
sine
of
the
meridian
of
the
place
betweene
the
pole
and
the
plane
:
So
the
sine
of
the
inclination
to
the
meridian
,
to
the
sine
of
the
height
of
the
pole
aboue
the
plane
.
Extend
the
compasses
from
the
sine
of
90
gr.
vnto
the
sine
of
18
gr.
the
arke
Pa
of
the
meridian
of
the
place
from
the
pole
to
the
plane
,
the
same
extent
wil
reach
from
the
sine
of
baP
the
inclination
of
the
plane
to
the
meridian
of
the
place
,
vnto
the
sine
of
17
gr.
26
m.
Or
As
the
sine
of
90
gr.
to
the
cosine
of
inclination
of
meridians
:
So
the
tangent
of
the
meridian
of
the
place
betweene
the
pole
and
the
plane
,
to
the
tangent
of
the
height
of
the
pole
aboue
the
plane
,
Extend
the
compasses
from
the
sine
of
90
gr.
vnto
the
sine
of
75
gr.
19
m.
the
complement
of
aPb
the
inclination
of
the
two
meridians
,
the
same
extent
will
reach
from
the
tangent
of
18
gr.
the
arke
Pa
of
the
generall
meridian
between
the
pole
and
the
plane
,
vnto
the
tangent
of
17
gr.
26
m.
And
such
is
Pb
the
height
of
the
pole
aboue
the
plane
;
and
such
must
be
the
height
of
the
style
aboue
the
substylar
.
6
This
distance
of
the
substylar
from
the
meridian
of
the
place
,
is
here
represented
by
ab
the
arke
of
the
plane
between
the
two
meridians
,
and
may
be
found
by
that
which
we
had
giuen
at
the
first
in
the
former
triangle
abP
.
For
As
the
sine
of
90
gr.
to
the
sine
of
the
inclination
to
the
meridian
:
So
the
tangent
of
the
meridian
of
the
place
betweene
the
pole
and
the
plane
,
vnto
the
tangent
of
the
substylar
from
the
meridian
of
the
place
.
Extend
the
compasses
from
the
sine
of
90
gr.
vnto
the
sine
of
14
gr.
the
complement
of
baP
,
the
inclination
of
the
plane
to
the
meridian
,
the
same
extent
will
reach
from
the
tangent
of
18
gr.
the
arke
of
the
generall
meridian
betweene
the
pole
and
the
plane
,
vnto
the
tangent
of
4
gr.
30
m.
And
such
is
the
arke
of
the
plane
betweene
the
two
meridians
;
and
such
must
be
the
distance
from
the
houre
of
12
to
the
substylar
.
Hor.
|
Ang.
Po.
|
Arc.
Pla.
|
Gr.
M.
|
Gr.
M.
|
7
|
89
41
|
88
57
|
8
|
74
41
|
47
35
|
9
|
59
41
|
27
9
|
10
|
44
41
|
16
31
|
11
|
29
41
|
9
41
|
12
|
14
41
|
4
30
|
|
M●rid
.
|
Substyl
|
1
|
0
19
|
0
6
|
2
|
15
19
|
4
42
|
3
|
30
19
|
9
56
|
4
|
45
19
|
16
52
|
5
|
60
19
|
27
45
|
6
|
75
19
|
48
51
|
7
The
distances
of
the
houre-lines
from
the
substylar
,
are
here
also
represented
by
those
arks
of
the
plane
,
which
are
intercepted
between
the
proper
meridian
and
the
houre-circles
.
For
in
these
triangles
the
angle
at
b
betweene
the
plane
and
the
proper
meridian
is
a
right
angle
,
the
side
P
b
is
the
height
of
the
pole
aboue
the
plane
,
and
then
the
angles
at
the
pole
between
the
proper
meridian
and
the
houre-circles
being
gathered
into
a
table
.
As
the
sine
of
90
gr.
to
the
sine
of
the
pole
aboue
the
plane
:
So
the
tangent
of
the
angle
at
the
pole
,
to
the
tangent
of
the
houre-line
from
the
substylar
.
geometric illustration
These
arkes
being
thus
found
,
will
serue
for
the
drawing
of
the
houre-lines
on
either
side
of
the
plane
:
but
supposing
it
to
be
the
vpper
side
,
I
consider
how
the
lines
doe
fall
in
the
fundamentall
diagram
,
and
accordingly
1
I
draw
an
occult
horizontall
line
DD
,
wherein
I
make
choice
of
the
center
C
,
and
thence
draw
an
occult
circle
for
the
horizon
of
the
plane
.
2
I
finde
a
chord
of
69
gr.
54
m.
the
arke
of
the
plane
betweene
the
horizon
and
the
meridian
,
and
inscribe
it
into
this
circle
from
D
vnto
a
,
and
there
draw
the
line
Ca
for
the
houre
of
12.
3
I
finde
a
chord
of
4
gr.
30
m.
the
arke
of
the
plane
betweene
the
two
meridians
,
and
inscribe
it
into
this
circle
from
a
vnto
b
,
and
there
draw
the
line
Cb
for
the
substylar
.
4
The
substylar
being
drawne
,
I
may
inscribe
the
chords
of
the
arkes
of
the
plane
from
the
substylar
,
and
draw
the
houre-lines
,
and
set
vp
the
style
as
in
the
former
planes
.
A
second
example
of
a
Plane
falling
betweene
the
pole
and
the
zenith
.
In
like
maner
if
in
our
latitude
a
plane
be
proposed
declining
from
the
verticall
24
gr.
20
m.
as
before
,
but
inclining
to
the
horizon
75
gr.
40
m.
Northward
,
the
vpper
face
being
open
to
the
Southwest
,
the
lower
to
the
Northeast
,
this
plane
shall
be
here
represented
by
the
circle
BFD
,
crossing
the
meridian
in
the
point
d
,
between
the
pole
and
the
zenith
,
and
the
proper
meridian
of
this
plane
,
by
the
pe●●endicular
arke
P
e.
Then
in
this
triangle
DN
d
kn●wing
the
side
DN
the
complement
of
the
declination
,
with
the
angle
of
inclinanation
to
the
horizon
at
D
,
and
the
right
angle
at
N
,
these
former
Canons
will
giue
Nd
the
arke
of
the
meridian
between
the
horizon
and
the
plane
to
be
74
gr.
20
m
;
and
therfore
Pd
the
arke
of
the
meridian
betweene
the
pole
and
the
plane
will
be
22
gr.
50
m.
the
angle
DdN
of
the
inclination
of
the
plane
to
the
meridian
,
will
be
found
to
be
66
gr.
29
m.
and
Dd
the
arke
of
the
plane
between
the
horizon
and
the
meridian
83
gr.
36
m.
Againe
,
in
the
triangle
Ped
knowing
the
side
Pd
the
arke
of
the
meridian
betweene
the
pole
and
the
plane
,
with
the
angle
of
inclination
to
the
meridian
at
d
,
and
the
right
at
e
,
the
angle
dPe
of
the
inclination
of
the
two
meridians
will
be
found
to
be
25
gr.
17
m.
and
Pe
the
height
of
the
pole
aboue
the
plane
to
be
20
gr.
50
m.
and
de
the
distance
of
the
substylar
from
the
meridian
about
9
gr.
32
m.
Declination
|
24
20
|
Inclination
|
75
40
|
Diff.
meri
.
|
83
36
|
dist.
substy
|
9
32
|
Alti
.
Styl
.
|
20
50
|
Hor.
|
Ang.
Po.
|
Arc.
Pla.
|
Gr.
M.
|
Gr.
M.
|
8
|
85
17
|
76
56
|
9
|
70
17
|
44
47
|
10
|
55
17
|
27
11
|
11
|
40
17
|
16
43
|
12
|
25
17
|
9
32
|
1
|
10
17
|
3
41
|
|
Merid
|
Substyl
|
2
|
4
43
|
1
40
|
3
|
19
43
|
7
16
|
4
|
34
43
|
13
50
|
5
|
49
43
|
22
46
|
6
|
64
43
|
37
0
|
7
|
79
43
|
62
58
|
Lastly
,
hauing
found
the
height
of
the
pole
aboue
the
plane
,
and
gathered
the
angles
at
the
pole
,
the
arks
of
the
plane
from
the
substylar
to
the
houre-lines
will
be
as
in
this
table
.
This
done
,
if
we
consider
how
the
lines
doe
fall
in
the
fundamentall
diagram
,
we
may
there
see
how
the
North
pole
is
eleuated
aboue
the
lower
face
,
&
the
South
pole
aboue
the
vpper
face
of
the
plane
,
and
accordingly
make
choice
of
a
center
,
draw
the
horizontall
,
the
meridian
,
the
substylar
,
&
the
houre-lines
,
and
set
vp
the
style
as
in
the
other
planes
.
A
third
example
of
a
Plane
inclining
to
the
Southward
.
If
in
our
latitude
a
plane
were
proposed
declining
from
the
verticall
24
gr.
20
m.
as
before
,
but
inclining
to
the
horizon
14
gr.
20
m.
Southward
,
the
vpper
face
being
open
to
the
Northeast
,
the
lower
to
the
South-west
,
this
plane
shall
be
here
represented
by
the
circle
BKD
crossing
the
meridian
in
the
point
f
betweene
the
equator
and
the
horizon
,
and
the
proper
meridian
of
this
plane
by
the
perpendicular
arke
Pg
let
downe
from
the
pole
to
the
plane
,
neare
the
houre
of
11
,
at
the
North
part
of
the
horizon
,
as
may
partly
appeare
by
the
nearest
extent
of
the
compasses
.
Then
in
the
triangle
BSf
,
knowing
the
side
BS
the
complement
of
the
declination
,
with
the
angle
of
inclination
to
the
horizon
at
B
,
and
the
right
angle
at
S
,
we
may
find
Sf
the
ark
of
the
meridian
betweene
the
horizon
and
the
plane
to
be
13
gr.
6
m.
And
therefore
Pf
the
arke
of
the
meridian
betweene
the
pole
and
the
plane
to
the
Southward
115
gr.
24
m.
but
64
gr.
36
m.
to
the
Northward
,
the
angle
B
fS
or
D
fN
of
the
inclination
of
the
plane
to
the
meridian
,
will
be
found
84
gr.
9
m
;
&
B
f
or
D
f
the
arke
of
the
plane
between
the
horizon
&
the
meridian
66
gr.
20
m.
Declination
|
24
20
|
Inclination
|
14
20
|
diff
.
merid
|
13
27
|
dist.
substy
|
12
8
|
Alt.
Styl
.
|
64
0
|
Hor.
|
Ang.
Po
|
Arc.
Pla.
|
Gr.
M.
|
Gr.
M.
|
6
|
76
33
|
75
6
|
7
|
61
33
|
58
56
|
8
|
46
33
|
43
30
|
9
|
31
33
|
28
55
|
10
|
16
33
|
14
58
|
11
|
1
33
|
1
25
|
|
Merid
|
Substyl
|
12
|
13
27
|
12
8
|
1
|
28
27
|
25
57
|
2
|
43
27
|
40
23
|
3
|
58
27
|
55
38
|
4
|
73
27
|
71
41
|
5
|
88
27
|
88
15
|
Again
,
in
the
triangle
Pgf
knowing
the
side
Pf
the
arke
of
the
meridian
betweene
the
pole
and
the
plane
,
with
the
angle
of
inclination
to
the
meridian
at
f
,
and
the
right
angle
at
g
,
the
angle
fPg
of
the
inclination
of
the
two
meridians
wil
be
found
to
be
13
gr.
27
m.
and
Pg
the
height
of
the
pole
aboue
the
plane
,
about
64
gr.
and
fg
the
distance
of
the
substylar
from
the
meridian
12
gr.
8
m.
Hauing
found
the
height
of
the
pole
aboue
the
plane
,
and
gathered
the
angles
at
the
pole
,
the
arkes
of
the
plane
from
the
substylar
to
the
houre-lines
will
be
found
as
in
this
table
.
This
done
,
if
we
consider
how
the
lines
do
fall
in
the
fundamentall
diagram
,
we
may
there
see
how
the
North
pole
is
eleuated
aboue
the
vpper
face
,
and
the
South
pole
aboue
the
lower
face
of
this
plane
,
and
accordingly
make
choice
of
the
center
,
draw
the
horizontall
,
the
meridian
,
the
substylar
,
and
the
houre-lines
,
and
set
vp
the
style
as
in
the
former
planes
.
CHAP.
XI
.
To
describe
the
Tropiques
and
other
circles
of
declination
in
an
equinoctiall
Plane
.
SVch
circles
as
are
parallell
to
the
equinoctiall
,
and
yet
fall
within
the
tropiques
,
may
be
described
on
any
plane
by
help
of
these
lines
of
proportion
,
but
after
a
different
maner
,
according
as
the
style
shall
be
either
perpendicular
,
or
parallel
to
the
plane
,
or
cut
the
plane
with
oblique
angles
.
In
an
equinoctiall
plane
where
the
style
is
perpendicular
to
the
plane
,
the
tropiques
and
other
circles
of
declination
will
be
perfect
circles
:
wherefore
consider
the
length
of
the
style
in
inches
and
parts
,
and
the
declination
of
the
circle
which
you
intend
to
describe
in
degrees
and
minutes
,
the
proportion
will
hold
,
As
the
tangent
of
45
gr.
to
the
length
of
the
style
:
So
the
cotangent
of
the
parallell
,
to
the
semidiameter
of
his
circle
.
geometric illustration
Or
if
it
were
required
to
proportion
the
style
to
the
plane
,
As
the
tangent
of
45
gr.
to
the
tangent
of
the
declination
:
So
the
semidiameter
of
the
plane
,
to
the
length
of
the
style
.
As
if
the
semidiameter
of
the
greatest
parallell
vpon
the
plane
were
but
six
inches
,
and
that
parallell
should
be
the
fift
degree
of
declination
:
extend
the
compasses
from
the
tangent
of
45
gr.
vnto
the
tangent
of
5
gr.
the
same
extent
wil
reach
in
the
line
of
numbers
from
6.00
vnto
about
0.53
,
which
shewes
that
the
length
of
the
style
must
be
53
parts
of
an
inch
diuided
into
100
;
then
the
length
of
the
style
being
knowne
,
the
semidiameter
of
the
other
circles
will
be
found
as
before
.
I
begin
here
with
the
fift
parallell
,
and
thence
proceed
vnto
the
tropique
,
because
the
shadow
of
the
rest
neare
the
equinoctial
,
would
be
ouerlong
,
and
the
equinoctiall
it selfe
cannot
be
described
.
The
parallels
of
North
declination
are
to
be
set
on
the
North
face
,
and
the
parallels
of
South
declination
on
the
South
face
of
the
plane
.
Neither
need
these
parallels
to
be
drawne
in
full
circles
,
but
onely
to
the
horizontall
line
,
which
shall
be
described
in
Cap.
xviij
.
Hauing
by
these
meanes
set
vp
the
style
to
his
true
height
,
and
drawne
the
circles
of
declination
,
if
we
shall
place
the
plane
so
as
it
shall
make
an
angle
with
the
horizon
equall
to
the
complement
of
the
latitude
,
and
then
turne
it
vntil
the
top
of
the
style
cast
the
shadow
vpon
the
parallell
of
declination
belonging
to
the
time
,
the
meridian
of
the
plane
will
shew
the
meridian
of
the
place
,
and
the
shadow
of
the
style
the
houre
of
the
day
,
without
the
help
of
a
magneticall
needle
.
CHAP.
XII
.
To
describe
the
Tropiques
and
other
circles
of
declination
in
a
polar
plane
.
IN
all
polar
planes
,
whether
the
be
parallel
to
the
meridian
or
to
the
circle
of
the
houre
of
6
,
or
otherwise
declining
,
the
equinoctiall
will
be
a
right
line
,
but
the
tropiques
and
other
circles
of
declination
wil
be
sections
hyperbolicall
,
and
be
thus
described
.
Consider
the
length
of
the
style
,
the
declination
of
the
parallell
,
and
the
angle
at
the
pole
betweene
the
substylar
and
the
houre-line
,
whereon
you
meane
to
describe
the
parallell
.
If
you
would
find
where
the
parallels
doe
crosse
the
substylar
;
As
the
tangent
of
45
gr.
to
the
tangent
of
declination
:
So
is
the
length
of
the
style
,
to
the
distance
of
the
parallell
from
the
equinoctiall
.
geometric illustration
As
in
the
example
of
the
polar
plane
,
where
the
length
of
the
style
BC
was
found
to
be
1
inch
61
cent
.
if
you
desire
to
know
the
distance
betweene
the
equinoctiall
and
the
tropique
vpon
the
substylar
line
:
extend
the
compasses
from
the
tangent
of
45
gr.
vnto
the
tangent
of
23
gr.
30
m.
the
same
extent
will
reach
in
the
line
of
numbers
from
1.61
vnto
0.70
;
and
therefore
the
distance
required
is
70
parts
of
an
inch
diuided
into
100.
The
like
reason
holdeth
for
all
other
parallels
of
declination
crossing
the
substylar
.
But
if
you
would
finde
where
the
parallels
doe
crosse
any
other
of
the
houre-lines
,
first
find
the
distance
betweene
the
axis
of
the
style
and
the
houre-line
,
then
the
distance
betweene
the
equinoctiall
and
the
parallell
,
both
these
may
be
represented
in
this
maner
.
On
the
center
B
and
any
semidiameter
BD
describe
an
occult
arke
of
a
circle
,
and
therein
describe
a
chord
of
23
gr.
30
m.
from
D
vnto
T
,
with
such
other
intermediate
declinations
as
you
intend
to
describe
on
the
plane
,
so
the
line
BD
shall
be
the
equator
,
and
BT
the
tropique
,
and
the
other
intermediate
lines
the
lines
of
declination
.
geometric illustration
That
done
,
consider
your
plane
,
which
for
example
may
be
either
the
meridian
or
the
declining
polar
plane
,
wherein
hauing
drawne
both
the
equator
,
and
the
houre-lines
as
before
,
first
take
out
the
height
of
the
style
,
and
prick
that
downe
in
this
equator
from
B
vnto
C
;
then
take
out
all
the
distances
,
betweene
B
the
top
of
the
style
and
the
seuerall
points
wherein
the
houre-lines
doe
crosse
the
equator
,
transfer
them
into
this
equator
BD
from
the
center
B
,
and
at
the
termes
of
these
distances
erect
lines
perpendicular
to
the
equator
,
crossing
the
lines
of
declination
,
and
note
them
with
the
number
of
the
houre
from
whence
they
were
taken
:
so
these
perpendiculars
shal
represent
those
houre-lines
,
and
the
seuerall
distances
between
the
equator
and
the
lines
of
declination
,
shall
giue
the
like
distances
between
the
equator
and
the
parallels
of
declination
vpon
your
plane
.
Vpon
this
ground
it
followeth
,
To
find
the
distance
betweene
the
axis
and
the
houre-lines
.
As
the
cosine
of
the
houre
from
the
substylar
,
is
to
the
sine
of
90
gr.
So
the
length
of
the
style
,
to
the
distance
between
the
axis
and
the
houre-line
.
geometric illustration
As
if
in
the
former
example
of
the
meridian
plane
,
where
BC
the
height
of
the
style
is
supposed
to
be
10
inches
,
it
were
required
to
find
the
distance
between
B
the
top
of
the
style
and
the
point
wherein
the
houre
of
11
in
the
morning
doth
crosse
the
equator
,
which
is
here
represented
by
B
5
,
because
it
is
the
fift
houre
from
the
substylar
,
whose
angle
at
the
pole
is
75
gr.
Extend
the
compasses
from
the
sine
of
15
gr.
the
complement
of
the
fift
houre
from
the
substylar
,
vnto
the
sine
of
90
gr.
the
same
extent
wil
reach
from
10.00
in
the
line
of
numbers
vnto
38.64
;
and
therefore
the
distance
B
5
between
the
axis
and
the
houre-line
,
is
38
inches
&
64
cent
.
and
may
be
called
the
secant
of
the
houre
.
Then
in
the
rectangle
B
5
T
,
hauing
the
side
B
5
,
and
the
angle
of
declination
at
B.
To
finde
the
distance
betweene
the
equinoctiall
and
the
parallell
.
As
the
tangent
of
45
gr.
to
the
tangent
of
the
declination
:
So
the
distance
betweene
the
axis
and
the
houre-line
,
to
the
distance
between
the
equinoctiall
and
the
parallell
.
Extend
the
compasses
from
the
tangent
of
45
gr.
vnto
the
tangent
of
23
gr.
30
m.
the
declination
of
the
tropique
,
so
the
same
extent
will
reach
in
the
line
of
numbers
from
38.64
,
the
distance
between
the
axis
and
the
fift
houre-line
vnto
16.80
;
and
therefore
the
distance
is
16
inches
and
80
cent
.
The
like
reason
holdeth
for
all
the
rest
,
which
may
be
gathered
and
set
downe
in
such
a
Table
as
this
which
followeth
.
Wherein
I
haue
set
downe
these
distances
for
seuerall
declinations
,
for
11
gr.
30
m.
for
16
gr.
55
m.
for
20
gr
12
m.
for
21
gr
41
m.
and
for
the
declination
of
the
Tropique
23
gr.
30
m.
which
may
be
applied
to
the
like
declinations
in
all
meridian
and
direct
polar
planes
.
As
in
the
former
example
of
the
polar
plane
,
where
BC
the
height
of
the
style
is
found
to
be
1
inch
61
cent
.
if
it
were
required
to
find
the
distance
betweene
B
the
top
of
the
style
and
the
points
wherein
the
houre-lines
of
7
in
the
morning
or
5
after
noon
,
do
crosse
the
equator
(
which
distances
,
I
called
the
secants
of
those
houres
.
)
either
you
may
extend
the
compasses
from
the
sine
of
15
gr.
the
complement
of
the
houre
from
the
substylar
vnto
the
sine
of
90
gr.
so
the
same
Hor.
|
Ang.
Po
|
Tangent
|
Secant
.
|
11
30
|
16
55
|
20
12
|
21
41
|
23
30
|
Gr.
M.
|
In.
Pat.
|
In.
Pa.
|
In.
Pa.
|
In.
Pa.
|
In.
Pa
|
In.
Pa.
|
In.
Pa
|
0
|
0
0
|
0
0
|
10
0
|
2
3
|
3
4
|
3
68
|
3
98
|
4
35
|
|
3
45
|
0
65
|
10
02
|
2
04
|
3
05
|
3
69
|
3
99
|
4
36
|
|
7
30
|
1
32
|
10
09
|
2
05
|
3
07
|
3
71
|
4
01
|
4
39
|
|
11
15
|
1
99
|
10
20
|
2
07
|
3
10
|
3
75
|
4
05
|
4
43
|
1
|
15
0
|
2
68
|
10
35
|
2
10
|
3
15
|
3
81
|
4
12
|
4
50
|
|
18
45
|
3
39
|
10
56
|
2
15
|
3
21
|
3
89
|
4
20
|
4
59
|
|
22
30
|
4
14
|
10
82
|
2
20
|
3
29
|
3
99
|
4
30
|
4
70
|
|
26
15
|
4
93
|
11
15
|
2
26
|
3
39
|
4
10
|
4
43
|
4
85
|
2
|
30
0
|
5
77
|
11
55
|
2
34
|
3
51
|
4
24
|
4
60
|
5
02
|
|
33
45
|
6
68
|
12
03
|
2
44
|
3
66
|
4
42
|
4
78
|
5
23
|
|
37
30
|
7
67
|
12
60
|
2
56
|
3
83
|
4
64
|
5
02
|
5
48
|
|
41
15
|
8
77
|
13
30
|
2
70
|
4
05
|
4
89
|
5
29
|
5
78
|
3
|
45
0
|
10
00
|
14
14
|
2
87
|
4
30
|
5
20
|
5
63
|
6
15
|
|
48
45
|
11
40
|
15
17
|
3
08
|
4
62
|
5
58
|
6
03
|
6
00
|
|
52
30
|
13
03
|
16
43
|
3
34
|
5
00
|
6
04
|
6
54
|
7
14
|
|
56
15
|
14
97
|
18
00
|
3
66
|
5
48
|
6
62
|
7
00
|
7
83
|
4
|
60
0
|
17
32
|
20
00
|
4
07
|
6
08
|
7
36
|
7
95
|
8
70
|
|
63
45
|
20
28
|
22
61
|
4
60
|
6
88
|
8
32
|
9
00
|
9
83
|
|
67
30
|
24
14
|
26
13
|
5
31
|
7
95
|
9
61
|
10
39
|
11
36
|
|
71
15
|
29
46
|
31
11
|
6
33
|
9
47
|
11
45
|
12
37
|
13
53
|
5
|
75
0
|
37
32
|
38
64
|
7
86
|
11
74
|
14
20
|
15
36
|
16
80
|
|
78
45
|
50
27
|
51
26
|
10
43
|
15
60
|
18
89
|
20
38
|
22
28
|
|
82
30
|
75
96
|
76
61
|
15
58
|
23
32
|
28
19
|
30
47
|
33
31
|
|
86
15
|
152
57
|
152
90
|
31
10
|
46
54
|
56
26
|
60
81
|
66
48
|
6
|
90
0
|
Infin
.
|
Infin
.
|
Infin
.
|
Infin
.
|
Infin
.
|
Infin
.
|
Infin
.
|
extent
wil
reach
in
the
line
of
numbers
from
1.61
the
length
of
the
style
,
vnto
6.21
,
according
to
the
former
Canon
.
Or
else
you
may
make
vse
of
the
former
Table
,
extending
the
compasses
in
the
line
of
numbers
from
10.00
the
length
of
the
style
in
the
Table
,
vnto
1.61
the
length
of
the
style
belonging
to
your
plane
,
so
the
same
extent
shall
reach
from
from
38.64
the
secant
in
the
Table
,
vnto
6.21
,
and
such
is
your
secant
required
,
the
distance
betweene
the
top
of
the
style
and
the
point
of
intersection
,
wherein
the
fift
houre
line
from
the
substylar
doth
crosse
the
equator
.
Again
,
the
same
extent
will
reach
from
16.80
the
distance
in
the
Table
belonging
to
the
fift
houre-line
betweene
the
equator
and
the
parallell
of
23
gr.
30
m.
declination
,
vnto
2.70
for
the
like
distance
vpon
your
plane
;
and
so
for
the
rest
,
which
may
be
gathered
and
set
downe
in
a
Table
.
Hor.
|
An.
Po
|
Tang
|
Secant
|
Trop
.
|
Gr.
M.
|
In.
P.
|
In.
P.
|
In
P.
|
12
|
0
0
|
0
0
|
1
61
|
0
70
|
11
1
|
15
0
|
0
43
|
1
63
|
0
72
|
10
2
|
30
0
|
0
93
|
1
85
|
0
80
|
9
3
|
45
0
|
1
61
|
2
27
|
0
99
|
8
4
|
60
0
|
2
79
|
3
22
|
1
40
|
7
5
|
75
0
|
6
00
|
6
21
|
2
70
|
That
done
,
and
the
equator
drawne
as
before
,
if
you
would
draw
the
tropiques
in
the
polar
plane
,
looke
into
the
Table
,
and
take
70
cent
.
out
of
the
line
of
inches
,
and
pricke
them
downe
in
the
substylar
on
either
side
of
the
equator
,
and
so
72
cent
.
on
the
first
houre
,
and
80
on
the
second
houre
,
and
2
inches
70
cent
.
on
the
fift
houre
from
the
substylar
,
and
the
rest
of
these
distances
on
their
seuerall
houre-lines
,
and
then
draw
a
crooked
line
through
all
these
points
,
so
as
it
makes
no
angles
,
the
line
so
drawne
shall
be
the
Tropique
required
.
In
like
maner
you
may
draw
any
other
parallell
of
declination
.
CHAP.
XIII
.
To
describe
the
Tropiques
and
other
circles
of
declination
in
such
a
Plane
as
is
neither
equinoctiall
nor
polar
.
IN
Planes
neither
equinoctiall
nor
polar
,
the
equator
will
be
a
right
line
,
the
tropiques
and
other
parallels
of
declination
will
be
conicall
sections
,
some
of
them
parabolicall
,
some
ellipticall
,
but
the
most
of
them
hyperbolicall
.
To
finde
the
points
of
intersection
of
these
parallels
with
the
houre-lines
,
wee
are
to
consider
,
first
the
length
of
the
axis
of
the
style
in
inches
and
parts
of
inches
;
secondly
the
height
of
the
style
aboue
the
plan●
;
●hirdly
the
angles
at
the
pole
betweene
the
proper
meridian
and
the
houre-circles
.
These
being
knowne
,
will
help
vs
to
find
,
first
the
angle
betweene
the
axis
and
the
houre-lines
on
the
plane
;
and
then
the
distance
between
the
center
and
the
parallels
:
both
these
may
be
represented
in
this
maner
.
geometric illustration
Let
the
triangle
ABC
be
made
equal
to
the
style
belonging
to
your
plane
,
AC
the
substylar
,
BC
the
axis
of
the
style
,
AB
the
length
of
the
style
perpendicular
to
the
plane
.
Then
hauing
drawn
the
line
BD
perpendicular
to
the
axis
on
the
center
B
,
&
any
semidiameter
BD
describe
an
occult
ark
of
a
circle
,
and
therein
inscribe
a
chord
of
23
gr.
30
m.
from
D
vnto
T
,
on
either
side
of
the
line
,
with
such
other
intermediate
declinations
as
you
intend
to
describe
on
the
plane
,
so
the
perpendicular
BD
shall
be
the
equator
,
and
BT
the
tropiques
,
and
the
other
intermediate
lines
the
parallels
of
declination
.
Wherefore
you
may
take
out
the
distance
C
♈
from
the
center
to
the
equator
,
and
pricke
it
downe
on
the
substylar
of
your
plane
from
the
center
at
C
vnto
♈
,
so
the
line
drawne
through
♈
perpendicular
to
your
substylar
,
shal
be
the
equator
of
your
plane
.
That
done
,
take
the
distance
of
each
houre-line
betweene
the
center
and
the
equator
of
your
plane
,
and
pricke
them
downe
in
the
equator
of
this
figure
,
from
the
center
at
C
,
noting
the
place
,
where
they
crosse
the
equator
,
with
the
number
belonging
to
the
houre
,
and
drawing
the
houre-lines
from
C
through
the
lines
of
declina●ion
.
Or
hauing
the
Sector
you
may
draw
an
occult
line
CE
perpendicular
to
the
axis
BC
,
and
therein
pricke
downe
the
tangent
of
the
height
of
the
style
aboue
the
plane
,
from
C
vnto
E.
Then
draw
the
line
EF
parallell
to
the
axis
,
crossing
the
substylar
produced
in
the
point
F
,
this
line
EF
will
be
the
line
of
sines
vpon
the
Sector
,
and
therein
you
may
pricke
downe
the
sines
of
the
complement
of
the
angles
at
the
pole
from
E
toward
F
,
and
draw
the
houre-lines
by
those
points
through
the
lines
of
declination
,
so
the
angles
at
C
between
the
axis
BC
and
those
houre-lines
,
shall
be
the
angles
betweene
the
axis
of
your
style
and
the
houre-lines
on
your
plane
,
and
the
seuerall
distances
betweene
the
point
C
and
the
lines
of
declination
,
shall
giue
you
the
like
distances
betweene
the
center
,
and
the
parallels
of
declination
vpon
the
houre-lines
in
your
plane
.
Vpon
this
ground
it
followeth
,
1
To
proportion
the
style
vnto
the
plane
.
Consider
the
height
of
the
style
aboue
the
plane
,
and
the
length
of
the
substylar
betweene
the
center
and
the
place
which
you
intend
for
the
tropique
.
If
it
be
the
tropique
which
is
farthest
from
the
center
,
adde
113
gr.
30
m
:
if
the
nearer
tropique
,
adde
66
gr.
30
m.
vnto
the
height
of
the
style
,
the
remainder
vnto
180
gr.
shall
giue
you
the
altitude
of
the
Sun
aboue
the
plane
when
he
commeth
to
that
tropique
.
As
in
our
latitude
the
height
of
the
style
aboue
an
horizontall
plane
is
51
gr.
30
m.
adde
vnto
this
113
gr.
30
m.
the
summe
is
165
gr.
which
being
taken
out
of
180
gr.
the
remainder
wil
be
15
gr.
and
such
is
the
altitude
of
the
Sun
aboue
this
plane
when
he
commeth
to
be
in
the
Winter
tropique
:
but
if
you
adde
66
gr.
30
m.
vnto
51
gr.
30
m.
the
remainder
to
180
gr.
will
be
62
gr.
And
such
is
the
altitude
of
the
Sunne
in
the
Summer
Tropique
.
Then
As
the
sine
of
66
gr.
30
m.
to
the
sine
of
the
Sunnes
altitude
:
So
the
length
of
the
substylar
line
,
to
the
length
of
the
axis
of
the
style
.
geometric illustration
As
in
the
first
example
of
the
declining
verticall
,
where
the
height
of
the
style
was
found
to
be
34
gr.
33
m.
and
is
here
represented
before
pag.
150.
by
the
angle
BC
♋
;
adde
to
this
height
113
gr.
30
m.
for
the
angle
CB
♋
,
the
sum
wil
be
148
gr.
3
m.
and
the
remainder
to
180
gr.
wil
be
31
gr.
57
m.
and
such
is
the
angle
B
♋
C
of
the
altitude
of
the
Sun
aboue
the
plane
,
when
he
cometh
to
be
in
the
tropique
of
♋
,
which
is
here
the
farthest
tropique
from
the
center
.
Then
supposing
the
length
of
the
substylar
line
between
the
center
and
the
place
which
is
fit
for
the
farthest
tropique
to
be
about
21
inches
,
extend
the
compasses
from
the
sine
of
66
gr.
30
m.
vnto
the
sine
of
31
gr.
57
m.
the
same
extent
will
reach
in
the
line
of
numbers
from
21
vnto
12.11
,
and
so
the
length
of
the
axis
of
the
style
should
be
12
inch
.
11
cent
.
Or
it
may
suffice
to
make
it
iust
12
inches
,
as
a
more
easie
ground
for
the
rest
of
the
worke
.
But
if
it
were
required
to
proportion
the
style
vnto
the
plane
,
so
as
it
may
cast
the
shadow
to
the
ful
length
of
the
substylar
line
at
all
times
of
the
yeare
,
you
may
then
consider
the
Sunne
in
the
tropique
,
which
is
to
be
set
nearest
vnto
the
center
,
and
adde
66
gr.
30
m.
vnto
34
gr.
33
m.
so
the
remainder
vnto
180
gr.
will
be
78
gr.
57
m.
And
if
you
extend
the
compasses
from
the
sine
of
66
gr.
30
m.
vnto
the
sine
of
78
gr.
57
m.
the
same
extent
will
reach
in
the
line
of
numbers
from
21
vnto
22.47
for
the
length
of
the
axis
of
the
style
.
2
Hauing
the
length
of
the
axis
,
and
the
height
of
the
style
aboue
the
plane
,
to
find
the
length
of
the
sides
of
the
style
.
The
style
of
a
plane
neither
equinoctiall
nor
polar
,
may
be
either
a
small
rod
of
iron
set
parallell
to
the
axis
of
the
world
,
or
perpendicular
to
the
plane
,
or
else
a
thin
plate
of
iron
or
brasse
made
in
forme
of
a
rectangle
triangle
BAC
,
with
the
base
BC
parallell
to
the
axis
of
the
world
,
the
side
AB
perpendicular
to
the
plane
,
&
the
side
AC
the
same
with
the
substylar
line
,
wherein
knowing
BC
,
and
the
angle
BAC
,
As
the
sine
of
90
gr.
to
the
length
of
the
axis
:
So
the
sine
of
the
height
of
the
style
,
to
the
length
of
the
perpendicular
side
:
And
so
the
cosine
of
the
height
of
the
style
,
to
the
length
of
the
substylar
side
.
Thus
in
the
former
example
,
the
length
of
the
axis
being
supposed
to
be
12
inches
,
and
the
height
of
the
style
34
gr.
33
m.
Extend
the
compasses
from
the
sine
of
90
gr.
(
or
else
from
the
sine
of
5
gr.
45
m.
)
vnto
12
in
the
line
of
numbers
,
the
same
extent
will
reach
from
the
sine
of
34
gr.
33
m.
vnto
6.80
in
the
line
of
numbers
for
the
length
of
the
perpendicular
side
,
and
from
the
sine
of
55
gr.
27
m.
vnto
9.88
for
the
length
of
the
substylar
side
.
3
To
find
the
distance
between
the
center
and
the
equator
vpon
the
substylar
line
.
This
is
here
represented
by
C
♈
,
and
may
be
found
by
resoluing
the
rectangle
triangle
CB
♈
.
As
the
cosine
of
the
height
of
the
style
,
is
to
the
sine
of
90
gr.
So
the
length
of
the
axis
,
to
the
distance
of
the
equator
from
the
center
.
Extend
the
compasses
from
the
sine
of
55
gr.
27
m.
vnto
the
sine
of
90
gr.
the
same
extent
will
reach
in
the
line
of
numbers
from
12
vnto
14.57
.
Wherefore
if
you
take
14
inch
.
57
cent
.
and
pricking
them
down
on
your
substylar
line
from
C
vnto
♈
,
draw
a
line
through
♈
,
crossing
the
substylar
at
right
angles
,
the
line
so
drawne
shall
be
the
equator
.
4
To
find
the
angles
contained
between
the
equator
and
the
houre-lines
vpon
your
plane
.
These
angles
made
by
B
♈
and
the
houre-lines
,
are
complements
of
those
which
are
at
C
,
betweene
BC
the
axis
and
those
seuerall
houre-lines
,
and
depend
vpon
the
angles
at
the
pole
,
between
the
proper
meridian
and
the
houre-circles
.
As
the
sine
of
90
gr.
to
the
cosine
of
the
angle
at
the
pole
:
So
the
cotangent
of
the
height
of
the
style
,
to
the
tangent
of
the
angle
between
the
equator
and
the
houreline
.
In
our
example
the
height
of
the
style
is
34
gr.
33
m.
and
the
proper
meridian
falleth
to
be
the
same
with
the
circle
of
the
second
houre
after
noone
,
whereupon
the
angle
at
the
pole
,
betweene
this
proper
meridian
,
and
the
circles
of
the
houre
of
1
on
the
one
side
,
and
3
on
the
other
side
,
wil
be
15
gr
;
so
between
this
meridian
and
the
houre-circles
of
12
and
4
,
the
angle
will
be
30
gr.
&c.
as
in
the
Table
.
Ho.
|
An.
Po
|
Arc.
Pla.
|
An.
Equ
|
C
♈
|
C
♋
|
C
♑
|
Gr.
M.
|
Gr.
M.
|
Gr.
M.
|
In.
P.
|
In.
P.
|
In.
P.
|
substy
|
0
0
|
0
0
|
55
27
|
14
57
|
20
80
|
11
21
|
1
3
|
15
0
|
8
38
|
54
30
|
14
74
|
21
36
|
11
25
|
12
4
|
30
0
|
18
8
|
51
30
|
15
33
|
23
44
|
11
40
|
11
5
|
45
0
|
29
33
|
45
45
|
16
75
|
29
06
|
11
76
|
10
6
|
60
0
|
44
30
|
36
0
|
20
00
|
50
84
|
12
77
|
9
7
|
75
0
|
64
42
|
20
36
|
34
10
|
Infin
.
|
15
82
|
8
8
|
90
0
|
90
0
|
0
0
|
Infinit
|
|
27
60
|
If
then
it
be
required
to
find
the
angle
,
which
the
houre-line
of
4
after
noone
doth
make
with
the
plane
of
the
equator
,
that
is
the
angle
C
4
B
contained
betweene
the
houre-line
C
4
and
the
line
B
4
,
drawne
from
the
top
of
the
style
vnto
the
intersection
of
the
houre-line
of
4
with
the
equator
.
Extend
the
compasses
from
the
sine
of
90
gr.
vnto
the
sine
of
60
gr.
the
complement
of
the
angle
at
the
pole
,
the
same
extent
wil
reach
from
the
tangent
of
55
gr.
27
m.
the
complement
of
the
height
of
the
pole
,
vnto
the
tangent
of
51
gr.
30
m.
and
such
is
the
angle
C
4
B
in
the
diagram
Pag.
150.
Or
in
crosse-worke
,
if
it
were
required
to
finde
the
angle
C
9
B
,
looke
into
the
Table
for
the
houre
of
9
,
and
there
you
shall
find
the
angle
at
the
pole
.
to
be
75
gr
;
and
if
you
extend
the
compasses
from
the
sine
of
90
gr.
vnto
the
tangent
of
55
gr.
27
m.
the
same
extent
will
reach
from
the
sine
of
15
gr.
the
complement
of
75
gr.
vnto
the
tangent
of
20
gr.
36
m.
and
such
is
the
angle
C
9
B
,
made
at
the
equator
betweene
the
line
B
9
drawne
from
the
top
of
the
style
,
and
the
houre-line
C
9
drawne
from
the
center
.
The
like
reason
holdeth
for
the
rest
,
which
may
be
found
and
set
downe
in
a
table
:
then
may
you
either
draw
these
angles
at
C
in
the
former
figure
more
perfectly
,
and
thence
finish
your
worke
,
or
else
proceed
5
To
find
the
distance
betweene
the
center
and
the
parallels
of
declination
.
The
distances
betweene
the
center
and
the
parallels
of
declination
,
may
be
found
by
resoluing
the
triangles
made
by
the
axis
BC
,
the
lines
of
declination
,
and
the
houre-lines
.
For
hauing
the
angles
at
the
equator
,
and
knowing
the
declination
of
the
parallell
,
if
the
parallell
shall
fall
betweene
the
equator
and
the
center
,
adde
the
declination
vnto
the
angle
at
the
equator
;
or
if
it
shall
fall
without
the
equator
,
take
the
declination
out
of
the
angle
at
the
equator
,
so
shall
you
haue
the
angle
at
the
parallell
.
Then
As
the
sine
of
the
angle
at
the
parallell
,
to
the
cosine
of
the
declination
:
So
the
length
of
the
axis
of
the
style
,
to
the
distance
between
the
center
and
the
parallell
.
Thus
in
our
example
,
the
angle
at
the
equator
belonging
to
the
houre
of
4
after
noone
,
was
found
before
to
be
51
gr.
30
m
:
if
you
would
find
the
distance
between
the
center
and
the
equator
,
extend
the
compasses
from
the
sine
of
51
gr.
30
m.
vnto
the
sine
of
90
gr.
the
complement
of
the
declination
,
the
same
extent
will
reach
in
the
line
of
numbers
,
from
12
vnto
15.33
,
and
such
is
the
distance
vpon
the
houre-line
of
4
between
the
center
and
the
equator
.
If
you
would
finde
the
distance
vpon
this
houre-line
,
between
the
center
and
the
inner
tropique
,
whose
declination
is
knowne
to
be
23
gr.
30
m.
adde
the
declination
to
the
angle
at
the
equator
,
so
the
angle
at
the
parallell
wil
be
75
gr.
wherefore
extend
the
compasses
from
the
sine
of
75
gr.
vnto
the
sine
of
66
gr.
30
m.
the
complement
of
the
declination
,
the
same
extent
will
reach
in
the
line
of
numbers
,
from
12
vnto
11.40
,
and
such
is
the
length
of
the
houre-line
of
4
betweene
the
center
and
the
tropique
of
♑
.
If
you
would
finde
the
distance
vpon
this
houre-line
between
this
center
and
the
tropique
of
♋
,
which
is
here
the
farthest
from
the
center
,
take
the
declination
out
of
the
angle
at
the
equator
,
so
the
angle
at
the
parallell
will
be
28
gr.
vnto
wherefore
extend
the
compasses
from
the
sine
of
28
gr.
vnto
the
sine
of
66
gr.
30
m.
the
same
extent
will
reach
in
the
line
of
numbers
,
from
12
vnto
23.44
,
and
such
is
the
distance
betweene
the
center
and
the
tropique
of
♋
vpon
this
houre-line
of
4.
The
like
reason
holdeth
for
all
the
rest
,
which
may
be
gathered
and
set
downe
in
a
table
.
That
done
and
the
equator
drawne
as
before
,
if
you
would
draw
the
tropique
of
♋
,
looke
into
the
table
,
and
there
finding
vnder
the
title
C
♋
the
distance
of
the
substylar
between
the
center
and
the
parallel
of
♋
to
be
20
inch
.
80
cent
.
take
20
inch
.
80
cent
.
out
of
the
line
of
inches
,
and
prick
them
downe
in
the
substylar
of
your
plane
from
C
vnto
♋
.
Or
if
either
the
center
fall
without
your
plane
,
or
the
extent
be
too
large
for
your
compasses
,
you
may
pricke
downe
the
difference
betweene
C
♈
and
C
♋
.
As
here
the
distance
C
♈
between
the
center
&
the
equator
is
14.57
,
the
distance
C
♋
20.80
,
the
difference
6.23
,
therfore
taking
6
inch
.
23
cent
.
pricke
them
downe
on
the
substylar
from
♈
vnto
♋
,
and
you
shall
haue
the
same
intersection
of
the
tropique
and
the
substylar
,
as
before
;
&
the
like
reason
holdeth
for
pricking
down
of
the
rest
of
these
distances
on
their
seuerall
houre-lines
.
Then
hauing
the
points
of
intersection
betweene
the
houre-lines
and
the
parallell
,
you
may
ioyne
them
all
in
a
crooked
line
without
making
of
any
angles
,
the
line
so
drawne
shall
be
the
tropique
required
.
And
after
this
maner
may
you
draw
any
other
parallell
of
declination
,
whereof
you
haue
examples
in
the
most
of
the
former
Diagrams
.
CHAP.
XIIII
.
To
describe
the
parallels
of
the
Signes
in
any
of
the
former
Planes
.
THe
equator
and
the
tropiques
before
described
,
do
shew
the
Suns
entrance
into
4
of
the
Signes
,
the
equator
into
♈
and
♎
,
the
one
tropique
into
♋
,
and
the
other
into
♑
,
the
rest
of
the
intermediate
Signes
will
be
described
in
the
same
maner
as
the
tropiques
,
if
first
we
know
their
declination
.
The
maner
of
finding
the
declination
not
onely
of
the
beginning
of
the
Signes
,
but
of
all
other
points
of
the
ecliptique
,
is
before
set
downe
in
2.
Prop.
Astronomicall
,
pag.
52.
by
which
you
may
finde
the
declination
of
the
beginning
of
♉
,
♍
,
♏
,
and
♓
to
be
11
gr.
30
m.
and
of
♊
,
♌
,
♐
and
♒
to
be
20
gr.
12
m.
If
then
you
inscribe
the
chords
of
11
gr.
30
m.
and
of
20
gr.
12
m.
into
the
former
figure
BDT
Pag.
145.
from
D
toward
T
,
the
lines
drawne
from
B
through
the
termes
of
those
chords
shall
be
the
Signes
required
.
And
with
these
declinations
,
the
height
of
the
style
,
and
the
length
of
the
axis
,
you
may
finde
the
angles
at
the
parallell
,
and
then
the
distances
between
the
center
and
the
parallell
,
which
being
pricked
downe
vpon
their
seuerall
houre-lines
shall
giue
you
the
points
of
intersection
,
by
which
you
may
draw
the
parallels
of
the
Signes
,
as
in
the
figures
belonging
to
the
polar
planes
.
CHAP.
XV.
To
describe
the
parallels
of
the
length
of
the
day
in
any
of
the
former
Planes
.
THe
length
of
the
day
will
alwayes
be
12
houres
long
when
the
Sunne
cometh
to
be
in
the
equator
,
and
this
holdeth
in
all
latitudes
;
but
at
other
times
of
the
yeare
the
same
place
of
the
Sunne
,
wil
not
giue
the
same
length
of
the
day
in
another
latitude
;
wherefore
the
latitude
being
known
,
we
are
first
To
find
the
declination
of
the
Sunne
agreeing
to
the
length
of
the
day
.
Consider
the
difference
betweene
the
length
of
an
equinoctiall
day
and
the
day
proposed
,
and
turne
the
time
into
degrees
and
minutes
.
As
the
sine
of
90
gr.
is
to
the
sine
of
halfe
the
difference
:
So
the
cotangent
of
the
latitude
,
to
the
tangent
of
the
declination
.
geometric illustration
If
then
you
inscribe
the
chords
of
these
arks
into
the
former
figure
BDT
,
the
lines
drawne
from
B
through
the
termes
of
these
arks
,
shall
be
the
lines
belonging
to
the
diurnall
arkes
,
and
the
seuerall
distances
betweene
them
and
the
point
C
giue
the
like
distances
betweene
the
center
and
the
parallels
of
the
length
of
the
day
vpon
the
houre-lines
in
your
plane
.
geometric illustration
CHAP.
XVI
.
To
draw
the
old
vnequall
houres
in
the
former
Planes
.
IT
was
the
maner
of
the
Ancients
to
divide
the
day
into
twelue
equall
houres
,
and
the
night
into
twelue
other
equal
houres
,
and
so
the
whole
day
and
night
into
24
houres
.
Of
these
24
,
those
which
belonged
vnto
the
day
,
were
either
longer
or
shorter
(
excepting
the
two
equinoctial
dayes
)
then
those
which
belonged
vnto
the
night
;
and
the
Summer
houres
alwayes
longer
then
the
houres
in
the
Winter
,
according
to
the
lengthening
of
the
dayes
,
whereupon
they
are
called
the
old
vnequall
(
and
by
some
the
Planetary
)
houres
.
geometric illustration
To
expresse
these
in
the
former
Planes
:
first
draw
the
common
houre-lines
,
the
equator
,
and
the
tropiques
,
as
before
:
then
describe
two
occult
parallels
of
the
length
of
the
day
,
one
for
9
houres
,
the
other
for
15
houres
;
for
so
you
may
draw
a
streight
line
for
the
first
vnequall
houre
through
5
ho.
45
m.
in
the
parallell
of
15
,
and
through
8
ho.
15
m.
in
the
parallel
of
9.
This
streight
line
shal
passe
directly
through
7
ho.
0
m.
in
the
equator
,
and
so
cut
off
a
twelfth
part
of
the
arks
aboue
the
horizon
,
both
from
these
two
parallels
and
the
equator
:
and
being
continued
vnto
the
tropiques
,
it
shall
also
cut
off
about
a
twelfth
part
from
them
,
and
all
the
rest
of
the
parallels
of
declination
,
without
any
sensible
error
.
In
like
maner
may
you
draw
the
second
vnequall
houre
through
7
ho.
in
the
parallell
of
15
,
through
8
ho.
in
the
equator
,
and
through
9
ho.
in
the
parallell
of
9
,
and
so
in
the
rest
,
as
in
this
Table
.
Horae
|
15
|
Aeq
|
9
|
Ho.
M.
|
Ho
|
Ho.
M.
|
0
|
4
30
|
6
|
7
30
|
1
|
5
45
|
7
|
8
15
|
2
|
7
0
|
8
|
9
0
|
3
|
8
15
|
9
|
9
45
|
4
|
9
30
|
10
|
10
30
|
5
|
10
45
|
11
|
11
15
|
6
|
12
0
|
12
|
12
0
|
7
|
1
15
|
1
|
0
45
|
8
|
2
30
|
2
|
1
30
|
9
|
3
45
|
3
|
2
15
|
10
|
5
0
|
4
|
3
0
|
11
|
6
15
|
5
|
3
45
|
12
|
7
30
|
6
|
4
30
|
And
of
these
vnequall
houres
you
haue
a
farther
example
in
the
diagram
belonging
to
the
polar
declining
plane
,
Pag.
130.
CHAP.
XVII
.
To
draw
the
houres
from
Sun
rising
and
sun
setting
in
the
former
Planes
.
TO
know
how
many
houres
are
past
since
the
Sun
rising
,
or
how
many
remaine
to
the
Sun
setting
;
first
draw
the
common
houre-lines
,
the
equator
,
and
the
tropiques
,
as
before
:
then
describe
two
occult
parallels
of
the
length
of
the
day
,
one
for
8
houres
,
and
the
other
for
16
houres
.
For
so
you
geometric illustration
may
draw
the
first
houre
from
the
Sun
rising
through
the
common
houres
of
5
in
the
parallell
of
16
,
of
7
in
the
equator
,
and
of
9
in
the
parallell
of
8.
In
like
maner
the
second
houre
from
Sun
rising
through
the
common
houres
of
6
in
the
parallell
of
16
,
of
8
in
the
equator
,
and
of
10
in
the
parallell
of
8.
And
so
the
rest
in
their
order
.
The
first
houre
before
Sun
setting
,
or
the
23
houre
from
the
last
Sun
setting
,
may
be
drawne
in
like
sort
through
the
common
houres
of
3
after
noone
in
the
parallell
of
8
,
of
5
in
the
equator
,
and
of
7
in
the
parallell
of
16.
The
second
houre
before
Sun
setting
,
or
the
22
houre
after
the
last
Sun
setting
through
the
common
houres
of
2
in
the
parallell
of
8
,
of
4
in
the
equator
,
and
of
6
in
the
parallell
of
16.
And
so
the
rest
in
the
like
order
,
whereof
you
haue
another
example
in
the
Diagram
belonging
to
the
declining
verticall
,
Pag.
116.
CHAP.
XVIII
.
To
draw
the
horizontall
line
in
the
former
planes
.
THe
common
houre-lines
do
commonly
depend
on
the
shadow
of
the
axis
,
but
the
parallels
of
the
Signes
,
and
of
the
length
of
the
day
,
the
houre-lines
from
Sun
rising
and
Sun
setting
,
with
many
others
,
depend
on
the
shadow
of
the
top
of
the
style
,
or
some
one
point
in
the
axis
,
which
here
signifieth
the
center
of
the
world
,
and
is
represented
by
the
point
B.
And
these
lines
so
depending
,
are
then
only
vseful
when
they
fall
betweene
the
two
tropiques
,
and
within
the
horizon
.
There
may
be
seuerall
horizontall
lines
drawne
vpon
euery
plane
,
as
I
shewed
before
in
finding
the
inclination
of
a
plane
;
but
the
proper
horizontall
line
which
is
here
meant
,
must
always
be
in
the
same
plane
with
B
the
top
of
the
style
;
so
that
in
an
horizontall
plane
there
can
be
no
such
horizontall
line
,
but
in
all
other
planes
it
may
be
found
by
applying
the
horizontall
leg
of
the
Sector
vnto
the
top
of
the
style
,
and
then
working
as
before
;
and
the
intersection
of
this
line
with
the
meridian
or
substylar
line
,
may
be
found
by
proportion
.
1
To
find
the
intersection
of
the
horizon
with
the
meridian
,
in
an
equinoctiall
plane
.
As
the
tangent
of
45
gr.
to
the
tangent
of
the
latitude
:
So
is
the
height
of
the
style
,
to
the
distance
between
the
style
and
the
horizontal
line
.
As
in
the
example
of
the
former
equinoctiall
plane
,
Pag.
142.
extend
the
compasses
from
the
tangent
of
45
gr.
vnto
51
gr.
30
m.
the
tangent
of
the
latitude
,
the
same
extent
will
reach
in
the
line
of
numbers
,
from
52
the
length
of
the
style
vnto
66
,
and
such
is
the
distance
betweene
the
style
and
the
horizontall
line
;
wherefore
I
take
66
parts
out
of
a
line
of
inches
,
and
pricke
them
downe
in
the
meridian
line
from
C
vnto
H
aboue
the
style
in
the
vpper
face
,
but
below
the
style
in
the
lower
face
of
the
plane
,
so
a
right
line
drawne
through
H
,
parallell
to
the
houre
of
6
,
shall
be
the
horizontall
line
.
2
To
find
the
intersection
of
the
horizon
with
the
meridian
,
in
a
direct
polar
plane
.
As
the
tangent
of
45
gr.
to
the
cotangent
of
the
latitude
:
So
the
length
of
the
style
,
to
the
distance
between
the
style
&
the
horizontal
line
.
As
in
the
example
of
the
former
polar
plane
,
Pag.
144.
extend
the
compasses
from
the
tangent
of
45
gr.
vnto
tangent
of
38
gr.
30
m.
the
complement
of
the
latitude
,
the
same
extent
will
reach
in
the
line
of
numbers
,
from
1.61
the
length
of
the
style
,
vnto
1.28
,
and
such
is
the
distance
vpon
the
meridian
betweene
the
style
and
the
horizontall
line
.
In
all
vpright
planes
,
whether
they
be
direct
verticall
,
or
declining
,
or
meridian
planes
,
the
horizontall
line
must
alwayes
be
drawne
through
A
the
foot
of
the
style
,
as
may
appeare
in
the
examples
before
,
Pag.
102.107.116
.
And
generally
in
all
planes
whatsoeuer
,
the
horizontal
line
must
be
drawne
through
the
intersection
of
the
equator
with
the
houre
of
6.
Or
if
that
intersection
fall
without
the
plane
,
yet
if
any
arks
of
the
length
of
the
day
be
drawne
on
the
plane
,
the
horizontall
line
may
be
drawne
through
their
intersections
,
with
the
houres
of
the
Suns
rising
or
setting
.
CHAP.
XIX
.
To
describe
the
verticall
circles
in
the
former
Planes
.
THe
vertical
circles
commonly
called
Azimuths
,
are
great
circles
drawne
through
the
zenith
,
by
which
we
may
know
in
what
part
of
the
heauen
the
Sunne
is
,
how
far
from
the
East
or
West
,
and
how
neare
vnto
the
meridian
.
In
all
vpright
planes
,
whether
they
be
direct
verticals
,
or
declining
,
or
meridian
planes
,
the
semidiameter
of
the
horizon
wil
be
the
same
with
AB
the
perpendicular
side
of
the
style
,
and
these
Azimuths
will
be
parallels
one
to
the
other
,
and
the
distance
of
each
Azimuth
,
from
the
foote
of
the
style
vpon
the
horizontall
line
,
may
be
found
in
this
manner
.
Consider
the
length
of
the
style
in
inches
and
parts
of
inches
,
and
the
distance
of
each
Azimuth
from
the
style
,
according
to
the
angle
at
the
zenith
in
degrees
and
minutes
.
As
the
tangent
of
45
gr.
to
the
tangent
of
the
azimuth
:
So
the
length
of
the
style
,
to
the
length
of
the
horizontall
line
between
the
style
and
the
azimuth
.
geometric illustration
As
if
it
were
required
to
draw
the
common
azimuths
on
the
South
face
of
the
verticall
plane
before
described
,
where
AB
the
length
of
the
style
may
be
supposed
to
be
10
inches
.
Here
the
plane
hauing
no
declination
,
the
style
is
in
the
plane
of
the
meridian
,
and
so
pointeth
directly
into
the
South
.
The
point
of
SbE
is
11
gr.
15
m.
distant
from
the
style
,
and
SSE
22
gr.
30
m.
and
the
rest
in
their
order
:
wherefore
extend
the
compasses
from
the
tangent
of
45
gr.
vnto
10
in
the
line
of
numbers
,
the
same
extent
will
reach
from
the
tangent
of
11
gr.
15
m.
vnto
1.99
in
the
line
of
numbers
for
the
length
of
the
tangent
line
,
betweene
the
style
and
the
point
SbE
,
and
from
the
tangent
of
22
gr.
30
m.
vnto
4.14
for
SSE
,
and
so
for
the
rest
,
as
in
this
Table
.
Azimuths
.
|
An.
Zen.
|
Tangen
|
Gr.
M.
|
In.
Pa.
|
South
|
0
0
|
0
0
|
SbE
|
11
15
|
1
99
|
SSE
|
22
30
|
4
14
|
SEbS
|
33
45
|
6
68
|
SE
|
45
0
|
10
00
|
SEbb
|
56
15
|
14
97
|
ESE
|
67
30
|
24
14
|
EbS
|
78
45
|
50
27
|
East
|
90
0
|
Infin
.
|
In
like
maner
in
the
first
example
of
the
declining
plane
,
where
the
style
standeth
according
to
the
declination
24
gr.
20
m.
distant
from
the
South
toward
the
West
.
The
next
point
of
SbW
is
but
13
gr.
5
m.
distant
from
the
style
;
and
the
second
of
SSW
onely
1
gr.
50
m.
and
the
third
of
SWbS
is
againe
9
gr.
25
m.
and
the
rest
in
their
order
.
Wherefore
hauing
before
found
the
length
of
the
style
to
be
6
inches
80
parts
,
extend
the
compasses
from
the
tangent
of
45
gr.
vnto
6.80
parts
in
the
line
of
numbers
,
the
same
extent
wil
reach
from
the
tangent
of
24
gr.
20
m.
vnto
3.07
in
the
line
of
numbers
for
the
length
of
the
tangent
line
between
the
style
&
the
South
,
and
from
the
tangent
of
13
gr.
5
m.
vnto
1.58
for
the
point
of
SbW
;
and
so
for
the
rest
,
as
in
this
Table
.
Azimuths
.
|
An.
Zen.
|
Tangen
|
Gr.
M
|
In.
Pa.
|
SEbE
|
80
35
|
41
00
|
SE
|
69
20
|
18
03
|
SEbS
|
58
5
|
10
91
|
SSE
|
46
50
|
7
25
|
SbE
|
35
35
|
4
86
|
South
|
24
20
|
3
07
|
SbW
|
13
5
|
1
58
|
SSW
|
1
50
|
0
22
|
The
foote
of
the
styl
|
SWbS
|
9
25
|
1
13
|
SW
|
20
40
|
2
57
|
SWbW
|
31
55
|
4
24
|
WSW
|
43
10
|
6
37
|
WbS
|
54
25
|
9
50
|
West
|
65
40
|
15
02
|
WbN
|
76
55
|
29
26
|
WNW
|
88
10
|
212
45
|
That
done
,
if
you
take
these
parts
out
of
a
line
of
inches
,
and
pricke
them
downe
in
the
horizontall
line
on
either
side
of
the
style
,
drawing
right
lines
perpendicular
to
the
horizon
through
these
intersections
,
but
so
as
they
may
be
contained
betweene
the
horizontall
and
the
tropiques
,
the
lines
so
drawne
shall
be
the
azimuths
required
.
In
an
horizontall
plane
these
azimuths
are
drawne
more
easily
.
For
here
the
perpendicular
side
of
the
style
is
the
same
with
the
axis
of
the
horizon
,
and
the
foote
of
the
style
is
the
verticall
point
,
in
which
all
the
azimuth
lines
doe
meete
as
their
circles
do
in
the
zenith
:
wherefore
let
any
circle
described
on
the
center
A
,
at
the
foote
of
the
style
,
be
diuided
first
into
foure
parts
,
beginning
at
the
meridian
,
and
then
each
quarter
subdiuided
either
into
eight
equall
parts
,
according
to
the
points
of
the
Mariners
compasse
,
or
into
90
gr.
according
to
the
Astronomical
diuision
;
if
you
draw
right
lines
through
the
center
and
these
diuisions
,
the
lines
so
drawne
shall
be
the
azimuths
required
.
In
all
other
planes
inclining
to
the
horizon
,
these
vertical
circles
will
meete
in
a
point
,
but
that
verticall
point
being
more
or
lesse
distant
from
the
foote
of
the
style
,
the
angles
at
this
point
will
be
vnequall
.
1
To
find
the
distance
between
the
foote
of
the
style
and
the
verticall
point
.
The
verticall
point
wherein
all
the
verticall
lines
do
meet
,
will
be
alwayes
in
the
meridian
,
directly
vnder
or
ouer
the
top
of
the
style
;
and
the
angle
betweene
the
perpendicular
side
of
the
style
and
the
verticall
line
,
will
be
equall
to
the
inclination
of
the
plane
to
the
horizon
.
Wherefore
As
the
tangent
of
45
gr.
to
the
tangent
of
the
inclination
of
the
plane
:
So
is
the
length
of
the
style
,
to
the
distance
betweene
the
foote
of
the
style
and
the
verticall
point
.
Thus
in
the
first
example
of
the
declining
inclining
planes
,
where
the
vpper
face
of
the
plane
looking
Southwest
,
the
declination
was
24
gr.
20
m.
the
inclination
36
gr
;
and
you
may
suppose
AB
the
length
of
the
style
to
be
6
inches
:
if
you
extend
the
compasses
from
the
tangent
of
45
gr.
vnto
geometric illustration
the
tangent
of
36
gr.
the
same
extent
will
reach
in
the
line
of
numbers
from
6.00
vnto
4.36
,
for
the
distance
AV
betweene
A
the
foote
of
the
style
and
V
the
verticall
point
.
2
To
find
the
distance
between
the
foote
of
the
style
and
the
horizontall
line
.
As
the
tangent
of
the
inclination
of
the
plane
,
is
to
the
tangent
of
45
gr.
So
the
length
of
the
style
,
to
the
distance
betweene
the
foote
of
the
style
and
the
horizontall
line
.
So
the
same
extent
of
the
compasses
as
before
,
wil
reach
in
the
line
of
numbers
from
6.00
vnto
8.26
for
the
distance
AH
betweene
the
foote
of
the
style
and
the
horizontall
line
.
Then
may
you
take
4
inches
36
cent
.
and
pricking
them
downe
from
A
the
foot
of
the
style
vnto
V
the
verticall
point
in
the
meridian
,
draw
the
line
VA
,
which
being
produced
shall
cut
the
horizon
in
the
point
H
with
right
angles
,
and
be
that
particular
azimuth
which
is
perpendicular
to
the
plane
.
Or
you
may
take
8
inches
26
cent
.
and
pricke
them
downe
in
the
former
line
VA
produced
from
A
vnto
H
,
and
so
draw
the
horizontall
line
through
H
perpendicular
vnto
VH
,
which
horizontall
line
being
produced
wil
crosse
the
equator
in
the
same
point
wherein
the
equator
crosseth
the
houre-line
of
6
,
vnlesse
there
be
some
former
error
.
3
To
find
the
angles
made
by
the
azimuth
lines
at
the
verticall
point
.
The
angles
at
the
zenith
depend
on
the
declination
of
the
plane
,
as
in
our
example
,
where
the
style
standeth
according
to
the
declination
24
gr.
20
m.
distant
from
the
South
toward
the
West
,
the
azimuth
of
10
gr.
from
the
meridian
Eastward
will
be
34
gr.
20
m.
the
azimuth
of
10
gr.
Westward
will
be
onely
14
gr.
20
m.
distant
from
the
style
,
and
so
the
rest
in
their
order
.
Or
if
you
would
rather
describe
the
common
azimuths
,
the
point
of
SbE
wil
be
35
gr.
35
m.
the
point
of
SbW
13
gr.
5
m.
distant
from
the
style
,
and
so
the
rest
in
their
order
.
Then
As
the
sine
of
90
gr.
to
the
cosine
of
the
inclination
of
the
plane
:
So
the
tangent
of
the
angle
at
the
zenith
,
to
the
tangent
of
the
angle
at
the
verticall
point
between
the
line
drawne
through
the
foot
of
the
style
and
the
azimuth
required
.
Wherefore
the
inclination
of
the
plane
in
our
example
being
36
gr.
extend
the
compasses
from
the
sine
of
90
gr.
vnto
the
sine
of
54
gr.
the
same
extent
shall
reach
in
the
line
of
tangents
,
from
24
gr.
20
m.
vnto
20
gr.
5
m.
for
the
angle
HVa
at
the
vertical
point
,
between
the
line
VH
drawne
through
A
the
foote
of
the
style
&
the
South
.
Againe
,
the
same
extent
will
reach
from
the
tangent
of
13
gr.
5
m.
vnto
10
gr.
38
m.
for
the
angle
belonging
to
SbW
;
and
so
for
the
rest
,
as
in
this
table
.
Azimuths
.
|
Ang.
Ze.
|
Ang.
Ve.
|
Gr.
M.
|
Gr.
M.
|
SEbE
|
80
35
|
78
25
|
SE
|
69
20
|
65
0
|
SEbS
|
58
5
|
52
25
|
SSE
|
46
50
|
40
46
|
SbE
|
35
35
|
30
3
|
South
|
24
20
|
20
5
|
SbW
|
13
5
|
10
39
|
SSW
|
1
50
|
1
29
|
|
Style
.
|
0
0
|
SWbS
|
9
25
|
7
38
|
SW
|
20
40
|
16
58
|
SWbW
|
31
55
|
26
45
|
WSW
|
43
10
|
37
11
|
WbS
|
54
25
|
48
30
|
West
|
65
40
|
60
48
|
WbN
|
76
55
|
73
58
|
WNW
|
88
10
|
87
44
|
These
angles
being
knowne
,
if
on
the
center
V
,
at
the
verticall
point
,
you
describe
an
occult
circle
,
and
therein
inscribe
the
chords
of
these
angles
from
the
line
VH
,
and
then
draw
right
lines
through
the
verticall
point
,
and
the
termes
of
those
chords
,
the
lines
so
drawne
shall
be
the
azimuths
required
.
The
like
reason
holdeth
for
the
drawing
of
the
azimuths
vpon
all
other
inclining
planes
,
wheof
you
haue
another
example
in
the
Diagram
belonging
to
the
meridian
incliner
,
Pag.
126.
Or
for
further
satisfaction
you
may
finde
where
each
azimuth
line
shall
crosse
the
equator
.
As
the
sine
of
90
gr.
to
the
sine
of
the
latitude
:
So
the
tangent
of
the
azimuth
from
the
meridian
,
to
the
tangent
of
the
equator
from
the
meridian
.
Extend
the
compasses
from
the
sine
of
90
gr.
vnto
the
line
of
our
latitude
51
gr.
30
m.
the
same
extent
will
reach
in
the
line
of
tangents
from
10
gr.
vnto
7.
gr.
50.
m.
for
the
intersection
of
the
equator
with
the
azimuth
of
10
gr.
from
the
meridian
.
Againe
,
the
same
extent
will
reach
from
20
gr.
vnto
15
gr.
54
m.
for
the
azimuth
of
20
gr.
And
so
the
rest
,
as
in
these
tables
.
Azim
.
|
Equat.
|
Gr.
M.
|
Gr.
M.
|
10
0
|
7
50
|
20
0
|
15
54
|
30
0
|
24
20
|
40
0
|
43
18
|
50
0
|
13
0
|
60
0
|
53
35
|
70
0
|
65
3
|
80
0
|
77
18
|
90
0
|
90
0
|
Azim
.
|
Equat.
|
Gr.
M.
|
Gr.
M.
|
11
15
|
8
51
|
22
30
|
17
58
|
33
45
|
27
36
|
45
0
|
38
2
|
56
15
|
49
30
|
67
30
|
62
6
|
78
45
|
75
44
|
90
0
|
90
0
|
By
which
you
may
see
that
the
azimuth
90
gr.
distant
from
the
meridian
,
which
is
the
line
of
East
and
West
,
will
crosse
the
equator
at
90
gr.
from
the
meridian
in
the
same
point
,
with
the
horizontall
line
and
the
houre
of
6.
And
that
the
azimuth
of
45
gr.
will
crosse
the
equator
at
38
gr.
2
m.
from
the
meridian
,
that
is
,
the
line
of
SE
will
crosse
the
equator
at
the
houre
of
9
and
28
m.
in
the
morning
,
and
the
line
of
SW
at
2
ho.
32
min.
in
the
afternoone
;
and
so
for
the
rest
,
whereby
you
may
examine
your
former
worke
.
CHAP.
XX.
To
describe
the
parallels
of
the
horizon
in
the
former
planes
.
THe
parallels
of
the
horizon
,
commonly
called
Almicanters
,
or
parallels
of
altitude
(
whereby
we
may
know
the
altitude
of
the
Sunne
aboue
the
horizon
)
haue
such
respect
vnto
the
horizon
,
as
the
parallels
of
declination
vnto
the
equator
,
and
so
may
be
described
in
like
maner
.
In
an
horizontall
plane
,
these
parallels
will
be
perfect
circles
;
wherefore
knowing
the
length
of
the
style
in
inches
and
parts
,
and
the
distance
of
the
parallell
from
the
horizon
in
degrees
and
minutes
.
As
the
tangent
of
45
gr.
is
to
the
length
of
the
style
:
So
the
cotangent
of
the
parallell
,
to
the
semidiameter
of
his
circle
.
Thus
in
the
example
of
the
horizontall
plane
,
Pag.
164.
if
AB
the
length
of
the
style
shall
be
5
inches
,
and
that
it
were
required
to
finde
the
semidiameter
of
the
parallell
of
62
gr.
extend
the
compasses
from
the
tangent
of
45
gr.
vnto
5.00
in
the
line
of
numbers
,
the
same
extent
will
reach
from
the
tangent
of
28
gr.
the
complement
of
the
parallell
vnto
2.65
,
and
if
you
describe
a
circle
on
the
center
A
to
the
semidiameter
of
2
inches
65
cent
.
it
shall
be
the
parallell
required
.
In
all
vpright
planes
,
whether
they
be
direct
verticals
,
or
declining
,
or
meridian
planes
,
these
parallels
will
be
conicall
sections
,
and
may
be
drawne
through
their
points
of
intersection
,
with
the
azimuth
lines
,
in
the
same
maner
as
the
parallels
of
declination
,
through
their
points
of
intersection
with
the
houre-lines
.
To
this
end
you
may
first
finde
the
distance
betweene
the
top
of
the
style
and
the
azimuth
;
and
then
the
distance
betweene
the
horizon
and
the
parallell
,
both
which
may
be
represented
in
this
maner
.
On
the
center
B
and
any
semidiameter
BH
,
describe
an
occult
arke
of
a
circle
,
and
therein
inscribe
the
chords
of
such
parallels
of
altitude
as
you
intend
to
draw
on
the
plane
,
(
I
haue
here
put
them
for
15.
30.
45
and
60
gr.
)
then
draw
right
lines
through
the
center
and
the
termes
of
those
chords
,
so
the
line
BH
shall
be
the
horizon
,
and
the
rest
the
lines
of
altitude
,
according
to
their
distance
from
the
horizon
.
geometric illustration
That
done
,
consider
your
plane
(
which
here
for
example
is
the
South
face
of
our
vertical
plane
,
p.
168
)
wherein
hauing
drawne
both
the
horizontall
&
verticall
lines
,
as
I
shewed
before
,
first
take
out
AB
the
length
of
the
style
,
&
pricke
that
downe
in
this
horizontall
line
from
B
vnto
A
;
then
take
out
all
the
distances
between
B
the
top
of
the
style
and
the
seuerall
points
wherein
the
verticall
lines
do
crosse
the
horizontal
,
transfer
them
into
this
horizontal
line
BH
,
from
the
center
B
,
and
at
the
terms
of
these
distances
erect
lines
perpendicular
to
the
horizon
,
noting
them
with
the
number
or
letter
of
the
azimuth
from
whence
they
were
taken
,
so
these
perpendiculars
shall
represent
those
azimuths
,
and
the
seuerall
distances
betweene
the
horizon
and
the
lines
of
altitude
shall
giue
the
like
distances
,
betweene
the
horizontall
and
the
parallels
of
altitude
vpon
the
azimuths
in
your
plane
.
Vpon
this
ground
it
followeth
,
1
To
find
the
distance
between
the
top
of
the
style
,
and
the
seuerall
points
wherein
the
azimuths
do
crosse
the
horizontall
line
.
Hauing
drawne
the
horizontall
and
azimuth
lines
as
before
,
looke
into
the
table
by
which
you
drew
them
,
and
there
you
shall
haue
the
angles
at
the
zenith
.
Then
As
the
cosine
of
the
angle
at
the
zenith
,
is
to
the
sine
of
90
gr.
So
the
length
of
the
style
,
to
the
distance
required
.
geometric illustration
Azimuths
.
|
Ang
Ze
|
Tangent
|
Secant
|
Par.
15.
|
Par.
30.
|
Gr.
M
|
Inch
P.
|
Inch
P.
|
Inch.
P.
|
Inch.
P.
|
South
.
|
0
0
|
0
0
|
10
00
|
2
68
|
5
77
|
SbE
|
11
15
|
1
99
|
10
20
|
2
73
|
5
90
|
SSE
|
22
30
|
4
14
|
10
82
|
2
90
|
6
24
|
SEbS
|
33
45
|
6
68
|
12
03
|
3
23
|
6
94
|
SE
|
45
0
|
10
00
|
14
14
|
3
80
|
8
16
|
SEbE
|
56
15
|
14
97
|
18
00
|
4
82
|
10
40
|
ESE
|
67
30
|
24
14
|
26
13
|
7
02
|
15
08
|
EbS
|
78
45
|
50
27
|
51
26
|
13
73
|
29
60
|
East
.
|
90
0
|
Infinit
|
Infinit
|
Infinit
|
Infinit
|
As
in
our
example
of
the
verticall
plane
,
where
AB
the
length
of
the
style
was
supposed
to
be
10
inches
,
extend
the
compasses
from
the
sine
of
78
gr.
45
m.
(
the
complement
of
11
gr.
15
m.
the
angle
at
the
zenith
,
belonging
to
SbE
and
SbW
)
vnto
the
sine
of
90
gr.
the
same
extent
wil
reach
from
10.00
the
length
of
the
style
,
vnto
10.20
for
the
distance
betweene
the
top
of
the
style
and
the
intersection
of
the
azimuth
SbE
with
the
horizontall
line
,
which
distance
may
be
called
the
secant
of
the
azimuth
,
and
may
serue
for
the
drawing
of
the
parallell
of
45
gr.
from
the
horizon
.
The
like
reason
holdeth
for
the
rest
of
these
distances
here
represented
in
the
line
BH
.
2
To
finde
the
distance
betweene
the
horizon
and
the
parallels
.
As
the
tangent
of
45
gr.
to
the
tangent
of
the
parallell
:
So
the
secant
of
the
azimuth
,
to
the
distance
required
.
As
if
it
were
required
to
draw
the
parallell
of
15
gr.
from
the
horizon
,
vpon
this
verticall
plane
;
extend
the
compasses
from
the
tangent
of
45
gr.
vnto
the
tangent
of
15
gr.
the
same
extent
will
reach
in
the
line
of
numbers
from
10.
00
the
secant
of
the
South
azimuth
vnto
2.68
,
and
therefore
the
distance
betweene
the
horizon
and
the
parallell
of
15
gr.
is
2
inches
68
cent
.
vpon
the
South
azimuth
.
Againe
,
the
same
extent
will
reach
from
10.20
the
secant
of
SbE
vnto
2.73
for
the
like
distance
belonging
to
SbE
and
SbW
;
and
so
for
the
rest
,
which
may
be
gathered
and
set
downe
in
the
table
.
That
done
,
and
the
horizon
and
azimuths
being
drawne
,
pricke
downe
10
inches
from
the
horizontall
line
vpon
the
South
azimuth
,
&
10
inches
20
cent
.
on
the
azimuths
of
SbE
and
SbW
,
and
10
inches
82
cent
.
on
the
azimuths
of
SSE
and
SSW
,
and
12
inches
3
cent
.
on
the
azimuth
of
SEbS
and
SWbS
,
and
so
the
rest
of
these
distances
on
their
seuerall
azimuths
:
then
if
you
draw
a
crooked
line
through
all
these
points
,
that
may
make
no
angles
,
the
line
so
drawne
shall
be
the
parallell
of
45
gr.
from
the
horizon
.
In
like
maner
may
you
draw
the
parallell
of
15
gr.
or
any
other
parallell
of
altitude
vpon
any
verticall
plane
.
If
the
plane
incline
to
the
horizon
,
after
we
haue
found
the
verticall
point
,
and
drawne
the
horizontall
line
,
we
are
farther
to
finde
the
length
of
the
axis
of
the
horizon
,
then
the
angles
betwixt
this
axis
and
the
azimuth
lines
,
and
so
the
seuerall
distances
betweene
the
parallels
and
the
verticall
point
,
all
which
may
be
represented
in
this
manner
.
On
the
center
B
,
and
any
semidiameter
,
describe
an
occult
quadrant
of
a
circle
,
and
therein
inscribe
the
chords
of
such
parallels
of
altitude
as
you
intend
to
draw
on
the
plane
,
drawing
right
lines
through
the
center
and
the
termes
of
these
chords
,
so
the
line
BH
shall
be
the
horizon
,
and
his
perpendicular
BV
the
axis
of
the
horizon
,
and
the
rest
the
lines
of
altitude
,
according
to
their
distance
from
the
horizon
.
geometric illustration
Or
hauing
the
Sector
you
may
draw
an
occult
line
VE
perpendicular
to
the
axis
VB
,
and
therein
prick
downe
the
tangent
of
the
complement
of
the
inclination
of
the
plane
from
V
vnto
E
:
then
draw
the
line
EF
parallell
to
the
axis
,
crossing
the
line
VH
produced
in
the
point
F
,
so
this
line
EF
will
be
as
the
line
of
sines
vpon
the
Sector
,
and
therein
you
may
pricke
downe
the
sines
of
the
complement
of
the
angles
at
the
zenith
from
E
towards
F
,
and
draw
the
verticall
lines
by
those
points
through
the
lines
of
altitude
,
so
the
angles
at
V
,
betweene
the
axis
VB
and
those
azimuth
lines
,
shall
be
the
angles
betweene
the
axis
of
the
horizon
and
the
azimuth
lines
on
your
plane
,
and
the
seuerall
distances
between
the
point
V
and
the
lines
of
altitude
,
shall
giue
the
like
distances
betweene
the
verticall
point
and
the
parallels
of
altitude
vpon
the
azimuths
in
your
plane
.
Vpon
this
ground
it
followeth
,
1
To
find
the
length
of
the
axis
of
the
horizon
.
The
verticall
point
is
alwayes
either
directly
ouer
or
vnder
the
top
of
the
style
,
and
the
distance
betweene
them
is
that
which
I
call
the
axis
of
the
horizon
,
which
may
thus
be
found
,
As
the
cosine
of
the
inclination
,
to
the
sine
of
90
gr.
So
the
length
of
the
style
,
to
the
length
of
the
axis
of
the
horizon
.
For
example
in
the
first
of
the
three
declining
inclining
planes
,
the
inclination
to
the
horizon
is
36
gr.
the
length
of
the
style
AB
six
inches
,
extend
the
compasses
from
the
sine
of
54
gr.
the
complement
of
the
inclination
vnto
the
sine
of
90
gr.
the
same
extent
wil
reach
in
the
line
of
numbers
from
6.00
vnto
7.42
,
&
such
is
VB
the
length
of
the
axis
required
.
2
To
find
the
angles
contained
between
the
horizon
and
the
verticall
lines
vpon
your
plane
.
The
angles
at
the
vertical
point
betweene
the
axis
of
the
horizon
and
the
azimuth
lines
vpon
your
plane
are
represented
in
this
figure
by
those
at
V
,
betwee●e
VB
and
the
azimuths
.
The
angles
betweene
the
horizon
and
the
azimuth
lines
being
complements
to
the
former
,
are
represented
either
by
those
which
are
made
by
VE
or
by
BH
,
and
the
azimuth
lines
which
are
drawne
from
V.
geometric illustration
In
our
example
where
the
inclination
to
the
horizon
is
36
gr.
and
the
angle
at
the
zenith
betweene
the
azimuth
at
the
style
and
the
meridian
,
is
according
to
the
declination
24
gr.
20
m.
extend
the
compasses
from
the
sine
of
90
gr.
vnto
the
tangent
of
36
gr.
the
same
extent
wil
reach
from
the
sine
of
65
gr.
40
m.
the
complement
of
the
angle
at
the
zenith
,
vnto
the
tangent
of
33
gr.
30
m.
for
the
angle
contained
between
the
horizon
and
the
South
part
of
the
meridian
line
.
Again
,
the
same
extent
wil
reach
from
the
cosine
of
35
gr.
35
m.
the
angle
at
the
zenith
belonging
to
SbE
vnto
the
tangent
of
30
gr.
3
m.
for
the
angle
betweene
the
horizon
and
the
azimuth
line
of
SbE
.
The
like
reason
holdeth
for
the
rest
,
which
may
be
found
and
set
downe
in
the
Table
.
Azimuths
.
|
Ang.
Ze.
|
Ang.
V.
|
Ang.
Ho
|
Horizon
|
11
18
|
26
34
|
45
0
|
Gr.
M.
|
Gr.
M.
|
Gr.
M.
|
Inch.
P.
|
Inch.
P
|
Inch.
P.
|
Inch.
P.
|
East
.
|
114
25
|
119
12
|
16
40
|
Infinite
.
|
|
38
60
|
11
05
|
EbS
|
103
5
|
106
2
|
19
20
|
210
24
|
22
40
|
9
00
|
E
SE
|
91
50
|
92
16
|
1
20
|
41
98
|
15
57
|
7
60
|
SE
bE
|
80
35
|
78
25
|
6
47
|
62
82
|
23
44
|
12
07
|
6
68
|
SE
|
69
20
|
65
0
|
14
23
|
29
87
|
16
79
|
10
12
|
6
00
|
SE
bS
|
58
5
|
52
25
|
21
0
|
20
70
|
13
61
|
8
99
|
5
79
|
SSE
|
46
50
|
40
46
|
26
25
|
16
68
|
11
90
|
8
31
|
5
53
|
SbE
|
35
35
|
30
3
|
30
35
|
14
58
|
10
90
|
7
90
|
5
42
|
South
|
24
20
|
20
5
|
33
30
|
13
44
|
10
32
|
7
66
|
5
35
|
SbW
|
13
5
|
10
39
|
35
17
|
12
84
|
10
02
|
7
55
|
5
33
|
SSW
|
1
50
|
1
29
|
35
59
|
12
62
|
9
90
|
7
47
|
5
31
|
|
Style
.
|
0
0
|
36
0
|
12
62
|
9
90
|
7
47
|
5
31
|
SWbS
|
9
25
|
7
38
|
35
37
|
12
74
|
9
96
|
7
50
|
5
32
|
SW
|
20
40
|
16
58
|
34
12
|
13
20
|
10
20
|
7
59
|
5
34
|
SWbW
|
31
55
|
26
45
|
31
40
|
14
13
|
10
67
|
7
81
|
5
39
|
WSW
|
43
10
|
37
11
|
27
55
|
15
85
|
11
50
|
8
15
|
5
49
|
WbS
|
54
25
|
48
30
|
22
55
|
19
05
|
12
94
|
8
73
|
5
66
|
West
|
65
40
|
60
48
|
16
40
|
25
87
|
15
51
|
9
60
|
5
96
|
W
bN
|
76
55
|
73
58
|
9
20
|
45
75
|
20
64
|
11
32
|
6
46
|
W
NW
|
88
10
|
87
44
|
1
20
|
318
88
|
33
27
|
14
18
|
7
25
|
NWbW
|
99
25
|
101
35
|
6
47
|
Infinit
.
|
92
40
|
19
60
|
8
48
|
NW
|
110
40
|
115
0
|
14
23
|
|
31
44
|
10
30
|
Then
may
you
either
draw
these
angles
at
V
in
the
former
figure
more
perfectly
,
and
thence
finish
your
worke
,
or
else
proceed
.
3
To
find
the
distance
between
the
verticall
point
and
the
parallels
of
the
horizon
.
These
distances
may
be
found
by
resoluing
the
triangles
in
the
last
figure
made
by
the
axis
,
the
lines
of
altitude
,
and
the
azimuth
lines
.
For
hauing
the
length
of
the
axis
and
the
angles
at
the
horizon
,
if
you
adde
the
distance
of
the
parallel
from
the
horizon
vnto
the
angle
at
the
horizon
,
you
shall
haue
the
angle
at
the
parallell
.
Then
As
the
sine
of
the
angle
at
the
parallell
,
to
the
cosine
of
the
altitude
:
So
the
length
of
the
axis
,
to
the
distance
betweene
the
verticall
point
and
the
parallell
.
Thus
in
our
example
if
it
were
required
to
finde
the
distance
vpon
the
stylar
azimuth
VH
,
betweene
the
verticall
point
and
the
horizon
,
you
haue
the
rectangle
triangle
VBH
,
wherein
the
angle
at
the
horizon
here
represented
by
BHV
is
(
equall
to
the
inclination
of
the
plane
)
36
gr.
and
BV
the
axis
of
the
horizon
betweene
the
plane
and
the
top
of
the
style
,
is
7
inches
42
cent
.
Wherefore
extend
the
compasses
from
the
sine
of
36
gr.
vnto
the
sine
of
90
gr.
the
complement
of
the
altitude
,
the
same
extent
will
reach
in
the
line
of
numbers
from
7.42
vnto
12
62
,
and
such
is
the
distance
of
the
pe●pendicular
azimuth
line
VH
between
the
verticall
point
and
the
horizon
.
In
like
maner
if
you
would
find
the
distance
vpon
the
meridian
between
the
verticall
point
and
the
horizon
,
extend
the
compasses
from
the
sine
of
33
gr.
30
m.
the
angle
at
the
horizon
,
to
the
sine
of
90
gr.
the
same
extent
will
reach
in
the
line
of
numbers
from
7.42
vnto
13.44
,
and
such
is
Va
the
distance
betweene
the
verticall
point
and
the
horizon
vpon
the
line
of
the
South
a●imuth
,
that
is
,
vpon
the
meridian
line
.
But
if
you
would
find
the
distance
vpon
the
meridian
betweene
the
verticall
point
and
any
other
parallell
of
the
horizon
,
as
vpon
the
parallell
of
26
gr.
34
m.
then
adde
these
26
gr.
34
m.
vnto
33
gr.
30
m.
the
angle
at
the
horizon
,
so
shall
you
haue
60
gr.
4
m.
for
BDV
the
angle
at
the
parallell
.
And
if
you
extend
the
compasses
from
the
sine
of
60
gr.
4
m.
vnto
the
sine
of
63
gr.
26
m.
the
complement
of
the
parallell
from
the
horizon
,
the
same
extent
will
reach
in
the
line
of
numbers
from
7.42
the
length
of
the
axis
,
vnto
7.66
,
and
such
is
the
distance
VD
between
the
verticall
point
and
the
parallel
of
26
gr.
34
m.
vpon
the
meridian
line
.
The
like
reason
holdeth
for
all
the
rest
,
which
may
be
gathered
&
set
down
in
the
table
.
That
done
,
and
the
horizon
drawne
as
before
,
if
you
would
draw
the
parallel
of
26
gr.
34
m.
from
the
horizon
,
looke
into
the
table
,
and
there
finding
vnder
the
title
of
the
parallel
of
26.34
,
the
distance
on
the
South
azimuth
line
to
be
7.66
,
take
7
inches
66
cent
.
out
of
a
line
of
inches
,
and
prick
them
down
on
the
meridian
of
your
plane
,
from
the
verticall
point
at
V.
Or
if
either
the
verticall
point
fall
without
your
plane
,
or
the
extent
at
any
time
be
too
large
for
your
compasses
,
you
may
pricke
downe
the
distance
betweene
the
horizon
and
the
parallel
.
As
here
the
distance
betweene
the
vertical
point
and
the
parallell
is
7.66
,
betweene
the
verticall
point
and
the
horizon
13.44
,
the
difference
between
them
5.78
is
the
distance
from
the
horizon
to
the
parallel
,
which
being
pricked
downe
vpon
the
meridian
,
shall
giue
the
same
intersection
as
before
.
And
the
like
reason
,
holdeth
for
the
pricking
down
the
rest
of
these
distances
on
their
seuerall
azimuths
.
Hauing
the
points
of
intersection
betweene
the
azimuths
and
the
parallell
,
you
may
ioyne
them
all
in
a
crooked
line
without
making
of
angles
,
the
line
so
drawne
shall
be
the
parallell
required
.
And
vpon
this
ground
it
followeth
,
To
describe
such
parallels
on
the
former
planes
,
as
may
shew
the
proportion
of
the
shadow
vnto
the
gnomon
.
The
proportion
of
a
mans
shadow
vnto
his
height
,
or
other
shadow
to
his
gnomon
set
perpendicular
to
the
horizon
,
may
be
shewed
by
parallels
to
the
horizon
,
if
they
be
drawne
to
a
due
altitude
,
which
may
thus
be
found
:
As
the
length
of
the
shadow
,
to
the
length
of
the
gnomon
:
So
the
tangent
of
45
gr.
to
the
tangent
of
the
altitude
.
As
if
it
were
required
to
finde
the
altitude
of
the
Sunne
when
the
shadow
of
a
man
shall
be
decuple
to
his
height
,
extend
the
compasses
from
10
vnto
1
in
the
line
of
numbers
,
the
same
extent
will
reach
in
the
tangent
of
45
gr.
vnto
the
tangent
of
5
gr.
42
m
;
which
shewes
that
when
the
Sun
commeth
to
the
altitude
of
5
gr.
42
m.
your
shadow
,
vpon
a
leuell
ground
,
will
be
ten
times
as
much
as
your
height
.
In
the
same
maner
you
may
finde
that
at
7
gr.
7
m.
of
altitude
your
shadow
wil
be
octuple
,
at
9
gr.
27
m.
sextuple
,
at
11
gr.
18
m.
quintuple
,
at
14
gr.
2
m.
quadruple
,
at
18
gr.
26
m.
triple
,
at
26
gr.
34
m.
double
to
your
height
,
at
33
gr.
41
m.
as
3
vnto
2
,
at
36
gr.
52
m.
as
4
vnto
3
,
at
38
gr.
40
m.
as
5
vnto
4
,
at
45
gr.
equall
,
at
51
gr.
20
m.
as
4
vnto
5
,
at
53
gr.
7
m.
as
3
vnto
4
,
at
56
gr.
19
m.
as
2
vnto
3
,
at
59
gr.
2
m.
as
3
vnto
5
,
at
63
gr.
26
m.
as
1
vnto
2
,
&c.
If
then
you
draw
a
parallell
to
the
horizon
at
5
gr.
42
m.
another
at
7
gr.
7
m.
and
so
the
rest
,
when
●he
shadow
o●
the
style
falleth
on
the
parallell
,
you
haue
the
proportion
,
and
thereby
may
you
know
the
shadow
by
the
height
,
and
the
height
by
the
shadow
,
whereof
you
haue
examples
Pag.
126.
and
137.
I
might
here
proceed
to
shew
the
description
of
th●●●●cles
of
position
,
the
Signes
of
the
Zodiacke
in
the
merid●●n
,
the
Signes
ascending
and
descending
,
with
such
o●●er
gnomonicall
conclusions
;
but
these
would
proue
superfluous
to
such
as
vnderstand
the
doctrine
of
the
Sphere
;
and
for
others
,
that
which
is
deliuered
may
suffise
for
ordinary
vse
,
it
being
my
intention
not
so
much
to
explane
the
full
vse
of
shadowes
(
whereof
I
haue
lately
giuen
a
large
example
in
an
other
place
)
as
the
vse
of
these
lines
of
proportion
,
that
were
not
extant
heretofore
.
An
Appendix
concerning
The
description
and
vse
of
a
small
portable
Quadrant
,
for
the
more
easie
finding
of
the
houre
and
Azimuth
.
CHAP.
I.
Of
the
description
of
the
Quadrant
.
HAuing
described
these
standing
planes
,
I
will
now
shew
the
most
of
these
conclusions
by
a
small
Quadrant
.
This
might
be
done
generally
for
all
latitudes
,
by
a
quarter
of
the
generall
Astrolabe
,
described
before
in
the
vse
of
the
Sector
,
pag.
58
;
and
particularly
for
any
one
latitude
,
by
a
quarter
of
the
particular
Astrolabe
,
there
also
described
,
pag.
63.
which
if
it
be
a
foote
semidiameter
,
may
shew
the
azimuth
vnto
a
degree
,
and
the
time
of
the
day
vnto
a
minute
;
but
for
ordinary
vse
this
smaller
Quadrant
may
suffice
,
which
may
be
made
portable
in
this
maner
.
1
Vpon
the
center
A
,
and
semidiameter
AB
,
describe
the
arke
BC
:
the
same
semidiameter
will
set
of
60
gr.
and
the
halfe
of
that
will
be
30
gr.
which
being
added
to
the
former
60
gr.
will
make
the
arke
BC
to
be
90
gr.
the
fourth
part
of
the
whole
circle
,
and
thence
comes
the
name
of
a
Quadrant
.
2
Leauing
some
little
space
for
the
inscription
of
the
moneths
and
dayes
,
on
the
same
center
A
,
and
semidiameter
AT
,
describe
the
arke
TD
,
which
shall
serue
for
either
tropique
.
3
Diuide
the
line
AT
in
the
point
E
,
in
such
proportion
,
as
that
AT
being
10000
,
AE
may
be
6556
,
and
there
draw
another
arke
EF
,
which
shall
serue
for
the
Equator
.
geometric illustration
5
This
part
of
the
ecliptique
may
be
diuided
into
three
Signes
,
and
each
Signe
into
30
A
Table
of
right
Ascensions
.
Gr.
|
♈
|
♉
|
♊
|
Gr.
M.
|
Gr.
M.
|
Gr.
M.
|
0
|
0
0
|
27
54
|
57
48
|
5
|
4
35
|
32
42
|
63
3
|
10
|
9
11
|
37
35
|
68
21
|
15
|
13
48
|
42
31
|
73
43
|
20
|
18
27
|
47
33
|
79
7
|
25
|
23
9
|
52
38
|
84
32
|
30
|
27
54
|
57
48
|
90
0
|
gr.
by
a
table
of
right
ascensions
,
made
as
before
pag.
60.
As
the
right
ascension
of
the
first
point
of
♉
being
27
gr.
54
m.
you
may
lay
a
ruler
to
the
center
A
&
27
gr.
54
m.
in
the
Quadrant
BC
,
the
point
where
the
ruler
crosseth
the
Ecliptique
,
shall
be
the
first
point
of
♉
.
In
like
maner
the
right
ascension
of
the
first
point
of
♊
being
57
gr.
48
m.
if
you
lay
a
ruler
to
the
Gr.
|
Parts
.
|
1
|
176
|
2
|
355
|
3
|
537
|
4
|
723
|
5
|
913
|
6
|
1106
|
7
|
1302
|
8
|
1503
|
9
|
1708
|
10
|
1917
|
11
|
2130
|
12
|
2348
|
13
|
2571
|
14
|
2799
|
15
|
3032
|
16
|
3270
|
17
|
3514
|
18
|
3763
|
19
|
4019
|
20
|
4281
|
21
|
4550
|
22
|
4825
|
23
|
5108
|
Tro
|
5252
|
center
A
,
and
57
gr.
48
m.
in
the
quadrant
,
the
point
where
the
ruler
crosseth
the
ecliptique
,
shal
be
the
first
point
of
♊
.
And
so
for
the
rest
:
but
the
lines
of
distinction
between
Signe
&
Signe
,
may
be
best
drawne
from
the
center
G.
6
The
line
ET
betweene
the
equator
and
the
tropique
,
which
I
call
the
line
of
declination
,
may
be
diuided
into
23
gr.
½
out
of
this
Table
.
For
let
AE
the
semidiameter
of
the
equator
be
10000
,
the
distance
betweene
the
equator
and
10
gr.
of
declination
may
be
1917
more
;
between
the
equator
and
20
gr.
4281
;
the
distance
of
the
tropique
from
the
equator
5252.
7
You
may
put
in
the
most
of
the
principall
starres
betweene
the
equator
and
the
tropique
of
♋
,
by
their
declination
from
the
equator
,
and
right
ascention
from
the
next
equinoctial
point
.
As
the
declination
of
the
wing
of
Pegasus
,
being
13
gr.
7
m.
the
right
ascension
358
gr.
34
m.
from
the
first
point
of
♈
,
or
1
gr.
26
m.
short
of
it
.
If
you
draw
an
occult
parallell
through
13
gr.
7
m.
of
declination
,
and
then
lay
the
ruler
to
the
center
A
,
and
1
gr.
26
m.
in
the
quadrant
BC
,
the
point
where
the
ruler
crosseth
the
parallell
shall
be
the
place
for
the
wing
of
Pegasus
,
to
which
you
may
set
the
name
and
the
time
when
he
cometh
to
the
South
,
in
this
maner
,
W.
Peg.
*
23
Ho.
54
M.
and
so
for
the
rest
of
these
fiue
,
or
any
other
starres
.
|
Ho.
M.
|
R.
Ascen
|
Decl.
M
|
Pegasus
wing
*
|
23
54
|
1
26
|
13
7
|
Arcturus
*
|
13
58
|
29
37
|
21
10
|
Lions
heart
*
|
9
48
|
32
58
|
13
45
|
Buls
eye
*
|
4
15
|
63
33
|
15
42
|
Vultures
heart
*
|
19
33
|
66
56
|
7
58
|
8
There
being
space
sufficient
between
the
equator
and
the
center
,
you
may
there
describe
the
quadrat
,
and
diuide
each
of
the
two
sides
farthest
from
the
center
A
into
100
parts
,
so
shall
the
Quadrant
be
prepared
generally
for
any
latitude
.
But
before
you
draw
the
particular
lines
,
you
are
to
fit
foure
tables
vnto
your
latitude
.
First
a
table
of
meridian
altitudes
for
diuision
of
the
circle
of
dayes
and
moneths
,
which
may
be
thus
made
:
Consider
the
latitude
of
the
place
and
the
declination
of
the
Sun
for
each
day
of
the
yeare
.
If
the
latitude
and
declination
be
alike
both
North
or
both
South
,
ad
the
declination
to
the
complement
of
the
latitude
;
if
they
be
vnlike
,
one
North
,
and
the
other
South
,
substract
the
declination
from
the
complement
of
the
latitude
,
the
remainder
will
be
the
meridian
altitude
belonging
vnto
the
day
.
Thus
in
our
latitude
of
51
gr.
30
m.
Northward
,
whose
complement
is
38
gr.
30
m.
the
declination
vpon
the
tenth
day
of
Iune
will
be
23
gr.
30
m.
Northward
,
wherefore
I
adde
23
gr.
30
m.
vnto
38
gr.
30
m.
the
summe
of
both
is
62
gr.
for
the
meridian
altitude
at
the
tenth
of
Iune
.
The
declination
vpon
the
tenth
of
December
will
be
23
gr.
30
m.
Southward
,
wherefore
I
take
these
23
gr.
30
m.
out
of
38
gr.
30
m.
there
wil
remain
15
gr.
for
the
meridian
altitude
at
the
tenth
of
December
;
and
in
this
maner
you
may
find
the
meridian
altitude
for
each
day
of
the
yeare
,
and
set
them
downe
in
a
table
.
Dies
.
|
0
|
5
|
10
|
15
|
20
|
25
|
30
|
Mo
|
Gr.
M
|
Gr.
M.
|
Gr.
M
|
Gr.
M.
|
Gr.
M.
|
Gr.
M
|
Gr.
M.
|
Ianuary
|
16
31
|
17
24
|
18
26
|
19
37
|
20
57
|
22
24
|
23
58
|
February
|
24
17
|
25
59
|
27
45
|
29
35
|
31
29
|
33
25
|
|
March
|
34
35
|
36
33
|
38
32
|
40
30
|
42
27
|
44
22
|
46
15
|
April
|
46
37
|
48
26
|
50
11
|
51
50
|
53
25
|
54
53
|
56
15
|
May
|
56
15
|
57
29
|
58
35
|
59
33
|
60
22
|
61
2
|
61
31
|
Iune
|
61
36
|
61
54
|
62
0
|
61
58
|
61
45
|
61
22
|
60
49
|
Iuly
|
60
49
|
60
6
|
59
14
|
58
13
|
57
4
|
55
48
|
54
24
|
August
|
54
7
|
52
36
|
50
59
|
49
17
|
47
31
|
45
41
|
43
49
|
September
|
43
26
|
41
30
|
39
33
|
37
36
|
35
38
|
33
41
|
31
46
|
October
|
31
46
|
29
53
|
28
3
|
26
16
|
24
35
|
22
59
|
21
29
|
Nouember
|
21
12
|
19
51
|
18
39
|
17
36
|
16
43
|
16
0
|
15
28
|
December
|
15
28
|
15
7
|
15
0
|
15
2
|
15
17
|
15
44
|
16
22
|
The
Table
being
made
,
you
may
inscribe
the
moneths
,
and
dayes
of
each
moneth
into
your
quadrant
,
in
the
space
left
below
the
tropique
.
For
lay
the
ruler
vnto
the
center
A
,
and
16
gr.
31
m.
in
the
quadrant
BC
,
there
may
you
draw
a
line
for
the
end
of
December
and
beginning
of
Ianuary
;
then
laying
your
ruler
to
the
center
A
,
and
24
gr.
17
m.
in
the
quadrant
,
there
draw
the
end
of
Ianuary
and
beginning
of
February
,
and
so
the
rest
,
which
may
be
noted
with
I
,
F
,
M
,
A
,
M
,
I
,
&c.
the
first
letters
of
each
moneth
,
and
wil
here
fall
betweene
15
gr.
and
62
gr.
The
second
Table
which
you
are
to
fit
,
may
serue
for
the
drawing
and
diuiding
of
the
horizon
.
For
drawing
of
the
horizon
.
As
the
cotangent
of
the
latitude
,
to
the
tangent
of
the
greatest
declination
:
So
the
sine
of
90
gr.
to
the
sine
of
intersection
,
where
the
horizon
shall
crosse
the
tropiques
.
So
in
our
latitude
of
51
gr.
30
m.
we
shall
find
the
horizon
to
cut
the
tropique
in
33
gr.
9
m
:
wherefore
if
you
lay
the
ruler
to
the
center
A
,
and
33
gr.
9
m.
in
the
quadrant
,
the
point
where
the
ruler
crosseth
the
tropique
shall
be
the
point
where
the
horizon
crosseth
the
tropique
.
And
if
you
finde
a
point
at
H
,
in
the
line
AC
,
whereon
setting
the
compasses
,
you
may
bring
the
point
at
E
,
and
this
point
in
the
tropique
both
into
a
circle
,
the
point
H
shall
be
the
center
,
and
the
ark
so
drawne
shall
be
the
horizon
.
Then
for
the
diuision
of
this
horizon
.
As
the
sine
of
90
gr.
to
the
sine
of
the
latitude
:
So
the
tangent
of
the
horizon
,
to
the
tangent
of
the
arke
in
the
quadrant
,
which
shall
diuide
the
horizon
.
So
in
our
latitude
of
51
gr.
30
m.
we
shall
finde
7
gr.
52
m.
belonging
to
10
gr.
in
the
horizon
,
and
15
gr.
54
m.
belonging
to
20
gr.
And
so
the
rest
,
as
in
this
Table
.
Ho
|
Gr.
M
|
Ho
|
Gr.
M
|
Ho
|
Gr.
M.
|
Ho
|
Gr.
M.
|
Ho
|
Gr.
M.
|
Ho
|
Gr.
M.
|
0
|
0
0
|
15
|
11
51
|
30
|
24
19
|
45
|
38
2
|
60
|
53
35
|
75
|
71
5
|
|
0
47
|
|
12
39
|
|
25
11
|
|
39
1
|
|
54
41
|
|
72
19
|
|
1
34
|
|
13
27
|
|
26
4
|
|
40
0
|
|
55
48
|
|
73
33
|
|
2
21
|
|
14
16
|
|
26
57
|
|
41
0
|
|
56
56
|
|
74
48
|
|
3
8
|
|
15
4
|
|
27
50
|
|
42
0
|
|
58
4
|
|
76
3
|
5
|
3
55
|
20
|
15
54
|
35
|
28
43
|
50
|
43
0
|
65
|
59
13
|
80
|
77
18
|
|
4
42
|
|
16
43
|
|
29
37
|
|
44
1
|
|
60
22
|
|
78
33
|
|
5
29
|
|
17
33
|
|
30
32
|
|
45
3
|
|
61
31
|
|
79
49
|
|
6
17
|
|
18
22
|
|
31
27
|
|
46
5
|
|
62
41
|
|
81
5
|
|
7
4
|
|
19
12
|
|
32
22
|
|
47
8
|
|
62
52
|
|
82
21
|
10
|
7
52
|
25
|
20
2
|
40
|
33
18
|
55
|
48
11
|
70
|
65
3
|
85
|
83
37
|
|
8
39
|
|
20
53
|
|
34
14
|
|
49
14
|
|
66
15
|
|
84
53
|
|
9
27
|
|
21
44
|
|
35
10
|
|
50
19
|
|
67
27
|
|
86
10
|
|
10
14
|
|
22
36
|
|
36
7
|
|
51
24
|
|
68
39
|
|
87
26
|
|
11
2
|
|
23
27
|
|
37
4
|
|
52
29
|
|
69
52
|
|
88
43
|
15
|
11
51
|
30
|
24
19
|
45
|
38
2
|
60
|
53
35
|
75
|
71
5
|
90
|
90
0
|
Wherefore
you
may
lay
the
ruler
to
the
center
A
,
and
7
gr.
52
m.
in
the
quadrant
BC
,
the
point
where
the
ruler
crosseth
the
horizon
shall
be
10
gr.
in
the
horizon
;
and
so
for
the
rest
:
but
the
lines
of
distinction
between
each
fift
degree
,
will
be
best
drawne
from
the
center
H.
The
third
Table
for
drawing
of
the
houre-lines
,
must
be
a
Table
of
the
altitude
of
the
Sunne
aboue
the
horizon
at
euery
houre
,
especially
when
he
cometh
to
the
equator
,
the
tropiques
,
and
some
other
intermediate
declinations
.
If
the
Sunne
be
in
the
equator
,
and
so
haue
no
declination
.
As
the
sine
of
90
gr.
to
the
cosine
of
the
latitude
:
So
the
cosine
of
the
houre
from
the
meridian
,
to
the
sine
of
the
altitude
.
Thus
in
our
latitude
of
51
gr.
30
m.
at
six
houres
from
the
meridian
the
Sunne
wil
haue
no
altitude
,
at
fiue
the
altitude
will
be
9
gr.
17
m
;
at
foure
18
gr.
8
m
;
at
three
26
gr.
7
m
;
at
two
32
gr.
37
m.
at
one
36
gr.
58
m
;
at
noone
it
will
be
38
gr.
30
m.
equall
to
the
complement
of
the
latitude
.
If
the
Sunne
haue
declination
,
the
meridian
altitude
wil
be
found
as
before
,
for
the
Table
of
dayes
and
moneths
.
If
the
houre
proposed
be
six
in
the
morning
or
six
at
night
.
As
the
sine
of
90
gr.
to
the
sine
of
the
latitude
:
So
the
sine
of
the
declination
,
to
the
sine
of
the
altitude
.
Thus
in
our
latitude
the
declination
of
the
Sunne
being
23
gr.
30
m.
the
altitude
will
be
found
to
be
18
gr.
11
m
:
the
declination
being
11
gr.
30
m.
the
altitude
will
be
9
gr.
If
the
houre
proposed
be
neither
twelue
nor
six
.
As
the
cosine
of
the
houre
from
the
meridian
,
to
the
sine
of
90
gr.
So
the
tangent
of
the
latitude
,
to
the
tangent
of
a
fourth
arke
.
So
in
our
latitude
and
one
houre
from
the
meridian
,
this
fourth
ark
will
be
found
to
be
52
gr.
28
m.
at
two
55
gr.
26
m.
at
three
60
gr.
39
m.
at
foure
68
gr.
22
m.
and
at
fiue
houres
from
the
meridian
78
gr.
22
m.
Then
consider
the
declination
of
the
Sunne
and
the
houre
proposed
;
if
the
latitude
and
declination
be
both
alike
,
as
with
vs
in
North
latitude
,
North
declination
,
and
the
houre
fall
between
noone
and
six
,
take
the
declination
out
of
the
fourth
arke
,
the
remainer
shall
be
your
fift
arke
.
But
if
either
the
houre
fall
between
six
and
midnight
,
or
the
latitude
and
declination
shall
be
vnlike
,
adde
the
declination
vnto
the
fourth
arke
,
and
the
summe
of
both
shall
be
your
fifth
arke
:
or
if
the
summe
shall
exceed
90
gr.
you
may
take
the
complement
vnto
180
gr.
This
fifth
arke
being
knowne
:
As
the
sine
of
the
fourth
arke
,
to
the
sine
of
the
latitude
:
So
the
cosine
of
the
fift
arke
,
to
the
sine
of
the
altitude
.
Thus
in
our
latitude
of
51
gr.
30
m.
Northward
,
the
Sunne
hauing
23
gr.
30
m.
of
North
declination
,
if
it
shall
be
required
to
finde
the
altitude
of
the
Sunne
for
seuen
in
the
morning
;
here
because
the
latitude
and
declination
are
both
alike
to
the
Northward
,
and
the
houre
proposed
falleth
betweene
noone
and
six
,
you
may
take
23
gr.
30
m.
the
arke
of
the
declination
out
of
78
gr.
22
m.
the
fourth
arke
belonging
to
the
fift
houre
from
the
meridian
,
so
there
will
remaine
54
gr.
52
m.
for
your
fift
arke
.
Then
working
according
to
the
Canon
,
you
shall
find
,
As
the
sine
of
78
gr.
22
m.
your
fourth
arke
,
to
the
sine
of
51
gr.
30
m.
for
the
latitude
:
So
the
sine
of
35
gr.
8
m.
the
complement
of
your
fift
ark
,
to
the
sine
of
27
gr.
17
m.
the
altitude
required
.
If
in
the
same
latitude
and
declination
,
it
were
required
to
finde
the
altitude
for
fiue
in
the
morning
,
here
the
houre
falling
betweene
six
and
midnight
;
if
you
adde
23
gr.
30
m.
vnto
78
gr.
22
m.
the
summe
will
be
101
gr.
52
m.
and
the
complement
to
180
gr.
will
be
78
gr.
8
m.
for
your
fifth
arke
.
Wherefore
As
the
sine
of
78
gr.
22
m.
to
the
sine
of
51
gr.
30
m.
So
the
cosine
of
78
gr.
8
m.
to
the
sine
of
9
gr.
32
m.
for
the
altitude
required
.
If
in
the
same
latitude
of
51
gr.
30
m.
Northward
,
the
Sunne
hauing
23
gr.
30
m.
of
South
declination
,
it
were
required
the
altitude
for
nine
in
the
morning
;
here
because
the
latitude
and
declination
are
vnlike
,
the
one
North
,
and
the
other
South
,
you
may
adde
23
gr.
30
m.
the
arke
of
declination
,
vnto
60
gr.
39
m.
the
fourth
arke
belonging
to
the
third
houre
from
the
meridian
,
so
shall
you
haue
84
gr.
9
m.
for
your
fift
arke
.
Wherefore
As
the
sine
of
60
gr.
39
m.
to
the
sine
of
51
gr.
30
m.
So
the
cosine
of
84
gr.
9
m.
to
the
sine
of
5
gr.
15
m.
for
the
altitude
required
.
And
so
by
one
or
other
of
these
meanes
you
may
finde
the
altitude
of
the
Sunne
for
any
point
of
the
ecliptique
at
all
houres
of
the
day
,
and
set
them
downe
in
such
a
Table
as
this
.
A
Table
for
the
altitude
of
the
Sunne
in
the
beginning
of
each
Signe
at
all
houres
of
the
day
,
calculated
for
51
gr.
30
m.
of
North
latitude
.
Ho.
|
♋
|
♊
♌
|
♉
♍
|
♈
♎
|
♓
♏
|
♒
♐
|
♑
|
Gr.
M.
|
Gr.
M.
|
Gr.
M.
|
Gr.
M.
|
Gr.
M.
|
Gr.
M.
|
Gr.
M.
|
12
|
62
0
|
58
42
|
50
0
|
38
30
|
27
0
|
18
18
|
15
0
|
11
1
|
59
43
|
56
34
|
48
12
|
36
58
|
25
40
|
17
6
|
13
52
|
10
2
|
53
45
|
50
55
|
43
12
|
32
37
|
21
51
|
13
38
|
10
30
|
9
3
|
45
42
|
43
6
|
36
0
|
26
7
|
15
58
|
8
12
|
5
15
|
8
4
|
36
41
|
34
13
|
27
31
|
18
8
|
8
33
|
1
15
|
|
7
5
|
27
17
|
24
56
|
18
18
|
9
17
|
0
6
|
|
|
6
6
|
18
11
|
15
40
|
9
0
|
0
0
|
|
|
|
5
7
|
9
32
|
6
50
|
|
|
|
|
11
37
|
4
8
|
1
32
|
|
|
|
|
|
21
40
|
Lastly
,
you
may
finde
what
declination
the
Sunne
hath
when
he
riseth
or
setteth
at
any
houre
.
As
the
sine
of
90
gr.
to
the
sine
of
the
houre
from
six
:
So
the
cotangent
of
the
latitude
,
to
the
tangent
of
the
declination
.
And
so
in
the
latitude
of
51
gr.
30
m.
you
shall
finde
that
when
the
Sun
riseth
,
either
at
fiue
in
the
Summer
,
or
seuen
in
the
Winter
,
his
declination
is
11
gr.
37
m
:
when
he
riseth
at
foure
in
the
Summer
,
or
eight
in
the
Winter
,
his
declination
is
21
gr.
40
m.
which
may
be
also
set
downe
in
the
Table
.
That
done
,
you
may
there
see
that
in
this
latitude
the
meridian
altitude
of
the
Sunne
in
the
beginning
of
♋
is
62
gr.
in
♊
58
gr.
42
m.
in
♉
50
gr.
in
♈
38
gr.
30
m.
&c.
But
the
beginning
of
♋
and
♑
is
represented
by
the
tropiques
TD
,
drawne
at
23
gr.
30
m.
of
declination
,
and
the
beginning
of
♈
and
♎
,
by
the
equator
EF.
If
you
draw
an
occult
parallell
betweene
the
equator
and
the
tropique
,
at
11
gr.
30
m.
of
declination
,
it
shall
represent
the
beginning
of
♉
,
♍
,
♏
,
and
♓
;
if
you
draw
another
occult
parallell
though
20
gr.
12
m.
of
declination
,
it
shall
represent
the
beginning
of
♊
,
♌
,
♐
,
and
♒
.
Then
you
may
lay
a
ruler
to
the
center
A
,
and
62
gr.
in
the
quadrant
BC
,
and
note
the
point
where
it
crosseth
the
tropique
of
♋
then
moue
the
ruler
to
58
gr.
52
m.
and
note
where
it
crosseth
the
parallell
of
♊
;
then
to
50
gr.
and
note
where
it
crosseth
the
parallell
of
♉
;
and
againe
to
38
gr.
30
m.
noting
where
it
crosseth
the
equator
;
so
the
line
drawne
through
these
points
shall
shew
the
houre
of
12
in
the
Summer
,
while
the
Sunne
is
in
♈
,
♉
,
♊
,
♋
,
♌
,
or
♍
.
In
like
maner
if
you
lay
the
ruler
to
the
center
A
,
and
27
gr.
in
the
quadrant
,
and
note
the
point
where
it
crosseth
the
parallel
of
♓
,
then
moue
it
to
18
gr.
18
m.
and
note
where
it
crosseth
the
parallell
of
♒
;
and
againe
to
15
gr.
noting
where
it
crosseth
the
tropique
of
♑
;
the
line
drawne
through
these
points
shall
shew
the
houre
of
12
in
the
Winter
,
while
the
Sunne
is
in
♎
,
♏
,
♐
,
♑
,
♒
&
♓
,
and
so
may
you
draw
the
rest
of
these
houre-lines
:
onely
that
of
7
from
the
meridian
in
the
Summer
,
and
5
in
the
Winter
,
will
crosse
the
line
of
declination
at
11
gr.
37
m.
and
that
of
8
in
the
Summer
,
and
4
in
the
Winter
at
21
gr.
40
m.
The
fourth
table
for
drawing
of
the
azimuth
lines
,
must
likewise
be
fitted
for
the
altitude
of
the
Sunne
aboue
the
horizon
at
euery
azimuth
,
especially
when
he
cometh
to
the
equator
,
the
tropiques
,
and
some
other
intermediat
declinations
.
If
the
Sunne
be
in
the
equator
,
and
so
haue
no
declination
:
As
the
sine
of
90
gr.
to
the
cosine
of
the
azimuth
from
the
meridian
:
So
the
cotangent
of
the
latitude
,
to
the
tangent
of
the
altitude
at
the
equator
.
Thus
in
our
latitude
of
51
gr.
30
m.
at
90
gr.
from
the
meridian
,
the
Sunne
will
haue
no
altitude
;
at
80
gr.
the
altitude
will
be
7
gr.
52
m
;
at
70
gr.
it
will
be
15
gr.
30
m
;
at
60
gr.
it
will
be
21
gr.
41
m.
If
the
Sunne
haue
declination
,
the
meridian
altitude
will
be
easily
found
as
before
,
for
the
table
for
dayes
and
moneths
.
And
for
all
other
azimuths
.
As
the
sine
of
the
latitude
,
to
the
sine
of
the
declination
:
So
the
cosine
of
the
altitude
at
the
equator
,
to
the
sine
of
a
fourth
arke
.
When
the
latitude
and
declination
are
both
alike
in
all
azimuths
from
the
prime
verticall
vnto
the
meridian
,
adde
this
fourth
arke
vnto
the
arke
of
altitude
at
the
equator
.
When
the
latitude
and
declination
are
both
alike
,
and
the
azimuth
more
then
90
gr.
distant
from
the
meridian
,
take
the
altitude
at
the
equator
out
of
this
fourth
arke
.
When
the
latitude
and
declination
are
vnlike
,
take
this
fourth
arke
out
of
the
arke
of
altitude
at
the
equator
,
so
shall
you
haue
the
altitude
of
the
Sunne
belonging
to
the
azimuth
.
Thus
in
our
latitude
of
51
gr.
30
m.
Northward
,
if
it
were
required
to
finde
the
altitude
of
the
Sunne
in
the
azimuth
of
60
gr.
from
the
meridian
,
when
the
declination
is
23
gr.
30
m.
Nothward
,
you
may
finde
the
altitude
at
the
equator
belonging
to
this
azimuth
to
be
21
gr.
41
m.
by
the
former
Canon
,
and
by
this
last
Canon
you
may
finde
the
fourth
arke
to
be
28
gr.
15
m.
Then
because
the
latitude
and
declination
are
both
alike
to
the
Northward
,
if
you
adde
them
both
together
,
you
shall
haue
49
gr.
56
m.
for
the
altitude
required
.
If
the
declination
had
been
23
gr.
30
m.
to
the
Southward
,
you
should
then
haue
taken
this
fourth
ark
out
of
the
ark
at
the
equator
,
which
because
it
cannot
here
be
done
,
it
is
a
signe
that
the
Sunne
is
not
then
aboue
the
horizon
.
But
if
you
take
the
arke
at
the
equator
out
of
this
fourth
arke
,
you
shall
haue
6
gr.
34
m.
for
the
altitude
of
the
Sunne
when
he
is
in
the
azimuth
of
60
gr.
from
the
North
,
and
120
gr.
from
the
South
part
of
the
meridian
.
The
like
reason
holdeth
for
the
rest
of
these
altitudes
,
which
may
be
gathered
and
set
down
in
a
table
.
Lastly
when
the
Sun
riseth
or
setteth
vpon
any
azimuth
,
to
find
his
declination
.
As
the
sine
of
90
gr.
to
the
cosine
of
the
latitude
:
So
the
cosine
of
the
azimuth
from
the
meridian
,
to
the
sine
of
the
declination
.
And
thus
in
our
latitude
of
51
gr.
30
m.
when
the
azimuth
is
80
gr.
from
the
meridian
,
the
declination
will
be
found
to
be
6
gr.
12
m
;
if
the
azimuth
be
70
gr.
the
declination
will
be
found
12
gr.
18
m
;
if
60
gr.
then
18
gr.
8
m.
And
so
for
the
rest
,
which
may
be
also
set
downe
in
the
Table
.
A
Table
for
the
altitude
of
the
Sunne
in
the
beginning
of
each
Signe
for
euery
tenth
azimuth
,
in
51
gr.
30
m.
of
North
latitude
.
Az.
|
♋
|
♊
♌
|
♉
♍
|
♈
♎
|
♓
♏
|
♒
♐
|
♑
|
Gr.
M
|
Gr.
M
|
Gr.
M.
|
Gr.
M
|
Gr.
M
|
Gr.
M.
|
Gr.
M.
|
0
|
62
0
|
58
42
|
50
0
|
38
30
|
27
0
|
18
18
|
15
0
|
10
|
61
43
|
58
24
|
49
38
|
38
4
|
26
30
|
17
45
|
14
25
|
20
|
60
51
|
57
28
|
48
33
|
36
46
|
25
0
|
16
5
|
12
41
|
30
|
59
52
|
55
52
|
46
40
|
34
34
|
22
27
|
13
15
|
9
45
|
40
|
57
10
|
53
29
|
43
55
|
31
21
|
18
48
|
9
14
|
6
34
|
50
|
54
3
|
50
12
|
40
11
|
27
5
|
13
58
|
3
57
|
0
6
|
60
|
49
56
|
45
53
|
35
23
|
21
41
|
8
0
|
|
|
70
|
44
40
|
40
25
|
29
27
|
15
13
|
1
0
|
|
|
80
|
38
11
|
33
46
|
21
29
|
7
52
|
|
|
|
90
|
30
38
|
26
10
|
14
45
|
0
0
|
|
|
|
100
|
22
27
|
18
2
|
6
45
|
|
|
|
6
12
|
110
|
14
14
|
9
58
|
|
|
|
|
12
18
|
120
|
6
34
|
2
30
|
|
|
|
|
18
8
|
That
done
,
if
you
would
draw
the
line
of
East
or
West
,
which
is
90
gr.
from
the
meridian
,
lay
the
ruler
to
the
center
A
,
and
30
gr.
38
m.
numbred
in
the
quadrant
from
C
toward
B
,
and
note
the
point
where
it
crosseth
the
tropique
of
♋
;
then
moue
the
ruler
to
26
gr.
10
m.
and
note
where
it
crosseth
the
parallell
of
♊
then
to
14
gr.
45
m.
and
note
where
it
crosseth
the
parallell
of
♉
then
to
0
gr.
0
m.
and
you
shall
finde
it
to
crosse
the
equator
in
the
point
F
;
so
a
line
drawne
through
these
points
,
shall
shew
the
azimuth
belonging
to
East
and
West
.
The
like
reason
holdeth
for
all
the
rest
.
These
lines
being
thus
drawne
,
if
you
set
two
sights
vpon
the
line
AC
,
and
hang
a
thread
and
plummet
on
the
center
A
,
with
a
bead
vpon
the
thread
,
the
foreside
of
the
quadrant
shall
be
fully
finished
.
Or
in
stead
of
the
fiue
starres
before
mentioned
,
you
may
place
the
Nocturnall
(
described
before
in
the
vse
of
the
Sector
,
pag.
60.
)
on
the
backside
of
the
Quadrant
,
and
so
also
it
will
be
fitted
both
for
day
and
night
.
CHAP.
II.
Of
the
vse
of
the
Quadrant
in
taking
the
altitude
of
the
Sunne
,
Moone
,
and
Starres
.
THe
Quadrant
is
the
fourth
part
of
a
circle
,
diuided
equally
into
90
gr.
and
here
numbred
by
10.20.30
.
&c.
vnto
90
gr.
each
degree
being
subdiuided
into
4.
Lift
vp
the
center
of
the
Quadrant
,
so
as
the
thread
with
the
plummet
may
play
easily
by
the
side
of
it
,
and
the
Sunne
beames
may
passe
through
both
the
sights
;
so
shall
the
degrees
cut
by
the
thread
,
shew
what
is
the
altitude
at
the
time
of
obseruation
,
as
may
appeare
by
this
example
.
Vpon
the
14
day
of
April
,
about
noone
,
the
Sun-beames
passing
through
both
the
sights
,
the
thread
fell
vpon
51
gr.
20
m.
and
this
was
the
true
meridian
altitude
of
the
Sun
for
that
day
in
this
our
latitude
of
51
gr.
30
m.
for
which
this
Quadrant
was
made
.
Againe
,
towards
three
of
the
clocke
in
the
afternoone
,
the
thread
fell
vpon
38
gr.
40
m.
and
such
was
the
Sunnes
altitude
at
that
time
.
CHAP.
III.
Of
the
Ecliptique
.
1
The
place
of
the
Sunne
being
giuen
to
finde
his
right
ascension
.
THe
Ecliptique
is
here
represented
by
the
arke
,
figured
with
the
characters
of
the
twelue
Signes
,
♈
,
♉
,
♊
,
&c.
each
Signe
being
diuided
vnequally
into
30
gr.
and
they
are
to
be
reckoned
from
the
character
of
the
Signe
.
Let
the
thread
be
laid
on
the
place
of
the
Sunne
in
the
Ecliptique
,
and
the
degrees
which
it
cutteth
in
the
Quadrant
shall
be
the
right
ascension
required
.
As
if
the
place
of
the
Sunne
giuen
be
the
fourth
degree
of
♊
,
the
thread
laid
on
this
degree
shall
cut
62
degrees
in
the
Quadrant
,
which
is
the
right
ascension
required
.
But
if
the
place
of
the
Sunne
giuen
be
more
then
90
gr.
from
the
beginning
of
♈
,
there
must
be
more
then
90
gr.
allowed
to
the
right
ascension
;
for
this
instrument
is
but
a
quadrant
:
and
so
if
the
Sunne
be
in
26
gr.
of
♋
,
you
shall
finde
the
thread
to
fall
in
the
same
place
,
and
yet
the
right
ascension
to
be
118
gr.
2
The
right
ascension
of
the
Sunne
being
giuen
,
to
find
his
place
in
the
Ecliptique
.
Let
the
thread
be
laid
on
the
right
ascension
in
the
Quadrant
,
and
it
shall
crosse
the
place
of
the
Sunne
in
the
Ecliptique
,
as
may
appeare
in
the
former
example
.
CHAP.
IIII.
Of
the
line
of
declination
.
1
The
place
of
the
Sunne
being
giuen
to
find
his
declination
.
THe
line
of
declination
is
here
drawne
from
the
center
to
the
beginning
of
the
Quadrant
,
and
diuided
from
the
beginning
of
♈
downward
into
23
gr.
30
m.
Let
the
thread
be
laid
,
and
the
beade
set
on
the
place
of
Sunne
in
the
ecliptique
;
then
moue
the
thread
to
the
line
of
declination
,
and
there
the
bead
shal
fall
vpon
the
degrees
of
the
declination
required
.
As
if
the
place
of
the
Sunne
giuen
be
the
fourth
degree
of
♊
,
the
beade
first
set
to
this
place
,
and
then
moued
to
the
line
of
declination
,
shall
there
shew
the
declination
of
the
Sunne
at
that
time
to
be
21
gr.
from
the
equator
.
2
The
declination
of
the
Sunne
being
giuen
,
to
find
his
place
in
the
Ecliptique
.
Let
the
thread
and
beade
be
first
laid
to
the
declination
,
and
then
moued
to
the
Ecliptique
.
As
if
the
declination
be
21
gr.
the
beade
first
set
to
this
declination
,
and
then
moued
to
the
ecliptique
,
shall
there
shew
the
fourth
of
♊
,
the
fourth
of
♐
,
the
26
of
♋
,
and
the
26
of
♑
;
and
which
of
these
foure
is
the
place
of
the
Sunne
,
may
appeare
by
the
quarter
of
the
yeare
.
CHAP.
V.
Of
the
circle
of
Moneths
and
Dayes
.
THis
circle
is
here
represented
by
the
arke
,
figured
with
these
letters
,
I
,
F
,
M
,
A
,
M
,
&c.
signifying
the
moneths
Ianuary
,
February
,
March
,
April
,
&c.
each
moneth
being
diuided
vnequally
,
according
to
the
number
of
the
dayes
that
are
therein
.
1
The
day
of
the
moneth
being
giuen
,
to
find
the
altitude
of
the
Sunne
at
noone
.
Let
the
thread
be
laid
to
the
day
of
the
moneth
,
and
the
degrees
which
it
cutteth
in
the
Quadrant
shall
be
the
meridian
altitude
required
.
As
if
the
day
giuen
be
the
15
of
May
,
the
thread
laid
on
this
day
shall
cut
59
gr.
30
m.
in
the
quadrant
,
which
is
the
meridian
altitude
required
.
2
The
meridian
altitude
being
giuen
,
to
finde
the
day
of
the
moneth
.
The
thread
being
set
to
the
meridian
altitude
,
doth
also
fall
on
the
day
of
the
moneth
.
As
if
the
altitude
at
noone
be
59
gr.
30
m.
the
thread
being
set
to
this
altitude
,
doth
fal
on
the
15
day
of
May
and
the
9
of
Iuly
;
and
which
of
these
two
is
the
true
day
,
may
be
knowne
by
the
quarter
of
the
yeare
,
or
by
another
dayes
obseruation
.
For
if
the
altitude
proue
greater
,
the
thread
wil
fall
on
the
16
day
of
May
and
the
8
of
Iuly
:
or
if
it
proue
lesser
,
the
thread
will
fall
on
the
14
of
May
and
the
10
of
Iuly
;
whereby
the
question
is
fully
answered
.
CHAP.
VI.
Of
the
Houre-lines
.
THat
arke
which
is
drawne
vpon
the
center
of
the
quadrant
by
the
beginning
of
declination
,
doth
here
represent
the
equator
:
that
arke
which
is
drawne
by
23
gr.
30
m.
of
declination
,
and
is
next
aboue
the
circle
of
moneths
and
dayes
,
representeth
the
tropiques
:
those
lines
which
are
betweene
the
equator
and
the
tropiques
,
being
vndiuided
and
numbred
at
the
equator
by
6.
7.
8.
9.
10.
11.
12.
at
the
tropique
by
1.
2.
3.
4.
&c.
do
represent
the
houre-circles
:
that
which
is
drawne
from
12
in
the
equator
to
the
middle
of
Iune
,
representeth
the
houre
of
12
at
noone
in
the
Summer
;
and
those
which
are
drawn
with
it
to
the
right
hand
,
are
for
the
houres
of
the
day
in
the
Summer
,
and
the
houres
of
the
night
in
the
Winter
.
That
which
is
drawne
from
12
in
the
equator
to
the
middle
of
December
,
representeth
the
houre
of
12
in
the
Winter
;
and
those
which
are
drawne
with
it
to
the
left
hand
,
are
for
the
houres
of
the
day
in
the
Winter
,
and
the
houres
of
the
night
in
the
Summer
;
and
of
both
these
,
that
which
is
drawne
from
11
to
1
,
serues
for
11
in
the
forenoone
,
and
1
in
the
afternoon
.
That
which
is
drawne
from
10
to
2
,
serues
for
10
in
the
forenoon
,
&
2
in
the
afternoon
:
for
the
Sun
on
the
same
day
is
about
the
same
height
two
houres
before
noon
,
as
two
houres
after
noone
.
The
like
reason
holdeth
for
the
rest
of
the
houres
.
1
The
day
of
the
moneth
,
or
the
height
at
noone
being
knowne
,
to
finde
the
place
of
the
Sunne
in
the
Ecliptique
.
The
thread
being
laid
to
the
day
of
the
moneth
,
or
the
height
at
noone
,
(
for
one
giues
the
other
by
the
former
proposition
)
marke
where
it
crosseth
the
houre
of
12
,
and
set
the
bead
to
that
intersection
;
then
moue
the
thread
till
the
bead
fall
on
the
ecliptique
,
and
it
shall
fall
on
the
place
of
the
Sun
.
As
if
the
day
giuen
be
the
15
of
May
,
or
the
meridian
altitude
59
gr.
30
m.
lay
the
thread
accordingly
,
and
put
the
bead
to
the
intersection
of
the
thread
with
the
houre
of
12
;
then
moue
the
thread
till
the
bead
fall
on
the
ecliptique
,
and
it
shall
there
shew
the
fourth
of
♊
,
the
fourth
of
♐
,
the
26
of
♋
,
and
the
26
of
♑
;
and
which
of
these
is
the
place
of
the
Sunne
,
may
appeare
by
the
quarter
of
the
yeare
,
or
another
dayes
obseruation
.
2
The
place
of
the
Sun
in
the
Ecliptique
being
knowne
,
to
finde
the
day
of
the
moneth
,
&c.
Let
the
thread
and
beade
be
first
laid
on
the
place
of
the
Sunne
in
the
Ecliptique
,
and
then
moued
to
the
line
of
12.
As
if
the
place
of
the
Sunne
giuen
be
the
fourth
of
♊
,
the
bead
being
laid
to
this
degree
,
and
then
moued
to
the
houre
of
12
,
in
the
Summer
,
the
thread
will
fall
on
the
15
day
of
May
,
and
the
9
of
Iuly
;
or
if
it
be
moued
to
the
houre
of
12
in
the
Winter
,
the
thread
wil
sall
on
the
6
of
Ianuary
and
the
16
of
Nouember
;
which
of
these
is
the
day
of
the
moneth
required
,
may
appeare
by
the
quarter
of
the
yeare
.
In
this
and
the
former
propositions
,
you
haue
two
wayes
to
rectifie
the
bead
,
by
the
place
of
the
Sunne
,
and
by
the
day
of
the
moneth
;
the
better
way
is
by
the
place
of
the
Sunne
,
for
in
the
other
the
Leap-yeare
may
breed
some
small
difference
.
There
is
yet
a
third
way
.
For
the
Sea-men
hauing
a
table
for
the
declination
on
each
day
of
the
yeare
,
may
set
the
bead
thereto
in
the
line
of
declination
.
3
The
houre
of
the
day
being
giuen
to
find
the
altitude
of
the
Sunne
aboue
the
horizon
.
The
bead
being
set
for
the
time
by
either
of
the
three
ways
,
let
the
thread
be
moued
from
the
houre
of
12
toward
the
line
of
declination
,
till
the
bead
fall
on
the
houre
giuen
;
and
the
degrees
which
it
cuts
in
the
Quadrant
,
shall
shew
the
altitude
of
the
Sunne
at
that
time
.
As
if
the
time
giuen
be
the
tenth
of
April
,
the
Sunne
being
then
in
the
beginning
of
♉
,
the
bead
being
rectified
,
you
shall
finde
the
height
at
noone
50
gr.
0
m.
at
11
in
the
morning
48
gr.
12
m.
at
10
but
43
gr.
12
m.
at
9
but
36
gr.
at
8
but
27
gr.
30
m.
at
7
but
18
gr.
18
m.
at
6
but
9
gr.
at
5
it
meeteth
with
the
line
of
declination
,
and
hath
no
altitude
at
all
,
and
therefore
you
may
think
it
did
rise
much
about
that
houre
.
Then
if
you
moue
the
thread
again
from
the
line
of
declination
toward
the
houre
of
12
,
you
shal
find
that
the
Sun
is
8
gr.
33
m.
below
the
horizon
at
4
in
the
morning
,
&
neare
16
gr.
at
3
,
and
21
gr.
51
m.
at
2
,
and
25
gr.
40
m.
at
1
,
and
27
gr.
at
midnight
.
4
The
altitude
of
the
Sunne
being
giuen
,
to
finde
the
houre
of
the
day
.
The
altitude
being
obserued
as
before
,
let
the
bead
be
set
for
the
time
,
then
bring
the
thread
to
the
altitude
,
so
the
bead
shall
shew
the
houre
of
the
day
.
As
if
the
10
of
April
hauing
set
the
bead
for
the
time
,
you
shall
finde
by
the
quadrant
,
the
altitude
to
be
36
gr.
the
bead
at
the
same
time
will
fall
vpon
the
houre-line
of
9
and
3
:
wherefore
the
houre
is
9
in
the
forenoone
,
or
3
in
the
afternoone
.
If
the
altitude
be
neare
40
gr.
you
shall
find
the
bead
at
the
same
time
to
fall
halfe
way
betweene
the
houre-line
of
9
and
3
,
and
the
houre-line
of
10
and
2
:
wherefore
it
must
be
either
halfe
an
houre
past
9
in
the
morning
,
or
halfe
an
houre
past
2
in
the
afternoone
;
and
which
of
these
is
the
true
time
of
the
day
,
may
be
soone
knowne
by
a
second
obseruation
:
for
if
the
Sunne
rise
higher
,
it
is
the
forenoone
;
if
it
become
lower
,
it
is
the
afternoone
.
5
The
houre
of
the
night
being
giuen
,
to
find
how
much
the
Sunne
is
below
the
horizon
.
The
Sunne
is
alwayes
so
much
below
the
horizon
at
any
houre
of
the
night
,
as
his
opposite
point
is
aboue
the
horizon
at
the
like
houre
of
the
day
;
and
therefore
the
beade
being
set
,
if
the
question
be
made
of
any
houre
of
the
night
in
the
Summer
,
then
moue
it
to
the
like
houre
of
the
day
in
the
Winter
;
if
of
any
houre
of
the
night
in
Winter
,
then
moue
it
to
the
like
houre
of
the
day
in
Summer
;
so
the
degrees
which
the
thread
cutteth
in
the
Quadrant
shall
shew
how
much
the
Sunne
is
below
the
horizon
at
that
time
.
As
if
it
be
required
to
know
how
much
the
Sunne
is
below
the
horizon
the
10
of
April
at
4
of
the
clock
in
the
morning
;
the
bead
being
set
to
his
place
according
to
the
time
in
the
Summer
houres
,
bring
it
to
4
of
the
clocke
in
the
afternoone
in
the
Winter
houres
,
and
so
shall
you
finde
the
thread
to
cut
8
gr.
and
about
30
m.
in
the
quadrant
;
and
so
much
is
the
Sunne
below
the
horizon
at
that
time
.
6
The
depression
of
the
Sunne
supposed
,
to
giue
the
houre
of
the
night
with
vs
,
or
the
houre
of
the
day
to
our
Antipodes
.
Here
also
because
the
Sun
is
so
much
aboue
the
horizon
at
all
houres
of
the
day
,
as
his
opposite
point
is
below
the
horizon
at
the
like
houre
of
the
night
;
therefore
first
set
the
bead
according
to
the
time
,
then
bring
the
thread
to
the
degree
of
the
Suns
depression
below
the
horizon
,
so
shall
the
bead
fall
on
the
contrary
houre-lines
,
and
there
shew
the
houre
of
the
night
in
regard
of
vs
,
which
is
the
like
houre
of
the
day
to
our
Antipodes
.
As
if
the
10
of
April
the
Sunne
being
then
in
the
beginning
of
♉
,
and
by
supposition
8
gr.
30
m.
below
the
horizon
in
the
East
,
it
be
required
to
know
what
time
of
the
night
it
is
;
first
set
the
bead
according
to
the
day
in
the
Summer
houres
,
then
bring
the
thread
to
8
gr.
30
m.
in
the
quadrant
,
so
shall
the
bead
fall
among
the
Winter
houres
,
on
the
line
of
4
of
the
clocke
in
the
afternoone
:
wherefore
to
our
Antipodes
it
is
4
of
the
clocke
in
their
afternoone
,
and
to
vs
it
is
then
4
of
the
clocke
in
the
morning
.
7
The
time
of
the
yeare
or
the
place
of
the
Sunne
being
giuen
,
to
find
the
beginning
of
day-breake
,
and
end
of
twi-light
.
This
proposition
differeth
little
from
the
former
:
for
the
day
is
said
to
begin
to
breake
,
when
the
Sun
cometh
to
be
but
18
gr.
below
our
horizon
in
the
East
,
and
twi-light
to
end
when
it
is
gotten
18
gr.
below
the
horizon
in
the
West
:
wherefore
let
the
bead
be
set
for
the
time
,
and
then
bring
the
thread
to
18
gr.
in
the
quadrant
,
so
shall
the
bead
fall
on
the
contrary
houre-lines
,
and
there
shew
the
houre
of
twilight
as
before
.
So
if
it
be
required
to
know
at
what
time
the
day
begins
to
breake
on
the
tenth
of
April
,
the
Sun
being
then
in
the
beginning
of
♉
;
first
set
the
bead
according
to
the
time
in
the
Summer
houres
,
and
then
bring
the
thread
to
18
gr.
in
the
quadrant
,
so
shall
the
bead
fall
among
the
Winter
houres
a
little
more
then
a
quarter
before
3
in
the
morning
;
and
that
is
the
time
when
the
day
begins
to
break
vpon
the
tenth
of
April
.
CHAP.
VII
.
Of
the
Horizon
.
THe
Horizon
is
here
represented
by
the
arke
drawne
,
from
the
beginning
of
declination
towards
the
end
of
February
,
diuided
vnequally
,
and
numbred
by
10.
20.
30.
40.
1
The
day
of
the
moneth
,
or
the
place
of
the
Sunne
being
knowne
,
to
finde
the
amplitude
of
the
Sunnes
rising
and
setting
.
Let
the
bead
rectified
for
the
time
,
be
brought
to
the
horizon
,
and
there
it
shall
shew
the
amplitude
required
.
As
if
the
day
giuen
be
the
15
of
May
,
the
Sunne
being
in
the
fourth
degree
of
♊
,
the
bead
rectified
and
brought
to
the
horizon
,
shall
there
fall
on
35
gr.
8
m.
such
is
the
amplitude
of
the
Sunnes
rising
from
the
East
,
and
of
his
setting
from
the
West
;
which
amplitude
is
alwayes
Northward
when
the
Sunne
is
in
the
Northerne
Signes
,
and
when
he
is
in
the
Southward
Signes
alwayes
Southward
.
2
The
day
of
the
moneth
,
or
the
place
of
the
Sunne
being
giuen
,
to
finde
the
ascensionall
difference
.
Let
the
bead
rectified
for
the
time
,
be
brought
to
the
horizon
,
so
the
degrees
cut
by
the
thread
in
the
quadrant
,
shall
shew
the
difference
of
ascensions
.
As
if
the
day
giuen
be
the
15
of
May
,
the
Sunne
being
in
the
fourth
degree
of
♊
,
let
the
bead
be
rectified
and
brought
to
the
horizon
;
so
shall
the
thread
in
the
quadrant
shew
the
ascensionall
difference
to
be
28
gr.
and
about
50
m.
Vpon
the
ascensionall
difference
depends
this
Corollarie
.
To
find
the
houre
of
the
rising
and
setting
of
the
Sun
,
and
thereby
the
length
of
the
day
and
night
.
The
time
of
the
Sunnes
rising
may
be
guessed
at
by
the
3
of
the
last
Cap.
but
here
by
the
ascensionall
difference
it
may
be
better
found
,
and
that
to
a
minute
of
time
.
For
if
the
ascensionall
difference
be
conuerted
into
time
,
allowing
an
houre
for
15
gr.
and
4
minutes
of
an
houre
for
each
degree
,
it
sheweth
how
long
the
Sunne
riseth
before
six
of
the
clock
in
the
Summer
,
and
after
six
in
the
Winter
.
As
if
the
day
giuen
be
the
15
of
May
,
the
Sunne
being
in
the
fourth
of
♊
,
and
his
ascensionall
difference
found
as
before
28
gr.
50
m
;
this
conuerted
into
time
,
maketh
1
ho.
and
somewhat
more
then
55
m.
of
an
houre
:
wherefore
the
Sun
at
that
time
,
in
regard
it
was
Summer
,
rose
1
ho.
and
full
55
m.
before
6
of
the
clocke
;
and
so
hauing
the
quantitie
of
the
semidiurnall
arke
,
the
length
of
the
day
and
night
need
not
be
vnknowne
.
CHAP.
VIII
.
Of
the
fiue
Starres
.
I
Might
haue
put
in
more
Starres
,
but
these
may
suffice
for
the
finding
of
the
houre
of
the
night
at
all
times
of
the
yeare
:
and
first
I
make
choice
of
Ala
Pegasi
,
a
starre
in
the
extremitie
of
the
wing
of
Pegasus
,
in
regard
it
wants
but
6
minutes
of
time
of
the
beginning
of
♈
;
but
because
it
is
but
of
the
second
magnitude
,
and
not
alwayes
to
be
seene
,
I
made
choice
of
foure
more
,
one
for
each
quarter
of
the
Ecliptique
,
as
of
Oculus
♉
the
Buls
eye
,
whose
right
ascension
conuerted
into
time
,
is
4
ho.
15
m
;
then
of
Cor
♌
the
Lions
heart
,
whose
right
ascension
is
9
ho.
48
m
;
next
of
Arcturus
,
whose
right
ascension
is
13
H.
58
m
;
and
lastly
of
Aquila
,
or
the
Vultures
heart
,
whose
right
ascension
is
19
H.
33
m.
These
fiue
starres
haue
all
of
them
Northern
declination
;
and
if
any
others
,
some
of
these
will
be
seene
at
all
times
of
the
yeare
.
The
vse
of
them
is
,
The
altitude
of
any
of
these
fiue
Starres
being
knowne
,
to
find
the
houre
of
the
night
.
First
put
the
beade
to
the
starre
which
you
intend
to
obserue
,
take
his
altitude
,
and
finde
how
many
houres
he
is
from
the
meridian
by
the
fourth
Prop.
of
the
sixt
Chap
;
then
out
of
the
right
ascension
of
the
starre
,
take
the
right
ascension
of
the
Sun
conuerted
into
houres
,
and
marke
the
difference
;
for
this
difference
being
added
to
the
obserued
houre
of
the
starre
from
the
meridian
,
shall
shew
how
many
houres
the
Sunne
is
gone
from
the
meridian
,
which
is
in
effect
the
houre
of
the
night
.
As
if
the
15
of
May
,
the
Sunne
being
in
the
fourth
of
♊
,
I
should
set
the
beade
to
Arcturus
,
and
obseruing
his
altitude
should
find
him
to
be
in
the
West
about
52
gr.
high
,
and
the
beade
to
fall
on
the
houre-line
of
2
afternoone
,
the
houre
would
be
11
bo.
50
m.
past
noone
,
or
10
m.
short
of
midnight
.
For
62
gr.
the
right
ascension
of
the
Sunne
,
conuerted
into
time
,
makes
4
ho.
8
m.
which
if
we
take
out
of
13
ho.
58
m.
the
right
ascension
of
Arcturus
,
the
difference
will
be
9
ho.
50
m.
and
this
being
added
to
2
ho.
the
obserued
distance
of
Arcturus
from
the
meridian
,
shewes
the
houre
of
the
night
to
be
11
ho.
50
m.
Another
example
wil
make
all
more
plaine
.
If
the
9
of
Iuly
the
Sunne
being
then
in
26
gr.
of
♋
,
I
should
set
the
beade
to
Oculus
♉
,
and
obseruing
his
altitude
should
find
him
to
be
in
the
East
about
12
gr.
high
,
and
the
bead
to
fall
on
the
houre-line
of
6
before
noone
,
which
is
18
ho.
past
the
meridian
,
the
houre
of
the
night
would
be
better
then
a
quarter
past
2
of
the
clocke
in
the
morning
.
For
118
gr.
the
right
ascension
of
the
Sun
,
conuerted
into
time
,
makes
7
ho.
52
m
;
this
taken
out
of
4
ho.
15
m.
the
right
ascension
of
Oculus
♉
,
adding
a
whole
circle
,
(
for
otherwise
there
could
be
no
substraction
)
the
difference
will
be
20
ho.
23
m.
and
this
being
added
to
18
ho.
which
was
the
obserued
distance
of
Oculus
♉
from
the
meridian
,
shewes
that
the
Sun
(
abating
24
ho.
for
the
whole
circle
)
is
14
gr.
23
m.
past
the
meridian
,
and
therefore
23
m.
past
2
of
the
clock
in
the
morning
.
CHAP.
IX
.
Of
the
Azimuth
lines
.
THose
lines
which
are
drawne
betweene
the
equator
and
the
tropiques
,
on
that
side
of
the
quadrant
which
is
nearest
vnto
the
sights
,
and
are
numbred
by
10.
20.
30.
&c.
doe
represent
the
azimuths
,
the
vttermost
to
the
left
hand
representeth
the
meridian
,
that
which
is
numbred
with
10
the
tenth
azimuth
from
the
meridian
,
and
that
which
is
numbred
with
20
the
twentith
,
and
so
the
rest
.
Those
lines
which
are
drawne
from
the
equator
to
the
left
hand
,
doe
shew
the
azimuth
in
the
Summer
;
and
those
other
to
the
right
hand
,
do
shew
the
same
in
the
Winter
.
The
vse
of
them
is
,
1
The
azimuth
whereon
the
Sunne
beareth
from
vs
being
knowne
,
to
find
the
altitude
of
the
Sun
aboue
the
horizon
.
First
let
the
bead
be
set
for
the
time
,
as
in
the
former
Chapter
,
then
moue
the
thread
vntill
the
bead
fall
on
the
azimuth
;
so
the
degrees
which
the
thread
cutteth
in
the
quadrant
,
shall
shew
the
altitude
of
the
Sun
at
that
time
.
Where
you
are
to
obserue
,
that
seeing
the
azimuths
are
drawne
on
the
right
side
of
the
quadrant
,
you
are
also
to
begin
to
number
the
degrees
of
the
Sunnes
altitude
from
the
right
hand
toward
the
left
.
As
if
the
sights
had
been
set
on
the
line
AB
,
and
you
had
turned
your
right
hand
towards
the
Sunne
in
obseruing
of
his
altitude
,
contrary
to
our
practise
in
the
former
Chapter
.
As
if
the
time
giuen
were
the
2
of
August
,
when
the
Sun
hath
about
15
gr.
of
North
declination
,
you
may
set
the
bead
for
the
time
,
so
you
shall
find
the
height
at
noone
when
the
Sunne
is
in
the
South
,
to
be
53
gr.
30
m.
when
he
is
10
gr.
from
the
South
53
gr.
10
m.
when
20
gr.
then
about
52
gr.
8
m.
when
30
gr.
then
50
gr.
20
m.
when
40
gr.
then
47
gr.
48
m.
when
50
gr.
then
44
gr.
12
m.
when
60
gr.
then
39
gr.
35
m.
when
70
gr.
then
33
gr.
50
m.
when
80
gr.
then
27
gr.
when
he
is
in
the
East
or
West
90
gr.
from
the
meridian
,
then
is
the
height
neare
19
gr.
20
m
;
when
he
comes
to
be
100
gr.
then
11
gr.
15
m.
when
110
gr.
then
3
gr.
20
m
;
and
before
he
commeth
to
the
azimuth
of
120
gr.
he
hath
no
altitude
.
For
the
Sunne
hauing
15
gr.
of
North
declination
,
will
rise
and
set
at
114
gr.
34
m.
from
the
meridian
.
2
The
altitude
of
the
Sunne
being
giuen
,
to
find
on
what
azimuth
he
beareth
from
vs
.
Let
the
beade
be
set
for
the
time
,
and
the
altitude
obserued
as
before
;
then
bring
the
thread
to
the
complement
of
that
altitude
,
so
the
bead
shall
shew
the
azimuth
required
.
As
if
the
second
of
August
,
hauing
set
the
beade
for
the
time
,
you
shall
find
the
altitude
of
the
Sun
to
be
19
gr.
20
m.
remoue
the
thread
vnto
70
gr.
40
m.
the
complement
of
the
altitude
;
or
,
which
is
all
one
,
to
19
gr.
20
m.
from
the
right
hand
toward
the
left
,
and
the
bead
will
fall
on
the
line
of
90
gr.
from
the
meridian
.
And
therefore
the
point
whereon
the
Sunne
beareth
from
vs
,
is
one
of
these
two
,
either
due
East
or
due
West
.
And
which
of
these
is
the
true
point
of
the
compasse
,
may
be
soone
knowne
by
a
second
obseruation
:
for
if
the
Sun
rise
higher
,
it
is
the
forenoone
;
if
it
be
lower
,
it
is
the
afternoone
.
By
knowing
the
azimuth
or
point
of
the
compasse
whereon
the
Sunne
beareth
from
vs
,
it
is
easie
to
finde
,
A
meridian
line
;
and
thereby
The
coasting
of
the
Countrey
.
The
site
of
a
building
.
The
variation
of
the
Compasse
.
As
if
the
second
of
August
in
the
afternoone
,
I
should
find
by
the
height
of
the
Sunne
that
he
beares
from
me
60
gr.
from
the
meridian
toward
the
West
;
then
there
being
90
gr.
belonging
to
each
quarter
,
the
West
will
be
30
gr.
to
the
right
hand
,
the
East
is
opposite
to
the
West
,
the
North
and
South
lie
equally
betweene
them
.
CHAP.
X.
Of
the
Quadrat
.
THe
Quadrat
hath
two
sides
diuided
,
the
other
two
sides
next
the
Center
may
be
supposed
to
be
diuided
,
each
of
them
into
100
equal
parts
:
of
the
sides
diuided
,
that
which
is
next
the
horizontall
line
containes
the
parts
of
right
shadow
,
the
other
next
the
sights
,
the
parts
of
contrary
shadow
.
The
vse
of
the
Quadrat
is
,
1
Any
point
being
giuen
,
to
finde
whether
it
be
leuell
with
the
eye
.
Lift
vp
the
center
of
the
quadrant
,
so
as
the
thread
with
the
plummet
may
play
easily
by
the
side
of
it
;
then
looke
through
the
sights
to
the
place
giuen
:
for
now
if
the
thread
shall
fall
on
AB
the
horizontall
line
,
then
is
the
place
giuen
leuel
with
the
eye
:
but
if
it
shal
fal
within
the
said
line
on
any
of
the
diuisions
,
then
it
is
higher
;
if
without
,
then
it
is
lower
then
the
leuell
of
the
eye
.
2
To
find
an
height
aboue
the
leuell
of
the
eye
,
or
a
distance
at
one
obseruation
.
Looke
through
the
sights
to
the
place
,
going
nearer
or
farther
from
it
,
till
the
thread
fall
on
100
parts
in
the
quadrat
or
45
gr.
in
the
quadrant
,
so
shall
the
height
of
the
place
aboue
the
leuell
of
the
eye
,
be
equall
to
the
distance
between
the
place
and
the
eye
.
geometric illustration measuring shadow cast by a tower
And
on
the
contrary
,
As
the
parts
cut
by
the
thread
are
to
100
:
So
the
height
vnto
the
distance
.
But
when
the
thread
shall
fall
on
the
parts
of
contrary
shadow
:
if
it
fall
on
50
parts
,
the
height
is
double
vnto
the
distance
;
if
on
25
,
it
is
foure
times
as
the
distance
.
For
as
oft
as
the
thread
falleth
on
the
parts
of
contrary
shadow
,
As
the
parts
cut
by
the
thread
are
vnto
100
:
So
is
the
distance
vnto
the
height
.
And
on
the
contrary
,
As
100
are
vnto
the
parts
cut
by
the
thread
:
So
is
the
height
vnto
the
distance
.
And
what
is
here
said
of
the
height
and
distance
,
the
same
may
be
vnderstood
of
the
height
and
shadow
.
3
To
find
a
height
or
a
distance
at
two
obseruations
.
As
if
the
place
which
is
to
be
measured
might
not
otherwise
be
approched
,
&
yet
it
were
required
to
find
the
height
BC
,
and
the
distance
:
first
if
I
make
choice
of
a
station
at
A
,
where
the
thread
may
fall
on
100
parts
in
the
quadrat
,
and
45
gr.
in
the
quadrant
,
the
distance
AB
will
be
equall
to
the
height
BC
;
then
if
I
go
farther
in
a
direct
line
with
the
former
distance
,
and
make
choice
of
a
second
station
at
D
,
where
the
thread
may
fall
on
50
parts
of
right
shadow
,
the
distance
BD
would
be
double
to
the
height
BC
:
wherefore
I
may
measure
the
difference
betweene
the
two
stations
A
and
D
,
and
this
difference
AD
will
be
equall
both
to
the
distance
AB
and
the
height
BC.
Or
if
I
cannot
make
choice
of
such
stations
,
I
take
such
as
I
may
,
one
at
D
,
where
the
thread
falleth
at
50
parts
of
right
shadow
;
the
second
at
E
,
where
it
falleth
on
40
parts
:
and
supposing
the
height
BC
to
be
100
,
I
find
that
As
50
parts
are
vnto
100
,
the
side
of
the
quadrat
:
So
100
the
supposed
height
,
vnto
200
the
distance
BD.
And
as
40
parts
,
at
the
second
station
,
vnto
100
:
So
100
the
supposed
height
,
vnto
250
the
distance
BE.
Wherfore
the
difference
between
the
stations
D
&
E
should
seeme
to
be
50
;
and
then
if
in
measuring
of
it
,
I
should
find
it
to
be
either
more
or
lesse
,
the
proportion
will
hold
,
as
from
the
supposed
difference
to
the
measured
difference
,
so
from
height
to
height
,
and
from
distance
to
distance
.
As
if
the
difference
between
the
two
stations
D
and
E
being
measured
,
were
found
to
be
30.
As
50
the
supposed
difference
,
vnto
30
the
true
difference
:
So
100
the
supposed
height
,
vnto
60
the
true
height
.
And
200
the
supposed
distance
,
vnto
120
the
true
distance
:
And
250
at
the
second
station
,
vnto
150
the
distance
BE.
The
like
reason
holdeth
in
all
other
examples
of
this
kind
:
and
if
an
Index
with
sights
were
fitted
to
turne
vpon
the
Center
,
it
might
then
serue
by
the
same
reason
for
the
finding
of
all
other
distances
.
FINIS
.