The
first
book
of
the
Catholique
Planisphear
.
Wherin
The
whole
Fabrick
or
making
thereof
is
plainly
Described
.
CHAP.
I.
Of
the
parts
of
the
Planisphear
.
And
of
the
Mater
,
his
matter
and
Lineaments
.
THis
Planisphere
is
made
up
of
five
parts
-
1
The
lying
plate
,
called
Mater
.
-
2
The
moving
plate
;
called
the
Rete
,
Reet
,
or
Net
.
-
3
The
Ring
or
Limb.
-
4
The
Label
.
-
5
The
Sights
.
The
Mater
is
a
round
plate
of
metal
or
past-board
flat
smooth
,
and
stiff
:
the
larger
the
better
.
And
if
you
will
have
the
circles
actually
drawn
for
every
degree
,
it
had
need
be
ten
Inches
at
least
in
Diameter
.
If
it
be
made
of
metal
(
as
Silver
Brass
or
Tin
and
Tin-glass
in
equal
quantity
melted
together
)
it
must
be
well
pollished
:
but
it
may
very
well
be
made
on
a
fair
pastboard
,
pasted
on
a
Messie
board
:
for
thereon
the
Lineaments
may
be
distinguished
with
inkes
of
several
colours
,
which
cannot
be
if
it
be
made
in
metall
.
The
Lineaments
of
the
Mater
(
though
they
be
fitted
to
represent
other
circles
of
the
Sphear
also
,
as
shall
be
shewed
,
yet
)
most
aptly
represent
the
Meridians
and
Parallels
:
and
therefore
so
we
call
them
here
,
while
wee
speak
of
the
Fabrique
.
Among
the
Parallels
the
two
Tropiques
and
the
two
Poler
circles
are
to
be
inserted
.
And
lastly
to
these
you
shall
add
the
Ecliptique
:
and
so
you
have
all
the
Lineaments
of
the
Mater
.
For
the
Declination
whereof
,
1
Upon
the
center
of
the
Mater
plate
,
describe
the
fundamental
circle
,
(
about
an
inch
within
the
edge
,
if
your
plate
be
12
inches
Diameter
)
that
so
a
convenient
space
may
be
left
for
the
Limb.
This
circle
shall
be
the
great
Meridian
passing
through
the
Poles
of
the
world
,
and
also
through
the
Zenith
and
Nadir
of
your
Countrey
;
and
is
the
bound
by
which
all
the
Lineaments
of
the
Mater
are
inclosed
.
Draw
two
Diameters
crossing
one
another
in
the
center
at
right
angles
,
and
dividing
this
Meridian
into
his
quarters
,
let
one
of
these
Diameters
be
A
B
the
Axtree-line
,
the
other
C
D
the
Equator
or
his
Diameter
:
divide
also
every
quarter
of
this
Meridian
into
90
equal
parts
or
degrees
.
This
Meridian
onely
of
all
the
circles
of
the
Mater
,
falleth
out
to
be
a
full
circle
in
this
projection
,
because
the
bisection
of
the
Sphear
is
supposed
here
to
be
made
in
the
plain
thereof
,
and
because
the
eye
is
supposed
to
be
the
Pole
thereof
,
and
so
equidistant
from
every
part
thereof
.
The
rest
of
the
Meridians
and
Parallels
their
Semicircles
are
in
this
Planisphere
fore-shortned
according
to
Optique
reason
,
(
as
shall
be
further
explained
Ch
2
)
because
they
all
either
are
great
circles
passing
through
the
eye-point
,
and
cutting
the
Meridian
at
Right
angles
,
as
the
Axtree-line
or
East
Meridian
,
the
Equator
,
and
the
Ecliptique
,
which
are
therefore
projected
upon
their
Diameters
and
become
straight
lines
;
or
else
lie
Oblique
to
the
eye
,
as
do
all
the
rest
of
the
Meridians
and
Parallels
,
which
are
all
of
them
projected
into
arches
less
then
a
semicircle
;
and
yet
every
one
of
them
is
to
be
accounted
a
semicircle
fore
shortned
,
and
to
be
divided
as
a
somicircle
into
180
degrees
.
2
To
describe
the
rest
of
the
Meridians
,
you
shall
lay
a
Ruler
from
the
Pole
A
to
the
several
degrees
of
the
semicircle
C
B
D
(
or
from
the
Pole
B
to
the
several
degrees
of
the
semicircle
C
A
D
,
for
all
comes
to
one
)
and
mark
where
the
Ruler
intersects
the
Equinoctial
line
C
D
for
every
degree
;
so
shall
you
have
the
Equinoctial
divided
into
180
deg.
(
viz.
90
degr.
on
one
side
the
center
,
and
90
degr.
on
the
other
)
through
which
divisions
or
degrees
the
severall
Meridians
shall
pass
,
and
all
of
them
must
meet
in
both
the
Poles
.
Therefore
having
three
points
given
for
every
Meridian
(
viz.
the
two
Poles
A
and
B
,
and
a
middle
point
in
the
Equator
C
D
,
bring
these
three
points
into
a
circle
,
and
you
shall
have
the
true
arch
of
every
Meridian
drawn
on
the
Planisphear
;
as
they
appear
in
this
Projection
:
which
how
to
do
shall
be
further
shewed
Chap.
3.
3
To
describe
the
Parallels
,
you
shall
first
divide
the
Axtree-line
A
B
into
180
deg.
(
as
you
divide
the
Equator
C
D
)
by
laying
your
Ruler
from
C
or
D
to
the
several
degrees
of
the
opposite
Semicircle
,
and
marking
where
the
Ruler
intersects
the
Axtree-line
A
B
;
or
with
your
Compasses
transfer
the
division
of
the
Diameter
C
D
to
the
Diameter
A
B
;
so
shall
you
have
three
points
given
for
every
Parallel
,
whereby
his
arch
may
be
drawn
:
as
for
example
,
the
tenth
Parallel
North
shall
be
drawn
from
the
tenth
degr.
of
the
Quadrant
D
A
,
counted
from
D
,
by
the
tenth
division
of
the
Semidiameter
E
A
,
counted
from
E
,
to
the
tenth
degr.
of
the
Quadrant
C
A
,
counted
from
C
:
and
so
of
the
rest
.
The
Tropiques
and
Poler
circles
are
described
as
the
Parallels
.
Yet
of
them
see
Chap.
6.
4
To
describe
the
Ecliptick
,
number
his
greatest
Declination
(
23
degr.
30
mi.
)
in
the
Meridian
from
C
,
towards
A
to
F.
and
again
from
D
toward
B
to
G
;
then
joyn
the
points
F
and
G
with
a
straight
line
which
shall
pass
through
the
center
of
the
Mater
and
be
the
Ecliptick
.
For
dividing
whereof
you
shall
only
transfer
the
division
of
the
Axtree-line
or
Equator
upon
this
Ecliptick
line
(
for
all
Diameters
have
like
divisions
)
and
you
shall
distinguish
every
tenth
and
30
degr.
by
longer
stroaks
,
and
shall
set
♑
at
G
;
♈
and
♎
at
the
center
;
♋
at
F
:
and
the
rest
in
their
order
.
CHAP.
II.
Of
the
reason
of
this
Declination
.
FOr
the
better
understanding
of
the
reason
of
this
Declination
,
either
take
or
suppose
to
your self
an
hollow
Sphear
or
Globe
,
of
equal
diameter
with
your
great
Meridian
before
drawn
.
Cut
this
Sphear
by
the
Meridian
into
two
dishes
or
Hemisphears
,
the
one
representing
the
Eastern
,
the
other
the
Western
Hemisphear
of
the
Heaven
:
So
in
each
of
these
Hemisphears
the
Meridian
is
an
whole
Circle
,
being
the
Base
of
the
Hemisphear
,
or
brim
of
the
dish
:
but
the
rest
of
the
Meridians
,
the
Parallels
,
Equinoctial
,
and
Ecliptick
,
are
all
bisected
by
the
great
Meridian
,
and
so
you
can
have
but
half
their
circles
in
the
Eastern
Hemisphear
,
and
the
other
half
is
in
the
Western
.
Next
suppose
also
the
plain
A
C
B
D
,
to
be
a
thin
plate
of
some
transparent
matter
,
as
clear
horn
glass
,
or
christal
;
and
this
plate
fitted
to
stick
in
the
dish
mouth
(
that
is
,
in
the
Meridian
of
the
Hemisphear
)
and
upon
the
center
E
a
straight
wyre
to
be
erected
perpendicular
to
the
plain
A
C
B
D
,
and
the
length
of
the
said
wyre
to
be
equal
to
the
semidiameter
E
A.
Now
place
your
eye
at
the
top
of
this
wyre
and
look
up
the
Lineaments
of
the
Hemisphear
through
the
glass
plate
,
and
observe
where
the
visual
lines
drawn
from
the
severall
points
of
the
Hemisphear
to
the
eye
cut
the
glass
and
what
kind
of
lines
and
arches
they
do
paint
upon
the
glass
,
and
you
shall
see
there
that
the
semicircles
of
the
Equator
,
the
East
Meridian
,
and
the
Ecliptick
,
will
be
depainted
on
the
glass
in
straight
lines
;
because
they
be
great
Circles
,
and
pass
through
the
eye
point
of
this
projection
)
and
the
lines
passing
between
your
eye
and
the
several
degrees
of
any
of
them
shall
divide
their
Diameters
upon
the
glass
into
such
parts
as
they
are
divided
by
the
precept
of
the
former
Chapter
.
For
the
top
of
the
wyre
(
here
supposed
to
be
the
West
point
of
the
Horizon
)
is
the
true
eye
point
from
which
the
Diameters
A
B
and
C
D
are
divided
,
and
from
which
the
whole
projection
of
the
Eastern
Hemisphear
is
made
at
one
view
upon
the
glass
But
because
this
concave
Hemisphear
with
his
wire
or
Axis
erect
,
cannot
easily
be
pictured
on
a
flat
,
therefore
to
supply
the
want
of
a
solid
Sche
m
you
may
consider
that
the
Axtree
line
A
B
is
not
onely
the
Diameter
of
the
East
Meridian
,
but
a
common
Diameter
to
all
the
Meridians
:
and
therefore
if
you
take
your
ey
point
at
C
or
D
,
and
by
your
eye
beam
or
a
ruler
laid
from
one
of
those
points
to
all
the
degrees
of
the
opposite
semicircle
do
divide
the
Axtree
line
,
it
shall
be
all
one
as
if
you
had
divided
him
from
the
top
of
the
wire
by
the
degrees
of
the
East
Meridian
which
passeth
through
the
bottom
of
the
concave
Hemisphear
.
For
all
great
Circles
of
the
Sphear
and
their
Diameters
likewise
being
equal
,
look
how
any
one
of
their
Diameters
is
divided
,
in
like
manner
they
shall
all
be
divided
,
if
the
eye
be
alike
situate
to
them
.
CHAP.
III.
How
to
find
the
centers
of
the
Meridians
five
several
wayes
.
THe
centers
of
the
Meridians
and
their
semidiameters
are
thus
found
,
-
1.
You
have
by
the
first
Chap.
three
points
given
by
which
they
must
be
drawn
(
viz.
)
the
two
Poles
,
and
a
third
point
in
the
Equator
)
and
how
to
bring
these
into
a
circle
or
arch
,
is
shewed
by
Euclid
.
4
,
5.
But
I
think
this
way
incomodious
for
this
purpose
.
-
2
A
better
way
is
to
get
the
centers
by
proffers
,
thus
,
Let
it
be
required
to
draw
the
twentieth
Meridian
whose
points
given
are
A
2
B.
Having
first
extended
infinitely
the
line
C
D
,
set
one
foot
of
your
Compasses
at
adventure
,
in
the
line
C
D
,
whereabout
you
guess
the
center
should
be
,
and
extending
the
other
foot
to
A
or
B
,
carry
it
about
at
the
same
extent
towards
2
,
and
if
it
touch
the
point
2
,
you
have
taken
the
true
center
,
and
may
draw
the
arch
as
is
required
,
if
your
Compass
over-reach
,
you
must
narrow
it
,
and
come
nearer
;
if
it
reach
short
,
widen
your
Compass
,
and
seek
your
center
further
,
till
by
tryal
you
light
upon
it
:
if
one
foot
of
your
Compass
stand
in
the
line
C
D
,
and
the
other
cut
the
middle
point
,
and
one
of
the
extream
points
A
or
B
,
it
ought
to
cut
the
other
also
;
if
your
plain
be
flat
,
and
the
line
C
D
straight
and
square
to
the
line
A
B
:
but
if
any
of
these
have
failed
,
you
shall
never
bring
the
three
points
into
an
arch
while
the
foot
of
your
compass
●andeth
in
the
line
C
D.
Therefore
in
such
case
set
one
foot
in
A
,
and
draw
with
the
other
foot
a
short
arch
crossing
the
line
C
D
;
then
set
the
standing
foot
in
B
,
and
with
the
running
foot
cross
the
short
arch
last
drawn
:
where
these
arches
cross
will
be
the
center
,
by
which
you
may
draw
an
arch
cutting
A
and
B
,
and
if
it
cut
2
also
,
you
have
your
desire
.
But
if
this
arch
over
reach
2
,
widen
your
Compass
:
if
it
come
short
of
2
(
the
middle
point
)
narrow
your
Compass
,
and
try
again
in
like
manner
till
you
can
compass
all
the
three
points
in
the
same
arch
.
3.
A
third
way
.
I
suppose
you
may
know
that
every
Meridian
cuts
the
Equator
twice
,
viz.
in
two
opposite
points
distant
180
degr.
one
from
another
:
as
,
for
example
,
the
Meridian
which
cutteth
the
Equator
in
60
degr.
of
Right
Ascension
,
cuts
it
again
in
the
opposite
degr
,
viz.
240.
Now
if
you
can
find
these
two
points
in
the
Equator
line
C
D
,
the
center
will
be
in
the
just
middle
betwixt
them
.
One
of
these
points
is
already
given
within
the
fundamental
Circle
;
the
other
without
,
is
thus
found
.
Prolong
your
Equator
infinitely
beyond
your
plate
both
wayes
,
and
divide
the
extension
thereof
by
like
reason
as
you
divided
his
Diameter
,
viz.
as
by
a
Ruler
laid
from
A
to
the
several
deg.
of
the
Quadrant
B
C
,
you
devided
the
Semidiameter
E
C
into
90
degr.
so
keeping
still
one
end
of
your
Ruler
fixed
at
A
,
and
carrying
about
the
other
end
thereof
to
the
severall
degrees
of
the
Quadrant
C
A
,
you
may
divide
the
excurrence
of
the
line
E
C
into
90
degr.
more
:
and
so
E
C
and
his
exccurrence
or
continuation
will
be
half
the
Equator
divided
into
his
degr.
and
E
D
with
his
excurrence
on
the
other
side
will
be
the
other
half
divided
by
like
reason
.
And
thus
the
whole
Equator
is
projected
in
one
straight
line
,
and
divided
into
degr.
also
.
Then
having
a
point
given
within
your
fundamental
circle
through
which
the
sixtieth
Meridian
must
pass
(
viz
the
60
deg.
of
the
Semidimeter
E
C
or
E
D
)
number
thence
over
the
center
to
180
deg.
and
there
is
the
point
where
the
other
semicircle
must
cross
,
and
the
middle
between
those
points
is
the
center
.
But
because
the
two
points
taken
in
the
quadrant
A
C
are
very
near
together
,
especially
towards
A
,
and
the
Ruler
also
will
cross
the
prolongation
of
the
line
E
C
very
obliquely
,
you
may
therefore
do
better
to
divide
this
line
into
his
degr.
by
a
Scale
of
Tangents
,
for
if
upon
the
Equinoctial
line
D
E
C
you
prick
down
from
E
both
wayes
the
Tangents
of
the
half
degr.
in
order
from
0
,
to
90
,
those
pricks
shall
be
the
whole
degr.
of
the
Equinoctial
line
in
this
Projection
,
to
be
numbred
from
E
both
wayes
to
180
deg.
where
the
Tangent
becomes
infinite
.
Thus
taking
A
E
for
Radius
,
E
D
is
the
Tangent
of
45
deg.
by
the
structures
:
yet
the
arch
or
Diameter
E
D
is
a
Quadrant
or
90
deg.
of
the
Equinoctial
in
this
Projection
,
because
the
Tangents
of
the
half
deg.
of
the
Quadrant
E
A
I
,
measure
out
the
whole
degr.
of
the
line
E
D
,
as
was
above-said
.
4
If
you
consider
well
what
hath
been
said
,
you
will
find
(
or
you
may
take
it
here
upon
trust
,
)
that
for
the
90
Meridians
to
be
drawn
between
C
and
E
,
half
the
centers
will
be
found
in
the
opposite
Semidiameter
E
D
,
and
the
other
half
without
D
in
the
said
Semidiameter
prolonged
.
And
that
every
second
division
of
the
line
E
D
from
E
toward
D
and
forwards
,
shall
be
the
centers
of
the
Meridians
which
cut
the
Semidiameter
C
E.
As
for
example
,
to
draw
every
fifth
Meridian
from
C
to
E
,
you
take
every
tenth
deg.
from
E
toward
D
for
the
centers
.
And
further
if
you
would
not
be
at
more
trouble
then
needs
,
to
divide
the
extension
of
the
Diameter
beyond
the
fundamental
circle
,
you
shall
but
do
thus
.
Begin
with
the
crookedest
Meridians
first
,
whose
centers
are
within
the
fundamental
Circle
,
and
first
pitching
one
foot
of
your
Compass
in
the
point
1
,
(
near
E
)
extend
the
other
foot
beyond
the
center
to
2
;
there
is
the
center
from
which
you
shall
draw
the
first
Meridian
A
1
B
:
and
also
turning
about
your
Compass
you
shall
make
a
marke
in
the
extension
beyond
D
at
1
,
where
the
other
Semicircle
of
this
Meridian
would
cross
the
Equator
.
So
for
the
next
Meridian
,
in
the
line
C
E
marked
with
2
,
your
center
is
beyond
E
at
4
,
and
after
you
have
drawn
his
arch
A
2
B
,
marke
with
your
compass
his
other
crossing
at
2
beyond
D
,
and
so
with
one
labour
you
shall
both
draw
the
45
crookeder
Meridians
,
and
also
make
the
out
lying
division
of
the
line
E
D
prolonged
:
of
which
division
every
second
or
even
number
will
be
a
center
to
some
of
the
straighter
Meridians
.
This
is
a
very
good
and
easie
way
:
and
this
way
Mr
Blagrave
alwaies
used
.
5
Or
lastly
You
may
frame
a
decimal
Scale
of
1000
or
10000
parts
,
equal
to
the
Semidiameter
of
the
Mater
;
by
which
Scale
with
the
help
of
the
Cannon
of
Triangles
,
you
may
presently
find
the
length
of
any
S
ne
Tangent
or
Secant
you
shall
desire
.
Now
look
what
inclination
any
Meridian
hath
to
your
fundamental
Circle
(
that
is
,
what
angle
they
make
between
them
)
the
Secant
of
that
inclination
is
the
Semidiameter
of
that
Meridian
;
and
the
Tangent
of
the
same
inclination
is
the
distance
of
his
center
from
E
the
center
of
the
Mater
.
For
example
,
the
Meridian
A
2
B
his
inclination
is
20
deg
(
for
the
angle
C
A
2
and
likewise
the
arch
C
2
which
measures
it
is
20
degr.
the
S
cant
of
20
degr
is
10641.
by
the
Cannon
of
Triangles
(
which
every
Mathematician
ought
to
have
at
hand
)
Take
with
your
Compasses
from
your
decimal
Scale
10641.
and
setting
one
foot
in
A
,
with
the
other
foot
cross
the
Semidiameter
E
D
;
in
that
crossing
is
the
center
:
or
take
with
your
Compasses
3639.
the
Tangent
of
20
degr.
and
set
it
from
E
toward
D
,
and
it
shall
give
you
the
same
center
at
4.
For
A
E
being
Radius
,
E
4
is
the
Tangent
,
and
A
4
the
Secant
of
20
deg.
by
the
structure
.
And
if
you
like
to
work
this
way
,
it
will
help
you
much
to
have
a
short
Cannon
of
Tangents
and
Secants
of
whole
degr.
of
the
Quadrant
gathered
into
one
page
:
which
Cannon
for
your
ease
is
here
annexed
.
A
Table
of
Tangents
and
Secants
to
every
degree
of
the
Quadrant
.
Degr
|
Tan.
|
Secan
|
1
|
00174
|
010001
|
2
|
00349
|
010006
|
3
|
00524
|
010013
|
4
|
00699
|
010024
|
5
|
00374
|
010038
|
6
|
01051
|
010055
|
7
|
01227
|
010075
|
8
|
01405
|
010098
|
9
|
01583
|
010124
|
10
|
01763
|
010154
|
11
|
01943
|
010187
|
12
|
02125
|
01022●
|
13
|
02308
|
010263
|
14
|
02493
|
010306
|
15
|
02679
|
010352
|
16
|
02867
|
010402
|
17
|
03057
|
010456
|
18
|
03249
|
010514
|
19
|
03443
|
010576
|
20
|
03639
|
010641
|
21
|
03838
|
010711
|
22
|
04040
|
010785
|
23
|
04244
|
010863
|
23.30
|
04348
|
010904
|
24
|
04452
|
010946
|
25
|
04663
|
011033
|
26
|
04877
|
011126
|
27
|
05095
|
011223
|
28
|
05317
|
011325
|
29
|
05543
|
011433
|
30
|
05773
|
011547
|
31
|
06008
|
011666
|
32
|
06248
|
011791
|
33
|
06494
|
011923
|
34
|
06745
|
012062
|
35
|
07002
|
012207
|
36
|
07265
|
012360
|
37
|
07535
|
012521
|
38
|
07812
|
012690
|
39
|
08097
|
012867
|
40
|
08390
|
013054
|
41
|
08692
|
013250
|
42
|
09004
|
013456
|
43
|
09325
|
013673
|
44
|
09656
|
013901
|
45
|
10000
|
014142
|
46
|
10355
|
014395
|
47
|
10723
|
014662
|
48
|
11106
|
014944
|
49
|
11503
|
015242
|
50
|
11917
|
015557
|
51
|
12348
|
015890
|
52
|
12799
|
016242
|
53
|
13270
|
016616
|
54
|
13763
|
017013
|
55
|
14281
|
017434
|
56
|
14825
|
017882
|
57
|
15398
|
018360
|
58
|
16003
|
018870
|
59
|
16642
|
019416
|
60
|
17320
|
02000
|
61
|
18040
|
020626
|
62
|
18807
|
021300
|
63
|
19626
|
022026
|
64
|
20503
|
022811
|
65
|
21445
|
023662
|
66
|
22460
|
024585
|
66.30
|
22998
|
025078
|
67
|
23558
|
025693
|
68
|
24750
|
026694
|
69
|
26050
|
027904
|
70
|
27474
|
029238
|
71
|
29042
|
030715
|
72
|
30776
|
032360
|
73
|
32708
|
034203
|
74
|
34874
|
036279
|
75
|
37320
|
038637
|
76
|
40107
|
041335
|
77
|
43314
|
044454
|
78
|
47046
|
048097
|
79
|
51445
|
052408
|
80
|
56712
|
057587
|
81
|
63137
|
063924
|
82
|
71153
|
071852
|
83
|
8144●
|
082055
|
84
|
9514●
|
095667
|
85
|
11430
|
114737
|
86
|
14300
|
143355
|
87
|
19081
|
191073
|
88
|
28636
|
286537
|
89
|
57289
|
572986
|
90
|
|
|
CHAP.
IIII.
To
find
the
Centers
of
the
Parallels
,
six
several
wayes
.
THe
first
way
,
but
the
worst
for
our
purpose
,
(
as
was
said
before
for
the
Meridians
)
is
by
the
fifth
Proposition
of
the
fourth
book
of
Euclid
;
to
find
the
Center
of
the
Circle
circumscribing
the
Triangle
made
by
the
three
points
given
.
2
A
better
way
is
by
profers
.
Take
this
upon
trust
:
that
as
you
found
the
Centers
of
all
the
Meridians
in
the
Equator
,
so
shall
you
find
the
Centers
of
all
the
Parallels
in
the
Axtree
line
prolonged
,
and
by
making
like
profers
as
you
were
taught
for
the
Centers
of
the
Meridians
,
(
Chap.
3.
)
you
may
quickly
find
the
Centers
of
the
Parallels
.
3
A
third
way
.
You
must
consider
that
the
Axtree
line
represents
the
East
Meridian
as
well
as
the
Axis
of
the
world
which
is
a
common
Diameter
to
all
the
Meridians
.
Also
that
every
Parallel
cuts
the
East
Meridian
(
as
it
doth
the
rest
)
in
two
points
Equidistant
from
the
Equinoctial
and
two
Equidistant
also
from
the
Poles
.
Therefore
having
one
point
already
given
in
the
Axtree
line
within
the
fundamental
Circle
where
the
Parallel
shall
cut
,
number
the
distance
from
this
point
to
the
next
Pole
,
and
number
also
the
same
distance
again
beyond
the
Pole
in
the
Axtree-line
prolonged
,
(
being
divided
also
as
you
were
taught
to
divide
the
Equator
line
Chap.
3.
and
at
the
end
of
this
number
shall
the
Parallel
out
the
Axtree
line
again
.
And
the
middle
between
these
two
sections
is
the
Center
.
For
example
,
the
50th
Parallel
is
40
degr.
distant
from
the
Pole
.
Count
therefore
in
the
Axtree
line
prolonged
40
degr.
beyond
the
Pole
,
and
there
is
the
utter
end
of
this
Parallels
Diameter
;
which
if
you
part
in
two
,
the
middle
at
G
is
the
Center
.
4
If
from
the
point
given
,
where
the
Parallel
cuts
the
great
Meridian
,
you
raise
a
Tangent
line
,
this
Tangent
shall
cut
the
Axtree
line
in
the
Center
of
the
Parallel
.
Example
,
The
said
50th
Parallel
cuts
the
great
Meridian
at
H
,
there
I
raise
the
Targent
H
G
perpendicular
to
the
Radius
E
H.
And
this
Tangent
as
you
see
cuts
the
Axtree
line
in
G
the
Center
of
the
Parallel
.
5
Hence
ariseth
a
fifth
way
.
For
it
appears
by
this
figure
that
the
Tangent
of
the
Parallels
distance
from
the
Pole
is
equal
to
his
Semidiameter
:
and
that
the
Secant
of
his
distance
from
the
Pole
is
equal
to
the
distance
of
his
Center
,
from
the
Center
of
the
great
Meridian
.
For
here
E
H
is
Radius
,
H
B
an
arch
of
40
degr.
H
G
the
Tangent
thereof
,
and
Semidiameter
of
the
Parallel
,
E
G
the
Secant
thereof
,
and
the
distance
of
the
Center
of
the
Parallel
from
the
Center
of
the
Meridian
.
And
all
this
is
evident
by
the
structure
in
the
Scheam
.
Wherefore
making
E
H
Radius
,
take
from
your
Scale
or
Sector
,
with
your
Compasses
the
Secant
of
the
Parallels
distance
from
the
Pole
,
and
set
it
from
E
in
the
Axtree
line
,
and
it
shall
end
in
the
Center
of
the
Parallel
.
Or
take
the
Tangent
of
the
Parallels
distance
from
the
Pole
,
and
set
it
from
the
point
of
his
Section
with
the
Meridian
toward
the
extension
of
the
Axtree
line
,
and
where
the
end
of
it
just
toucheth
the
Axtree
line
,
there
is
the
Center
.
6
For
want
of
a
Sector
,
or
other
fit
Scales
of
Tangents
and
Secants
you
may
do
thus
:
Set
one
foot
of
your
Compass
in
the
Center
E
,
and
extend
the
other
upon
the
Diameter
of
the
Equator
or
Axtree
line
,
to
twice
so
many
degr.
as
your
Parallel
is
distant
from
the
Pole
.
That
distance
is
the
very
Tangent
you
seek
.
For
example
,
for
the
40th
Parallel
from
the
Pole
,
I
number
from
E
toward
D
80
degr.
to
8.
now
E
8
is
the
Tangent
of
40
degr.
(
though
it
contain
just
twice
so
many
degr.
of
the
Circle
foreshortned
in
this
projection
:
as
hath
been
shewed
Chap.
3.
Sect.
3.
)
and
so
if
you
will
have
the
Secant
of
40
degr.
take
with
your
Compasses
the
length
from
8
(
where
the
Tangent
ends
)
to
A.
and
that
is
the
Secant
to
be
used
as
was
taught
in
the
last
Section
.
Thus
have
you
wayes
enough
for
finding
the
Centers
of
the
Meridians
and
Parallels
.
And
you
may
have
occasion
in
the
making
of
the
Instrument
to
use
most
of
them
one
time
or
other
.
However
,
the
knowledge
of
them
is
both
pleasant
,
and
usefull
for
the
right
understanding
of
this
and
other
Projections
of
the
Sphear
,
as
also
for
the
examination
of
your
work
,
when
you
shall
chance
to
doubt
of
it
.
CHAP.
V
,
How
to
draw
the
straighter
Meridians
and
Parallels
,
whose
Semidiameters
are
very
long
.
IT
may
trouble
you
very
much
to
draw
those
Meridians
and
Parallels
which
lie
near
to
the
Diameters
;
because
they
be
arches
of
great
Circles
and
require
Compasses
larger
then
you
can
well
get
,
or
manage
when
you
have
gotten
them
.
Till
you
come
to
the
80th
Meridian
from
the
Limb
,
a
Beam-compass
of
a
yard
long
will
reach
,
if
your
Mater
be
not
above
a
foot
Diameter
,
and
a
longer
Beam
you
cannot
well
manage
,
for
it
will
be
apt
to
tremble
with
it's
own
weight
,
and
draw
double
lines
,
though
it
be
made
very
thick
and
massie
.
But
the
89th
Meridian
will
require
a
Beam-compass
of
almost
ten
yards
long
:
For
his
Semidiameter
will
contain
the
Semidiameter
of
the
great
Meridian
57
times
.
Therfore
to
draw
the
10
last
Meridians
and
the
10
last
Parallels
,
you
may
help
your self
one
of
these
wayes
.
-
1.
Guido
Vbaldus
hath
devised
an
Instrument
for
this
purpose
,
consisting
of
three
rulers
in
form
of
an
obtuse
Triangle
.
The
description
and
use
thereof
you
may
see
in
Blagr
.
l.
4.
c
2
,
3.
and
in
Vbaldus
his
book
De
Theorica
Astrolabij
.
But
though
it
be
an
Ingenious
device
,
yet
I
have
found
by
experience
,
that
it
is
a
ticklish
Instrument
,
and
hardly
managed
,
for
which
reason
I
have
hanged
it
by
.
-
2
The
Bow
now
commonly
used
,
is
an
Instrument
not
so
artificial
,
but
more
tractable
and
steddy
then
the
former
.
It
is
made
of
too
steel
rulers
,
the
shorter
of
them
must
be
of
good
substance
,
as
three
quarters
of
an
inch
in
heighth
,
and
as
much
in
breadth
,
that
it
may
be
stiff
,
and
lie
flat
;
the
length
must
be
somewhat
more
then
the
Diameter
of
your
Instrument
:
The
other
may
be
an
inch
longer
,
of
the
same
heighth
but
much
narrower
,
that
it
may
be
bent
out
with
screws
into
an
arch
of
any
Circle
required
;
which
ruler
so
bent
,
being
laid
to
the
three
points
given
,
you
may
by
it
draw
the
arch
required
,
as
easily
as
you
draw
a
straight
line
by
a
straight
ruler
,
The
stiff
ruler
carries
the
screws
,
and
it
must
have
rivets
,
by
which
the
bending
ruler
may
be
staied
at
both
ends
,
while
it
is
bent
by
the
screws
.
See
the
figure
.
CHAP.
VI.
How
to
draw
the
Tropiques
,
and
Polar
Circles
,
and
to
finish
the
Mater
.
BEsides
the
180
Parallels
aboy
ementioned
,
you
have
four
more
to
draw
before
the
Mater
is
finished
.
viz.
the
two
Tropiques
,
and
the
two
Polar
Circles
;
of
which
the
Northern
is
called
the
Arctique
,
and
the
other
the
Antarctique
Circle
.
How
to
draw
these
you
are
sufficiently
instructed
Chap.
4.
if
you
know
but
their
Declination
,
for
they
be
Parallels
.
The
Tropique
of
Cancer
declineth
from
the
Equator
toward
the
North
Pole
23
degr.
30
min.
and
the
Tropique
of
Capricorn
declines
as
much
toward
the
South
Pole
.
The
Arctique
Circle
declines
Northward
66
degr.
30
min.
and
the
Antarctique
as
much
Southward
.
And
these
being
drawn
after
the
manner
of
the
other
Parallels
,
you
have
drawn
all
the
lineaments
of
the
Mater
,
And
the
better
to
adorn
and
distinguish
them
you
shall
with
your
Graver
hatch
every
fifteenth
Meridian
:
for
they
are
hour
lines
.
The
South
arch
of
the
great
Meridian
A
C
B
is
the
hour
of
Noon
:
and
his
North
arch
A
D
B
,
the
hour
of
Midnight
.
These
need
not
be
hatched
,
being
the
Semicircles
of
the
great
Meridian
,
or
fundamental
Circle
,
which
contains
all
but
the
Axtree
line
A
E
B
,
which
is
the
hour
line
of
the
sixes
,
and
the
rest
of
the
hour
lines
counted
from
him
both
waies
,
would
be
hatched
on
both
fides
,
to
shew
like
a
ragged
staff
,
for
distinction
sake
.
Also
every
fifth
Meridian
(
not
being
a
fifteenth
)
you
shall
make
a
pricked
line
,
not
punching
it
with
a
round
point
,
lest
you
make
your
plate
warp
,
but
making
many
short
strokes
cross
the
line
with
your
Graver
,
which
will
be
more
conspicuous
.
Every
tenth
Parallel
also
would
be
a
ragged
line
,
and
the
intermediate
fifths
pricked
lines
:
likewise
the
Tropiques
and
Polar
Circles
would
be
pricked
lines
.
Also
if
your
plate
be
large
,
you
may
set
figures
to
the
hour
lines
,
and
to
every
tenth
Meridian
at
the
Equator
:
but
if
your
plate
be
smal
the
divisions
of
the
Label
applied
upon
the
Equator
may
supply
the
lack
of
them
.
CHAP.
VII
.
Of
the
Reet
,
or
Nets
HAving
shewed
you
what
belongs
to
the
Fabrique
of
the
first
part
of
this
Instrument
called
the
Mater
:
A
few
words
more
will
instruct
you
how
to
make
the
Reet
,
whose
lineaments
are
for
the
most
part
the
same
.
The
Reet
is
a
round
plate
of
metal
,
or
pastboard
,
like
unto
the
Mater
,
but
of
less
Diameter
:
it
must
be
well
planished
and
polished
;
and
the
thinner
the
better
,
if
it
hold
working
it
would
not
be
thicker
then
a
shilling
,
being
of
a
foot
Diameter
.
It
is
called
the
Reet
or
Rete
,
that
is
the
Net
,
because
it
must
be
pierced
through
,
and
made
like
unto
a
Net
or
Lettess
;
that
the
lincaments
of
the
Mater
may
be
perceived
through
it
.
If
we
had
a
transparent
metall
,
much
labour
might
here
be
saved
.
A
clear
Lanthorn
horn
may
serve
for
a
smal
Instrument
,
but
for
large
Instruments
,
it
is
best
to
have
it
either
of
fine
pastboard
,
or
,
if
you
will
go
to
the
cost
,
of
metal
cancelled
;
as
shall
be
taught
.
1
For
the
delineation
of
the
Reet
,
first
draw
your
fundamenttall
Circle
equal
to
the
fundamental
Circle
of
the
Mater
leaving
a
border
or
Limb
without
,
of
such
breadth
as
may
receive
the
graduations
of
the
Circle
and
figures
set
to
them
,
which
breadth
may
be
three
tenths
of
an
inch
,
where
the
Reet
is
a
foot
in
Diameter
:
draw
likewise
two
Diameters
A
B
,
and
C
D
,
crossing
one
another
in
the
Center
E
at
right
Angles
,
and
dividing
the
Circle
into
his
four
Quadrants
,
which
you
shall
subdivide
again
into
90
degr.
apeece
,
as
you
did
in
the
Mater
.
2
You
shall
inscribe
two
arches
,
which
shall
represent
the
Semicircles
of
the
Ecliptique
,
which
shall
meet
at
the
points
C
and
D
of
the
Equator
,
and
the
middle
points
of
these
arches
shall
be
found
in
the
Diameter
A
B
,
thus
.
The
Diameter
A
B
being
divided
as
before
you
were
taught
to
divide
the
Diameters
of
the
Mater
,
number
from
A
toward
the
center
E
23
degr.
30
min.
and
there
make
the
point
F
,
for
♑
:
and
likewise
number
from
B
toward
E
23
degr.
30
min.
and
there
set
the
point
K.
for
♋
.
then
join
the
points
C
F
D
in
one
arch
,
and
the
points
C
K
D
in
another
arch
(
as
is
taught
Chap.
3
)
and
your
Ecliptique
is
drawn
.
But
now
you
must
make
him
a
narrow
Limb
inward
toward
the
Center
,
to
receive
the
scale
of
his
degr.
and
the
characters
of
the
Signes
.
And
to
divide
him
you
shall
do
thus
.
Number
in
the
Axtree
line
(
A
B
,
from
F
inwards
90
degr.
there
is
the
Pole
of
the
arch
C
F
D
,
to
this
Pole
fasten
one
end
of
your
ruler
(
having
an
ey-lid-hole
in
the
edge
for
that
purpose
)
and
carrying
about
the
other
end
over
the
several
degr.
of
the
Semicircle
C
A
D
,
you
shall
cut
the
arch
C
F
D
into
his
correspondent
degrees
.
As
if
you
lay
the
ruler
from
C
to
10
degr.
in
the
Limb
toward
A
,
it
shall
cut
the
Ecliptique
in
♎
10
,
and
so
of
the
rest
.
Likewise
for
the
other
Semicircle
C
K
D
,
find
his
Pole
90
degr.
from
K
toward
F
and
A
:
and
from
that
Pole
by
like
reason
,
you
shall
divide
the
Semicircle
C
K
D
by
the
divisions
of
the
Semicircle
C
B
D.
This
is
the
best
way
.
Or
you
way
divide
the
Ecliptique
by
a
Table
of
Right
Ascensions
,
thus
.
Lay
your
ruler
from
the
Center
E
to
27
deg.
54
min.
in
the
Limb
,
which
is
the
Right
Ascension
of
♉
0.
to
be
counted
from
D
towards
B
,
and
the
ruler
shall
at
the
same
time
cut
the
Ecliptique
in
♉
0
,
to
which
that
Right
Ascension
belongs
,
and
so
for
any
other
deg.
or
you
may
defer
the
dividing
of
the
Ecliptique
,
till
you
have
finished
and
cut
out
the
Reet
:
and
then
if
you
set
the
line
C
D
of
the
Reet
,
in
A
B
the
Axtree
line
of
the
Mater
,
the
Ecliptique
will
lie
among
the
Meridians
of
the
Mater
,
and
shall
be
so
divided
by
the
Parallels
of
the
Mater
,
as
the
Meridians
are
divided
by
them
.
But
my
advice
is
that
you
divide
your
Ecliptique
the
first
way
,
and
you
may
use
this
for
proof
of
your
work
at
last
.
3
The
rest
of
the
lineaments
of
the
Reet
are
the
Azimuths
,
to
be
drawn
as
the
Meridians
of
the
Mater
,
and
the
Almicanters
,
to
be
drawn
as
the
Parallels
.
Onely
you
shall
need
to
draw
but
half
the
Almicanters
,
and
the
Azimuths
but
half
way
,
leaving
one
half
of
your
Reet
viz.
E
C
B
D
blank
and
void
of
them
.
In
drawing
these
Azimuths
and
Almicanters
you
shall
be
carefull
to
skip
over
the
border
of
the
Ecliptique
,
leaving
it
fair
,
that
the
graduations
thereof
with
their
figures
set
to
every
tenth
degr.
and
the
characters
of
the
Signes
may
be
more
distinctly
seen
.
Also
you
shall
do
well
,
if
you
make
a
border
to
the
Axtree
line
on
the
Northside
,
that
is
toward
D
,
and
let
this
border
be
of
the
same
breadth
from
A
to
B
,
the
breadth
not
exceeding
one
fifth
of
an
inch
in
a
Reet
of
a
foot
Diameter
;
upon
which
border
you
may
make
a
scale
of
degrees
,
setting
figuresin
it
to
every
tenth
Almicanter
.
This
will
be
a
great
strength
and
Ornament
to
your
Reet
.
Below
the
Horizon
C
D
likewise
,
you
shall
make
a
Limb
or
border
for
the
Horizon
,
to
receive
his
graduations
:
this
may
be
a
quarter
or
three
tenths
of
an
inch
broad
,
where
the
Reet
is
a
foot
in
Diameter
:
and
upon
this
border
you
shall
set
figures
at
every
tenth
Azimuths
,
and
shall
number
them
both
wayes
,
from
the
Center
and
from
the
Meridian
.
4
You
shall
inscribe
so
many
of
the
fixed
Stars
as
your
Reet
may
well
receive
.
Which
to
do
you
must
know
their
Right
Ascensions
(
or
Culminations
)
and
also
their
Declinations
,
for
which
purpose
I
have
given
you
a
Table
of
110
of
the
more
notable
fixed
Stars
,
(
which
may
best
be
inserted
in
your
Reet
,
)
with
their
Right
Ascensions
and
Declinations
calculated
to
the
year
of
our
Lord
1671.
which
may
serve
for
40
years
before
and
after
without
any
considerable
error
.
To
inscribe
them
you
shall
first
number
the
Right
Ascension
of
the
Star
from
♈
0
,
that
is
from
D
upon
the
Limb
of
the
Reet
toward
B
,
and
at
the
end
of
that
number
fix
your
Label
,
(
which
by
this
time
should
be
made
and
pinned
on
the
Center
,
)
then
from
the
Limb
count
inwards
upon
the
Label
the
Stars
Declination
,
and
at
the
end
of
that
number
make
a
prick
in
your
Reet
close
to
the
edge
of
the
Label
,
there
is
the
Stars
place
.
Then
with
your
Graver
you
shall
make
there
the
shape
of
a
Star
,
with
4
,
5
,
or
6
points
,
according
as
the
magnitude
of
the
Star
deserves
:
and
let
one
point
be
longer
then
the
rest
;
and
let
it
point
outward
from
the
Center
,
if
the
Stars
Declination
be
North
,
but
inward
toward
the
Center
,
if
his
Declination
be
South
:
and
let
the
end
of
his
long
point
(
called
Apex
)
be
in
the
very
true
place
of
the
Star
.
But
if
your
Label
be
yet
unmade
,
then
take
the
measure
of
the
Stars
Declination
with
your
Compasses
upon
any
of
the
four
Semidiameters
of
the
Reet
,
(
measuring
it
from
the
Limb
inwards
,
)
then
lay
a
ruler
from
the
Center
to
the
Right
Ascention
of
the
Star
and
where
the
ruler
cuts
the
Limb
of
the
Reet
;
there
set
one
foot
of
your
Compasses
,
(
opened
as
before
)
and
with
the
other
make
a
prick
toward
the
Center
,
close
to
the
edge
of
your
ruler
,
and
there
is
the
Stars
place
in
your
Reet
.
5
Lastly
,
you
shall
cut
out
all
the
spaces
of
this
Reet
,
which
may
be
spared
;
remembring
alwaies
that
you
leave
uncut
the
borders
of
the
Ecliptique
,
Horizon
,
and
Axtree
line
,
and
be
very
carefull
that
you
cut
not
into
the
Center
of
your
Reet
,
but
leave
breadth
sufficient
about
the
Center
to
hold
the
Center-pin
,
which
must
joyn
Mater
Reet
and
Label
together
.
This
remembred
,
you
shall
cut
out
two
third
parts
of
the
spaces
of
the
Almicanters
,
beginning
from
the
Horizon
C
D
,
and
cutting
out
the
breadth
of
two
degrees
,
after
which
you
shall
leave
the
breadth
of
one
degree
;
and
then
cut
our
again
the
breadth
of
two
degrees
,
and
so
forward
.
But
for
the
greater
strength
and
ornament
of
your
Reet
,
and
for
ease
in
numbring
the
Azimuths
,
you
shall
at
every
15th
Azimuth
leave
a
string
of
the
breadth
of
one
degree
,
whole
from
the
Horizon
to
the
Pole
A
,
or
at
every
thirtieth
Azimuth
leave
such
a
string
going
quite
through
,
and
at
every
other
fifteenth
the
string
may
be
cut
off
when
it
comes
within
ten
degr.
of
the
Pole
,
because
there
the
spaces
of
the
Azimuths
be
very
narrow
and
close
together
.
And
where
among
those
Almicanters
and
Azimuths
you
have
any
Star
,
you
must
contrive
to
leave
him
standing
,
and
to
set
by
him
his
name
or
some
figure
by
which
you
may
know
him
again
.
But
you
are
to
content
your self
with
four
Stars
on
this
side
the
Horizon
,
because
you
will
want
convenient
room
.
On
the
other
side
you
may
have
more
,
and
room
also
to
writ
their
names
upon
strings
or
branches
left
for
that
purpose
;
which
you
may
contrive
into
some
voluntary
lettess-work
,
wherein
you
shall
not
much
regard
uniformity
of
the
Quadrants
,
but
to
make
the
Reet
as
open
as
you
can
,
provided
you
leave
it
of
sufficient
strength
.
Paste
this
on
fol.
17.
so
as
it
may
by
open
while
the
first
7.
Chapters
are
reading
diagram
To
cut
out
the
Reet
in
pastbord
is
much
easier
,
if
you
be
provided
of
sharp
knives
and
chesills
,
fitted
for
your
purpose
.
A
Table
of
the
Right
Ascensions
and
Declinations
of
110
of
the
more
notable
fixed
Stars
;
calculated
from
Tycho
his
Tables
:
rectified
for
the
year
of
our
Lord
1671.
|
As
.
|
mi.
|
D.
|
mi.
|
|
|
Andromeda
her
Head
,
|
357.
|
54.
|
27.
|
18.
|
N.
|
2.
|
(
Mirach
)
Girdle
.
|
12.
|
49.
|
33.
|
55.
|
N.
|
2.
|
Foot
(
Alamath
)
|
25.
|
57
,
|
40.
|
44.
|
N.
|
2.
|
Perseus
his
side
,
Algenib
.
|
44.
|
16.
|
48.
|
36.
|
N.
|
2.
|
Meadusaes
head
,
Algol
.
|
41.
|
46.
|
39.
|
39.
|
N.
|
3.
|
Henerichus
right
shoulder
.
|
85.
|
53.
|
44.
|
56.
|
N.
|
2.
|
Left
shoulder
,
Alhabot
.
|
73.
|
07.
|
45.
|
37.
|
N.
|
1.
|
Left
elbow
.
|
69.
|
29.
|
43.
|
15.
|
N.
|
4.
|
The
Kids
.
|
69.
|
57.
|
40.
|
33.
|
N.
|
4.
|
The
Kids
.
|
70.
|
51.
|
40.
|
42.
|
N.
|
4.
|
The
great
Wain
.
The
Wheels
.
|
160.
|
18.
|
58.
|
08.
|
N
,
|
2.
|
The
great
Wain
.
The
Wheels
.
|
160.
|
48.
|
63.
|
32.
|
N
,
|
2.
|
The
great
Wain
.
The
Wheels
.
|
173.
|
59.
|
55.
|
33.
|
N
,
|
2.
|
The
great
Wain
.
The
Wheels
.
|
179.
|
48.
|
58.
|
51.
|
N
,
|
2.
|
The
Horses
.
|
189.
|
53.
|
57.
|
47.
|
N.
|
2.
|
The
Horses
.
|
197.
|
37.
|
56.
|
41.
|
N.
|
2.
|
The
Horses
.
|
203.
|
37.
|
51.
|
00.
|
N.
|
2.
|
The
little
rain
,
the
Pole
Star
.
|
07.
|
53.
|
87.
|
34.
|
N.
|
2.
|
Last
Wheel
.
|
231.
|
14.
|
73.
|
15.
|
N
,
|
3.
|
Dragons
tongue
.
|
254.
|
36.
|
54.
|
55.
|
N.
|
4.
|
Head
first
,
|
260.
|
46.
|
52.
|
34.
|
N.
|
3.
|
Head
last
,
Ras
Aben.
|
267.
|
15.
|
51.
|
36.
|
N.
|
3.
|
Tail
|
167.
|
15.
|
71.
|
05.
|
N.
|
3.
|
Bootes
Arcturus
.
|
210.
|
13.
|
20.
|
58.
|
N.
|
1.
|
Engonasi's
Head
,
|
254.
|
12.
|
14.
|
50.
|
N.
|
3
|
Ophiucus
Head
.
|
259.
|
55.
|
12.
|
52.
|
N.
|
3.
|
Lyra
,
|
276.
|
27.
|
38.
|
30.
|
N.
|
1.
|
In
Ariadnes
brow
,
(
Mumir
)
|
230.
|
12.
|
27.
|
51.
|
N.
|
2.
|
The
Swans
bill
,
|
289.
|
23.
|
27.
|
18.
|
N.
|
3.
|
The
Eagles
Heart
,
|
293.
|
41.
|
08.
|
03.
|
N
|
2.
|
Tail
,
|
282.
|
36.
|
13.
|
25.
|
N
|
3.
|
The
Dolphins
tail
,
|
304.
|
24.
|
10.
|
14.
|
N
|
3.
|
Pegasus
his
mouth
,
(
Eniph
)
|
322.
|
03.
|
8.
|
24.
|
N
|
3.
|
Head
,
|
328.
|
24.
|
4
|
38.
|
N.
|
4.
|
Neck
,
|
336.
|
21.
|
9.
|
08.
|
N.
|
3.
|
Right
shoulder
(
Sheat
,
)
|
341.
|
59.
|
26.
|
18.
|
N.
|
2.
|
First
in
the
wing
,
(
Marcab
)
|
342.
|
07.
|
13.
|
28.
|
N.
|
2
,
|
Last
in
the
wing
,
|
359.
|
08.
|
13.
|
22.
|
N.
|
2.
|
The
Rams
head
,
|
27.
|
12.
|
21.
|
54.
|
N.
|
3.
|
Horn
,
|
23.
|
54.
|
17.
|
40.
|
N
|
4.
|
Horn
,
|
24.
|
08.
|
19.
|
12.
|
N.
|
4.
|
The
Bulls
North
eye
,
|
62.
|
23.
|
18.
|
26.
|
N.
|
3.
|
South
eye
,
(
Aldebaran
)
|
64.
|
17.
|
15.
|
48.
|
N
|
1.
|
Brightest
of
the
Pleiades
,
|
52.
|
00.
|
23.
|
03.
|
N
|
3.
|
Gemini
,
Head
of
Apollo
,
|
108.
|
24
|
32.
|
33.
|
N.
|
2.
|
Head
of
Hercules
,
|
111.
|
19.
|
28.
|
46.
|
N.
|
2.
|
The
first
foot
,
|
88.
|
48.
|
22.
|
33.
|
N
|
4.
|
The
first
foot
,
|
90.
|
48
|
22.
|
38.
|
N.
|
3.
|
The
third
foot
,
|
94.
|
41
|
16
|
38.
|
N.
|
2.
|
Cancer
,
Praecepe
,
|
125.
|
22.
|
20.
|
48.
|
N
|
ne
|
Leo
,
the
Lions
neck
,
|
150.
|
26.
|
21.
|
29.
|
N.
|
2.
|
Heart
,
(
Regulus
,
|
147.
|
43.
|
13.
|
33.
|
N
|
1.
|
Back
,
|
164.
|
08.
|
22.
|
20.
|
N
|
2.
|
Tail
,
|
173.
|
04.
|
16.
|
25.
|
N.
|
1.
|
Virgo
,
Spica
♍
,
Azimech
,
|
196.
|
56.
|
9.
|
31.
|
S.
|
1.
|
Libra
,
South
ballance
,
|
218.
|
13.
|
14.
|
37.
|
S.
|
2.
|
North
ballance
,
|
224.
|
52.
|
8.
|
07.
|
S.
|
|
Scorpio
,
in
his
forehead
the
lowest
,
|
234.
|
50.
|
25.
|
06.
|
S.
|
3.
|
Midst
,
|
235.
|
15.
|
21.
|
38.
|
S.
|
3.
|
Highest
,
|
236.
|
36.
|
18.
|
51.
|
S.
|
3.
|
Heart
,
(
Antares
,
)
|
242.
|
23.
|
25.
|
37.
|
S.
|
1.
|
Sagittaries
head
,
the
midst
,
|
281.
|
18.
|
22.
|
07.
|
S.
|
4
|
Capricoms
horn
,
the
highest
,
|
299.
|
58.
|
13.
|
29.
|
S.
|
3.
|
Tayl
,
|
320.
|
29.
|
18.
|
02.
|
S.
|
3.
|
Tayl
,
|
322.
|
15.
|
17.
|
32.
|
S.
|
3.
|
Aquarius
right
shoulder
,
|
327.
|
16.
|
1.
|
52.
|
S.
|
3.
|
left
shoulder
,
|
318.
|
07.
|
6.
|
56½
.
|
S
|
3.
|
Right
leg
,
(
Scheat
,
)
|
339.
|
22.
|
17.
|
31.
|
S.
|
3.
|
End
of
the
flood
(
Fomahant
,
)
|
339.
|
46.
|
31.
|
17.
|
S.
|
1.
|
The
Whale
,
Mandibula
,
|
41.
|
21.
|
2.
|
48.
|
N
|
2.
|
Mouth
,
the
midst
,
|
36.
|
36.
|
1.
|
50.
|
N
|
3.
|
Check
,
|
35.
|
44.
|
1.
|
07.
|
S.
|
3.
|
Belly
,
(
Baten
Kaitos
)
|
23.
|
52.
|
11.
|
54.
|
S.
|
3.
|
Tayl
,
Northern
,
|
0.
|
43.
|
10.
|
37.
|
S.
|
3.
|
Tayl
,
Southern
|
6.
|
45.
|
19.
|
48.
|
S.
|
2.
|
Orion
.
Right
shoulder
,
Bedalgieure
|
84.
|
23.
|
07.
|
18.
|
N
|
1.
|
Left
shoulder
Bellatrix
,
|
76.
|
54.
|
06.
|
01.
|
N
|
2.
|
Girdle
the
first
,
|
78.
|
52.
|
0.
|
35.
|
S.
|
2.
|
Second
,
|
79.
|
55.
|
1.
|
26.
|
S.
|
2.
|
Third
,
|
81.
|
04.
|
2.
|
09.
|
S.
|
2.
|
Left
foot
,
(
Rigel
)
|
74.
|
44.
|
8.
|
37.
|
S.
|
1.
|
Right
foot
,
|
83.
|
03.
|
9.
|
49.
|
S.
|
3.
|
Thegreat
Dogs
mouth
,
Alhaber
,
Sirius
,
|
97.
|
42.
|
16.
|
14½
|
S.
|
1.
|
Formost
foot
,
|
92.
|
08.
|
17.
|
49.
|
S.
|
2.
|
The
litle
dogs
neck
,
|
107.
|
22.
|
8.
|
54.
|
N
|
3.
|
Thigh
,
Procyon
,
|
110.
|
34.
|
6.
|
03.
|
N
|
2.
|
Hydra
,
the
heart
,
(
Alphard
)
|
137.
|
54.
|
7.
|
15.
|
S.
|
2.
|
Corvus
,
in
his
beak
,
the
first
of
the
□
,
|
177.
|
50.
|
22.
|
54
|
S.
|
4.
|
Second
of
the
□
,
|
179.
|
49.
|
15.
|
39.
|
S.
|
3.
|
Third
of
the
□
,
|
183.
|
16.
|
14.
|
39.
|
S.
|
3.
|
Fourth
□
,
|
184.
|
20.
|
21.
|
33.
|
S.
|
3.
|
CHAP.
VIII
.
Of
the
Ring
,
or
Limb
of
the
Mater
.
THe
third
part
of
this
Instrument
is
the
Ring
or
Limb
,
which
is
nothing
else
but
the
skirt
of
a
Circular
plate
equal
with
the
Mater
,
whose
middle
is
cut
out
by
a
lesser
concentrique
Circle
.
It
is
bounded
with
two
Parallel
Circles
,
the
outmost
must
touch
the
edge
of
the
Mater
round
the
inner
Circle
or
edge
of
this
Ring
:
it
must
be
a
little
less
then
the
Limb
of
the
Reet
,
that
it
may
take
hold
of
the
Reet
to
keep
it
flat
and
safe
from
harms
.
This
Ring
had
need
be
thicker
then
the
Reet
,
but
not
so
thick
as
the
Mater
;
and
for
breadth
,
about
1/10
part
of
the
Diameter
.
It
must
be
pinned
or
screwed
on
to
the
Mater
with
6
or
8
pinnes
or
screws
,
that
so
you
may
take
your
plates
asunder
when
need
is
to
cleanse
them
from
any
stain
or
dust
that
may
get
between
.
Let
the
pins
that
carry
the
naile-screws
be
riverted
in
the
Ring
,
and
chair
heads
so
filed
down
and
polished
,
that
they
be
not
seen
to
check
the
Label
:
and
holes
being
made
in
the
Mater
for
the
pins
to
pass
through
,
you
shall
have
smal
screws
of
what
fashion
you
like
best
,
to
turn
upon
them
on
the
backside
;
these
screws
would
be
made
all
of
a
length
,
and
may
serve
as
feet
to
bare
up
the
Instrument
about
a
third
part
of
an
inch
from
the
ground
,
that
it
be
not
scratched
and
be
readier
to
take
up
.
And
that
the
Reet
may
turn
more
pleasantly
under
the
Ring
,
and
lie
as
near
as
may
be
in
the
same
plain
with
the
Ring
,
you
shall
abate
half
the
thickness
of
the
upper
edge
of
the
Reet
,
about
a
barly-corn's
breadth
round
about
,
so
far
as
he
shall
run
under
the
Ring
and
likewise
aba
e
half
the
thickness
of
the
inner
edge
of
the
Ring
on
the
lower
side
where
he
clasps
down
the
Reet
,
(
which
a
good
Turner
knowes
now
to
do
;
)
or
you
may
make
a
shift
to
do
it
with
a
beam-compass
,
if
you
make
your
running
point
like
a
narrow
chesill
.
Your
Ring
thus
fitted
to
the
Mater
,
you
shall
set
one
foot
of
your
Compasses
in
the
Center
of
the
Mater
,
and
with
the
other
draw
near
the
inner
edge
of
the
Limb
a
Circle
about
8/100
of
an
inch
distant
from
the
edge
.
Also
opening
your
Compasses
about
⅙
of
an
inch
more
you
shall
draw
another
Parallel
Circle
:
and
laying
your
Ruler
or
Label
from
the
Center
to
the
several
degrees
of
the
fundamental
Circle
of
the
Mater
,
you
shall
draw
short
lines
for
every
degree
from
the
inner
edge
of
the
Ring
to
the
first
Circle
,
and
every
tenth
degree
you
shall
prolong
to
the
second
Circle
,
and
let
every
fifth
be
drawn
half
way
.
Between
these
two
Circles
also
you
shall
set
figures
to
every
tenth
degree
;
numbring
from
the
Equinoctial
line
C
D
to
the
Poles
on
both
sides
both
wayes
.
Also
without
the
second
Circle
you
shall
set
great
figures
for
the
several
Hours
,
setting
XII
at
C
,
and
thence
proceeding
in
order
to
the
right
hand
toward
B
,
at
15
degrees
set
I
,
at
30
degr.
set
II
,
at
45
degr
,
set
III
,
and
so
on
,
till
you
come
to
D
,
where
you
must
set
XII
.
Thence
you
shall
proceed
in
the
other
Semicircle
D
A
C
,
setting
I
,
II
,
III
,
and
so
on
in
order
,
till
you
be
come
round
.
And
remember
that
you
write
on
the
Ring
,
at
A
Oriens
,
at
C
Meridies
:
at
B
Occidens
,
and
at
D
Septentrio
.
CHAP.
IX
.
Of
the
Ephemeris
or
Calender
,
on
the
Ring
.
IF
there
be
space
enough
left
upon
the
Ring
without
the
Circles
of
the
degrees
and
Hours
,
you
may
fill
it
up
with
the
Ephemeris
of
the
Sun
in
this
manner
:
The
former
Scale
on
the
inner
edge
of
the
Ring
shall
serve
you
to
this
purpose
for
an
Ecliptique
;
and
you
may
set
to
him
the
Characters
of
the
Signes
,
if
you
will
,
at
every
thirtieth
degree
;
beginning
at
Oriens
and
there
setting
♈
,
and
♋
at
Meridies
,
and
the
rest
in
like
order
.
Then
draw
another
Scale
without
this
,
upon
the
Ring
,
consisting
of
two
spaces
.
In
the
inner
space
shall
be
the
Dayes
of
the
Year
:
in
the
outer
space
:
which
must
be
a
little
larger
)
shall
be
the
Names
of
the
moneths
in
their
order
.
And
to
divide
this
Scale
rightly
,
you
shall
do
thus
.
Go
to
some
Eshemeris
for
the
Leap
Year
that
next
comes
,
viz.
1660
,
(
or
rather
for
some
Leap
Year
about
20
Years
hence
,
that
your
Scale
may
serve
without
any
Prosthapheresis
,
for
40
Years
to
come
without
sensible
error
,
and
beginning
your
year
with
March
,
look
where
the
Sun
was
on
the
29
of
February
at
Noon
;
which
you
shall
find
to
be
♓
20
degr.
47
min
,
for
the
Year
1660.
Therefore
laying
the
Label
or
a
Ruler
from
the
center
to
♓
20
degr.
47
min.
in
the
inner
Scale
,
strike
a
long
stroke
through
your
outward
Scale
,
and
from
thence
begin
your
Year
,
writing
from
thence
toward
the
right
hand
March
1660.
Then
lay
the
Label
to
♓
21
degr.
47
min
,
(
which
is
the
Suns
place
on
the
first
day
of
March
at
noon
the
same
year
)
and
where
it
cuts
the
outer
Scale
,
mark
the
first
day
of
March
,
and
so
the
rest
in
order
.
And
to
the
first
day
of
every
moneth
,
you
shall
set
his
proper
Letter
which
belongs
to
him
in
the
Calender
;
as
to
the
first
of
March
you
shall
set
D
,
and
to
the
first
of
April
G
,
&c.
and
when
you
have
done
December
,
you
must
take
the
Suns
place
for
January
and
February
out
of
the
next
years
Ephemeris
,
viz.
1661
,
and
note
that
the
space
for
the
last
day
of
the
year
(
Febr.
28
)
will
fall
out
to
be
less
by
a
fourth
part
then
the
rest
,
by
reason
that
the
Sun
wants
almost
6
hours
to
finish
his
Circle
,
which
he
finishes
in
dayes
365
,
5
hours
48
minutes
.
And
for
this
cause
these
Scales
will
serve
you
to
find
the
Suns
place
at
noon
,
for
any
day
in
a
like
year
,
that
is
every
fourth
year
,
accounted
hence
,
either
backwards
or
forwards
;
which
year
shall
evermore
be
accounted
to
begin
from
Febr.
29.
and
may
be
accounted
the
first
year
after
Leap
year
,
because
the
Intercalation
was
February
25
next
before
.
Then
for
the
year
next
following
,
viz
,
1661.
(
beginning
March
1
and
being
second
from
the
Bissextile
or
Leap
year
)
these
Scales
shall
give
you
the
place
of
the
Sun
at
six
hours
after
noon
,
and
the
third
year
from
Bissextile
1662
(
beginning
as
before
March
1
)
these
Scales
shall
give
you
the
Suns
place
12
hours
after
noon
,
or
the
midnight
following
.
And
the
fourth
year
1663
being
Bissextile
,
these
Scales
shew
the
place
of
the
Sun
at
18
hours
after
noon
,
the
next
year
1664
,
being
the
first
after
Bissextile
,
and
beginning
(
as
aforesaid
)
March
1
,
is
the
very
same
year
for
which
your
Scale
is
made
and
gon
,
for
that
year
:
your
Scale
shewes
the
Suns
place
at
noon
again
.
But
because
the
Julian
years
are
bigger
then
the
true
Solar
years
by
almost
12
mi.
of
time
(
that
is
,
near
a
quarter
of
an
hour
)
in
which
time
the
Sun
moves
27
sec.
13
thirds
37
fourths
,
therefore
when
you
have
found
the
Suns
place
by
the
former
Scale
,
any
year
after
1660
,
look
how
many
years
are
passed
since
1660
and
so
many
times
you
must
add
27
sec.
13
thirds
37
fourths
,
(
that
is
almost
half
a
minute
)
to
the
Suns
place
found
:
and
for
years
past
●ou
must
subtract
as
much
,
that
you
may
find
the
Suns
place
exactly
.
This
Prosthapheresis
in
2
or
3
years
is
scarce
considerable
in
an
Instrument
,
but
in
10
years
there
will
be
4
minutes
32.
seconds
,
and
in
20
years
9.
minutes
5.
seconds
,
to
be
added
after
1660.
and
as
much
to
be
subtracted
in
like
number
of
years
preceding
the
year
1660.
to
which
this
Scale
is
supposed
to
be
framed
.
This
Ephemeris
or
Calender
M.
Blagrave
would
have
on
the
back-side
,
where
hee
would
also
have
a
Ruler
with
Sights
,
to
take
the
Altitude
of
the
Sun
or
Stars
.
But
this
will
be
found
incommodious
in
many
respects
,
both
in
the
framing
,
and
in
the
using
;
and
therefore
I
advise
that
nothing
be
set
on
the
back-side
but
the
Tables
of
the
Prime
,
Epact
,
and
Cycle
of
the
Sun
,
thereby
to
find
the
age
of
the
Moon
,
her
Conjunctions
and
Oppositions
,
and
the
moveable
Feasts
for
ever
,
Of
which
see
Chap.
11.
CHAP.
X.
Of
the
Label
and
Sights
.
THe
Label
is
a
Ruler
slit
in
the
midest
,
and
the
half
of
it
cut
away
to
the
Head
,
where
it
is
pinned
to
turn
upon
the
Center
,
and
reaching
to
the
outside
of
the
Limb.
The
Fiducial
edge
thereof
,
which
pointeth
upon
the
Center
,
must
be
graduated
,
like
to
the
semidiameters
of
the
Mater
and
Reet
,
into
90
degrees
,
to
be
numbred
either
inward
or
outward
.
The
fashion
of
it
may
be
understood
by
the
figure
without
more
words
.
To
this
Label
you
may
fit
Sights
,
either
fixed
or
moveable
,
as
you
like
best
,
for
observing
Altitudes
and
Azimuths
:
but
for
taking
.
Azimuths
you
had
need
have
one
tall
Sight
,
at
least
half
as
long
as
the
Label
:
and
then
it
had
need
be
moveable
,
to
take
off
at
pleasure
.
For
taking
the
Altitude
of
the
Sun
,
I
have
made
a
pair
of
moveable
Sights
,
to
slip
up
and
down
upon
the
edge
of
the
Planisphear
;
having
on
the
backside
springing
plates
of
brass
to
pintch
them
close
,
and
make
them
stick
where
you
set
them
.
These
are
commonly
to
be
set
at
C
and
D
the
ends
of
the
Equinoctial
line
.
At
A
in
the
Limb
and
in
the
Circle
next
unto
the
inner
edge
which
boundeth
the
strokes
of
the
severall
degrees
,
you
shall
drill
a
small
hole
,
through
which
you
may
put
a
thred
to
hang
a
plummet
on
.
The
Sun
then
shining
through
the
Sights
placed
at
C
and
D
,
the
plumb-line
shall
shew
his
Altitude
in
the
semicircle
B
C
A
,
you
beginning
to
number
from
B
,
and
observing
where
the
plumb-line
crosseth
the
Circle
in
which
the
hole
for
hanging
the
plumb-line
was
made
.
And
here
you
must
remember
that
because
the
plumb-line
falleth
not
from
the
Center
of
the
Planisphear
,
but
from
a
point
in
the
circumference
about
A
,
therefore
the
space
of
two
degrees
must
be
taken
but
one
degree
,
so
that
if
the
Plumb-line
fall
20
degr.
below
B
toward
C
,
the
Suns
Altitude
is
10
,
degrees
as
you
may
see
demonstrated
,
Euclid
.
3.20
.
and
Pitisc
.
Trigonem
.
1.53
.
And
thus
you
may
observe
the
Suns
Altitude
neer
the
Horizon
,
as
exactly
as
by
a
Quadrant
,
whose
semediameter
were
equal
to
the
diameter
of
your
Planisphear
.
But
if
the
Altitude
exceed
30
or
40.
degrees
then
will
the
Plumb-line
cut
the
limb
too
slope
and
have
too
much
play
to
your
trouble
:
For
remedy
whereof
you
shall
remove
the
Sight
at
D
towards
A
some
degrees
:
as
for
example
60
degrees
,
by
which
means
you
shall
abate
the
Suns
Altitude
30
degrees
,
which
30
degrees
must
be
added
to
the
Altitude
observed
:
as
for
example
,
the
Sights
are
placed
one
at
C
,
the
other
60
degrees
above
D
toward
A
,
and
the
Plumb-line
cuts
10
degrees
from
B
towards
C
,
I
say
,
then
is
the
Sun
5
degrees
high
and
30
degrees
more
,
in
all
35
degrees
:
in
like
manner
you
may
place
the
Sight
at
any
other
number
of
degrees
from
D
toward
A
,
as
you
shall
find
most
convenient
for
the
present
Altitude
;
remembring
always
that
how
many
degrees
soever
you
remove
the
Sight
,
half
so
many
are
to
be
added
to
the
Altitude
found
.
But
if
your
Reet
happen
to
run
so
far
under
the
Limb
,
that
you
cannot
make
a
center-hole
for
the
Plumbet
through
the
Limb
and
Mater
,
without
hindring
the
Reets
motion
,
then
may
you
have
a
small
plate
of
sheet
brass
,
in
fashion
of
an
Arm
or
Tongue
,
in
the
point
whereof
you
shall
have
a
Center-hole
drilled
,
and
this
plate
shall
be
so
joyned
with
a
sluce
or
screw
about
the
Limb
near
A
,
that
the
Center-hole
made
in
this
plate
may
lie
close
to
the
point
where
the
Center
should
have
been
boared
in
the
Limb
in
the
line
A
B
:
and
thus
you
may
put
it
on
,
and
take
it
off
,
at
pleasure
,
that
it
hinder
not
the
motion
of
the
Reet
or
Label
.
Of
the
fashion
of
the
Sights
see
more
Booke
4.2
.
CHAP.
XI
.
Of
the
perpetual
Calender
,
on
the
back-side
.
ON
the
back-side
of
your
Planisphear
you
may
set
the
Calender
following
,
which
consisteth
of
three
Tables
gathered
round
.
The
longest
would
be
set
outtermost
.
The
first
is
the
Table
of
the
Cycle
of
the
Sun
,
that
is
of
the
Sundays
Letter
.
This
is
here
placed
in
the
middle
.
It
is
a
Cycle
of
28.
years
,
in
which
time
the
Dominical
Letter
runs
all
his
changes
,
(
caused
by
the
one
odd
day
above
52
weeks
)
in
every
Common
year
,
and
two
odd
days
which
run
over
the
even
weeks
in
the
Leap
years
.
To
find
the
beginning
of
this
Cycle
,
add
to
the
year
of
our
Lord
9.
(
because
the
first
year
of
our
Lord
,
as
wee
commonly
acount
,
was
the
10
of
this
Cycle
)
and
divide
the
sum
by
28.
the
remainder
is
the
year
of
the
Cycle
running
;
and
if
nothing
remain
,
then
it
is
the
28.
or
last
year
.
So
you
shall
find
that
the
Cycle
now
running
shall
end
with
the
Julian
year
1671
as
in
the
Table
;
and
shall
begin
again
with
the
year
of
our
Lord
1672.
Thus
may
you
renew
the
Table
when
it
is
expired
;
or
make
this
very
Table
serve
you
for
ever
.
Example
.
Enter
this
Table
with
the
year
of
our
Lord
1656.
now
running
;
and
you
shaall
find
over
against
this
year
in
the
next
space
inwards
13.
shewing
you
that
it
is
the
13
year
of
the
Suns
Cycle
;
(
so
shall
the
28
year
forward
viz.
1684
be
again
the
13th
year
of
the
Cycle
next
comming
.
)
In
the
next
space
within
,
you
have
the
Dominical
Letters
F
and
E
,
(
because
it
is
Leap
year
)
F
shall
be
the
Dominical
Letter
till
you
come
past
the
Intercalary
day
,
which
is
the
day
following
the
24th
of
February
:
and
for
the
rest
of
this
year
the
Dominical
Letter
shall
be
E
;
(
for
the
Letters
always
change
backwards
)
also
you
shall
note
here
that
the
day
inserted
in
every
4th
year
is
not
February
29
,
but
February
25.
for
February
24.
(
being
6
Cal.
Martij
)
is
repeated
again
in
the
Leap
year
:
and
they
write
again
February
25.6
Cal.
Martij
:
and
the
25th
of
February
in
the
Leap
year
is
marked
with
the
same
letter
F
,
wherewith
the
24th
of
February
hapneth
always
to
be
marked
.
Hence
the
Leap
year
is
called
Annus
Bissextilis
:
and
note
that
by
the
Eclesiasticall
Law
S.
Matthias
day
,
which
is
February
24th
in
common
years
,
in
the
Bissextile
years
is
to
be
observed
on
February
25.
Nevertheless
in
our
Secular
Law
not
the
24th
and
25th
of
February
.
but
the
28th
and
29th
in
the
Leap
year
are
ordained
to
be
one
day
in
the
account
of
Law
:
as
by
Statutum
de
Anno
Bissextili
,
made
in
21
year
Henry
3.
may
appear
.
The
second
Table
is
of
the
Cycle
of
the
Moon
,
consisting
of
the
Prime
or
Golden
Number
,
and
the
Epact.
This
Table
contains
19
years
,
which
is
the
Annus
Metonicus
:
in
which
space
of
time
Meton
an
Astronomer
about
430
years
before
Christ
,
observed
the
Moon
to
finish
all
her
variations
.
So
that
every
19
year
the
mean
conjunctions
or
changes
should
happen
upon
the
same
days
of
the
moneth
,
that
they
did
happen
upon
19
years
before
;
onely
an
hour
and
half
sooner
.
Yet
beeause
every
19
years
contain
not
the
same
number
of
Leap
years
,
but
sometime
there
come
five
Leap
years
in
19
years
,
and
sometimes
but
four
,
therefore
there
may
happen
in
this
Period
of
Meton
an
error
of
an
whole
day
,
besides
the
hour
and
half
above
mentioned
.
For
remedy
whereof
Calippus
,
about
the
330
year
before
Christ
,
devised
to
quadruple
this
period
of
Meton
,
making
Period●s
Calippica
of
76
years
,
which
contains
just
19
quaternions
of
years
;
so
containing
always
the
same
number
of
Leap
years
and
days
.
This
period
is
therefore
more
perfect
then
Metons
:
for
after
this
period
of
76
years
,
the
Moon
runneth
over
the
same
course
for
her
conjunctions
and
oppositions
;
changing
in
the
same
year
of
the
period
always
,
upon
the
same
day
of
the
moneth
,
save
onely
that
shee
changeth
sooner
by
six
hours
in
the
latter
period
then
in
the
former
.
But
the
Church
still
retaineth
the
period
of
Meton
,
called
the
Prime
,
or
Golden
number
;
because
it
used
to
be
set
in
Golden
letters
in
the
Kalender
,
in
a
certain
artificiall
order
throughout
every
moneth
,
to
guide
you
to
the
day
of
the
Moons
Priming
or
Changing
:
so
may
you
find
it
in
red
letters
,
thus
,
Set
to
the
Kalender
printed
in
large
folio
with
the
Book
of
common
Prayer
,
in
the
time
of
the
late
Queen
Elizabeth
.
To
find
the
year
of
the
Prime
,
add
to
the
year
of
our
Lord
1
,
and
divide
the
sum
by
19.
the
remainder
is
the
year
of
the
Prime
,
or
if
nothing
remain
it
is
the
19th
or
last
year
.
Thus
you
may
find
the
present
year
1656
,
to
be
the
fourth
year
of
the
Prime
,
and
so
you
find
it
in
the
Table
.
The
Cycle
or
Period
now
running
,
ends
with
1671.
and
begins
again
with
the
year
following
.
The
Epact
or
Concurrent
is
set
against
the
Prime
in
the
next
space
inwards
:
and
finisheth
his
Cycle
in
the
same
time
.
It
was
devised
to
find
more
readily
the
day
of
the
change
,
and
age
of
the
Moon
.
The
way
to
find
it
is
this
.
Multiply
the
Golden
number
serving
for
your
year
by
11
,
and
divide
the
product
by
30
;
the
residue
is
the
Epact
for
your
year
.
Or
having
the
Epact
known
for
any
year
,
you
may
make
it
from
year
to
year
,
by
adding
11
to
the
Epact
of
the
year
foregoing
,
and
casting
away
30
when
the
sum
exceeds
30.
The
reason
whereof
is
,
that
the
Moon
changeth
12
times
in
354.
days
,
that
is
11
days
before
the
Sun
hath
gone
his
round
:
for
which
cause
the
changes
must
needs
happen
every
year
11
days
sooner
.
Observe
here
that
the
Prime
changeth
every
year
the
first
day
of
January
;
the
Epact
not
till
the
first
of
March
.
The
Dominical
letter
changeth
upon
the
first
of
January
going
one
letter
backward
yearly
;
and
in
every
Leap
year
it
changeth
again
on
the
Lords
day
next
after
February
24.
The
reason
whereof
is
this
:
The
Calender
is
marked
throughout
with
the
letters
of
the
week
A
B
C
D
E
F
G
,
and
the
same
day
of
the
year
is
always
marked
with
the
same
letter
,
now
if
the
year
contained
even
weeks
then
would
the
Dominical
letter
be
always
one
and
the
same
:
but
because
a
common
year
contains
52
weeks
and
one
day
,
therefore
the
last
day
of
the
year
must
be
marked
with
A
,
as
the
first
was
;
the
first
day
of
the
year
following
again
is
marked
with
A
:
now
put
case
the
last
of
December
marked
A
,
be
Sunday
in
the
year
1654.
the
day
following
viz
January
1
,
is
Munday
,
and
yet
marked
A
;
and
the
first
Sunday
in
1655
must
needs
fall
on
January
7
marked
with
G
:
and
so
G
became
Dominical
letter
for
1655.
as
A
was
for
1654.
and
as
those
2
days
marked
with
the
same
letter
A
,
coming
together
at
every
years
end
cause
a
change
of
the
Dominical
letter
,
so
in
the
Leap
year
the
Intercalary
day
February
the
25
,
being
marked
with
F
,
always
as
the
24th
is
,
the
coming
together
of
those
two
dayes
marked
with
one
letter
,
causeth
a
second
change
of
the
Dominical
letter
for
that
year
by
like
reason
.
The
third
Table
is
a
Table
to
find
Easter
for
ever
.
This
is
placed
outmost
because
it
is
the
longest
.
It
was
very
falsly
ordered
in
M.
Blagraves
book
:
and
so
it
is
in
Grostons
Tables
,
from
whence
I
suppose
he
transcribed
it
.
I
have
set
it
right
and
straight
,
and
taken
what
care
I
could
that
the
Printer
or
Graver
do
not
put
the
ranks
into
the
same
disorder
in
which
I
found
them
,
both
in
Blagrave
,
and
in
Grostons
Tables
,
printed
8
years
before
him
.
diagram
CHAP.
XII
.
Some
cautions
to
be
Observed
in
the
making
of
the
Instrument
.
THough
I
have
taught
the
making
of
the
Mater
first
as
being
the
base
and
principal
part
of
the
Instrument
;
yet
I
shall
advise
you
first
to
draw
and
cut
out
the
Reet
,
and
fit
the
Label
to
it
,
leaving
it
sufficient
length
to
reach
to
the
out-side
of
the
Mater
.
And
then
having
your
Ring
ready
fitted
to
screw
on
to
the
Mater
you
shall
drill
the
centers
of
the
Mater
,
Reet
,
and
Label
,
with
the
same
drill
;
and
fasten
them
together
with
a
Center-pin
well
fitted
to
the
bore
:
this
Center-pin
must
not
be
too
big
:
let
it
be
square
at
the
head
to
stick
in
the
backside
of
the
Mater
,
that
it
turn
not
;
and
let
the
other
end
have
a
male
screw
,
upon
which
you
shall
turn
a
female
screw
,
to
draw
the
plates
together
;
so
that
they
be
neither
too
loose
nor
stick
too
hard
.
Then
take
a
great
needle
,
or
such
a
round
point
of
hardned
steel
,
and
bearing
it
up
into
some
corner
of
the
Reet
close
to
the
inside
of
his
Limb
,
turn
about
both
Reet
and
Needle
upon
the
Mater
,
so
that
the
Needle
may
trace
out
the
great
Meridian
of
the
Mater
,
and
so
the
principall
Circles
of
the
Mater
and
Reet
shall
be
sure
to
be
Concentrique
,
and
to
argree
in
all
postures
of
the
Reet
:
whereas
if
you
draw
the
Meridian
of
the
Mater
before
you
have
bored
the
Center
,
and
fitted
on
the
Reet
,
you
shall
hardly
happen
to
bore
so
true
,
but
you
shall
find
the
Circles
to
be
a
little
Eccentrique
and
interfering
one
with
another
.
And
to
avoid
the
like
Eccentricity
in
the
Circles
of
the
Ring
,
it
were
best
to
have
a
sorry
Beam
Compass
made
onely
for
this
purpose
,
and
fitted
to
your
Center
pin
to
draw
them
by
:
yet
if
those
be
drawn
from
the
Center
of
the
Mater
unbored
,
there
will
be
no
perceiveable
error
,
if
you
devide
them
from
the
Limb
of
the
Reet
by
the
Label
when
all
is
pinned
together
.
These
things
done
,
quarter
the
Meridian
of
the
Mater
with
two
cross
Diameters
,
and
divide
them
as
is
above
directed
;
remembring
to
apply
your
Reet
to
the
Mater
often
,
to
see
how
the
divisions
of
the
Diameter
agree
,
and
how
the
Meridians
and
Parallels
which
you
are
drawing
on
the
Mater
agree
with
the
Azimuths
and
Almicanters
of
the
Reet
.
And
thus
by
comparing
your
plates
often
,
and
examining
the
lineaments
of
the
one
by
the
other
,
as
you
draw
them
,
you
shall
avoid
many
slips
and
mistakes
,
and
proceed
in
your
work
with
more
confidence
and
contentment
.
The
Third
Book
.
Of
the
Resolution
of
all
Spherical
Triangles
,
by
the
PLANISPHEAR
.
CHAP
,
I
,
Of
the
kinds
and
parts
of
Spherical
Triangles
.
IT
is
to
be
known
,
that
1
,
Spheri
.
Triang
.
are
-
Rectangular
-
Obliquangular
.
2
A
Rectangular
Triangle
is
that
which
hath
one
or
more
Right
Angles
.
3
A
Triangle
that
hath
three
right
Angles
hath
alwaies
his
three
sides
Quadrants
.
4
A
Triangle
that
hath
two
right
angles
hath
the
sides
opposite
to
those
angles
Quadrants
,
and
the
third
side
is
the
measure
of
the
third
angle
.
So
that
of
those
Triangles
which
have
more
Right
angles
,
seldome
ariseth
any
question
.
But
the
Right
angled
Triangle
with
one
Right
and
two
Acute
angles
,
is
that
which
comes
most
commonly
to
be
resolved
.
5
A
Right
angle
is
that
which
containeth
90
degrees
,
or
openeth
to
one
quarter
of
the
circumference
of
any
Circle
described
from
the
angular
point
.
6
All
Spherical
Triangles
not
Rectangled
are
called
Oblique-angled
.
And
if
they
have
one
angle
greater
then
a
Right
angle
,
they
be
called
Obtuse-angled
,
otherwise
they
be
Acute-angled
.
7
In
rectangled
Triangles
the
sides
including
the
Right
angle
be
called
Legs
,
the
side
subtending
it
is
called
Subtense
or
Hypotenusa
.
8
Either
Leg
of
a
rectangled
Triangle
may
be
made
Basis
,
(
if
you
will
imagine
him
to
lie
level
)
and
then
the
other
leg
shall
be
called
Cathetus
or
Perpendicular
.
9
In
Oblique
angled
Triangles
,
the
sides
comprehending
the
angle
given
or
sought
,
are
called
Legs
,
and
the
third
side
the
Base
.
10.
In
every
Spherical
Triangle
there
be
(
beside
the
Area
or
space
contained
)
six
containing
parts
viz.
three
sides
,
and
three
angles
:
of
those
six
there
must
be
three
alwaies
known
or
given
to
find
out
the
rest
.
CHAP.
II.
Of
the
16
Cases
of
Rectangled
Triangles
.
And
how
they
may
be
reduced
to
five
Problemes
.
IN
the
Rectangular
Spherical
Triangle
there
be
five
parts
onely
come
into
the
Question
;
the
three
sides
,
and
two
acute
angles
:
because
the
third
(
being
right
)
is
alwaies
known
.
Of
these
five
parts
any
two
being
given
the
rest
may
be
found
.
There
be
16
Cases
or
Problemes
about
Spherical
Triangles
.
six
for
finding
the
Legs
:
four
for
the
Hypotenusa
:
and
six
for
the
Angles
.
See
Gellibrand
.
Trigonom
.
Britan.
But
we
may
here
reduce
them
all
to
5
Problemes
,
because
our
Planisphear
resolveth
alwaies
two
of
them
at
once
.
For
by
two
parts
given
you
shall
presently
get
two
of
the
three
that
are
unknown
:
and
if
you
do
but
turn
the
Triangle
,
you
may
presently
have
the
third
also
:
as
shall
appear
in
this
book
.
The
Rectangular
Triangle
in
question
shall
be
marked
throughout
this
book
with
A
B
C
in
this
maner
:
so
that
the
Base
shall
be
marked
and
called
B
A
,
the
Cathetus
,
or
Perpendicular
Leg
C
A
,
and
the
Hypotenusa
.
B
C.
diagram
The
right
angle
A.
the
angle
at
the
Base
B.
the
angle
at
the
Cathetus
C.
And
note
that
in
the
4
first
Problemes
B
A
shall
be
alwaies
set
upon
the
Equinoctial
line
of
the
Mater
,
C
A
in
a
Meridian
,
B
C
on
the
Label
,
and
B
alwaies
at
the
Center
;
as
you
shall
find
in
the
four
next
Chapters
.
CHAP.
III.
PROBL.
I.
The
Legs
given
,
to
find
the
rest
.
diagram
Now
may
you
number
B
C
the
Hypotenusa
upon
the
Label
,
and
find
it
60.
degrees
,
the
angle
B
hath
his
measure
on
the
Limb
between
the
Equator
,
and
the
Label
23
degrees
½
.
Lastly
to
find
the
angle
C
do
but
turn
the
Triangle
,
setting
C
A
on
the
Equinoctial
,
and
B
A
on
a
Meridian
,
(
according
to
the
1.8
.
)
and
laying
the
Label
to
B
you
shall
find
B
C
60
,
degrees
as
before
,
and
the
measure
of
the
angle
C
between
the
Equator
and
the
Label
may
be
reckoned
on
the
Limb
77.
degr.
43.
min.
Note
here
that
in
this
and
all
other
like
cases
,
in
stead
of
the
Label
you
may
better
use
one
of
the
Semidiameters
of
the
Reet
;
for
they
have
the
same
graduation
,
and
lie
closer
to
the
Mater
.
CHAP.
IIII.
PROBL.
II.
A
Leg
and
the
Hypotenusa
given
to
find
the
rest
.
SEt
the
Hypotenusa
B
C
on
the
Label
from
the
Center
:
the
given
Leg
(
to
be
marked
C
A
)
in
one
of
the
Meridians
.
Example
.
In
the
former
Triangle
,
where
the
Hypotenusa
was
60.
degrees
,
the
Cathetus
20.
degrees
12.
minutes
.
I
number
on
the
Label
from
the
Center
60
degrees
,
and
there
make
a
prick
for
C
;
then
I
turn
this
prick
to
the
Parallel
of
20.
degrees
12.
minutes
,
and
the
Triangle
is
made
.
For
the
Meridian
cutting
this
prick
is
Cathetus
,
and
hath
betwixt
this
prick
and
the
Equator
20.
degrees
12.
minutes
,
as
the
Parallels
shew
.
Where
this
Cathetus
cuts
the
Equator
stands
the
right
angle
A
,
and
between
A
and
the
Center
lies
B
A
,
57.
degrees
48.
minutes
:
the
measure
of
B
is
on
the
limb
between
Meridies
and
the
Label
23
degrees
30.
minutes
.
And
for
the
angle
C
turn
the
Triangle
.
Set
C
now
at
the
Center
,
calling
it
B
,
and
you
may
find
this
angle
as
you
did
his
fellow
.
CHAP.
V.
PROBL.
III.
The
Hypotenusa
and
an
Angle
given
,
to
find
the
rest
.
SEt
the
Hypotenusa
on
the
Label
,
the
Angle
given
at
the
Center
.
Example
.
In
the
former
Triangle
the
Hypotenusa
was
60.
degrees
,
and
the
greater
angle
77.43
.
I
number
in
the
Limb
from
Meridies
toward
Oriens
77.
degrees
43.
minutes
,
and
there
I
set
the
Label
,
then
I
look
the
60.
degree
of
the
Label
numbred
from
the
Center
,
at
that
60
degree
is
C
where
the
Hypotenusa
and
Cathetus
meet
.
Here
therefore
the
Meridian
that
cuts
the
Label
in
60
degrees
makes
the
Cathetus
:
I
follow
him
down
to
the
Equator
and
find
his
length
57
degrees
48
minutes
,
and
from
the
point
where
he
cuts
the
Equator
I
go
straight
to
the
Center
,
and
find
20
degrees
12
minutes
the
length
of
B
A.
Lastly
for
the
angle
C
turn
the
Triangle
,
setting
C
at
the
Center
,
and
calling
it
B
;
and
you
shall
find
C
as
chap.
3.
For
whereas
your
Hypotenusa
is
60
,
and
your
Cathetus
20
degrees
12
minutes
,
lay
the
60
degree
of
the
Label
upon
the
20
Parallel
,
and
the
Label
shall
cut
in
the
Limb
23
degrees
30
minutes
,
the
measure
of
the
angle
C.
CHAP.
VI.
PROBL.
IIII.
A
Leg
and
an
Angle
given
to
find
the
rest
.
IF
the
Leg
be
conterminate
or
adjoyning
to
the
angle
given
,
then
make
the
given
Leg
Base
,
setting
it
upon
the
Equator
;
and
move
the
Label
from
the
Equator
toward
the
Pole
,
so
many
degr
,
as
the
given
angle
B
comes
to
.
Then
mark
what
Meridian
cuts
the
end
of
the
Base
,
that
Meridian
makes
the
Cathetus
:
follow
him
till
he
crosseth
the
Label
,
in
that
crossing
is
the
angle
C
of
your
Triangle
,
from
whence
you
reckon
the
length
of
the
Cathetus
to
the
Equinoctial
,
and
the
length
of
the
Hypotenusa
next
way
from
C
to
the
Center
.
And
now
having
all
the
sides
,
to
get
the
angle
C
you
shall
turn
the
Triangle
,
and
get
him
as
in
the
3
Chapter
.
But
if
the
Leg
be
opposite
to
the
Angle
given
,
make
the
given
Leg
Cathetus
,
that
the
angle
given
may
be
at
the
Center
.
Example
.
In
the
former
Triangle
I
have
given
the
less
angle
23
degrees
½
,
and
the
Leg
opposite
thereto
20.
degr.
12.
min.
I
open
therefore
the
Label
from
the
Equator
to
23
deg.
30.
min.
on
the
Lamb
,
and
mark
where
he
cutteth
the
20
⅕
Parallel
,
for
there
is
the
angle
C
of
the
Triangle
;
thence
you
shall
have
a
Meridian
for
the
Cathetus
going
to
the
Equator
,
whose
length
is
20
degrees
12
,
minutes
:
thence
in
the
Equator
to
the
Center
is
the
Base
57
degrees
47
minutes
;
and
thence
in
the
Label
to
C
again
,
is
the
Hypotenusa
60
degrees
.
And
now
having
all
the
sides
,
to
get
the
angle
C
,
you
shall
turn
the
Triangle
,
and
get
him
as
in
the
3
Chapter
.
CHAP.
VII
.
PROBL.
V.
The
Angles
given
to
find
the
Sides
.
THis
should
have
been
the
third
Probleme
,
considering
what
is
given
.
But
because
the
way
of
resolving
this
case
differeth
from
all
the
former
,
therefore
I
have
reserved
it
to
the
last
place
.
In
plain
Triangles
this
case
is
insoluble
,
But
in
Spherical
Triangles
it
may
be
resolved
on
the
Planisphear
,
two
wayes
.
1.
The
first
way
hath
M.
Elagrave
5
,
24.
He
sets
B
C
on
the
Limb
between
the
Pole
of
the
Mater
and
the
Vertex
of
the
Reet
,
B
A
and
C
A
one
on
a
Meridian
,
and
the
other
on
an
Azimuth
crossing
one
another
at
a
Right
angle
within
the
Limb.
Which
to
do
you
must
work
thus
.
Example
.
In
the
former
Triangle
,
the
Angles
are
given
A
90
,
B
23
,
degr.
30.
min.
C
77
,
degr.
43.
min.
Now
first
I
will
guess
that
the
Hypotenusa
is
40
degrees
,
and
setting
the
Zenith
40
degrees
from
the
Pole
,
that
arch
of
the
Limb
between
Pole
and
Zenith
I
take
for
my
Hypotenusa
,
yet
unknown
.
At
the
Pole
shall
be
B
,
and
at
the
Zenith
C.
Then
because
B
is
23
½
I
take
the
twentie-third
Meridian
from
my
Hypotenusa
which
is
on
the
Limb
,
and
between
that
and
the
24th
I
imagin
a
Meridian
which
shall
make
B
A
of
my
Triangle
.
Also
because
the
Angle
at
C
is
77
,
deg.
43.
min.
I
take
an
imaginary
Azimuth
near
the
78th
,
numbred
from
my
Hypotenusa
which
is
on
the
Limb
,
and
that
Azimuth
shall
make
C
A
of
my
Triangle
.
Now
have
I
the
Angles
B
and
C
,
and
three
arches
of
which
all
the
sides
of
my
Triangle
shall
be
made
:
but
whether
B
A
and
C
A
cross
at
Right
Angles
I
know
not
,
and
therefore
I
know
not
yet
certainly
the
length
of
any
side
.
Now
to
make
the
Angle
at
A
a
Right
Angle
,
mark
where
your
Azimuth
(
which
you
have
taken
for
C
A
)
cuts
the
Finiter
,
and
from
that
point
number
in
the
Finiter
toward
the
Center
to
90
degrees
,
and
there
is
the
Pole
of
your
Azimuth
;
(
viz.
12
degrees
17
minutes
from
the
Limb
)
make
a
prick
with
ink
at
that
Pole
,
and
then
look
whether
your
Meridian
23
½
(
which
you
took
for
B
A
)
cut
this
Pole
,
which
yet
he
doth
not
as
you
will
find
.
Therefore
turn
the
Reet
till
the
said
Meridian
do
cut
this
Pole
of
the
said
Azimuth
,
and
then
you
may
be
sure
that
Meridian
and
Azimuth
,
(
wherever
they
cross
,
)
do
make
Right
Angles
.
(
By
Pitisc
Trigonom.
1
,
57
)
Therefore
now
have
you
all
the
Angles
set
on
the
Planisphear
,
and
thereby
all
the
sides
found
,
viz.
B
C
in
the
Limb
60
,
degr.
B
A
in
the
Meridian
57
degrees
48
minutes
C
A
in
the
Azimuth
20
degrees
12
minutes
,
as
they
ought
to
be
.
2.
The
second
way
M.
Oughtred
useth
:
It
is
this
.
For
the
rectangled
Triangle
whose
three
Angles
be
given
,
you
shall
frame
another
oblique
Triangle
,
whose
sides
shall
be
equal
to
the
Angles
of
the
first
Triangle
.
And
so
the
Angles
of
the
second
Triangle
to
be
found
on
the
Planisphear
shall
be
equal
to
the
sides
of
the
first
Triangle
,
which
are
inquired
.
The
Triangle
which
serveth
for
an
Example
throughout
this
Book
hath
his
Angles
A
90
degrees
C
77
degrees
43
minutes
,
B
23
degrees
30
minutes
.
Here
to
find
the
sides
,
for
the
Angle
A
,
I
take
half
the
Axis
of
the
Mater
from
the
Center
to
the
Pole
,
and
that
shall
be
a
side
of
90
degrees
.
For
the
lesser
Angle
B
,
I
reckon
upon
the
Label
from
the
Center
a
side
of
23
degrees
30
minutes
,
and
at
the
end
thereof
make
a
prick
,
on
the
Label
.
I
turn
this
prick
upon
the
Parallel
from
the
Pole
77
degrees
43
minutes
,
and
the
Meridian
there
Crossing
shall
be
the
third
side
of
this
Triangle
.
Follow
this
Meridian
to
the
Equator
,
and
from
his
cutting
there
to
the
Center
is
the
measure
of
the
lesser
Angle
of
this
second
Triangle
,
which
is
equal
to
the
lesser
Leg
of
the
first
Triangle
.
20
degr.
12
minutes
:
likewise
the
arch
of
the
Limb
from
the
Pole
to
the
Label
,
is
the
measure
of
the
middle
Angle
of
the
second
Triangle
,
which
is
equal
to
the
middle
side
,
that
is
,
to
the
greater
Leg
of
the
first
Triangle
57
degrees
48
minutes
.
And
now
having
found
the
Legs
of
the
first
Rectangled
Triangle
you
may
by
the
first
Probleme
find
the
Hypotenusa
to
be
60
degrees
.
The
Rectangled
Triangle
used
in
the
first
way
,
and
the
oblique
Quadrantal
Triangle
used
in
this
second
way
,
shall
appear
in
such
formes
on
the
Planisphear
as
these
figures
following
do
express
.
The
first
Way
.
diagram
The
second
Way
.
diagram
CHAP.
VIII
.
How
to
represent
and
resolve
the
Cases
of
the
four
first
Problemes
of
Spherical
Triangles
,
divers
other
wayes
.
ONe
way
hath
been
shewn
for
representing
any
Rectangled
Spherical
Triangle
upon
the
Planisphear
,
by
the
Label
,
Equator-line
,
and
a
Meridian
,
and
thereupon
to
find
out
the
sides
and
Angles
of
any
such
Triangle
.
Now
for
Variety
sake
,
and
for
the
exercise
of
Learners
in
the
knowledge
of
the
Sphear
,
and
because
the
same
Angle
somtimes
may
be
more
distinctly
represented
in
one
part
of
the
Planisphear
then
in
another
,
I
have
thought
good
to
set
down
six
other
wayes
,
by
which
the
four
first
Cases
of
Spherical
Rectangled
Triangles
may
be
pictured
on
the
Planisphear
,
and
resolved
.
There
be
three
places
in
the
Planisphear
where
the
Angle
B
may
be
placed
,
whether
he
be
given
,
or
sought
.
-
1.
At
the
Center
,
and
there
his
quantity
is
measured
by
the
Label
or
any
Semidiameter
of
the
Reet
,
moving
upon
the
Ring
.
thus
was
B
placed
in
the
former
Chapters
,
and
shall
be
once
more
in
the
first
Variety
.
-
2.
At
either
of
the
Poles
of
the
Mater
,
where
by
the
Meridians
that
issue
thence
you
may
number
the
quantity
of
any
Angle
from
0
,
to
180.
-
3.
At
the
Zenith
or
Vertex
of
the
Reet
,
where
the
quantity
of
the
Angle
may
be
numbred
by
the
Azimuths
in
like
manner
.
CHAP.
IX
.
The
first
Variety
.
HEre
the
angle
B
shall
be
at
the
Center
as
before
;
B
A
on
the
Finiter
,
C
A
in
an
Azimuth
,
B
C
in
the
Axis
of
the
Mater
.
So
shall
you
have
your
Triangle
pictured
in
the
same
form
and
quantity
that
he
had
in
the
former
chapters
though
other
lines
be
here
used
.
And
to
resolve
the
four
first
Cases
of
Rectangled
Spherical
Triangles
with
these
Circles
,
you
shall
,
-
1.
In
the
first
Case
where
B
A
and
C
A
are
given
,
Number
B
A
from
the
Center
upon
the
Finiter
;
where
it
ends
,
you
shall
meet
an
Azimuth
upon
,
which
you
shall
number
C
A
toward
the
Zenith
;
in
the
top
of
C
A
make
a
prick
with
ink
for
C
,
and
then
turn
that
prick
to
touch
the
Axis
of
the
Mater
.
Thus
have
you
all
the
sides
in
view
,
and
the
measure
of
the
ngle
B
you
shall
find
upon
the
Limb
,
between
the
Pole
and
the
Finiter
.
And
for
C
you
must
turn
the
Triangle
as
before
hath
been
taught
.
-
2.
In
the
second
Case
.
B
C
and
C
A
given
,
Number
B
C
in
the
Axis
from
the
Center
,
and
at
his
end
for
C
make
a
prick
;
Then
for
C
A
count
to
what
Almicantar
he
will
rise
from
the
Finiter
,
and
turn
the
Reet
till
that
Almicantar
cut
the
prick
C
in
the
Axis
;
and
the
Azimuth
there
crossing
the
Axis
and
Almicantar
in
C
shall
make
C
A.
And
between
that
Azimuth
and
the
Center
shall
be
B
A
on
the
Finiter
.
B
shall
be
measured
as
before
.
C
shall
be
found
as
before
.
-
3.
In
the
third
Case
B
C
and
B
given
.
Set
the
Finiter
as
much
from
the
Pole
as
the
angle
B
comes
to
.
Then
number
B
C
in
the
Axis
from
the
Center
B
to
C.
Thence
turn
down
in
the
next
Azimuth
to
the
Finiter
and
you
make
C
A.
Thence
turn
to
the
Center
,
and
you
close
the
Triangle
with
B
A.
C
shall
be
found
as
before
.
-
4.
In
the
fourth
Case
,
if
B
and
B
A
be
given
.
Set
the
Finiter
to
the
angle
B
,
as
in
the
third
case
in
this
chap.
and
from
the
Center
upon
the
Finiter
number
B
A.
from
A
go
up
an
Azimuth
to
the
Axis
,
where
C
shall
stand
.
From
thence
go
to
the
Center
,
and
you
have
compassed
your
Triangle
,
and
all
is
shown
by
the
view
,
but
the
angle
C
,
which
may
be
found
as
before
.
But
if
B
and
C
A
be
given
,
set
the
Finiter
to
the
angle
,
as
in
the
third
Case
of
this
chap.
then
count
in
what
Almicantar
C
A
will
end
,
and
follow
this
Almicantar
to
the
Axis
,
where
they
meet
is
the
point
C.
And
the
Azimuth
that
cutteth
there
shall
cut
the
Finiter
in
the
place
of
A.
CHAP.
X.
The
second
and
third
Varieties
.
SEt
B
at
one
of
the
Poles
of
the
Mater
where
the
Meridians
meet
;
B
A
on
the
Limb
either
;
way
B
C
in
a
Meridian
C
A
on
the
Label
.
One
Example
of
the
third
Case
shall
suffice
.
B
and
B
C
are
given
.
B
is
77
degrees
43
minutes
;
Number
therfore
the
Meridians
from
the
Limb
till
you
come
past
77
and
almost
to
78
,
and
there
imagin
a
Meridian
to
be
drawn
for
the
Hypotenusa
of
your
Triangle
;
That
Meridian
maketh
an
angle
of
77
degrees
43
minutes
with
the
Limb
,
as
the
Hypotenusa
of
your
Triangle
doth
with
the
Base
.
And
because
the
Hypotenusa
B
C
is
60.
therfore
the
60th
Parallel
reckoned
from
the
Pole
shall
determin
his
length
,
and
cut
him
off
in
the
point
C.
Prick
the
point
C
(
that
is
,
the
crossing
of
the
77
43
Meridian
with
the
60th
Parallel
from
the
Pole
)
and
to
that
prick
lay
the
Label
:
so
that
part
of
the
Label
which
lieth
between
the
prick
and
the
Limb
shall
be
C
A
,
and
the
arch
of
the
Limb
between
the
Pole
and
the
Label
shall
be
B
A
of
the
Triangle
.
So
shall
all
be
known
but
C
,
which
also
may
be
found
if
you
turn
the
Triangle
as
before
.
Or
thirdly
,
using
only
the
Reet
and
Label
,
Set
B
at
the
Zenith
;
B
A
on
the
Limb
of
the
Reet
;
A
C
in
the
Label
;
B
C
in
an
Azimuth
,
and
you
shall
make
the
same
Triangle
on
the
Reet
,
that
you
made
last
on
the
Mater
.
CHAP.
XI
.
The
fourth
Variety
.
SEt
B
at
the
Zenith
of
the
Reet
.
B
A
upon
the
Limb
from
the
Zenith
to
one
of
the
Poles
of
the
Mater
,
C
A
in
the
Axtree-line
of
the
Mater
,
B
C
in
an
Azimuth
.
CHAP.
XII
.
The
fifth
Variety
.
SEt
B
at
one
of
the
Poles
of
the
Mater
,
B
A
upon
the
Limb
between
the
Pole
and
Zenith
,
C
A
in
the
Axis
of
the
Reet
,
B
C
in
a
Meridian
.
Note
here
,
that
if
you
set
the
Triangle
upon
the
Planisphear
either
of
those
two
last
wayes
,
you
shall
find
him
to
be
set
both
wayes
,
and
that
you
haue
your
Triangle
twice
found
,
or
two
Trangles
each
of
them
representing
the
Triangle
in
Question
;
one
is
toward
the
right
hand
and
the
other
toward
the
left
:
And
they
are
both
comprehended
between
the
Axletrees
of
the
Mater
and
Reet
;
and
the
arch
of
the
Limb
which
lie
:
between
the
two
Axletrees
is
Base
to
them
both
.
CHAP.
XIII
.
The
sixth
Variety
.
SEt
the
Zenith
line
of
the
Reet
in
the
Equinoctial
line
of
the
Mater
.
Then
set
B
at
the
Zenth
B
A
upon
the
Eqiunoctial
line
inwards
from
the
Limb
,
C
A
in
a
Meridian
,
B
C
in
an
Azimuth
.
Thus
have
you
various
wayes
for
describing
and
resolving
any
rectangular
Spherical
Triangle
upon
your
Planisphear
.
If
in
trying
one
way
you
find
the
points
of
your
Triangle
too
much
shadowed
with
the
Reet
,
or
that
the
sides
cross
one
another
too
obliquely
,
that
you
can
hardly
find
the
point
of
the
angle
,
then
may
you
trie
another
way
,
and
you
shall
likely
find
that
fault
amended
.
These
three
last
Chapters
you
shall
easily
understand
,
if
you
understand
the
former
Chapters
of
this
Book
.
And
therefore
I
thought
it
needless
to
use
any
further
examplification
.
CHAP.
XIV
.
Of
the
Solution
of
Oblique
angled
Spherical
Triangles
:
And
generally
of
all
Spherical
Triangles
.
THese
six
Chapters
following
might
well
have
been
placed
before
the
second
Chapter
.
For
howsoever
they
best
serve
Oblique
angled
Triangles
,
yet
are
the
Rules
general
,
and
may
serve
very
well
for
the
solution
of
all
Spherical
Triangles
whatsoever
.
But
I
like
this
order
well
enough
,
and
I
think
the
Reader
will
have
no
cause
to
dislike
it
.
There
be
twelve
Cases
of
Oblique-angled
Spherical
Triangles
.
But
for
the
Planisphear
they
are
here
reduced
to
six
.
And
they
be
all
(
unless
I
may
except
the
last
)
as
easily
resolved
upon
the
Planisphear
as
the
five
Cases
of
Rectangled
Triangles
.
Here
note
that
for
Oblique-angled
Triangles
,
in
all
the
Cases
following
,
one
side
shall
evermore
be
set
upon
the
Limb
between
one
of
the
Poles
of
the
Mater
and
the
Zenith
,
of
the
Reet
,
and
the
other
two
sides
shall
be
made
,
one
by
a
Meridian
,
and
the
other
by
an
Azimuth
;
at
the
meeting
whereof
is
the
Angle
C
,
which
onely
may
remain
unknown
after
any
Question
resolved
,
and
may
be
presently
found
by
turning
the
Triangle
,
as
before
it
hapned
in
the
Rectangled
Triangles
.
PROB.
1.
Three
Sides
given
,
to
find
the
Angles
.
SEt
one
Side
,
(
which
you
will
)
upon
the
Limb
between
the
Pole
and
Zenith
,
count
the
second
side
from
the
Pole
by
the
Parallels
,
and
count
the
third
side
from
the
Zenith
by
the
Almicanters
:
and
know
that
where
the
last
Parallel
cuts
the
last
Almicanter
,
there
is
the
point
of
the
third
Angle
C
:
the
Meridian
that
passeth
from
this
point
to
the
Pole
is
the
side
A
C
:
the
Azimuth
that
passeth
from
the
same
point
to
the
Zenith
is
the
side
B
C
:
and
the
third
side
A
B
is
on
the
Limb
between
the
Pole
and
Zenith
.
Now
may
you
count
the
Angles
at
the
Pole
and
Zenith
and
B
:
and
for
the
third
Angle
C
,
turn
the
Triangle
laying
one
of
the
other
sides
in
the
Limb
between
the
Pole
and
Zenith
,
and
you
shall
find
that
Angle
also
,
as
you
did
his
fellowes
.
Note
that
whereas
I
called
the
sides
of
Rectangled
Triangles
B
A.
C
A
,
and
B
C
,
that
is
Basis
,
Cathetus
,
and
Hypotenusa
,
I
choose
here
in
Oblique-angled
Triangles
,
to
transpose
the
letters
of
the
two
first
sides
for
diftinction
sake
,
calling
them
A
B
,
and
and
A
C
,
and
the
third
side
indifferently
either
B
C
or
C
B.
Example
.
Let
40
degrees
70
degrees
and
46
degrees
½
be
Sides
of
a
Triangle
,
whose
Angles
are
sought
.
Now
because
I
would
first
get
the
Angles
joyning
to
the
side
40
degrees
.
I
mark
that
side
A
B
,
and
set
A
B
upon
the
Limb
,
A
at
he
Pole
,
and
B
at
the
Zenith
;
which
I
remove
40
degrees
from
the
Pole
,
according
to
the
length
of
the
side
A
B.
Then
because
A
C
is
70
deg.
I
hold
one
finger
(
or
a
pin
)
upon
the
70
Parallel
from
the
Pole
,
and
because
B
C
is
46
½
I
hold
another
finger
on
the
46
½
Almicanter
counted
from
the
Zenith
,
and
look
where
this
Almicantar
crosseth
the
said
70
Parallel
,
there
is
C
of
my
Triangle
:
The
Meridian
that
comes
from
the
Pole
to
C
is
the
long
side
of
my
Triangle
A
C
;
I
count
then
from
the
side
A
B
on
the
Limb
how
many
Meridians
lie
between
A
B
and
A
C
,
and
I
find
that
A
C
is
just
the
45
Meridian
,
therefore
I
say
the
Angle
A
at
the
Pole
is
45
degrees
.
The
Azimuth
that
comes
from
the
Zenth
to
C
is
here
the
middle
side
of
my
Triangle
,
being
in
length
46½
I
count
from
the
side
A
B
of
my
Triangle
on
the
Limb
how
many
Azimuths
there
are
to
this
and
I
find
that
this
is
the
114
Azimuth
almost
,
therefore
the
Angle
B
at
the
Zenith
is
almost
114
degrees
(
exactly
113.30
.
minutes
.
)
Now
to
find
the
Angle
C
,
I
turn
the
Triangle
,
and
set
B
C
,
46
degrees
½
on
the
Limb
(
changing
the
letters
into
A
B
)
And
where
the
40
Parallel
crosseth
the
70
Almicanter
,
there
I
meet
with
the
39
Azimuth
,
which
shews
me
that
the
third
Angle
formerly
called
C
,
and
now
since
the
Triangle
turned
marked
B
is
39
degrees
(
exactly
38
deg
51
minutes
.
)
diagram
CHAP.
XV.
PROB.
2.
Two
Sides
and
an
Angle
comprehended
given
,
to
find
the
rest
.
SEt
the
Angle
given
at
the
Pole
and
set
one
of
the
given
sides
in
the
Limb
between
the
Pole
and
Zenith
,
the
other
given
side
you
shall
reckon
on
that
Meridian
which
is
distant
from
the
Limb
as
much
as
the
given
Angle
cometh
to
,
at
the
end
thereof
there
shall
meet
you
an
Azimuth
which
shall
make
the
third
side
of
your
Triangle
.
Example
.
In
the
former
Triangle
having
A
B
40.
A
C
70
and
the
Angle
comprehended
at
A
45
degrees
,
I
set
A
B
on
the
Limb
from
Pole
to
Zenith
,
then
because
the
Angle
A
is
45
degrees
I
take
the
45
Meridian
reckoned
from
A
B
,
and
thereof
I
take
70
degrees
(
counting
from
A
the
Pole
)
for
my
side
A
C
:
at
C
in
the
70
degree
of
this
Meridian
there
crosseth
an
Azimuth
which
makes
my
third
side
;
this
Azimuth
is
the
113
½
being
numbred
from
A
B
therefore
the
Angle
at
B
is
113
½
,
and
I
finde
between
B
and
C
in
this
Azimuth
46
½
for
the
length
of
the
side
B
C
:
onely
C
is
now
unknown
,
which
you
may
also
find
by
turning
the
Triangle
.
CHAP.
XVI
.
PROB.
3.
Two
Sides
and
an
Angle
opposite
to
one
of
them
given
,
to
find
the
rest
.
diagram
Example
.
I
have
given
A
B
37
degrees
45
min.
A
C
105
degrees
41
minutes
:
and
B
167
degrees
09
minutes
:
I
set
the
Zenith
B
37
degrees
45
minutes
from
the
Pole
A
,
then
because
B
is
167
degrees
9
minutes
,
I
count
the
Azimuths
from
the
side
A
B
to
107
degrees
,
and
farther
9
minutes
,
and
I
know
that
the
Azimuth
there
imagined
to
pass
(
set
between
167
,
and
168
)
shall
make
the
side
of
my
Triangle
B
C
;
but
yet
the
length
of
B
C
I
know
not
;
and
I
want
still
a
Meridian
for
the
side
A
C
opposite
to
the
angle
given
.
Now
because
A
C
his
length
is
given
(
105
degrees
41
minutes
,
though
the
Angle
A
be
yet
unknown
)
I
take
the
Parallel
105
degrees
41
minutes
,
numbred
from
the
Pole
A
,
and
where
this
Parallel
crosseth
the
167
Azimuth
,
there
I
am
sure
must
be
the
Angle
C
:
and
the
Meridian
passing
from
C
to
the
Pole
is
the
side
A
C
105
degrees
41
minutes
:
this
Meridian
lieth
between
the
12
and
13
number
from
A
B
,
and
sheweth
the
Angle
A
to
be
12
degrees
26
minutes
:
the
side
B
C
,
I
may
count
68
degrees
½
by
help
of
the
Almicantars
.
Now
have
I
three
sides
and
two
Angles
,
which
are
more
then
enough
to
find
the
Angle
C
,
when
the
Triangle
is
turned
.
Note
that
you
may
place
the
known
Angle
at
the
Pole
as
well
as
at
the
Zenith
,
and
it
may
be
needfull
so
to
do
when
the
Angle
C
of
your
Triangle
would
otherwise
fall
under
the
Limb
of
the
Zodiaque
.
Note
also
that
the
Angle
C
may
sometime
fall
under
the
Finiter
,
where
the
Azimuths
faile
.
As
if
you
had
set
the
Angle
167
degrees
at
the
Pole
,
the
opposite
side
105
degrees
41
minutes
had
been
set
in
an
Azimuth
,
and
C
had
been
beyond
the
Finiter
:
your
remedie
in
this
case
is
to
set
Nadir
in
the
place
of
Zenith
,
so
shall
C
fall
among
the
Azimuths
just
as
you
would
have
him
.
Example
.
Set
37
degrees
45
minutes
between
the
Pole
and
Nadir
(
a
b
)
count
the
Angle
given
at
the
Nadir
b
167
degrees
9
minutes
and
his
supplement
12
degrees
51
minutes
,
for
A
C
count
a
C
the
supplement
thereof
,
and
you
shall
find
b
C
111
½
,
whose
supplement
is
B
C
68
½
.
Note
thirdly
,
that
if
the
angle
given
in
this
chapter
be
a
cute
,
then
if
you
place
the
known
Angle
at
the
Zenith
,
the
Parallel
may
cross
the
Azimuth
twice
;
or
if
you
place
the
known
Angle
at
the
Pole
,
the
Almicanter
taken
to
find
out
the
opposite
side
,
may
cross
the
Meridian
twice
;
and
so
it
may
be
doubtfull
in
which
intersection
the
Angle
C
shall
be
found
:
That
you
may
discover
,
if
you
examine
which
agrees
best
with
the
other
parts
of
the
Triangle
being
turned
;
or
if
you
reduce
this
Triangle
to
two
Right-angled
Triangles
,
by
letting
fall
a
Perpendicular
.
Of
which
see
the
last
Chapter
.
CHAP.
XVII
PROB.
4.
Two
Angles
and
the
Side
comprehended
between
them
being
given
,
to
find
the
rest
.
SEt
the
side
given
between
the
Pole
and
Zenith
on
the
Limb
then
count
one
Angle
among
the
Meridians
,
the
other
among
the
Azimuths
,
and
where
the
Meridian
and
Azimuth
bounding
the
said
Angles
meet
,
there
is
the
point
of
the
Angle
C
,
and
all
is
known
but
the
Angle
C
,
which
you
may
find
also
,
if
you
turn
the
Triangle
.
Example
.
In
the
14th
Chap.
The
Angle
A
was
45
B
113
½
the
side
A
B
comprehended
40.
Having
set
the
Zenith
40
from
the
Pole
,
I
seek
the
45th
Meridian
from
A
B
,
and
the
113
½
Azimuth
from
A
B
,
and
where
they
cross
is
C.
Now
may
I
number
A
C
by
the
Parallels
70
and
B
C
by
the
Almicantars
46
½
.
C
may
now
be
found
by
any
of
the
3
former
Problemes
,
if
you
turn
the
Triangle
,
and
set
C
at
the
Pole
,
or
at
the
Zenith
.
CHAP.
XVIII
PROB.
5.
Two
Angles
and
a
Side
opposite
to
one
of
them
given
,
to
find
the
rest
.
SEt
the
Angles
given
,
as
A
,
and
B
,
at
the
Pole
and
Zenith
;
the
known
side
,
as
B
C
in
an
Azimuth
;
Count
among
the
Meridians
the
Angle
opposite
to
the
known
side
,
and
having
found
the
Meridian
that
boundeth
him
,
lay
a
finger
or
a
bodkin
point
thereon
;
then
count
the
other
Angle
among
the
Azimuths
,
and
when
you
come
to
the
Azimuth
that
boundeth
him
,
because
that
Azimuth
maketh
the
known
side
of
your
Triangle
,
you
shall
number
his
length
from
the
Zenith
,
and
at
the
end
therof
make
a
prick
,
then
turn
about
the
Reet
till
this
prick
in
the
Azimuth
touch
the
Meridian
before
found
;
and
then
is
your
Triangle
formed
on
the
Planisphear
,
and
all
is
known
:
but
the
Angle
C
to
be
found
as
in
the
former
Chapters
.
Example
.
Let
be
given
A
45
degrees
B
113
½
B
C
46
½
.
I
count
from
A
B
to
the
45th
Meridian
,
upon
which
I
lay
my
finger
,
that
he
get
not
away
for
he
must
make
my
side
A
C
,
then
I
look
the
113
½
Azimuth
(
from
A
B
)
to
stand
for
the
given
side
:
and
because
his
length
given
is
46
½
therfore
in
this
113
½
Azimuth
at
46
½
below
the
Zenith
I
make
a
prick
:
then
I
turn
the
Reet
till
this
prick
touch
the
45th
Meridian
,
there
at
that
touch
must
C
stand
;
thence
to
the
Pole
is
the
side
A
C
70
,
and
on
the
Limb
I
have
the
side
A
B
40.
C
is
to
be
had
by
turning
the
Triangle
,
as
in
every
of
the
former
Problemes
.
CHAP.
XIX
PROB.
6.
Three
Angles
given
to
find
the
Sides
.
THis
Case
comes
very
seldome
in
use
.
Yet
that
our
Method
of
Trigonometry
by
the
Planisphear
may
be
compleat
,
and
that
no
Probleme
that
is
soluble
may
be
left
here
unresolved
,
I
shall
shew
the
solution
of
this
Probleme
also
.
Mr
Blagrave
,
it
seemes
,
never
attempted
this
,
contenting
himself
that
he
had
found
the
way
to
resolve
this
Probleme
in
Rectingled
Triangles
,
which
also
he
had
once
given
over
as
impossible
.
Blagr
.
Book
5
,
24.
For
resolving
this
Probleme
it
is
to
be
known
that
if
you
go
to
the
Poles
of
the
3
great
Circles
wherof
your
Triangle
is
made
,
these
Poles
shall
be
the
angular
points
of
a
second
Triangle
;
and
the
two
lesser
sides
of
this
second
Triangle
shall
be
equal
to
the
two
lesser
Angles
of
your
first
Triangle
;
the
greatest
side
of
the
second
Triangle
shall
be
the
supplement
of
the
greatest
Angle
of
the
first
Triangle
(
that
is
,
shall
have
as
many
degrees
and
minutes
as
the
greatest
Angle
of
the
first
Triangle
wanted
of
180
degr.
)
see
Pi●●scus
Trigonometry
Lib.
1.
Prop.
61.
This
second
Triangle
therfore
(
all
whose
sides
are
known
from
the
Angles
of
the
first
)
you
shall
resolve
by
the
first
Probleme
of
Oblique
angled
Spherical
Triangles
.
Chap.
14.
And
having
by
that
Probleme
found
the
Angles
of
this
second
Triangle
,
know
that
the
2
lesser
Angles
of
the
second
Triangle
shall
be
severally
and
respectively
equal
to
the
two
lesser
sides
of
the
first
Triangle
.
(
and
the
least
Angle
to
the
least
side
,
the
middle
Angle
to
the
middle
side
)
and
the
greatest
Angle
of
this
second
Triangle
being
subtracted
out
of
180
degr
.
shall
leave
you
the
greatest
side
of
your
first
Triangle
.
diagram
diagram
Because
A
in
the
first
Triangle
is
45
degr.
therefore
in
the
second
Triangle
B
C
(
subtendeth
A
)
shall
be
45
degr.
And
because
C
in
the
first
Triangle
is
38
degr.
51
min.
therefore
in
the
second
Triangle
the
side
A
B
(
which
subtendeth
C
)
shall
be
38
degr.
51
min.
And
because
B
the
greatest
Angle
in
the
first
Triangle
,
is
113
½
therefore
in
the
second
Triangle
the
side
A
C
(
which
subtendeth
B
)
shall
be
the
supplement
thereof
,
viz.
66
½
.
Write
now
upon
the
sides
of
this
second
Triangle
the
quantities
of
the
sides
,
so
is
your
second
Triangle
ready
to
be
resolved
by
the
first
Probleme
of
Oblique-angled
Triangles
whereby
you
shall
find
the
Angles
of
the
second
Triangle
,
as
I
have
expressed
them
in
the
Scheam
.
A
46
,
26
min.
C
40
,
B
110
degrees
.
Now
lastly
I
say
these
Angles
of
the
second
Triangle
thus
found
,
give
me
the
sides
of
the
first
Triangle
,
which
I
seek
,
in
this
manner
.
In
the
second
Triangle
.
In
the
first
Triangle
.
-
-
A
is
46.26
.
Therefore
B
C
is
46.26
.
-
C
is
40.00
.
Therefore
A
B
40.00
.
-
B
is
110.00
Therefore
A
C
70.00
.
Supplement
of
110
degrees
.
And
thus
by
all
the
Angles
given
,
we
have
found
out
all
the
sides
,
which
was
required
.
diagram
CHAP.
XX.
How
to
reduce
an
Oblique
angled
Triangle
to
two
Rectangled
Triangles
,
by
letting
fall
a
Perpendicular
.
BEcause
the
third
Probleme
of
Oblique
angled
Triangles
cannot
be
resolved
by
the
Canon
of
Sines
and
Tangents
without
letting
fall
a
Perpendicular
,
and
because
in
that
case
the
crossing
at
the
Angle
C
is
oft
so
Oblique
that
you
cannot
define
the
Angular
point
certainly
,
and
because
in
the
Method
for
resolving
the
third
Probleme
one
of
the
sides
of
the
Triangle
hapneth
some
times
to
make
two
intersections
with
the
Parallels
or
Almicanters
,
and
there
may
be
doubt
which
of
these
intersections
is
to
be
taken
for
the
Angle
C.
Although
I
there
shewed
another
way
to
resolve
that
doubt
,
yet
I
will
shew
you
also
how
to
resolve
it
,
and
to
remedie
the
inconveniences
aforesaid
,
by
letting
fall
a
Perpendicular
.
And
it
shall
suffice
to
shew
you
this
in
one
example
,
which
if
you
mark
and
be
acquainted
with
the
four
first
Problemes
of
Rectangled
Spherical
Triangles
you
shall
be
able
to
do
it
in
any
other
needfull
case
whatsoever
.
Take
therefore
the
Triangle
of
Chap
16.
where
we
had
given
A
B
37.45
minutes
A
C
105
,
41.
minutes
.
B
167.
9
min.
you
must
observe
that
the
Perpendicular
ought
to
fall
from
the
end
of
a
known
side
,
and
to
subtand
some
known
Angle
,
which
here
cannot
be
,
because
both
the
Angles
at
the
Base
A
C
are
unknown
.
Continue
therefore
the
sides
A
C
and
B
C
to
Semicircles
,
and
you
shall
have
a
second
Triangle
N
P
C
,
in
which
,
N
P
is
equal
to
A
B
,
N
C
is
supplement
of
B
C.
P
C
suplement
of
A
C
:
N
supplement
of
B.
C
is
common
to
both
Triangles
.
diagram
In
the
Triangle
P
R
N
you
have
(
beside
the
Right
Angle
R
)
the
Angle
N
12.
51
minutes
,
(
supplement
of
B
)
and
the
Hypotenusa
N
P
37.
45
minutes
;
and
so
may
find
all
the
rest
of
this
Triangle
,
by
the
third
Probleme
of
Rectangular
Spharical
Triangles
,
viz.
P
R
7.
49
min.
½
N
P
R
79
,
49
min.
and
R
N
37
degrees
,
which
had
,
In
the
Triangle
P
R
C
,
by
the
second
Probleme
of
Rectangular
Spharical
Triangles
you
may
find
R
C
70
degrees
(
which
added
to
R
N
maketh
N
C
111
½
,
whose
supplement
is
C
B
68
½
)
C
8
degr.
½
,
C
P
R
88
degr.
which
added
to
N
P
R
maketh
the
whole
Angle
C
P
N
167
,
47.
which
being
subducted
out
of
180
degr.
leaveth
the
supplement
thereof
C
A
B
12.13
min.
as
I
find
it
by
my
Planisphear
;
and
by
exact
calculation
it
may
be
12
26
minutes
.
Thus
have
you
a
perfect
Method
of
resolving
all
Spherical
Triangles
by
the
Planisphear
.
The
end
of
the
Third
Book
.
The
fourth
Book
.
Shewing
the
Solution
of
the
SPHERICAL
PROBLEMES
,
Both
Astronomical
,
Astrological
,
and
Geographical
,
by
the
PLANISPHEAR
.
CHAP.
I.
The
Preface
.
THe
best
method
(
in
my
judgement
)
for
setting
down
the
Problemes
of
the
Sphear
is
,
to
set
them
in
such
order
,
that
the
former
may
be
Praecognita
to
the
latter
,
and
the
latter
presuppose
the
knowledge
of
the
former
.
This
most
Authors
have
used
.
But
this
method
here
aimed
at
,
perhaps
is
not
alwaies
kept
exactly
.
Because
where
one
Triangle
serves
to
resolve
divers
Problemes
,
I
was
willing
to
make
an
end
with
him
sometime
before
I
meddled
with
another
,
for
avoiding
the
multiplicity
of
Chapters
,
and
repetition
of
the
same
Schemes
.
THere
be
in
the
Sphearfive
famous
Triangles
,
by
the
knowledge
where
of
most
Astronomical
Problemes
are
resolved
,
insomuch
that
if
you
be
but
well
versed
in
the
general
Problemes
of
Trygonometry
set
down
in
the
former
Book
,
and
have
acquaintance
with
these
five
Triangles
in
the
Sphear
,
you
will
be
able
to
resolve
most
of
the
following
Problemes
without
any
further
help
.
Of
those
five
Triangles
three
are
Rectangled
,
which
shall
be
here
denominated
from
their
Hypotenusa's
.
1.
The
Ecliptical
Triangle
,
whose
Hypotenusa
is
an
arch
of
the
Ecliptick
,
his
Legs
are
arches
of
the
Equator
,
and
a
Meridian
:
he
serveth
especially
for
Questions
of
the
Suns
Longitude
,
Right
Ascension
,
and
Declination
,
with
some
others
,
See
this
Ch.
6
&c.
2.
The
Horizontal
Triangle
,
whose
sides
,
are
arches
of
the
Horizon
,
Equator
,
and
a
Meridian
.
He
serveth
especially
for
Questions
of
the
Suns
Amplitude
,
Ascensional
difference
,
and
Declination
,
and
of
the
Latitude
of
your
place
.
See
this
Chapter
14.
3.
The
Azimuthal
or
Parrallactical
Triangle
,
whose
sides
are
arches
of
an
Azimuth
,
the
Ecliptick
,
and
a
Circle
of
Longitude
:
he
serveth
especially
to
find
the
Moons
Parallaxes
in
Altitude
,
Longitude
,
and
Latitude
.
See
this
Chap.
64.
4.
The
other
two
are
Oblique
angled
.
One
I
use
to
call
the
Complemental
Triangle
,
because
all
his
sides
be
complements
,
viz.
the
Complement
of
Latitude
,
of
Declination
,
and
of
Altitude
.
He
serveth
.
cheifly
to
find
the
Altitude
,
Azimuth
,
and
Hour
.
See
this
Chap.
24.
5.
The
last
,
I
use
to
call
the
Polar
Triangle
,
because
one
side
of
him
is
evermore
the
distance
of
the
Poles
of
the
World
,
and
of
the
Ecliptick
(
23
degrees
½
)
his
other
sides
are
a
Meridian
,
and
a
Circle
of
Longitude
.
He
serveth
chiefly
to
find
the
Longitude
and
Latitude
,
the
Right
Ascension
and
Declination
of
the
Stars
.
See
this
Chap.
34.
CHAP.
II.
How
to
find
the
Altitude
of
the
Sun
or
Stars
,
by
Observation
,
with
the
Planisphear
.
Also
what
fashion
is
best
for
Sighst
.
THe
Planisphear
may
here
supply
the
office
of
a
Quadrant
(
which
is
the
fittest
and
most
common
Instrument
for
taking
Altitudes
)
For
the
Planisphear
is
divided
into
4
Quadrants
,
and
if
you
hang
a
plumb-line
at
the
Center
,
it
may
serve
any
of
them
.
Set
your
Sights
to
one
of
the
Semidiameters
of
the
Mater
,
and
turn
him
so
to
the
Sun
that
the
Sun
may
shine
through
the
Sights
;
then
shall
the
plumb-line
(
if
it
hang
Parrallel
to
the
Planisphear
,
neither
bearing
upon
it
,
nor
hanging
off
from
it
)
shew
in
the
lowest
Quadrant
of
the
Limb
the
degrees
of
Altitude
.
But
because
the
Quadrants
may
be
smal
,
I
have
shewed
you
a
way
how
to
make
them
serve
your
turn
as
well
as
if
they
were
of
double
Semidiameter
,
Book
1.10
.
whither
I
refer
you
.
My
Sights
for
the
Sun
and
Moon
I
have
devised
to
make
thus
.
Let
them
be
about
an
inch
square
for
a
Planisphear
of
a
foot
Diameter
:
And
in
the
middle
of
that
sight
next
you
(
which
must
be
a
thin
plate
)
let
a
very
smal
hole
be
drilled
quite
through
:
in
the
middle
of
the
Sight
next
the
Sun
,
bore
an
hole
as
big
as
a
Pease
,
or
bigger
,
whose
Center
must
answer
the
smal
hole
in
the
other
Sight
,
then
cross
the
Center
of
the
hole
in
the
Sight
next
the
Sun
with
an
hair
,
or
fine
thrid
,
so
that
the
thrid
may
run
level
or
Parrallel
with
the
Horizon
,
when
you
use
the
Sights
.
When
you
turn
the
Sights
toward
the
Sun
,
and
the
shadow
of
the
thrid
fall
,
upon
the
smal
hole
of
the
lower
Sight
,
you
shall
set
or
hold
a
white
paper
about
a
span
behind
the
lower
Sight
,
upon
which
paper
you
shall
perceive
a
smal
Image
of
the
Suns
body
,
and
likewise
of
the
thrid
cutting
through
the
midst
of
him
very
distinctly
.
And
here
you
shall
observe
that
the
image
of
the
thrid
moveth
upon
the
image
of
the
Sun
in
the
paper
,
contrary
to
the
motion
of
the
shadow
of
the
thrid
upon
the
lower
Sight
;
for
when
the
shadow
of
the
thrid
toucheth
the
bottom
of
the
Sight-hole
,
the
image
of
the
thrid
shall
touch
the
top
of
the
Suns
image
on
the
paper
,
and
contrarily
.
But
when
the
shadow
of
the
thrid
cutteth
the
middle
of
the
Sight-hole
,
then
shall
the
image
of
the
thrid
always
cut
the
middle
of
the
image
of
the
Sun
upon
the
paper
exactly
and
clearly
.
Also
you
shall
observe
that
though
the
Diameter
of
the
Sun
be
always
more
then
30
min.
yet
the
Diameter
of
the
image
cannot
be
observed
here
to
be
much
above
20
min.
as
you
may
measure
by
the
min.
of
the
Quadrant
which
the
plumb-line
passeth
over
,
while
the
image
of
the
thrid
passeth
over
the
image
of
the
Sun
:
whither
the
Diameter
of
the
Suns
image
on
the
paper
be
diminished
by
reason
of
the
thickness
of
the
plate
through
which
the
beams
pass
,
or
because
the
image
on
the
paper
is
smal
,
the
beginning
and
end
of
the
Obscuration
by
the
image
of
the
thrid
,
cannot
be
precisely
observed
,
for
the
present
,
I
leave
to
Optical
men
to
enquire
:
Also
what
is
the
reason
why
the
image
of
the
thrid
moveth
contrary
to
the
motion
of
his
shadow
,
is
a
question
of
some
difficulty
:
My
resolution
is
,
because
that
image
is
a
species
which
passeth
through
the
Sight-hole
with
the
species
of
the
Suns
body
:
For
when
the
shadow
of
the
thrid
falleth
upon
the
lower
part
of
the
Sight-hole
,
then
certainly
the
upper
part
of
the
Suns
body
is
above
the
obscuration
of
the
thrid
,
and
the
lower
edge
is
Eclipsed
at
the
Sight-hole
.
Now
the
wayes
of
the
Suns
body
thus
Eclipsed
on
the
lower
side
,
passing
through
the
Sight-hole
,
must
needs
be
there
decussated
,
so
that
the
wayes
or
beames
comming
from
the
lower
part
of
the
Sun
shall
make
the
higher
part
of
the
image
on
the
paper
,
and
contrarily
;
as
appeareth
when
an
Eclipse
of
the
Sun
is
observed
by
a
Telescope
,
or
by
a
smal
hole
,
letting
the
beames
into
a
dark
room
:
For
the
reason
here
and
there
is
the
same
.
I
have
used
those
Sights
for
the
Sun
and
Moon
almost
these
20
years
past
,
and
(
for
ought
I
could
ever
read
or
hear
)
they
are
of
my
own
invention
,
and
I
have
not
met
with
any
device
more
commodious
to
me
for
this
purpose
.
For
the
Moon
,
you
must
set
your
eye
to
the
lowest
Sight-hole
,
and
let
the
thrid
cut
the
middle
of
her
body
.
For
the
Stars
,
if
your
eye
cannot
discern
them
by
the
thrid
,
you
must
behold
them
by
the
edges
of
the
Sights
both
above
and
below
.
Or
if
you
would
observe
the
Stars
Altitude
by
some
larger
Instrument
,
I
advise
that
the
Sight
next
your
eye
be
a
broad
plate
4
or
5
inches
square
,
in
the
middle
whereof
you
shall
cut
a
window
whose
length
may
be
near
2
inches
,
and
his
breadth
or
height
about
an
inch
or
more
,
so
that
your
eye
may
be
well
shadowed
,
and
yet
have
free
scope
through
the
window
to
find
the
Star
.
Let
the
upper
Sight
be
a
Cylinder
or
ruler
set
Parrallel
to
the
lower
Sight
,
and
his
breadth
be
equal
to
the
window
almost
,
but
narrower
by
a
few
minutes
as
2
min.
or
4
min.
when
you
looking
through
this
window
can
see
the
Star
appear
alike
on
both
sides
,
the
upper
Sight
,
then
is
your
Instrument
right
set
,
and
the
plumb-line
shall
sh●w
you
his
Altitude
as
before
.
Note
that
for
all
curious
observation
of
the
Sun
or
Stars
,
your
Instrument
must
be
supported
with
a
Tripos
,
or
like
device
,
that
it
may
be
steddy
,
and
that
the
apparent
Altitudes
of
the
Sun
and
Moon
must
be
corrected
according
to
the
Table
of
Parrallax
and
Refraction
.
The
sixed
Stars
have
Refraction
,
but
no
Parrallax
sensible
.
The
quantity
of
the
Parrallax
is
to
be
added
,
and
the
quantity
of
the
Refraction
to
be
subtracted
alwayes
from
the
apparent
Altitude
found
,
so
shall
you
have
the
true
Altitude
.
Here
followeth
an
Abridgement
of
Lansbergius
Tables
of
Refraction
and
Parrallax
of
the
Sun
,
as
much
as
this
Instrument
may
need
,
for
the
rest
go
to
Lansbergius
or
Tycho
Brahe's
Tables
at
large
,
where
you
shall
find
the
Moons
Parrallax
in
the
Horizon
,
to
be
sometimes
51
minutes
,
sometimes
1
deg.
7
min.
at
70
degrees
of
Altitude
,
between
18
deg.
24
min.
Here
Refraction
is
as
the
Sun
.
Alt.
☉
|
Parall.
mi.
sec
.
|
Refr
.
mi.
see
|
0
|
2
|
18
|
34
|
00
|
5
|
2
|
18
|
14
|
00
|
10
|
2
|
16
|
8
|
15
|
15
|
2
|
13
|
6
|
00
|
20
|
2
|
10
|
4
|
33
|
25
|
2
|
05
|
3
|
12
|
30
|
2
|
00
|
1
|
51
|
35
|
1
|
53
|
0
|
54
|
40
|
1
|
46
|
|
|
45
|
1
|
38
|
|
|
50
|
1
|
29
|
|
|
55
|
1
|
19
|
|
|
60
|
1
|
09
|
|
|
65
|
|
58
|
|
|
70
|
|
47
|
|
|
CHAP.
III.
To
finde
a
Meridian
line
.
STrike
a
straight
line
upon
a
Table
or
any
Horizontal
plain
:
and
lay
your
Planisphear
so
that
one
of
the
Diameters
of
the
Mater
may
lie
in
that
line
.
Then
take
the
Suns
Altitude
:
the
Altitude
would
be
taken
at
least
2
hours
(
the
more
the
better
)
before
noon
:
and
note
,
that
if
you
take
it
between
29
and
30
degrees
you
shall
be
troubled
neither
with
Parrallax
nor
Refraction
,
because
the
Suns
Refract●●n
and
Parrallax
be
equall
at
the
Altitude
29
degrees
26
minutes
.
The
Altitude
taken
,
you
shall
immediately
lay
your
Planisphear
in
the
posture
aforesaid
;
and
turning
the
Label
to
the
Sun
,
make
a
prick
in
the
Limb
where
the
Label
cutteth
:
And
when
the
Sun
comes
to
the
same
Altitude
after-noon
,
your
Planisphear
laid
as
before
,
turn
your
Label
to
the
Sun
,
and
where
he
cuts
make
a
second
prick
in
the
Limb.
Then
divide
equally
the
Arch
of
the
Limb
comprehended
between
the
pricks
:
and
to
the
middle
thereof
lay
the
Label
,
and
it
shall
point
full
North
and
South
.
Look
then
through
your
Sights
;
and
if
you
see
any
Steeple
,
Pinacles
.
Chimney
,
Tree
,
or
such
mark
,
at
a
good
distance
in
the
line
of
Vision
,
you
may
note
him
for
a
South-mark
,
or
for
want
thereof
set
up
a
smaller
mark
neerer
hand
.
But
note
also
that
this
may
best
be
done
when
the
Sun
is
in
or
neer
the
Summer
Tropick
,
for
neer
the
Equator
he
changeth
his
Declination
so
fast
,
that
it
may
cause
you
an
error
of
a
few
minutes
,
unless
you
make
allowance
for
it
.
Note
,
that
all
lines
Parallel
to
your
Meridians
are
Meridians
.
2.
Another
way
.
Having
taken
the
Altitude
of
the
Sun
,
or
a
Star
,
at
a
good
distance
from
the
Meridian
;
presently
lay
your
Planisphear
flat
,
and
turn
the
Label
to
the
Sun
,
or
Star
,
as
before
.
Then
by
the
Altitude
taken
,
get
the
Azimuth
;
(
by
chap.
24
or
27
of
this
book
.
)
Then
remove
your
Labet
(
Eastward
;
if
the
Sun
or
Star
were
Westward
from
the
Meridian
;
or
Westward
if
the
Sun
or
Star
be
in
the
East
Hemisphear
;
)
so
many
degrees
as
the
Azimuth
cometh
to
,
and
your
Label
shall
be
in
the
Meridian
.
3.
A
third
way
.
When
the
great
Wain
is
seen
under
Cynosura
,
(
the
Pole
Star
)
observe
with
your
eye
the
distance
of
the
Thill-horse
,
called
Alioth
,
from
the
next
wheel
of
the
Wain
and
setting
that
distance
(
by
aime
)
in
5
parts
,
observe
by
a
plumb-line
when
Alioth
drawes
neer
to
be
in
the
same
Perpendicular
with
the
Pole
Star
.
For
when
he
wanteth
but
one
of
those
5
parts
to
come
into
the
Perpendicular
,
then
is
the
Pole-star
in
the
Meridian
over
the
Pole
in
our
age
:
at
other
times
of
the
night
the
Pole-star
may
be
4
degrees
wide
,
and
in
one
hour
neer
the
Meridian
he
changeth
his
Azimuth
above
one
degree
.
4.
A
fourth
way
.
Because
the
distance
of
the
Pole-star
from
the
Pole
is
now
2
degrees
30
minutes
,
and
the
Pole
is
in
the
circle
or
line
which
passeth
from
the
Pole-star
neer
Alioth
,
as
before
;
you
may
by
guess
cut
off
from
that
line
2
degrees
30
min.
and
in
that
Section
you
have
the
Pole
at
any
time
.
This
way
may
be
used
abroad
in
the
fields
,
where
you
cannot
stand
upon
exactness
;
and
herein
you
shall
miss
very
little
,
if
you
accustome
your self
to
observe
the
distances
of
the
Stars
about
the
Pole
.
CHAP.
IIII.
To
Observe
the
Azimuth
of
the
Sun
or
Stars
.
LAy
your
Planisphear
upon
an
Horizontal
plain
or
Level
,
and
his
Meridian
on
the
Meridian
line
of
your
Place
,
found
by
the
last
Chapter
.
Then
turn
your
Label
that
the
Sun
may
cast
the
shadow
of
one
Sight
upon
the
other
,
or
directly
towards
it
,
or
till
the
shadow
of
a
plumb
line
cut
both
the
Sights
alike
,
then
doth
the
Label
shew
the
Azimuth
in
the
Limb.
For
the
Stars
,
you
must
so
direct
the
Sights
by
your
eye
,
that
their
edges
may
touch
the
Visuall
line
that
comes
from
the
Star
to
your
eye
:
and
if
your
long
Sight
prove
too
short
,
turn
him
toward
your
eye
,
and
inlightning
the
shorter
Sight
by
a
candle
held
behind
you
,
mark
where
the
edge
of
the
long
Sight
cuts
both
the
edge
of
the
short
Sight
,
and
the
Star
;
for
there
is
your
Label
in
the
Azimuth
of
the
Star
,
which
you
may
count
on
the
Limb.
Note
that
if
you
seek
the
Azimuth
to
get
the
hour
,
you
shall
find
it
most
easily
when
the
Sun
or
Stars
are
neer
the
Horizon
:
and
then
you
shall
not
be
troubled
with
their
Refraction
.
But
there
is
most
use
of
observing
Azimuths
neer
the
Meridian
,
because
there
the
Azimuth
changeth
apace
,
the
Altitude
very
slowly
:
Yet
if
you
may
choose
,
choose
to
take
Altitudes
rather
then
Azimuths
(
so
you
come
not
within
2
or
3
hours
of
the
Meridian
)
because
the
Sights
serve
all
Altitudes
with
like
facility
,
and
you
may
sooner
have
a
true
plumb
line
any where
,
then
a
true
Horizontall
plain
,
and
a
true
Meridian
line
.
CAAP.
V.
To
find
the
Suns
Longitude
.
THe
Longitude
of
the
Sun
is
the
arch
of
his
distance
from
♈
0
in
the
Ecliptick
:
or
it
is
the
angle
made
at
the
Pole
of
the
Ecliptick
comprehended
between
the
circle
of
Longitude
passing
through
♈
0
,
and
another
Circle
of
Longitude
passing
through
the
center
of
the
Sun
:
for
the
said
arch
of
the
Ecliptick
is
always
the
proper
measure
of
this
Angle
.
And
because
the
Suns
center
never
hath
Latitude
,
therefore
for
the
Sun
you
shall
enquire
the
arch
;
but
contrarily
,
for
the
Stars
which
have
Latitude
,
you
shall
require
the
Angle
:
and
they
be
both
(
as
was
said
)
of
one
measure
.
The
Suns
Longitude
(
Arch
or
Angle
)
is
presently
found
by
the
Ephemeris
upon
the
Limb
of
your
Planisphear
,
for
if
you
lay
the
Label
upon
the
day
of
the
Moneth
,
it
shall
cut
the
degree
of
the
Signe
also
in
which
the
Sun
is
,
and
that
is
his
Longitude
:
in
doing
whereof
,
you
shall
observe
the
cautions
given
Lib.
1.8
.
to
which
I
refer
you
.
Note
here
,
that
the
Longitude
of
a
place
in
Geographie
is
the
Angle
at
the
Pole
of
the
World
,
comprehended
between
the
first
Meridian
(
passing
by
the
hither
side
of
S.
Michals
Island
,
which
is
the
neerest
of
the
Azores
)
and
the
Meridian
of
the
Place
:
and
this
Angle
hath
his
measure
in
the
Equator
.
CHAP.
VI.
The
Suns
Longitude
,
Declination
,
Right
Ascension
,
any
one
of
them
given
,
to
find
the
rest
in
the
first
Projection
.
WHat
the
Suns
Longitude
is
,
hath
been
shewed
chap.
5.
His
Declination
is
his
di●●ance
from
the
neerest
point
of
the
Equator
;
and
therefore
is
alwaies
measured
in
an
Arch
of
that
Meridian
which
hapneth
to
pass
through
the
center
of
the
Sun
,
and
always
cuts
the
Equator
at
right
Angles
,
as
do
all
the
Meridians
.
The
Right
Ascension
of
the
Sun
is
the
angle
at
the
Pole
of
the
World
comprehended
between
that
Side
of
the
Colurus
Equinoctiorum
which
cuts
the
intersection
of
the
Ecliptick
with
the
Equator
in
♈
0
,
and
the
arch
of
another
Meridian
which
passeth
through
the
center
of
the
Sun
.
And
note
,
that
this
angle
may
increase
above
180
degrees
,
even
to
360
degrees
,
though
every
angle
,
properly
so
called
be
less
then
180
degrees
,
and
never
more
then
90
degrees
comes
into
the
Triangle
:
for
if
you
number
backwards
or
forwards
from
either
of
the
Equinoctiall
points
,
you
shall
have
like
arches
of
Right
Ascension
answering
to
like
arhces
of
Longitude
and
Declination
;
so
that
having
found
the
Right
Ascension
in
any
one
Quadrant
,
or
the
complement
therof
,
you
shall
find
the
whole
Right
Ascension
from
♈
0
by
adding
one
,
two
,
or
three
whole
Quadrants
to
the
Right
Ascension
found
,
or
to
the
complement
therof
,
as
by
the
view
of
your
Planisphear
you
shall
presently
know
how
to
do
better
then
by
more
words
.
Otherwise
thus
.
The
Right
Ascension
of
the
Sun
is
an
arch
of
the
Equator
comprehended
between
the
Vernal
Equinox
and
that
point
of
the
Equator
which
riseth
with
the
Sun
in
a
right
Horizon
.
A
right
Horizon
is
where
the
Equator
passeth
through
the
Zenith
,
and
maketh
right
angles
with
the
Horizon
,
and
consequently
,
where
the
Poles
have
no
Elevation
:
For
from
that
posture
of
the
Sphear
in
which
the
Equator
riseth
upright
,
is
the
term
of
Right
Ascension
borrowed
:
I
would
,
if
I
might
,
call
it
rather
Equation
;
because
it
is
numbred
on
the
Equator
,
and
serves
for
the
Equation
of
naturall
days
,
and
may
as
easily
be
found
in
any
Sphear
as
in
a
right
Sphear
,
since
the
Horizon
of
a
right
Sphear
limits
the
Right
Ascension
only
because
that
Horizon
falls
in
with
a
Meridian
,
and
the
Meridians
do
limit
it
in
all
parts
and
postures
of
the
Equator
,
without
any
respect
to
the
Horizon
at
all
.
But
the
old
term
hath
so
long
inured
,
that
I
beleeve
it
will
not
be
changed
without
better
Authority
.
These
definitions
premised
,
you
shall
know
that
these
three
arches
,
viz.
of
Longitude
in
the
Ecliptick
,
of
Right
Ascension
in
the
Equator
,
and
of
Declination
in
a
Meridian
,
do
make
up
a
notable
Rectangled-Triangle
in
the
Sphear
,
The
Ecliptical
Triangle
.
like
unto
that
which
was
made
the
common
Example
,
in
all
the
five
Problemes
of
Rectangled-Triangles
.
Book
3
,
3.
&c.
diagram
But
to
see
your
Triangle
,
and
resolve
him
in
his
proper
lines
,
Go
to
the
Mater
of
your
Planisphear
,
and
take
him
there
in
the
first
Projection
.
There
number
60
the
Suns
Longitude
in
the
Ecliptick
line
of
the
Mater
from
the
Center
outward
.
Where
60
endeth
,
there
is
C
of
your
Triangle
,
and
the
Meridian
that
meets
you
there
is
C
A
the
arch
of
Declination
;
follow
him
to
the
Equator
,
and
you
shall
find
by
his
graduation
he
is
20
degr.
12
min.
Long.
thence
turn
in
the
Equator
to
the
Center
,
and
you
make
B
A
the
Right
Ascension
57
degr.
48
min.
so
have
you
the
true
picture
of
your
Triangle
in
his
proper
place
.
Observe
your
Triangle
now
,
and
you
may
see
A
is
a
right
angle
,
for
at
such
angle
all
the
Meridians
cut
the
Equator
.
B
is
23
½
,
for
such
an
angle
the
Ecliptick
dayly
maketh
with
the
Equator
,
as
the
arch
in
the
Limb
comprehended
between
them
shewes
.
Now
take
for
given
any
of
the
three
Sides
,
and
you
have
the
rest
.
Take
the
Longitude
for
given
(
and
be
it
60
degr.
as
before
,
or
70
degr.
or
what
you
will
)
and
you
may
find
the
Declination
,
and
Right
Ascension
as
before
.
Let
the
Right
Ascension
be
given
;
then
setting
a
needles
point
in
the
end
thereof
A
,
you
may
thence
in
a
Meridian
trace
out
the
Declination
C
A
to
the
Ecliptick
,
and
the
Longitude
B
C
thence
to
the
Center
,
every
Side
being
divided
into
his
whole
parts
or
degrees
.
If
the
Declination
be
given
,
say
,
Because
the
20th
Parrallel
almost
must
cut
off
C
A
(
the
arch
of
Declination
)
in
C
,
therefore
I
follow
the
Parallel
20
⅕
to
the
place
where
he
cutteth
the
Ecliptick
and
there
comes
the
Meridian
that
serves
my
turn
;
and
I
may
go
down
by
him
to
the
Equator
,
(
as
you
would
go
down
a
ladder
counting
the
rounds
or
degrees
as
you
go
)
and
so
on
,
round
my
Triangle
,
and
I
need
no
more
.
For
observe
it
when
you
will
in
the
use
of
this
Planisphear
,
if
you
can
find
the
way
to
go
round
your
Triangle
,
you
have
all
the
Sides
measured
to
your
hand
,
and
evermore
one
Angle
also
,
most
commonly
two
,
and
the
angle
C
onely
left
unknown
.
But
admit
the
Sun
be
in
♌
0
,
then
is
his
Longitude
120
,
degrees
,
and
he
is
come
back
from
the
Solstice
in
your
Planisphear
as
many
degr.
as
he
wanted
of
it
before
.
Here
the
Triangle
is
equal
to
the
former
,
and
resolved
in
like
manner
.
The
Declination
is
the
same
as
before
:
But
the
arches
of
Longitude
and
Right
Ascension
in
the
Triangle
are
supplements
of
the
true
Longitude
and
Right
Ascension
;
shewing
what
the
Sun
wants
of
the
Longitude
and
Right
Ascension
180
,
in
♎
0.
wherefore
subtract
the
Base
of
the
Triangle
57
degr.
48
min.
from
a
Semicircle
,
or
180
degr.
and
you
shall
leave
122
degr.
12
min.
the
Right
Ascension
of
♌
0.
or
number
in
the
Equator
from
the
Center
the
way
in
which
the
Right
Ascension
hath
increased
,
that
is
first
to
the
Limb
(
which
here
is
Colurus
Solstitiorum
)
90
degr.
then
back
again
to
A
the
Right
angle
of
your
Triangle
,
and
you
have
32
degr.
12
min.
to
be
added
thereto
.
The
Sum
is
122
degr.
12
min.
the
Right
Ascension
,
as
before
:
If
you
observe
this
Example
,
you
will
easily
perceive
,
that
when
the
Sun
is
past
♎
0.
the
Triangle
will
be
on
the
other
side
the
Center
,
and
between
♎
and
♑
you
must
add
to
the
Right
Ascension
and
Longitude
found
within
the
Triangle
180
degr.
and
in
the
last
Quadrant
between
♑
and
♈
(
where
the
Right
Ascension
again
increaseth
inwards
)
you
must
add
270
degr.
to
the
complement
of
Right
Ascension
found
in
the
Triangle
,
and
take
the
sum
,
or
else
subduct
the
Right
Ascension
found
in
the
Triangle
from
360
degr.
and
take
the
residue
for
the
Right
Ascension
.
CHAP.
VII
.
To
do
the
same
in
the
second
Projection
,
more
easily
.
IN
the
second
Projection
where
the
Center
is
the
Pole
of
the
World
,
and
the
Limb
Equator
,
you
shall
find
the
Ecliptick
,
fairly
drawn
upon
the
Reet
and
distinguished
into
his
quarters
and
degrees
.
Remember
now
from
the
former
chap.
that
the
Ecliptick
.
Equator
,
and
a
Meridian
,
must
make
your
Triangle
;
and
know
that
the
Label
supplieth
the
place
of
the
Meridians
.
If
the
Longitude
or
Right
Ascension
be
given
,
lay
the
Label
on
the
degree
given
(
in
the
Ecliptick
for
Longitude
,
or
in
the
Limb
of
the
Reet
for
Right
Ascension
)
and
your
Triangle
is
made
,
and
you
may
presently
see
your
desire
.
If
the
Declination
be
given
,
consider
in
what
quarter
of
the
Ecliptick
the
Sun
is
,
then
number
the
Declination
given
upon
the
Label
inwards
,
and
where
the
numbring
ends
make
a
prick
on
your
Label
,
then
move
the
Label
into
the
quarter
where
the
Sun
is
,
and
lay
the
prick
on
the
Ecliptick
there
,
and
your
Triangle
is
made
,
wherein
you
may
see
the
Longitude
and
Right
Ascension
desired
.
This
needeth
no
Example
.
CHAP.
VIII
.
To
find
the
Angle
at
the
Sun
,
made
between
the
Ecliptick
and
Meridian
.
THis
is
the
angle
C
of
the
former
Triangle
,
and
is
the
onely
part
which
cannot
be
found
in
the
former
posture
of
the
Triangle
,
neither
in
chap.
6
nor
7
,
but
is
easily
had
by
conversion
of
the
Triangle
,
as
you
may
remember
out
of
the
third
Book
.
Take
the
Triangle
of
chap.
6
,
and
make
the
Cathetus
Base
,
for
this
turn
:
and
by
the
1
or
2
Problemes
of
Rectangled
Triangles
,
you
may
find
this
angle
to
be
77
degr.
43
min.
CHAP.
IX
.
To
find
the
said
angle
of
the
Ecliptick
,
with
the
Meridian
,
by
the
Longitude
,
Declination
,
or
Right
Ascension
,
divers
other
wayes
.
IN
the
Meridional
Projection
do
thus
.
If
you
have
the
Longitude
given
,
count
the
distance
of
the
Sun
in
that
Longitude
from
the
next
Equinoctial
point
,
and
count
so
many
degrees
in
the
Arctick
Circle
from
the
Limb
inwards
:
to
the
end
of
this
numbring
,
lay
the
Label
,
and
between
the
Label
and
Equator
you
have
upon
the
Limb
the
lesser
angle
made
between
the
Ecliptick
and
Meridian
;
the
greater
angle
is
the
supplement
thereof
.
Also
between
the
Arctick
Circle
and
the
Limb
you
may
find
the
Declination
on
the
Label
,
which
is
more
then
was
required
.
If
you
have
the
Declination
given
,
count
it
on
the
Label
inwards
,
and
make
a
prick
where
the
number
ends
,
then
turn
this
prick
upon
the
Arctick
Circle
,
and
the
Label
sheweth
the
lesser
angle
in
the
Limb
,
as
before
.
Example
.
I
would
know
what
angle
the
Meridian
that
cutteth
the
Sun
in
♉
9
degr.
maketh
with
the
Ecliptick
.
I
number
therefore
in
the
Arctick
Circle
from
the
Limb
inwards
39
deg.
and
to
the
39th
degr.
I
say
the
Label
,
and
it
sheweth
in
the
Limb
the
angle
sought
71
degr.
20
min.
and
in
the
Label
the
Declination
of
♉
9
degr.
viz.
14.
32
minutes
:
this
is
a
good
way
.
But
that
the
Label
at
this
39th
degr.
cutteth
the
Pole
of
the
Ecliptick
(
as
Mr.
Blagrave
saith
Book
3
,
40.
)
is
not
true
:
either
Mr.
Blagrave
or
the
Printer
here
mistakes
.
For
the
Pole
of
the
Ecliptick
lies
14.
24
minutes
nearer
the
Axletree
,
as
you
shall
find
in
the
next
rule
.
2.
Another
way
.
Mark
what
is
the
Right
Ascension
of
the
point
proposed
,
being
counted
from
the
next
Equinoctial
point
(
as
of
♉
9
degr.
the
Right
Ascension
is
36.36
min.
)
count
so
many
degrees
in
the
Arctick
circle
from
the
Axeltree
:
at
the
end
of
this
number
is
the
Pole
of
the
Ecliptick
.
Lay
the
Label
to
him
,
and
you
shall
make
a
Quadrantal
Triangle
,
whose
Sides
shall
be
equal
to
the
Angles
of
the
former
Triangle
,
which
was
made
of
the
Longitude
,
Declination
,
and
Right
Ascension
,
of
the
point
proposed
:
for
the
Right
Angle
you
have
a
Radius
or
Quadrant
of
the
Axis
:
for
the
Angle
of
the
greatest
Declination
between
the
Equator
and
Ecliptick
23
½
,
you
have
the
arch
of
a
Meridian
between
the
Pole
of
the
Equator
and
the
Pole
of
the
Ecliptick
:
for
the
angle
sought
,
you
have
the
arch
of
the
Label
,
between
the
Pole
of
the
Ecliptick
and
the
Center
71.20
minutes
;
as
before
:
the
least
angle
of
this
Quadrantal
Triangle
is
at
the
Center
,
and
you
shall
find
his
measure
in
the
Limb
14.32
minutes
:
that
is
the
measure
of
the
least
Side
of
the
former
Triangle
,
viz.
the
Declination
of
the
point
proposed
.
Here
you
see
,
If
the
Declination
had
been
given
,
you
should
have
set
it
in
the
Limb
,
between
the
Pole
and
the
Label
,
and
so
had
you
made
the
same
Quadrantal
Triangle
,
and
might
have
found
on
the
Label
between
the
Arctick
Circle
and
the
Center
the
measure
of
the
angle
sought
:
and
likewise
in
the
Arctick
Circle
between
the
Label
and
the
Axtree-line
the
Right
Aseension
,
though
it
be
more
then
was
required
.
The
reason
hereof
you
may
learn
from
Book
3.7
.
CHAP.
X.
To
find
the
point
of
the
Ecliptick
in
which
the
Longitude
and
Right
Ascension
have
greatest
difference
.
Move
the
Label
.
on
the
Polar
circle
till
you
find
the
degrees
of
the
Label
between
the
Polar
circle
and
the
Limb
to
be
equal
to
the
degr.
of
the
Limb
between
the
Label
and
the
Pole
,
so
have
you
a
Rectangled
aeqaicrurall
Triangle
made
by
the
Limb
,
Label
,
and
the
Meridian
46
¼
;
like
to
that
in
the
second
Variety
,
Book
3.10
.
Here
the
angle
B
at
the
Pole
between
the
46
¼
Meridian
and
the
Limb
,
is
equal
to
the
Longitude
of
the
point
sought
46¼
,
and
either
Leg
is
equal
to
the
Declination
thereof
16
¼
:
Therefore
I
conclude
,
that
when
the
Sun
is
46
¼
in
Longitude
.
(
that
is
in
♉
16
¼
)
then
his
Longitude
hath
furthest
out
run
the
Right
Ascension
.
Subtract
now
the
Right
Ascension
of
♉
16
¼
,
which
is
43
¾
out
of
the
Longitude
46
¼
there
remains
2
deg
½
:
which
being
converted
into
Time
,
is
10
min.
the
greatest
inequality
of
Ascension
in
a
Right
Sphear
.
CHAP.
II.
To
find
the
Latitude
of
your
Place
,
or
the
Elevation
of
the
Pole
above
your
Horizon
,
by
the
Meridional
Altitude
,
and
Declination
of
the
Sun
.
Meridional
Projection
.
GEographers
call
the
distance
of
a
place
from
the
nearest
point
of
the
Equator
upon
Earth
,
the
Latitude
of
that
Place
,
as
the
Latitude
of
London
is
51
deg.
32
min.
from
the
Equator
Northward
:
the
Latitude
of
St
Thomas
Island
upon
the
coast
of
Africk
is
0
deg.
0
min.
because
the
middle
of
that
Island
lyeth
under
the
Equator
.
And
because
the
Latitude
of
your
Place
,
and
the
Elevation
of
the
Pole
above
your
Horizon
,
are
alwaies
equal
,
therefore
the
Elevation
of
the
Pole
is
oft
called
Latitude
of
the
Place
,
or
Latitude
simply
:
and
so
for
brevity
sake
we
shall
often
call
it
.
But
when
we
speak
of
the
Latitude
of
the
Moon
or
Stars
,
you
must
understand
Astronomers
thereby
mean
their
distance
from
the
neerest
point
of
the
Ecliptick
.
To
find
the
Latitude
of
your
Place
,
get
the
Suns
Declination
,
by
the
6
or
7th
.
and
his
Meridian
Altitude
by
the
second
of
this
Book
:
Then
find
the
parallel
of
the
Suns
Declination
,
North
or
South
as
the
Declination
is
,
and
where
it
toucheth
the
Limb
(
here
Meridian
)
there
is
the
point
where
you
observed
the
Sun
at
Noon
;
set
the
South
end
of
the
Finiter
so
many
degr.
below
this
point
as
the
Meridian
Altitude
had
,
then
is
your
Finiter
set
to
your
Latitude
,
and
you
shall
find
the
measure
of
it
between
the
Equator
and
the
Zenith
,
(
which
is
properly
the
Latitude
)
and
the
same
measure
shall
you
find
between
the
North
point
of
the
Finiter
and
the
North
Pole
,
where
it
is
more
properly
called
the
Elevation
of
the
Pole
.
Example
.
June
20
1651.
I
observed
the
Meridian
Altitude
of
the
Sun
,
here
at
Ecton
,
four
miles
Eastward
from
Northampton
,
60
degr.
59
min.
the
Longitude
of
the
Sun
was
then
♋
8
degr.
19
min.
½
,
his
Declination
23
degr
14
min.
Northward
.
Therefore
having
found
in
the
Limb
the
point
where
the
Parallel
23
degr.
14
min.
toucheth
above
the
Equator
,
I
put
the
South
end
of
the
Finiter
60
degr.
59
min.
below
that
point
,
toward
the
South
Pole
,
which
done
,
I
see
the
North
Pole
Elevated
above
the
Finiter
52
degr.
15
min.
and
the
Zenith
of
my
Horizon
likewise
to
be
removed
from
the
Equator
Northward
52
degr.
15
min.
which
is
the
Latitude
of
Ecton
.
Note
that
you
may
best
observe
the
Latitude
when
the
Sun
is
near
the
Summer
Tropick
;
for
then
you
shall
not
be
troubled
with
Refraction
;
and
then
the
Declination
varyeth
slowly
;
which
varyeth
almost
one
minute
every
hour
near
the
Equinoctial
.
CHAP.
XII
.
To
do
the
same
by
the
Meridian
Altitudes
of
the
Stars
about
the
Poles
.
MAny
of
the
Stars
near
the
Northern
Pole
may
be
seen
with
us
twice
in
the
Meridian
in
one
Winters
Night
:
that
is
,
one
while
above
the
Pole
,
and
12
hours
after
again
below
the
Pole
.
As
for
Example
,
the
Pole-star
,
called
Alrucabe
,
about
December
18
will
be
in
the
Meridian
above
the
Pole
at
6
of
the
clock
at
Night
,
and
at
6
next
morning
he
will
be
in
the
Meridian
below
the
Pole
.
Observe
both
the
Meridian
Altitudes
,
and
add
them
together
,
half
that
sum
is
the
Elevation
of
the
Pole
.
Example
.
I
observed
at
Ecton
the
greatest
Altitude
of
the
Pole-star
to
be
54
deg.
45
min.
and
his
least
Altitude
49
degr.
45
min.
the
sum
is
104
deg.
30
min.
the
half
52
degr.
15
min.
the
Latitude
of
Ecton
:
and
here
I
have
gotten
also
the
Pole-stars
distance
from
the
Pole
,
and
consequently
his
Declination
which
is
the
complement
thereof
,
for
the
Latitude
being
subducted
from
the
greater
Altitude
leaves
the
Stars
distance
from
the
Pole
2
degr.
30
min.
and
consequently
shewes
his
Declination
to
be
87
degr.
30
min.
which
is
39
min.
more
then
Gemma
Frisius
observed
it
,
Anno
Dom.
1547.
for
in
our
age
the
Pole-star
approcheth
about
1
min.
nearer
the
Pole
in
every
3
years
.
Note
that
these
Stars
which
are
distant
from
the
Pole
less
then
the
Latitude
,
and
more
then
the
complement
thereof
,
have
their
less
Meridian
Altitude
in
the
North
part
of
the
Meridian
,
and
their
greater
Meridian
Altitude
in
the
Southern
part
of
the
Meridian
beyond
the
Zenith
.
Wherefore
for
them
you
shall
take
the
complement
of
their
greater
Altitude
,
and
add
it
to
the
North
Quadrant
of
the
Meridian
,
and
if
to
that
sum
you
add
the
lesser
Altitude
,
the
half
thereof
shall
be
your
Latitude
.
But
the
nearer
any
Star
is
to
the
Pole
,
the
fitter
for
this
purpose
,
and
therefore
none
better
then
Alrucabe
,
who
is
the
nearest
of
all
.
CHAP.
XIII
.
To
find
the
Declination
of
the
Sun
or
Stars
,
by
their
Meridian
Altitude
,
and
the
Elevation
of
the
Pole
.
This
is
done
by
the
first
,
made
of
the
Meridional
Projection
,
where
having
set
your
Finiter
to
the
Elevation
of
the
Pole
,
or
your
Zenith
to
the
Latitude
,
(
for
as
hath
been
shewed
Chap.
11.
all
comes
to
one
,
and
in
doing
either
,
you
do
both
)
and
having
observed
the
Meridian
Altitude
of
the
Sun
or
Star
,
number
the
Altitude
observed
upon
the
Limb
of
the
Reet
on
the
South
or
North
side
of
the
Pole
,
according
as
the
Star
was
observed
to
be
,
and
there
shall
meet
you
on
the
Mater
his
Parallel
of
Declination
.
Example
.
I
observed
the
Suns
Meridian
Altitude
at
Ecton
,
20
deg.
I
look
therefore
where
the
20th
Almicanter
toucheth
the
Limb
,
(
the
Finiter
first
set
to
the
Latitude
)
and
there
meets
at
the
Limb
the
17
¾
Parallel
below
the
Equator
:
wherefore
I
say
,
the
Sun
declineth
17
degr.
45
min.
Southward
.
Again
,
I
observed
the
Star
Alhaiot
in
the
North
part
of
the
Meridian
6
degr.
42
min.
high
,
I
go
to
that
Almicanter
in
the
North
quarter
of
the
Reet
under
the
Pole
,
and
there
meeteth
at
the
Limb
the
Parallel
45
min.
½
of
North
Declination
.
CHAP.
XIV
.
To
find
the
Oblique
Asoension
and
Descension
,
and
the
Ascensional
difference
of
the
Sun
or
any
Star
:
by
his
Declination
,
and
the
Latitude
of
the
Place
,
Two
several
wayes
,
in
the
Horizontal
Triangle
.
THe
Oblique
Ascension
is
the
arch
of
the
Equator
which
riseth
with
the
Sun
or
any
Star
in
an
Oblique
Sphear
,
that
is
,
a
Sphear
wherein
the
Equator
maketh
an
Oblique
Angle
with
the
Horizon
.
This
arch
beginneth
alwayes
from
the
Vernal
Equinox
,
but
we
seek
the
latter
term
or
end
thereof
.
To
find
this
by
Calculation
,
we
use
to
find
first
the
Ascensional
difference
,
that
is
the
difference
of
the
Right
and
Oblique
Ascension
,
or
the
arch
of
the
Equator
comprehended
between
the
latter
termes
of
the
arches
of
the
Right
Ascension
and
Oblique
Ascension
of
the
Star
,
this
difference
for
North
Stars
,
we
subtract
from
the
Right
Ascension
,
and
the
remainder
is
the
Oblique
Ascension
;
but
for
South
Stars
we
add
it
to
the
Right
Ascension
to
make
the
Oblique
Ascension
:
and
for
Oblique
Descension
or
Setting
,
contrarily
,
we
add
the
Ascensional
difference
for
North
Stars
,
and
subtract
it
for
South
:
you
shall
see
all
plain
in
the
Meridional
Projection
of
the
Planisphear
,
and
the
first
Mode
thereof
,
where
the
Finiter
is
set
to
the
Latitude
.
Example
.
I
would
know
the
Oblique
Ascension
of
the
Sun
in
♋
0
and
the
Ascensional
difference
,
The
Declination
of
the
Sun
in
♋
0
is
23
degr.
30
min.
our
Latitude
52
degr.
15
min.
I
go
to
the
North
Parallel
23
degr.
½
,
which
is
the
Tropick
of
Cancer
,
on
the
Mater
,
and
following
him
to
the
Finiter
,
there
I
turn
in
the
Meridian
which
cutteth
there
,
and
go
down
to
the
Equator
under
the
Horizon
,
and
make
a
prick
here
;
I
say
,
is
the
Right
Ascension
of
the
Sun
in
Cancer
0
,
for
the
same
Meridian
cutteth
both
these
,
and
therefore
both
these
points
would
rise
at
once
in
a
Right
Sphear
,
where
the
Meridians
by
turns
successively
,
become
Horizon
:
but
counting
how
many
degrees
are
between
this
prick
and
the
rising
point
of
the
Equator
,
I
find
34
degr.
10
min.
this
is
the
arch
of
Ascensional
difference
,
which
being
subtracted
out
of
the
Right
Ascension
of
♋
0
(
which
by
Chap.
6
is
90
degr.
)
there
remaineth
the
Oblique
Ascension
55
degr.
50
min.
And
the
meaning
is
,
that
whereas
the
Sun
being
in
♋
0
in
a
Right
Sphear
,
riseth
with
the
90th
degree
of
the
Equator
,
in
our
Latitude
,
he
riseth
with
the
55
degr.
50
min.
of
the
Equator
:
the
difference
of
these
Ascensions
is
34
degr.
10
min.
add
this
difference
to
the
Right
Ascension
of
♋
0
,
and
it
maketh
124
degr.
10
min.
the
Oblique
Descension
,
for
the
124th
degree
of
the
Equator
setteth
with
♋
0
,
and
the
point
of
the
Suns
Right
Ascension
shall
in
North
Signes
Set
before
him
as
much
as
it
Riseth
after
him
,
and
in
South
Signes
shall
Set
after
him
,
as
much
as
it
Riseth
before
him
.
This
you
may
see
plainly
by
the
view
of
this
Projection
;
if
you
imagine
it
one
while
to
be
the
Eastern
Hemisphear
,
and
another
while
the
Western
Hemisphear
,
at
your
pleasure
.
The
Horizontal
Triangle
.
diagram
Take
in
the
Scheme
of
the
Horizontal
Triangle
annexed
,
so
many
Circles
of
your
Planisphear
as
you
shall
use
for
this
purpose
,
and
moreover
see
here
how
the
Ecliptick
should
lie
in
your
Planisphear
when
♋
0
is
rising
,
which
the
Planisphear
in
this
posture
cannot
express
.
♈
♋
Is
the
arch
of
the
Suns
Longitude
|
90
degr.
|
♈
A
Is
the
arch
of
the
Suns
Right
Ascension
|
90
degr.
|
♈
B
Is
the
arch
of
the
Oblique
Ascension
|
55
degr.
50
min.
|
B
A
Is
the
Ascensional
difference
|
34
degr.
10
min.
|
A
B
C
,
I
cal
the
Horizontal
Triangle
.
|
|
The
same
way
serveth
for
the
Stars
,
for
the
Stars
Parallel
of
Declination
followed
to
the
Finiter
,
shall
bring
you
to
C
of
the
Triangle
,
as
the
Suns
did
,
and
then
you
know
what
to
do
.
A
second
way
and
more
easie
and
pleasant
,
is
by
the
Equinoctial
Projection
.
Place
the
Sun
or
Star
upon
the
East
part
of
your
Horizon
,
(
in
the
North-east
quarter
,
if
the
Declination
be
North
;
but
in
the
South-east
quarter
,
if
the
Declination
be
South
;
as
you
had
direction
,
Book
2
,
2.
)
and
the
degrees
of
the
Limb
by
which
♈
0
is
gone
past
Oriens
,
or
the
six
a
clock
line
of
the
Mater
,
are
the
degrees
of
Oblique
Ascension
,
subduct
this
out
of
the
Right
Ascension
,
if
the
Star
be
North
,
or
out
of
this
subduct
the
Right
Ascension
,
if
the
Star
be
South
,
and
the
remainder
is
the
Ascensional
difference
.
But
this
subduction
is
made
to
your
hand
in
the
Planisphear
.
diagram
Take
the
former
Example
.
The
Latitude
here
is
52
degr.
15
min.
the
Suns
Declination
in
♋
0
,
is
23
degr.
30
min.
as
before
.
Now
See
in
this
second
figure
of
the
Horizontal
Triangle
A
B
C
,
how
the
Circles
lie
in
the
Planisphear
,
set
♋
0
on
the
Northeast
part
of
the
Horizon
at
C
,
and
you
have
before
your
eyes
.
♈
♋
the
Suns
Longitude
.
|
90
degr.
|
♈
A
the
Suns
Right
Ascension
.
|
90
degr.
|
♈
B
the
Suns
Oblique
Ascension
.
|
55
degr.
50
min.
|
B
A
the
Ascensional
difference
.
|
34
degr.
10
min.
|
CHAP.
XV.
The
Ascensional
difference
,
Declination
,
and
Amplitude
,
of
the
Sun
or
a
Star
,
and
the
Latitude
of
the
Place
,
any
two
of
them
given
,
to
find
the
rest
.
T
the
Amplitude
or
Ortive
Latitude
is
the
arch
of
the
Horizon
between
the
rising-point
of
a
Star
and
the
full
East
point
.
This
is
the
Hypotenusa
of
the
Horizontal
Triangle
,
expressed
in
both
the
Schemes
of
the
former
Chapt.
Now
I
told
you
Book
3.2
.
that
if
any
two
parts
of
a
Rectangled
Triangle
be
given
with
the
Right
angle
,
the
rest
may
be
easily
found
;
observe
then
your
Triangle
A
B
C
in
the
first
Scheme
of
the
former
Chapter
,
and
likewise
in
the
Meridional
Projection
of
your
Planisphere
,
you
shall
see
the
very
same
.
For
the
Finitor
being
set
to
the
Latitude
,
C
shall
be
where
the
Tropick
of
Cancer
cuts
the
Finiter
:
the
arch
of
the
Meridian
between
C
and
the
Equator
is
C
A
and
the
Declination
:
thence
in
the
Equator
to
the
center
is
A
B
the
Base
,
and
the
Ascensional
Difference
B
C
in
the
Horizon
is
the
Amplitude
;
B
is
the
complement
of
Latitude
;
A
is
90
degr.
C
is
unknown
,
and
we
need
it
not
,
else
,
if
you
have
read
the
third
Book
,
I
hope
you
can
find
him
.
Here
are
six
Cases
.
-
1.
Admit
now
that
the
Declination
and
Amplitude
be
given
,
put
the
term
of
the
Amplitude
(
I
mean
the
point
where
it
ends
,
counting
from
the
Center
)
upon
the
Parallel
of
the
Declination
,
and
your
Triangle
is
formed
,
and
thereby
the
Ascensional
difference
and
the
complement
of
Latitude
are
discovered
.
-
2.
Or
if
the
Declination
and
Ascensional
difference
be
given
,
number
the
Ascensional
difference
from
the
Center
downwards
in
the
Equator
:
Then
go
up
in
a
Meridian
as
many
degrees
as
the
Declination
comes
to
,
and
to
the
point
where
you
end
(
which
is
C
)
set
the
Finiter
,
so
he
is
placed
to
your
Latitude
,
and
the
Amplitude
also
is
shewn
.
-
3.
Or
if
the
Declination
and
Latitude
be
given
,
the
Finiter
being
set
to
the
Latitude
,
follow
the
Parallel
of
Declination
to
the
Finiter
there
is
C
,
thence
go
down
by
a
Meridian
to
A
in
the
Equator
,
thence
in
the
Equator
to
Bat
the
Center
,
thence
turn
by
the
Finiter
to
C
,
and
you
have
compassed
your
Triangle
,
and
therefore
have
all
known
but
C.
-
4.
If
the
Latitude
and
Ascensional
difference
be
given
,
the
Finiter
being
set
to
the
Latitude
,
count
from
the
Center
in
the
Equator
to
the
end
of
the
Ascensional
difference
,
there
is
A
:
Go
up
thence
in
a
Meridian
to
the
Finiter
;
there
is
C
:
Go
thence
in
the
Finiter
to
the
Center
:
there
is
B.
-
5.
If
the
Latitude
and
Amplitude
be
given
,
the
Finiter
being
set
to
the
Latitude
,
count
from
the
Center
(
B
)
in
the
Finiter
to
the
end
of
the
Amplitude
(
where
shall
be
C
)
go
down
thence
in
a
Meridian
to
the
Equator
,
(
where
is
A
)
thence
in
the
Equator
return
to
the
Center
B.
-
6.
If
the
Amplitude
and
Ascensional
difference
be
given
,
prick
the
end
of
the
Amplitude
numbred
in
the
Finiter
from
the
Center
,
and
prick
the
end
of
the
Ascensional
difference
,
numbred
in
the
Equator
from
the
Center
:
then
turn
about
the
Reet
till
some
one
of
the
Meridians
cut
both
these
pricks
,
and
that
shall
make
up
the
Triangle
.
Note
,
that
for
South
Stars
,
or
the
Sun
in
South
Signes
,
this
Triangle
lies
on
the
South-side
the
center
,
and
above
the
Finiter
;
but
for
North
Signes
it
lies
North
of
the
center
,
and
below
the
Finiter
.
CHAP.
XVI
.
To
do
the
same
in
the
Equinoctial
Projection
.
HEre
serves
the
second
figure
of
the
Horizontal
Triangle
in
Chap.
14.
where
B
A
is
the
Ascensional
difference
;
C
A
the
Declination
,
B
C
the
Amplitude
,
B
complement
of
the
Latitude
.
If
the
Latitude
and
Declination
be
given
,
number
the
Declination
on
the
Label
inwards
,
and
at
the
end
make
a
prick
,
turn
this
prick
to
the
Horizon
of
the
Mater
,
and
so
shall
the
outward
arch
of
the
Label
,
be
C
A
,
the
shorter
arch
of
that
Horizon
B
C
,
and
an
arch
of
the
Limb
B
A
of
your
Triangle
.
If
the
Latitude
and
Amplitude
be
given
,
do
as
in
this
Example
.
I
observed
Sirius
to
rise
27
¼
from
the
East
South-ward
my
Latitude
is
52
degr.
¼
.
I
go
to
the
52
¼
Meridian
on
the
Mater
,
reckoned
from
the
Center
on
the
South-side
,
because
the
Star
is
Southern
,
as
his
rising
shewes
.
This
52
¼
Meridian
being
my
Horizon
(
as
Book
2.2
.
)
I
number
in
him
the
Amplitude
of
Sirius
,
from
Oriens
toward
Meridies
27
¼
,
and
thereto
I
lay
the
Label
;
and
I
see
my
Horizon
cuts
the
Label
in
16¼
,
that
is
C
A
the
South
Decimation
of
Sirius
:
and
between
the
Label
and
Oriens
in
the
Limb
,
I
have
B
A
22
¼
,
his
Ascensional
difference
.
If
you
can
do
these
two
,
you
may
resolve
the
four
other
Cases
of
this
Chapter
with
like
facility
.
View
but
the
Scheam
in
the
Book
,
and
in
your
Planisphear
,
and
that
alone
will
instruct
you
.
CHAP.
XVII
.
To
find
the
Semi-diurnal
and
Semi-nocturnal
Arches
of
the
Sun
or
Stars
:
the
time
of
their
Rising
and
Setting
:
and
the
length
of
their
Day
and
Night
:
by
Declination
,
and
the
Latitude
of
the
Place
.
SEt
the
Finiter
to
the
Latitude
,
(
as-in
the
first
Mode
of
the
Meridional
Projection
.
)
Then
seek
the
Parallel
of
the
Declination
of
the
Sun
or
Star
,
North
,
or
South
,
as
it
hapneth
to
be
.
That
Parallel
shall
be
divided
by
the
Finiter
into
two
arches
:
the
arch
above
the
Finiter
is
the
Semi-diurnal
arch
,
in
which
you
may
count
the
time
of
Rising
and
Setting
,
and
the
Length
of
the
Day
:
that
below
is
the
Semi-nocturnal
arch
,
in
which
you
may
reckon
the
length
of
the
Night
;
or
if
your
Question
be
of
a
Star
,
the
time
he
spends
under
the
Horizon
.
Example
.
In
the
first
Scheme
of
the
14th
Chapter
,
D
E
is
the
Tropick
of
Cancer
,
that
is
the
23
½
Parallel
of
North
Declination
:
C
E
is
the
Semi-diurnal
arch
:
C
D
the
Semi-nocturnal
.
And
you
shall
find
in
the
Meridional
Projection
of
your
Planisphear
those
arches
are
divided
by
the
Meridians
;
and
the
arch
C
E
containeth
124
degr.
10
min.
which
turned
into
houres
and
minutes
,
(
accounting
every
degree
4
minutes
of
Time
,
and
every
15
degrees
an
houre
,
)
is
8
houres
16
min.
40
sec.
half
the
length
of
our
longest
Day
,
and
the
arch
C
D
containeth
55.
deg.
50
min.
that
is
,
three
houres
43
min.
20
sec.
half
the
length
of
our
shortest
Night
:
therefore
at
three
hours
43
min.
after
midnight
the
Sun
Riseth
in
the
Tropick
,
and
sets
so
much
before
midnight
,
that
is
,
at
eight
hours
16
min.
40
sec.
and
so
may
you
find
your
desire
in
any
other
Parallel
.
Example
.
2.
I
observe
that
Fomahant
his
Meridian
Altitude
is
but
6.30
min.
therefore
by
Chap.
13
he
declineth
Southward
31¼
.
I
would
know
how
long
he
shines
with
us
;
and
I
presently
see
in
the
Meridional
Projection
of
my
Planisphear
,
that
his
Parallel
hath
but
38
degr.
above
the
Horizon
;
that
is
,
he
will
set
two
hours
32
min.
after
he
is
South
;
and
the
whole
time
he
shines
in
our
Horizon
,
is
five
hours
four
minutes
.
Example
.
3.
Lyra
her
Declination
is
38.30
min.
North
;
and
I
see
his
Parallel
comes
within
45
min.
of
the
Horizon
,
in
the
North
part
of
the
Meridian
,
but
never
toucheth
it
:
therefore
I
conclude
that
Lyra
never
sets
with
us
at
all
.
CHAP.
XVIII
.
To
find
the
same
,
in
the
Equinoctial
Projection
.
TUrn
about
the
Reet
till
the
Suns
place
in
the
Ecliptick
,
or
the
point
of
the
Star
,
touch
your
proper
Horizon
:
and
that
on
the
North
side
,
if
the
Declination
be
North
,
or
on
the
South
side
,
if
it
be
South
.
Lay
the
Label
to
the
Sun
or
Star
in
the
Horizon
,
and
between
the
Label
and
Meridies
upon
the
Limb
you
shall
have
the
Semi-diurnal
arch
,
both
in
degrees
,
and
in
hours
and
minutes
.
And
you
shall
observe
that
those
Stars
whose
Declination
is
greater
then
the
complement
of
your
Latitude
(
as
Lyra's
was
in
the
last
Chap.
)
will
never
touch
the
Horizon
at
all
.
For
Stars
of
such
Declination
,
if
they
be
North
,
never
set
;
and
if
they
be
South
,
never
rise
at
our
Town
.
But
what
shall
I
do
if
the
Star
be
not
in
my
Reet
?
Then
will
I
number
his
Right
Ascension
on
the
Limb
of
the
Reet
,
and
having
thereto
laid
the
Label
,
I
will
number
his
Declination
upon
the
Label
from
the
Limb
inwards
,
and
where
it
ends
make
a
prick
by
the
edge
of
the
Label
,
in
the
Reet
,
for
him
:
for
there
is
the
place
of
the
Sta:
but
if
the
Stars
place
happen
to
be
in
a
window
of
the
Reet
,
where
the
Reet
is
perforated
,
then
I
will
make
the
prick
upon
the
Labels
edge
at
the
Stars
Declination
,
and
turn
that
prick
to
the
Horizon
.
I
may
pintch
the
Label
close
with
the
Reet
,
and
turn
both
together
,
which
will
be
the
handsomer
way
,
but
if
I
move
the
Labels
prick
alone
to
the
Horizon
,
it
is
sufficient
for
this
Probleme
,
which
needeth
no
more
words
.
CHAP.
XIX
.
To
find
the
beginning
and
end
of
Twilight
,
by
the
Suns
Declination
,
and
the
Latitude
of
the
Place
.
SEt
the
Planisphear
in
the
first
Mode
of
the
Meridional
Projection
;
then
turn
the
Planisphear
that
the
Zenich
may
be
downwards
,
and
the
Almicanters
mostly
below
the
Horizon
.
Then
go
to
the
18th
Almicanter
below
the
Horizon
:
and
wheresoever
the
Parallel
of
the
Suns
Declination
doth
cut
that
Almicanter
,
there
is
the
beginning
and
end
of
Twilight
:
and
because
every
Parallel
is
divided
by
the
Meridians
into
12
hours
,
or
180
degr.
(
every
15
degr.
being
one
hour
)
therefore
you
may
easily
count
how
far
the
point
where
Twilight
begins
,
is
distant
from
Midnight
,
or
from
Noon
,
or
from
Sun-rise
,
or
Sun-set
in
the
Horizon
.
Example
.
In
our
Latitude
52
degr.
15
min.
my
Planisphear
set
as
aforesaid
,
I
find
that
where
the
18th
Almicanter
cutteth
the
Equator
under
the
Horizon
,
there
cutteth
also
in
the
same
intersection
the
30th
Meridian
,
or
second
Hour
Circle
from
the
Axis
and
Center
;
by
which
I
gather
,
that
when
the
Sun
is
in
the
Equator
,
the
twilight
begins
two
hours
before
6
or
Sun-rising
,
and
ends
likewise
at
8
of
the
clock
at
Night
,
the
Sun
then
setting
,
(
as
you
may
see
)
at
6.
Likewise
where
the
Winter
Tropick
cuts
the
18th
Almicanter
,
there
cuts
also
the
first
Meridian
from
the
Axis
South-ward
;
shewing
that
in
the
depth
of
Winter
Twilight
begins
4
minutes
after
6
in
the
morning
,
and
lasteth
till
5
hours
56
minutes
after-noon
.
Likewise
I
see
that
about
the
beginning
of
♊
where
the
Sun
declineth
North-wards
about
20
degrees
,
the
Twilight
lasts
till
mid-night
,
and
that
from
that
time
till
the
Sun
comes
to
♌
(
that
is
,
from
May
11
to
July
11
,
or
thereabouts
,
)
we
have
no
dark
night
at
all
,
unless
the
Skie
be
Cloudy
,
for
in
all
that
time
the
Sun
is
never
found
above
18
degrees
under
the
Horizon
.
CHAP.
XX.
To
find
the
time
of
the
Cosmical
Rising
and
Setting
of
the
Stars
,
by
their
Declination
and
Right
Ascension
,
and
the
Latitude
of
the
Place
.
AStar
is
said
to
rise
Cosmically
when
he
riseth
at
the
same
instant
with
the
Sun
.
To
find
it
,
use
the
Equinoctial
Projection
:
Turn
the
Star
(
being
found
in
your
Reet
)
to
the
East
part
of
your
Horizon
,
and
look
what
degree
of
the
Ecliptick
cutteth
the
same
East
part
of
your
Horizon
;
for
when
the
Sun
comes
to
that
degree
,
the
Star
and
Sun
shall
rise
both
together
.
If
the
Star
be
not
in
your
Reet
,
put
him
in
with
ink
,
as
you
put
in
the
rest
,
Book
1.7
.
if
his
place
light
upon
a
window
,
or
hole
of
the
Reet
,
prick
him
on
the
edge
of
the
Label
,
and
hold
Reet
and
Label
close
together
,
while
you
turn
him
to
the
Horizon
.
Example
.
Sirius
I
have
among
40
other
principal
Stars
in
my
Reet
,
and
would
advise
you
not
to
be
without
him
,
for
he
is
a
little
Sun
in
a
Winters
Night
,
to
tell
you
how
the
time
passeth
;
he
is
called
by
the
Latines
both
Canis
,
and
Canicula
;
for
they
had
no
name
for
the
little
Dog
,
but
called
him
by
the
Greek
name
Procyon
,
as
Pliny
witnesseth
Lib.
18
Chap.
28
:
yet
I
have
seen
a
late
Writer
,
who
takes
upon
him
to
teach
the
Colledge
of
Physitians
both
Physick
and
Astrologie
,
before
he
hath
well
learned
either
of
them
:
who
in
his
Obtrectations
upon
the
Pharmacopaea
Lond.
in
the
Chapter
of
Vinum
Scilliticum
Galeni
,
betrayes
his
ignorance
herein
,
as
elswhere
in
100
other
things
,
which
I
could
shew
;
for
in
that
very
Chapter
pag.
147
of
his
fourth
Impression
,
in
24
short
lines
he
commits
5
absurd
errors
.
1.
He
makes
it
doubtfull
whether
Canis
be
to
be
taken
for
Sirius
,
or
Procyon
.
2.
He
goeth
about
to
teach
Galen
where
Squills
grow
;
and
that
there
is
no
hilly
ground
near
the
Sea
.
3.
He
supposeth
that
the
Acronycal
rising
of
the
Dog
(
which
hapneth
in
the
depth
of
Winter
)
is
a
fitter
time
to
gather
Squils
,
then
the
Heliacal
rising
,
which
hapneth
near
unto
the
Cosmical
,
in
the
heat
of
Summer
.
4.
He
either
supposeth
that
Squils
grow
in
the
Parallel
of
London
,
or
that
by
the
rising
of
the
Dog
at
London
men
should
gather
Squils
in
Greece
or
Spain
.
5.
He
tells
the
Colledge
that
both
the
Dogs
are
between
the
Equator
and
the
South
Pole
,
which
indeed
is
newes
,
if
it
were
true
.
Let
the
ingenuous
Reader
pardon
this
digression
,
and
I
proceed
.
This
Sirius
I
brought
to
our
Horizon
,
(
the
52
¼
)
and
found
that
there
riseth
with
him
in
the
Ecliptick
♌
18
½
.
in
like
manner
,
with
Procyon
riseth
♌
6
⅓
:
therefore
Sirius
riseth
Cosmically
with
us
August
1
:
and
Procyon
12
dayes
sooner
.
But
in
Greece
and
Spain
,
in
the
Latitude
38
degr.
Sirius
riseth
with
♌
4
½
,
that
is
a
fortnight
sooner
.
A
Star
is
said
to
Set
Cosmically
,
when
the
Sun
riseth
at
his
setting
.
Place
the
Star
therefore
on
the
West
part
of
your
Horizon
:
then
look
what
degree
of
the
Ecliptique
riseth
in
the
East
part
;
for
when
the
Sun
comes
to
that
degree
,
the
Star
shall
set
Cosmically
.
Example
.
I
brought
Sirius
to
the
South-west
part
of
our
Horizon
,
where
he
useth
to
set
.
And
in
the
South-east
part
I
saw
♏
23
degrees
in
the
Ecliptique
rising
:
therefore
when
the
Sun
is
in
♏
23.
(
which
is
about
November
,
5.
then
shall
Sirius
set
Cosmically
.
But
at
Athens
in
Lat.
37
¼
,
-
His
Cosmicall
Rising
is
☉
in
♌
4.
July
17
-
His
Cosmicall
Setting
is
☉
in
♐
9.
Nov.
20
-
The
Pleiades
in
our
Lat
Cosmically
Rise
.
☉
in
♉
12
½
Apr.
22.
-
The
Pleiades
in
our
Lat
Cosmically
Set
☉
in
♏
29
½
Nov.
11.
-
At
Athens
Cosmically
.
Rise
☉
in
♉
19
½
April
30.
-
At
Athens
Cosmically
.
Set
☉
in
♏
27
½
Novem.
9.
-
Arcturus
in
our
Lat.
Cosmically
.
Ri.
☉
in
♎
0
Sep.
13.
-
Arcturus
in
our
Lat.
Cosmically
.
Set.
☉
in
♋
4
June
15.
-
At
Athens
Cosmically
.
Rise
☉
in
♎
10
½
Sept.
23.
-
At
Athens
Cosmically
.
Set
☉
in
♊
6.
May
17.
CHAP.
XXI
.
To
find
the
time
when
any
Star
riseth
or
setteth
Acronycally
,
by
his
Declination
,
and
Right
Ascension
,
and
the
Latitude
of
the
Place
.
WHen
a
Star
Riseth
just
at
Sun-setting
,
he
is
said
to
rise
Acronically
.
To
find
the
time
,
turn
the
Star
to
the
East
part
of
the
Horizon
in
the
Equinoctial
Projection
,
and
mark
what
degree
of
the
Ecliptique
descendeth
in
the
West
:
for
when
the
Sun
comes
to
that
degree
,
the
Star
shall
rise
Acronically
.
Example
.
When
Sirius
toucheth
the
South
East
Quarter
of
our
Horizon
,
I
see
♒
18.
setting
.
Therefore
when
the
Sun
is
in
♒
18.
Sirius
riseth
Acronically
.
A
Star
setteth
Acronically
,
when
he
setteth
with
the
Sun
.
To
find
the
time
,
place
the
Star
setting
,
in
the
West-part
of
the
Horizon
,
and
see
what
degree
of
the
Ecliptique
setteth
with
him
:
for
when
the
Sun
is
in
that
degree
,
the
Star
shall
set
Acronically
.
Thus
in
our
Latitude
.
-
Sirius
Acronically
.
Riseth
☉
in
♒
18.
Jan.
27.
-
Sirius
Acronically
.
Seteth
☉
in
♉
23.
May
3.
-
At
Athens
Sirius
Acronically
.
Ri.
☉
in
♒
4.
Janu.
13.
-
At
Athens
Sirius
Acronically
.
Set.
☉
in
♊
9.
May
20.
-
Pleiades
Acronically
.
Riseth
☉
in
♏
13.
Octo.
26.
-
Pleiades
Acronically
.
Seteth
☉
in
♉
29
½
May
10.
-
At
Athens
Pleiades
Acroni
.
Riseth
☉
in
♏
19
½
Nov.
1.
-
At
Athens
Pleiades
Acroni
.
Seteth
☉
in
♉
27
½
May
8.
-
Arcturus
Acronically
.
Riseth
☉
in
♈
0.
March
10.
-
Arcturus
Acronically
.
Seteth
☉
in
♑
4
Dec.
15.
-
At
Athens
Arcturus
Acroni
.
Riseth
☉
in
♈
10.
½
Mar.
20.
-
At
Athens
Arcturus
Acroni
.
Seteth
☉
in
♐
6.
Nove.
18.
CHAP.
XXII
.
To
find
when
a
Star
riseth
or
setteth
Heliacally
.
AStar
riseth
Heliacally
when
he
geteth
out
of
the
beames
of
the
Sun
,
and
beginneth
to
be
seen
in
the
East
a
little
before
Sun
rise
.
And
a
Star
is
said
to
set
Heliacally
when
he
getteth
into
the
beams
of
the
Sun
,
and
beginneth
to
be
least
in
the
evening
by
reason
of
the
Suns
opproach
to
him
.
Those
Stars
which
you
see
nearest
the
East
Horizon
in
the
Morning
Twilight
are
Heliacall
Risers
;
and
those
which
you
see
nearest
the
Westpart
of
the
Horizon
,
in
the
evening
Twilight
are
Heliacall
Setters
.
For
this
no
exact
rule
can
be
given
,
for
all
men
have
not
like
quickness
of
sight
,
nor
all
Stars
like
brightness
,
nor
all
Climates
,
Countries
and
Dayes
of
the
Year
the
same
clearness
of
Air.
And
the
Moon
oft
times
augmenteth
the
Twilight
,
when
she
is
within
a
few
dayes
of
the
Change
,
and
keepeth
the
Stars
longer
Combust
.
Commonly
about
twenty
dayes
before
their
Acronicall
setting
they
come
within
the
Sun
beames
,
and
so
set
Heliacally
,
and
they
appear
again
,
(
that
is
,
rise
Heliacally
)
about
twenty
dayes
after
their
Cosmicall
rising
.
But
if
they
be
great
Stars
,
the
Air
clear
,
your
sight
good
,
the
angle
made
between
the
Ecliptique
and
the
Horizon
great
,
they
may
appear
sooner
:
and
later
in
the
contrary
Cases
.
According
to
this
rule
the
Pleiades
set
Heliacally
,
now
at
Athens
☉
in
♉
7.
and
rise
Heliacally
☉
in
♊
9.
so
they
should
be
Combust
there
32
dayes
:
but
because
they
be
Stars
of
less
Magnitude
,
we
may
perhaps
allow
them
40
dayes
as
Hesiod
did
in
his
time
,
in
the
beginning
of
his
Second
book
of
Weeks
and
Dayes
.
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
.
CHAP.
XXIII
.
To
find
the
Age
when
any
Astrologer
lived
,
and
what
time
of
the
Solar
year
the
Seasons
hapned
in
his
Country
,
by
knowing
his
Latitude
,
and
the
Rising
of
any
Star
in
his
time
.
THe
old
Grecians
,
and
after
them
the
Latines
,
before
Julius
Caesar
especially
,
designed
the
Seasons
of
the
Year
by
the
rising
and
setting
of
some
notable
Stars
.
Hesiod
begins
his
second
book
of
Weeks
and
Dayes
:
with
this
Georgical
Canon
.
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
.
That
is
,
when
the
Pleiades
rise
,
begin
to
Mow
,
and
to
Plow
when
they
set
;
And
in
the
same
Book
Vers.
182.
he
saith
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
.
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
.
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
.
That
is
,
60
dayes
after
the
Winter
Tropique
Arcturus
riseth
Acronically
,
and
then
appears
the
Swallow
,
the
Spring
being
then
new
begun
.
These
and
the
like
rules
were
the
Husband-mans
Almanack
.
by
which
they
measured
the
Solar
Year
,
and
the
return
of
the
Seasons
.
For
in
their
Civill
Year
,
consisting
of
Lunar
Moneths
,
by
reason
of
an
intercalary
Moneth
which
was
added
every
third
Year
,
and
somewhat
ofter
,
the
Seasons
could
happen
upon
the
same
day
of
the
moneth
yearly
,
but
sometimes
2
or
3
weeks
sooner
or
later
,
as
our
moveable
Feasts
do
.
The
rising
and
setting
of
the
fixed
Stars
keep
the
same
distance
Yearly
,
from
the
Equinoctial
and
Solstitiall
points
,
for
a
mans
age
near
enough
;
but
longer
those
rules
cannot
last
without
some
perceivable
error
:
for
in
100.
years
the
Stars
go
forward
in
Longitude
.
according
to
Tycho
.
1
degree
25
minutes
,
by
reason
whereof
the
risings
and
settings
of
the
Stars
happen
later
in
the
Year
,
about
a
day
and
half
every
100
years
in
the
same
Latitude
.
Now
if
you
would
note
in
what
Age
a
Star
had
such
a
rising
or
setting
,
in
such
a
Latitude
,
as
for
Example
.
In
what
age
Arcturus
rose
60
dayes
after
mid-winter
in
the
Latitude
of
Asera
in
Boeotia
near
Athens
,
whose
Latitude
is
37
¼
,
and
consequently
how
long
since
Hesiod
lived
,
in
whose
dayes
Arcturus
had
such
rising
,
you
shall
reason
thus
:
60.
dayes
after
the
Winter
Tropick
the
Sun
is
in
♓
1
degree
by
the
Ephemeris
for
in
those
60
dayes
near
his
Perigium
he
goeth
about
61
degrees
.
I
am
therefore
to
seek
when
Arcturus
did
rise
at
Athens
with
the
opposite
degree
of
the
Ecliptique
,
♍
1
degree
(
the
Sun
is
in
♓
1
degree
,
then
setting
over
against
it
.
)
I
seek
the
Longitude
and
Latitude
of
Arcturus
,
and
find
in
Tycho'es
Tables
that
Anno
Domi.
1600.
Arcturus
had
Longitude
♎
18.
39
minutes
,
Latitude
B.
31.02
.
minutes
:
then
I
will
suppose
that
Hesiod
lived
830.
years
before
Christ
(
for
there
some
Chronologers
place
him
,
but
without
any
good
proof
that
I
find
)
that
is
,
2430
years
before
Anno
Domi.
1600.
in
which
space
of
time
Arcturus
must
have
increased
his
Longitude
by
Tycho
'
Hypothesis
34
degrees
25
minutes
,
which
being
subducted
out
of
the
Longitude
which
Arcturus
had
Anno
Domi.
1600.
leaves
his
Longitude
for
the
year
before
Christ
830.
♍
14.
degree
14.
minutes
,
his
Latitude
was
then
and
ever
31.02
minutes
.
Now
from
this
Longitude
and
Latitude
,
I
get
his
Right
Ascension
,
and
Declination
,
by
Chap.
34.
of
this
Book
,
where
I
find
Ascension
180.
degrees
,
Declination
North
34.
degrees
15
minutes
:
those
had
,
I
place
Arcturus
in
my
Reet
according
to
that
Right
Ascension
and
Declination
(
as
was
taught
Book
1
7.
and
Book
4.18
.
)
and
by
Chapter
21.
I
find
♍
4
rising
with
him
;
and
at
the
same
time
♓
4.
setting
in
the
same
Herizon
of
Athens
.
But
I
ought
to
find
♓
1
degree
setting
in
Hesieds
time
.
Therefore
I
will
suppose
again
that
Hesiod
lived
1130
years
before
Christ
:
and
proceeding
as
upon
the
former
supposition
,
I
find
that
then
♒
29.
degrees
did
set
at
his
Acronical
rising
:
but
I
ought
to
find
♓
1
degree
rising
.
And
seeing
it
is
hereby
found
,
that
in
300
years
his
Acronicall
setting
varies
5.
degrees
,
or
dayes
;
I
take
the
proportional
part
of
that
time
,
and
lay
that
in
the
year
1010.
before
Christ
,
Arcturus
did
set
Acronically
60.
dayes
after
the
Winter
Trepique
:
and
then
lived
Hosiod
,
or
soon
after
.
For
being
an
Astrologer
himself
(
as
Pliny
tells
us
Lib.
18.25
.
saying
,
Hu●us
quaque
nomine
extat
Astrologia
,
)
it
is
likely
he
would
not
use
an
antiquated
rule
.
Arcturus
therefore
rose
Acronically
at
Athens
in
Hesiods
time
,
☉
in
♓
1
degree
,
that
is
,
about
Febr.
9.
of
our
Julian
year
,
as
it
now
goeth
:
then
the
Swallow
used
to
come
to
Athens
:
but
in
our
Age
he
riseth
Acronically
at
Athens
☉
in
♈
10
½
that
is
Mar.
20.
and
at
Ecton
or
Northampton
,
☉
♈
0.
that
is
,
Mar.
10.
By
this
you
may
see
that
the
old
Astrological
Rules
concerning
the
rising
and
setting
of
the
Stars
,
left
us
by
Hesiod
,
Cato
,
Aratus
,
Varro
,
Palladius
,
Virgil
,
Ovid
Pliny
,
Columella
,
Ptolomy
:
and
other
Ancient
Authors
cannot
serve
for
our
Age
,
nor
for
every
Latitude
;
and
the
best
use
we
can
make
of
them
,
is
to
find
the
Age
,
when
they
lived
.
Pliny
,
Lib.
18.26
.
saith
,
that
in
Caesurs
Calender
octavo
Calend
.
Martij
was
Hirundinis
adventus
&
jostero
die
Arcturi
exor
us
Vespertimus
.
Which
agrees
not
to
Caesars
time
.
Also
Lib.
2.47
.
he
saith
,
Ardentissimo
aestatis
tempore
exoritur
Caniculae
fidus
Sole
primem
partem
Leonis
ingrediente
,
qui
dies
est
15.
ante
Caelend
.
Augusti
(
that
is
,
July
18.
)
Rome
is
in
Latitude
42
,
degrees
,
Pliny
lived
about
70.
years
after
Christ
;
then
was
Canicula
(
that
is
Sarius
)
in
♊
17.
degrees
Latitude
,
39
½
,
Right
Ascension
,
79
½
Declination
,
South
,
16
⅔
,
and
did
rise
at
Rome
Cosmically
,
decimo
quinto
Calend.
Augusti
,
or
July
18.
as
thus
far
he
reports
truly
:
but
the
Sun
was
not
then
in
♌
1
deg.
as
Pliny
saith
but
in
♋
23
degrees
:
for
the
Sun
entred
♌
in
his
time
not
decimo
quinto
Calend
,
Augusti
,
but
octavo
Calend.
Augusti
.
The
Sun
in
those
dayes
entring
the
several
Signes
mostly
on
the
5
day
of
the
several
moneths
,
as
in
our
Age
about
the
11th
day
as
Astronomers
well
know
.
Pliny
seemes
to
have
taken
his
Astrologie
upon
trust
.
And
I
cannot
devise
what
should
lead
him
to
suppose
,
that
howsoever
the
Equinoxes
and
Solstices
in
his
time
hapned
octavo
Calend.
(
as
he
denyeth
not
)
yet
the
Sun
entred
into
a
new
Signe
about
the
Ides
of
every
Moneth
,
and
that
the
Equinoctial
and
Soistitial
points
were
in
Octavis
partibus
signorum
,
as
if
the
Sun
came
not
to
the
Equinoctial
till
he
came
to
the
8th
degree
of
Aries
.
See
Pliny
Book
18.
Chapter
25
,
26
,
27
,
28.
He
seemeth
to
distruct
the
Julian
Calender
,
and
to
adhear
more
to
the
account
used
by
Varro
de
Rerusticâ
Lib.
1.27
.
but
either
he
understood
neither
of
them
well
,
or
I
do
not
well
understand
him
.
Now
Sirius
riseth
in
our
Horizon
with
♌
18½
,
about
August
1
,
in
the
Declination
of
the
heat
,
who
in
Plinyes
time
rose
ardentissimo
astatis
tempore
.
And
our
Dog-dayes
if
we
follow
the
Dogs
rising
will
be
every
age
colder
and
colder
,
and
at
length
fall
in
Winter
.
It
were
better
to
reduce
them
to
the
Suns
entrance
into
Leo
,
or
to
Cancer
,
23
,
rather
as
they
were
in
Plinyes
time
:
and
to
count
the
ardentissimum
tempus
a
fortnight
before
and
a
fortnight
after
:
for
Sirius
was
not
by
the
Ancients
supposed
the
cause
of
the
sultry
heat
of
Summer
but
a
concomitant
signe
of
that
Season
,
whereof
the
Suns
continuance
in
the
North-Signes
was
the
cause
.
Would
you
know
also
when
they
began
to
Plow
and
to
Mow
in
Greece
in
Hesiods
Time
?
He
saith
,
when
the
Pleiades
rise
,
begin
to
Mow
,
and
to
Plow
when
they
set
.
The
Pleiades
(
I
mean
the
brightest
of
them
)
1010
years
before
Christ
,
were
in
♈
17.25
.
minutes
,
Latitude
4
degrees
North.
Declination
therefore
by
Chapter
34
)
11
degrees
,
Right
Ascension
14½
degrees
;
therefore
they
rose
Cosmically
at
Athens
or
Ascra
,
(
Hesiods
birth
Place
)
☉
in
♈
10
⅓
,
that
is
,
as
our
Julian
year
now
goeth
,
about
March
20.
The
Heliacall
rising
is
about
20.
dayes
after
the
Cosmicall
(
Chapter
22.
)
that
is
about
April
9.
Therefore
either
March
20.
at
the
Cosmical
rising
,
or
April
9.
at
the
Heliacall
rising
,
they
began
to
Mow
,
and
I
think
he
means
the
Cosmicall
;
the
Acronicall
rising
was
there
in
his
Age
☉
in
♎
10.
⅓
about
Sep.
23.
which
is
too
late
beyond
reason
.
Now
that
they
should
begin
Mowing
in
Greece
within
10
dayes
after
the
Equinoctial
is
not
strange
,
seeing
the
first
fruits
of
ripe
Corn
were
offered
at
Jerusalem
yearly
at
Easter
;
which
fell
ordinarily
15.
dayes
after
the
Equinoctial
,
or
thereabout
.
Duet
.
16.
And
in
Egypt
-
cum
falce
arva
visunt
Paulo
ante
Calendas
Apriles
,
mossis
autem
peragitur
Maio
,
saith
Pliny
18.18
.
viz.
Harvest
began
in
Egypt
a
little
before
April
,
and
April
then
began
8
,
dayes
after
the
Equinoctial
onely
.
The
Cosmicall
setting
of
the
Plaiades
at
Athens
,
in
Hesiods
time
1010.
years
before
Christ
was
☉
in
♎
18.
degrees
Octob.
1.
then
began
they
to
Plow
and
Sow
:
the
Egyptians
began
Novembri
mense
incipiente
Pliny
18.18
.
But
if
Hesiod
were
now
alive
at
Ascra
he
would
find
the
Plaiades
rise
Cosmically
,
with
♉
19
½
Alpril
30.
and
set
Cosmically
☉
in
♏
27
½
Nov.
9.
so
much
are
his
Georgique
rules
now
antiquated
,
and
serve
for
little
else
but
to
shew
how
many
Ages
ago
he
lived
;
and
how
the
Seasons
hapned
in
his
Age.
CHAP.
XXIV
.
The
Latitude
of
your
Place
,
the
Declination
,
Altitude
,
Azimuth
,
and
Hour
of
the
Sun
or
Stars
,
any
three
of
these
being
given
,
so
find
the
other
two
.
SEt
your
Planisphear
in
the
first
Mode
of
the
Meridional
Projection
,
The
Complemental
Triangle
.
and
you
shall
find
all
these
five
in
one
Oblique-angled
Triangle
;
which
I
use
to
call
the
Complemental
Triangle
;
because
it
consists
of
three
Sides
,
which
are
all
Complements
.
(
Others
may
call
it
as
they
please
.
)
A
B
in
the
Limb
between
the
Pole
and
Zenith
,
Complement
of
Latitude
.
A
C
in
a
Meridian
,
Complement
of
the
Declination
,
or
the
Supplement
of
that
Complement
.
B
C
in
an
Azimuth
,
Complement
of
the
Altitude
.
A
at
the
Pole
is
the
Angle
of
Horary
distance
from
the
Meridian
,
whose
full
measure
is
in
the
Equinoctial
line
;
but
because
every
Parallel
is
divided
by
the
Meridians
into
180.
degrees
as
the
Equator
is
,
and
every
5th
.
and
15th
.
Meridian
plainly
distinguished
from
the
rest
in
the
Fabrique
of
this
Instrument
,
therefore
you
may
easily
count
the
angle
of
the
Hour
in
any
Parallel
.
B
at
the
Zenith
,
is
the
angle
of
the
Azimuth
,
accounted
from
the
North
part
of
the
Meridian
:
his
full
measure
is
in
the
Finiterline
of
the
Reet
;
but
you
may
number
it
in
any
Almicanter
because
every
5th
.
and
15th
.
Azimuths
are
distinguished
on
the
Reet
,
as
the
Meridians
are
on
the
Mater
.
C
the
place
of
the
Sun
or
Star
,
in
the
meeting
of
the
Meridian
and
Azimuth
,
is
the
third
angle
,
which
commonly
is
neither
known
nor
enquired
;
but
it
may
be
found
when
you
please
,
by
turning
the
Triangle
,
as
hath
been
often
shewed
.
Now
if
you
be
versed
in
the
8
last
Chapters
of
the
third
Book
,
you
may
easily
find
any
of
the
requisites
of
this
Chapter
without
any
more
direction
.
Nevertheless
for
the
Learners
sake
,
I
shall
exemplifie
this
general
Probleme
,
in
the
4
next
Chapters
,
and
also
further
in
the
31
,
32
,
and
33.
Chapters
hereafter
following
.
See
the
Scheam
Chap.
26.
CHAP.
XXV
.
To
find
the
Altitude
and
Azimuth
of
the
Sun
or
Stars
,
at
any
time
proposed
;
the
Latitude
and
Declination
being
known
.
YOur
Planisphear
set
in
the
first
Mode
of
the
Meridional
Projection
,
as
in
the
former
Chapter
,
go
to
the
Parallel
of
the
Declination
of
the
Sun
or
Star
,
and
follow
him
through
all
the
Meridians
from
the
Finiter
to
the
Limb
;
(
which
is
the
Meridian
of
your
Place
;
)
and
thence
back
again
to
the
Finiter
,
and
you
shall
find
at
the
first
sight
what
Almicanter
and
Azimuth
cross
the
Parallel
in
any
point
proposed
:
and
so
have
you
the
Altitude
and
Azimuth
thereof
.
Example
.
June
10.
the
Sun
was
in
the
Tropique
of
Cancer
,
and
so
makes
his
diurnal
revolution
in
the
23
½
Parallel
of
Declination
;
I
follow
this
Parallel
,
the
Tropique
,
from
the
Horizon
upwards
,
and
having
gone
4
degrees
,
I
meet
the
ragged
arch
or
hour
line
of
4.
(
which
is
the
120th
.
Meridian
from
the
South
)
there
crosseth
the
second
Almicantar
,
and
the
three
and
fiftieth
Azimuth
from
the
North
;
whereby
I
learn
that
at
4.
in
the
Morning
,
June
10
,
the
Sun
is
2
degrees
high
;
and
in
Azimuth
from
the
North
53.
Thence
going
on
15
degrees
,
I
come
to
the
hour
circle
of
5.
where
cutteth
the
10th
.
Almicantar
almost
,
and
Azimuth
64
degrees
,
and
better
:
going
15
degrees
further
I
come
to
the
Axtree-line
,
which
is
the
hour
circle
of
6.
and
there
I
find
the
Suns
Altitude
18
degrees
,
and
his
Azimuth
from
the
North
75
degrees
,
&c.
And
look
what
Altitudes
and
Azimuths
I
find
at
4
,
5
,
6.
&c
,
in
the
Morning
,
the
same
I
find
at
the
afternoon
hours
,
that
have
like
distance
from
Noon
:
because
the
Eastern
and
Western
Hemisphears
are
alike
,
and
the
same
lines
serve
them
both
.
Thus
you
may
do
in
any
other
Parallel
,
and
for
any
Star
,
as
well
as
the
Sun
;
having
his
Declination
given
.
And
so
you
may
make
Tables
of
the
Suns
Altitude
and
Azimuth
,
at
every
hour
,
and
quarter
of
an
hour
,
if
you
please
,
for
every
day
throughout
the
year
:
and
that
as
fast
as
you
can
write
them
,
without
changing
the
posture
of
the
Planisphear
at
all
.
CHAP.
XXVI
.
The
Latitude
,
Altitude
,
and
Azimuth
given
,
to
find
the
Declination
,
and
the
Hour
.
EXample
.
Having
observed
the
Suns
Altitude
13.
degrees
,
and
his
Azimuth
from
the
South
Westward
28
½
in
our
Latitude
52
¼
.
my
Planisphear
set
in
the
same
manner
as
Chapter
25.
I
sought
out
the
13th
.
Almicantar
at
the
Limb
of
my
Reet
,
and
followed
him
inwards
till
I
came
between
Azimuth
28
,
and
29
,
there
I
met
the
30th
.
Meridian
,
and
the
20
¼
Parallel
of
Declination
,
by
which
I
gathered
that
it
was
2
of
the
clock
after
noon
,
and
that
the
Sun
declined
Southward
20
degrees
¼
.
Note
here
,
that
the
hour
of
a
Star
thus
found
,
is
not
the
hour
of
the
Night
,
unless
the
Star
happen
to
be
opposite
to
the
Sun
;
but
it
is
the
time
the
Star
lacketh
to
come
to
the
South
,
or
the
time
of
his
course
from
the
South
.
diagram
CHAP.
XXVII
.
The
Latitude
,
Declination
,
and
Altitude
,
given
,
to
find
the
Hour
,
and
Azimuth
.
HEre
the
three
sides
of
the
Complemental
Triangle
are
given
,
and
the
angles
A
and
B
sought
.
Example
,
March
the
10
in
the
Morning
the
Sun
being
in
the
Equinoctial
,
I
observed
his
Altitude
32
degrees
:
the
Finiter
being
set
to
my
Latitude
52
¼
.
as
before
,
I
went
to
the
32.
Almicantar
in
the
Reet
,
and
where
I
found
him
crossing
the
Equinoctial
line
of
the
Mater
,
there
I
conclude
was
the
place
of
the
Sun
at
the
time
of
my
observation
;
and
the
angle
C
of
my
Triangle
:
there
the
30
½
Meridian
passing
,
shewed
me
that
the
angle
A
at
the
Pole
was
30
½
,
or
,
that
it
wanted
half
a
degree
,
that
is
2
minutes
of
time
,
of
ten
of
the
clock
:
and
there
also
the
Azimuth
36
⅔
from
the
South
(
or
from
the
North
143
⅓
)
shewed
me
that
the
angle
B
at
the
Zenith
is
143
⅓
,
the
Azimuth
from
the
North
,
and
his
supplement
36
⅔
,
the
Azimuth
from
the
South
.
CHAP.
XXVIII
.
The
Declination
,
Altitude
,
and
Azimuth
of
the
Sun
given
,
to
find
the
Hour
,
and
Latitude
.
IN
the
Meridional
Projection
,
look
in
the
Reet
where
the
Almicantar
for
the
Altitude
given
,
and
the
Azimuth
given
do
cross
;
and
turn
the
point
of
the
Reet
where
they
cross
to
the
Parallel
of
the
Suns
Declination
upon
the
Mater
:
the
Meridian
that
cutteth
there
sheweth
the
hour
,
and
between
the
Finiter
and
the
Pole
,
or
between
the
Equator
and
the
Zenith
,
you
have
the
Latitude
in
the
Limb.
CHAP.
XXIX
.
To
find
the
Hour
of
the
Night
,
by
the
Northing
,
or
Southing
,
Rising
or
Setting
of
any
Star
.
USe
for
this
the
Equinoctial
Projection
.
And
if
the
Star
be
in
your
Reet
,
turn
him
to
the
North
or
South
of
the
Meridian
line
,
or
to
the
East
or
West
part
of
the
Horizon
in
your
Planisphear
,
as
you
see
him
in
the
Heaven
.
Then
turn
the
Label
to
the
Suns
place
in
the
Ecliptick
of
the
Reet
,
and
it
shall
shew
the
hour
in
the
Limb
:
but
if
the
Star
be
not
in
the
Reet
,
you
shall
supply
him
by
the
shift
used
Chap.
18.
Example
.
March
10th
.
I
saw
Sirius
setting
in
the
South-West
;
and
having
turned
him
to
the
same
place
in
my
Planisphear
,
I
laid
the
Label
to
♈
0.
which
was
the
Suns
place
for
that
day
;
and
it
cut
in
the
Limb
11.
hours
3
minutes
past
noon
.
Again
,
December
1.
Seeing
Lucida
Pleiadum
in
the
Meridian
,
I
turned
the
Star
till
he
touched
the
Meridian
line
of
the
Mater
;
then
laying
the
Label
to
♐
19.
the
place
of
the
Sun
,
I
found
it
was
10.
hours
17.
minutes
at
Night
,
The
same
Night
I
saw
Ras
Aben
,
or
the
brightest
in
the
Dragons
head
under
the
Pole
,
in
the
North
part
of
the
Meridian
;
wherefore
I
placed
him
on
the
Meridian
line
between
the
Center
and
Septentrio
:
and
the
Label
laid
to
♐
19.
shewed
me
it
was
12.
36.
minutes
;
that
is
more
then
half
an
hour
past
Midnight
.
CHAP.
XXX
.
The
time
of
Day
or
Night
given
,
to
find
in
what
Coast
any
Star
is
:
and
how
much
he
is
distant
from
the
Horizon
,
or
Meridian
.
LAy
the
Suns
place
to
the
hour
given
,
in
the
Equinoctial
Projection
,
then
may
you
presently
see
all
the
Stars
of
the
Reet
in
what
Coast
they
are
,
whether
under
the
Horizon
or
above
,
and
how
many
hours
they
lack
,
or
are
past
either
the
Horizon
or
Meridian
.
Example
.
Sitting
within
dores
at
seven
of
the
clock
on
Christmas
day
at
Night
,
I
desired
to
know
what
Stars
were
rising
,
and
what
near
the
Meridian
,
wherefore
laying
♑
14.
to
7.
of
the
clock
afternoon
,
I
saw
in
the
Reet
the
Rams
horn
,
a
little
past
South
.
The
Plaiades
wanted
1.
hour
28.
minutes
of
South
,
as
the
Label
shewed
me
in
the
Limb
;
♌
was
rising
,
but
Cor
♌
.
not
yet
up
.
I
would
know
now
what
he
wanted
of
rising
,
therefore
I
turned
forward
the
Reet
till
Cor
♌
came
to
the
Horizon
,
and
observed
how
many
degrees
of
the
Reet
passed
under
the
Label
(
or
by
any
point
of
the
Limb
)
while
the
Reet
turned
:
and
I
found
that
♈
0
(
and
so
any
other
point
)
moved
on
in
the
Limb
10
degrees
in
the
while
that
Cor
♌
was
comming
to
the
Horizon
.
Whereupon
I
understood
that
he
would
rise
40
minutes
after
.
CHAP.
XXXI
.
The
Time
,
and
Latitude
given
,
to
find
the
Altitude
,
and
Azimuth
of
any
Star
:
and
thereby
to
get
the
knowledge
of
the
Stars
.
FInd
the
hour
of
the
Star
,
by
the
former
Chapter
.
And
by
the
Label
observe
his
Declination
:
then
set
your
Planisphear
in
the
Meridional
Projection
,
and
in
the
Parallel
of
the
Stars
Declination
,
number
his
hour-distance
from
the
South
;
where
the
number
ends
,
set
a
needles
point
;
there
is
the
place
of
the
Star
;
and
the
Almicantar
and
Azimuth
that
cut
him
there
,
shew
your
desire
.
Example
.
I
would
know
the
cloudy
Star
Praesepe
,
in
the
breast
of
Cancer
(
which
indeed
is
a
glimmering
light
,
made
up
of
five
smal
and
bright
Stars
as
by
the
Telescope
appearech
)
This
Praesepe
I
found
by
the
last
Chapter
to
want
6.
hours
21.
min.
of
South
;
his
Declination
,
found
by
the
Label
,
is
21.
degrees
North.
Therefore
setting
the
Finiter
to
our
Latitude
52
¼
,
I
follow
the
21.
Parallel
of
North
Declination
,
from
the
Meridian
till
I
come
5
degrees
15
minutes
past
the
Axletree
(
because
I
found
him
6
hours
21
minutes
before
the
Meridian
)
there
the
17th
.
Azimuth
,
from
the
East
Northward
,
cutteth
the
Parallel
of
Praesepe
;
and
there
cutteth
him
also
the
13th
.
Almicantar
.
Now
to
find
him
out
,
I
lay
my
Planisphear
Horizontally
,
setting
the
Meridian
of
my
Planisphear
in
the
Meridian
of
the
Place
(
by
Chap.
3
,
and
4.
)
and
I
turn
my
Label
and
Sights
to
the
Azimuth
of
Praesepe
17
degrees
from
the
East
Northward
,
and
before
my
Sights
I
hang
up
a
Plumb-line
upon
a
Pole
,
to
keep
the
Azimuth
;
then
keeping
my
station
,
I
set
my
Planisphear
upon
his
edge
,
or
hang
him
upon
a
staf
with
a
socket
,
in
the
Azimuth
of
the
Star
;
so
that
the
Plumbet
show
Altitude
13
degrees
(
by
Chap.
1.
)
then
do
the
Sights
point
just
upon
Praesepe
,
and
would
teach
me
the
Star
,
if
I
did
not
know
him
before
.
CHAP.
XXXII
.
The
Latitude
of
the
Place
,
the
Declination
of
a
Star
,
with
his
Altitude
,
or
Azimuth
given
,
to
find
both
the
Hour
of
the
Star
,
and
the
Hour
of
the
Night
.
BY
this
Chapter
you
may
find
the
time
of
Night
,
at
any
time
,
by
any
Star
,
if
he
be
visible
above
the
Horizon
.
Use
the
first
Mode
of
the
Meridional
Projection
:
and
having
Observed
the
Altitude
or
Azimuth
of
the
Star
,
look
where
that
Altitude
or
Azimuth
cutteth
the
Parallel
of
the
Stars
Declination
,
there
cutteth
also
a
Meridian
which
sheweth
the
hour
of
the
Star
,
that
is
,
the
distance
of
a
Star
from
the
Meridian
in
hours
and
minutes
.
And
by
this
hour
of
the
Star
,
to
get
the
hour
of
the
Night
,
you
shall
place
the
Star
at
the
hour
found
,
in
the
Equinoctial
Projection
:
which
done
,
the
Label
laid
to
the
place
of
the
Sun
shall
shew
the
hour
of
the
Night
in
the
Limb.
Example
.
December
25.
I
observed
Procyon
to
be
full
East
,
his
Declination
North
,
is
6
degrees
3
minutes
.
In
the
Meridional
Projection
I
looked
where
the
Axis
of
the
Reet
(
which
is
the
East
Azimuth
)
cut
the
sixth
of
the
North
Parallels
,
and
I
found
the
intersection
1
degree
South-ward
from
the
Axtree
of
the
Mater
(
or
hour-line
of
six
)
there
also
cutteth
the
5
½
Almicantar
,
which
shewes
more
then
I
sought
,
that
Procyon
was
5
degrees
30
minutes
above
the
Horizon
.
Now
having
the
hour
of
the
Star
,
6
hours
,
4
minutes
,
before
the
Meridian
,
I
take
the
Equinoctial
Projection
,
and
having
laid
the
Label
one
degree
from
Oriens
South-ward
in
the
Limb
,
I
turn
Procyon
to
the
Label
,
which
sheweth
his
Hour
6
hours
4
minutes
,
and
leaving
him
there
,
I
turn
away
the
Label
to
♑
14
the
Suns
place
,
and
it
shewes
me
in
the
Limb
the
time
of
night
6
hours
26
min.
past
noon
.
And
the
same
I
might
have
found
,
if
instead
of
his
Azimuth
,
I
had
observed
his
Altitude
5
degree
½
,
the
crossing
of
that
Almicantar
which
the
6th
Parallel
would
have
given
me
the
same
hour
of
the
Star
,
and
further
,
his
Azimuth
,
undesired
.
CHAP.
XXXIII
.
Your
Latitude
known
,
and
the
Altitude
,
and
Azimuth
,
of
any
Star
,
Planet
,
or
Comet
,
observed
,
and
the
time
of
Night
:
how
to
find
his
Right
Ascension
,
and
Declination
.
THis
Case
differeth
little
from
the
Case
of
Chapter
26.
where
,
from
the
same
things
given
,
the
Declination
and
hour
was
required
.
For
the
hour
and
Right
Ascension
are
in
a
sort
the
same
thing
,
only
the
account
of
the
Right
Ascension
beginneth
alwayes
at
♈
0.
and
is
made
in
degrees
and
minutes
of
a
degree
.
The
account
of
the
Hour
beginneth
at
the
Meridian
,
and
is
made
in
hours
and
minutes
of
an
hour
.
Fifteen
whole
degrees
make
an
hour
,
and
consequently
15
minutes
of
a
Degree
make
one
minute
of
Time
,
for
in
every
minute
of
Time
,
there
passeth
the
Meridian
a
quarter
of
a
degree
of
the
Equinoctial
.
The
time
of
Night
is
here
further
required
to
be
given
;
which
may
be
had
by
Chap
29.
or
32.
The
rule
.
When
you
observe
the
Altitude
and
Azimuth
of
the
Star
,
observe
also
the
time
of
Night
,
by
Chapter
29.
or
32.
and
to
save
you
labour
herein
,
you
shall
do
best
to
observe
your
Altitude
and
Azimuth
,
when
some
known
Star
is
seen
just
in
the
Meridian
.
Then
with
your
Latitude
,
and
the
Altitude
,
and
Azimuth
of
the
Star
,
get
by
Chapter
26.
the
Declination
,
and
hour
of
the
Star
:
Then
in
the
Equinoctial
Projection
,
lay
the
degree
of
the
Sun
to
the
hour
of
the
Night
:
thence
turn
the
Label
to
the
hour
of
the
Star
,
and
you
have
his
Right
Ascension
in
the
Limb
of
the
Reet
,
between
♈
0
and
the
Label
.
Example
.
Put
case
I
would
find
the
Declination
and
Right
Ascension
of
Lucida
Pleiadum
.
The
Sun
being
in
Sagittarius
20.
December
2.
I
observed
that
when
Australis
caudae
Caeti
is
full
South
,
Lucida
Pleiadum
was
near
South-east
viz.
in
the
Azimuth
67.
from
the
Meridian
,
and
the
Altitude
of
the
said
Star
45.
0.
hence
,
by
Chapter
26.
I
find
his
Declination
23
degrees
North
,
and
the
hour-distance
from
the
Meridian
45
degr.
that
is
3
hours
before
noon
.
Then
in
the
Equinoctial
Projection
(
according
to
Chapter
29.
)
I
set
Australis
caud
.
Cati
in
the
Meridian
line
of
the
Mater
.
and
turning
the
Label
to
9.
of
the
clock
(
which
is
the
hour
of
the
Star
)
I
find
in
the
Limb
of
the
Reet
(
numbring
from
♈
0
,
to
the
Label
)
52
degrees
,
the
Right
Ascension
of
Lucida
Pleiadum
:
and
where
the
23
degrees
of
the
Label
now
touches
the
Reet
,
there
may
I
prick
the
Star
in
my
Reet
,
if
I
have
him
not
before
;
the
time
of
night
is
easily
seen
,
by
turning
the
Label
to
the
Suns
place
,
it
shewes
7.
hours
12
minutes
at
night
:
but
I
need
not
so
much
as
look
on
that
,
though
by
placing
Australis
caud
.
Caeti
in
the
South
,
I
have
the
time
implicitely
.
The
Proposition
therefore
,
might
have
been
thus
made
;
Your
Latitude
known
,
and
the
Altitude
and
Azimuth
of
an
unknown
Star
observed
,
just
at
the
time
when
any
known
Star
is
in
the
Meridian
;
to
find
both
the
Right
Ascension
,
and
Declination
of
the
Star
unknown
.
Note
also
,
that
if
you
observe
the
unknown
Star
in
the
Meridian
Azimuth
,
you
have
presently
his
Declination
,
by
Chapter
13.
and
the
Right
Ascension
of
Culmen
Caeli
,
is
the
Stars
Right
Ascension
.
CHAP.
XXXIIII
.
The
Declination
,
and
Right
Ascension
of
any
Star
given
,
to
find
his
Longitude
,
and
Latitude
.
LOok
the
Stars
place
in
the
Mater
,
(
which
is
the
intersection
of
the
Meridian
of
his
Right
Ascension
with
the
Parallel
of
his
Declination
)
and
make
a
prick
there
.
Then
your
Planisphear
being
set
in
the
second
Mode
of
the
Meridional
Projection
,
you
shall
presently
find
the
Longitude
and
Latitude
in
the
Reet
:
for
the
Azimuth
cutting
the
said
prick
,
shewes
his
Longitude
,
and
the
Almicantar
his
Latitude
.
Example
.
November
14.
1639.
I
observed
a
Star
of
the
third
Magnitude
in
the
Heart
of
Cetus
,
which
I
know
to
be
no
common
Star
,
because
I
had
never
noted
it
before
,
neither
could
I
find
it
in
the
Tables
of
Ptolemy
,
Tycho
,
or
any
other
:
the
Right
Ascension
thereof
was
30.
13
minutes
,
the
Declination
4.
50.
minutes
South
,
as
I
observed
by
2
way
which
hereafter
shall
be
shewed
Chapter
44.
I
made
therefore
a
prick
with
ink
in
the
Mater
of
my
Brass
Planisphear
,
where
the
30th
.
Meridian
(
numbred
from
the
Center
toward
my
right
hand
)
and
the
5th
.
Parallel
of
South
Declination
do
cross
;
regarding
also
the
odd
minutes
.
Then
as
soon
as
my
ink
was
drie
,
I
set
the
Finitor
in
the
Ecliptique
line
of
the
Mater
with
the
Zenith
South-wards
;
because
the
Latitude
of
the
Star
was
South
,
and
I
saw
the
26.
Azimuth
from
the
Axtree
line
cutting
the
prick
,
and
likewise
the
16th
.
Almicantar
cutting
about
10
minutes
below
the
prick
toward
the
Finiter
.
Therefore
because
in
this
Mode
the
Azimuths
be
Circles
of
Longitude
,
and
the
Almicantars
Parallels
of
Latitude
,
(
by
Book
2.1
.
)
I
conclude
the
Longitude
of
Cor
Caeti
.
was
♈
26.
and
his
Latitude
16.
10
minutes
South
.
When
first
I
observed
this
strange
Star
in
the
said
year
1639.
and
could
find
no
mention
of
it
in
the
Tables
of
Ptolemy
,
Copernicus
,
Stadius
,
Tycho
,
or
Maginus
,
I
did
thereof
advertise
my
very
good
friends
Dr.
John
Twysden
,
then
in
Kent
,
and
Mr
Samuel
Foster
,
Professor
of
Astronomy
in
Gresham
Colledge
,
then
at
London
;
who
thereupon
made
the
same
observation
of
the
Star
that
I
had
done
,
for
the
place
of
it
;
and
we
all
agreed
that
it
increased
in
light
,
and
was
above
the
third
Magnitude
in
December
1639.
and
that
it
had
no
perceivable
Parallax
.
And
as
I
was
thinking
to
publish
some
brief
advertisment
thereof
,
in
the
Latin
tongue
,
that
Astronomers
beyond
the
Seas
as
well
as
here
,
might
attend
the
observation
thereof
,
Mr
Foster
wrote
me
word
that
he
had
found
the
Star
pictured
in
Bayerus
his
Images
,
which
were
printed
Anno
Domi.
1616.
And
in
1640.
there
came
to
me
through
Dr
Twysdens
hands
a
Treatise
of
that
Star
,
then
newly
Printed
,
by
one
Phacylides
,
Professor
of
Logique
at
Franequers
,
whose
observations
agreed
with
ours
.
But
he
thought
this
Star
to
have
been
made
of
the
great
Eclipse
of
the
Moon
which
hapned
December
10.
1638.
in
the
foremost
foot
of
♊
.
wherein
we
were
not
of
his
mind
,
you
may
read
this
conceipt
in
his
Book
pag.
197.
This
Star
doth
often
appear
,
and
again
disappear
;
it
is
sometime
of
the
3d.
Magnitude
,
sometime
of
the
4th
.
I
have
seen
it
oft
in
the
Eastern
Hemisphear
,
seldome
in
the
Western
.
It
is
lost
sometimes
divers
weekes
together
:
this
year
I
could
never
see
it
,
till
February
2.
1656
/
7.
Such
as
have
leasure
for
the
Study
of
these
Arts
,
may
do
well
to
observe
it
,
and
to
search
the
reason
of
its
changes
:
for
which
purpose
I
thought
it
fit
to
give
this
notice
.
CHAP.
XXXV
.
The
Longitude
,
and
Latitude
,
of
any
Star
given
,
to
find
his
Right
Ascension
,
and
Declination
;
and
to
place
the
Stars
in
the
Mater
.
THis
Probleme
is
the
converse
of
the
precedent
.
The
Stars
are
registred
by
their
Longitudes
and
Latitudes
,
because
their
Longitudes
increase
equally
,
and
their
Latitudes
remain
the
same
.
And
so
the
Tables
are
easily
rectified
to
any
Age
,
by
Addition
or
Subtraction
of
a
few
degrees
or
minutes
of
Longitude
onely
:
but
the
Right
Ascension
and
Declination
of
the
Stars
happen
to
increase
and
decrease
very
unequally
,
and
must
therefore
be
calculated
from
Age
to
Age
,
from
the
Longitude
and
Latitude
whose
Tables
are
more
certain
.
Set
the
Planisphear
in
the
second
Mode
of
the
Meridional
Projection
,
as
in
the
former
Chapter
,
and
bearing
in
mind
that
the
Azimuths
here
are
Circles
of
Longitude
and
the
Almicantars
Parallels
of
Latitude
,
look
where
the
Longitude
and
Latitude
of
the
Star
meet
,
and
there
make
a
prick
in
the
Reet
;
and
look
what
Meridian
and
Parallel
of
the
Mater
cut
,
under
that
prick
they
shew
the
Right
Ascension
,
and
Declination
of
the
Star
.
Example
.
Eniph
.
Alpharats
,
that
is
,
Os
Pegasi
,
had
by
Tycho'es
Tables
An.
Dom.
1600.
Longitude
♒
26.
22
minutes
,
Latitude
22.07
½
.
North.
The
Finiter
set
to
the
Ecliptique
line
of
the
Mater
,
and
the
Zenith
toward
the
North
Pole
(
because
the
Stars
Declination
is
North
)
I
count
the
Longitude
of
the
Star
upon
the
Finiter
,
(
here
Ecliptique
)
thus
.
At
the
Center
say
I
,
is
♈
0.
thence
proceeding
rightward
to
the
Limb
,
I
say
,
here
is
♋
0.
whose
Right
Ascension
is
90.
thence
returning
to
the
Center
,
I
say
,
here
is
♎
0.
upon
the
Axis
of
the
Reet
and
Right
Ascension
180.
upon
the
Axis
of
the
Mater
;
thence
I
proceed
in
the
Finiter
to
the
other
side
of
the
Limb
,
and
say
here
is
♑
0.
bounded
by
the
Limb
of
the
Reet
and
Right
Ascension
270.
bounded
by
the
Limb
of
the
Mater
,
which
Limbs
here
fall
into
one
Circle
;
and
are
Colurus
Solstitiorum
:
these
numbers
I
keep
,
and
returning
back
in
the
Finiter
toward
the
Center
,
when
I
am
gone
30
degrees
,
I
say
,
here
begins
♒
.
and
going
on
26.
22.
minutes
further
,
I
say
,
thus
far
is
the
Star
gone
in
Longitude
.
Now
here
cuts
the
Finiter
(
by
this
account
)
the
Azimuth
35
⅓
.
from
the
Limb
;
in
this
Azimuth
I
number
the
Stars
Latitude
,
by
the
Almicantars
22.07
½
.
and
at
the
end
of
that
number
in
the
said
Azimuth
I
prick
the
Stars
place
.
And
here
I
see
the
8th
.
Parallel
of
North
Declination
upon
the
Mater
cutteth
him
,
and
the
Meridian
51
⅓
.
from
the
Limb
shewing
the
excess
of
his
Right
Ascension
above
270.
which
I
kept
before
.
Therefore
I
conclude
the
Right
Ascension
of
Eniph
.
Alpharats
,
Anno
Dom.
1600
,
was
321.
20
minutes
;
and
his
Declination
8.
deg.
North.
Past
this
on
fol.
102
so
as
it
may
ly
open
while
that
Chapter
is
Reading
.
diagram
Another
way
to
place
the
Stars
in
the
Mater
by
their
Declination
and
Horary-distance
from
the
Meridian
.
See
hereafter
Chapter
52.
CHAP.
XXXVI
.
The
Latitude
,
and
Declination
of
a
Star
given
,
to
find
his
Longitude
,
and
Right
Ascension
.
SEt
your
Planisphear
in
the
second
Mode
of
the
Meridional
Projection
,
turning
the
Zenith
Northward
or
Southward
as
the
Stars
Latitude
hapneth
to
be
North
or
South
.
Then
look
where
the
Parallel
of
the
Stars
Latitude
in
the
Reet
cutteth
the
Parallel
of
the
Stars
Declination
on
the
Mater
,
the
Azimuth
cutting
that
intersection
sheweth
the
Longitude
of
the
Star
;
and
the
Meridian
there
cutting
sheweth
his
Right
●scension
.
Example
.
The
Declination
of
Spica
♍
,
Anno
Dom.
1670.
will
be
9
½
.
South
,
the
Latitude
was
always
1.
59
minutes
South
.
Now
where
the
second
Almicantar
cutteth
the
9
½
.
Parallel
of
South
Declination
,
there
passeth
the
19th
¼
.
Azimuth
from
the
Axis
toward
my
left
hand
shewing
Spica's
Longitude
♎
19
¼
,
and
the
17th
.
Meridian
from
the
Axis
,
to
which
I
add
a
Semi-circle
(
because
♎
0.
is
at
the
Center
)
and
I
make
197
degrees
the
Right
Ascension
of
Spica
for
1670.
CHAP.
XXXVII
.
The
Longitude
,
and
Latitude
of
two
Stars
given
,
to
find
their
Distance
.
MAke
one
of
the
Poles
of
the
Mater
to
be
Pole
of
the
Ecliptique
,
for
this
turn
,
and
set
the
Star
which
hath
most
Latitude
at
his
distance
in
the
Limb
,
and
turn
the
Zenith
to
him
;
count
thence
by
the
Meridians
the
difference
of
Longitude
,
till
you
come
to
the
other
side
of
your
Triangle
;
and
in
that
side
number
either
the
Latitude
from
the
Equator
,
or
his
complement
from
the
Pole
;
at
the
end
of
this
number
is
the
other
Star
:
and
the
Azimuth
passing
from
him
to
the
Zenith
,
shewes
the
distance
.
This
is
done
by
the
second
Probleme
of
Obliquangled
Triangles
.
Book
3.15
.
Example
.
In
Tycho'es
Tables
for
1600.
Aldebarans
Longitude
is
♊
4.12
½
.
Latitude
5.
31.
min.
A.
Sirius
Longitude
♋
8.
35
½
.
Latitude
39.
30
½
.
A.
Difference
of
Longitude
34.
23.
I
number
therefore
39.
30
minutes
½
,
the
Latitude
of
Sirius
from
the
Equator
in
the
Limb
,
or
the
Complement
thereof
from
the
Pole
,
(
all
is
one
,
)
there
I
set
the
Zenith
to
stand
for
Sirius
;
then
because
Aldebaran
is
distant
from
Sirius
in
Longitude
34.
23.
minutes
,
I
take
the
34
½
,
Meridian
from
the
Zenith
,
and
where
the
5
½
Parallel
cutteth
him
,
there
say
I
,
is
Aldebaran
(
and
C
of
my
Triangle
)
and
the
Azimuth
passing
thence
to
the
Zenith
measureth
the
distance
of
the
Stars
46
degrees
almost
.
CHAP.
XXXVIII
.
The
Declination
,
and
Right
Ascension
of
any
two
Stars
given
,
to
find
their
distance
.
DO
here
with
the
Right
Ascension
and
Declination
as
you
should
do
with
the
Longitude
and
Latitude
,
by
the
former
Chapter
,
for
the
case
is
like
,
and
requireth
the
same
manner
of
working
.
CHAP.
XXXIX
.
The
Declination
of
a
Star
or
Planet
,
and
his
distance
from
a
known
Star
given
,
to
find
his
Right
Ascension
.
BEcause
this
Case
is
the
converse
of
the
precedent
,
and
soluble
by
the
first
Probleme
of
Obliquangled
Triangles
,
Book
3.
14.
an
Example
,
or
two
shall
suffice
.
Past
this
on
fol.
105
so
as
it
may
ly
open
while
that
Chapter
is
Reading
.
diagram
A
B
the
distance
of
Mandibulae
from
the
Pole
87.
20
minutes
,
I
set
between
the
Pole
and
Nadir
in
the
Limb
,
because
B
C
will
reach
beyond
the
Finitor
.
For
A
C
the
distance
of
the
Stars
.
I
seek
the
13th
.
Parallel
from
the
Pole
.
And
For
B
C
I
seek
the
94.
50
minutes
Almicantar
,
counted
from
the
Nadir
(
that
is
the
5th
.
almost
above
the
Finitor
)
and
where
the
said
Parallel
and
Almicantar
cross
,
there
is
Cor
Caeti
,
and
C
of
my
Triangle
:
through
it
there
cutteth
the
Azimuth
10
⅔
,
shewing
the
Difference
of
the
Right
Ascension
of
the
Stars
;
which
difference
I
subtract
out
of
the
Right
Ascension
of
Mandibula
,
because
he
was
further
East
;
and
there
remaineth
the
Right
Ascension
of
Cor
Caeti
30.
16
minutes
,
or
rather
13
minutes
.
And
I
have
here
also
numbred
by
the
Meridians
,
the
angle
A
at
Mandibula
120
degr
.
though
un-required
.
Another
Example
.
January
7.
1656
/
7
,
I
observed
by
my
Brass
Quadrant
of
12
inches
in
Radius
,
the
Meridian
Altitude
of
Jupiter
56.
20
minutes
,
out
of
which
subtracting
the
height
of
the
Equator
here
at
Ecton
37.
45
minutes
;
I
found
his
Declination
18.
35
minutes
North
;
his
distance
then
from
Lucida
Pleiadum
,
I
observed
by
my
Cross-staff
5.
12
minutes
,
and
from
Aldebaran
10.07
minutes
.
The
Complement
of
Decli
.
of
Lucida
Pleiadum
is
67.00
mi.
The
Complement
of
♃
his
Declination
was
observed
71.
25.
And
these
two
Complements
with
the
distance
of
♃
and
Lucida
Pleiadum
5.
12
minutes
,
make
a
Triangle
,
soluble
by
the
first
Probleme
of
Obliquangled
Triangles
;
whereby
you
may
find
the
angle
of
the
difference
of
Right
Ascension
of
Lucida
Pleiadum
and
♃
is
2.
56
minutes
;
which
added
to
the
Right
Ascension
of
Lucida
Pleiadum
(
because
♃
was
East-ward
)
maketh
54.
44
minutes
the
Right
Ascension
of
Jupiter
.
CHAP.
XL.
The
Latitude
of
a
Star
or
Planet
,
and
his
distance
from
a
known
Star
given
,
to
find
his
Longitude
.
DO
here
with
the
Longitude
and
Latitude
as
you
were
taught
to
do
with
the
Right
Ascension
and
Declination
,
in
the
former
Chapter
.
CHAP.
XLI
.
To
find
the
distance
of
two
Stars
by
their
Altitudes
,
and
their
difference
of
Azimuth
observed
at
the
same
time
.
THe
Complements
of
the
Altitudes
are
the
distances
of
the
Stars
from
the
Zenith
:
Set
one
of
the
Stars
at
the
Pole
,
and
set
the
Zenith
as
much
from
him
in
the
Limb
as
the
Complement
of
his
Altitude
comes
to
,
then
considering
what
difference
of
Azimuth
the
Stars
had
,
take
the
Azimuth
of
like
distance
from
the
Limb
(
beginning
from
that
side
of
the
Limb
where
the
Pole
aforesaid
is
)
and
in
that
Azimuth
reckon
from
the
Finitor
the
Altitude
of
the
other
Star
(
or
the
Complement
of
his
Altitude
from
the
Zenith
,
all
is
one
)
at
the
end
thereof
is
C
,
and
the
other
Star
;
and
the
Meridian
that
passeth
from
him
to
the
Pole
,
shewes
the
distance
of
the
Stars
.
This
case
is
so
like
that
of
Chapter
37.
that
he
who
knowes
one
may
know
the
other
also
.
CHAP.
XLII
.
To
find
the
Angles
of
Station
which
any
two
Stars
make
with
the
Pole
,
by
their
Right
Ascension
and
Declination
:
or
with
the
Pole
of
the
Ecliptique
,
by
their
Longitude
and
Latitude
:
or
with
the
Zenith
,
by
their
Altitude
and
Azimuth
.
THis
Case
agrees
with
the
second
Probleme
of
Oblique-angled
Triangles
.
Example
.
In
the
Triangle
of
Chapter
39.
made
between
the
Pole
of
the
World
,
Mandibula
Caeti
,
and
Cor
Caeti
,
I
would
know
the
angle
at
Mandibula
,
which
is
the
angle
of
his
Station
.
Place
the
Triangle
upon
your
Planisphear
as
in
Chapter
39.
where
the
angle
unsought
,
there
discovered
it self
to
be
125.
degrees
.
CHAP.
XLIII
.
To
find
whether
three
Stars
be
in
one
great
Circle
,
by
having
their
Longitude
and
Latitude
,
or
their
Right
Ascension
and
Declination
,
or
their
Azimuth
and
Altitude
known
.
EXample
.
I
would
know
whether
the
three
Stars
of
Orions
Girdle
be
in
the
same
great
Circle
.
Here
I
prick
them
down
,
and
draw
their
Circles
of
Longitude
to
meet
at
the
Pole
of
the
Ecliptique
;
so
have
you
two
Triangles
joyned
in
one
,
and
the
three
Stars
in
the
Base
of
it
.
Now
first
,
I
must
find
by
the
former
Chapter
what
angle
of
station
the
first
Star
hath
in
the
little
Triangle
P
A
B
,
and
then
what
angle
of
station
he
hath
in
the
whole
Triangle
P
A
C
,
and
if
these
two
angles
be
equal
,
then
be
the
Stars
all
in
one
great
Circle
,
otherwise
not
.
diagram
This
Probleme
may
be
of
use
to
find
how
the
tayle
of
a
Comet
pointeth
upon
the
Sun
,
or
upon
any
other
Planet
or
Star
,
below
the
Horizon
.
But
if
the
three
points
enquired
of
,
be
all
in
view
;
I
know
no
better
way
then
to
stretch
a
thrid
straight
at
a
reasonable
distance
from
your
eye
,
applying
it
to
the
Stars
;
for
if
the
same
straight
line
cut
them
all
,
they
be
all
in
one
great
Circle
,
otherwise
not
.
CHAP.
XLIV
.
If
a
Comet
or
Star
unknown
be
seen
in
a
straight
line
with
two
other
known
Stars
,
and
his
distance
from
one
of
the
known
Stars
be
observed
;
how
to
find
the
true
place
of
the
Comet
or
Star
unknown
.
diagram
Therefore
I
set
the
Nadir
of
my
Reet
as
far
from
the
Pole
as
P
is
from
A
,
and
so
between
them
on
the
Limb
is
the
side
P
A.
Then
for
the
side
P
B
it
reacheth
from
Nadir
beyond
the
Finitor
1.
16
minutes
,
therefore
in
the
1
¼
.
Almicantar
I
number
from
the
Limb
5.
35
minutes
,
the
difference
of
Right
Ascensions
for
the
Angle
A
P
B
,
and
where
the
1
¼
.
Almicantar
and
the
5
½
.
Azimuth
do
meet
,
there
is
B
for
Gena
:
thence
I
go
in
a
Meridian
to
the
Pole
at
A
,
and
as
I
go
I
number
the
distance
of
B
and
A
,
that
is
,
Gena
and
Mandibula
,
7
degrees
almost
;
and
I
observe
that
this
Meridian
is
the
125.
Meridian
from
the
Limb
;
so
much
is
A
,
the
angle
of
station
at
Mandibula
.
Now
I
say
,
in
this
Meridian
also
is
Cor
Ceti
,
because
he
is
in
a
right
line
with
the
other
two
Stars
which
are
cut
by
this
Meridian
:
and
he
is
13.
4
minutes
from
Mandibula
,
by
observation
;
therefore
I
run
from
the
Pole
A
so
many
degrees
in
this
Meridian
,
and
so
come
to
C
,
the
place
of
Cor
Ceti
,
and
there
cutteth
the
4.
50
minutes
Almicantar
,
shewing
the
Declination
of
it
,
and
the
Azimuth
10.
43
minutes
;
shewing
his
difference
of
Ascension
from
Mandibula
;
which
difference
I
subduct
from
the
Right
Ascension
of
Mandibula
(
because
Mandibula
is
further
East
,
)
and
there
remains
30.
13
minutes
,
the
Right
Ascension
of
Cor
Ceti
:
which
being
found
,
you
may
find
his
Longitude
26
degrees
,
Latitude
16.
10
minutes
,
by
the
34th
.
Chapter
:
Observe
how
your
Triangle
lies
in
the
Planisphear
,
where
Nadir
is
used
for
the
North
Pole
,
the
North
Pole
is
the
place
of
Mandibula
,
and
the
125.
Meridian
represents
the
great
Circle
cutting
the
three
Stars
.
CHAP.
XLV
.
The
distance
of
a
Planet
from
two
known
Stars
being
Observed
,
to
find
his
Longitude
and
Latitude
.
IT
is
true
that
Mr.
Blagrave
saith
,
Book
5.
25.
that
in
Questions
of
this
sort
it
is
harder
to
conceive
how
they
should
be
resolved
,
then
to
resolve
them
.
And
therefore
he
adviseth
to
draw
a
rude
Scheme
of
your
work
,
agreeable
to
the
Meridional
Projection
of
your
Planisphear
,
after
this
manner
.
December
the
28.
1656.
I
observed
somewhat
grosly
by
my
Cross-staff
that
♃
was
between
the
Hyades
and
the
Pleiades
,
distant
from
Aldebaran
9.
49
minutes
,
and
from
Lucida
Pleiadum
5.26
.
and
to
the
Southward
of
the
Stars
.
I
draw
therefore
a
rude
Scheam
representing
somewhat
near
the
posture
of
these
three
Stars
.
E
C
is
Ecliptique
,
and
P
his
South
Pole
,
P
C
the
Circle
of
Longitude
of
the
Westerly
Star
Lucida
Pleiadum
♉
25.
12
minutes
,
and
because
he
hath
North
Latitude
4
degrees
,
I
place
him
at
F
;
Aldebaran
,
whose
Longitude
is
♊
5.
Latitude
South
5.
31
minutes
,
I
place
somewhat
like
at
O
,
and
Jupiter
I
place
below
the
line
drawn
between
them
,
and
nearer
to
the
Pleiades
then
to
Aldebaran
,
as
I
observed
his
situation
in
the
Heaven
.
diagram
That
I
seek
now
here
,
is
P
♃
,
the
complement
of
Jupiters
Latitude
;
and
F
P
♃
,
his
difference
of
Longitude
from
Lucida
Pleiadum
.
First
,
in
the
great
Triangle
F
P
O
,
I
have
the
angle
P
,
the
difference
of
Longitude
between
Lucida
Pleiadum
and
Aldebaran
9
degrees
48
minutes
,
and
the
including
sides
P
O
84.29
.
(
Aldebaran
distance
from
the
South
Pole
)
and
P
F
94.
(
distance
of
the
Pleiades
from
the
South
Pole
)
and
hence
by
the
second
Problemes
of
Obliquangled
Triangles
Book
3.15
.
I
get
at
once
,
the
Base
O
F
,
distance
of
Lucida
Pleiadum
and
Aldebar
an
13.
45
minutes
,
and
the
angle
of
station
at
F
,
viz.
P
F
O
45.
28
minutes
.
2.
Then
in
the
Triangle
F
O
♃
,
whose
three
sides
are
now
known
,
I
get
the
angle
O
F
♃
(
by
the
first
Probleme
of
Oblique
Triangles
,
Book
3.14
.
)
35.
14
minutes
)
which
being
Subducted
from
the
angle
O
F
P
,
leaveth
the
angle
P
F
♃
10.
14
minutes
.
3.
In
the
Triangle
P
F
♃
,
having
now
the
angle
F
,
and
the
sides
including
it
,
I
get
the
third
side
P
♃
,
the
Complement
of
♃
Latitude
88.
39
minutes
(
by
Oblique
Problemes
2.
Book
3.15
.
)
And
lastly
the
three
sides
in
the
Triangle
P
F
♃
being
now
known
my
Planisphear
unmoved
will
shew
me
F
P
♃
58
minutes
(
by
the
Probleme
1
Oblique
Triangles
,
)
which
58
minutes
being
added
to
the
Longitude
of
Lucida
Pleiadum
maketh
up
the
Longitude
of
♃
♉
26.
10
minutes
,
and
his
Latitude
was
even
now
found
1.
21
minutes
South
.
CHAP.
XLVI
.
To
find
the
Culmen
Caeli
,
and
the
Altitude
thereof
,
at
any
time
proposed
.
CUlmen
Caeli
is
the
degree
of
the
Ecliptique
which
is
cut
by
the
Meridian
of
your
Place
.
Use
the
Equinoctial
Projection
,
where
having
laid
the
place
of
the
Sun
to
the
Hour
proposed
,
look
what
degree
of
the
Ecliptique
is
cut
by
the
Meridian
line
,
and
you
may
number
his
Altitude
from
your
proper
Horizon
.
Example
.
March
29.
1652.
I
laid
the
Suns
place
♈
19.
11
minutes
to
32
minutes
past
10.
of
the
Clock
before
noon
,
and
in
the
Meridian
I
saw
♓
25
½
Culminating
.
And
for
his
Altitude
I
looked
where
my
Horizon
cuts
the
South
part
of
the
Meridian
(
at
52
¼
from
the
Center
)
and
from
that
cutting
I
count
in
the
Meridian
to
the
Ecliptique
36
degrees
,
the
Altitude
of
Culmen
Caeli
.
But
note
,
That
if
♓
had
been
a
North
Signe
,
I
must
have
counted
first
to
the
Limb
37.
45
minutes
,
and
thence
back
again
to
Culmen
1
¾
.
in
39
½
.
CHAP.
XLVII
.
To
find
the
Ascendent
or
Horoscope
,
and
the
other
three
Principal
Houses
,
for
any
time
proposed
.
A
Strologers
divide
the
Heaven
into
twelve
Houses
,
of
which
,
four
are
principal
.
The
First
House
,
which
beginneth
at
the
Ascendent
or
Rising
point
of
the
Ecliptique
.
The
Fourth
,
which
beginneth
at
Imum
Caeli
,
or
Midnight
.
The
Seventh
,
which
beginneth
at
the
Descendent
point
of
the
Ecliptique
.
And
the
Tenth
,
which
beginneth
at
Medium
Caeli
,
or
Culmen
.
These
be
the
four
Cardinal
points
,
and
the
Ascendent
and
Descendent
,
and
likewise
the
Medium
and
Imum
Caeli
,
are
alwayes
opposite
one
to
the
other
,
so
that
one
being
known
,
the
other
is
known
also
.
To
find
these
points
,
use
the
Equinoctial
Projection
,
and
there
lay
the
Suns
place
to
the
hour
proposed
:
then
the
degree
of
the
Ecliptique
rising
in
your
Horizon
is
Ascendent
,
and
you
shall
see
the
same
degree
of
the
opposite
Signe
Descending
in
the
West
part
of
the
Horizon
;
and
look
what
degree
toucheth
the
South
part
of
the
Meridian
,
that
is
,
Medium
Caeli
,
and
the
same
degree
of
the
opposite
Signe
shall
be
in
Imo
Caeli
,
that
is
,
in
the
North
and
Subterranean
part
of
the
Meridian
.
Example
.
March
29.
1652.
I
observed
the
great
Eclipse
of
the
Sun
,
the
middle
whereof
hapned
at
Ecton
,
at
10
hours
32
minutes
04
seconds
before
noon
in
apparent
time
,
at
what
time
the
Sun
was
darkned
digits
11.
22
½
,
in
♈
19.
11
minutes
.
I
would
know
for
this
time
the
Figure
of
the
Heavens
.
Therefore
laying
the
Label
to
10.32
minutes
before
noon
,
and
bringing
♈
19.
11
minutes
to
the
Label
,
I
see
in
our
Horizon
♋
24.
7
minutes
rising
,
and
♑
24.7
minutes
setting
.
In
the
Meridian
above
the
Horizon
I
see
♓
25.
19
minutes
:
and
in
Imo
Caeli
,
toward
Septentrio
,
♍
25.
19
minutes
.
CHAP.
XLVIII
.
To
find
the
beginnings
of
the
other
eight
Houses
.
THere
be
six
great
Circles
,
by
which
the
twelve
Houses
are
distinguished
.
They
be
called
Circles
of
Position
:
and
so
they
call
the
rest
of
the
Circles
which
serve
to
subdivide
the
Houses
.
They
be
all
Horizons
to
some
Country
or
other
in
the
World
,
and
therefore
are
most
fitly
represented
by
the
Horizons
of
the
Mater
.
The
First
House
beginneth
alwayes
at
the
Ascendent
:
and
the
rest
follow
in
order
according
to
the
Sequel
of
the
Signes
.
But
Astrologers
are
not
well
agreed
about
their
situation
.
For
1.
Some
will
have
the
domifying
Circles
drawn
from
the
Poles
of
the
Ecliptique
through
every
30th
.
degree
thereof
,
as
Ptolemie
.
2.
Some
draw
them
from
the
Poles
of
the
World
,
through
every
30th
.
degree
of
the
Equator
;
as
Alcabitius
.
3.
Some
draw
them
from
the
intersections
of
the
Meridian
and
Horizon
,
through
every
30th
.
degree
of
the
Equator
;
as
Regiomontanus
.
4.
Some
draw
them
from
the
same
intersections
,
by
every
30th
.
degree
of
the
Prime
Vertical
or
East
Azimuth
;
as
Campanus
.
Yet
every
Astrologer
will
pretend
he
can
tell
you
your
Fortune
,
though
they
go
about
it
so
divers
wayes
,
that
they
may
be
all
false
,
and
but
one
of
them
can
be
true
:
and
no
Man
hath
shewed
any
better
reason
for
his
way
then
another
,
but
his
own
opinion
.
If
you
will
follow
the
first
or
second
way
,
the
matter
is
plain
.
For
in
the
first
way
every
30th
Circle
of
Longitude
reckoned
from
the
Ascendent
downwards
,
and
so
round
,
is
a
Domifying
Circle
:
and
likewise
every
30th
.
Meridian
from
the
Ascendent
is
a
Domifying
Circle
in
the
Second
way
.
And
if
you
know
but
what
Longitude
a
Star
hath
,
you
presently
find
in
what
House
he
is
,
after
the
first
way
.
And
if
you
know
the
Right
Ascension
of
a
Star
,
and
of
the
Ascendent
,
you
presently
find
in
what
House
he
is
,
after
the
second
way
.
But
if
you
will
use
the
third
way
(
now
commonly
used
)
you
shall
set
the
Zonith
line
of
the
Reet
to
the
Latitude
,
and
so
the
Azimuths
are
your
Circles
of
Position
;
then
look
what
Azimuth
cutteth
every
30th
degree
of
the
Equinoctial
,
that
is
a
Domifying
Circle
;
and
you
shall
reckon
here
from
the
Limb
,
which
shall
stand
for
the
beginning
of
the
10th
.
House
,
and
so
in
our
Horizon
52
¼
.
the
fourty
third
Azimuth
cutteth
the
30th
.
degree
of
the
Equator
,
serving
the
11th
.
and
third
House
:
and
the
Azimuth
70
½
cutteth
the
60th
.
degree
of
the
Equator
,
serving
the
12th
and
second
Houses
:
and
because
I
know
that
on
the
other
side
the
Center
the
Intersections
will
be
like
,
I
look
no
further
.
But
now
I
must
get
the
Depressions
of
these
Circles
under
the
Pole
in
this
manner
.
I
number
in
the
fourty
third
Azimuth
the
Latitude
of
my
Place
from
the
Zenith
;
to
the
end
of
which
number
I
lay
the
Label
,
and
I
see
the
Azimuth
cutting
on
the
Label
32
½
for
the
depression
of
that
Circle
.
And
in
like
manner
,
laying
the
Label
upon
the
52
degrees
¼
of
the
Azimuth
70
½
I
find
on
the
Label
his
depression
48
degrees
;
by
the
third
Probleme
of
Rectangl
.
Trangles
,
and
the
third
Variety
Book
3
.
5.1●
.
These
Horizons
therefore
I
choose
out
in
the
Mater
,
viz.
32
½
,
and
48.
for
these
with
the
Meridian
and
Horizon
of
my
Place
,
shall
serve
to
get
the
Houses
for
ever
,
in
my
Latitude
:
for
the
32
½
Horizon
shall
be
the
beginning
of
the
11th
.
and
third
Houses
;
and
the
48th
,
the
beginning
of
the
12th
and
Second
,
Thus
have
I
the
Circles
of
Position
of
the
11
,
12
,
2
,
and
3.
Houses
:
and
the
10th
.
and
first
,
are
had
by
the
former
Chapter
:
and
these
six
being
had
,
I
have
all
;
for
opposite
Hemisphears
are
alwayes
alike
,
and
one
description
serveth
both
.
CHAP.
XLIX
.
To
know
what
degree
of
the
Ecliptique
is
in
the
begining
of
every
House
.
DO
as
in
this
Example
.
By
Chap.
47.
I
had
the
degree
Culminating
in
the
middle
of
the
great
Eclipse
there
mentioned
,
♓
25
½
.
First
I
lay
♓
25
½
to
the
Axtree
line
at
6
in
the
morning
,
where
it
lieth
as
in
a
Right
Horizon
;
thence
I
move
it
30
degr.
South
ward
in
the
Limb
,
viz.
to
8.
of
the
clock
,
and
in
the
Horizon
of
the
11th
.
house
(
32
½
)
I
see
♉
7
degrees
setting
,
the
said
degrees
of
Culmination
:
30
degrees
further
,
viz.
to
10.
a
clock
,
I
see
in
the
Horizon
of
the
12th
.
House
(
48.
)
♊
24
½
.
And
setting
the
said
degree
of
Culmination
to
the
Noon-line
,
I
see
in
our
Horizon
(
52
¼
which
begins
the
first
House
)
♋
24
½
ascending
.
And
setting
the
said
degree
2
hours
further
on
,
I
see
in
the
Horizon
of
the
second
House
(
48
)
♌
13.
And
setting
the
said
degree
to
4
a
clock
,
I
see
in
the
Horizon
of
the
third
House
(
the
32
½
)
♍
1.
Thus
have
I
the
degrees
of
the
Ecliptique
in
the
beginning
of
6
Houses
,
and
the
6
Houses
opposxe
begin
with
the
same
degrees
of
the
opposite
Signes
.
CHAP.
L.
Another
way
to
find
what
degree
of
the
Ecliptique
is
in
the
beginning
of
every
House
,
and
thereby
to
set
a
Figure
more
easily
then
by
the
former
Chapter
.
The
figure
of
the
Heavens
.
March
29.
1652.
H.
10.
32.
a.m.
diagram
Your
Houses
being
thus
distinguished
on
the
Reet
,
get
the
degree
of
Culmination
,
and
the
Altitude
thereof
,
by
Chapter
46.
then
set
the
Zenith
under
the
North
Pole
,
so
much
as
the
Altitude
of
the
Culmination
comes
to
:
and
if
the
Ascendent
be
a
North
Signe
,
let
the
Pole
be
toward
your
left
Hand
;
and
contrary
if
it
be
a
South
Signe
:
so
shall
the
Axis
of
the
Reet
be
Horizon
,
and
the
Pole
Culmen
Caeli
.
Next
get
the
Ascendent
by
the
47th
.
and
his
Amplitude
by
the
16th
:
this
Amplitude
you
shall
number
in
the
Axletree
of
the
Reet
from
the
Center
alwayes
to
your
left
Hand
,
or
toward
Septentrio
;
and
mark
what
Meridian
there
cuts
the
Axletree
of
the
Reet
,
in
that
degree
of
Amplitude
;
that
Meridian
shall
be
your
Ecliptique
for
this
time
:
follow
him
up
to
the
Pole
,
and
you
trace
out
the
arch
of
the
Ecliptique
from
the
Ascendent
to
mid-heaven
:
and
if
you
go
down
in
his
match
to
the
like
degree
of
Amplitude
on
the
other
side
of
the
Center
,
there
is
the
Western
arch
of
the
Ecliptique
from
the
mid-heaven
to
the
Descendent
;
and
here
you
may
see
every
degree
of
the
Ecliptique
above
the
Horizon
,
and
in
what
House
it
is
,
without
any
more
coursing
after
them
.
Example
.
♓
25
½
was
Culminating
;
his
Altitude
36
degrees
;
the
Ascendent
had
♋
24.
whose
Amplitude
is
36
⅓
.
Setting
the
Zenith
therefore
36
degrees
to
the
right
Hand
under
the
Pole
,
I
number
in
the
Axtree-line
of
the
Reet
,
from
the
Center
to
my
left
Hand
the
Amplitude
of
the
Ascendent
36
⅓
.
there
cometh
the
23.
Meridian
from
the
Center
,
who
must
serve
for
the
Ecliptique
.
Now
because
it
is
troublesome
to
number
the
degrees
of
the
Signes
backward
,
I
will
begin
at
the
Descendent
36
⅓
from
the
Center
on
the
other
side
:
and
say
,
Here
is
♑
24
degrees
descending
,
(
because
♋
24.
was
ascending
,
)
hence
I
count
on
toward
the
Culmen
,
till
I
come
to
the
Azimuth
19
½
,
(
which
is
the
Domifyer
of
the
12th
.
and
8th
.
Houses
,
)
and
here
I
say
,
begins
the
8th
.
House
in
♒
13.
for
there
are
but
19
degrees
from
the
Descendent
hither
:
hence
I
count
to
the
47th
.
Azimuth
(
the
Domifyer
of
the
9th
.
and
11th
.
Houses
)
and
there
I
count
♓
1
degree
,
for
the
beginning
of
the
9th
.
House
:
hence
I
number
on
to
the
Pole
,
and
there
I
happen
on
♓
25
½
,
the
Culmen
and
beginning
of
the
10th
.
House
.
Thence
I
number
on
the
other
side
of
the
Maters
Axtree
,
in
the
twenty
third
Meridian
,
toward
the
Ascendent
,
and
I
find
the
47th
.
Azimuth
cuts
♉
7
degrees
for
the
beginning
of
the
11th
.
House
;
but
the
19
½
Azimuth
which
should
shew
me
the
12th
.
House
,
is
cut
off
by
the
Finitor
,
and
I
am
left
to
seek
him
else
where
:
And
to
find
him
I
need
but
turn
about
my
whole
Planisphear
(
the
Reet
unmoved
)
and
make
the
other
Pole
Culmen
for
this
turn
,
and
then
I
find
among
the
Azimuths
that
peece
of
my
Ecliptique
which
I
wanted
in
the
former
posture
;
and
I
may
reckon
on
him
between
the
Ascendent
and
the
Azimuth
19
½
,
29
½
.
and
thereby
see
that
♊
24
½
is
in
the
beginning
of
the
12th
.
House
:
so
have
I
6.
of
my
Houses
,
and
may
by
them
find
the
other
6
,
(
as
was
shewed
Chapter
48.
)
and
set
them
down
as
in
the
Figure
.
CHAP.
LI.
A
third
way
to
set
a
Figure
with
less
labour
.
LEt
the
Meridians
and
Azimuths
here
change
their
offices
in
which
they
served
in
the
former
Chapter
:
that
is
,
let
the
19
½
and
47th
.
Meridian
on
both
sides
the
Axis
of
the
Mater
be
Domifyers
;
and
let
the
23
Azimuth
be
Ecliptique
:
and
to
that
purpose
,
set
the
Zenith
above
the
Pole
,
according
to
the
Altitude
of
Culmen
36
degrees
,
and
make
the
Axis
of
the
Mater
Horizon
.
Then
beginning
as
you
did
before
at
the
Descendent
,
go
up
in
the
Ecliptique
till
you
come
to
the
Meridian
19
½
,
and
follow
the
Almicantar
that
there
cutteth
to
the
Limb
,
and
there
make
a
mark
for
the
8th
House
;
then
mark
where
the
same
Ecliptique
cuts
the
next
Domifyer
(
the
47th
.
Meridian
)
and
follow
the
Alm̄icantar
from
that
point
to
the
Limb
;
prick
there
the
9th
.
House
:
the
Zenith
is
the
10th
.
thence
go
toward
the
Ascendent
,
and
do
in
like
manner
;
making
pricks
for
the
11th
.
and
12th
.
Houses
:
also
in
the
Limb
of
the
Reet
at
the
end
of
that
Almicantar
which
cutteth
the
beginning
of
the
Houses
in
the
Ecliptique
.
Then
in
the
Zodiaque
of
the
Ring
,
look
the
degree
of
Culmination
,
and
set
the
Zenith
of
the
Reet
to
it
;
and
the
Label
laid
to
these
pricks
,
shall
shew
you
presently
in
the
Zodiaque
the
degrees
for
the
beginning
of
every
House
.
CHAP.
LII
.
How
to
place
any
Star
or
Planet
in
his
proper
House
.
IN
the
Equinoctial
Projection
,
get
the
Stars
Hour
distance
from
the
Meridian
,
thus
.
Lay
the
Suns
place
to
the
hour
proposed
:
then
turn
the
Label
to
the
Star
(
or
to
his
Right
Ascension
if
he
be
not
in
the
Reet
)
and
it
shall
shew
in
the
Limb
how
many
hours
and
minutes
the
Star
is
past
or
short
of
the
Meridian
:
get
also
the
Stars
Declination
North
or
South
,
by
the
Reet
,
or
by
the
35.
or
some
other
Chapter
;
and
where
the
Parallel
of
the
Stars
Declination
crosseth
the
hour
of
the
Star
in
the
Mater
,
there
is
his
place
for
this
turn
:
therefore
having
made
a
prick
with
ink
for
him
there
,
set
the
Zenith
line
to
your
Latitude
,
and
having
your
Domifying
Azimuths
marked
upon
the
Reet
(
as
Chapter
49.
30.
)
you
shall
presently
see
in
what
House
the
Star
is
.
Example
.
1652.
March
29
,
10.
hours
32
minutes
before
noon
,
I
would
know
in
what
House
the
Pleiades
are
.
The
hour
of
Lucida
Pleiadum
for
that
time
is
8.
14
minutes
after
midnight
:
the
Declination
is
23
degrees
North.
I
number
therefore
in
the
23.
Parallel
of
North
Declination
from
the
Atree
of
the
Mater
to
the
Meridian
33
½
,
there
is
the
place
of
Lucida
Pleiadum
,
where
I
prick
him
down
;
and
setting
the
Zenith
line
to
the
Latitude
,
I
find
the
39
Azimuth
or
Circle
of
Position
cuts
him
:
by
which
I
see
he
is
8
degrees
from
the
beginning
of
the
11th
.
House
,
for
that
begins
at
Azimuth
47
,
as
appeares
Chapters
49
,
50.
CHAP.
LIII
.
To
find
the
division
of
the
Houses
,
according
to
Campanus
.
CAmpanus
begins
the
Houses
at
every
30th
.
degree
of
the
East
Azimuth
,
accounting
from
the
Ascendent
in
the
Sequel
of
the
Signes
,
as
was
said
Chapter
48.
Therefore
if
you
will
use
his
way
,
set
the
Zenith
line
to
the
Latitude
,
and
the
Finitor
shall
become
the
East
Azimuth
;
and
every
30th
,
Azimuth
from
the
Limb
,
or
Axtree
line
,
is
a
Domifying
Circle
:
you
shall
therefore
in
stead
of
Azimuths
19
½
.
and
47.
(
which
are
Domifyers
after
Regiomontanus
for
our
Latitude
,
as
was
shewed
Chapters
49
,
50.
)
take
Azimuths
30.
and
60.
on
both
sides
the
Axtree
line
,
which
are
distinguished
to
your
hand
;
and
with
these
Domifyers
you
shall
work
in
all
respects
as
you
did
with
the
other
in
the
three
former
Chapters
.
CHAP.
LIIII
.
How
to
Direct
a
Figure
.
TO
Direct
is
to
turn
on
the
Sphear
,
till
some
Star
in
the
second
House
come
into
the
first
,
or
contrarily
:
and
so
observe
how
many
degrees
the
Equinoctial
is
moved
forward
or
backward
in
the
same
time
.
Place
the
Star
or
Planet
on
the
Reet
,
or
on
the
Label
,
(
as
Chapter
18.
is
taught
,
then
in
the
Equinoctial
Projection
(
as
Chapter
47.
)
set
the
Ascendent
at
your
Horizon
,
and
note
the
degree
Culminating
:
then
turn
on
the
Reet
forward
or
backward
till
the
Star
come
to
the
Horizon
:
then
lay
the
Label
on
the
degree
which
Culminated
before
;
and
mark
how
many
degrees
of
the
Limb
he
is
distant
from
noon
,
so
many
degrees
of
the
Equator
have
passed
the
Meridian
and
Horizon
:
which
Astrologers
take
to
signifie
so
many
years
before
the
effect
promised
by
the
said
Star
shall
happen
.
diagram
diagram
diagram
diagram
diagram
Appendix
,
Concerning
Judiciarie
Astrologers
.
SOme
learned
Artists
may
perhaps
think
that
these
8.
last
Chapters
pertaining
to
Astrologie
might
be
spared
;
and
I
think
so
too
:
but
that
I
foresee
they
may
be
of
use
to
such
as
would
examine
the
errors
and
fallacies
of
Astrologians
.
Astrology
of
old
was
no
more
then
Astronomy
,
an
Ingenuous
Science
,
leading
a
Man
to
the
knowledge
of
that
which
may
be
known
,
and
ought
to
be
Studied
,
the
greatuess
and
wisdome
of
God
manifested
in
his
works
.
But
Astrology
,
as
the
word
is
now
commonly
used
,
is
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
Act.
19.19
.
and
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
1.
Tim.
6.20
.
Teaching
Men
to
search
into
that
which
neither
can
be
known
,
nor
ought
to
be
Pryed
into
;
Contingencies
to
come
,
which
belong
to
God
onely
to
know
,
and
reveal
.
Esa.
41.23
.
Tell
us
things
to
come
,
(
saith
he
)
that
we
may
know
that
ye
are
Gods
.
And
Favorinus
an
Heathen
Philosopher
could
say
,
Tollitur
quod
maxime
inter
Deos
atque
homines
differt
,
si
homines
quaque
res
omnes
futuras
pranoscerent
.
This
kind
of
Astrology
God
derides
by
his
Prophets
,
and
forbids
to
his
People
,
Esa.
47.12
,
13.
He
saith
to
Babylon
,
where
this
Art
then
flourished
.
Stand
now
with
thine
Inchantments
,
and
with
the
Multitude
of
thy
Sorceries
,
wherein
thou
hast
laboured
from
thy
Youth
;
if
so
be
thou
shalt
be
able
to
profit
;
if
so
be
thou
mayest
prevail
.
Thou
art
wearied
in
the
Multitude
of
thy
Counsels
:
Let
now
the
Astrologers
,
the
Star-gazers
,
the
monethly
prognosticators
stand
up
,
and
save
thee
from
these
things
that
shall
come
upon
thee
.
Jerem.
10.2
.
Thus
saith
the
Lord
,
Learn
not
the
way
of
the
Heathen
,
and
be
not
dismayed
at
the
signes
of
Heaven
:
for
the
Heathen
are
dismayed
at
them
.
For
the
customes
of
the
people
are
vain
.
)
And
it
was
one
of
the
presages
of
their
Captivity
,
when
they
began
to
be
replenished
with
those
Eastern
Arts
,
and
to
be
South-sayers
,
like
the
Philistims
.
Esay
3.
I
have
seen
what
glosses
have
been
put
upon
some
of
these
Texts
:
and
how
the
judgements
of
Mr
Perkins
,
and
Mr
Gataker
,
are
slighted
by
some
late
pretenders
to
Astrolegy
;
whereat
I
wonder
not
:
having
seen
the
works
both
of
the
one
side
,
and
of
the
other
.
Judiciary
Astrology
hath
two
parts
,
The
Meteorological
,
and
the
Genethliacal
.
And
against
both
there
are
these
just
exceptions
,
1.
Whereas
about
the
time
of
Nabonassar
,
or
soon
after
,
the
Heaven
was
divided
upon
the
Poles
of
the
Ecliptique
into
12.
Spaces
,
called
Dodecatemoria
and
Signes
:
and
the
Astrologers
of
the
next
Ages
ascribed
to
these
spaces
certain
vertues
or
Powers
over
the
several
parts
of
Mans
Body
,
and
over
the
several
Countryes
and
Nations
of
the
World
:
especially
by
reason
of
the
qualities
they
supposed
to
be
in
the
fixed
Stars
;
which
then
occupied
those
spaces
:
their
Rules
are
still
observed
,
notwithstanding
that
those
Stars
are
removed
into
the
succeding
spaces
.
For
those
Stars
which
in
Nabonassars
time
were
in
the
Dodecatemorion
of
♈
,
are
all
removed
into
the
Dodecatemoria
of
♉
,
and
♊
;
and
those
of
♓
and
♒
are
come
in
their
roomes
:
for
the
fixed
Stars
are
found
to
pass
through
a
whole
Signe
,
that
is
,
a
12th
.
part
of
the
Compass
of
the
Heaven
,
in
2118.
Years
;
and
it
is
now
above
1900.
Years
since
Ptolemie
wrote
,
and
above
2400.
since
Nabonassar
began
his
Reign
.
2.
That
Astrologers
scarce
look
beyond
the
Zodiaque
for
their
Aspects
,
whereas
the
fixed
Stars
on
the
North
and
South
may
have
,
for
ought
they
know
,
as
operative
configurations
with
the
Planets
,
as
the
Planets
one
with
another
.
3.
That
their
own
Rules
are
so
manifold
,
and
the
Planets
and
fixed
Stars
and
their
Aspects
also
are
so
many
,
and
so
divers
in
qualities
and
degrees
,
that
it
is
impossible
to
judge
what
effect
the
mixture
of
those
influences
shall
produce
:
especially
considering
that
from
the
beginning
of
the
World
to
this
time
,
the
Heavenly
bodies
never
had
twice
the
same
posture
:
and
therefore
there
wants
experiment
to
build
the
Art
upon
.
For
the
Meteorological
part
,
I
was
much
helped
by
the
Studies
of
my
Father
,
who
for
10.
Years
did
most
curiously
observe
all
changes
of
Wind
and
Weather
,
while
〈◊〉
was
waking
;
and
noted
them
daily
over
against
the
dayes
in
Maginus
his
Ephemerides
.
His
conclusion
was
,
that
he
could
find
no
certainty
in
the
rules
of
Astrology
commonly
received
,
nor
frame
any
other
upon
his
experience
.
Yet
had
I
a
mind
to
try
a
little
further
,
and
about
the
Year
1631
and
forward
for
some
years
,
I
made
dayly
Observations
of
all
Winds
and
Weather
at
home
,
and
got
also
the
best
information
I
could
by
Letters
from
Friends
abroad
,
by
relation
of
Travellers
,
Journals
of
Sea-men
,
Curranto'es
(
as
they
were
then
called
)
Mercurius
Gallabelgicus
,
&c.
And
I
found
so
great
diversity
of
Weather
in
the
same
Climate
,
and
sometime
in
places
scarce
three
degrees
distant
in
Longitude
or
Latitude
,
that
I
dispaired
of
Prognesticating
Weather
,
till
I
could
learn
which
way
the
Wind
would
list
to
blow
.
This
among
my
Observations
is
memorable
,
that
the
Weather
Capt.
James
had
in
his
Voyage
,
Wiatering
and
return
from
the
Mid-land
Sea
of
America
,
was
very
divers
from
ours
,
and
oft
times
contrary
;
and
all
the
agreement
I
could
find
,
was
,
that
one
day
when
it
Thundred
in
America
very
sore
,
we
had
Thunder
18.
hours
after
:
yet
he
was
many
moneths
in
Charlton
Island
,
which
is
in
our
Latitude
;
where
his
Ship
was
so
Frozen
,
and
the
Sea
so
full
of
Ice
,
that
he
could
not
sail
till
May.
And
I
know
a
learned
Gentleman
that
used
for
a
recreation
at
Christmas
time
to
put
a
Dozen
new
Almanacks
into
the
hands
of
so
many
of
his
servants
,
and
set
them
to
read
in
order
the
several
judgements
of
the
Prognosticaters
about
the
Weather
;
wherein
he
found
as
good
sport
as
William
Duke
of
Mantua
did
in
reading
the
judgements
of
several
Astrologers
,
upon
a
Figure
erected
upon
the
Nativity
of
his
young
Mule
.
For
the
Genethliacal
part
,
since
Planets
signifie
diversly
according
to
the
Houses
in
which
they
are
,
it
were
very
needfull
for
Men
of
this
craft
,
first
to
demonstrate
how
the
Houses
ought
to
be
divided
,
before
they
undertake
to
give
any
judgement
:
but
herein
they
are
not
agreed
:
those
Stars
which
after
Ptolemy
are
in
one
House
,
and
signifie
thus
;
after
Regiomontanus
are
in
another
House
for
the
same
time
and
place
,
and
so
signifie
otherwise
.
Of
the
four
wayes
above
mentioned
Chapter
48.
I
cannot
say
which
is
best
:
but
that
of
Regiomontanus
which
is
now
most
followed
,
and
called
by
some
the
Rational
way
,
seemes
to
me
the
most
Irrational
of
all
,
because
of
the
very
unequal
division
of
the
Houses
:
for
let
a
man
Erect
a
Figure
for
Wardhuys
in
Norway
,
after
that
way
,
and
he
shall
find
the
4
Houses
next
the
Meridian
,
viz.
3
,
4
,
9
,
11.
hugely
great
,
each
of
them
containing
above
a
seventh
part
of
the
Sphear
;
and
the
4
Houses
next
the
Horizon
each
of
them
not
to
contain
a
twenty
fourth
part
;
he
shall
also
observe
that
all
the
Stars
within
48
degrees
of
either
Pole
(
which
take
up
a
full
third
part
of
the
Sphear
)
are
perpetually
confined
to
two
Houses
½
for
that
Parallel
;
though
in
warmer
Climates
some
of
them
have
a
little
more
liberty
;
but
Ptolomie
and
Alcabitius
gives
every
Star
the
Liberty
of
all
the
Houses
,
and
Campanus
Imprisons
but
a
few
.
I
could
never
see
any
good
reason
why
the
influences
of
the
Stars
should
make
more
impression
upon
the
Child
in
the
moment
of
his
Nativity
,
then
they
did
at
any
time
before
in
the
Womb.
If
mines
in
the
Earth
are
not
hidden
from
the
influences
of
the
Stars
,
neither
is
the
Child
in
the
Mothers
belly
.
If
the
Stars
have
such
operation
on
Men
in
their
Nativity
,
it
is
rationall
to
think
they
have
more
force
in
their
first
conception
,
when
the
matter
is
fluid
and
more
apt
to
receive
impression
,
then
when
the
Child
is
shaped
,
and
the
temper
more
confirmed
:
and
this
Johannes
Angeli
an
old
Astrologer
considering
,
framed
a
Table
to
discover
to
a
minute
the
time
of
conception
,
by
the
time
of
the
Birth
given
,
where
he
extends
the
time
of
gestation
to
40.
Weeks
,
and
(
as
I
remember
,
)
6
dayes
over
;
but
he
doth
not
tell
us
whether
he
made
that
Table
by
experience
,
or
divination
;
nor
how
many
experiments
he
had
used
for
the
triall
of
it
.
Favorinus
once
demanded
of
these
Genethliacal
men
,
who
pretend
to
know
mens
Fortunes
by
the
Positions
of
the
Stars
in
their
Nativity
.
How
it
comes
to
pass
that
thousands
of
people
,
of
both
Sexes
,
of
all
Ages
,
born
in
sundry
places
,
and
under
sundry
configurations
of
the
Stars
,
have
hapned
to
perish
in
the
same
hour
and
moment
,
and
by
the
same
misfortune
,
as
by
Shipwrack
,
Stormes
of
War
,
Earth-quakes
,
fall
of
Buildings
,
and
such
like
accidents
?
And
whether
this
fate
could
have
been
foreseen
in
all
the
Figures
Erected
for
their-several
Nativities
?
Which
demand
I
think
never
yet
received
any
good
answer
.
This
also
I
have
long
observed
that
no
Man
(
so
far
as
my
knowledge
and
intelligence
reacheth
)
are
so
Fortunate
in
setting
Figures
for
discovering
things
lost
,
or
Prognosticating
of
Life
and
Death
,
as
some
who
scarce
know
Charles
Wain
,
nor
any
Planet
in
the
Skie
beside
the
Sun
and
Moon
:
and
some
of
these
have
given
out
that
I
allowed
of
their
practises
,
before
ever
they
consulted
me
,
and
so
gave
me
occasion
to
make
this
digression
.
He
that
would
see
more
of
this
argument
may
read
what
my
Lord
Howard
,
sometime
Earle
of
Northampton
,
and
Mr
Perkins
have
written
thereupon
.
Pliny
Book
.
30.
Chapter
2.
hath
recorded
Nero's
experiment
and
judgement
of
this
Art
.
Species
magiae
plures
sunt
;
namque
&
ex
Aquâ
,
&
ex
Spheris
,
&
ex
Aere
,
&
Stellis
,
&
Lucernis
,
ac
Pelvibus
,
securibusque
,
&
multis
alijs
modis
divina
promittit
:
preterea
umbrarum
,
inferorumque
colloquia
:
quae
omnia
aetate
nostrâ
princeps
Nero
vana
falsaque
comperit
:
quippe
non
Cithorae
tragicique
cantus
libido
illi
major
fuit
,
fortuna
rerum
humanarum
summa
gestiente
in
profundis
animi
vitijs
:
Primumque
imperare
Dijs
concupivit
,
nec
quiequam
generosius
voluit
.
Nemo
unquam
ulli
artium
validius
favit
:
Ad
haec
,
non
opes
ei
defuere
,
non
vires
,
non
discendi
ingenium
;
aliaque
non
patiente
mundo
,
Immensum
&
indubitatum
exemplum
est
falsae
artis
,
quam
dereliquit
.
Nero
,
as
if
he
had
said
,
What
shall
the
Man
do
that
comes
after
the
King
?
And
a
little
after
,
Proinde
ita
persuesum
sit
,
intestabilem
,
irritam
,
inanem
esse
,
habentem
tamen
quasdam
veritatis
umbras
,
sed
in
his
veneficas
artes
pollere
,
non
Magicas
.
The
judgement
also
of
Favorinus
,
a
learned
Philosopher
,
is
worthy
to
be
here
rehearsed
,
as
it
is
reported
by
A.
Gellius
Noct.
Attic.
14.1
.
With
whose
words
I
will
end
this
digression
.
Cavendum
ne
qua
nobisisti
Sycophantae
ad
fidem
faciendam
irreperent
,
quod
viderentur
interdum
vera
effutire
aut
spargere
.
Non
enim
comprensa
(
inquit
)
neque
definita
,
neque
percepta
dicunt
,
sed
lubrica
atque
ambagiosa
conjectatione
nitentes
,
inter
falsa
atque
vera
pedetentim
quasi
per
tenebras
ingredientes
eunt
:
&
aut
multa
tentando
incidunt
repente
imprudentes
in
veritatem
;
aut
ipsorum
,
qui
eos
consulunt
,
multa
credulitate
ducente
perveniunt
callide
ad
ea
quae
verae
sunt
:
&
iccirco
videntur
in
praeteritis
rebus
quàm
in
futuris
veritatem
facilius
imitari
.
Ista
tamen
omnia
quae
aut
●em●re
aut
astute
vera
dicunt
,
prae
caeteris
quae
mentiuntur
,
pars
ea
non
est
millessima
.
And
a
little
after
,
Idem
Favorinus
deterrere
volens
ac
depellere
adolescentes
a
Genethliacis
istis
,
&
quibusdam
alijs
id
genus
,
qui
prodigiosis
artibus
futura
omnia
dicturos
se
pollicentur
,
nullo
pacto
adeundos
esse
consulendosque
bujusmodi
argumentis
concludebat
.
Aut
adversa
(
inquit
)
eventura
dicunt
,
aut
prospera
:
si
dicunt
prospera
,
&
fallunt
;
miser
fies
frustra
expectando
:
si
adversa
dicunt
,
&
mentiuntur
:
miser
fies
frustra
timendo
sin
vera
respondem
,
eaque
sunt
non
prospera
;
jam
inde
ex
animo
miser
fies
antequam
è
fato
fias
:
si
felicia
promittunt
,
eaque
eventura
sunt
,
tum
plane
duo
erunt
incommoda
:
&
expectatio
te
spesuspensum
fatigabit
:
&
futurum
gaudij
fructum
spes
tibijam
defloraverit
;
nullo
igitur
pacto
utendum
est
istiusmodi
hominibus
res
futuras
presagientibus
.
CHAP.
LV.
To
find
the
Angles
of
the
Ascendent
,
or
the
Angle
of
the
Ecliptique
with
the
Horizon
,
and
the
Altitude
of
the
Nonagesimus
gradus
,
at
any
time
.
IN
the
Equinoctial
Projection
set
the
Suns
place
to
the
time
proposed
,
and
get
the
Altitude
of
Culmen
Caeli
,
by
Chapter
46.
Then
,
if
the
Eastern
arch
of
the
Ecliptique
be
shorter
then
the
Western
,
you
shall
count
the
degrees
between
the
Ascendent
and
Mid-heaven
;
otherwise
count
from
the
Defcendent
to
Mid-heaven
.
Number
these
degrees
on
the
Label
from
the
Center
,
and
where
they
end
make
a
prick
;
which
prick
if
you
put
upon
the
Parallel
of
the
Altitude
of
Culmen
Caeli
,
you
shall
have
in
the
Limb
,
between
the
Finitor
and
the
Label
,
the
measure
of
the
lesser
angle
,
which
taken
out
of
180
degrees
leaveth
the
greater
angle
.
This
is
done
by
Probleme
2.
Rectang
.
And
note
that
the
lesser
angle
,
and
the
Altitude
of
the
Nonagesimus
gradus
be
alwayes
equal
.
Example
.
March
29.
1652.
10.
hours
32.
minutes
a.
m.
♓
25
½
was
in
Culmine
:
the
Meridian
Altitude
thereof
is
36.
Between
Culmen
and
the
Descendent
I
find
61
½
.
therefore
I
prick
the
degree
61
½
from
the
Center
in
the
Label
;
and
when
I
have
turned
that
prick
to
the
36.
Almicantar
,
the
Label
shewes
in
the
Limb
of
the
Reet
42
degrees
for
the
lesser
angle
of
the
Ecliptique
with
the
Horizon
(
exactly
41.
58
minutes
)
which
also
is
the
Altitude
of
the
Nonagesimus
gradus
:
the
greater
angle
is
138
degrees
2
minutes
.
Another
way
.
In
the
Equinoctial
Projection
:
lay
the
Label
on
the
Nonagesimus
gradus
;
and
observe
his
Declination
on
the
Label
,
and
his
Horary
distance
from
the
Meridian
.
Then
in
the
Meridional
Projection
and
his
first
mode
,
observe
where
that
Declination
and
Horary
distance
meet
on
the
Mater
,
and
the
Almicantar
touching
the
same
point
sheweth
the
Altitude
of
Nonagesimus
gradus
,
which
is
equal
to
the
angle
sought
.
Example
.
In
the
former
Case
,
where
♓
25
½
was
in
our
Meridian
,
♈
24.
was
Nonagesimus
gradus
;
the
Label
laid
to
it
shewed
me
his
Declination
9
½
almost
,
North
:
and
his
Horary
distance
from
the
Meridian
in
the
Limb
26.
20
minutes
.
then
the
Fimtor
being
set
to
the
Latitude
,
I
seek
the
Intersection
of
the
9
½
Parallel
of
North
Declination
with
the
26
⅓
Meridian
from
the
Limb
:
and
there
toucheth
the
42
Almicantar
;
shewing
the
Altitude
of
Nonagesimus
gradus
,
and
the
quantity
of
the
lesser
angle
sought
,
as
before
.
And
there
cometh
unasked
also
,
the
36
⅓
Azimuth
being
the
Azmuth
of
the
Nonagesimus
gradus
,
which
is
alwayes
equal
to
the
Amplitude
of
the
Ascendent
.
Other
wayes
,
See
Chapter
56.
and
57.
CHAP.
LVI
.
The
Ascendent
and
his
Amplitude
,
and
the
Altitude
of
Culmen
Caeli
given
;
so
to
represent
the
Ecliptique
,
that
you
may
presently
find
not
onely
the
Altitude
of
the
Nonagesimus
gradus
,
but
the
Altitude
and
Azimuth
of
every
degree
of
the
Ecliptique
at
one
view
.
SEt
your
Planisphear
in
the
third
Mode
of
the
Meridional
Projection
:
that
is
,
If
the
Ascendent
be
a
North
Signe
,
move
the
Finitor
from
Meridies
toward
the
North
Pole
,
till
the
North
Pole
be
elevated
above
the
Finitor
according
to
the
elevation
of
Culmen
Caeli
;
but
if
the
Ascendent
be
a
South
Signe
,
move
the
other
end
of
the
Finitor
from
Septentrio
toward
the
North
Pole
,
till
the
Pole
have
the
Elevation
of
Culmen
Caeli
.
Then
number
the
Amplitude
of
the
Ascendent
upon
the
Finitor
from
the
Center
to
your
left
hand
,
(
toward
Septentrio
)
and
take
the
Meridian
that
crosseth
there
for
the
Eastern
arch
of
the
Ecliptique
,
and
his
match
so
much
distant
from
the
Axtree
towards
Meridies
shall
be
the
Western
arch
;
so
do
the
Azimuths
and
Almicantars
of
the
Reet
shew
at
once
the
Altitude
and
Azimuth
of
every
degree
of
the
Ecliptique
.
Example
,
March
29.
1652.
10.
hours
32
min.
a.
m.
I
found
♓
25
½
Culminating
.
and
his
Meridian
Altitude
(
by
Chapter
46
)
36
degrees
,
the
Ascendent
♋
24.
(
by
Chapter
47.
)
and
his
Amplitude
36
⅓
(
by
Chapter
15
,
and
16.
)
the
Sun
being
then
Eclipsed
in
♈
19.
11
minutes
.
I
would
know
his
Altitude
and
Azimuth
,
and
likewise
the
Altitude
and
Azimuth
of
the
Nonagesimus
gradus
.
To
this
purpose
,
I
take
the
North
Pole
for
Culmen
,
and
set
the
Finitor
36.
below
him
toward
Meridies
;
and
from
the
Center
toward
my
left
hand
,
I
number
on
the
Finitor
the
Amplitude
of
the
Ascendent
36
⅓
,
there
cuts
the
twenty
third
Meridian
from
the
Axis
:
(
which
here
serveth
for
the
Eastern
arch
of
the
Ecliptique
:
)
the
degree
in
this
Ecliptique
here
cut
by
the
Finitor
is
the
Ascendent
♋
24.
thence
I
number
in
this
Ecliptique
South-ward
90
degrees
by
help
of
the
Parallels
,
and
so
I
come
to
♈
24
degrees
,
being
the
Nonagesimus
gradus
.
Here
the
42
Almicantar
toucheth
the
Nonagesimus
gradus
,
shewing
the
Altitude
thereof
,
and
here
also
cutteth
the
Azimuth
of
the
Nonagesimus
gradus
36
⅓
equal
to
the
Amplitude
of
the
Ascendent
,
as
it
is
alwayes
and
ought
to
be
;
so
as
that
you
might
have
found
the
Nonagesimus
gradus
by
this
Azimuth
with
less
numbring
.
Now
for
the
Sun
,
he
is
in
♈
19.
11
minutes
,
that
is
,
nearer
the
Meridian
then
the
Nonagesimus
gradus
by
almost
5.
degrees
:
I
count
therefore
4.
49
minutes
(
for
so
it
is
)
past
the
Nonagesimus
gradus
;
there
is
the
Sun
,
and
the
Almicantar
cutting
there
shewes
his
Altitude
41
⅔
,
and
his
Azimuth
is
shewn
by
the
29
Azimuth
some
what
near
.
Or
if
you
would
reckon
after
the
order
of
the
Signes
,
which
is
easier
,
begin
at
the
Descendent
,
where
is
♑
24.
thence
61
½
makes
♓
25
½
at
the
Pole
,
for
Culmen
Caeli
:
thence
in
the
Eastern
arch
to
the
Suns
place
I
make
85
degrees
11
minutes
;
and
4.49
minutes
further
is
♈
24.
the
Nonagesimus
gradus
.
CHAP.
LVII
.
To
do
the
same
another
way
,
by
the
Horizontal
Projection
,
very
plainly
.
TAke
the
Zenith
for
the
Ascendent
,
and
set
him
in
his
place
in
the
Limb
(
which
here
is
Horizon
)
so
much
from
Oriens
as
his
Amplitude
comes
to
;
and
that
toward
Septentrio
,
if
it
be
a
Northern
Signe
,
or
if
it
be
a
Southern
Signe
toward
Meridies
.
Then
number
upon
the
Meridian
line
from
the
Limb
inwards
the
Altitude
of
Culmen
Caeli
,
and
the
Azimuth
that
cutteth
there
shall
be
your
Ecliptique
in
this
Case
:
If
the
Azimuths
reach
not
the
Meridian
,
turn
about
the
Reet
,
and
set
Nadir
for
Ascendent
.
Lay
the
Label
to
any
degree
of
this
Ecliptique
,
and
the
degrees
of
the
Label
from
that
degree
to
the
Limb
shall
be
the
Altitude
thereof
:
and
between
the
Label
and
Meridies
in
the
Limb
the
Azimuth
thereof
.
Example
.
Because
in
the
Case
of
the
former
Chapter
I
foresee
that
the
Sun
will
be
past
the
Nonagesimus
gradus
,
and
so
in
the
West
Quadrant
of
the
Ecliptique
(
though
he
be
in
the
East
Quadrant
of
the
Horizon
)
therefore
I
set
Nadir
at
the
Amplitude
of
the
Ascendent
viz.
36
⅓
from
Oriens
North-ward
,
then
in
the
Meridian
line
I
number
from
the
Limb
inwards
36.
for
the
Altitude
of
Culmen
,
where
I
make
a
prick
,
and
say
,
Here
is
♓
25
½
Culminating
,
and
through
that
prick
passeth
the
42.
Azimuth
from
the
Limb
,
which
is
now
my
Ecliptique
;
and
by
that
I
see
that
the
angle
of
the
Ecliptique
which
the
Horizon
(
called
the
angle
of
the
Ascendent
,
and
alwayes
equal
to
the
Altitude
of
Nonagesimus
gradus
,
as
was
said
)
is
42
degrees
:
and
if
I
follow
this
Azimuth
to
the
Finitor
,
there
is
Nonagesimus
gradus
,
and
the
Altitude
thereof
42
degrees
counted
from
the
Limb
(
here
Horizon
)
the
Azimuth
thereof
lies
in
the
Limb
between
the
Finitor
and
the
Meridian
36
⅓
as
before
,
equal
to
the
Amplitude
of
the
Ascendent
,
I
number
also
from
♓
25
½
in
the
Meridian
23.
41
minutes
to
the
left
hand
still
,
and
there
I
have
♈
19.
11
minutes
,
the
Suns
place
,
which
cuts
on
the
Label
41
⅔
.
for
the
Altitude
of
the
Sun
there
,
and
the
Label
at
the
same
time
cutteth
in
the
Limb
about
29.
from
South
East-ward
for
the
Azimuth
of
the
Sun
:
and
after
the
same
manner
you
have
before
you
the
Altitude
and
Azimuth
of
every
other
degree
of
the
Ecliptique
for
the
time
proposed
.
CHAP.
LVIII
.
To
do
the
same
by
the
Nonagesimal
Projection
,
if
the
Altitude
of
Nonagesimus
gradus
be
first
given
instead
of
the
Altitude
of
Culmen
Caeli
.
SEt
your
Planisphear
in
the
Nonagesimal
Projection
(
by
Book
2.3
.
)
that
is
,
make
the
Limb
now
to
represent
the
Circle
of
Longitude
or
Azimuth
(
for
it
is
both
)
which
cutteth
the
Nonagesimus
gradus
,
and
make
the
Equinoctial
line
here
to
be
Horizon
:
and
from
the
Equinectial
line
number
in
the
Limb
the
Altitude
of
Nonagesimus
gradus
,
and
thereto
set
the
Finitor
,
so
shall
the
Finitor
be
Ecliptique
,
the
Nonagesimus
gradus
at
the
Limb
,
the
Ascendent
and
Descendent
at
the
Center
:
and
because
the
Equinoctial
line
is
Horizon
in
this
Projection
,
therefore
the
Meridians
become
Azimuths
,
and
the
Parallels
Almicantars
,
shewing
the
Altitude
and
Azimuth
of
every
degree
of
the
Ecliptique
,
if
you
reckon
as
you
ought
in
this
manner
.
Reckon
in
the
Equinoctial
line
(
here
Horizon
)
from
the
Center
the
Amplitude
of
the
Ascendent
,
to
the
right
Hand
,
if
it
be
a
North
Signe
,
and
contrarily
if
it
be
a
South
Signe
.
Where
this
Amplitude
ends
is
the
East
point
,
from
whence
you
shall
reckon
all
your
Azimuths
.
Count
thence
to
the
Limb
and
back
again
(
if
need
be
)
in
the
said
Equinoctial
line
,
till
you
have
made
90
degrees
,
there
is
your
Meridian
,
as
far
distant
from
the
Limb
,
as
the
East
point
was
from
the
Ascendent
.
Follow
this
Meridian
to
the
Finitor
,
and
there
he
shewes
you
Culmen
Caeli
,
and
the
Parallel
there
cutting
shewes
the
Altitude
thereof
.
Now
may
you
find
every
degree
of
the
Ecliptique
above
the
Horizon
,
if
you
know
but
what
Ascends
,
or
Descends
,
or
Culminates
;
and
of
every
such
degree
the
Parallels
shew
you
the
Altitude
,
and
the
Meridians
shew
his
Azimuth
,
if
you
begin
your
numbring
from
the
East
or
South
Azimuth
.
Example
.
When
♋
24
degrees
was
Ascending
(
as
in
the
Example
before
used
)
as
by
consequence
♈
24.
in
Nonagisimo
gradu
,
♂
was
in
♉
4.
45
minutes
,
and
had
but
3.
or
4.
minutes
South
Latitude
:
I
would
know
♂
his
Altitude
,
and
Azimuth
,
setting
go
the
Finitor
above
the
Equinoctial
line
42
degrees
(
which
is
the
Altitude
of
Nonagesimus
gradus
)
I
say
,
because
the
Nonagesimus
gradus
at
the
end
of
the
Finitor
in
the
Limb
,
is
♈
24.
therefore
I
must
count
back
10.
45
minutes
toward
the
Ascendent
for
Mars
,
and
there
the
Parallel
41
degrees
with
10
minutes
cutteth
the
Finitor
,
for
the
Altitude
of
♂
,
and
the
14th
.
Meridian
East-ward
from
the
Limb
gives
me
his
Azimuth
,
which
if
I
begin
to
reckon
from
the
East
point
,
falleth
out
to
be
almost
the
40th
.
Azimuth
from
the
East
.
Mars
his
Latitude
here
is
not
regarded
.
CHAP.
LIX
.
The
Nonagesimus
gradus
,
and
his
Altitude
and
Azimuth
given
,
as
in
the
former
Chapter
.
How
in
the
same
Projection
to
get
the
Altitude
and
Azimuth
of
any
Planet
or
Star
,
by
his
Longitude
and
Latitude
.
YOur
Palnisphear
set
as
in
the
former
Chapter
:
you
shall
number
the
Longitude
of
the
Star
upon
the
Finitor
,
(
here
Ecliptique
)
beginning
at
the
Descendent
or
Nonagesimus
gardus
;
and
in
the
Azimuth
serving
his
Longitude
,
count
his
Latitude
by
the
Almicantars
,
at
the
end
of
which
account
is
the
Stars
place
for
this
time
.
The
Parallel
cutting
there
shewes
his
Altitude
,
and
the
Meridian
cutting
there
shewes
his
Azimuth
,
if
you
count
from
the
East
point
as
you
were
taught
in
the
former
Chapter
.
Example
.
Lucida
Pleiadum
was
in
Longitude
♉
25.
10
minutes
,
Latitude
4
degrees
00
minutes
North.
Therefore
from
the
Nonagesimus
gradus
♈
24.
I
number
in
the
Finitor
toward
the
Ascendent
31.
10
minutes
,
and
there
is
the
Longitude
of
Lucida
Pleiaedum
;
in
the
Azimuth
that
cuts
here
I
go
up
Northward
4
degrees
,
and
there
I
make
a
prick
for
Lucida
Pleiadum
.
Now
the
Parallel
38
½
shewes
me
his
Altitude
,
and
the
48th
½
.
Meridian
from
the
Center
,
shewes
me
that
Lucida
Pleiadum
is
gone
48
½
in
Azimuth
from
the
Ascendent
,
but
from
the
East
point
onely
12
degrees
10
minutes
.
CHAP.
LX.
The
Altitude
and
Azimuth
of
any
Star
taken
,
and
either
the
Ascendent
,
Nonagesimus
gradus
,
or
Culmen
Caeli
known
:
How
by
the
same
Nonagesimal
Projection
to
find
the
Stars
Longitude
and
Latitude
.
IF
you
know
either
the
Ascendent
,
Nonagesimus
gradus
,
or
Culmen
Caeli
,
you
have
enough
to
put
your
Planisphear
in
the
Nonagesimal
Projection
,
by
the
former
Chapters
.
And
your
Planisphear
so
set
,
you
shall
seek
out
the
Meridian
,
which
standeth
for
the
Azimuth
in
which
you
observe
the
Star
;
and
therein
number
from
the
Equinoctial
line
the
Altitude
observed
:
the
Azimuth
and
Almicantar
cutting
there
shew
the
Longitude
and
Latitude
of
the
Star
inquired
.
If
the
Azimuths
reach
not
the
place
of
the
Star
,
turn
the
Reet
half
round
,
and
let
the
Zenith
and
Nadir
points
change
places
;
and
your
turn
is
served
.
Example
.
Febr.
13
,
1657/8.
I
observed
(
somewhat
near
)
that
♃
was
gone
West-ward
from
the
Meridian
in
Azimuth
14
degrees
,
and
that
his
Altitude
was
61
degrees
;
Sirius
was
then
in
the
Meridian
,
by
which
I
have
the
Ascendent
,
Culmen
,
and
Nonagesimus
gradus
,
any
or
all
of
them
given
.
For
when
in
the
Equinoctial
Projection
I
bring
Sirius
to
the
Meridian
line
,
it
is
all
one
as
if
I
had
set
the
Suns
place
to
the
hour
of
the
Night
(
by
Chapter
46.
)
and
I
see
there
Culminates
with
Sirius
♋
7.
10
minutes
,
whose
Meridian
Altitude
(
by
the
46.
)
is
61.
5
minutes
;
and
I
see
♎
5
½
ascending
in
my
Horizon
,
and
♈
5
½
descending
;
therefore
♋
5
½
is
Nonagesimus
gradus
,
which
is
90
degrees
distant
both
from
the
Ascendent
and
Descendent
:
his
Altitude
(
by
Chapter
55.
)
61.
10
minutes
almost
.
Therefore
I
set
the
Finitor
61.
10
minutes
above
Meridies
,
(
as
Chapter
58.
)
and
in
the
Finitor
at
the
Limb
I
count
♋
5
½
Nonagesimus
gradus
;
thence
I
go
inwards
in
the
Finitor
1.
40
minutes
where
I
come
to
♋
7.
10.
the
degree
of
Culmination
;
this
degree
is
cut
by
the
4th
.
Meridian
from
the
Limb
,
whereby
I
learn
that
this
4th
Meridian
will
be
the
Meridian
of
my
place
,
and
that
the
Amplitude
of
the
Nonagesimus
gradus
,
and
likewise
of
the
Ascendent
,
is
4
degrees
.
Now
to
place
♃
in
the
Mater
,
I
count
his
Azimuth
first
,
beginning
from
the
Meridian
of
my
Place
now
found
.
First
I
reckon
up
from
the
Culmen
Caeli
to
Nonagesimus
gradus
in
the
Limb
4
degrees
,
(
for
so
much
the
Nonagesimus
gradus
is
West
of
the
Meridian
)
and
thence
back
again
,
I
tell
to
the
10th
.
Meridian
from
the
Limb
,
which
maketh
the
14th
.
Azimuth
from
Medium
Caeli
,
in
which
Azimuth
I
observed
Jupiter
in
that
Meridian
;
used
here
for
the
14th
.
Azimuth
)
I
reckon
the
Altitude
of
Jupiter
from
he
Equinoctial
line
61
degrees
,
and
at
that
Altitude
I
make
therein
a
prick
for
the
place
of
♃
:
And
imediately
I
see
this
prick
standeth
a
quarter
of
a
degree
above
the
Finitor
,
shewing
the
Latitude
of
♃
15
minutes
North
,
and
it
is
cut
by
almost
the
5th
.
Azimuth
from
the
Limb
,
which
sheweth
me
that
the
Longitude
of
♃
is
4.
50
minutes
less
then
the
Longitude
of
Nonagesimus
gradus
;
and
therefore
that
♃
is
in
♋
0.
40
minutes
.
CHAP.
LXI
.
The
Latitude
and
Azimuth
of
a
Star
,
and
either
the
Ascendent
,
Nonagesimus
gradus
,
or
the
Culmination
given
,
to
find
his
Longitude
.
YOur
Planisphear
being
set
in
the
Nonagesimal
Projection
,
(
as
in
the
former
Chapter
)
seek
the
Meridian
that
serveth
for
the
Azimuth
of
the
Star
,
and
mark
where
it
cutteth
the
Almicantar
serving
for
the
Parallel
of
the
Stars
Latitude
.
The
Azimuth
cutting
there
shewes
the
Longitude
,
which
you
shall
reckon
from
the
Ascendent
,
or
Descendent
,
or
Nonagesimus
gradus
;
whose
Longitudes
are
known
,
as
was
shewed
in
the
former
Chapter
.
Example
.
Suppose
♃
his
Azimuth
observed
14
degrees
from
South
Westward
(
as
Chapter
60.
)
and
suppose
his
Latitude
known
15
minutes
North
:
(
Though
I
know
the
Tables
make
Jupiters
Latitude
here
divers
minutes
less
,
that
matters
not
to
our
purpose
here
)
I
say
where
the
10th
Meridian
(
which
by
the
former
Chapter
is
the
14.
Azimuth
in
this
posture
of
my
Planisphear
)
cutteth
the
Almicantar
0
¼
serving
for
Jupiters
Latitude
,
there
cutteth
an
Azimuth
which
gives
me
Jupiters
Longitude
,
as
in
the
former
Chapter
.
CHAP.
LXII
.
To
find
the
Parallactical
Angle
;
that
is
,
what
Angle
the
Azimuth
maketh
with
any
point
of
the
Ecliptique
,
by
the
Altitude
of
that
point
,
and
of
the
Nonagesimus
gradus
.
NUmber
on
the
Label
from
the
Center
the
Complement
of
the
Altitude
of
the
point
proposed
,
(
which
may
be
known
by
Chapter
56.
)
and
at
the
end
of
it
make
a
prick
,
and
having
(
by
Chapter
55.
or
otherwise
)
the
Altitude
of
Nonagesimus
gradus
,
turn
the
prick
you
made
on
the
Label
to
touch
the
Almicantar
,
which
is
Complement
of
that
Altitude
:
then
in
the
Limb
of
the
Reet
between
the
Finitor
and
the
Label
is
the
quantity
of
the
angle
.
Example
.
In
Chapter
56.
the
Altitude
of
the
Sun
in
the
middle
of
the
Eclipse
which
hapned
March
29.
1652.
10.
hours
32
minutes
a.
m.
was
41.
47
minutes
,
and
the
Altitude
of
Nonagesimus
gradus
41.
58
minutes
;
wherefore
I
make
a
prick
on
the
edge
of
the
Label
at
41
¾
counted
from
the
Limb
,
(
or
I
count
the
Complement
hereof
from
the
Center
,
and
make
the
prick
)
and
having
turned
that
prick
to
the
fourty
second
Almicantar
from
the
Zenith
,
I
find
the
Label
shewing
87
degrees
in
the
Limb
of
the
Reet
,
the
quantity
of
the
angle
.
But
because
the
Label
here
cutteth
the
Almicantar
so
slope
that
you
can
hardly
observe
the
just
point
of
Intersection
,
I
will
shew
you
another
way
.
The
Complement
of
the
Altitude
of
Culmen
Caeli
,
the
distance
of
the
point
proposed
from
Culmen
,
and
the
Complement
of
Altitude
of
the
said
point
,
make
a
Triangle
;
whose
3
sides
are
all
known
,
or
may
be
known
by
the
Chapters
foregoing
:
Therefore
by
the
first
Probleme
of
Obliquangled
Triangles
you
may
find
the
angle
.
diagram
CHAP.
LXIII
.
To
find
the
Parallax
of
Altitude
of
the
Sun
,
or
Moon
.
diagram
To
get
the
Parallax
for
any
Altitude
proposed
,
you
must
first
get
the
Horizontall
Parallax
out
of
some
Astronomical
Tables
:
for
it
varyes
according
to
the
Planets
distance
from
the
Earth
;
which
is
not
alwayes
the
same
;
yet
the
Suns
Horizontal
Parallax
you
may
alwayes
reckon
to
be
about
3
minutes
,
and
the
Moons
Horizontal
Parallax
to
be
at
the
least
50
minutes
,
and
at
the
most
68
minutes
.
This
had
,
I
number
the
Horizontal
Parallax
in
the
Limb
from
the
Equinoctial
line
,
and
thereto
lay
the
Label
;
and
number
the
Altitude
of
the
Planet
on
the
Label
from
the
Limb
,
and
the
Parallel
that
cuts
that
Altitude
shewes
the
Parallax
desired
.
And
note
here
,
that
for
every
minute
of
the
Horizontal
Parallax
you
may
reckon
5
,
10
,
or
20
,
times
so
many
,
so
that
your
Label
rise
not
beyond
10
degrees
in
the
Limb
;
so
shall
you
attain
the
minutes
more
exactly
.
Example
.
March
29.
1652.
the
Altitude
of
the
Sun
in
the
middle
of
the
Eclipse
was
41.
47
minutes
,
and
his
Horizontal
Parallax
according
to
Lantsbergius
,
2
minutes
18
seconds
,
for
which
I
number
2
degrees
18
minutes
from
the
Equinoctial
line
,
and
thereto
set
the
Label
;
and
so
I
find
the
43
¾
degrees
of
the
Label
to
cut
the
Parallel
1
degree
30
minutes
,
which
I
am
to
accompt
1
minute
30
seconds
,
the
Suns
Parallax
for
this
Altitude
.
Likewise
the
Moons
Horizontal
Parallax
according
to
Lantsbergius
,
was
then
62
minutes
,
her
Altitude
the
same
with
the
Suns
,
or
very
near
;
I
set
the
Label
therefore
to
make
an
angle
of
6
degrees
12
minutes
with
the
Equinoctial
,
and
so
I
find
the
Parallel
4
⅔
cutting
the
41
¾
degree
of
the
Label
;
which
shewes
the
Parallax
of
the
☽
in
that
Altitude
46
½
,
accounting
every
degree
10
minutes
,
as
here
I
had
appointed
them
to
signifie
.
CHAP.
LXIV
.
The
Parallactique
Angle
,
and
the
Parallax
of
Altitude
given
,
to
find
the
Parallax
of
Longitude
and
Latitude
.
IF
the
Azimuth
or
Circle
of
Altitude
make
no
angle
with
the
Ecliptique
,
but
be
co-incident
with
it
(
as
where
the
Ecliptique
cuts
the
Zenith
)
then
doth
the
Parallax
of
Altitude
vary
the
Longitude
only
;
and
so
much
as
the
Parallax
of
Altitude
is
,
so
much
is
the
apparent
Longitude
of
the
Planet
greater
then
the
true
Longitude
in
the
Eastern
Quadrant
of
the
Ecliptique
,
and
so
much
lesser
in
the
Western
.
If
the
Azimuth
make
a
Right
angle
with
the
Ecliptique
(
which
it
may
do
only
in
Nonagesimo
gradu
)
then
doth
the
Parallax
of
Altitude
vary
the
Latitude
onely
;
and
so
much
as
the
Parallax
of
Altitude
is
,
so
much
must
be
added
to
the
apparent
North
Latitude
,
or
subducted
from
the
apparent
South
Latitude
,
to
make
the
true
Latitude
of
the
Planet
North
or
South
.
If
the
Azimuth
cut
the
Ecliptique
with
Oblique
angles
(
as
most
commonly
it
hapneth
to
do
)
then
doth
the
Parallax
of
Altitude
vary
both
the
Longitude
and
Latitude
.
And
the
nearer
the
Planet
is
to
the
Nonagesimus
gradus
,
the
greater
is
the
Parallax
of
Latitude
,
and
the
Parallax
of
Longitude
less
:
and
contrarily
the
further
the
Planet
is
from
Nonagesimus
gradus
,
the
greater
is
the
Parallax
of
Longitude
,
and
the
Parallax
of
Latitude
the
less
.
The
Parallax
of
Altitude
is
alwayes
the
Hypotenusa
,
and
the
Parallax
of
Longitude
and
Latitude
are
the
legs
of
a
smal
Rectangled
Spherical
Triangle
,
The
Parallactical
Triangle
.
which
may
be
called
the
Parallactical
Triangle
,
and
the
leg
which
hath
the
Parallax
of
Longitude
is
a
segment
of
the
Ecliptique
,
or
of
a
Parallel
near
it
,
and
the
leg
which
hath
the
Parallax
of
Latitude
,
is
a
segment
of
a
Circle
of
Longitude
passing
through
the
apparent
place
of
the
Planet
,
and
through
the
Poles
of
the
Ecliptique
,
and
cutting
the
Ecliptique
,
or
his
said
Parallel
at
Right
angles
,
as
in
the
Figure
.
-
C
is
the
apparent
place
of
the
Planet
.
-
B
is
his
true
place
,
in
which
he
must
be
seen
from
the
Center
.
-
A
B
C
is
the
Parallactique
angle
.
-
B
C
is
the
Parallax
of
Altitude
.
-
B
A
the
Parallax
of
Longitude
.
-
C
A
the
Parallax
of
Latitude
.
diagram
Wherefore
by
the
third
Probleme
of
Rectangled
Triangles
,
Book
3.
5.
you
may
presently
get
both
the
legs
.
Example
.
The
Parallactical
angle
B
,
was
found
Chapter
62.
to
be
86.
41
minutes
,
and
the
Parallax
of
the
Moons
Altitude
,
(
Chapter
63.
)
to
be
46
½
minutes
for
the
same
time
.
Here
therefore
having
laid
the
Label
to
86.
41
minutes
from
the
Equinoctial
,
I
number
in
the
Label
from
the
Center
46
degrees
and
an
half
(
in
stead
of
46
minutes
and
an
half
the
Parallax
of
Altitude
)
and
I
find
that
the
46
⅓
Parallel
cutteth
the
said
46
½
,
degree
of
the
Label
;
by
which
I
know
that
the
Leg
C
A
for
the
Parallax
of
Latitude
is
46
⅓
very
near
,
(
for
the
☽
here
being
near
the
Nonagesimus
gradus
all
her
Parallax
almost
goes
into
Longitude
,
)
but
B
A
of
my
Triangle
is
covered
by
the
head
of
my
Label
.
Nevertheless
I
may
see
his
measure
in
any
of
the
Parallels
to
be
3
½
minutes
for
the
Parallax
of
Longitude
,
for
it
is
the
3
½
Meridian
from
the
Axis
,
which
cutteth
46
½
of
the
Label
:
and
if
I
had
not
this
shift
,
I
might
have
my
choice
of
other
shifts
,
shewed
Book
3.
the
8
,
9
,
10
,
11
,
12
,
and
13th
.
Chapters
.
The
Suns
Parallax
of
Altitude
Chapter
63
for
the
same
time
was
found
1
minute
½
.
therefore
laying
the
Label
to
the
Parallactique
angle
(
as
before
)
I
number
on
the
Label
for
the
side
B
C
(
the
Suns
Parallax
of
Altitude
,
being
1
minute
½
)
9
degrees
,
so
every
degree
here
signifieth
10
seconds
;
and
I
find
there
cutting
almost
the
9th
Parallel
,
shewing
me
that
C
A
the
Parallax
of
the
Suns
Latitude
is
1
½
minute
almost
(
that
is
almost
as
much
as
his
Parallax
of
Altitude
)
and
there
cutteth
also
the
Meridian
0
⅔
,
shewing
me
that
the
side
B
A
,
Parallax
of
Longitude
,
is
almost
7
seconds
.
The
Sun
therefore
(
though
he
never
have
Latitude
)
by
reason
of
his
Parallax
,
appeared
in
the
middle
of
this
Eclipse
to
have
South
Latitude
1
½
minute
,
the
Moons
true
Latitude
was
then
by
Lantsbergius
his
Tables
45
minutes
,
24
seconds
North
;
so
that
by
this
accompt
the
Sun
and
Moons
Centers
were
distant
in
Latitude
46
minutes
,
54
seconds
:
but
when
out
of
this
distance
you
have
subtracted
46
⅓
for
the
Moons
Parallax
of
Latitude
,
there
remains
34
seconds
for
the
apparent
distance
of
the
Centers
of
the
Sun
and
Moon
.
But
by
Observation
,
I
found
them
distant
1
minute
48
seconds
:
for
the
digits
Eclipsed
at
Ecton
,
were
11.
22
½
minutes
;
and
so
perhaps
might
I
have
found
by
my
Planisphear
,
or
some
what
near
,
had
it
been
large
enough
,
and
had
I
regarded
every
minute
and
second
precisely
in
setting
down
this
Example
,
which
were
more
then
needed
for
my
purpose
in
this
place
.
CHAP.
LXV
.
To
find
the
Moons
Latitude
,
by
her
distance
from
either
of
the
Nodi
,
called
Caput
,
and
Caudi
Draconis
.
AS
the
Ecliptique
crosseth
the
Equator
with
an
angle
of
23.
30
minutes
for
our
Age
,
so
the
Orbite
or
Circle
in
which
the
Moon
moveth
crosseth
the
Ecliptique
:
but
the
angle
of
Inclination
is
not
alwaies
nor
long
the
same
:
for
in
the
Conjunctions
of
the
Sun
and
Moon
the
angle
is
ever
5
00
minutes
,
and
increaseth
to
the
time
of
the
Quadrature
,
when
it
is
found
5
degrees
16
minutes
;
thence
it
decreaseth
to
the
Opposition
,
where
it
is
again
but
5
degrees
,
as
in
the
Conjunction
;
thence
it
increaseth
again
to
5
degrees
16
minutes
in
the
latter
□
,
and
again
thence
decreaseth
to
5
degrees
in
the
☌
.
Get
by
the
Astronomical
Tables
the
quantity
of
the
angle
made
between
the
Ecliptique
and
the
Orbite
of
the
Moon
,
(
which
in
all
Conjunctions
and
Oppositions
,
and
therefore
in
all
Eclipses
is
5
degrees
,
as
was
now
said
)
and
get
also
by
the
like
Tables
the
Moons
distance
from
the
nearest
of
the
Nodes
:
then
may
you
find
the
Moons
Latitude
,
by
the
Probleme
3.
Rectangled
Triangles
,
Book
3.
5.
just
as
you
use
to
find
the
Suns
Declination
,
by
his
Longitude
and
greatest
Declination
.
Example
.
The
☽
in
the
former
Case
was
distant
from
Caput
Draconis
by
Lantsbergius
Tables
8
degrees
43
minutes
.
I
lay
the
Label
from
the
Equinoctial
line
to
5
degrees
in
the
Limb
,
and
counting
in
the
Label
from
the
Center
8.
43.
I
see
there
the
Parallel
of
0
¾
(
that
is
,
45
minutes
,
or
46
minutes
)
crossing
for
the
Moons
Latitude
.
CHAP.
LXVI
.
To
find
the
Dominical
Letter
,
the
Prime
,
Epact
,
Easter
day
,
and
the
rest
of
the
moveable
Feasts
for
ever
,
by
the
Calender
,
discribed
Book
1.
11.
AN
Example
,
shall
serve
here
instead
of
a
Rule
.
For
the
Year
1657.
I
would
know
all
these
:
wherefore
I
seek
the
Year
1657.
in
the
Table
of
the
Suns
Cycle
,
and
over
against
it
,
I
find
14.
for
the
Year
of
the
Cycle
of
the
Sun
,
and
D
for
the
Dominical
Letter
.
And
note
here
,
that
every
Leap-year
hath
2
Dominical
Letters
(
as
1660.
hath
A
G
)
and
the
first
(
viz.
A
)
serveth
that
Year
till
February
25
,
and
the
second
(
G
)
for
the
rest
of
the
Year
.
And
note
that
these
letters
go
alwayes
backwards
when
you
count
forwards
(
as
B
A
,
then
G
F
,
&c.
not
F
G
,
and
then
A
B
)
as
you
may
see
by
the
Table
.
Then
in
the
Table
of
the
Cycle
of
the
Moon
,
I
have
for
the
Year
1657.
the
Prime
5.
the
Epact
25.
Those
had
,
I
go
to
the
Table
for
Easter
,
and
seek
there
in
the
first
rank
the
Prime
5.
and
under
it
in
the
middle
rank
stands
E
;
that
is
not
my
Dominical
Letter
;
therefore
I
seek
not
backward
,
but
alwayes
forward
in
the
middle
rank
,
till
I
come
to
my
Dominical
Letter
D.
and
under
it
I
find
in
the
third
rank
March
29.
upon
which
Easter
day
falls
this
Year
1657.
The
rest
of
the
moveable
Feasts
may
be
had
by
their
distances
from
Easter
,
which
are
alwayes
the
same
.
Onely
for
Advent
Sunday
,
remember
that
the
next
Sunday
after
November
26
is
Advent
Sunday
.
Read
Book
1.
11.
and
that
will
sufficiently
instruct
you
with
this
Example
.
CHAP.
LXVII
.
To
find
the
age
of
the
Moon
,
by
the
Epact.
REmember
first
that
the
Epact
begins
with
March
,
which
must
be
here
accounted
the
first
Moneth
:
Then
if
you
add
to
the
Epact
the
number
of
the
Moneth
current
,
and
the
number
of
the
day
of
the
Moneth
current
,
the
sum
or
the
excess
above
30
,
is
the
Moons
age
.
Example
.
January
20.
1656.
According
to
the
accompt
of
the
Church
of
England
,
(
who
begin
the
Year
with
March
25.
which
was
the
Equinoctial
day
about
Christ
time
)
the
Epact
is
14.
January
is
the
11th
Moneth
,
and
the
20th
day
is
proposed
;
now
add
14
,
11
,
and
20.
together
,
they
make
45.
out
of
which
I
take
30.
and
there
remains
15.
the
Moons
age
.
This
Rule
is
of
good
use
,
not
onely
to
find
the
age
of
the
Moon
,
and
so
her
changes
to
a
day
,
but
also
for
examining
of
Chronologie
,
where
the
time
is
most
certainly
reckoned
by
Eclipses
.
But
you
must
note
,
that
if
you
apply
this
Rule
to
the
Years
past
before
Anno
Dom
,
1600.
then
for
every
312.
Years
that
the
Year
proposed
precedes
Anno
Dom.
1600.
you
must
subtract
one
day
out
of
the
age
of
the
Moon
,
found
by
this
Rule
.
Example
.
●icius
lib.
1.
Reports
,
That
in
the
beginning
of
Tiberius
Caesar's
reigne
there
was
an
Eclipse
of
the
Moon
,
and
Temporarius
saith
,
that
whereas
Augustus
died
Aug.
29.
(
I
think
he
should
say
19.
)
this
Eclipse
hapned
Sep.
27.
I
would
know
whether
it
were
possible
for
an
Eclipse
to
happen
that
day
,
supposing
the
beginning
of
Tiberius
to
be
in
August
,
Anno
Dom.
14.
and
Anno
Periodi
Julianae
4727.
The
Prime
for
that
Year
is
15
,
and
the
Epact
15.
by
Book
1.
11.
add
now
to
the
Epact
,
for
September
7.
and
for
the
day
of
the
Moneth
27.
and
the
sum
is
49.
out
of
which
subducting
30.
I
leave
the
Moons
age
19.
but
because
Anno
Dom.
14.
precedes
Anno
Dom.
1600.
5.
times
312.
Years
,
therefore
out
of
19
I
subduct
5.
and
there
remaines
14.
the
age
of
the
Moon
,
corrected
for
September
27.
Anno
Dom.
14.
Therefore
it
was
about
the
full
Moon
:
and
it
is
possible
the
Moon
might
be
Eclipsed
then
,
as
Temporarius
saith
.
But
it
could
not
be
Eclipsed
September
27.
Anno
Dom.
13.
for
then
the
Epact
being
4.
the
age
of
the
Moon
by
the
same
Rule
was
3.
neither
could
it
happen
Sep.
27.
Anno
Dom.
15.
for
then
by
the
same
Rule
the
age
of
the
Moon
,
was
25.
at
what
age
the
Moon
was
far
from
her
opposition
to
the
Sun
:
and
therefore
could
not
be
Eclipsed
.
CHAP.
LXVIII
.
To
find
in
what
Parallel
and
Climate
a
Place
is
,
by
the
Latitude
given
.
PArallels
in
Geography
are
lesser
Circles
Parallel
to
the
Equator
,
and
passing
through
the
Zenith
of
a
Place
,
and
succeeding
one
another
at
such
distance
that
at
every
Parallel
the
length
of
the
day
is
varyed
a
quarter
of
an
hour
.
A
Climate
is
such
a
Parallel
as
altereth
the
length
of
the
day
half
an
hour
.
The
Parallels
and
Climates
begin
from
the
Equator
,
under
which
the
day
is
alwayes
equal
to
the
Night
,
and
each
12
hours
long
:
hence
they
count
the
Parallels
and
Climates
Northward
,
and
Southward
:
but
because
the
Earth
was
not
so
far
known
to
Ptolomy
and
the
Ancient
Geographers
,
as
it
hath
been
to
those
of
later
Times
,
therefore
there
is
great
difference
between
the
Ancient
and
later
Geographers
about
the
number
and
quantity
of
the
spaces
contained
by
them
:
as
among
others
Kerkerman
Syst.
Geography
,
lib.
1.
hath
shewed
.
Yet
may
they
easily
be
found
to
every
Mans
mind
,
by
the
Planisphear
in
the
Meridional
Projection
,
thus
.
Find
by
4.
17.
what
is
the
Semi-diurnal
arch
of
the
Sun
in
♋
,
out
of
which
,
take
6
hours
,
and
look
how
many
quarters
of
an
hour
the
double
of
the
residue
containeth
,
so
many
Geographical
Parallels
is
the
place
removed
from
the
Equator
,
and
half
so
many
Climates
.
Example
.
I
find
the
Semi-diurnal
arch
in
our
Latitude
to
be
8
hours
16
minutes
,
in
the
Tropique
of
Cancer
;
out
of
which
taking
6.
and
doubling
the
residue
,
I
have
4.
33.
which
is
more
then
9
half
hours
,
or
more
then
18.
quarters
:
so
much
our
longest
day
exceeds
12
hours
;
therefore
we
should
be
past
the
18th
Parallel
,
and
9th
Climate
,
viz.
in
the
beginning
of
the
10th
Climate
,
and
19th
Parallel
.
CHAP.
LXIX
.
The
Longitude
and
Latitude
of
two
Places
given
,
to
find
their
Distance
.
WHat
Longitude
and
Latitude
in
Geography
are
,
and
how
they
differ
from
Longitude
and
Latitude
in
Astronomy
,
hath
been
shewed
,
Book
4
,
5.11
.
If
the
places
differ
only
in
Latitude
,
and
have
one
Longitude
,
bearing
full
North
or
South
one
from
another
;
then
take
their
difference
of
Latitude
,
by
subducting
the
less
out
of
the
greater
,
if
the
places
have
both
North
Latitude
,
or
both
South
Latitude
:
or
take
the
sum
of
their
Latitude
,
if
one
be
North
,
and
the
other
South
.
Then
for
every
degree
of
this
difference
or
aggregate
number
,
you
may
reckon
69
½
English
.
miles
of
the
Statute
,
which
ordaineth
1760.
Yards
to
be
a
Mile
:
but
of
English
Miles
measured
by
common
estimation
,
there
go
not
above
60.
to
a
degree
;
so
that
every
such
Mile
that
you
Travel
North
or
South
shall
alter
your
Latitude
about
one
minute
.
If
they
differ
in
Longitude
onely
,
and
have
no
Latitude
,
but
be
both
under
the
Equator
,
you
shall
reckon
in
like
manner
for
every
degree
they
differ
in
Longitude
69
½
Miles
of
distance
.
In
all
other
Cases
you
have
a
Triangle
soluble
by
the
second
Probleme
of
Obliquangled
Triangles
:
of
which
Triangle
the
Complements
of
Latitude
make
the
two
comprehending
sides
,
and
the
difference
of
the
Longitudes
of
the
places
is
the
angle
comprehended
between
them
,
and
the
third
side
is
the
arch
of
the
distance
of
the
places
;
which
when
you
have
found
in
degrees
and
minutes
of
a
great
Circle
,
you
may
turn
into
Miles
as
before
:
mark
how
the
distance
of
two
Stars
is
found
by
their
Longitude
and
Latitude
given
,
Chapter
37.
in
the
same
manner
may
you
find
the
distance
of
two
Cities
or
Towns
.
Example
.
I
would
know
the
distance
of
London
from
Jerusalem
,
The
Complement
of
the
Latitude
of
London
is
38.
28
minutes
:
the
Complement
of
the
Latitude
of
Jerusalem
is
58.
05
minutes
:
the
difference
of
their
Longitudes
46.
0
minutes
.
I
set
the
Zenith
to
the
Latitude
of
London
in
the
Limb
,
that
is
38.
28
minutes
from
the
Pole
,
so
the
Limb
is
the
Circle
of
Longitude
in
which
London
standeth
,
then
I
seek
the
46
Meridian
from
that
side
of
the
Limb
where
Zenith
is
set
for
London
,
for
that
46
Méridian
is
the
Circle
of
Jerusalems
Longitude
:
(
because
the
difference
of
Longitude
is
46.
)
Now
because
Jerusalems
Latitude
is
31.
55.
and
the
Complement
thereof
,
or
distance
from
the
Pole
58.
5
minutes
,
I
walk
on
in
the
46
Meridian
till
I
come
where
the
58th
Parallel
from
the
Pole
crosseth
him
,
and
there
is
the
place
of
Jerusalem
:
the
Azimuth
that
goes
hence
to
the
Zenith
is
the
nearest
way
from
Jerusalem
to
London
:
what
Azimuth
this
is
I
regard
not
,
for
I
enquire
not
the
angle
at
London
,
but
I
observe
by
the
Parallels
how
many
degrees
there
be
in
him
between
the
places
of
Jerusalem
and
the
Zenith
,
and
I
find
38
degrees
20
minutes
:
which
being
resolved
into
Miles
is
2300.
Miles
of
common
estimation
,
but
Miles
of
the
Statute
2664.
the
distance
of
London
from
Jerusalem
.
CHAP.
LXX
.
The
Latitude
and
distance
of
two
Places
given
,
to
find
the
difference
of
Longitude
.
THe
Triangle
will
stand
as
in
the
former
Chapter
:
there
by
two
sides
and
the
angle
comprehended
you
sought
the
third
side
,
by
Probleme
2.
Obliquangled
Triangles
:
here
by
three
sides
given
,
you
seek
an
angle
by
Probleme
1
Obliquangled
Triangles
.
Make
the
Pole
,
Pole
:
and
set
the
Zenith
to
the
Latitude
of
one
of
the
places
,
as
you
did
London
.
(
Chapter
69.
)
38.
28
minutes
from
the
Pole
,
then
number
the
Complement
of
the
Latitude
of
the
other
place
from
the
Pole
,
by
the
Parallels
,
and
the
distance
of
the
two
places
from
the
Zenith
by
the
Almicantars
;
and
where
the
last
Parallel
and
last
Almicantar
meet
is
C
of
your
Triangle
:
(
see
Book
3.
14.
)
Now
count
how
many
Meridians
there
be
between
C
and
the
Limb
,
so
many
degrees
is
the
angle
at
the
Pole
sought
,
for
the
difference
of
Longitude
.
Example
.
Having
the
distance
of
London
from
the
Pole
38.
28
minutes
,
and
of
Jerusalem
from
the
Pole
58.
5
minutes
,
and
the
distance
of
London
from
Jerusalem
2300.
common
English
Miles
,
(
of
which
60.
make
a
degree
)
I
set
the
Zenith
for
London
38.
28
minutes
from
the
Pole
in
the
Limb
,
then
because
Jerusalem
is
distant
from
the
Pole
58.
5
minutes
,
I
go
to
the
58th
Parallel
from
the
Pole
,
and
lay
one
finger
or
the
point
of
a
bodkin
on
him
;
and
because
London
is
distant
from
Jerusalem
38
degrees
20
minutes
,
I
count
from
the
Zenith
to
the
Almicantar
38.
20
minutes
;
now
where
this
Almicantar
crosseth
the
Parallel
last
found
,
there
is
C
of
the
Triangle
,
and
the
place
of
Jerusalem
:
and
you
may
see
that
you
must
cross
46.
Meridians
before
you
can
go
thence
to
the
Zenith
in
the
Limb
;
which
sheweth
that
the
angle
at
the
Pole
for
the
difference
of
Longitude
is
46.
CHAP.
LXXI
.
To
find
what
degree
of
the
Ecliptique
Culminates
in
another
Country
,
at
any
time
proposed
,
if
the
difference
of
Longitude
be
known
.
IN
the
Equinoctial
Projection
,
Bring
the
Suns
place
to
the
hour
proposed
,
by
help
of
the
Label
:
and
in
the
Noon-line
you
see
presently
what
degree
Culminates
in
your
Country
.
(
as
Chapter
46
,
is
shewed
.
)
Now
to
know
this
for
another
Town
,
set
the
Label
so
many
degrees
from
the
Noon-line
as
the
difference
of
Longitude
requires
,
and
that
Eastward
,
if
the
place
proposed
be
East
,
or
Westward
if
it
bear
West
;
and
so
the
Label
shall
cut
the
degree
of
Culmination
,
for
the
place
proposed
.
Example
.
If
it
be
demanded
what
degree
is
Culminating
at
Jerusalem
March
10.
at
10.
a
clock
before
noon
,
I
will
set
the
Suns
place
♈
0.
to
the
hour
;
and
I
see
upon
the
Noon
line
,
which
is
our
Meridian
,
there
Culminates
♒
28
almost
.
Now
for
the
Meridian
of
Jerusalem
I
must
lay
the
Label
46
degrees
Eastward
,
that
is
from
Meridies
towards
Oriens
,
and
look
what
Star
or
degree
of
the
Ecliptique
is
then
cut
by
the
Label
,
that
is
then
Culminating
in
the
Meridian
at
Jerusalem
;
(
as
here
I
find
♈
17
½
.
)
for
in
this
Projection
the
Label
(
lay
him
where
you
will
)
is
a
Meridian
.
CHAP.
LXXII
.
To
find
what
a
Clock
it
is
in
another
Country
,
by
knowing
the
hour
at
Home
,
and
the
difference
of
Longitude
.
THis
is
done
easily
enough
without
an
Instrument
:
for
if
you
turn
the
difference
of
Longitude
into
hours
and
minutes
,
and
add
the
same
to
your
hours
for
any
place
which
lies
Eastward
,
or
subtract
the
same
for
any
place
which
lies
Westward
,
you
shall
make
the
hour
of
the
place
.
Example
.
The
difference
of
Longitude
between
London
and
Jerusalem
is
46.
or
being
converted
into
time
3
hours
4
minutes
:
therefore
adding
this
to
the
time
at
London
,
I
say
,
when
it
is
noon
at
London
,
it
is
4
minutes
past
3
a
clock
after
noon
at
Jerusalem
:
and
when
it
is
2
a
clock
at
London
,
it
is
5.
and
4
minutes
at
Jerusalem
.
If
you
will
do
it
by
the
Planisphear
,
you
shall
do
it
in
the
Equinoctial
Projection
,
thus
.
Whereas
the
Limb
of
your
Rect
is
graduated
into
360
degrees
,
if
you
distinguish
the
hours
also
at
every
15th
degree
,
beginning
at
the
Zenith
(
which
shall
be
12
)
and
numbring
thence
in
the
Limb
of
your
Reet
to
your
right
hand
or
Westward
,
1
,
2
,
3
,
&c.
then
shall
you
need
to
do
no
more
but
set
the
Zenith
to
the
difference
of
Longitude
,
East
or
West
from
your
Meridian
,
as
the
strange
place
happeneth
to
be
situate
:
for
then
the
Label
laid
to
the
hour
of
your
Country
in
the
Limb
of
the
Mater
,
shall
shew
the
hour
of
the
other
Country
in
the
Limb
of
the
Reet
.
And
so
the
Zenith
being
laid
to
60
degrees
Westward
,
which
is
the
Meridian
of
the
Isle
of
Barbados
the
Label
laid
to
Meridies
shall
cut
in
the
Limb
of
the
Reet
8
of
the
clock
before
noon
:
which
sheweth
me
that
when
it
is
noon
with
us
,
it
is
at
Barbados
but
eight
in
the
morning
.
The
end
of
the
Fourth
Book
.
CHAP.
LXXIII
.
The
Longitude
and
Latitude
of
one
Place
known
,
and
the
Rumb
and
distance
of
a
second
Place
,
to
find
both
the
Longitude
and
Latitude
of
the
second
Place
.
SEt
the
Zenith
to
the
Latitude
of
the
first
Place
,
then
seek
the
Azimuth
which
serveth
for
the
Rumb
of
the
second
Place
,
and
in
that
Azimuth
count
his
distance
from
the
Zenith
:
where
this
distance
ends
there
is
the
second
Place
,
whole
Latitude
is
shewn
you
by
the
Parallel
which
cutteth
him
,
and
the
Meridian
cutting
there
also
,
shews
his
Longitude
.
diagram
Example
.
Let
Z
be
London
,
and
because
Jerusalem
beareth
from
London
almost
S
b
E
or
77
½
from
South
Eastward
,
therefore
I
choose
the
Azimuth
77
½
Z
K.
therein
I
number
Jerusalems
distance
from
London
Z
I
2300.
miles
or
minutes
,
that
is
38.
20.
minutes
.
Now
in
the
Triangle
Z
P
I
,
I
may
find
P
I
the
complement
of
Jerusalems
Latitude
58.
05.
minutes
,
and
Z
P
I
the
difference
of
Longitude
46
,
which
must
be
added
to
the
Longitude
of
London
to
make
the
Longitude
of
Jerusalem
.
CHAP.
LXXIV
.
The
Latitudes
and
Distance
of
two
Places
given
,
to
find
the
Rumb
,
and
the
difference
of
Longitude
.
COunt
in
the
Meridian
from
P
(
the
Pole
)
the
complement
of
the
Latitude
of
the
first
place
,
and
thereto
set
Z
the
Zenith
.
Count
also
from
P
the
complement
of
the
Latitude
of
the
second
place
,
and
lay
your
finger
on
the
Parallel
at
which
your
number
ends
.
Count
also
from
Z
the
distance
of
the
places
in
Degrees
and
Minutes
,
and
note
the
Almicantar
at
which
this
number
ends
:
where
this
Almicantar
crosseth
the
aforesaid
Parallel
there
is
C
of
your
Triangle
;
(
but
here
marked
I
,
)
Look
what
Azimuth
cutteth
here
,
it
sheweth
the
Rumb
:
and
the
Meridian
that
cutteth
here
,
(
if
you
count
his
distance
from
the
Limb
)
shews
the
difference
of
the
Longitude
of
the
places
.
This
is
so
plain
from
Chapter
69
,
70
,
and
73
,
that
it
needeth
no
example
.
The
same
Scheam
serveth
these
4
,
Chapters
,
The
end
of
the
Fourth
Book
.
The
Fifth
Book
.
Shewing
the
way
to
resolve
all
GNOMONICAL
PROBLEMES
;
And
to
make
all
sorts
of
SVN
DYALS
,
very
easily
by
the
PLANISPHEAR
.
CHAP.
I.
The
Preface
.
Of
the
kinds
of
Dyals
.
ALthough
Gnomoniques
pertain
to
Astronomy
,
yet
I
think
it
not
amiss
,
for
the
ease
of
the
Reader
in
finding
them
,
to
place
the
Gnomonical
Problemes
in
a
distinct
Book
by
themselves
.
Suns
Dyals
may
be
reduced
to
two
sorts
.
Some
shew
the
hour
by
the
Altitude
of
the
Sun
,
as
Quadrants
,
Rings
,
Cylinders
,
&c.
for
the
making
whereof
you
must
know
the
Suns
Altitudes
for
every
day
,
or
at
least
every
10th
day
of
the
year
,
and
for
every
hour
of
those
dayes
:
which
Altitudes
you
may
find
immediately
upon
this
Planisphear
,
as
in
a
Table
made
to
your
hand
,
for
any
Latitude
,
by
Book
3.25
.
and
so
make
them
of
any
shape
according
to
your
mind
.
The
other
sort
shew
the
hour
by
the
shadow
of
a
Gnomon
,
or
Style
,
Parallel
to
the
Axis
of
the
World
:
and
of
those
I
treat
cheifly
in
this
Book
.
Those
be
all
Projections
of
the
Sphear
upon
a
plain
which
lies
Parallel
to
some
Horizon
or
other
in
the
World
.
And
if
upon
such
a
plain
the
Meridians
onely
be
projected
,
they
shall
suffice
to
shew
the
hour
,
without
projecting
the
other
Circles
,
as
the
Ecliptique
,
the
Equator
with
his
Parallels
of
Declination
,
the
Horizon
with
his
Almicantars
and
Azimuths
,
which
are
sometimes
drawn
upon
Dyals
,
more
for
ornament
,
then
for-necessity
.
CHAP.
II.
Theorems
premised
.
FOr
the
better
understanding
of
the
reason
of
Dyals
,
these
Theorems
would
be
known
.
-
1.
That
every
plain
whereupon
any
Dyal
is
drawn
,
is
part
of
the
plain
of
great
Circle
of
the
Heaven
:
which
Circle
is
an
Horizon
to
some
Country
or
other
:
that
the
Center
of
the
Dyal
represents
the
Center
of
the
Earth
and
World
,
and
the
Gnomon
which
casteth
the
shade
representeth
the
Axis
,
and
ought
to
point
directly
to
the
two
Poles
.
And
if
upon
the
Center
of
the
Dyal
you
fasten
a
Label
with
Sights
of
equal
Altitude
,
and
keeping
your
eye
in
the
line
of
the
Sights
turn
this
Label
round
,
you
shall
thereby
describe
in
the
Heavens
that
great
Circle
wherein
your
Dyal-plain
lies
,
and
see
where
it
cuts
our
Horizon
,
and
how
much
it
is
Elevated
above
it
on
one
side
,
and
depressed
on
the
other
.
-
2.
That
those
Dyal-plains
Geometrically
are
not
in
the
very
plains
of
great
Circles
;
for
then
they
should
have
their
Centers
in
the
Center
of
the
Earth
,
from
which
they
are
removed
almost
4000.
Miles
:
and
in
truth
they
lie
in
the
plains
of
Circles
Parallel
to
the
said
Horizons
,
but
so
near
them
,
that
Optically
they
seem
to
be
the
plains
of
those
Horizons
:
because
the
Semidiameter
of
the
Earth
beareth
so
smal
proportion
to
the
Suns
distance
,
that
the
whole
Earth
may
be
taken
for
one
point
or
Center
,
without
any
perceivable
error
.
-
3.
That
(
as
all
great
Circles
of
the
Sphear
,
so
)
every
Dyal-plain
hath
his
Axis
,
which
is
a
straight
line
passing
through
the
Center
of
the
plain
,
and
making
right
angles
with
it
:
and
at
the
ends
of
the
Axis
be
the
two
Poles
of
the
plain
,
whereof
that
above
our
Horizon
is
called
the
Pole
Zenith
,
and
the
other
the
Pole
Nadir
of
the
Dyal
.
-
4.
That
every
Dyal-plain
hath
two
faces
or
sides
:
and
look
what
respect
or
situation
the
North
Pole
of
the
World
hath
to
the
one
side
,
the
same
hath
the
South
Pole
to
the
other
,
and
these
two
sides
will
alwayes
receive
24
hours
;
so
that
what
one
side
wanteth
,
the
other
side
shall
have
;
and
the
one
is
described
in
all
things
as
the
other
.
-
5.
That
(
as
Horizons
,
so
)
Dyal-plains
are
with
respect
to
the
Equator
divided
into
1.
Parallel
or
Equinoctial
,
2.
Right
,
3.
Oblique
.
-
6.
A
Parallel
or
Polar
Dyal-plain
maketh
no
angles
with
the
Equator
:
but
lies
in
the
plain
of
it
,
or
Parallel
to
it
.
Such
Dials
are
Scioterica
Orthognomonica
,
that
is
,
have
the
Gnomon
erected
on
the
plain
at
Right
angles
,
as
the
Axis
of
the
World
is
upon
the
plain
of
the
Equator
:
because
the
Axis
and
Poles
of
the
Dyal
be
here
all
one
with
the
Axis
and
Poles
of
the
World
:
and
the
hour
lines
here
meet
all
at
the
Center
,
making
equal
angles
,
and
dividing
the
Dyal
Circle
into
24.
equal
parts
as
the
Meridians
do
the
Equator
,
in
whose
plain
the
Dyal
lies
.
-
7.
A
right
Horizon
or
Dyal-plain
cutteth
the
Equator
at
right
angles
,
and
so
cutteth
through
both
the
Poles
of
the
World
.
Therefore
such
Dyals
are
Paralielognomonical
:
that
is
have
the
Gnomon
Parallel
to
the
plain
,
and
so
the
hour
lines
and
the
hour
lines
all
Parallel
one
to
another
:
because
their
plains
though
infinitely
extended
will
never
cut
the
Axis
of
the
World
.
Yet
have
those
Dyals
a
Center
,
(
though
not
for
the
meeting
of
the
hour
lines
)
viz.
through
which
the
Axis
of
the
Dyal
Circle
passeth
,
cutting
the
plain
at
right
angles
,
and
cutting
also
(
near
enough
for
the
projecting
of
a
Dyal
)
the
Center
of
the
World
.
-
8.
An
Oblique
Horizon
or
Dyal-plain
cutteth
the
Equator
at
Oblique
angles
:
such
Dyals
are
Scalenognomonical
:
that
is
,
have
for
their
Gnomon
the
side
of
a
Triangle
whose
angles
vary
according
to
the
more
or
less
Obliquity
of
the
said
Horizon
:
and
the
Gnomon
shall
alwayes
make
an
angle
with
the
plain
of
so
many
degrees
as
the
Axis
of
the
World
maketh
with
the
plain
;
or
as
either
of
the
Poles
of
the
World
is
Elevated
above
the
plain
.
-
9.
Every
Oblique
Horizon
is
divided
by
the
Meridians
or
Hour
Circles
of
the
Sphear
into
24.
unequal
parts
:
which
parts
,
are
alwayes
lesser
as
they
are
scarer
to
the
Meridian
of
that
Horizon
or
plain
,
and
greater
as
they
are
further
off
:
and
on
both
sides
the
Meridian
of
the
plain
the
hour
Circles
which
are
equally
distant
in
Time
,
are
also
equally
distant
in
Space
:
whence
it
is
that
the
divisions
of
one
Quadrant
of
your
Dyal
plain
being
known
,
the
division
of
the
whole
Circle
is
likewise
known
.
-
10.
The
Hour-lines
in
an
Oblique
Dyal
are
the
Sections
of
the
plains
of
the
Hour-circles
of
the
Sphear
with
the
Dial
plain
.
And
because
the
plains
of
great
Circles
do
alwayes
cut
one
another
diagram
in
halfs
by
Diameters
,
which
are
straight
lines
passing
through
the
common
Center
,
therefore
lines
drawn
from
the
Center
of
the
Dyal
to
the
Intersections
of
the
Hour-circles
with
the
great
Circle
of
the
plain
,
shall
be
those
very
Sections
,
and
the
very
Hour-lines
of
the
Dyal
.
-
11.
Every
Dyal-plain
(
being
an
Horizon
to
some
place
in
the
Earth
,
as
was
said
Theorem
1.
)
hath
his
proper
Meridian
;
which
is
the
Meridian
cutting
through
the
●oles
of
the
plain
,
and
making
Right
angles
with
the
plain
.
If
the
Poles
of
the
Dyal-plain
lie
in
the
Meridian
of
our
place
;
then
is
the
Meridian
of
the
plain
all
one
with
the
Meridian
of
the
place
;
and
the
Gnomon
or
Style
shall
stand
erected
upon
the
Noon-line
,
or
line
of
12
a
clock
,
as
in
all
direct
Dyals
:
but
if
the
plain
decline
,
then
shall
the
substylar
or
line
wherein
the
Gnomon
standeth
,
which
is
the
Meridian
of
the
plain
,
vary
from
the
Noon-line
,
which
is
the
Meridian
of
the
place
:
and
this
variation
shall
be
East
if
the
Declination
of
the
plain
be
West
,
and
contrarily
:
because
the
visual
lines
by
which
the
Sphear
is
projected
on
Dyal-plains
,
do
all
like
the
beams
of
a
Burning-glass
intersect
or
cross
one
another
in
a
certain
point
of
the
Gnomon
(
to
be
assigned
at
pleasure
,
and
called
Nodus
)
and
so
do
all
place
and
depaint
themselves
on
the
Dyal-plain
beyond
the
Nodus
the
contrary
way
.
-
12.
Dyals
are
most
aptly
denominated
from
that
part
of
the
Sphear
where
their
Poles
lie
:
though
some
Authors
have
chosen
to
denominate
them
from
the
Circles
in
which
their
plains
lie
:
as
the
Dyal-plain
which
lieth
in
the
Equinoctial
or
Parallel
to
it
,
is
called
by
many
an
Equinoctial
plain
:
but
I
concur
with
those
who
would
rather
call
it
a
Polar-plain
;
because
the
Poles
thereof
are
in
the
Poles
of
the
World
.
CHAP.
III.
How
to
draw
an
Horizontal
or
Vertical
line
,
upon
any
plain
.
BEcause
in
the
Delineation
of
most
Dyals
the
Horizontal
or
the
Vertical
line
of
the
plain
must
first
be
drawn
,
before
you
can
place
the
hour-lines
;
I
will
shew
you
first
how
to
draw
either
of
them
.
Know
this
first
that
they
cross
one
another
at
Right
angles
in
the
Center
of
the
Dyal
:
and
therefore
if
you
can
draw
either
of
them
,
you
may
draw
the
other
also
.
Also
in
upright
plains
as
are
Walls
a
Quadrant
or
a
Square
with
a
Plumbet
applied
to
the
Wall
,
will
shew
you
how
to
draw
both
of
them
very
easily
.
Or
if
you
hang
a
Plumb-line
quiet
before
the
Wall
when
the
Sun
shineth
,
the
shadow
of
it
shall
be
a
Vertical
line
at
any
time
.
But
if
the
plain
incline
or
recline
,
you
shall
set
to
it
a
Square
with
a
Plumbet
,
and
thereby
first
draw
the
Horizontal
line
;
for
when
the
Plumbet
(
which
must
play
in
an
hole
)
hangs
Parallel
to
one
side
of
the
Square
,
a
line
drawn
by
the
other
shall
be
Horizontal
:
that
drawn
,
you
shall
lay
your
Square
flat
on
the
plain
,
and
draw
the
Vertical
from
any
point
of
the
Horizontal
at
Right
angles
by
your
Square
Another
way
is
this
.
Hang
your
Planisphear
by
the
handle
with
his
Plumbline
,
so
that
the
Plumbline
fall
upon
one
of
the
Diameters
;
then
setting
your
Label
and
Sights
to
the
other
Diameter
,
look
through
the
Sights
,
and
mark
where
the
visuall
line
cuts
the
plain
neer
to
one
side
,
and
there
make
a
prick
,
then
direct
your
Sights
to
the
other
side
of
the
plain
,
and
make
another
prick
,
(
your
Label
and
Plumbline
being
still
at
Right
angles
as
before
)
by
these
two
pricks
draw
a
line
,
and
it
shall
be
an
Horizontal
line
.
And
note
,
that
all
lines
Parallel
to
an
Horizontal
line
be
Horizontal
:
and
all
lines
Parallel
to
a
Vertical
line
be
also
Vertical
.
CHAP.
IV.
How
to
make
the
Polar
Dyal
,
and
how
to
place
it
.
THe
plain
of
the
Polar
Dyal
lieth
in
the
Equinoctial
:
where
the
12
cheife
Meridians
or
Hour-circles
divide
both
the
Equinoctial
and
this
plain
into
24
equal
parts
.
The
Gnomon
stands
upon
the
Center
at
Right
angles
with
the
plain
.
You
may
learn
to
make
him
only
by
the
6.
Theorem
of
the
second
Chapter
.
Take
your
Planisphear
in
the
Equinoctial
Projection
,
and
there
is
your
Dyal
ready
made
on
the
Limb
,
and
the
Hours
already
marked
.
Erect
now
a
wyer
or
thred
Perpendicular
upon
the
Center
,
or
hold
a
Square
to
the
Center
,
so
that
his
top
be
equally
distant
from
all
the
parts
of
the
Circle
:
and
there
is
your
Gnomon
placed
.
To
place
this
Dyal
do
thus
.
Having
(
by
Book
4.3
.
)
found
a
Meridian
line
,
if
you
cross
it
with
another
line
,
that
shall
be
an
East
or
West
line
.
Have
also
in
a
readiness
a
square
board
upon
which
you
may
fasten
your
Planisphear
or
Dyal-plate
with
pins
,
screws
,
or
wax
,
so
that
the
Noon-line
may
be
Parallel
to
two
sides
of
the
board
,
and
the
East-line
to
the
other
two
:
then
set
the
North
side
of
the
board
in
the
East
line
,
even
now
found
,
and
raise
the
South
side
by
a
Quadrant
to
the
height
of
the
Equinoctial
;
and
so
may
you
place
your
Dyal
in
any
Window
if
it
be
made
upon
a
loose
round
plate
;
but
if
the
plate
be
square
,
you
need
not
the
board
to
place
it
.
Note
that
both
faces
of
this
Dyal
must
be
divided
,
and
the
Gnomon
must
appear
on
both
sides
,
like
the
stick
in
a
Purre
(
or
whirligig
)
which
Children
use
:
otherwise
you
must
turn
him
upside
down
,
as
oft
as
the
Sun
passeth
the
Equinoctial
.
CHAP.
V.
How
to
make
the
South
Equinoctial
Dyal
,
or
Parallelognomonical
Dyal
direct
.
THe
Equinoctial
Dyal
we
call
that
which
hath
his
Poles
in
the
Equinoctial
Circle
:
of
which
there
be
three
kinds
.
-
1.
The
direct
or
South
Equinoctial
Dyal
,
which
faceth
the
Meridian
directly
,
not
looking
from
him
to
one
side
more
then
to
the
other
,
having
his
Poles
in
the
Intersections
of
the
Equinoctial
and
Meridian
.
-
2.
The
East
or
West
Equinoctial
Dyal
,
which
may
also
be
called
the
Equinoctial
Horizontal
Dyal
:
for
an
Horizontal
Dyal
declining
just
90
degrees
from
the
South
or
North
,
becomes
an
Equinoctial
Dyal
,
as
well
as
Horizontal
:
because
there
his
Poles
light
upon
the
Intersection
of
the
Horizon
with
the
Equinoctial
.
And
though
this
Dyal
be
of
kin
to
both
,
yet
his
Gnomon
shews
that
he
should
be
sorted
rather
with
the
Equinoctial
Dyals
then
with
the
Horizontal
:
for
he
is
Parallelognomonical
:
these
two
sorts
be
regular
,
having
their
Poles
in
the
four
notablest
points
of
the
Equator
:
the
third
is
somewhat
Irregular
,
but
may
be
brought
to
Rule
.
-
3.
The
Equinoctial
Dyal
declining
,
whose
Poles
happen
anywhere
else
in
the
Equator
between
the
Horizon
and
Meridian
.
Now
to
make
the
first
of
these
,
the
South
Equinoctial
Dyal
,
draw
first
an
Horizontal-line
upon
the
plain
,
and
cross
it
with
a
Vertical-line
,
by
Chapter
3.
The
Intersection
of
these
is
the
Center
of
your
Dyal
,
and
would
be
chosen
about
the
middle
of
the
plain
.
Now
your
Planisphear
being
fastned
to
a
square
board
,
as
in
the
former
Chapter
,
you
shall
set
the
North
side
of
the
said
board
in
the
Horizontal-line
of
the
plain
,
so
that
the
Axis
or
East
line
of
your
Planisphear
may
be
Parallel
thereto
,
and
the
Noon-line
(
or
Equator
line
of
the
Mater
)
may
point
directly
to
the
Center
,
making
Right
angles
with
the
plain
,
or
at
least
with
the
Horizontal
line
thereof
:
(
for
it
is
not
material
whether
the
board
be
upright
,
slope
,
or
lie
flat
upon
the
Dial
plain
)
then
placing
your
Sights
first
in
the
Noon
line
,
they
shall
point
to
the
Center
for
the
point
of
12.
thence
remove
your
Label
to
1.
of
the
diagram
clock
in
the
Limb
,
and
they
shall
point
out
in
the
Horizontal
line
of
the
plain
,
the
point
of
Intersection
for
one
a
clock
,
where
you
shall
make
a
prick
:
In
like
manner
remove
your
Label
to
1
,
2
,
3
,
4
,
and
5.
in
order
,
making
pricks
in
like
manner
▪
When
you
remove
your
Label
to
the
Axtree
line
for
6
you
shall
find
that
the
line
of
the
Sights
maketh
no
Intersection
with
the
plain
,
but
runneth
Parallel
to
it
;
because
the
Sun
is
then
in
the
Horizon
of
this
Dyal
,
where
he
projects
the
shadows
of
all
upright
things
infinite
.
And
as
you
found
the
points
for
the
hours
afternoon
,
so
may
you
by
like
reason
find
the
points
for
the
5.
morning
hours
,
and
their
quarters
also
,
if
you
please
:
which
had
,
if
by
these
points
found
you
draw
lines
Parallel
to
the
Vertical
line
of
the
plain
(
which
is
here
the
Meridian
of
the
plain
,
and
of
the
place
)
they
shall
be
the
true
hour-lines
.
And
the
Gnomons
edge
must
stand
over
the
Meridian
,
and
Parallel
to
it
,
at
the
same
distance
that
the
Axtree
line
of
your
Planisphear
was
situate
in
projecting
the
hour
points
.
If
you
cannot
fix
your
Planisphear
on
a
board
,
as
abovesaid
,
or
if
your
plain
require
a
Gnomon
of
a
greater
or
lesser
height
,
you
may
upon
any
board
presently
draw
so
much
of
your
Planisphear
as
serves
for
this
purpose
.
Or
to
say
more
breifly
,
Do
but
make
the
height
of
your
Gnomon
Radius
,
and
the
Tangents
of
15
,
30
,
45
,
60
,
75.
shall
give
the
distances
of
the
hour
points
in
the
Horizontal
line
,
on
both
sides
,
from
the
Center
of
the
Dyal
,
as
may
appear
in
the
Figure
.
CHAP.
VI.
How
to
make
the
East
Equinoctial
Dyal
,
or
the
West
.
THis
plain
is
a
Right
Horizon
of
those
People
who
dwell
under
the
Equator
,
distant
from
us
90
degrees
of
Longitude
;
as
the
South
Equinoctial-plain
of
the
last
Chapter
was
the
Horizon
of
those
who
dwel
under
the
Equator
,
in
the
same
Longitude
with
us
.
Therefore
these
Dyals
are
in
all
points
alike
:
onely
the
Substylar-line
which
in
the
South
Equinoctial
Dyal
is
12.
in
this
East
Equinoctial
Dyal
is
but
6.
in
the
Morning
for
our
Country
,
because
of
the
difference
of
Longitude
.
diagram
Lastly
,
draw
a
line
Parallel
to
the
Contingent
line
at
such
distance
as
the
plain
will
afford
,
as
the
line
E
I
,
and
to
this
you
shall
protract
your
hour
lines
,
drawing
them
from
every
point
of
the
Contingent
to
this
;
so
that
they
make
Right
angles
with
the
Contingent
and
with
this
Parallel
,
even
as
the
rounds
of
a
Ladder
do
with
the
sides
,
but
that
the
distance
of
the
rounds
of
a
Ladder
are
equal
,
and
these
distances
be
unequal
.
The
Gnomon
must
be
set
like
a
Bridge
Perpendicularly
over
the
6.
a
clock
hour-line
,
the
edge
that
casteth
shadow
being
Parallel
to
it
,
and
of
such
height
as
the
line
K
G
of
your
Planisphear
,
or
so
that
if
the
Gnomon
fall
,
his
edge
may
lie
in
the
line
of
3.
or
of
9.
of
the
clock
.
This
also
may
be
made
speedily
by
help
of
the
Tangents
,
as
the
South
Equinoctial
Dyal
.
For
the
West
Equinoctial
Dyal
,
it
is
made
like
the
East
in
all
points
;
onely
it
shews
but
the
after
Noon
hours
,
as
the
East
shews
but
those
of
the
fore
noon
.
When
you
have
drawn
on
paper
the
East
Dyal
,
and
set
it
by
guess
in
its
Situation
,
go
on
the
West
side
of
it
,
and
you
may
see
through
the
paper
the
picture
of
the
West
Dyal
,
and
so
will
the
back
side
of
the
West
Dyal
shew
you
the
true
picture
of
the
East
.
CHAP.
VII
.
How
to
make
the
Declining
Equinoctial
Dyal
.
ANy
Declining
plain
may
be
so
Reclined
that
he
shall
become
a
Right
Horizon
or
Equinoctial
plain
,
and
at
what
Reclination
this
shall
happen
you
may
easily
find
by
Chapter
19.
Latitude
52.
10
minutes
Declination
50.
00.
West
Reclination
26.32
.
D.
L.
43.16
.
Ascension
of
Noon-liee
61.59
diagram
Set
the
line
M
E
N
for
the
Axis
of
the
World
or
Gnomon
,
and
prop
him
up
over
the
line
P
Z
with
two
props
of
equal
height
,
and
Perpendicular
to
the
plain
,
and
make
the
point
E
(
which
standeth
Perpendicularly
over
Z
)
the
Center
of
the
World
;
then
from
this
Axis
or
Gnomon
mounted
in
the
Air
,
shall
the
hour-lines
be
projected
distinctly
,
and
all
of
them
shall
be
Parallel
to
the
Axis
and
one
to
another
,
as
it
hapneth
in
all
sorts
of
Equinoctial
Dyals
.
The
line
Z
P
shall
remain
now
onely
the
Meridian
of
the
plain
or
Substylar
.
And
to
find
the
hour-lines
you
shall
do
thus
.
Draw
through
Z
an
Equinoctial
or
Contingent
line
E
Z
,
making
Right
angles
with
the
Axis
M
E
N
or
Z
P
,
then
setting
Z
O
equal
to
Z
E
,
draw
upon
the
Center
O
(
with
an
extension
of
the
Compasses
,
)
the
arch
of
the
Equinoctial
b
Z
d
,
or
the
Parallel
arch
passing
by
D.
Then
number
in
this
arch
from
the
Substyle
(
Westward
if
your
plain
decline
West
,
or
Eastward
if
it
decline
East
)
the
difference
of
Longitude
,
and
where
it
ends
there
is
the
point
of
Noon
in
this
arch
:
from
that
point
begin
to
divide
the
said
arch
by
fifteens
of
degrees
,
or
24th
parts
of
the
whole
Circle
.
And
remember
,
that
when
you
come
to
90
degrees
from
the
Substyle
,
you
need
divide
no
further
,
for
the
Sun
is
no
longer
upon
this
plain
.
Also
you
may
leave
out
those
hours
at
which
the
Sun
is
alwayes
under
our
Horizon
,
as
the
hours
from
8.
at
Night
,
to
4
in
the
Morning
:
then
lay
a
Ruler
from
the
Center
O
to
every
one
of
these
divisions
of
the
Circle
,
and
where
the
Ruler
cuts
the
Contingent
,
there
make
points
for
the
hours
respectively
,
and
through
these
points
you
may
draw
the
hour-lines
Parallel
to
the
Substyle
,
of
what
length
you
please
;
and
mark
them
from
the
Noon
line
Eastward
1
,
2
,
3.
&c.
because
the
Suns
Diurnal
course
is
Westward
,
and
the
course
of
the
shadow
is
contrary
.
He
that
will
may
make
use
of
his
Planisphear
for
dividing
the
hours
,
as
was
taught
Chapter
4
,
and
5.
or
use
a
Quadrant
,
or
a
Scale
of
Chords
,
or
the
Tables
of
Tangents
with
a
Sector
,
or
a
Scale
of
equal
parts
.
But
it
needs
not
.
Note
that
this
Dyal
may
compare
with
the
hardest
:
however
Mr
Blagrave
and
other
Dyalists
have
omitted
it
,
as
seeming
easy
:
and
here
Wittekindus
,
(
to
whom
all
later
Dyalists
are
much
beholden
)
and
after
him
Fale
were
mistaken
,
using
the
Declination
of
the
plain
,
where
they
should
have
used
the
difference
of
Longitude
in
the
making
of
this
Dyal
.
CHAP.
VIII
.
Of
the
kinds
of
Oblique
Dyals
.
WHat
an
Oblique
Dyal
is
,
and
why
it
is
so
called
,
hath
been
shewed
Chapter
2.
They
be
-
Regular
.
-
Irregular
.
The
Regular
lie
in
some
notable
Circle
of
the
Sphear
;
as
,
1.
The
Vertical
Dyal
,
whose
plain
lieth
in
the
Horizon
:
for
which
cause
many
call
it
the
Horizontal
Dyal
.
2.
The
South
and
North
Horizontal
Dyal
,
whose
plain
lies
in
the
East
Azimuth
:
and
it
is
commonly
called
the
South
or
North
Erect
Direct
Dyal
.
As
for
the
East
and
West
Dyals
,
they
belong
to
another
place
,
as
was
said
Chapter
5.
The
Irregular
are
such
as
lie
Oblique
to
the
Horizon
,
as
Reclining
or
Inclining
Dyals
;
or
lie
Oblique
to
the
Meridian
,
as
Decliners
:
or
else
Oblique
to
both
,
as
Recliners
or
Incliners
Declining
;
which
are
esteemed
the
hardest
of
all
,
because
of
their
double
Irregularity
,
though
by
the
Planisphear
they
are
made
almost
as
easily
as
the
rest
.
The
Declination
of
a
plain
is
the
Azimuthal
distance
of
his
Poles
from
the
Meridian
of
the
Place
,
East
or
West
.
The
Reclination
is
the
distance
of
his
Poles
from
the
Zenith
and
Nadir
of
your
Place
.
Inclination
is
the
nearest
distance
of
the
Poles
of
the
plain
from
your
Horizon
.
And
whatsoever
the
Reclination
of
the
upper
face
of
a
plain
is
,
the
Inclination
of
the
lower
face
is
the
Complement
thereof
.
CHAP.
IX
.
How
to
make
the
Vertical
Dyal
.
IN
the
Meridional
Projection
the
Finitor
being
set
to
the
Latitude
of
your
Place
,
you
shall
see
the
Limb
which
is
your
Meridian
,
and
the
Axtree-line
which
is
the
sixt
hour-circle
,
dividing
the
Finitor
into
4
Quadrants
;
and
the
rest
of
the
Meridians
dividing
every
Quadrant
alike
.
Mark
now
at
what
degree
numbred
from
the
Limb
,
every
hour-circle
(
that
is
,
every
15th
Meridian
being
a
ragged
or
blacker
line
)
cutteth
the
Finitor
;
at
the
same
distance
shall
the
same
hour-circle
cut
the
Limb
of
your
Dyal
in
the
plain
.
diagram
Lastly
,
for
the
Gnomon
,
set
your
Compasses
to
the
Chord
of
the
arch
of
the
Poles
Elevation
in
the
Limb
:
that
is
,
measure
in
the
Limb
from
the
Pole
to
the
Finitor
,
and
setting
that
distance
in
the
Circle
of
your
Dyal
from
12.
either
way
,
make
a
point
,
through
which
if
you
draw
a
deleble
line
from
the
Center
,
you
have
between
this
line
and
the
line
of
12.
the
angle
of
your
Gnomon
,
by
which
when
you
have
shaped
him
,
you
must
set
him
upright
over
the
12
a
clock
line
,
with
the
point
of
the
said
angle
at
the
Center
,
and
all
is
done
.
CHAP.
X.
How
to
make
the
South
and
North
Horizontal
Dyal
.
THis
is
usually
called
the
Erect
Direct
Dyal
,
and
belongs
to
an
upright
Wall
looking
full
North
or
South
:
and
the
plain
of
it
lies
in
the
East
Azimuth
,
which
on
the
Planisphear
in
the
Meridional
Projection
is
represented
by
the
Axis
of
the
Reet
.
The
Finitor
set
to
the
Latitude
,
as
in
the
former
Chapter
,
mark
where
the
hour
Circles
cut
the
Axis
of
the
Reet
,
which
is
the
proper
Horizon
of
this
Dyal
;
you
shall
find
the
first
cutteth
9.
20
minutes
from
the
Meridian
,
the
second
19.
30
minutes
,
the
third
31.
30
minutes
,
the
fourth
46.
45
minutes
,
the
fifth
66.
24
minutes
;
the
sixt
90.
And
you
shall
see
the
North
Pole
depressed
under
this
plain
,
as
much
as
is
the
Complement
of
our
Latitude
,
and
the
South
Pole
as
much
Elevated
above
it
.
1.
Wherefore
for
the
South
Dyal
,
draw
an
Horizontal
line
about
the
top
of
your
Dyal
plain
,
which
shall
be
the
hour
of
Sixes
,
from
the
midst
whereof
let
fall
a
Perpendicular
,
which
shall
be
both
the
Vertical
and
the
Meridian
,
both
of
the
Place
,
and
of
the
Plain
,
wherein
the
Gnomon
must
stand
Elevated
37.
45.
minutes
or
the
Complement
of
your
Latitude
toward
the
South
Pole
.
diagram
Another
way
.
Because
the
Almicantars
may
oft
obscure
the
Intersections
of
the
Hour
Circles
with
the
Axis
,
you
may
avoid
that
inconvemence
,
if
you
reduce
this
Dyal
to
a
Vertical
Dyal
.
For
the
South
Horizontal
Dyal
being
the
very
Vertical
Dyal
of
those
People
that
live
90
degrees
Southward
from
us
,
that
is
,
in
South
Latitude
37.
45
minutes
,
if
you
set
the
Finitor
to
the
Latitude
37.
45
minutes
,
you
shall
see
the
sections
of
the
Hour
Circles
with
the
Finitor
more
●pparently
,
and
thereby
make
your
Dyal
.
2.
For
the
North
face
,
Imagine
you
had
for
you
.
Gnomon
a
wyre
thrust
aslope
through
the
Center
of
the
plain
from
the
South
side
Northward
and
you
will
presently
conceive
that
in
the
North
Dyal
the
Horizontal
or
6
a
clock
line
will
be
lowest
,
and
that
the
Gnomon
will
turn
upwards
toward
the
North
Pole
as
much
as
he
turned
downwards
on
the
other
side
:
and
that
all
the
hours
save
4
,
5
,
and
6.
in
the
Morning
,
and
6
,
7
,
and
8.
at
Night
may
be
left
out
in
our
Latitude
;
because
the
Sun
shineth
no
longer
upon
it
:
and
those
hour-distances
you
may
find
,
and
set
off
from
the
12
a
clock
line
,
or
from
the
6
a
clock
line
,
as
you
did
the
hours
of
like
distance
in
the
South
face
.
Another
general
and
pleasant
way
to
delineate
the
opposite
face
of
any
Dyal
,
see
hereafter
in
the
end
of
the
12th
Chapter
.
CHAP.
XI
.
How
to
Observe
the
Declination
of
any
Declining
Plain
.
diagram
A
B
is
a
Wall
or
plain
declining
East
by
the
arch
S
p
,
to
which
E
B
or
W
A
are
equal
,
for
so
much
as
the
Wall
bendeth
from
the
East
Azimuth
,
so
much
doth
his
Pole
at
p
decline
or
bend
from
the
Meridian
.
1.
Now
to
find
how
much
any
plain
declineth
,
and
so
in
what
Azimuth
he
lies
,
one
good
way
is
this
:
when
the
Sun
begins
to
inlighten
the
Wall
,
or
when
he
leaves
it
,
then
is
the
Sun
in
the
same
Azimuth
with
the
Wall
;
take
at
that
instant
his
Altitude
,
and
thereby
get
his
Azimuth
(
according
to
Book
4.27
.
)
and
that
is
the
Azimuth
of
the
Wall
.
2.
Another
way
,
First
draw
upon
the
Wall
an
Horizontal
line
,
by
Chapter
3.
then
your
Planisphear
being
fastned
to
a
Square
board
(
as
in
Chapter
4.
)
set
one
side
of
the
board
to
that
Horizontal
line
or
Parallel
to
it
and
fix
there
your
board
and
Planisphear
level
,
by
the
help
of
a
Square
set
under
him
like
a
bracket
,
the
place
your
Label
and
Sights
in
one
of
the
Diameters
of
your
Planisphear
,
and
mark
when
the
Sun
comes
into
the
line
of
the
Label
,
casting
the
shadow
of
one
Sight
upon
the
other
,
if
the
Label
be
then
in
the
Diameter
which
is
Parallel
to
the
Wall
,
then
is
the
Sun
at
that
time
in
the
Azimuth
of
the
Wall
:
if
the
Label
be
in
the
other
Diameter
which
is
Perpendicular
to
the
Wall
,
then
the
Sun
coming
to
it
is
in
the
Azimuth
of
the
Pole
of
the
plain
.
Now
having
the
hour
,
or
the
Altitude
of
the
Sun
get
his
Azimuth
(
by
4.27
,
)
the
same
is
the
Azimuth
of
the
Wall
or
plain
,
if
the
Label
were
Parallel
to
the
Wall
;
or
the
same
is
the
Azimuth
of
the
Pole
of
the
plain
(
that
is
the
very
Declination
)
if
the
Label
stood
Perpendicular
to
the
Wall
.
3.
Another
way
,
If
you
have
not
time
to
watch
till
the
Sun
come
into
the
Azimuth
of
the
Wall
or
the
Vertical
of
it
,
which
cutteth
the
Pole
thereof
,
then
get
the
Suns
Azimuth
by
the
said
Book
4.27
,
when
you
can
,
and
at
the
same
time
Observe
by
your
Label
the
Suns
Horizontal
distance
from
the
Pole
of
the
plain
,
and
by
comparing
these
together
you
may
easily
gather
the
Declination
of
the
Wall
:
as
in
Example
.
I
observed
the
Sun
to
be
gone
West
from
the
Pole
of
the
plain
70
degrees
,
and
by
the
Altitude
of
the
Sun
then
taken
,
I
found
his
Azimuth
60
degrees
:
here
I
reason
thus
,
The
Sun
is
gone
from
the
Pole
and
Vertical
of
the
Wall
70
degrees
,
and
from
the
Meridian
but
60
degrees
,
therefore
the
Meridian
lies
between
the
Pole
of
the
plain
and
the
Sun
and
because
☉
p
is
70.
and
☉
S
60.
therefore
S
p
the
Declination
of
the
plain
,
is
10
degrees
the
difference
of
70.
and
60
,
and
the
Declination
is
East
,
for
the
Sun
is
neerer
to
the
Meridian
then
to
the
Vertical
of
the
plain
:
and
thus
if
you
draw
a
rude
Scheam
of
your
Case
,
you
may
soon
reason
out
the
Declination
,
better
then
do
it
blind-fold
by
the
rules
commonly
given
.
And
by
these
two
last
wayes
you
may
take
the
Declination
,
not
only
of
upright
plains
,
but
of
Recliners
also
,
for
which
the
first
way
will
not
serve
,
CHAP.
XII
.
How
to
make
a
Declining
Horizontal
Dyal
.
diagram
But
Declining
Dyals
which
look
awry
from
our
Meridian
,
have
a
Meridian
of
their
own
,
which
is
called
the
Meridian
of
the
plain
,
and
the
Substyle
(
because
the
Style
or
Gnomor
stands
upon
it
)
and
is
indeed
the
Meridian
of
that
Place
,
where
this
Declining
Dyal
would
be
a
Vertical
Dyal
,
and
where
the
Substyle
would
be
the
Noon
line
:
and
to
this
Substyle
the
hours
of
the
plain
are
alwayes
so
conformed
,
that
the
nearer
they
be
to
the
Substyle
the
narrower
are
the
hour
spaces
,
and
contrarily
:
because
the
Meridians
do
so
cut
every
oblique
Horizon
,
that
is
,
thickest
near
the
Meridian
of
the
place
:
and
this
Declining
Dyal
(
being
a
stranger
with
us
)
followeth
the
fashion
of
his
own
Country
,
and
so
hath
his
narrowest
hour
spaces
near
his
own
Meridian
,
rather
then
ours
Now
as
that
is
the
Meridian
of
our
place
which
cutteth
our
Horizon
at
Right
angles
,
passing
through
his
Poles
Zenith
and
Nadir
,
so
the
Meridian
of
any
plain
is
that
which
cutteth
the
plain
at
Right
angles
,
and
passeth
through
his
Poles
.
You
may
find
all
these
requisites
in
the
Meridional
Projection
,
not
only
for
one
,
but
for
all
Declinations
,
lying
as
in
a
Table
before
you
,
with
admirable
ease
and
delight
;
for
there
is
no
Declining
Wall
or
Horizontal
plain
,
but
we
have
an
Azimuth
in
the
Reet
which
shall
picture
him
:
and
look
how
the
Meridians
divide
these
Azimuths
,
so
do
they
divide
the
Horizons
or
Circles
of
the
Declining
plains
.
The
Pole
of
any
Azimuth
is
found
in
the
Finitor
90
degrees
distant
from
him
;
the
Meridian
that
cuts
the
Pole
of
the
Azimuth
,
cuts
also
the
Azimuth
,
and
the
plain
thereby
represented
at
Right
angles
,
and
is
the
Meridian
of
the
plain
or
Substylar
,
(
Chapter
2.
Theorem
11.
)
and
the
degrees
of
that
Meridian
between
the
plain
and
the
next
Pole
of
the
World
are
the
Elevation
of
the
Pole
above
the
plain
:
and
so
the
Elevation
of
the
Gnomon
or
Style
,
and
the
arch
of
the
plain
comprehended
between
this
Meridian
of
the
plain
and
the
Limb
,
is
the
Declination
of
the
Gnomon
,
or
distance
of
the
Substyle
from
the
Meridian
,
or
distance
of
the
Meridian
of
the
plain
from
the
Meridian
of
the
Place
.
What
would
you
more
?
Example
.
If
a
Wall
Decline
East
30
degrees
.
I
say
,
because
the
face
of
the
Wall
looketh
30
degrees
from
the
South
Eastward
,
therefore
the
plain
,
which
lieth
90
degrees
from
his
Pole
,
is
in
the
30th
Azimuth
from
the
East
Northward
:
therefore
I
go
to
the
30th
Azimuth
from
the
East
line
of
Axis
,
counted
cither
way
,
and
take
that
Azimuth
and
his
Match
(
which
is
equally
distant
from
the
Axis
)
for
the
very
picture
of
my
Declining
plain
.
Then
seeking
the
Substyle
or
Meridian
of
the
plain
.
I
say
the
Pole
of
the
plain
is
in
the
Finitor
at
the
30th
Azimuth
from
Meridies
in
the
Limb
,
(
because
the
plain
it self
is
the
30th
Azimuth
beyond
the
Axis
)
the
Meridian
that
cuts
this
Pole
is
the
36
¼
(
exactly
36.
8
minutes
,
the
number
whereof
shewes
me
the
difference
of
Longitude
between
our
Country
and
the
Country
of
this
Dyal
.
This
36
¼
Meridian
,
being
the
Meridian
of
the
plain
,
I
follow
toward
the
Pole
,
and
find
him
cutting
both
the
arches
of
my
plain
on
both
sides
the
Axis
:
but
I
regard
the
cuttng
only
in
that
arch
which
is
nearest
to
the
Pole
,
because
there
the
angle
looks
more
like
a
Right
angle
,
and
there
is
the
nearest
distance
of
the
Pole
from
the
plain
,
and
there
I
see
the
hour
spaces
least
:
from
that
Intersection
therefore
,
I
reckon
in
the
same
Meridian
to
the
Pole
32
degrees
and
perhaps
a
minute
more
,
(
you
may
find
it
by
Calculation
,
)
this
is
the
Elevation
of
the
Pole
above
the
plain
,
and
of
the
Gnomon
likewise
:
also
from
the
same
Intersection
I
reckon
in
the
plain
to
the
Limb
or
Meridian
21.
degrees
10.
minutes
,
the
distance
of
the
Meridian
of
the
plain
from
the
Meridian
of
the
Place
:
the
same
is
the
Declination
of
the
Gnomon
,
or
of
his
Substyle
.
Then
for
the
hours
,
I
begin
at
the
Zenith
of
the
Reet
,
where
is
our
Meridian
,
and
numbring
first
toward
the
Substyle
,
I
seek
at
what
number
of
degrees
from
the
Zenith
the
hour
Circles
cut
my
plain
,
and
I
find
as
followeth
,
deg.
|
min.
|
11.
—
9.
|
35
|
10.
—
17.
|
54
|
9.
—
25.
|
54
|
8.
—
34.
|
22
|
7.
—
44.
|
17
|
6.
—
57.
|
10
|
5.
—
75.
|
4
|
Then
in
the
other
arch
of
the
plain
I
have
the
afternoon
hours
,
thus
.
deg.
|
min.
|
1
—
12.
|
10
|
2
—
28.
|
59
|
3
—
52.
|
26
|
4
—
80.
|
17
|
And
further
I
cannot
go
,
because
I
see
the
next
hour
is
above
90
from
the
Substyle
;
therefore
my
Dyal
receives
him
not
on
this
side
,
but
on
the
North
side
there
is
use
of
him
.
Now
to
draw
the
Dyal
,
I
consider
that
because
the
plain
declines
East
therefore
the
Gnomon
shall
decline
West
:
for
the
Dyal
being
such
a
projection
of
the
Sphear
wherein
all
the
Vi●ual
lines
cross
in
the
Nodus
of
the
Gnomon
,
and
thence
disperse
themselves
again
toward
the
plain
,
therefore
that
which
is
East
in
the
Sphear
will
be
expressed
West
on
the
plain
,
and
contrarily
,
(
as
was
shewed
Chapter
2.
Theorem
11.
)
Also
I
consider
that
howsoever
the
plain
be
turned
East
or
West
,
the
Gnomon's
place
is
fixed
,
because
it
is
a
part
of
the
Axis
of
the
World
,
or
a
line
Parallel
to
it
.
Now
therefore
if
I
turn
a
South
Dyal
,
and
make
him
Decline
East
,
and
hold
the
Gnomon
unmoveable
;
the
West
side
of
the
Dyal
will
approach
nearer
to
the
Gnomon
,
as
reason
and
sence
will
tell
me
:
likewise
the
hours
which
are
found
on
the
same
side
of
the
Meridian
or
Noon-line
with
the
Substile
must
be
set
the
same
way
with
it
from
the
Noon-line
in
the
Dyal
.
Therefore
having
drawn
an
Horizontal
line
E
W
on
the
Wall
,
from
the
Center
taken
at
A
,
I
let
fall
the
Perpendicular
A
B
for
the
Noon-line
,
then
upon
the
Center
A.
I
draw
a
blind
Semi-circle
with
the
Semi-diameter
of
my
Planisphear
,
or
of
some
Quadrant
,
as
E
B
W
and
therein
I
prick
down
the
Substyle
and
the
hours
,
after
the
manner
used
the
10th
Chapter
.
diagram
This
is
the
most
ready
way
to
delineate
the
opposite
face
of
any
Dyal
.
See
another
way
to
make
this
Declining
Horizontal
Dyal
,
Chapter
21.
CHAP.
XIII
.
How
to
Observe
the
Reclination
or
Inclination
or
any
Plain
.
WHat
Reclination
and
Inclination
are
,
hath
been
shewed
Chapter
8.
All
Reclining
and
Inclining
plains
have
their
Bases
or
Horizontal
Diameters
lying
in
the
Horiz
ontal
Diameter
of
some
Azimuth
:
but
the
top
or
Nonagesimus
gradus
of
the
plain
from
the
Horizon
leaneth
back
from
the
Zenith
of
your
Place
,
in
the
Vertical
of
the
plain
(
which
is
the
Azimuth
cutting
the
plain
at
Right
angles
)
so
much
as
the
Reclination
hapneth
to
be
;
and
the
Pole
of
the
plain
,
on
that
side
the
plain
inclines
to
,
is
sunk
as
much
below
the
Horizon
as
the
Nonagesimus
gradus
of
the
plain
is
sunk
below
the
Zenith
,
and
the
opposite
Pole
is
mounted
as
much
.
diagram
To
find
the
quantity
of
the
Reclination
you
shall
draw
a
Vertical
li●e
on
the
plain
,
by
Chapter
3.
and
thereto
apply
a
long
Ruler
which
may
over-shoot
the
plain
either
above
or
below
:
to
that
Ruler
apply
any
Semidiameter
of
your
Planisphear
,
or
of
any
Quadrant
;
and
the
degrees
between
that
Semidiameter
and
the
Plumb
line
shall
be
the
degrees
of
Reclination
.
Or
stick
up
in
the
Vertical
line
two
pins
of
equal
height
,
and
Perpendicular
,
and
placing
your self
either
above
or
below
the
plain
,
as
you
find
most
easy
,
direct
the
Sights
of
your
Planisphear
or
Quadrant
to
the
heads
of
the
two
pins
being
in
a
right
line
with
your
ey
;
and
the
Plumbet
shall
shew
the
Reclination
on
one
side
the
Quadrant
,
and
the
Inclination
(
which
is
alwayes
Complement
thereof
)
on
the
other
.
CHAP.
XIV
.
How
to
make
a
South
and
North
Reclining
Dyal
.
THe
Base
or
Horizontal
line
of
such
a
Dyal
lieth
in
the
East
Azimuth
,
and
his
Pole
in
the
Meridian
;
as
you
may
see
in
the
plain
of
the
former
Chapter
.
In
the
Meridional
Projection
having
set
the
Finitor
to
the
Latitude
,
count
from
the
Zenith
the
degrees
of
Reclination
Northward
or
South-ward
as
you
observed
it
to
be
,
and
remove
the
Zenith
so
many
degrees
the
same
way
,
then
shall
you
see
presently
which
Pole
is
Elevated
above
the
Zenith
line
,
(
for
that
is
the
picture
of
your
plain
)
and
how
much
:
to
which
Elevation
you
shall
make
your
Dyal
by
the
tenth
Chapter
,
remembring
to
turn
the
Gnomon
upwards
or
downwards
as
the
North
or
South
Pole
is
Elevated
above
the
face
of
your
plain
.
Example
.
The
plain
of
the
former
Chapter
was
a
North
plain
Reclining
Southward
50
degrees
,
that
is
,
almost
to
the
Equinoctial
:
when
the
Finitor
is
at
our
Latitude
the
Zenith
is
distant
from
the
North
Pole
the
Complement
thereof
37.45
.
toward
the
South
.
Now
I
must
put
the
Zenith
yet
50
degrees
more
Southward
,
because
my
plain
Reclines
so
much
that
way
,
and
I
see
that
then
the
North
Pole
is
Elevated
87.
45.
minutes
,
and
I
see
upon
the
Zenith
line
or
Axis
of
the
Reet
how
the
hour
Circles
cut
my
plain
almost
in
equal
spaces
:
if
this
plain
had
Reclined
but
2.
15
minutes
further
,
he
had
fallen
into
the
plain
of
the
Equinoctial
,
and
so
the
Dyal
would
have
been
a
Polar
Dyal
,
and
all
the
hours
would
have
had
equal
space
,
and
the
Gnomon
would
have
stood
Perpendicular
,
which
are
the
properties
of
a
Polar
Dyal
,
as
hath
been
shewed
Chapter
4.
For
the
opposite
face
of
this
Dyal
,
the
general
rule
given
Chapter
12.
may
suffice
.
CHAP.
XV.
How
to
make
an
East
or
West
Reclining
Dyal
.
AS
it
hath
been
shewed
Chapter
14
,
that
the
base
or
Horizontal
line
of
a
South
Recliner
lieth
alwayes
in
the
East
Azimuth
,
so
the
base
of
an
East
Recliner
lieth
alwayes
in
the
Meridian
of
the
Place
.
And
as
all
Declining
plains
lie
in
some
Azimuth
,
and
cross
one
another
in
the
Zenith
and
Nadir
,
by
Chapter
12.
so
these
Reclining
plains
lie
in
some
Circle
of
Position
,
and
cross
one
another
in
the
North
and
South
points
of
the
Horizon
:
which
being
considered
,
those
East
Recliners
shall
be
made
as
easily
as
the
Decliners
Chapter
12.
For
these
East
Recliners
be
in
very
deed
South
Decliners
to
those
that
live
90
degrees
from
us
Northward
or
Southward
;
and
have
one
of
the
Poles
Elevated
as
much
as
the
Complement
of
our
Latitude
:
for
the
Perpendicular
or
Plumb-line
of
those
People
is
Parallel
to
the
Horizontal
Diameter
of
our
Meridian
.
In
the
Meridional
Projection
,
set
the
Zenith
line
to
the
Latitude
,
and
then
are
the
Azimuths
Circles
of
Position
,
and
are
also
those
very
East
or
West
Reclining
plains
,
and
the
Zenith
line
is
the
base
or
Horizontal
line
to
them
all
,
and
to
the
Meridian
likewise
;
Take
any
of
these
Azimuths
,
and
see
how
the
Meridians
of
the
Mater
divide
him
,
so
shall
the
Dyal-plain
represented
by
it
be
divided
also
.
The
working
is
very
like
that
of
Chapter
12.
Compare
the
one
with
the
other
where
you
doubt
.
Example
.
I
have
an
East
plain
Reclining
45
degrees
,
to
which
I
would
make
a
Dyal
.
I
set
the
Zenith
line
to
the
Latitude
52.
15
minutes
and
going
to
the
arch
of
the
45
Azimuth
on
one
side
the
Zenith
line
,
and
his
match
so
many
degrees
distant
on
the
other
side
,
I
take
that
Circle
for
my
plain
,
his
Center
is
the
Center
of
the
Planisphear
:
the
Meridian
or
Noon
line
of
the
Place
in
these
Dyals
is
evermore
the
Axis
or
Zenith
line
of
the
Reet
,
for
he
is
drawn
to
the
Intersection
of
the
Meridian
of
the
Place
(
here
the
Limb
)
with
the
plain
.
The
Zenith
line
here
lies
Horizontal
,
therefore
the
Noon
line
in
these
East
Recliners
must
be
evermore
the
Horizontal
line
of
the
Dyal
,
as
in
all
Decliners
(
Chapter
12.
)
the
Noon
line
is
evermore
the
Vertical
or
Perpendicular
line
.
The
arch
of
my
plain
which
is
nearest
to
the
North
Pole
hath
his
Pole
in
the
Finitor
45
degrees
from
the
Center
Southward
,
and
there
this
Pole
is
cut
by
the
Meridian
31
¼
.
from
the
Axis
,
this
is
the
Meridian
of
my
plain
,
and
he
is
distant
from
the
Meridian
of
the
Place
58
¾
.
(
which
is
the
difference
of
Longitude
)
this
Meridian
I
follow
to
the
arch
of
the
plain
which
is
nearest
to
the
North
Pole
,
and
so
going
on
in
him
to
the
Pole
I
number
in
him
between
the
plain
and
the
Pole
34
degrees
,
which
is
the
Elevation
of
the
Pole
,
and
therefore
also
of
the
Gnomon
above
this
plain
:
and
between
this
Meridian
of
the
plain
and
the
Meridian
of
the
Place
at
Zenith
I
reckon
in
the
plain
42
½
for
the
Declination
of
the
Gnomon
.
diagram
Then
I
seek
my
hour
distances
,
and
I
find
the
first
hour-circle
from
the
Meridian
toward
the
Substyle
cuts
in
the
plain
14
½
,
therefore
taking
14
½
in
the
blind
arch
from
the
Noon
line
toward
the
Substyle
,
I
set
11.
for
so
it
is
,
and
not
1.
as
you
may
persently
perceive
,
if
you
hold
but
a
Book
or
a
Trencher
after
the
Situation
of
your
plain
,
somewhat
near
by
guess
,
and
consider
which
way
the
shadow
must
move
,
reason
will
tell
you
it
moves
downward
in
this
Dyal
from
11.
to
12.
&c.
then
I
see
the
second
hour-circle
cuts
the
plain
at
25
½
for
10.
the
next
at
34
⅔
.
for
9.
the
next
at
43.
for
8.
which
happneth
a
little
above
the
Substyle
,
as
he
ought
,
for
the
difference
of
Longitude
is
almost
60.
viz
58
¾
.
as
before
,
next
51
½
.
for
7.
next
61
⅓
for
6.
next
73
⅓
for
5.
next
88
½
for
4.
and
further
I
need
not
go
in
our
Country
.
Then
in
the
other
arch
of
the
plain
I
find
19
½
for
1.45
for
2.
71
for
3.
and
these
I
put
in
their
places
,
as
in
the
Figure
.
The
Gnomon
must
stand
square
upon
the
Substyle
,
at
an
angle
of
34
degrees
.
Note
that
the
Reclination
must
alwayes
be
reckoned
from
the
Limb
inwards
upon
the
Finitor
,
because
where
the
Finitor
touches
the
Limb
there
is
our
Zenith
for
this
turn
.
Inclination
is
reckoned
from
the
Zenith
line
,
which
here
is
both
the
Diameter
of
the
Horizon
,
and
Horizon
itself
.
For
the
Opposite
or
Inclining
face
of
this
Dyal-plain
,
use
the
general
way
I
shewed
you
,
Chapter
12.
that
is
,
strike
the
Substyle
and
all
the
hour
lines
through
the
Center
:
and
set
the
same
figures
to
every
hour
line
beyond
the
Center
which
he
had
on
this
side
,
and
set
the
Gnomon
upon
the
Substyle
downward
to
behold
the
South
Pole
,
and
it
is
done
.
And
so
by
the
Inclining
Dyal
,
if
you
had
him
first
drawn
,
you
might
presently
make
the
Recliner
.
CHAP.
XVI
.
How
to
find
the
Arches
and
Angles
that
are
requisite
for
the
making
of
the
Reclining
Declining
Dyal
.
BEfore
you
can
Intelligently
make
a
Reclining-Declining
Dyal
,
which
is
the
most
Irregular
of
all
,
having
two
Anomalies
,
viz.
Declination
and
Reclination
,
you
must
be
acquainted
with
3
Triangles
in
the
Sphear
,
wherein
certain
arches
and
angles
lie
which
are
neefull
to
be
known
.
I
advise
you
therefore
first
to
draw
(
though
it
be
but
by
aime
)
an
Horizontal
Projection
of
the
Sphear
,
such
as
here
I
have
drawn
for
a
South
Dyal
Declining
West
50
degrees
,
and
Reclining
60
degrees
in
the
Latitude
52.
10.
minutes
,
which
shall
be
our
Example
.
The
Circle
E
S
W
N
is
our
Horizon
N
S
our
Meridian
.
diagram
D
T
d
the
plain
,
Z
T
the
Reclination
thereof
.
D
d
the
Base
or
Horizontal
line
of
the
plain
.
V
u
the
Vertical
of
the
plain
cutting
it
right
in
T
,
and
cutting
the
Pole
thereof
at
H
:
for
u
is
the
Pole
of
a
plain
erected
upon
D
d
;
but
the
Pole
of
the
Reclined
plain
D
T
d
is
H.
S
u
or
N
V
the
Declination
of
the
plain
.
M
P
H
m
the
Meridian
of
the
plain
,
cutting
the
North
Pole
at
P
,
the
plain
in
Right
angles
at
R
,
and
the
Pole
thereof
at
H.
Now
see
your
three
Triangles
all
adjoyning
in
this
Scheam
,
viz.
D
N
O
and
O
R
P
Rectangled
at
N
and
R
,
and
P
Z
H.
Obtuse-angled
at
Z.
It
is
true
that
the
last
Triangle
alone
may
do
your
work
,
or
the
two
first
may
do
it
without
the
last
:
but
you
shall
do
well
to
be
acquainted
with
them
all
.
In
the
first
Triangle
D
N
O
you
have
given
D
N
40
degrees
the
Complement
of
the
plains
Declination
,
N
the
Right
angle
of
our
Meridian
with
our
Horizon
,
D
the
Complement
of
Reclination
,
whereby
you
may
find
D
O
the
Oblique
Ascension
of
our
Meridian
;
that
is
,
how
many
degrees
of
the
plain
the
noon-line
shall
lie
above
the
Horizontal
line
:
also
you
may
find
N
O
the
Perpendicular
Altitude
of
the
noon-line
,
or
the
Inclination
of
the
noon-line
of
the
Dyal
to
the
Horizon
;
(
where
you
shall
note
that
when
this
Altitude
of
the
noon-line
N
O
is
equal
to
N
P
the
Elevation
of
the
Pole
,
then
is
the
second
Triangle
P
R
O
quite
lost
in
the
point
P
,
and
the
plain
then
becometh
a
Declining
Equinoctial
plain
)
also
you
may
find
the
angle
O
called
the
Position-Reclination
for
a
reason
hereafter
to
be
shewed
;
Wittekindus
calls
it
,
Complementum
repetendum
,
because
he
means
to
have
a
Bout
with
it
again
,
to
find
other
arches
by
it
.
In
the
second
Triangle
O
R
P
you
have
given
O
,
as
before
,
for
this
angle
was
in
the
former
Triangle
,
or
his
equal
,
(
for
Anguli
precrucem
oppositi
sunt
aequales
)
R
the
Right
angle
of
the
plain
with
his
Meridian
:
O
P
the
position
Latitude
,
that
is
,
the
Latitude
of
that
Place
wherein
the
Reclining
plain
O
R
T
d
Q
,
shall
be
a
Circle
of
Position
:
this
is
given
if
you
subtract
N
O
the
Perpendicular
Altitude
of
the
noon-line
,
out
of
N
P
your
Latitude
,
(
this
O
P
is
Wittekindus
his
Differentia
Retenta
.
)
And
hence
may
be
found
O
R
the
Declination
of
the
Gnomon
,
or
distance
of
the
Meridian
of
the
plain
from
the
Meridian
of
the
Place
:
R
P
the
Elevation
of
the
Pole
above
the
plain
in
the
Plains
own
Meridian
:
P
the
angle
between
the
Meridian
of
the
plain
and
the
Meridian
of
the
Place
:
this
angle
is
called
the
difference
of
Longitude
,
because
it
shewes
how
far
the
places
are
distant
from
us
in
Longitude
,
wherein
this
Dyal
shall
be
a
Direct
Dyal
,
without
Declination
,
having
his
Gnomon
in
the
noon-line
of
the
Place
,
and
and
shews
also
how
many
degrees
of
the
Equinoctial
,
or
how
many
hours
and
minutes
there
are
between
our
Meridian
and
the
Meridian
of
the
plain
,
as
the
arch
O
R
shews
how
many
degrees
of
the
plain
come
between
the
said
Meridians
.
Let
this
be
well
observed
by
Learners
.
In
the
Third
Triangle
P
Z
H
,
you
have
given
P
Z
the
Complement
of
your
Latitude
,
Z
H
the
Complement
of
the
plains
Reclination
,
and
Z
the
Supplement
of
the
plains
Declination
.
Hence
may
be
found
H
P
whose
Complement
is
P
R
the
Elevation
of
the
Pole
above
the
plain
,
P
the
difference
of
Longitude
,
H
whose
measure
is
R
T
the
arch
of
the
plain
between
the
Meridian
of
the
plain
or
Substyle
and
the
Vertical
line
of
the
plain
,
the
Complement
whereof
is
R
D
the
Substyle's
distance
from
the
Horizontal
line
of
the
plain
.
Every
arch
and
angle
therefore
in
these
Triangles
is
given
,
or
may
presently
be
found
by
the
Problemes
of
Spherical
Triangles
,
Book
3.
But
I
shall
shew
you
a
short
and
pleasant
way
to
find
them
,
by
setting
the
whole
Scheam
at
once
on
your
Planisphear
,
where
you
shall
have
them
almost
all
at
one
view
.
For
this
purpose
I
use
my
Planisphear
in
the
Horizontal
Projection
.
But
note
,
that
I
make
use
onely
of
the
Reet
and
Label
,
and
one
of
the
Meridians
of
the
Mater
.
Thus
I
set
D
d
on
the
Axis
of
the
Reet
,
D
T
d
on
the
60th
Azimuth
from
the
Center
,
V
u
on
the
Finitor
,
Z
N
on
the
Label
,
fixing
it
in
the
Limb
of
the
Reet
40
degrees
from
the
Zenith
,
according
to
the
arch
D
N.
Then
in
the
Label
I
make
a
prick
with
ink
at
P
for
the
North
Pole
52.
15.
minutes
within
the
Limb
,
and
another
prick
in
the
Finitor
at
H
for
the
Pole
of
the
plain
30
degrees
from
the
Center
;
which
done
,
and
keeping
my
Label
fixed
to
my
Reet
,
I
turn
the
Reet
till
I
see
some
one
Meridian
cutting
both
the
pricks
P
and
H
,
(
as
the
15
Meridian
from
the
Axis
shall
do
in
this
Example
)
and
that
Meridian
shall
serve
for
the
Meridian
of
the
plain
for
this
time
.
And
by
this
time
I
see
my
three
Triangles
on
my
Planisphear
,
their
sides
divided
into
degrees
,
as
a
Carpenters
Rule
into
Inches
.
and
I
find
for
the
Latitude
52.
10.
minutes
,
that
D
O
the
Oblique
Ascension
of
the
Meridian
is
44.06
.
N
O
the
Altitude
of
the
Meridian
20.
22
minutes
.
O
P
the
Position
Latitude
31.
48.
minutes
.
O
R
the
Declination
of
the
Gnomon
(
13.
21.
minutes
)
from
the
Meridian
.
R
P
the
Elevation
of
the
Gnomon
29.
08.
minutes
,
and
P
H
Complement
thereof
.
H
the
distance
of
the
Meridian
of
the
plain
and
Vertical
thereof
,
you
may
see
by
the
arch
of
the
plain
which
measureth
the
angle
H
,
32.
32.
minutes
.
Now
have
you
nothing
to
ask
of
the
third
Book
,
but
the
angles
O
and
P
,
and
there
you
have
divers
wayes
to
find
that
.
O
the
Position-Reclination
is
67.
29.
minutes
.
P
the
difference
of
Longitude
26
,
00
½
.
minutes
.
CHAP.
XVII
.
How
to
find
the
Horary
distances
of
a
Reclining
Declining
Dyal
.
TAke
the
easiest
way
first
.
You
have
seen
Chapter
15.
how
easily
East
and
West
Reclining
Dyals
are
to
be
made
by
the
Planisphear
,
because
they
fall
out
to
be
Circles
of
Position
,
and
are
plainly
pictured
by
the
Azimuths
.
Now
I
will
shew
you
how
all
Reclining
Dyals
may
be
reduced
to
East
or
West
Recliners
,
for
some
Latitude
or
other
,
and
so
the
hour
distances
found
by
the
Method
of
the
15th
Chapter
.
The
Circles
of
Position
,
as
hath
been
shewed
,
do
all
cross
one
another
in
the
North
and
South
points
of
the
Meridian
.
Now
therefore
by
the
point
O
where
the
plain
cuts
our
Meridian
,
draw
a
new
Horizon
O
B
Q
C
,
and
then
shall
you
see
your
plain
in
that
Horizon
to
be
a
very
Circle
of
Position
.
But
now
we
are
gotten
into
a
new
Latitude
,
O
P
called
(
before
in
Chapter
16
)
the
Position
Latitude
,
and
we
have
here
a
new
Reclination
,
for
whereas
this
plains
Reclination
in
our
Latitude
is
Z
D
T
60
degrees
,
his
Position
Reclination
is
O
,
viz.
Z
O
T
,
or
P
O
R
67.
29.
minutes
.
In
the
making
of
this
Dyal
therefore
,
you
shall
forget
your
own
Latitude
,
and
the
plains
Reclination
in
your
Horizon
,
and
with
this
new
Latitude
and
Reclination
make
the
Dyal
after
the
manner
of
the
East
Recliner
Chapter
15.
not
regarding
the
Declination
at
all
:
For
the
Base
of
this
plain
is
now
fallen
into
the
Horizontal
line
of
the
Meridian
,
and
his
Declination
being
just
a
Quadrant
,
he
becomes
a
Regular
Plain
,
and
neither
his
Declination
nor
his
Reclination
shall
now
much
trouble
you
.
How
to
place
your
Noon-line
from
the
Horizontal
or
Vertical
line
of
the
plain
,
you
have
found
already
,
and
at
what
distances
every
Hour
shall
stand
from
the
Noon-line
in
the
plain
you
shall
thus
find
.
diagram
Set
the
Zenith
line
of
the
Reet
to
your
new
Latitude
P
O
31.
48
minutes
,
and
find
for
your
plain
the
Azimuth
67
½
from
the
Axis
,
because
P
O
R
is
67.
29
minutes
,
as
before
:
seek
his
Pole
in
the
Finitor
,
which
will
be
the
90th
degree
from
the
said
Azimuth
or
plain
,
and
you
shall
find
that
Pole
cut
by
the
26.
Meridian
:
this
therefore
is
the
Meridian
of
my
plain
,
and
shall
make
the
Sub-style
on
the
Dyal
:
his
distance
from
the
Meridian
of
the
Place
in
the
Equinoctial
is
26.
and
so
much
is
the
angle
O
P
R
the
difference
of
Longitude
,
as
before
:
then
follow
this
Meridian
of
your
plain
to
that
arch
of
the
plain
which
is
next
the
Pole
of
the
World
,
and
you
shall
number
from
the
plain
to
the
Pole
in
the
said
Meridian
29.
08
minutes
for
the
Elevation
of
the
Gnomon
P
R
,
as
before
:
and
from
this
Meridian
to
the
Meridian
of
the
Place
at
the
Zenith
you
shall
number
in
the
plain
13.
21
minutes
for
O
R
,
the
Declination
of
the
Gnomon
,
as
before
.
Now
the
crossing
of
the
plain
with
the
Meridian
of
the
Place
is
the
Noon-point
in
the
plain
,
and
that
in
this
case
is
alwayes
in
the
Zenith
,
and
I
see
the
rest
of
the
Meridians
cutting
the
plain
for
the
Morning
Hours
,
thus
.
Hor.
deg.
|
min.
|
8
—
68
|
28
|
9
—
41
|
23
|
10
—
22
|
28
|
11
—
9
|
35
|
12
—
0
|
00
|
And
for
the
Evening
Hours
in
the
other
arch
,
thus
.
Hor.
deg.
|
min.
|
1
—
7
|
56
|
Sub.
13
|
21
|
2
—
15
|
18
|
3
—
22
|
52
|
4
—
31
|
32
|
5
—
42
|
36
|
6
—
58
|
18
|
7
—
81
|
35
|
And
because
7
in
the
Morning
will
be
shewn
by
this
Dyal
in
the
Summer
;
to
find
the
distance
thereof
from
the
Noon-line
in
the
plain
,
I
set
Nadir
in
the
place
where
Zenith
was
before
,
and
so
I
see
the
7th
hour-circle
cutting
the
plain
at
98.
25.
minutes
.
But
this
is
more
then
I
need
to
do
:
for
having
once
found
12.
Hour-spaces
in
any
Dyal
,
I
can
make
any
of
the
rest
by
striking
the
Hour-line
of
the
same
denomination
through
the
Center
.
As
for
Example
.
If
I
prolong
the
7th
of
the
afternoon
Hour-lines
beyond
the
Center
,
I
there
make
the
line
of
7a
clock
in
the
Morning
.
CHAP.
XVIII
.
How
to
draw
the
Reclining
Declining
Dyal
.
FIrst
draw
an
Horizontal
line
,
as
A
B
and
upon
the
Center
A
describe
a
blind
Circle
equal
to
the
Limb
of
your
lanisphear
,
or
of
some
other
Circle
which
is
divided
to
your
hand
,
that
by
help
thereof
you
may
presently
divide
this
blind
Circle
into
any
parts
required
Then
set
in
this
blind
Circle
the
arch
D
O
the
Oblique
Ascension
of
the
Noon-line
44.
06
minutes
from
B
upwards
at
C
and
from
C
yet
upwards
(
as
the
Scheam
shews
you
)
set
the
arch
O
R
13.
21.
for
the
Declination
of
the
Gnomon
:
and
draw
the
lines
A
C
for
the
Noon-line
,
and
A
D
for
the
Sub-style
.
The
North
Face
.
The
South
Face
.
diagram
And
lastly
set
the
Gnomon
Perpendicular
upon
the
Sub-style
A
D
,
making
his
angle
at
A
equal
to
the
arch
P
R
,
found
above
to
be
29.
08.
minutes
,
and
your
Dyal
is
done
.
CHAP.
XIX
.
How
to
know
at
what
Reclination
any
Declination
Plain
shall
become
a
Declining
Equinoctial
Dyal-Plain
,
to
be
delineated
after
Chapter
7.
And
how
to
find
the
Oblique
Ascension
of
his
Meridian
or
Substyle
,
and
the
difference
of
Longitude
,
which
are
requisite
for
his
Delineation
.
THere
is
no
Declining
plain
but
at
some
certain
Reclination
cutteth
through
the
Poles
of
the
World
and
so
becometh
a
Right
Horizon
.
Therefore
to
find
whither
a
Declining
Reclining
plain
do
happen
to
be
a
Declining
Equinoctial
plain
,
you
shall
observe
what
the
Elevation
of
the
Noon-line
N
O
is
;
for
if
that
be
equal
to
N
P
the
Latitude
,
then
doth
the
plain
cut
the
Poles
,
otherwise
not
:
And
at
what
Reclination
any
Decliner
shall
cut
the
Poles
,
and
so
have
the
Altitude
of
his
Noon
line
equal
to
the
Latitude
,
you
shall
thus
find
.
Use
the
Reet
and
Label
in
the
Horizontal
Projection
as
you
did
Chapter
16.
that
is
,
set
the
Horizontal
line
of
the
plain
D
d
on
the
Axis
of
the
Reet
,
then
number
from
the
Zenith
in
the
Limb
of
the
Reet
the
Complement
of
the
Declination
,
and
thereto
lay
the
Label
,
and
having
made
a
prick
with
ink
in
the
Label
for
the
Pole
(
52
¼
from
the
Limb
)
mark
which
of
the
Azimuths
cutteth
that
Pole
for
he
sheweth
you
at
what
Reclination
that
Decliner
shall
cut
the
Pole
and
fall
into
the
plain
of
one
of
the
Maridians
.
And
now
you
shall
have
but
one
Triangle
to
resolve
,
viz
D
N
P
▪
(
for
the
whole
Triangle
P
R
O
of
Chapter
16.
is
swallowed
up
in
the
point
of
the
Pole
P
)
and
and
D
N
P
hath
all
his
sides
known
,
and
the
angle
D
at
first
Sight
;
and
for
P
you
may
find
him
if
you
turn
the
Triangle
as
Book
3.
hath
been
shewed
.
Example
.
I
would
make
an
Equinoctial
Dyal
in
the
West
Declination
50.
degrees
,
Lay
the
Label
therefore
50.
degrees
from
the
Finitor
,
or
40.
from
the
Zenith
,
and
so
the
Axis
of
the
Reet
represents
the
plain
Declining
50.
degrees
and
the
Label
represents
the
North
part
of
the
Meridian
,
and
now
I
see
the
Azimuth
26
½
from
the
Axis
cutting
the
Label
in
the
place
of
the
Pole
;
therefore
I
say
,
that
Azimuth
represents
the
Equinoctial
plain
which
belongs
to
this
Declination
.
And
now
I
see
the
Triangle
D
N
P
on
my
Planisphear
,
N
P
on
the
Label
is
the
Latitude
,
and
also
the
Altitude
of
the
Noon
line
52.
10.
minutes
,
D
N
Complement
of
the
Declination
40.
D
P
the
Oblique
Ascension
of
the
Noon
line
61.
59.
minutes
,
N
is
a
Right
angle
,
D
is
Complement
of
the
Reclination
63.
28.
minutes
,
(
whose
measure
in
the
Scheam
is
T
V
)
P
the
Complement
of
the
difference
of
Longitude
,
for
the
difference
of
Longitude
it self
in
the
Scheam
is
H
P
Z
,
and
the
Complement
thereof
Z
P
T
to
which
D
N
P
is
equal
,
by
the
Structure
for
it
is
Angulus
pre
decussim
oppositus
:
and
by
any
of
the
four
first
Problemes
of
Rectangled
Triangles
you
may
find
it
to
be
46.
44.
minutes
,
whose
Complement
is
43.
16.
minutes
,
the
difference
of
Longitude
.
The
Oblique
Ascension
of
the
Noon
line
,
and
the
difference
of
Longitude
thus
found
,
you
shall
have
enough
to
make
the
Dyal
by
Chapter
7.
CHAP.
XX.
An
Admonition
concerning
the
five
several
Cases
of
Declining
Recliners
.
BEcause
by
the
diversity
of
Declination
and
Reclination
,
the
figure
and
situation
of
the
three
Triangles
mentioned
Chapter
16.
is
so
charged
that
you
cannot
alwayes
find
them
on
the
sudden
,
unless
you
have
a
firm
comprehension
of
the
Sphear
in
your
head
and
in
the
Case
of
the
last
Chapter
the
middle
Triangle
is
quite
lost
,
having
all
his
sides
and
angles
contracted
into
the
very
point
of
the
Pole
;
therefore
I
have
thought
good
to
set
down
the
5
several
Cases
of
these
Dyals
in
so
many
several
Scheams
,
and
in
every
Scheam
to
mark
the
Triangles
with
the
same
Letters
,
that
what
Case
soever
shall
happen
to
be
proposed
;
you
may
have
a
Scheam
ready
to
direct
you
.
And
to
know
which
Scheam
shall
serve
to
express
the
situation
of
your
plain
,
take
these
Rules
-
1.
If
the
plain
Recline
North
below
the
Pole
,
so
that
the
arch
N
O
the
Perpendicular
Altitude
of
the
Noon
line
be
less
then
N
P
the
Elevation
of
the
North
Pole
,
then
the
first
Scheam
serves
your
Case
.
-
2.
If
the
plain
Recline
to
the
Pole
making
N
P
and
N
O
equal
,
you
shall
use
the
Second
cheam
.
-
3.
If
the
plain
Recline
not
so
far
as
the
Pole
,
but
make
N
O
greater
then
N
P
,
you
shall
use
the
Third
Scheam
.
-
4.
If
the
plain
Recline
Southward
,
then
instead
of
the
Triangle
D
N
O
you
shall
use
the
opposite
Triangle
d
S
O
where
if
S
O
be
greater
then
the
Elevation
of
the
Equator
or
equal
to
it
,
you
shall
use
the
4th
Scheam
.
And
if
it
be
less
,
you
shall
use
the
Fifth
.
And
note
that
in
the
Fifth
Case
you
may
best
do
you
work
by
the
Triangle
P
H
Z
alone
,
(
the
Triangle
P
R
O
being
here
too
big
)
setting
off
your
Sub-style
from
the
Vertical
by
the
measure
of
the
angle
H
,
or
of
the
arch
T
R
,
the
Noon
line
from
the
Substyle
.
CHAP.
XXI
.
How
to
make
the
Declining
Horizontal
Dyal
,
another
way
then
was
shewed
Chapter
12.
THough
you
have
in
the
former
Chapters
a
perfect
Method
for
the
making
of
all
sorts
of
Dyals
which
give
the
Hour
by
the
shadow
of
the
Axis
of
the
World
,
or
a
Gnomon
set
Parallel
to
it
:
yet
I
think
it
both
pleasant
and
profitable
for
the
Reader
to
see
some
other
ways
whereby
the
same
things
may
be
performed
.
diagram
2.
Having
found
these
two
,
You
shall
set
your
Planisphear
in
the
first
Mode
of
the
Meridional
Projection
:
that
is
,
Set
the
Finitor
to
the
Elevation
of
the
Pole
above
the
plain
,
so
shall
you
have
all
your
Hour
distances
distinguished
upon
the
Finitor
by
the
Meridians
.
But
here
you
must
carry
this
in
your
head
,
that
here
the
Limb
is
not
the
Meridian
of
your
Place
,
but
of
the
plain
;
and
then
to
find
the
Meridian
of
your
Place
,
number
from
the
Limb
in
the
Finitor
the
Declination
of
the
Gnomon
,
and
the
Meridian
cutting
there
is
the
Meridian
of
your
place
,
and
stands
for
Noon
:
therefore
every
15th
Meridian
numbred
from
hence
(
and
not
from
the
Limb
in
this
Case
)
is
an
Hour
line
for
your
Dyal
:
and
look
at
what
distance
from
the
Limb
they
cut
the
Finitor
,
at
the
same
distance
from
the
Sub-style
shall
the
Hour-lines
be
set
in
your
Dyal
plain
.
Example
.
I
have
a
South
plain
Erect
Declining
30.
degrees
Eastward
,
(
as
in
Chapter
12.
)
And
for
this
,
First
I
set
the
Zenith
30
degrees
from
Meridies
,
toward
Oriens
,
and
so
doth
the
Finitor
represent
my
plain
,
and
I
see
the
Azimuth
21.10
.
minutes
cutting
the
South
Pole
of
the
Horizontal
Projection
about
38
degrees
from
the
Center
;
from
the
Pole
to
the
Finitor
I
number
in
this
Azimuth
32.01
.
minutes
,
the
Elevation
of
the
Pole
above
the
plain
;
and
from
the
Intersection
of
this
Azimuth
with
the
Finitor
to
the
Center
where
the
Meridian
of
the
Place
meets
,
I
number
in
the
Finitor
21.10
.
minutes
,
so
much
is
the
Declination
of
the
Gnomon
or
of
his
Substyle
from
the
Noon
line
.
Secondly
,
setting
the
Finitor
to
the
Plain's
Latitude
32.01
.
minutes
,
I
number
from
the
South
point
of
the
Finitor
inwards
21.10
.
minutes
,
and
there
cuts
the
Meridian
36.08
.
minutes
from
the
Limb
,
shewing
me
the
difference
of
Longitude
,
or
Equincctial
distance
of
the
Meridian
of
the
Place
from
the
Meridian
of
the
plain
;
for
this
36
Meridian
is
the
Meridian
of
my
Place
,
and
therefore
I
mark
him
well
,
he
is
the
Vertical
of
my
Dyal
,
and
also
the
Noon
line
.
And
here
I
consider
that
the
Sub-style
will
be
Westward
from
him
upon
the
plain
,
because
it
Declines
Eastward
(
by
Chapter
2.
Theorem
11.
)
Therefore
beginning
at
Noon
,
where
the
36
Meridian
cutteth
the
Finitor
I
go
15.
toward
the
Limb
,
and
light
upon
the
21th
Meridian
from
the
Substyle
for
11
a
clock
,
and
he
cutteth
the
Finitor
at
11.35
.
minutes
from
the
Limb
:
so
the
10th
Hour
is
the
sixt
Meridian
on
this
side
the
Limb
,
and
cutteth
the
Finitor
3.16
.
minutes
:
the
9th
Hour
is
the
ninth
Meridian
beyond
the
Limb
,
as
you
come
back
again
,
and
cutteth
the
Finitor
at
4.44
.
minutes
from
the
Limb
or
Sub-style
on
the
other
side
(
that
is
,
Westward
of
the
Sub-style
in
the
Dyal
.
)
In
like
manner
you
may
gather
the
distances
of
the
other
Hour
lines
from
the
Sub-style
into
a
Table
,
and
thereby
plot
them
down
as
in
the
Figure
.
CHAP.
XXII
.
To
make
the
Reclining
Declining
Dyal
,
another
way
.
HAving
found
the
arches
and
angles
requisite
by
Chapter
16.
and
platted
down
your
Horizontal
and
Vertical
lines
,
and
placed
the
Noon
line
above
o●
below
the
Horizontal
line
,
according
as
the
arch
of
his
Oblique
Ascension
or
Descension
requireth
,
and
having
placed
also
the
Sub-style
in
his
due
situation
as
is
above
taught
,
you
may
easily
find
the
distances
of
the
several
Hours
from
the
Sub-style
,
as
you
found
them
in
the
former
Chapter
for
the
Declining
Horizontal
Dyal
.
For
when
you
have
set
the
Finitor
to
the
Latitude
of
your
plain
,
as
there
you
did
,
the
Limb
is
Sub-stylar
,
and
if
you
number
thence
in
the
Finitor
the
Declination
of
the
Gnomon
,
there
shall
meet
you
the
Meridian
of
the
Place
.
Here
you
shall
begin
,
and
take
every
15th
Meridian
forwards
and
backwards
for
an
Hour
line
,
and
observing
how
many
degrees
are
in
the
Finitor
between
the
Limb
and
every
one
of
these
Hour
lines
,
so
many
degrees
shall
you
place
that
Hour
line
from
the
Sub-style
in
the
plain
.
If
you
understand
the
former
Chapter
this
will
need
no
Example
.
CHAP.
XXIII
.
To
draw
the
proper
Hours
of
any
Declining
Dyal
.
EVery
Declining
plain
,
whether
it
Recline
or
not
,
hath
two
great
Meridians
much
spoken
of
.
1.
The
Meridian
of
the
plain
,
which
is
the
proper
Meridian
of
that
Country
to
whose
Horizon
the
plain
heth
Parallel
.
2.
The
Meridian
of
the
Place
,
which
is
the
Meridian
of
your
Country
,
in
which
you
set
up
this
Declining
plain
to
shew
the
Hours
;
and
so
either
of
these
Meridians
Dyals
may
be
conformed
.
How
to
draw
the
Hours
of
our
Country
on
such
a
plain
is
the
harder
work
,
because
the
plain
is
Irregular
to
our
Horiz
on
:
yet
I
suppose
I
have
made
the
way
very
easy
in
the
former
Chapters
.
But
to
draw
the
Hours
of
the
Country
to
which
the
plain
belongs
,
is
most
easy
.
For
if
you
take
the
Sub-stylar
for
the
Noon-line
,
and
the
Elevation
of
the
Pole
above
the
plain
for
the
Latitude
,
you
may
make
this
Dyal
in
all
points
like
the
Vertical
Dyal
,
after
the
precept
of
the
9th
Chapter
.
CHAP.
XXIV
.
To
know
in
what
Country
any
Declining
Dyal
shall
serve
for
a
Vertical
Dyal
.
IF
the
Dyal
Decline
East
,
add
the
difference
of
Longitude
(
found
as
above
Chapter
21.
)
to
the
Longitude
of
your
Place
,
and
the
sum
or
the
excess
above
360
is
the
number
of
the
Longitude
sought
.
If
the
Dyal
Decline
West
subtract
the
said
difference
of
Longitude
out
of
the
Longitude
of
your
Place
,
and
the
difference
is
the
Longitude
inquired
:
but
when
the
Longitude
of
your
Place
happens
to
be
less
then
the
difference
of
Longitude
you
must
add
to
it
360.
before
you
subtract
the
difference
of
Longitude
.
The
Elevation
of
the
Pole
above
the
plain
is
the
Latitude
of
the
Place
inquired
.
Example
.
The
Declining
plain
of
Chapter
12.
will
be
a
Vertical
plain
in
the
Longitude
61.
degrees
,
and
North
Latitude
32.
degrees
,
that
is
,
in
the
Mediterranean
Sea
between
Alexandria
and
the
Isle
of
Creet
.
And
the
Declining
Reclining
plain
of
Chapter
16
,
17
,
18.
is
Parallel
to
the
Horizon
of
those
that
sail
in
Longitude
359.
degrees
,
and
North
Latitude
29.
degrees
that
is
as
Terrestrial
Globes
and
Mapps
shew
me
,
between
the
Azores
and
Hesperides
.
CHAP.
XXV
.
To
set
a
Plain
Parallel
to
the
Horizon
of
any
Country
proposed
.
IF
you
can
get
the
Declination
and
Reclination
of
such
a
plain
,
you
have
enough
to
place
him
in
his
true
Situation
.
And
those
may
be
found
by
the
difference
of
Longitude
and
the
Latitude
of
the
strange
Country
,
(
which
are
in
this
Probleme
supposed
to
be
given
)
even
as
in
Chapter
16.
you
found
both
those
by
the
Declination
and
Reclination
given
.
Example
.
I
would
set
a
plain
Parallel
to
the
Horizon
of
Jerusalem
,
to
shew
me
what
time
the
Sun
Rises
and
Sets
there
any
day
of
the
Year
,
and
what
Hour
passeth
at
Jerusalem
at
any
time
of
our
day
.
First
I
seek
by
Geographical
Tables
or
Mapps
the
Longitude
and
Latitude
of
Jerusalem
,
and
I
find
that
Jerusalem
is
removed
Eastward
from
London
in
Longitude
47
degrees
,
and
that
the
Latitude
there
is
32
degrees
.
or
thereabouts
.
Therefore
in
the
Rectangled
Triangle
P
R
O
of
Chapter
16.
I
have
the
angle
P
47
degrees
difference
of
Longitude
,
also
the
side
P
R
the
Latitude
of
Jerusalem
32
degrees
,
and
hence
by
the
4th
Probleme
of
Rectangled
Triangles
Book
3.6
.
I
get
P
O
42.30
minutes
,
and
by
consequence
O
N
9.45
.
minutes
(
because
P
N
is
our
Latitude
)
and
I
get
also
the
angle
O
51.40
minutes
.
And
these
had
,
I
get
by
the
same
Probleme
in
the
adjoyning
Triangle
O
N
D
,
both
D
N
12.05
.
degrees
,
the
Complement
of
the
Declination
inquired
,
and
the
angle
D
39.23
.
Complement
of
the
Reclination
inquired
.
Wherefore
I
conclude
that
a
plain
which
shall
represent
here
the
Horizonof
Jerusalem
must
Decline
Eastward
77.55
.
minutes
,
and
Recline
Northward
50.37
.
minutes
.
Draw
upon
this
plain
the
proper
Hours
of
Jerusalem
,
by
Chapter
23.
and
know
that
when
the
Sun
leaveth
this
plain
ceasing
to
inlighten
the
upper
part
of
it
,
then
he
setteth
at
Jerusalem
,
and
look
how
many
Hours
and
minutes
the
Sun
setteth
after
noon
in
any
Country
,
so
many
Hours
and
minutes
he
rose
before
noon
.
CHAP.
XXVI
.
How
other
Circles
of
the
Sphear
besides
the
Meridians
may
be
Projected
upon
Dyals
.
THe
Projection
of
some
other
Circles
of
the
Sphear
beside
the
Meridians
though
it
be
not
necessary
for
finding
the
Hours
yet
may
be
both
an
ornament
to
Dyals
,
and
usefull
also
for
finding
the
Meridian
,
and
placing
the
Dyal
in
his
due
Situation
,
if
it
be
made
upon
a
moveable
Body
,
as
shall
be
shewed
Chapter
33.
The
Circles
fittest
to
be
projected
in
all
Dyals
for
those
purposes
are
the
Equator
with
the
Tropiques
and
other
his
Parallels
;
which
may
be
accounted
Parallels
of
Declination
,
as
they
pass
through
equal
degrees
,
as
every
5th
or
10th
of
Declination
:
or
Parallels
of
the
Signes
,
as
they
pass
through
such
degrees
of
Declination
as
the
Sun
Declineth
,
when
he
entreth
into
any
Signe
,
or
any
notable
degree
thereof
;
or
Parallels
of
the
length
of
the
day
,
as
they
pass
through
such
degrees
of
Declination
wherein
the
Sun
increaseth
or
decreaseth
the
length
of
the
day
by
Hours
or
half-Hours
.
Also
the
Horizon
with
his
Azimuths
and
Almicantars
are
an
ornament
to
Horizontal
and
Vertical
Dyals
;
and
are
likewise
use
full
for
projecting
the
Equator
and
his
Parallels
in
all
Dyals
.
My
purpose
is
to
be
breif
in
this
Treatise
of
the
Tumiture
here
following
because
I
hasten
to
an
end
.
I
shall
therefore
think
it
sufficient
if
I
shew
you
one
way
to
furnish
any
Dyal
with
the
Circles
of
the
Sphear
.
Leaving
you
to
devise
others
which
I
could
have
shewn
.
CHAP.
XXVII
.
How
to
describe
on
any
Dyal
the
proper
Azimuths
and
Almicantars
of
the
Plain
.
FRom
any
point
of
the
Gnomon
(
taken
at
pleasure
)
let
fall
a
Perpendicular
upon
the
Sub-style
;
that
Perpendicular
shall
be
part
of
the
Axis
of
the
plain
,
and
shall
be
reputed
Radius
to
the
Horizon
of
your
plain
.
The
top
of
this
Radius
in
the
Gnomon
is
called
Nodus
,
because
you
must
there
set
a
Knot
Bead
,
or
Button
to
give
shade
,
or
else
cut
there
a
notch
in
the
Gnomon
,
or
cut
off
the
Gnomon
in
the
Place
of
Nodus
,
that
the
end
may
give
the
shadow
for
those
lineaments
.
Let
not
your
Nodus
stand
too
high
above
the
plain
,
for
then
the
shadow
will
fall
beside
your
plain
for
too
great
a
part
of
the
plains
day
nor
let
it
stand
too
low
,
for
then
the
lineaments
will
run
too
close
together
.
A
mean
must
be
chosen
.
At
the
foot
of
this
Radius
take
your
Center
,
and
describe
a
Circle
on
the
plain
and
divide
it
into
equal
degrees
;
and
from
the
Center
draw
lines
through
those
degrees
infinitely
,
that
is
,
so
far
as
your
Dyal-plain
will
bear
;
these
lines
shall
be
the
Azimuths
of
the
Horizon
of
the
plain
,
and
shall
be
numbred
from
his
Meridian
or
Sub
style
.
diagram
And
to
divide
the
said
Azimuth
lines
you
use
the
Tables
of
Tangents
with
a
Scale
of
equal
parts
,
or
else
plot
the
Tangents
thus
on
paper
,
set
A
B
equal
to
the
Radius
of
your
Horizon
,
and
with
that
Radius
draw
the
Quadrant
A
B
C
,
or
A
b
c
,
and
divide
the
Quadrant
,
numbring
the
degrees
from
C
to
B
,
and
having
drawn
the
Tangent
B
D
,
or
B
d
,
Parallel
to
A
C
,
draw
lines
from
the
Center
through
the
several
degrees
to
the
said
Tangent-line
,
so
shall
this
Tangent-line
be
divided
for
your
purpose
:
and
from
it
you
may
transfer
the
divisions
to
your
plain
.
Now
if
your
plain
lie
in
the
Horizon
of
your
place
,
(
as
the
Vertical
plain
doth
)
these
Azimuths
and
Almicantars
may
be
of
some
use
to
shew
you
the
Altitude
and
Azimuth
of
the
Sun
for
any
time
.
See
them
in
the
Scheam
Chapter
30.
But
if
your
plain
lie
not
in
the
Horizon
of
your
place
,
then
you
shall
draw
the
said
Almicantars
or
Azimuths
,
or
so
many
of
them
as
you
shall
need
,
in
deleble
lines
,
because
here
they
serve
only
the
Horizon
of
the
plain
:
yet
shall
they
help
you
to
describe
the
Equator
and
his
Parallels
,
with
the
Horizon
of
your
Place
in
any
Dyal
:
and
when
they
have
done
this
,
unless
your
Dyal
be
Vertical
,
they
may
be
gone
.
CHAP.
XXVIII
.
How
by
help
of
the
proper
Azimuths
and
Almicantars
of
the
Plain
to
describe
the
Equator
and
his
Parallels
,
on
the
Polar
or
Orthognomonical
Dyal
.
IT
shall
suffice
here
to
shew
how
the
Parallels
of
the
Signes
may
be
described
,
because
the
Parallels
of
Declination
and
of
the
length
of
the
day
are
described
by
like
reason
.
And
know
that
in
the
Polar
plain
because
the
Gnomon
is
Perpendicular
to
the
plain
,
the
same
Gnomon
shall
serve
both
Hours
and
Azimuths
;
for
the
Hour-lines
be
Azimuths
in
this
plain
.
Note
also
that
the
Sun
is
never
Elevated
above
this
plain
more
then
he
Declineth
from
the
Equator
,
which
at
the
most
is
23
½
degrees
,
and
that
if
the
height
of
Nodus
be
above
a
sixth
part
of
the
Semidiameter
of
the
plain
,
the
ten
first
Almicantars
will
fall
beside
the
plain
.
A
sixth
part
therefore
must
serve
,
and
that
will
give
you
all
the
Altitudes
above
to
degrees
,
and
the
Parallels
of
the
Signes
whose
Declination
is
more
then
10.
Describe
therefore
the
Almicantars
here
,
as
you
were
taught
Chapter
27.
for
in
the
Hour
lines
you
have
already
every
15th
Azimuth
,
and
may
draw
more
if
you
please
.
diagram
CHAP.
XXIX
.
How
to
inscribe
the
Equator
and
his
Parallels
,
in
the
Equinoctial
or
Parallelognomonical
Dyal
.
IF
this
plain
Decline
not
,
the
Hour
lines
of
your
Country
will
serve
you
for
they
be
also
the
Hour
lines
of
the
plain
,
and
the
Noon-line
is
Sub
style
if
it
do
Decline
,
you
shall
draw
in
deleble
lines
the
proper
Dyal
of
the
plain
(
by
Chapter
23.
which
Declineth
not
.
And
having
here
the
Azimuths
or
Almicantars
of
the
plain
,
drawn
by
Chapter
27.
you
shall
observe
upon
your
Planisphear
at
what
Altitude
or
at
what
Azimuth
the
Parallels
cut
the
several
Hour
lines
,
and
where
the
like
Altitude
or
Azimuth
cuts
the
same
Hour
lines
upon
the
plain
,
you
shall
make
marks
,
and
through
those
marks
draw
the
Parallels
which
shall
be
all
Conical
sections
,
except
the
Equator
,
which
because
he
is
a
great
Circle
,
shall
be
a
straight
line
on
those
plains
;
and
in
all
other
plains
,
except
the
Polar
,
where
he
is
a
Circle
.
diagram
In
the
Equator
the
Azimuth
is
always
the
same
,
full
East
or
West
,
and
so
upon
your
plain
he
must
needs
be
a
straight
line
.
The
Altitudes
in
the
Equator
are
15
,
30
,
45
,
60
,
75
,
90.
The
Hours
alike
distant
from
the
Meridian
on
both
sides
are
alike
,
and
so
are
the
Parallels
alike
distant
from
the
Equator
alike
also
.
When
you
have
therefore
gathered
a
Table
out
of
your
Planisphear
for
the
Morning
Hours
of
the
North
Parallels
,
and
of
the
Equator
,
(
as
I
have
done
here
in
hast
for
the
Equator
,
and
Tropique
of
Cancer
)
you
may
by
that
Table
prick
down
the
Parallels
upon
one
quarter
of
your
Dyal
,
and
by
that
also
draw
the
rest
;
for
as
you
may
see
upon
your
Planisphear
,
all
the
4.
quarters
are
alike
.
Note
that
the
Azimuths
cut
the
Hour
lines
too
Obliquely
:
it
is
best
therefore
to
trust
to
the
Almicantars
,
and
so
shall
you
have
easier
and
surer
work
,
though
you
meddle
not
with
the
Azimuths
at
all
.
CHAP.
XXX
.
How
to
inscribe
the
Equator
and
his
Parallels
,
in
an
Oblique
or
Scalenognomonical
Dyal
.
IF
the
plain
neither
Decline
nor
Recline
,
and
so
be
a
Vertical
plain
,
the
Hour
lines
of
your
Place
will
serve
you
for
they
be
also
the
Hour
lines
of
the
plain
,
and
the
Noon
line
is
the
Sub-style
.
If
it
either
Decline
or
Recline
,
or
both
Decline
and
Recline
,
you
shall
draw
in
deleble
lines
the
proper
Dyal
of
the
plain
by
Chapter
23.
and
so
this
Dyal
shall
be
reduced
to
a
Vertical
Dyal
,
and
be
as
casily
furnished
with
the
Parallels
as
the
Vertical
:
and
when
you
have
by
the
deleble
Hour
lines
of
the
plain
inscribed
the
Parallels
,
you
may
wipe
out
those
Hour
lines
of
the
plain
,
and
let
the
Hour
lines
of
the
place
and
the
Parallels
stand
.
Having
therefore
drawn
the
Azimuths
or
Almicantars
of
your
plain
by
Chapter
27
,
take
your
Planisphear
in
the
Meridional
Projection
,
setting
the
Finitor
to
the
Latitude
of
your
plain
;
Then
find
your
Equator
and
Parallels
on
the
Mater
,
and
where
the
several
Hour
lines
intersect
them
above
the
Finitor
,
mark
what
Azimuth
or
rather
what
Almicantar
passeth
through
these
intersections
;
for
in
the
same
Azimuth
and
Almicantar
shall
the
Parallels
cut
the
Hour
lines
of
the
plain
upon
the
plain
.
Example
.
In
the
Vertical
Dyal
for
our
Latitude
52.15
.
minutes
,
I
set
the
Finitor
to
this
Latitude
,
and
going
first
to
the
Tropique
of
Cancer
,
I
begin
at
the
Limb
,
that
is
,
at
Noon
,
there
I
see
the
Azimuth
full
South
the
Altitude
61.15
.
minutes
,
at
1.
a
clock
:
Azimuth
27.
Altitude
59.
at
2.
Azimuth
49
½
.
Altitude
53
⅓
at
3.
Azimuth
67.
Altitude
45
½
.
&c.
Therefore
where
I
find
the
Noon
line
of
my
plain
cut
by
the
Almicantar
61
¼
,
I
make
a
prick
,
and
in
the
Hour
lines
of
11
and
of
1.
where
the
Azimuth
27
and
the
Almicantar
59
meet
,
I
make
pricks
,
and
where
the
Azimuth
49
½
and
the
Almicantar
53
⅓
do
meet
upon
the
Hour
lines
of
10.2
.
I
make
pricks
,
and
so
for
the
rest
.
Lastly
I
draw
with
an
even
hand
a
crooked
line
without
angles
through
those
pricks
,
and
that
shall
be
the
Parallel
or
Tropique
of
Cancer
:
and
in
like
manner
I
put
in
all
the
other
Parallels
,
and
the
Equator
in
the
midst
of
them
,
though
for
the
Equator
you
may
draw
him
more
speedily
by
striking
a
line
through
the
Center
of
the
Almicantars
making
Right
angles
with
the
Substyle
.
And
that
may
be
a
general
Rule
for
the
Equator
in
all
Dyals
which
have
a
Substyle
,
and
in
the
Polar
Dyal
where
there
is
no
Substyle
,
the
Equator
shall
be
a
Circle
,
as
before
is
shewn
.
Note
here
that
if
your
Dyal
be
great
,
and
you
have
not
points
enough
to
govern
you
in
the
draught
of
the
Conical
sections
you
may
draw
half-hour-lines
,
and
find
points
in
them
also
,
after
the
same
manner
.
diagram
CHAP.
XXXI
.
To
do
the
same
by
the
Hour-lines
of
the
Place
,
although
the
Plain
Decline
or
Recline
.
IF
you
like
not
to
draw
the
proper
Dyal
of
the
plain
where
it
Declines
or
Reclines
,
because
being
useless
in
your
Country
it
must
be
wiped
out
again
,
it
shall
suffice
you
to
find
the
Hour
lines
of
your
Country
upon
the
plain
by
Chapter
21
,
and
22
and
in
the
posture
your
Planisphear
hath
in
those
Chapters
to
observe
what
Almicantars
or
Azimuths
do
cross
those
Hour
lines
at
the
several
Hours
in
any
Parallel
,
and
thereby
make
pricks
upon
the
Hour
lines
of
your
Place
,
as
in
the
former
Chapter
you
did
upon
the
Hour
lines
of
your
plain
;
and
by
these
pricks
you
may
draw
your
Parallels
as
before
.
Note
that
if
you
work
this
way
,
you
shall
find
the
Suns
greatest
Altitude
to
be
in
the
Meridian
of
the
plain
or
Substyle
,
and
not
in
the
Noon-line
of
your
Place
;
whereat
you
must
not
wonder
:
so
if
the
Substyle
be
about
9
in
the
Morning
there
you
shall
find
the
Sun
at
highest
,
and
that
his
Altitude
decreaseth
from
thence
till
he
leaves
the
plain
.
CHAP.
XXXII
.
How
to
inscribe
the
Horizon
of
the
Place
,
with
his
Azimuths
and
Almicantars
,
in
the
Horizontal
Dyal
.
THe
Nodus
may
be
chosen
in
any
part
of
the
Gnomon
,
but
with
the
eaution
given
Chapter
27.
and
a
Perpendicular
falling
from
the
Nodus
on
the
Sub-style
shall
touch
the
Center
of
the
Azimuths
and
Almicantars
of
the
plain
,
as
hath
been
shewed
Chapter
27.
Here
you
have
no
use
of
those
Azimuths
and
Almicantars
:
but
through
the
Center
of
them
you
shall
draw
an
Horizontal
line
by
Chapter
3.
and
that
shall
be
Horizon
.
Now
if
your
plain
Decline
not
from
the
Meridian
,
and
so
this
Center
fall
upon
the
Noon
line
,
you
shall
divide
your
Horizon
both
wayes
from
the
Center
,
as
you
were
taught
to
divide
the
Azimuthal
lines
by
Tangents
,
Chapter
27.
and
shall
number
those
divisions
from
the
Center
on
both
sides
5
,
10
,
15
,
20.
&c.
and
from
the
several
points
so
made
for
5
,
10
,
15
,
&c.
In
the
Horizontal
line
let
fall
Perpendiculars
or
Vertical
lines
on
the
plain
,
and
they
shall
be
Azimuths
of
your
Place
.
But
if
your
plain
Decline
you
shall
divide
the
Horizontal
line
thus
.
Draw
a
short
Vertical
line
through
the
said
Center
downwards
,
by
Chapter
3.
which
shall
be
the
Verticle
of
your
plain
.
diagram
For
the
Almicantars
they
will
not
be
so
handsome
lines
,
but
if
you
will
have
them
,
do
thus
.
If
the
plain
Decline
not
,
set
the
Finitor
to
the
Latitude
of
your
Place
,
as
Chapter
10.
and
if
it
Decline
,
set
the
Finitor
to
the
Latitude
of
your
plain
as
Chapter
21.
Then
keeping
your
ey
above
the
Horizon
,
and
within
the
Tropiques
,
mark
what
Hour
lines
the
10th
Almicantar
(
for
Example
)
cutteth
,
and
what
Azimuth
there
with
him
cutteth
the
same
Hour
lines
also
,
and
in
the
interfections
of
the
same
Azimuths
and
Hour
lines
upon
your
plain
you
shall
make
marks
,
through
which
the
tenth
Almicantar
shall
be
drawn
:
and
so
of
the
rest
.
Note
here
That
your
Azimuths
and
Almicantars
must
not
be
drawn
beyond
the
Tropiques
,
nor
beyond
the
Horizon
:
neither
must
the
Hour
lines
,
if
the
Nodus
be
the
end
of
the
Gnomon
.
The
Scheam
shews
you
how
the
Azimuths
may
be
drawn
on
the
Dyal
of
Chapter
12.
and
21.
Declining
East
30
degrees
.
CHAP.
XXXIII
.
How
by
the
help
of
this
Furniture
to
place
any
moveable
Dyal-Plain
in
his
true
Situation
,
and
consequently
to
find
the
Meridian-line
of
the
Place
,
without
any
other
Instrument
then
the
Dyal
it self
.
SEt
the
Dyal
upon
a
level
Table
or
Board
,
and
turn
it
till
the
shadow
of
the
Nodus
touch
the
Suns
Parallel
,
Azimuth
,
or
Almicantar
,
any
or
all
of
them
:
but
the
Parallel
shall
best
guide
you
,
because
that
is
most
easily
known
by
memory
without
Observation
.
And
when
the
shadow
of
the
Nodus
toucheth
the
Suns
Parallel
it
shews
there
the
Hour
also
;
and
moreover
it
shews
the
Suns
Altitude
and
Azimuth
for
the
same
Time
if
the
Azimuths
and
Almicantars
also
be
drawn
upon
your
Dyal
.
But
you
shall
note
here
,
that
the
shadow
of
the
Nodus
may
touch
the
Parallel
at
like
distance
from
the
Sub-style
on
both
sides
.
Therefore
if
you
be
in
doubt
which
is
the
true
place
of
touching
(
as
you
may
well
doubt
when
the
shadow
cuts
the
Parallel
near
the
Sub-style
)
you
shall
Observe
a
while
whether
the
sh
dow
of
the
Radius
be
lengthning
or
shortning
:
If
it
shorten
,
the
Sun
is
not
come
to
the
Sub-style
,
and
so
the
earlyer
Hour
shewed
is
the
true
Hour
;
If
it
lengthen
,
the
Sun
is
past
the
Sub-style
,
and
the
later
Hour
is
the
true
Hour
.
And
when
the
Dyal
shews
the
true
Hour
,
the
Gnomon
and
the
plains
Parallel
thereto
do
point
North
and
South
.
And
here
you
may
see
,
that
the
further
the
Sun
is
from
the
Sub-style
,
the
more
easily
is
the
Dyal
placed
.
Thus
may
you
make
a
very
commodious
Polar
Dyal
,
to
stand
in
a
chamber
Window
,
and
to
remove
from
Window
to
Window
as
the
Sun
goes
,
which
shall
find
the
Meridian
line
it self
any where
,
in
the
4.
Summer
and
4.
Winter
Moneths
;
and
if
you
will
make
him
a
Limb
,
like
the
Limb
of
a
Box-lid
of
a
Cheese-fat
,
to
receive
the
Parallels
near
the
Equinoctial
,
which
else
fall
beyond
the
plain
,
he
shall
serve
for
all
the
Year
.
CHAP.
XXXIV
.
How
to
make
a
Vertical
Dyal
upon
the
Ceeling
of
a
Floor
within
Dores
,
where
the
Direct
Beams
of
the
Sun
never
come
.
THe
greatest
part
and
as
much
as
you
shall
use
of
the
Vertical
Dyal
described
Chapter
9.
may
by
Reflection
be
turned
upside
down
,
and
placed
upon
a
Ceeling
,
but
the
Center
will
be
in
the
Air
without
Dores
.
A
peece
of
a
Looking-glass
as
broad
as
a
Groat
or
Six-pence
set
level
,
or
a
Gally-pot
of
fair
Water
,
which
will
set
it self
level
,
being
placed
upon
the
sole
of
the
Window
shall
supply
the
use
of
the
Nodus
in
the
Gnomon
,
and
the
beams
of
the
Sun
being
Reflected
by
this
Glass
or
Water
shall
shew
the
Hours
upon
the
Ceeling
.
The
Planisphear
shall
help
you
to
make
this
Dyal
two
wayes
.
If
the
Window
Decline
not
much
from
the
South
,
you
may
make
it
most
easily
the
First
way
.
But
if
it
Decline
much
,
and
so
the
lines
fall
much
upon
the
partition
Walls
,
or
if
you
would
adorn
this
Dyal
with
the
Parallels
or
other
Circles
,
you
shall
use
the
Second
way
.
The
First
way
is
this
Draw
a
Meridian
line
upon
the
Floor
.
by
Book
4.3
.
so
that
it
may
point
upon
the
Perpendicular
,
which
you
shall
imagine
to
fall
from
the
Nodus
upon
the
plain
of
the
Floor
prolonged
.
And
this
may
be
most
easily
done
,
if
you
hang
a
Plumb-line
in
the
Window
dnecuy
over
the
Nodus
of
place
of
the
Glais
,
for
the
shadow
which
that
Plumb-line
gives
upon
the
Floor
at
Noon
,
is
the
Meridian
line
sought
;
and
by
a
Ruler
or
a
line
stretched
upon
it
you
may
prolong
it
as
far
as
you
shall
need
.
Then
let
a
Plumb
line
fall
from
the
Ceeling
upon
this
Meridian
line
of
the
Floor
,
and
behind
it
Northward
or
Southward
,
place
your
Ey
,
so
that
the
Plumb-line
may
hide
the
Meridian
line
of
the
Floor
from
your
Ey
:
then
keeping
your
head
steddy
,
cast
you
:
Ey
up
to
the
Ceeling
,
and
direct
One
to
make
two
points
at
a
good
distance
,
in
the
line
upon
the
Ceeling
which
the
Plumb-line
now
covereth
from
your
Ey
,
and
by
these
points
you
shall
draw
a
straight
Meridian
on
the
Ceeling
.
diagram
Then
having
fastned
one
end
of
a
Line
at
Nodus
,
let
Another
stretch
this
line
up
to
the
Meridian
on
the
Ceeling
,
and
let
him
move
his
hand
nearer
or
further
in
the
Meridian
till
you
find
by
a
Quadrant
that
this
line
pointeth
up
Northward
as
many
degrees
as
the
Elevation
of
the
Equator
is
in
your
Country
,
and
then
you
shall
cause
him
to
make
a
point
where
the
line
toucheth
the
Meridian
of
the
Cieling
,
and
through
that
point
you
shall
draw
the
Equinoctial
line
of
your
Dyal
,
cutting
the
said
Meridian
at
Right
angles
.
The
length
of
the
thred
from
the
Nodus
to
the
point
in
the
Meridian
where
the
Equinoctial
cuts
him
,
is
Radius
of
the
Equinoctial
:
to
that
Radius
you
shall
find
the
Tangents
of
15
,
30
,
45
,
60
,
75.
as
you
found
the
Co-tangents
Chapter
27.
(
knowing
that
the
Co-tangents
of
80
,
and
70.
be
the
Tangents
of
10
,
and
20
,
and
so
of
the
rest
)
and
beginning
in
the
Meridian
make
pricks
in
the
Equinoctial
line
,
at
the
end
of
the
Tangent
of
15.
Eastward
for
1.
and
Westward
for
11.
and
at
the
end
of
the
Tangent
of
30.
prick
Eastward
2.
and
Westward
10.
&c.
Then
by
Chapter
9.
seek
what
angles
the
Hour
lines
of
a
Vertical
Dyal
make
at
the
Center
,
which
in
our
Latitude
are
1.11.58
.
minutes
,
2.24.32
.
minutes
,
3.38.20
.
minutes
,
4.53.52
.
minutes
,
5.71.17
.
minutes
,
and
with
the
Complements
of
these
angles
shall
these
Hour
lines
cross
the
Equinoctial
:
so
the
Hour
line
of
1.
shall
Incline
to
the
Meridian
on
the
South
side
the
Equinoctial
line
,
and
shall
make
his
lesser
angle
with
the
Equinoctial
78.02
.
minutes
,
and
the
rest
as
in
the
Figure
.
The
Second
way
is
this
.
Fit
a
plain
smooth
Board
about
a
foot
Square
to
lie
level
from
the
fole
of
the
Window
inwards
,
and
near
the
outer
edge
thereof
make
a
Center
in
the
board
in
the
very
place
of
Nodus
,
or
a
little
under
it
,
remembring
that
the
Nodus
or
Center
of
the
Glass
must
be
set
so
much
higher
then
this
board
,
as
the
Center
of
your
Quadrant
is
placed
higher
in
the
Projecting
of
the
Dyal
.
Upon
that
Center
taken
in
the
board
describe
as
much
of
a
Circle
as
you
may
with
the
Semidiameter
of
your
Quadrant
;
which
Circle
shall
be
Horizon
:
Draw
here
from
the
Center
to
the
Horizon
inwards
a
Meridian
line
,
by
Book
4.3
.
and
where
it
cuts
the
Horizon
begin
to
graduate
the
Horizon
into
degrees
of
Azimuths
both
wayes
,
which
you
may
speedily
do
,
by
transferring
the
graduations
of
your
Quadrant
,
or
so
much
as
you
shall
need
,
to
this
Horizon
.
Next
you
must
devise
to
make
your
Quadrant
stand
firm
and
upright
upon
one
of
his
straight
sides
,
(
which
I
will
call
his
foot
for
this
time
)
and
that
you
may
thus
do
;
Take
a
short
peece
of
a
Ruler
or
sinal
Transom
,
and
saw
in
one
side
of
it
a
notch
Perperdicularly
,
in
which
notch
you
may
stick
fast
or
wedge
the
heel
or
the
toe
of
your
Quadrant
,
in
such
sort
that
his
foot
may
come
close
to
the
board
,
and
the
other
straight
side
or
leg
may
stand
Perpendicular
upon
it
.
Those
things
prepared
,
put
your
Planisphear
in
the
Meridional
Projection
,
with
the
Finitor
at
your
Latitude
,
and
first
observe
there
the
Altitudes
of
the
Sun
in
the
Meridian
,
which
in
Latitude
52.15
.
minutes
,
you
shall
find
in
the
Tropique
of
♋
61.15
.
minutes
,
in
the
Equator
37.45
.
minutes
,
and
in
the
Tropique
of
♑
14.15
.
minutes
.
Now
having
stuck
a
short
needle
in
the
Center
of
the
Horizon
,
close
to
which
you
must
alwayes
keep
the
Center
of
your
Quadrant
,
set
the
foot
of
your
Quadrant
in
the
Meridian
line
of
the
Board
,
and
from
the
Center
of
your
Quadrant
extend
a
thred
by
14.15
.
minutes
of
Altitude
straight
on
to
the
Cieling
(
the
thred
only
touching
the
plain
of
the
Quadrant
and
making
no
angle
with
it
,
but
held
Parallel
)
and
where
the
thred
thus
extended
touches
the
Cieling
make
a
point
,
then
the
Quadrant
unmoved
,
extend
the
thred
by
61.15
.
minutes
of
Altitude
,
and
make
another
point
as
before
,
and
between
these
two
points
draw
a
straight
line
,
and
that
shall
be
your
Meridian
,
and
shall
be
long
enough
for
your
use
:
then
extend
the
thred
by
37.45
.
minutes
of
Altitude
,
and
where
it
touches
this
Meridian
cross
the
Meridian
at
Right
angles
with
an
infinite
line
,
which
shall
be
the
Equator
.
Then
seek
upon
your
Planisphear
for
one
a
clock
,
and
you
shall
find
in
the
Tropique
of
♑
the
Suns
Azimuth
14.
and
his
Altitude
13.06
.
In
the
Tropique
of
♋
his
Azimuth
27½
and
his
Altitude
59.04
.
minutes
:
therefore
setting
the
foot
of
the
Quadrant
in
the
Azimuth
14.
from
the
Meridian
Eastward
,
I
extend
the
thred
by
13.06
.
of
Altitude
,
and
make
a
prick
in
the
Cieling
:
and
again
setting
the
foot
of
the
Quadrant
in
Azimuth
27
½
.
and
extending
the
thred
by
59.04
minutes
of
Altitude
,
I
make
another
prick
in
the
Cieling
,
and
the
straight
line
which
I
shall
draw
between
these
two
pricks
shall
be
all
the
Hourlines
of
One
,
and
so
of
the
rest
.
And
if
you
be
minded
to
have
the
other
Parallels
drawn
,
you
may
find
points
for
them
as
you
have
done
for
the
Tropiques
,
and
by
those
points
draw
them
.
And
note
that
two
points
made
in
the
Cieling
for
the
same
Hour
line
in
any
two
Parailels
,
or
in
the
Equator
and
any
Parallel
,
shall
suffice
to
direct
the
line
,
though
it
is
best
to
take
your
points
in
the
Tropiques
,
at
the
largest
distance
,
as
I
have
here
done
,
if
there
be
room
enough
on
the
Cieling
.
But
because
it
often
happens
that
part
of
your
Dyal
falls
beside
the
Cieling
,
and
the
plain
of
the
Cieling
and
of
the
Walls
is
often
interrupted
,
and
made
Irregular
by
Beams
,
Wal-plates
,
Corrishes
,
Wainscot
,
Chimney-peeces
,
and
such
like
bodyes
,
I
will
●hew
you
one
absolute
device
to
carry
on
your
Hour
lines
over
all
.
Extend
the
thred
for
any
Hour
line
to
the
Tropique
of
Cancer
●n
the
Cieling
,
as
you
where
taught
before
,
and
fix
it
there
,
and
extend
another
thred
in
like
manner
to
the
Tropique
of
Copricorn
,
where
ever
it
shall
happen
,
(
as
perhaps
beyond
the
middle
beam
,
or
quite
beyond
the
Cieling
upon
the
Wall
)
and
fix
that
thred
also
.
Then
place
your
Ey
so
behind
these
threds
that
one
of
them
may
cover
the
other
,
and
at
the
same
instant
where
the
upper
line
(
to
your
Sight
or
Imagination
)
cuts
the
Cieling
,
Beams
,
Wall
,
or
any
Regular
or
Irregular
body
,
above
the
end
of
the
lower
line
,
there
shall
the
Hour
line
pass
from
Tropique
to
Tropique
:
direct
any
By-stander
to
make
marks
as
many
as
you
shall
need
;
and
by
these
marks
draw
the
Hour
line
according
to
your
desire
.
If
the
arch
of
the
Horizon
between
the
Tropiques
be
within
view
of
your
Window
,
you
shall
draw
the
same
on
the
Wall
to
bound
the
Parallels
,
the
Horizons
Altitude
you
know
is
nothing
,
and
therefore
he
will
be
a
level
line
;
and
the
Suns
Azimuth
when
he
riseth
(
commonly
called
Amplitude
,
and
Ortive
Latitude
)
is
in
Cancer
40.40
.
minutes
East
Northward
,
and
in
Capricorn
as
much
Southward
;
and
these
will
be
reflected
to
the
contrary
coasts
on
the
Dyal
.
The
end
of
the
Fifth
Book
.