Elliptical or azimuthal horologiography comprehending severall wayes of describing dials upon all kindes of superficies, either plain or curved, and unto upright stiles in whatsoever position they shall be placed / invented and demonstrated by Samuel Foster ...
Foster, Samuel, d. 1652.
1654
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A40031
Wing F1632
ESTC R7034
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ocm 12251532
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Elliptical or azimuthal horologiography comprehending severall wayes of describing dials upon all kindes of superficies, either plain or curved, and unto upright stiles in whatsoever position they shall be placed / invented and demonstrated by Samuel Foster ...
Foster, Samuel, d. 1652.
Twysden, John, 1607-1688.
Wingate, Edmund, 1596-1656.
[6], 204, [3] p. : ill.
Printed by R. & W. Leybourn for Nicholas Bourn ...,
London :
1654.
"To the reader" signed: John Twysden, Edmund Wingate.
In four parts, paged continuously; pts. 2-4 each have special t.p.
Title page vignette.
Advertisement on p. [3] at end.
Reproduction of original in Harvard University Libraries.
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eng
Horology -- Early works to 1800.
Sundials.
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ELLIPTICAL
,
OR
AZIMUTHAL
Horologiography
.
Comprehending
severall
wayes
of
describing
DIALS
upon
all
kindes
of
Superficies
either
plain
or
curved
:
And
unto
upright
Stiles
in
whatsoever
position
they
shall
be
placed
.
Invented
and
Demonstrated
By
SAMVEL
FOSTER
,
Late
Professor
of
Astr●●●mie
in
Gresham-Colledge
.
LONDON
,
〈…〉
for
Nicholas
T
〈…〉
To
the
READER
.
Courteous
Reader
,
AMongst
other
Treatises
of
this
deceased
Author
,
of
which
,
in
their
due
time
,
we
intend
to
make
thee
partaker
,
we
have
,
in
the
first
place
,
made
choice
of
this
,
as
well
in
some
measure
to
keep
our
word
with
thee
,
as
also
to
stay
thy
expectation
till
other
things
can
be
made
ready
.
We
intend
not
to
spend
time
either
in
the
commendation
of
It
or
the
Author
,
being
w●ll
assured
that
our
words
will
not
,
where
the
matter
it self
is
〈…〉
capable
to
,
satisfie
therein
the
judicious
Reader
.
Onely
let
〈◊〉
say
thus
much
,
that
though
the
generall
Subject
of
this
Book
〈◊〉
Dialling
,
yet
't
is
handled
in
a
way
which
no
man
whosoever
,
that
we
know
of
,
hath
hitherto
fully
traced
.
For
by
it
thou
wilt
see
,
that
the
representing
the
true
Houre
by
the
shadow
made
by
the
Axis
of
the
World
is
but
one
of
those
infinite
wayes
which
may
be
invented
,
and
that
it
is
as
possible
to
do
the
same
thing
by
the
shadow
of
the
Axis
of
one
of
the
vertical
Circles
,
and
by
the
projection
of
one
Ellipsis
upon
a
plain
,
supply
the
place
of
all
the
Parallels
comprehended
within
the
Tropicks
.
'T
is
true
,
that
Mr.
Va●lezard
,
a
learned
Mathematician
,
we
think
yet
living
in
France
,
hath
some
yeers
since
published
a
short
Treatise
in
that
Language
,
in
which
he
shew●th
by
the
projection
of
an
Ellipsis
upon
the
plain
of
the
Horizon
,
and
by
the
help
of
an
upright
moveable
stile
to
finde
the
He●re
and
Azimuth
,
with
some
other
uses
of
the
same
.
But
this
Treatise
of
our
Author
is
very
different
from
that
,
and
most
of
the
things
here
handled
,
such
as
are
not
appliable
to
his
,
and
in
themselves
wholly
new
.
In
the
next
place
,
if
it
shall
seem
strange
to
any
that
amongst
other
things
,
as
well
of
this
,
as
of
different
natures
,
which
we
intend
shortly
to
Publish
;
we
begin
first
with
this
of
Dialling
(
a
Subject
upon
which
something
too
hath
already
been
published
by
our
Author
,
and
from
whence
some
might
perhaps
take
occasion
to
carp
at
him
,
as
either
unable
for
other
things
,
or
too
much
busied
in
this
)
we
desire
them
first
to
con●●der
how
difficult
this
Subject
hath
been
thought
by
the
〈◊〉
,
and
withall
what
large
Volumes
have
been
writ
by
som●
of
the
best
rank
of
later
Mathematicians
,
(
such
were
Clavius
,
Maigran
,
and
others
)
and
then
compare
their
wayes
with
what
they
shall
here
and
hereafter
(
God
willing
)
finde
in
our
Author
,
and
we
doubt
not
,
but
they
will
be
forced
to
yield
him
it
this
honour
,
to
have
made
that
Art
,
in
all
the
Cases
of
it
,
and
all
Circumstances
thereunto
belonging
,
more
easie
and
expedite
both
in
the
Understanding
and
Practice
,
and
with
much
more
brevity
than
any
that
have
gone
before
him
either
of
our
own
or
other
Nations
.
Lastly
,
we
advertize
thee
(
Reader
)
that
our
Authour
,
in
regard
of
his
great
and
long
infirmities
,
could
not
fit
either
this
,
or
any
other
of
his
Treatises
for
the
Presse
,
as
he
desired
and
intended
;
and
therefore
they
must
needs
want
much
of
that
accomplishment
which
otherwise
they
would
have
had
.
But
we
hope
,
notwithstanding
,
thou
wilt
finde
so
much
of
worth
in
them
as
they
now
are
,
whereby
thou
wilt
judge
them
(
as
we
do
)
fitter
to
be
made
publick
,
then
wholly
suppressed
.
John
Twysden
,
Edmund
Wingate
.
ERRATA
.
Page
85.
line
11.
for
Pag.
81.
read
Pag.
70.
There
are
divers
other
literall
faults
,
but
this
is
most
materiall
.
ELLIPTICAL
OR
AZIMVTHAL
Horologiography
.
Of
the
Elliptical
Dial
,
whose
Index
stands
perpendicular
to
the
plain
:
How
to
draw
and
divide
the
Ellipsis
upon
an
Horizontal
plain
,
or
any
other
plain
that
declineth
not
.
1.
LEt
the
two
right
lines
B
C
and
A
D
,
cut
each
other
at
right
angles
in
A.
Then
making
A
C
or
A
B
as
a
Radius
,
let
A
D
be
the
Right
Sine
of
the
Poles
elevation
above
the
plain
;
and
make
up
the
Rectangled
Parallelogram
B
E
F
C
,
and
continue
it
further
to
G
and
H.
2.
Divide
the
Radius
A
C
into
the
Sines
of
15
,
30
,
45
,
60
,
75
gr
.
at
a
b
c
d
e
,
which
stand
for
the
six
equall
houres
:
●o
that
A
a
may
be
the
Sine
of
one
houre
,
A
b
of
two
houres
,
A
c
of
three
houres
,
A
d
,
A
e
of
four
and
five
houres
.
And
in
the
same
sort
divide
the
other
lines
which
are
equall
to
A
C
,
namely
D
F
,
A
B
,
D
E
,
into
the
same
parts
,
at
a
b
c
d
e
,
and
at
l
m
n
o
p
:
and
draw
the
parallel
lines
a
a
,
b
b
,
&c.
l
l
,
m
m
,
&c.
Then
taking
A
D
,
or
C
F
,
or
B
E
,
for
a
Radius
,
set
on
again
the
Sines
of
the
six
equall
houres
from
B
and
C
,
at
f
g
h
i
k
,
and
as
many
as
shall
be
needfull
on
the
other
side
,
at
q
r
s
:
and
draw
the
parallel
lines
f
f
,
g
g
,
&c.
and
q
q
,
r
r
,
&c.
3.
First
,
If
a
curved
line
be
begun
at
D
or
12
,
and
carried
both
wayes
diagonally
through
the
succeeding
Parallelograms
,
the
same
line
shall
be
the
Ellipsis
required
.
Thus
you
see
it
carried
from
D
to
11
,
from
11
to
10
,
from
10
to
9
,
and
so
through
the
successive
opposite
angles
of
the
following
Parallelograms
,
at
8
,
7
,
6
,
5
,
4
,
3.
And
on
the
other
side
in
the
same
manner
,
from
D
,
diagonally
through
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9.
This
line
thus
regularly
described
without
angles
,
and
extravagant
inordinate
flexures
,
is
the
Ellipsis
here
intended
.
Secondly
,
If
from
the
center
A
,
through
each
of
the
angular
points
in
the
small
Parallelograms
,
certain
short
lines
be
drawn
,
(
as
in
the
figure
you
see
done
at
11
,
10
,
9
,
&c.
and
at
1
,
2
,
3
,
&c.
)
the
same
lines
shall
be
the
requisite
horary
divisions
of
the
Elliptical
line
.
4.
If
these
divisions
into
single
houres
shall
not
be
enough
,
but
that
more
exactnesse
is
required
,
into
halves
and
quarters
;
then
must
you
(
besides
the
Sines
of
whole
houres
at
a
,
b
,
&c.
l
,
m
,
&c.
)
put
in
the
right
Sines
of
the
intermediate
halves
and
quarters
of
the
equall
houres
;
which
will
be
easie
to
do
by
the
former
directions
.
Declinations
of
the
Sun
from
the
Aequinoctiall
.
Ianu.
Febru
.
March
Aprill
May
Iune
Iuly
Augu.
Septē
.
Octob.
Novē
.
Decēb.
1
21
49
13
57
3
34
8
30
18
01
23
11
22
09
15
15
4
29
7
10
17
36
23
07
2
21
39
13
37
3
11
8
52
18
16
23
14
22
00
14
57
4
06
7
33
17
52
23
12
3
21
29
13
17
2
47
9
13
18
31
23
18
21
51
14
39
3
43
7
56
18
08
23
16
4
21
18
12
57
2
23
9
34
18
46
23
21
21
42
14
21
3
20
8
18
18
24
23
19
5
21
07
12
36
2
00
9
56
19
00
23
24
21
33
14
02
2
57
8
40
18
40
23
22
6
20
56
12
15
1
36
10
17
19
14
23
26
21
23
13
43
2
34
9
02
18
55
23
25
7
20
44
11
54
1
12
10
38
19
27
23
28
21
13
13
24
2
10
9
24
19
10
23
27
8
20
32
11
33
0
48
10
59
19
40
23
29
21
02
13
05
1
47
9
46
19
24
23
29
9
20
19
11
12
0
24
11
20
19
53
23
30
20
51
12
45
1
24
10
08
19
38
23
30
10
20
06
10
50
Aequin
.
11
40
20
06
23
31
20
40
12
25
1
00
10
30
19
52
23
31
11
19
53
10
28
0
24
12
01
20
18
23
31
20
29
12
05
0
37
10
51
20
05
23
31
12
19
39
10
06
0
48
12
22
20
30
23
31
20
17
11
45
Aequinoct
.
14
11
12
20
18
23
31
13
19
25
9
44
1
12
12
42
20
42
23
31
20
05
11
25
10
11
33
20
31
23
31
14
19
10
9
22
1
36
13
02
20
53
23
30
19
53
11
04
0
34
11
54
20
43
23
30
15
18
55
9
00
2
00
13
21
21
04
23
28
19
40
10
43
0
58
12
15
20
55
23
28
16
18
40
8
38
2
23
13
40
21
14
23
26
19
27
10
22
1
22
12
36
21
06
23
25
17
18
25
8
15
2
47
14
00
21
24
23
24
19
13
10
01
1
46
12
57
21
17
23
22
18
18
10
7
52
3
11
14
19
21
34
23
20
18
59
9
40
2
10
13
17
21
27
23
19
19
17
54
7
29
3
34
14
38
21
43
23
17
18
45
9
19
2
33
13
37
21
37
23
16
20
17
37
7
06
3
57
14
57
21
52
23
14
18
30
8
58
2
56
13
57
21
47
23
12
21
17
20
6
43
4
20
15
15
22
01
23
11
18
15
8
36
3
19
14
16
21
57
23
08
22
17
03
6
20
4
43
15
32
22
10
23
06
18
00
8
14
3
42
14
35
22
06
23
03
23
16
46
5
57
5
06
15
49
22
18
23
01
17
45
7
52
4
06
14
54
22
15
22
58
24
16
28
5
34
5
29
16
06
22
25
22
56
17
30
7
30
4
29
15
13
22
23
22
52
25
16
10
5
11
5
52
16
23
22
32
22
51
17
14
7
08
4
52
15
32
22
32
22
45
26
15
52
4
47
6
15
16
40
22
39
22
45
16
58
6
46
5
15
15
51
22
38
22
38
27
15
34
4
24
6
38
16
57
22
45
22
39
16
42
6
24
5
38
16
09
22
45
22
31
28
15
15
4
01
7
00
17
13
22
51
22
32
16
25
6
01
6
01
16
27
22
51
22
24
29
14
56
7
22
17
29
22
57
22
25
16
08
5
38
6
24
16
45
22
57
22
16
30
14
37
7
45
17
45
23
02
22
17
15
51
5
15
6
47
17
02
23
02
22
08
31
14
17
8
08
23
07
15
33
4
52
17
19
21
59
Tangents
of
the
Suns
declination
from
the
Aequinoctiall
.
Ianu.
Febru
.
March
Aprill
May
Iune
Iuly
Augu.
Septē
.
Octob.
Novē
.
Decēb.
1
4005
2484
0623
1498
3251
4281
4071
2727
0784
1257
3170
4272
2
3971
2422
0554
1562
3300
4295
4042
2671
0717
1325
3222
4287
3
3936
2360
0485
1626
3348
4307
4012
2615
0650
1392
3275
4300
4
3900
2297
0416
1690
3396
4317
3981
2558
0584
1458
3326
4311
5
3863
2234
0347
1754
3443
4326
3949
3500
0517
1524
3377
4321
6
3825
2170
0278
1818
3489
4334
3916
2442
0450
1590
3427
4330
7
3785
2106
0209
1881
3534
4341
3882
2383
0383
1656
3476
4338
8
3744
2042
0140
1944
3576
4347
3847
2323
0316
1722
35●3
4345
9
3702
1978
0070
2007
3618
4350
3811
2263
0249
1788
3568
4350
10
3659
1913
Aequin
.
2069
3659
4351
3776
2202
0181
1853
3613
4352
11
3615
1847
0070
2131
3700
4352
3739
2141
0112
1917
3657
4352
12
3570
1781
0140
2193
3740
4352
3700
2079
Aequin
.
42
1981
3700
4351
13
3524
1715
0209
2254
3779
4351
3660
2017
29
2045
3742
4350
14
2477
1649
0278
2315
3817
4348
3618
1955
0099
2109
378●
4347
15
3429
1582
0347
2375
3854
4343
3575
1893
0169
2172
3821
4341
16
3380
1515
0416
2434
3889
4336
3532
1831
0239
2235
3859
4333
17
3330
1448
0485
2493
3922
4327
3487
1768
0309
2298
3896
4324
18
3280
1381
0554
2552
3952
4317
3442
1705
0378
2360
39●2
4314
19
3230
1314
0623
2610
3982
4306
3396
1642
0446
2422
3907
4302
20
3177
1246
0692
2668
4012
4294
3349
1578
0514
2483
4000
4288
21
3123
1178
0761
2725
4042
4281
3301
1513
0582
2543
4031
4273
22
3068
1110
0829
2780
4071
4266
3253
1448
0650
2603
4061
4256
23
3013
1042
0897
2834
4099
4250
3204
1382
0718
2662
4090
4238
24
2957
0974
0965
2888
4125
4232
3154
1316
0786
2721
4118
4217
25
2900
0906
1033
●941
4149
4213
3103
1250
0854
2779
4146
4195
26
2842
0837
1101
2994
4171
4193
3052
1184
0921
2837
4172
4172
27
2784
0768
1168
3047
4192
4172
3000
1118
0988
2895
4196
4148
28
2726
0699
1●35
3099
4212
4149
2947
1052
1055
2952
4218
4123
29
2667
1301
3150
4231
4124
2893
0985
1122
3008
4238
4096
30
2607
1367
3201
4249
4098
2838
0918
1189
3063
4256
4068
31
2546
1433
4266
2783
0851
3117
4038
SECT
.
I.
Of
the
Elliptical
Dial
,
where
the
Index
stands
perpendicular
to
the
Plain
:
How
to
draw
it
for
an
Horizontal
Plain
,
or
any
other
Plain
that
declineth
not
.
BEcause
here
is
no
declination
supposed
,
therefore
the
draught
will
be
the
more
easie
.
1.
Make
B
C
for
the
longer
Diameter
of
your
Ellipsis
,
and
count
one
halfe
of
it
,
that
is
A
B
or
A
C
as
the
Radius
,
and
through
the
point
A
,
draw
an
infinite
line
(
as
D
E
)
at
right
angles
to
B
C.
Then
for
the
shorter
Diameter
,
you
must
consider
the
elevation
of
the
Pole
above
your
plain
.
Looke
therefore
to
make
A
D
or
A
E
equall
to
the
Sine
of
the
Poles
elevation
,
which
Sine
must
be
estimated
to
the
Radius
A
B.
Thus
you
have
the
two
extream
diameters
.
2.
Through
the
four
points
D
B
E
C
you
may
describe
the
Ellipsis
,
either
by
Elliptical
Compasses
,
or
by
finding
many
points
through
which
it
is
to
passe
:
for
effecting
which
there
are
multitudes
of
wayes
prescribed
.
One
may
be
this
.
Describe
two
Circles
upon
the
two
extream
Diameters
B
C
and
D
E.
Divide
them
both
into
any
like
parts
,
as
B
f
h
n
p
q
C
,
and
V
i
k
E
o
t
X
;
and
from
each
couple
of
those
like
divisions
(
as
f
and
i
)
draw
f
g
i
g
,
(
the
first
parallel
to
A
n
,
the
second
parallel
to
A
B
)
cutting
each
other
at
right
angles
in
g
,
so
the
point
g
shall
be
one
of
the
points
through
which
the
Ellipsis
is
to
be
described
.
In
the
same
manner
,
from
h
and
k
must
be
drawn
two
lines
concurring
at
right
angles
in
m
,
which
will
therefore
be
another
point
through
which
the
Ellipsis
must
passe
.
So
again
from
p
and
o
,
t
and
q
,
lines
are
to
be
drawn
,
concurring
at
right
angles
in
s
and
r
,
which
are
two
other
points
of
passage
for
the
Ellipsis
.
After
this
manner
may
the
other
halfe
B
D
C
be
described
.
How
to
divide
the
Ellipsis
into
its
requisite
parts
.
1.
If
the
parts
B
f
h
n
p
q
C
were
all
equall
,
and
12
of
them
in
number
,
then
would
the
points
g
m
,
&c.
be
the
houre
points
.
And
accordingly
,
if
those
houres
were
subdivided
into
equall
halves
and
quarters
,
&c.
there
would
be
found
points
for
the
halves
and
quarters
,
&c.
of
the
houres
.
And
if
the
Ellipsis
be
first
drawn
,
then
the
exteriour
Circle
it selfe
by
lines
issuing
out
of
f
h
p
q
,
drawn
parallel
to
A
n
)
will
divide
the
Ellipsis
:
or
,
where
these
lines
cut
it
more
obliquely
(
as
at
g
and
r
,
&c.
)
there
the
lesser
Circles
equall
parts
(
at
i
and
t
,
&c.
)
will
cut
the
same
Ellipsis
at
more
right
angles
,
so
that
one
of
them
may
help
the
other
in
this
division
.
2.
Or
,
If
the
quadrants
B
n
C
,
or
V
E
X
(
or
of
any
other
Circle
described
upon
the
center
A
)
were
divided
into
Horizontall
Spaces
,
such
as
are
proper
for
this
Diall
(
as
you
see
them
done
upon
the
quadrants
B
y
C
)
and
a
Ruler
laid
from
the
center
A
did
transfer
those
divisions
into
the
Ellipsis
(
as
is
done
upon
that
halfe
of
it
which
is
noted
with
B
D
C
)
this
division
would
be
the
same
with
the
former
.
The
proportion
whereby
to
make
the
Table
of
Horizontall
Spaces
for
this
Ellipsis
,
is
inverse
to
that
which
is
for
common
Horizontall
Dials
:
thus
,
As
the
Sine
of
the
Poles
elevation
above
the
plain
,
Is
to
the
Radius
;
So
is
the
Tangent
of
15
gr
.
30
,
45
,
&c.
To
the
Tangent
of
the
spaces
of
every
houre
from
12
,
upon
the
plain
whereon
the
Ellipsis
is
described
.
And
according
to
these
spaces
must
every
houre
be
set
on
from
n
or
y
towards
B
or
C.
How
to
make
the
Zodiac
,
or
dayes
of
the
yeare
,
whereby
the
Ellipsis
and
Index
are
to
be
set
in
a
right
position
,
that
they
may
daily
stand
true
to
shew
the
houre
.
Whether
the
12
Signes
or
the
12
Moneths
be
fittest
for
use
,
is
left
to
every
mans
choise
.
Bu●
in
both
these
it
is
required
that
the
declinations
due
to
the
Signes
or
Dayes
be
known
,
because
these
are
to
be
inscribed
by
them
.
So
that
now
it
only
remains
to
be
shewed
what
kinde
of
Scale
that
must
be
,
out
of
which
the
foresaid
Declinations
are
to
be
taken
.
In
generall
,
the
Scale
must
be
a
Scale
of
Naturall
Tangents
.
In
particular
,
there
are
these
two
Rules
,
one
of
which
will
with
due
convenience
serve
for
all
kindes
of
Latitudes
,
or
elevations
.
You
may
use
which
is
most
sutable
to
your
purpose
.
The
first
for
great
Latitudes
.
Make
the
lesser
semidiameter
A
D
or
A
E
,
to
be
a
Tangent
of
the
Poles
elevation
above
the
plain
;
and
that
Scale
is
the
Scale
of
Tangents
out
of
which
the
Declinations
(
before
mentioned
)
must
be
taken
.
The
second
for
lesser
Latitudes
.
To
the
Radius
B
A
,
finde
the
Co-sine
of
the
plains
latitude
or
elevation
;
and
make
that
Co-sine
a
Radius
,
or
Tangent
of
45
gr
.
and
this
will
be
the
same
Scale
with
the
former
▪
By
these
wayes
the
Declinations
of
these
two
Tables
(
one
for
the
Signes
,
the
other
for
the
Dayes
of
the
Moneths
)
may
be
inserted
from
the
Equinoctiall
line
,
or
middle
of
the
Scale
,
by
the
graduations
of
such
a
Tangent
line
as
is
before
mentioned
.
Signes
.
Declin
.
Signes
.
♑
♋
0
23
31
30
5
23
26
25
10
23
9
20
15
22
41
15
20
22
2
10
25
21
12
5
♊
●
♒
♌
0
20
13
30
5
19
5
25
10
17
48
20
15
16
24
15
20
14
52
10
25
13
14
5
♉
♏
♓
♍
0
11
31
30
5
9
43
25
10
7
51
20
15
5
56
15
20
3
58
10
25
2
00
5
♈
♎
Moneth
Declin
.
Moneth
Declin
.
Moneth
Declin
.
Moneth
Declin
.
Ianu.
5
21
7
Aprill
5
9
56
Iuly
5
21
33
Octob.
5
8
40
10
20
6
10
11
41
10
20
41
10
10
29
15
18
55
15
13
20
15
19
39
15
12
15
20
17
37
20
14
55
20
18
30
20
13
56
25
16
10
25
16
23
25
17
14
25
15
32
31
14
17
30
17
45
31
15
33
31
17
18
Febr.
5
12
36
May
5
19
0
Aug.
5
14
1
Nove.
5
18
40
10
10
50
10
20
6
10
12
24
10
19
51
15
9
00
15
21
4
15
10
42
15
20
54
20
7
6
20
21
53
20
8
56
20
21
47
25
5
11
25
22
32
25
7
7
25
22
30
28
4
1
31
23
6
31
4
52
30
23
2
Mar.
5
2
0
Iune
5
23
24
Septe
.
5
2
56
Dece
.
5
23
22
10
Ae
1
10
23
31
10
1
0
10
23
31
15
2
0
15
23
28
15
Ae
57
15
23
28
20
3
57
20
23
14
20
2
55
20
23
11
25
5
52
25
22
50
25
4
51
25
22
45
31
8
7
30
22
17
30
6
46
31
21
58
Or
if
it
be
required
rather
to
set
them
on
by
a
Decimal
Scale
equally
divided
,
then
must
the
Tangents
of
these
Declinations
(
here
specified
)
be
used
;
and
the
Decimal
Scale
must
be
equal
to
the
Co-sine
of
the
plains
Latitude
,
which
Co-sine
must
be
limited
to
B
A
,
the
greater
Radius
of
the
Ellipsis
.
And
for
this
purpose
here
are
two
Tables
more
which
do
expresse
the
said
Tangents
.
Signes
.
Tang
Signes
.
Moneth
Tang
Moneth
Tang
Moneth
Tang
Moneth
Tang
♑
♋
0
4348
30
Ianu.
5
2862
Aprill
5
1751
Iuly
5
3949
Octob.
5
1524
5
4334
25
10
3659
10
2068
10
3775
10
1850
10
4276
20
15
3427
15
2370
15
3571
15
2171
15
4180
15
20
3175
20
2664
20
3346
20
2481
20
4047
10
25
2899
25
2940
25
3102
25
2780
25
3879
5
♊
●
31
2546
30
3201
31
2783
31
3115
♒
♌
0
3683
30
Febr.
5
2235
May
5
3443
Aug.
5
2496
Nove.
5
3378
5
3460
25
10
1914
10
3659
10
2199
10
3610
10
3211
20
15
1584
15
3852
15
1890
15
3819
15
2943
15
20
1246
20
4017
20
1572
20
3996
20
2655
10
25
0907
25
4149
25
1249
25
4142
25
2352
5
♉
♏
28
0702
31
4265
31
0851
30
4252
♓
♍
0
2038
30
Mar.
5
0349
Iune
5
4327
Septe
.
5
0512
Dece
.
5
4320
5
1712
25
10
Aeq
3
10
4348
10
0175
10
4348
10
1379
20
15
0349
15
4341
15
ae166
15
4341
15
1039
15
20
0690
20
4293
20
0509
20
4283
20
0693
10
25
1028
25
4210
25
0849
25
4193
25
0349
5
♈
♎
31
1426
30
4098
30
1187
31
4033
Another
way
to
describe
and
divide
the
same
Zodiac
,
or
Scale
of
Moneths
.
The
former
descriptions
do
suppose
that
the
yearly
course
of
the
Sun
is
to
be
set
in
a
streight
Scale
,
in
which
the
parts
neerest
to
the
two
Tropicks
will
be
exceeding
close
together
;
and
those
at
the
Aequinoctiall
or
middle
part
of
the
Scale
will
be
wide
.
But
if
it
shall
better
be
liked
that
the
parts
should
stand
distinctly
one
from
the
other
;
it
will
then
be
most
expedient
,
First
,
to
limit
out
the
whole
length
of
the
Scale
:
then
upon
that
length
(
as
a
diameter
)
to
describe
a
Circle
,
and
to
divide
it
as
is
here
under
shewed
.
How
to
divide
the
Annuall
Circle
into
its
requisite
parts
.
You
are
first
to
divide
the
Circle
(
it
selfe
,
or
one
equall
to
it
)
into
360
equall
degrees
,
and
by
them
to
divide
the
said
Circle
into
such
unequall
parts
as
the
numbers
of
degrees
and
minutes
expressed
in
the
Tables
will
require
.
The
two
Tables
are
the
Right
Ascensions
of
such
parts
of
the
Ecliptic
as
are
due
to
the
Suns
place
or
Declination
,
in
any
of
the
Signes
,
or
of
any
dayes
of
the
12
moneths
.
The
Tables
are
here
placed
.
The
manner
of
dividing
the
Circle
is
the
same
with
other
like
things
of
this
kinde
.
Mon
R.
Ascen
.
Mon
R.
Ascē
Mon
R.
Ascen
Mon
R.
Ascen
.
Ia
10
302
43
A
10
28
36
I.
10
119
47
O.
10
205
06
20
313
16
20
37
56
20
129
40
20
214
53
31
324
21
30
47
33
31
140
15
31
225
50
Fe
10
334
03
M
10
57
23
A
10
149
37
N
10
236
11
20
343
30
20
67
27
20
158
48
20
246
51
28
351
07
31
78
45
31
168
45
30
257
57
M
10
0
14
I.
10
89
05
S.
10
177
44
D
10
269
02
20
9
18
20
99
27
20
186
47
20
280
11
31
19
19
30
109
43
30
195
53
31
292
08
Signs
R.
Ascē
Signs
R.
Ascen
.
Signs
R.
Ascen
Signs
R.
Ascen
.
♈
10
9
11
♋
10
100
53
♎
10
189
11
♑
10
280
53
20
18
27
20
111
39
20
198
27
20
291
39
30
27
54
30
122
12
30
207
54
30
302
12
♉
10
37
34
♌
10
132
28
♏
10
217
34
♒
10
312
28
20
47
32
20
142
26
20
227
32
20
322
26
30
57
48
30
152
06
30
237
48
30
332
06
♊
10
68
21
♍
10
16●
33
♐
10
248
21
♓
10
341
33
20
79
07
20
170
49
20
259
07
20
350
49
30
90
00
30
180
00
30
270
00
30
360
00
The
beginning
of
the
equall
parts
or
degrees
,
must
be
upon
that
middle
Diameter
of
the
Circle
,
which
lies
parallel
to
the
two
six
a
clock
lines
,
or
perpendicular
to
the
twelve
a
clock
line
.
The
best
manner
of
dividing
the
Circle
is
as
in
the
preceding
figure
,
especially
if
the
fiduciall
Circle
be
cut
through
with
small
divisions
,
that
so
the
intersections
may
be
the
more
discernable
,
and
the
Circle
more
distinguishable
from
the
rest
;
and
more
cleere
from
mistakes
.
Concerning
the
Index
.
The
Index
must
stand
perpendicular
to
the
plain
,
and
must
,
according
to
the
time
of
the
year
,
be
neerer
or
further
from
the
Ellipsis
.
Now
,
Whether
the
Index
should
move
upon
the
plate
lying
still
,
or
the
elliptical
plate
move
towards
the
Index
fixed
upon
another
plate
,
must
be
left
to
every
mans
judgement
and
best
liking
.
But
the
mover
with
its
peculiar
Index
(
called
here
the
Zodiacall
Index
)
must
move
according
to
the
length
of
the
plains
Meridian
line
,
either
in
it
,
or
parallel
to
it
,
alwayes
so
that
the
fiduciall
edge
of
the
Dials
Index
may
ever
stand
in
the
Meridian
line
.
But
if
to
this
Ellipticall
Diall
it
be
thought
fit
to
joyn
an
ordinary
Horizontal
Dial
(
fitted
to
the
Elevation
of
the
Pole
above
the
plain
)
that
so
it
may
set
it selfe
true
North
and
South
,
which
by
this
double
kinde
of
Diall
it
will
do
;
for
the
houres
in
both
Dials
will
never
agree
to
be
the
same
till
it
stand
right
,
and
the
best
time
for
setting
it
,
is
when
the
Sun
is
in
the
East
or
West
Azimuth
,
the
worst
time
is
at
Noon
:
then
the
common
Horizontal
Dial
may
be
the
standing
plate
,
the
Index
of
the
Ellipsis
being
fixed
to
it
,
the
elliptical
plate
may
move
too
and
fro
upon
the
horizontal
fixed
plate
,
according
to
the
graduations
of
the
Zodiac
.
Of
the
place
for
the
Suns
Annual
course
or
Zodiac
.
Whichsoever
of
the
two
(
the
Ellipsis
or
Index
)
is
moveable
or
fixed
,
it
matters
not
.
In
both
wayes
it
is
necessary
to
set
the
Zodiac
right
,
which
must
thus
be
done
.
Let
the
fiduciall
edge
of
the
Index
be
placed
exactly
in
the
center
of
the
Ellipsis
.
And
being
there
set
,
let
the
place
of
the
line
or
threed
(
or
whatever
it
be
that
serves
for
the
Zodiacall
Index
)
be
marked
upon
the
standing
plate
.
Then
through
that
mark
or
point
draw
a
streight
line
parallel
to
the
line
of
the
two
sixes
,
or
perpendicular
to
the
line
of
twelve
.
The
same
line
is
to
be
esteemed
for
the
Aequinoctiall
:
and
from
thence
are
all
the
parts
of
the
Scale
(
or
Circle
)
to
be
insctibed
by
help
of
the
former
Tables
.
The
Vses
of
the
Elliptical
Dial.
The
Index
and
Ellipsis
being
used
alone
,
and
set
to
their
true
distance
,
by
the
help
of
the
Zodiac
,
the
Ellipsis
it selfe
with
its
upright
Index
,
will
shew
the
true
houre
of
the
day
.
But
it
is
supposed
hitherto
,
that
the
plain
be
either
direct
,
looking
full
North
or
South
,
or
else
Horizontal
:
and
in
every
such
plain
,
there
must
be
drawn
the
Meridian
line
of
the
place
or
plain
,
which
in
direct
flats
are
one
and
the
same
.
And
according
to
the
Coast
of
this
Meridian
line
must
the
moving
or
sliding
be
,
as
also
the
line
of
12
upon
the
Ellipsis
must
lie
in
or
parallel
to
it
.
This
way
therefore
requires
,
first
,
a
Meridian
line
to
be
drawn
,
whereby
to
place
the
Ellipsis
.
And
in
all
direct
plains
the
Meridian
line
is
the
same
with
the
Vertical
line
of
the
plain
.
But
if
to
this
Elliptical
Dial
,
be
adjoyned
a
common
Horizontal
Dial
with
an
Axis
,
then
there
needs
no
finding
of
a
Meridian
line
before
hand
,
because
they
two
betwixt
them
will
finde
one
,
and
consequently
will
place
themselves
in
a
true
position
.
Only
with
this
proviso
,
that
if
the
plain
be
not
Horizontal
,
it
must
yet
be
such
as
looketh
directly
North
or
South
.
This
was
intimated
before
.
There
may
other
uses
be
made
of
it
,
if
it
have
other
Scales
adjoyned
to
the
Zodiac
.
1.
If
a
Tangent
line
of
23½
gr
.
be
inscribed
according
to
the
length
of
the
Zodiac
,
then
when
the
situation
of
the
Ellipsis
to
the
Index
is
fitted
,
the
Zodiacal
Index
will
shew
(
upon
this
Tangent
Scale
)
the
Suns
Declination
.
2.
In
the
11
page
there
are
two
Rules
given
whereby
to
finde
that
Scale
of
Tangents
out
of
which
the
Zodiacal
Scale
is
to
be
divided
.
If
then
out
of
that
Scale
of
Tangents
so
found
and
limited
,
you
take
the
Co-tangent
of
your
Latitude
,
and
divide
that
length
as
a
Radius
or
whole
line
of
Sines
into
houres
and
parts
of
houres
,
and
put
in
so
many
of
them
as
will
reach
the
Suns
greatest
Declination
,
or
the
length
of
the
Zodiac
both
wayes
from
the
Equinoctiall
;
then
the
former
rectification
of
the
Zodiacal
Index
to
the
day
of
the
moneth
or
Suns
place
,
will
perfectly
shew
(
in
this
Scale
,
and
for
that
Latitude
to
which
all
the
work
is
made
)
the
Ascensional
difference
,
with
what
depends
thereon
:
namely
,
the
length
of
the
Day
and
Night
,
and
the
time
of
Sun
Rising
and
Setting
.
3.
The
Dial
being
fixed
in
a
true
position
,
the
Place
or
Coast
of
the
Sun
rising
and
setting
upon
the
plain
,
may
be
discerned
in
the
heavens
.
For
when
the
Ellipsis
is
rectified
,
and
the
time
of
the
Sun
rising
or
setting
is
known
(
as
before
)
then
with
your
eye
project
the
fiduciall
edge
of
the
upright
Index
upon
that
time
of
Sun
rising
or
setting
counted
in
the
Ellipsis
,
and
the
same
edge
will
shew
in
the
heavens
whereabout
the
Sun
will
ascend
or
descend
upon
the
plain
.
That
is
,
it
shews
the
Amplitude
of
Sun
rising
upon
the
plain
.
1.
Note
,
That
if
the
plain
be
Horizontall
,
then
the
two
last
Uses
serve
for
the
place
where
a
man
lives
.
But
if
the
plain
be
not
Horizontal
,
then
it
serves
not
for
the
place
,
but
is
proper
to
the
plain
it selfe
,
or
to
that
Horizon
or
Latitude
which
the
plain
represents
,
according
as
they
are
set
down
in
the
former
Rules
:
because
things
are
done
to
the
plains
Latitude
,
and
not
to
the
places
.
2.
Note
again
,
that
if
upon
plain●
that
are
not
Horizontal
,
you
would
yet
have
them
(
to
shew
the
second
of
these
last
preceding
conclusions
,
namely
,
)
to
shew
the
time
of
Sun
rising
or
setting
for
the
place
in
which
you
are
,
and
not
for
the
plains
,
then
must
you
take
(
not
the
Co-tangent
of
the
plains
Latitude
,
but
)
the
Co-tangent
of
the
Latitude
of
the
place
,
just
as
you
did
in
the
Horizontall
plain
,
and
then
the
Zodiacal
Index
being
rectified
will
effect
the
thing
required
for
the
place
,
and
not
for
the
plain
.
3.
Note
,
that
both
in
the
Horizontal
and
the
other
plains
,
there
may
be
a
peculiar
Scale
put
,
either
for
the
Amplitude
proper
to
those
plains
,
or
else
for
the
Amplitude
proper
to
the
place
,
if
a
Table
of
Amplitudes
be
computed
for
either
of
them
,
by
this
proportion
.
As
the
Radius
,
to
the
Sine
of
1
,
2
,
3
,
&c.
or
5
,
10
,
15
,
&c.
degr
.
of
Amplitude
;
So
the
Co-sine
of
the
poles
elevation
,
To
the
Sine
of
the
competent
declination
:
And
then
by
the
said
competent
declination
(
taken
in
the
Scale
of
Tangents
of
23½
equal
to
the
halfe
Zodiac
)
the
forenamed
Amplitudes
be
inscribed
.
Or
having
the
Declinations
,
you
may
look
out
their
Tangents
in
the
Canon
,
by
which
Tangents
the
said
Amplitudes
may
be
inferred
with
help
of
the
Decimall
Scale
made
to
the
Co-sine
of
the
plains
Latitude
,
as
was
before
mentioned
.
The
Amplitudes
proper
to
the
plain
are
of
no
great
use
,
unlesse
the
plain
do
justly
represent
the
Horizon
of
some
known
place
,
whose
Amplitude
you
desire
to
be
acquainted
withall
.
I
have
therefore
here
added
a
particular
Table
belonging
to
the
Latitude
of
London
51
gr
.
30
min.
that
by
it
the
said
Amplitudes
may
be
inserted
into
such
direct
Recliners
and
Incliners
as
shall
any
way
stand
in
the
said
Latitude
.
A
Table
of
Amplitudes
,
with
their
answerable
Declinations
,
for
the
Latitude
of
51
gr
.
30
min.
A.
Declin
A.
Declin
.
A.
Declin
.
A.
Decl●n
▪
1
0
37
11
6
43
21
12
53
31
18
42
2
1
14
12
7
26
22
13
29
32
19
16
3
1
52
13
8
03
23
14
04
33
19
49
4
2
29
14
8
40
24
14
40
34
20
22
5
3
07
15
9
16
25
15
15
35
20
55
6
3
44
16
9
53
26
15
50
36
21
28
7
4
21
17
10
29
27
16
25
37
22
0●
8
4
58
18
11
05
28
17
00
38
22
32
9
5
36
19
11
41
29
17
34
39
23
04
10
6
12
20
12
18
30
18
08
40
●3
35
Another
Table
of
Amplitudes
,
with
the
Tangents
of
their
answerable
Declinations
,
for
the
Latitude
of
51
gr
.
30
mi.
A.
Tan.
A.
Tan.
A.
Tan.
A.
Tan.
1
0109
11
1196
21
2289
31
3385
2
0217
12
1305
22
2398
32
3495
3
0326
13
1414
23
2507
33
3●04
4
0434
14
1524
24
2617
34
3712
5
0543
15
1632
25
2726
35
3822
6
0652
16
1741
26
2836
36
3932
7
0761
17
1851
27
2946
37
4040
8
0869
18
1960
28
3056
38
4149
9
0981
19
2069
29
3166
39
4258
10
1087
20
2180
30
3275
40
4365
So
far
of
this
Elliptical
Dial
,
as
it
is
to
be
described
upon
any
direct
plain
which
lies
under
the
same
Meridian
with
the
Meridian
of
the
place
,
and
which
declineth
not
from
it
.
SECT
.
II.
How
to
frame
the
Elliptical
Dial
to
other
Plains
which
are
not
direct
but
declining
:
To
an
Index
that
standeth
perpendicular
to
the
plain
.
IT
is
not
here
to
be
enquired
whether
the
plain
declining
,
be
erect
or
leaning
;
for
one
rule
serves
both
these
kindes
.
But
then
it
must
first
be
supposed
,
that
the
plains
situation
is
in
all
respects
known
,
how
much
and
which
way
it
declines
and
reclines
or
inclines
.
Secondly
,
These
three
things
must
further
be
found
(
either
by
calculation
or
otherwise
)
namely
,
1.
The
Poles
elevation
above
the
plain
.
2.
The
plains
difference
of
Longitude
.
3.
The
departure
of
the
substylar
(
which
is
the
plains
proper
Meridian
)
from
the
Vertical
line
of
the
plain
.
These
are
pre-requisites
to
that
which
comes
after
,
in
which
you
must
proceed
by
the
following
directions
.
How
to
limit
and
draw
the
Ellipsis
.
Having
the
substylar
or
proper
Meridian
drawn
upon
the
plain
,
in
its
true
position
,
and
also
knowing
the
elevation
of
the
Pole
above
the
plain
,
you
may
,
upon
the
substylar
line
,
set
out
your
shorter
diameter
,
and
on
a
line
drawn
perpendicular
thereto
,
you
may
set
off
the
longer
diameter
,
in
this
proportion
:
Let
the
longer
Diameter
be
as
the
Radius
,
the
shorter
as
the
Sine
of
the
plains
Latitude
taken
to
the
same
Radius
.
Or
,
the
longer
semidiameter
may
be
the
Radius
,
and
the
shorter
semidiameter
must
then
be
the
Sine
of
the
elevation
of
the
Pole
above
the
plain
.
Upon
these
two
extream
diameters
thus
limited
,
may
the
Ellipsis
be
described
with
Elliptical
Compasses
.
Or
otherwise
it
may
be
done
by
two
Circles
,
the
one
circumscribed
,
the
other
inscribed
,
both
divided
into
like
parts
,
and
so
points
found
through
which
the
Ellipsis
is
to
passe
,
as
in
the
former
direct
plains
was
prescribed
,
and
through
those
points
the
Ellipsis
may
,
with
a
stedfast
hand
be
described
.
In
right
Horizons
or
Polar
plains
there
will
be
no
Ellipsis
at
all
,
but
it
vanisheth
into
a
streight
line
only
.
How
to
divide
the
Ellipsis
into
such
requisite
parts
as
the
plain
shall
require
.
If
you
should
put
in
the
houres
proper
to
the
plain
there
will
be
no
difficulty
,
for
then
the
substilar
being
taken
for
the
line
of
12
,
the
houres
must
be
drawn
as
before
was
shewed
in
direct
plains
.
For
in
this
case
the
Dial
is
Horizontal
,
shewing
the
houre
of
the
day
to
all
places
that
lie
under
that
Meridian
under
which
the
plain
it selfe
lyeth
▪
But
If
you
would
put
in
the
houres
of
the
place
(
as
most
frequently
is
desired
)
then
you
are
to
work
(
for
the
division
of
the
Ellipsis
)
either
by
calculation
or
by
protraction
.
If
you
calculate
for
the
Ellipsis
,
you
must
first
forme
all
the
angles
at
the
Pole
as
the
manner
is
in
these
kindes
of
Dials
:
Then
you
are
to
invert
the
terms
of
proportion
from
the
common
way
of
calculating
ordinary
Dials
,
and
say
thus
:
As
the
Sine
of
the
Poles
elevation
above
the
plain
,
Is
to
the
Radius
;
So
is
the
Tangent
of
each
angle
at
the
Pole
,
To
the
Tangent
of
the
angular
distances
of
all
the
houres
from
the
substylar
line
of
the
plain
.
Supposing
therefore
that
you
had
your
Ellipsis
before
described
,
and
the
substilar
line
set
off
from
the
plains
Vertical
line
in
its
true
position
,
you
may
from
that
substylar
line
(
and
by
the
horary
spaces
found
by
calculation
)
set
in
every
houre
point
into
the
Ellipsis
,
by
help
of
any
Circle
described
upon
the
center
A.
As
you
see
here
are
two
Circles
,
the
circumscribed
and
the
inscribed
,
either
of
which
,
or
any
other
circle
else
,
will
serve
to
prick
on
the
houres
by
.
The
houre
points
are
here
set
upon
the
circumscribed
Circle
,
and
transferred
into
the
Ellipsis
by
a
Ruler
laid
to
A
,
the
common
center
of
the
Ellipsis
and
of
the
Circle
.
This
is
one
way
,
and
is
best
done
by
calculation
:
here
you
may
also
put
in
the
halves
,
and
quarters
,
and
halfe
quarters
of
houres
,
or
what
other
parts
you
shall
best
like
of
.
How
to
limit
the
Ellipsis
,
and
divide
it
,
without
Calculation
.
But
if
you
are
desirous
to
do
it
rather
by
protraction
,
then
must
you
work
somwhat
as
formerly
was
done
,
the
manner
I
will
in
briefe
shew
,
wherein
you
may
see
the
way
both
how
to
describe
the
Ellipsis
,
and
also
how
to
divide
it
,
all
under
one
.
1.
Having
prepared
all
necessary
requisites
before
hand
,
you
must
first
set
off
the
substilar
line
in
its
true
position
from
the
Vertical
line
,
which
to
do
,
I
here
suppose
already
known
.
So
in
the
following
figure
you
see
A
C
B
drawn
for
the
substilar
line
,
or
the
proper
meridian
of
the
plain
.
2.
Upon
this
plain
,
and
upon
some
point
of
the
Substylar
line
,
as
at
A
,
as
upon
a
center
,
describe
the
Circle
B
M
E
,
and
quarter
it
,
and
let
the
Semidiameter
of
that
Circle
be
counted
as
the
Radius
3.
Upon
one
of
the
quarters
,
set
off
the
Poles
elevation
as
from
E
to
F
,
and
draw
A
F
,
and
from
E
to
A
F
take
the
least
distance
,
as
E
G
,
and
with
that
distance
,
upon
the
former
center
A
,
describe
another
lesser
Circle
as
C
H.
4.
From
B
(
where
the
greater
Circle
cuts
the
substylar
)
set
B
M
equall
to
the
plains
difference
of
Longitude
:
and
from
M
divide
that
exterior
Circle
into
24
equal
parts
,
which
signifie
the
24
houres
;
and
by
a
Ruler
laid
to
the
common
center
A
,
transfer
them
into
the
lesser
Circle
,
as
M
H
,
10
I
,
6
K
,
&c.
5.
From
every
point
in
the
great
Circle
,
draw
lines
parallel
to
the
substylar
,
and
from
every
point
of
the
lesser
Circle
draw
lines
perpendicular
to
the
substylar
,
or
parallel
to
the
longest
diameter
P
A
R
,
and
so
each
couple
of
these
lines
shall
cut
each
other
at
right
angles
.
6.
Note
where
every
line
thus
drawn
from
any
point
of
the
greatest
Circle
,
meets
with
the
other
line
which
is
drawn
from
the
like
point
in
the
lesser
Circle
.
For
the
concourses
of
these
answerable
lines
are
the
points
here
required
.
That
is
,
Those
are
the
points
which
shew
both
where
the
Ellipsis
is
to
be
drawn
,
and
also
where
it
is
to
be
divided
into
its
requisite
parts
.
Thus
in
the
precedent
figure
:
M
O
and
H
O
meet
in
the
point
O
,
10
L
and
I
L
meet
in
the
point
L
,
6
N
and
K
N
meet
in
the
point
N
,
and
so
the
rest
will
meet
in
their
due
places
.
These
points
shew
the
way
where
through
the
Ellipsis
is
to
be
drawn
,
and
the
same
points
shew
where
the
houres
are
to
be
marked
out
.
The
like
may
be
done
for
quarters
and
halfe
quarters
of
houres
,
or
any
other
division
that
shall
be
best
liked
of
.
There
are
many
other
wayes
to
do
the
same
things
,
but
I
suppose
this
to
be
most
expedient
.
Note
:
that
1.
The
Index
in
these
Elliptical
Dials
must
stand
perpendicularly
upright
upon
the
plain
,
making
right
angles
with
it
every
way
,
as
it
was
ordered
to
do
upon
the
former
sort
of
direct
plains
.
2.
The
annual
course
of
the
Sun
must
be
limited
as
formerly
in
the
other
plains
,
that
is
,
making
the
greater
semidiameter
as
a
Badius
,
you
must
finde
the
Co-sine
of
the
Poles
elevation
above
the
plain
.
This
Co-sine
is
to
be
made
either
as
a
Tangent
of
45
gr
.
whereof
you
are
only
to
use
23½
:
Or
else
it
is
to
be
made
a
Decimal
Scale
.
By
both
these
you
have
Tables
and
Rules
how
to
compleat
the
Suns
annual
course
,
either
in
degrees
of
the
Signes
in
the
Zodiac
,
or
by
the
dayes
of
the
12
moneths
.
3.
Either
the
Index
must
move
and
the
Ellipsis
lie
still
,
or
contrarily
.
Every
man
in
this
must
do
as
his
invention
shall
best
suggest
.
And
that
motion
must
be
made
either
upon
the
substylar
line
,
or
else
parallel
to
it
,
which
way
soever
it
be
,
it
must
be
precisely
and
punctually
ordered
.
4.
All
other
things
concerning
the
time
of
Sun
rising
and
setting
,
the
Suns
Amplitudes
and
Declinations
,
may
here
(
in
the
same
manner
as
before
)
be
inserted
either
proper
to
the
plain
,
or
proper
to
the
place
,
as
shall
be
desired
.
5.
In
right
Horizons
or
Polar
plains
,
the
Ellipsis
closeth
quite
up
into
a
streight
line
:
and
so
the
division
of
it
is
only
by
the
exterior
Circles
parts
projected
upon
it
by
those
lines
that
are
drawn
parallel
to
the
substylar
.
Thus
far
of
the
Elliptical
Dial
,
as
it
is
to
be
described
upon
declining
plains
whose
Indexes
stand
perpendicular
to
the
plains
,
and
which
do
not
lie
under
the
Meridian
of
the
place
but
swerve
from
it
.
SECT
.
III.
Another
way
to
prick
down
the
Ellipsis
upon
an
Horizontal
Plain
.
Latitude
51
gr
.
30
min.
Hor
Ang.
Altit
.
Hor
XII
00
●0
38
30
XII
04
47
38
24
09
33
38
7
14
16
37
38
XI
.
18
54
36
58
I.
23
27
36
7
27
53
35
7
32
13
33
56
X.
36
25
32
37
II.
40
29
31
10
44
26
29
36
48
15
27
54
IX
.
51
57
26
7
III.
55
32
24
14
59
1
22
16
62
24
20
14
VIII
65
41
18
8
IV.
68
54
15
59
72
2
13
47
75
7
11
33
VII
78
9
9
17
V.
81
9
6
59
84
7
4
39
87
4
2
20
VI.
90
0
0
0
VI.
FOr
this
purpose
here
are
two
Tables
joyned
together
,
both
of
them
made
for
the
Latitude
of
London
,
51
gr
.
30
min.
the
like
to
which
every
man
may
calculate
for
his
own
place
.
The
first
of
them
is
a
Table
of
such
angles
as
are
made
by
the
hour
lines
(
coming
through
the
center
of
the
Ellipsis
)
with
the
Meridian
line
or
line
of
12.
And
it
was
made
by
that
rule
which
was
given
(
in
this
case
)
at
the
beginning
of
this
Treatise
.
Namely
,
As
the
Sine
of
your
Poles
elevation
(
which
is
here
51
gr
.
30
m.
)
is
to
the
Radius
;
So
is
the
Tangent
of
each
houre
and
their
quarters
(
counted
from
12
a
clock
,
)
To
the
Tangent
of
the
angle
required
.
The
second
Table
is
of
the
Altitudes
of
each
houre
and
quarter
,
in
the
Equinoctial
Circle
,
above
the
Horizon
;
and
it
is
calculated
by
this
Rule
.
As
the
Radius
,
To
the
Co-sine
of
your
Latitude
(
which
is
here
the
Sine
of
38
gr
.
30
min.
)
So
is
the
Sine
of
each
houre
and
their
quarters
,
(
counted
from
6
a
clock
,
)
To
the
Sine
of
the
altitude
which
is
here
sought
.
Then
having
computed
these
two
Tables
,
you
may
(
by
help
of
them
)
both
draw
and
divide
the
Ellipsis
into
its
true
houres
and
quarters
by
these
following
directions
.
First
,
draw
the
two
lines
6
6
and
D
E
,
crossing
each
other
at
right
angles
in
A
,
and
let
A
D
be
the
Meridian
,
and
6
A
6
the
two
six
a
clock
lines
.
Then
upon
A
as
a
center
describe
the
Circle
6
D
6
of
any
convenient
bignesse
;
and
upon
the
same
Circle
from
D
(
on
both
sides
of
it
)
set
on
the
houres
and
their
quarters
,
as
11
and
1
,
10
and
2
,
&c.
by
help
of
the
degrees
and
minutes
of
the
angles
set
to
every
houre
in
the
former
Table
.
And
through
every
of
these
points
inscribed
into
the
Circle
,
draw
lines
from
the
center
A
,
as
A
11
,
A
10
,
A
1
,
A
2
,
with
the
rest
of
the
houres
and
their
quarters
if
you
will.
Fourthly
,
looke
in
the
second
Table
for
the
altitudes
of
the
Equinoctiall
at
every
houre
.
Then
count
those
Altitudes
in
the
line
of
Sines
,
and
take
with
your
Compasses
the
several
distances
of
them
from
A
,
&
transfer
the
said
distances
from
the
center
A
to
every
houre
respectively
,
so
shall
those
intersections
give
you
the
points
through
which
the
Ellipsis
must
be
drawn
.
Thus
the
altitude
of
7
and
5
a
clock
(
which
is
9
gr
.
17
m.
)
being
taken
upon
the
line
of
Sines
from
A
towards
S
,
is
inserted
into
4
houres
at
the
note
m.
And
the
altitude
at
8
and
4
(
which
in
the
Table
is
18
gr
.
8
min.
)
being
taken
and
transferred
to
the
four
houres
at
n
,
do
give
four
points
more
.
The
rest
of
the
altitudes
give
(
each
of
them
)
two
houres
only
,
as
at
o
,
i
,
and
r
,
is
done
.
And
the
last
of
all
at
12
,
gives
one
only
point
at
t.
The
like
may
be
done
for
the
quarters
.
And
so
through
these
points
thus
found
the
Ellipsis
may
easily
be
drawn
,
and
the
lines
formerly
drawn
give
the
divisions
that
are
due
unto
it
.
Note
.
1.
This
is
propounded
only
for
Horizontal
plains
,
but
it
may
without
difficulty
be
applyed
to
any
other
direct
plain
.
The
trouble
that
is
,
comes
by
reason
that
the
foresaid
plains
have
a
different
elevation
of
the
Pole
,
from
that
elevation
that
belongs
to
the
Horizon
or
place
where
they
are
to
stand
:
and
consequently
there
will
be
required
two
new
Tables
for
that
elevation
which
is
proper
to
the
plain
;
the
calculation
of
which
will
easily
be
done
by
the
two
former
proportions
set
down
for
that
end
.
Note
further
that
by
these
two
Tables
you
may
prick
down
an
Ellipsis
upon
any
leaning
(
not
upright
)
plain
whatsoever
.
But
the
Index
must
then
lie
in
the
Zenith
line
of
the
place
,
(
not
of
the
plain
)
and
the
Ellipsis
(
or
Index
)
must
move
in
the
Meridian
of
the
place
(
not
of
the
plain
)
and
the
Zodiac
must
be
like
or
proportional
to
the
Zodiac
of
the
Horizon
,
but
augmented
,
for
the
most
part
,
in
this
proportion
.
As
the
Radius
,
To
the
Secant
of
that
angle
that
the
Meridian
line
(
upon
the
plain
)
makes
with
the
Horizon
;
(
which
angle
must
be
gott●n
by
a
Clinatorie
;
)
So
the
Radius
of
the
Horizontal
Zodiac
,
(
which
is
the
Co-sine
of
the
Latitude
,
)
To
the
Radius
of
the
Zodiac
proper
to
the
plain
:
which
Zodiac
must
be
set
according
to
the
line
of
12
upon
the
plain
.
2.
This
way
of
delineation
may
likewise
be
applyed
to
all
other
plains
which
decline
,
and
are
not
direct
.
But
in
these
will
be
found
more
difficulty
then
in
the
former
plains
which
declined
not
,
unlesse
it
be
required
to
put
in
the
houres
that
are
proper
to
the
plain
;
for
in
that
case
the
work
is
the
same
which
was
in
them
.
But
if
the
houres
of
the
place
are
to
be
inscribed
,
as
most
usually
they
are
,
then
there
will
be
some
trouble
,
by
reason
that
the
difference
of
the
plains
Longitude
seldome
falls
upon
any
just
houre
.
I
purpose
not
here
to
shew
the
way
,
it
being
such
as
will
prove
un-pleasant
to
the
unskilfull
,
and
such
as
have
knowledge
will
finde
it
out
quickly
.
Perhaps
the
pleasure
of
the
thing
done
,
will
recompense
their
labour
.
SECT
.
IV.
Here
follow
some
Vses
and
Varieties
of
this
Ellipticall
Diall
.
USES
.
FIrst
,
If
the
Index
stand
still
,
and
the
Ellipsis
be
made
moveable
;
you
may
upon
that
plate
which
is
immoveable
,
and
on
which
the
Index
standeth
,
describe
a
Circle
,
whose
center
must
be
the
foot
of
the
Index
.
And
drawing
a
Meridian
line
upon
the
said
immoveable
plate
(
just
under
the
Meridian
of
the
Ellipsis
,
and
)
from
the
foot
of
the
Index
you
may
divide
that
Circle
(
beginning
from
the
Meridian
line
)
into
360
gr
.
or
so
many
of
them
as
shall
be
needfull
.
By
these
degrees
you
shall
finde
the
Suns
Azimuth
by
the
shadow
of
the
upright
Index
,
and
by
the
divisions
of
the
Ellipsis
you
may
know
the
houre
of
the
day
.
So
both
Azimuth
and
Houre
are
shewed
upon
the
two
plates
by
one
upright
Index
.
This
kinde
of
Diall
must
be
set
in
its
true
position
(
for
it
will
not
set
it selfe
)
and
must
there
be
fixed
.
If
it
be
upon
an
Horizontall
,
or
any
direct
plain
,
you
may
draw
a
Meridian
line
thereon
,
and
either
draw
the
Azimuths
true
upon
it
,
and
so
make
the
Ellipsis
to
move
to
and
fro
according
to
that
Meridian
:
Or
else
,
if
both
your
plates
be
to
be
fixed
upon
any
Base
prepared
for
them
,
you
may
,
first
,
upon
that
Base
draw
a
true
Meridian
line
,
and
to
that
line
make
the
Meridian
of
the
Diall
to
agree
.
The
like
must
be
done
in
all
declining
flats
,
onely
you
must
be
carefull
to
make
the
shortest
Di●meter
of
the
Ellipsis
(
which
is
the
proper
Meridian
of
the
plain
)
to
agree
with
the
same
proper
Meridian
,
as
it
shall
first
be
drawn
upon
the
Base
in
its
true
position
.
2.
A
second
excellent
use
may
be
for
the
setting
of
moveable
Dials
,
such
as
are
made
to
stand
out
of
the
weather
,
and
are
only
to
be
set
in
the
Sun
at
pleasure
,
when
the
houre
shall
be
required
.
Of
this
kinde
are
Polyhedrall
Bodies
made
of
some
light
matter
,
and
Concaves
,
&c.
But
it
is
supposed
that
some
of
the
superficies
of
such
Bodies
,
(
&c.
)
have
the
houres
of
the
p●ace
described
upon
them
the
ordinary
way
,
by
the
Axis
of
the
Equinoctiall
,
and
that
this
Ellipsis
is
fitted
to
some
one
plain
(
which
principally
is
the
Horizontal
plain
(
because
that
alone
will
shew
all
the
houres
of
the
day
)
or
in
some
Horizontal
position
to
the
concave
,
or
any
such
Diall
)
in
its
just
position
and
alwayes
with
an
upright
Index
,
(
that
is
,
an
Index
perpendicular
to
the
plain
whereon
it
stands
)
which
is
not
alwayes
upright
in
respect
of
us
,
but
only
in
respect
of
the
plain
)
and
with
the
houres
also
of
the
place
.
These
things
being
so
fitted
,
you
may
place
your
Body
or
Concave
in
the
Sun
,
and
turn
it
about
till
you
discern
that
both
kindes
of
Dials
do
agree
to
shew
one
houre
,
which
when
they
do
,
then
all
stand
right
in
their
true
situation
,
which
will
never
else
happen
but
only
when
the
houres
do
in
both
agree
.
—
To
this
place
is
referred
the
Double
Horizontal
Diall
,
the
one
of
them
giving
the
houre
by
the
Axis
,
the
other
by
the
upright
Index
.
—
And
here
it
must
be
noted
,
that
the
worst
time
of
the
day
for
this
setting
the
Dials
in
a
true
position
,
is
at
the
neerest
to
noon
,
and
the
best
time
is
about
the
Suns
b●ing
in
the
East
and
West
Azimuth
.
3.
This
kinde
of
Diall
may
be
placed
(
in
a
moveable
posture
)
before
a
casement
in
any
window
that
hath
a
flat
board
upon
which
it
may
move
.
And
the
Index
to
it
may
be
an
upright
threed
(
precisely
upright
it
must
be
)
that
is
fixed
at
the
top
and
bottome
of
the
window
,
to
and
from
which
the
Ellipsis
must
move
in
a
Meridian
line
that
is
drawn
upon
the
board
of
the
window
,
&
from
or
justly
even
upon
the
threed
.
—
And
where
I
say
the
window
must
be
flat
,
it
is
to
be
understood
that
there
is
no
extraordinary
exactnesse
required
in
that
.
For
if
it
should
rise
or
incline
5
gr
.
to
the
Horizon
,
it
would
not
erre
one
minute
of
time
about
our
Latitude
;
provided
that
the
Meridian
be
truly
drawn
,
and
the
Index
perfectly
upright
,
in
which
all
possible
exactnesse
is
required
.
4.
If
any
choise
plain
reclining
,
or
however
situated
plain
that
shall
represent
the
Horizon
of
some
notable
place
,
be
fixed
as
a
Base
whereon
to
place
such
a
Diall
thus
fitted
,
and
drawn
to
the
proper
houres
of
that
same
place
:
this
double
Diall
being
placed
upon
that
Base
,
and
turned
about
thereon
,
will
shew
the
true
houre
at
that
place
when
both
of
them
do
agree
,
and
the
Meridian
of
it
will
shew
how
the
proper
Meridian
of
that
place
is
to
be
placed
,
and
which
way
drawn
.
5.
If
such
a
Recliner
(
or
Incliner
,
or
Erect
)
shall
be
furnished
with
such
a
double
Diall
to
some
intended
Declination
,
though
they
stand
and
be
framed
properly
to
the
plain
in
respect
(
chiefly
)
of
the
upright
Index
,
and
shall
(
notwithstanding
)
have
the
proper
houres
of
the
place
upon
both
Dials
:
and
if
further
such
a
Recliner
as
was
mentioned
,
have
an
horizontal
base
,
upon
which
it
may
be
turned
round
:
I
say
,
though
this
Body
be
horizontally
moveable
,
and
may
be
turned
any
way
,
yet
the
two
Dials
will
never
agree
to
shew
the
same
houre
,
untill
the
Base
stand
to
the
true
Declination
to
which
the
double
Diall
was
drawn
.
SECT
.
V.
Some
varieties
of
the
structure
of
it
do
here
follow
.
FIrst
,
Instead
of
the
Ellipsis
drawn
to
the
proper
Meridian
and
Latitude
of
the
plain
,
there
may
be
used
a
true
Circle
equally
divided
:
but
then
let
these
things
be
observed
.
First
,
That
the
Index
must
be
perpendicular
every
way
to
the
plain
on
which
the
Circle
is
elevated
,
standing
upright
upon
it
.
Secondly
,
That
the
motion
(
either
of
Index
or
Circle
)
must
be
according
to
the
proper
Meridian
of
the
plain
.
Thirdly
,
That
the
Zodiac
must
also
be
proper
to
the
plain
,
and
must
lie
upon
the
plain
or
Base
it selfe
,
and
must
be
the
same
with
that
which
should
serve
for
the
Ellipsis
.
But
then
,
Fourthly
,
this
equally
divided
Circle
must
be
elevated
over
the
proper
Meridian
of
the
plain
(
that
is
,
the
Base
of
it
must
rest
upon
the
proper
East
and
West
line
of
the
plain
)
so
much
it
must
be
elevated
as
the
elevation
of
the
Equinoctiall
above
that
plain
comes
unto
.
And
the
elevation
of
it
may
be
either
way
;
that
is
,
either
upon
the
South
Coast
,
and
then
it
will
be
answerable
to
the
Equinoctiall
in
the
heavens
;
or
else
it
may
be
towards
the
North
,
but
still
so
much
as
the
Equinoctials
(
not
the
Poles
)
elevation
comes
unto
,
and
in
some
cases
it
must
be
done
both
wayes
that
the
Sun
may
come
at
it
,
and
it
must
then
have
two
Indexes
.
Both
wayes
it
is
alike
true
.
It
must
be
divided
into
equall
houres
and
parts
,
and
so
much
of
it
must
stand
above
the
Base
or
plain
on
which
it
moves
,
as
the
longest
day
comes
unto
.
The
beginning
of
the
division
must
be
from
the
line
of
12
,
and
if
the
plain
be
not
direct
,
then
must
the
difference
of
Longitude
be
counted
in
the
circular
degrees
,
from
the
highest
point
,
or
from
the
Verticall
line
of
this
Equinoctiall
Circle
.
2.
This
Diall
is
not
so
tyed
to
a
plain
,
but
that
it
may
also
be
brought
to
an
un-even
Superficies
,
yet
so
,
that
the
motion
of
it
,
or
the
Zodiacal
Scale
must
be
upon
some
plain
.
But
to
shew
the
manner
how
this
should
be
performed
,
is
more
proper
for
those
Projecting
wayes
which
I
have
in
another
place
had
some
assayes
of
.
Only
remember
that
the
Horizontal
Diall
must
be
projected
,
not
by
the
Axis
,
but
by
the
Zenith
.
3.
A
double
Diall
depending
upon
this
kinde
,
may
be
made
to
set
it selfe
,
and
to
shew
the
houre
in
a
Craticular
forme
,
which
is
also
more
properly
to
be
handled
in
my
other
wayes
.
There
remains
yet
one
structure
more
,
by
which
it
is
fitted
for
some
uses
that
follow
after
,
and
is
made
a
portable
Instrument
,
not
at
all
to
be
used
in
observation
.
1.
MAke
a
right
angled
Parallelogram
,
as
A
B
C
D
,
whose
length
may
be
about
three
times
the
breadth
:
and
let
the
limbe
(
upon
which
the
graduations
stand
)
be
higher
then
the
Plain
or
Area
of
it
.
2.
Describe
a
Semiellipsis
to
your
own
Latitude
,
whose
longest
Radius
let
be
M
O
,
a
little
longer
than
A
B
,
that
when
the
center
M
is
left
out
and
cut
off
,
so
much
as
M
N
comes
to
(
which
may
be
about
halfe
a
quarter
of
an
houre
from
12
in
the
Ellipsis
)
the
remaining
breadth
N
O
,
or
E
F
,
or
G
H
,
may
be
just
fit
to
A
B
the
breadth
of
the
hollow
Area
,
in
which
this
lesser
rectangled
Parallelogram
E
F
G
H
may
be
moved
to
and
fro
as
there
shall
be
occasion
.
And
let
the
thicknesse
or
depth
of
this
moving
plate
be
the
same
with
the
thicknesse
of
the
limbe
of
the
lower
plate
.
The
division
of
the
Ellipsis
may
be
made
according
to
the
rules
before
given
.
3.
In
the
midst
of
this
lesser
plate
,
let
a
hole
be
pierced
quite
through
it
;
and
in
that
hole
let
a
threed
be
extended
,
as
I
K
,
which
is
to
serve
for
an
Index
,
by
which
the
said
Semiellipsis
is
to
be
rectified
.
And
therefore
it
is
best
that
the
threed
I
K
be
laid
just
in
the
longer
diameter
,
which
is
the
6
a
clock
line
.
4.
Take
the
lesser
Parallelogram
and
put
it
into
its
place
,
as
close
to
the
end
A
B
as
it
will
go
,
and
when
it
is
there
laid
,
observe
where
the
threed
lies
upon
the
lower
Area
,
and
draw
the
line
upon
which
it
lies
,
which
must
be
the
line
90
,
90
;
upon
which
the
Scales
(
that
are
inscribed
)
must
take
their
beginning
.
And
likewise
you
are
to
take
the
longer
Radius
O
M
,
and
as
the
plate
there
lies
,
set
one
foot
of
that
Radius
upon
O
,
and
let
the
other
reach
as
far
as
it
will
,
which
must
be
upon
the
rising
limbe
beyond
the
inner
edge
of
it
,
so
much
as
M
N
comes
to
:
in
that
point
you
are
to
make
one
center
:
so
again
,
turning
your
lesser
plate
about
,
you
must
lay
it
in
the
contrary
way
,
close
to
A
B
,
as
you
did
before
,
and
from
the
point
O
you
may
finde
the
second
center
:
both
these
centers
(
in
each
of
which
must
a
threed
be
fastned
)
will
happen
to
fall
in
the
former
line
90
,
and
90
,
neer
the
points
T
and
V.
Being
thus
far
prepared
,
you
may
take
away
the
lesser
plate
,
and
divide
the
Scales
upon
the
greater
plate
,
in
this
manner
.
5.
First
,
For
the
two
limbs
of
it
,
they
are
nothing
else
but
the
degrees
of
a
Circle
projected
thereon
from
the
two
centers
:
which
may
therefore
without
difficulty
be
done
either
from
the
degrees
of
a
Circle
,
or
else
by
Tangent
lines
,
as
any
man
will
easily
perceive
.
6.
Secondly
,
For
the
other
Scales
you
must
make
them
parallel
to
the
edge
of
the
limbs
.
There
are
three
of
them
in
number
:
The
longest
is
a
Scale
of
Tangents
:
The
middlemost
is
a
Scale
of
houres
,
made
in
the
form
of
a
line
of
Sines
:
The
third
is
a
Zodiac
,
or
Scale
of
the
year
.
The
Tangents
are
made
thus
.
Taking
M
O
as
Radius
,
finde
the
Co-sine
of
your
Latitude
agreeable
thereto
:
and
making
that
Co-sine
as
a
Radius
,
or
Tangent
of
45
degrees
,
all
the
other
Tangents
are
to
be
fitted
accordingly
.
Then
to
know
how
far
of
this
line
will
be
usefull
,
put
the
lesser
plate
again
into
its
place
,
as
close
to
the
end
C
D
as
it
will
go
,
so
shall
the
threed
I
K
shew
the
furthest
point
of
it
that
will
be
usefull
.
The
middle
Scale
of
Sines
is
made
thus
.
Count
the
complement
of
your
Latitude
upon
the
new
made
Tangent
line
,
and
from
the
beginning
of
the
Scale
to
that
point
,
make
a
line
of
Sines
for
six
houres
and
their
quarters
,
so
shall
this
Scale
be
fitted
.
The
Zodiac
or
Annual
Scale
is
equall
in
length
to
the
Tangent
of
23
gr
.
30
min.
and
the
manner
of
inscription
with
Tables
whereby
to
do
it
,
are
set
down
before
,
and
need
not
be
here
repeated
.
The
reason
why
this
Instrument
is
thus
contrived
.
The
intent
of
the
contrivance
is
,
that
it
should
shew
both
Azimuth
and
houre
,
and
that
therefore
there
should
be
a
Ci●●●
and
an
Ellipsis
sliding
over
it
,
both
of
them
made
to
one
center
,
so
that
the
center
should
be
in
the
midst
,
and
the
Circle
should
be
a
whole
entire
Circle
,
and
the
Ellipsis
also
whole
.
But
because
the
Ellipsis
sliding
upon
the
nether
plain
,
doth
somtimes
of
necessity
cover
the
center
,
so
as
that
there
can
be
no
threed
fixed
for
use
,
I
was
therefore
compelled
to
order
it
in
this
manner
that
is
before
exprest
,
that
the
motion
of
the
upper
plate
and
the
centers
migh●
stand
cleere
without
hin●●ing
each
other
.
Then
further
it
must
be
noted
,
that
the
Instrument
is
(
not
generall
to
all
,
but
)
particular
to
one
La●●tude
.
The
uses
of
it
do
here
follow
,
which
by
reason
of
the
Semiellipsis
will
not
be
so
easie
,
because
many
changes
are
required
.
In
generall
note
thus
much
.
1.
In
using
this
Instrument
you
must
alwayes
suppose
that
you
looke
upon
the
North.
2.
If
the
Declination
of
the
Sun
or
Star
be
North
,
hold
A
B
towards
you
.
If
South
,
then
hold
C
D
towards
you
.
3.
If
the
Sun
or
Star
be
Eastward
of
the
Meridian
,
let
the
morning
houres
be
towards
you
.
If
they
be
Westward
,
let
the
evening
houres
be
towards
you
.
4.
Alwayes
use
that
threed
which
is
on
the
same
side
with
the
center
of
the
Ellipsis
.
And
that
side
lin●
of
the
larger
plate
is
ever
to
be
taken
for
the
Meridian
,
and
consequently
the
other
side
must
be
esteemed
for
one
of
the
two
Coasts
of
East
or
West
.
5.
Becau●e
North
is
furthest
from
you
,
therefo●●
the
furthest
quarter
of
the
limbe
must
be
either
North-east
or
North-west
:
the
neerest
South-east
or
South-west
.
These
cautions
being
observed
,
the
conclusions
that
are
to
be
wrought
will
be
easie
.
1.
By
having
the
Azimuth
to
finde
the
Houre
.
First
,
Order
your
Instrument
by
the
former
cautions
,
according
to
the
present
case
of
your
observation
.
And
rectifie
the
upper
●late
(
by
help
of
the
threed
Index
)
either
to
the
Declination
of
the
Star
,
counted
in
the
Tangent
line
,
or
to
the
Suns
place
in
the
Zodiac
.
Then
count
the
Azimuth
given
upon
the
right
quarter
of
the
limbe
,
and
apply
the
threed
thereto
:
So
shall
the
threed
shew
(
upon
the
Ellipsis
)
the
houre
in
which
the
Sun
or
Star
is
.
If
the
work
be
for
the
Sun
,
then
the
houre
thus
shewed
is
the
Suns
houre
,
or
houre
of
the
day
.
If
the
work
be
for
a
Star
,
it
then
shews
the
Stars
houre
only
;
which
must
afterwards
be
reduced
into
the
Suns
houre
,
that
it
may
be
the
true
houre
of
the
night
.
This
change
may
most
easilie
be
done
by
a
common
Nocturnall
.
2.
By
having
the
Houre
in
which
the
Sun
or
any
Star
is
,
the
Azimuth
thereof
may
be
found
.
The
Instrument
being
rectified
;
lay
the
threed
upon
the
houre
(
counted
in
the
Ellipsis
)
the
same
threed
will
shew
the
Azimuth
in
the
graduations
upon
the
limbe
.
3.
To
finde
the
Ascensionall
difference
,
the
Semidiurnall
and
Seminocturnall
arkes
,
with
the
time
of
their
risings
and
settings
.
In
generall
note
this
:
If
the
Sun
or
Star
decline
Northward
,
then
the
Semidiurnall
arke
is
greater
than
six
houres
;
if
Southward
,
then
lesse
.
The
upper
plate
therefore
being
rectified
,
either
to
the
Suns
place
,
or
to
the
Stars
Declination
,
the
threed
will
shew
upon
the
middle
Scale
,
the
Ascensionall
difference
,
if
you
count
the
houres
how
many
they
are
from
6.
It
shews
the
Seminocturnall
arks
,
by
the
houres
of
setting
and
rising
being
rightly
taken
according
to
the
condition
of
North
or
South
declination
.
It
shews
the
time
of
Sun
rising
and
setting
,
according
to
the
houres
upon
which
it
lies
.
But
for
the
Stars
,
it
shews
only
upon
what
houre-circles
they
rise
and
set
;
but
the
true
houre
or
time
thereof
must
be
found
by
turning
the
Stars
houre
into
the
Suns
houre
,
which
(
as
I
said
before
)
will
be
done
with
most
ease
by
an
ordinary
Nocturnall
,
in
which
the
Stars
(
that
you
deal
with
)
are
placed
.
4.
To
finde
the
Amplitude
of
their
Risings
and
Settings
.
When
your
upper
plate
lies
so
rectified
,
lay
the
thre●●
upon
the
Ellipsis
,
to
the
Ascensionall
difference
,
or
S●midiurnall
arke
before
found
.
The
threed
so
laid
,
will
shew
the
Amplitude
in
the
limb
of
the
Instrument
.
Note
here
.
In
the
first
of
these
Propositions
I
suppose
the
Azimuth
,
in
the
second
I
suppose
the
Houre
to
be
given
by
observation
.
For
the
observing
of
these
two
,
you
may
make
in
any
convenient
window
,
both
an
Horizontall
Diall
,
with
a
threed
Axis
lying
in
the
Axis
of
the
world
,
and
an
Azimuth
Circle
with
a
threed
Axis
,
standing
upright
in
the
Zenith
line
.
The
divisions
sutable
to
both
these
,
may
be
inscribed
upon
the
board
of
the
window
;
and
for
the
Sun
,
the
shadow
of
the
threed
will
give
the
houre
of
it
upon
the
Horizontall
Diall
;
or
the
Azimuth
upon
the
Azimuthall
Circle
.
But
for
the
Stars
that
cast
no
shadows
,
you
may
make
sliding
sights
,
that
may
move
upon
the
edge
of
the
window
board
,
or
may
be
upon
the
bo●rd
it selfe
,
(
which
must
in
these
respects
be
level
,
or
at
least
neer
to
it
.
)
And
those
moving
sights
must
be
furnished
with
short
wires
or
pins
which
may
best
stand
upon
the
edge
of
the
sight
,
and
go
down
to
the
window
board
,
and
to
the
divisions
that
are
drawn
upon
it
,
which
may
be
both
for
sights
by
which
to
looke
at
the
standing
Index
and
Star
both
together
,
and
must
serve
as
Indexes
to
shew
the
houre
or
Azimuth
on
which
a
Star
is
at
any
time
.
Only
this
generall
caution
must
be
given
,
that
th●se
pins
be
set
parallel
to
the
threed
Axis
that
is
set
in
the
window
(
that
is
)
for
the
Azimuths
they
must
stand
upright
,
but
for
the
houre
they
must
lie
aslope
,
parallel
to
the
Axis
of
the
world
.
This
I
have
sometimes
done
,
and
this
any
man
may
at
his
pleasure
do
,
whereby
either
the
Houre
or
A●●muth
may
be
observed
,
and
when
any
one
of
them
is
known
;
this
Instrument
(
now
described
)
will
finde
the
other
.
In
such
like
cases
this
Instrument
will
be
delightfull
and
of
good
use
.
5.
To
finde
the
Declination
of
the
Sun.
Set
the
threed
of
the
upper
pla●e
upon
the
Suns
place
in
the
Zodiac
,
or
upon
the
day
of
the
moneth
in
the
Annual
Scale
,
the
same
th●eed
will
give
the
Declination
thereto
belonging
,
upon
the
Tangent
Scale
.
6.
How
to
make
an
Horizontall
Diall
by
this
Instrument
.
The
meaning
is
to
know
how
many
degrees
are
intercepted
between
the
Meridian
line
,
and
any
other
houre
line
.
There
are
two
wayes
to
do
it
.
The
first
is
the
converse
of
the
making
of
the
Instrument
.
For
if
the
threed
Index
be
laid
to
the
line
where
the
Scales
of
the
lower
plate
begin
,
so
that
the
line
of
six
do
lie
evenly
between
the
two
●●nters
where
the
threeds
are
fastned
,
t●en
the
center
threed
applyed
to
the
houres
severally
will
shew
how
much
they
are
gradually
distant
from
the
line
of
12.
But
you
must
here
remember
to
take
6
for
12
,
and
7
or
5
for
11
or
1
,
and
so
the
rest
of
the
houres
in
that
order
.
And
the
graduall
space●
or
distances
from
the
Meridian
or
line
of
12
,
must
be
counted
from
90
contrary
to
the
numeration
of
the
limb
.
Bu●
this
is
not
so
artificiall
as
this
that
now
follows
.
Set
the
moving
plate
to
any
houre
upon
the
middle
Scale
,
and
apply
the
center
threed
to
the
like
houre
in
the
Ellipsis
taken
on
that
side
of
it
that
is
furthest
from
●he
beginning
of
the
three
Scales
,
and
it
will
s●ew
you
(
upon
the
limb
)
the
angle
or
Horizontall
space
that
is
du●
to
that
houre
.
The
like
is
to
be
done
in
every
other
of
the
houres
.
SECT
.
VI.
AN
ADVERTISEMENT
Concerning
s●me
ot●er
uses
of
the
Instrument
that
was
last
described
.
SInce
the
writing
of
that
which
is
gone
before
,
other
things
there
are
which
came
into
my
thoughts
concerning
the
further
use
of
this
Instrument
.
It
were
more
expedient
therefore
that
the
Ellipsis
were
divided
into
houres
and
degrees
,
rather
then
into
quarters
and
half
quarters
of
houres
.
Which
division
(
as
also
the
description
of
it
)
may
by
protraction
be
performed
by
those
rules
that
are
given
before
.
But
if
it
be
thought
better
to
do
it
by
Tables
;
then
by
the
former
rules
of
calculation
given
for
this
work
,
you
may
frame
two
Tables
,
the
one
of
Angles
,
the
other
of
Altitudes
to
your
own
Latitude
,
such
as
in
the
following
Table
are
computed
to
●he
Latitude
of
London
.
B●
these
Tables
you
may
put
in
every
houre
and
third
degree
,
the
intermediate
degrees
may
be
equally
divided
.
The
numeration
will
be
either
by
15
,
30
,
45
,
&c.
according
as
the
signing
of
it
by
hours
wil
requir●
;
Or
●lse
besides
the
houres
you
may
set
on
10
,
20
,
30
,
&c.
in
small
figures
,
which
will
stand
without
hindring
the
numbers
(
or
numerall
letters
)
that
are
set
for
the
houres
.
Ad
Latitudinem
51
gr
.
30
min.
Hor
gr
.
Angles
Altit
.
gr
.
Hor
XII
00
00
00
38
30
00
XII
12
3
51
38
26
3
9
7
39
38
16
6
6
11
26
37
57
9
3
15
12
37
31
12
XI
.
00
18
54
36
58
00
I.
12
22
33
36
18
3
9
26
08
35
32
6
6
29
38
34
40
9
3
33
04
33
41
12
X.
00
36
25
32
37
00
II.
12
39
41
31
27
3
9
42
52
30
14
6
6
45
59
28
56
9
3
49
00
27
33
12
IX
.
00
51
57
26
07
00
III.
12
54
50
24
37
3
9
57
38
23
04
6
6
60
23
21
28
9
3
63
04
19
49
12
VIII
00
65
41
18
08
00
IV.
12
68
16
16
25
3
9
70
47
14
40
6
6
73
17
12
53
9
3
75
44
11
05
12
VII
00
78
09
09
17
00
V.
12
80
33
07
26
3
9
82
56
05
35
6
6
85
18
03
44
9
3
87
39
01
52
12
This
Ellipsis
thus
divided
,
may
represent
either
the
Horizon
,
or
else
any
of
the
(
Almicantars
or
)
parallel
circles
to
the
Horizon
;
&
the
divisions
of
it
must
then
signifie
the
Azimuths
.
And
if
it
be
too
big
(
especially
when
it
comes
to
represent
any
of
the
higher
Almicantars
)
you
may
adde
another
sliding
plate
of
the
same
breadth
that
the
former
was
,
na●●ly
M
O
here
equall
to
M
O
in
the
former
,
and
M
N
must
in
both
be
justly
equall
,
that
it
may
both
slip
in
the
former
cavity
,
so
as
just
to
fill
it
,
and
that
the
same
limbs
,
centers
,
and
threeds
,
may
in
both
agree
.
A
lesser
Ellipsis
(
I
say
)
must
be
described
,
which
though
it
be
lesse
,
yet
it
must
be
like
and
proportionall
to
the
former
.
Therefore
assuming
any
length
as
M
P
,
for
the
longer
Radius
,
you
must
(
to
that
Radius
)
make
M
R
and
M
S
equall●
to
the
Sine
of
your
Latitude
,
and
so
describe
and
divide
it
as
formerly
was
ordered
.
Then
again
,
to
this
new
Ellipsis
there
must
a
new
Scale
of
Tangents
be
made
,
whose
length
must
be
limited
as
the
former
was
,
in
this
manner
.
To
M
P
as
Radius
,
finde
the
Co-sine
of
your
Latitude
,
and
make
that
Co-sine
a
Radius
or
Tangent
of
45
gr
.
and
according
thereunto
,
continue
the
Scale
of
Tangents
as
far
as
it
will
go
.
It
must
begin
upon
the
same
line
that
the
other
Scale
of
Tangents
began
,
and
must
go
the
same
way
with
it
.
And
in
this
new
way
of
ordering
the
Scales
and
the
Ellipses
,
it
must
be
considered
what
the
other
parts
of
the
Instrument
do
signifie
The
end
of
the
lower
plate
,
upon
which
the
Scales
begin
,
must
be
accounted
for
the
North
,
the
other
end
C
D
is
to
be
taken
for
South
.
The
side
lines
are
the
Meridian
.
The
degrees
of
the
limbs
are
to
be
used
as
the
degrees
of
the
Equinoctiall
.
The
threed
applyed
to
those
degrees
are
the
Meridians
comming
from
the
Centers
,
which
are
the
North
Pole.
The
Tangents
are
the
degrees
of
Altitude
.
The
Ellipsis
notes
out
that
Almicantar
which
the
threed
Index
stands
unto
upon
the
Tangent
Scale
.
And
further
.
It
is
to
be
here
noted
,
that
if
the
Meridians
(
or
threeds
comming
from
the
center
)
were
rightly
divided
,
those
divisions
or
parts
of
it
should
signifie
the
degrees
of
Declination
from
the
Equinoctiall
.
But
the
inconvenience
of
it
is
,
that
because
one
Ellipsis
is
to
represent
every
Almicantar
,
therefore
the
Radius
cannot
possibly
be
at
all
times
the
same
in
length
,
but
must
varie
according
to
the
removing
or
severall
positions
of
the
said
Ellipsis
.
That
is
,
when
the
Ellipsis
signifies
the
Horizon
(
or
stands
at
the
beginning
of
the
Scales
,
without
any
altitude
at
all
)
then
the
Radius
M
O
in
the
first
moving
plate
,
or
M
P
in
the
second
moving
plate
,
is
the
Scale
of
Declinations
,
but
at
other
times
,
when
the
Ellipsis
signifies
any
Almicantar
,
then
must
the
Secant
of
that
Almicantar
(
of
Altitude
)
be
taken
as
the
Radius
for
that
particular
work
.
But
alwayes
the
Radius
(
of
what
length
soever
it
be
)
must
be
divided
as
a
whole
line
of
Sines
,
the
greatest
parts
beginning
towards
the
center
,
the
least
parts
ending
at
the
remotest
end
from
the
center
,
and
yet
then
again
,
the
numeration
of
the
parts
or
declinations
of
it
must
begin
at
the
remotest
end
,
and
must
end
in
the
center
at
the
Pole.
It
might
be
applyed
to
use
in
the
motions
of
the
Stars
,
but
that
would
be
troublesome
;
therefore
it
shall
suffice
to
make
it
usefull
only
for
the
Sun.
The
best
way
that
(
for
the
present
)
I
know
whereby
to
divide
this
Radius
(
thus
of
severall
lengths
)
is
this
.
Divide
the
length
from
the
center
T
to
C
into
90
Sines
(
the
greatest
parts
beginning
at
the
center
,
but
numbered
the
contrary
way
.
)
This
line
may
be
drawn
from
the
center
T
into
some
part
of
the
limbe
,
where
it
may
have
room
to
receive
divisions
.
Then
with
your
Compasses
take
the
Radius
T
V
,
equall
to
M
O
,
and
from
the
end
of
the
former
Scale
,
open
the
threed
to
the
least
distance
,
and
where
the
threed
stayes
,
there
make
a
mark
upon
the
limbe
V
D
,
for
there
is
a
new
Scale
to
begin
,
for
the
Secants
that
were
mentioned
before
.
Then
lay
the
threed
from
T
to
80
in
the
opposite
limb
,
and
move
the
Ellipsis
till
the
point
O
fall
just
under
the
threed
,
and
take
T
O
in
your
Compasses
,
and
put
in
that
distance
as
before
,
so
shall
you
put
in
10
gr
.
into
this
new
Scale
.
In
like
manner
lay
the
threed
to
70
,
60
,
&c.
and
bring
the
Ellipsis
till
the
point
O
lies
under
it
,
so
shall
T
O
(
in
these
severall
positions
)
give
the
lengths
to
be
inscribed
in
this
new
Scale
,
for
20
gr
.
and
30
gr
.
&c.
all
the
rest
must
be
done
in
like
manner
,
till
you
have
put
in
as
many
as
will
come
within
the
reach
of
the
limb
,
which
will
be
upon
60
gr
.
or
thereabout
.
The
like
should
also
be
done
for
the
lesser
Ellipsis
,
working
in
the
same
manner
by
the
threed
and
the
point
P.
These
are
called
the
new
inscribed
Scales
.
The
former
line
of
Sines
serves
to
them
both
indifferently
.
Now
when
things
are
thus
fitted
,
the
uses
of
them
will
be
such
as
in
particular
Astrolabes
is
vulgarly
known
.
Some
of
the
Uses
are
here
mentioned
.
1.
Having
the
Suns
Declination
and
Altitude
to
finde
the
Houre
and
Azimuth
.
First
,
For
North
Declinations
.
Set
the
Ellipsis
to
the
Altitude
counted
in
the
Tangent
Scale
.
Then
count
the
same
Altitude
in
the
new
inscribed
Scale
of
Secants
,
and
thereto
lay
the
threed
.
Afterwards
,
take
the
least
distance
from
the
Suns
declination
,
(
counted
in
the
fore-mentioned
inscribed
Scale
of
Sines
)
to
the
threed
.
Set
one
foot
of
that
distance
in
the
center
,
and
extend
the
other
till
it
crosse
the
Ellipsis
,
and
where
it
crosseth
,
thither
apply
the
threed
,
which
(
in
the
limb
)
will
shew
the
Houre
and
scruple
(
in
degrees
)
required
.
The
same
point
of
the
Compasses
doth
also
immediately
shew
(
upon
the
Ellipsis
)
the
Azimuth
sought
for
.
If
the
Compasses
do
crosse
the
Ellipsis
twice
(
as
somtimes
it
will
)
take
that
crossing
that
is
furthest
from
the
Meridian
.
Secondly
,
for
South
declinations
.
The
work
is
in
a
manner
the
same
:
only
you
must
note
that
the
threed
will
crosse
the
Ellipsis
twice
,
and
you
are
in
this
case
to
take
that
crossing
which
is
neerest
to
the
Meridian
.
By
supposing
the
Altitude
to
be
00
,
that
is
,
by
laying
the
Ellipsis
upon
the
line
T
V
,
and
using
the
Declination
as
before
,
you
may
finde
both
the
Amplitude
and
Ascensionall
difference
as
before
,
and
what
else
doth
thereon
depend
.
But
because
these
things
being
done
neer
the
Horizon
,
will
not
prove
good
,
especially
when
the
Declination
is
little
from
the
Equinoctiall
:
and
because
the
work
at
the
best
is
but
troublesome
,
I
shall
here
break
off
,
supposing
that
I
have
already
written
too
much
.
A
DEMONSTRATION
of
the
Ellipticall
Diall
upon
an
Horizontall
Plain
:
shewing
the
reason
why
the
same
Diall
,
by
an
upright
Index
should
shew
the
true
Houre
.
THE
reason
is
principally
deduced
from
the
Sphere
it self
:
and
secondly
,
from
the
Orthographicall
Projection
of
the
Sphere
upon
the
plain
of
the
Horizon
;
which
as
it
doth
represent
the
Sphere
it selfe
,
so
it
doth
perform
the
same
conclusions
with
the
like
certainty
that
the
Sphere
doth
;
and
that
upon
this
generall
ground
,
that
looke
what
Circles
,
and
in
what
parts
they
cut
each
other
upon
the
former
,
the
same
Circles
and
the
same
parts
of
those
Circles
do
upon
this
latter
cut
each
other
in
the
same
manner
.
I
shall
here
take
for
granted
,
that
the
Projecture
is
(
for
truth
)
in
all
respects
answerable
to
the
Sphere
it selfe
,
which
is
abundantly
made
good
by
those
that
have
treated
upon
this
subject
.
Then
from
th●s
Projection
I
shall
shew
how
the
Ellipticall
Diall
upon
the
Horizon
must
be
of
the
same
truth
with
it
.
And
therefore
these
things
following
must
be
considered
,
as
so
many
Lemmaes
to
prove
what
is
here
required
.
Note
,
that
in
no
form
of
Projection
besides
the
Orthographicall
,
this
kinde
of
Diall
can
be
true
;
that
is
,
that
no
figure
but
an
Ellipsis
will
do
it
,
nor
yet
all
Ellipses
,
or
Ellipses
divided
into
all
〈◊〉
,
as
in
some
Projections
the●
are
,
when
one
halfe
or
180
gr
.
on
one
side
,
are
greater
than
the
other
halfe
or
180
gr
.
on
the
other
side
:
these
will
not
do
here
,
but
only
such
as
whose
extream
Diameters
are
the
sixes
and
twelves
,
that
is
,
such
as
are
produced
by
infinite
Orthographicall
Projection
.
LEMMA
I.
First
,
The
Equinoctiall
Circle
,
and
all
the
Parallels
of
Declination
,
are
projected
upon
the
plain
of
the
Horizon
into
like
Ellipses
.
And
that
they
are
all
of
them
divided
(
by
the
Meridians
)
alike
or
proportionally
one
to
the
other
.
The
reason
of
this
may
be
conceived
upon
this
ground
.
Since
the
Equinoctiall
and
all
the
Parallels
are
Circles
of
a
just
parallel
situation
one
to
the
other
(
as
their
name
imports
:
)
therefore
on
what
point
soever
of
the
horizontall
Axis
the
eye
is
placed
,
the
said
Circles
must
be
Ellipses
(
excepting
only
if
the
eye
be
placed
upon
the
superficies
of
the
Sphere
,
for
then
,
and
only
then
,
they
will
be
all
perfect
Circles
,
)
as
all
that
write
upon
this
subject
do
sufficiently
make
good
:
and
consequently
,
when
the
eye
is
placed
upon
the
same
Axis
,
in
an
infinite
distance
,
as
this
Projection
supposeth
.
But
the
second
thing
is
more
to
our
purpose
,
and
that
is
,
that
all
their
degrees
and
quarters
,
and
other
like
parts
,
art
like
and
proportionall
one
to
the
other
(
not
all
the
parts
of
one
and
the
same
Ellipsis
are
like
,
that
is
equall
,
one
to
the
other
,
as
in
the
Circles
themselves
upon
the
Sphere
they
are
,
but
each
one
of
these
Ellipses
is
justly
like
any
other
of
them
,
both
in
the
whole
,
and
likewise
when
any
of
the
like
parts
of
one
be
compared
with
the
like
parts
of
the
other
,
as
one
quarter
,
&c.
with
another
.
)
And
the
reason
briefly
is
,
because
the
eye
is
supposed
(
in
this
Orthographical
Projection
)
to
stand
in
an
infinite
distance
from
the
plain
on
which
the
projection
is
made
;
and
because
it
doeh
stand
in
an
infinite
distance
,
therefore
all
these
Circles
are
alike
situated
to
the
eye
,
and
consequently
must
make
a
like
or
proportionall
Projection
of
every
of
those
parallels
with
their
like
situated
parts
.
But
if
the
eye
should
stand
at
a
finite
distance
,
then
these
parallels
would
not
have
a
like
situation
to
the
eye
,
neither
in
respect
of
distance
,
nor
in
respect
of
position
,
and
so
the
like
parts
of
the
parallels
could
not
be
like
one
to
the
other
in
the
Projection
,
because
of
their
different
position
.
But
in
the
Orthographicall
Projection
at
an
infinite
distance
all
things
will
be
alike
,
being
alike
projected
from
a
like
position
.
I
say
in
this
it
will
be
so
,
and
in
none
besides
this
.
COROLLARY
.
Whence
will
follow
:
That
the
Equinoctiall
Circle
,
or
Ellipsis
rather
,
may
represent
both
it selfe
,
and
likewise
any
other
of
the
Parallels
,
if
it
be
so
contrived
to
stand
from
the
Zenith
point
,
as
that
it
may
have
a
proportionall
situation
from
it
,
suiting
with
the
position
of
any
other
Parallel
which
it
is
to
represent
.
And
this
is
the
main
ground
upon
which
this
Diall
will
be
proved
to
be
true
.
LEMMA
II.
Secondly
,
Any
two
like
plain
Figures
may
be
alike
situated
.
to
any
point
assigned
;
so
as
that
from
that
point
any
two
right
lines
being
drawn
infinitely
,
shall
cut
proportionall
parts
from
those
two
like
figures
.
This
is
generall
to
all
like
figures
,
whether
they
be
two
or
more
;
plain
or
solid
:
but
for
our
purpose
it
will
be
enough
to
shew
it
to
be
true
in
like
Ellipses
.
One
way
for
the
making
of
one
Ellipsis
like
to
another
,
is
by
assuming
of
any
point
,
as
a
in
the
greater
Ellipsis
,
and
from
thence
drawing
as
many
lines
as
you
wil
,
as
a
b
,
a
c
,
you
must
divide
them
into
proportionall
parts
at
d
&
e
,
that
as
a
b
to
a
d
;
so
a
c
to
a
e
:
&
then
through
those
points
the
lesser
Ellipsis
may
be
drawn
,
which
must
be
like
to
the
greater
▪
and
alike
situated
to
the
point
a
,
because
all
those
lines
are
proportionall
,
and
the
subtenses
b
c
,
d
e
,
will
likewise
be
parallel
and
proportionall
.
And
if
the
lines
a
b
,
a
c
,
had
been
augmented
proportionally
(
as
here
they
were
diminished
)
there
would
have
been
a
greater
Ellipsis
described
,
like
to
the
other
.
COROLLARY
.
Hence
will
follow
▪
That
if
the
centers
of
two
like
Ellipses
be
laid
upon
one
line
,
as
upon
one
of
their
common
Diameters
,
and
from
any
point
assigned
(
unto
which
they
are
to
be
alike
situated
)
there
be
an
infinite
line
drawn
to
any
point
of
one
of
the
Ellipses
,
and
the
like
point
of
the
other
Ellipsis
be
brought
to
that
●●finite
line
(
the
center
of
it
still
keeping
upon
the
common
Diameter
,
)
those
Ellipses
in
that
situation
shall
stand
alike
posited
to
the
assigned
point
.
Because
there
are
two
lines
(
the
common
diameter
and
the
infinite
line
)
that
from
the
point
assigned
do
cut
like
parts
of
both
Ellipses
.
The
common
diameter
makes
them
alike
situate
in
respect
of
one
dimension
(
suppose
the
length
)
of
the
locus
planus
whereon
they
lie
,
and
the
infinite
line
limits
them
in
respect
of
the
other
dimension
(
suppose
the
breadth
)
of
the
same
locus
planus
.
So
that
they
are
quite
limited
in
respect
of
their
like
situation
to
the
point
,
and
can
(
in
this
respect
)
lie
in
no
other
place
for
this
individuall
position
.
LEMMA
III.
Thirdly
,
All
the
Azimuths
of
the
Horizon
are
(
in
this
Orthographicall
Projection
)
cast
into
streight
lines
meeting
all
in
the
Zenith
point
.
The
reason
is
,
because
all
these
Circles
do
crosse
each
other
upon
the
Axis
of
the
Horizon
(
or
upon
the
Zenith
line
,
)
upon
which
line
the
eye
is
imagined
to
be
placed
at
an
infinite
distance
.
Therefore
they
all
(
that
is
,
the
plains
of
them
infinitely
continued
)
do
crosse
through
the
eye
,
and
consequently
must
all
be
projected
into
streight
lines
.
And
their
concourse
being
upon
the
Zenith
line
,
and
the
same
Zenith
line
comming
into
the
eye
,
the
said
whole
line
,
and
their
whole
concourse
will
be
cast
or
resolved
into
one
single
point
,
answering
to
the
Zenith
point
of
the
Sphere
.
The
application
of
these
things
to
the
purpose
intended
.
First
,
In
the
Projection
it selfe
we
are
to
consider
these
things
.
If
it
be
furnished
with
parallels
and
houres
,
as
the
manner
is
,
and
laid
horizontally
with
the
Meridian
line
of
it
just
North
and
South
:
and
if
further
,
there
be
an
upright
Index
set
in
the
Zenith
point
at
Z
,
then
shall
the
shadow
of
that
erect
Index
represent
the
Azimuth
in
which
the
Sun
is
,
and
if
you
note
where
it
cuts
the
Parallel
wherein
the
Sun
(
for
that
day
)
is
,
the
same
shadow
will
shew
(
among
the
houres
of
that
parallel
)
the
time
of
the
day
.
This
conclusion
is
true
upon
any
Horizontal
Projection
(
the
Stereographicall
as
well
as
the
Orthographicall
)
for
the
streight
lines
or
Azimuths
whether
shadow
or
Index
comming
from
the
Zenith
,
and
cutting
through
any
houre
in
any
parallel
,
doth
shew
the
Azimuth
in
which
the
Sun
must
be
at
that
houre
for
the
day
of
that
parallel
.
And
further
,
because
all
the
Projection
is
made
upon
the
plain
of
the
Horizon
(
which
therefore
must
be
the
fundamentall
Circle
of
the
Projection
,
and
)
which
for
that
cause
is
,
and
must
be
(
as
all
fundamental
Circles
of
any
Projection
are
)
equally
divided
;
Therefore
an
upright
Index
standing
in
Z
the
center
of
it
,
and
so
answering
to
the
Zenith
line
of
the
Sphere
,
and
the
degrees
of
the
Horizontall
Circle
answering
to
the
degrees
of
the
Horizon
in
the
Sphere
.
I
say
the
shadow
of
that
Index
doth
really
shew
the
true
Azimuths
of
the
Sun
,
or
the
true
angles
of
position
that
the
Sun
at
any
time
maketh
with
the
Meridian
line
.
And
further
,
the
same
shadow
or
Azimuth
where
it
crosseth
the
Parallel
of
the
day
(
which
Parallel
is
divided
into
its
proper
parts
,
like
to
the
parts
of
the
Equinoctiall
,
by
the
Meridian
Circles
issuing
out
of
the
Poles
)
it
notes
out
also
the
houre
of
the
day
.
So
that
the
Projection
made
in
this
manner
with
an
upright
Index
and
divided
Parallels
,
will
(
by
the
shadow
of
that
Index
)
shew
the
true
houre
of
the
day
upon
that
parallel
that
is
proper
for
the
day
,
if
the
Meridian
of
the
Projection
lie
in
the
true
Meridian
of
the
Horizon
.
The
next
thing
to
be
shewed
,
is
how
the
Ellipticall
Equinoctiall
may
supply
the
use
of
all
the
Parallels
.
That
is
to
say
,
How
the
Equinoctiall
,
being
made
movaable
,
and
the
Index
standing
alwayes
still
,
may
be
fitted
to
represent
any
of
the
Suns
parallels
.
And
because
(
as
was
said
before
)
it
is
every
way
like
to
each
of
the
Parallels
,
it
will
therefore
be
only
required
to
give
some
rule
how
the
said
Equinoctiall
may
be
at
any
time
placed
in
a
like
position
to
the
Zenith
point
at
Z
,
that
any
parallel
hath
to
the
same
point
,
so
as
that
any
right
line
(
or
Azimuth
)
being
drawn
from
thence
may
cut
a
like
part
or
point
in
the
Equinoctiall
that
it
doth
in
the
Parallel
.
Let
the
Parallel
be
H
E
,
I
would
remove
the
Equinoctiall
Ae
A
so
,
as
that
it
might
have
a
like
position
to
the
point
Z
,
that
the
Parallel
H
E
hath
.
By
the
Corollary
of
the
first
Lemma
,
it
may
represent
it
.
By
the
Corollary
of
the
second
Lemma
,
the
way
of
it
will
be
easie
.
For
first
,
the
two
Ellipses
H
E
and
Ae
A
,
have
their
centers
upon
the
Meridian
line
Ae
H
Z
P
,
as
upon
one
of
their
common
Diameters
.
Secondly
,
the
Parallel
H
E
cuts
the
six
a
clock
Circle
,
or
Meridian
P
E
A
,
in
the
point
E
,
making
E
A
equall
in
radiall
degrees
to
H
Ae
(
for
the
Parallel
is
every
where
equidistant
to
the
Equator
,
and
the
Meridional
Arks
,
therefore
,
intercepted
between
them
,
such
as
are
E
A
and
H
Ae
,
must
be
equall
.
)
So
that
E
A
is
the
Declination
of
that
Parallel
from
the
Equinoctiall
.
If
therefore
the
line
(
or
Azimuth
)
Z
E
be
drawn
infinitely
through
the
point
of
6
in
the
Parallel
,
or
through
the
Declination
(
of
that
Parallel
)
counted
in
the
Circle
of
six
;
and
then
the
Equinoctiall
(
still
keeping
its
center
upon
the
common
Diameter
Ae
P
)
be
slipped
up
till
the
point
A
thereof
do
concur
with
the
infinite
line
(
drawn
before
)
at
G
;
in
thus
doing
,
I
say
that
the
Equinoctiall
is
in
a
like
situation
to
the
point
Z
,
that
the
Parallel
H
E
is
in
,
to
the
same
point
.
And
consequently
,
that
any
right
line
(
or
Azimuth
)
from
Z
,
will
cut
the
same
(
rather
like
)
parts
in
one
that
it
doth
in
the
other
.
As
by
the
Corollary
of
the
second
Lemma
will
appear
.
The
next
thing
to
be
enquired
is
,
what
the
line
A
G
is
,
or
how
it
must
be
found
and
estimated
.
For
this
purpose
,
consider
,
that
Z
P
E
and
Z
C
O
are
two
Triangles
rectangled
at
P
and
C
,
and
having
a
common
angle
at
Z
;
therefore
,
As
Z
P
Co-sine
of
the
Latitude
,
to
Z
C
the
Radius
;
So
P
E
,
the
Co-tangent
of
the
Declination
,
to
the
Tangent
of
C
O.
Or
,
As
the
Radius
Z
C
,
to
the
Co-sine
of
the
Latitude
Z
P
;
so
is
the
Tangent
of
E
A
,
the
Parallels
Declination
,
to
the
Tangent
of
A
O
,
that
is
,
to
the
line
A
G
,
which
is
the
Tangent
of
A
O
,
because
the
arke
A
O
is
expressed
in
its
full
quantity
▪
without
any
enlarging
or
fore-shortning
,
and
the
right
line
A
G
stands
to
it
as
a
Tangent
thereof
,
standing
at
the
end
of
the
Radius
Z
A.
Now
consider
,
that
if
Z
C
be
considered
as
Radius
,
then
must
the
Tangent
of
E
A
be
the
Tangent
of
the
Parellels
declination
,
estimated
to
the
same
Radius
.
And
because
Z
P
is
in
the
selfe
same
proportion
to
A
C
,
therefore
:
If
Z
P
(
the
Co-sine
of
the
Latitude
)
be
estimated
for
a
Radius
,
then
must
A
G
be
esteemed
as
the
Tangent
(
of
this
Parallels
declination
)
to
the
said
Radius
Z
P.
And
hence
it
will
follow
that
If
Z
P
the
Co-sine
of
the
Latitude
[
being
taken
(
as
that
Co-sine
)
to
the
Radius
Z
C
or
Z
A
,
which
is
the
longer
Radius
of
the
Ellipticall
Equinoctiall
]
be
counted
as
a
Radius
or
Tangent
of
45
gr
.
and
so
be
divided
and
continued
as
occasion
shall
be
,
It
is
to
be
noted
I
say
,
That
this
Scale
thus
limited
will
be
a
right
Scale
for
the
requisite
motion
of
the
Ellipticall
Equinoctiall
to
such
a
position
in
respect
of
the
Zenith
point
,
as
that
it
may
represent
any
parallel
,
whose
Declination
is
known
,
if
it
be
removed
to
the
same
Declination
counted
upon
that
Scale
.
And
being
set
in
that
like
posture
,
any
Azimuth
or
shadow
of
the
upright
Index
that
would
passe
through
any
houre
point
of
the
Parallel
,
will
also
passe
through
the
same
point
of
the
Equinoctiall
,
and
so
this
one
may
serve
for
them
all
.
This
gives
the
reason
of
making
the
Zodiac
or
Annuall
Scale
,
mentioned
in
the
former
Treatise
,
I
say
it
gives
the
reason
of
the
second
way
mentioned
for
lesser
Latitudes
.
For
if
either
the
Signes
of
the
Zodiac
,
or
the
moneths
of
the
year
be
put
in
according
to
their
declinations
taken
in
this
Scale
,
it
will
be
all
one
to
set
your
Ellipsis
to
the
Signe
or
Moneth
,
that
it
is
to
set
it
to
the
Declination
,
since
they
are
made
one
to
answer
to
the
other
.
Then
for
the
first
way
for
greater
Latitudes
:
because
,
As
the
Co-sine
of
the
Latitude
,
Is
to
the
Sine
of
the
Latitude
;
So
the
Radius
,
(
or
Tangent
of
45
gr
.
)
To
the
Tangent
of
the
Latitude
.
You
see
that
if
you
divide
the
Co-sine
of
your
Latitude
into
45
Tangents
;
or
the
Sine
of
your
Latitude
into
the
Tangent
of
your
Latitude
,
the
Scale
will
be
all
one
according
to
the
former
proportion
.
What
is
before
spoken
of
North
Parallels
of
Declination
,
is
the
same
in
South
also
,
and
one
Scale
(
for
the
kinde
of
it
)
serves
both
;
only
in
North
Declinations
the
Equinoctiall
goes
neerer
to
the
Zenith
point
,
in
South
it
goes
further
of
.
All
other
Scales
annexed
to
the
Ellipsis
have
their
dependance
wholly
upon
this
Scale
of
Declinations
,
and
will
not
need
any
further
explication
in
this
place
.
Then
again
,
it
matters
not
where
the
little
Scale
or
Zodiac
stands
,
so
that
it
give
the
just
quantity
of
removall
for
the
Equinoctiall
Ellipsis
.
And
so
also
,
it
is
no
matter
whether
the
Index
move
,
and
the
Equinoctiall
stand
,
or
contrarily
:
It
is
only
required
that
their
situation
be
such
as
may
make
the
foot
of
the
Index
stand
in
a
proportionall
position
to
the
station
of
the
Parallel
,
as
was
formerly
shewed
.
Further.
For
the
circular
year
or
Zodiac
,
the
ground
of
it
is
thus
.
The
Circle
is
supposed
to
be
divided
into
equall
parts
or
degrees
:
and
so
every
Sine
(
as
is
D
F
)
is
a
proper
Sine
to
any
arke
(
as
D
C
)
to
which
it
is
annexed
.
Again
,
A
B
in
another
consideration
is
to
be
esteemed
(
as
formerly
for
the
Semidiameter
of
the
Circle
,
so
now
again
)
for
the
Tangent
of
23½
degrees
,
whence
this
proportion
will
stand
:
As
the
Radius
A
B
,
Is
to
the
Sine
F
D
;
So
the
same
A
B
,
tangent
of
23½
▪
To
the
Tangent
A
E.
Which
is
the
Tangēt
belonging
to
the
Right
Ascension
C
D.
[
Because
in
the
Sphere
,
As
the
Radius
,
To
the
Sine
of
any
Right
Ascension
,
So
the
Tangent
of
23½
To
the
Tangent
of
the
Declination
answering
to
that
Right
Ascension
.
]
If
therefore
you
have
the
Right
Ascension
belonging
to
any
day
,
and
put
it
into
the
Circle
of
equall
degrees
,
as
C
D
here
is
,
the
Sine
of
that
Right
Ascension
,
viz.
F
D
,
will
be
equall
to
A
E
the
Tangent
belonging
to
that
Right
Ascension
,
if
B
A
be
taken
for
the
Tangent
of
23½
gr
.
So
that
this
way
of
putting
in
Moneths
and
Signes
is
the
same
in
effect
with
the
former
Zodiac
or
Annuall
course
of
the
Sun.
Thus
far
for
demonstration
of
what
was
hard
in
the
first
Section
,
Pag.
8
,
&c.
concerning
Horizontall
plains
.
What
is
added
more
in
the
first
Section
of
framing
it
to
other
plains
that
are
direct
,
is
the
same
with
this
.
For
there
is
no
difference
but
only
in
the
latitude
of
the
plain
,
which
is
no
re●ll
discrepance
from
the
former
,
they
both
going
upon
one
ground
,
and
therefore
no
more
to
be
said
here
of
such
plains
.
Sect.
2.
Pag.
22
,
&c.
Of
framing
it
to
Declining
Pl●ins
.
DEclining
plains
,
if
they
had
their
proper
houres
upon
them
,
would
al●o
(
in
this
Ellipticall
respect
)
be
the
same
with
the
former
direct
,
or
Horizontall
plains
.
The
distinction
is
,
because
they
are
made
to
another
Meridian
than
is
their
own
,
that
is
,
to
the
Meridian
of
the
place
,
and
all
the
account
of
houres
is
deduced
from
it
.
The
difference
therefore
of
these
from
the
former
,
is
only
this
,
that
the
houres
in
the
Ellipsis
are
not
evenly
fitted
to
the
quarters
or
diameters
of
the
Ellipsis
,
but
fall
to
stand
as
casually
they
may
.
Now
if
in
the
former
case
of
Horizonta●l
plains
,
you
do
but
suppose
the
Equinoctiall
and
Parallels
to
be
divided
from
any
casuall
Meridian
as
P
M
(
not
P
Ae
)
into
houres
and
parts
,
then
will
follow
still
the
same
things
in
substance
that
was
before
.
Namely
,
That
if
the
Equinoctiall
Circle
be
removed
according
to
A
G
(
which
must
be
supposed
to
be
suited
to
the
Latitude
of
the
place
as
well
as
to
the
Suns
parallel
of
declination
)
with
those
divisions
now
spoken
of
,
then
still
,
when
the
Equinoctiall
by
that
motion
hath
got
a
like
situation
to
the
Parallel
,
the
parts
of
one
will
be
answerable
to
the
parts
of
the
other
,
in
respect
of
the
shadows
or
Azimuths
that
are
cast
from
the
Zenith
point
and
upright
Index
standing
in
it
at
Z.
So
that
even
in
these
plains
also
,
the
same
houre
will
be
shewed
by
the
Equinoctiall
that
would
be
by
the
Parallel
.
And
so
the
ground
is
(
in
these
)
like
(
rather
the
same
)
in
substance
with
the
former
direct
plains
.
So
much
of
these
.
Sect.
3.
Pag.
29
,
&c.
THat
pricking
down
of
the
Horizontall
Diall
mentioned
in
this
Section
,
is
the
pricking
down
of
the
Equinoctiall
with
its
houres
or
parts
,
just
as
the
Orthographicall
Projection
it selfe
hath
them
.
The
first
Table
shews
what
Angles
or
upon
what
Azimuths
from
the
Meridian
every
houre
point
lies
.
The
second
gives
the
Altitudes
,
or
rather
the
distances
of
the
same
houre
points
from
the
Zenith
,
And
therefore
no
more
will
be
required
of
this
.
The
fourth
Section
requires
no
explication
nor
Demonstration
.
Sect.
5.
Pag.
37
,
&c.
FOr
the
varieties
it
must
be
known
,
that
the
substance
is
the
same
with
that
which
went
before
,
which
being
well
understood
will
give
light
enough
to
this
.
But
in
this
Section
there
is
mention
made
of
a
Circle
(
insteed
of
the
Horizontall
Ellipsis
)
elevated
to
the
height
of
the
Equinoctiall
.
The
reason
of
that
will
best
be
seen
out
of
the
Sphere
it selfe
,
For
there
we
know
,
that
the
Equinoctiall
and
all
the
Parallels
to
it
are
both
equally
divided
,
and
equally
elevated
,
and
so
being
all
alike
,
the
Equinoctiall
may
supply
the
room
of
each
and
all
of
them
.
Only
it
must
be
required
(
whether
the
Index
move
toward
the
Circle
;
or
the
Circle
towards
it
)
that
this
motion
(
if
the
Zodiac
be
upon
the
Horizontall
plain
)
be
limited
by
an
Horizontall
Zodiac
,
such
as
was
used
for
the
Ellipsis
upon
any
plain
,
(
which
plain
,
what
ever
it
be
,
I
now
count
as
an
Horizontall
plain
.
)
But
if
the
Index
be
made
to
move
upon
the
Equinoctiall
(
which
though
it
do
,
yet
it
must
still
lie
in
the
Zenith
line
,
not
perpendicular
to
the
circular
plain
,
but
making
an
angle
with
it
equall
to
the
Latitude
of
the
place
,
and
not
the
Latitude
of
the
plain
.
)
If
,
I
say
,
the
Index
be
made
to
move
upon
the
Equinoctiall
plain
,
then
must
a
new
Scale
be
made
like
to
that
upon
the
Horizontal
plain
,
but
somewhat
larger
;
that
is
,
it
must
be
augmented
so
as
that
the
parts
and
whole
of
the
Horizontall
Scale
,
to
the
parts
and
whole
of
this
Equinoctiall
Scale
;
must
be
as
the
Radius
,
to
the
Co-secant
of
the
Latitude
.
Or
thus
.
Because
,
As
the
Radius
,
to
the
Co-secant
of
the
Latitude
;
so
the
Sine
of
the
Latitude
to
the
Radius
:
and
so
again
,
the
Co-sine
of
the
Latitude
,
to
the
Co-tangent
of
the
Latitude
.
Therefore
,
whereas
on
the
Horizontall
plain
(
for
the
Ellipsis
thereon
described
)
you
looke
the
Co-sine
of
your
Latitude
,
and
make
it
a
Radius
or
Tangent
of
45
gr
.
you
must
in
this
take
the
Co-tangent
of
your
Latitude
,
and
make
it
a
Radius
or
Decimall
Scale
;
and
by
it
you
may
put
on
the
Zodiac
or
moneths
as
was
prescribed
before
.
This
Co
▪
sine
mentioned
before
was
taken
to
the
greater
Radius
,
of
the
Ellipsis
,
and
so
this
Co-tangent
here
mentioned
must
be
taken
to
the
(
same
greater
Radius
of
the
Ellipsis
,
if
the
Ellipsis
were
here
used
,
or
to
the
)
Radius
of
the
Equinoctiall
Circle
,
whose
Radius
must
be
conceived
to
be
the
same
with
the
greater
Radius
of
the
Ellipsis
.
For
from
this
Circle
projected
Orthographically
(
that
is
,
by
perpendiculars
let
fall
from
the
Periphery
to
the
subjacent
plain
)
is
the
Ellipsis
deduced
,
and
from
that
Ellipsis
(
again
)
is
this
Circle
raised
or
restored
.
And
so
the
reason
of
this
Circular
Diall
with
an
upright
Index
will
be
understood
well
enough
.
Then
whereas
it
is
said
that
it
makes
no
matter
which
way
this
Circle
be
raised
,
that
is
,
whether
toward
North
or
South
,
there
is
no
difficulty
in
this
,
for
which
way
soever
it
is
turned
,
if
the
plain
and
center
of
it
lie
in
a
just
distance
from
the
Index
or
Zenith
line
,
the
same
Index
must
shew
the
same
point
or
houre
,
because
in
both
wayes
they
are
situated
from
it
upon
the
same
Coast.
Sect.
6.
Pag.
47
,
&c.
THat
which
is
here
done
will
not
be
difficult
,
if
it
be
considered
that
what
is
aimed
at
,
hath
relation
to
Stofflerius
Astrolabe
:
and
that
here
one
Ellipsis
must
serve
to
represent
every
Almicantar
.
I
purpose
not
to
repeat
any
thing
that
is
before
said
:
and
the
rather
because
this
Structure
will
not
be
so
expedient
and
ready
in
some
uses
as
could
be
wished
.
They
that
desire
the
reason
of
it
,
may
fetch
it
out
of
this
that
hath
been
already
said
:
which
may
be
done
without
any
great
labour
.
SECT
.
VII
.
1.
How
to
draw
and
divide
the
Ellipsis
upon
any
Plain
,
to
an
Index
that
stands
upright
(
not
to
the
Plain
,
but
)
to
the
Zenith
line
of
the
place
,
or
perpendicular
to
the
plain
of
the
Horizon
.
THis
may
be
done
artificially
by
calculation
,
or
by
a
lineary
way
,
but
not
without
too
much
trouble
and
incumberance
,
which
would
deter
any
man
quite
from
putting
it
in
practise
.
I
shall
therefore
let
that
way
alone
,
and
fall
upon
another
more
feasible
and
delightfull
.
The
way
that
I
intend
,
is
partly
by
projection
of
such
lines
as
are
usefull
,
and
partly
by
inscription
of
such
points
as
the
Ellipticall
line
is
to
passe
through
,
as
shall
be
seen
hereafter
.
And
therefere
we
shall
not
need
to
looke
after
any
position
of
the
plain
in
respect
of
declination
or
leaning
.
2.
The
manner
of
the
work
.
1.
LEt
the
plain
be
A
B
C
,
and
the
upright
Index
A
D.
First
then
,
assume
any
point
in
the
Index
as
at
E
,
and
from
thence
(
by
the
projecting
Quadrant
,
or
some
Levell
)
cast
an
Horizontall
levell
line
as
F
G
upon
the
plain
;
or
if
it
cannot
be
cast
upon
the
plain
it selfe
,
set
any
board
(
for
a
time
)
that
may
receive
it
;
lying
in
the
same
Levell
with
the
assigned
point
at
E.
And
further
,
the
same
Horizontall
line
must
be
continued
and
returned
about
the
point
E
,
by
help
of
threeds
stretched
out
from
some
points
or
other
of
the
Horizontall
line
before
drawn
,
as
from
F
or
G
,
unto
two
supporters
set
up
(
for
a
while
)
for
this
purpose
(
either
upon
the
plain
or
otherwise
neer
to
it
)
as
is
to
be
seen
at
H
and
I
:
so
that
now
the
points
H
F
G
I
are
all
in
one
and
the
same
Horizontall
levell
to
one
another
,
and
likewise
to
the
first
point
of
the
Index
assumed
at
E.
2.
At
some
convenient
time
of
the
day
,
when
the
Azimuth
may
be
found
distinctly
,
observe
where
the
shadow
of
the
Index
cutteth
the
Horizontall
line
,
suppose
at
K
,
and
there
make
a
mark
,
and
immediately
take
the
Suns
Altitude
.
3.
By
the
Suns
Altitude
observed
,
compute
what
Azimuth
the
Sun
was
then
in
,
which
will
tell
you
what
Azimuth
the
observed
point
K
(
in
the
Horizontall
line
)
doth
represent
.
4.
Take
a
Pastboard
as
E
L
,
and
fit
it
to
the
Horizontall
line
of
the
plain
F
G
,
and
to
the
assumed
centerat
●
:
and
applying
it
to
its
proper
place
(
as
in
the
figure
is
represented
)
draw
a
line
from
the
point
K
to
the
center
E
,
quite
over
the
pastboard
.
And
then
knowing
what
Azim●●h
it
is
,
you
may
(
from
it
)
set
off
the
Meridian
in
the
true
Co●st
of
it
,
such
as
will
be
answerable
to
the
heavens
,
which
Meridian
line
suppose
to
be
P
E
upon
the
Pastboard
.
5.
You
are
next
of
all
,
from
this
Meridian
line
P
E
,
to
set
of
all
the
other
houres
of
an
Horizontall
Diall
(
not
the
common
Horizontall
Diall
,
but
)
according
to
su●h
numbers
or
arks
as
are
expressed
in
the
Table
of
Angles
,
Pag.
29.
)
And
when
this
is
done
upon
the
Pastboard
,
as
you
see
done
at
o
P
q
κ
s
t
u
:
Then
,
6.
Apply
the
Pastboard
to
its
former
position
upon
the
plain
,
and
from
the
center
E
project
the
houres
from
the
Pastboard
to
the
Horizontall
line
and
threeds
.
And
then
taking
away
your
Pastboard
,
you
may
draw
lines
from
A
to
such
points
as
are
in
the
Horizontall
line
i●
selfe
,
as
A
o
,
A
p
,
A
q
,
A
κ
,
A
s
,
A
t
,
A
u.
And
for
the
others
that
stand
not
upon
the
Horizontall
line
of
the
plain
,
but
upon
the
threeds
which
are
separate
from
the
plain
(
such
as
are
ε
ρ
π
)
you
may
project
them
by
the
Index
A
D
,
reposing
the
same
upon
the
said
points
(
ε
ρ
π
)
of
the
threed
;
and
marking
where
the
shadow
or
appearance
of
the
threed
traceth
the
plain
,
there
draw
the
lines
A
m
,
A
n
,
A
u.
And
so
all
the
lines
(
that
the
plain
is
capable
of
)
may
be
drawn
.
And
it
is
to
be
noted
,
that
these
lines
so
drawn
are
not
such
houre
lines
as
usually
are
upon
other
Dials
,
but
they
are
those
Azimuth
lines
in
which
the
Sun
is
at
every
houre
of
the
Equinoctiall
Circle
.
So
the
houres
and
quarters
may
be
put
in
by
the
Table
,
Pag.
29
,
and
the
Column
of
Angles
in
the
same
Table
.
7.
Looke
again
into
the
Table
Pag.
29
,
for
the
Column
of
Altitudes
;
for
by
that
must
the
Ellipsis
it selfe
be
desc●ibed
,
for
the
former
Azimuthall-ho●re
l●●es
will
give
it
the
true
divisions
into
its
requisite
parts
.
The
manner
of
the
division
is
this
.
Make
a
Scale
of
90
right
Sines
,
but
number
them
versedly
,
as
is
here
expressed
.
And
let
the
Scale
be
of
a
fit
●ignes●e
for
your
plain
,
which
your
own
judgement
must
direct
you
to
do
.
Out
of
this
Scale
(
setting
alwayes
one
foot
of
your
Compasses
in
the
point
of
90
)
take
such
Altitudes
as
are
(
in
the
Table
)
appropriated
to
every
houre
,
and
put
them
into
those
severall
houres
in
this
wise
.
Suppose
the
altitude
for
12
a
clock
were
to
be
put
in
,
which
altitude
is
38
gr
.
30
min.
with
my
Compasses
I
take
(
in
this
Scale
of
Sines
)
from
90
to
38
gr
.
30
min.
and
one
foot
of
th●t
length
I
put
into
the
line
of
12
,
namely
P
A
upon
the
plain
,
and
(
alwayes
keeping
it
upon
some
part
of
that
line
)
I
remove
it
thereon
,
neerer
to
,
or
further
from
the
center
A
,
till
the
other
foot
of
the
Compasses
being
turned
about
will
justly
touch
the
edge
(
or
fiduciall
line
)
of
the
Index
.
Then
I
diligently
observe
where
,
or
in
what
point
of
the
line
P
A
the
first
foot
of
my
Compasses
stayeth
,
for
that
is
the
point
through
which
the
Ellipsis
must
passe
upon
the
houre-line
of
12.
So
for
any
other
houre
(
and
the
intermediate
quarters
)
taking
their
altitudes
,
(
set
down
in
the
Table
)
out
of
the
same
Scale
of
Sines
,
and
inserting
them
in
the
same
manner
as
was
shewed
before
,
you
shall
finde
all
the
points
(
for
every
houre
and
quarter
)
through
which
the
Ellipsis
is
to
be
drawn
.
Through
these
points
therefore
you
must
draw
it
carefully
,
as
one
continued
line
,
without
any
breaches
or
angles
in
it
.
3.
Concerning
the
motion
that
is
to
be
made
either
by
the
Index
,
or
by
the
Ellipsis
it selfe
.
THe
motion
must
be
made
according
to
the
12
a
clock
line
,
which
is
alwayes
the
proper
Meridian
to
the
Index
or
Zenith
line
.
That
is
,
the
removall
or
sliding
must
be
either
in
the
Meridian
it selfe
,
or
else
parallel
to
it
.
Now
if
the
Diall
it selfe
be
made
to
slide
upon
the
plain
,
that
is
,
if
the
Ellipsis
be
to
be
described
upon
another
plate
that
s●all
slip
over
the
former
plain
,
then
must
this
plate
be
first
of
all
laid
fast
upon
the
plain
,
and
the
houres
and
Ellipticall
line
must
be
described
thereon
in
the
same
manner
that
was
shewed
before
.
And
this
plate
must
move
in
the
Meridian
line
,
that
is
,
the
12
a
clock
line
of
the
plate
(
being
imagined
to
be
continued
forth
right
)
must
move
through
the
fiduciall
edge
of
the
Index
.
And
so
again
,
If
the
Diall
be
drawn
upon
the
plain
it self
,
and
the
Index
be
made
to
move
(
the
Diall
it selfe
standing
still
)
then
must
care
be
had
that
the
fiduciall
edge
of
the
Index
do
precisely
move
in
the
Meridian
or
12
a
clock
line
,
and
that
the
same
Index
move
alwayes
precisely
upright
in
the
Zenith
line
.
4.
Concerning
the
Place
of
the
Suns
Annuall
course
or
Zodiac
.
THe
place
of
it
may
be
best
upon
the
plain
.
It
must
stand
(
if
it
be
inscribed
into
a
streight
line
)
parallel
to
the
line
of
12
,
and
needs
not
to
be
in
or
upon
the
said
line
.
The
mover
(
whether
plate
or
Index
)
must
have
another
peculiar
Index
in
it
,
called
he●e
the
Zodiacall
Index
,
by
which
it
is
to
be
rectified
according
to
the
time
of
the
year
,
If
the
Zodiac
be
put
into
a
Circle
(
as
is
mentioned
Pag.
15.
)
then
that
Diame●er
(
of
the
Annuall
Circle
)
which
passeth
from
Tropick
to
Tropick
,
must
be
either
in
or
parallel
to
the
line
of
12.
5.
Of
what
limitation
or
length
the
Zodiac
must
be
,
and
how
to
be
described
,
and
where
to
be
set
.
IT
must
be
regulated
by
the
former
Scale
of
Sines
mentioned
Pag.
73.
For
you
may
take
from
90
(
in
that
Scale
)
to
the
number
of
the
Latitude
of
your
place
(
not
complement
as
that
Scale
is
numbered
)
and
count
that
length
as
a
new
Radius
.
Unto
th●s
new
Radius
finde
the
Secant
of
the
Meridians
inclination
(
which
what
it
is
shall
be
presently
shewed
in
the
6
Proposition
following
)
and
then
make
this
Secant
to
be
a
Tangent
of
45
gr
.
and
out
of
that
Scale
of
Tangents
take
23½
gr
.
for
that
length
being
set
both
wayes
from
the
Equinoctiall
,
will
give
the
length
of
the
Zodiac
(
in
a
streight
line
,
or
the
Diameter
of
the
Zodiac
put
into
a
Circle
:
)
this
work
is
to
be
understood
for
the
describing
of
the
Zodiacall
Scale
upon
the
plain
it selfe
.
But
first
,
you
are
to
pitch
the
Equinoctials
place
,
which
is
that
place
only
upon
which
the
Zodiacall
Index
lyeth
,
when
the
mover
(
whether
plate
or
Index
)
is
placed
in
the
same
position
that
it
had
when
the
Ellipticall
line
was
described
.
Now
when
this
Tangent
of
45
gr
.
is
thus
found
,
you
may
take
the
Declinations
(
in
the
Tables
of
the
4
and
5
Pages
:
)
out
of
it
,
and
so
prick
on
the
Suns
yearly
course
.
Or
else
make
the
forementioned
Secant
a
Decimall
Scale
,
and
out
of
it
take
the
Tangents
noted
in
the
Tables
Pag.
6
,
7.
So
shall
you
make
the
same
Annuall
course
of
the
Sun
that
was
produced
the
former
way
.
Note
,
that
if
the
Index
be
made
to
move
,
it
is
not
necessary
that
it
should
move
upon
or
parallel
to
the
superficies
of
the
plain
,
but
may
be
made
to
move
either
horizontall
,
or
at
any
other
inclination
:
only
the
fiduciall
edge
must
necessarily
be
made
to
move
in
the
very
Meridian
line
,
and
must
also
be
(
in
all
motions
)
alwayes
upright
,
or
in
the
Zenith
line
of
the
place
;
and
therefore
must
slide
just
under
the
Meridian
line
.
Note
also
,
That
the
Zodiac
though
it
be
not
necessarily
confined
to
be
set
upon
the
superficies
of
the
plain
,
yet
it
will
be
most
conveniently
limited
thereunto
.
6.
What
the
Meridians
inclination
meaneth
,
and
how
to
finde
the
quantity
of
it
.
BY
the
Meridians
inclination
is
meant
what
angle
the
same
line
maketh
with
the
plain
of
the
Horizon
,
and
in
the
former
figure
Pag.
70
it
is
expressed
by
the
complement
of
the
angle
D
A
P
,
or
the
excesse
of
it
.
To
finde
it
,
you
may
either
apply
the
edge
of
an
Inclinatorie
square
to
the
said
Meridian
line
,
and
then
the
threed
(
being
drawn
perpendicularly
down
by
the
plummet
)
will
shew
the
said
elevation
(
or
inclination
)
of
the
Meridian
line
above
the
Horizon
,
in
the
degrees
of
the
limbe
,
if
the
same
degrees
be
taken
to
the
threed
from
that
side
of
the
Square
that
stands
perpendicularly
upon
the
Meridian
line
.
Or
else
another
way
may
be
by
protracting
or
measuring
the
angle
D
A
P
,
which
is
the
complement
of
the
former
required
inclination
.
It
may
be
done
thus
.
Set
one
foot
of
your
Compasses
at
A
,
and
let
the
other
foot
be
extended
to
any
point
of
the
Index
,
suppose
to
8
,
then
measure
the
same
length
A
8
upon
the
Meridian
line
,
which
let
be
A
7
,
and
to
this
length
fit
your
line
of
Chords
(
upon
the
Sector
,
or
some
like
opening
Scale
of
Chords
.
)
Then
take
the
length
from
7
to
8
,
and
measure
the
same
upon
your
Scale
of
Chords
,
so
shall
you
finde
the
quantity
of
the
angle
D
A
P
,
whose
complement
(
or
excesse
)
is
the
inclination
,
or
elevation
,
or
depression
of
the
Meridian
,
which
is
here
required
.
7.
Other
things
to
be
noted
concerning
the
Zodiacall
Scale
.
1.
IF
the
Diall
be
described
upon
an
irregular
superficies
,
such
as
is
not
flat
but
writhen
(
as
by
this
course
it
may
very
well
be
)
then
it
is
most
convenient
that
the
former
Scale
be
set
in
some
other
place
,
and
not
upon
the
superficies
whereon
the
Ellipsis
is
described
.
It
may
be
contrived
two
wayes
.
First
,
If
the
Diall
superficies
(
how
irregular
soever
it
be
on
that
face
whereon
the
houres
are
inserted
)
be
flat
at
the
bottome
,
and
be
made
to
move
(
the
Index
standing
still
)
upon
some
other
plain
below
it
)
then
the
best
way
will
be
to
make
the
Zodiacall
Index
upon
the
moving
bottome
of
it
,
and
to
describe
the
Zodiac
upon
the
other
plain
upon
which
the
motion
is
made
.
And
to
do
this
aright
,
you
must
project
the
Meridian
line
upon
that
nether
plain
,
and
finde
out
the
inclination
of
it
,
and
so
finde
the
Scale
by
the
Secant
of
that
inclination
,
just
as
you
were
before
directed
.
And
so
this
work
will
be
compleat
.
The
motion
of
dials
unequall
Superficies
must
be
in
or
parallel
to
this
last
projected
Meridian
line
,
alwayes
so
as
that
the
Meridian
of
the
Diall
must
passe
through
the
fiduc●all
edge
of
the
Index
.
Secondly
,
If
the
Index
be
made
to
move
(
the
Diall
standing
still
)
and
that
the
Diall
be
upon
some
unequall
super●icies
,
such
as
is
unfit
to
receive
the
Zodiac
,
then
the
foot
of
the
Index
may
have
it
inscribed
upon
it
.
Now
in
this
case
of
unequall
Superficies
,
it
is
supposed
that
this
foot
of
the
Index
cannot
move
parallel
to
that
uneven
Superficies
,
but
must
move
streight
forward
in
some
right
line
,
just
along
with
the
Meridian
line
.
In
this
case
you
must
finde
the
inclination
of
the
foot
of
the
Index
to
the
Horizon
,
that
is
,
what
inclination
a
line
drawn
upon
the
foot
of
the
Index
,
either
in
or
parallel
to
the
Meridian
line
,
hath
to
the
Horizon
,
which
must
be
done
by
some
Inclinatorie
Instrument
,
or
some
such
way
as
is
used
in
taking
the
Reclination
or
Inclination
of
Plains
.
And
when
this
is
done
,
you
are
to
finish
the
Zodiac
Scale
by
the
●ecant
of
that
inclination
,
in
the
same
manner
as
was
before
shewed
.
So
much
for
this
.
2.
For
the
other
Scales
of
the
Suns
Declination
,
Amplitude
,
Ascensionall
difference
,
and
graduall
motion
in
the
12
Signes
,
they
are
to
be
done
in
the
selfe
same
manner
,
and
by
the
same
Tables
as
before
(
without
any
difference
)
after
that
you
have
found
your
Tangent
Scale
or
Decimall
Scale
out
of
which
to
describe
them
.
8.
Another
Observation
.
ACcording
to
that
generall
observation
Pag.
36.
Note
here
;
That
if
to
an
Index
standing
in
the
Zenith
line
,
a
Plain
be
set
in
the
Equinoctiall
,
the
same
plain
shall
have
a
Circular
Diall
upon
it
,
equally
divided
.
Then
whether
the
said
Equinoctiall
plain
or
the
Index
move
,
if
they
be
made
to
move
upon
the
Horizontall
plain
,
the
Zodiac
for
the
Horizontall
plain
must
serve
.
But
if
the
upright
(
or
Zenith
line
)
Index
be
made
to
move
upon
the
Equinoctiall
plain
,
the
former
Horizontall
Zodiac
must
be
set
upon
the
Equinoctiall
plain
,
and
must
there
be
enlarged
above
the
Horizontall
Zodiac
,
that
is
,
every
part
of
the
Horizontall
Zodiac
must
be
made
greater
upon
the
Equinoctiall
plain
in
proportion
as
the
Radius
to
the
Co-secant
of
the
Latitude
.
That
is
,
the
Scales
that
made
the
Horizontall
,
being
taken
as
Radius
,
must
now
here
be
enlarged
to
be
as
Co-secant
of
your
Latitude
;
and
from
them
(
so
enlarged
)
must
the
parts
of
the
Zodiac
be
inscribed
upon
the
Equinoctiall
plain
.
Note
further
.
That
for
this
Equinoctiall
plain
which
is
to
descend
downwards
from
South
towards
North
,
you
may
insteed
of
it
set
another
plain
quite
contrarily
,
that
is
,
descending
from
North
towards
South
;
which
will
be
the
most
convenient
of
the
two
,
because
the
upper
face
of
this
will
give
the
houre
all
the
Seasons
of
the
year
,
whereas
the
other
will
be
only
for
Sommer
upon
the
upper
face
of
it
,
and
will
require
an
under
face
for
Winter
.
SECT
.
VIII
.
Hitherto
of
Ellipticall
Dials
to
all
Superficies
whether
plain
or
curved
,
whose
Indexes
stand
upright
in
the
Zenith
line
of
the
place
:
There
now
followeth
some
other
directions
how
the
same
thing
may
be
done
to
any
Superficies
,
and
to
an
Index
set
casually
in
any
position
whatsoever
.
But
first
are
premised
some
usefull
Propositions
tending
to
the
same
purpose
.
1.
An
Index
or
streight
line
being
set
casually
,
how
to
finde
the
Re
/
In-clination
and
declination
thereof
.
IF
it
stand
upright
,
it
is
free
from
both
those
accidents
,
and
falls
to
be
the
same
case
with
the
upright
Index
before
treated
of
.
But
if
it
lie
leaning
,
then
is
it
to
be
dealt
withall
in
this
place
.
And
first
for
the
re
/
in-clination
,
it
is
best
to
be
taken
by
some
inclinatory
(
square
or
other
)
Instrument
.
The
manner
of
the
work
is
the
very
same
that
is
performed
in
finding
the
re
/
in-clination
of
a
plain
:
because
the
fiduciall
edge
of
the
Index
is
(
here
)
like
the
Verticall
line
of
a
re
/
in-clining
plain
,
and
the
application
of
the
Square
must
here
be
the
same
to
this
edge
that
was
there
to
the
verticall
line
of
the
plain
,
and
the
degrees
of
re
/
in-clination
are
to
be
numbered
in
both
these
cases
alike
.
So
that
it
will
not
be
needfull
here
to
make
repetition
of
that
which
is
so
often
elswhere
declared
.
You
are
here
also
to
observe
whether
the
re
/
in-clination
be
towards
the
North
or
towards
the
South
.
To
this
new
plain
therefore
,
or
to
the
two
lines
A
B
,
C
D
,
apply
one
of
the
streight
edges
of
a
Quadrant
,
the
limbe
of
it
being
turned
alwayes
towards
the
Sun
,
and
observe
the
Horizontall
distance
of
the
Sun
,
as
is
usually
done
in
finding
the
declinations
of
plains
.
Only
here
take
this
caution
,
that
you
count
not
this
Horizontall
distance
from
that
side
of
the
Quadrant
which
is
perpendicular
to
the
plain
,
but
from
that
side
which
lies
in
the
line
E
D
,
or
upon
the
two
lines
A
B
,
B
C.
And
further
,
you
are
alwayes
to
account
this
Horizontall
distance
from
that
end
of
the
Quadrants
side
which
lookes
the
same
way
that
the
point
B
(
of
the
Index
,
which
is
furthest
remote
from
the
plain
A
C
)
doth
look
:
as
in
the
second
figure
is
exprest
,
where
the
Horizontall
distance
is
more
than
a
Quadrant
,
it
being
there
to
be
accounted
from
F
to
H.
Or
for
more
easie
conceit
,
you
may
alwayes
suppose
your
Quadrant
in
that
posture
to
be
continued
to
a
Semicircle
,
as
is
done
in
the
second
figure
:
and
then
count
your
Horizontall
distance
from
that
term
of
the
Semicircles
Diameter
which
respecteth
the
same
coast
of
the
heavens
with
that
point
of
the
Index
which
is
most
remote
from
the
plain
,
which
is
B
;
that
is
,
you
must
count
it
from
F
,
and
so
the
Horizontall
distance
will
be
F
G
H
,
more
than
a
Quadrant
,
in
this
particular
case
.
The
Horizontall
distance
being
thus
accounted
and
observed
,
you
are
immediately
to
take
the
Suns
Altitude
and
to
finde
the
Azimuth
.
Then
for
the
declination
of
the
Index
,
that
is
to
know
into
what
coast
of
the
world
,
that
is
into
what
Azimuth
of
the
Heavens
(
it
being
continued
from
the
plain
at
A
infinitely
forwards
toward
B
)
it
would
point
into
:
I
say
to
finde
this
,
you
have
only
the
same
work
to
do
which
is
usually
done
in
finding
the
declinations
of
plains
,
the
same
work
without
any
difference
at
all
.
And
therefore
I
shall
not
here
give
any
further
directions
in
this
particular
,
because
I
have
done
it
often
enough
in
other
places
,
to
which
the
Reader
may
have
recourse
.
And
so
I
suppose
the
position
of
the
Index
in
respect
of
re
/
in-clination
and
declination
to
be
fully
found
out
,
both
for
coast
and
quantity
;
by
which
two
things
known
,
we
are
further
to
enquire
what
Longitude
from
our
Meridian
,
and
what
Latitude
the
said
Index
pointeth
unto
,
which
will
be
the
next
Proposition
.
¶
Note
,
that
though
I
call
A
C
a
plain
,
yet
it
may
be
any
curved
Superficies
as
well
as
a
plain
,
for
the
Diall
will
be
described
upon
one
as
well
as
the
other
indifferently
.
2.
By
having
the
Re
/
In-clination
and
Declination
of
any
right
line
,
to
finde
the
Longitude
and
Latitude
thereof
.
Now
in
the
second
case
before
given
where
the
Index
points
downwards
,
or
under
the
Horizon
,
where
the
plain
is
inclining
,
and
so
looking
downward
,
we
may
shape
the
Projection
a
little
otherwise
,
and
the
work
then
will
be
as
easie
as
the
former
.
Let
n
E
S
W
be
the
Horizon
,
n
the
North
,
S
the
South
,
P
the
South
Pole
N
the
Nadir
point
opposite
to
the
Zenith
,
O
the
point
or
place
to
which
the
Index
respecteth
.
In
the
Triangle
N
P
O
,
are
given
,
N
P
the
complement
of
the
Latitude
of
the
place
,
N
O
the
inclination
of
the
Index
;
and
P
N
O
the
declination
of
the
same
Index
,
by
which
we
may
finde
P
O
,
whose
complement
is
the
South
Latitude
,
or
whose
excesse
is
the
North
Latitude
required
:
and
the
angle
at
P
will
be
found
,
which
is
the
difference
of
Longitude
here
sought
after
.
The
work
is
easie
to
them
that
understand
the
like
work
in
re
/
in-clining
plains
,
and
may
be
performed
either
by
calculation
,
or
else
Instrumentally
,
as
every
one
shall
like
best
of
.
And
this
is
all
that
needs
to
be
said
of
this
particular
.
3.
How
to
finde
a
Meridian
line
,
and
to
erect
a
true
Axis
of
the
World
from
the
foot
of
the
Index
.
THe
foot
of
the
Index
I
call
that
point
where
the
Index
enters
into
the
Diall
●●perficies
,
whatsoever
that
superficies
be
,
whether
plain
or
curved
.
As
in
the
two
former
figures
may
be
understo●●
by
the
point
A.
From
the
foot
or
po●●●at
A
,
set
some
upright
threed
which
may
represent
th●●●●enith-line
of
the
place
,
as
is
to
be
seen
in
that
Figure
Pag.
●●
.
and
is
there
represented
by
the
line
A
D.
Then
one
way
will
be
to
make
use
of
some
other
Meridian
line
,
observing
when
the
Sun
comes
to
it
,
and
at
the
same
moment
to
note
out
the
shadow
cast
by
the
line
A
D
,
for
that
is
the
Meridian
line
.
But
this
is
for
such
superficies
upon
which
the
Sun
comes
at
noon
:
and
it
ties
a
man
to
the
Meridian
moment
.
But
without
this
we
may
(
according
to
the
four
first
Precepts
)
set
off
the
Meridian
line
P
E
upon
the
Pastboard
,
and
by
the
Zenith-line
A
D
you
may
project
it
upon
all
objects
that
shall
stand
or
be
set
in
the
way
.
For
which
purpose
you
must
place
some
object
upon
that
Coast
of
A
D
upon
which
any
one
of
the
Poles
of
the
World
may
be
projected
,
and
also
elevated
above
your
Diall
superficies
.
And
upon
this
obiect
project
the
Meridian
neer
that
place
whereabout
you
conjecture
the
Axis
of
the
World
will
passe
through
.
Then
with
your
Semicircle
(
or
projecting
Quadrant
rather
)
project
the
Axis
of
the
World
,
as
is
usually
done
in
my
wayes
of
Dialling
,
and
by
that
means
you
shall
finde
a
point
in
the
formerly
projected
Meridian
,
which
shall
represent
the
Pole
of
the
World.
And
then
further
,
if
from
the
foot
of
the
Index
at
A
,
to
this
Pole
,
you
fasten
a
threed
,
the
same
threed
will
represent
the
true
Axis
of
the
world
comming
from
the
foot
of
the
Index
.
4.
Having
an
Axis
raised
from
the
foot
of
the
Index
,
how
to
finde
in
what
Longitude
and
Latitude
the
Index
it selfe
lyeth
,
by
a
way
easier
and
differing
from
the
former
,
without
looking
after
any
re
/
in-clination
or
declination
of
the
said
Index
.
THis
way
will
not
prove
(
perhaps
)
so
good
as
I
expected
,
especially
in
the
Longitude
,
and
therefore
the
former
may
be
used
.
The
former
way
that
I
used
was
troublesome
enough
for
effecting
of
what
was
by
it
intended
,
which
is
the
cause
that
I
shall
here
endeavour
to
give
another
more
easie
.
Let
the
Diall
superficies
be
supposed
to
have
upon
it
an
Index
,
an
Axis
truly
placed
,
issuing
from
the
foot
of
the
Index
,
with
a
Meridian
line
projected
also
to
,
or
directly
under
the
same
Axis
,
according
as
was
done
in
the
work
of
the
preceding
Proposition
.
All
the
difficulty
in
this
work
is
,
that
there
is
no
stable
place
whereon
to
fix
the
Compasses
for
measuring
of
the
distance
between
the
two
knots
m
and
n
:
but
you
may
hold
somthing
underneath
close
to
one
of
the
points
(
without
disturbing
the
threeds
place
)
on
which
setting
one
foot
of
your
Compasses
,
you
may
measure
the
distance
of
the
other
point
.
Or
else
,
because
it
is
supposed
that
the
threed
Axis
A
X
is
fixed
upon
some
solid
thing
at
X
,
you
may
open
your
Compasses
to
the
Radius
A
X
(
to
which
also
you
are
to
have
a
Radius
of
Chords
equall
)
and
set
the
same
extent
from
A
to
R
upon
the
Index
(
prolonging
the
Index
if
need
be
by
a
threed
till
it
come
to
be
of
a
competent
length
)
and
there
fasten
a
slipping
knot
.
Then
because
X
is
a
stable
point
,
you
may
take
the
extent
from
X
(
setting
one
foot
of
your
Compasses
in
that
point
)
to
R
,
and
measure
the
same
upon
your
Scale
of
Chords
,
where
it
will
give
the
former
angle
X
A
R
,
which
is
to
be
used
as
is
shewed
before
.
There
will
be
found
other
wayes
to
do
this
last
work
.
So
much
therefore
for
the
Latitude
.
Secondly
,
For
the
Longitude
of
the
Index
,
it
will
not
be
so
easily
had
as
the
other
was
.
The
way
that
best
likes
me
(
amongst
many
others
)
for
the
present
,
is
this
.
By
the
Precepts
,
Pag.
70.
you
must
make
some
observation
of
the
Sun's
Azimuth
(
and
so
must
you
also
do
by
the
last
precedent
Proposition
)
and
you
finde
what
Azimuth
that
is
too
.
You
may
also
(
further
)
project
it
upon
your
Diall
superficies
,
and
then
you
have
two
Azimuths
upon
the
said
superficies
,
namely
,
the
Meridian
,
and
the
observed
Azimuth
.
You
are
to
know
yet
further
,
what
Altitudes
the
Equinoctiall
hath
upon
these
two
Azimuths
.
And
for
the
Meridian
it
is
certain
enough
that
the
Altitude
there
,
is
equall
to
the
complement
of
your
Latitude
.
For
the
other
Azimuth
say
thus
.
As
the
Radius
,
Is
to
the
Co-tangent
of
your
Latitude
;
So
is
the
Co-sine
of
your
Azimuth
,
To
the
Tangent
of
the
Equinoctials
altitude
upon
that
Azimuth
Circle
.
After
this
you
must
make
some
mark
or
knot
upon
your
Axis
A
X
,
which
suppose
to
be
at
m
,
and
then
with
your
projecting
Quadrant
(
upon
your
Meridian
and
Azimuth
)
from
the
point
m
,
set
on
the
respective
Equinoctiall
Altitudes
,
as
the
manner
of
projecting
by
that
Instrument
useth
to
be
.
Next
,
you
must
project
this
Equinoctiall
Circle
all
over
the
Diall-superficies
(
and
upon
some
other
objects
where
need
shall
be
,
)
suppose
it
here
to
be
the
line
S
V.
And
furthermore
,
you
are
to
proiect
the
Axis
A
X
,
and
the
Index
A
B
,
one
upon
the
other
,
and
to
observe
the
line
that
they
both
(
in
that
position
)
do
make
upon
the
Diall-superficies
,
and
other
objects
if
need
be
.
I
say
you
must
,
first
,
principally
observe
this
line
,
for
it
is
the
proper
Meridian
line
to
the
Index
or
Zenith
line
A
B
,
because
it
passeth
(
quoad
superficiem
)
through
both
the
Axis
of
the
world
and
the
Zenith
line
.
Suppose
it
in
the
former
plain
to
be
A
☉
.
And
secondly
you
must
note
the
intersection
that
this
line
makes
with
the
Equinoctiall
line
which
was
now
projected
,
which
we
may
imagine
to
be
at
V
,
upon
some
object
laid
in
the
way
of
purpose
.
And
so
also
you
must
(
thirdly
)
take
notice
of
the
intersection
of
the
Equinoctiall
line
with
the
Meridian
line
,
which
suppose
at
S.
For
these
two
Equinoctiall
intersections
and
the
point
m
,
all
three
together
,
must
do
what
we
now
intend
.
The
thing
here
now
intended
is
,
to
know
what
angle
is
contained
between
the
two
superficies
X
A
B
,
and
X
A
T
,
which
are
two
Meridian
circles
,
or
(
which
is
the
same
)
the
angle
V
m
S
in
the
plain
of
the
Equinoctiall
,
which
angle
measures
the
said
inclination
of
the
two
forenamed
Meridians
,
as
it
doth
of
all
Meridians
.
Now
to
measure
this
angle
V
m
S
,
the
best
way
will
be
first
to
extend
two
threeds
,
one
from
V
to
m
,
and
the
other
from
S
to
m
,
crossing
each
other
at
m
,
and
they
may
be
continued
further
till
they
meet
with
some
object
(
in
the
way
standing
,
or
else
set
for
that
purpose
)
where
they
may
be
both
fixed
as
at
F
and
G.
Then
measure
from
S
to
m
,
and
from
V
to
m
,
which
is
the
shortest
,
suppose
S
m
to
be
the
shortest
,
and
V
m
the
longest
.
First
,
take
V
m
,
and
lay
it
down
in
a
line
,
as
there
is
done
.
Then
take
S
m
,
and
measure
it
on
the
same
line
from
M
to
S
or
D.
Thirdly
,
take
the
distance
from
V
to
D
out
of
this
line
,
and
set
it
from
V
to
D
,
upon
the
line
V
m
,
so
shall
m
S
and
m
D
be
equall
one
to
the
other
.
To
this
distance
therefore
m
S
or
m
D
as
a
Radius
,
open
some
line
of
Chords
,
then
take
the
Chord
or
subtense
from
S
to
D
(
S
is
a
stable
point
upon
which
you
may
firmly
set
your
Compasses
)
and
measure
the
same
length
upon
your
line
of
Chords
,
where
you
shall
see
how
many
degrees
the
angle
S
m
D
or
S
m
V
containeth
.
Now
all
this
businesse
is
to
finde
how
great
an
angle
is
contained
between
the
South
part
of
the
Meridian
of
the
place
,
and
the
Meridian
belonging
properly
to
the
Zenith
line
.
But
here
(
if
it
be
observed
well
)
the
angle
now
measured
is
made
between
the
proper
Meridian
and
the
North
part
of
the
Meridian
of
the
place
,
so
that
this
angle
must
be
the
supplement
(
to
a
Semicircle
)
of
what
is
required
.
The
supplement
therefore
of
the
angle
now
found
,
is
(
in
this
case
)
the
Longitude
or
Meridian
into
which
the
Zenith
line
A
B
pointeth
.
I
say
the
supplement
of
that
angle
is
the
proper
Meridian
of
the
Zenith
line
;
or
the
difference
of
Longitude
proper
to
that
line
,
from
the
Meridian
of
the
place
.
But
forasmuch
as
the
angle
V
m
S
doth
measure
the
inclination
or
angle
of
this
proper
Meridian
from
the
North
part
of
the
Meridian
of
the
place
,
and
that
the
shadow
of
the
Index
A
B
doth
first
come
to
the
proper
Meridian
A
O
before
it
comes
to
A
T
the
Meridian
of
the
place
:
we
may
therefore
in
such
case
say
,
that
the
angle
V
m
S
gives
the
difference
of
Longitude
;
and
that
the
said
difference
of
Longitude
lies
Eastward
from
the
South
part
of
the
Meridian
of
the
place
,
as
there
in
the
Figure
it
appears
to
lie
Westward
from
the
North
part
.
Suppose
that
the
difference
of
Longitude
were
39
gr
.
25
min.
and
the
proper
Latitude
40
gr
.
And
so
much
for
this
also
.
If
this
way
be
thought
not
feasible
enough
,
the
former
in
the
second
Proposition
may
be
used
;
yet
variety
in
all
kindes
is
delightfull
,
and
not
to
be
rejected
in
this
.
It
is
done
,
you
see
,
without
having
respect
to
any
declination
or
re
/
in-clination
,
which
the
former
was
founded
upon
.
5.
How
to
forme
the
Angles
at
the
Pole.
THis
Proposition
makes
way
for
computing
Tables
of
Horizontall
spaces
,
and
Equinoctiall
Altitudes
to
any
Longitude
or
Meridian
differing
from
the
Longitude
of
the
East
hour
generall
Table
West
hours
east
hou
generall
Table
west
hour
13
0
00
12
6
90
00
6
3
45
—
a
93
45
7
30
—
b
97
30
11
15
—
c
101
15
11
15
00
1-d
5
105
00
7
18
45
108
45
22
30
112
30
26
15
116
15
10
30
00
2
4
120
00
8
33
45
123
45
37
30
127
30
A
-
1
55
131
15
A
-
39
25
3
135
00
9
A
-
1
50
138
45
41
15
142
30
9
45
00
3
146
15
48
45
2
150
00
10
52
30
153
45
56
15
157
30
8
60
00
4
161
15
63
45
1
165
00
11
67
30
168
45
71
15
172
30
7
75
00
5
176
15
78
45
12
180
00
12
82
30
86
15
Angles
at
the
Pole.
Angles
at
the
Pole.
s
88
10
n
Proper
Merid.
00
00
-
3
t
84
25
n
p
01
50
r
v
80
40
n
9q
05
35
e
76
55
n
r
09
20
f
73
10
n
13
05
g
2
69
25
n
16
50
h
65
40
n
8
20
35
i
61
55
m
24
20
k
58
10
m
28
05
l
1
54
25
m
31
50
m
50
40
m
7
35
35
m
46
55
m
39
20
m
43
10
m
43
05
m
12
39
25
m
46
50
m
35
40
m
6
50
35
m
31
55
m
54
20
m
28
10
l
58
05
m
11
24
25
k
61
50
n
20
40
i
5
65
35
n
16
55
h
69
20
n
13
10
g
73
05
n
10
y
09
25
f
76
50
n
x
05
40
e
4
st
80
35
n
w
01
55
r
u
84
20
n
z
88
05
n
place
,
and
to
any
Latitude
differing
from
the
Latitude
of
your
place
.
After
that
the
difference
of
Longitude
is
known
,
you
are
first
to
frame
a
Table
of
angles
at
the
Pole
,
which
will
not
be
hard
to
perform
,
and
the
manner
of
it
will
be
seen
best
in
an
Example
,
being
altogether
like
the
computation
of
the
said
angles
at
the
Pole
for
the
houres
of
any
place
to
the
same
declining
plain
.
I
have
here
first
of
all
in
a
generall
Table
set
down
the
graduall
distances
of
houres
and
quarters
from
12
at
noon
or
mid-day
,
which
will
be
some
help
in
performing
the
work
.
The
houres
that
are
on
the
left
side
of
the
degrees
are
the
East
or
forenoon
houres
,
and
those
on
the
right
hand
are
the
West
or
afternoon
houres
.
Suppose
(
as
before
Prop.
4.
)
a
difference
of
Longitude
were
given
39
gr
.
25
min.
towards
the
East
,
and
it
were
required
to
know
what
angles
our
hours
(
namely
those
12
that
stand
neerest
to
it
,
six
upon
one
side
,
and
six
upon
the
other
)
with
their
intermediate
quarters
do
make
with
this
Longitude
or
Proper
Meridian
,
for
so
I
intend
to
call
it
.
I
first
take
the
given
Longitude
or
Proper
Meridian
,
39
gr
.
25
min
and
enter
it
among
the
East
houres
of
the
generall
Table
,
and
I
since
it
to
fall
in
at
A
;
that
is
,
between
a
quarter
and
halfe
an
houre
past
9
before
noon
,
and
I
finde
also
the
circumstant
numbers
to
be
37.30
the
lesser
,
and
41.15
the
greater
.
The
differences
between
these
two
numbers
and
the
Longitude
given
,
are
1
gr
.
55
min.
and
1
gr
.
50
min.
as
you
see
them
set
down
in
the
said
generall
Table
right
against
the
letter
A.
These
two
differences
are
to
be
first
placed
in
this
second
Table
of
Angles
at
the
Pole
,
as
you
see
in
the
Example
,
where
,
first
of
all
is
written
Proper
Meridian
00.00
separated
from
the
rest
as
signifying
only
the
place
where
it
is
to
stand
,
and
above
it
is
placed
(
the
difference
between
37.30
and
the
proper
Meridian
39.25
,
namely
)
1
gr
.
55
min.
So
again
,
below
it
is
placed
1
gr
.
50
min.
(
which
is
the
difference
of
39
gr
.
25
min.
the
proper
Meridian
from
the
next
greater
number
41
gr
.
15
min.
)
Now
to
these
two
radicall
numbers
,
I
adde
¼
½
¾
and
one
whole
houre
,
namely
a
,
b
,
c
,
and
thereby
produce
e
,
f
,
g
,
on
both
sides
the
proper
Meridian
.
Then
to
r
,
e
,
f
,
g
,
I
adde
15
gr
.
d
,
namely
,
the
number
of
one
houre
,
and
thereby
do
further
produce
h
,
i
,
k
,
l
,
on
both
sides
the
Proper
Meridian
,
as
before
.
And
afterwards
for
more
ease
,
to
the
numbers
r
,
e
,
f
,
g
,
h
,
i
,
k
,
l
,
I
adde
30
gr
.
or
two
houres
,
and
do
thereby
produce
on
both
sides
the
eight
numbers
noted
with
the
letter
m.
And
so
still
adding
30
gr
.
to
the
last
eight
numbers
noted
with
m
,
or
else
adding
60
gr
.
to
the
first
eight
numbers
noted
with
r
,
e
,
f
,
g
,
h
,
i
,
k
,
l
,
I
shall
make
the
eight
remaining
numbers
noted
with
the
letter
n
,
both
above
and
below
,
on
both
sides
the
Proper
Meridian
.
And
so
this
Table
of
Angles
at
the
Pole
is
compleated
,
for
if
it
should
(
in
the
same
manner
)
be
further
continued
,
the
next
numbers
(
above
)
would
be
91
gr
.
55
min.
and
(
below
)
91
gr
.
50
min.
both
greater
than
90
gr
.
beyond
which
there
will
be
neither
need
nor
expedience
to
go
.
This
way
of
forming
the
angles
I
thought
best
to
take
,
because
it
is
more
plain
and
easie
than
any
other
,
which
else
might
have
been
in
this
work
used
.
You
are
afterward
to
place
the
houres
(
in
this
last
Table
)
about
the
Proper
Meridian
,
just
as
you
see
their
order
to
be
about
the
letter
A
in
the
general
Table
,
and
as
you
see
they
are
in
this
particular
Table
of
Angles
at
the
Pole.
The
like
work
serves
for
West
Longitudes
.
Thus
you
have
the
angles
for
12
houres
and
their
quarters
,
the
same
angles
serve
for
the
other
12
houres
:
and
whatsoever
is
hereafter
computed
for
any
of
these
12
houres
,
must
be
understood
to
serve
also
for
the
12
opposites
,
and
so
the
work
will
be
done
for
the
whole
Circle
,
or
24
houres
of
the
naturall
day
.
6.
By
knowing
the
angles
at
the
Pole
,
and
the
Latitude
of
the
place
or
Horizon
,
how
to
finde
the
Horizontall
Spaces
thereto
belonging
.
WE
must
suppose
that
this
Longitude
or
Meridian
which
we
have
before
mentioned
,
is
proper
to
some
Horizon
or
other
,
above
which
also
the
elevation
of
the
Pole
must
be
imagined
known
,
which
we
may
suppose
to
be
40
gr
.
as
before
,
Prop.
4.
In
respect
of
this
plain
or
Horizon
it
is
,
that
the
Spaces
now
mentioned
are
called
Horizontall
Spaces
.
The
way
to
finde
them
for
this
Ellipticall
work
is
converse
to
that
which
is
usuall
in
declining
plains
.
Namely
thus
,
As
the
Sine
of
the
Poles
elevation
above
this
Horizon
or
Plain
,
Is
to
the
Radius
;
So
is
the
Tangent
of
each
of
these
angles
at
the
Pole
,
To
the
Tangent
of
the
Horizontall
space
belonging
to
each
of
those
foresaid
angles
.
So
having
found
all
these
Horizontall
spaces
severally
by
this
form
of
calculation
,
you
may
set
them
into
a
Table
,
as
here
you
see
it
done
in
the
first
of
the
two
main
Columns
.
They
signifie
the
distances
of
every
of
the
houres
and
quarters
from
the
proper
Meridian
,
and
are
to
be
accounted
as
numbered
in
the
degrees
of
the
said
plain
or
Horizon
.
7.
By
knowing
the
angles
at
the
Pole
,
and
the
Latitude
of
the
plain
or
Horizon
,
how
to
finde
the
Equinoctiall
Altitudes
or
depressions
(
above
or
under
the
same
Horizon
)
due
to
the
said
Angles
at
the
Pole
or
points
of
the
Equinoctiall
.
THe
proportion
by
which
this
is
to
be
effected
is
this
.
As
the
Radius
,
Is
to
the
Co-sine
of
the
Poles
elevation
above
the
plain
or
Horizon
;
So
is
the
Co-sine
of
any
of
the
Angles
specified
in
the
former
Table
of
Angles
at
the
Pole
;
To
the
Sine
of
the
Altitude
required
,
and
due
to
that
angle
,
or
houre
(
rather
)
in
the
Equinoctiall
Circle
.
So
in
the
former
example
.
The
Latitude
or
elevation
of
the
Pole
above
the
Horizon
was
40
gr
.
the
complement
of
that
is
50
gr
.
which
is
the
Meridian
altitude
and
depression
of
the
Equinoctiall
Circle
above
and
under
the
said
Horizon
or
plain
,
and
is
therefore
to
be
set
for
the
altitude
of
the
proper
Meridian
.
Now
,
As
the
Radius
,
Is
to
the
altitude
of
50
gr
.
So
is
the
Co-sine
of
every
angle
at
the
Pole
,
or
arke
of
the
Equinoctiall
in
the
former
Table
,
To
the
Altitude
due
thereunto
.
hours
Horizon
Spaces
altitudes
or
profu
.
hours
88
49
1
24
3
86
24
4
16
3
83
58
7
08
81
30
9
59
79
00
12
49
3
76
26
15
37
2
73
48
18
24
71
04
21
08
68
15
23
50
1
65
18
26
28
1
62
13
29
03
59
00
31
33
55
35
33
58
12
51
58
36
17
12
48
09
38
29
44
06
40
33
39
48
42
29
11
35
14
44
14
11
30
24
45
47
25
19
47
8
20
00
48
14
10
14
28
49
05
10
8
47
49
40
2
59
49
58
hours
Horizon
Spaces
.
altitudes
or
profun
hours
Proper
Merid.
00
00
50
00
2
51
49
58
9
8
39
49
40
9
14
21
49
06
19
53
48
16
25
13
47
09
8
30
18
45
49
8
35
08
44
16
39
42
42
31
44
00
40
36
7
48
04
38
32
7
51
53
36
20
55
30
34
01
58
55
31
36
6
62
09
29
06
6
65
14
26
32
68
11
23
53
71
00
21
12
5
73
44
18
28
5
76
22
15
41
78
56
12
53
81
27
10
03
4
83
55
7
12
4
86
21
4
20
88
46
1
28
And
note
that
r
e
f
g
h
,
&c.
below
the
propet
Meridian
in
the
Table
of
Angles
,
are
complements
to
n
n
n
n
n
,
&c.
at
the
top
of
the
Table
of
Angles
:
and
again
,
those
r
e
f
g
h
,
&c.
above
the
proper
Meridian
,
are
complements
to
the
lowest
n
n
n
n
n
,
&c.
successively
rising
in
order
one
after
the
other
.
So
that
,
As
the
Radius
,
Is
to
the
Sine
of
50
gr
.
So
is
the
Sine
of
p
q
r
,
&c.
To
the
Sines
of
the
Altitudes
belonging
to
s
t
v
,
&c.
And
so
again
,
the
Sines
of
w
x
y
,
&c.
to
the
Sines
of
the
Altitudes
belonging
to
z
u
st
,
&c.
And
accordingly
this
Table
of
Altitudes
and
depressions
is
calculated
.
If
the
Pole
be
elevated
upon
the
plain
or
Horizon
,
above
50
gr
.
it
will
be
sufficient
to
compute
these
two
Tables
to
halfe
houres
only
,
and
so
to
save
halfe
your
labour
.
But
in
lesser
elevations
it
is
best
to
do
it
to
quarters
,
as
here
is
done
n
this
Example
.
8.
How
to
finde
the
proper
Meridian
line
duly
belonging
to
any
Zenith
line
casually
placed
,
and
to
draw
it
upon
the
plain
.
THough
the
4.
Proposition
be
not
made
use
of
for
finding
the
Longitude
and
Latitude
(
which
of
due
pertaineth
to
any
Zenith
line
)
but
instead
of
it
the
second
precedent
be
thought
most
fit
to
be
used
;
yet
so
much
of
the
fourth
will
be
best
to
be
used
,
as
shall
concern
the
finding
out
of
the
proper
Meridian
line
.
That
is
,
you
must
raise
an
Axis
from
the
foot
of
the
Index
,
as
is
A
X
,
and
then
projecting
this
Axis
A
X
and
the
Index
A
B
one
upon
the
other
,
as
if
they
were
both
one
,
you
shall
thereby
also
project
(
by
them
both
together
)
the
Proper
Meridian
belonging
to
the
Zenith
line
A
B
,
such
as
(
in
the
figure
of
the
fourth
)
is
A
O
,
which
must
be
drawn
upon
the
plain
accordingly
.
The
manner
,
of
all
the
particular
workings
that
do
hereunto
tend
,
is
set
down
in
the
fourth
precedent
Proposition
,
and
therefore
will
not
here
again
need
to
be
repeated
.
9.
How
to
draw
and
divide
the
Elli●sis
into
houres
and
quarters
,
to
an
Index
casually
set
,
whose
Latitude
and
difference
of
Longitude
is
discovered
by
the
former
works
.
WHen
you
know
the
position
of
your
Index
in
respect
of
Longitude
and
Latitude
,
you
may
then
compute
two
Tables
to
the
same
Meridian
or
difference
of
Longitude
,
and
to
the
Latitude
of
your
Index
,
as
is
done
before
in
the
7
Proposition
,
one
of
which
Tables
is
of
Horizontall
Spaces
,
the
other
of
Equinoctiall
Altitudes
or
Depressions
above
and
under
that
Horizon
which
is
proper
to
the
Index
or
Zenith
line
casually
placed
.
By
these
two
Tables
the
work
will
be
done
in
such
manner
as
was
shewed
before
,
Prop.
2.
The
manner
of
the
work
is
this
.
1.
You
are
to
assume
some
point
in
your
Index
A
B
,
let
the
point
be
C
,
where
you
may
fasten
some
knot
of
threed
that
it
may
not
be
lost
again
.
2.
From
this
point
you
must
draw
an
Horizontall
line
,
not
in
the
levell
of
your
own
Horizon
,
but
in
that
Horizon
which
is
proper
to
that
Index
or
Zenith
line
A
B
:
that
is
,
it
must
lie
perpendicular
to
A
B
,
making
right
angles
(
in
every
part
of
it
)
to
that
line
,
and
must
have
respect
(
in
this
perpendicularity
)
to
the
point
C.
The
meaning
is
,
you
must
imagine
a
plain
to
passe
through
the
point
C
,
and
the
same
plain
to
be
perpendicular
to
the
line
A
B
;
or
that
the
line
A
B
is
a
perpendicular
surgent
line
to
the
said
plain
passing
through
the
point
at
C.
[
And
so
it
may
be
noted
,
that
if
other
Dials
be
described
by
the
Equinoctiall
Circle
and
not
by
the
Horizontall
,
the
work
of
drawing
the
Equinoctiall
perpendicular
to
the
Axis
will
be
difficult
,
but
though
no
great
accuratenesse
be
used
,
yet
the
work
will
be
perfect
enough
,
and
no
way
defective
for
the
losse
of
a
degree
or
two
in
the
perpendicularity
required
,
which
I
thought
good
here
also
to
note
,
because
I
have
omitted
it
in
all
my
other
Precepts
of
Projecting
Dials
.
]
Wherefore
you
may
do
it
by
some
Pastboard
,
applying
one
edge
of
it
to
any
line
projected
from
A
B
the
Index
,
as
to
A
D
,
and
keeping
the
edge
there
,
you
may
turn
the
flat
of
it
to
the
Index
A
B
,
and
draw
a
line
by
it
,
or
make
two
pricks
through
it
into
the
pastboard
,
whereby
a
line
may
be
drawn
,
but
above
all
note
the
point
C
upon
it
.
Then
to
this
line
of
the
Index
thus
drawn
,
and
from
the
noted
point
of
it
at
C
,
erect
a
perpendicular
:
So
applying
your
Pastboard
to
its
former
place
again
(
the
edge
of
it
lying
upon
A
D
,
and
the
flat
of
it
applyed
to
the
Index
A
B
,
and
the
point
in
it
,
noted
for
C
,
being
again
fitted
to
C
in
the
Index
,
I
say
thus
doing
)
you
may
note
where
the
last
drawn
perpendicular
doth
cut
the
line
A
D
(
which
must
be
extended
by
help
of
some
threed
if
need
be
)
suppose
at
E
:
at
the
point
E
(
then
)
you
must
say
that
one
point
of
the
proper
Horizontall
line
is
to
be
taken
.
Then
in
like
manner
you
must
seeke
another
Horizontall
point
:
first
by
projecting
a
line
from
the
Index
(
any
where
)
such
as
is
A
F
,
and
by
applying
one
edge
of
a
pastboard
to
that
line
,
and
the
plain
of
that
pastboard
to
the
Index
A
B
,
and
so
noting
the
point
C
,
and
drawing
or
marking
the
line
A
B
upon
it
,
&c.
as
was
done
before
.
So
you
shall
finde
another
point
of
the
same
Horizontal
line
or
plain
(
rather
)
which
suppose
to
fall
at
F.
Now
having
three
points
of
the
Horizontall
plain
at
C
,
E
,
and
F
,
(
which
I
suppose
not
to
lie
in
one
and
the
same
right
line
,
for
that
must
with
carefulnesse
be
avoided
here
)
you
may
project
some
part
of
that
line
upon
the
plain
,
as
E
F
,
the
rest
of
it
(
so
much
as
shall
be
found
usefull
)
may
be
made
up
with
returns
of
threed
,
and
regulated
or
kept
in
the
same
plain
by
projection
,
as
the
manner
of
working
that
way
useth
to
be
,
and
as
here
you
see
exprest
,
by
the
line
H
E
F
G
,
lying
in
the
same
plain
with
the
point
C
,
and
that
whole
plain
lying
perpendicular
to
the
Index
A
B.
3.
The
next
thing
to
be
done
is
the
drawing
of
the
houre
lines
upon
the
plain
.
And
the
first
thing
h●re
presupposed
to
be
done
,
is
the
drawing
of
the
proper
Meridian
,
performed
by
the
8
Prop.
That
(
I
say
)
is
supposed
as
already
done
before
any
of
this
work
is
begun
.
Let
the
proper
Meridian
be
A
V.
Having
then
made
Tables
(
by
the
7
Prop.
)
for
the
Horizontall
Spaces
of
your
hours
from
this
proper
Meridian
A
B
;
you
must
first
apply
a
pastboard
to
the
Horizontall
line
E
F
,
and
fit
the
center
of
it
to
C
:
and
upon
the
Pastboard
,
project
the
proper
Meridian
from
C
to
A
V
,
or
from
C
to
P.
And
then
by
the
Table
of
Horizontall
Spaces
in
the
7
Prop.
(
for
we
now
here
suppose
that
this
is
the
Diall
for
which
those
Tables
were
computed
)
you
may
(
upon
that
pastboard
)
set
off
all
the
houres
and
quarters
from
the
proper
Meridian
upon
this
pastboard
:
and
applying
the
pastboard
into
its
proper
place
,
namely
to
the
Horizontall
line
E
P
F
,
and
to
the
point
C
,
you
may
project
the
houres
and
parts
of
houres
from
the
Pastboard
to
the
horizontall
line
H
E
P
F
G
,
as
the
manner
in
this
way
of
Dialling
is
well
known
.
4.
Having
transferred
the
houre-points
into
the
Horizontall
line
,
you
may
(
by
help
of
your
Index
A
B
)
project
and
draw
the
houre
lines
upon
the
plain
,
which
we
will
suppose
done
,
because
the
manner
of
doing
it
is
the
same
with
that
which
was
done
before
for
upright
Indexes
.
5.
To
know
where
the
Ellipticall
line
must
come
,
or
to
finde
the
points
in
those
houre
lines
,
through
which
it
must
passe
,
we
must
work
in
the
same
manner
as
before
(
in
the
7
§
of
Pag.
7●
.
)
is
exprest
,
namely
thus
.
We
must
make
a
Scale
of
right
Sines
of
a
fit
length
,
and
number
them
Versedly
,
and
out
of
that
Scale
we
must
take
such
Altitudes
or
Profundities
(
which
you
will
,
one
or
both
)
as
the
Table
Pag
97.
giveth
,
which
Table
we
here
suppose
to
be
computed
for
this
Example
whereabout
we
now
are
.
And
taking
these
Altitudes
from
that
Scale
(
that
is
,
from
the
end
at
90
,
to
the
altitude
numbered
upon
the
parts
of
the
Scale
)
we
must
insert
them
into
their
respective
houres
to
which
they
belong
.
They
must
be
inserted
in
this
manner
.
Having
taken
any
altitude
out
of
the
Scale
,
and
found
the
houre
upon
which
it
must
be
placed
,
you
must
set
one
foot
of
that
extent
upon
the
houre
line
(
keeping
it
alwayes
thereon
,
but
)
removing
it
untill
the
other
foot
being
turned
about
,
may
only
touch
the
fiduciall
edge
of
the
Index
:
and
when
the
feet
of
the
Compasses
are
thus
fitted
,
you
must
note
upon
what
point
of
the
houre
line
the
foot
that
is
thereon
doth
stand
,
for
through
that
point
of
that
houre
must
the
Ellipticall
line
passe
.
The
same
manner
of
work
you
must
perform
upon
every
houre
,
untill
you
have
gone
through
12
of
them
,
which
do
make
up
halfe
the
houres
of
the
whole
Diall
.
And
if
you
strike
the
lines
through
the
center
,
you
shall
have
all
the
24.
And
looke
what
is
done
upon
any
one
houre
line
,
the
same
is
to
be
done
upon
the
opposite
.
That
is
,
looke
what
distance
(
upon
a
plain
)
the
Ellipsis
hath
from
the
center
upon
any
one
line
,
the
selfe
same
distance
from
the
center
must
the
opposite
line
have
.
But
if
the
description
be
made
upon
an
uneven
superficies
,
then
this
rule
may
not
,
perhaps
,
hold
:
yet
this
will
;
namely
,
Looke
what
distance
from
the
Index
(
the
least
or
perpendicular
Index
I
mean
)
any
point
of
the
Ellipsis
hath
,
the
same
perpendicular
distance
is
to
be
given
for
the
Ellipsis
upon
the
opposite
houre
line
.
And
by
this
means
you
may
put
in
as
many
houres
and
Ellipticall
points
as
you
please
.
And
through
these
points
you
are
to
draw
the
Ellipticall
line
.
10.
Concerning
the
motion
that
is
to
be
made
,
either
by
the
Index
,
or
by
the
Ellipsis
it selfe
.
THe
motion
must
be
either
in
,
or
else
parallel
to
,
the
proper
Meridian
,
and
not
elsewhere
.
Now
if
the
Diall
Superficies
be
made
to
slide
,
and
the
Index
stand
still
;
then
(
though
the
Diall
be
upon
an
unequall
superficies
,
yet
)
the
motion
or
sliding
must
be
upon
a
plain
.
And
it
must
be
upon
the
proper
Meridian
,
projected
by
the
Index
from
the
Diall
plate
,
unto
or
upon
that
said
plain
,
and
the
motion
must
be
in
or
parallel
to
it
.
And
it
must
be
noted
,
that
this
proper
Meridian
thus
projected
never
falls
directly
under
any
Index
that
is
(
not
direct
but
)
declining
,
but
it
falls
aside
from
it
,
according
as
the
Index
stands
aside
from
the
proper
Meridian
upon
the
Diall
superficies
,
and
as
it
shall
be
laid
by
projecting
it
.
But
if
the
Index
move
(
the
Diall
standing
still
)
then
the
fiduciall
edge
of
it
must
be
made
to
move
alwayes
according
to
,
and
in
the
very
proper
Meridian
line
it selfe
,
and
not
any
where
else
,
which
may
be
contrived
sundry
wayes
,
as
every
man
shall
invent
to
his
own
liking
.
Great
care
must
be
had
that
the
fiduciall
edge
of
the
Index
(
in
the
motion
of
it
)
keepe
alwayes
one
justly
parallel
situation
.
11.
Of
the
place
of
the
Suns
Annuall
course
or
Zodiac
.
THe
best
place
for
it
is
to
be
considered
of
according
to
that
which
moveth
.
If
the
Diall
Superficies
move
,
it
must
(
though
it
be
rough
(
as
I
said
before
)
yet
it
must
)
move
upon
a
plain
,
this
plain
therefore
in
this
case
is
fittest
to
receive
the
Zodiac
.
And
then
there
may
a
peculiar
point
or
threed
Index
(
serving
only
for
the
Zodiac
)
be
set
upon
the
Diall-moving-plate
,
whereby
it
may
be
rectified
according
to
the
time
of
the
year
.
If
the
Index
be
made
to
move
,
then
either
the
Zodiac
may
be
set
upon
the
foot
of
the
Index
which
guides
the
motion
of
it
,
and
the
peculiar
or
Zodiacall
Index
may
be
set
somewhere
where
it
may
stand
to
point
at
the
Zodiac
.
Or
else
the
Zodiac
it selfe
may
have
its
place
upon
some
part
of
the
standing
plate
or
body
,
and
the
Zodiacall
Index
may
be
placed
upon
the
foot
of
the
Diall
Index
;
from
whence
it
may
be
made
to
shew
the
Zodiacall
parts
,
as
occasion
shall
be
.
12.
How
the
Zodiac
is
to
be
limited
in
regard
of
length
,
and
how
to
be
described
and
set
in
its
true
place
.
THe
limitation
of
it
must
be
according
to
the
Scale
of
Sines
by
which
the
Ellipsis
is
described
,
which
Scale
is
mentioned
Pag
102
§
5.
and
Pag.
73
§
7
▪
The
Index
or
Diall
plate
(
that
is
the
mover
)
must
move
according
to
the
proper
Meridian
of
the
Diall
Superficies
,
and
precisely
so
,
as
that
fiducial
edge
of
the
Index
must
move
(
not
elsewhere
,
but
)
in
the
very
proper
Meridian
it self
.
This
is
,
if
the
Index
move
.
But
if
the
Diall
Superficies
move
,
then
the
proper
Meridian
thereof
(
or
the
Superficies
of
the
proper
Meridian
,
namely
the
Superficies
made
by
the
proper
Meridian
cutting
through
the
Diall
plate
,
which
is
of
any
thicknesse
)
must
move
directly
upon
,
or
through
,
the
fiduciall
edge
of
the
Index
.
These
things
are
intimated
before
.
And
that
the
Zodiac
is
to
be
described
upon
a
plain
,
though
the
Diall
it selfe
be
not
so
.
And
further
,
that
if
the
Diall
plate
move
,
(
though
it
be
not
a
plain
it selfe
,
yet
)
it
must
move
upon
a
plain
,
and
that
the
proper
Meridian
is
to
be
described
upon
the
same
plain
,
and
the
motion
to
be
directed
according
thereunto
.
These
things
are
often
inculcated
,
because
they
are
hard
to
be
conceived
,
and
had
need
of
the
better
consideration
for
that
reason
.
Now
further
.
1.
If
the
Diall
plate
be
supposed
thus
to
move
upon
a
plain
,
and
on
it
the
proper
Meridian
be
drawn
,
then
first
of
all
,
The
angle
is
to
be
inquired
that
is
made
between
the
Index
and
that
part
of
the
proper
Meridian
which
is
projected
upon
the
plain
whereon
the
motion
is
made
,
which
how
to
measure
will
be
a
hard
matter
to
give
rules
for
,
because
the
variety
of
cases
and
positions
of
one
to
the
other
will
be
so
various
.
It
is
first
to
be
supposed
,
that
it
makes
a
just
right
angle
with
it
,
and
consequently
that
the
Zodiac
is
described
upon
the
proper
horizon
of
the
Index
.
And
if
upon
this
supposition
the
Zodiac
be
to
be
limited
,
then
the
rule
will
be
the
same
with
the
former
given
in
the
like
case
,
namely
thus
.
Upon
your
Scale
of
Sines
(
by
which
you
described
your
Ellipticall
line
)
take
from
90
,
to
the
Latitude
of
the
Index
,
and
count
that
length
for
a
new
Radius
,
and
keepe
it
.
Then
when
you
have
found
the
forenamed
angle
(
of
finding
which
more
is
said
Prop.
13.
following
)
to
this
new
Radius
finde
the
Secant
of
the
complement
or
excesse
of
that
angle
:
this
length
or
Secant
will
be
the
Tangent
of
45
gr
.
or
the
Decimall
Scale
by
which
you
are
to
describe
the
Zodiac
on
both
sides
from
the
Equinoctiall
point
or
line
,
according
to
the
numbers
in
the
generall
Tables
made
for
this
purpose
,
Pag.
4
,
5
,
6
,
and
7.
[
For
placing
the
Equinoctiall
point
in
the
Zodiac
(
upon
which
all
the
other
parts
of
that
Scale
do
depend
)
you
must
set
all
in
the
same
posture
that
they
had
when
the
Diall
was
described
:
and
when
you
have
made
fit
place
for
the
Zodiac
,
and
set
on
some
peculiar
Zodiacall
Index
,
you
must
note
the
place
upon
which
the
said
Zodiacall
Index
pointeth
,
for
that
place
is
the
peculiar
place
for
the
Equinoctiall
,
from
whence
all
the
other
parts
of
the
Suns
annuall
course
must
be
set
on
.
]
2.
If
the
Index
be
made
to
move
in
any
depth
under
the
Diall
superficies
,
then
a
slit
must
be
made
in
the
same
Superficies
that
the
proper
Meridian
would
cut
,
and
the
Index
continued
therein
,
down
into
one
streight
line
till
it
meet
with
the
foot
whereon
the
Index
is
fixed
,
and
which
being
moved
,
carryeth
the
Index
along
with
it
.
And
the
Zodiac
must
be
described
either
upon
this
foot
(
the
Zodiacall
Index
being
made
to
stand
still
while
that
said
foot
with
its
divisions
moveth
)
and
that
also
parallel
to
(
or
in
)
the
proper
Meridian
projected
upon
it
:
or
else
the
foot
must
carry
the
Zodiacall
Index
,
and
the
Zodiac
must
be
described
upon
some
other
convenient
place
,
as
the
sides
(
&c.
)
of
that
forementioned
flat
.
And
here
,
for
limitation
of
the
Zodiacs
length
,
you
must
finde
the
Inclination
of
this
moving
foot
to
the
Index
or
Zenith
line
(
which
angle
of
inclination
is
mentioned
before
,
and
spoken
of
afterwards
in
the
13
Prop.
following
)
and
work
directly
as
you
did
before
by
finding
the
new
Radius
,
and
the
Secant
of
the
complement
(
or
excesse
)
of
the
angle
.
By
which
you
shall
finde
the
Tangent
of
45
gr
.
or
the
length
of
your
Decimall
Scale
,
out
of
which
the
Zodiac
may
be
described
by
the
Tables
▪
Pag.
4
,
5
,
&c.
as
is
mentioned
before
in
this
proposition
.
Thus
(
in
these
cases
)
is
the
Zodiac
to
be
set
in
its
true
place
,
thus
it
is
to
be
limited
and
d●scribed
.
For
Occurrences
of
other
sorts
of
Cases
than
are
here
mentioned
(
whereof
there
will
not
be
many
)
he
that
can
understand
to
do
these
things
aright
,
will
be
able
to
grapple
with
them
;
and
for
such
as
do
not
understand
what
is
here
said
,
their
best
course
is
to
let
these
difficulties
alone
.
13.
How
to
finde
what
angle
is
made
between
the
Index
and
that
part
of
the
proper
Meridian
which
is
projected
upon
the
plain
whereon
the
motion
is
made
,
or
which
drawn
upon
the
foot
of
the
Index
which
maketh
the
same
motion
.
THe
way
that
for
the
present
I
thinke
of
is
this
.
Apply
a
streight
edge
of
pastboard
to
the
projected
line
,
and
the
plain
of
the
same
pastboard
to
the
fiduciall
edge
of
the
Index
,
and
make
two
points
upon
the
pastboard
,
by
or
through
which
the
same
fiduciall
edge
passeth
.
Then
taking
away
the
Pastboard
,
draw
a
streight
line
through
those
two
points
or
marks
,
and
so
measure
the
angle
made
between
the
forenamed
edge
of
the
pastboard
and
this
right
line
.
If
it
be
so
that
the
line
crosseth
not
the
edge
of
the
Pastboar●
,
then
draw
such
a
parallel
line
to
the
edge
as
may
crosse
the
former
line
,
and
then
with
a
Scale
of
Chords
you
may
measure
the
angle
.
14.
Further
observations
concerning
the
motion
and
daily
fitting
of
the
Diall
and
Index
,
for
setting
them
true
.
THe
best
way
is
to
make
the
Diall
plate
move
,
and
the
Index
to
stand
still
(
in
these
obliquely
situated
Indexes
)
for
the
Zodiac
will
(
in
such
cases
)
be
most
easily
described
and
made
usefull
.
And
in
this
case
the
Index
may
be
also
set
fast
first
,
and
quite
finished
before
the
Diall
be
drawn
at
all
.
Then
also
the
Diall
will
be
drawn
more
easily
,
and
the
motion
of
the
Diall
plate
may
sooner
(
this
way
)
be
contrived
,
then
can
the
motion
of
the
Index
be
contrived
when
the
said
Index
is
to
move
and
the
Diall
stand
still
.
The
motion
(
as
hath
been
often
said
)
must
be
according
(
that
is
parallel
)
to
the
proper
Meridian
:
and
the
slit
(
for
the
fiduciall
edge
of
the
Index
)
may
be
so
contrived
that
the
fiduciall
edge
it selfe
(
which
is
best
to
be
a
fine
threed
)
may
be
also
the
proper
Index
,
and
the
Zodiac
may
be
described
upon
the
Diall-moving-plate
,
closely
contrived
to
the
threed
.
Or
thus
at
least
,
The
lenght
of
the
whole
Zodiac
may
be
so
limited
as
is
before
mentioned
,
and
then
if
you
desire
to
have
it
drawn
upon
some
other
place
of
the
plain
upon
which
the
Diall
superficies
moveth
,
and
the
same
Diall
plate
to
carry
the
peculiar
Index
for
the
Zodiac
;
then
(
I
say
)
by
these
two
terms
prefixed
to
the
length
of
the
Zodiac
(
by
the
fiduciall
edge
of
the
Index
)
having
reference
to
the
Diall-moving-plate
(
as
before
)
you
may
determine
the
length
of
the
same
Zodiac
upon
the
standing
plate
whereon
the
Diall
plate
moveth
,
for
the
new
Zodiacall
Index
set
fast
into
the
moving
plate
,
will
(
upon
the
standing
plate
)
shew
the
terms
of
the
same
Zodiac
suting
with
the
terms
upon
the
moving
plate
fitted
to
the
standing
Index
.
This
will
be
direction
enough
because
one
way
may
serve
as
well
as
many
.
Yet
I
doubt
not
to
set
down
many
more
which
would
be
without
end
,
if
they
were
all
put
together
,
for
inventions
in
this
kinde
would
be
too
too
many
.
2.
If
the
Index
be
thought
best
to
move
,
then
must
a
pattern
of
it
(
that
is
some
threed
)
be
first
set
up
.
And
to
this
vice-Index
must
the
Diall
be
described
upon
the
plain
(
or
whatever
curved
Superficies
it
be
.
)
And
the
Zodiac
is
best
to
be
fitted
either
upon
the
plain
whereon
the
foot
of
the
Index
is
to
move
(
for
a
plain
it
must
move
upon
,
whatever
be
the
Superficies
whereon
the
Diall
is
described
)
and
that
same
foot
to
carry
the
Zodiacall
Index
upon
it
;
or
else
upon
the
foot
of
the
Index
,
and
the
Zodiacall
or
peculiar
Index
to
stand
fixt
upon
the
plain
(
or
Diall
superficies
)
And
then
when
all
this
is
done
,
the
vice-Index
must
be
taken
away
and
a
true
substantiall
Index
put
in
the
place
of
it
.
And
for
that
purpose
,
you
were
best
,
before
you
take
it
away
,
to
finde
at
what
elevation
it
stood
from
the
plain
,
and
likewise
to
draw
the
perpendicular
just
under
it
.
For
by
these
two
helps
you
may
set
up
a
true
substantiall
Index
,
regulating
it
thereby
into
the
same
position
;
which
to
do
,
must
be
left
to
the
judgement
of
every
man
to
contrive
as
he
shall
see
requisite
.
But
in
both
these
wayes
care
must
be
had
what
inclination
the
Zodiacall
Scale
hath
to
the
Index
,
according
to
what
is
said
in
the
13
Prop.
3.
It
must
here
again
be
taken
for
a
Rule
,
thot
a
plain
,
if
it
lie
parallel
to
the
index
is
not
possibly
capable
of
these
kindes
of
Ellipticall
houres
,
but
the
Index
must
have
some
inclination
to
the
plain
,
and
make
some
angle
with
the
same
.
4.
All
these
Precepts
do
serve
to
make
such
Dials
as
are
upon
fixed
walls
or
Superficies
that
cannot
be
removed
.
But
if
it
be
to
be
done
for
moveable
Bodies
,
if
they
be
regularly
cut
,
then
the
declinations
of
every
of
the
plains
are
known
by
their
regularity
;
so
that
such
plains
of
them
as
are
capable
of
any
Diall
before
mentioned
(
for
the
Index
and
the
plain
must
not
be
parallel
in
any
case
,
they
may
be
perpendicular
)
may
have
such
Dials
upon
them
as
they
are
capable
of
,
either
to
Indexes
perpendicular
to
their
plains
,
or
to
Indexes
lying
in
the
Zenith
line
of
the
place
.
But
for
these
casuall
Indexes
upon
them
,
you
will
be
put
to
it
,
to
know
what
declinations
they
have
from
the
Meridian
upon
the
regular
body
(
for
I
suppose
that
there
is
a
Meridian
drawn
upon
some
one
of
the
plains
or
other
.
)
The
way
that
for
the
present
I
think
upon
is
this
.
Set
the
Body
upon
the
foot
true
in
respect
of
horizontall
or
upright
position
,
that
is
,
let
the
foot
of
it
stand
upon
some
just
horizontall
plain
,
though
in
respect
of
declination
it
stand
at
all
adventures
.
Then
from
the
fiduciall
edge
of
your
casuall
Index
,
let
fall
a
perpendicular
(
I
mean
perpendicular
to
the
Horizon
of
the
place
,
not
to
the
plain
)
and
to
that
perpendicular
point
,
from
the
point
of
the
Indexes
concourse
with
the
plain
,
draw
a
right
line
;
this
line
shall
represent
the
Azimuth
wherein
the
said
Index
lyeth
.
Then
again
,
with
your
eye
repose
a
perpendicular
hanging
threed
upon
the
two
points
(
namely
,
the
center
or
concourse
of
the
Index
with
the
plain
,
and
the
late
found
perpendicular
point
)
both
together
,
and
project
the
umbrage
of
the
threed
so
hanging
upon
the
Horizontall
plain
.
So
also
you
must
repose
the
shadow
of
some
one
of
the
Cocks
or
Axes
with
your
eye
,
upon
the
Meridian
or
line
of
12
,
and
then
also
the
perpendicular
threed
must
be
projected
upon
them
both
joyned
before
into
one
,
so
that
Threed
,
Stile
,
and
Meridian
must
now
be
all
as
one
line
;
and
the
threed
so
hanging
project
the
shadow
of
it
upon
the
same
Horizontall
plain
.
Now
these
two
lines
thus
projected
either
do
concur
and
make
an
angle
,
or
else
(
by
drawing
a
parallel
to
one
of
them
through
the
other
)
they
must
be
made
to
concur
,
and
looke
what
angle
they
make
at
their
concurrence
,
the
same
being
counted
the
right
way
is
the
angle
of
the
casuall
Indexes
declination
from
the
Meridian
line
.
Thus
the
Declination
is
found
.
Then
the
Reclination
may
be
found
as
before
is
declared
,
and
by
help
of
them
the
Longitude
and
Latitude
of
the
Index
may
be
had
,
and
so
the
Diall
made
by
the
precedent
Precepts
.
A
briefe
DEMONSTRATION
of
the
7th
.
and
8th
.
Sections
.
IN
the
7
§
all
are
made
to
the
Zenith
line
of
the
place
.
If
therefore
you
imagine
an
Horizontall
Ellipsis
to
be
described
to
that
Zenith
line
,
and
upon
that
Ellipsis
a
kinde
of
compressed
Cylinder
to
rise
upright
,
parallel
to
the
said
Zenith
line
,
with
houre
lines
raised
from
the
severall
points
of
the
Horizontall
Ellipsis
,
then
will
the
Zenith
line
shew
the
houre
upon
those
surgent
lines
:
and
the
same
Index
(
or
Zenith
line
)
must
shew
the
houre
among
tho●e
upright
lines
in
the
compressed
Cylinder
,
by
the
same
reason
that
it
doth
upon
the
Ellipsis
it selfe
,
upon
which
the
said
Cylinder
is
raised
,
And
if
so
,
then
it
matters
not
what
part
of
these
lines
is
taken
in
for
this
use
,
since
any
one
point
of
them
will
serve
to
do
the
work
.
The
way
therefore
that
is
used
for
the
projecting
of
these
houres
(
by
help
of
the
Table
in
29
Pag.
)
as
it
will
serve
to
finde
the
Ellipticall
points
upon
the
Horizontall
plain
it selfe
,
so
must
it
serve
to
finde
some
one
point
of
these
surgent
houre
lines
upon
any
other
Superficies
,
because
they
keepe
the
same
distances
alwayes
from
the
Zenith
line
(
from
whence
the
projection
of
them
is
made
,
and
whereon
they
depend
altogether
)
and
therefore
it
findes
(
upon
any
Superficies
)
the
points
of
those
lines
which
passe
through
,
or
do
intersect
the
Diall
Superficies
.
Therefore
it
is
that
these
Ellipticall
points
(
upon
all
Superficies
)
are
points
of
those
surgent
lines
rising
from
the
Horizontall
plain
,
and
that
the
Zenith
line
must
have
the
same
relation
and
situation
to
them
,
that
it
had
to
the
points
upon
the
Horizontall
plain
it selfe
,
And
consequently
,
that
this
way
must
be
of
the
same
truth
upon
any
Superficies
that
it
is
upon
the
Horizontall
.
So
much
for
any
upright
Index
,
set
upon
all
sorts
of
Superficies
,
with
the
houres
thereon
depending
.
In
the
8
§
.
are
handled
casuall
Indexes
;
which
if
well
considered
will
fall
to
be
the
same
with
them
in
the
7
§
.
For
the
Table
of
Angles
and
Altitudes
by
which
they
are
made
,
are
calculated
to
that
Horizon
which
is
proper
to
the
Index
,
considered
as
a
Zenith
line
.
And
so
from
that
Horizon
we
may
imagine
the
like
surgent
lines
to
rise
all
parallel
to
the
Index
or
Zenith
line
,
and
the
very
same
reason
to
hold
in
these
,
that
held
in
the
former
.
To
the
Reader
that
shall
have
the
view
of
this
first
draught
of
Precepts
.
THese
Rules
here
given
may
seeme
to
be
stuffed
with
many
impertinencies
,
and
some
need
lesse
difficulties
,
which
the
Author
acknowledgeth
wil●ingly
,
and
excuseth
,
by
reason
that
they
were
his
first
med●tat●ons
in
th●s
kinde
;
and
so
much
the
more
undigested
by
how
much
the
lesse
pra●●●se
hath
been
by
him
used
therein
.
The
truth
is
,
he
never
described
any
thing
sutable
to
the
Cases
of
these
two
last
Sections
.
And
if
the
Reader
be
any
way
able
to
discern
what
it
is
to
write
upon
a
Mathematicall
Subject
wherein
hath
preceded
no
reall
●epresentation
,
he
will
not
only
excused
difficulties
and
impert●nencies
in
the
trad●tion
,
but
will
wonder
if
there
be
not
some
miscarriages
in
point
of
truth
:
of
which
(
notwithstanding
)
the
Author
is
confident
this
Treatise
is
clear
.
S.
Foster
.
15.
Having
placed
a
Diall
plate
to
move
(
let
the
Coast
of
the
motion
be
casual●
)
how
to
fit
a
stedfast
Index
to
it
,
and
to
describe
an
Ellipticall
Diall
upon
the
said
moving
plain
.
THis
Case
was
forgotten
before
,
but
now
here
supplyed
.
You
must
observe
the
line
(
upon
the
immoveable
plate
)
which
the
midst
of
the
moving
plate
doth
describe
,
and
suppose
that
line
to
be
the
proper
Meridian
.
Then
from
some
convenient
point
of
that
proper
Meridian
,
raise
a
true
Axis
.
Project
the
Axis
upon
the
proper
Meridian
,
and
from
any
point
of
the
said
Meridian
raise
a
threed
(
or
Index
)
any
where
,
only
so
as
that
both
this
Threed
,
the
Axis
,
and
the
Proper
Meridian
may
all
three
appear
in
one
line
.
There
six
the
threed
,
and
then
finde
the
Longitude
and
Latitude
of
it
,
and
afterwards
describe
the
Diall
to
it
,
according
to
the
rules
given
before
.
16.
If
an
Index
should
be
set
up
and
made
moveable
upon
a
standing
plain
,
there
can
no
Diall
be
described
thereto
,
UNlesse
the
Index
be
made
of
wire
or
some
such
bending
substance
,
but
to
such
there
may
.
For
if
you
observe
what
streight
line
the
foot
makes
in
its
motion
,
you
must
count
that
as
the
proper
Meridian
,
and
so
setting
up
an
Axis
to
some
point
of
it
,
you
may
put
in
an
Index
into
that
foot
,
so
as
the
Axis
,
and
fiduciall
edge
of
the
Index
,
and
the
Proper
Meridian
,
may
all
three
appear
in
one
line
.
And
then
finish
your
work
as
is
before
directed
.
FINIS
.
CIRCULAR
Horologiography
.
SHEWING
,
How
to
make
an
Horizontall
Diall
in
a
Circle
equally
divided
,
to
shew
the
Hour
of
the
day
,
and
Azimuth
of
the
Sun.
Invented
and
written
by
Mr.
SAMVEL
FOSTER
,
Late
professor
of
Astronomie
in
Gresham-Colledge
.
London
,
Printed
for
Nicholas
Bourn
.
1654.
CIRCULAR
Horologiography
.
How
to
make
an
Horizontall
Diall
in
a
Circle
equally
divided
,
to
shew
the
Houre
of
the
day
,
and
Azimuth
of
the
Sun.
HEre
before
we
have
had
ELLIPTICAL
HOROLOGIOGRAPHY
,
now
shall
follow
CIRCULAR
HOROLOGIOGRAPHY
,
which
sheweth
how
to
make
a
Diall
in
a
perfect
Circle
equally
divided
into
houres
,
(
whereby
to
finde
the
houre
)
upon
any
plain
whatsoever
.
Divide
a
Circle
into
24
equall
parts
,
and
take
so
many
of
them
as
your
Horizon
hath
houres
in
the
longest
day
,
or
rather
so
many
as
the
degrees
of
your
greatest
Amplitude
(
East
and
West
)
from
the
South
do
arise
unto
;
which
here
at
London
will
be
neer
18
of
them
.
Then
divide
each
of
these
parts
into
15
,
which
for
the
Azimuth
will
signifie
degrees
,
for
the
houre
will
stand
for
four
minutes
of
time
apiece
.
The
altitude
of
the
Horary
Index
may
thus
be
found
.
Adde
your
Latitude
to
90
gr
.
halfe
that
summe
is
the
elevation
of
the
Horary
Index
above
the
Horizon
.
Thus
at
London
,
45
min.
which
is
the
altitude
or
elevation
required
.
Or
,
adde
halfe
the
complement
of
your
Latitude
(
which
is
19¼
)
to
your
Latitude
(
51½
)
the
sum
(
70¾
)
is
the
elevation
of
the
Index
.
The
standing
and
looking
of
it
.
It
must
stand
right
over
the
line
of
12
,
elevated
above
the
said
line
70¼
gr
.
and
must
looke
toward
(
but
not
into
)
the
North
Pole.
The
motion
of
the
Horary
Index
.
It
must
move
to
and
fro
,
directly
over
the
line
of
12.
Or
else
the
houres
must
move
to
and
from
it
,
according
to
the
line
of
12
,
so
as
that
the
same
line
may
alwayes
lie
under
the
foot
of
the
said
Index
.
The
motion
of
the
one
or
the
other
is
necessary
,
because
else
the
Circle
of
equall
parts
can
never
shew
the
true
houre
all
the
year
long
.
How
the
Zodiac
is
to
be
limited
,
and
laid
,
and
charactred
.
The
motion
of
the
Index
must
be
regulated
by
the
Zodiac
.
The
Zodiac
therefore
must
lie
either
in
,
or
else
parallel
to
the
Meridian
line
.
The
length
of
it
is
thus
to
be
limited
.
Count
the
Semidiameter
of
your
Circle
(
viz.
A
B
)
for
the
Radius
:
to
that
Radius
,
Either
1.
Make
the
Sine
of
70¾
(
taken
to
the
Radius
of
your
Circle
)
a
Tangent
of
70¾
;
and
to
the
Radius
of
that
Tangent
finde
the
Secant
of
19¼
(
the
complement
of
70¾
)
that
length
shall
be
the
Radius
of
the
degrees
,
or
the
Decimall
of
the
Tangents
of
the
Zodiac
,
to
be
inserted
by
the
Tables
Pag.
4
,
5
,
6
,
and
7.
2.
Or
else
,
Make
the
Sine
of
19¼
(
estimated
to
the
Semidiameter
of
the
Circle
A
B
as
Radius
)
a
Radius
,
and
to
that
Radius
finde
the
Secant
of
19
1
,
this
last
length
or
Secant
shall
be
the
quantity
of
the
Tangent
of
45
gr
.
or
of
the
Decimall
Scale
by
which
the
numbers
Pag.
4
,
5
,
&c.
are
to
be
inserted
.
And
in
these
Northern
Horizons
♋
and
♈
must
be
placed
so
that
12
may
be
neerest
to
the
Index
in
Summer
,
and
furthest
off
in
Winter
.
The
manner
how
to
fashion
the
Cock
which
holdeth
the
Index
.
The
fashion
may
be
seen
by
the
figure
A
C
D.
At
A
is
the
place
of
the
fiduciall
point
of
the
foot
of
the
Index
to
be
assigned
:
and
in
that
point
a
hole
must
be
pierced
,
and
a
threed
fixed
.
Then
the
Cock
must
have
two
holes
more
pierced
,
one
at
C
,
the
other
at
D
,
both
to
stand
perpendicularly
over
the
line
of
12.
That
at
C
must
be
so
placed
,
that
the
angle
C
A
B
may
be
70¾
gr
.
That
at
D
must
be
placed
so
,
that
D
A
B
may
be
an
angle
of
90
gr
.
How
to
place
this
Diall
for
use
.
YOu
must
either
fix
your
Diall
plate
in
the
Meridian
line
,
and
truly
Horizontall
▪
or
else
upon
an
Horizontall
or
levell
flat
you
must
draw
a
Meridan
line
,
whereby
to
place
it
upon
any
occasion
.
Then
,
To
finde
the
Houre
.
Make
use
of
the
Index
A
C
,
and
rectifie
the
foot
of
the
Index
to
the
requisite
place
in
the
Zodiac
,
(
either
to
the
day
of
the
moneth
,
or
the
degree
of
the
Signe
.
)
When
it
is
thus
rectified
and
set
,
the
shadow
of
the
threed
A
C
will
shew
the
houre
of
the
day
.
To
finde
the
Azimuth
.
Put
the
threed
from
A
to
D
,
and
let
A
D
be
your
Index
.
Then
(
alwayes
)
place
the
foot
of
the
threed
in
the
center
of
the
Circle
at
A
,
So
shall
the
threed
A
D
give
the
Azimuth
.
¶
Note
,
That
the
Scales
of
Declination
,
Amplitude
,
Ascensionall
difference
,
may
be
placed
by
the
Zodiac
,
and
used
as
is
before
shewed
in
the
Ellipticall
Dials
.
Another
way
to
make
the
same
Horizontall
Diall
equally
divided
,
to
finde
the
Houre
and
Azimuth
.
THe
former
way
maketh
the
Index
to
point
up
neer
to
the
elevated
Pole.
This
other
will
require
the
Index
to
point
towards
the
contrary
Coast
,
namely
,
that
of
Noon
or
Mid
day
,
or
toward
(
but
not
into
)
that
Coast
of
the
heavens
where
is
the
depressed
Pole
under
the
Horizon
,
but
still
the
Index
is
to
be
above
the
Horizon
.
The
altitude
of
the
horary
Index
above
the
Horizon
,
must
be
halfe
the
complement
of
the
Latitude
,
or
halfe
the
height
of
the
Equinoctiall
.
Here
at
London
19¼
gr
.
which
is
the
complement
of
the
former
wayes
elevation
70¾
T●e
standing
and
looking
of
it
,
must
be
right
over
the
line
of
12
,
drawn
out
at
competent
length
,
and
must
justly
point
into
the
Coast
of
12
mid
day
:
Or
towards
(
but
not
into
)
the
South
(
or
depressed
)
Pole.
It
must
move
as
the
former
did
.
The
Zodiac
is
to
be
laid
as
before
,
and
to
be
limited
and
charactered
thus
.
The
Semidiameter
of
your
Circle
being
taken
for
Radius
,
you
must
use
the
two
former
rules
in
this
manner
.
1.
Either
make
the
Sine
of
19¼
a
Tangent
of
19¼
,
and
to
the
Radius
of
that
Tangent
finde
the
Secant
of
(
the
complement
of
19¼
,
namely
)
70¾
,
that
length
shall
be
the
Radius
of
the
degrees
,
or
else
the
Decimall
Scale
of
the
Tangents
of
the
Zodiac
,
and
they
must
be
inserted
by
the
Tables
Pag.
4
,
5
,
&c.
2.
Or
else
,
make
the
Sine
of
70¾
(
estimated
to
the
Semidiameter
of
the
Circle
A
B
as
to
a
Radius
)
to
be
a
Radius
;
and
to
that
Radius
finde
the
Secant
of
70¾
,
this
last
length
or
Secant
,
shall
be
the
length
of
the
Tangent
of
45
gr
.
or
of
the
Decimall
Scale
,
by
which
the
numbers
in
the
Tables
Pag.
4
,
5
,
&c.
are
to
be
inserted
.
By
this
instance
for
the
Horizon
of
London
,
where
the
numbers
19¼
and
70¾
are
proper
,
may
the
like
work
be
performed
in
other
places
of
different
Latitudes
from
London
,
only
by
altering
the
numbers
,
according
as
the
proper
Latitudes
of
such
places
shall
require
.
And
in
all
Northern
Horizons
♋
(
or
the
longest
dayes
)
must
be
so
placed
that
the
point
of
12
in
the
Circle
may
then
be
neerest
to
the
Index
,
and
in
♑
(
or
the
shortest
dayes
)
it
may
be
furthest
of
.
But
in
Southern
Latitudes
,
the
places
of
♋
and
♑
are
quite
contrary
,
because
there
the
Sun
being
in
♑
makes
the
longest
day
,
and
in
♋
the
shortest
.
The
manner
of
fashioning
the
Index
,
of
placing
the
Diall
,
and
using
it
,
is
the
same
as
was
shewed
before
.
How
to
make
the
like
Houres
and
Azimuths
by
an
equally
divided
Circle
,
upon
all
Plains
whatsoever
.
YOU
are
first
to
finde
these
three
things
.
1.
How
much
,
and
which
Pole
is
elevated
above
your
plain
,
which
is
equall
to
the
Latitude
of
your
plain
.
2.
The
position
of
the
(
usuall
substylar
or
)
proper
Meridian
of
the
plain
.
3.
The
difference
of
the
Plains
Longitude
from
your
own
,
or
the
angle
at
the
Pole
made
between
yours
and
the
plains
Meridian
.
And
when
these
things
are
had
,
the
work
will
be
like
that
which
was
before
.
For
the
way
will
be
twofold
▪
as
there
it
also
was
.
The
first
way
.
Take
halfe
the
complement
of
the
plains
Latitude
,
and
adde
it
to
the
plains
Latitude
it selfe
,
the
summe
will
give
the
elevation
of
the
Index
above
the
plain
.
This
Index
is
to
point
towards
(
but
not
into
)
that
Pole
which
is
elevated
above
the
plain
.
The
other
circumstantiall
things
about
the
Index
are
as
those
before
,
if
insteed
of
the
line
of
12
there
mentioned
,
we
here
take
the
proper
Meridian
(
or
Substilar
drawn
in
its
right
position
)
of
the
plain
.
The
limitation
of
the
Zodiac
must
be
somewhat
as
before
.
That
is
,
you
must
count
the
Semidiameter
of
your
Diall
Circle
to
be
as
the
Radius
.
And
to
that
Radius
,
1.
Either
finde
the
Sine
of
the
elevation
of
the
Index
,
and
make
that
Sine
a
Tangent
of
the
same
elevation
of
the
Index
;
then
to
the
Radius
of
this
Tangent
finde
the
Co-secant
of
the
former
elevation
,
the
same
Co-secant
shall
be
the
Radius
or
Decimall
Scale
whereby
to
insert
those
dayes
or
Ellipticall
degrees
set
down
in
the
Tables
Pag.
4
,
5
,
&c.
2.
Or
else
to
that
same
Radius
of
the
Circle
,
finde
the
Co-sine
of
the
elevation
of
the
Index
,
and
to
this
being
made
a
Radius
,
finde
the
Co-secant
of
the
Indexes
elevation
,
and
so
shall
this
Co-secant
be
the
Radius
or
Decimall
Scale
as
before
.
The
second
way
.
The
altitude
of
the
Index
above
the
plain
is
the
halfe
complement
of
the
plains
Latitude
.
The
Index
must
(
in
this
way
)
point
towards
(
but
not
into
)
that
Pole
which
is
depressed
under
the
plain
(
I
mean
toward
that
Coast
of
the
world
,
but
must
still
remain
above
the
plain
.
)
The
standing
and
motion
of
the
Index
,
with
the
charactering
of
the
Zodiac
,
and
other
circumstantiall
matters
,
as
of
the
Cock
for
one
Index
,
and
the
usage
for
the
houre
,
must
be
agreeable
to
the
former
rules
given
before
.
The
limitation
of
the
Zodiac
must
be
done
by
the
two
former
rules
given
(
in
generall
terms
)
in
this
Page
.
How
to
divide
the
Circle
of
houres
.
It
must
be
divided
into
24
equall
parts
,
and
those
of
the
24
taken
into
use
as
will
be
serviceable
,
the
rest
left
out
.
Every
of
these
24
parts
may
be
divided
into
15
gr
.
or
into
quarters
and
halfe
quarters
(
for
here
in
such
plains
there
will
be
no
Azimuth
shewed
for
the
place
,
nor
well
for
the
plain
it selfe
,
and
therefore
it
is
best
to
omit
it
in
these
altogether
.
)
All
the
doubt
will
be
where
you
must
begin
your
division
.
From
the
intersection
of
the
proper
Meridian
(
of
the
plain
)
with
the
Circle
,
set
off
(
upon
the
same
Circle
,
and
to
the
true
Coast
)
an
arke
equall
to
the
plains
difference
of
Longitude
.
And
from
that
point
so
inserted
,
(
which
you
must
suppose
to
be
the
point
of
12
)
you
are
to
begin
the
division
of
your
Circle
.
The
numbring
of
the
houres
on
both
sides
12
will
not
be
hard
to
finde
,
in
respect
of
their
course
,
for
the
course
of
the
shadow
of
the
Index
will
give
that
.
NOTE
1.
In
all
Polar
plains
(
such
as
our
upright
East
and
West
,
&c.
)
the
Index
in
both
these
cases
or
wayes
(
here
before
given
)
will
be
elevated
45
gr
.
and
the
Radius
of
the
Diall
Circle
will
be
the
Radius
or
Decimall
Scale
by
which
the
Zodiac
Scale
is
to
be
inserted
out
of
the
Tables
,
Pag.
4
,
5
,
&c.
2.
If
at
any
time
the
Zodiac
prove
longer
than
the
Diameter
of
the
Horologicall
Circle
,
and
so
the
Index
do
go
without
the
Circle
;
at
such
time
as
it
is
without
,
the
shadow
of
the
Index
will
go
one
way
till
it
touch
the
Circle
,
and
then
back
again
the
same
way
it
formerly
came
.
A
briefe
DEMONSTRATION
of
these
Circular
wayes
of
making
DIALS
.
THe
Circular
wayes
have
dependance
,
and
are
deduced
out
of
the
precedent
Ellipticall
wayes
;
and
the
Cases
are
but
two
,
wherein
the
same
may
be
done
upon
any
Horizon
or
plain
,
as
may
be
perceived
by
the
former
Precepts
,
wherein
the
altitudes
of
the
Indexes
above
the
plain
are
ever
made
to
be
either
halfe
the
complement
of
the
Plains
Latitude
,
or
else
(
the
summe
of
the
Latitude
and
that
halfe
complement
,
which
summe
is
equall
to
)
the
complement
of
that
forenamed
halfe
complement
.
According
to
these
two
cases
here
are
two
Schemes
fitted
.
The
projections
are
made
upon
the
plain
of
the
Meridian
Circle
P
Z
H
X.
P
is
the
Pole
,
Z
the
Zenith
of
the
place
,
H
A
the
Horizon
of
the
place
,
Ae
A
the
Equinoctiall
Circle
,
D
A
and
R
A
are
drawn
in
the
middle
of
Z
P
and
Ae
H
or
Ae
O.
D
B
C
X
represents
one
Azimuth
or
Verticall
Circle
proper
to
the
plain
R
A
,
and
by
that
one
all
the
rest
of
them
may
be
understood
.
D
A
represents
the
Index
in
both
the
former
Cases
,
and
R
A
the
Horizontall
plain
(
not
of
the
place
but
that
Horizontall
plai●
which
is
)
properly
belonging
to
the
Index
or
Zenith
line
D
A.
The
first
Scheme
shews
the
first
of
the
two
former
Cases
,
the
second
shews
the
second
of
them
.
Their
Demonstrations
will
be
both
one
.
For
the
question
in
them
both
is
,
How
to
these
inclinations
of
the
Index
,
the
Diall
,
or
horary
line
,
that
falls
upon
the
Horizontall
plain
of
the
place
,
comes
to
be
a
perfect
Circle
.
First
,
therefore
consider
,
that
in
all
the
Ellipticall
Dials
,
the
horary
line
hath
relation
to
the
Equinoctiall
Circle
,
and
to
the
Index
or
line
that
is
to
give
the
shadow
(
whatsoever
the
Superficies
be
upon
which
the
projecture
is
made
,
whether
plain
or
otherwise
it
matters
not
,
as
appears
before
,
but
here
we
deale
only
with
plains
,
because
Ellipses
and
Circles
are
plain
Figures
.
Secondly
,
consider
,
that
through
the
severall
houre
points
of
the
Equinoctiall
Circle
,
right
lines
are
to
be
supposed
to
passe
infinitely
extended
,
till
they
meet
with
the
plain
whereon
the
projecture
is
to
be
designed
.
Which
issuing
out
of
these
infinite
lines
(
if
they
be
regularly
cloathed
about
with
an
inflected
Superficies
must
comprehend
a
Cylindricall
concave
,
either
round
or
compressed
according
as
the
forenamed
infinite
lines
are
either
perpendicular
to
the
plain
of
the
Equinoctiall
,
or
not
perpendicular
to
it
.
Thirdly
,
These
parallel
lines
so
drawn
through
the
houre
points
of
the
Equinoctiall
,
parallel
to
the
Index
,
must
all
fall
perpendicular
to
the
Horizontall
plain
which
is
properly
belonging
to
the
Index
as
a
Zenith
line
.
And
the
same
lines
upon
that
proper
Horizon
do
alwayes
make
an
Ellipsis
,
except
only
these
two
Cases
.
First
,
If
the
Horizon
be
the
same
with
the
Equinoctiall
Circle
,
and
the
Zenith
line
or
Index
the
same
with
the
Axis
of
the
world
,
for
then
those
lines
do
there
make
a
perfect
Circle
.
Secondly
,
If
the
Horizon
be
a
Polar
or
Meridian
Horizon
,
and
the
Index
or
Zenith
line
fall
into
the
Equinoctiall
(
having
no
Latitude
from
it
)
for
then
those
lines
do
all
fall
into
(
or
coincide
with
)
the
plain
of
the
Equinoctiall
,
and
consequently
do
make
(
upon
their
proper
Horizon
)
a
direct
right
line
:
●ea
more
,
because
the
said
lines
do
lie
all
upon
the
plain
of
the
Equinoctiall
,
and
are
all
drawn
out
of
the
equall
divisions
of
the
Equinoctiall
Circle
,
and
besides
are
all
parallel
one
to
the
other
;
therefore
they
make
(
in
this
case
)
a
double
line
of
Sines
,
as
is
said
before
in
the
fifth
note
Pag.
37.
Now
to
come
to
my
intended
purpose
,
which
is
,
to
prove
〈◊〉
Indexes
so
laid
(
as
is
mentioned
before
in
the
two
precedent
Cases
)
will
require
(
upon
the
Horizontall
plain
of
the
place
)
a
true
Circle
and
not
an
Ellipsis
.
Suppose
therefore
D
A
to
be
the
Index
(
according
to
one
of
the
two
precedent
conditions
)
then
must
R
A
be
the
proper
Horizon
to
that
Index
.
A●d
according
to
the
former
Doctrine
,
if
Ae
,
and
B
,
and
A
,
be
taken
for
three
of
the
houre
points
in
the
Equinoctiall
,
and
Azimuths
drawn
through
them
from
D
the
Verticall
point
of
the
Index
,
then
I
say
(
in
these
two
cases
and
not
otherwise
)
looke
how
the
Equinoctiall
is
divided
by
those
Circles
,
in
the
same
manner
is
H
O
the
Horizon
of
the
place
divided
.
But
because
the
Equinoctiall
is
ever
equally
divided
in
respect
of
houres
,
therefore
the
Circles
from
D
to
X
(
from
the
Zenith
of
the
proper
Horizon
R
A
to
the
Nadir
,
which
are
Azimuthall
Circles
)
shall
cut
the
Horizon
of
the
place
(
H
O
)
into
equall
parts
.
This
,
I
say
,
is
true
,
Because
D
Ae
B
and
X
H
C
are
equall
Sphericall
Triangles
in
three
of
their
quantities
[
For
D
Ae
is
equall
to
X
H
,
and
the
angles
at
D
and
X
are
equall
,
as
also
the
right
angles
at
Ae
and
H
]
and
therefore
equall
in
all
the
rest
,
if
like
be
compared
to
like
:
that
is
,
H
C
is
equall
to
Ae
B
;
and
consequently
A
C
equal
to
A
B
:
so
also
C
X
equall
to
B
D.
And
if
so
,
then
the
Chord
B
C
must
be
parallel
to
the
Index
or
Zenith
line
D
X
,
and
so
Ae
H
is
parallel
to
D
X
,
and
so
contrarily
(
in
these
two
Cases
)
right
lines
drawn
from
the
houre
points
in
the
Equinoctiall
Circle
,
parallel
to
the
Index
,
must
cut
(
upon
the
Horizon
of
the
place
)
equall
parts
,
equall
to
one
another
,
and
equall
to
the
houres
upon
the
Equinoctiall
Circle
.
That
is
to
say
,
they
cut
the
horizontall
Circle
it selfe
into
24
equall
parts
or
hours
.
And
consequently
(
in
these
two
Cases
)
those
forementioned
right
lines
must
designe
(
the
horizontall
Circle
it selfe
,
that
is
)
a
Circle
upon
the
horizontall
plain
of
the
place
,
though
upon
the
proper
Horizon
it selfe
(
R
A
)
they
must
designe
an
Ellipsis
(
falling
perpendicularly
thereon
)
according
to
the
third
observation
in
the
128
Pag.
And
from
that
Ellipsis
these
lines
B
C
,
H
Ae
,
may
be
conceived
as
surgent
lines
(
such
as
are
mentioned
before
Pag.
113
)
containing
a
compressed
Cylinder
,
which
Cylinder
being
cut
two
wayes
,
(
that
is
subcontrarily
)
will
again
reveile
the
originall
Circle
(
at
least
in
one
of
the
two
Sections
)
from
whence
it selfe
and
the
surgent
lines
tooke
their
forme
and
places
.
NOTE
.
1.
The
same
reason
holds
in
all
plains
,
that
in
them
also
there
may
be
found
such
a
Circular
Horologiography
.
For
they
are
Horizons
to
some
place
of
the
World
or
other
,
and
are
therefore
capable
of
the
same
accidents
that
the
forementioned
Horizon
of
a
particular
place
is
.
2.
That
these
Cases
are
the
same
(
in
a
manner
)
with
those
§
.
5.
For
here
a
new
Zenith
line
(
as
D
A
)
is
found
to
H
O
the
Horizon
of
the
place
,
such
a
Zenith
line
(
or
Index
)
as
may
lie
equidistant
from
the
Horizon
of
the
place
H
O
,
and
from
the
Aequinoctiall
Circle
Ae
A
:
And
there
there
is
a
plain
set
up
which
may
stand
so
,
as
that
the
Zenith
line
of
the
place
may
stand
equally
distant
from
the
Aequinoctiall
Circle
and
this
plain
.
In
both
,
the
Index
shands
equally
distant
from
the
Equinoctiall
,
and
from
the
plain
whereon
the
houre
points
are
to
be
inserted
.
For
the
limitation
of
the
Zodiac
.
The
first
work
in
the
limitation
of
it
is
for
the
Ellipsis
that
should
be
described
upon
the
proper
Horizon
,
and
the
ground
of
that
is
the
same
with
what
was
before
delivered
Pag.
75.
The
reason
of
both
will
be
plain
enough
out
of
the
foregoing
Diagrams
.
For
the
Zenith
line
D
A
,
if
it
were
to
move
upon
the
proper
Horizon
R
A
,
it
must
move
in
the
proper
horizontall
Zodiac
,
as
formerly
it
did
,
Pag.
17
,
18.
Which
Zodiac
is
to
be
made
as
before
,
and
here
in
the
first
work
is
shewed
:
the
reason
of
the
division
of
it
is
given
in
Pag.
61
,
62.
Here
is
only
to
be
shewed
the
reason
of
the
enlargement
.
If
the
Index
were
set
upon
the
proper
Horizon
R
A
,
then
it
must
move
thereon
(
perpendicularly
)
from
A
to
S
suppose
.
But
it
moves
not
upon
that
but
upon
H
O
the
horizontall
plain
of
the
place
,
and
yet
to
keepe
the
same
distances
in
motion
from
the
first
position
of
it
in
D
A
X
,
that
is
,
it
must
move
parallel
alwayes
to
D
A
,
but
not
upon
A
S
but
upon
A
H
,
which
to
do
,
it
must
not
have
the
parts
of
the
Zodiac
in
A
S
(
to
guide
upon
H
O
)
but
proportionall
ones
to
them
,
that
is
,
instead
of
A
S
must
be
taken
A
m
;
which
is
all
one
as
to
say
,
As
A
S
the
Radius
,
to
A
m
the
Cosecant
of
S
m
A
,
or
D
A
O
,
the
Indexes
elevation
(
above
the
Horizon
H
O
;
)
So
are
the
parts
of
A
S
the
prime
or
fundamentall
Zodiac
,
to
the
parts
of
A
m
,
the
secondary
or
inlarged
Zodiac
.
Thus
much
of
Circular
Dialling
.
APPENDIX
.
SEE
the
Figure
in
page
120.
To
that
Zodiacs
length
you
may
finde
such
a
Radius
as
shall
be
thereto
justly
competent
so
as
to
make
the
Ellipticall
Horizontall
Diall
R
T
S
;
and
having
made
that
Diall
,
the
upright
Index
D
A
will
give
the
hour
upon
it
,
as
the
slope
Index
A
C
will
give
the
houre
upon
the
Circle
.
And
these
two
being
severall
Indexes
and
Dialls
must
set
themselves
as
the
other
do
,
and
wil
not
stand
right
till
they
agree
.
Onely
this
cannot
be
so
punctuall
,
because
the
two
Indexes
A
C
and
A
D
are
no●
so
far
distan●●s
in
the
other
horizontall
Di●ll
.
But
the
Azimuth
will
here
be
still
shewed
,
as
was
mentioned
before
.
Having
the
Zodiac
limited
,
how
to
finde
the
two
extream
Diameters
of
the
Ellipsis
.
The
length
of
halfe
the
Zodiac
is
alwayes
to
be
esteemed
the
Tangent
of
23½
gr
.
Now
because
we
here
suppose
this
Zodiac
already
made
,
and
to
be
intended
for
an
horizontall
Diall
;
therefore
,
Either
thus
,
1
As
the
Tangent
of
23½
,
is
to
the
Tangent
of
your
Latitude
;
So
the
length
of
the
Semizodiac
upon
your
Diall
,
to
the
Sine
of
your
latitude
,
which
is
the
shorter
Semidiameter
A
T.
Or
else
thus
,
2
As
the
Tangent
of
23½
,
is
to
the
Radius
or
Tangent
of
45
gr
.
So
is
your
known
and
limited
Zodiac
,
to
the
Cosine
of
your
Latitude
.
The
Radius
,
to
which
Cosine
is
the
larger
Semidiameter
A
W.
Thus
having
fitted
your
Ellipsis
(
to
the
Zodiac
)
you
may
describe
and
divide
it
in
such
manner
,
as
was
formerly
shewed
,
pag.
1.
FINIS
.
RECTILINEAL
OR
DIAMETRAL
Horologiography
.
SHEWING
The
manner
how
to
describe
the
Houres
upon
a
finite
right
line
,
or
the
Diameter
of
a
Circle
,
and
to
fit
a
moveable
Index
unto
the
Diall
so
described
.
Invented
and
written
by
Mr.
SAMVEL
FOSTER
,
Late
professor
of
Astronomie
in
Gresham-Colledge
.
RECTILINEAL
OR
DIAMETRAL
Horologiography
.
Concerning
Dials
made
upon
a
finite
streight
line
,
with
a
moveable
Index
.
THis
may
seem
no
difficult
nor
new
thing
,
since
the
most
vulgar
known
way
of
pro●racting
Dials
is
either
by
one
contingent
line
(
which
is
a
right
line
;
)
or
else
by
two
Tangent
lines
,
as
by
the
Sector
is
usually
done
.
But
it
must
be
considered
,
that
the
contingent
line
is
infinite
,
and
this
line
(
here
mentioned
)
is
finite
.
Again
,
that
other
way
by
the
Sector
,
is
not
by
one
finite
line
,
but
by
two
at
the
least
;
whereas
this
is
by
one
finite
line
▪
Some
difference
there
is
.
And
besides
;
that
other
by
Tangents
may
be
as
well
done
upon
a
single
crooked
line
,
this
cannot
.
This
streight
line
is
mentioned
before
,
that
in
right
Horizons
or
Polar
plains
the
Ellipsis
shutreth
up
into
a
streight
line
.
And
it
is
true
,
that
this
falls
out
to
be
so
,
onely
in
right
Horizons
(
if
the
Index
stand
perpendicular
to
the
plain
,
as
is
before
intended
)
which
cases
happen
but
seldome
,
and
to
some
plains
onely
.
But
here
my
purpose
is
to
shew
how
it
may
be
done
to
all
plains
whatsoever
:
observing
only
this
one
condition
;
alwayes
to
set
the
Index
so
,
as
to
point
up
into
the
Equinoctiall
Circle
:
for
so
it
must
do
,
and
will
not
be
done
any
other
way
,
according
to
this
forme
of
casting
the
houres
.
There
are
two
Cases
which
will
offer
themselves
.
The
first
is
,
when
the
Index
pointeth
up
both
into
the
Equinoctiall
Circle
,
and
likewise
into
the
plains
proper
Meridian
Circle
.
The
second
is
,
when
it
points
up
into
the
Equinoctiall
Circle
of
the
heavens
(
for
so
it
must
ever
do
)
and
into
some
other
Meridian
Circle
,
such
as
is
not
proper
to
the
plain
.
¶
1.
How
to
make
a
Diall
upon
a
finite
streight
line
,
drawn
upon
any
plain
,
the
Index
being
so
ordered
that
it
point
up
into
the
Equinoctiall
Circle
,
and
into
the
plains
proper
Meridian
.
IN
this
Case
,
the
Index
is
to
lie
just
over
the
proper
Meridian
of
the
plain
,
and
is
to
be
elevated
so
much
above
that
line
as
the
height
of
the
Equinoctiall
above
that
plain
comes
unto
,
and
must
point
up
into
the
Equinoctiall
Circle
it selfe
.
It
must
therefore
here
be
supposed
that
the
situation
of
the
plain
,
with
such
things
as
are
thereon
depending
,
are
known
.
Such
are
these
that
follow
.
1
The
plains
reclination
or
inclination
and
declination
.
2
The
position
of
the
Substylar
,
or
plains
proper
meridian
3.
The
elevation
of
the
Pole
above
the
plain
.
4.
The
plains
difference
of
Longitude
from
your
Meridian
.
These
things
are
to
be
found
by
the
common
wayes
of
making
of
Dials
upon
plains
,
which
need
not
here
be
repeated
,
but
shall
be
supposed
to
be
already
known
,
and
placed
in
their
true
positions
upon
the
plain
.
Let
A
B
be
the
proper
Meridian
of
the
plain
truly
placed
,
and
A
D
the
Index
standing
over
it
,
and
pointing
up
into
the
Equinoctiall
.
Let
the
degrees
of
the
difference
of
the
plains
longitude
be
C
L
,
and
let
the
point
L
stand
from
C
upon
the
right
Coast
according
to
the
exigence
of
the
plain
.
Then
for
drawing
and
dividing
the
line
that
is
to
give
the
houres
,
do
thus
.
1.
Supposing
A
to
be
the
point
of
the
Equinoctiall
(
in
the
Zodiacall
motion
of
the
Index
)
through
that
point
draw
R
A
S
perpendicular
to
B
A
C
the
proper
Meridian
;
and
upon
A
as
the
center
,
and
to
any
convenient
extent
,
describe
the
Circle
R
C
S
,
in
which
,
C
L
is
to
be
set
according
to
the
difference
of
Longitude
,
as
was
before
mentioned
.
So
that
L
is
the
point
from
whence
you
must
begin
to
divide
your
Circle
,
for
from
L
is
the
point
of
the
houre
12
deduced
.
Therefore
,
2.
Divide
your
Circle
from
L
into
24
houres
at
a
,
b
,
c
,
d
,
e
,
f
,
&c.
l
,
m
,
n
,
o
,
&c.
and
from
each
of
these
points
let
perpendiculars
fall
upon
the
diameter
R
S
,
as
L
m
,
an
,
bp
,
&c.
the
same
Perpendiculars
shall
divide
the
Diameter
into
the
houres
required
:
m
shall
be
the
point
of
12
,
because
it
falls
from
L
,
the
point
of
the
difference
of
Longitude
:
n
and
o
shall
be
the
next
houres
to
12
on
both
sides
:
p
and
q
shall
be
the
second
houre
points
on
both
sides
12
,
and
so
the
rest
in
their
orders
.
But
then
in
these
Dials
,
when
the
difference
of
Longitude
falls
not
just
upon
an
hour
(
as
very
seldome
(
and
then
but
casually
)
it
doth
)
the
line
R
S
will
be
crosse
divided
as
you
see
it
in
this
example
,
for
e
R
u
is
one
hour
,
and
unequally
broken
and
set
upon
the
Diameter
R
S
:
so
t
S
o
is
the
space
of
5
houres
;
and
the
points
t
and
o
stand
crosse
or
distant
one
from
the
other
,
and
are
not
coincident
;
which
I
say
must
alwayes
happen
when
the
difference
of
Longitude
is
not
equall
to
one
or
more
just
full
houres
.
3.
In
these
crosse
houres
(
for
so
I
may
call
them
)
you
may
divide
each
side
of
the
line
severally
,
as
thus
:
Or
else
you
may
make
a
double
line
in
this
manner
,
supposing
the
fiduciall
line
to
be
the
white
that
lies
intire
and
undivided
betwixt
them
.
This
kinde
of
Diall
will
be
somwhat
difficult
in
the
use
,
because
of
this
crosse
division
of
the
houres
.
Yet
I
thought
best
not
to
omit
it
,
because
it
is
the
ground
of
the
next
Case
which
will
be
more
usefull
.
Concerning
the
Zodiac
and
the
Index
.
FOr
the
limitation
of
the
Zodiac
,
let
A
R
or
A
S
be
as
Radius
.
To
that
Radius
let
A
F
be
the
Tangent
of
23½
.
This
Tangent
is
to
be
augmented
according
to
the
proportion
that
the
Radius
bears
to
the
Secant
of
the
complement
of
the
angle
D
A
B
,
which
is
the
elevation
of
the
Equinoctiall
above
the
plain
:
That
is
,
As
the
Radius
,
To
the
Secant
of
the
Poles
elevation
above
the
plain
(
which
is
the
same
with
the
said
complement
;
)
So
is
the
Tangent
of
A
F
,
to
the
line
A
B
suppose
.
Therefore
now
A
B
is
the
length
of
halfe
vour
Zodiac
,
and
must
be
taken
as
a
Tangent
of
23½
gr
.
(
or
as
a
Scale
of
4348
/
10000
equall
parts
)
by
which
you
may
insert
the
Moneths
or
Signes
as
is
before
shewed
,
Pag.
4
,
5
,
&c.
These
must
be
set
on
and
charactered
as
the
nature
of
the
plain
shall
require
them
to
be
ordered
.
The
Index
though
it
move
,
yet
must
alwayes
lie
just
over
the
proper
Meridian
A
B
,
and
must
also
keepe
the
angle
D
A
B
unchanged
,
and
must
be
moved
from
B
to
A
,
and
so
to
B
,
as
the
time
of
the
year
shall
require
.
Or
the
Index
may
stand
and
the
Diall
move
upon
the
line
B
B
,
as
the
manner
of
these
kindes
of
Dials
may
be
,
and
which
have
been
too
often
mentioned
.
The
second
case
follows
.
¶
2.
How
to
make
a
Diall
upon
a
finite
streight
line
,
drawn
upon
any
plain
,
the
Index
being
so
ordered
,
that
it
point
up
into
the
Equinoctiall
Circle
,
and
into
a
just
hour
point
in
the
same
Circle
.
AS
before
,
so
here
,
it
must
be
supposed
that
the
situation
of
the
plain
with
the
other
things
(
before-mentioned
)
which
depend
thereupon
are
known
and
placed
upon
the
plain
.
And
the
matter
must
be
so
ordered
,
that
the
Index
must
both
looke
into
the
Equinoctiall
Circle
,
and
also
it
must
point
to
some
just
hour
therein
,
that
so
one
division
of
the
Diall
line
may
very
well
serve
for
accounting
the
houres
both
wayes
,
the
whole
houres
and
parts
upon
one
side
justly
corresponding
to
the
just
houres
and
parts
upon
the
other
side
.
This
will
be
for
more
expedite
use
than
the
former
way
could
be
,
and
no
such
trouble
nor
confusion
in
the
numeration
of
houres
and
parts
of
houres
,
as
will
appear
in
the
next
Example
.
In
this
Case
that
we
are
now
going
upon
,
we
must
consider
these
things
.
1.
How
the
Index
is
to
be
ordered
,
so
as
that
it
may
look
up
into
some
just
houre
point
of
the
Equinoctiall
?
2.
How
the
line
of
the
Diall
or
houres
(
the
horologicall
line
)
is
to
be
placed
,
and
limited
and
divided
?
3.
How
the
Zodiac
is
to
be
placed
and
limited
?
¶
First
,
for
the
Index
:
The
best
way
will
be
to
finde
what
Declination
and
Reclination
(
in
respect
of
the
Meridian
and
Zenith
of
the
plain
)
the
houre
point
(
into
which
the
Index
must
point
)
hath
.
And
that
is
done
by
these
two
Rules
.
1.
As
the
Radius
,
to
the
Sine
of
the
Equinoctials
altitude
above
the
plain
:
So
is
the
Co-sine
of
the
distance
of
that
houre
point
from
the
Meridian
of
the
plain
,
to
the
Co-sine
of
the
Reclination
;
or
to
the
Sine
of
the
elevation
of
the
Index
above
the
plain
.
2.
As
the
Radius
,
to
the
Sine
of
the
Poles
altitude
above
the
plain
:
So
is
the
Co-tangent
of
the
former
distance
(
of
the
intended
houre
point
from
the
Meridian
of
the
plain
,
)
to
the
Co-tangent
of
the
declination
of
the
Index
from
the
Meridian
of
the
plain
.
Or
thus
,
2.
As
the
Sine
of
the
Poles
altitude
above
the
plain
,
to
the
Radius
,
So
is
the
Tangent
of
the
former
distance
,
to
the
Tangent
of
the
declination
of
the
Index
from
the
plains
Meridian
.
Thus
,
]
suppose
A
B
were
the
proper
Meridian
of
the
plain
,
make
the
angle
B
A
C
on
the
right
Coast
,
and
equall
to
the
declination
of
the
Index
(
now
found
)
and
elevate
A
D
just
perpendicularly
over
A
C
,
to
the
angle
D
A
C
equall
to
the
elevation
of
the
Index
above
the
plain
,
and
so
you
shall
place
the
Index
in
a
just
position
,
that
it
shall
looke
up
into
a
just
houre
point
of
the
Equinoctiall
.
That
is
the
first
thing
.
¶
Secondly
.
The
Horologicall
line
is
to
be
set
for
position
as
it
was
before
in
the
precedent
Case
,
namely
,
perpendicular
to
the
proper
Meridian
,
as
in
the
former
and
this
Scheme
you
see
R
A
S.
Then
again
,
this
line
which
in
the
former
figure
was
equall
to
R
S
the
Diameter
of
the
Circle
(
or
two
assumed
Radiusses
)
must
now
in
this
case
be
enlarged
,
according
as
you
have
ordered
your
Index
to
wrye
from
the
proper
Meridian
of
the
plain
to
any
other
just
houre
;
as
in
this
figure
it
is
made
to
be
distant
from
that
proper
Meridian
unto
the
houre
of
12
:
that
is
,
it
is
supposed
to
be
turned
so
,
as
to
looke
into
our
12
a
clock
point
in
the
Equinoctiall
:
which
point
of
12
in
the
former
figure
was
noted
at
L
,
and
so
in
this
figure
let
it
be
supposed
again
.
I
say
,
let
it
be
supposed
that
(
because
the
former
Horologicall
line
was
crossely
divided
,
and
therefore
untoward
for
use
)
this
way
that
is
now
spoken
of
shall
be
to
take
away
that
inconvenience
in
use
which
was
in
the
former
.
And
therefore
suppose
the.
Style
or
Index
to
be
wryed
from
the
proper
Meridian
of
the
plain
to
the
Meridian
of
the
place
,
or
to
the
point
of
12.
It
may
as
well
be
put
to
any
other
just
houre
,
as
to
12
,
but
be
it
so
in
this
Example
,
and
let
the
point
be
L
selected
from
all
the
rest
.
From
the
selected
point
L
,
draw
the
diameter
L
A
M
,
and
from
the
selected
point
L
divide
the
Circle
into
24
equall
parts
or
houres
.
Then
from
each
two
houre
points
correspondently
distant
on
both
sides
the
Diameter
L
M
draw
right
lines
,
as
G
h
and
n
O
,
and
G
O
,
which
will
be
all
parallel
to
the
Diameter
L
M.
Draw
these
lines
(
I
say
)
and
continue
them
untill
they
meet
with
the
horologicall
line
R
A
S
,
sufficiently
prolonged
both
wayes
.
By
this
draught
thus
made
,
the
horologicall
line
R
A
S
shall
be
both
prolonged
to
h
O
,
and
also
divided
just
as
a
line
of
Sines
,
so
the
parts
of
it
are
in
the
same
proportion
that
the
parts
of
the
Diameter
G
A
G
are
,
which
are
Right
Sines
both
wayes
from
the
center
,
or
Versed
Sines
from
G
through
A
to
G.
¶
Note
that
here
(
as
formerly
)
you
may
assume
the
Circle
of
what
extent
you
list
,
and
shall
finde
most
fitting
your
purpose
.
It
is
plain
then
by
this
Example
,
First
,
that
the
Radius
of
the
Circle
A
R
,
is
to
be
enlarged
,
according
to
the
secant
of
the
Equinoctiall
distance
between
the
proper
Meridian
of
the
plain
,
and
that
just
hour
(
wha●ever
it
be
)
inro
which
the
Index
is
made
to
point
.
As
you
see
that
arke
to
be
here
L
C
,
or
G
R
,
or
G
S
,
whose
secant
is
A
h
,
or
A
o
,
in
respect
of
the
assumed
Radius
A
R.
And
secondly
,
that
h
o
thus
enlarged
is
divided
like
a
double
line
of
Sines
,
as
it
ought
to
be
.
And
for
numbering
the
houres
,
if
A
be
12
,
and
the
Index
look
toward
the
South
,
then
must
o
be
2
,
and
the
other
o
must
be
6
;
and
the
first
o
,
10
at
night
.
And
h
must
be
9
in
the
morning
;
the
other
h
,
6
in
the
morning
;
and
the
first
h
3
in
the
morning
.
But
if
the
Index
had
not
pointed
up
into
12
at
L
,
but
into
some
other
hour
as
P
,
then
must
a
Diameter
have
been
drawn
at
first
from
P
to
the
center
A
quite
through
,
and
all
the
houres
parallel
to
P
A
,
and
then
the
horologicall
line
would
have
been
thus
numbered
with
7
at
the
two
extreams
,
and
1
in
the
midst
at
A.
And
here
is
one
speciall
thing
to
be
noted
(
as
formerly
Pag.
125
)
but
this
here
is
alwayes
so
,
and
in
all
Oblique
Horizons
.
That
in
our
Climates
where
the
dayes
are
in
Summer
longer
than
12
houres
,
the
shadow
of
the
Index
doth
really
go
forward
and
backward
.
As
to
make
it
plain
.
If
the
Diall
were
described
upon
an
Horizontall
plain
(
for
instance
)
and
it
were
our
longest
day
,
then
the
shadow
of
the
Index
would
begin
at
4
a
clock
(
figured
below
the
line
)
and
would
go
thence
towards
the
left
hand
to
5
,
6
,
and
7.
At
7
it
would
return
back
towards
the
right
hand
through
8
,
9
,
10
,
11
,
12
,
1
,
2
,
3
,
4
,
5
,
6
,
untill
7
,
and
then
being
arrived
at
7
,
the
utmost
point
on
the
right
hand
,
it
will
come
back
again
towards
8
and
9
to
the
left
hand
.
This
for
the
placing
,
limiting
,
and
dividing
of
the
horologicall
line
.
¶
Thirdly
,
Concerning
the
Zodiac
how
it
must
be
placed
and
limited
,
consider
of
these
things
.
The
Zodiac
(
in
this
case
of
wrying
the
Index
from
the
plains
proper
Meridian
)
must
be
augmented
in
length
,
and
changed
in
situation
from
what
it
was
in
the
former
case
where
the
Index
looked
up
into
the
plains
proper
Meridian
.
Yet
I
will
here
first
begin
with
that
way
so
far
as
it
goes
,
and
then
proceed
to
what
is
further
needfull
.
Let
the
Figure
Pag.
137
be
here
repeated
.
To
the
Radius
A
R
,
the
Tangent
of
23
and
a
halfe
is
A
F
,
and
the
same
is
there
augmented
thus
.
As
the
Radius
,
To
the
Secant
of
the
Poles
elevation
above
the
plain
;
So
let
AF
be
supposed
to
be
,
To
AB
.
So
that
if
the
Index
were
pointing
directly
to
the
Meridian
point
(
in
the
Equinoctiall
)
which
is
proper
to
the
plain
,
then
the
businesse
was
at
an
end
;
for
A
B
must
be
the
length
of
the
Zodiac
,
and
standing
square
to
R
A
S.
But
now
we
here
suppose
the
Index
to
be
wryed
from
that
point
,
as
much
as
the
arke
C
L
comes
to
;
which
turning
aside
will
both
lengthen
the
Zodiac
again
,
and
turn
it
out
of
its
perpendicular
standing
to
R
A
S.
Which
thing
is
thus
to
be
effected
.
To
A
F
as
a
Radius
,
finde
the
Tangent
of
the
Poles
elevation
above
the
plain
;
suppose
that
Tangent
were
G
H.
Then
say
,
As
the
Radius
,
to
the
Tangent
of
C
L
the
quantity
of
wrying
,
so
G
H
to
X
Z.
Make
B
K
equall
to
X
Z
,
and
parallel
to
A
R
,
and
draw
A
K
,
so
shall
A
K
be
the
length
,
and
shall
also
give
(
in
this
position
)
the
situation
of
the
Zodiac
.
The
manner
how
to
divide
it
need
not
be
repeated
,
because
it
hath
been
often
set
down
heretofore
,
You
must
only
remember
to
place
this
line
upon
that
Coast
of
A
B
which
is
contrary
to
the
Coast
of
the
Indexes
deviation
or
wrying
from
the
proper
Meridian
point
of
the
plain
:
this
work
may
either
be
protracted
,
or
calculated
,
or
wrought
Instrumentally
,
as
shall
best
be
liked
of
.
Another
way
I
will
here
adde
to
do
the
same
work
,
which
may
perhaps
be
not
inferior
to
the
former
.
Let
all
things
preparatorie
in
the
former
Figure
,
be
here
again
repeated
,
R
A
the
Radius
,
and
A
F
the
Tangent
of
23
degrees
and
a
halfe
to
that
Radius
:
then
to
A
F
as
Radius
,
finde
the
Sine
and
Co-sine
of
the
Poles
elevation
above
the
plain
:
let
F
P
be
the
Sine
,
and
A
P
the
Co-sine
.
Make
A
m
equall
to
A
P
;
and
from
m
draw
the
line
m
n
,
making
the
angle
n
m
C
equall
to
the
angle
B
A
C
Pag.
142
,
and
standing
on
the
contrary
Coast
,
which
is
the
angle
of
the
declination
of
the
Index
from
A
C
in
this
Figure
,
the
proper
Meridian
of
the
plain
.
Then
say
,
As
the
Radius
,
is
to
the
Tangent
of
the
Reclination
of
the
Index
(
which
is
the
Complement
of
D
A
C
(
Pag.
142
)
the
elevation
of
the
Index
above
the
plain
,
)
So
is
the
Sine
F
P
to
a
fourth
quantity
,
suppose
the
same
to
be
D
G
:
make
m
n
equall
to
D
G
,
and
draw
the
streight
line
A
●
,
so
shall
A
n
in
this
Figure
be
equall
to
A
K
in
the
precedent
,
and
the
angle
n
A
C
be
(
here
)
equall
to
the
angle
K
A
B
in
the
former
.
So
that
A
n
must
be
the
length
of
the
halfe
Zodiac
,
and
will
lie
in
its
true
position
by
this
work
.
This
may
be
done
either
by
calculation
,
protraction
,
or
Instrument
,
as
all
other
works
may
also
be
.
Care
must
alwayes
be
had
to
place
these
Zodiacs
on
the
right
Coast
from
F
A
C
,
and
that
the
Zodiac
be
rightly
divided
and
charactred
according
to
the
Seasons
of
the
year
,
as
they
shall
have
respect
to
the
nature
of
the
plain
.
Or
thirdly
,
it
may
be
done
thus
.
First
,
1.
As
the
Radius
,
To
the
bine
of
the
Poles
elevation
above
the
plain
;
So
the
Tangent
of
the
Equinoctiall
discession
or
wrying
,
To
the
Tangent
of
the
Declination
of
the
Zodiac
;
which
is
the
angle
n
A
C
in
this
Figure
,
o●
B
A
K
in
the
former
.
2.
As
the
Co-sine
of
the
discession
,
or
Equinoctiall
wrying
,
Is
to
the
Radius
;
So
is
the
Tangent
of
the
Poles
elevation
above
the
plain
,
To
the
Tangent
of
another
Arke
.
3.
And
,
As
the
Radius
,
Is
to
the
Secant
of
that
arke
(
or
as
the
Co-sine
of
that
arke
to
the
Radius
;
)
So
is
the
Tangent
of
23½
gr
.
To
a
number
.
If
now
you
make
the
Radius
of
the
Circle
first
assumed
,
namely
R
A
or
A
S
to
be
a
Decimall
scale
,
and
out
of
that
Decimall
scale
do
take
this
number
,
the
same
shall
give
the
length
of
the
Zodiac
A
n
in
this
figure
,
or
A
K
in
the
former
.
The
division
of
this
Zodiac
now
it
is
thus
placed
and
limited
,
must
be
done
by
the
Tables
and
directions
in
the
4
and
5
Pages
.
For
A
n
must
be
taken
as
a
Tangent
of
23½
gr
.
or
as
the
number
4348
of
a
Decimall
scale
of
10000
parts
.
Thus
much
for
this
kinde
of
Diall
upon
a
streight
line
,
both
crossely
and
evenly
divided
.
A
DEMONSTRATION
of
the
former
way
.
IN
Pag.
28
,
Note
5
,
there
is
mention
made
of
Polar
plains
or
right
Horizons
,
that
in
them
,
the
Ellipsis
closeth
up
into
a
right
line
.
That
right
line
is
here
the
line
that
we
have
to
do
withall
.
For
we
here
suppose
alwayes
that
the
Index
pointeth
up
into
the
Equinoctiall
Circle
;
and
the
Index
we
alwayes
suppose
to
be
a
Zenith
line
,
and
therefore
the
Horizon
proper
to
that
Index
as
a
Zenith
line
,
must
needs
make
right
angles
w●th
the
Equinoctiall
Circle
,
and
must
therefore
be
a
right
Horizon
or
Polar
plain
.
When
the
Index
is
pitched
in
a
true
posture
,
all
the
houre
points
of
the
Equinoctiall
Circle
must
be
supposed
to
have
lines
issuing
from
them
all
parallel
to
the
Index
.
This
is
a
generall
truth
in
the
whole
course
of
the
Ellipticall
Dialling
,
Wherefore
,
First
of
all
,
These
lines
will
(
in
this
case
)
fall
all
upon
the
plain
of
the
Equinoctiall
;
and
because
the
common
section
of
any
two
plains
must
be
a
right
line
,
therefore
it
is
that
all
these
lines
so
issuing
,
will
fall
upon
one
and
the
same
streight
line
,
namely
,
whensoever
the
Equinoctiall
meets
with
or
cuts
any
other
plain
.
Hence
comes
it
to
be
Rectilineal
Horologiography
,
being
performed
upon
a
plain
.
Secondly
,
Because
they
are
all
parallel
to
the
Index
,
therefore
they
must
all
be
parallel
to
one
another
;
and
because
they
come
originally
from
a
Circle
,
and
are
parallel
,
they
will
all
also
keep
distance
one
frō
another
as
the
Sines
in
a
Circle
do
.
Thirdly
,
Because
the
Index
moves
alwayes
in
a
parallel
position
to
it selfe
,
and
would
shew
the
houre
upon
the
circular
divisions
of
the
Equinoctiall
Circle
(
at
all
times
of
the
year
,
that
is
,
when
this
Equinoctiall
is
made
to
represent
all
other
parallels
of
Declination
)
it
must
follow
that
the
Index
when
(
the
shadow
of
)
it
falleth
upon
any
one
such
point
,
it
must
also
fall
upon
the
whole
line
that
issueth
through
that
point
.
The
reason
is
,
because
the
line
is
alwayes
parallel
to
the
Index
,
therfore
is
it
touch
one
point
it
must
shadow
the
whol
.
Fourthly
,
Hence
will
follow
,
that
any
line
drawn
upon
the
plain
of
the
Equinoctiall
,
crossing
those
forementioned
parallel
lines
,
shall
be
a
line
of
Sines
(
or
a
double
line
of
Sines
,
if
you
will
rather
expresse
it
properly
)
and
if
it
passe
through
the
center
o●
the
Equinoctiall
Circle
(
where
the
Index
also
doth
alwayes
passe
)
then
is
such
a
line
one
of
the
Dials
that
we
here
are
to
deal
with
.
Because
every
plain
is
taken
for
a
great
Circle
cutting
the
Equinoctiall
in
the
center
.
And
the
divisions
of
this
line
(
the
Sinicall
divisions
)
must
be
the
houre
points
,
as
is
said
before
.
Now
all
that
we
have
here
to
say
is
the
making
good
of
the
former
rules
of
limiting
out
and
placing
the
houre
line
,
the
Zodiacall
line
,
and
the
Index
.
If
we
therefore
(
in
all
cases
)
suppose
the
Index
to
be
a
Zenith
line
,
and
then
an
Horizontal
plain
to
be
placed
perpendicular
to
it
,
this
plain
must
be
the
proper
Horizon
to
that
Index
(
or
Zenith
line
)
and
must
(
in
these
Cases
here
handled
)
go
through
the
Axis
of
the
world
(
a
part
of
which
I
must
call
the
primary
Zodiac
)
and
will
cut
the
Equinoctiall
plain
in
a
right
line
placed
perpendicularly
to
the
Index
,
which
I
call
the
primary
horizontall
line
.
And
again
,
all
that
is
done
in
other
plains
(
not
perpendicular
to
the
Index
)
must
be
imagined
primarily
(
or
originally
)
to
be
deduced
from
this
proper
Horizontall
plain
.
First
therefore
,
for
the
houre
line
.
If
the
Index
lie
in
the
proper
Me●idian
of
the
plain
(
as
is
supposed
in
the
first
of
the
former
Cases
)
then
the
Diameter
of
the
Circle
that
is
at
first
assumed
,
serves
for
the
horologicall
line
without
any
alteration
,
because
the
proper
Horizon
,
and
the
plain
upon
which
you
work
,
do
(
both
of
them
)
cut
the
Equinoctiall
in
one
and
the
same
line
,
which
is
to
be
the
horologicall
line
.
And
therefore
in
plain
reason
,
so
much
of
the
Equinoctiall
as
the
Index
is
wryed
from
this
proper
Meridian
of
the
plain
,
so
much
of
the
same
Equinoctiall
must
the
proper
Horizon
be
wryed
with
it
,
and
consequently
,
so
much
will
be
between
the
proper
horizontall
line
and
the
horologicall
line
of
the
plain
,
but
still
both
these
lines
will
continue
upon
the
same
plain
of
the
Equinoctiall
,
and
consequently
they
must
in
both
cases
be
one
and
the
same
line
,
namely
that
which
is
made
by
the
projection
of
the
Equinoctiall
Circle
upon
the
plain
,
on●y
it
must
b●
augmented
according
to
the
proper
horizontall
lines
departure
from
the
section
of
the
plain
with
the
Equinoctiall
,
that
is
,
according
to
the
wrying
of
the
Index
,
as
is
plain
enough
by
what
is
said
in
the
142
Pag
▪
And
so
for
the
division
of
that
line
in
the
same
Page
.
For
G
G
and
h
o
,
do
both
lie
in
one
and
the
same
plain
of
the
Equinoctiall
:
and
all
the
projecture
must
be
made
parallel
to
the
Index
,
whence
h
A
is
the
secant
(
to
G
A
Radius
)
of
R
G
or
C
L
,
the
wrying
of
the
Index
.
Then
for
the
Zodiac
,
you
are
to
deduce
it
from
the
Axis
which
is
perpendicular
to
the
Zenith
line
or
Index
:
and
from
so
much
of
it
as
containeth
the
Tangent
of
23½
gr
.
to
the
Radius
G
A.
Now
though
the
Index
do
keep
the
proper
Meridian
of
the
plain
,
yet
the
Axis
is
elevated
above
the
plain
.
And
because
the
projecture
of
that
Tangent
or
part
of
the
Axis
must
be
parallel
to
the
Index
,
therefore
look
what
elevation
the
Axis
or
Pole
of
the
World
hath
,
to
such
a
Secant
of
the
Radius
equall
to
the
Tangent
of
23½
before
mentioned
must
the
Zodiac
be
extended
.
By
which
means
the
Index
keeps
the
same
situation
and
distance
from
the
Equinoctiall
plain
that
it
would
do
if
it
moved
according
to
the
proper
and
primary
Tangents
upon
the
Axis
,
and
no
otherwise
.
And
so
doth
the
former
enlarging
of
the
horologicall
line
produce
the
like
effect
,
namely
,
that
the
Index
doth
upon
that
enlarged
line
shew
the
same
houres
that
it
would
do
upon
the
proper
horizontall
line
.
Then
for
the
Zodiac
to
a
wryed
Index
,
it
is
both
displaced
and
enlarged
too
,
according
as
right
lines
drawn
through
the
divisions
of
the
Axis
,
and
parallel
to
the
Index
,
would
make
a
projection
of
it
.
According
to
which
,
there
are
three
wayes
given
.
The
first
makes
A
F
equall
to
the
Tangent
of
23½
to
the
Radius
R
A
,
to
be
Radius
,
and
to
that
Radius
findes
F
H
,
and
to
that
again
as
Radius
,
finde
H
Z
the
Tangent
of
wrying
.
And
so
A
Z
is
the
Semizodiac
placed
and
enlarged
.
As
F
P
Radius
,
To
P
Z
the
Tangent
of
the
angle
P
F
Z
;
So
is
F
P
in
the
former
known
length
(
found
to
A
F
Tangent
of
23½
gr
.
)
To
P
Z.
Therefore
Z
is
the
point
terminating
the
extremity
of
the
projected
Zodiac
,
and
so
A
Z
must
be
the
Semizodiac
,
according
to
what
was
delivered
in
the
second
way
Pag.
147.
The
third
way
resolves
a
Sphericall
rectangled
Triangle
,
made
between
the
proper
Meridian
of
the
plain
P
A
,
the
plain
it selfe
A
B
,
and
the
Meridian
in
which
the
Index
lyeth
,
or
into
which
it
pointeth
,
P
B.
In
which
three
quantities
are
known
.
P
A
the
altitude
of
the
Pole
above
the
plain
.
The
angle
at
A
a
right
angle
.
And
the
angle
at
P
equall
to
the
arke
of
the
Equinoctiall
or
wrying
of
the
Index
from
the
proper
Meridian
of
the
plain
.
And
there
is
first
required
A
B.
As
the
Radius
P
n
,
To
the
Tangent
of
(
n
m
=
)
P
;
So
the
Sine
of
P
A
,
To
the
Tangent
of
A
B
,
=
n
A
C
,
Pag.
147
,
Then
secondly
,
the
arke
P
B
is
required
,
and
that
〈◊〉
wrought
by
the
second
work
,
Pag.
148
,
which
in
the
second
Figure
Pag.
153
,
may
be
understood
to
be
the
angle
F
A
Z
,
Then
because
A
F
Z
is
a
plain
Triangle
rectangled
at
F
,
and
the
angle
F
A
Z
is
now
known
,
and
the
side
A
F
(
equall
to
the
Tangent
of
23½
)
it
will
be
required
to
finde
A
Z
in
the
same
parts
.
Therefore
in
the
third
work
(
Pag.
148
)
it
is
said
,
As
A
F
,
Radius
,
To
A
Z
,
Secant
of
F
A
Z
;
So
is
A
F
in
parts
,
(
as
a
Tangent
of
23½
)
To
A
Z
in
the
same
parts
.
And
so
the
Zodiac
is
both
placed
and
limited
.
APPENDIX
.
IT
hath
been
forwerly
shewed
(
in
the
VIII
.
Section
)
how
to
make
the
Ellipticall
Diall
to
any
Index
however
placed
,
and
to
any
Superficies
flat
or
curved
.
That
Doctrine
will
shew
how
to
project
this
kinde
of
Rectilineall
Diall
(
and
the
former
Circular
Diall
too
)
onely
I
shall
adde
a
word
or
two
to
shew
that
that
way
will
produce
a
single
line
in
this
case
,
and
such
a
line
as
must
answer
to
a
double
line
of
Sines
.
1.
See
the
6
and
7
Propositions
Pag.
95
,
96.
The
first
of
them
is
to
finde
the
Horizontall
spaces
.
And
in
this
case
(
because
the
Index
lookes
alwayes
up
into
the
Equinoctiall
Circle
)
the
Latitude
of
the
Horizon
proper
to
this
Index
line
,
is
nothing
.
So
that
,
As
the
Sine
of
the
Poles
elevation
,
which
is
nothing
,
Is
to
the
Radius
;
So
is
the
Tangent
of
each
houre
from
the
proper
Meridian
of
the
Index
,
To
the
Tangent
of
an
infinite
length
:
[
For
so
it
must
be
because
the
first
term
is
00.
]
Which
infinite
Tangent
is
belonging
to
90
gr
.
therefore
it
will
follow
that
all
the
houres
and
each
of
them
are
90
gr
.
from
the
proper
Meridian
of
the
Index
:
that
is
,
that
they
are
all
coincident
into
one
line
,
which
is
perpendicular
to
the
proper
Meridian
of
the
Index
.
This
shews
that
this
Dial
will
be
a
single
line
,
and
that
by
the
former
generall
Doctrine
,
Section
VIII
.
2.
Then
for
the
second
thing
,
that
it
will
be
a
line
of
Sines
,
or
(
though
a
curved
line
,
yet
)
to
be
thence
deduced
or
divided
,
it
will
appear
by
the
seventh
Proposition
Pag.
96.
Because
,
As
the
Radius
,
To
the
Co-sine
of
the
Poles
elevation
;
which
here
is
the
Radius
;
So
the
Co-sine
,
&c.
To
the
same
term
again
.
For
if
the
two
first
terms
be
the
same
,
then
the
two
latter
must
be
equall
too
.
And
since
the
third
terms
are
a
line
of
Sines
,
the
fourth
terms
(
which
are
to
divide
the
Diall
line
)
being
the
same
with
the
third
,
will
be
also
such
as
will
make
a
line
of
Sines
.
So
this
is
made
good
likewise
.
And
there
may
other
wayes
be
found
to
effect
this
same
thing
upon
a
curved
Superficies
.
As
first
to
project
the
Equinoctiall
line
or
section
with
the
uneven
supersicies
,
upon
the
said
Superficies
:
and
when
that
is
done
,
to
lay
the
edge
of
some
boa●d
or
ruler
,
in
the
Superficies
of
the
Equinoctiall
,
with
the
Index
lying
in
the
plain
of
the
Equinoctiall
,
will
(
by
the
former
Equinoctiall
line
drawn
)
direct
you
in
,
and
afterwards
to
proceed
answerably
to
the
nature
of
the
Diall
,
&c.
But
what
need
I
to
mention
any
such
thing
thus
obscurely
.
They
that
understand
the
former
wayes
will
be
able
to
invent
many
more
,
and
so
I
ce●se
from
further
tro●bling
my selfe
or
the
Reader
.
One
Proposition
more
must
here
be
added
,
which
conduceth
to
this
Rectilineal
Horologiography
,
namely
:
How
to
lay
a
line
that
shall
point
up
to
any
assigned
place
or
point
in
the
heavens
.
THis
is
usefull
for
the
laying
of
an
Index
to
any
condition
that
shall
be
required
:
namely
,
so
as
to
cut
through
any
just
houre
,
either
in
the
Equinoctiall
Circle
,
or
any
other
Circle
in
the
heavens
.
The
first
work
must
be
to
finde
what
Reclination
and
Declination
the
said
point
shall
have
in
respect
of
our
own
Zenith
and
Meridian
.
This
is
an
easie
Sphericall
work
,
and
therefore
I
shall
omit
it
.
Only
note
,
that
the
Reclination
here
mentioned
is
the
distance
of
it
from
our
Zenith
;
and
the
declination
is
,
what
Azimuth
from
the
Meridian
the
said
point
fals
into
.
The
next
work
is
,
how
to
lay
or
stretch
a
line
to
any
given
Declination
and
Reclination
.
The
way
that
for
the
present
I
think
on
shall
be
this
.
For
the
houre
points
.
You
are
here
left
to
your
choise
to
take
which
(
of
three
wayes
)
you
like
best
.
Either
,
1.
These
points
in
the
Circle
may
serve
for
the
houre
points
without
any
more
draughts
,
and
then
this
will
be
another
Circular
Diall
,
differing
from
those
that
were
handled
before
in
Circular
Dialling
.
2.
The
perpendicular
Diameter
D
E
may
have
the
hours
inserted
by
drawing
lines
from
each
pare
of
houres
equally
distant
from
D
and
E
houre
points
of
the
Circle
,
parallel
to
the
Index
C
A
,
and
so
you
shall
have
the
Diameter
D
E
divided
like
a
double
line
of
Sines
(
unequally
)
at
12
,
11
,
1
,
2
,
3
,
and
10
,
9
,
8
,
&c.
3.
You
may
likewise
(
if
you
allow
not
the
two
other
wayes
)
draw
any
line
as
F
G
,
and
by
lines
drawn
through
the
houre
points
of
the
Circle
,
all
parallel
to
the
Index
C
A
,
the
same
line
F
G
shall
be
divided
as
a
double
line
of
Sines
,
at
12
,
1
,
2
,
&c.
and
11
,
10
,
9
,
&c.
and
will
shew
the
houre
as
the
other
do
,
and
the
two
extream
parallels
4
F
,
and
4
G
,
shall
limit
the
length
of
the
line
F
G.
☞
Upon
this
Circular
way
Note
thus
much
.
That
though
the
Index
A
C
should
not
lie
upon
a
just
houre
,
but
otherwise
accidentally
,
yet
because
the
houre
points
are
upon
a
Circle
,
the
houres
will
be
distant
from
each
other
,
and
not
breed
that
distraction
in
computing
that
was
before
noted
necessarily
to
fall
upon
the
streight
horologicall
line
.
And
so
you
are
left
at
liberty
to
set
your
Index
how
you
will
,
only
so
as
ever
to
looke
into
one
and
the
same
Meridian
Circle
.
But
you
must
then
be
carefull
to
place
some
one
houre
line
true
,
so
as
to
answer
to
some
just
houre
circle
in
the
heavens
,
which
upon
this
Equinoctiall
plain
(
when
the
Axis
is
once
truly
placed
)
will
be
easie
.
For
if
the
Axis
give
a
shadow
,
and
you
at
the
same
instant
time
(
noting
the
shadow
)
do
observe
the
houre
,
that
shadow
will
represent
the
same
houre
.
If
therefore
upon
that
shadow
line
(
as
a
Diameter
)
you
describe
a
Circle
,
you
may
(
from
the
intersections
of
the
said
line
,
with
the
Circle
)
set
off
any
one
full
and
just
houre
,
and
then
from
that
just
houre
you
may
divide
the
Circle
into
the
24
houres
.
This
equall
setting
of
the
Index
is
allowable
upon
the
Circle
,
but
not
upon
the
right
lines
D
E
or
F
G
,
for
in
them
there
will
be
confusion
of
parts
,
which
in
the
Circle
are
distant
.
Note
further
,
that
you
need
not
make
the
Center
of
the
horologicall
Circle
just
in
the
point
A
,
where
the
extremity
of
the
Index
which
came
down
upon
the
plain
,
but
you
may
make
it
in
any
point
where
the
Index
A
C
will
touch
the
plain
,
as
at
H
,
or
in
other
like
places
.
Concerning
the
Zodiac
.
You
must
account
the
Semidiameter
D
A
of
the
horologicall
Circle
to
be
as
Radius
.
And
to
that
length
(
and
none
other
)
you
must
make
the
length
of
the
Semizodiac
A
B
upon
the
Axis
to
be
the
Tangent
of
23
and
a
halfe
,
and
by
that
Scale
of
Tangents
you
must
put
in
the
Sines
or
moneths
as
hath
been
often
shewed
before
.
And
whereas
I
have
hitherto
said
that
the
Index
must
move
upon
the
Axis
,
and
agree
to
a
Zodiac
thereon
inscribed
,
it
must
be
understood
that
in
some
sense
this
is
necessary
,
but
in
some
sense
not
so
.
As
thus
,
The
Index
D
A
or
D
E
,
must
alwayes
move
perpendicular
A
X
the
Axis
.
A
C
the
Zodiac
.
A
D
the
Index
,
A
B
another
supporter
differing
from
the
Axis
.
to
the
Axis
A
X.
But
it
is
not
tied
to
move
upon
the
Axis
it selfe
.
It
may
move
according
to
any
line
that
lies
in
the
same
plain
with
the
Index
D
E
and
the
Axis
A
X
,
that
is
,
it
may
move
according
to
the
line
A
B
,
which
we
here
suppose
to
be
in
the
same
plain
or
Meridian
with
A
D
and
A
X.
This
condition
is
alwayes
requisite
,
and
it
must
ever
keep
one
&
the
same
angle
with
the
●ight
line
A
B
,
and
such
an
angle
that
may
keepe
it
alwayes
perpendicular
to
the
true
Axis
,
that
is
,
parallel
to
the
Equinoctiall
plain
.
And
this
being
observed
,
you
may
insert
the
Zodiac
upon
the
line
A
B
,
but
it
must
be
proportioned
from
the
Axis
A
X
,
thus
;
As
A
C
Radius
,
to
A
E
the
Secant
of
the
angle
C
A
E
,
so
is
A
C
the
Tangent
of
23
and
a
halfe
(
in
respect
of
the
Radius
of
the
horologicall
Circle
,
)
to
a
fourth
quantity
;
the
length
whereof
gives
A
E.
And
now
this
line
A
E
must
be
esteemed
as
a
Tangent
of
23
degrees
and
a
half
,
and
by
it
must
the
Semizodiac
or
halfe
year
be
put
in
,
as
hath
been
shewed
formerly
.
So
this
is
done
also
,
For
the
Index
will
move
upon
A
E
,
according
to
the
parts
upon
A
C
,
as
it
is
bound
to
do
.
Only
note
that
these
Dials
upon
the
Equinoctiall
plains
can
serve
but
halfe
a
year
upon
one
side
▪
&
therefore
for
the
other
half
year
you
must
do
the
like
work
upon
the
under
side
of
the
Equinoctiall
plain
.
The
work
I
say
is
like
without
any
reall
difference
,
and
therefore
there
will
need
no
more
words
concerning
it
.
For
DEMONSTRATION
of
these
things
.
THere
needs
nothing
to
be
added
more
than
what
hath
formerly
been
done
for
Rectilineal
Dials
.
For
what
is
done
there
in
the
proper
Horizon
(
proper
to
the
Index
I
mean
)
by
a
line
of
Sines
,
is
here
expressed
by
the
line
of
Sines
D
A
E
,
Pag.
162
:
only
there
,
all
the
lines
issuing
from
the
Equinoctiall
houre
points
did
meet
with
the
proper
Horizontal
plain
,
and
did
thereon
create
a
single
line
of
●80
Sines
,
and
so
the
Equinoctiall
Circle
standing
towards
the
eye
(
being
infinitely
removed
from
it
)
edgewise
is
projected
upon
his
own
diameter
,
and
appears
nothing
else
but
as
a
Diameter
(
or
streight
line
)
to
the
eye
.
But
here
the
same
forementioned
parallel
lines
issuing
from
the
houre
points
of
the
Equinoctiall
Circle
,
and
flowing
out
upon
the
plain
of
that
Circle
,
do
so
appear
in
full
view
to
the
eye
,
because
the
eye
stands
so
as
fully
to
view
the
said
plain
and
not
looking
edgewise
upon
it
.
When
the
plain
stood
edgewise
to
the
eye
,
the
eye
did
project
it
into
the
Diameter
,
but
now
we
conceive
the
eye
to
stand
only
to
looke
what
was
effected
upon
the
plain
by
the
former
Projection
.
And
further
.
Looke
what
projection
was
formerly
made
(
in
this
case
of
the
Indexes
pointing
into
the
Equinoctiall
Circle
)
upon
the
Indexes
proper
Horizon
,
and
was
there
received
by
a
streight
line
,
the
same
is
here
done
by
drawing
the
Diameter
D
A
E
,
Pag.
162
,
which
line
must
be
supposed
to
be
the
common
section
of
the
Indexes
proper
Horizon
with
the
Equinoctiall
plain
.
And
so
in
effect
both
these
Cases
are
nothing
different
.
Then
for
the
line
F
G
,
the
parts
of
it
,
and
the
points
of
the
Circle
,
and
the
points
of
D
E
must
necessarily
all
fall
into
one
houre
line
,
and
all
shew
the
true
houre
(
whichsoever
of
them
you
take
for
your
Horologicall
line
)
because
as
the
lines
issuing
from
the
houre
points
of
the
Equinoctiall
are
all
parallel
one
to
the
other
,
and
all
again
to
the
Index
,
therefore
if
any
part
of
the
shadow
of
the
Index
do
fall
upon
any
part
of
one
of
those
lines
,
the
whole
shadow
must
fall
upon
the
whole
line
,
and
consequently
the
correspondent
points
upon
D
E
and
F
G
and
the
Circle
,
will
all
joyntly
,
or
each
of
them
severally
,
give
the
houre
of
the
day
.
Other
things
are
plain
enough
.
Of
the
parallelisme
of
the
Index
and
Plain
.
It
must
further
be
noted
,
that
an
Index
may
be
made
to
point
up
into
the
Equinoctiall
Circle
,
and
yet
also
be
parallel
to
any
plain
whatsoever
.
For
every
plain
cuts
the
Equinoctiall
Circle
somwhere
:
Now
if
an
Index
be
made
to
lie
in
that
common
section
of
these
two
Circles
or
plains
,
then
will
it
lie
parallel
both
to
the
Equinoctiall
plain
and
to
the
other
plain
,
and
it
may
be
made
to
move
alwayes
parallel
to
them
both
,
if
it
be
moved
at
right
angles
to
any
line
that
shall
stand
perpendicular
to
the
said
common
Section
.
But
for
all
this
,
there
can
no
such
Diall
(
as
we
have
now
spoken
of
)
be
made
upon
any
other
plain
besides
the
Equinoctiall
.
I
say
there
can
no
parallelisme
of
the
Index
to
the
plain
be
(
in
this
kinde
of
Dialling
)
allowed
,
more
than
what
is
said
may
be
done
upon
the
Equinoctiall
plain
.
For
though
the
Index
may
be
parallel
to
the
Equinoctial
and
plain
both
,
yet
the
Equinoctiall
it selfe
must
make
angles
with
all
such
plains
.
And
because
the
Index
is
parallel
to
the
common
Section
,
therefore
all
the
houre
points
projected
by
parallel
lines
to
the
Index
,
must
be
upon
the
Equinoctiall
plain
it selfe
,
and
must
run
parallel
to
the
line
of
common
section
:
that
is
,
they
will
all
run
so
,
as
never
to
meet
so
much
as
with
the
line
of
common
section
(
which
line
is
upon
the
Equinoctiall
plain
)
much
lesse
ever
to
meet
with
any
part
of
the
plain
it selfe
,
which
is
every
where
distant
more
or
lesse
from
the
Equinoctiall
,
except
only
in
the
line
of
common
section
.
So
that
there
is
no
more
to
be
said
of
this
subject
.
FINIS
.
ELLIPTICAL
Horologiography
.
SHEWING
How
upon
any
plain
to
draw
an
Elliptical
Diall
,
to
an
Index
set
any
way
,
by
Sphericall
(
and
not
Projective
)
work
.
Invented
and
written
by
Mr.
SAMVEL
FOSTER
,
Late
professor
of
Astronomie
in
Gresham-Colledge
.
London
,
Printed
for
Nicholas
Bourn
.
1654.
ELLIPTICAL
Horologiography
.
HItherto
we
have
had
the
whole
businesse
of
Elliptical
Horologiography
,
so
far
as
that
more
cannot
seem
to
be
thought
of
,
or
required
.
Yet
,
Because
to
any
Index
set
eith●r
on
purpose
or
casually
,
the
houres
have
been
(
formerly
)
found
out
by
Projection
,
I
thought
fit
here
to
adde
another
way
(
but
for
plains
only
)
that
shall
not
need
that
projective
manner
of
working
.
And
so
the
Probleme
will
be
this
:
How
upon
any
plain
to
draw
an
Ellipticall
Diall
,
to
an
Index
set
any
〈◊〉
,
by
Sphericall
(
and
not
Projective
)
work
.
The
substance
of
what
I
shall
here
say
is
most
of
it
delivered
before
,
only
I
must
now
shew
what
Propositions
a●e
to
be
referred
hither
,
and
how
they
must
be
used
.
Things
that
are
pre-requisite
to
this
work
.
1.
You
must
know
the
situation
of
your
plain
,
that
is
,
what
Declination
and
Reclination
it
hath
in
respect
of
your
own
Horizon
.
This
is
to
be
done
the
ordinary
way
,
as
in
all
plains
is
usuall
.
2.
You
must
know
the
Declination
and
Reclination
of
your
Index
.
That
is
,
if
the
Index
be
set
casually
you
must
then
finde
out
the
Declination
and
Reclination
thereof
,
which
is
shewed
,
Pag.
80.
how
it
must
be
done
,
namely
,
by
observation
from
the
Sun
,
and
cannot
be
otherwise
performed
.
Or
if
you
choose
a
Longitude
and
Latitude
wherein
to
lay
your
Index
,
then
must
you
(
by
that
Longitude
and
Latitude
given
)
finde
what
Declination
and
Reclination
is
due
thereunto
,
in
respect
of
your
own
Horizon
:
and
according
to
this
Declination
and
Reclination
you
are
to
lay
your
Index
.
See
the
Schemes
Pag.
83
,
84
,
for
by
those
or
the
like
you
are
to
conceive
of
your
Sphericall
work
in
this
kinde
.
3.
Having
thus
found
the
situation
both
of
plain
and
Index
,
in
respect
of
your
own
Horizon
,
you
must
then
finde
what
situation
the
Index
hath
in
respect
of
the
plain
;
that
is
,
what
Declination
it
hath
from
the
proper
Meridian
of
the
plain
,
and
what
Reclination
it
hath
from
the
Plains
Zenith
.
This
Proposition
hath
not
been
handled
before
,
but
must
be
put
here
in
this
place
.
The
next
work
is
to
finde
what
situation
an
Index
(
set
up
upon
the
plain
)
shall
have
to
the
plain
,
and
the
fundamental
lines
thereon
described
,
viz.
the
Verticall
line
,
and
proper
Meridian
.
I
suppose
first
,
that
the
Declination
and
Reclination
of
the
Index
(
in
respect
of
your
own
Horizon
)
is
known
by
the
former
things
already
spoken
of
,
and
from
thence
you
may
gather
what
is
here
required
.
Thus
Let
the
Index
point
up
to
X
:
and
let
Z
X
(
the
reclination
from
your
Zenith
)
be
known
,
and
S
Z
X
(
the
declination
from
your
Meridian
)
be
also
known
.
Now
therefore
in
the
triangle
T
Z
X
,
there
are
three
things
known
.
1
T
Z
the
Complement
of
the
Plains
reclination
in
respect
of
your
Horizon
.
2
Z
X
the
complement
of
the
Indexes
reclination
in
respect
of
your
Horizon
.
3
Thirdly
,
the
angle
T
Z
X
,
which
is
the
difference
of
the
two
declinations
,
namely
of
the
plain
and
of
the
Index
.
By
these
all
the
rest
will
be
found
;
as
namely
,
1
T
X
,
the
reclination
of
the
Index
X
from
T
the
pole
of
the
plain
,
which
is
called
the
proper
reclination
of
the
Index
.
2
Z
T
X
,
or
V
Y
,
or
rather
V
M
,
which
shews
how
much
the
*
Subindicall
line
lies
from
the
Verticall
line
of
the
plain
(
in
degrees
of
a
Circle
described
upon
the
plain
)
and
consequently
,
because
P
T
Z
was
known
before
,
therefore
P
T
X
is
now
also
become
known
,
which
shewes
the
declination
of
the
*
Subindicall
line
from
the
plains
proper
Meridian
,
which
may
therefore
be
called
the
proper
declination
of
the
Index
.
3
The
angle
Z
X
T
(
if
it
be
of
any
use
here
)
shews
how
much
(
in
Azimuthal
position
)
your
Zenith
Z
lies
from
T
,
the
Zenith
of
the
plain
upon
that
Hotizon
whose
Zenith
line
is
the
same
with
the
Index
,
which
is
therefore
the
proper
Horizon
to
the
Index
.
Or
if
you
would
set
an
Index
so
as
to
fit
it
to
any
assigned
difference
of
Longitude
,
and
to
any
assigned
Latitude
also
,
you
must
work
thus
.
LEt
X
P
S
be
the
assigned
difference
of
Longitude
,
and
P
X
the
complement
of
the
Latitude
,
into
both
which
the
Index
shall
point
up
in
X.
Because
the
plains
difference
of
Longitude
Z
P
T.
is
known
before
,
therefore
the
angle
T
P
X
is
known
.
And
P
T
the
complement
of
the
plains
Latitude
is
also
before
known
.
Therefore
in
the
triangle
T
P
X
these
three
things
are
known
,
viz.
P
X
,
P
T
,
and
the
interjacent
angles
T
P
X.
By
which
three
you
may
finde
all
the
other
quantities
:
namely
,
P
T
X
,
or
B
Y
,
the
departure
of
the
Subindicall
line
from
the
plains
proper
Meridian
:
T
X
the
proper
reclination
of
the
Index
X
from
T
the
pole
of
the
plain
,
or
the
complement
of
it
(
X
Y
)
which
is
the
elevation
of
the
Index
above
the
plain
just
over
(
perpendicularly
over
in
respect
of
the
plain
)
the
Subindicall
line
:
with
the
angle
P
X
T
(
if
there
were
any
use
of
it
)
which
shewes
how
much
the
pole
of
the
plain
declines
from
the
proper
Meridian
of
the
Index
.
I
say
,
it
shewes
this
declination
upon
that
Horizon
(
whatever
it
should
be
)
which
shall
be
properly
belonging
or
perpendicularly
placed
to
the
said
Index
as
the
proper
Zenith-line
.
Thus
the
situation
of
the
Index
(
whether
casually
,
or
by
election
set
)
is
found
to
the
true
coasts
of
the
plain
it self
.
What
remains
to
be
done
.
THe
rest
of
the
work
for
making
the
Elliptical
Diall
it self
to
the
Index
assigned
,
and
for
limiting
and
dividing
the
Zodiacall
Scale
depends
upon
this
one
Rule
.
Namely
,
To
project
the
houre
points
of
the
Equinoctiall
,
and
the
Zodiacall
Scale
upon
the
Axis
(
which
Zodiacall
Scale
upon
the
Axis
it self
is
made
by
the
Tangents
of
23
gr
.
and
an
half
,
to
the
Radius
of
that
Sphere
wherein
you
shall
imagine
your
Circles
to
be
)
I
say
.
To
project
(
all
such
points
)
upon
the
plain
,
by
lines
passing
through
each
one
of
them
,
and
all
and
each
of
the
same
lines
to
be
drawn
parallel
to
the
Index
given
.
This
is
the
summe
of
what
is
here
now
to
be
done
,
and
of
what
hath
been
done
formerly
quite
through
this
Book
.
For
all
that
is
before
effected
upon
any
plain
or
curved
Superficies
,
is
nothing
more
then
what
I
have
now
said
;
namely
,
a
determining
of
the
true
places
of
every
such
(
houre
and
Zodiacall
)
point
,
as
they
shall
be
projected
upon
the
Dialls
Superficies
,
by
lines
going
through
them
,
and
going
parallel
also
to
the
Index
that
is
to
give
the
houre
.
For
so
,
the
Equinoctiall
Circle
will
very
well
supply
every
parallel
of
the
Sun
,
the
Index
and
horologicall
line
being
rectified
by
the
Zodiac
in
a
true
site
one
to
the
other
,
as
hath
been
often
mentioned
before
;
And
then
the
Index
,
casting
its
shadow
upon
the
houre
point
,
either
in
the
Equinoctiall
or
in
the
parallel
Circle
,
must
also
cast
it
upon
the
whole
projecting
line
,
which
passeth
th●ough
the
hour
point
,
(
because
that
line
is
made
alwayes
parallel
to
the
Index
)
and
consequently
upon
that
point
of
the
houre
,
which
(
by
the
said
line
)
is
carried
to
the
Diall-Superficies
.
And
so
for
the
points
of
the
Zodiac
,
they
are
also
projected
in
such
manner
,
as
that
the
Index
moving
in
them
(
however
they
may
seem
to
stand
awry
and
out
of
order
)
shall
alwayes
cut
the
Axis
of
the
World
(
if
it
were
there
placed
)
and
cut
it
according
to
the
Tangents
of
declination
thereon
expr●ssed
.
And
this
will
serve
for
a
brief
demonstration
of
that
work
which
is
●ow
to
come
afterwards
.
In
which
we
are
first
to
bring
down
the
houre
points
and
Zodiac
to
the
plain
considered
as
an
Horizon
,
and
the
projection
of
those
points
to
be
made
into
an
Ellipsis
,
by
lines
all
perpendicular
to
the
said
plain
,
that
is
,
parallel
to
the
plains
proper
Zenith
line
.
Secondly
,
we
are
to
project
or
transpose
these
first
(
both
Houre
and
Zodiacall
)
points
from
the
primary
Ellipsis
into
a
secondary
Ellipsis
,
such
as
the
position
of
the
true
intended
Index
will
require
:
And
so
also
the
points
of
the
Zodiac
must
be
dealt
withall
.
How
to
make
an
Ellipticall
Diall
upon
any
Plain
,
to
an
Index
perpendicular
to
the
Plain
.
THe
first
work
is
done
before
,
partly
by
Calculation
,
and
partly
by
protraction
,
Pag.
25
,
26
,
but
more
fully
and
better
,
Proposition
5
,
6
,
7
,
Pag.
92
,
95
,
96
,
by
which
you
make
two
Tables
,
one
of
Horizontal
spaces
,
the
other
of
Altitudes
.
So
that
I
shall
not
need
to
say
any
more
of
it
.
By
that
work
(
I
say
)
you
may
make
two
Tables
which
may
serve
to
prick
down
the
houre
points
,
which
points
you
shall
see
to
winde
about
the
center
in
forme
of
an
Ellipsis
.
1.
It
is
first
requisite
that
the
pr●per
Meridian
of
the
plain
be
rightly
situated
upon
the
plain
(
as
is
there
also
expressely
required
.
And
it
is
again
requisite
that
the
proper
line
of
the
two
sixes
(
proper
I
mean
to
the
plain
,
considered
as
an
Horizon
by
it selfe
)
be
also
drawn
:
which
line
(
as
in
all
Horizontal
Dials
is
usuall
)
must
be
perpendicular
(
upon
the
plain
)
to
the
proper
Meridian
.
And
the
points
also
of
the
two
sixes
must
be
carefully
set
on
,
whose
distance
from
the
center
is
the
longer
Radius
of
the
Ellipsis
.
The
reason
why
these
points
of
the
proper
Sines
must
be
had
,
is
,
because
the
●ew
Ellipsis
(
which
we
now
look
for
)
must
passe
through
them
both
.
So
also
the
just
point
of
the
proper
12
(
or
the
section
of
the
Ellipsis
with
the
plains
proper
Meridian
)
must
be
with
the
same
care
designed
upon
the
plain
.
And
the
Subindicall
line
must
be
drawn
in
its
true
position
upon
the
plain
,
according
to
the
proper
declination
of
the
Index
.
2.
After
this
is
done
,
you
must
call
to
minde
what
the
Declination
and
Reclination
of
your
Index
is
(
I
speak
now
of
the
proper
Declination
and
Reclination
before
mentioned
)
as
Pag.
172
is
mentioned
.
3.
Then
from
every
houre
point
lately
made
upon
the
plain
,
you
are
to
draw
right
lines
,
parallel
to
the
subindicall
line
,
and
draw
them
both
wayes
or
on
both
sides
of
each
houre
point
,
because
you
know
not
which
of
them
(
as
yet
)
you
are
to
make
use
of
for
the
pricking
down
of
your
New
or
Distorted
Ellipsis
.
4.
Take
the
larger
Radius
of
your
primary
Ellipsis
(
that
is
the
length
from
the
center
of
it
,
to
the
point
of
the
proper
six
a
clock
)
and
(
upon
the
Sector
,
for
that
Instrument
is
most
fit
for
this
use
or
practise
)
entering
it
upon
the
Tangent
of
45
degrees
,
open
your
Sector
to
tha●
length
:
then
take
over
from
the
reclination
of
the
Index
(
the
proper
reclination
I
now
speak
of
)
counted
on
the
same
Scale
of
Tangents
,
to
the
other
leg
,
and
reserve
this
length
.
That
is
,
As
the
Radius
,
to
the
Tangent
of
the
proper
reclination
of
the
Index
;
So
is
the
larger
Semidiameter
of
the
primary
Ellipsis
,
to
a
fourth
quantity
,
which
is
equall
to
the
●●rmer
reserved
length
,
and
to
that
large
Semidiameter
or
Radius
,
is
the
Tangent
of
the
Indexes
proper
Reclination
.
5.
This
reserved
length
must
again
be
entered
from
90
in
the
line
of
Sines
,
and
the
Sector
set
according
to
it
,
and
being
so
set
it
is
fit
for
the
rest
of
the
work
that
is
to
ensue
.
Because
this
proportion
we
are
to
go
upon
:
As
the
Radius
,
to
the
Tangent
of
the
Indexes
proper
reclination
(
which
two
terms
are
they
that
now
do
set
the
Sector
,
)
so
is
the
Sine
of
every
altitude
of
the
horary
points
in
the
Equinoctiall
,
(
which
altitudes
your
Table
of
altitudes
computed
by
the
forementioned
Proposition
,
Pag.
96
,
97
,
will
help
you
unto
,
)
so
is
the
Sine
of
every
(
horary-equinoctiall-points
)
altitude
,
to
the
Tangent
of
the
reclination
of
every
one
of
those
severall
houre
points
of
the
Equinoctiall
Circle
.
Note
that
the
reclinations
here
mentioned
are
,
in
respect
,
and
from
the
proper
Zenith
line
of
the
plain
.
And
note
also
that
these
Tangents
thus
found
,
are
to
be
estimated
to
the
larger
semidiameter
of
the
primary
Ellipsis
as
to
their
proper
Radius
.
6.
By
these
Tangents
so
found
,
you
are
to
set
on
the
new
houre
points
,
by
which
both
the
new
Ellipsis
is
to
be
described
and
also
divided
.
I
say
the
new
houre
points
are
to
be
set
on
upon
the
lines
drawn
through
each
of
the
old
houre-points
parallel
to
the
Subindicall
line
,
and
they
are
to
be
s●t
on
from
each
of
the
said
old
houre
points
in
those
lines
.
But
which
way
,
or
on
which
side
of
them
,
will
be
still
doubted
▪
as
was
before
mentioned
§
3.
And
to
take
away
this
doubt
,
it
will
be
best
to
consider
the
point
of
the
proper
12
upon
the
primary
Ellipsis
,
when
the
Sun
is
(
any
day
)
in
that
proper
12
point
of
the
heavens
,
then
must
the
upright
(
or
proper
Index
of
the
plain
)
cast
its
shadow
into
the
proper
12
point
of
the
primary
Ellipsis
.
Then
further
,
supposing
the
Sun
to
stand
still
there
,
imagine
how
your
slope
Index
is
to
lie
from
the
proper
Index
,
and
by
that
you
shall
easily
discern
which
way
the
inclination
of
that
Index
will
wry
the
proper
point
of
12
from
the
primary
Ellipsis
,
and
consequently
you
shall
be
able
(
thereby
)
to
judge
upon
which
part
of
that
line
which
is
drawn
through
the
point
of
proper
12
(
parallel
to
the
subindicall
line
)
the
houre
(
of
proper
12
)
shall
be
cast
into
the
secondary
Ellipsis
from
the
primary
.
This
point
being
set
right
,
you
may
place
all
the
houre
points
from
that
proper
12
to
the
two
proper
sixes
,
the
same
way
with
this
first
point
:
but
all
the
other
points
on
the
remoter
side
of
the
proper
12
,
must
go
the
contrary
way
,
as
you
will
easily
●erceive
,
the
very
figure
of
the
Ellipsis
will
require
it
should
so
be
.
As
here
.
R
S
is
the
Subindical
line
,
over
which
the
Index
leaneth
.
O
the
point
of
proper
12
upon
P
O
Q
the
primary
Ellipsis
.
O
A
,
9
B
,
●
C
,
&c.
parallel
lines
through
the
houre
points
in
the
primary
Ellipsis
,
parallel
I
say
to
the
Subindicall
line
R
S.
P
Q
the
two
proper
fixes
,
through
which
alwayes
the
distorted
Ellipsis
must
passe
.
Then
because
the
Index
leaneth
from
the
upright
Index
(
or
Zenith
line
of
the
plain
)
towards
S
,
therefore
the
proper
12
at
O
must
go
towards
S
,
viz.
to
A
:
and
therefore
also
9
B
,
8
C
,
&c.
11
G
,
&c.
must
go
the
same
way
till
you
come
to
the
two
proper
sixes
at
P
and
Q
,
and
then
the
houre
points
D
E
F
must
be
set
from
3
2
1
,
the
contrary
way
to
that
wherein
the
first
houres
were
placed
;
as
you
may
plainly
see
the
form
of
the
new
Ellipsis
will
require
.
For
it
cannot
possibly
be
made
to
turn
again
from
P
towards
M
,
if
it
must
make
up
an
Ellipsis
,
as
in
this
case
it
is
necessary
it
should
do
.
Another
way
followeth
.
A
second
way
to
make
the
Ellipsis
and
houres
to
a
slope
Index
upon
a
plain
:
and
without
trouble
of
making
that
Table
before
men●ioned
.
THis
way
will
be
more
expedite
than
the
former
.
I
shall
here
suppose
these
particulars
to
be
again
done
by
the
former
precepts
.
1.
That
the
plains
proper
Meridian
R
O
,
is
rightly
placed
.
2.
That
P
R
Q
is
drawn
perpendicular
thereunto
.
3.
That
the
Subindicall
line
R
S
is
also
rightly
placed
from
R
O
the
proper
Meridian
.
4.
That
R
P
or
R
Q
being
determined
for
the
Radius
,
R
O
and
R
K
are
each
of
them
the
Sines
of
the
plains
propet
Latitude
.
5.
From
the
points
O
and
K
(
which
are
the
two
proper
points
of
12
to
an
Index
standing
perpendicularly
to
the
plain
)
let
O
A
and
K
L
be
drawn
parallel
to
R
S
the
Subindicall
line
.
6.
Then
to
the
Radius
R
P
,
finde
the
Sine
of
the
plains
difference
of
Longitude
from
your
Meridian
[
because
I
purpose
here
to
shew
how
to
work
all
from
your
12
a
clock
line
,
though
there
is
no
obligation
to
that
houre
,
but
all
may
be
done
just
in
the
same
manner
from
any
other
houre
point
,
if
you
account
the
plains
difference
of
Longitude
from
some
one
of
them
,
which
is
easie
to
do
:
as
if
in
this
Example
the
difference
of
Longitude
from
12
be
36
gr
.
then
from
11
it
must
be
21
gr
.
from
10
it
must
be
6
gr
.
from
1
a
clock
it
must
be
51
gr
.
&c.
to
all
or
any
of
the
other
houres
]
which
here
for
example
sake
we
will
say
to
be
36
gr
.
With
that
Sine
set
off
R
T
,
and
draw
T
I
parallel
to
R
O
the
proper
Meridian
.
Then
again
to
the
Radius
R
O
,
finde
out
the
Co-sine
of
the
difference
of
Longitude
,
and
set
it
upon
the
line
T
I
,
from
T
to
I.
All
this
work
is
only
to
finde
the
point
of
12
in
the
primary
Ellipsis
,
as
you
see
it
expressed
in
the
former
Figure
at
12.
The
like
work
you
must
do
to
finde
the
point
of
one
of
the
houres
of
six
in
the
primary
Ellipsis
,
because
six
is
90
gr
.
from
12.
[
If
you
had
chosen
the
point
of
11
to
work
by
,
you
must
have
lookt
for
the
point
of
5
a
clock
,
because
that
point
is
the
90th
gr
.
from
11
,
if
10
,
then
4
,
&c.
]
That
is
,
to
the
Radius
R
O
finde
the
Sine
of
the
difference
of
Longitude
,
and
set
it
from
R
to
V
,
and
then
draw
V
X
parallel
to
P
R
the
line
of
the
plains
proper
six
.
Then
to
the
Radius
R
P
finde
the
Co-sine
of
the
said
difference
of
Longitude
.
and
set
it
upon
V
X
from
V
to
X
:
So
shall
X
be
the
point
of
your
own
six
a
clock
upon
the
proper
Ellipsis
;
as
you
may
see
in
the
Figure
is
expressed
.
Now
through
these
two
points
I
and
X
draw
right
lines
parallel
to
the
Subindical
line
R
S.
For
if
you
divide
R
12
(
both
wayes
)
as
a
line
of
Sines
(
R
e
and
R
f
being
Sines
of
15
gr
.
R
d
and
R
g
Sines
of
30
gr
.
&c.
R
12
and
R
12
Sines
of
90
gr
.
)
at
a
,
b
,
c
,
d
,
e
,
R
,
f
,
g
,
h
,
i
,
k
;
you
must
then
draw
right
lines
through
each
of
those
points
(
each
one
parallel
to
the
conjugate
diameter
6
R
6.
)
And
when
that
is
done
,
make
R
6
a
Radius
,
and
to
that
Radius
make
e
7
,
e
5
,
f
5
,
f
7
,
to
be
the
Sine
of
75
gr
.
or
one
houre
under
90.
Make
d
8
,
d
4
,
g
4
,
g
8
,
the
Sine
of
60
gr
.
or
two
houres
lesse
than
90.
So
c
9
,
c
3
,
is
the
Sine
of
three
houres
under
90
:
b
10
,
b
2
,
of
four
houres
under
90
:
a
11
,
a
1
,
is
the
Sine
of
five
houres
under
90
,
or
of
one
houre
.
The
like
you
may
do
for
halves
and
quarters
,
or
what
other
parts
you
desire
.
Thus
much
likewise
for
this
second
way
.
Alwayes
you
must
remember
to
put
all
things
upon
their
right
Coasts
.
Concerning
the
Zodiac
,
how
it
is
to
be
limited
and
placed
.
TO
omit
other
wayes
,
I
shall
here
pitch
upon
that
which
is
most
suitable
to
this
manner
of
working
upon
which
we
now
are
.
To
the
Radius
P
R
(
which
is
the
larger
Semidiameter
of
the
primary
Ellipsis
)
first
,
finde
the
Tangent
of
23
degrees
and
a
halfe
;
which
suppose
to
be
R
ξ
:
and
then
to
this
line
,
being
taken
as
a
new
Radius
,
finde
the
Co-sine
of
the
plains
Latitude
,
or
the
Sine
of
the
Equinoctials
altitude
,
which
suppose
to
be
the
length
R
W
,
which
is
also
to
be
set
from
the
center
R
upon
the
proper
Meridian
of
the
plain
,
as
here
it
is
from
R
to
W.
Then
through
the
point
W
,
draw
a
line
,
as
W
Z
,
parallel
to
the
subindical
line
R
S.
When
this
is
done
you
are
to
finde
the
Sine
of
the
Poles
elevation
above
the
plain
,
to
the
same
Radius
R
ξ
,
which
Sine
you
may
suppose
to
be
the
length
of
the
line
φ
ψ.
Then
say
again
;
As
the
Radius
,
to
the
Tangent
of
the
Indexes
reclination
;
so
φ
ψ
(
the
Sine
of
the
Poles
elevation
above
the
plain
)
to
a
fourth
line
;
which
line
let
be
represented
by
λ
ω.
Take
this
line
λ
ω
therefore
,
and
set
it
upon
W
Z
,
the
line
formerly
drawn
,
from
W
to
Z.
And
lastly
,
draw
the
line
R
Z
,
and
make
R
y
equall
to
it
,
so
shall
Z
y
be
the
Zodiac
in
its
true
length
and
position
:
and
R
Z
or
R
y
shall
give
the
halfe
of
it
,
which
halfe
is
to
be
estimated
as
a
Tangent
of
23
degrees
and
a
halfe
,
and
so
the
Zodiac
to
be
set
on
as
hath
formerly
been
often
shewed
.
You
are
here
to
take
care
that
all
things
be
set
in
their
right
position
.
In
this
example
,
Z
turns
from
W
towards
the
left
hand
,
because
it
is
here
supposed
that
the
North
Pole
is
elevated
upon
the
plain
,
and
so
the
Axis
riseth
out
of
R
above
the
plain
:
and
therefore
that
23½
tangent
degree
upon
that
Axis
a●e
above
the
plain
.
From
which
extream
point
of
23½
above
the
plain
,
you
may
suppose
a
line
to
issue
out
parallel
to
the
Index
,
which
inclines
from
R
towards
S
,
and
therefore
that
line
inclining
so
too
,
must
project
that
extream
point
from
W
towards
Z.
All
the
reason
of
the
work
depends
upon
that
which
hath
been
formerly
said
:
Namely
,
that
both
houre
points
of
the
Equinoctiall
,
and
declination
points
in
the
Axis
,
are
all
projected
from
their
true
positions
in
the
Sphere
(
upon
or
unto
a
plain
)
by
right
lines
passing
through
each
of
them
,
parallel
to
the
assigned
Index
.
I
shall
need
to
say
no
more
,
because
I
think
enough
(
if
not
too
much
)
hath
been
said
already
.
This
shall
serve
for
the
Effection
and
Demonstration
also
of
this
Proposition
or
Probleme
.
CORONIDIS
LOCO
:
I
shall
here
further
adde
;
How
upon
an
Horizontal
Plain
,
to
describe
an
Ellipticall
Diall
to
an
Index
lying
aslope
,
and
pointing
into
some
assigned
Longitude
and
Latitude
.
THis
I
shall
adde
as
a
Corollary
to
that
which
hath
last
of
all
been
delivered
in
general
for
all
Plains
and
Indexes
.
Out
of
which
Doctrine
this
(
as
a
particular
branch
)
is
deduced
.
It
containeth
not
any
new
thing
,
but
because
it
may
be
more
frequently
made
use
of
(
in
practice
)
than
other
Cases
,
therefore
I
adde
it
,
that
it
may
also
be
the
more
plain
and
easie
in
operation
.
It
is
(
for
more
ease
)
intended
that
the
Index
should
point
up
into
some
just
houre
Circle
,
and
to
have
some
just
degrees
of
Reclination
from
the
Zenith
of
the
Horizon
.
I
purpose
not
to
give
variety
of
wayes
,
for
they
are
shewed
in
the
last
precepts
now
given
which
are
generall
,
and
may
easily
be
brought
and
applyed
to
this
particular
Case
,
as
a
branch
of
that
generall
before
.
Suppose
in
our
Horizon
51
gr
.
30
min.
in
Latitude
,
and
to
two
Indexes
set
upon
an
Horizontall
plain
,
the
one
pointing
South-east
into
the
morning
Circle
of
7
a
clock
,
the
other
pointing
South-west
into
the
evening
Circle
of
5
a
clock
,
and
each
of
them
reclining
45
gr
.
it
were
required
to
make
two
Ellipticall
Dials
:
How
must
they
be
done
?
It
must
be
noted
that
the
making
of
one
will
make
the
other
,
because
they
are
to
be
set
upon
one
plain
,
and
their
differences
of
the
Indexes
Longitudes
,
is
in
each
5
houres
,
and
the
reclination
of
the
Index
in
each
45
degrees
.
Only
the
Coasts
in
respect
of
situation
must
be
changed
,
because
the
positions
of
the
Indexes
are
not
parallel
,
but
contrary
one
to
the
other
,
and
so
will
the
Ellipses
be
,
as
you
may
see
in
the
next
page
.
The
first
thing
to
be
done
,
is
,
By
having
the
difference
of
the
Indexes
Longitude
(
75
gr
.
)
and
the
reclination
of
the
Index
(
45
gr
.
)
to
finde
what
declination
the
Index
must
have
:
Or
to
finde
the
declination
of
the
subindicall
line
from
our
Meridi●n
.
When
this
work
is
done
;
Draw
A
R
B
for
the
Meridian
line
of
North
and
South
,
and
6
R
6
for
the
line
of
East
and
West
,
crossing
each
other
at
R
,
which
must
be
the
center
of
the
Diall
,
and
draw
the
Subindicall
line
R
S
60
gr
.
5
min.
from
A
R
upon
the
right
Coast
,
that
is
,
North-east
in
the
first
plain
,
and
North-west
in
the
other
plain
.
Then
making
6
R
for
Radius
,
finde
thereto
the
Sine
of
the
Latitude
of
the
place
51
gr
.
30
min.
which
let
be
the
line
R
C
,
and
set
it
upon
the
North
part
of
the
Meridian
line
A
B
(
I
call
it
the
Meridian
line
because
it
is
the
said
line
upon
the
Horizontal
plain
whereupon
these
descriptions
are
now
made
)
from
the
center
R
to
C.
Then
from
the
point
C
,
draw
C
D
parallel
,
and
the
same
way
with
R
S
the
Subindicall
line
.
So
shall
C
be
the
point
of
12
upon
the
primary
Ellipsis
which
should
be
drawn
,
if
an
Ellipticall
Diall
were
to
be
made
upon
the
Horizontall
plain
to
the
upright
Index
,
which
is
the
proper
Zenith
line
of
the
Horizontall
plain
.
After
this
,
to
the
same
Radius
R
6
,
finde
the
Co-sine
of
your
Latitude
,
or
the
Sine
of
the
Equinoctials
height
,
or
rather
(
in
this
particular
work
)
the
depth
of
it
under
the
Horizon
,
which
Sine
let
be
the
line
E
F.
Then
say
,
As
the
Radius
,
is
to
the
Tangent
of
45
gr
.
the
Indexes
reclination
;
so
is
the
Sine
E
F
,
to
a
fourth
;
which
fourth
will
be
(
in
this
Example
)
the
same
line
with
E
F
,
because
the
Tangent
of
45
(
the
Reclination
)
is
the
same
length
with
the
Radius
,
yet
I
will
call
it
the
fourth
quantity
.
Take
this
fourth
quantity
and
set
it
off
upon
the
line
C
D
,
from
C
to
D
,
so
shall
D
give
the
point
of
12
upon
the
distorted
Ellipsis
,
which
is
the
12
that
fitteth
with
the
reclining
Index
with
which
we
have
now
to
deal
.
Then
draw
(
from
R
to
D
)
the
line
R
D
,
both
wayes
,
as
D
R
G
is
done
.
By
these
meanes
we
have
now
got
two
conjugate
Diameters
.
And
now
the
rest
of
the
work
will
be
easie
.
For
you
are
onely
to
Divide
the
Semidiameters
R
D
and
R
G
(
from
R
towards
the
extremities
)
each
of
them
as
a
line
of
Sines
.
That
is
,
R
m
,
R
o
,
make
the
Sine
of
1
hour
or
15
gr
.
So
R
l
,
R
p
,
the
Sine
of
30
,
or
2
hours
:
R
k
,
R
q
,
the
Sine
of
45
gr
.
or
3
houres
:
R
i
the
Sine
of
60
gr
.
or
4
hours
:
R
h
the
Sine
of
5
houres
:
R
D
or
R
12
,
the
whole
Sine
or
Sine
of
90
gr
.
Then
through
each
of
these
points
at
h
,
i
,
k
,
l
,
m
,
R
,
o
,
p
,
q
,
draw
a
line
parallel
to
the
other
conjugate
Diameter
,
namely
,
to
6
R
6.
This
done
,
make
R
6
a
Radius
,
and
to
that
Radius
make
m7
,
m5
,
07
,
05
(
each
of
them
)
a
Sine
of
5
hours
or
75
degrees
.
Make
l
8
,
l
4
,
p
4
,
p
8
,
a
Sine
of
4
hours
or
60
gr
.
Make
k9
,
k3
;
q3
;
q
9
,
(
each
of
them
)
a
Sine
of
3
houres
or
45
gr
.
:
So
i
10
,
i●
,
must
be
the
Sine
of
2
houres
:
and
h
11
,
h
1
,
the
Sine
of
1
houre
or
15
gr
.
By
this
work
you
shall
(
upon
the
former
parallel
lines
drawn
through
h
,
i
,
k
,
l
,
m
,
o
,
p
,
q
,
)
finde
the
houre
points
of
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
,
11
,
12
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
;
In
which
you
are
to
signe
out
the
houre
points
,
and
through
which
you
are
to
draw
the
Ellipsis
,
as
you
see
the
figure
for
a
pattern
.
You
might
also
have
divided
each
of
the
Radiusses
,
noted
with
R
6
into
parts
correspondent
to
a
line
of
Sines
,
as
you
see
partly
done
at
the
letters
s
t
u
,
in
the
last
of
the
two
former
Scheams
)
as
at
s
t
u
is
expressed
,
and
through
those
points
you
must
draw
paralle●s
to
the
other
Diameter
D
R
G.
Then
to
the
Radius
D
R
make
s
7
,
s
5
,
the
Sine
of
15
gr
.
or
1
hour
:
t
8
,
t
4
,
the
Sine
of
30
gr
.
or
●wo
hours
:
u
9
,
u
3
,
the
Sine
of
45
gr
.
or
3
houres
;
and
so
forward
:
by
which
work
you
shall
finde
the
same
houre
points
(
3
,
4
,
5
,
6
,
7
,
8
,
9
,
&c.
)
that
you
did
before
,
and
so
you
may
finde
out
and
draw
the
Ellipsis
by
them
.
The
same
manner
of
work
is
to
be
performed
for
halves
and
quarters
of
the
severall
houres
.
How
the
Zodiac
is
to
be
placed
and
limited
.
IT
is
the
same
work
with
that
which
went
before
,
Pag.
185.
yet
I
will
here
insert
it
again
.
First
,
work
thus
.
As
the
Radius
,
is
to
the
Tangent
of
23½
gr
.
So
let
E
F
(
which
was
formerly
found
to
be
the
Sine
of
the
Equinoctials
altitude
above
the
Horizon
38
gr
.
30
mn
.
in
respect
of
the
Radius
R
6
,
and
must
now
again
be
here
used
)
be
to
a
fourth
quantity
,
which
suppose
to
be
F
W.
Take
therefore
that
length
F
W
,
and
set
it
from
R
to
X
on
the
North
side
(
in
this
example
:
)
and
through
X
draw
X
γ
parallel
to
the
Subindicall
line
R
S.
Then
say
again
.
As
the
Radius
,
is
to
the
Tangent
of
23
gr
.
and
an
half
;
So
let
R
C
(
which
was
before
found
to
be
the
Sine
of
the
Latitude
of
the
place
51
gr
.
and
an
half
in
respect
of
the
Radius
R
6
,
and
comes
now
to
be
here
used
again
)
be
to
a
fourth
quantity
,
which
suppose
to
be
F
z.
Take
therefore
F
z
,
and
set
it
upon
the
line
formerly
drawn
through
X
,
from
X
,
contrary
to
the
going
of
the
Subindical
line
,
namely
Southward
(
in
this
example
)
I
say
from
X
to
γ
Then
from
γ
through
R
,
draw
γ
R
π
,
and
make
R
π
equall
to
R
γ
,
So
shall
γ
π
be
the
Zodiac
both
placed
rightly
and
justly
limited
.
For
division
of
it
,
you
must
conceive
that
R
γ
or
R
π
is
a
Tangent
of
23
and
an
half
gr
.
the
Suns
greatest
declination
,
and
consequently
you
may
by
it
set
on
either
12
moneths
,
or
12
signes
,
as
you
will
;
the
manner
whereof
hath
been
often
shewed
in
this
Book
,
and
especially
Pag.
10
,
11.
and
Pag.
14
,
15
,
&c.
so
that
now
all
this
work
is
come
to
an
end
.
The
Index
must
slide
to
and
fro
upon
the
line
γ
π
(
or
the
Index
standing
,
the
Ellipsis
must
move
after
the
same
rate
)
γ
π
is
the
proper
Meridian
of
the
Index
.
And
if
your
work
(
when
you
make
this
Diall
,
according
to
this
case
here
before
put
)
be
exactly
true
,
you
shall
finde
the
Zodiacall
line
π
γ
to
fall
just
into
the
houre
lines
of
5
and
7
,
as
they
ought
to
do
,
because
the
difference
of
the
Indexes
Longitude
was
assumed
to
be
75
gr
.
from
the
South
Meridian
of
the
place
,
which
is
the
same
with
those
houres
.
The
signe
♋
must
be
placed
at
γ
,
and
♈
at
π
in
this
Diall
.
The
Index
must
lie
(
when
it
is
in
the
Center
R
)
just
over
the
line
R
S
,
and
must
alwayes
move
in
a
parallel
position
to
it self
.
And
it
must
in
this
example
recline
45
gr
.
or
rise
above
the
Horizon
(
or
above
R
S
making
an
angle
above
that
line
45
gr
.
also
.
But
though
it
lie
thus
,
yet
it
must
move
in
the
line
γ
π
,
and
it
must
be
so
set
on
,
that
when
you
project
the
fiduciall
edge
of
it
into
a
point
(
as
workmen
use
to
do
,
when
with
their
eye
they
tr●e
the
straightnesse
of
a
line
)
all
that
edge
may
at
once
justly
appear
to
run
into
the
line
γ
π
,
wherever
the
Index
be
set
in
the
Zodiac
.
This
is
to
be
done
carefully
.
I
have
here
set
two
of
them
,
whose
proper
Meridians
,
(
I
mean
proper
to
the
Index
,
which
are
the
lines
of
their
Zodiacs
,
viz.
γ
π
)
do
not
lie
in
one
and
the
same
right
line
,
or
parallelism
(
which
is
the
samenesse
of
po●ition
)
but
make
angles
one
to
the
other
.
This
is
done
,
because
they
should
not
lie
in
one
Meridian
:
for
by
that
meanes
they
will
alwayes
set
one
another
,
which
the
Horizontal
Diall
with
the
single
Ellipsis
to
an
upright
Index
will
not
do
at
all
times
.
These
two
,
I
say
,
will
set
themselves
:
and
if
you
adde
a
common
Horizontall
Diall
more
,
to
stand
between
them
,
as
you
see
done
there
▪
they
will
be
the
more
specious
and
usefull
for
setting
each
other
.
I
have
set
that
whose
Index
looks
North-east
upon
the
left
hand
,
you
may
set
it
on
the
right
hand
if
you
will
,
and
the
other
on
the
left
.
For
this
makes
no
materiall
change
.
The
Index
is
best
to
be
a
threed
.
Other
circumstances
I
remember
none
but
what
the
Workman
will
be
well
able
to
go
through
.
Another
way
for
the
description
of
an
Ellipticall
Diall
upon
the
Horizontall
plain
of
51
gr
.
30
min.
Latitude
,
whose
Zodiac
,
and
the
motion
of
the
Index
is
performed
upon
the
●oures
of
7
in
the
morning
,
and
of
5
in
the
afternoon
,
as
the
same
houres
are
drawn
upon
the
common
Horizontall
Diall
.
THis
way
was
formerly
written
,
and
though
it
need
not
to
have
been
here
placed
,
in
respect
the
way
last
given
is
the
best
,
yet
because
it
doth
somwhat
differ
from
what
was
set
down
before
,
I
thought
best
not
to
omit
it
:
because
it
is
delightfull
to
see
differ●●t
way
●●
meet
both
in
one
effect
.
1.
First
,
you
are
to
get
(
this
way
)
the
altitudes
of
each
houre
point
in
the
Equino●tiall
,
above
the
Horizon
,
by
this
R●le
.
As
the
Radius
,
to
the
Sine
of
38
gr
.
30
min.
So
●he
Sines
of
75
,
60
,
45
,
30
,
15
gr
.
to
so
many
4th
Sines
.
The
Co-sines
of
these
will
limit
out
the
points
(
upon
the
houres
of
the
Horizontall
Diall
formerly
drawn
)
through
which
the
Ellipsis
upon
the
Horizontall
plain
(
made
for
an
upright
Index
)
is
to
passe
:
as
is
shewed
before
in
the
28
,
29
,
30
,
and
31
,
Pages
,
which
points
in
the
figure
of
the
next
Page
may
be
represented
by
8
,
9
,
10
,
11
,
12
,
1
,
2
,
3
,
&c.
2.
As
the
Radius
,
to
the
Tangent
of
the
reclination
of
the
Index
;
So
are
the
4th
Sines
before
found
,
to
so
many
7th
terms
.
3.
Describe
the
regular
Horizontall
Ellipsis
,
as
you
see
P
O
Q
which
you
must
know
to
be
for
the
Latitude
of
51
gr
.
30
min.
and
that
P
Q
are
the
points
of
the
two
sixes
,
and
O
K
the
points
of
the
two
twelves
,
viz.
at
Noon
and
Midnight
:
and
also
that
all
the
houres
are
regularly
set
down
to
O
P
Q
K
,
&c.
Suppose
here
an
Index
to
gr
.
m.
Recline
Northwards
45
00
Differ
in
Longit.
S
E
/
W
75
00
Consequently
it
must
Decline
North
East
/
West
60
05
Have
North
Latitude
50
37
4.
From
each
point
there
set
down
,
draw
lines
parallel
to
the
Subindicall
line
R
S
,
as
O
A
,
8
C
,
9
B
,
&c.
the
angle
O
R
S
must
be
60
gr
.
5
min.
6.
Through
these
new
points
B
F
D
A
E
G
,
&c.
you
are
to
draw
a
new
Ellipsis
,
which
will
be
distorted
from
the
former
Ellipsis
first
drawn
.
Thus
the
Diall
it selfe
for
the
houre
points
is
limited
,
you
may
adde
halves
and
quarters
.
For
the
Zodiac
.
IN
this
Example
remember
,
that
the
Index
declineth
North
East
/
West
60
gr
.
5
min.
Reclineth
45
gr
.
Differeth
in
Longitude
South
East
/
West
75
gr
.
Hath
Latitude
50
gr
.
37
min.
1.
Because
the
difference
of
Longitude
is
75
gr
.
from
the
South
,
therefore
our
two
houres
of
7
in
the
morning
,
and
5
at
evening
,
are
the
proper
Meridians
upon
which
the
Indexes
must
slide
.
2.
For
that
cause
you
must
consider
how
much
of
those
two
Meridians
is
intercepted
between
the
Equinoctiall
and
our
Horizon
.
You
may
finde
it
by
this
Rule
.
As
the
Radius
,
is
to
the
Sine
of
15
gr
.
(
the
Conpl
.
of
75
,
)
So
is
the
Tangent
of
38
gr
.
30
min.
to
the
Tangent
of
11
gr
.
38
min.
which
is
the
arke
required
.
3.
Because
the
Latitude
of
the
Index
is
50
gr
.
37
min.
North
,
the
complement
of
it
must
be
39
gr
.
23
min.
and
so
much
under
the
Equinoctiall
points
of
7
and
5
(
and
upon
the
houre
circles
of
7
and
5
,
which
are
the
proper
Meridians
to
the
two
Indexes
)
doth
the
proper
Horizon
of
each
of
those
Indexes
cut
the
said
houres
of
7
and
5
upon
the
South-east
and
South-west
;
and
so
much
do
they
cut
the
same
Meridians
above
the
Equinoctiall
upon
the
South-west
and
South-east
.
So
that
upon
the
Southerly
Coast
,
that
portion
of
these
Meridians
which
is
comprehended
between
the
Equinoctiall
and
our
Horizon
,
namely
11
gr
.
38
min.
must
be
taken
from
39
gr
.
23
min.
and
the
remainder
will
be
27
gr
.
45
min.
which
is
the
arke
comprehended
between
the
proper
Horizon
of
the
Index
and
our
own
Horizon
,
which
upon
the
South
part
(
in
this
particular
Example
)
is
below
our
Horizon
,
on
the
North
part
above
.
4.
First
,
therefore
the
limitation
of
the
Semi-Zodiacs
length
must
be
to
that
Latitude
of
the
Index
or
proper
Horizon
50
gr
.
37
min.
And
then
secondly
,
the
said
Zodiac
first
found
and
limited
must
be
augmented
(
in
this
Example
)
in
proportion
of
the
Radius
to
the
Secant
of
27
gr
.
45
min.
In
this
manner
,
by
the
second
rule
given
at
the
beginning
of
this
Book
Pag.
11.
To
the
Radius
P
R
finde
the
Cofine
of
the
Latitude
of
the
Index
,
namely
the
Sine
of
39
gr
.
23
min.
which
suppose
to
be
N
T.
To
this
length
N
T
as
a
Radius
,
finde
out
the
Secant
of
the
forementioned
arke
27
gr
.
45
min.
which
suppose
to
be
N
V.
This
length
N
V
shall
give
the
Radius
or
Tangent
of
45
gr
.
which
is
the
Tangent
or
Decimall
Scale
,
out
of
which
(
either
of
them
)
you
are
to
make
your
Zodiac
by
the
numbers
of
those
Tables
in
the
6
and
7
Pages
.
To
this
Radius
or
Decimall
Scale
,
or
Tangent
of
45
gr
.
N
W
is
the
Tangent
of
23
gr
.
and
a
halfe
,
wherefore
the
Semizodiac
,
R
L
or
R
M
is
to
be
made
equall
in
length
to
N
W.
Thus
the
Zodiacs
length
is
limited
.
For
the
position
of
the
Zodiac
,
it
was
before
said
,
that
it
must
lie
in
the
line
of
7
in
the
morning
,
or
5
in
the
afternoon
.
So
in
this
last
figure
you
see
it
points
to
Z
,
which
is
7
in
the
morning
,
(
and
consequently
also
to
X
,
which
is
7
in
the
evening
)
in
the
distorted
Ellipsis
:
because
this
is
that
Diall
whose
Index
looks
North-east
:
you
see
it
do
so
likewise
in
the
first
figure
Pag.
189
,
as
in
the
second
figure
there
it
lies
to
5
in
the
evening
,
and
consequently
also
to
5
in
the
morning
.
Observe
in
the
last
figure
(
which
is
as
true
and
just
in
the
two
other
Pag.
189.
)
that
the
angle
P
R
Z
,
and
so
Q
R
X
,
is
equall
to
the
angle
made
(
upon
our
common
Horizontall
Diall
)
between
6
and
7
a
clock
,
namely
,
18
gr
.
54
min.
and
consequently
that
1
R
1
(
a
line
drawn
from
1
to
1
in
the
primary
Ellipsis
)
is
perpendicular
to
the
Zodiac
L
M
:
but
P
R
C
is
not
equall
to
the
angle
thereon
between
6
and
8
a
clock
;
nor
is
it
so
in
any
other
of
the
houres
.
Observe
again
,
that
if
a
line
be
drawn
parallel
to
O
K.
through
A
the
point
of
12
in
the
distorted
Ellipsis
,
till
it
meet
with
the
Zodiac
(
extended
)
at
y
;
observed
(
I
say
)
that
if
R
M
be
taken
as
a
Tangent
of
23½
gr
.
(
as
it
is
taken
in
this
work
)
then
will
R
y
be
the
Tangent
of
(
the
complement
of
your
Latitude
,
namely
)
38
gr
.
30
min.
And
this
must
be
so
,
because
if
the
Sun
should
decline
38
gr
.
30
min.
Southward
,
then
the
Sun
must
only
peep
up
upon
the
Horizon
at
12
,
and
consequently
the
shadow
of
the
Index
(
namely
the
point
of
it
that
intersects
with
your
Horizon
,
for
the
Sun
hath
no
altitude
at
that
place
and
time
)
must
run
full
from
South
to
North
to
shew
the
point
of
12
at
A.
Other
observations
might
be
made
,
but
I
go
no
further
.
The
reason
of
the
precedent
work
will
be
demonstrated
out
of
the
former
things
delivered
in
this
Book
:
especially
out
of
this
last
Treatise
,
which
begins
Pag.
171
,
and
ends
Pag.
186.
And
therefore
I
wholly
omit
to
adde
any
thing
more
to
this
particular
subject
.
They
that
understand
the
things
before
,
will
not
be
to
seeke
in
the
necessary
Demonstrations
of
these
particulars
.
A
Note
concerning
the
framing
of
Dials
to
finde
the
Azimuth
.
AS
the
Axis
of
the
world
is
to
the
Equinoctial
Circle
,
so
is
the
Zenith
line
to
the
Horizon
of
any
place
.
Whence
it
will
follow
:
That
whatsoever
is
in
this
Book
declared
concerning
the
projection
of
the
houre
points
of
the
Equinoctiall
Circle
upon
any
plain
,
&c.
the
same
may
be
applied
to
the
projection
of
Azimuths
or
points
of
the
Horizon
upon
any
plain
,
&c.
Onely
mutatis
mutandis
,
(
i.e.
)
instead
of
the
Equinoctiall
and
Poles
of
the
World
,
you
are
to
use
the
Horizon
and
the
Poles
of
it
,
(
i.e.
)
Zenith
and
Nadir
.
Then
for
the
Zodiac
(
which
depends
upon
the
Scale
or
Tangents
of
Declination
,
which
exceed
not
23
and
a
half
gr
.
you
are
to
use
(
in
the
Azimuthal
work
)
a
Scale
of
Tangents
(
rightly
limited
)
going
up
from
oo
to
so
much
altitude
as
you
mean
to
use
(
if
it
be
for
the
Suns
course
,
to
62
gr
.
here
at
London
.
)
As
there
you
rectifie
the
Index
to
the
Declination
of
the
Sun
:
So
here
you
must
(
answerably
)
rectifie
it
to
the
Suns
altitude
taken
by
observation
,
and
then
the
Index
being
set
to
the
altitude
gives
the
Azimuth
for
that
moment
and
no
more
.
COROLLARY
.
Therefore
,
to
finde
the
Azimuth
by
this
way
,
you
must
take
the
Suns
Altitude
,
and
so
finde
the
Azimuth
for
one
moment
onely
.
Quest.
But
why
is
there
so
little
said
here
of
this
?
Answ.
Because
it
is
a
thing
of
no
use
,
but
of
a
great
deal
of
trouble
:
for
the
altitude
is
to
be
observed
,
and
the
Index
set
thereto
every
moment
,
Quest.
Is
it
not
so
too
in
the
Dials
before
described
?
Answ.
No
;
for
there
the
Index
set
once
,
may
serve
in
that
place
one
whole
day
very
well
,
because
the
Suns
declination
doth
not
alter
much
in
one
day
.
Quest.
Why
then
is
this
Azimuthal
businesse
mentioned
at
all
?
Answ.
Because
the
Reader
might
see
,
that
the
Authour
of
this
Treatise
was
not
ignorant
of
it
:
yet
it
is
mentioned
also
that
if
any
delight
in
such
a
curiosity
he
may
to
his
liking
effect
it
.
¶
And
for
a
full
conclusion
note
,
that
,
All
the
former
Dials
may
be
made
in
a
Craticular
way
.
FINIS
.
AN
INDEX
Of
the
Chief
PARTICULARS
:
OF
the
Elliptical
Diall
,
with
an
Index
perpendicular
to
the
plain
.
Upon
an
Horizontall
,
or
any
other
direct
plain
,
see
from
Page
8
to
page
22
Upon
a
declining
plain
,
from
22
to
29
Another
way
to
prick
down
and
divide
the
Ellipsis
,
29
to
34
Some
uses
and
varieties
of
this
Elliptical
Diall
,
34
to
37
Some
varieties
of
the
structure
of
it
,
37
to
47
An
Advertisement
concerning
some
other
uses
of
the
last
description
,
47
to
53
A
demonstration
of
what
soever
went
before
,
53
to
69
Of
the
Elliptical
Diall
with
an
Index
standing
in
the
Zenith-line
.
Upon
any
plain
,
or
other
curved
Superficies
:
done
by
projection
,
from
Page
69
to
80
To
any
Superficies
,
and
to
an
Index
casually
set
:
done
also
by
projection
,
80
to
112
A
demonstrations
of
all
,
112
to
115
Circular
Horologiographie
,
performed
and
demonstrated
,
115
to
133
Rectilineal
Horologiography
,
how
done
,
and
demonstrated
,
133
to
169
Elliptical
Horologiography
,
169
Upon
all
plain
(
not
curved
)
Superficies
to
an
Index
(
casually
,
or
by
election
)
placed
any
way
:
performed
by
Spherical
operations
,
(
not
by
projection
,
and
demonstrated
,
169
to
201
A
Note
concerning
Azimuths
how
they
may
be
found
by
such
kinde
of
Dials
made
in
Analogie
to
the
former
,
201
All
the
former
works
may
be
craticularly
performed
,
202
FINIS
.
Courteous
Reader
,
be
pleased
to
take
notice
of
these
Books
following
,
which
are
very
usefull
and
necessary
for
all
Merchants
,
Tradesmen
,
Accomptants
,
they
are
sold
by
Nicholas
Bourn
at
the
South
entrance
of
the
Royall
Exchange
.
INtroduction
to
Merchants
Accompts
,
compiled
by
Iohn
Gollines
,
Student
in
the
Mathematicks
,
and
Professour
of
Merchants
Accompts
.
Tabula
Foeneratoria
,
or
Tables
for
the
forbearance
and
discompts
of
money
;
likewise
Tables
for
the
forbearance
,
discompt
and
purchase
of
Annuities
to
31
years
,
at
the
rate
of
6
per
Centum
,
per
Annum
,
Calculated
by
Roger
Clavell
,
Gent.
Student
in
the
Mathematicks
.
Manuel
of
Millions
or
Accompts
ready
cast
up
,
whereby
you
may
both
suddenly
and
truly
know
the
value
of
any
Commoditie
at
any
price
whatsoever
:
the
second
Edition
:
to
which
is
aded
a
second
Part
of
very
great
use
,
by
Richard
Hodgetts
.
Posthuma
Fosteri
,
the
description
of
a
Ruler
,
upon
which
is
inscribed
divers
Scales
,
and
the
uses
thereof
;
with
divers
Propositions
in
Astronomie
,
Navigation
and
Dialling
,
with
the
delineating
of
Horizontal
Dialls
,
by
Samuel
Foster
,
late
Professour
of
Astronomie
in
Gresham-Colle●ge
.
These
following
are
Books
of
other
sorts
:
Bread
for
the
poor
,
an
Advancement
of
the
English
Nation
;
promised
by
the
Inclosure
of
the
Wastes
and
Common
grounds
of
England
,
by
Adam
Moore
,
Gent.
Bills
of
Lading
in
all
Languages
,
for
shipping
forth
of
Merchants
goods
to
all
Countreys
of
Trading
.
Indentures
for
Virginia
,
Bermoodas
,
New-England
,
and
all
other
new
Plantations
.
A
Confutation
of
Athisme
,
by
Dr.
Dove
.
Bishop
Downame
,
on
the
Lords
prayer
.
Popular
Errours
in
Physick
,
by
Dr.
Primrose
.
Sweedish
Wars
compleat
in
four
Volumns
,
4o.
Youths
Guide
,
or
the
Fathers
Legacie
to
his
Son.
Dr.
Preston
,
of
Faith
and
Love.
Merchants
Mirrour
,
or
Directions
for
the
perfect
Booking
or
Surveying
of
his
Estate
;
Framed
by
way
of
Debitor
and
Creditor
,
after
the
so
termed
Italian
manner
;
and
Compiled
by
Richard
Dafforne
,
of
London
Accomptant
.
Certain
Sermons
,
preached
by
Dr.
Preston
at
Cambridge
and
London
upon
these
Texts
following
,
2d
to
the
Chronicles
7
,
14.5
of
the
Ephesians
,
32.
and
the
5
to
the
Ephesians
,
22
,
23
,
24.
called
the
Golden
Scepter
,
4o.
Dr.
Featly
,
Handmaid
to
Devotion
.
Surgions
Mate
,
by
Mr.
Woodall
,
Folio
.
Notes, typically marginal, from the original text
Notes for div A40031-e51860
*
Subindicall
line
I
call
that
line
upon
the
plain
which
lies
perpendicularly
under
the
Index
,
as
the
substilar
doth
under
the
Stile
.
I
mean
perpendicularly
in
respect
of
the
plain
,
not
of
the
Horizon
.
If
this
way
of
finding
the
two
points
I
and
X
,
seeme
not
so
expedite
,
you
must
remember
that
all
the
work
here
is
done
without
the
two
forementioned
calculated
Tables
.