THE
FIRST
PART
OF
THIS
BOOKE
,
Shewing
the
vse
of
the
First
side
of
the
Instrument
,
for
the
working
of
Proportions
both
simple
and
compounded
,
and
for
the
ready
and
easie
resolving
of
questions
both
in
Arithmetique
,
Geometrie
,
and
Astronomie
,
by
Calculation
.
CHAP.
I.
Of
the
Description
,
and
vse
of
the
Circles
in
this
First
side
.
1
THere
are
two
sides
of
this
Instrument
.
On
the
one
side
,
as
it
were
in
the
plaine
of
the
Horizon
,
is
delineated
the
proiection
of
the
Sphare
.
On
the
other
side
there
are
divers
kindes
of
Circles
,
divided
after
many
severall
waies
;
together
with
an
Index
to
be
opened
after
the
manner
of
a
paire
of
Compasses
.
And
of
this
side
we
will
speake
in
the
first
place
.
2
The
First
,
or
outermost
circle
is
of
Sines
,
from
5
degrees
45
minuts
almost
,
vntill
90.
Every
degree
till
30
is
divided
into
12
parts
,
each
part
being
5
min:
from
thence
vntill
50
deg.
into
sixe
parts
which
are
10
min:
a
peece
:
from
thence
vntill
75
degrees
into
two
parts
which
are
30
minutes
a
peece
.
After
that
vnto
85
deg.
they
are
not
divided
.
3
The
Second
circle
is
of
Tangents
,
from
5
degrees
45
min:
almost
,
untill
45
degrees
.
Every
degree
being
divided
into
12
parts
which
are
5
min:
a
peece
.
4
The
Third
circle
is
of
Tangents
,
from
45
degrees
untill
84
degrees
15
minutes
.
Each
degree
being
divided
into
12
parts
,
which
are
5
min:
a
peece
.
5
The
Sixt
circle
is
of
Tangents
from
84
degrees
till
about
89
degrees
25
minutes
.
The
Seventh
circle
is
of
Tangents
from
about
35.
min:
till
6
degrees
.
The
Eight
circle
is
of
Sines
,
from
about
35
minutes
til
6
degrees
.
6
The
Fourth
circle
is
of
Vnaequall
Numbers
,
which
are
noted
with
the
Figures
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
1.
Whether
you
vnderstand
them
to
bee
single
Numbers
,
or
Tenns
,
or
Hundreds
,
or
Thousands
,
&c.
And
every
space
of
the
numbers
till
5
,
is
divided
into
100
parts
,
but
after
5
till
1
,
into
50
parts
.
The
Fourth
circle
also
sheweth
the
true
or
naturall
Sines
,
and
Tangents
.
For
if
the
Index
bee
applyed
to
any
Sine
or
Tangent
,
it
will
cut
the
true
Sine
or
Tangent
in
the
fourth
circle
.
And
wee
are
to
knowe
that
if
the
Sine
or
Tangent
be
in
the
First
,
or
Second
circle
,
the
figures
of
the
Fourth
circle
doe
signifie
so
many
thousands
.
But
if
the
Sine
or
Tangent
be
in
the
Seventh
or
Eight
circle
,
the
figures
in
the
Fourth
circle
signifie
so
many
hundreds
.
And
if
the
Tangent
bee
in
the
Sixt
circle
,
the
figures
of
the
Fourth
circle
,
signifie
so
many
times
tenne
thousand
,
or
whole
Radij
.
And
by
this
meanes
the
Sine
of
23°
,
30′
will
bee
found
3987
:
and
the
Sine
of
it's
complement
9171.
And
the
Tangent
of
23°
,
30′
will
be
found
4348
:
and
the
Tangent
of
it's
complement
,
22998.
And
the
Radius
is
10000
,
that
is
the
figure
1
with
foure
cyphers
,
or
circles
.
And
hereby
you
may
finde
out
both
the
summe
,
and
also
the
difference
of
Sines
,
and
Tangents
.
7
The
Fift
circle
is
of
Aequall
numbers
,
which
are
noted
with
the
figures
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
0
;
and
every
space
is
divided
into
100
aequall
parts
.
This
Fift
circle
is
scarse
of
any
use
,
but
onely
that
by
helpe
thereof
the
given
distance
of
numbers
may
be
multiplied
,
or
divided
,
as
neede
shall
require
.
As
for
example
,
if
the
space
betweene
1
⌊
00
and
1
⌊
0833+
bee
to
bee
septupled
.
Apply
the
Index
vnto
1
⌊
0833+
in
the
Fourth
circle
,
and
it
will
cut
in
the
Fift
circle
03476+
;
which
multiplyed
by
7
makes
24333
:
then
againe
,
apply
the
Index
vnto
this
number
24333
in
the
Fift
circle
,
and
it
will
cut
in
the
Fouth
circle
1
⌊
7512+
.
And
this
is
the
space
betweene
1
⌊
00
and
1
⌊
0833+
septupled
,
or
the
Ratio
betweene
100
,
and
108⅓
seven
times
multiplied
into
it selfe
.
And
contrarily
,
if
1
⌊
7512
bee
to
bee
divided
by
7
:
Apply
the
Index
vnto
1
⌊
7568
in
the
fourth
circle
,
and
it
will
cut
in
the
fit
circle
24333
:
which
divided
by
7
giveth
03476+
.
Then
againe
vnto
this
Number
in
the
Fift
circle
apply
the
Index
,
and
in
the
Fourth
circle
it
wil
cut
vpon
1
⌊
0833+
for
the
Septupartion
sought
for
.
The
reason
of
which
Operation
is
,
because
this
Fift
circle
doth
shew
the
Logarithmes
of
Numbers
.
For
if
the
Index
be
applyed
unto
any
number
in
the
Fourth
circle
,
it
will
in
the
Fift
circle
cut
vpon
the
Logarithme
of
the
same
number
,
so
that
to
the
Logarithme
found
you
praefixe
a
Caracteristicall
(
as
Master
Brigs
termes
it
)
one
lesse
then
is
the
number
of
the
places
of
the
integers
proposed
(
which
you
may
rather
call
the
Graduall
Number
)
.
So
the
Logarithme
of
the
number
2
will
bee
found
0.30103
.
And
the
Logarithme
of
the
Number
43
⌊
6
will
bee
found
1.63949
.
Numbers
are
multiplied
by
Addition
of
their
Logarithmes
:
and
they
are
Divided
by
Substraction
of
their
Logarithmes
.
8
In
the
middest
among
the
Circles
,
is
a
double
Nocturnall
instrument
,
to
shew
the
hower
of
the
night
.
9
The
right
line
passing
through
the
Center
,
through
90
,
and
45
I
call
the
Line
of
Vnitie
,
or
of
the
Radius
.
10
That
Arme
of
the
Index
which
in
euery
Operation
is
placed
at
the
Antecedent
,
or
first
terme
'
,
I
call
the
Antecedent
arme
:
and
that
which
is
placed
at
the
consequent
terme
,
I
call
the
Consequent
Arme.
CHAP.
II.
Of
the
Operation
of
the
Rule
of
Proportion
:
and
also
of
Multiplication
,
and
Division
.
1.
Theoreme
.
IF
of
three
numbers
given
,
the
first
divide
the
second
and
the
quotient
multiply
the
third
,
the
product
shall
be
the
fourth
proportionall
,
to
the
three
numbers
given
.
Theoreme
.
If
of
three
numbers
given
,
the
second
divide
the
first
,
and
the
quotient
divide
the
third
;
this
later
quotient
shall
be
the
fourth
proportionall
,
to
the
three
numbers
given
.
Neither
is
it
materiall
whether
of
the
two
numbers
after
the
first
be
second
,
or
third
.
2
And
note
that
in
Reciprocal
proportion
,
that
terme
by
which
the
question
is
made
;
But
in
Direct
proportion
the
terme
that
is
homogeneal
thereto
,
is
the
first
terme
,
or
the
Antecedent
of
the
first
ratio
.
3
And
therefore
out
of
these
foundations
thus
layd
,
(
if
you
rightly
conceive
the
nature
of
the
Logarithmes
)
doth
follow
the
finding
out
of
the
fourth
proportionall
by
this
Instrument
:
whereof
this
is
the
Rule
.
Open
the
Armes
of
the
Instrument
to
the
distance
of
the
first
,
and
second
number
:
then
bring
the
Antecedent
arme
,
or
that
which
stood
vpon
the
first
number
vnto
the
third
,
and
so
the
consequent
arme
,
keeping
the
same
opening
,
will
shew
the
fourth
number
sought
for
.
In
which
operation
these
foure
things
are
diligently
to
be
considered
.
First
,
in
constituting
the
places
of
each
number
in
the
fourth
circle
;
whether
the
figures
written
in
the
spaces
doe
signifie
Vnites
,
Tenns
,
or
Hundreds
,
&c.
Secondly
,
if
that
arme
which
sheweth
the
fourth
proportionall
,
doe
reach
beyond
the
line
of
the
Radius
;
that
then
you
doe
account
the
fourth
in
a
new
circle
or
degree
.
Thirdly
,
whether
the
fourth
number
sought
,
ought
to
be
greater
,
or
lesser
then
the
third
.
For
if
a
fourth
number
bee
offered
greater
then
the
third
,
when
it
should
be
lesse
,
or
lesse
then
the
third
when
it
should
be
greater
;
it
is
a
signe
that
that
number
doth
appertaine
to
a
circle
of
another
degree
.
Fourthly
,
that
looke
what
true
distance
was
betweene
the
first
and
second
,
that
the
same
bee
supposed
betweene
the
third
and
the
fourth
,
and
also
on
the
same
part
.
4
And
for
because
Multiplication
and
Division
,
have
a
certaine
implicite
proportion
:
we
will
speake
of
them
in
the
first
place
.
5
In
Multiplication
,
As
an
Vnite
is
to
one
of
the
factores
(
or
numbers
to
be
multiplied
:
)
so
is
the
other
of
the
factores
,
to
the
product
.
And
the
product
of
two
numbers
shall
have
so
many
places
as
there
be
in
both
the
factores
,
if
the
lesser
of
them
exceede
so
many
of
the
first
figures
of
the
product
:
But
if
it
doe
not
exceede
,
it
will
have
one
lesse
.
6
And
in
Division
,
As
the
Divisor
is
to
an
Vnite
;
so
is
the
Dividend
,
to
the
Quotient
.
And
the
Quotient
shall
have
so
many
places
,
as
the
Dividend
hath
more
then
the
Divisor
,
if
the
Divisor
exceede
so
many
of
the
first
figures
of
the
Dividend
:
but
if
it
doe
not
exceede
,
it
shall
have
one
place
more
.
7
Wherefore
let
this
rule
bee
still
carefully
kept
in
minde
:
that
In
Multiplication
the
first
terme
of
the
implicite
proportion
is
evermore
1
:
And
in
Division
,
the
first
terme
is
the
Divisor
.
And
thus
much
concerning
the
operation
of
Proportion
,
Multiplication
,
and
Division
,
I
thought
meete
to
admonish
,
least
hereafter
in
Multiplying
,
or
Dividing
,
or
seeking
out
a
fourth
proportional
,
wee
be
constrained
to
repeate
the
same
things
many
times
over
.
8
An
example
of
Multiplication
.
How
many
pence
are
there
in
47li.
9
sh
?
For
because
1
shilling
containes
12
pence
,
and
1
pound
containes
20
shillings
,
that
is
240
pence
:
you
shall
multiply
47
by
240
,
and
9
by
12
,
and
then
adde
together
the
products
.
In
the
first
Multiplication
.
1
·
47
∷
240
·
11280
·
For
set
the
Armes
of
the
Index
at
1
and
47
in
the
fourth
circle
;
and
then
bring
the
Antecedent
arme
(
which
stood
at
1
)
vnto
240
,
and
the
Consequent
arme
will
shew
11280.
Againe
in
the
second
Multiplication
.
1
·
9
∷
12
·
108
·
For
set
the
two
Armes
of
the
Index
at
1
and
9
in
the
fourth
circle
;
and
bring
the
Antecedent
arme
unto
12
,
and
the
Consequent
arme
will
shew
108.
Lastly
,
adde
together
11280
and
108
and
the
summe
11388
will
be
the
number
of
pence
contained
in
the
said
summe
of
47
li
,
9sh.
9
An
example
of
Division
.
How
many
pounds
,
and
shillings
are
in
11388
pence
?
Divide
11388
by
240
:
the
division
is
thus
.
240
·
1
∷
11388
·
47
⌊
5
—
For
set
the
two
Armes
of
the
Index
at
240
,
and
1
in
the
fourth
circle
:
and
then
bring
the
Antecedent
arme
(
which
stood
at
240
)
unto
11388
;
and
the
Consequent
arme
will
shew
47
and
almost
an
halfe
.
But
how
many
shillings
that
excesse
doth
containe
wil
appeare
,
if
first
you
finde
by
Multiplication
that
11280
pence
are
contained
in
47
li:
which
subducted
from
11388
there
will
remaine
for
the
excesse
108
pence
.
Afterwards
by
division
you
may
seeke
how
many
shillings
are
in
108
pence
:
the
diuision
is
thus
.
12
·
1
∷
108
·
9.
For
set
the
Armes
of
the
Index
at
12
and
1
:
then
bring
the
Antecedent
arme
(
which
stood
at
12
)
unto
108
;
and
the
Consequent
arme
will
shew
9.
9
Any
Fraction
given
may
bee
reduced
into
Decimal
parts
,
thus
.
Set
the
Antecedent
arme
of
the
Index
at
the
Denominator
of
the
Fraction
given
,
in
the
fourth
circle
,
and
the
Consequent
arme
at
the
Numerator
,
then
keeping
the
same
distance
,
bring
the
Antecedent
arme
unto
1
,
and
the
consequent
arme
will
shew
the
decimal
parts
.
So
●40/●●●
is
0
⌊
75.
And
19/48
is
0
⌊
396
—
CHAP.
III.
Now
follow
certaine
examples
of
Proportion
.
Example
I.
IF
54
Elnes
of
Holland
bee
solde
for
96
shillings
,
for
how
many
shillings
was
9
elnes
sold
?
The
termes
given
are
54
·
96
∷
9
·
According
to
the
2
Chap.
3
Sect.
Set
one
of
the
armes
of
the
Index
at
the
Antecedent
terme
54
in
the
fourth
circle
,
and
the
other
arme
at
the
consequent
terme
96
:
then
keeping
that
distance
,
bring
the
Antecedent
arme
vnto
9
;
and
the
consequent
arme
beyond
the
line
of
the
Radius
will
shew
16
for
the
fourth
proportionall
,
according
to
the
considerations
in
the
2
Chap.
Sect.
3.
Therefore
54
·
96
∷
9
·
16
·
are
proportionals
.
And
16
shillings
is
the
price
of
9
Elnes
.
Example
II.
If
108
bushels
of
corne
be
sufficient
for
a
company
of
Souldiers
keeping
a
Fort
,
for
36
dayes
,
How
many
dayes
will
12
bushels
suffice
that
same
number
of
Souldiers
?
The
termes
giuen
are
108
·
36
∷
12
·
Set
one
Arme
of
the
Index
at
the
Antecedent
terme
108
in
the
fourth
circle
,
and
the
other
Arme
at
the
consequent
terme
12
,
(
being
mindfull
of
the
considerations
in
the
2
Chap.
3
Sect.
)
then
keeping
that
same
distance
,
bring
the
Antecedent
arme
vnto
36
;
and
the
consequent
arme
will
shew
4.
Therefore
108
·
36
∷
12
·
4
·
shall
bee
proportionals
.
And
4
is
the
number
of
dayes
sought
for
.
Example
III.
There
is
layd
vp
in
a
Fort
so
much
corne
as
will
suffice
for
756
Souldiers
which
keepe
that
Fort
,
for
196
dayes
:
how
many
Souldiers
will
that
same
corne
suffice
for
364
dayes
?
The
Proportion
is
reciprocall
,
therefore
the
termes
given
are
364
∶
756
∷
196
·
Set
one
Arme
of
the
Index
at
the
Antecedent
terme
364
,
and
the
other
Arme
at
the
consequent
196
:
and
keeping
the
same
distance
,
bring
the
Antecedent
arme
vnto
756
;
and
the
consequent
arme
will
shew
407+
:
And
therefore
for
so
many
Souldiers
will
the
corne
laid
vp
suffice
for
364
dayes
,
or
13
moneths
.
Example
IIII.
There
is
a
Tower
whose
height
I
would
measure
with
a
Quadrant
.
I
take
two
Stations
in
the
same
right
line
from
the
Tower
:
and
at
either
Station
having
observed
the
height
through
the
sights
,
I
finde
that
the
perpendicular
cutteth
in
the
nearer
Station
28
degrees
7
minutes
almost
:
and
in
the
further
Station
21
degr.
58
min.
almost
:
and
betweene
both
the
Stations
,
the
distance
was
76
feet
.
The
Rule
of
measuring
heights
by
two
Stations
is
contained
in
these
Theoremes
.
Theor.
As
the
difference
of
the
Tangents
of
the
arches
cut
in
either
station
,
is
to
the
distance
betweene
the
stations
;
so
is
the
Tangent
of
the
lesser
arch
,
to
the
nearer
distance
from
the
Tower
.
Againe
Theor.
As
the
Radius
is
to
the
Tangent
of
the
greater
arch
;
so
is
the
nearer
distance
found
,
to
the
height
.
And
therefore
because
by
the
1
Chap.
6
Sect.
the
Tangents
of
the
arches
28°
,
7′
—
,
and
21°
58′
—
are
5342
,
and
4032
whose
difference
is
1310
;
the
proportions
will
be
First
,
1310
·
76
∷
tang.
21°
,
58′
—
·
234
·
Wherefore
234
feet
is
the
nearer
distance
.
Second
Radius
·
tang.
28°
,
7′
—
∷
234
·
125
·
Wherefore
125
feet
is
the
height
sought
for
.
Example
V.
To
finde
the
Declination
of
the
Sunne
the
9th
day
of
May.
The
place
of
the
Sunne
for
every
day
,
may
be
had
nere
inough
out
of
this
Table
,
by
Adding
vnto
the
place
of
the
Sun
in
the
beginning
of
that
moneth
so
many
degrees
,
as
there
are
dayes
past
in
that
moneth
:
But
if
the
number
of
degrees
exceed
30
,
the
excesse
is
to
be
accounted
in
the
Signe
next
following
.
Wherefore
the
9th
of
May
the
place
of
the
Sun
is
♉
20+9
,
that
is
♉
29
:
which
is
59
degr.
distant
from
the
next
Aequinoctiall
point
.
The
place
of
the
Sunne
,
in
the
beginning
of
every
Moneth
.
Ianuary
|
♑
|
21
|
February
|
♒
|
22
|
March
|
♓
|
20
|
April
|
♈
|
21
|
May
|
♉
|
20
|
Iune
|
♊
|
19
|
Iuly
|
♋
|
18
|
August
|
♌
|
13
|
Septemb.
|
♍
|
18
|
October
|
♎
|
17
|
Novemb
|
♏
|
18
|
Decemb
|
♐
|
19
|
These
things
being
knowne
,
the
Rule
is
delivered
in
this
Theoreme
.
Theor.
As
the
Radius
is
to
the
sine
of
the
sunnes
distance
from
the
next
Aequinoctiall
point
;
so
is
the
sine
of
the
sunnes
greatest
declination
,
to
the
sine
of
the
declination
sought
for
.
The
proportion
will
be
Radius
·
sine
59°
∷
sine
2●°
,
30′
·
sine
19°
●9′·
And
so
much
is
the
Declination
sought
for
.
Example
VI.
To
finde
the
Right
ascension
of
the
Sunne
,
the
9th
day
of
May.
Seeke
the
place
of
the
Sunne
for
the
day
proposed
in
the
former
Table
;
and
the
Sunnes
distance
from
the
next
Aequinoctial
point
,
as
in
the
former
example
.
These
things
being
knowne
,
the
Rule
is
by
one
of
these
two
Theoremes
.
Theor.
As
the
Radius
,
is
to
the
sine
of
the
complement
of
the
sunnes
greatest
declination
;
so
is
the
the
tangent
of
the
sunnes
distance
from
the
next
Aequinoctiall
point
,
to
the
tangent
of
the
distance
of
the
right
ascension
of
the
sunne
,
from
the
same
Aequinoctiall
point
.
Or
Theor.
As
the
tangent
of
the
greatest
declination
of
the
Sunne
,
is
to
the
Radius
;
so
is
the
tangent
of
the
declination
of
the
sunne
for
the
time
proposed
,
vnto
the
sine
of
the
right
ascension
of
the
sunne
from
the
next
Aequinoctial
point
.
The
proportions
will
be
either
Radius
·
sine
66°
,
30′
∷
tang.
59°
·
tang
56°
46′
·
Or
,
tang.
23°
,
30′
·
Radius
∷
tang.
19°
,
●9′
·
sine
●6°
4●′
▪
Of
Continuall
proportion
,
Or
of
Progression
Geometricall
.
1
THE
Ratio
of
a
Progression
is
the
quotient
of
the
consequent
terme
divided
by
his
antecedent
.
And
therefore
in
the
Instrument
it
is
the
distance
taken
betweene
the
termes
in
the
fourth
circle
,
by
the
opening
of
the
Index
.
2
To
Double
,
Triple
,
or
Multiply
how
often
soever
any
Ratio
given
,
is
nothing
else
but
so
often
to
put
together
the
said
space
or
distance
between
the
termes
,
as
is
shewed
in
Chap
1.
Sect.
7.
As
for
example
,
if
the
Ratio
60
,
to
65
be
proposed
to
be
septupled
.
Set
the
Armes
of
the
Index
at
60
,
and
65
:
and
then
with
the
same
opening
,
bring
the
Antecedent
arme
which
was
at
60
,
vnto
1
,
and
the
consequent
arme
will
cut
1
⌊
08384
in
the
fourth
circle
,
and
03476
●
in
the
fift
circle
:
this
latter
number
being
multiplied
by
7
maketh
23444
;
vnto
which
number
in
the
fift
circle
applying
the
Index
,
it
will
in
the
fourth
circle
cut
1
⌊
●●●2
,
which
is
the
multiplied
number
sought
for
.
But
because
in
a
little
Instrument
,
the
arch
cut
in
the
fift
circle
,
cannot
be
estimated
exactly
:
and
a
small
error
in
the
beginning
often
repeated
,
by
multiplying
is
made
great
:
it
is
the
most
safe
way
,
to
take
the
Logarithmes
of
the
termes
of
the
Ratio
out
of
the
Canon
,
and
to
multiply
them
by
the
number
given
:
As
I
have
done
in
these
examples
.
Logarithmes
taken
out
of
the
Canon
.
100
|
2.0000000
|
104
|
2.0170333
|
105
|
2.0211893
|
106
|
2.0253059
|
106⅔
|
2.0280287
|
107
|
2.0293838
|
107½
|
2.0314085
—
|
108
|
2.0334238
—
|
108⅓
|
2.0347621
|
108½
|
2
.
0354297+
|
108¼
|
2.0364293
—
|
109
|
2.0374265
|
110
|
2.0313927
|
Or
if
the
Canon
be
wanting
;
you
may
come
neerer
the
marke
,
if
that
first
single
opening
of
the
Index
being
kept
,
and
the
Antecedent
arme
set
at
1
;
you
transferre
the
Antecedent
Arme
,
unto
that
place
which
the
consequent
arme
doth
cutt
;
and
the
consequent
arme
will
cut
the
same
space
duplicated
.
Then
holding
the
consequent
arme
in
that
same
place
,
open
the
Antecedent
arme
unto
1.
and
afterward
with
that
duplicated
opening
,
bring
the
Antecedent
arme
to
the
duplicated
space
,
and
the
consequent
arme
will
cut
the
space
quadrupled
.
Thirdly
,
bring
the
Anticedent
arme
unto
the
quadrupled
space
and
the
consequent
arme
,
keeping
that
duplicated
opening
,
will
cut
the
space
sextupled
.
Lastly
,
having
againe
taken
a
single
opening
,
bring
the
Antecedent
arme
unto
the
number
,
or
space
sextupled
;
and
the
consequent
arme
will
shew
the
Ratio
sought
for
septupled
.
And
this
manner
of
working
may
bee
observed
for
as
many
Multiplications
as
you
please
of
any
Ratio
given
.
3.
The
Ratio
,
and
first
terme
being
given
,
to
continue
the
same
unto
any
number
of
termes
.
Open
the
Armes
of
the
Index
,
the
one
unto
the
Antecedent
of
the
ratio
given
,
and
the
other
unto
the
consequent
:
then
the
same
opening
being
kept
,
bring
the
Antecedent
Arme
vnto
the
first
terme
given
,
and
the
consequent
arme
will
shewe
the
second
terme
:
againe
bring
the
Antecedent
arme
vnto
the
second
terme
found
,
and
the
consequent
arme
will
shew
the
third
.
After
that
bring
the
Antecedent
arme
unto
the
third
terme
found
,
and
the
consequent
arme
wil
shew
the
fourth
.
And
in
this
manner
you
may
proceed
as
farre
as
you
please
.
As
for
example
,
If
a
Progression
in
the
ratio
2
unto
5
,
beginning
at
8
,
Or
if
a
Progression
in
the
ratio
100
unto
108
,
beginning
at
5
,
is
to
be
instituted
;
the
termes
in
either
progression
will
be
as
followeth
.
1
|
2
·
5
∷
8
·
20
|
2
|
20
|
3
|
50
|
4
|
125
|
5
|
312
⌊
5
|
6
|
781
⌊
25
|
7
|
1953
⌊
125
|
1
|
100
·
108
∷
5
·
5
⌊
4
|
2
|
5
⌊
4
|
3
|
5
⌊
832
|
4
|
6
⌊
29856
|
5
|
6
⌊
8024448
|
6
|
7
⌊
346640384
|
7
|
7
⌊
93437161472
|
4
Theor
;
The
ratio
of
any
former
terme
,
in
a
row
of
concinuall
proportionals
,
vnto
any
of
the
termes
following
,
is
aequall
to
the
ratio
of
the
first
terme
unto
the
second
,
multiplied
into
it selfe
according
to
the
distance
of
that
latter
terme
from
the
former
.
As
for
example
,
The
Ratio
5
⌊
4
unto
6
⌊
●024448
which
is
the
third
terme
from
it
,
is
aequall
to
the
ratio
of
100
vnto
108
triplicated
,
Or
as
the
Cube
of
100
unto
the
Cube
of
108.
Wherefore
5
The
Ratio
,
and
the
first
terme
of
the
Progression
being
given
,
to
finde
out
any
other
terme
required
.
First
,
multiply
the
ratio
given
into
it selfe
,
according
to
the
distance
of
the
terme
sought
from
the
first
terme
,
by
the
2
sect:
then
say
As
1
is
to
that
mul●●ple
found
;
so
is
the
first
terme
,
to
the
terme
sought
for
.
Example
,
What
will
be
the
amount
of
26th
,
in
7
yeares
by
Interest
upon
Interest
,
after
the
rate
of
20
pence
in
the
pound
?
Because
1
pound
which
is
20
shillings
containeth
60
groates
,
and
20
pence
conteine
5
groates
,
the
rate
of
the
Interest
will
bee
60
unto
65
,
Or
100
unto
108
⌊
●●
×
●
But
the
first
terme
is
given
26
,
unto
which
there
are
to
be
acquired
7
other
termes
in
continuall
proportion
.
First
,
by
the
4
Chap:
and
2
Sect.
Let
the
ratio
given
be
septuplicated
,
that
is
multiplied
sevenfold
into
it selfe
,
which
will
be
1
⌊
552.
Then
set
the
consequent
arme
of
the
Index
vnto
the
septuplicate
number
1
⌊
552
,
and
open
the
Antecedent
arme
vnto
1
;
and
keeping
the
same
opening
,
bring
the
Antecedent
arme
unto
26
;
and
the
consequent
arme
will
shew
45
⌊
5312
li
the
amount
sought
for
.
6
The
Ratio
,
and
any
other
terme
of
the
Progression
being
given
,
to
finde
the
first
terme
.
First
multiply
the
Ratio
given
into
it selfe
according
to
the
difference
of
the
terme
given
from
the
first
terme
.
Then
say
As
that
multuple
is
unto
1
;
so
is
the
terme
given
,
unto
the
first
terme
.
Example
,
what
summe
in
7
yeares
did
amount
unto
45
⌊
5312
li
by
Interest
upon
Interest
after
the
rate
of
100
unto
108
⌊
33+
?
First
the
ratio
being
septuplicat
,
by
the
4
Chap.
2
Sect.
will
be
1
⌊
7512.
Then
setting
the
Antecedent
arme
of
the
Index
to
that
septuple
1
⌊
7512
,
open
the
consequent
arme
unto
1
:
and
keeping
the
same
opening
,
bring
the
Arme
unto
45
⌊
5312
li:
and
the
consequent
arme
will
shew
26
,
which
was
the
stocke
,
or
summe
of
money
,
from
which
that
amount
did
arise
.
7
The
Ratio
,
the
First
terme
,
and
any
other
terme
of
a
Progression
being
given
,
to
finde
how
many
places
the
terme
given
is
from
the
first
terme
.
First
say
,
As
the
first
terme
is
unto
1
;
so
is
the
other
terme
given
,
unto
the
ratio
multiplied
into
it selfe
according
to
the
distance
of
that
terme
from
the
first
.
Wherefore
according
to
1
Chap.
7
Sect.
and
3
Chap.
2
Sect.
by
helpe
of
the
fift
circle
,
see
how
often
the
ratio
given
,
is
contained
in
that
multuple
found
.
Example
,
In
how
many
yeares
did
26
li
,
by
Interest
upon
Interest
after
the
rate
of
100
unto
108
⌊
33+
+
increase
unto
45
⌊
5312
li
?
First
set
the
Antecedent
arme
of
the
Index
at
26
,
and
the
consequent
arme
at
1
:
and
keeping
the
same
opening
bring
the
Antecedent
arme
vnto
45
⌊
5312
,
and
the
consequent
arme
will
shew
1
⌊
7●12
:
to
which
in
the
fift
circle
answereth
24333.
Then
because
unto
the
consequent
terme
of
the
ratio
108
⌊
33+
+
there
agrees
in
the
fift
circle
03476
,
divide
24333
by
03476
,
and
the
quotient
will
be
7
,
the
number
of
yeares
sought
for
.
8
The
First
terme
,
and
any
other
terme
of
the
Progression
being
given
to
finde
the
ratio
of
the
Progression
.
First
say
,
As
the
first
terme
is
unto
1
:
so
is
the
other
terme
given
,
unto
the
ratio
multiplied
into
it selfe
according
to
the
distance
of
that
terme
from
the
first
.
Wherefore
according
to
Chap.
1.
Sect.
7
,
by
helpe
of
the
fift
circle
,
Let
the
multuple
found
be
divided
by
the
distance
of
the
terme
from
the
first
.
Example
,
26
li
by
Interest
upon
Interest
in
7
yeares
amounted
unto
45
⌊
5312
:
what
was
the
ratio
of
the
Interest
compared
unto
100
?
First
set
the
Antecedent
arme
of
the
Index
at
26
,
and
the
consequent
arme
at
1
:
and
then
keeping
the
same
opening
,
bring
the
Antecedent
arme
unto
45
⌊
5312
:
and
the
consequent
arme
will
shew
1
⌊
7●12
:
unto
which
in
the
fift
circle
answereth
24333
and
this
number
being
divided
by
7
will
give
03476+
:
unto
which
agreeth
in
the
fourth
circle
108
⌊
33+
+
the
consequent
terme
of
the
Interest
sought
for
.
9
Two
numbers
being
given
to
finde
as
many
Midle
proportionals
betweene
them
as
you
will
.
Divide
the
distance
of
the
greater
number
given
from
the
lesser
in
the
fourth
circle
iustly
taken
,
according
to
Chap.
1.
Sect.
7
,
by
helpe
of
the
fift
circle
into
equall
segments
,
one
more
then
are
the
number
of
Midle
proportionals
sought
for
.
All
these
segments
added
orderly
to
the
first
terme
,
doe
distinguish
the
termes
of
the
Progression
which
you
seeke
.
Example
,
Let
there
be
foure
Midle
proportionals
,
sought
out
betweene
8
and
19
⌊
90656.
Apply
the
Index
unto
8
in
the
fourth
circle
,
and
it
will
cut
in
the
first
circle
9039
:
also
set
the
Index
unto
19
⌊
91
—
in
the
fourth
circle
,
and
it
will
cut
in
the
fift
circle
2989
;
which
number
because
it
reaches
beyond
the
Vnite
●●ne
,
is
indeede
12989
,
according
to
Chap.
1.
Sect.
7
,
and
so
is
the
distance
iustly
taken
.
Then
subducting
9039
from
12989
,
there
will
remaine
3950
:
which
divided
by
4
+
1
,
the
quotient
will
give
790.
wherfore
9039
+
790
,
scil.
9829
in
the
fift
circle
doth
agree
with
9
⌊
6
in
the
fourth
circle
,
which
is
the
first
midle
proportionall
.
And
9039
+
twise
790
,
scil.
10619
in
the
fift
circle
doth
agree
with
11
⌊
52
in
the
fourth
circle
,
which
is
the
second
midle
proportionall
.
And
in
this
manner
13
⌊
824
,
and
16
⌊
5888
will
be
found
the
third
and
fourth
midle
proportionals
.
10
Theor.
If
from
the
Ratio
given
,
being
multiplied
in
it selfe
according
to
the
number
of
termes
,
you
subduct
1
,
and
multiply
the
remainder
,
by
the
antecedent
of
the
ratio
.
It
will
be
As
the
difference
of
the
termes
of
the
ratio
,
is
unto
the
product
;
so
is
the
first
terme
,
to
the
sum
of
the
Progression
.
As
for
example
if
the
Ratio
of
the
Progression
be
R
to
S:
and
the
difference
of
R
taken
out
of
S
be
D.
and
the
first
terme
of
the
Progression
α
:
and
the
whole
summe
of
the
termes
Z
it
shall
be
,
D
·
rat
.
multa
—
1
in
R
∷
α
·
Z.
Which
is
the
very
Theoreme
it selfe
expressed
in
Symboles
of
words
:
that
it
may
more
easily
be
fixed
in
the
phantasie
.
Which
proportion
also
wee
must
consider
,
doth
hold
both
alternly
,
and
conuersly
.
This
Theoreme
may
otherwayes
be
expressed
,
by
the
equality
which
the
product
of
the
two
midle
termes
hath
to
the
product
of
the
two
extreames
,
thus
Ratio
multiplicata
—
1
in
R
in
α
=
Z
in
D.
This
manner
of
setting
downe
Theoremes
,
whether
they
be
Proportions
,
or
Equations
,
by
Symboles
or
notes
of
words
,
is
most
excellent
,
artificiall
,
and
doctrinall
.
Wherefore
I
earnestly
exhort
every
one
,
that
desireth
though
but
to
looke
into
these
noble
Sciences
Mathematicall
,
to
accustome
themselues
unto
it
:
and
indeede
it
is
easie
,
being
most
agreeable
to
reason
,
yea
even
to
sence
.
And
out
of
this
working
may
many
singular
consectaries
be
drawne
:
which
without
this
would
,
it
may
be
,
for
ever
lye
hid
.
As
in
this
present
proportion
:
because
it
is
D
·
rat
.
multa
—
1
in
●
∷
α
·
Z
·
wherefore
Rat.
multa
—
1
in
R
·
D
∷
Z
·
α
·
And
α
·
Z
∷
D
·
rat
.
multa
—
1
in
R
These
are
exceeding
easie
:
but
this
following
is
more
difficult
,
and
requireth
attention
.
In
the
former
equation
it
was
Rat.
multa
—
1
in
R
in
α
=
Z
in
D
Now
because
Rat.
multa
—
1
in
R
in
α
,
and
Rat.
multa●●
in
R
in
α
—
R
in
α
,
and
Rat.
multa
in
R
—
R
in
α
,
and
Rat.
multa
in
α
—
α
in
R
,
are
equall
one
to
another
,
and
also
to
Z
in
D
,
these
aequations
shall
also
be
consectarious
.
and
and
and
And
besides
these
many
more
.
The
practise
whereof
I
leave
to
the
industry
of
the
studious
Reader
,
especially
having
delivered
the
whole
Art
of
such
operations
in
my
Clavis
Mathematica
.
Some
of
these
I
have
occasion
to
use
in
the
sections
following
.
11.
Therefore
the
Ratio
of
a
Progression
,
and
the
first
terme
,
and
the
number
of
termes
being
given
,
to
finde
the
summe
of
the
whole
progression
.
For
this
operation
the
rule
,
or
Theoreme
last
before
serveth
:
for
by
it
Example
.
If
an
Annuity
of
5
li
,
be
detained
7
yeares
,
what
will
be
the
amount
thereof
by
interest
upon
interest
after
the
rate
of
100
unto
108
?
Now
because
the
Amount
is
the
summe
of
the
Progression
,
whereof
the
first
terme
is
the
annuitie
,
Multiply
the
ratio
into
it selfe
according
to
the
number
of
yeares
,
by
Chap.
4.
Sect.
2
,
and
it
will
be
1
⌊
71382
:
from
which
if
you
subduct
an
vnite
,
there
remaines
0
⌊
71382
,
which
multiplied
by
100
maketh
71
⌊
382
:
then
set
the
Armes
of
the
Index
at
1
and
71
⌊
382
;
and
bring
the
Antecedent
arme
which
stood
at
1
,
unto
5
the
first
terme
:
and
the
consequent
arme
will
cut
356
⌊
915
(
that
is
Rat.
multa
—
1
in
R
in
α
)
and
lastly
this
number
being
divided
by
108
—
100
,
scil.
by
8
,
the
quotient
will
give
44
⌊
6●40
,
for
the
summe
of
the
whole
progression
:
and
so
much
is
the
amount
sought
for
.
12.
The
Ratio
,
the
Number
of
Termes
,
and
the
Summe
of
a
Progression
being
given
,
to
find
the
first
terme
.
By
the
converse
of
the
foregoing
Theoreme
,
it
is
manifest
,
that
Rat
∶
multa
—
1
in
R
·
D
∷
Z
·
α
·
The
declaration
in
words
was
in
the
10
Sect.
Example
.
If
an
Annuitie
detained
7
yeares
by
Interest
upon
Interest
,
after
the
rate
of
100
unto
108
,
did
increase
unto
44
⌊
6140li.
how
much
was
that
Annuitie
?
Multiply
the
Ratio
into
it selfe
according
to
the
number
of
yeares
(
per
cap.
4.
sect.
2.
)
and
the
product
will
be
1
⌊
7●●82
:
from
which
if
you
subduct
an
Vnite
,
there
remaines
0
⌊
7●●82
:
this
being
multipled
by
100
doth
make
71
⌊
●82
.
Then
say
71
⌊
382
·
8
∷
44
⌊
6140
·
5
·
for
the
First
terme
:
which
was
the
Annuitie
sought
for
.
13
The
Ratio
,
the
First
terme
,
and
the
Summe
of
the
Progression
being
given
,
to
find
the
Number
of
termes
.
By
the
Theoreme
in
the
10.
Sect.
it
was
α
·
Z
∷
D
·
rat
.
multa
—
1
in
R
·
Wherefore
set
the
Antecedent
arme
of
the
Index
unto
the
Antecedent
terme
of
the
ratio
,
and
the
consequent
arme
unto
the
Summe
of
the
Progression
:
and
with
that
same
opening
,
bring
the
Antecedent
arme
unto
the
difference
of
the
termes
of
the
ratio
;
and
the
consequent
arme
wil
shew
a
number
(
that
is
Rat.
mul.
—
1
in
R
;
)
which
if
you
divide
by
the
Antecedent
terme
of
the
ratio
,
and
unto
the
quotient
adde
an
Vnite
,
you
shall
have
the
ratio
multiplied
into
it selfe
according
to
the
number
of
termes
.
Therefore
taking
the
distance
betweene
the
termes
of
the
ratio
,
with
the
armes
of
the
Index
,
measure
by
helpe
of
the
fift
circle
(
per
Cap.
4.
Sect.
7.
)
how
often
that
distance
may
be
found
in
the
multiplied
ratio
:
for
so
many
are
the
termes
of
the
progression
.
Example
.
If
an
Annuitie
of
5
li
detained
by
Interest
upon
interest
after
the
rate
of
100
to
108
,
increased
unto
44
⌊
6140li.
How
many
yeares
was
the
Annuitie
detained
?
Set
the
Antecedent
arme
of
the
Index
at
5
,
and
the
consequent
arme
at
44
⌊
6140
,
and
with
the
same
opening
bring
the
Antecedent
arme
unto
8
,
and
the
consequent
arme
will
shew
71
⌊
382
(
that
is
Rat.
multa
—
1
in
R
:
)
this
number
being
divided
by
100
,
will
be
0
⌊
71382
:
and
if
unto
the
quotient
you
adde
1
,
you
shall
have
1
⌊
71●82
(
the
ratio
multiplied
into
it selfe
according
to
the
number
of
yeares
:
)
unto
which
in
the
fift
circle
answereth
2338
:
but
unto
108
in
the
fift
circle
there
answereth
0334.
divide
therefore
2338
by
0334
;
and
the
quotient
will
be
7
,
for
the
number
of
yeares
sought
for
.
Or
such
questions
may
be
more
easily
performed
by
this
Theoreme
,
which
the
industrious
Reader
may
by
himselfe
practise
.
14
Theor.
If
the
summe
of
the
whole
progression
be
divided
by
the
ratio
multiplied
into
it selfe
according
to
the
number
of
termes
,
the
quotient
will
be
the
first
terme
;
and
that
summe
given
will
be
the
last
terme
,
of
another
progression
,
having
the
same
ratio
but
one
terme
more
.
15
And
because
a
summe
of
mony
the
amount
whereof
in
any
number
of
yeares
given
,
by
Interest
upon
interest
,
doth
equall
an
Annuitie
so
long
detained
,
is
equivalent
to
the
same
Annuitie
;
and
the
amount
of
an
Annuitie
is
the
summe
of
a
Progression
continued
from
that
Annuitie
.
If
therefore
an
Annuitie
for
any
number
of
yeares
be
divided
by
the
ratio
multiplied
into
it selfe
according
to
the
number
of
yeares
;
the
quotient
will
be
the
just
price
of
an
Annuitie
to
endure
for
so
long
.
And
because
by
the
10
and
11
Sect.
it
hath
beene
shewed
that
.
Therefore
by
the
14
Sect.
will
be
equall
to
the
price
of
an
Annuitie
in
ready
money
,
which
shall
be
the
Rule
for
the
operation
following
.
Wherefore
also
Rat.
multa
—
1
in
R
in
α
=
Rat.
multa
in
D
in
Pret.
which
proportion
is
thus
enuntiated
in
words
.
Theor.
If
from
the
Ratio
multiplied
into
it selfe
according
to
the
Number
of
yeares
you
subduct
an
Vnite
,
and
the
remainder
be
multiplied
continually
by
the
Antecedent
of
the
Ratio
,
and
the
Annuitie
it selfe
:
And
againe
,
If
the
Ratio
multiplied
into
it selfe
,
according
to
the
number
of
yeares
be
multiplied
continually
by
the
difference
of
the
termes
of
the
Ratio
,
and
by
the
Price
:
both
those
products
will
be
equall
.
Example
.
An
Annuitie
of
5
li
,
to
endure
for
7
yeares
,
is
to
be
sold
:
what
is
it
worth
in
ready
money
,
after
the
Rate
of
100
unto
108
?
The
Ratio
multiplied
into
it selfe
according
to
the
number
of
yeares
(
per
Cap.
4
Sect.
2
)
is
1
⌊
7138
:
subduct
1
,
and
there
will
remaine
0
⌊
71382
:
which
multiplied
by
100
maketh
71
⌊
382.
Then
set
the
Antecedent
arme
of
the
Index
at
1
,
and
the
consequent
arme
71
⌊
382
:
and
keeping
that
same
opening
,
bring
the
Antecedent
arme
unto
5
;
and
the
consequent
arme
will
shew
356
⌊
915
;
keepe
this
(
for
it
is
.
After
that
set
the
Antecedent
arme
of
the
Index
at
1
,
and
the
consequent
arme
unto
the
multuple
ratio
1
⌊
7138
;
and
with
the
same
opening
bring
the
Antecedent
arme
unto
8
;
and
the
consequent
arme
will
cut
13
⌊
7106
,
keepe
this
number
also
(
for
it
is
Rat.
mul.
in
α
)
.
Lastly
,
place
the
Antecedent
arme
of
the
Index
at
13
⌊
7106
,
and
the
consequent
arme
at
1
;
and
with
the
same
opening
bring
the
Antecedent
arme
vnto
356
⌊
015
:
and
the
consequent
arme
will
shew
26
⌊
032li.
which
is
the
iust
Price
of
an
Annuitie
of
5
li
in
readie
money
.
16
And
by
the
last
praecedent
Theoreme
or
Rule
,
also
which
Theoreme
may
bee
enuntiated
in
words
as
was
there
shewed
.
Example
.
An
Annuitie
for
7
yeares
is
bought
for
26
⌊
032
li.
after
the
rate
of
100
unto
108
,
by
Interest
upon
interest
:
how
much
was
that
Annuitie
?
The
Ratio
multiplied
into
it selfe
for
the
number
of
7
yeares
(
per
Cap
:
4
,
Sect.
2
)
is
1
⌊
7138
:
which
multiplied
continually
by
8
,
and
by
the
price
,
doth
make
356
⌊
915.
Divide
this
number
found
,
by
71
⌊
382
,
(
which
is
the
multuple
ratio
)
it selfe
lesse
by
an
Vnite
,
and
multiplied
by
100
:
and
the
quotient
will
be
5
li
,
the
Annuitie
sought
for
.
17.
Also
by
Ratiocination
from
that
praecedent
rule
will
follow
this
proposition
which
is
thus
enuntiated
in
words
.
Theor.
If
the
product
of
the
Antecedent
of
the
ratio
multiplied
by
the
Annuitie
be
divided
by
it selfe
,
being
diminished
by
the
product
of
the
difference
of
the
termes
of
the
ratio
multiplied
by
the
Price
:
the
quotient
will
bee
equall
to
the
ratio
multiplied
into
if
selfe
according
to
the
number
of
yeares
.
As
If
the
ratio
be
100
unto
108
;
the
Annuitie
5
li:
and
the
price
thereof
26
⌊
032
li.
Set
the
Antecedent
arme
of
the
Index
at
1
,
and
the
consequent
arme
at
8
,
the
difference
of
the
termes
of
the
ratio
:
and
with
the
same
opening
bring
the
Antecedent
arme
unto
26
⌊
032
—
:
and
the
consequent
arme
will
shew
208
⌊
256
:
which
subducted
from
500
,
there
will
remaine
291
⌊
744
,
for
the
divisor
.
Set
therefore
the
Antecedent
arme
of
the
Index
at
the
divisor
291
⌊
744
,
and
the
consequent
arme
at
1
:
and
with
the
same
opening
,
bring
the
Antecedent
arme
unto
the
dividend
500
:
and
the
consequent
arme
will
shew
1
⌊
71●8
,
which
is
the
ratio
multiplied
into
it selfe
according
to
the
number
of
yeares
.
And
by
this
number
so
found
it
will
be
easie
(
by
helpe
of
the
fift
circle
)
the
ratio
of
the
Interest
being
given
,
to
finde
the
continuance
of
the
Annuitie
.
Example
.
An
Annuitie
of
5
il
,
was
bought
for
26
⌊
032
li
—
,
after
the
rate
of
100
unto
108
:
how
many
yeares
is
it
to
last
?
First
seeke
out
the
ratio
multiplied
into
it selfe
according
to
the
number
of
yeares
,
which
will
be
1
⌊
71●8
,
according
as
was
even
now
shewed
in
this
Sect.
to
this
in
the
fift
circle
there
answereth
2338
:
but
unto
108
there
answereth
in
the
fift
circle
0334.
Divide
therefore
2338
by
0334
;
and
the
quotient
will
be
7
for
the
number
of
yeares
sought
for
.
CHAP.
V.
Of
the
Quadrating
,
and
Cubing
of
Numbers
,
Sides
or
Rootes
:
and
of
the
Extraction
of
the
Quadrate
,
and
Cubic
side
,
or
roote
,
out
of
Numbers
,
or
Powers
given
.
1
IF
a
number
,
side
,
or
roote
be
multiplied
into
it selfe
;
the
product
will
be
a
Quadrat
.
And
if
a
quadrate
bee
multiplied
into
his
owne
side
,
or
roote
,
the
product
will
be
a
Cube
.
Wherefore
1
·
N
∷
N
·
Q
·
and
1
·
N
∷
Q
·
C
·
2
If
therefore
a
number
be
given
to
be
Quadrated
.
Set
the
Antecedent
arme
of
the
Index
at
1
,
and
the
consequent
arme
at
the
number
given
:
then
with
the
some
opening
bring
the
Antecedent
arme
to
the
number
given
;
and
the
consequent
arme
will
shew
the
Quadrat
thereof
.
And
the
number
of
figures
in
a
Quadrat
of
a
single
root
(
or
which
doth
not
exceede
9
)
is
easily
found
out
by
those
Rules
,
that
have
beene
before
delivered
concerning
Multiplication
.
But
if
a
side
or
root
consist
of
more
figures
then
one
;
for
each
figure
after
the
first
it
acquireth
two
more
places
of
figures
.
And
if
any
of
the
figures
of
the
root
given
be
decimal
parts
,
cut
off
from
the
Quadrat
found
,
twice
so
many
of
the
last
figures
for
decimals
.
Example
1.
The
Quadrat
of
the
side
7
is
required
.
Say
1
·
7
∷
7
·
49
Set
therefore
the
Antecedent
arme
of
the
Index
at
1
,
and
the
consequent
arme
at
7
;
and
with
that
opening
bring
the
Antecedent
arme
unto
7
;
and
the
consequent
arme
will
shew
49
which
is
the
Quadrat
sought
for
.
Example
11.
The
Quadrat
of
the
side
57
is
required
.
Set
the
Antecedent
arme
of
the
Index
at
1
;
and
the
consequent
arme
at
57
:
then
with
that
same
opening
bring
the
antecedent
arme
unto
57
;
and
the
consequent
arme
will
shew
3249
,
which
is
the
Quadrat
sought
for
,
consisting
of
foure
places
.
Example
111.
The
Quadrat
of
the
side
,
or
root
570
is
required
.
Having
found
as
before
,
3249
for
the
quadrat
of
the
side
57
:
put
thereunto
two
circles
:
and
it
will
bee
324900
,
the
quadrat
sought
for
.
Example
1111.
The
quadrat
of
the
side
574
is
required
.
Set
the
Antecedent
arme
of
the
Index
at
1
,
and
the
corsequent
arme
at
574
;
and
with
the
same
opening
bring
the
Antecedent
arme
unto
574
;
and
the
consequent
arme
will
shew
329476
,
the
quadrat
sought
for
consisting
of
sixe
figures
:
but
the
two
last
figures
cannot
at
all
be
discerned
by
the
Instrument
.
3
If
a
number
be
given
to
be
Cubed
.
Set
the
Antecedent
arme
of
the
Index
at
1
,
and
the
consequent
arme
at
the
number
given
;
and
with
that
same
opening
,
bring
the
Antecedent
arme
unto
the
number
given
;
and
the
consequent
arme
will
shew
the
Quadrat
;
then
bring
the
Antecedent
arme
unto
the
Quadrat
,
and
the
consequent
arme
with
that
same
opening
will
shew
the
Cube
of
that
side
given
.
The
number
figures
in
a
Cube
of
a
single
side
,
or
root
which
doth
not
exceede
9
,
is
easily
found
by
that
which
hath
beene
before
delivered
concerning
Multiplication
:
But
if
the
side
,
or
root
consist
of
more
figures
then
one
;
for
each
figure
after
the
first
it
obtaineth
three
more
places
of
figures
.
And
if
any
of
the
figures
of
the
root
given
be
decimal
parts
,
cut
off
from
the
Cube
found
thrice
so
many
of
the
last
figures
for
decimal
parts
.
Example
.
The
Cube
of
the
side
,
or
root
7
is
required
.
Say
,
1
·
7
∷
7
·
49
·
againe
1
·
7
∷
49
·
343
·
Set
therefore
the
Antecedent
arme
of
the
Index
at
1
,
and
the
consequent
arme
at
7
;
and
with
that
opening
bring
the
Antecedent
arme
unto
7
,
and
the
consequent
arme
will
shew
49
the
quadrat
thereof
:
Then
set
the
Antecedent
arme
at
49
;
and
the
consequent
arme
(
with
that
first
opening
)
will
shew
343
,
which
is
the
desired
Cube
of
the
side
proposed
.
Another
example
,
The
Cube
of
the
side
,
or
root
57
is
required
.
Set
the
Antecedent
arme
of
the
Index
at
1
,
and
the
consequent
arme
at
57
;
and
with
that
same
opening
bring
the
Antecedent
arme
unto
57
;
and
the
consequent
arme
will
shew
the
quadrat
3249.
Then
set
the
Antecedent
arme
at
3249
;
and
the
consequent
arme
keeping
the
former
opening
will
shew
185193
,
which
is
the
required
Cube
of
that
side
proposed
,
consisting
of
six
places
:
but
the
two
last
figures
cannot
be
known
by
the
Instrument
.
Example
III.
The
Cube
of
the
side
570
is
required
.
Hauing
found
as
before
the
Cube
of
the
side
,
or
root
57
to
be
185193
:
put
thereunto
three
circles
;
and
it
will
be
185
193000
the
Cube
sought
for
.
Examples
of
greater
Cubes
,
it
will
be
needlesse
to
set
downe
.
4
The
Extraction
of
the
Square
,
or
Quadrat
root
,
or
side
,
is
done
by
helpe
of
the
fift
circle
,
after
this
manner
.
Set
the
Index
at
the
Quadrat
proposed
;
and
of
that
number
which
it
cuts
in
the
fift
circle
,
take
halfe
:
then
set
the
Index
at
that
halfe
;
and
it
will
shew
in
the
fourth
circle
,
the
side
,
or
root
sought
for
.
But
you
must
know
that
if
the
number
which
is
the
Quadrat
proposed
,
have
onely
two
places
of
Integers
,
the
side
,
or
root
consisteth
of
one
figure
.
But
if
it
have
more
places
of
Integers
,
dividing
them
by
2
,
the
quotient
will
give
the
true
number
of
figures
in
the
root
,
if
it
measure
it
exactly
;
or
one
lesse
then
the
true
number
if
any
thing
remaine
.
Example
I.
The
side
,
or
root
of
the
Quadrat
49
is
required
.
Set
the
Index
at
49
in
the
fourth
circle
,
and
it
will
cut
in
the
fift
circle
6902
;
indeede
1.
6902
having
the
graduall
number
1
praefixed
,
because
1
in
the
fourth
circle
signifieth
10
,
one
circuition
thereof
being
finished
:
the
halfe
whereof
is
0.
8451.
Then
set
the
Index
at
0.
8451
,
in
the
fift
circle
;
and
it
will
cut
in
the
fourth
circle
7
,
the
side
,
or
root
sought
for
.
Example
II.
The
side
,
or
root
of
the
quadrat
3249
is
required
.
Set
the
Index
at
3249
in
the
fourth
circle
,
and
it
will
cut
in
the
fift
circle
5118
;
indeede
3.
5118
prefixing
the
gradual
number
3
,
because
1
in
the
fourth
circle
signifieth
1000
,
three
circuitions
thereof
being
finished
:
the
halfe
whereof
is
1.
7559.
Then
set
the
Index
at
7559
,
omitting
the
prefixed
graduall
number
1
;
and
it
will
shew
in
the
fourth
circle
57
the
side
sought
for
,
consisting
of
two
figures
,
because
1
in
the
fourth
circle
signifieth
10.
Example
III.
The
side
of
the
quadrat
329476
is
required
.
Set
the
Index
at
329476
in
the
fourth
circle
,
and
it
will
cut
in
the
fift
circle
5178
;
indeede
5.
5178
prefixing
the
graduall
number
5
,
because
1
in
the
fourth
circle
signifieth
100000
,
five
circuitions
thereof
being
past
over
;
the
halfe
whereof
is
2.
7589
;
Then
set
the
Index
at
7589
,
omitting
the
graduall
number
2
prefixed
thereto
;
and
it
will
shew
in
the
fourth
circle
574
the
side
,
or
root
sought
for
,
consisting
of
three
figures
,
because
1
in
the
fourth
circle
doth
signifie
100.
5
The
Extraction
of
the
Cubic
roote
,
or
side
is
done
by
helpe
of
the
fift
circle
after
this
manner
.
Set
the
Index
at
the
Cube
proposed
;
and
that
number
which
it
cuts
in
the
fift
circle
divide
by
3.
Then
set
the
Index
at
that
third
part
,
and
it
will
shew
in
the
fourth
circle
the
side
,
or
roote
sought
for
.
And
you
must
know
that
if
the
Cube
proposed
have
onely
three
places
of
Integers
,
the
side
,
or
roote
thereof
consisteth
of
one
figure
:
But
if
it
have
more
places
of
Integers
;
divide
them
by
3
,
the
quotient
will
give
the
true
number
of
the
figures
of
the
Root
,
if
it
measure
it
exactly
;
or
one
lesse
then
the
true
number
if
any
thing
remaine
.
Example
I.
The
side
,
or
roote
of
the
Cube
343
is
required
.
Set
the
Index
at
343
in
the
fourth
circle
,
and
it
will
cut
in
the
fift
circle
5353
;
indeede
2.
5353
prefixing
the
graduall
number
2
,
because
1
in
the
fourth
doth
signifie
100
,
two
circuitions
thereof
being
past
over
:
the
third
part
whereof
is
8451.
Then
set
the
Index
at
8451
,
and
it
will
shew
in
the
fourth
circle
7
,
the
side
sought
for
.
Example
II.
The
side
,
or
roote
of
the
Cube
185193
is
required
.
Set
the
Index
at
185193
in
the
fourth
circle
;
and
it
will
cut
in
the
fift
circle
2677
;
indeede
5.
2677
prefixing
the
graduall
number
5
,
because
1
in
the
fourth
circle
doth
signifie
100000
,
five
circuitions
thereof
being
past
over
:
the
third
part
whereof
is
1.
7599.
Then
set
the
Index
at
7599
,
omitting
the
prefixed
graduall
number
1
;
and
in
the
fourth
circle
it
will
shew
57
,
the
side
,
or
root
sought
for
,
consisting
of
two
figures
,
because
1
in
the
fourth
circle
doth
signifie
10.
Examples
of
greater
Cubes
it
will
be
needlesse
to
set
downe
.
CHAP.
VI.
Of
Duplicated
,
and
Triplicated
proportion
.
And
first
of
Duplicated
proportion
.
Theoreme
.
LIke
Plaines
are
in
a
Duplicated
ratio
,
that
is
,
As
the
Quadrats
of
their
homologal
sides
.
And
therefore
questions
in
the
which
the
sides
of
like
planes
are
compared
,
doe
appertaine
vnto
this
place
.
And
it
is
to
be
noted
,
that
if
three
numbers
be
given
,
in
which
As
the
quadrat
of
the
first
,
is
unto
the
quadrat
of
the
second
;
so
ought
the
third
to
be
unto
a
number
sought
for
.
Let
it
be
thus
done
,
As
the
first
number
,
is
to
the
second
;
so
is
the
third
to
afourth
;
And
againe
As
the
first
number
,
is
to
the
second
;
so
is
the
fourth
now
found
,
to
the
number
sought
for
.
Example
I.
There
are
two
like
rectangle
Areae
,
or
plaines
,
the
length
of
the
greater
,
is
40
feete
,
the
length
of
the
lesser
24
feet
:
each
of
them
paved
with
paving
tiles
;
the
greater
hath
1200
tiles
:
how
many
shall
the
lesser
have
?
The
Areae
,
or
plaines
are
one
to
the
other
,
as
the
quadrats
of
the
longitudes
given
.
And
the
proportion
is
direct
.
Say
therefore
1600
(
Q
∶
40
)
·
576
(
Q
∶
24
)
∷
1200
·
432
·
which
is
the
number
of
tiles
contained
in
the
pavement
of
the
lesser
Areae
,
or
plaine
.
Example
II.
How
many
Acres
of
woodland
measured
with
a
Perch
,
of
18
feete
,
are
there
in
73
Acres
of
champane
land
,
measured
with
a
Perch
of
16½
feete
?
The
measures
given
(
18
,
16½
)
being
reduced
into
their
lest
termes
,
are
as
12
unto
11
:
and
the
quadrats
of
these
numbers
,
are
,
144
,
and
121.
And
the
Proportion
is
Reciprocall
.
Say
therefore
144
(
Q
∶
12
)
·
121
(
Q
∶
11
)
∷
73
·
and
so
many
are
the
Acres
of
Wood
land
.
Of
Triplicated
proportion
.
3
Theor.
Like
Solids
are
in
a
Triplicated
ratio
,
that
is
,
As
the
Cubes
of
their
homologal
sides
.
And
therefore
questions
in
which
the
sides
of
like
solids
are
compared
,
doe
appertaine
unto
this
place
.
4
If
three
numbers
be
given
in
the
which
,
As
the
Cube
of
the
first
is
to
the
Cube
of
the
second
;
so
is
the
third
number
to
a
number
sought
for
.
Let
it
be
thus
done
,
As
the
first
number
is
to
the
second
;
so
is
the
third
to
a
fourth
:
Againe
,
As
the
first
is
to
the
second
;
so
is
the
fourth
now
found
unto
a
fift
.
And
thirdly
,
As
the
first
is
to
the
second
;
so
is
that
fift
to
the
number
sought
for
.
Example
.
If
4●/100lib.
of
gunpowder
,
suffice
to
charge
a
Gun
,
whereof
the
concave
diameter
is
inch
1½
.
How
many
pounds
of
powder
will
suffice
to
charge
a
Gunne
,
whose
concave
diameter
is
7
inches
?
The
capacities
are
one
to
another
,
as
the
Cubes
of
the
diameters
.
And
the
proportion
is
direct
.
Say
therefore
3
⌊
375
(
C
∶
1
⌊
5
)
·
343
(
C
∶
7
)
∷
0
⌊
43
·
4●●
+
wherefore
43
⌊
●lib.
of
Gunpowder
,
will
bee
needfull
to
be
had
.
Another
example
,
43
⌊
7lib.
of
Gunpowder
are
sufficient
to
charge
a
Gun
,
whose
diameter
in
the
concave
is
7
inches
:
now
there
is
another
sort
of
Gunpowder
,
much
more
strong
and
forcible
,
that
is
in
strength
unto
the
former
,
as
5
vnto
2
:
How
much
of
this
stronger
powder
,
will
suffice
to
charge
a
Gun
of
4
inches
diameter
?
Here
are
two
operations
:
the
first
seekes
out
,
how
much
of
that
stronger
powder
sufficeth
to
charge
a
Gun
of
7
inches
diameter
:
and
the
proportion
is
reciprocall
,
that
is
5
·
2
∷
43
⌊
7
·
15
⌊
48
The
second
operation
is
like
that
in
the
former
example
.
343
(
C
∶
7
)
64
(
C
∶
4
)
∷
15
⌊
48
·
3
⌊
●●●●
CHAP.
VII
.
Concerning
the
Measuring
of
Circles
,
Cones
,
Cylinders
,
and
Sphaeres
.
1
ARchimedes
in
a
peculiar
Treatise
found
the
proportion
of
the
Diameter
of
a
circle
to
the
Circumference
to
bee
a
very
smale
deale
greater
then
o●
7
unto
22
:
And
of
late
Ludolph
Van
Ceulen
insisting
in
the
same
steps
of
Archimedes
,
hath
more
precisely
found
it
to
be
of
1
vnto
3
⌊
141592653589793
but
for
our
Instrument
it
will
be
sufficient
to
take
the
ratio
of
1
vnto
3
⌊
1416
,
Or
of
0
⌊
3183+
unto
1
:
leaving
the
diligent
practizer
,
to
more
exactnesse
,
if
he
please
to
use
his
Pen.
And
note
that
the
Rules
following
are
set
downe
in
proportions
,
to
be
wrought
as
hath
been
taught
in
Chap.
2
,
Sect.
3.
Wherein
-
D
,
or
Diam
.
signifieth
the
Diameter
.
-
Dq
,
or
Q.
Diam
.
the
Quadrat
of
the
Diameter
.
-
Dc
,
or
C.
Diam
.
the
Cube
of
the
Diameter
.
-
R
,
or
Rad.
the
Radius
,
or
Semidiameter
.
-
P
,
or
Perif
.
the
Periferie
,
or
Circumference
.
-
Pq
,
the
Quadrat
of
the
Periferie
.
-
Long.
the
length
.
-
L
the
side
,
or
latus
-
Alt.
the
altitude
.
-
●
,
sheweth
that
the
two
magnitudes
betweene
which
it
is
set
,
are
to
be
multiplied
together
.
In
a
Circle
.
2.
The
Diameter
of
a
circle
being
given
,
to
finde
the
Periferia
.
Say
,
7
·
22
,
Or
1
·
3
⌊
1416
∷
Diam
·
Perif
·
Example
.
A
circle
is
given
,
the
Diameter
whereof
is
12
,
I
would
know
the
circumference
,
or
Periferia
of
it
.
Say
,
1
·
3
⌊
1416
∷
12
·
37
⌊
6992
·
3
The
Periferia
of
a
circle
being
giuen
,
to
finde
the
Diameter
.
Say
,
22
·
7
,
Or
1
·
0
⌊
3183
+
∷
Perif
·
Diam
·
4
The
Diameter
of
a
circle
being
giuen
,
to
finde
the
Area
.
Say
,
7
×
4
·
22
,
Or
1
·
0
⌊
7854
∷
Q
∶
Diam
·
Area
·
Or
els
1
·
3
⌊
1416
∷
Q
∶
Rad
·
Area
·
Example
.
A
circle
is
giuen
,
the
diameter
whereof
is
12
,
I
would
know
the
content
,
or
Area
of
it
.
Say
,
1
·
0
⌊
7854
∷
144
(
Q
∶
12
)
·
113
⌊
●●76
·
Or
1
·
3
⌊
1416
∷
36
(
Q
∶
6
)
·
113
⌊
0176
5
The
Area
of
a
circle
being
giuen
,
to
finde
the
Diameter
.
Say
,
22
·
7
×
4
,
Or
1
·
1
⌊
27324
∷
Area
·
Q
∶
Diam
·
Example
.
A
circle
is
giuen
,
the
content
whereof
is
113
⌊
0976
,
I
would
know
the
Diameter
of
it
.
Say
,
1
·
1
⌊
27324
∷
113
⌊
0976
·
144
·
6
The
Periferia
of
a
circle
being
giuen
,
to
finde
the
Area
.
Say
,
22
×
4
·
7
,
Or
1
·
0
⌊
0795775
∷
Q
∶
Perif
·
Area
·
7
The
Area
of
a
circle
being
giuen
,
to
finde
the
Periferia
.
Say
,
7
·
22
×
4
,
Or
1
·
12
⌊
●6637
∷
Area
·
Q
∶
Perif
·
In
a
Cone
.
8
The
side
of
a
right
Cone
,
and
the
Diameter
of
the
base
being
giuen
,
to
find
the
Superficies
.
Say
,
7
·
22
,
Or
1
·
3
⌊
1416
∷
½
D
×
L
·
Superf
·
Example
.
A
Cone
is
giuen
,
whereof
the
side
is
18
,
and
the
diameter
of
the
base
12
,
I
would
know
the
superficies
of
it
.
Say
,
1
·
3
⌊
1416
∷
108
(
½
D
×
L
)
·
33
⌊
●2928
·
9
The
Axis
,
or
height
of
a
right
Cone
,
and
the
Diameter
of
the
base
,
being
giuen
,
to
find
the
Soliditie
.
Say
,
7
×
4
·
22
,
Or
1
·
0
⌊
7854
∷
Dq
in
1/●
Axis
·
Soliditie
·
Example
.
A
Cone
is
giuen
,
whereof
the
Axis
is
18
,
and
the
Diameter
of
the
base
12
,
I
would
know
the
Solidity
.
Say
,
1
·
0
⌊
7854
∷
864
(
Dq
in
1/●
axis
)
·
678
⌊
2856
·
In
a
Cylinder
.
10
The
side
of
a
right
Cylender
and
the
Diameter
being
given
,
to
finde
the
Superficies
.
say
,
7
·
22
,
Or
1
·
3
⌊
1416
∷
Diam
∶
×
axem
·
Superfic
.
11
The
Side
of
a
right
Cilinder
,
and
the
Diameter
being
given
,
to
finde
the
Soliditie
.
Say
,
7
×
4
·
22
,
Or
1
·
0
⌊
7854
∷
Dq
×
L
·
Soliditie
·
12
The
Side
of
a
right
Cylinder
,
and
the
Circumference
P
,
being
given
,
to
finde
the
Soliditie
.
Say
,
22
×
4
·
7
,
Or
1
·
0
⌊
0795775
∷
Pq
×
L
·
solidit
·
In
a
Sphaere
.
13
The
Axis
,
or
Diameter
of
a
Sphaere
being
given
,
to
finde
the
Superficies
.
Say
,
7
·
22
,
Or
1
·
3
⌊
1416
∷
Dq
·
Superficies
·
14
The
Superficies
of
a
Sphaere
being
given
,
to
finde
the
Axis
.
Say
,
22
·
7
,
Or
1
·
0
⌊
31831
∷
Superfic
·
Dq
·
15
The
Segment
of
a
Sphaere
being
given
,
to
finde
the
Superficies
.
Say
,
7
·
22
,
Or
1
·
3
⌊
1416
∷
Q
∶
chord
of
½
Segm.
·
Superfic
·
16
The
Axis
,
or
Diameter
of
a
Sphaere
being
giuen
,
to
finde
the
Soliditie
.
Say
,
7
×
6
·
22
,
Or
1
·
0
⌊
5236
∷
Dc
·
Soliditie
·
Example
.
A
Sphaere
is
giuen
,
whereof
the
Axis
is
12
,
I
would
know
the
solidity
of
it
.
Say
,
1
·
0
⌊
5236
∷
1728
(
Dc
)
·
59●
⌊
6208
·
17
The
Soliditie
of
a
Sphere
being
giuen
,
to
finde
the
Axis
.
Say
,
22
·
7
×
6
,
Or
1
·
1
⌊
90986
∷
Soliditie
·
Dc
·
Example
.
A
Sphaere
is
giuen
,
the
Solidity
whereof
is
590
⌊
6208
,
I
would
know
the
Axis
thereof
.
Say
,
1
·
1
⌊
90986
∷
590
⌊
6208
·
1728
(
Dc
)
·
18
A
Segment
of
a
Sphere
being
giuen
to
finde
the
Soliditie
.
Say
,
First
.
As
the
altitude
of
the
other
Segment
,
is
to
the
altitude
of
the
Segment
giuen
:
so
is
that
altitude
of
the
other
Segment
increased
by
halfe
the
Axis
,
vnto
a
fourth
.
Then
againe
say
,
As
7×3
is
to
22
,
Or
as
1
is
to
10472
:
so
is
the
product
of
the
quadrat
of
halfe
the
chord
of
the
Periferia
of
that
Segment
,
multiplied
by
that
fourth
,
to
the
Soliditie
.
Viz.
7
×
3
·
22
,
Or
1
·
1
⌊
0472
∷
Q
∶
chorde
in
quartam
·
Solidit
·
19
For
note
that
a
Sphaere
,
is
equall
to
two
Cones
,
having
their
height
and
the
diamerer
of
their
base
,
the
same
with
the
Axis
of
the
Sphaere
.
Or
which
is
all
one
,
A
Sphaere
is
two
third
parts
,
of
a
Cylinder
,
having
the
height
and
the
diameter
of
the
base
the
same
with
the
Axis
of
the
Sphaere
.
CHAP.
VIII
.
Concerning
Plaine
,
and
Solide
Measures
.
1
THe
diuiding
of
the
Carpenters
ruler
into
Inches
,
and
halfe
,
and
quarters
,
and
halfe
quarters
of
Inches
,
that
is
of
every
Inch
into
eight
parts
,
is
most
inartificiall
,
and
vnfit
for
measuring
,
by
reason
of
the
manifold
denominations
,
which
must
be
brought
into
one
,
and
is
hard
to
bee
done
of
them
that
are
vnskilfull
.
I
would
wish
therefore
that
euery
Inch
were
diuided
into
10
parts
,
or
rather
that
the
foot
were
diuided
into
100
parts
,
which
is
best
of
all
:
for
then
there
will
neede
no
reduction
.
And
all
other
divisions
must
bee
reduced
vnto
this
,
by
these
Rules
following
.
2.
If
the
measures
be
taken
vpon
a
Ruler
diuided
into
Inches
and
tenth
parts
of
an
Inch
,
first
take
out
all
the
whole
feet
,
and
then
diuide
the
Inches
remaining
,
with
their
decimall
parts
if
there
be
any
by
12.
Example
.
How
many
feet
and
decimall
parts
of
a
foot
,
are
in
Inches
17
⌊
3
?
First
take
out
the
whole
foot
which
is
12
Inches
,
and
there
will
remaine
Inches
5
⌊
3
:
which
being
diuided
by
12
,
you
shall
haue
442
thousand
parts
almost
:
wherefore
Inches
17
⌊
3
,
is
feet
1
⌊
442
—
.
And
contrariwise
,
feet
1
⌊
442
—
,
shall
be
reduced
into
Inches
17
⌊
●
,
being
multiplied
by
12.
3.
If
the
measure
bee
taken
vpon
a
Ruler
diuided
into
inches
and
halfe
quarters
,
that
is
each
inch
into
8
parts
,
First
you
must
reduce
the
eight
parts
into
decimals
of
Inches
,
by
diuiding
the
number
of
parts
given
by
8
the
Denominator
thereof
:
and
afterward
by
the
former
Rule
,
reduce
the
Inches
,
and
decimall
parts
,
into
decimall
parts
of
a
foot
.
Example
.
How
many
feet
,
and
decimall
parts
of
a
foot
,
are
in
7
inches
,
and
5
eight
parts
?
First
diuide
the
5
eight
parts
by
8
,
and
you
shall
haue
625
thousand
parts
:
which
being
put
to
7
inches
,
will
make
inches
7
⌊
625
:
Again
diuide
these
by
12
,
as
was
shewed
in
the
former
rule
:
and
the
whole
measure
will
be
feete
0
⌊
635.
And
contrariwise
feet
0
⌊
635
,
will
be
reduced
into
inches
7
⌊
625
,
being
multiplied
by
12.
4.
I
must
aduise
all
those
that
haue
occasion
,
to
measure
Plaines
,
or
Solids
,
to
make
themselues
very
perfect
in
this
kind
of
Reduction
(
because
most
Rulers
they
shall
ordinarily
meet
withall
,
are
diuided
into
inches
,
and
halfe
quarters
)
which
will
be
very
easie
to
them
,
if
they
doe
but
remember
,
that
In
diuision
the
first
terme
of
the
proportion
implyed
,
is
the
Diuisor
it selfe
:
but
in
Multiplication
,
the
first
terme
is
euermore
1.
as
hath
beene
shewed
in
Chap:
2
,
Sect:
7.
And
therefore
,
presuming
on
the
diligence
of
the
Practiser
herein
,
I
shall
not
neede
in
this
kind
of
measuring
,
to
speake
any
more
of
inches
,
but
of
feet
and
decimall
parts
of
feet
,
as
if
the
Ruler
were
so
diuided
.
Of
Plaine
measures
.
5.
A
Parallelogram
,
or
foure
sided
rectangle
Superficies
,
being
proposed
,
to
find
the
length
of
a
Superficiall
foote
.
Take
with
your
Ruler
the
breadth
thereof
in
feet
,
and
decimals
of
a
foot
:
and
by
the
breadth
so
taken
diuide
1.
the
quotient
shall
be
the
length
of
a
superficiall
foot
.
Example
.
A
boord
is
feet
1
⌊
17
broad
,
how
much
thereof
will
make
a
foot
?
Diuide
1
by
1
⌊
17
,
the
quotient
will
bee
0
⌊
855
almost
:
so
much
shall
the
length
of
a
foot
be
,
which
multiplying
the
parts
by
12
,
will
giue
inches
10
⌊
26.
And
againe
those
parts
multiplied
by
8
,
will
giue
2
eight
parts
of
an
inch
.
Example
,
II.
In
tileing
,
or
healing
they
vse
to
reckon
by
the
Square
,
which
is
10
foot
euery
way
,
in
all
100
●eet
.
There
is
roofe
,
feet
16
⌊
2●
broad
,
how
much
thereof
maketh
a
Square
?
Diuide
100
by
16
⌊
25
the
quotient
will
be
6
⌊
●●4
almost
:
so
much
shall
be
the
length
of
one
square
;
which
multiplying
the
parts
by
12
,
will
be
6
feet
,
and
inches
1
⌊
8●●
—
almost
.
And
againe
those
parts
multiplied
by
8
,
will
giue
somewhat
more
then
6
eight
parts
of
an
inch
.
Example
,
III
:
In
pauing
they
vse
to
reckon
by
the
yard
,
which
is
3
feet
euery
way
,
in
all
9
feet
.
There
is
a
roome
to
bee
paued
,
which
is
feet
17
⌊
35
broad
;
how
much
thereof
maketh
a
yard
?
Diuide
9
by
17
⌊
35
,
the
quotient
will
be
0
⌊
519
almost
,
the
length
of
one
yard
.
6
:
A
fower
sided
rectangle
Superficies
,
with
all
the
opposite
sides
parallell
being
proposed
,
to
find
the
content
.
Take
with
your
Ruler
both
the
breadth
and
length
of
and
multiply
the
one
number
into
the
other
.
Example
,
A
board
is
feet
1
⌊
17
broad
,
and
feet
16
⌊
32
long
:
how
many
feet
doth
it
containe
in
all
?
Multiply
16
⌊
3●
by
1
⌊
17
the
product
will
be
feete
19
⌊
●
almost
:
the
whole
quantity
of
that
board
.
Example
.
II.
A
certaine
barne
tiled
,
hath
the
breadth
of
the
roofe
feet
,
16
⌊
2●
,
and
the
length
of
the
barne
is
feet
47
,
how
many
squares
of
tiling
hath
it
?
Double
the
length
(
that
you
may
haue
both
sides
of
the
roofe
)
and
it
will
be
94
,
which
being
multiplied
by
16
⌊
25
,
will
giue
feet
1527
⌊
●
.
Againe
diuide
those
feete
by
100
,
so
shall
you
haue
squares
15
⌊
275.
Example
.
III.
A
certaine
hall
paued
hath
the
breadth
feet
17
⌊
35
,
and
the
length
feet
30
⌊
5
,
how
many
yards
doth
it
containe
?
Multiply
17
⌊
35
by
30
⌊
5
,
the
product
will
be
feet
229
⌊
175.
Diuide
these
by
9
and
the
quotient
will
bee
yards
58
⌊
797.
7.
A
fower
sided
Superficies
with
the
two
sides
of
length
onely
parallel
being
proposed
,
to
find
the
content
thereof
.
Take
with
your
Ruler
the
length
of
the
two
parallell
sides
thereof
:
adde
both
those
numbers
together
,
multiply
halfe
that
summe
,
by
the
breadth
of
the
Superficies
taken
the
nearest
way
over
,
and
the
product
will
bee
the
content
thereof
.
Example
.
A
Trapezium
,
or
fower
sided
figure
is
proposed
,
hauing
two
sides
thereof
parallell
,
the
length
of
the
longer
parallel
side
is
feete
18
⌊
75
and
the
length
of
the
shorter
side
is
feete
14
⌊
4●
,
the
breadth
thereof
being
taken
the
nearest
way
over
is
feete
12
⌊
5
,
I
would
know
how
many
feete
are
contained
in
the
whole
Superficies
?
The
length
of
the
parallell
sides
are
feet
18
⌊
●5
,
and
14
⌊
45
,
which
added
together
make
33
⌊
●
,
halfe
whereof
is
16
⌊
●
,
which
multiplied
by
the
breadth
12
⌊
●
,
the
product
will
be
207
⌊
5
,
so
many
feet
are
contained
in
the
superficies
of
the
Trapezium
proposed
.
8.
A
fower
sided
Superficies
which
hath
none
of
the
sides
parallel
,
as
also
euery
plaine
figure
of
more
sides
the
fower
being
proposed
,
must
with
Diagoniall
lines
bee
diuided
into
triangles
.
And
note
that
euery
such
figure
containeth
so
many
triangles
as
it
hath
sides
,
abating
two
out
of
the
number
.
Then
those
triangles
are
to
be
measured
seuerally
as
followeth
.
9
To
find
the
content
,
or
Area
of
a
Triangle
.
Take
the
perpendicular
height
,
or
neerest
distance
betweene
the
base
or
knowne
side
,
and
the
angle
opposite
:
and
by
that
height
multiply
halfe
the
base
,
or
multiply
the
whole
base
by
halfe
that
perpendicular
height
:
and
the
product
shall
be
the
content
,
of
that
Triangle
.
But
if
it
be
an
Aequilater
triangle
:
say
,
1000
·
433
⌊
0127
∷
the
side
of
the
triangle
·
Area
·
10.
To
find
the
content
of
a
segment
of
a
Circle
,
whereof
the
Periferia
is
giuen
in
degrees
and
decimall
parts
.
First
say
,
As
100000
,
is
to
1745
⌊
32025
:
so
is
the
Arch
in
degrees
,
to
the
Arch
in
the
diuisions
of
the
Radius
.
keepe
this
number
found
.
Againe
by
the
6
Sect:
1
chap
:
find
out
the
true
sinus
of
the
Arch
given
.
Then
take
the
difference
of
these
two
numbers
found
,
by
subducting
the
Sinus
out
of
the
Arch.
And
lastly
multiply
halfe
that
difference
by
the
Radius
100000
,
the
product
shall
be
the
content
of
that
segment
.
11
The
chord
of
any
arch
,
together
with
the
Radius
,
or
semidiameter
of
the
whole
circle
being
giuen
,
to
finde
out
the
Arch
it selfe
.
Say
,
At
the
Radius
giuen
,
is
to
halfe
the
chord
(
reckoned
in
the
fourth
circle
)
so
is
100000
,
to
the
sinus
of
halfe
the
arch
(
to
be
reckoned
in
the
first
,
or
eighth
circle
)
.
Wherfore
double
the
arch
found
,
and
so
haue
you
the
arch
of
the
chord
proposed
.
12
To
finde
a
Quadrat
,
or
Square
equall
to
a
superficies
giuen
.
First
,
seeke
out
(
as
hath
beene
taught
)
the
content
of
that
superficies
:
then
take
the
quadrat
roote
thereof
by
Chap.
5
,
Sect.
4.
Of
Solid
measures
.
13
In
a
Columne
,
or
Cylinder
,
hauing
the
base
,
to
finde
how
much
of
it
maketh
a
foot
solid
.
By
a
Columne
I
meane
a
solid
body
arising
from
a
plaine
base
,
the
angular
lines
whereof
are
parallell
,
and
equall
:
and
if
the
angular
lines
make
right
angles
with
the
base
,
it
is
a
right
Columne
,
and
the
length
is
the
height
thereof
:
but
if
they
make
oblique
angles
,
it
is
an
Oblique
Columne
,
and
the
length
is
not
the
height
,
but
the
height
is
a
perpendicular
line
let
downe
from
the
top
of
the
Columne
vnto
the
base
,
extended
if
neede
bee
:
as
in
the
Diagrame
,
the
Solid
ABCDEFGH
,
is
an
Oblique
Columne
,
because
the
angular
line
EB
,
standeth
obliquely
vpon
the
side
of
the
base
BA
,
and
indeede
vpon
the
base
it selfe
.
Wherefore
the
height
of
it
shall
be
equall
to
the
line
FP
,
let
fall
from
the
top
vnto
the
base
extended
.
diagram
And
after
this
manner
also
a
Cylinder
,
and
a
Pyramid
,
and
a
Cone
is
esteemed
either
right
,
or
oblique
,
and
the
height
taken
accordingly
.
First
therefore
the
base
is
to
be
found
,
of
what
fashion
soeuer
it
is
,
as
hath
euen
now
beene
shewed
,
either
in
this
Chapter
,
or
in
the
last
before
:
and
then
diuide
1
by
that
same
base
:
the
quotient
shall
bee
,
the
height
of
a
Section
thereof
,
which
is
equall
to
one
foot
solid
.
Example
.
A
Columne
,
or
peece
of
tymber
,
whose
sides
are
all
parallel
,
hath
the
breadth
feete
1
⌊
75
,
and
the
thicknesse
thereof
is
feet
1
⌊
25
:
which
multiplied
together
the
product
will
be
2
⌊
1875.
Diuide
therefore
1
,
by
2
⌊
1875
:
and
the
quotient
shall
be
0
⌊
457143
almost
.
And
so
much
is
the
height
of
a
solid
foot
,
of
that
peece
of
timber
.
14
Hauing
the
Base
,
and
the
height
of
a
Columne
,
or
Cylinder
,
to
finde
the
whole
content
.
Multiply
the
base
into
the
height
,
and
the
product
shall
be
the
content
.
Example
.
A
Columne
hath
the
base
feet
2
⌊
1875
,
and
the
height
thereof
is
feet
17
⌊
34
,
how
many
feet
are
contained
in
the
whole
?
Multiply
the
base
2
⌊
1875
,
by
17
⌊
34
and
the
product
will
be
36
⌊
086875
,
so
many
solid
feet
are
contained
in
that
Columne
.
And
in
this
very
manner
may
you
finde
the
content
of
a
Cylinder
,
hauing
either
the
diameter
,
or
circumference
giuen
,
together
with
the
height
.
15
To
measure
tapering
timber
,
the
base
,
or
bases
thereof
,
together
with
the
height
being
giuen
.
A
Tapering
peece
of
timber
,
according
as
the
base
thereof
is
right
lined
,
or
circular
,
is
either
a
Pyramide
or
a
Cone
,
or
else
a
segment
of
one
of
these
two
:
If
it
be
a
compleat
Pyramide
,
or
Cone
,
it
hath
but
one
base
,
Multiply
that
base
by
⅓
of
the
height
,
and
the
product
shall
be
the
content
.
But
if
it
be
the
segment
of
a
Pyramide
,
or
Cone
,
First
finde
out
the
bases
at
both
the
ends
,
and
multiply
the
one
by
the
other
,
and
out
of
the
product
thereof
extract
the
Quadrat
root
:
then
adde
together
both
the
bases
,
and
that
quadrat
root
,
and
multiply
the
Aggregate
thereof
by
⅓
of
the
height
,
the
product
shall
be
the
content
.
Example
.
There
is
a
tapering
peece
of
timber
,
the
height
whereof
is
feet
12
⌊
6
,
and
the
breadth
of
the
base
at
the
greater
end
is
feet
1
⌊
75
,
and
the
thicknes
is
feet
1
⌊
32
,
which
multiplied
together
,
the
product
will
be
feet
2
⌊
3●
for
the
greater
base
:
the
breadth
of
the
base
at
the
lesser
end
is
feet
1
⌊
2
,
and
the
thicknesse
there
,
is
feete
0
⌊
91
,
which
multiplied
together
,
the
product
will
be
feet
1
⌊
052
for
the
lesser
base
.
Multiply
the
bases
together
,
the
product
will
be
2
⌊
52252
,
the
quadrat
root
whereof
is
1
⌊
588
almost
,
to
which
if
you
adde
the
summes
of
both
the
bases
,
the
aggregate
will
be
4
⌊
99
,
which
being
multiplied
by
⅓
of
the
height
,
scil.
4
⌊
2
,
the
product
will
be
feet
20
⌊
958
,
the
content
of
the
peece
of
timber
.
16
Wherfore
that
vulgar
manner
which
Carpenters
vse
in
measuring
of
tapering
timber
,
is
not
true
:
for
if
a
peece
of
timber
be
tapering
,
they
measure
it
in
the
very
midle
,
and
take
the
base
,
or
Section
there
,
multiplying
it
by
the
whole
length
.
Which
their
manner
of
working
,
Isay
,
is
erroneous
.
For
first
by
practise
a
content
will
be
giuen
,
euer
lesse
then
the
true
content
found
according
to
the
former
Sect
:
which
way
of
working
is
infallibly
true
,
as
is
Analitically
demonstrated
,
in
my
Clauis
mathematica
,
Cha.
20
,
Sect:
15.
And
secondly
Isay
,
that
the
product
of
that
middle
base
,
or
Section
,
multiplied
by
the
length
,
shall
bee
lesse
then
the
true
content
,
by
foure
Pyramides
,
hauing
for
their
bases
,
a
rectangle
vnder
halfe
the
difference
of
the
thicknesse
at
the
ends
,
and
a
quarter
of
the
difference
of
the
breadths
:
and
are
as
long
as
halfe
the
peece
of
timber
:
Or
which
is
all
one
,
by
a
Parallelepipedon
,
vnder
halfe
the
differences
of
the
breadthes
,
and
thicknesses
,
at
both
ends
,
and
a
third
part
of
the
whole
length
.
diagram
Which
that
I
may
shew
,
suppose
one
quarter
of
a
tapering
peece
of
timber
giuen
,
sawed
in
●unner
,
at
halfe
the
breadth
,
and
halfe
the
thicknesse
be
ABCDGFEH
:
the
midle
Section
is
IK●M
.
Measure
vpon
the
greater
base
BN
,
and
CO
,
equall
to
AF
,
and
BY
,
and
NZ
,
equall
to
FE
.
Diuide
CT
,
and
DO
to
the
midst
,
in
the
points
X
,
and
Q
,
and
draw
the
lines
PQ
,
NO
parallel
to
BC
,
and
YZ
,
SX
parallel
to
BG
.
Measure
also
FER
,
equall
to
GS
,
or
KL
.
And
so
the
Parallelegram
BPTX
,
shall
be
equall
to
the
middle
Section
IKLM
.
Lastly
,
draw
the
lines
LO
,
LV
,
LZ
,
and
KP
,
KN
,
and
MX
,
MY
.
I
say
that
in
this
one
quarter
of
the
peece
of
timber
,
A
solid
hauing
the
bases
equall
to
IKLM
,
and
the
length
AB
(
which
is
the
vsuall
measuring
of
Carpenters
)
is
lesse
then
the
true
content
,
by
the
Pyramide
DSVOL
,
in
which
DO
,
or
SV
,
is
halfe
the
difference
of
thicknes
,
and
DS
,
or
VO
,
one
quarter
of
the
difference
of
breadths
in
both
ends
:
and
the
height
of
it
equall
to
halfe
the
length
.
For
the
two
solides
AFEHIKLM
,
BNZYIKLM
,
are
apparantly
equall
,
againe
the
two
solide
wedges
GSTPKL
,
PTVNKL
are
equall
,
and
also
the
two
solide
wedges
OCXVLM
,
VXYZLM
are
equall
.
Now
if
you
turne
ouer
the
wedge
GSTPKL
,
vnto
the
lesser
part
AFEHIKLM
,
it
will
ouer-reach
in
thicknesse
,
at
the
lesser
end
the
quantity
of
ER
,
and
if
you
also
turne
ouer
the
wedge
OCXVLM
,
vnto
the
lesser
part
,
you
shall
find
it
to
fill
vp
the
former
ouer-reaching
,
and
to
make
an
exact
Parallelepipedon
,
the
bases
or
ends
whereof
are
equall
to
the
middle
Section
IKLM
.
but
ouer
and
aboue
those
two
wedges
turned
ouer
,
you
shall
haue
left
the
Pyramide
DSVOL
,
which
was
to
be
proued
.
And
in
like
manner
if
a
round
tapering
solide
,
or
peece
of
timber
,
be
measured
by
the
middle
Section
,
or
circular
base
,
I
say
,
That
the
product
thereof
multiplied
by
the
length
,
shall
be
lesse
then
the
true
content
,
by
a
Cylinder
,
the
diameter
of
whose
base
,
is
equall
to
halfe
the
difference
of
the
diameters
of
the
two
bases
:
and
the
length
is
one
third
part
of
the
whole
length
.
CHAP.
IX
.
Concerning
the
Measuring
,
or
Gauging
of
Vessells
.
AWine
,
or
Beere
vessell
,
whether
Pipe
,
Hogshead
,
Barrell
,
Kilderkin
,
or
Pirkin
,
and
such
like
,
is
in
forme
of
a
Sphaeroides
,
hauing
the
two
ends
equally
cut
off
:
and
accordingly
may
be
measured
thus
.
Measure
the
two
diameters
of
the
Vessell
,
in
inches
,
or
else
in
tenth
parts
of
a
foote
,
the
one
at
the
bung
hole
,
the
other
at
the
head
,
and
also
the
length
within
.
And
by
the
diameters
found
,
finde
out
the
circles
;
then
adde
together
two
third
parts
,
of
the
greater
circle
,
and
one
third
part
of
the
lesse
:
Lastly
,
multiply
the
aggregate
by
the
length
:
so
shall
you
haue
the
content
of
the
Vessell
,
either
in
Cubic
inches
,
or
cubic
tenth
parts
of
a
foot
.
Example
.
Suppose
a
vessell
,
having
the
Diameter
at
the
bung
32
inches
,
and
at
the
head
18
inches
,
and
the
length
40
inches
.
The
quadrat
of
32
is
1024.
and
the
quadrat
of
18
is
324
:
Say
then
euermore
,
1
·
0
⌊
●236
∷
1024
·
536
⌊
166
,
●/3
the
circle
at
the
bung
·
1
·
0
⌊
●618
∷
324
·
84
⌊
823
,
⅓
circle
at
the
head
.
Or
else
by
Chap.
7
,
Sect.
4.
The
aggregate
of
those
two
circles
is
620
⌊
989
,
which
being
multiplied
by
40
,
the
length
giueth
24839
⌊
56
cubic
inches
for
the
whole
content
of
that
vessell
.
2
Mr.
Edm
:
Gunter
in
his
second
booke
of
the
Crosse
Staffe
,
Chap.
4.
pretending
to
shew
the
manner
of
gauging
Wine
vessels
,
beginneth
with
these
words
.
The
Vessels
which
are
heere
measured
,
are
supposed
to
bee
Cylinders
,
or
reduced
into
Cylinders
by
taking
the
meane
,
betweene
the
Diameter
at
the
head
,
and
the
Diameter
at
the
boungue
,
after
the
usuall
manner
.
And
according
to
this
supposition
,
teacheth
to
finde
a
Gauge
point
,
For
a
gallon
of
wine
,
in
that
his
imagined
Cylindriacall
vessell
.
Because
his
words
are
cautelous
,
and
subterfugious
,
wee
must
a
little
examine
them
;
for
if
his
way
bee
true
,
my
Rule
before
set
downe
,
though
grounded
vpon
demonstration
,
cannot
stand
.
Well
then
,
that
reduction
of
a
wine
Vessell
into
a
Cylinder
,
is
either
true
,
or
false
;
if
it
bee
true
what
neede
those
ambiguities
of
Vessels
which
are
here
measured
:
and
are
supposed
to
be
,
&c.
and
after
the
vsuall
manner
?
if
false
why
is
it
not
noted
,
but
deliuered
as
a
Rule
to
confirme
an
error
.
And
what
meaneth
,
the
meane
betweene
the
Diameter
at
the
head
,
and
the
Diameter
at
the
boungue
?
is
it
the
meane
Geometricall
,
or
Arithmeticall
,
that
is
the
meane
proportionall
,
or
that
which
equally
differeth
from
both
?
such
shifting
is
vnworthy
an
Artist
.
First
therefore
,
Let
it
be
the
meane
in
respect
of
difference
,
which
is
equall
to
halfe
the
summe
of
the
two
Diameters
:
I
say
that
the
Vessell
cannot
truely
be
reduced
to
a
Cylinder
by
such
a
meane
Diameter
.
For
seeing
it
is
most
apparent
that
such
a
Vessell
,
is
greater
in
the
middle
then
at
the
ends
,
the
boords
,
or
sides
thereof
,
shall
from
the
middle
to
the
ends
,
goe
either
streight
,
and
so
the
Vessel
shall
be
as
it
were
,
two
equall
segments
of
Cones
,
set
base
to
base
:
Or
else
arching
,
and
so
the
Vessel
shall
(
as
before
I
sayd
,
and
is
commonly
taken
for
a
truth
)
bee
a
Sphaeroides
,
hauing
the
two
ends
equally
cut
of
.
If
it
be
considered
as
two
segments
of
Cones
:
the
measure
by
that
meane
Diameter
,
or
midle
Section
is
quite
false
,
as
hath
beene
demonstrated
in
the
former
Chapter
,
Sect:
16
,
17.
and
will
be
giuen
lesse
then
the
true
content
,
although
the
sides
goe
straight
:
much
more
then
,
if
the
sides
goe
arching
;
for
that
conuexnesse
,
must
needs
yeeld
a
greater
capacity
.
And
therefore
in
neither
can
that
manner
of
Gauging
be
true
.
Againe
,
if
the
meane
Diameter
be
vnderstood
to
be
the
meane
proportionall
betweene
the
two
Diameters
,
it
is
much
more
false
,
for
betweene
any
two
numbers
,
the
meane
Geometricall
is
lesse
then
the
meane
Arithmeticall
.
Thus
much
I
haue
thought
good
in
this
place
ingenuously
to
signifie
to
the
inexpert
Learner
,
that
hee
might
not
beguile
himselfe
with
a
prejudged
opinion
.
3.
I
haue
shewed
the
measuring
of
vessels
by
the
cubic
inch
:
but
our
vsuall
reckoning
is
by
the
Gallon
,
and
parts
thereof
.
Wee
must
therefore
doe
the
best
we
can
,
to
inquire
the
true
quantitie
of
a
Gallon
in
inch
measure
,
which
will
bee
difficult
to
doe
exactly
,
both
because
the
Standards
vsually
are
not
streight
sided
,
but
a
little
arching
,
neither
doe
they
agree
perfectly
one
with
another
:
but
what
partly
by
experience
,
both
mine
owne
,
and
others
,
which
hath
come
to
my
sight
;
and
partly
by
reasoning
shall
seeme
to
me
most
probable
,
I
will
not
refuse
to
set
downe
.
4
Our
English
Gallon
is
vnderstood
to
bee
either
in
Ale
measure
,
or
Wine
measure
:
and
these
two
measures
not
a
little
differing
.
And
first
we
will
inquire
about
our
Ale
measure
.
I
my selfe
haue
measured
Bushels
,
and
P●cks
,
which
haue
exactly
beene
fitted
to
the
Standards
,
and
haue
still
in
my
account
found
a
Gallon
to
containe
better
then
270
cubic
inches
,
indeed
much
about
272
,
or
273
as
precisely
as
I
could
measure
in
a
vessell
not
truely
regular
.
Also
my
worthy
friend
Master
William
Twine
,
who
hath
vndergone
great
paines
and
charge
,
in
finding
out
the
true
content
of
our
English
measures
,
gaue
vnto
me
two
severall
measures
of
an
Ale-gallon
,
and
those
in
due
consideration
but
little
differing
.
The
one
was
found
out
by
a
brassen
vessell
made
in
manner
of
a
Parallelepipedon
,
the
base
whereof
was
exactly
sixe
inches
square
,
and
the
sides
diuided
into
inches
and
twentieth
parts
:
into
which
vessell
he
powring
out
a
standard
Gallon
of
Queene
Elizabeth
,
filled
with
water
,
found
it
therein
to
ari●e
vnto
7
inches
,
and
6
tenth
parts
:
which
being
computed
maketh
cubic
inches
273
⌊
6.
The
other
was
found
by
taking
the
dimensions
of
that
standard
Gallon
,
which
was
made
in
forme
of
a
segment
of
a
Cone
,
but
that
the
sides
were
a
little
arching
:
the
dimensions
were
thus
;
the
Diameter
of
the
top
was
inches
6
⌊
5
:
the
Diameter
of
the
bottome
was
inches
5
⌊
●
:
and
the
height
of
it
was
inches
9
⌊
8
:
which
being
cast
vp
by
Chap:
8
,
Sect:
15
,
will
be
found
to
containe
cubic
inches
268
⌊
85
:
differing
from
the
former
,
only
cubic
inches
4¾
:
which
difference
might
well
arise
through
the
curuitie
of
the
sides
.
These
measures
he
did
not
only
take
himselfe
,
but
to
giue
me
satisfaction
,
shewed
me
the
experience
in
the
said
Vessell
and
Standard
:
but
the
truth
is
,
I
obserued
the
Standard
,
besides
the
arching
of
the
sides
,
to
bee
not
exactly
circular
within
,
nor
the
brimme
of
an
even
height
,
nor
the
bottome
plaine
:
and
in
taking
the
height
of
the
water
in
the
Vessell
,
our
sight
was
not
able
to
estimate
the
ascent
thereof
so
precisely
,
that
a
spoonefull
of
water
,
more
or
lesse
,
could
breed
any
sensible
difference
.
What
therefore
shall
wee
doe
in
this
difficultie
?
indeede
looke
to
the
first
ground
,
and
principle
of
our
English
measuring
,
from
Barley
cornes
.
For
the
length
of
3
Barley
cornes
taken
out
of
the
middle
of
the
eare
is
an
Inch
,
or
Vncia
,
that
is
a
twelfth
part
of
a
foot
.
3
feete
make
a
Yard
:
and
16
feet
and
an
halfe
,
that
is
5
yards
and
an
halfe
,
a
Perch
,
with
which
wee
measure
our
land
;
for
40
perches
is
a
Furlong
,
and
8
furlongs
an
English
mile
:
and
againe
40
square
perches
is
a
Roodland
,
4
roodland
make
an
Acre
.
So
then
a
Perch
which
is
feet
16½
,
or
yards
5½
is
as
it
were
the
beginning
of
all
land
measure
in
length
:
and
a
square
Perch
which
is
feete
272¼
,
is
as
it
were
the
beginning
of
all
land
measure
in
the
superficiall
content
.
Now
therefore
seeing
in
Vessels
a
gallon
is
as
it
were
the
beginning
of
Vessell
measure
(
for
a
Pottell
is
but
a
diminatiue
of
it
,
and
a
quart
,
the
quarter
)
it
is
not
vnlikely
that
our
wise
Ancestors
had
such
a
consideration
also
in
solide
measures
,
that
as
a
square
Perch
(
the
beginning
of
Superficiall
land
measure
)
did
containe
372
square
feete
and
a
quarter
;
so
a
Gallon
(
the
beginning
of
vessell
measure
)
should
containe
272
Cubic
inches
and
a
quarter
.
And
the
rather
seeing
that
the
ancient
Geographers
,
diuide
a
foote
into
4
Palmes
,
a
palme
being
3
of
our
inches
,
as
3
feete
are
a
yard
.
So
that
as
the
side
of
a
square
Perch
consisteth
of
yards
5½
,
a
Gallon
also
should
consist
,
of
a
number
of
Cubic
inches
the
square
side
whereof
is
palmes
5½
.
Wherefore
sauing
the
exact
truth
when
it
shall
appeare
,
and
in
the
meane
time
the
more
probable
reasons
of
other
men
,
I
make
bold
to
tender
this
my
coniecture
,
to
the
censures
of
more
diligent
Inquirers
,
That
the
measure
of
an
English
Ale
gallon
should
be
a
square
Vessell
of
inch
16½
,
or
Palmes
5½
every
way
,
and
1
inch
deepe
:
that
is
272
⌊
2●
Cubic
inches
.
5.
And
this
my
opinion
may
peraduenture
receiue
some
confirmation
by
the
inquiry
of
an
English
wiue
Gallon
.
M.
Henry
Briggs
that
learned
Geometrician
,
and
my
very
louing
friend
,
made
an
experiment
,
with
a
Cubicall
vessell
,
which
was
12
inches
euery
way
,
which
hauing
filled
with
water
carefully
measured
,
found
it
to
containe
7
gallons
and
an
halfe
wanting
a
moment
,
as
hee
himselfe
long
since
,
being
then
of
Gresham
Colledge
,
signified
to
me
.
Now
if
it
had
contained
exactly
7
gallons
,
and
an
halfe
,
a
wine
gallon
should
haue
beene
230
⌊
4
cubic
inches
,
but
because
it
wanted
a
little
,
the
gallon
must
be
somewhat
bigger
,
for
which
moment
therefore
if
you
will
put
6
hundred
parts
of
an
inch
,
the
wine
gallon
shall
containe
231
cubic
inches
.
Againe
M.
Gunter
in
the
place
before
mentioned
sheweth
,
that
the
common
opinion
is
that
at
London
,
a
Cylindriacall
vessell
,
whose
diameter
is
38
inches
,
and
length
66
inches
,
doth
containe
324
gallons
:
wherefore
by
this
account
a
gallon
should
bee
231
cubic
inches
almost
exactly
,
which
in
both
so
neerely
agreeing
,
wee
may
well
conclude
,
That
an
English
wine
gallon
doth
containe
231
cubic
inches
.
It
is
also
commonly
receiued
,
that
the
reason
of
the
greatnesse
of
an
Ale
gallon
aboue
the
Wine
gallon
is
,
that
because
of
the
frothing
of
the
Ale
or
Beere
,
the
quantity
becommeth
lesse
,
and
therefore
such
liquors
that
did
not
so
yeeld
froth
,
as
Wine
,
Oyle
,
and
the
like
,
should
in
reason
haue
a
lesser
measure
.
If
then
we
compare
these
two
gallons
together
,
we
shall
finde
that
272
⌊
25
·
231
∷
16
⌊
5
·
14
·
which
abatement
might
to
our
Ancestors
,
in
apportioning
those
measures
,
seeme
to
be
reasonable
.
6
To
finde
how
many
Cubic
tenth
parts
of
a
foot
are
in
a
gallon
,
both
of
beere
,
and
Wine
:
or
also
in
any
number
of
Cubic
inches
.
Because
there
be
in
a
Cubic
foote
,
1000
Cubic
tenth
parts
,
and
1728
Cubic
inches
,
say
,
1728
(
C
∶
12
)
·
1000
(
C
∶
10
)
∷
272
⌊
25
·
157
⌊
●521
—
cubic
tenth
parts
of
a
Beere
gallon
.
And
1728
(
C
∶
12
)
·
1000
(
C
∶
10
)
∷
231
·
133
⌊
6803
—
cubic
tenth
parts
in
a
Wine
gallon
.
Also
1728
·
1000
∷
24839
⌊
56
·
14374
⌊
746
,
cubic
tenth
parts
,
are
in
the
vessell
measured
in
Sect.
1.
7
And
according
to
these
measures
so
found
,
you
may
easily
finde
the
content
of
a
pint
,
or
quart
,
or
peck
,
or
bushell
,
which
two
last
are
to
be
reckoned
in
Ale
measure
.
8
The
Content
of
a
Vessell
,
being
giuen
in
cubic
inches
,
or
in
cubic
tenth
parts
of
a
foot
,
to
finde
how
many
gallons
it
containeth
.
This
is
easily
done
if
you
diuide
the
content
giuen
in
inches
,
by
272
⌊
●5
for
Ale
measure
:
and
by
231
,
for
Wine
measure
.
But
if
the
content
be
giuen
in
decimall
parts
of
a
foote
diuide
it
by
157
⌊
5521
—
for
Ale
measure
;
and
by
133
⌊
6803
—
for
Wine
measure
.
Example
.
How
many
wine
gallons
are
in
a
vessell
containing
24839
⌊
56
cubic
inches
,
or
14374
⌊
746
cubic
tenth
parts
of
a
foote
.
Diuide
24839
⌊
56
by
231
,
or
diuide
14374
⌊
746
by
133
⌊
68
,
and
the
quotient
shall
be
107
⌊
●3
wine
gallons
.
CHAP.
X.
Concerning
the
Comparison
of
sundry
Metals
,
in
quantity
and
weight
.
1
IF
foure
pieces
of
Metalls
,
whereof
the
third
is
of
the
same
kinde
with
the
first
,
and
the
fourth
of
the
same
kind
with
the
second
,
are
proportionall
,
their
grauities
also
,
or
weights
,
shall
be
proportionall
.
2.
If
there
bee
foure
pieces
of
metall
,
whereof
the
third
is
of
the
same
kind
with
the
first
,
and
the
fourth
of
the
same
kinde
with
the
second
:
and
the
first
and
second
be
of
equall
greatnesse
,
and
the
third
and
fourth
of
equall
weight
;
the
weights
of
the
first
and
second
,
shall
be
reciprocall
to
the
magnitudes
of
the
third
and
fourth
.
3.
Two
Sphaeres
of
the
same
matter
are
in
weight
,
as
the
cubes
of
their
Diameters
,
are
in
magnitude
.
Et
contra
.
4.
Pieces
of
Metall
if
they
bee
of
equall
magnitude
,
haue
their
weights
in
direct
proportion
,
as
is
here
set
downe
:
but
if
they
be
of
equall
weight
,
they
haue
their
magnitudes
in
proportion
reciprocall
:
According
to
the
experiments
of
Marinus
Ghetaldi
,
in
his
tractate
called
Archimedes
promotus
.
Gold
.
|
3990
|
Brasse
.
|
1890
|
Arg.
Viu
.
|
2850
|
Iron
.
|
1680
|
Lead
.
|
2415
|
Tinne
.
|
1554
|
Siluer
.
|
2030
|
|
|
5.
To
finde
the
Weight
of
a
Sphaere
of
Tinne
,
hauing
the
Diameter
1
Inch
,
or
else
●
tenth
part
of
a
foot
.
Take
a
piece
of
Tinne
,
and
turne
it
exactly
in
a
Lathe
,
into
a
Cylinder
,
hauing
both
the
Diameters
of
its
base
,
and
also
the
length
equall
,
to
the
Diameter
of
the
Sphere
giuen
.
Then
weight
that
Cylinder
,
that
you
may
haue
the
weight
thereof
in
graines
:
And
lastly
,
take
two
third
parts
of
the
whole
number
of
graines
:
for
the
weight
of
the
Sphaere
.
After
this
manner
Marinus
Ghetaldi
found
a
Cylinder
of
1
inch
,
or
twelfth
part
of
a
foot
thick
,
and
long
to
weigh
1824
graines
:
whereof
●
is
1216
,
the
weight
of
a
Sphaere
of
that
thicknesse
.
Againe
if
you
say
,
1000
(
C
∶
10
)
·
1728
(
C
∶
12
)
∷
1216
·
2101
⌊
248
You
shall
haue
the
weight
of
a
Sphaere
whose
Diameter
is
one
tenth
part
of
a
foot
.
Wherefore
also
a
Cubed
inch
of
Tinne
weigheth
2322
⌊
4
—
,
and
a
Cubed
tenth
part
of
a
foote
weigheth
4013
⌊
1
—
And
note
that
Mar:
Ghetaldi
vseth
the
ancient
Roman
foot
,
which
by
the
measure
set
downe
in
his
booke
seemeth
to
be
very
little
lesse
,
then
our-vsuall
English
foot
,
if
not
exactly
the
same
.
Note
also
that
he
diuideth
one
pound
,
into
12
ounces
,
and
euery
ounce
into
24
Scruples
,
and
euery
Scruple
into
24
graines
:
So
that
an
ounce
with
him
weigheth
576
graines
:
and
a
pound
6912.
Whereas
our
English
pound
of
Iroy
weight
by
Assize
,
or
Gold
smiths
weight
,
is
but
5760
graines
,
and
our
ounce
480.
But
whether
the
Raguzean
graine
,
be
the
same
with
our
English
,
I
leaue
to
be
tryed
by
the
diligent
Practizer
.
6
To
finde
the
Weight
of
a
Sphaere
of
Tinne
,
at
any
other
Diameter
assigned
.
Multiply
the
Cube
of
the
Diameter
giuen
by
1216
,
if
it
be
in
inch
measure
;
or
by
2101
⌊
248
,
if
the
measure
be
by
decimall
parts
of
a
foot
:
and
the
product
will
be
the
weight
of
that
Sphaere
.
And
contrariwise
to
find
the
Diameter
of
a
Sphaere
of
Tinne
,
by
the
weight
giuen
in
graines
.
Diuide
the
weight
giuen
in
graines
by
1216
if
you
would
haue
inch
measure
:
or
by
2101
⌊
248
if
you
would
measure
by
decimall
parts
of
a
foot
,
and
the
quotient
shall
be
the
Cube
of
the
Diameter
.
7.
To
finde
the
Weight
of
a
Sphaere
of
any
Mettall
,
at
any
Diameter
giuen
,
either
in
Inch
measure
,
or
in
decimall
parts
of
a
foot
.
First
by
Sect.
6
,
seeke
the
weight
of
a
Sphaere
of
Tinne
,
at
that
Diameter
:
then
by
Sect.
4
,
say
,
As
the
proportionall
number
of
Tinne
,
is
to
the
proportionall
number
of
that
other
Metall
:
so
is
the
weight
of
the
Sphaere
of
Tinne
now
found
,
to
the
weight
of
the
Sphaere
proposed
.
Example
.
Suppose
a
Sphaere
of
Iron
,
whose
Diameter
is
3
inches
;
what
shall
be
the
weight
thereof
?
First
,
the
weight
of
a
Sphaere
of
Tinne
,
of
3
inches
Diameter
,
will
be
found
to
be
32832
graines
.
Then
say
,
1554
·
1680
∷
32832
·
35494
⌊
054
graines
the
weight
of
the
Sphaere
proposed
.
8
To
finde
the
Diameter
of
a
Sphaere
of
any
Metall
in
inch
measure
,
or
decimall
parts
of
a
foot
,
the
weight
thereof
being
giuen
.
First
by
the
contrary
of
Sect.
6
,
seeke
the
Cube
of
the
Diameter
of
a
Sphere
of
Tinne
of
that
weight
.
Then
by
Sect.
4
,
say
reciprocally
.
As
the
proportionall
number
of
that
other
Metall
,
is
to
the
proportionall
number
of
Tinne
:
so
is
the
Cube
of
the
Diameter
now
found
,
to
the
Cube
of
the
Diameter
of
the
Sphaere
proposed
.
Example
.
A
Sphaere
of
Iron
weigheth
35494
⌊
0●4
grains
:
how
many
inches
is
the
Diameter
thereof
?
First
,
the
Cube
of
the
Diameter
of
a
Sphere
of
Tinne
of
35494
⌊
054
graines
weight
,
will
be
29
⌊
18910695
,
then
say
,
reciprocally
,
1680
·
1554
∷
29
⌊
18910691
·
27
—
The
Cubic
root
whereof
is
3
,
the
Diameter
of
a
Sphaere
of
Iron
of
that
weight
proposed
.
CHAP.
XI
.
Concerning
the
Ordering
of
Soldiers
,
in
any
kinde
of
rectangular
forme
of
battaile
.
1
BAttailes
are
considered
either
in
respect
of
the
number
of
men
,
or
in
respect
of
the
forme
of
ground
.
As
asquare
battaile
of
men
is
that
which
hath
an
equall
number
of
men
,
both
in
Rank
and
File
,
though
the
ground
on
which
they
stand
,
bee
longer
on
the
File
,
then
on
the
Ranke
.
And
a
square
battaile
of
ground
is
that
which
hath
the
Ranke
as
long
as
the
F●le
,
though
the
men
in
Ranke
be
more
then
in
File
.
2.
In
respect
of
the
number
of
men
,
it
is
called
either
asquare
battaile
,
or
a
double
battaile
,
or
a
battaile
of
the
grand
front
,
which
is
quadruple
,
or
a
battaile
of
any
proportion
,
of
the
number
in
Ranke
,
to
the
number
in
File
.
3.
If
it
bee
asquare
battaile
of
men
:
Extract
the
quadrat
root
out
of
the
whole
number
of
men
,
and
the
same
shall
be
the
number
of
Souldiers
,
to
be
set
in
a
Ranke
.
Example
.
576
Souldiers
are
to
bee
martialled
in
a
square
battaile
,
that
so
many
may
be
in
Ranke
,
as
in
File
.
Take
the
quadrat
root
of
576
,
which
is
24
:
the
same
shall
be
the
number
to
be
placed
in
a
Ranke
.
4.
If
it
be
a
double
battaile
of
man
:
Extract
the
quadrat
root
out
of
halfe
the
number
of
men
,
and
the
same
doubled
shall
bee
the
number
of
Souldiers
to
bee
set
in
a
Ranke
.
Example
.
1458
Souldiers
,
are
to
be
placed
in
a
double
battaile
;
so
that
twise
so
many
may
be
in
Rank
as
in
File
.
Take
halfe
the
given
number
1458
,
which
is
729
,
the
quadrat
root
whereof
is
27
:
double
it
and
you
shall
haue
54
men
to
be
placed
in
a
ranke
.
5.
If
it
be
a
quadruple
battaile
,
which
is
called
of
the
great
front
:
Extract
the
quadrat
root
out
of
one
quarter
of
the
number
of
men
,
and
the
same
quadrupled
shall
be
the
number
of
Souldiers
to
be
set
in
a
ranke
.
Example
.
1024
Souldiers
are
to
bee
martialled
into
a
battaile
of
the
grand
front
,
so
that
fower
times
so
many
be
in
ranke
as
in
file
.
Take
one
quarter
of
1024
the
number
given
,
which
is
256
,
the
quadrat
root
whereof
is
16
:
quadruple
it
,
and
you
shall
haue
64
men
to
be
placed
in
a
ranke
.
6.
If
a
battaile
bee
required
of
any
other
forme
,
that
is
,
if
a
Ratio
be
given
,
according
to
which
the
number
of
men
in
Ranke
,
shall
be
to
the
number
in
file
.
Multiply
the
two
termes
of
the
Ratio
given
:
Then
say
As
the
product
is
to
the
quadrat
of
the
terme
which
is
for
the
ranke
,
or
As
the
terme
which
is
for
the
file
,
is
to
the
terme
which
is
for
the
ranke
:
so
is
the
whole
number
of
Souldiers
,
to
the
quadrat
of
the
number
of
men
to
be
placed
in
a
ranke
.
Example
.
1944
Souldiers
,
are
to
be
martialled
so
,
that
the
number
of
the
ranke
,
be
to
the
number
of
the
file
,
as
8
vnto
3
,
that
is
for
8
men
in
ranke
,
3
are
to
be
set
in
file
.
First
multiply
the
two
termes
of
the
Ratio
8
and
3
,
the
product
whereof
is
24
,
also
quadrat
8
,
the
terme
of
the
ranke
,
which
will
be
64.
Then
say
,
3
·
8
∷
1944
·
5184
·
out
of
which
extract
the
quadrat
root
72
,
and
it
will
giue
you
the
true
number
of
the
ranke
.
7.
In
respect
of
the
forme
of
ground
,
the
battaile
is
either
a
square
of
ground
,
or
longer
one
way
then
the
other
.
For
the
distance
,
or
order
of
Souldiers
martialled
in
array
,
is
distinguished
either
into
Open
order
,
or
Order
.
Open
order
is
when
the
very
centers
of
their
places
,
are
distant
7
feet
asunder
,
both
in
ranke
and
file
.
Order
is
when
the
centers
of
their
places
,
are
distant
3
feet
and
a
halfe
in
ranke
,
and
so
much
in
file
.
Or
else
3
feet
and
a
halfe
in
ranke
,
and
7
feet
in
file
:
which
last
order
,
and
whatsoeuer
order
else
there
is
,
in
which
the
distance
of
the
rankes
one
from
another
is
greater
,
then
the
distance
of
files
,
causeth
that
a
square
of
men
,
maketh
not
a
square
of
ground
,
but
the
ground
is
longer
on
the
file
then
on
the
ranke
.
8
If
it
be
a
square
battaile
of
ground
,
the
centers
of
the
distances
being
feet
3½
in
ranke
,
and
7
feet
in
file
.
Because
3½
is
halfe
of
7
,
the
ratio
of
the
distances
,
is
as
1
unto
2.
And
seeing
the
number
in
ranke
,
to
the
number
in
file
,
is
reciprocall
to
the
distances
,
the
ratio
of
the
number
of
men
in
ranke
,
to
the
number
of
men
in
file
,
shall
be
as
2
unto
1.
And
so
the
Rule
shall
be
the
same
with
that
in
Sect.
6
,
namely
,
As
the
terme
of
the
file
,
is
to
the
terme
of
the
ranke
;
so
is
the
whole
number
of
Souldiers
,
to
the
true
number
of
the
ranke
.
Example
,
1352
Souldiers
,
are
to
bee
set
in
a
square
of
ground
,
that
their
distances
may
be
feet
3½
in
ranke
,
and
7
feet
in
file
.
The
Ratio
of
the
ranke
to
the
file
,
shall
reciprocally
be
,
as
7
to
3½
,
that
is
as
2
to
1.
Say
therefore
1
·
2
∷
1352
·
2704
,
the
quadrat
roote
whereof
52
is
the
number
of
men
to
be
set
in
a
ranke
.
9
If
a
battaile
wherein
the
distance
in
ranke
is
vnequall
to
that
in
file
,
be
longer
one
way
then
the
other
,
according
to
any
Ratio
giuen
:
there
is
to
be
considered
a
double
ratio
,
one
reciprocall
in
respect
of
the
distances
,
the
other
according
to
the
forme
of
the
ground
.
Wherefore
to
finde
the
Ratio
of
men
in
rank
,
to
the
men
in
file
,
Multiply
the
two
termes
of
the
ranke
,
for
the
ranke
,
and
the
two
termes
of
the
file
,
for
the
file
.
And
then
the
Rule
shall
bee
the
same
with
that
in
Sect.
6
,
namely
,
As
the
terme
of
the
file
,
is
to
the
terme
of
the
ranke
:
so
is
the
number
of
Souldiers
,
to
the
quadrat
of
the
true
number
of
the
ranke
.
Example
.
10290
souldiers
,
are
to
be
set
in
a
battaile
,
so
that
they
may
stand
onely
3
feet
asunder
in
ranke
,
and
7
feet
in
file
,
and
the
length
of
the
ground
for
the
ranke
,
to
the
length
of
the
ground
for
the
file
,
shall
haue
the
ratio
of
5
vnto
2.
First
in
respect
of
the
distances
,
the
Ratio
of
Rank
to
file
,
reciprocally
is
as
7
vnto
3.
Secondly
,
in
respect
of
the
ground
,
the
ratio
of
rank
,
to
file
,
is
as
5
to
2.
Wherefore
by
multiplication
of
like
termes
,
the
true
ratio
of
rank
to
file
shall
be
7
×
5
to
3
×
2
,
that
is
as
35
to
6.
Say
therefore
6
·
35
∷
10290
·
60025
the
quadrat
root
whereof
is
245
,
the
number
of
men
to
be
set
in
ranke
.
10
If
1000
Souldiers
,
may
be
lodged
in
a
square
,
of
300
feete
,
how
many
feete
must
the
side
of
a
square
be
,
which
will
serue
to
lodge
5000
?
Say
,
1000
·
5000
∷
300
×
300
·
450000
·
the
quadrat
roote
whereof
671
—
is
the
square
side
sought
for
.
And
this
is
the
order
for
resolution
of
all
other
questions
of
this
sort
.
CHAP.
XII
.
A
collection
of
the
most
necessarie
Astronomicall
operations
,
1
BEfore
wee
deliuer
the
Rules
of
such
operations
,
it
will
not
be
inconuenient
,
to
set
downe
certaine
Reductions
,
wherof
we
may
haue
frequent
vse
.
To
reduce
sexagesime
parts
into
decimals
.
Diuide
the
sexagesimes
giuen
by
60.
Example
.
How
many
decimals
are
34′
,
12″
?
Here
are
required
two
reductions
,
first
of
the
seconds
into
decimals
of
minutes
:
then
of
the
minutes
with
their
decimals
,
into
decimals
of
degrees
,
Thus
60
·
1
∷
12″
·
0
⌊
2
Againe
60
·
1
∷
34
⌊
′
·
0
⌊
57°
Wherefore
34′
,
12″
are
equall
to
0
⌊
57
of
a
degree
.
And
contrariwise
to
reduce
decimall
parts
of
degrees
in
sexagesimes
.
Multiply
the
decimall
part
giuen
by
60.
Example
.
How
many
sexagesime
parts
are
0
⌊
57°
?
1
·
60
∷
0
⌊
57°
·
34
⌊
●′
Againe
1
·
60
∷
0
⌊
2
·
12″
To
reduce
houres
into
degrees
.
Multiply
the
houres
with
their
decimall
parts
by
15.
Example
.
How
many
degrees
are
8
Ho
,
34′
,
12″
;
that
is
by
the
former
reduction
8
⌊
57
Ho
?
thus
1
·
15
∷
8
⌊
57
·
128
⌊
55
Wherefore
Houres
8
,
34′
,
12″
doe
containe
128
⌊
55
degrees
.
And
contrariwise
to
reduce
degrees
into
houres
.
Diuide
the
degrees
with
their
decimall
parts
by
15.
Example
.
How
many
houres
are
in
degrees
128
⌊
55
?
15
·
1
∷
128
⌊
55
·
8
⌊
57
2
It
is
to
be
vnderstood
,
that
if
foure
numbers
are
proportionall
,
their
Order
may
be
so
transposed
,
that
each
of
those
termes
,
may
bee
the
last
in
proportion
.
In
this
manner
,
-
I.
As
the
first
is
to
the
second
;
so
is
the
third
to
the
fourth
.
-
II.
As
the
third
is
to
the
fourth
;
so
is
the
first
to
the
second
.
-
III.
As
the
second
is
to
the
first
;
so
is
the
fourth
to
the
third
.
-
IIII.
As
the
fourth
is
to
the
third
;
so
is
the
second
to
the
first
.
Wherfore
euery
proportion
doth
implicitly
containe
foure
Orders
,
two
descending
,
and
two
ascending
,
as
may
be
seene
by
their
combinations
:
By
one
of
which
orders
,
if
of
foure
proportionall
numbers
,
any
three
be
giuen
,
that
other
which
is
vnknowne
,
may
be
found
out
.
Example
.
To
finde
out
any
of
these
,
termes
|
1
As
the
Sine
of
the
complement
of
the
suns
declination
,
|
2
is
to
the
Sine
of
the
compl
.
of
his
altitude
;
|
3
So
is
the
Sine
of
the
Sunnes
Azumith
from
the
meridian
,
|
4
to
the
Sine
of
the
horary
distance
from
the
meridian
.
|
If
the
first
,
second
,
and
third
termes
be
giuen
,
the
fourth
shall
be
found
out
by
the
I
order
.
If
the
first
,
third
,
and
fourth
termes
be
giuen
,
the
second
shall
be
found
out
by
the
II
order
.
If
the
first
,
second
,
and
fourth
termes
bee
giuen
,
the
third
shall
be
found
out
by
the
III
order
.
If
the
second
,
third
,
and
fourth
termes
be
giuen
,
the
first
shall
be
found
out
by
the
IIII
order
.
3.
To
finde
out
any
one
of
these
.
termes
|
1
As
the
Radius
,
or
totall
Sine
|
2
is
to
the
Sine
of
the
distance
,
or
longitude
of
the
Sunne
in
the
Ecliptic
,
from
the
next
Aequinoctial
point
:
|
3
So
is
the
Sine
of
the
Sunnes
greatest
declination
(
which
is
the
angle
of
the
Ecliptic
with
the
Aequinoctial
)
,
|
4
To
the
Sine
of
the
Sunnes
declination
in
that
longitude
.
|
4.
To
finde
out
any
one
of
these
.
termes
|
1
As
the
Radius
,
|
2
is
to
the
Sine
of
the
Sunnes
right
ascension
,
from
the
next
aequinoctiall
point
:
|
3
So
is
the
tangent
of
the
Sunnes
greatest
declination
,
|
4
to
the
tangent
of
the
Sunnes
declination
in
that
place
.
|
5.
To
finde
out
any
one
of
these
.
termes
|
1
As
the
Radius
|
2
is
to
the
Sine
of
the
compl
.
of
the
Sunnes
greatest
declination
:
|
3
So
is
the
tangent
of
the
longitude
of
the
Sunne
from
the
next
aequinoctiall
point
.
|
4
to
the
tangent
of
the
Right
ascension
of
the
Sunne
,
from
the
same
aequinoctial
point
.
|
6.
To
finde
out
any
one
of
these
.
termes
|
1
As
the
Radius
|
2
is
to
the
Sine
of
the
compl
.
of
the
longitude
of
the
Sunne
from
the
next
aequinoctiall
point
;
|
3
So
is
the
tangent
of
the
Sunnes
greatest
declination
,
|
4
to
the
tangent
of
the
compl
.
of
the
angle
of
the
Ecliptic
with
the
Meridian
.
|
7.
To
finde
out
any
one
of
these
.
termes
|
1
As
the
Radius
,
|
2
is
to
the
Sine
of
the
Sunnes
greatest
declination
:
|
3
So
is
the
Sine
of
the
compl
.
of
the
Sunnes
right
ascension
from
the
next
aequinoctial
point
,
|
4
to
the
Sine
of
the
compl
.
of
the
Angle
of
the
Ecliptic
with
the
Meridian
.
|
8.
To
finde
out
any
one
of
these
.
termes
|
1
As
the
Sine
of
the
compl
.
of
the
Poles
height
,
|
2
is
to
the
Radius
;
|
3
So
is
the
Sine
of
the
Sunnes
declination
,
|
4
to
the
Sine
of
the
Sunnes
Amplitude
ortiue
,
that
is
the
arch
of
the
horizon
from
the
place
of
the
Sunnes
rising
or
setting
to
the
true
East
,
or
West
point
.
|
9.
To
find
out
any
one
of
these
.
termes
|
1
As
the
Radius
,
|
2
is
to
the
Sine
of
the
Sunnes
greatest
amplitude
ortiue
,
which
is
in
the
Tropick
:
|
3
So
is
the
Sine
of
the
longitude
of
the
Sunne
from
the
next
aequinoctial
point
,
|
4
to
the
Sine
of
the
Sunnes
amplitude
ortiue
.
|
Or
also
of
these
.
termes
|
1
As
the
Sine
of
the
compl
.
of
the
Poles
height
,
|
2
is
to
the
Sine
of
the
compl
.
of
the
Sunnes
greatest
declination
:
|
3
So
is
the
Sine
of
the
Sunnes
longitude
from
the
next
aequinoctial
point
,
|
4
to
the
Sine
of
the
Sunnes
amplitude
ortiue
.
|
10.
To
finde
out
any
one
of
these
.
termes
|
1
As
the
Radius
,
|
2
is
to
the
tangent
of
the
height
of
the
Pole
.
|
3
So
is
the
tangent
of
the
Sunnes
declination
,
|
4
to
the
Sine
of
the
Sunnes
Ascensional
difference
.
|
11.
To
finde
out
any
one
of
these
.
termes
|
1
As
the
Radius
|
2
is
to
the
Sine
of
the
height
of
the
Pole
:
|
3
So
is
the
tangent
of
the
Sunnes
amplitude
ortiue
,
|
4
to
the
tangent
of
the
Sunnes
ascensionall
difference
,
|
12.
To
finde
out
any
one
of
these
.
termes
|
1
As
the
Sine
of
the
compl
.
of
the
Sunnes
declination
,
|
2
is
to
the
Radius
:
|
3
So
is
the
Sine
of
the
compl
.
of
the
Sunnes
amplitude
ortiue
,
|
4
to
the
Sine
of
the
compl
.
of
the
Sunnes
ascensionall
difference
.
|
13.
To
finde
out
any
one
of
these
.
termes
|
1
As
the
tangent
of
the
height
of
the
pole
,
|
2
is
to
the
Radius
:
|
3
So
is
the
tangent
of
the
Sunnes
declination
,
|
4
to
the
Sine
of
the
Suns
horary
distance
from
the
Meridian
,
being
due
East
or
West
.
|
14.
To
finde
out
any
one
of
these
.
termes
|
1
As
the
Sine
of
the
height
of
the
pole
|
2
is
to
the
Radius
:
|
3
So
is
the
Sine
of
the
Sunnes
declination
,
|
4
to
the
Sine
of
the
Sunnes
altitude
being
due
East
,
or
West
.
|
15.
To
find
out
any
one
of
these
.
termes
|
1.
As
the
Radius
,
|
2
is
to
the
Sine
of
the
height
of
the
Pole
:
|
3
So
is
the
Sine
of
the
Sunnes
declination
,
|
4
to
the
Sine
of
the
Sunnes
altitude
aboue
the
Horizon
at
sixe
of
the
Clocke
.
|
16.
To
find
out
any
one
of
these
.
termes
|
1
As
the
Radius
,
|
2
is
to
the
Sine
of
the
complement
of
the
Poles
height
.
|
3
So
is
the
tangent
of
the
Sunnes
declination
.
|
4
to
the
tangent
of
the
Sunnes
Azumith
from
the
North
meridian
,
at
6
of
the
Clocke
.
|
17
The
hower
of
the
Sunnes
Rising
,
and
setting
is
found
out
by
the
Ascensionall
difference
.
For
if
you
reduce
the
degrees
of
the
Ascensionall
difference
,
into
howers
,
it
will
shew
you
how
much
the
Sunne
riseth
,
or
setteth
before
,
or
after
6
a
Clock
.
18.
The
Oblique
ascension
also
of
the
Sunne
is
found
out
by
the
Ascensionall
diff●rence
.
For
if
you
subduct
the
Sunnes
Ascensionall
difference
,
out
of
the
right
ascension
of
the
Sunne
,
from
the
beginning
of
Aries
,
for
the
sixe
Northerne
signes
which
are
♈
,
♉
,
♊
,
♋
,
♌
,
♍
,
or
if
you
adde
it
thereto
,
for
the
sixe
Southerne
Signes
,
which
are
♎
,
♏
,
♐
,
♑
,
♒
,
♓
,
you
shall
have
the
Sunnes
oblique
ascension
.
19.
The
declination
of
the
Sunne
,
and
his
Altitude
aboue
the
Horizon
at
any
time
,
together
with
the
height
of
the
Pole
being
giuen
,
to
find
the
hower
of
the
day
.
Say
,
As
the
Radius
,
is
to
the
Sine
of
the
complement
of
the
Sunnes
declination
.
So
is
the
Sine
of
the
compl
.
of
the
Poles
height
,
to
a
fourth
number
.
Keepe
it
,
Then
out
of
the
Sunnes
distance
from
the
North
pole
,
subduct
the
complement
of
the
Pole
;
and
of
that
remaines
,
and
the
complement
of
the
Sunnes
altitude
,
take
both
the
Summe
,
and
also
the
difference
.
And
say
againe
,
As
the
fourth
before
kept
,
is
to
the
Sine
of
halfe
that
summe
:
so
is
the
Sine
of
halfe
the
difference
,
vnto
a
number
which
being
multiplied
by
the
Radius
,
is
equall
to
the
quadrat
,
of
the
Sine
of
halfe
the
Angle
of
the
Sunnes
horary
distance
from
the
Meridian
.
20.
The
declination
of
the
Sunne
,
and
his
altitude
aboue
the
Horizon
at
any
time
together
with
the
height
of
the
Pole
being
given
,
to
find
the
Sunnes
Azumith
.
Say
,
As
the
Radius
is
to
the
Sine
of
the
compl
.
of
the
Suns
altitude
.
So
the
Sine
of
the
compl
.
of
the
Poles
height
,
is
to
a
fourth
,
Keepe
it
,
Then
out
of
the
complement
of
the
Sunnes
altitude
,
subduct
the
complement
of
the
Pole
;
and
of
that
remaines
,
and
the
distance
of
the
Sunne
from
the
North
Pole
,
take
both
the
Sunne
,
and
also
the
difference
:
and
say
againe
,
As
the
fourth
before
kept
,
is
to
the
Sine
of
halfe
the
summe
:
So
is
the
Sine
of
halfe
the
difference
,
vnto
a
number
which
being
multiplied
by
the
Radius
,
is
equall
to
the
quadrat
of
the
Sine
of
halfe
the
angle
of
the
Sunnes
horizontall
distance
from
the
Meridian
.
21.
To
find
the
length
of
the
Crepusculum
,
or
Twilight
.
Betweene
the
light
of
the
day
,
and
the
darkenesse
of
the
night
,
the
Twilight
is
set
by
the
wise
Creator
;
that
wee
here
vpon
the
earth
might
not
in
an
instant
,
passe
from
one
extreame
into
another
,
but
by
successiue
degrees
.
The
Twilight
is
nothing
else
but
the
refraction
of
the
Sunnes
beames
,
in
the
densitie
of
the
aire
.
And
Pet-Nonnius
to
find
the
length
of
the
Twilight
,
watched
the
time
after
Sunne
set
,
when
the
twilight
in
the
West
was
shut
in
,
so
that
no
more
light
appeared
there
,
then
in
any
other
part
of
the
sky
neere
the
Horizon
:
then
by
one
of
the
knowen
fixed
Stars
,
having
taken
the
true
hower
of
the
night
,
found
by
many
obseruations
,
that
at
the
time
of
shutting
in
the
Twilight
,
the
Sunne
was
vnder
the
Horizon
18
degrees
,
and
vntill
the
Sunne
was
gone
so
low
,
the
Twilight
continued
.
Say
therefore
As
the
Radius
is
to
the
Sine
of
the
compl
.
of
the
Sunnes
declination
:
So
the
Sine
of
the
compl
.
of
the
heigth
of
the
Pole
,
is
to
a
fourth
,
Keepe
it
.
Then
out
of
the
Sunnes
distance
from
the
South
Pole
,
subduct
the
complement
of
the
Pole
;
and
of
that
remaines
,
and
degrees
62
,
take
both
the
Summe
and
also
the
difference
;
and
say
againe
,
As
the
fourth
kept
,
is
to
the
Sine
of
halfe
the
summe
:
so
is
the
Sine
of
halfe
the
difference
,
to
a
number
which
being
multiplied
by
the
Radius
is
equall
to
the
quadrat
of
the
Sine
of
halfe
the
angle
of
the
Sunnes
distance
at
the
ending
of
the
Twilight
,
from
the
high
Noone
next
to
it
.
Wherefore
if
out
of
the
whole
angle
conuerted
into
howers
,
you
subduct
halfe
the
diurnall
arch
,
or
the
hower
of
the
Sunnes
setting
,
you
shall
haue
the
true
length
of
the
Crepusculum
,
or
Twilight
.
22.
To
find
the
length
of
the
least
Crepusculum
in
the
yeare
.
The
Sunne
being
in
the
winter
Tropic
maketh
the
longest
Crepusculum
,
of
the
whole
winter
halfe
yeare
,
and
from
thence
,
as
the
dayes
increase
,
the
Crepuscula
doe
decrease
vntill
they
come
to
bee
shortest
,
which
is
in
a
certaine
Parallel
,
betweene
that
Tropic
,
and
the
Aequinoctiall
:
the
declination
whereof
is
thus
found
out
.
As
the
tangent
of
the
complement
of
the
Pole
,
is
to
the
Sine
thereof
:
So
is
the
tangent
of
9
degrees
,
to
the
Sine
of
the
declination
of
the
Parallell
,
in
which
the
Sunne
maketh
the
shortest
Crepusculum
of
the
whole
yeare
.
23
But
before
the
Crepusculum
come
to
bee
shortest
,
there
is
another
Parallel
,
in
which
the
Crepusculum
is
equall
to
that
in
the
Aequinoctiall
:
the
declination
wherof
is
thus
found
out
.
As
the
Radius
is
to
the
Sine
of
the
altitude
of
the
Pole
:
So
is
the
Sine
of
18
degrees
to
the
Sine
of
declination
of
the
Parallel
in
which
the
Sunne
maketh
the
Twilight
equall
to
that
in
the
Aequinoctiall
.
24.
If
an
Arch
of
the
Ecliptic
,
be
equall
to
his
Right
ascension
,
one
end
thereof
beeing
knowne
,
to
find
out
the
other
end
.
Say
,
As
the
Sine
of
the
Compl.
of
the
declination
of
the
arch
giuen
.
is
to
the
Radius
:
so
is
the
Sine
of
the
compl
.
of
the
greatest
declination
,
to
the
Sine
of
the
compl
.
of
the
other
and.
25.
To
find
the
poynt
of
one
quadrant
of
the
Ecliptic
,
wherein
the
difference
of
longitudes
cease
to
be
greater
,
then
the
differences
of
the
right
ascensions
.
Multiply
the
Sine
of
the
complement
of
the
greatest
declination
,
by
the
Radius
,
and
out
of
the
product
extract
the
quadrat
root
:
the
same
shall
bee
the
Sine
of
the
complement
of
the
declination
sought
for
.
26.
To
find
the
quantitie
of
the
angles
,
which
the
circles
of
the
12
Howses
make
with
the
Meridian
.
Say
As
the
Radius
is
to
the
Tang.
of
60
degr.
for
the
11th
,
9th
,
5th
,
and
third
howers
,
or
to
the
Tang.
of
30
deg.
for
the
12th
,
8th
,
6th
,
and
second
howses
;
so
is
the
Sine
of
the
complement
of
the
Pole
,
to
the
tang
.
of
the
compl
.
of
any
howse
with
the
Meridian
.
And
note
that
on
the
Easterne
part
of
the
vpper
hemisphaere
,
there
are
three
circles
of
Howses
,
the
Horoscope
,
which
is
also
the
Horizon
,
and
next
to
that
is
the
circle
of
the
12th
Howse
,
then
the
circle
of
the
11th
Howse
.
On
the
Westerne
part
also
,
are
three
circles
of
Howses
,
the
circle
of
the
7th
Howse
,
which
also
is
the
Horizon
,
and
next
thereto
the
circle
of
the
8th
Howse
,
then
the
circle
of
the
9th
Howse
.
But
the
circle
of
the
10th
Howse
,
is
the
very
vpper
Meridian
it
selfe
.
Contrary
Howses
are
1
and
7
,
2
,
and
8
:
3
and
9
:
4
and
10
:
5
and
11
:
6
and
12.
27.
Resolue
the
whole
time
from
the
Noone
last
past
into
degrees
(
by
multiplying
the
howers
with
their
decimall
parts
by
15
,
according
to
Sect
:
1
)
which
adde
vnto
the
right
Ascension
of
the
Sunne
:
and
you
shall
haue
the
right
ascension
of
the
point
of
the
Aequator
in
the
vpper
Meridian
,
which
is
called
the
Right
ascension
of
Medium
coeli
.
28.
Adde
99
degrees
to
the
Right
ascension
of
Med.
Caeli
:
and
it
shall
be
the
degree
of
the
Aequator
then
rising
vpon
the
East
Horizon
.
29.
If
the
first
quadrant
of
the
Aequator
doe
arise
,
the
beginning
of
γ
is
distant
from
the
meridian
Eastward
,
so
much
as
is
the
distance
of
the
Right
ascension
of
Med.
coeli
,
from
360.
But
if
the
second
qua●rant
of
the
Aequator
doe
arise
,
the
beginning
of
γ
,
is
distant
from
the
Meridian
Westward
,
so
much
as
is
the
distance
of
☉
,
from
the
Right
ascension
of
Med.
coeli
.
And
in
both
of
them
the
lower
angles
of
the
Ecliptick
with
the
Meridian
,
on
the
East
side
is
obtuse
,
and
on
the
West
side
acute
:
and
the
90th
degree
of
the
Ecliptick
,
commonly
called
non
agesimus
gradus
,
is
on
the
East
part
.
30.
If
the
third
quadrant
of
the
Aequator
doe
arise
,
the
beginning
of
♎
is
distant
from
the
Meridian
Eastward
,
so
much
as
is
the
distance
of
the
Right
ascension
of
Med.
coeli
from
180.
But
if
the
fourth
quadrant
of
the
Aequator
doe
arise
,
the
beginning
of
♎
,
is
distant
from
the
Meridian
Westward
,
so
much
as
is
the
distance
of
180
,
from
the
Right
ascension
of
Med.
coeli
.
And
in
both
of
them
the
lower
angle
of
the
Ecliptick
with
the
Meridian
,
on
the
East
side
is
acute
,
and
on
the
West
side
obtuse
:
and
on
the
90th
degree
is
on
the
West
part
.
31.
The
point
of
the
Ecliptick
culminant
in
the
Meridian
,
which
is
called
Medium
coeli
,
or
Cor
coeli
,
and
is
the
cuspis
of
the
10th
house
,
may
be
found
by
Sect.
5.
32.
The
declination
of
the
said
culminant
point
,
may
be
found
by
Sect.
3.
Wherefore
also
by
adding
or
subducting
that
declination
,
to
,
or
from
the
eleuation
of
the
Aequator
,
(
which
is
the
complement
of
the
Pole
)
the
Altitude
of
Med.
coeli
may
be
had
.
33.
The
Angle
of
the
Ecliptick
with
the
Maridian
,
may
be
found
by
Sect.
7.
34.
To
finde
the
Altitude
of
the
90
degr.
Or
the
Angle
of
the
Ecliptick
with
the
horizon
.
As
the
Radius
is
to
the
Sine
of
the
compl
.
of
the
altitude
of
Med:
coeli
.
So
is
the
Sine
of
the
angle
of
the
Ecliptick
with
the
Meridian
,
to
the
Sine
of
the
compl
.
of
the
angle
sought
for
.
35.
To
finde
the
Azumith
of
90
degr.
which
is
also
the
Amplitude
ortiue
of
the
Ascendent
,
or
Horoscopus
.
As
the
Radius
,
is
to
the
Sine
of
the
Altitude
of
Med.
coeli
.
So
is
the
tang
.
of
the
Angle
of
the
Ecliptick
with
the
Meridian
,
to
the
tang
.
of
the
compl
.
of
the
distance
of
that
Azumith
from
the
Meridian
.
36.
To
finde
the
Horoscopus
,
or
Ascendent
degree
of
the
Ecliptick
,
Or
the
Cuspis
of
the
first
house
.
The
Distance
of
the
Azumith
of
90
degrees
from
the
Meridian
,
is
equall
to
the
Amplitude
ortiue
of
the
Ascendent
degree
.
Wherefore
the
Ascendent
degree
or
the
Ecliptic
,
may
thence
bee
found
,
by
Sect:
8
,
or
9.
Or
else
thus
.
As
the
Radius
is
to
the
Sine
of
the
complement
of
the
angle
of
the
Ecliptic
with
the
Meridian
:
So
is
the
tang
.
of
the
complement
of
the
altitude
of
Med.
Coeli
,
to
the
tangent
of
the
distance
of
Med.
coeli
from
the
Ascendent
degree
.
37.
To
find
the
parts
of
the
angle
of
the
Ecliptic
with
the
Meridian
,
cut
with
an
arch
perpendicular
to
the
Circle
of
any
of
the
Howses
.
Say
As
the
Radius
is
to
the
Sine
of
the
compl
.
of
the
altitude
of
Med.
coeli
:
so
is
the
tangent
of
the
circle
of
any
Howse
with
the
Meridian
,
to
the
tang
:
of
the
compl
:
of
the
part
of
that
angle
,
which
is
next
the
Meridian
Then
subduct
that
part
found
out
of
the
whole
Angle
for
the
remaining
or
latter
part
.
38.
To
find
the
Distance
of
the
cuspis
of
any
howse
,
from
Med:
coeli
.
Say
As
the
Sine
of
the
compl
.
of
the
later
part
of
the
angle
of
the
Ecliptic
with
the
Meridian
,
is
to
the
Sine
of
the
compl
:
of
the
former
part
of
that
angle
:
So
is
the
tang
▪
of
the
altitude
of
Med:
coeli
,
to
the
tang
:
of
the
distance
of
the
cuspis
of
that
Howse
sought
for
.
39.
To
find
the
Altitude
of
the
Pole
aboue
any
of
the
circles
of
the
Howses
.
First
find
out
the
Angle
which
the
circle
of
the
Howse
proposed
maketh
with
the
Meridian
,
by
Sect:
23
:
And
then
say
.
As
the
Radius
is
to
the
Sine
of
the
angle
of
the
circle
of
the
Howse
with
the
Meridian
:
So
is
the
Sine
of
the
height
of
the
Pole
aboue
the
Horizon
of
the
place
,
to
the
Sine
of
the
height
of
the
Pole
aboue
that
circle
of
position
.
40.
The
longitude
,
and
latitude
of
any
fixed
Starre
being
given
,
to
find
out
the
Right
ascension
,
and
Declination
thereof
.
The
angle
which
the
Circle
of
the
Sunnes
longitude
maketh
with
the
Meridian
,
at
the
Pole
of
the
Ecliptic
,
I
call
the
Angle
of
longitude
.
And
the
angle
which
the
Circle
of
the
Sunnes
Right
ascension
,
maketh
with
the
Meridian
at
the
Pole
of
the
world
,
I
call
the
Angle
of
right
Ascension
.
The
condition
and
quantitie
of
which
two
angles
,
is
thus
found
out
.
In
Starres
of
the
Northerne
latitude
If
the
longitude
be
in
the
I
quadrant
of
the
Ecliptic
:
subduct
it
out
of
90
:
the
remaines
will
bee
the
angle
of
longitude
,
acute
.
And
the
Angle
of
Right
ascension
,
being
found
,
must
be
added
vnto
270.
If
the
longitude
bee
in
the
II
quadrant
:
subduct
90
out
of
it
:
the
remaines
will
bee
the
angle
of
longitude
,
acute
.
And
the
Angle
of
right
ascension
being
found
must
be
taken
out
of
270.
If
the
longitude
bee
in
the
III
quadrant
:
subduct
90
out
of
it
,
the
remaines
will
be
the
angle
of
longitude
,
obtuse
.
And
the
Angle
of
right
ascension
being
found
,
must
be
taken
out
of
270.
If
the
longitude
bee
in
the
IIII
quadrant
,
subduct
it
out
of
90+360
:
the
remaines
will
bee
the
angle
of
longitude
,
obtuse
.
And
the
Angle
of
right
ascension
being
found
,
must
be
added
vnto
270.
In
Starres
of
the
Southerne
latitude
If
the
longitude
be
in
the
I
quadrant
,
subduct
270
out
of
it
+
360
:
the
remaines
will
be
the
angle
of
longitude
,
obtuse
.
And
the
Angle
of
right
ascension
being
found
,
must
be
taken
out
of
90.
If
the
longitude
bee
in
the
II
Quadrant
,
subduct
it
out
of
270
:
the
remaines
will
bee
the
angle
of
longitude
,
obtuse
.
And
the
Angle
of
right
ascension
being
found
,
must
bee
added
to
90.
If
the
longitude
be
in
the
III
quadrant
:
subduct
it
out
of
270
;
the
remaines
will
be
the
angle
of
longitude
,
acute
.
And
the
Angle
of
right
ascension
being
found
,
must
bee
added
to
90.
If
the
longitude
be
in
the
IIII
quadrant
:
Subduct
270
out
of
it
:
the
remaines
wil
be
the
angle
of
longitude
,
acute
.
And
the
Angle
of
right
ascension
being
found
,
must
bee
subducted
out
of
90×
360.
Then
say
,
As
the
Radius
,
or
totall
Sine
,
is
to
the
Sine
of
the
complement
,
or
excesse
of
the
angle
of
longitude
:
So
is
the
tang
.
of
the
compl
.
of
the
latitude
,
to
the
tang
.
of
the
first
base
.
If
the
angle
of
longitude
bee
obtuse
;
vnto
the
first
base
found
,
adde
the
greatest
declination
deg.
23½
:
and
the
summe
shall
be
the
second
base
;
and
the
angle
of
right
ascension
shall
be
acute
.
But
if
the
angle
of
longitude
be
acute
;
out
of
the
first
base
subduct
the
greatest
declination
:
and
the
remaines
shall
be
the
second
base
.
And
the
angle
of
right
ascension
shall
be
obtuse
.
Or
else
out
of
the
greatest
declination
of
the
Sun
,
subduct
the
first
base
;
and
the
remaines
shall
be
the
second
base
:
and
the
angle
of
right
ascension
shall
be
acute
.
Say
againe
,
As
the
Sine
of
the
second
base
,
is
to
the
Sine
of
the
first
base
:
So
is
the
tang
.
of
the
angle
of
longitude
,
to
the
tang
.
of
the
angle
of
right
asce●sion
.
Whence
by
adding
or
subducting
as
was
before
deliuered
in
the
conditions
of
those
angles
,
shall
be
giuen
the
Right
ascension
of
that
Starre
sought
for
.
Lastly
say
As
the
tang
.
of
the
second
base
,
is
to
the
Radius
:
So
is
the
Sine
of
the
compl
.
or
excesse
of
the
angle
of
right
ascension
to
the
tang
.
of
Declination
.
Where
note
that
if
the
second
base
exceede
90
degr
:
the
declination
found
,
shall
not
be
of
the
same
kinde
,
that
the
latitude
is
,
but
in
the
contrary
Hemisphaere
.
A
Table
of
the
Right
Ascensions
,
and
Declinations
,
of
40
of
the
cheifest
fixed
Starres
,
Calculated
for
the
yeare
of
our
Lord
.
1650.
Names
of
Starres
|
Right
Ascension
|
Declination
|
mag
.
|
The
Polar
Starre
|
7°
|
4′7
|
80°
|
2′7
|
N
|
2
|
Andromedaes
girdle
|
12
|
32
|
33
|
48
|
N
|
2
|
the
former
horne
of
♈
|
23
|
38
|
17
|
37
|
N
|
4
|
the
bright
starre
in
the
head
of
♈
|
26
|
56
|
21
|
48
|
N
|
3
|
the
iawe
of
the
Whale
|
41
|
03
|
2
|
42
|
N
|
2
|
Medusaes
head
|
41
|
27
|
39
|
35
|
N
|
3
|
the
eye
of
the
Bull
|
64
|
00
|
15
|
46
|
N
|
1
|
the
Goat
starre
|
72
|
44
|
45
|
35
|
N
|
1
|
the
former
shoulder
of
Orion
|
76
|
38
|
4
|
59
|
N
|
2
|
the
latter
shoulder
of
Orion
|
84
|
07
|
7
|
18
|
N
|
2
|
the
great
dogge
starre
|
97
|
27
|
16
|
13
|
S
|
1
|
the
higher
head
of
♊
|
108
|
01
|
32
|
35
|
N
|
2
|
the
lesser
Dogge
starre
|
110
|
17
|
6
|
06
|
N
|
2
|
the
lower
head
of
♊
|
111
|
00
|
28
|
49
|
N
|
2
|
the
Cribb
,
or
Manger
|
125
|
4
|
20
|
52
|
N
|
Neb.
|
the
heart
of
Hydra
|
137
|
39
|
7
|
10
|
S
|
2
|
the
heart
of
the
Lion
|
147
|
27
|
13
|
39
|
N
|
1
|
In
the
loynes
of
the
Lion
|
163
|
54
|
22
|
26
|
N
|
2
|
In
the
tayle
of
the
Lion
|
172
|
49
|
16
|
32
|
N
|
1
|
In
the
girdle
of
Virgo
|
189
|
32
|
5
|
20
|
N
|
3
|
Names
of
Starres
|
R.
Ascen
.
|
Declination
|
mag
.
|
Aliot
|
189°
|
36
|
57
|
53
|
N
|
2
|
Vindemiatrix
|
191
|
15
|
15
|
51
|
N
|
3
|
Spica
Virginis
|
196
|
44
|
9
|
17
|
S
|
1
|
Arcturus
|
209
|
56
|
21
|
34
|
N
|
1
|
the
Southerne
ballance
|
217
|
56
|
14
|
32
|
S
|
2
|
the
Northerne
ballance
|
224
|
31
|
8
|
2
|
S
|
2
|
the
bright
star
in
the
serpents
neck
|
231
|
49
|
7
|
35
|
N
|
3
|
the
heart
of
Scorpius
|
242
|
4
|
25
|
34
|
S
|
1
|
the
head
of
Hercules
|
254
|
40
|
14
|
51
|
N
|
3
|
the
head
of
Ophiucus
|
259
|
41
|
12
|
52
|
N
|
3
|
bright
starre
in
the
Harp
|
276
|
17
|
38
|
30
|
N
|
1
|
bright
starre
in
the
Vultur
|
293
|
27
|
8
|
1
|
N
|
2
|
vpper
horne
of
♑
|
299
|
30
|
13
|
32
|
S
|
3
|
the
left
hand
of
♒
|
307
|
10
|
10
|
43
|
S
|
4
|
the
left
shoulder
of
♒
|
318
|
18
|
7
|
2
|
S
|
3
|
the
mouth
of
Pegasus
|
321
|
49
|
8
|
18
|
N
|
3
|
the
right
shoulder
of
♒
|
326
|
59
|
1
|
58
|
S
|
3
|
Fomahant
|
339
|
29
|
31
|
23
|
S
|
1
|
In
the
vpper
wing
of
Pegasus
|
341
|
53
|
13
|
21
|
N
|
2
|
In
the
tip
of
the
wing
of
Pegasus
|
358
|
52
|
13
|
15
|
N
|
2
|
41.
The
longitude
,
and
latitude
of
any
two
Starres
being
giuen
,
to
finde
their
distances
.
If
the
Starres
haue
both
the
same
longitude
,
differing
onely
in
latitude
;
the
difference
of
latitude
,
is
the
distance
of
the
starres
.
And
if
they
differ
onely
in
longitude
hauing
the
same
latitude
;
Say
As
the
Radius
,
is
to
the
Sine
of
halfe
the
difference
of
longitude
:
So
is
the
Sine
of
the
compl
.
of
the
latitude
giuen
,
to
the
Sine
of
halfe
the
distance
of
the
Starres
.
But
if
they
differ
both
in
longitude
,
and
latitude
,
whether
the
latitudes
be
both
of
the
same
kinde
,
or
one
Northerne
,
and
the
other
Southerne
.
Take
the
difference
of
both
the
starres
,
from
the
pole
of
the
Ecliptick
,
toward
which
the
starre
hauing
the
greater
latitude
is
.
And
say
,
As
the
tang
.
of
the
compl
.
of
the
lesser
distance
from
the
pole
,
is
to
the
Radius
:
So
is
the
Sine
of
the
compl
.
or
excesse
of
the
difference
of
longitudes
,
to
the
tang
.
of
the
first
base
.
Take
this
first
base
out
of
the
greater
distance
from
the
pole
,
and
the
remaines
shal
be
the
second
base
,
Then
say
,
As
the
Sine
of
the
compl
.
of
the
first
base
,
is
to
the
Sine
of
the
compl
.
of
the
second
base
:
So
is
the
Sine
of
the
compl
.
of
the
lesser
distance
from
the
pole
,
to
the
Sine
of
the
compl
.
of
the
distance
of
the
two
Starres
.
If
any
man
will
take
paines
to
calculate
(
by
this
last
Rule
)
the
distances
of
some
noted
starrs
of
the
first
,
second
,
and
third
magnitudes
,
round
about
the
heauens
,
which
are
not
aboue
5
,
or
6
degrees
,
at
the
most
,
one
from
the
other
:
and
shall
keepe
them
written
in
his
booke
:
they
may
serue
as
a
Rule
,
or
Instrument
,
whereby
he
may
reasonably
estimate
with
his
eye
,
the
distance
of
any
Planet
,
or
Comet
,
or
other
apparition
from
a
knowen
fixed
starre
,
not
very
farre
remote
:
by
comparing
the
distance
which
hee
would
know
with
some
of
those
knowen
distances
which
he
shall
find
,
either
to
be
equall
,
or
else
to
haue
some
proportion
thereto
.
42.
The
longitude
,
and
latitude
of
any
two
Cities
being
giuen
,
to
find
their
distance
.
The
manner
of
the
operation
is
the
very
same
with
the
former
,
vnto
which
therefore
I
referre
the
Reader
:
onely
will
note
,
that
in
the
heauens
,
the
longitude
and
latitude
is
taken
in
respect
of
the
Ecliptic
,
which
being
the
way
of
the
Sunne
,
all
the
starrs
in
their
proper
motion
,
haue
reference
vnto
it
,
as
vnto
their
measure
and
rule
.
But
in
the
Earth
the
principall
Circle
is
the
Equinoctiall
,
diuiding
it
into
the
Northerne
,
and
Southerne
he
misphaeres
.
And
therefore
in
the
earth
,
the
longitude
,
and
latitude
is
reckoned
by
the
Aequinoctiall
.
The
distance
of
two
places
vpon
the
Earth
,
being
found
in
degrees
,
may
bee
conuerted
into
English
miles
,
by
taking
60
miles
for
euery
degree
,
and
one
mile
for
every
minute
.
43.
To
find
at
what
hower
a
fixed
starre
commeth
into
the
Meridian
any
day
.
Seeke
the
Right
ascension
of
the
Sunne
,
for
that
day
,
by
Sect
〈◊〉
;
and
subduct
it
out
of
the
Right
ascension
of
the
Starre
.
And
reduct
the
degrees
remaining
into
howers
,
by
Sect:
1.
The
same
shall
shew
how
long
time
from
the
Noone
before
,
the
same
starre
shall
come
into
the
Meridian
.
Wherefore
if
at
any
time
of
the
night
,
a
Starr
whose
Right
ascension
is
knowne
,
be
in
the
Meridian
,
the
hower
of
the
night
is
easily
found
.
44.
The
height
of
any
knowne
Starre
aboue
the
Horizon
,
being
by
any
means
given
,
to
find
the
hower
of
the
night
.
First
seeke
out
the
hower
of
that
starrs
comming
into
the
Meridian
the
same
day
,
by
Sect.
43.
Againe
seeke
out
the
horary
distance
of
that
starr
from
the
Meridian
,
according
to
Sect:
19.
And
then
if
the
starr
bee
on
the
East
side
,
not
yet
come
to
the
Meridian
,
take
the
difference
of
those
two
numbers
;
but
if
the
starre
bee
past
the
Meridian
,
take
the
Summe
of
them
,
for
the
houre
of
the
night
.
45.
The
height
of
the
Pole
being
given
to
find
the
comming
of
any
fixed
Starre
,
in
the
due
East
,
or
West
.
Say
As
the
Radius
is
to
the
tang
:
of
the
starres
declination
:
So
is
the
tang
:
of
the
compl
:
of
the
Pole
,
to
the
Sine
of
the
compl
.
of
the
Starres
horary
distance
from
the
Meridian
.
46.
The
height
of
the
Pole
being
giuen
,
to
find
the
Altitude
of
any
fixed
starr
aboue
the
Horizon
,
being
due
East
or
West
.
Say
,
As
the
Sine
of
the
height
of
the
Pole
,
is
to
the
Radius
:
so
is
the
Sine
of
the
Starrs
declination
,
to
the
Sine
of
the
Altitude
,
at
due
East
or
West
.
47.
By
the
Altitudes
of
any
two
knowne
fixed
starrs
taken
when
they
are
both
in
the
same
Azumith
,
to
find
the
height
of
the
Pole
.
First
say
,
As
the
Sine
of
the
difference
of
the
starrs
altitudes
,
is
to
the
Sine
of
the
difference
of
their
Right
ascensions
:
so
is
the
sine
of
the
neerer
starrs
distance
from
the
apparent
Pole
,
to
the
Sine
of
an
angle
to
be
kept
.
Againe
compare
the
furthest
starrs
distance
from
the
Pole
with
the
distance
from
the
Zenith
,
and
say
As
the
Radius
is
to
the
Sine
of
the
compl
.
of
the
Angle
kept
:
so
is
the
tang
:
of
the
lesser
of
the
compared
arches
,
to
the
tang
:
of
the
first
base
.
Subduct
the
first
base
out
of
the
greater
of
the
two
compared
arches
;
and
the
remaines
shall
bee
the
second
base
.
Then
lastly
say
,
As
the
Sine
of
the
complement
of
the
first
base
,
is
to
the
Sine
of
the
compl
.
of
the
second
base
:
so
is
the
Sine
of
the
compl
.
of
the
lesser
of
the
two
compared
arches
,
to
the
Sine
of
the
height
of
the
Pole
.
48.
To
find
out
the
horizontall
Parallax
of
the
Moone
.
First
the
distance
of
the
Moone
from
the
Center
of
the
earth
must
be
knowne
in
Semidiameters
of
the
earth
:
which
vnto
them
that
are
acquainted
with
the
Theorie
of
the
Planets
,
is
not
very
difficult
.
And
whereof
peraduenture
,
I
may
hereafter
teach
the
practise
,
by
most
easie
and
exact
instruments
,
which
I
haue
long
since
framed
.
Say
,
As
the
distance
of
the
Moone
,
from
the
center
of
the
earth
,
is
to
the
Semidiameter
of
the
earth
:
So
is
the
Radius
,
to
the
Sine
of
the
Moones
horizontall
parallax
in
that
distance
.
49.
The
horizontall
Parallax
of
the
Moone
being
given
,
to
find
her
Parallax
in
any
apparent
altitude
.
As
the
Radius
,
is
to
the
Sine
of
the
altitude
of
the
Moone
:
so
is
the
Sine
of
the
horizontall
Parallax
,
to
the
Sine
of
the
Parallax
in
that
altitude
.
50.
The
place
of
the
Moone
in
the
Ecliptic
hauing
little
or
no
latitude
(
as
in
the
Eclips
of
the
Sunne
)
together
with
her
Parallax
of
altitude
being
giuen
,
to
find
the
Parallaxes
of
her
longitude
,
and
latitude
.
If
the
Moone
bee
in
the
90th
degree
of
the
Ecliptic
:
shee
hath
no
Parallax
of
longitude
,
and
the
Parallax
of
latitude
,
is
the
very
Parallax
in
that
altitude
.
But
if
the
Moone
be
not
in
the
90th
deg.
say
,
As
the
Radius
is
to
the
tang
.
of
the
angle
of
the
Ecliptick
with
the
horizon
:
So
is
the
Sine
of
the
compl
.
of
the
distance
of
the
Moone
,
from
the
Ascendent
,
or
descendent
degree
of
the
Ecliptick
,
to
the
tang
.
of
the
compl
.
of
the
angle
of
the
Ecliptick
with
the
Azumith
of
the
Moone
.
Againe
say
,
As
the
Radius
is
to
the
Sine
of
that
angle
:
So
is
the
Parallax
of
the
Moones
altitude
,
to
the
Parallax
of
her
latitude
.
Lastly
say
,
As
the
Radius
is
to
the
Sine
of
the
compl
.
of
that
former
angle
:
So
is
the
Parallax
of
the
Moones
altitude
,
to
the
Parallax
of
her
longitude
.
which
is
to
bee
added
to
the
true
motion
of
the
Moone
,
if
she
be
on
the
East
part
of
the
90th
degree
of
the
Ecliptick
:
or
to
be
subducted
out
of
it
,
if
she
be
on
the
West
part
.
Many
other
Astronomicall
and
Geographicall
problemes
might
be
added
.
But
because
it
is
impossible
to
set
downe
all
,
which
may
be
of
vse
,
at
some
time
or
other
:
I
haue
in
the
next
Chapter
deliuered
briefly
the
doctrine
of
triangles
fitted
vnto
practise
:
with
all
the
seuerall
cases
belonging
thereto
.
CHAP.
XIII
.
Of
Trigonometria
,
or
the
manner
of
calculating
both
Plaine
,
and
Sphaericall
triangles
.
And
first
concerning
certaine
generall
notions
,
and
rules
necessary
thereto
.
IN
euery
triangle
both
Plaine
,
and
Sphaerical
,
the
greater
side
subtendeth
the
greater
angle
.
And
the
greater
angle
hath
the
greater
side
opposite
vnto
it
.
Also
the
greater
angle
lyeth
to
the
lesser
side
,
and
the
greater
side
hath
the
lesser
angle
lying
vnto
it
.
In
euery
plaine
triangle
,
any
two
angles
being
giuen
,
the
third
is
also
giuen
:
and
one
of
the
angles
being
giuen
,
the
summe
of
both
the
other
two
is
giuen
.
For
all
the
three
angles
together
,
are
equall
to
two
right
angles
,
that
is
to
180
degrees
.
In
a
plaine
rectangle
triangle
,
one
of
the
acure
angles
is
the
complement
of
the
other
.
Where
note
that
when
the
complement
is
named
without
any
other
addition
,
it
is
meant
of
the
arch
,
which
is
wanting
of
a
quadrant
of
that
circle
,
or
90
degrees
.
In
like
manner
the
excesse
is
meant
of
the
arch
,
which
is
aboue
a
quadrant
.
But
when
it
is
said
the
complement
to
a
semicircle
,
it
is
vnderstood
of
so
many
degrees
as
will
make
vp
180.
But
in
a
Sphaericall
rectangle
triangle
,
one
of
the
oblique
angles
is
alwayes
greater
then
the
compl
:
of
the
other
.
If
two
arches
together
make
vp
a
Semicircle
,
the
excesse
of
the
greater
arch
,
is
equall
to
the
complement
of
the
lesser
.
The
same
Right
Sine
,
and
the
same
Tangent
,
and
Secant
,
doth
belong
both
to
the
arch
it selfe
,
and
also
to
the
complement
of
it
to
a
Semicircle
.
But
their
versed
Sines
differ
:
For
the
versed
Sine
of
an
arch
lesse
then
a
quadrant
,
is
equall
to
the
difference
of
the
Radius
,
and
the
Sine
of
the
complement
of
that
arch
:
and
the
versed
Sine
of
an
arch
greater
then
a
quadrant
,
is
equall
to
the
summe
of
them
.
And
the
versed
Sine
is
thus
found
out
,
As
the
Radius
is
to
the
Sine
of
halfe
the
arch
;
so
is
the
Sine
of
that
halfe
arch
,
to
halfe
the
versed
Sine
of
the
whole
arch
.
In
a
right
angled
triangle
both
Plaine
,
and
Sphaericall
,
one
of
the
sides
containing
the
right
angles
,
is
called
the
Base
,
and
the
other
the
Cathetus
:
and
the
fide
subtending
the
right
angle
is
the
Hypotenusa
.
And
know
that
euery
rectangled
triangle
,
is
most
fitly
noted
with
the
letters
ABC
;
so
that
BA
may
be
the
Base
,
and
CA
the
Cathetus
,
and
BC
the
Hypotenusa
:
and
B
the
angle
at
the
base
,
and
C
the
angle
at
the
Cathetus
,
and
A
the
right
angle
.
Likewise
euery
oblique-angled
triangle
with
the
letters
BCD
;
so
that
out
of
the
angle
C
a
perpendicular
CA
,
being
let
downe
,
it
may
in
the
base
BD
distinguish
the
two
cases
BA
and
DA
,
which
are
the
bases
of
the
two
particular
triangles
into
which
it
is
cut
.
And
in
noting
the
triangles
with
letters
,
obserue
diligently
,
that
if
any
angle
be
giuen
together
with
one
of
the
sides
including
it
,
the
same
angle
be
noted
with
B
;
and
the
side
with
BC.
If
both
the
angles
at
the
Base
BD
be
acute
,
the
perpendicular
CA
shal
fall
within
the
triangle
:
And
BD
=
BA
+
DA
that
is
BD
is
equall
to
the
summe
of
BA
and
DA.
And
if
the
angle
B
be
obtuse
,
the
perpendicular
CA
shall
fall
without
the
triangle
,
beyond
the
obtuse
angle
B
:
And
BD
=
DA
−
BA
,
that
is
BD
is
equall
to
the
excesse
of
DA
aboue
BA
or
if
the
angle
B.
be
obtuse
,
the
perpendicular
CA:
shall
fall
without
the
triangle
beyond
the
obtuse
angle
D
:
And
BD
=
BA
−
DA
,
that
is
BD
is
equall
to
the
excesse
of
BA
aboue
DA.
The
lesser
case
being
still
taken
from
the
greater
angle
.
And
note
that
this
signe
+
,
or
pl
(
that
is
plus
)
sheweth
that
the
magnitude
before
which
it
is
set
,
is
affirmed
and
positiue
in
nature
;
and
therefore
to
bee
added
.
And
that
this
signe
—
or
mi
(
that
is
minus
)
sheweth
the
magnitude
before
which
it
is
placed
,
is
denied
,
and
priuatiue
in
nature
;
and
therefore
to
be
substracted
,
as
you
may
see
in
those
former
examples
.
Againe
,
some
magnitudes
are
taken
seuerally
and
apart
;
as
s
BA
,
that
is
the
Sine
of
the
Base
;
s
co
BC
,
that
is
the
Sine
of
the
complement
of
the
Hypotenusa
;
t
B
,
that
is
the
tangent
of
the
angle
ABC
at
the
base
;
t
co
C
,
or
t
co
ABC
,
the
tangent
of
the
complement
of
the
angle
at
the
Cathetus
:
So
also
VqZ
,
that
is
the
quadrat
side
of
the
plaine
Z.
And
some
magnitudes
are
taken
vniuersally
,
and
then
they
are
included
in
pricks
:
as
:
that
is
the
Sine
of
halfe
the
arch
,
which
is
composed
of
the
summe
of
the
two
arches
DC
,
and
BD
,
abating
thereout
the
arch
BC.
So
also
Vq
∶
Z
×
X
∶
that
is
the
quadrat
side
of
the
two
plaines
Z
and
X
put
together
:
also
Vq
∶
Q
in
R
∶
Or
Vq
∶
Q
×
R
∶
that
is
the
quadrat
side
of
a
rectangular
plaine
,
the
two
sides
whereof
are
the
lines
Q
and
R
,
or
some
fourth
proportionall
already
found
,
and
the
Radius
,
or
Semidiameter
,
which
is
the
totall
Sine
.
For
by
the
signe
in
,
or
×
,
I
vse
to
expresse
multiplication
.
When
any
triangle
is
giuen
to
be
resolued
by
Trigonometrie
,
note
the
parts
thereof
(
either-sides
or
angles
)
which
are
giuen
and
knowne
,
with
a
little
line
drawne
crosse
each
such
part
:
and
note
the
vnknowne
part
which
is
sought
for
with
a
little
circle
.
And
if
a
triangle
Sphaericall
(
bee
it
right
angled
or
oblique-angled
)
proposed
hath
two
sides
each
of
them
seuerally
greater
then
a
quadrant
:
you
shall
in
resoluing
thereof
,
keepe
the
least
side
with
the
least
angle
opposed
to
it
:
and
for
the
two
other
both
sides
,
and
angles
,
take
the
complements
of
them
to
a
semicircle
.
Lastly
,
if
a
triangle
with
all
the
three
angles
giuen
,
be
required
to
be
conuerted
into
a
triangle
hauing
the
three
sides
giuen
.
You
shall
for
the
greatest
angle
of
the
triangle
proposed
,
&
for
the
greatest
side
subtending
it
,
take
the
complements
to
a
semicircle
;
keeping
the
other
two
lesser
angles
,
with
their
subtendent
sides
as
they
are
.
THE
CALCVLATION
OF
PLAINE
right-angled-triangles
.
I.
diagram
BC
·
BA
∷
R
·
〈…〉
B
(
●C
)
·
II.
diagram
BA
·
CA
∷
R
·
●B
(
●●●
C
)
∶
Of
plaine
Oblique-angled
triangles
.
III.
diagram
●B
·
DC
∷
●D
·
BC
∷
●C
·
BD
·
and
here
it
is
necessary
to
be
knowne
,
whether
the
angle
sought
for
be
greater
,
or
less
then
a
right
angle
,
or
90
deg.
IIII.
diagram
First
seek
the
angle
D
,
by
the
III
;
then
both
the
angles
Band
D
being
subducted
out
of
180
,
you
shall
haue
180
−
B
−
D
=
C
·
V.
diagram
First
seeke
the
angle
D
,
by
the
III
;
then
both
the
angles
B
and
D
being
subducted
out
of
180
,
Say
●B
·
DC
∷
s
∶
180
−
B
−
D
·
BD
·
VI.
diagram
Let
the
side
BD
be
greater
then
the
side
BC
:
First
,
then
for
the
other
two
angles
:
—
the
greater
—
the
lesser
.
VII
.
diagram
Let
the
side
BD
be
greater
then
the
side
BC
:
First
,
the
angles
C
and
D
are
to
be
sought
,
by
the
VI.
and
the
side
DC
,
by
the
III.
VIII
.
diagram
Take
the
greatest
side
BD
for
the
base
:
and
let
the
side
BC
,
be
greater
then
the
side
DC
.
First
say
,
BD
·
BC
+
DC
∷
BC
−
DC
·
Q
(
viz
BD
−
2DA
)
·
then
Nextly
seeke
the
angles
B
and
D
,
by
the
III.
Lastly
180
−
B
−
D
=
C.
THE
CALCVLATION
OF
Sphaericall
right-angled
,
and
quadrantall
triangles
.
I.
diagram
R
·
●B
∷
sBC
·
sCA
·
II.
diagram
R
·
sB
∷
●●●BA
∶
s
●●CA
·
III.
diagram
R
·
●●●BA
∷
●●●CA
∶
●●●BC
∶
IIII.
diagram
R
·
sBA
∷
tB
·
tBA
·
R
·
t
coB
∷
tCA
·
sBA
·
R
·
sBA
∷
t
coCA
·
t
coB
·
V.
diagram
R
·
s
coB
∷
tBC
·
tBA
·
R
·
t
coBC
∷
tBA
·
●
coB
·
R
·
●
coB
∷
t
coBA
·
t
coBC
·
VI.
diagram
R
·
●
coBC
∷
●B
·
●●●C
·
VII
.
diagram
If
a
triangle
BCD
be
quadrantall
,
hauing
one
side
BC
equall
to
a
quadrant
;
vpon
the
pole
D
describe
an
arch
of
a
great
circle
CA
,
cutting
the
side
DB
extended
in
A
:
and
so
making
a
right-angled
triangle
ABC
without
the
other
.
This
outward
right-angled
triangle
shall
be
resolued
in
steed
of
the
quadrantall
proposed
.
Of
Sphaericall
Oblique-angled
triangles
.
VIII
.
diagram
sB
∶
●DC
∷
●D
·
sBC
∷
●C
·
sBD
·
and
in
these
it
is
necessary
to
bee
knowne
whether
the
terme
sought
for
be
greater
then
a
quadrant
,
or
not
.
The
same
also
is
to
be
knowne
in
the
tenne
rules
next
following
,
if
the
sides
BC
and
DC
are
both
giuen
.
IX
.
diagram
First
,
R
·
s
coB
∷
tBC
·
●BA
·
then
,
s
coBA
·
s
coDA
∷
●
coBC
·
s
coDC
·
X.
diagram
First
,
R
·
s
coB
∷
tBC
·
tBA
·
then
●DA
·
sBA
∷
●B
tD
·
XI
.
diagram
First
,
R
·
s
coB
∷
●BC
·
●BA
then
,
s
coBC
·
●●●
DC
∷
●●●BA
·
s
●●DA
·
XII
.
diagram
First
,
R
·
s
coB
∷
tBC
·
tBA
·
then
,
tD
·
tB
∷
●BA
·
sDA
·
XIII
.
diagram
First
,
R
·
●
coBC
∷
●B
·
t
coBCA
·
then
,
s
coDCA
·
s
coBCA
∷
tBC
·
tDC
·
XIIII
.
diagram
First
,
R
·
s
coBC
∷
tB
·
t
coBCA
·
then
,
sBCA
·
sDCA
∷
s
coB
·
s
coD
·
XV.
diagram
First
,
R
·
s
coBC
∷
tB
·
t
coBCA
·
then
,
tDC
·
tBC
∷
s
coBCA
·
s
coDCA
·
XVI
.
diagram
First
,
R
·
s
coBC
∷
tB
·
t
coBCA
·
then
,
s
coB
·
s
coD
∷
●BCA
·
sDCA
·
XVII
.
diagram
First
,
R
·
sBD
∷
sBC
·
QI
·
then
,
See
what
QII
cutteth
in
the
fift
circle
,
which
is
of
equall
diuisions
:
and
thereto
adde
the
Radius
,
by
setting
I
before
that
number
.
Diuide
the
whole
into
two
equall
parts
:
and
reckoning
one
halfe
in
that
fift
circle
,
set
the
Index
to
it
,
and
it
shall
in
the
first
circle
cut
the
Sine
of
halfe
the
angle
B.
XVIII
.
diagram
If
all
the
three
angles
be
giuen
:
conuent
the
triangle
into
another
hauing
all
the
three
sides
giuen
:
and
resolue
the
same
for
the
triangle
proposed
.
CHAP.
XIIII
.
Of
the
Nocturnall
Dials
.
THere
are
in
the
Instrument
,
two
seuerall
Nocturnall
Dyals
.
The
innermost
of
them
is
fitted
to
the
starre
in
the
rump
of
the
great
Beare
,
commonly
called
Aliot
.
The
other
is
composed
of
12
seuerall
starres
:
whose
names
you
shall
finde
written
within
,
neere
to
the
center
.
The
outermost
circle
of
the
Nocturnall
Dyall
is
diuided
into
twise
12
houres
:
each
houre
being
subdiuided
into
quarters
,
and
are
noted
with
figures
belonging
to
the
houres
,
as
may
be
seene
in
the
Instrument
.
The
middlemost
circle
of
the
Nocturnall
is
diuided
into
12
moneths
,
hauing
their
names
written
:
each
moneth
being
distinguished
into
tenth
dayes
with
longer
lines
;
and
into
fift
dayes
with
shorter
lines
.
And
if
the
Instrument
be
large
enough
,
each
day
of
the
monethes
throughout
the
yeare
,
may
be
noted
with
pricks
.
In
the
innermost
circle
are
the
diuisions
and
names
of
12
fixed
starres
:
which
are
these
.
-
The
bright
star
in
the
head
of
♈
.
-
the
Bulls
eye
-
the
latter
shoulder
of
Orion
-
the
little
dogge
-
the
heart
of
the
Lyon
-
the
tayle
of
the
Lyon
-
Spica
Virginis
-
the
North
ballance
-
the
head
of
Ophiuchus
-
the
heart
of
the
Vultur
-
the
mouth
of
Pegasus
-
the
tip
of
the
wing
of
Pegasus
.
To
finde
out
the
houre
of
the
night
by
Aliot
.
Seeke
the
day
of
the
moneth
in
the
annuall
circle
of
the
Nocturnall
:
and
apply
the
Index
thereto
:
marke
what
houre
it
cutteth
,
in
the
houre
circle
.
Remember
this
houre
for
all
that
day
:
then
at
night
when
you
would
finde
out
the
houre
,
hold
vp
your
Instrument
by
the
handle
,
and
moue
it
vp
and
downe
,
till
you
see
the
pole
starre
through
the
middle
hole
,
and
the
starre
called
Aliot
by
the
limbe
:
Set
the
Index
or
labell
to
Aliot
,
and
marke
what
houre
the
labell
cutteth
,
for
if
vnto
this
houre
you
adde
the
houre
kept
in
minde
for
that
day
,
the
summe
of
both
shall
giue
you
the
true
houre
of
the
night
:
so
that
you
cast
out
12
houres
,
from
the
said
summe
if
it
shall
chance
to
be
more
.
Example
.
If
on
the
15th
of
Nouember
you
would
find
the
houre
of
the
night
by
the
starre
Aliot
.
Apply
the
Index
to
the
day
of
the
moneth
,
and
it
will
cut
in
the
houre
circle
8
and
an
halfe
:
then
suppose
the
Index
being
set
to
the
starre
Aliot
,
as
aboue
taught
,
doth
cut
in
the
houre
circle
10.
these
two
numbers
being
added
together
,
the
summe
wil
be
18½
,
out
of
which
subduct
12
;
and
the
remaines
6½
,
will
be
the
true
houre
of
the
night
.
To
finde
out
the
houre
of
the
night
by
the
Inner
Nocturnall
Dyall
.
To
performe
this
it
is
necessary
that
you
know
the
true
Meridian
of
the
place
wherein
you
are
,
and
can
finde
it
out
by
night
,
which
you
may
thus
doe
.
Hauing
a
Meridian
line
drawne
in
some
window
,
or
other
conuenient
place
(
as
is
shewed
in
the
Second
part
of
this
booke
,
Vse
19
)
stick
vp
therein
a
long
needle
perpendicularly
,
and
watch
till
the
Sunne
casteth
the
shadow
of
the
needle
,
vpon
the
Meridian
line
.
Or
else
in
a
true
Sunne
Dyall
obserue
when
the
shadow
falls
iust
on
12
a
clock
,
for
then
is
the
Sunne
in
the
Meridian
.
Wherefore
goe
instantly
into
some
place
about
your
house
where
you
may
see
some
marke
,
either
a
chimney
,
or
the
corner
of
an
house
,
or
else
some
tree
,
or
such
like
,
directly
betweene
you
and
the
Sunne
:
then
haue
you
the
true
Meridian
.
Or
otherwise
you
may
in
a
cleare
night
goe
into
some
plaine
place
neere
your
house
,
and
setting
vp
a
straight
pole
perpendicularly
on
the
ground
,
goe
a
good
distance
from
it
Southwards
;
and
then
moue
vp
and
downe
,
till
you
see
the
top
of
your
pole
,
directly
betweene
your
eye
,
and
the
North
polar
starre
:
then
set
vp
another
pole
perpendicularly
betweene
your
feet
,
so
that
both
your
poles
,
and
the
Polar
starre
,
may
be
in
one
right
line
.
And
then
going
backe
againe
to
your
first
pole
,
looke
what
knowne
starre
is
directly
ouer
your
last
pole
,
for
that
starre
is
in
the
Meridian
.
You
may
therefore
instantly
goe
to
some
conuenient
place
,
and
take
a
marke
whereby
you
may
at
all
times
know
the
Meridian
as
is
afore
taught
.
When
therefore
at
any
time
of
the
night
you
would
know
what
a
clocke
it
is
,
goe
to
that
place
where
you
stood
,
and
looking
directly
ouer
your
marke
,
see
if
any
of
the
12
fixed
starres
,
bee
in
the
Meridian
;
or
if
none
of
them
be
therein
,
obserue
which
two
of
them
are
on
either
side
thereof
,
and
what
part
of
that
space
is
in
the
Meridian
.
Then
goe
into
the
light
,
and
take
your
instrument
,
and
set
the
Index
to
that
starre
,
or
point
which
you
saw
in
the
Meridian
:
marke
what
houre
it
cutteth
,
for
that
same
houre
being
added
to
the
houre
,
which
the
day
of
the
moneth
sheweth
,
shall
giue
you
the
true
houre
of
the
night
:
so
that
you
cast
out
12
houres
,
from
the
said
summe
,
if
it
shall
chance
to
be
more
.
Example
.
Suppose
the
fifth
of
December
,
that
the
middle
point
of
the
space
betweene
the
bright
starre
in
the
head
of
Aries
,
and
the
Bulls
eye
,
bee
in
the
Meridian
.
Set
the
Index
to
the
middle
point
of
the
space
betweene
those
two
starres
in
the
Instrument
:
and
it
will
cut
in
the
houre
circle
2
and
an
halfe
:
then
againe
set
the
Index
to
the
fifth
of
December
,
and
in
the
houre
circle
it
wil
cut
7
:
which
added
vnto
2
and
an
halfe
,
giueth
9
and
an
halfe
,
for
the
true
houre
of
the
night
.
Another
example
.
Suppose
the
19th
of
December
,
that
one
third
part
of
the
space
betweene
the
Bulls
eye
,
and
the
right
shoulder
of
Orion
,
be
in
the
Meridian
.
Set
the
Index
to
one
third
part
of
that
space
in
the
Instrument
,
and
it
will
cut
in
the
houre
circle
4
and
halfe
a
quarter
almost
:
againe
,
set
the
Index
to
the
19th
of
December
,
and
in
the
houre
circle
it
will
cut
6
,
which
being
added
vnto
4
,
and
halfe
a
quarter
almost
,
giueth
10
and
almost
halfe
a
quarter
for
the
houre
of
the
night
.
diagram
THE
SECOND
PART
OF
THIS
BOOKE
.
Shewing
the
vse
of
the
Second
side
of
the
Instrument
,
for
the
working
of
most
questions
,
which
may
be
performed
by
the
Globe
:
And
the
declination
of
Dyals
,
vpon
any
kinde
of
Plaine
.
VPon
the
second
side
of
the
Instrument
,
is
delineated
the
proiection
of
the
vpper
Hemisphaere
vpon
the
plaine
of
the
Horizon
:
The
Horizon
it selfe
is
vnderstood
to
bee
the
innermost
circle
of
the
limbe
:
and
is
diuided
on
both
sides
,
from
the
points
of
East
,
and
West
into
degrees
,
noted
with
10
,
20
,
30
,
&c.
vnto
90.
And
the
center
of
the
Instrument
is
the
Zenith
,
or
Vertical
point
.
Within
the
Horizon
,
the
middle
straight
line
,
or
Diameter
pointing
North
and
South
,
is
the
Meridian
,
or
12
a
clock
line
:
and
the
other
shortarching
lines
,
on
both
sides
of
it
are
the
houre
lines
,
distinguished
accordingly
by
their
figures
.
These
houre
lines
should
indeede
bee
drawne
through
the
whole
plaine
,
crossing
one
another
in
the
Pole
of
the
world
:
but
that
the
Instrument
may
be
more
faire
,
they
are
onely
drawne
short
.
And
because
diuers
excellent
vses
,
doe
require
the
totall
delineation
of
the
houre
circles
,
I
haue
in
a
seuerall
paper
,
inscribed
intirely
,
both
the
houre
lines
,
and
also
two
other
circles
betweene
them
,
containing
euery
one
fiue
degrees
.
(
But
if
the
Instrument
were
large
enough
to
receiue
them
,
it
were
best
if
euery
degree
had
his
circle
:
and
so
euery
15
circle
should
bee
an
houre
line
.
)
And
of
the
parallels
,
there
needes
no
more
,
but
the
Aequinoctial
,
and
both
the
Tropics
.
For
as
much
as
there
will
be
great
vse
of
this
paper
Instrument
;
I
haue
in
the
24
Vse
shewed
the
manner
of
making
it
:
so
that
any
that
1●
ingenuous
,
and
ready
handed
may
himselfe
delineate
one
sufficient
enough
to
serue
his
turne
,
for
any
eleuation
.
The
two
arches
which
crosse
the
houre
lines
meeting
on
both
sides
in
the
points
of
intersection
of
the
sixe
a
clocke
lines
with
the
Horizon
,
are
the
two
Semicircles
of
the
Ecliptick
,
or
Annuall
circle
of
the
Sunne
:
the
vpper
of
which
arches
serueth
for
the
Summer
halfe
yeare
,
and
the
lower
for
the
Winter
halfe
yeare
:
and
are
therefore
diuided
in
365
dayes
:
which
are
also
distinguished
into
12
moneths
with
longer
lines
,
hauing
their
names
set
downe
:
and
into
tenthes
,
and
fifthes
with
shorter
lines
:
and
the
rest
of
the
dayes
with
pricks
:
as
may
plainly
bee
seene
in
the
Instrument
.
And
this
is
for
the
ready
finding
out
of
the
place
of
the
Sunne
euery
day
:
and
also
for
shewing
of
the
Sunnes
yearely
motion
:
because
by
this
motion
the
Sunne
goeth
round
about
the
heauens
in
the
compasse
of
a
yeare
,
makeing
the
foure
parts
,
or
seasons
thereof
.
Namely
the
Spring
,
in
that
quarter
of
the
Ecliptick
which
beginneth
at
the
intersection
on
the
West
side
of
the
Instrument
,
and
is
therefore
called
the
Vernall
intersection
.
Then
the
Summer
in
that
quarter
of
the
Ecliptick
which
beginneth
with
the
intersection
of
the
Meridian
in
the
highest
point
next
the
Zenith
.
And
after
that
Autumne
in
that
quarter
of
the
Ecliptick
,
which
beginneth
at
the
intersection
on
the
East
side
of
the
Instrument
,
and
is
therefore
called
the
Autumnall
intersection
.
And
lastly
,
the
Winter
in
that
quarter
of
the
Ecliptick
,
which
beginneth
at
the
intersection
with
the
Meridian
,
in
the
lowest
point
next
the
Horizon
.
But
besides
this
yearely
motion
,
the
Sunne
hath
a
Diurnall
or
dayly
motion
,
whereby
it
maketh
day
and
night
with
all
the
diuersities
,
and
inequalities
thereof
:
which
is
expressed
by
those
other
circles
drawne
crosse
the
houre
lines
:
the
middlemost
whereof
being
grosser
then
the
rest
,
meeting
with
the
Ecliptick
in
the
points
of
the
Vernall
,
and
Autumnall
intersections
,
is
the
Aequinoctiall
:
and
the
rest
on
both
sides
of
it
,
are
called
the
Parallels
,
or
Diurnall
arches
of
the
Sunne
:
the
two
outermost
whereof
are
the
Tropics
,
because
in
them
the
Sunne
hath
his
furthest
digression
,
or
Declination
from
the
Aequinoctiall
,
which
is
degrees
23½
:
and
thence
beginneth
againe
to
returne
to
the
Aequinoctiall
.
The
vpper
of
the
two
Tropics
next
the
center
(
in
this
our
Northerne
Hemisphaere
)
is
the
Tropick
of
Cancer
:
and
the
Sunne
being
in
it
,
is
highest
into
the
North
,
making
the
longest
day
of
Summer
.
And
the
lower
next
the
Horizon
,
is
called
the
Tropick
of
Capricorne
;
and
the
Sunne
being
in
it
,
is
lowest
into
the
South
,
making
the
shortest
day
of
Winter
.
Betweene
the
two
Tropics
,
and
the
Aequinoctiall
infinite
such
parallel
circles
are
vnderstoode
to
bee
contained
:
for
the
Sunne
is
what
point
soeuer
of
the
Ecliptick
it
is
caried
,
describeth
by
his
lation
,
a
circle
parallel
to
the
Aequinoctial
.
Yet
those
parallels
which
are
in
the
Instrument
,
though
drawne
but
to
euery
second
degree
of
Declination
,
may
be
sufficient
to
direct
the
eye
,
in
imagining
and
tracing
out
,
through
euery
day
of
the
whole
yeare
in
the
Ecliptick
,
a
proper
circle
,
which
may
be
the
Diurnall
arch
of
the
Sunne
for
that
day
.
For
vpon
the
right
estimation
of
that
imaginary
parallel
,
doth
the
manifold
vse
of
this
Instrument
especially
rely
:
because
the
true
place
of
the
Sunne
,
all
that
day
,
is
in
some
part
,
or
point
of
that
circle
.
Wherefore
for
the
better
conceiuing
,
and
bearing
in
minde
thereof
,
euery
fift
parallel
,
is
herein
made
a
little
grosser
then
the
rest
.
I
Vse
.
And
thus
by
the
eye
,
and
view
only
,
to
behold
and
comprehend
the
course
of
the
Sunne
,
both
for
his
Annuall
and
Diurnall
motion
,
may
be
the
first
vse
of
this
Instrument
.
II
Vse
.
To
take
the
height
of
the
Sun
aboue
the
Horizon
.
Set
vp
the
pinne
,
(
which
is
therefore
made
fit
for
the
hole
at
the
center
)
perpendicular
in
the
center
:
and
put
the
Indices
on
both
sides
,
downe
vpon
the
Meridian
,
that
they
with
their
waight
,
may
not
sway
the
Instrument
any
way
as
it
hangeth
:
then
with
a
threed
put
into
the
hole
aboue
in
the
handle
,
hang
it
perpendicularly
,
bearing
the
edge
toward
the
Sunne
,
that
the
pinne
may
cast
a
shadow
,
vpon
the
degrees
in
the
limbe
:
for
that
degree
which
the
shadow
of
the
pinne
cutteth
in
the
limbe
,
is
the
height
of
the
Sunne
aboue
the
Horizon
,
at
that
present
.
III.
Vse
.
To
find
the
Declination
of
the
Sunne
euery
day
.
Looke
the
day
of
the
moneth
proposed
in
the
Ecliptic
,
and
marke
how
many
degrees
the
prick
shewing
that
day
,
is
distant
from
the
Equinoctiall
,
either
on
the
Summer
,
or
Winter
side
,
viz
North
,
or
South
.
Example
.
I.
What
will
the
Declination
of
the
Sunne
be
,
vpon
the
11th
day
of
August
?
Looke
the
11th
day
of
August
:
and
you
shall
find
it
in
the
sixt
Circle
aboue
the
Aequinoctiall
:
now
because
each
Parallel
,
standeth
(
as
hath
beene
said
before
)
for
2
degrees
,
the
Sunne
shall
that
day
decline
North-wards
12
degrees
.
Example
.
II.
What
Declination
hath
the
Sunne
,
vpon
the
24th
day
of
March
?
Looke
the
24th
day
of
March
,
and
you
shall
find
it
,
betweene
the
second
,
and
third
Northerne
parallels
,
as
it
were
an
halfe
and
one
fift
part
more
of
that
distance
from
the
second
:
reckon
therefore
4
degrees
for
the
two
Circles
,
and
one
degree
for
the
halfe
space
:
so
shall
the
Sunnes
declination
bee
5
degrees
,
and
about
one
fift
part
of
a
degree
Northward
,
that
same
day
.
Example
.
III.
What
Declination
hath
the
Sunne
vpon
the
13th
day
of
Nouember
?
Looke
the
13th
day
of
Nouember
,
and
you
shall
find
it
below
the
Equinoctiall
,
tenne
parallels
and
about
one
quarter
,
which
is
20
degrees
,
and
an
halfe
South-wards
.
So
much
is
the
Declination
.
And
according
to
these
examples
iudge
of
all
the
rest
.
IIII.
Vse
.
To
find
the
Right
ascension
of
the
Sunne
euery
day
.
Imagine
an
hower
line
through
the
day
of
the
moneth
giuen
,
and
marke
in
what
point
it
will
cr●sse
the
Aequinoctiall
:
then
lay
a
Ruler
,
or
a
streight
Scroule
of
paper
,
to
the
Pole
of
the
world
(
noted
in
the
Instrument
with
P
W
)
and
that
same
point
.
For
the
Ruler
shall
in
the
innermost
Circle
of
the
limbe
,
of
the
South
side
,
cut
the
Right
ascension
of
the
Sunne
for
that
day
,
to
be
reckoned
from
the
West
,
to
the
point
of
intersection
,
for
the
first
,
or
vpper
Semicircle
of
the
Ecliptic
:
or
from
the
East
together
with
180
,
for
the
second
,
or
lower
Semicircle
of
the
Ecliptic
.
V.
Vse
.
To
find
the
longitude
of
the
Sunne
,
or
in
what
degree
of
the
Signe
he
is
euery
day
.
The
Pole
of
the
first
Semicircle
of
the
Ecliptic
is
noted
P
I.
and
the
Pole
of
the
second
Semicircle
is
noted
P
I
I.
Lay
a
Ruler
,
or
a
streight
Scroule
of
paper
,
to
the
day
of
the
moneth
,
and
the
proper
Pole
of
the
Semicircle
of
the
Ecliptic
,
in
which
it
is
:
for
the
Ruler
shall
in
the
innermost
Circle
of
the
limbe
,
on
the
South
side
,
cut
the
degree
of
the
Sunnes
place
in
the
Ecliptic
,
reckoning
it
in
the
same
manner
as
you
did
in
finding
the
Sunnes
Right
ascension
:
and
the
Arch
thus
found
is
called
the
longitude
of
the
Sunne
.
which
may
bee
expansed
into
signes
,
by
reckoning
on
the
limbe
,
from
the
West
to
South
♈
,
♉
,
♊
,
and
from
South
to
East
♋
,
♌
,
♍
:
then
backe
againe
from
East
to
South
♒
,
♏
,
♐
;
and
lastly
from
South
to
West
♑
♒
,
♓
,
allowing
30
degrees
,
for
each
of
those
twelue
signes
.
VI.
Vse
.
To
find
the
Diurnall
Arch
,
or
Circle
of
the
Sunnes
course
euery
day
.
The
Sunne
euery
day
by
his
motion
(
as
hath
beene
said
)
describeth
a
Circle
parallell
to
the
Aequinoctiall
,
which
is
either
one
of
the
Circles
in
the
Instrument
,
or
some-where
betweene
two
of
them
.
First
then
seeke
〈◊〉
of
the
moneth
;
and
if
it
fall
vpon
one
of
those
〈◊〉
,
that
is
the
Circle
of
the
Sunnes
course
that
same
day
:
But
if
it
fall
betweene
any
two
of
those
Parallel●
,
imagine
in
your
minde
,
and
estimate
with
your
eye
,
another
Parallel
through
that
point
,
betweene
those
two
Parallels
,
keeping
still
the
same
distance
from
each
of
them
.
As
in
the
first
of
the
three
former
Examples
,
The
circle
of
the
Sunnes
course
,
vpon
the
11th
day
of
August
,
shall
be
the
very
sixt
Parallel
aboue
the
Aequinoctiall
towards
the
Center
.
In
Example
II.
The
Circle
of
the
Sunnes
course
vpon
the
24th
day
of
March
,
shall
bee
an
imaginary
Circle
betweene
the
second
,
and
third
Parallels
,
still
keeping
an
halfe
of
that
space
,
and
one
fift
part
more
of
the
rest
from
the
second
.
In
Example
III.
The
Circle
of
the
Sunnes
course
vpon
the
13th
day
of
Nouember
,
shall
be
an
imaginary
Circle
,
betweene
the
tenth
,
and
eleauenth
Parallels
,
below
the
Aequinoctiall
,
still
keeping
one
quarter
of
that
space
from
the
tenth
.
VII
.
Vse
.
To
find
the
Rising
and
Setting
of
the
Sunne
euery
day
.
Seeke
out
(
as
was
last
shewed
)
the
imaginary
Circle
or
Parallel
of
the
Sunnes
course
,
for
that
day
,
and
marke
the
point
where
it
meeteth
with
the
Horizon
,
both
on
the
East
and
West
sides
thereof
,
for
that
is
the
very
point
of
the
Sunnes
rising
,
and
setting
that
same
day
,
and
the
hower
lines
which
are
on
both
sides
of
it
,
by
proportioning
the
distance
reasonably
,
according
to
15
minuts
,
for
the
quarter
of
the
hower
,
will
shew
the
hower
of
the
Sunnes
rising
,
on
the
East
side
,
and
the
Sunnes
setting
on
the
West
side
.
VIII
.
Vse
.
To
know
the
reason
,
and
manner
,
of
the
Increasing
,
and
Decreasing
of
the
dayes
and
nights
throughout
the
whole
yeare
.
When
the
Sunne
is
in
the
Aequinoctiall
,
it
riseth
,
and
setteth
at
6
a
Clock
,
for
in
the
instrument
,
the
intersection
of
the
Aequinoctiall
,
and
the
Ecliptic
with
the
Horizon
,
is
in
the
6
a
clock
Circle
on
both
sides
.
But
if
the
Sunne
bee
out
of
the
Aequinoctiall
,
declining
toward
the
North
,
the
intersections
of
the
Parallel
of
the
Sunne
with
the
Horizon
,
is
before
6
in
the
Morning
,
and
after
6
in
the
Euening
:
and
the
diurnall
Arch
of
the
Sunne
,
greater
then
12
howers
;
and
so
much
more
great
,
the
greater
the
Northerne
Declination
is
.
Againe
if
the
Sun
be
declining
toward
the
South
,
the
intersections
of
the
Parallel
of
the
Sunne
with
the
Horizon
,
is
after
6
in
the
Morning
,
and
before
6
in
the
Euening
;
and
the
diurnall
Arch
lesser
then
12
howers
;
and
by
so
much
lesser
,
the
greater
the
Southerne
Declination
is
.
And
in
those
places
of
the
Ecliptic
in
which
the
Sun
most
speedily
changeth
his
Declination
,
the
length
also
of
the
day
is
most
altered
,
and
where
the
Ecliptic
goeth
most
parallel
to
the
Aequinoctiall
changing
the
Declination
but
little
,
the
length
of
the
day
also
is
but
little
altered
.
As
for
example
,
when
the
Sunne
is
neere
vnto
the
Aequinoctiall
,
on
both
sides
,
the
dayes
increase
,
and
also
decrease
suddenly
and
apace
:
because
in
those
places
,
the
Ecliptic
inclineth
to
the
Aequinoctiall
in
a
manner
like
a
streight
line
,
making
sensible
declination
.
Againe
when
the
Sunne
is
neere
his
greatest
Declination
,
as
in
the
height
of
the
Summer
,
and
the
depth
of
Winter
,
the
dayes
keepe
for
a
good
time
,
as
it
were
at
one
stay
,
because
in
those
places
the
Ecliptic
is
in
a
manner
parallel
to
the
Aequinoctiall
,
scarce
altering
the
declination
:
and
because
in
those
two
times
of
the
yeare
,
the
Sunne
standeth
as
it
were
still
,
at
one
declination
;
they
are
called
the
Summer
Solstice
,
and
the
Winter
Solstice
.
And
in
the
meane
spaces
,
the
neerer
every
place
is
to
the
Aequinoctiall
,
the
greater
is
the
diversitie
of
dayes
.
Wherefore
we
may
hereby
plainely
see
,
that
the
common
receiued
opinion
,
that
in
every
moneth
,
the
dayes
doe
equally
increase
,
is
erroneous
.
Also
wee
may
see
that
in
Parallels
equally
distant
from
the
Aequinoctiall
,
the
day
on
the
one
side
,
is
equall
to
the
night
on
the
other
side
.
IX
.
Vse
.
To
find
the
Ascensionall
difference
of
the
Sunne
every
day
.
Seeke
out
the
time
of
the
Suns
Rising
,
or
Setting
that
same
day
(
by
the
VII
Vse
)
and
see
how
much
it
differeth
from
sixe
a
clocke
,
then
conuert
the
same
difference
into
degrees
(
as
was
taught
in
1
Part.
Chap.
12.
Sect.
1.
)
by
multiplying
the
howers
with
their
decimall
parts
,
by
15.
And
so
haue
you
the
Ascensionall
difference
for
that
day
.
X.
Vse
.
To
find
out
the
Oblique
ascension
of
the
Sunne
every
day
.
Seeke
out
the
Sunnes
Right
ascension
(
by
the
IIII
Vse
)
and
the
Ascensionall
difference
(
by
the
IX
Vse
:
)
And
if
the
Sunne
be
in
the
first
Semicircle
of
the
Ecliptic
,
Subduct
the
Ascensionall
difference
,
out
of
the
Right
ascension
:
But
if
the
Sunne
be
in
the
second
Semicircle
of
the
Ecliptic
,
adde
the
Ascensionall
difference
to
the
Right
ascension
:
and
you
shall
haue
the
oblique
ascension
.
XI
.
Vse
.
To
find
how
farre
the
Sunne
riseth
,
and
setteth
,
from
the
true
East
and
West
points
,
which
is
called
the
Sunnes
Amplitude
ortiue
,
and
occasiue
.
Seeke
out
(
as
was
shewed
in
the
VI.
Vse
)
the
imaginary
Circle
,
or
Parallel
of
the
Sunnes
course
,
and
the
points
of
that
Circle
in
the
Horizon
,
on
the
East
,
and
West
sides
,
cutteth
the
degree
of
the
Amplitude
Ortiue
,
and
occasiue
.
XII
.
Vse
.
To
find
the
length
of
every
day
and
night
.
Double
the
hower
of
the
Sunnes
setting
,
and
you
shall
haue
the
length
of
the
day
;
and
double
the
hower
of
the
Sunnes
rising
,
and
you
shall
haue
the
length
of
the
night
.
XIII
.
Vse
To
find
the
true
place
of
the
Sunne
,
vpon
the
Instrument
,
which
answereth
to
the
point
,
wherein
the
Sunne
is
in
the
heauens
:
and
is
the
ground
of
all
the
questions
following
.
Take
with
your
Instrument
the
height
of
the
Sunne
,
and
reckon
it
on
the
moueable
Index
,
or
Labell
:
and
then
moue
the
said
Labell
,
till
you
find
the
height
of
the
Sunne
,
exactly
to
fall
vpon
the
Parallel
of
the
Sunne
for
that
day
,
on
the
East
side
if
it
bee
in
the
Fore-noone
,
and
on
the
West
side
,
if
it
bee
in
the
After-noone
;
the
point
of
intersection
,
where
the
Index
,
or
Labell
crosseth
the
Parallel
,
in
that
point
of
the
Sunnes
altitude
,
shall
bee
the
true
place
of
the
Sunne
on
the
Instrument
.
XIIII
.
Vse
.
To
find
the
Hower
of
the
day
.
The
true
place
of
the
Sunne
on
the
Instrument
(
found
out
as
was
last
shewed
)
sheweth
among
the
hower
lines
the
true
hower
of
the
day
.
XV.
Vse
.
To
find
out
the
Azumith
or
verticall
Circle
in
which
the
Sunne
is
,
or
the
Horizontall
distance
of
the
Sunne
from
the
Meridian
.
The
Index
or
Labell
fastned
at
the
Center
,
is
a
moueable
Azumith
:
apply
therefore
the
edge
thereof
,
vnto
the
true
place
of
the
Sunne
on
the
instrument
(
found
out
as
was
shewed
by
XIII
.
Vse
.
)
And
marke
what
point
of
the
Horizon
,
or
Limbe
,
the
same
edge
of
the
Labell
cutteth
;
reckon
how
many
degrees
of
the
H●rizon
,
are
intercepted
betweene
that
point
,
and
the
Meridian
line
,
or
South
point
,
either
on
the
East
,
or
West
side
:
and
that
Arch
shall
be
the
Horizontall
distance
sought
for
,
whereby
is
shewed
the
Azumith
of
the
Sunne
at
that
instant
:
and
consequently
the
Angle
which
the
verticall
Circle
,
or
Azumith
of
the
Sunne
maketh
with
the
Meridian
.
XVI
.
Vse
.
The
Azumith
of
the
Sunne
being
knowne
,
to
find
out
the
Altitude
of
the
Sunne
,
and
the
Hower
of
the
day
.
Set
the
edge
of
the
Labell
to
the
Azumith
given
,
and
marke
in
what
point
the
same
edge
crosseth
the
Parallel
of
the
Sunne
for
that
day
:
that
point
of
intersection
sheweth
the
height
of
the
Sunne
aboue
the
Horizon
,
vpon
the
Labell
:
and
also
it
sheweth
the
hower
of
the
day
among
the
hower
lines
.
XVII
.
Vse
.
To
find
at
what
hower
the
Sunne
commeth
to
be
full
East
,
or
West
every
day
in
Summer
.
Apply
the
edge
of
the
Labell
,
vnto
the
East
,
or
West
points
of
the
Limbe
,
and
marke
in
what
point
,
the
said
edge
cutteth
the
Parallel
of
the
Sunne
for
that
day
,
for
that
same
point
among
the
hower
lines
,
shall
shew
the
time
of
the
Sunnes
comming
to
be
full
East
,
or
West
in
that
day
,
and
likewise
of
what
altitude
the
Sunne
will
be
aboue
the
Horizon
,
at
that
time
of
his
being
full
East
,
or
West
.
XVIII
.
Vse
.
To
find
the
height
of
the
Sunne
at
high
Noone
every
day
,
and
likewise
at
every
other
hower
.
Marke
in
what
point
the
Parallel
of
the
Sunne
for
that
day
,
cutteth
the
line
of
that
hower
,
for
which
you
would
know
the
Sunnes
altitude
:
And
vnto
that
point
of
intersection
,
apply
the
edge
of
the
moueable
Labell
,
or
Index
:
and
thereon
shall
you
find
,
the
very
degree
of
the
Sunnes
altitude
,
at
that
hower
.
By
this
XVIII
Vse
,
and
by
the
XVI
,
are
made
the
Quadrants
,
described
by
Gemma
Frizius
,
Munster
,
Clauins
,
Mr.
Gunter
,
and
others
:
and
also
all
manner
of
Rings
,
Cylinders
,
&
inuumerable
other
topicall
Instruments
,
for
the
finding
out
of
the
hower
,
and
other
like
conclusions
.
And
likewise
the
reason
,
of
finding
the
hower
of
the
day
,
by
a
mans
shadow
,
or
by
the
shadow
of
any
Gnomon
,
set
vp
perpendicular
to
the
Horizon
,
or
else
parallel
to
it
.
XIX
.
Vse
.
To
find
out
the
Meridian
line
,
and
the
points
of
the
compasse
without
a
Magneticall
needle
,
yea
more
exactly
then
with
a
needle
.
Take
the
height
of
the
Sunne
,
by
the
shadow
of
the
pinne
:
and
apply
the
same
height
,
reckoned
on
the
Index
,
or
Labell
to
the
parallel
of
the
Sunne
for
that
day
,
whereby
you
haue
the
true
place
of
the
Sunne
,
in
the
instrument
,
as
hath
beene
shewed
in
the
XIII
.
Vse
.
Then
keeping
both
the
Labell
at
that
point
,
and
the
pinne
vpright
in
the
Center
,
hold
,
or
set
your
instrument
parallel
to
the
plaine
of
the
Horizon
,
with
the
pinne
toward
the
Sunne
,
and
moue
it
gently
,
till
the
shadow
of
the
pinne
shall
fall
,
exactly
vpon
,
the
fiduciall
edge
of
the
Labell
.
For
then
the
Meridian
line
of
the
instrument
,
shall
be
in
the
true
Meridian
of
the
place
:
and
the
foure
quarters
of
the
instrument
,
shall
looke
into
the
foure
cardinall
points
,
of
East
,
West
,
North
,
and
South
.
Wherefore
if
with
a
bodkin
,
you
make
a
prick
at
each
end
of
the
Meridian
of
your
instrument
where
it
standeth
:
and
with
a
Ruler
draw
a
line
through
them
:
the
same
shall
bee
the
Meridian
of
that
place
.
This
is
a
most
excellent
practise
,
for
finding
out
the
Meridian
in
any
place
,
and
is
in
an
instant
performed
,
and
that
easily
.
And
hereby
you
may
examine
the
Variation
of
the
Compasse
.
And
also
exactly
place
any
Sunne
Dyall
.
XX.
Vse
.
Consid●rations
for
the
vse
of
the
instrument
in
the
night
.
In
such
questions
as
concerne
the
night
,
or
the
time
before
Sun
rising
,
and
after
Sunne
setting
,
the
instrument
representeth
the
lower
Hemisphaere
,
wherin
the
Southerne
Pole
is
eleuated
.
And
therefore
the
Parallels
,
which
are
aboue
the
Aequinoctiall
,
shall
bee
for
the
Southerne
,
or
Winter
Parallels
,
and
those
beneath
the
Aequinoctiall
,
for
the
Northerne
,
or
Summer
parallels
.
And
the
East
shall
be
accounted
for
West
,
and
the
West
for
East
:
and
the
North
shall
bee
accounted
for
South
,
and
the
South
for
North
:
contrary
to
that
which
was
before
,
when
the
Instrument
represented
the
vpper
Hemisphaere
.
XXI
.
Vse
.
To
find
how
many
degrees
the
Sunne
is
vnder
the
Horizon
,
at
any
time
of
the
night
.
Seeke
the
declination
of
the
Sunne
for
the
day
proposed
:
and
at
the
same
declination
,
on
the
contrary
side
,
imagine
a
Parallel
for
the
Sunne
that
night
:
and
marke
what
point
of
it
is
in
the
very
hower
and
minute
proposed
:
then
set
the
Index
,
or
Labell
to
that
point
of
the
Parallel
,
and
it
will
shew
you
thereon
the
degree
of
the
Sunnes
depression
vnder
the
Horizon
.
XXII
.
Vse
.
To
find
out
the
length
of
the
Crepusculum
,
or
Twilight
euery
day
.
Because
the
question
concerneth
the
night
time
,
you
must
seeke
out
the
Sunnes
Parallel
,
for
the
night
,
on
the
other
side
of
the
Aequinoctiall
,
hauing
the
same
declination
with
that
which
the
day
of
the
moneth
sheweth
:
then
moue
about
the
Labell
,
vntill
the
said
Parallel
cutteth
the
edge
thereof
in
the
18th
deg.
on
the
West
side
for
the
Morning
Twilight
,
and
on
the
East
side
for
the
Evening
Twilight
,
of
the
same
day
.
And
note
that
in
the
height
of
Summer
,
the
Twilight
in
our
Horizon
,
continueth
all
night
long
:
because
the
same
goeth
not
vnder
the
Horizon
,
full
18
degrees
XXIII
.
Vse
.
To
find
the
Declination
of
any
Wall
,
or
Plaine
.
Take
a
board
hauing
one
streight
edge
,
and
a
line
drawne
perpendicular
vnto
that
edge
:
apply
the
streight
edge
vnto
the
Wall
,
at
what
time
the
Sunne
shineth
theron
,
holding
the
board
parallel
to
the
plaine
of
the
Horizon
:
and
hang
vp
a
thread
with
a
plummet
,
so
that
the
shadow
of
the
thread
may
fall
on
the
board
,
crossing
that
perpendicular
line
.
Then
take
with
your
Instrument
the
height
of
the
Sunne
,
and
instantly
make
two
pricks
,
in
the
shadow
of
the
thread
on
the
board
,
a
good
way
distant
one
from
the
other
:
and
laying
a
Ruler
to
those
two
pricks
,
draw
a
line
,
which
line
shall
be
the
Azumith
of
the
Sunne
,
on
the
board
:
againe
with
the
height
of
the
Sunne
lastly
taken
,
find
out
on
your
instrument
,
the
Azumith
of
the
Sunne
;
or
the
Angle
which
the
Sunnes
Azumith
maketh
with
the
Meridian
,
(
by
the
XV.
Vse
.
)
And
on
the
board
taking
the
intersection
of
the
shadow
line
with
the
perpendicular
for
the
Center
,
describe
a
Circle
equall
to
the
innermost
Circle
of
the
Limbe
:
(
which
you
may
easily
doe
,
if
you
set
one
foot
of
your
compasses
vpon
the
East
,
or
West
point
,
and
extend
the
other
foot
vnto
60
degrees
,
on
the
same
innermost
Circle
,
for
this
distance
is
equall
to
the
Radius
thereof
.
)
Againe
with
your
compasses
,
take
of
the
Arch
betweene
the
Azumith
of
your
Instrument
,
and
the
Meridian
,
and
set
that
on
the
Circle
of
the
board
,
that
way
that
the
true
South
is
:
and
through
the
end
of
that
Arch
measured
on
the
board
,
draw
a
streight
line
for
the
Meridian
.
Lastly
take
with
your
compasses
,
the
Arch
intercepted
between
the
Meridian
on
the
board
,
and
the
perpendicular
line
,
and
by
applying
it
to
the
in
most
Circle
of
the
limbe
,
from
the
East
,
or
West
points
see
how
many
degrees
it
containeth
:
for
that
is
the
declination
of
the
Wall
.
Or
else
you
may
find
the
Meridian
vpon
the
board
,
by
XIX
Vse
.
If
the
Angle
of
the
Meridian
with
the
perpendicular
,
on
the
board
,
be
a
right
Angle
,
the
Wall
is
direct
East
,
or
West
.
But
if
the
Meridian
fall
vpon
the
perpendicular
,
or
be
parallel
there
to
making
no
Angle
with
it
,
the
Wall
is
direct
North
,
or
South
.
XXIIII
.
Vse
.
The
Art
of
Dyalling
.
And
first
how
to
make
the
Instrument
in
paper
,
promised
in
the
beginning
of
this
second
part
.
For
the
Delincation
of
this
instrument
in
paper
,
it
will
bee
necessary
first
to
shew
the
manner
how
the
Semidiameter
is
to
bee
graduated
,
or
diuided
into
degrees
:
and
how
the
Centers
,
and
Semidiameters
,
of
the
seuerall
kinds
of
Arches
are
to
be
found
.
Vpon
halfe
a
sheet
of
strong
large
Dutch
paper
,
(
the
larger
,
the
better
)
draw
two
streight
lines
,
making
a
right
Angle
neere
one
of
the
corners
,
the
one
through
the
length
,
and
the
other
through
the
breadth
of
the
paper
;
which
two
lines
I
therefore
call
the
longer
,
and
the
shorter
perpendicular
.
Vpon
the
right
Angle
point
,
being
the
Center
,
with
a
Semidiameter
equall
to
that
by
which
you
intend
to
delineate
your
instrument
,
describe
a
quadrant
of
a
Circle
:
and
on
the
point
where
it
meeteth
with
the
shorter
perpendicular
,
draw
a
long
tangent
line
parallel
to
the
longer
pependicular
.
Divide
the
Quadrant
into
90
degrees
,
among
which
from
the
beginning
at
the
shorter
perpendicular
,
reckon
the
eleuation
of
the
Pole
,
for
which
you
will
make
your
instrument
,
and
applying
a
Ruler
to
the
end
thereof
,
and
to
the
Center
,
where
the
Ruler
cutteth
the
tangent
line
make
a
prick
.
And
taking
with
your
compasses
the
distance
from
the
Center
to
that
prick
,
measure
it
vpon
the
shorter
perpendicular
:
this
shall
be
the
Semidiameter
of
the
sixt
hower
Circle
.
At
the
end
thereof
draw
another
long
line
parallel
also
to
the
longer
perpendicular
.
Then
out
of
the
Center
vnto
the
second
parallel
through
every
degree
of
the
quadrant
,
draw
fine
streight
lines
,
cutting
also
the
first
Parallel
.
The
intersection
of
those
lines
with
the
first
Parallel
,
shall
be
The
scale
of
centers
of
Arches
.
And
their
intersection
with
the
second
Parallel
shall
be
The
scale
of
centers
of
hower
Circles
.
And
the
segments
of
those
lines
,
intercepted
betweene
the
Center
,
and
the
first
Parallel
,
shall
be
the
Semidiameters
of
Arches
:
and
the
whole
lines
betweene
the
Center
and
the
second
Parallel
,
shall
bee
The
Semidiameters
of
hower
Circles
.
And
that
you
may
know
for
what
Circle
,
every
Center
,
and
Semidiameter
serueth
,
you
shall
note
every
fift
line
from
the
beginning
,
with
the
figures
5
,
10
,
15
,
20
,
&c.
Set
vnder
the
second
Parallel
,
vnto
90
which
will
fall
vpon
the
longer
perpendicular
:
that
so
you
may
readily
find
the
Center
,
and
Semidiameter
of
any
Circle
required
.
Againe
divide
the
first
45
degrees
of
the
Quadrant
in
the
middest
:
and
applying
your
Ruler
to
the
Center
,
and
to
every
one
of
those
halfe
divisions
,
where
in
each
place
the
Ruler
cutteth
the
first
Parallel
,
or
tangent
line
,
make
a
prick
.
So
shall
you
haue
vpon
the
tangent
line
betweene
the
shorter
perpendicular
and
the
midlemost
line
45
,
a
third
scale
,
which
is
,
The
scale
of
90
degrees
,
for
the
graduating
of
the
Semidiameter
of
your
instrument
on
the
paper
:
In
which
you
shall
also
distinguish
every
fift
degree
,
with
figures
set
vnder
the
tangent
line
.
Having
thus
prepared
your
paper
of
scales
with
lines
neatly
and
exactly
drawne
,
keepe
it
by
you
to
haue
it
still
in
a
readinesse
for
the
making
,
and
vsing
of
the
Instrument
in
paper
.
The
making
whereof
is
thus
.
Take
with
your
compasses
the
Semidiameter
of
the
Quadrant
in
your
paper
of
scales
:
and
therewith
vpon
a
peice
of
strong
Dutch
paper
,
Describe
the
Horizontall
Circle
:
which
you
shall
cut
into
two
Semicircles
with
a
Meri●ian
line
drawne
through
the
Center
:
divide
them
into
Quadrants
in
the
points
of
East
,
and
West
:
and
each
Q●adr●nt
into
90
degrees
to
be
marked
with
figures
iust
as
is
done
in
the
Instrument
.
Then
with
your
compasses
take
the
eleuation
of
the
pole
vpon
the
scale
of
degrees
in
your
paper
:
&
set
it
vpon
the
Meridian
line
from
the
Center
which
way
you
please
:
that
shall
bee
the
intersection
of
the
Aequinoctiall
with
the
Meridian
.
Also
reckon
the
complement
of
the
height
of
the
Pole
,
vpon
the
scale
of
Centers
of
Arches
,
and
with
your
compasses
take
the
distance
from
the
end
therof
to
the
Center
:
the
same
shall
bee
the
Semidiameter
of
the
Aequinoctiall
,
to
bee
drawne
from
the
East
point
of
the
Horizon
through
the
point
of
intersection
with
the
Meridian
vnto
the
West
point
.
Againe
take
with
your
compasses
vpon
the
scale
of
degrees
in
your
paper
the
complement
of
the
height
of
the
Pole
:
and
set
it
vpon
the
Meridian
on
the
other
side
of
the
Center
from
the
Aequinoctiall
:
there
shall
bee
the
Pole
of
the
Aequinoctiall
,
or
of
the
World
,
in
which
all
the
houre
lines
shall
crosse
one
another
.
diagram
Moreover
with
your
compasses
,
take
the
distance
betweene
the
Center
and
the
second
Parallel
in
your
paper
,
which
is
the
Semidiameter
of
the
sixt
houre
Circle
:
and
set
it
on
the
Meridian
from
the
Pole
beyond
the
Aequinoctiall
:
that
shall
be
the
Center
of
the
sixt
houre
Circle
:
vpon
which
you
may
draw
the
same
Circle
,
from
the
East
point
of
the
Horizon
through
the
Pole
to
the
West
point
.
Then
through
the
center
of
the
sixt
houre
Circle
erect
a
line
perpendicular
to
the
Meridian
,
extending
it
infinit●y
on
both
sides
of
the
Meridian
:
and
in
that
line
both
wayes
,
pricke
downe
the
Centers
of
the
horary
Circles
,
out
of
the
scale
in
paper
:
And
lastly
opening
your
compasses
from
every
one
of
those
Centers
vnto
the
Pole
severally
,
describe
all
the
horary
Circles
,
or
at
least
every
fift
of
them
,
and
so
is
your
paper
instrument
perfectly
finished
.
The
vse
of
this
instrument
on
paper
is
,
that
lines
,
and
arches
may
bee
designed
vpon
it
with
a
fine
pennicell
of
blacke
lead
,
and
afterward
be
wiped
out
againe
.
Wherefore
it
will
bee
needfull
for
him
that
will
vse
this
instrument
,
to
all
the
purposes
thereof
,
to
get
a
good
paire
of
large
compasses
with
three
points
,
●ne
sharpe
,
another
for
inke
,
a
third
for
blacke
Lead
.
And
I
suppose
it
would
doe
well
to
fast
●n
over
your
instrument
a
peice
of
thinne
oyled
paper
,
through
which
the
lineaments
may
be
conspicuous
:
and
vpon
it
to
trace
such
lines
,
and
arches
as
you
haue
occasion
to
vse
:
that
so
your
instrument
may
be
kept
cleane
,
and
last
longer
.
For
as
much
as
in
delineating
the
horary
Circles
,
which
are
within
30
degrees
of
the
Meridian
,
the
Semidiameters
will
be
too
long
for
your
compasses
:
you
may
in
that
streight
thus
helpe
your selfe
.
First
say
,
As
the
Radius
,
is
to
the
Sine
of
the
Eleuation
of
the
Pole
;
So
is
the
tang
:
of
the
distance
of
any
Horary
Circle
from
the
Meridian
,
suppose
25
,
or
20
,
or
15
,
or
10
,
or
5
degrees
,
to
the
arch
of
the
Horizon
betweene
the
Meridian
,
and
that
horary
Circle
.
Reckon
this
distance
on
the
limbe
of
your
Instrument
from
both
ends
of
the
Meridian
,
and
marke
it
.
Thus
doe
for
the
25th
,
20th
,
15th
,
10th
,
and
5th
horary
Circle
on
both
sides
of
each
end
of
the
Meridian
.
Then
in
any
peice
of
cleane
paper
,
through
the
middest
of
the
longer
way
,
draw
aline
:
and
toward
one
end
which
(
I
call
the
upper
ende
)
crosse
the
same
with
a
perpendicular
line
exactly
equall
to
the
Diameter
of
your
Instrument
,
the
point
of
Intersection
being
the
center
.
Take
with
your
Compasses
out
of
the
paper
of
scales
,
the
semidiameter
of
60.
degr
:
(
which
you
may
well
doe
for
an
ordinary
instrument
)
:
and
setting
one
foot
on
either
end
of
the
Diameter
,
that
point
wherein
the
other
foot
shall
cutt
the
first
long
line
,
make
your
Center
,
and
thereon
draw
an
Arch
through
both
ends
of
the
Diameter
,
and
cutting
the
vpper
part
of
the
first
long
line
:
this
Arch
is
equall
to
that
horary
Circle
,
which
is
distant
from
the
Meridian
30
degrees
the
complement
of
60.
diagram
Divide
each
halfe
of
the
Diameter
into
3
equall
parts
,
with
4
points
,
and
from
every
of
those
points
vnto
the
Arch
draw
lines
parallel
to
the
first
long
line
.
And
having
divided
euery
one
of
those
fiue
parallel
lines
intercepted
betweene
the
Diameter
and
the
Arch
into
6
equall
parts
,
for
the
6
times
5
degrees
which
remaine
to
the
Meridian
,
draw
through
those
divisions
from
the
ends
of
the
Diameter
(
with
a
smooth
and
even
hand
)
the
Arches
25
20
,
15
,
10
and
5.
Those
Arches
you
may
transferre
from
the
paper
to
your
instrument
in
this
manner
.
Rubb
the
backe
side
of
the
paper
against
the
Arches
,
with
fine
powder
of
blacke
lead
:
then
applying
the
paper
with
Arches
to
your
instrument
,
that
the
ends
of
the
Diameter
may
exactly
fall
vpon
the
two
opposite
markes
,
in
the
limbe
of
your
instrument
,
which
serue
for
the
horary
Circle
that
you
would
draw
,
either
25
,
20
,
15
,
10
,
or
5
,
trace
over
that
Arch
with
the
point
of
any
hard
peice
of
wood
sharpened
:
and
the
blacke
lead
on
the
backe
side
will
vpon
the
instrument
leaue
the
print
of
that
Arch.
XXV
.
Vse
.
To
set
an
vpright
Wall
or
plaine
vpon
the
instrument
:
and
to
find
how
many
houres
the
Sunne
shall
shine
thereon
at
some
time
of
the
yeare
.
The
situation
of
Wals
,
or
Plaines
is
considered
either
in
respect
of
the
Meridian
,
or
of
the
Horizon
.
And
vnto
both
it
is
either
perpendicular
,
or
oblique
,
or
parallel
.
The
plaine
perpendicular
to
the
Meridian
,
is
that
which
standeth
directly
North
,
or
South
:
which
if
it
be
also
perpendicular
to
the
Horizon
,
is
called
North
,
or
South
direct
vpright
.
But
if
it
stoope
from
the
Zenith
forward
,
it
is
called
North
,
or
South
inclining
:
if
backeward
,
it
is
called
North
or
South
reclining
.
And
note
that
in
a
stooping
Plaine
that
side
which
is
toward
the
Horizon
,
is
inclining
,
and
that
which
is
toward
the
Zenith
is
reclining
.
The
Plaine
oblique
to
the
Meridian
is
that
which
standeth
not
directly
North
,
or
South
,
but
declineth
one
side
into
the
East
,
and
the
other
into
the
West
:
and
is
therefore
called
Declining
Eastward
,
or
Westward
,
according
as
either
side
of
the
Plaine
looketh
:
As
if
an
vpright
Wall
being
Southerne
,
declineth
from
the
South
into
the
East
,
it
is
called
South
declining
Eastwards
vpright
.
But
if
it
be
not
vpright
,
it
is
called
South
declining
Eastward
,
and
inclining
,
or
reclining
.
The
Plaine
parallel
to
the
Meridian
,
is
that
which
looketh
directly
East
,
or
West
;
and
accordingly
,
hath
his
denomination
,
whether
it
bee
Vpright
,
Inclining
,
or
Reclining
.
The
Plaine
Parallel
to
the
Horizon
,
is
called
Horizontall
:
and
is
represented
by
the
instrument
it selfe
,
or
at
least
by
the
inner
most
Circle
of
the
limbe
thereof
.
And
note
that
the
Arch
of
Declination
,
is
reckoned
from
the
next
East
,
or
West
point
.
And
that
the
Arch
of
Inclination
,
or
Reclination
is
reckoned
from
the
Zenith
,
or
the
complement
of
it
from
the
Horizon
.
So
that
euery
vpright
Plaine
is
vnderstood
to
passe
through
the
Zenith
,
which
in
the
instrument
is
the
Center
.
And
thus
having
shewed
the
seuerall
affections
of
Plaines
,
wee
will
now
proceed
to
shew
the
manner
how
to
set
them
vpon
the
Instrument
.
A
Direct
North
,
or
South
vpright
Plaine
,
is
represented
in
the
instrument
by
a
line
drawne
through
the
Center
from
the
East
point
to
the
West
,
which
is
also
the
Horizontall
intersection
of
the
Pla●ne
.
And
by
it
you
shall
see
that
the
Southerne
side
or
face
of
the
plaine
is
open
to
all
the
houres
betweene
sixe
in
the
morning
and
sixe
in
the
evening
.
And
that
about
London
,
the
Northerne
side
,
onely
in
the
Summer
enioyeth
the
Sunne
from
his
rising
till
after
seven
in
the
morning
:
and
from
before
5
a
clocke
in
the
afternoone
,
till
his
setting
.
A
direct
East
,
or
West
upright
plaine
,
is
represented
in
the
Instrument
by
the
Meridian
,
which
is
also
the
Horizontall
intersection
of
the
plaine
.
And
in
it
you
shall
see
that
all
the
forenoone
houres
are
open
to
the
East
side
:
and
all
the
afternoone
houres
to
the
West
side
.
A
Declining
Plaine
is
thus
set
upon
the
Instrument
,
reckon
on
the
Horizon
the
arch
of
Declination
,
from
the
East
,
or
West
point
:
and
at
the
end
draw
a
line
through
the
Center
vnto
the
opposite
point
of
the
Horizon
:
So
that
each
side
thereof
may
be
open
to
that
point
,
either
East
,
or
West
,
into
which
the
Declination
is
supposed
.
That
line
so
drawne
throught
the
center
is
the
Horizontall
intersection
of
the
plaine
,
and
representeth
the
plaine
it selfe
,
if
it
bee
vpright
.
For
example
,
there
is
about
London
an
vpright
Wall
declining
Eastwards
35
degrees
:
which
I
would
set
vpon
the
Instrument
.
Hold
the
Southerne
part
of
the
Instrument
to
you
,
and
reckon
from
the
East
backward
into
the
North
vpon
the
Horizon
35
degrees
:
there
draw
a
line
through
the
Center
:
this
line
shall
not
onely
vpon
the
South
side
represent
a
Southerne
Plaine
declining
Eastward
35
degrees
.
But
also
vpon
the
North
side
shall
represent
a
Northerne
Plaine
declining
Westward
35
deg.
And
moreover
it
will
appeare
that
on
the
Southerne
side
shall
bee
drawne
the
houres
from
almost
4
a
clocke
in
the
morning
,
till
3
in
the
afternoone
.
And
that
in
the
Northerne
side
shall
bee
drawne
vpon
one
side
4
a
clocke
in
the
morning
onely
:
and
vpon
the
other
side
all
the
houres
from
3
in
the
afternoone
till
Sunne
set
.
And
●o
consequently
the
declination
of
an
vpright
wall
,
or
Window
being
given
,
it
may
be
found
at
what
houre
the
Sunne
vpon
any
day
in
the
yeare
will
come
to
that
Wall
,
or
Window
,
and
when
it
will
goe
from
it
.
As
in
the
former
example
,
There
is
about
London
a
Northerne
wall
declining
Westward
35
deg.
I
would
know
at
what
time
of
the
day
the
Sunne
will
begin
to
shine
vpon
it
on
the
24th
day
of
March.
Set
the
Index
at
35
deg.
from
West
toward
South
:
and
because
that
day
the
Sunnes
Declination
is
6
degrees
Northward
;
Looke
at
what
houre
the
sixt
Parallel
aboue
the
Aequinoctiall
toward
the
Center
meeteth
with
the
Index
so
placed
:
and
you
shall
find
it
at
3
●
clock
in
the
Afternoone
.
Wherefore
at
that
time
the
Sunne
will
begin
to
shine
vpon
the
Wall
that
same
day
.
The
Poles
of
every
vpright
Wall
are
in
the
Horizon
90
deg.
that
is
a
quarter
of
a
Circle
,
distant
from
the
line
representing
the
Plaine
.
Wherefore
if
vpon
that
line
in
the
Center
you
erect
a
perpendicular
,
the
ends
therof
in
the
Horizon
shall
be
the
poles
of
that
Plaine
:
and
are
so
farre
distant
from
the
North
and
South
points
,
as
the
Plaine
it selfe
is
from
the
East
,
and
West
.
XXVI
.
Vse
.
To
set
an
Inclining
,
and
Reclining
Wall
,
or
Plaine
vpon
the
Instrument
:
and
to
find
how
many
houres
the
Sunne
shall
shine
thereon
,
at
some
time
of
the
yeare
.
When
you
haue
an
Inclining
,
or
Reclining
Plaine
to
be
described
on
the
Instrument
.
First
the
Horizontall
intersection
is
to
be
set
thereon
,
as
if
it
were
vpright
;
together
with
the
line
perpendicular
thereto
,
in
which
are
the
Poles
of
the
Plaine
:
according
as
was
taught
in
the
XXV
Vse
.
Then
vpon
the
scale
of
degrees
in
your
paper
,
reckon
the
arch
of
Inclination
,
or
Reclination
;
and
with
your
compasses
take
,
&
set
it
in
your
Instrument
vpon
the
line
perpendicular
to
the
Horizontall
intersection
of
your
Plaine
,
from
the
Center
that
way
into
which
the
Inclination
,
or
Reclination
tendeth
:
the
same
shall
bee
the
vppermost
point
of
your
Plaine
.
Againe
,
with
your
Compasses
take
the
Complement
of
inclination
,
or
reclination
,
both
upon
the
scale
of
degrees
,
and
also
upon
the
scale
of
centers
of
arches
in
your
paper
:
and
set
both
spaces
upon
the
same
perpendicular
line
,
but
on
the
other
side
of
the
center
(
extended
if
need
be
)
:
At
the
shorter
of
those
spaces
shal
be
the
pole
of
your
plaine
:
and
at
the
longer
of
them
shal
be
the
Center
of
it
.
Lastly
,
setting
one
foot
of
your
Compasses
in
the
center
of
your
Plaine
,
and
extending
the
other
foot
to
the
vppermost
point
,
describe
in
your
Instrument
an
Arch
of
a
Circle
:
which
if
you
haue
done
well
,
will
exactly
fall
vpon
the
ends
of
the
Horizontal
intersection
of
your
Plaine
.
That
Arch
shall
represent
your
Plaine
,
inclining
vpon
the
lower
side
,
which
is
toward
the
Horizon
,
or
Limbe
:
but
reclining
vpon
the
vpper-side
,
which
is
toward
the
Zenith
,
or
Center
.
And
so
either
side
shall
shew
in
what
hower
lines
the
Sunne
,
at
some
time
of
the
yeare
,
will
shine
vpon
it
:
that
in
delineating
a
Dyall
thereon
,
it
may
not
be
combered
with
vnnecessary
houre
lines
.
For
Example
,
suppose
that
the
former
Plaine
,
which
with
the
South
declined
Eastward
35
deg.
doe
also
incline
41
deg.
30
min.
Wherefore
also
with
the
North
side
it
shall
decline
Westward
35
deg.
and
Recline
41
deg.
30
min.
Describe
this
plaine
vpon
your
Instrument
with
an
Arch
of
a
Circle
,
found
out
as
was
taught
last
before
.
And
it
will
appeare
that
vpon
the
Inclining
side
shall
be
drawne
all
the
houre
lines
from
almost
4
in
the
morning
,
to
4
in
the
afternoone
:
And
vpon
the
Reclining
side
shall
bee
drawne
first
4
and
5
in
the
morning
:
and
then
beginning
at
0
a
clocke
,
all
houres
to
Sunne
set
.
And
note
that
in
all
Northerne
plaines
,
the
North
Pole
is
eleuated
:
and
in
all
Southerne
plaines
the
South
Pole
is
eleuated
.
Except
such
North
inclining
,
and
South
reclining
Plaines
,
that
in
the
Instrument
fall
below
the
North
Pole
,
betweene
it
and
the
Horizon
:
For
in
them
the
contrary
Pole
is
eleuated
.
And
also
that
a
direct
East
and
West
plaine
,
if
it
Recline
,
hath
the
North
Pole
eleuated
:
and
if
it
Incline
,
the
South
Pole
.
XXVII
.
Vse
.
The
Plaine
being
set
vpon
the
Instrument
,
to
find
the
distances
of
the
houre
lines
,
and
Substile
from
the
Horizontall
Intersection
.
And
also
the
height
of
the
Stile
aboue
the
Substile
.
Euery
Dyall
either
hath
a
Center
in
which
all
the
houre
lines
,
together
with
the
Substile
,
and
Stile
doe
meet
:
or
else
it
hath
no
center
,
&
so
they
are
al
parallel
one
to
another
.
If
the
Plaine
being
set
vpon
the
Instrument
,
cutteth
the
Pole
of
the
Aequinoctiall
(
that
is
the
point
in
which
all
the
horary
Circles
crosse
the
Meridian
)
the
Dyal
to
be
drawne
vpon
that
Plaine
shall
haue
no
Center
.
But
if
it
cutteth
not
the
Pole
,
the
Dyal
shall
haue
a
Center
.
And
of
these
Dyals
with
Centers
wee
will
first
intreate
:
as
being
most
proper
for
the
vse
of
the
Instrument
.
Behold
therefore
the
Pole
of
your
plaine
heedily
what
horary
Circle
it
falleth
vpon
:
Or
if
it
fall
betweene
any
two
,
the
distance
of
each
being
reasonably
apportioned
,
imagine
a
horary
Circle
passing
through
it
.
Marke
in
what
point
that
horary
Circle
,
either
real
,
or
imagined
,
doth
cut
the
Plaine
,
that
same
point
shall
bee
the
place
of
the
Substile
in
your
Plaine
:
and
the
height
of
the
Stile
aboue
it
,
is
the
Arch
of
that
horary
Circle
intercepted
betwixt
the
Pole
of
the
Aequinoctial
,
and
the
point
of
the
Substile
noted
in
the
Plaine
.
Therefore
applying
a
Ruler
to
the
Pole
of
your
Plaine
,
carry
it
about
vnto
all
the
intersections
of
the
Plaine
with
the
houre
Circles
,
and
the
substile
seuerally
:
and
where
in
euery
place
the
Ruler
shall
cut
the
innermost
Circle
of
the
Limbe
,
there
make
a
visible
marke
:
For
the
arches
of
the
Limbe
intercepted
betweene
the
Horizontall
points
of
your
Plaine
,
and
euery
one
of
those
marks
,
shall
be
the
distance
intended
to
be
sought
.
But
for
the
Horizontall
plaine
,
the
ends
of
the
houre
Circles
in
the
Limbe
of
your
instrument
,
doe
giue
the
distance
without
any
more
adoe
.
Concerning
the
height
of
the
Stile
aboue
the
Substile
:
It
is
apparent
by
the
instrument
,
that
in
a
Horizontal
dyal
,
the
6
a
clocke
line
lyeth
directly
East
and
West
:
and
the
Meridian
perpendicular
to
it
,
directly
North
and
South
.
And
that
the
Meridian
is
the
Substile
.
And
that
the
height
of
the
Stile
aboue
the
Substile
,
is
equall
to
the
height
of
the
Pole
in
that
place
.
It
is
also
apparent
,
that
in
all
direct
North
and
South
Dyals
,
the
6
a
clocke
Line
is
drawne
parallel
to
the
Horizon
,
and
the
Meridian
perpendicular
to
it
:
And
that
the
Meridian
is
the
substile
.
And
that
the
height
of
the
Stile
aboue
the
substile
,
if
the
Plaine
bee
vpright
,
is
equall
to
the
complement
of
the
height
of
the
Pole
.
But
if
it
be
North
inclining
,
or
South
Reclining
,
it
is
equall
to
the
difference
,
of
the
height
of
the
Pole
,
and
the
Arch
of
Inclination
,
or
Reclination
.
And
if
the
Plaine
bee
North
Reclining
,
or
South
Inclining
,
it
is
equall
to
the
Summe
of
the
complement
of
the
height
of
the
Pole
and
of
the
Arch
of
Inclination
,
or
Reclination
.
And
if
the
Plaine
fall
vpon
the
Aequinoctiall
,
the
stile
shall
stand
vp
perpendicular
vpon
it
in
the
Center
:
and
the
hower
lines
shall
be
drawne
all
at
equall
Angles
,
viz.
15
degrees
one
from
another
.
In
such
Dyals
as
haue
not
the
Meridian
for
the
substile
,
the
height
of
the
Stile
aboue
the
Substile
is
thus
sound
by
the
Instrument
.
It
was
shewed
before
that
the
height
of
the
stile
aboue
the
substile
is
the
Arch
of
the
horary
Circle
through
the
Pole
of
the
Plaine
,
intercepted
betweene
the
pole
of
the
Aequinoctiall
,
and
the
point
noted
in
the
Plaine
for
the
substile
.
Therefore
from
the
Horizontall
points
,
or
intersections
of
that
horary
Circle
reckon
90
degrees
both
wayes
:
and
thereto
through
the
Center
draw
an
obscure
line
:
in
which
line
shall
be
both
the
Inclination
of
that
horary
Circle
,
which
is
the
distance
of
the
intersection
,
or
vppermost
point
thereof
from
the
Center
:
and
also
the
Pole
.
Then
with
your
compasses
take
that
distance
,
or
Inclination
:
and
setting
it
vpon
the
scale
of
degrees
in
your
paper
,
see
how
many
degrees
it
containeth
vpon
that
scal●
.
Againe
vpon
the
same
scale
take
with
your
compasses
the
complement
of
that
inclination
or
distance
,
which
being
set
vpon
the
obscure
line
on
the
other
side
of
the
Center
,
shall
shew
the
Pole
of
that
horary
Circle
.
Lastly
applying
a
Ruler
to
the
Pole
of
that
horary
Circle
,
and
both
to
the
Pole
of
the
Aequinoctiall
,
and
to
the
point
of
the
substile
in
the
Plaine
seuerally
:
marke
where
in
both
places
the
Ruler
cutteth
the
innermost
limbe
of
your
instrument
:
For
the
degrees
of
the
limbe
intercepted
betwixt
those
markes
,
shall
be
the
height
of
the
stile
,
aboue
the
substile
,
which
was
sought
for
.
And
by
this
which
hath
beene
taught
you
shall
find
that
in
an
vpright
Dyall
declining
,
as
before
35
deg.
from
the
South
into
the
East
,
or
from
the
North
into
the
West
,
the
substile
shall
fall
vpon
that
horary
Circle
,
which
is
about
3
deg.
after
9
a
clock
in
the
morning
:
and
the
stile
eleuated
aboue
the
substile
about
31
deg.
And
also
that
in
a
South
Dyall
declining
Eastward
35
deg.
and
inclining
41
deg.
30
min:
Or
in
a
North
declining
Westward
35
deg.
and
reclining
41
deg.
30
min:
the
substile
shall
fall
vpon
that
horary
Circle
which
is
about
8
degrees
after
6
a
clock
:
and
the
stile
eleuated
aboue
the
substile
about
63
deg.
30
minutes
.
XXVIII
.
Vse
.
The
making
of
all
manner
of
plaine
Dyals
with
Centers
.
I
haue
already
shewed
how
to
find
out
in
our
Instrument
the
distances
of
all
the
houre
lines
,
and
the
substile
,
from
the
Horizontal
intersection
.
Now
the
delineating
of
a
Dyal
is
nothing
else
but
to
transferre
those
distances
out
of
your
Instrument
into
the
Dyal
plaine
,
euery
one
in
his
due
situation
:
and
then
through
them
,
out
of
the
Center
,
to
draw
such
houre
lines
as
shall
be
of
vse
,
together
with
the
substile
.
The
due
situation
of
those
distances
vpon
the
Dyal
plaine
,
dependeth
on
the
true
placing
of
the
Meridian
,
or
12
a
clocke
line
:
for
that
being
truely
described
,
all
the
rest
will
be
easie
enough
.
First
therefore
I
will
shew
the
manner
how
the
Meridian
,
or
12
a
clocke
line
,
is
to
be
described
.
Take
in
your
Dyal
some
point
for
the
Center
,
where
you
shall
thinke
fit
:
and
through
it
draw
a
line
parallel
to
the
plaine
of
the
Horizon
.
Crosse
it
in
the
Center
with
a
perpendicular
line
.
And
hauing
opened
your
compasses
to
the
length
of
the
Semidiameter
of
your
paper
Instrument
,
describe
on
the
Center
a
Circle
equall
to
the
innermost
Limbe
thereof
.
In
which
Circle
the
line
parallel
to
the
Horizon
is
for
the
Horizontal
intersection
:
and
the
other
for
the
line
perpendicular
to
it
:
and
the
Circle
it selfe
representeth
the
plaine
:
Marke
therein
the
East
and
West
sides
of
the
Plaine
with
E
and
W.
In
the
Horizontal
,
and
in
al
North
and
South
direct
Plaines
,
both
vpright
,
and
stooping
;
and
in
all
vpright
declining
plaines
,
the
Meridian
is
perpendicular
to
the
Line
parallel
to
the
Horizon
.
In
North
●●clining
,
and
South
reclining
plaines
,
the
Meridian
is
to
bee
drawne
on
that
side
of
the
Dyal
plaine
either
East
,
or
West
,
into
which
the
declination
is
:
But
in
North
inclining
and
South
reclining
,
on
the
contrary
side
.
And
if
the
plaine
bee
Northerne
,
the
Meridian
shall
be
aboue
the
Line
parallel
to
the
Horizon
:
and
if
the
plaine
be
Southerne
,
it
shal
be
vnder
it
.
And
if
the
contrary
Pole
be
eleuated
,
it
shall
be
drawne
through
the
Center
into
the
opposite
Quadrant
of
the
Circle
in
your
Dyal
plaine
.
Lastly
in
a
direct
East
and
West
plaine
,
both
inclining
and
reclining
,
the
Meridian
is
the
same
with
the
line
parallel
to
the
Horizon
.
Wherefore
with
your
compasses
take
the
distance
in
the
limbe
of
your
Instrument
,
from
the
next
Horizontall
point
,
vnto
the
marke
of
the
Meridian
;
and
measure
it
vpon
the
Circle
of
the
Dyal
plaine
,
in
that
part
,
and
on
that
side
,
according
as
in
consideration
of
the
eleuated
Pole
,
and
of
the
qualitie
of
the
Plaine
,
was
shewed
to
be
agreeable
.
And
at
the
end
of
that
arch
,
through
the
Center
,
draw
a
line
for
the
Meridian
.
Againe
with
your
compasses
take
the
distances
in
the
limbe
of
your
Instrument
,
betweene
the
marke
of
the
Meridian
,
and
the
markes
of
all
the
houre
Lines
seuerally
:
and
setting
them
vpon
the
Circle
of
the
Dyal
plaine
orderly
from
the
Meridian
,
the
Forenoone
houres
on
the
West
side
of
it
,
and
the
Afternoone
houres
on
the
East
side
:
at
the
end
of
euery
one
of
those
arches
draw
the
houre
Lines
:
and
distinguish
them
with
their
proper
figures
accordingly
.
Lastly
fasten
the
stile
in
the
Center
,
so
that
it
may
hang
perpendicular
vnto
the
plaine
in
the
Substile
,
at
the
iust
height
.
And
because
the
stile
in
every
Dyal
is
vnderstood
to
be
a
segment
of
the
Axis
of
the
world
which
is
a
line
imagined
to
passe
from
the
North
to
the
South
Pole
through
the
Center
of
the
earth
;
the
stile
being
rightly
placed
shal
still
with
the
end
point
towards
the
eleuated
Pole
,
that
is
vpward
from
the
Center
,
if
the
North
Pole
be
eleuated
;
or
downeward
from
the
Center
if
the
South
Pole
be
eleuated
.
XXIX
.
Vse
.
The
making
of
all
manner
of
plaine
Dyals
not
hauing
Centers
.
If
the
plaine
represented
on
the
Instrument
(
as
was
taught
before
in
the
XXV
and
XXVI
Vses
)
cut
the
Pole
of
the
Aequinoctiall
,
it
is
an
horary
Circle
,
either
one
of
them
which
are
drawne
in
the
Instrument
,
or
falling
betweene
some
two
of
them
:
and
the
Dyall
plaine
it selfe
shall
not
crosse
the
axis
of
the
world
,
but
lye
parallel
to
it
,
without
any
Angle
of
eleuation
.
And
therefore
such
a
Dyal
can
haue
no
Center
:
But
the
stile
,
the
substile
,
and
all
the
houre
lines
shall
be
parallel
one
to
another
.
Every
such
Plaine
represented
on
the
Instrument
,
Either
,
First
it
is
the
Meridian
of
the
place
,
the
Horizontall
intersection
whereof
is
the
12
a
clocke
Line
drawne
from
North
to
South
:
and
the
Dyall
made
thereon
,
is
a
direct
East
,
or
West
vpright
Dyal
:
In
which
the
substile
is
distant
from
the
Line
,
in
the
Circle
of
the
Dyall
plaine
parallel
to
the
Horizon
,
with
an
Arch
equall
to
the
eleuation
of
the
Pole
,
and
vpward
toward
the
Pole
.
And
is
also
the
6
a
clocke
line
in
your
Dyal
.
The
rest
of
the
houre
lines
are
thus
described
.
Draw
through
the
substile
,
in
any
point
,
a
long
Line
at
right
Angles
:
that
line
shall
bee
the
Aequinoctial
intersection
vsually
called
the
Contingent
line
:
And
taking
a
conuenient
distance
for
the
stile
to
hang
parallel
over
the
substile
(
according
to
the
greatnesse
of
your
Dyall
plaine
)
measure
it
vpon
the
substile
from
the
Aequinoctiall
intersection
:
and
vpon
the
end
of
that
measure
,
describe
halfe
a
Circle
for
the
Aequinoctiall
it selfe
.
Diuide
each
Quadrant
thereof
from
the
substile
,
into
6
equall
parts
,
or
houres
.
Then
applying
a
Ruler
to
the
Center
,
and
to
euery
one
of
those
diuisions
seuerally
,
where
in
euery
place
the
Ruler
shall
cut
the
long
line
of
Aequinoctiall
intersection
,
make
pricks
:
and
through
those
pricks
draw
the
houre
lines
,
parallel
to
the
Substile
,
or
6
a
clocke
line
:
distinguishing
so
many
of
them
as
bee
needfull
,
with
their
figures
:
that
is
all
the
Forenoone
houres
on
the
East
plaine
,
and
all
the
Afternoone
houres
on
the
West
plaine
.
But
in
these
Dyals
there
is
no
12
a
clock
line
,
it
being
infinitly
distant
from
the
Substile
.
Lastly
hang
the
stile
directly
over
the
Substile
,
and
parallel
to
it
,
at
the
distance
formerly
taken
.
And
thus
are
your
East
,
and
West
Dyals
finished
.
Or
Secondly
,
it
is
the
sixt
houre
Circle
,
the
Horizontal
intersection
whereof
is
the
line
of
East
,
and
West
:
and
the
Dyal
made
thereon
is
direct
North
inclining
,
or
South
reclining
,
with
an
Arch
equal
to
the
complement
of
the
height
of
the
Pole
.
And
the
parallel
to
the
Horizon
is
the
Aequinoctial
intersection
:
and
the
line
perpendicular
to
it
is
the
12
a
clocke
line
,
and
also
the
Substile
.
The
rest
of
the
houre
Lines
,
from
7
a
clocke
in
the
morning
,
to
5
in
the
euening
,
are
thus
described
.
Take
a
conuenient
distance
for
the
Stile
from
the
Substile
,
measuring
it
vpon
the
Substile
from
the
Aequinoctial
intersection
:
and
on
the
end
of
that
space
describe
the
Semicircle
of
the
Aequinoctial
,
to
bee
diuided
on
both
sides
of
the
Substile
into
6
houres
:
through
euery
one
of
which
out
of
the
Center
,
a
Ruler
being
applyed
;
at
the
points
of
the
seuerall
intersections
of
the
Ruler
with
the
Aequinoctial
intersection
,
draw
the
houre
Lines
parallel
to
the
Substile
,
or
12
a
clocke
Line
:
distinguishing
them
with
their
figures
,
namely
11
,
10
,
9
,
8
,
7
,
on
the
West
side
:
and
1
,
2
,
3
,
4
,
5
,
on
the
East
side
:
but
in
these
Dyals
there
is
no
sixe
a
clocke
Line
,
it
being
infinitly
distant
from
the
Substile
.
Lastly
hang
the
Stile
directly
over
the
Substile
,
and
parallel
to
it
,
at
the
distance
formerly
taken
.
Or
Thirdly
,
it
is
North
inclining
,
or
South
reclining
,
and
also
declining
:
in
which
.
As
the
tangent
of
the
Eleuation
of
the
Pole
,
is
to
the
Radius
;
So
is
the
Sine
of
the
compl
:
of
Declination
,
to
the
tang
:
of
the
compl
:
of
Inclination
or
Reclination
.
The
Plaine
being
set
vpon
the
Instrument
by
the
Arches
of
Declination
and
stooping
thereof
(
as
hath
beene
taught
in
XXV
●
Vse
)
shall
cut
the
pole
of
the
Aequinoctiall
.
Apply
therefore
a
Ruler
to
the
Pole
of
the
plaine
,
and
to
the
Pole
of
the
Aequinoctial
;
and
the
point
in
which
it
cutteth
the
Limbe
,
marke
for
the
substile
:
which
is
to
bee
transferred
vnto
the
Circle
of
the
Dyal
plaine
,
by
taking
the
distance
betweene
that
point
,
and
the
next
Horizontal
intersection
,
and
setting
it
on
that
Circle
from
the
line
parallel
to
the
Horizon
,
vpward
if
the
plaine
be
North
:
or
downe-ward
if
the
plaine
bee
South
:
and
on
that
side
which
is
contrary
to
the
Declination
.
The
substile
being
thus
sound
,
draw
a
long
line
perpendicular
to
it
,
for
the
Aequinoctial
intersection
.
And
taking
a
conuenient
distance
for
the
stile
from
the
substile
,
measure
it
vpon
the
substile
from
the
Aequinoctiall
intersection
:
and
on
the
end
of
that
space
describe
the
Semicircle
of
the
Aequinoctial
.
Then
looke
in
your
Instrument
how
many
degrees
of
the
Aequinoctial
are
intercepted
betweene
the
Meridian
,
and
the
Arch
representing
your
Plaine
:
and
reckoning
the
same
number
of
degrees
vpon
the
Aequinoctial
of
the
Dyal
plaine
,
from
the
substile
towards
the
side
of
Declination
,
there
make
a
marke
for
the
Meridian
point
thereof
:
in
which
you
must
begin
to
diuide
the
Aequinoctial
semicircle
into
houres
both
wayes
:
And
that
being
diuided
,
apply
a
ruler
to
the
center
,
and
to
euery
one
of
the
diuisions
:
and
at
the
points
of
the
seuerall
intersections
of
the
ruler
with
the
Aequinoctial
intersection
,
draw
the
houre
lines
parallel
to
the
substile
.
Set
12
at
that
houre
line
which
was
drawne
at
the
intersection
through
the
Meridian
point
of
the
Equinoctial
:
and
11
,
10
,
9
,
8
,
&c.
on
the
West
side
:
and
1
,
2
,
3
,
4
,
&c
,
on
the
East
side
.
Lastly
,
hang
the
stile
directly
ouer
the
substile
,
and
parallel
to
it
,
at
the
distance
formerly
taken
.
XXX
Vse
.
How
by
Sines
and
tangents
to
calculate
the
places
of
the
Meridian
,
and
Substile
,
and
the
height
of
the
Stile
aboue
it
:
and
the
distance
of
the
Meridian
of
the
Aequinoctiall
from
the
Substile
;
together
with
the
places
of
houre
lines
,
both
by
calculation
,
and
also
Geometrically
.
I
haue
already
taught
the
making
of
all
manner
of
plaine
Dyals
most
easily
by
the
Instrument
,
for
the
same
height
of
the
Pole
.
But
if
any
man
either
want
an
Instrument
,
or
else
desireth
greater
exactnesse
,
I
will
also
here
shew
how
to
performe
the
same
by
calculation
,
on
the
other
side
of
the
Instrument
.
In
a
plaine
erect
Dyall
declining
.
As
the
Radius
is
to
the
Sine
of
doclination
;
So
is
the
tang
.
of
the
compl
.
of
the
Poles
height
.
to
the
tang
.
of
the
distance
of
the
substile
from
the
Meridian
.
In
plaine
Dyalls
both
declining
and
also
inclining
,
and
reclining
.
As
the
Radius
is
to
the
Sine
of
inclination
reclination
So
is
the
tang
:
of
Declination
,
to
the
tang
.
of
the
compl
.
of
the
distance
of
the
Meridian
,
from
the
line
parallel
to
the
horizon
.
Againe
As
the
Radius
is
to
the
Sine
of
the
compl
.
of
Declination
;
So
is
the
tang
.
of
the
compl
.
of
the
Poles
height
,
to
the
tang
.
of
Base
I.
If
the
Dyall
be
South
reclining
,
or
North
inclining
,
the
summe
of
Base
I
,
and
of
the
complement
of
Inclination
or
reclination
shall
be
Base
II.
But
if
the
Plaine
be
South
inclining
,
or
North
reclining
,
the
difference
of
Base
I
,
and
of
the
complement
of
inclination
,
or
reclination
shall
bee
Base
II.
Then
say
thirdly
,
As
the
Sine
of
the
compl
.
of
Base
I
,
is
to
the
Sine
of
the
compl
.
or
excesse
of
Base
II
;
So
is
the
Sine
of
the
height
of
the
Poles
to
the
Sine
of
the
height
of
the
stile
aboue
the
substile
.
Fourthly
,
As
the
Sine
of
the
compl
.
of
the
height
of
the
stile
aboue
the
substile
,
is
to
the
Sine
of
Declination
;
So
is
the
Sine
of
the
compl
.
of
the
height
of
the
Pole
,
to
the
Sine
of
the
compl
.
of
the
distance
of
the
substile
from
the
line
parallel
to
the
horizon
.
Againe
As
the
Radius
is
to
the
Sine
of
the
compl
.
of
Declination
;
So
is
the
Sine
of
the
compl
.
of
the
Poles
height
,
to
the
Sine
of
the
height
of
the
stile
aboue
the
substile
.
Thirdly
As
the
Sine
of
the
Poles
height
is
to
the
Radius
;
So
is
the
tang
.
of
Declination
,
to
the
tang
.
of
the
Distance
of
the
Meridian
of
the
Aequinoctiall
from
the
substile
.
And
this
distance
is
euer
lesse
then
90
degrees
.
In
a
plaine
East
and
West
Inclining
and
Reclining
Dyall
.
As
the
Radius
is
to
the
Sine
of
the
compl
.
of
inclination
reclination
So
is
the
tang
.
of
the
height
of
the
Pole
,
to
the
tang
.
of
the
distance
of
the
substile
from
the
Meridian
.
Againe
As
the
Radius
is
to
the
Sine
of
inclination
reclination
So
is
the
Sine
of
the
height
of
the
Pole
,
to
the
Sine
of
the
height
of
the
stile
aboue
the
substile
.
Thirdly
,
As
the
Sine
of
the
compl
.
of
the
Poles
height
;
is
to
the
Radius
;
So
is
the
tang
.
of
the
compl
.
of
inclination
reclination
to
the
tang
.
of
the
distance
of
the
Meridian
of
the
Aequinoctiall
from
the
substile
.
And
this
distance
is
euer
greater
then
90
degrees
.
Fiftly
As
the
Sine
of
the
compl
.
of
the
height
of
the
stile
aboue
the
substile
,
is
to
the
Sine
of
Declination
;
So
is
the
Sine
of
the
compl
.
of
inclination
reclination
to
the
Sine
of
the
distance
of
the
Meridian
of
the
Aequinoctiall
from
the
substile
.
And
note
that
in
South
reclining
,
and
North
inclining
Plaines
,
if
Base
II
be
lesse
then
a
quadrant
,
the
contrary
pole
is
eleuated
aboue
the
Plaine
:
And
if
Base
II
be
equall
to
a
quadrant
,
the
Plaine
doth
cut
the
Pole
of
the
Aequinoctiall
.
Now
concerning
the
placing
of
the
substile
vpon
the
Dyall
plaine
(
as
I
haue
already
in
the
XXVIII
Vse
shewed
for
the
Meridian
)
Wee
are
to
know
,
First
that
the
substile
is
to
be
drawne
vpward
from
the
line
parallel
to
the
horizon
,
if
the
Plaine
be
Northerne
;
or
downeward
from
it
,
if
it
bee
Southerne
.
Except
in
North
reclining
,
and
South
inclining
Dyalls
,
in
which
the
Base
I
exceedeth
the
complement
of
inclination
,
and
reclination
:
for
in
them
it
is
quite
contrary
.
And
secondly
that
the
substile
is
to
be
drawne
in
the
contrary
side
from
the
Declination
.
But
in
North
inclining
,
and
South
reclining
Dyalls
,
in
which
the
contrary
Pole
is
eleuated
,
the
substile
must
be
drawne
through
the
center
into
the
opposite
quadrant
of
your
Dyall
circle
.
Lastly
,
the
houre
lines
in
all
manner
of
plaine
Dyalls
,
are
thus
to
be
found
.
If
the
substile
and
houre
bee
both
on
the
same
side
of
the
Meridian
:
the
arch
of
the
Aequinoctiall
betweene
the
substile
,
and
the
houre
line
,
shall
bee
equall
to
the
difference
of
the
two
distances
,
namely
of
the
houre
line
from
noone
,
and
of
the
Meridian
of
the
Aequinoctiall
from
the
substile
.
But
if
the
substile
bee
vpon
one
side
of
the
Meridian
,
and
the
houre
on
the
other
side
:
it
shall
be
equall
to
the
summe
thereof
.
Then
say
As
the
Radius
is
to
the
Sine
of
the
height
of
the
stile
aboue
the
substile
;
So
is
the
tang
.
of
the
arch
of
the
aequinoctiall
,
betweene
the
substile
and
the
houre
line
,
to
the
tang
.
of
the
arch
of
the
circle
of
your
Dyall
plaine
,
betweene
the
substile
and
that
lower
line
.
Or
else
you
may
without
calculation
Geometrically
inscribe
the
houre
lines
in
Dyals
hauing
centers
(
for
how
to
doe
it
in
Dyals
not
hauing
centers
,
I
haue
already
shewed
in
the
XIX
Vse
)
thus
.
Describe
in
your
Dyall
plaine
a
line
for
the
stile
,
at
the
same
height
or
distance
from
the
substile
,
that
the
true
stile
ought
to
haue
.
Take
also
in
the
substile
(
as
in
reason
you
shall
see
fit
)
a
point
,
and
through
it
draw
at
right
angles
a
long
line
,
for
the
contingent
,
or
Aequinoctiall
intersection
.
Againe
from
the
same
point
let
fall
a
perpendicular
vnto
the
stile
:
the
length
of
this
perpendicular
is
the
nearest
distance
betweene
that
point
and
the
stile
:
and
it
is
also
the
distance
of
the
center
of
the
Aequinoctiall
from
that
point
:
measure
it
therefore
vpon
the
substile
,
the
contrary
way
from
the
center
of
the
Dyall
:
and
hauing
thus
the
center
of
the
Aequinoctiall
,
describe
therevpon
toward
the
contingent
line
one
halfe
of
the
Aequinoctiall
circle
:
which
if
the
substile
be
the
Meridian
,
or
12
a
clock
line
of
your
Dyall
,
you
must
begin
to
diuide
into
houres
at
the
substile
:
But
if
the
substile
and
Meridian
of
your
Dyall
be
seuerall
lines
,
apply
a
ruler
to
the
center
of
the
Aequinoctial
,
and
to
the
intersection
of
the
12
a
clock
line
with
the
contingent
,
and
there
draw
a
line
:
this
line
shal
bee
the
Meridian
of
the
Aequinoctial
:
at
which
you
must
begin
to
diuide
the
Aequinoctiall
circle
into
houres
,
both
wayes
.
Then
applying
a
ruler
vnto
the
center
of
the
Aequinoctial
&
euery
one
of
those
diuisions
,
where
the
ruler
in
euery
place
shal
cut
the
contingent
line
,
there
make
a
marke
:
and
lastly
,
through
euery
one
of
those
marks
from
the
center
of
the
Dyall
,
draw
the
houre
lines
themselues
.
And
if
in
any
houre
line
it
shall
happen
,
that
the
ruler
so
applied
will
not
reach
to
intersecat
the
contingent
line
:
you
may
thus
help
your selfe
.
Which
rule
also
may
serue
you
to
find
the
Meridian
of
the
Aequinoctiall
,
as
often
as
the
intersection
of
the
Meridian
of
the
Dyall
with
the
contingent
,
falleth
without
your
paper
or
plaine
.
Draw
the
houre
line
as
farre
as
it
will
goe
.
And
take
with
your
Compasses
the
distance
of
the
intersection
point
of
the
contingent
with
the
substile
,
both
from
the
center
of
the
Dyal
,
&
from
the
center
of
the
Aequinoctial
.
And
taking
at
all
aduenture
a
point
in
the
contingent
line
,
on
that
side
in
which
the
houre
line
is
,
measure
from
that
point
on
the
contingent
,
both
those
distances
:
and
at
the
ends
of
them
both
draw
two
lines
parallel
to
the
substile
,
crossing
the
contingent
.
Then
applying
a
ruler
to
the
point
,
which
you
tooke
at
all
aduenture
,
and
to
the
intersection
of
the
parallel
,
which
hath
the
distance
of
that
center
,
whence
the
houre
line
giuen
proceedeth
,
with
that
houre
line
:
where
the
ruler
shall
cut
the
other
parallel
,
make
a
prick
:
and
measure
the
distance
betweene
that
prick
and
the
contingent
,
vpon
the
former
parallel
,
on
the
other
side
of
the
contingent
.
Lastly
,
out
of
the
proper
center
through
the
end
of
that
measure
,
draw
a
line
:
which
shall
be
that
you
desire
.
An
example
of
this
Geometricall
way
of
delineating
th●
houre
lines
you
shall
finde
in
the
description
of
a
South
vpright
Dyall
declining
35
degr.
and
reclining
degr.
41
min.
30.
by
considering
whereof
these
rules
will
be
found
exceeding
plainely
set
downe
:
As
also
all
the
other
rules
and
obseruations
here
deliuered
,
to
one
that
is
any
whit
pregnant
and
ingenious
,
will
neede
no
other
exemplification
,
then
the
inspection
of
the
instrument
it selfe
,
and
of
these
seuerall
Dyalls
following
FINIS
.