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perspectiue
,
and
Hydrography
Arthur
Hopton
N.
O.
Simon
Waterson
Lisippus
.
Appelles
Geodeticall
Staffe
Topographical
Glasse
Topographicall
Glasse
Plaine
Table
Geodeticall
Staffe
Catonians
,
Peripatitians
,
Academians
Epicurians
.
Anaximander
I
praise
not
my
Ancesters
for
their
knowledge
,
but
for
that
they
desired
knowledge
.
Compasses
,
Sea-cards
G●l●risius
Astr●l●be
I.
Blagra●e
Theodelitus
.
&c.
Plaine
Table
Homogeneall
Mechanicall
Plaine
Table
Plaine
Table
Topographicall
Glasse
Geodeticall
Staffe
What
distinctions
right
and
oblique
lines
do
admit
.
Conchales
,
Ouales
,
Lenticulares
,
&c.
Lines
compared
amongst
themselues
be
diuersly
affected
:
for
either
they
be
perpendiculars
,
paralels
,
or
contrarie
.
It
followeth
to
speake
of
Lineaments
,
and
first
of
Angles
.
Of
Triangles
and
Figures
31
Superficies
Isopleuron
,
Isosceles
Right
Angles
Oblique
Angles
Acutangled
triangle
Isopleurons
,
Isosceles
Orthogoniumes
,
Ambligoniumes
Orthogonium
Isopleuron
Oxigonium
Isosceles
Oxigonium
Scalenum
Paralelogram
equilater
52
Obliquangled
Paralelogrames
57
Curuilines
superficies
Peripher
,
Sectores
Tangents
,
Secants
,
Radius
Lib.
7.
chap.
2.
Geodeticall
Staffe
Ouales
,
Lenticulares
,
&c.
Right
Angle
or
perpendicular
is
created
by
the
extention
of
a
right
line
from
both
the
sections
of
two
arches
of
equall
or
inequall
Diameters
,
the
centers
whereof
remaining
in
the
line
assigned
,
and
this
creation
of
right
Angles
or
Perpendiculars
is
generall
.
A
p
c
d
e
c
g
h
f
g
h
f
f
p
a
a
b
a
b
e
f
a
b
e
f
a
b.
Graduator
intersect
a
b
c
d
c
,
d
a
a
b
d
a
c
d.
A
b
c
d
e
f
c
i
h
d
e
i
h
c
f
g
b
a
c
A
B
c
d
e
f
a
b.
A
B
d
a
c
c
b
d
a
b
A
b
c
1.
Prop.
a
b
b
c
e
d
f
d
a
b
c
a
b
f
g
i
h
d
b
a
c
p
q
t
u
e
g
i
f
h
i
f
h
f
i
f
f
h
e
g
i
f
h
e
g.
a
b
c
d
Isosceles
Triangle
Isosceles
Triangles
See
the
ensuing
figures
in
the
folded
sheet
.
p
q
p
q
t
u
p
q
29.
½
p
q
a
b
c
d
See
the
ensuing
figures
in
the
folded
sheet
.
Index
i
31.
Chap.
r
r
p
q
r
r
See
the
precedent
figures
in
the
folded
sheet
.
Indexes
g
Circles
are
as
the
squares
which
are
made
of
their
diameters
,
and
that
their
diameters
are
as
the
Circumferences
.
d
c
b
r
o
r
r
r
Geodeticall
staffe
b
c
11
10
9
8
7
a
b
a
b
b
d
c
f
d
e
a
d
e
d
f
a
b
a
i
b
f
d
d
e
a
i
b
a
i
b
a
i
b
m
n
o
m
n
o
p
q
m
n
o
3
6
9
m
n
o
Index
g
Two
strings
are
good
to
one
bowe
.
b
m
h
g
g
k
g
f
g
m
k
l
h
b
b
c
C
g
g
h
p
q
4
Indexes
Crocus
Martis
a
c
b
d
Index
f
g
h
i
Index
f
g
Theodelitus
,
Topographicall
instrument
,
Geometricall
Quadrant
,
Geometricall
Quadrant
Topographicall
Instrument
Geometricall
Quadrant
4
Indexes
We
will
take
Maister
Digges
his
own
example
,
a
b
c
are
the
markes
in
the
field
to
be
measured
,
d
the
first
station
,
where
you
shall
set
the
center
of
your
instrument
,
his
Diameter
or
line
where
the
diuisions
take
beginning
pointing
directly
to
a
,
so
doe
e
f
g
the
visuall
lines
running
by
the
angles
of
position
of
the
instrument
vnto
all
the
angles
or
markes
obserued
,
expresse
my
obseruations
at
the
first
station
d.
Lastly
doth
h
note
90.
degrees
,
which
directeth
to
the
second
station
m.
so
is
d
m
my
stationary
line
,
which
must
be
measured
,
and
is
300.
yard
:
s
whereby
I
gather
a
Table
,
thus
:
Then
going
to
the
second
station
m
,
where
ye
shall
now
place
the
center
of
your
instrument
,
the
line
where
the
degrees
take
beginning
,
pointing
iust
from
m
to
d
:
so
do
the
visuall
lines
i
k
l
,
running
to
the
markes
before
noted
,
cut
new
angles
of
position
,
which
you
must
collect
as
before
in
a
Table
,
thus
:
Now
if
ye
marke
diligently
where
each
match
lines
do
crosse
one
the
other
,
there
is
the
true
proportion
of
such
places
,
and
so
by
drawing
right
lines
from
those
intersections
,
if
it
were
a
field
,
or
such
like
,
you
haue
the
bounds
and
limits
thereof
.
And
if
the
distance
betwixt
any
two
places
or
markes
bee
required
,
seeke
the
space
betwixt
any
two
places
or
markes
bee
required
,
seeke
the
space
betwixt
the
two
stations
d
m
,
whicg
as
I
formerly
said
is
300.
yards
,
I
diuide
d
m
into
18.
equall
parts
,
and
demande
the
distance
betiwxt
a
b
,
which
conteines
11
of
those
18
parts
.
Then
seeing
I
am
ignorant
what
number
of
yards
be
conteined
in
those
11
parts
,
I
fly
to
the
rule
of
proportion
,
saying
a
b
c
d
e
f
g
f
,
g
a
g
e
,
g
d
,
g
c
,
g
b
,
h
a
,
h
f
,
h
●
,
h
d
,
h
●
,
h
b
,
a
f
,
f
c
,
e
d
,
d
c
,
c
b
,
b
a
To
doe
this
Proposition
without
calculation
.
f
e
g
h
f
e
12.
a
f
,
e
d
,
d
c
,
c
b
,
b
a
g
f
,
g
a
,
g
e
,
g
d
,
g
c
,
g
b
,
h
a
,
h
f
,
h
e
,
h
c
,
h
d
,
h
b
,
f
d
,
f
c
,
f
b
,
a
d
,
a
c
,
&c.
Geodeticall
Staffe
Geodeticall
Staffe
TO
performe
this
,
you
must
ascend
to
the
top
of
some
high
Hill
,
Eliffe
,
or
Tower
,
from
whence
you
may
directly
behold
the
situation
of
the
Country
,
as
it
lyeth
adiacient
round
about
in
your
Horizon
,
we
hold
the
Semidiamer
of
the
Horizon
to
conteine
Stadias
,
and
so
farre
may
one
see
:
for
we
must
alwaies
when
wee
be
at
this
worke
,
imagine
our selues
to
be
in
the
center
of
the
Horizon
,
and
thence
necessarily
must
see
to
the
Circumference
,
which
is
limited
from
vs
by
the
Semidiameter
.
But
to
proceede
,
your
place
being
appointed
,
there
set
vp
the
vpon
his
staffe
,
ordering
it
in
such
sort
,
by
helpe
of
the
Needle
,
that
the
foure
Semidiameters
thereof
may
point
iust
East
,
Weast
,
North
and
South
,
euery
one
pointing
correspondantly
into
his
like
quarter
of
the
heauen
:
then
turne
the
with
the
sights
to
euery
Towne
,
Village
,
or
Hauen
,
or
whatsoeuer
you
desire
to
place
in
the
Map
,
espying
through
the
sights
,
the
middle
,
or
most
notable
marke
in
euery
of
them
,
as
commonly
the
Stéeple
,
if
it
be
a
Church
Towne
,
noting
at
euery
of
those
places
the
degrees
cut
by
the
in
great
circle
,
and
also
the
parts
of
the
degrees
,
which
are
properly
called
angles
of
position
,
and
collect
you
a
Table
of
your
first
station
thereby
.
Then
casting
your
eye
round
about
,
search
some
mountaine
or
lofty
place
from
whence
againe
you
may
view
all
these
places
and
appoint
that
to
be
your
second
station
then
turne
thereto
the
&
note
also
the
degree
cut
,
this
done
repaire
vnto
your
second
station
formerly
found
,
where
situate
your
in
al
respects
as
he
was
at
the
first
station
,
turning
the
and
sight
about
,
still
obseruing
all
such
markes
you
saw
before
,
and
note
againe
the
degrée
cut
,
or
angles
of
position
,
writing
the
name
of
euery
place
and
his
angle
by
it
,
so
haue
you
collected
a
second
table
,
which
is
for
your
second
station
.
These
things
so
done
take
a
skinne
of
vellam
,
royall
paper
,
or
what
you
please
,
and
in
some
place
thereof
appoint
your
first
station
,
about
which
describe
a
circle
which
you
must
diuide
into
degrées
,
beginning
in
that
quarter
of
the
world
in
which
the
beginning
of
the
degrée
in
your
Instrument
were
placed
,
or
else
protracting
them
by
your
as
you
be
taught
betaking
either
of
the
waies
from
the
center
of
this
circle
,
to
euery
degree
noted
in
your
first
table
,
there
must
right
lines
be
produced
infinitely
,
noting
to
euery
of
them
the
name
of
his
place
,
now
protract
the
line
of
your
second
station
,
according
to
the
degrée
cut
,
and
vpon
that
line
describe
another
circle
,
which
vse
in
all
respects
as
you
did
the
former
,
taking
direction
from
the
table
,
obserued
at
your
second
station
.
To
conclude
,
diligently
note
the
concurse
or
intersection
of
euery
like
lines
making
there
on
some
marke
,
as
☉
with
the
name
of
the
place
correspondent
,
and
so
you
haue
finished
.
Now
to
know
how
farre
euery
of
these
townes
&c.
bée
distant
from
other
,
doe
thus
:
measure
the
distance
betwixt
your
stations
by
your
or
this
Instrument
,
as
you
shall
bée
after
taught
,
or
by
any
other
Instrument
,
to
you
seeming
best
,
and
diuide
your
stationary
line
,
or
line
included
betwxit
the
center
of
the
circles
into
so
many
equall
parts
as
there
be
miles
,
furlongs
,
or
scores
,
betwéene
your
stations
,
this
line
so
diuided
,
measure
the
distance
of
any
place
thereby
,
as
you
doe
with
an
ordinary
Scale
in
a
mappe
,
taking
the
distance
of
any
two
places
with
your
compasse
,
and
applying
the
widnesse
to
the
diuided
line
,
for
so
many
equall
parts
as
bee
then
included
betwixt
the
féete
of
your
compasse
,
so
many
miles
,
scores
&c.
is
it
betwéene
the
two
places
according
to
the
denomination
of
the
diuisions
in
the
stationary
line
.
Cordocke
hill
Hopton
Castle
Knookin
Castle
Wrekin
hill
Hopton
Castle
f
p
Hopton
C●stle
,
f
e
Montgomery
,
f
d
Knookin
Castle
,
f
m
,
Ludlow
f
g
f
g
THis
kind
of
worke
although
it
be
something
more
tedious
and
difficult
then
the
former
,
yet
hath
it
in
it selfe
a
most
exact
and
certaine
operation
:
you
must
in
performance
hereof
ascend
the
top
of
some
high
mountaine
,
hill
,
or
such
like
,
whence
you
may
directly
behold
all
the
adiacent
townes
within
the
circuit
of
that
Horizon
,
and
also
from
that
hill
espie
some
other
mountaine
,
to
whose
sumunity
the
view
of
all
the
foresaid
adiacent
townes
be
subiect
.
This
so
done
,
make
the
first
hill
a
center
,
and
the
other
a
terme
,
of
one
of
the
sides
of
euery
angle
,
and
so
with
your
Instrument
by
the
Chap.
or
any
other
Instrument
take
the
true
quantity
of
the
angle
that
euery
towne
maketh
with
these
two
hils
,
and
note
the
same
downe
in
some
Table
booke
:
this
so
done
,
get
to
the
next
hill
,
and
there
againe
obserue
in
like
manner
the
quantitie
of
euery
angle
euen
vpon
this
hill
as
you
did
vpon
the
fornier
:
finally
,
get
the
true
distance
betwixt
the
top
of
the
two
hils
,
so
haue
you
whereby
get
the
other
angle
with
the
two
lines
vnknowne
,
and
then
place
euery
towne
in
his
due
place
,
as
you
shall
be
better
taught
in
the
Example
.
g
a
b
f
a
b
a
b
c
b
a
g
a
b
g
b
a
a
g
b
g
a
b
a
g
b
g
b
a
a
g
g
b
g
b
a
g
a
b
a
g
b
a
g
g
b
a
b
b
g
g
a
b
a
g
b
,
viz.
a
g
a
f
,
a
e
&c.
d
f
,
d
e
,
b
d
,
&c.
multiply
the
signe
of
the
angle
,
conteining
the
line
sought
,
by
the
line
knowne
,
and
diuide
the
product
by
the
line
of
the
angle
containing
the
said
knowne
line
.
a
b
h
i.
c
d
b
,
&c.
Topographicall
Glasse
Geometricall
quadrant
,
Hypsometricall
Scale
Gemma
Frisius
,
Orontius
,
&c.
s
k
k
l
Therefore
if
you
say
a
l
l
r
a
c
c
b
Which
is
no
other
then
if
you
should
say
.
a
l
l
r
a
c
b
c
I
As
if
you
should
say
.
d
p
p
v
t
o
o
d.
p
v
o
x
t
o
o
d
d
p
p
v
w
p
r
l
p
d
d
o
o
x
p
v
o
x
v
w
w
p
a
d
d
c
t
x
t
o
a
d
d
c
p
v
o
x
w
p
r
l
p
v
o
f
o
x
d
x
o
b
●
c
x
l
t
o
a
d
d
c.
d
v
p
d
b
c
v
w
w
p
b
z
z
c
b
z
z
c
a
d
a
c
d
z
a
b
a
b
c
Therefore
to
conclude
v
w
w
p
a
d
a
c.
b
p.
Topographical
Glasse
Geodeticall
Staffe
And
if
the
distance
of
c
a
be
required
,
multiply
211273
the
secant
of
the
angle
b
c
a
,
by
73
,
so
haue
you
15422929
,
which
parted
by
the
totall
signe
,
leaueth
154
22929/100000
perches
your
desire
,
and
thus
must
you
deale
with
any
other
like
question
.
Geometricall
Quadrant
A
is
the
place
whose
distance
is
tequired
,
b
the
marke
where
my
instrument
was
first
disposed
,
whence
(
as
in
the
last
Chapter
)
I
depart
orthogonally
to
c
,
the
Index
cutting
37.
parts
,
and
about
a
halfe
in
the
right
side
of
the
Geometricall
parts
.
Now
the
distance
of
b
c
is
found
73.
yards
,
wherefore
I
increase
60.
by
73.
so
haue
5080.
which
diuide
by
the
parts
of
the
square
cut
,
as
by
37.
and
better
,
so
haue
you
135.
yards
,
with
certaine
odde
parts
more
,
the
distance
of
b
a.
Or
by
diuiding
3600.
the
square
of
60.
by
the
parts
of
the
Quadrant
cut
,
as
37.
and
better
,
the
quotient
shall
produce
you
a
a
proportionall
number
:
which
number
part
by
b
c
73.
the
quotient
whereof
is
the
longitude
of
a
b
,
as
before
.
And
take
this
note
with
you
,
that
the
Index
will
neuer
cut
in
the
parts
of
the
Quadrant
vpon
the
left
hand
,
vnlesse
the
longitude
sought
be
shorter
then
your
stationary
line
;
I
meane
,
vnlesse
the
line
knowne
,
or
measured
,
bee
longer
then
the
line
sought
.
And
further
note
,
that
the
more
parts
you
diuide
the
sides
of
your
scale
into
,
the
easier
and
truer
shall
you
worke
.
Another
way
these
kindes
of
longitudes
be
performed
,
and
that
is
by
protracting
after
obseruation
of
the
angles
,
which
for
that
it
is
already
set
downe
,
I
omit
it
heere
,
onely
you
may
here
protract
with
a
circle
readily
diuided
,
as
hereafter
you
shall
be
taught
,
if
your
staffe
bee
wanting
.
LEt
it
be
supposed
that
you
were
standing
in
an
open
field
,
and
a
certaine
Castle
and
a
Fort
shewed
you
,
to
neither
of
which
you
could
approach
,
and
yet
vpon
some
occasion
were
required
to
deliuer
the
distance
betwixt
the
same
.
The
first
thing
in
performance
hereof
that
you
are
to
doe
,
is
to
plant
your
Glasse
at
the
place
where
you
meane
to
make
obseruation
,
and
then
the
vpon
the
beginning
of
the
degrées
,
turne
the
Instrument
&
all
about
vntill
you
espy
the
Fort
vpon
your
left
hand
:
the
Planisphere
resting
,
conuey
the
to
the
other
Castle
vpon
the
right
hand
,
noting
the
degrées
cut
by
the
fiduciall
edge
of
the
Now
are
you
to
view
some
other
marke
for
a
second
station
vpon
your
right
hand
,
whereunto
turne
the
vntil
through
the
sight
you
espy
the
same
,
noting
againe
the
degrées
cut
but
by
the
and
it
will
bee
the
better
to
let
the
vpon
these
last
degrées
bee
about
for
the
neerer
that
the
stationary
line
,
running
from
your
first
station
to
the
Fort
vpon
the
left
hand
be
to
conteine
a
right
angle
,
the
better
it
is
.
)
Now
leauing
some
apparant
marke
where
the
center
of
your
Glasse
was
,
take
vp
your
Glasse
,
&
beare
the
same
to
the
place
appointed
for
your
second
station
:
and
hauing
planted
him
there
,
turne
the
Glasse
,
the
vpon
the
beginning
of
the
degrées
,
vntill
through
the
sight
you
espye
the
marke
or
man
left
at
your
first
station
:
the
resting
,
conuey
the
vnto
the
Fort
vpon
your
right
hand
,
noting
the
degrées
cut
next
to
the
other
Castle
or
Turret
more
rightwards
,
noting
the
degrées
cut
:
finally
,
measure
the
distance
betwixt
both
your
stations
,
all
which
note
downe
,
as
well
the
angles
obserued
at
your
first
station
,
as
those
obserued
at
your
second
,
as
also
the
quantity
of
the
line
that
is
included
betwixt
both
those
stations
,
and
protracting
the
same
vpon
paper
,
by
your
Scale
and
compasses
,
you
haue
finished
.
c
a
d
c
a
b
a
b
c
d
b
a
a
b
l
i
e
f
k
i
i
e
g
g
c
h.
a
g
c
d.
Geometrical
Quadrant
Topographicall
Glasse
4.
Hypsometricall
Scale
,
Geometricall
Quadrant
Geometricall
Quadrant
In
taking
of
these
altitudes
you
must
note
that
you
meddle
with
no
part
of
the
same
,
but
that
which
is
aboue
the
leuell
of
your
eye
,
the
which
leuell
you
shall
bee
taught
hereafter
to
obserue
.
Hauing
taken
the
angle
of
Altitude
,
you
may
worke
this
proposition
by
signes
&c.
according
as
you
be
largely
taught
in
the
7
booke
of
my
Staffe
called
Trigonometria
,
Prob
,
1.
Geodeticall
Staffe
,
To
take
all
kinde
of
altitudes
by
the
Glasse
.
Chap.
4.
We
haue
obserued
the
angle
a
b
d
to
bee
29.
degrees
40.
min.
and
a
c
d
to
be
46.
degrees
,
the
Tangent
of
the
Complement
of
29.
deg.
40.
min.
is
175556.
and
of
46.
degrees
,
96568.
Now
subtract
96568.
from
175556.
and
there
rests
78988.
Then
multiply
100000.
the
totall
signe
by
90.
the
distance
betwixt
.
both
your
stations
(
for
so
many
feet
I
found
it
by
measure
)
and
there
g
c
g
c
a
f
b
g.
B
c
is
a
Tower
,
and
you
standing
at
a
are
required
to
deliuer
part
of
the
Altitude
,
as
c
h
,
first
therefore
I
obserue
the
angle
h
a
b
,
11
degrees
,
then
c
a
b
27
degrees
,
next
I
measure
the
distance
a
b
,
which
I
find
178
feete
,
these
had
,
I
take
the
Tangent
of
11
degres
,
viz.
19080
from
50952
,
the
Tangent
of
27
degrees
,
the
remainer
is
31872
,
which
increased
by
172
produceth
5481984
,
which
being
parted
by
100000
the
quotient
is
54
½
1/●
9/●
8/●
4/●●
●●/●
feete
,
30
,
360
IN
performing
hereof
,
you
must
obserue
the
angle
of
Altitude
made
at
both
the
markes
in
Altitude
,
whose
distance
is
required
,
noting
the
parts
of
the
cut
by
the
moueable
sight
,
and
also
note
if
the
parts
cut
at
both
your
obseruations
were
in
the
side
towards
which
is
the
right
side
,
or
in
the
side
fromwards
which
is
the
left
side
,
or
if
at
the
one
obseruation
the
parts
cut
were
in
one
side
,
and
the
other
in
the
contrary
side
.
If
the
parts
cut
at
both
the
angles
were
in
the
left
side
(
which
is
furthest
from
subtract
the
lesser
from
the
greater
,
and
with
the
which
remaines
augment
your
stationary
line
,
which
parted
by
the
whole
side
as
by
leaueth
the
desired
Longitude
Now
if
the
parts
cut
at
both
lines
were
in
the
right
side
towards
by
the
last
Chapter
,
reduce
those
parts
into
parts
proportionall
and
then
worke
as
in
the
last
difference
.
Or
if
the
parts
cut
,
be
at
one
line
in
the
left
and
at
another
in
the
right
,
then
must
you
reduce
the
parts
cut
by
the
right
side
into
parts
proportional
,
as
in
the
last
Chapter
,
which
done
(
as
in
the
first
difference
)
deduct
the
lesser
from
the
greater
,
&
the
product
increase
by
your
stationary
distance
,
which
diuided
by
the
whole
side
,
yeelds
your
desire
,
and
so
must
you
deale
,
if
the
one
section
were
made
in
in
the
midst
betwixt
the
right
and
left
side
.
B
c
b
z
,
a.
z
a
c
k
l
a
c
b
z.
o
s
s
p
b
z
b
d
c
b
d
a
b
a
d
a
b
d
b
d
c
a
b
d
b
c
d
c
a
a
d
c
b
a
87
b
c
a
,
74.
c
a
b
c
b
a
c
a
b
c
a.
b
a
b
d
c
a
c
e
b
d
4
c
e
7
c
e
b
d.
euery
line
of
leuell
doth
make
right
angles
with
the
perpendicular
,
and
euery
perpendicular
pointeth
to
the
center
of
the
earth
,
f
b
c
g
Aqua
ascendit
,
quantum
descendit
A
is
a
place
where
water
is
found
,
and
the
question
is
,
to
know
if
it
will
be
brought
to
the
Tower
at
b
,
which
is
distant
from
a
4
miles
,
I
plant
my
Instrument
therefore
at
c
,
so
that
the
diameter
p
q
lye
paralell
,
and
also
equall
in
heights
to
a
the
spring
head
,
then
looking
through
q
p
towards
b
,
I
cannot
see
the
same
by
reason
of
a
certaine
hill
,
that
is
betwixt
me
and
b
,
therefore
I
obserue
a
marke
in
that
hill
through
the
sight
,
as
at
c
,
where
againe
Note
that
some
hold
pipes
of
earth
baked
,
to
be
better
then
lead
,
and
some
hold
pipes
of
alderwood
,
firre
,
pine
tree
,
of
such
wood
that
hath
rosen
in
it
to
be
better
then
the
former
.
And
if
the
ground
lye
reasonable
leuell
,
so
that
you
conuey
the
water
by
trenches
,
order
the
said
trenches
so
by
helpe
of
a
plumbe
,
that
the
water
may
haue
currant
4
½
inches
at
euery
miles
end
,
then
fill
the
gutter
with
pibble
stones
a
foote
or
more
deepe
,
and
vpon
them
throw
earth
,
so
will
the
water
run
more
cleere
to
the
place
appointed
.
Note
lastely
,
that
it
is
best
,
if
you
bring
your
water
by
pipes
,
to
let
it
come
by
many
croked
turnings
,
and
sometimes
to
fall
directly
downewards
,
and
then
againe
to
rise
by
little
and
little
,
&
by
this
meanes
some
thinke
one
may
force
the
waters
issue
to
be
aboue
the
head
of
the
spring
,
so
that
in
pipes
you
shall
not
need
the
foresaide
abatement
.
But
now
whereas
in
conueiance
of
waters
by
this
way
,
whether
it
be
waters
to
houses
or
new
riuers
,
sometimes
happely
you
shall
meete
with
a
deepe
valley
,
out
of
which
you
cannot
get
the
water
by
ditches
,
and
to
finde
leuel
ground
you
cannot
,
without
going
a
great
compasse
:
and
hauing
found
it
happely
cannot
haue
liberty
for
to
cut
a
trench
through
the
same
:
such
a
matter
and
such
a
difference
I
saw
in
bringing
the
new
riuer
from
wards
Ware
to
London
,
for
remedy
whereof
if
the
floud
be
great
,
you
must
erect
arches
in
manner
of
a
bridg
,
which
may
extend
it selfe
ouer
the
valley
euen
from
the
one
banke
to
the
other
,
and
if
the
water
be
but
for
a
house
,
postes
may
serue
to
beare
the
same
:
the
like
must
you
do
,
if
you
meete
with
a
riuer
,
brooke
or
such
like
.
M.
L.
doth
teach
you
how
to
solder
your
pipes
of
earth
or
wood
,
&
that
is
with
vnquenched
lime
and
hogs-grease
,
or
with
rosen
and
white
of
egges
,
or
with
lime
,
white
of
egges
,
and
fylings
of
Iron
.
a
b
c
a
c
d
e
d
e
Geodetia
Chap.
53.
Lib.
5.
Geodeticall
Staffe
Chap
,
2.
Iacobs
Staffe
f
a
h
i
k
l
a
b
c
h
i
k
i
g
h
viz.
115
h
k
h
k
l
230
h
m
l
h
i
k
l
230
h
g
g
h
g
n
g
i
,
g
k
g
l
g
o
12
g
k
24
g
p
6
Geodeticall
Staffe
For
as
the
Secant
is
to
the
Radius
,
so
is
the
Radius
to
the
signe
of
the
Complement
of
the
said
Secant
,
by
the
seuenth
booke
of
the
Geodeticall
Staffe
,
called
Trigonemetria
.
If
the
grounds
descend
,
as
b
d
,
go
to
b
,
and
then
worke
as
before
.
But
you
shall
not
need
to
make
this
reducement
in
the
fields
,
vntill
you
come
to
protract
.
And
note
further
,
that
if
the
ground
ascend
as
a
b
,
and
also
descend
as
b
d
,
then
must
you
in
your
protracting
adde
the
Horizontall
line
a
c
,
and
d
c
together
,
and
protract
that
in
stead
of
the
lines
a
b
and
b
d
,
so
that
by
what
is
said
before
,
you
may
gather
that
the
line
which
you
measured
in
the
field
,
to
be
40.
pearches
,
and
rising
30.
degrees
high
,
if
you
will
worke
truely
,
should
be
laid
downe
not
altogether
35.
pearches
.
Compendium
I.
Regula
.
●or
the
signe
that
is
in
the
first
place
,
put
the
Secant
of
the
Complement
thereof
;
Et
voti
compos
eris
.
Compendium
II.
Regula
.
For
the
Tangent
placed
in
the
first
place
,
take
the
Tangent
of
the
Complement
;
Et
negotium
confectum
erit
.
Compendium
III.
Regula
.
For
the
secant
put
in
the
first
place
,
take
the
signe
of
the
Complement
,
so
shall
you
haue
a
true
proportion
,
the
Radius
being
in
the
first
place
Trigonometria
▪
a
b
c
b
c
b
c
b
c
a
d
East
,
West
,
North
Let
the
proposition
be
to
describe
England
,
and
therein
to
situate
such
townes
as
shall
be
required
in
that
bignesse
,
as
is
here
set
downe
according
to
my
Scale
:
the
first
thing
that
I
doe
,
hauing
appointed
my
Card
,
I
crosse
the
same
with
two
right
lines
at
right
angles
,
appointing
at
the
end
thereof
the
foure
quarters
of
the
world
,
as
in
the
Type
:
then
I
find
out
some
towne
that
I
coniecture
(
by
conference
with
my
tables
)
lieth
about
the
midst
of
the
land
,
which
let
(
for
examples
sake
)
be
Middle
Wiche
,
a
towne
situate
in
Cheshire
,
the
which
towne
I
place
vpon
the
very
intersection
of
the
foresaid
lines
,
and
thereby
write
the
name
of
the
same
,
searching
in
my
tables
for
some
other
towne
that
lieth
direct
East
,
West
,
North
,
or
South
;
I
find
none
:
therefore
I
take
some
other
towne
,
as
Bristow
,
drawing
a
line
according
to
the
position
of
the
same
,
getting
also
from
your
tables
the
distance
of
that
towne
from
Middle
Wiche
,
as
97
miles
,
to
which
widenesse
vpon
your
Scale
open
the
feete
of
your
compasse
,
and
then
place
that
widenesse
vpon
the
line
of
position
for
Bristow
,
placing
the
one
foote
in
the
marke
made
for
Middle
Wiche
,
making
a
note
with
the
other
in
the
said
line
of
position
,
where
write
Bristow
.
These
two
townes
so
placed
,
let
vs
now
go
situate
Northampton
:
first
by
my
tables
I
finde
the
distance
of
Middle
Wiche
from
Northampton
,
and
according
to
that
distance
,
the
one
foote
resting
in
the
marke
for
Middle
Wiche
,
with
the
other
I
strike
the
portion
of
any
arch
:
the
like
I
doe
with
the
distance
of
Northampton
from
Bristow
,
as
in
the
12
Chapter
is
plaine
.
Now
the
intersection
of
these
two
arches
is
the
true
place
of
Northampton
,
where
make
a
marke
for
the
towne
,
and
write
the
name
thereof
by
the
said
marke
,
and
so
proceed
,
limiting
all
the
townes
,
ports
,
angles
,
and
nookes
in
the
Island
in
their
proper
places
,
as
you
may
sufficiently
gather
by
the
former
demonstration
.
Hauing
finished
,
in
some
void
place
you
may
appoint
,
the
Mariners
compasse
,
as
in
the
Card
before
,
and
this
compasse
will
serue
you
for
many
necessarie
vses
,
as
it
is
not
vnknowne
to
men
seene
that
way
.
a
Dane
,
and
diligent
obseruer
of
the
motions
of
the
celestiall
bodies
,
found
at
last
by
conferring
daily
practise
and
experience
with
the
optikes
,
that
the
Sunne
seemed
to
vs
eleuated
higher
vpon
the
verticall
circle
,
then
indeed
he
is
,
and
his
reason
hee
drawes
from
for
they
appoint
the
heauens
the
Elements
&c.
to
be
themselues
most
together
,
whe●
the
Sunne
is
●●●test
to
th●
Horizon
,
which
eleuating
themselues
by
little
and
〈◊〉
,
successiuely
,
at
length
meerely
vanish
and
are
nothing
●ale
ob●●●●●s
as
in
the
ensuing
Table
of
the
Sunnes
re●●●ction
the
grea●est
eleuation
of
the
vapours
,
according
to
the
said
optiks
of
of
which
the
●●●●●●ter
of
the
earth
conten●●●
〈◊〉
which
〈◊〉
●●●●ce
Ge●maino
●●les
,
but
heauing
the
ample
discourse
hereof
to
the
●●pa●●
of
where
the
desirou●
may
roade
at
large
,
but
vs
onely
returne
vnto
the
pr●●ticall
vse
thereof
,
so
shall
it
bee
●●●ine
by
〈◊〉
of
this
refraction
caused
by
the
thickenesse
of
the
vapours
a●●ue
the
Horizon
,
that
the
sight
of
the
Sunne
is
altered
so
that
he
seemes
to
rest
sooner
and
set
slower
thou
indeed
〈◊〉
doth
,
insomuch
that
the
center
of
his
body
seeming
to
touch
the
Horizon
,
the
same
is
then
minutes
below
the
Horizon
,
neither
doth
the
magnitude
thereof
hinden
the
same
,
although
hee
seeme
〈◊〉
●●ggat
at
the
setting
th●●
when
he
is
in
the
〈◊〉
wherby
●e
may
〈◊〉
,
that
by
occasi●●
of
this
refrac●●●●
this
who●●
〈…〉
Horizon
a●
vising
or
●●●ting
whe●●●●●
there
〈◊〉
not
any
pa●●
thereof
abo●e
the
sa●●
and
〈…〉
●●●tificial
,
continuing
from
Sunne
to
Sunne
,
bee
longer
according
to
the
apparent
rising
and
setting
thereof
,
then
those
our
Astronomicall
calculations
for
the
Poles
eleuation
,
But
to
set
apart
further
discourse
hereof
,
behold
the
Tablet
,
he
vse
whereof
is
this
:
Séeke
the
Altitude
of
the
Sunne
in
one
of
the
rowes
vnder
Alti
.
☉
,
answering
to
which
vpon
the
right
hand
vnder
the
title
of
refraction
is
the
minute
and
section
of
the
Sunnes
refraction
,
which
must
be
abated
from
the
Altitude
instrumentally
obserued
,
because
the
refraction
doth
alwaies
cause
the
sun
to
seeme
higher
then
indeed
he
is
.
Of
the
Paralaxes
of
the
Sunne
.
Tycho
Brahe
Alt.
G.
Max.
Med.
4
½
1610.
Sol
29
½
Index
f
aequilater
paralelogram
a
d
b
c
m
n
k
l
a
d
The
distance
a
b
is
required
,
first
thererfore
I
plant
my
Table
at
b
,
then
working
as
before
,
I
finde
c
my
second
station
,
and
so
draw
a
line
to
point
from
b
to
a
,
and
another
from
b
to
c.
Next
,
I
measure
the
line
b
c
,
and
finde
it
7●
.
ya●d●
,
which
I
lay
downe
vpon
my
paper
with
my
Scale
and
Compasses
.
Lastly
,
I
note
the
degrees
cut
by
the
South
end
of
the
needle
,
which
let
be
40.
This
done
,
I
go
to
c
,
and
there
againe
plant
the
Table
,
as
before
:
So
do
I
make
the
stationary
line
protracted
,
point
iust
to
b
,
and
then
noting
the
degrees
cut
a
gaine
by
the
needle
,
I
finde
them
40.
as
before
,
which
argues
I
haue
well
planted
my
Table
.
To
conclūde
,
I
place
the
fiduciall
edge
of
my
rule
vpon
c
,
mouing
●he
other
end
,
vntill
it
intersect
with
the
line
representing
a
b
;
therefore
by
my
Scale
I
measure
the
line
representing
b
a
,
so
haue
I
the
distance
of
b
a
135.
yards
,
by
the
same
Scale
might
you
haue
expressed
c
a.
Geodeticall
Staffe
Geodeticall
Staffe
a
b
a
c
a
b
a
c
50.
●
d
a
b
a
b
d
c
a
b
d
c
131
f
e
,
e
d
,
d
c
,
g
h
,
10
f
e
12
i
a
,
i
b
,
i
c
,
i
d
,
i
k
,
i
e
,
i
f
,
i
g
,
i
h
i
a
27
i
b
9
¾
f
g
h
i
k
l
m
n
o
p
q
r
s
,
s
f
g
h
i
k
l
m
n
o
p
q
r
a
b
b
c
a
h
a
i
causa
breuitatis
.
The
South
end
of
the
Needle
cuts
like
degrees
in
the
card
at
euery
station
,
as
at
the
first
station
.
lib.
6.
Geodetia
,
Chap.
7.
defi
.
5.
The
figure
is
a
b
c
d
e
f
g
,
the
angle
where
the
instrument
is
planted
is
a
,
whence
I
drawe
lines
into
euery
angle
,
as
a
b
,
a
c
,
a
d
,
a
e
,
a
f
,
and
a
g
,
then
I
measure
the
line
a
b
,
and
lay
it
downe
,
then
I
measure
the
line
b
c
,
and
note
where
it
intersects
with
a
c
,
as
at
c
,
then
I
measure
c
d
,
and
note
where
it
intersects
with
d
,
&
so
I
go
round
about
the
field
,
and
produce
a
figure
proportionall
to
the
figure
proposed
.
def.
4.
Geodetia
C
d
f
g
h
i
k
My
first
station
is
a
,
whence
I
obserue
the
angles
d
e
f
k
l
m
,
my
second
station
is
b
,
whence
I
draw
lines
to
point
to
as
many
of
the
angles
I
obserued
at
my
first
station
as
you
can
see
as
b
d
,
d
e
,
b
f.
and
so
noting
the
intersection
of
matchy
lines
,
draw
the
lines
d
e
and
e
f
,
which
is
so
much
of
the
hedges
that
you
haue
obserued
:
now
the
Instrument
vnremoued
at
b
,
I
espy
as
many
more
angles
from
b
as
I
well
can
,
as
g
h
and
i
,
and
so
draw
lines
to
represent
b
g
b
h
and
b
i.
Lastly
,
I
espy
some
other
place
whence
I
may
see
all
these
three
former
angles
:
but
the
way
to
finde
your
third
station
c
is
thus
,
vpon
some
point
on
the
paper
,
representing
some
angle
in
the
field
as
e
,
laying
there
the
edge
of
your
ruler
,
mouing
the
other
end
vntill
you
obserue
through
the
sights
the
angle
e
,
then
note
where
the
edge
of
your
rule
and
the
line
b
c
intersecte
,
as
at
c
,
so
shall
you
finde
the
true
place
where
your
instrument
stands
,
your
instrument
resting
situate
at
c
,
in
all
respects
as
at
the
other
stations
,
draw
lines
to
point
from
c
to
g
h
and
i
,
and
so
note
the
intersection
of
these
lines
,
with
their
matchy
lines
drawn
from
b
,
so
haue
you
another
part
of
the
perimeter
,
by
drawing
lines
from
one
intersection
to
another
,
as
g
h
,
and
h
i
:
and
for
that
you
may
see
from
c
to
all
the
rest
of
the
angles
k
l
and
m
,
obserued
at
a
,
therefore
I
draw
lines
to
point
from
c
to
k
,
to
l
and
m
,
and
so
noting
the
intersections
as
before
,
and
drawing
lines
I
haue
included
a
figure
proportionall
and
like
to
the
proposed
figure
.
a
b
obseruare
decorum
Geodeticall
Staffe
5.10.15
.
&c.
Topographicall
Glasse
Of
the
Sights
Of
the
shorter
Sight
5.10.15
.
&c.
Of
the
longer
Sight
5
,
10
,
15
1
,
2
,
3
,
4
,
100
,
102
,
103
,
104
102
103
Of
the
slit
in
the
sight
.
Of
the
Index
Of
the
Staffe
The
6
of
October
in
the
Morning
,
I
made
obseruation
according
vnto
the
first
difference
,
where
hauing
planted
my
Instrument
paralell
,
and
spied
the
Sunne
through
the
hole
in
the
vane
,
and
by
the
pins
head
,
I
found
the
vane
to
out
12
degrees
vpon
the
sight
,
and
the
South
end
of
the
needle
to
cut
19
●/60
degrees
,
which
shewed
me
that
the
Sunne
was
1
:
degrees
high
,
and
that
he
wanreth
58
degrees
of
the
Meridian
,
for
the
South
point
cutting
29
degrees
,
&
20
minutes
,
I
multiply
the
same
by
3
there
commeth
58
,
which
sheweth
me
there
is
so
many
degrees
included
betwixt
the
Sun
and
the
Meridian
,
and
so
of
the
rest
.
I
find
the
South
point
cut
45
degrees
,
I
conclude
the
thing
seene
is
North-east
.
In
all
right-lined
Triangles
the
proportion
of
the
one
side
vnto
the
other
,
is
such
as
the
signe
representing
the
angles
be
.
The
sides
of
opposite
angles
bee
direct
proportionall
to
their
signes
.
Iacobs
Staffe
Geodeticall
Staffe
Geodeticall
Staffe
Geodetical
Staffe
Let
the
number
of
angles
be
8.
from
which
take
6.
which
is
a
lesse
,
then
multiply
60.
by
6.
and
the
product
will
be
360.
which
agr●●ing
with
the
totall
of
all
your
quantities
of
angles
added
together
,
is
one
argument
that
the
plat
will
close
.
Suppose
the
parts
of
the
Hipothenusall
diuisions
cut
,
to
be
4
and
the
line
measured
to
be
30
pearches
,
now
you
are
to
finde
a
number
to
beare
like
proportion
to
30
,
as
100
beareth
to
104
which
you
shall
find
to
be
28
1/5
1/1
,
so
that
the
line
measured
by
the
cheine
to
be
30
pearches
,
must
be
laid
downe
28
1/1
1/3
pearches
in
your
protracting
.
But
for
asmuch
as
these
calculations
be
tedious
in
the
field
,
your
best
way
is
to
note
the
Hipothenusall
parts
cut
,
and
then
reduce
them
when
you
come
home
.
Let
the
different
angles
from
the
Horizon
be
taken
18
degr.
and
the
line
measured
20
perches
,
first
count
18
degrees
in
the
limbe
,
then
thereunto
bring
the
edge
of
the
Index
,
next
count
the
line
measured
viz.
20.
pearches
vpon
the
Index
fromwards
the
center
,
so
shall
you
there
see
the
19
perpendicular
counted
from
the
center
intersect
,
which
sheweth
that
the
line
measured
20
perches
,
must
be
protracted
19.
And
if
the
length
of
the
line
measured
exceed
30
pearches
,
and
be
lesse
then
60
,
then
take
halfe
the
number
vpon
the
Index
,
and
the
perpendicular
will
answere
to
halfe
the
length
of
the
Horizontall
line
,
but
if
the
line
exceed
60
,
take
then
¼
¾
&
c.
&
the
perpendicular
will
answer
proportionally
.
30
,
60
,
90
In
Superuis
.
manerij
praed.
ibid.
fact
.
xiij
.
&
xiiij
.
diebus
Septembris
anno
regni
Dom.
nost.
Iacobi
Dei
gran
Ang.
Scotiae
,
Fran
.
&
Hibern
.
Reg.
fidei
defensor
.
&c.
viz.
Angl.
Franc.
&
Hibern
.
sexto
,
&
Scotiae
xlij
.
Per
B.
G.
gent.
virtute
commissionis
dicti
Domin
.
Reg.
extra
Scaccar
.
suum
sibi
direct
.
continetur
inter
alia
,
vt
sequitur
.
viz.
R.
G.
A.
Hos
iiij
.
acr
l.
acr
xx
.
acr
xiij
.
ac
xij
.
s.
ij
.
d.
iij.
l.
l.
s.
3.
l.
x.
s.
G.
I.
4.
l.
30.
s.
vij
.
l.
iiij
.
l.
iij.
l.
xlij
.
s.
xx
.
s.
xxij
.
s.
xxx
.
s.
x.
s.
3.
d.
xx
.
s.
ix
.
l
iij.
l.
vj.
l.
iij.
l.
To
make
a
plat
or
map
,
and
place
a
Sea-Card
therein
.
THis
Chapter
is
easily
performed
,
if
you
doe
but
call
to
mind
how
to
séeke
the
true
proportion
of
any
field
,
Island
,
or
such
like
,
euen
as
you
be
taught
in
the
Chapter
:
but
indéed
I
hold
this
Chapter
(
for
that
it
is
to
be
performed
only
by
two
stations
)
best
to
be
wrought
sinically
,
so
shall
you
place
and
situate
the
angles
more
truely
then
you
can
by
the
intersection
of
lines
,
for
because
thereby
you
be
taught
to
find
out
the
true
distance
of
euery
angle
from
your
station
;
in
so
much
that
if
you
doe
but
obserue
the
quantitie
of
euery
angle
,
and
againe
protract
the
same
truely
,
you
cannot
erre
any
thing
at
all
:
and
for
that
it
may
be
desired
of
many
,
I
will
not
leaue
for
their
sakes
to
prosecute
the
same
with
an
Example
.
c
d
e
f
g
b
a
a
b
g
,
a
b
f
,
a
b
c
,
a
b
c
,
a
b
d
a
b
b
a
c
,
b
a
d
,
b
a
e
,
b
ag
,
b
a
f
a
b
d
b
a
d
b
d
a.
a
d
b
a
b
d
a
b
a
d
a
d
b
a
b
a
b
d
a
d.
a
d
a
e
,
a
f
a
g
i
k
k
i
k
i
l
b
a
c
k
i
m
i
m
k
i
n
,
k
i
p
,
k
i
o
a
c
i
l
i
l
a
d
i
m
i
n
,
i
p
i
o
i
n
i
o
i
o
l
m
n
o
p.
Geodeticall
Staffe
l
m
n
o
p
l
m
n
,
l
n
o
,
l
o
p
l
o
l
n
p
q
,
o
r
m
r
as
the
square
of
the
small
diuisions
vpon
the
one
side
,
is
to
the
square
of
the
number
of
pearches
,
responding
to
the
same
side
,
so
is
the
superficiall
content
in
the
former
small
diuisions
,
vnto
the
true
superficiall
content
in
pearches
.
The
figure
protracted
in
plano
,
is
l
m
n
o
p
,
Now
viewing
into
what
regular
figures
it
may
best
be
resolued
,
into
which
(
as
you
may
perceiue
by
the
pricked
lines
)
it
is
readily
conuerted
into
three
Tryangles
,
by
two
right
lines
issuing
from
l
to
n
and
o
,
next
I
diuide
some
one
side
as
l
p
into
60
equall
parts
,
and
so
by
that
diuided
line
,
as
by
a
scale
I
measure
how
many
of
those
diuisions
is
in
the
base
l
o
,
and
l
m
,
so
do
I
find
the
one
to
conteine
88
and
the
other
92
of
those
parts
,
next
vpon
those
bases
I
let
fall
perpendiculars
,
which
in
like
manner
I
measure
,
as
p
q
25
,
o
r
59
,
and
m
r
53
,
now
do
I
multiply
25
in
halfe
88
,
so
haue
I
1100
,
then
59
in
halfe
92
,
and
there
is
made
2714
,
and
lastly
53
in
halfe
92
,
so
is
the
product
2438
(
or
since
l
n
was
a
diagonal
,
or
a
base
common
to
both
the
Tryangles
,
you
might
haue
multiplyed
o
r
and
m
r
in
l
n
and
all
had
bene
one
)
the
which
three
aggregated
sums
adde
together
,
so
haue
you
6252
produced
,
the
whole
content
of
l
m
n
o
p
,
by
the
which
multiply
16900
(
the
square
of
130
perches
the
length
of
the
hedge
in
the
Parke
c
g
)
and
the
product
is
105658000
,
the
which
parted
by
3600
the
square
of
60
,
leaueth
the
number
of
perches
viz.
29340
24/6
perches
,
the
iust
content
of
the
plat
which
you
may
reduce
into
acres
by
the
24
Chap.
of
the
Art
of
Geodetia
,
or
if
you
please
,
you
need
not
to
protract
this
figure
at
all
,
but
measure
all
the
diagonals
&
perpendiculars
by
the
3
Axiome
of
Trigonometria
,
as
if
you
would
measure
the
diagonall
l
n
,
here
haue
you
two
sides
knowne
i
l
and
i
n
comprehending
an
angle
knowne
,
therefore
worke
by
the
said
third
.
Axiome
:
the
like
might
you
do
to
l
m
,
m
n
or
l
o
,
&c.
and
then
the
area
is
easily
found
by
the
20
Chapter
of
Geodetia
,
without
regard
of
perpendiculars
.
Hauing
the
3
sides
of
euery
tryangle
,
you
may
lay
them
downe
seuerally
vpon
some
smoth
boords
or
such
like
,
and
so
by
your
scale
and
compasse
,
after
the
common
order
,
find
the
perpendiculars
,
without
regard
to
the
quantity
of
the
angles
,
for
the
3
sides
being
knowne
,
the
true
intersection
lymits
the
right
proportion
.
The
proposed
field
is
c
d
e
f
g
h
,
which
I
obserued
at
two
statitions
a
and
b
and
so
when
I
come
to
protract
,
note
the
intersection
of
matchy
or
like
lines
as
in
such
a
case
I
am
wont
,
but
for
that
I
perceiue
the
section
at
g
and
h
falleth
out
very
acute
,
therefore
(
as
is
said
)
I
espy
a
third
station
,
I
noting
the
angle
i
b
a
and
b
a
i
,
whereby
is
gotten
the
distance
of
i
my
second
station
,
then
do
I
go
vnto
i
,
and
there
againe
obserue
the
angle
b
i
f
and
b
i
g
and
b
i
h
,
or
any
other
angle
that
by
the
section
of
lines
issuing
from
the
two
first
stations
would
proue
acute
,
so
shall
the
lines
issuing
from
i
and
b
make
a
more
perfect
intersection
,
so
that
when
you
come
to
protract
,
you
shall
finde
all
the
acute
sections
corrected
,
for
whereas
the
lines
a
h
,
and
b
h
made
an
acute
intersection
at
h
,
the
section
of
i
h
and
b
h
do
correct
and
reforme
the
same
,
finding
out
the
true
point
of
intersection
,
and
so
of
any
other
:
and
in
your
obseruations
in
the
field
you
may
well
know
which
angles
in
protracting
will
proue
acute
,
for
if
you
adde
the
gles
g
b
a
,
and
g
a
b
,
or
h
b
a
and
b
a
h
together
,
the
totall
taken
from
180
leaueth
the
angle
g
or
h
,
of
whose
acuity
you
may
iudge
.
Thus
by
meanes
of
3
or
4
stations
,
or
more
,
if
occasion
be
,
may
you
finde
the
true
plat
of
any
proposed
plaine
,
by
the
intersection
of
lines
,
neuer
measuring
but
onely
one
line
in
the
whole
worke
,
certainely
no
proposition
performed
by
the
intersection
of
lines
is
better
then
this
,
if
your
stations
be
respondently
taken
and
the
obseruations
truely
made
.
Gem.
Frisius
A
is
a
certaine
Turret
,
whose
distance
is
required
from
b
,
where
I
plant
my
first
staffe
,
departing
thence
a
certaine
distance
Orthogonally
,
as
to
c
163
yards
,
placing
my
second
staffe
at
c
,
d
is
my
third
staffe
placed
backewards
from
b
158
yardes
in
a
right
line
with
a
b
:
e
is
the
fourth
staffe
placed
Orthogonally
from
d
,
and
in
a
right
line
with
a
c
,
which
is
measured
200
yards
,
so
that
you
must
depart
squire-wise
from
d
towards
e
,
vntill
you
be
in
a
right
line
with
c
a
,
as
is
said
:
these
staues
so
set
,
and
their
distances
measured
,
deduct
b
c
163
the
first
number
,
from
de
200
,
the
third
number
,
so
haue
you
37
your
Diuisor
,
then
multiply
the
third
number
200
by
158
the
second
number
,
and
there
is
produced
31600
,
which
parted
by
your
diuidend
37
leaueth
854
2/37
yards
,
the
distance
betwixt
d
and
a
,
whence
take
d
b
158
,
so
haue
you
the
distance
of
a
b
696
2/37
yards
,
making
●96
yards
and
a
little
better
then
2
inches
.
Let
the
distance
of
the
base
of
the
wall
i
g
be
23.
paces
,
whose
square
is
529.
the
altitude
of
the
wall
aboue
your
feet
g
h
be
10.
paces
,
the
square
whereof
is
100.
which
added
to
529.
maketh
629.
the
square
roote
whereof
is
25
4/60
pearches
and
better
,
the
THis
Chapter
is
very
exact
,
so
the
obseruations
be
well
made
;
and
indéed
is
but
wrested
from
the
chapter
of
my
booke
of
the
howbeit
it
is
set
downe
much
after
the
same
methode
I
shall
deliuer
now
,
by
Maister
our
countriman
,
and
others
In
performance
whereof
you
must
haue
any
kinde
of
triangle
prepared
:
which
had
,
set
vp
a
staffe
,
whereunto
apply
your
angle
in
such
sort
that
the
one
of
the
containing
sides
point
directly
vnto
the
marke
or
Castle
vpon
the
left
hand
:
the
triangle
so
resting
,
looke
by
the
containing
side
vpon
the
right
hand
,
causing
one
to
set
vp
a
second
and
third
staffe
in
a
right
line
therewith
,
and
the
further
the
second
staffe
is
distant
from
the
first
,
the
better
it
is
:
the
triangle
yet
remaining
,
as
before
,
at
your
first
staffe
,
make
a
marke
in
the
side
subtending
the
angle
at
your
staffe
,
in
such
sort
that
it
may
bee
in
a
right
line
with
the
said
angle
,
and
the
second
marke
vpon
your
right
hand
;
so
haue
you
performed
all
your
obseruations
at
your
first
staffe
,
onely
as
you
come
thence
measure
the
distance
from
your
first
staffe
to
the
third
staffe
.
Next
repaire
vnto
your
second
staffe
,
where
situate
your
triangle
,
in
all
respects
as
he
was
before
at
the
first
staffe
:
then
must
you
depart
towards
the
desired
distance
in
a
right
line
,
vntil
such
time
that
you
come
direct
betwéene
the
Tower
or
marke
that
was
vpon
your
left
hand
and
your
third
staffe
,
and
there
set
vp
your
fourth
staffe
.
Next
go
vnto
your
second
staffe
,
where
your
triangle
yet
remaines
,
and
looking
from
the
angle
at
your
staffe
by
the
marke
made
before
,
in
the
side
subtending
the
said
angle
,
your
eye
will
direct
you
in
a
right
line
,
which
you
must
continue
on
,
vntill
you
bee
also
in
a
right
line
with
the
second
and
third
staffe
,
&
there
set
vp
your
first
staffe
.
This
so
ordered
,
measure
the
distance
betwixt
the
second
and
third
staffe
,
the
quantity
of
which
measure
reserue
for
your
diuisor
.
Lastly
increase
the
distance
betwéene
your
first
and
third
staffe
in
the
distance
betwixt
the
fourth
&
fift
staffe
,
the
product
whereof
part
by
your
reserued
diuisor
,
so
doth
the
quotient
yéeld
you
the
desired
distance
,
as
Geometrically
might
be
proued
.
c
e
d
e
c
e
f
g
d
e
a
b.
Triangular
aequiangula
habent
latera
circa
aequales
angulos
proportionalia
,
&
contra
,
Eucl.
4.
p.
6.
Ramus
.
lib.
7.
pag.
9.
b
c
a
b
c
d
e
a
c
d
a
e
b
d
b
e.
And
as
in
this
chapter
,
so
likewise
in
most
other
like
conclusions
may
you
so
order
your
obseruations
that
you
may
auoyde
diuision
,
according
as
in
the
end
of
the
84.
Chapter
.
If
you
desire
the
distance
b
e
,
multiply
the
distance
of
a
e
,
by
the
distance
of
your
third
and
fourth
staffe
,
parting
the
product
by
the
second
and
fourth
staffe
,
as
by
the
distance
of
c
d
,
so
doth
the
quotient
yeeld
the
distance
b
e.
And
for
that
your
Geodeticall
Staffe
will
take
or
deliuer
any
angle
,
or
represent
any
tryangle
,
hee
may
aptly
performe
this
Chapter
according
vnto
this
kind
of
methode
.
Let
vs
suppose
we
saw
two
Ships
,
the
one
in
chase
with
the
other
,
and
we
were
required
to
know
whether
the
Ship
in
pursuit
did
win
any
thing
of
the
Ship
pursued
,
and
if
he
did
,
when
the
Ship
chased
should
be
ouertaken
,
first
therefore
I
take
their
distance
,
which
I
will
suppose
400
yards
,
so
resting
a
quarter
of
an
houre
(
which
is
15
minuts
)
I
obserue
their
distance
againe
which
I
find
300
yards
,
lastly
I
subtract
300
from
400
,
the
remainder
is
100
,
therefore
I
multiply
400
by
15
,
the
product
is
6000
which
parted
by
100
leaueth
60
minutes
,
the
time
how
long
it
will
bee
before
the
Ship
pursued
shall
beouertaken
by
the
Ship
pursuing
,
which
is
one
houre
.
In
the
like
manner
may
you
deale
with
Ships
or
Nauies
approaching
towards
any
porte
,
hauen
or
such
like
.
Geodeticall
Staffe
Topographicall
Glasse
Topographicall
Glasse
Q.
Scaeuola
Aristotle
2.
Meteor
,
cap.
6.
Magirus
,
l.
6.
c.
9
d.
13.
Cape
Misenum
C.
Martius
Iean
Liebault
Nose
bleed
,
Veruaine
,
wild
peniroyall
,
Venus
haire
,
Cammomill
,
Dogs
tooth
,
foxtaile
,
trifoly
,
Cinkefoile
,
Millefoile
,
Coliander
,
Geodeticall
Staffe
I
Had
thought
to
haue
concluded
this
Booke
which
the
diuers
nature
of
grounds
,
with
the
artificiall
mending
of
them
,
and
the
killing
of
Gorse
that
aboundeth
vpon
cold
clayie
ground
,
and
Ferne
in
a
sandy
and
hote
soyle
,
or
broome
flourishing
in
barren
ground
,
hote
and
drye
,
or
mosse
spreding
in
a
cold
ground
,
all
these
,
&
more
I
thought
once
to
haue
handled
,
but
my
minde
altering
,
and
withall
bethinking
my selfe
of
a
Gentleman
of
worship
,
and
my
kinsman
,
one
Maister
Whorde
,
that
had
bestowed
great
cost
and
trauell
in
dreyning
of
grounds
,
hauing
now
(
as
he
told
me
)
brought
(
to
his
great
cost
)
the
dreyning
of
grounds
to
such
a
perfect
head
,
and
easie
methode
that
he
was
able
to
trebble
the
ranknesse
of
his
ground
onely
by
waters
,
with
the
one
quarter
of
the
charges
it
was
in
the
beginning
:
and
indéede
as
the
inuention
is
new
and
excellent
,
though
it
hath
béene
stumbled
at
by
many
,
so
no
doubt
but
in
time
hee
will
bee
perswaded
to
publish
to
the
world
(
for
the
good
of
his
country
,
and
credit
of
himselfe
)
the
forme
and
methode
of
the
worke
,
to
which
it
would
doe
well
to
adde
what
I
intended
to
haue
spoken
of
in
this
place
which
made
me
the
rather
referre
it
to
a
Gentleman
,
so
honest
,
and
well
experienced
.
But
to
the
matter
.
Wheras
there
be
many
annueties
which
some
desire
to
buy
,
and
some
to
sell
,
behold
the
insuing
Table
,
which
telleth
you
what
pound
anuety
is
worth
for
any
time
vnder
yeares
,
according
to
pound
in
the
hundred
,
and
if
you
would
apply
the
same
vnto
any
other
summe
more
then
pound
,
vse
the
rule
of
proportion
,
as
I
taught
you
in
the
booke
of
my
which
made
me
repeate
it
here
againe
,
because
the
Table
is
wanting
there
.
Thus
much
(
good
Reader
)
haue
I
said
of
the
vse
of
my
not
doubting
,
ioyning
this
booke
and
the
together
,
but
that
you
haue
the
ample
vse
of
all
Instruments
,
now
extant
,
whereby
you
shall
not
be
to
séeke
further
instructions
,
I
meane
such
instruments
that
are
now
in
most
request
,
and
by
the
better
opinions
most
commended
,
and
therefore
for
this
time
,
I
will
conclude
my
Booke
,
wishing
my selfe
present
euer
to
explaine
any
thing
that
to
the
yong
Practitioner
shall
séeme
obscure
.
SENECA .
Nullum est tam magnum beneficium , quod non vilificare malignitas possit : nullum tam angustum , quod non bonus interpres extendat .
The
end
of
the
Topographicall
Glasse
.
A
b
is
a
growing
tree
,
whose
square
is
required
,
that
is
,
how
much
square
that
tree
will
beare
on
euery
side
when
he
is
squared
:
first
therefore
I
gird
the
tree
about
with
a
gut-string
as
farre
from
the
ground
as
I
can
reach
,
as
at
r
,
then
doe
I
note
the
length
of
the
line
that
girded
the
tree
,
as
c
d
,
which
I
diuide
into
4
equall
parts
,
reseruing
one
of
the
said
4
parts
,
as
c
e
,
then
doe
I
take
the
legges
of
my
staffe
,
and
fit
the
length
of
c
e
ouer
in
110
and
1●0
parts
amongst
the
equall
diuisions
:
the
legges
resting
there
,
I
take
the
distance
ouer
from
70
to
70
;
the
length
of
which
line
I
keepe
,
as
l
m
,
and
turning
the
other
side
of
the
legges
vpward
I
place
l
m
ouer
in
60
and
60
,
and
so
taking
the
distance
ouer
from
90
to
90
(
the
legges
beeing
not
stirred
)
you
haue
the
line
n
o
the
one
side
of
the
square
that
the
tree
will
beare
,
which
applying
to
your
ruler
diuided
into
feete
and
inches
,
you
shall
find
2
●/5
foot
the
iust
square
,
which
is
equall
to
f
h
i
g
the
4
side
of
the
tree
beeing
squared
.
c
d
d
b
A
b
c
d
is
the
end
of
a
peece
of
timber
,
c
d
the
longer
side
,
d
b
the
shorter
,
therefore
I
take
the
length
of
c
d
and
d
b
and
ioynt
them
together
in
v
,
making
one
right
line
thereof
,
as
r
s
:
next
I
part
r
s
into
two
equall
parts
at
w
,
then
placing
the
one
foote
of
my
compasse
in
w
,
extending
the
other
to
s
or
r
t
,
describe
the
semicircle
r
t
s
:
lastly
,
vpon
v
where
the
two
lines
were
ioyned
together
raise
a
perpendicular
v
t
,
which
is
equal
vnto
p
q
,
I
conclude
the
square
made
of
the
line
t
v
is
equall
to
c
d
b
a
,
and
thus
of
any
such
other
.
Geodeticall
Staffe
,
fol.
142.
Or
the
length
of
the
perpendicular
b
m
is
the
square
falling
vpon
b
c
at
right
angles
.
44
Chap.
Metamorphosis
7.
Geodeticall
Staffe
b
d
a
c
a
b
c
h
i.
A
Solid
or
Cubicall
foote
of
tymber
doth
containe
cubicall
inches
,
for
so
many
foure
square
inches
may
be
taken
out
of
one
cubicall
foot
,
I
meane
such
Inches
that
are
square
euery
way
like
vnto
a
dye
,
now
hauing
the
square
of
any
péece
of
tymber
giuen
,
square
the
same
,
diuiding
by
the
product
,
so
doth
the
quotient
shew
you
how
much
of
the
length
of
the
tymber
must
be
taken
to
make
a
square
foote
,
by
the
which
diuide
the
whole
length
or
altitude
of
the
tymber
,
so
doth
the
quotient
acquaint
you
how
many
foote
of
tymber
is
in
the
péece
.
a
b
a
b
THis
kind
of
measure
taketh
no
regard
to
the
square
feete
in
the
tymber
,
but
vnto
the
solid
capacity
thereof
,
but
for
that
it
is
not
much
pertinent
to
the
requiring
rather
numerall
then
instrumentall
operation
I
will
be
the
more
briefe
.
When
you
haue
any
péece
of
tymber
,
stone
,
pillar
or
such
like
,
whose
solid
content
is
required
by
the
rules
taught
in
the
sixt
book
,
part
of
my
seeke
the
superficiall
contents
of
the
end
of
the
tymber
,
the
which
augment
in
the
altitude
or
length
of
the
same
,
so
is
the
product
your
desire
.
Euclide
27
D.
11
,
&
29
,
D.
11
,
&c.
tetraedrons
,
hexaedrons
cubes
,
Octaedrons
,
Dodecaedrons
E.
l.
12.
p.
11.
E.
l.
12.
p.
11.
R.
l.
vlt.
pa.
7.
E.
l.
12.
p.
5.
and
6.
6
,
8
E.
l.
12.
p.
7.
E.
l.
1.
p.
29
,
30
,
&
31.
1
RAmus
l.
20.
p.
5.
4
:
3
R.
l.
26.
p.
5.2
5
Prop.
2
E.
li.
12.
p.
18.
SENECA .
Plus caeteris dedit , quia sine spe recipiendi dedit .