THE
FIRST
PART
OF
REGVLAR
FORTIFICATION
,
DEMONSTRATED
BY
THE
DECEASED
SAMVEL
MAROLOIS
,
And
Reviewed
,
corrected
&
augmented
BY
ALBERT
GIRARD
,
SANMIELOIS
.
GOd
having
found
good
to
create
Man
,
did
with
all
prepare
his
affection
for
societie
,
that
by
this
meanes
he
might
the
longer
preserue
him
,
and
so
make
him
become
more
stronge
&
able
.
According
to
which
&
to
the
same
end
,
there
were
first
built
howses
,
and
lodgings
,
to
keepe
men
from
the
injurie
of
the
aire
,
as
windes
,
stormes
,
raine
,
excessiue
heate
,
and
colde
,
as
the
situation
of
their
dwellings
,
and
the
commodiousnesse
of
those
places
did
afford
.
And
as
by
divine
blessing
,
and
their
owne
industrie
,
they
attayned
unto
many
Earthly
riches
,
as
famillies
,
cattell
,
and
other
goods
:
so
by
the
instinct
of
nature
,
they
sought
alwaies
to
strengthen
&
defend
themselues
with
the
best
fortifications
they
could
,
that
thereby
they
might
not
onely
preserue
their
bodies
against
the
violence
of
their
Adversaires
;
but
also
their
goods
,
kindred
,
and
Associates
.
To
this
end
(
as
we
may
obserue
as
well
in
ancient
histories
,
as
also
at
this
day
in
some
East
and
West
Indian
Nations
)
they
droue
in
piles
of
timber
,
and
made
Empalements
closse
one
to
an
other
,
that
enclossing
themselues
so
narrowly
together
,
they
might
be
the
better
able
to
defend
and
resist
the
furie
of
their
Ennemies
.
But
in
processe
of
time
,
the
defects
of
such
enclossure
being
well
marked
,
they
began
afterwards
to
make
their
Fortifications
of
a
more
solid
,
firme
,
and
durable
matter
,
as
stone
,
brick
,
earth
and
such
like
.
And
though
experience
taught
them
,
that
these
their
Empalements
of
wood
,
were
not
able
to
●esist
fyre
,
&
diuerse
other
accidents
,
which
might
befall
them
:
yet
it
is
certaine
,
they
used
these
and
the
like
fortifications
,
till
they
found
the
harme
&
conveniencie
,
which
they
receiued
by
them
.
But
necessitie
,
the
Inventrix
of
Arts
,
found
that
walls
built
with
stones
and
bricks
were
better
.
These
were
in
use
and
traine
,
till
such
time
also
,
as
their
imperfections
were
knowne
,
and
that
by
the
force
of
man
,
and
Engines
they
were
diuerse
wayes
attempted
;
yea
many
times
beaten
downe
&
overthrowne
:
which
made
them
seeke
from
time
to
time
to
remedie
their
defects
:
for
perceiving
that
their
walls
were
heretofore
subject
to
be
beaten
downe
(
by
Rames
)
and
other
ancient
Engines
;
they
made
them
(
upon
the
right
line
of
the
wall
)
Spures
,
and
roundles
to
hinder
the
beating
of
them
downe
,
and
to
preserue
them
the
better
from
a
breach
,
and
tumbling
downe
,
they
gaue
them
the
Talude
,
that
is
,
a
slooping
towards
the
inside
of
the
place
,
that
they
might
be
the
more
able
to
resist
the
violence
&
force
of
these
Engines
.
Finally
these
walls
were
made
after
diverse
manners
:
first
in
a
round
forme
,
which
,
as
on
the
one
side
,
it
was
held
the
strongest
;
because
these
Engines
beating
against
the
wall
crushed
the
stones
,
&
made
them
stick
the
closser
together
,
in
regard
the
circle
exteriour
was
greater
then
the
interiour
,
and
that
the
Engines
could
not
unfasten
,
loose
and
breake
them
but
with
great
difficultie
.
So
on
the
other
side
,
it
was
impossible
to
defend
such
walls
,
when
approches
was
made
unto
them
;
because
no
part
of
them
could
be
discouvered
,
or
flancked
:
so
that
afterward
they
built
them
in
a
square
forme
,
with
small
squares
in
their
angles
,
for
the
defense
of
their
Curtaines
.
Then
made
them
also
Demycircles
with
angles
interiour
,
and
exteriour
,
and
lastly
triangular
wise
,
on
the
one
side
to
resist
the
force
of
the
furious
Canon
,
used
at
this
day
;
and
on
the
other
side
,
that
they
might
be
the
more
capable
to
defend
themselues
by
discovering
euery
part
of
the
wall
,
euen
to
the
very
foundation
.
And
as
invention
of
attempting
is
growne
to
the
highest
degree
in
these
dayes
,
by
reason
of
the
longe
experience
of
the
warrs
in
these
parts
(
which
is
the
Schole
of
all
military
actions
)
and
the
abilitie
,
and
capacitie
of
the
Assaillants
&
Defendants
:
So
am
I
of
the
opinion
,
that
the
Fortifications
made
in
these
Low
Countryes
,
are
the
strongest
,
exactest
,
&
perfectest
,
which
can
be
invented
,
and
which
haue
bene
made
and
practized
,
not
by
a
simple
Generall
,
but
by
one
of
the
greatest
Captaines
of
the
World
,
endowed
with
a
singular
courage
and
spirit
,
an
excellent
Mathematician
,
and
is
not
onely
a
Prince
of
a
great
howse
;
but
also
experienced
&
beaten
in
all
militaire
actions
,
and
stratagems
;
aswell
offensiue
,
as
defensiue
:
of
which
Fortification
,
we
haue
undertaken
to
treate
briefly
at
this
present
,
and
as
succinctly
as
possibly
may
be
.
Of
the
definitions
.
I.
Forasmuch
then
as
the
definitions
of
Fortification
are
by
the
dayly
use
of
armes
growne
so
common
:
it
were
in
vaine
for
me
(
in
my
opinion
)
to
make
any
further
explication
thereof
:
yet
to
satisfie
the
ignorant
,
we
will
marke
out
the
angles
,
and
sides
of
a
Fortresse
,
by
the
Letters
of
the
Alphabeth
,
and
opposite
to
the
said
letters
,
ye
shall
finde
their
names
,
and
appellations
,
as
we
may
note
by
the
figures
1.
and
2
,
following
.
Icnographie
,
or
ground-markeing
.
1.
Figure
.
-
N.
O.
The
side
of
the
Polygone
,
that
is
many
angles
.
-
N.
D.
The
line
of
the
gorge
,
-
D.
C.
The
line
of
the
flanke
,
-
B.
N.
The
Capitall
line
.
-
B.
C.
Q.
R.
The
Moate
.
-
P.
The
Raveline
,
or
halfe
mone
.
-
Q.
S.
The
covert
way
.
-
T.
S.
The
Parapett
thereof
.
-
B.
I.
The
line
of
defence
.
-
D.
K.
The
Curtaine
.
-
K.
F.
The
Parapett
.
-
K.
M.
The
Rampart
.
-
A.
N.
The
semy-diameter
.
-
V.
C.
The
flanke
lengthned
.
-
C.
N.
D.
The
angle
forming
the
flank
.
-
B.
C.
D.
The
angle
of
the
shoulder
.
Orthographie
,
or
the
Profile
.
2.
Figure
.
-
A.
B.
The
foote
or
basis
of
the
Rampart
.
-
G.
H.
The
heigth
of
the
Rampart
.
-
H.
B.
The
talud
(
or
slooping
)
of
the
infiel
of
the
Rampart
.
-
A.
Y.
The
Talud
on
the
outside
of
the
Rampart
,
or
scharfe
.
-
Z.
D.
The
foote
of
the
Parapett
.
-
Z.
E.
The
Parapett
itselfe
.
-
D.
F.
The
foote-banke
.
-
F.
G.
The
Terra-plaine
or
bredth
of
the
Rampart
.
-
K.
A.
The
way
for
the
round
,
or
the
falsebray
.
-
I.
K.
The
foote
banke
thereof
.
-
I.
M.
The
Parapett
of
the
falsebray
.
-
M.
N.
The
Scharfe
.
-
P.
O.
N.
M.
The
moate
.
-
P.
O.
The
Counterscharfe
.
-
P.
Q.
The
Covert
way
.
-
R.
Q.
The
foote
banke
thereof
.
-
T.
S.
R.
The
Parapett
of
the
Covert
way
.
The
other
names
,
which
haue
neede
of
explanation
shal
be
declared
in
their
due
places
.
Before
we
come
to
instruct
yow
particularly
in
the
Art
of
Fortification
;
we
will
briefely
treate
of
the
calculation
thereof
:
In
which
supputation
,
ye
shall
haue
first
sett
downe
the
knowne
termes
,
and
under
them
the
disposition
of
the
Characters
or
Letters
:
beginning
with
a
square
fortresse
with
foure
angles
,
or
Bulwarks
,
and
proceede
on
to
a
Dodecagone
,
a
fortresse
with
twelue
angles
,
or
Bulwarks
,
makeing
upon
every
Polygone
,
3
or
4
Trialls
,
that
afterwards
one
may
choose
the
best
of
them
:
and
because
the
angles
will
not
be
much
altered
by
the
diversitie
of
the
trials
:
I
haue
thought
good
to
giue
this
generall
rule
for
them
following
.
It
is
a
thing
generally
received
of
all
men
,
that
a
square
fortresse
with
foure
Bulwarks
,
is
not
so
good
as
a
Pentagonall
with
fiue
angles
,
nor
a
Pentagonall
so
stronge
as
an
Exagonall
with
sixe
,
and
so
consequently
of
the
rest
.
If
the
cause
thereof
be
sought
out
;
one
may
obserue
that
this
proceedes
from
the
smallnesse
of
their
angles
;
as
not
being
able
to
beare
such
a
body
of
a
Bastion
,
as
the
subsequent
Polygones
:
so
that
a
square
fortresse
for
this
reason
,
wil
be
more
defective
,
then
the
Pentagonall
,
and
this
lesse
defensive
then
the
Hexagonall
,
and
so
well
the
rest
following
even
to
a
Dodecagone
,
which
hath
the
angle
of
the
Bastion
right
,
which
is
the
cause
,
that
constraines
one
to
make
the
angles
flanked
lesser
,
then
the
reason
of
building
well
doth
require
,
that
is
,
the
flanks
too
litle
,
the
Gorge
too
narrow
,
and
the
line
of
defence
too
longe
.
To
encrease
then
proportionally
the
angles
of
Fortresses
,
according
as
the
angle
of
their
Polygone
augmenteth
;
we
will
take
then
the
halfe
of
their
angles
,
and
adding
thereunto
15
degrees
,
the
summe
wil
be
the
angle
of
the
Bulwarke
,
which
we
terme
the
angle-flanked
,
and
if
the
angle-flanked
be
substracted
from
the
angle
of
the
Polygone
,
there
will
remayne
the
double
of
the
angle
flanking
interiour
,
which
double
being
substracted
from
180
degrees
,
`then
will
remayne
the
angle
flanking
exteriour
,
called
the
Tenaille
,
and
if
ye
add
to
the
angle
flanking
interiour
90
degrees
,
then
the
summe
wil
be
the
angle
of
the
Shoulder
.
To
finde
out
the
angle
of
the
Polygone
from
the
number
of
its
substract
,
namely
2
,
the
remaynder
must
be
multiplyed
by
2.
and
the
product
wil
be
the
number
of
the
right
angles
,
which
such
a
Polygone
containeth
,
as
ye
maye
see
by
this
exemple
following
.
Or
thus
.
And
by
the
same
rule
ye
shall
finde
the
angles
of
the
Subsequent
Polygones
,
beginning
from
a
square
Fortresse
to
a
Dodecagone
.
|
4.
|
5.
|
6.
|
7.
|
8.
|
9.
|
10.
|
11.
|
12
|
|
90.
|
72.
|
60.
|
51
3/7.
|
45.
|
40.
|
36.
|
32
8/●●
.
|
30
the
angl
.
of
the
center
.
|
|
90.
|
108.
|
120.
|
128
4/7.
|
135.
|
140.
|
144.
|
147
●/11.
|
150
the
angle
of
the
Polig
.
|
|
45.
|
54.
|
60.
|
64
2/7.
|
67
1/2.
|
70.
|
72.
|
73
7/11.
|
75
The
halfe
.
|
|
15.
|
15.
|
15.
|
15.
|
15.
|
15.
|
15.
|
15.
|
15.
|
|
Sum.
|
60.
|
69.
|
75.
|
79
2/7.
|
82
1/2.
|
85.
|
87.
|
88
7/11.
|
90
The
angle
flanked
.
|
|
Remaines
30.
|
39.
|
45.
|
49
2/7.
|
52
1/2.
|
55.
|
57.
|
58
7/11.
|
60
the
double
of
the
ang.
|
|
180.
|
180.
|
180.
|
180.
|
180.
|
180.
|
180.
|
180
|
180
The
flanke
interiour
.
|
|
150.
|
141.
|
135.
|
130
5/7.
|
127
1/2.
|
125.
|
123.
|
121
4/11.
|
120
The
flank
exteriour
.
|
|
15.
|
19
1/2.
|
22
1/2.
|
24
9/14.
|
26
1/4.
|
27
1/2.
|
28
1/2.
|
29
7/22.
|
30
The
flank
interiour
.
|
|
90.
|
90.
|
90.
|
90.
|
90.
|
90.
|
90.
|
90.
|
90
|
|
105.
|
109
1/2.
|
112
1/2.
|
114
9/14.
|
116
1/4.
|
117
1/2.
|
118
1/2.
|
119
1/22.
|
120
the
ang.
of
the
should
.
|
And
seing
the
angle
flanked
of
a
Dodecagone
is
right
,
and
able
to
resist
a
batterie
,
which
is
also
made
alwayes
with
right
angles
to
shake
the
more
the
face
of
the
Bulwarke
;
one
must
fortifie
the
Polygones
,
which
are
aboue
the
Dodecagone
,
with
a
right
angle
,
that
the
line
of
defence
may
come
the
more
into
the
curtaine
,
that
one
may
giue
the
more
fire
upon
it
:
but
the
Polygones
,
which
are
under
a
Dodecagone
,
must
be
fortified
,
according
to
the
precedent
Table
,
and
the
calculation
thereof
shal
be
made
hereafter
.
Some
times
we
augment
,
aswell
the
angles
of
the
Bulwarkes
,
as
the
Octogone
,
a
Fortresse
with
8
angles
or
Bulwarks
,
with
a
right
angle
,
and
those
aboue
are
alwayes
right
,
&
those
under
diminishing
to
the
square
fortresse
(
which
hath
the
angle
of
its
Bulwarke
onely
of
60
degrees
.
)
According
to
which
the
Bulwarks
are
somewhat
larger
,
and
the
gorges
and
flanke
greater
then
the
former
,
but
the
second
flanks
lesser
.
Now
for
the
finding
out
of
every
angle
,
ye
must
doe
this
following
,
where
ye
may
observe
,
that
in
the
manner
aboue
sayd
,
the
angles
flanking
interiour
are
the
fourth
part
of
the
angle
flanked
,
or
the
1/●
of
the
angle
of
the
Polygone
.
|
IIII.
|
V.
|
VI.
|
VII
.
|
VIII
|
the
angle
|
|
90.
|
108.
|
120.
|
128
4/7.
|
135.
|
the
angle
|
of
the
Polygone
.
|
|
60.
|
72.
|
80.
|
85
5/7.
|
90.
|
the
angle
|
Flanked
.
|
|
90.
|
72.
|
60.
|
51
3/7.
|
45.
|
the
angle
|
of
the
center
added
thereunto
.
|
|
150.
|
144.
|
140.
|
137
1/7.
|
135.
|
the
angle
|
Flanking
exteriour
.
|
|
30.
|
36.
|
40.
|
42
●/7.
|
45.
|
the
angle
|
double
of
the
angl
.
flank
.
interiour
.
|
|
15.
|
18.
|
20.
|
21
3/7.
|
22
●/2.
|
the
angle
|
Flanking
interiour
,
|
|
90.
|
90.
|
90.
|
90.
|
90.
|
the
angle
|
which
is
the
flanke
alwayes
|
|
105.
|
108.
|
110.
|
111
3/7.
|
112
1/2.
|
the
angle
|
of
the
shoulder
.
|
In
the
same
manner
also
may
be
made
right
the
angle
flanked
of
the
Decagone
,
a
Fortresse
of
ten
Bulwarks
,
where
ye
must
note
also
,
that
before
wee
proceede
further
,
we
will
make
use
in
the
supputation
following
of
the
tenths
or
decinall
numbers
;
which
though
it
gives
some
imperfection
:
yet
seing
the
things
,
which
we
omit
therein
are
of
no
great
consequence
,
it
were
ridiculous
to
make
any
further
search
thereof
;
considering
likewise
that
the
tables
of
Sines
,
tangents
,
and
secants
,
are
one
and
the
same
;
I
thought
it
good
therfore
to
make
use
of
these
following
.
THE
I.
QUESTION
.
The
1.
Figure
,
&
1.
Plate
.
Let
there
be
made
upon
a
Square
a
fortification
of
foure
Bulwarkes
,
so
that
the
line
of
the
Gorge
be
7
parts
:
DI
the
curtaine
21
,
and
IF
,
which
is
the
flanke
of
5
,
and
from
the
angle
of
the
flanke
is
drawne
the
line
of
defence
by
the
angle
of
the
Shoulder
,
to
giue
it
a
face
.
The
question
is
how
many
the
angles
will
be
,
and
every
line
of
the
same
fortresse
,
when
as
the
line
of
defence
will
take
up
600
feete
.
Now
the
length
of
a
foote
is
sett
downe
in
the
25
modell
of
Geometrie
,
noted
by
1
,
and
is
divided
into
12
ynches
,
&
the
ynch
into
10
equall
parts
,
and
is
the
same
foote
,
whereof
12
makes
a
rodd
,
which
his
Excell
,
useth
in
all
his
Fortifications
.
ALBERT
GIRARD
.
THe
Authour
hath
here
aboue
so
disposed
his
calculations
,
that
in
stead
of
explaining
them
briefly
,
he
confounds
them
teadiously
,
as
if
heretofore
there
had
bene
noe
certaine
rule
sett
downe
in
writing
,
to
calculate
lines
and
angles
as
is
ordinarily
done
by
the
Trigometrie
of
plaine
Triangles
,
although
there
haue
bene
a
number
of
Authours
,
which
haue
treated
of
them
,
th'
one
after
one
manner
,
th'
other
after
an
other
;
and
the
most
part
of
them
commixed
with
longe
discourses
,
which
moved
me
not
longe
since
,
to
putt
into
light
some
tables
of
Sines
in
a
portable
volume
,
with
the
most
succinct
method
that
possibly
could
,
touching
the
supputation
of
such
plaine
triangles
,
reducing
them
into
foure
diverse
cases
,
where
I
did
insert
in
their
places
,
some
of
my
owne
inventions
to
the
purpose
,
knowne
as
I
suppose
to
none
others
heretofore
;
so
that
it
cannot
be
,
but
having
3
knowne
termes
,
in
a
triangle
,
but
that
one
maye
knowe
the
other
three
,
or
one
of
them
onely
,
which
one
desires
,
as
the
Reader
may
knowe
it
;
the
manner
&
order
thereof
being
much
more
facile
,
&
easier
to
conceiue
,
then
the
reading
of
our
Authour
in
his
former
editions
,
being
obscure
,
troublesome
,
and
hard
to
be
attayned
unto
.
For
this
reason
the
Learners
of
this
science
,
are
required
to
be
forewarned
before
they
come
to
the
reading
of
this
booke
;
&
are
advertized
hereby
,
that
when
it
is
sayd
,
a
triangle
hath
three
termes
,
he
must
understand
knowne
,
or
given
,
and
that
when
I
say
that
a
triangle
right-angle
hath
three
termes
,
that
then
I
am
bound
but
to
shew
two
of
them
;
seing
that
this
word
(
right-angle
)
presupposeth
that
the
triangle
hath
a
right
angle
,
to
weet
of
90
degrees
:
moreover
,
that
touching
this
present
question
,
these
7
,
21
,
5
parts
,
demonstrate
the
reason
of
the
lines
ND
,
DI
,
IF
,
and
not
the
quantitie
of
the
same
in
feete
,
as
is
necessary
to
finde
in
the
manner
following
.
To
finde
out
the
Angles
.
IN
the
triangle
right
angle
CDI
,
the
side
CD
,
is
to
DI
,
as
5
to
21
,
(
for
CD
being
of
5
parts
;
then
DI
wil
be
the
21
thereof
)
therefore
the
angle
DIC
,
wil
be
found
for
13
:
24
it
s
double
CLD
26
:
48
,
and
its
adjunct
CLF
flanking
exteriour
,
wil
be
of
153
:
12.
Now
in
the
figure
quadrilatere
,
or
of
foure-sides
,
ABLH
;
the
angle
exteriour
BLH
is
alwayes
equall
to
the
3
interiours
B
,
A
&
H
;
now
A
90
,
the
angle
of
the
center
,
being
taken
from
BLH
,
153
:
12
,
there
will
remaine
the
two
Demy-flankes
ABL
,
&
LHA
63
:
12.
for
one
entire
flank
ZBC
:
&
seing
that
the
angle
of
the
Shoulder
BCD
,
is
exteriour
in
the
triangle
CDI
,
it
wil
be
equall
to
the
two
interiours
D.
90
.
and
1.
13
;
24
,
and
therefore
the
Shoulder
C
,
wil
be
103.
24.
To
finde
out
the
Lengths
.
IN
a
triangle
ambligone
BNI
,
the
side
BI
is
600
feete
,
&
the
angles
are
found
,
therefore
the
other
termes
shall
be
found
,
to
wit
BN
the
Capitall
line
196
,
64
,
&
NI
,
444
,
62
,
but
ND
is
the
third
of
DI
(
for
ND
being
7
,
DI
is
21
by
the
Hypotheses
)
therefore
the
fourth
part
of
the
found
NI
,
wil
be
ND
,
111
,
15
the
Gorge
,
&
the
rest
wil
be
333
,
46
,
for
DI
,
the
Curtaine
,
now
DI
to
DC
is
as
21
,
to
5
:
therefore
one
may
say
,
if
21
giues
me
5
,
how
many
DI
333
,
46
,
giue
me
?
facit
for
CD
79
,
40.
&
so
CI
wil
be
343
,
84
,
which
taken
from
BI
600
,
there
will
remaine
BC
257
,
16
,
for
the
face
.
Finally
,
the
triangle
right-angle
VDH
hath
three
termes
,
DH
600
,
and
the
angle
H
equall
to
HDI.
13
:
24
(
because
of
the
parallels
DI
;
VH
)
then
yow
shall
know
VD
139
,
05
,
&
VH
583
,
67
,
from
which
side
substract
VP
(
equall
to
Demy
DI
)
166
,
73
,
there
will
remaine
PH
416
,
94
,
whereof
the
double
wil
be
for
BH
833
,
88
,
and
PA
wil
be
416
,
94
,
for
it
is
equall
to
PH
,
and
thus
must
ye
doe
with
the
rest
following
.
The
3.
Figure
,
1.
Plate
.
Let
there
be
a
square
Fortresse
,
whereof
the
curtaine
DI
being
4
parts
;
the
flanck
CD
shall
haue
1
of
them
,
and
also
the
Gorge
1
:
The
defence
running
from
the
angle
of
the
flanke
,
formeth
the
face
,
&
the
length
BI
is
of
600
feete
,
one
requires
the
greatnesse
of
the
other
lines
of
the
same
Fortresse
.
To
finde
out
the
Angles
.
THe
triangle
right-angle
CDI
hath
three
termes
,
as
the
reason
of
the
sides
CD
,
1
,
to
DI
,
4
,
therefore
the
flancking
interiour
CDI
wil
be
14
degrees
&
2
minutes
,
whereunto
add
D
90
degrees
,
then
the
Shoulder
BCD
wil
be
104
:
2
;
now
if
ye
double
the
angle
CID
,
it
wil
be
28
:
4
,
CMD
,
and
its
adjunct
CMF
,
151
:
56
for
the
angle
flancking
exteriour
,
from
which
substract
the
angle
of
the
center
A
90
:
there
will
remaine
61
:
56
for
the
angle
flancked
entire
B
,
and
thus
much
for
the
angles
.
For
the
sides
.
THe
triangle
BNI
,
hath
three
termes
,
to
witt
,
BI
600
feete
,
and
the
angles
B
,
30
:
58.
&
N
,
135.
I
,
14.
2
,
so
one
shall
finde
the
lines
BN
205
,
76
&
NI
436
,
59
;
but
in
regard
that
ND
to
DI
is
as
1
,
to
4
,
according
to
the
Hypotese
,
then
ND
will
be
the
fifth
of
NI
,
where
will
follow
that
ND
wil
be
87
,
32
(
and
DC
as
much
)
and
DI
the
rest
349
,
27.
Now
seing
that
in
the
triangle
right-angle
CDI
,
the
sides
CD
,
DI
are
notified
,
one
shall
finde
the
Hypotenuse
CI
of
360
,
which
taken
from
BI
600.
there
remaines
the
face
BC
240.
Finally
,
in
the
triangle
right-angle
VDH
,
which
hath
3
termes
DH
600
,
the
angle
H
,
equall
to
CID
14.2
,
ye
shall
haue
VD
145
,
49
:
also
VH
582.
09
,
from
which
take
VP
174
,
64.
(
equall
to
the
demy-curtaine
DI
)
there
will
remaine
PH
407
,
45
,
also
PA
its
equall
:
then
in
the
triangle
right-angle
APH
,
ye
shall
finde
AH
576
,
22
from
whence
take
BN
or
HO
,
the
capitall
line
205
,
76
,
there
will
remaine
AO
370
,
46
,
&
in
takeing
VD
,
from
AP
,
there
will
remaine
the
perpendicular
of
the
Center
,
A
upon
the
middest
of
the
curtaine
261
,
96.
Now
BH
wil
be
114
,
90.
as
being
the
double
of
PH.
4.
Figure
.
&
1.
Plate
.
In
this
square
Fortresse
,
the
defence
BI
is
600
feete
,
and
the
angle
flanked
60
degrees
,
whereof
DBC
is
the
fourth
part
,
which
is
15
degrees
,
the
question
is
what
the
quantity
of
the
parts
of
such
a
Fortresse
wil
be
?
For
the
angles
.
SEing
that
the
line
flanked
is
60
,
then
in
the
triangle
BNI
the
angle
B
wil
be
30
,
&
N
135
,
(
the
adjunct
of
ANI
45
:
)
whereof
the
angle
remayning
BIN
wil
be
15
degrees
:
therefore
the
triangle
BDI
wil
be
Isosceles
,
that
is
,
two
sides
a
like
,
such
that
DBI
is
aswell
15
;
as
BID
;
The
Shoulder
C
wil
be
105
,
and
in
the
triangle
DMI
,
the
angles
upon
the
basis
are
each
of
them
15
degrees
,
the
remaynder
M
will
be
then
150
for
the
flanking
exteriour
BMH.
For
the
sides
.
THe
triangle
Isoceles
BDI
hath
three
termes
,
the
defence
BI
600
feete
,
and
the
pointed
angles
every
of
them
15
degrees
.
Therefore
DI
the
Curtaine
wil
be
310
,
584.
Also
the
triangle
CDI
,
hath
three
termes
,
the
angle
D
right
;
I
15.
and
the
curtaine
DI
,
then
CD
the
flanke
83
,
217
,
&
CI
321
,
535
,
which
being
taken
from
BI
600
,
there
will
remaine
BC
the
face
278
,
465.
Moreover
in
the
triangle
IBN
,
the
angles
B
30
,
I
,
15
,
&
the
defence
600
:
so
then
BN
the
Capitall
line
wil
be
219
,
623
,
&
NI
424.
268.
from
whence
DI
being
taken
there
will
remaine
the
gorge
ND
,
113.
684.
Moreover
,
the
triangle
BIL
having
three
termes
,
L
right
,
B
equall
to
CID
15.
and
BI
600
,
then
I
L
wil
be
155
,
292
,
&
BL
579
,
558
,
and
takeing
from
it
PL
155.
292.
(
which
is
the
halfe
of
the
curtaine
)
there
will
remaine
BP
424
,
266
for
AP
;
also
it
s
double
being
BH
848
,
532
,
one
may
also
easely
knowe
BA
600.
The
5.
Figure
.
&
2.
Plate
.
ALBERT
GIRARD
.
Heitherto
of
the
defence
drawne
from
the
angle
of
the
flanke
,
but
in
those
following
,
there
is
a
second
flank
,
and
the
distance
of
the
angle
of
the
flank
,
euen
to
the
angle
flanked
,
which
is
called
Fichant
.
The
5
figure
is
the
dessigne
or
draught
of
a
square
Fortresse
,
whereof
the
line
of
defence
fichant
DH
maketh
600
feete
,
the
angle
flanked
60
degrees
,
the
line
HB
is
divided
into
7
equall
parts
;
whereof
the
one
which
is
betweene
the
Characters
1.
2.
is
subdiuided
into
5
equall
parts
,
and
from
the
center
B
,
is
made
the
Arch
,
4
N
cutting
the
capitall
line
at
N
,
from
which
point
is
drawne
NDZ
parallel
to
BH
,
and
from
the
point
V
(
which
is
in
the
character
2
)
the
perpendicular
VD
,
the
question
is
how
many
these
lines
,
and
the
angles
of
such
a
Fortresse
,
will
make
?
For
the
lines
in
parts
indetermined
.
FOrasmuch
as
BH
contayneth
35
such
parts
as
BN
9
;
BV
10
,
VH
25
,
ye
shall
haue
in
the
triangle
right-angle
BTN
termes
enough
to
knowe
BT
,
or
TN
6
,
364
,
and
TV
or
,
ND
3
,
636
:
Also
the
triangle
right-angle
DVH
,
had
three
termes
,
VD
equall
to
TN
,
6364
,
&
VH
25
,
then
DII
wil
be
25
,
797
parts
or
the
same
DH
is
600
foote
.
We
must
then
calculate
according
to
this
reason
,
the
lines
aboue
mentioned
to
bring
them
into
feete
,
saying
,
as
followeth
.
For
the
lines
in
parts
determined
,
to
wit
,
brought
into
feete
.
THe
25
,
797
parts
,
make
600
foote
,
how
many
then
will
BH
35.
come
to
?
There
wil
be
for
BH
814
,
04
foote
,
likewise
VL
,
or
DZ
the
curtaine
may
be
made
soe
;
BH
7
parts
make
814
,
04
foote
,
how
many
VL
3
?
it
will
come
for
the
curtaine
to
348
,
87
,
&
for
BN
the
Capitall
line
ye
shall
say
,
if
35
giues
814
,
04
:
how
many
9
?
ye
shall
haue
for
BN
209.32
:
and
so
of
the
other
:
for
TN
,
or
BT
,
or
VD
wil
be
148.
02
,
&
TV
,
or
the
Gorge
ND
84
57.
and
seing
that
BH
is
knowne
,
also
VL
the
halfe
of
the
remainder
wil
be
BV
232
,
58.
In
the
triangle
right-angle
BVC
,
the
angle
B
wil
be
15
degrees
:
(
for
PBO
is
45
,
&
CBO
30
:
)
and
BV
being
knowne
,
therefore
BC
the
face
wil
be
240
,
78
,
and
VC
62
,
32
,
which
being
taken
from
VD
,
there
will
remaine
CD
the
flanck
85.
7
;
and
therfore
the
triangle
right-angle
CDI
,
shall
haue
3
termes
,
the
angle
C
75
,
&
CD
85
,
7.
soe
DI
wil
be
319
,
84
,
CI
331
,
15
,
likewise
BI
flancking
571
,
93.
&
I
,
Z
29.
03
for
the
second
flancke
.
Now
OP
is
knowne
as
being
the
halfe
of
BH
.
Now
of
the
Pentagons
,
or
Fortresses
with
fiue
angles
or
Bulwarks
.
6.
Figure
.
&
2.
Plate
.
In
the
Pentagone
,
a
Fortresse
with
fiue
Bulwarks
,
KFBDL
,
let
the
line
,
KL
be
63
rods
,
&
divided
into
7
equall
parts
,
whereof
the
Capitall
line
KA
is
2
of
them
,
also
LE
,
and
from
the
Character
2
(
or
G
)
let
GB
be
the
perpendicular
.
Also
the
angle
flancked
is
69
degrees
,
according
to
the
precedent
table
,
how
many
then
wil
be
the
lines
and
the
angles
of
such
a
Fortresse
?
ALB.
GIRARD
.
Before
wee
come
to
the
Construction
hereof
,
take
notice
,
that
KA
in
this
figure
is
not
answerable
to
the
length
of
KG
,
yet
ye
must
suppose
them
to
be
equall
,
aswell
as
the
cipher
1
,
and
the
letter
H
,
are
two
differing
points
,
which
would
haue
bene
the
better
discerned
,
if
the
figure
had
bin
well
made
,
which
may
serue
as
a
fore-warning
for
some
figures
following
,
upon
which
one
ought
not
to
stand
so
much
as
upon
the
Suppositions
,
or
Hypoteses
of
the
Propositions
.
The
triangle
right-angle
KHA
hath
three
termes
,
KA
18
rodd
,
the
angle
AKH
54
degrees
,
which
being
of
KH
wil
be
10
,
5802
,
&
HA
or
GB
14
,
5623.
Now
if
ye
take
away
KH
from
KG
,
there
will
remaine
for
AB
the
Gorge
,
7.
4198.
Item
the
curtaine
BD
wil
be
27
rodd
,
being
equall
to
GM
.
And
because
the
angle
flancked
is
69
degrees
,
if
ye
add
thereunto
the
angle
of
the
Center
,
which
is
72
,
ye
shall
haue
the
angle
flancking
exteriour
,
141
degrees
,
the
halfe
of
its
adjunct
for
the
interiour
19
1/2
degrees
.
The
triangle
right-angle
KGF
,
hauing
3
termes
,
KG
18
,
GKF
19
1/2
:
the
face
KF
wil
be
19
,
0953
,
and
FG
6.
3741
,
which
being
taken
from
GB
14
,
5623.
there
will
remayne
FB
(
the
flanck
)
〈…〉
,
then
the
triangle
right-angle
FBI
may
be
knowne
,
seing
that
FB
is
found
,
and
the
angle
〈…〉
1/2
the
complement
of
1
;
therefore
BI
wil
be
23.1227
,
which
taken
from
BD
27
,
there
will
remayne
ID
for
the
second
flanck
3
,
8773
,
also
FI
wil
be
found
to
be
24
,
5297
,
whereunto
add
KF
19.
0953
,
and
ye
shall
haue
KI
,
the
defence
flancking
43.
6250
,
finally
the
triangle
right-angle
LGB
,
hauing
three
termes
,
to
wit
LG
45
rodd
,
and
GB
14
,
5623
;
one
may
easely
know
BL
,
and
the
distance
of
the
Center
of
the
Fortresse
by
K
,
because
that
KP
is
31
1/2
rodd
,
and
the
angle
AKP
54
degrees
.
The
7.
Figure
,
&
2.
Plate
.
In
a
Fortresse
Pentagonall
,
BOV
,
let
AB
the
deffence
flancking
be
50
rodd
,
the
Flanck
ED
9
rodd
,
and
the
angle
of
the
Bulwarke
72
degrees
.
How
many
then
will
the
other
parts
of
the
same
make
,
when
the
second
flanck
AG
makes
fiue
rodd
.
THe
angle
of
the
Polygone
is
108
,
the
halfe
is
54
for
FBQ
,
to
which
add
CBF
36
degrees
by
the
Hypoteses
,
you
shall
haue
90
for
CBQ
,
but
CF
is
paralell
to
BQ
therefore
BCA
right
,
the
triangle
BCA
wil
be
then
the
right-angle
,
and
BA
of
50
,
and
the
angle
B
of
72
degrees
,
then
ye
shall
finde
BC
,
or
QD
to
be
of
15
,
451
,
and
CA
47
,
553
,
and
the
angle
A
18
degrees
.
Also
in
the
triangle
right-angle
EDA
,
the
angle
A
is
giuen
,
and
the
side
ED
9
rodd
,
by
the
Hypotese
,
then
EA
wil
be
29
,
12463
,
&
DA
27
,
69912
;
now
if
ye
take
EA
from
DA
,
there
will
remaine
BE
the
face
20
,
87537
,
and
if
ye
substract
DA
from
AC
,
there
will
remaine
CD
,
or
BQ
,
19.
854.
Also
the
triangle
right-angle
BCF
,
hath
3
termes
,
BC
and
the
angles
;
in
fine
BF
the
capitall
line
wil
be
19
,
097
&
CF
,
11
,
226
,
which
being
taken
from
CD
,
there
will
remaine
FD
8
,
628
,
the
Gorge
;
now
if
to
DA
,
ye
add
AG
5
rodd
,
the
second
flanck
,
ye
shall
haue
the
curtaine
DG
32.
69912
,
to
which
add
twise
BQ
,
and
ye
shall
haue
BV
72
,
40711
its
half
BK
36
,
20355
,
and
so
the
triangle
right-angle
BKO
,
hauing
3
termes
,
the
angle
KBO
wil
be
54
degrees
,
ye
shall
finde
BO
61
,
593099
,
and
KO
49
,
82984.
Finally
VB
,
BQ
being
knowne
,
then
the
triangle
right
angle
DQV
,
hauing
3
termes
giuen
VQ
,
&
QD
;
ye
shall
finde
the
fichant
DV
easely
.
ALB.
GIRARD
.
IF
one
will
consider
the
difference
which
is
betweene
this
operation
,
and
that
of
our
Authour
published
in
his
former
Editions
,
he
shall
finde
that
this
question
is
defectiue
,
for
I
haue
added
thereunto
(
that
the
second
flanke
AG
is
rodd
)
seing
that
in
all
such
questions
,
where
there
is
a
second
flank
,
he
ought
to
haue
set
downe
5
knowne
termes
&
neither
more
nor
lesse
,
without
the
one
depending
on
the
other
,
as
in
this
present
,
the
name
of
the
Fortresse
,
to
wit
,
Pentagonall
is
a
terme
,
in
the
second
place
there
is
the
defence
;
then
the
flanck
,
the
angle
flancked
,
and
the
second
flancke
,
which
are
fiue
termes
:
where
ye
must
note
,
that
when
there
are
reasons
in
the
proposition
,
and
that
though
a
reason
hath
two
numbers
,
neverthelesse
it
is
but
one
terme
;
but
where
there
is
no
second
flanck
(
as
in
the
figures
of
the
first
plate
,
then
4
termes
will
suffice
.
)
Finally
,
my
Authour
had
so
ordained
his
Supputation
,
that
in
stead
of
makeing
an
addition
,
or
a
substraction
;
he
made
the
rules
of
three
very
great
,
so
that
ye
must
imagine
it
was
easier
for
me
to
change
all
,
then
to
recorrect
it
,
hauing
no
other
respect
,
but
to
the
explication
of
the
figures
,
and
as
much
as
is
lawfull
and
possible
to
shew
his
intention
,
which
I
am
bound
to
doe
:
Moreouer
,
if
peradventure
same
one
should
finde
this
manner
of
operation
strange
,
which
I
〈…〉
heitherto
,
it
is
requisite
for
him
to
know
also
(
be
it
spoken
under
●●●●ection
)
that
he
understandeth
not
much
,
if
he
doth
not
practize
it
soe
himselfe
in
other
Subjects
,
yea
though
I
had
not
sett
downe
one
onely
number
;
which
I
should
haue
done
already
heretofore
,
if
I
had
not
had
regard
to
the
obscuritie
,
which
students
might
pretend
to
finde
,
which
hauing
cleared
heitherto
,
ye
may
goe
on
with
the
more
assurance
in
the
rest
following
:
and
must
also
know
,
that
the
8
figure
was
left
without
explication
by
the
Authour
.
The
8.
9.
10
Figure
.
&
3.
Plate
.
In
this
figure
Pentagonall
,
the
angle
ELO
is
divided
into
2
equall
parts
by
LD
,
the
flanck
FB
9
rodd
,
the
angle
flancked
69
degrees
,
and
the
curtaine
30
rodd
,
how
many
then
will
the
other
lines
&
angles
be
?
THe
angle
of
the
Polygone
is
parted
in
the
midst
by
KA
,
and
the
angle
AKF
is
knowne
,
also
the
rest
shal
be
FKG
,
or
its
equall
FB
,
whence
followes
that
the
triangle
right-angle
FBI
shall
haue
3
termes
FB
,
and
the
angles
;
therefore
the
other
3
termes
wil
be
knowne
,
also
the
whole
line
DB
,
and
the
part
BI
,
ergo
the
rest
ID
is
the
second
flanke
.
The
triangle
KBI
hath
three
termes
,
the
angle
found
,
BI
&
K
(
the
fourth
part
of
the
flanck
by
consequence
the
3
termes
remayning
are
knowne
,
and
the
difference
of
the
found
lines
KI
,
IF
wil
be
for
the
face
KF
;
which
makes
that
in
the
triangle
right-angle
KGF
wil
be
knowne
KG
.
GF
;
and
if
to
KG
doubled
,
ye
add
the
curtaine
,
ye
shall
haue
KL
;
also
KG
to
the
curtaine
is
for
GL
,
or
XB
if
to
GF
ye
add
FB
,
ye
shall
haue
LX
,
and
therefore
in
the
triangle
right-angle
LB
wil
be
knowne
;
seing
that
LX
,
XB
are
found
.
Moreouer
,
in
the
triangle
right-angle
LXE
,
the
angle
E
is
a
demy
Polygone
,
and
the
line
LX
being
knowne
,
ye
shall
haue
LE
the
Capitall
line
,
and
XE
taken
from
XD
(
equall
to
KG
)
there
will
remayne
DE
the
Gorge
.
-
BI
25
,
4151
-
FI
26
,
9617
-
ID
4
,
5848
-
KI
51
,
2795
-
KF
24
,
3178
-
KG
22
,
9229
-
GF
8
,
1175
-
KL
75
,
8459
-
XB
52
,
9233
-
LX
17
,
1175
-
LB
55
,
6227
-
LE
21
,
1583
-
XE
12
,
4366
-
DE
10
,
4863
The
9.
Figure
,
which
is
the
8
,
9.
10.
&
3
Plate
.
In
the
former
figure
Pentagonall
,
there
are
other
Hypoteses
BL
,
fichant
60
rodd
,
the
angle
DLM
36
degrees
,
and
45
minutes
,
DM
17
rodd
,
and
the
angle
OLE
parted
in
the
midst
by
LM
.
THe
angle
MLE
is
54
:
from
which
take
DLM
36
,
45
;
there
will
remaine
17
,
15
;
for
DLE
its
quadruple
or
fourefould
for
the
angle
flancked
69
:
ye
must
first
calculate
the
traingles
right-angles
MDL
,
QEL
,
whereby
is
found
DE
the
Gorge
,
LE
the
Capitall
line
,
and
ML
in
the
triangle
right-angle
LGB
;
the
line
of
defence
LB
600
,
and
GB
17
;
therefore
ye
shall
finde
GL
,
which
added
to
KG
equall
to
ML
,
ye
shall
haue
〈…〉
take
ML
,
from
it
,
there
will
remaine
GM
for
the
curtaine
BD
,
in
the
triangle
MLO
,
ye
shall
find
the
face
OL
,
&
MO
,
which
taken
from
17
,
there
remaines
OD
for
the
flanck
,
&
the
triangle
right-angle
ODC
may
be
knowne
,
because
the
angle
C
,
is
equall
to
MLO
19
1/2
,
then
CO
&
CD
being
notified
,
ye
shall
find
CL
&
BC
,
and
from
the
Center
of
the
Fortresse
wil
be
knowne
the
distances
towards
K
,
P
,
A.
-
DE
10
,
41454
-
LE
21
,
01319
-
ML
22
,
76572
-
GL
57
,
54128
-
KL
80
,
30700
-
BD
34
,
77556
-
OL
24
,
15101
-
MO
8
,
06179
-
OD
8
,
93821
-
OC
26
,
77657
-
CD
25
,
24075
-
CL
50
,
92758
-
CB
9
,
53481
-
A
(
†
)
47
,
30000
The
10.
Figure
,
which
is
the
8
,
9.
10.
In
the
figure
Pentagonall
,
let
LB
fichant
be
60
rodd
,
and
LC
the
defence
flancking
50
,
92758
fifth-parts
.
Also
BC
the
second
flancke
9
,
53481
,
and
the
angle
OLE
parted
in
the
midst
by
LD
.
THe
triangle
BCL
hauing
the
three
sides
giuen
,
then
ye
shall
finde
the
angle
C
,
or
the
lengthning
of
the
basis
BC
unto
the
perpendicular
X
,
as
followeth
here
,
the
basis
BC
,
9
,
53481
giueth
me
the
Summe
of
both
the
other
sides
,
110
,
92758
,
how
much
then
will
their
difference
9
,
07242
giue
?
it
will
come
to
105
,
54816
,
from
which
take
the
basis
(
because
that
the
perpendicular
falls
without
,
which
is
seene
,
when
the
sayd
quotient
is
more
then
the
basis
)
there
will
remaine
one
number
,
whereof
the
halfe
is
CX
48
,
00668
,
this
being
done
ye
must
calculate
the
triangle
right-angle
CLX
,
by
which
ye
shall
know
the
angle
C
,
or
LMO
,
which
taken
from
LME
,
the
remainder
wil
be
OLE
,
34
1/2
,
and
the
angle
flancked
wil
be
the
double
of
it
69
;
also
ye
shall
haue
LX
,
or
MD
,
and
then
shall
ye
calculate
the
triangle
MDL
(
for
the
angle
MLD
is
knowne
)
seing
it
is
equall
to
MLO
19
1/2
,
and
OLD
17
1/2
:
Also
the
triangle
LQE
,
whereby
ye
shall
finde
the
Gorge
,
the
Capitall
line
,
&
CD
,
then
by
the
triangle
ODC
is
found
the
flanke
,
&
after
that
the
face
:
&
if
to
CX
aboue
sayd
,
ye
put
BC
being
giuen
,
there
will
come
out
BX
,
or
GL
,
to
which
add
ML
by
addition
,
then
by
substraction
ye
shall
haue
KL
,
and
the
curtaine
BD
,
whereof
the
numbers
agree
with
those
of
the
precedent
question
.
The
11.
Figure
.
3.
Plate
.
In
this
figure
Hexagone
,
that
is
,
a
Fortresse
with
sixe
angles
or
Bulwarks
,
let
the
Curtaine
be
sesquialtere
,
or
halfe
as
much
againe
to
the
face
,
&
the
face
to
the
flancke
doubled-sesquialtere
,
the
angle
flancked
75
degrees
,
and
DP
distant
from
the
points
70
rodd
.
Wee
must
finde
out
the
rest
.
SEsquialtere
,
or
one
and
a
halfe
,
is
as
3
in
respect
of
2
;
twise
sesquialtere
is
as
5
to
2
:
so
that
BH
,
DC
,
CB
wil
be
as
15
,
10
,
4.
whereunto
if
yee
add
4
cifphers
to
each
of
them
,
DC
wil
be
100000
sines
of
the
right-angle
.
Now
ADF
is
60
,
and
ADC
37
1/2
,
then
CDF
wil
be
22
1/2
degrees
;
therefore
CF
38268
,
and
DF
92388
also
KP
;
but
BH
,
or
FK
is
150000
,
then
DB
wil
be
334776
parts
,
which
make
70
rodd
by
the
Hypoteses
,
wee
will
bring
then
these
lines
into
Rodds
,
according
to
this
reason
following
;
because
334776
parts
make
70
Rodd
.
How
many
partes
.
they
will
come
to
.
-
150000
31
,
364
BH
-
100000
20
,
909
DC
-
40000
8
,
364
CB
-
38268
8
,
002
CE
-
92388
19
,
318
DF
And
seing
that
PD
,
DF
are
knowne
,
FP
wil
be
also
;
and
by
FC
,
CB
,
wil
be
knowne
FB
,
or
AE
,
16.
366
,
now
by
the
47
pro.
1
of
Euclide
BP
fichant
wil
be
53
,
259
:
And
if
one
calculates
the
triangles
CBG
,
DEA
,
ye
shall
finde
DG
,
42
,
765
;
and
DE
,
9
,
448
,
ergo
EF
,
or
AB
the
Gorge
wil
be
9
,
87
;
the
distance
from
the
Center
to
D
,
is
equall
to
DP
,
70
rodd
.
The
12.
Figure
.
&
4.
Plate
.
Let
DP
be
72
rod
,
the
angles
of
the
Bulwarks
are
72
degrees
,
the
curtaine
BH
32
rod
,
the
flancks
of
8
,
3638
,
fourth
of
rodds
,
how
many
then
wil
be
the
other
parts
of
this
Fortresse
Hexagonall
?
Seing
that
DP
is
72
,
&
BH
,
or
FK
32
,
then
the
halfe
of
the
rest
wil
be
20
for
DF
,
and
also
the
angle
ABF
being
60
degrees
,
and
ADC
37
1/2
,
then
CDF
wil
be
22
1/2
degrees
:
whereby
ye
shall
know
then
the
triangle
right-angle
CDF
.
Secondly
,
FC
found
,
with
CB
giuen
,
will
make
knowne
FB
,
but
PF
is
52
,
then
the
triangle
right-angle
BFP
hath
the
sides
BF
,
FP
knowne
,
&
BP
wil
be
knowne
in
the
triangles
right-angles
ADE
,
CBG
,
their
termes
will
suffice
to
make
the
rest
to
be
understood
.
when
ye
haue
DA
,
its
halfe
wil
be
for
DE
,
which
taken
from
DF
,
there
will
remaine
AB
.
-
DG
21
,
64780
-
FB
16
,
64800
-
BP
54
,
60000
-
AD
19
,
22344
-
DE
9
,
61172
-
AB
10
,
38828
-
GH
11
,
80804
-
DG
43
,
50349
The
13.
Figure
.
In
this
Figure
Hexagonall
,
the
second
flanke
is
in
the
flanke
as
6
to
7
,
the
flanke
hath
its
gorge
as
7
to
10
,
the
Gorge
to
the
line
of
the
Polygone
,
as
2
to
9.
the
question
is
how
many
they
,
&
the
other
dimensions
wil
be
,
when
as
DH
makes
60
rodd
.
IF
ye
make
GH
6
,
then
HO
wil
be
7
,
&
HI
10
,
the
Curtaine
wil
be
25
;
Then
ye
shall
calculate
the
triangles
CBG
,
DAG
by
which
ye
haue
the
angle
G
,
or
his
equall
CDF
,
&
the
face
D●
and
then
in
the
triangle
DCF
,
the
lines
DF
,
FC
,
FB
,
FP
,
and
finally
BF
44
,
93555
,
which
make
〈◊〉
▪
for
which
reason
one
must
reduce
the
parts
into
rodds
:
but
this
reason
may
be
sett
downe
more
easie
,
in
takeing
but
7
,
48926
,
which
make
1000000
parts
;
Now
if
ye
had
rather
worke
it
by
multiplication
,
without
division
,
then
ye
must
take
but
100000
parts
,
which
make
13352
rodd
,
and
so
ye
shall
finde
-
DC
25
,
38302
-
CB
9
,
34672
-
BA
13
,
35246
-
DP
81
,
01484
-
BH
33
,
38115
-
The
angle
flancked
79.
33.
The
14.
Figure
,
the
1.
of
the
two
.
In
this
Fortresse
Hexagonall
,
let
EH
be
60
rodd
,
and
EI
,
BC
,
KH
equall
,
the
flanke
KC
8
rodd
,
it
behoues
us
to
finde
out
the
rest
.
BY
the
help
of
the
rule
of
Algeber
,
ye
shall
finde
that
the
angle
EIF
wil
be
69
degrees
,
4
1/2
minuts
;
then
in
the
triangle
IBC
,
the
angle
I
,
&
the
side
IB
8
rodd
being
giuen
,
ye
shall
finde
BC
20
,
9229
for
the
face
EI
;
also
IC
wil
be
22
,
4002
,
then
EC
will
make
43
,
3231
,
the
rest
is
easie
,
without
the
second
flanck
.
The
15.
Figure
.
This
figure
hath
for
the
flanke
10
rodd
;
the
rest
being
like
to
the
former
they
shal
be
put
in
order
in
the
Tables
following
.
The
16.
Figure
.
&
5.
Plate
.
The
flanke
of
this
Heptagone
,
that
is
,
A
Fortresse
with
seuen
angles
,
or
Bulwarks
,
maketh
10
rodd
,
the
distance
of
the
points
72
rodd
,
and
the
angle
flanked
80
degrees
,
how
many
then
will
the
rest
make
,
when
as
the
second
flanke
is
10
rodd
.
SEing
that
the
angle
of
the
Bulwarke
makes
80
degrees
,
and
the
angle
of
the
Polygone
128
:
34
,
17
the
difference
divided
in
the
halfe
wil
be
the
angle
flanking
interiour
24
,
17
,
18
1/2
,
the
flanck
CB
10
rodd
,
whence
the
triangle
CBG
wil
be
knowne
,
&
by
adding
GH
giuen
to
BG
,
ye
shall
haue
the
curtaine
32
,
1623
,
which
taken
from
DP
72
rod
,
and
then
take
the
halfe
of
the
rest
,
ye
shall
finde
DF
,
and
also
DC
.
CF
,
for
the
face
DC
21
,
8525
,
&
DG
46
,
1666
,
also
in
the
triangle
DAE
;
ye
shall
haue
DE
,
&
consequently
EF
,
or
AB
.
Moreouer
ye
shall
finde
DH
in
the
triangle
DKH
,
to
be
55
,
4345.
The
17.
Figure
,
&
5.
Plate
.
This
Heptagone
hath
the
flanke
of
9
rodd
,
the
angle
flanked
79
degrees
25.
43
,
DP
●2
rodd
,
and
the
choice
of
the
second
flanke
.
ALB.
GIRARD
.
HAuing
calculated
the
angles
as
in
the
former
.
OM
wil
be
21
,
6436
,
and
HM
,
19
6836
,
let
us
make
the
second
flanke
7
,
3164
,
and
then
the
curtaine
wil
be
26
;
DF
,
23
;
the
face
OP
25
,
2901.
MP
46
,
9337.
Marolois
had
giuen
here
80
degrees
,
but
tooke
no
more
then
79
,
25
,
43
,
for
the
angle
flanked
,
which
he
did
without
premeditation
.
The
18.
Figure
.
&
5.
Plate
.
In
this
present
Heptagone
the
angle
flanked
is
79
degrees
,
the
gorge
12
rodd
,
&
the
curtaine
32
rodd
,
how
much
will
the
other
lines
&
angles
make
of
this
Fortresse
.
TO
resolue
this
question
,
ye
must
suppute
from
the
triangle
CBG
,
the
lines
CG
,
GB
,
and
ye
shall
haue
GB
the
second
flanke
;
now
if
to
BG
,
ye
add
AB
,
ye
shall
haue
AG
,
then
the
triangle
DAG
wil
be
knowne
;
afterward
the
triangle
DAE
,
and
finally
the
triangles
DCF
,
and
DHK
.
ALB.
GIRARD
.
NOte
that
in
this
Question
Marolois
had
sett
downe
the
angle
flanked
79
and
2
seuenths
,
yet
did
not
followe
this
number
,
but
79
degrees
:
the
same
errour
also
was
committed
in
his
supputation
of
the
former
,
and
in
the
14
figure
.
One
cannot
gesse
well
his
Supposition
:
neuerthelesse
that
needes
not
to
stay
the
reader
;
for
I
haue
sett
downe
the
question
,
as
it
seemes
he
would
haue
propounded
it
,
but
the
worst
is
,
he
made
2
figures
,
and
one
cannot
understand
well
of
which
of
either
of
them
he
would
speake
,
but
we
will
speake
more
thereof
hereafter
;
howsoeuer
those
which
are
most
intricate
(
aswell
by
reason
of
the
faults
escaped
in
the
impression
of
the
former
Editions
,
as
by
the
errour
of
his
disciples
,
which
did
calculate
them
)
shal
be
partly
omitted
,
and
shall
giue
no
impediment
,
but
that
the
rest
may
giue
contentment
to
the
Readers
,
for
that
which
they
are
desirous
to
finde
out
in
this
booke
:
(
in
the
20
figure
following
the
Letters
ye
were
sent
unto
were
wholly
repugnant
.
)
The
19.
Figure
.
In
this
Heptagone
let
the
angle
flanked
be
80
degrees
,
&
the
angle
from
the
Capitall
line
,
&
from
the
imagined
DB
(
to
wit
ADB
)
let
be
22
½
degrees
,
DP
82
rodd
,
&
the
flank
CB
10
rodd
.
FRom
the
angle
of
the
Polygone
128
,
34
,
17
,
take
the
flanked
80
,
the
halfe
of
theremainder
wil
be
24
,
17
,
9
:
for
the
angle
flanking
interiour
DGA
:
beginn
then
the
triangles
CBG
,
DBG
.
(
whereof
the
angle
in
D
maketh
17
½
)
ye
shall
finde
BG
22
,
1621
,
CG
24
,
3138
;
&
BD
30
,
3118
,
then
GD
wil
be
49
,
1096
,
and
DC
the
face
24
,
7958
,
afterward
the
triangle
BDF
,
the
line
BF
wil
be
found
to
be
20
,
1982
,
and
DF
22
,
6014
,
whereof
the
double
taken
from
DP
82
,
there
will
remaine
for
the
curtaine
BH
36
,
7972
,
and
consequently
for
the
second
flanke
GH
,
14
,
6350.
Now
knowing
the
triangle
DEA
,
ye
shall
finde
the
Capitall
,
also
the
gorge
AB
12
,
8744.
If
in
the
triangle
DKH
,
yee
seeke
DH
,
then
ye
shall
finde
it
to
be
62
,
7388.
Now
the
reason
of
the
Demy-diameter
or
middle
line
in
the
side
of
this
Heptagone
,
is
as
1000000000
,
to
867767478.
The
20.
Figure
.
This
present
Octogone
,
that
is
a
Fortresse
of
8
angles
or
Bulwarks
,
hath
the
distance
DP
divided
into
7
equall
parts
,
whereof
DF
,
&
FB
make
each
of
them
2
,
&
BP
fichant
is
60
rod
,
the
angle
flanked
82
½
degrees
.
THe
line
BF
being
2
,
and
FP
5
,
then
the
square
of
BP
wil
be
29
,
and
seing
that
BP
is
60
rodd
,
its
square
wil
be
3600
rod
,
by
which
reason
ye
may
knowe
one
part
of
the
7
of
DP
,
saying
,
if
29
giues
me
3600
,
how
many
will
1
giue
me
(
the
square
of
an
other
part
?
)
it
will
come
to
124
,
1379310345
,
whereof
the
square
roote
substracted
wil
be
11
,
14172
,
and
its
double
22
,
28344
for
DF
,
or
FB
,
the
triple
is
33
,
42516
for
the
curtaine
BH
,
and
the
seventhfould
77
,
99204
for
DP
.
Moreouer
,
the
angle
flanking
interiour
,
or
its
equall
FDC
wil
be
26
,
15
;
so
that
DC
wil
be
24
,
84581
,
also
FC
10
,
98907
,
the
flanke
then
wil
be
11
,
29437
(
for
FB
was
knowne
)
in
the
triangle
right-angle
DRG
,
the
angle
D
is
also
26.
15
;
and
RC
equall
to
FB
,
then
DG
the
line
of
defence
flanking
,
wil
be
50
,
38218
;
finally
,
the
Capitall
wil
be
found
24
,
11937
,
and
the
gorge
AB
13
,
05342
,
if
ye
calculate
the
triangle
DEA
.
The
21.
Figure
.
In
this
Octogone
let
the
face
be
24
rod
,
the
flanke
12
,
&
the
curtaine
36
rod
,
&
the
angle
flanked
,
right
,
it
is
required
how
many
the
other
lines
&
angles
wil
be
?
The
angles
ADF
67
½
,
&
ADC
45
,
will
make
knowne
CDF
to
be
22
½
Now
DC
is
24
rod
,
by
consequence
DF
,
FC
,
DP
,
FB
wil
be
knowne
;
also
seing
that
FB
is
found
(
or
EA
)
ye
haue
also
the
triangle
DEA
,
by
which
ye
shall
finde
DA
,
AB
,
&
shall
finde
DG
in
the
triangle
CBG
:
for
hauing
CG
,
ye
add
unto
it
DC
)
finally
DH
in
the
triangle
DKH
.
-
DP
80
,
34624
-
DG
55
,
35756
-
DH
61
,
91032
-
GH
7
,
02948.
-
AB
13
,
39837.
-
AI
62
,
39837.
The
22.
Figure
.
In
this
present
Octogone
,
the
flanke
is
11
rod
,
the
angle
flanked
82
½
degrees
,
the
line
DP
76
rod
,
&
the
second
flanke
to
be
choosen
,
how
many
then
wil
be
the
other
parts
of
this
Fortresse
?
THe
angle
of
the
Octogone
is
knowne
,
and
the
angle
flanked
,
also
their
halfe
ADF
,
ADC
,
therefore
the
rest
,
and
the
triangle
CBG
wil
be
knowne
,
and
BG
wil
be
found
22
,
3058
,
suppose
the
second
flanke
GH
is
9
,
6942
,
then
BH
32
rod
,
likewise
as
much
for
FK
;
but
DP
is
76
,
therefore
DF
makes
22
,
and
after
ye
haue
calculated
the
triangles
DCF
,
DAE
,
ye
shall
finde
DC
,
24
,
52978
,
DG
49
,
40045
,
AB
12
,
94981
,
and
finally
in
the
triangle
DKH
the
fichant
DH
,
58
,
25283.
The
23.
Figure
.
Let
the
angle
flanked
be
82
½
of
this
present
Octogone
,
the
line
of
defence
DG
50
rod
,
DP
76
,
and
CAB
the
halfe
of
the
angle
flanked
.
AFter
ye
haue
carryed
AC
,
which
was
forgetten
,
if
ye
count
the
triangles
DAG
,
ACG
,
ye
shall
finde
the
Capitall
,
the
face
,
and
the
gorge
,
and
BG
,
then
the
triangles
CBG
,
DCF
,
afterward
ye
shall
haue
CB
,
CF
,
DF
,
and
therfore
FK
,
or
the
curtaine
,
and
the
second
flanke
,
then
DH
,
58
,
3498.
-
The
Capitall
23
,
93655
-
The
Gorge
12
,
84323
-
The
Flanke
11
,
26362
-
The
second
flanke
9
,
15324
-
The
face
24
,
53340
-
The
Curtaine
31
,
99346
The
24.
Figure
.
Let
there
be
an
Enneagone
,
A
fortresse
with
nine
Angles
or
Bulwarks
,
(
that
is
with
9
sides
)
whereof
the
angle
flanked
is
85
degrees
,
the
face
24
rod
,
the
flanke
12
,
and
the
curtaine
36
rod.
THe
angle
of
the
Polygone
is
140
degrees
,
seeke
the
triangles
,
DCF
,
CBG
,
DEA
,
DKH
,
and
then
ye
shall
finde
the
lines
GH
12
,
94824
:
DP
78
,
57648
:
AB
12
,
88709.
The
25.
Figure
.
As
in
the
tables
aboue
let
the
angle
flanked
of
the
Enneagone
be
85
degrees
,
the
fichant
60
rod
,
and
DP
being
7
parts
,
then
let
DF
be
two
of
them
,
afterward
FB
the
perpendicular
2
,
for
the
makeing
of
the
flanke
CB
,
wee
require
the
rest
.
DK
being
5
parts
,
and
KH
2.
the
square
DH
wil
be
29.
which
makes
3600
,
whereof
1
makes
124
,
13793103
for
the
square
of
one
part
,
its
root
substracted
wil
be
11
,
14172
for
one
part
,
whereof
the
double
wil
be
for
DF
,
or
FB
,
and
the
triple
for
FK
,
or
the
curtaine
33
,
42516
BH
,
and
after
ye
haue
calculated
the
triangles
DFC
,
CBG
,
DEA
,
ye
shall
finde
the
other
lines
DC
25
,
1219
,
DG
,
48
,
25855
,
GH
12
,
9026
,
BC
wil
be
10
,
68335
,
the
seuen
parts
of
the
number
aboue
sayd
wil
be
for
DP
,
77
,
99204.
The
26.
Figure
.
In
this
present
Enneagone
,
let
the
angle
flanked
be
85
degrees
,
the
defence
50
rod
,
the
other
60
,
&
the
gorge
in
the
flanke
as
4
to
3.
the
rest
is
required
.
IF
ye
take
away
the
angle
ADC
42
½
from
ADF
70
,
the
angles
GDR
will
remaine
,
and
DG
makes
50
rod
,
then
the
triangle
DGR
wil
be
knowne
,
to
wit
,
DR
,
&
RG
,
or
KH
23.
0875
,
which
will
make
ye
knowe
the
triangle
DKH
(
for
DH
is
60
)
then
RK
,
or
GH
wil
be
11
,
0301.
Moreouer
,
in
the
triangle
DEA
the
side
EA
is
equall
to
KH
,
by
DE
&
DR
,
ye
shall
haue
AG
35
,
94735
,
then
in
carrying
AC
,
ye
shall
seeke
the
angle
A
from
the
triangle
ABC
,
setting
downe
AB
4
,
and
BC
3
parts
,
according
to
the
Hypatese
,
ye
shall
finde
then
the
angle
A
of
36
degrees
,
52
min.
&
12.
seconds
.
Then
let
us
goe
to
the
triangle
ACG
,
hauing
AG
knowne
,
the
angle
A
&
G
,
equall
to
CDF
27
½
degrees
,
to
finde
CG
afterward
the
face
,
and
consequently
DF
,
FK
,
EF
or
AB
the
Gorge
,
which
wil
be
14
,
72808
,
therefore
if
4
giues
3
,
how
many
then
AB
?
ye
shall
haue
BC
for
the
flanke
11
,
04606
,
&
DP
78
,
51183
,
DA
24
,
56925.
The
27.
Figure
.
The
distance
DP
being
7
parts
,
DF
&
FB
each
of
them
2
parts
,
the
defences
50
&
60
rod
,
and
the
angle
flanked
85
degrees
,
how
many
will
the
other
dimensions
be
of
such
a
Fortresse
non
angular
?
ALB.
GIRARD
.
THis
question
is
impossible
to
resolue
being
exceeding
,
seing
that
there
is
a
condition
in
it
more
then
one
desires
,
&
which
is
worst
,
repugnant
to
the
others
,
for
which
fault
the
Authour
may
be
excused
,
seing
that
in
his
time
,
there
were
no
such
advertisements
giuen
as
wee
haue
giuen
thereof
in
the
beginning
of
the
Trigonometrie
,
cited
in
the
first
question
going
before
,
the
which
though
they
may
seeme
to
be
of
litle
consequence
to
some
;
yet
one
must
acknowledg
,
that
those
that
knowe
them
shall
not
fall
into
the
like
errours
,
as
these
which
may
be
explained
thus
;
There
are
two
reasons
giuen
DP
to
DF
,
and
DF
to
FB
(
a
reason
of
equality
)
two
lines
of
defence
the
angle
flanked
,
&
the
name
of
the
figure
of
nine-side
figure
,
which
are
sixe
termes
,
yet
one
needes
but
5
(
as
ye
shall
finde
it
noted
in
the
7
figure
,
where
the
question
was
defectiue
,
and
of
some
others
afterward
)
finally
,
the
proofe
of
this
may
be
seene
in
makeing
comparaison
of
this
with
the
25
figure
,
where
the
same
question
is
propounded
,
and
where
ye
shall
finde
that
the
defence
flanking
ought
to
be
48
,
25855
,
and
here
he
will
haue
it
50
,
which
is
absurd
,
as
is
said
.
The
28.
Figure
.
In
this
present
Decagone
,
a
Fortresse
with
ten
angles
or
Bulwarks
,
let
the
angle
flanked
be
87
degrees
,
the
Gorge
in
the
flanke
in
reason
sesquitertia
,
the
defences
50
and
60
rod
,
it
is
required
how
many
the
other
parts
thereof
will
make
?
THe
reason
sesquitertia
is
,
as
4
to
3
,
for
AB
to
BC
,
then
the
imaginall
angle
BAC
wil
be
36
degrees
,
52
minutes
,
12
seconds
.
Moreouer
,
the
angle
ADC
being
43
½
then
CDF
wil
be
28
½
,
which
is
an
angle
of
the
triangle
GDR
,
and
which
may
be
knowne
,
seing
that
DG
is
50
rod
,
therefore
GR
23
,
858
,
or
its
equall
KH
,
and
for
as
much
as
DH
is
60
,
then
DK
wil
be
knowne
,
and
also
DR
withall
,
and
so
ye
shall
finde
RK
for
the
second
flanke
:
likewise
the
triangle
DEA
wil
be
knowne
,
then
ED
and
DR
will
make
knowne
ER
,
or
AG
,
&
consequently
ye
shall
haue
the
triangle
ACG
for
the
angle
A
was
found
aboue
,
and
the
angle
G
is
28
½
degrees
,
then
the
face
wil
be
26
,
11334
,
and
hauing
found
out
DF
,
ye
shall
finde
EF
,
or
the
gorge
15
,
197
:
Also
FK
for
the
curtaine
32
,
10821
;
DP
78
,
00619.
The
29.
Figure
.
In
this
Decagone
the
angle
flanked
maked
87
degrees
,
the
defence
fichant
60
rod
,
the
flanke
12
rod
,
and
the
gorge
16
rod
,
it
is
required
how
much
the
other
parts
make
?
THe
quadrangle
ABCD
hauing
fiue
termes
giuen
,
ye
shall
finde
the
other
parts
;
also
the
triangle
DFC
,
whereby
ye
haue
FB
,
or
KH
,
&
consequently
the
triangle
DKH
,
and
hauing
FD
&
DK
,
their
summe
and
difference
wil
be
for
DP
,
78
,
65
,
and
BH
30
,
32633
;
the
face
shall
be
27
,
49377
:
DG
52
,
64265.
The
30.
Figure
.
In
this
Decagone
,
let
the
curtaine
be
36
rod
,
and
the
flanke
12
,
the
face
24
,
and
the
reason
of
the
Bulwarke
in
the
flanking
interiour
,
as
58
to
19.
THe
halfe
of
58
is
29
,
then
the
angles
ADG
to
AGD
,
or
CDF
,
wil
be
as
29
to
19
,
therefore
setting
downe
ADF
to
CDF
,
it
wil
be
as
48
to
19
;
but
ADF
is
72
degrees
,
then
CDF
wil
be
28
½
degrees
,
&
then
the
angle
flanked
87
degrees
,
the
rest
is
easie
,
for
the
triangles
DCF
,
CBG
will
make
knowne
CF
,
FD
;
DG
is
49
,
14888
,
DH
61
,
68324
;
DP
78
,
18336
,
AD
24
,
52336.
The
31.
Figure
.
Let
there
be
a
Decagone
,
whereof
the
face
is
24
rod
,
the
flanke
12
,
and
the
curtaine
36
,
how
many
will
those
parts
make
,
when
the
defence
flanking
is
doubled
to
the
Capitall
.
IF
DG
be
sett
downe
2.
then
DA
wil
be
1.
Now
the
angle
DAG
is
108
degrees
,
then
the
angle
flanking
interiour
G
wil
be
28.
23.
38
,
and
the
angle
flanked
87
,
12
,
44●
the
rest
is
easie
,
&
is
found
in
the
same
manner
,
as
the
end
of
the
former
,
&
therefore
Marolois
leaues
its
so
for
this
reason
.
The
32.
Figure
.
In
this
present
Vndecagone
,
a
fortresse
with
eleuen
Bulwarks
,
let
the
face
be
24
rod
,
the
flanke
12
,
the
curtaine
36
,
and
the
capitall
DA
to
AG
as
5
to
7
,
the
unknowne
parts
are
required
.
SEt
downe
DA
5
,
and
AG
7
,
and
the
angle
DAG
is
106
degrees
,
and
4
eleuenths
:
that
is
106
,
21
,
49
,
and
hence
ye
shall
know
the
angles
remayning
ADC
,
43
,
55
,
48
,
and
the
other
or
CDF
29
,
42
,
23
,
therefore
in
the
triangle
CDF
,
the
angle
D
is
then
so
,
and
DC
24
rod
,
and
then
DF
,
DK
,
DP
wil
be
knowne
,
likewise
FC
,
FB
,
BP
,
&
AD
,
DE
&
EF
for
the
gorge
.
Here
is
nothing
but
that
which
is
ill
calculated
by
the
Autheur
,
or
rather
by
his
disciples
,
as
from
the
beginning
(
without
all
doubt
)
seekeing
to
help
themselues
with
the
figure
put
here
under
,
which
was
needlesse
;
supposing
that
they
had
skill
in
Trigonometrie
.
I
will
onely
set
downe
here
the
reason
of
the
raid
(
or
semy-diameter
)
in
the
side
of
the
Vndecagone
inscribed
in
the
Circle
,
is
as
100000000000000
to
563●46511368285
.
So
that
ye
may
take
here
the
reason
or
proportion
as
precisely
as
ye
will
.
The
33.
Figure
.
In
this
Vndecagone
let
the
face
of
the
curtaine
be
as
2
to
3
:
&
the
gorge
in
the
flanke
as
4
to
3
,
the
distance
of
the
points
of
the
Bastions
75
rod
,
and
the
angle
flanked
88
degrees
,
38
min.
11
seconds
,
the
question
is
how
ye
shall
finde
out
the
other
dimensions
?
SEing
that
ADC
,
and
ADF
are
knowne
,
the
rest
CDF
wil
be
also
,
being
29
,
19
,
6
,
&
setting
downe
DC
2
parts
,
(
then
the
curtaine
,
or
FK
wil
be
3
)
the
triangle
CDF
wil
be
knowne
by
the
parts
,
namely
DF
1
,
74384
,
as
much
also
is
KL
,
and
FK
3
,
then
DP
wil
be
(
in
parts
)
6
,
48768
,
which
make
by
the
Hypotese
75
rod
,
&
by
this
reason
,
ye
shall
finde
out
the
face
,
and
the
curtaine
,
saying
,
if
6
,
48768
parts
make
75
rod
,
how
many
then
will
make
aswell
2
,
as
3
?
DC
23
,
12075
,
&
BH
34
,
68112
as
much
makes
FK
,
which
taken
from
75
,
the
halfe
of
the
rest
wil
be
for
DF
,
ye
shall
finde
then
also
FC
,
afterward
AB
,
being
set
downe
upon
4.
BC
wil
be
3
,
then
the
triangle
BAC
wil
be
36
,
52
,
12
;
which
taken
from
DAB
,
there
will
remaine
DAC
,
&
so
the
triangle
DAC
shall
haue
3
termes
,
ye
shall
then
seeke
DA
,
or
AC
,
by
which
ye
shall
come
to
haue
AB
13
,
7969
,
&
BC
10
,
34743
,
then
in
the
triangle
CBG
,
ye
shall
finde
BG
,
afterward
DR
,
or
CG
being
knowne
ye
shall
haue
DG
41
,
54579
,
GH
11
,
57496
,
and
finally
DH
the
fichant
58
,
96636.
The
34.
Figure
.
Moreouer
for
this
Vndecagone
let
the
angle
flanked
be
88
degrees
38
,
11
,
&
the
face
to
the
curtaine
as
2
to
3
,
and
the
gorge
in
the
flanke
as
8
to
5.
&
the
fichant
60
rod
,
how
many
will
all
these
lines
make
?
YE
must
first
finde
out
the
Quadrangle
ABCD
,
whereof
the
angles
are
knowne
,
&
sett
downe
AB
,
8
parts
;
then
BC
wil
be
5
,
&
DC
being
found
,
ye
shall
finde
DF
,
FC
,
also
BH
,
seing
that
DC
is
to
BH
as
2
to
3
;
hauing
DK
,
and
KH
ye
shall
haue
DH
in
parts
,
which
make
60
rod
,
by
which
reason
ye
haue
the
lines
required
,
AB
,
14
,
6
,
CB
9
,
127
,
DG
42
,
374
,
DC
23
,
7363
,
BH
35
,
604
,
&
DP
76
,
99675.
Vpon
the
35.
Figure
.
ALB.
GIRARD
.
THis
question
is
defectiue
,
in
regard
it
hath
but
foure
termes
knowne
,
DP
70
rod
,
&
CB
10
,
the
reason
of
DC
to
BG
is
as
10
to
9
,
and
the
name
of
the
figure
Undecagonall
,
which
are
but
4
termes
,
he
comes
to
the
〈◊〉
▪
setting
downe
the
face
20
rod
,
to
see
if
there
be
not
in
it
repugnanci●
,
as
if
the
question
were
exceeding
,
so
that
he
in
this
search
committeth
the
faulth
,
which
is
called
petitio
principij
.
Which
is
spoken
not
to
defame
the
Authour
,
but
to
shew
,
how
this
proceeded
from
hence
,
that
in
this
time
many
thought
that
Geometrie
hath
attained
to
her
highest
degree
;
though
wee
haue
had
but
the
A
,
B
,
C
of
it
:
till
that
some
undertakeing
the
restauration
thereof
,
haueing
in
part
made
it
flourish
againe
,
could
neuerthelesse
escape
the
blame
of
some
envious
&
ignorant
men
in
this
divine
science
,
as
happens
often
,
haue
let
passe
the
most
difficult
of
the
Analitica
,
which
ought
to
adorne
those
,
that
make
profession
thereof
,
who
contrarywise
setting
downe
the
limits
of
their
pretensions
from
the
beginning
of
their
course
;
by
emulating
one
an
other
,
content
themselues
to
grope
together
without
learning
to
goe
forward
with
a
solide
,
and
a
firme
pace
in
a
faire
way
.
The
36.
Figure
.
In
this
Dodecagone
,
let
the
face
be
24
,
the
flanke
and
the
curtaine
36
rod
,
the
angle
flanked
right
,
how
many
will
the
remayning
lines
make
?
LEt
DF
be
found
,
also
FC
,
ye
shall
haue
DK
,
&
KH
for
to
haue
DH
,
afterward
in
the
triangle
DGR
,
hauing
the
angles
&
RG
,
ye
shall
finde
DG
.
Finally
ye
shall
finde
DE
in
the
triangle
DEA
,
for
to
haue
EF
,
or
the
gorge
:
Now
seing
that
GCB
is
60
degrees
,
then
of
necessity
GC
wil
be
24
,
as
being
asmuch
againe
as
CB
,
then
DG
wil
be
48
,
and
GR
,
or
KH
the
halfe
is
12
,
therefore
DH
wil
be
61
,
648
,
DP
,
77
,
56944.
The
37.
Figure
.
In
this
present
Dodecagone
,
let
the
angle
flanked
be
right
,
the
defences
flanking
,
&
the
fichant
45
,
&
60
rod
,
the
gorge
in
the
flanke
is
as
4
to
3.
SEing
that
the
angle
DCF
is
60
degrees
.
DH
45
wil
be
the
double
of
GR
22
½
rod
,
DH
,
HK
being
knowne
,
DK
wil
be
55
,
621
,
DR
38
,
971
,
then
the
second
flanke
16
,
65.
Also
DE
wil
be
found
by
the
triangle
DAE
,
&
taken
from
DR
,
there
will
remaine
AG
the
basis
of
the
triangle
ACG
,
the
angle
A
thereof
wil
be
36
,
52
,
11
,
because
of
the
reason
giuen
,
and
G
60
degrees
,
then
the
sayd
triangle
ACG
will
make
knowne
CG
21
,
4933
,
&
also
the
face
23
,
5067
(
for
DG
is
45
)
the
halfe
of
GC
is
CB
10
,
7466
,
which
multiplyed
by
4/3
,
ye
shall
haue
BA
14
,
3289
,
DP
75
,
979
,
BH
35
,
2637.
The
38.
Figure
.
Let
the
curtaine
be
36
rod
,
the
line
of
defence
flanking
45
,
the
angle
BAC
36
,
45
,
&
the
flanked
right
.
How
many
will
the
other
parts
of
this
Fortresse
Dodecagonall
come
to
?
THe
triangle
DAG
,
afterward
ACG
wil
be
knowne
,
and
also
CBG
,
DCF
,
DKH
,
whereby
DC
wil
be
found
23
,
974
,
GR
22
½
(
the
halfe
of
DG
)
10
,
5127
,
DH
61
,
0593.
DP
,
77
,
52516
,
AB
14
,
733.
The
39.
Figure
.
In
this
Dodecagone
the
angle
BAC
makes
37
degrees
,
the
flanked
right
,
the
face
24
rod
,
the
fichant
62
,
how
many
makes
then
the
other
lines
?
THe
line
DC
24
shal
be
the
double
of
CF
12
;
the
angle
BAC
being
giuen
,
the
triangle
DAC
,
wil
be
knowne
,
therefore
DA
,
AC
,
and
the
triangles
DAE
,
ACB
DGR
,
DHK
wil
be
knowne
;
and
consequently
the
lines
required
,
as
AB
,
14
,
6179.
BC
,
11
,
01525
,
BH
36
,
7851
;
DP
78
,
3547.
DA
23
,
82709
;
He
failed
herein
to
calculate
them
a
new
together
:
but
the
reason
of
this
was
,
that
wee
agree
not
,
seing
that
in
the
construction
,
in
stead
of
37
for
the
angle
BAC
,
(
as
he
sett
downe
)
he
tooke
38.
All
questions
comming
after
this
are
defectiue
euen
to
the
discourse
,
which
endeth
the
order
of
the
same
questions
;
but
seing
I
know
whither
it
tends
,
I
will
explane
them
,
in
adding
thereunto
the
things
,
that
were
wanting
passing
ouer
the
figures
40
,
41
,
and
so
we
will
beginn
with
the
42
,
as
followeth
.
The
42.
Figure
.
In
a
square
Fortresse
with
foure
Bulwarks
,
let
the
angle
flanked
be
60
degrees
,
the
angle
forming
the
flanke
FID
40
degrees
,
the
face
24
rod
,
hauing
the
reason
to
the
curtaine
as
4
to
5
,
the
other
lines
are
required
.
Seeke
the
triangles
DAL:
AID
,
IFD
,
DFP
,
ye
shall
finde
also
PH
,
for
FH
wil
be
30
:
IO
48
,
45532
:
AK
76
,
6464
,
IF
9
,
22766
:
FD
,
7
,
74298
,
IA
19
,
73479
,
the
fichant
AH
54
,
98265.
The
43
Figure
.
Is
a
Pentagone
,
the
angle
flanked
69
degrees
,
the
angle
forming
the
flanke
●0
degrees
,
the
curtaine
to
the
face
as
5
to
4
,
the
face
24
rod
;
wee
desire
to
know
the
other
lengths
.
Ye
must
doe
as
before
,
and
then
there
wil
be
no
difference
in
the
operation
,
Touching
the
number
of
the
names
of
the
figures
in
this
9
plate
,
ye
shall
finde
them
marked
about
the
angle
flanking
exteriour
,
the
reason
of
the
face
to
the
curtaine
is
marked
on
the
point
of
the
Bastions
,
the
length
aswell
of
the
face
,
as
of
the
curtaine
next
themselues
,
and
must
be
proposed
as
the
two
former
,
hauing
alwayes
the
angle
forming
the
flanke
of
40
degrees
:
the
opening
of
the
flanked
angles
,
is
according
to
the
manner
of
the
first
table
,
to
wit
15
degrees
more
then
the
demy-angle
of
the
Polygone
,
sauing
that
in
the
figures
2
,
43
,
&
2
,
44
,
afterward
an
other
time
the
45
,
46
,
he
taketh
for
the
angles
flanked
the
⅔
of
the
angle
of
the
Polygone
,
according
to
the
second
table
in
the
beginning
placed
before
:
Of
these
things
ye
shall
finde
2
tables
in
the
end
of
this
booke
,
setting
downe
the
faces
to
the
curtaines
,
as
2
to
3
,
the
faces
24
rod
,
which
make
the
former-flankes
40
degrees
,
and
the
flanked
according
to
the
two
manners
abouesaid
,
where
ye
must
knowe
,
that
the
lines
of
defence
fichant
are
about
60
,
61
rod.
The
said
tables
are
both
calculated
〈◊〉
;
because
of
the
errours
escaped
in
the
other
editions
.
Moreouer
,
after
the
Dodecagone
ye
shall
haue
ouer
and
aboue
following
a
Fortification
upon
a
right
line
,
which
is
called
upon
a
right
curtaine
.
Now
let
us
marke
what
the
Authour
saith
.
Note
1.
NOw
not
to
take
so
much
paines
in
remembering
the
diverse
proportions
of
the
face
to
the
curtaine
,
whereof
the
Fortresse
quadrangular
,
&
Pentagonall
Fortresse
is
as
4
to
5
,
and
the
Exagonall
as
3
to
4
,
it
will
not
be
amisse
(
seing
that
the
line
of
defence
will
beare
it
)
to
make
also
the
said
Fortresses
of
the
same
proportion
,
as
the
subsequent
,
to
wit
,
in
reason
sesquialtere
,
as
appeareth
by
the
3
figures
of
the
9
plate
,
quoted
by
the
numbers
2
,
42.
2
,
43.
2
,
44.
where
the
lines
of
defence
doe
not
much
exceede
60
rod
,
which
is
as
farre
as
a
musket
can
well
carry
,
&
therefore
one
ought
not
to
exceede
this
number
;
because
that
alwayes
from
the
flanke
reciprocally
the
entrance
into
the
moate
must
be
defended
,
which
is
often
done
,
and
most
commodiously
by
the
Muskettiers
,
because
that
Canon
cannot
so
suddenly
(
by
reason
of
the
weight
)
be
brought
thither
,
many
good
occasions
are
neglected
,
for
which
cause
they
are
preferred
before
Canon
,
and
in
regard
a
Musket
will
but
carrie
some
700
feete
point-blanke
,
Bulwarkes
ought
to
be
made
noe
further
one
from
another
,
for
otherwise
the
line
of
defence
would
be
too
long
,
which
should
cause
imperfection
.
Note
2.
IF
in
stead
of
takeing
the
face
24
rod
,
for
the
line
of
the
Polygone
exteriour
take
80
rod
,
and
the
rest
according
to
the
former
proportion
,
the
parts
wil
be
brought
very
neere
together
,
as
appeareth
by
the
precedent
Examples
,
according
to
the
figures
of
the
10
plate
,
marked
with
the
numbers
51.
52.
53.
54.
55.
56.
57.
58
,
59
,
whereof
we
haue
made
no
calculation
;
because
they
are
easely
done
by
the
former
supputations
.
Note
3.
THat
in
all
our
designes
and
calculations
,
we
are
resolued
to
use
but
one
measure
onely
,
which
is
in
the
25
plate
of
our
Geometrie
upon
the
rule
of
the
instrument
marked
number
1
,
which
is
a
foote
deuided
into
12
equall
parts
,
whereof
12
foote
makes
a
Rheneland
rodd
in
the
territoire
of
Leyden
.
Designes
&
plat
formes
of
diverse
Fortifications
.
WE
might
according
to
the
former
supputations
giue
diuerse
constructions
of
the
platformes
of
Fortifications
;
but
considering
the
great
diversity
of
rules
,
(
which
often
causes
confusion
,
and
that
time
will
not
permit
us
)
wee
will
giue
but
one
simple
and
generall
rule
for
them
,
which
is
this
following
.
The
9
Plate
,
and
the
42
,
43
,
44
,
45
,
46
,
47
,
48
,
49
,
50
Figures
.
FIrst
,
if
the
face
be
giuen
(
suppose
it
to
be
24
rod
)
then
ye
must
doe
as
followeth
:
Ye
shall
draw
a
right
line
infinite
AC
from
the
point
A
,
the
angle
CAB
shal
be
made
according
to
the
forme
of
the
Polygone
,
upon
which
A
,
B
shal
be
sett
downe
the
24
rod
abouesayd
,
as
from
A
to
D
,
and
by
D
is
drawne
the
line
,
which
formeth
the
angle
of
the
Center
of
the
Bulwarke
40
degrees
,
then
must
ye
sett
downe
upon
A
,
C
and
A
,
B
,
the
reason
of
the
curtaine
to
the
face
that
by
the
same
ye
may
draw
the
line
infinite
A
,
E
,
makeing
from
the
point
D
,
a
parallell
with
A
,
C
,
cutting
the
said
AE
,
at
G
,
by
which
point
is
drawne
the
line
,
that
formeth
the
other
face
G
,
K
,
in
such
sort
that
the
angle
G
,
K
,
A
,
wil
be
equall
to
the
angle
D
,
A
,
K
,
and
so
the
figure
wil
be
drawne
,
according
to
the
said
proportion
.
The
10.
Plate
,
&
the
51
,
52
,
53
,
54
,
55
,
56
,
57
,
58
,
59
,
Figures
.
SEcondly
,
if
the
line
from
angle
to
angle
be
giuen
AB
(
for
example
80
rod
)
ye
shall
then
proportion
out
the
face
to
the
curtaine
sesquialtera
,
as
2
to
3.
Then
in
conformitie
of
the
table
of
angles
set
downe
before
,
wil
be
made
the
angle
CAB
,
according
to
its
forme
,
to
wit
,
in
a
square
of
15
degrees
,
in
a
Pentagone
of
89
½
degrees
,
in
the
Exagone
22
½
degrees
,
and
so
consequently
of
the
rest
,
then
for
the
forming
of
the
abouesaid
proportion
of
the
face
to
the
curtaine
,
which
is
(
as
we
haue
said
)
as
2
to
3.
aswell
in
a
square
Fortresse
as
in
a
Pentagone
,
and
others
following
ye
shall
take
upon
a
scale
of
a
reasonable
greatnesse
2.
which
shal
be
sett
from
A
to
C
,
and
3
from
A
to
D
,
suppose
that
it
be
at
C
&
D
,
then
from
the
point
D
wil
be
made
the
arch
G
of
the
distance
of
2
,
(
to
wit
CA
)
&
from
C
of
the
distance
of
D
,
A
,
3
,
wil
be
made
the
arches
,
which
cutt
betweene
each
other
at
G
,
by
which
a
right
line
being
drawne
to
A
,
where
the
same
cutting
the
line
FB
,
as
here
at
F
ye
haue
the
face
of
the
Bulwarke
,
and
by
that
the
curtaine
wil
be
knowne
.
To
knowe
also
the
line
of
the
gorge
,
and
the
flanke
,
and
all
the
other
parts
,
ye
must
make
the
angle
forming
the
gorge
,
and
the
flanke
of
40
degrees
,
as
is
here
the
angle
KIE
,
cutting
the
diameter
P
,
A
,
at
I
,
from
which
point
I
a
right
line
parallel
being
drawne
from
A
,
B
,
as
is
I
,
N
,
ye
shall
drawe
the
line
perpendicular
to
be
E
,
K
,
which
shal
be
the
flanke
,
and
K
,
I
,
the
line
of
the
gorge
of
this
Fortresse
,
and
then
ye
haue
the
thing
required
.
In
like
manner
ye
shall
finde
,
and
marke
out
the
other
Fortresses
,
according
to
their
seuerall
formes
,
takeing
heede
that
as
here
ye
take
2
&
3
,
ye
must
in
the
others
also
take
2
&
3
,
to
place
them
aswell
upon
A
,
B
,
as
upon
A
,
C
,
and
the
rest
being
the
same
;
as
is
in
the
former
construction
it
wil
be
needelesse
,
to
giue
yow
here
any
further
particular
instruction
for
them
.
The
3
,
50.
Figure
,
and
the
9.
Plate
.
ALBERT
GIRARD
.
THirdly
,
When
upon
the
line
of
the
Polygone
interiour
AB
,
ye
desire
to
construe
,
or
explane
a
Fortresse
,
hauing
5
termes
,
to
wit
,
the
name
of
the
figure
,
the
angle
of
the
bastion
M
,
The
angle
forming
the
flanke
GBH
,
and
the
reason
of
the
face
DG
,
to
the
curtaine
HL
,
conceiuing
therein
that
AB
is
giuen
.
This
figure
is
in
stead
of
the
2
,
50
,
being
worth
nothing
.
I
am
so
constrained
to
make
a
new
figure
,
and
a
new
explanation
.
Let
then
NBA
be
equall
to
the
demy-angle
of
the
Polygone
,
and
let
the
angle
forming
the
flanke
be
HBG
,
LAK
,
so
ye
shall
haue
the
intersection
or
cutting
betweene
C
,
then
DBF
the
halfe
equall
to
the
demy-angle
flanked
,
and
finding
the
point
F
,
so
that
AB
,
to
BF
is
the
reason
of
the
curtaine
to
the
face
,
and
hauing
drawne
CF
,
cutting
NB
at
D
,
and
makeing
DG
a
paralell
with
FB
,
meeting
with
BC
,
at
G
,
ye
haue
finally
GH
the
perpendicular
with
BA
,
and
doing
the
like
also
on
the
other
side
,
ye
shall
haue
all
the
parts
required
;
the
demonstration
is
manifest
,
seing
that
C
is
the
common
point
of
the
like
figures
FBA
,
DGK
:
then
as
FB
,
to
BA
:
so
DG
the
face
,
wil
be
to
GK
,
or
HL.
The
11.
Plate
,
and
60
,
61
,
62
,
63
,
64
,
65
,
66
,
●7
,
●●
,
Figures
.
IF
there
be
a
question
to
proportion
out
the
face
to
the
curtaine
,
and
the
gorge
to
the
flanke
,
when
the
line
EF
is
giuen
,
ye
shall
doe
as
the
figures
in
appearance
demonstrate
to
the
eye
,
namely
,
ye
shall
first
proportion
out
the
face
to
the
curtaine
by
the
former
rules
,
and
make
it
soe
that
the
curtaine
of
a
square
Fortresse
be
300
foote
,
and
the
face
HF
,
250
foote
,
hauing
reason
as
5
to
6
,
according
to
which
ye
shall
finde
the
point
G
,
which
is
the
angle
of
the
shoulder
.
To
finde
then
afterward
the
gorge
&
the
flanke
,
ye
shall
make
the
couert
line
GH
,
which
must
be
diuided
into
4
equall
parts
(
in
foure
,
because
ye
would
haue
the
reason
of
the
gorge
in
the
flanke
as
4
to
3
)
whereof
ye
must
place
three
of
them
from
H
to
I
,
and
then
drawe
I
,
G
,
cutting
through
the
semy-Diameter
of
the
Polygone
,
which
is
here
squared
into
A
,
and
then
the
line
AB
being
drawne
it
wil
be
parallell
with
EF
,
and
so
ye
haue
the
side
of
the
Polygone
interiour
,
upon
which
the
2
perpendiculars
being
drawne
GC
,
HD
,
ye
haue
the
2
flankes
,
and
consequently
the
essentiall
parts
of
a
square
fortification
knowne
:
ye
must
also
obserue
the
same
in
the
other
figures
following
.
If
the
line
EF
,
which
is
the
side
of
the
Polygone
exteriour
(
or
the
distance
of
the
angles
of
the
Bulwarks
)
be
not
giuen
,
but
onely
the
line
AB
,
the
side
of
the
Polygone
interiour
,
ye
shall
seeke
out
in
the
tables
a
Fortresse
of
such
a
forme
,
whereof
the
face
to
the
curtaine
is
as
5
to
6
,
and
shall
worke
it
by
the
rule
of
proportion
,
or
else
let
it
be
made
after
the
former
manner
,
which
is
much
more
easie
,
then
all
the
figures
of
the
11.
Plate
.
If
such
a
side
of
a
Polygone
interiour
,
giueth
such
a
side
of
a
Polygone
exteriour
,
what
will
such
a
side
interiour
giue
?
&
ye
shall
haue
the
thing
required
;
to
wit
,
the
Polygone
exteriour
,
whereby
one
may
easely
know
the
other
parts
of
the
Fortresse
,
when
the
line
AB
of
the
Polygone
interiour
is
giuen
.
But
if
in
the
tables
the
proportion
be
not
found
,
make
first
upon
the
line
AB
the
triangle
ARB
,
whereof
the
halfe
of
the
basis
hath
the
like
reason
to
the
perpendicular
,
as
ye
will
haue
proportioned
out
the
gorge
of
the
flanke
,
demonstrated
by
this
exemple
as
4
to
3
,
which
wil
be
done
in
setting
downe
upon
the
said
line
AB
foure
equall
parts
(
4
because
ye
desire
to
haue
proportioned
out
the
gorge
to
the
flanke
,
as
4
to
3
)
to
wit
from
A
to
D
,
raising
out
of
the
point
D
a
perpendicular
,
and
put
3
of
the
said
parts
from
D
to
I
,
then
doing
the
like
on
the
other
side
,
by
this
meanes
wil
be
formed
the
triangle
ARB
,
and
from
the
points
A
&
B
,
the
angles
flanking
interiour
wil
be
made
according
to
the
forme
of
the
Polygone
,
(
which
is
a
Pentagone
)
conformable
to
the
table
of
angles
described
heretofore
,
as
appeareth
by
the
letters
SBA
and
TAB
,
which
cutt
the
lines
infinite
A
,
L.
L
,
B
at
the
points
S
,
T
,
and
the
line
S
,
&
T
being
drawne
,
the
angles
T
,
S
,
B
,
and
S
,
T
,
A
wil
be
equall
to
the
angles
S
,
B
,
A
,
and
T
,
A
,
B
,
as
appeareth
by
the
28
proposition
of
the
first
part
of
Euclide
.
According
to
which
the
reason
of
the
face
being
putt
to
the
curtaine
,
upon
T
,
S
,
and
T
,
A
,
namely
,
the
face
upon
TA
,
and
the
curtaine
upon
T
,
S
;
ye
haue
the
like
distances
,
and
from
the
points
I
&
X
,
the
arches
which
cuts
betweene
them
in
a
certaine
place
,
by
which
intersection
the
line
T
,
V
,
shal
be
drawne
cutting
the
line
S
,
B
,
at
V
,
then
the
right
line
V
,
L.
being
drawne
passing
through
the
center
L
,
&
cutting
A
,
R
at
G
,
ye
shall
haue
all
the
parts
of
this
Fortresse
:
for
hauing
sett
downe
the
distance
R
,
G
,
from
R
to
H
,
from
the
same
point
G
is
drawne
a
paralell
to
S
,
B
,
as
E
,
G
,
cutting
the
Diagonall
line
L
,
A
,
at
E
,
the
like
ye
shall
finde
for
F
,
and
consequently
all
the
other
parts
of
the
said
Fortresse
Pentagonall
.
The
same
must
ye
also
understand
of
the
other
figures
following
,
from
the
figure
60
,
to
the
figure
68.
Also
ye
may
finde
the
point
V
,
in
setting
downe
upon
T
,
A
,
2
equall
parts
,
such
as
ye
require
,
and
from
the
extremity
or
utmost
end
,
hauing
made
it
a
parallell
to
S
,
T
,
and
upon
the
same
,
3
of
those
parts
,
and
drawne
from
the
extremity
the
line
TV
,
the
said
point
will
consequently
be
knowne
thereby
.
NOte
that
the
proportion
giuen
here
,
betweene
the
face
and
the
curtaine
is
not
so
much
to
relye
fast
upon
it
,
as
to
show
that
the
generall
rule
sett
downe
here
before
in
the
9
and
10
plates
,
take
place
also
in
all
other
reasons
,
which
may
be
propounded
.
For
otherwise
wee
are
of
the
opinion
;
that
the
former
figures
would
rather
be
accepted
then
these
here
,
because
that
the
reason
of
the
curtaine
to
the
face
is
(
as
wee
haue
said
aboue
,
sesquialtera
as
well
in
square
Fortresse
,
as
of
Pentagonall
,
and
others
following
.
Which
for
the
facility
,
&
simplicity
together
with
the
goodnesse
of
them
ought
to
be
preferred
before
the
figures
60
,
61
,
62
,
63
,
64
,
65
,
66
,
67
,
and
68
,
of
the
11
Plate
aboue
said
.
Whereof
the
face
to
the
curtaine
is
(
as
the
exemples
shew
)
of
diuerse
reasons
;
in
such
sort
,
that
hence
forward
one
ought
to
resolue
,
that
the
reason
of
the
curtaine
to
the
face
ought
in
all
formes
of
Fortification
to
be
sesquialtera
,
and
the
face
24
rod
,
each
rod
containing
12
foote
,
the
length
whereof
is
sett
downe
in
the
25
plate
of
our
Geometrie
,
noted
with
the
character
or
the
figure
1
,
&
is
diuided
into
12
ynches
,
which
rod
is
used
in
the
Fortifications
of
the
United
Provinces
,
to
the
end
that
the
line
of
defence
may
not
exceede
much
aboue
60
rod
,
which
is
about
as
farre
as
a
Musket
can
well
beare
,
the
gorge
to
the
flanke
may
be
made
by
the
rule
giuen
in
the
11
plate
,
according
to
the
reason
giuen
,
but
forasmuch
as
it
is
more
convenient
to
make
the
angle
forming
the
flanke
GAC
(
plate
the
11
)
of
40
degrees
which
giueth
the
reason
almost
as
6
to
7
,
in
my
opinion
one
ought
to
rest
thereupon
,
and
so
ye
shall
haue
a
generall
rule
for
all
manner
of
Fortifications
,
whither
they
be
Quadrialtera
,
Pentagonall
,
Hexagonall
,
or
the
others
following
,
as
ye
may
perfectly
understand
by
that
which
shal
be
said
hereafter
.
In
the
meane
while
note
that
I
giue
here
the
reason
of
the
gorge
to
the
flanke
as
7
to
6
,
which
ought
to
be
understood
in
Fortresses
without
Casemates
,
but
if
it
were
my
intention
to
make
some
of
them
,
I
would
then
alter
somewhat
of
the
said
proportion
.
The
manner
how
to
describe
succinctly
the
designes
or
Plots
of
some
regular
Fortifications
.
The
12.
Plate
,
and
the
69.
Figure
.
ALthough
one
may
(
by
that
which
we
haue
taught
)
sufficiently
now
understand
the
order
,
and
Method
requisite
to
be
held
in
all
manner
of
Fortifications
:
yet
I
haue
thought
it
good
to
trace
out
here
some
from
among
the
rest
,
to
make
the
louers
of
this
Art
to
conceiue
the
better
my
intention
,
and
will
begin
with
a
regular
Pentagone
,
whereof
the
angle
of
the
center
maketh
72
degrees
,
and
the
angle
of
the
Polygone
108
degrees
,
and
seing
the
angle
flanked
according
to
the
table
before
mentioned
,
maketh
69
degrees
:
It
wil
be
easie
to
finde
the
angle
C
,
A
,
D
,
which
is
alwayes
equall
to
the
angle
flanking
interiour
,
and
shall
finde
the
same
to
be
19
½
degrees
,
setting
then
your
graduate
Instrument
upon
19
½
degrees
,
it
makes
the
angle
CAD
,
and
drawing
first
the
covert
line
infinite
AB
,
takeing
upon
the
scale
24
parts
,
or
rod
:
and
sett
then
from
A
to
C
,
drawing
from
the
point
C
a
perpendicular
upon
A
,
B
,
as
is
CD
afterward
hauing
sett
on
DE
the
length
of
the
curtaine
,
which
is
here
onely
34
rod
(
because
the
false
brayes
makes
the
lines
of
defence
too
great
)
&
then
the
distance
AD
,
from
B
to
E
,
raysing
from
the
E
the
〈◊〉
cular
EF
,
being
equall
to
D
,
C
,
and
FB
drawne
out
,
the
two
faces
wil
be
knowne
;
for
to
knowe
the
center
of
the
Bulwarke
,
shal
be
made
of
2
lines
GA
,
&
GB
the
Demy-diameter
of
the
Polygone
,
then
your
instrument
being
sett
upon
40
degrees
,
ye
shall
finde
the
angle
HKA
,
cutting
the
said
demy-diameter
at
the
same
point
H
:
from
which
point
the
line
HI
being
drawne
;
ye
shall
haue
the
side
of
the
Polygone
interiour
upon
which
out
of
the
points
C
,
and
F
,
shal
be
drawne
the
perpendiculars
CL
&
FM
,
which
will
forme
the
flanks
,
and
the
gorges
of
the
said
Fortresse
Pentagonall
,
in
the
inside
of
the
side
of
the
Polygone
interiour
shal
be
drawne
a
parallell
of
5
½
rod
for
the
thicknesse
of
the
rampart
,
as
N
,
O
N
,
R
,
and
RS
,
and
ye
must
draw
for
the
parapett
of
the
rampart
a
parallell
of
20
feete
(
also
in
the
inside
of
the
said
side
of
the
Polygone
HI
)
and
on
the
outside
thereof
ye
shall
drawe
a
parallell
of
20
foote
for
the
falsebray
as
LX
,
and
yet
more
outward
20
other
foote
for
its
parapet
,
&
so
all
the
other
parts
wil
be
traced
out
,
which
are
within
the
moate
;
for
seing
that
there
is
here
noe
Casemates
,
the
falsebray
is
carried
not
onely
about
the
curtaines
;
but
also
about
the
flankes
,
and
the
faces
of
the
Bulwarks
,
in
such
sort
,
that
the
falsebray
serueth
as
a
Casemats
to
the
said
flankes
,
the
entrance
or
sallies
ought
in
my
opinion
to
be
made
in
the
midst
of
the
curtaines
,
as
covertly
as
possibly
may
be
;
the
Bulwarkes
are
made
either
massiue
,
or
hollow
from
Earth
;
at
this
present
wee
haue
drawne
them
out
as
being
hollow
from
earth
,
&
so
the
superficies
,
or
plaine
in
the
midst
of
the
bulwarke
N
,
R
,
S
,
H.
is
of
the
same
heigth
,
as
the
rest
of
the
enclosure
of
the
Pentagone
is
,
the
moate
shal
be
10
rod
broad
,
which
is
traced
out
as
followeth
:
In
the
point
A
,
or
the
angle
of
the
Bulwarke
,
shal
be
made
an
arch
of
the
distance
of
10
rod
,
&
in
the
point
V
make
an
other
arch
of
somewhat
a
lesser
extention
then
the
former
about
10
or
12
foote
,
then
drawe
upon
the
back
thereof
a
covert
line
infinite
;
the
like
must
be
done
on
the
otherside
of
the
Bulwarke
,
and
so
from
place
to
place
,
with
covert
lines
being
drawne
as
abouesaid
,
the
intersections
of
them
will
bring
forth
the
angles
of
the
said
moate
,
aswell
on
the
interiour
as
on
the
exteriour
parts
,
ye
must
make
upon
the
brinke
of
the
moate
exteriour
a
parallell
of
20
or
24
foote
;
for
the
bredth
of
the
covert
way
,
and
besides
somewhat
more
towards
the
exteriour
part
,
ye
shall
make
a
parallell
of
50
foote
for
the
thicknesse
of
the
parapet
thereof
,
which
shall
goe
downe
sloaping
being
onely
raised
towards
the
covert
way
,
or
the
Coridor
of
6
foote
,
and
diminishing
litle
by
litle
to
the
furthest
part
of
the
50
foote
,
&
so
ye
shall
haue
finished
the
Icnographia
,
or
ground-draught
of
this
Fortresse
Pentagonall
,
the
streets
betweene
the
ramparts
and
the
housen
ought
to
be
30
foote
,
that
with
the
more
convenience
they
may
be
entrenched
in
the
time
of
neede
,
and
that
the
souldiers
may
stand
there
in
battle-ray
.
The
streets
ought
to
be
made
of
24
or
30
foote
,
and
for
the
market
place
every
side
shal
be
made
12
rod
,
and
of
the
same
forme
,
as
the
Polygone
is
,
which
is
fortified
,
being
a
Pentagone
,
in
which
the
towne-house
is
to
be
made
,
&
other
publick
buildings
,
&
thus
we
haue
finished
the
forme
thereof
.
The
70.
Figure
.
THe
Profile
or
draught
shal
be
made
in
this
manner
:
first
a
privie
line
is
drawne
from
the
utmost
part
thereof
,
then
ye
shall
beginne
to
lay
out
the
breath
of
the
ramparts
,
moates
and
wayes
,
&
as
wee
haue
sett
downe
in
our
Icnographia
for
the
bredth
of
the
rampart
66
foote
,
which
make
5
½
rod
,
we
will
take
upon
the
skale
joyned
to
the
Profile
66
foote
,
for
to
place
them
upon
the
said
covert
line
as
from
A
to
B
:
then
as
the
way
of
the
rounds
,
which
we
otherwise
terme
the
falsebray
,
is
counted
20
foote
,
which
one
may
take
upon
the
said
skale
,
&
remoue
them
upon
the
said
covert
line
,
as
from
B
to
I
,
and
the
same
20
foote
from
I
to
M
,
for
the
bredth
of
the
parapet
,
then
ye
shall
make
an
edge
MN
of
6
foote
,
which
is
called
in
flemish
the
Teen
,
or
the
toe
of
the
foote
of
the
parapet
,
the
better
to
prevent
the
falling
of
the
same
into
the
moate
:
from
whence
ye
must
take
the
bredth
of
the
moate
,
which
shal
be
120
foote
;
for
so
much
makes
the
10
rod
as
from
N
to
Q
,
from
Q
to
R
,
ye
must
drawe
out
20
or
24
foote
for
the
covert
way
,
and
from
R
to
T
50
foote
for
the
parapet
;
and
so
ye
haue
all
the
dimensions
required
.
For
their
heigths
aboue
AB
ye
shall
make
a
parallell
of
14
foote
,
which
shal
be
the
height
of
the
rampart
,
and
underneath
a
parallell
of
10
foote
for
the
depth
of
the
moate
,
the
talud
or
slooping
interiour
of
the
rampart
A
,
V
,
ought
to
be
14
foote
,
and
the
talud
exteriour
XB
,
must
be
the
halfe
of
its
heigth
,
which
is
7
foote
,
and
the
trapeze
,
or
unequall
sides
giuen
A
,
D
,
C
,
B
,
being
the
profile
of
the
rampart
:
for
its
parapet
ye
shall
take
20
foote
from
C
to
E
,
and
from
the
point
E
draw
a
perpendicular
of
6
foote
,
which
shal
be
the
height
of
the
said
parapet
,
and
upon
the
said
perpendicular
EG
shal
be
layd
out
4
foote
to
drawe
a
parallell
to
DC
,
cutting
the
line
B
,
H
,
at
H
,
which
shal
be
the
heigth
of
the
said
parapet
in
this
place
,
and
for
the
superiour
part
ye
shall
draw
the
line
GH
.
Your
foote
banke
shal
be
made
as
followeth
:
from
E
is
drawne
the
line
FE
of
3
foote
,
and
1
½
foote
high
,
then
from
the
upper
part
of
the
footebanke
make
the
line
G
2.
in
such
sort
,
that
the
said
foot-banke
shall
not
be
broader
in
heigth
then
2
foote
,
and
shall
haue
the
talud
1
foote
,
which
shal
be
the
talud
interiour
of
the
said
parapet
.
The
parapet
of
the
falsebray
IM
shal
be
made
as
is
aboue
said
;
to
wit
,
from
the
point
I
,
is
raised
the
perpendicular
IK
of
6
foote
,
which
shal
be
the
height
of
the
said
parapet
,
and
from
I
towards
B
draw
3
foote
for
the
footebanke
,
being
a
foote
and
a
halfe
high
,
and
so
drawe
out
the
line
K2
,
that
the
bredth
of
the
upper
part
of
the
footebanke
be
not
(
as
wee
haue
said
)
aboue
2
foote
,
in
such
sort
that
the
Talud
interiour
(
after
ye
haue
drawne
out
the
said
line
K
2
)
of
the
said
parapet
shal
be
but
a
foote
:
for
from
the
point
I
,
the
foote
banke
is
3
foote
broad
,
but
in
the
point
3
it
is
found
to
be
but
2
foote
broad
.
For
the
moate
first
draw
out
NO
,
which
is
the
brinke
interiour
thereof
,
hauing
the
talud
,
as
much
as
is
the
depth
of
the
moate
,
to
wit
,
of
10
foote
,
likewise
the
line
Q
,
P
,
is
taken
,
which
is
the
brinke
exteriour
of
the
said
moate
,
the
parapet
of
the
covert
way
Q
,
P
,
is
made
by
rising
from
the
point
R
,
the
perpendicular
line
RS
of
6
foote
,
takeing
from
the
said
point
R
towards
the
left
hand
3
foote
for
the
bredth
of
the
footbanke
,
and
to
make
it
a
foote
and
a
halfe
high
;
To
giue
the
more
firmenesse
to
the
said
parapet
,
there
shal
be
made
from
the
upper
part
of
the
point
S
an
oblique
line
upon
the
said
footebanke
,
which
is
(
as
is
before
said
)
three
foote
broad
;
in
such
sorte
,
that
the
said
line
declining
a
foote
upon
the
said
footebanke
,
maketh
the
said
footebanke
to
be
noe
broader
then
2
foote
.
Finally
a
right
line
is
drawne
from
S
to
T
,
which
makes
the
slooping
of
the
said
parapet
.
Note
.
WE
haue
made
the
rampare
no
higher
then
14
foote
,
upon
the
upper
part
whereof
is
made
a
parapet
of
6
foot
high
,
all
upon
a
plaine
field
,
which
is
a
convenient
heith
for
a
place
,
that
hath
noe
hils
or
heigths
about
it
,
but
in
case
there
be
any
hils
,
that
doe
commaund
the
said
place
,
it
wil
be
then
necessarie
to
raise
the
rampart
somewhat
higher
,
then
is
here
described
,
answerable
to
the
heigth
of
the
said
hill
,
which
otherwise
would
commaund
the
Fortresse
,
&
for
the
better
blinding
of
such
places
,
sometimes
one
makes
upon
the
upper
part
of
the
rampart
an
other
parapet
about
7
foote
broad
,
or
there
abouts
&
6
foote
high
,
which
Parapet
serueth
so
much
the
better
to
blinde
the
said
place
,
and
to
discouver
the
fields
about
it
,
but
when
an
Ennemy
hath
approched
neere
unto
such
places
,
then
the
said
parapet
can
doe
them
no
further
seruice
,
so
that
necessity
will
compell
them
to
take
it
away
for
their
more
safegard
,
and
to
lye
under
the
said
parapet
of
20
foote
broad
:
for
Canon
shott
comming
every
time
to
pierce
the
said
parapet
being
but
7
footebroade
;
they
cannot
safely
lodge
behinde
it
,
without
extreame
danger
;
and
therefore
is
necessary
to
be
razed
,
to
make
use
of
the
other
,
which
is
able
to
resist
the
force
of
Canon
.
Wee
haue
not
made
here
in
this
79
figure
any
Ravelins
or
halfe
moones
,
which
are
loose
peeces
on
the
further
side
of
a
moate
,
neither
any
Catts
,
or
high
platformes
to
plant
ordinance
upon
,
which
are
usually
made
in
divers
places
within
Forts
,
and
upon
Bulwarks
;
because
we
intend
to
speake
of
them
hereafter
.
In
the
meane
while
it
wil
be
good
to
note
,
that
the
said
Cats
,
Ravelins
,
and
other
workes
,
may
be
applyed
to
this
present
Fortification
,
but
in
this
place
is
omitted
for
the
avoyding
of
prolixity
,
because
we
are
minded
to
treate
of
them
briefly
in
an
other
place
.
The
description
of
the
designe
or
the
plate
forme
of
a
Fortresse
Hexagonall
.
The
13.
Plate
,
and
71.
Figure
.
LEt
there
be
giuen
a
Fortresse
Hexagonall
to
be
fortified
,
whereof
the
face
AC
maketh
24
rod
,
and
the
angle
flanked
80
degrees
,
according
to
which
the
angle
flanking
interiour
will
make
20
degrees
,
and
the
exteriour
140
degrees
,
and
let
the
curtaine
be
30
rod
,
which
giueth
the
reason
of
the
face
to
the
curtaine
,
as
3
to
4.
To
doe
this
,
ye
shall
draw
the
covert
line
infinite
,
A
,
B
,
&
by
the
helpe
of
an
instrument
graduate
,
the
other
angle
CAD
of
20
degrees
(
of
20
,
because
that
the
angle
flanking
interiour
,
which
is
alwayes
equall
to
it
maketh
here
20
degrees
)
by
meanes
of
the
line
indefinite
AC
,
upon
which
ye
make
the
length
of
the
face
24
rod
,
as
from
A
to
C
,
from
which
point
C
,
the
perpendicular
CD
being
drawne
upon
the
line
A
,
B
,
shal
be
placed
from
D
,
the
length
of
the
curtaine
,
which
is
here
32
rod
as
from
D
to
E
,
finally
the
distance
AD
from
E
to
B
,
and
the
perpendicular
EF
the
distance
of
CD
as
from
E
to
F
,
drawing
the
line
FB
,
ye
haue
the
other
face
,
so
that
all
the
parts
of
the
reason
giuen
are
described
;
and
for
to
finde
the
curtaine
,
ye
must
make
first
the
angles
GAB
,
&
GBA
of
60
degr
.
seing
the
whole
angle
of
the
Polygone
maketh
120
degr
.
by
the
lines
AG
,
GB
,
which
cutting
through
on
an
other
at
G
,
is
the
center
of
the
Polygone
,
and
seing
the
gorge
,
when
as
ye
would
make
casemates
in
them
or
in
a
bulwarke
where
Cats
or
mounts
are
raised
,
they
had
neede
of
a
larger
extent
,
then
otherwise
;
we
suppose
that
it
were
requisite
,
to
make
Casemates
in
them
,
to
that
end
we
make
the
angle
HKA
,
which
otherwise
might
be
but
40
degrees
onely
of
35
,
according
to
which
the
gorge
in
the
flanke
wil
be
almost
as
4
to
3
,
or
somewhat
more
by
reason
of
the
line
H
,
K
,
cutting
the
line
Diagonall
AG
at
H
,
from
which
point
H
the
line
HN
being
drawne
,
parallell
to
AB
,
ye
shall
haue
the
Polygone
interiour
,
upon
which
the
lines
CL
,
and
FM
,
being
drawne
in
length
the
lines
DC
to
L
,
and
EF
to
M
;
in
so
doing
all
the
essentiall
parts
of
the
said
Fortresse
,
wil
be
described
;
Now
to
continew
the
same
draught
or
platforme
in
every
place
ye
shall
make
from
the
center
G
a
privie
circle
from
the
distance
GB
,
and
shal
be
sett
upon
the
privie
circumference
the
distance
AB
,
which
being
the
6
part
of
the
said
figure
Hexagonall
,
the
said
circumference
will
contayne
justly
still
fiue
parts
,
which
finally
will
come
to
end
at
A.
In
the
like
manner
is
the
privie
circle
made
out
of
the
same
center
G
,
and
from
the
distance
G
,
N
,
upon
the
circumference
thereof
shal
be
set
the
line
of
the
Polygone
interiour
HN
,
afterward
ye
must
onely
coppie
out
the
rest
.
Moreouer
,
the
parapet
wilbet
20
foote
,
which
shal
be
made
in
the
inside
of
ACL
.
MFB
,
and
as
the
way
of
the
rounds
,
or
the
falsebray
,
is
on
the
outside
of
the
body
of
the
Fortresse
,
we
will
make
parallells
towards
the
moate
of
20
foote
broad
,
&
then
20
foote
more
for
the
parapet
thereof
,
on
the
outside
of
the
parapet
,
before
ye
come
to
the
moate
,
is
made
an
edge
,
or
a
toe
of
6
or
8
foote
,
to
keepe
the
said
parapet
from
falling
into
the
moate
,
which
is
made
of
140
,
or
150
foote
or
thereabouts
,
as
necessitie
,
and
the
bottome
requires
;
for
it
must
be
fitted
according
to
this
consideration
,
because
the
ground
lying
lowe
,
it
will
not
beare
much
digging
before
ye
come
to
water
,
and
therefore
in
this
place
one
is
forced
to
make
the
moate
to
gett
earth
enough
for
the
rampart
,
but
when
the
ground
lies
high
,
then
ye
may
digg
very
deepe
before
ye
come
to
the
water
,
&
so
by
this
meanes
ye
may
haue
asmuch
earth
,
as
possibly
ye
can
well
use
,
&
seing
without
it
cannot
be
but
hurtfull
,
it
wil
be
better
not
to
make
the
moate
so
broad
,
that
ye
may
receiue
noe
hindrance
or
hurt
by
the
store
of
Earth
that
it
affordeth
;
which
may
be
cast
upon
the
outside
:
For
to
beleeue
that
the
ramparts
being
raised
higher
then
the
dimension
giuen
in
the
former
figure
to
wit
of
14
;
or
at
least
15
foote
,
that
they
were
better
as
some
haue
mainteyned
,
yet
experience
hath
showne
many
times
the
contrarie
,
for
an
ennemie
hauing
once
approched
to
the
brinke
of
the
moate
,
the
bredth
of
the
parapet
hinders
one
from
makeing
any
defence
upon
it
,
which
notwithstanding
is
most
necessarie
:
because
the
neerer
an
ennemie
comes
with
his
approches
towards
us
,
we
ought
the
more
to
defend
our selues
,
and
this
being
committed
by
our
default
,
it
is
too
manifest
that
one
ought
to
take
heede
of
it
,
and
to
remedie
this
errour
by
the
former
way
:
namely
in
makeing
the
rampart
of
the
heigth
abouesaid
.
Some
are
of
the
opinion
by
a
simple
commaund
that
one
ought
to
heigthen
the
ramparts
aboue
the
said
14
or
15
foote
,
and
not
the
Bulwarks
,
for
which
they
haue
some
reason
;
for
indeede
the
Bulwarkes
make
the
greatest
defence
,
and
when
an
ennemy
is
become
master
of
the
said
bulwarke
,
there
were
a
meanes
to
commaund
the
said
bulwarke
more
absolutely
,
so
that
an
ennemy
could
not
hinder
the
use
of
them
,
if
he
raised
not
his
works
aboue
the
said
rampart
,
to
hinder
the
defences
of
the
besieged
,
which
they
might
make
with
the
said
ramparts
so
raised
by
a
simple
commaund
,
which
in
some
manner
might
hinder
the
intents
&
approches
of
the
besiegers
,
the
more
easie
would
the
entrenchements
be
made
,
and
seing
the
more
the
bulwarks
are
raised
,
the
more
one
ought
to
use
diligence
about
the
entrenchment
;
hence
would
follow
this
difficulty
,
that
the
comming
to
the
rampart
or
the
bulwarke
,
would
not
be
so
easie
,
as
otherwise
,
neither
could
one
make
any
great
defence
from
such
ramparts
by
reason
of
their
too
much
heigth
,
in
such
sort
that
one
bulwarke
ought
to
defend
an
other
,
&
in
such
a
case
the
second
flanke
would
be
of
noe
use
after
an
ennemy
is
entred
into
the
moate
,
at
which
time
it
is
then
most
needfull
to
giue
the
greatest
resistance
that
may
be
,
seing
that
when
they
are
gotten
ouer
the
moate
,
and
taken
in
the
foot
of
the
bulwarke
,
then
the
courage
of
the
besiegers
begins
to
encrease
,
&
the
besieged
to
faile
them
,
for
it
is
too
apparant
,
that
many
times
resistance
failes
in
such
and
the
like
accidents
.
Betweene
two
bulwarkes
are
commonly
made
ravelins
or
halfe
moones
,
which
are
Quadrangular
figures
,
euery
face
containing
12
,
15
,
or
sometimes
20
rod
,
which
beginn
at
the
brinke
of
the
moate
:
so
that
its
angle
interiour
,
or
the
tenaille
O
,
lies
just
upon
the
brinke
and
angle
of
the
moate
,
the
reason
why
their
faces
P
,
Q
,
and
Q
,
R
,
are
made
of
a
greater
distance
,
is
that
the
deffence
may
be
the
better
:
for
otherwise
their
could
be
noe
defence
made
but
from
one
part
of
the
face
of
the
bulwarke
,
which
otherwise
would
not
be
sufficient
to
hinder
the
assaults
of
an
ennemie
,
which
ravelins
or
loose
peeces
li●
so
open
and
are
raised
aboue
the
plaine
field
some
foure
foote
,
that
one
may
the
easier
discouer
the
said
field
,
and
so
also
to
prevent
the
inconveniences
which
might
happen
,
when
one
would
giue
fire
from
them
upon
the
besiegers
,
(
if
the
said
ravelin
had
not
bene
raised
)
which
would
haue
annoyed
those
which
were
under
the
covert
way
.
Upon
the
said
Ravelin
being
so
raised
,
is
made
a
parapet
of
20
foote
broad
,
and
6
foote
high
,
which
wil
be
able
to
resist
Canon
,
the
moate
whereof
ought
to
be
made
50
,
or
60
foote
,
and
as
deepe
,
as
possibly
may
be
about
the
said
moate
,
ye
must
make
also
a
cover
way
of
some
20
foote
broade
,
as
hath
bene
noted
before
;
Afterward
ye
must
make
a
parapet
of
50
foote
broade
&
six
foote
high
,
ending
at
the
end
of
the
50
foote
towards
the
plaine
field
;
as
the
profile
doth
show
in
the
72
figure
,
and
to
the
intent
ye
may
the
better
understand
my
meaning
,
we
haue
made
the
said
Hexagone
in
perspectiue
,
with
the
falsebrayes
,
Ravelins
,
covert
wayes
,
and
their
parapets
apart
,
as
appeareth
by
the
second
71
figure
,
but
ye
must
note
,
that
the
said
Ravelins
must
be
in
respect
of
Ramparts
,
which
haue
a
double
height
,
to
see
in
them
the
more
perfection
,
also
a
double
height
,
to
make
them
show
the
better
:
for
otherwise
they
would
seeme
too
little
for
the
reason
abouesaid
against
our
intention
.
The
72.
Figure
.
THe
Profile
shal
be
made
as
in
the
12
Plate
of
the
former
figure
70
,
by
drawing
a
privie
line
infinite
,
and
takeing
upon
it
all
the
dimensions
,
as
first
the
bredth
of
the
ramparts
,
the
parapets
,
the
falsebrayes
,
its
parapet
,
the
bredth
of
the
moate
,
the
covert
way
,
and
the
other
parts
of
the
said
profile
;
and
seing
wee
haue
here
ordained
Ravelins
,
wee
will
make
this
profile
from
the
midst
of
the
rampart
,
passing
also
through
the
midst
of
the
Raveline
,
that
ye
may
the
better
understand
our
meaning
.
According
to
which
ye
shall
take
the
bredth
of
the
Rampart
68
foote
from
A
to
C
,
from
C
to
D
20
foote
,
from
D
to
E
20
foote
,
and
from
E
to
F
6
foote
,
the
bredth
of
the
moate
shal
be
150
foote
,
as
from
F
to
G
,
which
ought
to
be
broader
,
but
seing
the
distance
falls
out
too
litle
,
it
shal
be
made
but
of
150
foote
,
the
Raveline
ought
to
haue
in
this
place
180
foote
,
but
seing
the
ground
faile
us
,
wee
haue
made
the
dimension
thereof
180
foote
,
but
this
hindrance
would
not
permit
us
,
to
make
it
of
that
competent
greatnesse
as
it
ought
,
which
I
speake
to
giue
noe
stop
to
the
Reader
,
which
Ravelin
is
raised
4
foote
aboue
the
plaine
field
from
6
to
T
,
upon
which
a
parapet
is
made
,
whereof
the
basis
shal
be
20
foote
broad
,
and
6
foote
high
,
to
wit
its
bredth
from
Q
to
R
,
and
the
height
from
Q
to
S
,
the
footebankes
are
made
of
the
same
bredth
&
heigth
,
&
therefore
I
will
make
here
no
mention
of
them
,
neither
of
the
taluds
or
sloopings
of
Ramparts
,
Parapets
,
&
moates
;
aswell
the
interiour
,
as
the
exteriour
parts
are
made
according
to
the
nature
of
the
soile
;
for
by
how
much
the
ground
is
leane
&
sandie
,
by
so
much
ye
ought
to
giue
it
the
talud
,
&
by
that
meanes
ye
shall
hinder
the
falling
downe
of
the
workes
,
for
which
reason
often
times
ye
must
giue
on
the
exteriour
side
,
being
a
sandie
ground
,
as
much
talude
as
heigth
,
and
if
the
said
Ravelin
be
also
raised
4
foote
aboue
the
plaine
field
,
me
thinks
it
will
not
be
amisse
,
to
raise
the
rampart
thereof
,
which
is
15
foote
,
some
6
foote
higher
to
commaund
the
better
the
said
Raueline
,
&
leaue
the
Bulwarks
of
the
height
of
15
foote
,
that
all
inconveniences
may
be
prevented
:
on
the
outside
of
the
Raveline
is
made
the
edge
HI
of
6
foote
,
and
a
moate
of
50
foote
,
then
the
covert
way
20
foote
,
&
the
parapet
LN
60
foote
,
with
a
footebanke
of
the
ordinary
bredth
of
3
foote
in
that
manner
,
as
the
72
figure
demonstrateth
.
The
description
of
the
plote
of
a
Fortresse
Heptagonall
.
The
14.
Plate
,
and
73.
Figure
.
LEt
there
be
giuen
to
be
fortified
a
Heptagone
,
whereof
the
side
of
the
Polygone
exteriour
maketh
63
rod
,
and
the
angle
flanked
80
degrees
.
To
doe
this
,
ye
shall
first
divide
the
said
side
AB
into
7
equall
parts
,
as
ye
see
by
the
points
1.
2.
3.
4.
5.
6.
7.
and
from
the
points
of
the
5
and
2
parts
the
perpendiculars
D
,
N
,
and
E
,
O
shal
be
drawne
being
equall
to
A
,
D
,
and
E
,
B
,
to
wit
,
either
of
them
2
seuenth
parts
of
the
line
A
,
B
,
then
ye
shall
draw
from
the
extremities
,
or
the
utmost
parts
of
these
perpendiculars
,
the
privie
line
infinite
H
,
I.
which
shal
be
the
side
of
the
Polygone
interiour
,
and
by
consequence
N
,
O
the
curtaine
.
The
angles
C
,
A
,
B
,
and
C
,
B
,
A
,
shal
be
made
by
the
helpe
of
a
graduate-instrument
,
as
our
compasse
is
,
or
an
other
;
either
of
64
½
degrees
;
in
such
sort
that
the
intersection
of
the
lines
A
,
C.
C
,
B
,
which
is
C
,
shal
be
the
center
of
the
said
Fortresse
Heptagonall
.
Afterward
I
,
B
,
8
,
of
the
40
degr
.
which
is
the
halfe
of
the
angle
flanked
,
that
ye
may
haue
the
whole
angle
of
80
degr
.
according
to
the
Hypotheses
,
and
where
the
said
lines
cutt
the
said
perpendiculars
D
,
N
,
and
E
,
O
,
in
the
points
2.
&
8
,
ye
shall
haue
the
faces
A
2.
&
B
8
,
the
flanks
N.
2.
&
O.
8
,
and
also
the
gorges
H
,
N.
&
O
,
I
,
the
Parapets
,
Ramparts
,
Falsebrayes
&
for
the
other
parts
of
this
Fortresse
,
ye
shall
doe
the
like
as
hath
ben
taught
in
the
former
figure
Hexagonall
,
the
faces
in
this
present
Raveline
shal
be
made
of
15
rod
,
the
moate
of
10
rod
,
and
the
moate
to
the
Ravelin
of
15
foote
,
then
about
it
shal
be
made
a
covert
way
of
20
foote
,
&
a
parapet
of
60
foote
broad
descending
(
as
we
haue
said
before
)
sloopingly
.
According
to
this
forme
the
flanke
wil
be
then
made
of
a
reasonable
greatnesse
,
but
the
gorge
so
much
the
lesser
,
which
may
be
made
so
,
seing
ye
intend
not
to
make
any
flankes
in
the
Casemates
,
but
in
case
ye
would
,
ye
must
then
make
the
flanke
the
lesser
,
and
consequently
the
gorge
the
greater
,
to
the
end
ye
may
haue
space
enough
for
to
make
within
them
the
said
Casemates
.
The
like
is
to
be
understood
when
ye
meane
to
make
a
Catt
upon
the
Bulwarke
,
for
the
gorge
being
so
litle
(
besides
other
inconveniences
)
it
is
impossible
to
make
there
such
a
bodie
,
in
reserving
sufficient
space
for
the
flanks
,
which
me
thinks
in
such
places
,
that
haue
such
narrow
gorges
,
it
were
better
to
make
the
said
Cats
upon
the
Curtaine
,
yet
so
that
they
be
noe
hindrance
to
the
Rounds
going
along
the
Ramparts
,
for
which
reason
they
ought
to
be
made
more
inward
,
and
so
that
they
may
lie
as
neere
the
rampart
as
possibly
may
be
,
the
better
to
discouver
,
and
commaund
the
fields
about
them
.
The
74.
75.
Figures
.
THe
figure
74
is
the
profile
of
the
Rampart
,
Falsebray
,
its
Parapet
,
moate
,
covert
way
,
and
its
parapet
,
whereof
the
length
,
bredth
,
heigth
and
depth
are
marked
out
in
the
said
figure
74
,
which
is
a
section
of
the
Bulwarck
,
as
the
figure
75
,
the
section
of
the
midst
of
the
Rampart
,
and
because
all
the
profile
cannot
commodiously
be
drawne
out
,
we
haue
represented
but
a
part
of
the
moate
,
the
forme
of
the
Raveline
,
its
heigth
,
and
the
forme
of
its
parapet
,
the
bredth
and
the
depth
of
its
side
,
with
their
taluds
or
sloopings
,
the
covert
way
with
its
parapet
&
footebanke
,
as
ye
may
exactly
note
by
the
figures
hereunto
annexed
.
The
2.
of
the
14.
Plate
.
ANd
for
the
better
facilitating
of
that
〈◊〉
we
haue
spoken
of
this
present
figure
73.
we
haue
added
hereunto
the
platforme
of
the
Fortresse
of
Coverden
in
Frizland
,
whereby
ye
may
the
better
understand
our
intention
,
being
accounted
the
master
peece
,
and
the
most
regular
,
and
royall
Fort
in
the
Low-Countries
,
hauing
a
False-braye
,
Rauelins
,
or
half
moones
,
covertwayes
,
with
which
wee
will
put
an
end
to
the
description
of
our
regular
Fortresses
.
A
succint
description
of
some
other
works
in
the
said
Heptagone
,
which
are
of
an
other
manner
of
makeing
then
the
former
.
The
15.
Plate
,
&
76.
Figure
.
LEt
the
side
of
the
Polygone
interiour
be
B
,
C
,
upon
which
ye
would
haue
a
part
of
a
Fortresse
Heptagonall
made
,
whereof
the
face
to
the
curtaine
is
as
3
to
4.
and
the
gorge
to
the
flanke
as
13
to
9.
To
doe
this
,
ye
must
doe
as
we
haue
taught
yow
in
the
11.
Plate
,
or
in
the
9.
Plate
,
and
Figure
3.
50
,
where
ye
shall
finde
the
proportion
requisite
:
on
the
outside
of
the
same
face
,
shal
be
made
the
parallells
each
of
20
foote
,
aswell
for
the
parapet
,
as
for
the
falsebray
,
the
like
is
to
be
done
in
the
flanks
,
and
curtaines
,
and
on
the
inside
a
parallell
of
20
foote
,
with
an
other
parallell
of
72
foote
for
the
bredth
of
the
Rampart
in
the
basis
,
in
the
same
manner
as
this
76
figure
sheweth
,
on
the
outside
ye
shall
make
a
parallell
of
10
rod
for
the
bredth
of
the
moate
,
and
upon
the
brinke
of
it
to
the
tenaille
,
ye
shall
lay
out
the
Ravelins
,
as
we
haue
said
in
the
former
figure
73
,
in
the
angles
of
the
Bulwarks
ye
shall
make
also
Ravelins
,
such
as
ye
see
here
marked
with
I
,
K
,
L
,
M
,
which
shal
be
defended
by
the
Ravelins
E
,
F
,
G
,
H
,
&
the
better
to
defend
the
said
ravelins
,
which
are
in
the
angles
of
the
Bulwarks
,
ye
shall
make
the
other
works
N
,
O
,
P
,
&
Q
,
which
are
called
Horne-works
,
which
are
made
in
such
a
manner
,
that
the
moates
of
these
Horneworks
comes
to
answere
to
the
Falsebray
,
which
is
in
the
flanks
of
the
Bulwarks
:
so
that
their
bredth
S
,
T
,
maketh
about
32
rod
,
of
which
distance
the
two
demy-Bulwarks
V
,
X
,
are
made
(
whereof
the
face
,
and
the
curtaine
are
of
a
like
greatnesse
)
according
to
the
rule
prescribed
in
the
10.
Plate
,
the
flanks
wil
be
found
,
makeing
from
the
angle
of
the
shoulder
,
two
lines
perpendicular
upon
the
curtaine
,
which
is
directly
opposite
to
it
.
euen
as
the
figures
77
,
78
demonstrate
.
The
utmost
angles
of
the
said
Horneworks
ought
not
to
be
further
from
the
curtaine
of
the
Fortresse
,
then
a
Musket
can
carry
point-blanke
,
which
is
60
rod
or
there
abouts
,
and
if
the
Rauelins
EF
,
GH
,
are
not
made
in
the
tenailles
,
then
ye
may
cut
the
said
Hornworke
,
as
is
here
marked
out
in
the
78
figure
:
the
bredth
of
the
moate
shal
be
24
or
25
foote
:
The
Rampart
here
is
noe
otherwise
then
the
parapet
of
the
same
bredth
.
The
depth
of
the
moate
may
be
made
6
foote
in
case
the
ground
lies
low
;
but
otherwise
the
deeper
it
is
made
it
wil
be
so
much
the
better
,
the
bredth
will
not
hinder
it
,
though
it
were
made
but
36
foote
;
for
according
to
the
same
bredth
&
depth
ye
may
enlarge
the
Rampart
,
aswell
in
height
as
in
bredth
,
upon
which
afterward
ye
may
make
a
parapet
as
great
as
it
can
well
beare
:
but
when
haste
requireth
,
I
would
make
noe
Rampart
,
but
onely
a
Parapet
of
24
foote
thick
,
as
wee
haue
made
here
being
6
foote
high
,
and
a
footbanke
of
3
or
4
foote
.
The
figure
79
marketh
out
a
Profile
,
which
is
a
section
or
cutt
drawne
through
the
midst
of
a
curtaine
passing
through
the
midst
of
the
Ravelines
G
,
and
traversing
through
the
Horneworke
P
,
and
the
Rampart
and
the
moate
.
The
true
dimensions
thereof
are
clearely
expressed
in
the
said
Profile
,
by
the
helpe
of
the
scale
;
joyned
to
it
:
as
also
by
the
meanes
of
the
Alphabeticall
letters
,
shewing
the
feete
to
be
12
ynches
,
whereof
the
length
is
〈…〉
in
the
25
Plate
of
our
Geometrie
.
The
benefit
of
such
workes
are
well
knowne
,
when
they
are
made
in
places
of
advantage
for
deffence
,
or
where
noe
men
,
victualls
,
and
amunition
of
warre
are
wanting
,
as
also
where
the
ground
is
of
a
reasonable
largenesse
,
to
wit
at
the
least
32
rod
,
that
the
faces
of
the
said
horneworkes
may
be
about
12
rod
,
which
is
the
least
length
that
one
can
giue
to
such
workes
,
against
the
attempt
,
which
may
be
giuen
by
an
Army
:
&
as
a
great
assault
,
cannot
be
resisted
with
a
few
men
,
but
up
a
small
roome
:
so
is
it
manifest
that
the
greatnesse
&
largenesse
of
such
a
place
,
must
be
answerable
.
Therefore
in
my
opinion
,
it
is
a
thing
repugnant
to
the
rule
of
Fortification
,
to
make
such
Horneworkes
in
the
angles
of
Bulwarkes
,
where
they
are
so
straightned
.
For
Demy-bulwarkes
cannot
(
by
reason
of
their
smallnesse
)
be
well
mainteyned
and
defended
,
and
on
the
other
side
,
their
two
wings
cannot
be
defended
from
the
maine
Fortresse
,
but
with
great
disadvantage
,
as
wee
intend
to
discusse
:
thereof
more
at
large
hereafter
in
the
plateforme
of
Gulick
.
How
Citadels
,
or
Castles
may
be
joyned
to
Townes
,
or
Townes
to
Castles
,
The
16.
Plate
,
and
the
81.
82.
Figures
.
FOr
to
build
a
Castle
(
or
a
Citadell
)
to
a
Towne
,
either
to
help
to
defend
it
the
better
,
or
to
curbe
it
:
ye
must
finde
out
first
the
most
advantagious
place
:
in
case
there
be
a
river
,
then
ye
shall
build
your
Castle
upon
the
side
of
it
,
where
it
may
best
commaund
,
with
all
the
advantage
that
possibly
may
be
,
also
takeing
heede
that
it
be
built
in
such
a
place
,
that
the
Castle
may
receiue
noe
disadvantage
thereby
,
but
that
the
Rampart
of
the
Towne
where
the
letters
A
,
G
are
,
be
the
weakest
part
;
to
the
end
that
by
this
meanes
,
ye
may
frustrate
the
desire
of
the
inhabitans
of
the
said
towne
,
from
opposing
themselues
against
the
said
Castle
:
but
if
ye
feare
noe
such
inconvenience
,
ye
may
make
the
places
A
,
G
,
as
strong
as
the
other
:
saving
that
the
Ramparts
in
the
said
angles
be
noe
thicker
in
the
angles
A
,
G
(
which
are
angles
of
imperfect
Bulwarkes
)
then
in
other
places
,
and
not
massiue
or
full
of
earth
,
as
the
Bulwarkes
B
,
&
C
,
D
,
are
.
Let
there
be
a
towne
then
contayning
8
angles
,
whereof
7
of
them
are
marked
out
with
these
Alphabeticall
letters
A
,
B
,
C
,
D
,
E
,
F
,
&
G
,
and
that
you
are
desirous
to
make
a
Pentagonall
Citadell
.
For
the
doing
whereof
ye
must
first
make
a
mapp
of
the
said
towne
Octogonall
:
in
case
it
be
not
made
,
according
to
the
methode
giuen
heretofore
in
our
Geometrie
,
which
is
here
marked
out
by
these
letters
A
,
B
,
C
,
D
,
E
,
F
,
&
G
,
with
the
Falsebrays
,
Parapets
,
Moates
,
Covert
way
,
&
its
Parapet
.
Then
upon
a
paper
of
the
same
scale
,
whereof
the
mapp
of
the
said
towne
is
made
,
being
traced
out
in
a
Pentagone
with
all
the
outworkes
,
such
as
the
figure
marked
H
,
I
,
K
,
L
,
M
,
demonstrats
,
according
to
the
method
giuen
yow
before
in
the
10
and
11
Plates
,
then
the
same
figure
Pentagonall
must
be
cutt
perfectly
,
and
so
fitted
to
the
towne
,
remouing
it
so
longe
till
yow
haue
at
length
found
out
the
most
advantagious
place
to
make
there
your
Castle
Pentagone
,
takeing
heede
that
the
tenailles
N
and
O
,
fall
out
right
,
to
answere
the
curtaines
P
,
O
,
and
Q
,
N
,
to
the
end
that
the
entrance
into
the
towne
may
be
the
more
commodious
,
&
that
the
towne
in
the
places
N
,
R
,
S
,
T
,
O
,
be
not
discouered
without
,
whereof
one
ought
to
haue
a
singular
care
:
for
there
where
the
said
angles
of
the
tenaille
,
be
not
well
joyned
to
the
Ramparts
,
those
places
will
alwayes
be
badly
defended
in
the
said
joynings
,
as
ye
may
obserue
in
the
Castle
of
Gulick
,
which
platforme
followes
next
.
It
is
true
;
that
when
ye
make
these
angles
in
the
curtaines
,
that
the
parts
N
,
R
,
S
,
T
,
O
,
take
up
a
great
deale
of
the
towne
,
which
may
be
prevented
in
setting
the
Castle
more
outward
,
and
makeing
the
said
curtaines
meete
neerer
the
faces
of
the
Bulwarks
of
the
Castle
.
But
the
side
of
the
towne
N
,
R
,
S
,
T
,
O
,
will
in
noe
wise
be
guarded
,
as
the
Platforme
of
the
Castle
of
Gulick
sheweth
:
so
that
necessity
will
require
,
for
the
preventing
of
such
an
inconveniencie
,
to
make
some
Ravelins
,
or
some
other
works
without
the
said
place
,
as
the
commodiousnesse
of
the
ground
may
afforde
,
which
would
make
mee
resolue
rather
to
approue
of
the
forme
of
the
80
figure
,
then
of
the
other
,
were
it
not
for
some
cause
more
urgent
,
then
is
here
alledged
,
which
reason
would
moue
mee
to
change
my
opinion
.
If
the
towne
or
Cittie
A
,
B
,
C
,
D
,
E
,
F
,
G
,
were
greater
,
then
were
it
vnnecessary
to
take
so
much
care
for
the
place
,
which
the
Castle
taketh
up
:
for
then
ye
might
make
the
curtaines
P
,
O
,
&
Q
,
N
,
longer
,
to
wit
some
50
rod
,
that
the
flanks
V
,
X
,
and
Y
,
Z
,
might
not
come
so
neere
the
Bulwarks
E
,
M
:
from
whence
one
may
easely
discouer
the
Falsebray
of
the
Castle
,
for
the
preventing
of
which
inconvenience
,
it
wil
be
good
to
raise
the
Bulwarkes
of
the
Castle
,
which
looks
into
the
towne
,
somewhat
higher
aboue
the
Ramparts
thereof
,
but
the
towne
being
of
a
small
circuit
(
as
wee
haue
said
)
one
shal
be
constrained
to
use
for
it
the
same
figures
as
are
demonstrated
by
the
figures
80.
&
81.
If
one
would
built
a
towne
to
a
Castle
,
first
ye
must
obserue
diligently
the
site
of
the
place
,
and
if
it
may
be
made
regular
,
to
take
care
,
how
many
Inhabitants
this
place
will
containe
,
the
nature
and
quality
of
them
,
and
whether
the
towne
may
in
processe
of
time
be
enlarged
:
which
ye
shal
be
able
to
judge
,
when
ye
see
the
countrie
that
lies
about
it
,
what
wayes
,
avenues
and
passages
there
are
to
come
to
it
,
as
also
to
consider
the
goodnesse
of
the
aire
,
and
of
the
soile
of
all
which
things
(
as
Vitruvius
saith
)
one
ought
to
haue
a
most
speciall
care
.
According
to
all
which
one
ought
to
gouerne
themselues
:
for
to
enclosse
a
great
peece
of
ground
,
where
there
is
none
,
or
little
appearance
of
Inhabitants
,
this
were
rather
to
ruinate
,
then
to
edifie
,
to
which
takeing
heede
then
,
ye
must
resolue
of
the
Circuit
,
or
Capacity
,
which
ye
would
giue
to
the
said
towne
,
and
suppose
ye
would
make
the
forme
thereof
Octogonall
,
and
regular
:
ye
must
trace
out
then
an
Octogonall
Fortresse
regular
,
according
to
the
former
method
,
with
covert
wayes
,
and
what
ye
require
more
upon
the
same
scale
,
as
you
made
your
Castle
Pentagonall
,
then
ye
must
cutt
the
said
forme
Octogonall
,
and
joyne
it
to
the
said
Castle
,
removing
it
so
long
,
till
yow
haue
found
out
&
chosen
out
(
with
all
circumspection
requisite
)
the
most
convenientest
place
,
and
staying
there
ye
may
trace
out
the
said
towne
,
as
the
figures
80
and
81
demonstrate
,
joyned
to
the
Castle
.
But
as
we
haue
said
,
it
will
loose
much
of
its
capacity
,
because
the
curtaines
of
the
towne
will
come
to
meete
with
the
angles
of
the
tenailles
of
the
Castle
in
the
points
marked
O
,
and
N.
The
17.
Plate
,
and
the
82.
83.
Figure
.
IF
ye
will
not
tye
your selfe
so
precisely
to
this
forme
Octogonall
,
ye
may
make
the
towne
much
better
by
the
meanes
which
wee
will
now
speake
of
.
Let
first
be
drawne
the
lines
D
,
D
,
and
A
,
A.
&c.
right
perpendiculars
upon
the
midst
of
the
curtaines
N
,
M
,
and
K
,
Y
,
of
some
60
rod
,
or
there
abouts
,
then
let
there
be
made
the
perpendicular
line
infinite
in
the
midst
,
and
upon
the
curtaine
D
,
E
;
in
which
ye
haue
the
center
of
the
circle
,
which
makes
the
circle
of
all
the
figure
,
and
suppose
ye
desire
to
make
a
forme
regular
of
nine
Bulwarks
,
whereof
the
angles
flanked
,
are
distant
one
from
an
other
80
rod
,
from
the
utmost
end
Q
,
shal
be
made
an
arch
,
in
such
sort
,
that
within
the
circuit
thereof
the
distance
N
,
P
,
is
contained
9
times
,
which
shall
make
the
point
of
the
center
C
,
C.
then
upon
the
circumference
of
the
said
arch
the
said
9
points
,
being
marked
as
they
are
noted
here
by
the
letters
Q
,
R
,
S
,
T
,
V
,
X
,
Y
,
Z
,
&c
ye
shall
make
the
angles
interiour
flanking
,
according
to
their
forme
,
that
ye
may
haue
the
angle
flanked
right
,
&
that
ye
may
the
better
make
the
angle
flanking
,
or
the
tenaille
pointed
and
small
,
which
is
accounted
the
best
,
as
that
skilfull
,
and
diligent
man
M.
Iohn
Errard
of
Barleduke
,
Ingenier
to
the
King
of
France
hath
obserued
well
,
and
to
know
how
much
the
said
angle
flanking
will
make
,
ye
must
substract
the
halfe
of
the
angle
from
the
Bulwarke
,
or
flanked
from
the
halfe
of
the
angle
of
the
Polygone
,
the
rest
shal
be
the
angle
flanking
interiour
.
But
if
the
angle
of
the
Polygone
be
lesser
then
150
degrees
,
which
is
the
angle
of
a
Dodecagone
,
ye
shall
then
governe
your selfe
according
to
the
table
giuen
before
,
takeing
heede
that
the
angle
of
the
Polygone
agreeth
most
with
the
angle
to
be
fortified
,
and
in
its
columne
is
found
the
angle
flanking
interiour
,
necessary
to
the
said
angle
,
by
which
ye
may
come
to
the
angle
of
the
tenaille
,
&
seing
in
this
exemple
wee
haue
set
downe
the
distance
from
angle
to
angle
,
as
wel
of
the
Pentag
.
as
of
the
towne
80
rod
,
wee
wil
set
downe
upon
the
lines
,
which
make
the
tenailles
2
equall
parts
,
&
from
the
said
2
parts
in
the
line
which
formeth
the
angle
flanking
,
a
parallell
line
being
drawne
to
that
of
80
rod
,
&
placed
upon
the
3
equall
parts
,
a
privie
line
infinite
,
must
be
drawne
through
the
said
3
point
,
&
through
the
angle
,
and
where
the
same
comes
to
cutt
the
other
side
of
the
tenaille
in
the
point
O
,
ye
shall
haue
the
shoulder
,
through
which
the
line
A
,
B
,
being
drawne
,
makeing
the
angle
forming
the
flanke
40
degrees
,
ye
shall
haue
the
center
of
the
Bulwarks
justas
wee
haue
taught
heretofore
in
the
10
&
11
plates
.
If
the
Castle
had
bin
made
with
foure
angles
,
the
construction
would
haue
ben
the
same
,
and
if
the
lines
D
,
D
,
Q
,
and
A
,
A
,
&c.
had
bene
somewhat
more
then
60
rod
,
the
forme
would
haue
bene
better
,
for
though
we
haue
made
the
lines
of
defence
,
to
be
60
rod
in
the
former
Fortresses
,
there
is
an
other
consideration
in
this
present
building
;
in
regard
the
Bulwarkes
Q
,
&c.
advance
themselues
towards
the
angles
Q
,
&c.
from
whence
the
said
curtaines
Q
,
E
,
E.
&
B
,
B
&c.
maye
be
sufficiently
defended
,
when
they
doe
not
exceede
aboue
70
rod
,
the
moates
,
covert
wayes
,
falsebrayes
and
other
workes
,
must
be
made
in
that
forme
as
we
haue
sett
them
downe
before
in
the
16.
plate
.
How
to
describe
the
order
which
is
obserued
in
makeing
of
a
platforme
,
and
of
some
other
appendances
.
THe
draught
being
made
upon
a
paper
,
ere
ye
make
a
platforme
of
it
,
ye
ought
first
to
ouercast
,
aswell
the
lines
,
as
the
body
of
the
Fortresse
,
in
that
manner
as
we
haue
showne
yow
before
in
our
Geometrie
,
that
ye
may
know
pertinently
all
the
particular
dimensions
,
&
so
committ
noe
errour
,
in
makeing
a
plateforme
thereof
:
Before
ye
lett
it
out
,
according
to
our
Method
giuen
in
the
conclusion
of
the
practize
of
Geometrie
,
which
being
done
before
the
Inginier
begins
the
worke
,
he
shall
declare
the
particular
conditions
,
which
he
would
haue
obserued
in
the
makeing
and
finishing
of
this
worke
,
as
the
time
when
they
are
to
beginn
,
how
long
it
wil
be
before
it
be
finished
,
how
many
men
he
must
ordinarily
haue
to
worke
,
what
instruments
they
must
use
for
the
driving
in
of
piles
,
and
the
laying
of
the
foundation
,
how
many
foote
,
and
what
edge
they
are
to
giue
to
the
wall
,
or
the
parapet
of
the
falsebray
,
the
thicknesse
of
the
parapet
,
and
how
broad
the
falsebray
ought
to
be
,
how
much
the
talud
(
or
slooping
)
on
the
inside
and
on
the
outside
must
be
▪
and
that
according
to
the
goodnesse
or
weaknesse
of
the
matter
upon
which
one
builds
,
how
many
bundles
of
brush
in
a
sand●e
Earth
he
is
to
lay
in
the
parapet
of
the
falsebray
,
and
in
the
Rampart
,
the
height
,
the
Talud
or
slooping
,
both
on
the
inside
,
&
on
the
outside
,
the
bredth
and
depth
of
the
moate
,
&
its
slooping
,
and
generally
and
specially
all
things
touching
the
makeing
of
such
a
Castle
,
whereof
he
hath
made
his
platforme
.
And
some
dayes
before
they
come
to
the
place
,
or
that
the
forme
be
drawne
out
,
they
shall
sett
up
billets
in
the
next
adjoyning
townes
to
giue
notice
to
the
Worke
masters
,
that
upon
such
and
such
a
day
,
one
is
resolued
to
hire
and
let
out
such
&
such
works
,
to
him
that
will
take
them
on
upon
the
least
price
:
and
before
the
day
prefixed
the
said
Inginier
shall
come
thither
himself
with
some
one
of
the
Deputies
,
or
some
other
that
hath
the
managing
of
that
worke
,
to
draw
a
draught
of
the
said
Fortresse
,
and
to
prepare
all
things
necessarie
against
that
day
assigned
.
Then
tho
day
being
come
,
all
the
Worke-masters
&
Undertakers
shall
come
together
,
either
upon
the
place
where
the
worke
is
to
be
made
,
or
else
in
some
other
place
where
the
conditions
and
couenants
are
read
,
according
as
they
will
haue
the
worke
hastned
:
this
done
,
they
shall
demaund
,
who
will
take
it
on
for
the
least
money
,
then
one
of
the
Workemasters
will
say
for
so
much
,
an
other
for
so
much
lesser
,
which
is
done
till
noe
man
will
take
it
on
at
a
lesser
rate
,
then
underneath
they
signe
unto
the
articles
of
the
conditions
,
that
I
N
,
N
,
haue
undertaken
the
worke
upon
such
and
such
conditions
,
for
the
summe
of
so
much
money
:
sometimes
ye
shall
haue
two
or
three
Undertakers
joyne
together
and
undertake
the
whole
worke
,
all
of
them
signing
to
the
articles
,
as
also
the
Commissioners
,
and
Inginier
must
doe
on
their
part
:
then
they
beginn
the
worke
,
and
commonly
the
Undertakers
are
bound
by
the
said
articles
and
Contracts
,
to
deliuer
all
the
materialls
necessarie
for
the
said
worke
,
which
they
receiue
from
the
Master
of
the
Magazin
there
or
else
,
where
giving
him
a
note
under
their
hands
for
the
restitution
of
them
back
againe
,
when
the
worke
is
finished
,
and
to
make
good
whatsoeuer
they
either
breake
or
loose
.
Then
the
Workemaster
placeth
his
men
in
order
,
as
he
knowes
the
quality
of
the
worke
requireth
best
,
some
to
digg
,
and
cast
up
the
Earth
,
others
to
fill
Karrs
,
and
Wheele-barrowes
,
others
to
levell
the
ground
which
is
pricked
out
;
for
in
the
beginning
it
wil
be
necessary
to
carry
the
Earth
away
,
with
a
Horse
and
a
Karre
,
which
is
cast
up
on
the
outside
of
the
moate
upon
the
basis
of
the
Rampart
,
&
not
with
Wheele-barrowes
as
they
must
doe
at
the
last
,
when
the
worke
growes
high
,
and
that
ye
are
come
to
the
depth
of
the
moate
,
for
then
ye
cannot
so
well
use
Horse
and
Karrs
;
by
reason
of
the
difficulties
knowne
to
all
men
,
namely
,
that
the
treading
of
the
horses
and
the
Karrs
spoyle
the
groundworke
,
and
cannot
be
laid
so
well
,
as
when
men
brings
the
Earth
up
in
Wheele-barrowes
upon
away
ascending
with
plancies
,
as
hath
bene
found
by
experience
often
times
by
the
undertakers
of
such
workes
.
Of
the
Foundations
.
2.
The
17.
Plate
.
WHen
the
foundation
is
bad
,
it
wil
be
your
best
course
to
driue
in
piles
,
along
the
foote
of
the
Rampart
,
just
where
the
said
Rampart
beginneth
,
some
piles
25
,
or
30
foote
,
or
else
yong
Alders-trees
,
about
foure
or
fiue
yeares
growth
along
the
worke
about
18
or
20
foote
long
,
closse
one
to
an
other
,
and
the
closser
the
better
,
and
though
this
be
a
great
charge
,
and
a
long
worke
;
yet
this
is
the
best
foundation
ye
can
choose
:
But
as
we
haue
said
oftentimes
,
seing
it
hapneth
that
one
cannot
stay
so
long
,
ye
must
then
take
bundles
of
brush
,
and
binde
them
up
in
fagots
,
&
lay
them
crossewise
,
driving
them
into
the
Earth
with
stakes
to
fasten
them
together
,
that
they
may
lie
firme
,
and
if
ye
lay
the
foundation
in
a
ditch
or
a
quagmire
,
ye
must
make
use
of
such
fagots
bound
and
tyed
up
together
with
cords
,
&
if
it
be
in
a
river
,
then
ye
must
binde
brickbats
,
or
stones
betweene
two
bundles
,
to
make
them
sinke
to
the
botome
:
ye
may
make
them
as
long
,
or
as
short
as
ye
please
,
and
so
necessity
requires
,
till
ye
haue
filled
up
the
water
with
these
sinkers
,
upon
which
yow
must
lay
your
foundation
,
then
ye
must
also
driue
in
some
piles
about
18
or
20
foote
longe
,
(
according
to
the
depth
)
to
make
them
lie
the
more
firme
and
solid
together
.
Now
these
bundles
are
made
in
the
forme
of
Saucidges
,
as
here
followeth
.
First
,
ye
driue
into
the
ground
stakes
of
the
heigth
&
bredth
as
ye
desire
to
make
these
Saucidges
,
either
great
or
small
,
the
one
a
foote
high
,
the
other
a
foote
and
a
halfe
,
and
sometimes
two
foote
,
then
ye
lay
betweene
these
stakes
bulrushes
,
boughs
or
small
bundles
of
brush
bound
fast
together
,
and
in
the
midst
of
them
ye
must
lay
brick-bats
,
when
ye
sinke
them
in
a
river
,
but
of
Earth
if
ye
will
lay
them
as
a
foundation
in
a
moate
all
along
the
said
worke
,
then
ye
binde
the
said
bundles
with
stones
,
or
Earth
in
them
,
fastning
them
with
withs
in
the
same
manner
,
as
ye
binde
up
a
fagot
,
or
a
bavin
,
and
in
this
forme
are
your
Saucidges
,
or
sinkers
made
,
which
being
done
,
the
letters
A
are
bound
one
to
another
,
as
strong
as
possible
may
be
,
as
ye
may
perceiue
by
the
letter
B
in
the
2.
17.
Plate
.
Then
ye
sinke
these
downe
to
the
bottome
,
that
ye
may
afterward
build
your
Fortresse
upon
them
,
according
to
the
former
platforme
,
whereof
a
Plate
is
made
before
,
which
groundworks
among
the
Ingeniers
of
the
united
Provinces
are
much
used
,
because
these
parts
are
maritimate
,
moorish
,
and
subject
to
bad
foundations
,
so
that
in
some
places
the
foundations
cost
more
then
the
rest
of
the
building
it self
,
yea
in
diuerse
places
of
these
Countries
,
the
Inhabitants
are
driuen
to
keepe
their
grounds
from
the
violence
of
the
Sea
and
Rivers
,
by
such
and
such
like
meanes
and
remedies
,
to
their
great
and
unspeakeable
charges
,
for
one
would
haue
thought
it
had
ben
impossible
for
them
so
many
yeares
to
doe
it
,
since
the
times
that
these
Provinces
were
united
,
had
not
the
superintendants
,
&
Dikegraues
used
great
care
and
diligence
in
continuing
the
same
preservation
for
the
good
of
the
inhabitants
,
and
common
weale
,
whereof
wee
will
speake
more
at
large
hereafter
.
When
the
worke
is
raised
aboue
the
water
,
and
that
the
Earth
in
the
moate
diminisheth
,
the
Worck-master
shall
then
beginn
to
lay
his
edge
,
or
border
upon
the
line
drawne
out
for
the
Rampart
.
Now
the
line
being
drawne
out
upon
the
ground
,
as
ye
see
in
the
line
C.
by
the
2.
of
the
17.
Plate
,
being
fiue
or
six
ynches
broad
,
and
as
deepe
,
which
is
done
by
the
Ingenier
himself
to
take
away
all
excuse
from
the
Work-masters
,
that
he
may
likewise
carrie
a
right
line
,
aswell
on
the
inside
as
on
the
outside
of
the
Rampart
,
as
also
for
the
moate
,
and
the
other
parts
of
the
said
Fortresse
,
represented
in
the
plat-forme
:
Besides
if
the
condition
of
the
undertaker
containes
,
that
the
outside
of
the
Rampart
must
be
raised
with
sodds
,
which
are
ordinarily
4
or
5
ynches
square
,
and
14
or
15
ynches
long
diminishing
towards
the
inside
,
as
ye
may
see
here
by
the
figure
D
,
to
the
end
that
betweene
the
said
rowes
of
sodds
,
one
may
lay
a
litle
earth
to
settle
them
the
better
together
to
the
bodie
of
the
Rampart
.
Moreouer
,
they
are
so
laid
together
that
the
rowes
aboue
must
lie
euen
in
the
midst
of
the
joynture
of
the
rowes
beneath
,
makeing
them
settle
together
according
as
the
condition
beares
,
and
as
the
articles
are
which
the
undertaker
hath
signed
unto
.
Now
for
his
better
ease
,
he
must
make
use
of
a
triangular
Instrument
,
such
a
one
as
the
letter
F
demonstrats
,
being
〈◊〉
●●de
A
,
B
,
about
2
,
or
3
foote
more
or
lesse
,
according
to
the
quality
of
the
worke
,
and
the
rest
so
consequently
:
For
as
ye
desire
to
haue
the
Talud
,
either
great
or
little
,
so
ye
must
enlarge
the
line
A
,
G
,
or
lessen
it
:
If
it
be
needefull
to
lay
bundles
of
brush
to
strengthen
your
works
of
the
Fortresse
,
ye
must
lay
them
so
that
the
ends
touch
the
former
sodds
,
to
wit
the
one
row
half
a
foote
aboue
the
other
,
and
betweene
each
of
them
halfe
a
foote
of
Earth
,
continuing
the
worke
so
untill
it
be
finished
:
upon
the
top
of
the
Earth
of
the
Rampart
,
ye
shall
raise
the
parapet
with
such
a
Talud
and
bredth
,
as
the
conditions
imports
,
in
the
same
manner
as
before
,
raising
it
with
sodds
,
as
we
haue
said
.
If
ye
make
your
Fortresse
in
a
fatt
and
a
clayie
soile
,
then
ye
may
(
in
stead
of
sodds
)
make
a
cruste
of
3
,
or
4
foote
,
or
more
,
stamping
it
well
with
a
stamper
made
to
that
end
,
and
of
such
an
inclination
as
your
agreement
beares
,
in
which
crust
ye
may
sett
a
certaine
kinde
of
grasse
,
called
in
Flemish
Queeckcruyt
,
in
Latin
Gramen
,
&
in
English
Meddow
grasse
,
which
roote
is
of
such
a
nature
,
that
it
will
spread
itfelfe
ouer
the
whole
Rampart
,
and
binde
the
Earth
fast
together
:
so
that
it
will
make
the
said
crust
of
a
long
durance
,
and
almost
perpetuall
:
on
the
outside
of
the
said
brust
,
ye
may
sowe
Oats
or
Hay-seed
:
they
use
also
to
sett
upon
the
Bulwarke
,
roots
of
seuen-leaued-grasse
,
which
is
likewise
very
good
:
but
the
leaues
doe
not
spread
themselues
ouer
the
exteriour
superficies
,
or
this
crust
,
as
the
abouesaid
grasse
doth
:
for
this
reason
some
yeares
since
his
Excellencie
hath
found
it
good
,
to
repaire
all
the
Fortifications
of
such
a
crust
without
sodds
;
because
experience
hath
taught
us
,
that
the
said
sodds
bindes
not
the
rest
of
the
Earth
,
so
well
as
the
said
crust
doth
,
which
being
moist
by
that
meanes
joyneth
so
much
the
better
the
rest
of
the
wall
or
Rampart
together
,
which
is
of
very
great
use
,
and
is
quickly
done
.
The
talud
of
the
Rampart
is
made
somewhat
greater
,
that
one
may
stand
an
end
the
better
upon
it
,
as
ye
may
see
by
the
profiles
here
before
,
aswell
of
Ramparts
as
Falsebrayes
.
This
being
done
,
ye
must
make
the
parapet
of
the
covert-way
of
earth
digged
out
of
the
moate
,
whereof
good
store
was
layd
on
the
outside
of
the
moate
,
to
that
purpose
:
and
the
said
undertaker
ought
to
that
end
make
an
exact
calculation
of
the
quantity
of
Earth
,
which
is
necessary
for
that
use
.
But
seing
oftentimes
the
undertakers
are
not
skillfull
in
casting
up
such
accounts
:
It
were
good
that
the
Ingenier
himself
should
make
an
exact
calculation
,
not
onely
of
this
part
,
but
also
of
the
whole
worke
,
which
may
be
done
in
that
manner
,
as
wee
haue
showne
in
our
Geometrie
,
or
else
in
this
easie
manner
following
.
The
second
17.
Plate
,
and
Letters
G
,
H
,
R.
LEt
the
12
part
of
the
Fortresse
be
Dodecangular
G
,
and
AB
the
face
24
rod
;
the
flanke
AD
12
,
and
the
curtaine
36.
and
for
the
Rampart
H.
Let
the
talud
interiour
be
15
foote
,
and
the
height
asmuch
,
the
talud
exteriour
10
foote
,
the
bredth
upon
the
Rampart
50
foote
,
which
makes
that
the
superficies
of
the
Profile
of
the
said
Rampart
wil
be
933
1/3.
Moreouer
,
for
the
Profile
of
the
parapet
,
which
is
K
99
1/4.
the
heigth
EC
6
foote
,
HF
,
4.
AB
for
the
footbanke
a
foote
and
a
halfe
;
AC
3.
BD
2.
CG
20.
HG
2
1/2.
the
solidity
of
the
Rampart
&
the
parapet
is
required
.
Of
the
Rampart
.
THe
Icnographie
sheweth
,
that
CM
10
foote
is
the
talud
exteriour
,
and
HN
15
foote
the
interiour
,
the
angle
B
is
right
in
the
Dodecagone
,
and
in
the
Polygones
upon
it
,
ye
must
remember
once
for
all
,
that
if
the
Profile
on
all
sides
hold
one
&
the
same
dimensions
,
to
wit
in
the
faces
,
flanks
,
&
〈◊〉
the
angles
BAD
shal
be
cutt
by
halues
from
the
lines
HB
,
EA
,
KD
.
Also
the
angle
of
the
Polygone
being
150
,
&
the
flanked
right
;
the
angle
flanking
interiour
wil
be
30
degrees
,
as
it
is
marked
,
and
the
shoulder
120
degrees
,
which
makes
that
the
triangles
right-angles
are
as
followes
:
those
which
haue
their
angles
pointed
of
45
degrees
,
shal
be
Isosceles
,
or
like
angles
,
BM
,
MC
upon
the
basis
each
of
them
10
foote
,
the
lines
C
3.
3
G
,
each
50
foote
,
BX
&
XH
,
each
75
foote
,
GN
,
NH
15
foote
,
and
the
like
in
the
triangles
hauing
their
Hypotenouses
at
KD
.
Moreouer
in
the
triangle
ORA
,
in
the
shoulder
RO
10
,
OA
5
,
7735
,
the
line
marked
5
A
,
43
,
30125
,
therefore
takeing
the
lines
BX
,
and
5
A
from
BA
24
rod
,
or
288
foote
,
there
will
remaine
HF
169
,
69875.
Likewise
AD
144
foote
,
PD
,
or
RV
,
wil
be
138
,
2265
,
and
by
adding
to
it
FV.
FR
will
come
to
148
,
2265
,
the
line
A
12
is
equall
to
A
5.
taken
from
AD
,
there
will
remaine
100
,
69857
for
T
8
,
to
which
add
8
I
,
60
,
TI
will
come
to
160
,
69875
,
and
LI
,
169
,
359
,
add
to
the
said
TI
,
IZ
,
15
,
ye
shall
haue
EK
175
,
69875.
The
demy-curtaine
13
D
,
makes
216
,
then
9
F
,
226
,
also
11
K
291
,
&
10
I
,
276
:
so
that
if
ye
add
CR
272
,
2265
,
RF
148
,
2265
,
and
F
9.
which
is
226
,
ye
shall
haue
646
,
453
for
CFR
9.
the
circuit
exteriour
.
Ye
shall
finde
also
GL
193
,
359
,
&
LI
,
169
,
35904
:
&
10
I
,
276.
whereof
the
summe
for
GLI
10
,
wil
be
638
,
718
the
circuit
interiour
,
which
being
added
to
the
exteriour
abouenamed
,
will
amount
to
1285
,
171
,
its
halfe
is
642
,
5855
,
which
multiplyed
by
G3
,
the
distance
of
the
parallells
50
,
will
come
to
32129
,
275
for
the
superficies
,
betweene
the
abouesaid
circuits
,
and
then
multiply
by
15
the
heights
,
ye
shall
haue
the
solidity
of
the
wall
,
without
the
taluds
481939
,
125.
CRF
9.
10.
ILGC
,
which
had
bene
too
short
as
here
followeth
.
The
Authour
hauing
made
this
aboue
with
so
many
lines
,
if
from
the
midst
of
LR
,
he
had
drawne
a
perpendicular
line
upon
the
face
,
and
cutt
af
from
it
20
,
2071
its
double
substracted
from
the
summe
of
the
abouesaid
perpendicular
,
and
from
BA
,
AD
,
D
13
,
which
is
683
,
the
remainder
wil
be
642
,
5856
,
which
multiplyed
by
15
,
it
wil
be
the
same
as
aboue
.
Of
the
Taluds
or
slooping
walls
or
Ramparts
.
THe
Taluds
without
the
Pyramids
,
which
are
in
the
angles
,
shal
be
calculated
thus
,
their
Profile
is
a
triangle
,
which
multiplyed
by
the
length
,
or
else
as
Demy-parallelipedes
,
the
Basis
multiplyed
by
the
heigth
,
afterward
the
halfe
taken
,
add
to
it
then
CR
272
,
2265
,
RV
138
,
2265
,
with
D
13
,
which
is
216
,
the
summe
626
,
453
multiplyed
by
MC
,
10
,
the
superficiall
basis
wil
be
6264
,
53
;
likewise
HE
169
,
69875
,
EY
160
,
69875
&
I
10
,
which
is
276
,
makeing
together
606
,
3975
,
which
being
multiplyed
by
IY
15
,
will
make
9095
,
9625
for
the
other
superficiall
basis
,
together
with
the
former
will
make
15360
,
4925
(
because
they
are
of
a
like
heigth
)
by
which
heigth
of
15
being
multiplyed
,
then
the
halfe
will
come
to
115203
,
69375
for
the
taluds
,
aswell
interiour
,
as
exteriour
besides
the
Pyramides
.
Of
Pyramids
,
and
double
Pyramids
of
the
Taluds
.
THere
are
3
Pyramides
in
the
angles
convexe
B
,
A
,
K
,
and
three
double
Pyramids
in
the
angles
concave
,
H
,
E
,
D
,
which
I
say
because
Marolois
makeing
no
distinction
of
them
,
taketh
two
things
for
one
:
As
for
exemple
,
the
body
upon
VFXD
is
1000
foot
cubicall
,
and
he
〈…〉
but
500
,
and
so
of
the
two
others
.
The
Superficies
of
the
triangle
BMC
,
is
50
from
ROAP
,
is
the
right-angle
RO
,
OA
57
,
735
,
and
YIZK
225
,
and
then
the
double
superficies
of
GNH
225
(
according
to
that
as
we
haue
said
before
touching
the
Authours
fault
)
ESLT
its
double
Superficies
is
259
,
8057
,
&
the
double
superficies
VFXC
,
of
200
,
whereof
the
summe
is
1071
,
5425
,
which
multiplyed
by
the
tierce
of
the
common
heights
,
ye
shall
haue
for
the
abouesaid
fixe
bodies
5087
,
7125
,
(
whereas
Marolois
hath
but
3375
,
69375
)
to
which
if
ye
add
,
aswell
the
Taluds
115203
,
69375
,
as
the
Rampart
481934
,
125
,
the
summe
wil
be
602230
,
5312
,
for
the
soliditie
of
the
Rampart
,
besides
the
parapet
.
Those
which
will
not
take
the
paines
to
haue
it
done
soe
exactly
,
but
are
contented
to
come
somewhat
neere
unto
it
,
shall
add
unto
it
BAD
13
,
which
is
648
,
with
HEK
11
,
being
636
,
396
,
and
the
halfe
of
the
summe
multiplyed
by
the
profile
of
the
Rampart
937
1/2
ye
shall
haue
602060
,
which
comes
the
neerest
unto
602230
,
found
out
aboue
(
being
onely
170
too
litle
)
for
Marolois
found
out
but
600518
,
with
much
adoe
,
hauing
1712
too
litle
.
Of
a
Parapet
.
WE
haue
before
sett
downe
the
dimensions
of
a
Parapet
as
ye
may
see
by
the
letter
K
,
newly
cutt
,
with
its
Rampart
H
,
in
this
plate
,
&
ye
must
not
thinke
it
strange
,
that
the
Parapet
is
made
upon
a
greater
measure
,
then
that
of
the
Rampart
,
the
reason
is
,
because
there
are
noe
more
dissections
,
or
cutting
of
,
and
that
wee
haue
not
roome
to
make
that
which
we
would
within
the
some
Plate
.
Now
if
ye
desire
to
calculate
the
Profile
as
the
Authour
hath
done
,
the
whole
as
the
brevitie
of
the
Rampart
showes
aboue
mentioned
,
its
situation
is
termined
on
the
outside
by
CRF
9.
and
on
the
inside
by
a
parallell
line
,
betweene
the
abouesaid
&
GLI
10
,
distant
from
the
other
20
foote
,
and
for
the
avoyding
of
the
confusion
which
the
plurality
of
lines
might
cause
,
we
will
take
for
the
said
paralell
,
the
line
GLI
10
,
and
therefore
3
G
,
or
7
S
,
will
make
20
,
or
else
the
line
LT
;
ye
shall
finde
by
the
triangles
,
that
TR
wil
be
5
,
369.
add
hereunto
C
3
,
which
is
20
,
and
the
summe
of
CR
knowne
aboue
,
taken
from
272
,
2265
,
there
will
remaine
for
GL
246
,
8675
,
then
from
RF
148
,
2265
,
substracting
the
length
equall
to
TR
,
5
,
359
,
there
will
remaine
142
,
8675
,
afterward
add
thereunto
8
V
,
or
its
equall
20
,
ye
shall
haue
162
,
8675
for
LI
;
also
F
9
,
being
226
,
then
I
10
wil
be
246
;
then
the
summe
of
CRF
9
,
(
added
to
GLI
10
)
and
the
halfe
taken
,
ye
shall
haue
for
the
middle
length
of
the
parapett
651
,
094.
For
the
superficies
of
the
Profile
of
the
Parapet
K
,
the
triangle
EIF
17
1/2
:
and
DOE
2
1/4
,
the
trapeze
,
(
or
unequally
side
)
IFGC
75
;
and
BOCA
4
1/2
,
the
summe
is
99
1/4
,
for
the
Superficies
of
the
Profile
,
which
multiplyed
by
the
middle
length
aboue
said
,
ye
shall
haue
64621
,
0795
for
the
solidity
of
the
parapet
,
which
is
much
too
much
;
for
it
ought
to
be
as
I
haue
calculated
it
with
its
dissections
63976
,
9475
,
to
which
add
602230
,
5312
from
the
Rampart
ye
shall
haue
666207
,
478
,
for
the
whole
solidity
of
the
Rampart
and
Parapet
,
whereof
the
double
1332414
,
956
,
wil
be
the
twelfth
part
of
the
Circuit
,
which
multiply
by
12
,
ye
shall
haue
the
totall
summe
15988979
,
472
,
and
diuided
by
144
,
ye
shall
haue
111034
,
68
plyntides
,
or
as
Marolois
calls
them
Chevilles
,
peggs
or
pynnes
,
for
the
whole
circuit
of
the
Dodecagone
.
Seing
then
there
are
9252
,
88
pinnes
in
the
12
part
of
this
Fortresse
Dodecagonall
,
the
Ingenier
may
conjecture
,
for
how
much
he
may
let
out
the
same
worke
,
which
is
about
16
,
20
,
25
or
30
pinnes
,
according
to
the
situation
of
the
place
,
the
ground
and
other
considerations
,
and
after
that
the
undertakers
shall
offer
to
make
it
according
to
the
calculation
,
made
thereof
,
they
may
take
on
the
said
worke
upon
condition
,
that
they
shall
finish
it
in
such
a
time
,
&
with
so
many
men
as
is
specified
in
the
contract
,
and
soe
at
that
instant
they
beginn
the
worke
,
&
finish
it
according
to
their
bargaines
and
conditions
,
as
the
greatnesse
of
the
worke
requires
,
and
the
time
limited
,
for
which
also
the
undertaker
is
bound
to
haue
as
many
worke
men
upon
it
,
as
shall
be
found
needfull
for
the
finishing
of
the
said
worke
:
&
for
the
better
assurance
thereof
,
it
is
requisite
to
know
how
many
pinnes
a
man
may
spit
,
and
lay
upon
a
wheele-barrow
in
a
day
:
for
those
that
are
skilled
therein
,
say
that
a
man
if
he
works
his
best
,
he
may
dig
&
lay
up
4
1/2
pinnes
in
a
day
,
but
ordinarily
men
suppose
that
one
can
doe
three
pinnes
euery
day
,
without
takeing
too
much
paines
,
according
to
which
,
one
may
easely
make
his
calculation
how
many
spadmen
he
must
use
to
make
up
that
worke
in
such
a
time
prefixed
,
for
men
which
shall
doe
nothing
else
,
but
spit
and
digg
up
the
Earth
&
the
others
to
carrie
it
away
upon
wheele-barrowes
to
the
Rampart
,
for
ye
must
haue
as
many
men
to
carry
up
the
Earth
in
the
wheel-barrowes
,
as
that
the
spiters
may
be
continually
kept
in
worke
without
any
cessation
or
stay
,
to
the
intent
the
Work-master
may
finish
his
worke
within
the
time
limited
.
Note
.
WE
say
here
,
that
a
man
may
in
case
of
necessity
spit
in
a
day
4
1/2
pinnes
,
and
commonly
three
in
a
day
with
leasure
,
but
it
is
to
be
understood
,
this
must
be
in
a
good
and
fast
ground
,
which
wil
be
beare
spiting
without
falling
or
breaking
;
but
when
the
Earth
is
leane
and
sandie
,
it
wil
be
impossible
for
a
man
to
doe
so
much
,
&
so
by
consequence
one
cannot
know
any
certaine
rule
.
The
1.
Maxime
.
ACcording
to
which
plotts
and
calculations
all
other
places
aswell
regular
as
irregular
may
be
fortified
,
and
hold
this
for
a
maxime
,
that
the
place
so
fortified
,
as
we
haue
taught
aboue
,
wil
be
better
then
the
others
.
The
2.
Maxime
.
THat
the
line
of
defence
which
comes
from
the
angle
of
the
flanke
to
the
angle
of
the
Bulwarke
,
called
the
line
of
deffence
fichant
ought
not
to
be
much
longer
then
60
rod
,
because
it
is
as
farre
as
a
Musket
can
well
carry
,
but
if
ye
will
defend
the
face
with
Canon
,
then
it
may
be
a
100
,
or
120
foote
longe
.
The
3.
Maxime
.
THat
lesser
the
angle
of
the
tenaille
is
,
it
is
so
much
the
better
,
because
the
faces
looke
more
directly
one
upon
another
,
therefore
ye
may
see
that
in
the
plots
of
the
Dodecagone
aboue
said
:
ye
must
make
the
angle
of
the
Bulwarke
onely
of
90
degrees
,
that
the
angle
of
the
tenaille
may
be
shutt
the
closser
.
The
4.
Maxime
.
BY
how
much
the
flanke
and
the
gorge
are
greater
and
spacious
,
by
so
much
they
are
the
fitter
for
defence
.
The
5.
Maxime
.
THe
further
the
defence
is
made
in
the
curtaine
,
it
is
the
better
,
because
ye
may
bring
the
more
men
to
giue
fire
upon
it
,
for
the
defence
of
the
faces
.
Therefore
in
the
plots
of
the
Dodecagone
abouesaid
,
ye
must
see
that
the
angle
of
the
Bulwarke
be
made
of
90
degrees
,
to
the
end
ye
may
giue
fire
the
larger
.
The
6.
Maxime
.
That
the
angle
flanked
ought
to
be
noe
lesse
then
60
,
and
not
aboue
90
degrees
.
The
7.
Maxime
.
THat
the
angle
of
the
Polygone
,
which
ye
intend
to
fortify
,
ought
not
to
be
lesse
then
90
degrees
,
but
the
greater
it
is
,
it
wil
be
so
much
the
better
.
The
8.
Maxime
.
THat
in
all
Fortresses
wee
will
make
the
face
to
be
24
rod
,
and
the
curtaine
36
,
according
to
which
the
lines
of
the
Polygones
exteriour
will
make
about
60
rod
,
and
the
lines
of
the
Polygones
interiour
60
rod.
In
a
Heptagone
and
the
Polygones
underneath
,
the
Polygones
exteriour
wil
be
more
then
80
rod
,
and
aboue
lesse
then
80.
The
Polygones
interiour
which
are
beneath
the
said
Heptagone
,
are
lesse
then
60
rod
,
and
aboue
more
then
60
,
the
one
encreasing
,
the
other
decreasing
,
untill
that
the
angles
of
the
Polygones
come
to
be
180
degrees
,
and
then
ye
haue
the
Polygones
exteriour
,
and
interiour
equall
,
and
each
of
them
70
rod
,
the
line
of
defence
fichant
in
the
whole
about
60
rod
,
&
the
angle
forming
the
flanke
of
40
degrees
.
The
9
Maxime
.
THat
the
angle
flanking
interiour
ought
to
be
noe
lesse
then
15
degrees
,
but
the
greater
the
better
they
are
.
The
10.
Maxime
.
THat
the
angle
of
the
flanke
be
alwayes
right
,
the
angle
of
the
shoulder
must
be
at
the
least
105
degrees
,
and
the
greater
the
better
.
The
11.
Maxime
.
IF
it
be
needfull
to
fortifie
any
Polygone
,
either
regular
or
irregular
,
whereof
the
sides
are
lesse
then
is
here
spoken
of
,
then
ye
must
proportion
out
the
gorges
,
flankes
and
faces
,
according
to
the
forme
of
the
angle
,
as
ye
shall
se
in
a
table
following
.
Herewith
wee
will
put
an
end
to
our
first
part
of
Fortification
,
and
will
come
now
to
the
fortifying
of
places
irregular
.
FINIS
.