The
diffinition
THat
moste
auncient
and
learned
playe
,
called
the
Philosophers
game
,
beinge
in
Greeke
termed
〈◊〉
〈◊〉
〈◊〉
〈◊〉
〈◊〉
,
is
as
much
to
saye
in
Englishe
,
as
the
battell
of
numbers
.
Numbers
be
either
euen
or
odde
,
wherefore
the
euen
parte
is
against
the
odde
,
either
parte
hauinge
a
kyng
,
whych
being
taken
of
the
aduersaryes
part
,
and
a
triumphe
celebrated
within
his
campe
,
the
game
is
ended
.
¶
Of
diuerse
kyndes
of
playinge
.
AMonge
the
dyuerse
kyndes
of
playinge
thys
game
,
we
shall
sette
forth
three
sortes
,
of
whiche
the
reader
maye
chose
whether
of
them
he
lyketh
beste
.
And
of
all
those
three
,
we
shall
gyue
suche
shorte
and
easye
rules
,
that
no
man
(
althoughe
he
were
altogether
ignoraunt
in
Arithmetike
)
shall
fynde
the
game
so
hard
,
but
that
he
may
learne
to
playe
it
.
¶
Of
the
partes
of
thys
Game
.
HE
that
wyll
learne
thys
game
,
any
of
the
three
waies
,
muste
firste
be
enstructed
of
these
sixe
partes
.
The
table
as
the
fielde
.2
.
the
menne
and
the
numbers
of
them
as
the
hoste
.3
.
the
placynge
of
them
,
as
the
encampinge
.4
.
the
order
of
playe
and
remouynge
the
men
,
as
the
marchynge
and
fyghtynge
5.
the
manner
and
lawes
of
conqueryng
and
taking
.6
.
and
last
of
al
the
triumphe
after
the
victorye
,
¶
Of
these
partes
in
the
fyrst
kynd
of
playng
.
The
table
muste
be
a
playne
borde
conteynynge
.128
.
squares
that
is
.8
.
in
breadth
and
.16
.
in
length
sette
forthe
in
two
dyuerse
collours
.
Or
for
a
plainer
vnderstandynge
,
the
table
is
a
doble
chesse
bord
,
as
it
were
two
chessebordes
ioyned
together
,
the
length
of
twoo
,
the
breadth
of
one
,
whereof
thys
is
an
example
.
¶
Of
the
men
.
THe
men
be
in
number
.48
.
Wherof
.24
.
be
of
one
side
&
must
be
knowen
by
one
colour
,
and
.24
.
on
the
other
syde
,
whyche
also
must
be
marked
with
a
contrarye
colour
,
as
White
and
Blacke
,
Blew
and
Redde
,
or
what
colours
els
you
lyke
best
.
But
in
the
colering
these
.3
.
thinges
must
be
obserued
,
the
bottome
or
lower
part
of
euery
man
(
excepte
the
two
kinges
)
muste
be
marked
wyth
hys
aduersaryes
colour
,
that
when
he
is
taken
,
he
maye
chaunge
hys
coate
and
serue
hym
vnto
whome
he
is
prisoner
.
The
kynges
because
they
consist
of
all
thrée
sortes
,
as
it
is
knowen
by
the
learned
speculation
of
the
numbers
,
beare
the
fashion
of
all
thre
kinds
,
his
foundations
are
two
squares
,
on
which
are
sette
,
two
triangles
&
vpon
them
rounds
.
But
this
difference
is
betwene
the
kinges
,
that
the
king
of
the
euen
nubers
,
hath
a
pointed
toppe
,
the
king
of
the
odde
numbers
is
not
pointed
,
the
cause
dependeth
vpon
the
consideration
of
there
numbers
by
which
they
arise
into
piramidall
fashion
.
The
third
thing
considered
in
the
men
,
is
the
number
that
must
be
written
or
grauen
vpon
them
which
to
learne
plainely
for
practise
marke
these
short
rules
.
There
be
of
eche
kynde
of
men
,
two
rankes
or
orders
.
The
first
ranke
or
order
of
roundes
be
the
digites
euen
or
odde
namely
of
the
euen
.2.4.6.8
.
of
the
odde
.3.5.7.9
.
The
second
order
of
rounds
are
found
by
multiplyinge
these
digites
by
themselues
as
.2
.
times
.2
.
is
.4.3
.
times
.3
.
is
.9
.
Of
the
euen
they
be
.4.16.36.64
.
of
the
odde
they
be
.9.25.49
,
81.
The
first
order
of
the
triangles
are
found
by
addinge
two
of
the
roundes
together
one
of
the
firste
order
and
another
of
the
seconde
order
,
as
.2
.
and
.4
.
make
sixe
3.
and
.9
.
make
twelue
,
on
the
euen
syde
they
are
these
.6.20.42.72
.
on
the
odde
syde
.12.30.56.90
.
The
second
order
of
triangles
be
made
by
addynge
one
to
euery
one
of
the
first
order
of
roundes
,
and
then
multiplying
that
number
in
hym selfe
:
as
.2
.
is
one
of
the
firste
order
of
roundes
,
thereto
adde
one
,
that
is
.3
.
then
.3
.
tymes
.3
.
is
.9
.
a
triangle
of
the
seconde
order
,
on
the
euen
syde
.
Likewise
to
thre
a
round
on
the
odde
side
,
adde
.1
.
so
is
it
.4
.
then
.4
.
tymes
.4
.
is
.16
.
On
the
euen
parte
,
they
be
.9.25.49.81
.
on
the
odde
parte
.16.36.64.100
.
The
first
order
of
squares
(
in
whyche
are
contayned
the
kynges
)
be
made
by
addynge
two
triangles
together
,
one
of
the
fyrste
order
,
and
another
of
the
seconde
,
as
.6
.
and
.9
.
make
.15
.
likewyse
12.
and
.16
.
make
.28
.
Amonge
the
euen
they
be
.15.45
.
and
.91
.
the
Kynge
.153
.
amonge
the
odde
they
be
.28.66.120
.
and
.190
.
the
Kynge
.
The
last
order
of
squares
be
found
,
by
dobling
of
euery
one
of
the
firste
order
of
roundes
,
and
after
adding
one
,
last
of
all
by
multipling
that
number
in
it self
,
as
twise
.2
.
is
.4
.
and
.1
.
added
is
.5
.
so
.5
.
times
5.
is
.25
.
likewyse
twyse
.3
.
is
.6.1
.
added
is
7.
then
.7
.
tymes
.7
.
is
.49
.
These
be
on
the
euen
syde
.25.81.169.289
.
And
of
the
odde
syde
.49.121.225.361
.
These
numbers
must
be
sette
vppon
the
men
both
on
the
vpper
side
,
&
also
on
the
nethersyde
.
Except
one
of
the
kinges
,
which
must
with
the
whole
number
of
their
pyramis
,
be
marked
,
onely
on
the
bottome
.
Because
the
sydes
muste
haue
other
numbers
,
namely
the
highest
point
of
the
euen
kyng
,
must
haue
.1
.
the
rounde
next
vnder
him
marke
with
.4
.
the
vpermost
triangle
with
.9
.
the
nethermost
with
.16
.
The
vpper
most
square
muste
haue
.25
.
The
nethermost
square
shall
haue
.36
.
The
king
of
the
odde
vpon
his
head
,
whiche
is
a
rounde
,
not
pointed
hath
.16
.
vpon
his
first
triangle
.25
.
on
the
second
triangle
.36
.
vppon
the
fyrste
square
.49
.
vpon
the
lowest
square
.64
.
¶
The
reason
of
these
numbers
and
the
knowledge
of
their
proportione
.
FOr
them
that
seke
the
speculation
of
these
numbers
,
rather
then
the
practise
for
playing
,
and
haue
some
sight
in
the
sciens
of
Arithmetike
,
some
thyng
must
be
sayde
of
proportion
.
For
this
purpose
there
be
three
kyndes
of
proportion
.
Multiplex
,
superparticuler
and
superpartiens
.
¶
Of
multiplex
.
MULTIPLEX
proportion
,
is
when
a
great
number
conteyneth
a
lesse
number
manye
tymes
,
and
leaueth
nothinge
,
as
.8
.
conteyneth
.2
fower
tymes
and
nothing
remaineth
.16
.
conteineth
.4
.
&c
,
this
proportion
semeth
best
to
agree
with
roundes
because
the
one
number
conteyneth
the
other
and
nothynge
remaineeth
as
the
fyrste
order
〈◊〉
roundes
be
.
The
second
order
be
these
.
doble
.
quadruple
.
sextuple
.
occuple
.
proportion
.
triple
.
quintuple
.
septupl
.
noncuple
¶
Of
superparticuler
proportion
.
SUper
particuler
proportion
is
when
a
greater
number
contayneth
a
lesser
with
one
part
of
it
,
which
may
measure
the
whole
,
as
.12
.
contayneth
.9
.
and
3.
whiche
is
a
thyrde
parte
of
nine
.6
.
conteyneth
.4
.
and
.2
.
that
is
one
halfe
to
4.
Thys
proportion
beinge
the
cheife
,
next
vnto
multiplex
,
is
beste
figured
by
a
trianguler
forme
,
whyche
hathe
fewest
lynes
and
angles
next
vnto
a
circle
.
For
the
manner
of
thys
proportion
consider
thys
figure
.
sesquialter
.
sesquiquart
.
sesqui
.
sext.
sesqu
.
oct.
sesquiter
.
sesquiquint
.
sesquisept
.
sesquinona
.
¶
Superpartiens
proportion
.
THE
superpartiens
proportion
is
when
the
greater
number
conteineth
the
lesser
and
mo
partes
of
it
then
one
as
.15
.
conteyneth
.9
.
and
.6
.
whiche
is
two
thirdes
of
.9
.
lyke
wyse
.28
.
coteineth
.16
.
and
.12
.
that
is
3/4
;
of
.16
.
This
proportion
conteineth
diuers
parts
beside
the
whole
number
therfore
is
wel
figured
in
the
square
,
which
also
conteyneth
more
corners
and
sides
.
For
the
maner
of
their
proportion
consyder
thys
table
.
The
first
order
of
squares
.
suꝑparticulres
added
being
the
squares
.
The
second
order
followeth
.
Thirde
|
fyft
|
seuenth
|
ninth
|
5.
|
9.
|
13.
|
17.
|
10.
|
36.
|
78.
|
136.
|
superbipartiens
tertias
.
|
suꝑquadrupartiens
quintas
.
|
suꝑsextupartiens
septimas
.
|
suꝑoctupartiens
nonas
|
Fourth
|
sixth
|
eight
|
tenth
|
7.
|
11.
|
15.
|
19.
|
21.
|
55.
|
105.
|
171.
|
supertripartiens
.
quartas
.
|
suꝑquintupartiēs
.
septimas
.
|
suꝑseptupartiens
.
Octauas
.
|
suꝑnonpartiens
.
decimas
.
|
¶
Of
the
kings
.
THE
kinges
conteine
in
them
suche
numbers
,
as
beyng
all
added
together
,
make
the
whole
piramidall
number
,
the
lowest
square
of
the
euen
,
is
36.
which
riseth
of
the
multiplying
of
.6
.
in
it selfe
.
The
next
square
that
must
be
lesse
,
is
.25
.
arisinge
by
the
multiplyinge
of
fyue
in
it self
and
so
followeth
.16
.
of
.4
then
.9
.
of
.3
.
laste
.4
.
of
.2
.
and
single
.1
.
all
these
added
together
,
make
.91
.
After
the
same
maner
consisteth
the
king
of
odde
.
The
lowest
square
is
.64
.
arisinge
of
.8
.
multiplied
in
himselfe
.
The
next
.49
.
of
7.
times
.7
.
then
.36
.
of
.6
,
25.
of
.5
.
and
.16
.
of
4.
these
numbers
make
the
whole
pyramidall
number
.190
.
which
because
it
riseth
not
to
the
poynct
of
one
,
oughte
not
to
be
sharpe
poyncted
,
as
hathe
beene
sayde
before
.
¶
Of
the
placing
,
encamping
or
setting
in
araie
.
TO
retorne
againe
to
the
plaine
and
easye
playing
of
this
game
,
next
to
the
armie
&
their
armour
,
follow
ether
the
order
of
their
battel
or
encamping
.
Whiche
bicause
it
is
more
playne
and
easely
seen
which
the
eye
,
then
learned
by
the
eare
,
I
referre
thee
vnto
the
table
where
the
battell
is
appoynted
in
suche
order
as
thys
kynde
of
Playe
requireth
.
¶
Of
the
marchinge
or
remouing
of
the
men
.
THE
battell
beyng
duely
placed
,
it
followeth
next
,
to
know
the
maner
of
marching
&
remouing
,
for
euery
kynd
of
men
,
hath
their
proper
kynde
of
motion
,
and
fyrste
we
muste
speake
of
the
roundes
.
¶
The
motyon
of
the
roundes
.
THE
roundes
muste
moue
into
the
space
that
is
next
vnto
them
corner
wyse
,
as
in
the
table
,
from
the
space
A.
to
any
of
these
.
B.C.D.
or
.
E.
¶
Of
the
triangles
.
THE
triangles
passe
three
spaces
counting
that
in
which
they
stande
for
one
,
and
that
into
whych
they
do
remoue
for
another
,
that
is
leaping
ouer
one
space
.
As
from
the
space
.
A.
he
maye
remoue
into
any
of
these
spaces
.
F.G.H.
or
.
I.
this
is
the
motion
of
the
triangle
in
marchyng
or
takyng
.
But
in
flying
he
maye
remoue
the
Knyghtes
draught
of
the
chesse
,
as
from
.
A.
into
.
X.
or
.
W.
&c.
¶
Of
the
Squares
.
THe
Squares
remoue
into
the
fourth
place
from
them
,
that
is
leaping
ouer
two
,
right
forwarde
or
sydelong
,
as
from
▪
the
place
of
.
A.
to
any
of
these
spaces
.
L.
N.
P.
R.
flyinge
they
maye
remoue
after
the
Knyghts
draught
,
but
that
they
must
passe
foure
spaces
,
as
from
.
P.
to
.
Y.
or
.
T.
&c.
And
this
for
the
marchinge
and
remouyng
of
the
men
,
where
note
,
that
with
theyr
flying
draughte
they
can
take
no
man
,
but
if
neede
be
helpe
to
besiege
a
man
.
¶
Of
the
Kyngs
marching
.
THe
kings
because
thei
beare
the
forme
of
al
the
thre
kynds
,
may
remoue
any
of
all
theyr
draughts
when
they
list
,
into
the
nexte
with
the
rounde
,
into
the
thyrde
with
the
triangle
,
and
into
the
fourth
with
the
square
,
and
finally
in
all
poyntes
lyke
the
Queene
at
the
Chesse
,
sauing
that
he
can
not
passe
aboue
foure
spaces
at
the
most
.
¶
Of
the
maner
of
taking
.
THe
men
may
be
taken
sixe
wayes
,
namely
by
Equalitie
,
Obsidion
,
Addition
,
Substraction
,
Multiplication
and
Diuision
,
and
also
if
you
wyll
,
and
soagree
,
by
Proportion
-
Arithmeticall
.
-
Geometricall
.
-
Musicall
.
¶
Of
Equalitie
.
BY
equality
men
may
be
taken
,
when
one
man
after
hys
motion
,
seeth
hys
enemye
beyng
of
the
same
number
that
he
is
,
standing
in
such
place
as
he
may
remoue
into
,
then
maye
he
take
awaye
hys
enemye
and
not
remoue
into
his
place
,
as
in
this
example
.9
.
a
triangle
of
the
euen
army
,
after
he
hath
remoued
,
espyeth
.9
.
a
rounde
of
the
odde
armye
,
hym
may
he
take
vp
and
not
remoue
into
his
place
.
But
if
.9
.
the
triangle
,
espye
nine
the
rounde
,
before
he
remoue
,
standing
in
his
draught
,
he
maye
take
hym
vp
and
remoue
into
his
place
.
These
men
may
be
taken
by
equalitie
9.16.25.36.49.64.81
.
because
they
are
found
in
both
the
armies
,
and
in
asmuch
as
anye
man
taken
beinge
torned
wyth
hys
bottome
vpward
,
&
that
beareth
hys
aduersaries
colloure
,
may
serue
his
enemye
on
whose
syde
he
is
taken
,
there
maye
yet
be
taken
by
equalitie
.4
.
and
6.
¶
Of
taking
by
obsidion
.
BY
obsidion
anye
man
maye
be
taken
euen
the
kinge
him selfe
,
if
he
be
so
compassed
with
4.
men
,
that
hys
law
full
draught
be
hindered
,
as
for
example
the
round
standing
in
the
place
of
.1
.
and
4.
men
of
what
kynd
it
skylleth
not
,
occupying
the
places
of
.2.3.4.5
.
after
you
haue
set
your
last
man
in
hys
place
may
be
taken
vp
,
also
if
a
triangle
be
enclosed
,
as
in
.
a
.
with
any
foure
men
standing
in
.
b.c.d.e
.
he
may
be
taken
,
euen
so
may
a
square
be
taken
.
Also
Triangles
and
squares
may
be
beseged
,
it
al
the
foure
men
or
any
of
them
,
the
rest
standyng
nearer
,
doe
stande
in
the
thyrde
or
fourth
space
from
them
so
that
they
haue
no
waye
to
remoue
,
as
a
triangle
or
square
standing
in
.
A.
may
be
beseged
by
.4
.
men
or
anye
of
them
(
the
reste
standynge
nearer
)
in
.
F.G.H.I.
Also
a
square
standyng
in
.
A.
maye
be
taken
by
〈◊〉
,
yf
the
fower
men
or
some
of
them
(
the
rest
standing
nearer
)
doe
stande
in
L.N.P.R.
and
this
is
sufficient
for
Obsidion
,
by
which
euery
man
may
be
taken
in
maner
and
forme
as
it
hath
bene
taught
.
Of
taking
by
Addition
.
WHen
two
numbers
are
so
brought
that
they
fynde
one
of
theyr
enemies
,
which
is
as
muche
as
bothe
they
beyng
added
together
,
standing
in
such
place
as
bothe
they
might
remoue
into
,
they
shall
take
hym
vp
,
without
remouing
into
his
place
,
so
soone
as
the
latter
of
those
two
is
set
downe
,
but
if
the
aduersaries
men
be
in
their
daunger
before
they
remoue
,
one
of
them
whether
the
player
lyst
,
shal
be
remoued
into
the
place
of
that
man
which
is
taken
by
Addition
.
As
for
example
.12
.
the
triangle
is
in
.
A.
if
you
can
bring
sixe
the
rounde
,
to
stande
in
.
B.
and
.6
.
the
triangle
to
stande
in
.
G.
because
.6
.
and
.6
.
being
added
make
.12
.
and
bothe
maye
remoue
to
.
A.
you
maye
take
vp
the
triangle
.12
.
by
addition
.
Also
.120
.
the
square
standing
in
.
P.
and
.49
.
the
rounde
standing
in
.
B.
or
elles
.49
.
the
square
standing
in
.
L.
which
being
added
together
make
.69
.
which
standeth
in
.
A.
shal
take
the
sayde
square
.169
.
by
Addition
.
¶
Of
taking
by
Substraction
.
WHen
two
men
do
so
stande
,
that
the
lesser
beyng
substracted
out
of
the
greater
,
the
number
remaining
,
is
all
one
with
the
aduersaries
man
that
standeth
in
bothe
their
draughtes
,
so
soone
as
the
latter
is
set
in
his
place
,
he
may
take
awaye
the
aduersarie
,
not
remouing
into
his
place
,
vnlesse
he
finde
him
so
before
he
remoue
:
as
for
as
example
,
2.
the
rounde
standing
in
.
B.
&
.9
.
the
triangle
standing
in
.6
.
shall
take
theyr
aduersarie
.7
.
standynge
in
.
A.
for
.2
.
out
of
.9
.
remayneth
.7
.
Another
example
.
The
rounde
.2
.
standyng
in
.
A.
maye
be
taken
by
.30
.
the
Triangle
standynge
in
H.
and
the
square
.28
.
standinge
in
.
P.
for
take
.28
.
out
of
.30
.
and
their
remaineth
.2
.
¶
Of
takynge
by
multiplycation
.
WHen
two
numbers
stande
so
,
that
being
multiplied
one
by
the
other
,
the
producte
is
all
one
with
their
aduersaryes
man
standynge
in
bothe
their
draughts
,
they
may
take
that
man
so
sone
as
the
latter
is
placed
.
And
if
they
lye
so
before
thei
remoue
,
being
so
left
of
the
aduersarie
,
one
of
them
shal
succede
in
his
place
that
is
taken
,
as
in
example
.
The
rounde
.3
.
standeth
in
.
D.
and
5.
standeth
in
.
E.
these
two
shal
take
the
square
15.
standynge
in
.
A.
because
three
tymes
fyue
is
.15
.
another
example
.
The
rounde
2.
standing
in
.
B.
and
the
triangle
.6
.
standynge
in
.
I.
shall
take
their
enemye
the
triangle
.12
.
standing
in
.
A.
by
multiplycation
for
.2
.
tymes
.6
.
is
.12
.
¶
Of
takyng
by
diuision
.
BY
diuision
a
manne
maye
be
taken
,
when
twoo
of
hys
enemyes
doe
so
stand
,
that
one
of
them
beyng
deuided
by
the
other
,
the
product
is
the
same
that
their
enemye
is
,
standynge
in
their
draught
,
immediatly
after
the
latter
is
placed
,
the
enemye
may
be
remoued
.
If
he
were
left
in
their
daunger
before
remouyng
,
one
of
them
may
remoue
into
his
place
,
an
example
.
The
round
4.
standyng
in
.
D.
and
the
triangle
.20
.
standing
in
.
F.
may
take
the
aduersarie
.5
.
standing
in
.
A.
by
diuision
,
bycause
.4
.
in
.20
.
is
conteyned
.5
.
tymes
.
Another
example
,
the
round
.5
.
standyng
in
B.
and
the
triangle
30.
standynge
in
.
F.
maye
take
their
enemye
.6
.
standynge
in
A.
for
.5
.
in
30.
is
conteyned
.6
.
tymes
.
¶
Of
the
takynge
of
the
kynges
.
THe
game
is
neuer
wonne
,
vntyll
the
King
be
taken
.
The
Kings
(
as
hath
bene
sayde
)
may
remoue
anye
way
,
so
they
passe
not
the
fourth
space
.
They
can
not
be
taken
by
equalitie
.
But
by
obsidion
the
whole
kyng
maye
be
taken
away
.
Also
his
whole
number
at
ones
,
that
is
.91
.
or
.190
.
by
Addition
,
by
Subtraction
,
by
Multiplication
,
or
by
Diuision
.
Also
he
maye
be
taken
by
partes
,
when
any
of
hys
syde
numbers
maye
be
taken
then
leseeth
he
that
draughte
,
as
when
anye
of
hys
square
numbers
is
gone
he
can
not
remoue
the
square
draught
,
and
so
of
the
rest
,
tyl
nothyng
of
him
be
left
,
then
muste
he
be
taken
away
,
and
the
triumph
prepared
.
¶
The
lawe
of
prisoners
.
WHen
any
is
taken
captiue
,
he
must
be
tourned
with
his
conquerers
collor
vpward
&
placed
in
the
hindermost
space
of
his
victors
campe
,
and
from
thens
being
remoued
must
fight
against
his
conquerours
enemies
,
and
serue
him
also
to
make
his
triumphe
.
❧
A
Table
to
take
any
of
the
men
,
by
addition
subtraction
,
multiplication
or
diuision
.
Addition
.
|
subtraction
|
Addition
.
|
Substract
.
|
|
1
|
|
|
4
|
|
8
|
56
|
64
|
|
30
|
|
1
|
2
|
3
|
4
|
4
|
8
|
|
9
|
|
30
|
36
|
66
|
1
|
3
|
4
|
4
|
5
|
9
|
9
|
36
|
45
|
30
|
42
|
72
|
1
|
4
|
5
|
4
|
8
|
12
|
9
|
72
|
81
|
30
|
90
|
120
|
1
|
5
|
6
|
4
|
12
|
16
|
9
|
81
|
90
|
30
|
91
|
121
|
1
|
6
|
7
|
4
|
16
|
20
|
9
|
91
|
100
|
|
36
|
|
1
|
7
|
8
|
4
|
45
|
49
|
|
12
|
|
36
|
36
|
72
|
1
|
8
|
9
|
|
50
|
|
12
|
16
|
28
|
36
|
45
|
81
|
1
|
15
|
16
|
|
7
|
12
|
12
|
30
|
42
|
36
|
64
|
100
|
1
|
120
|
121
|
5
|
15
|
20
|
|
15
|
|
|
42
|
|
|
2
|
|
5
|
20
|
25
|
15
|
30
|
45
|
42
|
49
|
91
|
2
|
3
|
5
|
5
|
25
|
30
|
15
|
49
|
64
|
|
45
|
|
2
|
4
|
6
|
|
6
|
|
15
|
66
|
81
|
45
|
noth
.
|
2
|
5
|
7
|
6
|
6
|
12
|
|
16
|
|
|
49
|
|
2
|
6
|
8
|
6
|
9
|
15
|
16
|
20
|
36
|
49
|
72
|
121
|
2
|
7
|
9
|
6
|
30
|
36
|
16
|
56
|
72
|
49
|
120
|
169
|
2
|
28
|
30
|
6
|
36
|
42
|
16
|
153
|
169
|
|
56
|
|
2
|
64
|
66
|
6
|
66
|
72
|
|
20
|
|
56
|
64
|
120
|
|
3
|
|
|
7
|
|
20
|
25
|
45
|
56
|
169
|
225
|
3
|
4
|
7
|
7
|
8
|
15
|
20
|
36
|
56
|
|
64
|
|
3
|
5
|
8
|
7
|
9
|
16
|
20.
|
100
|
120
|
64.
|
225.
|
289
|
3
|
6
|
9
|
7
|
42
|
49
|
|
25
|
|
|
72
|
|
3
|
9
|
12
|
7
|
49
|
56
|
25
|
56
|
81
|
72.
|
81.
|
153.
|
3
|
12
|
15
|
|
8
|
|
25
|
66
|
91
|
72.
|
153.
|
225.
|
3
|
42
|
45
|
8
|
12
|
20
|
|
28
|
|
72.
|
28.
|
361.
|
|
|
|
8
|
20
|
28
|
28
|
36
|
64
|
81
|
nothing
|
|
|
|
8
|
28
|
36
|
28
|
72
|
100
|
|
|
|
Addition
.
|
Multiplication
&
Diuision
.
|
|
90
|
|
|
2
|
|
5
|
9
|
45
|
90.
|
100.
|
190
|
2
|
3
|
6
|
5
|
20
|
100
|
|
91
|
|
2
|
4
|
8
|
5
|
45
|
125
|
91
|
nothing
|
2
|
6
|
12
|
|
6
|
|
|
100
|
|
2
|
8
|
16
|
6
|
6
|
36
|
100.
|
noth
.
|
2
|
15
|
30
|
6
|
7
|
42
|
|
120
|
|
2
|
28
|
56
|
6
|
12
|
72
|
120
|
169.
|
289
|
2
|
36
|
72
|
6
|
15
|
90
|
|
121
|
|
2
|
45
|
90
|
6
|
20
|
120
|
121.
|
169.
|
190
|
|
3
|
|
|
7
|
|
153
|
|
|
3
|
4
|
12
|
7
|
8
|
56
|
169
|
|
|
3
|
5
|
15
|
|
8
|
|
190
|
noth
.
|
3
|
12
|
36
|
8
|
9
|
72
|
225
|
|
|
3
|
15
|
45
|
8
|
15
|
120
|
289
|
|
|
3
|
30
|
90
|
|
|
|
361
|
|
|
|
4
|
|
|
9
|
|
|
|
|
4
|
4
|
16
|
9
|
9
|
81
|
|
|
|
4
|
5
|
20
|
9
|
25
|
225
|
|
|
|
4
|
7
|
28
|
|
|
|
|
|
|
4
|
9
|
36
|
|
|
|
|
|
|
4
|
16
|
64
|
|
|
|
|
|
|
4
|
25
|
100
|
|
|
|
|
|
|
4
|
30
|
120
|
|
|
|
|
|
|
|
5
|
|
|
|
|
|
|
|
5
|
6
|
30
|
|
|
|
By
this
Table
any
man
though
he
haue
small
or
no
skyll
in
Arithmeticke
,
maye
learne
to
playe
at
this
game
,
and
in
playinge
learne
some
parte
of
Arithmeticke
.
¶
Of
takynge
by
proportion
.
IF
the
Gamesters
be
disposed
,
they
maye
take
men
also
by
proportion
,
Arithmeticall
,
Geometrical
,
or
Musicall
.
But
because
it
is
not
necessarily
required
that
they
shoulde
so
do
,
I
wyll
fyrst
prosecute
the
maner
of
triumph
,
in
which
also
they
maye
learne
to
take
by
proportion
,
as
afterwarde
shal
be
seene
.
For
when
they
can
ioyne
two
or
three
of
their
men
to
one
of
their
aduersaries
men
in
such
order
as
the
triumph
is
set
,
so
that
those
three
or
foure
numbers
haue
anye
of
these
three
proportions
they
maye
take
their
aduersaries
man
.
¶
Of
the
triumphe
.
WHen
the
King
is
taken
,
the
triumph
must
be
prepared
to
be
set
in
the
aduersaries
campe
.
The
aduersaries
campe
is
called
al
the
space
,
that
is
betwene
the
first
front
of
his
men
,
as
they
were
first
placed
,
vnto
the
neither
ende
of
the
table
,
conteyning
.40
.
spaces
or
as
some
wil
.48
▪
When
you
entend
to
make
a
triumph
you
must
proclaime
it
,
admonishing
your
aduersarie
,
that
he
medle
not
with
anye
man
to
take
hym
,
whiche
you
haue
placed
for
youre
triumphe
.
Furthermore
,
you
must
bryng
all
your
men
that
serue
for
the
triumph
in
their
direct
motions
,
and
not
in
theyr
flying
draughtes
.
To
triumphe
therefore
,
is
to
place
three
or
foure
men
within
the
aduersaries
campe
,
in
proportion
Arithmeticall
,
Geometricall
or
Musicall
,
as
wel
of
your
owne
men
,
as
of
your
enemyes
men
that
be
taken
,
standing
in
a
right
lyne
,
direct
or
crosse
,
as
in
.
D.A.B.
or
els
5.1.3
.
if
it
consist
of
three
numbers
,
but
if
it
stande
of
foure
numbers
,
they
maye
be
set
lyke
a
square
two
agaynst
two
,
as
in
.
E.B.D.C.
or
.2.3.4.5
.
and
after
the
same
maner
muste
you
set
them
so
that
your
aduersaries
man
make
the
thyrde
or
fourth
,
when
you
take
by
proportion
.
¶
Of
dyuers
kyndes
of
triumphes
.
THere
be
thre
kyndes
of
triumphes
a
great
triumphe
,
a
greater
triumphe
,
and
the
greatest
and
moste
noble
of
all
.
¶
Of
the
great
triumph
.
THe
great
triumph
standeth
in
proportion
,
eyther
Arithmeticall
,
Geometrical
,
or
Musicall
onely
.
¶
Of
Arithmeticall
proportion
.
ARithmeticall
proportion
,
is
when
the
mydle
number
differeth
as
muche
from
the
first
,
as
from
the
thyrde
,
that
is
to
saye
,
when
the
thyrde
hath
so
many
more
,
from
the
seconde
,
as
the
seconde
hath
from
the
firste
,
as
.2.4.6
.
Here
,
two
,
is
the
distans
,
for
.4
.
excedeth
2.
by
two
,
&
.6
.
is
more
then
foure
by
.2
.
¶
A
rule
to
fynde
out
Arithmeticall
proportion
betwene
the
firste
and
the
laste
.
WHen
you
haue
the
first
and
the
last
if
you
woulde
finde
out
the
midle
in
proportion
.
Adde
the
first
&
the
last
together
,
and
deuide
the
whole
into
2.
for
the
halfe
is
the
midle
in
proportion
as
I
woulde
knowe
what
is
the
midle
number
in
proportion
betwene
.5
.
and
25.
first
I
adde
.5
.
to
.20
.
that
is
.30
.
the
half
of
thirtie
is
.15
.
whiche
is
midle
in
proportion
betwene
.5
.
and
.30
.
so
haue
I
.5.15.35
.
in
Arithmeticall
proportion
.
¶
A
table
of
al
the
Arithmetical
proportions
that
be
in
this
game
.
2
|
3
|
4
|
6
|
7
|
8
|
28
|
64
|
100
|
2
|
4
|
6
|
6
|
9
|
12
|
30
|
36
|
42
|
2
|
5
|
8
|
6
|
36
|
66
|
42
|
49
|
56
|
2
|
7
|
12
|
7
|
8
|
9
|
42
|
66
|
90
|
2
|
9
|
16
|
7
|
16
|
25
|
49
|
169
|
289
|
2
|
15
|
28
|
7
|
64
|
121
|
56
|
64
|
71
|
2
|
16
|
30
|
9
|
12
|
15
|
72
|
81
|
90
|
3
|
4
|
5
|
9
|
45
|
81
|
|
49.
|
|
3
|
5
|
7
|
9
|
81
|
153
|
|
|
|
3
|
6
|
9
|
12
|
16
|
20
|
|
|
|
3
|
9
|
15
|
12
|
20
|
28
|
|
|
|
4
|
5
|
6
|
12
|
42
|
72
|
|
|
|
4
|
6
|
8
|
12
|
66
|
120
|
|
|
|
4
|
8
|
12
|
15
|
20
|
25
|
|
|
|
4
|
12
|
20
|
15
|
30
|
45
|
|
|
|
4
|
20
|
36
|
15
|
120
|
225
|
|
|
|
4
|
30
|
56
|
16
|
36
|
56
|
|
|
|
5
|
6
|
7
|
20
|
25
|
30
|
|
|
|
5
|
7
|
9
|
20
|
28
|
36
|
|
|
|
5
|
15
|
25
|
20
|
42
|
64
|
|
|
|
5
|
25
|
45
|
28
|
42
|
56
|
|
|
|
¶
Of
Geometricall
proportion
.
GEometricall
proportion
,
is
when
the
seconde
hath
that
proportion
to
the
first
,
that
the
thyrde
hath
to
the
seconde
,
as
.2.4.8
.
as
.4
.
excedeth
.2
.
by
2.
so
.8
.
excedeth
.4
.
by
.4
.
¶
A
rule
to
fynde
the
mydle
number
in
Geometricall
proportion
.
MUltiplie
the
firste
by
the
thyrde
,
and
of
the
product
fynde
out
the
roote
square
,
for
that
is
the
midle
,
if
the
numbers
haue
anye
roote
square
in
whole
numbers
.
The
roote
square
is
a
number
multiplied
in
it selfe
,
wherefore
you
muste
seeke
such
a
number
,
as
multiplied
in
it selfe
,
maketh
the
producte
of
the
fyrst
and
the
thyrde
number
multiplied
one
by
the
other
.
As
.20
.
multiplied
by
.45
.
is
.900
.
the
roote
is
.30
.
square
,
whych
multiplyed
in
it selfe
is
.900
.
But
yf
you
lyste
not
to
take
suche
paynes
,
here
is
a
Table
that
maye
serue
your
tourne
for
Geometricall
proportion
to
be
vsed
in
this
game
.
¶
A
table
for
Geometricall
proportion
.
2
|
4
|
8
|
16
|
36
|
81
|
2
|
12
|
72
|
20
|
30
|
45
|
3
|
6
|
12
|
25
|
30
|
36
|
4
|
6
|
9
|
25
|
45
|
81
|
4
|
8
|
16
|
36
|
42
|
49
|
4
|
12
|
36
|
36
|
66
|
121
|
4
|
16
|
64
|
36
|
90
|
225
|
4
|
20
|
100
|
49
|
56
|
64
|
5
|
15
|
45
|
49
|
91
|
169
|
9
|
12
|
16
|
64
|
72
|
81
|
9
|
15
|
25
|
64
|
120
|
225
|
9
|
45
|
225
|
81
|
90
|
100
|
16
|
20
|
25
|
81
|
153
|
289
|
16
|
28
|
49.
|
|
27.
|
|
¶
Of
Musicall
proportion
.
MUusicall
proportion
is
when
the
differences
of
the
first
and
last
from
the
middes
,
are
the
same
,
that
is
betwene
the
first
and
the
last
,
as
.3.4.6
.
betwene
.3
.
and
.4
.
is
.1
.
betwene
.4
.
and
6.
is
.2
.
the
whole
difference
is
.3
.
which
is
the
difference
betwene
.6
.
and
.3
.
the
first
and
the
last
.
¶
A
rule
to
fynde
the
first
,
when
you
haue
the
two
last
.
MUltiplie
the
seconde
by
the
thyrd
,
deuide
the
producte
by
the
distans
and
the
thyrde
number
,
and
the
quotient
is
the
first
,
as
hauynge
.6
.
and
12.
I
would
fynde
the
first
,
6.
tymes
.12
.
is
72
,
the
difference
betwene
.6
.
and
.12
.
is
6
,
whiche
added
to
.12
.
is
.18
,
deuide
.72
.
by
18.
the
quotient
is
.4
.
so
haue
you
.4.6.12
.
in
Musicall
proportion
.
¶
To
finde
the
midle
betwene
the
first
and
the
last
.
MUltiplie
the
first
by
the
last
,
then
double
the
producte
,
and
deuide
the
whole
by
the
first
and
the
laste
added
together
,
the
quotient
is
then
the
mydle
number
.
As
hauyng
.6
.
and
.12
.
I
woulde
knowe
the
mydle
in
Musicall
proportion
.
First
I
multiplie
one
by
the
other
,
the
product
is
.72
.
that
doubled
is
144
,
this
deuided
by
.18
.
which
is
the
addition
,
of
.6
.
and
12.
geueth
the
quotient
8.
so
haue
I
.6.8.12
.
in
musicall
proportion
.
And
thus
must
you
worke
to
fynde
out
the
thyrde
in
musicall
proportion
.
But
if
you
had
rather
playe
then
worke
,
this
table
folowing
shall
serue
your
torne
.
¶
A
table
of
Musicall
proportion
.
2
|
3
|
6
|
3
|
4
|
6
|
3
|
15
|
16
|
4
|
6
|
12
|
4
|
7
|
28
|
5
|
8
|
20
|
5
|
9
|
45
|
6
|
8
|
12
|
7
|
12
|
42
|
8
|
15
|
120
|
9
|
15
|
45
|
9
|
16
|
72
|
12
|
15
|
20
|
15
|
20
|
30
|
5
|
45
|
225
|
30
|
36
|
45
|
30
|
45
|
49
|
72
|
90
|
120
|
|
17.
|
|
¶
Of
the
greater
triumphe
.
THe
greater
victorie
is
,
when
foure
numbers
be
broughte
together
,
whiche
agree
in
two
proportions
,
either
Arithmeticall
and
Geometricall
,
or
elles
Arithmeticall
and
Musicall
,
or
elles
Geometricall
and
Musicall
.
Of
these
three
coniunctions
the
greater
triumph
consisteth
,
of
the
which
the
table
foloweth
.
❧
A
table
of
Arithmeticall
,
and
Geometricall
proportion
.
2
|
3
|
4
|
8
|
9
|
12
|
15
|
16
|
2
|
4
|
6
|
8
|
9
|
12
|
15
|
25
|
2
|
4
|
6
|
9
|
9
|
12
|
16
|
20
|
2
|
4
|
5
|
8
|
9
|
45
|
81
|
225
|
2
|
7
|
12
|
72
|
9
|
25
|
45
|
81
|
2
|
9
|
12
|
16
|
9
|
12
|
16
|
20
|
2
|
12
|
42
|
72
|
9
|
15
|
20
|
25
|
3
|
6
|
9
|
12
|
9
|
8
|
153
|
289
|
3
|
4
|
6
|
9
|
12
|
16
|
20
|
25
|
3
|
9
|
15
|
25
|
15
|
16
|
20
|
25
|
4
|
5
|
6
|
9
|
15
|
20
|
30
|
45
|
4
|
6
|
8
|
9
|
16
|
20
|
25
|
30
|
4
|
6
|
9
|
12
|
16
|
36
|
56
|
81
|
4
|
6
|
8
|
16
|
20
|
25
|
30
|
45
|
4
|
12
|
20
|
36
|
30
|
36
|
42
|
49
|
4
|
8
|
12
|
16
|
36
|
42
|
40
|
56
|
4
|
8
|
12
|
36
|
42
|
49
|
56
|
64
|
4
|
8
|
16
|
28
|
49
|
56
|
64
|
72
|
4
|
12
|
20
|
100
|
49.
|
91.
|
169.
|
289
|
4
|
16
|
28
|
49
|
56
|
64
|
72
|
81
|
4
|
16
|
28
|
64
|
64
|
72
|
81
|
90
|
4
|
20
|
36
|
100
|
72
|
81
|
90
|
100
|
5
|
9
|
15
|
25
|
|
|
|
|
5
|
15
|
25
|
45
|
|
52.
|
|
5
|
25
|
45
|
81
|
|
|
|
|
6
|
9
|
12
|
16
|
|
|
|
|
7
|
16
|
20
|
25
|
|
|
|
|
7
|
49
|
91
|
169
|
|
|
|
|
8
|
9
|
12
|
16
|
|
|
|
|
8
|
64
|
120
|
225
|
|
|
|
|
Arithmeticall
and
musicall
proportion
.
|
Geometricall
and
musicall
proportion
together
.
|
3
|
4
|
5
|
6
|
2
|
3
|
6
|
12
|
3
|
4
|
5
|
15
|
3
|
4
|
6
|
9
|
3
|
4
|
6
|
9
|
3
|
4
|
6
|
12
|
3
|
5
|
7
|
25
|
3
|
6
|
8
|
12
|
3
|
5
|
9
|
15
|
4
|
6
|
12
|
36
|
3
|
9
|
15
|
45
|
4
|
7
|
28
|
49
|
3
|
4
|
6
|
8
|
5
|
9
|
15
|
45
|
4
|
5
|
6
|
12
|
5
|
9
|
45
|
225
|
4
|
6
|
12
|
15
|
5
|
9
|
45
|
81
|
4
|
6
|
12
|
20
|
9
|
12
|
16
|
72
|
4
|
12
|
15
|
20
|
9
|
15
|
25
|
45
|
5
|
7
|
9
|
45
|
9
|
15
|
45
|
225
|
6
|
7
|
8
|
12
|
9
|
25
|
45
|
225
|
8
|
15
|
120
|
225
|
15
|
20
|
30
|
45
|
9
|
12
|
15
|
45
|
20
|
30
|
36
|
45
|
9
|
12
|
15
|
20
|
25
|
45
|
81
|
225
|
9
|
15
|
30
|
45
|
|
16.
|
|
|
9
|
15
|
45
|
81
|
|
|
|
|
12
|
15
|
20
|
25
|
|
|
|
|
15
|
20
|
25
|
30
|
|
|
|
|
15
|
20
|
30
|
45
|
|
|
|
|
15
|
30
|
36
|
45
|
|
|
|
|
15
|
30
|
45
|
90
|
|
|
|
|
30
|
36
|
42
|
45
|
|
|
|
|
72
|
81
|
90
|
120
|
|
|
|
|
|
25.
|
|
|
|
|
|
¶
Of
the
greatest
triumphe
.
THe
greatest
triumph
is
of
Arithmeticall
,
Geometricall
,
and
Musicall
proportions
all
ioyned
together
.
Arithmeticall
,
Geometricall
,
and
Musicall
proportions
,
all
together
.
2
|
3
|
4
|
6
|
6
|
8
|
12
|
16
|
2
|
3
|
6
|
9
|
6
|
12
|
15
|
20
|
2
|
4
|
6
|
12
|
7
|
12
|
42
|
72
|
2
|
5
|
8
|
20
|
8
|
15
|
64
|
120
|
2
|
7
|
12
|
42
|
8
|
15
|
120
|
225
|
2
|
9
|
16
|
72
|
12
|
15
|
16
|
20
|
3
|
4
|
6
|
8
|
12
|
15
|
20
|
25
|
3
|
4
|
6
|
9
|
15
|
20
|
36
|
45
|
3
|
5
|
9
|
15
|
15
|
30
|
45
|
90
|
3
|
5
|
15
|
25
|
|
30
|
|
|
3
|
9
|
15
|
45
|
|
|
|
|
4
|
6
|
8
|
12
|
|
|
|
|
4
|
6
|
9
|
12
|
|
|
|
|
4
|
7
|
16
|
28
|
|
|
|
|
4
|
7
|
28
|
49
|
|
|
|
|
5
|
6
|
25
|
45
|
|
|
|
|
5
|
9
|
45
|
81
|
|
|
|
|
5
|
25
|
45
|
225
|
|
|
|
|
5
|
15
|
25
|
45
|
|
|
|
|
6
|
8
|
9
|
12
|
|
|
|
|
And
thus
is
the
first
kynd
of
playing
at
an
end
.
And
this
is
sufficient
to
teach
you
to
play
,
but
if
you
would
learne
to
play
conningly
,
you
must
vse
to
playe
often
,
so
shall
you
learne
better
then
by
anye
preceptes
or
rules
.
¶
Of
the
seconde
kynde
of
playinge
at
the
Philosophers
game
.
THere
is
in
this
kynde
of
playing
to
be
considered
,
the
table
,
the
men
,
the
marking
of
them
,
the
setting
of
them
in
araye
,
their
marching
,
their
lawes
of
taking
,
and
the
maner
of
triumphynge
.
¶
Of
the
Table
.
THe
Table
is
the
same
that
was
first
described
,
namely
a
double
chesbord
.
¶
Of
the
men
.
THe
men
be
as
before
in
number
48.23
.
on
a
syde
,
and
two
contrarye
kynges
of
euen
and
of
odde
.
They
must
be
of
diuers
colours
,
as
hath
bene
sayde
,
the
bottome
of
euery
one
must
haue
his
enemies
colour
,
and
his
owne
mark
of
number
,
differing
in
this
poinct
from
the
former
playing
,
that
the
enemies
men
taken
,
may
serue
onely
to
celebrate
a
triumphe
,
but
not
to
fight
on
his
syde
that
taketh
them
.
¶
Of
the
markyng
of
the
men
.
THey
be
marked
with
the
same
numbers
,
that
haue
bene
shewed
before
and
therefore
so
are
to
be
founde
out
as
is
taught
before
.
But
they
be
marked
besyde
their
numbers
,
with
cossicall
signes
,
which
be
signes
vsed
in
the
rule
called
regula
cossa
,
or
algebra
,
betokening
rootes
,
quadrats
,
cubes
,
fouresquared
quadrats
,
sursolides
,
&
quadrates
of
cubes
.
All
these
.6
.
signes
must
be
conteyned
in
thys
game
.
The
signe
-
of
the
roote
.¹
-
of
the
quadrate
.²
-
of
the
cube
,
or
solide
quadrat
.³
-
of
the
fouresquared
quadrat
.⁴
-
of
the
sursolide
.⁵
-
of
the
squared
cube
.⁶
EUery
number
maye
be
taken
for
a
roote
,
as
.2
.
this
number
multiplyed
in
it self
is
a
square
as
.4
.
The
quadrat
or
square
multiplied
by
the
roote
geueth
a
cube
or
solide
square
,
as
.4
.
multiplied
by
.2
.
geueth
.8
.
that
is
a
cube
.
Multiplie
the
cube
by
the
roote
,
so
haue
you
a
squared
quadrat
,
as
.8
.
by
.2
.
geueth
16.
which
is
a
quadrate
of
a
quadrate
.
Multiplie
the
square
or
quadrat
of
quadrat
by
the
roote
,
and
the
product
is
the
sursolyde
,
as
.2
.
tymes
.16
.
is
.32
.
whiche
is
a
sursolide
.
Multiplie
the
sursolide
by
the
roote
,
and
the
product
is
the
quadrate
of
a
cube
,
as
.2
.
times
.32
.
is
.64
.
which
is
a
quadrat
of
a
cube
.
So
haue
you
the
roote
quadrat
,
cube
,
quadrat
of
quadrat
,
sursolide
,
quadrat
of
cube
.2.4.8.16.32.64
.
So
.2
.
referred
to
.4
.
is
a
roote
of
a
square
,
referred
to
.8
.
it
is
a
roote
of
a
cube
.2
.
referred
to
.16
.
is
the
roote
of
a
foure
squared
quadrate
.2
.
referred
to
.32
.
is
the
roote
of
a
sursolide
.2
.
referred
to
.64
.
is
the
roote
of
a
quadrate
of
a
cube
.
These
numbers
muste
haue
the
proper
cossicall
signes
.
Also
one
number
hauing
diuers
relations
,
may
haue
diuers
cossical
signes
,
as
9.
referred
to
.81
.
being
roote
,
hathe
the
signe
of
a
roote¹
,
but
beyng
referred
to
.3
.
it
hath
the
signe
of
a
quadrate
,
for
it
is
a
quadrate
of
.3
.
and
is
thus
signed
.²
.
and
so
of
the
rest
that
haue
like
relation
.
¶
The
marking
of
the
men
.
THe
first
order
of
roundes
in
bothe
numbers
,
must
haue
the
signe
of
the
roote
vpon
them
al
after
this
maner
.
8¹
6¹
4¹
2¹
9¹
7¹
5¹
3¹
THe
second
order
of
roundes
founde
out
as
before
,
be
not
all
marked
with
cossicall
signes
,
but
onely
.4
.
and
.9
.
with
the
roote
,
and
.81
.
with
the
quadrate
.
The
rest
haue
none
,
because
amonge
their
aduersaries
men
there
is
none
that
can
be
cossicall
roote
to
them
in
such
maner
as
this
game
requireth
.
64.
36.
16
4¹
81²
49
25
9¹
THe
first
order
of
triangles
(
hauyng
the
same
numbers
that
haue
bene
taught
before
)
do
all
lack
the
cossicall
signes
,
except
onely
.6
.
which
is
signed
with
the
roote
.
72.
42.
20
6¹
81
56
30
12
THe
seconde
order
of
triangles
,
haue
all
excepte
one
(
whiche
is
the
number
of
.100
.
)
their
cossicall
signes
,
as
9.
bothe
of
the
roote
and
of
the
quadrate
,
25.36
.
and
.49
.
haue
the
signe
of
the
quadrate
.64
.
of
the
quadrate
and
the
cube
,
and
also
the
quadrat
of
cube
.16
.
and
.81
.
of
the
quadrate
,
and
the
foure
squared
quadrate
.
81²
.⁴
49²
25²
9¹
100
64¹
.³
36²
16²
In
the
firste
order
of
squares
,
onely
15.
is
marked
with
the
roote
,
all
the
rest
doe
want
theyr
cossicall
sygnes
in
thys
game
.
153.
91
45
15¹
190
120
66
28
THe
seconde
order
of
squares
hath
.3
.
numbers
marked
with
cossicall
signes
,
that
is
.25
.
and
.225
.
wyth
the
signe
of
the
quadrate
.81
.
is
marked
with
the
sygne
of
the
quadrate
and
the
foure-squared
quadrate
.
289.
169.
81²
.⁴
25.²
361.
225.²
121.
49.
And
thus
haue
you
all
the
men
that
be
marked
with
cossicall
sygnes
.
¶
The
setting
in
aray
.
THe
teachers
of
this
kynde
of
playing
,
doe
not
so
well
allowe
,
the
former
kynde
of
placing
or
any
other
,
as
the
naturall
placing
of
euery
man
vnder
him
of
whome
he
aryseth
.
So
thei
conteyne
.6
.
ranks
in
length
,
extending
to
the
furthermoste
edge
of
the
Table
after
this
sorte
.
¶
The
marching
or
mouing
.
THe
men
maye
remoue
euery
way
,
into
voyde
places
,
forwarde
,
backewarde
,
towarde
both
sydes
,
directe
or
cornerwyse
.
So
that
the
rounde
men
remoue
into
the
next
space
,
the
triangles
into
the
third
place
,
and
the
squares
into
the
fourth
place
,
accompting
that
place
in
which
they
stande
for
one
.
Also
euery
man
sauyng
the
two
kynges
to
besiege
his
enemie
,
or
to
flye
from
the
siege
him self
,
may
remoue
the
knights
draught
in
chesse
,
but
neither
take
anye
man
(
except
it
be
by
siege
)
nor
erect
a
triumphe
by
suche
motions
.
The
kynges
moue
euen
as
squares
,
but
that
they
haue
not
the
flyinge
draughte
.
It
is
compted
lawefull
amonge
suche
as
wyll
so
agree
,
that
the
Triangles
and
Squares
,
maye
remoue
into
voyde
places
,
thoughe
the
spaces
betwene
be
occupyed
of
other
men
.
¶
The
maner
of
takyng
.
THe
men
may
be
taken
seuen
ways
by
Obsidion
,
by
Equalitie
,
by
Addition
,
by
Substraction
,
by
Multiplication
,
by
Diuision
,
and
by
Cossicall
Sygnes
.
¶
Of
takynge
by
Obsidion
.
ALl
men
maye
be
taken
by
Obsidion
when
by
foure
men
they
be
letted
of
theyr
ordinarie
draughte
,
as
hath
bene
taught
before
.
¶
Of
takynge
by
Equalitie
.
BY
Equalitie
maye
these
men
take
or
be
taken
,
as
hathe
bene
sayde
before
,
9.16.25.36.49.64.81
,
as
yf
after
you
haue
played
your
.9
.
you
see
youre
aduersaryes
.9
.
stande
in
your
mans
draught
,
you
may
take
him
vp
not
remouyng
into
his
place
,
vnlesse
you
espye
him
standing
in
your
draught
before
you
playe
,
then
muste
you
take
him
vp
and
remoue
into
his
place
.
¶
Of
takynge
by
Addition
.
THe
takyng
by
Addition
is
all
one
with
the
first
kynde
of
play
,
in
all
respectes
,
sauing
that
some
require
the
men
that
shoulde
take
by
Addition
to
stande
in
the
next
spaces
to
him
that
is
taken
,
either
directly
,
or
cornerwyse
,
but
the
former
waye
is
better
.
Of
taking
by
substraction
.
THat
whiche
was
sayde
in
the
first
kinde
of
substraction
and
that
whiche
was
last
sayde
of
Addition
may
be
bothe
referred
hyther
.
For
this
substraction
differeth
not
from
the
former
,
but
for
the
opinion
of
them
,
that
would
haue
the
two
takers
stande
onelye
in
the
nexte
spaces
to
hym
that
is
taken
.
¶
Of
takyng
by
Multiplication
.
TAkyng
by
multiplication
doth
differ
.
For
in
this
kynde
of
playng
,
it
is
thus
.
When
your
man
standeth
so
,
that
beyng
lesser
then
your
aduersaries
man
,
you
may
multiplie
your
man
by
the
voyde
spaces
betwene
them
,
and
the
product
is
all
one
with
the
aduersarye
,
you
maye
take
hym
vp
,
not
remouynge
into
his
place
,
except
you
espye
hym
so
,
before
you
remoue
your
man
.
¶
Of
takynge
by
Diuision
.
LYkewise
by
Diuision
,
yf
your
man
beyng
greater
then
the
aduersarye
,
stande
so
,
that
beyng
deuyded
by
the
voyde
spaces
,
the
quotient
is
all
one
with
the
aduersarye
,
you
maye
take
hym
vp
,
not
remouyng
into
hys
place
,
vnlesse
you
see
hym
so
standynge
before
you
drawe
.
¶
Of
taking
by
Cossicall
signes
.
BY
Cossicall
sygnes
anye
man
that
hath
these
signes
,²
.
meeting
with
his
roote
in
his
ordinary
draught
that
hath
this
signe¹
.
taketh
him
vp
,
or
elles
is
taken
of
him
,
without
remouyng
into
his
place
,
except
he
maye
take
him
before
he
remoue
.
¶
Of
the
Kynges
,
and
their
taking
.
THe
King
of
the
euen
must
be
foure
square
,
hauyng
sixe
steppes
,
euery
one
losser
then
other
,
on
one
syde
he
muste
haue
on
him
these
rootes
.1.2.3.4.5.6
.
on
the
other
syde
the
quadrates
arising
of
these
rots
,
that
is
1.4.9.16.25.36
.
¶
The
King
of
the
odde
men
,
muste
haue
but
fyue
steppes
,
that
is
.4.5.6.7.8
.
lackyng
the
rootes
that
he
can
not
ende
in
.1
.
The
quadrates
of
hys
rootes
be
these
.16.25.36.49.64
.
These
muste
be
so
set
on
,
that
the
least
must
be
hyghest
and
the
greatest
lowest
.
¶
The
Kinges
be
taken
by
Obsidion
,
or
yf
theyr
Pyramidall
number
,
be
taken
by
anye
of
the
aforsayde
meanes
.
Also
yf
by
suche
meanes
you
can
take
all
his
quadrates
one
after
another
.
¶
The
priuilege
of
the
King.
IF
anye
of
the
Kynges
quadrates
be
taken
,
he
maye
redeme
it
by
anye
of
his
men
hauing
the
same
number
,
and
muste
remoue
into
hys
place
,
whiche
redemed
hym
.
But
yf
he
haue
none
of
the
same
number
,
he
maye
redeme
hym
for
anye
man
of
hys
,
that
his
aduersarye
wyll
chuse
,
and
lykewyse
remoue
into
hys
place
by
whome
he
is
redemed
.
¶
A
table
to
take
the
men
by
Multiplication
and
Diuision
.
euen
against
od
spaces
.
|
euen
.
spaces
.
od
|
euen
.
spaces
.
od
|
2
|
8
|
16
|
3
|
5
|
15
|
6
|
2
|
12
|
8
|
8
|
64
|
5
|
5
|
25
|
8
|
2
|
16
|
4
|
9
|
36
|
9
|
5
|
45
|
15
|
2
|
30
|
9
|
9
|
81
|
12
|
6
|
72
|
45
|
2
|
90
|
25
|
9
|
225
|
7
|
7
|
49
|
4
|
3
|
12
|
9
|
10
|
90
|
5
|
9
|
45
|
4
|
4
|
16
|
|
21.
|
|
9
|
9
|
81
|
9
|
4
|
36
|
od
against
euen
spaces
.
|
3
|
12
|
36
|
16
|
4
|
64
|
3
|
14
|
42
|
6
|
5
|
30
|
3
|
2
|
6
|
|
17.
|
|
20
|
5
|
100
|
36
|
2
|
72
|
|
|
|
2
|
6
|
12
|
3
|
3
|
9
|
|
|
|
15
|
6
|
90
|
5
|
3
|
15
|
|
|
|
20
|
6
|
120
|
12
|
3
|
36
|
|
|
|
4
|
7
|
28
|
5
|
4
|
20
|
|
|
|
8
|
7
|
56
|
9
|
4
|
36
|
|
|
|
Spaces
.
|
16
|
4
|
64
|
|
|
|
¶
For
Diuision
.
euen
against
od
spaces
.
|
od
against
euen
spaces
.
|
¶
To
take
by
cossical
signes
|
6
|
2
|
3
|
12
|
2
|
6
|
2
|
16.⁴
|
72
|
2
|
36
|
16
|
2
|
8
|
2
|
64.⁶
|
15
|
3
|
5
|
30
|
2
|
15
|
3
|
81.⁴
|
36
|
3
|
12
|
90
|
2
|
45
|
|
|
9
|
3
|
3
|
12
|
3
|
4
|
3
|
9.²
|
20
|
4
|
5
|
16
|
4
|
4
|
4
|
16.²
|
36
|
4
|
9
|
36
|
4
|
9
|
4
|
64.³
|
64
|
4
|
16
|
64
|
4
|
16
|
5
|
25.²
|
15
|
5
|
3
|
100
|
4
|
25
|
|
|
25
|
5
|
5
|
22
|
5
|
45
|
6
|
36.²
|
45
|
5
|
9
|
30
|
5
|
6
|
7
|
49.²
|
42
|
6
|
7
|
100
|
5
|
20
|
8
|
64.²
|
72
|
6
|
12
|
12
|
6
|
2
|
9
|
81.²
|
49
|
7
|
7
|
36
|
6
|
6
|
15
|
225.²
|
72
|
8
|
9
|
90
|
6
|
15
|
|
|
45
|
9
|
5
|
120
|
6
|
20
|
|
|
81
|
9
|
9
|
28
|
7
|
4
|
|
|
36
|
12
|
3
|
56
|
7
|
8
|
|
|
91
|
13
|
7
|
16
|
8
|
2
|
|
|
42
|
14
|
3
|
64
|
8
|
8
|
|
|
|
|
|
120
|
8
|
15
|
|
|
|
20.
|
|
3
|
9
|
4
|
|
|
|
|
|
81
|
9
|
9
|
|
|
|
|
|
225
|
9
|
25
|
|
|
|
|
|
90
|
10
|
9
|
|
|
|
|
|
66
|
11
|
6
|
|
|
|
|
|
28
|
14
|
2
|
|
|
|
|
|
|
27.
|
|
|
|
¶
Of
the
triumph
.
THe
triumph
is
after
the
Kynge
be
cleane
taken
away
,
to
be
created
in
the
aduersaries
campe
,
as
well
of
your
owne
men
as
of
your
aduersaries
men
that
be
taken
,
or
of
both
in
proportion
as
hath
bene
shewed
before
,
and
proclaimed
that
those
men
ons
placed
,
may
not
be
taken
,
as
it
was
declared
sufficiently
,
and
no
difference
betwene
the
triumphes
,
sauyng
that
some
wyll
not
alowe
a
triumphe
but
of
foure
numbers
,
and
two
proportions
at
the
lest
.
All
three
for
the
greater
victorie
,
makynge
but
two
kynds
of
triumphes
.
¶
Here
foloweth
the
thyrd
kynde
of
playing
at
the
Philosophers
game
.
THere
must
also
in
this
thyrd
kynde
be
considered
the
table
,
the
men
,
their
markyng
,
the
order
of
theyr
battell
,
the
motions
,
their
taking
,
and
last
of
all
theyr
triumphing
.
The
table
is
the
same
that
hath
bene
twyse
already
discribed
.
Yet
some
wyll
not
haue
it
so
longe
,
but
at
the
lest
it
must
conteyne
.10
.
squares
in
length
and
alwayes
.8
.
in
breadeth
.
The
longest
is
best
.
¶
Of
the
men
.
THe
men
be
.48
.
as
it
hath
bene
said
of
two
contrary
collor
,
the
head
and
bottom
all
of
one
collor
,
because
men
ons
taken
be
no
more
occupyed
in
thys
kynde
of
playing
.
¶
The
inscription
and
fashion
.
THe
fashion
is
as
hath
bene
last
declared
both
of
the
men
,
and
of
the
kynges
,
the
inscription
of
numbers
the
same
,
but
without
cossical
signes
.
¶
Of
the
order
of
the
battell
.
THe
order
of
battell
is
after
the
firste
maner
,
but
not
so
farre
from
the
bordes
end
,
namely
the
.4
.
squares
standynge
in
the
plattes
nearest
to
the
bordes
end
the
rest
accordingly
ioyned
to
them
,
as
in
the
firste
kynde
of
playing
.
¶
Of
their
motions
.
THe
men
moue
frowarde
and
backward
,
to
the
right
hand
,
and
to
the
left
hande
,
but
not
cornerwise
,
except
the
gamesters
so
agree
,
the
rounds
into
the
next
space
,
the
triangles
into
the
thyrde
,
and
the
squares
into
the
fourth
,
the
kyngs
moue
as
squares
.
And
these
be
their
ordinary
draughts
in
marching
.
¶
Of
their
taking
.
THey
are
taken
by
encountering
,
by
eruption
,
by
laying
wayght
,
and
by
Obsidion
.
¶
Of
takyng
by
encountering
.
TO
take
by
encountering
is
to
take
by
Equalitie
,
as
hath
bene
twyse
before
declared
.
¶
Of
taking
by
eruption
.
TO
take
by
eruption
is
when
a
lesse
number
beyng
multiplied
by
the
spaces
that
are
betwene
him
&
hys
aduersary
,
the
product
is
asmuch
as
his
aduersary
,
he
may
take
his
enemie
awaye
whether
he
stand
directly
from
him
or
cornerwise
.
For
men
that
may
be
taken
by
eruption
looke
in
the
table
of
takyng
by
multiplication
in
the
second
kynd
of
playing
.
¶
Of
takyng
by
deceypt
or
lying
weyght
.
TO
take
by
deceypt
or
lying
weight
,
is
to
take
by
addition
,
not
as
before
when
the
aduersary
standeth
within
the
draught
of
two
men
which
being
added
make
the
iuste
number
of
the
aduersary
,
but
when
the
.2
.
numbers
that
are
to
be
added
,
stande
in
the
next
spaces
to
the
aduersarie
.
For
to
take
by
deceipt
,
looke
in
the
table
that
was
set
forth
for
takyng
by
addition
in
the
first
kynde
of
playinge
.
¶
Of
taking
by
Obsidion
.
BY
Obsidion
all
men
may
be
taken
,
when
foure
men
besiege
the
aduersarye
,
standynge
in
the
foure
nexte
spaces
about
him
directly
,
or
cornerwise
,
the
man
so
besieged
can
not
escape
,
because
he
can
not
remoue
cornerwyse
,
therefore
maye
be
taken
vp
,
so
soone
as
the
last
of
the
foure
is
set
in
his
place
.
In
all
thrée
kyndes
of
playing
no
Obsidion
can
be
of
any
man
with
some
of
his
fellowes
,
but
all
foure
muste
be
hys
aduersaries
.
In
this
thyrde
kynde
,
these
men
can
be
none
otherwyse
taken
but
by
Obsidion
.
Namely
amonge
the
euen
.2.4.4.135
.
among
the
odde
.3.5.7.190
.
In
all
maner
of
taking
this
is
to
be
noted
,
that
we
muste
not
place
the
man
which
taketh
in
place
of
him
that
is
taken
,
but
when
he
maye
be
taken
before
we
drawe
,
then
shall
we
remoue
our
man
into
his
place
.
¶
The
priuilege
of
the
king
.
THe
king
standeth
for
so
many
men
as
he
hath
steppes
,
that
is
the
euen
for
.6
.
the
odde
for
.5
.
if
anye
of
these
(
except
the
lowest
and
greatest
)
be
taken
the
king
may
redeme
hym
,
by
any
man
of
his
that
is
of
the
same
number
.
If
he
haue
none
of
the
same
number
,
he
maye
redeme
him
by
any
of
his
men
that
hys
aduersary
wyll
chuse
.
But
if
his
lowest
square
be
taken
,
no
ransom
will
delyuer
him
.
Also
if
the
whole
kyng
at
ons
that
is
the
whole
number
of
Pyramis
be
taken
,
he
can
not
be
redemed
.
¶
Of
the
triumphe
.
TO
take
awaye
the
tediousnes
of
long
play
from
them
that
be
yonge
beginners
,
wryters
of
this
game
haue
inuented
diuers
kyndes
of
shorte
victories
,
wherefore
they
deuide
victory
into
proper
and
common
.
Of
the
proper
victorie
need
nothing
here
be
spoken
,
for
all
things
thereto
belonging
are
sufficiently
set
forth
in
the
first
kynd
of
playing
.
¶
Of
the
common
victory
.
THe
common
victorie
(
they
say
)
is
after
fyue
maners
,
for
men
contende
either
for
bodies
,
goods
,
quarelles
,
honour
,
or
els
for
both
quarels
&
honor
.
¶
Uictory
of
bodies
,
VIctory
of
bodies
is
only
to
take
a
certain
number
of
men
,
as
if
the
gamesters
agree
,
that
he
which
first
taketh
4.02.5.02.6.02.10
.
men
&c
,
shall
wyn
the
game
.
¶
Uictorye
of
goods
.
VIctorie
of
goods
,
is
to
take
a
certain
number
without
respect
of
the
men
.
As
if
it
be
couenanted
,
that
he
which
first
taketh
men
amounting
to
the
number
of
.100
.
or
.200
.
shall
haue
the
victorie
.
¶
Uictory
of
quarell
.
VIctorie
of
quarell
is
when
neither
the
men
,
nor
the
number
,
but
the
characters
of
the
number
be
considered
.
As
if
it
be
determined
that
he
which
first
taketh
.100
.
in
.8
.
characters
not
regarding
in
how
many
men
they
stande
,
shall
winne
.
As
.2.4.6.8.24.64
.
so
you
haue
.100
.
in
.8
.
characters
it
skilleth
not
,
although
there
be
more
then
.100
.
as
in
this
example
there
is
more
then
.100
.
by
.4
.
¶
Uictorie
of
honour
.
VIctorie
of
honour
,
is
when
a
determined
number
is
made
in
a
determined
number
of
men
,
as
if
it
be
determined
that
he
whiche
first
commeth
to
.100
.
in
.8
.
men
,
shall
winne
the
game
.
As
in
these
.2.4.6.8.4.16.45.15
.
And
though
there
were
somwhat
more
then
100.
so
it
be
in
.8
.
men
,
it
skilleth
not
.
¶
Of
victorie
of
honour
and
quarell
.
THe
victorie
of
honour
and
quarell
,
is
when
one
obteyneth
the
decreed
number
,
in
the
decreed
number
of
men
and
the
decreed
number
of
characters
:
as
let
.100
.
be
the
decreed
number
8.
the
determined
number
of
men
,
and
9.
the
determined
number
of
characters
,
He
that
obteyneth
.2.4.6.8.4.6.9.64
.
obteineth
the
victorie
of
honour
and
quarell
.
It
shal
be
no
hinderance
though
.8
.
men
and
.9
.
caracters
conteyne
somwhat
more
then
.100
.
so
that
there
be
not
.100
.
vpon
one
man
,
as
in
the
victorie
before
.
¶
Uictorie
of
standers
.
THey
haue
inuented
another
victorie
,
that
is
of
standerdes
,
by
counterfeyting
two
armies
,
one
of
the
Christians
,
another
of
the
Turkes
.
The
whyte
men
,
that
is
the
euen
hoste
,
conteyneth
.1312
.
footemen
(
not
compting
the
rootes
of
squares
expressed
in
the
kynges
)
let
the
first
and
last
be
captaines
and
let
them
deuide
the
whole
armye
into
.10
.
standerds
so
euery
standerd
shall
haue
.130
.
men
,
besyde
the
two
captaines
and
the
ten
standerd
bearers
.
The
black
men
,
that
is
the
odde
armie
(
except
the
kings
rootes
)
be
.1752
.
The
two
captaynes
and
ten
standerd
bearers
taken
out
,
there
remayneth
.1740
.
souldyers
,
to
euery
standerd
.174
.
He
that
wynneth
more
standers
hath
the
victorye
.
If
the
euen
hoste
wyne
.348
.
men
he
hath
obtayned
two
standerds
if
he
wynne
.522
he
hath
gotten
thre
standerds
and
forth
of
the
rest
.
Yf
the
odde
armye
wynne
.260
.
they
wyn
two
standerds
.390
.
three
standerds
and
so
of
the
rest
.
¶
A
Table
of
the
victorye
of
standerds
.
One
standerd
of
the
euen
,
conteyneth
.
|
130.
|
Two
standerds
.
|
260.
|
Three
standerds
.
|
390.
|
Foure
standerds
.
|
520.
|
Fyue
standerds
.
|
650.
|
Sixe
standerds
.
|
780.
|
Seuen
standerds
.
|
910.
|
Eyght
standerds
.
|
1040.
|
Nyne
standerds
.
|
1170.
|
Tenne
standerds
.
|
1300.
|
One
standerd
of
the
odde
,
conteyneth
.
|
174.
|
Two
standers
.
|
348.
|
Thrée
standerds
.
|
522.
|
Foure
standerds
.
|
696.
|
Fyue
standerds
.
|
870.
|
Sixe
standerds
.
|
1044.
|
Seuen
standerds
.
|
1218.
|
Eyght
standerds
.
|
1392.
|
Nyne
standerds
.
|
1566.
|
Tenne
standerds
.
|
1740.
|
YOu
maye
vse
anye
of
these
syxe
kyndes
of
common
victorie
,
in
euery
one
of
the
three
kynds
of
playing
.
FINIS
.