¶
The
fyrste
parte
is
of
Numeration
.
NVmeration
is
a
maner
of
expressynge
of
numbers
by
certayn
figures
which
are
called
figurs
of
Algorisme
,
the
which
be
tenne
,
as
in
this
example
.
i.
|
ii
.
|
iii.
|
iiii
.
|
v.
|
vi
.
|
vii
.
|
viii
.
|
ix
.
|
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
0
|
Of
the
which
nyne
be
significatyue
,
the
tenth
called
a
siphre
,
sygnifieng
nothyng
of
it selfe
,
but
only
set
before
the
other
significatyue
fygurs
augmentyth
theyr
signification
.
In
numeration
by
this
crafte
ye
must
euermore
begynne
at
the
ryghte
syde
of
the
boke
,
and
so
towardes
the
lyft
syde
,
as
in
this
example
.
k
|
i
|
h
|
g
|
f
|
e
|
d
|
c
|
b
|
a
|
3
|
2
|
0
|
4
|
6
|
7
|
5
|
2
|
8
|
9
|
This
fygure
9
vnder
a
,
standeth
in
the
fyrste
place
.
8
vnder
b
,
standeth
in
the
seconde
place
,
and
so
forthe
to
the
ende
,
so
that
3
vnder
k
,
standeth
in
the
laste
place
.
By
these
tenne
figures
all
maner
of
number
possyble
to
be
excogitate
,
may
clearely
and
playnely
be
expressed
:
whiche
all
be
it
,
that
of
them selfe
they
sygnifie
but
symple
and
lyttell
number
,
as
ye
se
afore
,
yet
according
to
the
diuersite
of
the
place
they
stande
in
,
dyuersly
doth
theyr
signification
amount
.
Wherfore
in
numeration
ye
muste
note
twoo
thynges
,
the
fygure
significatiue
,
and
the
place
it
standeth
in
for
the
signification
of
the
fygure
dependeth
vpon
the
number
of
the
place
it
standeth
in
:
For
example
,
this
fygure
8
standynge
alone
,
or
in
the
fyrste
place
signifyeth
but
.viii.
but
yf
he
stande
in
the
seconde
place
,
as
here
80
he
signifyeth
.viii.
tymes
tenne
,
whiche
is
called
.iiii.
score
.
yf
he
stand
in
the
thyrde
place
,
as
here
800
he
signifyeth
.viii.
hundreth
.
&c.
Therfore
ye
must
know
perfectely
the
signifycation
of
euery
place
,
before
ye
can
perfectely
number
.
wherfore
vnderstand
ye
,
that
the
first
place
is
a
place
of
vnitees
,
so
that
a
figur
standynge
in
it
,
signifieth
no
more
then
though
he
stand
alone
.
The
second
place
is
a
place
of
tennes
.
The
thyrd
is
a
place
of
hūdrydes
.
The
fourth
place
is
a
place
of
thousandes
.
The
fifth
place
,
a
place
of
tenne
thousandes
.
The
.vi.
place
a
place
of
hundreth
thousandes
.
The
.vii.
place
,
is
of
thousand
thousandes
,
which
is
called
a
myllion
.
The
.viii.
place
,
is
of
tenne
myllions
.
The
.ix.
is
a
place
of
hundreth
myllyons
.
The
.x.
of
thousand
myllions
.
The
.xi.
of
ten
thousande
myllions
.
The
xii
.
of
a
hundreth
thousande
myllions
.
The
.xiii.
of
a
thousand
thousand
myllion
,
whiche
is
called
myllyon
vpon
myllyon
.
And
so
forth
infinitely
,
euery
place
ensuenge
,
signifieth
.x.
tymes
as
moch
as
the
place
goynge
before
.
This
muste
thou
knowe
perfitely
what
euery
place
gyueth
and
signifyeth
:
for
the
place
gyueth
denomination
,
and
the
figure
standyng
in
the
same
place
expresseth
how
many
of
the
same
denomination
is
to
be
vnderstand
:
as
in
example
ye
shal
more
playnly
perceyue
In
this
summe
3400872619
this
figure
●
standyth
in
the
.iiii.
place
,
nowe
by
your
rule
afore
,
the
.iiii.
place
is
a
place
of
thousandes
,
then
this
fygure
2
standynge
in
the
same
place
gyueth
vs
to
wytte
,
that
it
is
two
thousande
,
Lykewyse
this
fygure
8
standeth
in
the
.vi.
place
,
nowe
by
your
rule
afore
spoken
of
,
the
.vi.
place
is
of
hunderth
thousandes
:
then
this
figure
8
situat
in
the
same
place
receyueth
denomination
of
the
place
,
and
representeth
to
vs
viii
.
hundreth
thousandes
.
Lyke
wyse
this
fygure
1
standeth
in
the
second
place
and
for
bycause
the
second
place
is
a
place
of
tennes
,
therfore
this
figure
1
standyng
there
is
bounde
to
the
signification
of
the
place
,
and
so
signifyeth
one
tenne
:
yf
a
fygure
of
4
stode
there
,
it
shuld
signify
.iiii.
tennes
,
that
is
fourty
,
and
so
forth
.
Then
for
a
farther
declaracyon
of
the
foresaide
summe
,
and
all
other
lyke
summes
.
This
figure
9
standynge
in
the
fyrste
place
,
signifyeth
but
hym selfe
,
that
is
.ix.
This
fygure
1
standynge
in
the
seconde
place
,
by
cause
the
seconde
place
is
euer
a
place
of
tennes
,
sygnyfyeth
one
tenne
.
The
fygure
6
,
standynge
in
the
thyrde
place
,
bycause
the
thyrde
place
is
a
place
of
hundrethe
,
doth
signifye
.vi.
hundreth
:
the
fygure
2
in
the
fourth
place
,
signifieth
.ii.
thousande
:
the
figure
7
for
bycause
it
standethe
in
the
fyfte
place
,
and
that
place
is
a
place
of
tenne
thousandes
,
it
signifieth
vii
.
tymes
tenne
thousande
,
the
which
is
iii.
score
thousande
and
ten
:
the
figure
8
in
the
.vi.
place
fignifieth
.viii.
hundreth
thousand
:
the
sypher
0
that
standeth
in
the
vii
.
place
signifieth
nothynge
,
but
onely
maketh
vp
a
place
that
the
fygures
significatyue
folowynge
maye
encrease
there
signification
.
Lyke
iudgement
is
of
the
sypher
standynge
in
the
.viii.
place
:
in
the
ix
.
place
standeth
the
figure
of
4
,
and
this
place
is
a
place
of
hundreth
myllyons
:
therfore
this
fygure
4
there
signifie
the
iiii
,
C.
myllions
.
In
the
.x.
place
standeth
the
fygure
3
,
and
this
place
is
a
place
of
thousande
myllions
:
therfore
it
signifyeth
.iii.
thousande
myllions
.
So
the
hole
summe
is
,
thre
thousand
mylions
.iiii.
C.
millions
.viii.
C.
thousand
,
thre
score
thousandes
.xii.
thousand
.vi.
hundreth
,
and
.xix.
Now
to
exercyse
your selfe
in
numeration
numbre
with
your selfe
these
summes
folowynge
,
&
you
shal
be
perfecte
ynough
.
Million
.
Mil.
Mil.
Mil.
x.
|
M.
|
C.
|
x.
|
M.
|
C.
|
x.
|
one
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
2
|
2
|
2
|
2
|
2
|
2
|
2
|
2
|
3
|
3
|
3
|
3
|
3
|
3
|
3
|
3
|
4
|
4
|
4
|
4
|
4
|
4
|
4
|
4
|
5
|
5
|
5
|
5
|
5
|
5
|
5
|
5
|
6
|
6
|
6
|
6
|
6
|
6
|
6
|
6
|
7
|
7
|
7
|
7
|
7
|
7
|
7
|
7
|
8
|
8
|
8
|
8
|
8
|
8
|
8
|
8
|
9
|
9
|
9
|
9
|
9
|
9
|
9
|
9
|
Furthermore
thou
muste
note
that
there
be
in
algorisme
thre
maner
of
numbres
,
Diget
number
,
Article
,
and
Composte
.
The
digette
number
,
is
all
maner
of
numbres
,
whiche
are
vnder
.x.
as
these
.
The
article
number
is
,
all
numbers
whiche
are
of
.x.
as
these
.
10
|
20
|
30
|
40
|
50
|
60
|
70
|
80
|
90
|
The
compounde
number
is
all
maner
of
numbres
which
are
compound
or
made
of
the
dyget
&
article
togyther
,
as
folow
.
11
|
12
|
13
|
14
|
15
|
16
|
17
|
18
|
19
|
21
|
22
|
23
|
24
|
25
|
26
|
27
|
28
|
29
|
31
|
32
|
33
|
34
|
35
|
36
|
37
|
38
|
39
|
And
so
forth
of
al
other
.
This
is
sufficient
for
the
knowledge
of
numbre
in
Algorisme
.
¶
The
seconde
parte
called
Addition
.
ADdition
is
a
collection
of
diuers
and
sundrye
summes
,
into
one
totall
summe
,
whiche
contaynyth
as
much
in
hym
as
al
the
other
summes
,
beynge
before
sundry
.
In
addition
are
two
nombres
to
be
consyderyd
,
the
one
is
,
the
numbres
which
muste
be
adioyned
togyther
:
the
other
is
the
numbers
which
redoundeth
of
theyr
addition
together
,
whiche
otherwyse
is
called
the
totall
summe
.
Then
when
ye
wyll
adde
many
summes
togyther
,
fyrst
write
them
fayre
the
one
dyrectely
vnder
the
other
,
so
that
the
fyrst
fygure
of
the
one
,
be
ryght
vnder
the
fyrst
of
the
other
,
&
the
seconde
vnder
the
seconde
:
euery
place
correspondent
vnder
other
:
that
done
drawe
a
lyne
vnder
al
these
seueral
summes
,
as
is
to
se
in
the
example
folowyng
.
And
whan
ye
wyll
adde
your
numbers
togyther
,
begyn
at
the
fyrst
places
of
your
summes
&
adde
all
the
figures
that
ye
se
in
the
fyrst
places
of
all
your
summes
togither
,
and
that
that
commeth
of
that
addition
,
se
whether
it
be
digette
number
,
article
,
or
compost
:
yf
it
be
but
dyget
,
set
the
dyget
benethe
the
lyne
,
directely
vnder
the
same
first
places
:
yf
it
be
article
put
a
cypher
beneth
the
lyne
,
right
vnder
the
same
fyrst
place
,
&
reserue
the
article
to
be
added
to
the
nexte
places
of
thy
summes
&
there
do
likewyse
,
yf
it
be
compost
,
set
the
dyget
vnder
the
lyne
right
vnder
the
same
place
,
&
reserue
the
article
in
your
mynde
lykewise
to
be
added
to
the
nexte
places
of
thy
summes
:
when
the
figures
standyng
in
the
last
places
of
your
summes
be
adioyned
togyther
,
yf
any
article
or
articles
remayne
,
set
them
downe
nexte
to
the
fygure
ye
sette
last
before
vnder
the
same
lyne
:
as
by
examples
shall
appere
.
The
fyrste
summe
.
|
|
6
|
7
|
8
|
9
|
4
|
The
seconde
summe
|
|
3
|
4
|
5
|
6
|
7
|
The
thyrde
summe
|
|
2
|
3
|
4
|
5
|
6
|
The
fourth
summe
|
|
7
|
8
|
9
|
3
|
4
|
The
fyfth
summe
|
|
6
|
7
|
4
|
2
|
5
|
The
syxthe
summe
|
|
3
|
4
|
3
|
2
|
2
|
Summa
totalis
|
3
|
0
|
6
|
5
|
9
|
8
|
Your
fygures
set
after
this
sorte
,
adde
all
the
fygures
that
ye
fynde
in
the
fyrste
places
of
all
the
summes
together
,
begynnynge
at
the
nethermost
sayenge
,
2
&
5
is
7
,
and
4
that
is
11
,
and
6
that
is
17
,
and
7
that
is
24
,
ans
4
that
t
s
28.
This
is
the
hole
summe
of
the
figurs
added
togyther
founde
in
the
fyrst
places
,
the
whiche
number
is
composte
:
wherfore
,
as
is
in
your
rule
,
ye
must
set
the
dyget
ryghte
vnder
the
same
place
,
benethe
the
lyne
,
the
whiche
is
8
,
&
kepe
the
articles
in
your
mynd
,
which
is
2.
Now
to
the
seconde
place
,
towarde
the
lyfte
hande
,
say
,
2
that
I
haue
in
mynde
and
2
is
4
,
and
2
maketh
6
,
and
3
is
9
,
and
5
is
14
,
and
6
is
20
,
and
9
is
29
,
now
set
the
9
vnder
2
,
&
kepe
2
in
mynde
,
and
adde
them
to
the
fyrste
fygure
of
the
thyrde
place
,
that
is
3.
Now
say
2
and
3
is
5
,
and
4
is
9
,
and
9
is
18
,
and
4
is
22
,
and
5
is
27
,
and
8
is
35.
Now
set
5
vnder
3
,
and
kepe
●
in
mynde
.
Now
to
the
fourthe
place
,
towarde
the
lyfte
hande
where
4
standeth
,
now
3
that
ye
haue
in
mynde
and
4
is
7
,
&
7
is
14
,
and
8
is
22
,
and
3
is
25
,
&
4
is
29
,
and
7
is
36
,
set
6
vnder
4
,
and
kepe
3
,
and
adde
that
3
to
the
vndermost
fygure
of
the
syxt
summe
that
is
3
,
and
say
3
and
3
is
6
,
and
6
is
12
,
and
7
is
19
,
and
2
is
21
,
and
3
is
24
,
and
6
is
30.
All
the
figures
of
this
place
added
togyther
as
ye
se
,
maketh
article
number
wherfore
accordyng
to
your
rule
set
a
cyfer
0
vnder
that
place
benethe
the
lyne
,
&
the
artycle
whiche
is
3
nexte
to
the
same
cyfer
,
&
al
is
finished
.
And
all
these
summes
thus
collected
togither
maketh
306598.
An other
example
of
addition
.
|
1
|
0
|
0
|
6
|
6
|
7
|
8
|
4
|
5
|
|
6
|
0
|
0
|
0
|
3
|
1
|
9
|
5
|
0
|
|
5
|
0
|
0
|
5
|
4
|
5
|
1
|
6
|
1
|
|
|
|
|
8
|
0
|
1
|
2
|
0
|
2
|
|
|
|
|
|
|
6
|
4
|
2
|
1
|
2
|
2
|
0
|
2
|
0
|
5
|
2
|
5
|
7
|
9
|
Begyn
fyrst
as
ye
dyd
before
,
at
the
fyrst
places
,
addyng
them
all
togyther
,
begynnynge
at
the
nethermost
,
sayenge
,
1
and
2
is
3
,
and
1
is
4
,
and
5
is
9
,
this
is
the
hole
summe
of
the
fygures
standynge
in
the
fyrst
place
,
the
whiche
is
diget
number
,
and
therfore
accordynge
to
the
rule
,
set
it
ryghte
vnder
the
same
place
benethe
the
lyne
:
then
procede
to
the
second
place
and
begyn
at
the
nether
ende
sayenge
,
2
and
6
is
8
,
and
5
is
13
,
and
4
is
17
,
these
number
is
compost
number
,
therfore
set
the
diget
righte
vnder
that
place
beneth
the
lyne
,
which
is
7
,
reseruynge
the
artycle
in
your
mynde
,
and
so
to
the
thyrde
place
,
sayeng
,
1
that
I
haue
in
my
mynde
and
4
that
is
5
,
&
2
is
7
,
and
1
is
8
,
and
9
is
17
,
and
8
is
●5
,
this
number
also
is
compost
,
wherfo●e
set
the
diget
5
vnder
that
thyrde
place
,
&
reserue
the
article
2
in
mynde
to
the
nexte
place
,
then
to
the
next
places
sayeng
,
2
that
I
haue
in
mynde
&
6
is
8
,
&
1
is
9
,
&
5
is
14
,
&
1
is
15
,
&
7
is
22
,
this
is
also
compost
,
therfore
set
the
diget
2
vnder
that
.iiii.
place
,
&
reserue
the
article
2
to
the
next
places
,
then
to
the
fifth
place
sayeng
2
that
I
haue
in
mynde
&
4
is
6
,
&
3
is
9
,
&
●
is
15
,
this
is
also
compost
number
,
set
the
diget
5
vnder
the
fyfth
place
,
and
kepe
the
article
in
mynde
,
to
the
.vi.
place
sayeng
,
1
the
I
haue
in
mynde
&
8
is
9
&
5
is
14
,
and
6
is
20
this
is
article
number
,
therfore
accordynge
to
the
rule
set
a
syfer
vnder
the
place
benethe
the
lyne
,
&
kepe
the
article
in
mynde
and
cum
to
the
.vii.
place
,
in
the
whiche
places
for
bycause
thou
fyndest
nothyng
but
cyphers
to
the
whith
thou
myghtes
adioyne
thy
article
reserued
,
the
which
was
2
,
therfore
vnder
the
same
.vii.
place
set
that
same
reserued
2
,
and
then
com
to
the
.viii.
place
and
there
fyndest
thou
nothyng
but
cyphers
wherfore
vnder
the
same
place
set
beneth
the
lyne
a
cypher
,
accordyng
to
the
rule
,
then
c●●●e
to
the
.ix.
places
&
say
5
and
6
is
11
,
&
1
is
12
,
the
which
is
compost
number
,
therfore
set
the
diget
which
is
2
vnder
the
line
&
reserue
the
article
in
mynd
,
which
is
1
,
now
for
bycause
there
is
no
mo
places
whervnto
ye
myght
adde
this
reserued
article
therfore
according
to
your
rule
ye
shal
set
it
downe
next
vnto
the
figure
that
ye
dyd
set
vnder
the
lyne
laste
,
as
is
in
your
example
.
This
.ii.
examples
were
sufficient
ynough
to
the
redynesse
of
addition
,
how
be
it
yet
that
it
may
be
the
playner
I
wyl
subscribe
an
other
example
.
1
|
4
|
6
|
9
|
9
|
0
|
0
|
0
|
3
|
8
|
2
|
9
|
0
|
2
|
0
|
0
|
0
|
1
|
0
|
9
|
1
|
6
|
0
|
0
|
|
1
|
0
|
2
|
0
|
0
|
0
|
0
|
5
|
5
|
1
|
0
|
0
|
8
|
0
|
0
|
Adde
the
fyrst
place
togyther
.
Fyrste
there
thou
fyndest
nothyng
but
cyfers
,
wherfore
sette
a
cyfer
vnder
the
lyne
,
and
so
lykewyse
in
the
seconde
place
.
In
the
thyrde
place
thou
fyndest
6
and
●
whiche
maketh
8
,
the
whiche
for
bycause
it
is
dygette
number
,
set
it
vnder
that
place
benethe
the
lyne
.
In
the
.iiii.
place
is
1
and
9
which
maketh
10
,
and
for
bycause
that
this
is
article
number
,
set
a
cyfer
vnder
that
place
benethe
the
line
,
and
reserue
the
article
to
the
next
place
sayenge
,
1
that
I
haue
in
my
mynde
and
2
is
3
,
and
9
is
12
,
and
9
is
21
,
and
9
is
30
this
is
also
article
number
,
wherfore
set
a
cyfer
vnder
that
place
beneth
the
lyne
,
and
reserue
the
article
3
in
mynde
to
the
nexte
place
.
Then
come
to
the
.vi.
place
:
sayenge
,
3
that
I
haue
in
my
mynde
and
2
is
5
,
and
6
is
11
,
this
is
composte
numbre
therfore
I
set
the
dyget
which
is
1
ryght
vnder
that
place
beneathe
the
lyne
,
and
reserue
the
article
1
to
the
ne●te
place
,
sayenge
1
and
1
is
2
,
and
1
is
3
,
and
8
is
11
,
&
4
is
15
,
this
is
also
composte
,
therfore
sette
the
dyget
5
vnder
the
lyne
,
and
adde
the
article
reserued
to
the
figure
in
the
ne●te
place
sayenge
,
1
and
3
is
4
,
and
1
is
5
,
this
is
dyget
numbre
,
therfore
sette
it
vnder
the
lyne
,
and
all
is
done
.
¶
Certayne
examples
to
practyse
youre selfe
in
,
touchynge
the
exercyse
of
Addition
.
|
1
|
6
|
7
|
6
|
8
|
9
|
0
|
0
|
|
3
|
6
|
2
|
1
|
9
|
8
|
8
|
0
|
|
9
|
2
|
0
|
0
|
0
|
0
|
3
|
2
|
|
1
|
1
|
1
|
1
|
6
|
8
|
4
|
1
|
|
1
|
9
|
4
|
2
|
1
|
3
|
2
|
6
|
1
|
7
|
5
|
5
|
2
|
6
|
9
|
7
|
9
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
3
|
4
|
5
|
6
|
2
|
8
|
9
|
1
|
0
|
0
|
2
|
0
|
1
|
0
|
1
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
3
|
8
|
9
|
2
|
1
|
0
|
0
|
0
|
9
|
2
|
3
|
9
|
7
|
3
|
9
|
0
|
|
9
|
0
|
9
|
0
|
2
|
0
|
1
|
0
|
0
|
0
|
|
2
|
6
|
5
|
1
|
2
|
6
|
0
|
0
|
0
|
0
|
|
|
|
2
|
5
|
4
|
3
|
2
|
0
|
7
|
1
|
|
|
|
2
|
1
|
6
|
5
|
4
|
0
|
0
|
0
|
|
|
|
|
|
9
|
6
|
2
|
0
|
0
|
1
|
|
|
|
|
|
|
|
1
|
0
|
0
|
0
|
1
|
1
|
7
|
8
|
9
|
5
|
1
|
0
|
0
|
7
|
2
|
|
6
|
4
|
0
|
0
|
0
|
0
|
|
8
|
6
|
0
|
0
|
0
|
0
|
|
9
|
9
|
8
|
0
|
0
|
0
|
|
7
|
8
|
0
|
0
|
0
|
0
|
|
5
|
9
|
0
|
0
|
0
|
0
|
|
1
|
0
|
0
|
0
|
0
|
3
|
3
|
9
|
6
|
8
|
0
|
0
|
3
|
|
9
|
0
|
0
|
0
|
1
|
6
|
|
2
|
0
|
5
|
8
|
1
|
2
|
|
9
|
0
|
1
|
0
|
0
|
0
|
|
0
|
0
|
9
|
6
|
1
|
9
|
|
1
|
0
|
0
|
0
|
0
|
0
|
9
|
0
|
1
|
6
|
4
|
5
|
7
|
3
|
0
|
0
|
0
|
2
|
6
|
8
|
1
|
0
|
6
|
0
|
9
|
8
|
6
|
1
|
0
|
0
|
0
|
9
|
3
|
9
|
1
|
0
|
0
|
0
|
9
|
3
|
9
|
2
|
0
|
0
|
0
|
9
|
1
|
7
|
1
|
0
|
6
|
4
|
6
|
6
|
9
|
¶
Of
the
proue
of
Addition
.
¶
For
the
proue
of
addition
,
ye
shal
make
a
crosse
after
the
fashyon
that
foloweth
.
And
then
ye
shall
come
fyrst
to
the
addible
summes
,
and
plucke
out
all
the
9
that
ye
fynde
there
,
and
the
reste
what
so
euer
it
be
,
that
wyll
not
make
9
set
it
at
the
vpper
syde
of
the
crosse
.
Then
come
to
the
totall
summe
vnder
the
lyne
,
and
lykewyse
deduck
all
the
9
that
ye
can
fynde
there
and
that
that
remayneth
,
not
able
to
make
9
set
it
●t
the
vndermost
parte
of
the
crosse
,
and
yf
it
be
lyke
the
remenant
of
the
addyble
numbres
which
standith
in
the
vpper
parte
of
the
crosse
,
your
worke
was
good
,
yf
not
it
was
naught
,
as
by
example
ye
shall
the
better
perceyue
.
¶
An
example
of
the
proue
.
A
|
1
|
5
|
0
|
6
|
7
|
0
|
B
|
3
|
3
|
0
|
4
|
2
|
8
|
C
|
5
|
8
|
1
|
0
|
9
|
8
|
¶
Nowe
for
to
make
the
proue
of
this
numbres
,
ye
shall
begynne
at
the
fyrst
fygure
that
ye
haue
made
,
in
sayenge
,
8
and
0
is
8
and
2
is
10
,
take
awaye
9
then
there
resteth
1
,
than
1
and
7
is
8
,
and
4
is
12
,
take
awaye
9
reste
3
,
than
3
and
6
is
9
,
than
to
the
two
cyphers
of
nothynge
that
nothynge
do
sygnifie
,
thanne
3
and
5
is
8
and
3
is
11
,
take
awaye
9
reste
2
,
thanne
2
and
2
is
4
,
this
4
it
behoueth
you
to
put
at
the
nether
ende
of
the
crosse
,
than
come
to
the
place
of
C.
vnder
the
lyne
and
say
8
,
ye
shall
leue
the
9
and
the
cypher
0
that
is
nothynge
worthe
,
and
adioyne
1
therto
and
make
it
9
,
and
leue
that
,
thanne
8
and
5
is
13
,
take
awaye
9
reste
4
,
whiche
4
ye
shal
put
at
the
vpper
ende
of
the
crosse
and
than
is
your
proue
god
,
for
bothe
the
endes
be
lyke
as
ye
se
in
this
fygure
of
the
crosse
.
And
at
the
two
other
endes
ye
shall
put
two
0
0
in
certifyeng
that
of
them
cometh
nothynge
.
¶
An
other
example
.
A
|
|
7
|
8
|
9
|
1
|
5
|
4
|
3
|
2
|
6
|
li.
|
B
|
|
4
|
9
|
3
|
0
|
0
|
6
|
7
|
1
|
5
|
li.
|
C
|
|
2
|
0
|
9
|
9
|
3
|
4
|
7
|
8
|
4
|
li.
|
D
|
|
4
|
6
|
0
|
6
|
4
|
5
|
5
|
3
|
0
|
li.
|
E
|
|
9
|
3
|
6
|
4
|
5
|
8
|
7
|
7
|
8
|
li.
|
F
|
|
4
|
4
|
5
|
1
|
9
|
3
|
0
|
0
|
1
|
li.
|
G
|
3
|
3
|
3
|
4
|
3
|
9
|
3
|
1
|
3
|
4
|
li.
|
¶
We
shall
saye
semblably
1
and
8
is
9
and
alwayes
leue
them
,
than
0
that
dothe
nothynge
,
than
4
and
5
ben
9
,
than
6
,
then
we
shall
retourne
to
the
tenthes
,
and
shal
fynde
0
that
doth
nothynge
,
than
7
that
mak●th
9
teste
4
,
than
3
ben
7
,
than
8
ben
9
reste
6
,
then
1
is
7
,
then
2
is
9
,
then
0
that
is
nothynge
worth
,
then
7
and
5
is
9
rest
●
then
7
is
9
reste
1
,
then
7
is
8
and
3
is
9
reste
1.
Thanne
we
come
to
the
place
of
hundrethes
,
&
adioyne
the
2
to
the
3
that
is
5
than
8
is
9
rest
4
,
and
so
consequently
vnto
the
ende
.
And
yf
peraduenture
we
find
this
fygure
9
bycause
of
the
brefnes
,
we
shall
leue
it
.
And
shall
fynde
at
the
ende
9
,
therfore
we
shall
put
at
the
ende
of
the
crosse
0
in
sygnifienge
that
there
is
nothynge
aboue
9.
And
so
shall
we
do
in
the
number
of
G.
and
we
shall
fynde
lyke
9
for
the
whiche
semblably
we
shall
put
0.
And
so
is
the
addition
good
&
well
made
.
¶
The
proue
As
touchynge
of
addition
in
broken
numbers
,
ye
shall
fynde
that
vnder
the
tytle
of
Reduction
hereafter
.
¶
Of
Subtraction
,
the
thyrde
parte
.
SVbtraction
is
a
maner
of
debatyng
or
subducyng
a
lesse
summe
out
of
a
greater
:
or
lyke
of
lyke
shewyng
what
remayneth
.
In
subtraction
are
two
numbers
,
the
fyrste
is
the
number
abatyd
,
the
seconde
,
the
number
abatynge
.
¶
Then
when
ye
wyll
subtrahe
any
one
number
out
of
an
other
.
Fyrste
ye
shall
wryte
the
number
to
be
abatyd
,
and
vnder
it
directely
fygure
vnder
fygure
,
and
place
vnder
place
wrytte
the
abatoure
,
and
benethe
these
two
summes
drawe
a
lyne
,
then
begyn
your
subtraction
at
the
fyrste
places
,
and
subduce
the
figure
standynge
in
the
fyrst
place
of
the
abatour
of
the
fyrste
figure
standynge
in
the
fyrste
place
of
the
nomber
to
be
abatyd
:
and
the
rest
that
remayneth
after
the
abatement
set
it
ryght
vnder
the
same
place
benethe
the
lyne
:
and
so
do
lyke
wyse
in
the
second
the
thyrd
,
and
al
other
places
.
And
when
ye
haue
all
done
,
the
number
that
shal
remayne
vnder
the
lyne
,
shall
be
that
,
that
remayneth
after
the
subduction
of
the
abatour
of
the
number
abatyd
.
As
for
example
.
Lent
|
8
|
3
|
4
|
5
|
6
|
Payd
|
4
|
1
|
1
|
3
|
1
|
Reste
|
4
|
2
|
3
|
2
|
5
|
¶
I
lent
a
man
83456
li.
of
the
which
he
hath
payed
me
41111
li.
agayne
now
I
desyre
to
know
how
mych
remayneth
.
Then
accordynge
to
the
rule
,
fyrste
I
sette
the
lente
money
,
and
ryght
vnder
that
I
sette
the
repayed
monie
,
figure
vnder
figure
,
and
place
vnder
place
:
as
ye
se
by
the
example
Vnder
bothe
these
sommes
I
must
draw
a
lyne
.
Begyn
to
subtrahe
the
vnder
sum
out
of
the
vpper
,
sayenge
,
1
out
of
6
remayneth
5
,
this
5
that
remayneth
accordynge
to
the
rule
set
vnder
the
same
place
beneth
the
lyne
:
then
to
the
seconde
place
plucke
3
out
of
5
remayneth
2
,
set
that
vnder
the
lyne
:
then
to
the
thyrd
place
,
plucke
1
out
of
4
remayneth
3
,
set
that
vnder
the
lyne
:
then
to
the
fourth
place
,
take
1
oute
of
3
remayneth
2
,
sette
it
vnder
the
lyne
:
then
in
the
fyfth
place
,
take
4
out
of
8
remayneth
4
,
set
that
also
vnder
the
lyne
,
&
so
thou
hast
fynyshed
:
Then
thou
shalte
vnderstande
that
it
which
is
vnder
the
lyne
is
the
remanent
of
the
monie
not
yet
paid
.
¶
An
other
example
.
8
|
7
|
6
|
6
|
0
|
li.
|
6
|
7
|
5
|
6
|
0
|
li.
|
2
|
0
|
1
|
0
|
0
|
li.
|
¶
Begyn
at
the
fyrst
place
sayenge
,
0
out
of
0
remayneth
nothynge
,
sette
the
fygure
of
nothyng
vnder
the
lyne
:
then
to
the
seconde
place
6
out
of
6
,
remayneth
nothynge
,
sette
the
cypher
vnder
the
lyne
:
then
to
the
thyrde
place
5
out
of
6
remayneth
1
,
set
1
vnder
the
lyne
:
then
to
the
.iiii.
place
7
out
of
7
remayneth
nothynge
,
set
the
figure
of
nothynge
vnder
the
lyne
,
then
to
the
fyfte
place
,
take
6
out
of
8
remayneth
2
,
set
that
vnder
the
lyne
,
and
thus
thou
hast
done
.
Then
20100
remayneth
yet
to
be
payde
.
¶
Nowe
thou
shalt
noote
,
that
somtyme
it
chaunseth
that
the
fygure
standyng
benethe
is
greatter
thenne
the
fygure
standynge
aboue
hym
in
the
sum
from
whom
subduction
is
made
.
In
this
case
thou
shalte
in
thy
mynde
put
tenne
,
to
the
fygure
in
the
vpper
summe
,
and
then
subtrahe
the
nether
fygure
oute
of
the
same
,
sette
the
remanaunt
vnder
the
lyne
,
and
for
the
same
tenne
the
which
thou
dyddest
put
to
the
vpper
fygure
to
make
hym
gretter
,
thou
shalte
adde
one
to
the
nexte
fygure
standynge
in
the
nether
summe
,
and
then
subtrahe
that
lykewyse
out
of
the
figure
aboue
hym
,
yf
the
fygure
aboue
be
bygger
then
the
figure
benethe
with
his
addition
other
elles
equall
,
and
that
remayneth
set
it
vnder
the
lyne
,
as
ye
dyd
in
the
other
example
.
If
the
figure
aboue
be
lesse
then
the
figure
benethe
,
then
do
to
hym
as
ye
dydde
to
the
other
before
:
that
is
to
saye
adde
tenne
to
hym
:
and
so
forth
in
all
other
places
.
Where
the
nether
fygure
of
the
abatour
is
greatter
thenne
the
vpper
figure
frome
whence
it
sholde
be
abatyd
:
as
by
this
example
ye
shall
more
clerely
perceyue
.
¶
An
example
.
5
|
7
|
2
|
9
|
5
|
4
|
9
|
0
|
4
|
8
|
7
|
6
|
5
|
2
|
9
|
7
|
|
8
|
5
|
3
|
0
|
1
|
9
|
3
|
¶
Begyn
your
subtraction
sayenge
,
7
out
of
0
that
can
not
be
,
therfore
for
bycause
that
7
standynge
in
the
nether
summe
is
more
than
the
fygure
standynge
in
the
fyrst
place
of
the
vpper
summe
,
ye
muste
adde
a
tenne
,
then
deduc
your
7
out
of
10
and
there
remayneth
3
,
then
come
to
the
seconde
place
and
for
the
ten
that
ye
borowed
in
youre
mynde
and
a●●yd
it
the
fygure
in
the
fyrst
place
to
make
〈◊〉
●ygge
ynough
for
the
figure
vnder
it
to
〈◊〉
subduced
oute
of
it
,
for
the
same
tenne
I
say
ye
shall
put
to
the
next
fygure
in
the
nether
place
of
the
nether
summe
1
,
then
saye
9
and
1
is
10
,
then
subduce
this
10
out
of
the
figure
of
9
standynge
aboue
it
in
the
vppar
summe
and
that
ye
can
not
,
therfore
do
as
ye
dyd
before
in
the
fyrste
place
,
putte
10
to
the
9
in
your
mynde
sayenge
10
and
9
is
19
,
thenne
deduck
the
10
beneth
out
of
the
19
aboue
,
&
there
remayneth
9
to
be
set
vnder
the
lyne
:
then
to
the
fygure
standynge
in
the
thyrde
place
in
the
nether
summe
,
putte
1
for
the
ten
that
ye
borowed
in
your
mynd
the
whiche
ye
addyd
to
9
in
the
seconde
place
of
the
vpper
summe
to
make
it
greater
:
sayenge
1
and
2
is
3
,
subtrahe
that
3
out
of
4
aboue
it
,
remayneth
1
to
be
sette
vnder
the
lyne
.
Then
to
the
fourth
place
,
take
5
out
of
5
,
remayneth
nothynge
,
set
a
fygure
of
nothynge
vnder
the
lyne
,
and
come
to
the
.v.
place
,
take
6
out
of
9
remayneth
;
to
be
set
vnder
the
line
,
so
to
the
.vi.
place
,
take
7
out
of
2
that
can
not
be
,
therfore
put
to
the
same
2
accordynge
to
thy
rule
10
,
and
then
it
is
12
,
then
subduce
7
out
of
12
remayneth
5
to
be
set
vnder
the
lyne
:
and
for
the
same
10
that
thou
borowdeste
in
thy
mynde
to
put
to
the
fygure
of
2
in
the
vpper
summe
,
thou
shalt
adde
1
to
the
fygure
standyng
in
the
nexte
farther
place
in
the
nether
sum
,
comming
to
the
same
place
whiche
is
the
seuenth
place
sayeng
8
and
1
the
whiche
I
haue
too
sette
to
hym
is
9
,
then
9
out
of
7
that
I
can
not
,
wherfore
lykewyse
agayne
I
must
helpe
the
same
7
with
a
tenne
and
then
it
is
17
out
of
that
now
subtrake
your
9
and
remayneth
8
to
be
sette
vnder
the
lyne
:
nowe
as
ye
haue
done
before
in
all
other
places
for
the
10
here
borowed
&
adioyned
,
then
adde
1
to
the
nexte
figure
standynge
in
the
seuenth
place
of
the
nether
number
sayeng
,
4
&
1
is
5
,
then
subduce
this
5
out
of
the
5
aboue
and
remayneth
nothynge
,
wherfore
sette
a
fygure
of
nothynge
benethe
the
lyne
,
&
so
ye
haue
done
.
¶
How
be
it
ye
shall
noote
that
when
ye
haue
a
cypher
to
be
wrytten
in
the
laste
place
of
any
summe
,
ye
shall
not
wryte
it
,
for
in
the
last
place
it
signifyeth
nothyng
of
it selfe
,
neyther
dothe
it
augmente
the
signification
of
any
of
the
other
.
¶
Yet
one
other
example
wyl
we
set
and
then
make
an
ende
of
Subtraction
.
5
|
0
|
0
|
0
|
0
|
8
|
1
|
0
|
0
|
7
|
1
|
0
|
0
|
|
4
|
8
|
4
|
0
|
5
|
7
|
4
|
8
|
0
|
0
|
8
|
7
|
|
5
|
1
|
6
|
0
|
2
|
3
|
5
|
2
|
7
|
0
|
1
|
3
|
¶
Ye
shal
begynne
sayenge
,
7
out
of
0
that
can
not
be
,
for
ye
canne
not
take
7
out
of
nothing
,
wherfore
as
ye
haue
done
alwayes
in
the
example
afore
,
put
tenne
to
that
cypher
,
and
that
maketh
10
,
then
deduc
your
7
out
of
it
nowe
,
and
remayneth
3
to
be
sette
vnder
the
lyne
:
then
for
this
ten
that
ye
adde
to
the
fygure
in
the
fyrst
place
of
your
vpper
number
,
set
1
to
the
fygure
standynge
in
the
seconde
and
nexte
place
of
the
nether
number
sayenge
●
and
1
is
9
,
then
9
out
of
the
cypher
aboue
that
can
not
be
,
therfore
as
ye
dyd
before
make
that
0
,
10
,
and
then
subduce
youre
9
out
of
this
added
10
remayneth
1
to
be
set
benethe
the
lyne
:
then
for
this
10
lykewyse
,
that
ye
borowed
in
the
second
place
of
your
vpper
number
,
ye
shall
sette
one
to
the
next
fygure
standynge
in
the
thyrd
and
nexte
place
of
the
nether
summe
,
sayenge
,
1
and
the
0
is
one
,
thenne
take
that
1
out
of
1
aboue
hym
,
remayneth
nothyng
,
set
a
fygure
of
nothynge
benethe
the
lyne
then
to
the
.iiii.
place
take
the
syfer
0
out
of
7
aboue
remayneth
7
stylle
:
to
be
sette
vnder
the
lyne
.
So
to
the
.v.
place
take
8
out
of
0
that
ye
can
not
,
therfore
put
10
to
the
sypher
and
then
subduce
it
,
and
remayneth
2
sette
that
vnder
the
lyne
:
for
this
tenne
adde
1
to
the
nexte
figure
in
the
.vi.
place
,
whiche
is
4
,
then
4
and
1
is
5
,
and
5
out
of
0
that
ye
can
not
,
then
make
0
10
&
take
the
5
out
of
it
remayneth
5
to
be
sette
vnder
:
then
for
the
borowed
ten
,
lykewise
sette
to
the
next
fygure
in
the
.vii.
place
of
the
nether
nomber
1
,
sayeng
,
1
and
7
make
8
,
&
8
out
of
1
that
can
not
be
,
therfore
put
ten
to
that
1
and
then
10
and
1
is
11
out
of
this
11
deduce
your
8
remayneth
3
to
be
set
vnder
the
lyne
:
then
for
this
10
to
the
nexte
fygure
in
the
.viii.
of
the
nether
som
set
1
sayeng
,
5
and
1
is
6
then
6
out
of
8
remaineth
2
,
then
to
the
.ix.
place
,
take
0
out
of
0
remayneth
also
0
,
set
that
vnder
the
lyne
,
in
the
.x.
place
take
4
out
of
0
that
can
not
be
therfore
put
10
to
that
0
and
subduce
your
4
,
remayneth
6
,
then
to
the
fygure
in
the
nexte
place
whiche
is
the
.xi.
put
1
saieng
8
and
1
is
9
,
then
9
out
of
0
,
that
can
not
be
,
therfore
put
ten
to
it
,
and
then
subtrahe
your
9
out
of
10
remayneth
1
,
sette
it
vnder
the
lyne
:
for
this
borowed
tenne
put
one
agayne
to
the
nexte
fygure
which
is
4
,
sayenge
4
and
1
is
5
,
5
out
of
0
that
can
not
be
,
therfore
lykewyse
agayne
make
it
10
,
and
then
take
5
out
of
it
,
remayneth
5
,
then
agayne
for
your
borowed
10
put
1
to
the
nexte
place
:
but
for
bycause
there
be
no
mo
places
and
therfore
subtrahe
it
alone
out
of
the
figure
aboue
,
sayenge
,
1
out
of
1
remayneth
nothynge
,
therfore
nothynge
is
to
be
set
vnder
the
lyne
,
not
so
moche
as
a
0
,
for
bycause
it
is
in
the
laste
place
.
So
then
the
summe
vnder
the
lyne
is
the
remayne
that
remayneth
after
the
subtraction
of
the
lower
summe
out
of
the
vpper
summe
.
¶
Here
after
foloweth
the
proue
of
Subtraction
.
¶
The
proue
of
Subtraction
.
The
proue
whether
ye
haue
subtrahed
well
or
no
,
ye
muste
adde
the
remayne
to
the
nombre
payde
,
and
yf
they
twayne
added
together
do
make
the
fyrste
summe
lent
completelye
then
is
it
well
subtrahyd
,
yf
not
,
it
is
not
wel
subtrahyd
,
as
by
the
laste
example
ye
maye
well
perceyue
:
for
by
the
rule
of
addition
,
adde
3
to
7
therof
commeth
10
,
set
the
sypher
vnder
the
lyne
and
reserue
the
article
to
the
next
place
forth
accordynge
to
the
rule
of
Addition
,
and
thou
shalte
se
this
twoo
summes
added
togyther
to
come
to
the
fyrste
lente
summe
:
and
this
of
Subtraction
shal
be
sufficient
.
¶
Of
Multiplication
.
MVltiplication
is
a
maner
of
encreasing
or
augmenting
one
sum
by
another
In
this
feat
of
multiplication
are
.iii.
numbres
to
be
noted
the
multiplyed
nomber
,
the
multiplyer
,
&
the
numbre
that
redowndeth
of
the
multiplication
of
the
multyplyed
nomber
by
the
multypliar
,
as
in
example
.
Multiply
this
number
4
by
3
&
therof
come
12
,
4
is
the
number
multiplied
:
3
is
the
number
multiplyer
,
12
the
thyrde
number
that
redounded
of
the
multiplycation
of
one
of
these
number
by
the
other
then
for
more
experience
and
redy
workynge
in
this
kynde
of
operation
ye
shal
perfectely
knowe
by
memory
the
multiplication
of
one
dyget
by
an
other
,
the
whiche
ye
shall
haue
here
in
this
table
folowynge
,
of
the
whiche
one
dygette
ye
shall
loke
for
in
the
hed
of
the
table
,
and
the
other
in
the
lefte
syde
of
the
table
.
¶
Here
after
foloweth
the
Table
.
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
2
|
4
|
6
|
8
|
10
|
12
|
14
|
16
|
18
|
20
|
3
|
6
|
9
|
12
|
15
|
18
|
21
|
24
|
27
|
30
|
4
|
8
|
12
|
16
|
20
|
24
|
28
|
32
|
36
|
40
|
5
|
10
|
15
|
20
|
25
|
30
|
35
|
40
|
45
|
50
|
6
|
12
|
18
|
24
|
30
|
36
|
42
|
48
|
54
|
60
|
7
|
14
|
21
|
28
|
35
|
42
|
49
|
56
|
63
|
70
|
8
|
16
|
24
|
32
|
40
|
48
|
56
|
64
|
72
|
80
|
9
|
18
|
27
|
36
|
45
|
54
|
63
|
72
|
81
|
90
|
10
|
20
|
30
|
40
|
50
|
60
|
70
|
80
|
90
|
100
|
¶
By
this
table
ye
shal
sufficiently
lerne
to
multyply
one
dygette
by
an
other
.
As
for
example
,
yf
ye
wyll
multiplye
9
by
5
,
loke
for
the
9
,
at
the
heed
of
the
table
,
and
for
5
the
multiplyer
at
the
lefte
syde
of
the
table
,
then
with
thy
fynger
descende
downe
from
the
place
where
9
standeth
tyll
thou
come
before
the
place
where
the
5
standeth
and
there
in
the
same
angle
,
thou
shalte
fynge
45
,
and
that
cometh
of
5
tymes
9
,
&
so
do
lyke
wyse
of
other
.
¶
There
is
also
a
proper
rule
for
the
multiplication
of
one
dygette
by
an
other
,
&
it
is
this
,
when
thou
wylte
multyply
one
dyget
by
an
other
noote
the
distaunce
of
the
greater
diget
from
10
and
by
the
same
distaunce
multiplye
the
lesse
dyget
or
equall
,
&
that
that
procedeth
of
it
deducte
out
of
that
article
whom
the
lesse
number
doth
denominate
,
and
the
reste
is
it
that
ye
seake
for
,
as
for
example
:
yf
ye
wyll
multyply
7
by
5
,
fyrste
se
the
distaunce
betwene
7
which
is
the
greater
number
and
5
,
and
that
is
3
,
by
this
3
multiply
5
,
and
that
is
15
then
subduce
this
15
out
of
the
artycle
that
5
the
lesse
number
doth
denominate
,
whiche
is
50
,
then
remayneth
35
,
that
is
5
tymes
7
:
so
lykewyse
shall
ye
doo
yf
the
multyplyar
and
the
multiplyed
be
lyke
.
How
be
it
moost
ready
it
is
to
know
without
boke
very
perfytely
the
multyplication
of
euery
dyget
one
in
an
other
.
Now
when
ye
wyll
multyply
any
one
number
the
one
by
the
other
.
Fyrst
wryte
fayre
your
number
to
be
multyplyed
,
and
vnder
it
the
multiplicatour
,
beneth
both
these
summes
,
ye
shal
drawe
a
lyne
.
Then
shal
ye
consyder
whether
your
multiplier
be
a
dyget
or
articly
,
other
elles
compost
number
.
I●
it
be
dyget
number
ye
shall
begynne
to
multyply
by
the
dyget
the
fygure
or
dygette
standynge
in
the
fyrs●e
place
of
the
number
to
be
multiplied
,
and
that
that
commeth
of
it
,
if
it
be
but
a
dygette
set
it
vnde
the
lyne
ryght
vnder
the
same
place
and
then
procede
forther
to
the
nexte
place
and
multiply
the
figure
standynge
in
that
place
by
the
same
multiplier
,
and
that
that
redowndeth
of
it
,
yf
it
be
a
dygette
●ette
it
lykewyse
vnder
the
lyne
r●ghte
vnder
the
same
place
,
and
soo
do
lykewyse
in
euery
place
folowynge
,
vnto
suche
tyme
as
all
the
figures
standynge
in
euery
place
,
be
multyplyed
:
then
that
the
whiche
shal
be
founde
vnder
the
lyne
is
the
summe
commyng
of
the
multiplication
of
this
two
numbers
,
the
one
by
the
other
:
as
by
example
ye
shall
the
better
perceyue
.
¶
If
ye
wyll
multyplye
this
sum
2314
by
this
2●
ye
shall
sette
your
figurs
after
this
sorte
,
as
ye
se
them
.
Begyn
youre
multiplication
sayenge
2
tymes
4
is
8
,
sette
that
8
vnder
the
lyne
,
then
come
to
the
next
place
and
say
,
2
tymes
1
is
2
,
sette
it
vnder
the
lyne
,
then
to
the
thyrde
place
,
2
tymes
3
is
6
,
set
that
vnder
the
lyne
,
so
to
the
fourth
place
,
2
tymes
2
is
4
,
set
that
vnder
the
lyne
also
,
&
then
thou
hast
done
:
so
that
this
number
4
28
vnder
the
lyne
,
is
it
that
commeth
of
the
multiplication
of
this
summe
2314
by
this
number
2.
But
yf
it
be
so
that
in
the
multiplication
of
any
fygure
in
the
number
multyplicable
,
by
the
multiplyer
that
it
which
redoundyth
of
it
be
article
number
,
then
ye
shall
set
a
cypher
beneth
the
lyne
ryght
vnder
the
same
place
where
the
multiplycation
is
,
and
reserue
the
artycle
to
be
addyd
to
the
number
that
procedeth
of
the
multiplication
of
the
figure
in
the
nexte
place
by
the
aforesayd
multiplyar
,
the
whiche
lykewise
yf
it
amounte
to
an
article
do
likewyse
as
I
byd
you
to
do
in
the
fyrste
place
:
but
yf
it
be
number
composte
,
then
shall
ye
set
the
dyget
vnder
the
same
place
beneth
the
line
,
and
reserue
the
article
to
be
addyd
lykewyse
as
is
before
sayde
of
article
nomber
,
as
in
this
example
.
|
8
|
1
|
4
|
1
|
6
|
4
|
2
|
|
|
|
|
|
|
|
5
|
4
|
0
|
7
|
0
|
8
|
2
|
1
|
0
|
¶
yf
ye
wyll
multyply
this
nomber
8141642
by
this
fygure
5.
Begyn
at
the
fyrst
place
sayenge
5
tymes
2
is
10
,
nowe
for
bycause
that
this
number
is
article
,
ye
shal
accordynge
to
the
rule
before
,
sette
the
cypher
vnder
the
lyne
,
and
reserue
the
article
1
to
be
added
to
the
nomber
that
precedeth
of
the
multyplycation
of
the
nexte
fygure
standynge
in
the
next
place
of
the
summe
multiplycable
,
by
the
multiplyer
:
so
then
come
to
the
nexte
place
sayeng
,
5
tymes
4
is
20
,
to
this
20
adde
1
for
that
article
that
ye
reserued
,
and
that
maketh
21
,
therfore
bycause
that
this
is
a
composte
number
therfore
sette
the
digette
vnder
the
lyne
benethe
the
same
place
,
and
reserue
the
article
to
the
nexte
place
:
then
come
to
the
.iii.
place
sayenge
5
tymes
6
is
30
,
to
this
adde
the
article
2
which
ye
reserued
in
the
place
next
goinge
before
,
and
then
it
is
32
sette
the
dygette
vnder
the
lyne
as
ye
dyd
before
reseruing
the
article
to
the
next
place
then
come
to
the
.iiii.
place
saieng
,
5
times
1
is
5
to
this
adde
the
article
reseruyd
,
whiche
is
3
and
that
maketh
8
,
set
this
digette
nomber
vnder
the
line
,
and
then
come
to
the
v.
place
sayenge
,
5
tymes
4
is
20
,
nowe
for
bycause
that
this
nomber
is
article
set
0
vnder
that
place
beneth
the
lyue
reseruynge
the
article
2
to
be
addyd
vnto
the
next
place
:
then
comme
to
the
.vi.
place
sayenge
,
5
tymes
1
is
5
,
to
this
adde
the
article
2
reserued
and
then
it
is
7
,
set
it
vnder
the
lyne
:
them
to
the
.vii.
place
,
sayenge
,
5
tymes
8
is
40
,
now
for
bycause
it
is
an
article
nomber
ye
shall
sette
a
sypher
vnder
the
lyne
,
and
reserue
the
article
4
too
the
nexte
place
,
and
for
as
moche
as
there
is
no
mo
places
,
ye
shall
set
this
4
vnder
the
lyne
nexte
vnto
the
0
that
ye
sette
downe
laste
,
and
then
ye
haue
done
.
¶
When
that
your
multiplyer
is
composte
or
article
,
then
shall
ye
take
the
fyrst
fygure
of
your
multyplyer
,
and
by
hym
shall
ye
multiplye
all
the
fygures
of
the
multiplycable
numbers
,
settyng
alwaye
that
that
amounted
of
it
benethe
the
lyne
as
ye
dyd
before
.
And
when
ye
haue
multiplyed
the
number
multiplycable
by
the
fyrst
fygure
of
the
multyplyar
:
then
multiplye
it
agayne
by
the
seconde
figure
of
the
multiplyer
,
settyng
euermore
the
fyrst
fygure
of
the
number
multiplycate
,
dyrectely
vnder
the
figure
multiplycatour
,
in
what
place
so
euer
it
stande
:
and
the
number
multiplycable
is
multiplyed
by
al
the
figures
of
the
multiplicatour
,
then
make
a
stryke
vnder
them
all
,
addyng
al
the
numbers
multyplycate
together
as
they
stande
,
and
that
which
procedeth
of
that
addition
is
the
number
multiplycable
nowe
multyplyed
by
the
hoole
number
multyplycatour
,
as
by
this
example
ye
shall
playnly
perceyue
.
|
|
|
2
|
3
|
4
|
5
|
|
|
|
1
|
2
|
3
|
4
|
|
|
|
9
|
3
|
8
|
0
|
|
|
7
|
0
|
3
|
5
|
|
|
4
|
6
|
9
|
0
|
|
|
2
|
3
|
4
|
5
|
|
|
|
2
|
8
|
9
|
3
|
7
|
3
|
0
|
¶
If
ye
wyll
multiply
this
number
2345
by
this
number
1234
,
set
them
fyrste
as
ye
se
here
2
,
vnder
them
drawe
a
lyne
:
then
begyn
with
the
fyrste
figure
of
the
multiplycatour
,
whiche
is
4
,
and
by
hym
fyrste
accordynge
to
the
rule
multyplye
all
the
multiplicable
number
through
out
,
sayenge
4
tymes
5
is
20
,
sette
the
cypher
vnder
the
lyne
reseruynge
the
artycle
2
to
the
nexte
place
:
then
to
the
seconde
place
,
4
tymes
4
is
16
,
to
that
put
youre
reserued
article
2
and
it
is
18
,
set
the
dygette
8
vnder
the
lyne
reseruynge
the
artycle
1
:
then
to
the
thyrde
place
,
4
tymes
3
is
12
and
1
reserued
from
the
place
before
that
is
13
,
set
the
dyget
3
vnder
the
lyne
,
reseruynge
the
article
1
,
then
to
the
.iiii.
place
,
4
tymes
2
is
8
and
1
reserued
is
9
,
set
that
dygette
9
vnder
the
lyne
,
and
soo
haste
thou
multiplyed
this
nomber
multiplicable
by
the
seconde
figure
of
multiplicatour
,
Now
then
accordynge
to
the
rule
afore
,
multiplye
the
multiplicable
nomber
by
the
seconde
fygure
of
the
multiplycatour
sayenge
,
3
tymes
5
is
15
sette
the
dyget
5
vnder
the
lyne
,
accordynge
to
the
rule
,
which
byddeth
to
sette
euermore
the
fyrst
fygure
of
the
number
multiplycate
vnder
the
place
where
the
fygure
multiplicatour
doth
stande
:
as
here
nowe
thou
multipliest
the
multiplicable
by
the
second
figure
of
the
multiplicatour
,
whiche
is
3
,
than
say
3
times
5
is
15
set
this
diget
5
vnder
the
lyne
,
and
beneth
the
fyrst
nomber
multiplicate
ryght
vnder
the
fygure
multiplicatour
,
as
thou
seeste
in
the
example
,
and
reserue
the
article
1
:
then
to
the
seconde
place
of
the
multiplycable
,
3
tymes
4
is
12
,
and
1
that
is
reserued
is
13
set
the
dyget
3
vnder
the
lyne
,
as
ye
se
in
the
example
,
&
reserue
the
article
1
,
and
so
to
the
.iii.
place
3
tymes
1
is
9
and
1
reserued
is
10
,
set
a
sypher
vnder
the
lyne
&
reserue
the
article
1
:
so
to
the
.iiii.
place
sayenge
2
tymes
3
is
6
and
1
reserued
is
7
set
it
vnder
the
lyne
,
thus
haue
ye
done
youre
multiplication
by
the
seconde
fygure
of
the
multiplicatour
3.
Then
take
the
.iii.
fygure
of
multiplicatour
whiche
is
2
,
and
multiply
also
al
the
nombers
multiplicable
by
hym
sayenge
2
tymes
5
is
10
set
the
sypher
beneth
the
line
right
vnder
the
place
where
this
figure
2
the
multiplicator
standith
,
as
ye
se
in
the
example
:
and
reserue
the
article
1
,
then
to
the
seconde
place
2
tymes
4
is
8
,
and
1
reserued
is
9
,
set
that
9
vnder
the
lyne
:
then
to
the
.iii.
place
,
2
tymes
3
is
6
,
set
that
vnder
the
lyne
:
so
to
the
.iiii.
place
sayenge
2
tymes
2
is
4
set
that
4
vnder
the
lyne
.
Now
begin
to
multiply
with
the
fourth
and
last
fygure
of
the
multyplycatour
,
sayeng
1
tymes
5
is
5
sette
the
5
vnder
the
lyne
as
I
warened
ye
before
,
and
as
ye
se
in
the
example
,
then
to
the
seconde
place
1
tymes
4
is
4
,
set
that
4
vnder
the
lyne
,
then
1
tymes
3
is
3
,
set
that
3
vnder
the
lyne
,
then
1
tymes
2
is
2
set
that
2
vnder
the
lyne
&
ye
haue
done
your
multiplication
:
then
muste
ye
adde
accordynge
to
your
rule
afore
all
this
syngle
multiplied
number
togyther
,
and
that
the
whiche
commeth
of
the
addition
is
the
number
that
commeth
of
the
multiplication
of
this
number
2345
multiplycable
by
the
number
1234
,
multiplicatour
.
Then
come
to
the
fyrste
place
,
and
se
what
is
there
,
&
there
ye
shall
fynde
a
0
,
set
it
vnder
the
lyne
,
&
so
to
the
seconde
place
:
there
ye
shall
fynd
5
and
8
which
is
13
,
set
the
dygette
3
vnder
the
lyne
reseruynge
the
article
1
to
be
addyd
to
the
next
place
:
then
come
to
the
.iii.
place
,
there
is
0
,
3
and
3
whiche
is
6
to
that
adde
the
reserued
1
and
that
is
7
,
set
that
7
vnder
the
lyne
,
nowe
to
the
.iiii.
place
,
5
,
9
,
0
,
and
9
maketh
23
,
sette
the
3
vnder
the
lyne
,
reserue
the
article
2
,
so
to
the
.v.
place
4
,
6
,
and
7
,
is
17
,
to
that
adde
the
reserued
2
,
whiche
maketh
19
,
sette
the
9
vnder
the
lyne
,
and
kepe
the
article
1
in
mynde
then
to
the
.vi.
place
3
and
4
is
7
,
and
1
reserued
is
8
set
it
vnder
the
lyne
,
then
too
the
.vii.
place
,
there
fynde
ye
but
2
,
wherfore
set
it
vnder
the
lyne
,
and
then
haue
ye
done
:
so
that
this
summe
vnder
the
lyne
2893730
is
the
hole
number
multiplicate
¶
An other
example
of
multiplication
.
A
|
|
|
|
|
|
|
6
|
4
|
2
|
6
|
0
|
0
|
3
|
B
|
|
|
|
|
|
|
|
5
|
0
|
2
|
0
|
0
|
0
|
|
|
|
|
|
|
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
|
|
|
|
|
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
|
|
|
|
|
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
|
|
|
|
|
1
|
2
|
8
|
5
|
2
|
0
|
0
|
6
|
|
|
|
|
|
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
|
|
|
|
|
|
3
|
2
|
1
|
3
|
0
|
0
|
1
|
5
|
|
|
|
|
C
|
3
|
2
|
2
|
5
|
8
|
5
|
3
|
5
|
0
|
6
|
0
|
0
|
0
|
¶
Your
figures
set
after
this
sorte
,
A
is
the
multiplicable
nomber
.
B
is
the
number
multiplicatour
.
C
is
the
number
multiplicate
,
which
commeth
of
the
addition
of
all
the
seuerall
numbers
togyther
standyng
betwen
the
lynes
.
Begyn
then
your
worke
,
takynge
the
fyrst
fygure
of
B
,
the
multiplycatour
which
is
0
&
by
hym
multyply
all
the
figures
of
A
,
the
multiplicable
and
that
that
procedeth
of
it
set
vnder
the
lyne
as
ye
se
:
and
so
to
the
seconde
fygure
of
the
multiplicatour
which
is
also
0
multiplie
all
the
fygures
of
A
by
it
lyke
wyse
,
and
set
that
whiche
cometh
of
it
vnder
the
lyne
,
ryght
vnder
the
second
place
where
the
multiplicant
figure
standeth
:
then
to
the
thyrde
fygure
whiche
also
is
0
multyply
all
the
multyplycable
nomber
A
,
and
set
that
whiche
cometh
of
it
ryghte
vnder
the
thyrd
place
benethe
the
lyne
,
as
ye
se
playne
in
your
example
:
for
of
the
multiplycation
euermore
by
syphers
commeth
nothynge
but
cyphers
.
Nowe
to
the
.iiii.
place
of
B
,
the
multiplicatour
,
there
shal
ye
fynde
the
fygure
2
,
multiply
then
all
A
the
multiplicable
nomber
by
this
fygure
2
sayeng
2
tymes
3
is
6
set
that
6
vnder
the
lyne
ryght
vnder
the
place
where
the
multiplicatour
2
standeth
,
as
it
appeareth
in
youre
example
:
then
to
the
seconde
place
,
2
tymes
0
is
nothynge
,
sette
that
0
vnder
the
lyne
nexte
the
afore
sayde
6
,
and
so
to
the
thyrde
place
,
2
tymes
0
is
nothynge
,
sette
the
fygure
of
nothynge
downe
vnder
the
lyne
,
and
so
to
the
fourthe
place
,
2
tymes
6
is
12
sette
the
dygette
2
vnder
the
lyne
and
reserue
the
article
1
to
the
nexte
place
:
then
come
to
the
.v.
place
,
2
tymes
2
is
4
and
1
that
I
reserued
is
5
,
set
that
5
vnder
the
lyne
:
nowe
come
to
the
syxthe
place
,
sayenge
2
tymes
4
is
8
sette
that
8
vnder
the
lyne
:
so
to
the
seuenthe
place
,
2
tymes
6
is
12
,
sette
the
dygette
2
beneth
the
lyne
,
and
reserue
the
artycle
1
to
be
sette
in
the
nexte
and
laste
place
as
ye
se
in
the
example
.
Thus
haue
ye
multiplyed
A
the
multiplycable
by
.iiii.
fygures
of
B
the
multiplycatour
,
therfore
nowe
take
0
th●
v.
fygure
of
the
multiplicatour
:
and
by
also
multiplye
all
the
fygures
of
A
,
t●
multiplicable
,
and
thereof
shall
come
a●
syphers
to
be
set
vnder
the
lyne
,
as
ye
se●
here
in
the
copye
.
Then
to
the
sy●the
fygure
of
B
,
the
multyplycatoure
wh●che
is
5
,
by
this
5
also
multyplye
all
t●
fygures
of
A
the
multiplitable
sayen●
5
tymes
3
is
15
sette
that
5
beneth
the
ly●
ryghte
vnder
the
syxthe
place
wher●
the
multiplicatour
standeth
,
as
is
to
se
in
the
coopie
:
and
reserue
the
article
to
the
nexte
place
,
then
come
to
the
second
place
and
say
5
tymes
0
is
nothynge
,
sette
the
1
whiche
ye
reserued
in
your
mynde
vnder
the
lyne
,
and
so
to
the
thyrde
place
sayeng
5
tymes
0
is
nothynge
,
set
the
0
vnder
the
lyne
:
then
to
the
fourth
place
sayenge
,
5
tymes
6
is
30
,
set
the
cypher
0
vnder
the
lyne
,
reseruyng
the
article
3
vnto
the
next
place
:
then
come
to
the
v.
place
sayeng
,
5
tymes
2
is
10
,
and
3
that
I
reserued
is
13
●●t
the
dyget
3
vnder
the
lyne
,
and
reserue
●e
article
1
to
be
added
to
the
nexte
place
:
to
the
.vi.
place
sayenge
5
tymes
4
is
20
●d
1
reserued
is
21
set
1
the
dygette
vnder
●e
lyne
reseruyng
2
the
article
to
the
next
●ace
:
then
to
the
.vii.
and
laste
place
say●●ge
5
tymes
6
is
30
and
2
that
was
reser●d
is
32
set
the
dyget
vnder
the
lyne
and
serue
3
the
article
to
be
set
in
the
nexte
&
●e
place
beneth
the
lyne
as
ye
maye
se
●he
example
,
&
so
is
all
finished
:
Then
●er
all
these
particular
summes
drawe
●eke
,
and
adde
all
them
together
,
settynge
euer
that
whiche
commeth
of
the
addycyon
,
vnder
the
lyne
,
as
is
in
the
example
:
the
whiche
shall
amounte
vnto
this
summe
,
3225853506000
,
&
this
is
it
that
commeth
of
the
multiplicacyon
of
the
sum
A
,
by
the
sum
B.
¶
Sertayne
examples
of
multiplicacyon
in
the
whyche
ye
may
exercyse
youre selfe
to
be
the
more
practizyd
in
it
.
|
A
|
|
|
|
|
|
|
|
3
|
4
|
5
|
2
|
3
|
6
|
7
|
¶
To
multiplye
by
,
|
B
|
|
|
|
|
|
|
|
8
|
8
|
9
|
2
|
5
|
3
|
9
|
|
|
|
|
|
|
|
|
3
|
1
|
0
|
7
|
1
|
3
|
0
|
3
|
|
|
|
|
|
|
|
1
|
0
|
3
|
5
|
7
|
1
|
0
|
1
|
|
|
|
|
|
|
|
1
|
7
|
2
|
6
|
1
|
8
|
3
|
5
|
|
|
|
|
|
|
|
|
6
|
9
|
0
|
4
|
7
|
3
|
4
|
|
|
|
|
|
|
|
|
1
|
0
|
7
|
1
|
3
|
0
|
3
|
|
|
|
|
|
|
|
2
|
7
|
6
|
1
|
8
|
9
|
3
|
6
|
|
|
|
|
|
|
|
2
|
7
|
6
|
1
|
8
|
9
|
2
|
9
|
|
|
|
|
|
|
Sum.
|
|
3
|
0
|
7
|
0
|
0
|
3
|
0
|
8
|
1
|
8
|
9
|
8
|
1
|
9
|
|
|
6
|
4
|
9
|
7
|
0
|
To
mul.
|
|
|
|
|
1
|
3
|
|
1
|
9
|
4
|
9
|
1
|
0
|
|
6
|
4
|
9
|
7
|
0
|
|
Sum.
|
8
|
4
|
4
|
6
|
1
|
0
|
|
|
|
|
7
|
4
|
3
|
2
|
To
mul.
|
|
|
|
|
3
|
2
|
4
|
|
|
|
2
|
9
|
7
|
2
|
8
|
|
|
1
|
4
|
8
|
6
|
4
|
|
|
2
|
2
|
2
|
9
|
6
|
|
|
Sum.
|
2
|
4
|
0
|
7
|
9
|
6
|
8
|
¶
As
for
the
multiplycacyon
by
squares
is
nother
worth
the
writynge
nor
the
readynge
:
And
where
as
in
other
copies
is
set
duplacyon
,
triplacion
,
and
quadruplacyon
,
all
that
is
superfluous
,
for
so
muche
as
it
conteyned
vnder
the
kynde
of
multiplicacyon
:
and
they
that
are
experte
in
this
feete
,
maye
ryghte
well
perceaue
it
.
¶
The
profe
of
multyplycacion
.
THe
profe
of
multiplicacyon
maye
be
by
two
meanes
.
By
the
subducyng
out
of
all
the
9
:
and
the
second
way
is
by
particion
.
As
concernyng
the
fyrste
waye
:
ye
shall
fyrste
make
a
crosse
,
then
beholde
the
multiplicable
nomber
,
and
subdue
oute
of
it
all
the
nynes
,
and
that
that
remayneth
not
able
to
make
9
sette
it
at
the
vpper
ende
of
the
crosse
:
then
come
to
the
multyplycator
,
and
do
lykewise
in
hym
,
and
that
which
remayneth
all
the
9
beynge
subducyd
,
set
it
at
the
vnder
parte
of
the
crosse
:
then
multiplye
the
fygure
standynge
in
the
vpper
part
of
the
crosse
by
the
fygure
standyng
in
the
nether
part
of
the
crosse
,
and
out
of
the
same
that
commeth
of
it
take
9
as
ofte
as
ye
can
:
and
that
that
remayneth
not
able
to
make
9
,
set
it
at
the
right
syde
of
the
crosse
:
then
come
to
the
totall
sum
multiplicate
,
and
subduce
all
the
6
out
of
hym
lykewyse
,
&
that
whiche
remaineth
not
able
to
make
9
,
set
it
at
the
lefte
syde
of
the
crosse
,
and
if
it
be
lyke
the
fygure
standinge
at
the
ryghte
syde
of
the
crosse
,
then
is
it
well
,
otherwyse
it
is
not
well
.
¶
An
example
.
A
|
|
|
|
|
7
|
9
|
6
|
3
|
B
|
|
|
|
|
1
|
8
|
5
|
2
|
|
|
|
|
1
|
5
|
9
|
2
|
6
|
|
|
|
3
|
9
|
8
|
1
|
5
|
|
|
|
6
|
3
|
7
|
0
|
4
|
|
|
|
|
7
|
9
|
6
|
3
|
|
|
|
C
|
1
|
4
|
7
|
4
|
7
|
4
|
7
|
6
|
¶
To
knowe
whether
the
sum
C
,
be
the
very
sum
which
commeth
of
the
multyplycacion
of
A
,
by
B.
then
firste
subduce
all
the
9
that
ye
fynde
in
the
multiplycable
A
,
and
the
rest
set
it
at
the
vpper
ende
of
the
crosse
,
whiche
ye
shall
fynde
to
be
7
:
Then
to
the
multiplicatour
B.
do
lykewise
and
se
what
remaineth
,
and
there
remayneth
also
7
,
set
the
also
of
the
nether
ende
of
the
crosse
:
then
multiply
this
7
standynge
in
the
vpper
ende
by
7
standynge
in
the
nether
ende
:
and
therof
commeth
49
when
thou
hast
taken
all
the
9
out
of
this
49
there
wyll
remayne
4
,
the
whiche
thou
shalt
set
at
the
ryghte
syde
of
the
crosse
,
Then
come
to
C
,
the
totall
sum
of
the
multiplication
&
there
lykewise
take
out
all
the
9
that
ye
fynde
there
:
and
the
reste
not
sufficient
to
make
9
set
it
at
the
lefte
syde
of
the
crosse
,
the
which
thou
shalte
fynde
to
be
4
,
and
for
bycause
that
this
4
to
be
set
at
the
left
syde
is
lyke
the
figure
standyng
in
the
righte
syde
(
for
that
is
4
also
)
therfore
this
multiplycacion
is
good
and
wel
made
:
&
so
lykewyse
in
all
other
examples
.
¶
The
profe
by
Partition
is
to
diuide
the
total
summe
C
by
the
multiplicator
B
and
yf
the
quocient
be
iust
A
than
is
it
wel
multiplyed
other
els
not
.
But
this
waye
can
ye
not
practyse
,
vnto
such
time
as
ye
haue
learned
the
feate
of
Particion
.
¶
Of
particion
the
fourth
Kynde
of
Algorisme
.
PArticion
is
a
parte
of
algorisme
,
by
the
whiche
ye
maye
easyly
diuyde
any
greater
sum
by
a
lesse
or
equall
shewynge
howe
oftentimes
the
diuisor
is
contayned
in
the
nomber
diuysyble
.
¶
In
this
feate
of
particion
be
.iiii.
nombers
to
be
noted
:
the
nomber
diuysyble
,
the
nomber
diuisor
,
the
quotient
,
and
the
remayne
yf
there
be
any
.
¶
Before
you
come
to
particion
it
shall
be
very
nedefull
and
necessarie
for
you
,
ryght
perfytely
to
know
the
table
of
multiplycacion
of
dygettes
:
whiche
is
set
in
the
chapter
of
multiplycacion
:
For
vnlesse
that
ye
know
that
perfitely
ye
shal
stycke
greatly
not
only
in
multiplicacion
,
but
also
in
this
feate
of
particion
,
and
that
exactely
had
in
memorye
,
the
rest
shall
be
farre
easyar
.
As
for
example
.
yf
ye
wyll
knowe
howe
often
tymes
7
is
contayned
in
68
ymagyn
by
&
by
that
this
7
shulde
be
contayned
8
tymes
:
then
yf
ye
knowe
without
the
booke
perfytly
the
foresayde
table
ye
shall
se
that
8
tymes
7
is
but
56
ergo
7
is
contayned
more
then
8
tymes
in
68
,
ymagen
then
and
suppose
it
to
be
9
tymes
in
68
,
then
by
the
table
se
what
9
tymes
7
is
,
and
thou
shalte
se
that
it
is
64
wherfore
thou
mayst
conclude
that
in
68
,
7
is
conteyned
9
tymes
and
4
ouer
.
No.
|
|
|
|
quocient
.
|
|
15077
|
|
3
diuisor
.
|
The
proue
.
¶
To
diuyde
this
nomber
45231
by
3
the
3
is
diuisor
.
Fyrste
ye
shall
set
downe
your
nombers
to
be
diuyded
,
and
atthe
ende
of
that
nomber
on
the
righte
hande
ye
shall
make
a
streke
,
wherin
ye
shall
set
your
quocient
,
and
then
set
downe
youre
diuisor
which
is
3
vnder
the
fygure
that
standeth
at
the
vttermost
ende
at
the
lyft
hand
that
is
vnder
4
,
and
than
say
howe
many
tymes
3
maye
I
haue
in
4
,
ones
3
and
3
remayneth
ouer
,
set
1
within
the
stryke
and
that
1
that
remayneth
set
ouer
4
then
stryke
the
duysor
3
with
a
dasshe
of
your
pen
,
and
set
the
dyuisor
3
vnder
the
fygure
5
,
then
ioyne
the
artycle
1
to
the
dygnet
5
,
and
it
is
15
,
then
saye
howe
manye
tymes
3
maye
I
haue
in
15
,
5
tymes
3
,
set
that
5
in
the
stryke
nexte
to
the
fygure
1
and
close
vp
the
article
1
and
the
dyggette
5
with
a
cyfer
0
ouer
eyther
of
them
,
and
then
strike
the
diuisor
3
with
a
dasshe
of
your
pen
,
and
set
the
diuysor
3
vnder
the
thyrde
fygure
2
,
and
se
howe
many
tymes
3
ye
may
haue
out
of
2
none
therfore
sette
downe
a
cyfer
0
within
the
stryke
nexte
to
the
fygure
5
and
strike
out
your
diuisor
with
a
dasshe
of
youre
pen
,
and
set
the
dyuysor
3
vnder
the
fourth
fygure
3
then
ioyne
the
article
2
to
the
dyggette
3
and
that
makethe
23
than
se
howe
many
tymes
3
ye
maye
haue
in
23
,
7
tymes
,
and
3
remayneth
,
sette
that
7
within
the
stryke
nexte
to
the
cyfer
,
and
the
2
that
remayneth
set
ouer
the
fourth
figure
3
&
close
vp
the
article
2
with
a
cyfer
0
,
then
stryke
out
the
diuisor
and
set
it
vnder
the
fyrste
figure
1
at
the
ryghte
hande
,
then
ioyne
the
article
2
to
the
dygget
1
and
it
makethe
21
,
than
se
howe
manye
tymes
3
ye
maye
haue
in
21
,
7
tymes
and
nothynge
remayneth
,
than
set
the
7
within
the
stryke
and
close
the
article
2
with
a
cyfer
0
ouer
eche
of
them
and
strike
oute
the
deuysor
with
a
dashe
of
your
pen
,
&
so
the
thyrde
part
of
45231
is
15077.
¶
The
seconde
example
.
No.
|
|
|
|
6
diuysor
.
|
|
quocyent
.
|
|
390
5/0
part
.
|
The
proue
¶
To
dyuyde
this
nomber
2345
by
●
,
the
6
is
the
dyuysor
,
begynne
your
diuision
at
the
lyfte
hande
,
as
is
sayde
in
the
fyrst
example
,
and
sette
youre
dyuisor
vnder
the
thyrd
figure
3
,
for
ye
may
not
haue
6
oute
of
2
,
and
therefore
saye
howe
manye
tymes
6
maye
ye
haue
in
23
,
3
and
5
remaynethe
,
set
the
3
within
the
stryke
,
&
the
dygget
5
that
remnyneth
set
it
ouer
the
second
fygure
,
and
close
the
artycle
2
with
a
cypher
0
ouer
it
,
and
then
strike
oute
the
diuisor
with
a
dasshe
of
your
pen
,
&
set
your
diuysor
agayne
vnder
the
thyrde
fygure
4
,
and
then
ioyne
the
artycle
5
to
the
dygget
4
,
and
it
is
54
,
then
se
how
many
tymes
6
ye
maye
haue
in
54
,
9
and
nothinge
remayneth
,
sette
the
9
within
the
strike
and
close
vp
the
article
5
and
the
dygget
4
with
a
cypher
0
ouer
eyther
of
them
,
and
stryke
out
the
diuysor
with
a
dasshe
of
your
pen
,
and
set
the
diuisor
vnder
the
fygure
5
and
say
howe
many
tymes
6
may
ye
haue
out
of
5
,
no
tymes
,
therfore
set
down
a
cypher
0
with
the
stryke
,
and
lette
the
5
stande
&
stryke
out
the
dyuysor
with
a
dasshe
of
your
penne
,
and
so
the
6
parte
of
2345
is
390
,
and
the
5
that
remayneth
sette
at
the
ende
of
the
quocient
in
this
maner
.
⅚
&
so
the
quocyent
is
390
⅚
¶
To
dyuyde
by
2
or
3
fygures
,
or
by
as
many
as
pleaseth
you
.
¶
Fyrst
set
downe
your
nomber
to
be
diuyded
and
your
diuisor
vnder
it
,
begynnynge
at
the
lefte
syde
at
suche
a
place
as
ye
may
take
the
laste
fygure
of
your
deuysor
in
the
laste
ende
,
and
then
se
howe
ofte
ye
may
haue
that
fygure
in
the
figure
aboue
it
,
and
that
set
aparte
for
your
quocyent
,
with
the
whiche
quocient
ye
shall
multiply
euery
fygure
by
it selfe
of
your
diuisor
and
that
that
commeth
of
the
multyplycacyon
,
ye
shall
abate
of
the
fygure
right
ouer
it
,
puttynge
out
that
other
fygure
,
and
set
the
rest
aboue
it
,
and
so
worke
with
euery
fygure
by
it selfe
throughout
the
diuysor
.
Then
renewe
your
diuisor
1
fygure
forwarde
towarde
youre
ryghte
hande
,
as
before
is
rehersed
,
and
so
contynue
your
worde
folowynge
to
the
fyrst
fygure
of
your
nomber
to
be
diuided
.
Then
it
is
to
be
noted
that
yf
it
happe
that
your
multiplied
number
that
ye
shulde
abate
be
more
then
the
number
ouer
it
,
then
for
a
generall
rule
,
ye
shall
not
take
your
dyuysor
out
of
the
fygure
aboue
it
excepte
that
it
maye
sufficientlye
yelde
ynough
to
all
the
abatementes
of
the
resydue
,
as
more
playnly
shall
apere
in
the
example
folowynge
.
No.
|
|
|
|
quocient
,
|
|
343
7/12
|
|
|
|
12
dyuysor
|
|
|
The
proue
.
¶
Fyrst
set
downe
this
nomber
4123
,
and
diuyde
it
be
your
dyuysor
12
,
begyn
youre
worke
at
youre
lyfte
hande
,
settynge
the
article
1
of
your
deuisor
vnder
4
,
and
the
dygget
2
vnder
the
thyrde
fygure
1
,
and
than
se
how
many
tymes
the
artycle
1
of
your
diuysor
ye
many
haue
in
the
4
ouer
it
,
ye
wolde
saye
4
tymes
1
,
but
that
can
not
be
bycause
there
ye
may
not
haue
the
quocyent
4
multiplyed
with
the
dygget
2
of
your
diuisor
,
for
therof
commeth
8
,
and
then
that
8
ye
may
not
take
out
of
1
ouer
the
dygget
2
Therefore
say
againe
howe
manye
tymes
2
maye
ye
haue
in
4
,
3
tymes
and
1
remayneth
,
set
the
3
within
the
stryke
for
the
quocyent
,
&
the
1
that
remayneth
set
ouer
4
,
and
strike
out
the
article
2
of
your
diuisor
with
your
pen
.
Then
multiplye
the
quocyent
3
with
the
dygget
2
of
your
diuisor
,
and
therof
commeth
6.
Then
ioyne
the
article
1
that
remayneth
,
&
the
digget
1
,
and
it
is
11
therout
take
6
&
there
remayneth
5
,
set
the
5
ouer
the
thyrde
fygure
1
,
and
close
vp
the
article
1
ouer
4
with
a
figure
ouer
it
,
and
stryke
out
the
dygget
2
of
your
diuisor
againe
but
one
fygure
forwarde
,
as
thus
:
set
the
article
1
vnder
the
thyrde
figure
1
is
the
No.
and
the
diget
2
vnder
the
second
figure
2
&
there
se
howe
many
tymes
1
I
maye
haue
in
5
that
remayneth
,
4
tymes
,
&
yet
there
remayneth
1
whiche
must
be
set
ouer
5
,
and
stryke
out
the
article
1
with
your
pen
.
Then
multyply
the
dygget
2
of
your
deuisor
with
the
quocient
4
and
it
is
8
,
then
ioyne
the
article
1
that
remayneth
,
and
the
dygget
2
in
No.
togither
&
it
is
12
,
then
take
the
8
out
of
the
12
,
and
there
remayneth
4
,
set
that
4
ouer
the
seconde
fygure
2
in
the
No.
and
close
vp
the
article
1
with
a
cyfer
0
ouer
it
and
stryke
out
the
digget
2
of
your
diuysor
with
your
pen
.
Then
renewe
youre
diuysor
againe
as
before
is
sayde
,
&
set
the
article
1
vnder
the
seconde
fygure
No.
and
thense
how
many
tymes
1
I
haue
in
4
,
that
remayneth
3
tymes
and
1
remayneth
,
set
that
3
within
the
stryke
for
the
quocient
,
&
the
1
that
remayneth
set
ouer
the
4
,
and
stryke
out
the
article
1
of
your
dyuysor
with
your
pen
.
Then
multiply
the
quocient
3
with
the
digget
2
of
your
diuysor
,
and
it
is
6
,
then
ioyne
the
article
1
that
remayneth
,
and
the
dygette
5
in
No.
and
it
is
13.
Then
take
6
oute
of
13
and
there
remayneth
7
,
set
that
7
ouer
the
dygget
3
in
the
No.
and
close
vp
the
artycle
1
with
a
cyfer
0
ouer
it
,
and
stryke
oute
the
dygget
2
of
your
diuysor
,
and
then
the
12
parte
of
4123
is
for
the
quocyent
343
,
and
the
7
that
remayneth
shall
be
sette
at
the
ende
of
your
quocyent
,
as
thus
7/16
¶
ye
shall
note
that
in
these
two
examples
the
quocient
standeth
in
the
myddes
betwyxte
the
two
lynes
,
and
the
nomber
to
be
deuyded
standeth
nexte
aboue
the
vppermoste
lyne
,
and
the
deuysor
standethe
nexte
vnder
that
nether
lyne
.
But
than
ye
must
marke
that
there
be
two
dyuysors
,
one
is
called
the
diuisor
currant
,
bycause
it
is
alwayes
remouable
toward
the
right
hande
in
the
operation
,
and
also
it
is
stryken
out
,
and
this
dyuysor
standethe
alwaye
vnder
the
nether
lyne
of
the
quocyent
.
The
other
diuysor
is
called
the
diuysor
permanent
,
for
he
is
not
remoued
nor
blotted
as
the
other
is
,
but
standethe
alwaye
permanent
on
the
lyfte
hande
dyrectly
againste
the
nomber
that
is
to
be
deuided
.
And
iust
ouer
hym
there
standeth
the
remayne
of
the
whole
number
whiche
remayne
can
not
be
deuyded
by
the
deuysor
,
and
therfore
it
is
set
ouer
the
deuysor
permanent
with
a
strike
betwyxte
:
as
ye
maye
se
in
the
fyrste
ensample
,
where
1
is
remaynynge
,
and
6
is
deuysor
.
¶
For
as
muche
as
in
this
ensample
we
can
not
take
4
whiche
is
the
diuisor
,
oute
of
3
,
therfore
we
shall
set
4
vnder
5
&
saye
howe
many
tymes
4
haue
ye
in
35
,
ye
haue
8
tymes
4
,
and
there
restethe
3
,
ye
shall
set
the
8
betwyxte
the
two
lynes
,
and
the
3
aboue
5
,
then
efface
the
5
&
the
4
,
then
we
shall
sette
,
4
vnder
0
&
saye
,
in
30
how
many
tymes
4
,
7
tymes
,
set
7
betwene
the
lynes
at
the
right
syde
by
the
8
,
and
there
resteth
2
,
whiche
we
shall
set
aboue
0
,
and
efface
0
,
than
set
4
vnders
,
and
say
in
28
howe
many
tymes
4
,
7
,
&
there
resteth
nothyng
,
set
7
betwene
the
lynes
by
the
7
,
than
set
4
vnder
0
&
saye
howe
many
tymes
4
in
0
,
there
is
none
,
therefore
set
0
betwene
the
lynes
,
then
shall
we
saye
in
9
howe
manye
tymes
4
,
2
tymes
,
set
than
2
betwene
the
lynes
,
and
restethe
1
which
we
shall
set
aboue
9
and
efface
9
than
12
howe
manye
tymes
4
,
3
tymes
,
set
than
3
betwene
the
lynes
by
2
,
&
there
resteth
nothynge
.
Than
in
3
that
is
the
laste
fygure
howe
many
tymes
4
,
no
tymes
therfore
at
the
ende
of
the
figure
ye
shall
set
the
3
thus
¾
and
it
is
made
.
|
|
3
|
2
|
|
|
3
|
|
|
|
A
|
3
|
5
|
0
|
8
|
0
|
9
|
2
|
3
|
3
|
C
|
|
8
|
7
|
7
|
0
|
2
|
3
|
|
40
|
B
|
|
4
|
0
|
0
|
0
|
0
|
0
|
0
|
|
|
|
|
4
|
4
|
4
|
4
|
4
|
|
|
¶
Example
whan
the
diuisor
is
an
artycle
,
it
behoueth
to
do
semblably
,
in
sayenge
,
in
3
howe
many
tymes
4
,
no
tymes
and
therfore
we
shall
sette
4
vnder
5
,
and
0
vnder
0
,
and
saye
howe
many
tymes
4
in
35
,
8
tymes
,
sette
8
betwene
the
two
lynes
vnder
5
,
and
there
resteth
3
whiche
we
shall
sette
ouer
5.
Then
set
the
3
that
standeth
ouer
5
and
the
0
together
,
and
that
is
30
,
than
saye
howe
many
tymes
4
in
30
,
7
and
alwayes
so
to
the
ende
.
And
than
we
shall
set
4
vnder
2
,
and
0
vnder
3
,
and
saye
in
12
in
takynge
the
1
that
shall
rest
of
the
summe
afore
and
shall
be
aboue
9
,
and
the
2
that
is
after
9
,
howe
many
tymes
4
,
3
tymes
4
,
and
thenne
set
3
in
the
number
of
C
,
agaynste
2
,
&
than
shall
we
cease
,
for
there
remayneth
all
onely
3
to
be
parted
by
40
,
nowe
we
shall
not
make
0
vnder
3
as
is
afore
,
but
at
the
ende
we
shall
set
3
thus
3/40
¶
Ensample
whan
the
diuysor
is
compost
,
as
in
this
fygure
afore
,
we
shall
say
in
35
that
ben
nere
A
,
howe
many
tymes
4
that
are
in
the
number
of
B
,
8
tyme
4
set
that
8
betwene
the
two
lynes
in
the
place
of
C
,
and
there
resteth
3
whiche
we
shall
sett
aboue
5
and
efface
35
of
A
,
and
4
of
B
,
then
shall
we
say
in
multiplyenge
the
8
of
C
,
by
the
second
fygure
of
B
,
that
is
2
,
we
shall
saye
2
tymes
8
bene
16
,
than
abate
16
of
the
number
of
A
agaynst
the
same
42
,
and
there
be
3
whiche
is
ouer
5
and
0
of
the
number
,
that
be
worth
30
and
we
shall
saye
,
of
30
abate
16
and
there
restethe
14
,
of
the
whyche
14
we
shall
sette
●
ouer
3
,
and
efface
3
,
and
4
aboue
0
and
efface
0
,
than
shall
we
set
the
diuisor
somwhat
forward
,
the
4
agaynste
0
,
that
shall
be
effaced
,
and
2
agaynste
8
,
and
saye
in
14
demonstrynge
1
that
shall
be
aboue
3
and
4
aboue
0
howe
many
tymes
4
,
3
tymes
,
set
the
3
benethe
the
lynes
in
the
number
of
C.
and
there
restethe
2
whiche
we
shall
sette
ouer
4
,
and
efface
4
,
than
shall
we
saye
againe
in
multiplyenge
the
3
of
C
,
by
the
seconde
figure
of
B
,
that
is
2
we
shall
saye
than
2
tymes
3
bene
6
and
of
8
that
is
againste
it
we
shall
abate
6
and
there
shall
reste
2
,
whiche
we
shall
set
ouer
8
,
and
efface
8
,
and
alwayes
so
vnto
the
ende
,
&
when
we
come
to
the
2
laste
fygurs
of
A
,
&
that
we
wold
deuyde
them
by
42
,
we
maye
not
,
for
the
firste
that
is
but
2
shal
be
effaced
with
1
that
standethe
aboue
5
and
because
that
we
maye
take
there
nothynge
we
shall
set
0
agaynste
2
of
A
,
in
the
number
of
C
,
betwene
the
lynes
,
and
so
it
is
done
,
&
there
shall
reste
3
to
be
deuyded
by
4●
,
and
that
3
shall
be
set
at
the
ende
of
the
particion
as
thus
3/42
and
it
is
finished
.
¶
And
it
is
to
be
knowen
that
as
many
fygures
as
foloweth
the
firste
fygure
in
the
number
of
B
,
shall
be
multiplyed
by
them
of
the
number
of
C
,
then
the
multyplycacion
that
therof
shall
come
ye
shall
abate
in
the
nomber
of
A
,
as
in
this
ensample
shall
more
playnly
appere
.
The
proue
.
¶
In
this
ensample
in
the
number
of
B
,
that
is
the
deuysor
,
be
many
figures
,
and
therfore
we
shall
saye
,
in
3
of
A
,
how
many
tymes
2
of
B
,
1
tyme
,
sette
that
1
vpon
C
and
1
that
remaynethe
of
3
,
ouer
3
,
and
than
shall
we
come
to
the
4
of
B
and
to
1
of
C
,
and
multyplye
them
in
sayenge
,
1
tymes
4
is
4
,
whyche
4
we
shall
abate
of
the
number
of
A
,
in
takynge
1
aboue
3
and
5
after
3
,
that
shal
be
worthe
15
and
therof
we
shall
abate
4
and
there
resteth
11
,
and
for
the
more
shorteste
waye
of
5
onlye
abate
4
,
and
sette
the
1
that
remayneth
aboue
5
,
&
there
restethe
alwayes
11
,
then
shall
we
come
to
the
3
of
B
,
and
to
1
of
C
,
and
make
all
onely
the
multiplycacion
in
saynege
1
tymes
3
bene
3
,
thanne
of
10
abate
3
in
demonstrynge
1
ouer
5
and
0
after
,
and
then
there
resteth
7
,
whiche
we
shall
sette
ouer
0
,
then
bycause
of
the
cyfer
0
maye
nothynge
come
,
we
shall
leue
it
,
and
goo
to
the
nexte
fygure
and
saye
1
tymes
5
that
is
at
the
laste
ende
of
B
ben
5
but
in
so
muche
that
we
maye
nothynge
abate
of
0
that
is
againste
it
in
the
number
A
we
shall
borowe
of
the
figure
afore
that
is
3
onelye
one
and
efface
the
8
,
and
sette
the
7
aboue
the
8
,
and
the
1
that
we
shall
holde
shal
be
worth
10
to
the
regarde
of
the
number
that
we
be
in
,
then
we
shal
saye
of
10
abate
5
there
restethe
5
whiche
we
shal
set
aboue
0
,
then
shall
we
auaunte
our
partetour
consequently
vnder
the
other
fygurs
folowynge
,
that
is
to
saye
tyll
the
laste
of
B
,
be
set
vnder
the
laste
of
A
,
and
then
ye
maye
not
auaunce
them
any
further
bycause
ye
be
come
to
the
endes
of
both
the
numbers
.
¶
The
proue
of
diuision
or
particyon
is
made
in
this
maner
:
ye
shall
fyrst
make
a
crosse
,
as
ye
dyd
before
in
multiplycacion
and
abate
the
9
of
the
particyon
,
and
sette
the
reste
at
the
lyfte
ende
of
the
crosse
semblably
of
the
thyrde
nomber
that
is
betwixte
the
two
lynes
,
and
set
the
rest
at
the
right
ende
of
the
same
crosse
,
and
yf
there
be
no
thynge
reste
set
0
,
Then
multyply
the
two
nombers
of
fygure
,
for
they
2
be
dyggettes
that
one
by
that
other
,
&
therof
abate
all
the
9
,
yf
there
be
nothynge
in
the
fyrste
number
,
or
yf
ye
may
not
diuide
it
ioyne
it
with
the
same
that
shall
come
thereof
.
And
so
the
rest
that
maye
not
make
9
sette
it
at
the
ende
vnder
the
crosse
.
Then
shall
we
come
to
the
fyrst
number
&
semblably
do
away
the
9
therof
,
&
set
the
reste
aboue
the
crosse
,
&
if
that
aboue
&
that
beneth
be
lyke
,
the
particion
is
good
,
or
yf
not
,
it
is
fals
.
And
for
to
vnderstande
it
better
we
make
proues
by
the
ensamples
aforesayde
.
proue
¶
For
the
fyrst
we
shall
take
the
partetour
that
is
4
and
set
it
at
the
lyfte
syde
of
the
crosse
,
than
shall
we
abate
the
9
of
the
thyrde
nomber
,
&
there
reste
8
whiche
8
we
shall
sette
at
the
ryght
ende
of
the
crosse
and
multiplye
it
by
4
and
thereof
cometh
32
wherof
resteth
5
then
adiouste
them
with
the
2
farthynges
that
we
myghte
not
deuyde
,
and
they
shall
make
7
,
the
whiche
7
we
shall
set
vnder
the
crosse
,
than
shal
we
abate
the
9
of
the
fyrste
number
that
ben
the
farthynges
,
and
there
shall
reste
7
whiche
7
we
shall
set
at
the
vpper
ende
of
the
crosse
,
and
so
ben
the
two
endes
lyke
and
it
is
well
made
.
¶
Reduction
.
REduction
is
a
kynde
of
Algorisme
by
the
which
ye
be
taught
to
reduce
numbers
of
lesse
denominacion
or
value
to
numbers
of
more
denominacion
or
value
:
or
yf
the
case
requyre
it
,
numbers
of
great
denominacion
to
the
numbers
of
lesse
value
.
Example
of
the
fyrst
.
20.
li.
63.
s.
44.
d
10.
far
.
Thus
reduce
the
farthinges
to
pens
,
&
the
pens
to
shelynges
,
and
the
shelynges
to
poundes
:
and
then
this
summe
is
23
li.
6
s.
10
d̔
,
and
2
far
.
so
haue
you
reduced
the
lesse
summe
to
the
more
.
Example
to
reduce
the
more
to
the
lesse
.
Take
the
same
example
agayne
,
and
reduce
the
20
li.
63
s.
44.
d
10.
far
,
all
in
to
farthinges
,
and
it
wyll
make
22410
farthinges
.
Fyrst
reducynge
the
powndes
to
shelynges
,
then
to
pens
,
&
all
that
pens
to
farthinges
:
wherefore
it
shall
be
verye
necessary
for
you
to
knowe
what
thynge
your
number
dothe
signifie
,
whether
wayght
,
mony
measure
,
or
tyme
:
and
to
be
expert
in
all
maner
of
accomptes
:
it
shal
be
necessary
for
you
to
knowe
all
maner
of
wayghtes
,
coynes
,
measures
,
and
tyme
.
Example
in
englysh
money
4
farthinges
make
1
d
12.
d
maketh
a
shelynge
20
shelynges
maketh
a
pounde
.
¶
In
weyght
,
and
fyrst
of
troy
weyght
,
euery
pownde
hath
12
ounces
,
and
euerye
ounce
20
peny
weyght
,
and
euerye
penye
weyght
20
graynes
.
&c.
¶
The
haperdepeys
pownd
hath
16
ounces
,
an
ounce
8
drammes
,
the
dramme
3
scruples
,
the
scruple
20
graynes
.
¶
Of
measure
,
the
yarde
hathe
3
foot
,
the
foote
12
ynches
,
the
ynche
3
barley
cornes
of
length
.
¶
Of
tyme
,
the
yere
hath
365
dayes
,
the
daye
24
houres
,
the
houre
hath
60
minutes
,
euery
minute
60
secondes
,
euerye
seconde
60
thyrde
,
euerye
thyrde
60
quartes
,
euery
quartes
60
fyftes
,
euerye
fyfte
60
,
syxtes
,
and
so
forth
infinitely
.
¶
To
reduce
the
more
summe
to
the
lesse
.
¶
When
thou
wylte
reduce
the
more
to
the
lesse
,
loke
howe
many
tymes
the
lesse
is
contayned
in
the
more
,
and
by
that
number
multiply
the
number
of
the
more
and
that
that
commeth
of
the
multyplycacyon
sheweth
the
more
reducyd
to
the
lesse
.
Example
.
I
wold
reduce
8
d
to
farthynges
loke
howe
many
tymes
a
farthynge
is
contayned
in
a
peny
,
and
that
is
as
ye
knowe
4
tymes
,
then
multiplye
accordynge
to
the
rule
8
by
4
,
and
that
maketh
32
whiche
be
32
farthynges
,
and
so
8
d̔
,
maketh
32
farthynges
.
¶
An
other
example
.
Here
is
a
summe
of
28
li.
and
6
s.
I
wolde
haue
this
powndes
,
which
is
of
more
denomynacyon
reduced
to
the
shelynges
,
whiche
be
of
lesse
denominacion
:
then
loke
fyrste
howe
ofte
a
shelynge
is
contayned
in
a
pound
,
and
that
is
20
tymes
,
for
20
s.
makethe
a
li.
multiply
then
the
28
li.
by
20
,
thereof
commeth
560
,
whiche
be
all
shelynges
:
to
this
putte
the
other
6
shelynges
and
so
all
is
566
shelynges
.
¶
But
ye
shall
note
that
where
there
be
any
summe
of
meane
denominacyons
betwene
the
more
to
be
reduced
and
the
lesse
to
whome
reduccyon
is
made
:
then
shall
it
be
easy
ar
to
reduce
fyrste
the
more
to
the
meane
,
and
so
by
the
meane
to
the
lesse
.
¶
The
example
.
43
li.
19
s.
20
d
4
farthynges
.
yf
ye
wyll
reduce
all
this
summes
to
the
farthinges
:
then
shall
it
be
better
for
you
to
reduce
the
powndes
fyrste
to
shelynges
and
then
beynge
shelynges
to
reduce
them
to
pens
,
and
at
the
laste
to
farthynges
:
so
by
your
rule
43
pownde
maketh
860
shellynges
,
to
that
adde
the
19
shellynges
,
it
maketh
879
,
then
reduce
this
879
shellynges
to
pens
:
loke
fyrste
howe
many
pens
are
contayned
in
a
shelynge
,
and
that
is
12
,
multyply
879
by
12
therof
commeth
10548
whiche
be
all
pens
,
to
this
adde
your
20
pens
,
&
that
maketh
10568
d
then
reduce
this
pens
to
farthynges
,
se
how
many
farthinges
be
in
a
peny
that
is
4
,
multyply
10568
by
4
commeth
4●272
to
these
adde
the
4
farthynges
and
that
maketh
42279
farthinges
.
Thus
haue
ye
reduced
43
li.
19
s.
20.
d̔
,
4
farthynges
the
more
by
the
meane
to
the
lesse
.
¶
To
reduce
the
lesse
to
the
more
.
¶
Fyrste
marke
howe
manye
tymes
the
more
doth
contayne
the
lesse
:
and
by
that
number
dyuyde
the
lesse
,
and
the
quocient
sheweth
the
lesse
reduced
to
the
more
.
Example
.
I
wolde
haue
this
sum
5600
s.
reducyd
into
powndes
:
for
howe
manye
tymes
a
pounde
doth
contayne
a
shelyng
that
is
20
tymes
,
thenne
diuyde
5600
by
20
the
quocyent
shal
be
280
,
whiche
be
powndes
:
so
that
5600
s.
reduced
to
powndes
maketh
280
li.
&
so
lykewyse
in
all
other
rekeninges
.
¶
When
summes
of
dyuers
denomynacyons
come
in
addition
to
be
addyd
togither
,
then
begynnynge
at
the
summes
of
least
denomination
:
adde
them
ouer
togyther
tyll
such
tyme
as
they
make
a
number
of
the
nexte
denominacion
,
and
that
that
remayneth
not
able
to
make
any
number
of
greater
denominacion
,
set
it
vnder
the
lyne
&
procede
to
the
nexte
summe
of
greater
denomination
,
to
the
whiche
adde
the
number
of
the
same
denomination
reducyd
out
of
the
sum
before
of
the
lesse
denominacion
,
so
procedynge
to
the
ende
.
¶
Ensample
.
li.
|
s.
|
d
|
far
.
|
1680
|
10
|
5
|
3
|
8200
|
29
|
7
|
2
|
1008
|
3
|
10
|
3
|
¶
Begyn
at
the
least
whiche
be
farthynges
:
sayenge
3
and
2
ben
5
,
and
3
ben
8
this
8
farthynges
make
2
pens
,
therfore
take
this
pens
and
adde
them
to
the
nexte
sum
which
is
of
the
same
denomination
,
sayenge
2
and
10
be
12
d
which
is
1
shyllynge
the
7
and
5
be
12
whiche
also
makethe
a
shelynge
,
so
emonge
these
pens
ye
haue
●
shelynges
to
be
added
to
the
nexte
order
of
shelynges
,
sayenge
2
and
3
be
5
and
9
be
14
,
put
the
diggette
4
vnder
the
lyne
,
and
reserue
the
article
1
to
the
nexte
place
sayenge
1
and
2
be
3
and
1
be
4
,
sette
that
4
vnder
the
lyne
also
and
then
is
it
44
s.
the
whiche
reduced
to
poundes
makethe
2
li.
and
4
s
,
remayneth
vnder
the
tytle
of
shelynges
:
then
put
that
2
li.
to
the
other
powndes
,
and
so
haste
thou
done
in
reduccion
of
the
summes
of
lesse
value
to
the
greatest
sum
,
wh●ch
be
powndes
.
And
this
is
sufficiently
entreatyd
of
reduccion
.
¶
Here
foloweth
of
progressyon
.
PRogressyon
shewethe
the
nomber
whan
it
begynnethe
at
1
or
at
2
in
mountynge
alwayes
by
one
,
&
one
,
as
dothe
this
number
123456789.
Now
yf
ye
wyll
knowe
the
valour
of
the
numbers
firste
ye
must
regarde
two
thinges
,
that
is
to
wyte
,
yf
the
number
procede
continually
without
leuynge
any
thing
betwyxte
as
here
1234567
or
if
it
leue
any
thynge
betwyxte
as
here
,
13576.
Secondly
ye
must
consider
yf
the
number
be
euen
or
odde
.
And
after
these
two
consyderacyous
,
then
by
foure
rules
that
here
foloweth
ye
maye
knowe
the
valour
of
eche
whole
number
.
❧
The
fyrste
rule
is
whan
one
number
procedeth
in
mountinge
alwayes
contynually
in
the
begynnyng
,
then
if
it
ende
in
an
euyn
number
,
than
shal
we
take
the
halfe
of
that
euyn
number
,
and
by
it
we
shall
multiply
the
odde
number
that
commeth
of
the
euen
number
,
as
ye
maye
se
in
this
ensample
folowynge
.
¶
Ensample
.
¶
yf
ye
wyll
knowe
howe
much
this
number
is
worth
,
than
multyply
the
halfe
of
8
that
is
4
and
the
number
that
is
after
8
is
9
,
and
then
thereof
commeth
36
,
and
so
muche
is
the
summe
worth
,
and
thus
may
ye
do
with
all
suche
lyke
questions
.
¶
An
other
example
.
The
proue
.
¶
For
to
multyply
this
number
7
wherin
the
greatest
and
the
more
halfe
is
4
ye
muste
multyply
7
by
4
and
it
is
28
,
and
so
muche
in
the
hole
summe
,
¶
The
thyrde
is
,
yf
a
number
procede
not
continually
,
and
ende
in
an
euen
number
ye
shall
take
the
halfe
of
the
sayde
number
that
is
euen
,
and
by
hym
multiplye
the
same
that
is
nexte
comyng
after
the
same
halfe
,
and
in
thus
doynge
,
ye
shall
haue
the
summe
of
the
same
●umber
.
¶
An
example
.
The
proue
¶
Yf
ye
wyll
knowe
how
much
this
number
is
worth
,
than
take
the
halfe
on
8
that
is
4
,
then
multyplye
by
the
4
the
number
whiche
foloweth
,
that
is
5
in
sayenge
4
tymes
5
is
20
,
and
so
muche
is
worth
the
whole
summe
.
¶
The
fourth
is
whan
the
sayd
number
procedeth
not
contynually
,
then
yf
it
ende
¶
Here
foloweth
the
rules
,
and
fyrst
the
rule
of
thre
.
MVltiply
by
the
contrary
&
diuide
by
the
semblaunt
or
lyke
.
This
rule
maye
be
vnderstande
in
two
maners
.
Fyrste
multiplye
the
same
that
ye
wyll
bye
by
his
contrary
,
that
is
to
witte
,
by
the
price
,
and
diuide
by
the
semblaunte
that
is
to
witte
,
by
as
muche
as
ye
haue
bought
:
or
thus
,
multiplye
ihe
price
by
his
contrary
,
that
is
to
witte
,
by
the
same
that
thou
wylt
bye
,
and
diuide
it
by
his
semblaunt
,
that
is
that
same
that
ye
haue
bought
.
And
note
ye
why
it
is
called
the
rule
of
thre
,
for
with
thre
nombers
certayne
ye
may
knowe
and
fynde
the
fourth
nomber
vncertayne
.
And
it
is
a
rule
ryght
no
table
and
necessarye
in
the
fayct
of
marchaundyse
.
For
to
haue
knowlege
of
this
rule
,
it
behoueth
to
set
some
rules
dyfferent
in
maner
of
questyons
,
and
fyrke
in
measures
longe
.
¶
The
rule
of
hole
numbers
.
¶
Yf
9
elles
of
cloth
cost
25
crones
,
how
moch
shall
cost
15
by
the
price
.
Answer
.
It
behoueth
you
to
set
the
somme
,
that
is
to
wyte
,
25
crones
.
And
than
ye
shal
multiply
by
his
contrary
,
that
is
to
wite
,
by
15
that
ben
375
,
and
than
diuide
them
by
that
semblaunt
,
that
is
to
w●te
,
by
9
,
and
therof
cometh
41
crones
and
an
halfe
,
and
there
remaineth
1
crone
and
and
halfe
,
the
which
ye
shall
make
in
sꝪ
.
and
there
ben
54
sꝪ
.
the
whiche
ye
shall
deuide
by
9
and
therof
cometh
6
sꝪ
.
Therfore
ye
maye
answere
that
the
15
elles
shall
cost
41
crones
and
an
half
and
6
sꝪ
.
Nowe
yf
ye
wyll
make
the
proue
it
behoueth
you
to
forme
your
question
thus
yf
15
elles
cost
.
41
crones
and
an
half
and
6
shyllynges
,
howe
moch
shal
coste
9
elles
by
the
price
.
Then
it
behoueth
you
fyrst
to
multiplye
the
6
sz
,
by
9
and
that
been
54
then
it
behoueth
you
to
make
thereof
crones
,
that
is
1
crone
an
an
halfe
,
&
then
ye
shall
multiplye
the
41
crones
and
an
halfe
by
9
,
and
they
ben
373
crones
and
an
halfe
,
and
then
set
thereto
1
crone
and
an
halfe
,
and
they
be
375
crones
whiche
ye
shall
deuyde
by
15
,
that
ben
25
,
the
which
25
ben
the
price
of
9
elles
,
and
so
the
rule
is
good
,
and
thus
ye
may
do
of
all
other
semblable
.
☞
The
seconde
rule
of
hole
nombers
with
numbers
broken
semblable
.
¶
yf
10
elles
and
2
thyrde
partes
of
cloth
cost
35
franc
.
howe
muche
shall
coste
14
elles
by
the
pryce
.
Answer
.
For
to
knowe
this
rule
and
other
semblable
:
it
behoueth
you
to
reduce
the
elles
bought
,
and
them
that
ye
wyll
bye
all
into
thyrdes
bycause
of
them
that
be
bought
,
in
sayenge
thus
,
3
tymes
10
ben
30
,
and
set
thereto
2
thyrdes
,
that
is
than
32
thyrdes
.
Then
it
behoueth
you
to
make
deuysion
by
32
,
and
than
ye
shall
reduce
the
14
,
elles
in
to
1
thyrde
,
in
sayenge
3
tymes
14
bene
42.
Then
42
shall
be
the
multiplicator
.
Nowe
sette
the
somme
,
that
is
to
wyte
35
franc
.
the
whiche
multiplyed
by
42
be
1470
the
whiche
dyuyded
by
32
thereof
cometh
45
franc
.
and
an
halfe
,
and
there
resteth
14
fran
,
the
which
ye
shal
reduce
to
shyllynges
,
and
than
deuyde
theym
by
32
and
therof
commeth
8
shyllynges
,
and
an
halfe
,
there
resteth
8
shyllynges
,
and
than
shall
ye
make
them
in
pens
,
and
dyuyde
them
by
32
,
and
therof
cometh
3
pens
,
therfore
ye
maye
aunswere
that
the
14
elles
of
clothe
shall
cost
45
francz
and
an
halfe
8
shyllynges
and
an
halfe
and
3
pens
.
¶
For
to
make
the
proue
it
behoueth
you
to
make
your
worke
by
the
contrarye
,
for
it
behoueth
you
to
multyplye
the
somme
that
the
14
elles
cost
by
the
deuisor
,
and
dyuyde
it
by
the
multiplycatour
.
Therefore
sette
the
somme
vppon
the
lyfte
syde
,
and
fyrste
multiplye
the
3
d̔
,
by
32
,
&
whan
they
be
multiplyed
ye
shal
make
of
them
shyllynges
,
and
then
ye
shall
multyplye
the
8
sꝪ
.
and
the
halfe
by
32
,
and
then
make
therof
francz
.
And
then
ye
shall
multyply
the
45
francz
and
the
halfe
by
32
,
and
deuyde
them
by
42
,
and
so
ye
shall
knowe
yf
the
rule
be
well
made
.
¶
The
thyrde
rule
of
hole
numbers
with
dyuers
minutes
.
¶
yf
4
elles
and
2
thyrdes
of
clothe
coste
10
crones
,
how
moch
shall
coste
6
elles
&
2
quarters
by
the
pryce
,
For
to
know
this
rule
,
it
behoueth
you
fyrst
to
reduce
the
4
elles
and
2
thyrdes
thus
,
3
tymes
4
ben
.
12
And
than
ye
shall
adioyne
the
2
thyrdes
,
&
than
it
is
14.
And
than
the
elles
that
ye
wyll
bye
,
ye
shall
reduce
theym
in
to
one
fourthe
thus
,
4
tymes
6
ben
24.
And
then
set
the
2
quarters
therto
and
than
there
is
26
quarters
.
And
than
ye
shall
multiplye
that
one
by
that
other
,
that
is
to
wytte
the
nombrant
of
the
fyrste
by
the
denominant
of
the
seconde
,
in
sayenge
4
tymes
14
ben
56.
And
those
56
shal
be
the
deuysor
.
Than
multiply
the
numbrant
of
the
seconde
,
by
the
denominant
of
the
fyrste
in
sayenge
3
tymes
26
ben
7●
,
and
those
78
shall
be
the
multiplicator
.
And
therfore
set
10
crones
and
multiplye
them
by
78
,
and
deuide
them
by
57.
And
ye
shall
fynde
that
the
6
elles
and
2
quarters
cost
11
crones
and
an
holfe
,
15
shyllynges
&
5
pens
.
And
there
resteth
8.
The
example
.
Deuisor
|
|
|
M
|
59
|
142
|
242
|
78
|
10
crones
|
246
|
2
|
|
|
3
|
4
|
|
¶
If
4
elles
⅔
coste
10
crones
,
6
elles
2/4
shall
coste
13
crones
and
an
halfe
15
shyllynges
5
pens
,
there
resteth
8.
For
to
make
the
proue
it
behoueth
you
to
worke
the
contrary
,
for
it
behoueth
you
to
multiplye
the
somme
by
the
diuisour
,
that
is
to
wyte
,
by
59
,
and
make
diuision
by
the
multiplicatour
,
that
is
to
wyte
,
by
78
and
ye
shall
fynde
10
,
otherwise
yf
there
be
more
or
lesse
the
rules
he
false
.
¶
The
fourth
rule
conteining
hole
nombers
to
the
marchaundyse
that
ye
haue
bought
and
minutes
to
the
same
that
ye
wyll
by
.
¶
yf
8
elles
of
cloth
cost
15
crownes
,
how
much
shal
cost
two
quarters
by
the
price
.
For
to
know
this
rule
ye
must
reduce
the
8
elles
into
quarters
,
in
sayenge
4
tymes
8
ben
32
then
32
shall
be
the
deuysor
,
and
the
2
quarters
shal
be
the
multiplycator
.
Now
set
the
15
crones
and
multyply
them
by
2
quarters
,
and
diuide
by
32
and
ye
shall
fynde
that
the
2
quarters
coste
0
crones
and
the
halfe
15
shyllynges
and
an
halfe
3
pens
.
For
to
make
the
proue
ye
must
worke
the
contrary
,
for
ye
shall
multiply
the
somme
that
the
2
quarters
coste
,
that
is
to
wytte
,
0
crones
,
and
the
halfe
15
sꝪ
.
and
an
halfe
3
pens
by
32
and
dyuyde
them
by
2.
¶
The
rule
of
rounde
measures
,
that
is
to
wyte
,
measure
of
corne
of
wyne
and
oyle
.
¶
Fyrste
it
behoueth
you
to
presuppose
and
knowe
the
measures
of
corne
.
¶
One
muy
is
worth
12
septiers
¶
One
septiers
is
worth
4
mynotes
.
¶
One
mynot
is
worth
3
busshelles
.
¶
One
busshell
is
worthe
4
quarters
,
¶
The
measures
of
wyne
.
¶
One
muy
of
wyne
holdeth
36
septyers
¶
The
septyer
holdeth
4
quarters
.
¶
The
quarte
holdeth
2
pyntes
.
¶
The
pynt
holdeth
2
choppynes
.
¶
The
choppyne
2
halfe
septiers
.
¶
The
halfe
septer
2
possions
.
¶
The
fyrst
rule
.
¶
If
the
muy
of
corne
cost
10
francz
,
how
moche
is
worth
the
busshel
.
Answere
.
For
to
knowe
this
rule
ye
muste
knowe
how
many
busshels
ben
in
1
muy
,
Therefore
multiply
the
muy
by
12
,
and
than
by
4
,
&
than
by
3
,
whiche
ben
144
busshelles
,
the
whiche
shall
be
the
deuysor
of
10
francz
therfore
deuyde
10
by
144.
And
therof
commeth
1
sꝪ
4
,
d
and
an
halfe
,
resteth
24.
d
Therfore
the
bushell
costeth
1
sꝪ
.
4
pens
&
an
halfe
,
resteth
24
d
¶
The
seconde
rule
.
¶
To
the
contrary
,
yf
the
busshell
cost
1
sꝪ
.
how
muche
shall
coste
a
thousand
and
4
hondreth
Muys
by
the
pryce
.
Answere
,
For
to
knowe
this
rule
,
it
behoueth
you
to
make
all
the
Muys
in
busshels
.
And
there
be
201600
bushels
,
the
whiche
it
behoueth
you
to
multiplpe
by
2
,
and
there
be
403200
,
and
of
them
ye
shall
make
crones
.
Therfore
deuyde
by
36
,
and
there
ben
1●20●
crones
.
Therfore
ye
maye
aunswere
that
yf
the
bushell
cost
2
sꝪ
.
a
thousaunde
and
4
hondreth
muys
shall
cost
enleuen
thousand
and
2
hundreth
crones
,
and
thus
ye
may
do
of
all
other
semblable
.
¶
The
thyrde
rule
.
¶
yf
the
septier
of
corne
be
worth
1
francz
&
the
lote
of
peny
torneys
weyght
12
ounces
,
how
mych
ought
it
to
weygh
whan
the
septer
is
worth
15
tornoys
.
Answere
.
Multiply
the
fyrst
nomber
by
the
seconde
,
that
is
to
wytte
,
20
by
12
,
and
deuyde
it
by
15
,
&
ye
shall
fynd
that
it
ought
to
wegh
16
ounces
.
And
thus
ye
may
do
of
al
other
lyke
.
¶
If
the
muy
of
wyne
be
worth
12
francz
howe
muche
ought
the
pynte
to
be
worth
Answere
.
For
to
knowe
this
questyon
,
it
behoueth
you
to
reduce
the
12
muys
into
septyers
,
from
septiers
into
quartes
,
and
from
quarters
into
pyntes
,
and
that
ben
188
pyntes
.
And
than
ye
shall
reduce
the
12
francz
in
to
sꝪ
.
that
ben
240
,
and
that
into
pens
,
that
bene
2880
pens
,
the
whiche
behonethe
you
to
deuyde
by
288
&
it
cometh
to
10
d
Therfore
yf
the
muye
of
wyne
cost
12
francz
,
the
pynte
is
worthe
10
d
But
it
is
requisite
that
the
Tauerner
haue
some
gaynes
yf
ye
sell
12
d
the
pynt
.
I
demaunde
howe
moche
shall
he
wynne
vpon
the
muy
:
Answer
.
He
selleth
it
2
d
more
than
it
is
worth
therefore
multyply
288
pyntes
by
2
and
they
be
576
,
the
whiche
ye
may
deuyde
by
12
and
ther
shal
be
48
sꝪ
.
Therfore
may
ye
answere
that
he
getteth
48
sꝪ
.
vpon
the
muy
.
❧
yf
the
muy
cost
10
francz
how
mych
is
worth
the
pynt
.
Answere
.
It
behouethe
you
to
do
as
is
aboue
sayde
:
and
ye
shal
fynd
that
it
is
worth
8
pens
and
1
thyrde
.
¶
yf
the
pynt
cost
6
pens
,
how
much
shal
cost
12
muys
by
the
pryce
.
Aunswere
.
It
behoueth
you
to
know
how
many
pyntes
ben
in
a
muy
,
that
is
288
,
multyplye
12
muys
by
288
that
is
3456
,
pyntes
.
And
than
multiply
the
pyntes
by
6
that
bene
207
,
6
,
of
whome
ye
shall
make
sꝪ
by
diuysion
.
and
there
ben
10728
sz
,
and
of
shyllynges
ye
shall
make
francz
.
Therefore
ye
shall
make
diuysyon
by
26
,
ye
shall
fynde
86
francz
8
sꝪ
.
Therefore
ye
maye
aunswere
that
the
12
muys
shall
coste
86
francz
8
sꝪ
.
¶
In
so
myche
as
competently
we
haue
tracted
of
the
rule
of
thre
in
the
fayct
of
measures
,
it
is
expediente
that
we
tracte
therof
in
the
faycte
of
weyghte
.
¶
yf
an
hondreth
poundes
of
peper
cost
20
sꝪ
.
how
muche
shall
cost
6
pound
by
the
price
:
Answere
.
For
to
knowe
this
question
,
ye
must
multyply
by
the
contrarye
and
deuyde
by
the
semblaunt
,
that
is
to
wyte
,
multiplye
by
6
and
dyuyde
by
10
,
and
ye
shal
fynde
that
the
6
poundes
shal
cost
1
francz
,
and
4
sꝪ
.
To
make
the
proue
ye
must
multiplye
by
100
and
dyuyde
by
6.
Now
I
demaunde
of
the
6
poundes
cost
1
franc
4
sz
,
how
moche
is
worth
the
ounce
.
For
to
knowe
this
ye
shal
make
the
poundes
in
ounces
,
the
whiche
ben
96
ounces
,
&
then
make
the
money
in
pens
,
the
which
ben
288
d
the
whiche
ye
shall
deuyde
by
96
and
therof
commeth
3
pens
,
therefore
the
ounce
shall
coste
3
pens
.
¶
If
one
li
of
saffron
cost
3
francz
and
an
halfe
,
how
muche
is
worth
the
ounce
.
Answere
.
It
behoueth
you
to
knowe
that
in
a
pounde
ben
16
ounces
,
therfore
deuyde
the
3
francz
,
and
the
half
by
16
and
ye
shal
fynde
that
the
ounce
is
worth
4
sꝪ
.
4
d
and
an
halfe
,
&
thus
ye
maye
do
of
other
lyke
.
¶
If
4
pounde
of
saffron
cost
16
fran
.
6
sꝪ
8
d
howe
much
shall
cost
3
quartrones
by
the
pryce
.
For
to
knowe
this
rule
,
ye
shall
reduce
the
4
li.
in
thyrdes
and
shall
saye
●
tymes
4
ben
12
,
and
1
thyrde
ben
14
then
ye
shall
multiply
by
4
,
and
shall
say
4
tymes
14
ben
56
the
deuysor
,
than
for
the
seconde
number
we
shall
say
,
3
tymes
3
ben
9
fourthes
or
quarters
,
the
whiche
9
shall
be
the
multiplyer
.
Now
set
the
16
francz
6
sꝪ
8
d
tournoys
,
&
multiplye
them
by
6
and
deuyde
them
by
16
,
and
therof
cometh
2
fran
.
and
an
halfe
,
2
sꝪ
.
6
d
therfore
ye
may
answere
that
the
3
quarters
shal
cost
2
fran
.
&
an
halfe
2
sꝪ
.
6
d
For
to
make
the
proue
ye
must
worke
by
the
contrary
in
multyplyenge
by
the
diuysor
,
that
is
to
wyt
by
56
and
make
diuysyon
by
9
,
and
so
maye
ye
do
of
other
semblable
.
¶
If
one
pound
of
tyn
cost
9
blances
,
how
many
hundreth
shall
I
haue
for
a
thousand
and
4
hundreth
francz
.
It
behoueth
you
to
knowe
how
moch
is
worth
the
hunderth
by
9
blances
the
pound
.
And
ye
shal
fynde
that
there
is
12
francz
and
an
halfe
.
Now
make
diuysion
of
1400
frances
by
12
frances
and
an
halfe
,
ye
shall
fynde
112.
Therfore
ye
maye
saye
that
I
shall
haue
112
pound
of
tynne
for
1400
frances
,
And
also
as
we
haue
made
this
rule
,
ye
maye
do
in
all
other
marchandyses
,
as
in
lead
,
yron
,
spyces
peper
,
suger
.
And
as
we
haue
done
of
poundes
ye
maye
do
of
quartrons
,
ounces
,
&
al
other
weyghtes
,
¶
A
rule
whiche
is
without
tyme
THre
marchauntes
put
theyr
monye
togyther
for
to
haue
gaynes
,
the
whiche
haue
boughte
suche
marchaundyse
as
hath
cost
125
francz
,
whereof
the
fyrste
hath
laide
15
francz
.
The
seconde
64
sꝪ
.
and
the
thyrde
36
,
fꝪ
.
And
they
haue
goten
54
franc
.
of
clere
gaynes
.
I
demaund
how
shall
they
deuide
it
,
so
that
eche
man
haue
gaines
accordynge
to
the
moneye
that
he
hath
layd
downe
.
Answer
.
In
all
suche
rules
and
questions
ye
shall
multiplye
eche
one
after
the
money
that
he
had
layde
,
therfore
multiplye
the
gaines
for
the
fyrst
by
25
and
deuide
by
125
,
that
is
the
diuisor
commune
.
For
the
seconde
multiplye
the
gaynes
by
64
,
and
dyuyde
by
125
the
diuisor
commune
.
And
for
the
thirde
multiply
the
gaines
by
36
,
and
dyuyde
125
the
deuysor
commune
.
And
for
to
fynd
the
deuysor
commune
,
ye
shal
set
togither
the
multiplycatours
,
that
is
to
wyte
25
,
64
and
36
which
is
125
the
deuysor
commune
.
And
so
shal
ye
do
in
al
rules
of
company
Nowe
ye
maye
fynde
&
knowe
how
moch
eche
one
hath
of
gaines
,
and
ye
may
se
it
by
the
ensaumple
here
present
.
The
fyrste
hathe
10
fꝪ
and
halfe
2
sꝪ
.
The
seconde
hathe
27
fꝪ
&
halfe
2
sꝪ
.
and
halfe
5
d
&
halfe
,
resteth
2
d
and
halfe
,
The
third
hath
15
fꝪ
and
halfe
5
sꝪ
.
resteth
60
pens
.
256436
|
125
|
Multiplicatour
|
Dyuysour
.
|
And
they
haue
yet
to
be
deuyded
among
them
of
restes
62
d
and
and
halfe
.
¶
For
to
make
the
proue
it
behoueth
you
to
diuyde
the
restes
,
and
than
reduce
all
togithers
,
and
ye
shall
fynde
the
somme
diuided
,
for
all
the
rules
of
company
ben
proued
by
addicion
of
sommes
.
¶
The
seconde
rule
of
hole
tyme
.
¶
Foure
marchauntes
laye
money
together
for
wynninge
for
a
certaine
tyme
,
of
whome
the
fyrste
hath
laide
10
fꝪ
.
for
two
yere
.
The
second
20
fran
.
for
3
yere
.
The
thyrde
100
francz
for
one
yere
.
And
the
fourth
hath
laide
40
franc
.
for
4
yere
:
and
they
haue
gayned
454
fran
.
I
demaunde
how
moch
eche
one
ought
to
haue
of
winnynge
after
the
money
that
he
hath
layd
,
&
after
the
tyme
that
he
hath
holden
his
money
in
gayne
for
company
.
Answere
.
For
to
knowe
this
rule
and
other
semblable
,
ye
shall
multiplye
the
monye
that
eche
one
hathe
layde
by
the
tyme
that
he
hath
holden
it
in
companye
.
Example
.
The
fyrste
hath
layde
10
fran
.
for
2
yere
,
therfore
it
behoueth
you
to
multyplye
10
by
2
in
sayenge
2
tymes
10
ben
20.
For
the
seconde
3
tymes
20
ben
60.
For
the
thyrde
1
tyme
100
is
an
100.
For
the
fourth
4
tymes
40
ben
160.
&
then
it
behoueth
you
to
fynd
a
dyuysor
commune
,
for
eche
hath
his
multiplycator
,
that
is
to
wytte
,
the
same
that
he
hath
layde
,
and
for
to
fynde
it
ye
shall
sette
together
all
the
multiplycatours
,
that
is
to
wytte
the
20
,
60
,
100
,
160
the
whiche
maketh
340
,
therfore
these
340
shall
be
the
dyuysor
commune
to
all
,
thenne
howe
muche
eche
one
oughte
to
haue
ye
maye
se
by
the
ensample
here
folowynge
454
sꝪ
.
The
fyrste
hath
26
francz
and
halfe
4
sꝪ
.
one
peny
,
resteth
140.
d̔
,
The
seconde
hath
80
frances
2
sꝪ
4
d
reste
80
pens
The
thyrde
hath
134
frances
1
sꝪ
.
5
pens
,
reste
20
pens
.
The
fourth
hath
213
franc
.
and
an
halfe
2
sꝪ
and
a
halfe
5
d
rest
.
100
pens
.
20
,
60
,
100
,
160
,
|
340
|
Multiplycatours
.
|
Dyuysour
.
|
Of
rest
they
haue
to
deuide
one
peny
.
¶
The
rule
of
company
where
as
is
hole
tyme
and
partes
of
tyme
.
THre
marchauntes
lay
money
in
company
for
to
haue
gaynes
thereby
,
of
whome
the
fyrste
hathe
layde
30
frances
for
two
yeres
.
The
seconde
hath
layde
400
fran
.
for
one
yere
&
thre
monethes
.
And
the
thyrde
hath
layde
60
fran
.
for
thre
yeres
&
two
monethes
.
And
they
haue
gayned
with
this
money
44
franc
.
I
demaund
howe
they
shal
deuyde
it
to
the
ende
,
that
eche
one
haue
his
right
after
the
money
and
the
tyme
that
they
haue
sette
and
holden
for
to
gayne
.
Answer
.
For
this
rule
&
all
other
semblable
,
ye
shall
multiply
the
tyme
by
the
money
,
as
we
haue
saide
aboue
but
for
as
much
as
there
be
monethes
ye
must
set
&
reduce
all
the
tyme
of
eche
one
in
monethes
,
and
also
yf
there
were
anye
dayes
ye
shulde
set
all
the
tyme
in
dayes
.
The
fyrste
hathe
layde
30
frances
for
2
yeres
,
in
2
yeres
ben
24
monethes
,
therefore
multyplye
30
by
24
there
bene
720
,
and
these
720
shal
be
the
multyplycatoure
of
the
fyrste
.
The
seconde
hathe
layde
40
fran
.
for
1
yere
and
3
monethes
,
in
one
yere
ben
12
monethes
,
and
3
dothe
make
15
monethes
,
multyplye
40
by
15
,
they
make
600
whiche
is
the
multiplycatoure
of
the
seconde
.
The
thyrde
hathe
layde
60
francz
for
3
yeres
and
2
monethes
,
3
yeres
ben
worth
36
monethes
and
2
bene
38
monethes
.
Nowe
multyplye
60
by
38
,
and
there
ben
2280
,
which
shall
be
the
multiplicatour
of
the
thyrd
.
Now
for
to
haue
a
dyuysor
comune
,
ye
shall
set
togyder
all
the
multiplycatours
that
is
3600
the
dyuysour
comune
.
They
haue
to
deuyde
44
francz
.
The
fyrste
hath
8
francz
,
and
halfe
6
sꝪ
rest
●
.
The
seconde
hath
7
frances
.
6
sꝪ
.
and
half
.
rest
0.
The
thyrde
hath
27
francz
and
halfe
7
sꝪ
.
4
pens
,
rest
0.
720
,
600
,
2280
,
,
|
360●
.
|
Multiplycatours
|
Dyuysour
|
¶
A
rule
of
dyuers
syluer
and
dyuers
tyme
.
THre
marcaūhtes
haue
made
companye
togyder
of
whom
the
fyrst
hath
layd
10
francz
.
4
shyllynges
for
2
monethes
.
The
seconde
hath
layd
15
fran
.
for
one
yere
.
And
the
thyrde
hath
layde
6
francz
7
sꝪ
for
8
monethes
,
and
they
haue
goten
of
this
money
24
francz
.
How
they
shall
dyuyde
it
after
the
money
and
after
this
tyme
I
demaunde
.
Answere
.
For
to
knowe
this
rule
and
all
other
semblable
it
behoueth
you
to
reduce
the
money
of
euery
man
in
shyllynges
.
And
all
the
tyme
in
monethes
.
And
then
multiplye
the
money
by
the
tyme
.
Ensample
.
The
fyrste
hath
layde
10
francz
that
ben
200
sꝪ
.
and
4
ben
204
the
whyche
ye
shall
multyplye
by
2
monethes
,
and
they
shall
be
408
the
multiplicatour
of
the
fyrste
.
The
seconde
hath
layd
15
francz
for
one
yere
,
and
in
15
francz
ben
300
sꝪ
.
and
in
one
yere
ben
12
monethes
,
therefore
multiplye
300
by
12
,
and
there
shall
be
3600
the
multyplyecatour
of
the
seconde
.
The
thyrde
hathe
layde
6
francz
7
shyllynges
,
and
in
6
fran
.
ben
120
sꝪ
.
and
7
ben
127
sꝪ
.
for
8
monethes
,
therfore
multiply
127
by
8
,
and
they
shall
be
1016
the
multiplicator
of
the
thyrd
.
And
for
to
haue
the
deuysor
comune
,
ye
muste
reduce
togyther
all
the
multiplicators
,
&
that
shall
be
the
deuysor
commune
,
as
ye
may
se
by
the
example
folowynge
.
They
haue
24
frances
of
wynnynge
.
The
fyrst
hath
2
francz
and
halfe
8
sꝪ
.
and
halfe
5
pens
and
halfe
restethe
1360
pens
.
The
seconde
shall
haue
17
francz
,
3
sꝪ
.
and
halfe
d
resteth
1952
pens
.
The
thyrde
shall
haue
4
sꝪ
.
&
halfe
,
7
sz
,
0
pens
,
&
halfe
resteth
17112
pens
.
408
,
3600
,
1016
|
5024.
|
Multiplicatours
.
|
Dyuysor
.
|
And
they
haue
to
deuyde
1
d
of
the
restes
For
to
make
the
proue
ye
shall
reduce
togyther
the
thre
sommes
that
they
haue
had
.
And
yf
there
be
more
or
lesse
the
rule
is
euyll
made
.
¶
Here
foloweth
the
rule
of
company
of
factours
with
marchauntes
seruauntes
.
OF
this
rule
of
factours
ye
may
make
3
rules
in
maner
of
questyons
that
fall
amonge
marchauntes
.
Example
,
8
marchauntes
5
factours
,
and
3
seruauntes
or
varlettes
haue
made
companye
togyther
,
and
haue
clerely
gotten
150
fran
.
wherof
the
factours
oughte
to
haue
the
halfe
of
the
marchauntes
,
and
the
seruauntes
the
thyrde
parte
of
the
factours
,
how
shall
they
dyuyde
these
150
franc
.
Answer
For
all
suche
rules
and
questions
it
behoueth
you
to
fynde
a
nomber
wherin
is
an
halfe
and
a
thyrde
,
and
that
shal
be
6
,
and
these
6
shal
be
for
the
marchaunt
.
And
the
halfe
of
6
ben
3
,
that
shall
be
for
the
factours
,
and
the
thyrd
part
of
the
factours
is
1
whiche
shall
be
for
the
seruauntes
.
And
than
ye
shall
multyplye
the
one
by
the
other
,
that
is
to
wytte
,
the
personages
by
theyr
nomber
,
6
tymes
8
ben
48
,
and
these
48
shall
be
the
multiplicator
of
the
marchauntes
.
And
than
there
ben
5
factours
,
that
haue
3
and
3
tymes
5
ben
15
,
and
than
there
ben
3
seruauntes
that
haue
1
,
and
1
tymes
3
is
3
,
&
therfore
the
factours
shall
ye
multiplye
by
15
and
the
seruauntes
by
3.
Nowe
for
to
fynd
the
dyuysor
comune
ye
shall
set
together
all
the
multyplycatours
,
that
is
to
wyte
,
48
,
15
,
3
,
whiche
bene
66
these
66
shall
be
the
dyuysor
commune
.
Example
they
haue
to
dyuyde
150
francz
.
The
marchauntes
haue
109
franc
.
1
sꝪ
.
and
halfe
3
d̔
,
and
halfe
,
resteth
21
d
The
factours
haue
34
fran
.
1
sꝪ
.
and
halfe
3
d
and
halfe
,
resteth
21
d
The
seruauntes
haue
6
fran
.
and
halfe
6
sꝪ
.
4
d
rest
24
pens
.
48
,
15
,
3
|
66
|
Multyplycatours
|
Dyuysor
.
|
¶
They
haue
to
deuyde
1
peny
of
restes
For
to
make
the
proue
ye
shall
dyuyde
al
the
restes
by
the
dyuysor
comune
.
And
than
ye
shall
reduce
all
togither
,
for
to
haue
150
frances
.
¶
The
rule
of
factours
the
which
gate
the
halfe
of
the
gayne
and
of
the
princypall
.
¶
And
other
rule
in
maner
of
a
question
a
marchaunt
hath
gyuen
50
franc
.
to
his
factor
by
suche
couenant
that
he
gouerne
them
for
10
yeres
.
And
at
the
ende
of
the
tyme
,
that
is
to
wyte
,
at
the
ende
of
10
yeres
.
And
at
the
ende
of
10
yeres
,
they
shall
dyuyde
the
gaine
and
the
principall
It
hapneth
that
the
factour
wyll
go
his
waye
at
the
ende
of
6
yeres
,
and
he
fyndeth
that
he
hathe
gayned
a
thousande
francz
.
I
demaunde
how
ought
the
sayde
factour
to
be
paied
,
and
how
moch
ought
the
said
marchaunte
to
haue
:
Answere
.
ye
ought
to
regarde
howe
moche
he
shold
haue
gained
in
those
10
yeres
that
he
sholde
haue
holden
them
in
gayne
as
he
had
promysed
.
Therfore
ye
maye
forme
the
questyon
,
yf
6
haue
gotten
a
thousande
:
howe
muche
shall
be
the
gaynes
of
10.
Multyplye
1000
by
10
and
dyuyde
by
6
and
ye
shall
fynde
that
he
shoulde
haue
gutten
1666
fran
,
and
an
halfe
3
sꝪ
.
2
pens
.
Of
the
whiche
gaynes
the
marchaunt
ought
to
haue
the
halfe
that
ben
833
francz
.
6
shyllynges
and
halfe
and
1
peny
.
And
than
take
vp
those
833
fran
.
6
shylynges
and
halfe
1
penye
of
1000
francz
that
he
hath
gayned
,
and
there
remayneth
166
frances
,
13
shyllynges
5
pens
for
the
factour
.
Nowe
ye
maye
aunswere
that
the
marchaunt
shall
haue
of
the
gaynes
833
francz
6
shyllynges
,
and
halfe
18.
And
the
halfe
of
the
princypall
,
that
is
to
wytte
of
50
,
that
is
25
and
there
ben
852
francz
.
6
sꝪ
.
and
halfe
1
d
And
the
factoure
shall
haue
of
gaine
166
francz
.
13
sꝪ
.
5
pens
.
And
of
the
princypall
25
that
ben
191
fran
.
13
sꝪ
5
d
And
thus
maye
ye
do
of
all
other
semblable
.
And
it
is
proued
by
the
reductyon
of
the
two
sommes
gayned
.
☞
The
thyrd
rule
of
factours
with
couenauntes
,
that
the
factour
shall
gayne
the
halfe
of
the
princypall
.
¶
An
other
rule
of
company
of
factours
&
marchauntes
with
conuenaunt
that
the
factours
shall
gaine
the
halfe
of
the
princypall
and
not
of
the
gayne
.
Example
.
A
marchaunt
gyueth
vnto
his
factoure
400
fran
.
that
he
shall
gouerne
them
for
6
yeres
,
&
at
the
ende
of
the
tyme
the
halfe
of
the
princypal
shal
be
to
the
factour
.
It
happeneth
the
factour
wyll
go
his
waye
at
the
ende
of
2
yeres
,
&
hath
gayned
200
fran
.
I
demaunde
how
ought
the
factour
to
be
payed
.
Answere
.
ye
ought
to
regard
howe
moche
he
sholde
haue
gayned
yf
he
had
serued
all
his
tyme
,
and
for
to
fynd
it
ye
maye
worke
by
the
rule
of
thre
,
for
ye
must
multiplye
by
his
contrarye
,
that
is
to
wytte
by
6
,
and
diuyde
by
his
semblaunt
that
is
to
knowe
,
by
2
,
in
saienge
yf
2
haue
gayned
200
frances
:
howe
muche
shall
9
gayne
,
and
ye
shall
fynde
that
he
should
haue
gotten
900
franc
,
and
he
gayned
but
200
franc
.
wherefore
he
oughte
to
make
a
gaine
400
fran
.
to
the
marchaunt
:
and
he
ought
to
haue
the
half
of
the
principal
,
that
ben
200
frances
,
therfore
he
oweth
200
vnto
the
marchaunte
,
and
so
he
hath
loste
all
his
tyme
,
and
200
fran
.
of
aduauntage
for
the
marchaunt
ought
nothyng
to
lose
lyke
as
he
had
accomplyshed
all
his
time
¶
The
thyrde
rule
of
chaunges
for
to
vse
deceyte
or
fraude
.
¶
Two
marchauntes
wyl
chaunge
theyr
marchaundyse
,
&
the
one
begyled
the
other
the
one
hath
peper
,
and
that
other
cloth
.
He
that
hath
peper
wyll
sell
for
25
franc
.
the
hondereth
by
chaunge
,
whyche
is
no
more
worthe
than
20
fran
.
in
syluer
contented
.
I
demaunde
for
how
moch
ought
the
other
to
sell
vnto
him
the
elle
of
hys
cloth
,
that
is
worth
but
15
sꝪ
.
to
kepe
hym selfe
from
losse
.
Answere
.
For
the
rule
of
thre
ye
maye
saye
thus
,
yf
20
frances
of
content
gyue
me
25
fran
.
at
the
chaunge
howe
muche
shall
gyue
me
15
of
contente
.
It
behoueth
you
to
multiplye
the
25
by
15.
whiche
ben
373
,
the
whiche
ye
shall
dyuyde
by
20
and
therof
commeth
18
sꝪ
.
9
d
therfore
ye
maye
say
that
he
shall
sell
the
elle
of
clothe
for
18
shyllynges
9
d
And
thus
may
ye
do
of
all
other
.
¶
Two
marchauntes
wyll
chaunge
theyr
marchaundyse
,
of
whom
that
one
hath
100
pounde
of
wolle
,
that
is
no
more
worthe
but
15
crones
.
And
he
wyll
chaunge
wyth
an
other
in
a
pyce
of
cloth
that
is
worthe
21
crones
,
and
he
wyl
gyue
hym
the
wolle
for
17
crones
.
I
demaunde
for
how
moche
ought
the
other
to
sell
the
peyce
of
clothe
to
the
ende
that
he
be
not
betromped
●
Answer
.
By
the
rule
of
thre
whan
15
are
worth
17
demaunde
howe
moche
shall
be
worth
21
Dyuyde
by
15
and
ye
shall
fynde
the
same
that
ye
requyre
.
¶
Two
marchantes
wyl
chaunge
theyr
marchaundyse
,
and
the
one
defraude
that
other
that
hath
peper
,
and
wyll
sell
it
24
fran
.
the
hondreth
by
chaunge
.
whiche
is
no
more
worth
but
20
frances
in
moneye
content
,
and
he
wyll
haue
the
halfe
in
money
contente
.
I
demaunde
for
howe
mych
ought
the
other
to
sel
the
elle
of
his
cloth
that
is
no
more
worth
but
15
sꝪ
.
Answere
,
ye
must
take
awaye
the
money
content
that
the
other
demaundeth
,
that
ben
12
franc
.
for
the
iust
price
,
&
of
the
whiche
he
wyll
sell
ouer
.
Therefore
take
awaye
,
and
withdrawe
12
of
20
franc
.
which
is
the
iust
yrice
,
and
there
rest
8
fran
.
for
8
and
4
ben
12
,
And
ye
maye
saye
by
the
rule
of
thre
,
yf
8
gyue
me
12
,
what
shall
gyue
me
15
sꝪ
.
whiche
is
the
iuste
pryce
of
the
clothe
,
multyply
12
by
15
and
dyuyde
by
8
,
and
therof
cometh
22
sꝪ
.
6
d
And
therefore
the
marchant
ought
to
sell
the
elle
of
his
cloth
after
22
sz
,
6
d
els
he
shold
haue
losse
And
thus
ye
ought
to
do
of
al
maners
of
chaunges
and
barathes
,
for
yf
he
that
hath
the
peper
,
demaunded
but
the
thyrde
or
the
fourth
or
2
or
3
,
abate
all
onely
the
same
that
he
shal
demaunde
,
and
then
by
the
rule
,
as
is
said
.
And
note
ye
well
that
yf
he
wil
multiplye
shyllynges
,
ye
shall
haue
shyllynges
.
And
of
crones
ye
shall
haue
crones
,
and
of
frances
ye
shall
haue
frances
And
in
lyke
maner
of
all
other
.
¶
Here
foloweth
many
rules
&
questions
to
haue
the
more
knowledge
of
the
science
of
arismetryke
,
and
the
fyrste
is
of
collectes
and
tallyages
.
TEnne
men
owe
vnto
the
kynge
of
collecte
and
tallyage
244
fran
.
I
demaunde
howe
shall
they
diuyde
them
to
the
ende
that
eche
one
paye
after
the
valour
of
his
goodes
,
for
it
is
reason
that
more
be
payed
by
the
riche
then
by
the
poure
.
For
he
that
is
more
endowed
with
goodes
is
more
holden
vnto
god
and
to
the
prince
.
Answer
.
It
behoueth
to
know
how
muche
eche
one
is
worth
in
his
goodes
,
and
in
his
possessions
.
-
The
fyrste
is
wourth
100
francz
-
The
seconde
is
wourth
400
franc
.
-
The
thyrde
is
wourth
154
franc
.
-
The
fourth
is
wourth
1000
franc
.
-
The
fyfte
is
wourth
1150
franc
.
-
The
syxte
is
wourth
40
franc
.
-
The
seuenth
is
wourth
440
franc
.
-
The
eyght
is
wourth
80
franc
.
-
The
nynth
is
wourth
600
franc
.
-
The
tenth
is
wourth
360
francz
.
Nowe
it
behoueth
you
to
fynde
the
multyplycator
and
the
deuisor
.
The
multyplycator
shal
be
eche
one
by
hym self
,
and
so
for
the
fyrste
it
behoueth
you
to
multyplye
by
100
,
for
the
seconde
by
400.
for
the
thyrde
by
154
,
and
so
must
ye
do
of
the
other
:
And
for
to
fynde
the
diuisor
,
ye
shall
sette
togyther
all
the
multiplicatours
,
as
100
,
400
,
154
&c.
and
all
that
togither
shall
be
the
diuisor
commune
,
whiche
is
4464
Therfore
multyply
the
collecte
,
that
is
to
wytte
,
244
for
eche
one
his
valour
,
and
dyuyde
by
4464
,
or
by
the
halfe
that
is
2232
,
and
then
ye
shall
wryte
howe
moche
eche
one
ought
to
paye
.
Example
.
The
fyrste
sholde
paye
5
franc
.
9
shyllynges
3
d
and
halfe
reste
1464.
The
seconde
sholde
pay
12
franc
.
17
shyllynges
3
pens
,
re●●eth
1●92
.
The
thyrde
sholde
paye
8
frances
8
sꝪ
.
4
pens
resteth
660.
The
fourth
sholde
paye
54
frances
15
shyllynges
2
pens
,
resteth
1248.
The
fyfte
sholde
paye
62
frances
17
sꝪ
.
2
pens
resteth
96
The
syxte
sholde
paye
2
frances
3
sꝪ
.
and
halfe
2
pens
and
halfe
,
resteth
1032.
The
seuenth
shulde
paye
24
frances
1
shylling
0
pens
,
resteth
192
The
eyght
shoulde
paye
4
frances
7
sꝪ
.
5
pens
resteth
2064.
The
nynth
sholde
paye
23
frances
15
sꝪ
.
10
pens
and
halfe
,
resteth
2088
The
tenth
shoulde
paye
27
frances
6
sꝪ
.
7
pens
resteth
624
¶
And
they
haue
to
dyuyde
2
pens
and
halfe
of
restes
.
Then
whan
ye
haue
all
dyuyded
and
wryte
the
somme
and
the
restes
,
ye
shall
set
togyder
all
the
restes
,
and
diuide
them
by
the
deuysor
comune
,
or
by
the
halfe
.
And
yf
there
be
more
or
lesse
.
the
rule
is
not
well
made
,
for
the
remaynant
of
all
ought
to
be
dyuyded
by
the
diuisor
comune
.
And
the
proue
of
this
rule
is
reduccyon
.
And
marke
well
this
rule
for
it
is
right
good
vnto
the
countre
where
all
the
goodes
be
praysed
by
al
the
townes
and
castels
,
as
it
is
in
manye
places
of
Daulphyne
,
and
of
Prouence
¶
The
rule
of
the
mylnes
.
¶
One
man
hath
thre
mylnes
of
whom
one
gryndeth
eche
day
5
septyers
of
corne
and
the
other
gryndeth
7
and
the
thyrde
8.
There
commeth
a
marchaunt
that
wyll
haue
gronden
one
hundreth
septyers
of
corne
,
I
demaund
how
ought
the
mylner
to
dyuyde
the
corne
to
the
mylnes
to
the
ende
that
eche
one
haue
assone
done
as
an
other
.
Answere
.
For
to
know
this
questyon
and
rule
.
ye
must
fynde
the
dyuysor
and
the
multiplicator
,
the
multyp
.
shall
be
eche
one
by
him selfe
,
and
the
deuysor
shal
be
the
thre
multiplycatours
set
togyther
that
ben
20.
Therfore
yf
ye
wyll
knowe
howe
muche
corne
ought
to
be
layde
vppon
the
fyrste
mylne
,
ye
muste
multyplye
the
100
septiers
of
corne
by
5
&
deuyde
by
20
,
whiche
shall
be
25
septiers
,
that
shal
be
layde
vppon
the
fyrst
mylne
.
And
for
the
seconde
ye
shall
multiply
100
by
7
and
dyuyde
by
20
,
and
there
shall
be
35
septiers
,
the
whiche
ye
shall
putte
vppon
the
seconde
mylne
,
and
for
the
thyrde
ye
shall
multyplye
100
by
8
and
dyuyde
by
20
&
there
shal
be
40
septiers
,
whiche
ye
shall
put
vppon
the
thyrde
mylne
.
And
thus
may
ye
do
of
all
other
semblable
.
It
maye
be
made
otherwyse
,
set
togyther
the
sommes
that
the
thre
mylnes
grynd
that
is
20
,
and
by
the
rule
of
thre
ye
shall
say
,
yf
20
gyue
me
an
100
,
howe
moche
shall
gyue
me
5
or
7
or
8.
And
it
is
proued
by
addition
.
Example
.
100
The
fyrst
shall
haue
25
septiers
.
The
seconde
35
septiers
.
The
thyrde
40
septiers
,
7
,
5
,
8
,
|
20
|
Multiplicatours
.
|
Dyuysour
.
|
The
rule
and
questyon
of
a
shepeherde
or
pastour
.
FOure
men
haue
300
shepe
or
moutons
,
of
whome
the
fyrst
hath
an
100
shepe
/
the
seconde
40
,
the
thyrde
150
and
the
fourthe
10
And
they
gyue
vnto
a
shepherde
for
to
kepe
these
shepe
25
fꝪ
.
for
a
yere
.
I
demaunde
how
ought
the
one
to
paye
of
the
25
fran
.
after
the
shepe
that
he
hath
.
And
how
long
time
ought
eche
one
to
haue
hym
at
comense
or
meat
.
Answer
.
For
to
knowe
this
rule
and
al
other
semblable
,
it
behoueth
you
to
fynde
the
multyplycator
and
the
deuysor
,
the
multyplycator
of
the
fyrste
shall
be
100
,
of
the
seconde
40
,
of
the
thyrde
150
,
and
of
the
fourth
10
,
&
than
set
togyther
all
these
som̄es
the
whiche
ben
300
the
dyuysour
comune
.
Or
ye
may
make
it
by
the
rule
of
thre
in
sayeng
,
yf
300
gyue
me
25
,
howe
moche
shall
gyue
me
100
or
40
or
150
or
10
,
&
alwayes
dyuyde
by
300
and
thus
of
all
other
rules
.
¶
Ensample
of
the
fyrst
.
And
for
to
know
in
howe
many
houres
this
vessell
shall
voyde
,
ye
shall
set
togyder
the
thre
nombers
,
that
is
to
wyt
,
1
,
2
,
5
which
ben
8
and
that
8
is
the
deuisor
therfore
dyuyde
60
by
8
,
and
ye
shall
fynd
that
in
7
houres
and
an
halfe
it
shal
be
emptye
And
thus
may
ye
do
of
al
other
semblable
¶
The
rule
and
question
of
zarasins
for
to
cast
them
within
the
see
.
THere
is
a
galle
vpon
the
see
wherein
be
thyrty
marchauntes
,
that
is
to
wit
15
crysten
men
,
and
15
sarazyns
,
ther
falleth
great
tempest
where
vpon
it
behoueth
them
to
cast
all
the
marchaundyse
in
to
the
see
,
and
yet
for
all
that
they
be
not
in
surete
from
perysshynge
,
for
the
galle
is
feble
and
weke
,
so
that
by
ordynaunce
made
by
the
patrone
,
it
is
necessary
that
there
be
caste
into
the
see
the
halfe
of
the
thyrty
machaūtes
,
but
the
sarazyns
wyll
not
be
cast
in
,
nor
also
the
christiens
:
then
by
an
apoyntment
made
,
they
shall
sette
them
downe
vpon
a
rowe
.
&
then
counte
them
vnto
9
and
he
that
sholde
fall
vpon
the
9
to
be
caste
into
the
see
,
how
wolde
ye
set
them
that
none
of
the
chrystyens
shold
be
caste
into
the
see
.
Answer
.
ye
shall
ordeyne
them
after
these
meters
folowyng
.
Post
.iiii.
quique
da
post
duos
vnum
colloca
Tres
numerabis
,
postea
vnum
collocabis
Vnum
dic
panther
,
&
duo
consequenter
,
Duos
post
ponas
&
,
iii.
siml
hic
apponas
Semel
dic
ann
bis
.
post
.ii.
vnum
terminabꝭ
Primi
christiani
,
sunt
saracenique
secundi
.
That
is
to
wytte
,
4
christiens
5
sarazins
2
christyens
1
sarazyn
,
3
christyens
1
sarazin
●
christien
2
sarasyns
,
2
christiens
3
sarazyns
,
1
chrystyen
2
sarazyns
,
2
christiens
1
sarazyn
.
Or
for
to
know
it
more
shortely
ye
may
worke
by
this
verse
folowynge
,
by
the
nomber
of
the
vouels
.
Populeam
virgam
matrem
regina
tenebat
¶
The
rule
and
question
of
a
testament
.
A
Man
hath
made
his
testament
,
the
which
hath
lefte
his
wyfe
great
,
and
hath
ordeyned
in
his
testame●t
that
yf
she
brought
forth
a
sone
,
he
shold
haue
two
partes
of
his
goodes
,
that
is
to
wyt
,
of
1200
crones
,
and
his
wyfe
the
other
part
,
and
yf
she
brought
forth
a
doughter
,
then
the
moder
shold
haue
two
partes
,
and
the
doughter
the
other
parte
.
It
happeneth
whan
the
man
is
dede
,
the
wyfe
bryngeth
forthe
a
sone
and
a
doughter
.
I
demande
how
shall
they
deuyde
the
1200
crones
.
Answere
.
ye
shall
set
1
for
the
doughter
,
and
3
for
the
mother
,
for
the
mother
ought
to
haue
two
partes
agaynst
the
doughter
,
and
set
4
for
the
sone
for
he
ought
to
haue
two
partes
agaynste
the
mother
.
Therfore
ye
shall
multiply
the
1200
crones
by
4
for
the
sone
,
by
2
for
the
mother
,
and
by
1
for
the
doughter
.
And
for
to
fynde
the
dyuisor
ye
shall
set
togyther
1
,
2
,
and
4
,
whyche
ben
7
,
therfore
dyuyde
by
7.
Example
.
-
4
The
sone
shall
haue
685
crones
&
an
halfe
,
7
sꝪ
.
8
d
&
halfe
,
resteth
a
halfe
d̔
-
2
The
mother
shall
haue
342
crones
,
&
and
halfe
12
shyllynges
and
halfe
,
4
d̔
Resteth
2
pens
.
-
2
The
doughter
shall
haue
171
crones
.
15
sꝪ
.
5
pens
Resteth
1
d
7
Multiplicators
Deuysor
.
They
haue
to
dyuyde
an
halfe
peny
.
¶
The
rule
and
question
for
to
buylde
.
And
fyrste
for
the
place
.
A
Man
hath
a
ground
that
is
in
length
100
yardes
,
and
in
brethe
70
yardes
,
where
as
he
wyll
edyfye
and
buylde●
houses
,
of
lengthe
5
yardes
,
and
4
brethe
I
demaunde
how
many
houses
shall
he
haue
vpon
that
ground
.
Answer
.
ye
shall
multyply
the
lenghte
by
the
breth
in
sayenge
70
tymes
100
ben
7000
,
an
be●he
house
must
haue
5
yardes
of
length
,
and
4
of
brede
/
multyplye
that
one
by
the
other
,
and
they
make
20
,
whiche
20
shall
be
the
dyuysor
commune
,
therfore
deuyde
7000
by
20
,
&
ye
shall
fynde
that
there
shall
be
350
houses
Note
well
this
rule
.
¶
The
rule
&
questyon
of
the
walles
.
A
Manne
wyll
make
a
wall
32
fote
in
lenghte
,
and
2
of
thycknes
,
and
the
heyght
25
fote
,
and
eche
fote
shal
cost
the
makynge
2
sꝪ
.
I
demaunde
how
moch
shall
cost
the
makynge
of
all
the
wall
.
Answere
.
For
to
know
this
rule
,
ye
shall
multyply
the
lengthe
by
the
thyckenes
in
sayenge
2
tymes
32
ben
64
/
&
then
ye
shal
multyplye
it
by
heyghte
in
sayenge
25
tymes
64
ben
1600
,
and
than
multyply
by
the
pryce
,
that
is
to
wytte
by
2
shyllynges
the
which
ben
3200
shyllynges
,
wherof
ye
shal
make
francz
,
therfore
dyuyde
them
by
20
and
they
ben
160
francz
.
And
so
moch
shal
coste
the
makyng
of
the
wall
.
¶
The
rule
and
questyon
of
the
couerynge
.
¶
If
ye
wyll
haue
a
house
couered
with
tyelles
,
ye
must
knowe
how
many
tyelles
behoueth
you
to
haue
vnto
the
length
of
a
lygne
,
and
how
many
to
the
bredde
.
Example
.
If
the
house
hadde
nede
of
54
for
the
length
,
and
34
for
the
brethe
,
I
demaunde
howe
many
shold
be
requysy●e
vnto
all
the
house
.
Answere
.
Multyplye
the
length
by
the
brethe
in
sayenge
34
tymes
54
ben
1836
tyelles
,
and
so
many
must
ye
haue
to
couer
the
house
.
The
rule
&
questyon
of
a
graden
.
¶
A
louer
dyd
entre
into
a
garden
for
to
gather
apples
for
his
lady
,
and
vnto
the
sayd
gardyn
ben
thre
gates
,
and
in
eche
gate
is
a
porter
,
and
whan
he
shall
ysue
after
that
he
hath
gathered
the
apples
,
he
must
gyue
the
halfe
of
his
apples
&
one
,
to
the
fyrste
porter
,
and
whan
he
is
at
the
second
porter
,
he
must
gyue
vnto
hym
the
halfe
and
one
/
and
to
the
thyrde
porter
the
halfe
and
one
,
&
whan
he
is
forth
he
hath
no
more
but
one
apple
to
gyue
vnto
his
lady
paramour
.
I
demaunde
how
many
apples
had
he
gathered
.
Answere
.
He
had
one
,
aple
whan
he
was
forthe
,
set
to
it
one
,
and
than
it
is
2
,
and
then
double
the
2
and
it
is
4
,
therfore
he
hadde
4
at
the
thyrde
porter
.
Then
to
this
4
set
1
&
that
is
5
,
and
then
double
them
and
that
is
10
therfore
he
hath
10
apples
at
the
seconde
porter
,
to
this
10
sette
1
and
it
is
11
,
double
them
,
and
that
be
22
apples
.
Therfore
ye
maye
say
that
he
had
gathered
22
apples
.
¶
The
rule
and
questyon
of
a
ladder
or
stayre
.
¶
I
haue
sene
a
stayre
that
had
100
step
,
pes
,
in
the
fyrst
steppe
was
1
douffe
in
the
seconde
step
2
,
in
the
thyrde
3
,
in
the
fourth
4
,
and
so
vnto
100
,
I
was
demaunded
how
many
douffes
were
in
al
the
stayre
.
I
answered
5050.
Probacyon
I
wyll
gyue
you
certayne
of
all
nomber
that
do
procede
naturally
,
that
is
to
wyt
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
,
And
infinitly
as
ye
wyll
,
for
all
nomber
naturall
is
ended
in
nomber
euen
or
in
nomber
not
euen
,
yf
it
be
ended
in
nomber
euen
,
than
by
the
halfe
therof
multyply
the
nomber
not
ouen
,
that
encloseth
it
Example
.
1
,
2
,
3
,
4
,
wyll
ye
know
what
all
amounteth
vnto
in
sayenge
2
tymes
5
ben
10
,
for
2
is
the
halfe
of
4
,
and
5
is
the
nomber
not
euen
that
encloseth
4
,
And
yf
the
nomber
ende
in
nomber
not
euen
.
As
by
ensample
,
1
,
2
,
3
,
4
,
5
,
wyll
ye
knowe
what
all
amounteth
vnto
.
Multyply
5
by
his
greater
halfe
,
that
is
3
,
sayenge
3
tyme
5
ben
15
,
And
thus
shall
ye
alwayes
doo
in
what
nomber
so
euer
it
be
euen
or
not
euen
&c.
¶
The
rule
and
question
of
two
men
.
IF
two
menne
go
by
one
waye
,
and
that
they
go
in
to
any
farre
place
,
and
procede
in
such
wyfe
,
that
the
one
procede
eche
daye
certayne
nomber
of
my
les
,
that
is
to
saye
4
and
6
more
or
lesse
.
And
that
other
man
goeth
encreasynge
the
fyrste
daye
one
myle
,
the
seconde
daye
two
,
the
thyrde
thre
,
and
so
encreasynge
after
progressyon
.
Be
ye
all
certayne
that
in
some
day
the
one
ouertaketh
the
other
.
It
is
demaunded
in
what
daye
,
and
how
many
myles
they
shall
go
.
Answere
.
Double
the
nomber
of
his
mylles
that
goeth
eche
day
an
egall
nomber
of
myles
And
of
the
nomber
dowble
take
awaye
one
vnyte
,
and
the
remanant
shall
shewe
you
what
daye
they
shall
mete
eyther
other
¶
Example
.
We
shall
set
it
that
the
one
goeth
a
daye
6
myles
,
double
that
and
it
is
12
,
and
fro
that
12
withdrawe
one
vnyte
,
as
it
is
sayd
in
the
rule
,
and
there
remayneth
11
,
that
is
the
nomber
of
the
day
that
they
shal
mete
togyther
.
And
for
to
know
the
nomber
of
the
myles
that
they
haue
gone
.
Multyplye
11
by
6
in
sayenge
6
tymes
11
ben
66
myles
that
they
haue
gone
.
Thus
ye
may
knowe
it
by
the
rule
of
progression
contynued
,
11
is
a
nomber
not
euen
,
be
it
therfore
multiplyed
by
the
greater
halfe
that
is
to
wytte
by
6
in
sayenge
11
tymes
6
or
6
tymes
11
ben
66.
And
also
one
onely
nomber
amounte
by
progressyon
,
&
by
multyplycation
,
wherby
it
appereth
that
vpon
the
elleuenth
day
they
mete
eche
other
,
&
haue
gone
66
myles
.
¶
The
rule
and
questyon
of
the
women
that
bare
apples
to
the
market
.
THre
women
bare
apples
well
&
honestly
trymmed
to
the
market
,
of
whome
the
one
bare
50
the
other
30
,
and
the
thyrde
10
,
theyr
housboundes
were
brethern
and
gaue
commaundement
to
them
that
they
shold
make
as
good
market
one
as
an
other
,
that
is
,
that
they
sell
all
after
one
pryce
,
&
that
the
one
brynge
as
moch
money
home
as
the
other
.
I
demaunde
howe
that
maye
be
done
.
Answer
.
It
is
possible
.
For
fyrste
there
cometh
a
marchaunt
to
her
that
hath
50
appels
:
and
sayth
to
her
howe
many
for
one
peny
,
and
she
answered
7
and
so
she
maketh
7
d
of
her
50
appels
and
hath
remaynynge
one
apple
.
The
other
solde
after
the
same
pryce
.
And
she
that
had
30
apples
solde
hers
for
4
d.
and
she
had
remaynyng
3
apples
.
The
other
that
had
10
apels
solde
hers
for
4
d
and
she
had
remaynge
2
appels
.
And
then
came
there
anothe
marchaunt
that
gaue
3
d
for
an
apple
.
And
so
eche
one
bare
home
10
d̔
as
ye
se
in
this
ensample
.
And
thus
may
ye
do
of
all
other
semblable
.
❧
The
rule
and
questyon
of
the
bagge
A
Marchaunt
hath
a
bagge
that
weygheth
19
ounces
of
thre
mettalles
,
wherof
7
ounces
ben
of
golde
8
of
syluer
,
and
4
of
copre
.
And
he
wyll
take
therout
5
ounces
.
I
demaunde
howe
moche
of
golde
,
howe
moch
of
syluer
,
and
howe
moche
of
copre
is
in
these
5
ounces
.
Answere
,
ye
shall
multyplye
the
5
for
to
knowe
the
golde
by
7
,
for
the
syluer
by
●
and
for
the
coper
by
4.
And
for
to
fynde
the
deuysor
,
ye
shall
sette
all
the
multiplycators
togyther
,
that
ben
19
,
therfore
deuyde
by
19.
The
answere
is
in
this
ensample
5
ounces
.
-
7
Of
golde
1
ounce
an
halfe
8
pens
5
grayne
.
Resteth
1
pens
.
-
8
Of
syluer
2
ounces
,
2
pens
and
halfe
,
1
halfe
grayne
.
Resteth
2
pens
.
-
4
Of
coper
1
ounce
1
peny
6
graynes
.
Resteth
6.
¶
Now
set
the
remeynaunt
togyther
and
dyuyde
it
by
the
dyuysor
commune
,
that
is
19.
And
it
is
1
halfe
grayne
.
¶
The
rule
and
question
of
the
bell
.
IN
a
chyrche
is
made
a
bell
,
and
there
in
is
put
30
pounde
of
golde
50
li.
of
syluer
,
100
of
tynne
,
and
102
of
coper
.
Whan
the
bell
is
made
there
remayneth
40
pound
in
one
pyece
,
that
they
wyll
sell
I
demaunde
how
mych
is
there
of
golde
howe
mych
of
syluer
,
how
moch
of
tynne
,
and
how
moche
of
copre
.
Answere
.
ye
shall
do
as
aboue
is
sayd
of
the
bagge
for
ye
shall
multiply
40
eche
one
by
hymselfe
,
and
dyuyde
by
282.
¶
Example
.
-
30
Of
golde
4
pounde
4
ounces
4
pens
,
1
grayne
.
Resteth
6
-
50
Of
syluer
7
pounde
1
ounce
11
pens
,
9
graynes
and
an
halfe
.
Resteth
57.
-
100
Of
tynne
14
pounde
2
ounces
&
halfe
10
pens
,
and
halfe
7
graynes
.
Rest
114
pens
.
-
102
Of
copre
14
pounde
7
ounces
11
pens
and
an
halfe
5
graynes
and
halfe
.
Resteth
105.
Multyplycators
,
282
Dyuysor
commune
And
all
dyuyded
they
haue
of
restes
1
grayne
.
¶
This
rule
is
proued
by
reduction
,
set
to
the
same
that
remayneth
/
and
dyuyde
by
the
dyuysor
commune
,
&
therof
cometh
1
grayne
.
¶
The
rule
and
questyon
to
chaunge
golde
into
syluer
.
A
Marchaunt
hath
100
trancz
in
gold
and
he
goth
vnto
a
chaunger
&
sayth
,
I
haue
100
francz
in
pyeces
of
golde
I
wolde
haue
the
money
therof
in
small
pyces
,
that
is
to
wytte
,
of
2
pens
,
of
3
pens
of
4
pens
,
of
5
pens
,
of
6
pens
,
of
8
pens
,
4
of
10
pens
,
&
I
wolde
haue
as
many
pyecz
of
one
as
of
an
other
.
I
demaunde
howe
many
pyeces
of
euery
money
oughte
the
chaunger
to
gyue
hym
.
Answere
,
ye
must
set
togither
all
these
nombers
2
,
3
,
4
,
5
,
6
,
8
,
and
10
,
that
ben
2
the
deuysor
commune
and
then
ye
must
make
of
the
francz
pens
that
is
24000
pens
,
whiche
ye
shal
deuyde
by
32
and
there
been
750
pyeces
of
eche
money
,
and
thus
ye
may
do
of
all
other
semblable
.
¶
The
rule
and
question
of
cloth
of
dyuers
colours
I
haue
a
pyece
of
clothe
wherof
the
thyrde
part
is
whyte
,
the
fourth
part
blacke
,
and
8
elles
of
graye
.
I
demaunde
howe
moch
hath
it
of
lenghte
.
Answer
.
Set
12
,
for
in
12
ye
shall
fynde
one
thyrde
and
one
fourth
,
the
thyrde
and
the
fourth
of
12
is
7
,
and
there
remayneth
5
/
therfore
forme
the
rule
of
thre
,
yf
5
,
be
comen
of
12
,
of
how
moche
shall
come
58
multyply
12
,
by
3
that
ir
96
,
&
dyuyde
by
5
,
and
therof
cometh
19
elles
and
1
fyfte
,
therfore
ye
maye
answere
that
the
pyece
of
clothe
hath
of
lenght
19
elles
and
one
fyfte
.
¶
The
rule
and
questyon
of
spyceryes
.
¶
A
bourgesse
sayd
vnto
h●s
seruaunte
holde
these
13
frances
,
and
go
and
bye
the
peper
that
costeth
15
sꝪ
.
the
pounde
,
and
sugre
that
costeth
18
sꝪ
.
the
pounde
,
&
of
fyne
spyces
that
costeth
9
sꝪ
.
the
pounde
,
and
gynger
that
costeth
13
sꝪ
.
the
pounde
,
and
cloues
that
costeth
10
sꝪ
.
the
pounde
,
and
brynge
me
as
many
poundes
of
one
as
of
another
.
I
demaunde
how
many
poundes
oughte
the
apotycarye
to
gyue
hym
for
13
francz
.
Answer
.
ye
shal
set
al
the
pryces
togyther
15
,
18
,
9
,
13
and
10
,
that
ben
65
which
shall
be
the
dyuysor
,
&
then
ye
shall
make
the
francz
in
shyllynges
,
that
is
260
shyllynges
.
And
than
ye
shall
deuyde
by
65
,
&
therof
cometh
4
pound
/
therfore
ye
maye
answer
,
that
he
ought
to
gyue
hym
4
poundes
of
all
these
spyceryes
.
¶
The
rule
and
question
of
the
egges
.
A
yonge
mayden
bayreth
egges
to
the
market
for
to
sell
and
hyr
meteth
a
yonge
man
that
wold
play
with
hyr
in
so
mich
that
he
ouerthroweth
&
breketh
the
egges
euery
one
,
&
wyl
not
paye
for
them
.
The
mayde
doeth
hym
to
be
called
afore
the
iudge
.
The
iudge
condempneth
hym
to
pay
for
the
egges
,
but
the
iudge
knoweth
not
howe
manye
egges
there
were
.
An
that
he
demaundeth
of
the
mayde
,
she
answereth
that
she
is
but
yonge
,
and
can
not
well
compte
,
but
she
and
hyr
moder
had
ordeyned
and
dysposed
them
by
2
and
2
&
there
remayned
1
egge
.
Than
by
3
and
3
&
there
remayned
1
/
than
by
4
and
4
and
there
remayned
1
/
than
by
5
&
5
,
&
there
remayned
1
/
than
by
6
and
6
&
there
remayned
1
,
and
at
the
last
by
7
and
7
and
there
remayned
none
/
I
demaunde
how
many
egges
there
were
.
Answer
.
721.
And
for
to
proue
it
,
multyply
the
nombers
one
by
another
in
sayenge
2
tymes
3
ben
6
,
4
times
6
ben
24
,
5
tymes
24
ben
120
,
6
tymes
120
ben
720
,
and
set
therto
1
that
remayned
always
&
than
they
ben
721
that
which
ye
shal
deuyde
by
7
/
&
there
remayneth
nothyng
/
and
so
she
hadde
721
egges
.
And
fater
this
ensample
maye
the
iudge
iuge
the
yonge
man
to
pay
.
¶
The
rule
and
question
of
money
forgotten
with
a
chaungeour
.
AN
aduocate
hath
gyuen
to
a
chaun
,
geour
money
,
&
hath
forgotten
howe
moch
.
For
to
know
howe
moche
and
for
to
haue
all
his
money
,
he
fyndeth
subtyltye
that
ensueth
,
he
sayeth
to
one
of
his
sonnes
,
of
whom
he
hath
many
go
vnto
suche
a
chaungeour
&
bryng
me
a
france
and
the
tenth
part
of
the
money
that
I
delyuered
hym
,
and
so
was
it
done
And
an
other
tyme
he
sayd
vnto
another
sone
,
go
vnto
the
chaungeour
&
brynge
me
2
frances
and
the
tenth
part
of
the
remaynant
,
and
so
he
sayd
vnto
all
,
but
vnto
the
last
he
said
vnto
the
chaungeour
,
and
brynge
me
all
the
remaynant
of
the
money
,
and
so
was
it
done
,
and
as
mouche
brought
the
one
as
the
other
.
I
demaunde
how
moche
mony
he
hadde
,
how
many
sonnes
,
&
how
moch
money
eche
one
of
them
brought
.
Answer
For
this
thre
questyons
pose
the
nomber
that
they
all
brought
,
that
is
to
wytte
,
the
tenth
ben
10
,
and
of
10
take
one
and
there
do
remayne
9
,
therfore
ye
may
say
that
he
had
9
sonnes
,
and
eche
one
brought
9
sꝪ
.
And
for
to
knowe
how
moche
he
had
gyuen
to
the
chaungeour
,
ye
must
multypyl
9
by
hym selfe
,
and
it
is
81.
Therfore
he
had
delyuered
81
frances
to
the
chaungeour
.
For
to
make
the
proue
lay
81
and
take
vp
for
the
fyrst
sone
1
and
the
tenth
parte
of
the
remaynant
,
and
in
lyke
maner
ye
muste
do
of
all
other
.
¶
The
rule
and
question
,
of
tyme
.
&c.
¶
A
man
sayth
yf
I
hadde
as
moch
more
of
tyme
as
I
haue
,
and
the
halfe
,
the
thyrd
and
the
fourth
of
my
time
that
I
haue
set
to
.
I
sholde
haue
of
yeres
50
,
I
demaund
what
age
he
hath
.
Answere
.
Laye
12
,
for
in
12
ye
fynde
an
halfe
,
a
thyrd
,
and
a
fourth
And
then
set
there
ones
as
moch
,
&
that
ben
24
,
than
set
therto
1
halfe
,
1
thyrde
,
&
●
fourth
of
12
,
and
they
ben
37
,
and
thenne
fourme
thy
question
.
yf
37
be
comen
of
12
of
howe
moch
shall
come
50.
Multyply
12
by
50
and
dyuyde
by
37
,
and
ye
shall
fynde
that
he
hath
16
yeres
78
dayes
and
a
halfe
10
houres
resteth
2.
¶
The
rule
and
questyon
for
to
deuyde
dystrybucyons
.
IN
a
churche
bell
12
channons
9
prestes
and
6
clerkes
they
haue
to
dyuyde
a
dystrybucyon
of
400
frnces
,
werof
the
chanons
haue
3
,
the
prestes
2
and
the
clerkes
●
,
I
demaunde
howe
moche
shall
haue
the
chanons
,
how
moche
the
prestes
and
howe
moche
the
clerkes
.
Answere
.
Multyplye
one
nomber
by
an
other
in
sayenge
3
tymes
12
ben
36
that
is
the
multyplycatour
for
the
chanous
,
2
tymes
6
ben
18
,
the
multyplycatour
for
the
prestes
,
1
tyme
6
ben
6
,
the
multyplycatour
for
the
clerkes
.
Howe
moche
eche
one
oughte
to
haue
ye
may
se
in
the
ensample
by
the
deuysor
.
Set
togyder
al
the
multyplycatours
&
they
ben
60
,
the
deuysor
comune
.
36
|
240
|
frances
|
38
|
120
|
frances
|
6
|
40
|
frances
|
Multiplycatours
|
Deuysor
|
60
|
¶
The
rule
and
question
of
the
speyre
.
A
Speyr
is
the
halfe
and
the
thyrde
part
within
the
water
,
and
9
fote
with
oute
.
I
demaunde
howe
myche
of
lenghte
hath
the
spere
.
Answere
.
Set
6
,
for
in
6
is
founde
a
halfe
and
a
thyrde
the
halfe
and
the
thyrde
of
6
ben
5
,
and
there
remaneth
1
,
forme
the
rule
of
three
yf
1
be
comen
of
6
,
of
howe
many
shall
come
9
multiplye
6
by
9
,
and
they
ben
54
,
deuyde
them
by
1
and
they
ben
54
,
therfore
ye
may
answere
that
the
spere
hath
54
fote
of
length
,
the
halfe
is
27
,
&
the
thyrd
is
18
,
and
there
been
45
fote
within
the
water
,
and
9
without
that
is
54.
And
so
maye
ye
do
of
all
other
semblable
,
as
of
a
toure
.
¶
The
rule
and
question
of
two
men
that
went
that
one
agaynst
that
other
.
TWo
men
begin
to
go
and
take
theyr
iourney
that
one
agaynst
that
other
vpon
one
daye
and
in
one
houre
.
For
that
one
that
goeth
fro
Parys
,
to
London
and
goeth
euery
daye
7
myles
,
that
other
goeth
frome
Lyon
to
Parys
,
and
goethe
eche
daye
9
myles
,
and
from
Lyon
vnto
Parys
ben
80
myles
.
I
demaunde
howe
longe
tyme
shal
it
be
or
they
mete
.
Answer
Set
togyder
the
myles
that
they
go
in
one
daye
,
the
is
to
wyt
,
7
and
9
ben
16
,
forme
now
the
rule
of
16
come
of
1
daye
,
of
howe
moche
shall
come
80
that
they
haue
to
go
,
multyplye
80
by
1
&
it
is
80
they
whiche
he
may
deuyde
by
16
&
therof
cometh
5
,
therfore
in
5
days
they
mete
.
The
proue
is
,
for
he
the
from
Parys
to
Lyon
goeth
in
5
dayes
goeth
35
myles
/
&
that
other
45
the
which
ben
80
myles
.
¶
The
rule
and
questyon
of
a
catte
.
THere
is
a
catte
at
the
fote
of
a
tre
the
lenght
of
300
fote
,
this
catte
goeth
vp
warde
eche
daye
17
fote
,
and
descendeth
the
nyght
12
fote
.
I
demaunde
in
howe
longe
tyme
shall
she
be
at
the
toppe
,
Answer
.
Take
vp
and
abate
the
nyght
of
the
day
,
that
is
12
of
17
and
ther
remayneth
5
,
therfo●e
the
catte
mounteth
eche
daye
5
fote
/
deuyde
now
300
by
5
and
therof
cometh
60
dayes
then
she
shall
be
at
the
toppe
.
And
thus
ye
maye
do
of
all
other
semblable
.
For
of
this
rule
ye
may
make
4
questyons
,
as
it
appereth
in
the
practyse
therof
.
¶
The
rule
and
questyon
of
20
scolers
.
IF
●0
scolers
owe
vnto
theyr
host
5
d
tourneyes
,
how
oughte
they
to
paye
,
so
that
eche
one
pay
his
duty
&
gyue
the
money
of
his
purse
.
How
moche
shall
eche
one
paye
.
Answere
.
Eche
one
shall
pay
1
peny
Parys
,
and
the
hoste
shall
rethurne
vnto
him
agayne
1
peny
tournoys
and
so
eche
one
shall
paye
the
4
parte
of
a
tournoys
.
¶
The
rule
and
question
of
pylgrymes
.
¶
Twenty
pylgrymes
,
that
is
to
wytte
,
men
,
women
,
and
lyttell
chylderne
,
haue
spended
in
drinke
20
pens
,
wherof
the
men
paye
3
,
pens
,
the
women
2
pens
,
and
the
lytell
chyldren
halfe
pens
.
I
demaunde
how
many
men
,
&
how
many
women
,
and
how
many
chyldren
be
there
,
for
to
pay
this
20
pens
,
so
that
there
be
20
persons
.
Answer
.
There
shall
be
one
man
.
5
women
,
and
14
chyldren
.
¶
The
rule
&
queston
of
a
chauntour
.
¶
A
chauntour
hathe
eche
daye
of
rente
fro
the
courte
of
the
prynce
12
sꝪ
.
the
which
is
payed
by
knyghtes
,
damoyselles
,
and
squyres
of
whome
the
knyghtes
paye
2
sꝪ
the
damoysels
6
pens
,
and
the
squyers
3
pens
.
I
demaunde
how
many
knyghtes
how
many
damoysels
,
&
how
many
esquyres
ought
there
to
be
,
to
paye
this
12
sꝪ
.
so
that
there
be
12
persones
.
Answere
.
There
must
be
5
knyghtes
,
1
damoysell
,
and
6
esquyres
.
¶
The
rule
and
questyon
for
to
dyuyne
,
¶
If
ye
wyll
cause
your
felowe
to
byleue
that
ye
shall
dyuyne
howe
many
pyeces
of
syluer
he
hath
in
his
ryght
hande
say
vnto
him
that
he
put
as
many
pyeces
in
that
one
hande
as
in
that
other
.
And
than
that
he
take
fyue
from
the
lyft
hande
to
the
ryghte
hande
and
than
that
he
put
forth
of
the
ryght
hande
into
the
lyfe
hand
as
many
pieces
as
he
hath
remayning
in
the
lyfte
hand
.
And
there
shall
remayne
10
in
the
ryght
hande
.
¶
The
rule
and
questyon
of
thre
saynctes
.
A
Holy
hermyte
is
entred
within
a
churche
wherin
there
ben
thre
sayntes
:
that
is
to
wyt
/
saynt
Peter
,
saynt
Paule
,
and
saint
Francoys
/
this
hemite
cometh
fyrst
to
saynt
Peter
and
sayth
to
him
in
a
maner
of
his
orayson
,
I
pray
the
that
it
pleas
the
to
double
me
the
great
blances
that
I
haue
in
my
purse
,
and
I
shall
gyue
the
6
,
and
so
was
it
done
.
Than
came
he
te
saynt
Paule
&
sayde
to
hym
,
please
it
the
to
double
me
the
great
blances
that
I
haue
in
my
purse
and
I
shall
gyue
the
6
&
so
was
it
done
.
Then
came
he
to
saynt
Francoys
&
sayde
,
yf
it
wold●
please
to
double
me
the
great
blances
that
I
haue
in
my
purse
I
shall
gyue
the
6
,
and
so
was
it
done
,
and
nothynge
had
he
remaynyng
.
I
demaund
howe
many
greate
blances
had
he
in
his
purse
.
Answe
.
He
had
5
and
1
fourth
.
And
for
to
know
it
double
them
and
they
ben
10
and
an
halfe
,
and
then
ye
must
giue
6
to
saynt
Peier
,
and
there
remayneth
4
&
an
halfe
,
double
them
.
&
they
ben
9
And
then
gyueth
he
6
to
saynt
Paule
,
&
then
there
remayneth
3
double
them
and
there
ben
6
and
that
6
gyueth
he
to
saynt
Francoys
,
and
so
he
hat
nothynge
remaynynge
.
¶
Here
folowe
dyuers
other
proper
rules
and
questyons
.
A
Lorde
hyreth
a
seruaunt
,
the
whiche
he
sholde
gyue
euery
yere
●0
nobles
and
a
gowne
,
and
the
same
seruaunt
dwelleth
7
monethes
with
hym
,
and
then
they
varye
in
so
moch
that
his
lorde
gaue
hym
lycence
to
go
his
way
.
And
sayth
,
go
thy
wayes
out
of
my
house
and
take
thy
gowne
with
the
,
and
then
I
am
nothyng
in
thy
dette
.
Now
I
demaund
what
was
the
gowne
worth
,
wyll
ye
know
that
,
then
marke
how
many
monethes
7
is
lesse
than
a
yere
,
that
is
5
monethes
lesse
.
And
had
the
seruaunt
taryed
so
long
yet
by
his
mayster
than
shold
he
haue
had
the
gowne
&
10
nobles
.
Therfore
saye
thus
5
monethes
gyueth
10
nobles
,
what
giueth
7.
Make
it
after
the
rule
of
thre
/
&
it
cometh
14
nobles
.
¶
Of
thre
felowes
or
yonge
men
.
¶
Thre
felowes
play
togyther
the
one
to
wynne
the
others
money
.
For
the
one
had
more
money
than
the
other
.
And
the
fyrst
casteth
,
that
the
one
of
them
thre
leseth
iust
so
mych
money
as
the
other
two
hadde
.
Than
casteth
the
second
and
leseth
also
as
mych
as
the
other
two
hadde
Then
casteth
the
thyrde
and
leseth
also
iust
as
mych
as
the
other
two
had
.
And
than
was
the
money
iust
deuyded
,
&
had
eche
lyke
moch
.
Now
I
demaunde
how
moche
had
eche
or
they
began
to
playe
,
&
how
moche
money
that
eche
had
whan
they
played
.
Wyl
ye
know
that
,
then
marke
how
many
persons
dyde
playe
,
&
adde
1
to
them
,
as
here
adde
1
to
3
maketh
4.
So
many
nobles
had
the
fyrst
.
Now
dowble
4
cometh
8
,
&
subtra
1
from
8
rest
7
:
so
many
nobles
had
the
seconde
.
Then
double
7
cometh
14
,
therof
subtra
3
reste
13
,
so
many
nobles
had
the
thyrd
.
An other
question
.
A
Man
byeth
46
pounde
of
saffron
for
30
pounde
,
what
shall
cost
63
poundes
of
saffron
.
wyll
ye
knowe
that
,
then
multyply
the
30
poundes
with
the
63
poundes
of
saffron
,
cometh
1890.
Now
deuyde
them
with
46
cometh
41
poundes
and
4/49
parte
of
a
pounde
to
paye
for
the
63
poundes
of
saffron
.
Now
wyll
ye
knowe
howe
many
shyllynges
that
4/46
parte
of
a
li.
is
.
than
multyply
4
by
20
,
for
20
sꝪ
maketh
a
li.
cometh
80
sꝪ
.
deuyde
them
with
46
commeth
1
sz
,
and
4/64
part
of
a
sꝪ
.
Now
wyll
ye
knowe
how
many
pens
that
34/64
parte
of
a
shyllynge
is
,
them
multiply
34
with
12
12
pens
maketh
a
sz
,
cometh
408.
Dyuyde
them
wyth
46
cometh
8
pens
and
40/46
part
of
a
pen̄y
.
Now
wyl
ye
knowe
how
many
farthynges
that
●●/4●
parte
of
a
peny
is
,
then
multypplye
40
with
4
,
for
4
farthynges
maketh
a
peny
,
comet
160
farthynges
.
Now
deuyde
them
with
49
,
cometh
3
farthynges
and
22/46
parte
of
a
farthynge
,
Thus
done
ye
shall
fynde
that
63
li.
of
,
saffron
coste
41
li.
1
sꝪ
.
8
farthynges
and
22/46
parte
of
a
farthynge
.
¶
Item
a
165
poundes
of
alome
coste
2
poundes
5
shyllynges
6
pens
9
fa●thynges
:
what
shal
cost
22
poundes
of
alome
.
If
ye
wyll
soyle
this
question
,
than
make
of
your
poundes
shyllynges
&
adde
therto
the
odde
5
shyllynges
:
commeth
45
sꝪ
.
Then
make
of
the
45
sꝪ
.
penes
,
and
adde
6
pens
,
commeth
546
pens
,
than
make
of
your
pens
farthynges
,
and
adde
therto
the
9
odde
farthynges
,
cometh
2193
farthynges
.
Now
multyply
the
farthynges
with
22
cometh
48246
farthinges
.
Now
deuyde
them
with
165
cometh
592
and
66/165
part
of
a
farthynge
,
for
so
many
farthynges
shall
coste
22
li.
of
alome
.
Nowe
wyll
ye
knowe
how
many
pens
that
the
forewryten
farthynges
make
,
then
dyuyde
them
with
4
,
for
4
farthynges
make
a
peny
.
Then
wyll
ye
knowe
how
many
shyllynges
that
they
make
,
then
deuide
the
pens
with
12
,
for
12
pens
maketh
a
sꝪ
.
Thus
done
ye
shall
fynde
that
22
pound
of
alome
cost
●
sꝪ
.
3.
d
1
farthyng
,
and
it
is
done
.
¶
An
other
questyon
.
¶
A
marchaunt
hath
bought
a
bagge
of
peper
,
I
saye
not
how
heuy
,
but
whan
he
gyueth
for
a
pound
of
peper
12
pens
,
then
remayneth
hym
yet
37
d
And
when
that
he
gyueth
for
a
pounde
of
peper
15
pens
,
then
he
lacketh
44
pens
to
paye
for
the
peper
.
Now
I
demaund
how
heuy
the
bagge
of
peper
was
,
and
how
moch
money
that
the
marchaunt
had
.
For
to
know
this
&
suche
other
lyke
questyon
,
ye
shall
take
and
subtra
12
from
15
and
there
resteth
3
,
whiche
3
shal
be
your
deuysor
.
Then
shall
ye
adde
44
and
37
togyther
/
and
that
maketh
81.
Then
muste
ye
deuyde
81
with
2
,
&
therof
cometh
27
,
so
many
pounde
wayeth
the
bagge
of
peper
.
Now
wyll
ye
knowe
how
moche
money
the
marchaunt
hadde
,
then
must
ye
multiplye
12
with
27.
and
adde
37
therto
,
or
multyply
15
with
27
and
subtra
44
,
comith
361
,
so
many
pens
hadde
the
marchaunt
.
An other
questyon
.
A
Dronkart
drynketh
a
barell
of
bere
in
the
space
of
14
days
,
and
when
his
wyfe
drinketh
with
him
than
they
drincke
it
out
within
10
daies
.
Now
I
demaunde
in
what
space
that
his
wyfe
shold
drincke
that
barell
of
bere
alone
.
For
to
soyle
this
questyon
&
suche
other
lyke
,
ye
shall
fyrst
subtra
the
lee●t
drynker
from
the
more
that
is
10
from
14
and
ther
remayneth
4
,
&
that
is
your
deuisor
.
Now
saye
4
gyueth
10
what
gyueth
14
Make
it
after
the
golden
rule
,
and
ye
shall
fynde
that
she
shold
drynke
it
in
35
dayes
.
¶
Here
endeth
the
introduction
of
awgrym
for
the
pen
.
¶
Here
begynneth
the
introduction
for
to
lerne
to
reken
with
the
counters
,
with
diuers
rules
belongynge
to
the
same
.
diagram of counters
C.
thousand
,
X.
thousand
.
Thousand
Hondreth
Ten.
One
.
FOr
as
moch
as
there
ben
many
persones
that
ben
vnlerned
,
and
can
not
write
,
yet
neuertheles
the
crafte
or
scyence
of
awgrym
and
rekenynge
is
nedefull
for
them
to
know
,
wherfore
I
shal
hereafter
declare
&
write
of
this
science
in
the
best
&
shortest
wyse
that
maye
be
possible
,
how
that
ye
shal
order
your selfe
in
rekenynge
and
to
caste
counter
.
¶
Fyrste
ye
shall
vnderstand
that
in
the
crafte
of
awgrym
be
9
letters
or
fygures
that
men
may
laye
and
wryte
all
maner
of
sommes
withall
.
Therfore
fyrst
of
all
a
man
muste
knowe
in
this
crafte
or
scyence
for
to
laye
9
counters
in
the
places
of
that
9
syfers
,
for
they
must
laye
euermore
styll
for
a
remembraunce
,
so
that
ye
maye
remember
your
place
by
theym
.
And
ye
muste
laye
them
the
one
ryghte
aboue
the
other
,
that
is
to
saye
,
in
the
fyrst
place
euery
counter
standeth
for
one
,
and
the
nethermoste
counter
is
the
fyrste
place
,
in
the
seconde
place
euery
counter
standeth
for
10
In
the
thyrde
place
for
a
100.
In
the
fourth
place
for
a
thousande
.
In
the
fyfte
place
for
10
thousande
.
In
the
syxth
place
for
100
thousande
.
In
the
seuenth
place
for
a
myllyon
.
In
the
eyghte
place
for
10
myllyons
.
In
the
nynth
place
for
a
hondreth
myllyons
.
In
the
tenth
place
for
a
thousande
myllyons
,
and
so
forth
infinitely
.
And
note
well
that
euery
counter
that
is
layde
betwen
the
lygnes
,
betokeneth
euermore
5
tymes
more
thenne
the
counter
that
lyeth
in
the
place
nexte
vnde
hym
/
that
is
to
saye
the
fyrste
counter
lyenge
alone
aboue
the
fyrste
place
betokeneth
5
the
counter
lyenge
alone
betwene
the
seconde
and
the
thyrd
lyer
and
place
,
standeth
fore
50
,
aboue
the
thyrde
place
5
honderth
,
aboue
the
fourth
5
thousand
aboue
the
fyfte
place
50
thousande
,
aboue
the
syxt
place
500
thousande
,
aboue
the
seueth
5
myllyons
,
aboue
the
eyght
50
myllyons
,
aboue
the
9
place
500
myllyons
,
aboue
the
tenth
place
fyue
thousand
myllyons
.
But
yf
ye
wyll
the
more
surer
know
your
places
it
is
necessary
for
you
to
marke
euery
place
with
a
marke
,
as
to
lay
a
counter
or
some
other
thyng
which
shal
euer
lay
styl
and
in
no
wyse
be
remeued
/
but
ye
muste
take
hede
yf
ye
laye
counters
for
the
marke
of
your
places
,
that
ye
laye
them
not
to
nygh
the
counter
that
ye
must
worke
with
all
,
lest
that
ye
take
the
one
for
the
other
,
but
lay
them
as
ye
se
them
marked
in
the
ensamples
folowyng
.
And
whan
ye
haue
layd
markes
and
know
the
order
of
your
places
,
ye
may
adde
,
and
subtra
,
multyply
and
deuyde
what
nombers
ye
lyst
,
that
is
to
say
,
to
cast
and
to
abate
at
your
pleasure
.
¶
Item
whan
there
lye
2
counters
betwene
two
lyers
,
take
hym
vp
and
laye
1
besyde
the
nexte
lyer
aboue
them
.
And
when
there
lye
5
counters
besyde
any
lyer
take
them
vp
and
laye
1
in
the
next
space
aboue
them
.
¶
Of
addycyon
.
ADdycyon
is
none
other
thyng
but
to
set
togyther
2
or
3
nombers
and
to
make
of
them
a
totall
somme
,
as
in
the
ensample
folowynge
.
¶
There
is
a
man
whyche
owyth
20
li.
18
pounde
,
100
pounde
,
50
pounde
,
and
69
pounde
.
Now
yf
ye
wyll
knowe
howe
.
mych
that
al
these
sommes
maketh
togyther
Then
for
the
first
somme
ye
must
lay
two
countres
besyde
the
seconnde
lyer
,
for
the
two
stande
for
20
/
that
is
for
the
fyrste
somme
.
Now
for
the
seconde
somme
laye
1
counter
besyde
the
seconde
lyer
,
for
that
is
10
/
and
lay
1
counter
betwyxt
the
nethermost
&
the
seconde
lyer
,
for
that
1
standeth
for
5
/
and
then
lay
3
counters
besyde
the
nethermost
lyer
/
and
they
all
together
make
18.
Now
for
the
thyrde
som
ye
shall
laye
1
counter
besyde
the
thyrde
lyer
/
for
that
is
an
100.
For
the
fourthe
somme
laye
1
counter
besyde
the
thyrde
and
the
seconde
lyer
,
that
is
50.
Now
for
the
fyfte
somme
lay
1
counter
betwyxte
the
thyrde
and
the
seconde
,
and
1
besyde
the
seconde
lyer
,
and
1
bewene
the
seconde
and
the
nedermost
,
and
4
besyde
the
ne
,
thermost
lyer
,
and
that
maketh
togyther
69
/
and
in
so
doynge
ye
shall
fynde
that
all
the
forewryten
sommes
make
togither
247
as
ye
shall
se
in
the
fygure
folowyng
And
euermore
for
a
generall
rule
remembre
your
places
,
for
euery
counter
that
lyeth
besyde
the
fyrst
lyer
standeth
but
for
1
,
in
the
second
place
euery
counter
standeth
fof
10
,
in
the
thyrde
place
for
100
as
is
afore
rehersed
.
diagram of counters
¶
Wyll
ye
proue
whither
ye
haue
added
well
or
not
,
than
subtra
all
your
sommes
one
after
an
other
.
And
in
lyke
wise
as
ye
do
with
this
ensample
so
ye
shall
do
with
all
other
of
addicion
,
❧
Of
subraction
.
SVbtraction
is
,
if
ye
wyll
withdrawe
any
summe
from
an
other
summe
,
ye
must
knowe
two
numbers
,
that
is
to
wytte
,
the
nomber
that
ye
wyl
withdrawe
,
and
the
nomber
where
fro
ye
wyll
withdrawe
.
An
ensample
.
There
is
a
man
that
oweth
you
9756
poundes
,
and
there
vppon
he
hath
payed
you
5989
poundes
,
Now
yf
ye
wyl
knowe
what
there
resteth
then
set
downe
your
summe
that
he
oughte
you
,
&
therof
withdrw
the
summe
that
he
hath
payed
you
,
and
that
the
remayneth
is
the
som
that
he
doth
yet
owe
you
,
as
ye
more
playnly
maye
se
it
in
the
ensamples
hereafter
folowynge
.
diagram of counters
And
when
ye
haue
set
your
deth
/
that
is
to
say
975●
poun̄des
vnder
this
maner
as
afore
shewed
.
Then
yf
ye
wyll
knowe
the
rest
,
them
take
therof
that
ye
haue
payed
as
5989
poundes
.
Now
for
to
do
this
/
ye
shall
fyrst
take
vp
the
counter
that
lieth
betwene
the
fourth
&
fyft
lyer
for
that
is
5000.
Then
take
vp
one
of
the
counters
which
lyeth
besyde
the
fourth
lyer
,
&
that
is
a
thousand
/
&
ye
shold
take
away
but
900
,
therfore
ye
must
lay
downe
1
counter
agayne
besyde
the
thyrde
lyer
,
that
is
a
hondreth
.
Then
take
vp
one
of
the
counters
that
lyeth
besyde
the
thyrde
lyer
,
whyche
is
a
hondreth
/
and
ye
sholde
take
vp
but
80
,
therfore
ye
muste
laye
2
counters
besyde
the
seconde
lyer
/
that
is
20
and
80
,
that
ye
haue
take
vp
maketh
100
,
then
take
vp
one
of
the
counters
that
lyeth
besyde
the
seconde
lyer
/
that
is
10
/
and
ye
sholde
haue
take
awaye
but
9
,
therfore
ye
must
laye
one
counter
besyde
the
nethermost
lyer
,
that
is
1
and
the
9
that
ye
haue
subtrahed
or
take
vp
,
maketh
10
,
and
there
remayneth
3767
pounde
det
,
and
stande
thus
.
diagram of counters
Wyll
ye
proue
whyther
ye
haue
subtrahed
well
or
not
,
then
adde
therto
that
ye
haue
payed
,
and
yf
the
somme
come
then
so
great
as
it
was
afore
,
then
is
your
subtraction
true
,
els
not
.
Multiplication
.
MVltiplycation
is
nothynge
e●les
but
to
multiplye
one
nomber
by
an
other
,
as
thus
,
to
know
what
is
6
tymes
9
or
6
tymes
12
and
suche
lyke
.
And
in
multiplycation
ye
must
consyder
two
nombers
,
that
is
to
wyt
,
the
nomber
that
ye
wyll
multyply
,
and
the
nomber
wherby
ye
wyll
multiply
,
and
ye
muste
worke
in
multiplycatyon
after
this
maner
:
Fyrste
ye
shall
laye
downe
the
lesser
nomber
,
which
is
6
,
and
this
6
is
the
nomber
that
shall
be
multyplyed
,
and
the
9
is
the
nomber
that
ye
shall
mutiply
withall
And
ye
shall
laye
the
nomber
that
shal
be
multiplied
,
at
the
right
syde
of
your
lyers
&
when
ye
worke
your
multiplycation
,
ye
shall
laye
them
at
the
lyfte
syde
,
as
in
this
ensample
here
after
folowyng
shall
more
playnely
appere
.
diagram of counters
¶
Fyrst
ye
muste
laye
downe
the
lesser
nomber
,
whych
is
4
,
as
in
this
ensample
,
as
ye
se
them
layde
here
on
the
ryght
hande
of
the
lyers
.
And
when
that
ye
haue
thus
done
,
ye
muste
take
vp
one
counter
and
laye
9
for
it
on
the
other
syde
of
the
markes
,
that
is
to
wyt
/
at
the
lyfte
syde
.
And
after
that
take
vp
another
counter
/
and
laye
also
9
for
it
/
and
so
forthe
for
euery
counter
that
ye
take
vp
ye
must
laye
9
for
it
at
the
other
syde
.
And
when
that
ye
haue
so
wrought
your
worke
,
it
wyll
come
iuste
to
36
,
as
ye
se
the
counters
before
layde
on
the
lyfte
hande
of
the
lygnes
.
¶
And
yf
ye
wyll
multiplplye
by
greater
numbers
as
thus
/
to
know
what
is
24
tymes
14.
Fyrste
laye
14
on
the
ryghte
hand
of
your
lyers
or
markers
,
as
this
ensample
folowynge
sheweth
.
place
nexte
aboue
your
fynger
/
and
reke●
euery
chone
of
them
for
2
whiche
maketh
iuste
6
,
then
10
and
6
maketh
16
/
as
the
fygure
before
shewed
/
when
ye
haue
so
done
,
then
take
away
your
fynger
,
and
for
euery
one
of
the
3
counters
that
lyeth
in
the
fyrste
place
on
the
ryght
syde
lay
16
on
the
lyft
syde
/
and
then
take
them
of
the
ryghte
syde
away
/
and
ye
shall
se
that
the
nomber
shall
iuste
come
to
128
,
as
the
ensample
before
shewed
.
And
thys
wyse
ye
muste
reken
all
counters
that
lyeth
in
the
spaces
yf
the
multplycacyon
shal
be
truely
made
.
¶
An
other
ensample
¶
For
to
knowe
howe
many
grotes
be
in
4563
nobles
.
Fyrst
ye
shall
set
downe
the
lesse
nomber
,
that
is
the
nomber
that
ye
shall
multiylye
,
as
this
fygure
folowynge
playnely
here
sheweth
,
diagram of counters
¶
Now
for
to
make
of
these
nobles
grotes
,
ye
muste
multyplye
them
with
20
,
for
20
grotes
maketh
a
noble
.
Nowe
for
to
multiply
this
nomber
/
euermore
ye
must
set
downe
the
nomber
that
ye
wyll
multyplye
at
the
ryght
syde
at
your
markes
and
set
youre
fynger
agaynste
the
marke
that
ye
begynne
at
,
for
your
fynger
shall
be
a
remembraunce
to
you
for
that
place
where
youre
fynger
standeth
is
the
fyrste
place
and
damneth
all
the
places
vnderneth
hym
.
¶
Now
for
to
make
grotes
of
●hese
4563
nobles
.
Fyrste
ye
shall
set
your
fynger
agaynst
the
fourth
lyer
,
&
take
vp
one
of
the
foure
counters
that
lyeth
agaynste
the
sayd
fourth
lyer
,
&
lay
two
counters
besyde
the
nerte
lyer
aboue
that
,
where
your
fynger
standeth
,
for
that
is
the
seconde
place
frome
youre
fynger
,
and
the
two
counters
so
layde
standeth
for
20
that
is
one
noble
,
and
lyke
as
ye
haue
done
with
this
one
counter
,
so
shall
ye
do
wyth
the
other
3
folowynge
.
Then
take
vp
the
counter
that
lyeth
betwene
the
thyrd
and
the
fourth
lyer
and
laye
two
counters
in
the
next
space
aboue
that
,
and
that
is
also
20
or
elles
ye
may
take
it
vp
and
laye
one
counter
besyde
the
seconde
lyer
,
for
the
place
where
your
fynger
standeth
and
that
is
also
20.
Tken
take
vp
the
counter
that
lyeth
betwyxt
the
seconde
and
the
thyrde
lyer
,
and
laye
2
in
the
nexte
space
aboue
that
/
then
take
vp
the
counter
that
lyeth
besyde
the
second
lyer
,
and
lay
two
counters
besyde
the
next
lyer
aboue
that
same
.
Then
set
your
fynger
agaynst
the
fyrste
lyer
,
and
take
vp
one
of
the
3
counters
and
laye
2
counters
for
it
besyde
the
nexte
lyer
aboue
that
,
&
as
ye
haue
done
with
that
,
so
must
ye
do
with
the
othertwo
,
and
then
ye
shall
fynde
that
4563
nobles
maketh
91290
grotes
,
and
standeth
thus
as
the
ensample
hereafter
sheweth
.
And
as
ye
haue
done
with
these
forewyten
ensample
of
multyplcation
,
so
shal
ye
do
with
all
other
of
multiplycacyon
.
diagram of counters
¶
Wyll
ye
know
or
proue
whither
ye
haue
multyplied
well
or
not
,
then
deuide
the
grotes
,
that
is
91260
by
20
,
&
yf
the
somme
come
to
stand
as
it
was
afore
,
then
ye
haue
multyplyed
well
.
And
thus
alwaye
ye
maye
make
your
proue
vpon
all
maner
of
multyplycatours
.
❧
Of
dyuysyon
.
DYuysyon
is
to
deuyde
a
somme
thorough
an
other
somme
:
and
in
this
dyuysyon
must
be
knowen
two
nombers
,
that
is
the
nober
that
ye
wyll
deuyde
,
and
the
nober
wherby
ye
wyll
deuyde
it
,
as
to
know
how
many
tymes
ye
maye
haue
a
small
nomber
out
of
a
great
as
by
ensample
,
yf
ye
wyl
deuyde
336
,
by
14
as
in
this
ensample
hereafter
.
Fyrste
laye
336
on
the
ryght
hand
of
your
lyers
,
&
then
set
your
fynger
at
the
hyest
place
where
any
counter
lyeth
,
for
as
I
shewed
you
before
,
the
damneth
all
the
other
places
beneth
,
so
the
then
there
as
your
fynger
is
,
is
the
fyrste
place
.
And
then
loke
ye
may
take
14
from
that
place
,
whiche
ye
can
nat
do
,
for
euery
counter
standeth
but
for
one
,
bycause
your
fynger
is
there
,
therfore
ye
must
remoue
your
fynger
to
the
nexte
place
benethe
where
the
other
3
counters
lye
/
and
then
loke
yf
ye
maye
take
14
from
that
place
whyche
ye
maye
do
ryght
well
/
for
these
3
counters
at
youre
fynger
standeth
but
for
,
3
and
the
other
3
counters
aboue
standeth
for
30
,
and
then
se
howe
many
tymes
14
ye
may
haue
out
of
33
and
so
many
counters
ye
must
laye
on
the
other
syde
iust
agaynste
your
fynger
/
that
is
to
saye
,
ye
maye
haue
28
out
of
33
,
that
is
two
tymes
14
out
of
33
/
and
therefore
ye
must
take
vp
28
and
laye
2
counters
on
the
diagram of counters
other
syde
agaynste
your
fynger
,
&
than
ye
can
haue
14
no
more
,
than
ye
muste
remeue
your
fynger
to
the
next
place
beneth
,
&
then
reken
that
place
at
your
fynger
to
be
the
fyrst
place
,
as
ye
dyd
before
:
&
then
loke
how
many
tymes
ye
maye
haue
14
from
that
place
,
which
ye
can
not
,
for
that
counter
at
your
finger
standeth
but
for
1
,
&
the
other
in
the
space
aboue
standeth
but
for
10
whiche
is
in
all
but
11
,
therfore
ye
muste
remoue
youre
fynger
to
the
nexte
place
beneth
/
and
then
ye
shal
se
that
that
nomber
is
56
out
of
the
whyche
ye
may
well
take
out
4
tymes
14
whyche
maketh
iuste
56
therfore
ye
must
take
vp
56
and
lay
4
counters
on
the
other
syde
agaynste
your
fynger
/
and
then
take
away
your
fynger
,
and
ye
shall
se
that
that
nomber
that
ye
haue
layde
on
the
lyfte
syde
of
your
markes
cometh
iust
to
24
as
the
ensample
before
shewed
,
and
then
ye
haue
your
question
soyled
;
for
yf
ye
deuyde
336
by
14
it
cometh
iust
to
24
for
24
tymes
14
maketh
iuste
336
,
as
I
haue
shewed
you
before
in
the
rule
of
multyplycation
.
And
lykewyse
as
ye
haue
deuyded
this
nomber
,
ye
may
do
with
all
other
nombers
.
And
yf
ye
wyll
proue
whither
ye
haue
well
deuyded
or
not
,
take
the
nomber
that
cometh
of
your
diuisyon
,
and
multyplye
it
with
the
smal
nomber
that
is
your
deuysor
,
and
adde
that
remayneth
therto
yf
there
be
any
,
and
than
it
wyll
come
iust
to
the
great
nomber
that
was
the
nomber
to
be
deuided
.
And
lykewyse
yf
ye
wyll
prone
whyther
ye
haue
truely
multyplyed
or
no
,
take
the
great
nomber
that
cometh
of
your
multiplycation
,
and
deuyde
hym
by
the
nomber
that
is
to
be
multyplyed
,
and
it
wyl
come
iust
to
the
thyrde
nomber
that
was
your
multyplyer
,
diagram of counters
¶
yf
ye
desyre
to
knowe
how
many
grotes
ben
in
79992
pens
.
Fyrst
ye
shall
set
downe
your
pens
as
ye
se
in
this
fygure
afore
,
and
ye
shall
deuyde
them
with
4
for
4
d̔
maketh
a
grote
.
Now
to
the
operacyon
therof
,
whan
that
ye
haue
set
downe
youre
pens
,
as
the
fygure
afore
shewed
/
than
set
your
fynger
at
the
hyest
counter
.
and
se
yf
ye
may
haue
4
from
that
place
/
which
of
a
suretie
ye
can
not
,
for
there
lyeth
but
one
,
and
it
standeth
but
for
one
,
by
cause
you
fynger
standeth
there
therfore
ye
shal
remeue
your
fynger
and
sette
it
agaynst
the
fyfte
lyer
,
and
se
yf
ye
maye
take
awaye
4
the
whiche
ye
may
do
,
for
there
lyeth
7.
Now
your
fynger
standeth
agaynste
the
fyfte
lyer
,
therfore
ye
shall
take
vp
the
counter
that
lyeth
in
the
next
space
aboue
your
fynger
,
for
that
counter
is
5
,
and
ye
sholde
take
vp
but
4
therfore
ye
shall
take
it
vp
and
lye
it
besyde
the
fyfte
lyer
of
the
ryght
syde
,
and
lay
1
on
the
lyft
syde
also
beside
the
fyfte
lyer
,
then
se
yf
ye
can
haue
4
any
more
from
the
place
that
which
ye
can
not
,
therfore
remeue
your
finger
and
set
it
against
the
fourth
lier
,
and
then
se
how
many
tymes
4
,
ye
maye
haue
out
of
39
ye
may
haue
9
tymes
4
,
and
for
this
9
tymes
4
ye
shall
lay
1
counter
in
the
space
betwene
the
fourth
and
fyfte
lyers
,
and
4
besyde
the
fourth
lyer
,
&
that
make
9
And
as
ye
haue
done
with
these
so
shall
ye
do
with
all
other
folowyng
,
&
whan
ye
haue
fynished
your
worke
ye
shal
fynde
the
79992
pens
make
19908
gretes
as
ye
may
se
playnly
here
in
this
fygure
.
diagram of counters
¶
The
proue
.
¶
If
ye
wyl
proue
whyther
ye
haue
deuyded
well
or
not
,
then
multyply
the
grotes
with
4
,
for
4
pens
maketh
a
grote
,
and
yf
the
some
come
to
stande
as
it
dyde
afore
then
ye
haue
deuyded
well
.
¶
Item
when
ye
deuyde
ony
somme
with
d
yf
there
remayneth
any
thing
it
is
pens
And
yf
ye
deuyde
by
shyllynges
,
yf
there
remayneth
any
thynge
it
is
shyllynges
.
And
as
ye
haue
done
with
these
forewryten
examples
so
maye
ye
do
with
al
other
.
¶
The
golden
rule
.
¶
Regula
aurea
is
called
the
golden
rule
for
lyke
as
golde
passeth
all
other
metall
so
this
rule
passeth
all
other
rules
in
awgrym
.
And
to
the
operacyon
of
this
rule
muste
alwayes
be
noted
thre
thynges
or
thre
nombers
,
of
the
whiche
two
of
them
must
be
lyke
of
names
and
of
kynde
,
that
is
to
wytte
,
the
fyrste
and
the
thyrde
nomber
,
&
alwayes
ye
shall
multyply
the
second
nomber
with
the
thyrde
,
&
that
that
cometh
of
the
multiplycation
is
the
nomber
to
be
to
deuyded
by
the
fyrst
nomber
that
is
general
dyuysor
/
and
the
quocient
of
the
dyuysor
,
sheweth
a
nomber
of
solucyon
of
name
and
kynde
of
the
mydle
nombers
,
as
in
these
ensamples
folowynge
shall
appere
.
¶
If
a
man
byeth
40
egges
for
20
pens
how
many
for
12
pens
,
yf
ye
wyll
soyle
this
question
,
ye
must
mutyplye
the
seconde
and
the
thyrde
nomber
togyther
/
and
the
producte
or
the
somme
that
commeth
of
that
multyplycation
,
ye
shall
dyuyde
by
the
fyrste
nomber
,
lyke
as
here
is
shewed
in
this
ensample
,
when
ye
bye
40
egges
for
20
pens
.
what
shal
one
pay
for
12
egges
ye
shall
multyply
20
with
12
,
commeth
240
,
the
whiche
ye
shal
deuyde
with
40
cometh
6
pens
,
and
so
moche
shall
ye
paye
for
the
12
egges
.
And
thus
ye
maye
do
with
al
other
suche
questyons
.
❧
An
other
qustyon
.
¶
Item
a
100
apples
cost
12
pens
,
what
shal
one
paye
for
87
,
ye
shall
multyply
12
,
with
87
,
cometh
●044
,
the
whiche
ye
shall
deuide
with
a
100
cometh
10
pens
44/10
parte
of
a
d
for
the
87
appels
.
Wyll
ye
knowe
how
many
farthynges
that
the
44
hondreth
parte
of
a
peny
is
wdrth
,
then
multiplie
44
with
as
many
farthynges
as
a
peny
is
worth
,
that
is
4
cometh
176
the
whiche
ye
shall
deuyde
with
100
cometh
1
farthynge
and
67/100
parte
of
a
farthinge
.
❧
Item
165
pounde
of
waxe
cost
2
li.
5.
sꝪ
.
6.
d
9
mytes
.
what
shall
coste
22
l.
For
to
soyle
this
question
and
suche
other
lyker
,
fyrst
ye
muste
make
of
poundes
sz
,
&
adde
therto
the
odde
5
sꝪ
.
the
whiche
stande
in
this
question
,
and
the
come
togyther
to
45
sꝪ
.
then
make
of
the
sꝪ
.
pens
,
and
adde
thereto
the
odde
6
pens
that
standeth
in
the
questyon
cometh
546
d
than
make
of
the
pens
mytes
,
and
24
mytes
is
a
peny
,
and
therto
adde
the
odde
9
mytes
,
that
standeth
in
the
question
commeth
togyther
to
23113
mytes
,
and
that
is
the
totall
some
of
all
the
li.
sꝪ
.
d
,
&
mytes
togyther
Nowe
make
it
after
the
rule
and
say
165
pounde
of
wax
cost
13113
mytes
,
what
shall
coste
●2
pounde
,
fyrste
multiplye
the
myldelmost
and
the
laste
togyther
,
that
is
ye
shall
multyplye
the
mytes
with
the
last
,
that
is
with
22
,
and
it
shall
come
to
388486
,
deuyde
them
with
165
and
it
shall
come
to
1748
mytes
and
6/165
parte
of
a
myte
,
so
many
mytes
shal
the
22
pounde
of
waxe
coste
.
Now
wyll
ye
knowe
how
many
pens
that
the
forewryten
mytes
make
:
then
deuyde
them
with
24
,
for
24
mytes
maketh
2
peny
Then
wyll
ye
know
how
many
sꝪ
.
that
the
pens
make
,
then
deuyde
them
with
12
,
for
12
d
maketh
a
sꝪ
.
And
thus
doing
ye
shal
fynd
that
the
22
li.
of
wax
shal
cost
6
sꝪ
.
20
mytes
&
66/165
patt
of
1
myte
and
it
is
done
.
¶
Item
when
there
standeth
1
in
the
fyrst
place
,
As
1
goos
cost
3
d
what
shall
cost
28.
ye
shal
multyply
the
myddels●e
with
the
last
in
sayeng
3
tymes
28
is
84
so
many
pens
cost
28
gese
,
and
it
is
fynyshed
.
¶
Item
in
the
contrary
as
whan
that
1
cometh
in
the
latter
ende
,
as
here
in
this
ensample
20
capons
cost
23
pens
/
what
shall
cost
1
capon
.
For
to
soyle
this
question
ye
shall
deuide
the
myddelmost
with
the
first
that
is
23
with
20
cometh
1
d
and
1/20
part
of
a
peny
,
that
is
0
mytes
&
⅘
part
of
1
myte
for
1
capon
,
☞
Item
17
elles
and
½
cost
14
nobles
and
1
/
part
of
a
noble
,
what
shall
cost
32
elles
and
2/●
part
.
For
to
soyle
this
questyon
and
such
lyke
/
fyrst
ye
must
breke
the
first
and
last
broken
togyther
crossewyse
in
sayeng
1
times
4
is
4
,
set
the
4
by
17
elles
Then
say
3
tymes
2
is
6
/
set
the
6
by
the
22
elles
.
Then
multyplye
bothe
the
nombers
of
your
fractyons
togyther
/
in
sayenge
4
tymes
2
is
8
the
whiche
ye
shall
set
vnder
4
and
6
/
then
it
standeth
thus
:
as
in
the
rule
of
thre
/
yf
17
elles
&
4/8
parte
of
an
elle
cost
14
nobles
&
⅓
part
of
a
noble
/
what
shal
cost
32
elles
and
●/●
part
of
an
elle
.
Fyrst
mulyply
the
fyrste
hole
nomber
with
the
nethermost
of
his
broken
that
is
17
with
8
cometh
136
&
therto
ye
shall
adde
that
4
that
standeth
aboue
8
comeeh
140
,
the
whiche
ye
shal
multyply
with
the
3
that
standeth
vnder
the
seconde
broken
com●th
430
that
is
your
deuysor
,
then
multiply
14
with
the
3
that
standeth
vnder
1
and
adde
that
1
therto
cometh
43
that
is
your
multiplicatour
,
then
multyply
32
with
the
nethermost
figure
of
the
broken
,
that
is
with
8
and
adde
therto
the
same
6
that
standeth
aboue
the
same
8
cometh
262.
Now
set
it
in
the
rule
of
thre
in
sayenge
420
elles
cost
43
nobles
,
what
shal
cost
262
,
multiply
the
second
with
the
thyrd
and
then
deuyde
that
that
cometh
of
that
multiplycation
with
the
fyrste
,
and
ye
shall
fynde
that
the
32
elles
of
cloth
cost
26
nobbles
16
d
34
mytes
and
150/420
parte
of
a
myte
.
¶
Item
whan
that
there
standeth
at
the
begynnyge
a
hole
nomber
with
a
broken
and
in
the
seconde
and
thyrde
place
no
broken
,
as
here
.
36
elles
and
1/1
coste
●
li.
what
shall
coste
16
elles
.
For
to
soyle
this
question
,
ye
must
multyply
the
fyrst
hole
nomber
with
the
vndermoste
fygure
of
his
broken
,
that
is
36
with
2
,
and
adde
the
1
thereto
that
standeth
aboueth
2
commynge
73
,
and
that
is
youre
deuysor
,
then
multiplye
8
li.
also
with
the
vndermoste
fygure
of
the
broken
,
that
is
to
wyth
with
2
,
and
it
cometh
to
16
,
then
multyplye
16
withe
16
cometh
to
259
the
whiche
ye
shal
deuyde
with
37
,
and
it
wyll
shewe
you
that
there
shall
be
to
paye
for
the
foresayd
clothe
3
pounde
10
shyllynges
1
peny
17
mytes
,
and
2●/73
parte
of
a
myte
.
¶
Item
when
ther
standeth
in
the
fyrste
nor
in
the
seconde
no
broken
nomber
,
but
in
the
latter
ende
a
hole
nomber
with
a
broken
,
as
here
1●
ounces
of
grayne
coste
14
sꝪ
.
what
for
9
ounces
of
grayne
and
one
thyrde
.
For
to
knowe
this
ye
shall
multiplye
the
fyrst
14
with
3
that
is
your
deuysor
,
then
multiplye
9
with
3
and
adde
that
1
therto
the
standeth
aboue
3
cometh
28
,
then
set
it
thus
4●
gyueth
14
what
gyueth
18
,
make
if
forth
after
the
rule
and
ye
shal
fynde
that
there
is
to
paye
9
shyllynges
4
pens
for
the
9
ounces
of
grayne
and
one
thyrde
parte
.
¶
Item
when
that
ye
fynde
neythere
at
the
beginnynge
nor
at
the
latter
ende
no
b●oken
nomber
,
but
in
the
myddes
a
hole
with
a
broken
as
here
.
A
man
bought
48
shepe
for
64
crones
and
●/4
what
shall
coste
18
shepe
,
For
to
soyle
this
questyon
ye
muste
multyplye
48
with
4
cometh
192
that
is
your
deuysor
,
then
multyplye
64
with
4
,
and
adde
therto
the
2
that
standdeth
aboue
4
,
cometh
258
,
the
whiche
ye
shall
multyplye
with
18
cometh
494●
the
which
ye
shall
dyuynde
with
192.
And
thus
ye
shal
fynde
that
ye
muste
paye
for
the
16
shepe
24
crones
4
stuners
&
36
mytes
¶
Item
when
ye
fynde
no
broken
at
the
begynnynge
but
in
the
seconde
and
thyrd
one
hole
with
a
broken
.
As
7
elles
for
6
pounde
¼
what
shall
cost
16
elles
&
●/●
For
to
soyle
this
question
ye
must
multyplye
the
two
vndermost
borken
nombers
togyther
,
in
sayenge
3
tymes
4
is
12
,
the
whiche
ye
shall
multiplye
with
7
cometh
84
that
is
the
dyuysor
.
Then
multyply
eche
with
his
broken
cometh
25
and
46
the
which
ye
shall
multiplye
one
with
the
other
,
cometh
1225
,
the
whiche
ye
shall
deu●●e
with
84
,
and
the
solucyon
shal
be
to
paye
14
li
12
sꝪ
.
18
d
¶
when
that
ye
fynde
at
the
begynnynge
and
the
myddes
a
hole
with
a
broken
,
and
at
the
latter
end
standyng
a
hole
without
a
broken
as
9
elles
&
●/4
for
5
pounde
⅝
what
shall
cost
15
elles
multiplye
9
with
4
and
adde
therto
3
cometh
59
,
the
whiche
ye
shall
multiplye
with
8
cometh
312
,
that
is
your
deuysor
then
multyply
5
with
8
and
adde
therto
the
5
that
standeth
aboue
8
cometh
45
the
which
ye
shall
multyply
with
4
that
standeth
vnder
the
fyrste
broken
,
cometh
180.
Now
set
it
in
the
rule
of
thre
in
sayeng
312
gyues
180
what
gyueth
15
,
make
i●
after
the
rule
and
it
cometh
8
li.
13
sꝪ
.
22
mites
and
1●/59
of
a
myte
.
☞
The
rule
of
company
.
¶
There
be
3
marchauntes
or
companyons
the
whiche
laye
togyther
theyr
money
in
marchaundyse
,
and
eche
to
wynne
after
his
inlayenge
,
wherof
he
fyrst
layd
in
170
crones
.
The
seconde
60
crones
The
thyrde
40
crones
and
therwith
they
haue
wonne
50
crones
besyde
al
vncost
Now
I
demaunde
howe
moche
that
eche
shall
haue
after
his
inlayeng
.
Now
for
to
soyle
this
question
and
all
suche
other
rules
of
company
,
ye
muste
make
of
theyr
money
that
they
haue
layd
in
a
totall
somme
cometh
250
/
now
saye
250
gyueth
50
what
gyueth
150
make
it
after
the
rule
of
ther
and
it
cometh
to
the
fyrste
man
30
crones
wynnynge
.
Nowe
for
knowe
what
the
seconde
hath
wonne
,
ye
shall
saye
250
gyueth
50
what
gyueth
60
make
it
after
the
rule
,
and
ye
shall
fynde
that
the
seconde
hath
wonne
12
crones
,
wyll
ye
knowe
what
the
thyrde
hath
,
then
saye
250
giueth
50
what
gyueth
40
make
it
after
the
rule
,
and
ye
shall
fynde
that
the
thyrde
hath
wonne
8
crones
.
And
thus
shal
ye
do
with
all
other
rules
of
company
.
The
rule
of
company
with
tyme
.
¶
Thre
felowes
doth
marchaundyse
togyther
,
wherof
the
fyrst
layeth
in
50
crones
for
4
monethes
.
The
seconde
80
crones
for
2
monethes
.
The
thyrde
100
crones
for
5
monethes
,
&
withall
this
mony
they
haue
wonne
6
crones
besyde
all
vncost
payde
.
Now
I
demaunde
what
eche
hath
wonne
with
his
money
,
for
to
know
this
ye
muste
mulyplye
eche
mannes
money
with
his
tyme
,
that
is
for
the
fyrste
50
with
4
cometh
200
,
set
the
as
thoughe
he
hath
layde
in
so
muche
,
for
the
seconde
multyply
80
with
2
cometh
160
set
it
also
as
though
he
had
layde
in
so
moche
,
for
the
thyrde
multyplye
10●
with
5
cometh
500
,
set
that
also
as
though
he
had
layde
in
so
moche
.
Nowe
adde
the
3
nombers
togythere
and
then
make
it
after
the
rule
of
companye
and
then
shal
ye
fynde
what
eche
hath
wonne
with
his
money
.
The
rule
of
baterynge
.
¶
Two
marchaunte
men
wyll
chaunge
theyr
wares
togyther
and
the
one
hath
a
fyne
blake
clothe
the
whiche
is
43
elles
longe
,
and
he
wyll
gyue
the
elle
no
lesse
than
18
pens
.
The
other
marchaunt
hath
peper
,
and
he
wyll
sell
the
pound
no
lesse
than
13
d
Now
I
demaunde
howe
many
pound
of
peper
the
fyrst
marchaunt
shall
haue
for
his
43
elles
of
clothe
.
For
to
soyle
this
question
ye
shall
say
13
gyueth
43
what
gyueth
18
,
make
it
after
the
rule
of
3
and
ye
shall
fynde
that
the
fyrste
shall
haue
for
his
clothe
59
pounde
of
peper
8
ounces
12
engelsche
and
4/13
parte
of
an
englysche
☞
Of
a
watte
.
¶
A
wat
ronneth
in
the
fylde
and
ouerronneth
in
one
mynute
/
there
be
60
in
an
houre
/
12
roddes
of
grounde
.
And
a
grahounde
beynge
her
enemye
/
foloweth
her
and
ouerronneth
in
one
mynute
15
roddes
of
grounde
.
But
or
the
grahonde
began
to
ronne
the
hare
had
ronne
200
roddes
of
londe
.
Now
is
to
be
demaunded
in
howe
many
minutes
and
how
many
roddes
of
londe
was
the
hare
taken
.
For
to
soyle
this
question
and
suche
lyke
,
ye
shal
subtra
the
lesse
ronnynge
out
of
the
more
that
is
12
out
of
15
,
and
there
remayneth
3
and
therwith
ye
shall
deuyde
the
space
that
the
hare
hath
ronne
afore
or
the
grahounde
began
to
ronne
/
that
is
200
roddes
,
And
in
so
doynge
ye
shall
fynde
that
the
grahounde
ouertoke
the
hare
in
the
66
mynutes
and
⅔
partes
of
a
mynute
that
is
one
houre
and
6
mynues
and
⅔
of
a
mynute
,
wyll
ye
knowe
how
many
roddes
that
the
grahonde
dyd
ronne
or
that
he
toke
the
hare
/
then
multiply
66
and
2/●
with
15
cometh
3000/●3
the
whyche
ye
shal
deuyde
with
3
cometh
1000
hole
so
many
roddes
dyd
the
hounde
ronne
or
that
he
toke
the
hare
.
¶
The
rule
of
two
felowes
.
¶
Two
felowes
went
togyther
out
of
a
towne
,
and
the
one
goeth
euery
daye
12
myle
,
and
the
other
goethe
the
fyrste
daye
but
1
myle
,
and
the
seconde
daye
2
myles
the
thryde
daye
3
myles
/
and
so
forth
euere
daye
one
myle
more
.
Now
I
demaund
in
howe
many
days
/
and
how
many
myle
went
he
or
that
he
ouertoke
his
felowe
.
For
to
soyle
this
questyon
,
ye
shal
double
the
myles
of
hym
that
wente
euery
daye
lyke
moche
,
that
is
12
and
2
tymes
12
is
24
,
therof
ye
shall
subra
the
one
myle
that
the
other
goeth
the
fyrste
day
,
and
there
resteth
23
,
vpon
the
same
daye
was
the
fyrste
man
ouertaken
of
his
companyon
?
wyll
ye
knowe
in
howe
many
myle
,
then
multiply
23
with
12
cometh
276
for
so
many
myles
went
he
or
he
ouertoke
hym
.
☞
A
man
hath
a
golden
crowne
of
34
stuuers
,
and
a
phylyppus
gulden
of
15
stuuers
,
and
a
ducate
of
28
stuuers
,
and
with
this
money
he
goeth
to
the
chaunger
and
wyll
haue
for
it
negenmannekens
crownes
of
9
mites
,
&
of
3
mytes
and
of
2
mites
and
halfe
mytes
.
Now
I
demaunde
how
moche
that
he
shall
receyue
of
eche
for
the
foresayde
golde
and
receyue
of
eche
lyke
moche
.
For
to
soyle
this
question
and
such
lyke
then
make
of
all
the
greate
money
that
he
wyl
chaunge
mytes
,
for
that
is
the
leste
coyne
that
he
wyll
haue
/
and
cometh
9072
mytes
,
then
loke
howe
many
mytes
that
al
the
smal
pens
be
worth
that
he
wyl
haue
,
that
is
25.
Nowe
deuyde
the
greate
somme
of
the
mytes
,
that
is
to
witte
,
9072
withe
25
and
ye
shall
fynde
that
he
muste
haue
of
eche
362
and
1●/25
and
it
is
done
,
¶
Of
four
carpenters
.
¶
Foure
carpentes
wyll
make
a
house
wherof
the
fyrst
taketh
vppon
him
to
make
it
him selfe
alone
in
a
yere
.
The
seconde
wyll
make
it
in
two
yeres
,
The
thyrd
wyl
make
it
in
3
yeres
.
And
the
fourth
in
4
yeres
/
Nowe
I
demaunde
yf
all
these
4
wrought
vpon
that
howse
in
what
space
wolde
they
4
make
that
howse
.
wyll
ye
know
that
,
then
say
the
fyrst
wolde
make
it
in
one
yere
that
were
12
tymes
in
12
yere
The
seconde
in
2
yere
,
that
were
6
tymes
in
12
yere
.
The
thyrd
in
thre
yere
that
were
4
tymes
in
12
yere
.
And
the
fourth
in
foure
yere
that
were
thre
tymes
in
12
yere
.
Nowe
somme
them
all
together
that
is
12
,
6
,
4
,
3
,
cometh
25
,
therwith
deuyde
12
cometh
12/25
prat
of
a
yere
Now
yf
ye
wyll
know
how
many
dayes
that
it
is
,
then
multyplye
12
wthy
365
for
so
many
dayes
be
in
a
yere
/
&
that
that
cometh
of
that
multyplycacyon
deuyde
it
by
25
cometh
175
dayes
and
5/25
parte
of
a
daye
.
¶
The
rule
of
false
posytyons
,
by
the
whiche
all
maner
of
difficult
and
harde
questions
maye
easelye
be
dyssoluyd
&
fyrste
of
one
salfe
posytion
.
NOw
shal
ye
knowe
how
by
false
posytions
or
coniectures
one
or
two
ye
shall
fynde
owt
the
very
trewthe
of
that
the
which
ye
seake
fore
,
and
fyrste
ye
shall
vnderstande
how
to
fynd
the
trewth
of
a
question
proposed
by
one
coniecture
or
posytyon
.
When
that
any
question
is
put
furthe
vnto
you
,
too
be
assoylyd
,
of
the
whiche
one
part
is
knowyn
and
the
other
vnknowyn
.
Answere
to
that
question
by
and
by
,
with
youre selfe
at
all
auenture
,
and
then
consyder
with
your self
whether
ye
haue
made
ryghte
answere
or
no
,
yf
not
,
loke
what
proportion
is
betwene
your
coniecture
and
that
that
folowyth
of
your
coniecture
and
the
same
proportion
is
betwene
the
thing
knowen
and
that
the
partaynyth
vnto
the
selfe
thyng
beynge
yet
vnknowen
.
As
by
example
ye
shal
more
playnly
perceaue
,
¶
A
certayne
way
farynge
man
commyng
by
the
way
,
founde
so
many
crownes
.
that
the
seconde
,
the
thyrde
,
&
the
fourth
parte
of
them
addyd
togyther
made
50
,
I
demaunde
what
summe
he
founde
.
To
make
answere
to
this
question
by
one
posycyon
,
ymagyne
some
summe
that
hath
these
partes
in
it
,
that
is
to
say
:
a
seconde
a
thyrde
,
and
a
fourth
part
,
and
be
it
12
,
whose
seconde
part
or
halfe
is
6
,
thyrde
parte
4
,
the
fourth
parte
3
,
whiche
all
addyd
togyther
643
,
make
13.
but
the
sum
that
he
founde
,
the
second
,
thyrde
,
and
the
fourth
of
it
made
50.
wherfore
12
is
not
the
som
he
founde
,
therfore
this
position
is
false
/
&
yet
by
this
false
shall
ye
come
to
the
lyght
of
treweth
,
by
the
helpe
of
the
rule
of
thre
.
For
loke
what
proporcyon
is
betwene
the
seconde
,
the
thyrde
,
and
the
fourthe
part
of
12
addyd
togyther
,
the
whiche
maketh
13
,
and
12
whose
partes
ther
be
,
the
same
proporcyon
is
betwene
50
whiche
is
the
seconde
,
the
thyrde
,
and
the
fourthe
parte
addyd
togyther
of
the
nomber
vnknowen
,
&
the
same
vnknowen
nomber
it selfe
.
Then
saye
thus
with
thy selfe
:
yf
13
whiche
contayneth
the
foresayd
partes
in
them
,
come
of
12
of
whome
come
50
,
then
set
them
thus
13
,
12
,
50
,
then
by
the
rule
of
thre
multyplye
50
by
12
,
and
therof
commeth
600
dyuyde
the
same
by
the
fyrste
nomber
13
and
in
the
quotient
thou
shalt
fynde
46
2/13
the
whiche
was
the
summe
of
the
corwnes
the
which
the
man
founde
:
of
the
whiche
summe
the
halfe
part
is
23
1/13
the
thyrd
part
is
15
1/1
,
the
fourth
parte
11
7/1●
the
whiche
partes
addyd
togyther
make
iuste
50.
Thus
thou
seest
how
that
by
one
false
posicion
or
coniecture
with
the
helpe
of
the
rule
of
thre
,
this
question
is
soone
dissoluyd
.
¶
An
other
questyon
.
¶
Fynde
me
a
number
in
the
which
5
is
⅔
that
is
to
say
,
two
thyrd
partes
of
hym
Answer
.
ymagyne
any
number
ye
lyste
3
that
hath
thyrdes
in
it
,
as
be
it
6
,
then
loke
what
is
the
thyrde
parte
of
6
,
that
is
2.
then
two
of
this
thyrde
partes
of
6
makyth
4
wherfore
this
posicyon
is
false
,
yet
by
this
false
posicyon
with
the
helpe
of
the
rule
of
thre
,
thou
shalt
fynde
out
the
trewth
,
after
this
maner
.
If
4
be
the
⅔
parte
of
6
to
whome
is
5
⅔
partes
serche
by
the
rule
of
thre
,
thou
shalt
fynde
it
5
⅓
¶
An
other
question
.
¶
What
number
is
that
in
the
which
after
that
the
thyrde
,
the
fourth
,
&
the
fyfte
parte
be
deductyd
out
of
it
,
there
shall
yet
remayne
24
?
Answer
.
ymagen
any
nomber
that
hath
a
thyrde
,
a
fourth
,
&
a
fyfte
in
hym
.
As
for
example
:
say
it
is
60
,
then
subtrahe
out
of
hym
his
third
,
his
fourth
and
his
fyfte
part
:
and
thou
shalt
fynde
remayne
but
13.
Lo
how
muche
thou
hast
myssed
thou
shuldest
haue
founde
suche
a
nomber
,
in
the
which
after
the
foresayd
partes
were
subtrahyd
shuld
remayne
24
and
here
remayneth
but
13
,
yet
proue
by
the
rule
of
thre
,
and
thou
shalt
fynde
the
trewe
numbe
.
If
13
remayne
after
the
substraccyō
of
the
aforesayde
partes
is
60
what
number
is
that
out
of
the
which
after
lyke
substraccyō
of
his
thyrde
,
fourth
,
&
fyfth
,
parte
shall
remayne
24
proue
by
the
rule
&
thou
shalt
fynde
it
110
●/1●
whose
thyrde
part
is
36
12/13
the
fourth
27
10/1●
the
fyfte
22
⅔
which
all
added
together
make
86
20/●●
the
which
deductyd
out
of
110
10/13
shall
remayne
24.
These
and
dyuerse
other
questions
before
rehersyd
by
the
same
crafte
one
false
posycyon
maye
soone
be
assoyled
.
Now
wyll
I
shewe
you
howe
to
dyssolue
all
maner
of
questions
how
dyffyculte
so
euer
they
be
by
two
false
posycyons
,
For
by
one
false
posycyon
ye
shall
〈◊〉
answere
to
all
maner
of
questions
,
but
two
false
posions
,
what
euer
questyon
it
be
/
it
may
soone
haue
solucyon
.
How
to
answere
by
two
false
posycyons
.
INnumerable
questyons
do
chaunce
in
numbres
,
the
whiche
though
they
can
not
be
dyssoluyd
by
one
posycion
or
coniecture
yet
shall
it
not
mysse
but
be
assoyled
by
two
posicions
:
in
the
whiche
maner
ye
must
dilygentlye
note
howe
farre
aboue
the
trueth
or
vnder
bothe
posycyons
do
fall
:
For
by
the
obseruacyon
of
.ii.
coniectures
how
nere
they
be
to
the
treuthe
,
&
the
dyfference
of
the
errors
whiche
ensue
of
the
posycyons
,
the
verite
commeth
to
lyght
which
may
be
done
.iii.
wayes
:
one
waye
by
the
rule
of
bothe
more
or
bothe
lesse
.
Another
waye
by
the
rule
of
the
one
more
and
the
other
lesse
¶
when
bothe
posycions
be
more
then
the
veryte
or
bothe
lesse
,
then
subduce
the
lesse
errour
out
of
the
more
errour
,
&
that
that
remayneth
shal
be
the
diuysor
:
then
multyplye
the
fyrste
errour
by
the
seconde
posycion
,
&
the
latter
errour
by
the
fyrste
posycion
,
and
then
this
.ii.
numbers
beynge
multyplyed
,
deduct
the
lesse
out
of
the
more
,
and
that
that
remayneh
diuide
it
by
the
foresayde
diuisor
,
and
the
quocient
shall
shewe
the
verite
.
Example
¶
Thre
marchauntes
diuided
a
100
crownes
so
that
the
seconde
shulde
haue
3
crownes
more
then
the
fyrste
,
and
the
thyrde
4
mo
than
the
second
:
I
demaunde
now
how
many
crownes
eche
of
them
receaued
?
Answer
.
Fyrste
make
saynte
Androwes
crosse
,
as
ye
se
hereafter
,
then
coniecture
what
ye
lyst
,
as
for
example
,
Say
the
fyrst
had
33
,
and
then
muste
the
seconde
haue
36
,
and
the
thyrde
marchaunt
40
,
whiche
sum
gathered
togyther
maketh
109
but
ye
had
but
100
to
dyuyde
,
wherfore
ye
haue
myssed
,
and
your
posycion
redowndyth
to
more
then
the
very
sum
by
9
,
whiche
came
of
your
fyrste
posycion
33
wherfore
set
the
fyrst
posycion
33
at
the
vpper
ende
of
the
crosse
on
the
left
syde
of
the
cosse
,
and
the
errour
which
hath
ensued
of
that
at
the
foote
of
the
crosse
on
the
same
syde
,
as
ye
se
in
the
example
.
And
for
bycause
that
this
coniecture
came
to
more
then
the
trueth
,
therfore
set
this
letter
M.
in
the
space
betwene
the
vpper
ende
of
the
crosse
and
the
nether
.
And
for
as
muche
as
in
this
fyrste
coniecture
ye
haue
erred
thus
much
,
coniecte
agayne
and
spupose
that
the
fyrste
marchaunt
had
31
then
muste
the
seconde
haue
34
,
the
thyrde
38
,
all
these
collecte
make
103
,
so
that
nowe
ye
haue
erred
agayne
,
your
posycion
beynge
to
much
/
so
that
your
errour
is
3
,
and
for
because
that
this
seconde
posycion
is
more
then
the
veryte
as
the
fyrste
was
,
set
the
posycion
32
at
the
vpper
ende
of
the
crosse
on
the
ryght
syde
,
&
the
errour
3
the
foote
of
the
crosse
on
the
same
syde
,
&
put
this
letter
M.
betwene
the
space
to
sygnyfey
more
.
Bothe
these
pasycion
then
be
more
then
the
verite
,
wherfore
accordynge
to
the
rule
fyrst
subduce
the
lesse
errour
3
at
the
fote
of
the
ryghte
syde
of
the
crosse
from
the
greater
errour
at
the
foote
of
the
lefte
syde
of
the
same
crosse
,
remayneth
●
to
be
set
in
the
space
betwene
both
the
feete
as
ye
se
:
which
shal
be
the
dyuysor
.
Then
accordynge
to
the
rule
,
multiplye
the
fyrst
posycyon
whiche
is
33
by
the
errour
of
the
seconde
posycyon
which
is
3
,
and
therof
cometh
99
,
then
the
seconde
posicion
31
by
the
errour
9
of
the
fyrste
posycion
,
and
therof
commeth
279
,
then
deducte
the
lesse
sum
99
out
of
this
more
summe
279
,
remayneth
180
,
dyuyde
this
summe
by
the
dyfference
of
the
errours
whiche
is
6
,
and
the
quocyent
shal
be
30
whiche
is
the
trewe
posycyon
:
For
the
fyrste
man
hauynge
30
,
the
seconde
must
haue
33
,
and
thyrde
37
,
whiche
all
set
togyther
,
make
iust
100.
Thus
wonderfull
craftly
by
these
.ii.
false
posycyons
the
trewe
&
iust
posycyon
is
brought
to
lyght
.
☞
An
example
when
bothe
posycyons
come
to
lesse
then
the
veryte
.
WHen
bothe
posycyons
come
to
lesse
then
is
the
veryte
,
the
whiche
is
all
one
matter
with
the
other
,
as
ye
shall
perceyue
by
the
same
example
agayne
.
As
suppose
ye
had
couiected
that
the
fyrst
had
receued
27
,
then
most
the
seconde
receaue
30
,
and
the
thyrde
34
,
whiche
addyd
togyther
make
91
whiche
is
lesse
by
●
then
the
sum
100
,
the
which
sholde
be
deuided
amonge
them
.
Set
then
this
fyrste
false
posycion
27
at
the
vpper
ende
of
the
crosse
on
the
lefte
syde
,
and
the
errour
ensuynge
of
that
at
the
foote
of
the
same
crosse
on
the
same
syde
.
And
for
bycause
the
this
posicyon
come
to
lesse
then
the
verite
therfore
sette
this
letter
L.
for
lesse
,
in
the
space
betwene
the
posicion
and
the
errour
,
as
ye
se
in
the
example
folowinge
.
Then
coniecte
agayn
and
suppose
that
the
fyrst
marchaunte
had
receaued
29
then
by
that
rekennynge
the
seconde
sholde
receaue
3●
the
thyrde
36
,
which
all
set
togyther
make
97
,
so
that
yet
this
posycion
commeth
not
to
the
veryte
100
,
but
lacketh
3
of
it
,
wherfore
set
this
posycion
29
at
the
vppar
ende
of
the
crosse
at
the
ryghte
syde
,
and
this
errour
3
at
the
foote
of
the
crosse
,
and
in
the
space
betwene
the
posycion
and
the
errour
set
this
letter
L
,
for
lesse
.
Nowe
for
so
much
as
bothe
these
posicyons
be
lesse
then
the
veryte
,
worke
as
ye
dyd
before
,
accordynge
to
the
rule
,
subduce
the
lesse
errour
3
out
of
the
greater
errour
9
,
remayneth
6
for
the
dyuisor
to
be
set
in
the
space
betwene
the
two
errours
.
Then
multiplye
the
fyrste
posycion
27
by
the
second
errour
2
commeth
81
,
then
the
seconde
coniecture
29
by
the
fyrst
errour
9
commeth
261
then
deducte
81
out
of
261
remayneth
180
whiche
diuyded
by
6
the
diuysor
afore
sayde
,
the
quotient
shal
be
30
whiche
is
the
iust
and
very
coniecture
,
the
whiche
ye
sholde
haue
coniected
.
Thus
ye
haue
had
sufficient
example
of
this
fyrst
rule
of
bothe
more
,
and
both
lesse
.
¶
Here
after
folweth
the
example
of
bothe
lesse
.
¶
Here
foloweth
the
rule
of
one
more
,
and
the
other
lesse
.
WHen
the
one
posycyon
amounteth
to
more
then
the
veryte
,
and
the
other
lesse
then
the
veryte
,
then
adde
the
errours
togyther
and
that
addyd
number
shal
be
the
diuysor
.
Then
multyplye
the
fyrste
posycyon
by
the
seconde
errour
,
and
the
seconde
posyon
by
the
fyrste
errour
,
and
that
that
commeth
of
bothe
these
multyplycatours
adde
them
togyther
also
,
then
diuyde
this
addyd
number
by
the
addyd
errours
the
dyuysor
aforesayd
,
&
the
quocyent
sheweth
the
true
posycion
.
¶
The
example
,
¶
we
wyll
take
the
fyrste
case
agayne
,
and
suppose
that
the
fyrste
marchaunts
haue
receaued
32
crownes
,
then
must
the
seconde
receue
35
,
&
the
thyrde
39
,
whiche
all
addyd
togyther
make
106.
wherfore
that
posycion
is
false
and
to
much
bye
set
the
posycion
32
at
the
vpper
ende
of
the
crosse
/
and
the
errour
6
at
the
nether
ende
of
the
crosse
in
the
space
betwene
,
ye
shall
set
this
letter
M
,
for
more
.
And
for
by
case
that
this
posycion
hath
excedeth
the
veryte
,
couiecte
agayne
lesse
,
and
suppose
that
the
fyrste
hath
receaued
29
,
then
must
the
seconde
receaue
32
,
the
thyrde
36
all
addyd
togyther
maketh
97
,
whiche
is
lesse
then
the
veryte
by
3
,
wherfore
set
this
false
posycion
29
at
the
vppar
ende
of
the
ryght
syde
of
the
crosse
,
and
the
errour
3
at
the
nether
foote
of
the
crosse
,
in
the
space
betwene
set
this
leter
L
,
for
lesse
.
Of
this
two
false
posicyons
the
one
is
more
then
the
trueth
the
other
is
lesse
,
wherfore
accordynge
to
the
rule
adde
both
the
errours
6
and
3
togyther
,
that
maketh
9
for
the
diuisor
:
then
multyplye
the
fyrste
posycion
32
by
the
seconde
errour
3
,
which
maketh
●6
.
and
the
seconde
posycyon
19
by
the
fyrste
errour
6
,
whiche
maketh
174
,
and
that
that
en●ueth
of
bothe
these
multyplycations
adde
it
togither
,
and
it
maketh
270
diuyde
this
addyd
number
by
the
addyd
errours
,
whiche
was
9
,
and
the
quocyent
shal
be
30
whiche
is
the
trewe
posycyon
as
ye
may
proue
.
¶
The
example
.
❧
Thus
maye
ye
dissolue
all
other
maner
of
questyons
,
whiche
haue
ben
set
before
in
this
boke
,
without
great
payne
or
studye
,
¶
Finis
.