THE
DIFFINITION
of
Nomber
.
NOmber
is
as
much
to
say
,
as
a
multitude
compound
of
manye
vnityes
,
as
two
is
compounde
of
two
vnities
,
three
is
compounde
of
three
vnities
,
foure
of
foure
vnities
,
fyue
of
fyue
vnities
ten
of
ten
,
fourtene
of
fourtene
,
fyftene
of
fyftene
,
twentie
of
twentie
vnities
,
&c.
And
therefore
an
vnitie
is
no
nomber
,
but
the
beginninge
and
originall
of
number
,
as
yf
you
doe
multiplye
or
diuide
an
vnitie
by
it selfe
,
it
is
resolued
into
it selfe
without
increase
.
But
it
is
in
nomber
otherwise
,
for
there
can
be
no
nomber
,
howe
great
so
euer
it
bee
,
but
that
it
maye
continuallye
be
encreased
by
addynge
euermore
one
vnitie
vnto
the
same
.
Numeration
.
Cap
.i.
NUumeration
is
the
art
wherby
to
expresse
and
declare
the
value
of
any
summe
proposed
,
and
is
of
two
kyndes
,
the
one
gathereth
the
value
of
a
sum
proposed
,
&
the
other
expresseth
anye
summe
conceyued
by
oue
figures
and
places
,
for
the
value
is
one
thinge
,
and
the
figures
are
another
thynge
:
and
that
commeth
partlye
by
the
diuersitye
of
fygures
,
but
chiefelye
of
the
places
wherein
they
be
ordeclye
sei
.
And
fyrste
marke
,
that
there
are
but
ten
fygures
or
characters
whiche
are
vsed
in
Arithmeticke
,
wherof
nine
of
them
are
called
signifiynge
fygures
,
and
the
tenth
is
called
a
ciphar
,
whiche
is
made
lyke
an
0.
and
of
it selfe
signifieth
nothinge
,
but
it
beynge
ioyned
with
any
of
the
other
figures
,
encreaseth
their
value
,
and
these
be
they
.
-
1
one
.
-
2
two
.
-
3
thre
.
-
4
four
.
-
5
fyue
.
-
6
syre
.
-
7
seuen
.
-
8
eyght
.
-
9.
nyne
.
Also
you
shal
vnderstande
that
euery
one
of
these
fygures
hath
two
values
:
One
is
alwaye
certaine
and
hathe
his
signification
of
his
owne
forme
,
and
the
other
is
vncertayne
whiche
be
taketh
of
his
place
.
A
place
is
called
the
scate
or
roome
that
a
figure
standeth
in
,
and
howe
many
fygures
so
euer
are
written
in
one
summe
,
so
many
places
hathe
the
whole
value
thereof
.
And
that
is
called
the
firste
place
(
whiche
nexte
is
towarde
the
ryghte
hande
)
of
anye
summe
,
and
so
recknynge
by
order
towarde
the
lefte
hande
,
so
that
,
that
place
is
laste
whiche
is
nexte
the
lefte
hande
.
And
contrarywise
,
when
you
expresse
the
value
of
the
fygures
in
anye
summe
you
muste
begynne
at
the
lefte
hande
,
and
so
recken
towarde
the
ryght
hande
.
Euerye
of
these
nine
fygures
,
(
whiche
are
called
signifiynge
fygures
)
hath
hys
owne
simple
valew
when
he
is
founde
alone
,
or
in
the
fyrste
place
of
anye
sume
.
In
the
seconde
place
towarde
the
lefte
hande
;
he
betokeneth
his
one
valewe
ten
times
.
As.
70.
is
,
seuen
times
ten
:
that
is
to
saye
seuentie
,
80.
is
viii
.
times
ten
:
that
is
to
saye
eyghtie
.
In
the
thirde
place
euery
fygure
betokeneth
his
owne
valew
a
hundreth
times
.
As.
700.
in
that
thirde
place
betokeneth
,
a
hundreth
tymes
.
7.
that
is
to
saye
,
seuen
hundred
:
In
the
fourthe
place
euerye
fygure
betokeneth
hys
owne
valew
a
thousande
times
.
As.
7000.
is
seuen
thousande
,
and
8000.
is
eyghte
thousande
.
These
foure
fyrste
places
muste
be
had
perfectlye
in
minde
,
ye
and
that
by
harte
,
for
by
the
knowledge
of
them
you
maye
expresse
all
kinde
of
nombres
howe
great
so
euer
they
be
.
In
the
fyfte
place
euery
figure
by
tokeneth
his
owne
valewe
tenne
thousande
times
.
As
70000.
is
ten
tymes
seauen
thousande
,
that
is
to
sayen
seauentie
thousand
:
In
the
sytte
place
euerye
fygure
standeth
for
his
owne
valew
,
a
hundreth
thousande
tymes
.
As
700000
,
is
seauen
hundreth
thousande
.
The
vii
.
place
a
M.
M.
times
,
or
a
million
:
as
7000000
,
is
vii
.
M.
M.
or
vii
.
millions
.
And
the
viii
.
place
.x.
M.
M.
times
,
or
ten
millions
,
so
that
euery
place
towarde
the
lefte
hande
,
excedeth
the
former
ten
times
.
But
nowe
for
the
easye
readynge
,
and
Redye
expressynge
orderlye
of
anye
summe
proposed
you
shall
practise
this
maner
,
folowing
.
And
for
example
I
propone
this
nomber
765432658.
in
the
whiche
are
nine
places
.
In
the
fyrste
place
is
8.
and
betokeneth
,
but
eyghte
,
in
the
seconde
place
is
5.
and
betokeneth
,
x.
times
fyue
that
is
fyfte
,
in
the
thirde
place
is
syxe
,
and
betokeneth
a.
C.
tymes
syxe
,
that
is
.vi.
C.
In
the
forthe
place
is
.
2.
and
that
is
twoo
.
M.
And
3.
in
the
fyfte
places
is
x.
M.
tymes
3.
that
is
xxx
.
M.
So
4.
in
the
syxie
place
is
a
C.
thousand
times
.
4.
that
is
.iiii.
C.
M.
then
fyue
in
the
seuenth
place
is
a.
M.
M.
times
5.
that
is
v.
M.
or
rather
5.
millyons
.
And
6.
in
the
viii
.
place
is
.vi.
times
x.
millions
,
that
is
lx.
millions
.
And
laste
of
all
.vii.
in
the.
ix
place
,
is
vii
.
C.
millions
now
foloweth
the
practise
.
Fyrste
put
a
prick
ouer
the
fourthe
fygure
,
and
so
ouer
the
seuenthe
,
and
lykewyse
ouer
the
tenthe
.
And
also
ouer
the
13
,
16.
or
19
if
you
had
so
many
,
and
so
stil
leauinge
twoo
fygures
betwene
euerye
twoo
prickes
and
those
roomes
from
one
pricke
to
an
other
are
called
ternaries
,
then
you
muste
pronounce
euery
three
fygures
from
one
pricke
to
an
other
as
though
they
were
written
alone
from
the
reste
.
And
at
the
ende
of
theyr
valewe
,
adde
so
manye
tymes
thousande
,
as
your
nomber
hathe
prickes
(
that
is
to
saye
,
yf
there
be
but
1.
pricke
,
it
is
but
one
M.
yf
two
prickes
a
M.
M.
or
els
a
million
yt
3.
prickes
a.
M.
M.
M.
or
a
M.
million
,
and
so
consequentlye
of
all
other
fygures
folowynge
.
)
Then
come
lykewyse
to
the
nexte
three
fygures
,
and
sounde
them
as
yf
they
were
aparte
from
the
reste
,
and
adde
to
their
valewe
so
manye
times
thousandes
as
there
are
pryckes
betwene
them
and
the
fyrst
place
of
your
whole
nomber
.
And
so
do
by
they
next
three
fygures
folowyng
and
of
all
the
reste
likewise
as
in
example
.
451234678567.
The
fyrste
pricke
is
ouer
.
8.
in
the
fourth
place
,
whiche
is
the
place
of
a.
M.
the
Seconde
pricke
is
ouer
4.
in
the
vii
.
place
,
whiche
is
the
place
of
a
M.
M.
or
one
million
,
the
third
pricke
is
ouer
the
.x.
place
whiche
is
the
place
of
a
M.
M.
M.
or
of
a
M.
million
,
as
in
the
former
example
.
Then
for
the
expressynge
of
this
nomber
by
the
valewe
of
euerye
fygure
,
accordynge
to
the
place
wherin
they
stande●
,
you
shall
fyrste
begynne
at
the
laste
prycke
ouer
one
,
and
take
yt
and
the
other
twoo
fygures
.
5.
and.
4.
whiche
do
folowe
him
&
valew
them
alone
and
they
are
iiij
.
C
li.
MMM
.
or
els
CCCC
li.
M.
millions
.
Then
take
the
other
iij.
fygures
from
one
to
the
nerte
pricke
,
and
valewe
them
as
if
they
were
a
parte
from
the
other
,
and
they
are
.
234.
which
are
.
CCxxxiiij
.
million
,
or
234.
MM.
Then
come
to
the
thyrde
pricke
ouer
8.
and
take
the
other
ii
.
fygures
behinde
it
,
and
recken
them
lykewyse
as
yf
they
were
alone
,
and
they
are
vi
C
lxxviii
.
M.
And
laste
of
all
come
to
the
other
twoo
fygures
whiche
remaine
,
that
is
.
567.
and
they
are
fyue
Clxvii
.
Thus
the
whole
sume
of
these
fygures
,
is
iiii
.
Cli.
M.
ii
Cxxxiiii
.
Millions
,
viC
.lxxviii.
M.
vClxvii
,
as
before
.
Note
also
that
whole
nomber
is
deuided
¶
And
here
foloweth
a
briefe
rehersall
of
the
order
and
Denominatours
of
the
places
.
And
this
shal
be
sufficient
for
Numeration
.
The
denominatours
.
of
the
places
.
|
|
The
order
of
the
places
.
|
|
M.
of
Millions
.
|
4
|
Tenth
.
|
|
C.
of
Millions
.
|
3
|
Nyneth
.
|
|
X.
of
Millions
.
|
2
|
Eyghteth
.
|
|
Millions
.
|
1
|
Seuenth
.
|
|
C.
of
Thousandes
.
|
0
|
Sixte
.
|
|
X.
Thousandes
.
|
1
|
Fyfthe
.
|
|
Thousandes
.
|
8
|
Fourth
.
|
|
Hundrethes
.
|
3
|
Thyrde
.
|
|
Tenthes
.
|
4
|
Seconde
.
|
|
Vnities
.
|
5
|
Fyrste
.
|
Addition
in
whole
nomber
Chap.
2.
ADdicion
is
as
muche
as
to
bringe
together
twoo
summes
or
more
into
one
,
as
if
there
were
due
to
anye
man
223
,
li.
by
seme
one
body
,
and
334
.li.
by
another
,
and
431.
by
another
,
&
you
woulde
knowe
howe
many
pound
is
due
to
the
same
man
in
all
,
these
iii.
summee
shall
you
set
downe
orderly
the
one
vnder
the
other
,
writinge
the
greatest
summe
highest
and
the
nexte
to
the
greatest
vnder
it
,
and
the
least
sum
vnder
the
laste
,
in
suche
sorte
that
the
firste
fygure
of
the
one
summe
by
directly
vnder
the
firste
figure
of
the
other
,
and
the
seconde
vnder
the
seconde
,
an
so
furth
in
order
,
when
you
haue
thus
done
drawe
vnder
them
a
straight
line
,
&
then
wyll
they
stande
thus
.
Nowe
begynne
alwayes
at
the
fyrste
places
towarde
yours
ryghte
hande
,
and
put
together
the
.iii.
firste
figures
of
these
.iii.
summes
,
and
looke
what
commeth
of
them
,
write
that
vnder
them
beneath
the
lyne
,
as
in
sayinge
.
3.
4.
and
1.
beinge
put
together
do
make
8.
write
8.
vnder
.
3.
as
thus
.
And
then
go
to
the
second
figures
and
doe
likewise
:
as
in
sayinge
.
2.
3.
and
3.
maketh
8.
write
8.
vnder
.
2.
as
here
you
see
.
And
lykewise
doe
wyth
the
fygures
that
be
in
the
thirde
place
,
in
sayinge
.
2.
3.
and
4.
are
.
9.
put
.
9.
vnder
them
and
so
wyll
your
whole
sum
appeare
thus
:
whereby
you
maye
perceyue
that
those
three
summes
beyng
added
together
do
make
988
li.
and
this
is
the
art
of
addicion
accordinge
to
his
simplicitye
,
if
the
sum
of
anye
place
doe
not
excede
a
diget
nomber
.
But
in
case
the
sum
of
anye
one
place
can
not
be
expressed
by
one
figure
but
by
two
you
shal
put
the
fyrste
of
those
figures
vnder
the
line
,
and
kepe
the
other
in
your
minde
for
to
adde
it
vnto
the
firste
figure
of
the
nexte
place
.
And
if
the
same
nexte
place
can
not
be
aualued
but
by
two
fygures
,
you
muste
in
lyke
maner
put
the
fyrste
of
those
fygures
vnder
the
lyne
,
and
reserue
the
second
for
the
other
place
nexte
after
,
and
thus
must
you
do
from
one
place
to
another
vntyll
you
haue
come
to
the
laste
place
,
where
in
case
you
doe
fynde
that
the
summe
be
of
twoo
fygures
,
you
muste
set
them
both
downe
because
it
is
the
ende
of
that
woorke
,
as
in
this
example
.
where
the
firste
figures
are
,
3.
1.
5.
6.
whiche
added
togither
maketh
15.
and
for
that
,
that
15.
is
of
twoo
figures
,
I
do
put
the
firste
fygure
5.
vnder
the
line
,
and
kepe
the
seconde
fygure
(
which
is
.
〈◊〉
)
in
my
minde
,
the
whiche
I
muste
adde
with
the
nexte
fygures
of
the
seconde
place
,
that
is
to
saye
with
2.
9.
4.
and.
5.
the
whiche
togither
make
21.
I
writte
1.
vnder
the
line
for
the
seconde
fygure
of
that
addition
,
that
is
to
saye
after
.
5.
&
I
kepe
2.
to
be
added
vnto
the
third
place
the
whyche
wyth
the
other
fygures
1.
8.
3.
4.
do
make
18.
therefore
I
put
.
8.
nexte
after
.
1.
in
the
third
place
vnder
the
line
,
and
kepe
1.
to
be
added
vnto
the
fygures
of
the
fourth
place
,
whiche
is
with
2.
7.
2.
2.
the
whiche
with
the
1.
that
I
kepe
doe
make
14.
I
set
downe
4.
for
the
fourth
fygure
(
vnder
the
line
)
that
is
to
saye
,
after
8.
and
I
kepe
1.
to
be
added
vnto
the
fygures
of
the
fyfte
place
,
the
whiche
is
7.
6.
3.
8.
with
the
1.
that
I
kepe
maketh
.
25.
I
put
.
in
the
fyfte
place
vnder
the
line
nexte
after
,
4.
and
kepe
2.
in
mynde
to
be
adoed
wyth
the
fygures
of
the
syxie
place
,
that
is
with
6.
4.
9.
6.
and
that
2.
whiche
I
kepe
,
maketh
27.
I
write
downe
7.
vnder
the
line
in
the
syxte
place
,
and
I
kepe
2.
whiche
I
adde
wyth
the
fygures
in
the
seuenth
place
,
and
they
make
13.
I
put
downe
3.
vnder
the
lyne
in
the
seuenth
place
,
and
adde
1.
vnto
the
fygures
in
the
eyght
place
and
they
are
10.
I
doe
put
0.
vnder
the
lyne
in
the
eyghte
place
,
and
then
I
adde
1.
vnto
the
nynethe
place
,
that
is
to
saye
with
4.
7.
and
they
make
12.
the
whiche
12.
I
writte
at
length
vnder
the
lyne
bycause
it
is
the
ende
of
this
addicion
,
and
this
is
to
be
done
of
all
suche
lyke
.
And
for
the
easyer
vnderstandinge
of
that
whiche
we
haue
spoken
of
addition
,
you
may
e●amen
these
two
other
examples
folowinge
,
in
the
whiche
the
fyrste
hath
these
nombers
.
3570.
2763.
579.
28.
whiche
beinge
added
toghther
doe
make
this
nomber
6940
,
and
in
the
seconde
example
dothe
resolute
this
nomber
51683.
by
addynge
together
of
these
nombers
,
47630.
3756
,
272
,
25
,
as
here
vnder
written
.
¶
Of
substraction
in
whole
nomber
.
chap.
3.
SUbstraction
teacheth
howe
you
shal
abate
one
lesser
number
from
a
greater
,
and
what
there
doth
reremaine
after
that
you
shall
haue
abated
the
same
,
I
speake
not
of
the
abating
of
one
egall
nomber
,
from
an
other
egall
vnto
it
,
for
the
facilitie
therof
requireth
no
Rule
.
In
substraction
are
founde
three
nombers
,
the
one
is
that
,
from
the
whiche
the
substraction
is
made
.
The
seconde
is
the
nomber
,
that
is
to
be
substracted
,
and
the
thirde
is
the
nomber
whiche
remaineth
after
the
substraction
is
ended
.
As
when
I
woulde
substract
.
25.
from
40.
The
40.
is
the
nomber
from
the
which
the
substraction
is
made
,
25.
is
the
nombre
to
be
substracted
,
and.
15.
is
the
nomber
whiche
remaineth
after
you
haue
done
the
substraction
,
here
foloweth
the
practise
.
You
shall
put
the
lesser
nomber
vnder
the
greater
in
such
sorte
,
that
euerye
figure
of
the
one
numbre
maye
aunswere
vnto
euerye
fygure
of
the
other
orderlye
,
and
then
drawe
a
right
lyne
vnder
those
twoo
numbers
,
as
you
dyd
in
Addition
.
Then
muste
you
beginne
at
the
right
hande
and
take
the
fyrste
figure
of
the
vndermoste
number
,
and
abate
that
from
the
first
figure
of
the
vppermost
numbre
,
and
that
whiche
remaineth
you
muste
set
vnderneth
the
line
right
vnder
that
figure
which
you
haue
substracted
.
Then
afterwarde
take
likewise
the
seconde
figure
of
the
neither●
moste
snumbre
,
and
abate
that
also
from
the
seconde
fygure
of
the
higher
numbre
.
The
thirde
from
the
thirde
,
and
soe
fourth
of
all
the
reste
tyll
you
come
to
the
ende
,
puttynge
alwayes
the
remaine
of
euerye
fygure
vnder
the
line
in
hys
order
,
example
.
I
will
subbstracte
.
2345.
from
9876.
after
that
I
haue
put
them
doune
accordynge
to
the
maner
aforesaied
.
I
take
firste
.
5.
from
.
6.
and
there
resteth
.
1.
the
whiche
I
sette
vnder
the
lyne
right
againste
.
5.
Secondlye
I
abate
.
4.
from
.
7.
and
there
resteth
.
3.
the
whiche
I
set
in
the
seconde
place
vnder
the
lyne
,
nexte
after
.
1.
Thirdly
,
I
abate
.
3.
from
.
8.
and
there
resteth
15.
The
whiche
I
put
vnder
the
line
in
the
thirde
place
,
finallye
I
doe
abate
.
2.
from
.
9.
and
there
resteth
.
7.
the
whiche
I
putte
vnder
the
lyne
in
the
fourth
and
laste
place
,
and
thus
is
this
substraction
ended
,
by
the
whiche
there
resteth
.
753.
But
when
twoo
fygures
of
one
likenes
doe
chaunce
to
meete
,
so
that
the
one
muste
bee
abated
from
the
other
,
as
if
I
shoulde
abate
seuen
from
seuen
,
there
remaineth
nothyng
,
and
then
muste
I
set
a
ciphre
.
0.
vnder
the
line
.
But
when
the
figure
whiche
is
to
bee
abated
,
dothe
excede
the
figure
whiche
is
ouer
hym
,
so
that
it
can
not
be
taken
out
of
the
same
figure
.
Then
muste
you
abate
the
neather
fygure
from
,
10.
And
that
whiche
dothe
remain
you
shall
adde
vnto
the
same
figure
whiche
is
vppermost
.
And
the
sum
whiche
commeth
therof
shall
you
set
vnder
the
line
.
But
when
so
euer
you
do
borow
any
such
.
10.
of
the
ouer
nomber
:
you
muste
adde
.
1.
vnto
the
nexte
neather
fygure
folowynge
whiche
is
to
bee
abated
.
And
there
is
nothinge
elles
to
be
done
in
substraction
.
Example
I
will
substract
93576.
from
.
4037479.
after
that
I
haue
placed
my
two
nombers
,
as
I
ought
to
do
,
I
do
fyrst
.
Abate
.
6.
from
9.
and
there
resteth
.
3.
then
I
put
the
3.
vnder
the
line
righte
against
.
6.
And
secondlye
,
I
abate
.
7.
from
7.
And
there
resteth
nothinge
.
I
doe
put
a
cipher
.
0.
vnder
the
line
right
againste
.
7.
in
the
seconde
place
.
Then
I
come
to
the
third
place
where
I
finde
.
5.
whiche
I
cannot
abate
from
the
fygure
ouer
hym
whiche
is
but
4
:
therfore
I
do
abate
it
from
10.
as
before
I
taught
,
and
there
resteth
.
5.
the
whiche
I
doe
adde
wyth
the.
4.
whiche
is
ouer
hym
,
and
that
maketh
.
9.
I
put
9.
in
thirde
place
vnder
the
line
for
the
thirde
fygure
,
fourthlye
for
the.
10.
whiche
I
borowed
I
adde
one
vnto
the
nerte
neather
fygure
whiche
is
3.
and
they
make
4.
the
whiche
I
doe
abate
from
the
ouer
fygure
.
7.
and
there
resteth
3.
I
put
.
3.
vnder
the
lyne
for
the
fourth
fygure
.
And
then
I
come
to
the
fyfte
place
where
I
do
fynde
.
9.
whiche
I
can
not
abate
from
the
fygure
ouer
hym
whiche
is
but.
3.
but
I
abate
9.
from
10.
and
there
resteth
.
1.
the
whiche
I
do
adde
with
3.
and
they
make
,
4.
I
put
.
4.
vnder
the
lyne
for
the
fyfte
fygure
.
And
if
it
were
not
,
for
that
I
did
laste
borowe
.
10.
the
substraction
shoulde
haue
bene
ended
.
But
for
bycause
that
I
muste
(
for
euerye
suche
tenne
that
I
borowe
)
alwayes
adde
.
1.
vnto
the
nexte
neather
figure
folowyng
,
I
muste
therefore
proceade
vnto
the
substraction
.
And
for
that
,
that
there
is
noe
other
figure
folowyng
in
the
neather
number
,
it
shall
suffice
to
haue
kepte
the
vnitee
,
and
to
abate
it
from
the
nerte
ouer
figure
.
But
I
find
there
.
0.
and
I
cannot
abate
.
1.
from
.
0.
therefore
I
abate
it
from
.
10.
and
there
resteth
nine
whiche
I
doe
putte
vnder
the
lyne
in
the
sixte
place
,
fynallie
for
the
tenthe
,
which
I
borowed
,
I
kéepe
.
1.
in
mind
.
The
whiche
I
doe
abate
from
.
4.
and
there
remaineth
thre
,
the
whiche
I
do
〈…〉
tte
vnder
the
lyne
in
the
seauenth
place
after
nine
.
And
the
operation
is
thus
ended
.
¶
Another
Example
.
But
if
there
were
many
numbers
to
be
substracted
from
one
numbre
alone
,
then
muste
you
firste
adde
those
numbers
together
according
vnto
the
doctrine
of
the
chapitre
goyng
before
,
and
then
to
make
your
substraction
as
aboue
saied
.
As
if
I
would
abate
these
thre
summes
.
123.
234.
456.
from
.
98925
first
I
do
adde
the.
3
summes
into
one
and
thei
are
813.
The
which
I
do
abate
from
98925.
and
there
resteth
98112.
¶
Of
multiplication
.
Chapiter
.
4.
IN
multiplication
there
are
thre
numbers
to
bee
noted
,
that
is
to
saie
the
numbre
whiche
is
to
bee
multiplied
,
the
whiche
wee
will
call
the
multiplicande
:
and
the
numbre
by
the
whiche
we
multiplie
,
we
call
the
multiplier
,
or
multiplicator
.
And
the
third
numbre
is
that
which
commeth
of
the
multiplication
of
the
one
by
the
other
,
which
is
called
the
product
.
As
when
I
would
knowe
how
much
mounteth
.
10.
multiplied
by
9.
that
is
to
saie
how
much
are
.
10.
tymes
9.
I
find
that
thei
ar
worth
90.
than
10.
is
the
multiplicāo
.
9.
is
the
multipter
.
And
90.
is
called
the
producte
:
Then
for
to
multiplye
,
is
none
other
thinge
,
but
to
fynde
a
nombre
whiche
conteineth
the
multiplicande
so
many
tymes
,
as
the
multiplyer
conteineth
vnities
.
As
10.
multiplyed
by
9.
do
make
90.
as
before
said
.
And
90.
conteyneth
10.
so
manye
tymes
,
as
9.
conteineth
vnities
,
that
is
to
saye
nyne
times
.
In
multiplycation
,
it
forceth
not
muche
whiche
of
the
twoo
nombers
by
the
multiplicande
,
nor
whiche
be
the
multiplier
.
For
10.
multiplyed
by
9.
maketh
as
manye
as
9.
multiplied
by
.
10.
yet
neuertheles
it
shal
be
more
commodious
that
the
lesser
nomber
be
alwayes
the
multiplier
.
And
for
that
,
that
the
multiplication
of
fygures
the
one
by
the
other
,
is
the
moste
chiefe
and
necessaryest
kynde
,
whereby
to
knowe
howe
to
worke
in
the
multiplication
of
compounde
nombers
,
and
that
euery
man
hath
it
not
at
the
fingers
ende
:
I
wyll
therfore
gyue
you
heare
certein
easye
wayes
of
multiplication
of
diget
nombers
.
When
you
woulde
multiplye
twoo
simple
fygures
,
or
digits
the
one
by
the
other
,
abate
eche
of
those
diget
nombers
from
10.
Then
multiplye
the
twoo
remaines
the
one
by
the
other
.
And
if
the
summe
doe
excede
10.
writte
onely
the
firste
fygure
,
and
kepe
the
other
to
be
added
to
the
nexte
operation
,
whiche
is
thus
.
Adde
your
two
simple
figures
togither
:
And
of
the
which
resulteth
of
that
addicion
,
take
onelye
the
fyrste
fygure
,
vnto
the
whiche
you
muste
adde
the
vnitye
whiche
you
kept
before
.
And
that
shal
be
the
seconde
fygure
of
the
summe
which
you
do
seeke
.
Example
,
I
woulde
multiply
7.
by
6.
I
take
7.
from
10.
and
there
resteth
3.
likewise
I
abate
6.
from
10.
&
there
resteth
4.
then
I
saye
thus
3.
times
4.
make
12.
I
writte
2.
for
my
first
fygure
,
&
I
kepe
1.
in
my
minde
,
then
I
adde
6.
with
7.
and
they
are
13.
of
the
whiche
I
caste
awaie
the
seconde
figure
.
1.
and
I
take
onelie
the
firste
fygure
.
3.
vnto
the
whiche
I
adde
the
vnitie
whiche
I
kepte
,
and
they
make
4.
whiche
I
wryte
in
the
seconde
place
after
.
2.
And
thus
I
finde
.
42.
whiche
is
the
valuer
of
7.
multiplied
by
.
6.
Otherwise
,
and
all
commeth
to
one
effecte
,
sette
doiune
youre
two
dig●●e
numbers
the
one
right
ouer
the
other
,
and
right
against
euery
or
them
towarde
the
right
hande
wryt
&
hys
owne
distaunce
from
.
10.
Then
multiplie
the
two
differences
together
,
the
figure
whiche
commeth
thereof
,
shall
you
set
doune
vnder
both
the
differences
.
But
if
there
be
two
fygures
sette
doune
but
the
fyrste
,
and
keepe
the
other
in
youre
mynde
,
afterwardes
abate
(
from
one
of
the
two
digit
numbers
)
the
difference
of
the
other
digite
number
that
is
to
saie
,
croswise
.
And
vnto
the
remayne
adde
the
fygure
whiche
you
kepte
,
and
that
shall
bée
the
seconde
nomber
,
and
thus
you
shall
haue
your
multiplication
.
Example
of
the
like
figures
,
that
is
to
saie
,
of
7.
multiplied
by
.
6.
the
distaunce
of
.
7.
vnto
.
10.
is
.
3.
And
the
distaunce
of
.
6.
from
.
10.
is
.
4.
I
sette
them
doune
crossewaies
,
as
you
see
:
And
then
I
saie
.
3.
tymes
.
4.
are
.
12.
I
sette
doune
.
2.
and
keepe
one
in
my
mynde
,
then
I
abate
.
4.
from
.
7.
or
els
.
3.
from
.
6.
it
forceth
not
from
whiche
of
theim
:
and
there
resteth
alwaies
.
3.
vnto
the
whiche
I
adde
the
vnitie
,
whiche
I
kepte
in
my
mynde
,
and
thei
are
fower
,
whiche
shall
bee
the
seconde
figure
of
the
multiplication
.
And
thus
I
finde
that
seuen
,
multiplied
by
sixe
,
maketh
fowertle
and
twoo
,
as
in
the
other
operation
.
This
practise
hath
no
place
where
the
twoo
digette
noumbers
(
dooe
not
exceade
tenne
)
by
addyng
theim
together
,
and
then
is
multiplication
easye
ynoughe
without
any
rule
.
An other
waye
to
knowe
the
multiplication
of
symple
numbers
,
is
by
this
table
folowyng
:
the
vse
whereof
is
thus
.
Fyrste
you
shall
vnderstande
,
that
the
nombers
from
1.
and
so
downewardes
to
9.
set
in
the
left
part
or
hanging
margin
of
this
table
do
betoken
the
multipliers
of
all
simple
nombers
.
And
the
elements
or
fygures
beyng
put
highest
in
euery
square
roome
drawynge
towarde
youre
ryght
hande
right
against
euery
of
the
multipliers
,
doe
signifie
,
the
multiplicands
,
vnto
the
multipliers
of
the
hangyng
margin
.
And
the
lower
or
inferiour
nombers
in
euerye
square
roome
,
do
betoken
the
product
of
that
multiplication
,
which
is
made
in
multiplyinge
the
vpper
nomber
ouer
it
,
wyth
the
figure
in
the
hanging
margin
,
answering
directly
vnto
the
sayd
square
:
as
by
example
.
THE
TABLE
OF
multiplication
by
all
the
diget
nombers
.
|
1
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
|
2
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
|
|
4
|
6
|
8
|
10
|
12
|
14
|
16
|
18
|
|
|
3
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
|
|
|
9
|
12
|
15
|
18
|
21
|
24
|
27
|
|
|
|
4
|
4
|
5
|
6
|
7
|
8
|
9
|
|
|
|
|
16
|
20
|
24
|
28
|
32
|
36
|
|
|
|
|
5
|
5
|
6
|
7
|
8
|
9
|
|
|
|
|
|
25
|
30
|
35
|
40
|
45
|
|
|
|
|
|
6
|
6
|
7
|
8
|
9
|
|
|
|
|
|
|
36
|
42
|
48
|
54
|
|
|
|
|
|
|
7
|
7
|
8
|
9
|
|
|
|
|
|
|
|
4
|
56
|
63
|
|
|
|
|
|
|
|
8
|
8
|
9
|
|
|
|
|
|
|
|
|
64
|
72
|
|
|
|
|
|
|
|
|
9
|
9
|
|
|
|
|
|
|
|
|
|
81
|
|
|
|
|
|
|
|
|
Firste
,
because
one
doeth
not
multiplie
,
I
sette
in
the
vpper
margin
the
figures
from
.
1.
to
.
9.
bothe
in
the
hier
and
also
in
the
inferiour
rowes
,
for
one
in
the
hangyng
margine
,
multiplied
by
one
the
vpper
nomber
in
the
firste
square
bryngeth
but
one
,
so
likewise
twoo
beyng
the
higher
nomber
in
the
seconde
square
of
the
vpper
margine
,
multiplied
by
.
1.
in
the
hangyng
margine
,
bryngeth
twoo
for
the
lower
nomber
in
the
seconde
square
of
the
vpper
margine
,
for
one
tymes
one
maketh
but
one
,
and
one
tymes
twoo
maketh
twoo
,
then
one
tymes
three
maketh
.
3.
and
one
tymes
fower
maketh
fower
,
and
so
continewyng
toward
the
right
hande
,
vntill
you
come
to
the
figure
of
9.
whiche
is
one
tymes
.
9.
maketh
.
9.
Then
after
multiplie
twoo
of
the
hangyng
margine
by
twoo
,
the
vpper
nomber
of
the
square
,
nexte
towarde
the
right
hande
,
and
that
maketh
fower
,
whiche
is
the
producte
of
twoo
,
multiplied
by
twoo
,
whiche
fower
is
set
vnder
the
twoo
,
for
twoo
times
twoo
are
fower
,
and
twoo
tymes
thred
maketh
sixe
,
then
twoo
tymes
fower
maketh
8.
and
twoo
tymes
.
5.
maketh
.
10.
and
so
conunewyng
vnto
twoo
tymes
.
9.
whiche
maketh
.
18.
The
like
is
to
bee
doen
with
the
thirde
rowe
,
and
so
likewise
of
all
the
residue
.
Example
,
I
would
knowe
what
is
the
producte
of
.
9.
multiplied
by
.
8.
I
seke
in
the
hangyng
margin
the
multiplier
eight
,
and
emongest
the
squares
directly
againste
eight
,
drawyng
toward
the
right
hande
,
I
seke
the
multiplicande
.
9.
in
the
higher
rowe
,
and
I
finde
the
producte
right
vnder
.
9.
to
bee
.
72.
Then.
72.
is
the
nomber
which
commeth
of
the
multiplication
of
.
9.
by
eight
,
and
so
is
to
bee
vnderstande
of
all
the
reste
of
the
table
,
whiche
table
muste
bee
of
all
menne
learned
by
harte
,
or
as
thei
saie
without
booke
,
whiche
beyng
learned
,
you
shall
the
better
attaine
to
the
resle
of
multiplication
.
To
come
nowe
vnto
the
practise
of
multiplication
,
when
you
woulde
multiplye
twoo
nombers
,
the
one
by
the
other
,
you
muste
set
them
downe
after
the
same
maner
as
you
dyd
in
addicion
,
and
in
substraction
.
That
is
to
saye
,
the
first
fygure
of
the
multiplier
,
vnder
the
firste
figure
of
the
multiplicande
,
the
second
vnder
the
seconde
,
and
the
third
vnder
the
third
,
yf
there
be
so
manye
,
and
then
drawe
a
ryghte
lyne
vnder
them
,
as
in
the
other
operations
goynge
before
.
After
you
shall
multiplie
all
the
fygures
of
the
multiplicande
by
the
multiplyer
,
and
set
downe
the
fygures
(
commynge
of
any
suche
multiplication
)
vnder
the
lyne
euery
one
in
order
.
Example
,
I
woulde
multiplye
123.
by
.
3.
that
is
to
saye
,
I
woulde
knowe
howe
muche
amounteth
iij.
tymes
one
hundreth
,
twentie
and
three
.
The
twoo
nombers
beynge
placed
in
suche
order
as
is
before
saied
,
you
muste
beginne
towardes
the
right
hande
:
and
say
thus
three
tymes
thrée
are
nine
,
wryte
downe
9.
vnder
the
line
right
againste
3.
for
the
firste
figure
:
secondlye
by
the
same
thrée
you
muste
multiplie
the
seconde
fygure
2.
and
they
make
6
:
putte
downe
6.
after
the
9.
vnder
the
line
:
Thirdly
by
the
same
3.
you
shall
multiplie
the
last
figure
.
1.
and
they
are
but
3.
set
doune
3
after
6.
for
the
thirde
and
laste
figure
.
And
thus
is
that
worke
ended
:
wherby
you
shal
finde
,
that
123.
being
multiplied
by
3.
maketh
369.
But
when
that
of
the
multiplication
of
one
figure
by
an
other
:
The
sum
whiche
commeth
thereof
shal
be
of
two
figures
,
as
it
happeneth
moste
often
,
then
shall
you
wryte
downe
the
firste
figure
,
and
kepe
the
other
figure
to
bee
added
vnto
the
multiplication
of
the
next
figure
.
Example
.
Syxe
menne
haue
gained
(
euery
one
of
them
)
345.
crownes
,
I
woulde
knowe
howe
manye
Crownes
,
thei
hadde
in
all
.
Fyrste
I
multiplye
6.
times
fiue
are
.
30.
I
wright
.
0.
vnder
the
line
,
and
keepe
.
3.
to
bee
added
to
the
next
multiplication
:
Secondlye
I
saie
6.
tymes
four
,
are
.
24.
vnto
the
whyche
I
adde
3.
whyche
I
reserued
.
And
thei
make
27.
I
wryt
.
7.
in
the
second
place
vnder
the
line
,
and
I
kéepe
2.
to
be
added
to
the
next
multiplication
,
thirdlie
I
saie
sire
tymes
.
3.
are
18.
vnto
the
whiche
I
adde
the.
2.
whiche
I
kepe
,
and
thei
make
.
20.
the
whiche
I
wright
all
doune
for
because
that
is
the
laste
woorke
.
And
so
I
fynde
that
345.
beyng
multiplied
by
.
6.
doe
make
2070.
Whan
the
multiplier
is
of
manie
figures
you
muste
multiplie
al
the
whole
multiplycande
by
euerye
one
of
those
figures
,
&
wryte
the
productes
euerie
one
vnder
his
owne
figure
.
Example
.
I
woulde
knowe
howe
manye
dayes
are
paste
from
the
natiuitie
of
Iesus
Christe
vntill
the
yeare
1560.
full
complete
.
I
haue
to
multiplie
.
1560.
by
365.
which
are
the
daies
of
one
whole
yere
.
The
leape
yeares
not
beyng
reckened
,
which
haue
euerie
one
of
them
.
366.
daies
.
First
by
the
figure
.
5.
I
multiplie
all
the
higher
fygures
,
saiyng
thus
.
5.
tymes
.
0.
maketh
.
0.
I
wryte
.
0.
vnder
the
lyne
for
the
firste
fygure
,
and
because
I
kéepe
nothyng
for
the
next
place
.
I
procede
and
saye
.
5.
tymes
6.
are
.
30.
I
sette
0.
vnder
the
lyne
for
the
seconde
fygure
,
and
I
keepe
.
3.
to
bee
added
to
the
next
multiplication
,
thirdlie
I
say
5.
tymes
.
5.
are
.
25.
The
whiche
with
3.
that
I
keepe
are
.
28.
I
sette
doune
8.
and
keepe
.
2.
to
bée
added
with
the
next
multiplication
.
Then
comming
vnto
the
fourth
and
last
fygure
,
I
saie
fiue
tymes
.
1.
are
.
5.
the
whiche
with
2.
that
I
reserued
are
7.
I
putte
7.
for
the
laste
fygure
of
thys
fyrste
operation
by
the
figure
.
5.
with
the
which
figure
we
haue
no
more
to
doe
.
And
therfore
I
cancel
thesame
.
8.
with
a
little
strik
through
it
,
to
signifie
that
wée
haue
finished
with
that
figure
.
And
for
as
muche
that
in
multiplication
there
is
alwaies
as
manie
simple
operations
,
as
the
multiplier
conteineth
figures
.
There
resteth
yet
twoo
operations
to
be
made
.
I
come
then
vnto
the
seconde
operation
,
whiche
is
by
the
fygure
6.
by
the
whiche
I
muste
againe
multiply
all
the
figures
of
the
multiplicand
as
I
did
by
5
and
the
first
figure
,
which
shal
be
produced
,
you
muste
putte
one
ranke
more
lower
then
the
figures
of
the
operation
euen
nowe
made
by
fiue
:
not
right
vnder
the
firste
fygure
of
the
multiplier
fiue
,
but
vnder
sixe
:
that
is
to
say
:
one
place
more
forward
then
the
fiue
towarde
the
left
hande
,
and
one
rancke
more
lower
then
the
firste
operation
:
and
you
shall
put
afterwarde
euerie
of
the
other
figures
whiche
commeth
of
the
same
multiplication
in
their
order
:
thirdlye
you
muste
make
the
multiplication
by
the
thirde
figure
and
that
whiche
shall
come
thereof
you
muste
sette
in
hys
rancke
,
as
here
vnder
you
see
.
And
nowe
wee
néede
make
no
further
discourse
hereof
,
because
that
hee
whiche
canne
doe
the
firste
multiplication
by
fiue
,
may
as
easely
doe
all
the
others
.
It
shall
therefore
suffice
to
sette
here
vnder
the
examples
.
Nowe
,
if
you
will
knowe
howe
muche
the
operations
thus
placed
doe
amount
vnto
,
which
in
value
are
but
one
number
:
you
must
adde
those
thre
numbers
together
,
but
not
after
the
same
maner
as
wee
haue
done
in
the
chapiter
of
addition
,
the
first
figure
of
the
first
ranck
with
the
firste
figure
of
the
second
ranck
,
and
of
the
thirde
:
but
you
muste
adde
them
in
the
same
sort
as
you
shall
finde
them
situated
or
placed
:
that
is
to
saie
,
the
firste
figure
of
the
fyrste
rancke
alone
by
it selfe
the
seconde
of
that
ranck
with
the
first
of
the
seconde
rancke
.
The
thirde
of
the
firste
rancke
wyth
the
seconde
fygure
of
the
thyrde
rancke
:
and
so
of
all
other
as
hereafter
doth
appeare
.
And
thus
the
1560
yeares
doe
containe
fyue
hundred
sixty
and
nine
thousand
four
hundred
daies
not
counting
herein
the
dayes
of
the
leape
yeares
,
whiche
are
heare
in
numbre
320.
then
the
whole
summe
of
the
daies
shoulde
be
569790.
¶
Another
Example
.
The
summe
of
multiplication
,
when
you
would
multiplie
any
number
by
10.
you
muste
onelye
adde
one
cipher
vnto
all
the
number
.
As
345.
multiplied
by
10.
maketh
3450.
yf
you
will
multiplie
by
.
100.
Adde
vn-the
whole
number
two
cyphers
.
00
:
yf
by
1000.
adde
.
000.
And
to
be
briefe
,
when
the
laste
figure
of
the
multiplier
is
.
1.
and
all
the
rest
be
ciphers
,
adde
so
many
ciphers
to
your
multiplicande
,
as
ther
shal
be
found
in
your
multiplier
But
yf
in
multipliyng
,
the
laste
fygure
were
not
one
,
but
that
there
were
onelye
certayne
Cyphers
in
the
beginnynge
:
and
that
the
other
were
signifiynge
fygures
,
and
likewise
those
of
that
multiplicande
,
then
shall
you
putte
those
cyphers
a
part
,
and
multiplie
the
signifiyng
figures
of
the
one
by
the
signifiyng
figures
of
the
other
.
Than
adde
vnto
the
product
of
that
multiplication
,
all
the
ciphers
whiche
you
dyd
before
put
a
part
.
As
if
I
would
multiply
.
46000
by
3500.
I
put
apart
the
thrée
ciphers
of
the
fyrste
,
and
twoo
ciphers
of
the
seconde
numbers
.
And
then
I
multiply
fourtie
sixe
by
35.
and
thereof
commeth
2610
:
vnto
the
whiche
I
add
the
00000.
and
then
the
whole
producte
will
be
.
161000000.
Of
diuition
the
5.
chapiter
.
DIuition
or
partition
is
,
to
seeke
how
many
times
one
numbre
dothe
containe
an
other
for
in
thys
operation
at
first
required
two
numbers
for
the
findyng
out
of
the
thirde
.
The
firste
numbre
is
called
the
diuidende
or
numbre
whiche
is
to
bee
deuided
,
and
that
muste
bee
the
greater
numbre
,
the
other
numbre
is
called
the
diuisor
,
and
that
is
the
lesser
.
And
the
thirde
numbre
which
we
seeke
is
called
the
quotient
.
As
if
I
woulde
deuide
36.
by
9.
the
diuidend
shal
be
36.
and
the
diuisour
is
9.
And
for
because
that
nine
is
conteined
in
36.
foure
times
,
that
is
to
saie
,
that
4
:
times
9.
do
make
36.
The
quotient
shal
be
4.
as
in
marking
how
many
tymes
9.
is
conteined
in
36.
The
practise
.
Write
doune
firste
the
diuidende
in
the
hygher
number
,
and
the
diuisor
vnderneth
,
in
suche
sorte
,
that
the
first
fygure
of
the
diuisour
towarde
the
lefte
hande
be
vnder
the
first
of
the
diuidend
,
and
euerie
figure
of
the
same
diuisour
vnder
hys
lyke
,
that
is
to
say
,
the
fyrste
vnder
the
firste
,
the
seconde
vnder
the
seconde
,
the
third
vnder
the
thyrde
,
and
so
consequentlye
of
the
other
,
yf
there
bée
anie
more
,
whiche
is
contrarie
to
the
other
three
kyndes
before
specified
,
but
you
muste
consyder
if
all
the
lower
fygures
of
the
diuisor
,
maye
be
taken
out
of
the
higher
figures
of
the
diuidence
,
by
the
order
of
substraction
.
The
whiche
if
you
canne
not
doe
,
then
muste
you
sette
the
fyrste
fygure
of
the
Dyuysour
(
towarde
the
lefte
hande
)
vnder
the
seconde
figure
of
the
diuidende
,
and
soe
consequently
the
reste
,
if
anie
bée
to
be
sette
doune
euerie
one
of
them
vnder
his
like
as
before
is
saied
.
And
than
drawe
a
line
betwene
the
diuidence
and
the
diuisour
.
And
at
the
ende
of
them
an
other
croked
line
,
behinde
the
whiche
towarde
the
right
hand
,
shal
be
set
your
quotient
.
As
by
this
example
following
wher
the
diuisor
is
but
of
1.
figure
If
you
woulde
diuide
860.
by
.
4.
you
muste
sette
doune
.
4.
vnder
the.
8.
with
a
line
betwene
them
as
herevnder
you
may
see
.
And
then
you
muste
seeke
howe
many
times
the
diuisour
in
contained
in
the
higher
numbre
,
or
diuidende
aunswering
to
him
,
as
in
this
our
example
I
muste
seke
how
many
tymes
4.
is
contained
in
.
8.
in
the
whiche
I
finde
2.
tymes
,
then
I
wryte
doune
.
2.
aparte
behinde
the
crooked
lyne
,
as
you
se
,
whiche
shall
be
the
firste
figure
of
the
quotient
to
come
,
secondlye
by
this
figure
(
beeynge
thus
putte
aparte
)
I
must
multiply
the
diuisor
:
and
vnder
the
same
multiplication
.
I
muste
sette
that
number
whiche
commcth
of
the
same
multiplication
,
as
two
tymes
foure
doe
make
eight
,
whiche
eight
I
doe
set
vnder
the
four
,
whiche
is
the
diuisour
.
Thirdlie
I
doe
substract
the
producte
of
the
saied
multiplication
(
of
the
quotient
by
the
diuisor
)
from
the
higher
numbre
corespondant
to
the
same
,
as
if
I
abate
8.
from
8.
there
rematneth
nothyng
,
and
then
I
cancell
or
stryke
out
that
whiche
is
doen
as
you
see
.
In
these
three
operations
is
comprehended
the
art
of
diuition
.
The
which
ar
to
be
obseraed
from
point
to
point
,
for
there
is
no
diuersity
in
the
finishyng
of
the
same
whiche
is
thus
.
I
muste
remoue
mye
diuisor
one
place
nerer
towarde
my
right
hande
,
as
in
procedyng
with
.
oure
example
,
I
remoue
my
diuisour
4.
whiche
was
vnder
.
8.
and
I
set
it
vnder
.
6.
then
I
seeke
howe
manye
tymes
.
4.
is
conteined
in
.
6.
where
I
finde
but
one
tyme
,
then
I
sette
.
1.
behinde
the
croked
line
behinde
.
2.
afterwarde
by
this
laste
and
new
figure
.
1.
I
multiplie
the
diuisour
.
4
and
that
maketh
but.
4
(
for
an
vninitie
whiche
is
but.
1.
encreaseth
nothyng
)
I
abate
4.
from
the
higher
figure
.
6.
and
there
resteth
.
2.
the
whiche
2.
I
sette
ouer
the
6.
and
I
cancell
the.
6.
for
so
must
you
do
when
there
resteth
any
thing
after
you
haue
made
the
substraction
.
Thirdlye
for
that
there
yet
remaineth
an
other
,
figure
in
the
diuidend
,
I
remoue
againe
the
diuisour
,
and
I
sett
vnder
the
cipher
.
0.
Then
I
seeke
how
many
tymes
foure
is
in
the
higher
number
whiche
is
.
10.
where
I
fynde
5.
times
,
I
put
.
5.
behinde
the
crooked
iyne
for
the
third
and
last
figure
of
the
quotient
.
Then
by
the
same
5.
I
multiplie
the
diuisour
4.
and
that
maketh
20.
the
whiche
I
abate
from
the
higher
number
,
and
there
resteth
nothing
.
And
so
is
thys
diuisyon
ended
:
and
I
haue
founde
that
.
860.
being
diuided
by
foure
,
bringeth
for
the
quotient
215.
that
is
to
saie
,
that
.
4.
is
conteined
in
860.
twoo
hundred
and
fiftene
tymes
.
This
is
the
moste
eastest
workynge
that
is
in
diuision
,
but
that
whiche
foloweth
,
appertayneth
to
the
whole
and
perfecte
vnderstandyng
of
the
same
.
When
the
firste
figure
of
your
diuisor
towarde
your
lefte
hande
is
greater
then
the
firste
of
diuidende
,
you
must
not
place
the
firste
figure
of
your
diuisor
right
vnderneth
the
first
of
your
diuidend
,
but
vnder
the
2.
figure
of
the
same
diuidende
,
nerer
to
your
right
hand
,
as
before
is
saied
.
When
the
diuisour
is
of
manye
figures
,
and
that
you
haue
to
séeke
howe
manye
tymes
it
is
contained
in
the
higher
numbre
(
for
the
more
easier
workyng
)
you
must
not
seeke
to
abate
the
diuisour
all
at
one
tyme
,
but
you
muste
see
and
marke
howe
maye
tymes
the
firste
figure
of
the
same
towarde
the
leste
hande
is
contained
in
the
higher
numbre
aunsweringe
to
the
saied
numbre
,
and
then
to
work
after
the
same
maner
as
is
before
taught
.
Erample
.
I
haue
.
316215.
crownes
to
be
deuided
amonge
.
45.
menne
for
to
make
my
diuision
,
I
muste
not
putte
the
firste
figure
of
the
diuisour
whiche
is
.
4.
vnder
the
firste
of
the
deuidende
,
whiche
is
.
3.
because
that
.
4.
is
greater
number
then
.
3.
And
further
,
I
can
not
take
.
4.
out
of
.
3.
wherfore
I
muste
sette
the.
4.
vnder
the
seconde
figure
of
the
higher
number
whiche
is
.
1.
and
the
fygure
.
5.
of
the
diuisour
next
right
vnder
the
6.
as
you
maie
sée
.
I
muste
firste
séeke
,
home
many
tymes
45.
ie
contayned
in
316.
whiche
is
but
parte
of
the
diuidende
,
wherefore
for
the
more
casie
workyng
I
nede
but
to
seke
how
many
tymes
4.
is
conteined
in
31.
&
because
I
may
haue
it
7.
tymes
I
put
.
7.
behinde
the
croked
line
,
as
is
afore
saied
,
then
by
.
7.
I
multiply
all
the
diuisour
.
45.
and
they
are
315
:
the
whiche
I
set
vnder
the
same
diuisour
,
the
fyrste
figure
vnder
the
fyrste
.
And
the
other
in
order
towarde
the
lefte
hande
.
Then
I
substract
thre
hundred
fiftine
,
from
the
higher
number
.
316.
and
of
this
fyrst
working
there
remaineth
but.
1.
the
whiche
I
sette
ouer
the
6
and
I
cancell
the
315
,
and
the
other
figures
3
,
1
,
6
,
also
the
diuisour
:
and
then
it
will
stande
thus
.
And
when
I
come
to
remoue
the
diuisour
,
and
that
I
muste
séeke
howe
manye
tymes
it
is
contained
in
the
higher
numbre
,
if
I
se
that
I
can
not
finde
it
there
,
that
is
to
saie
that
if
the
higher
numbre
be
lesser
than
the
diuisonr
,
as
it
is
in
this
example
,
then
must
I
put
a
cipher
in
the
quotient
behind
the
croked
line
,
and
if
ther
remaine
anye
fygures
in
the
diuidende
whiche
are
not
finished
,
I
must
remoue
t
〈…〉
diuisor
againe
nerer
towarde
my
right
hande
by
one
place
,
for
to
finde
a
newe
figure
in
the
quociente
.
As
in
this
our
example
,
for
after
that
I
haue
remoued
the
diuisor
,
I
seke
how
many
tymes
.
45.
is
conteined
in
.
12.
and
because
I
can
not
haue
.
45.
in
.
12.
I
put
a
0.
behinde
the
croked
line
after
7.
then
without
multipliyng
or
abatyng
,
I
remoue
again
the
diuisor
nerer
toward
my
right
hand
,
and
I
seke
how
many
tymes
4
(
which
is
the
first
figure
of
the
diuisour
)
is
in
the
higher
nomber
,
that
is
to
saie
,
in
.
12.
whereas
I
find
it
3.
times
I
putte
.
3.
behinde
the
croked
line
,
for
the
thirde
figure
of
the
quocient
:
then
by
.
3.
I
multiply
the
diuisor
.
45.
and
thereof
commeth
135.
But
here
is
to
bee
noted
,
that
if
it
happen
that
the
figure
beyng
laste
founde
,
whiche
is
put
in
the
quotient
,
doe
produce
or
bryng
for
the
a
greater
number
(
in
multipliyng
al
the
diuisor
by
the
same
)
then
that
whiche
is
ouer
the
saied
diuisor
:
you
must
then
make
the
same
figure
of
your
quotient
(
whiche
you
dooe
put
doune
)
lesser
by
one
.
and
after
that
you
haue
cancelled
the
firste
multiplication
,
you
muste
make
a
newe
.
And
the
same
must
be
so
doen
as
often
tymes
:
as
(
in
decreasyng
the
same
)
it
produceth
a
lesser
number
,
or
at
the
leaste
,
a
number
egall
to
that
,
whiche
is
ouer
it
.
As
in
the
laste
work
for
because
that
the
diuisour
,
beyng
multiplied
by
.
3.
bryngeth
for
the
135.
whiche
amounteth
more
then
121.
the
same
producte
must
be
cancelled
.
And
likewise
the
figure
.
3.
whiche
I
did
put
in
the
quotient
,
must
be
chaunged
into
a
figure
of
.
2.
Then
by
the
saied
.
2.
I
must
multiplie
the
diuisor
.
45.
&
therof
commeth
90.
the
whiche
I
abate
from
121.
and
there
remaineth
.
31.
And
then
will
the
somme
stande
thus
.
And
here
is
also
to
be
noted
,
that
the
But
in
case
that
after
the
diuision
is
ended
,
there
doe
remaine
any
thyng
in
the
diuidende
,
as
moste
often
times
there
doeth
:
I
must
then
sette
that
remaine
aparte
behinde
the
croked
line
,
after
the
entier
quotiente
,
and
the
diuisor
,
right
vnder
thesame
remaine
,
with
a
line
betwene
them
bothe
,
as
in
this
diuision
followyng
,
where
there
remaineth
.
3.
in
the
last
woorke
of
the
same
.
And
we
shall
see
what
the
same
doeth
signifie
,
when
we
shall
treate
of
fractions
,
or
broken
nombers
.
In
somme
,
all
the
whole
practise
of
diuision
,
maie
be
kept
in
remembraunce
by
three
letters
,
that
is
to
saie
:
S.
M.
A
whiche
three
letters
dooe
signifie
to
seeke
,
to
multiplie
,
to
abate
.
Firste
,
I
must
seeke
how
many
tymes
the
diuisor
is
conteined
in
the
higher
nomber
:
then
,
by
the
quotiente
(
whiche
I
finde
)
I
must
multiplie
the
diuisor
:
finally
,
I
must
abate
the
producte
of
that
multiplicacion
,
from
the
higher
nomber
to
thesame
correspondent
,
that
is
to
saie
:
out
of
the
diuidende
aunsweryng
to
the
diuisor
.
And
further
,
besides
this
kinde
of
woorkyng
in
diuision
.
The
whiche
is
reguler
and
common
:
I
will
here
put
an
other
maner
of
woorkyng
verie
easie
.
The
whiche
shall
serue
for
suche
diuisions
as
are
difficil
to
be
wrought
.
That
is
to
witte
,
when
the
nomber
to
bee
diuided
is
verie
greate
,
and
the
diuisor
greate
also
,
and
it
shall
serue
againe
for
to
auoide
errour
in
supputacion
,
and
for
the
placyng
of
fewer
figures
in
the
quotiente
:
and
consequently
it
shall
saue
muche
labor
vnto
them
,
whiche
as
yet
haue
not
muche
studied
in
this
art
.
The
practise
wherof
is
thus
,
as
followeth
.
I
haue
to
deuide
.
7894658.
by
643.
In
the
firste
place
,
you
shall
vnderstande
,
that
although
the
firste
figure
of
the
diuisor
towarde
your
lefte
hande
,
maie
bee
founde
many
tymes
in
the
higher
number
,
as
.
10.
tymes
,
12.
tymes
or
more
:
yet
is
it
so
,
that
you
must
neuer
putte
but
one
figure
onely
at
a
tyme
in
your
quotient
.
And
thus
you
shall
at
no
tyme
put
any
number
in
your
quotient
,
whiche
exceadeth
the
figure
of
.
9.
that
is
to
saie
,
any
number
beyng
greater
then
9.
for
to
come
then
vnto
your
practise
,
write
donne
your
diuisour
one
tyme
:
and
behinde
it
towarde
youre
righte
hande
,
drawe
a
line
doune
straighte
,
and
right
against
thesame
diuisor
behinde
the
line
put
this
figure
.
1.
Then
double
your
saied
diuisour
,
and
right
againste
the
same
(
beyng
doubled
)
put
behinde
the
line
the
figure
of
.
2.
After
adde
vnto
the
same
number
(
whiche
you
doubled
)
your
saied
diuisour
,
and
right
againste
the
same
producte
,
behinde
the
line
putte
the
figure
of
3.
And
vnto
this
thirde
producte
,
you
muste
adde
againe
your
diuisor
:
and
right
againste
the
same
producte
,
behinde
the
line
sette
the
figure
.
4.
And
this
muste
you
dooe
,
vntill
you
come
to
the
figure
of
.
9
:
in
suche
sorte
that
euery
of
the
productes
doe
surmounte
so
muche
his
former
noumber
,
as
all
the
diuisoure
dooeth
amounte
vnto
:
placyng
at
the
right
side
of
euery
producte
behinde
the
line
,
the
noumber
whiche
signifieth
howe
muche
he
is
in
order
.
That
is
to
saie
,
righte
againste
the
fifte
producte
,
you
muste
putte
.
5.
right
againste
the
sixte
producte
,
you
must
put
.
6
:
and
so
likewise
of
all
the
other
.
Example
of
the
diuisour
proponed
,
643.
firste
,
I
write
doune
.
643.
and
right
against
the
same
behinde
the
line
,
I
put
.
1.
secondly
,
I
double
643.
and
they
make
.
1286.
and
righte
againste
hym
behinde
the
line
I
put
.
2.
Thirdly
,
vnto
that
same
.
1286.
I
adde
the
diuisor
.
643.
and
thei
are
1929.
and
right
againste
the
same
I
set
.
3.
Fourthly
,
vnto
the
saied
1929.
I
adde
the
diuisor
.
643.
and
thei
are
2572.
and
right
againste
the
same
I
putte
.
4.
And
thus
must
you
dooe
alwaies
by
encreasyng
so
muche
euery
producte
,
as
the
diuisor
doeth
amount
vnto
,
vntill
you
haue
so
doen
nine
tymes
,
as
you
see
in
this
present
table
.
This
beyng
dooen
,
you
muste
sette
downe
your
diuisor
vnder
the
diuidende
,
after
the
same
maner
as
is
before
declared
:
that
is
to
saie
.
643.
vnder
the
three
firste
figures
of
the
diuidende
,
towarde
your
righte
hande
,
which
are
.
789
:
Then
must
you
seke
how
many
times
.
643.
are
conteined
in
.
789
:
And
for
to
knowe
the
same
,
I
looke
in
my
foresaied
table
,
if
I
maie
there
finde
the
same
nombers
.
789.
the
whiche
is
not
there
:
Therefore
I
must
take
a
lesser
nomber
,
the
nereste
to
it
in
quantitie
,
that
I
can
finde
in
the
table
,
the
whiche
is
.
643.
whiche
nomber
hath
againste
it
on
the
righte
hande
of
the
line
this
diget
.
1.
Then
I
take
the
said
.
1.
and
I
put
it
behind
the
croked
line
,
for
the
firste
figure
of
the
quotiente
.
Then
I
doe
abate
.
643.
from
.
789
and
there
remaineth
.
146.
the
which
I
putte
ouer
the.
789.
and
I
cancell
the.
789.
and
thus
is
the
firste
operacion
ended
.
Then
I
sette
forewarde
the
diuisour
,
one
figure
nerer
to
my
right
hande
,
and
I
seeke
a
newe
quotient
,
as
I
sought
this
,
where
I
finde
the
higher
number
ouer
my
diuisour
to
bée
.
1464.
The
whiche
I
doe
seeke
in
the
table
,
and
because
I
can
not
finde
it
there
,
I
take
a
lesser
number
,
the
nigh●st
to
it
that
I
can
finde
,
and
that
is
.
1286
:
whiche
number
hath
againste
it
this
digette
.
2.
I
putte
.
2.
for
the
seconde
figure
of
the
quotient
behinde
the
line
,
and
I
dooe
abate
.
1286.
from
.
1464.
and
there
remaineth
.
178
Thirdly
,
I
remone
forward
the
diuisor
,
as
before
,
and
I
finde
the
higher
number
to
be
.
1786.
and
that
the
next
lesser
number
to
it
in
my
table
,
is
againe
.
1286.
I
putte
the
refore
ones
againe
.
2.
in
the
quotient
for
the
thirde
figure
:
and
I
abate
.
1286.
from
.
1786.
and
there
remaineth
.
500.
Fourthly
,
I
set
forward
the
diuisor
and
the
higher
number
ouer
it
,
is
.
5005
and
the
next
lesser
number
to
it
in
my
table
is
.
4501.
right
against
the
which
noumber
is
.
7.
I
putte
.
7.
in
the
quotiente
,
for
the
fowerth
figure
.
And
after
that
I
haue
abated
.
4501.
from
5005.
there
remaineth
.
504.
Finally
,
I
remoue
forward
my
diuisor
vnto
the
laste
place
:
and
I
finde
the
higher
number
to
bee
.
5048.
And
the
nexte
lesser
noumber
to
it
in
my
table
,
is
.
4501.
I
sette
.
7.
againe
in
the
quotient
,
for
the
fifte
and
laste
figure
.
Then
I
take
.
4501.
from
.
5048.
and
there
remaineth
.
547.
whiche
must
be
put
at
the
ende
of
the
whole
quotieute
with
the
diuisor
vnder
it
,
and
a
line
betwene
them
,
in
this
maner
folowing
.
¶
The
somme
of
deuision
.
WHen
you
would
deuide
any
number
by
.
10.
you
muste
take
awaie
the
laste
figure
nexte
towardes
your
right
hand
,
and
the
rest
shal
be
the
quotient
.
As
if
you
would
diuide
.
46845
by
10.
take
awaie
the.
5.
&
then
.
4684.
shal
be
the
quotiente
,
and
the.
5.
shal
be
the
number
that
doeth
remain
.
Likewise
when
you
would
diuide
any
number
by
.
100.
take
awaie
the
twoo
laste
figures
towardes
your
righte
hande
,
and
if
you
would
diuide
by
.
1000
take
awaie
thre
figures
,
if
by
.
10000.
take
awaie
fower
figures
.
And
so
of
all
other
,
when
the
first
figure
of
the
diuisor
toward
the
lefte
hande
,
shal
be
onely
.
1.
and
the
reste
of
the
same
diuisour
beyng
but
ryphers
.
¶
Here
foloweth
the
proofes
of
addition
,
substraction
,
multiplication
,
and
diuision
.
¶
The
proofe
of
addition
.
WHen
you
would
proue
,
whether
your
addition
boe
well
made
,
consider
the
figures
of
the
numbers
,
whiche
bee
added
,
euery
one
in
his
simple
value
:
not
hauyng
any
regarde
to
the
place
where
he
standeth
,
but
to
recken
hym
as
though
he
were
alone
by
hymself
,
and
then
recken
them
all
,
one
after
an
other
,
casting
awaie
fram
them
the
number
of
9.
as
ofte
as
you
maie
.
And
after
youre
discourse
made
,
kepe
in
minde
thesame
figure
whiche
remaineth
after
the
nines
be
taken
awaie
,
or
sette
thesame
in
a
voide
place
at
the
vpper
ende
of
a
line
.
For
if
your
addition
be
well
made
,
the
like
figure
will
remaine
,
after
that
you
haue
taken
awaie
all
the
nines
,
out
of
the
totalle
somme
of
thesame
addition
,
as
often
as
you
may
ther
finde
any
:
as
in
this
addition
which
here
you
se
.
There
remaineth
.
2.
for
eche
part
¶
The
proofe
of
substraction
.
ADde
the
number
whiche
you
doe
substracte
,
with
that
number
which
remaineth
after
the
substraction
is
made
:
and
if
the
totall
somme
of
that
addition
,
bee
like
vnto
the
number
from
the
whiche
the
substraction
was
made
,
you
haue
dooen
well
,
otherwise
not
:
as
in
this
example
dooeth
appeare
,
where
you
see
the
number
whiche
is
to
bee
substracted
,
is
.
3584.
and
the
number
whiche
doeth
remaine
,
is
1879.
the
whiche
twoo
sommes
beyng
added
together
,
doe
make
5463.
which
is
like
to
the
higher
number
,
out
of
the
which
the
substraction
was
made
,
as
before
is
sated
.
¶
The
proofe
of
multiplication
.
THe
proofe
of
multiplication
,
is
made
by
the
help
of
diuision
,
for
if
you
diuide
the
number
produced
of
the
multiplication
,
by
the
multiplier
:
you
shal
finde
the
higher
number
,
whiche
is
the
multiplicande
.
¶
The
proofe
of
diuision
.
TO
knowe
if
your
diuision
be
wel
made
:
you
must
multiplie
all
the
quotiente
by
your
diuisor
,
and
if
any
thyng
remained
after
youre
diuision
was
made
.
The
same
shall
you
adde
vnto
the
producte
,
whiche
commeth
of
the
multiplication
:
and
you
shal
finde
the
like
number
vnto
your
diuidende
if
you
haue
wel
diuided
:
otherwise
not
¶
Of
progression
,
the
vi
.
Chapiter
.
PRogression
Arithmeticall
,
Then
if
you
will
adde
quickly
the
number
of
any
progression
,
you
shall
doe
thus
,
firste
tell
how
many
numbers
there
are
,
and
write
their
somme
doune
by
it self
,
as
in
this
example
,
2.
5.
8.
11.
14.
where
the
numbers
are
5
as
you
maie
see
,
therefore
you
muste
sette
doune
.
5.
in
a
place
alone
,
as
I
Then
shall
you
adde
the
first
number
,
and
the
laste
together
,
whiche
in
this
example
are
.
14.
and.
2.
and
thei
make
.
16.
take
halfe
thereof
,
whiche
is
.
8.
and
multiplie
it
by
the.
5.
whiche
I
noted
in
the
margente
,
for
the
nomber
of
the
places
,
and
the
somme
whiche
amounteth
of
that
multyplication
,
is
the
iuste
somme
of
all
those
figures
added
together
,
as
in
this
example
:
8.
multiplied
by
.
5.
dooe
make
.
40.
and
that
is
the
somme
of
al
the
figures
.
An other
example
of
parcelles
that
are
euen
,
as
thus
.
1.
2.
3.
4.
5.
6.
in
this
example
you
must
likewaies
note
doun
the
number
of
the
places
,
as
before
is
taught
,
and
then
adde
together
the
last
nomber
and
the
first
.
And
the
somme
whiche
commeth
of
that
addition
,
shall
you
multiplie
by
halfe
the
nomber
of
the
places
,
whiche
before
are
noted
,
and
that
,
whiche
resulteth
of
the
same
multiplication
,
is
the
whole
somme
of
al
those
figures
,
as
in
this
former
example
,
where
the
nomber
of
the
places
is
6.
I
note
the
6.
apart
,
and
then
I
adde
Progression
Geometricall
is
,
when
Likewaies
.
3.
9.
27.
81.
243.
are
in
triple
proportion
.
And.
2.
8.
32.
128.
512.
are
in
proporcion
quadruple
.
That
is
to
saye
,
in
the
firste
example
,
where
the
proporcion
is
double
,
euery
nomber
containeth
the
other
.
2.
tymes
.
In
the
seconde
example
of
triple
proporcion
,
the
noumbers
exceade
eche
other
thre
times
.
And
in
the
third
example
,
the
nombers
exceade
eche
other
fower
times
,
and
thus
you
se
that
progression
Arthmeticalle
,
differeth
from
Progression
Geometricalle
for
that
,
that
in
the
Arithmeticall
.
The
excesse
is
only
in
quantity
,
but
in
the
Geometricall
,
the
excesse
is
in
proporcion
.
Nowe
if
you
will
easelie
finde
the
somme
of
any
soche
nombers
,
you
shal
dooe
thus
,
consider
by
what
noumber
thei
be
multiplied
,
whether
by
.
2.
3.
4.
5.
or
any
other
,
and
by
the
same
nomber
you
must
multiply
the
last
somme
in
the
progression
.
And
from
the
producte
of
the
same
multiplication
,
you
shall
abate
the
first
nomber
of
the
progression
.
And
that
whiche
remaineth
of
the
saied
multiplicacion
,
you
shall
diuide
by
.
1.
lesse
then
was
the
nomber
,
by
the
which
I
did
multiplie
.
And
the
quotient
shall
shew
you
the
some
of
all
the
nombers
in
any
Progression
.
As
in
this
exaumple
.
5.
15.
45.
135.
405.
whiche
are
in
triple
proporcion
:
now
muste
you
multiplie
.
405.
by
.
3.
and
thei
are
.
1215.
from
the
which
you
shal
abate
the
first
nomber
of
the
progression
,
whiche
is
.
5.
and
there
resteth
1210.
the
whiche
you
shall
diuide
by
the
nomber
lesse
by
.
1.
then
by
the
which
you
did
multiplie
,
that
is
to
saie
,
by
.
2.
and
you
shall
finde
in
the
quocient
605.
whiche
is
the
totall
somme
of
the
nombers
of
that
progression
.
Likewise
.
4.
16.
64.
256.
1024.
whiche
are
in
proportion
quadruple
:
therfore
multiplie
1024.
by
.
4.
and
thereof
commeth
4096.
from
the
whiche
abate
the
first
nomber
.
4.
and
there
resteth
.
4092
:
the
whiche
you
must
deuide
by
.
3.
and
you
shall
finde
in
your
quotient
.
1364
whiche
is
the
total
somme
of
that
progression
,
and
this
shal
be
sufficient
for
progression
.
¶
The
.vij.
Chapiter
treateth
of
the
Rule
of
.
3.
called
the
golden
Rule
.
THE
rule
of
three
is
the
chiefest
,
the
moste
profitable
,
and
the
moste
excellente
rule
of
all
the
rules
of
Arithmetike
.
For
al
other
rules
haue
nede
of
it
,
and
it
passeth
all
the
other
,
for
the
whiche
cause
it
is
saied
,
that
the
Philosophers
did
name
it
the
golden
rule
.
And
after
others
opinion
and
iudgemente
,
it
is
called
the
rule
of
proportions
of
nombers
.
But
now
in
these
daies
,
by
vs
it
is
called
the
rule
of
thrée
,
be
cause
it
requireth
three
nombers
in
his
operacion
.
Of
the
whiche
three
nombers
,
the
twoo
first
are
set
in
a
certain
proportion
.
And
in
suche
proportion
as
thei
be
stablished
,
this
rule
serueth
to
finde
out
vnto
the
third
nomber
,
the
fourth
nomber
to
hym
proportioned
,
in
suche
sort
as
the
seeonde
is
proportioned
vnto
the
firste
.
Not
for
that
,
that
the
fower
noumbers
,
nor
yet
the
three
,
are
or
bee
proportionall
,
or
set
in
one
proportion
,
but
suche
proportion
,
as
is
from
the
firste
to
the
seconde
,
ought
to
bee
from
the
thirde
vnto
the
fowerth
,
that
is
to
sate
,
if
the
seconde
noumber
dooe
contain
the
firste
,
twoo
tymes
or
more
,
so
many
tymes
shall
the
fourth
nomber
containe
the
thirde
.
And
note
well
that
the
firste
noumber
,
and
the
thirde
in
euery
rule
of
three
,
oughte
and
must
bee
alwaies
semblable
,
and
of
one
condition
.
And
the
second
nomber
,
and
the
fowerth
muste
likewise
bee
of
one
semblaunce
and
nature
.
And
are
dissemblaunte
,
and
contrarie
to
the
other
twoo
noumbers
:
that
is
to
saie
to
the
firste
,
and
the
thirde
.
And
if
you
dooe
multiplie
the
firste
by
the
fowerth
.
And
the
seconde
noumber
by
the
thirde
.
The
twoo
multiplications
will
bee
egall
.
Likewise
if
you
diuide
the
one
sembleaunte
by
the
other
,
that
is
to
saie
,
the
thirde
noumber
by
the
firste
.
And
likewise
the
one
dissembleaunt
by
the
other
:
that
is
to
saie
,
the
fowerth
nomber
by
the
second
(
which
are
dissembleaunt
to
the
other
twoo
nombers
)
your
twoo
quocientes
will
be
egall
The
stile
of
this
rule
in
thus
,
you
¶
Example
.
IF
,
8.
be
worthe
.
12.
what
are
14.
worth
after
the
rate
,
or
els
if
.
8.
require
.
12.
for
his
proportionall
,
what
will
.
14.
demaunde
?
The
whiche
three
noumbers
maie
conuenientlie
bee
sette
in
soche
order
,
as
hereafter
doeth
appere
.
If.
8
.
-12.-14
.
multiplie
the
thirde
nomber
.
14.
by
the
seconde
,
whiche
is
.
12.
And
thereof
commeth
(
for
the
whole
producte
of
this
multiplication
.
168.
the
whiche
(
as
the
rule
teacheth
)
you
muste
diuide
by
the
first
nomber
,
that
is
to
saie
by
.
8.
and
therof
commeth
.
21.
And
so
muche
are
the
14.
worthe
.
This
is
the
waie
,
whiche
is
moste
vsed
.
Otherwise
diuide
.
8.
by
.
12.
whiche
you
can
not
doe
,
for
thei
are
8/72.
wherefore
abreuie
.
8/12.
and
thei
are
2/9.
for
your
quotient
,
then
diuide
the
thirde
nomber
.
14.
by
the
saied
⅖
,
and
you
shall
haue
.
21.
as
before
.
Orels
diuide
the
seconde
nomber
.
12.
by
the
firste
nomber
.
8.
thereof
commeth
.
1
½
.
the
whiche
1
½
you
shall
multiplie
by
14.
and
therof
will
come
.
21.
as
is
aboue
saied
,
and
thus
muste
you
dooe
of
all
other
.
And
although
,
that
the
nombers
of
this
rule
maie
bee
founde
in
three
differences
,
for
sometymes
thei
are
whole
nombers
and
broken
together
,
sometymes
broken
and
broken
together
,
and
somtymes
all
whole
nombers
,
if
thei
bee
whole
nombers
,
you
muste
doe
none
otherwise
,
then
you
did
in
the
last
example
.
But
in
case
thei
be
broken
noumbers
,
or
broken
and
whole
noumbers
together
,
the
maner
and
waie
to
dooe
theim
,
recetueth
a
certaine
variacion
,
and
difficultee
,
according
to
the
varietie
of
the
noumbers
,
the
whiche
operation
easeiy
to
do
,
and
vnuariable
,
this
rule
teacheth
.
The
three
noumbers
beyng
sotte
doune
,
according
vnto
the
order
of
the
whole
noumbers
aforesaied
,
without
any
broken
nomber
,
let
.
1.
bee
put
alwaies
vnderneath
euery
whole
nomber
,
with
a
line
betwene
them
fraction
wise
,
as
thus
8/1.
and
that
.
1.
is
denominatour
to
euery
soche
whole
nomber
.
When
you
haue
whole
nomber
and
broken
,
thei
must
bee
reduced
and
added
with
their
broken
nomber
,
and
if
there
bée
broken
nomber
without
any
whole
nomber
,
thesame
broken
must
remain
in
their
estate
.
A
Rule
.
This
beynge
deen
,
you
shall
multiplie
the
denominatour
of
the
firste
nomber
,
by
the
numeratour
of
the
second
,
and
the
producte
thereof
againe
by
the
numerator
of
the
third
nomber
.
And
so
shall
you
haue
the
diuidende
,
or
nomber
whiche
muste
bee
diuided
,
then
multiplie
the
numeratour
of
the
first
nomber
,
by
the
denominator
of
the
seconde
,
and
the
product
thereof
by
the
denominator
of
the
third
nomber
,
and
that
which
commeth
of
this
multiplication
shal
be
your
diuisor
.
Then
diuide
the
nomber
,
whiche
is
to
bee
diuided
,
by
the
diuisour
,
and
you
shall
finde
the
fowerth
nomber
that
you
seke
.
Of
the
which
maner
and
fashions
of
the
rule
of
.
3.
are
diuers
kindes
,
wherof
the
first
is
of
.
3.
whole
nombers
,
as
was
the
last
example
,
and
here
foloweth
the
second
If.
15.
poundes
doe
buy
me
two
clothes
,
how
many
clothes
wil.
300.
poundes
buye
me
of
thesame
pr●e
,
that
the
twoo
clothes
did
cost
,
sette
doune
your
three
nombers
thus
.
Lib:
Clothes
.
Lib.
And
than
as
you
see
,
multiplie
the
thirde
nomber
,
which
is
.
300.
l●
.
by
.
2.
whiche
is
the
second
nomber
,
and
therof
commeth
600.
the
which
.
600.
you
must
diuide
by
the
first
nomber
.
15.
and
you
shal
find
in
your
quocient
40.
whiche
is
.
40.
clothes
,
and
so
many
clothes
shall
I
buye
for
.
300
.l.
as
appereth
by
practise
here
aboue
written
.
And
here
you
muste
marke
that
the
first
nomber
&
the
thirde
in
this
question
be
of
one
denominacion
,
and
like
wise
the
second
&
the
fowerth
which
you
haue
found
are
of
one
semblaunce
:
but
in
case
that
the
first
nomber
and
the
third
in
any
question
:
bee
not
of
like
denominacion
,
you
muste
in
workyng
bryng
them
into
one
,
as
in
this
example
folowing
.
If.
12.
nobles
do
gaine
me
6.
nobles
,
how
many
nobles
wil.
48.
poundes
gatne
me
:
Here
you
se
that
the
denominacion
of
the
first
nomber
is
nobles
,
and
the
denominacion
of
the
thirde
,
is
poundes
,
wherefore
,
before
you
doe
procede
to
worke
by
the
rule
of
three
,
you
muste
firste
turne
the
poundes
into
nobles
in
multipliyng
.
48.
poundes
by
three
nobles
and
they
make
.
144.
nobles
,
for
that
there
is
in
euery
pounde
of
money
.
3.
nobles
,
or
otherwise
if
you
will
,
you
maie
bryng
the
first
nomber
beyng
.
12
nobles
,
into
poundes
,
by
diuiding
them
by
.
3.
and
thus
shal
your
first
and
third
nombers
,
bee
brought
into
one
denominacion
.
Then
shal
you
sette
doune
your
.
3.
nombers
in
order
thus
.
If.
12.
nobles
doe
gaine
me
.
6.
nobles
,
what
shall
.
144.
nobles
gaine
?
the
which
.
144.
are
the
nobles
which
are
in
.
48.
〈◊〉
.
Then
multiplie
the
third
nomber
.
144.
by
the
seconde
nomber
6.
and
thereof
cometh
864.
the
whiche
you
muste
diuide
by
.
12.
nobles
,
and
thereof
commeth
.
72.
nobles
.
But
here
it
maie
perchaunce
make
some
menne
masse
,
to
see
all
the
three
nombers
in
this
rule
of
three
,
to
bee
of
one
denomination
,
whiche
can
not
otherwise
bee
dooen
,
if
you
reduce
the
third
nomber
,
to
the
denominacion
of
the
firste
.
But
if
you
will
reduce
the
firste
nomber
,
to
the
denomination
of
the
thirde
,
that
is
to
saie
the.
12.
nobles
into
poundes
,
then
shall
the
firste
and
the
third
nombers
onely
agree
in
one
denominacion
,
and
the
fowerth
nomber
which
you
seke
,
shal
be
of
the
same
denominacion
as
is
the
seconde
,
as
in
the
former
example
.
If.
12.
nobles
doe
yelde
me
.
6.
nobles
,
what
will
.
48.
poundes
yeld
me
:
first
you
shall
diuide
12.
nobles
by
three
to
bryng
theim
in
poundes
,
and
thei
shall
bee
like
to
the
thirde
nomber
,
whiche
is
also
poundes
,
then
will
thei
stande
thus
.
There
is
yet
a
more
exacte
waie
to
woorke
in
this
rule
of
three
,
whiche
is
thus
.
You
muste
marke
if
the
third
&
first
nombers
in
the
rule
of
three
,
maie
bee
bothe
diuided
by
one
like
diuisor
:
the
which
after
you
haue
diuided
them
,
you
shall
write
doune
eche
of
the
quocientes
orderly
,
in
the
saied
rule
of
.
3.
euery
one
of
theim
in
his
owne
place
,
as
though
those
were
twoo
of
the
nombers
of
your
question
,
and
not
chaungyng
the
middle
noumber
,
that
is
to
saie
the
seconde
,
as
thus
,
if
.
50.
Croudes
doe
buye
me
.
44.
yardes
of
clothe
,
howe
manye
yardes
shall
I
haue
for
120
,
here
you
maye
see
that
the
thirde
and
the
firste
nombers
,
maie
be
diuided
by
.
10.
whiche
in
the
thirde
nomber
is
.
12.
tymes
,
and
in
the
firste
.
5.
times
.
Wherefore
you
shall
put
.
12.
for
the
thirde
nomber
in
the
rule
of
three
▪
in
stede
of
120.
and
5.
for
the
first
nomber
in
stede
of
50.
and
let
.
44.
remain
still
in
the
middest
for
the
second
nomber
,
after
this
sorte
as
foloweth
,
and
then
worke
by
the
rule
as
before
.
Multiplie
.
44.
by
.
12.
and
thereof
commeth
.
528.
diuide
thesame
.
528.
by
5.
and
you
shall
finde
in
your
quocient
105.
⅕
.
and
euen
so
many
yardes
should
you
haue
founde
,
if
you
had
wrought
the
rule
of
thre
,
by
the
first
nombers
proposed
.
There
is
yet
certaine
other
varieties
,
in
woorkinge
by
the
rule
of
three
,
but
for
that
they
require
the
knowledge
of
fractions
,
and
because
thei
are
not
so
easte
as
this
first
waie
,
whiche
is
common
,
therefore
content
your selues
with
this
same
,
vntil
you
haue
tasted
the
fractions
,
the
whiche
by
gods
helpe
I
intende
to
fet
forth
in
seconde
part
of
this
boke
,
incontinently
after
that
I
haue
firste
taught
you
the
backer
rule
of
three
.
The
backer
rule
of
thre
is
so
called
:
because
it
requireth
a
contrary
woorkyng
to
that
,
whiche
doeth
the
rule
of
three
directe
,
where
of
wee
haue
nowe
treated
.
For
in
the
directe
rule
of
three
the
greater
the
thirde
noumber
is
,
so
moche
the
greater
will
the
fourth
be
.
But
here
in
this
backer
rule
it
is
contrariwise
,
for
the
greater
the
third
nomber
is
,
so
moche
lesser
wil
the
fowerth
bee
.
Then
,
where
as
in
the
rule
of
.
3.
directe
,
the
third
noumber
is
multiplied
by
the
seconde
,
and
the
producte
thereof
diuided
by
the
firste
.
Here
you
muste
multiplie
the
seconde
noumber
by
the
firste
,
and
diuide
the
producte
of
the
same
by
the
third
,
and
the
nomber
whiche
commeth
in
the
quotiente
,
answereth
to
the
question
.
For
suche
practise
commeth
often
tymes
in
vse
:
In
suche
sorte
,
that
if
you
woorke
thesame
by
the
rule
of
three
directe
(
not
hauyng
a
regarde
vnto
the
Proposition
of
the
question
)
you
should
then
committe
an
euidente
and
open
errour
.
¶
Example
.
If.
15.
shillynges
worthe
of
Wine
wil
serue
for
the
ordinary
of
.
46
men
when
the
Tonne
of
Wine
is
worthe
12.
poundes
:
for
howe
many
menne
will
the
same
.
15.
shillynges
suffice
,
when
the
Tonne
of
Wine
is
worthe
but.
8.
poundes
?
It
is
certain
,
that
the
lower
the
price
is
,
that
the
Tonne
of
Wine
doeth
coste
,
and
so
many
more
persones
will
the
saied
.
15.
shillynges
in
Wine
suffice
.
Therefore
set
doune
your
nombers
thus
,
if
.
12.
poundes
suffice
.
46.
menne
,
how
many
will
.
8.
poundes
suffice
,
you
muste
multiplie
46.
by
.
12.
and
thereof
commeth
.
552.
the
whiche
.
552.
you
shall
diuide
by
.
8.
and
thereof
commeth
.
69.
and
vnto
69.
menne
will
the
saied
.
15.
shillynges
worthe
in
Wine
suffice
,
when
the
Tonne
of
Wine
is
worthe
but
eight
poundes
,
as
hereafter
dooeth
appeare
by
practise
.
Likewise
,
a
messenger
maketh
a
iourney
in
.
24.
daies
,
when
the
daie
is
but
,
12.
houres
long
:
how
many
daies
shal
he
be
vpon
the
same
iourney
,
when
the
daie
is
.
16
houres
in
length
?
Here
you
must
perceiue
,
that
the
more
houres
are
in
a
daie
,
the
fewer
daies
will
the
messenger
bee
in
goyng
his
iourney
.
Therefore
write
doune
your
nombers
thus
,
as
here
you
maie
se
.
And
then
multiplie
24.
daies
by
.
12
houres
,
and
thereof
commeth
.
288.
diuide
the
same
.
288.
by
the
thirde
nomber
.
16.
and
you
shall
finde
18.
the
whiche
is
.
18.
daies
,
and
in
so
many
daies
will
the
messenger
make
his
iourney
when
the
daie
is
.
16.
houres
long
.
Likewise
,
when
the
Bushell
of
wheate
doeth
coste
.
3.
shillynges
,
the
penie
loafe
of
bread
waieth
.
4.
lib.
I
demaunde
what
the
same
penie
loafe
shall
weye
,
when
the
bushell
of
wheate
is
worthe
but
twoo
shillynges
:
here
is
to
bee
considered
,
that
the
better
cheape
the
Wheate
is
,
the
heauier
shall
the
pennie
loafe
wiie
,
and
therefore
write
doune
your
.
3.
nombers
,
thus
.
Then
multiplie
.
4.
lib.
whiche
is
the
seconde
nomber
,
by
the
first
nomber
3.
and
thei
make
.
12.
the
which
.
12
you
shall
diuide
by
the
thirde
nomber
.
2.
and
thereof
commeth
.
6.
lib.
and
so
muche
muste
the
penie
loafe
of
breade
waye
,
when
the
bushell
of
wheate
is
worthe
but
twoo
shillynges
,
as
maie
appere
.
And
nowe
,
accordyng
to
my
former
promise
,
shall
followe
the
seconde
parte
of
Arithmeticke
,
whiche
teacheth
the
workyng
by
Fractions
.
¶
Here
endeth
the
first
parte
of
Arithmeticke
.
¶
The
seconde
parte
of
Arithmeticke
,
whiche
treateth
of
Fractions
or
broken
nombers
.
¶
The
firste
Chapiter
treateth
of
Fractions
,
or
broken
nombers
,
and
the
difference
thereof
.
BRoken
nomber
is
as
muche
as
a
parte
or
many
partes
of
one
,
whereof
there
are
two
noumbers
with
a
line
betwene
them
bothe
:
That
is
to
saie
,
the
one
whiche
is
aboue
the
line
,
is
cal
led
the
numerator
.
And
the
other
vnderneath
the
line
,
is
called
the
denominator
:
as
by
example
,
three
quarters
,
whiche
must
bee
set
doune
thus
,
4
:
whereof
.
3.
whiche
is
the
higher
nomber
aboue
the
line
,
is
called
the
numerator
,
and.
4.
whiche
is
vnder
the
line
is
called
the
denominator
.
And
it
is
al
waies
conuenient
,
that
the
numerator
be
lesse
in
nomber
,
then
the
denominator
.
For
if
the
numerator
,
and
the
denominator
were
egall
in
value
:
then
should
thei
represente
a
whole
nomber
,
thus
,
as
1/1
,
2/2
,
3/3
,
whiche
are
whole
nombers
:
by
reason
that
the
numerators
of
these
,
and
al
suche
like
maie
bee
diuided
by
their
denominators
,
and
their
quotientes
will
alwaies
be
but.
1.
But
in
case
that
the
numerator
do
excede
his
denominator
,
then
it
is
more
then
I
whole
:
as
20/18
,
is
more
than
a
whole
nomber
by
2/18
,
other
diffinition
dooeth
not
hereunto
appertaine
.
Furthermore
it
is
to
bée
vnderstande
,
that
the
middest
of
all
broken
nombers
,
is
the
iuste
halfe
of
.
1.
whole
,
as
5/12
,
7/14
,
8/16
,
9/18
and
other
like
,
are
the
halfes
of
one
whole
nomber
,
wherof
doeth
growe
,
and
come
forthe
2.
progressions
natural
:
the
one
progrediyng
by
augmentyng
,
or
encreasyng
,
as
these
.
½
⅔
¾
⅘
⅚
6/7
⅞
8/9
9./10
&c.
And
thei
doe
proceade
infinitely
,
and
wil
neuer
reche
to
make
a
whole
nomber
,
thus
1/1.
And
the
other
progression
,
dooeth
progrede
by
diminishyng
or
decreasyng
,
as
thus
.
½
⅓
¼
⅕
⅙
1/7
⅛
1/9
1/10
&c.
And
these
dooe
proceade
infinitely
,
and
shall
neuer
come
to
make
a.
○
.
whiche
signifieth
nothyng
,
but
shall
euer
reiaine
some
certaine
nomber
whatsoeuer
,
whereby
it
doeth
appere
that
broken
nombers
are
infinite
.
¶
The
seconde
Chapiter
treateth
of
the
reducyng
or
bringyng
together
of
twoo
nombers
,
or
many
broken
dissemblyng
,
vnto
one
broken
semblyng
.
REduction
,
is
as
muche
as
to
bryng
together
,
or
to
put
in
sembleaunce
,
twoo
or
many
noumbers
dissembling
one
from
the
other
,
in
reducyng
them
vnto
a
common
denominator
.
For
bicause
the
diuersitie
and
difference
of
the
broken
numbers
,
doe
come
of
the
denominators
part
,
or
of
diuers
denominators
,
and
for
the
vnderstanding
hereof
,
there
is
a
general
rule
whose
operation
is
thus
.
Multiple
the
Denominators
the
one
by
the
other
,
and
so
you
shall
haue
a
new
denominator
common
to
al
,
the
whiche
denominator
diuide
by
the
perticuler
denominators
,
and
multiplye
euerie
quotient
by
his
numerator
and
so
you
shall
haue
newe
numerators
,
for
the
numbers
whiche
you
woulde
reduce
,
as
appeareth
by
thys
example
followyng
.
¶
Reduction
in
common
denomination
.
IF
you
wil
reduce
3/3
and
⅘
together
,
you
must
firste
multiplie
If
you
will
reduce
½
,
⅔
,
¾
,
⅚
,
together
,
you
must
multiplie
all
the
denominators
,
the
one
by
the
other
,
that
is
to
saie
,
2.
by
.
3.
maketh
.
6.
then
.
6.
by
4.
and
mounteth
.
24.
Laste
of
all
.
24.
by
.
6.
and
thereof
commeth
.
144.
for
the
common
denominator
.
Then
,
for
the
firste
diuide
.
144.
by
the
denominator
.
2.
and
thereof
commeth
.
72.
the
whiche
multiplie
by
the
numerator
1.
and
it
is
stil
72.
set
that
ouer
the
½
and
it
is
27/144
,
for
the
½
:
Then
deuide
144
by
the
second
denominator
3.
and
therof
commeth
48
:
the
whiche
multiplie
by
the
seconde
numerator
2.
and
they
are
96.
whiche
set
ouer
the
⅔
and
they
make
96/144
,
for
the
⅔
:
Then
diuide
144.
by
the
thirde
denominator
4.
&
thereof
commeth
36.
the
whiche
multiplie
by
the
thirde
numerator
3.
and
thei
make
108.
whiche
sette
ouer
the
¾
.
and
thei
are
108/144
for
the
5/4.
Finally
diuide
.
144.
by
the
last
denominator
.
6.
and
thereof
commeth
24
:
the
whiche
multiplie
by
the
laste
numerator
5.
and
thereof
commet
120
Whiche
sette
ouer
the.
⅚
.
and
thei
are
120/144
,
for
the
⅚
,
as
appeareth
here
by
pratise
.
¶
Reduction
of
broken
nombers
of
broken
.
IF
you
will
reduce
the
broken
¶
An
other
example
of
the
same
reduction
,
and
of
the
seconde
reduction
.
IF
you
will
reduce
.
⅔
.
of
.
5/4.
of
⅘
.
the
5/4.
of
5/7
:
and
the
½
.
of
the
½
.
of
the.
⅔
.
of
the.
⅓
.
Firste
,
it
behoueth
you
of
euery
partie
of
the
broken
nombers
,
to
make
of
eche
of
them
one
broken
,
as
by
the
third
reduction
is
taughte
:
That
is
to
saie
,
in
multipliyng
the
numerators
by
numerators
,
and
denominators
by
denominators
:
firste
,
for
the
firste
parte
,
whiche
is
⅔
.
of
¼
.
of
⅘
,
you
muste
as
is
said
before
,
multiplie
.
2
by
.
1.
and
then
by
.
4.
and
you
shall
haue
.
8.
for
the
numerator
,
likewise
multiplie
.
3.
by
.
4.
and
the
producte
by
.
5.
and
you
shall
haue
.
60.
for
the
denominator
,
so
thei
make
.
8/60.
whiche
beyng
abreuied
,
are
2/15
for
the
first
parte
,
that
is
to
saie
,
for
the
⅔
of
〈◊〉
of
⅘
,
secondly
for
the
¾
of
5/7
multiplie
likewise
the
numerator
.
3.
by
.
5
maketh
.
15.
for
the
numerator
,
and
multiplie
.
4.
by
.
7.
maketh
.
28.
for
the
denominator
,
and
then
thei
bee
.
15/28.
for
the
seconde
parte
,
that
is
to
saie
,
for
the
¾
of
5/7.
Thirdly
,
for
the
½
of
½
of
⅔
of
⅓
.
multiplie
the
numerators
th
one
by
the
other
,
that
is
to
saie
.
1.
by
.
1.
and
then
by
.
2.
and
laste
by
.
1.
and
all
maketh
but
2
for
the
numerator
,
likewise
multiplie
.
2.
by
.
2.
maketh
.
4.
and
4.
by
3.
maketh
.
12.
and
then
.
12.
by
.
3.
maketh
.
36.
for
the
denominator
,
and
thei
are
2/36
,
whiche
beyng
abreuied
maketh
1/18.
for
the
thirde
parte
,
that
is
to
saie
,
for
½
of
the
½
.
of
⅔
of
⅓
.
Last
of
al
,
take
the
2/15
the
15/28.
and
the
1/18.
and
reduce
them
according
to
the
order
of
the
second
reduction
,
and
you
shall
finde
1000/7●60
.
for
the
2/15.
And
4050/●500
for
the
15/28.
And
426/7560
for
1/18
:
and
thus
are
broken
nombers
of
broken
,
reduced
,
as
appeareth
by
by
practise
.
¶
Reduction
of
broken
nombers
,
and
the
partes
of
broken
together
.
IF
you
will
reduce
⅓
.
and
the
½
of
⅓
.
together
,
to
bryng
them
into
one
broken
nomber
,
you
muste
firste
sette
doune
the
⅓
.
and
½
.
as
appeareth
in
the
margent
with
a
crosse
betwene
theim
,
and
then
multiply
the
twoo
denominators
.
The
one
by
the
other
,
that
is
to
saie
,
2.
by
3.
maketh
.
6
set
that
vnder
the
crosse
,
then
multiplie
the
firste
numerator
,
one
by
the
laste
denominator
twoo
,
and
that
maketh
.
2.
vnto
the
whiche
adde
the
laste
numerator
one
,
and
thei
be
three
,
whiche
set
aboue
your
crosse
,
so
you
shall
finde
that
the
⅕
and
the
½
of
●/3.
doe
make
3/6.
whiche
beyng
abreuied
doeth
make
⅓
,
which
is
as
much
as
the.
⅓
.
and
the.
½
.
of
.
⅓
.
Likewise
if
you
will
reduce
the
⅔
,
and
the
¼
.
of
⅓
.
you
must
doe
as
before
,
set
doune
the
⅔
and
¼
with
a
crosse
betwene
them
,
&
then
multiplie
the
twoo
denominators
,
the
one
by
the
other
,
that
is
to
saie
.
3.
by
.
4
maketh
.
12.
whiche
set
vnder
the
crosse
as
you
see
in
the
margent
,
and
then
multiplie
the
firste
numerator
2
by
the
last
denominator
.
4.
and
thereof
commeth
.
8.
whereunto
adde
the
laste
numerator
.
1.
and
that
maketh
:
9.
whiche
sette
ouer
the
crosse
,
so
shall
you
finde
that
the
⅔
.
and
the
¼
of
⅓
are
worthe
9/●
,
whiche
abreuied
,
dooe
make
¼
,
as
appeareth
by
example
in
the
margent
.
¶
Reduction
of
whole
nombers
and
broken
together
into
a
Fraction
.
IF
you
will
reduce
whole
nomber
In
case
you
haue
whole
number
and
broken
to
bee
reduced
,
with
broken
you
muste
bringe
the
whole
number
into
his
broken
,
in
multiplyinge
it
by
the
denominator
of
the
broken
number
going
therwith
,
and
adde
thereunto
the
numerator
of
the
saide
broken
number
,
as
in
the
laste
example
,
and
then
reduce
that
broken
number
wyth
the
other
broken
,
as
here
appereth
by
this
example
.
Reduce
10.
⅔
&
4/7
togither
,
first
bring
10⅔
al
into
thirds
,
as
by
the
syxt
reduction
,
and
you
shal
finde
〈◊〉
,
then
reduce
the
32/3
and
4/7
togither
,
by
the
fyrste
reduction
,
and
you
shall
finde
224/21
for
the
32/3
:
and
〈◊〉
for
〈◊〉
as
appereth
here
by
practise
.
Also
in
case
you
haue
in
bothe
partes
of
your
Reduction
,
as
well
whole
number
as
broken
,
you
must
alwayes
put
the
whole
into
their
broken
(
as
by
the
syxte
reduction
)
of
either
part
.
¶
Example
.
If
you
wyll
reduce
12.
¼
wyth
14.
⅔
to
bringe
them
into
one
denomination
,
first
bringe
the
12.
¼
all
into
fourthes
,
and
you
shall
fynde
49/4
:
then
likewise
reduce
14.
⅔
all
into
thirdes
,
and
you
shall
haue
4●/3
,
for
the
14.
⅔
,
then
reduce
〈◊〉
and
44/3
togither
,
by
the
order
of
the
firste
Reduction
,
and
you
shall
fynde
〈◊〉
for
the
〈◊〉
.
And
〈◊〉
for
the
14.
⅔
as
here
by
practise
dothe
plainlye
appere
.
¶
The
thirde
Chapter
treateth
of
abbreuiation
of
one
greate
broken
number
into
a
lesser
broken
.
ABbreuiation
is
asmuch
as
to
set
downe
,
or
to
write
a
broken
number
by
figures
of
lesse
signification
,
&
not
diminishing
the
value
thereof
.
The
whiche
to
doe
,
there
is
a
rule
whose
operation
is
thus
,
diuide
the
numerator
and
likewise
the
denominator
,
by
one
whole
number
,
the
greatest
that
you
maye
fynde
in
the
same
broken
number
,
and
of
the
quotient
of
that
numerator
,
make
it
the
numerator
,
and
likewise
of
that
of
the
denominator
,
make
it
your
denominator
,
as
by
example
.
1.
If
you
wyl
abbreuiat
54/8●
,
you
shal
vnderstande
that
the
greatest
whole
number
that
you
maye
take
,
by
the
which
you
maye
diuide
the
numerator
&
denominator
is
27
,
which
is
the
half
of
the
numerator
,
&
that
is
a
whole
number
,
for
you
can
not
take
a
whole
number
oute
of
the
denominator
.
81.
but
that
there
will
bee
either
more
or
lesse
than
a
whole
number
,
therefore
if
you
diuide
54.
by
27.
you
shall
finde
2.
for
the
numerator
,
likewise
if
you
diuide
81.
by
27.
you
shall
finde
3.
for
the
denominator
.
then
put
.
2.
ouer
the
3.
with
a
line
betwene
them
,
and
you
shall
find
⅔
and
thus
by
this
rule
the
53/81
are
abreuted
vnto
⅔
,
as
appeareth
in
the
margent
,
and
so
is
to
be
vnderstande
all
other
.
¶
The
forme
&
maner
how
to
finde
oute
the
greater
number
,
by
the
which
you
mai
wholy
diuide
,
the
numerator
&
denominator
(
to
th
ende
that
you
may
abreuiat
them
)
is
thus
.
First
,
diuide
the
denominator
by
hys
numerator
,
and
if
anye
number
doe
remaine
,
let
your
diuisor
be
diuided
by
the
same
number
,
and
so
you
must
continue
vntill
you
haue
so
diuided
that
there
maye
nothing
remaine
,
then
is
it
to
be
vnderstande
,
that
your
last
diuisor
(
wherat
you
did
ende
,
and
that
o
did
remaine
after
your
last
diuision
)
is
the
greatest
number
,
by
the
whiche
you
must
abreuiat
,
as
you
did
in
the
laste
example
,
but
in
case
that
your
last
diuisor
be
1.
it
is
a
token
that
the
same
number
can
not
bee
abreuied
.
Example
,
of
54/81
diuide
81.
(
whiche
is
the
denomination
)
by
54.
which
is
his
numerator
,
and
there
resteth
27.
then
diuide
54.
by
27.
and
there
remaineth
nothinge
,
wherefore
your
last
diuisor
27.
is
the
number
,
by
the
whiche
you
must
abreuiat
54/81
as
in
the
laste
example
is
specifyed
.
¶
An
other
stile
of
abbreuiation
.
2.
Mediate
the
numerator
,
and
also
the
denominator
of
your
fraction
in
case
the
noumbers
be
euen
,
that
is
to
saye
,
take
alwayes
the
halfe
of
the
numerator
,
and
likewise
of
the
denominator
,
and
of
the
mediation
or
halfe
of
the
numerator
,
make
the
numerator
,
also
of
½
the
denominator
,
make
your
denominator
,
and
so
continue
as
often
as
you
may
in
takinge
alwayes
the
½
of
the
numerator
,
and
semblablie
of
the
denominator
,
or
els
see
if
you
may
abbreuiate
the
numbers
which
doe
remaine
,
by
3.
by
4.
by
5.
6.
7.
8.
9.
or
by
10.
for
you
must
abbreuiate
them
as
often
as
you
can
by
any
of
the
saide
numbers
,
and
it
is
to
bée
noted
,
that
with
whatsoeuer
number
of
these
,
you
doe
abbreuiate
the
Numerator
of
your
Fraction
,
by
the
same
you
muste
abbreuiate
likewyse
the
Denominator
,
so
continuynge
vntil
they
can
no
more
bee
abbreuied
.
And
it
is
to
bee
vnderstande
that
if
the
Numerator
and
the
Denominator
be
euen
numbers
,
as
you
maye
know
when
the
fyrste
fygure
is
an
euen
number
,
or
a
〈◊〉
,
them
maye
you
perceaue
if
both
the
Numerator
and
the
Denominator
may
be
abbreuied
by
10.
by
8
by
4.
or
by
2.
although
that
some
times
they
maye
bee
abbreuied
by
three
.
And
if
they
be
odde
numbers
,
then
muste
you
consider
if
they
maye
bee
abbreuied
by
9.
by
7
by
.
5.
or
by
3
:
but
when
the
first
number
,
as
well
of
the
Numerator
,
as
of
the
Denominator
are
euen
numbers
,
then
may
you
well
knowe
that
suche
numbers
maye
bee
abbreuied
be
2.
as
is
aforesaide
.
And
if
you
adde
the
fygures
of
the
Numerator
togither
,
in
suche
manner
as
you
doe
in
makynge
the
proofe
by
nyne
in
whole
Numbers
:
that
is
,
if
you
fynde
9.
〈◊〉
appeareth
that
you
maye
abb
〈…〉
that
number
by
9.
And
lykewise
by
3.
and
sometimes
by
6.
if
you
fynde
6.
it
maye
bee
abbreuied
by
6.
and
alwaies
by
3.
if
you
finde
3.
it
is
a
signe
that
you
may
abreuiate
by
3.
And
by
whatsoeuer
nomber
that
you
doe
abreuiate
the
numerator
,
by
the
same
must
you
abreuiate
likewise
the
denominator
,
and
if
the
first
figures
of
the
same
nomber
be
.
5.
or
0.
you
maye
abreuiate
them
by
5.
but
if
the
firste
fygures
be
both
0.
they
may
be
abreuied
by
10.
in
cutting
awaye
the
twoo
Cyphers
thus
,
as
〈◊〉
whiche
maketh
2/●
,
&
sometimes
by
100.
thus
,
as
〈◊〉
in
cutting
away
the
foure
ciphers
after
this
sorte
,
〈◊〉
and
then
the
100/200
doe
make
½
,
and
after
this
maner
haue
I
set
here
diuers
examples
,
althoughe
that
all
numbers
cannot
be
abreuied
by
this
rule
,
that
is
to
saye
,
all
those
whiche
maye
bee
well
abreuyed
by
the
fyrste
rule
aforesayde
.
3.
Furthermore
you
shall
vnderstande
that
sometimes
it
happeneth
,
that
all
the
fygures
of
the
numerator
are
egall
vnto
them
of
the
denominator
,
which
when
it
so
happeneth
,
you
maye
then
take
one
of
them
of
the
numerator
,
and
also
one
of
them
of
the
denominator
,
and
it
shall
bee
abreuyed
as
555/888
,
beinge
abreuiated
after
this
maner
commeth
to
⅝
.
And
yet
it
happeneth
sometimes
,
that
two
,
or
manye
fygures
of
the
numerator
are
proportioned
vnto
two
,
or
many
fyguree
of
their
denominators
and
the
other
fygures
of
the
same
number
doe
beholde
the
one
the
other
in
thys
proportion
?
Then
may
you
take
twoo
or
many
fygures
,
as
well
of
the
numerator
as
of
the
denominator
,
and
by
this
maner
the
same
number
shall
bee
abbreuied
,
as
4747/〈…〉
whiche
beinge
abbreuied
by
this
rule
,
do
come
to
〈…〉
.
4.
Also
it
happeneth
somtimes
that
you
woulde
abbreuiate
one
number
vnto
the
semblaunce
or
likenesse
of
another
.
And
for
to
knowe
if
the
same
maye
by
abbreuied
,
and
also
by
what
number
it
maye
bee
abbreuied
,
you
must
diuide
the
numerator
of
the
one
number
,
by
the
numerator
of
the
other
,
and
likewise
the
denominator
of
the
one
,
by
the
denominator
of
the
other
,
for
in
case
that
after
euery
diuision
there
doe
remaine
0.
and
that
the
twoo
quotiens
be
〈◊〉
all
,
then
is
one
of
them
the
number
by
the
whiche
the
saide
fraction
must
be
abbreuied
,
as
by
example
of
11●/〈…〉
.
I
woulde
knowe
if
they
maye
be
abbreuied
vnto
5/9
,
and
for
to
doe
this
,
you
must
diuide
115.
by
5.
and
you
must
diuide
207.
by
9.
and
there
will
come
into
bothe
the
quotients
23.
by
the
which
it
appeareth
that
this
number
may
be
abbreuied
by
23.
¶
The
4.
Chapter
treateth
of
the
assembling
of
two
or
many
broken
numbers
togither
,
as
by
example
.
FOr
to
adde
broken
numbers
togither
,
there
is
a
generall
rule
,
which
is
thus
,
if
the
numbers
be
vnlike
that
one
to
the
other
,
you
must
reduce
than
into
a
common
denomination
,
whiche
after
you
haue
reduced
them
,
you
must
then
adde
both
the
numerators
togither
,
&
set
the
product
of
the
saide
addition
ouer
the
crosse
,
&
diuide
the
same
by
the
common
denominator
,
as
by
this
example
folowinge
.
1.
If
you
wyll
adde
⅔
wyth
¾
,
you
muste
fyrst
reduce
the
twoo
fractions
bothe
into
one
denomination
,
accordinge
to
the
introduction
of
the
fyrste
reduction
,
that
is
to
saye
,
in
multiplyinge
the
denominator
of
the
firste
fraction
whiche
is
3
,
by
the
denominator
of
the
other
fraction
whiche
is
foure
,
and
they
make
12.
for
your
common
denominator
,
the
which
12.
set
vnder
the
crosse
,
them
multiplie
that
fyrst
numerator
2.
by
the
last
denominator
4.
and
thereof
commeth
8.
whiche
sett
ouer
the
⅔
,
and
then
multiply
the
last
numerator
3.
by
the
fyrste
denominator
3.
and
therof
commeth
9.
whiche
you
must
set
ouer
the
¾
,
then
adde
the
numerator
8.
with
the
numerator
9.
&
they
make
17.
which
set
ouer
the
crosse
,
and
then
your
fraction
wyll
be
17/12
whiche
is
the
addition
of
the
⅔
wyth
¾
.
And
bycause
your
numerator
17.
is
greater
than
his
denominator
12.
therfore
you
must
diuide
17.
by
12.
and
thereof
will
come
1.
and
5.
remaining
,
which
5.
are
worth
●/12
,
and
so
muche
are
the
⅔
added
with
¾
as
doth
appere
.
¶
Addition
in
broken
.
numbers
.
2.
Also
if
you
will
adde
½
,
⅔
,
¾
,
⅘
,
togither
,
you
muste
fyrste
adde
the
〈◊〉
and
⅔
together
,
accordynge
to
the
doctrine
of
the
laste
rule
,
and
you
shall
finde
7/6
:
then
adde
¾
and
⅘
togither
by
the
saide
last
chapter
,
and
they
make
〈◊〉
.
Then
finally
adde
the
7/6
(
whiche
came
of
the
½
and
⅔
added
together
)
with
31/20
,
&
you
shall
fynde
by
the
foresaide
addition
that
they
amounte
vnto
326/120
,
Wherfore
diuide
326.
by
120.
&
therof
commeth
2.
and
86.
remaineth
whiche
is
86/120
of
one
whole
,
&
thei
being
abreuied
do
make
43/60
:
&
thus
the
½
,
⅔
,
¾
,
⅘
,
added
together
doe
amount
to
2.
4●/●●
,
as
here
vnder
doth
appere
.
¶
Addition
of
broken
numbers
of
broken
.
3.
Furthermore
,
if
you
will
adde
the
broken
numbers
of
broken
togither
,
as
to
adde
the
⅔
of
¾
of
⅘
with
the
⅚
of
½
of
〈◊〉
:
first
you
must
reduce
the
numbers
accordyng
to
the
order
of
the
fourth
reduction
,
in
multiplying
the
numerator
of
the
fyrst
3.
fractions
,
the
one
by
the
other
,
and
of
the
producte
make
your
numerator
,
&
likewise
you
must
multiplie
the
denominators
of
the
foresaid
thre
fractions
,
the
one
by
the
other
and
of
the
product
make
your
denominator
,
and
you
shal
finde
24/60
for
the
first
three
broken
numbers
,
which
being
abbreuied
do
make
⅖
,
then
reduce
the
other
3.
fractions
,
by
the
said
fourth
reduction
,
in
multiplyinge
the
numerators
by
numerators
,
&
denominators
,
by
denominators
,
as
you
did
by
the
first
3.
broken
numbers
,
&
you
shal
finde
25/96
then
must
you
adde
y●
⅖
whiche
came
of
the
fyrste
3.
broken
numbers
,
&
25/56
whiche
are
of
the
last
3.
fractions
,
both
togither
,
by
the
instruction
of
the
first
addition
&
you
shall
find
317/480
whiche
cannot
be
abreuied
,
but
is
the
product
of
the
addition
:
so
muche
are
the
⅔
of
¾
of
⅘
added
with
the
⅚
of
〈◊〉
of
⅝
as
hereafter
by
practise
doth
euidently
appere
.
¶
Addition
of
broken
number
the
partes
of
broken
together
.
4.
Likewise
if
you
will
addde
the
½
,
and
the
½
of
⅓
with
the
⅘
and
●/4
of
〈◊〉
,
you
must
reduce
the
11/32
by
the
fyft
reduction
and
therof
commeth
3/6
for
the
〈◊〉
&
〈◊〉
,
of
one
of
the
saide
thirdes
,
then
reduce
the
⅘
and
¼
by
the
saide
fift
reduction
,
and
thereof
commeth
〈◊〉
.
Last
of
all
adde
the
⅚
and
17/20
togither
according
to
the
firste
rule
of
addition
,
and
you
shall
fynde
20●/〈…〉
which
beinge
diuided
bringeth
1.
&
〈◊〉
part
remaining
,
whiche
abrouied
maketh
41/60
and
thus
you
doe
perceaue
that
the
2/1
&
〈◊〉
added
with
the
⅘
and
¼
doe
amounte
vnto
1.
〈◊〉
as
hereafter
by
practise
doth
plainly
appere
.
¶
Addition
of
whole
nomber
and
broken
together
.
5.
Also
if
you
will
adde
.
12.
⅘
.
with
20.
⅚
,
you
maie
(
if
you
will
)
adde
.
1●
.
and.
20.
together
,
and
thei
make
.
32.
and
then
adde
the
twoo
broken
nombers
together
,
that
is
to
saie
⅘
,
and
⅚
,
by
the
order
of
the
firste
addition
,
and
thei
make
49/〈…〉
:
therefore
diuide
49.
by
〈◊〉
,
and
thereof
commeth
1.
and
〈◊〉
partes
remaine
,
whiche
.
1.
you
must
adde
vnto
the.
●2
and
the
whole
addition
wil
be
.
33
〈◊〉
,
or
otherwise
,
you
maie
reducte
12.
⅘
.
into
the
likenesse
of
a
fraction
by
the
sixt
reduction
,
and
thei
will
bee
〈◊〉
,
and
likewise
by
thesame
reduction
,
reduce
.
2
〈…〉
,
and
thei
be
〈◊〉
,
then
adde
〈◊〉
with
the
〈◊〉
,
by
the
firste
addition
,
and
you
shall
finde
.
〈◊〉
.
Therefore
diuide
●●09
.
by
.
●0
,
and
thereof
commeth
〈◊〉
.
as
before
,
and
as
by
practise
of
thesame
bothe
the
waies
,
doeth
here
vnder
appeare
.
¶
The
.v.
Chapiter
treateth
of
Substraction
in
broken
nombers
.
IF
you
will
substracte
⅔
from
2/4
you
must
firste
reduce
bothe
the
fractions
into
a
common
denomination
by
the
fyrst
reduction
,
and
you
shall
finde
8/12
for
the
⅔
,
and
9/12
for
the
¾
.
Therefore
abate
the
numerator
8.
from
the
numerator
9.
and
there
remaineth
1/12
as
may
appere
here
by
practise
.
2.
But
if
you
haue
a
broken
nomber
,
to
bee
substracted
from
a
whole
nomber
,
you
must
borow
one
of
the
whole
nomber
,
and
resolue
it
into
a
fraction
of
like
denomination
,
as
is
the
fraction
,
which
you
would
abate
from
the
same
whole
nomber
,
and
then
abate
the
saied
fraction
there
from
,
and
you
shall
finde
what
doeth
remaine
,
as
by
this
example
.
If
you
abate
⅘
from
.
8.
you
must
borowe
out
of
the
said
.
8.
and
resolue
it
into
fiftes
like
vnto
the
fraction
,
because
it
is
4.
fiftes
,
that
.
1.
will
bee
5.
fiftes
thus
5/5.
therefore
abate
⅘
.
from
5/5.
and
there
will
remaine
⅘
,
and
substract
that
.
1.
whiche
you
borowed
from
8.
and
there
doeth
remain
.
7.
and
also
the
⅕
.
Thus
the
⅘
being
substracted
from
.
8.
doeth
leaue
.
7.
⅕
,
as
by
practise
doeth
plainly
appere
3.
If
you
will
substract
broken
nomber
,
from
whole
nomber
,
and
broken
beyng
together
:
thus
,
as
if
you
would
substract
¾
.
from
.
6.
⅚
,
you
maie
by
the
first
substraction
,
abate
¾
.
from
⅚
,
and
there
will
remaine
1/12
,
and
the
6.
doeth
still
remaine
whole
,
because
that
5/4.
beyng
abated
from
.
6.
⅚
.
leaueth
.
6.
1
/
12.
as
appeareth
by
practise
.
Likewise
if
you
will
abate
⅔
,
from
14.
⅖
,
you
muste
firste
reduce
.
14.
⅖
.
all
into
fiftes
by
the.
6.
reduction
,
and
thei
bee
〈◊〉
,
then
reduce
⅔
.
into
a
common
denomination
,
by
the
firste
reduction
,
and
you
shall
finde
10/15.
for
the
⅔
:
and
21●/15
for
the
70/5
:
then
substracte
the
numerator
.
10.
of
the
firste
fraction
,
from
the
numerator
.
216.
of
the
seconde
fraction
,
and
there
remaineth
206/15.
Therfore
diuide
206.
by
.
15.
and
thereof
commeth
13.11/15
,
and
so
muche
remain
of
this
substraction
,
as
maie
appeare
.
4.
If
you
will
substracte
whole
nomber
and
broken
,
from
whole
and
broken
,
as
thus
,
if
you
will
substract
.
9¼
.
from
.
20.
½
.
you
must
reduce
.
9.
¼
.
into
fowerthes
,
and
likewise
the.
20
½
.
into
halfes
by
the
sixt
reduction
:
and
you
shall
find
37/4
for
the.
9.
¼
.
And
42/2.
for
the
20.
●/2.
Then
reduce
57/4
and
31/2
into
one
denomination
,
accordyng
vnto
the
first
reduction
and
you
shall
finde
74/8
for
the
37/4
,
and
●64/●
for
the
41/2
than
abate
the
numerator
of
●65/8
and
there
remaineth
90/8
then
diuide
90.
by
8.
and
thereof
commeth
11.
½
whiche
is
the
remaine
of
this
substraction
.
¶
Substraction
of
broken
nombers
of
broken
.
5.
If
you
will
substracte
,
the
½
of
⅔
of
⅗
.
from
the
⅚
.
of
¾
.
of
⅞
,
you
muste
firste
bryng
the
½
.
of
⅔
.
of
⅗
.
into
one
fraction
by
the.
3.
reduction
,
and
the
⅚
of
¾
of
⅞
likewise
into
one
fraction
by
the
same
reduction
,
and
you
shal
finde
6/30.
for
the
firste
.
3.
broken
nombers
,
whiche
beyng
abreuied
dooe
make
.
⅕
:
and
for
the
other
3.
broken
nombers
,
you
shall
finde
105/192
:
whiche
beyng
likewise
abreuted
dooe
make
35/64.
then
you
shall
substracte
⅕
.
from
.
35/64.
by
the
instruction
of
the
firste
substraction
,
in
reducyng
bothe
the
fractions
into
a
common
denomination
,
as
before
is
dooen
,
and
you
shall
finde
remainyng
〈◊〉
/
3●0
,
as
maie
appeare
by
example
.
¶
The
sixt
Chapiter
is
of
multiplication
in
broken
nombers
.
FIrste
,
for
to
multiplie
in
broken
nomber
,
there
is
a
rule
,
whiche
is
thus
,
multiplie
the
numerator
of
the
one
fraction
,
by
the
numerator
of
the
other
.
And
then
diuide
that
fraction
if
you
maie
,
or
els
abreuiate
it
,
and
you
haue
do●en
:
but
if
there
be
whole
nomber
and
broken
together
,
you
muste
reduce
the
whole
nombers
into
broken
,
and
adde
thervnto
the
numerator
of
his
broken
,
and
then
multiplie
,
as
is
before
saied
,
as
also
hereafter
by
examples
shal
more
plainly
appeare
.
1.
If
you
will
multiplie
⅔
by
¾
,
you
muste
multiplie
the
numerator
.
2.
by
the
numerator
.
3.
and
therof
commeth
6.
for
the
numerator
.
Likewise
multiplie
the
denominators
th
one
by
the
other
,
that
is
to
saie
.
3.
by
.
4.
and
thereof
commeth
.
12.
for
the
denominator
,
so
that
this
multiplication
commeth
to
6/12
,
whiche
beyng
abreuied
doe
make
½
.
and
so
muche
amounteth
the
multiplication
of
the
⅔
.
by
¾
as
by
practise
.
2.
Likewise
,
if
you
will
multiplie
a
broken
nomber
by
whole
nomber
,
or
whole
nomber
by
broken
,
whiche
is
all
one
,
as
⅘
.
by
.
18
or
els
.
18.
by
⅘
,
you
must
set
.
1.
vnder
.
18.
thus
.
18/1
:
and
then
multiplie
.
18.
by
the
numerator
4.
and
therof
commeth
.
72.
the
whiche
diuide
by
the
denominator
.
5.
and
thereof
commeth
.
14.
⅖
.
for
the
whole
multiplication
,
or
otherwise
abate
from
.
18.
his
●/5.
parte
,
whithe
is
.
3.
⅗
,
and
there
remaineth
.
14.
⅖
.
as
aboue
.
3.
Also
if
you
wil
multiplie
a
whole
nomber
,
by
whole
nomber
&
broken
,
or
els
whole
nomber
&
broken
by
a
whole
nomber
,
whiche
is
all
one
.
As
by
example
.
If
you
will
multiplie
.
15.
by
.
16.
¾
.
or
els
.
16.
¾
by
.
15
First
reduce
.
16
¾
all
into
fourthes
,
in
multipliyng
16.
by
the
deneminator
of
¾
whiche
is
4.
&
therof
commeth
64
whereunto
adde
the
numerator
.
3.
&
it
maketh
67/4
,
whiche
multiplie
by
15/1
,
accordyng
vnto
th
instruction
of
the
laste
example
,
and
you
shall
find
the
product
of
this
multiplication
to
be
251
¼
as
by
practise
doth
here
appere
4.
And
if
you
will
multiple
a
broken
nomber
,
by
whole
nomber
and
broken
,
or
els
whole
nomber
and
broken
by
a
broken
.
As
by
example
,
if
you
will
multiplie
¼
.
by
.
18.
⅔
,
or
els
18
⅔
.
by
¼
,
whiche
is
all
one
:
you
muste
reduce
the
whole
noumber
into
his
broken
by
the
sixt
reduction
.
And
you
shall
finde
,
56/3
,
whiche
you
shall
multiplie
by
the
¼
,
after
the
doctrine
of
the
first
multiplication
,
that
is
to
saie
:
in
multipliyng
the
numerator
.
56.
by
the
Numerator
of
¼
,
which
is
1.
And
it
is
still
56.
because
1.
doth
neither
multiplie
nor
deuide
.
And
likewise
you
muste
multiplie
the
Denominator
.
3.
by
the
Denominatour
.
4.
and
it
maketh
.
12.
then
diuide
.
56.
by
.
12.
and
thereof
commeth
.
4.
⅔
.
And
so
muche
amounnteth
the
multiplication
of
the
〈◊〉
.
⅔
.
multiplied
by
¼
,
as
by
example
.
5.
If
you
will
multiplie
whole
nomber
and
broken
,
with
whole
and
broken
,
you
muste
firste
put
either
whole
nomber
into
his
broken
,
accordyng
to
the
instruction
of
the
sixte
reduction
,
and
〈◊〉
multiplie
the
one
numera
〈◊〉
the
other
,
and
of
the
producte
make
your
numerator
.
And
likewise
multiplie
the
denominators
,
the
one
by
the
other
,
and
thereof
make
the
denominator
,
then
diuide
the
numerator
,
by
the
denominator
,
and
the
quotient
shal
be
the
encrease
of
this
multiplication
.
¶
Example
.
If
you
would
multiplie
.
12.
⅘
.
b
〈…〉
.
¾
:
firste
by
the
sixte
reduction
,
the.
12.
⅘
will
make
64/5
,
and
the.
6.
¾
.
will
make
27/4
then
multiplie
the
numerator
.
64.
by
the
numerator
.
27.
and
thereof
commeth
.
1728.
for
the
numerator
.
And
then
you
must
multiplie
the
denominator
.
5.
by
the
denominator
.
4.
and
thei
doe
make
.
20.
then
diuide
.
1728.
by
.
20.
and
thereof
commeth
.
86.
⅖
.
for
the
whole
multiplication
,
as
by
example
.
6.
If
you
will
multiplie
one
broken
nomber
by
many
broken
nombers
,
thus
:
As
to
multiplie
⅔
.
by
5/7.
and
by
4/9
:
you
muste
multiplie
the
numerators
of
all
the
fractions
,
the
one
by
the
other
,
and
of
the
product
make
the
numerator
,
that
is
to
saie
:
2.
by
.
5.
and
thei
be
.
10.
then
.
10.
by
.
4.
and
thei
bee
40.
for
the
numerator
.
Likewise
you
must
multiplie
the
denominators
,
the
one
by
the
other
,
that
is
to
saie
.
3.
by
.
7.
maketh
.
21.
then
.
21.
by
.
9.
maketh
.
189
for
the
denominator
:
then
sette
.
40.
ouer
the.
189.
with
a
line
betwene
them
,
and
thei
make
40/189.
And
so
muche
amounteth
the
whole
multiplication
of
the
⅔
,
multiplied
by
3/7
and
4/9
,
as
by
example
folowyng
.
And
thus
is
to
be
vnderstande
of
all
suche
like
.
¶
The
.vii.
Chapiter
treateth
of
diuision
in
broken
nombers
.
NOte
,
that
in
diuision
of
broken
nombers
,
you
muste
sette
your
diuisor
downe
firste
,
nexte
vnto
the
lefte
hande
,
and
the
diuidende
or
nomber
,
whiche
is
to
bee
diuided
alwaies
towarde
the
right
hande
.
And
then
multiplie
crosse wise
,
that
is
to
saie
,
the
numeratour
of
youre
Diuisour
by
the
Denominator
of
the
Diuidende
,
and
the
producte
shal
bee
the
Denominatour
,
whiche
afterwarde
shall
bee
your
Diuisour
.
And
likewise
you
muste
multiplie
the
Denominatour
of
your
firste
noumber
,
that
is
to
saie
of
your
Diuisour
:
By
the
Numeratour
of
the
Diuidende
,
whiche
afterwarde
shall
bee
the
Diuidende
,
and
that
muste
bee
sette
ouer
the
Crosse
,
and
the
Denominatour
vnder
the
Crosse
,
then
shall
you
diuide
the
Numeratour
by
the
Denominatour
,
if
any
maie
bee
diuided
,
if
not
,
you
muste
abreuiate
theim
,
as
hereafter
by
examples
shall
more
plainly
appeare
.
1.
If
you
will
diuide
¾
.
by
⅔
,
you
muste
sette
the
Diuisour
(
whiche
is
⅖
)
nexte
to
the
lefte
hande
,
and
the
diuidende
¾
.
towarde
your
right
hande
,
with
a
crosse
betwene
them
:
as
maie
appere
by
this
example
in
the
margente
.
Then
you
shall
multiplie
the
numer
at
or
of
the
⅔
,
whiche
is
.
2.
by
the
denominator
of
the
¾
.
which
is
4.
and
thereof
commeth
.
8.
which
shal
be
your
newe
diuison
:
set
that
.
8.
vnder
the
〈…〉
osse
,
as
the
denominator
,
then
multiplie
the
numerator
of
the
diuidende
,
that
is
to
saie
,
of
the
¾
whiche
is
3.
by
the
denominator
of
the
diuisour
,
that
is
to
wit
,
of
the.
⅔
.
whiche
is
.
3.
set
that
ouer
the
crosse
,
and
it
is
.
9.
for
the
numerator
,
whiche
shal
be
now
the
diuidende
,
or
nomber
to
be
diuided
.
Then
finally
,
you
shall
diuide
.
9.
by
.
8.
and
thereof
commeth
into
the
quotient
.
1.
2/8
and
so
often
times
is
⅔
.
conteined
in
¾
,
as
dooeth
appeare
before
in
the
margente
.
But
in
case
you
would
diuide
⅔
.
by
.
¾
,
you
muste
likewise
sette
your
diuisor
¾
nexte
to
your
left
hande
,
as
is
before
said
.
And
then
procede
,
as
is
aboue
declared
,
&
you
shall
finde
that
⅔
diuided
by
¾
bringeth
into
the
quotient
2/9
,
whiche
can
not
be
diuided
nor
abbreuied
,
wherfore
it
appereth
that
⅔
diuided
by
¾
bringeth
but
2/9
of
one
vnitie
into
the
quotient
as
doth
appere
.
2.
Likewise
if
you
wil
diuide
a
broken
number
by
a
whole
nomber
,
or
els
a
whole
number
by
a
broken
,
as
to
diuide
¾
by
13.
you
shall
put
1.
vnder
13.
and
it
wil
be
13/1
which
is
your
diuisor
,
set
that
toward
your
left
hande
,
and
then
multiply
13.
by
4.
according
to
the
first
diuision
,
&
therof
commeth
52.
for
the
denominator
,
set
that
vnder
the
crosse
&
multiply
3.
by
1.
which
is
3.
for
the
numerator
,
that
set
ouer
the
crosse
,
and
it
is
3/52
as
appeareth
in
the
margent
.
But
if
you
will
diuide
13.
by
¾
then
set
the
¾
nexte
your
left
hand
and
put
one
vnder
13.
as
in
the
last
example
,
&
it
is
●/3
set
that
toward
your
right
hande
thus
,
as
appeareth
in
the
margent
,
and
then
worke
according
to
the
doctrine
of
the
first
diuision
,
&
you
shall
finde
that
13.
being
diuided
by
¾
bringeth
into
the
quotient
52/4
,
then
diuide
52.
by
3.
and
therof
commeth
17.
●/3
,
and
so
oftentimes
is
¾
conteined
in
13.
as
doth
appere
.
3.
And
if
you
wil
diuide
whole
number
by
whole
number
and
broken
,
or
els
whole
number
and
broken
by
whole
number
,
as
to
diuide
20.
by
5.
⅚
,
you
shall
reduce
5.
⅚
into
his
broken
by
the
sixt
reduction
,
&
it
maketh
35/●
for
your
diuisor
,
then
put
1.
vnder
20.
And
it
wyll
bee
20/1
,
then
shall
you
multiply
35.
by
1.
and
20.
by
6.
as
is
taught
in
the
other
diuisions
,
and
you
shall
finde
220/3
:
then
diuide
12.
by
35.
and
you
shall
fynde
in
your
quotient
3.
3/●
&
so
many
tymes
is
5.
●/6
conteined
in
20.
as
in
the
margent
doth
appere
.
But
if
you
will
diuide
5.
3/6
by
20.
you
must
diuide
35.
by
120.
which
you
can
not
,
wherefore
you
shall
abbreuiate
35/120
,
and
thereof
commeth
.
7/24.
4.
If
you
will
diuide
a
broken
number
by
whole
number
and
broken
,
or
els
a
whole
number
and
broken
,
by
a
broken
number
.
As
to
diuide
¾
by
13.
⅔
,
you
muste
reduce
13.
⅔
,
into
hys
broken
,
by
the
syxte
reduction
And
they
be
41/3
for
your
diuisor
,
then
multiplye
41.
by
4.
&
they
make
164.
for
your
denominator
,
likewise
multiplye
3.
by
3.
and
they
make
9.
for
the
numerator
,
and
then
will
your
summe
be
9/104.
But
yf
you
will
diuide
13.
⅔
by
¾
then
you
must
diuide
164.
by
9.
and
you
shall
fynde
18.
2
/
9.
5.
If
you
will
diuide
whole
number
and
broken
,
by
whole
number
&
broken
,
as
to
diuide
7.
¾
by
13.
⅔
you
must
reduce
the
whole
numbers
into
their
broken
,
by
the
doctrine
of
the
sixt
reduction
,
&
you
shall
fynde
31/4
for
the
7.
¾
,
&
41/3
for
the
13.
⅔
.
Then
sett
downe
41/3
towarde
the
left
hande
bycause
it
is
your
diuisor
,
and
the
31/4
towarde
the
right
hande
,
and
multiplie
41.
by
4.
for
your
denominator
,
and
thereof
commeth
164.
Likewise
multiply
31.
by
3.
for
your
numerator
,
and
it
amounteth
to
93.
the
whiche
diuision
will
bee
thus
〈◊〉
as
before
doth
appere
.
But
if
you
will
diuide
13⅔
.
by
7.
¾
you
muste
contrariwyse
to
the
other
example
,
diuide
164.
by
93.
and
you
shall
fynde
in
the
quotient
1.
71
/
93.
6.
The
broken
numbers
of
broken
,
muste
be
diuided
in
suche
maner
as
broken
numbers
are
,
&
there
is
no
difference
,
sauing
onely
that
of
many
broken
numbers
you
must
make
but
two
broken
numbers
,
that
is
to
saye
the
diuisor
,
and
the
diuidend
,
or
number
that
is
to
be
diuided
,
example
.
If
you
will
diuide
the
¾
of
⅗
of
½
,
by
the
2/2
of
4/7.
For
the
fyrst
,
the
¾
of
⅗
of
½
are
9/4.
by
the
thirde
reduction
:
and
the
⅔
of
4/7
are
by
the
same
Reduction
8/21
,
then
haue
you
8/21
for
your
diuisor
,
&
9/40
for
your
number
to
bee
diuided
,
then
multiply
8.
by
40.
which
maketh
320.
set
that
vnder
the
crosse
and
multiply
9.
by
21
:
&
thereof
commeth
189.
which
set
ouer
the
crosse
for
the
numerator
,
and
they
make
189/120
for
this
diuision
as
doth
appere
.
But
if
you
woulde
diuide
1/21
by
9/200.
you
must
worke
contrary
to
the
laste
example
,
that
is
to
saye
,
you
must
diuide
320.
by
189.
And
therof
commeth
in
the
quotient
1.
131
/
189.
¶
The
eyght
Chapter
treateth
of
duplation
,
triplation
,
and
quadruplation
of
all
broken
numbers
.
IF
you
wyll
double
any
broken
number
,
you
shall
diuide
the
same
by½
:
likewise
if
you
wyll
triple
any
fraction
you
muste
diuide
it
by
⅔
.
And
for
to
quadruple
any
broken
number
,
you
shall
diuide
it
by
¼
,
and
so
is
to
be
vnderstande
of
all
other
.
Example
of
duplation
.
IF
you
will
double
⅜
you
shal
diuide
3/●
by
½
,
and
thereof
commeth
6/8
,
which
being
abbreuied
are
¾
:
as
by
example
.
Or
otherwise
,
in
case
the
denominator
of
any
fraction
bee
an
euen
number
,
you
may
take
halfe
the
sayde
denominator
,
without
anye
other
operation
,
and
the
numerator
to
abyde
still
the
numerator
,
vnto
the
said
halfe
of
the
denominator
of
the
Fraction
,
as
by
the
other
example
before
rehearsed
:
that
is
to
say
of
〈◊〉
,
take
½
of
8.
which
is
4.
and
that
is
the
denominator
,
and
3.
remaineth
stil
numerator
to
4.
and
it
maketh
¾
and
so
of
all
other
.
But
in
case
the
denominator
bee
an
odde
nomber
,
that
is
to
say
,
not
euen
,
then
you
may
multiply
the
numerator
by
2.
or
els
double
the
numerator
,
whiche
is
al
one
thing
;
and
that
fraction
shall
bee
doubled
.
Example
,
if
you
will
double
●/5
you
must
onely
multiply
the
numerator
3.
by
2.
&
they
be
6.
whiche
maketh
that
fraction
to
be
〈◊〉
,
the
which
6.
being
diuided
by
5.
bringeth
1.
⅕
and
so
much
is
the
double
of
3/●
.
Example
of
Triplation
.
If
you
wil
triple
⅗
you
muste
diuide
●/5
by
●/3
and
thereof
commeth
9/5
whiche
beinge
diuided
bringeth
1⅘
,
or
otherwise
,
bicause
the
denominator
is
an
odde
number
you
maye
multiplie
the
numerator
3.
by
3.
and
therfore
commeth
9.
which
maketh
〈◊〉
as
before
.
Example
of
quadruplacion
.
If
you
will
quadruple
⅘
,
you
shall
diuide
⅘
by
¼
and
thereof
commeth
⅘
which
16.
being
diuided
by
5.
bringeth
3●/5
,
or
otherwise
,
bicause
the
denominator
of
the
fraction
is
an
odde
number
,
you
shall
multiplie
the
numerator
of
the
〈◊〉
that
is
to
say
4.
by
4.
and
therof
commeth
16.
the
whiche
diuide
by
5.
and
you
shall
finde
3.
●/5
as
before
,
and
this
sufficeth
for
duplacion
,
tripiacion
and
quadruplacion
.
¶
The
9.
Chapter
treateth
of
the
proofes
of
broken
numbers
.
And
first
of
Reduction
.
IF
you
doe
abbreuiate
the
broken
numbers
whiche
bee
reduced
,
you
shall
retourne
them
into
their
firste
estate
:
as
by
example
,
if
you
reduce
●/5
wyth
⅘
you
shall
fynde
10/〈…〉
and
12/〈…〉
,
then
abbreuiate
10/15
and
you
shall
fynde
⅔
,
abbreuiate
likewise
〈◊〉
and
thereof
commeth
⅘
as
before
.
The
proofe
of
Abbreuiation
.
IF
you
doe
multiplie
that
number
whiche
you
haue
abbreuied
by
that
or
those
numbers
,
by
the
whiche
you
haue
abbreuied
them
,
you
shall
returne
them
againe
into
their
firste
estate
.
Example
,
if
you
wyll
abbreuiate
32/48
by
16.
in
takyng
y●
〈◊〉
part
both
of
the
numerator
,
and
also
of
the
denominator
,
you
shall
finde
⅔
,
the
proofe
is
thus
,
you
must
multiplye
bothe
the
numerator
&
denominator
of
2/●
by
16.
that
is
to
say
,
three
by
16.
maketh
48.
for
the
denominator
,
&
2.
by
16.
maketh
32.
for
the
numerator
,
then
set
the
numerator
32.
ouer
the
denominator
48
and
they
be
32/48
as
before
.
The
proofe
of
Addition
.
If
you
doe
substract
one
of
the
numbers
,
or
manye
of
them
(
which
you
haue
added
)
from
the
totall
summe
,
there
shal
remaine
the
other
,
or
others
;
Example
:
if
you
do
adde
1/●
with
¼
you
shall
fynde
7/12.
The
proofe
is
,
is
if
you
substract
⅓
from
7/12
you
shall
fynde
remaining
the
other
number
whiche
is
2/4
,
or
els
yf
you
doe
substract
¼
from
〈◊〉
there
will
remains
the
other
nomber
,
which
is
1/●
.
The
proofe
of
Substraction
.
If
you
do
adds
that
number
whiche
remaineth
,
with
the
number
whyche
you
did
substract
,
you
shall
fynde
the
totall
summe
,
oute
of
the
which
you
made
the
abateinēt
:
or
otherwise
,
if
you
adde
the
twoo
lesser
nombers
togither
,
you
shall
finde
the
greater
.
Examplet
if
you
doe
abate
or
substracte
¼
from
⅓
there
will
remaine
●/12.
The
profe
is
thus
:
you
muste
adde
1/12
&
¼
togither
,
and
you
shall
fynde
●/3
,
whiche
is
the
greatest
nomber
.
The
proofe
of
multiplication
.
If
you
diuide
the
producte
of
the
whole
multiplication
,
by
the
multiplicator
,
you
shal
fynde
in
your
quotient
,
the
multiplicande
or
nomber
by
the
which
you
haue
multiplyed
:
or
els
if
you
diuide
the
totall
some
which
is
come
of
the
multiplication
,
by
the
multiplicande
:
you
shall
finde
in
the
quotient
the
multiplicator
.
Example
,
if
you
multiply
〈◊〉
by
⅘
,
the
product
of
this
multiplication
will
bee
8/15.
The
proofe
is
thus
:
you
shal
diuide
8/15
by
the
multiplicator
⅘
,
and
therof
commeth
⅔
.
Or
els
diuide
8/15
by
⅔
&
you
shall
finde
the
⅘
which
is
the
multiplicator
.
The
proofe
of
Diuision
.
If
you
doe
multiplye
the
quotient
by
the
diuisor
,
you
shall
finde
the
number
which
you
did
diuide
,
that
is
to
say
,
your
diuidende
.
Example
:
if
you
diuide
⅔
by
¾
,
your
quotient
will
bee
2/9
the
proofe
is
thus
,
you
must
multiply
8/9
by
¾
,
and
thereof
commeth
24/36
whiche
being
abbreuiated
art
⅔
whiche
is
your
diuidende
,
&
by
this
maner
all
whole
numbers
haue
their
proofes
as
well
as
broken
numbers
.
¶
The
tenth
Chapter
treateth
of
certaine
questions
done
by
broken
numbers
.
And
first
by
Reduction
.
FInde
twoo
numbers
,
where
of
the
2/●
of
the
one
number
may
bee
egal
vnto
the
〈◊〉
of
the
other
.
Aunswere
:
you
shall
reduce
2/7
&
⅜
crossewise
,
and
you
shall
finde
16.
ouer
the
⅔
and
21.
ouer
the
⅜
,
which
are
the
two
numbers
that
you
seeke
:
for
the
⅜
of
16.
are
6.
and
so
are
the
2/●
of
21.
lykewise
6.
wherefore
you
may
perceiue
that
the
the
⅜
of
16.
which
are
6.
are
egall
vnto
the
2/7
of
21.
whiche
is
also
6.
2.
Finde
twoo
numbers
,
wherof
the
2/●
of
the
one
may
be
double
to
the
¼
of
the
other
.
Aunswere
:
double
¼
&
you
shal
haue
2/4
,
which
being
abbreuiated
is
½
:
them
reduce
⅔
&
½
crossewise
,
&
you
shall
finde
4.
ouer
the
⅔
&
three
ouer
the
½
which
are
the
two
numbers
that
you
seeke
.
For
the
2/1
of
3.
which
is
2.
is
double
vnto
the
¼
of
4.
which
is
but
1.
3.
Finde
two
numbers
whereof
the
⅔
and
the
¼
of
the
one
,
maye
bee
egall
vnto
the
¼
&
⅕
of
the
other
.
Aunswere
:
Adde
the
⅓
and
¼
togither
,
and
they
make
7/12
then
adde
¼
and
●/5
togither
,
&
they
are
6/20
,
then
reduce
7/17
&
9/20
crosse-wise
,
&
you
shall
haue
140.
ouer
the
7/12
&
108.
ouer
the
9/20
,
whiche
are
the
two
numbers
that
you
seke
.
For
63.
whiche
are
the
7/12
of
108.
are
also
the
9/20
of
140.
4.
Finde
two
numbers
,
wherof
the
●/2
the
and
the
¼
of
the
one
of
them
,
maye
by
egall
vnto
the
⅕
the
⅙
and
1/7
of
the
other
number
.
Aunswere
:
first
you
must
adde
½
,
⅓
,
and
¼
togither
,
&
they
make
13/12
:
then
adde
⅕
,
⅙
and
1/7
togither
,
&
they
make
207/210.
Then
reduce
13/12
and
107/210
crossewise
,
as
by
the
fyrste
question
of
reduction
,
and
you
shall
finde
2730.
ouer
the
13/12
and
1284.
ouer
the
107/210
,
whiche
are
the
twoo
nombers
that
you
seeke
:
for
1391
which
is
the
½
the
⅓
and
¼
of
1284.
is
lyke
to
the
11/56
&
¼
of
2730
,
which
is
also
1391.
5.
Finde
three
nombers
,
whereof
the
⅖
of
the
first
,
the
4/7
of
the
seconde
,
&
the
4/9
of
the
thirde
,
maye
be
egall
the
one
to
the
other
.
Aunswere
:
set
downe
the
23/57
and
〈◊〉
,
and
then
multiplie
the
Denominator
of
the
⅖
that
is
to
saye
5.
by
the
Numerators
of
the
other
twoo
Fractions
,
that
is
to
saye
,
by
the
Numerator
of
3/7
,
and
by
the
Numerator
of
4/9
,
whiche
is
3.
and
4.
And
thereof
commeth
60.
for
your
fyrste
nomber
,
then
shall
you
multiplye
the
Denominator
of
the
3/7
whiche
is
7.
by
the
Numerators
of
⅕
and
4/9
,
that
is
to
say
by
2.
and
4.
and
thereof
commeth
56.
for
the
seconde
number
:
Then
multiplie
the
Denominator
of
4/9
,
that
is
9.
by
the
Numerators
of
⅖
and
●/7
that
is
by
2.
and
by
3.
and
thereof
commeth
54.
for
the
third
nomber
And
thus
the
⅖
of
60.
which
is
24.
is
likewise
the
3/7
of
56.
whiche
is
the
second
nomber
and
the
4/9
of
54.
whiche
is
the
thirde
nomber
.
6.
Finde
three
nombers
,
of
whiche
the
fyrste
and
the
seconde
maye
bee
in
suche
proporcion
as
½
and
⅓
,
and
the
seconde
and
thirde
in
suche
proportion
as
¼
and
⅕
.
Aunswere
:
reduce
½
and
⅓
crossewise
,
and
you
shall
haue
3.
ouer
the
½
and
2.
ouer
the
⅓
,
then
reduce
¼
and
⅕
in
lyke
maner
,
and
you
shall
fynde
5.
ouer
the
¼
and
4.
ouer
the
3/●
.
Then
say
by
the
Rule
of
three
,
●●
5.
do
gyue
me
4.
what
shall
twoo
gyue
me
,
whiche
is
the
seconde
proportionall
,
multiply
the
seconde
nomber
4.
by
the
thyrde
nomber
twoo
,
and
therof
commeth
eyght
,
the
whiche
diuide
by
the
first
number
5.
and
therof
commeth
1.
⅗
for
the
third
proportionall
,
and
you
shal
fynde
that
3.
2.
1.
⅗
are
the
three
nombers
proportionall
that
I
demaunde
,
or
els
15.
10.
&
8.
in
whole
numbers
.
1.
What
number
is
that
,
vnto
the
whiche
if
you
doe
adde
13.
the
whole
amounteth
to
31.
Aunswere
:
rebate
13.
from
31.
and
there
wyll
remaine
18.
which
is
the
number
that
you
seeke
.
2.
What
number
is
that
,
vnto
the
which
if
you
adde
⅖
the
addition
wyll
be
⅚
.
Answere
:
abate
⅖
from
⅚
,
and
there
will
remaine
13/30
,
whiche
is
the
number
that
you
desire
.
3.
What
number
is
that
,
whereunto
if
you
adde
7.
⅔
,
the
whole
addition
will
be
12.
¼
.
Aunswere
:
abate
7.
⅔
from
12.
¼
.
&
the
remaine
will
be
4
7/12
which
is
the
number
that
you
desire
to
know
.
4.
What
number
is
that
,
whereinto
if
you
adde
the
¾
of
it selfe
,
that
is
to
say
,
of
the
number
that
your
séeke
,
the
whole
addition
may
be
⅚
.
Aunswere
:
Here
followeth
a
generall
rule
for
all
suche
like
questions
.
First
,
of
.
3.
whiche
is
the
numeratour
of
.
¾
.
make
still
the
numeratour
,
and
likewise
of
.
3.
and.
4
together
,
whiche
is
bothe
the
numerator
,
and
the
denominator
of
the.
¾
.
make
your
denominatour
,
so
you
shall
finde
.
3/7
,
then
take
the
3/7
of
⅚
,
whiche
is
15/42
,
or
5/14
;
and
substracte
theim
from
⅚
,
and
there
will
remaine
10/21
,
whiche
is
the
nomber
that
you
seke
.
5.
What
nomber
is
that
,
vnto
the
whiche
if
you
adde
his
owne
⅔
.
that
is
to
saie
⅔
.
of
it self
,
the
whole
addition
shall
bee
.
20.
Aunswere
:
do●e
as
in
the
laste
question
:
of
the
numeratour
of
⅔
that
is
to
saie
,
of
.
2.
make
stil
your
numerator
.
And
likewise
of
the
numerator
.
2.
and
the
denominator
.
3.
of
the
⅔
,
make
of
them
bothe
,
your
denominator
,
and
you
shall
finde
⅖
,
then
take
the
⅖
.
of
.
20.
whiche
are
.
8.
And
abate
them
from
.
20.
and
there
will
remain
12.
whiche
is
the
nomber
that
you
desire
,
and
so
is
to
bee
dooen
of
all
suche
like
reasons
.
1.
What
nomber
is
that
,
from
the
whiche
if
you
dooe
abate
.
17.
the
reste
maie
bee
.
19.
Aunswere
:
adde
.
17.
and
19.
together
,
and
you
shall
finde
.
36.
whiche
is
the
nomber
that
you
seke
.
2.
What
nomber
is
that
,
from
the
whiche
if
you
abate
⅗
,
the
rest
maie
bée
⅛
.
Aunswere
:
adde
⅗
and
⅛
together
,
and
you
shall
finde
29/40
,
whiche
is
the
nomber
that
you
demaunde
.
3.
What
nomber
is
that
,
from
the
whiche
if
you
deduct
.
13.
½
the
rest
maie
be
.
5.
5
/
7.
Aunswere
:
adde
.
13.
½
and.
5.
5
/
7.
together
,
and
there
of
commeth
.
19.
1¾
,
whiche
is
the
nomber
that
you
seke
.
4.
What
nomber
is
that
,
from
the
whiche
if
you
substracte
his
⅖
,
the
reste
maie
bee
.
12.
Aunswere
:
and
a
rule
for
5.
What
nomber
is
that
,
from
the
whiche
if
you
doe
abate
his
¾
,
the
reste
maie
bee
8/●
.
Aunswere
:
from
the
denominator
of
¾
,
whiche
is
.
4.
substracte
his
numerator
.
3.
and
there
resteth
.
1.
Thus
of
¾
.
you
haue
made
.
3/1.
Then
multiplie
5/●
by
8/9
,
and
therof
commeth
2.
⅔
,
the
whiche
adde
vnto
8/9
,
and
you
shall
haue
.
3.
5/9
,
whiche
is
the
nomber
that
you
seke
.
6.
What
nomber
is
that
,
from
the
whiche
if
ye
abate
his
⅘
,
the
rest
maie
be
.
12.
⅔
.
Aunswere
:
Doe
as
you
did
in
the
laste
question
,
and
you
shall
finde
that
the
⅘
.
will
bee
4/1.
And
therefore
multiplie
.
12.
⅔
by
4/1
,
and
thereof
commeth
.
50
⅔
,
the
whiche
adde
vnto
.
12
⅔
,
and
you
shall
finde
.
93.
〈◊〉
,
for
the
nomber
that
you
demaunde
.
And
thus
of
all
like
questions
.
1.
What
nomber
is
that
,
which
beyng
multiplied
by
13.
the
whole
.
Mul●iplication
shal
m●●nte
to
.
221.
Aunswere
:
〈…〉
de.
221.
by
.
13.
and
thereof
comnieth
.
17.
whiche
is
the
nomber
that
you
seeke
.
2.
What
nomber
is
that
,
whiche
beyng
multiplied
by
.
15.
the
whole
multiplication
wil
amount
to
¾
.
Aunswere
:
diuide
¼
.
by
15/1.
and
thereof
commeth
1/20.
whiche
is
the
noumber
,
that
you
seeke
.
3.
What
nomber
is
that
,
whiche
beeyng
multiplied
by
.
21.
the
whole
multiplication
will
bee
:
16.
⅘
.
Aunswere
:
diuide
.
16.
⅘
.
by
.
21/5
,
and
you
shall
finde
⅘
,
whiche
is
the
nomber
that
you
demaund●
.
4.
What
nomber
is
that
,
whiche
beyng
multiplied
by
¾
.
the
multipliplication
will
amounte
to
.
18.
Aunswere
:
diuide
18/1
by
¾
,
and
there
of
commeth
.
24.
whiche
is
the
nomber
that
you
desire
to
knowe
.
5.
What
nomber
is
that
,
whiche
if
it
bee
multiplied
by
⅔
.
the
whole
multiplication
will
bee
.
●/4.
Aunswere
:
diuide
¼
by
⅔
.
and
the
quotient
will
bee
⅜
,
whiche
is
the
nomber
that
you
require
to
knowe
.
6.
What
noumber
is
that
,
whiche
beyng
multiplied
by
⅝
,
the
product
of
that
multiplication
will
bee
.
16.
⅔
.
Aunswere
:
diuide
.
16.
by
⅔
.
by
〈◊〉
.
and
thereof
commeth
.
26.
⅔
,
whiche
is
the
nomber
that
you
seeke
.
¶
Here
ensueth
other
necessarie
questions
,
whiche
are
wrought
by
multiplication
in
broken
nombers
.
I
Demaunde
howe
muche
the
⅝
of
.
20.
shillynges
are
worth
or
what
are
the
⅝
.
of
.
20.
shillynges
.
Aunswere
:
you
muste
multiplie
⅝
by
1●/5
,
and
the
product
will
bee
100/8
,
therefore
diuide
.
100.
by
.
8.
and
thereof
commeth
.
12.
½
,
whiche
is
to
saie
.
12.
shillynges
.
6.
pence
,
and
so
muche
are
the
5/●
.
of
.
20.
shillynges
worthe
.
2.
I
demaunde
what
the
¼
of
⅚
of
〈◊〉
pounde
of
money
are
worthe
,
that
is
to
sa●e
,
of
.
20.
shillynges
.
Aunswere
:
multiplie
.
¾
.
by
⅚
.
And
thereof
commeth
5/2.
Then
take
the
5/●
of
20.
shillinges
,
as
in
the
last
question
goyng
before
and
you
shall
finde
.
12.
shillynges
.
6.
pence
,
and
so
muche
are
the
¾
of
⅚
,
of
20
shillynges
worthe
.
3.
I
demaunde
what
the
⅔
of
.
8.
pence
●/2
are
worth
.
Aunswere
:
multiplie
.
8.
●/2
by
⅔
,
or
els
⅔
by
.
8.
&
●/2
whiche
is
all
one
,
and
you
shall
finde
34/6.
Then
diuide
34
by
.
6.
and
your
quotient
will
bee
fiue
pence
⅔
,
and
so
muche
are
the.
〈◊〉
.
of
.
8.
pence
〈◊〉
〈…〉
the.
4.
What
are
the
¾
.
of
.
14.
pence
〈◊〉
,
Aunswers
multiplie
.
14.
⅗
by
¾
,
and
therof
commeth
219/20.
Therefore
diuide
219.
by
.
20.
and
your
quotiente
will
bee
.
10.
pence
19/20.
and
so
muche
are
the
●/4
of
.
14.
●/5.
5.
How
many
〈◊〉
of
sworth
〈◊〉
●●●contain●r
in
.
7.
⅔
.
Aunswer
multiplie
,
7.
⅔
by
4/●
(
because
one
whole
containeth
.
4.
quarters
)
and
thereof
commeth
,
30.
⅔
,
and
so
many
quarters
are
in
the.
7.
⅔
.
that
is
to
saie
.
30.
quarters
,
and
⅔
of
a
quarter
.
6.
Howe
many
thirdes
are
in
¾
and
●/2
,
that
is
to
saie
in
.
3
,
quarters
,
and
●/2
of
one
quarter
,
whiche
are
⅞
by
the
〈◊〉
reduction
.
Aunswere
:
multiplie
7/●
by
〈◊〉
(
for
because
that
in
one
whol●
are
contained
.
3.
thirdes
)
and
thereof
wil
come
⅔
and
⅝
of
a
third
,
and
so
many
thirdes
are
in
¾
and
½
or
in
⅞
,
whiche
is
al
one
¶
Question
doen
by
diuision
in
broken
nomber
.
WHat
nomber
is
that
,
which
beyng
diuided
by
.
17.
the
quotiente
will
bee
.
13.
Aunswere
:
multiplie
.
17.
by
.
13.
And
thereof
commeth
.
221.
whiche
is
the
nomber
that
you
seke
.
2.
What
nomber
is
that
,
which
beyng
diuided
by
¾
,
the
quotient
will
be
21.
Aunswere
:
multiplie
23/1.
by
●/4
and
therof
commeth
63/4.
Then
diuide
.
〈◊〉
by
4
and
thereof
commeth
.
15.
¾
,
whiche
is
the
nomber
that
you
seke
.
3.
What
nomber
is
that
,
which
beyng
diuided
by
.
⅛
,
the
quotiente
will
bee
⅔
.
Aunswere
:
multiplie
2/1.
by
⅛
.
and
thereof
commeth
2/24
,
whiche
beeyng
abreuiated
are
1/12.
for
the
nomber
,
whiche
you
require
.
4.
What
nomber
is
that
,
whiche
beyng
diuided
by
⅘
.
the
quotient
will
be
.
16.
⅔
?
Aunswere
:
multiplie
.
16.
⅔
,
by
⅘
,
and
thereof
commeth
200/15.
Therefore
diuide
.
200.
by
.
15.
and
thereof
commeth
.
13.
●/3
,
whiche
is
the
nomber
that
you
desire
to
finde
.
5.
What
nomber
is
that
,
which
beyng
diuided
by
.
13.
⅓
,
the
quotient
will
bee
.
20.
Aunswere
:
multiplie
〈◊〉
,
by
.
13
〈◊〉
and
thereof
commeth
900/3
,
then
diuide
800.
by
.
3.
and
thereof
commeth
.
266
〈◊〉
for
the
nomber
,
whiche
you
seeke
,
6.
What
nomber
is
that
,
whiche
it
it
hee
diuided
,
by
.
12.
½
,
the
quotiente
wil
be
7/●
.
Answere
:
multiplie
7/●
by
.
12
½
and
thereof
commeth
175/〈…〉
,
then
diuide
175
by
.
16.
and
thereof
commeth
.
10
15/●●
for
the
nomber
whiche
you
desire
.
¶
Other
necessarie
questions
dooen
by
diuision
in
broken
nomber
.
I
Demaunde
what
parte
.
30.
is
of
.
70.
Aunswer
:
diuide
.
30.
by
70.
whiche
you
can
not
,
for
thei
are
30/7●
,
but
abreuiate
them
and
thei
are
3/7.
Thus.
30.
are
the
3/7.
of
.
70.
2.
I
demaunde
what
parte
.
10.
is
of
16
⅔
.
Aunswere
:
diuide
.
10/〈…〉
.
by
.
16
⅔
,
and
thereof
commeth
30/50
,
whiche
beyng
abreuiated
are
〈◊〉
.
And
thus
10.
is
found
to
bee
⅗
of
.
16.
2
/
●
.
3.
More
,
what
parte
is
.
25.
of
.
5/●
.
Aunswere
:
diuide
5/●
.
by
.
25/10
,
and
thereof
commeth
5/20●
,
whiche
beyng
abreniated
is
.
1/40.
And
thus
.
5/●
.
is
but
the
1/40.
of
.
25.
4.
More
,
⅚
what
parte
are
thei
of
⅞
Aunswere
:
diuide
⅚
by
⅞
,
and
you
shal
finde
4●/42
whiche
abreuiated
are
20/21.
5.
More
,
⅘
what
parte
are
thei
of
.
13.
〈◊〉
.
Aunswer
:
diuide
⅘
by
.
13.
⅓
.
and
you
shall
finde
12/200
,
whiche
beyng
abbreuiated
are
●/50.
And
thus
4/1
are
the
3/50.
of
13.
⅓
.
6.
More
.
12.
½
.
what
parte
are
thei
of
.
Aunswere
:
diuide
.
12.
½
.
by
30/●
,
and
you
shall
finde
.
●5/60
,
whiche
beyng
abreuiated
are
●/12
,
and
thus
.
12.
½
,
are
the
5/12
of
.
30.
7.
More
.
16.
●/3
,
what
parte
are
thei
of
.
57.
1
/
7.
Aunswere
:
diuide
.
16.
⅔
by
.
57
●/7.
and
therof
commeth
550/1200
,
whiche
beyng
abreuiated
are
7/24
,
and
thus
.
16.
⅔
.
are
the.
7/24
of
.
57.
1
/
7.
8.
More
,
●/4
and
⅔
of
●/4
,
or
three
quarters
,
and
⅖
of
one
quarter
,
what
part
are
thei
of
.
1.
Aunswere
:
reduce
5/4
,
and
the
⅔
of
●/4
,
into
one
broken
by
the
first
reduction
,
and
you
shall
finde
11/12.
And
thus
the
●/4
,
and
●/3
of
¼
are
the
11/12
of
one
whole
.
9.
More
,
of
what
nomber
are
.
9.
the
〈◊〉
.
Aunswere
:
diuide
.
9.
by
3/3
,
and
thereof
commeth
.
13.
●/2
,
whiche
is
the
nomber
whereof
.
9.
are
the
〈◊〉
.
10.
More
,
of
what
nomber
are
⅘
the
●/4.
Aunswere
:
diuide
⅖
by
¾
,
and
thereof
commeth
8/15
,
whiche
is
the
nomber
wherof
2/●
are
the
¾
of
the
same
nomber
.
11.
More
,
of
what
nomber
are
.
5.
¾
.
the
●/7.
Aunswere
:
diuide
.
15.
¾
by
3/●
,
and
you
shall
finde
.
13.
5/12
,
whiche
is
the
nomber
whereof
.
5
,
●/4
are
the
3/7.
12.
More
.
9.
〈◊〉
what
parte
are
thei
of
.
33.
Aunswere
:
diuide
.
9.
2/●
by
33.
½
.
And
thereof
commeth
18/20●
:
and
thus
9.
〈◊〉
are
the
58/201
of
33.
½
as
appereth
.
¶
The
thirde
parte
treateth
of
certaine
brief
rules
,
called
rules
of
practise
,
with
diuerse
necessarie
questions
,
profitable
for
Marchauntes
.
¶
The
first
Chapiter
.
SOme
there
be
,
which
doe
cal
these
rules
of
practise
br
〈…〉
rules
,
for
that
by
them
many
questions
maie
be
doen
with
quicker
expedition
,
then
by
the
rule
of
thre
.
There
bée
others
,
whiche
call
them
the
small
multiplication
,
for
because
that
the
producte
,
is
alwaies
lesse
in
quantitie
,
then
the
nomber
whiche
is
to
be
multiplied
.
This
practise
commeth
not
in
vse
,
but
onely
emong
small
kindes
of
nombers
,
whiche
haue
ouer
theim
,
other
nombers
that
are
greater
.
And
this
beyng
well
considered
,
is
no
other
thyng
,
but
to
conuerte
lesser
and
perticuler
kindes
of
nomber
,
into
greater
,
the
whiche
maie
bee
dooen
by
the
meanes
of
diuision
,
in
taking
the
half
,
the
third
,
the
fowerth
,
the
fift
,
or
suche
other
partes
of
the
somme
,
whiche
is
to
bee
multiplied
,
as
the
multiplier
is
part
of
his
greater
kinde
,
&
that
which
commeth
thereof
is
worthe
as
muche
(
not
in
quantitie
,
but
in
his
owne
forme
)
as
if
you
did
multiplie
simplie
the
twoo
sommes
,
the
one
by
thother
,
and
for
the
better
vnderstandyng
of
suche
conuersions
,
you
must
haue
respecte
to
one
of
these
twoo
considerations
.
The
first
is
,
when
one
would
demaunde
this
question
.
At
6.
pence
the
yarde
of
Cotton
,
what
are
.
18.
yardes
worth
by
the
price
?
It
is
manifest
that
thei
are
worthe
.
18.
peeces
of
.
6.
pence
the
pece
,
or
.
18.
halfe
shillynges
,
which
must
be
tourned
into
shillynges
,
in
takyng
the
halfe
of
.
18.
shillynges
,
and
thei
make
.
9.
shillinges
.
Or
otherwise
you
muste
consider
,
that
at
.
1
shillyng
the
yarde
,
the.
18.
yardes
are
worth
.
18.
shillynges
,
wherefore
at
.
6.
pence
thei
shal
be
but
halfe
so
muche
,
for
.
6.
pence
is
but
the
½
of
.
1.
shiliyng
.
Therefore
you
must
take
½
of
.
18.
and
thei
make
9
shillinges
,
whiche
are
worth
as
much
as
.
108.
pence
,
that
is
to
saie
,
as
.
18.
times
.
6.
pence
.
2.
Firste
,
if
you
will
multiplie
any
nomber
,
after
this
maner
by
pence
,
whereof
the
nomber
of
thesame
pence
doe
not
extende
vnto
.
12.
and
therof
to
bryng
shillinges
into
the
product
:
you
must
knowe
the
certaine
partes
of
.
12
whiche
are
these
:
that
is
to
saie
,
6
,
4.
3.
2.
and.
1.
For.
6.
is
the
½
of
.
12.
and.
4.
is
the
〈◊〉
.
of
.
12
:
3.
is
the
¼
:
2
is
the
⅙
:
and
1.
is
the.
●/12.
Then
for
.
6.
pence
,
whiche
is
the
halfe
of
.
1.
shillyng
,
you
must
take
the
½
of
all
the
nomber
,
whiche
is
to
be
multiplied
.
And
that
which
commeth
thereof
,
shal
be
shillynges
,
if
there
do
remaine
.
1.
it
is
.
6.
pence
.
For
fower
pence
you
must
take
the
●/4
of
all
the
nomber
,
that
is
to
bée
multiplied
:
and
if
any
vnities
doe
remaine
thei
shal
be
thirdes
of
a
shillyng
,
euery
one
beyng
in
value
.
4.
pence
.
For.
3.
pence
you
muste
take
the
,
¼
.
of
all
the
somme
:
if
any
vnities
dooe
remaine
,
thei
shall
bee
fowerthes
of
a
shillyng
,
euery
one
beyng
worthe
three
pence
.
For
2
pence
you
must
take
the
⅙
of
all
the
the
somme
,
and
if
any
vnities
doe
remaine
,
thei
shall
bee
sixe
partes
of
a
shillyng
,
beyng
euery
one
of
theim
worthe
twoo
pence
.
For.
1.
penie
,
take
the
1/12
of
the
whole
somme
,
if
any
vnities
remaine
,
thei
are
.
12.
partes
of
a
shillyng
,
eche
of
them
beyng
in
value
.
1.
penie
,
as
by
these
examples
folowing
doeth
plainly
appeare
.
Here
you
maie
se
in
the
first
example
,
that
.
59.
yardes
,
at
.
6.
pence
the
yarde
is
worthe
.
29.
shillynges
.
6.
in
taking
the
½
of
.
59.
And
in
the
seconde
example
,
the.
82.
yardes
at
.
4.
pence
the
yarde
is
worthe
.
27.
shillynges
.
4.
pence
,
in
takyng
the
⅓
of
.
82.
Likewise
,
in
the
third
example
.
97
yardes
,
at
three
pence
the
yarde
,
bringeeh
.
24.
shillynges
.
3.
pence
,
in
takyng
the
¼
of
.
97.
Also
in
the
●owerth
example
.
346.
yardes
,
a●
.
2.
pence
the
yarde
,
maketh
.
57.
shillynges
eighte
pence
,
in
taking
the
1/●
of
.
346.
And
final
-
in
the
fyft
example
.
343.
yardes
,
at
I.
d.
the
yarde
,
amount
to
28.
shill.
7.
d.
in
taking
the
1/12
of
343.
And
so
is
to
be
done
of
all
suche
lyke
,
when
the
nomber
of
the
pence
,
is
any
of
the
certaine
partes
of
12.
But
if
the
nomber
of
the
pence
be
not
a
certain
parte
of
12.
you
muste
reduce
them
into
some
certaine
partes
of
12.
and
after
the
foresayd
maner
you
shall
make
two
or
three
productes
as
neede
shall
require
,
and
adde
them
togither
into
one
summe
as
5.
d.
may
be
reduced
into
4.
&
I.
or
els
into
3.
&
2.
wherfore
if
you
wil
work
by
4.
&
by
I
:
you
muste
for
4.
d.
take
fyrst
the
⅓
.
of
the
nomber
,
that
is
to
bée
multiplied
,
and
for
I.
d.
take
the
1/12
,
or
rather
for
I.
d.
you
may
take
the
¼
of
the
producte
whiche
did
come
of
the
4.
d.
bycause
that
I.
d.
is
the
¼
of
4.
d.
But
if
you
wyll
worke
by
.
3.
and
you
shal
take
for
3.
d.
the
¼
of
the
nomber
whiche
is
to
bee
multiplied
:
and
likewise
for
2.
d.
the
⅙
of
the
same
nomber
,
adding
togyther
both
the
productes
.
The
totall
summe
of
those
two
nombers
shall
be
the
solution
to
the
question
.
And
in
like
maner
is
to
be
done
of
all
other
.
As
by
these
former
folowyng
may
appeare
.
Here
in
this
same
example
where
it
is
demaunded
(
at
5
pence
the
yard
)
howe
much
are
nine
and
fourty
yardes
worth
?
Firste
for
four
pence
,
I
take
the
●/3
of
49.
s.
and
thereof
commeth
16.
s.
4.
d.
then
for
1.
d.
I
take
the
¼
of
the
same
product
,
that
is
to
say
,
of
16.
s.
4.
d.
and
that
bringeth
.
4.
shill.
1.
d.
these
.
twoo
summes
added
togither
,
do
make
20.
s.
5.
d.
And
so
much
are
the
49.
yardes
worth
at
5.
d.
the
yarde
.
For
7.
d.
take
the
⅓
and
the
¼
of
the
whole
summe
whiche
is
to
be
multiplied
,
and
adde
them
togither
,
that
is
to
say
,
for
4.
d.
the
〈◊〉
and
for
3.
d.
the
¼
:
bycause
4.
d.
is
the
〈◊〉
of
12.
d.
and
3.
d.
is
the
¼
as
in
the
second
example
before
doth
appeare
:
Where
the
question
is
thus
,
at
7.
d.
the
yarde
what
are
54.
yardes
worth
?
Firste
for
4.
d.
I
take
the
〈◊〉
of
54
:
and
they
make
18.
s.
Likewise
for
3.
〈◊〉
.
I
take
the
〈◊〉
of
54.
and
they
are
13.
s.
6.
d.
Then
I
adde
18.
s.
and
13.
s.
6
d.
togither
,
so
both
amount
to
●1.
s.
6.
〈◊〉
and
so
muche
are
the
54
yardes
worth
at
5.
d.
the
yarde
.
Otherwise
for
7.
d.
take
first
the
½
,
of
the
whole
summe
for
6.
d.
Then
for
1.
d.
take
the
⅙
of
the
same
product
,
and
adde
them
togither
,
so
shall
you
haue
the
like
summe
as
before
.
For
eight
pence
you
must
first
take
●/3
of
the
whole
summe
for
4.
pence
,
and
another
●/●
for
other
4.
d.
and
adde
than
togither
as
in
the.
3.
example
doth
euidently
appeare
.
Where
the
question
is
thus
,
at
8.
d.
the
yarde
,
what
are
40
yardes
worth
?
Firste
for
4.
d.
I
take
the
⅓
of
40.
which
is
13.
s.
4.
d.
Againe
,
I
take
another
⅓
for
the
other
4
pence
which
is
also
13.
shillings
&
4.
pence
.
These
twoo
summes
being
added
togither
,
do
make
26.
shillings
8.
pence
,
and
so
muche
are
the
40.
yards
worth
at
8.
pence
the
yard
,
as
in
the
third
example
abouesayd
doth
appeare
.
Otherwayes
,
for
eyght
pence
you
may
take
first
the
½
of
the
whole
summe
for
6.
d.
Then
for
2.
d.
you
shal
take
the
●/3
of
the
product
,
which
did
come
of
the
said
½
,
and
adde
them
togither
,
so
shall
you
haue
likewise
the
solution
to
the
question
.
As
in
the
same
third
example
of
40.
yardes
,
I
take
first
the
½
of
40.
for
6.
d.
and
thereof
commeth
20.
shil
.
then
for
2.
d.
I
take
●/3
of
the
saide
product
,
that
is
to
say
of
20.
s.
which
bringeth
6.
〈◊〉
.
8.
d.
these
two
summes
(
20.
s
and
6.
s.
8.
d.
)
I
adde
togither
and
they
make
26.
s.
8.
d.
as
before
.
For
9.
d.
you
must
take
the
●/2
&
the
¼
of
the
whole
summe
,
and
adde
them
togither
:
or
els
for
6.
d.
take
fyrst
½
of
the
whole
summe
,
then
for
3.
d.
take
the
●/2
of
the
same
product
,
bicause
3.
d.
is
the
halfe
of
6.
d
:
And
6.
d.
added
with
3.
d.
bringeth
9.
d.
as
by
the
fourth
example
,
where
it
is
demaunded
after
this
sort
:
at
9.
d.
the
yarde
,
what
are
73.
yardes
worthe
.
First
for
6.
d.
I
take
the
½
of
73.
and
therof
commeth
36.
s.
6.
d.
then
for
3.
d.
I
take
½
of
the
same
36.
shil.
6.
d.
which
is
18.
s.
3.
d.
these
twoo
summes
doe
I
adde
togither
,
&
they
make
54.
shil.
9.
d.
as
in
the
said
fourth
example
is
euident
.
For
10.
d.
take
first
the
½
,
then
the
⅓
of
the
whole
summe
,
&
adde
them
togither
For
11.
●●take
fyrst
⅓
for
4.
pence
,
secondely
,
another
⅓
for
other
4.
d.
and
thirdly
¼
for
3.
d.
of
all
the
whole
summe
.
and
adde
them
togither
.
Or
els
for
11.
d
take
first
the
½
then
then
⅓
of
the
whole
summe
,
and
finallye
the
¼
of
the
laste
producte
,
addinge
them
togither
.
3.
Lykewise
by
the
same
reason
,
when
you
wil
multiply
(
by
shillings
)
anye
nomber
that
is
vnder
xx
.
s.
you
shall
haue
in
the
product
poundes
,
if
you
knowe
the
certaine
partes
of
20
:
which
are
these
.
10.
5.
4.
2.
&
.
1.
For
10.
is
the
½
of
20.
5.
is
the
¼
part
:
4
is
the
½
.
2.
is
the
1/10
:
and
1.
is
the
1/20.
Then
for
10.
s.
which
is
the
½
of
a
pounde
:
you
muste
take
the
●/2
of
the
nomber
,
whiche
is
to
bee
multiplied
,
and
you
shal
haue
poundes
in
the
product
.
If
there
doe
remaine
1
,
it
shal
be
worth
ten
shillings
.
For
5.
shillinges
you
muste
take
the
¼
of
the
nomber
whiche
is
to
bee
multiplied
,
&
if
there
do
remaine
any
vnities
,
they
shall
be
foure
partes
of
a
pound
,
euery
one
being
in
value
5.
s.
For
4.
〈◊〉
.
you
must
take
the
●/5
of
the
number
whiche
is
to
bee
multiplyed
.
And
if
there
do
remaine
any
vnities
,
they
shall
be
fift
parts
of
a
pound
,
euery
one
being
worth
foure
shillings
.
For
2.
shillings
you
must
take
the
1/10
of
the
nomber
that
is
to
be
multiplied
.
Wherefore
,
if
you
wyll
take
the
1/10
of
any
nomber
,
you
muste
separate
the
last
figure
of
the
same
number
whiche
is
nerest
your
righte
hande
,
from
all
the
other
fygures
.
For
all
the
other
figures
whiche
doe
remaine
towarde
your
lefte
hande
,
from
the
same
fygure
,
which
is
separated
,
shall
bee
the
sayde
1/1●
of
a
pounde
:
and
that
separated
fygure
,
towarde
your
right
hand
shall
be
so
many
peeces
of
2.
shillings
the
peece
:
the
whiche
fygure
must
be
doubled
,
to
make
therof
shillings
,
as
by
these
examples
appeareth
.
Herevpon
doth
depend
an
other
exact
way
for
to
multiply
by
shillings
(
if
the
nomber
of
shillings
be
euen
)
which
is
thus
:
you
shall
take
〈◊〉
the
nomber
of
the
same
shillings
,
and
conuert
them
into
peeces
of
2.
shillings
.
Then
by
the
nomber
of
this
halfe
,
you
must
firste
multiply
the
last
figure
towarde
your
right
hande
,
of
the
nomber
which
is
to
be
multiplied
:
And
if
ther
be
any
tennes
in
the
same
product
,
those
must
you
reserue
in
your
minde
:
But
if
(
wyth
the
same
of
els
without
the
same
)
you
doe
finde
any
diget
nomber
,
the
same
diget
nomber
shall
you
double
,
&
put
it
in
the
place
of
shillings
:
Then
must
you
proceede
to
the
multiplication
of
the
other
figures
,
adding
vnto
the
product
the
tennes
whiche
you
before
reserued
:
and
therof
shall
come
pounds
.
Nowe
,
for
your
better
vnderstanding
of
this
which
hath
bene
said
and
by
the
way
of
example
,
I
will
propone
vnto
you
this
question
.
At
8.
shillings
the
grosse
,
what
are
●7
.
grosse
worth
after
the
rate
?
Firste
in
this
example
I
take
halfe
the
nomber
of
Shillinges
,
as
before
is
taught
,
that
is
to
say
of
eighte
shillings
,
which
is
foure
shillinges
,
this
4.
shil
.
I
put
apart
,
behinde
a
crooked
line
righte
againste
97.
towardes
the
left
hand
,
as
here
you
may
see
and
as
here
after
dooeth
appeare
by
diuers
examples
.
Nowe
in
the
first
example
,
where
it
is
demaunded
,
at
8.
〈◊〉
.
the
grosse
,
what
are
97
grosse
?
First
the
½
of
8.
s.
whiche
is
4.
s.
being
set
apart
behind
the
crooked
line
,
as
before
is
said
:
them
I
multiplye
the
97
by
4.
saying
first
,
4.
times
7.
is
28.
I
double
the
diget
number
8.
and
that
maketh
16
,
the
which
16
,
I
do
put
vnder
the
line
,
in
the
place
of
shillings
&
I
kepe
the
tennes
in
my
mind
,
which
here
are
2.
For
20.
are
two
times
ten
:
Then
secondly
,
I
multiply
9.
by
the
sayd
4
,
and
thereof
commeth
36
:
whervnto
I
adde
the
2.
tennes
,
whiche
before
I
reserued
,
and
they
make
38.
Therfore
I
put
38
,
vnder
the
lyne
in
the
place
of
poundes
,
and
the
whole
summe
will
be
38
.li.
16.
s.
Thus
much
are
the
97.
grosse
worth
,
at
eight
shillings
the
grosse
:
the
like
is
to
be
done
of
all
other
.
As
of
12.
shillings
in
multipliyng
by
6.
Likewise
of
6.
shillings
if
you
multiply
by
3.
also
of
14.
if
you
multiply
by
7.
And
so
of
all
euen
nombers
after
the
same
maner
.
For
1.
Shilling
you
must
take
the
½
of
the
1/10
parte
of
any
nomber
that
is
to
be
multiplied
.
And
if
any
thyng
do
remaine
,
they
are
shil
.
Thus
by
thys
maner
shil
.
are
conuerted
into
poundes
:
for
it
is
euen
like
,
as
if
you
did
diuide
them
by
20.
s.
as
by
this
example
in
the
margent
doth
appeare
.
Wher
it
is
demaunded
at
1.
s.
the
yard
,
the
peece
,
or
any
other
thing
,
what
are
350.
worth
?
First
I
separate
the
laste
fygure
of
359.
nexte
to
my
ryght
hand
,
which
is
the
0
with
a
line
betwene
it
and
the
figure
5.
Then
I
make
a
line
vnder
the
3
0
,
and
I
take
the
½
of
35
,
after
this
maner
:
saiyng
the
½
of
3.
is
I.
and
I
remaineth
,
whiche
remaine
signifieth
10.
in
that
second
place
.
Then
I
put
I.
vnder
the
line
against
3
,
&
I
proceede
to
the
rest
,
saiyng
:
the
halfe
of
15.
is
7.
(
which
15.
came
of
the
I.
that
remayned
,
and
of
the
5.
in
the
first
place
)
I
put
7.
vnder
the
line
right
against
5
,
and
they
make
17
li.
The
I.
which
did
last
remaine
,
is
10.
s.
Therfore
I
put
10.
s
aparte
vnder
the
line
,
and
the
whole
summe
is
17
.li.
19.
s.
so
much
are
350.
worth
at
I.
s.
the
peece
.
But
when
the
nomber
of
shillings
is
not
some
certaine
parte
of
20.
shil
.
you
must
then
conuert
the
same
nomber
of
shillings
,
into
the
certain
parts
of
20.
and
make
twoo
or
thrée
products
,
as
nede
shall
require
,
the
whiche
must
bee
added
togither
after
this
maner
following
.
For
3.
shillings
you
must
firste
take
for
2.
shil
.
the
1/10
of
the
nomber
that
is
to
be
multiplied
,
them
for
1.
shilling
you
must
take
the
½
of
the
producte
whiche
did
come
of
the
same
1/10
part
:
and
adde
those
twoo
summes
togither
,
as
appeareth
by
this
example
following
.
At
3.
s.
the
peece
of
any
thing
,
what
shall
684
peeces
coste
mee
after
the
rate
.
First
,
for
2
shillings
I
take
the
1/10
of
684
,
which
is
68
:
in
separating
the
last
figure
4
,
whiche
4
I
must
double
,
&
they
be
8.
I
set
eight
shillings
aparte
from
the
place
of
poundes
,
and
then
I
haue
68.
poundes
8.
s.
for
the
1/10
parte
,
that
is
to
say
,
for
the
2.
s.
secondlye
,
for
1.
shil
.
I
take
the
½
of
the
product
,
that
is
to
saye
:
of
68
.li.
8.
s.
whiche
is
34
.li.
4.
s.
and
I
put
the
same
vnder
the
68
.li.
8.
shil
.
Then
finally
,
I
adde
those
two
summes
together
,
that
is
to
saie
,
68
.li.
8.
s.
and
34
.li.
4.
shil
.
so
they
make
102
.li.
12.
s.
and
so
much
are
the
684.
peeces
worth
at
3.
shillings
the
peece
,
as
may
appere
in
the
margent
.
For
6.
shll
.
take
3/●●
of
the
nomber
whiche
is
to
be
multiplied
:
that
is
to
say
,
first
1/10
,
then
double
the
product
of
the
same
1/10
and
adde
them
together
.
Or
otherwise
for
4.
s.
take
first
the
⅕
of
the
nomber
that
is
to
be
multiplied
,
then
take
the
½
of
the
product
which
is
for
two
.
s.
and
adde
them
togither
.
Or
els
take
for
5.
shill.
the
¼
of
the
whole
summe
,
then
for
1.
shil
.
the
1/●●
of
the
product
and
adde
them
togither
.
Likewise
for
7.
shil
.
take
firste
for
5.
shil
.
the
¼
then
for
2.
shillings
take
the
1/10
of
the
nomber
whiche
is
to
be
multiplied
,
and
adde
them
togither
.
For
eyght
shillings
take
the
⅖
at
two
sundry
times
,
that
is
to
say
,
first
⅖
for
4.
shil
.
and
then
as
muche
more
for
other
4.
shil
.
and
adde
them
togyther
.
For
9.
shil
.
take
first
the
¼
and
lykewise
the
⅓
of
the
nomber
that
is
to
be
multiplied
,
and
adde
them
togither
.
For
11.
shil
.
take
first
½
for
10.
s.
Then
for
1.
shil
.
take
the
1/10
of
the
producte
,
&
adde
them
togither
.
For
12.
shil
.
take
first
the
½
for
10.
shil
then
for
2.
s.
take
the
⅖
part
of
the
product
,
and
adde
them
togither
,
For
13.
shil
.
take
the
¼
then
the
⅕
,
and
againe
another
⅕
of
the
nomber
which
is
to
be
multiplied
.
And
adde
the
productes
togither
,
that
is
to
saye
:
fyrste
for
5.
shil
.
take
the
¼
,
then
for
4.
shil
.
take
the
⅕
.
And
againe
,
another
⅕
for
the
other
4
shil
.
and
assemble
the
three
productes
,
the
like
is
to
be
done
of
all
others
,
when
the
price
of
the
thynge
which
is
valued
,
is
onely
of
shillings
.
And
as
by
these
examples
followyng
doth
plainly
appeare
.
4.
Likewise
in
multipliyng
by
pence
you
shall
haue
(
at
the
firste
instaunte
)
poundes
in
the
producte
,
in
case
you
knowe
the
certaine
partes
of
the
1/10
of
a
pounde
,
or
of
.
24.
pence
,
whiche
are
these
,
12.
pence
.
6.
4.
3.
and.
2.
For.
12.
is
the
½
of
.
24
:
8.
is
the
⅕
:
6.
is
the
¼
.
4.
is
the
⅙
:
3.
is
the
⅛
:
2.
the
1/12
:
but
for
.
12.
pence
,
whiche
is
1.
shillyng
:
wee
haue
before
made
mention
thereof
.
For
8
pence
you
muste
take
the
⅓
of
the
1/10
,
and
the
reste
whiche
are
the
peces
of
8
pence
,
muste
bee
doubled
to
make
of
them
peeces
of
.
4.
pence
.
And
of
the
same
nomber
beyng
doubled
,
you
muste
take
the
⅓
,
whiche
will
bee
shillynges
,
and
if
there
doe
yet
remain
any
thing
,
thei
are
thirds
of
a
shilling
being
in
value
4
pence
the
pece
.
For
6
pence
take
the
¼
of
the
1/12
,
and
of
that
whiche
remaineth
,
you
muste
take
the
½
,
whiche
shall
be
shillynges
,
if
there
doe
yet
remaine
.
1.
it
shall
bee
in
value
.
6.
pence
.
For
4
pence
you
must
take
the
¼
of
the
1/10
,
and
of
that
whiche
resteth
,
take
the
⅓
to
make
therof
shillynges
,
if
any
thyng
dooe
yet
remaine
,
thei
are
thirdes
of
a
shillyng
,
beyng
in
value
.
4.
pence
the
pece
.
For
3
pence
take
the
⅕
of
the
1/10
,
and
of
that
whiche
remaineth
,
take
the
¼
,
to
make
of
theim
shillynges
:
if
any
thyng
doe
yet
remaine
,
thei
are
fourethes
of
a
shillyng
,
euery
one
of
theim
beyng
worthe
3.
pence
.
For
2
pence
take
the
1/12
of
the
1/10
:
and
of
that
whiche
resteth
take
the
⅙
,
the
whiche
are
shi
.
if
there
do
still
remain
any
thing
,
thei
shal
be
sixte
partes
of
a
shilling
,
euery
one
being
in
value
2
d.
For
1
,
penie
it
is
impossible
with
ease
,
to
bryng
of
pence
,
poundes
(
into
the
producte
)
vpon
the
totall
somme
:
But
firste
you
must
bryng
them
into
shillinges
by
the
order
of
the
seconde
rule
of
this
chapiter
,
and
then
afterwarde
you
shall
conuerte
them
into
pounds
,
if
nede
so
require
.
As
by
these
examples
followyng
may
appere
.
But
if
the
number
of
pence
,
be
not
a
certaine
parte
of
24.
pence
.
Then
must
you
bring
them
into
the
certain
partes
of
24.
and
make
thereof
diuers
productes
,
which
must
be
added
together
,
as
shall
hereafter
appeare
.
For.
5.
pence
you
shall
first
take
for
3.
pence
,
then
for
.
2.
pence
,
and
adde
them
together
,
accordyng
to
the
instruction
of
the
laste
rule
.
Or
els
firste
take
for
4.
pence
,
and
then
for
.
1.
penie
.
For.
7.
pence
,
first
take
for
.
4.
pence
then
for
.
3.
d
,
and
adde
them
together
.
For
9
pence
,
firste
take
for
6
pence
,
then
for
3
pence
adoyng
them
together
.
For.
10.
pence
,
firste
take
for
6.
d.
then
for
.
4.
d
,
and
adde
them
together
.
For.
11.
pence
take
firste
for
.
8.
pence
then
for
.
3.
pence
,
and
adde
them
together
:
as
by
these
examples
followyng
doeth
appeare
.
5.
If
you
will
multiplie
any
nomber
by
shillynges
and
pence
,
beyng
bothe
together
,
you
must
take
first
for
the.
s.
accordyng
to
the
instruction
of
the
third
rule
of
this
chapiter
,
then
take
for
the
pence
,
after
the
order
of
the
●owerth
rule
before
mentioned
:
but
if
there
be
any
certaine
partes
of
.
1.
pounde
,
containyng
bothe
shillynges
and
pence
,
then
for
suche
partes
you
shal
take
the
like
parte
of
the
nomber
that
is
to
bee
multiplied
,
as
the
nomber
is
part
of
1
.li.
the
whiche
certain
partes
are
these
,
6.
s.
8.
d
:
3.
s.
4.
d
:
2.
s.
6.
d
:
and.
1.
s.
8.
d
:
For
6.
s.
8.
d.
is
the
⅓
of
a
li.
3.
s.
4.
d.
is
the
⅙
of
a
li.
2.
s.
6.
d.
is
the
⅛
:
and
1.
s.
8.
d.
is
the
1/12
:
then
for
6.
s.
8.
d.
you
muste
take
the
⅓
of
the
nomber
that
is
to
bee
multiplied
:
and
if
any
thyng
dooe
remain
,
thei
are
thirdes
of
a
pound
,
euery
one
being
worthe
6
s.
8.
pence
.
For
3.
s.
4
d.
you
must
take
the
⅙
if
any
thing
doe
remain
,
thei
are
6
partes
of
a
li.
euery
one
beyng
in
valor
3.
s.
4.
d.
For
2.
s.
6.
pence
,
you
must
take
the
⅛
▪
if
any
thing
be
remainyng
,
the●
are
8
partes
of
a
li.
euery
one
being
worth
2.
〈◊〉
.
6.
d.
For.
1.
shillyng
.
8.
pence
,
you
shall
take
the
●/12
,
if
there
dooe
any
thyng
remaine
,
thei
are
twelfth
partes
of
a
.li.
euery
one
beyng
valued
at
.
1.
s.
8.
d.
6.
Here
shall
you
accustome
your self
,
to
multiplie
by
all
sortes
of
sommes
,
beyng
compounde
of
shillynges
,
and
pence
,
whiche
maie
come
to
practise
.
As
thus
,
for
.
1.
shillyng
.
1.
penie
:
for
.
1.
shillyng
.
2.
pence
:
for
.
1.
shillyng
3.
pence
:
for
.
1.
shillyng
4.
pence
.
Likewise
for
2.
shillynges
1.
penie
:
2.
shilly
.
2.
pence
:
2.
shillynges
.
3.
pence
.
2.
s.
4.
d
And
so
of
all
other
:
consideryng
moreouer
,
many
subtile
abreuiations
,
whiche
happen
oftentymes
,
that
are
easie
to
bee
conceiued
.
As
thus
:
at
11.
s.
3.
d.
after
that
I
haue
taken
firste
the
½
for
10.
shillynges
.
Then
for
1.
shillyng
.
3.
pence
.
I
take
the
⅛
of
the
product
,
because
1.
shillyng
.
3.
pence
is
the
⅛
of
10
shillynges
,
in
takyng
the
saied
⅛
of
the
producte
.
And
by
this
meanes
,
when
ye
haue
taken
one
producte
,
ye
maie
oftentymes
vpon
the
same
,
take
an
other
more
briefly
,
then
vpon
the
some
that
is
to
be
multiplied
,
whiche
thyng
you
must
foresée
.
7.
But
if
you
will
multiplie
,
by
poundes
,
shillinges
&
pence
being
together
.
Firste
you
muste
wholly
multiplie
by
poundes
.
Then
take
for
the
shillyngs
&
pence
,
as
in
the.
5.
rule
of
this
chapiter
is
plainly
declared
.
And
as
by
these
examples
folowyng
maie
appeare
.
8.
So
these
rules
do
serue
,
both
to
buy
&
sel
,
at
suche
a
price
the
ell
,
the
yard
,
the
pece
,
the
li.
waight
or
any
other
thing
:
how
much
such
a
thing
.
Likewise
thei
are
very
necessary
to
conuert
all
peces
of
gold
&
siluer
into
li.
for
I
may
aswel
saie
,
at
.
4.
s.
8.
d.
the
Frenche
croune
,
what
are
135
crounes
worth
?
As
to
say
at
.
4.
s.
8.
d.
the
yarde
of
clothe
,
what
are
135.
yardes
worthe
?
9.
When
any
1
of
the
sommes
(
which
is
to
be
multiplied
)
is
compounde
of
many
denominations
:
&
thother
is
of
one
figure
alone
:
then
shall
ye
multiplie
al
the
denominations
of
thother
somme
,
by
the
same
one
figure
,
beginning
first
with
the
somme
,
which
is
least
in
value
towards
your
right
hand
,
&
bryng
the
product
of
those
pence
into
shi
.
and
the
product
of
the
shillinges
into
poundes
as
by
this
example
doeth
appeare
.
10.
But
(
〈…〉
of
the
nombers
whiche
are
to
be
multiplied
)
there
bee
with
it
a
broken
nomber
,
you
muste
(
accordyng
to
his
denominator
)
take
one
or
many
partes
of
the
nomber
,
as
nede
doeth
require
:
and
sette
the
nomber
whiche
commeth
thereof
,
vnder
the
productes
,
addyng
the
same
together
.
As
thus
:
At
5.
pounde
7.
shillynges
8.
pence
the
grosse
,
what
shall
.
34.
grosse
½
cost
?
First
you
shall
multiplie
5.
pound
.
7.
s.
8.
pence
by
.
34.
grosse
,
saiyng
.
5.
tymes
.
34.
dooe
make
170.
pound
then
for
.
6.
shilly
.
8.
pence
,
take
the
●/3
of
.
34.
whiche
is
.
11.
pounde
.
6.
shillynges
,
8.
pence
.
Thirdly
,
for
.
1.
shillyng
,
take
.
34.
shillynges
,
whiche
is
1
pounde
14.
shillynges
.
0.
Lastly
,
for
the
½
grosse
,
you
muste
take
½
of
the.
5.
pounde
7.
shillynges
8
pence
,
whiche
is
.
2.
pounde
.
13.
s.
10.
d.
And
then
adde
them
all
together
,
so
you
shall
finde
that
the.
34.
grosse
½
at
.
5
.li.
7.
shillynges
.
8
pence
is
worthe
.
185.
pounde
.
14.
shillynges
.
6.
pence
,
as
appeareth
in
the
margent
.
And
as
in
this
last
example
,
you
did
take
the
halfe
of
the
money
,
(
whiche
one
grosse
was
worthe
)
for
the
½
grosse
Because
that
one
grosse
beyng
worth
5.
pounde
.
7.
shillynges
.
8
,
pence
,
the
½
grosse
must
be
worthe
halfe
so
muche
So
likewise
,
if
you
haue
⅓
of
a
grosse
or
of
any
other
thyng
,
you
muste
take
the.
⅓
.
of
the
price
,
that
one
grosse
is
worthe
.
Semblable
,
for
the.
¼
.
of
any
thyng
you
shall
take
the
¼
of
the
price
,
that
one
is
worth
,
and
of
all
other
fractions
,
as
by
these
examples
folowing
doth
appeare
.
11.
If
you
will
make
the
proofe
of
these
rules
aforsaid
,
you
must
first
abate
the
somme
of
money
(
which
the
fraction
of
the
multiplicand
doeth
importe
)
from
the
total
somme
.
And
diuide
the
rest
of
the
poundes
of
the
said
totall
some
,
by
the
whole
multiplicande
,
the
fraction
onely
accepted
.
And
if
any
thing
dooe
remain
after
the
diuision
is
made
,
that
remain
shal
be
multiplied
by
20
&
vnto
the
producte
of
that
multiplication
,
you
shal
adde
the
shi
.
which
remained
of
the
rest
of
the
totall
some
.
Again
,
if
anything
do
remain
after
the
same
diuision
,
you
must
multiply
the
same
by
12
&
vnto
the
product
adde
the
pence
of
the
total
some
that
remained
if
any
be
left
And
thus
if
ye
haue
truely
wrought
,
you
shall
finde
again
the
higher
some
of
your
question
,
that
is
to
saie
,
the
price
that
one
grosse
,
or
any
other
thyng
is
worthe
,
whereof
you
demaunde
.
Or
otherwise
,
reduce
the
remaine
of
the
totall
somme
(
the
value
of
the
money
that
the
fraction
is
worthe
,
beyng
firste
deducted
)
all
into
pence
,
in
multipliyng
the
poundes
by
.
20.
and
the
shillynges
by
.
12.
addyng
thereunto
the
shillynges
and
pence
,
which
are
i●igned
with
the
remaine
of
the
saied
totall
somme
,
if
any
suche
be
,
then
diuide
those
pence
by
the
foresaied
nomber
that
is
to
be
multiplied
,
the
fraction
of
the
same
nomber
being
also
abated
.
So
shall
you
finde
the
price
that
one
pece
,
I
grosse
,
or
any
other
thyng
is
valued
at
.
As
in
the
firste
example
goyng
before
,
where
the
totall
some
is
.
201.
pounde
.
10.
shillynges
,
from
the
whiche
I
dooe
first
abate
the
price
of
the
halfe
grosse
,
whiche
is
.
2.
pound
3.
shillynges
.
4.
pence
,
the
reste
is
199
.li.
6.
shillynges
8.
pence
,
whiche
being
reduced
into
pence
,
bringeth
47840.
d.
I
diuide
thesame
by
46.
and
thereof
commeth
1040.
d.
Then
I
diuide
that
1040.
d.
by
12
,
and
thei
bryng
86.
s.
8.
d.
that
is
to
saie
4
.li.
6.
s.
8.
pence
,
which
is
the
price
that
one
grosse
,
or
any
other
thyng
did
coste
,
as
in
that
firste
example
doeth
eppeare
.
12.
The
like
is
to
bee
doen
of
any
maner
of
thyng
,
that
is
solde
by
the
hundred
,
or
by
Kintall
.
As
thus
:
at
12.
pounde
seuen
shillynges
six
pence
the
hundreth
pounde
waight
:
what
shall
.
374.
pounde
waight
coste
.
You
shall
first
multiplie
twelue
pound
seuen
shillinges
,
sixe
pence
by
thre
:
that
is
to
saie
,
by
three
hundreth
.
Then
for
50
.li.
waight
,
you
shall
take
the
½
of
12
.li.
7.
s.
6.
d.
bicause
50
.li.
is
the
½
of
100
.li.
Likewise
for
20.
pound
waight
,
which
is
the
⅕
of
100
.l.
take
the
⅕
of
12
.li.
7.
shil.
6.
d.
lastly
for
4
.li.
waight
take
the
⅕
of
the
laste
product
.
This
done
,
you
muste
adde
all
these
productes
into
one
summe
,
whiche
will
make
the
summe
of
64
.li.
5.
s.
7.
d.
⅘
,
as
by
this
exāply
aboue
written
doth
appeare
.
The
proofe
is
made
by
reducinge
the
totall
summe
into
pence
.
And
to
diuide
the
product
by
the
nomber
that
is
to
be
multiplyed
,
that
is
to
saie
by
374
likewise
diuide
the
quotient
produced
of
that
first
diuision
by
12.
so
shall
you
finde
againe
the
higher
summe
12
.li.
7
shil.
6.
d.
whiche
is
the
price
of
100
.li.
wayght
,
as
before
.
13.
Also
the
like
maye
be
done
of
our
vsuall
waight
here
in
Englande
(
whiche
is
112
.li.
for
euerye
hundred
pounde
waight
)
.
in
case
you
knowe
the
certaine
parts
of
a
hundred
,
that
is
to
say
,
of
112
.li.
waight
,
whiche
are
these
56
.li.
28
.li.
14
.li.
7
.li.
For
56
.li.
is
the
½
of
112.
28
.li.
is
the
¼
of
112
.li.
14
.li.
is
the
⅛
,
and
7
.li.
is
the
1/16.
Therfore
,
for
56
.li.
take
the
½
of
the
summe
of
money
,
that
the
112.
pound
waight
is
worth
.
For
28
.li.
take
the
¼
of
the
summe
of
money
that
the
112
.li.
is
worth
.
For
14
li.
take
the
⅛
of
the
summe
that
the
C.
is
worth
.
For
7
.li.
take
the
1/16
of
the
summe
of
money
that
the
C
,
is
worth
.
As
thus
:
at
3
.li.
6.
s.
8.
d.
the
hundreth
pounds
waight
,
that
is
to
saye
,
the
112
,
li.
What
shall
24.
C.
3.
quar.
21
,
li.
cost
after
the
ra●e
?
Fyrst
,
you
shall
multiply
24
,
hundreth
by
3.
whiche
is
the
3
.li.
&
thereof
commeth
72
.li.
then
for
6.
s
,
8
,
d.
whiche
is
the
⅓
of
20.
s.
you
shall
take
y●
⅓
of
24
which
is
8
,
li.
for
24.
nobles
maketh
8
,
li.
afterwarde
,
for
the
3.
quarters
of
the
C.
you
shall
first
for
the
56
.li.
take
●the
½
of
3
.li.
6.
s.
8
,
d.
bicause
56.
li
is
the
½
of
the
C.
&
thereof
commeth
1
.li.
13.
shil.
4.
d.
then
for
28
.li.
(
whiche
is
the
quar.
of
a
C.
)
you
shall
take
the
¼
of
3
.li.
6.
s.
8.
d.
or
els
the
½
of
the
product
,
whiche
came
of
56
.li.
which
is
16
,
s.
8
,
d
likewise
for
14
.li.
take
the
⅛
of
3
.li.
6
,
s.
8.
d.
whiche
is
8
,
d.
4
,
d.
or
els
the
½
of
the
producte
of
28
.li.
which
is
all
one
:
lastly
for
7
.li.
take
the
1/16
of
3
,
li.
6.
s.
8
,
d.
or
els
the
½
of
the
product
,
that
came
of
14
li.
and
therof
commeth
4
,
s.
2
,
d.
Then
adde
al
these
products
togither
:
&
the
totall
summe
wil
be
83
.li.
2.
s.
6.
d.
so
muche
are
the
24
.c.
3.
quar.
21
.l.
waight
worth
after
3
,
li.
6.
s.
8.
d.
y●
C.
as
appereth
in
the
margent
.
The
proofe
hereof
is
made
,
lyke
to
to
the
other
proofes
aforesaide
,
saning
that
where
in
those
proofes
,
you
aba●e
the
price
of
the
money
,
that
the
fraction
was
worthe
,
from
the
totall
summe
:
here
in
this
example
(
and
in
suche
other
like
)
you
muste
abate
the
price
of
money
,
that
the
odde
waight
amounteth
vnto
(
ouer
and
aboue
the
iust
hundrethes
)
from
the
saide
totall
summe
,
the
rest
thereof
shall
you
conuert
into
pence
,
diuidinge
the
product
of
the
multiplication
by
the
iuste
number
of
the
hundrethes
,
so
shall
you
finde
the
pence
the
one
hundreth
is
worthe
,
whiche
you
shall
bringe
into
poundes
by
the
order
of
diuision
,
&
so
all
other
.
¶
The
second
Chapter
treateth
of
the
rule
of
three
compounde
,
which
are
foure
in
nomber
.
THere
belongeth
to
the
fyrst
&
seconde
partes
of
the
rule
of
thre
compound
alwaies
fyue
numbers
:
whereof
(
in
the
first
part
of
the
rule
of
three
compo●nde
the
seconde
nomber
and
the
〈◊〉
,
are
alwayes
of
one
semblaunce
,
and
●ke
denomination
:
whose
rule
is
thus
,
multiply
the
first
nomber
by
the
seconde
,
&
that
shal
be
your
diuisor
:
then
multiplie
the
other
three
nombers
the
oneby
the
other
to
be
your
diuidende
.
Example
,
of
this
first
part
:
if
100.
crowns
in
12.
monthes
,
do
gaue
15
.li.
what
will
60.
crownes
gaine
.
in
8.
monthes
?
Answere
,
first
multiplie
100.
crownes
by
12.
monthes
,
&
therof
commeth
1200.
for
your
diuisor
:
then
multiply
15
.li.
by
60.
crownes
,
&
by
8.
monthes
&
you
shal
haue
7200.
diuide
7200.
by
1200
,
&
therof
commeth
6
.li.
so
many
li.
wyl
6.
crownes
gaine
in
8.
monthes
:
this
question
may
be
done
by
the
double
rule
of
3.
that
is
to
say
by
the
rule
of
3.
at
2.
times
,
but
yet
this
rule
of
3
compound
is
more
brief
.
2.
In
the
seconde
part
of
the
rule
of
third
compound
,
the
3.
nomber
is
like
vnto
the
fift
,
wherof
the
rule
is
thus
:
multiplie
the
3.
nomber
by
the
4.
the
product
shal
be
your
diuisor
:
then
multiply
the
first
nomber
by
the
seconde
,
&
the
product
therof
by
the
fift
,
that
whiche
nomber
shal
be
your
diuidend
,
or
nomber
that
is
to
be
diuided
:
as
by
example
,
When
60.
crownes
in
8
monthes
do
gaine
6
.li.
in
howe
many
monthes
wil
:
100.
crownes
gain
.
15
.li.
Aunswer
Multiply
the
thirde
nomber
6.
by
the
fourth
nomber
100
,
&
ther
of
commeth
600
then
multiplye
the
first
nomber
60.
by
the
second
nomber
8.
&
by
the
fift
nomber
15
thereof
will
come
7200.
then
diuide
7200.
by
600.
&
the
quotient
wil
be
12
:
in
so
many
monthes
will
100.
crownes
gaine
15
.li.
This
question
may
likewise
be
done
by
the
double
rule
of
3.
3.
In
the
thirde
part
of
the
rule
of
3.
compound
,
there
may
be
5.
nombers
or
more
:
&
in
this
rule
the
first
nomber
&
the
last
alwayes
dissemblaunt
the
one
to
thother
:
&
the
question
is
from
the
last
nomber
vnto
the
first
,
wherof
the
rule
is
thus
:
multiply
that
nomber
which
you
woulde
know
by
those
nombers
which
do
giue
the
value
,
&
diuide
the
product
of
the
same
,
by
the
multiplication
of
the
nombers
which
are
already
valued
,
as
by
example
.
If
4.
deniers
Parisis
,
be
worth
5.
deniers
arnols
,
&
10.
deniers
tournois
,
be
world
12.
deniers
of
Sauoy
,
I
demaund
how
many
deniers
Parisis
are
8.
deniers
of
Sauoy
worth
▪
Aunswere
:
Multiply
8.
deniers
of
Sauoy
(
which
is
the
nomber
that
you
would
know
)
by
4.
deniers
parisis
,
&
by
10
deniers
t
〈…〉
is
whiche
are
the
nombers
that
giue
the
value
,
&
they
make
320
:
then
multiplie
5.
denters
tournois
,
by
12
deniers
of
sauoy
(
which
are
the
nombers
alredye
valued
)
and
they
make
60
:
lastly
diuide
320.
by
60
and
you
shall
finde
5.
deniers
⅓
parisis
,
so
muche
are
the
denies
of
Sauoye
worth
.
4.
In
the
fourthe
parte
of
the
rule
of
three
compounde
:
the
first
nomber
and
the
last
are
always
semblant
and
of
one
denomination
,
and
the
question
of
this
rule
,
is
alwayes
from
the
last
nomber
to
the
last
sauing
one
.
Whereof
there
is
a
rule
which
is
thus
.
You
must
multiplye
that
nomber
whiche
you
woulde
knowe
,
by
the
nombers
that
are
alreadye
valued
,
and
diuide
the
product
of
the
same
,
by
the
multiplication
which
commeth
of
the
nombers
that
giue
the
value
,
as
by
example
If
4
:
deniers
Parisis
,
bee
worth
5.
Deniers
Lournois
,
and
10.
Deniers
Lournois
,
be
worthe
12.
Deniers
of
Sauoy
,
I
demaunde
how
many
Deniers
of
Sauye
.
are
15.
Deniers
Parisis
worth
.
Aunswere
:
Multiplye
15.
Deniers
Parisis
that
you
woulde
knowe
,
by
5.
Deniers
Lournois
,
&
by
12.
Deniers
of
Sauoye
,
which
are
the
nombers
alreadye
valued
,
and
they
make
900
Diuide
the
same
by
4
times
10.
which
are
the
nombers
that
doe
giue
the
value
,
and
you
shal
finde
22.
Deniers
½
of
Sauoye
,
so
much
are
the
15.
Deniers
Parisis
worth
.
The
thirde
Chapter
treateth
of
questions
of
the
trade
of
Marchaundise
.
IF
31.
Deuonsh
.
dosens
do
cost
me
100
.li.
15.
shil
.
What
shal
4.
dosens
cost
?
Aunswere
:
first
bring
the
100
.li.
15.
shill.
all
into
shillings
,
in
multiplying
the
100
.li.
by
20.
adding
to
the
product
the
15.
shill.
and
thereof
commeth
2015.
shill.
then
multiplye
2015.
by
the
thirde
number
4.
and
diuide
the
product
by
31.
and
the
quetietit
wil
be
260.
s.
The
which
diuide
a-againe
by
20.
&
therof
commeth
13
.li.
If
foure
Dosens
be
worth
13.
pound
What
are
31.
Dosens
worthe
by
the
price
?
Aunswere
:
Multiply
31.
by
13.
&
therof
commeth
403.
The
whiche
you
shall
diuide
by
4.
and
thereof
commeth
100
.li.
¾
,
which
¾
are
15.
s.
and
so
much
are
31.
Dosens
worth
as
before
.
If
49
elles
be
worth
2
,
li.
4
,
s.
11
,
d.
what
are
18
elles
worthe
by
the
price
?
First
you
must
bring
2
.li.
4
,
s.
11
,
d.
all
into
pence
,
in
multipliyng
2
.li.
by
20.
maketh
40.
adde
thereto
4
,
shil
.
they
make
44.
s.
the
which
multiply
by
12.
d.
&
they
make
528
,
d.
wherunto
adde
11.
d
all
is
539
,
d.
the
whiche
539
,
d.
muste
be
your
second
nomber
in
the
rule
of
3.
then
multiply
539
by
18
,
&
therof
commeth
9792.
diuide
the
same
by
49
,
&
you
shall
haue
in
your
quotient
198
,
d.
the
whiche
diuide
by
12.
&
you
shall
finde
16
s.
6
d.
so
much
are
the
18.
elles
worth
.
IF
18.
elles
be
worth
16
s.
6.
d.
what
are
49.
elles
worth
by
the
price
?
Auns
.
bring
19.
s.
6.
d.
into
pence
,
in
multiplying
19.
by
12.
and
thereof
commeth
198.
d.
with
the
6.
d.
adde
dto
it
,
then
multiplye
198
by
49.
the
product
will
be
2702.
The
which
diuide
by
18.
elles
and
therof
commeth
539.
d.
Then
diuide
539.
d.
by
12.
and
the
product
therof
by
20.
So
shall
you
haue
2
.li.
4.
sh.
11.
d.
so
muche
are
the
49.
elles
worth
.
If
a
yarde
of
Ueluet
cost
19.
s.
what
shalt
¾
of
a
yarde
cost
?
Aunswere
:
sette
down
your
nombers
thue
.
If
1/1
|
19/1
|
3/●
,
Then
multiplye
1.
times
16.
by
3.
and
therof
commeth
57
,
for
your
diuidende
,
or
nomber
to
be
diuided
.
The
whiche
57.
you
shall
diuide
by
1
,
times
1
,
foure
times
;
which
are
4
,
and
your
quotient
wil
be
14
s.
¼
,
which
¼
is
worth
3.
d.
so
muche
are
the
¼
of
a
yarde
worth
after
19.
shil
.
the
yarde
,
as
by
practise
followeth
.
Or
otherwise
by
the
rules
of
practise
first
for
2/4
of
a
yarde
whiche
is
½
of
a
yarde
,
you
muste
take
the
½
of
19
s.
which
is
9
,
s.
6
,
d.
then
for
¼
,
take
the
½
of
the
product
,
that
is
to
saye
,
of
9
,
s.
6
,
d.
and
therof
commeth
4
,
s.
9.
d.
adde
these
nombers
togither
,
&
you
shall
haue
14
,
s.
3
,
d.
as
aboue
is
sayd
,
and
as
appeareth
here
in
the
margent
.
IF
¾
of
a
yarde
of
Ueluet
do
cost
14.
shil.
3.
d.
What
shall
1.
yarde
coste
,
set
your
nombers
downe
thus
:
if
¾
|
14
¼
|
1/1.
Reduce
14
,
¼
into
a
traction
,
and
they
wil
be
57/4
them
multiply
57.
by
1.
4
,
times
,
&
thereof
commeth
228.
for
your
diuidend
.
Likewise
multiply
1.
times
4
,
3
,
times
,
&
therof
commeth
12
,
for
your
diuisor
:
then
diuide
228
by
12.
&
your
quotient
will
be
19.
shil
.
so
much
is
the
yarde
of
veluet
worth
.
Or
otherwise
by
the
rule
of
practise
:
you
shall
take
the
⅓
parte
of
14
,
sh.
3.
d.
and
adde
it
with
the
same
14
,
sh.
3
,
d.
and
you
shall
haue
19
,
shill.
as
before
.
If
one
elle
of
Hollonde
clothe
be
worth
5
,
s.
what
are
⅔
worth
after
the
rate
?
Aunswere
:
say
thus
if
1/1
|
5/1
|
⅔
.
Then
multiply
2
times
5
,
one
time
,
and
therof
commeth
10
,
for
your
diuidende
:
likewise
multiply
three
times
1
one
time
,
they
make
3
,
for
your
diuisor
,
then
diuide
10
,
by
3.
&
thereof
commeth
30.
s.
⅓
which
●/3
is
worth
4.
pence
,
&
so
much
are
the
〈◊〉
of
an
ell
worth
.
Or
otherwise
,
by
the
rule
of
practise
:
take
first
the
⅔
of
5.
s.
for
the
⅔
of
an
ell
,
which
is
1.
s.
8.
d.
Likewise
,
for
the
other
⅔
of
an
ell
take
againe
another
⅔
of
5.
s.
which
is
also
1.
sh.
8
,
d.
and
adde
them
together
,
and
so
shall
you
haue
3.
s.
4.
d.
as
before
.
If
of
an
ell
of
Hollande
cloth
doe
cost
me
3.
s.
4.
d.
what
shal
the
el
cost
?
Aunswere
:
set
down
your
summe
thus
,
if
〈◊〉
3
〈◊〉
‑
〈◊〉
.
Firste
reduce
31/●
all
into
thirds
,
and
it
will
be
10/3.
Then
multiply
1.
times
10.
3
,
times
,
and
thereof
commeth
30.
for
your
diuidende
.
Likewise
multiplie
1.
times
3.
2
times
,
your
quotient
wil
be
6.
then
diuide
30
,
by
6.
&
you
shall
haue
5
,
s.
so
much
is
the
ell
of
Hollande
clothe
worthe
.
Or
otherwise
by
practise
,
take
the
½
of
.
3.
s.
4.
whiche
is
.
1.
s.
8.
d.
and
adde
it
to
the
same
3.
s.
4.
d.
and
thereof
will
come
.
5.
as
before
.
For
the
⅓
of
.
5.
s.
is
as
muche
as
the
½
of
3.
s.
4.
pence
,
whiche
was
the
price
that
the
⅔
of
an
elle
did
coste
,
as
appereth
.
If
one
elle
coste
me
.
17.
s.
what
shal
15.
elles
⅔
part
cost
?
Whiche
⅛
is
halfe
a
quarter
of
an
elle
.
Aunswere
:
safe
of
〈◊〉
‑
〈◊〉
15
.
½
.
First
reduce
15.
1/●
into
.
8.
partes
,
and
thei
make
121/8
,
then
multiplie
121.
by
.
17.
1.
tyme
,
and
thereof
commeth
2057.
for
your
diuidende
.
Likewise
multiplie
.
8.
times
.
1.
1.
time
,
and
your
quotiente
will
bee
.
8.
for
your
diuisor
,
then
diuide
.
2057.
by
.
8.
and
you
shall
find
.
257.
shillynges
⅛
,
which
is
.
12.
〈◊〉
.
17.
shillynges
.
1.
penie
½
,
and
so
muche
are
the.
15.
elles
⅛
worthe
,
as
by
practise
doeth
appere
.
Or
otherwise
,
for
10.
s.
take
the
½
of
15
,
which
is
7
li.
10.
s.
then
for
5
s.
take
the
½
of
7
li.
10s
:
which
is
3
li.
15
s
:
thirdly
,
for
2
:
s
:
take
⅕
of
7
li:
10
:
s
:
because
the
½
of
10.
s
:
is
2
:
s
:
Fourthly
,
for
the
⅛
of
the
ell
,
you
shal
take
the
〈◊〉
of
17
:
s
:
whiche
is
2
:
s
:
1
:
d
:
½
:
Lastlie
,
adde
all
these
sommes
together
,
and
then
shall
you
finde
12
.li.
17
:
s
:
1
:
d
:
½
as
before
,
and
as
appereth
more
plainly
in
the
margent
.
If
25
elles
bee
worthe
2
,
li:
3
:
s
:
4
:
d
:
what
are
18
elles
¾
worth
by
the
price
?
Answer
:
first
put
3.
s.
4.
d.
into
the
part
of
a
li.
and
you
shall
haue
⅙
,
then
saie
,
if
25/1
geue
me
2
li.
⅙
what
shall
18¾
geue
:
put
the
whole
nomber
into
his
broken
,
and
then
multiplie
1
tymes
13
by
75
:
the
product
will
be
975
the
which
you
shall
diuide
by
25
tymes
6
:
4
tymes
,
which
maketh
600.
Then
diuide
975
by
600
,
and
your
quotient
will
be
1
li.
and
375.
remaineth
,
the
which
375.
you
shal
multiply
by
20.
thereof
commeth
7500.
diuide
the
same
by
600.
youre
quotient
wil
be
12.
s.
&
300.
remaineth
that
which
abreuiated
bringeth
½
whiche
is
6.
d.
thus
the
18.
elles
¾
are
worth
1
.li.
12.
s.
6.
d.
as
by
practise
appeareth
.
Or
otherwise
by
the
rules
of
practise
:
for
because
that
12.
elles
½
is
the
½
of
25.
elles
,
therefore
take
the
½
of
2
.li.
3.
s.
4.
d.
which
is
.
1
.li.
1.
s.
8.
d.
then
for
9.
elles
¼
take
the
¼
of
2
.li.
3.
s.
4.
d.
or
else
the
½
of
the
last
product
(
that
is
to
safe
of
1
.li.
1.
s.
8.
d.
)
whiche
is
all
one
,
and
adde
them
together
,
so
shall
you
haue
1
.li.
12.
s.
6.
d
:
as
before
.
If.
15.
yardes
be
worthe
.
32.
s.
what
are
half
a
yard
,
or
half
a
quarter
,
or
els
〈◊〉
of
a
yarde
worthe
.
Aunswere
:
saie
,
if
〈◊〉
geue
32/●
what
will
⅝
geue
?
Multiplie
〈◊〉
times
32.
by
5
,
and
diuide
the
product
by
.
15.
tymes
.
〈◊〉
.
s.
and
4
,
remaineth
,
whiche
is
⅓
of
a
shi
.
that
is
to
saie
4.
d
and
so
muche
are
5/2
of
a
yarde
worthe
.
Otherwise
,
see
what
the
yarde
is
worthe
,
after
the
maner
aforesaied
in
the
other
examples
,
and
you
shall
find
that
the
yarde
is
worthe
,
2.
s.
1.
d.
⅗
of
the
whiche
nomber
take
first
the
½
for
4/6
,
whiche
is
1.
s.
0.
d.
⅘
,
of
the
whiche
nomber
,
take
the
¼
for
the
other
〈◊〉
,
whiche
is
3.
d.
⅕
,
adde
these
.
2.
nombers
together
,
and
you
shall
finde
the
⅝
to
be
worthe
1.
s.
4.
d.
as
before
is
saied
.
If
13
els
〈◊〉
,
be
worthe
27.
s.
what
are
10
elles
⅔
worthe
by
the
price
?
Aunswere
:
saie
if
13⅚
geue
27/〈…〉
which
shal
10
⅔
geue
:
putte
the
whole
nombers
into
their
broken
,
and
you
shall
finde
83/6
,
27/1
,
end
32/3.
Then
multiplie
6
tymes
27
by
32
and
thereof
commeth
5184
the
which
nomber
you
shall
diuide
by
83
times
〈◊〉
thre
times
,
and
you
shall
finde
20.
s.
〈◊〉
whiche
is
worth
9.
d.
〈◊〉
/
85
part
of
a
peny
.
If
two
yardes
½
be
worthe
4.
s.
8.
d.
what
are
8
yardes
¼
worthe
?
Aunswer
put
the
8.
d.
into
the
part
of
a
shillyng
,
which
wil
be
⅔
,
then
reduce
the
whole
nombers
into
their
broken
,
and
thei
will
stand
thus
,
5/2
,
14/3
,
33/4
,
then
multiplie
twoo
tymes
14
by
33
,
and
diuide
the
producte
by
.
5.
tymes
3.
4.
tymes
,
and
you
shall
finde
15.
s.
4.
d.
⅘
,
so
muche
are
the
eight
yardes
¼
worthe
.
If
one
Kersey
bee
worthe
2
.li.
6.
s.
8.
d.
how
many
Kerseis
shal
I
buy
for
36
.li.
3.
s.
4
d.
after
the
rate
:
Answere
:
put
6.
s.
8.
d.
into
the
parte
of
a
li.
and
you
shall
haue
2
li.
⅓
for
the
first
nomber
in
the
rule
of
3.
and
1
elle
for
the
seconde
nomber
:
then
put
3
s.
4
d
:
into
the
part
of
a
li.
and
you
shal
finde
36
li.
3/6
for
the
3
nomber
,
then
will
your
3
nombers
in
the
rule
of
3
stand
thus
.
2⅓
|
1/1
|
36⅙
.
Therefore
reduce
the
whole
nombers
into
their
broken
,
and
you
shall
haue
7/3
|
1/1
|
217/6
.
Then
multiplie
1
by
217
,
and
thereof
will
come
651
for
your
diuidende
.
Likewise
,
multiplie
7
tymes
1
by
6
,
and
the
producte
thereof
will
be
42.
Then
diuide
651
by
42.
and
you
shall
finde
15
½
.
So
many
Kerseis
of
2
pounde
,
6
shillinges
8
pence
the
pece
shall
you
haue
for
.
36.
pounde
,
3.
shillynges
4
pence
.
¶
The
fowerth
Chapiter
treateth
of
losses
and
gaines
,
in
the
trade
of
marchaundise
.
IF
13
yardes
⅓
be
worth
22
li.
10
shillynges
,
how
shall
I
sell
the
yard
to
gaine
⅓
,
or
to
make
of
3.
4
?
whiche
is
all
one
?
Answere
:
saie
by
the
rule
of
3
,
if
3
be
come
of
4.
or
if
3.
yelde
4.
what
will
22.
½
:
multiplie
and
diuide
,
and
you
shal
find
30
.li.
Then
saie
again
by
the
rule
of
3.
if
13
yardes
1/〈…〉
doe
geue
30
.li.
aswell
of
principall
as
of
gain
:
what
will
1
yard
be
worthe
by
the
price
?
Multiplie
and
diuide
,
and
you
shall
finde
2.
pounde
5
shil
.
and
for
that
price
must
the
yard
be
sold
to
gaine
the
⅓
,
or
to
make
of
3.
4.
Or
otherwise
,
take
the
⅓
part
of
22.
●i
.
10.
s.
which
is
7
.l.
10.
s.
that
shal
you
adde
with
22
pounde
10
shillynges
,
&
you
shall
haue
30
pound
as
before
.
Then
diuide
30.
by
13.
⅓
.
and
you
shall
finde
2
pounde
5
shillynges
,
as
aboue
is
saied
.
If
one
yarde
bee
worthe
27.
s.
6.
d.
for
how
muche
shall
16
yardes
⅔
bee
solde
,
to
gaine
2.
s
vpon
the
pounde
of
money
,
that
is
to
saie
:
open
20.
s.
Answere
:
adde
2
vnto
20
&
you
shall
haue
22
,
then
saie
:
if
20.
s.
of
principall
,
doe
geue
22.
s.
aswel
of
principal
as
gain
:
howe
muche
will
27.
s.
6.
d.
principall
yeld
.
Multiplie
and
diuide
&
you
shall
find
30.
s.
¼
:
then
saie
again
by
the
rule
of
3
,
if
1.
yard
do
geue
me
30.
s
¼
(
which
is
aswell
the
principall
as
the
gaine
)
what
shal
16
yardes
⅔
geue
me
?
Multiplie
and
diuide
,
and
you
shall
finde
25
.li.
4.
s.
2.
d.
For
the
same
price
shall
the
16
yardes
⅔
be
solde
to
gaine
after
the
rate
of
2.
s
vpon
the
pound
of
money
,
or
in
20.
s.
whiche
is
all
one
.
If
10
yardes
⅔
be
worthe
25
.li.
10.
s
For
how
muche
shall
2
yards
¼
be
sold
to
gaine
after
10
.li.
vpon
the
100
.li.
of
money
?
Answere
:
saie
if
100
of
principall
yeld
110
aswell
principall
as
gain
how
muche
will
25
.li.
10.
s.
yelde
me
?
Multiplie
and
diuide
,
and
you
shall
finde
28
.li.
1.
s.
Then
saie
if
10
yardes
⅔
doe
yeld
me
28
.li.
1.
s.
aswell
of
principall
as
of
gaine
,
how
muche
shall
.
2.
yardes
¼
yelde
me
?
Multiplie
and
diuide
,
and
you
shall
finde
5
.li.
18.
s.
4.
d.
1/●2
,
for
so
muche
shall
the
twoo
yardes
be
solde
,
to
gaine
after
10
.li.
vpon
the
100
.li.
of
money
.
And
although
that
in
these
questions
of
gaine
and
losse
,
sometymes
the
firste
nombers
is
not
like
vnto
the
third
nomber
,
that
is
to
saie
,
of
the
same
denomination
:
as
one
would
saie
:
if
20.
shillynges
gaine
.
2.
shillynges
,
what
shall
.
50
.li.
gaine
?
Or
25
.li.
&c.
Or
if
20
.li.
doe
gaine
2
.li.
What
shall
25.
s.
gaine
me
,
or
what
shall
.
27.
s.
gaine
?
Yet
neuerthelesse
,
the
rule
is
not
therfore
false
.
For
if
20.
s.
dooe
gaine
2.
s
:
20
.li.
shall
gaine
2
.li.
and
20.
d.
shall
gaine
2.
d.
likewise
20
crounes
shall
gaine
2
crounes
,
and
so
of
all
other
:
therefore
it
is
to
bee
vnderstande
,
that
the
first
nomber
in
these
reasons
,
is
presupposed
to
be
semblable
to
the
thirde
.
When
one
marchaunt
selleth
wares
to
an
other
,
and
he
geueth
to
the
buyer
2
vpon
15
:
how
muche
shall
the
buyer
gain
vpon
the
100
after
the
rate
?
Answere
:
saie
if
15
geue
17
,
what
shall
100
geue
?
Multiplie
and
diuide
,
and
you
shall
finde
113⅓
,
so
the
buyer
getteth
after
the
rate
of
13⅓
vpon
the
100.
If
1.
Northe
dosen
cost
me
3
li.
5
s.
&
I
sel
the
same
again
for
3
.li.
12.
s.
6
d.
how
muche
do
I
gain
vpon
the
li.
of
money
after
the
rate
.
Answere
saie
if
3
.li.
¼
do
geue
3.
li
⅝
,
what
shall
20/8
geue
,
put
the
whole
nomber
into
their
broken
,
&
you
shall
haue
13/4
,
29/8
,
20/1
,
then
multiplie
4
tymes
29
,
by
20
,
&
therof
commeth
2320
for
your
nomber
that
is
to
be
diuided
,
likewise
multiplie
13
times
8.
1
time
,
&
therof
commeth
104.
Then
diuide
2320
by
104
,
and
you
shall
finde
22
shillinges
4/13.
So
I
shall
get
2
shillynges
4/13
vpon
20
shill.
or
vpon
the
li.
of
money
.
If
a
yarde
of
clothe
coste
me
7
shillynges
8
pence
,
and
afterward
I
deliuer
out
13
yardes
¼
,
for
4
pounde
13.
s.
4
pence
.
I
would
know
whether
I
do
winne
or
lose
,
and
how
muche
vpon
the
100
pounde
of
money
?
Answere
:
se
first
at
7
shillynges
8
pence
the
yarde
,
what
the
13
yardes
¼
shall
cost
,
and
you
shall
finde
5
pound
1
shil.
7
pence
.
And
I
solde
them
but
for
4
pounde
13
shil.
4
pence
,
so
that
I
doe
lose
vpon
the
13
yardes
¼
the
somme
of
8
shilly
.
3
pence
.
Then
for
to
knowe
how
muche
is
lost
vpon
the
100
:
saie
by
the
rule
of
3
,
●●
5.
pound
1
shil.
7.
pence
doe
lose
8
shil.
3.
d
What
will
100
lose
?
First
put
1
shill.
7
pence
into
the
part
of
a
pounde
,
and
it
will
be
19/240.
Likewise
put
8
shill.
3.
d.
into
the
part
of
a
li.
and
it
is
33/80.
Then
will
your
nomber
stande
thus
5.
19/240
,
33/80
,
100/●●
,
put
the
whole
into
his
broken
,
and
then
multiplie
and
diuide
,
so
you
shall
finde
8
.li.
1184/9752
,
whiche
is
worthe
2.
shillynges
.
5.
d.
169/1219
and
so
muche
is
lost
vpon
the
100
.li.
of
money
.
More
,
if
12
yardes
½
of
Scarlet
bes
solde
for
30
pounde
15
shillynges
,
vpon
the
whiche
is
gained
after
the
rate
of
〈◊〉
1/9
vpon
the
100.
I
demaunde
what
the
yarde
did
coste
at
the
firste
.
Aunswere
:
from
30
pounde
15
shillynges
,
substracte
his
1/10
part
,
whiche
is
3
.li.
1.
s
6.
d.
and
there
resteth
27
.li.
13.
s.
6.
d.
the
whiche
number
multiplied
by
.
2.
bryngeth
55
.li.
7.
s.
of
the
which
is
2
pounde
,
one
shillyng
and
●ower
pence
.
Then
take
againe
the
⅕
of
the
saied
〈◊〉
pounde
1
shillyng
4
pence
,
whiche
is
2
pounde
●ower
shillynges
three
pence
,
9/25.
And
so
muche
did
the
elle
cost
at
the
firste
penie
.
More
,
if
15
yardes
¾
of
Scarlette
doe
cost
me
32
pound
13.
s.
4.
pence
.
And
I
sell
the
yarde
again
for
2
pound
,
whether
dooe
I
winne
or
lose
,
and
howe
muche
vpon
the
pounde
of
money
.
Answere
:
loke
what
the
15
yards
¾
are
worthe
at
2
.li.
the
yarde
,
and
you
shall
find
that
thei
are
worth
31
.l.
10.
s.
But
thei
did
cost
32
.li.
13.
s.
4.
d.
so
that
there
is
lost
vpon
the
whole
1
.li.
3
,
s
4.
d.
Then
to
knowe
how
muche
is
lost
vpon
the
li.
saie
by
the
rule
of
3
,
if
32
li.
⅛
doe
lose
1
li.
⅙
:
what
will
⅓
lose
?
That
is
to
saie
:
what
will
1
li.
lose
?
Reduce
the
whole
nombers
into
their
broken
,
and
then
multiplie
and
diuide
,
so
shal
you
finde
21/●8●
parte
of
a
li.
Then
multiplie
21
by
240.
because
so
many
pence
are
in
a
li.
and
deuide
the
producte
by
588.
so
shall
you
finde
8.
d.
⅗
⅜
⅝
which
beyng
abreuiated
doe
make
4/7
,
and
thus
you
se
that
8.
d.
4/7
is
lost
vpon
the
li.
of
money
.
If
1.
yarde
of
cloth
of
tissue
be
solde
for
3
.li.
15.
s.
whervpon
is
lost
after
the
rate
of
10.
s.
vpon
the
100.
I
demaund
what
12.
yardes
½
of
the
same
tissue
did
cost
?
Aunswere
:
adde
vnto
3
.li.
15.
hys
owne
1/10
part
,
whiche
is
7.
s.
6.
d.
and
al
amounteth
to
4
.li.
2.
s.
6.
d.
then
looke
what
the
12.
yardes
½
wil
amount
vnto
,
after
4
.li.
2.
s.
6.
d.
&
you
shall
finde
that
thei
will
come
to
51
.li.
11.
s.
3.
d.
so
muche
did
the
12.
yardes
½
coste
.
More
,
if
I
sell
1
Wilshire
white
for
6
li.
12
s.
whereupon
I
doe
gain
after
the
rate
of
2
shil
.
vpon
the
li.
of
money
,
that
is
to
saie
,
vpon
20
s.
I
demaunde
what
11
peeces
of
the
same
whites
did
cost
me
?
Answere
:
abate
from
6
pound
12
s.
(
whiche
is
132
s
)
his
1/●●
parte
,
and
thereof
commeth
12
s.
and
there
remaineth
120
s.
or
.
6
li.
Then
se
at
6
.li.
the
clothe
,
what
the
11
clothes
are
worthe
and
you
shall
finde
them
to
be
worthe
66
li.
so
muche
did
the
11
clothes
coste
.
If
I
sel
10
elles
½
of
Holland
for
12
s.
6
d.
wherupon
I
doe
lose
after
the
rate
of
2
s.
vpon
the
li.
of
money
.
I
demaund
what
the
ell
did
cost
me
?
Answere
:
saie
by
the
rule
of
:
3
,
if
18
geue
20
:
s.
what
will
20
s
:
6
:
d
:
geue
?
Multiplie
and
diuide
,
and
you
shall
find
25
s
:
Then
dinide
25
s.
by
10
½
,
and
thereof
commeth
2
shil.
4.
d.
4/7.
So
muche
did
the
ell
cost
If
I
sell
1
clothe
for
5
li.
wherupon
I
do
lose
10
vpon
the
100
,
I
demaunde
how
muche
I
should
lose
or
gaine
vppon
the
100
,
in
case
I
had
sold
the
same
for
5
li.
10.
s.
Answere
:
saie
,
if
90
yelde
100
,
how
muche
will
5
li.
geue
.
Multiplie
and
diuide
,
and
you
shall
finde
5
li.
5/9
:
then
saie
again
by
the
rule
of
3
,
if
5/9
come
to
5
½
,
what
will
100
come
vnto
?
Multiplie
and
diuide
,
and
you
shal
finde
99
li.
whiche
beyng
abated
from
100.
there
will
remain
1
pounde
,
and
so
muche
is
loste
vpon
the
100.
¶
The
fifte
Chapiter
treateth
of
lengthes
&
breadthes
of
tapistrie
,
and
other
clothes
.
IF
a
peece
of
Tapistri●e
bee
5
elles
¾
longe
,
and
4
elles
⅔
in
bredth
,
how
many
ells
square
doeth
the
same
pece
containe
?
Answere
:
multiplie
the
length
by
the
breadth
,
that
is
to
saie
5
¾
by
4
⅔
,
and
therof
commeth
26
elles
⅚
,
so
many
elles
square
doth
the
same
peece
conteine
.
More
,
if
a
peece
of
Tapistrie
doe
conteine
32.
elles
square
,
and
the
same
being
in
length
6
elles
¼
.
I
demaunde
howe
many
elles
in
breadth
the
same
peece
doth
conteine
.
Auns
.
diuide
32.
elles
by
6
¼
and
thereof
commeth
5.
3
/
28
:
So
manye
elles
dothe
the
same
péece
conteine
in
breadth
.
More
,
a
péece
of
clothe
beynge
13.
yardes
⅓
in
length
,
and
5
,
quarters
½
in
breadth
,
how
many
yardes
of
⅔
and
½
broade
will
the
same
peece
make
?
Aunswere
:
see
what
parte
of
a
yarde
,
the
5/4
and
½
be
,
and
you
shall
finde
that
they
make
1
yarde
⅜
.
Then
multiplye
13
yardes
〈◊〉
by
1
yarde
⅜
and
you
shall
haue
18.
yardes
⅓
in
square
the
whiche
you
must
diuide
by
⅔
&
½
that
is
to
saye
by
〈◊〉
,
(
bicause
the
⅔
,
½
beinge
brought
into
1
fraction
maketh
5/〈…〉
)
and
you
shall
finde
22.
yardes
:
So
many
yardes
of
〈◊〉
&
〈◊〉
large
doth
the
same
peece
conteine
.
More
,
a
marchant
hath
bought
4.
yardes
⅔
of
cloth
beinge
syre
quarters
〈◊〉
broade
to
make
hym
a
gowne
the
whiche
he
will
line
thoro●out
,
wyth
black
Say
of
three
quarters
of
a
yarde
broad
,
I
demaund
how
much
Say
he
must
bye
?
Aunswere
:
Multiplye
the
length
of
the
cloth
,
by
the
breadth
,
that
is
to
say
4
⅔
by
1
⅝
,
(
which
is
the
syx
quarters
½
)
and
thereof
commeth
7.
yardes
7/12
,
the
which
diuide
by
¾
and
you
shal
finde
ten
yardes
1/9.
So
many
yardes
of
Say
must
he
haue
to
line
the
same
4
yardes
⅔
of
clothe
of
6.
quart.
〈◊〉
broad
.
More
,
at
6
,
s.
8
,
d.
the
elle
square
,
what
shall
a
peece
of
tapistrie
cost
me
,
which
is
fiue
elles
½
long
and
4
,
elles
¼
broade
?
Aunswere
,
multiply
5
,
½
by
4
,
〈◊〉
and
therof
commeth
23
elles
⅜
square
:
then
say
by
the
rule
of
three
,
if
one
elle
square
cost
me
6
,
s.
8
,
d.
what
shal
23
⅜
cost
?
Multiplye
and
diuide
,
and
you
shall
finde
7
li.
15.
s.
10
d.
so
muche
the
said
peece
of
tapistrie
did
cost
.
Or
otherwise
,
by
the
rules
of
practise
,
take
the
⅓
of
23.
⅜
:
and
you
shall
finde
7
.li.
15.
s.
10.
d.
as
aboue
is
sayde
.
More
,
a
peece
of
Hollande
clothe
conteining
42.
elles
⅔
Flemish
,
howe
many
elles
Englishe
doe
they
make
?
Here
must
you
fyrst
note
that
100.
els
Flemish
,
do
make
but
60.
elles
Englishe
,
and
so
consequentlye
fiue
elles
Flemishe
do
make
but
3.
els
English
.
Therfore
say
by
the
rule
of
3.
if
5.
elles
flemish
doe
make
three
elles
English
,
how
many
elles
English
will
42
,
els
⅔
Flemish
make
.
Multiply
and
diuide
so
shall
you
finde
25.
elles
⅗
Englishe
,
and
so
many
elles
Englishe
doth
42.
⅔
Flemish
conteine
,
the
lyke
is
to
bee
done
of
all
others
.
More
,
I
haue
boughte
a
peece
of
Tapistrie
,
being
5.
elles
¾
longe
,
and
4.
elles
⅔
broade
measure
of
Flaunders
,
I
demaunde
howe
manye
elles
square
it
maketh
Englishe
measure
?
Aunswere
:
First
,
forasmuch
as
three
elles
english
are
worth
5
elles
flemishe
,
therefore
put
3
elles
english
into
hys
square
,
in
multipliyng
3.
by
himself
whiche
maketh
9
:
likewise
multiplye
5.
in
hymselfe
squarely
,
and
it
wil
be
25.
Then
multiplye
5¾
whiche
is
the
length
of
the
peece
,
by
4⅔
which
is
the
breadth
,
&
therof
commeth
26
elles
⅚
square
,
then
say
by
the
rule
of
thrée
,
if
25
elles
square
of
flemishe
measure
,
be
worth
9
elles
square
of
englishe
measure
,
what
are
26
elles
flemish
⅚
worth
?
multiply
and
diuide
,
and
you
shall
finde
that
they
are
worthe
nine
elles
33/50
square
of
english
measure
.
More
at
3.
s.
6.
d
:
the
ell
flemish
what
is
the
englishe
ell
worth
after
the
rate
.
Answere
:
saye
if
5.
elles
flemishe
bee
worth
three
ells
english
,
what
is
1
ell
flemishe
worth
?
multiply
and
diuide
,
&
you
sh●ll
fiynde
3/●
of
an
englishe
ell
.
Then
saye
by
the
rule
of
3
,
if
〈◊〉
of
an
englishe
ell
,
be
worth
3
s.
6
d.
what
is
1.
englishe
ell
worth
?
multiplye
and
diuide
,
and
you
shall
finde
5
s.
10.
d.
so
much
shall
the
englishe
ell
be
worth
.
More
at
6
s.
8
d.
the
flemishe
elle
square
,
what
is
the
english
ell
worth
.
Aunswere
:
say
by
the
aforesaid
reason
,
if
25
elles
flemishe
square
,
be
worth
9.
elles
square
englishe
,
what
is
one
ell
square
flemishe
worth
?
multiply
and
diuide
,
&
you
shall
finde
9/25
of
a
square
englishe
elle
:
Then
saye
,
if
9/25
of
an
englishe
elle
be
worth
6
s.
8.
d.
what
is
one
square
elle
englishe
worth
?
multiplie
and
diuide
,
and
you
shall
fynde
18.
s.
6.
d.
●/9
,
so
muche
shal
one
englishe
elle
square
be
worth
.
¶
The
sixt
Chapter
treateth
of
the
reducing
of
the
paumes
of
Genes
into
englishe
yardes
,
wherof
foure
Paumes
maketh
one
english
yarde
.
I
Haue
bought
97.
paumes
〈◊〉
of
Genes
veluet
,
&
I
would
know
howe
manye
yardes
they
will
make
?
Aunswere
,
Diuide
97.
〈◊〉
by
4.
and
you
shall
haue
24.
yardes
〈◊〉
.
So
manye
yardes
doe
the
97.
paumes
●/2
conteine
.
Or
otherwise
,
take
some
other
nomber
at
your
pleasure
,
as
20
,
paumes
,
which
doe
make
fiue
yardes
,
and
then
say
by
the
rule
of
three
,
if
20/1
paumes
,
giue
5/1
yardes
,
what
will
97.
½
giue
?
Multiplye
and
diuide
,
and
you
shall
finde
24.
yardes
⅜
as
before
.
More
,
at
two
shillings
7
,
d.
y●
pau
〈…〉
of
Genes
,
what
wil
the
English
yarde
be
worth
after
the
rate
?
Aunswere
,
say
by
the
rule
of
three
,
if
¼
of
an
English
yarde
bee
worthe
twoo
shillings
7/12.
What
is
〈◊〉
yarde
worth
?
Multiplie
&
diuide
,
and
you
shall
finde
ten
shillings
4.
d.
So
muche
is
the
Englishe
yarde
worth
.
Or
otherwise
,
multiply
4.
paumes
(
which
is
one
yarde
)
by
two
shillings
7
,
pence
,
and
you
shal
finde
10
,
s.
4.
d.
as
before
.
If
257.
Paumes
½
bee
worth
20
.li.
16.
s.
8.
d.
What
is
one
yarde
worthe
after
the
rate
?
Annswere
,
saye
:
by
the
rule
of
3
,
if
257
,
½
paumes
be
worthe
20.
5/●
,
what
are
4/●
paumes
worth
.
Multiply
and
diuide
,
and
you
shall
fynde
100/309
part
of
a
pounde
,
which
is
worthe
6
s.
5.
pence
,
〈◊〉
:
so
much
is
one
yarde
worthe
.
¶
The
vii
.
Chapter
treateth
of
marchaundise
solde
by
waight
.
AT
9.
d.
½
the
ounce
,
what
is
the
li.
waight
worth
?
Answere
,
say
if
3/●
giue
9.
½
what
will
〈◊〉
gyue
multiply
and
diuide
,
&
you
shal
finde
12.
s.
8
d.
so
much
is
the
yarde
worth
?
Or
otherwise
,
by
the
rules
of
practise
for
syre
pence
,
take
the
½
of
16.
which
is
8.
s.
then
for
3.
d.
take
the
½
of
16.
s.
whiche
is
4.
s.
Finally
,
for
the
halpenye
,
take
16.
ob
.
which
are
8.
d.
adde
all
these
nombers
togither
and
you
shal
finde
12.
s.
8.
d.
as
before
.
More
,
at
10
d.
½
the
ounce
,
what
are
112
.li.
waight
worth
after
the
rate
?
Aunswere
reduce
.
112
.li.
into
ounces
:
in
multipliyng
.
112
.li.
by
16.
ounces
&
you
shall
haue
1792.
ounces
,
then
say
by
the
rule
of
3.
if
1/●10½1722
●10½1722/1
:
Multiplie
and
diuide
,
and
you
shall
finde
18816
d.
whiche
do
make
78
.li.
8.
s.
and
so
much
are
the
112
.li.
worth
after
10.
d.
½
the
ounce
.
At
12.
s.
8.
d.
the
li.
waight
,
what
is
the
ounce
worth
:
Aunswere
:
put
12.
s.
8
d.
into
pence
,
and
you
shall
haue
152.
pence
:
then
say
by
the
rule
of
3.
if
16.
ounces
cost
152.
d.
what
shall
1.
ounce
coste
,
multiplye
and
diuide
,
and
you
shall
finde
9.
d.
1/1
,
so
much
is
the
ounce
worth
.
Or
otherwise
,
take
the
¼
of
12
s.
8.
d.
for
4
ounces
,
and
thereof
commeth
3.
s.
2.
d.
then
for
one
ounce
,
take
the
¼
of
3.
s.
2
d.
and
you
shall
haue
9.
d.
½
as
before
.
At
32
.li.
10.
s.
the
quintall
,
that
is
to
saye
,
the
100
.li.
waight
what
is
1
.li.
waight
worthe
after
the
same
rate
?
Aunswere
:
Put
32
.li.
12.
s.
al
into
shillings
and
you
shal
haue
650.
s.
Then
say
,
by
the
rule
of
three
,
if
〈…〉
650
〈…〉
.
multiply
and
diuide
,
and
you
shall
finde
,
6
s.
6.
d.
so
muche
is
the
11.
worthe
.
If
one
pound
waight
of
saffron
do
cost
me
18.
s.
8
d.
what
shall
355
.li.
10.
ounces
cost
me
by
the
same
price
?
Aunswere
,
saye
by
the
rule
of
3.
if
½18
⅔355⅝
.
Multiply
and
diuide
,
and
you
shall
finde
331
.li.
18.
s.
4.
d.
so
much
are
the
355
.li.
ten
ounces
worth
.
Brief
rules
of
waight
.
WHo
that
multiplieth
the
pence
that
1
.li.
waight
is
worth
by
〈◊〉
,
&
diuideth
the
product
therof
by
.
12.
hee
shall
finde
howe
manye
pounds
in
mony
the
quintal
is
worth
that
is
to
saye
,
howe
much
the
100
.li.
waight
is
worth
.
And
contrariwise
he
that
multiplieth
the
pounds
of
money
that
the
100
waight
is
worth
by
12.
and
deuide
the
product
by
5.
and
you
shall
fynde
howe
many
pence
the
pound
waight
is
worth
.
¶
Example
.
AT
seuentene
pence
the
pounde
waight
,
what
is
the
100
pounde
waight
worth
?
Aunswere
,
Multiplie
17.
by
5
,
and
thereof
commeth
85.
diuide
the
same
by
12.
and
you
shall
finde
7.
pound
1/12
,
whiche
1/12
is
worth
one
shilling
and
eight
pence
.
So
much
is
the
100
pounde
waight
worth
.
More
,
at
13
,
li.
the
100
li.
waight
,
what
is
one
pounde
waight
worth
?
Aunswere
,
Multiplie
13.
by
12.
amounteth
to
156.
the
whiche
diuide
by
5.
and
you
shal
finde
31.
d.
●/5
which
is
2.
s.
7
d.
1/●
and
so
muche
is
one
pounde
waight
worth
.
The
lyke
is
to
be
done
of
yardes
,
elles
,
or
of
any
other
measure
,
when
we
recken
but
fyue
score
to
the
hundred
.
Briefe
Rules
for
measure
.
Who
that
multiplieth
the
pence
that
one
ell
is
worthe
,
by
6.
And
diuideth
the
product
by
12.
he
shall
fynde
how
many
poundes
in
money
the
120.
elles
are
worth
,
which
120
elles
wee
count
but
for
a
C.
And
contrariwise
,
be
that
multiplieth
the
poundes
in
money
that
the
120
elles
are
worth
by
12
,
and
diuideth
the
multiplicotion
by
6
,
shall
fynde
howe
many
pence
the
ell
is
worthe
.
¶
Example
.
At
ten
pence
the
ell
,
what
are
120.
elles
worth
:
Answere
,
multiplie
10.
d.
by
6.
and
thereof
commeth
60
:
The
which
diuide
by
12.
and
you
shall
find
fyue
pounde
,
so
many
pounds
in
money
are
120.
ells
worth
at
10.
d.
the
ell
.
More
.
at
9.
pounde
,
the
120
,
elles
,
what
is
one
elle
worthe
?
Aunswere
,
Multiplie
nine
pound
by
twelue
,
and
therof
commeth
108
,
the
which
diuide
by
6.
and
you
shal
finde
18.
d.
so
much
is
one
elle
worth
.
The
like
is
to
be
done
of
alll
maner
of
wares
,
which
are
sold
after
120.
for
the
hundred
.
¶
Briefe
Rules
for
●ure
hundreth
waight
here
at
London
,
whiche
is
after
112
.li.
for
the
C.
WHo
that
multiplieth
the
pence
that
one
pounde
waight
is
worthe
by
28.
and
diuideth
the
product
by
60.
shall
finde
howe
many
pounds
in
money
the
112
.li.
waight
is
worth
.
ANd
contrariwise
,
he
that
multiplied
the
poundes
in
money
that
the
112
,
li.
is
worth
by
6●
,
and
diuideth
the
product
by
28.
shal
finde
howe
manye
pence
one
li.
waight
is
worth
.
Example
.
AT
nine
pence
the
pounde
waight
,
what
is
the
112
li.
waight
worth●
Aunswere
:
multiplie
6.
〈◊〉
.
by
28
,
and
thereof
commeth
2
,
2
,
the
whiche
diuide
by
60
,
and
you
shal
finde
4
.li.
12/60
which
beinge
abbreuiated
is
〈◊〉
of
a
pounde
,
whiche
is
worthe
4
,
s.
And
thus
the
〈◊〉
.li.
2.
is
worth
4.
pound
4
,
shil
.
At
8
li.
the
112
,
li.
waight
,
what
is
1
.li.
waight
worth
?
Aunswere
,
Multiplie
8
li.
by
60
,
and
thereof
commeth
480
,
the
whiche
diuide
by
28.
and
you
shall
finde
17.
d.
〈◊〉
:
so
muche
is
1
,
li.
waight
worth
.
¶
The
.viij.
Chapter
treateth
of
tares
and
allowances
of
marchaundise
solde
by
waight
:
AT
12
li.
the
100.
suttell
,
what
shall
987
li.
suttell
be
worth
?
in
giuing
4
li.
waight
vppon
euery
100
for
tret
?
Answere
,
adde
4
li.
vnto
100.
&
you
shall
haue
104.
Then
say
by
the
rule
of
thre
,
if
104.
be
worth
1●
.li.
what
are
987
li.
waight
worthe
?
multiply
&
diuide
,
&
you
shal
finde
113
,
li.
〈◊〉
which
is
worthe
17.
s.
8
,
d.
4/●●
.
So
much
shal
the
987
.li.
waight
be
worth
.
At
6
,
s.
8.
d.
the
pound
waight
what
shall
345
,
li.
½
be
worth
in
giuing
4
,
li.
waight
vpon
euery
100
,
for
the
tret
.
Aunswere
,
see
first
by
the
rule
of
three
,
what
the
100
pound
is
worth
saying
,
if
〈◊〉
6⅓100/1
Multiplie
and
diuide
,
and
you
shall
finde
33
.li.
⅓
then
adde
4
.li.
vnto
100
,
&
they
are
104.
then
say
again
by
the
rule
of
3.
if
a
104
.li.
be
sold
for
33
,
li.
⅓
for
how
muche
shall
345
.li.
½
be
sole
?
multiply
&
diuide
,
and
you
shal
finde
110
.l.
14.
s.
8.
d.
12/13
So
much
shal
the
345
.li.
●/2
be
worth
,
at
6.
s.
8.
d.
the
pound
,
in
giuing
4.
vpon
the
100.
More
,
if
100
bee
worth
36.
s.
8
d.
what
shall
780
.li.
bee
worth
in
rebating
4
li.
vpon
euery
100
,
for
Tare
&
Close
?
Answere
,
Multiply
780.
by
4.
and
therof
commeth
3120.
The
which
diuide
by
100.
and
you
shal
haue
31
,
li.
⅕
abate
31.
⅕
from
780.
and
there
wyll
remaine
748⅘
.
Then
say
by
the
rule
of
three
,
if
100/1
do
cost
36.
⅔
,
what
will
748.
⅘
cost
after
the
rate
?
Multiplie
&
diuide
so
shall
you
finde
274.
s.
6.
d.
●8/25
,
and
so
much
shall
the
780
.li.
cost
,
in
rebatinge
4
.li.
vpon
euery
100.
for
Tare
and
Cloffe
.
More
,
whether
doth
he
lose
more
that
giueth
5
li.
vpon
the
100.
or
he
that
rebateth
5
.li.
vpon
the
100.
for
care
and
cloffe
?
Answere
.
First
,
note
that
hee
which
giueth
5
.li.
vpon
the
100.
giueth
105.
for
100
:
and
he
which
rebateth
5
.li.
vpon
the
100.
giueth
the
100.
for
95.
Therefore
saye
by
the
rule
of
3.
if
105.
be
giuen
for
100
:
for
how
much
shal
the
100.
be
giuen
?
Multiply
and
diuide
&
you
shal
finde
65.
5/2●
:
and
be
whiche
rebateth
5.
vpon
the
100.
maketh
but
95.
of
100
:
so
that
he
loseth
5.
vpon
the
100
&
the
other
which
giueth
5.
vpon
the
100
loseth
but
4.
16/21
vpon
the
100.
Thus
he
that
rabateth
5.
vpon
the
100.
loseth
more
by
5/21
vppon
the
100.
than
the
other
whiche
gaue
5.
vppon
the
100.
for
tare
and
clo●fe
.
If
100.
of
Allam
do
〈…〉
me
.
26.
s.
8.
d.
how
shall
I
sell
the
li.
waight
to
gaine
after
the
rate
of
10.
vpon
the
100.
Answere
,
put
26.
s.
8.
d.
al
into
pence
.
&
you
shal
haue
320.
d.
Then
say
by
the
rule
at
3.
if
100.
giue
110
what
shal
320
giue
multiplie
and
diuide
,
and
you
shall
finde
352.
d.
Then
say
,
if
100
li.
be
worth
352
d.
what
is
〈◊〉
,
li.
multiplye
&
diuide
,
and
you
shall
haue
3
,
d.
26/50
whiche
26/50
is
worth
½
,
and
1/25
of
½
.
That
is
to
saye
,
the
pounde
waight
shal
be
worth
3.
d.
½
,
1/25
of
a
halfe
pennye
,
in
gaining
10.
vpon
the
100.
If
one
pound
waight
doe
cost
me
,
6
s.
10
,
d.
and
I
sell
the
same
for
7
,
s.
2
d.
I
demaund
howe
much
I
should
gaine
vpon
the
100
li
,
of
money
after
the
rate
?
Answere
,
say
by
the
rule
of
3.
if
6
,
⅚
yelde
7.
2/6
what
will
105/1
yelde
?
Put
the
whole
nombers
into
theyr
broken
,
the
〈…〉
ul●iplie
and
diuide
,
&
you
shall
finde
10436/41
from
the
whiche
substract
100
,
and
there
resteth
4
.li.
36/41
so
muche
is
gained
vpon
the
hundred
pounde
of
money
after
the
rate
.
More
,
if
one
pound
do
cost
me
5
,
s.
4.
d.
and
I
sell
the
same
againe
for
4
,
s.
9
d.
I
demaunde
how
much
I
shal
lose
vpon
the
100.
pounde
of
money
?
saye
,
if
5
,
⅓
doe
giue
but
4
,
¾
,
what
shall
100/1
geue
?
Putte
the
whole
nomber
into
their
broken
.
Then
multiplie
and
diuide
,
and
you
shall
finde
89
1/10.
the
whiche
you
must
substract
from
100.
and
there
will
remain
.
10.
pound
15/●6
,
so
muche
is
lost
vpon
the.
100.
pound
of
money
.
More
,
if
the
pounde
waight
doe
cost
me
.
3.
shillynges
.
2.
pence
:
and
I
sell
it
againe
for
.
3.
shillynges
.
4.
pence
,
how
muche
shall
I
gaine
vpon
.
20.
shillin
.
Aunswere
:
saie
if
3⅙
geue
4⅓
,
what
shall
20/1
geue
.
Multiplie
and
diuide
and
you
shall
finde
27.
shillinges
7/19
:
out
of
the
whiche
abate
.
20.
shillynges
,
and
there
will
remaine
.
7.
shillynges
7/19
,
whiche
is
.
4.
pence
4/19
:
and
so
muche
is
gained
vpon
the
pound
of
money
,
that
is
to
saie
vpon
20.
shillinges
.
If
the
pounde
waight
doe
coste
me
4.
shillinges
.
4.
pence
,
and
sell
it
again
for
.
3.
shillinges
.
2.
pence
.
I
demaunde
how
muche
I
shall
lose
vpon
the
pound
of
money
?
That
is
to
saie
,
vpon
.
20.
shillynges
.
Answere
:
saie
,
if
,
4.
⅓
geue
but
3⅙
,
what
will
20/1
geue
,
multiplie
and
diuide
,
and
you
shall
finde
.
14.
s.
〈◊〉
,
the
whiche
you
must
abate
from
.
20
shillynges
,
and
there
will
remaine
.
5.
shillinges
5/13
,
whiche
〈◊〉
,
is
worthe
.
4.
d.
〈◊〉
of
a
penie
,
and
so
muche
is
lost
vpon
the
pounde
of
money
.
¶
The
ninth
Chapter
treateth
of
certain
questions
,
doen
by
the
double
rule
,
and
also
by
the
rule
of
three
compounde
.
WHen
the
quarter
of
Wheate
doeth
cost
6.
s.
8.
d
the
loafe
of
breade
waiyng
.
20.
ounces
is
solde
for
a
ob.
I
demaunde
that
if
the
quarter
of
Wheate
did
cost
10
shillinges
,
for
how
muche
shall
the
loafe
of
bread
be
sold
,
that
waieth
.
16.
ounces
?
Answere
:
by
the
first
part
of
the
rule
of
3.
compounde
,
whiche
is
mentioned
in
the
third
parte
of
this
boke
,
and
in
the
seconde
Chaptier
of
the
same
.
Therefore
saie
by
thesame
firste
parte
of
the
rule
of
.
3.
compounde
,
if
6⅔20/1½10
1½10/116
116/1.
Then
multiplie
the
firste
nomber
by
the
seconde
,
and
the
producte
thereof
shal
be
your
diuisor
.
Likewise
multiplie
the
other
three
nombers
,
the
one
by
the
other
,
and
the
producte
thereof
shal
be
your
diuidende
:
As
thus
,
firste
multiplie
.
6⅔
by
.
20.
and
thereof
commeth
400/〈…〉
for
your
diuisor
,
then
multiplie
½
by
10/1
,
and
the
producte
thereof
by
16/1
,
so
you
shall
haue
〈◊〉
for
youre
nomber
that
is
to
bee
diuided
,
then
diuide
150/●
by
400/●●
,
and
thereof
commeth
480/800.
the
whiche
beeyng
abbreuiated
,
bryngeth
〈◊〉
.
of
a
penie
:
and
for
that
price
muste
the
loafe
of
breade
bee
solde
,
whiche
waieth
sixtene
ounces
,
and
the
quarter
of
Wheate
beeyng
worthe
tenne
shillynges
.
Or
otherwise
by
the
rule
of
three
,
at
twoo
tymes
.
Firste
saie
if
●0/1
ounces
geue
½
,
what
will
16/1
ounces
geue
?
Multiplie
and
diuide
,
and
you
shall
finde
2/●
,
of
a
pennie
.
Then
saie
againe
,
if
.
6.
〈◊〉
dooe
geue
me
⅖
,
what
will
10/2
geue
?
Multiplie
and
diuide
,
and
you
shall
finde
⅗
of
a
penie
,
as
afore
is
saied
.
When
the
cariage
of
one
hundreth
waighte
of
Marchaundise
.
50.
miles
doeth
cost
.
5.
shillinges
,
what
shall
the
cariage
of
.
500.
waight
cost
me
for
.
16.
mile
?
Aunswere
.
By
the
firste
parte
of
the
rule
of
.
3.
compounde
,
saiyng
,
if
10050550016
Multiplie
.
100.
by
50.
the
produ●●e
will
be
.
5000.
whiche
shal
bee
your
diuisor
.
Then
multiplie
5.
tymes
.
500.
by
.
16.
and
thereof
commeth
.
40000.
for
your
diuidende
.
Therefore
diuide
.
40000.
by
.
5000.
and
you
shall
finde
.
8.
shillynges
,
so
muche
shall
coste
the
cariage
of
.
500.
waight
.
16.
miles
.
Or
otherwise
by
the
double
rule
of
three
,
that
is
to
saie
,
by
the
rule
of
three
at
twoo
tymes
:
firste
safe
if
.
50.
miles
dooe
paie
.
5.
shillinges
,
what
shall
.
16.
miles
paie
?
Multiplie
and
diuide
,
and
you
shall
finde
.
1.
shilling
⅗
.
Then
saie
againe
,
if
.
100.
waight
doe
coste
me
.
1
:
shilling
3/●
,
what
shall
.
500.
waighte
coste
?
Multiplie
and
diuide
,
and
you
shall
finde
.
8.
shillinges
,
as
before
.
When
the
cariage
of
.
800.
pounde
waight
of
Marchaundise
.
84.
miles
,
doeth
cost
me
.
6.
shillinges
,
how
many
miles
maie
I
haue
.
64.
pound
waight
carted
for
.
3.
shillinges
.
4
pence
.
Aunswere
,
by
the
second
part
of
the
rule
of
3.
compounde
:
saie
if
〈◊〉
|
84/1
|
6/1
|
64/1
|
3
⅓
.
Then
multiplie
the
fowerth
nomber
64/1
by
the
third
nomber
⅙
,
and
therof
commeth
〈…〉
for
your
diuisor
.
Likewise
multiplie
.
3⅓
by
100/1
,
and
by
14/1
,
and
you
shal
haue
in
the
product
24000/●
.
then
diuide
●●000/3
by
384/1
and
you
shall
find
.
72.
miles
11/12
of
a
mile
.
So
many
miles
shal
the.
64.
pounde
waight
be
caried
,
for
three
shillinges
.
4.
pence
.
Otherwise
by
the
rule
of
.
3.
at
twoo
tymes
:
first
saie
,
if
.
100.
waight
do
cost
me
.
6.
shillinges
,
what
shall
.
64.
p●und
waight
cost
?
Multiplie
and
diuide
.
and
you
shall
finde
.
3.
shillinges
21/25.
Then
saie
,
if
3
21/25.
hee
paied
for
.
84.
miles
carriage
:
for
how
many
miles
shal
〈◊〉
.
shillynges
⅓
bee
paied
?
Multiplie
and
diuide
,
and
you
shall
finde
72
miles
1●/12.
If
100
horses
in
100.
daies
do
spend
180.
quarters
of
otes
:
how
many
quarters
of
Otes
will
.
300.
horses
spende
in
.
150.
daies
?
Aunswere
:
by
the
firste
parte
of
the
rule
of
three
compounde
:
multiplie
.
180.
tymes
.
350.
by
.
150.
and
diuide
the
product
by
.
100.
tymes
.
100
and
you
shall
finde
.
945.
quarters
.
So
many
quarters
of
Otes
will
350.
horses
spende
in
.
150.
daies
.
Or
otherwise
by
the
rule
of
three
at
twoo
tymes
:
first
saie
,
if
100.
daies
doe
yelde
me
.
180.
quarters
of
Otes
:
what
shall
.
150.
daies
yelde
:
multiplie
and
diuide
,
and
you
shall
finde
.
270.
quarters
:
then
saie
againe
,
if
.
100.
horses
doe
spende
.
270.
quarters
of
otes
,
how
many
quarters
of
otes
wil.
350.
horses
spende
?
Multiplie
and
diuide
,
as
you
shall
finde
.
945.
quarters
as
before
.
¶
The
tenth
Chapiter
treateth
of
the
rule
of
fellowship
,
with
out
any
time
limited
.
THE
rule
of
Fellowship
is
thus
:
you
must
sette
doune
eche
mannes
somme
of
money
that
he
laieth
into
companie
,
euery
one
directly
vnder
the
other
,
the
whiche
you
shall
adde
altogether
,
and
the
totall
somme
of
all
their
whole
stocke
beyng
thus
assembled
,
shal
be
your
common
diuisor
,
to
the
findyng
out
of
euery
mannes
part
of
the
gaine
.
Then
shall
you
multiplie
the
gaine
,
or
els
the
losse
,
by
eche
mannes
portion
of
money
that
be
laied
in
,
and
diuide
the
productes
by
the
saied
diuisor
:
so
shall
you
haue
in
your
quotient
euery
mannes
parte
of
the
gaine
,
or
els
of
the
losse
,
if
any
thyng
be
lost
.
¶
Example
.
1.
Twoo
Marchauntes
haue
made
companie
together
,
the
firste
laied
in
fiue
hundred
pounde
.
The
second
put
in
three
hundred
pounde
,
and
with
occupiyng
thei
haue
gained
.
64.
pound
:
I
demaunde
how
muche
eche
man
shal
haue
of
thesame
gaines
,
accordyng
to
the
money
that
he
laied
in
.
Aunswere
:
Adde
.
500.
and.
300.
bothe
together
,
which
are
the
percelles
that
thei
laied
in
,
and
thereof
commeth
.
800.
for
your
diuisor
:
then
saie
by
the
rule
of
three
,
if
300
.li.
(
whiche
is
their
stocke
)
do
gain
64
.li.
what
shall
.
500
.li.
gain
?
(
which
is
the
first
mannes
money
that
he
laied
in
)
multiplie
and
deuide
,
and
you
shal
finde
.
40
.li.
for
the
first
mans
parte
of
the
gaine
:
then
saye
if
.
800.
giue
.
64.
what
will
.
300.
giue
?
Multiply
and
diuide
,
and
you
shall
finde
24
.li.
for
the
seconde
mannes
parte
of
the
gaine
.
Or
otherwise
,
put
500
.li.
which
is
the
first
mans
money
that
he
laied
in
,
ouer
the
800
li.
whiche
is
the
whole
,
stocke
,
and
you
shal
haue
509/800
which
being
abbreuiated
,
do
make
⅝
,
and
suche
part
of
the
gain
shal
the
first
man
take
,
that
is
to
say
⅝
of
64
.li.
which
is
40
.li.
And
consequentlie
,
by
the
same
manner
,
the
seconde
shal
take
the
⅜
of
64.
which
is
24.
pound
for
his
part
of
the
gaine
as
before
.
2.
Twoo
Marchauntes
haue
compainied
together
,
the
first
put
in
640
.li.
and
he
taketh
⅝
partes
of
the
gaine
.
I
demaunde
what
the
seconde
Marchaunte
layed
in
?
Aunswere
.
Seeyng
that
the
first
Marchaunt
taketh
⅝
of
the
gaine
,
it
followeth
that
the
second
must
haue
⅜
which
is
the
rest
,
and
ther fore
saie
by
the
rule
of
thrée
,
if
⅝
of
the
gaine
,
whiche
the
first
manne
taketh
,
did
saie
into
the
stocke
〈◊〉
.
How
muche
shall
the
⅜
of
the
gaine
laie
in
,
whiche
is
the
seconde
mannes
gaine
?
Multitiplie
and
diuide
,
and
you
shall
finde
384.
pounde
,
so
muche
ought
the
seconde
man
to
laie
into
companie
.
3.
Twoo
Marchauntes
haue
companied
together
,
the
first
manne
laied
in
.
640.
pounde
,
and
the
seconde
hath
laied
in
so
muche
,
that
he
muste
haue
60.
pounde
for
his
part
of
.
100.
pounde
whiche
thei
haue
gained
.
I
demaunde
howe
muche
the
seconde
man
did
laie
into
companie
?
Answere
:
seyng
that
the
seconde
man
taketh
.
60.
pounde
of
the
gaine
:
it
followeth
that
the
firste
must
haue
but.
40.
pounde
.
Therfore
saie
by
the
rule
of
.
3.
if
.
40.
pounde
doe
laie
in
.
640.
pounde
,
what
shall
.
60.
pounde
laie
in
?
Multiplie
and
diuide
,
and
you
shall
finde
.
260
.li.
so
muche
did
the
seconde
Marchaunt
laie
in
.
4.
Twoo
Marchauntes
haue
companied
together
,
the
firste
laied
in
.
83.
pounde
6.
shillinges
.
8
:
pence
,
the
seconde
put
in
.
170.
duckettes
:
and
thei
haue
gained
100.
li
of
that
which
the
first
man
must
haue
60
pounde
.
I
demaund
what
the
ducket
was
worth
?
Answer
seyng
that
the
first
man
must
haue
60
pounde
,
it
followeth
that
the
seconde
muste
haue
.
40.
pounde
,
therefore
saie
by
the
rule
of
three
,
if
.
60.
pounde
of
gaine
that
the
first
man
taketh
did
lay
in
.
83
.li.
6
s.
8.
pence
of
principall
,
how
muche
shall
.
40
.li.
of
gain
put
it
,
mulplie
and
diuide
,
and
you
shall
finde
55
.li.
5/9
:
so
muche
are
the
170.
duckets
worth
.
Then
put
55
.li.
5/9
into
shillings
and
you
shall
haue
1111.
s.
5/9
then
to
knowe
what
the
ducket
is
worth
,
saie
by
the
rule
of
thrée
,
if
a
〈◊〉
giue
1111.
〈◊〉
,
what
will
〈◊〉
giue
?
Multiplie
and
deuide
,
and
you
shall
finde
6.
s.
6.
d.
12/51
,
so
muche
is
the
ducket
worth
.
5.
Twoo
Marchauntes
haue
companied
together
,
the
second
man
laied
in
more
by
30.
poundes
then
did
the
firste
man
:
and
they
gained
120.
poundes
of
the
whiche
the
firste
man
ought
to
haue
50.
pounds
.
I
demaund
what
eche
of
them
did
laie
in
.
Aunsware
,
from
120.
pound
abate
50.
pounde
and
ther
resteth
70.
pound
for
the
second
mans
parte
:
so
that
by
thys
meanes
the
seconde
manne
(
because
he
laide
in
30.
pound
more
then
the
first
man
did
)
taketh
20
.l.
more
of
the
gain
:
&
therfore
say
by
the
rule
of
3.
if
20
.li.
of
gaine
did
laie
in
.
30.
pounde
of
principall
,
how
muche
shall
.
50.
pounde
laie
in
Multiplie
and
diuide
,
and
you
shall
finde
.
75.
pounde
,
so
muche
did
the
first
man
laie
in
,
and
consequently
the
seconde
laied
in
.
105.
pounde
.
6.
Twoo
Marchauntes
haue
companied
together
,
the
second
hath
laied
in
twise
so
muche
as
the
first
man
did
,
and.
10.
pounde
more
:
and
thei
gained
100.
pounde
,
of
the
whiche
,
the
firste
ought
to
haue
.
32.
pounde
for
his
part
:
I
demaunde
how
muche
eche
of
them
did
laie
into
companie
?
Answere
.
If
it
were
not
for
the.
10.
pound
that
the
seconde
manne
laied
in
more
:
he
should
haue
had
but.
64.
pounde
of
the
gaine
,
which
is
the
double
of
the
first
mannes
parte
.
But
because
he
laied
in
.
10
.li.
more
,
he
hath
●ower
pounde
more
of
the
gaine
,
and
therfore
saie
by
the
rule
of
.
3.
if
.
4.
pound
of
gaine
did
laie
in
10
pound
of
principall
(
whiche
was
ouer
and
aboue
the
double
of
the
first
mannes
laiyng
in
)
what
shall
32
.li.
of
gaines
laie
in
?
whiche
is
the
first
mannes
part
of
the
gaines
that
he
taketh
.
Multiplie
and
diuide
,
and
you
shall
finde
80
.li.
for
the
firste
mannes
laiyng
in
:
and
consequently
.
170.
pounde
for
the
second
mannes
portion
that
he
laid
in
.
4.
Twoo
Marchauntes
haue
companied
together
,
and
thei
haue
gained
100.
pound
of
the
which
the
first
must
haue
after
the
rate
of
10
vpon
the
100
pound
,
and
the
second
must
haue
after
the
rate
of
15
pound
vpon
the
100
.li.
I
demaunde
howe
muche
eche
of
theim
ought
to
haue
?
Answere
.
But
10
.li.
for
the
first
mannes
laiyng
in
,
and.
15
li.
for
the
second
mannes
laiyng
in
Adde
10
.li.
and.
15.
pound
together
,
and
thei
make
25.
pound
.
Then
put
10.
ouer
25
and
it
is
10/25
,
whiche
beyng
abreuiated
are
⅖
.
Therefore
he
that
taketh
10
li.
vpon
the
100.
it
must
haue
the
2/●
of
the
gaine
,
whiche
is
40
.li.
Then
put
.
1●
.
ouer
25.
and
it
is
15/25
,
which
being
abreuiated
are
3/●
.
Therfore
the
second
must
haue
3/●
of
the
100
.li.
which
is
.
60
.li.
8.
Twoo
Marchautes
haue
companied
together
,
the
first
laied
in
46
.li.
18.
shilling
.
and
the
seconde
laied
in
33.
pounde
2.
s.
so
thei
haue
gained
30.
pound
.
I
demaunde
how
much
euerie
man
shal
haue
for
his
part
of
the
gain
Answere
:
Adde
46.
pound
18.
shilling
and
33.
pound
2.
shilling
both
togither
and
you
shall
finde
80.
pound
for
your
common
diuisor
:
then
saie
if
80.
pound
which
is
all
their
stocke
do
gain
30
.li.
what
wil
46.
6/10
gaine
,
whiche
is
the
firste
mans
laiyng
in
:
Multiplie
and
deuide
,
and
you
shall
finde
17.
pounde
11.
shillyng
9.
pence
for
the
firste
mans
part
of
the
gaine
.
Then
saie
again
,
if
80.
pound
do
gain
30.
pound
what
wil
33.
pound
1/10
gain
,
whiche
was
the
seconde
mans
laiyng
in
:
multiplie
&
deuide
,
and
you
shall
finde
12
.li.
8.
s.
3.
d.
for
the
seconde
mans
part
of
the
gain
.
And
after
the
same
maner
shal
you
doe
,
in
case
there
were
three
or
foure
Marchauntes
that
woulde
companie
together
:
Addyng
al
their
summes
of
money
(
which
thei
lay
into
the
stocke
)
into
one
totall
summe
:
whiche
shal
be
your
common
deuisor
:
and
then
work
with
the
reste
,
as
is
taught
in
the
former
questions
of
the
rule
of
company
.
9.
Three
Marchautes
haue
companied
togither
,
the
first
laide
in
I
know
not
how
muche
:
the
seconde
did
put
in
20.
peeces
of
clothe
,
and
the
thirde
hath
laide
500.
pounde
.
So
at
the
ende
of
their
companie
,
their
gaines
amounted
vnto
a
thousande
pounde
,
wherof
the
firste
manne
ought
to
haue
350.
pounde
,
and
the
seconde
muste
haue
foure
hundred
pounde
.
Now
I
demaunde
how
muche
the
first
man
did
laye
in
,
and
for
how
muche
the
twentie
péeces
of
clothe
were
put
into
companie
?
Aunswere
.
Seeyng
that
the
fyrste
and
the
seconde
Marchautes
must
haue
750
.li.
for
their
partes
of
the
gain
.
Then
the
thirde
manne
muste
haue
the
reste
of
the
thousande
pound
which
is
250
.li.
And
theerfore
saie
by
the
rule
of
.
3.
〈◊〉
250.
of
gaine
,
be
come
of
500.
pound
of
principall
:
of
how
muche
shall
come
350
pounde
of
gaine
?
whiche
the
firste
man
taketh
,
multiplie
and
diuide
,
and
you
shall
finde
700.
pound
.
So
muche
did
the
first
man
laie
in
then
saie
if
.
250
pound
of
gaine
be
come
of
.
500.
pound
of
principall
,
of
how
muche
will
come
400.
pounde
,
whiche
is
the
gain
that
the
second
man
taketh
.
Multiplie
and
diuide
,
and
you
shal
finde
.
800.
pound
For
so
muche
were
the.
20.
peeces
of
clothe
laied
into
companie
.
10.
Three
Marchauntes
haue
gained
.
100
.li.
the
firste
must
haue
the
½
,
the
seconde
must
haue
⅓
:
and
the
third
must
haue
¼
.
I
demaunde
how
muche
euery
manne
must
haue
of
the
gaine
?
Answere
:
reduce
½
,
⅓
,
¼
,
into
a
common
denomination
,
after
the
order
of
the
second
reduction
in
fractions
,
and
you
shall
finde
12/24
,
for
the
½
,
8/24
,
for
the
½
:
and
6/●●
,
for
the
¼
:
Then
take
twelue
for
the
firste
mannes
laiyng
in
,
8.
for
the
second
mannes
laying
in
:
and
6
for
the
thirde
mannes
laying
in
.
The
whiche
three
nombers
beinge
added
together
shall
be
your
common
diuisor
,
whiche
do
make
25.
Then
multiply
100
by
12
for
the
firste
man
,
by
8
,
for
the
second
man
,
and
by
6
for
the
third
man
.
And
diuide
euery
multiplication
by
26.
So
shall
you
fynde
46
li.
2/13
for
the
fyrste
mannes
part
of
the
gaine
.
30
.li.
10/13
for
the
second
mannes
parte
:
and
23
li.
1/1●0
for
the
third
mannes
parte
.
11.
Two
marchauntes
haue
gained
100
li.
the
firste
muste
haue
½
and
5
li.
more
:
the
second
must
haue
⅓
and
4
li.
more
:
I
demaunde
howe
muche
ethe
of
them
shall
haue
?
Aunswere
,
From
100
abate
5
and
4.
so
ther
wil
remain
91.
then
take
the
½
of
100
l.
which
is
50
li.
for
the
first
mans
laying
in
:
Likewise
,
take
⅓
of
100
li.
for
the
seconde
mans
laying
in
,
which
is
33
li.
⅓
.
Then
adde
50
li.
and
33
li.
⅓
togither
,
and
you
shall
haue
83
li.
⅓
for
youre
common
diuisor
,
then
multiply
91.
pound
by
50.
and
diuide
by
83.
⅓
:
and
thereof
commeth
54
pound
,
⅗
vnto
the
whiche
nomber
adde
5
,
and
all
is
59
.li.
⅗
for
the
first
mans
part
.
Likewise
multiplye
.
91.
by
33.
⅓
:
and
diuide
by
83.
⅓
:
and
you
shal
finde
36
li.
⅖
vnto
the
which
adde
4
:
and
you
shal
haue
fourty
pound
,
〈◊〉
for
the
seconde
mans
part
.
12.
Two
Marchauntes
haue
gayned
a
hundred
pound
,
the
firste
muste
haue
the
½
lesse
by
4
pound
,
the
second
must
haue
⅓
:
lesse
by
2
pounde
.
I
demaund
how
much
eche
of
them
shall
haue
?
Answere
,
Adde
4
&
2.
with
100.
&
they
make
106.
Then
take
as
before
is
saide
●0
pounde
,
for
the
first
man
,
&
33.
⅓
for
the
seconde
,
adde
them
bothe
togither
,
&
they
be
83.
⅓
whiche
shal
be
your
diuisor
.
Then
multiplie
106.
by
50.
and
diuide
the
product
by
83
⅓
,
so
thereof
commeth
63
.li.
⅗
.
From
the
whiche
abate
the
foure
pounde
lesse
that
the
fyrste
man
taketh
,
and
then
is
there
remaining
59.
pounde
,
⅗
for
his
parte
.
Likewise
multiplie
106
by
33.
⅓
and
diuide
by
83⅓
&
you
shall
finde
42
.li.
⅖
:
from
the
which
abate
2
.li.
lesse
and
there
remaineth
40.
pounde
,
⅖
for
the
seconde
mans
part
.
¶
The
rule
of
Felowship
with
time
.
THe
money
that
euery
man
layeth
in
,
muste
be
multiplied
by
the
time
that
it
remaineth
in
company
:
and
of
that
whiche
commeth
thereof
you
shall
make
their
newe
layings
in
for
eche
of
them
:
and
then
multiplye
the
gaines
by
euery
one
of
them
seuerally
,
the
whiche
you
shall
diuide
by
all
their
new
layings
in
adding
togither
,
and
you
shall
haue
proportionally
eche
mannes
parte
of
the
gaine
according
to
his
laying
in
.
¶
Example
.
1.
Two
Marchauntes
haue
companied
togither
,
the
first
hath
put
in
the
fyrst
of
Ianuary
450.
pounde
,
the
second
did
lay
in
the
2.
of
May
.
750.
pound
And
at
the
yeres
ende
,
they
had
gayned
100
li.
I
demaunde
howe
muche
eche
of
them
shall
haue
of
the
gayne
?
Answere
:
forasmuche
as
the
firste
did
put
450
.li.
the
fyrste
of
January
:
his
money
remained
in
company
12.
monethes
,
and
therefore
multiply
450.
by
12
monethes
,
and
therof
commeth
5400.
for
his
newe
laying
in
.
And
the
seconde
layed
in
his
7
,
0
li.
but
at
the
first
daye
of
Maye
:
so
that
his
money
remained
in
companye
but
8
monethes
.
Therefore
multiplye
his
750
li.
by
8.
and
therof
commeth
6000
for
his
new
laying
in
:
Thē●dde
5400.
with
6000.
and
they
make
11400
for
your
common
diuisor
:
Then
multiply
100
li.
which
is
the
gaynes
by
5400
,
and
diuide
the
product
by
11400.
and
therof
commeth
48
li.
7/19
for
the
first
mannes
part
of
the
gayne
.
Likewise
multiplye
100.
by
6000
,
and
diuide
the
producte
by
11400
,
and
you
shal
finde
52.
12/19
and
so
muche
must
the
second
man
haue
for
his
parte
of
the
gayne
.
2.
Two
marchaunts
haue
companied
togither
,
the
first
hath
put
in
the
first
of
Ianuary
640
.li.
The
seconde
can
lay
in
nothing
vntil
the
first
of
Aprill
.
I
demaunde
how
much
he
shall
then
laye
in
,
to
the
ende
that
he
maye
take
halfe
the
gaines
?
Aunswere
,
Multiply
640
li.
by
12.
monethes
that
his
money
abideth
in
the
companye
,
and
therof
commeth
7680
li.
for
his
layinge
in
.
And
so
muche
oughte
the
seconde
mannes
layinge
in
to
be
,
for
because
he
taketh
½
of
the
gaine
:
But
for
that
,
that
be
putteth
in
nothinge
vntill
the
first
of
Aprill
,
his
money
can
be
in
company
no
longer
than
9
monethes
.
And
therefore
diuide
8680
by
9
,
and
therof
commeth
753
li.
⅓
So
much
oughte
the
seconde
marchaunt
to
laye
in
the
first
of
Aprill
,
to
the
ende
that
he
maye
take
the
one
moy●y
of
the
gaynes
.
3.
Three
Marchayntes
haue
companied
togither
.
the
firste
layed
in
the
firste
of
Marche
100
li.
The
seconde
laide
in
the
first
of
Iune
so
muche
money
,
that
of
the
gaine
,
he
muste
haue
the
⅓
parte
:
and
the
thirde
laide
in
the
first
of
Nouember
so
muche
money
,
that
of
the
gaines
he
muste
haue
likewise
⅓
and
thei
continued
in
company
,
vntil
the
next
Marche
folowing
.
I
demaunde
howe
muche
the
seconde
and
the
thirde
Marchauntes
did
laye
in
?
Answere
,
Multiply
100.
which
the
firste
man
did
lay
in
,
by
12.
monethes
that
his
money
continued
in
companie
,
and
therof
commeth
1200.
for
his
laying
in
:
and
so
muche
ought
the
seconde
and
the
thirde
marchaunt
eche
of
them
to
lay
in
:
Bicause
they
parte
the
gaynes
by
thirdes
.
But
for
that
,
that
the
seconde
Marchaunt
putteth
in
nothinge
tyll
the
first
of
Iune
,
hys
money
can
bee
in
companye
but
nine
monethes
.
Therefore
diuide
1200.
by
nine
monethes
,
and
therof
commeth
133.
⅓
.
And
so
muche
ought
the
seconde
Marchaunt
to
laye
in
:
Then
,
forasmuch
as
the
thirde
Marchaunt
,
dyd
laye
in
nothing
vntil
the
first
of
Nouember
:
His
money
abideth
in
companye
but
the
space
of
foure
monethes
.
Therefore
diuide
1200.
by
4.
and
thereof
commeth
three
hundred
pounde
.
And
so
much
ought
the
thirde
marchaunt
to
lay
into
companye
.
4.
Three
Marchauntes
haue
companied
togither
,
the
fyrste
layde
in
the
fyrst
of
Ianuary
a
hundred
Duckettes
.
The
seconde
hath
layed
in
fyftye
pounde
,
the
fyrste
of
Marche
:
And
the
thirde
putte
in
a
Iewell
the
fyrste
of
Iulye
:
And
at
the
yeares
ende
,
they
had
gained
foure
hundred
crownes
:
of
the
whiche
,
the
fyrste
marchaunt
must
haue
fifty
crownes
,
and
the
seconde
muste
haue
80.
I
demaunde
what
the
Duc●et
was
worth
and
at
what
price
the
Iewell
was
valued
,
whiche
the
thirde
Marchaunte
layde
in
?
Aunswere
:
the
firste
mannes
money
is
1200
as
afore
is
sayde
,
and
he
taketh
50
crownes
of
the
gayne
:
therefore
say
,
if
fifty
crownes
of
gayne
be
come
of
1200
,
whice
was
his
stock
,
of
how
much
shall
come
80.
crownes
of
gaine
that
the
seconde
man
taketh
?
multiplye
and
diuide
,
and
you
shall
finde
1920.
for
the
second
marchaunts
laying
in
.
Then
say
again
,
if
50
crownes
be
come
of
1200.
stocke
:
of
howe
much
shal
come
270.
crownes
,
which
the
thirde
man
taketh
of
the
gayne
?
Multiply
and
diuide
,
&
you
shall
finde
6480.
for
the
third
marchauntes
laying
in
.
Then
diuide
1920
,
whiche
is
the
seconde
mannes
laying
in
,
by
10.
monethes
that
his
money
did
continue
in
company
,
and
you
shall
fynde
192
Duckets
,
which
are
worth
50
.li.
bicause
he
layed
in
50
li.
Then
diuide
192
Duckets
by
the
sayde
50
,
li.
(
being
reduced
into
shillinges
)
and
thereof
commeth
5.
shillings
2.
pence
,
½
.
So
muche
was
the
Ducket
worth
:
Finallye
,
diuide
6480.
(
whiche
is
the
third
mannes
layinge
in
)
by
6.
monethes
that
his
Iewel
remained
in
companye
,
and
you
shall
finde
1080
Duckets
:
and
for
that
price
was
the
Iewell
put
in
company
.
5.
Three
Marchauntes
haue
companied
togither
:
the
first
layed
in
the
first
of
Ianuary
100
li.
and
the
firste
of
April
he
hath
taken
backe
againe
20
.li.
The
second
hath
layed
in
the
firste
of
Marche
60
.li.
and
afterward
he
dyd
put
in
more
100
li.
the
first
of
August
.
The
third
layd
in
the
first
of
Iuly
150
.li.
And
the
first
of
October
he
did
take
backe
againe
50
li.
And
at
the
yeres
end
,
they
found
that
they
had
gained
160
li.
I
demaunde
how
muche
euery
man
shall
haue
?
Aunswere
,
Multiply
100
li.
which
the
first
man
layed
,
by
12
monethes
,
and
therof
commeth
1200.
li
from
that
nomber
abate
9
times
20
which
are
180.
and
there
wil
remaine
1020
,
for
the
first
mans
laying
in
.
Then
multiplie
60.
which
the
seconde
man
layde
in
,
by
ten
&
you
shall
haue
600
vnto
the
which
adde
5.
times
one
hundred
,
which
are
500.
so
all
amounteth
to
1100.
for
the
second
mans
laying
in
Afterwardes
,
multiplie
150.
pounde
,
which
the
thyrde
man
hath
layed
in
,
by
6.
monethes
,
and
therof
commeth
900.
from
the
which
nomber
abate
three
times
50.
and
they
are
150
:
so
there
resteth
750.
for
the
thyrde
mans
laying
in
.
Then
procede
wyth
the
reste
,
as
in
the
firste
Question
of
the
rule
of
felowship
with
tyme
,
in
adding
1020
,
1100.
and
750.
altogither
,
whiche
shall
be
your
Diuisor
:
Then
multiplie
160
by
1020.
by
1100.
and
by
750.
&
diuide
at
euery
time
by
your
Diuisor
,
which
is
by
all
theyr
layinges
in
added
togither
,
and
they
make
2870
,
so
you
shal
fynde
56.
241/287
:
for
the
fyrste
man
,
61.
93/●87
for
the
seconde
,
and
41.
233/287
for
the
thirde
man
.
6.
Two
Marchaunts
haue
companied
togither
,
the
fyrste
hath
put
in
960.
pounde
,
for
the
space
of
12.
monthes
,
and
he
ought
to
haue
8.
pounde
vpon
the
hundred
pound
of
the
gaine
.
The
second
hath
layed
in
1120
.li.
for
the
space
of
eight
monthes
,
&
he
ought
to
haue
after
12.
pounde
vpon
the
100.
pound
of
the
gaine
.
And
at
the
yeares
ende
,
they
haue
gained
eyght
hundred
pounde
.
I
demaunde
how
much
eche
of
them
shall
haue
of
the
gaine
.
Answere
,
multiplie
960.
that
the
first
man
did
lay
in
,
by
12.
monethes
,
and
the
product
thereof
,
multiplie
againe
by
8.
and
you
shall
haue
62160.
for
the
fyrst
mans
laying
in
:
then
multiplye
the
1120.
that
the
seconde
hath
layed
in
by
eyght
monethes
,
and
that
whiche
commeth
thereof
you
shall
multiplie
againe
by
12.
and
you
shal
finde
10●520
.
for
the
seconde
mans
layinge
in
,
then
proceede
with
the
rest
,
as
in
the
first
Question
of
the
Rule
of
Felowshippe
,
and
as
in
the
laste
Exaumple
,
and
you
shall
finde
3993/13
for
the
first
man
:
and
430
.li.
10/13
for
the
second
man
.
¶
The
rule
of
company
,
betwene
Marchauntes
and
their
Factours
.
7.
The
estimation
of
the
bodye
or
persone
of
a
Factour
,
is
in
suche
proportion
to
the
stocke
,
whiche
the
Marchaunt
layeth
in
:
as
the
gaine
of
the
sayd
Factour
is
vnto
the
gayne
of
the
sayd
Marchaunt
.
As
thus
:
if
a
Marchaunt
do
put
into
the
handes
of
hys
Factour
200
li.
to
employe
,
and
he
to
haue
halfe
the
profise
,
the
persone
of
the
sayd
Factour
shal
be
esteemed
200
li.
And
if
the
Factour
do
take
but
the
●/3
of
the
gaine
,
he
shoulde
haue
but
½
so
much
of
the
gaine
as
the
marchaunt
taketh
,
which
should
take
⅔
wherfore
the
persone
of
the
Factour
is
estemed
but
the
½
of
that
which
the
Marchaunt
layeth
in
,
that
is
to
say
100
li.
And
if
the
Factour
did
take
the
⅖
of
the
gaine
,
then
the
Marchaunt
shall
take
the
residue
,
which
are
⅗
of
the
gaine
wherefore
the
gayne
of
the
Mayster
vnto
that
of
the
Factoure
is
in
suche
proportion
as
3
vnto
2.
Then
if
you
will
knowe
the
estimation
of
the
persone
of
the
Factour
,
say
if
3
giue
me
2
what
wil
200
giue
?
Multiply
200
by
2
and
diuide
be
3
so
you
shall
finde
133
●/3
Otherwise
,
consider
that
the
Factour
taketh
the
⅗
of
that
whiche
the
Marchaunte
taketh
.
And
therfore
take
the
⅔
of
200
,
and
you
shall
fynde
133.
⅓
as
before
:
and
so
muche
is
the
persone
of
the
Factoure
esteemed
to
bee
worthe
.
8.
And
if
the
Marchaunt
should
deliuer
vnto
his
Factoure
200
.li.
and
the
Factour
would
laye
in
40
.li.
and
his
person
,
to
the
ende
he
might
haue
the
halfe
of
the
gaine
:
I
demaund
for
how
much
shall
his
person
be
esteemed
Ans.
abate
40
li.
from
200
li.
and
ther
will
remaine
160
.li.
And
at
so
much
shall
his
person
be
estemed
.
And
if
the
factour
woulde
take
the
⅔
of
the
gaine
,
his
person
with
his
40
pounde
,
shall
bee
esteemed
twise
as
muche
as
the
stock
that
the
marchant
layeth
in
,
which
shoulde
haue
but
⅔
of
the
gaine
:
for
⅔
vnto
⅓
,
is
in
double
proportion
.
Therefore
double
twoo
hundred
pounde
,
therof
commeth
400
.li.
from
the
which
abate
40
.li.
&
there
will
remaine
360
.li.
And
if
the
Factour
would
take
but
the
⅓
of
the
gaine
,
that
shall
bee
but
the
½
of
⅔
whiche
the
marchaunt
taketh
:
then
the
estima●ion
of
his
person
,
with
his
layinge
in
should
be
esteemed
but
the
halfe
of
the
whiche
the
marchaunt
layeth
in
:
take
therfore
the
●/2
of
200
.li.
whiche
is
100
.li.
from
the
which
abate
fourty
pound
and
the
rest
whiche
is
60
.li.
is
the
estimation
of
his
person
.
9.
If
it
so
chaunce
that
for
to
make
traffick
of
240
.li.
the
person
of
the
factour
should
be
so
estemed
,
that
he
shoulde
haue
but
the
¼
of
the
gaine
,
and
yet
he
would
haue
the
⅔
,
I
demaunde
howe
much
he
shoulde
put
in
of
readye
money
,
besides
his
person
?
Aunswere
,
seing
that
his
person
gaineth
the
¼
,
all
the
whole
layinge
in
,
shall
gaine
the
rest
that
is
to
saye
the
¾
:
nowe
for
bycause
¼
is
the
⅓
of
¾
therfore
his
person
shal
be
estemed
the
⅓
of
all
the
layinge
in
.
Take
then
the
⅓
of
240.
and
you
shall
haue
80.
for
the
estimation
of
his
person
,
and
for
that
,
that
he
wyll
haue
the
halfe
of
the
gaine
,
you
shal
adde
80.
with
240
.li.
and
therof
commeth
320.
of
the
which
take
the
halfe
,
which
is
160.
and
from
the
same
you
shal
abate
the
80.
and
there
wyll
remaine
other
80.
which
he
ought
to
lay
in
of
readye
money
,
and
the
marchaunt
muste
lay
in
the
ouerplus
,
whiche
amounteth
to
160
.li.
10.
A
marchaunt
hath
deliuered
to
his
Factour
1200
.li.
to
gouerne
them
in
the
trade
of
Marchandise
vpon
such
condition
that
hee
for
hys
seruyce
shall
haue
the
⅓
of
the
gaine
if
any
thing
be
gayned
,
or
of
the
losse
if
any
thing
be
lost
:
I
demaund
for
how
much
hys
person
was
estemed
?
Answere
,
seeing
that
the
Factoure
taketh
the
⅓
of
the
gaine
,
hys
persone
ought
to
bee
esteemed
as
muche
as
½
of
the
stock
whiche
the
Marchaunte
layeth
in
,
that
is
to
say
the
½
of
1200
li.
which
is
900
l.
The
reason
is
,
bycause
the
⅓
of
the
gayne
that
the
Factoure
taketh
,
is
the
●/2
of
the
⅔
of
the
gaine
that
the
Marchaunt
taketh
.
11.
A
Marchaunt
that
deliuereth
vnto
his
Factour
1200
li.
and
the
Factour
layeth
in
500
li.
and
his
person
:
Now
,
bicause
he
laieth
in
500
li.
and
his
persone
,
it
is
agreed
betwene
them
that
be
shal
take
the
⅖
of
the
gain
:
I
demaunde
for
how
much
his
persone
was
esteemed
?
Aunswere
,
Forasmuche
as
the
Factour
taketh
the
⅖
of
the
gaine
,
he
saketh
the
⅔
of
that
which
the
March●ūt
taketh
,
for
⅖
are
the
⅔
of
⅗
:
and
therefore
the
Factours
laiyng
in
ought
to
be
800.
pound
,
which
is
the
⅔
of
1200.
pounde
,
that
the
marchaunt
laied
in
:
Then
abate
500.
pounde
,
whiche
the
Factour
did
lay
in
,
from
800.
pounde
,
whiche
should
be
his
whole
stock
and
there
remaineth
thrée
hundred
pound
for
the
estimation
of
his
persone
.
12.
More
,
a
marchaunt
hath
deliuered
vnto
hys
factoure
a
thousande
pounde
vpon
suche
condition
,
that
the
factour
for
his
paines
and
seruice
,
shal
haue
the
gaines
of
2.
hundred
pound
,
as
though
he
had
laide
so
muche
in
of
redy
money
:
I
demand
what
the
portion
of
the
gaine
,
the
saide
Factour
shall
take
?
Aunswere
:
Sée
what
parte
the
two
hundreth
pounde
(
whiche
the
Factour
layed
in
)
is
of
one
thousande
twoo
hundreth
whiche
is
the
whole
stocke
of
their
companie
,
and
you
shal
finde
that
it
is
the
⅙
,
and
such
parte
of
the
gaine
shall
the
Factour
take
.
But
in
case
that
in
making
the
couenauntes
,
it
were
agreed
that
the
Factour
should
haue
the
gaine
of
two
boundred
pounde
of
the
stocke
,
whiche
the
marchaunte
layeth
in
,
that
is
to
saie
of
the
thousande
pounde
.
Then
shoulde
the
Factour
take
the
⅕
part
of
the
gaine
.
For
twoo
hundred
pounde
is
the
⅕
of
a
thousande
pounde
.
¶
The
11
Chapiter
treatteth
of
the
Rules
of
barter
.
TWoo
Marchaunts
will
chaunge
their
marchandise
,
the
one
with
the
other
.
The
one
of
them
hath
clothe
of
7.
s.
1.
d.
the
yarde
to
selle
for
readie
money
,
but
in
barter
he
wil
sel
it
for
8.
s.
4.
d.
The
other
hath
Sinamon
of
4.
s.
7.
d.
the
li.
to
sell
for
readie
money
.
I
demaunde
how
he
shall
sell
it
in
barter
to
the
ende
he
be
no
loser
?
Aunswere
,
saie
,
if
7.
1/12
(
whiche
is
the
price
that
the
yarde
of
clothe
is
worth
in
readie
money
(
be
solde
in
barter
for
8.
⅓
for
what
shal
4.
7/12
be
solde
in
barter
which
4.
7/12
is
the
price
that
the
l.
of
Synamon
is
worth
in
readie
money
,
reduce
the
whole
numbers
in
to
their
broken
,
and
then
multiply
and
diuide
,
and
you
shall
finde
5.
s.
4.
d.
●2/1●
parts
,
of
a
penie
,
and
for
so
much
shall
he
sell
the
pounde
of
Synamon
in
barter
.
2.
Two
Marchaunts
will
chaunge
their
marchaundise
the
one
with
the
other
,
the
one
of
them
hath
Chamlets
of
two
pounde
18.
s.
4.
pence
the
péece
to
sell
for
ready
money
,
and
in
barter
he
will
sell
the
péece
for
4
.li.
3
,
s.
4.
d.
the
other
hath
fine
cappes
of
35.
s.
10.
d.
the
dossen
to
sell
in
barter
.
I
demaund
what
the
dosen
of
caps
did
cost
in
redy
money
?
Answere
:
saie
if
4
.li.
3.
s.
4.
d.
whiche
is
the
ouerprice
of
the
péece
of
Chamlet
,
become
of
2
.li.
18.
s.
4.
pence
whiche
was
the
iust
price
of
the
same
,
of
what
shal
come
35.
s.
10.
d.
whiche
is
the
ouerprice
of
the
dossen
of
cappes
?
Multiplie
and
deuide
,
and
you
shall
finde
25.
shil.
1.
d.
and
so
muche
are
the
dossen
of
caps
worthe
in
redie
money
.
3.
Two
Mharchants
will
chaunge
their
marchaundise
the
one
with
the
other
:
the
one
of
them
hath
Fustians
of
18.
s.
4.
d.
the
peece
to
sell
for
readye
money
,
and
in
barter
hee
will
sell
the
pece
for
26.
s.
8.
pence
.
The
other
hath
tapistrie
of
15.
d.
the
ell
to
sel
for
readie
money
,
and
in
barter
he
will
sell
it
for
20.
pence
the
ell
:
I
demaunde
whiche
of
them
gaineth
,
and
how
muche
vppon
the
hundred
pounde
of
money
?
Aunswere
:
saye
if
18.
s.
⅓
(
which
is
the
iuste
price
of
the
peece
of
F●itian
)
bee
solde
in
barter
for
16.
s.
⅔
:
for
howe
muche
shall
1.
s.
¼
(
whiche
is
the
iuste
price
of
the
ell
of
Tapistry
)
be
solde
in
barter
?
Multiplie
and
diuide
,
and
you
shall
finde
21.
d.
9/11.
And
he
doeth
ouersel
it
but
for
20.
d.
so
that
of
21.
d.
9/11
:
bée
maketh
but
20.
d.
And
therefore
saye
by
the
rule
of
three
,
yf
the
seconde
marchaunte
,
of
21
9/11
,
do
make
but
20/1
how
muche
shall
he
lose
vpon
the
100/1
?
Multiplie
and
deuide
,
and
you
shall
find
21.
⅔
,
the
which
being
abated
from
a
hundred
there
wil
remaine
8.
⅓
.
And
after
the
rate
of
8.
⅓
.
doeth
the
seconde
marchant
lose
vpon
the
100.
And
consèquently
the
firste
marchant
,
of
20.
d.
maketh
21.
d.
9/11
:
and
therfore
say
again
by
the
rule
of
thre
,
if
the
first
marchant
of
10/11
,
doe
make
21.
9/11
how
much
shall
he
gaine
vpon
100/1
?
Multiplie
and
diuide
,
and
you
shall
finde
102
.li.
1/11.
Thus
the
firste
gaineth
after
the
rate
of
nine
pounde
.
1/11
vpon
the
hundred
pounde
of
money
.
For
your
better
vnderstandyng
of
these
questions
,
you
muste
note
that
when
one
Marchante
gaineth
of
an
other
after
the
rate
of
ten
pound
vpon
the
hundred
pounde
hee
gaineth
the
〈◊〉
of
his
owne
principall
,
and
the
other
whiche
loseth
after
the
rate
of
9.
1/11
vppon
the
hundred
he
loseth
the
1/11
of
his
principal
.
And
it
may
be
proued
thus
:
When
one
marchaunt
will
sell
hys
wares
vnto
an
other
,
whiche
wares
stand
him
but
in
100
.li.
and
he
wil
sel
them
for
110
.li.
he
of
his
100
.li.
maketh
110
.li.
wherfore
he
gaineth
after
10.
li
vpon
the
100.
whiche
is
the
1/1●
of
his
principall
,
and
the
other
whiche
byeth
wares
for
110
.li.
that
coste
but
110
.li.
of
the
110.
pound
he
maketh
but
100
.li.
And
therfore
say
by
the
rule
of
three
,
if
110.
become
of
100.
of
howe
muche
shall
come
100
?
Multiplie
and
deuide
,
and
you
shall
finde
90.
10/11
,
the
whiche
abate
from
100
:
and
there
resteth
9.
1/11
whiche
is
the
1/11
of
his
principall
that
the
second
leseth
vpon
the
100.
as
afo●●
is
saied
.
And
therefore
,
who
so
that
will
know
what
one
marchant
gaineth
of
another
,
either
after
the
rate
of
tenne
vpon
the
hundred
,
whiche
is
the
1/11
of
hys
principall
,
or
else
after
the
rate
of
twentie
vpon
the
hundred
which
is
the
⅕
,
or
of
anye
other
parte
,
and
that
he
would
likewise
knowe
what
parts
the
other
loseth
of
his
principall
:
hee
muste
take
for
the
numeratour
of
the
broken
number
of
hym
that
loseth
,
as
much
as
for
him
that
gaineth
,
then
adde
the
numerator
and
the
denominator
(
of
the
broken
number
of
hym
that
gaineth
)
both
togither
,
and
make
therof
the
denominator
of
the
broken
number
of
hym
that
loseth
,
and
then
shall
you
haue
the
parte
of
hym
that
loseth
,
as
by
example
,
of
him
that
gaineth
after
ten
pound
vpon
the
100
.li.
which
is
the
1/10
of
his
principall
:
take
the
numerator
whiche
is
1.
and
make
that
the
numeratoure
of
the
broken
number
of
hym
that
loseth
,
then
adde
1.
whiche
is
the
numerator
of
the
fraction
of
hym
that
gaineth
with
tenne
,
whiche
is
his
denominator
,
and
you
shall
haue
11.
for
the
denominatour
of
the
fraction
of
hym
that
loseth
.
Then
put
one
ouer
the
11.
and
you
shal
haue
1/11.
Thus
it
appeareth
when
one
marchaunt
gaineth
of
an
other
after
ten
vpon
the
hundred
,
he
gaineth
the
1/10
of
his
principall
,
and
the
other
loseth
9.
1/11
whiche
is
the
1/11
of
hys
principall
.
And
if
he
woulde
gaine
after
twentie
vppon
the
hundred
whiche
is
the
⅕
of
hys
principall
,
the
other
shoulde
lose
sixtine
⅖
whiche
is
the
⅙
of
hys
principall
,
and
so
is
to
bee
vnderstande
of
al
other
fractions
.
4.
Two
marchaunts
wil
chaunge
their
marchaundise
the
one
with
the
other
,
the
one
of
them
hath
Seies
of
20.
s.
&
10.
d.
the
peece
to
sel
for
readie
moneye
,
and
in
barter
he
will
sell
the
peece
for
23.
s.
4
d.
and
yet
he
wil
gain
moreouer
after
ten
pounde
vppon
the
hundred
pounde
.
The
other
hath
wol
of
50.
s.
the
hundred
to
sell
for
readye
money
.
I
demaunde
how
he
shall
sell
the
C.
of
woll
in
barter
;
Answere
:
say
if
20.
s.
10.
d.
whiche
is
the
iuste
price
of
the
peece
of
Seye
,
be
solde
in
barter
for
23.
s.
4.
d.
for
how
much
shall
50.
s.
(
which
is
the
iust
price
of
y●
C.
of
woll
)
be
solde
in
barter
?
Multipliye
&
diuide
,
and
you
shal
finde
56.
s.
Then
for
because
the
first
marchant
gaineth
after
10.
li
vpon
the
C
.li.
he
maketh
of
his
c
.li.
110
.li.
and
consequently
the
second
marchant
maketh
of
110
.li.
but
100
.l.
And
therefore
saie
,
if
the
seconde
marchaunt
of
110.
doe
make
but
100
how
much
shal
he
mak
of
56
:
Multiply
and
diuide
&
you
shal
finde
50.
s.
20.
d.
10/12
of
a
penie
,
and
for
so
muche
shall
he
selle
the
hundred
of
woll
in
barter
.
5.
More
,
two
Marchauntes
wyll
chaunge
their
marchaundise
,
the
one
with
the
other
,
the
one
of
theim
hath
Taffeta
,
of
16
crownes
the
peece
to
sell
for
readie
money
,
and
in
barter
he
will
sel
the
peece
for
twenty
crownes
,
and
yet
he
wyll
gaine
moreouer
after
ten
pounde
,
vpon
the
hundred
pound
.
The
other
hath
ginger
of
3.
s
9.
d.
the
pounde
waight
,
to
sell
in
barter
.
I
demaunde
what
the
pounde
did
coste
in
readie
moneye
?
Answere
:
saye
if
twenty
crownes
which
is
the
surprice
of
the
peece
of
Laffeta
,
become
of
16.
crownes
the
iuste
price
,
of
how
muche
shall
come
.
3.
s.
9.
d.
whiche
is
the
price
of
the
ouerselling
the
pounde
of
Ginger
?
Multiply
and
diuide
and
you
shal
finde
3.
s.
Then
,
for
because
that
the
Marchaunt
of
Laffeta
will
gaine
after
the
rate
of
ten
vppon
the
hundred
▪
say
if
100.
doe
giue
100.
what
shal
3.
s.
giue
?
Multiplie
and
deuide
and
you
shall
finde
thre
shillinges
thre
pence
〈◊〉
and
so
much
did
the
pound
of
Ginger
coste
in
readite
money
.
6.
More
twoo
marchauntes
will
chaunge
their
marchaundise
the
one
with
the
other
,
the
one
of
theim
hath
Worsteds
of
25
shil
.
the
péece
to
sel
for
ready
money
,
and
in
barter
he
wil
sell
the
péece
for
33.
shill.
4.
pence
and
yet
he
loseth
after
ten
vpon
the
hundred
:
the
other
hath
waxe
of
3
pound
6.
shil.
8.
pence
the
hundred
to
sell
for
readye
money
.
I
would
know
how
he
should
sell
his
ware
in
barter
?
Aunswere
:
say
if
25.
shil
.
whiche
is
the
iuste
price
of
the
peece
of
Worsted
be
solde
in
barter
for
three
and
thirtie
shill.
4.
pence
,
for
how
much
shall
thre
pound
6.
shil.
8.
d.
be
solde
,
whiche
is
the
iust
price
of
the
hundred
of
waxe
.
Multiply
and
diuide
,
and
you
shall
find
foure
pound
4/9
whiche
is
eight
shillinges
ten
pente
,
⅔
then
for
because
that
the
Marchaunt
of
Worsteds
,
loseth
after
ten
vppon
the
hundred
:
Of
a
hundred
he
maketh
but
sixtie
.
And
thererefore
,
saie
:
If
ninetie
geue
a
hundred
what
giueth
foure
pounde
.
4/9
?
Multiply
and
diuide
,
and
you
shall
finde
foure
76/81
whiche
is
worth
eightine
shillinges
nine
pence
5/27
,
and
for
so
much
shall
he
sell
the
one
hundreth
of
Waxe
in
barter
.
7.
More
,
twoo
Marchauntes
wyll
chaunge
their
marchaundise
the
one
with
the
other
,
the
one
of
them
hath
Worsteds
of
5.
pounde
six
shillinges
,
eight
pence
the
peece
to
sell
for
readie
moneye
,
and
in
barter
he
will
sell
the
peece
for
6.
pounde
,
13.
shillinges
.
4.
d.
and
yet
he
loseth
after
tenne
vpon
the
hundred
,
and
the
other
hath
Muske
of
two
shillings
.
nine
pence
⅓
the
pound
waight
,
to
sell
in
barter
?
I
demaunde
what
the
pound
did
cost
in
readie
money
?
Answere
:
say
if
6
pound
.
⅔
whiche
is
the
ouerprice
of
the
peece
of
Worsted
,
become
of
5.
pound
,
1/9
whiche
is
the
iust
price
of
the
same
,
of
how
much
shall
come
twoo
shillinges
9.
pence
.
⅓
.
Multiplye
and
deuide
and
you
shall
finde
2.
2
/
9.
whiche
is
twoo
pence
⅔
then
for
because
that
the
marchaunte
of
Worsteds
loseth
after
tenne
vpon
the
hundred
,
of
a
hundred
he
maketh
but
90.
and
therfore
saie
if
a
hundred
giue
but
90.
how
muche
shall
2.
s.
2/9
giue
?
Multiply
and
diuide
and
you
shal
find
twoo
shillynges
and
so
muche
coste
the
pound
of
Muske
in
readie
money
.
Other
Rules
of
Barter
,
wherein
is
giuen
some
parte
in
readie
money
.
WHen
a
Marchaunt
ouerselleth
hys
marchaundise
and
he
will
giue
also
some
part
of
hys
ouerprice
in
readie
money
as
in
the
½
the
⅓
or
the
¼
&c.
He
must
substract
the
same
part
of
money
from
the
iuste
price
,
and
also
from
the
ouerprice
of
his
marchandise
:
and
the
two
numbers
that
remaine
after
the
substraction
is
made
,
shal
be
that
twoo
firste
numbers
in
the
rule
of
three
and
the
iuste
price
of
the
second
marchant
shal
be
the
thirde
,
to
know
how
much
he
shal
ouersel
the
part
of
his
marchandise
8
:
Two
Marchaunts
wil
chaunge
their
marchaundise
the
one
with
the
other
the
one
of
theim
hath
fine
woll
at
fiue
pounde
the
hundreth
,
to
sell
for
readie
moneye
,
and
in
barter
hee
will
sell
it
for
sixe
pounde
,
and
yet
hee
will
haue
the
●/3
in
readye
money
.
The
other
hath
clothe
of
13.
shillinges
four
pence
to
sel
for
ready
money
.
I
would
knowe
howe
he
shall
sell
the
same
in
barter
?
Aunswere
:
take
the
⅓
of
6
.li.
whiche
is
the
ouerprice
of
the
100.
of
wolle
,
and
you
shall
haue
2.
pound
the
whiche
abate
from
5
.li.
whiche
is
the
iust
price
of
the
100.
of
woll
and
from
6
.li.
which
is
the
ouerprice
,
and
there
shal
rest
3
.l.
and
4
.l.
for
the
2
first
numbers
in
the
rule
of
thrée
,
then
take
1●
.
s.
4.
d.
whiche
is
the
iust
price
of
a
yarde
of
clothe
for
the
thirde
number
:
Then
multiplie
and
deuide
and
you
shall
finde
17.
shill.
9.
d.
⅓
.
for
so
muche
shall
the
seconde
selle
his
clothe
in
barter
.
9.
More
,
twoo
Marchauntes
will
chaunge
their
marchaundise
the
one
with
the
other
,
the
one
of
them
hath
waxe
of
thrée
pound
.
6.
s.
8.
d
the
C.
to
sell
for
readie
money
,
and
in
barter
he
will
sell
the
same
for
4
.li.
3.
s.
4.
d.
and
yet
he
will
haue
the
¼
in
ready
money
,
and
the
other
hath
fine
Crimson
sattine
of
15.
s.
the
yarde
to
sell
in
barter
.
I
demaund
what
it
is
worth
in
ready
money
.
Aunswere
.
Take
the
¼
of
4
.li.
3.
s.
4.
d.
and
abate
it
from
4
.li.
3.
s.
4.
d.
and
from
thrée
pounde
6.
s.
8.
pencs
,
and
there
resteth
3
.li.
2.
s.
6.
d.
and
2
.li.
5.
shillinges
10.
pence
for
the
two
first
numbers
in
the
rule
of
thrée
,
and
15.
s
for
the
thirde
number
whiche
is
the
ouerprice
of
the
yard
of
sattine
.
Then
multiply
and
diuide
,
and
you
shal
find
11.
shillinges
.
And
soe
muche
did
the
yarde
of
sattine
cost
in
readie
money
.
10.
Two
Marchants
will
chaunge
their
marchandise
the
one
with
the
other
,
the
one
of
them
hath
tinne
of
50.
shillings
the
hundred
to
sell
for
ready
money
,
and
in
barter
he
wil
sell
it
for
three
pounde
6.
s.
8.
d
and
he
wil
gaine
after
ten
vppon
the
hundred
,
and
yet
he
wil
haue
the
one
half
in
readie
money
:
and
the
other
hath
leade
of
3.
half-pence
the
li.
to
sell
for
readie
money
.
I
demaund
how
he
shall
sell
the
pounde
in
barter
?
Aunswere
:
See
first
at
tenne
vpon
the
hundred
,
what
the
thre
pound
⅓
will
come
vnto
,
and
you
shall
finde
that
they
will
come
to
3
.li.
⅔
,
which
is
13.
s.
4.
d.
of
the
which
,
the
halfe
which
he
demaundeth
in
readie
money
,
is
36
shillinges
and
8.
pence
the
whiche
beyng
abated
from
fiftie
shillinges
,
and
also
from
three
pounde
13.
shillinges
four
pence
,
there
shal
rest
13.
shillings
foure
pence
,
and
one
pound
sixtine
.
s.
eight
pence
,
for
the
twoo
first
numbers
in
the
rule
of
three
,
whiche
you
muste
put
al
into
halfepence
,
and
thre
halfepence
for
the
thirde
number
,
and
then
multiplye
and
diuide
,
and
you
shall
finde
4.
pence
⅛
,
and
for
so
muche
shall
hée
sell
the
pounde
of
leade
in
barter
.
11.
More
twoo
marchauntes
will
chaunge
their
marchaundise
the
one
with
the
other
,
the
one
of
theim
hath
steele
of
16.
s.
8.
d.
the
hundred
waight
to
sell
for
readie
money
,
and
in
barter
he
wil
sell
it
for
25.
s.
and
yet
he
loseth
after
tenne
vpon
the
hundred
,
but
hee
will
haue
the
½
in
readie
moneye
,
the
other
hath
yron
of
6.
shill.
8.
pence
the
dundred
to
sell
in
barter
,
I
demaunde
what
it
did
coste
in
readye
money
?
Aunswere
:
saie
if
a
hundred
come
but
to
90.
how
muche
shall
25.
s.
come
to
?
Multiplie
and
deuide
,
and
you
shall
finde
22.
s.
6.
d.
of
the
whiche
number
,
take
½
which
is
11.
s.
3.
d.
and
substract
it
from
22.
s.
6.
d.
and
from
16.
s.
8.
d.
and
there
shall
rest
11
s.
3.
d.
and
5.
s.
5.
pence
for
the
two
first
numbers
in
the
rule
of
thrée
,
and
6.
shil.
8.
pence
which
is
the
ouerprice
of
a
hundred
of
yron
for
the
thirde
number
,
then
multiplie
and
deuide
,
and
you
shall
finde
.
3.
s.
2.
pence
,
½4/7
:
and
so
muche
did
the
hunof
yron
coste
in
readie
moneye
.
12.
More
,
twoo
marchauntes
wyll
chaunge
their
marchaundise
,
the
one
with
the
other
,
the
one
of
them
hath
sayes
of
200
shill.
10
d.
the
pece
to
sell
for
ready
money
,
and
in
barter
he
wil
sell
the
pece
for
21
shillings
,
and
he
wyll
haue
the
¼
in
readie
money
:
The
other
hath
cappes
of
35
shill.
the
dosen
to
tell
for
readye
money
:
but
he
wyll
gayne
after
10
vppon
the
100.
I
demaunde
howe
he
shall
sell
the
same
cappes
in
barter
?
Aunswere
,
saye
,
if
100
be
worthe
110
:
What
shall
35.
shill.
be
worthe
,
which
is
the
iust
price
of
the
dosen
of
cappes
?
Multiplye
and
diuide
,
and
you
shall
fynd
38
shill.
6.
d.
Then
take
the
¼
of
25.
whiche
is
6
sh.
3.
d.
and
substracte
it
from
20
shill.
10.
d.
and
from
25.
sh.
and
there
shall
rest
14
sh.
7.
d.
&
18
shill.
9.
d.
for
the
two
firste
numbers
in
the
rule
of
three
,
and
38
shill.
6.
d.
whiche
is
the
iuste
price
with
his
gaine
of
the
dosen
of
cappes
,
for
the
thyrd
nomber
:
then
multiplie
and
diuide
,
and
you
shall
finde
49.
shill.
6.
d.
and
for
so
muche
shall
he
sell
the
dosen
of
cappes
in
barter
.
¶
The
12
Chapter
treateth
exchaunging
of
money
from
one
place
to
another
.
FIrste
,
you
muste
note
,
that
at
Andwerpe
they
vse
to
make
their
accomptes
by
Deniers
de
gros
,
that
is
to
saye
by
pence
Flemishe
,
whereof
12
doe
make
1
shill.
Flemishe
,
and
20
shil
.
Flemish
do
make
1
li.
de
gros
.
1.
If
I
deliuer
in
Flaunders
,
500
li.
Flemishe
,
at
19
shil.
6.
de
gros
that
is
to
saye
,
at
19s
.
wil.
6.
d.
Flemish
,
to
receiue
20
shil
at
London
:
I
demaund
howe
much
I
shall
receyue
sterlynge
at
London
for
the
sayde
500
li.
Flemishe
?
Aunswere
:
Saye
,
yf
19
½
geue
20/1●
what
wyll
50●/1
geue
?
Multiplie
and
diuide
,
and
you
shal
finde
512
li.
16.
sh.
4.
d.
12/13
of
pennye
.
And
so
muche
sterling
shall
I
receyue
in
London
for
my
500
li.
Flemishe
.
2.
If
I
deliuer
in
London
375
li.
sterlyng
,
to
receyue
in
Andwerpe
21
shill.
9.
d.
de
gros
,
that
is
to
saye
Flemishe
,
for
euery
pounde
sterlinge
.
I
demaunde
howe
many
poundes
Flemishe
I
shall
receiue
in
Andwerpe
,
for
the
sayd
375
pound
sterling
?
Answere
,
say
if
23/●
giue
21
¾
:
what
will
375/●
giue
?
Multiplie
and
diuide
,
and
you
shall
finde
407
.li.
So
many
poundes
Flemishe
shall
I
receyue
for
the
sayde
375
.li.
sterlinge
in
Andwerpe
.
3.
If
I
take
vp
monie
at
Andwerpe
after
19
sh.
6
d.
Flemishe
to
pay
for
the
same
at
London
20
s.
ster
.
&
when
the
daye
of
paiment
is
come
,
I
am
forced
to
rechaunge
the
same
,
and
to
take
vp
money
againe
here
in
London
to
repay
the
same
,
so
that
for
20
shillings
,
which
I
take
vp
here
,
I
must
repaye
,
19
shill.
9.
at
Andwerpe
.
I
demaunde
whether
I
do
winne
or
lose
,
and
how
muche
vpon
the
100
pound
of
money
?
Aunswere
,
Saye
,
if
19
●/4
giue
19
●/●
,
what
wyll
100/●
giue
?
Multiplie
and
diuide
,
and
you
shall
finde
9858/79
,
the
whiche
beinge
abated
from
100
there
will
remayne
121/79.
And
so
muche
doe
I
lose
vpon
the
100
li.
of
money
.
4.
If
I
take
vp
at
London
20.
shill.
ster
.
to
paye
at
Andwerpe
21.
shill.
8.
d.
Flemishe
,
and
when
the
day
of
payment
is
come
.
I
am
constrained
to
take
vp
money
againe
at
Andwerpe
wherewith
to
repay
the
foresayde
summe
:
and
there
I
doe
receyue
22.
shill.
Flemishe
to
paye
20.
shill.
at
London
.
Nowe
I
demaund
whether
I
do
winne
or
losse
and
howe
much
vpon
the
100
pounde
of
money
after
the
rate
.
Answere
:
say
,
if
212/2
giue
22/1.
What
wil
100/1
giue●
Multiplie
and
diuide
,
and
you
shall
finde
1017/13
,
from
the
whiche
abate
100
,
and
there
will
remaine
1
〈…〉
and
so
muche
shall
I
gaine
vpon
the
100
li.
of
money
.
The
exchaunce
from
London
into
Fraunce
,
is
not
lyke
as
it
is
into
Flaunders
but
is
deliuered
by
the
Frenche
crowne
,
whiche
is
worth
50
souse
Tournois
the
pece
.
And
here
muste
you
note
that
in
Fraunce
they
make
theyr
accompte
by
Deniers
Tournoys
,
whereof
12
maketh
one
souse
Tournois
,
and
20.
souse
.
Tournois
maketh
1.
pound
Tournois
,
whiche
thei
call
a
Liure
,
and
the
Frenche
Crowne
is
currant
amonge
Marchauntes
for
51
souse
Tournois
,
but
by
excaunge
it
is
otherwyse
,
for
they
wyll
deliuer
but
50
souse
.
Tournois
,
whiche
is
2
li.
10
souse
.
Tournois
for
a
crowne
,
or
at
suche
price
as
the
takers
vp
of
money
can
agree
with
the
deliuerer
.
As
by
Example
.
5.
If
I
deliuer
340
pound
ster
.
here
in
London
after
6.
shill.
4.
d.
ster
.
the
crowne
,
to
receyue
at
Roan
or
at
Paris
50
souse
Tournois
for
euerye
crowne
,
I
woulde
knowe
howe
manye
liures
Tournoys
I
shall
receiue
there
for
my
340
li.
ster
.
Aunswere
,
say
if
6
sh.
⅓
do
giue
me
2
li.
Tournois
what
will
6100/1
sh.
giue
(
whiche
is
the
340
li.
reduced
into
shillings
)
multiplie
and
diuide
,
&
you
shall
finde
2684
Liores
4/19
whiche
is
worthe
4.
shill.
4/19
Tournois
,
and
so
much
shal
I
receiue
in
Roan
or
Paris
for
my
340
li.
sterl.
6.
If
I
deliuer
in
Paris
or
Roan
,
or
els
where
in
Fraunce
1250
liures
Tournyis
,
at
50
souse
tournois
the
crowne
,
to
recyue
for
euerye
suche
crowne
6
shillings
3.
d.
sterling
at
London
.
I
demaunde
howe
much
sterling
money
I
shall
receyue
at
London
for
my
1250
.li.
Tournois
.
Aunswere
:
saye
,
if
twoo
li.
½
doe
giue
me
6
shillinges
¼
,
what
wyll
1250/1
giue
?
Multiplie
and
diuide
,
and
you
shal
finde
3125
shyllynges
sterlynge
,
whiche
maketh
156
,
pounde
,
fyue
shillyngs
sterlynge
.
And
so
many
poundes
shal
I
receyue
at
London
,
for
the
sayde
1250
pounde
Tournois
,
after
6
shillinges
3
pence
for
euery
crowne
.
¶
The
13
Chapter
treateth
of
the
Rule
of
Alligation
.
THe
Rule
of
Alligation
is
so
named
,
for
that
it
teacheth
to
alligate
or
binde
togither
diuers
percelles
of
sundrie
prices
,
and
to
knowe
howe
muche
you
muste
take
of
euerye
percell
,
accordynge
to
the
nombers
of
the
Question
.
Example
.
1
A
Goldsmythe
hath
three
sortes
of
Golde
.
The
fyrste
is
worth
thirtye
Crownes
the
pounde
weyghte
:
The
secende
is
worthe
36
Crownes
.
And
the
thyrde
is
worthe
45.
crownes
,
and
of
these
three
sortes
he
will
make
a
Scepter
of
sixe
pound
weight
,
whiche
shal
be
worthe
40
,
Crownes
the
pounde
.
I
demaunde
howe
muche
he
muste
take
of
euery
sorte
?
Answere
:
first
you
must
set
downe
the
nombers
whereof
you
wyl
make
the
Alligation
(
which
are
30.
36.
38
,
and
45
)
orderly
the
one
vnder
the
other
,
as
if
you
shoulde
make
of
them
an
addicion
:
and
the
common
nomber
whereunto
you
will
reduce
them
,
shall
you
set
on
the
left
hand
,
which
common
nomber
in
this
example
is
40.
Then
marke
what
summes
bee
lesser
,
then
that
common
nomber
,
and
whiche
bee
greater
,
and
wyth
a
draught
of
your
penne
,
euermore
lynke
two
nombers
togither
,
so
that
the
one
be
lesser
then
that
common
nomber
,
and
the
other
greater
then
he
.
For
2
greater
nor
twoo
smaller
nombers
maye
not
bee
linked
together
,
for
they
wyll
eyther
be
lesser
,
or
els
greater
then
the
common
nomber
:
but
one
greater
nomber
,
and
one
smaller
maye
be
so
mixed
,
that
they
will
make
the
common
nomber
.
And
two
greater
or
twoo
smaller
nombers
,
can
neuer
make
the
common
nomber
in
dewe
order
,
as
here
after
shall
appeare
.
After
that
you
haue
thus
lynked
them
,
then
make
howe
muche
eche
of
the
lesser
nombers
is
smaller
then
the
common
nomber
,
and
that
difference
shall
you
set
against
the
greater
nombers
,
whiche
bee
linked
with
those
smaller
,
eche
of
them
with
his
matche
styll
on
the
righte
hande
.
And
lykewise
you
muste
set
the
excesse
of
the
greater
nombers
againste
the
lesser
which
be
combyned
with
them
.
Then
shall
you
adde
all
those
differences
into
one
summe
,
whiche
shal
be
the
first
nomber
in
the
rule
of
three
,
and
the
second
nomber
shal
be
the
whole
massye
pece
that
you
will
haue
of
all
the
perticulers
,
the
thirde
summe
shal
be
eche
difference
by
it selfe
,
and
by
them
shall
you
finde
out
the
fourth
nomber
declarynge
the
luste
portion
of
euerye
perticuler
in
that
mixture
,
as
nowe
by
the
former
example
,
I
wyll
make
it
plaine
.
Here
in
this
former
example
,
you
see
that
I
haue
set
downe
the
seuerall
prices
,
whiche
be
30
,
36.
42.
45
,
and
haue
linked
together
30
,
with
45.
&
36.
with
42.
The
common
price
40
,
I
haue
set
on
the
lefte
syde
,
and
the
difference
of
it
from
euerye
seuerall
price
,
I
haue
set
on
the
ryght
hande
,
agaynste
that
summe
wyth
the
whiche
it
is
lynked
.
So
the
difference
of
30
from
40
is
10
,
whiche
I
set
against
45
,
that
he
is
lynked
wythall
,
and
the
difference
of
45
,
aboue
40
is
5.
whiche
I
haue
set
against
30.
So
lykewayes
,
the
differences
of
42
aboue
40
,
is
2
,
that
I
haue
sette
againste
36.
And
the
difference
betwen
36
and
40
(
whiche
is
4
)
I
haue
sette
againste
42.
Then
I
adde
all
those
differences
togither
and
they
make
21
,
whiche
I
make
the
fyrste
nomber
in
the
Rule
of
three
,
and
6
the
seconde
nomber
,
which
is
the
weyght
of
the
Scepter
of
Golde
,
and
the
thyrde
nomber
shall
bee
euerye
perticuler
difference
.
Then
I
worke
by
the
Rule
of
three
:
sayinge
yf
21.
(
whiche
is
the
differences
added
together
)
doe
geue
me
6
pounde
,
whiche
is
the
weyghte
of
the
Scepter
,
what
shall
5
gyue
,
whiche
is
the
first
difference
?
Multiplie
and
diuide
,
and
you
shall
fynde
one
pounde
1/7
:
so
muche
muste
I
haue
of
the
fyrste
price
.
Then
doe
likewise
with
the
reste
and
you
shall
finde
4/7
of
the
seconde
price
,
1
li.
1/7
of
the
thirde
price
,
and
2
.li.
6/7
of
the
fourth
,
the
whiche
4
summes
beinge
added
together
,
do
make
6
li.
whiche
is
the
totall
that
I
would
haue
.
And
now
to
proue
if
the
prices
do
agre
,
you
shall
do
thus
:
Firste
multiplie
this
totall
summe
6
by
the
common
price
40
and
it
will
make
240
Crownes
,
whiche
you
shall
kepe
be
it selfe
.
And
afterward
multiplie
euery
seuerall
summe
of
weight
by
the
price
belongynge
to
the
same
weyght
,
and
if
that
sume
do
agree
with
the
first
that
you
kepte
by
it selfe
,
then
is
youre
worke
well
done
,
as
here
1
li.
3/7
is
the
weyght
of
that
sorte
of
golde
whiche
is
of
30
Crownes
price
.
Then
multiplie
30.
by
1
li.
3/●
,
and
it
maketh
42.
crownes
,
6/7
,
which
you
shall
set
downe
.
Then
multiplie
4/7
(
whiche
is
the
weyght
of
the
seconde
sorte
of
golde
)
by
36
,
which
is
the
price
of
the
same
,
&
therof
commeth
20
crownes
4/2
:
so
again
1
.li.
⅓
multiplyed
by
42
,
doth
make
48
crownes
.
And
laste
of
all
2
pounde
.
〈◊〉
multiplied
by
45.
maketh
128
crownes
4/7.
Al
these
added
together
dothe
make
240
crownes
,
agreable
to
the
former
summe
of
40
multiplied
by
6
:
And
thus
I
maye
affirme
that
this
worke
is
well
done
.
2.
A
Tauerner
hathe
foure
sortes
of
wine
,
of
foure
seuerall
prices
,
the
firste
of
8
pence
the
Gallonde
,
the
seconde
of
tenne
pence
the
Gallonde
,
the
third
of
15
pence
,
and
the
fourht
of
18
pence
.
And
he
will
mingle
one
punchen
with
all
these
sortes
,
so
that
the
Gallonde
shal
be
worthe
but
twelfe
pence
.
I
demaunde
howe
many
Gallondes
he
muste
take
of
euery
sorte
?
Aunswere
:
Firste
suppose
the
punchen
to
holde
some
certayne
measure
,
as
to
conteyne
84
Gallondes
and
then
the
(
forme
wyll
bee
after
this
sorte
,
as
you
see
here
after
folowynge
.
¶
The
14
chapter
treateth
of
the
rule
of
falshode
,
or
false
positions
.
THe
Rule
of
falsehod
is
so
named
not
for
that
it
teacheth
any
deceyte
of
falsehode
,
but
that
by
fayned
nombers
taken
at
all
aduentures
,
it
teacheth
to
finde
out
the
true
nomber
that
is
demaunded
.
And
this
(
of
all
the
vulgare
Rules
whiche
are
in
practise
)
is
the
most
excellent
:
this
rule
hath
two
parts
,
the
one
is
of
one
false
position
alone
:
the
other
is
of
two
positions
as
here
after
shall
appeare
.
Those
questions
whiche
are
done
by
false
positions
,
haue
theyr
operations
,
in
a
maner
like
vnto
that
of
the
rule
of
three
,
but
onely
that
in
the
rule
of
three
,
we
haue
three
nombers
knowen
,
and
here
in
this
rule
wee
haue
but
one
(
I
meane
that
commeth
in
operation
)
vnto
the
likenes
whereof
we
muste
diuise
two
other
,
the
one
multiplying
,
and
the
other
diuidyng
,
as
by
example
.
1.
I
haue
deliuered
to
a
banker
a
certein
summe
of
pounds
in
money
,
to
haue
of
him
by
the
yeare
6
li.
vpon
the
100
li.
And
at
the
ende
of
10
yeres
he
payed
me
500
pounde
for
all
bothe
principall
an
gaine
.
I
demaund
howe
muche
was
the
principall
summe
that
I
deliuered
at
the
fyrste
.
Here
you
see
that
there
are
diuers
termes
:
but
the
chiefe
to
worke
with
all
is
500
li.
whiche
commeth
of
the
other
nombers
,
that
is
to
saye
,
of
10
,
and
100
for
of
them
is
compounde
the
tenour
of
the
question
,
the
practise
whereof
is
thus
.
Let
vs
faine
a
nomber
at
pleasure
,
and
with
the
same
let
vs
make
oure
discourse
,
euen
as
though
it
were
the
principalll
summe
that
we
seke
for
.
As
by
Example
.
Suppose
that
I
deliuered
hym
at
the
firste
200
pounde
,
the
whiche
were
worthe
to
me
in
10
yeares
120
pounde
after
the
rate
of
6
vpon
the
hundred
pound
.
Then
120
pounde
added
with
200
pounde
.
Do
make
but
320
li.
and
I
must
haue
500
li.
Thus
you
see
that
I
haue
three
termes
for
the
rule
of
three
:
the
one
which
shall
conteine
the
Question
the
other
two
,
which
I
haue
200
,
&
320
:
in
such
sorte
,
that
320
,
ought
ta
haue
suthe
proporcion
to
200
,
as
500
hat
vnto
the
nomber
that
I
seeke
:
that
is
to
say
,
vnto
the
true
principal
summe
,
then
must
I
haue
recourse
vnto
the
rule
of
three
,
after
this
sorte
,
saying
.
If
320.
pound
become
of
200.
pounde
of
howe
muche
shall
come
500.
pound
Multiply
500
:
by
200.
and
thei
are
100000.
the
whiche
you
shall
diuide
by
320.
pounde
and
thereof
commeth
312.
pounde
.
½
whiche
is
the
summe
that
I
deliuered
at
the
firste
,
and
thus
,
this
rule
hath
some
congruence
with
the
double
rule
of
three
.
2.
I
haue
a
Cesterne
with
3.
vnegal
cockes
conteining
60.
pipes
of
water
:
And
if
the
greatest
cocke
bée
opened
,
the
water
will
auoyde
cleane
in
one
houre
,
at
the
seconde
yt
will
anoyde
in
twoo
houres
,
and
at
the
thirde
it
will
require
three
houres
.
Nowe
I
demande
in
what
space
wil
it
auoide
,
all
the
cocks
beyng
set
open
.
Suppose
that
it
wyll
auoide
in
halfe
an
houre
,
that
is
to
saie
,
in
30.
minutes
.
Then
muste
there
auoide
at
the
firste
cocke
,
the
½
,
whiche
is
thirtie
pipes
,
and
by
the
seconde
cocke
the
¼
whiche
is
15
,
pipes
,
and
by
the
thirde
cocke
the
⅙
,
that
is
tenne
pypes
,
all
the
whiche
summes
beeyng
added
together
doeo
make
fiftie
fiue
pipes
,
but
it
shoulde
be
sixtie
pipes
.
Therefore
I
saie
by
the
rule
of
thrée
,
if
fiftie
fiue
pipes
doe
voide
in
thirtie
minutes
:
in
how
manye
minutes
will
sixtie
pipes
voide
?
Multiplie
and
diuide
,
and
you
shall
finde
thirtie
twoo
minutes
40/55.
And
in
that
space
will
the
water
auoide
if
all
the
cockes
be
set
open
.
¶
Of
the
rule
of
two
false
positions
.
THe
summe
of
this
rule
of
two
false
positions
is
thus
,
when
any
question
is
proponed
appertaining
to
this
rule
.
First
imagine
any
number
at
your
pleasure
,
which
you
shal
name
the
first
position
,
and
with
the
same
shall
you
worke
in
stede
of
the
true
number
,
as
the
question
doth
import
,
and
if
you
sée
that
you
haue
missed
.
Then
is
the
last
number
of
the
worke
either
to
great
or
to
litle
,
that
shal
you
note
for
to
be
the
first
errour
,
in
the
which
you
haue
missed
with
the
signe
of
more
,
or
lesse
,
whiche
signes
shal
be
noted
with
these
figures
,
4●
:
—
This
figure
4
:
betokeneth
more
,
and
this
plaine
line
—
signifieth
lesse
,
that
is
to
saie
the
one
signifieth
to
muche
&
the
other
to
little
:
then
begin
again
,
&
take
an
other
number
,
whiche
shal
be
the
second
position
,
and
worke
by
the
question
as
before
,
if
you
haue
missed
againe
,
note
the
excesse
or
want
,
for
that
is
the
seconde
errour
.
Then
shall
you
multiplie
the
first
position
by
the
second
errour
crossewise
,
and
againe
the
seconde
position
by
the
first
errour
(
and
this
must
alwaies
be
obserued
)
&
keepe
the
two
productes
:
then
if
the
signes
be
both
like
,
that
is
to
saie
,
either
both
to
muche
,
or
both
to
little
,
abate
the
lesser
producte
from
the
greater
and
likewise
,
you
shall
substract
the
lesser
error
from
the
greater
,
&
by
the
remaine
of
those
errours
,
you
shal
diuide
the
residue
of
the
products
:
the
quotient
shal
be
the
true
number
that
you
séeke
for
.
But
if
the
twoo
signes
be
vnlyke
,
that
is
to
say
,
the
one
to
muche
and
the
other
to
lyttle
,
then
shall
you
adde
those
productes
together
so
shall
you
also
adde
bothe
the
errours
together
,
and
by
the
summe
of
those
errours
,
diuide
the
totall
summe
of
both
the
products
:
the
quotient
shall
be
likewise
the
true
number
that
the
question
seketh
,
and
this
is
the
wholerule
,
as
by
example
.
●3
.
A
manlying
at
the
point
of
death
,
saied
that
he
had
in
a
certain
Coffer
a
hundred
duckets
,
the
which
he
to
3.
of
his
frends
by
him
named
,
after
this
sorte
.
The
first
must
haue
a
certain
portion
,
the
2.
must
haue
twise
so
manie
as
the
first
abatyng
8.
Duckets
&
the
third
muste
haue
3.
times
so
many
as
the
first
,
lesse
by
15
Duckets
.
Nowe
,
I
demaunde
howe
many
euerie
of
them
must
haue
.
Answere
:
First
I
doe
imagine
that
the
firste
man
had
thirtie
Duckets
,
then
by
the
order
of
the
question
,
the
seconde
shoulde
haue
fiftie
twoo
:
and
the
third
seuentie
fiue
.
These
three
summes
beyng
added
together
doe
make
157
:
and
I
shoulde
haue
but
a
hundred
soe
that
this
fiirste
errour
is
to
much
by
fiftie
seuen
,
then
I
note
a
parte
the
first
position
thirtie
,
with
hys
errour
fiftie
seuen
,
to
muche
after
this
sorte
thirtie
.
457.
Therefore
I
prosecute
my
worke
and
I
suppose
that
the
firste
hadde
twentie
foure
,
then
by
the
orderder
of
the
question
,
the
seconde
should
haue
40.
and
the
third
,
fiftie
seuen
,
these
three
summes
beyng
added
together
,
doe
ma●●
121.
and
I
must
haue
but
a
hundred
,
so
the
seconde
errour
,
is
to
muche
by
21.
Therefore
I
note
24.
421.
vnder
the
thirtie
.
457.
as
you
maye
see
in
the
margent
of
the
nexte
side
following
.
Then
I
multiplie
crossewaies
,
thirty
(
which
is
the
firste
position
)
by
twelue
which
is
the
seconde
errour
,
and
therof
commeth
630.
likewise
I
multiplie
twentie
and
four
(
which
is
the
second
position
)
by
fiftie
seuen
,
whiche
is
the
firste
errour
,
and
I
finde
1368
:
then
because
the
signes
of
the
errours
are
bothe
like
:
that
is
to
saye
muche
,
I
must
therfore
substracte
630.
from
1368.
and
ther
wil
remain
731
which
is
the
diuidende
:
again
I
must
substract
the
lesser
error
from
the
greater
,
that
is
to
wit
,
21.
out
of
57.
and
there
will
remain
36.
whiche
shal
bee
my
diuisor
.
Thys
done
I
diuide
738
by
36
and
the
quotient
will
bee
20.
〈◊〉
.
The
which
20.
½
is
the
iust
number
of
the
duckets
that
the
first
man
had
for
his
parte
,
so
consequently
the
seconde
manne
had
thirtie
three
Duckets
,
and
the
thirde
fourty
six
½
,
as
by
proofe
in
the
margent
maie
appeare
.
The
like
number
wil
also
appeare
.
in
case
the
errours
wer
both
to
little
,
as
in
makyng
the
twoo
positions
by
18.
and
20.
where
you
shal
finde
the
two
errours
both
to
little
,
the
firste
will
bée
to
lyttle
by
15.
and
the
seconde
to
little
by
3.
as
by
perusing
this
worke
in
the
margent
you
shal
well
perceiue
.
Againe
if
one
of
the
errours
were
to
muche
,
and
the
other
to
little
,
yet
shall
I
haue
the
true
number
,
as
before
:
As
if
the
twoo
positions
were
24.
and
20.
I
shall
finde
that
the
firste
errour
will
be
21.
to
muche
,
and
the
seconde
wil
be
three
to
little
:
Therfore
I
multiplie
twentie
foure
by
thirtie
,
crossewaies
,
thereof
commeth
72.
Likewise
I
multiplie
twentie
by
21.
the
product
will
be
420
:
These
twoo
summes
72.
and
420
,
I
adde
both
together
because
the
signes
of
the
errours
bee
vnlyke
,
and
thei
mak
.
492.
the
whiche
shal
bee
my
diuidende
,
and
againe
,
I
addde
the
lesser
errour
three
,
with
the
greater
errour
21.
and
they
mak
24.
for
my
diuisor
,
then
diuiding
foure
hundred
ninetie
two
by
twenty
foure
,
the
quotiet
wil
be
twentie
½
:
as
in
the
margent
doth
plainlie
appeare
.
And
now
because
you
shall
not
forget
this
parte
of
the
rule
,
learne
thys
briefe
remembraunce
followyng
.
The
signes
both
like
substraction
doe
require
And
vnlike
signes
addition
will
desire
The
meanyng
whereof
is
thus
yf
both
the
errours
haue
lyke
sygnes
,
then
muste
the
diuidende
and
the
diuisor
bee
made
by
substraction
,
as
is
taught
before
,
and
if
those
sygnes
bee
vnlike
,
then
muste
I
by
addition
gather
the
diuidende
,
and
the
diuisor
,
as
I
haue
done
in
this
laste
example
.
4.
A
man
hath
twoo
siluer
cups
of
vnegall
waight
,
hauynge
to
them
bothe
,
but
one
couer
,
the
waight
whereof
is
fiue
ounces
,
if
the
couer
be
putte
to
the
lesser
cuppe
,
it
wyll
bee
in
double
proportion
vnto
the
waight
of
the
greater
,
and
the
couer
béeyng
put
to
the
greater
cuppe
,
will
bee
in
triple
proportion
,
vnto
the
waight
of
the
lesser
.
I
demaunde
what
was
the
waight
of
euery
cuppe
.
Suppose
that
the
lesser
cuppe
dyd
wayghe
seauen
ounces
,
then
wyth
the
couer
it
muste
waigh
twelue
,
and
this
waight
shoulde
be
in
double
proportion
vnto
the
greater
,
therefore
the
greater
muste
waigh
six
ounces
adde
vnto
it
5
ounces
for
the
couer
,
all
wil
be
11.
ounces
,
but
it
should
be
21
for
to
haue
it
in
triple
proportion
,
vnto
7.
which
representeth
the
weght
of
the
lesser
cuppe
So
that
this
first
errour
is
to
little
by
10.
which
you
shall
note
after
seuen
in
this
sorte
.
7
.
-10
.
After
you
shall
suppose
some
other
number
,
as
9.
and
make
the
like
work
as
before
,
so
shal
you
finde
15.
to
little
,
for
the
second
errour
,
which
you
shall
put
behind
nine
,
and
then
worke
with
the
reste
as
aboue
is
sayed
,
and
you
shall
finde
that
the
lesser
cuppe
wayed
three
ounces
,
and
consequentlye
the
greater
foure
ounces
.
5.
One
man
demaunded
of
another
in
a
mornynge
what
a
clocke
it
was
,
the
other
made
hym
thys
aunswere
,
if
you
doe
〈◊〉
(
saieth
hee
)
the
½
of
the
houres
whiche
he
passte
synce
mindnight
,
with
the
⅓
of
the
houres
whiche
are
to
come
vntill
noone
,
you
shall
haue
the
iuste
houre
,
that
is
to
saie
,
you
shall
knowe
what
a
clocke
it
was
:
Suppose
that
it
was
4●
clocke
in
the
mornynge
,
so
shoulde
there
remaine
8.
vntill
none
:
then
I
take
the
●/4
of
4.
whiche
is
1.
and
the
⅔
of
8.
whiche
is
5.
⅓
,
and
I
adde
them
together
,
so
I
finde
6.
⅓
and
I
supposed
but
4.
therfore
thys
first
errour
is
to
muche
by
2.
〈◊〉
,
which
I
note
after
my
position
thus
4.
†
2.
●/3
:
then
againe
I
suppose
an
other
number
,
that
is
to
saye
nine
,
soe
shoulde
remaine
but
3.
houres
vntill
noone
.
,
I
take
the
¼
of
9.
and
the
⅔
of
3.
whiche
is
2.
¼
and
2.
these
I
adde
together
and
they
make
4.
¼
:
but
I
supposed
that
it
was
9.
therfore
the
second
errour
is
4.
¾
to
little
whiche
I
note
behinde
my
position
thus
.
9.
†
4.
2
/
4.
And
then
I
multiplie
crossewyse
,
as
before
is
taught
,
and
bicause
th●
sygnes
of
the
errours
are
vnlike
,
that
is
to
saye
,
the
one
to
muche
,
and
the
other
to
lyttle
,
therefore
in
thys
woorke
I
muste
adde
the
products
,
and
they
will
bee
fourtie
.
Likewise
I
adde
the
errours
,
and
thei
be
seuen
,
〈◊〉
.
Then
I
diuide
fourtie
by
seuen
1/12
,
and
thereof
commeth
fiue
houres
11/17
,
and
that
houre
it
was
in
the
mornyng
.
¶
The
fiueth
Chapiter
treateth
of
sportes
,
and
pastime
,
done
by
number
.
IF
you
would
know
the
number
that
any
man
doth
thinke
or
imagine
in
hys
minde
,
as
though
you
coulde
deuine
.
Bydde
hym
triple
the
same
number
,
then
of
the
product
let
hym
take
the
〈◊〉
if
the
number
be
euen
,
or
els
the
greater
halfe
,
if
the
same
bee
odde
,
then
bid
him
triple
againe
the
saied
½
:
after
say
to
him
that
he
put
awaie
,
if
he
can
36.
27.
18.
or
.
9.
from
the
laste
number
beyng
tripled
:
that
is
to
saie
,
cause
hym
subtellye
to
put
awaie
nine
as
many
times
as
is
possible
and
kepe
the
number
secretly
:
and
when
ye
can
no
more
take
away
9.
then
to
know
if
that
yet
there
remaine
anye
nomber
,
bid
hym
abate
three
and
twoo
or
one
,
if
he
can
:
this
doone
,
see
how
manie
tymes
nine
you
haue
caused
hym
to
abate
,
for
the
which
keepe
you
in
mind
so
manie
tymes
2.
and
if
that
you
knowe
that
hee
had
any
thing
remaining
beside
the
nines
,
the
same
shall
also
note
vnto
you
〈◊〉
.
Suppose
that
he
thought
6.
whiche
beyng
tripled
is
18.
wherof
the
●/2
is
9.
the
triple
of
that
is
27
:
now
cause
hym
to
abate
18
,
or
9.
or
27.
and
againe
9
:
but
then
he
will
saie
vnto
you
that
he
cannot
,
bid
hym
then
abate
3.
or
2.
or
1.
●e
wyl
saie
also
that
he
can
not
:
wherfore
consideryng
that
you
haue
made
hym
to
abate
3.
tymes
nine
iustly
,
you
shall
tell
him
that
he
thought
6.
for
3.
tymes
2.
maketh
6.
If
he
had
thought
5.
the
triple
therof
is
fiftene
,
wherfore
the
greater
½
is
8.
the
triple
of
that
maketh
24.
which
conteineth
2.
times
9.
thei
ar
worth
4.
and
the
remain
signifieth
1.
the
which
added
togither
make
5.
which
is
the
nomber
that
he
thought
.
2.
If
in
any
companie
,
one
of
them
hath
a
ringe
vpon
his
finger
,
and
you
woulde
know
by
maner
of
deuining
,
who
hath
the
same
and
vpon
what
finger
and
what
ioint
:
cause
the
persons
to
sit
doune
in
order
,
&
keepe
likewise
an
order
of
their
fingers
:
then
separate
your self
from
them
in
some
certaine
place
,
and
saie
vnto
one
of
the
lookers
on
,
that
hee
double
the
number
(
markyng
the
order
)
of
hym
that
hath
the
ring
:
and
vnto
the
double
bid
hym
ad
5.
and
then
cause
him
to
multiply
this
addition
by
5
and
vnto
the
product
bid
hym
adde
the
number
of
the
finger
of
the
person
whiche
hath
the
ring
:
bee
if
that
the
same
last
sum
did
amounte
to
89.
then
afterward
say
to
him
that
he
put
after
the
same
laste
number
towarde
his
right
hand
a
figure
signifiyng
vppon
whiche
of
the
iointes
hee
hath
the
ring
.
As
if
it
be
vpon
the
third
iointe
,
lette
hym
put
3.
after
89.
and
it
will
bee
893
:
this
doen
,
you
shal
aske
him
what
number
hee
keepeth
,
from
the
whiche
you
shal
abate
250.
and
you
shal
haue
thrée
figures
remainyng
at
the
least
.
The
fyrste
towarde
youre
lefte
hande
shall
signifie
the
number
of
the
person
whiche
hath
the
ring
.
The
second
or
middle
figure
shall
represente
the
number
of
the
finger
.
And
the
last
figure
towarde
your
right
hande
,
shall
be
token
the
number
of
the
ioynte
.
As
if
the
number
whiche
hée
dyd
keepe
were
893.
from
that
you
shall
abate
250.
and
there
will
remaine
64●
Which
do
note
vnto
you
,
that
the
sixt
person
hath
the
ringe
vpon
the
fourth
finger
,
and
vpon
his
thirde
iointe
.
But
note
that
when
you
haue
made
your
substraction
,
if
there
do
remaine
a
cipher
in
the
place
of
tennes
,
that
is
to
saie
in
the
seconde
place
,
you
muste
then
abate
from
that
figure
whiche
is
in
the
place
of
hundreds
,
that
is
to
saie
from
the
figure
which
is
next
your
left
hand
,
&
that
shal
be
worth
ten
tenthes
,
signifiyng
the
tenth
finger
:
as
if
there
should
remain
703.
you
must
saye
that
the
sixte
person
)
vpon
his
tenth
finger
and
vpon
his
third
ioint
)
hath
the
ring
3.
And
after
the
same
maner
,
if
a
man
do
cast
thrée
dice
,
you
maie
know
the
poincts
of
euery
one
of
them
,
for
if
you
do
cause
him
to
doube
the
poincts
of
one
die
,
&
vnto
that
double
to
ad
5.
and
the
same
sum
to
multiplye
by
5.
&
vnto
the
product
ad
the
poincts
of
one
of
the
other
dice
,
and
behind
the
number
towarde
the
right
hand
,
to
put
the
figure
which
signifieth
the
poinctes
of
the
last
die
,
&
then
shal
you
aske
him
what
number
he
kepeth
〈◊〉
the
which
abat
250
〈◊〉
ther
wil
rema
〈…〉
figures
,
which
do
note
vnto
you
the
pointes
of
euery
die
.
4.
Likewise
if
3.
of
your
companions
,
to
saie
Peter
,
Iames
,
and
Ihon
,
that
would
(
in
your
absence
)
giue
them selfe
euerie
one
a
contrarie
name
:
as
for
example
:
Peter
woulde
be
called
a
king
,
Iames
a
duke
,
and
Ihon
a
countie
:
And
I
woulde
deuine
whiche
of
them
is
called
the
kinge
,
whiche
the
duke
,
and
which
the
county
.
Take
24.
stones
,
or
other
peces
whatsoeuer
,
and
geue
vnto
Peter
.
1.
vnto
Iames
2.
and
vnto
Ihon.
3.
or
otherwise
.
But
marke
well
vnto
whiche
of
them
you
haue
1.
vnto
which
2.
and
vnto
whom
3.
Then
leuing
the
18.
stones
(
before
them
)
that
are
ramainyng
,
you
shall
absent
your selfe
from
their
sighte
,
or
elles
tourne
your
face
from
them
,
saiyng
thus
vnto
them
,
whosoeuer
nameth
himselfe
a
kinge
:
for
euerie
stone
that
I
gaue
him
lette
hym
take
one
of
the
residue
,
and
he
that
nameth
hymselfe
a
duke
foreuerie
stone
that
I
gaue
hym
lette
hym
take
twoo
of
them
that
remain
,
and
he
that
calleth
hym selfe
a
countie
,
for
euerie
stone
that
I
gaue
hym
let
,
hym
take
foure
,
this
beyng
done
approache
nere
them
,
and
marke
towe
man●e
stones
are
remainyng
:
and
know
this
that
there
can
not
remaine
any
other
,
but
one
of
these
sire
,
1
,
2
,
3
,
5
,
6
,
7
,
for
the
whiche
sire
numbers
we
haue
chosen
to
euerie
of
them
a
seuerall
name
,
whiche
are
these
:
Angeli
,
Beati
,
qualiter
,
Messias
,
Israel
,
p●etas
:
eche
of
them
containyng
three
●oweiles
,
a
,
e
,
i
,
whiche
doe
shewe
the
names
by
order
:
that
is
to
say
,
a
,
sheweth
whiche
is
the
king
,
e
,
telleth
which
is
the
duke
,
i
,
sheweeth
which
is
the
coum
〈◊〉
folewynge
the
order
how
,
and
to
whom
you
haue
geuen
1.
stone
,
to
which
〈◊〉
.
and
to
which
〈◊〉
.
Then
there
doe
remaine
but
1
stone
,
the
first
name
Augeli●
by
these
three
vowels
a
,
e.
i
,
sheweth
that
Peter
is
the
kinge
,
Iames
the
doke
,
and
Ihon
the
county
.
And
if
there
doe
remaine
2
stones
,
the
second
name
Bea●
,
shall
shewe
you
by
these
three
vowels
,
e
,
a
,
i
,
that
Peter
is
the
duke
.
Iames
the
king
,
and
Ihon
the
countie
.
And
so
of
the
other
as
by
this
Table
dooeth
plainly
appeare
.
FINIS
.